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\section{Introduction}
The gauge/gravity duality \cite{Maldacena:1997re,Witten:1998qj,Aharony:1999ti} has been employed to holographically compute quantum information theoretic quantitites and has thereby helped us to understand the bulk-boundary relations. Among various observables of quantum information theory, entanglement entropy (EE) has been the most fundamental thing to study as it measures the correlation between two subsystems for a pure state. EE has a very simple definition yet sometimes it is notoriously difficult to compute. However, the holographic computation of entanglement entropy which can be denoted as the Ryu-Takayanagi (RT) prescription, is a remarkably simple technique which relates the area of a codimension-2 static minimal surface with the entanglement entropy of a subsystem \cite{Ryu:2006bv,Ryu:2006ef,Nishioka:2009un}. The RT-prescription along with its modification for time-dependent scenario (HRT prescription \cite{Hubeny:2007xt}) has been playing a key role for holographic studies of information theoretic quantitites as the perturbative calculations which could not be done on the field theoretic side due to its strongly coupled nature, can now be performed in the bulk side since it is of weakly coupled nature.\\
Another important information theoretic quantity which has gained much attention recently is the computational complexity. The complexity of a quantum state represents the minimum number of simple operations which takes the unentangled product state to a target state \cite{watrous2008quantum,aaronson2016complexity,Hashimoto:2018bmb}. There are several proposals to compute complexity holographically. Recently, several interesting attempts has been made to define complexity in QFT \cite{Jefferson:2017sdb,Chapman:2017rqy,Khan:2018rzm,Doroudiani:2019llj}. In context of holographic computation, initially, it was suggested that the complexity of a state (measured in gates) is proportional to the volume of the Einstein-Rosen bridge (ERB) which connects two boundaries of an eternal black hole \cite{Susskind:2014moa,Susskind:2014rva}
\begin{eqnarray}\label{CV1}
C_V(t_L,t_R)=\frac{V_{ERB}(t_L,t_R)}{8\pi R G_{d+1}}
\end{eqnarray}
where $R$ is the AdS radius and $V_{ERB}(t_L,t_R)$ is the co-dimension one extremal volume of ERB which is bounded by the two spatial slices at times $t_L$ and $t_R$ of two CFTs that live on the two boundaries of the eternal black hole. Another conjecture states that complexity can be obtained from the bulk action evaluated on the Wheeler-DeWitt patch \cite{Brown:2015bva,Brown:2015lvg,Goto:2018iay}
\begin{eqnarray}\label{CA}
C_A = \frac{I_{WDW}}{\pi \hbar}~.
\end{eqnarray}
The above two conjectures depends on the whole state of the physical system at the boundary. In addition to these proposals, there is another conjecture which depends on the reduced state of the system. This states that the co-dimension one volume enclosed by the co-dimension two extremal RT surface is proportional to the complexity
\begin{eqnarray}\label{HSC}
C_V=\frac{V(\Gamma_A^{min})}{8\pi R G_{d+1}}~.
\end{eqnarray}
This proposal is known as the holographic subregion complexity (HSC) conjecture in the literature \cite{Alishahiha:2015rta,Carmi:2016wjl,Karar:2017org}. Recently, in \cite{Gangopadhyay:2020xox}, it was shown that there exists a relation between the universal pieces of HEE and HSC. Furthermore, the universal piece of HSC is proportional to the sphere free energy $F_{S^p}$ for even dimensional dual CFTs and proportional to the Weyl $a$-anomaly for odd dimensional dual CFTs.\\
In recent times, much attention is being paid to the study of entanglement entropy and complexity for mixed states. For the study of EE for mixed states, the entanglement of purification (EoP) \cite{Terhal_2002} and entanglement negativity $\mathcal{E}$ \cite{Vidal:2002zz} has been the promising candidates. In the subsequent analysis, our focus will be on the computation of EoP. Consider a density matrix $\rho_{AB}$ corresponding to mixed state in Hilbert space $\mathcal{H}$, where $\mathcal{H}=\mathcal{H}_A \tens \mathcal{H}_B$. Now the process of purification states that one can construct a pure state $\ket{\psi}$ from $\rho_{AB}$ by adding auxillary degrees of freedom to the Hilbert space $\mathcal{H}$
\begin{eqnarray}
\rho_{AB} = tr_{A^{\prime}B^{\prime}}\ket{\psi}\bra{\psi};~\psi \in \mathcal{H}_{AA^{\prime}BB^{\prime}}=\mathcal{H}_{AA^{\prime}} \tens \mathcal{H}_{BB^{\prime}}~.
\end{eqnarray}
Such states $\psi$ are called purifications of $\rho_{AB}$. It is to be noted that the process of purification is not unique and different procedures for purification for the same mixed state exists. In this set up, the definition of EoP ($E_P$) reads \cite{Terhal_2002}
\begin{eqnarray}\label{EoP}
E_P(\rho_{AB}) = \mathop{min}_{ tr_{A^{\prime}B^{\prime}}\ket{\psi}\bra{\psi} }S(\rho_{AA^{\prime}});~\rho_{AA^{\prime}} = tr_{BB^{\prime}}\ket{\psi}\bra{\psi}~.
\end{eqnarray}
In the above expression, the minimization is taken over any state $\psi$ satisfying the condition $\rho_{AB} = tr_{A^{\prime}B^{\prime}}\ket{\psi}\bra{\psi}$, where $A^{\prime}B^{\prime}$ are arbitrary. In this paper we will compute the holographic analogy of EoP, given by the minimal area of entanglement wedge cross section (EWCS) $E_W$ \cite{Takayanagi:2017knl}. However, there is no direct proof of $E_P=E_W$ duality conjecture yet and it is mainly based on the following properties of $E_P$ which are also satisfied by $E_W$ \cite{Terhal_2002,Takayanagi:2017knl}. These properties are as follows.
\begin{eqnarray}\label{prop}
&(i)&~E_P(\rho_{AB}) = S(\rho_{A})=S(\rho_{B});~\rho_{AB}^2=\rho_{AB}\nonumber\\
&(i)&~\frac{1}{2} I(A:B) \leq E_P(\rho_{AB}) \leq min\left[S(\rho_{A}),S(\rho_{B})\right] \nonumber~.
\end{eqnarray}
In the above properties, $I(A:B)=S(A)+S(B)-S(A\cup B)$ is the mutual information between two subsystems $A$ and $B$. Further, there exists a critical separation length $D_c$ between $A$ and $B$ beyond which there is no connected phase for any $l$. At $D_c$, $E_W$ probes the phase transition of the RT surface $\Gamma_{AB}^{min}$ between connected and disconnected phases. The disconnected phase is characterised by the condition mutual information $I(A:B)=0$. Some recent very interesting observations in this direction can be found in \cite{Espindola:2018ozt,Agon:2018lwq,Umemoto:2018jpc,Hirai:2018jwy,Bao:2019wcf,Kusuki:2019evw,Jeong:2019xdr,Umemoto:2019jlz,Harper:2019lff,Jokela:2019ebz,BabaeiVelni:2019pkw,Ghodrati:2019hnn,Boruch:2020wbe,BabaeiVelni:2020wfl,Chakrabortty:2020ptb}. Further, recently several measures for mixed states dual to EWCS has been proposed. Some of them are odd entropy \cite{Tamaoka:2018ned}, reflected entropy \cite{Dutta:2019gen,Chu:2019etd} and logarithmic negativity \cite{MohammadiMozaffar:2017chk,Kudler-Flam:2018qjo}. On the other hand, recently the study of complexity for mixed states has gained appreciable amount of attention \cite{Alishahiha:2018lfv,Caceres:2018blh,Agon:2018zso,Caceres:2019pgf}. Similar to the case of EE for mixed state, the concept of `purification' is also being employed in this context \cite{Agon:2018zso,Camargo:2020yfv}. The purification complexity is defined as the minimal pure state complexity among all possible purifications available for a mixed state. Preparing a mixed state on some Hilbert space $\mathcal{H}$, starting from a reference (pure) state involves the extension of the Hilbert space $\mathcal{H}$ by introducing auxillary degrees of freedom \cite{Caceres:2018blh,Caceres:2019pgf}. In this set up, a quantity denoted as the mutual complexity $\Delta C$ has been prescribed in order to probe the concept of purifiaction complexity \cite{Alishahiha:2018lfv,Caceres:2018blh,Caceres:2019pgf,Agon:2018zso}. The mutual complexity $\Delta C$ satisfies the following definition
\begin{eqnarray}
\Delta\mathcal{C} = \mathcal{C}(\rho_{A})+ +\mathcal{C}(\rho_{B})-\mathcal{C}(\rho_{A\cup B})~.
\end{eqnarray}
In this paper, we will incorporate the HSC conjecture in order to compute the complexities $\mathcal{C}(\rho_{A})$, $\mathcal{C}(\rho_{B})$ and $\mathcal{C}(\rho_{A\cup B})$. We compute $\Delta\mathcal{C}$ in two different set ups. In one set up, we consider two disjoint subsystems $A$ and $B$ of width $l$ on the boundary Cauchy slice $\sigma$, separated by a distance $x$. We then compute the mutual complexity between these two subregions. The other set up, we consider that the boundary Cauchy slice $\sigma$ is a collection of two adjacent subsystems $A$ and $B$ of width $l$ with $A \cap B =0$ (zero overlap) and $A^c=B$. In this set up we compute the mutual complexity between a subregion $A$ and the full system $A \cup A^c$.\\
The paper is organized as follows. In Section \ref{sec2}, we briefly discuss the aspects of the bulk theory which in this case is a hyperscaling violating geometry. We then consider a single strip-like subsystem and holographically compute the EE in Section \ref{sec3}. We also make comments on the thermodynamical aspects of the computed HEE by computing the entanglement Smarr relation satisfied by the HEE. Furthermore, we holographically compute the relative entropy in order to obtain the Fisher information metric. In Section \ref{sec4}, we consider two strip-like subsystems and holographically compute the EoP by using the $E_P=E_W$ conjecture. We briefly study the temperature dependent behaviour of EWCS along with the effects of $z$ and $\theta$ on the $E_W$. The Butterfly velocity $v_B$ corresponding to the hyperscaling violating geometry is computed in \ref{Bvel}. We then compute the HSC corresponding to a single strip-like subsystem in Section \ref{sec5}. In Section \ref{sec6}, we holographically compute the mutual complexity $\Delta C$ by incorporating the HSC conjecture for the BTZ black hole and the hyperscaling violating geometry. We consider two different set ups to study the mutual complexity. We then conclude in Section \ref{sec7}. We also have an Appendix in the paper.
\section{Bulk theory: Hyperscaling violating geometry}\label{sec2}
We shall start our analysis with a bulk hyperscaling violating spacetime geometry. The solution corresponds to the following effective action of Einstein-Maxwell-scalar theory \cite{Ogawa:2011bz,Huijse:2011ef}
\begin{eqnarray}\label{1}
S_{bulk} = \frac{1}{16\pi G}\int d^{d+1}x \sqrt{-g} \left[\left(R-2\Lambda\right)-W(\phi)F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)\right]
\end{eqnarray}
where $F_{\mu\nu}$ is the Faraday tensor associated with the gauge field $A_{\mu}$, $\phi$ is the scalar field associated with the potential $V(\phi)$ and $W(\phi)$ is the coupling. Extremization of this action leads to the following black hole solution \cite{Huijse:2011ef}\footnote{Note that this geometry is different in form than the one considered in \cite{BabaeiVelni:2019pkw}.}
\begin{eqnarray}\label{2}
ds^2= \frac{R^2}{r^2}\left[-\frac{f(r)}{ r^{\frac{2(d-1)(z-1)}{(d-\theta-1)}}}dt^2 + r^{\frac{2\theta}{d-\theta-1}}\frac{dr^2}{f(r)}+\sum_{i=1}^{d-1}dx_i^2\right]~.
\end{eqnarray}
The lapse function $f(r)$ has the form $f(r)= 1-\left(\frac{r}{r_h}\right)^{(d-1)(1+\frac{z}{d-\theta -1})}$ where $r_H$ is the event horizon of the black hole. The Hawking temperature of black hole is obtained to be
\begin{eqnarray}\label{12}
T_H =\frac{(d-1)(z+ d-\theta -1)}{4\pi (d-\theta -1)}\frac{1}{r_{h}^{z(d-1)/(d-\theta -1)}}~.
\end{eqnarray}
The above mentioned metric is holographic dual to a $d$-dimensional non-relativistic strongly coupled theory with Fermi surfaces. The metric is associated with two independent exponents $z$ and $\theta$. The presence of these two exponents leads to the following scale transformations
\begin{eqnarray}\label{3}
x_i &\rightarrow& \xi~x_i\nonumber\\
t &\rightarrow& \xi^z~t\nonumber\\
ds &\rightarrow& \xi^{\frac{\theta}{d-1}}~ds\nonumber~.
\end{eqnarray}
This non-trivial scale transformation of the proper spacetime interval $ds$ is quite different from the usual AdS/CFT picture. The non-invariance of $ds$ in the bulk theory implies violations of hyperscaling in the boundary theory. Keeping this in mind, $\theta$ is identified as the hyperscaling violation exponent and $z$ is identified as the dynamical exponent. In the limit $z=1$, $\theta=0$, we recover the SAdS$_{d+1}$ solution which is dual to a relativistic CFT in $d$-dimensions and in the limit $z\neq1$, $\theta=0$, we obtain the `Lifshitz solutions'.\\
The two independent exponents $z$ and $\theta$ satisfy the following inequalities
\begin{eqnarray}\label{11}
\theta \leq d-2,~z \geq 1+\frac{\theta}{d-1}~.
\end{eqnarray}
The `equalities' of the above mentioned relations holds only for gauge theories of non-Fermi liquid states in $d=3$ \cite{PhysRevB.82.075127}. In this case, $\theta =1$ and $z=3/2$. For general $\theta=d-2$, logarithmic violation of the `Area law' of entanglement entropy \cite{Srednicki:1993im} is observed. This in turn means for $\theta = d-2$, the bulk theory holographically describes a strongly coupled dual theory with hidden Fermi surfaces. Some studies of information theoretic quantities for the above mentioned hyperscaling violating geometries can be found in \cite{MohammadiMozaffar:2017nri,Alishahiha:2018tep}.
\section{Holographic entanglement Smarr relation}\label{sec3}
To begin our analysis, we consider our subsystem $A$ to be a strip of volume $V_{sub}=L^{d-2}l$, where $-\frac{l}{2}<x_1<\frac{l}{2}$ and $-\frac{L}{2}<x_{2,3,..,d-1}<\frac{L}{2}$. The amount of Hawking entropy captured by the above mentioned volume reads
\begin{eqnarray}\label{10}
S_{BH}=\frac{L^{d-2} l}{4G_{d+1} r_{h}^{d-1}}~.
\end{eqnarray}
It is to be noted that the thermal entropy of the dual field theory is related with the temperature\footnote{The thermal entropy of the dual field theory is basically the Hawking entropy of the black hole given in eq.(\ref{12}).} as $S_{th}\propto T^{\frac{d-1-\theta}{z}}$. For $\theta=d-2$, it reads $S_{th}\propto T^{\frac{1}{z}}$. This result is observed for compressible states with fermionic excitations.
We parametrize the co-dimension one static minimal surface as $x_1 = x_1(r)$ which leads to the following area of the extremal surface $\Gamma_A^{min}$
\begin{eqnarray}\label{4}
A(\Gamma_A^{min}) = 2R^{d-1}L^{d-2}r_t^{\left(\frac{p}{p-\theta}\right)-p}\sum_{n=1}^{\infty}\frac{1}{\sqrt{\pi}}\frac{\Gamma(n+\frac{1}{2})}{\Gamma(n+1)}\alpha^{np\left(1+\frac{z}{p-\theta}\right)}\int_{0}^{1}du\frac{u^{\frac{\theta}{p-\theta}-p+np\left(1+\frac{z}{p-\theta}\right)}}{\sqrt{1-u^{2p}}};~u=\frac{r}{r_t},~\alpha=\frac{r_t}{r_h}
\end{eqnarray}
where $r_t$ is the turning point and $p=(d-1)$ which we have introduced for the sake of simplicity. By substituting the area functional (given in eq.(\ref{4})) in the RT formula, we obtain the HEE \cite{Ryu:2006bv}
\begin{eqnarray}\label{5}
S_{E} &=& \frac{A(\Gamma_A^{min})}{4G_{d+1}}\nonumber\\
&=& \frac{2L^{d-2}}{4G_{d+1}\left(\frac{p}{p-\theta}-p\right)} \left(\frac{1}{\epsilon}\right)^{p-\left(\frac{p}{p-\theta}\right)}+\frac{L^{d-2}r_t^{\left(\frac{p}{p-\theta}\right)-p}}{4G_{d+1}}\sum_{n=0}^{\infty}\frac{\Gamma(n+\frac{1}{2})}{\Gamma(n+1)}\frac{\alpha^{np\left(1+\frac{z}{p-\theta}\right)}}{p}\frac{\Gamma\left(\frac{\frac{p}{p-\theta}-p+np\left(1+\frac{z}{p-\theta}\right)}{2p}\right)}{\Gamma\left(\frac{\frac{p}{p-\theta}+np\left(1+\frac{z}{p-\theta}\right)}{2p}\right)}\nonumber~.\\
\end{eqnarray}
The relationship between the subsystem size $l$ and turning point $r_t$ reads (with the AdS radius $R=1$)
\begin{eqnarray}\label{6}
l = r_t^{\frac{p}{p-\theta}} \sum_{n=0}^{\infty}\frac{\Gamma(n+\frac{1}{2})}{\Gamma(n+1)}\frac{\alpha^{np\left(1+\frac{z}{p-\theta}\right)}}{p}\frac{\Gamma\left(\frac{\frac{p}{p-\theta}+p+np\left(1+\frac{z}{p-\theta}\right)}{2p}\right)}{\Gamma\left(\frac{\frac{p}{p-\theta}+2p+np\left(1+\frac{z}{p-\theta}\right)}{2p}\right)}~.
\end{eqnarray}
We now proceed to probe the thermodynamical aspects of HEE. It can be observed from eq.(\ref{5}) that the exprssion of $S_{E}$ contains a subsystem independent divergent piece which we intend to get rid by defining a finite quantity. We call this finite quantity as the renormalized holographic entanglement entropy ($S_{REE}$). From the point of view of the dual field theory this divergence free quantity represents the change in entanglement entropy under an excitation. In order to obtain $S_{REE}$ holographically, firstly we need to compute the HEE corresponding to the asymptotic form ($r_h\rightarrow\infty$) of the hyperscaling violating black brane solution given in eq.(\ref{2}). This yields the following expression
\begin{eqnarray}\label{7}
S_{G}= \frac{2L^{d-2}}{4G_{d+1}\left(\frac{p}{p-\theta}-p\right)} \left(\frac{1}{\epsilon}\right)^{p-\left(\frac{p}{p-\theta}\right)} -\frac{2^{p-\theta}L^{d-2}}{4G_{d+1}\left(p-\frac{p}{p-\theta}\right)} \left(\frac{p-\theta}{p}\right)^{p-\theta-1}\frac{\pi^{\frac{p-\theta}{2}}}{l^{p-1-\theta}}\left(\frac{\Gamma\left(\frac{p+\frac{p}{p-\theta}}{2p}\right)}{\Gamma\left(\frac{\frac{p}{p-\theta}}{2p}\right)}\right)^{p-\theta}~.
\end{eqnarray}
We now subtract the above expression (which represents the HEE corresponding to the vacuum of the dual field theory) from $S_{E}$ (given in eq.(\ref{5})) in order to get a finite quantity $S_{REE}$. This can be formally represented as
\begin{eqnarray}\label{8}
S_{REE} = S_{E} - S_{G}~.
\end{eqnarray}
On the other hand, the internal energy $E$ of the black hole can be obtained by using the Hawking entropy (given in eq.(\ref{10})) and the Hawking temperature (given in eq.(\ref{12})). The computed expression of $E$ can be represented as
\begin{eqnarray}\label{13}
E = \left(\frac{p-\theta}{z+p-\theta}\right)S_{BH}T_H~.
\end{eqnarray}
This is nothing but the classical Smarr relation of BH thermodynamics. In \cite{Saha:2019ado}, it was shown that the quantity $S_{REE}$ and the internal energy $E$ satisfies a Smarr-like thermodynamic relation corresponding to a generalized temperature $T_g$. In this set up, this relation reads \cite{Saha:2020fon}
\begin{eqnarray}\label{9}
E= \left(\frac{p-\theta}{z+p-\theta}\right)S_{REE}T_g~.
\end{eqnarray}
It is remarkable to observe that the relation given in eq.(\ref{9}) has a striking similarity with the classical Smarr relation of BH thermodynamics, given in eq.(\ref{13}). In the limit $r_t \rightarrow r_h$, the leading term of the generalized temperature $T_g$ produces the exact Hawking temperature $T_H$ whereas in the limit $\frac{r_t}{r_h}\ll1$, the leading term of $T_g$ reads \cite{Saha:2020fon}
\begin{eqnarray}\label{14}
\frac{1}{T_g} = \Delta_1 l^z
\end{eqnarray}
where the detailed expression of $\Delta_1$ reads
\begin{eqnarray}
\Delta_1 = \frac{2\pi^{3/2}}{p}\left(\frac{1}{\frac{p}{p-\theta}-p+p\left(1+\frac{z}{p-\theta}\right)}\right)\left(\frac{p}{\sqrt{\pi}}\frac{\Gamma\left(\frac{p+\frac{p}{p-\theta}}{2p}\right)}{\Gamma\left(\frac{2p+\frac{p}{p-\theta}}{2p}\right)}\right)^{1+z}\left(\frac{\Gamma\left(\frac{p+\frac{p}{p-\theta}+p(1+\frac{z}{p-\theta})}{2p}\right)}{\Gamma\left(\frac{2p+\frac{p}{p-\theta}+p(1+\frac{z}{p-\theta})}{2p}\right)}\right)\nonumber~.
\end{eqnarray}
From eq.(\ref{14}) it can be observed that in the UV limit, $T_g$ shows the similar behaviour as entanglement temperature $T_{ent}$ (proportional to the inverse of subsystem size $l$) \cite{Bhattacharya:2012mi}.
\subsection{Relative entropy and the Fisher information metric}
We now proceed to compute the Fisher information metric for the hyperscaling violating geometry using the holographic proposal. The Fisher information metric measures the distance between two quantum states and is given by \cite{Banerjee:2017qti}
\begin{eqnarray}\label{29}
G_{F,\lambda\lambda} = \langle\delta\rho~\delta\rho\rangle^{(\sigma)}_{\lambda\lambda}=\frac{1}{2}~Tr\left(\delta\rho \frac{d}{d(\delta\lambda)}\log(\sigma+\delta\lambda\delta\rho)\vert_{\delta\lambda=0}\right)
\end{eqnarray}
where $\sigma$ is the density matrix and $\delta\rho$ is a small deviation from the density matrix. On the other hand, there exists a relation between the Fisher information metric and the relative entropy $S_{rel}$ \cite{Lashkari:2015hha}. This reads
\begin{eqnarray}\label{30}
G_{F,mm} = \frac{\partial^2}{\partial m^2}S_{rel}(\rho_m\vert\vert\rho_0);~S_{rel}(\rho_m\vert\vert\rho_0)=\Delta\langle H_{\rho_0}\rangle-\Delta S~.
\end{eqnarray}
In the above expression, $\Delta S$ is the change in the entanglement entropy from vacuum state, $\Delta\langle H_{\rho_0}\rangle$ is the change in the modular Hamiltonian and $m$ is the perturbation parameter. In this set up, we holographically compute the relative entropy $S_{rel}(\rho_m\vert\vert\rho_0)$. We consider the background is slightly perturbed from pure hyperscaling violating spacetime while the subsystem volume $L^{d-2}l$ is fixed. Then the inverse of the lapse function $f(r)$ (given in eq.(\ref{2})) can be expressed as
\begin{eqnarray}\label{31}
\frac{1}{f(r)}=1+mr^{p\left(1+\frac{z}{p-\theta}\right)}+m^2r^{2p\left(1+\frac{z}{p-\theta}\right)}
\end{eqnarray}
where $m=\left(\frac{1}{r_H}\right)^{p\left(1+\frac{z}{p-\theta}\right)}$ is the holographic perturbation parameter. Since we consider a perturbation to the background geometry and also consider that the subsystem size $l$ has not changed, we can express the turning point in the following perturbed form
\begin{eqnarray}\label{32}
r_t = r_t^{(0)} + m r_t^{(1)} + m^2 r_t^{(2)}
\end{eqnarray}
where $r_t^{(0)}$ is the turning point for the pure hyperscaling violating geometry and $r_t^{(1)}$, $r_t^{(2)}$ are the first and second order corrections to the turning point. We now write down the subsystem length $l$ upto second order in perturbation as
\begin{eqnarray}\label{33}
\frac{l}{r_t^{\frac{p}{p-\theta}}} = a_0 + m a_1r_t^{p\left(1+\frac{z}{p-\theta}\right)}+m^2a_2r_t^{2p\left(1+\frac{z}{p-\theta}\right)}
\end{eqnarray}
where
\begin{eqnarray}
a_0= \frac{\sqrt{\pi}}{p}\frac{\Gamma\left(\frac{\frac{p}{p-\theta}+p}{2p}\right)}{\Gamma\left(\frac{\frac{p}{p-\theta}}{2p}\right)},~a_1=\frac{\sqrt{\pi}}{2p}\frac{\Gamma\left(\frac{\frac{p}{p-\theta}+p+p\left(1+\frac{z}{p-\theta}\right)}{2p}\right)}{\Gamma\left(\frac{\frac{p}{p-\theta}+p\left(1+\frac{z}{p-\theta}\right)}{2p}\right)},~a_2=\frac{3\sqrt{\pi}}{8p}\frac{\Gamma\left(\frac{\frac{p}{p-\theta}+p+2p\left(1+\frac{z}{p-\theta}\right)}{2p}\right)}{\Gamma\left(\frac{\frac{p}{p-\theta}+2p\left(1+\frac{z}{p-\theta}\right)}{2p}\right)}\nonumber~.
\end{eqnarray}
Using eq.(\ref{32}) in eq.(\ref{33}) and keeping in mind the consideration that $l$ has not changed, we obtain the forms of $r_t^{(0)}$, $r_t^{(1)}$ and $r_t^{(2)}$
\begin{eqnarray}\label{34}
r_t^{(0)} = \left(\frac{l}{a_0}\right)^{\frac{p-\theta}{p}},~r_t^{(1)}=-\left(\frac{p-\theta}{p}\right)\left(\frac{a_1}{a_0}\right)\left(r_t^{(0)}\right)^{1+p\left(1+\frac{z}{p-\theta}\right)},~ r_t^{(2)}= \xi \left(r_t^{(0)}\right)^{1+2p\left(1+\frac{z}{p-\theta}\right)}
\end{eqnarray}
where
\begin{eqnarray}
\xi = \left(\frac{p-\theta}{p}\right)\left[\left(\frac{2p-\theta}{2p}\right)\left(\frac{a_1}{a_0}\right)^2+(p-\theta)\left(1+\frac{z}{p-\theta}\right)\left(\frac{a_1}{a_0}\right)^2-\left(\frac{a_2}{a_0}\right)\right]~.
\end{eqnarray}
On a similar note, the expression for area of the static minimal surface upto second order in perturbation parameter $m$ can be obtained from eq.(\ref{4}). We then use eq.(\ref{32}) to recast the expression for the area of the minimal surface in the form
\begin{eqnarray}\label{35}
A(\Gamma_A^{min}) = A(\Gamma_A^{min})^{(0)}+mA(\Gamma_A^{min})^{(1)}+m^2 A(\Gamma_A^{min})^{(2)}~.
\end{eqnarray}
It has been observed that at first order in $m$, $S_{rel}$ vanishes \cite{Lashkari:2015hha} and in second order in $m$ it reads $S_{rel}=-\Delta S$. In this set up, it yields
\begin{eqnarray}\label{36}
S_{rel} = - m^2\frac{A(\Gamma_A^{min})^{(2)}}{4G_{d+1}}=m^2\frac{L^{d-2}}{4G_{d+1}}\Delta_2 \left(\frac{l}{a_0}\right)^{1+2z+(p-\theta)}
\end{eqnarray}
where
\begin{eqnarray}
\Delta_2= 2p\left(\frac{p-\theta}{p}\right)\left(\frac{a_1^2}{a_0}\right)+p\left(p-\frac{\theta}{p-\theta}\right)\left(\frac{p-\theta}{p}\right)^2\left(\frac{a_1^2}{a_0}\right)-\left(\frac{2pa_2}{2p(1+\frac{z}{p-\theta})-p+\frac{p}{p-\theta}}\right)-2pa_0\xi\nonumber~.
\end{eqnarray}
By substituting the above expression in eq.(\ref{30}), the Fisher information is obtained to be
\begin{eqnarray}\label{37}
G_{F,mm}= \frac{L^{d-2}}{2G_{d+1}}\Delta_2 \left(\frac{l}{a_0}\right)^{1+2z+(p-\theta)}\propto l^{d+2z-\theta}~.
\end{eqnarray}
\noindent In the limit $z=1$ and $\theta=0$, the above equation reads $G_{F,mm}\propto l^{d+2}$ which agrees with the result obtained in \cite{Karar:2019bwy}. The Fisher information corresponding to the Lifshitz type solutions can be found in \cite{Karar:2020cvz}.
\section{Entanglement wedge cross-section and the $E_P = E_W$ duality}\label{sec4}
We now proceed to compute the holographic entanglement of purification by considering two subsystems, namely, $A$ and $B$ of length $l$ on the boundary $\partial M$. From the bulk point of view, $\partial M$ is the boundary of a canonical time-slice $M$ made in the static gravity dual. Furthermore, $A$ and $B$ are separated by a distance $D$ so that the subsystems does not have an overlap of non-zero size ($A\cap B =0$). Following the RT prescription, we denote $\Gamma_A^{min}$, $\Gamma_B^{min}$ and $\Gamma_{AB}^{min}$ as the static minimal surfaces corresponding to $A$, $B$ and $AB$ respectively. In this set up, the domain of entanglement wedge $M_{AB}$ is the region in the bulk with the following boundary
\begin{eqnarray}\label{16}
\partial M_{AB} = A \cup B \cup \Gamma_{AB}^{min}~.
\end{eqnarray}
It is also to be noted that if the separation $D$ is effectively large then the codimension-0 bulk region $M_{AB}$ will be disconnected. We now divide $\Gamma_{AB}^{min}$ into two parts
\begin{eqnarray}\label{17}
\Gamma_{AB}^{min} = \Gamma_{AB}^{A} \cup \Gamma_{AB}^{B}
\end{eqnarray}
such that the boundary $\partial M_{AB}$ of the canonical time-slice of the full spacetime $M_{AB}$ can be represented as
\begin{eqnarray}
\partial M_{AB} = \bar{\Gamma}_A \cup \bar{\Gamma}_B
\end{eqnarray}
where
$\bar{\Gamma}_A = A \cup \Gamma_{AB}^{A}$ and $\bar{\Gamma}_B = B \cup \Gamma_{AB}^{B}$. In this set up, it is now possible to define the holographic entanglement entropies $S(\rho_{A \cup \Gamma_{AB}^{A}})$ and $S(\rho_{B \cup \Gamma_{AB}^{B}})$. These quantities can be computed by finding a static minimal surface $\Sigma^{min}_{AB}$ such that
\begin{eqnarray}
\partial \Sigma^{min}_{AB} = \partial \bar{\Gamma}_A = \partial \bar{\Gamma}_B~.
\end{eqnarray}
There can be infinite number possible choices for the spliting given in eq.(\ref{17}) and this in turn means there can be infinite number of choices for the surface $\Sigma^{min}_{AB}$. The entanglement wedge cross section (EWCS) is obtained by minimizing the area of $\Sigma^{min}_{AB}$ over all possible choices for $\Sigma^{min}_{AB}$. This can be formally written down as
\begin{eqnarray}\label{18}
E_W(\rho_{AB}) = \mathop{min}_{\bar{\Gamma}_A \subset \partial M_{AB}}\left[\frac{A\left(\Sigma^{min}_{AB}\right)}{4G_{d+1}}\right]~.
\end{eqnarray}
We now proceed to compute $E_W$ for the holographic dual considered in this paper. As we have mentioned earlier, EWCS is the surface with minimal area which splits the entanglement wedge into two domains corresponding to $A$ and $B$. This can be identified as a vertical, constant $x$ hypersurface. The time induced metric on this constant $x$ hypersurface reads
\begin{eqnarray}
ds_{ind}^2= \frac{R^2}{r^2}\left[ r^{\frac{2\theta}{p-\theta}}\frac{dr^2}{f(r)}+\sum_{i=1}^{d-2}dx_i^2\right]~.
\end{eqnarray}
By using this above mentioned induced metric, the EWCS is obtained to be
\begin{eqnarray}\label{15}
\begin{aligned}
E_W =& \frac{L^{d-2}}{4G_{d+1}}\int_{r_t(D)}^{r_t(2l+D)}\frac{dr}{r^{d-1}\sqrt{f(r)}}\nonumber\\
=& \frac{L^{d-2}}{4G_{d+1}}\sum_{n=0}^{\infty}\frac{1}{\sqrt{\pi}}\frac{\Gamma(n+\frac{1}{2})}{\Gamma(n+1)}\left[\frac{r_t(2l+D)^{np(1+\frac{z}{p-\theta})-p+1}-r_t(D)^{np(1+\frac{z}{p-\theta})-p+1}}{np(1+\frac{z}{p-\theta})-p+1}\right]\left(\frac{1}{r_h}\right)^{np(1+\frac{z}{p-\theta})}~.
\end{aligned}
\end{eqnarray}
As mentioned earlier, the above expression of $E_W$ always maintains the following bound
\begin{eqnarray}\label{bound}
E_W \geq \frac{1}{2}I(A:B);~I(A:B)=S(A)+S(B)-S(A\cup B)
\end{eqnarray}
where $I(A:B)$ is the mutual information between two subsystems $A$ and $B$. On the other hand, in \cite{Yang:2018gfq} it was shown that there exists a critical separation between $A$ and $B$ beyond which there is no connected phase for any $l$. This in turn means that at the critical separation length $D_c$, $E_W$ probes the phase transition of the RT surface $\Gamma_{AB}^{min}$ between connected and disconnected phases. The disconnected phase is characterised by the fact that the mutual information $I(A:B)$ vanishes, which in this case reads
\begin{eqnarray}
2S(l)-S(D)-S(2l+D)=0~.
\end{eqnarray}
The above condition together with the bound given in eq.(\ref{bound}) leads to the critical separation length $D_c$ \cite{Ben-Ami:2014gsa}.\\
\noindent We now write down the expression for $E_W$ (given in eq.(\ref{15})) in terms of the parameters of the boundary theory. This we do for the small temperature case ($\frac{r_t(D)}{r_h}\ll \frac{r_t(2l+D)}{r_h} \ll 1 $) and for the high temperature case $(\frac{r_t(D)}{r_h}\ll 1, \frac{r_t(2l+D)}{r_h}\approx1)$.
\subsection{$E_W$ in the low temperature limit}
In the limit $\frac{r_t(D)}{r_h}\ll \frac{r_t(2l+D)}{r_h} \ll 1 $, it is reasonable to consider terms upto order $m$ (where $m= \frac{1}{r_h^{p(1+\frac{z}{p-\theta})}}$) in the expression for $E_W$ (given in eq.(\ref{15})). On the other hand for low temeprature considerations, it is possible to perturbatively solve eq.(\ref{6}) which leads to the following relationship between a subsystem size $l$ and its corresponding turning point $r_t$
\begin{eqnarray}\label{19}
r_t(l) = l^{\frac{p-\theta}{p}}\left(\frac{p}{\sqrt{\pi}}\frac{\Gamma\left(\frac{2p+\frac{p}{p-\theta}}{2p}\right)}{\Gamma\left(\frac{p+\frac{p}{p-\theta}}{2p}\right)}\right)^{\frac{p-\theta}{p}}\times \left[1- \Delta_3 l^{(p-\theta)\left(1+\frac{z}{p-\theta}\right)}T^{\left(1+\frac{p-\theta}{z}\right)}
\right]
\end{eqnarray}
where
\begin{eqnarray}
\Delta_3 &=& \left(\frac{p-\theta}{2p}\right)\left(\frac{4\pi}{p(1+\frac{z}{p-\theta})}\right)^{1+\frac{p-\theta}{z}} \left(\frac{\Gamma\left(\frac{2p+\frac{p}{p-\theta}}{2p}\right)}{\Gamma\left(\frac{p+\frac{p}{p-\theta}}{2p}\right)}\right)\left(\frac{\Gamma\left(\frac{p+\frac{p}{p-\theta}+p(1+\frac{z}{p-\theta})}{2p}\right)}{\Gamma\left(\frac{2p+\frac{p}{p-\theta}+p(1+\frac{z}{p-\theta})}{2p}\right)}\right)\left(\frac{p}{\sqrt{\pi}}\frac{\Gamma\left(\frac{2p+\frac{p}{p-\theta}}{2p}\right)}{\Gamma\left(\frac{p+\frac{p}{p-\theta}}{2p}\right)}\right)^{(p-\theta)\left(1+\frac{z}{p-\theta}\right)}\nonumber~.
\end{eqnarray}
Now by using eq.(\ref{19}) for $r_t(2l+D)$ and $r_t(D)$, we obtain the expression for $E_W$ in the low temperature limit to be
\begin{eqnarray}\label{20}
E_W = E_W^{T=0} - \frac{L^{d-2}}{4G_{d+1}}\Delta_4\left[(2l+D)^{\left(\frac{p-\theta}{p}\right)\left(1+\frac{pz}{p-\theta}\right)}-D^{{\left(\frac{p-\theta}{p}\right)\left(1+\frac{pz}{p-\theta}\right)}}\right]T^{1+\left(\frac{p-\theta}{z}\right)}+...
\end{eqnarray}
where the detailed expression for $\Delta_4$ is given in the Appendix. The first term in eq.(\ref{20}) is the EWCS at $T=0$. This reads
\begin{eqnarray}\label{21}
E_W^{T=0} = \frac{L^{d-2}}{4(p-1)G_{d+1}} \left(\frac{\sqrt{\pi}}{p}\frac{\Gamma\left(\frac{p+\frac{p}{p-\theta}}{2p}\right)}{\Gamma\left(\frac{2p+\frac{p}{p-\theta}}{2p}\right)}\right)^{(p-\theta)-\left(\frac{p-\theta}{p}\right)}\left[\left(\frac{1}{D}\right)^{(p-\theta)-\left(\frac{p-\theta}{p}\right)}-\left(\frac{1}{2l+D}\right)^{(p-\theta)-\left(\frac{p-\theta}{p}\right)}\right]~.\nonumber\\
\end{eqnarray}
It can be observed from eq.(\ref{20}) that the EWCS is a monotonically decreasing function of temperature $T$ (as $1+\frac{p-\theta}{z}>0$). We now compute the critical separation length $D_c$ at which $E_W$ probes the phase transition of the RT surface $\Gamma_{AB}^{min}$ between the connected and disconnected phases. This to be obtained from the condition
\begin{eqnarray}\label{MI}
2S(l)-S(D)-S(2l+D)=0~.
\end{eqnarray}
The general expression for the HEE of a strip of length $l$ is given in eq.(\ref{5}). Now similar to the above computation, in the limit $\frac{r_t(D)}{r_h}\ll \frac{r_t(2l+D)}{r_h} \ll 1$, we consider terms upto $\mathcal{O}(m)$ in eq.(\ref{5}). By using this consideration, eq.(\ref{MI}) can be expressed as
\begin{eqnarray}
\beta_1\left(\frac{2}{l^{p-\theta-1}}-\frac{1}{D^{p-\theta-1}}-\frac{1}{(2l+D)^{p-\theta-1}}\right)+\frac{\beta_2}{r_h^{p(1+\frac{z}{p-\theta})}}\left(2l^{1+z}-D^{1+z}-(2l+D)^{1+z}\right)=0
\end{eqnarray}
where
\begin{eqnarray}
\beta_1 &=& \frac{\sqrt{\pi}}{4p} \frac{\Gamma\left(\frac{\frac{p}{p-\theta}-p}{2p}\right)}{\Gamma\left(\frac{\frac{p}{p-\theta}}{2p}\right)}\left(\frac{\sqrt{\pi}}{p}\frac{\Gamma\left(p+\frac{\frac{p}{p-\theta}}{2p}\right)}{\Gamma\left(\frac{2p+\frac{p}{p-\theta}}{2p}\right)}\right)^{p-\theta-1}\nonumber\\
\beta_2&=&\Delta_3\left(p-\frac{p}{p-\theta}\right)\beta_1+\frac{\sqrt{\pi}}{8p}\left(\frac{p}{\sqrt{\pi}}\frac{\Gamma\left(\frac{p+\frac{p}{p-\theta}}{2p}\right)}{\Gamma\left(\frac{2p+\frac{p}{p-\theta}}{2p}\right)}\right)^{1+z}\left(\frac{\Gamma\left(\frac{p+\frac{p}{p-\theta}+p(1+\frac{z}{p-\theta})}{2p}\right)}{\Gamma\left(\frac{2p+\frac{p}{p-\theta}+p(1+\frac{z}{p-\theta})}{2p}\right)}\right)~.\nonumber
\end{eqnarray}
By solving the above equation we can find out the critical separation length $D_c$ (where we substitute $\frac{D}{l}=k=constant$). It is worth mentioning that in the above computations of $E_W$ and $I(A:B)$, we have considered terms upto $\mathcal{O}(m)$ which is the leading order term for thermal correction. Similarly, one can incorporate next-to leading order terms or more to get a more accurate result.
\begin{figure}[!h]
\begin{minipage}[t]{0.48\textwidth}
\centering\includegraphics[width=\textwidth]{EWL1.eps}\\
{\footnotesize Effect of hyperscaling violating exponent $\theta$ (we set $z=1$)}
\end{minipage}\hfill
\begin{minipage}[t]{0.48\textwidth}
\centering\includegraphics[width=\textwidth]{EWL2.eps}\\
{\footnotesize Effect of dynamical exponent $z$ (we set $\theta=0.2$)}
\end{minipage}
\caption{Effects of $\theta$ and $z$ on $E_W$ and $I(A:B)$ at low temperature (with $d=3$, $k=0.4$, $L=1$ and $G_{d+1}=1$)}\label{fig2}
\end{figure}
\subsection{$E_W$ in the high temperature limit}
We now consider the limit $r_t(2l+D)\rightarrow r_h$ and $\frac{r_t(D)}{r_h}\ll 1$. This in turn means the static minimal surface associated with the turning point $r_t(2l+D)$ wraps a portion of the event horizon $r_h$. In the large $n$ limit, the infinite sum associated with the turning point $r_t(2l+D)$ goes as $\approx \frac{1}{\sqrt{\pi}}\left(\frac{1}{n}\right)^{3/2}\left(\frac{r_t(2l+D)}{r_h}\right)^{np(1+\frac{z}{p-\theta})}$ which means it is convergent ($\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}=\xi(\frac{3}{2})$). Further, we are considering $\frac{r_t(D)}{r_h}\ll 1$ and in this limit it is reasonable to keep terms only upto order $m$ in the infinite sum associated with $r_t(D)$. In this set up, the expression for EWCS reads
\begin{eqnarray}\label{22}
E_W(T)=E_W^{T=0}+\frac{L^{d-2}}{4G_{d+1}}\Delta_4 D^{z+\left(\frac{p-\theta}{p}\right)}T^{1+\left(\frac{p-\theta}{z}\right)}-\frac{L^{d-2}}{4G_{d+1}}\Delta_5 T^{\left(\frac{p-\theta}{z}\right)-\left(\frac{p-\theta}{pz}\right)}+...
\end{eqnarray}
where the temperature independent term (EWCS at $T=0$) is being given by
\begin{eqnarray}\label{23}
E_W^{T=0}=\frac{L^{d-2}}{4(p-1)G_{d+1}D^{(\frac{p-\theta}{p})(p-1)}} \left(\frac{\sqrt{\pi}}{p}\frac{\Gamma\left(\frac{p+\frac{p}{p-\theta}}{2p}\right)}{\Gamma\left(\frac{2p+\frac{p}{p-\theta}}{2p}\right)}\right)^{(p-\theta)-\left(\frac{p-\theta}{p}\right)}~.
\end{eqnarray}
The expressions for $\Delta_4$ and $\Delta_5$ are given in the Appendix. Now we procced to compute the critical separation length $D_c$ in the high temperature configuration. In the limit $r_t \rightarrow r_h$, the computed result of $S_E$ (HEE of a strip like subsystem with length $l$), given in eq.(\ref{5}) can be rearranged in the following form\footnote{The first term of the expression is nothing but the thermal entropy of the boundary subsystem given in eq.(\ref{10}).}
\begin{eqnarray}
S_{E} = \frac{L^{d-2}l}{4G_{d+1}r_h^p}+\frac{L^{d-2}}{4G_{d+1}r_h^{p-\frac{p}{p-\theta}}}\sum_{n=0}^{\infty}P_n;~P_n =\left(\frac{1}{\frac{p}{p-\theta}-p+np(1+\frac{z}{p-\theta})}\right) \frac{\Gamma\left(n+\frac{1}{2}\right)}{\Gamma\left(n+1\right)} \frac{\Gamma\left(\frac{p+\frac{p}{p-\theta}+np(1+\frac{z}{p-\theta})}{2p}\right)}{\Gamma\left(\frac{2p+\frac{p}{p-\theta}+np(1+\frac{z}{p-\theta})}{2p}\right)}\nonumber~.\\
\end{eqnarray}
By using the above form of HEE, we can write down eq.(\ref{MI}) in the following form
\begin{eqnarray}
\frac{\sum_{n=0}^{\infty}P_n}{r_h^{p-\frac{p}{p-\theta}}}-\frac{D}{4r_h^{p-\frac{p}{p-\theta}}}-\frac{\beta_1}{D^{p-\theta-1}}-\frac{\beta_2}{r_h^{p(1+\frac{z}{p-\theta})}}D^{1+z}=0~.
\end{eqnarray}
\begin{figure}[!h]
\begin{minipage}[t]{0.48\textwidth}
\centering\includegraphics[width=\textwidth]{EWH1.eps}\\
{\footnotesize Effect of hyperscaling violating exponent $\theta$ (we set $z=1$)}
\end{minipage}\hfill
\begin{minipage}[t]{0.48\textwidth}
\centering\includegraphics[width=\textwidth]{EWH2.eps}\\
{\footnotesize Effect of dynamical exponent $z$ (we set $\theta=0.2$)}
\end{minipage}
\caption{Effects of $\theta$ and $z$ on $E_W$ and $I(A:B)$ at high temperature (with $d=3$, $L=1$ and $G_{d+1}=1$)}
\label{fig3}
\end{figure}
In Fig(s).(\ref{fig2}) and (\ref{fig3}), we have graphically represented the effects of $z$ and $\theta$ on the EWCS and holographic mutual information (HMI) for both low and high temperature case respectively. For the low temperature case (Fig.(\ref{fig2})) we have chose the separation length $D$ between the subsystems to be $D = 0.4~l$. From the above plots it can be observed that the EWCS always maintains the bound $E_W > \frac{1}{2} I(A:B)$. The HMI continously decays and approaches zero at a particular critical separation length $D_c$. This critical separataion $D_c$ decreases with increasing $z$ and $\theta$. On the other hand $E_W$ shows a discontinous jump at $D_c$. Upto $D_c$, $E_W$ has a finite cross-section due to the connected phase whereas beyond this critical separation length $E_W$ vanishes due to the disconnected phase of the RT surface $\Gamma_{AB}^{min}$.
\section{Butterfly velocity}\label{Bvel}
In this section we shall discuss about information spreading in the dual field theory from the holographic point of view. In context of quantum many body physics, the study about the chaotic nature of the system (response of the system at a late time after a local perturbation at initial time) can be characterized by the following thermal average
\begin{eqnarray}\label{40}
C(x,t) = \langle\left[W(t,x),V(0)\right]^2\rangle_{\beta}
\end{eqnarray}
where $V(0)$ is a generic operator acting at the origin at earlier time and $W(t,x)$ is a local operator acting at position $x$ at later time $t$. The butterfly effect is usually governed by such commutators. It probes the dependency of a late time measurement on the initial perturbation. The time at which the commutator grows to $\mathcal{O}(1)$, is known as the Scrambling time \cite{Hayden:2007cs}. The study of Butterfly effect in context of AdS/CFT naturally occurs as the black holes are observed to be the fastest Scramblers in the nature \cite{Sekino:2008he}. For large $N$ field theories, eq.(\ref{40}) grows as
\begin{eqnarray}
C(x,t) = \frac{K}{N^2} \exp[\lambda_L\left(t-\frac{|x|}{v_B}\right)]+\mathcal{O}\left(\frac{1}{N^4}\right)
\end{eqnarray}
where $K$ is a constant, $\lambda_L$ is the Lyapunov exponent which probes the growth of chaos with time and $v_B$ is the Butterfly velocity. The Butterfly velocity $v_B$ probes the speed at which the local perturbation grows (the speed of the growth for chaos). This velocity composes a light cone for chaos and outside this light cone, the perturbation does not effects the system. Further, $v_B$ has the interpretation of effective Lieb-Robinson velocity (state-dependent) $v_{LR}$ for strongly coupled field theories \cite{Roberts:2016wdl}. The Lyapunov exponent safisfies a upper bound \cite{Maldacena:2015waa}
\begin{eqnarray}
\lambda_L \leq 2\pi T~.
\end{eqnarray}
One can consider acting of a local operator (perturbation) on a thermal state of a CFT. At the initial stages, the information about the nature of the operator can be obatined by operating another local operator at position $x$. However, due to the Scrambling property, this information about the initial perturbation will spread out in a larger and larger region with time.\\
In context of gauge/gravity duality, $v_B$ can be calculated by using the subregion duality \cite{Bousso:2012sj}. The static black hole in the bulk represents the initial thermal state in the dual field theory. One can add a local perturbation in this set up which eventually falls into event horizon. The time-like trajectories of the local perturbation in the bulk can be probed, by using the co-dimension two RT surfaces \cite{Shenker:2013pqa,Mezei:2016wfz}. In both of these scenario, there shall be a smallest subregion which will contain enough information (at later time $t$) about the local perturbation. It is aasumed that the bulk dual to this smallest subregion of the boundary is the entanglement wedge \cite{Czech:2012bh}. The Butterfly velocity represents the rate at which these subregions increases. The holographic computation requires only the near-horizon data about the dual gravitational solution. By following the approach given in \cite{Mezei:2016wfz}, a expression for the Butterfly velocity $v_B$ corresponding to a general black brane geometry has been computed in \cite{DiNunno:2021eyf}. For hyperscaling violating geometry (given in eq. (\ref{2})), the expression for $v_B$ is obtained to be
\begin{eqnarray}\label{41}
v_B = \sqrt{\frac{1}{2}\left(1+\frac{z}{d-\theta-1}\right)} \left[\frac{4\pi}{(d-1)\left(1+\frac{z}{d-\theta-1}\right)}\right]^{1-\frac{1}{z}} T^{1-\frac{1}{z}} \propto T^{1-\frac{1}{z}}.
\end{eqnarray}
In the AdS limit $z\rightarrow 1, \theta\rightarrow 0$, it reduces to the well-known result $v_B = \sqrt{\frac{d}{2(d-1)}}$ \cite{Mezei:2016wfz}. It is known that $v_B$ is a model dependent parameter which in this case captures the collective effects of $z$ and $\theta$. It is also to be noted that for hyperscaling violating backgrounds, $v_B$ is related to the Hawking temperature (temperature of the dual thermal field theory) \cite{Mezei:2016wfz,Blake:2016wvh}.
\section{Holographic Subregion Complexity}\label{sec5}
The Quantum complexity (QC) can be realised in the following way. Considering a simple (unentangled) product state $\ket{\uparrow\uparrow...\uparrow}$ as a reference state, QC is defined as the minimum number of $2$-qubit unitary operation required to prepare a target state $\ket{\psi}$ from the reference state. In this section we study the holographic subregion complexity (HSC) proposal \cite{Alishahiha:2015rta}. This conjecture states that the volume enclosed by the co-dimension two static minimal surface (RT surface) with the boundary coinciding with that of the subsystem, is dual to the complexity of that subregion. For the hyperscaling violating geometry, this co-dimension one volume reads
\begin{eqnarray}\label{24}
V(\Gamma^{min}_A) &=& 2L^{d-2} r_t^{\frac{p+\theta}{p-\theta}-p} \int_{\epsilon/r_t}^{1}du \frac{u^{\frac{2\theta - p}{p-\theta}-p}}{\sqrt{f(u)}}\int_{u}^{1} dk \frac{k^{p+\frac{\theta}{p-\theta}}}{\sqrt{f(k)}\sqrt{1-k^{2p}}}.
\end{eqnarray}
We now use the above volume to obtain the HSC. This is given by (setting AdS radius $R=1$) \cite{Alishahiha:2015rta}
\begin{eqnarray}\label{25}
C_V(A)&=& \frac{V(\Gamma^{min}_A)}{8\pi G_{d+1}}\nonumber\\
&=&\frac{L^{d-2}l}{8\pi G_{d+1} \left(p-\frac{\theta}{p-\theta}\right)\epsilon^{\left(p-\frac{\theta}{p-\theta}\right)}}+\frac{L^{d-2}r_t^{\frac{p+\theta}{p-\theta}-p}}{8\pi G_{d+1}} \sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\bar{V} \left(\frac{r_t}{r_h}\right)^{(m+n)p\left(1+\frac{z}{p-\theta}\right)}
\end{eqnarray}
where
\begin{eqnarray}
\bar{V}= \frac{\Gamma(n+\frac{1}{2})\Gamma(m+\frac{1}{2})\Gamma\left(\frac{(m+n)p(1+\frac{z}{p-\theta})+1+\frac{2\theta}{p-\theta}}{2p}\right)}{\sqrt{\pi}\Gamma(n+1)\Gamma(m+1)\Gamma\left(\frac{(m+n)p(1+\frac{z}{p-\theta})+1+p+\frac{2\theta}{p-\theta}}{2p}\right)}\left[\frac{1}{p\left(mp(1+\frac{z}{p-\theta})+\frac{\theta}{o-\theta}-p\right)}\right]\nonumber~.
\end{eqnarray}
The first term in eq.(\ref{25}) is the divergent piece of the HSC. In the subsequent analysis we will denote it as $C^{div}$. We now consider the small temperature limit, that is $\frac{r_t}{r_h}\ll 1$. In the spirit of this limit, we keep terms up to $\mathcal{O}(m)$ in eq.(\ref{25}). This in turn means that we are interested only in the leading order temperature corrections to the HSC. These considerations lead to the following expression of HSC
\begin{eqnarray}
C_V(A) = C^{div}+ \frac{L^{d-2}}{8\pi G_{d+1}} \bar{V}_{(0)}r_t^{\frac{p+\theta}{p-\theta}-p} + \frac{L^{d-2}}{8\pi G_{d+1}} \left[\bar{V}_{(1)}+\bar{V}_{(2)}\right]r_t^{\frac{p+\theta}{p-\theta}-p} \left(\frac{r_t}{r_h}\right)^{p\left(1+\frac{z}{p-\theta}\right)}
\end{eqnarray}
where $\bar{V}_{(0)} = \bar{V}|_{(n=0,m=0)}$, $\bar{V}_{(1)} = \bar{V}|_{(n=1,m=0)}$ and $\bar{V}_{(2)} = \bar{V}|_{(n=0,m=1)}$. We now use eq.(\ref{19}) in order to express the above expression in terms of the subsystem size $l$. This is obtained to be
\begin{eqnarray}\label{39}
C_V(A) = C^{div}+C_1l^{(\frac{p+\theta}{p})-(p-\theta)} + C_2 l^{(\frac{p+\theta}{p})+z}T^{1+(\frac{p-\theta}{z})}
\end{eqnarray}
where the expressions for $C_1$ and $C_2$ are given in the Appendix. As we have mentioned earlier, $C^{div}$ represents the divergent piece of HSC whereas the second term is the temperature independent term. The lowest order temperature correction occurs in the third term. This result for the HSC will be used in the next section to compute the mutual complexity between two subsystems.
\section{Complexity for mixed states: Mutual complexity ($\Delta\mathcal{C}$)}\label{sec6}
The complexity for mixed states (purification complexity) is defined as the minimal (pure state) complexity among all possible purifications of the mixed state. This in turn means that one has to optimize over the circuits which take the reference state to a target state $\ket{\psi_{AB}}$ (a purification of the desired mixed state $\rho_A$) and also need to optimize over the possible purifications of $\rho_A$. This can be expressed as
\begin{eqnarray}
\mathcal{C}(\rho_{A})= \mathrm{min}_{B}~ \mathcal{C}(\ket{\psi_{AB}});~ \rho_A = \mathrm{Tr}_B \ket{\psi_{AB}}\bra{\psi_{AB}}
\end{eqnarray}
where $A^c=B$. Recently a quantity denoted as the `mutual complexity ($\Delta\mathcal{C}$)' has been defined in order to compute the above mentioned mixed state complexity \cite{Alishahiha:2018lfv,Caceres:2018blh}. The computation of $\Delta\mathcal{C}$ starts with a pure state $\rho_{AB}$ in an extended Hilbert space (including auxillary degrees of freedom), then by tracing out the degrees of freedom of $B$, one gets the mixed state $\rho_{A}$. On the other hand, tracing out the degrees of freedom of $A$ yields $\rho_B$. These computed results then can then be used in the following formula to compute the mutual complexity $\Delta\mathcal{C}$ \cite{Alishahiha:2018lfv}
\begin{eqnarray}\label{26}
\Delta\mathcal{C} = \mathcal{C}(\rho_{A})+ +\mathcal{C}(\rho_{B})-\mathcal{C}(\rho_{A\cup B})~.
\end{eqnarray}
The mutual complexity $\Delta\mathcal{C}$ is said to be subadditive if $\Delta\mathcal{C}>0$ and superadditive if $\Delta\mathcal{C}<0$.\\
We now choose to follow the subregion `Complexity=Volume' conjecture to compute the quantities $\mathcal{C}(\rho_{A})$, $\mathcal{C}(\rho_{B})$ and $\mathcal{C}(\rho_{A \cup B})$. Similary one can follow the `Complexity=Action' conjecture or $C=V2.0$ conjecture \cite{Couch:2016exn} to compute these quantities. We consider two different set up to probe the mutual complexity $\Delta\mathcal{C}$. In the first scenario, we consider two disjoint subsystems $A$ and $B$ of width $l$ on the boundary Cauchy slice $\sigma$. These two subsystems are separated by a distance $x$. We then compute the mutual complexity between these two subregions. Next we consider that the boundary Cauchy slice $\sigma$ is a collection of two adjacent subsystems $A$ and $B$ of width $l$ with $A \cap B =0$ (zero overlap) and $A^c=B$. In this set up we compute the mutual complexity between a subregion $A$ and the full system $A \cup A^c$.
\subsection{Case 1: Mutual complexity between two disjoint subregions}
In this set up, we assume the two subsystems $A$ and $B$ of width $l$ on the Cauchy slice $\sigma$. The separation length between $A$ and $B$ is $x$. We want to see how the rate of complexification of these two subsystems get affected when we introduce correlation (classical and quantum) between these two subregions. In \cite{Auzzi:2019vyh}, the authors have used the `Complexity=Action' conjecture to study the mutual complexity between two subsystems and in \cite{Ghodrati:2019hnn} `Complexity = Volume' conjecture was incorporated to probe $\Delta\mathcal{C}$ between two subregions. Here we follow the approach given in \cite{Ghodrati:2019hnn}.
\subsubsection{BTZ black hole}
We first compute the HSC corresponding to a single subsystem $A$ of length $l$ for the BTZ black hole. The mentioned black hole geometry is characterized by the following metric \cite{Banados:1992wn,Saha:2018jjb}
\begin{eqnarray}
ds^2 = \frac{R^2}{z^2}\left[-f(z) dt^2+\frac{dz^2}{f(z)}+dx^2\right];~f(z)=1-\frac{z^2}{z_h^2}~.
\end{eqnarray}
Following the prescription of $C=V$ conjecture, the HSC corresponding to a single subsystem $A$ of length $l$ in the dual field theory is obtained by computing the co-dimension $1$ volume enclosed by the co-dimension two RT-surface. This leads to the following
\begin{eqnarray}
\mathcal{C}(\rho_{A})&=& \frac{2R}{8\pi R G_{2+1}} \int_{0}^{z_t} \frac{x(z)}{z^2 \sqrt{f(z)}} dz\nonumber\\
&=& \frac{2R}{8\pi R G_{2+1}} \int_{0}^{z_t} \frac{1}{z^2 \sqrt{f(z)}} dz \int_{z}^{z_t} \frac{du}{\sqrt{f(u)}\sqrt{(\frac{u}{z_t})^2-1}}\nonumber\\
&=& \frac{1}{8\pi} \left(\frac{l}{\epsilon}-\pi\right)~(setting~ G_{2+1}=1)
\end{eqnarray}
where $\epsilon$ is the cut-off introduced to prevent the UV divergence and $z_t$ is the turning point of the RT surface. It is to be observed that the computed result of HSC in this case is independent of the black hole parameter (that is, event horizon $z_h$). This is a unique feature of AdS$_3$ as the subregion complexity in this case is topological. We now consider two subsystems $A$ and $B$ of equal length $l$, separated by a distance $x$. In this set up, the connected RT surface is governed by two strips of length $2l+x$ and length $x$ \cite{Ben-Ami:2016qex}. This leads to the following \cite{Ghodrati:2019hnn,Ben-Ami:2016qex}
\begin{eqnarray}\label{38}
\mathcal{C}(A \cup B) = \mathcal{C}(2l+x) - \mathcal{C}(x)~.
\end{eqnarray}
Note that when the separation $x$ between the two subsystems vanishes, then
\begin{eqnarray}
\mathcal{C}(l\cup l)=\mathcal{C}(2l)~.
\end{eqnarray}
By substituting eq.(\ref{38}) in eq.(\ref{26}), the mutual complexity is obtained to be
\begin{eqnarray}
\Delta\mathcal{C}
&=& 2 \mathcal{C}(l) - \mathcal{C}(2l+x) + \mathcal{C}(x)= - \frac{1}{4}~.
\end{eqnarray}
It can be observed that the mutual complexity is less than zero. This implies that the complexity is superadditive. In the above computation of $\mathcal{C}(A\cup B)$, we have considered only the connected RT surface. This is true as long as we work within the limit $\frac{x}{l}\ll 1$. However, there can be disconnected configuration also in which $\mathcal{C}(A\cup B) = \mathcal{C}(A) + \mathcal{C}(B)$ \cite{Ben-Ami:2016qex}. This in turn means that when the separation length $x$ is large enough, mutual complexity between two subregions $\Delta\mathcal{C}$ is zero. This is similar to mutual information.
\subsubsection{Hyperscaling violating geometry}
By considering the same set up, we now compute the mutual complexity between $A$ and $B$ for the hyperscaling violating geometry. The HSC corresponding to a single strip of length $l$ is given in eq.(\ref{25}). We use this result to compute the complexities $\mathcal{C}(l)$, $\mathcal{C}(2l+x)$ and $\mathcal{C}(x)$. This leads to the following expression of mutual complexity
\begin{eqnarray}
\Delta\mathcal{C}&=&2 \mathcal{C}(l) - \mathcal{C}(2l+x) + \mathcal{C}(x)\nonumber\\
&=& \frac{L^{d-2}}{8\pi G_{d+1}} \sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\bar{V}\Big[2r_t(l)^{\frac{p+\theta}{p-\theta}-p} \left(\frac{r_t(l)}{r_h}\right)^{(m+n)p\left(1+\frac{z}{p-\theta}\right)}+r_t(x)^{\frac{p+\theta}{p-\theta}-p} \left(\frac{r_t(x)}{r_h}\right)^{(m+n)p\left(1+\frac{z}{p-\theta}\right)}\nonumber\\
&&-r_t(2l+x)^{\frac{p+\theta}{p-\theta}-p} \left(\frac{r_t(2l+x)}{r_h}\right)^{(m+n)p\left(1+\frac{z}{p-\theta}\right)}\Big]
\end{eqnarray}
where $r_t(l)$, $r_t(2l+x)$ and $r_t(x)$ corresponds to the turning points of the RT surfaces associated to $l$, $2l+x$ and $x$ respectively. It is to be observed that the divergent pieces of HSC cancels out which yields a finite result. We now consider the small temperature limit $\frac{r_t(x)}{r_h}\ll\frac{r_t(l)}{r_h}\ll\frac{r_t(2l+x)}{r_h}\ll1$ and keep terms up to $\mathcal{O}(m)$. This in turn yields the following expression for mutual complexity
\begin{eqnarray}
\Delta\mathcal{C}&=&C_1\left[2l^{(\frac{p+\theta}{p})-(p-\theta)} -(2l+x)^{(\frac{p+\theta}{p})-(p-\theta)}+x^{(\frac{p+\theta}{p})-(p-\theta)}\right]\nonumber\\
&+&C_2\left[2l^{z+\frac{p+\theta}{p}} -(2l+x)^{z+\frac{p+\theta}{p}}+x^{z+\frac{p+\theta}{p}}\right]T^{1+\left(\frac{p-\theta}{z}\right)}~.
\end{eqnarray}
Similar to the study of mutual information, we can also point out a critical separation length $x_c$ at which the above expression is zero.
\begin{figure}[!h]
\begin{minipage}[t]{0.48\textwidth}
\centering\includegraphics[width=\textwidth]{mutualCt.eps}\\
{\footnotesize Effect of hyperscaling violating exponent $\theta$ (we set $z=1$)}
\end{minipage}\hfill
\begin{minipage}[t]{0.48\textwidth}
\centering\includegraphics[width=\textwidth]{mutualCz.eps}\\
{\footnotesize Effect of dynamical exponent $z$ (we set $\theta=0.2$)}
\end{minipage}
\caption{Effects of $\theta$ and $z$ on $\Delta\mathcal{C}$ (with $d=3$, $k=0.4$, $L=1$ and $G_{d+1}=1$)}
\label{fig4}
\end{figure}
In Fig.(\ref{fig4}), we have graphically represented the computed result of $\Delta\mathcal{C}$. In the above plots, we have introduced $k=\frac{x}{l}$ in order to compute the critical separation length $x_c$ at which $\Delta\mathcal{C}=0$. These plots also represent the collective effects of $z$ and $\theta$ on the mutual complexity.
\subsection{Case 2: Mutual complexity between two adjacent subsystems}
We now consider that the boundary Cauchy slice $\sigma$ is composed of two adjacent subsystems $A$ and $B$ of width $l$. Further, we assume $A \cap B =0$ (zero overlapping) and $A^c=B$ and the full system (on $\sigma$) is in a pure state. In this set up we compute the mutual complexity between a subregion $A$ and the full system $A \cup A^c$ \cite{Agon:2018zso,Caceres:2019pgf}.
\subsubsection{BTZ black hole}
We now proceed to compute the mutual complexity between $A$ and $B=A^c$. In this set up, the connected RT surface is composed of one strip of length $2l$. This leads to the following expression for mutual complexity
\begin{eqnarray}
\Delta \mathcal{C} &=& 2\mathcal{C}(l)-\mathcal{C}(2l)\nonumber\\
&=&\frac{1}{8 \pi} \left[\frac{l}{\epsilon}-\pi+\frac{l}{\epsilon}-\pi-\frac{2l}{\epsilon}+\pi\right]\nonumber\\
&=& - \frac{1}{8}.
\end{eqnarray}
Similar to the disjoint subregion case, mutual complexity in this set up is also superadditive. This in turn means that the complexity of the state corresponding to the full system is greater than the sum of the complexities of the states in the two subsystems.
\subsection{Hyperscaling violating geometry}
We now proceed to compute the mixed state complexity for the hyperscaling violating geometry. By using the HSC result given in eq.(\ref{25}), $\Delta\mathcal{C}$ in this set up reads
\begin{eqnarray}\label{27}
\Delta\mathcal{C} &=& \frac{1}{8\pi G_{d+1}} \left[V(\Gamma^{min}_A)+V(\Gamma^{min}_B)-V(\Gamma^{min}_{A \cup B})\right]\nonumber\\
&=&\frac{L^{d-2}}{8\pi G_{d+1}}\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\bar{V}\left[2r_t(l)^{\left(\frac{p+\theta}{p-\theta}\right)-p}\left(\frac{r_t(l)}{r_h}\right)^{(m+n)p\left(1+\frac{z}{p-\theta}\right)}-r_t(2l)^{\left(\frac{p+\theta}{p-\theta}\right)-p}\left(\frac{r_t(2l)}{r_h}\right)^{(m+n)p\left(1+\frac{z}{p-\theta}\right)}\right]~.\nonumber\\
\end{eqnarray}
We now consider the limit $\frac{r_t(l)}{r_h}\ll \frac{r_t(2l)}{r_h} \ll 1 $. In this limit, we keep terms up to order $\mathcal{O}(m)$ in the above expression. This in turn leads to the following expression
\begin{eqnarray}\label{28}
\Delta\mathcal{C} = \left[2-2^{(\frac{p+\theta}{p})-(p-\theta)}\right]C_1l^{\left(\frac{p+\theta}{p}\right)-\left(p-\theta\right)}+\left[2-2^{z+\left(\frac{p+\theta}{p}\right)}\right]C_2 l^{z+\left(\frac{p+\theta}{p}\right)}T^{1+\left(\frac{p-\theta}{z}\right)}~.
\end{eqnarray}
We observe that similar to the BTZ case, the above result for $\Delta C$ is less than zero. This in turn means that the mutual complexity computed using the HSC conjecture yields a superadditive result.
\section{Conclusion}\label{sec7}
In this paper, we compute the entanglement entropy and complexity for mixed states by using the gauge/gravity correspondence. We start our analysis by considering a hyperscaling violating solution as the bulk theory. This geometry is associated with two parameters, namely, hyperscaling violating exponent $z$ and dynamical exponent $\theta$. It is dual to a non-relativistic, strongly coupled theory with hidden Fermi surfaces. We then consider a single strip-like subsystem in order to compute the HEE of this gravitational solution. We observe that the computed result of HEE along with the internal energy $E$, satisfies a Smarr-like thermodynamics relation associated with a generalized temeperature $T_g$. This thermodynamic relation naturally emerges by demanding that the generalized temperature $T_g$ reproduces the Hawking temperature $T_H$ as the leading term in the IR ($r_t \rightarrow r_h$) limit. In UV limit ($\frac{r_t}{r_h}\ll 1$), it is found that $T_g \propto \frac{1}{l^z}$, that is, $T_g$ is inversely proportional to subsystem size $l$. This behaviour is compatible with the definition of entanglement temperature given in the literature. We then holographically compute the relative entropy $S_{rel}$, by incorporating the perturbative approach. Using this the Fisher information metric is computed. We find that in this case the power of $l$ carries both the exponents $z$ and $\theta$. We then consider two strip-like subsystems $A$ and $B$ separated by a length $D$, in order to compute the EWCS ($E_W$) which is the holographic analogy of EoP. We compute $E_W$ for both low and high temperature conditions. In both cases, there is a temperature independent term (denoted as $E_W^{T=0}$) which is independent of the hyperscaling violating exponent $z$ but depends on the dynamical exponent $\theta$. On the other hand for a large enough value of $D$ (critical separation length $D_c$), the RT surface $\Gamma_{AB}^{min}$ becomes disconnected and $E_W$ should vanish. This in turn means that $E_W$ probes the phase transition between the connected and disconnected phases of the RT surface $\Gamma_{AB}^{min}$. We evaluate these critical separation point $D_c$ by using the property that at $D_c$ the mutual information between $A$ and $B$ becomes zero as they become disconnected. This behaviour for $I(A:B)$ and $E_W$ is shown in Fig(s).(\ref{fig2}, \ref{fig3}) for both low and high temperature cases. We observe that $E_W$ always satisfies the property $E_W > \frac{1}{2} I(A:B)$. We then discuss the property of information spreading by computing the Butterfly velocity. We observe that the computed expression of Butterfly velocity explicitly depends on the temperature of the dual field theory. We then compute the HSC by considering again a single strip-like subsystem. The complexity for mixed state is computed by following the concept of mutual complexity $\Delta C$. We have used the HSC conjecture to compute the $\Delta C$ for both BTZ black hole and hyperscaling violating geometry. We have studied the mutual complexity by considering two different set ups. Firstly, we consider two disjoint subsystems $A$ and $B$ of width $l$, separated by a length $x$ on the boundary Cauchy slice $\sigma$. Computation of $\Delta\mathcal{C}$ in this set up probes rate of complexification of these two subsystems when we consider correlation (both classical and quantum) between them. Next we consider a single subsystem $A$ in such a way that $A^c=B$ and $A \cap B=0$. We then measure the mutual complexity between a subsystem $A$ and the full system $A\cup A^c$. We observe that the computed result of mutual complexity is superadditive that is $\Delta C <0$. This in turn means that the complexity of the state corresponding to the full system is greater than the sum of the complexities of the states in the two subsystems. We observe that for BTZ black hole $\Delta \mathcal{C}$ is independent of temeprature however for hyperscaling violating solution it contains a temperature independent term as well as a temperature dependent term. It is to be kept in mind that this nature of $\Delta C$ is observed for the HSC conjecture and similarly one can use `Complexity=Action' conjecture \cite{Alishahiha:2018lfv,Auzzi:2019vyh} or `CV2.0' conjecture to compute $\Delta C$ in this context \cite{Agon:2018zso}.
\section*{Acknowledgements}
A.S. would like to acknowledge the support by Council of Scientific and Industrial Research (CSIR, Govt. of India) for Senior Research Fellowship.
\section*{Appendix}
In this appendix, we give the expressions of quantities that appear in the main text. These are as follows.
\begin{eqnarray*}
\Delta_4&=& \Delta_3 \left(\frac{p}{\sqrt{\pi}}\frac{\Gamma\left(\frac{2p+\frac{p}{p-\theta}}{2p}\right)}{\Gamma\left(\frac{p+\frac{p}{p-\theta}}{2p}\right)}\right)^{(p-\theta)\left(\frac{1-p}{p}\right)}-\frac{\left(\frac{4\pi}{p(1+\frac{z}{p-\theta})}\right)^{1+\frac{p-\theta}{z}}}{2\left(p(1+\frac{z}{p-\theta})-p+1\right)}\left(\frac{p}{\sqrt{\pi}}\frac{\Gamma\left(\frac{2p+\frac{p}{p-\theta}}{2p}\right)}{\Gamma\left(\frac{p+\frac{p}{p-\theta}}{2p}\right)}\right)^{\left(\frac{p-\theta}{p}\right)\left(1+\frac{pz}{p-\theta}\right)}\nonumber\\
\Delta_5 &=& \frac{\sqrt{\pi}}{(p-1)}\left(\frac{4\pi}{p(1+\frac{z}{p-\theta})}\right)^{(p-1)(\frac{p-\theta}{pz})}\frac{\Gamma\left(\frac{1+\frac{pz}{p-\theta}}{p(1+\frac{z}{p-\theta})}\right)}{\Gamma\left(\frac{2-p+\frac{pz}{p-\theta}}{2p(1+\frac{z}{p-\theta})}\right)}\nonumber\\
\end{eqnarray*}
\begin{eqnarray*}
C_1 &=& \frac{L^{d-2}}{8\pi G_{d+1}} \bar{V}_{(0)} \left(\frac{p}{\sqrt{\pi}}\frac{\Gamma\left(\frac{2p+\frac{p}{p-\theta}}{2p}\right)}{\Gamma\left(\frac{p+\frac{p}{p-\theta}}{2p}\right)}\right)^{(\frac{p+\theta}{p})-(p-\theta)}\nonumber\\
C_2 &=& \left[(\bar{V}_{(1)}+\bar{V}_{(2)})\left(\frac{p}{\sqrt{\pi}}\frac{\Gamma\left(\frac{2p+\frac{p}{p-\theta}}{2p}\right)}{\Gamma\left(\frac{p+\frac{p}{p-\theta}}{2p}\right)}\right)^{(\frac{p+\theta}{p})+z}-\left(\frac{p+\theta}{p-\theta}-p\right)\bar{V}_{(0)}\Delta_3\left(\frac{p}{\sqrt{\pi}}\frac{\Gamma\left(\frac{2p+\frac{p}{p-\theta}}{2p}\right)}{\Gamma\left(\frac{p+\frac{p}{p-\theta}}{2p}\right)}\right)^{(\frac{p+\theta}{p})-(p-\theta)}\right]\nonumber\\
&&\times \frac{L^{d-2}}{8\pi G_{d+1}} \left[\frac{4\pi}{p(1+\frac{z}{p-\theta})}\right]^{1+(\frac{p-\theta}{z})}~.\nonumber
\end{eqnarray*}
\bibliographystyle{hephys}
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\section{Introduction}\label{intro}
The evolution of a collisionless stellar system is determined by the
collisionless Boltzmann equation (CBE)
\begin{equation}
\frac{\partial F(\vec{x},\vec{v},t)}{\partial t} +
\vec{v}\cdot\frac{\partial F(\vec{x},\vec{v},t)}{\partial
\vec{x}}+ \frac{\partial V(\vec{x},t)}{\partial
\vec{x}}\cdot\frac{\partial F(\vec{x},\vec{v},t)}{\partial
\vec{v}}=0.
\end{equation}
Here, $F(\vec{x},\vec{v},t)$ is the distribution function (DF), which
gives the stellar phase-space density at location $(\vec{x},\vec{v})$
and time $t$. The motion of the collisionless fluid is controlled by
the (positive) binding potential $V(\vec{x},t)$. The direct numerical
integration of the CBE in six-dimensional phase space is in general
impossible because under the CBE the DF develops ever finer structures
owing to phase mixing or chaotic mixing. However, numerical schemes
which smooth out such fine structure (whereby violating the CBE) are
possible but taxing
\citep{2013ApJ...762..116Y,2014MNRAS.442.3073S,2015MNRAS.450.3724C}.
A much more popular method for modelling collisionless stellar
dynamics is an $N$-body simulation:~a Monte-Carlo approach, which
integrates the CBE via the method of characteristics. The DF is
represented by a collection of phase-space points, called
``particles'', and each particle is evolved through phase space along
its characteristic curve by solving the first-order differential
equations
\begin{subequations}
\begin{align}
\frac{\d \vec{x}}{\d t} &= \vec{v}, \\ \frac{\d \vec{v}}{\d t}
&= \frac{\partial V(\vec{x},t)}{\partial \vec{x}}.
\end{align}
\end{subequations}
In the case of gravitational forces, the binding potential can be
written as
\begin{subequations}
\label{eqs:V}
\begin{align}
V(\vec{x},t) &= G \int \frac{\rho(\vec{x}',t)\,\d
\vec{x}'}{|\vec{x}-\vec{x}'|} \\ \label{eq:V:unsoft}
&\approx G \sum_i \frac{m_i}{|\vec{x}-\vec{x}_i|}
\end{align}
\end{subequations}
with $\rho$ the stellar mass density and $m_i$ and $\vec{x}_i$ the
mass and position, respectively, of the $i^{\text{th}}$
particle. Using the approximation~{\eqref{eq:V:unsoft}} to the
gravitational field, diverging accelerations may occur in close
encounters (`collisions') between particles. Such collisions are an
artefact of the much smaller $N$ in the simulation than in the
simulated system. This problem is generally solved by ``softening''
gravity, when the $1/r$ potential in equations~\eqref{eqs:V} is
replaced by a non-diverging form,
\begin{equation}
\label{eq:soft:kern}
\frac{1}{r} \;\to\; \pot(r) = \frac{1}{\eps}
\phi\left(\frac{r}{\eps}\right),
\end{equation}
where $\pot(r)$ is the softening Green's function, $\eps$ the
\emph{softening length}, and $\phi$ the dimensionless \emph{softening
kernel}. For suitable functions $\phi$ this modifies the
inter-particle interactions such that accelerations remain bounded
and strongly deflecting encounters are avoided.
Unfortunately, this force modification results in a \emph{bias} of
gravity and hence changes the character of the physical problem being
addressed by the $N$-body simulation. In practice, a balance must be
found between too much softening, causing force bias, and too little
softening, allowing strong encounters that render the $N$-body
dynamics intractable and collisional
\citep{1996AJ....111.2462M}. \cite{2001MNRAS.324..273D} has derived
asymptotic relations in the context of spherically symmetric systems,
which can be used to inform the choice of the softening parameters
(kernel and length) such as to minimise the resulting mean-square
gravity error.
However, it remains unclear what the optimal choice of these
parameters is in terms of accurately modelling the dynamics, rather
than merely minimising the gravity error. The goal of this series of
papers is to investigate this question by considering stellar
dynamical problems which invoke non-trivial dynamics but are still
simple enough that accurate solutions of the CBE are
available. Specifically, we consider unstable two- and
three-dimensional systems, whose eigenmodes can be accurately obtained
from linear perturbation theory. In this first paper, we focus on the
two-armed (multiplicity $m=2$) spiral-shaped eigenmodes of
two-dimensional razor-thin disc galaxies in the limit of $N\to\infty$
(which can be achieved with computationally cheap linear mode analysis
without $N$-body simulations).
Research into the origin and longevity and/or transience of spiral
structure in disc galaxies has by and large relied on $N$-body
simulations
\citep{1971ApJ...168..343H,1977ARA&A..15..437T,sellwood89,sellwood91,
2011MNRAS.410.1637S,ovh13,2014ApJ...785..137S} and on
(semi-)analytical mode analysis using the first-order CBE as a
starting point
\citep{1976PhDT........26Z,kalnajs77,polyfrid,palmer,b9,1997MNRAS.291..616P,1998MNRAS.300..106E,jalali05,jalali07,bintrem,poly15}. Here,
we use linear theory to investigate how gravitational softening affects
the growth of small-amplitude eigenmodes in $N$-body simulations.
We introduce the various softening techniques explored by us in
section \ref{sec:soft:2D}. Our implementation of gravitational
softening in linear mode theory is layed out in section
\ref{pystab}. The properties of the unsoftened axially symmetric disc
models are discussed in section \ref{themodel}. Our results are
presented in section \ref{sec:results} and their implications are
discussed in section \ref{sec:dis}.
\section{Gravitational softening in two spatial dimensions} \label{sec:soft:2D}
The usual motivation for softening in two-dimensional $N$-body
simulations is to account for the finite thickness of the stellar
disc, which is neglected in the razor-thin limit
\citep{1971Ap&SS..14...73M}. In this interpretation, the softening
length $\eps$ is no longer a numerical but a \emph{physical} parameter
and the modification of gravity and, consequently, of the dynamics is
deliberate, because real galaxies are not razor-thin
\citep{1992MNRAS.256..307R,1997A&A...324..523R}.
However, as we are interested in the errors introduced by the
softening-induced modification of gravity, we cannot adopt this
interpretation but must consider the razor-thin disc as the desired
physical model, whose modes one may attempt to recover with an
$N$-body simulation.
\subsection{Gravity bias} \label{sec:soft:error}
When inserting the softening kernel~\eqref{eq:soft:kern} into
equation~\eqref{eq:V:unsoft}, we obtain the softened potential
\begin{equation}
\label{eq:V:soft}
V(\vec{x}) \approx \hat{V}(\vec{x}) = \sum_i\frac{Gm_i}{\eps}
\phi\left(\frac{|\vec{x}-\vec{x}_i|}{\eps}\right).
\end{equation}
Here, $\hat{V}(\vec{x})$ can be interpreted as an \emph{estimate},
based on the masses and positions of the simulation particles, for the
true potential $V(\vec{x})$. The mean-square error made by this
estimate can be decomposed into a variance,
\begin{equation}
\mathrm{var}_{\vec{x}}(\hat{V}) = \left\langle \left[ \left\langle
\hat{V}(\vec{x})\right\rangle-\hat{V}(\vec{x})\right]^2 \right\rangle,
\end{equation}
and a bias,
\begin{equation}
\mathrm{bias}_{\vec{x}}(\hat{V}) = \left\langle\hat{V}(\vec{x})\right\rangle-V(\vec{x}).
\end{equation}
Here, $\langle\cdot\rangle$ denotes the ensemble average of a quantity
over all $N$-body realisations, in the limit $N\to\infty$, of the
underlying smooth density distribution. The variance measures the mean
amplitude of the random fluctuations of a softened $N$-body potential
around its ensemble mean. In other words, it measures the graininess
of the $N$-body potential:~the error made by \emph{not softening
enough}. The bias measures the deviation of the ensemble mean of the
softened $N$-body potential from the underlying smooth potential. This
is the error made by \emph{softening too much}.
For the situation of three-dimensional $N$-body simulations,
\cite{2001MNRAS.324..273D} has derived analytical asymptotic relations
for these quantities. An adaptation of his derivations to an $N$-body
simulation of a razor-thin disc with surface density $\Sigma(\vec{x})$
gives
\begin{equation}
\label{eq:bias:V:2D}
\mathrm{bias}_{\vec{x}}(\hat{V}) =
a_0\,\eps\,G\,\Sigma(\vec{x}) +a_2\,\eps^3
G\,\vec{\nabla}^2\Sigma(\vec{x}) + \mathcal{O}(\eps^5),
\end{equation}
for the bias on the potential $V$ (see Appendix \ref{app1} for a
derivation). The coefficients $a_n$ depend only on the functional form of
the softening kernel:
\begin{equation}
a_n =
\frac{2\pi}{2^n\,([n/2]!)^2}\int_0^\infty\left[1-u\phi(u)\right]u^n\d
u. \label{an}
\end{equation}
This is different for three-dimensional systems, where this bias
asymptotes as $\eps^2$ at lowest order. Here, in two dimensions, the
gravity biases are proportional to $\eps$. This is a direct
consequence of the reduced number of dimensions. Thus, in
two-dimensional $N$-body simulations, the gravity bias is in general
significantly stronger than in three-dimensional $N$-body
simulations\footnote{\label{foot:bias:3D} Except in three-dimensional
simulations of disc galaxies with scale height $h\apprle\eps$, since
in three dimensions
\begin{subequations}
\label{eq:bias:3D}
\begin{align}
\mathrm{bias}_{\vec{x}}(\hat{V}) &\approx
-a_0^{\mathrm{3D}}\eps^2G\rho(\vec{x})
\\ \mathrm{bias}_{\vec{x}}(\hat{\vec{a}}) &\approx
-a_0^{\mathrm{3D}}\eps^2G\vec{\nabla}\rho(\vec{x})\propto\eps^2/h
\end{align}
\end{subequations}
\citep[][eqs.~10]{2001MNRAS.324..273D} with $\rho(\vec{x})$ the
spatial density.}.
Note that the integrand in equation (\ref{an}) is always well-behaved
in the limit $u \to 0$ but can be problematic for large $u$. If
$1-u\phi(u) \propto u^{-p}$ for large $u$ then only coefficients $a_n$
with $n<p-1$ are finite; the rest come out infinite. In this case,
the Taylor series~\eqref{eq:bias:V:2D} does not converge, but the
terms at $n<p-1$ still provide a useful approximation, only the
remainder grows faster than $\eps^{2\lceil (p-1)/2\rceil}$.
\begin{table*}{
\caption{\label{tab:soft} Characteristics of the various
softening methods used in this study. Here, $\phi$,
$\varrho$, and $\sigma$ take argument $u=r/\eps$ and are the
dimensionless kernels for, respectively, potential, spatial
and surface density, defined in
equations~\eqref{eq:soft:kern} and
\eqref{eqs:soft:dens:surf}. The coefficients $a_0$ and $a_2$
determine the gravitational biases (see
equation~\ref{eq:bias:V:2D}). Here, $\varepsilon_0$
quantifies the scaling of the kernels with non-zero $a_0$ to
a common level of gravity bias and $\varepsilon_F$
quantifies the scaling of the kernels to a common maximum
inter-particle force. A dagger indicates softening methods
with compact support, where the formul\ae{} for $\phi$ and
$\varrho$ only apply at $r<\eps$ or
$t\equiv1-u^2\in[0,1]$. For these methods, no sensible
razor-thin surface-density kernel $\sigma(u)$ can be
provided. }
\begin{tabular}{lllllcccc}
\multicolumn{2}{c}{name}
& $\phi(u)$ & $\varrho(u)$ & $\sigma(u)$ & $a_0$ & $a_2$ & $\varepsilon_0/\eps$ & $\varepsilon_F/\eps$ \\
\hline
& Newton & $\displaystyle\frac{1}{u}$
& $\displaystyle\delta_{\mathrm{3D}}(u)$
& $\displaystyle\delta_{\mathrm{2D}}(u)$
& 0 & 0 & 0 & 0
\\[2ex]
P$_0$ & Plummer & $\displaystyle\frac{1}{\sqrt{1+u^2}}$
& $\displaystyle\frac{3}{4\pi}\frac{1}{(1+u^2)^{5/2}}$
& $\displaystyle\frac{1}{2\pi}\frac{1}{(1+u^2)^{3/2}}$
& $2\pi$ & $\infty$ & 1 & 1
\\[3ex]
Q$_2$ &
$\begin{array}{l}\text{2$^\mathrm{nd}$\,modified}\\ \text{Kuz'min}\end{array}$
& $\displaystyle\frac{3+\tfrac52u^2+u^4}{(1+u^2)^{5/2}}$
& $\displaystyle\frac{15}{8\pi}\frac{4-3u^2}{(1+u^2)^{9/2}}$
& $\displaystyle\frac{3}{4\pi}\frac{4-u^2}{(1+u^2)^{7/2}}$
& 0 & $\displaystyle-\frac{\pi}{3}$ & --- & 2.568
\\[3ex]
$^{\dag}$F$_3$ & Ferrers $n=3$
& $1+\frac12t+\frac38t^2+\frac5{16}t^3+\frac{35}{128}t^4$
& $\displaystyle \frac{315}{64\pi}t^3$
& ill-behaved
& $\displaystyle\frac{63\pi}{128}$
& $\displaystyle\frac{7\pi}{1024}$
& $\displaystyle\frac{63}{256}$ & 2.309
\\[3ex]
$^{\dag}$L$_2$ &
$\begin{array}{l}\text{2D modified}\\ \text{Ferrers $n=2$}\end{array}$
& $1+\frac12t+\frac38t^2+\frac{5}{2}t^3$
& $\displaystyle\frac{105}{8\pi}(1-2u^2)t$
& ill-behaved
& $\displaystyle0$
& $\displaystyle-\frac{7\pi}{384}$
& --- & 3.711
\\
\hline
\end{tabular}
}
\end{table*}
\subsection{Softening kernels} \label{sec:soft:kernels}
\begin{figure}
\includegraphics[trim=15 15 10 10,clip,width=0.475\textwidth]{Id}
\caption{Top panel:~the dimensionless interaction potential of the
softening recipes listed in Table~\ref{tab:soft}. Bottom panel:~the
corresponding interparticle forces. The dimensionless interaction
potentials (top) and inter-particle forces (bottom) of these
softening kernels are listed in Table \ref{tab:soft}. The kernels
are scaled according to equations~\eqref{eq:eps:scaling} and
\eqref{eq:eps:scale:force} to obtain a common maximum inter-particle
force (as is obvious in the bottom panel), see also Section
\ref{sec:soft:scale}. Note that kernels with $a_0=0$ (Q$_2$ and
L$_2$) have super-Newtonian gravity at intermediate scales to
compensate for the sub-Newtonian behaviour at $r\to0$.
\label{fig:Id}}
\end{figure}
The functional forms and other properties of the softening kernels
used in this study are listed in Table~\ref{tab:soft}. Their
interparticle interaction potentials and forces are plotted in Figure
\ref{fig:Id}. In particular, we list the dimensionless interparticle
interaction potential $\phi$ and the corresponding three-dimensional
and two-dimensional dimensionless density kernels, denoted by
$\varrho$ and $\sigma$, respectively, assigned to each point particle.
In three dimensions, gravitational softening is equivalent to
estimating the spatial mass density as a superposition of spheres with
density distribution $\rho(r)$ placed at the particle positions. In
two dimensions, one can consider softening as a way of smoothing the
overall mass distribution as a superposition of razor-thin discs with
density distribution $\Sigma(r)$ at the particle positions. For a
softening kernel $\phi$, these spherical or razor-thin surface density
distributions are given by
\begin{equation}
\rho(r) = \frac{m}{\eps^3} \varrho\left(\frac{r}{\eps}\right)
\qquad\text{and}\qquad \Sigma(r) = \frac{m}{\eps^2}
\sigma\left(\frac{r}{\eps}\right), \label{eq:soft:rS}
\end{equation}
where the dimensionless density and surface-density kernels are given
by
\begin{subequations}
\label{eqs:soft:dens:surf}
\begin{align}
\label{eq:soft:dens:kernel}
\varrho(u) &= -\frac{1}{4\pi u^2}\frac{\d}{\d
u}\left(u^2\frac{\d\phi}{\d u} \right), \\[1ex]
\label{eq:soft:surf:kernel}
\sigma(u) &= -\frac{1}{\pi^2} \int_{u}^\infty\frac{\d
x\,}{\sqrt{x^2-u^2}} \frac{\d}{\d x}\int_0^{x} \frac{x\,\d
t}{\sqrt{x^2-t^2}}\,\frac{\d\phi(t)}{\d t},
\end{align}
\end{subequations}
respectively \citep{bintrem,1999ASPC..165..325K}.
Note that the softened force is that between a softened particle (with
density distribution $\rho(r)$ or $\Sigma(r)$ given by
eqn. (\ref{eq:soft:rS})) and a point particle and {\em not} the force
between two softened particles \citep{2012MNRAS.425.1104B}. This
remark applies to all softening techniques.
\subsubsection{Plummer softening P$_0$:~infinite support, $a_0 \ne 0$} \label{sec:soft:P0}
In three dimensions, this popular softening method corresponds to
estimating the spatial mass density as a superposition of Plummer
spheres \citep{1911MNRAS..71..460P} with scale radius $\eps$ at the
particle positions. In two dimensions, Plummer softening amounts to
smoothing the overall mass distribution as a superposition of
razor-thin Kuz'min discs \citep{Kuzmin56}.
Plummer softening modifies the gravitational interaction at all
interparticle separations and asymptotes to the Newtonian interaction
only for infinitely large particle separations. As a result,
$a_0^{\mathrm{3D}}=\infty$ and the gravity bias~\eqref{eq:bias:3D} in
this case grows faster than $\propto\eps^2$. In two dimensions, this
method's non-zero $a_0$ indicates that the gravity bias increases
linearly with softening length.
\subsubsection{Modified Kuz'min softening Q$_2$:~infinite support, $a_0 =0$} \label{sec:soft:Q2}
One can modify the Plummer kernel such that $a_0$, $a_2$, {\ldots},
$a_k=0$, for any chosen even $k$, in order to significantly reduce the
gravity bias. For instance, we introduce the class of modified Kuz'min
potentials, with an interaction potential given by
\begin{equation}
\phi(u) = \frac{K_n(u^2)}{\left( 1+u^2 \right)^{n+1/2}},
\end{equation}
here $K_n(u^2) = \sum_{i=0}^n c_i u^{2i}$ is a polynomial of degree
$n$ in $u^2$ with coefficients $c_i$. We refer to the $n^{\rm th}$
member of this class as Q$_n$. Clearly, the choice $n=0$ yields the
Plummer kernel, or:~Q$_0$=P$_0$.
The extra degree of freedom that comes with the choice $n=1$, can be
exploited to make $a_0=0$. For instance, for this Q$_1$ kernel, one
finds that
\begin{align}
a_0 &\propto \lim_{x \rightarrow \infty} \int_0^x \left( 1 -
\frac{c_0 u + c_1 u^3}{\left( 1+u^2 \right)^{3/2}} \right)\mathrm{d}u \nonumber \\
& =\lim_{x \rightarrow \infty} \left\{
(1-c_1)x + 2c_1-c_0 + {\cal O}\left( \frac{1}{x} \right) \right\}
\end{align}
and the choice $c_0=2$, $c_1=1$ makes $a_0=0$. Thus, the `Q$_1$'
method is defined by the interparticle potential \beqn \phi(u) =
\frac{2+u^2}{(1+u^2)^{3/2}} \neqn and by the corresponding 3D and 2D
density distributions
\begin{equation}
\varrho(u) = \frac{3}{4\pi}\frac{4-u^2}{(1+u^2)^{7/2}}
\,\,\text{ and }\,\, \sigma(u) =
\frac{3}{2\pi}\frac{1}{(1+u^2)^{5/2}}. \label{eq:soft:Q1}
\end{equation}
In order to achieve $a_0=0$, this softening
method compensates with slightly super-Newtonian forces at $r\apprge
1.2\eps$ for the substantially sub-Newtonian accelerations at small
separations. Unfortunately, while we now have $a_0=0$, the Q$_1$
softening kernel still has a diverging second-order coefficient $a_2$,
such that the gravity bias grows faster than $\eps^3$.
The extra degree of freedom provided by coefficient $c_2$ of kernel
Q$_2$ can be used to provide $a_2$ with a finite value. Indeed, for
this kernel, we find that
\begin{align}
a_0 &\propto \lim_{x \rightarrow \infty} \int_0^x \left( 1 -
\frac{c_0 u + c_1 u^3+ c_2 u^5}{\left( 1+u^2 \right)^{5/2}} \right)\mathrm{d}u \nonumber \\
&= \lim_{x \rightarrow \infty} \left\{ (1-c_2)x + \frac{1}{3} ( 8c_2-2c_1-c_0) + {\cal O}\left( \frac{1}{x} \right)
\right\}
\end{align}
and
\begin{align}
a_2 &\propto \lim_{x \rightarrow \infty} \int_0^x \left( 1 -
\frac{c_0 u + c_1 u^3+ c_2 u^5}{\left( 1+u^2 \right)^{5/2}} \right)u^2\mathrm{d}u \nonumber \\
&= \lim_{x \rightarrow \infty} \left\{ \frac{1}{3}(1-c_2)x^3 + \left( \frac{5c_2}{2}-c_1\right)x
\right. \nonumber \\ & \left. \hspace{8em}- \frac{2}{3} ( 8c_2-4c_1+c_0 )
+ {\cal O}\left( \frac{1}{x} \right)\right\}.
\end{align}
Demanding $a_0$ to be zero and $a_2$ to be finite, leads to $c_2=1$,
$c_1=5/2$, and $c_0=3$. All properties of this Q$_2$ kernel are listed
in Table~\ref{tab:soft}.
Of course, this game can be continued to obtain $a_2=0$, then finite
$a_4$, then $a_4=0$, etc. However, the functional form of the
resulting interaction potentials becomes increasingly complex.
\subsubsection{Ferrers $n=3$ softening F$_3$:~compact support, $a_0 \ne 0$} \label{sec:soft:F3}\label{sec:soft:cubic:spline}
An increasingly popular method is cubic spline softening, which in
three-dimensional $N$-body simulations corresponds to replacing each
particle by a cubic-spline smoothing kernel as is widely used in
Smoothed Particle Hydrodynamics (SPH) codes \citep[see
e.g.][]{1992ARA&A..30..543M} and was introduced as a gravitational
softening kernel for $N$-body/SPH codes by
\cite{1989ApJS...70..419H}. Its main advantages in this context are
its exactly Newtonian behaviour beyond the softening length and its
dual use as hydrodynamics smoother and gravity softener. However, its
interparticle potential and 3D density distribution are numerically
rather unattractive due to their complex, piecewise continuous
functional forms.
Here and in the remainder, we will use the phrase ``compact support''
to indicate that a kernel yields exactly Newtonian forces outside the
softening kernel. In three dimensions, it immediately follows from
Newton's first theorem that the corresponding density distribution is
zero outside the kernel, i.e. $\varrho=0$ at $u>1$. In two dimensions,
this is not the case. In fact, all softening kernels with exact
Newtonian gravity at separations $r>\eps$ have poorly behaved
corresponding razor-thin disc profiles
$\sigma(u)$~\eqref{eq:soft:surf:kernel}, with infinite spatial extent
and negative values.
We here opt for the so-called Ferrers softening methods, labeled
`F$_n$', whose interaction potentials are polynomials of degree $n+1$
in the variable $t=1-u^2$ inside the softening length and that behave
exactly Newtonian at separations $r>\eps$. In three dimensions, they
correspond to replacing each particle with a \cite{Ferrers} sphere of
order $n$. For $n=0$ this is just a homogeneous sphere. Higher-order
models have spherical densities that are simple polynomials, with $n$
continuous derivatives, in the variable $t=1-u^2$.
For this paper, we investigate member $n=3$ from this family, with
properties listed in Table~\ref{tab:soft}, as an example of a
softening method with compact support but with $a_0\ne 0$.
\subsubsection{Modified Ferrers softening L$_2$:~compact support, $a_0 = 0$} \label{sec:soft:L1}
It is possible to modify the Ferrers softening methods to obtain
$a_0=0$ in order to reduce their gravitational bias while retaining
the attractive property of having compact support. Here, we test the
method labeled 'L$_2$', or `2D modified Ferrers $n=2$', in
Table~\ref{tab:soft}. Its name derives from the fact that it's based
on the F$_2$ kernel, which has as an interaction potential
$\phi(u)=1+\frac{1}{2}t+\frac{3}{8}t^2+\frac{5}{16}t^3$, but where the
coefficient of the last term is tuned to make $a_0=0$. This leads to
the L$_2$ interaction potential
$\phi(u)=1+\frac{1}{2}t+\frac{3}{8}t^2+\frac{5}{2}t^3$.
This kernel achieves its desirable properties by
having super-Newtonian accelerations for a limited range of
separations close to and inside of $r=\eps$.
\subsection{Softening scale} \label{sec:soft:scale}
The softening length and kernel are only defined up to a
re-scaling:~the softened potential \eqref{eq:V:soft} is invariant under the
transformation
\begin{equation}
\label{eq:eps:scaling}
\varepsilon \to a\varepsilon
\qquad\text{and}\qquad
\phi(q) \to a\phi(aq)
\end{equation}
with \emph{scaling factor} $a$. This implies that the parameter
$\varepsilon$ has no natural scale by itself and comparing different
kernels at the same $\varepsilon$ is meaningless. Therefore, some other
measure is required for such a comparison. One such measure valid for
all softening kernels is the force-scaling as
\begin{equation}
\label{eq:eps:scale:force}
\varepsilon_{F} = \varepsilon / \sqrt{-\phi'_{\max}},
\end{equation}
where $-\phi'_{\max}$ denotes the maximum value of the derivative of
$\phi$ \citep{2010MNRAS.401..791S}. The ratios $\varepsilon_{F}/\varepsilon$, scaled to unity for the
Plummer kernel, are listed in Table \ref{tab:soft} for the kernels
considered in this study.
For kernels with $a_0\neq0$, another natural measure of the softening
length is
\begin{equation}
\label{eq:soft:scale}
\varepsilon_0\equiv \frac{a_0}{2\pi}\eps =
\int_0^\infty\left[1-r\pot(r)\right]\d r,
\end{equation}
which measures the actual scale of the bias irrespective of any
re-scaling. With this definition, $\varepsilon_0=\eps$ for Plummer
softening. For other softening methods used in this study,
$\varepsilon_0$ is given in Table~\ref{tab:soft}.
Likewise, softening techniques with zero $a_0$ but non-zero $a_2$ can
be scaled to a common level of gravity bias via the softening length
transformation
\begin{equation}
\varepsilon_2 = \left|
\frac{3a_2}{\pi} \right|^{1/3} \varepsilon. \label{a2scal}
\end{equation}
With this definition, $\eps=\varepsilon_2$ for Q$_2$ softening.
\section{Softened gravity in stability analysis}
\label{pystab}
\subsection{Linear mode theory} \label{linmodthe}
We use {\sc pyStab}, a {\sc Python}/{\sc C++} computer code, to
analyse the stability of a razor-thin stellar disc embedded in an
axially symmetric gravitational potential. The details of the
mathematical formalism behind this code and of its implementation can
be found in \citet{b9}, \citet{dury08}, and \citet{dv16} so we will
not repeat these here. An axially symmetric disc galaxy model is
characterized by a distribution function $F_0(E,J)$, with $E$ the
specific binding energy and $J$ the specific angular momentum of a
stellar orbit, and a mean gravitational potential $V_0(r)$. In the
remainder, we will refer to this unperturbed axially symmetric state
as the ``base state'' of the system. Note that this base state is only
a correct solution of the CBE when employing the Newtonian
gravitational interaction (but see below).
For any given base state, {\sc pyStab} can retrieve those complex
frequencies $\omega$ for which a spiral-shaped perturbation of the
form
\begin{equation}
V_{\rm pert}(r,\theta,t) = V_{\rm
pert}(r)\,\mathrm{e}^{\mathrm{i}( m\theta-\omega t)}
\end{equation}
constitutes an eigenmode. Here, $(r,\theta)$ are polar coordinates in
the stellar disc, $m$ is the multiplicity of the spiral pattern,
$\Omega_p=\Re\{\omega\}/m$ its pattern speed, and $\Im\{\omega\}$ its
growth rate. A general perturbing potential can always be expanded in
such modes and, owing to the linear approximation, these can be
studied independently from each other.
In essence, {\sc pyStab} solves the first-order CBE to find the
response distribution function $f_{\rm resp}(r,\theta, v_r,
v_\theta,t)$ produced by a given perturbation $V_{\rm
pert}(r,\theta,t)$. This response distribution function generates
the response density
\begin{equation}
\Sigma_{\rm resp}(r,\theta,t) = \int \sub{f}{resp}(r,\theta,
v_r, v_\theta,t) \,\d v_r\d v_\theta
\end{equation}
which in turn {gives rise to the} response {softened} gravitational
potential
\begin{equation}
\label{Vprp}
V_{\rm resp}(\vec{x}) = G \int\Sigma_{\rm
resp}(\vec{x}')\,\pot(|\vec{x}-\vec{x}'|)\,\d^2\!\vec{x}',
\end{equation}
where the integral runs over the whole surface of the stellar disc
{and $\pot(r)$, defined in equation~\eqref{eq:soft:kern}, is the
softened Green's function for} gravitational interactions{,
replacing the Newtonian $1/r$}.
Eigenmodes are then identified by the fact that
\begin{equation}
V_{\rm pert}(r,\theta,t) \equiv V_{\rm resp}(r,\theta,t)
\end{equation}
and \textsc{pyStab} employs a matrix method \citep{kalnajs77} to find
them. The perturbing potential $V_{\rm pert}$ is expanded in a basis
of potentials, $V_\ell$. The response to each basis potential, denoted
by $V_{\ell,\rm resp}$, can likewise be expanded in this basis as
\begin{equation}
V_{\ell,\rm resp} = \sum_{k} {\mathcal C}_{k \ell} V_{k}.
\end{equation}
If the perturbation is an eigenmode, then the ${\mathcal C}$ matrix
can be shown to possess a unity eigenvalue
\citep{b9,dury08,dv16}. This feature is exploited by {\sc pyStab} to
identify the eigenmodes.
The formalism contains a number of technical parameters, such as the
number of orbits on which phase space is sampled (here we use $n_{\rm
orbit}(n_{\rm orbit}+1)/2$ orbits with $n_{\rm orbit}=600$ in the
allowed triangle of turning point -- or pericentre/apocentre --
space), the number $n_{\rm Fourier}$ of Fourier components in which
the periodic part of the perturbing potential is expanded (here we use
$n_{\rm Fourier}=80$), the number of potential-density pairs (PDPs)
that is used for the expansion of the radial part of the perturbing
potential and density (we use 44 PDPs), and the shape and extent of
the PDP density basis functions. As in \citet{dv16}, we use PDP
densities of the form
\begin{equation}
\label{basefunc}
\Sigma_\ell(r) = \Sigma_0(r)
\exp\left(-\frac{1}{2}\left(\frac{r-r_\ell}{\sigma_\ell}\right)^2\right)
\end{equation}
where the average radii $r_\ell$ cover the relevant part of the
stellar disc and are evenly spaced on a logarithmic scale so the
resolution is highest in the inner regions of the disc. The widths
$\sigma_\ell$ are automatically chosen such that consecutive basis
functions are sufficiently unresolved to represent any smooth
function. The position of these PDP density basis functions can be
tuned to achieve a high spatial resolution there where the eigenmodes
live. The corresponding PDP potentials are obtained via
\begin{equation}
\label{Vpdp}
V_{\ell}(\vec{x}) = G \int \Sigma_{\ell}(\vec{x}')
\,\pot(|\vec{x}-\vec{x}'|)\,\d^2\!\vec{x}'.
\end{equation}
\subsection{Introducing gravitational softening}
Since we want to validate our approach by comparing particular results
with published work based on numerical simulations, we mimic the
strategies employed by simulators when setting up and performing
$N$-body simulations of disc galaxies aimed at mode analysis. Usually,
an initial condition is generated by sampling stellar particles from
the distribution function $F_0(E,J)$ evaluated using the Newtonian
gravitational potential $V_0(r)$, independent of the gravitational
softening that is employed later on when evolving the particles
through time. Moreover, the axially symmetric force field of the base
state is subsequently evaluated correctly, i.e. without softening,
either by directly using the analytical expression for the potential
$V_0$ or by adding a small correction to the softened gravitational
field derived from the particles. Only the non-axisymmetric force
field of the growing waves is softened
\citep{1995ApJ...451..533E,2001ApJ...546..176S,2012ApJ...751...44S}.
This allows a simulator to sample particles from the correct DF
evaluated in the correct potential so that at least the initial
conditions of a simulation correspond to the intended base state and
the particle dynamics in the axially symmetric force field is followed
correctly.
Therefore, we only implement gravitational softening in the response
potential $V_{\rm resp}(r,\theta,t)$, but not in the axially symmetric
base state potential $V_0(r)$. Using this strategy, eqn. (\ref{Vprp}),
and a fortiori eqn (\ref{Vpdp}), is the only place where the softened
gravitational interaction enters the computation of the modes. It is
therefore straightforward to insert interaction potentials other than
the Newtonian one into a mode analysis code. The gravity bias
introduced in Section \ref{sec:soft:error} must then be regarded as a
measure for the fidelity with which the softened response potential
resembles the Newtonian one. Thus, we can use linear stability theory
to emulate the results expected in the large $N$ limit from $N$-body
simulations of disc galaxies. In this paper, we investigate the effect
on the eigenmodes in disc galaxy models from the P$_0$, Q$_2$, F$_3$,
and L$_2$ softening methods listed in Table~\ref{tab:soft}.
\section{The base states}\label{themodel}
Below, we give the essential details of the two base states that we
employ for this study. We also list the frequencies of the known
eigenmodes of these base states computed for a Newtonian interparticle
interaction.
\subsection{The isochrone disc model} \label{isochrone}
The isochrone disc is characterized by the cored density profile
\begin{equation}
\Sigma_0(r) = \frac{Mb}{2\pi r^3}\left( \ln
\frac{r+\sqrt{r^2+b^2}}{b} -\frac{r}{b} \right)
\end{equation}
which self-consistently generates the gravitational potential
\begin{equation}
V_0(r) = \frac{GM}{b+\sqrt{b^2+ r^2}},
\end{equation}
\citep{1959AnAp...22..126H,1976ApJ...205..751K}. Here, $M$ is the
total mass of the stellar disc and $b$ its scale-length. As shown by
\cite{1976ApJ...205..751K} {and} \citet{1995ApJ...451..533E}, a family
of distribution functions that generate this potential-density pair
is given by
\begin{equation}
F_0(E,J) = \left[ \frac{E}{V_0(0)} \right]^{{m_K}-1}g_{m_K}(x)
\end{equation}
with $m_K$ an integer, $x = J\sqrt{2E}/GM$, and
\begin{align}
g_{m_K}(x) &= \frac{2^{m_K}}{2\pi V_0(0)}\Bigg[x\frac{\d
\tau_{m_K}}{\d x} - \frac{{m_K}({m_K}-3)}{2} \tau_{m_K}(x)
\nonumber \\ &\phantom{=\frac{2^{m_K}}{2\pi V_0(0)}\Bigg[} +
\int_0^1 \tau_{m_K}(\eta x)\,\eta^{m_K}\frac{\d
^2P_{{m_K}-1}}{\d \eta^2} \d\eta \Bigg].
\end{align}
Here, $P_m$ is the Legendre polynomial of degree $m$ and
\begin{align}
\tau_{m_K}(x) &= -\frac{M}{16\pi
b^2}\frac{(1-x^2)^{3-{m_K}}}{x^3(1+x^2)} \nonumber
\\ & \hspace{4em} \times \left[ 2x + (1+x^2) \ln\frac{1-x}{1+x}
\right].
\end{align}
We adopt $m_K=12$ for this study. The Legendre polynomial can be
evaluated explicitly, allowing the integral featuring in the
expression for the distribution function to be evaluated in closed
form. However, the resulting expression is numerically very unstable
for small $x$. Therefore, we opted to simply evaluate the integral
numerically. This distribution function is only used to populate
orbits with positive angular momentum.
Counter-rotating stars have been added according to the prescription
given in \citet{1995ApJ...451..533E} which, unfortunately,
necessitates going back and forth between energy, angular momentum and
the radial action variable, $J_r$:
\begin{equation}
F'_0(E,J) =\left\{ \begin{array}{ll} \frac12 F_0(E',0) \qquad
& \text{if $J<0$} \\[1ex] F_0(E,J) - \frac12 F_0(E',0)
\qquad & \text{if $J>0$}
\end{array}
\right.
\end{equation}
with $E'$ the energy corresponding to a radial action $J_r+|J|$ and
zero angular momentum. Fortunately, analytical conversion formul\ae{}
between energy, angular momentum, and radial action exist for the
isochrone disc \citep{bintrem}.
We will focus here on the bisymmetric ($m=2$) modes of this
model. \citet{1997MNRAS.291..616P} provide the frequencies of three
modes of this base state model, choosing units such that $G=M=b=1$, as
\begin{equation}
\begin{array}{r@{=}r@{\,+\,}r@{\,\mathrm{i}}}
\omega_1 & 0.59 & 0.21 \\[0.5ex] \omega_2 & 0.46 & 0.14
\\[0.5ex] \omega_3 & 0.26 & 0.05
\end{array}
\end{equation}
while \citet{jalali05} find
\begin{equation}
\begin{array}{r@{=}r@{\,+\,}r@{\,\mathrm{i}}}
\omega_1 & 0.584 & 0.217 \\[0.5ex] \omega_2 & 0.468 & 0.148
\end{array}
\end{equation}
for the two main modes\footnote{\label{foot:acc} Accuracy estimates
for mode frequencies derived from linear stability computations are
hard to obtain since many numerical parameters come into
play. Judging from the differences between published mode
frequencies and from our own limited experiments with varying the
values of the employed numerical parameters (described in Section
\ref{linmodthe}) we estimate the mode frequencies to be accurate to
about the percent level. }.
\subsection{The Mestel disc} \label{mestel}
The \citet{1963MNRAS.126..553M} disc has a cusped total surface
density given by
\begin{equation}
\Sigma_0(r) = \Sigma_0 \frac{r_0}{r}
\end{equation}
which self-consistently generates a gravitational potential of the
form
\begin{equation}
V_0(r) = -v_0^2\ln \left( \frac{r}{r_0} \right)
\end{equation}
with the surface density scale given by $\Sigma_0 = v_0^2/2\pi G r_0$.
Here, $v_0$ is the value of the disc's constant circular velocity. A
central hole is cut out of this disc model by multiplying its
distribution function \citep{1977ARA&A..15..437T}
\begin{equation}
f(E,J) = \frac{\Sigma_0 v_0^q}{\sqrt{2^q \pi} \Gamma\left(
\frac{q+1}{2}\right) \sigma^{q+2} } \left( \frac{J}{r_0 v_0}
\right)^q \mathrm{e}^{E/\sigma^2},
\end{equation}
where $q$ is a real number, with a cut-out function of the form
\begin{equation}
H_{\rm cut}(J) = \frac{x}{1+x}
\end{equation}
with $x=\left( J/r_0 v_0 \right)^n$ (obviously, this also slightly
suppresses the distribution function at larger $J$-values). Outside
the central cut-out region, the disc's constant radial velocity
dispersion is given by $\sigma = v_0/\sqrt{1+q}$.
Here, we will focus on the $q=6$, $n=4$ member of this model family
and adopt units such that $G=v_0=r_0=1$. Its dominant bisymmetric mode
is then expected to have a frequency
\begin{equation}
\omega_1 \approx 0.88+0.13\,\mathrm{i},
\end{equation}
as shown, e.g., by
\citet{1977ARA&A..15..437T,1997PhDT.........1R,1998MNRAS.300..106E,poly15}. This
mode owes its existence to the inner cutout:~if the angular momentum
cut-off is not sufficiently steep, i.e. if $n$ is too small, there is
no eigenmode. The idea is that incoming trailing wave packets are
(partially) reflected from this sharp inner edge to travel back
outwards as leading waves. Overreflection, or swing amplification, at
the evanescent zone around the corotation resonance
\citep{1976ApJ...205..363M,toomre81} sends amplified trailing waves
back inwards. Inside this resonance cavity, growing modes can occur
\citep{1998MNRAS.300..106E}.
\section{Results} \label{sec:results}
\subsection{The isochrone disc}
\subsubsection{Unsoftened gravity} \label{bigL}
For the two most rapidly growing $m=2$ modes of the $m_K=12$ isochrone
disc, we find frequencies
\begin{equation}
\begin{aligned}
\omega_1 &= 0.582+0.215\;\mathrm{i}\\
\omega_2 &= 0.466+0.146\;\mathrm{i},
\end{aligned}
\end{equation}
in good agreement with published values.
However, the third mode listed by \citet{1997MNRAS.291..616P} showed
up in our analysis as only the fifth fastest growing mode, with a
frequency
\begin{equation}
\begin{array}{r@{=}r@{\,+\,}r@{\;\mathrm{i}}}
\omega_3 & 0.272 & 0.053
\end{array}.
\end{equation}
The two interloping modes at
\begin{equation}
\begin{array}{r@{=}r@{\,+\,}r@{\;\mathrm{i}}}
\omega'_3 & 0.384 & 0.103 \\[0.5ex] \omega'_4 & 0.323 & 0.075
\end{array}
\end{equation}
have not been described in the literature before. We confirmed that
they are robust to changes of the numerical parameters in the code
(resolution in phase space, number of Fourier modes, etc.) and that
they exert a zero total torque on the disc, as they should, and
therefore see no reason to discard them as spurious \citep{poly15}.
\subsubsection{Softened gravity}
\begin{figure}
\includegraphics[trim=0 12.5 0
0,clip,width=0.49\textwidth]{Isochrone_allsoft}
\caption{Pattern speeds $\Re\{\omega\}=m\Omega_p$ (top panel)
and growth rates $\Im\{\omega\}$ (bottom panel) of the two
dominant $m=2$ modes of the $m_K=12$ isochrone disc, with
unsoftened frequencies of $\omega_1=0.582+0.215\,{\mathi}$
and $\omega_2=0.466+0.146\,{\mathi}$ for different softening
recipes (Plummer P$_0$, modified Kuz'min Q$_2$, Ferrers
F$_3$, and modified Ferrers L$_2$). The softening lengths
are scaled according to equations~\eqref{eq:eps:scaling} and
\eqref{eq:eps:scale:force} to obtain a common maximum
inter-particle force. The bullets are data taken from the
$N$-body simulations using Plummer softening reported in
\citet{1995ApJ...451..533E}. The horizontal grey lines indicate the
Newtonian mode frequency.
\label{fig:Isochrone_allsoft}}
\end{figure}
For this base state, published information on how the properties of
the two main $m=2$ modes change with softening in a numerical
simulation exists. \citet{1995ApJ...451..533E} use a polar grid code
with 120,000 particles, a fixed time step, and a polar grid of 128
azimuthal and 85 radial nodes to simulate the $m_K=12$ isochrone disc
from quiet-start initial conditions \citep{1983JCoPh..50..337S} using
Plummer softening with different softening lengths. The results of
these simulations are shown in Fig. \ref{fig:Isochrone_allsoft} as
black data points. Both the pattern speed and the growth rate of the
two dominant modes appear to be declining functions of softening
length. As \citet{1995ApJ...451..533E} note:~``[\ldots] it is clear
that both parts of the eigenfrequency are strongly affected by even
moderate softening.'' The dependence of the mode frequencies on
softening length is markedly non-linear which hampers a simple
extrapolation to zero softening length.
Overplotted in Fig. \ref{fig:Isochrone_allsoft} are the results from
our linear stability analysis with {\sc pyStab}, using different
softening prescriptions. Clearly, our results for Plummer softening
agree rather well with those presented in
\citet{1995ApJ...451..533E}:~both the pattern speed and growth rate
are non-linearly declining functions of the softening length (scaled
according to equations~\eqref{eq:eps:scaling} and
\eqref{eq:eps:scale:force} to a common maximum inter-particle
force). The drop is steepest for small softening lengths and becomes
shallower for larger $\eps$. Especially for larger $\eps$-values,
numerical simulations and linear mode analysis predict the same
behaviour for frequency as a function of softening length. We
tentatively attribute the deviations between the simulations and
linear theory at small $\eps$-values to variance, i.e.~to the gravity
error caused by not softening enough (cf. paragraph
\ref{sec:soft:error}).
Using the other softening recipes, the pattern speed and growth rate
likewise decline with increasing softening length but they do so much
less dramatically and with smaller deviation from a linear dependence
on softening length than when using Plummer softening. Moreover, it
appears that having a zero gravity bias parameter $a_0$ induces a much
stronger effect than having compact support. This is exemplified in
this case by the Q$_2$ method (infinite support, $a_0=0$) yielding
results much closer to the Newtonian ones than the F$_3$ method
(compact support, $a_0>0$). Methods that combine compact support with
having $a_0=0$, like the L$_2$ method, appear vastly superior, with
very little deviation between the retrieved mode frequencies and the
correct, Newtonian values.
However, we refer the reader to section \ref{sec:dis} for a discussion
of how to correctly interpret this apparent success.
\subsection{The Mestel disc}
\begin{figure}
\includegraphics[trim=0 12.5 0
0,clip,width=0.49\textwidth]{Mestel_allsoft}
\caption{Pattern speed $\Re\{ \omega \}=m\Omega_p$ (top panel) and
growth rate $\Im\{ \omega \}$ (bottom panel) of the dominant $m=2$
mode of the $q=6$, $n=4$ Mestel disc for different softening recipes
(Plummer P$_0$, modified Kuz'min Q$_2$, Ferrers F$_3$, and modified
Ferrers L$_2$). The softening lengths are scaled according to
equations~\eqref{eq:eps:scaling} and \eqref{eq:eps:scale:force} to
obtain a common maximum inter-particle force. The datapoints are
derived from the $N$-body simulations by
\citet{2001ApJ...546..176S}. The horizontal grey lines indicate the
Newtonian mode frequency.
\label{fig:Mestel_allsoft}}
\end{figure}
\subsubsection{Unsoftened gravity}
{\sc pyStab} retrieves the dominant mode of the $q=6$, $n=4$ Mestel
disc, along with a number of much slower growing modes. Since the
frequencies of these minor modes are sensitive to the choice of
numerical parameter values, they are most likely spurious
\citep{poly15}. Using unsoftened gravity, we find the dominant mode to
have a frequency $\omega=0.876+0.128\;\mathrm{i}$, which is in good
agreement with the values reported by \citet{1977ARA&A..15..437T,
1997PhDT.........1R,poly15} and which were computed using different
mode analysis techniques and codes.
\subsubsection{Softened gravity}
In Fig. \ref{fig:Mestel_allsoft}, we show how the pattern speed (top
panel) and growth rate (bottom panel) of the dominant mode of this
base state change with increasing softening length (scaled according
to equations~\eqref{eq:eps:scaling} and \eqref{eq:eps:scale:force} to
a common maximum inter-particle force) using the softening recipes
listed in Table~\ref{tab:soft}.
Overplotted in this figure, we show the frequency estimates of
\citet{2001ApJ...546..176S} for this Mestel disc model, based on
$N$-body simulations with a particle-mesh code employing 2.5 million
particles, and a grid of 256 azimuthal and 200 radial nodes. The 5
simulations presented here all start from exactly the same initial
conditions but are evolved using different Plummer softening
lengths. The agreement with our linear mode analysis is not as good as
in the case of the isochrone disc. As reported by
\citet{2001ApJ...546..176S}, there is a $\sim 10$~\% scatter between
the measured frequencies of simulations with resampled initial
conditions at a constant particle number. This may be why the
simulation datapoints do not converge to the Newtonian linear-mode
result for zero softening length. Moreover, particle noise may have
negatively affected the frequency measurements. Still, the trend
followed by these simulations is in qualitative agreement with our
results:~both the pattern speed and the growth rate decrease with
increasing softening length.
As for the isochrone disc, softening methods with $a_0=0$ (like Q$_2$)
stay much closer to the Newtonian mode frequency than methods with
compact support but non-zero $a_0$ (like F$_3$) for a given value of
the softening length $\varepsilon$. Methods that combine compact
support with $a_0=0$ (like L$_2$) generally outperform the others.
Again, we refer the reader to section \ref{sec:dis} for a discussion
of how to correctly interpret this apparent success.
\section{Discussion} \label{sec:dis}
\subsection{Scaling to the same level of gravity bias}
\begin{figure*}
\includegraphics[trim=0 12.5 0
0,clip,width=0.49\textwidth]{Isochrone_a0}
\includegraphics[trim=0 12.5 0
0,clip,width=0.49\textwidth]{Mestel_a0}
\caption{Pattern speed $\Re\{ \omega \}=m\Omega_p$ (top
panels) and growth rate $\Im\{ \omega \}$ (bottom panels) of
the dominant $m=2$ mode of the $m_K=12$ isochrone disc (left
panels) and the $q=6$, $n=4$ Mestel disc (right panels) as a
function of the scaled softening length $\varepsilon_0$,
which allows for a direct comparison of the softening
kernels with non-zero $a_0$ (i.e. P$_0$ and F$_3$) at the
same level of gravity bias. The bullets are data taken from
the $N$-body simulations using Plummer softening reported in
\citet{1995ApJ...451..533E} and
\citet{2001ApJ...546..176S}. The horizontal grey lines
indicate the Newtonian mode frequencies.
\label{fig:Isoa0}}
\end{figure*}
As mentioned in paragraph \ref{sec:soft:scale}, the softening length
and kernel are only defined up to a re-scaling and we advocate the
scale \beqn \varepsilon_0 = \frac{a_0}{2\pi}\eps \neqn to bring
methods with non-zero $a_0$ to a common gravity bias level.
In Fig. \ref{fig:Isoa0}, we show the retrieved frequencies of the
modes of the isochrone and Mestel discs as a function of
$\varepsilon_0$ for the two softening methods with non-zero gravity
bias parameter $a_0$ (i.e. P$_0$ and F$_3$). Clearly, the differences
between both softening methods, which are so striking in Figures
\ref{fig:Isochrone_allsoft} and \ref{fig:Mestel_allsoft}, now largely
disappear. At a given $\varepsilon_0$-value, all $a_0 \ne 0$ methods
perform almost equally well. Thus, it is always possible to re-scale
the softening length of one $a_0 \ne 0$ softening technique such that
it approximately matches the performance of another $a_0 \ne 0$
method. No exact matching is possible because of the higher order
terms in the expansion of the gravity bias.
\begin{figure*}
\includegraphics[trim=0 12.5 0
0,clip,width=0.49\textwidth]{Isochrone_a2}
\includegraphics[trim=0 12.5 0
0,clip,width=0.49\textwidth]{Mestel_a2}
\caption{Pattern speed $\Re\{ \omega \}=m\Omega_p$ (top
panels) and growth rate $\Im\{ \omega \}$ (bottom panels) of
the dominant $m=2$ mode of the $m_K=12$ isochrone disc (left
panels) and the $q=6$, $n=4$ Mestel disc (right panels) as a
function of the scaled softening length $\varepsilon_2$,
which allows for a direct comparison of the softening
kernels with zero $a_0$ (i.e. Q$_2$ and L$_2$) at the same
level of gravity bias. The horizontal grey lines indicate
the Newtonian mode frequencies.
\label{fig:Isoa2}}
\end{figure*}
As can be seen in Figure \ref{fig:Isoa2}, softening techniques with
$a_0=0$ but non-zero $a_2$ can be scaled to a common level of gravity
bias using the transformation (\ref{a2scal}), allowing for higher
order terms in the expansion (\ref{eq:bias:V:2D}) for the gravity
bias.
Based on these results, it seems fair to say that softening strategies
with $a_0=0$ generally yield more accurate (i.e. Newtonian-like) mode
frequencies than strategies with $a_0 \ne 0$ because the gravity bias
of the latter grows linearly with softening length $\varepsilon$ while
for the former it grows much more slowly, as $\varepsilon^3$. However,
within each of these classes of softening techniques, there is no
particular reason to favour one method over another provided they are
compared at (approximately) the same level of gravity bias.
\subsection{Physical interpretation}
We define the two-dimensional Fourier transform $\widehat{\pot}(k)$ of
the interparticle interaction potential $\pot(r)$ as
\begin{equation}
\pot(\vec{r}) = \frac{1}{(2\pi)^2}\int \widehat{\pot}(k) {\mathe}^{{\mathi}
\vec{k}.\vec{r}} \mathrm{d}\vec{k}.
\end{equation}
Based on Poisson's equation, using separation of variables it is
straightforward to show that the radial part of the gravitational
response potential, which we denote here by $V_m(r)$, generated by an
$m$-armed spiral response density of the form
\begin{equation}
\Sigma_m(r,\theta) = \Sigma_m(r) {\mathe}^{{\mathi} m \theta}
\end{equation}
can be retrieved from the relation
\begin{equation}
{\mathcal H}_m \left\{ V_m(r) \right\} = - G
\widehat{\pot}(k) {\mathcal H}_m\left\{ \Sigma_m(r) \right\}
\end{equation}
with ${\mathcal H}_m$ the Hankel transform of order $m$ \citep[see
e.g.][]{bintrem}. The Hankel transform of order $m$ of a function
$f(r)$ is defined as
\begin{equation}
{\mathcal H}_m\left\{ f \right\}(k) = \int_0^\infty f(r) J_m(kr) r
\mathrm{d}r
\end{equation}
with $J_m(x)$ a Bessel function of the first kind.
Here, we will use the notation $\widehat{\pot}_N(k)$ for the Fourier
transform of the Newtonian interaction potential, with
\begin{equation}
\widehat{\pot}_N(k) = \frac{2\pi }{k}.
\end{equation}
Likewise, for the interaction potentials listed in
Table~\ref{tab:soft} we find that:
\begin{align}
\widehat{\pot}_{P_0}(k) &= 2\pi\frac{{\mathe}^{-k\varepsilon}}{ k} \label{kp0}
\\ \widehat{\pot}_{Q_2}(k) &= 2\pi\left(1+k\varepsilon + \frac{1}{2}
(k\varepsilon)^2\right)\frac{ {\mathe}^{-k\varepsilon}}{ k}. \label{kq2}
\end{align}
For the softening techniques with compact support, F$_3$ and L$_2$, no
simple analytical expression exists for the Fourier transform of their
interaction potentials but they can easily be obtained numerically.
\begin{figure}
\includegraphics[trim=5 15 10 10,clip,width=0.465\textwidth]{hank}
\caption{The suppression factor $\widehat{\pot}/\widehat{\pot}_N$ as a
function of the dimensionless wave number $k\varepsilon_F$, where
the softening lengths are scaled according to
equations~\eqref{eq:eps:scaling} and \eqref{eq:eps:scale:force} to
obtain a common maximum inter-particle force.
\label{fig:hank}}
\end{figure}
\begin{figure}
\includegraphics[trim=5 15 10 10,clip,width=0.475\textwidth]{hank_a0}
\includegraphics[trim=5 15 10 10,clip,width=0.475\textwidth]{hank_a2}
\caption{The suppression factor $\widehat{\pot}/\widehat{\pot}_N$ as a
function of the re-scaled dimensionless wave numbers
$k\varepsilon_0$ (top panel), for the softening methods with $a_0
\ne 0$, and $k\varepsilon_2$ (bottom panel), for the softening
methods with $a_0 = 0$.
\label{fig:hank02}}
\end{figure}
For a given response density $\Sigma_m$, the softened response
potential $V_m^\varepsilon$ and the unsoftened response potential
$V_m^0$ are connected as
\begin{equation}
{\mathcal H}_m\left\{ V_m^\varepsilon \right\} =
\frac{\widehat{\pot}}{\widehat{\pot}_N} {\mathcal H}_m\left\{ V_m^0
\right\}
\end{equation}
with $\varepsilon$ the softening length. We plot the $k$-dependent
suppression factor $\widehat{\pot}/\widehat{\pot}_N$ that links the
Fourier transforms of the softened and unsoftened response potentials
in Figure \ref{fig:hank}. The most striking consequence of
gravitational softening is the suppression of the small-scale,
i.e. large wavenumber $k$, structure in the Fourier transform of the
response potential.
In Appendix \ref{app1}, we show how this suppression factor is
connected to the gravity bias. More specifically, we prove that the
even coefficients in the series expansion of the suppression factor
around zero $k$ are directly proportional to the even coefficients in
the series expansion of the gravity bias around zero $\varepsilon$ (in
case the latter exist).
Clearly, desiging a softening kernel to
have vanishing bias coefficients is equivalent to designing an
interaction kernel whose suppression factor
$\widehat{\pot}/\widehat{\pot}_N$ is increasingly close to unity for
small wavenumbers $k$.
However, just as it is not sensible to compare the various softening
strategies as we did in Figures \ref{fig:Isochrone_allsoft} and
\ref{fig:Mestel_allsoft}, it makes little sense to compare the
suppression factors as a function of $k \eps_F$. It is more meaningful
to compare the suppression factors at the same level of gravity bias,
i.e. as a function of $k \eps_0$ for the softening methods with $a_0
\ne 0$, and as a function of $k\varepsilon_2$ for the softening
methods with $a_0 = 0$. This comparison is shown in Figure
\ref{fig:hank02} and restates our previous conclusions. As a function
of the re-scaled wave number $k \varepsilon_0$, which places the P$_0$
and F$_3$ kernels on an equal gravity bias footing, the suppression
factors of these two softening techniques behave remarkably
similar. In fact, Plummer softening leads to less suppression for
large $k\varepsilon_0$ than F$_3$ softening. This agrees with Figure
\ref{fig:Isoa0} in which Plummer softening is shown to stay closer to
the correct, Newtonian result than F$_3$ softening at an equal level
of gravity bias. The Q$_2$ kernel, in turn, leads to less suppression
than the L$_2$ kernel and, as can be seen in Figure \ref{fig:Isoa2},
it also leads to slightly better frequency estimates.
This suppression of the response potential will likely lead to an
increased stability of the model galaxy. This expectation is borne out
by studying the stability of axially symmetric WKBJ waves under
Plummer softening, where a Toomre $Q$-value $Q<1$ now separates
growing from stationary waves \citep{1971Ap&SS..14...73M}. This
analysis has been extended to include general softening kernels by
\citet{1994A&A...286..799R,1997A&A...324..523R}. The physical
background of our results and the expected influence of softening on
$m=0$ WKBJ waves are, therefore, already well understood. Here, we
took this work further by studying general eigenmodes beyond the WKBJ
approximation and by going to $m=2$ patterns.
\section{Conclusions}
We use linear perturbation theory to investigate how different recipes
for gravitational softening, as employed in numerical $N$-body
simulations of razor-thin disc galaxies, affect predictions for the
properties of the latter's spiral eigenmodes. We specifically focus on
the frequencies, i.e. pattern speeds and growth rates, of two-armed
modes in the linear regime. We warn the reader that our approach does
not take into account the effects of approximate force evaluations
\citep{1986Natur.324..446B}, finite-$N$
\citep{2001MNRAS.324..273D,2012ApJ...751...44S,2015A&A...584A.129F},
stochasticity \citep{2009MNRAS.398.1279S}, implicit softening
contributed by the grid in particle-mesh codes
\citep{1994A&A...286..799R}, etc. whose respective magnitudes,
moreover, may depend on the amount of gravitational softening.
We have tested our linear mode analysis approach by comparing the
behaviour of the frequencies of the dominant $m=2$ modes of an
isochrone disc and of the Mestel disc as a function of Plummer
softening length with those found in the $N$-body simulations reported
by \citet{1995ApJ...451..533E} and
\citet{2001ApJ...546..176S}. Overall, we found reasonably good
agreement between linear theory and numerical simulations, also in the
softened regime.
We argue that the only meaningful way of comparing softening kernels
is to scale them to the same gravity bias level. In this paper, we
show how this scaling can be achieved, based on the results of
\citet{2001MNRAS.324..273D}. Thus, it is always possible to re-scale
the softening length of one softening technique such that it matches
the performance of another method with the same dependence of gravity
bias on softening length.
We have shown that softening methods with a vanishing lowest-order
term in the expansion of the gravity bias as a function of softening
length (in two dimensions, this is a linear term; in three dimensions,
this term is quadratic in the softening length) and whose gravity bias
therefore grows slowly with increasing softening length (e.g. the
Q$_2$ and L$_2$ methods discussed in this paper) provide more accurate
mode frequency estimates than methods with a non-zero lowest-order
term (e.g. the P$_0$ and F$_3$ methods). Softening methods with zero
lowest-order term compensate the sub-Newtonian forces deep inside the
kernel with super-Newtonian forces near $r\sim \eps$.
Kernels with compact support, in the sense that they yield exactly
Newtonian forces outside of the softening kernel, perhaps somewhat
counter-intuitively, do not necessarily provide more accurate
frequency estimates than kernels with infinite extent. For instance,
when compared at a common level of gravity bias, the Plummer kernel
(P$_0$) provides more accurate frequency estimates than the F$_3$
kernel. Likewise, the Q$_2$ kernel outperforms the L$_2$ kernel in
this regard.
The relative merit of a softening kernel can be judged from its
suppression of the small-scale, i.e. large wavenumber $k$, structure
in the Fourier transform of its response potential. The stronger this
suppression, measured at a given level of gravity bias, the more the
mode frequency estimates deviate from their Newtonian values.
As a guide to simulators, we provide an example of how a softening
technique, in this case Plummer softening, can be used as a basis for
developing new softening kernels whose gravity biases grow more slowly
with increasing softening length. These then provide much more
accurate estimates for mode frequencies than Plummer softening does.
Generally, the use of gravitational softening lowers the exponential
growth rate of spiral modes. Therefore, strongly softened $N$-body
simulations may risk ``losing'' some of these modes as their growth
rates are overtaken by that of e.g. swing amplified noise
\citep{1995A&AT....7..317R}.
\section*{Acknowledgements}
We wish to thank the organizers of the workshop ``The secular
evolution of self-gravitating systems over cosmic ages'' at the
Institut d'Astrophysique de Paris, May 24-27 2016, where this work was
initiated. We are grateful to Prof. Jerry Sellwood for his very
helpful insights into setting up and performing $N$-body simulations
and to Prof. Scott Tremaine for his helpful suggestions. JBF
acknowledges support from Program number HST-HF2-51374 which was
provided by NASA through a grant from the Space Telescope Science
Institute, which is operated by the Association of Universities for
Research in Astronomy, Incorporated, under NASA contract
NAS5-26555. SDR acknowledges financial support from the European
Union's Horizon 2020 research and innovation programme under the Marie
Sk{\l}odowska-Curie grant agreement No 721463 to the SUNDIAL ITN
network. WD acknowledges support by STFC grant ST/N000757/1.
\bibliographystyle{mn2e}
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\section{Introduction}
Many recent advances in deep neural networks have led to significant improvement in the quality of abstractive summarization \cite{radford2019language, gehrmann-etal-2019-generating, lewis2019bart}.
Despite this progress,
there are still many limitations facing neural text summarization \cite{kryscinski-etal-2019-neural},
the most serious of which is their tendency to generate summaries that are not factually consistent with the input document;
a factually consistent summary only contains statements that can be derived from the source document. Recent studies show that about 30\% of the summaries generated by neural network sequence-to-sequence models suffer from fact fabrication \cite{AAAI1816121}. Unfortunately, the widely used ROUGE score is inadequate to quantify factual consistency \cite{kryscinski-etal-2019-neural}.
Factual inconsistency can occur at either the entity or the relation level.
At the entity level, a model generated summary may contain named-entities that never appeared in the source document. We call this the entity \emph{hallucination} problem. For example, consider the following model generated summary:
\begin{quotation}
\emph{People in Italy and the Netherlands are more likely to consume fewer cups of coffee than those in the \underline{UK}, a study suggests.}
\end{quotation}
``UK'' never appeared in the input source document (taken from the test set of the XSUM dataset \cite{narayan-etal-2018-dont}). In fact, the source document mentioned a study involving people in Italy and Netherlands; ``UK'' was a result of model hallucination.
Another type of inconsistency occurs when the entities indeed exist in the source document but the relations between them are not in the source document. This type of inconsistency is much harder to identify.
Open Information Extraction (OpenIE) and dependency parsing tools have been used \cite{AAAI1816121} to identify the underlying relations in a summary, but
are not yet accurate enough for practical use.
Ultimately, these researchers
relied on manually classifying generated summaries into \emph{faithful}, \emph{fake}, or \emph{unclear}.
In this paper, we propose a set of simple metrics to quantify factual consistency at the entity-level.
We analyze the factual quality of summaries produced by the state-of-the-art BART model \cite{lewis2019bart} on three news datasets. We then propose several techniques including data filtering, multi-task learning and joint sequence generation to improve performance on these metrics. We leave the relation level consistency to future work.
\section{Related work}
Large transformer-based neural architectures combined with pre-training have set new records across many natural language processing tasks \cite{NIPS2017_7181, devlin-etal-2019-bert, radford2019language}. In particular, the BART model \cite{lewis2019bart} has shown superior performance in many text generation tasks including abstractive summarization. In contrast to encoder-only pre-training such as in BERT \cite{devlin-etal-2019-bert} or decoder-only pre-training such as in GPT-2 \cite{radford2019language},
BART is an encoder-decoder transformer-based neural translation model jointly pre-trained to reconstruct corrupted input sequences of text.
Several authors have pointed out the problem of factual inconsistency in abstractive summarization models \cite{kryscinski-etal-2019-neural, kryciski2019evaluating, AAAI1816121, welleck-etal-2019-dialogue}.
The authors in \cite{kryciski2019evaluating} proposed to train a neural network model to classify if a summary is factually consistent with a given source document, similar to a natural language inference task. In the dialogue generation setting, authors in \cite{DBLP:journals/corr/abs-1911-03860} proposed using unlikelihood to surpress logically inconsistent responses. Our work is complementary to such existing approaches as we focus on simple entity-level metrics to quantify and improve factual consistency. Our goal of improving entity-level metrics of summaries is also related to controllable abstractive summarization \cite{fan-etal-2018-controllable}, where a list of named-entities that a user wants to see in the summary can be passed as input to influence the generated summary. In contrast, our goal is to \emph{predict} which entities are summary-worthy while generating the summary that contains them. In this view we are trying to solve a more challenging problem.
\section{Entity-level factual consistency metrics} \label{sec:entity}
We propose three new metrics that rely on off-the-shelf tools to perform Named-Entity Recognition (NER). \footnote{We use Spacy \cite{spacy2}.}
We use $\mathcal{N}(t)$ and $\mathcal{N}(h)$ to denote the number of named-entities in the target (gold summary) and hypothesis (generated summary), respectively. We use $\mathcal{N}(h\cap s)$ to denote the number of entities found in the generated summary that can find a match in the source document. If a named-entity in the summary consists of multiple words, we consider it a match as long as any n-gram of the named-entity can be found in the source document. This is meant to capture the situation where the named-entity can be shortened; for example, ``Obama '' is a match for ``Barack Obama'' and ``Harvard'' is a match for ``Harvard University''. When the match is at the unigram level, we make sure that it is not a stop word such as ``the''. We also make the match case-insensitive to accommodate casing variances.
\paragraph{Precision-source:}
We propose precision-source ($\mathbf{prec}_s$) to quantify the degree of hallucination with respect to the source: $\mathbf{prec}_s = \mathcal{N}(h\cap s) / \mathcal{N}(h).$
It is simply the percentage of named-entities in the summary that can be found in the source. Low $\mathbf{prec}_s$ means hallucination is severe.
We first evaluate the $\mathbf{prec}_s$ score on the ground truth summaries of the 3 datasets: Newsroom \cite{newsroom}, CNN/DailyMail \cite{nallapati-etal-2016-abstractive} and XSUM \cite{narayan-etal-2018-dont}.
\begin{table}[h]
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}
\hline
\multirow{2}{*}{} & \multicolumn{3}{c|}{Newsroom} & \multicolumn{3}{c|}{CNNDM} & \multicolumn{3}{c|}{XSUM} \\ \hline
& \multicolumn{1}{c|}{train} & \multicolumn{1}{c|}{val} & \multicolumn{1}{c|}{test} & \multicolumn{1}{c|}{train} & \multicolumn{1}{c|}{val} & \multicolumn{1}{c|}{test} & \multicolumn{1}{c|}{train} & \multicolumn{1}{c|}{val} & \multicolumn{1}{c|}{test} \\ \hline
avg. $\mathcal{N}(t)$ & 2.08 & 2.10 & 2.09 & 4.36 & 5.09 & 4.87 & 2.08 & 2.06 & 2.08 \\ \hline
avg. $\mathcal{N}(t \cap s)$ & 1.88 & 1.90 & 1.90 & 4.21 & 4.92 & 4.70 & 1.64 & 1.64 & 1.64 \\ \hline
$\mathbf{prec}_s$ (\%) & 90.6 & 90.6 & 90.5 & 96.5 & 96.7 & 96.6 & 79.0 & 79.5 & 79.3 \\ \hline
\end{tabular}
}
\vspace{-1mm}
\caption{Average number of named-entities and the $\mathbf{prec}_s$ scores (\%) in the ground truth summary.}
\label{table:groundTruthSummaryStats}
\end{table}
Table \ref{table:groundTruthSummaryStats} shows that among the three datasets, the ground truth summaries in XSUM have the lowest $\mathbf{prec}_s$ score.
This is because the ground truth summaries in the XSUM dataset often use the first sentence of the article as the summary;
the source document is constructed to be the rest of the article and may not repeat the named-entities that appeared in the summary. We hypothesize that the hallucination problem is largely caused by the training data itself. Thus, we propose to perform entity-based data filtering to construct a ``clean'' version of these datasets as described next.
\paragraph{Entity-based data filtering:}
For each dataset, we apply Spacy NER on the gold summary to identify all the named-entities. \footnote{We ignore certain types of entities such as date, time, numerals because they tend to have large variations in representation and are difficult to determine a match in the source document. The appendix contains more details.}
If any of the entities cannot find a match in the source document, we discard the sentence that contains the entity from the ground truth summary. If the ground truth summary consists of only one sentence and it needs to be discarded, we remove the document-summary pair from the dataset.
This way, we ensure that our filtered dataset does not contain hallucination of entities ($\mathbf{prec}_s =1$) in the ground truth summary. The dataset size before and after the filtering is shown in Table \ref{table:datasetStats}. About a third of examples are filtered out for XSUM. Again, this is because of the way XSUM dataset is constructed as mentioned in the previous paragraph. As we shall see in Table \ref{table:dataFilterResult}, entity-based data filtering reduces hallucination of the trained model and the effect is especially significant in the XSUM dataset.
\begin{table*}[h]
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{|l|r|r|r|r|r|r|r|r|r|}
\hline
\multirow{2}{*}{} & \multicolumn{3}{c|}{Newsroom} & \multicolumn{3}{c|}{CNNDM} & \multicolumn{3}{c|}{XSUM} \\ \cline{2-10}
& \multicolumn{1}{c|}{train} & \multicolumn{1}{c|}{val} & \multicolumn{1}{c|}{test} & \multicolumn{1}{c|}{train} & \multicolumn{1}{c|}{val} & \multicolumn{1}{c|}{test} & \multicolumn{1}{c|}{train} & \multicolumn{1}{c|}{val} & \multicolumn{1}{c|}{test} \\ \hline
\thead{original} & \thead{922,500 (1.58)} & \thead{100,968 (1.60)} & \thead{100,933 (1.59)} & \thead{287,112 (3.90)} & \thead{13,368 (4.13)} & \thead{11,490 (3.92)} & \thead{203,540 (1.0)} & \thead{11,301 (1.0)} & \thead{11,299 (1.0)} \\ \hline
\thead{after filtering} & \thead{855,975 (1.62)} & \thead{93,678 (1.64)} & \thead{93,486 (1.64)} & \thead{286,791 (3.77)} & \thead{13,350 (3.99)} & \thead{11,483 (3.77)} & \thead{135,155 (1.0)} & \thead{7,639 (1.0)} & \thead{7,574 (1.0)} \\ \hline
\end{tabular}
}
\vspace{-1mm}
\caption{Number of examples in three datasets together with the average number of sentences in the ground truth summary (in parentheses) before and after entity-based filtering.}
\label{table:datasetStats}
\end{table*}
\paragraph{Precision-target and recall-target:} Although the precision-source ($\mathbf{prec}_s$) metric quantifies the degree of entity hallucination with respect to the source document, it does not capture the entity-level accuracy of the generated summary with respect to the ground truth summary. To get a complete picture of the entity-level accuracy of the generated summary, we propose the precision-target ($\mathbf{prec}_t$) score: $\mathbf{prec}_t = \mathcal{N}(h\cap t) / \mathcal{N}(h),$
where $\mathcal{N}(h\cap t)$ is the number of named-entities in the generated summary that can find a match in the ground truth summary; and the recall-target ($\mathbf{recall}_t$) score: $\mathbf{recall}_t = \mathcal{N}(h\cap t) / \mathcal{N}(t),$
where $\mathcal{N}(t)$ is the number of named-entities in the ground truth summary. We compute the F1 score as $F1_t=2 \cdot \mathbf{prec}_t \cdot \mathbf{recall}_t / (\mathbf{prec}_t + \mathbf{recall}_t)$.
\begin{table*}[ht]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline
& \thead{training \\ data} & \thead{Rouge1} & \thead{Rouge2} & \thead{RougeL} & \thead{macro \\ $\mathbf{prec}_s$} & \thead{micro \\ $\mathbf{prec}_s$} & \thead{macro \\ $\mathbf{prec}_t$} & \thead{micro \\ $\mathbf{prec}_t$} & \thead{macro \\ $\mathbf{recall}_t$} & \thead{micro \\ $\mathbf{recall}_t$} & \thead{macro \\ $F1_t$} & \thead{micro \\ $F1_t$} \\ \hline
\multirow{4}{*}{Newsroom} & original & 47.7{\scriptsize $\pm$0.2} & 35.0{\scriptsize $\pm$0.3} & 44.1{\scriptsize $\pm$0.2} & 97.2{\scriptsize $\pm$0.1} & 97.0{\scriptsize $\pm$0.1} & 65.4{\scriptsize $\pm$0.3} & 62.9{\scriptsize $\pm$0.4} & 70.8{\scriptsize $\pm$0.3} & 68.5 {\scriptsize $\pm$0.2} & 68.0{\scriptsize $\pm$0.2} & 65.6{\scriptsize $\pm$0.3} \\
& + filtering & 47.7{\scriptsize $\pm$0.1} & 35.1{\scriptsize $\pm$0.1} &44.1 {\scriptsize $\pm$0.1} & 98.1{\scriptsize $\pm$0.1} & 98.0{\scriptsize $\pm$0.0} & 66.5{\scriptsize $\pm$0.1} & 63.8{\scriptsize $\pm$0.1} & 70.2 {\scriptsize $\pm$0.2} & 67.7{\scriptsize $\pm$0.3} & 68.3{\scriptsize $\pm$0.1} & 65.7{\scriptsize $\pm$0.1} \\
& + classification & 47.7{\scriptsize $\pm$0.2} & 35.1{\scriptsize $\pm$0.1} & 44.2{\scriptsize $\pm$0.2} & 98.1{\scriptsize $\pm$0.1} & 98.0{\scriptsize $\pm$0.0} & 67.2{\scriptsize $\pm$0.4} & 64.2{\scriptsize $\pm$0.4} & 70.3{\scriptsize $\pm$0.2} & 67.8{\scriptsize $\pm$0.4} & 68.7{\scriptsize $\pm$0.3} & 65.9{\scriptsize $\pm$0.4} \\
& ~~ JAENS & 46.6 {\scriptsize $\pm$0.5} & 34.3{\scriptsize $\pm$0.3} & 43.2{\scriptsize $\pm$0.3} & {\bf 98.3}{\scriptsize $\pm$0.1} & {\bf 98.3}{\scriptsize $\pm$0.1} & {\bf 69.5}{\scriptsize $\pm$1.6} & {\bf 67.3}{\scriptsize $\pm$1.2} & 68.9{\scriptsize $\pm$1.5} & 66.8{\scriptsize $\pm$1.6} & {\bf 69.2}{\scriptsize $\pm$0.1} & {\bf 67.0}{\scriptsize $\pm$0.2} \\ \hline
\multirow{4}{*}{CNNDM} & original & 43.7{\scriptsize $\pm$0.1}& {\bf 21.1}{\scriptsize $\pm$0.1} & 40.6{\scriptsize $\pm$0.1} & 99.5{\scriptsize $\pm$0.1} & 99.4{\scriptsize $\pm$0.1} & 66.0{\scriptsize $\pm$0.4} & 66.5{\scriptsize $\pm$0.4} & 74.7{\scriptsize $\pm$0.7} & 75.4{\scriptsize $\pm$0.6} & 70.0{\scriptsize $\pm$0.2} & 70.7{\scriptsize $\pm$0.3} \\
& + filtering & 43.4{\scriptsize $\pm$0.2} & 20.8{\scriptsize $\pm$0.1} & 40.3{\scriptsize $\pm$0.2} & {\bf 99.9}{\scriptsize $\pm$0.0} &{\bf 99.9}{\scriptsize $\pm$0.0} & 66.2 {\scriptsize $\pm$0.4} & 66.6{\scriptsize $\pm$0.3} & 74.1{\scriptsize $\pm$0.6} & 74.9{\scriptsize $\pm$0.6} & 69.9{\scriptsize $\pm$0.2} & 70.5{\scriptsize $\pm$0.2} \\
& + classification &43.5{\scriptsize $\pm$0.2} & 20.8{\scriptsize $\pm$0.2} & 40.4{\scriptsize $\pm$0.2} & {\bf 99.9}{\scriptsize $\pm$0.0} & {\bf 99.9}{\scriptsize $\pm$0.0} & {\bf 67.0}{\scriptsize $\pm$0.6} & {\bf 67.5}{\scriptsize $\pm$0.5} & 74.7{\scriptsize $\pm$0.2} & 75.5{\scriptsize $\pm$0.1} & 70.6{\scriptsize $\pm$0.3} & 71.3{\scriptsize $\pm$0.3} \\
& ~~ JAENS &42.4 {\scriptsize $\pm$0.6} & 20.2{\scriptsize $\pm$0.2} & 39.5{\scriptsize $\pm$0.5} & {\bf 99.9}{\scriptsize $\pm$0.0} & {\bf 99.9}{\scriptsize $\pm$0.0} & {\bf 67.9}{\scriptsize $\pm$0.7} & {\bf 68.4}{\scriptsize $\pm$0.6} & {\bf 75.1}{\scriptsize $\pm$0.7} & {\bf 76.4}{\scriptsize $\pm$0.7} & {\bf 71.3}{\scriptsize $\pm$0.2} & {\bf 72.2}{\scriptsize $\pm$0.2} \\ \hline
\multirow{4}{*}{XSUM} & original & {\bf 45.6}{\scriptsize $\pm$0.1} & {\bf 22.5}{\scriptsize $\pm$0.1} & {\bf 37.2}{\scriptsize $\pm$0.1} & 93.9{\scriptsize $\pm$0.1} & 93.6{\scriptsize $\pm$0.2} & 74.1{\scriptsize $\pm$0.2} & 73.3{\scriptsize $\pm$0.2} & 80.1{\scriptsize $\pm$0.1} & 80.3{\scriptsize $\pm$0.3} & 77.0{\scriptsize $\pm$0.1} & 76.6{\scriptsize $\pm$0.2} \\
& + filtering & 45.4{\scriptsize $\pm$0.1} & 22.2{\scriptsize $\pm$0.1} & 36.9{\scriptsize $\pm$0.1} & 98.2{\scriptsize $\pm$0.0} & 98.2{\scriptsize $\pm$0.1} & 77.9{\scriptsize $\pm$0.2} & 77.3{\scriptsize $\pm$0.2} & 79.4{\scriptsize $\pm$0.2} & 79.6{\scriptsize $\pm$0.2} & 78.6{\scriptsize $\pm$0.1} & 78.4{\scriptsize $\pm$0.2} \\
& + classification & 45.3{\scriptsize $\pm$0.1} & 22.1{\scriptsize $\pm$0.0} & 36.9{\scriptsize $\pm$0.1} & 98.3{\scriptsize $\pm$0.1} & 98.2{\scriptsize $\pm$0.1} & 78.6{\scriptsize $\pm$0.3} & {\bf 78.0}{\scriptsize $\pm$0.3} & 79.5{\scriptsize $\pm$0.3} & 79.8{\scriptsize $\pm$0.4} & {\bf 79.1}{\scriptsize $\pm$0.1} & {\bf 78.9}{\scriptsize $\pm$0.1} \\
& ~~ JAENS &43.4{\scriptsize $\pm$0.7} & 21.0{\scriptsize $\pm$0.3} & 35.5 {\scriptsize $\pm$0.4}& {\bf 99.0}{\scriptsize $\pm$0.1} & {\bf 99.0}{\scriptsize $\pm$0.1} & 77.6{\scriptsize $\pm$0.9} & 77.1{\scriptsize $\pm$0.6} & 79.5{\scriptsize $\pm$0.6} & 80.0{\scriptsize $\pm$0.5} & 78.5{\scriptsize $\pm$0.2} & 78.5{\scriptsize $\pm$0.1} \\ \hline
\end{tabular}%
}
\vspace{-1mm}
\caption{Comparison of models trained using original data, with entity-based data filtering, with an additional classification task and with JAENS. Scores are all in percentages, averaged over 5 runs and shown with standard deviations. We bold the numbers that are significantly better in the sense that the means are separated by at least the standard deviations. We report both the micro and macro averages of our proposed entity-level scores. In all datasets, data filtering leads to higher $\mathbf{prec}_s$ scores, indicating that entity hallucination can be alleviated by this simple technique. In addition, data filtering generally improves other entity level metrics: $\mathbf{prec}_t$, $\mathbf{recall}_t$ and $F1_t$. Adding the classification task (multi-task) or JAENS to data filtering further improves the performance on $\mathbf{prec}_t$ and $\mathbf{recall}_t$ and therefore the overall entity-level $F1_t$.}
\label{table:dataFilterResult}
\vspace{-1mm}
\end{table*}
\section{Multi-task learning:}
In addition to entity-based data filtering, we also explore another method to further improve the summarization quality. In particular, we incorporate an additional task of classifying
summary-worthy named-entities in the source document.
A summary-worthy named-entity in the source document is one that appears in the ground truth summary and thus, is a salient entity, worthy of inclusion in the generated summary. Intuitively, if we can identify these summary-worthy named-entities using the encoder representation, we may potentially increase the entity-level precision and recall metrics as well as the overall quality of the summary. We achieve this by adding a classification head to the encoder of BART.
To prepare for the classification label, we first identify the named-entities in the ground truth summary and find the matching tokens in the source document. We then assign the (B)eginning-(I)nside-(O)utside labels to each token of the source document to denote if the token is beginning, inside or outside of a summary-worthy named-entity, respectively.
During training, we simply add the classification loss for each token at the encoder to the original sequence-to-sequence loss.
More precisely, let $\{\left(x^i, y^i\right)\}_{i=1}^N$ be a dataset of $N$ examples where $x^i=x^i_1, \dots, x^i_{ts(i)}$ are the tokens of the $i$th source document and $y^i=y^i_1, \dots, y^i_{tt(i)}$ are the tokens of the target (ground truth summary). The standard sequence-to-sequence training minimizes the maximum log likelihood estimation (MLE) loss:
\begin{equation*}
\mathcal{L}^i_{\text{MLE}} (\theta, x^i, y^i) = - \sum_{t=1}^{tt(i)} \log p_{\theta}(y^i_t | x^i, y^i_{<t}).
\end{equation*}
With summary-worthy entity classification, each example has an additional sequence of BIO labels $z^i=z^i_1, \dots, z^i_{ts(i)}, z^i_t \in \{0,1,2\}$. By adding an additional fully connected layer on top of the BART encoder, we obtain the classification loss
\begin{equation*}
\mathcal{L}^i_{\text{BIO}} (\theta(\text{enc}), x^i, z^i) = - \sum_{t=1}^{ts(i)} \log p_{\theta(\text{enc})} (z^i_t |x^i).
\end{equation*}
Finally, we can minimize the joint loss $ \mathcal{L}^i_{\text{Multitask}} = \mathcal{L}^i_{\text{MLE}} + \alpha \mathcal{L}^i_{\text{BIO}},$
where $\alpha$ is a hyper parameter. We choose $\alpha$ between 0.1 to 0.5 via the validation sets.
\section{Joint Entity and Summary Generation:}
We also explore another generative approach to promote entity-level precision and recall metrics. In particular, instead of just generating the summary, we train the BART model to generate the sequence of summary-worthy named-entities, followed by a special token, and then the summary. We call this approach JAENS (Join sAlient ENtity and Summary generation). Similar to the multi-task learning approach discussed earlier, JAENS encourages the model to jointly learn to identify the summary-worthy named-entities while learning to generate summaries. Since the decoder generates the salient named-entities first, the summaries that JAENS generate can further attend to these salient named-entities through decoder self-attention.
\section{Experiment results}
We use the pre-trained BART-large model in the Fairseq library \cite{ott2019fairseq} to fine-tune on the 3 summarization datasets.\footnote{Our code is available at \rurl{https://github.com/amazon-research/fact-check-summarization}}
The appendix contains additional details of experimental setup.
In Table \ref{table:dataFilterResult}, we
show the effect of the entity-based data filtering. For each dataset, we train two separate models: using the training data before and after entity-based data filtering as shown in Table \ref{table:datasetStats}. We evaluate both models on the ``clean'' test set after entity-based data filtering. We choose this filtered version of the original test set because we only want to measure entity-level consistency against the correct set of entities; using the unfiltered dataset means we could count a hallucinated entity as correct. We observe improvements of $\mathbf{prec}_s$ across all three datasets trained using the filtered subset of data. For example in XSUM, the $\mathbf{prec}_s$ is increased from 93.6\% to 98.2\%, indicating a significant reduction in entity hallucination. In addition, the entity-based data filtering generally improves other entity-level metrics as well. Even with less training data, the entity-based data filtering is able to maintain the ROUGE scores quite well. For XSUM, about 34\% of the training data is filtered out (c.f. Table \ref{table:datasetStats}), which explains the more noticable impact on the ROUGE scores.
The results in Table \ref{table:dataFilterResult} suggest that entity-level data filtering is a simple yet effective approach to achieve higher entity-level factual consistency as well as general summarization quality. In Table \ref{tab:dataFilterResultQualitative} we provide qualitative examples where the model trained on the original data produces hallucination and the entity-level data filtering removes such hallucination.
Table \ref{table:dataFilterResult} shows that adding the classification task (multi-task) futher increases the $\mathbf{prec}_t$ and $\mathbf{recall}_t$ metric and therefore the overall entity-level $F1_t$ on top of the improvements from data filtering. Similar gains can be observed with JAENS, which out-performs the multi-task approach on CNNDM and Newsroom datasets. The result confirms our intuition that the summaries in JAENS can benefit from attending to the generated salient entities in terms of the entity level metrics. However, the additional complexity during decoding may have hurt the ROUGE scores.
For the interested readers, we also evaluated the PEGASUS \cite{pmlr-v119-zhang20ae} models for the ROUGE and entity level metrics on these three datasets in the appendix.
\paragraph{Accuracy of entity level metrics:} As our entity level metrics are based on automatic NER tools and heuristics matching rules, errors in both steps can lead to inaccuracy in the metrics.
By manually checking 10 random ground truth summaries together with the source documents in the validation split of XSUM dataset, we found that all of the named entities are correctly identified by the NER tool and the matchings are correct. Therefore, we believe that even our current NER tool and matching rule already produce high accuracy in practice.
\begin{table*}[ht]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{p{0.25\linewidth}|p{0.25\linewidth}|p{0.25\linewidth}|p{0.25\linewidth}}
\thead{Before data filtering} & \thead{After data filtering} & \thead{With classification} & \thead{Ground truth summary} \\ \hline
People in Italy and the Netherlands are more likely to consume fewer cups of coffee than those in the \underline{UK}, a study suggests.
& The desire to drink coffee may be encoded in our DNA, according to scientists.
& People with a particular gene are more likely to consume fewer cups of coffee, a study has suggested.
& Researchers have identified a gene that appears to curb coffee consumption. \\ \hline
A cathedral in \underline{Surrey} is set to be restored after more than £5m was raised to pay for repairs and improvements.
& A £7m project to save a Grade II-listed cathedral from demolition is set to go ahead.
& A cathedral which has been threatened with demolition is set to be saved by a £5m fundraising campaign.
& A 1960s-built cathedral that was "at serious risk of closure" has raised more than 90\% of its £7m target for urgent repairs and development. \\ \hline
More than 800,000 chemists in the Indian \underline{capital, Delhi}, have gone on strike in protest against online drug sales.
& More than 800,000 chemists in India will go on strike on Wednesday to protest against illegal online drug sales.
& More than 800,000 chemists in India are set to go on strike on Wednesday in a row over the sale of drugs online.
& At least 800,000 pharmacies in India are on a one-day strike, demanding an end to online drug sales which they say is affecting their business. \\ \hline
Police officers in \underline{Pembrokeshire} are to be issued with body-worn cameras. & Police officers in Powys are to be issued with body-worn cameras in a bid to improve transparency in the force.
& Police officers in Powys are to be issued with body cameras in a bid to improve transparency in the force.
& A police force has begun the rollout of body cameras for 800 officers and community support officers. \\ \hline
Wales midfielder \underline{Becky Lawrence} has been speaking to \underline{BBC Sport} about her time as a player-manager with Melbourne City. & It's been a great few weeks for me as a player-manager and now I'm heading home to Wales ahead of the Cyprus Cup.
& It's been a very busy few weeks for me as I'm heading home to Wales ahead of the Cyprus Cup.
& I have certainly had worse 24 hours in my life than winning the Grand Final with Melbourne City and then being named in the Wales squad for the Cyprus Cup.
\end{tabular}%
}
\vspace{1mm}
\caption{Generated and ground truth summary examples from the test set of XSUM. The first three columns are generated from the model trained without entity-based data filtering, with entity-based data filtering and with the additional classification task, respectively. The right column contains the ground truth summaries. The hallucinated named-entities are underscored. Proposed data filtering overcomes hallucination in these examples.}
\label{tab:dataFilterResultQualitative}
\end{table*}
\section{Conclusion}
In this paper we study the entity-level factual consistency of the state-of-the-art summarization model. We propose precision-source score $\mathbf{prec}_s$ to quantify the degree of entity hallucination. We also propose additional metrics $\mathbf{prec}_t$ and $\mathbf{recall}_t$ to measure entity level accuracy of the generated summary with respect to the ground truth summary. We found that the ground truth summaries of the XSUM dataset contain a high level of entity hallucination. We propose a simple entity-level data filtering technique to remove such hallucination in the training data. Experiments show that such data filtering leads to significant improvement in $\mathbf{prec}_s$. ($\mathbf{prec}_s$ increases from below 94\% to above 98\% in XSUM for example.)
We futher proposed a multi-task learning and a joint sequence generation approach to further improve the entity-level metrics.
Overall, combining our proposed approaches significantly reduces entity hallucination and leads to higher entity level metrics with minimal degradation of the ROUGE scores.
\bibliographystyle{acl_natbib}
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\section{Introduction}
\section{Theoretical Description}
We use the chiral Lagrangian in the vector formulation for spin-1 fields
of Refs.~\refcite{Ecker:1988te}-\refcite{Prades:1993ys}.
In the even-intrinsic-parity sector one has
\begin{equation}
\Lagr_{\gamma V} = - \frac{e f_V F^{\mu\nu}}{2} \bigl(
\hat{\rho}^0_{\mu\nu} + \frac{1}{3}\hat{\omega}_{\mu\nu} -
\frac{\sqrt{2}}{3}\hat{\phi}_{\mu\nu} \bigr)\, ,
\label{eq:lagr_gV}
\end{equation}
where $F^{\mu\nu} = \partial^\mu B^\nu - \partial^\nu B^\mu$, and
$\hat{V}_{\mu\nu} = \partial_\mu \hat{V}_\nu - \partial_\nu \hat{V}_\mu$ in terms of the vector fields
for the vector mesons $V \equiv \rho^0, \omega, \phi$.
The interactions in the odd-intrinsic-parity sector read
\begin{equation}
\Lagr_{V\gamma P} = - \frac{4 \sqrt{2} e h_V }{3f_\pi}
\epsilon^{\mu\nu\alpha\beta} \partial_\mu B_\nu
\bigl( \rho^0_\alpha + 3\omega_\alpha +
3\varepsilon_{\omega\phi}\phi_\alpha \bigr)\partial_\beta\pi^0 \, ,
\label{eq:lagr_VgP}
\end{equation}
\begin{equation}
\Lagr_{VVP} = - \frac{4 \sigma_V}{f_\pi}
\epsilon^{\mu\nu\alpha\beta} \pi^0
\partial_\mu \bigl( \omega_\nu + \varepsilon_{\omega\phi} \phi_\nu \bigr)
\partial_\alpha \rho^0_\beta \, .
\label{eq:lagr_VVP}
\end{equation}
The $\epsilon^{\mu \nu \alpha \beta}$ is the totally antisymmetric
Levi-Civita symbol, pion decay constant is $f_{\pi} = 92.4$~MeV.
The terms with $\eta$ and $\eta'$ mesons and the G-parity-violating $\phi\omega\pi^0$ vertex are neglected.
The model parameters $f_V, h_V, \sigma_V$ are related via the special short-distance constraint:\cite{Prades:1993ys,Knecht:2001xc}
$\sqrt{2} h_V - \sigma_V f_V = 0 \, $ .
The radiative decays of the light vector resonance $V$ into the pseudoscalar meson $P$
and photon are widely used as electromagnetic probes of the flavor content of the mesons.\cite{Achasov:2000wy}$^-$\cite{Usai:2011zza}
For example, these decays provide an access to the value of the coupling constant $h_V$ via the partial width
\begin{equation}
\Gamma (\omega\to\pi^0\gamma) = \frac{4\alpha M_{\omega}^3 {h_V}^2}{3f_\pi^2}
{\biggl( 1 - {\frac{m_{\pi}^2}{M_{\omega}^2}} \biggr)} ^3
\label{eq:partial_width}
\end{equation}
Using PDG data\cite{Beringer:1900zz} for the $\omega \to \pi^0 \gamma$ decay we obtain $h_V=0.03753$.
As it is seen from Eqs.~(\ref{eq:lagr_VgP})~and~(\ref{eq:lagr_VVP}), the transition $\phi \to \pi^0 \gamma$ is related
to the small parameter $\varepsilon_{\omega\phi}$, responsible for the $u\bar{u}+d\bar{d}$
component in the physical $\phi$ meson. Thus one can compare the $\omega\to\pi^0\gamma$ and $\phi \to \pi^0 \gamma$ widths
and find the value for the $\omega\phi$ mixing parameter
$\varepsilon_{\omega\phi}=(5.79\pm 0.17)\times 10^{-2}$.
The transition form factors can be extracted from the decay line shape $\frac{d\,\Gamma(V \to P \gs)}{d\,Q^2}$
and the cross section $\frac{d\,\sigma(e^+e^- \to \omega \pi^0)}{d\,Q^2}$.
Experimentally, only the normalized form factors are known $F_{V\to P\gs} (Q^2=0) = 1$.
We include direct $\omega \pi^0 \gamma$ coupling and mediated $\omega \pi^0 \rho$ with
subsequent $\gamma\rho$ conversion, contributing to the
Dalitz decay $\omega \to \pi^0 \mu^+ \mu^-$.
According to the Lagrangian terms from Eqs.~(\ref{eq:lagr_gV})-(\ref{eq:lagr_VVP}),
the form factor reads
\begin{equation}
F_{\omega \pi^0 \gs} (Q^2) =
1 - \frac{\sigma_V f_{\rho}(Q^2)}{\sqrt{2} h_V} Q^2 D_{\rho} (Q^2) \, .
\label{eq:from_factor}
\end{equation}
An additional energy dependence of the EM coupling $f_{\rho}(Q^2)$
arises due to higher-order corrections\cite{Klingl:1996by};
$ D_{\rho} (Q^2) = \bigl[ Q^2 - M_{\rho}^2 - \Pi_{\rho} (Q^2) \bigr] ^{-1}\,$
is $\rho$-meson propagator.
\begin{figure}[h]
\begin{minipage}[h]{0.6\linewidth}
\center{\includegraphics[width=\linewidth]{plot_im_self-energy}}
\end{minipage}
\quad
\quad
\begin{minipage}[h]{0.25\linewidth}
\center{\includegraphics[width=\linewidth]{pi_omega}
\includegraphics[width=\linewidth]{pi_pi}}
\end{minipage}
\caption{The dominant loop contributions to the $\gamma\rho$ vertex.}
\label{fig:imP}
\end{figure}
In the region of interest the most important contribution
to the self-energy operator $\Pi_{\rho} (Q^2)$ consist of the $\pi^+\pi^-$ and $\pi^0\omega$ corrections.\cite{Ivashyn:2006gf}
In the following we include only the imaginary parts of the loop contributions (see Fig.~\ref{fig:imP}).
This will be the dominant term for the energy-dependent width
$\Gamma_{tot,\rho} (Q^2) = - M_{\rho}^{-1}\ {\rm Im}\, \Pi_{\rho} (Q^2) \, $,
which is important only for the $\rho$ meson within the scope
of current research in the region of momenta $0<\sqrt{Q^2}<1.4$~GeV overlapped with $\rho$ resonance.
The ways to include the real part of the self-energy are discussed, e.g. in Ref.~\refcite{Klingl:1996by}.
The modified EM coupling in terms of the loop corrections
\begin{equation}
f_{\rho}(Q^2) = f_{V} - \frac{\imath}{e\,Q^2}
\sum_c {\rm\, Im} \Pi_{\gamma(c)\rho}(Q^2) \, ,
\label{eq:mod_coupl}
\end{equation}
where $c = (\pi\pi , \pi^0\omega)$ stands for the dominant loop contributions.
The ``bare'' constant $f_V$ is real-valued.
The modified coupling constant $f_{\rho}(Q^2)$ at $Q^2 = M_{\rho}^2$ has to describe
the leptonic decay width of $\rho$ meson:
\begin{equation}
\Gamma (\rho^0\to e^+e^-) = \frac{e^4 M_\rho} {12\pi}
{\bigl| f_{\rho}(Q^2 = M_{\rho}^2) \bigr|}^2 \ .
\label{eq:lept_dec}
\end{equation}
Eqs.~(\ref{eq:mod_coupl})~and~(\ref{eq:lept_dec}) allow us to find the bare coupling $f_V=0.202$.
\section{Results and Conclusions}
\begin{figure}[h]
\begin{minipage}[h]{0.565\linewidth}
\center{\includegraphics[width=\linewidth]{plot_omega_ff}}
\end{minipage}
\quad
\begin{minipage}[h]{0.39\linewidth}
\center{\includegraphics[width=\linewidth]{plot_phi_ff}}
\end{minipage}
\caption{The $\sqrt{Q^2}$ behavior of the $\omega \to \pi^0 \gs$(left) and $\phi \to \pi^0 \gs$(right) transition form factors.}
\label{fig:plot}
\end{figure}
The $\omega \to \pi^0\gs$ form factor (normalized) is shown in Fig.~\ref{fig:plot},~left.
We can notice that our model agrees with data slightly better than the model of Ref.~\refcite{Terschluesen:2010ik},
but for both models the problematic region is near $0.6$~GeV.
This fact makes the problem of $F_{V\to P\gs}$ modeling very important.
Also our results for the $\phi \to \pi^0\gs$ form factor (Fig.~\ref{fig:plot},~right) is compared with obtained
in Ref.~\refcite{Schneider:2012ez}. This process is not measured yet and represents strong experimental interest
(e.g., see materials by M. Mascolo in this proceedings).
Although the proposed approach is in qualitative agreement with the data from $e^+e^- \to \omega \pi^0$ at high energies,
the $\gamma V$ vertex modification is not enough to reduce the discrepancy with NA60 data in the region $\sqrt{Q^2} > 0.4$~GeV.
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\section{INTRODUCTION}
Many overcontact binaries are thought to have evolved from initially detached binaries by angular momentum loss (AML) via magnetic torques
from stellar winds and to end with coalescence of both components into single stars (Bradstreet \& Guinan 1994; Pribulla \& Rucinski 2006).
In this scenario, the orbital angular momentum is tidally coupled to the spin angular momentum. In order for the spin-orbit coupling
to work efficiently, the initial orbital periods should be shorter than about 5 days. Circumbinary objects in multiple systems may remove
angular momentum from the central pairs via Kozai oscillation (Kozai 1962; Pribulla \& Rucinski 2006) or a combination of the Kozai cycle
and tidal friction (Fabrycky \& Tremaine 2007) and form the initial tidal-locked detached binaries with a short orbital period.
The statistical study of Pribulla \& Rucinski (2006) suggests that a large percentage of close binares reside in triple or multiple systems.
The presence of a third body orbiting an eclipsing close binary causes a periodic variation of the eclipsing period due to the increasing
and decreasing light-travel times (LTT) to the observer (Irwin 1952, 1959). The LTT effect can be studied by a detailed analysis of
eclipse timing diagram, showing the differences between the observed ($O$) and the calculated ($C$) timings of minimum light {\it versus}
time (e.g., Lee et al. 2009a, 2013). In order to advance this subject, we have been observing short-period eclipsing binaries,
such as overcontact and near-contact systems. In this work, we choose the W UMa-type binary EP And, because the list of eclipse timings
goes back far enough to understand the binary's period behavior.
Since the discovery of its variability by Strohmeier et al. (1955), EP And (TYC 2827-17-1, GSC 2827-17, 2MASS J01422933+4445424;
$V$=+11.90, $B$--$V$=+0.57) has been the topic of several investigations, but its properties are relatively poorly known compared to
other short-period binaries. Most recently, Manzoori (2012) reviewed the observational history of the system and presented the first
comprehensive photometric study. The author analyzed the photometric data from both the AAVSO (American Association of Variable Stars Observers)
International Database\footnote {http://www.aavso.org/} and the WASP (Wide Angle Search for Planets) public archive (Butters et al. 2010),
and concluded that EP And is an A-type (defined observationally by Binnendijk 1970) overcontact binary with a mass ratio of $q$=0.395,
an orbital inclination of $i$=81$^\circ$.5, a temperature difference between the components of $\Delta T$=120 K, and a fill-out factor
of $f$ = 16.6 \%. Here, $q$ is the ratio of the secondary'mass to that of the primay star (eclipsed at the primary minimum) and
$f$=($\Omega_{\rm in}$--$\Omega$)/($\Omega_{\rm in}$--$\Omega_{\rm out}$), where the potentials $\Omega_{\rm in}$ and
$\Omega_{\rm out}$ define the inner and outer critical surfaces in Roche geometry and $\Omega$ is the potential corresponding
to the surface of the overcontact binary. From the analysis of eclipse timings including five epochs determined from the AAVSO data,
he suggested that the orbital period of EP And can be sorted into a secular period decrease caused by mass transfer between
the components, an LTT effect due to the orbit of a third body with a period of 41.2 yr and a minimum mass of 0.15 M$_\odot$, and
the effect of magnetic activity with a cycle length of 11.7 yr.
In order to obtain a unique set of photometric solutions and to examine whether the Manzoori's (2012) suggestion is appropriate
for the orbital period change, we have studied in detail the long-term photometric behavior of EP And from all available data.
Our results from both the light-curve synthesis and the orbital period study show that EP And is probably a multiple system.
This study follows the following structure. In section 2, we present our new photometric observations of EP And.
Absolute dimensions of the eclipsing pair are determined from light-curve modeling in section 3. In section 4, we present
an orbital period study of the eclipse timing diagram and determine two significant periods in the timing measurements.
In the last section we give a discussion highlighting that the two periodic variations are most likely attributed to the presence
of circumbinary companions.
\section{NEW LONG-TERM CCD PHOTOMETRY}
New CCD photometric observations of EP And were obtained between 2007 and 2012, using CCD cameras and a $BVR$ filter set attached
to the 61-cm reflector at Sobaeksan Optical Astronomy Observatory (SOAO) in Korea. The observations of the first two seasons were
carried out to secure complete multiband light curves and the others to collect additional eclipse timings. A summary of
the observations is listed in Table 1, where we present observing interval, numbers of nights, CCD type, and field of view (FOV).
The instruments and reduction methods for the SITe 2K and FLI IMG4301E CCD cameras are the same as those described by
Lee et al. (2007, 2011). The observations of both 2011 and 2012 seasons were obtained with a PIXIS: 2048B CCD camera that offers
cooling down to $-70\rm ^o$ C. The e2v CCD42-40 chip has 2048$\times$2048 pixels and a pixel size of 13.5 $\mu$m.
With the conventional IRAF package, we processed the CCD frames to correct for bias level and pixel-to-pixel inhomogeneities of
quantum efficiency (flat field correction) and applied simple aperture photometry to obtain instrumental magnitudes.
Following the procedure described by Lee et al. (2010), TYC 2827-103-1 ($V\rm_T$=10.99, ($B$--$V$)$\rm_T$=+0.64) and TYC 2827-72-1
($V\rm_T$=10.98, ($B$--$V$)$\rm_T$=+1.16), imaged on the chip at the same time as the variable (V), were chosen as comparison (C)
and check (K) stars, respectively. From the 2007 and 2008 seasons, a total of 3,027 individual observations were obtained in
the three bandpasses (1007 in $B$, 1012 in $V$, and 1008 in $R$) and a sample of them is listed in Table 2. The light curves are
plotted in the top panel of Figure 1 as the (V$-$C) magnitude differences {\it versus} orbital phase, which was computed according
to the quadratic ephemeris for our binary model (Model 2) determined later in this article with the Wilson-Devinney synthesis code
(Wilson \& Devinney 1971, hereafter W-D). The differences ('07-'08) between the two seasons are shown in the middle panel and
the magnitude differences between the check and comparison stars appear in the bottom panel. The 1$\sigma$-values of the dispersion
of the (K$-$C) differences are $\pm$0.016 mag, $\pm$0.012 mag, and $\pm$0.013 mag from $B$ to $R$ bandpasses, respectively,
for the 2007 season and $\pm$0.009 mag, $\pm$0.005 mag, and $\pm$0.005 mag for the 2008 season. Whereas the reference stars were
evidently constant, the light curves display season-to-season light variability, with evident changes in the phase interval
encompassing primary eclipse.
\section{LIGHT-CURVE SYNTHESIS AND ABSOLUTE DIMENSIONS}
As shown in Figure 1, our observations display a typical light curve of an overcontact system and a flat bottom at primary minimum,
indicating that the smaller primary star is totally occulted by the secondary. This would mean that EP And belongs to the W-type
of W UMa stars. On the contrary, Pribulla et al. (2001) and Manzoori (2012) analyzed their $BV$ light curves and both the AAVSO
and WASP datasets, respectively, and classified the binary system as a member of the A-type category with the mass ratios of 0.34
and 0.395. The light curves of 2008 present equal light levels at the quadratures (Max I and Max II) within about 0.002 mag,
while those of 2007 show the O'Connell effect with Max I brighter than Max II by about 0.014, 0.011, and 0.009 mag for
the $B$, $V$, and $R$ bandpasses, respectively. The effect is usually interpreted as spot activity on the component stars and
the seasonal light variations most likely arise from the variability of the spots with time presumably produced by a magnetic dynamo.
In order to obtain a consistent solution of EP And, we simultaneously solved all avaliable light curves using contact mode 3 of
the W-D synthesis code and with a weighting scheme identical to that for the eclipsing binary GW Gem (Lee et al. 2009b). Table 3
lists the light-curve sets for EP And analyzed in this paper and the standard deviations ($\sigma$) of a single observation.
Although the binary parameters have been reported by Pribulla et al. (2001) and Manzoori (2012), their solutions with $q<$1.0
do not correspond to our high-precision observations of 2008, showing a total eclipse at primary minimum and hence implying $q>$1.0.
To resolve this confusion, we analyzed the light curves of EP And in a manner similar to that for the overcontact systems AR Boo
(Lee et al. 2009c) and GW Cep (Lee et al. 2010) using the so-called $q$-search procedure. In the computation of our solutions,
the surface temperature of the larger, and presumably more massive, star was held fixed at 6,360 K from Flower's (1996) table,
according to ($B-V$)=$+$0.57$\pm$0.02 given by Terrell et al. (2012) and $E$($B-V$)=$+$0.09 calculated following
Schlegel et al. (1998). The logarithmic bolometric ($X$, $Y$) and monochromatic ($x$, $y$) limb-darkening coefficients were
interpolated from the values of van Hamme (1993) in concert with the model atmosphere option. Before the light curves are analyzed,
the AAVSO times were transformed from JD into HJD. Further, two LTT effects proposed in the following section were applied to
the observed times of all individual points: HJD$_{\rm new}$=HJD$_{\rm obs}$--($\tau_{3}$+$\tau_{4}$). The quantities shifted by
the two LTTs are 0.00722$\sim$0.00871 d for 2001, 0.00733$\sim$0.00831 d for 2007, and 0.00026$\sim$0.00052 d for 2008.
In this paper, we refer to the primary and secondary stars as those being eclipsed at Min I and Min II, respectively.
Terrell \& Wilson (2005) showed that the mass ratio for a totally-eclipsing overcontact system can be accurately determined from
a light-curve analysis. Thus, we conducted the detailed $q$-search procedures for both all datasets and only the 2008 light curves,
permitting no perturbations such as a third light ($l_3$) or a spot. As displayed in Figure 2, the $q$-search results indicate
a minimum value of the weighted sum of the squared residuals ($\Sigma$) around $q$=2.75. This value corresponds to an occultation
at primary minimum and indicates that EP And is a W-type overcontact binary. To obtain an unperturbed solution (Model 1),
we analyzed all light curves by treating the initial value of $q$ as a free parameter. The results are listed in columns (2)--(3)
of Table 4. Then, we reanalyzed the EP And curves by considering a third light ($l_3$) as an additional adjustable parameter
because it had been suggested by our period study later. New results with the third light source are given as Model 2 in Table 4
and the light residuals from this binary model are plotted in Figure 3. There is a statistically significant difference in
the mass ratios between the two models due to the third light effect. In a formal sense, as shown by the entries on the last line
of the table, the third-light model gives a smaller value of $\Sigma W(O-C)^2$.
As indicated by Figure 3, the model light curves describe the observations of the 2001 and 2008 seasons quite well, but not those
of the 2007 season. The non-modelled light could be explained by a magnetic cool spot on either of the component stars. Model spots
were added to fit the small light variations of 2007 by adjusting only the spot and luminosity parameters among the Model 2 parameters.
Final results are given in Table 5, and the residuals from the cool-spot model on the secondary star are plotted as the plus symbols
in the middle panels of Figure 3. From these displays, we can see that the spot model does fit the asymmetries in the light maxima
acceptably but it is difficult to distinguish between the cool-spot models because there is no $\Sigma$ differences among them.
Our light-curve solutions indicate that EP And is a totally-eclipsing W-type overcontact binary with a fill-out factor of about 28 \%
and with a small temperature difference of 27 K between the components and that $l_3$ contributes 2--3 \% light in all bandpasses.
Absolute dimensions for EP And can be estimated from the photometric solutions (Model 2) in Table 4 and from Harmanec's (1988) relation
between temperature (spectral type) and stellar mass. We assumed the more massive secondary star to be a normal main-sequence one
with a spectral type of about F6 and computed the physical properties for the system listed in Table 6. The luminosity ($L$) and
bolometric magnitudes ($M_{\rm bol}$) were computed by adopting $T_{\rm eff}$$_\odot$=5,780 K and $M_{\rm bol}$$_\odot$=+4.73
for solar values. For the absolute visual magnitudes ($M_{\rm V}$), we used the bolometric corrections (BCs) appropriate for the temperature
of each component from the expression between $\log T_{\rm eff}$ and BC given by Torres (2010). Using an apparent visual magnitude of
$V$=+11.90 (Terrell et al. 2012) at maximum light, the computed light ratio at phase 0.25, and the interstellar absorption of $A_{\rm V}$=0.28,
we calculated an approximate distance to the system of about 470 pc. In the mass-radius, mass-luminosity, and Hertzsprung-Russell diagrams
from Hilditch et al. (1988), the locations of both components of EP And conform to the general pattern of overcontact binaries.
\section{ORBITAL PERIOD STUDY}
From the SOAO observations, 29 new times of minimum light and their errors were determined with the weighted means for the timings
in each filter by using the method of Kwee \& van Woerden (1956). In addition, 61 eclipses were newly derived by us from the WASP data
and five timings from the AAVSO data. The AAVSO times from Manzoori (2012) may be not HJD but JD. For a period study of EP And,
we have collected a total of 414 timings (58 photographic plate, 216 visual, 140 photoelectric and CCD) including our measurements.
All photoelectric and CCD timings are listed in Table 7, wherein the second column gives the HJED (Heliocentric Julian Ephemeris Date)
timings transformed to the terrestrial time scale (Bastian 2000). Because most earlier timings from Kreiner et al. (2001) were published
without error information, we calculated the standard deviations of the scatter bands of the timing residuals to provide mean errors
for the observational methods, as follows: $\pm$0.0101 d for photographic plate, $\pm$0.0076 d for visual, and $\pm$0.0012 d for PE
and CCD minima. Relative weights were then scaled from the inverse squares of these values consistent with the errors and weights
for the PE and CCD timings.
The orbital period of EP And was studied for the first time by Qian \& Yuan (2001). From a quadratic least-squares fit, they reported
a period increase with a rate of $+$1.16$\times$10$^{-7}$ d yr$^{-1}$. Recently, Manzoori (2012) claimed that two periodicities of
41.2 yr and 11.7 yr, superimposed on the upward parabolic variation, exist in the timing residuals. As the first step for
ephemeris computations, we applied a periodogram analysis to the complete dataset using the \texttt{PERIOD04} program (Lenz \& Breger 2005).
As can be seen from Figure 4, two frequencies of $f_1$=0.0000424 cycle d$^{-1}$ and $f_2$=0.00149 cycle d$^{-1}$ were detected
corresponding to 23,585 d (64.6 yr) and 671 d (1.8 yr), respectively. Thus, the two periods were used to provide an initial guess
for the Levenberg-Marquart (LM) fitting procedure (Press et al. 1992). The oscillations were assumed to be due to a combination of
two LTT effects caused by the third and fourth bodies in the system and all times of minimum light were fitted to the following
two-LTT ephemeris:
\begin{eqnarray}
C_1 = T_0 + PE + \tau_3 + \tau_4.
\end{eqnarray}
Here, $\tau_{3}$ and $\tau_{4}$ are the LTT due to two additional companions orbiting the eclipsing pair (Irwin 1952, 1959) and each
includes five parameters ($a_{12}\sin i$, $e$, $\omega$, $n$, $T$). The LM technique was applied to solve for the twelve parameters
of the ephemeris and the results are listed in columns (2)--(3) of Table 8, together with related quantities. Our absolute dimensions
presented in Table 6 have been used for these and subsequent calculations.
The $O$--$C_1$ diagram constructed with the linear terms of the two-LTT ephemeris is plotted in the top panel of Figure 5, where
the solid and dashed curves represent the full contribution and the $\tau_{3}$ orbit, respectively. The middle panel displays
the PE and CCD residuals from the complete ephemeris and the bottom panel represents the $\tau_{4}$ orbit. As displayed in the figure,
all times of minimum light currently agree with the two-LTT ephemeris satisfactorily. The successful fit to the times of minimum light
with the two LTT orbits tempted us to try to discover a secular term which might be hidden in the two periodic variations.
Because EP And is in an overcontact configuration with common convective envelope, a parabolic variation should be produced by
mass transfer between both components and/or by AML due to magnetic stellar wind. Therefore, a more general fit to the times of
minimum light was made by adding a quadratic term to the two-LTT ephemeris:
\begin{eqnarray}
C_2 = T_0 + PE + A E^2 + \tau_3 + \tau_4.
\end{eqnarray}
The calculations using the LM method converged quickly to yield the results given in columns (4)--(5) of Table 8.
The $O$--$C_2$ diagram constructed with the linear light elements are drawn at the top of Figure 6 with the solid curve due to
the sum of the non-linear terms and the dashed parabola due to the quadratic term of equation (2). The second to bottom panels
are plotted in the same sense as in Figure 5. The timing residuals from the full ephemeris appear as $O$--$C_{\rm 2,full}$
in the fifth column of Table 7. Figure 7 shows the PE and CCD residuals phased with the $\tau_{4}$ cycle (1.8340 yr) listed
in Table 8. This ephemeris resulted in a smaller $\chi^2_{\rm red}$=0.981 than the two-LTT ephemeris ($\chi^2_{\rm red}$=1.102).
Its long-term period ($\tau_4$) is short compared to that of the two-LTT ephemeris by about 21 yr, while the short-term periods
($\tau_3$) for the two ephemerides are in excellent agreement with each other.
If it is assumed that the orbits of the two circumbinary objects are coplanar with that of the eclipsing pair of EP And
($i_{3,4}$=83\fdg3), the masses of the third and fourth bodies are $M_3$=0.25 M$_\odot$ and $M_4$=0.90 M$_\odot$, respectively.
If they are main-sequence stars, the radii and temperatures are calculated to be $R_3$=0.26 R$_\odot$ and $T_3$=3039 K, and
$R_4$=0.92 R$_\odot$ and $T_4$=5082 K from the empirical relation of Southworth (2009). The third and fourth bodies would
contribute about 0.1 \% and 11.6 \%, respectively, to the total bolometric luminosity of the quadruple system. Because
our light-curve solutions in Tables 4 and 5 detected only $l_3$ of 2--3 \% in all bandpasses, the putative fourth object
have to be very under-luminous in comparison to the binary components and may be a compact star. Alternatively, it is possible
that the fourth body might be a binary itself. This could reduce the luminosity for the given total mass of 0.90 M$_\odot$
and implies that EP And should be a quintuple system. The semi-amplitude of the expected systemic radial velocity changes of
the eclipsing pair due to the third and fourth components would be about 1.5 km s$^{-1}$ and 11.8 km s$^{-1}$, respectively.
Hence, the M-type third companion is very difficult to reveal because of the low systemic velocity change and
the large orbital period suggested by the LTT model, while the massive circumbinary object might be easily detected with
high-resolution spectroscopy. On the other hand, the eclipse timing variations might be partly caused by the perturbative effect
of the fourth component added to the geometrical LTT effect because its LTT period is very short (Borkovits et al. 2003, 2011).
One such example is IU Aur: \" Ozdemir et al (2003) showed that the third companion with a period of 293.3 d cause the non-negligible,
dynamical contribution to the $O$--$C$ curve. We computed the semi-amplitude of the fourth-body dynamic perturbation on the motion
of the overcontact binary to be 0.000013 d and found that its contribution is not significant.
The positive coefficient of the quadratic term ($A$) listed in Table 8 yields a secular period increase with a rate of
+5.09$\times$10$^{-8}$ d yr$^{-1}$, corresponding to a fractional period change of +1.39$\times$10$^{-10}$. This value agrees
well with the value of $+$1.40$\times$10$^{-10}$ calculated with our W-D binary code, independently of the eclipse timings.
Under the assumption of conservative mass transfer, this gives a continuous mass transfer from the less massive primary to
the secondary component at a modest rate of 3.41$\times$10$^{-8}$ M$_\odot$ yr$^{-1}$. The observed value is small by a factor of
about 50\% compared with the predicted rate of 6.77$\times$10$^{-8}$ M$_\odot$ yr$^{-1}$ calculated by assuming that the primary
transfers its present mass to the secondary on a thermal time scale. Thus, the possible explanation of the parabolic variation
might be some combination of non-conservative mass transfer and AML due to magnetic braking.
\section{DISCUSSION AND CONCLUSIONS}
In this article, we presented and analyzed new long-term CCD observations of EP And, together with historical data collected
from the literature. The light curves display a total eclipse at primary minimum and season-to-season light variability.
The asymmetric light curves in 2007 were modeled by a magnetic cool spot on either of the component stars. Our detailed study
of the light curves and the orbital period represent EP And to be a quadruple (or a quintuple) system with a W-type overcontact binary.
Because the period is increasing and the mass is transferring from the primary star to the more massive secondary, the eclipsing pair
may presently be in an expanding state evolving from a overcontact to a non-contact configuration as it undergoes thermal relaxation
oscillations (Lucy 1976; Lucy \& Wilson 1979).
In principle, the periodic variations in the eclipse timing residuals can be expected because of stellar activity variations
of a magnetically active star, as was initially proposed by Applegate (1992) and later modified by Lanza et al. (1998). With
the modulation periods ($P_{3,4}$) and amplitudes ($K_{3,4}$) listed in columns (4)--(5) of Table 8, the model parameters
for each cycle were calculated from the Applegate formulae and are listed in Table 9, where the rms luminosity changes
($\Delta m_{\rm rms}$) converted to magnitude scale were obtained with equation (4) in the paper of Kim et al. (1997).
In the table, the primary component with 1.2 L$_\odot$ and the secondary with 2.7 L$_\odot$ exhibit
the predicted luminosity variations of 55.4 L$_\odot$ and 34.8 L$_\odot$, respectively, for the short-term cycle. The variations
of the gravitational quadrupole moment ($\Delta Q$) for the long-term cycle are two orders of magnitude smaller than typical values
of $10^{51}-10^{52}$ for close binaries (Lanza \& Rodono 1999). Moreover, it is difficult for the model to produce perfectly
smooth and tilted periodic components in the eclipse timing variation. These suggest that Applegate mechanism cannot explain
the observed period modulations of EP And. On the other hand, a single periodic variation could be attributed to the rotation
of the apsidal line of the binary orbit due to tidal forces between the two binary components. However, our light-curve analysis
suggest that the binary orbit is circular which in turn excludes timing variations from apsidal precession.
As can be seen in Figures 5--7, all times of minimum light agree quite well by interpreting the observed LTT signal as caused
by a third and fourth body surrounding the eclipsing pair. The possible existence of the circumbinary objects is consistent
with the suggestion of Pribulla \& Rucinski (2006) that most W UMa systems exist in multiple systems.
We have carried out a stability study for the orbit parameters in Table 8, under the assumption that the eclipsing pair
can be replaced by a single massive object with mass equal to the combined mass of the two binary components. In general,
our methodology on assessing the orbital stability is similar to the work presented in Hinse et al. (2012) who considers
the stability of a similar system (SZ Herculis) of two circumbinary M-type companions (Lee et al. 2012).
The dynamical stability test suggests that the two proposed companions are on highly unstable orbits. In order to reconcile
the apparent contradiction we are therefore left with two options. Either i) we discard our two-companion interpretation
or ii) our LTT model is in lack of important physics that is not included in the present analysis/model. In light of
the above given arguments we do not favour to discard the two companion interpretation as presented in this work.
Of the possible causes of the periodic variations, the Applegate effect and apsidal motion can be ruled out.
The most reasonable explanation of both cycles is a pair of the LTT effects driven by the presence of circumbinary companions.
In future work we plan a re-analysis of the data with an improved model that formulates the LTT effect in Jacobi coordinates
and includes mutual gravitational interactions. Mutual interactions between the companions is a highly non-linear process and
could result in a significant different orbital architecture consistent with the observed timing data. The two-Kepler assumption
might be inadequate when larger companion masses are involved. This system is an obvious candidate for
future photometric follow-up programs for further characterisation (Pribulla et al. 2012).
Assuming our LTT interpretation is correct, then the outer components may have played an important role in the formation and
evolution of the inner eclipsing pair, which would cause it to evolve into an overcontact configuration by AML via magnetic braking
and ultimately to coalesce into a single rapid-rotating star. High-resolution photometry and spectroscopy will help to identify and
understand the orbital period variation of the binary system and to determine the absolute parameters and evolutionary status of the
multiple system better than is possible with photometry alone.
\acknowledgments{ }
We would like to thank the staff of the Sobaeksan Optical Astronomy Observatory for assistance during our observations.
We appreciate the careful reading and valuable comments of the reviewer Dirk Terrell. This research has made use of
the Simbad database maintained at CDS, Strasbourg, France. We have used data from the AAVSO International Database and
the WASP public archive in this research. The WASP consortium comprises of the University of Cambridge, Keele University,
University of Leicester, The Open University, The Queen's University Belfast, St. Andrews University and the Isaac Newton Group.
Funding for WASP comes from the consortium universities and from the UK's Science and Technology Facilities Council. This work was
supported by the KASI (Korea Astronomy and Space Science Institute) grant 2013-9-400-00. T.C.H. acknowledges financial support from
the Korea Research Council for Fundamental Science and Technology (KRCF) through the Young Research Scientist Fellowship Program.
\newpage
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\section{Introduction}
In recent years, several new excited charmed baryon states have been
observed by the BaBar, Belle and CLEO Collaborations, such as the
$\Lambda_c(2765)^+$, $\Lambda_c^+(2880)$, $\Lambda_c^+(2940)$,
$\Sigma_c^+(2800)$, $\Xi_c^+(2980)$, $\Xi_c^+(3077)$,
$\Xi_c^0(2980)$, $\Xi_c^0(3077)$
\cite{ShortRV1,ShortRV2,ShortRV3}, and re-vivified the interest in the
charmed baryon spectrum. On the other hand, the QCD sum rules is a powerful
theoretical tool in studying the
ground state heavy baryon states \cite{SVZ79,PRT85}. In the QCD sum
rules, the operator product expansion is used to expand the
time-ordered currents into a series of quark and gluon condensates
which parameterize the long distance properties of the QCD vacuum.
Based on the quark-hadron duality, we can obtain copious information
about the hadronic parameters at the phenomenological side
\cite{SVZ79,PRT85}. There have been several works on the masses of
the heavy baryon states with the full QCD sum rules and the QCD sum
rules in the heavy quark effective theory
\cite{Bagan93,Bagan922,Bagan921,M-Nielsen07,Huang0805,M-Huang0811,Narison0904,Shuryak82,Grozin92,
Dai961,Dai962,Huang02,Zhu00,HuangCS,HuangMQ,M-Liu07}.
In Refs.\cite{Wang0704,Wang0809,Wang0910}, we study the
${\frac{1}{2}}^+$ heavy baryon states $\Omega_Q$, $\Xi'_Q$ and
$\Sigma_Q$ and ${\frac{3}{2}}^+$ heavy baryon states $\Omega_Q^*$,
$\Xi^*_Q$ and $\Sigma^*_Q$ with the full QCD sum rules, and observe
that the pole residues of the ${\frac{3}{2}}^+$ heavy baryon states
from the sum rules with different tensor structures are consistent
with each other, while the pole residues of the ${\frac{1}{2}}^+$
heavy baryon states from the sum rules with different tensor
structures differ from each other greatly. Those pole residues are
important parameters in studying the radiative decays $\Omega_Q^*\to
\Omega_Q \gamma$, $\Xi_Q^*\to \Xi'_Q \gamma$ and $\Sigma_Q^*\to
\Sigma_Q \gamma$ \cite{Wang0910,Wang0909}, we should refine those
parameters to improve the predictive ability.
In Ref.\cite{Oka96}, Jido et al introduce a novel approach based on
the QCD sum rules to separate the contributions of the
negative-parity light flavor baryon states from the positive-parity
light flavor baryon states, as the interpolating currents may have
non-vanishing couplings to both the negative- and positive-parity
baryon states \cite{Chung82}. Before the work of Jido et al, Bagan
et al take the infinite mass limit for the heavy quarks to separate
the contributions of the positive and negative parity heavy baryon
states unambiguously \cite{Bagan93}.
In Ref.\cite{Wang0912}, we follow Ref.\cite{Oka96} and re-study the
masses and pole residues of the ${\frac{1}{2}}^+$ flavor sextet
heavy baryon states $\Omega_Q$, $\Xi'_Q$ and $\Sigma_Q$ by
subtracting the contributions of the negative parity heavy baryon
states with the full QCD sum rules. In this article, we use the same
approach to study the ${1\over 2}^{\pm}$ flavor antitriplet heavy
baryon states ($\Lambda_c^+$, $\Xi_c^+,\Xi_c^0)$ and ($\Lambda_b^0$,
$\Xi_b^0,\Xi_b^-)$.
The article is arranged as follows: we derive the QCD sum rules for
the masses and the pole residues of the heavy baryon states
($\Lambda_c^+$, $\Xi_c^+,\Xi_c^0)$ and ($\Lambda_b^0$,
$\Xi_b^0,\Xi_b^-)$ in Sect.2; in Sect.3, we present the
numerical results and discussions; and Sect.4 is reserved for our
conclusions.
\section{QCD sum rules for the $\Lambda_Q$ and $\Xi_Q$ }
The ${1\over 2}^+$ flavor antitriplet heavy baryon states
($\Lambda_c^+$, $\Xi_c^+,\Xi_c^0)$ and ($\Lambda_b^0$,
$\Xi_b^0,\Xi_b^-)$ can be interpolated by the following currents
$J_\Lambda(x)$ and $J_\Xi(x)$
respectively,
\begin{eqnarray}
J_\Lambda(x)&=& \epsilon^{ijk} u^T_i(x)C\gamma_5 d_j(x) Q_k(x) \, , \nonumber \\
J_\Xi(x)&=& \epsilon^{ijk} q^T_i(x)C\gamma_5 s_j(x) Q_k(x) \, ,
\end{eqnarray}
where the $Q$ represents the heavy quarks $c$ and $b$, the $i$,
$j$ and $k$ are color indexes, and the $C$ is the charge conjunction
matrix.
The corresponding negative-parity heavy baryon states can be
interpolated by the currents $J_{-} =i\gamma_{5} J_{+}$ because
multiplying $i \gamma_{5}$ to the $J_{+}$ changes the parity of the
$J_{+}$ \cite{Oka96}, where the $J_{+}$ denotes the currents
$J_\Lambda(x)$ and $J_\Xi(x)$. The correlation functions are defined
by
\begin{eqnarray}
\Pi_{\pm}(p)&=&i\int d^4x e^{ip \cdot x} \langle
0|T\left\{J_{\pm}(x)\bar{J}_{\pm}(0)\right\}|0\rangle \, ,
\end{eqnarray}
and can be decomposed as
\begin{equation}
\Pi_{\pm}(p) = \!\not\!{p} \Pi_{1}(p) \pm \Pi_{0}(p)\, ,
\end{equation}
due to Lorentz covariance. The currents $J_{\pm}$ couple to both
the positive- and negative-parity baryon states \cite{Chung82},
i.e. $ \langle{0}|J_{\pm}| B^{-}\rangle \langle
B^{-}|\bar{J}_{\pm}|0\rangle = - \gamma_{5}\langle 0|J_{\mp}|
B^{-}\rangle \langle B^{-}| \bar{J}_{\mp}|0\rangle
\gamma_{5}$, where the $B^{-}$ denote the negative parity baryon states.
We insert a
complete set of intermediate baryon states with the same quantum
numbers as the current operators $J_{+}(x)$ and $J_{-}(x)$ into the
correlation functions $\Pi_{\pm}(p)$ to obtain the hadronic
representation \cite{SVZ79,PRT85}. After isolating the pole terms of
the lowest states, we obtain the following result \cite{Oka96}:
\begin{eqnarray}
\Pi_{\pm}(p) & = & \lambda_+^2 {\!\not\!{p} +
M_{+} \over M^{2}_+ -p^{2} } + \lambda_{-}^2
{\!\not\!{p} - M_{-} \over M_{-}^{2}-p^{2} } +\cdots \, ,
\end{eqnarray}
where the $M_{\pm}$ are the masses of the lowest states with parity
$\pm$ respectively, and the $\lambda_{\pm}$ are the corresponding
pole residues (or couplings).
If we take $\vec{p} = 0$, then
\begin{eqnarray}
\rm{limit}_{\epsilon\rightarrow0}\frac{{\rm Im} \Pi_{\pm}(p_{0}+i\epsilon)}{\pi} & = &
\lambda_+^2 {\gamma_{0} + 1\over 2} \delta(p_{0} - M_+) +
\lambda_{-}^{2} {\gamma_{0} - 1\over 2} \delta(p_{0} - M_{-})+\cdots \nonumber \\
& = & \gamma_{0} A(p_{0}) \pm B(p_{0})+\cdots \, ,
\end{eqnarray}
where
\begin{eqnarray}
A(p_{0}) & = & {1 \over 2} \left[ \lambda_+^{2}
\delta(p_{0} - M_+) + \lambda_-^{2} \delta(p_{0} -
M_{-})\right] \, , \nonumber \\
B(p_{0}) & = & \pm{1 \over 2} \left[ \lambda_+^{2}
\delta(p_{0} - M_+) - \lambda_-^{2} \delta(p_{0} -
M_{-})\right] \, ,
\end{eqnarray}
the contribution $A(p_{0}) + B(p_{0})$ ($A(p_{0}) - B(p_{0})$)
contains contributions from the positive parity (negative parity)
states only for the $\Pi_+(p_0)$, while the contribution $A(p_{0}) +
B(p_{0})$ ($A(p_{0}) - B(p_{0})$) contains contributions from the
negative parity (positive parity) states only for the $\Pi_-(p_0)$.
We calculate the light quark parts of the correlation functions
$\Pi_{\pm}(p)$ in the coordinate space and use the momentum space
expression for the heavy quark propagators, then resort to the
Fourier integral to transform the light quark parts into the
momentum space in $D$ dimensions, take $\vec{p} = 0$, and use the
dispersion relation to obtain the spectral densities $\rho^A(p_0)$
and $\rho^B(p_0)$ (which correspond to the tensor structures
$\gamma_0$ and $1$ respectively) at the level of quark-gluon degrees
of freedom, finally we introduce the weight functions
$\exp\left[-\frac{p_0^2}{T^2}\right]$,
$p_0^2\exp\left[-\frac{p_0^2}{T^2}\right]$, and obtain the
following sum rules,
\begin{eqnarray}
\lambda_{\pm}^2\exp\left[-\frac{M_{\pm}^2}{T^2}\right]&=&\int_{\Delta}^{\sqrt{s_0}}dp_0
\left[\rho^A(p_0)
+\rho^B(p_0)\right]\exp\left[-\frac{p_0^2}{T^2}\right] \, ,
\end{eqnarray}
\begin{eqnarray}
\lambda_{\pm}^2M_{\pm}^2\exp\left[-\frac{M_{\pm}^2}{T^2}\right]&=&\int_{\Delta}^{\sqrt{s_0}}dp_0
\left[\rho^A(p_0)
+\rho^B(p_0)\right]p_0^2\exp\left[-\frac{p_0^2}{T^2}\right] \, ,
\end{eqnarray}
where
\begin{eqnarray}
\rho^A_{\Xi_Q}(p_0)&=&\frac{3p_0}{128\pi^4}\int_{t_i}^1dt
t(1-t)^2(p_0^2-\widetilde{m}_Q^2)^2+\frac{p_0m_s\left[\langle\bar{s}s\rangle-2\langle\bar{q}q\rangle\right]}{16\pi^2}\int_{t_i}^1
dt t\nonumber\\
&&+\frac{p_0}{128\pi^2}\langle \frac{\alpha_sGG}{\pi}\rangle
\int_{t_i}^1 dt t-\frac{m_Q^2}{768\pi^2}\langle
\frac{\alpha_sGG}{\pi}\rangle \int_{t_i}^1 dt \frac{(1-t)^2}{t^2}
\delta
(p_0-\widetilde{m}_Q)\nonumber \\
&&+\frac{m_s\left[3\langle\bar{q}g_s\sigma
Gq\rangle-\langle\bar{s}g_s\sigma Gs\rangle \right]}{192\pi^2}\delta
(p_0-m_Q) +\frac{
\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{12}\delta(p_0-m_Q) \,
,
\end{eqnarray}
\begin{eqnarray}
\rho^B_{\Xi_Q}(p_0)&=&\frac{3m_Q}{128\pi^4}\int_{t_i}^1dt
(1-t)^2(p_0^2-\widetilde{m}_Q^2)^2+\frac{m_sm_Q\left[\langle\bar{s}s\rangle-2\langle\bar{q}q\rangle\right]}{16\pi^2}\int_{t_i}^1
dt \nonumber\\
&&+\frac{m_Q}{128\pi^2}\langle \frac{\alpha_sGG}{\pi}\rangle
\int_{t_i}^1 dt +\frac{m_Q}{192\pi^2}\langle
\frac{\alpha_sGG}{\pi}\rangle
\int_{t_i}^1 dt \frac{(1-t)^3}{t^2}\nonumber \\
&&-\frac{m_Q}{768\pi^2}\langle \frac{\alpha_sGG}{\pi}\rangle
\int_{t_i}^1 dt \frac{(1-t)^2}{t}\widetilde{m}_Q
\delta(p_0-\widetilde{m}_Q)\nonumber \\
&&+\frac{m_s\left[3\langle\bar{q}g_s\sigma
Gq\rangle-\langle\bar{s}g_s\sigma Gs\rangle \right]}{192\pi^2}\delta
(p_0-m_Q) +\frac{
\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{12}\delta(p_0-m_Q) \,
,
\end{eqnarray}
\begin{eqnarray}
\rho^A_{\Lambda_Q}(p_0)&=&\frac{3p_0}{128\pi^4}\int_{t_i}^1dt
t(1-t)^2(p_0^2-\widetilde{m}_Q^2)^2+\frac{ \langle\bar{q}q\rangle^2}{12}\delta(p_0-m_Q)\nonumber\\
&&+\frac{p_0}{128\pi^2}\langle \frac{\alpha_sGG}{\pi}\rangle
\int_{t_i}^1 dt t -\frac{m_Q^2}{768\pi^2}\langle
\frac{\alpha_sGG}{\pi}\rangle \int_{t_i}^1 dt \frac{(1-t)^2}{t^2}
\delta (p_0-\widetilde{m}_Q) \, ,
\end{eqnarray}
\begin{eqnarray}
\rho^B_{\Lambda_Q}(p_0)&=&\frac{3m_Q}{128\pi^4}\int_{t_i}^1dt
(1-t)^2(p_0^2-\widetilde{m}_Q^2)^2 +\frac{
\langle\bar{q}q\rangle^2}{12}\delta(p_0-m_Q) \nonumber\\
&&+\frac{m_Q}{128\pi^2}\langle \frac{\alpha_sGG}{\pi}\rangle
\int_{t_i}^1 dt +\frac{m_Q}{192\pi^2}\langle
\frac{\alpha_sGG}{\pi}\rangle
\int_{t_i}^1 dt \frac{(1-t)^3}{t^2}\nonumber \\
&&-\frac{m_Q}{768\pi^2}\langle \frac{\alpha_sGG}{\pi}\rangle
\int_{t_i}^1 dt \frac{(1-t)^2}{t}\widetilde{m}_Q
\delta(p_0-\widetilde{m}_Q) \, ,
\end{eqnarray}
where $\widetilde{m}_Q^2=\frac{m_Q^2}{t}$,
$t_i=\frac{m_Q^2}{p_0^2}$, the $s_0$ are the threshold parameters,
$T^2$ is the Borel parameter, $\Delta=m_Q+m_s$ and $m_Q$ in the
channels $\Xi_Q$ and $\Lambda_Q$ respectively.
\section{Numerical results and discussions}
The input parameters are taken to be the standard values $\langle
\bar{q}q \rangle=-(0.24\pm 0.01 \,\rm{GeV})^3$, $\langle \bar{s}s
\rangle=(0.8\pm 0.2 )\langle \bar{q}q \rangle$, $\langle
\bar{q}g_s\sigma Gq \rangle=m_0^2\langle \bar{q}q \rangle$, $\langle
\bar{s}g_s\sigma Gs \rangle=m_0^2\langle \bar{s}s \rangle$,
$m_0^2=(0.8 \pm 0.2)\,\rm{GeV}^2$ \cite{Ioffe2005,LCSRreview},
$\langle \frac{\alpha_s GG}{\pi}\rangle=(0.012 \pm
0.004)\,\rm{GeV}^4 $ \cite{LCSRreview},
$m_s=(0.14\pm0.01)\,\rm{GeV}$, $m_c=(1.35\pm0.10)\,\rm{GeV}$ and
$m_b=(4.7\pm0.1)\,\rm{GeV}$ \cite{PDG} at the energy scale $\mu=1\,
\rm{GeV}$.
The value of the gluon condensate $\langle \frac{\alpha_s
GG}{\pi}\rangle $ has been updated from time to time, and changes
greatly \cite{NarisonBook}.
At the present case, the gluon condensate makes tiny contribution, the updated value $\langle \frac{\alpha_s GG}{\pi}\rangle=(0.023 \pm
0.003)\,\rm{GeV}^4 $ \cite{NarisonBook} and the standard value
$\langle \frac{\alpha_s GG}{\pi}\rangle=(0.012 \pm
0.004)\,\rm{GeV}^4 $ \cite{LCSRreview} lead to a difference less
than $15\,\rm{MeV}$ for the masses.
In the conventional QCD sum rules \cite{SVZ79,PRT85}, there are two
criteria (pole dominance and convergence of the operator product
expansion) for choosing the Borel parameter $T^2$ and threshold
parameter $s_0$. We impose the two criteria on the heavy baryon
states to choose the Borel parameter $T^2$ and threshold parameter
$s_0$, the values are shown in Table 1. From Table 1, we can see
that the contribution from the perturbative term is dominant, the
operator product expansion is convergent certainly. In this article,
we take the contribution from the pole term is larger than $45\%$
($49\%$) for the positive (negative) parity baryon states, the
uncertainty of the threshold parameter is $0.1\,\rm{GeV}$, and the
Borel window is $1\,\rm{GeV}^2$.
In calculation, we neglect the contributions from the perturbative
corrections. Those perturbative corrections can be taken into
account in the leading logarithmic
approximations through anomalous dimension factors. After the Borel transform, the effects of those
corrections are to multiply each term on the operator product
expansion side by the factor, $ \left[ \frac{\alpha_s(T^2)}{\alpha_s(\mu^2)}\right]^{2\Gamma_{J}-\Gamma_{\mathcal
{O}_n}} $,
where the $\Gamma_{J}$ is the anomalous dimension of the
interpolating current $J(x)$ and the $\Gamma_{\mathcal {O}_n}$ is the anomalous dimension of
the local operator $\mathcal {O}_n(0)$. We carry out the operator product expansion at a special energy
scale $\mu^2=1\,\rm{GeV}^2$, and set the factor $\left[
\frac{\alpha_s(T^2)}{\alpha_s(\mu^2)}\right]^{2\Gamma_{J}-\Gamma_{\mathcal
{O}_n}}\approx1$, such an approximation maybe result in some scale
dependence and weaken the prediction ability. In this article, we
study the flavor antitriplet $\frac{1}{2}^{\pm}$ heavy baryon states
systemically, the predictions are still robust as we take the
analogous criteria in those sum rules.
Taking into account all uncertainties of the relevant parameters,
we obtain the values of the masses and pole residues of
the flavor antitriplet $\frac{1}{2}^{\pm}$ heavy baryon states ($\Lambda_c^+$,
$\Xi_c^+,\Xi_c^0)$ and ($\Lambda_b^0$, $\Xi_b^0,\Xi_b^-)$, which are
shown in Figs.1-2 and Table 2. In this article, we calculate the
uncertainties $\delta$ with the formula
\begin{eqnarray}
\delta=\sqrt{\sum_i\left(\frac{\partial f}{\partial
x_i}\right)^2\mid_{x_i=\bar{x}_i} (x_i-\bar{x}_i)^2}\, ,
\end{eqnarray}
where the $f$ denote the
hadron mass $M$ and the pole residue $\lambda$, the $x_i$ denote
the input QCD parameters $m_c$, $m_b$, $\langle \bar{q}q \rangle$,
$\langle \bar{s}s \rangle$, $\cdots$, and the threshold parameter
$s_0$ and Borel parameter $T^2$. As the partial
derivatives $\frac{\partial f}{\partial x_i}$ are difficult to carry
out analytically, we take the approximation $\left(\frac{\partial
f}{\partial x_i}\right)^2 (x_i-\bar{x}_i)^2\approx
\left[f(\bar{x}_i\pm \Delta x_i)-f(\bar{x}_i)\right]^2$ in the
numerical calculations.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline\hline & $T^2 (\rm{GeV}^2)$& $\sqrt{s_0} (\rm{GeV})$&pole&perturbative\\
\hline
$\Lambda_c({\frac{1}{2}}^+)$ &$1.7-2.7$ &$3.1$& $(46-83)\%$ &$(47-73)\%$\\ \hline
$\Xi_c({\frac{1}{2}}^+)$ &$1.9-2.9$ &$3.2$ & $(46-79)\%$&$(59-77)\%$\\ \hline
$\Lambda_b({\frac{1}{2}}^+)$ &$4.3-5.3$ &$6.5$& $(46-67)\%$&$(58-72)\%$\\ \hline
$\Xi_b({\frac{1}{2}}^+)$ &$4.4-5.4$ &$6.5$& $(45-64)\%$&$(62-73)\%$\\ \hline
$\Lambda_c({\frac{1}{2}}^-)$ &$2.2-3.2$ &$3.4$& $(49-77)\%$&$(70-84)\%$\\ \hline
$\Xi_c({\frac{1}{2}}^-)$ &$2.4-3.4$ &$3.5$ & $(49-75)\%$&$(76-86)\%$\\ \hline
$\Lambda_b({\frac{1}{2}}^-)$ &$4.7-5.7$ &$6.7$& $(49-67)\%$&$(69-80)\%$\\ \hline
$\Xi_b({\frac{1}{2}}^-)$ &$5.0-6.0$ &$6.8$& $(49-65)\%$&$(75-83)\%$\\ \hline\hline
\end{tabular}
\end{center}
\caption{ The Borel parameters $T^2$ and threshold parameters $s_0$
for the heavy baryon states, the "pole" stands for the contribution
from the pole term, and the "perturbative" stands for the
contribution from the perturbative term in the operator product
expansion.}
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline\hline & $T^2 (\rm{GeV}^2)$& $\sqrt{s_0} (\rm{GeV})$&
$M(\rm{GeV})$&$\lambda
(\rm{GeV}^3)$&$M(\rm{GeV})[\rm{exp}]$\cite{PDG}\\\hline
$\Lambda_c({\frac{1}{2}}^+)$ &$1.7-2.7$ &$3.1\pm0.1$& $2.26\pm0.27$&$0.022\pm0.008$ &2.28646\\ \hline
$\Xi_c({\frac{1}{2}}^+)$ &$1.9-2.9$ &$3.2\pm0.1$ & $2.44\pm0.23$&$0.027\pm0.008$&2.4678/2.47088\\ \hline
$\Lambda_b({\frac{1}{2}}^+)$ &$4.3-5.3$ &$6.5\pm0.1$& $5.65\pm0.20$&$0.030\pm0.009$&5.6202\\ \hline
$\Xi_b({\frac{1}{2}}^+)$ &$4.4-5.4$ &$6.5\pm0.1$& $5.73\pm0.18$&$0.032\pm0.009$&5.7924\\ \hline
$\Lambda_c({\frac{1}{2}}^-)$ &$2.2-3.2$ &$3.4\pm0.1$& $2.61\pm0.21$&$0.035\pm0.009$ &2.5954\\ \hline
$\Xi_c({\frac{1}{2}}^-)$ &$2.4-3.4$ &$3.5\pm0.1$ & $2.76\pm0.18$&$0.042\pm0.009$&2.7891/2.7918\\ \hline
$\Lambda_b({\frac{1}{2}}^-)$ &$4.7-5.7$ &$6.7\pm0.1$& $5.85\pm0.18$&$0.042\pm0.012$&?\\ \hline
$\Xi_b({\frac{1}{2}}^-)$ &$5.0-6.0$ &$6.8\pm0.1$& $6.01\pm0.16$&$0.051\pm0.012$&?\\ \hline
\hline
\end{tabular}
\end{center}
\caption{ The masses $M(\rm{GeV})$ and pole residues
$\lambda(\rm{GeV}^3)$ of the heavy baryon states.}
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline\hline
&\cite{M-Nielsen07}&\cite{M-Huang0811}&\cite{M-Liu07}&This work\\ \hline
$\Lambda_c({\frac{1}{2}}^+)$& &$2.31\pm 0.19$ & $2.271^{+0.067}_{-0.049} $& $2.26\pm0.27$ \\ \hline
$\Xi_c({\frac{1}{2}}^+)$& $2.5\pm0.2$&$2.48 \pm 0.21$ & $2.432^{+0.079}_{-0.068}$& $2.44\pm0.23$ \\ \hline
$\Lambda_b({\frac{1}{2}}^+)$& &$5.69 \pm 0.13$ & $5.637^{+0.068}_{-0.056}$& $5.65\pm0.20$ \\ \hline
$\Xi_b({\frac{1}{2}}^+)$& $5.75\pm0.25$&$5.75 \pm 0.13$ & $5.780^{+0.073}_{-0.068}$& $5.73\pm0.18$ \\ \hline
$\Lambda_c({\frac{1}{2}}^-)$& &$2.53\pm 0.22$ & & $2.61\pm0.21$ \\ \hline
$\Xi_c({\frac{1}{2}}^-)$& &$2.65 \pm 0.27$ & & $2.76\pm0.18$ \\ \hline
$\Lambda_b({\frac{1}{2}}^-)$& &$5.85 \pm 0.15$ & & $5.85\pm0.18$ \\ \hline
$\Xi_b({\frac{1}{2}}^-)$& &$5.95 \pm 0.16$ & & $6.01\pm0.16$ \\ \hline
\hline
\end{tabular}
\end{center}
\caption{ The masses $M(\rm{GeV})$ of the heavy baryon states
from the QCD sum rules.}
\end{table}
\begin{figure}
\centering
\includegraphics[totalheight=4cm,width=5cm]{mass-Lambda-C.EPS}
\includegraphics[totalheight=4cm,width=5cm]{mass-Xi-C.EPS}
\includegraphics[totalheight=4cm,width=5cm]{mass-Lambda-B.EPS}
\includegraphics[totalheight=4cm,width=5cm]{mass-Xi-B.EPS}
\includegraphics[totalheight=4cm,width=5cm]{Nmass-Lambda-C.EPS}
\includegraphics[totalheight=4cm,width=5cm]{Nmass-Xi-C.EPS}
\includegraphics[totalheight=4cm,width=5cm]{Nmass-Lambda-B.EPS}
\includegraphics[totalheight=4cm,width=5cm]{Nmass-Xi-B.EPS}
\caption{ The masses $M$ of the heavy baryon states, the $A$, $B$, $C$ and $D$ correspond
to the ${\frac{1}{2}}^+$ heavy baryon channels $\Lambda_c$, $\Xi_c$, $\Lambda_b$ and $\Xi_b$ respectively, while
the $E$, $F$, $G$ and $H$ correspond to the
${\frac{1}{2}}^-$ heavy baryon channels $\Lambda_c$, $\Xi_c$,
$\Lambda_b$ and $\Xi_b$ respectively. }
\end{figure}
\begin{figure}
\centering
\includegraphics[totalheight=4cm,width=5cm]{residue-Lambda-C.EPS}
\includegraphics[totalheight=4cm,width=5cm]{residue-Xi-C.EPS}
\includegraphics[totalheight=4cm,width=5cm]{residue-Lambda-B.EPS}
\includegraphics[totalheight=4cm,width=5cm]{residue-Xi-B.EPS}
\includegraphics[totalheight=4cm,width=5cm]{Nresidue-Lambda-C.EPS}
\includegraphics[totalheight=4cm,width=5cm]{Nresidue-Xi-C.EPS}
\includegraphics[totalheight=4cm,width=5cm]{Nresidue-Lambda-B.EPS}
\includegraphics[totalheight=4cm,width=5cm]{Nresidue-Xi-B.EPS}
\caption{ The pole residues $\lambda$ of the heavy baryon states, the $A$, $B$, $C$ and $D$ correspond
to the ${\frac{1}{2}}^+$ heavy baryon channels $\Lambda_c$, $\Xi_c$, $\Lambda_b$ and $\Xi_b$ respectively, while
the $E$, $F$, $G$ and $H$ correspond to the ${\frac{1}{2}}^-$
heavy baryon channels $\Lambda_c$, $\Xi_c$, $\Lambda_b$ and $\Xi_b$
respectively. }
\end{figure}
In Ref.\cite{Bagan922}, Bagan et al study the masses and couplings
(pole residues) of the heavy baryon states $\Sigma_b^*$,
$\Sigma_c^*$, $\Lambda_b$ and $\Lambda_c$ with the full QCD sum
rules, and observe that the $\Sigma_{b(c)}^*$ and $\Sigma_{b(c)}$
have degenerate masses within the uncertainties, while the
$\Lambda_{b(c)}$ is lighter than the $\Sigma_{b(c)}$. In
Refs.\cite{Huang0805,M-Huang0811}, Zhang et al perform a systematic
study of the masses of charmed and bottom baryon states with the
full QCD sum rules, where the vacuum condensates up to dimension
six are taken into account; while in Ref.\cite{M-Nielsen07}, Duraes
et al study only the masses of the heavy baryon states $\Xi_c$,
$\Xi_b$ and $\Omega_b$. In Ref.\cite{M-Liu07}, Liu et al perform a
systematic study of the masses of the bottom baryon states up to
$\frac{1}{m_Q}$ in the heavy quark effective field theory using the
QCD sum rules.
In the heavy quark limit, $M_Q=m_Q+\bar{\Lambda}+\mathcal
{O}\left(\frac{1}{m_Q}\right)$. We can calculate the bound energy
$\bar{\Lambda}$ and the $\frac{1}{m_Q}$ corrections of the lowest
heavy baryon states $\Lambda_Q$ with the QCD sum rules, compare
them with the experimental data $M_Q$, and determine the heavy quark
masses $m_Q$, which always suffer from large uncertainties, then use
the $m_Q$ as basic input parameters to calculate other heavy baryon
masses, one can consult Refs.\cite{Dai961,Dai962,Huang02} for
example. In Table 3, we also present the predictions from the full
QCD sum rules and the QCD sum rules in the heavy quark effective
theory (where systematic studies are preformed)
\cite{M-Nielsen07,M-Huang0811,M-Liu07}.
From Tables 2-3, we can see that the present and other theoretical
predictions are all in good agreement with the experimental data for
the positive parity baryon states. The negative parity baryon states
$\Lambda_b$ and $\Xi_b$ are not observed yet, we make reasonable
predictions for their masses to confront with experimental data in
the future at the LHCb \cite{LHC}.
The fractions
\begin{eqnarray}
R&=& \frac{\int_{\Delta}^{\sqrt{s_0}}dp_0\left[\rho^A(p_0) -\rho^B(p_0)\right]\exp\left[-\frac{p_0^2}{T^2}\right]}
{\int_{\Delta}^{\sqrt{s_0}}dp_0\left[\rho^A(p_0)
+\rho^B(p_0)\right]\exp\left[-\frac{p_0^2}{T^2}\right]}
\end{eqnarray}
are less than $4.5\%\,(6.0\%)$ and $0.6\%\,(0.8\%)$ in the positive
(negative) parity charmed baryon and bottom baryon channels,
respectively. So the contaminations from the negative (or positive)
parity baryon states are very small. In
Refs.\cite{Wang0809,Wang0910}, we study the ${\frac{1}{2}}^+$ flavor
sextet heavy baryon states $\Omega_Q$, $\Xi'_Q$ and $\Sigma_Q$
with the full QCD sum rules, and observe that the pole residues
from the sum rules with different tensor structures ($\!\not\!{p}$
and $1$) differ from each other greatly. In Ref.\cite{Wang0912}, we
re-study the masses and pole residues of the ${\frac{1}{2}}^+$
flavor sextet heavy baryon states $\Omega_Q$, $\Xi'_Q$ and
$\Sigma_Q$ by subtracting the contributions of the negative parity
heavy baryon states, and find that the predictions for the masses
and pole residues are improved considerably. In the present case, we
can choose the tensor structures $\!\not\!{p}$ or $1$ or
$\gamma_0+1$ freely to study the masses and pole residues.
The pole residues of the $\frac{1}{2}^+$
and $\frac{3}{2}^+$ heavy baryon sextets $B_6$ and $B^*_6$ have been
calculated in our previous works
\cite{Wang0704,Wang0910,Wang0912,Wang1002}. Once reasonable values
of the pole residues $\lambda_{\Lambda}$ and $\lambda_{\Xi}$ of the
$\frac{1}{2}^+$ heavy baryon antitriplet $B_{\bar{3}}$ are obtained,
we can take them as basic input parameters and study the strong
decays $\Sigma^*_Q\rightarrow\Lambda_Q\pi$,
$\Sigma_Q\rightarrow\Lambda_Q\pi$ and $\Xi^*_Q\rightarrow \Xi_Q \pi$
and the radiative decays $B^*_6 \to B_{\bar{3}}\gamma$ and $B_6 \to
B_{\bar{3}}\gamma$ in a systematic ways with the light-cone QCD sum
rules or the QCD sum rules in external field, and confront the
predictions with the experimental data in the future at the BESIII,
$\rm{\bar{P}ANDA}$ and LHCb \cite{LHC,BESIII,PANDA}.
The strong
decays $\Sigma^*_c(2520)\rightarrow\Lambda_c\pi$ and
$\Sigma_c(2455)\rightarrow\Lambda_c\pi$ saturate approximately the
widths of the $\Sigma^*_c(2520)$ and $\Sigma_c (2455)$
respectively, while the
strong decays $\Xi^*_c(2645)\rightarrow \Xi_c \pi$ are seen
\cite{PDG}. From our previous works \cite{Wang0912,Wang1002}, we can
see that the corresponding strong decays
$\Sigma^*_b\rightarrow\Lambda_b\pi$,
$\Sigma_b\rightarrow\Lambda_b\pi$ and $\Xi^*_b\rightarrow \Xi_b \pi$
are kinematically allowed; although the bottom baryon states
$\Xi^*_b$ have not been observed experimentally
yet.
In Refs.\cite{Wang0910,Wang0909}, we perform systematic studies for
the radiative decays $B^*_6 \to B_{6}\,\gamma$ with the light-cone
QCD sum rules as the strong decays $B^*_6 \to B_{6}\,\pi$ are
forbidden due to the unavailable phase space, while the radiative
channels are not phase space suppressed and become relevant;
although the electromagnetic strength is weaker than that of the
strong interaction. The radiative decays $B^*_6 \to
B_{\bar{3}}\gamma$ and $B_6 \to B_{\bar{3}}\gamma$ are important
processes in testing the heavy quark symmetry and the chiral
symmetry, for example, the $\Xi'_c$ and $\Omega_c^*$ are governed by
the radiative decays.
\section{Conclusion}
In this article, we study the ${1\over 2}^{\pm}$ flavor antitriplet
heavy baryon states ($\Lambda_c^+$, $\Xi_c^+,\Xi_c^0)$ and
($\Lambda_b^0$, $\Xi_b^0,\Xi_b^-)$ by subtracting the contributions
from the corresponding ${1\over 2}^{\mp}$ heavy baryon states with
the QCD sum rules, obtain the masses which are in good agreement
with the experimental data and make reasonable predictions for the
unobserved ${1\over 2}^-$ bottom baryon states. In calculation, we
observe that the contaminations from the negative (or positive)
parity baryon states are very small, one can choose the tensor
structures $\!\not\!{p}$ or $1$ or $\gamma_0+1$ freely to study the
masses and pole residues. Once reasonable values of the pole
residues $\lambda_{\Lambda}$ and $\lambda_{\Xi}$ are obtained, we
can take them as basic input parameters and study the strong decays
and radiative decays in a systematic
ways with the light-cone QCD sum rules or the QCD sum rules in
external field, and confront the predictions with the experimental
data in the future at the BESIII, $\rm{\bar{P}ANDA}$ and LHCb.
\section*{Acknowledgements}
This work is supported by National Natural Science Foundation,
Grant Number 10775051, and Program for New Century Excellent Talents
in University, Grant Number NCET-07-0282, and the Fundamental
Research Funds for the Central Universities.
|
train/arxiv
|
BkiUdHQ5qhLA_wzE88_L
| 5
| 1
|
\section{Introduction}
Certain concepts in classical mechanics cannot be carried over to the quantum regime no matter how useful they may be. Transition state theory (TST)\cite{eyring-jcp3,wigner-jcp5} in its rigorous definition\cite{wigner-jcp5,pechukas-r1,pechukas-jcp58,chandler-jcp68,miller-acc26} is tied to such concepts of classical mechanics, namely, the definability of deterministic trajectories in the phase space. Therefore, no exact analogue of the classical TST can be found in the quantum regime.\cite{miller-acc26,mclafferty-cpl27} Nonetheless, attempts to develop a quantum TST (QTST) have remained an ongoing and challenging theoretical subject. This is because there is a significant benefit in calculating a rate constant without carrying out any actual real time quantum dynamics calculation. The insight available from a quantum correction factor for the TST also serves as an important motivation. However, a major conceptual issue to be settled in developing a QTST is that distinct starting points of the rate formulation,\cite{mclafferty-cpl27,yamamoto-jcp123,miller-jcp79,affleck-prl46,gillan-jp-c20,voth-jcp91,hansen-jcp,stuchebrukhov-jcp95,cao-jcp105,pollak-jcp108} which all lead to the same TST rate expression in the classical limit, can produce different results in the genuine quantum regime.
In fact, this non-uniqueness of QTST is not unexpected. It rather reflects the fundamental principles of quantum mechanics, and should be recognized as an intrinsic quantum feature. In this sense, it is imperative to understand clearly the assumptions and approximations involved in a given QTST because such an understanding offers a correct assessment of the implications and utility of that particular QTST.
In a recent work,\cite{jang-jcp144} \rc{ we analyzed a new kind of QTST formulated by Hele and Althorpe (HA).}\cite{hele-jcp138} Our analysis offered a context for this theory within the general consensus on the QTST, as described above, by clarifying key assumptions implicit in HA-QTST. First, we pointed out that the starting expression of HA-QTST\cite{hele-jcp138,althorpe-jcp139} was missing a clear physical origin tied to response or scattering theories, unlike those introduced by Yamamoto\cite{yamamoto-jcp123} and by Miller and coworkers.\cite{miller-jcp79} Second, we analyzed\cite{jang-jcp144} an apparent approximation HA employed for the evaluation of their flux-side correlation function,\cite{hele-jcp138} and have also provided an alternative and exact evaluation. The resulting form based on our evaluation suggested that HA-QTST and ring polymer molecular dynamics (RPMD) TST\cite{craig-jcp122,craig-jcp123} are different in general.
An important issue that was not discussed in detail in our work\cite{jang-jcp144} was the concept of a {\it general dividing surface defined in the imaginary time path integral space}, which was introduced as a new means to optimize RPMD-TST by Richardson and Althorpe\cite{richardson-jcp131} and had also previously been considered by others\cite{cao-jcp105,mills-cpl278} in the context of improving the path integral QTST (PI-QTST).\cite{voth-jcp91} It is important to first note that the introduction of such a dividing surface is a departure from the original formulation of the RPMD rate theory by Craig and Manolopoulos\cite{craig-jcp122,craig-jcp123} because not every dividing surface in the imaginary time path integral space can be expressed as a ring polymer average of classical ones, namely, an equal weight linear combination of classical dividing surfaces over all the beads of the ring polymer. For this reason, our previous analysis\cite{jang-jcp144} was limited to the centroid dividing surface, an obvious ring polymer averaged quantity. For this latter case, we have confirmed that HA-QTST reproduces the PI-QTST rate expression above the crossover temperature,\cite{voth-jcp91} to which the RPMD-TST expression proposed a number of years later was also shown to be equivalent.\cite{craig-jcp123} \rc{We note that this outcome is not an exact derivation of PI-QTST or RPMD-TST because neither of these formulations of QTST are exact except in certain limits.}
In a more recent article,\cite{hele-jcp144} HA provided further analysis of the general expression obtained in our work,\cite{jang-jcp144} and presented a result that seems to suggest again the equivalence between HA-QTST and RPMD-TST\cite{richardson-jcp131} for general dividing surfaces. However, as will be shown here, this is not always true.
Before presenting our new analysis addressing the issue of general path integral dividing surfaces, we want to stress that our assessment\cite{jang-jcp144} of HA-QTST as being one of many non-unique versions of QTST remains intact. In particular, our assessment of the assumption implicit in employing the $t=0_+$ limit of a specially constructed flux-side time correlation function, namely, one that neglects a certain part of the Feynman diagram and thus in effect assumes that certain operators can commute, requires no further clarification. Without such an assumption, and without any coarse-graining in time, the exact $t=0_+$ limit of the flux-side correlation function will always vanish in the quantum limit.\cite{costley-cpl83,wolynes-prl47} With this issue clarified, we will here focus our attention mostly on the subject of general path integral dividing surfaces. These general dividing surfaces arise from the specially constructed time correlation function within the approximation of HA-QTST.
An important condition underlying the analysis by HA in their recent work\cite{hele-jcp144} is that the dividing surface is a smooth function of imaginary time, an important assumption that \rc{should not be overlooked.} We will show here that this analysis does not hold true for more general \rc{ and} yet cyclically invariant dividing surfaces in the imaginary time path integral space. \rc{This result} confirms our previous conclusion that HA-QTST\cite{hele-jcp138} is not an exact derivation of RPMD-TST\cite{craig-jcp122,craig-jcp123,richardson-jcp131} in general because an exact quantum mechanical identity should be independent of the nature of the dividing surface.
This paper is organized as follows. Section II provides a brief overview of HA-QTST and our recent analysis, and summarizes its relationship to RPMD-TST. Section III provides a detailed evaluation of our final expression for HA-QTST for two types of cyclically invariant paths, and demonstrates our central point that HA-QTST does not lead to RPMD-TST for general cyclic paths. Section IV provides concluding remarks.
\section{Summary of HA-QTST and its relationship with RPMD-TST}
The formulation of HA-QTST,\cite{hele-jcp138} for the case where the population function is the step function $\Theta(x)$ \rc{and assuming one dimensional reaction coordinate}, starts from the following generalized Kubo-transformed side-side correlation function (GKSCF):
\begin{eqnarray}
&&\tilde C_{ss}(t) = \int d{\bf q}\int d \boldeta\int d{\bf z}\ \rho({\bf q},\boldeta){\mathcal G}({\bf q}, \boldeta,{\bf z};t)\nonumber \\
&&\hspace{1in} \times \Theta(\left (f({\bf q})-d\right) \Theta \left (f({\bf z})-d\right) \ , \label{eq:c_sst}
\end{eqnarray}
where $f({\bf q})$ represents the dividing surface\endnote{\rc{Although $f({\bf q})$ is defined in the space of imaginary time path integral, this may not always be a true dividing surface of the multidimensional ring polymer potential energy profile. Rather, this can be viewed as a some kind of path integral generalization of the classical dividing surface in general.}} defined in the space of imaginary time path integral, and
\begin{equation}
\rho({\bf q},\boldeta)=\prod_{k=1}^P\langle q_{_k}-\frac{\eta_{_k}}{2}|e^{-\beta \hat H/P}|q_{_{k+1}}+\frac{\eta_{_{k+1}}}{2} \rangle \ , \label{eq:def-rho}
\end{equation}
with $\beta=1/(k_BT)$, and
\begin{eqnarray}
&&{\mathcal G}({\bf q}, \boldeta,{\bf z};t)=\prod_{k=1}^P \langle q_{_k}+\frac{\eta_k}{2}|e^{it\hat H/\hbar}|z_k\rangle \nonumber \\
&&\hspace{1.in} \times \langle z_k|e^{-it\hat H/\hbar}|q_{k}-\frac{\eta_k}{2}\rangle \ . \label{eq:def-g}
\end{eqnarray}
As mentioned in the Introduction and in our previous work,\cite{jang-jcp144} the quantum dynamical meaning of Eq. (\ref{eq:c_sst}) is not clear as it is not derived from, {\it e.g.}, any response theory. The primary motivation provided by HA\cite{hele-jcp138} for using this time correlation function is that the $t=0_+$ limit of its time derivative is nonzero and positive, although the value of the exact time correlation function at time $t=0$ is zero.\cite{jang-jcp144} \rc{The purely} mathematical basis for the classical-like singularity in the correlation function that imparts its nonzero value at $t\rightarrow 0_+$ is easy to understand, \rc{but its physical meaning is much less clear in the context of quantum mechanics.} HA then utilized this singular behavior of the time derivative of GKSCF to define the following rate expression:\cite{hele-jcp138}
\begin{eqnarray}
k_{HA}Z_a=\left . -\frac{d}{d t} \tilde C_{ss}(t)\right |_{t=0_+} \ ,\label{eq:dc_sst}
\end{eqnarray}
and evaluated this by employing a short time approximation for real time propagators, $e^{\pm it\hat H/\hbar}$, within the definition of ${\mathcal G}({\bf q}, \boldeta, {\bf z};t)$, Eq. (\ref{eq:def-g}). \rc{In this procedure, a natural classical-like variable defined as}
\begin{equation}
p_k=m (z_k-q_k)/t \ , \label{eq:pk_def}
\end{equation}
\rc{emerges and plays an important role in the derivation of the final rate expression by HA.\cite{hele-jcp138}}
However, on the other hand, we have shown\cite{jang-jcp144} through partial integration that Eq. (\ref{eq:dc_sst}) can be expressed as
\begin{eqnarray}
&&k_{HA}Z_a= \frac{1}{(2\pi \hbar)^P}\int d{\bf q}\int d \boldeta \int d{\bf p} \rho({\bf q},\boldeta) \nonumber \\
&&\hspace{.2in}\times\exp (i{\bf p} \cdot \boldeta /\hbar) \Theta \left ({\bf p}\cdot \nabla f({\bf q}) \right) \delta (f ({\bf q})-d) \nonumber \\
&&\hspace{.2in}\times \sum_{k=1}^P \frac{\partial f({\bf q})}{\partial q_k} \frac{1}{2m}\left (\tilde p_{k,+}+\tilde p_{k,-} \right) \ , \label{eq:dc_sst-10+2}
\end{eqnarray}
where $\tilde p_{k,+}$ and $\tilde p_{k,-}$ are average imaginary time momenta defined as follows:
\begin{eqnarray}
&&\bar p_{k,+}= \frac{\langle q_{k-1}-\eta_{k-1}/2|e^{-\beta\hat H/P}\hat p|q_k+\eta_k/2\rangle}{\langle q_{k-1}-\eta_{k-1}/2|e^{-\beta\hat H/P}|q_k+\eta_k/2\rangle} \ ,\label{eq:p+}\\
&&\bar p_{k,-}=\frac{\langle q_k-\eta_k/2|\hat p e^{-\beta\hat H/P} |q_{k+1}+\eta_{k+1}/2\rangle }{\langle q_{k}-\eta_k/2|e^{-\beta \hat H/P} |q_k+\eta_k/2\rangle} \ .\label{eq:p-}
\end{eqnarray}
No approximation has been made in deriving Eq. (\ref{eq:dc_sst-10+2}). When evaluated up to only the leading order of $\beta/P$,
\begin{eqnarray}
&&\bar p_{k,+}\approx \frac{imP}{\hbar\beta}\left (q_{k-1}-q_{k}-\frac{\eta_{k-1}}{2}-\frac{\eta_k}{2} \right )\ ,\label{eq:p+app}\\
&&\bar p_{k,-}\approx \frac{imP}{\hbar \beta}\left (q_{k}-q_{k+1}-\frac{\eta_k}{2}-\frac{\eta_{k+1}}{2}\right) . \label{eq:p-app}
\end{eqnarray}
Equation (\ref{eq:dc_sst-10+2}) is in contrast with the expression derived by HA,\cite{hele-jcp138} which amounts to replacing the imaginary time momenta $\tilde p_{k,+}$ and $\tilde p_{k,-}$ with $p_k$ defined by Eq. (\ref{eq:pk_def}). \rc{ As noted in our earlier work,\cite{jang-jcp144} this is clearly an approximation.}
In our \rc{earlier} work,\cite{jang-jcp144} we then showed that Eq. (\ref{eq:dc_sst-10+2}) can be evaluated further employing a normal mode transformation used by HA.\cite{hele-jcp138} To this end, it is convenient to introduce\cite{hele-jcp138,jang-jcp144} the following functions:
\begin{eqnarray}
&&T_{k}({\bf q})=\frac{1}{\sqrt{B_P({\bf q})}}\frac{\partial f({\bf q})}{\partial q_k} \ , \label{eq:tk-def}\\
&&B_P({\bf q})=\sum_{k=1}^P \left (\frac{\partial f({\bf q})}{\partial q_k}\right )^2 \ , \label{eq:bnq} \\
&&\tilde \eta_0({\bf q})=\sum_{k=1}^P \eta_k T_{k}({\bf q}) \ .
\end{eqnarray}
Then, employing Eqs. (\ref{eq:p+app})-(\ref{eq:p-app}), and performing integrations over all the modes of $\boldeta$ and ${\bf p}$ perpendicular to the vector formed by $T_k$'s,
we have obtained the following expression:\cite{jang-jcp144}
\begin{eqnarray}
&&k_{HA}Z_a= \frac{P}{2\pi \hbar \beta}\int d{\bf q} \int d \tilde \eta_{_0}({\bf q}) \rho({\bf q},\boldeta_0)\delta (f ({\bf q})-d)\nonumber \\
&&\times \sum_{k=1}^P \frac{\partial f({\bf q})}{\partial q_k} \frac{T_{k-1}({\bf q})+2T_{k}({\bf q})+T_{k+1}({\bf q})}{4} \ ,\label{eq:dc_sst-10+6}
\end{eqnarray}
where $(\boldeta_0)_k=\tilde \eta_0({\bf q})T_k({\bf q})$ and $\rho({\bf q},\boldeta_0)$ can be approximated as
\begin{eqnarray}
\rho({\bf q},\boldeta_0)\approx \rho({\bf q},0) \exp\left \{-\frac{mP}{2\beta\hbar^2}\tilde \eta_0^2({\bf q})-g_{_P}({\bf q})\frac{\tilde \eta_0({\bf q})}{\hbar}\right\} \ .
\end{eqnarray}
In the above expression, $\rho({\bf q},0)$ is the conventional imaginary time path integral representation of the diagonal position element of the canonical density operator \rc{defined as}
\begin{equation}
\rho({\bf q},0)=\left (\frac{mP}{2\pi\beta \hbar^2}\right)^{P/2} \prod_{k=1}^P e^{-\epsilon V(q_k)-\frac{m}{2\epsilon\hbar^2}(q_k-q_{k+1})^2} \ ,
\end{equation}
with the cyclic boundary condition $q_{_{P+1}}=q_1$, and
\begin{equation}
g_{_P}({\bf q})=\frac{mP}{2\beta\hbar}\sum_{k=1}^P (q_{k+1}-q_k) T_k({\bf q}) \ . \label{eq:gnq-1}
\end{equation}
In their recent work,\cite{hele-jcp144} HA argued that Eq. (\ref{eq:dc_sst-10+6}) becomes equivalent, in the limit of $P\rightarrow \infty$, to the following version of the RPMD-TST expression:\cite{richardson-jcp131}
\begin{equation}
k_{RP}Z_a= \left(\frac{P}{2\pi m\beta}\right)^{1/2}\int d{\bf q}\ \sqrt{B_P({\bf q})} \rho({\bf q},0)\delta (f ({\bf q})-d) \ , \label{eq:rp-tst}
\end{equation}
where $B_P({\bf q})$ is the normalization factor defined by Eq. (\ref{eq:bnq}) and, through Eq. (\ref{eq:tk-def}), can also be expressed as
\begin{equation}
\sqrt{B_P({\bf q})}= \sum_{k=1}^P \frac{\partial f({\bf q})}{\partial q_k} T_{k}({\bf q}) \ .\label{eq:bp-rel}
\end{equation}
In order for Eqs. (\ref{eq:dc_sst-10+6}) and (\ref{eq:rp-tst}) to be equivalent in the limit of $P\rightarrow \infty$, as claimed by HA,\cite{hele-jcp144} contributions of $T_{k+1}({\bf q})-T_k({\bf q})$ and $g_{_P}({\bf q})$ in the former should become negligible compared to others in that limit. To be more precise, the two become equivalent if the following conditions are satisfied:
\begin{eqnarray}
&&\lim_{P\rightarrow \infty} (T_{k+1}({\bf q})-T_k({\bf q}))\sqrt{P} =0 \ ,\label{eq:equiv-1}\\
&&\lim_{P\rightarrow \infty} g_{_P}({\bf q})/\sqrt{P}=0 \ .\label{eq:equiv-2}
\end{eqnarray}
For the type of smooth dividing surfaces assumed by HA,\cite{hele-jcp138,hele-jcp144} the above conditions are satisfied. However, in the following section, we show that the analysis does not hold true for more general cyclically invariant dividing surfaces in the imaginary time path integral space.
\section{Dependence of HA-QTST on Dividing Surface}
Because of the cyclic invariance of the summation in Eq. (\ref{eq:gnq-1}), $g_{_P}({\bf q})$ can also be expressed as follows.
\begin{eqnarray}
&&g_{_P}({\bf q})=\frac{mP}{2\beta \hbar} \sum_{k=1}^P q_k \left ( T_{k-1}({\bf q}) -T_k({\bf q})\right) \nonumber \\
&&\hspace{.2in}=\frac{mP}{2\beta\hbar} \frac{1}{\sqrt{B_P({\bf q})}} \sum_{k=1}^P q_k \left (\frac{\partial f({\bf q})}{\partial q_{k-1}}-\frac{\partial f ({\bf q})}{\partial q_k}\right) \ ,\label{eq:gnq-2}
\end{eqnarray}
\rc{where Eq. (\ref{eq:tk-def}) has been used in the second equality.}
The main argument made by HA\cite{hele-jcp144} is that $T_{k+1}({\bf q})-T_k({\bf q}) =O(P^{-1})$ and that $\lim_{P\rightarrow \infty} g_{_P}({\bf q})=0$, for which Eqs. (\ref{eq:equiv-1}) and (\ref{eq:equiv-2}) are satisfied and Eq. (\ref{eq:dc_sst-10+6}) reduces to Eq. (\ref{eq:rp-tst}) after Gaussian integration over $\tilde\eta_0({\bf q})$. However, such an agreement is true only for a certain simple class of dividing surfaces in which the contribution of the quantum kinetic energy is vanishingly small, and hence so are the quantum path fluctuations. For general cyclically invariant dividing surfaces, this is not true. This can be shown by demonstrating results for two specific examples of the dividing surface as described below.
\subsection{Dividing surface containing a quadratic form in the Fourier modes of the imaginary time path}
\rc{Generalizing the dividing surfaces introduced by Althorpe and coworkers,\cite{richardson-jcp131,hele-jcp138} we here consider a class of cyclically invariant dividing surfaces defined as}
\begin{eqnarray}
f({\bf q})&=&\frac{\cos\phi}{P}\sum_{j=1}^P q_j+\frac{\sqrt{2}\sin \phi}{P} L_n({\bf q}) \ , \label{eq:fq-1}
\end{eqnarray}
where $\phi$ \rc{is a real phase factor that can be} determined or optimized separately, $n$ is a nonnegative integer less than or equal to $P$, and
\begin{eqnarray}
L_n({\bf q})= \left (\sum_{j=1}^P \sum_{j'=1}^Pe^{i2\pi n (j-j')/P} q_jq_{j'} \right)^{1/2} \ .
\end{eqnarray}
The above function corresponds to the norm of the $n$th Fourier mode of the cyclic imaginary time path. \rc{It is worthwhile to note here that there can be additional restriction for $\phi$ in order for $f({\bf q})$ to be a genuine dividing surface in the path integral space. An obvious condition is that $\cos \phi$ cannot be zero or close to it because the information on the average position of ring polymers gets lost in such case. With this issue clarified and under the assumption that a physically appropriate choice of $\phi$ can be made, below we examine whether the conditions of Eqs. (\ref{eq:equiv-1}) and (\ref{eq:equiv-2}) are indeed satisfied for any choice of $n$. }
\rc{First, taking derivative of Eq. (\ref{eq:fq-1}) with respect to $q_k$, we find that}
\begin{eqnarray}
\frac{\partial f ({\bf q})}{\partial q_k} =\frac{\cos \phi}{P}+\frac{\sqrt{2}\sin \phi}{P L_n({\bf q})}\sum_{j=1}^N \cos \left (\frac{2\pi n(k-j)}{P}\right) q_j \ . \label{eq:partial_f}
\end{eqnarray}
\rc{Using the above expression in Eq. (\ref{eq:bnq}), we also fine that}
\begin{equation}
B_P({\bf q})=\frac{1}{P} \ .
\end{equation}
Therefore, based on the definitions of Eqs. (\ref{eq:tk-def}) and (\ref{eq:gnq-1}), and after employing standard trigonometric manipulation, \rc{we obtain the following general expressions:}
\begin{eqnarray}
&&T_{k-1}({\bf q})-T_{k} ({\bf q})\nonumber \\
&&\hspace{.2in}=\sqrt{\frac{2}{P}}\frac{\sin \phi}{L_n({\bf q})}\left \{ \sin (\frac{2\pi n}{P})\sum_{j=1}^P \sin \left (\frac{2\pi n (k-j)}{P}\right) q_j\right . \nonumber \\
&&\hspace{.4in} \left .-2\sin^2(\frac{\pi n}{P})\sum_{j=1}^P \cos \left (\frac{2\pi n (k-j)}{P}\right) q_j\right\} \ , \label{eq:dtk_q}\\
&&g_{_P}({\bf q})=-\frac{m\sin \phi}{\beta\hbar} \sqrt{2P}\sin^2(\frac{\pi n}{P})L_n({\bf q})\ . \label{eq:gnq-new}
\end{eqnarray}
\rc{Employing} the fact that $L_n({\bf q})=O(P)$, \rc{from Eq. (\ref{eq:dtk_q})}, we identify the following three types of scaling behavior depending on the value of $n$:
\begin{equation}
T_{k-1}({\bf q}) -T_k ({\bf q})=\left \{ \begin{array}{lll} O(P^{-1/2})&,& \mbox{for }n =O(P) \\
O(P^{-1})&,& \mbox{for }n=O(P^{1/2}) \\
O(P^{-3/2})&, & \mbox{for }n =O(1)
\end{array} \right . \label{eq:td-order}
\end{equation}
Similarly, \rc{from Eq. (\ref{eq:gnq-new})}, \rc{we find} the following three types of scaling behavior:
\begin{equation}
g_{_P}({\bf q})=\left \{ \begin{array}{lll} O(P^{3/2})&,& \mbox{for }n =O(P) \\
O(P^{1/2})&, &\mbox{for }n=O(P^{1/2}) \\
O(P^{-1/2})&, & \mbox{for }n =O(1)
\end{array} \right . \label{eq:gn-order}
\end{equation}
The results shown in Eqs. (\ref{eq:td-order}) and (\ref{eq:gn-order}) confirm that the conclusion\cite{hele-jcp144} by HA is true only for $n=O(1)$. In fact, this limitation was noted briefly in the Appendix A of Ref. \onlinecite{hele-jcp144}, but has not been stated anywhere else. For the case of $n=O(P^{1/2})$, the term involving $g_{_P}({\bf q})$ does not vanish but remains finite. Therefore, the difference between HA-QTST and RPMD-TST can be accounted for by a finite factor. For $n=O(P)$, both the terms involving $T_{k-1}({\bf q})-T_{k}({\bf q})$ and $g_{_P}({\bf q})$ cause HA-QTST to be different from RPMD-TST. In fact, in this case, it is not clear whether HA-QTST results in a finite value. Neglecting these cases amount to projecting the imaginary time path integral space onto the subspace where the quantum kinetic energy is not fully accounted for, which it seems cannot be overlooked in any QTST that aspires to more fully incorporate quantum effects.
\rc{As an example, let us consider the case where $P$ is even and $n=P/2$. Equation (\ref{eq:partial_f}) in this case becomes
\begin{eqnarray}
\frac{\partial f({\bf q})}{\partial q_k}=\frac{\cos \phi}{P}-(-1)^k\frac{\sqrt{2}\sin \phi}{2P} \frac{(Q_1-Q_2)}{\sqrt{(Q_1-Q_2)^2}}\ ,
\end{eqnarray}
where $Q_1=\sum_{j=1}^{P/2} q_{2j-1}$ and $Q_2=\sum_{j=1}^{P/2}q_{2j}$.
Using the above expression in Eq. (\ref{eq:bnq}), we also find that
\begin{eqnarray}
B_P({\bf q})=\sum_{k=1}^P \left (\frac{\partial f}{\partial q_k}\right)^2=\frac{1}{P}\left (\cos^2 \phi+\frac{1}{2}\sin^2\phi\right) \ .
\end{eqnarray}
Using the above expression in Eq. (\ref{eq:tk-def}), we can calculate that
\begin{eqnarray}
|T_{k+1}({\bf q})-T_k({\bf q})|=\frac{1}{\sqrt{2P}} \frac{|\sin \phi|}{(\cos^2\phi+\sin^2 \phi/2)^{1/2}} \ . \label{eq:dtq-ex1}
\end{eqnarray}
In addition, from Eq. (\ref{eq:gnq-2}), we also find that
\begin{eqnarray}
g_{_P}({\bf q})=-\frac{m}{2\beta\hbar} \frac{\sqrt{P}\sin \phi}{(\cos^2\phi+\sin^2\phi/2)}|Q_2-Q_1| = O(P^{3/2}) \ . \label{eq:gp-ex1}
\end{eqnarray}
The above two expressions, Eqs. (\ref{eq:dtq-ex1}) and (\ref{eq:gp-ex1}), serve as explicit demonstration of the scaling behavior of Eqs. (\ref{eq:td-order}) and (\ref{eq:gn-order}) for $n=P/2$ and thus prove that both conditions of Eqs. (\ref{eq:equiv-1}) and (\ref{eq:equiv-2}) are indeed violated.
}
\begin{figure}
\includegraphics[width=3.2in]{Jang_HA-QTST2_Fig.eps}
\caption{Dependences of the absolute value of Eq. (\ref{eq:dtk-ex1}) with $k=2$ and $\alpha=0$ on the number of discretization $P$ in Log-Log scale for three specific examples of $n=1$, $P^{1/2}$, and $P/4$.}
\end{figure}
\rc{As an alternative and more general example, let us consider a specific realization of the cyclic path ${\bf q}$ along which $f({\bf q})$ remains constant as follows:}
\begin{equation}
q_j=q_0+\sqrt{2}A_n \sin( \frac{2\pi n j}{P}+\alpha)\ ,
\end{equation}
\rc{where $q_0$, $A_n$, and $\alpha$ are fixed parameters.} For this choice, it can be shown that
\begin{eqnarray}
&&f({\bf q})=q_0\cos\phi +A_n\sin \phi \ , \label{eq:fbq-ex1}\\
&&L_n({\bf q})=\frac{A_nP}{\sqrt{2}} \ , \label{eq:ln-ex1}\\
&&\frac{\partial f({\bf q})}{\partial q_k}=\frac{\cos \phi}{P}+\frac{\sqrt{2}\sin \phi}{P} \sin (\frac{2\pi n k}{P}+\alpha) \ . \label{eq:par-fq-ex1}
\end{eqnarray}
Inserting Eq. (\ref{eq:ln-ex1}) into Eq. (\ref{eq:gnq-new}), we find that
\begin{equation}
g_{_P}({\bf q})=-\frac{m\sin \phi}{\beta\hbar} A_n \sin^2 (\frac{\pi n}{P}) P^{3/2} \ ,
\end{equation}
which clearly follows the scaling behavior of Eq. (\ref{eq:gn-order}). Equation (\ref{eq:dtk_q}) \rc{for this cyclic path} can also be calculated and simplifies to
\begin{eqnarray}
&&T_{k-1}({\bf q})-T_k({\bf q})=\frac{2\sqrt{2}}{\sqrt{P}}\left \{ \sin (\frac{2\pi n}{P})\cos(\frac{2\pi n k}{P} +\alpha)\right .\nonumber \\
&&\hspace{1 in}\left .-\sin^2(\frac{\pi n}{P}) \sin(\frac{2\pi n k}{P}+\alpha)\right\} \ . \label{eq:dtk-ex1}
\end{eqnarray}
The above expression follows the scaling behavior shown in Eq. (\ref{eq:td-order}). Figure 1 shows this behavior more clearly for $k=2$ and for three specific choices of $n=1$, $\sqrt{P}$, and $P/4$ assuming $\alpha=0$.
Ultimately, what contributes to the difference between HA-QTST and RPMD-TST is the difference of the summation in Eq. (\ref{eq:dc_sst-10+6}) and that in Eq. (\ref{eq:rp-tst}). This can also be calculated explicitly as follows:
\begin{eqnarray}
&&\frac{1}{4}\sum_{k=1}^P \frac{\partial f ({\bf q})}{\partial q_k}\left \{ (T_{k-1}({\bf q})-T_k({\bf q}))+(T_{k+1}({\bf q})-T_k({\bf q}))\right\} \nonumber \\
&&\hspace{.5in}=\frac{1}{\sqrt{P}} \sin^{2}(\frac{\pi n}{P})\left \{ 3\cos^2(\frac{\pi n}{P})-1\right\} \ .
\end{eqnarray}
The above expression shows that the actual difference between the two sums decreases further for $n=O(1)$ and $n=O(P^{1/2})$ due to some cancellation, but the scaling behavior predicted from Eq. (\ref{eq:td-order}) remains the same for $n=O(P)$.
\subsection{Dividing surface containing a quadratic form defined in terms of the imaginary time path coordinates}
We here consider another kind of cyclically invariant dividing surface defined as
\begin{eqnarray}
f({\bf q})&=&\frac{\cos\phi}{P}\sum_{j=1}^P q_j+\frac{\sin \phi}{R(n)} D_n({\bf q}) \ , \label{eq:fq-2}
\end{eqnarray}
where
\begin{equation}
D_n({\bf q})=\left (\sum_{j=1}^P(q_j-q_{j+n})^2\right)^{1/2} \ .
\end{equation}
Because of the cyclic boundary condition for $q_j$, this function is invariant with respect to cyclic permutation. In Eq. (\ref{eq:fq-2}), $R(n)$ is an appropriate normalization factor that makes $D_n({\bf q})/R(n)$ an order of unity. For example, for $n=1$, $R(1)=O(1)$. For $n\sim P/2$, $R(n)=O(\sqrt{P})$.
For the dividing surface defined above,
\begin{equation}
\frac{\partial f({\bf q})}{\partial q_k}=\frac{\cos\phi}{P}+\frac{\sin \phi}{R(n)D_n({\bf q})} (2q_k-q_{k+n}-q_{k-n}) \ ,
\end{equation}
and it can be shown that
\begin{eqnarray}
&&B_P({\bf q})=\frac{\cos^2\phi}{P}\nonumber \\
&&+\frac{\sin^2\phi}{R(n)^2D_n({\bf q})^2}\sum_{k=1}^P(2q_k-q_{k+n}-q_{k-n})^2 \ , \\
&&T_{k-1}({\bf q})-T_k({\bf q})=\frac{\sin \phi}{R(n)\sqrt{B_P({\bf q})}D_n({\bf q})}\Big \{ 2(q_{k-1}-q_k)\nonumber \\
&&\hspace{.4in} -(q_{k-1+n}-q_{k+n})-(q_{k-1-n}-q_{k-n})\Big \} \ ,\\
&&g_{_P}({\bf q})=\frac{mP\sin \phi}{2\beta\hbar R(n)\sqrt{B_P({\bf q})}D_n({\bf q})}\sum_{k=1}^P q_k \Big \{ 2(q_{k-1}-q_k)\nonumber \\
&&\hspace{.4in} -(q_{k-1+n}-q_{k+n})-(q_{k-1-n}-q_{k-n})\Big \} \ .
\end{eqnarray}
For the case where $n=1$, $D_1({q}) =O(1)$ and $R(1)=O(1)$. Therefore, assuming that $q_{j-1}-q_j=O(P^{-1/2})$ for $j=k$, $k-n$, and $k+n$, we obtain the following estimates.
\begin{eqnarray}
&&B_P({\bf q})=O(1)\ , \\
&&T_{k-1}({\bf q})-T_k({\bf q})=O(P^{-1/2})\ , \\
&&g_{P}({\bf q})=O(P^{3/2})
\end{eqnarray}
This is similar to the case of $n=O(P)$ for the dividing surface in the previous subsection, for which HA-QTST may not be well defined.
On the other hand, for $n\sim P/2$, $D_n({\bf q}) =O(P^{1/2})$ and $R(n)=O(P^{1/2})$. As a result,
\begin{eqnarray}
&&B_P({\bf q})=O(P^{-1}) \ ,\\
&&T_{k-1}({\bf q})-T_k({\bf q})=O(P^{-1}) \ , \\
&&g_P({\bf q})=O(P) \ .
\end{eqnarray}
The behavior shown above is a mixture of those for $n=O(P)$ and $n=O(P^{1/2})$ for the dividing surface in the previous subsection, and also renders HA-QTST ill-defined.
The analysis in this subsection thus demonstrates that, for the case where the dividing surface function mixes different Fourier modes of the imaginary time path in a nonlinear manner, the assumption of the analysis by HA\cite{hele-jcp144} does not hold true in general. As a result, our exact evaluation\cite{jang-jcp144} of HA-QTST\cite{hele-jcp138} expression becomes different from the RPMD-TST expression.\cite{richardson-jcp131}
\section{Concluding Remarks}
In this work, we have presented a new and detailed analysis of the effect of general cyclic imaginary time path integral dividing surfaces on the corresponding result of HA-QTST.\cite{hele-jcp138} What is believed to make HA-QTST unique is the fact that it is defined for an arbitrary form of cyclically invariant dividing surface constructed in the space of the imaginary time path integral. However, HA-QTST in its present form does not offer any independent and self-consistent prescription of the best choice out of the infinite number of dividing surface possibilities available, and its implementation practice has been confined to the dividing surfaces of the type in Eq. (\ref{eq:fq-1}) with $n=1$ followed by a limited variational optimization. This is different from the instanton theory,\cite{affleck-prl46,gillan-jp-c20} which \rc{provides a definition} of the dividing surface as the saddle point within the semiclassical approximation for the imaginary time action and can also be \rc{connected to} the PI-QTST.\cite{cao-jcp105,jang-jpca103-2} Although HA-QTST has been shown to work well for the kind of dividing surface given by Eq. (\ref{eq:fq-1}) with $n=1$, and it becomes equivalent to RPMD-TST for $n=O(1)$, the analysis presented in this work clarifies that different outcomes are expected for more general dividing surfaces. Along with our previous work,\cite{jang-jcp144} which was focused more on the underlying quantum mechanical expressions and approximations inherent in HA-QTST, the present work again emphasizes the major challenges in developing a unique and well defined QTST while utilizing the full quantum free energy expression defined in the imaginary time path integral space.
Regardless of our analyses presented here and in a recent work,\cite{jang-jcp144} one can \rc{justifiably} argue that HA-QTST\cite{hele-jcp138} or RPMD-TST\cite{craig-jcp123} for a smooth dividing surface defined in the imaginary time path integral space is useful \rc{in practice.} However, one should not misinterpret \rc{this degree of usefulness} as a quantum mechanical validation of some key assumptions and approximations behind HA-QTST and RPMD-TST.
We conclude this work by offering a few \rc{additional} insights into RPMD-TST and HA-QTST.
First, it is relatively easy to understand RPMD-TST.\cite{craig-jcp123} As is clear from Eq. (\ref{eq:rp-tst}), RPMD-TST for a general dividing surface \rc{is the} relative thermal weight for the dividing surface multiplied by the average ``classical" flux across it. Therefore, this can be seen \rc{as an} extension of the PI-QTST above the crossover temperature\cite{voth-jcp91} for a general dividing surface defined in the imaginary time path integral space. The fact that this limit is obtained from the RPMD rate theory\cite{craig-jcp122,craig-jcp123} is the result of the principles governing classical mechanics and does not \rc{have a quantum} mechanical origin.
\rc{The implications inherent in HA-QTST\cite{hele-jcp138} are more subtle as} its starting point, GKSCF, already \rc{takes} both classical and ensemble concepts and encodes them into the theory. Unlike ring polymer averages defined in the RPMD rate theory, which can be considered as ensemble averages of actual physical observables, the GKSCF defines a function of imaginary time paths (a dividing surface in the path integral space) as the argument of the population measurement. Can this kind of population be measured? Such a measurement requires measuring positions for the quantum canonical ensemble and then giving a weight only when a nonlinear (in general) function constructed from the measured positions meets the criterion of the population. This is not in general an ensemble average of a physical measurement but rather a subset of the ensemble of measured values that are selected and weighted in certain way. Even if such a measurement were possible, assuming a time correlation of such a quantity exists implies that it does not disturb the initial quantum state, which is why its time derivative has a finite $t=0_+$ limit. Thus, GKSCF \rc{is different} from the original exact Kubo-transformed time correlation function\cite{kubo-jpsj-12,kubo-jpsj-12-2} which represents a well-defined response of a quantum ensemble following a physical perturbation.
The fact that HA-QTST reduces to RPMD-TST for only smooth dividing surfaces, but may \rc{ not be} well defined for general cyclic dividing surfaces in the path integral space as demonstrated in this work, also reveals an additional \rc{ feature} of the theory. This result means that HA-QTST becomes equivalent to RPMD-TST only when the effect of operator ordering along the dividing surface of the imaginary time path integral becomes vanishingly small. In practice, confining the \rc{ dividing} surface to smooth functions ensures that it remains close to the instanton trajectory below the crossover temperature, which may be sufficient. \rc{Interestingly}, HA-QTST, in its practice of using a smooth dividing surface, may be viewed as a heuristic approximation that interpolates the PI-QTST above the crossover temperature\cite{voth-jcp91} and the instanton theory below the crossover temperature,\cite{affleck-prl46} but without a change in the pre-exponential factor \rc{ (a change that can be argued to be ``needed"\cite{cao-jcp105})}. Alternatively, \rc{ HA-QTST might be considered to be} a semi-analytical approximation for a numerical \rc{ approach developed some years ago} that proposes\cite{mills-cpl278} to calculate the exact saddle point of the action in the full imaginary time path integral space.\cite{cao-jcp105}
\acknowledgments
SJ acknowledges the support for this research from the National Science Foundation (CHE-1362926) and the Office of Basic Energy Sciences, Department of Energy (DE-SC0001393). GAV acknowledges the support of the National Science Foundation (NSF) through grant CHE-1465248. \rc{ We thank David Reichman for discussion in the early stage of this work and Eli Pollak for helpful comments and suggesting detailed consideration of the case $n=P/2$.}
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\section{\label{sec:model}Drift-diffusion model}
For a quantitative interpretation of the data we have extracted the
spin signal values from all the measurements of Fig.~\ref{fig:meas}
and plotted them in Fig.~\ref{fig:simu} against the DC field for the
three gate voltages. For the theoretical description, we adopt the
drift-diffusion model introduced by Yu and Flatt\'{e}
\cite{YuFlatte1,YuFlatte2}. In steady-state, the drift-diffusion
equation for the spin imbalance $n_s$ reads:
\begin{equation}\label{Eq:DriftDiff}
D \nabla^2 n_s + \mu E \nabla n_s - \frac{n_s}{\tau_{sf}}=0,
\end{equation}
where $D$ is the diffusion constant, $\tau_{sf}$ is the spin
relaxation time, $\mu$ is the carrier mobility and $E$ is the DC
electric field. The term $\mu E$ represents the drift velocity
$v_d$ and its magnitude compared to the Fermi velocity $v_F =
10^6$~m/s is important for the contribution of the spin drift to
the spin transport. Similar to the spin diffusion length
$\lambda_{sf} = \sqrt{D\tau_{sf}}$, we define a spin drift length
$\lambda_d = D/v_d$. The symmetric diffusion and asymmetric drift
effects add up to form a spin transport characterized by a pair of
length scales, named \emph{upstream} and \emph{downstream} lengths
$\lambda_{\pm}$ \cite{YuFlatte1}:
\begin{equation}\label{Eq:UpDownStream}
\frac{1}{\lambda_{\pm}} = \pm \frac{1}{2} \frac{1}{\lambda_d} +
\sqrt{\frac{1}{4} \frac{1}{\lambda_d^2} + \frac{1}{\lambda_{sf}^2}
}.
\end{equation}
The general solution to the spin imbalance equation
\ref{Eq:DriftDiff} in the direction $x$ parallel to the spin
transport is
\begin{equation}\label{Eq:Sol}
n_s(x) = A \exp{(+\frac{x}{\lambda_+})} +
B\exp{(-\frac{x}{\lambda_-})},
\end{equation}
where A and B are determined by the boundary conditions. Since we
take the electrode F4 as the spin source, only the decaying
solution with respect to the injection point need to be
considered. Therefore, we can write the spin signal ($R_{SV}$)
dependence versus the applied DC field in the form:
\begin{equation}\label{Eq:Rdependence}
R_{SV} = R_0 \exp{(-\frac{L}{\lambda_{\pm}})},
\end{equation}
where $R_0$ is the spin signal that would be measured at the
injection point, $\lambda_{\pm}$ carries the electric field
dependence (Eq.~\ref{Eq:UpDownStream}) and $L \simeq 1.5\mu$m is
the distance between F3 and F4.
In the following we compare the drift-diffusion equation for the
spin signal (Eq.~\ref{Eq:Rdependence}) with the spin drift
measurements in the high carrier density regime. From the
conductivity measurements presented in Fig.~2b we extract $\mu
\simeq 0.25 $~m$^{2}$/Vs using $\sigma=ne\mu$, where the carrier
concentration $n$ is calculated from the sample's response to gate
voltage (fig. 2b), using the method described in
Refs.~\onlinecite{RiseofG},\onlinecite{Tan2007Dirac}. This gives a
drift velocity of $v_d \simeq \pm 1.7 \cdot 10^4$ m/s
for $E \simeq \pm 68$~kV/m. The diffusion constant of $D \simeq
0.02$~m$^2$/s is obtained using the Einstein relation
$\sigma=Ne^2D$, where the density of states $N$ is determined
according to Ref.~\onlinecite{Tombros2007}. Using
Eq.~\ref{Eq:UpDownStream} and a spin relaxation length
$\lambda_{sf} \simeq 2 \mu$m \cite{Tombros2007}, we calculate the
magnitude of the drift effect manifesting in the up/downstream
lengths as $\lambda_+ = 1.3 \mu$m, $\lambda_- = 3.0 \mu$m for E
=$\pm34$~kV/m and $\lambda_+ = 0.9 \mu$m, $\lambda_- = 4.4 \mu$m
for E =$\pm68$~kV/m. Note, that at E =$\pm68$~kV/m the
characteristic length scale of the drift-diffusion spin transport
is about two times increased/decreased compared to the diffusion
process. The value of $R_0$ we estimate from the spin valve
measurements in zero electric field when
$\lambda_{\pm}=\lambda_{sf}$ and $R_0$ is given by
$R_0=R_{SV}\exp{(L/\lambda_{sf})}$. By taking the length of the
drift region $L=1.5\mu m$, $\lambda_{sf}=2\mu m$ and the measured
$R_{SV}$ at E =0 we can determine $R_0$. Now, we are able to
calculate the drift effect according to Eq.~4, all parameters
being known. The theoretical curves (solid lines in
Fig.~\ref{fig:simu}) are in excellent agreement with the
experimental data. There are no free parameters in the calculation
and, in fact, these measurements could be used to extract
$\lambda_{sf}$, if it would be unknown.
\begin{figure}[h!]
\includegraphics[width=8.5cm]{Fig4.eps}
\caption{\label{fig:simu} Spin valve signal for high carrier
densities (circles), and near the Dirac neutrality point (stars).
The lines represent a calculation (with no free parameters) of the
drift effect based on the drift-diffusion equation.}
\end{figure}
At the Dirac point, due to the symmetry, no drift effect is
expected. However, in our measurements we are probably slightly
away from the Dirac point. Here we cannot apply our above
analysis. It remains an open theoretical question to what the
drift velocity should be at the charge neutrality point.
In conclusion, we studied spin transport manipulation in graphene,
by carrier drift under the action of a DC electric field. For high
charge carrier densities ($n \simeq 3.5 \times 10^{16}$~m$^{-2}$)
depending on the direction of the applied DC field and the nature
of the carriers we were able to modify the effective spin
relaxation length by factor of 4.8 which resulted in a modulation
of the spin valve signal of about 300\%. The spin-drift
measurements are described well by a drift-diffusion model. The
control over the drift velocity we demonstrated opens new
possibilities for exploring spin related phenomena in other types
of graphene devices, such as graphene p-n junctions.
We would like to acknowledge B. Wolfs, S. Bakker for technical
asistance. This work was financed by MSC$^{plus}$, NanoNed, NWO
(via a 'PIONIER' grant) and FOM.
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\section{Introduction}
\label{sec:intro}
Graph representation learning analyzes complex structured data by representing node attributes and relationships in a low-dimensional vector space. In recent years, it has attracted increasing attention owing to the prevalent presence of graph-structured data. The use of deep neural networks, particularly graph neural networks (GNNs), has further facilitated the ability of graph representation learning to represent nodes. For example, GNNs have been used to study social media~\citep{Fan2019WWW, Sankar2021WC}, protein interactions~\citep{Gainza2020Nature}, traffic flow forecasting~\citep{Lan2022ICML}, and neuroscience~\citep{Bessadok2022TPAMI}.
Many applications of graph representation learning involve temporal interactions, yet most existing methods do not consider such dynamics. As \cite{Xu2020ICLR} pointed out, ignoring the temporal evolution in dynamic graphs can result in suboptimal performance. In certain scenarios, the dynamic structure holds key insights into the system. For example, when using a pandemic model to predict the spread of infection, the evolution of social relationships due to human events (\textit{e}.\textit{g}.,~ immigration, travel, education) must be taken into account~\citep{Zhong2021COVID}. In another example, malignant cells within tumors secrete proteins that influence neighboring stromal cells and create an environment conducive to their growth and metastasis~\citep{Podhajcer2008CMR}. For more examples, see~\cite{Kazemi2020JMLR}.
In this article, we focus upon the realm of representation learning for dynamic graphs, where explicitly modeling both the time- and node-dependent interactions is generally required to better represent temporal evolution and the dynamic nature of the data. Despite the importance, it can be rather challenging to capture both of these dynamics effectively, mainly when the changes are continuous-time and nonlinear. While learning representations on dynamic graphs is a relatively new field, prior works are limited to discrete-time dynamic graphs, which are represented as a sequence of snapshots of the graph~\citep{Goyal2018arXiv, Kumar2019KDD, Pareja2020AAAI, Chen2021arXiv}. Recently, Ordinary Differential Equations (ODEs) have been incorporated into graph neural networks for continuous-time dynamic scenarios~\citep{Zang2020KDD, Yan2021arXiv, Choi2022AAAI}. Nevertheless, these approaches still rely on a static graph structure or segments of static graphs for dynamic inference.
To address these limitations, we propose a novel and unified framework for dynamic graph representation learning that can handle both \emph{the structural dynamics} and \emph{the intrinsic dynamics of nodes} simultaneously, wherein the node embeddings are considered to undergo a dynamic evolution over time, in a manner similar to the concept of Neural Controlled Differential Equations (Neural CDEs)~\citep{Kidger2020CDE} built for time series tasks. Neural CDEs are a powerful concept that has desired calibration ability with subsequent data, robustness to missing values and a memory-efficient property based on adjoint-based backpropagation. We extend this concept to dynamic graphs, which we refer to as the Graph Neural Controlled Differential Equation (GN-CDE) model. The main difference is that GN-CDE creates a continuous path for temporal interactions among two nodes, allowing both structural dynamics and intrinsic dynamics to be naturally incorporated into the integration process. This makes GN-CDE not simply an extension of Neural CDEs, as it can handle the causal effects inherent in dynamic graph structures while the original Neural CDEs cannot. Excitingly, the capability of adjusting the predicted trajectories with incoming even partially observed data and training via adjoint backpropagation from Neural CDEs still hold for our model, making it a promising method for practical usage. It is worth noting that, GN-CDE is a flexible framework that we can leverage to tackle node attribute prediction, dynamic node classification and temporal link prediction tasks with minor modifications. Besides, it can easily be extended to more complex graph structures (such as directed graphs and knowledge graphs). To demonstrate the superiority of our method, we further experimentally evaluate it on node attribute prediction tasks with the underlying graph structure evolving and our method can achieve favorable results across different setups.
The contributions of this work are summarized below.
\begin{itemize}[noitemsep]
\item We propose a generic model GN-CDE which expresses the graph structural dynamics via creating graph paths into our controlled differential equation for dynamic graphs, allowing modeling the continuously evolving process of node embeddings.
\item Two alternative approaches are presented, and their theoretical properties are analyzed for comparative purposes.
\item We further propose an approximation for the integration process that not only results in a more efficient implementation, but also allows for more flexibility in message passing between graphs.
\item Experimental results verify that our proposed method can achieve better results than other methods across different graph tasks.
\end{itemize}
\subsection{Related Works}
\label{sec:related_work}
\noindent\textbf{Graph Embedding Learning.}~
Early works for graph representation learning include graph factorization approaches~\citep{Belkin2001NIPS, Ahmed2013ICWWW} and random walk-based methods~\citep{Perozzi2014KDD, Tang2015WWW}. With the success of deep learning, Graph Neural Networks which learn node representations by aggregating neighborhood features at each layer, achieve outstanding performance in various tasks~\citep{Welling2016ICLR, Hamilton2017NIPS, Velivckovic2017arXiv}. However, all the above methods are only limited to node representation learning with a static graph structure.
In the real world, graphs are inherently dynamic rather than static. For example, the interactions of users can change from time to time for e-commerce and social platforms. Existing works for tackling such dynamics can be roughly categorized into three categories. The first type focuses on capturing temporal information along with local structure so as to enhance the expressive ability of the model~\citep{Yan2018AAAI, Pareja2020AAAI, Ma2020SIGIR, Lan2022ICML}. Specifically, it can be achieved via learning new parameters for each snapshot and maintaining the shareable temporal information. The second type investigates the efficient update schemes when the graph structure changes~\citep{Kumar2019KDD, Rossi2020arXiv, Chen2021arXiv}, wherein the interaction events are classified in a fine-grained manner, and the nodes only update when affected by some events. Very recently, differential equations have been incorporated with graph neural networks for dynamic graphs~\citep{Zang2020KDD, Yan2021arXiv, Choi2022AAAI}. These methods are capable of processing irregularly sampled observations and inferring the continuous dynamics of nodes. It is this last one that is of most interest to us here. Unlike these methods limited to a fixed graph structure only, we propose a graph neural controlled differential equations model to incorporate the graph structural dynamics when integrating over graphs. As a result, the requirement for a static graph can be eliminated naturally.
\noindent\textbf{Neural Differential Equations.}~Neural differential equations are an attractive option for modeling temporal dynamics on hidden representations via using a neural network to parameterize the vector field~\citep{Chen2018ODE, Kidger2020CDE}. The majority of existing work aims to utilize such integration procedure to stimulate neural networks with infinite depth, thus the representation ability can be strengthened. For example, Neural ODEs are analog to a continuous version of ResNet~\citep{Chen2018ODE}. Neural CDEs correspond to Recurrent Neural Network~\citep{Kidger2020CDE}. Recently, some works devoted to using Neural ODEs combined with GNNs to characterise the continuous message-passing flow of node representations~\citep{Xhonneux2020ICML, Poli2021arXiv}. However, these methods are not built for dynamic graphs, nor can they tackle structural dynamics.
\subsection{Paper organization and notations}
The rest of this paper is structured as follows. In Section~\ref{sec:preliminary}, we briefly introduce the background knowledge. In Section~\ref{sec:method}, we describe our main model, compare it with the competitors, and declare an approximation for efficient computation. The application in several representative graph representation learning tasks is also presented. In Section~\ref{sec:experiments}, we illustrate the empirical performance of our model in node attribute prediction task. Finally, we conclude in Section~\ref{sec:conclusion}. The complete proofs will follow thereafter in the appendix.
Before continuing, we introduce several notations used throughout the paper. First of all, we use lower-case letters to denote scalars, bold lower-case letters to denote vectors, and
bold upper-case letters to denote matrices. For a matrix $\mathbf{X}$, we represent the $i$-th row of $\mathbf{X}$ as $\mathbf{X}^{(i)}$, and the element at the $i$-th row and $j$-th column as $\mathbf{X}^{(i,j)}$. We use $\odot$ to represent element-wise (Hadamard) multiplication.
\section{Preliminary}
\label{sec:preliminary}
This section briefly reviews the basic definitions and common manners to learn graph embeddings, and then presents two typical neural differential equations.
\subsection{Graph Embedding Learning}
\label{sec:graph_emb_learn}
\noindent\textbf{Static graph.}~A static graph only contains a fixed topological structure. Let a static graph represented as $\mathcal{G}=\{\mathcal{V}, \mathcal{E}\}$ where $\mathcal{V}$ is the set of nodes, and $\mathcal{E} \subseteq \mathcal{V}\times \mathcal{V}$ is the set of edges. Let $v_i \in \mathcal{V}$ denote a node and $e_{ij} \in \mathcal{E}$ denote an edge between node $v_i$ and $v_j$, $i,j \in \{1,...,|\mathcal{V}|\}$. Then the topology of the graph can be represented by an adjacency matrix $\mathbf{A} \in \mathbb{R}^{|\mathcal{V}| \times |\mathcal{V}|}$ where $\mathbf{A}^{(i,j)}=1$ if $e_{ij} \in \mathcal{E}$ otherwise 0. In most complex scenarios, the graph is equipped with a node attribute matrix $\mathbf{F}=\{\mathbf{F}^{(i)}\}_{i=1}^{|\mathcal{V}|}, \mathbf{F}^{(i)} \in \mathbb{R}^m$ and edge feature matrix $\mathbf{E}=\{\mathbf{E}^{(i,j)}\}_{i,j=1}^{|\mathcal{V}|}, \mathbf{E}^{(i,j)} \in \mathbb{R}^w$. Graph embedding learning for static graphs is to create an embedding $\bm{z}{(v_i)}$ for each node $v_i$ following a specified aggregation rule such that the specific local topology and node intrinsic information can be captured, formally
\begin{equation*}
\bm{z}{(v_i)}=\sum_{j,~\mathbf{A}^{(i,j)}=1} h(\text{msg}(\mathbf{F}^{(i)}, \mathbf{F}^{(j)}, \mathbf{E}^{(i,j)}),\mathbf{F}^{(i)}),
\end{equation*}
where $\text{msg}$ and $h$ are predefined or learnable functions.
\noindent\textbf{Dynamic graphs.}~According to the interval of observations, dynamic graphs can be roughly categorized into \emph{discrete-time dynamic graphs} and \emph{continuous-time dynamic graphs}~\citep{Kazemi2020JMLR}. A discrete-time dynamic graph comprises a chronological sequence of static graph snapshots regularly sampled according to a fixed time interval while a continuous-time dynamic graph consists of graph snapshots that are irregularly sampled. Dynamic graphs contain structural dynamics arising from edge addition or deletion, node addition or deletion events, and node intrinsic dynamics caused by node or edge feature transformations in different time stamps. As a result, the adjacency matrix $\mathbf{A}_t$ can vary at different time stamps $t$. In this work, we embark on representation learning for continuous-time dynamic graphs where the observations are irregularly sampled. We start with undirected graphs ($\mathbf{A}_t$ is symmetric) without time-varying node attributes and edge features, then we discuss the extensions to more subtle graph structures. It should be noted that our developed method can naturally deal with tasks on discrete-time dynamic graphs as well.
\subsection{Neural Differential Equations}
\label{sec:neural_diff_equ}
\noindent\textbf{Neural ordinary differential equations (Neural ODEs).}~
Neural ordinary differential equations ~\citep{Chen2018ODE} are the continuous-depth analogue to residual neural networks. Let $f_\theta: \bm{x} \rightarrow \bm{y}$ be a function mapping with some learnable parameters $\theta$, and $\zeta_\theta$ and $\ell_\theta$ are two linear maps. Neural ODEs are defined as
\begin{equation}
\label{eq:ode}
\bm{z}_t = \bm{z}_0 + \int_0^t f_{\theta} (\bm{z}_s) \mathrm{d}s \quad \mathrm{and} \quad \bm{z}_0 = \zeta_\theta(\bm{x}),
\end{equation}
here $\bm{y} \approx \ell_\theta(\bm{z}_T)$ can be utilized to approximate the desired output. In this formula, the solution $\bm{z}_t$ is determined by the initial condition on $\bm{z}_0$ when $\theta$ has been learned. There exists no direct way to modify the trajectory given subsequent observations, let alone tackle structural dynamics in the data generation procedure, making the plain Neural ODEs not suitable for dynamic graph setups.
\noindent\textbf{Neural controlled differential equations (Neural CDEs).}~~Neural controlled differential equations ~\citep{Kidger2020CDE} are the continuous-time analogue to recurrent neural networks and provide a natural method for modeling temporal dynamics with neural networks.
Provide an irregularly sampled time series $\bm{x}=((t_0, \bm{x}_{t_0}), (t_1, \bm{x}_{t_1}), ..., (t_N, \bm{x}_{t_N}))$, with each $t_k \in \mathbb{R}$ the time stamp of the observation $\bm{x}_{t_k} \in \mathbb{R}^v$ and $t_0<\cdot\cdot\cdot<t_N$. Let $X:[t_0, t_N] \rightarrow \mathbb{R}^{v+1}$ be a continuous function of bounded variation with knots at $t_0, ..., t_N$ such that $X_{t_k}=(t_k, \bm{x}_{t_k})$. Let $f_\theta: \mathbb{R}^w \rightarrow \mathbb{R}^{w \times (v+1)}$ and $\zeta_\theta: \mathbb{R}^{v+1} \rightarrow \mathbb{R}^w$ are neural networks depending on their own learnable parameters $\theta$. Then Neural CDEs can be defined as
\begin{equation}
\label{eq:cde}
\bm{z}_t = \bm{z}_{t_0} + \int^t_{t_0} f_{\theta}(\bm{z}_s) \mathrm{d}X_s \quad \mathrm{for}~t\in(t_0,t_N],
\end{equation}
where $\bm{z}_{t_0} = \zeta_\theta(t_0, \bm{x}_{t_0})$ and $\bm{z}_t$ are the solution of the CDE. A key difference from Neural ODEs is the interpolation of observations to form a continuous path $X_s$, thus the dependency upon the time-varying data can be naturally incorporated into the integration process and the trajectory of the system can adapt according to the subsequent observations.
\section{Main Results}
\label{sec:method}
In this section, we first present the embedding learning problem under evolving graphs. Then we introduce our proposed differential model. After that, we provide the applications to several representative graph-related tasks (\textit{e}.\textit{g}.,~node attributes prediction, dynamic node classification, temporal link prediction).
\subsection{Problem Setup}
Consider a dynamic graph generated following an underlying continuous procedure that we only observe a sequence of irregularly sampled graph snapshots $\mathcal{G}=\{(t_0, G_{t_0}),...,(t_N, G_{t_N})\}$, with each $t_k \in \mathbb{R}$ the time stamp of the observed graph $G_{t_k}$ and $t_0<\cdot\cdot\cdot <t_N$. Among these observations, a graph snapshot $G_{t_k}=\{\mathcal{V}, \mathcal{E}\}$ is comprised of nodes $\mathcal{V}=\{v_1,...,v_{|\mathcal{V}|}\}$ and edges $\mathcal{E} \subseteq \mathcal{V}\times \mathcal{V}$ (we assume all snapshots share a common node set and edge set, and omit the subscript for simplicity). Commonly, we can represent the graph topological information for graph $G_{t_k}$ via a time-specified adjacency matrix $\mathbf{A}_{t_k}\in \mathbb{R}^{|\mathcal{V}| \times |\mathcal{V}|}$ that each interaction $e_{ij} \in \mathcal{E}$ is valued in $\mathbf{A}_{t_k}$ where $i,j \in \{1,...,|\mathcal{V}|\}$. Our goal is to learn a non-linear dynamical system on the dynamic graph $\mathcal{G}$ based on the observations, formally the dynamics follow the form:
\begin{equation}
\label{eq:dynamic_graph}
\mathbf{Z}_t = \mathbf{Z}_{t_0} + \int^t_{t_0} f(\mathbf{Z}_s)\mathrm{d}\mathbf{X}_s \quad \mathrm{for}~t\in(t_0,t_N],
\end{equation}
where $\mathbf{Z}_t=\{\bm{z}_t{(v_i)}\}_{i=1}^{|\mathcal{V}|}, \mathbf{Z}_t\in\mathbb{R}^{|\mathcal{V}| \times d}$ is output node embedding matrix, $\mathbf{X}$ is an input signal path defined on $[t_0,t_N]$ which comprises the evolving topology of $\mathcal{G}$. The subscript notation here refers to function evaluation over time. When there exists an assigning function $\tau:\mathbb{R}^d\rightarrow\mathbb{R}^c$ on $\mathcal{G}$ such that $\mathbf{Y}_t=\tau(\mathbf{Z}_t), \mathbf{Y}_t \in \mathbb{R}^{|\mathcal{V}| \times c}$, this becomes by now an increasingly popular problem with applications in several machine learning tasks such as node attribute prediction~\citep{Gao2016Nature}, dynamic node classification~\citep{Kumar2019KDD}, temporal link prediction~\citep{Nguyen2018WWW}~\textit{etc}.~Very recently, some works incorporate graph convolutional networks with ODEs for the continuous inference~\citep{Zang2020KDD, Yan2021arXiv, Choi2022AAAI}, however, they degrade to an oversimplified setup where the neighborhood for nodes remains unchanged over time, making the proposed methods impractical for usage since the structural change could yield an unignorable effect on node embeddings. To make this problem solvable and the designed methods practical, we make the following assumption:
\begin{assumption}[Continuity]
The evolving path $\mathbf{z}:[t_0,t_N]\rightarrow \mathbb{R}^d$ of each node embedding is absolutely continuous.
\end{assumption}
This continuity assumption is standard for enabling differential equations~\citep{Chen2018ODE, Kidger2020CDE} and widely used by current dynamic graph approaches \citep{Zang2020KDD, Wang2020CIKM, Yan2021arXiv, Choi2022AAAI}.
\subsection{GN-CDE: Graph Neural Controlled Differential Equations}
\label{sec:gn_cde}
Before introducing our differential equations, we need to prepare a continuously evolving path for graph structure first. Specifically, for a dynamic graph $G_{t_k}$ collected at time stamp $t_k$ endowed with adjacency matrix $\mathbf{A}_{t_k}$, we augment each interaction $e_{ij}$ in $\mathbf{A}_{t_k}$ by time stamp as $\hat{e}_{ij}=(t_k, e_{ij}) \in \mathbb{R}^2$ where $i,j \in \{1,...,|\mathcal{V}|\}$ such that all these processed interactions can be represented by a time augmented adjacency matrix as $\hat{\mathbf{A}}_{t_k}=(t_k, \mathbf{A}_{t_k}) \in \mathbb{R}^{|\mathcal{V}| \times |\mathcal{V}|\times 2}$. After that, we interpolate each possible interaction among two nodes independently utilizing the discrete observations to form a continuous path, this can be represented as $\hat{\mathbf{A}}:[t_0, t_N] \rightarrow \mathbb{R}^{|\mathcal{V}| \times |\mathcal{V}|\times 2}$ such that $\hat{\mathbf{A}}_{t_k}=(t_k, \mathbf{A}_{t_k})$. {In this paper, we assume $\hat{\mathbf{A}}$ to be piecewise twice continuously differentiable with bounded second derivative, thus many interpolation schemes can be employed~\citep{Morrill2022TMLR}.}
Then, let $\zeta_{\theta}: \mathbb{R}^{|\mathcal{V}| \times |\mathcal{V}|\times 2} \rightarrow \mathbb{R}^{|\mathcal{V}| \times d}$ and $f_{\theta}: \mathbb{R}^{|\mathcal{V}| \times d} \times \mathbb{R}^{|\mathcal{V}| \times |\mathcal{V}|} \rightarrow \mathbb{R}^{|\mathcal{V}| \times d} \times \mathbb{R}^{|\mathcal{V}| \times |\mathcal{V}|\times 2}$ be two graph neural networks. We can define our controlled differential equation for dynamic graphs as
\begin{equation}
\label{eq:gn_cde}
\mathbf{Z}_t = \mathbf{Z}_{t_0} + \int^t_{t_0} f_{\theta}(\mathbf{Z}_s, \mathbf{A}_s)\mathrm{d}\hat{\mathbf{A}}_s \quad \mathrm{for}~t\in(t_0,t_N],
\end{equation}
where $\mathbf{Z}_{t_0}=\zeta_\theta(t_0, \mathbf{A}_{t_0})$ is treated as the initial value to avoid translational invariance. One can utilize another linear function $\ell_\theta$ to acquire the final prediction as $\tilde{\mathbf{Y}}_t=\ell_\theta(\mathbf{Z}_t), \tilde{\mathbf{Y}}_t\in\mathbb{R}^{|\mathcal{V}| \times c}$. The notation ``$f_{\theta}(\mathbf{Z}_s, \mathbf{A}_s)~\mathrm{d} \hat{\mathbf{A}}_s$'' in Eq.~\ref{eq:gn_cde} represents a matrix-matrix product. Our formula differs from the standard Neural CDE presented in~\citet{Kidger2020CDE} in that we highlight the causal effect of graph structural dynamics via the defined path $\hat{\mathbf{A}}$, making it more suitable for the dynamic graphs scenarios, while Neural CDEs concentrate more on sequential relationships across observations rather than the graph structural dynamics. Based on this modification, $\mathbf{A}_s$ in $f_{\theta}(\mathbf{Z}_s, \mathbf{A}_s)$ can strengthen such causal effect for the vector field update, and the derivative $\mathrm{d} \hat{\mathbf{A}}_s$ can indicate the magnitudes and directions of the instantaneous change for the interactions.
Given $f_{\theta}$ implemented as a Graph Neural Network with the formula $f_{\theta}(\mathbf{Z}_s, \mathbf{A}_s)=\sigma(\mathbf{A}_s \mathbf{Z}_s^{(l)} \mathbf{W}^{(l)})$ where $\mathbf{W}^{(l)}$ is parameters for $l$-th layer GCN and $\sigma$ is a rectified linear unit (ReLU) with a Lipschitz constant 1. In practice, we can leverage a regularized adjacency matrix of $\mathbf{A}_s$ to stabilize the algorithm learning~\citep{Welling2016ICLR}. Obviously, $f_{\theta}$ is global Lipschitz continuous, and we have the following theorem
\begin{theorem}
The solution $\mathbf{Z}_t$ in Eq.~\ref{eq:gn_cde} exhibits global existence and uniqueness.
\end{theorem}
The proof is straightforward by using Picard–Lindelöf theorem~\citep{Coddington1955ODE}. We provide some theoretical comparisons of different interpolation schemes for our Graph Neural CDE model in Appendix~\ref{sec:app_inter}.
\noindent\textbf{Evaluating.}~Provided $\hat{\mathbf{A}}$ as piecewise twice continuously differentiable with bounded second derivative, Eq.~\ref{eq:gn_cde} can be rewritten as
\begin{equation}
\label{eq:gn_cde_deri}
\mathbf{Z}_t = \mathbf{Z}_{t_0} + \int^t_{t_0} f_{\theta}(\mathbf{Z}_s, \mathbf{A}_s)\frac{\mathrm{d}\hat{\mathbf{A}}_s}{\mathrm{d}s}\mathrm{d}s \quad \mathrm{for}~t\in(t_0,t_N],
\end{equation}
where $\mathbf{Z}_{t_0}=\zeta_\theta(t_0, \mathbf{A}_{t_0})$. This equation can be interpreted and solved as an ordinary differential equation, and one can solve our proposed GN-CDE model using the same technique for Neural ODEs \citep{Kidger2020CDE}. Algorithm~\ref{alg:inference} depicts the continuous inference procedure of GN-CDE using an ODE solver.
\begin{algorithm2e}[!tbp]
\caption{Continuous inference of GN-CDE algorithm}
\label{alg:inference}
\SetAlgoLined
\DontPrintSemicolon
\KwIn{Sequentially observed topological structures of a dynamic graph $\{(t_0, \mathbf{A}_{t_0}),...,(t_N, \mathbf{A}_{t_N})\}$, initial function $\zeta_\theta$, vector field $f_\theta$ and decoder $\ell_\theta$}
\SetKwBlock{Begin}{Initializing}{end}
\Begin{
$\hat{\mathbf{A}}$: Interpolate the time-augmented adjacency matrix\;
$\mathbf{Z}_{t_0} \leftarrow \zeta_\theta(t_0, \mathbf{A}_{t_0})$;
}
\SetKwBlock{Begin}{Continuously inferring}{end}
\Begin{
$\mathbf{Z}_{t} \leftarrow \textbf{\rm{ODESolve}}(\mathbf{Z}_{t_0}, \hat{\mathbf{A}}, t_0, t_N, f_\theta) $ following Eq.~\ref{eq:gn_cde_deri}\;
$\Tilde{\mathbf{Y}}_t \leftarrow \ell_\theta(\mathbf{Z}_t)$\;
}
\Return $\mathbf{Z}_{t}$, $\Tilde{\mathbf{Y}}_t$
\end{algorithm2e}
\subsection{Properties}
\noindent\emph{Robustness to missing values.}~GN-CDE is capable of processing partially observed data. This is because each channel may independently be interpolated between observations to create $\hat{\mathbf{A}}_s$ in exactly the same manner as before.
\noindent\emph{Expressivity.}~Compared to representing the evolving graph structure via learnable parameters $\mathbf{W}$ within GCN layers~\citep{Yan2018AAAI} or introducing additional neural network layers~\citep{Choi2022AAAI}, our method can maintain the representation capability of GCN layers without introducing additional computational cost.
\noindent\emph{Calibration.}~Providing additional observations at intermediate time steps can help calibrate the inferred trajectory. This property is inherited from controlled differential equations~\citep{Kidger2020CDE}.
\noindent\emph{Memory-efficient.}~Our model can continuously incorporate incoming data without interrupting the differential equation, as a result, memory-efficient adjoint backpropagation may be performed for model training.
\noindent\textbf{Discussion.}~The most similar work to this paper is STG-NCDE~\cite{Choi2022AAAI} which employs Neural CDEs to learn the temporal and spatial dependencies for traffic forecasting tasks. Although this method is capable of tackling an unfixed graph structure via leveraging some learnable parameters to approximate the structure, it induces additional computational burden and cannot be extended to continuously evolving dynamic graphs. On the contrary, we define a generic framework that can incorporate the graph structural dynamics into the integration procedure naturally, thus no need to introduce additional parameters.
Actually, STG-NCDE can be included in our framework presented for dynamic node classification task (See Section~\ref{sec:method_expansion}) by ignoring the dynamic structure and edge features.
\subsection{Comparison to Alternative Models}
In this section, we compare and discuss our framework with two alternatives that also combine dynamic graph structure with differential equations.
\vspace{0.3cm}
\noindent\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {1}}}~Neural ODE
\vspace{0.3cm}
One choice for the alternative of Eq.~\ref{eq:gn_cde} could be directly incorporating the graph structure into the vector field of Neural ODEs presented in Eq.~\ref{eq:ode} and define the graph neural ODE model as
\begin{equation}
\label{eq:gn_cde_abb1}
\mathbf{Z}_t = \mathbf{Z}_{t_0} + \int^t_{t_0} f_{\theta}(\mathbf{Z}_s, \mathbf{A}_{\lfloor s \rfloor})\mathrm{d}s \quad \mathrm{for}~t\in(t_0,t_N],
\end{equation}
where $\mathbf{Z}_{t_0}=\zeta_\theta(t_0, \mathbf{A}_{t_0})$, $\mathbf{A}_{\lfloor s \rfloor}=\mathbf{A}_{t_k}$ if $t_k \leq s < t_{i+1}$.
\begin{theorem}
\label{theo:gn_ode}
Any equation of the form $\mathbf{Z}_t = \mathbf{Z}_{t_0} + \int^t_{t_0} f_{\theta}(\mathbf{Z}_s, \mathbf{A}_{\lfloor s \rfloor})\mathrm{d}s$ may be represented exactly by a Graph Neural Controlled Differential Equation of the form $\mathbf{Z}_t = \mathbf{Z}_{t_0} + \int^t_{t_0} f_{\theta}(\mathbf{Z}_s, \mathbf{A}_s)\mathrm{d}\hat{\mathbf{A}}_s$. However, the converse statement is not true.
\end{theorem}
\begin{proof}
The main proof idea is to build an intermediate differential model to bridge these two formulas. From Theorem~\ref{theo:gn_cde_linear} we know that, given a differential equation in the form $\mathbf{Z}_t = \mathbf{Z}_{t_0} + \int^t_{t_0} f_{\theta}(\mathbf{Z}_s)\mathrm{d}\hat{\mathbf{A}}_s$, it can be represented by $\mathbf{Z}_t = \mathbf{Z}_{t_0} + \int^t_{t_0} f_{\theta}(\mathbf{Z}_s, \mathbf{A}_s)\mathrm{d}\hat{\mathbf{A}}_s$. After that, we only need to proof any equation in the form $\mathbf{Z}_t = \mathbf{Z}_{t_0} + \int^t_{t_0} f_{\theta}(\mathbf{Z}_s, \mathbf{A}_{\lfloor s \rfloor})\mathrm{d}s$ can be represented by $\mathbf{Z}_t = \mathbf{Z}_{t_0} + \int^t_{t_0} f_{\theta}(\mathbf{Z}_s)\mathrm{d}\hat{\mathbf{A}}_s$ which has been proofed by~\cite{Kidger2020CDE}. See Appendix~\ref{sec:app_proof_gn_ode} for more detailed proof.
\end{proof}
According to Theorem~\ref{theo:gn_ode}, although the GN-ODE model can also take the dynamic graph structure into the vector field computational procedure, its representation ability is inferior to our proposed GN-CDE model.
\vspace{0.3cm}
\noindent\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {2}}}~Neural CDE
\vspace{0.3cm}
Another alternative of Eq.~\ref{eq:gn_cde} could be implementing the vector field without $\mathbf{A}_s$ as input following the standard Neural CDEs presented in Eq.~\ref{eq:cde} which emphasizes the linear dependency on $\mathrm{d}\hat{\mathbf{A}}_s$. We formulize this as
\begin{equation}
\label{eq:gn_cde_abb2}
\mathbf{Z}_t = \mathbf{Z}_{t_0} + \int^t_{t_0} f_{\theta}(\mathbf{Z}_s)\mathrm{d}\hat{\mathbf{A}}_s \quad \mathrm{for}~t\in(t_0,t_N],
\end{equation}
where $\mathbf{Z}_{t_0}=\zeta_\theta(t_0, \mathbf{A}_{t_0})$.
\begin{theorem}
\label{theo:gn_cde_linear}
Any equation of the form $\mathbf{Z}_t = \mathbf{Z}_{t_0} + \int^t_{t_0} f_{\theta}(\mathbf{Z}_s)\mathrm{d}\hat{\mathbf{A}}_s$ can be represented exactly by a Graph Neural Controlled Differential Equation of the form $\mathbf{Z}_t = \mathbf{Z}_{t_0} + \int^t_{t_0} f_{\theta}(\mathbf{Z}_s, \mathbf{A}_s)\mathrm{d}\hat{\mathbf{A}}_s$ and vice versa.
\end{theorem}
\begin{proof}
Here we sketch the proof idea. Since the vector field $f_\theta$ in $\mathbf{Z}_t = \mathbf{Z}_{t_0} + \int^t_{t_0} f_{\theta}(\mathbf{Z}_s, \mathbf{A}_s)\mathrm{d}\hat{\mathbf{A}}_s$ takes both the $\mathbf{Z}_s$ and $\mathbf{A}_s$ as input, we can construct a stacked input $\bm\beta_s = \begin{bmatrix} \mathbf{Z}_s \\ \mathbf{A}_s\end{bmatrix}$ for alternative such that the newly built controlled differential equation on $\bm\beta_s$ is equivalent to the Neural CDE formulation presented in Eq.~\ref{eq:gn_cde_abb2}. The full proof of Theorem~\ref{theo:gn_cde_linear} is detailed in Appendix~\ref{sec:app_proof_gn_cde_linear}.
\end{proof}
Although the dynamic graph can be learned by these two forms of CDE according to Theorem~\ref{theo:gn_cde_linear}, they own different preferences during model learning stage. In the experimental part, we find that GN-CDE with non-linear dependency (as shown in Eq.\ref{eq:gn_cde}) performs better compared to the linear dependency variant (Eq.\ref{eq:gn_cde_abb2}). We conjecture that this is due to the fact that explicitly incorporating dynamic graph structure into the vector field allows for more precise control of information flow among nodes over time.
\subsection{Approximation of GN-CDE}
\label{sec:simplification}
Directly implementing Eq.~\ref{eq:gn_cde_deri} by following \citet{Kidger2020CDE} would result in an undesirable computational burden due to the high dimensional output of $f_\theta$. For ease of computation and to ensure scalability to large neural networks, we consider some approximations to simplify this procedure via leveraging the message passing mechanism among graphs and the universal approximation property of graph neural networks. Given the naive graph as homogeneous and isotropic, the derivative on time stamp $s$ can be shared by all interactions, and this enables the alternative between $\mathrm{d}\Hat{\mathbf{A}}_s$ and $\mathrm{d}\mathbf{A}_s \mathrm{d}s$ that the output dimension will be halved. Afterward, we fuse $\mathrm{d}\mathbf{A}_s$ with $\mathbf{A}_s$ in the vector field using a transformation matrix and produce a new adjacency matrix $\Tilde{\mathbf{A}}_s$ that indicates the instantaneous structural change of graph, this yields two advantages: 1) the representations diffusion procedure by learnable parameters can be approximated via an adjusted graph structure; 2) the dimension of output for $f_\theta$ can be largely reduced, from $\mathbb{R}^{|\mathcal{V}| \times d} \times \mathbb{R}^{|\mathcal{V}| \times |\mathcal{V}|\times 2}$ to $\mathbb{R}^{|\mathcal{V}| \times d}$. Formularly, for a vector field parameterised by a $L$-layers graph neural network, the approximated equation that can be implemented much more efficiently as follows
\begin{equation}
\label{eq:gn_cde_imp}
\mathbf{Z}_t = \mathbf{Z}_{t_0} + \int^t_{t_0} \sigma \Big( \Tilde{\mathbf{A}}_s \mathbf{Z}_s^{(L)} \mathbf{W}^{(L)}\Big)\mathrm{d}s \quad \mathrm{for}~t\in(t_0,t_N],
\end{equation}
where $\Tilde{\mathbf{A}}_s=\begin{bmatrix} \mathbf{A}_s \\ \frac{\mathrm{d}\mathbf{A}_s}{\mathrm{d}s} \end{bmatrix} \mathbf{W}^{(DR)}$, $\mathbf{W}^{(DR)} \in \mathbb{R}^{2|\mathcal{V}| \times |\mathcal{V}|}$ is a transformation matrix for the fusion. Besides, $\mathbf{Z}_s^{(L)}$ can be acquired iteratively following the rule: $\mathbf{Z}_s^{(l)} = \sigma\big(\Tilde{\mathbf{A}}_s \mathbf{Z}_s^{(l-1)} \mathbf{W}^{(l-1)}\big)$ for $l\in\{1, ..., L\}$, and $\sigma$ is ReLU activation function.
\begin{theorem}
\label{theo:gn_cde_approx}
Eq.~\ref{eq:gn_cde_imp} is a valid approximation of our Graph Neural Controlled Differential Equation model with the form $\mathbf{Z}_t = \mathbf{Z}_{t_0} + \int^t_{t_0} f_{\theta}(\mathbf{Z}_s, \mathbf{A}_s)\mathrm{d}\hat{\mathbf{A}}_s$.
\end{theorem}
\begin{proof}
The full derivation of Theorem~\ref{theo:gn_cde_approx} is deferred to Appendix~\ref{sec:app_approx}, here we present several key steps. Consider our GN-CDE model presented in Eq.~\ref{eq:gn_cde_deri}, it can be implemented using a L-layers graph neural network directly as
\begin{equation}
\label{eq:gn_cde_direct}
\mathbf{Z}_t = \mathbf{Z}_{t_0} + \int^t_{t_0} \sigma(\mathbf{A}_s \mathbf{Z}_s^{(L)} \mathbf{W}^{(L)}) \mathbf{W}^{(\mathrm{BR})} \frac{\mathrm{d}\mathbf{A}_s}{\mathrm{d} s} \mathrm{d}s,
\end{equation}
where $\mathbf{Z}_s^{(l)} = \sigma(\mathbf{A}_s \mathbf{Z}_s^{(l-1)} \mathbf{W}^{(l-1)}) ~\mathrm{for}~l\in\{1, ..., L\}$. In Eq.~\ref{eq:gn_cde_direct}, the parameter matrices $\mathbf{W}^{(L)}$ and $\mathbf{W}^{(\mathrm{BR})}$ within $\sigma(\mathbf{A}_s \mathbf{Z}_s^{(L)} \mathbf{W}^{(L)}) \mathbf{W}^{(\mathrm{BR})} \frac{\mathrm{d}\mathbf{A}_s}{\mathrm{d} s}$ can be combined and we obtain a new integrand $\sigma(\mathbf{A}_s \mathbf{Z}_s^{(L)} \mathbf{W}^{(\mathrm{LBR})} \frac{\mathrm{d}\mathbf{A}_s}{\mathrm{d} s})$. After that, we introduce a transformation matrix $\mathbf{W}^{(DR)}$ such that
\begin{equation}
\sigma \Big( \begin{bmatrix} \mathbf{A}_s \\ \frac{\mathrm{d}\mathbf{A}_s}{\mathrm{d}s} \end{bmatrix} \mathbf{W}^{(DR)} \mathbf{Z}_s^{(L)} \mathbf{W}^{(L)}\Big) \approx \sigma(\mathbf{A}_s \mathbf{Z}_s^{(L)} \mathbf{W}^{(\mathrm{LBR})} \frac{\mathrm{d}\mathbf{A}_s}{\mathrm{d} s})
\end{equation}
Finally, we utilize the universal approximation theorem \citep{Scarselli2008TNN} to verify that this approximation is valid.
\end{proof}
Besides the theoretical analysis, we also provide an empirical comparison of this simplified version in Section~\ref{sec:experiments} to assess its performance.
\subsection{Applications}
\label{sec:method_expansion}
\noindent\textbf{Node attributes prediction.}~For the node attributes prediction task, we are equipped with a node attributes matrix $\mathbf{F}_{t_k} \in \mathbb{R}^{|\mathcal{V}| \times m}$ containing $m$-dimensional attributes of the nodes for model training. When new edges and nodes emerge at time stamp $t_k$, $\mathbf{A}_{t_k}$ and $\mathbf{F}_{t_k}$ will evolve accordingly under the effects of graph structural dynamics and intrinsic dynamics of nodes. Our task is to predict the node attributes,~\textit{i}.\textit{e}.,~to predict $\mathbf{F}_t$ at unseen time $t$ based on the previous observations. To achieve this goal, we need to learn informative node representations $\mathbf{Z}_t\in\mathbb{R}^{|\mathcal{V}| \times d}$ that can be used for the prediction of the nodes attributes $\mathbf{F}_t$. The objective is to minimize the following expected loss
\begin{equation}
\min_{f_\theta, \ell_\theta}~\mathbb{E}_{t} [\mathrm{Loss}(\mathbf{F}_t, \tilde{\mathbf{Y}}_t)] \quad \mathrm{for}~t\in(t_0,T],
\end{equation}
where $\tilde{\mathbf{Y}}_t=\ell_\theta(\mathbf{Z}_t)$ is the prediction based on $\mathbf{Z}_t$ which is inferred by Eq.~\ref{eq:gn_cde}. We use the squared error to measure the mismatch between $\mathbf{F}_t$ and $\tilde{\mathbf{Y}}_t$. Moreover, $T$ can be $T \leq t_N$ which corresponds to interpolation prediction or $T > t_N$ corresponds to extrapolation prediction.
\noindent\textbf{Dynamic node classification.}~The task of node classification is leveraging the collected information at time stamps $\{t_0, ..., t_N\}$ to predict the label of nodes $\tilde{\mathbf{Y}}_t$ at time stamp $t, t>t_N$. The objective can be represented as
\begin{equation}
\min_{f_\theta, \ell_\theta}~\mathbb{E}_{t} [\mathrm{Loss}(\mathbf{Y}_t, \tilde{\mathbf{Y}}_t)] \quad \mathrm{for}~t\in(t_N,T],
\end{equation}
where $\tilde{\mathbf{Y}}_t=\ell_\theta(\mathbf{Z}_t)$ is the prediction based on $\mathbf{Z}_t$ and $\ell_\theta$ is implemented as a MLP with a softmax activation function to obtain the class probability. We can use the squared loss or cross-entropy to measure the prediction error.
Commonly, this works on attributed graphs, which means the node attribute matrix $\mathbf{F}_{t_k}$ and edge feature matrix $\mathbf{E}$ are provided in advance. Thus, the controlled differential equation can be written as
\begin{equation}
\label{eq:gn_cde_node}
\mathbf{Z}_t = \mathbf{Z}_{t_0} + \int^t_{t_0} f_{\theta}\Big(\mathbf{Z}_s, \mathbf{A}_s, \mathbf{F}_s, \mathbf{E}\Big) \mathrm{d}\hat{\mathbf{A}}_s \quad \mathrm{for}~t\in(t_0,t_N],
\end{equation}
where $\mathbf{Z}_{t_0}=\zeta_\theta(t_0, \mathbf{A}_{t_0}, \mathbf{F}_{t_0}, \mathbf{E})$. In this equation, we need to interpolate the time augmented adjacency matrix as before and the node attribute matrix $\mathbf{F}_{t_k}$ to conduct the integral. In addition, when $\mathbf{F}_{t_k}$ or $\mathbf{E}$ is not given, we can disable the corresponding terms in Eq.~\ref{eq:gn_cde_node}.
\noindent\textbf{Temporal link prediction.}~This task is to predict the existence of an edge $e_{ij}$ at time $t, t>t_N$. The node attribute matrix $\mathbf{F}_{t_k}$ and edge feature matrix $\mathbf{E}$ are also provided, thus we can follow the same setup designed for dynamic node classification tasks. To acquire the link probability among two nodes, we apply a MLP over the concatenation of the corresponding nodes' embeddings.
\iffalse
\begin{table*}[!tbp]
\caption{Quantitative evaluation of prediction accuracy between GN-CDE and other baselines on node attribute prediction tasks. We run each method 10 times and report the average $\ell_1$ loss value with standard deviation. Here, the sum of extrapolation and interpolation results is presented for evaluating the performance among the whole dynamic procedure. Best results are printed in boldface.}
\label{tab:exp_attribute}
\begin{center}
\begin{footnotesize}
\begin{tabular}{ccccccc}
\hline
\toprule[1pt]
Model & Algorithms & Grid & Random & Power Law & Small World & Community \\
\hline
\multirow{7}*{\shortstack{Heat \\ Diffusion}} & Neural ODE & 0.612 $\pm$ 0.131 & 0.784 $\pm$ 0.141 & 0.640 $\pm$ 0.110 & 0.526 $\pm$ 0.054 & 0.699 $\pm$ 0.165 \\
& Neural CDE & 1.028 $\pm$ 0.395 & 1.745 $\pm$ 0.317 & 1.780 $\pm$ 0.619 & 1.052 $\pm$ 0.159 & 1.889 $\pm$ 0.240 \\
& STG-NCDE & 0.861 $\pm$ 0.541 & 1.676 $\pm$ 0.260 & 2.311 $\pm$ 0.450 & 2.491 $\pm$ 0.313 & 1.922 $\pm$ 0.216 \\
& GN-ODE & 1.091 $\pm$ 0.344 & 0.629 $\pm$ 0.116 & 1.154 $\pm$ 0.189 & 1.093 $\pm$ 0.123 & 1.408 $\pm$ 0.091 \\
& GN-CDE-linear & 0.962 $\pm$ 0.306 & 1.601 $\pm$ 0.471 & 1.642 $\pm$ 0.313 & 1.201 $\pm$ 0.179 & 1.857 $\pm$ 0.312 \\
\cdashline{2-7}
& GN-CDE-missing & 0.451 $\pm$ 0.143 & 0.527 $\pm$ 0.128 & 0.685 $\pm$ 0.224 & 0.564 $\pm$ 0.115 & 0.557 $\pm$ 0.118 \\
& GN-CDE & \textbf{0.369} $\bm\pm$ \textbf{0.134} & \textbf{0.521} $\bm\pm$ \textbf{0.202} & \textbf{0.630} $\bm\pm$ \textbf{0.135} & \textbf{0.484} $\bm\pm$ \textbf{0.127} & \textbf{0.457} $\bm\pm$ \textbf{0.112} \\
\midrule[0.6pt]
\multirow{7}*{\shortstack{Gene \\ Regulation}} & Neural ODE & 2.659 $\pm$ 0.804 & \textbf{1.072} $\bm\pm$ \textbf{0.149} & 0.891 $\pm$ 0.079 & 1.528 $\pm$ 0.274 & \textbf{1.301} $\bm\pm$ \textbf{0.101} \\
& Neural CDE & 3.431 $\pm$ 0.894 & 2.464 $\pm$ 3.022 & 1.819 $\pm$ 0.252 & 3.339 $\pm$ 0.715 & 2.864 $\pm$ 1.532 \\
& STG-NCDE & 6.554 $\pm$ 0.621 & 9.285 $\pm$ 0.808 & 3.917 $\pm$ 1.010 & 4.920 $\pm$ 1.247 & 8.278 $\pm$ 2.597 \\
& GN-ODE & 3.153 $\pm$ 0.562 & 3.732 $\pm$ 1.066 & 2.549 $\pm$ 0.226 & 2.252 $\pm$ 0.430 & 4.685 $\pm$ 0.759 \\
& GN-CDE-linear & 2.967 $\pm$ 0.245 & 6.107 $\pm$ 3.202 & 2.764 $\pm$ 0.162 & 2.302 $\pm$ 0.591 & 5.325 $\pm$ 0.500 \\
\cdashline{2-7}
& GN-CDE-missing & 1.489 $\pm$ 0.345 & 2.353 $\pm$ 0.652 & 1.323 $\pm$ 0.684 & 1.534 $\pm$ 0.572 & 2.726 $\pm$ 1.145 \\
& GN-CDE & \textbf{1.388} $\bm\pm$ \textbf{0.262} & 2.193 $\pm$ 0.550 & \textbf{0.886} $\bm\pm$ \textbf{0.072} & \textbf{1.331} $\bm\pm$ \textbf{0.323} & 1.737 $\pm$ 0.260 \\
\bottomrule[1pt]
\end{tabular}
\end{footnotesize}
\end{center}
\end{table*}
\fi
\begin{figure}[htbp]
\centering
\begin{minipage}{\linewidth}
\centering
\includegraphics[width=1.0\linewidth]{./figs/train_curves_heat.pdf}
\vspace{-0.7cm}
\caption{\textbf{Heat Diffusion:}~The test errors of Neural ODE, Neural CDE and our GN-CDE models with respect to the optimization iteration count under five different graph structures: (a) grid, (b) Random, (c) power law, (d) small world and (e) community. We run each method 10 times and report the average $\ell_1$ loss value with standard deviation. Here, the sum of extrapolation and interpolation results is presented for evaluating the performance among the whole dynamic procedure.}
\label{fig:train_curve_heat}
\vspace{0.4cm}
\end{minipage}
\\
\begin{minipage}{\linewidth}
\centering
\includegraphics[width=1.0\linewidth]{./figs/train_curves_gene.pdf}
\vspace{-0.7cm}
\caption{\textbf{Gene Regulation:}~The test errors of Neural ODE, Neural CDE and our GN-CDE models with respect to the optimization iteration count under five different graph structures: (a) grid, (b) Random, (c) power law, (d) small world and (e) community. Here, the sum of extrapolation and interpolation $\ell_1$ errors averaged over 10 runs is presented for performance evaluation.}
\label{fig:train_curve_gene}
\vspace{0.4cm}
\end{minipage}
\\
\begin{minipage}{\linewidth}
\centering
\includegraphics[width=0.68\linewidth]{./figs/node_attribute_colt.pdf}
\vspace{-0.4cm}
\caption{Visualisation of learned dynamics for heat diffusion over dynamic graphs. Our GN-CDE model fits the dynamics for the whole progress accurately.}
\label{fig:node_attribute_viz}
\end{minipage}
\end{figure}
\section{Experimental Results}
\label{sec:experiments}
In this section, we conduct a comprehensive set of experiments on node attribute prediction tasks to validate the effectiveness of our proposed Graph Neural CDE model. More details of dataset construction and experimental setup can be found in the supplementary material.
\noindent\textbf{Data.}~We consider two representative dynamic models: heat diffusion dynamics and gene regulatory dynamics. The underline networks own 400 nodes and are initialized as Grid network, Random network, Power-law network, Small world network, and Community network respectively, then some edges are randomly dropped or added occasionally to simulate the dynamic environments. After that, we irregularly sample 120 snapshots from the continuous-time dynamics to form the whole observations. The standard data splits presented in~\cite{Zang2020KDD} are employed where 80 snapshots for training, 20 snapshots for testing the interpolation prediction task and 20 snapshots for testing the extrapolation prediction task.
\noindent\textbf{Setup.}~
The methods for comparison include Neural ODE (presented in Eq.~\ref{eq:gn_cde_abb1}) and Neural CDE (presented in Eq.~\ref{eq:gn_cde_abb2}). We use the natural cubic spline method as the interpolation scheme for all CDE-based methods since the interpolated path is smooth enough as twice differentiable~\citep{Kidger2020CDE}. Moreover, for a fair comparison, we keep the neural network architecture of the vector field part to be the same for all baselines for different benchmarks. We implement the vector field as one GCN layer with the dimension of output node embeddings $d=20$. Then, all models are trained using Adam optimizer for $2,000$ iterations, and the initial learning rate is set as $0.01$. Finally, $\ell_1$ loss is used as the evaluation metric. We report the results as the average value of 10 runs with the standard deviation shown aside.
\noindent\textbf{Results.}~
The test results of our method and competitive baselines throughout model optimization procedure for the node attribute prediction in heat diffusion and gene regulation dynamical systems are depicted in Fig.~\ref{fig:train_curve_heat} and Fig.~\ref{fig:train_curve_gene}, respectively. We can observe that our GN-CDE model consistently outperforms other baselines, achieving a significantly lower test error across a variety of dynamic networks. In addition, compared to Neural ODE and Neural CDE, GN-CDE spends less time converging to stable results. These results demonstrate the capability and efficiency of our model in modeling the whole dynamics under evolving graph structures no matter for interpolation or for extrapolation tasks. It is worth noting that Neural ODE and Neural CDE exhibit comparable results in most scenarios, with the exception of heat diffusion prediction under grid and random structures, we conjecture the reason that Neural ODE primarily emphasizes on the structural dynamics in the vector field while Neural CDE primarily focuses on the dynamics in differential term, and only taking one aspect into account is insufficient for the representation learning on dynamic graphs.
In Fig.~\ref{fig:node_attribute_viz}, we present a visualization of the ground truth dynamics and the learned dynamics by different methods in a heat diffusion dynamical system evolving with changing grid networks in order to gain insight into the the diverse learning behaviors of these methods. In the first panel of Fig.~\ref{fig:node_attribute_viz}, we can observe several irregularities existing in the node attribute surface of the ground truth, which are attributed to the instantaneous structural changes during energy diffusion to neighboring nodes. However, both Neural ODE and Neural CDE fail to precisely fit the surface, let alone accurately capturing these structural changes. On the contrary, our GN-CDE model demonstrates excellent fitting of the dynamics across the entire time span.
\section{Conclusion}
\label{sec:conclusion}
In this paper, we propose a novel generic differential equations-based framework GN-CDE for representation learning on continuous-time dynamic graphs. GN-CDE creates graph paths into the controlled differential equation that the graph structural dynamics can be naturally incorporated when conduction integration. With this framework, we can apply it to solve different tasks on dynamic graphs with minor modifications. Experimental results on node attribute prediction tasks across different underlying graph structures demonstrate the superiority of our proposed method compared to other baselines.
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\section{Introduction}
\noindent
Determining the strange content of the nucleon has been a long-standing interest of nuclear and particle physicists.
The scalar strange content of the nucleon can be related to kaon-nucleon scattering and phenomenologically to the possible condensation of kaons in dense nuclear environments~\cite{Kaplan:1986yq,Nelson:1987dg}.
The strange content of the nucleon may also play an important role in the scattering of dark-matter particles off nuclei.
The general low-energy coupling of dark matter to nuclei has recently been worked out systematically using low-energy effective field theory~\cite{Fitzpatrick:2012ix,Fitzpatrick:2012ib}.
The spin-independent coupling is the simplest and has hence received the most attention historically.
The spin-independent elastic scattering of dark matter off a nucleon is proportional to the square of the scalar matrix elements $\langle N | m_q \bar{q} q |N \rangle$ for quarks of flavor $q$.~\cite{Bottino:1999ei,Bottino:2001dj,Kaplan:2000hh,Ellis:2008hf,Ellis:2009ai,Giedt:2009mr,Freytsis:2010ne,Hill:2011be,Cheung:2012qy}.
There are no direct experimental means of measuring these matrix elements.
The heavy quark $q=\{c,b,t\}$ matrix elements can be computed from perturbative QCD and are reasonably well known~\cite{Shifman:1978zn,Kryjevski:2003mh}.
The light quark $q=\{u,d\}$ matrix elements can be reasonably determined from $\pi N$ scattering~\cite{Koch:1982pu,Gasser:1990ce,Hoferichter:2012wf}.
The scalar strange-quark matrix element presents the most theoretical challenge to determine reliably and has contributed one of the largest uncertainties in dark-matter detection experiments~\cite{Bottino:1999ei,Bottino:2001dj,Ellis:2008hf} (cancellations between different contributions to potential dark-matter--matter cross sections lead to even larger uncertainty than previously appreciated~\cite{Hill:2011be}).
There have been estimates using baryon chiral perturbation theory and $SU(3)$ symmetry~\cite{Borasoy:1996bx} as well as constraints with earlier lattice calculations~\cite{Frink:2005ru}.
For these reasons, there has been a resurgent interest in determining $m_s \langle N | \bar{s} s | N \rangle$ using lattice QCD, beginning with the work in Refs.~\cite{Young:2009zb,Giedt:2009mr}.
It is more common in the context of dark-matter searches to normalize this quantity by the nucleon mass,
\begin{equation}\label{eq:fs}
f_s = \frac{m_s \langle N | \bar{s} s | N \rangle}{m_N}\, .
\end{equation}
There are two typical approaches used to determine this quantity from lattice QCD.
The scalar strange-quark matrix element can be directly computed
or one can take advantage of the Feynman-Hellmann theorem;
\begin{equation}\label{eq:FH}
m_s\langle N | \bar{s} s | N \rangle = m_s \frac{\partial m_N}{\partial m_s} \, .
\end{equation}
Most groups use the direct method~\cite{Takeda:2010cw,Babich:2010at,Bali:2011ks,Dinter:2012tt,Gong:2012nw,Oksuzian:2012rzb,Engelhardt:2012gd},
one group uses a hybrid approach which involves elements of both methods~\cite{Toussaint:2009pz,Freeman:2012ry},
and some groups use the Feynman-Hellmann method~\cite{Young:2009zb,Durr:2011mp,Horsley:2011wr,Semke:2012gs,Oksuzian:2012rzb,Shanahan:2012wh,Ren:2012aj,Jung:2013rz}.
For a recent review of the scalar strange content of the nucleon, see Ref.~\cite{Young:2013nn}.
The present work utilizes the Feynman-Hellmann theorem which has the following distinct advantages over the direct method: it is numerically less expensive and the ground state contributions to the two-point correlation functions can be significantly more reliably determined than plateaus in direct matrix element calculations with equal computing resources.
We begin by presenting details of our lattice calculation in Sec.~\ref{sec:latt_details}
and then present the determination of $m_s\langle N | \bar{s} s | N \rangle$ in Sec.~\ref{sec:sbars}.
We have found a quantitative comparison of various lattice QCD calculations of this quantity lacking in the literature. Given its important phenomenological role, we were compelled to compile such a comparison, which we provide in Sec.~\ref{sec:results}, along with the results of the present work.
While lattice calculations of $f_s$ still need improvement, there is a welcoming consistency in the determination of this quantity from a wide variety of lattice calculations.
\section{Details of the Lattice Calculation and Numerical Results\label{sec:latt_details}}
\noindent
The present work utilizes mixed-action lattice QCD calculations with domain-wall fermion~\cite{Kaplan:1992bt,Shamir:1992im,Shamir:1993zy,Shamir:1998ww,Furman:1994ky}
propagators computed on the $n_f = 2+1$ asqtad-improved~\cite{Orginos:1998ue,Orginos:1999cr}
rooted, staggered sea-quark configurations generated by the MILC Collaboration~\cite{Bernard:2001av,Bazavov:2009bb},
(with hypercubic-smeared~\cite{Hasenfratz:2001hp,DeGrand:2002vu,DeGrand:2003sf,Durr:2004as}
gauge links to improve the chiral symmetry properties of the domain-wall propagators),
a strategy initiated by the LHP Collaboration~\cite{Renner:2004ck,Edwards:2005kw,Edwards:2005ym,Hagler:2007xi,WalkerLoud:2008bp,Bratt:2010jn}.
A principal motivation for this choice is the good chiral symmetry properties of the domain-wall action, while utilizing the less numerically expensive lattice configurations of the staggered action.
It has been shown that the chiral symmetry properties of the valence domain-wall fermions highly suppresses sources of chiral symmetry breaking from the sea-quark action~\cite{Chen:2005ab,Chen:2006wf,Chen:2007ug,Chen:2009su}.
This has led to a number of important results, including
a determination of the kaon bag parameter $B_K$~\cite{Aubin:2009jh};
the charmed and static baryon spectrum~\cite{Lin:2009rx,Liu:2009jc};
charmed meson interactions with pions and kaons~\cite{Liu:2012zya};
hyperon axial charges~\cite{Lin:2007ap};
a number of results from the NPLQCD Collaboration including two-hadron scattering lengths~\cite{Beane:2005rj,Beane:2007xs,Beane:2006gj,Beane:2007uh,Beane:2006mx,Beane:2006gf,Torok:2009dg};
multi-meson interactions, condensates and the three-pion interaction~\cite{Beane:2007es,Detmold:2008fn,Detmold:2008yn};
as well as a number of others~\cite{Beane:2006pt,Beane:2006fk,Beane:2006kx,Detmold:2008bw}.
There have been a few other choices for mixed actions all utilizing overlap~\cite{Narayanan:1992wx,Narayanan:1994gw} valence-fermions on a variety of sea-quark configurations.
These include Wilson sea-fermions~\cite{Durr:2007ez},
twisted-mass sea-fermions~\cite{Cichy:2010ta,Cichy:2012vg},
domain-wall sea fermions~\cite{Li:2010pw,Lujan:2012wg},
and HISQ sea fermions~\cite{Basak:2012py}.
Mixed-action calculations are inherently unitarity violating with partially quenched effects only vanishing in the continuum limit.
It is therefore imperative to compare numerical results with the scaling violations predicted from the mixed-action effective field theory~\cite{Bar:2002nr,Bar:2003mh,Bar:2005tu,Golterman:2005xa,Tiburzi:2005is,Chen:2005ab,Prelovsek:2005rf,Aubin:2006hg,Chen:2006wf,Jiang:2007sn,Orginos:2007tw,Chen:2007ug,Aubin:2008wk,Chen:2009su}.
This has been undertaken to an exploratory extent with baryons~\cite{WalkerLoud:2008bp,Jenkins:2009wv}, but the only systematic studies have been with the $a_0$ correlator~\cite{Prelovsek:2005rf,Aubin:2008wk}, which is highly contaminated by the unitarity violating effects and a recent determination of low-energy constants in the two-flavor chiral Lagrangian for pions~\cite{Beane:2011zm}.
Despite the limited study of discretization effects, there are reasons to believe they are small for many quantities~\cite{Chen:2005ab,Chen:2006wf,Chen:2007ug,WalkerLoud:2008bp,WalkerLoud:2008pj}.
\subsection{Parameters of the lattice QCD calculation}
\noindent
The present calculation utilizes the Feynman-Hellmann theorem to determine the scalar strange-quark matrix element in the nucleon, Eq.~\eqref{eq:FH},
limiting the work to a small set of available ensembles.
Details of the various ensembles and parameters are collected in Table~\ref{tab:latt_params}.
\begin{table}[t]
\caption{\label{tab:latt_params}
Parameters used in the present work. For some of the calculations, the time direction was chopped at $t=32$ with Dirichlet boundary conditions (denoted by volumes with $\times32$). For the MILC configurations, the notation m010m030 (and similar) means the input quark mass values are $bm_l=0.010$ and $bm_s = 0.030$ for the light and strange sea quarks respectively.
}
\begin{ruledtabular}
\begin{tabular}{cccccccccc}
$\beta$& $m_{sea}$& V& $M_5$& $L_5$
& $bm_l^{dwf}$& $bm_l^{res}$& $bm_s^{dwf}$& $bm_s^{res}$
& $N_{src}\times N_{cfg}$\\
\hline
\multicolumn{10}{c}{$b \approx 0.125~\texttt{fm}$ ensembles}\\
6.75& m010m030& $20^3\times64$& $1.7$& $16$& 0.0138& 0.001564(03)& 0.081& 0.000892(2)& $53\times 328$\\
6.76& m010m050& $20^3\times64$& $1.7$& $16$& 0.0138& 0.001566(11)& 0.081& 0.000913(2)& $\phantom{0}4\times 656$\\
6.76& m010m050& $20^3\times32$& $1.7$& $16$& 0.0138& 0.001552(27)& 0.081& 0.000913(2)& $24\times 769$\\
6.79& m030m030& $20^3\times64$& $1.7$& $16$& 0.0478& 0.001052(04)& 0.081& 0.000809(4)&
$30\times367$\\
6.81& m030m050& $20^3\times32$& $1.7$& $16$& 0.0478& 0.001013(06)& 0.081& 0.000862(7)& $24\times 564$\\
\hline
\multicolumn{10}{c}{$b \approx 0.09~\texttt{fm}$ ensembles}\\
7.08& m0031m0186& $40^3\times96$& $1.5$& $12$& 0.0035& 0.000431(3)& 0.0423& 0.000236(2)& $1\times 356$ \\
7.08& m0031m031& $40^3\times96$& $1.5$& $12$& 0.0035& 0.000428(3)& 0.0423& 0.000233(2)& $1\times 422$\\
\end{tabular}
\end{ruledtabular}
\end{table}
There are two sets of ensembles at the $b\approx0.125$~fm lattice spacing with fixed light-quark mass and strange-quark masses that straddle the physical strange-quark mass. These are denoted by the sets $m_{sea} = \{\textrm{m010m030, m010m050}\}$ and $m_{sea} = \{\textrm{m030m030, m030m050}\}$, respectively.%
\footnote{The notation $m_{sea} =\textrm{m010m030}$ means the light quark has an input light quark mass value in lattice units of $bm_l = 0.010$ and the strange-quark input mass value is $bm_s = 0.030$.}
On the $b\approx0.09$~fm ensembles there are two sets, with fixed light quark mass and strange-quark masses straddling the physical strange-quark mass. In this work, preliminary results are presented only for one of these sets with $m_{sea} = \{\textrm{m0031m0186, m0031m031}\}$.
The values of the domain-wall quark masses, the fifth-dimensional extent $L_5$, and the domain-wall mass $M_5$ were taken from the NPLQCD production runs~\cite{Beane:2011zm}.
\subsection{Results of the lattice calculation}
\noindent
The light- and strange-quark propagators were computed with a Gaussian-smeared source~\cite{Frommer:1995ik,Pochinsky:1997} and both smeared (SS) and point (PS) sinks.
Correlation functions were then constructed with the quantum numbers of the pion and proton.
The pion masses were determined with a fully correlated simultaneous fit to the SS and PS correlation functions, with a single cosh used for both correlators,
\begin{align}
C_{(XS)}(t) \sim A_{(XS)}\ e^{-m_\pi T/2}\ \cosh ( m_\pi (t - T/2))\, ,
\end{align}
where $X=S,P$. In all cases, the configurations are binned until the statistical uncertainty of the extracted masses stopped changing appreciably.
To determine the fitting systematic, the length of the time extent used in the fit and the starting time were varied over a wide range, with a minimum plateau length of $\sim0.5$~fm.
For each fit, the $Q$ value is used as a weight, where
\begin{equation}\label{eq:Q}
Q \equiv \int_{\chi^2_{min}}^{\infty} d \chi^2\ \mc{P}(\chi^2,d)\, ,
\end{equation}
with the probability distribution function for $\chi^2$ with $d$ degrees of freedom,
\begin{equation}
\mc{P}(\chi^2,d) = \frac{1}{2^{d/2} {\Gamma}(d/2)} (\chi^2)^{d/2 - 1} e^{-\chi^2 / 2}\, .
\end{equation}
The central value is determined from the weighted sum,
\begin{equation}
\bar{m} = \frac{\sum_i m_i Q_i}{\sum_j Q_j}\, .
\end{equation}
In many cases, the systematic is approximately Gaussian, and so the 16\% and 84\% quantiles are used to determine the systematic uncertainties.
The choice to use the $Q$ values as weights is simply motivated. $Q$ ranges from $[0,1]$ with a value of $1$ indicating the fit function and resulting parameters perfectly describe the correlation function over the range of fit.
It also allows one to compare fits with different model functions (e.g. single and double state fits). While not the only choice for determining a fitting systematic, it is a convenient and useful choice.
The results of these fits are plotted over a representative window in time along with cosh-style effective masses,
\begin{equation}
m^{\cosh}_{eff}(t,{\tau}) = \frac{1}{{\tau}} \cosh^{-1} \left(
\frac{C(t+{\tau}) + C(t-{\tau})}{C(t)} \right)
\end{equation}
in Figs.~\ref{fig:mpi_coarse} and \ref{fig:mpi_fine}. The (black) squares are from the PS correlation functions while the (colored) open circles are from the SS correlation functions.
\begin{figure}
\includegraphics[width=0.8\textwidth]{figures/pi_m010m030_20x64}
\includegraphics[width=0.8\textwidth]{figures/pi_m010m050_20x64}
\includegraphics[width=0.8\textwidth]{figures/pi_m030m030_20x64}
\includegraphics[width=0.8\textwidth]{figures/pi_m030m050_20x32}
\caption{\label{fig:mpi_coarse}Pion mass effective mass plots on the $b\approx0.125$~fm ensembles.}
\end{figure}
\begin{figure}
\includegraphics[width=0.8\textwidth]{figures/pi_m0031m031_LS12}
\includegraphics[width=0.8\textwidth]{figures/pi_m0031m031_ml0035}
\caption{\label{fig:mpi_fine}Pion mass effective mass plots on the $b\approx0.09$~fm ensembles.}
\end{figure}
The right-side bar in each plot displays the mass probability distribution function determined from
\begin{equation}
\mc{P}_i(m) = \frac{Q_i}{\sum_j Q_j}\, .
\end{equation}
In all plots, the inner (colored) band represents the statistical uncertainty in the fit while the outer band represents the statistical and fitting systematic added in quadrature.
In the case of the $m_{sea} = \textrm{m030m050}$ ensemble, only results with Dirichlet boundary conditions in time are available. For this case, the correlation functions are fit simultaneously with a single exponential,
\begin{align}\label{eq:corr_exp}
C_{(XS)}(t) \sim A_{(XS)}\ \exp ( - m_\pi t)\, ,
\end{align}
and compared with the standard effective mass,
\begin{equation}
m^{\ln}_{eff}(t,{\tau}) = \frac{1}{{\tau}} \ln \left( \frac{C(t)}{C(t+{\tau})} \right)\, .
\end{equation}
Clearly, in this case, the ability to explore the fitting systematic is more limited. For reasons discussed in Ref.~\cite{Beane:2011zm}, the dip in the effective mass is not believed to represent a lower ground state energy, but rather contaminations from the Dirichlet boundary condition.
The results are collected in Table~\ref{tab:mass_results}.
\begin{table}
\caption{\label{tab:mass_results}Computed pion and nucleon masses on the various ensembles. Additionally, the value of $r_1/b$ used to convert to physical units is provided, obtained from Refs.~\cite{Bazavov:2009bb,Bernard:r1b}.}
\begin{ruledtabular}
\begin{tabular}{cccccc}
$\beta$& $m_{sea}$& V& $bm_\pi$& $bm_N$& $\frac{r_1}{b}(bm_l^\textrm{phy},bm_s^\textrm{phy},\beta)$ \\
\hline
6.75& m010m030& $20^3\times64\times16$& $0.22178(33)({}^{54}_{28})$& $0.7177(18)({}^{19}_{26})$& 2.711(4) \\
6.76& m010m050& $20^3\times64\times16$& $0.22285(28)({}^{46}_{37})$& -- & 2.739(3) \\
6.76& m010m050& $20^3\times32\times16$& -- & $0.7311(19)({}^{36}_{26})$& 2.739(3) \\
6.79& m030m030& $20^3\times64\times16$& $0.37323(27)(20)$& $0.8653(17)({}^{27}_{33})$& 2.821(7) \\
6.81& m030m050& $20^3\times32\times16$& $0.37493(26)({}^{24}_{11})$& $0.8740(18)({}^{36}_{32})$& 2.877(4)\\
7.06& m0031m0186& $40^3\times96\times12$& $0.10192(38)({}^{59}_{55})$& $0.4621(64)({}^{99}_{85})$& 3.687(4)\\
7.08& m0031m031& $40^3\times96\times12$& $0.10165(35)({}^{84}_{76})$& $0.4603(48)({}^{79}_{74})$& 3.755(4)
\end{tabular}
\end{ruledtabular}
\end{table}
The proton masses are trickier to determine as the signal-to-noise ratio decays exponentially in time~\cite{Lepage:1989hd},
\begin{equation}
\lim_{t\rightarrow \infty} \frac{S(t)}{N(t)} = A\, e^{-(m_N - \frac{3}{2}m_\pi)t}\, .
\end{equation}
The mass determined in a given fit from $t_i$ to $t_f$ is then susceptible to larger fitting systematics.
It is not uncommon for the effective mass plateau to shift by order one standard deviation and form a new plateau, either higher or lower at times when the statistical fluctuations grow appreciably.
It is therefore important to develop a systematic analysis algorithm that both takes advantage of the precise statistical fluctuations at early times while allowing for the possibility that the late-time fluctuations represent the true ground state.
In Ref.~\cite{Beane:2009kya}, it was demonstrated that correlation functions determined with $\mc{O}(10^5)$ reasonably statistically independent sources on $\mc{O}(10^4)$ Monte Carlo trajectories, a variety of analysis methods could be used all producing consistent results.
With fewer \textit{measurements}, not all methods work as well.
One technique which works better than others is the Matrix-Prony method~\cite{Prony} (similar to the variational method which has gained popularity lately), as described in Refs.~\cite{Beane:2009kya,Beane:2009gs}.
The general idea is to find linear combinations of correlation functions which isolate various eigenstates and allow for a determination of the masses starting from earlier Euclidean times.
The Matrix-Prony method is well suited to matrices of correlation functions that are neither square nor positive-definite, as is often the case in lattice QCD calculations.
One begins with the \textit{ansatz} that the (vector) of correlation functions can be described with a transfer matrix,
\begin{equation}\label{eq:MP_ansatz}
y(t+{\tau}) = \hat{T}({\tau}) y(t)\, ,
\end{equation}
where in our case $y(t)$ is composed of just two correlation functions,
\begin{equation}
y(t) = \begin{pmatrix} C_{PS}(t)\\ C_{SS}(t) \end{pmatrix}\, .
\end{equation}
It is useful to factorize the transfer operator $\hat{T}({\tau}) = M^{-1}({\tau}) V$ and multiply on the right by the transpose vector to form the matrix equation,
\begin{equation}\label{eq:MP_2}
M({\tau}) y(t+{\tau})y^T(t) = V y(t) y^T(t)\, .
\end{equation}
To be useful, Eq.~\eqref{eq:MP_ansatz} must be satisfied over a range of time,
\begin{equation}\label{eq:MP_3}
M({\tau}) \sum_{t=t_0}^{t_0+{\Delta} t} y(t+{\tau})y^T(t) = V \sum_{t=t_0}^{t_0+{\Delta} t} y(t) y^T(t)\, .
\end{equation}
A solution to Eq.~\eqref{eq:MP_3} is given by
\begin{align}
&M({\tau}) = \left( \sum_{t=t_0}^{t_0+{\Delta} t} y(t+{\tau}) y^T(t) \right)^{-1}\, ,&
&V = \left( \sum_{t=t_0}^{t_0+{\Delta} t} y(t) y^T(t) \right)^{-1}\, .&
\end{align}
In order to guarantee the inverse can be found, enough times must be summed over to ensure the corresponding matrices are of full rank.
One then solves the eigenvalue equation for the principal correlators,
\begin{align}
&\hat{T}({\tau}) q_n = ({\lambda}_n)^{\tau} q_n\, ,&
&\textrm{with } {\lambda}_n = e^{-E_n}\, .&
\end{align}
A point that differentiates the Matrix-Prony method from other variational methods is the sum over time slices in Eq.~\eqref{eq:MP_3}.
Most variational methods pick a reference time at which to perform the diagonalization of the correlation functions, whereas with Matrix-Prony, one must sum over a number of time slices greater than or equal to the number of correlation functions.
Moreover, one can increase confidence in the subsequent analysis by maximizing ${\Delta} t$ in Eq.~\eqref{eq:MP_3}.
The original ansatz \eqref{eq:MP_ansatz} is satisfied if over the range of time, $t_0$ to $t_0+{\Delta} t$, the resulting principal correlation functions are well described by a single exponential.
\begin{figure}
\includegraphics[width=0.8\textwidth]{figures/prot_m010m030_20x64}
\includegraphics[width=0.8\textwidth]{figures/prot_m010m050_20x32}
\includegraphics[width=0.8\textwidth]{figures/prot_m030m030_20x64}
\includegraphics[width=0.8\textwidth]{figures/prot_m030m050_20x32}
\caption{\label{fig:mp_coarse}Proton mass and representative effective mass plots on the $b\approx0.125$~fm ensembles.}
\end{figure}
\begin{figure}
\includegraphics[width=0.8\textwidth]{figures/prot_m0031m0186}
\includegraphics[width=0.8\textwidth]{figures/prot_m0031m031}
\caption{\label{fig:mp_fine}Proton mass and representative effective mass plots on the $b\approx0.09$~fm ensembles.}
\end{figure}
In this work, to determine the fitting systematic, the choices of $t_0$ and ${\Delta} t$ are varied over a wide range, with ${\Delta} t \gtrsim 0.5$~fm. For each choice, the ground state principal correlation function is fit with a single exponential, Eq.~\eqref{eq:corr_exp}, over ranges of time $t_i - t_f$, chosen independently of $t_0$ and ${\Delta} t$.
The initial and final times in the fit are also varied over a wide range under the constraint $t_f - t_i \gtrsim 0.5$~fm.
For each fit, the $Q$ value is recorded along with the statistical uncertainty of the fit. The various fits are then averaged with the weight similar to that of the pions, but also suppressed by the statistical uncertainty of the fit;
\begin{align}
&\bar{m} = \frac{\sum_i m_i w_i}{\sum_j w_j}&
&\textrm{with } w_i = \frac{Q_i}{{\sigma}_i}\, .&
\end{align}
In this way, the plateaus at later times, with larger uncertainties, and hence larger $Q$ values, do not dominate the determination of the fitting systematic.
The resulting fits are displayed along with effective mass plots of representative Matrix-Prony determinations of the ground state principal correlation function in Figs.~\ref{fig:mp_coarse} and \ref{fig:mp_fine}.
In these figures, the colored effective mass points correspond to the time window over which the Matrix-Prony method is applied in the representative choice of times $t_0$ and ${\Delta} t$, while the gray effective mass points lie outside this region.
As is evident, the resulting systematic mass-probability distribution tends not to be Gaussian. For simplicity, we still take the 16\% and 84\% quantiles to define the systematic uncertainty. The inner colored bands represent the statistical uncertainty, and the outer gray bands represent the statistical and systematic uncertainties added in quadrature.
\subsection{Scale setting}
\noindent
To convert from lattice units to physical units we use the scale setting procedure described in Ref.~\cite{Beane:2011zm}.
The dimensionless lattice results are converted into $r_1$ units with $\frac{r_1}{b}(bm_l,bm_s,\beta)$ determined by the MILC Collaboration on each ensemble.
But importantly, it is not the value computed on a given ensemble that is used; it is rather the values that have been extrapolated to the physical light- and strange-quark mass point, $\frac{r_1}{b}(bm_l^\textrm{phy},bm_s^\textrm{phy},\beta)$, which have also been determined by the MILC Collaboration~\cite{Bazavov:2009bb,Bernard:r1b}, listed here in Table~\ref{tab:mass_results}.
While depending upon reference quark mass values, this amounts to a quark-mass independent scale setting procedure, such that all remaining light- and strange-quark mass dependence of the computed observables is that of interest.
The MILC Collaboration has also determined the physical value of $r_1$,
\begin{equation}\label{eq:r1_phys}
r_1^\textrm{phy} = 0.31174(20)~\textrm{fm}\, ,
\end{equation}
which is used to then convert all values into physical units, Table~\ref{tab:mass_results_mev}.
\begin{table}
\caption{\label{tab:mass_results_mev}Computed masses and decay constants converted to MeV with $r_1 = 0.31174(20)$~fm.}
\begin{ruledtabular}
\begin{tabular}{ccccc}
$\beta$& $m_{sea}$& V& $m_\pi$~[MeV]& $m_N$~[MeV] \\
\hline
6.75& m010m030& $20^3\times64\times16$& $380.5(.6)({}^{.9}_{.5})$& $1231(3)({}^{3}_{4})$ \\
6.76& m010m050& $20^3\times64\times16$& $386.3(.5)({}^{.8}_{.6})$& -- \\
6.76& m010m050& $20^3\times32\times16$& -- & $1267(3)({}^{6}_{5})$ \\
6.79& m030m030& $20^3\times64\times16$& $666.4(.5)(.4)$& $1545(3)({}^{5}_{6})$ \\
6.81& m030m050& $20^3\times32\times16$& $682.7(.5)({}^{.4}_{.2})$& $1591(3)({}^{7}_{6})$\\
7.06& m0031m0186& $40^3\times96\times12$& $237.8(0.9)(1.3)$& $1078(15)({}^{26}_{22})$\\
7.08& m0031m031& $40^3\times96\times12$& $241.6(0.8)({}^{2.0}_{1.8})$& $1094(11)({}^{19}_{18})$\\
\end{tabular}
\end{ruledtabular}
\end{table}
There is an important additional advantage to this method of scale setting. To invoke the Feynman-Hellmann theorem, the change in the nucleon mass with respect to a change in the strange-quark mass must be undertaken with all other parameters held fixed~\cite{Toussaint:2009pz,Freeman:2012ry}.
The MILC Collaboration chose to make slight changes in the coupling $\beta$ while changing the light quark masses.
Our scale setting procedure allows us to asses the quantitative significance of the slightly different values of ${\beta}$ used on the pairs of ensembles, \{m010m030,m010m050\}, \{m030m030,m030m050\} and \{m0031m0186,m0031m031\}.
For each pair, the relative difference in the values of ${\beta}$ was less than 1\% ($\frac{{\beta}^{(2)}-{\beta}^{(1)}}{{\beta}^{(2)}+{\beta}^{(1)}} < 0.01$) and the corresponding relative difference in the values of $\frac{r_1}{b}(bm_l^\textrm{phy},bm_s^\textrm{phy},\beta^{(i)})$ are also less than 1\%.
While strictly speaking, the change in $m_s$ was not undertaken with all other parameters held fixed, the effect of this change is contained well within the other uncertainties on the determined values of $m_s \langle N| \bar{s} s | N \rangle$, as detailed in the next section.
\section{The Strange Scalar Matrix Element in the Nucleon\label{sec:sbars}}
\noindent
As discussed in the Introduction, there are a few methods for determining the scalar strange-quark matrix element in the nucleon. These include a direct calculation of the matrix element employed by some groups~\cite{Babich:2010at,Takeda:2010cw,Bali:2011ks,Dinter:2012tt,Gong:2012nw,Oksuzian:2012rzb,Engelhardt:2012gd}, an indirect determination through the Feynman-Hellmann theorem~\cite{Young:2009zb,Durr:2011mp,Horsley:2011wr,Semke:2012gs,Shanahan:2012wh,Ren:2012aj,Jung:2013rz}, Eq.~\eqref{eq:FH
\footnote{The first attempt to determine the strange content of the nucleon from lattice QCD with the Feynman-Hellmann method utilized $SU(3)$ baryon $\chi$PT analysis of $b\approx0.125$~fm MILC results~\cite{Frink:2005ru} resulting in a value consistent with zero.},
and a hybrid approach~\cite{Toussaint:2009pz,Freeman:2012ry}.
This work utilizes the Feynman-Hellmann method.
For each light quark mass ensemble, we have a determination of the nucleon mass at values of the strange-quark mass which straddle the physical strange-quark mass.
These results, Table~\ref{tab:mass_results_mev}, can be used to interpolate to the physical value of the strange-quark mass, Taylor expanding about $bm_s^\textrm{phy}$, and determine the two quantities
\begin{align}
&m_N(m_s^\textrm{phy})\, ,&
&\frac{\partial m_N(m_s)}{\partial m_s} \Big|_{m_s^\textrm{phy}}\, .&
\end{align}
\begin{table}
\caption{\label{tab:ms_sbars}Extracted values of $m_N(m_s^\textrm{phy})$ and $m_s^\textrm{phy} \langle N | \bar{s} s | N \rangle$. The first uncertainty is statistical, the second fitting systematics and the third is from the uncertainty on the determination of $m_s^\textrm{phy}$.}
\begin{ruledtabular}
\begin{tabular}{ccc}
$m_\pi$~[MeV]& $m_N(m_s^\textrm{phy})$~[MeV]& $m_s^\textrm{phy} \langle N | \bar{s} s | N \rangle$~[MeV] \\
\hline
$383.4(.6)({}^{.9}_{.6})$& $1241(2)(3)(1)$& 62(8)(11)(1) \\
$674.6(.5)(.4)$& $1556(2)(4)(2)$& $79(8)(13)(2)$ \\
\hline
$240(1)(2)$& $1090(11)(17)(1)$& $50(40)(65)(1)$
\end{tabular}
\end{ruledtabular}
\end{table}
To apply the Feynman-Hellmann theorem with all parameters except $m_s$ held (approximately) fixed, the following approximation for the derivative is used,
\begin{equation}\label{eq:mssbars_mev}
m_s \langle N| \bar{s} s | N \rangle [\textrm{MeV}] =
\frac{ \frac{r_1}{b}^{(2)} bm_N^{(2)} - \frac{r_1}{b}^{(1)} bm_N^{(1)}}
{\frac{r_1}{b}^{(2)} bm_s^{(2)} - \frac{r_1}{b}^{(1)} bm_s^{(1)}}
\times
\frac{\frac{r_1}{b}^{(2)} + \frac{r_1}{b}^{(1)}}{2}
b m_s^\textrm{phy}
\times
\frac{197.3 \textrm{ MeV fm}}{r_1^\textrm{phy}[\textrm{fm}]}\, ,
\end{equation}
where $\frac{r_1^{(i)}}{b}$ denotes the value of $\frac{r_1}{b}(bm_l^\textrm{phy},bm_s^\textrm{phy},\beta^{(i)})$ for the given ensemble with all parameters except $bm_s$ held approximately fixed and $r_1^\textrm{phy}[\textrm{fm}]$ is taken from Eq.~\eqref{eq:r1_phys}.
The MILC Collaboration has determined values of the strange-quark mass to be $bm_s^\textrm{phy} = 0.0350(7)$ and $bm_s^\textrm{phy} = 0.0261(5)$ on the $b\approx0.125$~fm and $b\approx0.09$~fm ensembles respectively~\cite{Aubin:2004ck,Bazavov:2009bb}.
The resulting values of $m_N(m_s^\textrm{phy})$ and $m_s^\textrm{phy} \langle N | \bar{s} s | N \rangle$ are collected in Table~\ref{tab:ms_sbars} and the resulting interpolations are displayed in Fig.~\ref{fig:mn_vs_ms}.
\begin{figure}
\includegraphics[width=0.8\textwidth]{figures/mN_vs_ms_ms030}
\includegraphics[width=0.8\textwidth]{figures/mN_vs_ms_ms050}
\includegraphics[width=0.8\textwidth]{figures/mN_vs_ms_fine}
\caption{\label{fig:mn_vs_ms}Nucleon mass versus the strange-quark mass on the $b\approx0.125$~fm and $b\approx0.09$~fm ensembles. The vertical dashed lines represent the 68\% confidence interval for the determination of $bm_s^\textrm{phy}$ on the $b\approx0.125$~fm and $b\approx0.09$~fm ensembles. The conversion to $r_1$ units is performed as in Eq.~\eqref{eq:mssbars_mev} using $\frac{1}{2}(\frac{r_1}{b}^{(1)} + \frac{r_1}{b}^{(2)})$ for each pair of ensembles.}
\end{figure}
In these figures, the vertical dashed lines represent the 68\% confidence interval for the determination of $bm_s^\textrm{phy}$ on the $b\approx0.125$~fm and $b\approx0.09$~fm ensembles.
The uncertainty on $bm_s^\textrm{phy}$ is included in the analysis and represented by the third uncertainty in Table~\ref{tab:ms_sbars}.
The conversion to $r_1$ units is performed as in Eq.~\eqref{eq:mssbars_mev} using $\frac{1}{2}(\frac{r_1}{b}^{(1)} + \frac{r_1}{b}^{(2)})$ for each pair of ensembles.
The estimated correction due to the difference in ${\beta}$ on the pairs of ensembles is at the same level as the uncertainty arising from the determination of $bm_s^\textrm{phy}$, which are at least an order of magnitude smaller than the statistical or other systematic uncertainties.
On the $b\approx0.125$~fm ensembles, a precise determination of the scalar matrix element is obtained. However, on the $b\approx0.09$~fm ensembles, the results are too imprecise to determine a nonzero value.
\subsection{Chiral extrapolation}
\noindent
The results for $m_s <N|\bar{s}s|N>$ must be extrapolated to the physical value of the pion mass.
In Ref.~\cite{Chen:2002bz}, the two-flavor extrapolation formula for this matrix element was determined at next-to-leading order (NLO) in the chiral expansion,
\begin{equation}
\langle N | \bar{s} s | N \rangle =
<N|\bar{s}s|N>^0
-\frac{g_{\pi N{\Delta}}^2}{4\pi^2 f^2} \left( <N|\bar{s}s|N>^0 - <{\Delta}|\bar{s}s|{\Delta}>^0 \right)
\mc{J}_{m_\pi}^{\Delta}
+\tilde{E}_s \frac{m_\pi^2}{8\pi^2 f^2}\, ,
\end{equation}
where $\langle H | \bar{s} s | H \rangle^0$ represent the leading-order (LO) contribution to the scalar strange matrix element in the hadron $H$, $g_{\pi N{\Delta}}$ is the axial pion-nucleon-delta coupling appearing in the $SU(2)$ baryon chiral Lagrangian, $\mc{J}_{m_\pi}^{\Delta}$ is a chiral loop function nonanalytic in the pion mass and the delta-nucleon mass splitting (${\Delta}=m_{\Delta} - m_N$) and $\tilde{E}_s$ is a low-energy constant appearing at NLO.
In the large-$N_c$ expansion, the LO matrix elements for the nucleon and the delta are both $\mc{O}(N_c^{-1})$, but there is no cancellation at this order~\cite{Jenkins:1995gc}, so one does not expect a strong cancellation between these NLO contributions
\footnote{See also Ref.~\cite{Cherman:2012eg} for further discussion on the baryon masses in the large $N_c$ counting.}
In principle, one should use the partially quenched formula, also provided in Ref.~\cite{Chen:2002bz}, and convert it to the relevant mixed-action formula~\cite{Chen:2007ug} to perform the extrapolation.
However, clearly the most significant shortcoming of the present work is the limited number of light quark mass points.
With nonzero results at only a single lattice spacing, the mixed-action extrapolation cannot be performed regardless.
The best that can be done with the present results is a simple, effectively zero degree of freedom extrapolation using the formula,
\begin{equation}\label{eq:sbars_simple}
m_s \langle N | \bar{s} s | N \rangle = c_0 + c_2 m_\pi^2\, .
\end{equation}
While this will not result in a precise and accurate determination of the scalar strange matrix element, it will provide a good guide to the approximate value at the physical point.
While not a rigorous expectation, it has been found that matrix elements of the nucleon tend to have very mild pion mass dependence; see for example the recent review~\cite{Lin:2012ev}.
Performing this simplistic pion mass extrapolation, using the isospin averaged $m_\pi^\textrm{phy} = 138.0$~MeV, we obtain
\begin{equation}
m_s^\textrm{phy} \langle N | \bar{s} s | N \rangle \Big|_{m_\pi^\textrm{phy}} =
54 \pm 11 \pm 17 \textrm{ MeV}\, .
\end{equation}
The extrapolation is displayed in Fig.~\ref{fig:ms_sbars_v_mpisq}.
\begin{figure}
\includegraphics[width=0.8\textwidth]{figures/ms_sbars_vs_mpisq}
\caption{\label{fig:ms_sbars_v_mpisq}Light quark extrapolation of $ms\langle N| \bar{s} s | N \rangle$ versus $m_\pi^2$. The location of the vertical dashed line is given by $(m_\pi^\textrm{phy})^2$.}
\end{figure}
Given the limited ability to perform the chiral extrapolation, we also explore the light quark mass dependence of $f_s = m_s \langle N | \bar{s} s | N \rangle / m_N$ to improve the estimate of systematic uncertainties.
It has been observed that the nucleon mass displays a remarkably linear dependence on the pion mass~\cite{WalkerLoud:2008bp,WalkerLoud:2008pj}.
For this reason, the following two extrapolation functions are used to estimate extrapolation systematics:
\begin{subequations}
\begin{align}
f_s &= f_s^{(0)} + f_s^{(2)} m_\pi^2\, ,\label{eq:f_s_mpisq}\\
f_s &= f_s^{(0)} + f_s^{(1)} m_\pi\, ,\label{eq:f_s_mpi}
\end{align}
\end{subequations}
yielding the results
\begin{subequations}
\begin{align}
f_s &= 0.049 \pm 0.009 \pm 0.013\, ,\\
f_s &= 0.049 \pm 0.012 \pm 0.018\, ,
\end{align}
\end{subequations}
respectively.
These extrapolations are displayed in Fig.~\ref{fig:fs_extrap}. The quantity $f_s$ is observed to have negligible light quark mass dependence.
\begin{figure}
\includegraphics[width=0.48\textwidth]{figures/fs_v_mpisq}
\includegraphics[width=0.48\textwidth]{figures/fs_v_mpi}
\caption{\label{fig:fs_extrap}Extrapolation of $f_s$. The location of the vertical dashed line in each plot is determined from $m_\pi^\textrm{phy}$.}
\end{figure}
These results can be compared with the extrapolation of $m_s \langle N | \bar{s} s | N \rangle$ by converting with the isospin averaged nucleon mass $m_N^\textrm{phy} = 938.9$~MeV.
In Table~\ref{tab:final_extrap}, these three different extrapolation results are collected.
Additionally, a correlated weighted average is performed. To perform the correlated average, Gaussian distributions of the results in Table~\ref{tab:ms_sbars} are created independently for each light quark mass point, with $N_{Gauss} = 10^4$ in all cases. For each sample, all three extrapolations are performed, preserving the correlations between the fits, with inverse weights given by the statistical and systematic uncertainties on the individual mass points.
For each sample, these three results are then averaged with weights given by the inverse uncertainties from the individual analyses (quoted in Table~\ref{tab:final_extrap}).
This yields the final result
\begin{subequations}
\begin{align}
m_s \langle N | \bar{s} s | N \rangle & = 48 \pm 10 \pm 15 \textrm{ MeV}\, ,\\
f_s &= 0.051 \pm 0.011 \pm 0.016\, .
\end{align}
\end{subequations}
\begin{table}
\caption{\label{tab:final_extrap}Extrapolated values of $m_s \langle N | \bar{s} s | N \rangle$ and $f_s$. These results are averaged in a weighted and correlated fashion described in the text.}
\begin{ruledtabular}
\begin{tabular}{cccc}
Quantity Extrapolated& Extrapolation Function& $m_s^\textrm{phy} \langle N | \bar{s} s | N \rangle$~[MeV]& $f_s$\\
\hline
$m_s^\textrm{phy} \langle N | \bar{s} s | N \rangle$ & Eq.~\eqref{eq:sbars_simple}&
$56 \pm 12 \pm 17$& $0.059 \pm 0.012 \pm 0.019$ \\
$f_s$& Eq.~\eqref{eq:f_s_mpisq} & $47 \pm\phantom{0}9\pm 13$& $0.050 \pm 0.009 \pm 0.014$ \\
$f_s$& Eq.~\eqref{eq:f_s_mpi} & $47 \pm 12 \pm 17$& $0.050 \pm 0.012 \pm 0.018$ \\
\hline
Correlated Average & -- & $49\pm10\pm15$& $0.053\pm0.011\pm0.016$
\end{tabular}
\end{ruledtabular}
\end{table}
\section{Results and Discussion\label{sec:results}}
\noindent
For the present work, the Feynman-Hellmann theorem was invoked to determine the strange content of the nucleon through a change $m_N$ as the strange-quark mass is varied
\begin{equation*}
m_s \langle N | \bar{s} s | N \rangle = m_s \frac{\partial m_N}{\partial m_s} \, .
\end{equation*}
By taking care to set the scale using values of $r_1/b$, which were extrapolated to the physical values of the light- and strange-quark masses, the nucleon mass variation was determined with all other parameters held constant (with precision better than 1\%), as is required for a proper determination of this quantity~\cite{Toussaint:2009pz,Freeman:2012ry}.
There are several groups who have used the Feynman-Hellmann theorem~\cite{Young:2009zb,Durr:2011mp,Horsley:2011wr,Oksuzian:2012rzb,Semke:2012gs,Shanahan:2012wh,Ren:2012aj,Jung:2013rz}
as well as more determinations with a direct calculation of the matrix element~\cite{Babich:2010at,Takeda:2010cw,Bali:2011ks,Dinter:2012tt,Gong:2012nw,Oksuzian:2012rzb,Engelhardt:2012gd} and results from a hybrid approach~\cite{Toussaint:2009pz,Freeman:2012ry}.
Before making a detailed comparison with other works, we first highlight advantages and disadvantages of the present work.
The distinct advantage of using the Feynman-Hellmann theorem over direct methods is that the ground state plateau of the nucleon can be significantly more reliably determined than the plateau for the matrix element calculation with equal computing resources; see the plots of ratio determinations in any of Refs.~\cite{Babich:2010at,Takeda:2010cw,Bali:2011ks,Dinter:2012tt,Gong:2012nw,Oksuzian:2012rzb,Engelhardt:2012gd} (the direct calculation requires a vacuum subtraction, adding substantial statistical noise).
The disadvantage of most groups employing the Feynman-Hellmann theorem is the reliance upon $SU(3)$ baryon $\chi$PT~\cite{Young:2009zb,Semke:2012gs,Shanahan:2012wh,Ren:2012aj}, which is known to not have a converging expansion for the nucleon mass~\cite{WalkerLoud:2008bp,Jenkins:2009wv,Torok:2009dg,Ishikawa:2009vc,WalkerLoud:2011ab}.
Therefore, it is not clear that the full extrapolation systematic has been properly addressed in those works
\footnote{The work in Ref.~\cite{Durr:2011mp} also uses $SU(3)$ baryon $\chi$PT, but uses a variety of other extrapolation methods, resulting in a conservative estimate of their uncertainties.}
This concern is substantiated by the discrepancy between independent $SU(3)$ baryon $\chi$PT analyses and their determination of $f_s$~\cite{Young:2009zb,Shanahan:2012wh,Semke:2012gs,Ren:2012aj}
\footnote{Despite these criticisms, we point out in Ref.~\cite{Shanahan:2012wh}, a striking agreement is found between baryon mass results extrapolated from one set of lattice calculations~\cite{WalkerLoud:2008bp,Ishikawa:2009vc}, with $SU(3)$ baryon $\chi$PT, and then used to predict results from a completely independent calculation~\cite{Bietenholz:2011qq}. Moreover, independent verification of the consistency of various lattice calculations of the ground state baryon spectrum and $SU(3)$ baryon $\chi$PT has been found~\cite{Semke:2011ez,Semke:2012gs,Lutz:2012mq}.
}
For further discussion on the convergence problems using $SU(3)$ baryon $\chi$PT specifically for the scalar strange content of the nucleon, see Ref.~\cite{Alarcon:2012nr}.
The current work does not suffer from this issue.
The most severe limitation of the present work is the small number of light quark mass points (two) for which there is a nonzero determination of $m_s \langle N | \bar{s} s | N \rangle$. Given the significant numerical cost of the domain-wall propagators on the $b\approx0.09$~fm ensemble with $m_\pi \simeq 240$~MeV, it is not clear how soon a more precise determination will be obtained at this point.
Given the very mild light quark mass dependence observed in this work, and in nucleon matrix elements in general, we believe the present determination offers a reliable estimate of the scalar strange content of the nucleon, but neither a precise nor demonstrably accurate value. Our final result is
\begin{align*}
m_s \langle N | \bar{s} s | N \rangle & = 49 \pm 10 \pm 15 \textrm{ MeV}\, ,\\
f_s &= 0.053 \pm 0.011 \pm 0.016\, .
\end{align*}
\subsection{Lattice QCD comparison and average}
\noindent
Given the phenomenological importance of the scalar strange content of the nucleon, see for example Refs.~\cite{Bottino:1999ei,Bottino:2001dj,Kaplan:2000hh,Ellis:2008hf,Ellis:2009ai,Giedt:2009mr,Freytsis:2010ne,Hill:2011be,Cheung:2012qy}, it is prudent to review the limitations of the present determination and to compare and contrast these results to other lattice QCD determinations.
There are two results which use the same MILC ensembles with staggered valence quarks~\cite{Toussaint:2009pz,Freeman:2012ry} and one determination with the same mixed-action scheme but a direct determination~\cite{Engelhardt:2012gd}.
It is interesting to first compare our results with these.
Reference~\cite{Freeman:2012ry} (an update of \cite{Toussaint:2009pz}) quotes only the value of $\langle N | \bar{s} s | N \rangle$ in $\overline{\textrm{MS}}\ (2 \textrm{ GeV})$.
To convert this number into the dimensionful, renormalization scheme invariant quantity, we take the ratio of quoted values $m_s \langle N | \bar{s} s | N \rangle / \langle N | \bar{s} s | N \rangle$ from Ref.~\cite{Toussaint:2009pz}, which amounts to $m_s[\overline{\textrm{MS}}\ (2 \textrm{ GeV})] = 86$~MeV.
Alternatively, we could use the strange-quark mass determination of HPQCD~\cite{Mason:2005bj} (updated by MILC~\cite{Bazavov:2009tw}), $m_s[\overline{\textrm{MS}}\ (2 \textrm{ GeV})] = 89.0(4.8)$~MeV, but within uncertainties, these are the same.
Comparing to these works, as well as the mixed-action calculation, good agreement is found:
\begin{equation}
m_s \langle N | \bar{s} s | N \rangle [\textrm{MeV}]=
\left\{ \begin{array}{ll}
59\pm6\pm8 & \textrm{Ref.~\cite{Toussaint:2009pz}}\\
54\pm5\pm6 & \textrm{Ref.~\cite{Freeman:2012ry}}\\
43\pm8\pm6 & \textrm{Ref.~\cite{Engelhardt:2012gd}}\\
49\pm10\pm15 & \textrm{ present work}
\end{array}
\right. \, .
\end{equation}
In the literature, there is currently no determination of $f_s$ that considers all the available results from lattice QCD, and so we take the opportunity to provide one here
\footnote{There is a recent review on the topic in Ref.~\cite{Young:2013nn}, but a lattice average is not provided.}
We use an approach similar to the FLAG working group of FLAVIANET, which has provided lattice determinations of various quantities important to low-energy hadronic physics~\cite{Colangelo:2010et}.
In particular, the FLAG working group has developed a scheme to judge the confidence to place in various determinations, based upon standards such as the lightest pion mass used, whether or not a continuum limit has been performed, and whether the infinite volume limit has been performed.
For each criterion, a green star ({\large{\color{green}{$\star$}}}) is awarded to results that meet the strictest constraints, an orange circle ({\large{\color{orange}{$\bullet$}}}) is given to results with room for improvement and a red square ({\scriptsize{\color{red}{$\blacksquare$}}}) to those with room for significant improvement.
This provides a useful guide to people outside the lattice community and motivation for those in the community to improve their results.
Using the standards of Ref.~\cite{Colangelo:2010et}, most results for $f_s$ receive an orange circle. There is one group that receives the green star, and the rest receive a red square.
The results with a red square suffer either from too few light quark mass points to make a reliable chiral extrapolation or they rely too heavily on $SU(3)$ baryon $\chi$PT.
There are two analyses that we promote from a red square to an orange circle because while they rely heavily on $SU(3)$ baryon $\chi$PT, they have demonstrated a remarkable consistency of their analysis with four or more independent lattice calculations~\cite{Semke:2012gs,Shanahan:2012wh}.
We exclude results that are either not published or not in an arXiv e-print posting (as results in conference proceedings often undergo larger-than-quoted systematic changes).
We further exclude results which have not been extrapolated to the physical value of the light-quark mass, and results calculated without dynamical strange quarks ($n_f=2$) are not included in the average.
To convert results from $m_s \langle N| \bar{s} s | N \rangle$ to $f_s$, we use $m_N = 938.9$~MeV.
These results are displayed in Fig.~\ref{fig:fs_compare}.
For the scalar strange content of the nucleon, the current state of results is such that a simple weighted average of good (green star) results can not be performed in a meaningful way.
As can be seen in Fig.~\ref{fig:fs_compare}, there is good consistency between most of the results.
There are not a large number of orange circle results, so we chose to include all results in the average.
Moreover, we believe despite their red-square assignment, these results offer valuable information which should not be ignored at this time.
\begin{figure}
\includegraphics[width=0.7\textwidth]{figures/fs_compare}
\caption{\label{fig:fs_compare}Comparison and average of lattice QCD calculations of $f_s$ as described in the text.
Only values that have been extrapolated to the physical quark masses are used. Results that quote $m_s \langle N | \bar{s} s | N \rangle$ are normalized by $m_N = 938.9$~MeV to convert to $f_s$.
The quoted uncertainties are taken as the statistical and systematic uncertainties added in quadrature from a given reference. $n_f=2+1$ indicates a dynamical strange quark as well as up and down. $SU(3)$ is used to indicate results that rely heavily on $SU(3)$ baryon $\chi$PT.
Some results are excluded for various reasons but displayed to demonstrate their consistency:
\cite{Toussaint:2009pz} was updated in \cite{Freeman:2012ry},
the $n_f=2$ results~\cite{Takeda:2010cw,Bali:2011ks} were not averaged with the $n_f=2+1$,
the results in \cite{Dinter:2012tt} were preliminary and not extrapolated to the physical pion mass,
the results in \cite{Gong:2012nw,Jung:2013rz} are preliminary and only exist in a conference proceedings.
All excluded results are presented as quoted in the literature, with no attempt to perform chiral extrapolations
}
\end{figure}
A simple weighted average, using the quoted uncertainties as the inverse weights, produces an unbelievably small final uncertainty.
This also ignores the fact that systematic uncertainties are typically non-Gaussian, and in the case of lattice QCD calculations, not cleanly separable from the statistical uncertainties.
Moreover, it does not account for the quality of the results, judged using the rubric of the FLAG working group.
In an attempt to include all these issues, the following \textit{ad hoc} procedure is used to perform a weighted average of all the results (presented in Figure~\ref{fig:fs_compare}):
\begin{enumerate}[i)]
\item for each of the $N_{latt}=11$ results, $f_i \pm {\sigma}_i^\pm$, an independent random sample is generated with a sample size of $N_{dist}=10^4$, drawn from a uniform distribution between the quoted uncertainties,
\begin{align*}
&\textrm{for i in range($N_{latt}$):}\\
&\quad\textrm{for j in range($N_{dist}$):}\\
&\quad\quad f_{i,j} = \textrm{random.uniform}(f_i - {\sigma}_i^-, f_i + {\sigma}_i^+)
\end{align*}
\item for each random sample, a weighted average of all results is performed, with weight
\begin{equation}\label{eq:avg_weights}
w_i = y_i / {\sigma}_i\, ,
\end{equation}
where ${\sigma}_i$ is the symmetric uncertainty, ${\sigma}_i = 0.5*({\sigma}_i^+ + {\sigma}_i^-)$ from a given result, and we arbitrarily chose $y_i = 1,2,3$ for the red square, orange circle and green star, respectively. An extra multiplicative reduction of $0.5$ is assigned to results which rely heavily on $SU(3)$ baryon $\chi$PT,
\begin{align*}
&\textrm{for j in range($N_{dist}$):}\\
&\quad \bar{f}_j = \frac{\sum_i w_i\ f_{i,j}}{\sum_{i^\prime} w_{i^\prime}}
\end{align*}
The choice to weight with $1/{\sigma}_i$ instead of $1/{\sigma}_i^2$ is partly motivated from the non-Gaussian behavior of the systematic uncertainties that typically dominate the lattice results.
\item the mean and 99\% confidence intervals of the resulting distribution are quoted, see Fig.~\ref{fig:fs_compare}
\end{enumerate}
A principal concern one should have about this average is the choice of weights used, Eq.~\eqref{eq:avg_weights}.
To help judge the stability of the average presented here, a variety of different weights are chosen, and the subsequent averages are compared and presented in Table~\ref{tab:fs_weights}.
The different choices in weights result in very consistent values.
This is a statement about the consistency of the values of $f_s$ from a variety of lattice QCD calculations, and it is this striking consistency that leads us to believe a lattice average with the present results is meaningful (despite the shortcomings of most of the individual results).
The resulting lattice average, quoted at the 99\% confidence interval to be conservative, is
\begin{align}\label{eq:fs_avg}
m_s \langle N | \bar{s} s | N \rangle &= 40 \pm 10 \textrm{ MeV}\, ,\nonumber\\
f_s &= 0.043 \pm 0.011\, .
\end{align}
\begin{table}
\caption{\label{tab:fs_weights} Value of $f_s$ determined with various weights as described in text. The right-most value (with $w_i = y_i/{\sigma}_i$) is the value taken in this work to represent the lattice average.}
\begin{ruledtabular}
\begin{tabular}{c|ccccc|c}
$w_i$ &$1/{\sigma}_i^2$& $y_i/{\sigma}_i^2$& $1/{\sigma}_i$& $y_i^2/{\sigma}_i$& 1& $y_i/{\sigma}_i$ \\
\hline
$f_s(68\%)$ &0.0458(31)& 0.0470(35)& 0.0442(36)& 0.0420(55)& 0.0487(63)& 0.0428(41)
\end{tabular}
\end{ruledtabular}
\end{table}
As was first discussed in Refs.~\cite{Young:2009zb,Giedt:2009mr}, there is now compelling evidence from lattice QCD that the value of the scalar strange content of the nucleon is substantially smaller than previously estimated and does not play as significant a role in dark-matter searches as previously thought~\cite{Bottino:1999ei,Bottino:2001dj,Ellis:2008hf,Hill:2011be}.
This has potential implications for the importance of spin-dependent dark-matter searches as discussed in Ref.~\cite{Freytsis:2010ne}.
For a recent review of the lattice QCD determinations of the scalar strange content of the nucleon, see Ref.~\cite{Young:2013nn}.
\subsection{Estimating the heavy quark matrix elements}
\noindent
Knowledge of $f_u$, $f_d$ and $f_s$ can be used to determine the values of $f_c$, $f_b$ and $f_t$~\cite{Shifman:1978zn,Kryjevski:2003mh}.
In Ref.~\cite{Kryjevski:2003mh}, these heavy quark matrix elements were computed using perturbative QCD to $\mc{O}({\alpha}_s^3)$, finding%
\footnote{We have updated the values of the quark masses used in Ref.~\cite{Kryjevski:2003mh} to the current PDG values~\cite{Beringer:1900zz}.}
\begin{align}
&f_c = 0.08896(1-x_{uds})\, ,&
&f_b = 0.08578(1-x_{uds})\, ,&
&f_t = 0.08964(1-x_{uds})\, ,&
\end{align}
where
\begin{equation}
x_{uds} = f_u + f_d + f_s\, .
\end{equation}
The light-quark matrix elements are given by the pion-nucleon sigma term $m_N(f_u+f_d) = {\sigma}_{\pi N}$, which has also been determined from lattice QCD. As can be seen in Ref.~\cite{Young:2013nn}, the determination by the BMW Collaboration~\cite{Durr:2011mp} not only would have the only green-star ranking but also is a good approximation for the average of all lattice QCD calculations of this quantity, with a value ${\sigma}_{\pi N} = 39({}^{+18}_{-8})$~MeV. Combining this with our estimate for $f_s$ yields a value $x_{uds} = 0.085({}^{+.022}_{-.014})$, and values of the heavy-quark matrix elements
\begin{align}
&f_c = 0.0814({}^{+12}_{-20})\, ,&
&f_b = 0.0785({}^{+12}_{-19})\, ,&
&f_t = 0.0820({}^{+13}_{-20})\, ,&
\end{align}
or in dimensionful units
\begin{align}
m_c \langle N | \bar{c} c | N \rangle &= 76({}^{+11}_{-19}) \textrm{ MeV},
\nonumber\\
m_b \langle N | \bar{b} b | N \rangle &= 74({}^{+11}_{-18}) \textrm{ MeV},
\nonumber\\
m_t \langle N | \bar{t} t | N \rangle &= 77({}^{+12}_{-19}) \textrm{ MeV}.
\end{align}
The resulting charm-quark matrix element is in good agreement with the direct lattice QCD calculations of this quantity~\cite{Freeman:2012ry,Gong:2013vja}.
\acknowledgments
\noindent
PMJ would like to especially thank S.~Beane for many helpful conversations and for suggesting this project.
PMJ also thanks the hospitality of LBNL where some of this work was completed.
AWL would like to thank R.~Lebed for clarifying some subtleties in the large $N_c$ expansion. AWL would also like to thank J.~Ruderman for helpful conversations.
We thank our fellow members of the NPLQCD Collaboration for providing some of the numerical results used in the present work and for helpful comments.
We thank C.~Bernard for providing the updated values of $r_1/b$ for the MILC Collaboration.
We thank J.~Ruderman for the motivation to compare all lattice results of this quantity.
Numerical calculations for the present work were performed with the
\texttt{CHROMA} software suite~\cite{Edwards:2004sx}.
We acknowledge computational support from the USQCD SciDAC project, LLNL, the Argonne Leadership Computing Facility at Argonne National Laboratory (Office of Science of the DOE, under Contract No. DE-AC02-06CH11357). Calculations were also performed on Endeavour, a UNH computing cluster.
The work of PMJ was supported in part by NSF Grant No PHY1206498.
The work of AWL was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Divisions of Nuclear Physics, of the U.S. DOE under Contract No. DE-AC02-05CH11231.
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\section{Introduction} \label{sec:intro}
In a turbulent fluid, kinetic energy ($e_{\rm kin}$) is expected to cascade from the driving scale to the viscous dissipation scale at a constant rate, i.e., $\mathrm{d} e_{\mathrm{kin}}/{\mathrm{d} t} \propto \rho v^{2}/{t} \propto \rho v^{3}/{l}\, =\,\text {const}$, where $\rho$, $v$, and $l$ are respectively the density, velocity, and length scale of the fluid elements, and $t$ is the energy transfer timescale. Under this assumption, a scaling relation can be drawn between $\rho$, $v$, and $l$, i.e., $\rho^{1/3}v\propto l^{1/3} $. This simple dimensional analysis offers profound insights into the turbulent structures of an incompressible fluid, where $v\propto l^{1/3} $ \citep{Kolmogorov1941}. Such a relation is found in X-ray observations of nearby galaxy clusters, where the one-component velocity amplitude scales with wavenumber ($k\equiv l^{-1}$) to the $-1/3$ power \citep{Zhuravleva2014, Zhuravleva2019}. Additionally, X-ray line widths have been used to probe the level of turbulence in galaxy clusters, e.g., \citet{Sanders2011} examined a sample of 62 systems, half of which have line width upper limits below $700\kms$. Recent X-ray calorimeter observations also found a low line-of-sight velocity dispersion (Gaussian $\sigma$) of $164\pm10\kms$ in the core of the Perseus cluster \citep{Hitomi2016}. Both of these observational results indicate that the level of turbulence in the ICM is mild and subsonic in a relaxed galaxy cluster.
The subsonic structure of the ICM does not preclude the presence of supersonic flows in galaxy clusters, particularly the outflows driven by the central AGN, which are believed to heat the surrounding medium and quench star formation \citep{Fabian2012, McNamara2012}. Recent spectral analysis of the spiral galaxy M81 found direct evidence for hot AGN-driven winds with line-of-sight velocities of $\pm2800\kms$ based on the Fe\,$\textsc{xxvi}$ Ly$\alpha$ line shifts \citep{Shi2021}.\footnote{Note that the wind in this low-luminosity AGN is likely from a hot accretion flow, not radiatively driven. Both a wind and a mass-loaded relativistic jet may drive the outflows explored in this work \citep[for a discussion of these outflow-driving mechanisms, see][]{Qiu2021}.} In a rich cluster environment, however, direct detection of such outflows is more difficult due to the X-ray emission from the massive ICM. On the other hand, recent simulations of galaxy clusters and AGN feedback indicate that cold gas may fragment out of a supersonic, radiatively cooling outflow \citep{Qiu2020, Qiu2021a}, giving rise to extended filamentary H$\alpha$ nebulae, such as observed in the Perseus cluster \citep{Conselice2001, GM2018}. Therefore, the velocity structure of the $10^4$\,K gas offers a unique opportunity for probing the original outflow properties in galaxy clusters.
\begin{figure*}[t!]
\centering
\includegraphics[width=1\linewidth]{Fig_1.pdf}
\caption{The spatial distribution of the projected H$\alpha$ emissivity (columns 1, 3, 5) and the emissivity-weighted line-of-sight velocity (columns 2, 4, 6) for the three simulation runs with different initial outflow velocity, $v_{\rm out}$, projected along $\theta=90^\circ$ (viewing angle $\theta$ between sightline and initial outflow direction). The initial outflow direction and the length scale are shown in the two bottom right panels. The top, middle, and bottom rows show three representative evolution epochs at $t=20$, $30$, and $40\,$Myr, respectively. A horseshoe-shaped outer shell consisting of H$\alpha$-emitting gas is produced for the outflow with $v_{\rm out}=1500\kms$ at $t=20$\,Myr \citep[albeit smaller in size compared to the Perseus horseshoe filament;][]{Conselice2001}.}
\label{fig:projections}
\end{figure*}
\begin{figure*}[t!]
\centering
\includegraphics[width=\linewidth]{Fig_2.pdf}
\caption{Line-of-sight VSF of the simulated cold gas from different viewing angles ($\theta$; between sightline and initial outflow direction, indicated by the different line colors), for outflows with different initial velocities ($v_{\rm out} = 1500,\ 1000,\ 500\,\kms$), at two epochs of evolution ($t=20,\ 30\,$Myr). The scaling relations of $v\propto l^{1/3}$ and $v\propto l^{1/2}$ are plotted for comparison. The locations of the scaling relations are shifted in each column to approximately bracket the VSF lines. An additional scaling of $v\propto l^1$ is plotted in the lower middle panel, which shows that the VSF is dominated by the velocity gradient for large separations. Note that the cut-off of the lines at separations beyond a few kiloparsecs is constrained by the size of each filament complex, without the contribution from nearby filaments to extend the separation bins beyond 10\,kpc, such as shown in Fig.~\ref{fig:obs}.}
\label{fig:sim}
\end{figure*}
\begin{figure*}[t!]
\centering
\includegraphics[width=\linewidth]{Fig_3.pdf}
\caption{Left: Line-of-sight velocity distribution of the H$\alpha$ nebula in the Perseus cluster \citep[][with the systemic velocity shifted to $5264\kms$]{GM2018}, divided into 9 azimuthal regions. Only pixels with velocity errors $<30 \kms$ are plotted and used in the VSF calculation. The arc in Region 2 separates the inner filament complex lacking a radial gradient, and the outer horseshoe filament. An X-ray shock front is detected across Regions 4 and 5, which also indicates supersonic flows \citep{Fabian2003a}. Right: The VSF calculated from pixels in each region. The scaling relations of $v\propto l^{1/3}$ (subsonic), $v\propto l^{1/2}$ (supersonic), and $v\propto l^1$ (gradient) are plotted for comparison. The VSF of the entire nebula from \citet{Li2019d}, which represents an average of all 9 regions, is shown by the solid black line. Data points below the velocity error threshold ($30\kms$) and the seeing limit (0.4\,kpc) are marked with shaded regions.}
\label{fig:obs}
\end{figure*}
Ideally, for supersonic flows where the kinetic energy of the fluid elements may be thermalized, the turbulent velocity structure is expected to follow a steeper slope \citep[e.g., $v\propto l^{1/2}$;][]{Burgers1948,Federrath2013}, due to the partial energy cascade from large to small scales. Therefore, in this work we test this hypothesis by examining the velocity structure of the cold gas in the simulated outflow. In Section~\ref{sec:sim} we describe the simulation parameters and the turbulent structure of the simulated cold gas. In Section~\ref{sec:obs} we examine the turbulent structure of the H$\alpha$ filaments in the Perseus cluster using observations taken with SITELLE \citep{GM2018}. Finally, in Section~\ref{sec:discussion}, we discuss the implications of our work and conclude.
\section{Simulations of Cold Filaments}\label{sec:sim}
\subsection{Simulation setup}\label{sec:setup}
In order to study the turbulent velocity structure of the cold gas, we perform a set of simulations where cold gas cools and fragments out of an AGN-driven outflow. The simulation setup is similar to that presented in \citet{Qiu2021a}. In each simulation, a spherical gas clump is launched from the center of the cluster, with temperature $T_{\rm out}$, outflow velocity $v_{\rm out}$, and mass $M_{\rm out}$. Initially, the clump is in thermal pressure equilibrium with the surrounding ICM. After launch, the gas in the outflow cools radiatively as it travels in the idealized cluster environment modeled on the Perseus cluster \citep{Qiu2018}, with the highest resolution $\sim60$\,pc. The fluid elements comprise of electron, hydrogen and helium species coupled to the non-equilibrium chemistry/cooling solver in the code \texttt{Enzo} \citep{Bryan2014}, which allows us to characterize the H$\alpha$ emission from the recombination process \citep[see also][]{Qiu2019}. We caution that the shock smoothing length of the Zeus solver \citep{Stone1992} employed in this work is around 120\,pc, so we bin the pixels $2\times2$ and only focus on the analyses above this scale.
Based on the observational constraints on the cold gas mass, we fix the initial mass $M_{\rm out}=10^8\ms$ of the outflowing gas clump. In the previous simulations, we vary $T_{\rm out}$ around $10^7$\,K, and $v_{\rm out}$ from $1200-2000\kms$ to study the dynamical and morphological evolution of the emergent cold gas and compare it with observed filaments. In this new set of simulations, in order to focus on the supersonic/subsonic nature of the originating outflow, we fix $T_{\rm out}=10^7$\,K, and vary the initial velocity from supersonic to subsonic regime, $v_{\rm out}=1500,\,1000,\ {\rm or}\ 500\kms$. Note that the sound speed for $10^7$\,K plasma is $\approx 500\kms$, so the Mach number of the outflows explored in this work is between 1 and 3.
For the two cases where $v_{\rm out}=1500,\,1000\kms$, the outflows are launched 1\,kpc away from the cluster center. Due to the slow velocity that inhibits the spatial reach of the outflow, for the last case where $v_{\rm out}=500\kms$ we launch the outflow 10\,kpc away from the cluster center. With this scenario we intend to explore a slower, subsonic phase of the outflow, in a hypothetical scenario where it emerges from the central 10\,kpc with a temperature of $10^7$\,K. Even though this outflow would in reality also be launched from the cluster center as an initially faster and hotter outflow, in this case we focus only on its decelerated, subsonic component, for simplicity (because the properties of a mixture of supersonic and subsonic turbulence may be difficult to interpret). On the other hand, this case also allows us to test the scenario where filaments originate outside of the cluster core due to thermal instabilities of the hot ICM moving subsonically \citep[e.g.,][]{Li2015,Wang2019}.
\subsection{Cold gas distribution}\label{sec:dist}
As demonstrated in \citet{Qiu2021a}, the radiative cooling time of the $10^7$\,K outflow in initial pressure equilibrium with the ambient ICM is $t_{\rm cool} \lesssim 10$\,Myr. This is a few times larger than the sound crossing time $t_{\rm cross} \equiv \Delta l/c_{\rm s}\approx 2$\, Myr\,($\Delta l$/kpc), where $\Delta l \sim$\,kpc is the typical width of each observed filament complex, $c_{\rm s}$ is the sound speed. Before $10^4$\,K cold gas forms, a turning point therefore must exist around a few$\times 10^6$\,K, where $t_{\rm cool}<t_{\rm cross}$. In this transition phase, the plasma cools faster than the timescale required for the turbulent eddies to propagate though the length scale $\Delta l$, allowing the emergent cold gas to preserve the ``frozen-in'' turbulent structure.
In Fig.~\ref{fig:projections}, we show the spatial distribution of the projected H$\alpha$ emissivity, as well as the emissivity-weighted line-of-sight velocity for the three velocity cases at three evolutionary epochs. In all cases, the cold gas fragments out of the radiatively cooling outflow after $\sim10$\,Myr. As discussed in \citet{Qiu2021a}, the morphology of the cold gas may take both longitudinal and transverse shapes depending on the initial outflow properties, such as the outer horseshoe-shaped shell at $t=20$\,Myr for $v_{\rm out}=1500\kms$. Note that in the slowest outflow case, it takes much longer for the low-temperature plasma to stretch and form elongated cold gas filaments under cluster gravity, unless the initial thermally unstable region is already spatially extended. We refer the readers to our previous work on the dynamical and morphological evolution of the cold gas, which can be described reasonably well by a 1D model comprising radiative cooling and ICM ram pressure \citep{Qiu2020, Qiu2021a}. In this work, we instead focus on the small-scale turbulent structure that can be extracted from the cold gas velocity.
\subsection{Turbulent velocity structure}\label{sec:VSF}
In order to characterize the turbulent structure of the cold gas in the simulations, in Fig.~\ref{fig:sim} we compute the first-order velocity structure function (VSF). The simulated outflow is first projected along different viewing angles ($\theta$) with respect to the initial outflow direction to obtain both the line-of-sight velocity ($v_{\rm los}$) map and the H$\alpha$ emissivity map (the case where $\theta=90^\circ$ is shown in Fig.~\ref{fig:projections}). $v_{\rm los}$ is weighted by the H$\alpha$ emissivity in each line integral. After projection, pixels from the $v_{\rm los}$ map are selected to compute the VSF if the corresponding projected H$\alpha$ emissivity is $>4\times10^{-5}\ergs\,{\rm cm}^{-2}$, a threshold consistent with observations of the Perseus cluster filaments \citep[][assuming isotropic emission from the redshift $z=0.01756$ of the Perseus cluster]{GM2018}. For all possible pair combinations of the selected pixels, we compute the $v_{\rm los}$ difference $\delta v$ and the projected separation $l$. We then divide the pixel pairs into $\Delta l \approx 0.1$\,kpc bins based on their separation $l$, before taking the average of the absolute velocity difference, $\langle |\delta v^n| \rangle=\langle | v_i-v_j|^n \rangle$ ($n=1$ for the first-order VSF) in each separation bin. Bins with less than 10 pixel pairs are removed from the final plot.
In Fig.~\ref{fig:sim}, we present the VSF of the simulated cold gas with different initial velocities ($v_{\rm out}=1500,\,1000,\ {\rm or}\ 500\kms$), for projections along varying viewing angles $\theta$, at two epochs of evolution ($t=20,\,30$\,Myr). The timestamps represent $\Delta t\approx10,\, 20 $\,Myr after the cold gas formation. The driving scale of the turbulence, where the VSF peaks or flattens, is on the order of a few kpc, corresponding to the size of the initial outflow with coherent velocity (which later develops into varying speeds on smaller scales due to turbulence). In all cases at the early epoch ($t=20$\,Myr), the VSF below 1\,kpc follows the power indices ($\gamma$) expected of super- and sub-sonic turbulence at small-to-intermediate viewing angles ($\theta\lesssim80^\circ$), i.e., $\gamma\approx1/2$ for $v_{\rm out}=1500,\,1000\,\kms$, and $\gamma\approx1/3$ for $v_{\rm out}=500\kms$. At these viewing angles, the line-of-sight velocity ($v_{\rm los}$) is dominated by the component parallel to $v_{\rm out}$, which is the driving direction of turbulence. For small $\theta\lesssim 30^\circ$, the slope slightly steepens for all cases around 1\,kpc, when the viewing angle is close to the outflow direction. This may be contributed by the steep velocity gradient for small $\theta$, which features a scaling relation $v\propto l$, as discussed in more detail in Section~\ref{sec:obs} below. On the other hand, for large $\theta\gtrsim80^\circ$, $\gamma$ flattens to 0 because the $v_{\rm los}$ is nearly perpendicular to the outflow direction, which results in a $v_{\rm los}$ distribution similar at all separation scales (e.g., the $v_{\rm los}$ distribution shown in Fig.~\ref{fig:projections}). For the average viewing angle, however, the power index $\gamma$ of the cold gas VSF below a few kiloparsecs signifies the property of the plasma from which it originates, either supersonic for $\gamma\approx 1/2$, or subsonic for $\gamma\approx1/3$.
After the cold clumps fragment out of the plasma, their interaction is primarily gravitational, both with the background potential and with each other. This inviscid interaction inevitably hinders the energy cascade and leads to the flattening of the velocity structure at smaller scales. In the simulations, this trend gradually develops after $t=20$\,Myr (or 10\,Myr after cold gas formation). Note that the sound speed for the cold clumps below $10^4$\,K is $c_{\rm s}\lesssim 10\kms$, indicating that the gas may still be traveling supersonically. However, the turbulent structure likely develops at scales much smaller than those probed both in our simulations and in observations of galaxy clusters \citep[for an example of the turbulence on parsec scales, see the velocity structure and the small-scale flattening probed by the stellar velocity in the Orion Complex;][]{Ha2021}.
This trend of flattened VSF is seen in the later epoch of the simulated cold gas. In the lower panels of Fig.~\ref{fig:sim}, where some cold gas clumps have existed for more than 20\,Myr, their relative motion has ceased developing into smaller scales, e.g., forming binary-like systems. The VSF flattens significantly compared with the early epoch, with $\gamma$ ranging between 1/3 and 1/2 in the cases where $v_{\rm out}=1500,\,1000\kms$ ($\gamma$ decreases with larger $\theta$), and $\gamma$ approaching 0 in $v_{\rm out}=500\kms$ at separations between 0.3 and 1\,kpc. Realistically, however, the cold gas will likely continue to form molecules or fuel star formation, dropping out of the gas phase probed by H$\alpha$ emission. Therefore, the lack of $\gamma\leq 1/3$ in observations (see Fig.~\ref{fig:obs}) indicates that the lifetime of the H$\alpha$ emitting gas should be shorter than 20\,Myr, before gravitational interactions flattens the VSF.
\section{Comparison between Simulated and Observed Filaments in Perseus}\label{sec:obs}
\begin{figure*}[t!]
\centering
\includegraphics[width=\linewidth]{FIg_4.pdf}
\caption{Left two panels: Line-of-sight velocity distribution of the simulated filament ($v_{\rm out}=1000\kms$, $t=40$\,Myr, $\theta=77^\circ$) and the observed northern filament in the Perseus cluster. Right: Comparison of the velocity gradient over the length of the simulated and observed filaments ($-8.0\kms\,{\rm kpc}^{-1}$). Data points show the flux-weighted average velocity in each length bin of 1\,kpc. Solid lines show the linear fit to the data sets. Points with deviations larger than $0.5\,\sigma$ from the fit are omitted, which removes bins with few pixels at both ends and the middle of the observed filament.}
\label{fig:gradient}
\end{figure*}
In this section we examine filament observations in the Perseus cluster taken with the optical imaging Fourier transform spectrometer SITELLE at the Canada-France-Hawaii Telescope \citep[CFHT;][]{GM2018}. These data were taken at a spectral resolution of $R=1800$ in a filter covering $647-685$\,nm. SITELLE's angular resolution is 0.321''$\times$0.321'', and the data cubes are binned by a factor of $2\times2$ (to an effective pixel resolution of 0.642''$\times$0.642'') to increase the signal-to-noise ratio. Three lines ([NII]$\lambda$6548, H$\alpha$, and [NII]$\lambda$6584) were simultaneously fit in each pixel using Gaussian functions (convolved with the instrument line shape), with typical central velocity shift errors between $5\kms$ and $20\kms$. Compared with the map published in \citet{GM2018}, this map includes several small faint filaments that were previously cut off in the west (right) region. The same H$\alpha$ flux cut ($3\times10^{-17}\,{\rm erg\,s}^{-1}\,{\rm cm}^{-2}\,{\rm pixel}^{-1}$) is applied to the maps for the analysis.
The steeper slope of the cold gas VSF in the Perseus cluster was first discovered by \citet{Li2019d}, who examined the entire filament nebula and found a peak in the VSF at $\sim10$\,kpc, indicating that the filaments may be lifted by the AGN-inflated X-ray cavities \citep[e.g.,][]{Revaz2008, McNamara2016}. However, unlike AGN jets or winds that drive supersonic flows \citep{Qiu2020, Hillel2019}, the buoyantly rising cavities often travel at subsonic speeds in the centers of galaxy clusters \citep[see, e.g., the bubble speed estimates in][]{Birzan2004}, unlikely to display steeper slopes without additional damping mechanisms such as magnetic fields \citep[e.g.,][]{Wang2020, Mohapatra2021a}. On the other hand, the entire nebula consists of multiple generations of filaments driven by past AGN activity at different epochs, each filament should retain structural information about their origin. Therefore, we examine the VSF of filament complexes divided into 9 azimuthal regions in Fig.~\ref{fig:obs} to compare with our supersonic outflow model. We note that precisely separating individual outflows is often challenging due to the projection of overlapping filaments. Therefore, this division represents a simple approach to probe the turbulence likely driven by the same AGN outburst in a particular direction. We also caution that the seeing limit of the observation is about 1.1'' ($\approx0.4$\,kpc, FWHM), so we focus our comparison above this length scale.
In Fig.~\ref{fig:obs}, we plot the VSFs for the azimuthal regions, each showing a steep slope with $\gamma\approx1/2$ at scales $\lesssim 2$kpc. The steep slops indicate that the turbulence in these filaments are supersonic, in agreement with the supersonic outflows modeled in our simulations. The only exception occurs in Region 2, where $1/3< \gamma <1/2$.\footnote{Note there is a flattened ``ghost'' X-ray cavity located near the outer horseshoe filament, which may suggest that the filament is lifted by the rising bubble \citep{Fabian2003}. We have further separated the inner and outer filaments in Region 2, as indicated by the arc in Figure~\ref{fig:obs}, and found that the VSF slope of the horseshoe filament is still $1/2$. The flattening of the VSF in Region 2 is therefore due to the inner filament complex.} The inner filament complex (below the arc located at $r=19$\,kpc in Fig.~\ref{fig:obs}) in this region lacks a radial gradient, indicating that the viewing angle $\theta$ may be perpendicular to the direction of motion, therefore the VSF may be contaminated by the uniform $v_{\rm los}$ distribution discussed in Section~\ref{sec:sim}. Nevertheless, the VSF slopes below the driving scale of $\sim2$\,kpc indicate the ubiquity of supersonic turbulence in the Perseus cluster. Compared with the simulation results presented in this work, we find that the supersonic turbulence structure is a direct indicator that the cold gas fragmented out of a fast outflow driven by recent AGN activity.
Beyond the driving scale $\sim 2$\,kpc, however, the VSF depends primarily on the radial gradient of the filament complex in each region, which has a characteristic slope of 1, as indicated by the linear relation $v\propto l$. For Regions 1 and 6 with large radial gradients, the VSF continues to rise, while for Region 4 with a small radial gradient, the VSF drops immediately. The average behavior, however, leads to the extension of the $\gamma\approx1/2$ slope from $2-10$\,kpc noted in \citet{Li2019d}. As a case study, the velocity gradient of the northern filament in Region 3 is calculated in Fig.~\ref{fig:gradient}. While the gradient of $-8.0\kms\,{\rm kpc}^{-1}$ does not significantly contribute to the VSF below $\sim2$\,kpc, it yields a velocity difference of $80\kms$ at the separation of 10\,kpc, comparable to the VSF amplitude in Region 3 ($\sim100\kms$ at 10\,kpc). In the case of Region 8, the mixed radial gradient, likely due to overlapping filaments at different epochs of evolution, steepens the VSF slope significantly at separations near $\sim1$\,kpc. We therefore caution that the turbulent VSF slope shall only be extracted at spatial separations minimally affected by the radial velocity gradient.
Given the dependence of $v_{\rm los}$ on the viewing angle $\theta$, we also try to constrain $\theta$ for the most extended northern filament in Region 3 of the Perseus cluster. As shown in Fig.~\ref{fig:gradient}, there is a smooth velocity gradient along the length of the northern filament, with the bottom half redshifted and the top half blueshifted. A similar morphology and velocity gradient can be found in the simulated filament with $v_{\rm out} = 1000\kms$ at $t=40$\,Myr. Compared with the simulation model, the mixed velocity shifts indicate that the bottom half contains an older generation of gas that fragmented out of the outflow and has started to fall back, while the gas in the top half is still rising in the cluster potential. The same line-of-sight velocity gradient can be reproduced in the simulation when the viewing angle $\theta=77^\circ$. If the northern filament is similarly inclined, the angle indicates a factor of $1/\sin\theta\approx1.03$ increase to the filament extent, and a factor of $1/\cos \theta\approx4.4$ increase to the velocity. This boosts the maximum inflow/outflow speeds to $\approx900\kms$, significantly larger than commonly observed of the filaments.
\section{Conclusions}\label{sec:discussion}
In this paper we perform hydrodynamical simulations to study the turbulent velocity structure of the cold gas that fragments out of radiatively cooling hot outflows. By varying the initial velocities, we find that the cold gas turbulent velocity structure depends on the supersonic or subsonic nature of the original outflow, and can be used in observations to probe the interactions between AGN and the ICM. The main findings are summarized below:
1. For hot outflows with a short radiative cooling timescale $t_{\rm cool}$, the turbulent velocity structure may be ``frozen'' in the emerging cold gas, when the dynamical sound crossing time $t_{\rm cross}>t_{\rm cool}$. The preserved velocity structure therefore can be an indicator of the characteristic velocity of the plasma out of which the cold gas fragments. For viewing angles $\theta\lesssim80^\circ$ with respect to the outflow direction, the slope of the first-order velocity structure function is $\geq1/2$ for the cold gas that originates from supersonic flows, or $\approx1/3$ for subsonic flows.
2. For individual filaments in both the simulations and the observations of the Perseus cluster, the VSF slope extends from a few hundred parsecs up to the driving scale of a few kiloparsecs, corresponding to the size of the original outflow, as well as the width of the filament complex. Both the velocity structure function slope and the driving scale can therefore be used to constrain properties of the AGN-driven outflows.
3. Beyond the driving scale, the VSF slope is contributed primarily by the velocity gradient along the length of the filament. Because the projected gradient depends on the viewing angle $\theta$, this contribution may steepen the VSF slope for small $\theta$ (large gradient) at separations $\sim 1$\,kpc. By varying $\theta$ of the simulated filament to compare with the velocity gradient of the most extended northern filament in the Perseus cluster, we infer that the viewing angle of the northern filament is $\approx77^\circ$. This angle implies that the maximum inflow/outflow speed in the northern filament is $\approx900\kms$, which corroborates the argument that supersonic flows are needed to reproduce the observed turbulent structure of the filaments.
4. Gravitational interactions of the cold gas result in the flattening of the VSF slope in the simulations over 20\,Myr, which indicates the stagnation of the energy cascade without additional mechanisms to transfer the kinetic energy or dissipate the cold gas. The lack of flattened VSF in the observed Perseus filaments suggests that the H$\alpha$-emitting gas is short-lived with a lifespan $<20$\,Myr, during which time it either fuels the molecular gas growth and the scattered intracluster star formation, or becomes recycled to the ICM by local heating processes.
We note that while the filament VSF provides a powerful diagnostic tool for understanding the origin of the filaments and the dynamic interaction between the AGN and the ICM, (a) the supersonic/subsonic nature of the originating outflow, (b) the steepening from velocity gradient, (c) the flattening due to gravitational interactions, as well as (d) the modulation by the viewing angle all contribute to the final VSF of the filaments. The modeling presented in this work therefore provides a viable method to disentangle these (sometimes competing) parameters for observed filaments. Through the dynamical modeling and evolution time measurement of different generations of filaments, the simulations can also constrain the duty cycle and reconstruct AGN activities in the recent few$\times10$\,Myr located at the cores of galaxy clusters.
\hspace{1cm}
\noindent This work is supported by the National Natural Science Foundation of China (12003003, 12073003, 11721303, 11991052, 11950410493), the China Postdoctoral Science Foundation (2020T130019), the National Key R\&D Program of China (2016YFA0400702), and the High-Performance Computing Platform of Peking University.
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\section{Introduction}
The discovery of the Higgs boson with a mass near 125~GeV \cite{Aad:2012tfa, Chatrchyan:2012ufa} has important consequences for physics beyond the Standard Model, especially supersymmetry. In the MSSM, it implies that the stops must either be very heavy or have a large trilinear coupling (``$A$-term") with the Higgs \cite{Hall:2011aa, Heinemeyer:2011aa, Arbey:2011ab, Arbey:2011aa, Draper:2011aa, Carena:2011aa, Cao:2012fz, Christensen:2012ei, Brummer:2012ns}. The large $A$-term scenario is more interesting from several points of view. It is less fine-tuned and it allows for lighter ($\sim$~1~TeV) stops that are still within reach of the LHC. It also presents an interesting model-building challenge -- prior to the discovery of the Higgs, mechanisms for generating the $A$-terms from an underlying model of SUSY-breaking mediation were not well-explored.
In the framework of gauge mediated SUSY-breaking (GMSB) (for a review and original references, see \cite{Giudice:1998bp}), the problem of how to obtain large $A$-terms becomes especially acute. In GMSB, the $A$-terms are always negligibly small at the messenger scale. If the messenger scale is sufficiently high and the gluino sufficiently heavy, a sizable weak scale $A$-term with relatively light stops may be generated through RG-running \cite{Draper:2011aa}. However, this setup is in strong tension with electroweak symmetry breaking (EWSB) \cite{GGMinprogress}. This strongly motivates extending gauge mediation with additional MSSM-messenger couplings
that generate $A$-terms through threshold corrections at the messenger scale.
In all models for $A$-terms considered since the observation of a Higgs boson at 125 GeV \cite{Kang:2012ra,Craig:2012xp,Evans:2013kxa,Craig:2013wga,Knapen:2013zla,Abdullah:2012tq,Kim:2012vz,Byakti:2013ti,Calibbi:2013mka,Jelinski:2013kta,Galon:2013jba,Fischler:2013tva,Ding:2013pya,Calibbi:2014yha}, the focus has been on generating $A$-terms at one-loop level through weakly coupled messengers. Integrating out the messengers produces one or more of the following K\"ahler operators
\begin{equation}\label{AtermKahler}
\frac{1}{16\pi^2 } {1\over M} X^\dagger H_u^\dagger H_u\,\, , \quad\quad \frac{1}{16\pi^2 } {1\over M} X^\dagger Q_3^\dagger Q_3\,\, ,\quad\quad \frac{1}{16\pi^2 } {1\over M} X^\dagger \overline{u}_3^\dagger \overline{u}_3
\end{equation}
Here $X$ is a field that spontaneously breaks SUSY, and $M$ is the messenger scale. After substituting $\langle X\rangle = \theta^2 F_X$ and integrating out the auxiliary components of the MSSM fields, one obtains the desired $A$-term
\begin{equation}
\mathcal{L}\supset y_t A_t H_u Q_3\bar{u}_3 \quad , \quad A_t \sim {1 \over 16\pi^2}{F_X\over M}
\end{equation}
This setup has the advantage that the $A$-terms come out parametrically the same size as the other soft masses in GMSB (one-loop gaugino masses, two-loop scalar mass-squareds). However, one-loop $A$-terms from (\ref{AtermKahler}) introduce a host of complications as well. First and foremost is the ``$A/m^2$ problem" \cite{Craig:2012xp}: in addition to the $A$-terms, one also generates a scalar mass-squared at one-loop, completely analogous with the more well-known $\mu/B_\mu$ problem. A one-loop scalar mass-squared would overwhelm the GMSB contributions and lead to serious problems with fine-tuning and/or EWSB. Previous solutions to the $A/m^2$ problem include taking the messengers to be those of minimal gauge mediation \cite{Craig:2012xp}, or having the hidden sector be a strongly-coupled SCFT \cite{Craig:2013wga,Knapen:2013zla}.
In this paper, we will explore a new solution to the $A/m^2$ problem: models where the $A$-terms are generated {\it at tree-level in the MSSM-messenger couplings}. The advantage with this approach is that there is simply no $A/m^2$ problem to begin with, since at worst any accompanying sfermion mass-squareds would be tree-level as well. An added benefit of this approach is that it will lead us to a consider an interesting new operator for the $A$-terms: one which arises in the effective \emph{superpotential}, rather than in the K\"ahler potential. As we will see, this superpotential operator will have qualitatively different effects on the MSSM soft terms as compared to K\"ahler potential operators.
The basic setup is quite simple. To generate a tree-level $A$-term, either the Higgs or stops must mix with the messengers in the mass-matrix. For example, consider the superpotential
\begin{equation}\label{exampleH}
W = X' H_u \tilde{\phi} + \lambda_u^{ij}\phi Q_i \overline{u}_j + M \tilde{\phi}\phi
\end{equation}
Here $X'$ is another spurion for SUSY-breaking, and $\phi$, $\tilde\phi$ are heavy messenger fields. Upon integrating out the messengers at the scale $M$, one generates the effective superpotential operator
\begin{equation}\label{eq:effsuperpotential}
W_{eff}\supset -{ \lambda_u^{ij}\over M }X' H_u Q_i \overline{u}_j
\end{equation}
Note that because of the SUSY non-renormalization theorem, $W_{eff}$ can {\it only} arise at tree-level, so it is perfectly suited for our purposes. In order to produce an $A$-term of the correct size, one must have\footnote{Note that this is a loop factor smaller than the usual GMSB relation. A smaller $F$-term satisfying this hierarchy can easily be dynamically generated using weakly-coupled messengers, see e.g.\ \cite{Komargodski:2008ax}. In this paper we will simply assume that $F_{X'}$ of the right size can be obtained somehow and not explore it any further.}
\begin{equation}\label{eq:FXrel}
{F_{X'}\over M}\sim \mathcal{O}({\rm TeV})
\end{equation}
The tree-level $A$-term originating from (\ref{eq:effsuperpotential}) is minimally flavor violating (MFV), provided that the operator in (\ref{eq:effsuperpotential}) generates the full up-type Yukawa coupling of the MSSM. For this to work, $X'$ should acquire a lowest component vev of size $\sim M$.
The interesting complication in these models comes from the fact that when integrating out the messengers, in addition to the superpotential operator (\ref{eq:effsuperpotential}), a K\"ahler potential operator is also generated at tree-level. For example, in the model (\ref{exampleH}), one generates the term:
\begin{equation} \label{listofmass}
K_{eff} \supset {1\over M^2} X'^\dagger X' H_u^\dagger H_u
\end{equation}
(For a more general treatment of the K\"ahler operators, see appendix A.)
This leads to a soft mass for $H_u$ of roughly the same order as the $A$-term:
\begin{equation}\label{littleamhexample}
\delta m_{H_u}^2 = - \frac{y_t^2}{|\lambda_u^{33}|^2} |A_t|^2
\end{equation}
For $\lambda_u^{33}\lesssim 1$, this represents a large, irreducible contribution to $m_{H_u}^2$, and correspondingly to the fine-tuning of the electroweak scale.
This is another manifestation of the ``little $A/m^2$ problem" encountered in \cite{Craig:2012xp}, whereby a large $A$-term was accompanied by an equally large sfermion mass-squared. In \cite{Craig:2012xp}, the situation was even worse, because the contribution was irreducible with a fixed coefficient:
\begin{equation}\label{littleamh}
\delta m_{H_u}^2 = \abs{A_t}^2
\end{equation}
There both the $A$-terms and the irreducible contribution to $m_{H_u}^2$ (\ref{littleamh}) originated from integrating out the auxiliary components of the MSSM fields in the first K\"ahler operator in (\ref{AtermKahler}).
Since we are starting instead with the effective superpotential operator (\ref{eq:effsuperpotential}), the coefficient in (\ref{littleamhexample}) is free to vary in our present models. Importantly, however, we will see that the sign in (\ref{littleamhexample}) is always \emph{negative}, such that (\ref{littleamhexample}) does not jeopardize electroweak symmetry breaking, in contrast to the relation in (\ref{littleamh}).
In this paper, we will consider various ways to alleviate the fine-tuning problem introduced by the little $A/m^2$ problem (\ref{littleamhexample}). Clearly, if $\lambda_{33}$ is taken to be large (e.g.\ $\lambda_{33}\sim 3$), then the little $A/m^2$ problem is ameliorated. This requires a UV completion at a relatively low scale. We will provide such a UV completion in this paper, using a novel application of Seiberg duality \cite{Seiberg:1994pq, Seiberg:1994bz}.
Alternatively, one can consider non-MFV models obtained from (\ref{exampleH}) by exchanging the role played by $H_u$ with $\bar u_3$:\footnote{Because these models are not MFV, one should worry about the potential constraints from precision flavor and CP observables. This is beyond the scope of this work (see however \cite{flavorinprogress}). We will assume for simplicity (as in \cite{Evans:2013kxa}) that the coupling $\kappa$ is real and fully aligned with the third generation. We will also focus on the $\bar u_3$ model because then the flavor violation is limited to the up-squark sector and the constraints are much weaker.}
\begin{equation}
W = X' \bar{u}_3 \tilde{\phi}_u + \kappa H_u Q_3 \phi_u + M \tilde{\phi}_u \phi_u\label{exampleU}
\end{equation}
For this model the expression analogous to (\ref{littleamhexample}) contains $m_{\bar u_3}^2$ instead of $m_{H_u}^2$. As in \cite{Evans:2013kxa}, the fine-tuning is greatly reduced with respect to the perturbative MFV case because the stop contribution to $m_{H_u}^2$ is diluted by a loop factor. Moreover, the situation is even better than in \cite{Evans:2013kxa}, because in that case there were still sizeable two-loop contributions to $m_{H_u}^2$, whereas here the contribution is solely to the squarks.
An important thing to note about the framework for generating tree-level $A$-terms presented in this paper is that it can in principle be tacked on to any mediation mechanism for the rest of the MSSM soft terms; the framework itself does not lead to a particularly compelling choice. This is in contrast to the one-loop models considered previously, whereby the $A$-term messengers also contributed to the MSSM soft spectrum through minimal gauge mediation, and thus GMSB was the most economical choice. Moreover, the tree-level $A$-term module does not affect the overall phenomenology much; the one essential difference occurs in the non-MFV models, where the stops can be split by several TeV due to the non-MFV analogue of (\ref{littleamh}).
For simplicity and concreteness, in this paper we will couple our models to minimal gauge mediation (MGM) \cite{Dine:1993yw,Dine:1994vc,Dine:1995ag}. We will see that after imposing the Higgs mass constraint, the models are typically out of reach of Run I LHC; however they will be accessible (especially the lightest stop) at 14 TeV LHC. Finally, we will estimate the fine tuning in these models and show that they achieve essentially the best tuning possible in the MSSM (percent level).
The remainder of this paper is organized as follows: Since no strongly coupled UV completion is needed for the non-MFV models, we discuss those first in section 2, as well as their phenomenology when coupled to minimal gauge mediation. In section 3 we analyze the MFV example in a similar way. In section 4, we UV complete the MFV model using Seiberg duality. Finally, in the conclusions we list some potential future directions suggested by our work. A general discussion of the little $A/m^2$ problem and Landau poles in models for tree-level $A$-terms is left for appendix \ref{app:generalization}.
\section{A non-MFV model}
\label{sec:NMFV models}
As discussed in the introduction, the non-MFV model (\ref{exampleU}) has a less severe version of the little $A/m^2$ problem, and thus does not need an immediate UV completion, unlike the MFV model (\ref{exampleH}). Since the story is simpler here, let us start by analyzing the non-MFV model in detail. Apart from the issues of flavor alignment discussed in the introduction, the form of the renormalizable superpotential (\ref{exampleU}) is the most general that couples the spurion, messengers and MSSM fields up to terms that are irrelevant for our purposes (powers of the spurion $X'$ and a small soft mass for the messenger pair from $X'\phi_{u}\tilde{\phi}_{u}$).
After diagonalizing the mass matrix and integrating out $\phi_u, \tilde{\phi}_u$ at the messenger scale $M$, we obtain the IR effective theory
\begin{eqnarray}\begin{aligned}
W_{eff}&\supset -\kappa \frac{X'}{M} H_u Q_3 \bar{u}_3\\
K_{eff}&\supset \frac{X^{'\dagger} X'}{M^2} \bar{u}_3^\dagger \bar{u}_3
+\frac{\kappa^2}{M^2} H_u^\dagger H_u Q_3^\dagger Q_3
\label{eq:NMFVeff}
\end{aligned}\end{eqnarray}
The irrelevant operator induced in the low energy superpotential leads to an $A$-term for the corresponding MSSM fields after substituting $\langle X'\rangle = \theta^2 F_{X'}$. However, an additional contribution to $m_{\bar{u}_3}^2$ from the first term in the K\"ahler potential is also induced, such that
\begin{equation}
\delta m_{\bar{u}_3}^2 = -\frac{y_t^2}{\kappa^2} A_t^2
\label{eq:littleAmu}
\end{equation}
Note that the contribution to $m_{\bar{u}_3}^2$ is negative, so to avoid a tachyonic right handed stop, it must be cancelled off by additional contributions at the messenger scale (e.g.\ from GMSB) or from MSSM renormalization group running from the messenger scale down to the weak scale. If $\kappa\sim1$, the fine tuning from (\ref{eq:littleAmu}) is comparable to the fine tuning from the $A$-term itself, since both enter the running of $m_{H_u}^2$ in exactly the same fashion. Taking $\kappa >1$ therefore does not substantially improve the overall fine tuning of the model. One major improvement relative to the non-MFV models considered in \cite{Evans:2013kxa} is that there are no sizeable contributions generated to $m_{H_u}^2$ from integrating out the messengers.
To study the phenomenology of a model with tree-level $A$-terms and a 125~GeV Higgs, we must add our tree-level $A$-term module (\ref{exampleU}) to an underlying model for the rest of the MSSM soft masses. While in principle any model could be used, GMSB is a particularly well-motivated choice given the SUSY flavor problem. So for simplicity and concreteness, let us now specialize to the case of minimal gauge mediation (MGM) with ${\bf 5}\oplus {\bf \bar 5}$ messengers \cite{Dine:1993yw,Dine:1994vc,Dine:1995ag}.
The parameter space of our model is as follows. The MGM sector of the model is characterized by four parameters: messenger index $N_{m}$, $\tan{\beta}$, messenger scale $M$ and SUSY-breaking mass scale $\frac{F_X}{M}$, where $F_X$ is the highest component vev of the SUSY breaking spurion. We take the masses of the additional messengers in (\ref{exampleU}) to be the same scale $M$ for simplicity. We consider $\mu$ and $B_{\mu}$ to be determined by the EWSB conditions and we remain agnostic about their origin. Finally, our model contains additional parameters $\frac{F_{X'}}{M}$, which sets the scale for the tree level contribution to $A_t$, and the coupling $\kappa$ (see (\ref{exampleU})).
A low messenger scale $M=250$ TeV and a large messenger number $N_m=3$ are motivated by the simultaneous requirements of reducing the tuning from the RG while allowing a large enough SUSY scale to be achieved for the Higgs mass. (A different choice of messenger number does not alter the phenomenology heavily, for reasons that will be explained later.) We take $\tan\beta = 20$ to saturate the tree level bound of the Higgs mass and $\kappa=1$ for simplicity and perturbativity. With these choices, the parameter space of our models reduces to $(\frac{F_{X'}}{M}, \frac{F_{X}}{M})$. (Recall that we must take $\frac{F_{X'}}{M}\sim\frac{1}{16\pi^2}\frac{F_{X}}{M}$ to achieve $A$-terms comparable to the GMSB soft masses.) To make contact with the IR observables, we can trade $(\frac{F_{X'}}{M}, \frac{F_{X}}{M})$ by the IR values of $A_t$ and the mass of the lightest stop $m_{\tilde{t}_1}$ or the mass of the lightest stau $m_{\tilde{\tau}_1}$. This parametrization is especially relevant for the LHC phenomenology, since $\tilde t_1$ and $\tilde \tau_1$ are the lightest colored particle and the NLSP respectively, as will be seen shortly.
\begin{figure}
\centering
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{nmfv55barstop.eps}
\label{fig:nmfvstop}
\end{subfigure}%
\hfill
~
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{nmfv55barstau.eps}
\label{fig:nmfvstau}
\end{subfigure}
\caption{Contours of the Higgs mass (black), geometric mean of the stop masses (blue) and tuning (dashed), in the $(A_t,m_{\tilde{t}_1})$ (left) and $(A_t,m_{\tilde{\tau}_1})$ (right) planes. The shaded region on the $(A_t, m_{\tilde{\tau}_1})$ plane corresponds to points with tachyonic stops. The black dot on both figures corresponds to the same point in parameter space, with a spectrum presented in figure \ref{fig:NMFVspec}. All quantities are evaluated at $M_{SUSY}$.}
\label{fig:nmfv}
\end{figure}
To generate the IR spectrum we use \texttt{SOFTSUSY 3.5.1} \cite{Allanach:2001kg}. Fine tuning $\Delta_{FT}$ is calculated according to the measure introduced in \cite{Evans:2013kxa}, given by
\begin{eqnarray}\begin{aligned}
\label{eq:tuning}
&\Delta_i \equiv \frac{\partial \log m_z^2}{\partial \log \Lambda_i^2}&\Lambda_i \in \{g_1^2 \frac{F_{X}}{M}, g_2^2\frac{F_{X}}{M}, g_3^2\frac{F_{X}}{M},\frac{F_{X'}}{M}, \kappa \frac{F_{X'}}{M},\mu\}\\
&\Delta_{FT} \equiv \max \Delta_i. &
\end{aligned}\end{eqnarray}
The results are presented in figure \ref{fig:nmfv} where we show contours of the Higgs mass, tuning and $M_{SUSY}$, both in the $( A_t,m_{\tilde{t}_1})$ and $(A_t, m_{\tilde{\tau}_1})$ planes. Note that $M_{SUSY}$ is significantly larger than $m_{\tilde{t}_1}$. This is because the two stop soft masses are split due to the negative contribution to $m_{\bar{u}_3}^2$ in (\ref{eq:littleAmu}). In the gray shaded region the GMSB contribution is insufficient to cancel this negative contribution, and the spectrum is invalidated by a stop tachyon. The main source of tuning in this model is the running effect due to the colored spectrum or the $A$-term. From the Higgs and tuning contour lines in both figures, we see that the model is able to reproduce the Higgs mass, while keeping fine tuning to the percent level (which is basically the best that can be achieved in the MSSM). Moreover, the Higgs mass can be reproduced in interesting parts of parameter space, where there is both a light colored particle $m_{\tilde{t}_1}$ and a light slepton $m_{\tilde{\tau}_1}$.
A typical spectrum for the model is presented in figure \ref{fig:NMFVspec}, which corresponds to the black dot indicated in the two different planes presented in figure \ref{fig:nmfv}.\footnote{We choose our benchmark point here and in the next subsection to have $m_h=124$~GeV in order to account optimistically for the theory uncertainty on the Higgs mass calculation.} In general, the spectrum across the parameter space of our model is basically that of MGM with $N_{mess}=3$ (gaugino unification, colored sparticles heavier than electroweak sparticles, right-handed stau NLSP, etc.). There are, however, two key differences. First, in order to counteract the large negative contribution (\ref{eq:littleAmu}) to the right-handed stop, the MGM scale $\frac{F_{X}}{M}$ is considerably larger than would otherwise be the case. This results in the other colored sparticles being essentially decoupled. It also results in a higher gravitino mass, which explains \cite{Ambrosanio:1997bq} why slepton co-NLSPs do not occur in figure \ref{fig:nmfv}. Second, the right-handed sleptons are a bit lighter than in MGM due to the effects of running induced by the split stops. Amusingly, this effect of running means that the stau is the NLSP even for lower $N_{mess}$, unlike in MGM, where lowering $N_{mess}$ leads to bino NLSP.\footnote{Note that if we exchange the roles of $\bar{u}_3$ and $Q_3$ in (\ref{exampleU}), a negative soft mass for $Q_3$ would be induced instead, leading to a heavier $\tilde{\tau}_1$ through running. In this case, it could be possible to have a bino NLSP even for $N_m>1$.}
Due to the split spectrum, the largest sparticle pair production cross sections at LHC correspond to $\tilde{t}_1$ and the right-handed sleptons. Pair production of stops leads to a decay chain with jets, leptons and missing energy. When right handed sleptons are directly pair produced, the decay chain will include relatively soft leptons (due to the moderate splitting of the right handed sleptons and the stau), taus and missing energy. Of course the direct pair production of staus will lead to taus and missing energy.
Of the above signatures, the most spectacular one is given by the decay of pair produced stops, which can contain two jets, 4 leptons (from the decay of the bino to RH sleptons and RH sleptons to stau), and two $\tau$ jets plus missing energy. A search with a similar topology was carried out in \cite{Aad:2014mra}, where a limit on the total strong production cross section of $\sim1$ fb was obtained. This limit can be used to set an approximate bound on our parameter space, by comparing with our model's tree level total strong production cross section, which we obtain using {\sc MadGraph} \cite{Alwall:2014hca}. This leads to excluding stops roughly below $800$ GeV in the parameter space presented in figure \ref{fig:nmfv}, which corresponds to staus heavier than $150$ GeV.
\begin{figure}[t]
\begin{center}
\includegraphics[width=12cm]{pointNMFVmodel.eps}
\caption{Spectrum for the point marked with the dot in figure \ref{fig:nmfv}. The Higgs mass is $m_h=124$ GeV, with $A_t=-2.9$ TeV. $\tilde{\tau}_1$ is $17$ GeV lighter than the right handed sleptons. The Higgsino mass is $\mu=1.05$ TeV. Fine tuning is $\sim 1/400$.}
\label{fig:NMFVspec}
\end{center}
\end{figure}
The spectrum presented in figure \ref{fig:NMFVspec} is inaccessible to the LHC run at $8$ TeV, but it will become accessible at $14$ TeV. The total SUSY cross section of such point at the $14$ TeV LHC is $~8$ fb, while the total tree level colored production cross section is $~2$ fb. Relevant searches will be the updated versions of multilepton or GMSB-inspired searches as \cite{Chatrchyan:2014aea} and \cite{Aad:2014mra}.
\section{An MFV model}
\label{sec:MFV}
Next we will turn to the MFV model (\ref{exampleH}). Apart from the issues of UV completions to be discussed in the next section, this model is slightly more complicated than the non-MFV model because here we would like to generate the MSSM up-type Yukawas and the $A$-terms from the same operator. To achieve this, it is necessary to turn on a lowest component vev for $X'$, which implies that one must re-diagonalize the messenger mass matrix prior to integrating out the messengers. For later convenience, we will redefine $X'$ so that its lowest component vev is separated out and denoted by $X_0'$. Then (\ref{exampleH}) becomes
\begin{equation}\label{minimal MFV}
W = (X'_0+X')H_u \tilde \phi + \lambda_u^{ij} \phi Q_i \bar{u}_j + M\phi \tilde \phi \end{equation}
with $\langle X'\rangle=F_{X'} \theta^2$. The form of (\ref{minimal MFV}) is the most general allowed by a $\mathbb{Z}_3$ symmetry, as detailed in table \ref{tab:Z3}, which also allows for a $\mu$-term and down type Yukawas,
\begin{equation}
\label{downtypeYuk}
\delta W = \mu' H_u H_d + \lambda_d^{ij} H_d Q_i \bar{d}_j + \lambda_e^{ij} H_d L_i \bar{e}_j
\end{equation}
We will not discuss the down sector Yukawas any further.
\begin{table}[t]
\begin{center}
\begin{tabular}{|c|cccccc|}\hline
&$X'$&$Q,\bar{u},\bar{d},L,\bar e$ &${H}_u$& $H_d$& $\phi$ &$\tilde{\phi}$ \\\hline
$\mathbb{Z}_3$&1/3&2/3&1/3&2/3&2/3&1/3 \\\hline
\end{tabular}
\end{center}
\caption{Charge assignments securing (\ref{minimal MFV}) and (\ref{downtypeYuk}).}
\label{tab:Z3}
\end{table}
After diagonalizing the mass matrix and integrating out the heavy messenger states, we are left with the supersymmetric effective action:
\begin{eqnarray}\begin{aligned}\label{MFVeffWK}
W_{eff}&\supset y_u^{ij}\left(1+\cot\theta_H\cos\theta_H\frac{X'}{M'}\right) H_u Q_i \bar{u}_j +\mu\left(1+\sin \theta_H \frac{X'}{M'}\right) H_u H_d \\
K_{eff}&\supset \frac{\cos^2 \theta_H}{M'^2}X^{'\dagger} X' H_u^\dagger H_u+ \frac{\cot^2 \theta_H}{M'^2}y_u^{il} {y_u^{jk}}^{*} Q_i^\dagger u_l^\dagger Q_j u_k \end{aligned}\end{eqnarray}
where
\begin{eqnarray}\begin{aligned}
& M'=\sqrt{X^{'2}_0+M^2},\qquad \sin \theta_H=\frac{X'_0}{M'},\qquad y_u^{ij}=-\lambda_u^{ij}\sin\theta_H, \qquad \mu= \mu'\cos\theta_H
\label{eq:rotationMFV}
\end{aligned}\end{eqnarray}
and we have everywhere expanded in $\mu'\ll M,X_0$, keeping only the lowest nonzero order.
In (\ref{MFVeffWK}), the first term in the effective superpotential leads to an $A$-term proportional to the up-type Yukawas. The second term in the effective K\"ahler potential is an MFV interaction suppressed by the messenger scale, so it is safe from flavor constraints \cite{Isidori:2010kg}. Meanwhile, the first term in $K_{eff}$ represents a contribution to the soft mass of $H_u$:\footnote{The second term in the effective superpotential (\ref{MFVeffWK}) gives rise to $B_\mu= \mu A_t\tan^2 \theta_H$ at the messenger scale. While this is parametrically of the right size for EWSB, it has the incorrect sign to lead to the large $\tan\beta$ EWSB condition $B_\mu\approx 0$ at the weak scale. Thus a more complete model that also aspires to explain the origin of $\mu$ and $B_\mu$ must include additional contributions to these parameters.}
\begin{equation}
\delta m^2_{H_u}
= - |A_t|^2\tan^2 \theta_H
\label{eq:MFVsoftmass}
\end{equation}
This is a manifestation of the little $A/m_H^2$ problem.
Note that this contribution is negative, so it is not dangerous for electroweak symmetry breaking, unlike what was found in the K\"ahler potential models \cite{Craig:2012xp}. However, if $\tan \theta_H\gtrsim 1$ it still represents a major contribution to fine-tuning. Taking $\tan \theta_H\ll 1$ would alleviate this fine-tuning problem, but at the cost of enlarging the underlying coupling $\lambda_u^{33}$ according to (\ref{eq:rotationMFV}). This leads to a Landau pole at low scales and a UV completion becomes necessary. Such a UV completion is the subject of section \ref{sec:fullycomposite}, in which we use Seiberg duality \cite{Seiberg:1994pq, Seiberg:1994bz} to realize the large coupling $\lambda^{33}_u$.
As in the previous section, to generate the rest of the soft masses we specialize to the case of MGM. The parameter space is essentially the same as before, namely the MGM sector is described by $N_m$, $\tan\beta$, $M$ and $\frac{F_{X}}{M}$, while our effective theory contains $\frac{F_{X'}}{M}$ which sets the scale for the tree level contribution to $A_t$, and a coupling $\lambda_u^{33}$. Again, we consider $\mu$ and $B_{\mu}$ to be determined by the EWSB conditions. We fix most of the parameters to the same values as before -- $N_m=3$, $\tan\beta=20$ and $M=250$~TeV -- for essentially the same reasons.
Finally, we consider two values for $\lambda_u^{33}$: $\lambda_u^{33}=1$ is chosen to illustrate the perturbative case, while $\lambda_u^{33}=3$ is studied since it has a beneficial effect on decreasing tuning. With these choices, the parameter space of our model reduces to $(\frac{F_{X'}}{M}, \frac{F_{X}}{M})$, which we can trade for the IR values of the $A$-term $A_t$ and the gluino mass $M_{\tilde{g}}$.
In figure \ref{fig:MFVhiggs} we show contours of the Higgs mass, tuning and $M_{SUSY}$ in the $(M_{\tilde{g}}, A_t)$ plane for the two choices of $\lambda_u^{33}$. In both figures \ref{fig:k1} and \ref{fig:k3} a large Higgs mass can be achieved with moderate values of $M_{SUSY}$ thanks to the large $A$-terms. In figure \ref{fig:k1} however, the $\mu$-term is very large and induces sizable negative contributions to $m_h$ through the stau and sbottom sectors. This implies that a higher $M_{SUSY}$ is needed to obtain the correct Higgs mass. (see e.g. \cite{Carena:1995wu}.) The main source of tuning can be either the large induced Higgs soft mass from (\ref{eq:MFVsoftmass}) or, for large $M_{SUSY}$, the running effect. We immediately see from figure \ref{fig:k1} that the first of these sources represents a serious tuning problem for $\lambda_u^{33}=1$, in which case for a $125$ GeV Higgs we obtain a typical tuning of $\sim 10^{-4}$. In figure \ref{fig:k3} we see the beneficial effect of considering a larger value for $\lambda_u^{33}$. This choice suppresses the fine tuning induced by (\ref{eq:MFVsoftmass}), in such a way that a $125$ GeV Higgs can be achieved while keeping tuning to the one part in $\sim 500$ level.
\begin{figure}[t!]
\centering
\begin{subfigure}{0.5 \textwidth}
\psfrag{Y}[Bc]{\small $M_{\tilde{g}}$ \normalsize}
\psfrag{X}[lc]{ \small $A_t$ \normalsize}
\centering
\includegraphics[width=7cm]{mfvksmall.eps}
\caption{$\lambda_u^{33}=1$}
\label{fig:k1}
\end{subfigure
\begin{subfigure}{0.5 \textwidth}
\psfrag{Y}[Bc]{\small $M_{\tilde{g}}$ \normalsize}
\psfrag{X}[lc]{ \small $A_t$ \normalsize}
\centering
\includegraphics[width=7cm]{mfvklarge.eps}
\caption{$\lambda_u^{33}=3$}
\label{fig:k3}
\end{subfigure}
\caption{Contours of the Higgs mass (black), geometric mean of the stop masses (blue) and tuning (dashed), for two choices of $\lambda_u^{33}$ with $N_m=3$, $\tan\beta=20$, $M=250$ TeV. Different Higgs mass contours are presented to account for the uncertainty in the theoretical Higgs mass calculation. The shaded region corresponds to tachyonic stops/staus. The dot on the figure on the right corresponds to the point in parameter space with the spectrum presented in figure \ref{fig:largekappa}. The parameter space below the red line on the same figure is excluded by \cite{Aad:2014mra}. All quantities are evaluated at $M_{SUSY}$.}\label{fig:MFVhiggs}
\end{figure}
In figure \ref{fig:largekappa} we present a typical spectrum for the model with $\lambda_u^{33}=3$, which corresponds to the black dot in figure \ref{fig:k3}. This model is even more similar to MGM with stau NLSP than the one presented in the previous subsection, since there is no negative contribution to the right-handed stop to counteract. The only difference now with MGM is the large $A$-term, which has a minor effect on the rest of the spectrum primarily through the RG. The MGM collider signatures here are potentially spectacular. If colored superpartners are accessible to collider experiments they will lead to a long decay chain including jets, leptons and missing energy. As in our non-MFV model, searches that look for jets, tau final states and large missing energy can be sensitive to this spectrum when the strong production is accessible. In particular ATLAS search \cite{Aad:2014mra} analyses a similar spectrum and their results apply directly to our case, setting strong bounds on parts of the parameter space. For $\tan\beta=20$, gluinos of up to $1.6$ TeV are excluded, which corresponds to a total strong production cross section of $\sim1.5$ fb at tree level \cite{Alwall:2014hca}.
Multilepton searches could also be a leading probe of this model, especially when the colored sparticles are too heavy to be produced. The stau NLSP scenario considered in \cite{Chatrchyan:2014aea} can be sensitive to our case, but since in our spectrum $\tilde{m}_{e_R}-\tilde{m}_{\tau_1}\sim 20$ GeV and $150 \text{ GeV}<\tilde{m}_{\tau_1}$, the obtained bounds are not currently relevant for us. However, updates of these searches in Run II of the LHC can be very interesting for our models.
\begin{figure}[t]
\begin{center}
\includegraphics[width=12cm]{pointMFVmodel.eps}
\caption{Spectrum for the point shown in figure \ref{fig:k3}. The Higgs mass is $m_h=124$ GeV, with $A_t=-2.7$ TeV. $\tilde{\tau}_1$ is $32$ GeV lighter than the right handed sleptons. The Higgsino mass is $\mu=1.3$ TeV. Fine tuning is $\sim 1/400$.}
\label{fig:largekappa}
\end{center}
\end{figure}
\section{A composite model from Seiberg duality}
\label{sec:fullycomposite}
As discussed in the previous section, the little $A/m_H^2$ problem in the MFV model (\ref{eq:MFVsoftmass}) necessitates a large value for $\lambda_u^{33}$, and the theory has a Landau pole at a low scale. One way to explain physics above the Landau pole is to build composite models that naturally provide $|\lambda_{u}^{33}|\gg 1$ due to the underlying strong interactions. In general, characterizing such a strongly coupled UV completion is challenging at best, however in the context of supersymmetric gauge theories we can make use of Seiberg duality \cite{Seiberg:1994pq, Seiberg:1994bz}. We embed the model of section \ref{sec:MFV} in the magnetic side of the duality, where the fields $Q_3, \bar{u}_3$ and $\phi$ will be composite degrees of freedom. Since it is conceptually simpler, we first discuss the electric side of the duality. In a second stage we discuss the mapping to the composite degrees of freedom on the magnetic side, and we complete the model by adding in a number of spectator fields.
\subsection{Electric theory}
\label{subsec:electric}
The electric theory is defined by SQCD with $N_c=2$ colors and $N_f=3$ flavors. Since the fundamental of the electric gauge group $SU(2)_E$ is pseudo-real, this theory is invariant under an $SU(6)$ global symmetry. It is therefore convenient to parametrize its degrees of freedom with a single matter field $q^i_a$ in the fundamental of $SU(2)_E$ and $SU(6)$. The standard model gauge group can be embedded in the global symmetry as follows
\begin{equation}\label{embedding}
SU(6)\supset SU(5)\supset SU(3)_c \times SU(2)_L \times U(1)_Y
\end{equation}
With this matter content, the global symmetry is anomalous. In section \ref{subsec:spectators} we will introduce some spectator fields to cancel the gauge anomalies and give vector-like masses to some exotics. Note that because the global symmetry contains $SU(5)$, grand unification is manifest in this model from the outset. Concretely, the fundamental of $SU(6)$ trivially decomposes as
\begin{equation}
\mathbf{6}=\mathbf{5}\oplus \mathbf{1}
\end{equation}
where the $\mathbf{5}$ further decomposes into standard model representations in the conventional way. The quantum numbers of $q_a^i$ are summarized in table \ref{tab:electricmattercontent}.
In addition to hypercharge $U(1)_Y$, the breaking pattern in (\ref{embedding}) allows for an additional global symmetry which we will denote by $U(1)_G$. As will be seen in section \ref{subsec:magnetic}, it is necessary to consider the MSSM baryon number to be part of the global symmetries for proton stability. It will also be seen that baryon number has a unique embedding in $U(1)_G$ and $U(1)_Y$ given by:
\begin{equation}\label{baryongen}
B=\frac{4}{5} Y+\frac{1}{10} G \quad\quad \mathrm{with} \quad\quad \begin{array}{l}Y=\text{diag}(-\frac{1}{3},-\frac{1}{3},-\frac{1}{3},\frac{1}{2},\frac{1}{2},0)\\G=\text{diag}(1,1,1,1,1,-5)\end{array}
\end{equation}
Note that both the electric and magnetic theories have a $\mathbb{Z}_{N_f}$ discrete symmetry that is leftover from the anomalous global $U(1)$ symmetry. As we will discuss in the next subsection, we will identify this $\mathbb{Z}_3$ with the one of table \ref{tab:Z3}.
\begin{table}[t]\centering
\begin{tabular}{|c|c|c|ccc|c|c|}\hline
GUT&field&$SU(2)_E$& $SU(3)_c$ & $SU(2)_L$ & $U(1)_Y$&$\mathbb{Z}_3$&$B$\\\hline
\multirow{2}{*}{$\mathbf{5}$}&$q_c$ & $\square$ & $\square $&1&$-\frac{1}{3}$&$\frac{1}{3}$&$-\frac{1}{6}$\\
&$q_L$ & $\square$ & $1$& $\square$&$\frac{1}{2}$&$\frac{1}{3}$&$\frac{1}{2}$\\\hdashline
$\mathbf{1}$&$q_S$ & $\square $ & $1 $&1&$0$&$\frac{1}{3}$&$-\frac{1}{2}$\\\hline
\end{tabular}
\caption{Matter content of the electric theory. $q=q_c\oplus q_L\oplus q_S$ form a fundamental of the $SU(6)$ global symmetry. \label{tab:electricmattercontent} }
\end{table}
\subsection{Magnetic theory}
\label{subsec:magnetic}
This theory s-confines in the IR and has a weakly-coupled magnetic dual description in terms of the mesons and baryons of the electric theory as described in table \ref{tab:magneticmattercontent}. These gauge invariants $q^iq^j$ transform as the antisymmetric tensor $\mathbf{15}_A$ of the global $SU(6)$. Under $SU(5)$ this decomposes as
\begin{equation}
\mathbf{15}_A= \mathbf{10}_A\oplus \mathbf{5}.
\end{equation}
\begin{table}[!t]\centering
\begin{tabular}{|c|c|ccc|c|c|c|}\hline
GUT &field& $SU(3)_c$ & $SU(2)_L$ & $U(1)_Y$&composite&$\mathbb{Z}_3$&$B$\\\hline
\multirow{3}{*}{$\mathbf{10}$}&$\color{black}{Q_3}$ & $\square$&$\square$&$1/6$&$q_c q_L$&$2/3$&1/3\\
&$\color{black}{\overline u_3}$ & $\overline{\square}$&$1$&$-2/3$&$q_c q_c$&$2/3$&-1/3\\
&$E'$& $1$&$1$&$1$&$q_L q_L$&$2/3$&1\\\hdashline
\multirow{2}{*}{$\mathbf{5}$}&$\phi$ & $1$&$\square$&$1/2$&$ q_L q_S$&$2/3$&0\\
&$d'$ & $\square$&$1$&$-1/3$&$q_c q_S$&$2/3$&-2/3\\\hline
\end{tabular}
\caption{Matter content of the magnetic side of the duality. All fields fill out complete GUT multiplets. Since $E'$ carries baryon number, it cannot be identified with a right handed lepton. \label{tab:magneticmattercontent}}
\end{table}
The resulting $SU(5)$ representations allow us to identify $Q_3$, $\bar{u}_3$ and $\phi$ with composite degrees of freedom. Note that the baryon numbers of $Q_3$ and $\bar{u}_3$ uniquely determined the coefficients of $U(1)_Y$ and $U(1)_G$ in (\ref{baryongen}). The rest of the composite fields are $E'$ and $d'$, of which $E'$ has the same gauge quantum numbers as right handed leptons, but non-zero baryon number.
The confining electric gauge group dynamically generates a superpotential in the magnetic dual, given by
\begin{eqnarray}\begin{aligned}\label{pfaff}
W_{\mathrm{mag}} &= \frac{1}{\Lambda^3}\mathrm{Pf} (q^i q^j)\\
&= \kappa ( \phi {Q}_3 {\bar{u}}_3 - {Q}_3{Q}_3 d'+d' \bar{u}_3 E')\label{eq:magpotential}
\end{aligned}\end{eqnarray}
where $\mathrm{Pf}$ is the Pfaffian of the antisymmetric matrix $q^i q^j$, and we used the mapping to the magnetic theory in the second line.
The coupling $\kappa$ descends from the strong dynamics in the electric theory and can be large (for concreteness we assumed $\kappa \sim 3$ in section \ref{sec:MFV}). From the last two operators in (\ref{eq:magpotential}) it should also be clear that rapid, dimension 6 proton decay would be introduced if one were to identify $E'$ with one of the MSSM leptons. The $B$ and $\mathbb{Z}_3$ charges for the composite fields are fixed by those of the electric quarks in table \ref{tab:electricmattercontent}.
\subsection{Complete model with spectators}
\label{subsec:spectators}
Let us now weakly gauge a $SU(3)_c \times SU(2)_L \times U(1)_Y$ subgroup of the global symmetry. To cancel anomalies, fill out complete GUT multiplets, and match the field content of the magnetic theory to the model of section \ref{sec:MFV}, we add a number of fundamental fields, which are all spectators as far as the Seiberg duality is concerned. Among these spectators are all three $\bar d$, $L$ and $\bar e$ generations of the MSSM, as well as the first two generations of the $Q$ and $\bar u$ sectors. Finally, the $H_u$ and $H_d$ are spectators as well, but do not come in complete GUT multiplets. This is nothing other than the usual doublet-triplet splitting problem in models with grand unification. The spectators and their quantum numbers are introduced in table \ref{tab:spectatormattercontent}. Aside from the usual baryon number, we also assign the $\mathbb{Z}_3$ charges for the spectator fields such that the symmetry in table \ref{tab:Z3} is realized. In addition to the fields we introduced so far, one may choose to add up to three pairs of conventional, $\mathbf{5}$-$\mathbf{\overline{5}}$ gauge mediation messengers without spoiling perturbative gauge coupling unification.\footnote{We hereby assume that any uncalculable threshold corrections at the compositeness scale are negligible.}
\begin{table}[!t]\centering
\begin{tabular}{|c|c|ccc|c|c|}\hline
GUT &field& $SU(3)_c$ & $SU(2)_L$ & $U(1)_Y$&$\mathbb{Z}_3$&$B$\\\hline
\multirow{2}{*}{$\mathbf{\overline 5}$}&$\tilde{\phi}$ & $1$&$\square$&$-1/2$& $1/3$&0\\
&$\overline d'$ & $\overline{\square}$&$1$&$1/3$&$1/3$&2/3\\\hdashline
\multirow{2}{*}{$\mathbf{\overline 5}$}&$\color{black}{L_3}$ & $1$&$\square$&$-1/2$&$2/3$&0\\
&$\color{black}{\overline{d}_3}$ & $\overline{\square}$&$1$&$1/3$&$2/3$&-1/3\\\hdashline
\multirow{3}{*}{$\mathbf{10}$}&$Q'$ & $\square$&$\square$&$1/6$&$1/3$&1/3\\
&$\overline U'$ & $\overline{\square}$&$1$&$-2/3$&$1/3$&-1/3\\
&$\color{black}{\bar{e}_3} $& $1$&$1$&$1$&$2/3$&0\\\hdashline
\multirow{3}{*}{$\mathbf{\overline{10}}$}&$\overline Q'$ & $\overline\square$&$\square$&$-1/6$&$2/3$&-1/3\\
&$ U'$ & ${\square}$&$1$&$2/3$&$2/3$&1/3\\
&$\overline E' $& $1$&$1$&$-1$&$1/3$&-1\\\hdashline
\multirow{1}{*}{$$}&$H_u$ & $1$&$\square$&$1/2$& $1/3$&0\\\hdashline
\multirow{1}{*}{$$}&$H_d$ & $1$&$\square$&$-1/2$& $2/3$&0\\
\hline
\end{tabular}
\caption{Spectators of the Seiberg duality required to cancel anomalies and fill out complete GUT multiplets. Primed fields have heavy vector-like masses and are integrated out at the duality scale. The first two generations are also spectators but are not shown here for simplicity. \label{tab:spectatormattercontent}}
\end{table}
All the non-MSSM fields have vector-like masses. Some arise from Yukawa interactions in the electric theory, while others are mass terms:
\begin{eqnarray}\begin{aligned}\label{eq:electricmass}
W_{\mathrm{elec}}&\supset y_{d'} q_c q_S \overline{d}'+y_{E'} q_L q_L \overline{E}' + M_{Q'}Q' \bar Q' + M_{U'}U'\bar U'\\
&\qquad \rightarrow W_{\mathrm{mag}}\supset y_{d'} \Lambda d' \overline{d}'+y_{E'} \Lambda E' \overline{E}'+ M_{Q'}Q' \bar Q' + M_{U'}U'\bar U'
\end{aligned}\end{eqnarray}
Those that are Yukawas in the electric theory are naturally of the same size as the compositeness scale $\Lambda$, and so for unification we must also take $M_{Q'}\sim M_{U'}\sim \Lambda$.
We can see that it is possible to reproduce the model in (\ref{minimal MFV}) by adding interactions between spectators and the composites and between spectators themselves if we allow the following interactions
\begin{eqnarray}\begin{aligned}\label{eq:spectpotential}
\delta W &=(X_0'+X') H_u \tilde{\phi} + \tilde{\lambda}_{u}^{ij} \phi {Q}_i {\bar{u}}_j + M \phi \tilde{\phi}
\end{aligned}\end{eqnarray}
where $i,j$ identify quark fields in the gauge eigenbasis. To avoid clutter, we suppressed the mass terms that are introduced in (\ref{eq:electricmass}), as well as the $\mu$-term and the down and lepton Yukawas. This superpotential is generic if we impose the $\mathbb{Z}_3$ symmetry of tables \ref{tab:magneticmattercontent} and \ref{tab:spectatormattercontent}.
As noted earlier, the first and second generations of the MSSM matter fields are all elementary and spectators as far as the Seiberg duality is concerned. Since $\phi$ is a composite operator in the electric theory, all up-type Yukawa couplings (other than the top Yukawa) must arise from irrelevant operators in the electric theory. (Recall that the ${\Bbb Z}_3$ symmetry of table \ref{tab:Z3} forbids the usual up-type Yukawa couplings $H_u Q \bar u$.) For instance
\begin{eqnarray}\begin{aligned}\label{eq:irrelev1}
\frac{1}{\Lambda_{UV}^2}(q_L q_S)(q_c q_L)\bar u_2\quad&\rightarrow \quad\frac{\Lambda^2}{\Lambda_{UV}^2}\phi Q_3\bar u_2\\
\frac{1}{\Lambda_{UV}}(q_L q_S)Q_2\bar u_2\quad&\rightarrow\quad \frac{\Lambda}{\Lambda_{UV}}\phi Q_2\bar u_2\label{eq:irrelev2}
\end{aligned}\end{eqnarray}
where $\Lambda_{UV}$ is a cut-off scale of the electric theory. In the notation of section \ref{sec:MFV} this yields:
\begin{eqnarray}\begin{aligned}
\label{eq:yukawa}
\lambda_u^{ij}&= \kappa \delta^{i3}\delta^{3j}+\tilde \lambda_u^{ij}\sim\left(\!\!\begin{array}{ccc}0&0&0\\0&0&0\\0&0&\kappa\end{array}\!\!\right)+
\left(\!\!\begin{array}{ccc}\epsilon&\epsilon&\epsilon^2\\\epsilon&\epsilon&\epsilon^2\\\epsilon^2&\epsilon^2&\epsilon^3\end{array}\!\!\right)
\end{aligned}\end{eqnarray}
with $\epsilon\sim \Lambda/\Lambda_{UV}\ll 1$. The composite sector therefore naturally provides a partial explanation of the texture of the up-type Yukawa matrix.
Since $Q_3$ is a composite degree of freedom, it also predicts $\epsilon \sim y_b \sim 0.1$, but the rest of the hierarchies in $y_d$ and $y_\ell$ are not explained.
Upon integrating out the messenger fields, the analysis further reduces to what was presented in section \ref{sec:MFV}. There is one exception, in the sense that the model is no longer manifestly MFV since the third generation was given a special treatment. In particular a non-MFV dimension six operator is generated in the K\"ahler potential from integrating out $d'$ in (\ref{eq:magpotential})
\begin{equation}
\delta K_{\mathrm{eff}}\sim {1\over \Lambda^2}(Q_3 Q_3)^\dagger (Q_3 Q_3)\sim {1\over \Lambda^2} (u_3 d_3)^\dagger (u_3 d_3).
\end{equation}
By rotating $Q_3$ to the mass eigenbasis, this operator can in principle couple quarks of different generations. However note that this operator does not introduce any new CP phase into the model and it does not contribute to FCNC processes at tree level. Moreover it is suppressed by the duality scale that is above the messenger scale $\gtrsim 100$~TeV. The effects in the first two generation quarks are further suppressed by powers of $\epsilon$ coming from (\ref{eq:yukawa}). For instance, the operator contributing to $K$-$\bar K$ mixing receives an additional suppression of $\sim \epsilon^8$. Therefore we conclude that it is consistent with the bounds from flavor observables \cite{Isidori:2010kg}.
\section{Conclusions}
In this paper, we presented a new mechanism to generate large $A$-terms through tree-level superpotential operators. We provided explicit examples of both MFV and non-MFV models. In contrast to the conventional setups with one-loop $A$-terms through K\"ahler potential operators, our tree-level mechanism does not induce any dangerously large soft masses and is therefore manifestly free from the $A/m^2$ problem.
Generically, a soft mass of the same order as the $A$-term is nevertheless still generated. For the non-MFV example this contribution greatly increases the splitting between the stop mass eigenstates, but otherwise does not significantly impact the phenomenology or the fine tuning. For the MFV case, the soft mass could potentially lead to disastrous levels of fine tuning, but it can be brought under control by the existence of strong dynamics near the messenger scale. We provide an example of such a composite sector which has a description in terms of Seiberg duality and which explicitly allows for gauge coupling unification.
Some potential future directions suggested by this work include:
\begin{itemize}
\item For concreteness, we focused on an MGM setup as a first example, but we emphasize that tree-level $A$-terms are merely a module that can be added to any mechanism for mediating SUSY breaking. In particular, it would be interesting to study whether the mechanism can naturally be embedded in more realistic models of dynamical supersymmetry breaking. In addition one could generalize $X'$ beyond the spurion limit, and study the effects of its dynamics on the phenomenology.
\item In the non-MFV case it may be interesting to embed the tree-level $A$-term into a full fledged theory of flavor.
\item In the MFV case, we saw that the $A$-term module generated a contribution to $B_\mu$ which unfortunately was of the wrong sign for EWSB. An interesting opportunity here would be to construct a complete model that produces both tree-level $A$-terms and $B_\mu$, perhaps along the lines of the models constructed in \cite{Komargodski:2008ax}.
\item Finally, the emergence of large $A$-terms from a composite sector in the MFV case may open a new avenue towards constructing a realistic model where large $A$-terms are generated at the TeV scale, hence further reducing the fine-tuning.
\end{itemize}
\section*{Acknowledgments}
We thank David E.~Kaplan for discussions that originally inspired this work. We also thank Nathaniel Craig, Jared Evans, Diego Redigolo, Arun Thalapillil and Scott Thomas for useful discussions. We are grateful to Ben Allanach for advice regarding \texttt{SOFTSUSY 3.5.1} and to Diego Redigolo for comments on the manuscript. The work of A.B.\ and D.E.\ is supported by DOE-ARRA-SC0003883 and DOE-SC0010008. D.E. is further supported by CONICYT Becas Chile and the Fulbright Program. The work of D.S.\ is supported by a DOE Early Career Award and a Sloan Foundation Fellowship. This manuscript has been authored by an author (SK) at Lawrence Berkeley National Laboratory under Contract No. DE-AC02-05CH11231 with the U.S. Department of Energy. The U.S. Government retains, and the publisher, by accepting the article for publication, acknowledges, that the U.S. Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for U.S. Government purposes.
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\section{Introduction}
Quantum measurement is a probabilistic process where the
detector's outcome is correlated with the system being in
a certain state. In turn, the detector's back-action onto the
system affects it according to the specific outcome of the
measurement~\cite{Helstrom}.
In a projective measurement this is described
by the wave-function collapse, in which the outcome of the
measurement is an eigenvalue of the measured observable,
and the system after the measurement is projected onto the
eigenstate corresponding to the obtained eigenvalue~\cite{von-neumann}.
The implications of quantum measurement become particularly interesting when applied to yet another key feature
of quantum mechanics--—to classically forbidden processes.
A striking example of such a forbidden process is that of
tunneling under a potential barrier, where a particle appears
on the other side of a barrier that it classically could not
surmount. Whereas this appearance indicates that a tunneling
event occurred, a direct observation of the particle during
its virtual passage under the barrier is required as additional
verification of this mechanism.
Indeed, such a proposition stands in direct conflict with
quantum measurement: if the particle is measured to be under
the barrier, it would collapse midway and not tunnel to the
other side. Additionally, in order for the particle to collapse in
the position of the barrier, it must obtain energy and appear
above the barrier. As a result, one may conclude that such a
measurement would correspond to effectively increasing the
potential barrier, and blocking tunneling altogether. Still there
is a constant interest in accessing the properties of a particle
tunneling under a barrier (e.g., its traversal time), with a variety
of approaches~\cite{Condon1931,Wigner1955,Buttiker1982,Sokolovski1987,Steinberg1995}.
Beyond the scheme of projective measurement, a better
suited and more realistic approach in detecting the virtual state
under the barrier is that of a modification of the systems's
state in a continuous process accompanied by a gradual
acquisition of information by the detector~\cite{Clerk2010}.
In particular,
in a weak measurement regime, as opposed to a strong
projective measurement, the detector's outcome corresponds
to the measured state of the system, but the back-action does
not disturb it much. This allows for nontrivial effects in
conditional (postselected) measurements, e.g., the appearance
of weak values~\cite{Aharonov:1988a}, and utilization of the measurement outcome
in quantum feedback circuits \cite{Korotkov:2001a,Korotkov:2001b}.
Such effects have been successfully employed in practical problems including
precision measurements~\cite{Hosten:2008,Dixon:2009,Starling:2009,Brunner:2010,Starling:2010b,Zilberberg:2011}, quantum state discrimination~\cite
{Zilberberg2013} or quantum state stabilization~\cite{Korotkov:2001b}. Moreover,
weak measurements have also been successfully employed
in the study of coherent quantum transport under a potential
barrier~\cite{Buttiker1983,Steinberg1995}, as well as transport through many-body virtual states~\cite{Romito2014}.
The detection of a tunneling process via weak measurements can be directly explored in electronic solid-state devices.
The typical system under consideration is that of transport
between two leads across a quantum dot. By tuning the
capacitance of the dot, one can position its eigenenergies
relative to the chemical potential of the leads, such that an
addition of charge onto the dot is unfavored by Coulomb
interactions. As a result, transport through the dot is classically
blocked~\cite{Aleiner2002}. In this regime, the transport through the dot
happens via cotunneling processes with a virtual (classically
forbidden) occupation of the dot~\cite{Glazman2005}.
Charge detection in quantum dots using a quantum point
contact (QPC) that is capacitively coupled to the dot is
experimentally well established~\cite{Field1993,Elzerman2003,DiCarlo2004,Harbusch2010,Gasparinetti2012,Granger2012,clemens2012,clemens2013,zumbuhl2014}. The QPCs can be tuned to be close to the quantum limit, making them
sensitive to the dot's charge fluctuations. Theoretically QPC
detectors are well understood, and a proper formalism has
been developed to describe the many-electron macroscopic
classical signal therein in response to local charges~\cite{gurvitz,Korotkov:2001b}.
Yet these descriptions are developed for classical transport
through the dot~\cite{gurvitz}, or for coherent charge oscillations in
isolated systems, e.g., double quantum dots~\cite{gurvitz,Korotkov:2001b,Romito:2008}.
In a recent work, measurement of cotunneling was addressed in the
regime of weak coupling between a dot and a QPC~\cite{Romito2014}. Using
a weak value approach extended to deal with the interacting dot
and a large bandwidth of the QPC, the short cotunneling time
was resolved. This led to a vanishingly small cotunneling time
in the regime of a diffusive 2D dot with sufficiently closed
contacts.
In this work, we extend the standard model of QPC transport~\cite{gurvitz,Korotkov:2001b} to treat the partial measurement of virtual
occupation of the dot in the cotunneling regime. In our
scheme, we treat the interaction between quantum dot (QD)
and QPC exactly by utilizing a rate equation formalism, which is perturbative in the tunneling and exact in the regime
of an almost pinched-off QPC. We illuminate and study
three mechanisms of back-action that a detector induces
onto coherent transport, namely (i) increased phase space for
QPC-assisted transport, e.g., inelastic processes in which an
electron enters the dot with energy $\epsilon$ and leaves the dot with energy $\epsilon' \neq \epsilon$ while another electron crosses the QPC such
that the total energy is conserved, (ii) reduced elastic transport
(decoherence), and (iii) widening of the dot energy levels.
Thus, we can address some of the questions introduced above:
we find that due to (i) and contrary to the prediction above,
the measurement amplifies the current through the dot and
does not block it. Nonetheless, mechanism (ii) implies that
transport would be blocked if the dot is incorporated inside an
interferometer.
Alongside these main results, we find that the detector
signal shows nonmonotonous behavior as a function of the dot
variables, which we attribute to sensitivity to the directionality
of transport through the dot. Additionally, we determine the
correlation between the QPC transport and the successful
cotunneling passage through the quantum dot. Interestingly,
despite the fact that the regime considered here is different
from that of previous approaches which established such a
relationship~\cite{Steinberg1995,Romito2014}, we can use the correlated-currents signal
to extract the time of the cotunneling process. The obtained
time is compared with the simple Heisenberg's uncertainty
expectation of $\tau_{\rm{cot}} \sim \hbar/\Delta \mathcal{E}$, which is set by the inverse of the
``energy-debt'' during the virtual transport.
Our results are relevant for contemporary transport experiments~\cite{Harbusch2010,clemens2012}, where different mechanisms of QPC backaction are discussed and an estimation of the cotunneling time
is obtained from the linewidth of the differential conductance
through the dot. Note that, differently from other approaches
dealing with the many-body physics in the cotunneling system
and detector~\cite{Romito2014}, our formalism addresses the regime where
single cotunneling events are correlated with single electron
signals through the QPC.
The paper is structured as follows: in Sec.~\ref{model}, we write down the model of a dot measured by a QPC. In Sec.~\ref{method}, we derive the essential tools and methods for describing the measurement of cotunneling in its wake. Section \ref{result}, details the resultant interplay between dot and QPC, and its implications on measurable currents on correlations, as well as the ability of measuring the cotunneling time. In Sec.~\ref{conclude}, we conclude and discuss possible future directions. We provide a comprehensive Appendix \ref{rigorous}, which details the microscopic calculation required to establish the formalism used in Sec.~\ref{method}.
\section{Model}
\label{model}
Our setup is divided into two components: the system, and the detector that measures it,
\begin{align}
H=H_{\text{sys}}+H_{\text{det}}+H_{\text{int}}\, ,
\end{align}
where $H_{\text{sys}}$ describes the system, $H_{\text{det}}$ describes the detector, and $H_{\text{int}}$ describes the interaction between them. For simplicity, we consider a spinless problem. The results can be directly extended in some regimes to the spinful case.
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{Fig1.pdf}
\caption[]{\label{Fig:1}
A sketch of the setup: a single-level dot is tunnel-coupled to two leads, $S$ and $D$ with amplitudes $t_S$ and $t_D$, respectively. The respective dot-leads chemical potentials are $\mu_S$ and $\mu_D$, corresponding to a voltage bias $eV_{SD}=\mu_S-\mu_D$. A quantum point contact (QPC) is capacitively coupled to the dot, i.e., transport between the leads $L$ and $R$ is governed by a tunneling amplitude $\Omega$ when the dot is empty, and $\Omega-\delta \Omega$ when the dot is occupied. The respective QPC-leads chemical potentials are $\mu_L$ and $\mu_R$, corresponding to a voltage bias $eV_{LR}=\mu_L-\mu_R$.
}
\end{figure}
The system consists of a single level quantum dot, that is
tunnel-coupled to two electronic leads, the source $S$, and the
drain $D$ (see Fig. \ref{Fig:1}).
By applying a voltage bias, $eV_{SD}=\mu_S-\mu_D$, between the source
and the drain chemical potentials ($\mu_S$, $\mu_D$), we can measure
the transport properties of the dot. We assume, henceforth, that $eV_{SD} \geq 0$.
The quantum dot is capacitively coupled to another lead, the gate
lead.
Varying the gate-voltage, $V_g$, on the gate lead, controls the number
of electrons in the dot.
The Hamiltonian that describes the system is
\begin{equation}
\label{eq: H}
H_{\text{sys}}=H_{SD}+H_{\text{dot}}+H_T\, ,
\end{equation}
where
\begin{align}
H_{SD}&=\sum_{k,\alpha=S,D}\epsilon_{k,\alpha} c_{k,\alpha}^\dagger c_{k,\alpha}\, ,\\
H_{\text{dot}}&=\epsilon_d d^\dagger d\, ,\\
\label{eq: Ht}
H_{T}&=\sum_{k,\alpha=S,D}t_\alpha c_{k,\alpha}^\dagger d + \text{H.c.}\, .
\end{align}
The operator $c_{k,\alpha}$ annihilates an electron with momentum $k$
and energy $\epsilon_{k,\alpha}$ in the lead $\alpha\in\{S,D\}$, $d$
annihilates an electron on the dot with energy $\epsilon_d$, which is
modulated by $V_g$.
We have assumed that the tunneling coefficients, $t_\alpha$, between
the lead $\alpha$ and the dot are independent of the energy.
As long as $\mu_S > \epsilon_d >\mu_D$, transport occurs via sequential
tunneling processes through the dot.
When the dot level is outside the energy window provided by the leads,
$\mu_D < \mu_S < \epsilon_d$ or $\mu_S > \mu_D > \epsilon_d$, and at low temperature, $T \ll \min\left\{\left|\mu_D-\epsilon_d\right|,\left|\epsilon_d-\mu_S\right|\right\}$, sequential tunneling is exponentially suppressed.
Nonetheless, a small current is still detected in the drain.
This small current is carried by the so-called cotunneling processes,
in which electrons from the source virtually tunnel through the dot
into the drain~\cite{Averin1990,Glazman2005}.
By virtually we mean that the tunneling into the dot is classically
forbidden by energy conservation, but the overall cotunneling
process is energy conserving.
In the present work we focus on this cotunneling regime.
We now turn to describe the detector. The detector is chosen to be a QPC because of its non-invasive nature \cite{Korotkov:2001a}. The QPC consists of two leads, left and right, that are tunnel-coupled to each other, see Fig.~\ref{Fig:1}. It is described by the following Hamiltonian
\begin{align}
H_{\text{det}}=&\sum_l \mathcal{E}_l a_l^\dagger a_l+\sum_r \mathcal{E}_r a_r^\dagger a_r\nonumber\\
&+\sum_{l,r}\Omega\left(a_l^\dagger a_r+\text{H.c.}\right)\, ,
\end{align}
where $a_{l}$ ($a_r$) annihilates an electron on the left (right) lead with momentum $l$ ($r$) and energy $\mathcal{E}_l$ ($\mathcal{E}_r$). The tunneling amplitude between the left and right leads, $\Omega$, is assumed to be energy independent. The QPC is out-of-equilibrium, namely, the chemical potentials of the left and right leads are different, $\mu_L=eV_{LR}/2$, $\mu_R=-eV_{LR}/2$.
The QPC is capacitively coupled to the quantum dot, i.e.~the tunneling amplitude, $\Omega$, is modulated by the charge on the dot
\begin{equation}
H_{\text{int}}=-\sum_{l,r}\delta\Omega \; d^\dagger d\left(a_l^\dagger a_r+\text{H.c.}\right)\, .
\end{equation}
The typical time scales for a tunneling event in the QPC to occur, are $\mathcal{D}^{-1}$ and $\tilde{\mathcal{D}}^{-1}$ for an empty and an occupied dot, respectively~\cite{gurvitz}. They are given by $\mathcal{D}=2\pi\Omega^2\rho_L\rho_R eV_{LR}/\hbar$ and $\tilde{\mathcal{D}}=2\pi\tilde{\Omega}^2\rho_L\rho_R eV_{LR}/\hbar$ where $\tilde{\Omega}\equiv\Omega-\delta\Omega$. For simplicity, henceforth, we assume the same density of states in the left and right leads, $\rho_L=\rho_R\equiv\rho$.
Note that a single-level quantum dot model corresponds to a quantum dot with appreciable level
spacing, larger than the other energy scales in the problem. Such a system can
be experimentally realized in semiconductors. Similarly, the QPC detector can be easily tuned to detect single electron tunneling in the sequential limit~\cite{Sukhorukov2007}, and to weak coupling in the cotunneling regime~\cite{clemens2012,clemens2013}. Furthermore, this model provides a valid effective description of multi-level quantum dots in the limit where the Thouless energy is the largest energy scale~\cite{Aleiner2002}.
\section{Methods}\label{method}
The currents through the dot and the QPC are carried by tunneling processes between the source and the drain via the dot, and from the left lead to the right lead in the QPC. In order to obtain these currents and their cross-correlations, we use a rate equation formalism~\cite{sakurai1985,Korotkov:1994,Koch:2006} and calculate the rates for these tunneling processes (see Appendix~\ref{rigorous}).
We assume that all the tunnel-couplings are weak, namely, $\rho_\alpha |t_\alpha|^2\ll eV_{SD}$ and $\rho|\Omega|^2\ll eV_{LR}$, and treat them perturbatively. Within this approach the interaction term, that makes the QPC transport dependent on the dot-occupancy, is treated exactly.
We focus on cotunneling rates, in which the occupation of the dot changes only virtually. Cotunneling can occur either between two different leads (e.g.~, source to drain) or back and forth between a lead and the dot (e.g.~, source to source). During these processes electrons can tunnel through the QPC. Therefore, the QPC current is sensitive to the virtual changes in the dot-occupancy. We denote by $W_{\alpha\alpha'}^{n}$ the cotunneling rate from lead $\alpha$ to lead $\alpha'$ (where $\alpha,\alpha'\in\{S,D\}$), during which $n$ electrons pass through the QPC.
\subsection{Toy model}
To study the interplay between the QPC and the dot we introduce a
simplified version of our model. In this simplified toy model, we wish to have, at most, one electron that tunnels through the
QPC during a cotunneling process through the dot, namely,
$W_{\alpha\alpha'}^n$ is negligible for $n>1$.
Thus, only five rates remain relevant in the analysis of the currents
and cross-correlations: $W_{SD}^0$, $W_{SD}^1$, $W_{DS}^1$, $W_{SS}^1$, and
$W_{DD}^1$ [see Figs.~\ref{Fig:2a} and \ref{Fig:2}(a)-(d), respectively].
The rate $W_{SD}^0$ describes processes in which an electron
co-tunnels through the dot while no tunneling events through the QPC
take place.
Therefore, this rate contributes only to the current through the
dot. The rates $W_{SD}^1$ and $W_{DS}^1$ describe processes in which, in addition to
cotunneling through the dot, an electron tunnels across the QPC.
They contribute both to the current through the dot, and to the current
through the QPC.
Hence, they generate cross-correlations between these currents. The rates
$W_{SS}^1$ and $W_{DD}^1$ are of processes that contribute to the
current through the QPC, but not to the current through the dot.
Here, the electron tunnels back and forth between the dot and the same lead.
In order to work in the regime of our toy model, we take the following assumptions: First, we work in the zero temperature limit and assume that no tunneling events occur in the QPC when the dot is occupied, i.e., we set
$\tilde{\Omega}=0$. As a result, transport through the QPC occurs only alongside
cotunneling events, i.e.~only during the time interval in which the
dot is virtually empty, $\tau_{\rm cot}$.
We further take the limit where $\tau_{\rm cot}$ is much shorter than
the typical tunneling time in the QPC, i.e.~$\mathcal{D}\tau_{\rm
cot}\ll 1$.
The cotunneling time, $\tau_{\rm cot}$, is related by Heisenberg's uncertainty principle to the difference
between the energy of the initial (or final) state, and the energy of
the virtual interim state, e.g.~$\epsilon-\epsilon_d$ and
$\epsilon'-\epsilon_d$ in Fig.~\ref{Fig:2}.
Of all cotunneling processes, the process with the longest
cotunneling time is the one with the lowest energy difference,
namely, for $\epsilon'=\mu_D$ (assuming $\mu_S>\mu_D$).
Hence, the assumption above implies that
$\hbar \mathcal{D}\ll\mu_S-\epsilon_d,\;\mu_D-\epsilon_d~$.
\begin{figure}
\centering
\includegraphics[width=0.5\columnwidth]{Fig2a}
\caption[]{\label{Fig:2a}
Sketch of the rate $W_{SD}^0$ of cotunneling through the dot accompanied by no tunneling through the QPC [cf.~Eq.~\eqref{eq: Wsd0}]. Similar to Fig.~\ref{Fig:1}, the upper part (brown) depicts the dot-system and the lower (blue) the QPC. }
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Fig2}
\caption[]{\label{Fig:2}
Sketches of the relevant cotunneling rates in our toy model that involve an electron tunneling through the QPC [cf.~Eq.~\eqref{integralForm}]. Similar to Fig.~\ref{Fig:1}, the upper part (brown) depicts the dot-system and the lower (blue) the QPC. (a) The rate $W_{SD}^1$. (b) The rate $W_{DS}^1$. (c) The rate $W_{SS}^1$. (d) The rate $W_{DD}^1$. }
\end{figure}
Note that this chosen working point of the QPC puts the detector in a regime
different from the one typically considered in the context of weak
measurements.
In a typical weak measurement one assumes to work at finite
$\mathcal{D}$ and $\tilde{\mathcal{D}}= \mathcal{D}- \delta \mathcal{D}$.
The measurement process is then characterized by a measurement time,
$\tau_M \sim \mathcal{D}/\delta \mathcal{D}^2$, needed to resolve the
occupancy of the dot.
One can then work in a regime with $\tau_{\rm{cot}} \mathcal{D} \gg
1$ and $\tau_{\rm{cot}} \delta \mathcal{D} \ll
1$, such that the detector's bandwidth is larger than the cotunneling
time we are interested in ($\mathcal{D} \gg 1/\tau_\textrm{cot}$), but the
measurement during such a time remains weak ($\tau_{\rm{cot}} \ll
\tau_M$).
In our case $\delta \mathcal{D}=\mathcal{D}$, and as $\tau_{\rm{cot}} \mathcal{D} \ll
1$ we do not have a large bandwidth in the detector.
Nonetheless, due to the very same choice, $\tilde{\mathcal{D}}=0$, corresponding to a blocked transport through the QPC, the current in the QPC is strongly correlated with a cotunneling through the dot, and one can expect that virtual transport through the dot could be resolved. In other words, the point $\tilde{\mathcal{D}}=0$ is peculiar since, as soon
as an electron tunnels through the QPC, one can infer with certainty
the happening of a cotunneling event through the quantum dot.
This is not a weak measurement in the standard sense, but resembles more a partial-collapse measurement~\cite{Katz2006,Korotkov:2007,Katz08,Zilberberg2012,Zilberberg2013}. Nonetheless, the rarity of such joint tunneling events generates limited back-action, and the detector can be used as a non-destructive detector of the virtual occupation of the dot. In any case, this parameters' regime has the advantage of
simplifying the formalism, and rendering the processes involved in the
detection clearer~\cite{bandwidth}.
Experimentally, the small bandwidth of the QPC should not serve as a hurdle, as there are sufficiently many cotunneling events in the dot to obtain a measurable signal. We envision that tuning in to this measurement regime corresponds to starting from a conductance plateau in the QPC in the presence of a full dot that is decoupled from leads, i.e.~, no virtual charge fluctuation on the dot occurs. Increasing the coupling of the dot to its leads allows for cotunneling. If at the same time a QPC current is generated, it should correspond to the mechanism reported here.
\subsubsection{Derivation of the rates}
\label{derivationOfRates}
In Appendix \ref{tunnelRates}, we rigorously derive the rates
$W_{SD}^0$, $W_{SD}^1$, $W_{DS}^1$, $W_{SS}^1$, and $W_{DD}^1$.
Here, we present a simpler, and intuitive way, to calculate these
rates.
We start by incorporating one of the effects of the coupling with the QPC by bestowing a finite width to the energy level of the dot, i.e.~the single level gains an effective width $\hbar(\mathcal{D}-\tilde{\mathcal{D}})/2$, which is proportional to the measurement strength. Note that in our toy model the measurement strength is, therefore, equivalent to $\hbar\mathcal{D}/2$. Hence, the cotunneling rate is simply the standard cotunneling rate through a single-level dot with energy $\epsilon_d$ and width $\hbar\mathcal{D}/2$. This is indeed the effect obtained by the microscopic calculation, which includes the dynamics of the QPC in Appendix \ref{tunnelRates}.
The total cotunneling rate $W_{SD}^0$, is the sum over the rates of all possible cotunneling processes (namely, integration over all incoming energies):
\begin{align}
&W_{SD}^0=\frac{2\pi}{\hbar}\int_{\mu_D}^{\mu_S}d\epsilon\; \rho_S \rho_D \left|\frac{t_St_D}{\epsilon-\epsilon_d-i\hbar\mathcal{D}/2}\right|^2\label{eq: Wsd0}\\
&=\frac{\Gamma_S \Gamma_D}{ \pi \mathcal{D}} \left[\tan^{-1}\left(\frac{2(\mu_S-\epsilon_d)}{\hbar\mathcal{D}}\right) -\tan^{-1} \left(\frac{2(\mu_D-\epsilon_d)}{\hbar\mathcal{D}}\right)\right]\;,\nonumber
\end{align}
where $\Gamma_\alpha\equiv2\pi\rho_\alpha|t_\alpha|^2/\hbar$. For later use, we also denote the total coupling $\Gamma=\Gamma_S+\Gamma_D$.
The cotunneling events through the dot, which are accompanied by a tunneling event through the QPC, consist of three stages (see Fig.~\ref{Fig:2}): First, the electron in the dot tunnels out into the source or the drain. Second, a tunneling event occurs in the QPC. Last, a new electron tunnels into the empty dot. Here too, we include an effective width, $\hbar\mathcal{D}/2$ to the energy of the dot. The total tunneling rate is given by
\begin{widetext}
\begin{align}
W_{\alpha\alpha'}^1&=\frac{2\pi}{\hbar}\int_{-\infty}^{\mu_{\alpha}}d\epsilon\int_{\mu_{\alpha'}}^{\infty}d\epsilon'\int_{-\infty}^{\mu_L}d\epsilon_l\int_{\mu_R}^{\infty}d\epsilon_r\; \rho_S \rho_D \rho_L \rho_R
\left|\frac{t_{\alpha}t_{\alpha'}\Omega}{(\epsilon'-\epsilon_d-i\hbar\mathcal{D}/2)(\epsilon_r+\epsilon'-\epsilon_l-\epsilon_d-i\hbar\mathcal{D}/2)}\right|^2\delta(\epsilon+\epsilon_l-\epsilon'-\epsilon_r)\nonumber\\
&=\frac{\hbar^2 \Gamma_{\alpha} \Gamma_{\alpha'} \mathcal{D}}{4 \pi^2 e V_{LR}}\int_{\mu_{\alpha'}-eV_{LR}}^{\mu_{\alpha}} d\epsilon\int_{\mu_{\alpha'}}^{\epsilon+eV_{LR}} d\epsilon'\frac{ (eV_{LR}+\epsilon -\epsilon ')}{\left[(\epsilon-\epsilon_d)^2+\frac{\hbar^2\mathcal{D}^2}{4}\right] \left[(\epsilon'-\epsilon_d)^2+\frac{\hbar^2\mathcal{D}^2}{4}\right]}\, .
\label{integralForm}
\end{align}
\end{widetext}
The integrals in Eq.~\eqref{integralForm} are solvable. Out of space consideration, we choose to present these rates in integral form.
Here we notice the full extent of the interaction of the QPC with the dot; the QPC affects the cotunneling through the dot in two fashions: First, as previously noted, it gives a finite width to the energy level of the dot. Second, it allows new QPC-assisted cotunneling processes through the dot, where electrons are emitted from lead $\alpha$ \emph{inelastically} into lead $\alpha'$, provided that the energy difference is compensated by the QPC.
When the dot and the QPC are decoupled ($\Omega=0$) all the cotunneling processes are elastic, namely, an electron at energy $\epsilon$ in the lead $\alpha$, can be emitted into lead $\alpha'$ only at the same energy, $\epsilon$. In our case, the dot and the QPC are coupled ($\Omega\neq0$), and the two systems can exchange energy. Therefore, an electron with energy $\epsilon$ in lead $\alpha$ can end up in lead $\alpha'$ having a different energy $\epsilon'\neq\epsilon$. This is enabled by an electron with energy $\epsilon_l$ which is transmitted from the left lead of the QPC into the right lead with energy $\epsilon_r\neq \epsilon_l$ keeping the total energy conserved $\epsilon+\epsilon_l=\epsilon'+\epsilon_r$. Examples of inelastic cotunneling processes are schematically depicted in Fig.~\ref{Fig:2}. The total cotunneling rate $W_{\alpha\alpha'}^1$, is the sum over the rates of all possible cotunneling processes (namely, integration over all energies in the relevant leads of the dot and QPC systems).
At zero temperature, the maximal energy that a QPC-electron can lose
by tunneling is $\mu_L-\mu_R=eV_{LR}$ (assuming that
$\mu_L>\mu_R$). Hence, the maximal energy that a dot-electron can gain
is $eV_{LR}$, namely, $\epsilon-\epsilon'\leq eV_{LR}$ [see
Figs.~\ref{Fig:2}(b)-(d)]. Similarly, the maximal energy that the dot-electron
can lose is $eV_{SD}$ [see Fig.~\ref{Fig:2}(a)]. Increasing the applied bias voltages (both on
the dot and on the QPC) therefore increases the phase space for
QPC-assisted cotunneling processes. Hence, the total rate,
$W_{\alpha\alpha'}^1$, increases with the applied bias voltages.
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{Fig3}
\caption[]{\label{Fig:3}
Plots of (a) $W_{SD}^0$ and (b) $W_{SD}^1$ as a function of $\hbar\mathcal{D}/(\mu_D-\epsilon_d$) with bias voltages $eV_{ SD}=4\hbar\Gamma$, $eV_{ LR}=3.9\hbar\Gamma$, and $\Gamma_S=\Gamma_D$. The different curves are for different dot-gating $\epsilon_d$ such that: $\mu_{ D}-\epsilon_d=4\hbar\Gamma$ (dark blue); $\mu_{ D}-\epsilon_d=5\hbar\Gamma$ (medium-dark blue); $\mu_{ D}-\epsilon_d=6\hbar\Gamma$ (bright blue). We highlight the area in which our toy model is relevant, namely, where $W_{SD}^0$ slightly decreases, $W_{SD}^1$ is far from peaking, and $W_{SD}^n$ for $n>1$ are negligible. Two effects are seen here: (i) The higher the dot-energy is, the higher the cotunneling rates are; (ii) as $\mathcal{D}/(\mu_D-\epsilon_d)$ increases it is more probable for electrons to pass through the QPC during the time-window in which the dot is empty. This is seen in the decrease in $W_{SD}^0$ vs. the increase in $W_{SD}^1$. At the point in which $W_{SD}^1$ peaks, the probability for two electrons to tunnel through the QPC during a cotunneling event becomes relevant.}
\end{figure}
In Fig.~\ref{Fig:3}, we plot the rates $W_{SD}^0$ and $W_{SD}^1$ as a function of $\hbar\mathcal{D}/(\mu_D-\epsilon_d)$ for different values of $\mu_D-\epsilon_d$ and at fixed $V_{ SD}$, $V_{LR}$. In this configuration, $\mu_D-\epsilon_d$ affects all possible cotunneling rates. Lowering $\epsilon_d$ suppresses the rate of cotunneling processes, leading also to shorter cotunneling times $\tau_{\rm cot}$. This is seen in Fig.~\ref{Fig:3} by the overall decrease in magnitude of both $W_{SD}^0$ and $W_{SD}^1$ as $\epsilon_d$ is lowered. As $\mathcal{D}$ increases the probability for an electron to tunnel through the QPC during the time window $\tau_{\rm cot}$ grows. Hence, $W_{SD}^0$ constantly decreases with $\mathcal{D}$ while $W_{SD}^1$ increases.
At a certain point, $W_{SD}^1$ peaks and decreases as well. At this point the probability for two electrons to tunnel through the QPC during a cotunneling event becomes relevant. We highlight the regime in which our toy model is valid, i.e.~, the regime in which $W_{SD}^0$ slightly decreases while $W_{SD}^1$ increases but is still far from peaking. Note, however, that even beyond the highlighted regime, in the weak QPC-tunnel-coupling regime the rates $W_{SD}^n$ with $n>1$ appear as contributions of $O(\Omega^{2n})$.
We have assumed throughout this section an additional finite width $\hbar\mathcal{D}/2$ of the dot energy level. We stress that this is a result of the formal derivation of Eqs.~\eqref{eq: Wsd0} and \eqref{integralForm} that appears in Appendix \ref{tunnelRates}.
\subsubsection{Currents and correlations}
Using the rates, Eqs.~\eqref{eq: Wsd0} and \eqref{integralForm}, we are able to express the currents and cross-correlations of our setup~\cite{Korotkov:1994,Koch:2006}. The average currents of the dot and the QPC are given by
\begin{align}\label{dot_current_eq}
&\langle I_{\rm dot}\rangle=e\left(W_{SD}^0+W_{SD}^1-W_{DS}^0-W_{DS}^1\right)~,\\
&\langle I_{\rm QPC}\rangle=e\left(W_{SD}^1+W_{DS}^1+W_{SS}^1+W_{DD}^1\right)~.\label{qpc_current_eq}
\end{align}
The zero-frequency cross-correlation between these currents,
\begin{equation}
\label{correlazioni}
S=2\int_{-\infty}^{\infty}d\tau \left[\langle I_{\rm dot}(t+\tau)I_{\rm QPC}(t)\rangle-\langle I_{\rm dot}\rangle\langle I_{\rm QPC}\rangle\right]~,
\end{equation}
is given by
\begin{equation}
\label{correlazioni2}
S=2e^2(W_{SD}^1-W_{DS}^1)~.
\end{equation}
\section{Results}
\label{result}
In this section, we analyze the resulting currents through the dot and
the QPC, and their cross-correlations.
Also, we relate these quantities to a conditional partial measurement of the occupation of the dot, akin to null weak values~\cite{Zilberberg2013}. This quantity enables the determination of $\tau_{\rm cot}$.
\subsection{Current through the dot; measurement back-action}
The current through the dot is carried by cotunneling processes, in
which the occupation of the dot is virtually changed for a short
time.
The intermediate evolution state, where the occupation of the dot is
changed, does not preserve energy.
Thus, ideally a strong measurement of the charge on the dot would
destroy cotunneling processes.
Given that the current through the QPC is affected by the charge on
the dot, it can be used to measure this charge.
Naively, one would expect that turning on a weak coupling between the
QPC and the dot, will slightly reduce the current through the dot.
It turns out that the opposite is true.
As seen in Fig.~\ref{Fig:4}, by increasing the coupling between the dot and the QPC (increasing $\Omega$ in our toy model), the current through the dot is even slightly \emph{enhanced}.
Figure~\ref{Fig:4} depicts the current through the dot as a function
of the coupling to the QPC, $\Omega$, and as a function of the QPC
voltage-bias, $V_{LR}$.
The current is enhanced both by increasing $\Omega$ and by increasing
$V_{LR}$, or alternatively, by increasing $\mathcal{D}$, and
$V_{LR}$.
Note that the regime in which our model is valid is $d\equiv
\hbar\mathcal{D}/(\mu_D - \epsilon_d)\lesssim 0.3$, where up to one electron is transfered in the QPC during the virtual cotunneling time of the dot.
Therefore, the area in which $I_{\rm dot}$ peaks and decreases is outside the scope of our model's validity.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Fig4}
\caption[]{\label{Fig:4}
Plots of $I_{\rm dot}$ [cf.~Eq.~\eqref{dot_current_eq}] as a function of $eV_{LR}/(\mu_D-\epsilon_d$) and $\rho\Omega$ with $eV_{ SD}=4\hbar\Gamma$, $\mu_{ D}-\epsilon_d=5\hbar\Gamma$, and $\Gamma_S=\Gamma_D$. (a) A density plot of $I_{\rm dot}$. The $I_{\rm dot}$ along the vertical (green) and horizontal (blue) mesh lines is plotted in (b) and (c), respectively. The dashed lines represent equal-$d\equiv \hbar\mathcal{D}/(\mu_D - \epsilon_d)$ lines. We can see that the current increases with the coupling $\mathcal{D}$ of the QPC. However, the effect does not depend exclusively on $\mathcal{D}$, but it also directly depends on the QPC parameters: the tunnel-coupling $\Omega$ and the voltage-bias $V_{LR}$.}
\end{figure}
The current through the dot has two parts, a ``coherent'' part and
``incoherent'' part.
The coherent part is carried by coherent cutunneling processes, where
electrons tunnel through the dot without changing the state of the
QPC, the total rate of these processes is $W_{SD}^0$.
The incoherent part is carried by cotunneling processes through the
dot that are accompanied by changes in the state of the QPC, namely,
tunneling through the QPC.
The rates for these processes are $W_{SD}^1$ and $W_{DS}^1$.
As discussed in Sec.~\ref{derivationOfRates}, increasing $\Omega$ or
$V_{LR}$, and correspondingly $\mathcal{D}$, increases the probability
for tunneling through the QPC, and hence decreases $W_{SD}^0$ and
increases $W_{SD}^1$.
Additionally, $V_{LR}$ increases also the phase-space for inelastic
QPC-assisted cotunneling processes through the dot. Hence $V_{LR}$
increases $W_{SD}^1$ (and $W_{DS}^1$) even for a fixed $\mathcal{D}$.
Hence, the growth of $I_{\rm dot}$ in Fig.~\ref{Fig:4} is a result of the
availability of phase-space for inelastic QPC-assisted cotunneling
processes in $W_{SD}^1$ being higher than the decrease in $W_{SD}^0$
as a function of these parameters.
The strong measurement nature of the incoherent channel is highlighted
by the fact that, in our simplified toy model, having current through
the QPC destroys the coherence of a cotunneling process.
If the dot would have been embedded into the arm of a Mach-Zehnder
interferometer a current in the QPC could serve as a \emph{strong}
which-path measurement and would reduce the interference signal.
In this case, the interference signal would be proportional to the
decreasing $W_{SD}^0$.
\subsection{Current through the QPC; measurement signal}
The current through the QPC is sensitive to the virtual changes in the
charge on the dot during cotunneling processes.
In the discussed toy model, QPC-current pulses (electron tunneling)
occur only alongside cotunneling processes in the dot.
The cotunneling processes can be either between two leads (from the
source to the drain and vice versa), or back and forth between the dot
and one of the leads (source-to-source or drain-to-drain). We find, here, a non-monotonous signal that is dependent on the directionality of the dot-transport.
In Fig.~\ref{Fig:5}, we plot $I_{\rm QPC}$ as a function of the two
bias voltages: $V_{SD}$ on the dot, and $V_{LR}$ on the QPC.
Increasing $V_{LR}$ has two effects: First, it increases the phase
space for elastic QPC-tunneling processes, where
$\epsilon_l=\epsilon_r$ (see Fig.~\ref{Fig:2} for clarification of the
notations); Second, it increases the phase space for inelastic processes with
$\epsilon_l>\epsilon_r$.
While the first effect is relevant only for source-to-drain tunneling
through the dot, the second effect is relevant for all four
possibilities of cotunneling through the dot (from any lead $\alpha$
to any lead $\alpha'$).
The amount of energy that can be lost in the QPC as a compensation
for inelastic cotunneling through the dot is bound from above by
$eV_{LR}$. Hence, in cotunneling processes that involve a single lead,
raising $V_{LR}$ allows for more energetic particles-hole excitations
to appear in the lead [see Figs.~\ref{Fig:2}(c) and \ref{Fig:2}(d)].
Similarly, drain-to-source cotunneling consumes energy and therefore
increasing $V_{LR}$ enlarges the availability of drain-to-source
cotunneling [see Fig.~\ref{Fig:2}(b)].
To conclude, the rates $W_{\alpha\alpha'}^1$ (all the four
combinations) increase with $V_{LR}$ and hence $I_{\rm QPC}$ increases
with $V_{LR}$.
Raising the bias voltage on the dot, $V_{SD}$, increases the phase
space for both elastic and inelastic source-to-drain tunneling, and
hence, it increases the rate $W_{SD}^1$.
The rate $W_{DS}^1$ however, is suppressed by a raised $V_{SD}$ as the
drain-to-source cotunneling processes require higher amount of energy
for larger $V_{SD}$.
For a fixed $V_{LR}$, the available energies for inelastic for
drain-to-source cotunneling through the dot are bounded from
above. Hence, increasing $V_{SD}$ reduces the phase space for such
processes, where for $V_{SD}>V_{LR}$ such cotunneling becomes impossible.
The increase of $V_{SD}$ has an additional effect; for a fixed $\mu_D$
it reduces the rate $W_{SS}^1$.
The probability for a cotunneling process to occur, depends on the
energy difference between the dot and the available energies in the
lead. Increasing $\mu_S-\epsilon_d$, increases the difference between
the energy of the electron in the dot and the available energies in
the source [see Fig.~\ref{Fig:2}(c)], making the source-to-source
cotunneling processes less probable.
Since $\mu_D$ is fixed, the drain-to-drain tunneling processes are not
affected by $V_{SD}$ and hence, for a fixed $\mu_D$, $W_{DD}^1$ is
independent of $V_{SD}$.
To conclude, $W_{SD}^1$ grows with $V_{SD}$, while $W_{DS}^1$ and $W_{SS}^1$
are reduced by its increase, and $W_{DD}^1$ is not affected by it.
As a result, $I_{\rm QPC}$ generally grows with $V_{SD}$ since the
dominant processes are QPC-tunneling processes accompanied by
source-to-drain cotunneling through the dot.
Yet, for a small bias voltage on the dot, $V_{SD}\ll
\mu_D-\epsilon_d$, the four rates $W_{SD}^1$, $W_{DS}^1$, $W_{SS}^1$,
and $W_{DD}^1$ are roughly of the same magnitude and are determined
mostly by $V_{LR}$.
Raising $V_{SD}$ in this configuration causes an increase in
$W_{SD}^1$ alongside a decrease in $W_{DS}^1$ and $W_{SS}^1$, causing the
current through the QPC to slightly decrease.
$I_{\rm QPC}$ starts to increase when $V_{SD}$ is further
increased. This can be shown in Fig.~\ref{Fig:5}(b), where for a
relatively large $V_{LR}$, a small decrease in $I_{\rm QPC}$ appears
as $V_{SD}$ grows, before the overall current starts to increase.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Fig5}
\caption[]{\label{Fig:5}
Plots of $I_{\rm QPC}$ [cf.~Eq.~\eqref{qpc_current_eq}] as a function of $eV_{LR}/(\mu_D-\epsilon_d$) and $eV_{SD}/(\hbar\Gamma)$ with $\rho\Omega=0.15$, $\mu_{ D}-\epsilon_d=5\hbar\Gamma$, and $\Gamma_S=\Gamma_D$. (a) A density plot of $I_{\rm QPC}$. The $I_{\rm QPC}$ along the vertical (green) and horizontal (blue) mesh lines is plotted in (b) and (c), respectively. Naturally, the current increases with the the QPC voltage-bias $V_{LR}$. As the voltage on the dot $V_{SD}$ is increased the following occurs: (i) The cotunneling to the drain and back $W_{DD}^1$ remains constant; (ii) The cotunneling to the source and back $W_{SS}^1$ becomes less probable; (iii) The cotunneling from drain to source $W_{DS}^1$ decreases, and disappears once $V_{SD}>V_{LR}$; (iv) Additional cotunneling processes from source to drain in $W_{SD}^1$ are allowed. The latter features dominate the behavior of $I_{\rm QPC}$. We see initially a slight decrease as $W_{DS}^1$ vanishes with an overall increase due to $W_{SD}^1$. }
\end{figure}
\subsection{Cross-current correlation; focusing on source-drain processes}
We finally arrive to describe the sensing of virtual change in the charge on the dot
during cotunneling processes from source to drain.
Here, we are interested in the cross-current correlation between the
dot and the QPC. This correlation eliminates processes that contribute to the QPC-current from cotunneling
processes to a specific lead and back, as they do not generate a current through the dot.
The correlation function, $S$, in Eq.~(\ref{correlazioni}) is plotted in Fig.~\ref{Fig:6}, as a function of the system's parameters, $V_{SD}$ and $\mu_D-\epsilon_d$.
Its behavior stems from the same effects considered above for the currents, $I_\textrm{dot}$ and $I_{QPC}$.
The correlation is suppressed at $V_{SD}\ll V_{LR}$ because of counter-propagating QPC-assisted processes. Once the drain-to-source processes ($W_{DS}^1$) are suppressed for $V_{SD}\geq V_{LR}$, the correlation increases monotonically with $V_{SD}$ because of the increased phase space for cotunneling events. Also, quite intuitively, the deeper the dot's energy level is, the smaller is the probability for a cotunneling event to occur, and thus the smaller $S$ becomes. As to the effects of the detector on the correlation, that we do not plot, a larger coupling to the QPC or a larger $V_{LR} $ leads to a stronger current across the detector, which directly reflects in a larger signal in the correlation.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Fig6}
\caption[]{\label{Fig:6}
Plots of $S$ [cf.~Eqs.~\eqref{correlazioni} and \eqref{correlazioni2}] as a function of $eV_{SD}/(\hbar\Gamma)$ and $(\mu_D-\epsilon_d)/(\hbar\Gamma)$ with $\rho\Omega=0.15$, $eV_{LR}=1.5\hbar\Gamma$, and $\Gamma_S=\Gamma_D$. (a) A density plot of $S$. The $S$ along the vertical (green) and horizontal (blue) mesh lines is plotted in (b) and (c), respectively. As the voltage on the dot $V_{SD}$ is increased the following occurs: (i) The cotunneling from drain to source $W_{DS}^1$ decreases, and disappears once $V_{SD}>V_{LR}$; and (ii) additional cotunneling processes from source to drain in $W_{SD}^1$ are allowed. Hence, $S$, initially, increases with $V_{SD}$ slowly, and only afterward increases as $W_{SD}^1$. The deeper the dot level is, the shorter the virtual time in which the dot is empty. As a result, (i) cotunneling processes through the dot become less probable, and (ii) it becomes less probable for a QPC electron to manage to tunnel during this time window, leading to the descent of $S$. }
\end{figure}
\subsection{Weak values; cotunneling time}
The current through the QPC is generated while the dot is virtually empty, i.e.~$I_{\textrm{QPC}}=\sum_i I_{\textrm{QPC}}^{\{i\}}$ is generated by processes $i$ for which the dot is virtually empty for time $\tau_{\rm cot}^{\{i\}}$ with $I_{\textrm{QPC}}^{\{i\}}\sim e\mathcal{D}\tau_{\rm cot}^{\{i\}}$.
Since the correlation function, $S$, isolates the contributions to the QPC current arising from cotunneling between the two distinct leads, it encodes information on the physical properties of these cotunneling processes, i.e.~one can extract the time the dot is virtually empty, when restricting to current-generating cotunneling events.
This can be defined via a weak value procedure~\cite{Aharonov:1988a}, where the detector's signal is postselected by retaining only processes of successful cotunneling current from source to drain.
In this framework the cotunneling current through the dot plays the role of a postselection operator~\cite{Romito2014}.
This leads to the average over cotunneling times of current-generating events~\cite{Romito2014} [cf.~Eq.~(\ref{correlazioni})]
\begin{align}
\bar{\tau}_{\textrm{SD}} &=
\frac{ \int_{- \infty} ^{\infty}ds [ \langle I_{\textrm{QD}}
(t) I_{\textrm{QPC}} (t-s) \rangle - \langle I_{\textrm{QD}}
\rangle \langle I_{\textrm{QPC}} \rangle]}{e\mathcal{D} \langle I_{\textrm{QD}}\rangle}\nonumber\\
&=\frac{S}{2e\mathcal{D} \langle I_{\textrm{QD}}\rangle}\,.
\label{tau}
\end{align}
The obtained $\tau_\textrm{SD}$ is plotted in Fig.~\ref{Fig:7} as a function of $eV_{SD}$ and $\mu - \epsilon_d/(\hbar\Gamma)$.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Fig7}
\caption[]{\label{Fig:7}
Plots of $\bar{\tau}_{\textrm{SD}}$ [cf.~Eq.~\eqref{tau}] as a function of $eV_{SD}/(\hbar\Gamma)$ and $(\mu_D-\epsilon_d)/(\hbar\Gamma)$ with $\rho\Omega=0.15$, $eV_{LR}=1.5\hbar\Gamma$, and $\Gamma_S=\Gamma_D$. (a) A density plot of $\bar{\tau}_{\textrm{SD}}$. The $\bar{\tau}_{\textrm{SD}}$ along the vertical (green) and horizontal (blue) mesh lines is plotted in (b) and (c), respectively. Interestingly, as $V_{SD}$ increases, $\bar{\tau}_{\textrm{SD}}$ increases as well, i.e. it appears that the increased phase-space adds processes with a slower time into the average. As expected, the deeper the dot level is, the shorter $\bar{\tau}_{\textrm{SD}}$ becomes. }
\end{figure}
We note that such a quantity generally depends on the detector's parameters as well.
However, the effects of the detector are minimal in the weak measurement regime, which, as discussed in Sec.~\ref{method}, corresponds to $\hbar \mathcal{D}\ll eV_{LR} \ll \mu_D -\epsilon_d$. This allows us to define an intrinsic (dot-dependent only) cotunneling time when $V_{LR} \to 0$. The evaluation of $\bar{\tau}_\textrm{SD}$ that is reported in Fig.~\ref{Fig:7} is obtained within such a regime, and its values do not depend on the detector parameters much.
In Fig.~\ref{Fig:7}, we see that $\bar{\tau}_{\textrm{SD}}$ increases with $V_{SD}$, and becomes shorter the lower the dot-level is. The decay of the cotunneling time as a function of $\mu_D-\epsilon_d$ can be intuitively understood in terms of the energy-time uncertainty principle: at equilibrium, the virtual hole that is excited in the dot sets $\hbar/ (\mu_D-\epsilon_d)$ as the typical time scale for the occupation of the virtual state.
\subsubsection{Discussion}
\label{discussion}
We wish to make a more quantitative evaluation of our obtained cotunneling time [cf.~$\bar{\tau}_{SD}$ in Eq.~\eqref{tau} and Fig.~\ref{Fig:7}]. Due to the finite bias voltage on the dot, in Eq.~\eqref{tau} we have a weighted average over the cotunneling times of different cotunneling processes. Within a simple comparative model, this can be taken into account by averaging the cotunneling times predicted by Heisenberg's uncertainty principle, $\tau_h=\hbar/ (\epsilon_h-\epsilon_d)$, over all cotunneling processes, each characterized by a different virtual energy, $\epsilon_h-\epsilon_d$, where $\mu_D< \epsilon_h < \mu_S$. Each cotunneling process occurs with probability $P(\epsilon')=2\pi\rho_S \rho_D \left|t_St_D/(\epsilon'-\epsilon_d-i\hbar\mathcal{D}/2)\right|^2$ [cf.~Eq.~\eqref{eq: Wsd0}]. The averaged cotunneling time is then
\begin{widetext}\begin{align}
\bar{\tau}_h=\frac{\int_{\mu_D}^{\mu_S} P(\epsilon') \tau_h(\epsilon')}{\int_{\mu_D}^{\mu_S} P(\epsilon')}=\frac{1}{\hbar \mathcal{D}}\frac{\ln\left[(\hbar \mathcal{D})^2 + 4 (\mu_S-\epsilon_d)^2\right]-\ln\left[(\hbar \mathcal{D})^2 + 4 (\mu_D-\epsilon_d)^2\right] + 2 \ln[\mu_D-\epsilon_d]- 2 \ln[\mu_S-\epsilon_d]}{ \arctan\left[2 (\mu_S-\epsilon_d)/(\hbar \mathcal{D})\right]+\arctan\left[2 (\mu_D-\epsilon_d)/(\hbar \mathcal{D})\right]}\,.
\label{tauEff}
\end{align}\end{widetext}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Fig8}
\caption[]{\label{Fig:8}
LogLog plots of $\bar{\tau}_{\textrm{SD}}$ and $\bar{\tau}_{h}$ [cf.~Eqs.~\eqref{tau} and \eqref{tauEff}] as a function of $(\mu_D-\epsilon_d)/(\hbar\Gamma)$ with $\rho\Omega=0.15$, $eV_{LR}=1.5\hbar\Gamma$, and $\Gamma_S=\Gamma_D$. The plots of $\bar{\tau}_{\textrm{SD}}$ have an overall shorter time (blue) than those of $\bar{\tau}_{h}$ (red). For both plotted times the curves are for $V_{SD}=2.5 \hbar \Gamma$ (light), $V_{SD}=5.5 \hbar \Gamma$ (medium-light), and $V_{SD}=8.5 \hbar \Gamma$ (dark). In the inset, we plot the approximate slopes of the LogLog curves, defined as $(\partial \ln\bar{\tau})/[\partial \ln(\mu_D-\epsilon_d)]$. The overall magnitude difference between the two time models can be attributed to an unknown prefactor in the time taken from the energy-time uncertainty principle. The other differences can be attributed to an inherent discrepancy between the two models, as the former incorporates phase-space contributions of slower QPC-assisted processes.}
\end{figure}
The resulting $\bar{\tau}_h$ is plotted in Fig.~\ref{Fig:8}, where it is compared to $\bar{\tau}_\textrm{SD}$.
We see the following main differences: (i) there is, approximately, an overall order of magnitude between the two times, which can be attributed to an unknown prefactor in the used $\tau_h (\epsilon')$ from the energy-time uncertainty principle, (ii) whereas $\bar{\tau}_h$ decreases almost as $1/(\mu_D-\epsilon_d)$ (it exhibits almost a linear decrease in a LogLog plot) with a slight dependence on $V_{SD}$, the $\bar{\tau}_\textrm{SD}$ deviates from such a $1/(\mu_D-\epsilon_d)$ descent and shows a stronger dependence on $V_{SD}$, (iii) the dependence on $V_{SD}$ is opposite in the two cases. With increasing $V_{SD}$, in $\bar{\tau}_h$ additional faster cotunneling processes are added into the average and it becomes shorter, whereas in $\bar{\tau}_\textrm{SD}$ the average time becomes longer. The latter two differences demonstrate the inherent discrepancy between our simplistic model for $\bar{\tau}_h$, that takes into account only the finite width of the dot level, versus the physical model of $\bar{\tau}_\textrm{SD}$ that incorporates also an increased back-action from the detector by adding phase-space for slower QPC-assisted processes.
\section{Conclusions and Outlook}
\label{conclude}
We have presented a model for a direct detection of
electron cotunneling through a single-level quantum dot.
The detector has been modeled by
a charge sensing QPC that is capacitively coupled to the dot.
In the regime where the transport through the QPC
is fully blocked when the dot is occupied, we obtain a simplified model
that allows us to incorporate a microscopic description of the QPC into the rate equation formalism of
cotunneling through the quantum dot.
We have, thus, determined the current through the QPC (detector's signal), the cotunneling current in the dot (including the detector's back-action) as well as their correlations (related to weak values and the cotunneling time).
We found that the detector's back-action consists of three different mechanisms: (i) a broadening of the dot energy levels, (ii) a suppression of elastic (coherent) charge transfer processes, and (iii) an increase of phase-space due to QPC-assisted transport. In particular, the latter mechanism is responsible for an increase of the cotunneling current upon increasing the QPC voltage bias. The QPC current generally increases also as a function of the dot voltage bias corresponding to the increased probability of cotunneling processes. However, a counter-intuitive \emph{decrease} of $I_\textrm{QPC}$ with the voltage bias across the dot is predicted at small voltage biases. In this regime, the dominant effect of the increase of voltage bias is a suppression of the source-to-source, drain-to-drain, and drain-to-source cotunneling processes, which in turn result in a suppressed current through the QPC. Importantly, the current-current correlations allow us, via a weak value based approach, to access the cotunneling time. We find that the cotunneling time obtained from such a direct measurement can be compared to the time estimated from the energy-time uncertainty principle, after taking into account proper averaging of all possible cotunneling events.
Let us, finally, comment on the validity and applicability of our simplified model.
Our model essentially relies on the assumptions that (i) transport through the QPC is fully blocked for an occupied dot, and that (ii) only processes of single electron transfer through the QPC during a cotunneling event are considered. The former assumption is used in order to obtain time-independent cotunneling rates in the presence of the QPC. Introducing a Markovian thermalization time in the QPC leads or cutting the infinite time integration in the cotunneling rates by sequential tunneling rates should cure this constraint.
The latter condition appears to be quite strong: in order to detect a cotunneling event of duration $\tau_\textrm{cot}\sim \hbar/(\mu_D-\epsilon_d)$, the detector is expected to have a large bandwidth $\gtrsim \hbar/\tau_{\rm cot}$, corresponding to a large number of charge-transfer events in the QPC per single cotunneling event~\cite{Romito2014}.
In our model we consider the opposite regime. Nonetheless, the detector is equally sensitive to the short-lived virtual cotunneling states, at the working point defined by (i), since cotunneling processes are the only possible mechanism of activating a current signal in the QPC. This implies that any weak signal in the detector corresponds to obtaining full information on the happening of cotunneling, which makes our measurement of a partial-collapse type (i.e., a strong measurement that happens with a small probability). Hence, unlike in a weak measurement case, the measured cotunneling time is affected by back-action.
Even in its simplified form, it is important to stress that our simplified theoretical model provides a valid description for experiments at the pinched-off working point of the QPC.
In this working point, the only approximation is the assumption that the relevant rates are those corresponding to single electron transfer across the QPC.
Though this is the key simplification, it is also a realistic physical approximation.
What our approximation neglects are the coherences between subsequent electrons tunneling across the QPC, which decay very fast due to the relaxation processes of electrons in the bulk leads. Hence, contributions from a higher number of transported QPC-electrons per cotunneling process are small (cf.~Fig.~\ref{Fig:3}). In other words, the virtual cotunneling time is so short that the assumption $\mathcal{D}\tau_{\rm cot}\ll 1$ is physically sensible.
\textit{Spinful electrons.} In order to highlight the interplay between measurement and coherent transport, we have assumed that the electrons are spin
less. Whereas spin physics do not play an important role in QPC
transport, in the dot system the Kondo effect~\cite{hewson1997kondo} could
qualitatively change our predictions.
Our treatment is immediately applicable for polarized electrons, e.g. in
the presence of a strong magnetic field, which could be realized in
experiments. Moreover, also in the case of complete spin degeneracy,
where Kondo physics takes place, our approach remains valid as long as
the Kondo temperature $T_k \sim \sqrt{\rho_\alpha |t_\alpha|^2 U}
e^{-U/(2\rho_\alpha |t_\alpha|^2)}$, with $U$ the charging energy on
the dot, is smaller than the different energy scales of the system,
e.g.~, $T_K \ll T\ll eV_{SD}$. In this regime, the system is not
sensitive to the Kondo physics and our treatment is essentially
correct. Note that this specific limit can be achieved by making the
tunneling between leads and dot arbitrarily weak. Importantly,
extending our analysis to the Kondo regime is extremely interesting,
as it features an interplay between transport through a truly
many-body correlated virtual state and weak measurement.
Since our model can describe realistic configurations~\cite{Field1993,Elzerman2003,DiCarlo2004,Harbusch2010,Gasparinetti2012,Granger2012,clemens2012,clemens2013,zumbuhl2014}, it becomes interesting to generalize it to include experimentally relevant effects, e.g. finite temperature, many-level dots, spin physics, and to extend the present results to the more general regime of weak measurement.
\acknowledgments
We would like to thank S. A. Gurvitz, Y. Oreg, T. Ihn, C. R\"{o}ssler, K. Ensslin, and D. Z\"{u}mbuhl for useful discussions. We acknowledge the support from DFG, 4710/1-1, and Swiss
National Science Foundation (SNSF).
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train/arxiv
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BkiUdkQ5qhLACG6g_bTQ
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\section{Introduction}
\subsection{Context}
A wide range of methods are available for constructing models of
rational $G$-spectra and calculating with them. In various ways they
are based on assembling information from the geometric $H$-fixed
points as $H$ varies through the closed subgroups of $G$. From the
point of view of transformation groups, the most natural thing to use
as input for the contribution at $H$
is the homology of the Borel construction on the geometric $H$-fixed
points. We construct an abelian category from this data, call it the
{\em torsion model}, and prove that it gives an effective means of calculation.
In the case of the circle group $G$ two methods are discussed in
\cite{s1q}. The first is based on the standard abelian model $\cA
(G)$ and the second on the torsion model $\cAt (G)$, which is the
rank 1 case of the model constructed here. Even in the rank 1 case, one can see the technical advantages of the standard model.
The standard model is of injective dimension
1 and monoidal, and the torsion model is of injective dimension 2 and
cannot be monoidal. Because of this, the focus when considering other
groups has so far been on the standard model. Indeed,
the full form of the torsion model has only been considered
previously in the case of the circle group.
Nonetheless, the torsion model also has some advantages. As described
above,
the ingredients are very natural from the point of view of
transformation groups. In fact they are also rather natural from the
point of view of algebraic geometry, where they mirror the Cousin
complex. Partly for these reasons, when identifying the algebraic model of a spectrum
in the standard model it is often useful to approach through the
torsion model. From the point of view of commutative algebra,
the standard model is based on complete and Noetherian objects and their
localizations whilst the torsion model is based on torsion and
Artinian objects. This often means the vector spaces concerned
tend to be smaller. For any of these reasons it is valuable to
develop a torsion model.
The purpose of the present paper is to document the torsion model and
its homological algebra when $G$ is a torus and to construct the Adams spectral
sequence based on the torsion model. This is a preliminary step
towards developing a model category of the same flavour as the torsion
abelian category, since it suggests the general form of such a
model. Ongoing work with Balchin, Pol and Williamson
considers torsion models in the much more general context
of tensor triangulated categories. The paper \cite{Torsion1} is
concerned with 1-dimensional Noetherian Balmer
spectrum, and a special case gives an actual torsion model in the rank
1 case. Work on higher dimensional Noetherian Balmer spectra is
underway, and it is hoped that it will provide a model which in the
particular case of rational $G$-spectra for a torus $G$ will be a model categorical
lift of the torsion abelian model considered here.
\subsection{Main results}
We will define (Section \ref{sec:At}) an abelian category $\cAt(G)$ built from
torsion modules over the polynomial rings $H^*(BG/K)$ for all
subgroups $K$. The category $\cAt(G)$ is rather easy to work with: it
has enough injectives (Section \ref{sec:algAt}), it is of finite
injective dimension (Proposition \ref{prop:id}) and
it is straightforward to make calculations. The category is
precisely designed to be the codomain of a
homology theory $\piAts : \Gspectra \lra \cAt(G)$ (Section \ref{sec:piAt}), and the main
theorem (Theorem \ref{thm:AtASS}) states that there is a finite Adams
spectral sequence
$$\Ext_{\cAt (G)}^{*,*}(\piAts (X), \piAts (Y))\Rightarrow [X,Y]^G_*$$
strongly convergent for any rational $G$-spectra $X$ and $Y$.
\subsection{Notation}
The models assemble data from various subgroups, and it enormously
aids readability to have consistent and suggestive notation. The
ambient torus is $G$, and we generally let containment follow the
alphabet as in $G\supseteq H \supseteq K\supseteq L$.
One of the features of rational $G$-spectra is that one often needs to
group all subgroups with the same component together. We often write
$\Kt$ for a subgroup with identity component $K$ and so forth. We also
write $\cF$ for the family of finite subgroups of $G$ and $\cF/K$ for
the family of finite subgroups of $G/K$ (which is in bijection to the
set of subgroups $\Kt$ of $G$ with identity component $K$.
The piece of data corresponding to a subgroup $K$ is built from the fixed
point set on which $G/K$ acts, so we write $V(G/K)$ and index on the
quotient group. We combine this with the above conventions, so that
$V(\cF/K)$ collects the data for $V(G/\Kt)$ for all subgroups $\Kt\in
\cF/K$.
\subsection{Organization}
Section \ref{sec:semifreecircle} recapitulates the rank 1 semifree
case and should sensitize the reader to issues that will
arise. Section \ref{sec:At} defines the abelian torsion model. Section
\ref{sec:piAt} describes the homology functor on $G$-spectra taking
values in $\cAt (G)$. Section \ref{sec:algAt} begins the algebraic
study of $\cAt (G)$. Section \ref{sec:inj} identifies sufficiently
many injectives, giving a description involving local cohomology of
various localized polynomial rings and residue maps between them.
In Section \ref{sec:id} we show that for a non-trivial torus, the injective dimension of
the torsion model is $\leq 2r$. Section \ref{sec:indcorep} explains how to pick
out the contribution from a specific subgroup by giving a
corepresentation theorem. This then gives us all we need to construct
a finite and strongly convergent Adams spectral sequence based on the
homology of the Borel constructions of fixed points as stated: the
pieces are assembled in Section \ref{sec:AtASS}.
\section{The circle group}
\label{sec:semifreecircle}
The special case when $G$ is the circle group (i.e., the rank $r=1$)
was covered in \cite[Chapter 6]{s1q}. Nonetheless, it will be useful to run
through the arguments, partly because the treatment is a little
condensed in \cite{s1q}, and partly to motivate the general constructions we
will need later.
To make the algebraic structures clearer, we work here with semifree
$G$-spectra (i.e., those with geometric isotropy in $\{1, G\}$) and
their model. Thus only
$G$ and the trivial group $1$ will play a role and diagrams are much
smaller. The case of general $G$-spectra is covered along with other
ranks below. The basic algebraic ingredient is a map of rings $\cOcF\lra
\cEi_G\cOcF$ where $\cE_G$ is a multiplicatively closed set of $\cOcF$.
This will be introduced in general later, but for semifree $G$-spectra it
is the map of rings $k[c]\lra k[c,c^{-1}]$ for a field $k$ and an element
$c$ of degree $-2$. We write $t=k[c,c^{-1}]$ for brevity and we will take
$k=\Q$ for the topological applications.
\subsection{Homological algebra}
The objects of the semifree torsion abelian model $\cAsft (G)$ are
$X=(t\tensor V \lra T)$ where $V$ is a $k$-module, $t=k[c,
c^{-1}]$,
and $T$ is a torsion $k[c]$-module and we have a $k[c]$-map between
them. In view of the adjunction $\Hom_{k[t]}(t\tensor V,
T)=\Hom_k(V, \Hom_{k[t]}(t,T))$, it is sometimes more convenient to
consider the equivalent adjoint form of the torsion
abelian category, with objects $\tilde{X}=( V \lra T^t)$ where
$T^t=\Hom_{k[c]}(t, T)$. Note that maps on the torsion parts $A^t\lra
B^t$ are required to be of the form $\theta^t$ for a map $\theta :
A\lra B$, and cokernels are $\cok^t (A^t\lra B^t)=\cok(A\lra
B)^t$, which may not be the same as the ordinary cokernel of the module map
$A^t \lra B^t$.
Of course we have objects $\ft_G(V)=(t\tensor V \lra 0)$ and
$\ft_1(T)=(0\lra T)$. The former is
injective and for $T\neq 0$ the latter is not injective even if $T$ is an injective
$k[c]$-module.
It is easy to see that $\ft_G$ is right adjoint to evaluation at $G$, and there is a
functor $\at_1$ right adjoint to evaluation at the trivial subgroup
$1$.
This is more obvious in the adjoint form where we have
$\att_1(T)=(id: T^t\lra T^t)$. Thus $\at_1(T)=(ev: t\tensor T^t \lra
T)$. It is immediate from the adjunction that $\at_1(I)$ is injective
if $I$ is an injective torsion $\cOcF$-module.
\begin{lemma}
The injective dimension of $\cAsft(G)$ is $\leq 2$.
\end{lemma}
\begin{proof} Choose a resolution $0\lra T \lra I \lra
J\lra 0$ of $T$ by torsion injective $k[c]$-modules.
Now take the maps (i) $X \lra \at_1(I)$, which is $T\lra I$ at $G/1$, and
(ii) $X \lra \at_G(V)$, which is the identity at $G/G$. This gives a
monomorphism
$X\lra \at_1(I)\oplus \at_G(V)$. The cokernel $C$ is $J$ at
$G/1$, and if we suppose it is $V'$ at $G/G$ we obtain a resolution
$$0\lra X \lra \at_1(I)\oplus \at_G(V)\lra \at_1(J)\oplus \at_G(V')\lra
\at_G(V'')\lra 0. $$
\end{proof}
To see the injective dimension is exactly 2, we need to make a
calculation.
\begin{lemma}
We have
$$\ExtAt^s (\ft_G(k), \at_1(T))=
\dichotomy {\Ext_{k[t]}^{1}(t,T)& s=2}
{0 & \mbox{otherwise}}$$
\end{lemma}
\begin{proof}
If $T$ is a torsion $k[c]$-module with injective resolution
$$0\lra T \lra I \lra J \lra 0$$
then Hom and Ext are given by the exact sequence
$$0\lra T^t \lra I^t\lra J^t \lra \Ext_{k[t]}(t, T)\lra 0.$$
We may now write down a resolution of $\at_1(T)$:
$$0\lra \at_1(T) \lra \at_1(I)\lra \at_1(J)\lra
\at_G(T^{\dagger t}) \lra 0$$
where $T^{\dagger t}=\Ext^1_{k[t]}(t,T)$.
The answer is clear by applying $\HomAt(\ft_1(k), \cdot )$ since if
$\tilde{Y}=(\tilde{q}: W\lra U^t)$ we have
$$\HomAt(\ft_1(k), X) =\ker (\tilde{q}: V\lra U^t) .$$
\end{proof}
One needs to think a little to identify a torsion module $T$ with nonzero
$\Ext^1_{k[c]}(t, T)$. However
$$\Ext^1(t, T)= R\lim \left[ \cdots \stackrel{c} \lra
\Sigma^{-4}T\stackrel{c}\lra \Sigma^{-2}T\stackrel{c}\lra T\right]$$
so we see $T=\bigoplus_{s\geq 1}\Sigma^{2s} k[c]/c^s$ will do.
\begin{remark}
\label{rem:suminjnotinj}
The sum of injectives need not be injective. Indeed if
we take
$$I=\left[ \bigoplus_s\Sigma^{2s}k[c]\right]^*=\prod_s\Sigma^{2s}k[c]^*,$$
we see that the map
$$\bigoplus_s \Hom (t, \Sigma^{2s}k[c]^*) \lra \Hom(t, \bigoplus_s
\Sigma^{2s}k[c]^*)=\Hom (t, I)$$
is not an isomorphism. Choosing an element $\delta$ not in the image
(such as the map diagonal on nonzero entries in each degree) we see
that there is no solution to the problem
$$\xymatrix{0\rto &f_1(k[c]^*)\rto
\dto^{\delta}&a_1(k[c]^*)\ar@{..>}[dl]\\
&\bigoplus_s\Sigma^{2s}a_1(k[c]^*)&
}$$
and hence $\bigoplus_s\Sigma^{2s}a_1(k[c]^*)$ is not injective.
\end{remark}
\subsection{The Adams spectral sequence}
Writing $\mbox{sf-$G$-spectra}$ for the homotopy category of semi-free
rational $G$-spectra, we begin by defining the homology theory
$$\piAts: \mbox{sf-$G$-spectra} \lra \cAt^{sf} (G). $$
For this, we consider the isotropy separation sequence
$$\efp \lra S^0 \lra \etf \lra \Sigma \efp$$
and for a semifree $G$-spectrum $X$ we take
$$\piAt_*(X)=\pi^G_*(\etf \sm D\efp \sm X \lra \Sigma \efp\sm D\efp \sm X). $$
Of course we are more precisely using
$$\pi^G_*(\etf\sm D\efp \sm X)\cong t \tensor \pi_*(\Phi^GX)=:t\tensor V_X$$
and, since $D\efp \sm \efp \simeq \efp$, and $X$ is semifree,
$$\pi^G_*(\Sigma \efp\sm D\efp \sm X)\cong \pi^G_*(\Sigma EG_+\sm X)\cong
H_*^{G}(\Sigma^2 X) =: T_X$$
(where $H^G_*$ denotes homology of the Borel construction) so that
$$V_X=H_*(\Phi^GX)$$
and
$$T_X=\Sigma^2 H_*^G(X). $$
We thus have a map
$$d: [X,Y]^G_*\lra \Hom_{\cAt(G)}(\piAts (X), \piAts(Y)). $$
We will now proceed by the usual method towards an Adams spectral
sequence.
The first question is whether we can realise enough
injectives. We have made a start since $\ft_G(k)=\at_G(k)$. Next we
would like to realize $\at_1(k[x]^*)$. We could use Brown representability as in
the general case below (Lemma \ref{lem:rinj}), but by way of
variation, we give here a construction in terms of
well known objects.
It is standard that $\piAts(EG_+)=H_*^G(\Sigma EG_+)=\Sigma k[t]^*$
and therefore (care about
suspensions) $\piAts(EG_+)=\Sigma^2 \ft_1(k[x]^*)$.
We break $\at_1(k[x]^*)$ down using the short exact sequence
$$0\lra \ft_1(k[c]^*)\lra \at_1(k[c]^*)\lra \ft_G((k[c]^*)^t)\lra
0. $$
It is reassuring to note that $(k[c]^*)^t\cong t$, but in fact it is
more helpful to retain the functional form $(k[x]^*)^t$. In any case, we see that $\at_1(k[x]^*)$ is the
fibre of a map
$$\ft_G(t)\lra \Sigma \ft_1(k[c]^*), $$
so (with care again about suspensions) the realization should be the fibre of a map
$$\etf [t]\lra \Sigma^{-1}EG_+, $$
where $[t]$ indicates the use of a module of graded coefficients.
We may calculate the maps $\etf\lra EG_+$ using the exact
sequence
$$\cdots \lra [\Sigma EG_+, EG_+]^G\lra [\etf, EG_+]^G\lra [S^0,
EG_+]^G\lra \cdots$$
A priori it is long exact, but the outer two groups are in odd
degrees, so it is short exact. Since $S^{\infty z}$ is a smash factor of
$\etf$, multiplication by $c$ is an
isomorphism in the middle we see $[\etf, \Sigma^{-1}EG_+]^G=t$.
Indeed, we may take the map
$$t\tensor (k[c]^*)^t \lra \Sigma^{-1}k[c]^*$$
to be evaluation. Following through the isomorphisms we see
$$a_1(k[c]^*)=\piAts( \fibre (ev: \etf[t]\lra \Sigma^{-1}EG_+)). $$
\begin{thm} \cite[Theorem 6.6.2]{s1q}
For rational semifree $G$-spectra $X$ and $Y$ there is an Adams spectral sequence
$$\ExtAsft^{*,*} (\piAts (X), \piAts (Y))\Rightarrow [X,Y]^G_*. $$
This is a finite, strongly convergent spectral sequence.
\end{thm}
\begin{proof}
As usual we need only show that enough injectives are realizable and
that the $d$-invariant is an isomorphism when $Y$ is one of the
realizable injectives. Convergence is clear since $\piAts$ is detects
contractibility and commutes with telescopes.
Now we need to show
$$[X, \at_1(k[c]^*)]^G_*\lra \HomAt (\piAts (X), \at_1
(k[c]^*)) =\Hom_{k[t]}(T(X), k[x]^*) =\Hom_{k}(T(X), k)$$
This is straightforward since we can see immediately that $[f_G(k),
\at_1(k[x]^*)]^G_*=0$. That in turn means we may assume $X$ is free
and hence it suffices to take $X=G_+$. Now we only need to check that
$$d: [G_+, \ft_1(k[c]^*]^G_*=[G_+, EG_+]^G_*\lra
\Hom_{k[t]}(\pi^G_*(G_+), \pi^G_*(EG_+))$$
is an isomorphism. The idea is to use the cofibre sequence $G_+\lra
S^0\lra S^z$ and connectivity: this is implemented in \cite[Lemma 6.3]{gfreeq}.
\end{proof}
\section{The torsion model}
\label{sec:At}
We now begin work on the general case, so that $G$ is a torus of rank $r$.
We will write down the category $\cAt (G)$ directly,
because this emphasizes the algebraic simplicity but many features
will appear mysterious. We will return to explain the form of the
definition in Section \ref{sec:piAt}: the category is precisely
designed as the appropriate receptacle for a torsion-based homology
theory on rational $G$-spectra.
\subsection{Inflation systems and Euler classes}
We begin with a diagram of rings and some localizations. For the
present we will restrict this to the context of our applications.
Starting with our torus $G$, we consider the poset of {\em connected}
subgroups $H$ ordered by inclusion.
We then have a diagram
$$\cO_{\cF /}: \connsub(G)^{op}\lra \crings$$
to graded commutative rings. If $K \subseteq H$ the map $\cOcFH\lra
\cOcFK$ is called {\em inflation}, so we call $\cO_{\cF/}$ the {\em
inflation diagram} of rings.
For each $K\subseteq H$ we have a multiplicatively closed subset
$\cE_{H/K}\subseteq \cOcFK$, and these are compatible in the sense
that, for $H \supseteq K \supseteq L$, the inflation of $\cE_{H/K}\subseteq \cOcFK$ lies in
$\cE_{H/L}\subseteq \cOcFL$, and in fact
$$\cE_{H/L}=\langle \cE_{H/K}, \cE_{K/L} \rangle . $$
A $\cOcFK$-module $M$ is {\em torsion} if $\cEi_{H/K}M=0$ for every
$H> K$.
There are two examples to bear in mind from equivariant homotopy
theory. The first is easier to understand, and is often helpful as a
warm-up. The second is
the motivating example, and the notation we have
used for an abstract inflation functor with Euler classes comes from
it.
\begin{example}
The connected group inflation functor is the diagram of rings with
$\cOcFK=H^*(BG/K)$ and $\cE_H=\{ e_1(V)\st V^H=0\}$. Here $e_1(V)\in
H^{|V|}(BG)$ is the classical Euler class.
\end{example}
\begin{example}
\label{eg:fullisotropy}
The full isotropy topological example has
$\cOcFK=\prod_{\Kt} H^*(BG/\Kt)$ (with the product over all subgroups
$\Kt$ with identity component $K$), and
$\cE_H=\{ e(V)\st V^H=0\}$. Here $e(V)\in \cOcF$ has $F$-component
the Euler class $e_F(V)=e_1(V^F)\in H^{|V^F|}(BG/F)$.
The ring $\cOcFK$ has one idempotent for each subgroup $e_{\Kt}$, and
hence any $\cOcFK$-module $M$ has summands $e_{\Kt}$. The nature of
the Euler classes means that any torsion module is a sum of these
pieces: $M=\bigoplus_{\Kt} e_{\Kt}M$.
If $K$ is a subgroup of $H$ the inflation map $\cOcFH\lra \cOcFK$
requires the observation that for each subgroup $\Kt$ with identity
component $K$ there is a unique subgroup $\Ht$ (namely $\Ht=\Kt\cdot
H$) with with identity component $H$ with $\Kt $ cotoral in $\Ht$.
\end{example}
\subsection{The definition}
In the presence of an inflation diagram of rings, and an Euler system
of multiplicatively closed subsets we may define a torsion category.
\begin{defn}
Objects of the abelian torsion model $\cAt(G)$ are cochain complexes
\begin{multline*}
\cEi_G\cOcF \tensor V(\cF/G)\stackrel{h^0}\lra
\bigoplus_{\codim(H)=1}\cEi_H\cOcF \tensor_{\cOcFH} V(\cF/H)\stackrel{h^1}\lra \\
\bigoplus_{\codim(K)=2}\cEi_K\cOcF \tensor_{\cOcFK} V(\cF/K)\stackrel{h^2}\lra \cdots
\stackrel{h^{r-2}}\lra
\bigoplus_{\codim(L)=r-1}\cEi_L\cOcF \tensor_{\cOcFL} V(\cF/L)\stackrel{h^{r-1}}\lra
V(\cF/1)
\end{multline*}
where the sums are over connected groups of the stated sort and $V(\cF/K)$ is a torsion $\cOcFK$-module.
The decomposition into direct sums and tensor products
is a given part of the structure so the morphisms of $\cAt(G)$ are
given by $\cOcFK$-maps $V(\cF/K)\lra V'(\cF/K)$ that give a map of cochain
complexes.
It is convenient to formalize this a little further.
\begin{itemize}
\item For each connected subgroup $K$ we have a torsion $\cOcFK$-module
$V(\cF/K)$, which we call the {\em $\cF/K$ torsion component} of the object.
\item If $K\supseteq L$ there is an inflation map $\cOcFK \lra \cOcFL$
and an upward {\em vertical} map
$$v_{\cF/K}^{\cF/L}: V(\cF/K)\lra \cEi_K \cOcFL\tensor_{\cOcFK}V(\cF/K)$$
of $\cOcFK$-modules.
\item If $K\supseteq L$ there is an inflation map $\cOcFK \lra \cOcFL$
and rightward {\em horizontal} structure maps
$$h_L^K: \cEi_K \cOcFL \tensor_{\cOcFK} V(\cF/K) \lra V(\cF/L)$$
of $\cOcFL$-modules
\item For each connected subgroup $M$ the horizontal maps for
subgroups containing $M$ form a cochain complex $C_{G/M}(\cF/M)$
\begin{multline*}
\cEi_{G/M}\cOcFM \tensor V(\cF/G)\stackrel{h^0_{G/M}}\lra
\bigoplus_{\codim(H/M)=1}\cEi_{H/M}\cOcFM \tensor_{\cOcFH} V(\cF/H)\stackrel{h^1_{G/M}}\lra \\
\cdots
\lra \bigoplus_{\codim(L/M)=\dim(G/M)-1}\cEi_{L/M}\cOcFM
\tensor_{\cOcFL} V(\cF /L)
\stackrel{h^{\dim(G/M)-1}_{G/M}}\lra V(\cF/M)
\end{multline*}
of $\cOcFM$-modules.
\end{itemize}
\end{defn}
\begin{remark}
\newcommand{\LI}{\mathcal{LI}}
(i) We note that there are no suspensions on the objects $V(\cF /H)$. At
present it seems odd to even comment on this, but it will be the basis
of discussion when we return to consider $G$-spectra.
(ii) There is no direct relationship between the different modules
$V(\cF/K)$. We can define a localized inflation diagram $\LI$ of module categories
$$\LI_G: \connsub (G)\lra \cat$$
where $\LI_G(G/H)=\cOcFH\mod$ and if $L\subseteq K$ then
$$\LI_G(\pi_{G/K}^{G/L}): \LI_G(G/K)=\cOcFK\mod \lra \cOcFL\mod =\LI_G(G/L)$$
is defined by
$$\LI_G(\pi_{G/K}^{G/L}) (M) =\cEi_{K/L}\cOcFL\tensor_{\cOcFK}M. $$
In this context $V$ is a section of the diagram $\LI_G$.
\end{remark}
\begin{example}
When $G$ is a circle, the diagram is very simple: an object is given
by a diagram
$$\xymatrix{
\cEi_G \cOcF \tensor V(\cF/G)\rto & V(\cF/1)\\
V(\cF/G)\uto &
}$$
where $V(\cF/G)$ is a graded $\Q$-vector space and $V(\cF/1)$ is a torsion $\cOcF$-module.
\end{example}
\begin{example}
In rank 2 we may still display the diagram.
In short form, $\cAt (G)$ is the category of cochain complexes of $\cOcF$-modules
$$\cEi_G\cOcF \tensor V(\cF/G) \stackrel{h^0}\lra \bigoplus_H\cEi_H \cOcF
\tensor_{\cOcFH}V(\cF/H) \stackrel{h^1}\lra V(\cF/1), $$
where $V(\cF/G)$ is a $\Q$-module, $V(\cF/H)$ is a torsion $\cOcFH$-module
and $V(\cF/1)$ is a torsion $\cOcF$-module.
More explicitly objects are actually part of a larger diagram with this as the top horizontal:
$$
\xymatrix{
\cEi_G\cOcF \tensor V(\cF/G)\rto^(.4){h^0} & \bigoplus_H\cEi_H \cOcF
\tensor_{\cOcFH} V (\cF/H)\rto^(.7){h^1} & V(\cF/1)\\
\cEi_{G/H}\cOcFH \tensor V(\cF/G) \uto^j \rto^(.6){h^G_H} & V(\cF/H) \uto^j & \\
V(\cF/G) \uto^i &&
}$$
The top row is the cochain complex $C_G(\cF/1)$ of $\cOcF$-modules, the
second row (which is really just one of the countably many second rows
corresponding to codimension 1 connected subgroups $H$) is the cochain complex $C_{G/H}(\cF/H)$
of $\cOcFH$-modules and the third row is the cochain complex $C_{G/G}(\cF/G)$ of
$\Q$-modules, since $\cO_{\cF/G}=\Q$. We require that $V(\cF/G)$ is a
$\Q$-module,
$V(\cF/H)$ is a torsion $\cOcFH$-module in the sense that
$\cEi_{G/H}V(\cF/H)=0$ and $V(\cF/1)$ is a torsion $\cOcF$-module in the
sense that $\cEi_H V(\cF/1)=0$ for all circle subgroups $H$.
\end{example}
\subsection{Maps into sums}
Since the map from a sum to a product is a monomorphism, a map
$\theta: A\lra \bigoplus_i B$ is determined by the components
$\theta_i: A\lra B_i$. If $A$ is finitely generated, only finitely
many of these are non-zero, but in general
$$\Hom (A, \bigoplus_i B_i) =\ilim_{\alpha} \Hom (A_\alpha,
\bigoplus_i B_i)
=\ilim_{\alpha} \bigoplus_i \Hom (A_\alpha, B_i), $$
where $A_\alpha$ runs through finitely generated submodules of $A$.
We say that $\{ \theta_i\}_i$ is {\em locally finite} if it lies in
this subgroup.
We may consider what this means for the horizontal maps
$$h: \bigoplus_K \cEi_K \cOcFL \tensor_{\cOcFK} V(\cF/K) \lra \bigoplus_L
V(\cF/L). $$
Such a map $h$ is freely and uniquely determined by the components
$$h(K) : \cEi_K \cOcFL \tensor_{\cOcFK} V(\cF/K) \lra \bigoplus_L
V(\cF/L)$$
by the universal property of the first sum. This map $h(K)$ in turn is
determined by the maps
$$h_L^K: \cEi_K \cOcFL \tensor_{\cOcFK} V(\cF/K) \lra
V(\cF/L)$$
but for each fixed $K$, the collection $h^K_*$ (where * runs through
the subgroups $L\subseteq K$) is subject to the
condition of being locally finite.
In general it is probably easier to consider the map $h$ as a whole
rather than decomposing it into factors.
\subsection{Extra idempotents}
The full-isotropy topological example of Example
\ref{eg:fullisotropy} has the feature that the ring
$\cOcFK$ contains idempotents for each subgroup $\Kt$ with identity
component $K$. It is a consequence of the torsion condition that the
natural map from the sum of idempotent pieces gives an isomorphism
$$V(\cF/K)=\bigoplus_{\Kt}\Vt (G/\Kt),$$
with $\Vt (\Kt)$ a torsion $H^*(BG/\Kt)$-module. This is the reason
for our notation, since of course we usually find $V (\cF/ K)\neq
V(G/K)$.
This also allows us to explain the quotient notation: for comparison
with subgroups we refer to quotients call it $V(G/H)$ after the
ambient group. The point is that if $G\supseteq H \supseteq M$ we have a
canonical isomorphism $(G/M)/(H/M)=G/H$.
This means that there is a second layer of sums available for
decomposition, and we may consider the idempotent pieces
$\htt_{\Lt}^{\Kt}$. These also determine the map $h$, but these are
now subject to two separate sets of local finiteness conditions.
In fact the first condition is that $\htt_{\Lt}^{\Kt}$ is only nonzero
when $\Lt$ is cotoral in $\Kt$. Then for a fixed subgroup $\Kt$, the
collection $\htt^{\Kt}_*$ (where $*$ runs through subgroups $\Lt$
cotoral in $\Kt$) is locally finite. Furthermore, for each fixed $K$
we may write $h^K_{\Lt}=\bigoplus_{\Kt}\htt^{\Kt}_{\Lt}$ and then the
collection $h^K_*$ is locally finite.
\subsection{Support and geometric fixed points}
The most visible feature of an object of $\cAt(G)$ is where it is
non-zero.
\begin{defn}
For an object $X$ of $\cAt(G)$ the {\em connected support} is defined by
$$\supp_c (X)=\{ H \subseteq G \st V(\cF/H)\neq 0\}.$$
In the full isotropy example we may also define the {\em support}
$$\supp (X)=\{ \Ht \subseteq G \st \Vt (G/\Ht)\neq 0\}.$$
\end{defn}
\begin{remark} It is clear that in the full isotropy example the
connected support can be recovered from the support
$$\supp_c(X)=\{ H \st \mbox{ there is a subgroup } \Ht \in \supp
(X) \mbox{ with identity component } H \}.$$
\end{remark}
For a connected subgroup of $G$ there is a functor
$$\Phi^K: \cAt(G)\lra \cAt (G/K). $$
obtained by picking out the part of the diagram below $K$ in the sense
that if $H\supseteq K$
$$(\Phi^K X) ((G/K)/(H/K))=X(G/H). $$
\begin{example}
For example if
$$X=[\cEi_G\cOcF \tensor V(\cF/G) \stackrel{h^0}\lra \bigoplus_H\cEi_H \cOcF
\tensor_{\cOcFH}V(\cF/H) \stackrel{h^1}\lra V(\cF/1)]$$
we have
$$\Phi^HX=\left[ \cEi_{G/H}\cOcFH\tensor V(\cF/G)\lra V(\cF/H) \right]. $$
\end{example}
It is evident that
$$\supp (\Phi^HX)=\supp (X)\cap \{ H \st H\supseteq K\}, $$
where we identify the subgroups of $G/H$ as those containing $K$.
\section{The torsion model and the torsion homology functor}
\label{sec:piAt}
We return to the topology which motivated the definition of
$\cAt(G)$. Thus $G$ is a torus of rank $r$ and we consider $G$-spectra
with arbitrary geometric isotropy. In this section we will define
the homology functor $\piAts: \Gspectra \lra \cAt (G)$.
\subsection{Isotropy separation}
First we recall the filtration
$$\emptyset \subset \cF_{\leq 0}\subset \cF_{\leq 1}\subset
\cF_{\leq 2} \subset \cdots \subset \cF_{\leq r}=\All$$
of the set of closed subgroups, where $\cF_{\leq s}=\{ K \st \dim (K)\leq s\}$. Taking universal
spaces we have the diagram
$$\xymatrix{
\ast =E(\emptyset)_+ \rto & E(\cF_{\leq 0})_+\rto \dto &E(\cF_{\leq
1})_+\rto \dto &E(\cF_{\leq
2})_+\rto \dto
&\cdots \rto&E(\cF_{\leq r})_+=E\All_+=S^0 \dto \\
&E\lr{0}&E\lr{1}&E\lr{2}&&E\lr{r}
}$$
where $E\lr{s}=\cofibre(E(\cF_{\leq r-1})_+\lra E(\cF_{\leq r})_+)$.
Composing the vertical maps with the connecting maps, we obtain the
sequence of maps
$$\xymatrix{
E\lr{r}\rto&\Sigma E\lr{r-1}\rto &
\cdots \rto&\Sigma^{r-1} E\lr{1}\rto &\Sigma^r E\lr{0}.
}$$
This is a cochain complex in the sense that the composite of two
adjacent maps is nullhomotopic.
Finally, by use of idempotents in Burnside rings, we have a rational splitting
$$E\lr{s}\simeq \bigvee_{\dim (\Kt)=s}E\lr{\Kt} \simeq \bigvee_{\dim
(K)=s} \siftyV{K}\sm E\cF/K_+ , $$
where $\Kt$ runs through all subgroups of dimension $s$ and $K$ runs
through connected subgroups of dimension $s$. As usual,
$E\lr{\Kt}=\cofibre(E[\subset \Kt]_+\lra E[\subseteq \Kt]_+)$ (with
geometric isotropy the singleton $\{ \Kt\}$), and
$\siftyV{K}=\bigcup_{V^K=0}S^V$ (with geometric isotropy consisting of
exactly those
subgroups containing $K$).
\begin{example}
If $r=2$ we may write this in more familiar terms
$$\xymatrix{
\ast \rto & \efp \rto \dto &\epp \rto \dto &S^0\dto \\
&\efp &\bigvee_H \siftyV{H}\sm E\cF/H_+&\siftyV{G}
}$$
where $\cF$ is the family of finite subgroups and $\cP$ is the family
of proper subgroups. This in turn gives a sequence
$$\siftyV{G}\lra \Sigma \siftyV{H}\sm E\cF/H_+ \lra \Sigma^2 \efp$$
in which the composite is null.
\end{example}
\subsection{Homotopy of pure strata}
The first approximation to $\cAt (G)$ is obtained by taking the
homotopy of the isotropy separation filtration. One can give a formula for the homotopy of
the subquotients in terms of the homology of the Borel construction.
\begin{lemma}
$$\piGs(E\lr{s}\sm
X)=\Sigma^{r-s}\bigoplus_{\dim(\Kt)=s}H^{G/\Kt}_*(\Phi^{\Kt}X)$$
where the sum is over all subgroups $\Kt$ of dimension $s$. The term
$H^{G/\Kt}_*(\Phi^{\Kt}X)$ is a torsion $H^*(BG/\Kt)$-module.
\end{lemma}
The $E_1$-term of the spectral sequence of the filtration is the
homotopy of the sequence
$$\xymatrix{
E\lr{r}\sm X\rto&\Sigma E\lr{r-1}\sm X\rto &
\cdots \rto&\Sigma^{r-1} E\lr{1}\sm X\rto &\Sigma^r E\lr{0}\sm X
}$$
namely
$$\xymatrix{
\piGs(E\lr{r}\sm X)\rto \ar@{=}[d]
&\piGs(\Sigma E\lr{r-1}\sm X)\rto \ar@{=}[d]&
\cdots \rto&\piGs(\Sigma^r E\lr{0}\sm X) \ar@{=}[d]\\
H_*(\Phi^G X)\rto &
\Sigma^2 \bigoplus_{\dim(\Ht)=r-1}H^{G/\Ht}_*(\Phi^{\Ht} X)\rto &
\cdots \rto&
\Sigma^{2r} \bigoplus_{\dim(F)=0}H^{G/F}_*(\Phi^{F} X)
}$$
In geometric terms this is the direct analogue of the Cousin complex. In order to get
something more algebraic we need to smash the whole thing with $D\efp$
first.
\begin{lemma}
\label{lem:htpypureDefp}
$$\piGs(E\lr{s}\sm D\efp \sm X)=\Sigma^{r-s}\bigoplus_{\dim(\Kt)=s}
\cEi_K\cOcF\tensor_{H^*(BG/\Kt)} H_{G/\Kt}^*(\Phi^{\Kt}X), $$
where the sum is over all subgroups $\Kt$ of dimension $s$. The term
$H^{G/\Kt}_*(\Phi^{\Kt}X)$ is a torsion $H^*(BG/\Kt)$-module.
\end{lemma}
\begin{remark}
If $T$ is a module over $H^*(BG/\Kt)$ then
$$\cOcF \tensor_{H^*(BG/\Kt)}T=[e_{\Kt}\cOcF ]\tensor_{H^*(BG/\Kt)}T$$
where $e_{\Kt}$ is the idempotent supported on finite subgroups
cotoral in $\Kt$.
\end{remark}
\subsection{Collecting subgroups by identity component}
In this section so far, we have treated all subgroups of the same dimension
equally. However the behaviour of Euler classes (specifically the fact
that $e(\alpha^n)=ne(\alpha)$, and we are working rationally) means that it becomes
important to collect together the subgroups according to their
identity component. The convention is that letters $G, H, K, ...$
will be connected subgroups, and $\Ht$ is a subgroup with identity
component $H$, $\Kt$ is a subgroup with identity component $K$, and so
forth.
\subsection{The homology functor}
It should be apparent that the structures in $\cAt(G)$ mirror those in
topology. The following definition should therefore not be a
surprise.
\begin{defn}
$$\piAts: \Gspectra \lra \cAt (G)$$
is defined by taking $C_G(\cF)$ to be $\piGs (D\efp \sm \cdot)$ to the sequence
$$\xymatrix{
E\lr{r}\sm X\rto&\Sigma E\lr{r-1}\sm X\rto &
\cdots \rto&\Sigma^{r-1} E\lr{1}\sm X\rto &\Sigma^r E\lr{0}\sm X
}$$
Specifically
$$V_X(\cF/H)=\bigoplus_{\Ht}V(G/\Ht)$$ and
$$V_X(G/\Ht)=\pi^{G}_* (\Sigma^{\dim (G/H)} E\lr{\Ht} \sm X)
=\Sigma^{2\dim(G/H)}H^{G/\Ht}_*(\Phi^{\Ht}X)$$
\end{defn}
\begin{lemma}
This definition does give a functor to $\cAt(G)$.
\end{lemma}
\begin{proof}
The fact that the modules are torsion and the maps are of the correct
form is the content of Lemma \ref{lem:htpypureDefp}.
It is clear that $C_G(\cF)$ is a cochain complex since the composite of two
morphisms is null-homotopic.
Finally, we need to observe that we have the appropriate comparison
map $C_{G/K}(\cF/K)\lra C_{G/L}(\cF/L)$ when $L\subseteq K$. It evidently suffices to
treat the case $L=1$, and at the point $G/K$ we need the vertical map
$$\xymatrix{
\cEi_K \cOcF\tensor_{\cOcFK}V(\cF/K) \rrto^{\cong}&&\piGs
(\siftyV{K}\sm D\efp \sm E\cF/K_+\sm X) \\
V(\cF/K) \rto^{\cong}\uto & \pi^{G/K}_*(E\cF/K_+\sm \Phi^K X) \rto^{\cong}&
\piGs
(\siftyV{K}\sm E\cF/K_+\sm X) \uto
}$$
It is apparent this is induced by the map $S^0\lra D\efp$.
All of the justifications are natural in $X$, so this does give a
functor.
\end{proof}
\begin{remark}
Note the conventions on suspensions. In transformation groups, the modules $H_*^{G/K}(\Phi^KX)$
occur naturally. We could have arranged that they occur in the model
without suspensions, but for general groups this would lead to
confusion. Instead, with our conventions, we find that in $\piAts(X)$
we have
$$\Vt_X (G/\Kt)=\Sigma^{2\dim (G/\Kt)} H^{G/\Kt}_*(\Phi^{\Kt}X)), $$
so that the contribution from a subgroup occurs suspended by twice its codimension.
For example in rank 1, we take the homotopy of
$$D\efp \sm \etf \sm X \lra \Sigma \efp \sm X$$
giving the object
$$\cEi_G\cOcF \tensor H_*(\Phi^GX)\lra
\bigoplus_F\Sigma^2H_*^{G/F}(\Phi^FX)$$
Thus we have
$$V_X(G/G)=H_*(\Phi^GX),
V_X(G/1)=\Sigma^2\bigoplus_FH^{G/F}_*(\Phi^FX). $$
\end{remark}
\begin{remark}
One of the attractive features of the torsion model is that it
directly reflects the geometric isotropy. It is immediate that
$$\cI_G(X)=\supp (\piAts (X)). $$
\end{remark}
\section{Algebra of the torsion model}
\label{sec:algAt}
We now turn to an algebraic study of the torsion model. Since our
purpose is to define an Adams spectral sequence, it is not surprising
that this is mostly about homological algebra.
\subsection{Skyscraper objects}
We begin by giving a name to the most obvious and elementary construction.
\begin{defn}
If $T$ is a torsion $\cOcFH$-module, we wrtite $\ft_H(T)$
for the object
$$\ft_H(T)(\cF/K)=
\dichotomy{\cEi_H\cOcF \tensor_{\cOcFH} T& \mbox{ if } K=H}
{0& \mbox{ if } K\neq H}
$$
\end{defn}
Evidently if $T\neq 0$, the object $\ft_H(T)$ corresponds to a $G$-spectrum with
connected geometric isotropy $\{H\}$. In particular
$$\piAts (E\langle \Ht\rangle)=\ft_H(H_*(BG/\Ht)), $$
so their importance is clear. However, even though $H_*(BG/\Ht)$ is an
injective $\cOcFH$-module, from an algebraic point of view this
object is not particularly simple.
\begin{lemma}
For a torsion $\cOcFK$-module $T$,
$$\HomAt ( X, \ft_K(T))=\Hom_{\cOcFK }(C_KX , T)$$
where
$$C_KX=\cok (\bigoplus_{H>K} \cEi_{H/K}\cOcFK\tensor_{\cOcFH}V_X(\cF/H) \lra V_X(\cF/K))$$
\end{lemma}
We see that $C_GX=V_X(\cF/G)$, so that $\ft_G(W(\cF/G))$ is always
injective, but for $H\neq G$ the object
$\ft_H(T)$ is never injective unless it is zero.
\subsection{Adjoint form}
It is invaluable to have a right adjoint to evaluation at a particular
subgroup $H$, and in writing these down we would like to work with
adjoint form of torsion diagrams. The basic idea is as in the rank 1
case, but this is complicated by the fact that there are
infinitely many connected subgroups in most dimensions.
\begin{example}
We make this explicit in the case $G$ has rank 2. Suppose then that
$$X=\left[ \cEi_G \cOcF\tensor V\stackrel{h^0}\lra\bigoplus_H \cEi_H\cOcF
\tensor_{\cOcFH} T(H)\stackrel{h^1}\lra S\right]$$
where $V$ is a $\Q$-vector space $T(H)$ is a torsion $\cOcFH$-module
and $S$ is a torsion $\cOcF$-module (against our general principle,
we are abbreviating $T(\cF/H)=T(H))$. We consider first the case when
only finitely many of the $T(H)$ are non-zero. The adjoint form is
then
$$\xymatrix{
V\rto &\bigoplus_H \Hom_{\cOcFH}(\cEi_{G/H}\cOcFH, T(H)) \rto
&\Hom_{\cOcF} (\cEi_G\cOcF, S)
}$$
In long form, $X$ is given by a diagram
$$
\xymatrix{
\cEi_G\cOcF \tensor V\rto^(.37){h^0} & \bigoplus_H\cEi_H \cOcF
\tensor_{\cOcFH} T(H) \rto^(.78){h^1} & S\\
\cEi_{G/H}\cOcFH \tensor V \uto^j \rto^(.56){h^G_H} & T(H) \uto^j & \\
V\uto^i &&
}$$
We convert this to adjoint form
$$\xymatrix{
&&S\dto \\
&\bigoplus_H T(H)\rto \dto & \Hom_{\cOcF} (\cEi_H\cOcF, S)\dto \\
V\rto &\bigoplus_H \Hom_{\cOcFH}(\cEi_{G/H}\cOcFH, T(H)) \rto &\Hom_{\cOcF} (\cEi_G\cOcF, S)
}$$
where, for the purpose of understanding the bottom right entry, it is worth noting
\begin{multline*}
\Hom_{\cOcF}(\cEi_G\cOcF, S)
=\Hom_{\cOcF}(\cEi_H\cOcF\tensor_{\cOcFH}\cEi_{G/H}\cOcFH, S) =\\
\Hom_{\cOcFH}(\cEi_{G/H}\cOcFH, \Hom_{\cOcF}(\cEi_H\cOcF, S))
\end{multline*}
When infinitely many of the $T(H)$ are non-zero, the map from $V$ maps
into the product, but it is a locally finite map. Accordingly, it is
still sufficient to describe the second map as coming from the sum.
\end{example}
It is possible in principle to describe the adjoint form in
full detail. However our narrow purpose is to define a right adjoint
to evaluation, so we will just
record what the adjoint form does on components.
\begin{defn}
Given a flag
$$G\supset H_{r-1} \supset H_{r-2} \supset \cdots \supset
H_1 \supset H_0=1$$
of connected subgroups, the corresponding component of the torsion
model is the cochain complex
\begin{multline*}
\cEi_G \cOcF \tensor V(\cF/G)\lra
\cEi_{H_{r-1}} \cOcF \tensor_{\cO_{\cF/H_{r-1}}} V(\cF/H_{r-1})\\
\lra
\cEi_{H_{r-2}} \cOcF \tensor_{\cO_{\cF/H_{r-2}}} V(\cF/H_{r-2})\lra
\cdots \lra
\cEi_{H_{1}} \cOcF \tensor_{\cO_{\cF/H_{1}}} V(\cF/H_{1})\lra
V(\cF/1)
\end{multline*}
Its adjoint form is
\begin{multline*}
V(\cF/G)\lra
\Hom_{\cO_{\cF/H_{r-1}}}(\cEi_{G/H_{r-1}}\cO_{\cF_{H_{r-1}}},
V(\cF/H_{r-1})) \\
\lra \Hom_{\cO_{\cF/H_{r-2}}}(\cEi_{G/H_{r-2}}\cO_{\cF_{H_{r-2}}},
V(\cF/H_{r-2})) \lra \cdots \\
\lra
\Hom_{\cO_{\cF/H_{1}}}(\cEi_{G/H_{1}}\cO_{\cF/H_1},
V(\cF/H_{1})) \lra
\Hom_{\cO_{\cF}}(\cEi_{G}\cOcF,
V(\cF/1))
\end{multline*}
\end{defn}
\begin{remark}
Even if $V(\cF/H)$ is $\cF/H$-torsion, it does not follow that
$\Hom_{\cOcFH}(\cEi_{G/H}\cOcFH, V(\cF/H))$ is $\cF/H$-torsion.
\end{remark}
\subsection{Right adjoints to evaluation}
From an algebraic point of view the simplest behaviour comes from
objects that let us calculate in terms of rings rather than diagrams
of rings. For this we use right adjoints to evaluation at a subgroup.
The idea is that the adjoint forms are constant
above a given point. This is correct if the subgroup in question is of
codimension $\leq 1$, but not in general, because of the need to use
torsion objects and the distinction between sums and
products.
\begin{example} If $G$ is of rank 2, the
functors representing evaluation are as follows.
$$\at_G(V)=f_G(V)=\left[ \cEi_G \cOcF \tensor V\lra 0 \lra 0\right]$$
$$\at_H(T(H))=\left[ \cEi_G\cOcF\tensor
\Hom_{\cOcFH}(\cEi_{G/H}\cOcFH, T(H))\lra
\cEi_H\cOcF\tensor_{\cOcFH} T(H) \lra 0\right] $$
$$\at_1(S)=\left[ \cEi_G\cOcF\tensor
\Gamma_{\Sigma}\Hom_{\cOcF} (\cEi_G\cOcF, S)\stackrel{h^0}\lra
\bigoplus_H \cEi_H\cOcF\tensor_{\cOcFH}\Gamma_{\cF /H}\Hom_{\cOcF}
(\cEi_{H}\cOcF, S)
\stackrel{h^1}\lra
S\right]$$
where $\Gamma_{\cF/H}$ indicates torsion $\cOcFH$-modules and $\Gamma_{\Sigma}$ refers to the elements mapping into the
the torsion submodules, and into the sum rather than the
product. These two phenomena occur in rank 2 for the first time.
\end{example}
Equipped with this example we can define the objects $\at_L(T)$.
\begin{defn}
\label{defn:atL}
Given a torsion $\cOcFL$-module $T$, we may define an object
$\at_L(T)$ by taking its torsion components to be
$$V(\cF /H)=\dichotomy
{\Gamma_\Sigma \Gamma_{\cF/H} \Hom_{\cOcFL}(\cEi_{H/L}\cOcFL , T)
& \mbox{ if } K\supseteq L }
{0 & \mbox{ if } K\not\supseteq L }
$$
The functor $\Gamma_{\cF/H}$ takes the torsion submodule and
$\Gamma_\Sigma$ takes the submodule mapping into the sum; we will
describe them fully in the next subsection and prove these properties
(Lemma \ref{lem:Gamma}), but for the present we
need only know they give natural submodules of the Hom functor.
\end{defn}
\begin{lemma}
The functor $\at_L$ is right adjoint to evaluation at $L$:
$$\HomAt (X, \at_L(T))=\Hom_{\cOcFL} (V_X(\cF/L), T)$$
\end{lemma}
\begin{proof}
We describe the counit and unit of the adjunction. The triangular
identities will follow from those of the Hom-Tensor adjunction.
The counit $ev_L \at_L(T)\lra T$ is the identity. For the unit
$$X\lra \at_L(V_X(\cF/L)),$$
let us suppose that $L$ is of codimension $t$ and that $r\geq s\geq
t$. In the display, $H$ runs through subgroups of codimension $s+1$
containing $L$ and $K$ runs through subgroups of codimension $s$
containing $L$.
$$\xymatrix{
\bigoplus_{H} \at_L(T)(\cF/H)
\ar@{=}[d] \rto &
\bigoplus_{K} \at_L(T)(\cF/K)
\ar@{=}[d] \\
\bigoplus_{H\geq L} \cEi_{H/K}\cOcFL\tensor_{\cOcFH} \Gamma_{\Sigma}\Gamma_{\cF/H} \Hom_{\cOcFL}(\cEi_{H/L} \cOcFL, T)
\rto &
\bigoplus_{K\geq L} \Gamma_{\Sigma}\Gamma_{\cF/L} \Hom_{\cOcFL}(\cEi_{K/L} \cOcFL, T)
}$$
\end{proof}
\subsection{The torsion subfunctor}
Certain limit constructions do not preserve torsion objects, or do not
preserve the property of mapping into the sum of components, so it is
useful to formalize a bigger category in which the constructions can
be made, together with a right adjoint for returning to $\cAt(G)$.
\begin{defn}
The category $\cAhatt(G)$, has objects $X$ consisting
collections $\{ V(\cF/K)\}_K$ with $V(\cF/K)$ an $\cOcFK$-module with
structure maps
$$h^K_L: \cEi_{K/L}\cOcFL \tensor_{\cOcFK} V(\cF /K)\lra V(\cF/L).$$
\end{defn}
Taking components gives an obvious inclusion functor
$$i: \cAt(G)\lra \cAhatt(G). $$
\begin{lemma}
\label{lem:Gamma}
The inclusion $i$ has a right adjoint $\Gamma: \cAhatt(G)\lra \cAt(G)$.
\end{lemma}
\begin{proof}
The functor $\Gamma$ is the composite $\Gamma=\Gamma''\Gamma'$.
The functor $\Gamma'$ takes the torsion submodule $\Gamma_{\cF/K}
V(\cF/K)$ at $K$. This is compatible with the structure maps, since any
$\cF/K$-torsion element is $\cF/L$-torsion for $L\leq K$.
The functor $\Gamma''$ is defined by taking $\Gamma_\Sigma$ at each
point, where $\Gamma_\Sigma$ can be defined by induction on the
dimension of $H$ using the pullback square
$$\xymatrix{
\Gamma_{\Sigma}\Gamma_{\cF/H}
\Hom_{\cOcFL}(\cEi_{H/L}\cOcFL , T)\dto \rto &
\Hom_{\cOcFK}(\cEi_{H/K}\cOcFK,
\bigoplus_{K\leq H}
\Gamma_\Sigma \Gamma_{\cF/K}\Hom_{\cOcFL}(\cEi_{K/L}\cOcFL , T))\dto \\
\Hom_{\cOcFL}(\cEi_{H/L}\cOcFL , T) \rto &
\Hom_{\cOcFK}(\cEi_{H/K}\cOcFK, \prod_{K\leq H}
\Hom_{\cOcFL}(\cEi_{K/L}\cOcFL , T))
}$$
\end{proof}
In particular this lemma formalizes Definition \ref{defn:atL}.
\section{Injectives}
\label{sec:inj}
We need to show there are enough injectives and describe them in
a way that allows us to do computations. Since $\at_K$
is a right adjoint, it is obvious that, for any torsion injective
$\cOcFK$-module $I$, the object
$\at_K(I)$ is injective in the torsion model. Indeed, this is the
main reason for introducing the $\at_K$ construction.
Next, we know that there are enough injective torsion
$\cOcFK$-modules, which can be constructed as sums of those of the form
$H_*(BG/\tK)$, where $\tK$ runs through subgroups with identity
component $K$.
Finally it is both illuminating and convenient to give a more explicit
description of the functors $\at_K(H_*(BG/\tK))$. After some
recollections about Gorenstein rings in Subsections \ref{subsec:GorD}
and \ref{subsec:GorDllp}, we give the description in Subsection \ref{subsec:atK}.
\subsection{Enough injectives}
\label{subsec:sumsinj}
We wish to construct enough injectives by taking products of those with
explicit constructions.
\begin{cor}
\label{cor:prods}
The category $\cAt(G)$ has products.
\end{cor}
\begin{proof}
Given objects $X_i$ of $\cAt(G)$ we wish to say that the $K$-torsion
component of the product $\prod_i X_i$ is $\prod_i
V_i(\cF/K)$. This lies in $\cAhatt(G)$, so by Lemma \ref{lem:Gamma} we
may apply the right adjoint
$\Gamma$ to obtain an object of $\cAt(G). $
\end{proof}
\begin{lemma}
\label{lem:enoughinj}
There are enough injectives which are products of
injectives of the form $a_K(I)$ for torsion injective $\cOcFK$-modules $I$.
\end{lemma}
\begin{remark}
Exactly as in the rank 1 semifree case (Remark \ref{rem:suminjnotinj}), an arbitrary sum of
injectives need not be injective.
\end{remark}
\begin{proof}
For each subgroup $\Kt$, there are enough torsion injective
$H^*(BG/\Kt)$-modules formed as direct sums of
$H_*(BG/\Kt)$. This means that for any $X$ there is a monomorphism
$V(\cF/K) \lra I(\cF/K))$ where $I(\cF/K)$ is a sum of
$\cOcFK$-modules $H_*(BG/\Kt)$. There is a corresponding map
$X \lra \at_K(I(\cF/K))$ monomorphic at $K$, and hence a monomorphism
$$X\lra \prod_k \at_K(I(\cF/K)).$$
\end{proof}
\subsection{Gorenstein duality}
\label{subsec:GorD}
One particular case of the $\at_K$ construction is especially
important: the one supplying enough injectives. Indeed, we know that
there are enough injective torsion $\cOcFK$-modules of the form
$H_*(BG/\tK)$, where $\tK$ runs through subgroups with identity
component $K$. It is illuminating to have a description of the entries
in $\at_K(H_*(BG/\tK))$ in simple terms. We are able to do this using
the fact that polynomial rings are Gorenstein. The basic idea is very
simple, but there are some issues to highlight along the way.
The ring $H^*(BG)$ is Gorenstein and
$$H^*_{\fm}(H^*(BG))=H^r_{\fm}(H^*(BG))\cong \Sigma^r\Hom_k(H^*(BG),
k)=\Sigma^r H_*(BG). $$
There are two notable things about this. First,
the isomorphism is not natural in the sense that it should be twisted
by the determinant in order to be natural for ring isomorphisms. Second, the $\Hom_k$ consists of {\em graded} maps. If
we used all maps we would obtain a completion of $H_*(BG)$, and we
would need to pass to cellularizations to recover $H_*(BG)$ itself.
We need something more general. For a Gorenstein local
ring $R$ of dimension $r$ we have
$$H^*_{\fm}(R)=H^r_{\fm}(R)\cong E_R(k)$$
where $E_R(k)$ is the injective hull of the residue field $k$. If $R$
happens to be a $k$-algebra, we may form $\Ehat_R(k)=\Hom_k(R,k)$, and
the map $k=\Hom_k(k,k)\lra \Hom_k(R,k)=\Ehat_R(k)$ extends to an
embedding $E_R(k)\subseteq \Ehat_R(k)$. However we note that if $R$ is
of countably infinite dimension as a $k$-vector space $\Ehat_R(k)$ will be of
uncountable dimension, whilst $E_R(k)$ will be of countable
dimension, so $E_R(k)\neq \Ehat_R(k)$ in general.
Accordingly the best we can hope for is that we have a monomorphism
$$H^r_{\fm}(R)\lra \Hom_k(R,k)=\Ehat_R(k)$$
with the image being a copy of $E_R(k)$. When $R$ is polynomial, we can
recover this since we have a residue map
$$\res: H^r_{\fm}(R)\tensor \Omega^r_{R/k}\lra k, $$
where $\Omega^r_{R/k}$ is the module of K\"ahler differentials, with
$\Omega^r_{R/k}\cong R$. This
gives rise to
$$H^*_{\fm}(R)=H^r_{\fm}(R)\lra \Hom_k(\Omega^r_{R/k},
k)\cong \Ehat_R(k) $$
\subsection{Gorenstein duality for linear-localized polynomial rings}
\label{subsec:GorDllp}
Our situation has all the features described in Subsection
\ref{subsec:GorD}
but is not quite standard since
the rings are not local. For a connected subgroup $K$ of codimension $s$, we need to consider the ring
$R=\cEi_KH^*(BG)$.
We think of $H^*(BG)$ as polynomial functions on the affine space
$TG=\spec(H^*(BG))$. Corresponding to the short exact sequence
$$1\lra K \lra G \lra G/K\lra 1$$
we have maps
$$H^*(BK)\lla H^*(BG)\lla H^*(BG/K)$$
and a fibration
$$TK \lra TG \lra TG/K. $$
The ring $H^*(BG/K)$ is thus naturally a
subring of $H^*(BG)$. Choosing particular degree $-2$ generators
$x_i$, we have $H^*(BG/K)=k[x_1, x_2, \ldots , x_s]$ and the kernel of
restriction to $K$ is the ideal $\fm_{G/K}=(x_1, \ldots ,x_s)$.
To go further we choose a splitting. If $H^*(BK) = k[y_1, y_2, \ldots
, y_t]$, we have $H^*(BG)\cong k[x_1, \ldots , x_s, y_1, \ldots
, y_t]$. The linear forms in $\cE_K$ are precisely those forms
involving some $y_i$ (or, intrinsically, those not in the image of
$H^*(BG/K)$). We therefore have a natural map $\cEi_KH^*(BK)\lla
\cEi_K H^*(BG)$ which
plays the role of the map from $R$ to its residue field.
Since $K$ is connected, we may choose a splitting of the inclusion $K
\lra G$ and hence make $R=\cEi_KH^*(BG)$ an algebra over
$\vark = \cEi_KH^*(BK)$.
The ring $R$ consists of meromorphic functions on $T G$ regular on
$TK$ but with denominators that are products of linear
forms.
\begin{lemma}
\label{lem:GorD}
We have a natural embedding
$$H^*_{\fm_{G/K}}(\cEi_K H^*(BG))=H^{2s}_{\fm_{G/K}}(\cEi_K H^*(BG))\lra
\Sigma^{2s}\Hom_k(\cEi_K H^*(BG),
k), $$
giving an isomorphism
$$H^{s}_{\fm_{G/K}}(\cEi_K H^*(BG))\lra
\Sigma^{2s}\Gamma_{\cF /K}\Hom_{\vark} (\cEi_K H^*(BG),
\vark ), $$
\end{lemma}
\begin{proof}
Choose a splitting $H^*(BG)\cong H^*(BG/K)\tensor
H^*(BK)$, and note that
$$H^*_{\fm_{G/K}}(H^*(BG))=H^*_{\fm_{G/K}}(H^*(BG/K))\tensor H^*(BK). $$
As in Subsection \ref{subsec:GorD}
$H^*_{\fm_{G/K}} (H^*(BG/K)) =H^{s}_{\fm_{G/K}} (H^*(BG/K))$
has a basis consisting of monomials $x_1^{i_1}x_2^{i_2}\cdots
x_{s}^{i_{s}}$ with all exponents negative. Thus the top degree basis element is
$(x_1x_2\cdots x_s)^{-1}$ and the map to $\Hom_k(H^*(BG/K), k)$
is given by residues. This gives an isomorphism
$$H^*_{\fm_{G/K}}(H^*(BG/K))\cong \Gamma_{\fm_{G/K}}\Hom_k(H^*(BG/K),
k). $$
With our usual convention of taking graded Hom, the
$\Gamma_{\fm_{G/K}}$ could be omitted since $H^*(BG/K)$ is finite
dimensional in each degree.
Now tensor this with $H^*(BK)$ and localize:
\begin{multline*}
\cEi_K H^*_{\fm_{G/K}}(H^*(BG))=
\cEi_K \left [ H^*(BK)\tensor H^*_{\fm_{G/K}}(H^*(BG/K))\right] \cong \\
\cEi_K \left[ H^*(BK)\tensor \Gamma_{\fm_{G/K}}\Hom_k(H^*(BG/K),
k)\right]
\end{multline*}
We continue to write $\cE_K\subseteq H^*(BG)$ for the multiplicatively
closed set of $K$-essential representations of $G$. Having chosen a
splitting of $K\lra G$, we need a different notation for those arising
from representations of $K$, so write $\cE'_K=\cE_K\cap H^*(BK)$.
Now if $M$ is an $\fm_{G/K}$-power torsion module then inverting
$\cE_K$ is the same as inverting $\cE'_K$ (indeed, if
$u$ is invertible and $x\in \fm_{G/K}$ then $u+x$ is invertible).
Accordingly, $\cEi_K H^*_{\fm_{G/K}}(H^*(BG))$ is a free module over
$k_K=\cEi_K H^*(BK)$ on the monomial basis of negative powers above.
Thus $\cEi_K H^*_{\fm_{G/K}}(H^*(BG))$ embeds in
$\Hom_{k_K}(\cEi_K H^*(BG), k_K) $.
Now consider the $\Gamma_{\cF /K}$-torsion in a module $M$. It has a
filtration
$$0\subseteq \ann (\fm_{G/K}, M)\subseteq \ann (\fm_{G/K}^2,
M)\subseteq \ann (\fm_{G/K}^3, M) \subseteq \cdots \subseteq \Gamma_{\cF/K}M.$$
The subquotients are modules over $\cEi_K H^*(BG)/\fm_{G/K}=\cEi_K
H^*(BK)$.
Taking $M=\Hom_{k_K}(\cEi_KH^*(BG), k_K)$ we prove by induction on $a$
that $\ann (\fm_{G/K}^{a+1}, M)/\ann (\fm_{G/K}^{a}, M)$ is a free
module on the negative monomials of total degree $-s-a$. The
monomials are independent, so it remains to show they span. We know
$\cEi_K H^*(BG)$ is free over $k_K$ on the monomials, so we need only
observe that a function annihilated by $\fm_{G/K}^a$ is zero on
monomials of degree $\geq a+1$.
\end{proof}
\subsection{Explicit description of generating injectives}
\label{subsec:atK}
Using the work of the last two subsections we can give an
attractive description of $\at_L(H_*(BG/\Lt))$ showing that it
captures global geometry.
\begin{lemma}
\label{lem:injasloccoh}
For a subgroup $\Kt$, the torsion part of $\at_L(H_*(BG/\Lt))$ is given by
$$V(G/\Kt)=\dichotomy
{\Sigma^{-2\dim(G/K)}H^{\dim(G/K)}_{\fm_{G/\Kt}}(\cEi_{K/L}H^*(BG/\Lt)) & \mbox{ if
$\Lt$ is cotoral in $\Kt$ }}
{ 0 & \mbox{ otherwise }} $$
\end{lemma}
\begin{proof}
The vanishing is clear. Let $s=\dim (G/K)$. By Lemma \ref{lem:GorD},
we have an isomorphism
$$H^{s}_{\fm_{G/K}}(\cEi_K H^*(BG/\Lt))\stackrel{\cong}\lra
\Sigma^{2s}\Gamma_{\cF/K}\Hom_{\vark} (\cEi_{K/L} H^*(BG/\Lt),
\vark ). $$
The result follows form Definition \ref{defn:atL}, once we take
account of the effect of $\Gamma_\Sigma$.
First, we observe that the embedding maps into $\Gamma_\Sigma$.
We have already seen that
an element $h=\lambda/e(V)\tensor f/x_1^{i_1} x_2^{i_2}\cdots x_s^{i_s}$
automatically maps to a torsion element. It remains to observe that it is only nonzero
for finitely many $L$. Indeed, if $L$ is of codimension 1 in $K$ and
$\Lt$ is cotoral in $\Kt$ then $H^*(BG/\Lt)=k[x_1, \ldots , x_s, z]$
for some independent linear form $z$. Then $h$ will only map to a
nonzero element if it is not regular, and this only happens if $z$
occurs amongst the finitely many linear factors of $e(V)$.
Since the map is surjective by Lemma \ref{lem:GorD} it follows that
$\Gamma_\Sigma$ is the identity on this object.
\end{proof}
The structure maps are given by the relative residue maps.
\begin{lemma}
For cotoral inclusions $\Ht\supseteq \Kt \supseteq \Lt$ the structure
map
\begin{multline*}
h^{\Ht}_{\Kt}: \cEi_{H/L}H^*(BG/\Lt )\tensor
\Sigma^{-2\dim(G/H)}H^{\dim (g/H)}(\cEi_{H/L}H^*(BG/\Lt))\\
\lra
\Sigma^{-2\dim(G/K)}H^{\dim(G/K)}(\cEi_{K/L}H^*(BG/\Lt))
\end{multline*}
is given by the relative residue.\qqed
\end{lemma}
\section{Injective dimension}
\label{sec:id}
For the {\em convergence} of the Adams spectral sequence it is
enough to know $\cAt(G)$ has finite injective dimension. This is an
easy consequence of the finite injective dimension of torsion modules
over a polynomial ring, and the proof is given in Proposition \ref{prop:id}.
For the {\em use} of the Adams Spectral Sequence it is enough to be able to
calculate with $\cAt(G)$, and for this purpose one does not need to
know the exact injective dimension either, so it will suffice to
give a finite upper bound.
\subsection{An upper bound for the injective dimension}
It is very easy to give an upper bound, simply using the fact that the
injectives $\at_L(I)$ are only non-zero at $K$ when $K$ contains
$L$. Indeed, we may argue by induction on the codimension of support
that an object $X$ with support in codimension $\leq c$ is of finite
injective dimension. This is obvious for $c=0$ since $\at_G(V)$ is
always injective. Supposing that $X$ has support in codimension $c$
and that objects with support of lower codimension are of finite
injective dimension then we may
construct the start of an injective resolution by focusing on
subgroups $L$ of codimension $c$ and starting an injective resolution
$$0\lra X \lra \bbI_0\lra \bbI_1\lra \cdots \lra \bbI_c$$
so that for each subgroup $L$, the torsion components $V(\cF/L)$ give
an injective resolution of $V_X(\cF/L)$. The cokernel of
$\bbI_{c-1}\lra \bbI_c$ has support in codimension $\leq c-1$ and it
is therefore of finite injective dimension by induction. This gives a
bound quadratic in $c$. With an additional fact we can get a
linear bound.
\begin{lemma}
\label {lem:atIcomponents}
If $I$ is an injective torsion $\cOcFK$-module then the components of
$\at_K(I)$ are all injective.
\end{lemma}
\begin{proof}
We have seen the component at $H$ is zero unless $K\leq H$ and
the component at $H$ is
$$\Gamma_\Sigma \Gamma_{\cF/H}\Hom_{\cOcFK}(\cEi_{H/K}\cOcFK, I)).$$
This is formed from the injective $I$ by applying three right
adjoints, $ \Hom_{\cOcFK}(\cEi_{H/K}\cOcFK, \cdot )), $
$\Gamma_{\cF/K}$ and $\Gamma_\Sigma$ so the result is also
injective.
\end{proof}
\begin{prop}
\label{prop:id}
An object $X$ of codimension $c$ has injective dimension $\leq 2c$. In
particular, the injective dimension of $\cAt(G)$ is $\leq 2r$.
\end{prop}
\begin{proof} We will give the argument for a general object (i.e.,
with support in codimension $\leq r$), but clearly it applies to any
codimension.
For an object $X$ of $\cAt(G)$, we describe how to construct an injective resolution
$$0\lra X \lra \bbI_0\lra \bbI_1\lra \cdots \lra \bbI_{2r}\lra 0$$
of length $2r$, and we write $X=X_0$ and for $i\geq 1 $
we take $X_i=\cok (X_{i-1} \lra \bbI_{i-1})$.
There are two halves to the construction
\begin{itemize}
\item For $X_0, \ldots , X_r$ the injective dimensions of the individual
components is steadily reduced: noting that if $K$ is of codimension
$c$ then the injective dimension of torsion $\cOcFK$-modules is $c$,
we ensure that $X_s(\cF/K)$ is either injective or of injective
dimension $\leq c-s$. Thus the torsion components of $X_r$ are
all injective.
\item For $X_{r}, X_{r+1}, \ldots , X_{2r} $ we retain the
property that all torsion components are injective but we ensure
steadily more of them are zero, so that $X_{r+i}(\cF/K)=0$ for
$\dim (K)<i$.
\end{itemize}
This is rather straightforward. We construct the resolution
recursively. We take $X=X_0$ and for $s\geq 0$ we suppose we
have constructed up to $X_s$ in the exact sequence
$$0\lra X \lra \bbI_0\lra \cdots \lra \bbI_{s-1}\lra X_s\lra 0 $$
For each $K$ we choose an
injective torsion $\cOcFK$-module $I_s(K)$ and a monomorphism $i_s(K): X_s (\cF/K)\lra
I_s(K)$. If $X_s(\cF/K)$ is already injective, we take
$I_s(K)=X_s(\cF/K)$ and $i_s(K)$ to be the identity.
Now take
$\bbI_s=\prod_K \at_K(I_s(K))$ and define
$i_s : X_s\lra \bbI_s$ by ensuring the $\at_K(I_s(K))$ component
corresponds to $i_s(K)$ under the adjunction
$$\Hom_{\cAt(G)} (X_s, \at_K(I_s(K)))=\Hom_{\cOcFK} (X_s(\cF/K), I_s(K)).$$
This ensures the map $i_s$ is a monomorphism and we take
$X_{s+1}=\cok (i_s: X_s\lra \bbI_s)$.
By Lemma \ref{lem:atIcomponents}, the value of $\bbI_s$ at $\cF/K$ is
injective, so that if $X_s(\cF/K)$ is not already injective,
$\id (X_{s+1}(\cF/K)) =\id (X_s(\cF/K))-1$. This deals with the first
half of the construction. For the second half we suppose that that
$s=r+i$ with $i\geq 0$ and $X_r, X_{r+1}, \ldots, X_{r+i}$ have support as required,
so that in particular $X_{r+i}$ has support in dimension $\geq i$.
By construction $\bbI_{r+i}$ also has support in dimension $\geq i$,
and if $K$ is of dimension $i$, the only factor of $\bbI_{r+i}$ not zero at $K$ is
$\at_K(I_{r+i}(K))$, so that
$$X_{r+i+1}(\cF/K)=\cok \left[X_{r+i}(\cF/K)\lra
\bbI_{r+i}(\cF/K)=I_{r+i}(K)
\right]=0 $$
as required. We note that when $i=r$ the conclusion is $X_{2r+1}=0$.
\end{proof}
\subsection{Local duality}
\label{subsec:localduality}
The purpose of this section is to observe that if $T$ is an Artinian
torsion module then $a_L(T)$ is of injective dimension $\leq \dim (G/L)$ as
one might expect. The proof is straightforward using local duality.
One might view this as saying that most of the objects of $\cAt(G)$
that we need to consider have injective dimension $\leq r$. However
this is a bit misleading, since even simple operations like infinite sums may give objects of
higher injective dimension.
\begin{prop}
\label{prop:artid}
If $T$ is an Artinian $\cOcFL$-module then $\id_{\cOcFL} (T)=\id_{\cAt(G)} (\at_L(T))$.
\end{prop}
\begin{proof}
Since $T$ is Artinian, we may choose
a resolution by Artinian injective torsion modules
$$0\lra T \lra I_0\lra I_1\lra \cdots \lra I_d\lra 0. $$
We will show that
$$0\lra \at_L(T) \lra \at_L(I_0)\lra \at_L(I_1)\lra \cdots \lra \at_L(I_d)\lra 0$$
is exact, which gives the desired conclusion.
The point is that if we take $I=H_*(BG/L)$ then by Lemma
\ref{lem:injasloccoh} the torsion
component of $\at_L(I)$ at $\cF/K$ is
$H^t_{\fm_{G/K}}(\cEi_KH^*(BG/L))$, so we need to see that this
process is exact.
For this, we apply local duality. The idea is that the
injective $\cOcFK$-module resolution $I_\bullet$ can be recognized as the local
cohomology of the dual of a free $\cOcFK$-module resolution $F_\bullet$ of a dual
module $N$. By a similar process after localization, the values at
other levels are then recognized as local
cohomology of localizations of $N$. The exactness of
$I_\bullet$ then gives exactness of these.
The key here is that for the polynomial ring $P=k[x_1, \ldots , x_s]$ we have
$$H^*_\fm(P)=H^s_\fm(P)=\Sigma^{2s} \Hom_k(P,k). $$
As usual the $\Hom_k$ refers to graded maps, and if we used ungraded
maps we would insert $\Gamma_\fm$ to achieve the same end. From the
variance we can see this cannot be natural, but we can easily correct
that: for arbitrary free $P$-modules $F$ we have a natural isomorphism
$$\Hom_k( H^s_\fm (F), k)=\Sigma^{-2s} \Hom_P (F, P). $$
Thus if $F_\bullet$ is a free resolution of a
$P$-module $N$ we have
$$\Hom_k( H^{s-i}_\fm (N), k)=\Sigma^{-2s} \Ext^{i}_P (N, P) $$
Note that we have natural maps
$$H^s_\fm (F)\lra \Sigma^{2s}\Hom_k (\Hom_k (H^s_\fm(F), k),k) \mbox{ and }
F\lra \Sigma^{2s}\Hom_P (\Hom_P (F,P), P), $$
both of which are isomorphisms if $F$ is of finite rank. Thus if
$M$ is Artinian, the injective resolution $I_\bullet$ of $M$
determines a free resolution $F_\bullet$ of $N=\Hom_k(M,k)$.
Similarly $\cEi_K F_\bullet$ is a free resolution of
$\cEi_K\Hom_k(M,k)$, and then by taking
$\Hom_{k_K}(\cdot , k_K)$ we obtain an
injective resolution of $\Hom_{k_K}(\cEi_K M, k_K)$.
By local duality this is
$$\at_L(I_{\bullet}(G/K))=\Sigma^{-2t}H^{t}_{\fm_{G/K}}(\Hom_{\cEi_K
P}(\cEi_KF_\bullet, k_K)) , $$
showing that the complex is exact as required.
\end{proof}
\subsection{Attainment of the bound}
\label{subsec:attained}
The aim of this subsection (not currently achieved!) is to establish
the exact injective dimension of $\cAt(G)$ by writing down an object
of injective dimension equal to the upper bound established in
Proposition \ref{prop:id}. We showed in Proposition \ref{prop:artid}
that if $T$ is Artinian $\at_L(T)$ has the same injective dimension as $T$ does, and
hence all objects of this type have injective dimension $\leq r$.
We will show that for $r\geq 1$ there are objects $X$ with codimension
$1$ that are of injective dimension 2. This only establishes the injective
dimension of $\cAt(G)$ for the circle group. The same argument
shows that there is an object of codimension $c$ with injective
dimension $c+1$ in general, so that we only know that the injective
dimension of $\cAt(G)$ lies between $r+1$ and $2r$.
\begin{lemma}
If $X$ has support in codimension 1 then
\begin{itemize}
\item $X$ is injective if and only if $V(\cF/G)\lra \Hom_{\cOcFH}( \cEi_{G/H}
\cOcFH, V(\cF/H))$ is surjective for all $H$ of codimension 1.
\item $X$ is of injective dimension $\leq 1$ if and only if
$$\Ext_{\cOcFH}^1(\cEi_{G/H}\cOcFH, V(\cF/H))= 0 \mbox{ for all subgroups
$H$ of codimension 1}$$
\end{itemize}
\end{lemma}
\begin{lemma}
\label{lem:aspherical}
$$\Ext^s_{\cOcFH}(\cEi_H\cOcF , T)=
\trichotomy{\Hom_{\cOcFH}(\cEi_H\cOcF , T) \mbox{ if } s=0}
{\ilim^1(T, \cE_H) \mbox{ if } s=1}
{0 \mbox{ if } s\geq 2}$$
\end{lemma}
For a commutative ring $R$ and a
multiplicatively closed set $\cE$, we have
$$\Ext_R^1(\cEi R, M)=\ilim^1 (M,\cE). $$
We make some observations about vanishing and non-vanishing of this.
\begin{lemma}
Suppose $R$ is a commutative ring and $\cE$ is a multiplicatively
closed subset. There is an isomorphism
$$\ilim^1_{\cE} (eM)\cong M_{\cE}^\wedge/M. $$
There is an epimorphism
$$\ilim^1_{\cE} (M,e)\lra R^1\ilim_{\cE} (eM)$$
\end{lemma}
\begin{proof}
For the first statement we have the inverse system
$$\xymatrix{0\rto &eM \rto & M\rto &M/eM \rto &0\\
0\rto &efM \uto \rto & M\uto^{=} \rto &M/efM \uto \rto &0
}$$
where the maps are inclusions and projections.
Taking inverse limits gives a six term exact sequence with the last
two zero.
For the second we have a inverse system
$$\xymatrix{
0\rto &\ann_M (e) \rto & M\rto^e &eM \rto &0 \\
0 \rto &\ann_M(ef) \rto \uto^f& M\rto^{ef}\uto^f &efM \uto \rto &0
}$$
The last map in the six term exact sequence is an epimorphism.
\end{proof}
In particular if we may take $R=k[x_1, \ldots , x_r]$ to be a polynomial
ring on generators of degree $-2$. Now take $M=\bigoplus_{n\geq 0}
\Sigma^{2n}R/\fm^n$
and $\cE=\langle x_1\rangle$, and we see that $M$ is not
$\cE$-complete and hence $R^1\ilim(M,\cE)\neq0$.
This is enough to construct an object of $\cAt(G)$ of injective
dimension $r+1$ when $r\geq 1$. Indeed we may take
$M=\bigoplus_n \Sigma^{2n}k[c_1]/c_1^n$. We then find
$$\Gamma_{\cF/L}\Ext^1(\cEi_L\cOcF, M)\neq 0. $$
Since the module is annihilated by $c_2, \ldots , c_r$ it has
injective dimension $r-1$. Hence $\at_1(M)$ has injective dimension
$\geq 2+(r-1)=r+1$.
One might expect to be able to iterate this construction, but this
doesn't work, at least not in a simple way.
\begin{lemma}
For any commutative ring $R$, any $R$-module $M$ and any countable
multiplicatively closed subsets $\cE_1, \cE_2$ we have
$$\Ext_R^s(\cEi_1 R, \Ext_R^t(\cEi_2R, M))=0 \mbox{ for } a+b\geq 2. $$
\end{lemma}
\begin{proof}
In view of the adjunction
$$\Hom (A, \Hom(B, M))\cong \Hom (A\tensor B, M), $$
there is a spectral sequence
$$\Ext^i_R(\cEi_1R, \Ext^j_R(\cEi_2 R, M))\Rightarrow
\Ext^{i+j}_R(\cEi_1\cEi_2R, M). $$
\end{proof}
The following result is trivial, and probably irrelevant.
\begin{lemma}
For any commutative ring $R$, and any countable
multiplicatively closed subsets $\cE$ if $M$ is $\cE$-divisible then
$$\Ext^1_R(\cEi R, M)=0. $$
\end{lemma}
\begin{proof}
The divisibility proves that $(M,e)$ is Mittag-Leffler and hence
$R^1\ilim (M,e)=0$.
\end{proof}
\section{Ind-corepresenting evaluation}
\label{sec:indcorep}
\newcommand{\Kts}{\tilde{K}^*}
Given an object $X$ of $\cAt (G)$, we would like to be able to
determine its torsion modules $V(\cF/K)$ by considering maps from
objects of $\cAt(G)$, in the same way that homotopy groups are given
by maps out of a sphere. Although this is not possible as it stands,
we can recover $V(\cF/K)$ as a colimit of such values. Since
$V(\cF/K)=\bigoplus_{\Kt}V(G/\Kt)$ we focus on a single subgroup $\Kt$.
This phenomenon should be rather familiar.
Given a torsion $k[c]$-module $T$ we may recover $T$ as
$T=\Hom_{k[c]}(k[c], T)$, but if we are restricted to using torsion
modules in the domain this is not available to us. However,
$\ann (c^n,T)=\Hom_{k[c]}(k[c]/c^n, T)$, so
$$T=\bigcup_n \ann (c^n,T)=\colim_n \Hom_{k[c]}(k[c]/c^n, T).$$
In this sense the module is ind-corepresented by the inverse system $\{ k[c]/c^n\}_n$.
The situation with $\cAt (G)$ is a little more complicated, so we
start with the circle group in the semifree case as in Section \ref{sec:semifreecircle}.
\begin{example}
By the discussion above we see immediately that evaluation at $G/1$ is ind-corepresented by $\{ (0\lra
k[c]/c^n)\}_n$. This is an object familiar from the topology since
$\pi^G_*(DS(nz)_+)=k[c]/c^n$, where $D$ denotes Spanier-Whitehead
duality, so that $\piAts(DS(nz)_+)=(0\lra
k[c]/c^n)$. In fact the representing inverse system is the homotopy of
the dual of the direct system $S(z)_+\subset S(2z)_+\subset S(3z)_+\subset \cdots$.
Evaluation at $G/1$ is ind-corepresented by the inverse system $\{ t\tensor k \lra
k[c,c^{-1}]/c^nk[c])\}_n$. To define a map
$$e: \colim_n \Hom \left( [t\tensor k\lra k[c,c^{-1}]/c^nk[c]], [t\tensor
V\lra T]\right) \lra V$$
we proceed as follows. Consider a map $\theta$ from the $n$th term
$$
\xymatrix{
t\tensor k \dto \rto^{1\tensor \theta(G/G)}&t\tensor V\dto^q\\
k[c,c^{-1}]/c^n k[x]\rto^(.7){\theta(G/1)}&T
}$$
and note first that it is determined by $\theta (G/G)$. This is
because the vertical structure
map in the domain is an epimorphism. Thus $\theta $ is determined by
evaluation at $1\tensor \iota $. This is compatible with the maps in
the inverse system as $n$ varies (since all are the identity at $G/G$), and so we may
define $e([\theta ])=\theta (G/G) ( \iota)$ and obtain a well defined
injective map $e$. Every element in the image of $q$
is divisible, so that if $q(1\tensor v)$ is annihilated by $c^s$ there
is a map $\theta$ with evaluation $v$ if $n\geq s$, and since $T$ is
torsion the map $e$ is surjective.
Again the objects are geometrically familiar since
$\piAts (S^{-nz})=(t\tensor \Q \lra k[c,c^{-1}]/c^nk[c])$.
In fact the representing inverse system is the homotopy of
the dual of the system $S^0\subset S^z \subset S^{2z} \subset S^{3z}\subset \cdots$
\end{example}
In principle we could continue purely algebraically, but the
topological motivation will lead us efficiently to a solution.
Recall that
\begin{multline*}
\piAts (X)(G/\Kt)=\piGs(\siftyV{K}\sm EG/\Kt_+\sm X)\\
=[S^0, \colim_{V^K=0}S^V \sm \colim_n EG/\Kt^n_+\sm X]^G
=\colim_{V^K=0,n}[S^{-V} \sm DEG/\Kt^n_+, X]^G
\end{multline*}
This suggests that if we choose a nice filtration of the
universal space, we could use
$$B_{\Kt} (V,n)=\piAts (S^{-V}\sm DEG/\Kt_+^n)$$
to give ind-corepresenting objects.
We choose to be very explicit so as to make this more easily
digested, but pay the price of making unnecessary choices. We have a short exact sequence
$$1\lra \Kt/\Lt \lra G/\Lt \lra G/\Kt\lra 1$$
giving a short exact sequence of commutative $k$-algebras
$$k \lla H^*(B\Kt/\Lt) \lla H^*(BG/\Lt) \lla H^*(BG/\Kt)\lla k. $$
This is natural, but because $\Kt /\Lt$ is a torus we may go further
and choose a splitting.
Thus the projection $G/\Lt\lra G/\Kt$ is split, and we may choose $\Kts\supseteq \Lt$ so that the composite $\Kt /\Lt \lra G/\Lt\lra G/\Kts$
is an isomorphism, and hence $G/\Lt =G/\Kt \times G/\Kts$.
We choose one dimensional representations
$\alpha_1, \alpha_2, \ldots, \alpha_s$
with kernels $K(\alpha_i)=\ker(\alpha_i)$ so that
$$G/\Kt =G/K(\alpha_1)\times G/K(\alpha_2)\times \cdots \times
G/K(\alpha_s)$$
Similarly, we choose $\beta_1, \beta_2, \ldots, \beta_t$
so that
$$\Kt/\Lt \cong G/\Kts =G/K(\beta_1)\times G/K(\beta_2)\times \cdots \times
G/K(\beta_t). $$
This allows us to have explicit models for universal spaces:
$$EG/\Kt=S(\infty \alpha_1)\times S(\infty \alpha_2)\times \cdots \times
S(\infty \alpha_s) \mbox{ and }
EG/\Kts=S(\infty \beta_1)\times S(\infty \beta_2)\times \cdots \times
S(\infty \beta_s). $$
As filtration, we take
$$EG/\Kt^n=S(n \alpha_1)\times S(n \alpha_2)\times \cdots \times
S(n \alpha_s). $$
Now take $x_i=e(\alpha_i), y_j=e(\beta_j)$, so that
$$H^*(BG/\Lt)\cong H^*(BG/\Kt)\tensor H^*(B\Kt /\Lt)\cong k[x_1, \ldots, x_s, y_1,
\ldots, y_s]$$
We also write
$$\fm_{G/\Kt}=(x_1, \ldots, x_s), \fm_{G/\Kt}^{[n]}=(x_1^n, \ldots, x_s^n), $$
and note
$$H^*(BG/\Kt^n)=H^*(BG/\Kt)/\fm_{G/\Kt}^{[n]}
\mbox{ and } H^*(BG/\Kt^n)=\ann
(\fm_{G/\Kt}^{[n]}, H_*(BG/\Kt)). $$
It is obvious that $H^*(BG/\Kt^n)$ is cyclic, but we note that it is
also self-dual. This then gives the important fact that
$H_*(BG/\Kt^n)$ is also cyclic, and generated by the
element dual to $(x_1x_2\cdots x_s)^n$ if we use the monomial
basis. We note that this is the Euler class of
$(\alpha_1\oplus \cdots \oplus \alpha_s)^{\oplus n}$. One might say
$$H_*(BG/\Kt^n)=\frac{1}{(x_1x_2\cdots x_s)^n} \cdot H^*(BG/\Kt^n).$$
\begin{remark}
This corresponds to a Spanier-Whitehead duality statement. Since
$D(S(V)_+)\simeq \Sigma^{1-V}S(V)_+$ we
see that
$$D(EG/\Kt^n_+)\simeq \Sigma^{s-(\alpha_1\oplus \cdots \oplus
\alpha_s)^{\oplus n}}EG/\Kt^n_+ . $$
\end{remark}
Finally, we note that any representation $V$ of $\Lt$ with $V^{\Kt}=0$
can be written as a sum of monomials in the $\alpha_i$ and $\beta_j$
where each monomial involves at least one $\beta_j$. Accordingly,
$$\cE_{G/\Kt}=\{ \sum_i \lambda_i x_i +\sum_j\mu_j y_j \st (\mu_1,
\ldots , \mu_t)\neq (0, \ldots , 0)\} . $$
\begin{remark}
Since our construction has involved a lot of choice, it is worth
considering what is intrinsic once $\Kt\supseteq \Lt$ is chosen.
First of all, $H^*(BG/\Kt)=k[x_1, \ldots , x_s]$ is an intrinsic
subalgebra of $H^*(BG/\Lt)$. From the geometric point of view, writing
$TG:= \spec (H^*(BG))$ we are considering the projection $TG/\Lt\lra
TG/\Kt$. Similarly the ideal $\fm_{G/\Kt}$ is intrinsic, and $T\Kt/\Lt$
is its zero set, giving the fibration
$$T\Kt/\Lt \lra TG/\Lt\lra TG/\Kt. $$
Finally, the multiplicatively closed subset $\cE_{K/L}$ is generated
by elements of $H^2(BG/\Lt)$ not in $\fm_{G/\Kt}$ and is also intrinsic:
inverting $\cE_{K/L}$ is localization at $TK/L=T\Kt/\Lt$.
Accordingly, the $H^*(BG/\Lt)$-module
$\cEi_{K/L}H^s_{\fm_{G/\Kt}}(H^*(BG/\Lt))$ (the localized $T\Kt/\Lt$-local
cohomology) is intrinsic.
\end{remark}
\begin{prop}
\label{prop:indcorep}
Evaluation at $G/\Kt$ is ind-corepresented by objects $B_{\Kt}(V,n)$ as $V$
varies through representations with $V^K=0$ and $n\geq 0$. Here
$B_{\Kt}(V,n)(G/\Lt)=0$ unless $\Lt$ is cotoral in $\Kt$ and in that
case we have
$$B_{\Kt}(V,n) (G/\Lt)= S^{-V^{\Lt}}\sm H^*(BG/\Kt^n)\tensor H_*(B\Kt/\Lt)$$
as $V$ varies through representations with $V^K=0$ and $n\geq 0$.
\end{prop}
\begin{proof}
We suppose $X$ is an object of $\cAt(G)$ with torsion module $V
(G/\Kt)$ at $G/\Kt$. To define a map
$$e:\colim_{V^K=0,n} \Hom_{\cAt(G)} (B_{\Kt}(V,n) , X) \lra V(G/\Kt)$$
we pick a representative $\theta : B_{\Kt}(V,n) \lra X$ of an
element of the domain. First
consider the $G/\Kt$ level, and note that since $V^{\Kt}=0$, there is
no dependence on $V$. We take
$e([\theta ])=\theta (G/\Kt)(\iota)$ where $\iota\in
H^0(BG/\Kt)/\fm_{\Kt}^{[n]}$ represents the unit. Since the maps in the
inverse sytem as $n$ varies are the identity in this
degree, this does not depend on
the choice of representative. Accordingly, the map $e$ is well
defined.
To see $e$ is injective we note that the structure maps of $B_{\Kt}(V,n)$ are
surjective. First of all, if $V=0$ the maps
$B_{\Kt}(0, n)(G/\Kt) \lra B_{\Kt}(0, n)(G/\Lt)$ are surjective.
We may as well suppose $\Kt/\Lt$ is a circle, since the general case
is a composite of such cases. Here it is easy to check that (writing $z$
for the natural representation of $\Kt/\Lt$),
$$S^{\infty z}\sm DE\Kt/\Lt_+ \lra \Sigma S(\infty z)_+ \sm
DE\Kt/\Lt_+ \simeq \Sigma S(\infty z)_+$$
is surjective and the general case is obtained by tensoring up.
Now as $V$ varies we use the fact that $e(V)$ is an isomorphism in the
domain and (since $V^K=0$) it is a monomorphism in the codomain.
Finally, because $V (G/\Kt)$ is torsion, for any element $x$ of degree
$i$ in $V (G/\Kt)$ we can find an $n$ so that $(x_1x_2\ldots x_s)^nx=0$.
There is therefore a map
$$\Sigma^iH^*(BG/\Kt^n)\cong \Sigma^iH^*(BG/\Kt)/\fm_{\Kt}^{[n]}
\lra V(G/\Kt)$$
with $\iota$ mapping to $x$. Next, if $\Kt$ is of codimension $c$, by
definition of $\cAt(G)$, the element $1\tensor x$ at level $G/\Kt$ only has non-zero
image at $G/L$ for finitely many connected codimension $c+1$ subgroups
$L$. For each of these we find $V_L$ so that $q_L(1\tensor x)$
is annihilated by $e(V_L)$, and take $V=\bigoplus_L V_L$. This shows
that $1\tensor x$ is in the image of a map $B_{\Kt}(V,n)\lra X$, and
so $e$ is surjective.
\end{proof}
Having ind-corepresented evaluation at $G/\Kt$ we should describe how
the structure maps are ind-corepresented. Indeed, if $\Lt$ is cotoral
in $\Kt$, we have maps
$$
\xymatrix{
\cEi_{\Kt/\Lt}H^*(BG/\Lt)\tensor_{H^*(BG/\Kt)}V(G/\Kt) \ar@{=}[d]\rto&
V(G/\Lt)\ar@{=}[d]\\
\colim_{V^K=0,m}\cEi_{\Kt/\Lt}H^*(BG/\Lt)\tensor_{H^*(BG/\Kt)}\Hom(B_{\Kt}(V,m), X) \rto&
\colim_{W^L=0,n}\Hom (B_{\Lt}(W,n), X)
}$$
In other words, given $x\in H^*(BG/\Lt)$, a representation $U$ of
$G/\Lt$ with $U^K=0$ and a representative map $\theta : B_{\Kt}(V,
m)\lra X$ we need a representative of the colimit in the
codomain. Since all maps are linear in $H^*(BG/\Lt)$, it suffices to
treat the case $x=1$.
\begin{lemma}
Noting that $W^L=0$ implies, $W^K=0$,
The structure map is ind-corepresented by precomposing $\theta$ with
$$B_{\Lt}(W, n)\lra B_{\Kt}(W,n). $$
Mutliplication by $e(U)$ is given by $B_{\Kt}(V, m)\lra
B_{\Kt}(U\oplus V, m)$, so division by $e(U)$ is effected by shifting
filtration by $U$.
\end{lemma}
\begin{proof}
We only need to give the answer at $G/\Mt$ where $\Mt$ is cotoral in
$\Lt$ since otherwise the codomain is zero. At this level we are
looking at
$$S^{-V^{\Mt}}\sm H^*(BG/\Lt^n)\tensor
H_*(B\Lt/\Mt)\lra \Sigma^2
S^{-V^{\Mt}}\sm H^*(BG/\Kt^n)\tensor H_*(B\Kt/\Mt). $$
We may suppose $\Kt/\Lt$ is a circle. So the only effect is that
$H^*(B\Kt/\Lt)$ is removed from the cohomology in the domain and replaced by
$\Sigma^2 H_*(B\Kt/\Lt)$ in the homology of the codomain. As in the
proof of Proposition \ref{prop:indcorep}, writing $z$
for the natural representation of $\Kt/\Lt$ we note that it is
$$S^{\infty z}\sm DE\Kt/\Lt_+ \lra \Sigma S(\infty z)_+ \sm
DE\Kt/\Lt_+ \simeq \Sigma S(\infty z)_+$$
\end{proof}
\begin{remark}
This gives another approach to constructing injectives.
Indeed, by definition, if $I$ is an injective $H^*(BG/\Lt)$-module,
then $\at_L(I)$ has the property
$$\Hom_{\cAt(G)}(X, \at_L (I))=\Hom_{H^*(BG/\Lt)}(V(G/\Lt), I)$$
so if $\Lt$ is cotoral in $\Kt$, we must have
\begin{multline*}
\at_L(I)(G/\Kt)=
\colim_{V^K=0, n}\Hom_{H^*(BG/\Lt)}(B_{\Kt}(V, n)(G/\Lt), I)\\
=\colim_{V^K=0, n}\Hom_{H^*(BG/\Lt)}(S^{-V^L}\tensor
H^*(BG/\Kt^n)\tensor H_*(B \Kt/\Lt), I).
\end{multline*}
We now take the generating torsion injective $I=H_*(BG/\Lt)$ so that
$\Hom_{H^*(BG/\Lt)}(\cdot , I)=\Hom_k (\cdot ,k)$. As noted above
$$H^*(BG/\Kt^n)\tensor H_*(B \Kt/\Lt)
\cong \Sigma^{s-na}H_*(BG/\Kt^n)\tensor H_*(B \Kt/\Lt) $$
where $a=|x_1x_2\cdots x_s|$. Accordingly the value is
$$S^{+V^L}\tensor \Sigma^{na-s}
H^*(BG/\Kt^n\times B\Kt/\Lt). $$
The colimit is
$$\cEi_{K/L}H_*(BG/\Kt)\tensor
H^*(B\Kt/\Lt)=\cEi_{K/L}H^s_{\fm_{\Kt}}(H^*(BG/\Lt)). $$
This agrees with the previous construction by Lemma
\ref{lem:injasloccoh}.
Indeed, it was the calculation by ind-corepresentability that
first alerted the author to the connection between injectives and
local cohomology, and hence led to the formulation of Lemma
\ref{lem:injasloccoh}.
\end{remark}
\section{The Adams spectral sequence}
\label{sec:AtASS}
The paper has been leading up to the construction of a method of
calculation based on the abelian torsion model.
\begin{thm}
\label{thm:AtASS}
There is an Adams spectral sequence
$$\ExtAt^{*,*} (\piAts (X), \piAts (Y))\Rightarrow [X,Y]^G_*. $$
This is a finite, strongly convergent spectral sequence.
\end{thm}
There is a standard method for constructing an Adams spectral
sequence: we first establish that enough
injectives are realizable and that the Adams spectral sequence applies
to them, and that the homology theory detects triviality.
\begin{lemma}
\label{lem:rinj}
For any injective $\cOcFL$-module $I$, there is a $G$-spectrum
$A_L(I)$ realizing $\at_L(I)$ in the sense that
$\piAts(A_L(I))=\at_L(I)$ and $\piAts$ gives an isomorphism
$$\piAts: [X, A_L(I)]^G_*\stackrel{\cong}\lra \HomAt(\piAts (X), \at_L(I)). $$
\end{lemma}
\begin{proof}
For an injective, torsion $\cOcFL$-module $I$, the functor
$$X\longmapsto A_L(I)_G^*(X) =\HomAt(\piAts (X), \at_L(I))
=\Hom_{\cOcFH}(V_X(\cF /L), I)$$
is exact because $I$ is injective and is therefore a cohomology
theory on $G$-spectra, so by Brown Representability there is a $G$-spectrum $A_L(I)$ so that
$$[X, A_L(I)]^G_*=\HomAt(\piAts (X), \at_L(I)). $$
The isomorphism $\piAts (A_L(I))=\at_L(I)$ follows from Proposition \ref{prop:indcorep}.
\end{proof}
\begin{lemma}
\label{lem:piAtsfaithful}
If $\piAts(X)=0$ then $X\simeq *$.
\end{lemma}
\begin{proof}
If $\piAts (X)=0$ then $H^*_{G/\Kt}(\Phi^{\Kt}X)=0$ for all subgroups
$\Kt$. The geometric isotropy of $X$ is therefore empty and $X$ is
contractible by the Geometric Fixed Point Whitehead Theorem.
\end{proof}
\begin{proof}[of Theorem \ref{thm:AtASS}]
As usual we need only show that enough injectives are realizable,
that the spectral sequence is correct for maps into these spectra and
that the spectral sequence is convergent.
In more detail, we take an injective resolution of $\piAts (Y)$:
$$0\lra \piAts(Y)\lra I_0\lra I_1\lra \cdots \lra I_n\lra 0. $$
This is finite by Proposition \ref{prop:id}, and by Lemma \ref{lem:enoughinj} we may assume
each $I_s$ is a sum of injectives $\at_K(I)$ for subgroups $K$
and injective $\cOcFK$-modules $I=H_*(BG/\Kt)$ where $\Kt$ has
identity component $K$.
By Lemma \ref{lem:rinj} this is realizable by a tower
$$\xymatrix{
Y\ar@{=}[r]&Y_0\dto &Y_1\lto\dto &Y_2\lto\dto &\cdots\lto & Y_n\lto\dto &Y_{n+1}\lto\\
&\bbI_0&\Sigma^{-1}\bbI_1&\Sigma^{-2}\bbI_2&&\Sigma^{-n}\bbI_n&
}$$
which is built inductively, starting with $Y\lra \bbI_0$ realizing
$\piAts(Y)\lra I_0$ and taking $Y_1$ to be the fibre. Once $Y_s$ has
been defined as the fibre of $Y_{s-1}\lra \Sigma^{-s+1}\bbI$, we see
that $\piAts(Y_s)=\Sigma^{-s}(\im (I_{s-1}\lra
I_s))$. In particular, $\piAts(Y_{n+1})=0$ and $Y_{n+1}\simeq * $ by Lemma
\ref{lem:piAtsfaithful}.
We obtain the spectral sequence by applying $[X, \cdot]^G_*$ to the
tower. By Lemma \ref{lem:rinj} the $E_1$ term is $\HomAt(\piAts
(X), I_\bullet)$ and therefore the $E_2$-term is as stated. Strong
convergence is clear because the filtration is finite.
\end{proof}
|
train/arxiv
|
BkiUbSU5qX_AYzDLsCKO
| 5
| 1
|
\section{Introduction}
Gamma-Ray Bursts (GRBs) are brief, intense flashes of gamma-rays originating at cosmological distances,
and they are the most luminous objects in the universe. The emitted radiation energy is dominant in the
0.1--1 MeV energy range.
It has been established that the prompt gamma-ray
emission is produced in the relativistic jets. However, in spite of extensive observational and theoretical
efforts, several key questions concerning the nature of the central engines of the relativistic jets and
the jets themselves remain poorly understood (for recent reviews, see \cite{gehrels12,sinoue13,amati13}).
In fact, some of these questions are very difficult or
even impossible to answer with the light-curve and spectral information currently collected from the
optical to GeV gamma-rays.
On the other hand, polarization information can lead to unambiguous answers to these questions.
In particular, polarimetric observations of GRBs can address the prompt emission mechanism, magnetic
composition, and geometric structure of GRB jets \cite{toma09,lazzati06}.
The polarimetric data taken together with the light-curve and spectral data may reveal the driving mechanism
of the jets and the nature of the central engines.
Recently, we have measured gamma-ray polarizations in the 70--300 keV range
with $\sim 3\sigma$ confidence levels, for the first time, in three bursts
with the Gamma-ray Burst Polarimeter (GAP) onboard the small solar-power-sail demonstrator IKAROS
\cite{yonetoku11,yonetoku12}. We review the observational results of the GAP as well as other instruments
with discussing some theoretical implications.
We also report a strict observational verification of $CPT$ invariance in the photon sector as a result of
the GRB polarization measurements \cite{toma12}.
\section{Observations}
There were several reports of detections of linear polarization with low significance. Coburn \& Boggs (2003)
\cite{coburn03} reported detection of strong polarization from GRB 021206 with RHESSI solar satellite.
However, independent authors analyzed the same data, and concluded that no polarization signals were
confirmed \cite{rutledge04,wigger04}. INTEGRAL-SPI and -IBIS data showed detections of polarization with
$\sim 2\sigma$ confidence level from GRB 041219 \cite{kalemci07,mcglynn07,gotz09}. However, the results
of SPI and IBIS for the brightest pulse of GRB 041219 appear inconsistent with each other, i.e.,
the SPI teams detected strong polarization of $\Pi = 98 \pm 33\%$ and $\Pi = 63^{+31}_{-30}\%$ with $2\sigma$
statistical level, while the IBIS team reported a strict upper limit of $\Pi < 4 \%$ (although their results
for the other temporal intervals are consistent). Therefore, the previous
reports of the gamma-ray polarimetry for GRBs are all controversial.
IKAROS is a small solar-power-sail
demonstrator \cite{kawaguchi08}, and successfully launched on 21 May 2010. The GAP onboard IKAROS
is fully designed to measure linear polarization in prompt emission of GRBs in the energy range of
70--300 keV. The GAP's high axial symmetry in shape and high gain
uniformity are keys for reliable measurement of polarization and avoiding fake modulation due to
background gamma-rays. These realized the quite small systematic uncertainty of $\simeq 1.8\%$ level
\cite{yonetoku11PASJ}.
\begin{table}[t]
\begin{center}
\begin{tabular}{|l|cccc|}
\hline
Event name & $\Pi$ & $2\sigma$ limit & Detection significance & PA change \\
\hline
GRB 100826A & $27 \pm 11$\% & $>6\%$ & $2.9\sigma$ & yes \\
GRB 110301A & $70 \pm 22$\% & $>31\%$ & $3.7\sigma$ & no \\
GRB 110721A & $84^{+16}_{-28}$\% & $>35\%$ & $3.3\sigma$ & no \\
\hline
\end{tabular}
\caption{Polarimetric data of the three GRBs obtained with GAP. The Polarization degrees
$\Pi$ are shown with $1\sigma$ error. The `2$\sigma$ limit' means the lower limit on $\Pi$
at the $2\sigma$ statistical significance level. The `detection significance' means the significance
levels for $\Pi > 0\%$.}
\label{tab:data1}
\end{center}
\end{table}
\begin{table}[t]
\begin{center}
\begin{tabular}{|l|ccc|}
\hline
Event name & $T_{90}$ [s] & fluence [${\rm erg}\; {\rm cm}^{-2}$] & $E_p$ [keV] \\
\hline
GRB 100826A & $\simeq 150$ & $(3.0\pm 0.3)\times 10^{-4}$ & $606^{+134}_{-109}$ \\
GRB 110301A & $\simeq 5 $ & $(3.65\pm 0.03)\times 10^{-5}$ & $106.8^{+1.85}_{-1.75}$ \\
GRB 110721A & $\simeq 24$ & $(3.52\pm 0.03)\times 10^{-5}$ & $393^{+199}_{-104}$ \\
\hline
\end{tabular}
\caption{Light-curve and spectral data of the three GRBs taken from the GCN circulars.
$E_p$ is the photon energy of the time-averaged $\nu F_{\nu}$ spectrum.}
\label{tab:data2}
\end{center}
\end{table}
The GAP detected the linear polarization of the prompt emission of GRB
100826A, GRB 110301A, and GRB 110721A. The polarimetric data as well as the light-curve and
spectral data of these three bursts are summarized in Table~\ref{tab:data1} and \ref{tab:data2}.
The polarization degrees $\Pi > 0\%$ at $\sim 3 \sigma$ confidence level, and these are
the most convincing detections of polarization of GRB prompt emission so far.
See Yonetoku et al. (2011; 2012) and Toma et al. (2012) \cite{yonetoku11,yonetoku12,toma12}
for more details on the data analysis.
We see that there are cases with and without a significant change of the polarization angle (PA).
GRB 100826A, with long duration $T_{90} \sim 100\;$s, shows a PA change, while GRB 110301A
and 110721A, with short duration $T_{90} \sim 10\;$s, shows no PA change.
On the other hand, the polarization is detected both for the GAP observed energy range
$< E_p$ (GRB 100826A and GRB 110721A) and for $>E_p$ (GRB 110301A). The time-averaged fluxes
(the fluences divided by $T_{90}$) of the three bursts are all
$\sim 3\times 10^{-6}\;{\rm erg}\; {\rm cm}^{-2}\; {\rm s}^{-1}$, which is very high.
We note that no spectroscopic redshifts were determined for these three bursts.
The polarimetric data of GRB 041219A as well as the recent report on GRB 061122
with IBIS onboard INTEGRAL appear consistent with the GAP results listed above. GRB 041219A shows PA
changes, and GRB 061122 has $\Pi \gtrsim 30\%$ at $2\sigma$ significance level
\cite{gotz09,gotz13}.
\section{Implications for Emission Mechanism}
Here we present some theoretical implications for the emission mechanism, focusing on the
above results that there are cases with and without a significant PA change, and that
$\Pi > 30\%$ for the cases of no PA change.
For the light-curve and spectral dataset of GRB prompt emission in the 0.1--1 MeV energy range,
the synchrotron emission model and the photospheric quasi-thermal emission model
have been actively debated (for reviews, see \cite{gehrels12,sinoue13,toma11}).
In general, the relativistic jets are optically thick in the vicinity of the central engine,
and the photospheric emission is released at the photosphere. The jets are optically thin
outside the photosphere and can emit the synchrotoron radiation through the internal shock
dissipation and/or the magnetic energy dissipation. The long-standing problem is which emission
is dominant in the 0.1--1 MeV energy range, synchrotron or photospheric.
\subsection{Synchrotron Models}
First, let us discuss the synchrotron models. They include three different models in respect of
the magnetic field structure in the emitting region. For all the models, we assume that the electrons
have an isotropic pitch angle distribution and a non-thermal energy spectrum, and thus the polarization
degree from a local emitting point is $\Pi^{\rm syn}_{\rm max} = -\gamma_{\rm B}/(-\gamma_{\rm B}+2/3)$, where
$\gamma_{\rm B}$ is the photon power-law index \cite{rybicki79}.
Typical bursts have $\gamma_{\rm B} \simeq -1$ for $E < E_p$ and
$\gamma_{\rm B} \lesssim -2$ for $E > E_p$.
\subsubsection{SO model}
One type of the magnetic field structure is a helical field,
which may be advected from the central engine. Such globally ordered fields
can produce high-$\Pi$ emission \cite{lyutikov03,granot03,toma09}. We call this the ``SO model''
(synchrotron, ordered-field model).
As the simplest model, we may assume an instantaneous emission from a thin spherical shell moving
radially outward with a bulk Lorentz factor $\Gamma \gg 1$ and an opening angle $\theta_j$, where
the emissivity is uniform over the shell.
\begin{figure}[t]
\begin{minipage}{0.5\hsize}
\begin{center}
\includegraphics[scale=0.6]{f1.eps}
\end{center}
\end{minipage}
\begin{minipage}{0.5\hsize}
\begin{center}
\includegraphics[scale=0.6]{f2.eps}
\end{center}
\end{minipage}
\caption{
Left: Polarization degrees as functions of $q=\theta_v/\theta_j$ in the SO model, where
$\theta_v$ is the viewing angle of the line of sight and $\theta_j$ is the jet opening angle.
$y_j \equiv (\Gamma \theta_j)^2$, where $\Gamma$ is the bulk Lorentz factor of the jet.
Typical parameters are adopted for the emission spectrum (see \cite{toma09} for details).
Right: Schematic picture of the jet with the toroidal component of the magnetic fields (thin lines).
Only a fraction of the emitting shell,
$\theta < \Gamma^{-1}$ around the line of sight is bright because of the relativistic beaming effect.
}
\label{fig:SO}
\end{figure}
Figure~\ref{fig:SO} (left) shows $\Pi$ calculated in this model as a function of the viewing angle
$\theta_v$ in respect of the jet axis for different values of $y_j \equiv (\Gamma \theta_j)^2$.
Since we consider the spherical shell, only a fraction of the shell with $\theta < \Gamma^{-1}$
around the line of sight is bright because of the relativistic beaming effect. This bright region
is small for the cases of $\Gamma^{-1} \ll \theta_j$ (i.e., $y_j \gg 1$) (see Fig~\ref{fig:SO} right).
For the on-axis case, i.e., $\theta_v < \theta_j$, the direction of the
magnetic field is quite ordered in the bright region, one has high polarization degree,
$\Pi \sim 40\%$.
Another important point is that the PA does not vary for a fixed $\theta_v$ but different $\Gamma$.
Therefore, the observed significant PA change may suggest that the emission is not uniform over the
shell, but consists of multiple patches with characteristic angular size much smaller than jet
opening angle, $\theta_p \ll \theta_j$ \cite{yonetoku11} (see Figure~\ref{fig:SOpatch}).
In the case of $\Gamma^{-1} \sim \theta_j$,
it is natural that one sees multiple patches with different magnetic field directions, and
observes significant PA changes. On the other hand, if $\Gamma^{-1} \ll \theta_j$, one only sees
a limited range of the curved magnetic fields, which leads to no significant PA change even if the
emission is patchy. In such a scenario, GRB 100826A corresponds to the case of $\Gamma^{-1} \sim \theta_j$,
while the other two bursts with no PA change correspond to the case of $\Gamma^{-1} \ll \theta_j$.
We may consider an alternative scenario in which the initially ordered helical fields get distorted
during the energy dissipation phase, making different field directions within the bright region of
$\theta < \Gamma^{-1}$ \cite{zhang11}.
The PA changes can naturally occur in this scenario, but when the emission duration is short,
the PA change does not necessarily occur.
Another scenario is that the GRB jets consist of multiple shells which have globally ordered
transverse (not helical or toroidal) magnetic fields with a different direction for each
shell. It has been recently claimed that such impulsive shells can be accelerated to relativistic
speeds \cite{granot12}. In this scenario also, the PA changes naturally occur for long duration bursts with
large number of emitting shells, but do not necessarily occur for short duration bursts with
small number of emitting shells.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.6]{f3.eps}
\end{center}
\caption{
Schematic picture of the patchy emission in the SO model. The left and right ones correspond
to the cases of $\Gamma^{-1} \sim \theta_j$ and $\Gamma^{-1} \ll \theta_j$, respectively.
The thick arrows represent the polarization vectors.
}
\label{fig:SOpatch}
\end{figure}
\subsubsection{SR model}
The collisionless shocks formed in the jet may produce sizable magnetic fields with random directions
on plasma skin depth scales through e.g., the Weibel instability \cite{medvedev99,gruzinov99}.
Synchrotron emission from such fields can have high $\Pi$, provided that the field directions
are not isotropically random, reflecting the direction of the shock propagation direction.
In many studies, the extreme case is assumed, i.e.,
the field directions are confined in the plane parallel to the shock front
\cite{granot03,nakar03,toma09}. We call this the ``SR model'' (synchrotron, random-field model).
In this model, the radiation propagating in the direction parallel to the shock front is maximally
polarized in the comoving frame of the emitting fluid. Such radiation is observed as that from
the points with $\theta = \Gamma^{-1}$ around the line of sight. As a result, the local polarization
vectors are axisymmetric around the line of sight (see Figure~\ref{fig:SR} right).
If the jet is observed from an off-axis angle,
$\theta_v \gtrsim \theta_j$, all the polarization vectors are not canceled and the net polarization remains.
Figure~\ref{fig:SR} (left) shows $\Pi$ calculated in this model as a function of the viewing angle
$\theta_v$ in respect of the jet axis for different values of $y_j \equiv (\Gamma \theta_j)^2$.
In this model with the uniform emissivity over the shell, a high $\Pi$ can be obtained only when
$\theta_v \sim \theta_j + \Gamma^{-1}$. For this configuration, one cannot have the PA change
even for a fixed $\theta_v$ but different $\Gamma$. For $\theta_v < \theta_j$, one can have the PA change with
varying $\Gamma$, but $\Pi$ is very low.
The observed PA change with high $\Pi$ may suggest the patchy emission structure in this model \cite{yonetoku11}.
If the emission is patchy, $\Pi$ can be high even for $\theta_v < \theta_j$, and one can have
the PA changes (see also \cite{lazzati09}). The characteristic angular size of the patches may be
hydrodynamically constrained to be $\theta_p \gtrsim \Gamma^{-1}$. In this model, however, one requires
fine tuning that the observed patches should be dominated by those with $\theta_{vp} \sim \theta_p + \Gamma^{-1}$
to have $\Pi \gtrsim 30\%$, where $\theta_{vp}$ is the viewing angle of the patch.
The patches observed with $\theta_{vp} \lesssim \theta_p$ decrease the net $\Pi$.
On the other hand, the bursts we observed are all very bright, which implies that some patches are seen with
$\theta_{vp} \lesssim \theta_p$. Therefore, the SR model is not favored to explain the observed $\Pi \gtrsim 30\%$.
\begin{figure}[t]
\begin{minipage}{0.5\hsize}
\begin{center}
\includegraphics[scale=0.6]{f4.eps}
\end{center}
\end{minipage}
\begin{minipage}{0.5\hsize}
\begin{center}
\includegraphics[scale=0.6]{f5.eps}
\end{center}
\end{minipage}
\caption{
Left: Polarization degrees as functions of $q=\theta_v/\theta_j$ in the SR model
(see \cite{toma09} for details).
Right: Schematic picture of the jet. The net polarization property is determined by the
bright emission from the points with $\theta \sim \Gamma^{-1}$ around the line of sight,
whose polarization vectors (represented by the thick arrows) are axisymmetric.
}
\label{fig:SR}
\end{figure}
\subsubsection{SH model}
The internal shocks may also produce strong magnetic fields with random directions on hydrodynamic
scales, much larger than the plasma skin depth scales, through e.g., the Richtmyer-Meshkov instability
\cite{inoue11,gruzinov99}. We call this the ``SH model" (synchrotron model with random fields on
hydrodynamic scales).
If the field directions are isotropically random, the net polarization degree is $\Pi \sim
\Pi^{\rm syn}_{\rm max}/\sqrt{N}$, where $N$ is the number of independent patches with coherent
field in the bright region with $\theta \sim \Gamma^{-1}$ around the line of sight, and
the PA change can be naturally realized. Unlike the SR model, the emission from patches seen
with small $\theta_{vp}$ can have high $\Pi$, so that this model is in agreement with the high brightness of the bursts.
By utilizing the MHD simulations of internal shocks with initial density fluctuations,
Inoue et al. (2011) \cite{inoue11} deduced $N \sim 10^3$ from the typical scale of the
coherent magnetic fields, which did not appear to be
consistent with the observed $\Pi \gtrsim 30\%$. However, the recent detailed analysis of the
numerical simulation suggests that the magnetic fields perpendicular to the shock front are
selectively amplified, which might increase the net $\Pi$ \cite{inoue13}. The aim of this recent
simulation is to explain the radially aligned fields observed in some young supernova remnants,
e.g., \cite{reynoso13}, in which the shock velocity is non-relativistic, although probably the properties
of the amplified fields may not be different in the mildly-relativistic case like the internal shocks
of jets (T. Inoue, private communication).
\subsection{Photospheric Emission Model}
The photospheric emission model assumes that the emission at $E \gtrsim E_p$ is the quasi-thermal
radiation from the photosphere (see \cite{beloborodov11} and references therein).
The emission at $E < E_p$ may be a superposition of many
quasi-thermal components with different temperatures \cite{ryde10,toma11,mizuta11} or contribution of
the synchrotron emission \cite{vurm11}. The quasi-thermal radiation can have high $\Pi$ when the
radiation energy is smaller than the baryon kinetic energy at the photosphere \cite{beloborodov11}.
In this case, the angular distribution of the radiative intensity is beamed towards the expansion
direction {\it in the fluid frame} around the photosphere, and the last electron scatterings produce the
linear polarization. Then the photons scattered to the transverse directions are maximally polarized
in the fluid frame. This situation is the same as the SR model. As a result, the observed polarization
vectors are symmetric around the line of sight. The polarization degree of emission from a given
direction is determined by the brightness distribution below the photosphere (the last scattering
points are widely distributed below the photosphere) and the photon anisotropy before the last
scatterings, which provide $\Pi \leq \Pi^{\rm qt}_{\rm max} \sim 40\%$ \cite{beloborodov11}.
Similar to the SR model, the observed PA change may suggest the patchy emission structure.
The angular size of the patches may be hydrodynamically constrained to be $\theta_p \gtrsim \Gamma^{-1}$.
Only the patches with $\theta_{vp} \sim \theta_p + \Gamma^{-1}$ provide a high $\Pi$ and the bright patches
with $\theta_{vp} \lesssim \theta_p$ give $\Pi \ll \Pi^{\rm qt}_{\rm max}$. The observed high brightness of
the three bursts disfavors the quasi-thermal emission.
However, it is possible that the synchrotron emission contributes to the net polarization at $E < E_p$
\cite{vurm11}.
One may consider the SO or SH model for this synchrotron component. This interpretation may be valid
for GRB 100826A and GRB 110721A, for which the GAP energy range is below $E_p$.
\section{Observational Test of $CPT$ Invariance}
Lorentz invariance is the fundamental symmetry of Einstein's theory of relativity. However, in quantum
gravity such as superstring theory, loop quantum gravity, and Ho\v{r}ava-Lifshitz gravity, Lorentz
invariance may be broken either spontaneously or explicitly
(see \cite{toma12,myers03,gleiser01,mukohyama10} and references therein).
Dark energy, if it is a rolling scalar
field, may also break Lorentz invariance spontaneously. In the absence of Lorentz invariance, the
$CPT$ theorem in quantum field theory does not hold, and thus $CPT$ invariance, if needed, should be
imposed as an additional assumption. Hence, tests of Lorentz invariance and those of
$CPT$ invariance can independently deepen our understanding of the nature of space-time.
If $CPT$ invariance is broken then group velocities of photons with right-handed and left-handed circular
polarizations should differ slightly, leading to birefringence and a gradual PA rotation of linear
polarization. This is quite similar to the Faraday rotation effect on the photons propagating through
the ordered magnetic fields \cite{rybicki79}, although the $CPT$ invariance violation effect is stronger
in higher energy range, in the opposite way to the Faraday rotation effect.
The $CPT$ invariance violation effect is very tiny, but its accumulation from a celestial object
through the long distances may be significant. Thus a test of such an effect can be performed with the
polarization of the GRB prompt emission, which originates at the cosmological distances and is
bright in high-energy gamma-rays.
As we explicitly show below, the reliable measurement of gamma-ray linear polarization presented above
enables us to obtain a strict limit on $CPT$ violation. In order to do this, source distances of the
three GRBs are required to be estimated, but unfortunately their redshifts are not determined.
Instead, we use a well-known distance indicator for GRBs, $L_p = 10^{52.43\pm 0.33} \times
[E_p (1+z)/355\;{\rm keV}]^{1.60\pm 0.082}\;{\rm erg}\; {\rm s}^{-1}$, where $L_p$ is the peak luminosity
and $z$ is the source redshift \cite{yonetoku04,yonetoku10}. Once we measure $E_p$ and peak flux
we can calculate a possible redshift. This correlation equation includes systematic uncertainty
caused by the data scatter. Possible redshifts are then estimated to be
$0.71 < z < 6.84$, $0.21 < z < 1.09$, and $0.45 < z < 3.12$ with $2\sigma$ confidence level
for GRB 100826A, GRB 110301A, and GRB 110721A, respectively.
\subsection{Limit on $CPT$ violation}
In the effective field theory approach, Lorentz violating (LV) effects suppressed by $E/M_{\rm Pl}$
arise from dimension-5 LV operators \cite{myers03}, where $M_{\rm Pl} = (\hbar c/G)^{1/2} = 1.22 \times 10^{19}\;$GeV
is the Planck mass.
Hereafter we shall adopt the unit with $\hbar = c = 1$. In the photon sector
they manifest as the Lorentz- and $CPT$-violating dispersion relation of the form
\begin{equation}
E_{\pm}^2 = p^2 \left( 1 \pm 2\xi \frac{p}{M_{\rm Pl}} \right),
\label{eq:dispersion}
\end{equation}
where $\pm$ denotes different circular polarization states and $\xi$ is a dimensionless parameter.
If $\xi \neq 0$, then Eq.~(\ref{eq:dispersion}) leads to slightly different group velocities for
different polarization states. Hence, the polarization vector of a linearly polarized wave rotates
during its propagation \cite{gleiser01}. The rotation angle in the infinitesimal time interval
$dt$ is $d\theta = (E_+ - E_-)dt/2 \simeq \xi p^2 dt/M_{\rm Pl}$. Substituting $p=(1+z)k$,
$dt = -dz/[(1+z)H]$ and $H^2 = H_0^2 [\Omega_m (1+z)^3 + \Omega_\Lambda]$, the rotation angle
during the propagation from the redshift $z$ to the present is expressed as
\begin{equation}
\Delta \theta (k,z) \simeq \xi \frac{k^2}{M_{\rm Pl} H_0} \int^{z}_0 \frac{(1+z')dz'}
{\sqrt{\Omega_m (1+z')^3 + \Omega_\Lambda}}.
\end{equation}
Here $k$ is the comoving momentum, $H_0 = 1.51 \times 10^{-42}\;$GeV, $\Omega_m = 0.27$,
and $\Omega_\Lambda = 0.73$.
For large $\Delta \theta(k,z)$ in the GAP energy range ($E_{\rm min} = 70\;$keV and
and $E_{\rm max} = 300\;$keV), the polarizations at different energies are canceled, and
the net polarization integrated over the GAP energy range is significantly depleted and
cannot be as high as the observed level.
The detection of highly polarized gamma-ray photons in the GAP energy range thus implies that
$|\Delta \theta(E_2,z) - \Delta \theta(E_1,z)| \leq \pi/2$, where we consider that
a certain proportion of the total number of photons are included in $E_1 < k < E_2$.
In order to obtain an upper bound on
$|\xi|$ from this inequality, we set $E_1 = E_{\rm min}$ and determine $E_2$ by
$\int^{E_2}_{E_{\rm min}} E^{\gamma_{\rm B}} dE/\int^{E_{\rm max}}_{E_{\rm min}} E^{\gamma_{\rm B}} dE = \Pi$,
where $\Pi$ is the net polarization degree over the GAP energy range $E_{\rm min} \leq k \leq E_{\rm max}$.
This prescription for $E_{1,2}$ corresponds to an ideal situation in which the detected signal
has 100\% of the polarization degree and uniform polarization direction over the range
$E_{\rm min} \leq k < E_2$, but has no polarization in the range $E_2 \leq k \leq E_{\rm max}$.
With more realistic momentum-dependencies of the polarization degree and direction, $E_2$ would
be higher and, hence, the bound on $|\xi|$ would be tighter.
For GRB 110721A, the $2\sigma$ lower limits $\gamma_{\rm B} > -0.98$ and $\Pi > 35\%$ for the
GAP energy range lead to $E_2 \simeq 120\;$keV.
Setting $z>0.45$ in $|\Delta \theta(E_2,z) - \Delta \theta(E_{\rm min},z)| \leq \pi/2$, we obtain
the constraint as $|\xi| < 7 \times 10^{-15}$.
More accurate constraints are obtained by requiring that $\sqrt{Q^2+U^2}/N > \Pi$, where
$N = \int^{E_{\rm max}}_{E_{\rm min}} E^{\gamma_{\rm B}} dE$, $Q=\int^{E_{\rm max}}_{E_{\rm min}}
E^{\gamma_{\rm B}} \Pi_i \cos[2\Delta\theta(E,z)]$, and $Q=\int^{E_{\rm max}}_{E_{\rm min}}
E^{\gamma_{\rm B}} \Pi_i \sin[2\Delta\theta(E,z)]$ with the intrinsic polarization degree $\Pi_i = 1$.
Using $\Pi > 0.35$ and $\gamma_{\rm B} > -0.98$, we obtain the constraint from GRB 110721A as
\begin{equation}
|\xi| < 2 \times 10^{-15},
\end{equation}
which is tighter than the above rough estimate.
From GRB 100826A and GRB 110301A, we obtain weaker constraints \cite{toma12}.
See \cite{toma12} for comparison of this result to the constraints from other experiments.
\section{Summary and Discussion}
As for the prompt emission mechanism,
the SO model can explain all of the GAP results, although even this model does not easily produce
polarization as high as $\Pi > 40\%$.
The SR model requires the off-axis viewing angles of the emission regions, for which the brightness
is low because of the relativistic beaming effect. The observed high brightness of the three bursts
disfavors this model.
The SH model is also disfavored if the turbulent field directions are isotropic and $N \sim 10^3$
as suggested by the MHD simulations of \cite{inoue11}. However, the field directions might be
anisotropic as claimed by the recent detailed analysis of the simulations \cite{inoue13}.
More careful studies are needed.
The polarization properties of the quasi-thermal emission from the photosphere
are quite similar to those of the SR model, and thus it is also disfavored. However, it is possible
that the contribution of the synchrotron emission from the photosphere at $E<E_p$ \cite{vurm11} based on the
SO or SH model might reproduce the observational results.
In general, the ordered helical magnetic fields are advected from the
central engine. If the SR, SH, or quasi-thermal model is valid, the energy density of such an ordered field component is
sub-dominant, while if the SO model is valid, it is dominant and also controls the dynamics.
In other words, the SO model implies that the jets are dominated by the Poynting flux rather than the
baryon kinetic energy flux, and they are driven electromagnetically rather than by
the thermal pressure.
However, it should be noted that the SO model assumes globally ordered fields in the emission region.
On the other hand, the observed dim X-ray afterglows just after the prompt emission (i.e., the
so-called X-ray plateau) have suggested that prompt emission has very high efficiency
(even $>90\%$ for some bursts) \cite{ioka06,zhang07}, which means that the energy dissipation,
usually involving field distortion, occurs globally. Reconciling the high $\Pi$ with the high
radiation efficiency looks to be a dilemma, which will have to be resolved in more quantitative modeling.
Anyway, more accurate observational data and the statistical study of such data are needed and essential
to solve the problems, cf., \cite{toma09}. There are several polarimeter mission concepts, such as
PETS \cite{mcconnell13} and TSUBAME \cite{yatsu12}.
There is a possibility that some bursts would be detected more reliably with $\Pi \gg 40\%$, which
suggests the Compton drag model \cite{eichler03,lazzati04,toma09}.
The polarization spectrum would be also valuable. The composite model of the quasi-thermal
plus synchrotron emissions \cite{vurm11} may show a specific tendency in the polarization spectra.
We have obtained the tight constraint on the $CPT$ invariance parameter, $|\xi| < 2 \times 10^{-15}$,
from the reliable measurement of the gamma-ray polarization with GAP \cite{toma12}. Tighter constraint,
$|\xi| < 3 \times 10^{-16}$, has been recently obtained for GRB 061122 with spectroscopic redshift
by using the INTEGRAL-IBIS data \cite{gotz13}.
In the effective field theory approach \cite{myers03}, there is only one operator that leads to a
linear energy dependence of the speed of light in vacuum, and it is the dimension-5 $CPT$-odd LV
operator considered above (see Eq.~\ref{eq:dispersion}).
Constraints on the same operator from observation of energy dependence
of GRB light curve, as performed for GRB 090510 \cite{abdo09}, are much weaker than those
from observation of polarization such as ours. For this reason, it is natural
to interpret the light-curve observation as limits on the dimension-6 LV operator.
The observation of GRB 090510 puts the lower bound on the quantum gravity mass scale as
$M_{QG,2} > 10^{11}\;$GeV. This is consistent with the natural expectation that the quantum gravity
mass scale is of the order of the Planck mass.
\Acknowledgements
I am grateful to D. Yonetoku, T. Murakami, S. Gunji, T. Mihara, S. Mukohyama, T. Inoue for useful discussions. This work is partly supported by JSPS Research Fellowships for Young Scientists
No. 231446.
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\section{Introduction}
The majority of stars form within star clusters \citep[e.g.,][]{2003ARA&A..41...57L}, and our solar system is also thought to be formed within a cluster from many perspectives \citep[e.g.,][]{2010ARA&A..48...47A}.
For example, some of trans-Neptunian objects including (90377) Sedna have large heliocentric eccentricity and perihelion distance \citep[e.g.,][]{2004ApJ...617..645B}, and the plausible scenario for their formation is a close encounter with a passing star in the Sun's birth cluster \citep[e.g.,][]{2004AJ....128.2564M, 2006Icar..184...59B}.
Parts of presolar grains, whose isotopic compositions are significantly anomalous by comparison with the average solar composition, originate from core-collapse supernovae (CCSNe; e.g., \cite{2016ARA&A..54...53N}), and the chromium isotopic heterogeneity found in various chondritic meteorites might be the evidence of the injection of dust grains from a CCSN \citep[e.g.,][]{2014M&PS...49..772S, 2021ApJ...908...64F}.
Injection of dust grains into the solar system from a nearby CCSN is also a classical scenario for the origin of short-lived radionuclides (SLRs) found in meteorites \citep[e.g.,][]{2008ApJ...688.1382T, 2010ApJ...711..597O}.
SLRs are radioactive nuclides that were present in the early solar system but now extinguished, and their existence in the early solar system is recorded in meteorites as excesses of the daughter isotopes.
\citet{2009GeCoA..73.4922H} reviewed the stellar nucleosynthetic processes in the context of stellar evolution, and they concluded that massive stars in the range of $20$--$60 m_{\sun}$ ($m_{\sun}$ is the solar mass) would be plausible sources of SLRs when we assumed the direct injection from CCSNe.
The range of the progenitor mass was discussed based on the amount of production for several SLRs including $^{26}{\rm Al}$ and $^{60}{\rm Fe}$.
The oldest dust particles condensed in the solar system are calcium--aluminum-rich inclusions (CAIs) 4.567 billion years ago based on their lead isotopic ages \citep[e.g.,][]{2002Sci...297.1678A}.
Although the majority of primitive (i.e., unmelted) CAIs are uniformly enriched in $^{26}{\rm Al}$ \citep[e.g.,][]{2012E&PSL.331...43M}, some unusual CAIs show the order-of-magnitude lower initial abundance of $^{26}{\rm Al}$.
Platy hibonite crystals (PLACs) are classified into those unusual CAIs, and they are regarded as the oldest CAIs based on their high condensation temperature and large nucleosynthetic anomalies \citep[e.g.,][]{1998ApJ...509L.137S, 2016GeCoA.189...70K}, although their origin is still under debate \citep[e.g.,][]{2020E&PSL.53516088L, 2022arXiv220311169D}.
CAIs with fractionation and unidentified nuclear effects (FUN CAIs) have also low and varied initial abundance of $^{26}{\rm Al}$, and they are thought to be formed prior to normal CAIs \citep[e.g.,][]{2017GeCoA.201....6P}.
The coexistence of $^{26}{\rm Al}$-rich and $^{26}{\rm Al}$-poor CAIs would be the evidence of the direct injection of $^{26}{\rm Al}$-enriched dust grains into the solar nebula (or the solar molecular cloud core) in the epoch of CAI formation \citep[e.g.,][]{1998ApJ...509L.137S, 2013PNAS..110.8819H}.
As the duration of CAI formation (i.e., the formation age spread of CAIs) is a few $10^{5}$ years or less \citep[e.g.,][]{2006ApJ...646L.159T, 2012Sci...338..651C, 2020GeCoA.279....1K}, the direct injection event would have occurred in the first $10^{5}$ years of the solar system formation.
We note that the initial abundance of $^{26}{\rm Al}$ recorded in CAIs is not altered even if additional injection of $^{26}{\rm Al}$-enriched materials into the solar nebula occurred after the condensation of CAIs.
This timescale is one order of magnitude shorter than the duration of star formation in star clusters \citep[e.g.,][]{2019Natur.569..519K, 2021PASJ...73.1074F}, and we can expect that at least one CCSN would occur in the Sun's birth cluster during its star formation period.
The condition for acquiring at least one CCSN in the birth cluster has been studied in previous studies \citep[e.g.,][]{2010ARA&A..48...47A, 2019A&A...622A..69P}.
They evaluated the minimum number of stars in the birth cluster based on the probability for existing at least one massive star whose mass is large enough to trigger a CCSN with ejection of SLR-rich dust grains.
For example, \citet{2010ARA&A..48...47A} demonstrated that the probability for hosting at least one massive star whose mass is larger than $25 m_{\sun}$ reaches 50\% when the number of stars in the cluster is approximately 800.
We note, however, that these estimates did not consider the timing of CCSNe in the cluster, and the existence of massive stars in the cluster does not support the occurrence of CCSN events during its star formation period.
This is because there is a finite time lag for the birth of a massive star and its explosion.
As both the time lag (i.e., the lifetime of a massive star) and the duration of star formation in a cluster are several to tens of million years \citep[e.g.,][]{1992A&AS...96..269S}, the lifetime of a massive star could have a large impact on the evaluation of the number of stars in the birth cluster.
Figure \ref{fig.1} shows the schematic of the direct injection of SLRs into the early solar system within the birth cluster.
Stars continuously formed within the birth cluster, and the duration of star formation, $t_{\rm SF}$, would be several million years.
Massive stars those were born in the cluster would trigger CCSNe when they finished their lifetime, $t_{\star}$.
A large number of solar-type stars also formed in the cluster.
To explain the coexistence of $^{26}{\rm Al}$-rich and $^{26}{\rm Al}$-poor CAIs in the early solar system, a direct injection of SLR-rich dust grains from a CCSN should occur during CAI formation in the solar system.
As the duration of CAI formation is a $10^{5}$ years and negligibly shorter than $t_{\rm SF}$, the necessary condition to form the solar system is that at least one CCSN occurs in the birth cluster within $t_{\rm SF}$.
\begin{figure*}
\begin{center}
\includegraphics[width=0.8\textwidth]{fig1.pdf}
\end{center}
\caption{
Schematic of the direct injection of SLRs into the early solar system within the birth cluster.
Massive stars those were born in the cluster would trigger CCSNe when they finished their lifetime, $t_{\star}$.
To explain the coexistence of $^{26}{\rm Al}$-rich and $^{26}{\rm Al}$-poor CAIs in the early solar system, a direct injection of SLR-rich dust grains from a CCSN should occur during CAI formation in the solar system.
The necessary condition to form the solar system is that at least one CCSN occurs in the birth cluster within the duration of star formation, $t_{\rm SF}$.
}
\label{fig.1}
\end{figure*}
In this paper, we revisit the number of stars in the Sun's birth cluster from the point of view of direct injection of SLRs from a CCSN to the early solar system.
We calculated the probability for acquiring at least one CCSN within the finite duration of star formation in the birth cluster.
We found that the estimated number of stars in the birth cluster is significantly larger than that previously considered, especially for the cases that the duration of star formation is far less than 10 million years; e.g., for $t_{\rm SF} = 5~{\rm Myr}$, the probability is 50\% when the number of stars in the cluster is approximately $2 \times 10^{4}$, which is 25 times larger than the estimated number by \citet{2010ARA&A..48...47A}.
\section{Model}
In this section, we briefly describe basic equations used in this study.
We calculate the expected number of CCSN events during the star formation period of the birth cluster using probabilistic approach.
Our model is similar to that considered in \citet{2010ARA&A..48...47A}.
The novel point of this study is that we consider the impacts of finite duration of star formation and lifetime of massive stars on the expected number of CCSN events.
\subsection{Initial mass function}
It is known that the maximum stellar mass in the birth cluster, $m_{\rm max}$, depends on the total cluster mass, $M_{\rm cl}$ \citep[e.g.,][]{2006MNRAS.365.1333W}.
\citet{2019A&A...622A..69P} provides an approximated relation between $m_{\rm max}$ and $M_{\rm cl}$ as follows:
\begin{eqnarray}
{\log_{10} \frac{m_{\rm max}}{m_{\sun}}} & = & - 0.76 + 1.06{\log_{10} \frac{M_{\rm cl}}{m_{\sun}}} \nonumber \\
& & - 0.09{\left( {\log_{10} \frac{M_{\rm cl}}{m_{\sun}}} \right)}^{2},
\label{eq.m_max}
\end{eqnarray}
where $m_{\sun}$ is the solar mass.
The minimum stellar mass, $m_{\rm min}$, is set to be $m_{\rm min} = 0.07 m_{\sun}$.
The initial mass function of stars in the birth cluster, ${\xi ( m_{\star} )}$, is given by a broken power-law mass function \citep[e.g.,][]{2002Sci...295...82K, 2021arXiv211210788K}, where $m_{\star}$ is the stellar mass.
For low-mass stars of $m_{\rm min} < m_{\star} < m_{\rm tra}$, the initial mass function is given by
\begin{equation}
{\xi ( m_{\star} )} = k {\left( \frac{m_{\star}}{m_{\rm min}} \right)}^{- 1.3},
\end{equation}
where $m_{\rm tra} = 0.5 m_{\sun}$ is the transition mass for the broken power-law and $k$ is the constant.
For high-mass stars of $m_{\rm tra} \leq m_{\star} < m_{\rm max}$, the initial mass function is given by
\begin{equation}
{\xi ( m_{\star} )} = k {\left( \frac{m_{\rm tra}}{m_{\rm min}} \right)}^{- 1.3} {\left( \frac{m_{\star}}{m_{\rm tra}} \right)}^{- 2.3}.
\end{equation}
The initial mass function satisfies the following equation:
\begin{equation}
\int_{m_{\rm min}}^{m_{\rm max}} {\rm d}m_{\star}~m_{\star} {\xi ( m_{\star} )} = M_{\rm cl}.
\end{equation}
The expected total number of stars in the birth cluster, $N_{\rm cl}$, is
\begin{equation}
\int_{m_{\rm min}}^{m_{\rm max}} {\rm d}m_{\star}~{\xi ( m_{\star} )} = N_{\rm cl}.
\end{equation}
Figure \ref{fig.M} shows the relation between $m_{\rm max}$ and $N_{\rm cl}$.
As massive stars of $20 m_{\sun} < m_{\star} < 60 m_{\sun}$ can trigger CCSNe that might provide SLRs to the early solar system, $N_{\rm cl}$ for a cluster that hosts a progenitor of a plausible CCSN is $N_{\rm cl} \gtrsim 600$.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{fig2.eps}
\end{center}
\caption{
The relation between the maximum stellar mass in the birth cluster, $m_{\rm max}$, and the total number of stars, $N_{\rm cl}$.
}
\label{fig.M}
\end{figure}
\subsection{Time distribution of star formation}
Each star in the birth cluster forms at different time within the finite duration.
The formation rate of stars whose mass is $m_{\star}$ and the birth time is $t$, ${n_{\rm SF} ( m_{\star}, t )}$, is given by
\begin{equation}
{n_{\rm SF} ( m_{\star}, t )} = {\xi ( m_{\star} )} {f ( m_{\star}, t )},
\end{equation}
where ${f ( m_{\star}, t )}$ is the normalized time distribution of star formation, which is defined as
\begin{equation}
\int_{0}^{t_{\rm SF}} {\rm d}{t}~{f ( m_{\star}, t )} = 1,
\label{eq.f}
\end{equation}
and $t_{\rm SF}$ is the duration of star formation in the birth cluster.
In this study, we assume that ${f ( m_{\star}, t )}$ is given as the following mass- and time-independent equation for simplicity:
\begin{equation}
{f ( m_{\star}, t )} = \frac{1}{t_{\rm SF}}.
\label{eq.fb}
\end{equation}
We note that ${f ( m_{\star}, t )}$ must depend on both $m_{\star}$ and $t$ in reality.
Direct numerical simulations of the star cluster formation in giant molecular clouds \citep[e.g.,][]{2020MNRAS.497.3830F, 2021MNRAS.506.5512F} are necessary to discuss the detail of ${f ( m_{\star}, t )}$.
\citet{2021MNRAS.506.5512F} calculated the temporal evolution of the total stellar mass in their numerical simulations of star cluster formation.
The total stellar mass increased with time and saturated at several times of free-fall timescale of star-forming molecular clouds.
From their Figure 7, ${f ( m_{\star}, t )}$ takes the maximum at around the free-fall timescale, although the timing must depend on the physicochemical properties of star-forming molecular clouds including the initial density profile and the metallicity in reality (see Section \ref{sec.duration}).
\subsection{Lifetime of stars}
The lifetime of stars depends on the stellar mass \citep[e.g.,][]{1992A&AS...96..269S}.
Assuming that the lifetime of the star, $t_{\star}$, is given by the sum of durations of H-burning and He-burning, we obtained an empirical relation between $t_{\star}$ and $m_{\star}$ for non-rotating massive stars by using the numerical results shown in \citet{2012A&A...537A.146E};
\begin{equation}
\frac{t_{\star}}{1~{\rm Myr}} = 3.25 {\left( {\log_{10} \frac{ m_{\star}}{4.69 m_{\sun}}} \right)}^{-1.80} + 1.22.
\end{equation}
Figure \ref{fig.LT} shows $t_{\star}$ as a function of $m_{\star}$, and we found that $t_{\star} > 4~{\rm Myr}$ for massive stars of $20 m_{\sun} < m_{\star} < 60 m_{\sun}$, i.e., the progenitors of plausible CCSNe \citep[e.g.,][]{2009GeCoA..73.4922H}.
It is important to note that no CCSNe that provide SLRs to the early solar system would be triggered within $t_{\rm SF}$ if the lifetime of the most massive star in the cluster, $t_{\star, {\rm max}}$, is longer than $t_{\rm SF}$.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{fig3.eps}
\end{center}
\caption{
The relation between the lifetime of stars (i.e., the time lag from the birth of a star to trigger CCSN), $t_{\star}$, and the stellar mass, $m_{\star}$.
We also show the main-sequence lifetime (i.e., the duration of H-burning) as reference.
Circle plots are data taken from \citet{2012A&A...537A.146E}, and the solid curve is the empirical fitting.
}
\label{fig.LT}
\end{figure}
\section{Results}
When we assume that the situation that a CCSN occurred at the timing of the birth of the Solar System, we need (at least) one CCSN event within the duration of star formation in the birth cluster.
In this section, we discuss the probability for acquiring at least one CCSN within the duration of star formation.
\subsection{Probability for acquiring at least one CCSN within the duration of star formation}
The event rate of CCSNe whose progenitor mass is $m_{\star}$ and the explosion time is $t$, ${n_{\rm SN} ( m_{\star}, t )}$, is given by
\begin{equation}
{n_{\rm SN} ( m_{\star}, t )} = {n_{\rm SF} ( m_{\star}, t - t_{\star} )}.
\label{eq.nSN}
\end{equation}
Then the total event rate of CCSNe at $t$, ${p_{\rm SN} ( t )}$, is
\begin{equation}
{p_{\rm SN} ( t )} = \int_{20 m_{\sun}}^{60 m_{\sun}} {\rm d}m_{\star}~{n_{\rm SN} ( m_{\star}, t )}.
\end{equation}
The cumulative number of CCSNe within the duration of star formation in the birth cluster, $Z_{\rm SN}$, is given as follows:
\begin{equation}
Z_{\rm SN} = \int_{0}^{t_{\rm SF}} {\rm d}{t}~{p_{\rm SN} ( t )}.
\end{equation}
Then the probability for acquiring at least one CCSN, $P_{\rm SN}$, would be given by (see Appendix \ref{appA})
\begin{equation}
P_{\rm SN} = 1 - \exp{\left( - Z_{\rm SN} \right)}.
\end{equation}
Figure \ref{fig.P} shows $P_{\rm SN}$ and its dependence on $N_{\rm cl}$ and $t_{\rm SF}$.
The minimum number of $N_{\rm cl}$ for $P_{\rm SN} > 0$ depends on $t_{\rm SF}$; this is because $t_{\star, {\rm max}}$ depends on $N_{\rm cl}$, and $P_{\rm SN} = 0$ when $t_{\rm SF} \leq t_{\star, {\rm max}}$.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{fig4.eps}
\end{center}
\caption{
Probability for acquiring at least one core-collapse supernova with mass in the range of $20$--$60 m_{\sun}$ within the duration of star formation, $P_{\rm SN}$.
Here we assume that $m_{\rm max}$ depends on $N_{\rm cl}$ and it is given by Equation (\ref{eq.m_max}).
}
\label{fig.P}
\end{figure}
The birth cluster of the Sun needs to satisfy the condition of $P_{\rm SN} > 0$ and probably meets the condition $P_{\rm SN} \gtrsim 0.5$ \citep[e.g.,][]{2010ARA&A..48...47A}.
We note that the threshold value of $P_{\rm SN} = 0.5$ was arbitrarily chosen in \citet{2010ARA&A..48...47A}, although 50\% would be a fair threshold from the perspective of whether at least one CCSN event is likely to occur within $t_{\rm SF}$ in a star cluster or not.
If $P_{\rm SN} \ll 1$ for a star cluster, no solar-type stars would have experienced a direct injection of SLR-rich materials in the star cluster.
We acknowledge, however, that one of solar-type stars could have experienced a direct injection even if $P_{\rm SN}$ for each star cluster is $P_{\rm SN} \ll 1$ when we consider dozens of star clusters.
Our results indicate that the total number of stars in the birth cluster would be $N_{\rm cl} \gtrsim 2 \times 10^{4}$ if $t_{\rm SF} = 5~{\rm Myr}$, although it depends on the shape of ${f ( m_{\star}, t )}$.
When the duration of star formation is $t_{\rm SF} = 12~{\rm Myr}$, the required condition for $N_{\rm cl}$ is mitigated: $N_{\rm cl} \gtrsim 2 \times 10^{3}$.
Considering the relation between timings of star formation and explosion (see Equation (\ref{eq.nSN})), $P_{\rm SN}$ depends on the shape of ${f ( m_{\star}, t )}$.
If ${f ( m_{\star}, t )}$ is low at $t \sim 0$ and increases with time for massive stars of $20$--$60 m_{\sun}$, $P_{\rm SN}$ is lower than that for time-independent ${f ( m_{\star}, t )}$.
Recent astronomical observations of nearby star-forming regions \citep[e.g.,][]{2020A&A...642A..87K, 2022A&A...658A.114K} revealed that low-mass stars were born in filamentary structures while massive stars were born in hubs formed via merging of multiple filaments.
We speculate that ${f ( m_{\star}, t )}$ is low at $t \sim 0$ for massive stars if star clusters formed from hub--filament systems, although it should be tested in future numerical simulations and astronomical observations.
\subsection{Impacts of the initial mass function on $P_{\rm SN}$}
In the previous section, we considered that $m_{\rm max}$ depends on $N_{\rm cl}$ as shown in Figure \ref{fig.M}.
However, some of previous studies did not consider this dependence but they assumed that $m_{\rm max}$ as a constant.
In \citet{2010ARA&A..48...47A}, $m_{\rm max}$ is set to $100 m_{\sun}$.
Here we show how the assumption of $m_{\rm max}$ affects the probability for acquiring at least one CCSN within the duration of star formation.
Figure \ref{fig.P_c} shows $P_{\rm SN}$ and its dependence on $N_{\rm cl}$ and $t_{\rm SF}$.
In contrast to Figure \ref{fig.P}, $P_{\rm SN}$ is nonzero for any $N_{\rm cl}$ when $t_{\rm SF}$ is longer than the lifetime of stars whose mass is $60 m_{\sun}$ ($\simeq 4~{\rm Myr}$).
We found that $P_{\rm SN}$ barely changes with the assumption of $m_{\rm max}$ for the cases of $N_{\rm cl} \gtrsim 4 \times 10^{3}$.
This $N_{\rm cl}$ corresponds to the condition for $m_{\rm max} = 60 m_{\sun}$.
As both mass and number of stars in the cluster is dominated by smaller stars, the number of massive stars with mass in the range of $20$--$60 m_{\sun}$ barely depends on the assumption of $m_{\rm max}$ when $m_{\rm max} > 60 m_{\sun}$.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{fig5.eps}
\end{center}
\caption{
Probability for acquiring at least one core-collapse supernova with mass in the range of $20$--$60 m_{\sun}$ within the duration of star formation, $P_{\rm SN}$.
Here we assume that $m_{\rm max} = 100 m_{\sun}$ and it does not depend on $N_{\rm cl}$.
}
\label{fig.P_c}
\end{figure}
We also calculate the probability for existing at least one massive star whose mass is large enough to trigger a plausible CCSN.
The expected number of progenitors formed in the birth cluster, $Z_{\rm exist}$, is given by
\begin{equation}
Z_{\rm exist} = \int_{20 m_{\sun}}^{60 m_{\sun}} {\rm d}m_{\star}~{\xi ( m_{\star} )}.
\end{equation}
Then the probability for existing at least one progenitor, $P_{\rm exist}$, would be given by
\begin{equation}
P_{\rm exist} = 1 - \exp{\left( - Z_{\rm exist} \right)}.
\end{equation}
We note that $P_{\rm SN} \to P_{\rm exist}$ for $t_{\rm SF} \to \infty$ by definition.
The classical estimation by \citet{2010ARA&A..48...47A} corresponds to the dashed line of Figure \ref{fig.P_c}, whereas our results those take into consideration the finite duration of star formation are the solid lines in Figure \ref{fig.P}.
We found that $P_{\rm exist} > 0.5$ is achieved when $N_{\rm cl} > 500$ and $m_{\rm max}$ is independent of $N_{\rm cl}$.
In contrast, when we consider both finite $t_{\rm SF}$ and variable $m_{\rm max}$, $P_{\rm SN} > 0.5$ is achieved only when $N_{\rm cl} \gg 1000$.
For example, $N_{\rm cl} \gtrsim 2 \times 10^{4}$ if $t_{\rm SF} = 5~{\rm Myr}$ and this is 40 times larger than the value obtained from a classical way.
In particular, the impact of finite $t_{\rm SF}$ on the estimate of $N_{\rm cl}$ is significant as shown in Figures \ref{fig.P} and \ref{fig.P_c}.
\subsection{Plausible parameters of the Sun's birth cluster in $N_{\rm cl}$--$t_{\rm SF}$ plane}
Figure \ref{fig.map} is the contour chart of $P_{\rm SN}$ in $N_{\rm cl}$--$t_{\rm SF}$ plane.
We found that $P_{\rm SN} = 0$ outside the shaded area of Figure \ref{fig.map}.
The boundary (dashed line) consists of three regions:
\begin{enumerate}
\item $N_{\rm cl} = N_{\rm cl}|_{m_{\rm max} = 20 m_{\sun}} = {\rm const.}$, where $N_{\rm cl}|_{m_{\rm max} = 20 m_{\sun}}$ is the number of stars in the cluster for $m_{\rm max} = 20 m_{\sun}$,
\item $t_{\rm SF} = t_{\star, {\rm max}}$, where $t_{\star, {\rm max}}$ is the lifetime of the most massive star in the cluster,
\item $t_{\rm SF} = t_{\star}|_{m_{\star} = 60 m_{\sun}} = {\rm const.}$, where $t_{\star}|_{m_{\star} = 60 m_{\sun}}$ is the lifetime of stars whose mass is $60 m_{\sun}$.
\end{enumerate}
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{fig6.eps}
\end{center}
\caption{
Contour chart of $P_{\rm SN}$ in $N_{\rm cl}$--$t_{\rm SF}$ plane.
We note that $P_{\rm SN} = 0$ outside the shaded area.
}
\label{fig.map}
\end{figure}
As $t_{\star}|_{m_{\star} = 60 m_{\sun}} \simeq 4~{\rm Myr}$ (see Figure \ref{fig.LT}), $P_{\rm SN}$ sharply decreases when $t_{\rm SF} \to 4~{\rm Myr}$.
For example, if we set $N_{\rm cl} = 10^{4}$, we obtained that $P_{\rm SN} = 2 \times 10^{-3}$ for $t_{\rm SF} = 4~{\rm Myr}$, $P_{\rm SN} = 0.3$ for $t_{\rm SF} = 5~{\rm Myr}$, and $P_{\rm SN} = 0.7$ for $t_{\rm SF} = 6~{\rm Myr}$.
The condition for $P_{\rm SN} = 0.1$ is not significantly different from the condition for $P_{\rm SN} = 0$ in $N_{\rm cl}$--$t_{\rm SF}$ plane.
This corresponds to the sharp increase of $P_{\rm SN}$ around $P_{\rm SN} = 0$ as shown in Figure \ref{fig.P}, and this is the feature for the model with variable $m_{\rm max}$.
We therefore regard the shaded region as the possible parameters of the Sun's birth cluster.
When we focus on the condition for $P_{\rm SN} > 0.5$ (magenta solid line), the plausible number of stars in the Sun's birth cluster would be larger than $2 \times 10^{3}$ when $t_{\rm SF} < 10~{\rm Myr}$.
Moreover, the plausible number would be $N_{\rm cl} > 2 \times 10^{4}$ when much shorter timescale of $t_{\rm SF} < 5~{\rm Myr}$ is assumed.
\section{Discussion}
\label{sec.discussion}
\subsection{On the duration of star formation in the birth cluster}
\label{sec.duration}
Understanding the duration of star formation in the birth cluster is crucial for determining the plausible $N_{\rm cl}$.
\citet{2021PASJ...73.1074F} studied the formation of massive star cluster that satisfies $N_{\rm cl} > 10^{4}$, and they found that the duration of star formation in star clusters has a significant variation.
For a large cluster formed in a numerical simulation of \citet{2021PASJ...73.1074F}, the duration of star formation is $t_{\rm SF} > 5~{\rm Myr}$, while another small cluster showed a very short duration of approximately 1 million years (see their Figure 25).
The maximum value of $t_{\rm SF}$ would be evaluated from the sum of free-fall timescale and disruption timescale of star-forming giant molecular clouds due to expansion of \ion{H}{II} bubbles \citep[e.g.,][]{2020MNRAS.497.3830F}.
As the efficiency of photoionization feedback strongly depends on the initial surface density of star-forming giant molecular clouds \citep[e.g.,][]{2021MNRAS.506.5512F}, $t_{\rm SF}$ may also depend on the initial surface density, and thus it should depend on the cluster radius.
Future studies on this point is needed to evaluate the physical parameters of the Sun's birth cluster.
\subsection{Dynamical constraints on the birth cluster}
Not only the existence of SLRs but also dynamical properties of solar system bodies give crucial constraints on $N_{\rm cl}$.
\citet{2009ApJ...696L..13P} and \citet{2010ARA&A..48...47A} reviewed the constraints on the number of stars ($N_{\rm cl}$) and radius ($R_{\rm cl}$) of the Sun's birth cluster.
Close encounters of stars within a distance of 100 au can easily destroy the planetary system, and the sharp truncation of the Kuiper belt at approximately 50 au could be reproduced when the distance of the closest encounter, $b$, is in the range of 150--200 au \citep[e.g.,][]{2001Icar..153..416K}.
Considering the dynamical evaporation of the star cluster, \citet{2001Icar..153..416K} estimated the relation among $b$, $N_{\rm cl}$ and $R_{\rm cl}$ as follows \citep[see also][]{2001Icar..150..151A}:
\begin{equation}
b \sim 200 {\left( \frac{N_{\rm cl}}{2 \times 10^{3}} \right)} {\left( \frac{R_{\rm cl}}{2~{\rm pc}} \right)}^{-1}~{\rm au}.
\end{equation}
\citet{2019MNRAS.482..732C} listed $N_{\rm cl}$ and $R_{\rm cl}$ for various star clusters and star-forming regions which host class II young stellar objects.
They reported that $N_{\rm cl} \simeq 2000$ and $R_{\rm cl} \simeq 1~{\rm pc}$ for the Orion Trapezium cluster, and $N_{\rm cl} \simeq 12700$ and $R_{\rm cl} \simeq 52~{\rm pc}$ for the Lupus cloud star-forming region.
Using these values, we obtained $b$ for these two regions: $b \sim 100~{\rm au}$ for the Orion Trapezium cluster and $b \sim 800~{\rm au}$ for the Lupus cloud star-forming region.
A large cluster with $N_{\rm cl} \sim 10^{4}$ and $R_{\rm cl} \simeq 10~{\rm pc}$ might be plausible from the points of view of dynamical and cosmochemical constraints.
We note that the estimation above is based on the assumption that stars are distributed homogeneously within a cluster.
However, in reality, stars in clusters would be initially distributed in a fractal \citep[e.g.,][]{2004A&A...413..929G}.
\citet{2019A&A...622A..69P} argued that $b$ tends to be small when a cluster with an initial fractal distribution is considered.
\subsection{On the timing of injection event}
There are two major scenarios for the direct injection of SLRs from a nearby CCSN: injection to the molecular cloud core \citep[e.g.,][]{1977Icar...30..447C, 2002ApJ...575.1144V, 2008ApJ...686L.119B, 2012ApJ...745...22G} and to the circumstellar disk \citep[e.g.,][]{1977Icar...32..255C, 2000ApJ...538L.151C, 2007ApJ...662.1268O, 2010ApJ...711..597O}.
Here we briefly review these scenarios.
Molecular cloud cores are the progenitors of protostars, and the gravitational collapse of molecular cloud cores forms stars and their circumstellar disks.
The duration of this stage is typically several $10^{5}$ years, and it corresponds to the class 0/I young stellar objects \citep[e.g.,][]{2011AREPS..39..351D}.
The injection to the molecular cloud core is regarded not only as the event for SLR enrichment but also as the trigger for the collapse of the protosolar molecular cloud core \citep[e.g.,][]{2014MNRAS.444.2884L, 2021ApJ...921..150K}.
Another scenario is the direct injection of dust particles to the earliest phase of the circumstellar disk around the young Sun corresponding to class II young stellar objects.
\citet{2010ApJ...711..597O} found that the injection efficiency depends on the size of dust particles, and 0.1--1 micron sized dust particles could be implanted to the circumstellar disk.
Nearby CCSNe would have a great impact not only on the abundance of SLRs but also the structural evolution of the circumstellar disk.
\citet{2018A&A...616A..85P} numerically investigated interactions between a supernova (both irradiation and blast wave) and the solar circumstellar disk.
They concluded that the observed misalignment between the Sun's equator and the ecliptic plan \citep[e.g.,][]{2005ApJ...621L.153B} and the sharp truncation of the Kuiper belt \citep[e.g.,][]{2001ApJ...549L.241A} could be explained by a nearby CCSN event.
We note that the timing of injection event could also affect the isotopic heterogeneity found in various stable isotopes.
\citet{2021ApJ...908...64F} mentioned that injection of clumpy supernova ejecta into the circumstellar disk around the young Sun could be reasonable from the point of view of the nucleosynthetic chromium isotopic variation recorded in carbonaceous chondrites, although we cannot rule out the possibility that the injection event has occurred in the molecular cloud core stage.
\subsection{Injection of SLRs from Wolf-Rayet winds}
Wolf-Rayet (WR) stars are massive evolved stars whose surface composition has been altered by mass loss and internal mixing.
The minimum initial mass for a star to become a WR star is approximately $25 M_{\sun}$ at the solar metallicity, although the threshold mass depends on its rotation \citep[e.g.,][]{2012A&A...542A..29G, 2014ARA&A..52..487S}.
For nonrotating WR stars, the mass loss due to WR winds starts after their main-sequence and the lifetime in the WR phase is less than 0.5 million years \citep[e.g.,][]{2003A&A...404..975M, 2012A&A...542A..29G}.
WR winds might play an important role as injectors of SLRs \citep[e.g.,][]{1997A&A...321..452A, 2006A&A...453..653A, 2012A&A...545A...4G}.
As WR winds are enriched in $^{26}{\rm Al}$ but depleted in $^{60}{\rm Fe}$, several studies \citep[e.g.,][]{2015ApJ...802...22T} regarded WR stars as one of the most plausible sources for the origin of SLRs in the early solar system (see Appendix \ref{appB}).
There are two types of scenarios for introducing SLRs from WR stars: direct injection to solar nebula by WR winds \citep[e.g.,][]{2019A&A...622A..69P} and enrichment of SLRs before the formation of the parental molecular cloud core \citep[e.g.,][]{2012A&A...545A...4G, 2017ApJ...851..147D}.
In the latter scenario, the Sun would be born in the dense shell around a WR star together with a huge numbers of solar siblings.
We note that the time interval of the start of the WR phase and the following CCSN explosion would be approximately 0.5 million years or less.
This timescale is comparable to the timescale for collapse of the molecular cloud core.
Therefore, we can imagine that both WR wind and CCSN could contribute to the enrichment of SLRs in the early solar system.
We should investigate what would be happen if both WR wind and CCSN occurred within the Sun's birth cluster in future studies.
\subsection{Alternative scenarios without direct injection events}
In this study, we focus on the situation that SLRs in the early solar system are directly injected from nearby sources.
We note, however, that several studies proposed that sequential star formation events and/or self-enrichment in parental molecular clouds may also explain the abundance of SLRs in the early solar system \citep[e.g.,][]{2012A&A...545A...4G, 2015A&A...582A..26G, 2018MNRAS.480.4025F, 2021NatAs...5.1009F, 2022arXiv220311169D}.
Although these scenarios can naturally explain the abundance of SLRs recorded in normal CAIs, we point out that these scenarios would have difficulty in reproducing the coexistence of $^{26}{\rm Al}$-poor unusual CAIs and $^{26}{\rm Al}$-rich normal CAIs.
\citet{2016ApJ...826...22K} performed giant molecular cloud scale numerical simulations that can trace the star formation history and abundance of $^{26}{\rm Al}$ in star-forming gas.
They reported that $^{26}{\rm Al} / ^{27}{\rm Al}$ ratio of accreting gas onto a newborn star is spatially and temporally homogeneous during its class 0/I phase.
This might indicate that all CAIs have same initial value of $^{26}{\rm Al} / ^{27}{\rm Al}$ in the framework of these scenarios; it seems inconsistent with meteoritical evidence.
\citet{2016ApJ...826...22K} and \citet{2020E&PSL.53516088L} speculated that FUN CAIs and PLACs might be formed via selective thermal processing of presolar dust grains.
Their scenario requires that CAIs formed via partial evaporation and recondensation with spatial partitioning; however, whether the CAI formation region satisfies these requirements or not is unclear.
Future studies on the particle dynamics in the innermost region of the solar nebula would be needed.
\section{Summary}
Several pieces of evidence indicate that our solar system has formed within a star cluster.
Injection of SLR-rich dust grains into the solar system from a nearby CCSN is a classical scenario for the origin of SLRs found in meteorites including $^{26}{\rm Al}$ and $^{60}{\rm Fe}$.
Although the majority of primitive CAIs are uniformly enriched in $^{26}{\rm Al}$, some unusual CAIs (i.e., FUN CAIs and PLACs) show the order-of-magnitude lower initial abundance of $^{26}{\rm Al}$.
The coexistence of $^{26}{\rm Al}$-rich and $^{26}{\rm Al}$-poor CAIs is usually interpreted as the evidence of the direct injection of $^{26}{\rm Al}$-enriched dust grains into the solar system in the epoch of CAI formation.
As the duration of CAI formation is approximately $10^{5}$ years, the direct injection event would have occurred in the first $10^{5}$ years of the solar system formation.
Therefore, at least one CCSN would occur in the Sun's birth cluster during its star formation period that would be typically several million years (Figure \ref{fig.1}).
The condition for acquiring at least one CCSN in the birth cluster has been studied in previous studies \citep[e.g.,][]{2010ARA&A..48...47A}.
They evaluated the minimum number of stars in the birth cluster based on the probability for existing at least one massive star whose mass is large enough to trigger a CCSN with ejection of SLR-rich dust grains.
However, those studies did not consider the timing of CCSNe in the cluster, and the existence of massive stars in the cluster does not support the occurrence of CCSN events during its star formation period.
In this paper, we revisited the number of stars in the Sun's birth cluster from the point of view of direct injection of SLRs from a CCSN to the early solar system.
We calculated the probability for acquiring at least one CCSN within the finite duration of star formation in the birth cluster (Figure \ref{fig.P}).
We found that the estimated number of stars in the birth cluster, $N_{\rm cl}$, is significantly larger than that previously considered, especially for the cases that the duration of star formation, $t_{\rm SF}$, is far less than 10 million years.
The plausible number of stars in the Sun's birth cluster would be $N_{\rm cl} > 2 \times 10^{3}$ when $t_{\rm SF} < 12~{\rm Myr}$ (Figure \ref{fig.map}), although it depends on many assumptions.
Moreover, the plausible number would be $N_{\rm cl} > 2 \times 10^{4}$ when much shorter timescale of $t_{\rm SF} < 5~{\rm Myr}$ is assumed.
In contrast, the minimum number of stars without consideration of the timing of explosion, i.e., the lower limit of $N_{\rm cl}$ obtained from a classical way, is approximately 500 (Figure \ref{fig.P_c}).
In other words, our novel constraint is an order of magnitude higher than previous estimates.
Understanding the evolution of the solar nebula in the birth cluster is essential to unveiling how the Earth formed.
The effects of nearby massive stars on disk evolution (e.g., external photoevaporation and stellar feedback from winds and supernovae; \cite{2019MNRAS.482..732C}) should depend on $N_{\rm cl}$, $t_{\rm SF}$, and the geometry of the cluster.
Future studies on the disk evolution around solar-type stars in a large star cluster taking the impact of finite $t_{\rm SF}$ into account would be crucial.
We acknowledge that whether the source of SLRs is a CCSN or a WR star is still under debate.
Moreover, whether a direct injection event contribute to the SLR enrichment of the solar system or not is also controversial.
We discussed these topics in Section \ref{sec.discussion}.
These uncertainty on the origin of SLRs must have a great impact on the evaluation of $N_{\rm cl}$ and $t_{\rm SF}$ for the Sun's birth cluster.
We need to discuss this point in depth in future studies.
\begin{acknowledgements}
The authors wish to express their cordial thanks to the referee Anthony G.\ A.\ Brown for constructive comments.
The authors thank Michiko S.\ Fujii, Takashi J.\ Moriya, and Ryota Fukai for useful comments and discussions.
S.A.\ was supported by JSPS KAKENHI Grant No.\ JP20J00598.
E.K.\ was supported by JSPS KAKENHI Grant No.\ 18H05438.
This work was supported by the Publications Committee of NAOJ.
\end{acknowledgements}
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\section{\label{sec:intro}Introduction}
Although the existence of naked singularities in Nature is the subject of intense debate nowadays, it has been well established that Einstein field equations for the gravitational field allow solutions that can be interpreted as describing naked singularities. In particular, black hole solutions are characterized by the existence of naked singularity counterparts \cite{kerr}. However, it seems that the particular choice of the physical parameters, which is necessary for the formation of naked singularities, is difficult to be realizable in Nature. Indeed, a rotating naked singularity needs a specific angular momentum that must be greater that its mass (in geometric units), a condition that probably cannot be fulfilled in realistic configurations because it would imply such a high angular velocity that the object would destroy itself before reaching it \cite{mtw}.
There is, however, a simpler way to generate naked singularities, namely, by considering mass distributions with quadrupole moment \cite{quev11}. Indeed, from the point of view of multipole moments, the uniqueness theorems prove that black holes can have only mass monopole and angular momentum dipole \cite{heusler}. Consequently, the addition of a quadrupole to a mass distribution, even in the static case, would imply that the corresponding gravitational field describes a naked singularity. Consequently, a simple shape deviation from spherical symmetry in a mass distribution leads to the appearance of naked singularities.
In previous works \cite{quev11,Quevedo2016,Boshkayev2016,Boshkayevpxa}, we used a particular static quadrupolar solution \cite{zipoy,voorhees,malafarina05,solutions}
to study the physical properties of naked singularities.
There are several solutions of Einstein field equations that can be used to describe the exterior gravitational field of a static mass distribution with quadrupole moment \cite{aqs18}. In the limiting case of vanishing quadrupole, they reduce to the spacetime of a Schwarzschild black hole.
An interesting characteristic of all of them is that the outermost naked singularity is a sphere with radius $r=2m$, where $m$ is the mass of the gravitational source. This could be interpreted intuitively as if the presence of the quadrupole causes the destruction of the regular horizon turning it into a singular hypersurface.
In this work, we show that this is not always the case. Indeed, we will derive a new solution, whose singularity is located on a sphere of radius $r=m$. This means that in this case the quadrupole completely destroys the regular horizon at $r=2m$, but generates a new special hypersurface at $r=m$ which contains a singularity. To derive this new solution, we use the fact in realistic situations we expect that compact objects deviates only slightly from spherical symmetry. This implies that the quadrupole can be considered as a small quantity. With this in mind, we investigate a particular approximate line element, which is valid only up to the first order in the quadrupole parameter. Then, we find the general solution of the corresponding Einstein vacuum field equations and show that a particular solution is characterized by a curvature singularity located on a sphere of radius $r=m$.
This work is organized as follows. In Sec. \ref{sec:exq}, we consider a line element that is especially adapted to the study of interior and exterior solutions. We derive the general field equations for the case of vacuum gravitational fields. In Sec. \ref{sec:gvs}, we find the most general solution that is linear in the quadrupole moment. We select a particular case that is characterized by the presence of naked singularity at a distance $r=m$ from the origin of coordinates.
We also calculate the Newtonian limit of the new approximate solution and show that it corresponds to a mass distribution with a small quadrupole. In Sec. \ref{sec:geo}, we investigate the motion of test particles in the spacetime described by the approximate solution. In general, we find that the quadrupole affects the behavior of Schwarzschild orbits. By analyzing the behavior of free falling particles we show that no effects associated with the presence of repulsive gravity can be detected in contrast to repulsive effects found previously in the case of exact solutions with quadrupole. We conclude that repulsive gravity in naked singularities is a non-linear phenomenon.
Finally, Sec. \ref{sec:con} contains a summary of our results.
\section{\label{sec:exq} Line element and field equations}
The search for and investigation of physically meaningful solutions of Einstein equations begins with the choice of an appropriate line element. In the case of static gravitational fields of deformed mass distributions, one can assume that the fields preserve axial symmetry. Moreover, if we are interested in describing the gravitational field outside as well as inside the mass distribution, it is convenient to choose a line element that can be used in both cases.
In a previous work \cite{approxi_2021}, we found interior perfect-fluid solutions that can be matched with exterior vacuum solutions under the assumption that the quadrupole moment is small. From a physical point of view, this implies that the mass distribution is only slightly deformed. We were able to find a particular line element that can be used to search for interior and exterior approximate solutions. It can be written as
\begin{eqnarray}
ds^2 =&& e^{2 \nu} (1+qa) dt^2 - (1+qc+qb)\frac{dr^2}{1-\frac{2\tilde m}{r}} \nonumber \\ &&-
(1+qa+qb)r^2d\theta^2
-(1-qa)r^2\sin^2\theta d\varphi^2\ ,
\label{apin1}
\end{eqnarray}
where the set of $(t,r,\theta,\varphi)$ can be interpreted as polar coordinates in the limiting case $q\rightarrow 0$. Moreover, the functions $\nu=\nu(r)$, $a=a(r)$, $c=c(r)$, $\tilde m=\tilde m(r)$, and $b=b(r,\theta)$ are arbitrary.
In the particular case $q=0$, the above line element can be used to describe the exterior Schwarzschild solution
\begin{equation}
\tilde m = m =const,\ e^{2\nu} = 1-\frac{2m}{r}\ ,
\end{equation}
and the interior perfect fluid Schwarzschild metric with
\begin{equation}
\tilde m = 4 \pi \rho R^{3} \ , \
p = \frac{3m}{4\pi R^3} \frac{\left[ f(r) - f(R) \right ] }
{\left[ 3 f(R) - f(r) \right ]}\ ,
\end{equation}
with
\begin{equation} e^{2\nu} = \bigg[ \frac{3}{2} f\left(R \right) -\frac{1}{2} f\left(r \right) \bigg]^2,\ \
f\left(r \right)= \sqrt{1-\frac{2mr^2}{R^3}} \end{equation}
where $\rho = const$ and $p=p(r)$ are the density and pressure of the fluid, respectively.
Furthermore, for $q\neq 0$ the line element (\ref{apin1}) contains the approximate (up to the first order in $q$) quadrupolar metric ($q-$metric), which has been interpreted as the simplest generalization of the Schwarzschild metric that includes a quadrupole moment \cite{quev11}. In this limit, $q$ has been interpreted as the quadrupole parameter.
Here, we will use the advantages of the line element (\ref{apin1}) to search for more general approximate solutions. Then, the Einstein vacuum field equations, up to the first order in $q$, can be written as
\begin{equation}
\tilde m_{,r} = 0 \quad {\rm i.e.} \quad \tilde m = m = const.\ ,
\end{equation}
\begin{equation}
\nu_{,r}=\frac{ {m}}{r \left(r-2 {m} \right)}\ ,
\label{nuv}
\end{equation}
\begin{equation}
(r-m)(a_{,r}-c_{,r}) + (a-c) = 0\ ,
\label{acv}
\end{equation}
\begin{eqnarray} \nonumber
2 &&r \left(r-2m \right)a_{,rr} + \left(3r-m \right) a_{,r} + \left(r-3m \right) c_{,r} \\ &&- 2 \left(a-c \right)=0,
\end{eqnarray}
\begin{eqnarray} \nonumber
r &&\left(r-2m \right) b_{,rr}+ b_{,\theta \theta} + \left(r-m \right) b_{,r} -2 \left(r-2m \right) c_{,r} \\ &&+ 2 \left(a-c \right)=0 ,
\end{eqnarray}
\begin{eqnarray} \nonumber
&& \left( r^2 -2 mr + m^2 \sin^2 \theta \right) b_{,\theta} + 2 r \left(r-2m \right) \nonumber \\ && \times \left( m a_{,r}- a+c \right) \sin \theta \cos \theta =0 ,
\end{eqnarray}
\begin{eqnarray} \nonumber
&&\left( r^2 -2 m r + m^2 \sin^2 \theta \right) b_{,r}+ 2 \left(r-2 m \right) \nonumber \\ && \times \left(r- m \sin^2 \theta \right) a_{,r}+2 \left(r-m \right) \left(a-c \right) \sin^2 \theta =0 ,
\label{eqb}
\end{eqnarray}
where a comma represents partial differentiation with respect to the corresponding coordinate. For simplicity, we replaced the solution of the first equation $\tilde m = m = const.$ in the remaining equations.
\section{\label{sec:gvs} Approximate solutions}
We now investigate the system of partial differential equation (\ref{nuv})-(\ref{eqb}).
Equations (\ref{nuv}) and (\ref{acv}) can be integrated and yield
\begin{equation}
\nu = \frac{1}{2} \ln\left(1-\frac{2m}{r}\right) + \alpha_1 \ ,\quad a-c = \frac{\alpha_2 m^2}{(r-m)^2} \ ,
\label{sol_nuac}
\end{equation}
where $\alpha_1$ and $\alpha_2$ are dimensionless integration constants. It turns out that
the remaining system of partial differential equations can be integrated in general and yields
\begin{equation}
a= - \frac{\alpha_2 m}{r-m} + \frac{1}{2}\left(\alpha_3 - {\alpha_2}\right) \ln\left(1-\frac{2m}{r}\right) + \alpha_4\ ,
\label{vsola}
\end{equation}
\begin{equation}
c= - \frac{\alpha_2 m r}{(r-m)^2} + \frac{1}{2}\left(\alpha_3 - {\alpha_2}\right) \ln\left(1-\frac{2m}{r}\right) + \alpha_4\ ,
\label{vsolc}
\end{equation}
\begin{eqnarray} \nonumber
&b=& \frac{2\alpha_2 m}{r-m} - \left(\alpha_3 - {\alpha_2}\right)
\bigg[ \ln 2 \\ &+& \ln\left(1-\frac{2m}{r} + \frac{m^2\sin^2\theta}{r^2} \right)\bigg] + \alpha_5 \ ,
\label{vsolb}
\end{eqnarray}
where $\alpha_3$, $\alpha_4$, and $\alpha_5$ are dimensionless integration constants. We can see that the general approximate exterior solution with quadrupole moment is represented by the 5-parameter family of solutions (\ref{sol_nuac})--(\ref{vsolb}).
In this general solution, the additive constants $\alpha_4$ and $\alpha_5$ can be chosen such that at infinity the solution describes the Minkowski spacetime in spherical coordinates. This means that non asymptotically flat solutions are also contained in the above general solution.
To partially investigate the physical meaning of this solution, we calculate the Kretschmann scalar $K= R_{abcd} R^{abcd}$. We obtain
\begin{eqnarray} \nonumber
&K=& \frac{48 m^2}{r^6} \Bigg \{1+ q \Bigg[
\left( \alpha_2 -\alpha_3 \right) \Big[ \ln \left( 1- \frac{2m}{r}\right) \nonumber \\
&-& 2 \ln \left( 1- \frac{2m}{r} + \frac{m^2}{r^2} \sin^2{\theta} \right) - 2\ln2 \Big] \nonumber \\
&+\alpha_{2}& A_{1}+ \alpha_{3} A_{2} -2 \left( \alpha_4 + \alpha_5 \right) +\alpha_{2} A_{3} \Bigg] +\mathcal{O}(q^2)\Bigg\}
\end{eqnarray}
where
\begin{eqnarray} \nonumber
A_{1}&=&-\frac{1}{6 \left( m^2 \sin^2 \theta-2mr +r^2 \right) \left(m-r \right)^3 m^2} \nonumber \\ &\times& \bigg[ \left(12m^3-11m^2r-14m r^2+11r^3 \right)m^4 \cos^2 \theta \nonumber \\ &-&\left(12m^3-17m^2r-2m r^2+5 r^3\right) m^2 \left(m-r\right)^2 \bigg]
\end{eqnarray}
\begin{eqnarray}
A_{2}= \frac{2 m^2 \sin^2 \theta - 3 m r+r^2}{ m^2 \sin^2 \theta-2mr +r^2}
\end{eqnarray}
\begin{eqnarray}
A_{3}= - \frac{1}{6} \frac{r \big[ 7 \left(m-r \right)^2-2 r^2\big]}{ \left(m-r \right)^3}
\end{eqnarray}
where the term proportional to $q^2$ has been neglected. We can see that this approximate spacetime is characterized by the presence of three different curvature singularities located at
\begin{equation}
r=0\ ,\quad r=m\, \quad r=2m\ , \quad r=m(1\pm \cos\theta)\ .
\end{equation}
The geometric structure of the curvature singularities of the solutions (\ref{sol_nuac}), (\ref{vsola}), (\ref{vsolc}) and (\ref{vsolb}) is illustrated in Fig.\ref{fig1}.
\begin{figure}%
\includegraphics[scale=0.4]{fig_sing_q2.eps}%
\caption{Curvature singularities of the general metric.
\label{fig1}%
\end{figure}
Another interesting particular case corresponds to the choice
\begin{equation}
\alpha_1 =0 \ ,\quad \alpha_3 = {\alpha_2}\ ,\quad \alpha_4= 0\ ,\quad \alpha_5 = 0\ ,
\end{equation}
which leads to the line element
\begin{eqnarray} \nonumber
ds^2 &= & \left(1-\frac{2m}{r}\right)\left(1-\frac{q\alpha_2 m}{r-m}\right) dt^2 \nonumber \\ & & -\left[1+ \frac{q\alpha_2 m (r-2m)}{(r-m)^2} \right] \frac{dr^2}{1-\frac{2m}{r}} \nonumber \\ & & - \left(1 + \frac{q\alpha_2 m}{r-m} \right) r^2(d\theta^2+\sin^2\theta d\varphi^2)
\label{appsol1}
\ .
\end{eqnarray}
This is an asymptotically flat approximate solution with parameters $m$, $q$ and $\alpha_2$. Since $\alpha_2$ appears always in combination with $q$, it can be absorbed in the definition of $q$. We, therefore, set $\alpha_2=1$ without loss of generality.
The singularity structure can be found by analyzing the Kretschmann invariant, which in this case reduces to
\begin{equation}
K= \frac{48 m^2}{r^6}\left(1+ q \frac{r-4m}{r-m} + \mathcal{O}(q^2)\right)\ .
\label{Kretsch_q}
\end{equation}
We see that there are only two singularities, which are located at $r=0$ and $r=m$. This is an interesting property of this solution because all the remaining solutions contained in (\ref{sol_nuac})--(\ref{vsolb}) are singular on the hypersurface $r=2m$. Moreover, all the known exact solutions with quadrupole turn out to be singular at $r=2m$ \cite{aqs18}. To our knowledge, the solution (\ref{appsol1}) is the only one in which the outermost singularity is located inside the sphere with $r=2m$. This means that the spacetime is well defined behind in the interval $r\in (m,2m]$. We are interested in investigating the properties of this spacetime near the singularity $r=m$.
To further analyze the physical meaning of the solution (\ref{appsol1}), we calculate the corresponding Newtonian limit. To this end, we perform a coordinate transformation of the form $(r,\theta)\rightarrow (R,\vartheta)$ defined by the equations \cite{approxi_2021,Mashhoon2018}
\begin{eqnarray} \nonumber
r&=& R \bigg[1- q \frac{m}{R} \big[ 1+ \frac{m}{R} \left( \beta_{1} + \sin^2 \vartheta \right) \\ &+& \frac{m^2}{R ^2} \left( \beta_{2} - \sin^2 \vartheta \right)+ .. \big] \sin^2 \vartheta \bigg]\label{trans_r} ,
\end{eqnarray}
and
\begin{equation}
\theta = \vartheta -q \frac{m^2}{R^2} \left(1+ 2 \frac{m}{R} + . . . \right) \sin \vartheta \cos \vartheta
\label{trans_theta},
\end{equation}
where the $\beta_1$ and $\beta_2$ are constants and we have neglected terms of the order higher that $m^3 / R^3 $. Inserting the above coordinates into the metric (\ref{appsol1}), we obtain the approximate line element
\begin{eqnarray} \nonumber
ds^2&=& \left( 1+ {2 \Phi} \right)dt^2 - \frac{dR^2 }{1+ {2 \Phi}} \\ & -& U \left( R, \vartheta \right) R^2 \left( d\vartheta^2 + \sin^2 \vartheta d \varphi^2 \right),
\label{Newline}
\end{eqnarray}
with
\begin{equation}
\Phi= -\frac{GM}{R}+ \frac{GQ}{R^3} P_{2}\left( \cos \vartheta \right),
\end{equation}
\begin{equation}
U \left( R, \vartheta \right)= 1- 2 \frac{GM}{R^3} P_2(\cos \vartheta),
\end{equation}
where $P_2(\cos \vartheta)$ is the Legendre polynomial of degree 2, and we have chosen the free constants as $\alpha_{2}=2$, $\beta{_1}=1/3 $, and $\beta{_2}=5/3 $.
We recognize the metric (\ref{Newline}) as the Newtonian limit of general relativity, where $\Phi$ represents the Newtonian potential. Moreover, the constants
\begin{equation}
M= \left( 1+q \right)m, \ \ \ Q=\frac{2}{3} q m^3,
\end{equation}
can be interpreted as the Newtonian mass and quadrupole moment of the corresponding mass distribution.
\section{\label{sec:geo} Motion of test particles}
Consider the trajectory $x^{a}(\tau)$ of a test particle with 4-velocity $u^{a} =dx^{a}/d\tau = {\dot x}^{a}$.
The 4-moment $p^a = \mu {\dot x}^{a} $ of the particle can be normalized so that the equations and constraint for geodesics are given as
\begin{equation}
\ddot{x}^{a}+ {\Gamma^{a}}_{bc} \dot{x}^{b} \dot{x}^{c}=0
\label{eq_geodesics}
\end{equation}
\begin{equation}
g_{a b} \dot x^{a} \dot x^{b} = \epsilon,\label{eq1:normomentum}
\end{equation}
where $\epsilon =0, 1, -1$ for null, timelike, and spacelike curves, respectively \cite{Pugliese:2010ps}.
For the approximate metric (\ref{appsol1}) we obtain from (\ref{eq_geodesics}) that geodesics are determined, in general, by the following set of equations
\begin{eqnarray}
\ddot{t}+ m \bigg[
\frac{2}{r \left(r-2m \right)} + \frac{q}{\left( r-m\right)^2} \bigg]\dot{t} \dot{r}=0 \ ,
\label{eq_time}
\end{eqnarray}
\begin{eqnarray} \nonumber
&& \ddot{r}+ \bigg[
1 + \frac{q\left( 6m^2-6mr+r^2\right)}{2\left( r-m\right)^2}
\bigg]\frac{m \left(r-2m \right)\dot{t}^2}{r^3}
\nonumber \\
&&- \frac{m}{2} \bigg[ \frac{2}{r \left(r-2m \right)}+ \frac{q \left( r-3m\right)}{\left( r-m\right)^3}
\bigg] \dot{r}^2 \nonumber \\
&&- \bigg[ r-2m - \frac{qm \left(r-2m \right)^2}{ \left(r-m \right)^2} \bigg]
\dot{\theta}^2 \nonumber \\
&& - \left(r-2m \right) \sin^2 \theta \bigg[
1- \frac{qm \left(r-2m \right)}{\left( r-m\right)^2}
\bigg]
\dot{\varphi}^2 =0 \ ,
\end{eqnarray}
\begin{equation}
\ddot \theta -\sin \theta \cos \theta \dot{\varphi}^2+ \left( \frac{2}{r}-\frac{qm}{ \left( r-m\right)^2}\right) \dot{r} \dot{\theta} =0 .
\label{eq_theta}
\end{equation}
\begin{eqnarray}
\ddot {\varphi} + \Bigg[
\left( \frac{2}{r} - \frac{qm}{\left( r-m\right)^2} \right) \dot{r} + \frac{2 \cos \theta}{\sin \theta} \dot{\theta}
\bigg]\dot{ \varphi}=0 \ ,
\label{eq_phi}
\end{eqnarray}
The 4-moment $p^a = \mu {\dot x}^{a} $ of the particle can be normalized so that
\begin{eqnarray}
g_{a b} \dot x^{a} \dot x^{b} = \epsilon,\label{eq:normomentum}
\end{eqnarray}
where $\epsilon =0, 1, -1$ for null, timelike, and spacelike curves, respectively \cite{pqr11a}. Then, for the approximate metric (\ref{appsol1}) we obtain from (\ref{eq:normomentum}) that
\begin{eqnarray}
\left(1- \frac{q m^2}{\left(r-m \right)} \right){\dot r}^2 = {\tilde E}^2 - \Phi^2,\label{eq:genequat}
\end{eqnarray}
where
\begin{eqnarray} \nonumber
V_{eff}& =& \Phi^2 = \left(1- \frac{2m}{r} \right) \bigg[
r^2 \dot{\theta}^2 + \left( \epsilon + \frac{2l^2}{r^2 \sin^2 \theta } \right) \\ &\times& \left(1- \frac{qm}{r-m} \right)- \frac{l^2}{r^2 \sin^2 \theta}
\bigg]
\label{veff}
\end{eqnarray}
is the effective potential
and we have used the expression for the energy $E=\mu \tilde E$ and the angular moment $l=\mu \tilde l$ of the test particle which are constants of motion
\begin{equation}
E = g_{a b}\xi^{a}_{t}p^{b}= \left(1-\frac{2m}{r}\right)\left(1-\frac{q m}{r-m}\right) \mu {\dot t},
\label{cons_m_E}
\end{equation}
\begin{equation}
l = - g_{a b}\xi^{a}_{\varphi}p^{b} = \left(1+ \frac{q m}{r-m}\right) r^2 \sin^2\theta \mu {\dot \varphi},
\label{cons_m_l}
\end{equation}
associated with the Killing vector fields $\xi_t= \partial_t$ and $\xi_{\varphi}= \partial_{\varphi}$, respectively. For the sake of simplicity we set $\mu=1$ so that $\tilde E = E$ and $\tilde l = l$.
Figure \ref{fig2} illustrates the behavior of the effective potential in terms of the parameter $q$ for $\theta = \pi/2$.
The effective potential of the Schwarzschild spacetime is also shown for comparison. For positive (negative) values of $q$, the effective potential at a given point outside the outer singularity is always less (greater) than the Schwarzschild value. This indicates that the distribution of orbits on the equator of the metric (\ref{apin1}) can depend drastically on the value of $q$.
\begin{figure}%
\includegraphics[width=7cm, height=5cm]{fig_eff2.eps}%
\caption{The effective potential for timelike geodesics on the equatorial plane as a function of the radius for different values of the quadrupole parameter. Here we set $l^2=20$ for concreteness. }%
\label{fig2}%
\end{figure}
\subsection{Circular orbits}
\label{sec:cir}
We will now investigate the properties of circular orbits on the equatorial plane, $\theta=\pi/2$ of the gravitational described by the approximate metric (\ref{apin1}).
Circular orbits correspond to the limiting case $\dot r=0$. Their stability properties are determined by the extrema of the effective potential.
Following the conventional stability analysis of circular orbits involving a potential function, from Eq.(\ref{veff}), we obtain
\begin{eqnarray} \nonumber
&V_{eff,r}=&
\frac{2 \epsilon m}{r^2} + \frac{2 \left(3m-r \right)l^2 }{r^4}
+\frac{qm}{r^2 \left(r-m \right)^2} \nonumber \\
&\times& \bigg[
\left(2 m^2 -4mr +r^2 \right) \epsilon \nonumber \\ &+& \frac{2\left(6m^2-10mr+3r^2 \right) l^2}{r^2}
\bigg] \ ,
\end{eqnarray}
and
\begin{equation}
V_{eff,rr}= B_1 +B_{2}
\end{equation}
where $B_{1}$ and $B_{2}$ are given by
\begin{equation}
B_{1}= \frac{m \left( 36m^4 -90m^3 r+74m^2r^2-22mr^3+2 r^4\right)}{r^3 \left(3m^2-4mr +r^2 \right)^2} \ ,
\end{equation}
\begin{equation}
B_{2}= \frac{mq \left( 36m^4 -114m^3 r+74m^2r^2-17mr^3+r^4\right)}{r^3 \left(3m^2-4mr +r^2 \right)^2}
\end{equation}
where we set $\epsilon=1$ and replaced the value of the angular momentum
\begin{eqnarray} \label{l2co} \nonumber
&l^2&= \frac{\epsilon m r^2}{\left(r-m\right)\left(r-3m\right)^2} \bigg[
2 \left(3m^2-4mr+r^2\right) \\&+&q \left( r^2-6m^2\right) \bigg],
\label{cir_l2}
\end{eqnarray}
which can be derived from the condition $V_{eff,r} =0$.
The numerical analysis of the stability condition, $V_{eff,rr} > 0$, is depicted in Fig.\ref{fig_sta_Vrr}.
\begin{figure}%
\includegraphics[width=7cm, height=7cm]{fig_sta_Vrr.eps}%
\caption{
Stability analysis of circular orbits with radius $r/m$ for different values of the quadrupole parameter $q$. Green (yellow) points represent stable (unstable) orbits. The red region lies inside the singularity. }%
\label{fig_sta_Vrr}%
\end{figure}
The green region contains only stable orbits whereas the yellow region corresponds to unstable orbits. For comparison, we include the limiting values of the Schwarzschild spacetime.
We see that the quadrupole parameter $q$ changes the value of the minimum allowed radius ($3m$) and of the inner most stable circular orbit radius ($6m$) of the Schwarzschild metric. In fact, the quadrupole leads to the appearance of a second stable region below and over the radius $3m$, which is not present in the Schwarzschild limiting case. This region can reach the value of $r/m \approx 1 $, approaching the singularity which is located at $r=m$. Moreover, within the spacetime determined by the interval $r\in (m, 2m]$, we notice that most of this region allows the existence of stable circular orbits with a disjoint region of instability for positive values of $q$.
Furthermore, the energy of test particles on circular orbits can be expressed as
\begin{equation}
{E} ^2= \frac{\epsilon \left(r-2m \right)^2}{r\left( r-3m\right)} \bigg[
1- \frac{mq \left(r-6m \right)}{2 \left( r-m\right)\left(r-3m \right)}
\bigg].
\end{equation}
In Fig. \ref{fig_sta_E2l2}, we plot the regions in which the energy $E^2$ and angular momentum $l^2$ are both positive or negative simultaneously.
The red region denotes all the radii that are not allowed for circular orbits because either the squared of the energy or of the angular momentum are negative.
A comparison with Fig. \ref{fig_sta_Vrr}
shows that the region contained between the singularity $r/m=1$ and around $r/m=3$ is allowed for circular orbits by the stability condition but is excluded by the energy and angular momentum conditions.
We conclude that the effect of the quadrupole on the properties of circular orbits is as follows. A positive quadrupole leads to an increase of the minimum allowed radius whereas a negative quadrupole generates the opposite effect. This means that only in the case of an oblate object, test particles are allowed to exist on orbits closer to the singularity, which is situated at $r=m$.
\begin{figure}
\includegraphics[width=6cm,height=5cm,scale=0.1]{fig_sta_E2l2.eps}
\caption{
{Energy $E^2$ and angular momentum $l^2$ of test particles on circular orbits in terms of the radius $r/m$ and the quadrupole parameter $q$.
}
}
\label{fig_sta_E2l2}
\end{figure}
Finally, we consider the angular velocity
\begin{eqnarray} \nonumber
&\Omega(r)=& \dot\varphi= \frac{1}{r} \bigg[ \frac{m}{r-3m} \bigg]^{1/2} \\ &&\times \bigg[ 1 + \frac{q \left( 6m^2-4mr+r^2\right)}{4 \left(r-m \right)\left(r-3m \right)} \bigg],
\end{eqnarray}
and the period
\begin{eqnarray} \nonumber
T(r) &=& \int \frac{\dot t}{\dot\varphi} d\varphi = 2\pi \frac{dt}{d\varphi} \\ &=& 2\pi r^{3/2} m^{-\frac{1}{2}} \bigg[ 1- \frac{q \left(r-4m \right)}{4 \left(r-m \right)}\bigg],
\end{eqnarray}
of circular orbits.
The behavior of the angular velocity and period are depicted in Fig. (\ref{fig ometa_T}). We can see that the influence of the quadrupole on the value of the angular velocity increases as the radius of the orbit approaches the value of $r\approx 3m$. This agrees with the behavior of the stability condition and the energy and angular momentum shown in Figs. \ref{fig_sta_Vrr} and \ref{fig_sta_E2l2}, respectively. For a given orbit radius, the angular velocity increases (decreases) for positive (negative) values of the quadrupole. Notice that close to the singularity located at $r=m$, there is a region in which the angular velocity is a well behaved function of $q$ and $r$. This region corresponds to the stability region that was also found in Fig. \ref{fig_sta_Vrr}.
\begin{figure}
\includegraphics[width=6cm,height=5cm,scale=0.1]{fig_Omeg_r2.eps}
\includegraphics[width=6cm,height=5cm,scale=0.1]{fig_T_r.eps}
\caption{Angular velocity and period of circular orbits in terms of the radius $r/m$ and the quadrupole parameter $q$. }
\label{fig ometa_T}
\end{figure}
\subsection{Bounded and unbounded orbits}
\label{sec:unb}
We now study the influence of the quadrupole on the trajectories of massive test particles, moving along unbounded paths on the equatorial plane.
The geodesics for different values of the quadrupole parameter are given in Figs. \ref{figі3}-\ref{fig4c},
where the radial coordinate is dimensionless ($r/m$).
We consider first unbounded Schwarzschild trajectories with non-zero initial radial velocities under the influence of the quadrupole.
In this case, we see that for the chosen initial angular and radial velocities all the particles escape from the gravitational field of central slightly deformed body. This is illustrated Fig.\ref{figі3}.
The direction along which the particle escapes to infinity depends on the value of the quadrupole. It is worth noticing that, in principle, this effect could be used to measure the quadrupole of the central mass distribution.
\begin{figure}%
\includegraphics[scale=0.19]{fig_unboun2.eps}%
\quad
\includegraphics[scale=0.19]{fig_unboun1.eps}%
\quad
\includegraphics[scale=0.19]{fig_unboun3.eps}%
\caption{Influence of the quadrupole on unbounded orbits.
Initial conditions:
$ \varphi(0) = \pi/2, \ r(0) =7, \ \dot{\varphi}(0)= 0.07145$, $\dot r(0)=0 $ (top, left);
$ \varphi(0) = 0, \ r(0) =20, \ \dot{\varphi}(0)= 0.04305$, $\dot r(0)=-2.5 $
(top, right);
$ \varphi(0)=0 $, $r(0)=20$, $\dot{\varphi}(0)=0.04305$, $\dot{r}(0)=-2.489$ (bottom) ;} \label{figі3}
\end{figure}
In Fig. \ref{fig4c}, we consider a Schwarzschild bounded orbit
with zero initial radial velocity and the same non-zero
value for the initial angular velocity.
The left panel shows the Schwarzschild geodesic. The central and right panels illustrate the influence of a negative and positive small quadrupole, respectively.
We conclude that that the small quadrupole does not affect the bounded character of the geodesic, but it does drastically modify the morphology of the trajectories.
\begin{figure}%
\includegraphics[scale=0.19]{fig_boun1.eps}%
\quad
\includegraphics[scale=0.19]{fig_boun2.eps}%
\quad
\includegraphics[scale=0.19]{fig_boun3.eps}%
\caption{Influence of the quadrupole on Schwarzschild bounded orbits with vanishing initial radial velocity ($\dot r(0)=0)$.
The initial conditions are $\varphi(0)=0$, $r(0)=7$, and $\dot{\varphi}(0)=0.08$ for all the trajectories. }
\label{fig4c}%
\end{figure}
We conclude that the quadrupole always affects the Schwarzschild trajectories. The explicit modifications depend on the properties of the original Schwarzschild trajectory and the value of the quadrupole parameter.
\subsection{Radial geodesics and repulsive gravity}
\label{sec:rad}
We now study the free fall of test particles. To this end, we consider the geodesics equations in their most general form as given in Eqs.(\ref{eq_time})-(\ref{eq_phi}). The important point is that in all the cases to be considered all the initial spatial velocities are assumed to vanish. The starting point can be chosen arbitrarily, but some special values of the angle $\theta$ are of interest, namely, the symmetry axis $\theta=0$, the equatorial plane $\theta=\pi/2$, and some other different value that we can choose as $\theta=\pi/4$. In fact, in \cite{Boshkayev2016}, it was shown that by analyzing the behavior of radial geodesics one can detect the presence of repulsive gravity \cite{defelice89,lq12,lq14}
A free falling particle will continue its motion along the radial direction, unless a force acts on it and changes the original radial direction. This phenomenon has been reported in the case of an exact solution with quadrupole moment in \cite{Boshkayev2016}.
We will now consider the same situation in the case of the approximate solution we are analyzing in this work. The result of the integration of the geodesic equations
(\ref{eq_time})-(\ref{eq_phi}) is illustrated in Fig. \ref{fig_radial_3d}. We see that, in fact, free falling particles move along the original directions ($\theta =0$, $\theta=\pi/4$, and $\theta=\pi/2$), independently of the value of quadrupole parameter $q$. This means that no repulsive gravity effects can be detected in the case of the approximate metric. Taking into account also the result of \cite{Boshkayev2016}, we conclude that repulsive effects are non-linear, i.e., they appear only in the case of an exact quadrupolar metric.
\begin{figure}%
\includegraphics[width=3.7cm, height=4cm]{fig_3d_0.eps
\quad
\includegraphics[width=3.7cm, height=4cm]{fig_3d_4pi.eps
\quad
\includegraphics[width=3.7cm, height=4cm]{fig_3d_pi2.eps}%
\caption{Free fall of test particles with vanishing initial velocities. On the axes $\theta = 0$ (top, left), $\theta = \pi/4$ (top, right), and $\theta = \pi/2$ (bottom). The sphere represents the singularity at the radius $r=m$ (top).
The trajectories of the free falling test particles do not depend on the value of the parameter $q$.
For concreteness, we set the energy of the test particle as $E=1$.
}
\label{fig_radial_3d}%
\end{figure}
\section{\label{sec:con}Conclusions and remarks}
In this work, we have derived a family of approximate vacuum solutions of Einstein equations with quadrupole. Among all the solutions contained in this family we choose a particular one, which presents a naked singularity at the hypersurface $r=m$, instead of $r=2m$ as in other quadrupolar metrics. To our knowledge this is the only metric with such a singularity.
By applying an appropriate coordinate transformation, we found the Newtonian limit of the approximate solution and showed that it corresponds to the gravitational potential of a mass with quadrupole.
Then, we investigated the motion of test particles along circular orbits. We established that a positive quadrupole leads to an increase of the minimum allowed radius whereas a negative quadrupole generates the opposite effect. This means that only in the case of an oblate object, test particles are allowed to exist on orbits closer to the singularity, which is situated at $r=m$. In the case of bounded and unbounded trajectories, we found that
the quadrupole always affects the Schwarzschild trajectories. The explicit modifications depend on the properties of the original Schwarzschild trajectory and the value of the quadrupole parameter.
Finally, we analyzed radial geodesics that have been used previously to detect effects of repulsive gravity in an exact quadrupolar metric. However, in the case of the approximate solution no repulsive effects were found. We conclude that repulsive gravity in the the quadrupolar naked singularities is a non-linear phenomenon.
\section*{Acknowledgments}
This work was partially supported by Ministry of Education and Science (MES) of the Republic of Kazakhstan (RK), Grant No. BR10965191, and by UNAM-DGAPA-PAPIIT, Grant No. 114520,
Conacyt-Mexico, Grant No. A1-S-31269.
S.T. acknowledges the support through the postdoctoral fellowship program of Al-Farabi Kazakh National University.
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\section{Introduction}
The nuclear fission phenomenon, discovered in 1938, is still very attractive to experimentalists and theoreticians. Accurate reproduction of such observables as the mass and the total kinetic energy yields of fission fragments or the multiplicities of emitted neutrons is a good test of any modern theoretical model. A review of the existing fission models can be found, e.g., in Refs. \cite{MJV19,RMo13,SJA16,BBB20,MSc17,ACD21}. Readers interested in the theory of nuclear fission can find more details in the textbook \cite{KPo12}.
The present research is a continuation of our previous works \cite{PIN17,PDH20,PBK21,LCW21,KDN21} in which the fission fragment mass-yields of nuclei from different mass regions were obtained using the Fourier shape parametrization \cite{SPN17}. In the present paper, we extend our 3D Langevin model by adding a mode responsible for the charge equilibration of the fragments and a Master type equation describing the multiplicity of emitted neutrons. In addition, an innovatory, better adopted to make the fission calculation on a grid, the Fourier-over-Spheroid (FoS) shape parametrization (see also \cite{PNB17}) is used. One has to stress here that the nuclear shapes generated by FoS are equivalent to those given by the Fourier parameterization \cite{SPN17}. The main features of our model are described in Section 2, while the neutron emission from the fragments is discussed in Section 3, followed by the Summary.
\section{Nuclear shape parametrization}
The potential energy surfaces of fissioning nuclei are obtained using the macro-micro model. The macroscopic part of the energy is evaluated according to the Lublin-Strasbourg-Drop (LSD) formula \cite{PDu09}, while the microscopic energy corrections are calculated using the Yukawa-folded single-particle potential \cite{DPB16}.
The surface of the fissioning nucleus is described in the cylindrical coordinates $(\rho,\varphi,z)$ by the following formula:
\begin{equation}
\rho^2(z,\varphi)=\frac{R_0^2}{c}\,f\left(\frac{z-z_{\rm sh}}{z_0}\right)
{1-\eta^2\over 1+\eta^2+2\eta\cos(2\varphi)} ~,
\label{rhos}
\end{equation}
where $\rho(z,\varphi)$ is the distance from the $z$-axis to the surface.
Function $f(u)$ defines the shape of the nucleus having half-length $c=1$:
\begin{equation}
\begin{array}{rl}
f(u)&=1-u^2-\left({a_4\over 3}-{a_6\over 5}+\dots\right)
\cos\left({\pi\over 2}u\right)-a_3\sin(\pi\,u) \\[2ex]
&-a_4\cos\left({3\pi\over 2}u\right)
-a_5\sin(2\pi\,u)-a_6\cos\left({5\pi\over 2}u\right)-\dots~,
\end{array}
\label{fos}
\end{equation}
where $-1\leq u \leq 1$. The first two terms in $f(u)$ describe a sphere, the third ensures volume conservation for arbitrary deformation parameters $\{a_3,\;a_4,\;\dots\}$. The parameter $c$ determines the elongation of the nucleus keeping its volume fixed, while $a_3$ and $a_4$ describe the reflectional asymmetry and the neck size, respectively. The half-length is $z_0=cR_0$, where $R_0$ is the radius of a sphere with the same volume. The $z$-coordinate varies in the range $-z_0+z_{\rm sh}\leq z\leq z_0+ z_{\rm sh}$. The shift $z_{\rm sh} = -3/(4\pi) z_0 (a_3-a_5/2+\dots)$ places the nuclear center of mass at the origin of the coordinate system. The parameter $\eta$ describes a possible elliptical, non-axial deformation of a nucleus.
The formula (\ref{rhos}) is completely equivalent to those based on the Fourier expansion and described in Refs.~\cite{SPN17}. Here, the deviation from a sphere with radius $\rho=1$ is firstly expanded in the Fourier series, and subsequently this deformed object of the length $2R_0$ is scaled to the elongation equal to $2cR_0$. The formula (\ref{rhos}) is more adapted to calculation of the PES made on a mesh in the multidimensional deformation parameter ($c,a_3,a_4,...,a_n$) space, since the range of variability of the $a_k$ coefficients does not depend on the elongation $c$. In addition, the mass ratio of the fragments, their relative distance, and the radius of the neck between them, measured in $z_0$ units, do not depend on the elongation of the nucleus. It is also worth noticing that for the reflection symmetric shapes, the geometrical scission points appear when $a_4=a_4^{\rm sc}={3\over 4}+{6\over 5}a_6\dots$ independently of the elongation $c$. Such properties of the present FoS shape parametrization make it very useful for all kinds of calculations related to nuclear fission.
A typical PES of a fissioning nucleus is shown in Fig.~\ref{Fig1}. It is a projection of the 4D PES onto the $(c,a_4)$ plane, i.e., each energy point in the $(c,a_4)$ map of $^{236}$U is minimized with respect the non-axial $\eta$ and reflectional $a_3$ deformation parameters. The ground state (g.s.), first (A), and second (B) saddle points, as well as the exit points from the fission barrier leading to the asymmetric (C) and symmetric (D) fission valleys, are marked. The upper value of the neck-parameter $a_4=0.72$ corresponds to the neck radius approximately equal to the nucleon radius. The non-axial degree of freedom is important at a smaller elongation of the nucleus until the neighborhood of the second saddle. At larger deformation, their effect is negligible, which allows us to restrict the Langevin calculations to 3D when discussing the dynamics of fission. Moreover, the role of the higher-order deformation parameters $a_5$ and $a_6$ is rather small even in the region of well-separated fission fragments as it was shown in Ref.~\cite{KDN21}.\\[-5ex]
\begin{figure}[htb]
\includegraphics[width=12.5cm]{U236e24.pdf}\\[-4ex]
\caption{Potential energy surface of $^{236}$U on the ($c,\,a_4$) plane. Each point is minimized with respect to the non-axial ($\eta$) and the reflectional ($a_3$) deformations.}
\label{Fig1}
\end{figure}
\subsection{Mass and TKE yields obtained within 3D Langevin calculation}
The Langevin equation governs the dissipative fission dynamics. In the generalized coordinates ($\{q_i\},~~i=1,2,...,n$) it has the following
form \cite{KPo12}:
\begin{equation}
\begin{array}{l}
{dq_i\over dt} = \sum\limits_{j} \; [{\cal M}^{-1}(\vec q\,)]_{i\, j} \; p_j \\
{dp_i\over dt} = - {1\over 2} \sum\limits_{j,k} \,
{\partial[{\cal M}^{-1}]_{jk}\over\partial q_i}\; p_j \; p_k
-{\partial V(\vec q)\over\partial q_i}
- \sum\limits_{j,k} \gamma_{ij}(\vec q) \;
[{\cal M}^{-1}]_{jk} \; p_k + F_i(t) \,\,,
\end{array}
\label{LGV}
\end{equation}
Here $V(\vec q\,)=E_{\rm pot}(\vec q\,)-a(\vec q\,)T^2$ is the free-energy of
fissioning nucleus having temperature $T$ and the single-particle level density
$a(\vec q\,)$. In the present calculation the inertia ${\cal M}_{jk}$ and the friction $\gamma_{ij}$ tensors are evaluated in the irrotational flow and the wall approximation, respectively, as described in Refs.~\cite{BNP19,KDN21}.
\begin{figure}[htb]
\includegraphics[width=0.5\textwidth]{u235t-m.pdf}
\includegraphics[width=0.5\textwidth]{u235t-m-r.pdf}\\[-5ex]
\caption{Fission fragment mass-yield of $^{235}$U + n$_{\rm th}$ as a function
of the mass of the fragment. The l.h.s. figure shows the yields obtained assuming the constant (solid line) and the proportional to the surface of the nucleus (dashed line) pairing strength. Similar results obtained assuming two different neck radii (radius of nucleon or alpha-particle) at which the fission occurs are shown in the r.h.s. panel. The experimental data are taken from Ref. \cite{DKT69}.}
\label{Fig2}
\end{figure}
The vector $\vec F(t)$ stands for the random Langevin force, which couples the collective dynamics to the intrinsic degrees of freedom and is defined as:
\begin{equation}
F_i(t) \!\!=\!\! \sum_{j} g_{ij}(\vec q\,) \; G_j(t) \,\,,
\label{rforce}
\end{equation}
where $\vec G(t)$ is a stochastic function whose strength
$g(\vec q\,)$ is given by the diffusion tensor ${\cal D}(\vec q\,)$
defined by the generalized Einstein relation:
\begin{equation}
{\cal D}_{ij} \!\!=\!\!T^*\gamma_{ij} \!\!=\!\! \sum_{k} g_{ik} \; g_{jk}~,~~~
{\rm where}~~~T^*=E_0/{\rm tanh}\left({E_0\over T}\right)~.
\label{Eirel}
\end{equation}
Here $E_0$= 1 MeV is the zero-point collective energy. The temperature $T$ is obtained from the thermal excitation energy ($E^*$) equal to the difference between the initial ($E_{\rm init}$ ) and the total collective energy, beeing the sum of the kinetic ($E_{\rm kin}$) and potential ($V$) energies of the fissioning nucleus at a given deformation point ($\vec q$):\\[-2ex]
\begin{equation}
a(\vec q\,)T^2=E^*(\vec q\,)=E_{\rm init}-(E_{\rm kin}+V)~.
\label{temp}
\end{equation}
Running thousands of random Langevin trajectories, which end at the scission configuration, one can estimate the distribution of the mass and total kinetic energies of the fragments. An example of our estimates of the fission fragment mass yields obtained in the thermal neutron-induced fission of $^{235}$U is shown in Fig.~\ref{Fig2}.\\[-4ex]
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.5\textwidth]{u235t-tke.pdf}\\[-3ex]
\caption{Total kinetic energy of fission fragments of $^{235}$U + n$_{\rm th}$ as a function of the fragment mass. The solid and the dashed lines correspond to the constant and the proportional to the surface of nucleus pairing strength, respectively. The experimental data (+) are taken from Ref. \cite{ATJ20}.}
\end{center}
\label{Fig3}
\end{figure}
The fission fragments' total kinetic energy (TKE), shown in Fig. 3, is approximately given by their Coulomb repulsion energy. This energy is equal to the difference between total Coulomb energy of the nucleus at the scission configuration and Coulomb energies of both deformed fragments:
\begin{equation}
{\rm TKE}={3e^2\over 5r_0}\left[{Z^2\over A^{1/3}}B_{\rm Coul}(\vec q_{\rm sc})
-{Z_{\rm h}^2\over A_{\rm h}^{1/3}}\,B_{\rm Coul}(\vec q_{\rm h})
-{Z_{\rm l}^2\over A_{\rm l}^{1/3}}\,B_{\rm Coul}(\vec q_{\rm l}) \right]~.
\label{Ekin}
\end{equation}
It is undoubtedly a more accurate estimate of the fission-fragment kinetic energy than the frequently used its point-charge approximation: $E_{\rm kin}=e^2 Z_{\rm h}Z_{\rm l}/R_{12}$, where $R_{12}$ is the distance between the fragment mass-centers.
\subsection{On the charge equilibration at scission}
Knowing the fragment deformation at scission, it is possible to find a
preferred charge for each fragment. In the majority of Langevin-type calculations, one assumes that the ration $N/Z$ of the fragments is the same as the one of the fissioning nucleus. Looking at the proton and neutron microscopic density distributions, one could obtain better estimates, which is rather hard to do. Below, we propose a simple model for the proton-neutron equilibrium at scission based on the LD and the pairing correlation energy. Such charge equilibration can be determined by looking at the change of the total energy of the fissioning system with the charge number of the heavy fragment $Z_{\rm h}$:
\begin{equation}
\begin{array}{rl}
E(Z,A,Z_{\rm h};A_{\rm h},\vec q_{\rm h},\vec q_{\rm l})
&=E_{\rm LD}(Z_{\rm h},A_{\rm h};\vec q_{\rm h}
+\,E_{\rm LD}(Z-Z_{\rm h},A-A_{\rm h});\vec q_{\rm l})\nonumber\\[+1ex]
&+\,e^2Z_{\rm h}(Z-Z_{\rm h})/R_{12}-E_{\rm LD}(Z,A;0)~,
\end{array}
\label{echeq}
\end{equation}
where $Z, A$, and $Z_{\rm h}, A_{\rm h}$ are the charge and mass numbers of the mother nucleus and the heavy fragment, respectively. The mass as well as the deformation parameters of the heavy ($A_{\rm h},\,\vec q_{\rm h}$) and the light fragments ($A_{\rm l},\,\vec q_{\rm l}$) are fixed by the shape of the nucleus at scission corresponding to the end of each Langevin trajectory.
The total energy as a function of the fragment charge number is shown in the l.h.s. of Fig.~\ref{Fig4}.
\begin{figure}
\includegraphics[width=0.5\textwidth]{qeqlsd.pdf}\hfill
\includegraphics[width=0.5\textwidth]{qeqplsd.pdf}\\[-4ex]
\caption{Energy of $^{240}$Pu at scission as a function of the heavy fragment charge number in the LSD mass formula \cite{PDu09} (l.h.s.) and the Wigner distribution probability of the fragment charge number (r.h.s.).}
\label{Fig4}
\end{figure}
The distribution of the heavy-fragment charge number can be estimated using a Wigner function corresponding to the energy $E$ given by Eq.~\ref{echeq} for different
values of $Z_{\rm h}$:
\begin{equation}
W(Z_{\rm h})=\exp\{-[E(Z_{\rm h})-E_{\rm min}]^2/E_{\rm W}^2]~,
\label{Wigner}
\end{equation}
which gives the distribution probability of the fragment charge shown in
the r.h.s. of Fig.~\ref{Fig4}. $E_{\rm min}$ in Eq.~\ref{Wigner} is the lowest discrete energy as a function of $Z_{\rm h}$. Furthermore, the following random number decides about the charge number $Z_{\rm h}$ of the heavy fragment, with $Z_{\rm l}=Z-Z_{\rm h}$. The energy $E_{\rm W}$ should be comparable with the energy distance $\hbar\omega_0$ between harmonic oscillator shells since we have here a single-particle (proton) transfer between the touching fragments.
\begin{figure}[htb]
\includegraphics[width=0.5\textwidth]{u235tz-m.pdf}
\includegraphics[width=0.5\textwidth]{chargeeq.jpg}\\[-4ex]
\caption{The fission fragment yield of $^{235}$U + n$_{\rm th}$ (l.h.s.) as a function of the fragment charge. The solid and the dashed lines correspond respectively to the constant and the proportional to the surface of nucleus pairing strength. The experimental data (+) are taken from Ref. \cite{MSc17}. The correlation between the fragment mass and charge is displayd in the r.h.s. figure.}
\label{Fig5}
\end{figure}
The above charge equilibration effect has to be considered at the end of each Langevin trajectory when one fixes the integer fragment mass and charge numbers of the fission fragments. The resulting fission fragment yield is compared with the data \cite{MSc17} in the l.h.s. part of Fig.~\ref{Fig5}, while the correlation between fragment mass and charge is displayed in the in the r.h.s. figure.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.6\textwidth]{u235t-nm.pdf}\\[-8ex]
\end{center}
\caption{Multiplicity of neutrons emitted by the fission fragments of $^{235}$U + n$_{\rm th}$ as a function of the mass of the fragment. The solid and the dashed lines correspond respectively to the constant and the proportional to the surface of nucleus pairing strength. The experimental data (*) are taken from Ref.~\cite{GHO18}.}
\label{Fig6}
\end{figure}
\section{Neutron emission from the fission fragments}
The maximal energy of a neutron emitted from a fragment (mother) can be
obtained from the energy conservation law:
\begin{equation}
\epsilon_{\rm n}^{\rm max}= M_{\rm M}+E_{\rm M}^*-M_{\rm D}-M_{\rm n}~,
\label{En}
\end{equation}
where $M_{\rm M},\,M_{\rm D},\,M_{\rm n}$ are the mass excesses of mother and daughter nuclei, and of the neutron, respectively. These data can
be taken from a mass table \cite{KWH21}.
The thermal excitation energy of the daughter nucleus is:
\begin{equation}
E_{\rm D}^*=\epsilon_{\rm n}^{\rm max}-\epsilon_{\rm n}~.
\label{Eexc}
\end{equation}
Here $e_{\rm n}$ is the kinetic energy of emited neutron.
One assumes that the thermal energy of a fragment $E^*_i$ in the scission point is proportional to its single-particle level density:
\begin{equation}
{E^*_{\rm l}\over E^*_{\rm h}}= {a(Z_{\rm l},A_{\rm l};{\rm def}_{\rm l})\over a(Z_{\rm h},A_{\rm h};{\rm def}_{\rm h})}~~~~
{\rm and}~~E^*=a(Z,A;{\rm def})\,T^2=E^*_{\rm l}+E^*_{\rm h}~.
\label{Eteq}
\end{equation}
The deformation energy of each fragment can be evaluated in the LD model:
\begin{equation}
E_{\rm def}^{(i)}\approx E_{\rm LD}(Z_i,A_i,{\rm def}_i)
- E_{\rm exp}(Z_i,A_i,{\rm g.s.})~.
\label{Edefi}
\end{equation}
The total excitation energy ($E^{(i)}_{\rm exc}$) of fragment $i$ is then the sum of its thermal and its deformation energy:
\begin{equation}
E_{\rm exc}^{(i)}=E_{\rm def}^{(i)}+E_i^*~.
\label{Eexci}
\end{equation}
This energy is converted into heat due to the presence of the friction
force, which allows to evaluate the effective temperature $T_i$ of each
fragment:
\begin{equation}
E_{\rm exc}^{(i)}=a(i)\,T_i^2~,
\label{Ti}
\end{equation}
where $a(i)$ is the single-particle level density of the $i$-th fragment.
These data enable us to estimate the number of neutrons emitted from each fragment.
The neutron emission probability is given by the Wei{\ss}kopf formula \cite{Del86}:
\begin{equation}
\Gamma_{\rm n}(\epsilon_{\rm n})={2\mu\over\pi^2\hbar^2\rho_{\rm M}
(E^*_{\rm M})}\int\limits_0^{\epsilon_{\rm n}}\sigma_{\rm inv}(\epsilon)\,
\epsilon\,\rho_{\rm D}(E^*_{\rm D})\,d\epsilon~.
\label{Wei}
\end{equation}
Here $\mu$ is the reduced mass of the neutron, $\sigma_{\rm inv}$ is the
neutron inverse cross-section \cite{DFF80}:
\begin{equation}
\sigma_{\rm inv}(\epsilon)=[0.76+1.93/A^{1/3} + (1.66/A^{2/3}-0.050)/\epsilon]\,\pi\,(1.70A^{1/3})^2~,
\label{ncross}
\end{equation}
while $\rho_{\rm M}$ and $\rho_{\rm D}$ are, respectively, the level densities of mother and daughter nuclei:
\begin{equation}
\rho(E)={\sqrt{\pi}\over 12a^{1/4}E^{5/4}}\exp(2\sqrt{aE})~,
\label{lden}
\end{equation}
Here the single-particle level density parameters $a$ of the mother and the daughter are taken here from Ref.~\cite{Ner02}.
Our estimates of the neutron multiplicity obtained for $^{235}$U + n$_{\rm th}$
is compared in Fig.~\ref{Fig6} with the experimental data \cite{GHO18}.
\section{Summary}
We have shown that our 3D model based on the Fourier-over-Spheroid shape parametrization describes reasonably the main features of the low-energy fission of atomic nuclei. Typical results for the thermal neutron induced fission of $^{235}$U shown above illustrate well the quality of our approach. The following conclusions can be drawn from our investigation:\\[-4ex]
\begin{itemize}
\item Fourier expansion of nuclear shape offers a very effective way
of describing the shapes of fissioning nuclei both in the vicinity of the
ground-state as well as in the scission point.\\[-4ex]
\item The potential energy surfaces are evaluated in the macro-micro model using
the LSD formula for the macroscopic part of the energy and the
Yukawa-folded single-particle to obtain the microscopic energy
correction.\\[-4ex]
\item It was shown that a 3D Langevin model, which couples the fission,
neck and mass asymmetry modes, describes the main features
of the fragment mass and kinetic energy yields.\\[-4ex]
\item The distribution of the multiplicity of neutrons emitted by the fragments
are reproduced well in our model.\\[-2ex]
\end{itemize}
Further calculations for wider mass and excitation energy ranges of fissioning nuclei are in progress.\\[3ex]
\parindent 0pt
{\bf Acknowledgments}\\[1ex]
We acknowledge discussions with F. A. Ivanyuk and C. Schmitt, and thank A. G\"o\"ok and A. Al-Adili for providing experimental data. This work was supported by the National Science Centre, project No. 2018/30/Q/ST2/00185.
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\section{Introduction}
\subsection{Computational questions on Kronecker coefficients}
The \emph{Kronecker coefficients} $g_{\lambda, \mu, \nu}$ are the structure constants in the decomposition of a tensor product of irreducible representations of the symmetric group into irreducible representations:
\begin{equation}
V_{\mu} \otimes V_{\nu} = \bigoplus\limits_{\lambda} g_{\lambda, \mu, \nu} V_{\lambda}.
\end{equation}
Consequently, they can be expressed using Schur functions
\begin{equation} \label{eq:schur}
s_{\lambda}[XY] = \sum_{\mu, \nu} g_{\lambda, \mu, \nu} s_{\mu}[X]s_{\nu}[Y],
\end{equation}
where $X := (x_1, \dots, x_m), Y := (y_1, \dots, y_n), XY := (x_1y_1, x_1y_2, \dots, x_my_n)$.
Here, the Schur functions are indexed by partitions $\lambda, \mu, \nu$ with at most $mn,m,n$ parts respectively. The number of parts of a partition $\alpha$ is its \emph{length} and is denoted here by $\ell(\alpha)$. When we focus on the set of partitions of length at most $k$, we may pad the partitions with $0$'s so that they are all sequences of length $k$.
Since their introduction in 1938 by Murnaghan, the Kronecker coefficients have proved to be among the most intriguing objects in algebraic combinatorics. After several decades of research, many open questions about the Kronecker coefficients remain. They are all non-negative integers, but have no known combinatorial interpretation. The Littlewood-Richardson $c_{\mu, \nu}^{\lambda}$ coefficients describe the decomposition of a tensor product of irreducible representations of the \emph{general linear group} into irreducible representations. This difference could appear small, yet the Littlewood-Richardson $c_{\mu, \nu}^{\lambda}$ coefficients have several combinatorial interpretations. One might view Kronecker coefficients as a generalization of the Littlewood-Richardson coefficients, hence the resistance to a clear interpretation is surprising, particularly in view of the publicity they have received\footnote{``One of the main problems in the combinatorial representation theory of the symmetric group is to obtain a combinatorial interpretation for the Kronecker coefficients." - Stanley \cite{Stan99}}.
The basic problem of computing the Kronecker coefficient $g_{\lambda, \mu, \nu}$ for general partitions $\lambda, \mu, \nu$ is $\#$P-hard (in the bitlength of the size of the partitions) and contained in GapP~\cite{BuIk08}\footnote{Problems in GapP can be expressed as the difference of two functions which are in $\#$P.}. Baldoni, Vergne and Walter distribute code~\cite{BaVeWa17} for use with Maple mathematical software to compute Kronecker coefficients for partitions $\lambda, \mu, \nu$ of bounded lengths ($\ell(\lambda), \ell(\mu), \ell(\nu) \leq 3$; and $\ell(\lambda) \leq 6, \ell(\mu) \leq 2, \ell(\nu) \leq 3$). There are at least two packages that handle partition sets without a bound on length, such as~\cite{ChDoWa12} and the SF Maple package of Stembridge~\cite{Stem05}. These two packages are prohibitably computationally expensive except for some small, or particular cases.
Even the problem of approximating the Kronecker coefficients is non-trivial and very few useful bounds are known. Pak and Panova~\cite[Corollary 3.4]{PaPa14} determine a bound for partitions $\lambda, \mu, \nu$ with $\ell(\lambda) \leq l, \ell(\mu) \leq m, \ell(\nu) \leq n$:
\begin{equation} \label{eq:pp-bound1}
g_{\lambda, \mu, \nu} \leq \prod_{i=1}^{l} \binom{\lambda_i - i + mn}{mn - i}.
\end{equation}
More recently, in~\cite{PaPa20}, they obtained the following bound in $N = |\lambda| = |\mu| = |\nu|$ via contingency tables
\begin{equation} \label{eq:pp-bound2}
g_{\lambda, \mu, \nu} \leq \left(1 + \frac{lmn}{N}\right)^N\left(1 + \frac{N}{lmn}\right)^{lmn}.
\end{equation}
Remark both of these bounds are polynomial in the length of the partitions - however, the degree is generally not optimal. For example, when $l = 4, m=2, n=2$, we find that if $\lambda = (\frac{N}{4}, \frac{N}{4}, \frac{N}{4}, \frac{N}{4})$, both bounds are of the order $O(N^{16})$ whereas the actual growth is (more precisely) $O(N^2)$~\cite{BrOrRo09}.
By $h(u)$, we denote the hook-length of the box~$u$ in the Ferrers diagram of $\lambda$. The hook length formula~\cite[Corollary 3.2]{PaPa20} also gives a bound:
\begin{equation}
g_{\lambda, \mu, \nu} \leq \min(f^{\lambda}, f^{\mu}, f^{\nu}),
\end{equation}
where $f^{\alpha} := \frac{k!}{\prod\limits_{u \in [\lambda]} h(u)}$ for a partition $\alpha$ of length $k$.
Some progress has been made to understand conditions on $\lambda, \mu, \nu$ for which $g_{\lambda, \mu, \nu}=0$. Denote by $k\alpha$ the partition obtained by multiplying each part of $\alpha$ by $k$. Littlewood-Richardson coefficients satisfy a \emph{saturation property}:
\[c_{\lambda, \mu}^{\nu} = 0 \iff c_{k\lambda, k\mu}^{k\nu} = 0.\]The Kronecker coefficients do not satisfy such a property universally:
\[g_{(1, 1), (1, 1), (1, 1)} = 0,\quad\text{ but }\quad g_{(2, 2),(2, 2),(2, 2)} = 1.\]
Even deciding ``$g_{\lambda, \mu, \nu} = 0$?" is NP-hard~\cite{IkMuWa17}. There are numerous vanishing conditions known- typically expressed as inequalities in the parts of $\lambda, \mu, \nu$ which guarantee that the coefficient $g_{\lambda, \mu, \nu}$ is zero. A classical result of Murnaghan and Littlewood (appearing for example in \cite{JaKe81}) is that for any non-zero Kronecker coefficient $g_{\lambda, \mu, \nu}$, it follows that $\overline{\lambda} \leq \overline{\mu} + \overline{\nu}$, where $\overline{\gamma}$ is the partition obtained by deleting the first part of partition $\gamma$. Consider the set of points constructed by concatenating partitions of fixed length. Those points that come from partitions giving a non-zero Kronecker coefficient have a nice geometry. Specifically,
\begin{equation}
Kron_{l,m,n} := \{(\lambda, \mu, \nu)\in \mathbb{Z}^{l+m+n} : g_{\lambda, \mu, \nu} \neq 0,\ \ell(\lambda) \leq l,\ \ell(\mu) \leq m, \ell(\nu) \leq n\}
\end{equation}
is a finitely generated semigroup in $\mathbb{Z}^{l+m+n}$ that generates a rational polyhedral cone. Following Manivel~\cite{Man15}, we call this cone the \emph{Kronecker polyhedron} and denote it $PKron_{l,m,n}$. In~\cite{Kl04} the cone $PKron_{l,m,n}$ is computed explicitly for small values of $l,m,n$, and it seems the number of inequalities increases rapidly. While this set is theoretically computable for any positive integers $l,m,n$, it is quickly computationally infeasible to do so. Another set of vanishing conditions valid for triples of partitions of any lengths were given recently by Ressayre in~\cite[Theorems 1 \& 2]{Re19}.
A classic result of Murnaghan states that for partitions $(\lambda, \mu, \nu)$ the sequence $\left(g_{\lambda + (k), \mu + (k) \nu + (k)}\right)_{k \geq 0}$ eventually stabilizes. Since then, many other partition triples $\alpha, \beta, \gamma$ with this property have been identified-- that is, the values of the sequence $\left(g_{\lambda + k\gamma, \mu + k\alpha, \nu + k\beta}\right)_{k \geq 0}$ stabilize for any choice of $\lambda, \mu, \nu$. Such triples $(\alpha, \beta, \gamma)$ are called \emph{stable triples}.
Stabilization phenomenon have been studied in \cite{BrOrRo11, Man15, Man15-2, Pell21, Stem14}.
Applications of Kronecker coefficients extend beyond the realm of algebraic combinatorics. The \emph{Geometric Complexity Theory} (GCT) program, developed by Mulmuley and Sohoni, with the goal of solving $P$ versus $NP$, relies heavily on the computation of Kronecker coefficients as one of its main ingredients (see \cite{BlMuSo15, IkMuWa17, Mu11}). More specifically, problems of positivity (as discussed in the Appendix of \cite{BrOrRo09-2} entitled \emph{Erratum to the saturation hypothesis (SH) in
``Geometric Complexity Theory VI''} and contributed by Mulmuley) related to the previously described saturation problems play an important role.
Kronecker coefficients appear in quantum computing where they encode the relationship between composite systems and their subsystems \cite{ChDoWa12, ChHaMi07, ChMi06}. As in the case of GCT, being able to determine the positivity of Kronecker coefficients is useful. In the context of quantum computing, non-zero Kronecker coefficients correspond to \emph{admissible spectral triples} which play an important role in the study of bipartite quantum states in quantum information theory \cite{Ch06}.
\subsection{Kronecker coefficients and vector partition functions} \label{sec:KCandVP}
Here, we address many of these fundamental questions on Kronecker coefficients using a detailed analysis of Eq.~\eqref{eq:schur}. The first step is to deduce an expression for $g_{\lambda, \mu, \nu}$ using coefficient extraction of multivariate Taylor series of rational functions. This formulation allows us to represent Kronecker coefficients as a signed sum of \emph{vector partition function} evaluations. Let~$A$ be a~$d \times n$ matrix~$A$ with integer entries satisfying \[\operatorname{ker}(A) \cap \mathbb{R}^n_{\geq 0} = \{\BF{0}\}.\] Denote the columns of $A$ by $\BF{a}_1, \dots, \BF{a}_n$. The \emph{vector partition function} $p_A : \mathbb{Z}_{\geq 0}^m \to \mathbb{Z}_{\geq 0}$ is the counting function
\[
p_A(\BF{b}) := \# \{\BF{x} \in \mathbb{N}^d : A\BF{x} = \BF{b} \}.
\]
Geometrically, $p_A(\BF{b})$ is the discrete volume of the polyhedron defined by the solutions of $A\BF{x}=\BF{b}$ and the inequalities $x_i \geq 0$ for $i = 1,\dots,d$. The generating function formulation is in terms of the coefficient of the term $\BF{x}^{\BF{b}}$ in the Taylor series expansion of a product of geometric series:
\begin{equation}
p_A(\BF{b}) = [\BF{x}^{\BF{b}}] \prod\limits_{j=1}^{n} \frac{1}{{1-\BF{x}}^{\BF{a}_j}},
\end{equation}
with the convention that for vectors $\BF{u},\BF{v} \in \mathbb{Z}^d$, $\BF{u}^{\BF{v}}$ denotes the product $\prod\limits_{i=1}^{d} u_i^{v_i}$. The rational function $\prod\limits_{j=1}^{n} \frac{1}{{1-\BF{x}}^{\BF{a}_j}}$ is called the \emph{vector partition generating function}.
Sturmfels~\cite{Stur94} determined that the vector partition function is a piecewise quasi-polynomial whose domains of quasi-polynomiality are convex polyhedral cones called \emph{chambers} of a fan called the \emph{chamber complex} of $A$ (defined by Zelevinsky, Alekseevskaya, and Gelfand \cite{AlGeZe87}).
Barvinok gave an algorithm which allows one to compute $p_A(\BF{b})$ in polynomial-time for fixed dimension $n$ of the the polytope $A\BF{x} = \BF{b}, \BF{x} \geq \BF{0}$ \cite{Barv94} . An adapted version of this algorithm (the Barvinok-Woods algorithm) has been implemented in C (the implementation is named \emph{Barvinok}) and is publicly available \cite{KoVeWo08}.
We reformulate the expression for~$g_{\lambda, \mu, \nu}$ given in~\cite[Theorem 26]{MiRoSu21} as Theorem~\ref{theo:vpf-to-kron} below. The main ingredients in this approach are:
\begin{enumerate}
\item a matrix $A^{m,n}$ and its vector partition function $p_{A^{m,n}}$;
\item vectors $\alpha, \beta$;
\item linear functions $r_s, r_t$;
\item linear functions $l_s(\cdot; \sigma), l_t(\cdot; \sigma)$ defined for each $\sigma \in \mathfrak{S}_{mn}$.
\end{enumerate}
The quantities $\alpha, \beta$, and the linear functions $r_s, r_t, l_s, l_t$ (which all depend on $m,n$) are explicitly given in the discussion after Theorem \ref{theo:vpf-to-kron} and defined (implicitly) in \cite{MiRoSu21}. The matrices $A^{m,n}$ are given implicitly in~\cite{MiRoSu21}; we give explicitly only the cases $m=2$, $n =3,4$ used in our work. For given $m,n$ and $\sigma$, we call the function in the parts of $\lambda, \mu, \nu$, $\mathbf{b}^{m,n}(\lambda, \mu, \nu; \sigma) := (r_s(\mu, \nu) + \alpha - l_s(\lambda; \sigma),\ r_t(\mu, \nu) + \beta - l_t(\lambda; \sigma))$ the \emph{vector partition function input of $\sigma$}. Additionally, we refer to the quantity
\begin{equation}
\operatorname{sgn}(\sigma)\ p_{A^{m,n}}\biggr(\mathbf{b}^{m,n}(\lambda, \mu, \nu; \sigma)\biggr)
\end{equation}
as the \emph{contribution of the alternant term associated to $\sigma$}.
In general it will be clear to which $m,n$ we refer, but we explicitly state this when needed.
\begin{theo} \label{theo:vpf-to-kron}
Let~$m,n$ be positive integers. Then for any partitions $\lambda, \mu, \nu$ with $\ell(\lambda) \leq mn,\ \ell(\mu) \leq m,\ \ell(\nu) \leq n$, we have
\begin{equation} \label{eq:vpf-to-kron}
g_{\lambda, \mu, \nu} = \sum_{\sigma \in \mathfrak{S}_{mn}} \operatorname{sgn}(\sigma)\ p_{A^{m,n}}\biggr(\mathbf{b}^{m,n}(\lambda, \mu, \nu; \sigma)\biggr).
\end{equation}
\end{theo}
The following expressions are valid for all integers $u,v$ with $1 \leq u \leq m-1$ and $1 \leq v \leq n-2$. The components of the vectors $\alpha, \beta$ are:
\begin{align*}
\alpha_0 &= \frac{1}{2} \, {\left(n m + n - m - 2\right)} {\left(n - 1\right)} {\left(m - 1\right)} \\
\alpha_u &= \frac{1}{2} \, {\left(u^{2} n - 2 \, u n m + 2 \, n m^{2} - u^{2} + u - n - 2 \, m + 2\right)} {\left(n - 1\right)}\\
\beta_v &= \frac{1}{12} \, {\left(8 \, n^{2} m^{2} - 6 \, v n m + 5 \, n^{2} m - 10 \, n m^{2} + 6 \, v^{2} - 12 \, v n + 6 \, v m - 19 \, n m + 2 \, m^{2} + 18 \, v + 14 \, m\right)} {\left(m - 1\right)}.
\end{align*}
The components of the vectors $r_s(\mu, \nu)$ and $r_t(\mu, \nu)$ are:
\begin{align*}
r_s(\mu, \nu)_0 &= |\nu| - \nu_1 + \binom{n-1}{2} \\
r_s(\mu, \nu)_u &= \sum_{i=u+1}^{m} \mu_{i} + |\nu| - \nu_1 + \binom{m-u}{2} + \binom{n-1}{2} \\
r_t(\mu, \nu)_v &= \sum_{i=2}^{m} (i-1)\mu_{i} + (m-1)\sum_{j=2}^{v+1} \nu_{j} + m\sum_{j=v+2}^{n} \nu_{j} + \binom{m}{3} + (m-1)\binom{n-1}{2} + \binom{n-v-1}{2}.
\end{align*}
The components of the vectors $l_s(\lambda; \sigma)$ and $l_t(\lambda; \sigma)$ are:
\begin{align*}
l_s(\lambda; \sigma)_0 &= \sum_{i=m+1}^{mn} \biggr(\lambda_{\sigma(i)} + \delta_{\sigma(i)}\biggr) \\
l_s(\lambda; \sigma)_u &= \sum_{i=u+1}^{m+u(n-1)} \biggr(\lambda_{\sigma(i)} + \delta_{\sigma(i)}\biggr) + 2\sum_{i=m+u(n-1) + 1}^{mn} \biggr( \lambda_{\sigma(i)} + \delta_{\sigma(i)}\biggr) \\
l_t(\lambda; \sigma)_v &= \sum_{i=2}^{m} (i-1)\biggr(\lambda_{\sigma(i)} + \delta_{\sigma(i)}\biggr) + (m-1)\sum_{i=m+1}^{m+v} \biggr( \lambda_{\sigma(i)} + \delta_{\sigma(i)} \biggr) + m\sum_{i=m+v+1}^{m+n-1} \biggr( \lambda_{\sigma(i)} + \delta_{\sigma(i)}\biggr)\\
&+ \sum_{i=1}^{m-1}\sum_{j=1}^{v} (i+m-1) \biggr( \lambda_{\sigma(m+i(n-1)+j)} + \delta_{\sigma(m+i(n-1)+j)} \biggr) + \sum_{i=1}^{m-1}\sum_{j=v+1}^{n-1} (i+m)\lambda_{m+i(n-1) + j}.
\end{align*}
The identity permutation in $\mathfrak{S}_{mn}$ is denoted by $Id$. It is convenient to have an explicit derivation in the case when $\sigma = Id$:
\begin{align*}
l_s(\lambda; Id)_0 &= \sum_{i=m+1}^{mn} \lambda_i + \frac{1}{2} \, {\left(n m - m - 1\right)} {\left(n - 1\right)} m \\
l_s(\lambda; Id)_u &= \sum_{i=u+1}^{m+u(n-1)}\lambda_i + 2\sum_{i=m+u(n-1) + 1}^{mn} \lambda_i + \frac{1}{2}(m-u)\biggr(2mn - u - m - 1\biggr) \\
&+ (n-1)^2m - \binom{n}{2} + \frac{1}{2}(n-1)(m-1)\biggr(n(m-1) - m\biggr)\\
&+ \frac{1}{2}(n-1)(m-u)\biggr(mn - u(n-1) - m - 1\biggr) \\
l_t(\lambda; Id)_v &= \sum_{i=2}^{m} (i-1)\lambda_{i} + (m-1)\sum_{i=m+1}^{m+v} \lambda_{i} + m\sum_{i=m+v+1}^{m+n-1} \lambda_{i}\\
&+ \sum_{i=1}^{m-1}\sum_{j=1}^{v} (i+m-1)\lambda_{m+i(n-1)+j} + \sum_{i=1}^{m-1}\sum_{j=v+1}^{n-1} (i+m)\lambda_{m+i(n-1) + j} \\
&+ \frac{m}{12} \left( 8\,{m}^{2}-3\,m+1 \right) {n}^{2} -m\left( m+1 \right)
\left( 10\,m+6\,v-1 \right) n \\
&+2\,m\left( 3\,{v}^{2}+3\,vm+2\,{m}^{2}+6\,v+3\,m+1 \right).
\end{align*}
Notably, for all $0 \leq u\leq m-1$, the constant term (with respect to $\lambda_1, \dots, \lambda_{mn},\ \mu_1, \dots, \mu_m,\ \nu_1, \dots, \nu_n$) of the $i$\textsuperscript{th} coordinate of $r_s(\mu, \nu) + \alpha - l_s(\lambda; Id)$ is $0$, and for all $1 \leq v \leq n-2$, the constant term of the $j$\textsuperscript{th} coordinate of $r_t(\mu, \nu) + \beta - l_t(\lambda; Id)$ is also $0$. In other words both $r_s(\mu, \nu) + \alpha - l_s(\lambda; Id)$ and $r_t(\mu, \nu) + \beta - l_t(\lambda; Id)$ are linear forms whose variables are the parts of $\lambda, \mu, \nu$ (and thus so is the vector partition function input $\BF{b}^{m,n}(\lambda, \mu, \nu; Id)$).
The expression in Eq.~\eqref{eq:vpf-to-kron} writes the Kronecker coefficient $g_{\lambda, \mu, \nu}$ as a signed sum of permutations. The single term associated with the identity permutation is the largest, and can be used to derive properties about the Kronecker coefficient. Specifically, for partitions $\lambda, \mu, \nu$ with $\ell(\mu) \leq m,\ \ell(\nu) \leq n\ \ell(\lambda) \leq mn$, the \emph{atomic Kronecker coefficient} $\tilde{g}^{m,n}_{\lambda, \mu, \nu}$ is the coefficient obtained by taking only the contribution of the alternant term corresponding to the identity permutation in Eq.~\eqref{eq:vpf-to-kron} - that is,
\begin{equation}\label{eq:atomic}
\tilde{g}^{m,n}_{\lambda, \mu, \nu} := p_{A^{m,n}}\biggr(\mathbf{b}^{m,n}(\lambda, \mu, \nu; Id)\biggr).
\end{equation}
Atomic Kronecker coefficients were introduced in~\cite{MiRoSu21}, where it was proven that in the $m=n=2$ case they provide an upper bound for the Kronecker coefficients. These authors also conjecture that they provide an upper bound in general~\cite{MiRoSu18}. This seems to be justified in each computation we have made (in the $m=2, n=3,4$ cases). Interestingly, the atomic Kronecker coefficients depend on the values $m,n$ and not just the indexing partitions. As an example (given also in \cite{MiRoSu21}), consider $\lambda = (12, 7, 4, 1), \mu = (12, 12), \nu = (12, 12)$. If we set, $m=n=2$, the atomic Kronecker coefficient $\tilde{g}^{2,2}_{\lambda, \mu, \nu}$ is $32$ - however, by by padding $\lambda$ and $\nu$ with zeroes (i.e. representing $\lambda, \nu$ as $\lambda = (12, 7, 4, 1, 0, 0), \nu = (12, 12, 0)$), we find that the atomic Kronecker coefficient $\tilde{g}^{2,3}_{\lambda, \mu, \nu}$ in this case is $8793$. The atomic Kronecker coefficients are expressed using a single partition function, which is polynomial time computable for a fixed dimension. However, the dimension grows very quickly as a function of $m,n$.
\subsection{Summary of contribution}
In this work we apply Theorem \ref{theo:vpf-to-kron} to study some of the main questions of Kronecker coefficients: exact computation, vanishing conditions, stability, and upper bounds. In \cite{MiRoSu21} the authors focused on the $m=n=2$ case; we adapt the main ideas of that article to general $m,n$. Section \ref{sec:VPF} describes the pertinent aspects of their work to this article.
Once an expression of the vector partition function $p_{A^{m,n}}$ as a piecewise quasi-polynomial has been computed, the complexity of using this form to determine the Kronecker coefficient comes from the large number $(mn)!$ of terms in the sum. Significantly fewer than the $(mn)!$ terms are needed (either due to vanishing or cancellation): when $m=n=2$ only $7$ of the $24$ terms are needed, and when $m=2, n=3$ at most $482$ are needed. However, we do not know how many terms are needed in general for a given $m,n$.
Using this to compute $g_{\lambda, \mu, \nu}$ is efficient for small $m$ and $n$, and we have developed a \emph{Sagemath} tool for computing Kronecker coefficients $g_{\lambda, \mu, \nu}$ for $l \leq 8, m \leq 2, n \leq 4$. The exact formulas are given in Section \ref{sec:explicit-computation}. This section is split into two subsections - in \ref{sec:2-3-6-case} we describe the more restricted case $\ell(\mu) \leq 2, \ell(\nu) \leq 3, \ell(\lambda) \leq 6$, and in \ref{sec:2-4-8-case} we describe the general case.
In Section \ref{sec:gen-bravyi-vanish}, we show vanishing conditions (conditions on $\lambda, \mu, \nu$ ensuring that the coefficient in question is $0$) on the atomic Kronecker coefficient give vanishing conditions for the Kronecker coefficients. We subsequently deduce explicit conditions. These are given in Theorem \ref{theo:gen-bravyi}. For each $m,n$ we obtain a set of $m + n - 2$ conditions for partitions $\lambda, \mu, \nu$ with $\ell(\mu) \leq m,\ \ell(\nu) \leq n,\ \ell(\lambda) \leq mn$. Our conditions have the advantage of being easy to compute and implement practically.
By considering the set of partition triples $(\lambda, \mu, \nu)$ satisfying the equation
\begin{equation} \label{eq:atomic-origin}
\BF{b}^{m,n}(\lambda, \mu, \nu; Id) = \BF{0}
\end{equation}
we obtain a stable face of the Kronecker cone $PKron_{m,n,mn}$. Additionally, each $(\lambda, \mu, \nu)$ satisfying the above equation has the property that $g_{\lambda, \mu, \nu} = 1$, and moreover the partition triple is stable (Theorem \ref{theo:stability}). Eq.~\eqref{eq:atomic-origin} is natural to consider from the point of view of the expression for the Kronecker coefficient given in Eq.~\eqref{eq:vpf-to-kron}. In this case, the contribution of the alternant term associated to the identity permutation is $1$, and the contribution of all other alternant terms is $0$ (and so the atomic Kronecker coefficient and Kronecker coefficient are both equal to $1$). These results are described in Section \ref{sec:stability}.
The atomic Kronecker coefficient can be bounded from above using binomial coefficients (Theorem \ref{theo:atomic-bounds}). By bounding each of the terms of Eq.~\eqref{eq:vpf-to-kron} we are able to obtain upper bounds for the Kronecker coefficients which seem to be best known in certain cases. This is described in Section \ref{sec:kc-bounds}, and the main results are Corollaries \ref{cor:kc-bounds-general} and \ref{cor:kc-bounds}.
Finally, in Section \ref{sec:conclusion} we summarize some open questions.
\section{Vector partition functions and Kronecker coefficients}
\label{sec:VPF}
The central formula, Eq.~\eqref{eq:vpf-to-kron}, was developed by Mishna, Rosas and Sundaram~\cite{MiRoSu21}. It is deduced from the formula using Schur polynomials, determinant formulas for Schur polynomials and, a variable substitution. We reproduce some of the details here to establish notation.
\subsection{From Schur polynomials to vector partition generating functions}
We recall the \emph{staircase partition} $\delta^{(k)} = (k-1, k-2, \dots, 1, 0)$. For $\lambda$ with $\ell(\lambda) \leq k$, the \emph{alternant} $a_{\lambda}(x_1, \dots, x_k)$ is the anti-symmetric polynomial
\begin{equation}
a_{\lambda}(x_1, \dots, x_k) := \det(x_i^{\lambda_j})_{1 \leq i,j \leq k}.
\end{equation}
An expression for the Kronecker coefficients involving alternants is
\begin{equation}\label{eq:KCusingAlt}
\frac{a_{\delta_n[X]}a_{\delta_n[Y]}}{a_{\delta_{mn}[XY]}}a_{\lambda + \delta_{mn}}[XY] = \sum_{\mu, \nu} g_{\lambda, \mu, \nu}S(a_{\mu + \delta_{m}}[X])S(a_{\nu + \delta_{n}}[Y]),
\end{equation}
where $X = (1, x_1, \dots, x_{m-1}), Y = (1, y_1, \dots, y_{n-1}), XY = (1, x_1, \dots, x_{m-1}, y_1, \dots, y_{n-1}, x_1y_1, x_1y_2, \dots, x_{m-1}y_{n-1})$, and
\begin{equation}
S(a_{\alpha}(z_1, \dots, z_k)) = \prod_{i=1}^{k}{z_i^{\alpha_i}},
\end{equation}
for a partition $\alpha$ of length at most $k$.
The ratio of alternants $\frac{a_{\delta_m}[X]a_{\delta_m}[Y]}{a_{\delta_{mn}}[XY]}$
simplifies to the rational function
\begin{equation}
\frac{a_{\delta_m}[X]a_{\delta_n}[Y]}{a_{\delta_{mn}}[XY]} = \frac{1}{\mathcal{A}\mathcal{B}\mathcal{C}\mathcal{D}\mathcal{E}\mathcal{F}}
\end{equation}
with the following polynomials:
\begin{align}
\mathcal{A} &= \prod_{j=1}^{n} \prod_{i=1}^{m} (x_i - y_j)\\ \label{eq:A}
\mathcal{B} &= \prod_{j=1}^{n}\prod_{i=1}^{m} (1-x_iy_j) \\
\mathcal{C} &= \prod_{i=1}^{m-1} x_i^{n-1} \prod_{j=1}^{n-1} y_j^{m-1} \prod_{i=1}^{m-1} (1 - x_i) \prod_{j=1}^{n-1} (1-y_j)^{m-1} \\
\mathcal{D} &= \prod_{k=1, k \neq i}^{m-1} \prod_{i=1}^{m-1}\prod_{j=1}^{n-1} (x_k - x_iy_j) \prod_{k=1, k \neq i}^{m-1} \prod_{i=1}^{m-1}\prod_{j=1}^{n-1} (y_k - x_iy_j) \\
\mathcal{E} &= \prod_{j \neq l = 1}^{n-1}\prod_{1 \leq i < k \leq m-1} (x_iy_j - x_ky_l) \\
\mathcal{F} &= \prod_{i=1}^{m-1} x_i^{\binom{n-1}{2}} \prod_{j=1}^{n-1} y_j^{\binom{m-1}{2}} \prod_{1 \leq i < k \leq m-1}(x_i - x_k)^{n-1} \prod_{1 \leq j < l \leq n-1} (y_j-y_l)^{m-1}. \label{eq:F}
\end{align}
After the variable substitution
\begin{align}
x_i &= s_1s_2\dots s_i(t_1t_2\dots t_{n-2})^i \quad \text{ for } 1 \leq i \leq m-1, \label{eq:s-change} \\
\text{and } y_j &= s_0s_1\dots s_{m-1} (t_1t_2\dots t_{n-2})^{m-1} t_1t_2\dots t_{j-1} \quad \text{ for } 1 \leq j \leq n-1 \label{eq:t-change}
\end{align}
the rational function $\frac{1}{\mathcal{A}\mathcal{B}\mathcal{C}\mathcal{D}\mathcal{E}\mathcal{F}}$ can be written as the product
\[
\mathbf{s}^{\alpha}\mathbf{t}^{\beta} F_{m,n}(s_0, s_1, \dots, s_{m-1}, t_1, \dots, t_{n-2})
\]
where $F_{m,n}$ is a vector partition generating function in the variables $s_0, s_1, \dots, s_{m-1}, t_1, \dots, t_{n-2}$. After the variable substitution, the terms $S(a_{\mu + \delta_m}[X])$ and $S(a_{\nu + \delta_n}[Y])$ become $\mathbf{s}^{r_s(\mu, \nu)}$ and $\mathbf{t}^{r_t(\mu, \nu)}$ respectively. Finally, the term of the determinant \[a_{\lambda + \delta_{mn}}(1, s_0, \dots, s_{m-1}, t_1, \dots, t_{n-2})\] corresponding to permutation $\sigma$ becomes~$\mathbf{s}^{l_s(\lambda; \sigma)}\mathbf{t}^{l_t(\lambda; \sigma)}$.
For a monomial $M$ and variable $x$, by $\deg_x(M)$ we denote the exponent of $x$ in the monomial $M$.
\begin{prop} \label{rem:var-change}
Let $u \in \{1, s_0, \dots, s_{m-1}, t_1, \dots, t_{n-2}\}$. Then
\begin{multline}
\deg_u(1) \leq \deg_u(x_1) \leq \dots \leq \deg_u(x_{m-1}) \\ \leq \deg_u(y_1) \leq \dots \leq \deg_u(y_{n-1}) \\ \leq \deg_u(x_1y_1) \leq \deg_u(x_1y_2) \leq \dots \leq \deg_u(x_{m-1}y_{n-1}).
\end{multline}
\end{prop}
\subsection{The vector partition functions $p_{A^{m,n}}$}
By $\mathcal{P}_A(\BF{b})$ we denote the set $\mathcal{P}_A(\BF{b}) := \{\BF{x} \in \mathbb{Z}_{\geq 0}^n : A\BF{x} = \BF{b}\}$ of vector partitions of $\BF{b}$, so that $p_A(\BF{b})$ is the cardinality of $\mathcal{P}_A(\BF{b})$. By exploiting some of the properties of the matrices $A^{m,n}$ given in Corollary 30 of~\cite{MiRoSu21} (Properties 1--5 in the list below), we can deduce properties of the corresponding vector partition functions $p_{A^{m,n}}$ without explicitly computing the associated piecewise quasi-polynomials:
\begin{enumerate}[label=(\roman*)]
\item each entry of $A^{m,n}$ is a non-negative integer;
\item the largest entry of $A^{m,n}$ is $2m -1$;
\item the number of columns of $A^{m,n}$ is ${\binom{mn}{2}} - {\binom{n}{2}} - {\binom{m}{2}}$;
\item the number of rows of $A^{m,n}$ is $m + n - 2$;
\item each of the standard basis vectors appears as a column of $A^{m,n}$, and so its rank is $m + n - 2$.
\end{enumerate}
\begin{comment}
Given a partition $\lambda$ with $\ell(\lambda) \leq k$, the \emph{alternant} $a_{\lambda}(x_1, x_2, \dots, x_k)$ is the anti-symmetric polynomial
\begin{equation}
a_{\lambda}(x_1, x_2, \dots, x_k) := det(x_i^{\lambda_j})_{1 \leq i,j, \leq k},
\end{equation}
and the \emph{Schur polynomial} $s_{\lambda}(x_1, x_2, \dots, x_k)$ is the symmetric polynomial
\begin{equation}
s_{\lambda}(x_1, x_2, \dots, x_k) := \frac{a_{\lambda + \delta^{(k)}}(x_1, x_2, \dots, x_k)}{a_{\delta^{(k)}}(x_1, x_2, \dots, x_k)}.
\end{equation}
The Schur polynomials form an integral basis for the algebra of symmetric functions, and the \emph{Kronecker coefficients $g_{\lambda, \mu, \nu}$} are the structure constants in the (Kronecker coproduct) equation
\begin{equation}
s_{\lambda}
\end{equation}
\end{comment}
\section{Explicit computation of Kronecker coefficients}
\label{sec:explicit-computation}
When the partition lengths are sufficiently small, it is computationally feasible to determine the vector partition functions needed to compute individual Kronecker coefficients $g_{\lambda, \mu, \nu}$. We provide explicit formulas for two cases here, starting from Eq.~\eqref{eq:vpf-to-kron}, rewritten below:
\[ g_{\lambda, \mu, \nu} = \sum_{\sigma \in \mathfrak{S}_{mn}} \operatorname{sgn}(\sigma)\ p_{A^{m,n}}\biggr(\mathbf{b}^{m,n}(\lambda, \mu, \nu; \sigma)\biggr).\]
We compute $p_{A^{m,n}}$ first for $m=2,n=3$, then $m=2,n=4$ (the $m=n=2$ case appears in \cite{MiRoSu21}). Remark that, to compute a coefficient, it is best to minimize the choice of $m$ and $n$ that bound the lengths of $\mu$ and $\nu$. The first optimization comes from trying to identify which terms in the sum are zero. Recall, in the $m=n=2$ case, only $7$ of the terms are needed since of the original $4!=24$ terms in the right hand side: $13$ of them always evaluate to zero for partitions $\lambda, \mu, \nu$, and another $4$ of them cancel pairwise. To eliminate terms in other cases, we consider restrictions imposed by positivity in the linear algebra system, and the partition inequalities on the parts of the partitions.
\subsection{Exact expressions for $g_{\lambda, \mu, \nu}$ when $\ell(\lambda) \leq 6,\ \ell(\mu) \leq 2,\ \ell(\nu) \leq 3$} \label{sec:2-3-6-case}
The matrix $A^{2,3}$ is determined in \cite[Example 5]{MiRoSu21}:
\begin{equation}
A^{2,3} =
\begin{bmatrix}
1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\
0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 2 & 2 \\
0 & 0 & 1 & 1 & 1 & 1 & 2 & 1 & 2 & 2 & 3 \\
\end{bmatrix}
\end{equation}
Using \em Barvinok \em it is straightforward to determine that corresponding vector partition function $p_{A^{2,3}}$ is of degree 8 and has 34 chambers. At most $482$ of the $720$ terms of the alternant $a_{\lambda + \delta_6}$ yield a non-zero contribution to the Kronecker coefficient computation. The most non-zero terms we have found for any given coefficient is $288$. This occurs for $\mu = (99, 99),\ \nu = (66, 66, 66),\ \lambda = (87, 87, 24, 0, 0, 0).$ It is less clear how to find cancelling pairs as in the $m=n=2$ case, so this remains a place for potential optimization - each term represents a vector partition function evaluation, which in the worst case means searching through all chambers. The formula is as follows.
\begin{prop}
Let $\lambda, \mu, \nu$ be partitions with $\ell(\lambda) \leq 6,\ \ell(\mu) \leq 2,\ \ell(\nu) \leq 3$. Then the Kronecker coefficient is given by
\begin{multline}
g_{\lambda, \mu, \nu} = \sum_{\sigma \in \mathfrak{S}_6} \operatorname{sgn}(\sigma) \, \\p_{A^{2,3}}\left(\nu_2 + \nu_3 + 6 - l_s(\lambda; \sigma)_1,\quad \mu_2 + \nu_2 + \nu_3 + 11 - l_t(\lambda; \sigma)_1,\quad \mu_2 + \nu_2 + 2\nu_3 + 13 - l_t(\lambda; \sigma)_2 \right).
\end{multline}
and the atomic Kronecker coefficient is given by
\begin{dmath}
\tilde{g}^{2,3}_{\lambda, \mu, \nu} = p_{A^{2,3}}\left(\nu_2 + \nu_3 - \lambda_3 - \lambda_4 - \lambda_5 - \lambda_6,\quad \mu_2 + \nu_2 + \nu_3 - \lambda_2 - \lambda_3 - \lambda_4 - 2\lambda_5 - 2\lambda_6, \\ \quad \mu_2 + \nu_2 + 2\nu_3 - \lambda_2 - \lambda_3 - 2\lambda_4 - 2\lambda_5 - 3\lambda_6 \right).
\end{dmath}
\end{prop}
The implementation of Baldoni, Vergne and Walter \cite{BaVeWa17} takes on the order of 1 minute to compute a $2,3,6$ Kronecker coefficient whereas our implementation takes on the order of 10 microseconds. However, they are able to compute dilated Kronecker coefficients and, more generally, expressions that hold over the entire chamber, while our code does not do either.
\subsection{Exact expressions for $g_{\lambda, \mu, \nu}$ when $\ell(\lambda) \leq 8,\ \ell(\mu) \leq 2,\ \ell(\nu) \leq 4$} \label{sec:2-4-8-case}
It is straightforward to determine~$A^{2,4}$ following the same method
\begin{equation}
A^{2,4} = \begin{bmatrix}
0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 \\
0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 2 & 2 \\
0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 2 & 2 & 2 & 2 & 3 & 3 \\
1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 2 & 1 & 2 & 2 & 2 & 3
\end{bmatrix}.
\end{equation}
The corresponding vector partition function $p_{A^{2,4}}$ is of degree $17$ with $4328$ chambers. It took roughly 20 days to compute it on the Compute Canada \em Cedar\em\ research cluster. The vector partition function is available in .sobj format and in .txt format. The .txt format is the raw output from \em Barvinok\em.
Out of the $8! = 40320$ terms of the alternant $a_{\lambda + \delta_8}$, at most $28322$ yield a non-zero contribution to the Kronecker coefficient. It is not apparent if they can be grouped for cancellation as in the $m=n=2$ case.
\begin{prop}
Let $\lambda, \mu, \nu$ be partitions with $\ell(\lambda) \leq 8,\ \ell(\mu) \leq 2,\ \ell(\nu) \leq 4$. Then the Kronecker coefficient is given by
\begin{dmath}
g_{\lambda, \mu, \nu} = \sum_{\sigma \in \mathfrak{S}_8} \operatorname{sgn}(\sigma)
p_{A^{2,4}}(\nu_2 + \nu_3 +\nu_4 + 15 - l_s(\lambda; \sigma)_1, \mu_2 + \nu_2 + \nu_3 + \nu_4 + 24 - l_s(\lambda; \sigma)_2, \\ \mu_2 + \nu_2 + 2\nu_3 + 2\nu_4 + 32 - l_t(\lambda; \sigma)_1, \mu_2 + \nu_2 + \nu_3 + 2\nu_4 + 27 - l_t(\lambda; \sigma)_2),
\end{dmath}
and the atomic Kronecker coefficient is given by
\begin{dmath}
\tilde{g}^{2,4}_{\lambda, \mu, \nu} = p_{A^{2,4}}(\nu_2 + \nu_3 +\nu_4 - \lambda_3 - \lambda_4 - \lambda_5 - \lambda_6 - \lambda_7 - \lambda_8, \\ \mu_2 + \nu_2 + \nu_3 + \nu_4 - \lambda_2 - \lambda_3 - \lambda_4 - \lambda_5 - 2\lambda_6 - 2\lambda_7 - 2\lambda_8, \\ \mu_2 + \nu_2 + 2\nu_3 + 2\nu_4 - \lambda - \lambda_3 - 2\lambda_4 - 2\lambda_5 - 2\lambda_6 - 3\lambda_7 - 3\lambda_8, \\ \mu_2 + \nu_2 + \nu_3 + 2\nu_4 - \lambda_2 - \lambda_3 - \lambda_4 - 2\lambda_5 - 2\lambda_6 - 2\lambda_7 - 3\lambda_8)
\end{dmath}
\end{prop}
\begin{eg}
This formula gives the same result for the following example, take from~\cite{BaOr05}.
For $\lambda = (6, 4, 4, 1)$, $\mu = (12, 3)$, $\nu = (5, 4, 3, 3)$, we compute that $g_{\lambda, \mu, \nu} = 4$. The authors of \cite{BaOr05} computed this via a combinatorial rule - in this case the Kronecker coefficient is counting combinatorial objects called \em Kronecker tableaux\em.
\end{eg}
\begin{eg}
Let $\lambda = (57, 57, 57, 33, 33, 33, 10, 0),\ \mu = (140, 140),\ \nu = (70, 70, 70, 70)$, we compute that $g_{\lambda, \mu, \nu} = 391$. We were unable to compute this example with the package \em SF\em\ (it ran into a memory error after using $203146718216$ bytes), nor the \em Sagemath\em\ symmetric functions package (which also ran into a memory error). It cannot be computed by the Maple package of Baldoni, Vergne, and Walter \cite{BaVeWa17} which specifically handles the cases $\ell(\lambda), \ell(\mu), \ell(\nu) \leq 3$ and $\ell(\lambda) \leq 6, \ell(\mu) \leq 2, \ell(\nu) \leq 3$.
\end{eg}
\begin{comment}
Pak and Panova bound for previous example is $8.43 \times 10^{35}$.
Our bound is $5.37 \times 10^{14}$.
\end{comment}
The $A^{3,3}$ matrix is straightforward to compute, it has 4 rows and 30 columns. However obtaining the piecewise quasi-polynomial representation of the vector partition function was not computationally feasible: we had no results after roughly 30 days on the Compute Canada research cluster Cedar at which time the computation was terminated by the server.
\section{Vanishing conditions}
\label{sec:gen-bravyi-vanish}
A key to our analysis is a dominance property of vector partition functions. We use this property to prove Theorem~\ref{theo:gen-bravyi}, a generalization of some non-vanishing conditions for the Kronecker coefficients given in \cite{Bra04}. Let $\BF{u}, \BF{v} \in \mathbb{R}^k$. We say that $\BF{u}$ \emph{dominates} $\BF{v}$ if $\BF{u}_i \geq \BF{v}_i$ for each $1 \leq i \leq k$, and we denote this by $\BF{u} \succeq \BF{v}$.
\begin{lem} \label{lem:vpf-dominance}
Let $m,n$ be positive integers, if $\BF{a} \succeq \BF{b}$, then $p_{A^{m,n}}(\BF{a}) \geq p_{A^{m,n}}(\BF{b})$
\end{lem}
\begin{proof}
Each of the standard basis vectors $\BF{e}_1, \dots, \BF{e}_{m+n-2}$ is a column of $A^{m,n}$. Without loss of generality assume that columns $1, \dots, m+n-2$ are the standard basis vectors $\BF{e}_1, \dots, \BF{e}_{m+n-2}$ in the same order. It follows that any vector partition $\BF{x} \in S_{A^{m,n}}(\BF{b})$ can be mapped to a unique vector partition $\BF{x'} \in S_{A^{m,n}}(\BF{a})$ by taking $\BF{x'}_i := \BF{x}_i + (a_i-b_i)\BF{e}_i$ for each $1 \leq i \leq m + n - 2$. This forms an injective map from~$\mathcal{P}_A(\BF{b})$ to~$\mathcal{P}_A(\BF{a})$.
\end{proof}
\begin{lem} \label{lem:poset-structure}
Let $m,n$ be positive integers. Let $\sigma_1, \sigma_2 \in \mathfrak{S}_{mn}$ such that
\[
(l_s(\lambda; \sigma_1), l_t(\lambda; \sigma_1)) \succeq (l_s(\lambda; \sigma_2), l_t(\lambda; \sigma_2))
\]
for all partitions $\lambda$ with $\ell(\lambda) \leq mn$. Then
\[
p_{A^{m,n}}\biggr(\BF{b}^{m,n}(\lambda, \mu, \nu; \sigma_1)\biggr) \leq p_{A^{m,n}}\biggr(\BF{b}^{m,n}(\lambda, \mu, \nu; \sigma_2)\biggr)
\]
for all partitions $\lambda, \mu, \nu$ with $\ell(\lambda) \leq mn, \ell(\mu) \leq m, \ell(\nu) \leq n$.
\end{lem}
\begin{proof}
Multiplication by $-1$ reverses domination. The domination of one vector over another is preserved if we subtract the same vector from both sides, and if we add a positive vector to the larger one. Thus, for any partitions $\lambda, \mu, \nu$ with $\ell(\lambda) \leq mn, \ell(\mu) \leq m, \ell(\nu) \leq n$, we find that
\begin{align*}
\BF{b}^{m,n}(\lambda, \mu, \nu; \sigma_2) &= \biggr(r_s(\mu, \nu), r_t(\mu, \nu)) + (\alpha, \beta) - (l_s(\lambda; \sigma_2), l_t(\lambda; \sigma_2)\biggr) \\
&\succeq\,%
\biggr(r_s(\mu, \nu), r_t(\mu, \nu)) + (\alpha, \beta) -(l_s(\lambda; \sigma_1), l_t(\lambda; \sigma_1)\biggr) \\ &= \BF{b}^{m,n}(\lambda, \mu, \nu; \sigma_1).
\end{align*}
Then by Lemma \ref{lem:vpf-dominance} we have that $p_{A^{m,n}}\biggr(\BF{b}^{m,n}(\lambda, \mu, \nu; \sigma_2) \biggr) \geq p_{A^{m,n}}\biggr(\BF{b}^{m,n}(\lambda, \mu, \nu; \sigma_1) \biggr)$ as required.
\end{proof}
The previous lemma induces a poset structure on $\mathfrak{S}_{mn}$ via the relation $\sigma_2 \geq \sigma_1$ if and only if
\[
(l_s(\lambda; \sigma_1), l_t(\lambda; \sigma_1)) \succeq (l_s(\lambda; \sigma_2), l_t(\lambda; \sigma_2))
\]
for all partitions $\lambda$ with $\ell(\lambda) \leq mn$. Figure \ref{fig:poset-22} illustrates the poset for the $m=n=2$ case, showing only the permutations associated to the 7 alternant terms which contribute to the Kronecker coefficient.
The poset in the figure is the \em dependency digraph for the monomials in $P_{\lambda}$\em\ given in \cite[Figure 4]{MiRoSu21}. However, there the poset is computed by comparing the contributions of the alternant terms as opposed to the vector partition function inputs. Our approach allows us to compute the posets for larger $m,n$ when comparing the contributions is infeasible (either due to the large number of chambers or the difficulty of computing the vector partition function as a piecewise quasi-polynomial).
In the following lemma we show that the identity permutation is a maximal element of the poset for any positive integers $m,n$ (in fact it is unique, and thus the maximal element).
\begin{figure}
\centering
\begin{tikzpicture}[
myNODE/.style={rectangle, minimum size=7mm, draw=gray},
]
\node[myNODE] (ID) {1234};
\node[myNODE] (1324) [below=of ID] {1324};
\node[myNODE] (2134) [right=of 1324] {2134};
\node[myNODE] (1243) [left=of 1324] {1243};
\node[myNODE] (3124) [below=of 2134] {3124};
\node[myNODE] (2143) [below=of 1324] {2143};
\node[myNODE] (1342) [below=of 1243] {1342};
\draw[->] (3124.north) -- (2134.south);
\draw[->] (3124.north) -- (1324.south);
\draw[->] (2143.north) -- (2134.south);
\draw[->] (2143.north) -- (1243.south);
\draw[->] (1342.north) -- (1243.south);
\draw[->] (1342.north) -- (1324.south);
\draw[->] (2134.north) -- (ID.south);
\draw[->] (1324.north) -- (ID.south);
\draw[->] (1243.north) -- (ID.south);
\end{tikzpicture}
\caption{The poset of contributing alternant terms in the $m=n=2$ case. Each alternant term is given by its permutation in one line notation.}
\label{fig:poset-22}
\end{figure}
\begin{lem} \label{lem:domination}
For all $\sigma \in \mathfrak{S}_{mn}$ and partitions $\lambda$ of length at most $mn$,
\[(l_s(\lambda; \sigma), l_t(\lambda; \sigma)) \succeq (l_s(\lambda; Id), l_t(\lambda; Id)).\]
\end{lem}
\begin{proof}
The alternant $a_{\lambda + \delta_{mn}}$ is the determinant of the matrix $(z_i^{\lambda_j})_{1 \leq i,j \leq mn}$ where $z_i$ is the $i$\textsuperscript{th} variable in $XY$. The $k$\textsuperscript{th} coordinate of $(l_s(\lambda; \sigma), l_t(\lambda; \sigma))$ is
\begin{equation} \label{eq:k-coordinate}
(l_s(\lambda; \sigma), l_t(\lambda; \sigma))_k = \sum_{i = 1}^{mn} \left(\lambda_i + mn - i \right)\deg_u(z_{\sigma^{-1}(i)})
\end{equation}
where $u$ is the $k$\textsuperscript{th} element of $(1, s_0, \dots, s_{m-1}, t_1, \dots, t_{n-2})$. Since $(\lambda_1 + mn - 1, \lambda_2 + mn - 2, \dots, \lambda_{mn})$ is a monotonically decreasing sequence, and $\deg_u$ monotonically increasing over $(1, x_1, \dots, x_{m-1}, y_1, \dots, y_{n-1}, \\ x_1y_1, \dots, x_my_n)$, the above expression is minimized for the term obtained by the change of variables from
\[
1^{\lambda_1 + mn - 1}x_1^{\lambda_2 + mn - 2}\dots\ (x_my_n)^{\lambda_{mn}}
\]
corresponding to the identity permutation.
\end{proof}
Combining the previous two lemmas yields the following result relating the atomic Kronecker coefficient $\tilde{g}^{m,n}_{\lambda, \mu, \nu}$ with the Kronecker coefficient $g_{\lambda, \mu, \nu}$, from which vanishing conditions (given in Theorem \ref{theo:gen-bravyi}) can be derived.
\begin{lem} \label{lem:atomic-vanish}
Let $\lambda, \mu, \nu$ be partitions with $\ell(\lambda) \leq mn, \ell(\mu) \leq m, \ell(\nu) \leq n$ for some positive integers $m,n$.
If $\tilde{g}^{m,n}_{\lambda, \mu, \nu} = 0$, then $g_{\lambda, \mu, \nu} = 0$.
\end{lem}
\begin{proof}
If $\tilde{g}^{m,n}_{\lambda, \mu, \nu}=0$ then for any $\sigma\in \mathfrak{S}_{mn}, $
\begin{eqnarray*}0=\tilde{g}^{m,n}_{\lambda, \mu, \nu} &=& p_{A^{m,n}}\biggr(\BF{b}^{m,n}(\lambda, \mu, \nu; Id \biggr)\quad\text{by Eq.~\eqref{eq:atomic}}, \\
&\geq& p_{A^{m,n}}\biggr(\BF{b}^{m,n}(\lambda, \mu, \nu; \sigma \biggr) \quad\text{by Lemmas~\ref{lem:poset-structure} and \ref{lem:domination}}, \\
&\geq& 0, \\
\end{eqnarray*}
and thus $p_{A^{m,n}}\biggr(\BF{b}^{m,n}(\lambda, \mu, \nu; \sigma \biggr) \ = 0$. Since it is true for all $\sigma$, all the terms in the sum in Eq.~\eqref{eq:vpf-to-kron} vanish, and so $g_{\lambda, \mu, \nu} = 0$.
\end{proof}
Since $\tilde{g}^{m,n}_{\lambda, \mu, \nu}$ is given by a single vector partition function evaluation $p_{A^{m,n}}(\mathbf{b})$, we know that it is~$0$ exactly when $\mathbf{b}$ is not in the cone generated by the columns of $A^{m,n}$. This occurs if and only if $b_i < 0$ for some $1 \leq i \leq m+n-2$. Since $\BF{b} = (r_s(\mu, \nu) + \alpha - l_s(\lambda, Id), r_t(\mu, \nu) + \beta - l_t(\lambda, Id))$ for the atomic Kronecker coefficient $\tilde{g}^{m,n}_{\lambda, \mu, \nu}$, we get a set of vanishing conditions for the Kronecker coefficient $g_{\lambda, \mu, \nu}$. We express the conditions using the contrapositive (i.e. we give conditions imposed on $\lambda, \mu, \nu$ if the Kronecker coefficient is non-zero) since the set of $\lambda, \mu, \nu$ satisfying them forms a cone.
\begin{theo} \label{theo:gen-bravyi}
Let $m,n$ be positive integers and $\lambda, \mu, \nu$ be partitions with $\ell(\lambda) \leq mn, \ell(\mu) \leq m, \ell(\nu) \leq n$. If $g_{\lambda, \mu, \nu} \neq 0$ then each of the following inequalities hold:
\begin{equation}
\sum_{k=1}^{m} \lambda_k \geq \nu_1;
\end{equation}
For all $a$ satisfying $1 \leq a \leq m-1$:
\begin{equation}
\sum_{k=1}^{a} \lambda_k - \sum_{k=m+n}^{m + (a+1)(n-1)} \lambda_k \geq \nu_1 - \sum_{k=a+1}^{m} \mu_k
\end{equation}
For all $b$ satisfying $1 \leq b \leq n-2$:
{\small\begin{equation}
m\lambda_1 + \sum_{k=2}^{m} (m-k+1)\lambda_k + \sum_{k=m+1}^{m+b} \lambda_k - \sum_{i=1}^{m-1}\sum_{j=1}^{b}(i-1) \lambda_{m+i(n-1)+j} - \sum_{i=1}^{m-1}\sum_{j=b+1}^{n-1}i\lambda_{m+i(n-1) + j} \geq m\nu_1 + \sum_{k=2}^{b+1} \nu_k - \sum_{k=2}^{m}(k-1)\mu_k.
\end{equation}}
\end{theo}
\begin{rem}
When $m=n=2$, Theorem \ref{theo:gen-bravyi} reduces to vanishing conditions given by Bravyi in \cite{Bra04}. This case was worked out explicitly in \cite[Proposition 5]{MiRoSu21}.
\end{rem}
An inequality $n \cdot x \leq 0$ is \emph{essential} for a cone $\tau$ if $\{x : n \cdot x = 0\} \cap \tau$ is a facet of $\tau$, and each $p \in \tau$ satisfies the inequality ($n \cdot p \leq 0$ for all $p \in \tau$).
\begin{rem}
Klyachko~\cite{Kl04} gives the full list of $41$ essential inequalities in the $m=2, n=3$ case. In this case, none of our inequalities appear on Klyachko's list. Thus, while our inequalities are easy to compute and use practically, regrettably none are essential inequalities for the cone $PKron_{2,3,6}$. Thus, one should not expect, for general $m,n$, that the inequalities given by Theorem \ref{theo:gen-bravyi} are essential.
\end{rem}
\begin{comment}
\begin{eg} \label{eg:23vanish}
Let $\lambda, \mu, \nu, $ be partitions with $\ell(\mu) \leq 2, \ell(\nu) \leq 3, \ell(\lambda) \leq 6$. If at least one of the conditions below are met
\begin{enumerate}
\item $\lambda_3 + \lambda_4 + \lambda_5 + \lambda_6 \leq \nu_2 + \nu_3$,
\item $\lambda_2 + \lambda_3 + \lambda_4 + 2\lambda_5 + 2\lambda_6 \leq \mu_2 + \nu_2 + \nu_3$,
\item $\lambda_2 + \lambda_3 + 2\lambda_4 + 2\lambda_5 + 3\lambda_6 \leq \mu_2 + \nu_2 + 2\nu_3$,
\end{enumerate}
then $g_{\lambda, \mu, \nu} = 0$.
\end{eg}
\end{comment}
Ressayre determined two sets of vanishing conditions for the Kronecker coefficients for any lengths $l,m,n$ which are essential,~\cite[Theorems 1 \& 2]{Re19}.
\begin{theo}[Ressayre~\cite{Re19}]
Let $e, f$ be two positive integers, and let $\lambda, \mu, \nu$ be partitions of $N$ with
\begin{equation}\label{eq:Ress}
l(\mu) \leq e+1, \ l(\nu) \leq f+1,\ l(\lambda) \leq e + f + 1
\end{equation}
If $g_{\lambda, \mu, \nu} \neq 0$, then
\[
N + \lambda_1 + \lambda_{e+j} \leq \mu_1 + \nu_1 + \nu_j
\]
for all $2 \leq j \leq f+1$.
\end{theo}
These are quite strong, although there is likely a smaller error in these conditions, given the following example we found.
\begin{eg}
Upon setting $e=1, f=3, n=4$ and $j=4$ in Eq.~\eqref{eq:Ress}, $\ell(\mu) \leq 2, \ell(\nu) \leq 4, \ell(\lambda) \leq 5$ and the Kronecker coefficient $g_{\lambda, \mu, \nu}$ should be $0$ if, furthermore,
\begin{equation}\label{eq:ress2}
|\lambda| + \lambda_1 + \lambda_5 > \mu_1 + \nu_1 + \nu_4.
\end{equation}
Consider $\lambda = (1, 1, 1, 1, 0), \mu = (2, 2), \nu = (2, 2)$. Inequality~\ref{eq:ress2} is satisfied, but $g_{\lambda, \mu, \nu} = 1$, not $0$. A second example is given by $\lambda = (4), \mu = (2, 2), \nu = (2,2)$.
\end{eg}
\begin{comment}
Consider alpha = (2, 2), beta = (2, 2, 0, 0), gamma = (1, 1, 1, 1, 0) (so that e=1, f=3, n=4). Remark that the Kronecker coefficient g_{alpha, beta, gamma} is 1, hence non zero. Using j=4 in Eq (7)
yields the inequality
n + gamma1 + gamma5 <= alpha1 + beta1 + beta4.
=> 4+1+0 <= 2+2+0 => 5<=4.
This example is one of an infinite family:
alpha=(2+k, 2) beta=(2+k,2,0,0) gamma = (k+1, 1, 1,1,0)
It also doesn't seem to hold for alpha = (2, 2), beta = (2, 2, 0, 0), gamma = (4, 0, 0, 0, 0). These examples appear in the paper "The stability of the Kronecker product of Schur functions" by Briand, Orellana, Rosas Journal of Algebra 331 (2011) 11–27 (page 12).
\end{comment}
\section{A stable face of the Kronecker polyhedron}
\label{sec:stability}
By considering a set of conditions implying that the atomic Kronecker coefficient and Kronecker coefficient are both equal to $1$, we are able to obtain a stable face of the Kronecker polyhedron $PKron_{m,n,mn}$ for each $m,n$. Moreover, each partition triple $(\lambda, \mu, \nu)$ satisfying these conditions is a stable triple. We note that elements of this approach appear in \cite{MiRoSu18} for the case $m=n=2$.
\begin{prop} \label{prop:origin}
If $\BF{b}^{m,n}(\lambda, \mu, \nu; Id) = \BF{0}$, then $
g_{\lambda, \mu, \nu} = \tilde{g}^{m,n}_{\lambda, \mu, \nu} = 1.$
\end{prop}
\begin{proof}
When $\BF{b}^{m,n}(\lambda, \mu, \nu; Id) = \BF{0}$, $\BF{b}^{m,n}(\lambda, \mu, \nu; \sigma)$ has at least one negative coordinate for each $\sigma \in \mathfrak{S}_{mn},\ \sigma \neq Id$, and so
\begin{align}
g_{\lambda, \mu, \nu} &= \tilde{g}^{m,n}_{\lambda, \mu, \nu} \\
&= p_{A^{m,n}}(\BF{b}^{m,n}(\lambda, \mu, \nu; Id)) \\
& = 1.
\end{align}
\end{proof}
The condition $\BF{b}^{m,n}(\lambda, \mu, \nu; Id) = \BF{0}$ yields $m+n-2$ equations involving the parts of $\lambda, \mu, \nu$. By also including the equations $|\lambda| = |\mu| = |\nu|$, we obtain relatively simple expressions for each part of $\mu$ and $\nu$ in the parts of $\lambda$.
\begin{prop} \label{prop:rewrite-eqns}
Let $\lambda, \mu, \nu$ be partitions of the same positive integer $N$ with $\ell(\mu) \leq m, \ell(\nu) \leq n, \ell(\lambda) \leq mn$.
Then $\BF{b}^{m,n}(\lambda, \mu, \nu; Id) = \BF{0}$ if and only if $(\lambda, \mu, \nu)$ satisfy the following equations:
\begin{align}
\mu_u &= \lambda_u + \sum\limits_{i=m+(u-1)(n-1)+1}^{m + u(n-1)} \lambda_i &\text{ for } u=1, \dots, m \label{eq:stab-mu} \\
\nu_1 &= \sum\limits_{i=1}^{m} \lambda_i & \label{eq:stab-nu1} \\
\nu_v &= \sum\limits_{i=0}^{m-1} \lambda_{m+(n-1)i + v-1} & \text{ for } v=2,\dots,n \label{eq:stab-nu2}.
\end{align}
\end{prop}
The proof of this appears in Appendix~\ref{sec:proof-rewrite-system}. The following result follows directly from Propositions \ref{prop:origin} and \ref{prop:rewrite-eqns}.
\begin{cor} \label{cor:atomic=1}
Let $\lambda, \mu, \nu$ be partitions of $N$ with $\ell(\mu) \leq m, \ell(\nu) \leq n, \ell(\lambda) \leq mn$, such that $\lambda, \mu, \nu$ satisfy Eqs.~\eqref{eq:stab-mu}--\eqref{eq:stab-nu2}. Then $g_{\lambda, \mu, \nu} = \tilde{g}^{m,n}_{\lambda, \mu, \nu} = 1$.
\end{cor}
We can say more about the partition triples $(\lambda, \mu, \nu)$ satisfying Eqs.~\eqref{eq:stab-mu}--\eqref{eq:stab-nu2}. We follow \cite{Man15-2} for notation. A triple of partitions $(\lambda, \mu, \nu)$ is called \em weakly stable \em if $g_{k\lambda, k\mu, k\nu} = 1$ for each positive integer $k$. Recall that a triple of partitions $(\lambda, \mu, \nu)$ is \em stable\em\ if for any partitions $\alpha, \beta, \gamma$ the sequence $(g_{\alpha + k\lambda, \beta+k\mu, \gamma + k\nu})_{k \geq 0}$ stabilizes.
For given positive integers $l,m,n$, the \em weight lattice\em\ $W_{l,m,n}$ is the sublattice of $\mathbb{Z}^{l+m+n}$ defined by the equations $|\lambda| = |\mu| = |\nu|$. In \cite{Man15-2}, Manivel defines a \em stable face\em\ of the cone $PKron_{l,m,n}$ to be a face of $PKron_{l,m,n}$ whose intersection with $W_{l,m,n}$ is a subset of $SKron_{l,m,n}$ - the set of all weakly stable triples $(\lambda, \mu, \nu)$ with $\ell(\lambda) \leq l,\ \ell(\mu) \leq m, \ell(\nu) \leq n$. A stable face is \em maximal\em\ if it is maximal in $SKron_{l,m,n}$.
Note that the set of triples $(\lambda, \mu, \nu)$ satisfying Eqs.~\eqref{eq:stab-mu}--\eqref{eq:stab-nu2} along with the partition inequalities (for any partition $\alpha$ of length $k$, $\alpha_1 \geq \alpha_2 \geq \dots \geq \alpha_k \geq 0$) generate a cone $\tau_{m,n}$. By Corollary \ref{cor:atomic=1}, each $\lambda, \mu, \nu$ in the intersection $\tau_{m,n} \cap W_{mn, m, n}$ is weakly stable. In fact, as the next theorem shows, they are actually stable.
\begin{theo} \label{theo:stability}
Each triple $\lambda, \mu, \nu$ satisfying Eqs.~\eqref{eq:stab-mu} -- \eqref{eq:stab-nu2} is a stable triple. Moreover, the cone $\tau_{m,n}$ is a stable face of $PKron_{mn,m,n}$.
\end{theo}
The proof of the previous theorem is given in Appendix \ref{sec:proof-stable-triples}. It relies on the connection between \emph{additive tableaux} and stable faces given in \cite[Propositions 7 and 9]{Man15}.
\begin{eg}
Let $\lambda = (10, 8, 5, 3, 2, 2), \mu = (17, 12), \nu = (18, 7, 5)$. One can check that $\lambda, \mu, \nu$ satisfy Eqs~\eqref{eq:stab-mu}--\eqref{eq:stab-nu2}. Further we have checked that $g_{k\lambda, k\mu, k\nu} = 1$ for all positive integers $k$ computing the quasi-polynomial $g_{k\lambda, k\mu, k\nu}$ via the code of Baldoni, Vergne and Walter. We now give an example to illustrate the stability of $\lambda, \mu, \nu$. For $\alpha = (34, 27, 20, 12, 4, 3), \beta = (70, 30), \nu = (43, 39, 18)$, the sequence $(g_{\alpha + k\lambda, \beta + k\mu, \gamma + k\nu})_{k \geq 1}$ stabilizes at $44729$ at $k = 6$. The sequence from $k=0$ to $6$ is $2566$, $18028$, $36174$, $43896$, $44638$, $44713$, $44729$.
\end{eg}
\begin{comment}
\begin{proof}
Clearly the intersection of $\gamma_{m,n}$ with the weight lattice is a subset of $SKron_{m,n,mn}$, so it remains to show that it is a face of the \em Kronecker polyhedron\em\ $PKron_{a,b,c}$. Note that $PKron_{a,b,c}$ lies in the subspace $|\lambda| = |\mu| = |\nu|$, so $PKron_{a,b,c} \cap H = PKron_{a,b,c} \cap H \cap \{(\lambda, \mu, \nu) : |\lambda| = |\mu| = |\nu|\}$ for any hyperplane $H \subseteq \mathbb{R}^{m + n + mn}$. For $i=1,\dots,m+n-2$, let $H_i$ be the hyperplane defined by the equation $b^{m,n}(\lambda, \mu, \nu; Id)_i = 0$. Then $H_i$ is a separating hyperplane for the cone $PKron_{a,b,c}$ since $PKron_{a,b,c} \subseteq \{(\lambda, \mu, \nu) : b^{m,n}(\lambda, \mu, \nu; Id)_i \geq 0 \}$. Therefore
\begin{align*}
\gamma_{m,n} &= PKron_{m,n,mn} \cap H_1 \cap \dots \cap H_{n+m-2} \cap \{(\lambda, \mu, \nu) : |\lambda| = |\mu| = |\nu|\} \\
&= PKron_{m,n,mn} \cap H_1 \cap \dots \cap H_{n+m-2}
\end{align*}
is a face of $PKron_{m,n,mn}$ as required, and so $\gamma_{m,n}$ is a stable face of $PKron_{m,n,mn}$. Since $\gamma_{m,n}$ is defined by $m+n$ independent linear equations, it has codimension $m+n$ in $\mathbb{R}^{m + n + mn}$, and thus codimension $m+n-2$ in the subspace $|\lambda| = |\mu| = |\nu|$. Therefore the co-dimension of the face $\gamma_{m,n}$ of $PKron_{m,n,mn}$ is $m+n-2$.
\end{proof}
\end{comment}
We note that the stable face $\tau_{m,n}$ is not maximal in general. For example, $\tau_{3,3}$ is contained in the stable faces $F_2^-, F_5^-, F_7^-$ and $F_8$ from \cite[Example 2]{Man15-2}. In particular, $F_5^-$ is the (maximal) stable facet defined by the intersection of $PKron_{3,3,9}$ and the equation
\begin{equation}
\mu_2 + 2\mu_3 + 2\nu_2 + 3\nu_3 = \lambda_2 + 2\lambda_3 + 2\lambda_4 + 3\lambda_5 + 3\lambda_6 + 4\lambda_7 + 4\lambda_8 + 5\lambda_9
\end{equation}
which is $b^{3,3}(\lambda, \mu, \nu; Id)_4 = 0$.
\begin{comment}
\begin{conj}
Let $m$ be a positive integer. Then the set
\[
\{(\lambda, \mu, \nu) : \ell(\lambda) \leq 3m, \ell(\mu) \leq m, \ell(\nu) \leq 3,\ b^{m,n}(\lambda, \mu, \nu; Id)_{m+1} = 0\} \cap PKron_{m,3,3m}
\]
forms a stable facet of $PKron_{3m,m,3}$.
\end{conj}
The relaxation tableau here given by the sequences $x = (0, m-1, m)$ and $y = (0, 1, \dots, m-1)$ seems to produce a stable facet since there are exactly $m+n-3 = m$ pairs of repeated numbers in the resulting relaxation tableau.
\end{comment}
We remark also that a couple well-known results are implied by Theorem \ref{theo:stability}. When $\lambda, \mu, \nu$ are each rectangular partitions of lengths $mn,m,n$ respectively (that is $\lambda_1 = \dots = \lambda_{mn}$,\ $\mu_1 = \dots = \mu_m$,\ $\nu_1 = \dots = \nu_n$), the Kronecker coefficient is $1$ (and the triple ($\lambda, \mu, \nu$) is stable). It is straightforward to check that $\lambda, \mu, \nu$ satisfy Eqs.~\eqref{eq:stab-mu}--\eqref{eq:stab-nu2}. The case $\mu = \lambda$ and $\ell(\nu) = 1$ (so $\nu = (|\lambda|)$) also satisfies the same equations (and again the Kronecker coefficient in this case is $1$, and the partition triple ($\lambda, \mu, \nu$) is stable).
\section{Upper bounds for Kronecker coefficients} \label{sec:kc-bounds}
The atomic Kronecker coefficients are given by a single vector partition function evaluation $p_{A^{m,n}}(\mathbf{b})$. By constructing a companion matrix to $A^{m,n}$, we are able to obtain a simpler vector partition function for which the evaluations can be computed by hand and whose evaluations bound $p_{A^{m,n}}$ from above. By bounding each of the terms of Eq.\eqref{eq:vpf-to-kron}, we are then able to obtain upper bounds for the Kronecker coefficients.
\subsection{A bound in terms of atomic Kronecker coefficients}
In~\cite{MiRoSu21}, Mishna, Rosas and Sundaram show that in the $m=n=2$ case, the atomic Kronecker coefficient $\tilde{g}^{2,2}_{\lambda, \mu, \nu}$ bounds the corresponding Kronecker coefficient $g_{\lambda, \mu, \nu}$ from above, and in~\cite{MiRoSu18} they conjecture that this is the case in general. Since we do know that the atomic Kronecker coefficient is the largest term in the sum, we can use this to give a general weaker bound.
\begin{prop} \label{prop:atomic-bounds-kc}
Let $\lambda, \mu, \nu$ be partitions with $\ell(\mu) \leq m, \ell(\nu) \leq n, \ell(\lambda) \leq ln$. Then
\begin{equation}
g_{\lambda, \mu, \nu} \leq \frac{(mn)!}{2} \tilde{g}^{m,n}_{\lambda, \mu, \nu}.
\end{equation}
\end{prop}
\begin{proof}
Splitting the sum in Eq.~\eqref{eq:vpf-to-kron} in two, one for the permutations with positive sign, and one for the permutations with negative sign, we bound each of the negative sign terms above by $0$ and each of the positive terms by the atomic term (by Lemmas \ref{lem:vpf-dominance} and \ref{lem:domination}).
\end{proof}
\subsection{Estimating atomic Kronecker Coefficients}
We can approximate the partition function of a matrix $A$ by replacing its columns with standard basis vectors so that the rank is preserved. Partition functions of such matrices are easy to write using binomial coefficients. Lemma~\ref{lem:col-replace} describes the replacement process and Proposition~\ref{prop:binomial-bound} is the resulting bound. The following proposition sets up Lemma~\ref{lem:col-replace}.
\begin{prop} \label{prop:col-replace}
Let $A$ be a $d \times n$ matrix with integer entries and $\ker(A) \cap \mathbb{R}^{m}_{\geq 0} = \{\BF{0}\}$. Let $\BF{c}$ be a column of $A$, and let $\BF{c'}$ be a $1 \times n$ vector. Let $A'$ be the matrix obtained by replacing column $\BF{c}$ with $\BF{c'}$. If $p_{A'}(\BF{c}) \geq p_{A}(\BF{c})$, then
\[
p_{A'}(\BF{b}) \geq p_{A}(\BF{b})
\]
for all $\BF{b} \in \mathbb{Z}_{\geq 0}^{d}$.
\end{prop}
\begin{proof}
Let $j$ be the index at which column $\BF{c}$ appears in A (and thus column $\BF{c'}$ appears in $A'$).
Partition the set of vector partitions $\mathcal{P}_A(\BF{b})$ of $\BF{b}$ into $U_1 := \{\BF{x} : A\BF{x} = \BF{b}, x_j = 0\}$ and $U_2 := \{\BF{x} : A\BF{x} = \BF{b}, x_j > 0\}$. The set $U_1$ is equal to the set $\{x : A'\BF{x} = \BF{b}, x_j = 0\}$. Also $\BF{c} \in \{A'\BF{x} : x_j > 0\}$ since $p_{A'}(\BF{c}) \geq p_{A}(\BF{c})$, and so $|U_2| \leq |\{x : A'\BF{x} = \BF{b}, x_j > 0\}|$. Thus $p_A(\BF{b}) = |U_1| + |U_2| \leq p_{A'}(\BF{b})$ as required.
\end{proof}
The following Lemma describes how to replace columns of $A^{m,n}$ with standard basis vectors via the previous proposition.
\begin{lem} \label{lem:col-replace}
Let $A$ be a $d \times n$ matrix with non-negative integer entries and each standard basis vector $\BF{e}_1, \dots, \BF{e}_d$ appearing as a column of $A$. Let $\BF{c}$ be a column of $A$, and let $I = \{k : c_k > 0, 1 \leq k \leq n\}$ be the set of non-zero coordinates of $\BF{c}$. Let $E^{(i)}$ denote the matrix obtained by replacing column $\BF{c}$ with $\BF{e}_i$ for some $i \in I$. Then
\[
p_{E^{(i)}}(\BF{b}) \geq p_A(\BF{b})
\]
for all $\BF{b} \in \mathbb{Z}_{\geq 0}^{d}$.
\end{lem}
\begin{prop} \label{prop:binomial-bound}
Let~$E$ be a~$d \times n$ matrix such that the columns of $E$ are formed by taking $i_j$ copies of each standard basis vector $\BF{e}_i$ where $i_1, \dots, i_d \geq 0$. Then
\[
p_A(\BF{b}) = \prod_{i=1}^{k} {\binom{b_i + i_j - 1}{i_j - 1}}.
\]
\end{prop}
\begin{proof}
For each component $i$ we must take a total of $b_i$ copies of the standard basis vector $\BF{e}_i$. We can think of this problem as distributing $b_i$ balls to the $i_j$ different columns of $A$ which are the copies of $\BF{e}_i$. This is counted by the $i$\textsuperscript{th} term in the given product of binomial coefficients.
\end{proof}
By application of Lemma~\ref{lem:col-replace} and Proposition \ref{prop:binomial-bound} we obtain binomial coefficient bounds for the atomic Kronecker coefficients, and thus the Kronecker coefficients as well. The technical details of the proof appear in Appendix~\ref{sec:appendix2} where we work out explicitly which columns have which non-zero coordinates. Note that there are many choices of column replacements that can be made, and different choices provide better bounds for certain choices of $\lambda, \mu, \nu$. The formulation of Theorem \ref{theo:atomic-bounds} represents a single choice whose advantage is that it is relatively simple to explain.
\begin{theo} \label{theo:atomic-bounds}
Let $m,n$ be positive integers, and $\lambda, \mu, \nu$ be partitions with $\ell(\lambda) \leq mn, \ell(\mu) \leq m,\ \ell(\nu) \leq n$. Then:
\begin{equation} \label{eq:atomic-bounds}
\tilde{g}^{m,n}_{\lambda, \mu, \nu} \leq {\binom{b_1 + c_1}{b_1}}{\binom{b_2 + c_2}{b_2}}{\binom{b_{m+n-2} + c_3}{b_{m+n-2}}}\prod_{i=3}^{m} {\binom{b_i + f_1(i)}{ b_i}}\prod_{j=1}^{n-3} {\binom{b_{m+j} + f_2(j)}{b_{m+j}}}.
\end{equation}
where
$\BF{b} = (b_1, \dots, b_{m+n-2}) = \mathbf{b}^{m,n}(\lambda, \mu, \nu; Id)$ and
\begin{align}
c_1 &= (m^2-1)(n-1) - 1\\
c_2 &= (m-1)(n-1)^2 - 1 \\
c_3 &= {\binom{m-1}{2}}(n-1) + (m-1) - 1 \\
f_1(i) &= 2{\binom{n-1}{2}}(i-2) - 1 \\
f_2(j) &= (n- j - 1)(m-1) - 1
\end{align}
\end{theo}
\begin{cor} \label{cor:kc-bounds}
Theorem \ref{theo:atomic-bounds} in combination with Proposition~\ref{prop:atomic-bounds-kc} gives:
\[ g_{\lambda, \mu, \nu} \leq \frac{(mn)!}{2}R_1\] where $R_1$ is the expression on the right-handside of Inequality~\eqref{eq:atomic-bounds}.
\end{cor}
We also give a weaker general bound which depends only on $m,n$ and the size $N$ of the partitions $\lambda, \mu, \nu$. We do this by bounding the coordinates of $\mathbf{b}^{m,n}$ by multiples of $N$.
\begin{cor} \label{cor:atomic-kc-bounds}
Let $m,n$ be positive integers, and $\lambda, \mu, \nu, $ be partitions of $N$ with lengths at most $mn,m, n$ respectively.
\begin{align}
\tilde{g}^{m,n}_{\lambda, \mu, \nu} \leq {\binom{N + c_1}{N}}{\binom{2N + c_2}{2N}}{\binom{(2m-1)N + c_3}{(2m-1)N}} \prod_{i=3}^{m}{\binom{2N + f_1(i)}{2N}} \prod_{j=1}^{n-3}{\binom{(2m-1)N + f_2(j)}{ (2m-1)N}}.
\label{eq:atomic-easybound}
\end{align}
where $c_1, c_2, c_3, f_1, f_2$ are as in Theorem \ref{theo:atomic-bounds}.
\end{cor}
\begin{proof}
Recall that for each component of $\mathbf{b}^{m,n}(\lambda, \mu, \nu; Id) = (r_s(\mu, \nu) + \alpha - l_s(\lambda; Id), r_t(\mu, \nu) + \beta - l_t(\lambda; Id)$ the constant terms cancel. Therefore each $b_i$ of Theorem \ref{theo:atomic-bounds} is bounded above by linear combination in the parts of $\mu, \nu$ appearing. Explicitly, we find that $b_1 \leq N$, $b_i \leq 2N$ for $2 \leq i \leq m$ and $b_{m+j} \leq (2m-1)N$ for $1 \leq j \leq n-2$.
\end{proof}
As before, combining the previous result with Proposition \ref{prop:atomic-bounds-kc}, we obtain the following bound for the Kronecker coefficients.
\begin{cor} \label{cor:kc-bounds-general}
Let $m,n$ be positive integers, and $\lambda, \mu, \nu$ be partitions of $N$ of lengths at most $mn,m,n$ respectively. Then
\[ g_{\lambda, \mu, \nu} \leq \frac{(mn)!}{2}R_2\] \label{eq:kc-easybound} where $R_2$ is the expression on the right-hand side of Inequality~\eqref{eq:atomic-easybound}.
\end{cor}
The bound given in line~\eqref{eq:kc-easybound} is $O(N^d)$, where $d$ is the difference between the number of columns and rows of $A^{m,n}$ - that is:
\begin{equation}
d = {\binom{mn}{2}} - {\binom{n}{2}} - {\binom{m}{2}} - n - m + 2
\end{equation}
whereas the bound given in \cite{PaPa20} is $O(N^{(mn)^2})$. Note that this analysis holds for the case when $\ell(\mu) = m, \ell(\nu) = n, \ell(\lambda) = mn$. For example, the bound given by Pak and Panova is stronger if $\ell(\lambda) = \ell(\mu) = \ell(\nu) = m$ since in this case
their bound is $O(N^{m^3})$, while our bound is $O(N^{{\binom{m^2}{2}} - 2{\binom{m}{2}} - 2m + 2})$. If $\ell(\mu) = m,\ \ell(\nu) = n$ are fixed, we find that the exponent $x$ given by our $O(N^x)$ expression is smaller when
\begin{equation} \label{eq:exponent-compare}
\ell(\lambda) > \frac{mn}{2} - \biggr(\frac{m^2 + n^2 + m + n - 4}{2mn}\biggr) - \frac{1}{2}
\end{equation}
and larger when the inequality is flipped. Note that for $\ell(\mu), \ell(\nu) \geq 2$ (i.e. $m,n \geq 2$), the expression on the right-hand side of \eqref{eq:exponent-compare} is smaller than $\frac{mn}{2}$.
\begin{comment}
For any $m,n$, we find that the bound given by Inequality \eqref{eq:kc-easybound1} is stronger than the bound given in Inequality \eqref{eq:pp-bound2} as $N \to \infty$. Namely, we compute that
\begin{equation}
\lim_{N \to \infty} \frac{\frac{(mn)!}{2}{N + {\binom{mn}{2}} - {\binom{m}{2}} - {\binom{\binom{n}{2}} - (m-1)(n-1) - 1}{N}}{\binom{(2m-1)N + m-2}{(2m-1)N }^{n-3}}}{\biggr(1 + \frac{(mn)^2)}{N}\biggr)^N\biggr(1 + \frac{N}{(mn)^2}\biggr)^{(mn)^2}} = 0
\end{equation}
\end{comment}
\begin{comment}
In the case that $\lambda = \mu = \nu = (3,2,1), (3,2,1), (3,2,1)$. We get $\tilde{g}^{3,3}_{\lambda, \mu, \nu} \leq 16380.$ Pak and Panova's Conjecture 6.2 gives $602e^{-O(m)}$ where $m=9$ in this case.
\end{comment}
We give the explicit bound in the $m=n=3$ case for which there is no efficient computational tool.
\begin{cor} \label{cor:atomic-bounds} For all partitions $\lambda, \mu, \nu$ of $N$ with $\ell(\mu), \ell(\nu) \leq 3, \ell(\lambda) \leq 9$.
\begin{equation}
g_{\lambda, \mu, \nu} \leq \frac{9!}{2} {\binom{b_1 + 15}{15}}{\binom{b_2 + 7}{7}}{\binom{b_3 + 1}{1}}{\binom{b_4 + 3}{3}}
\end{equation}
where
\begin{align*}
b_1 &= \nu_2 + \nu_3 -\lambda_{4} - \lambda_{5} - \lambda_{6} - \lambda_{7} - \lambda_{8} - \lambda_{9} \\
b_2 &= \mu_{2} + \mu_{3} + \nu_{2} + \nu_{3} -\lambda_{2} - \lambda_{3} - \lambda_{4} - \lambda_{5} - 2\lambda_{6} - 2\lambda_{7} - 2\lambda_{8} - 2\lambda_{9} \\
b_3 &= \mu_3 + \nu_2 + \nu_3 -\lambda_{3} - \lambda_{4} - \lambda_{5} - \lambda_{6} - \lambda_{7} - 2 \, \lambda_{8} - 2 \, \lambda_{9} \\
b_4 &= \mu_{2} + 2 \, \mu_{3} + 2 \, \nu_{2} + 3 \, \nu_{3} -\lambda_{2} - 2 \, \lambda_{3} - 2 \, \lambda_{4} - 3 \, \lambda_{5} - 3 \, \lambda_{6} - 4 \, \lambda_{7} - 4 \, \lambda_{8} - 5 \, \lambda_{9}.
\end{align*}
\end{cor}
\begin{eg}
Table~\ref{tab:bound} presents bounds on $g_{\lambda, \mu, \nu}$ where
$\lambda = (15, 15, 15, 10, 10, 10, 10, 10, 5)$, $\mu = (35, 35, 30)$, $\nu = (40, 30, 30)$.
\begin{table}[h]
\begin{tabular}{lcc}\toprule
Source & Bound \\ \midrule
Corollary \ref{cor:kc-bounds} & $1.42 \cdot 10^{16}$\\
Corollary \ref{cor:kc-bounds-general} & $5.38 \cdot 10^{45}$ \\
Pak and Panova, Inequality \ref{eq:pp-bound1} \cite{PaPa14} & $2.84 \cdot 10^{27}$\\
Pak and Panova, Inequality \ref{eq:pp-bound2} \cite{PaPa20} & $1.13 \cdot 10^{54}$\\ \bottomrule
\end{tabular}
\bigskip
\caption{Upper bound comparison for $g_{15^3\,10^5\,5,35^2\,30,40\,30^2}$}
\label{tab:bound}
\end{table}
The bound given by Inequality~\ref{eq:pp-bound1} by Pak and Panova is better on some examples. From our experience our bound is the better choice when~$\lambda$ is close to rectangular due to the large coefficients on small parts of~$\lambda$.
\end{eg}
\section{Conclusion and open problems} \label{sec:conclusion}
The partition function approach to Kronecker functions is elementary, yet provides a useful structure to calculate values and upper bounds. The notion of the atomic Kronecker Coefficient is very useful to determine vanishing conditions, bounds, and also to generate stable triples. We summarize a few open problems.
\tocless\subsection*{\bf 1.} Proving that the atomic Kronecker coefficient $\tilde{g}^{m,n}_{\lambda, \mu, \nu}$ is an upper bound for the Kronecker coefficient $g_{\lambda, \mu, \nu}$ for general partitions. Proving this would allow us to remove the factorial growth in the lengths $m,n$ (the $\frac{(mn)!}{2}$ term).
\tocless\subsection*{\bf 2.} Determine which alternant terms make a non-zero contribution to the Kronecker coefficient, and find cancelling terms (i.e. determine the minimal number of $\sigma \in \mathfrak{S}_{mn}$ needed to sum over in Eq.~\eqref{eq:vpf-to-kron}). This would speed up the computation of the Kronecker coefficients since it reduces the number of vector partition function evaluations necessary.
\tocless\subsection*{\bf 3.} For $\lambda, \mu, \nu$ with $\ell(\mu), \ell(\nu) \leq 2, \ell(\lambda) \leq 4$ one can compute $g_{\lambda, \mu, \nu}$ via Theorem~\ref{theo:vpf-to-kron} with $m=n=2$. One can also apply this computation with $m=2,\ n=3$ (or any choice of $m,n$), however in this case one gets many more alternant terms. Is there a way to exploit this in order to simplify the expression in the $m=2,\ n=3$ case (and in general)?
\tocless\subsection*{\bf 4.} Explore the structure of the poset in the discussion preceding Lemma \ref{lem:domination} for general $m,n$.
\tocless\subsection*{\bf 5.} Compute $p_{A^{3,3}}$ as a piecewise quasi-polynomial.
\section{Acknowledgments}
We are extremely grateful to have been given access to the computational resources provided by WestGrid (www.westgrid.ca) and Compute Canada Calcul Canada (www.computecanada.ca) We also thank John Hebron for important technical assistance locally.
Both authors benefited from the financial support of the the National Science and Engineering Research Council (NSERC) of Canada, via NSERC Discovery Grant R611453. We also wish to explicitly acknowledge the important support of Mercedes Rosas and Sheila Sundaram throughout this project. Their feedback, commentary and insights have been essential.
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\section{Introduction}
Codimension $1$ holomorphic foliations on complex projective spaces
are a central theme in complex dynamics.
An important numerical invariant of such a foliation $\mathscr{F}$ is its degree $\deg(\mathscr{F})$, defined as the number
of tangencies of a general line with $\mathscr{F}$.
Codimension $1$ foliations on $\p^n$ with low degree present special behavior, and
those with $\deg(\mathscr{F})\le 2$ have been classified.
Codimension $1$ foliations on $\p^n$ with $\deg(\mathscr{F})\le 1$ were classified in \cite{jouanolou}.
If $\deg(\mathscr{F})=0$, then $\mathscr{F}$ is induced by a pencil of hyperplane sections, i.e., it is the relative tangent
sheaf to a linear projection $\p^n\dashrightarrow \p^1$.
If $\deg(\mathscr{F})=1$, then
\begin{itemize}
\item either $\mathscr{F}$ is induced by a pencil of hyperquadrics containing a double hyperplane, or
\item $\mathscr{F}$ is the linear pullback of a foliation on $\p^{2}$ induced by a global
holomorphic vector field.
\end{itemize}
Codimension $1$ foliations of degree $2$ on $\p^n$ were classified in \cite{CLN}.
The space of such foliations has $6$ irreducible components, and much is known about them.
In particular, when $n\ge 4$, their leaves are always covered by rational curves.
In this paper we extend this classification to arbitrary complex projective manifolds.
In order to do so, we reinterpret the degree of a foliation $\mathscr{F}$ on $\p^n$
as a numerical invariant defined in terms of its \emph{canonical class} $K_{\mathscr{F}}:=-c_1(\mathscr{F})$.
For a codimension $1$ foliation $\mathscr{F}$ on $\p^n$, $\deg(\mathscr{F})=n-1+\deg(K_{\mathscr{F}})$.
So, foliations with low degree are precisely those with $-K_{\mathscr{F}}$ most positive.
\begin{defn}[\cite{fano_fols}]
A \emph{Fano foliation} is a holomorphic foliation $\mathscr{F}$ on a complex projective manifold $X$
such that $-K_{\mathscr{F}}$ is ample.
The \emph{index} $\iota_{\mathscr{F}}$ of $\mathscr{F}$ is the largest integer dividing $-K_{\mathscr{F}}$ in $\textup{Pic}(X)$.
\end{defn}
It follows from \cite[Theorem 0.1]{bogomolov_mcquillan01} that the leaves
of a Fano foliation $\mathscr{F}$ are always covered by positive dimensional rationally connected algebraic subvarieties of $X$.
Recent results suggest that the higher is the index of $\mathscr{F}$, the higher is the dimension of these subvarieties.
In order to state this precisely, we define the \emph{algebraic} and \emph{transcendental} parts of
a holomorphic foliation.
\begin{defn}
Let $\mathscr{F}$ be a holomorphic foliation of rank $r_{\mathscr{F}}$ on a normal variety $X$.
There exists a normal variety $Y$, unique up to birational equivalence, a dominant rational map with connected fibers $\varphi:X\dashrightarrow Y$,
and a holomorphic foliation $\mathscr{G}$ on $Y$
of rank $r_{\mathscr{G}}=r_{\mathscr{F}}-\big(\dim(X)-\dim(Y)\big)$ such that the following holds (see \cite[Section 2.4]{loray_pereira_touzet}).
\begin{enumerate}
\item $\mathscr{G}$ is purely transcendental, i.e., there is no positive dimensional algebraic subvariety through a general point of $Y$ that is tangent to $\mathscr{G}$; and
\item $\mathscr{F}$ is the pullback of $\mathscr{G}$ via $\varphi$ (see \ref{pullback_foliations} for this notion).
\end{enumerate}
The foliation on $X$ induced by $\varphi$ is called the \emph{algebraic part} of $\mathscr{F}$, and its rank is the \emph{algebraic rank} of $\mathscr{F}$,
which we denote by $r_{\mathscr{F}}^a$. When $r_{\mathscr{F}}^a=r_{\mathscr{F}}$, we say that $\mathscr{F}$ is \emph{algebraically integrable}.
\end{defn}
\begin{thm} \label{Thm:ADK}
Let $\mathscr{F}$ be a Fano foliation of rank $r_{\mathscr{F}}$ on a complex projective manifold $X$.
Then $\iota_{\mathscr{F}}\le r_{\mathscr{F}}$, and equality holds only if $X\cong \p^n$ \textup{(\cite[Theorem 1.1]{adk08})}.
In this case, by \cite[Th\'eor\`eme 3.8]{cerveau_deserti},
$\mathscr{F}$ is induced by a linear projection $\p^n \dashrightarrow \p^{n-r_{\mathscr{F}}}$.
In particular, $r_{\mathscr{F}}^a=r_{\mathscr{F}}$.
\end{thm}
In analogy with the case of Fano manifolds, we define \emph{del Pezzo foliations} to be
Fano foliations $\mathscr{F}$ with index $\iota_{\mathscr{F}} = r_{\mathscr{F}}-1\ge 1$.
Del Pezzo foliations were investigated in \cite{fano_fols} and \cite{codim_1_del_pezzo_fols}.
By \cite[Theorem 1.1]{fano_fols}, if $\mathscr{F}$ is a del Pezzo foliation on a complex projective manifold $X$, then $r_{\mathscr{F}}^a=r_{\mathscr{F}}$, except
when $X\cong \p^n$ and $\mathscr{F}$ is the pullback under a linear projection of a transcendental foliation on $\p^{n-r_{\mathscr{F}}+1}$ induced by a global
vector field, in which case $r_{\mathscr{F}}^a=r_{\mathscr{F}}-1$.
The following is the complete classification of codimension $1$ del Pezzo foliations
on complex projective manifolds.
For manifolds with Picard number $1$, the classification was obtained in \cite[Proposition 3.7]{lpt3fold}, while
\cite[Theorem 1.3]{codim_1_del_pezzo_fols} deals with mildly singular varieties of arbitrary Picard number.
\begin{thm}[{\cite[Theorem 1.3]{codim_1_del_pezzo_fols}}] \label{Thm:codim1_dP}
Let $\mathscr{F}$ be a codimension $1$ del Pezzo foliation on an $n$-dimensional complex projective manifold $X$.
\begin{enumerate}
\item Suppose that $\rho(X)=1$. Then either $X\cong\p^n$ and $\mathscr{F}$ is a degree $1$ foliation, or
$X\cong Q^n\subset \p^{n+1}$ and $\mathscr{F}$ is induced by a pencil of hyperplane sections.
\item Suppose that $\rho(X) \ge 2$. Then there exist
\begin{itemize}
\item an exact sequence of vector bundles on $\p^1$, $0\to \mathscr{K}\to \mathscr{E}\to \mathscr{V} \to 0$;
\item a foliation by curves $\mathscr{C}$ on $\p_C(\mathscr{K})$, generically transverse to the natural projection $p:\p_C(\mathscr{K})\to \p^1$,
induced by a nonzero global section of $T_{\p(\mathscr{K})}\otimes q^*\det(\mathscr{V})^*$;
\end{itemize}
such that $X\cong \p_C(\mathscr{E})$, and $\mathscr{F}$ is the pullback of $\mathscr{C}$ via the induced relative linear projection
$\p_C(\mathscr{E})\dashrightarrow \p_C(\mathscr{K})$.
Moreover, one of the following holds.
\begin{enumerate}
\item $(\mathscr{E},\mathscr{K})\cong \big(\mathscr{O}_{\p^1}(2)\oplus \mathscr{O}_{\p^1}(a)^{\oplus 2},\mathscr{O}_{\p^1}(a)^{\oplus 2}\big)$
for some positive integer $a$.
\item $(\mathscr{E},\mathscr{K})\cong \big(\mathscr{O}_{\p^1}(1)^{\oplus 2}\oplus \mathscr{O}_{\p^1}(a)^{\oplus 2},\mathscr{O}_{\p^1}(a)^{\oplus 2}\big)$
for some positive integer $a$.
\item $(\mathscr{E},\mathscr{K})\cong\big(\mathscr{O}_{\p^1}(1)\oplus \mathscr{O}_{\p^1}(a)\oplus \mathscr{O}_{\p^1}(b),\mathscr{O}_{\p^1}(a)\oplus \mathscr{O}_{\p^1}(b)\big)$ for distinct
positive integers $a$ and $b$.
\end{enumerate}
\end{enumerate}
\end{thm}
Next we define \emph{Mukai foliations} as Fano foliations $\mathscr{F}$ with index $\iota_{\mathscr{F}} = r_{\mathscr{F}}-2\ge 1$.
When $X=\p^n$ and $r_{\mathscr{F}}=n-1\ge 3$, the Mukai condition is equivalent to $\deg(\mathscr{F})=2$.
One checks from the classification in \cite{CLN} that $r_{\mathscr{F}}^a\ge r_{\mathscr{F}}-2$.
The aim of this paper is to classify codimension $1$ Mukai foliations on complex projective manifolds $X\not\cong\p^n$.
The classification is summarized in the following two theorems.
\begin{thm}\label{main_thm_rho=1}
Let $\mathscr{F}$ be a codimension $1$ Mukai foliation on an $n$-dimensional complex projective manifold $X\not\cong\p^n$ with
$\rho(X)=1$, $n\ge 4$. Then the pair $(X,\mathscr{F})$ satisfies one of the following conditions.
\begin{enumerate}
\item $X\cong Q^n\subset \p^{n+1}$ and $\mathscr{F}$ is one of the following.
\begin{enumerate}
\item $\mathscr{F}$ cut out by a general pencil of hyperquadrics of $\p^{n+1}$ containing a double hyperplane.
In this case, $r_{\mathscr{F}}^a= r_{\mathscr{F}}$.
\item $\mathscr{F}$ is the pullback under the restriction of a linear projection $\p^{n+1}\dashrightarrow \p^2$ of a foliation on $\p^{2}$ induced by a
global vector field. In this case, $r_{\mathscr{F}}^a\ge r_{\mathscr{F}}-1$.
\end{enumerate}
\item $X$ is a Fano manifold with $\rho(X)=1$ and index $\iota_X=n-1$, and $\mathscr{F}$ is induced by a pencil in $|\mathscr{O}_X(1)|$, where $\mathscr{O}_X(1)$
is the ample generator of $\textup{Pic}(X)$. In this case, $r_{\mathscr{F}}^a= r_{\mathscr{F}}$.
\end{enumerate}
\end{thm}
\begin{rem}
In case (2), by Fujita's classification (see Section~\ref{Fano_manifolds_high_index}), $X$ is isomorphic to one of the following:
\begin{itemize}
\item A cubic hypersurface in $\p^{n+1}$.
\item An intersection of two hyperquadrics in $\p^{n+2}$.
\item A linear section of the Grassmannian $G(2,5)\subset\p^9$ under the Pl\"ucker embedding.
\item A hypersurface of degree $4$ in the weighted projective space $\p(2,1,\ldots,1)$.
\item A hypersurface of degree $6$ in the weighted projective space $\p(3,2,1,\ldots,1)$.
\end{itemize}
\end{rem}
\begin{thm}\label{main_thm_rho>1}
Let $X$ be an $n$-dimensional complex projective manifold with $\rho(X)>1$, $n\ge 4$.
Let $\mathscr{F}$ be a codimension $1$ Mukai foliation on $X$.
Then one of the following holds.
\begin{enumerate}
\item $X$ admits a $\p^{n-1}$-bundle structure $\pi:X\to \p^1$, $r_{\mathscr{F}}^a= r_{\mathscr{F}}$,
and the restriction of $\mathscr{F}$ to a general fiber of $\pi$ is induced by a
pencil of hyperquadrics of $\p^{n-1}$ containing a double hyperplane.
\item There exist
\begin{itemize}
\item a complete smooth curve $C$, together with an exact sequence of vector bundles on $C$
$$
0\ \to \ \mathscr{K} \ \to \ \mathscr{E} \ \to \ \mathscr{V} \ \to \ 0,
$$
with $\mathscr{E}$ ample of rank $n$, and $r:=\textup{rank}(\mathscr{K})\in \{2,3\}$;
\item a codimension $1$ foliation $\mathscr{G}$ on $\p_C(\mathscr{K})$, generically transverse to the natural projection $p:\p_C(\mathscr{K})\to C$,
satisfying $\det(\mathscr{G}) \cong p^*\big(\det(\mathscr{V})\big)\otimes \mathscr{O}_{\p_C(\mathscr{K})}(r-3)$ and $r_{\mathscr{G}}^a\ge r_{\mathscr{G}}-1$;
\end{itemize}
such that $X\cong \p_C(\mathscr{E})$, and $\mathscr{F}$ is the pullback of $\mathscr{G}$ via the induced relative linear projection
$\p_C(\mathscr{E})\dashrightarrow \p_C(\mathscr{K})$. In this case, $r_{\mathscr{F}}^a\ge r_{\mathscr{F}}-1$.
\item $X$ admits a $Q^{n-1}$-bundle structure $\pi:X\to \p^1$, $r_{\mathscr{F}}^a= r_{\mathscr{F}}$,
and the restriction of $\mathscr{F}$ to a general fiber of $\pi$ is induced by a
pencil of hyperplane sections of $Q^{n-1}$.
More precisely, there exist
\begin{itemize}
\item an exact sequence of vector bundles on $\p^1$
$$
0\ \to \ \mathscr{K} \ \to \ \mathscr{E} \ \to \ \mathscr{V} \ \to \ 0,
$$
with natural projections $\pi:\p_{\p^1}(\mathscr{E})\to \p^1$
and $q:\p_{\p^1}(\mathscr{K})\to \p^1$;
\item an integer $b$ and a foliation by rational curves $\mathscr{G}\cong q^*\big(\det(\mathscr{V})\otimes \mathscr{O}_{\p^1}(b)\big)$ on $\p_{\p^1}(\mathscr{K})$;
\end{itemize}
such that $X\in \big|\mathscr{O}_{\p(\mathscr{E})}(2)\otimes \pi^*\mathscr{O}(b)\big|$, and
$\mathscr{F}$ is the pullback of $\mathscr{G}$ via the restriction to $X$ of the relative linear projection $\p_{\p^1}(\mathscr{E}) \dashrightarrow \p_{\p^1}(\mathscr{K})$.
Moreover, one of the following holds.
\begin{enumerate}
\item $(\mathscr{E},\mathscr{K})\cong (\mathscr{O}_{\p^1}(a)^{\oplus 2}\oplus\mathscr{O}_{\p^1}^{\oplus 3},\mathscr{O}_{\p^1}(a)^{\oplus 2})$
for some integer $a\ge 1$, and $b=2$ (n=4).
\item $(\mathscr{E},\mathscr{K})\cong (\mathscr{O}_{\p^1}(a)^{\oplus 2}\oplus\mathscr{O}_{\p^1}^{\oplus 2}\oplus \mathscr{O}_{\p^1}(1),\mathscr{O}_{\p^1}(a)^{\oplus 2})$
for some integer $a\ge 1$, and $b=1$ (n=4).
\item $(\mathscr{E},\mathscr{K})\cong (\mathscr{O}_{\p^1}(a)^{\oplus 2}\oplus\mathscr{O}_{\p^1}\oplus \mathscr{O}_{\p^1}(1)^{\oplus 2},\mathscr{O}_{\p^1}(a)^{\oplus 2})$
for some integer $a\ge 1$, and $b=0$ (n=4).
\item $\mathscr{K}\cong \mathscr{O}_{\p^1}(a)^{\oplus 2}$ for some integer $a$, and
$\mathscr{E}$ is an ample vector bundle of rank $5$ or $6$ with $\deg(\mathscr{E})=2+2a-b$ ($n\in\{4,5\}$).
\item $\mathscr{K}\cong \mathscr{O}_{\p^1}(a)\oplus \mathscr{O}_{\p^1}(c)$ for distinct integers $a$ and $c$, and
$\mathscr{E}$ is an ample vector bundle of rank $5$ or $6$ with $\deg(\mathscr{E})=1+a+c-b$ ($n\in\{4,5\}$).
\end{enumerate}
\item There exist
\begin{itemize}
\item a smooth projective surface $S$, together with an exact sequence of $\mathscr{O}_S$-modules
$$
0\ \to \ \mathscr{K} \ \to \ \mathscr{E} \ \to \ \mathscr{Q} \ \to \ 0,
$$
where $\mathscr{K}$, $\mathscr{E}$, and $\mathscr{V}:=\mathscr{Q}^{**}$ are vector bundles on $S$, $\mathscr{E}$ is ample of rank $n-1$, and
$\textup{rank}(\mathscr{K})=2$;
\item a codimension $1$ foliation $\mathscr{G}$ on $\p_S(\mathscr{K})$, generically transverse to the natural projection $q:\p_{S}(\mathscr{K})\to S$,
satisfying $\det(\mathscr{G})\cong q^*\det(\mathscr{V})$ and $r_{\mathscr{G}}^a\ge 1$;
\end{itemize}
such that $X\cong \p_{S}(\mathscr{E})$, and
$\mathscr{F}$ is the pullback of $\mathscr{G}$ via the induced relative linear projection $\p_C(\mathscr{E})\dashrightarrow \p_C(\mathscr{K})$.
In this case, $r_{\mathscr{F}}^a\ge r_{\mathscr{F}}-1$.
Moreover, one of the following holds.
\begin{enumerate}
\item $S\cong\p^2$, $\det(\mathscr{V})\cong\mathscr{O}_{\p^2}(i)$ for some $i\in \{1,2,3\}$, and $4\le n\le 3+i$.
\item $S$ is a del Pezzo surface $\not\cong\p^2$, $\det(\mathscr{V})\cong \mathscr{O}_S(-K_S)$ , and $4\le n\le 5$.
\item $S\cong \p^1\times \p^1$, $\det(\mathscr{V})$ is a line bundle of type $(1,1)$, $(2,1)$ or $(1,2)$, and $n=4$.
\item $S\cong \mathbb{F}_e$ for some integer $e \ge 1$, $\det(\mathscr{V})\cong\mathscr{O}_{\mathbb{F}_e}(C_0+(e+i)f)$,
where $i\in\{1,2\}$, $C_0$ is the minimal section of the natural morphism $\mathbb{F}_e\to\p^1$,
$f$ is a general fiber, and $n=4$.
\end{enumerate}
\item $n=5$, $X$ is the blowup of one point $P\in \p^5$, and $\mathscr{F}$ is induced by a pencil
of hyperplanes in $\p^5$ containing $P$ in its base locus.
\item $n=4$, $X$ is the blowup of $\p^4$ at $m\le 8$ points in general position on a plane $\p^2 \cong S\subset \p^4$,
and $\mathscr{F}$ is induced by the pencil of hyperplanes in $\p^4$ with base locus $S$.
\item $n=4$, $X$ is the blowup of a smooth quadric $Q^4$ at $m\le 7$ points in general position on a
codimension $2$ linear section $Q^2 \cong S\subset Q^4$,
and $\mathscr{F}$ is induced by the pencil of hyperplanes sections of $Q^4\subset \p^5$ with base locus $S$.
\end{enumerate}
\end{thm}
\begin{rem} \
\begin{itemize}
\item Foliations $\mathscr{G}$ that appear in Theorem~\ref{main_thm_rho>1}(2)
are classified in Proposition~\ref{proposition:P-bdle_over_curve_(2)}.
\item Foliations $\mathscr{G}$ that appear in Theorem~\ref{main_thm_rho>1}(4) are classified in Remark~\ref{rem:flat_connection},
Proposition~\ref{proposition:P-bdle_over_S_hirzebruch},
Proposition~\ref{proposition:P-bdle_over_S_plane}, and
Proposition~\ref{proposition:P-bdle_over_S_plane_2}.
\end{itemize}
\end{rem}
This paper is organized as follows.
In Section~\ref{section:foliations}, we review basic definitions and results
about holomorphic foliations and Fano foliations.
Section~\ref{section:rho=1} is devoted to Mukai foliations on manifolds with Picard number $1$.
First we show that if $X$ is a manifold with $\rho(X)=1$ admitting a codimension $1$ Mukai foliation $\mathscr{F}$, then $X$
is a Fano manifold with index $\iota_X\ge \dim(X)-1$ (Lemma~\ref{lemma:bound_on_index2}).
The proof of Theorem~\ref{main_thm_rho=1} relies on the classification of such manifolds, which is reviewed
in Section~\ref{Fano_manifolds_high_index}.
The proof distinguishes two cases, depending on whether or not $\mathscr{F}$ is semi-stable.
When $\mathscr{F}$ is not semi-stable, then it contains a codimension $2$ del Pezzo subfoliation $\mathscr{G}$.
Such foliations are classified in Section~\ref{section:del_pezzo_codim2}
(Theorem~\ref{thm:codim2_delPezzo}).
Section~\ref{section:rho>1} is devoted to Mukai foliations on manifolds with Picard number $>1$.
The existence of a codimension $1$ Mukai foliation $\mathscr{F}$ on a manifold $X$ with $\rho(X)>1$
implies the existence of an extremal ray in $\overline{\textup{NE}}(X)$ with large length.
We use adjunction theory to classify all possible contractions of such extremal rays
(Theorem~\ref{tironi}).
In order to prove Theorem~\ref{main_thm_rho>1},
we analyze the behavior of the foliation $\mathscr{F}$
with respect to the contraction of a large extremal ray.
This is done separately for each type of contraction.
Section~\ref{subsection:P-bdles/curves} deals with projective space bundles over curves.
Section~\ref{subsection:Q-bdles/curves} deals with quadric bundles over curves.
Section~\ref{subsection:P-bdles/surfaces} deals with projective space bundles over surfaces.
Birational contractions are treated in Section~\ref{subsection:proof_main_rho>0}.
\
\noindent {\bf Notation and conventions.}
We always work over the field ${\mathbb C}$ of complex numbers.
Varieties are always assumed to be irreducible.
We denote by $\textup{Sing}(X)$ the singular locus of a variety $X$.
Given a sheaf $\mathscr{F}$ of $\mathscr{O}_X$-modules on a variety $X$, we denote by $\mathscr{F}^{*}$ the sheaf $\mathscr{H}\hspace{-0.1cm}\textit{om}_{\mathscr{O}_X}(\mathscr{F},\mathscr{O}_X)$.
If $r$ is the generic rank of $\mathscr{F}$, then we denote by $\det (\mathscr{F})$ the sheaf $(\wedge^r \mathscr{F})^{**}$.
If $\mathscr{G}$ is another sheaf of $\mathscr{O}_X$-modules on $X$, then we denote by $\mathscr{F}[\otimes]\mathscr{G}$ the sheaf $(\mathscr{F}\otimes\mathscr{G})^{**}$.
If $\mathscr{E}$ is a locally free sheaf of $\mathscr{O}_X$-modules on a variety $X$,
we denote by $\p_X(\mathscr{E})$ the Grothendieck projectivization $\textup{Proj}_X(\textup{Sym}(\mathscr{E}))$,
and by $\mathscr{O}_{\p}(1)$ its tautological line bundle.
If $X$ is a normal variety and $X\to Y$ is any morphism, we denote by
$T_{X/Y}$ the sheaf $(\Omega_{X/Y}^1)^*$. In particular, $T_X=(\Omega_{X}^1)^*$.
If $X$ is a smooth variety and $D$ is a reduced divisor on $X$ with simple normal crossings support,
we denote by $\Omega_X^1(\textup{log }D)$ the sheaf of differential $1$-forms with logarithmic poles
along $D$, and by $T_X(-\textup{log }D)$ its dual sheal $\Omega_X^1(\textup{log }D)^*$. Notice that
$\det(\Omega_X^1(\textup{log }D))\cong\mathscr{O}_X(K_X+D)$.
We denote by $Q^n$ a (possibly singular) quadric hypersurface in $\p^{n+1}$.
Given line bundles $\mathscr{L}_1$ and $\mathscr{L}_2$ on varieties $X$ and $Y$, we denote by $\mathscr{L}_1\boxtimes\mathscr{L}_2$ the line bundle
$\pi_1^*\mathscr{L}_1\otimes \pi_2^* \mathscr{L}_2$ on $X\times Y$, where $\pi_1$ and $\pi_2$ are the projections onto $X$ and $Y$,
respectively.
Let $L$ be a Cartier divisor on a projective variety.
We denote by $\textup{Bs}(L)$ the base locus of the complete linear system $|L|$.
\
\noindent {\bf Acknowledgements.}
Much of this work was developed during the authors' visits to IMPA and Institut Fourier.
We would like to thank both institutions for their support and hospitality.
\section{Preliminaries}\label{section:foliations}
\subsection{Foliations}
\begin{defn}
A \emph{foliation} on a normal variety $X$ is a (possibly zero) coherent subsheaf $\mathscr{F}\subsetneq T_X$ such that
\begin{itemize}
\item $\mathscr{F}$ is closed under the Lie bracket, and
\item $\mathscr{F}$ is saturated in $T_X$ (i.e., $T_X / \mathscr{F}$ is torsion free).
\end{itemize}
The \emph{rank} $r_{\mathscr{F}}$ of $\mathscr{F}$ is the generic rank of $\mathscr{F}$.
The \emph{codimension} of $\mathscr{F}$ is defined as $q_{\mathscr{F}}:=\dim(X)-r_{\mathscr{F}}\ge 1$.
The inclusion $\mathscr{F}\hookrightarrow T_X$ induces a nonzero map
$$
\eta: \ \Omega_X^{r_{\mathscr{F}}}=\wedge^{r_{\mathscr{F}}}(\Omega_X^1) \to \wedge^{r_{\mathscr{F}}}(T_X^*)
\to \wedge^{r_{\mathscr{F}}}(\mathscr{F}^*) \to \det(\mathscr{F}^*).
$$
The \emph{singular locus of $\mathscr{F}$} is the singular scheme of this map.
I.e., it is the closed subscheme of $X$ whose ideal sheaf is the image of
the induced map $\Omega^{r_{\mathscr{F}}}_X[\otimes] \det(\mathscr{F})\to \mathscr{O}_X$.
A closed subvariety $Y$ of $X$ is said to be \emph{invariant} by $\mathscr{F}$ if
it is not contained in the singular locus of $\mathscr{F}$, and
the restriction $\eta_{|Y} : {\Omega^{r_{\mathscr{F}}}_X}_{|Y}\to \det(\mathscr{F}^*)_{|Y}$ factors through the natural map
${\Omega^{r_{\mathscr{F}}}_X}_{|Y}\to\Omega^{r_{\mathscr{F}}}_Y$.
\end{defn}
\begin{say}[Foliations defined by $q$-forms] \label{q-forms}
Let $\mathscr{F}$ be a codimension $q$ foliation on an $n$-dimenional normal variety $X$.
The \emph{normal sheaf} of $\mathscr{F}$ is $N_\mathscr{F}:=(T_X/\mathscr{F})^{**}$.
The $q$-th wedge product of the inclusion
$N^*_\mathscr{F}\hookrightarrow (\Omega^1_X)^{**}$ gives rise to a nonzero global section
$\omega\in H^0\big(X,\Omega^{q}_X[\otimes] \det(N_\mathscr{F})\big)$
whose zero locus has codimension at least $2$ in $X$.
Such $\omega$ is \emph{locally decomposable} and \emph{integrable}.
To say that $\omega$ is locally decomposable means that,
in a neighborhood of a general point of $X$, $\omega$ decomposes as the wedge product of $q$ local $1$-forms
$\omega=\omega_1\wedge\cdots\wedge\omega_q$.
To say that it is integrable means that for this local decomposition one has
$d\omega_i\wedge \omega=0$ \ $\forall i\in\{1,\ldots,q\}$.
Conversely, let $\mathscr{L}$ be a reflexive sheaf of rank $1$ on $X$, $q\ge 1$, and
$\omega\in H^0(X,\Omega^{q}_X[\otimes] \mathscr{L})$ a global section
whose zero locus has codimension at least $2$ in $X$.
Suppose that $\omega$ is locally decomposable and integrable.
Then one defines
a foliation of rank $r=n-q$ on $X$ as the kernel
of the morphism $T_X \to \Omega^{q-1}_X[\otimes] \mathscr{L}$ given by the contraction with $\omega$.
These constructions are inverse of each other.
\end{say}
\begin{say}[Foliations described as pullbacks] \label{pullback_foliations}
Let $X$ and $Y$ be normal varieties, and $\varphi:X\dashrightarrow Y$ a dominant rational map that restricts to a morphism $\varphi^\circ:X^\circ\to Y^\circ$,
where $X^\circ\subset X$ and $Y^\circ\subset Y$ are smooth open subsets.
Let $\mathscr{G}$ be a codimension $q$ foliation on $Y$ defined by a twisted $q$-form
$\omega\in H^0\big(Y,\Omega^{q}_Y[\otimes] \det(N_\mathscr{G})\big)$.
Then $\omega$ induces a nonzero twisted $q$-form
$\omega_{X^\circ}\in
H^0\Big(X^\circ,\Omega^{q}_{X^\circ}[\otimes] (\varphi^\circ)^*\big(\det(N_\mathscr{G})_{|Y^\circ}\big)\Big)$ which in turn defines a codimension $q$ foliation $\mathscr{F}^\circ$ on $X^\circ$. We say that
the saturation $\mathscr{F}$ of $\mathscr{F}^\circ$ in $T_X$
\emph{is the pullback of $\mathscr{G}$ via $\varphi$},
and write $\mathscr{F}=\varphi^{-1}\mathscr{G}$.
Suppose that $X^\circ$ can be taken so that $\varphi^\circ$ is an equidimensional morphism. Let
$(B_i)_{i\in I}$
be the (possibly empty) set of hypersurfaces in $Y^\circ$ contained in the set of critical values of $\varphi^\circ$ and invariant by $\mathscr{G}$. A straightforward computation shows that
\begin{equation}\label{pullback_fol}
N_{\mathscr{F}^\circ} \ \cong \ (\varphi^\circ)^*N_{\mathscr{G}_{|Y^\circ}}
\otimes\mathscr{O}_{X^\circ}\Big(\sum_{i\in I}\big((\varphi^\circ)^*B_i\big)_{red}-(\varphi^\circ)^*B_i\Big).
\end{equation}
Conversely, let $\mathscr{F}$ be a foliation on $X$, and suppose that the general fiber of $\varphi$ is tangent to $\mathscr{F}$. This means that, for a general point $x$ on a general fiber $F$ of $\varphi$,
the linear subspace $\mathscr{F}_x\subset T_xX$ determined by the inclusion $\mathscr{F}\subset T_X$
contains $T_xF$. Suppose moreover that $\varphi^\circ$ is smooth
with connected fibers. Then, by \cite[Lemma 6.7]{fano_fols}, there is a holomorphic foliation $\mathscr{G}$ on $Y$
such that $\mathscr{F}=\varphi^{-1}\mathscr{G}$.
Suppose that
$X^\circ$
can be taken so that $\textup{codim}_X(X\setminus X^\circ)\ge 2$.
Denote by $T_{X/Y}$ the saturation of $T_{X^\circ/Y^\circ}$ in $T_X$, and by $\varphi^*\mathscr{G}$ an extension of
$(\varphi^\circ)^*\mathscr{G}_{|Y^\circ}$ to $X$.
Then \eqref{pullback_fol} gives
\begin{equation} \label{K_pullback_fol}
\det(\mathscr{F})\cong \det(T_{X/Y})[\otimes] \det(\varphi^*\mathscr{G}).
\end{equation}
\end{say}
\begin{defn}
Let $\mathscr{F}$ be a foliation on a normal projective variety $X$.
The \textit{canonical class} $K_{\mathscr{F}}$ of $\mathscr{F}$ is any Weil divisor on $X$ such that $\mathscr{O}_X(-K_{\mathscr{F}})\cong \det(\mathscr{F})$.
\end{defn}
\begin{say}[Restricting foliations to subvarieties] \label{restricting_fols}
Let $X$ be a smooth projective variety, and $\mathscr{F}$ a
codimension $q$ foliation on $X$ defined by a twisted $q$-form $\omega\in H^0\big(X,\Omega^{q}_X\otimes \det(N_\mathscr{F})\big)$.
Let $Z$ be a smooth subvariety with normal bundle $N_{Z/X}$.
Suppose that the restriction of $\omega$ to $Z$ is nonzero.
Then it induces a nonzero twisted $q$-form
$\omega_Z\in H^0\big(Z,\Omega^{q}_Z\otimes \det(N_\mathscr{F})_{|Z}\big)$,
and a codimension $q$ foliation $\mathscr{F}_Z$ on $Z$.
There is a maximal effective divisor $B$ on $Z$ such that
$\omega_Z\in H^0\big(Z,\Omega^{q}_Z\otimes \det(N_\mathscr{F})_{|Z}(-B)\big)$.
A straightforward computation shows that
$$
\mathscr{O}_Z\big(K_{\mathscr{F}_Z}\big) \ \cong \ \det(N_{Z/X})({K_{\mathscr{F}}}_{|Z}-B).
$$
\end{say}
\begin{defn}
Let $X$ be normal variety. A foliation $\mathscr{F}$ on $X$ is said to be
\emph{algebraically integrable} if the leaf of $\mathscr{F}$ through a general point of $X$ is an algebraic variety.
In this situation, by abuse of
notation we often use the word \textit{leaf} to mean the closure in $X$ of a leaf of $\mathscr{F}$.
\end{defn}
\begin{say}[{\cite[Lemma 3.2]{fano_fols}}]\label{lemma:leaffoliation} \label{notation:family_leaves}
Let $X$ be normal projective variety, and $\mathscr{F}$ an algebraically integrable foliation on $X$.
There is a unique irreducible closed subvariety $W$ of $\textup{Chow}(X)$
whose general point parametrizes the closure of a general leaf of $\mathscr{F}$
(viewed as a reduced and irreducible cycle in $X$). In other words, if
$U \subset W\times X$ is the universal cycle, with universal morphisms
$\pi:U\to W$ and $e:U\to X$,
then $e$ is birational, and, for a general point $w\in W$,
$e\big(\pi^{-1}(w)\big) \subset X$ is the closure of a leaf of $\mathscr{F}$.
We call the normalization $\tilde W$ of $W$ the \emph{space of leaves} of $\mathscr{F}$,
and the induced rational map $X\dashrightarrow \tilde W$
a \emph{rational first integral for $\mathscr{F}$}.
\end{say}
We end this subsection with a useful criterion of algebraic integrability for foliations.
\begin{thm}[{\cite[Theorem 0.1]{bogomolov_mcquillan01}, \cite[Theorem 1]{kebekus_solaconde_toma07}}] \label{thm:BM}
Let $X$ be a normal complex projective variety, and $\mathscr{F}$ a foliation on $X$.
Let $C \subset X$ be a complete curve disjoint from the singular loci of $X$ and $\mathscr{F}$.
Suppose that the restriction $\mathscr{F}_{|C}$ is an ample vector bundle on $C$.
Then the leaf of $\mathscr{F}$ through any point of $C$ is an algebraic variety, and the
leaf of $\mathscr{F}$ through a general point of $C$ is rationally connected.
\end{thm}
\subsection{Fano foliations}
\begin{defn}
Let $\mathscr{F}$ be a foliation on a normal projective variety $X$.
We say that $\mathscr{F}$ is a \emph{Fano foliation} (respectively \emph{$\textup{\textbf{Q}}$-Fano foliation})
if $-K_{\mathscr{F}}$ is an ample Cartier (respectively $\textup{\textbf{Q}}$-Cartier) divisor on $X$.
The \emph{index} $\iota_{\mathscr{F}}$ of a Fano foliation $\mathscr{F}$ on $X$
is the largest integer dividing $-K_{\mathscr{F}}$ in $\textup{Pic}(X)$.
We say that a Fano foliation $\mathscr{F}$ is a \emph{del Pezzo foliation} if $\iota_{\mathscr{F}} = r_{\mathscr{F}}-1$.
We say that it is a \emph{Mukai foliation} if $\iota_{\mathscr{F}} = r_{\mathscr{F}}-2$.
\end{defn}
The existence of a $\textup{\textbf{Q}}$-Fano foliation on a variety $X$ imposes strong restrictions on $X$.
\begin{thm}[{\cite[Theorem 1.4]{codim_1_del_pezzo_fols}}]\label{Thm:KX-KF_not_nef}
Let $X$ be a klt projective variety, and $\mathscr{F}\subsetneq T_X$ a $\textup{\textbf{Q}}$-Fano foliation.
Then $K_X-K_\mathscr{F}$ is not pseudo-effective.
\end{thm}
Suppose that a complex projective manifold $X$ admits a Fano foliation $\mathscr{F}$.
By Theorem~\ref{Thm:KX-KF_not_nef},
$K_X$ is not pseudo-effective, and hence $X$ is uniruled by \cite{bdpp}.
So we can consider a \emph{minimal dominating family of rational curves} on $X$.
This is an irreducible component $H$ of $\rat(X)$ such that
\begin{itemize}
\item the curves parametrized by $H$ sweep out a dense subset of $X$, and
\item for a general point $x\in X$, the subset of $H$ parametrizing curves through $x$ is proper.
\end{itemize}
To compute the intersection number $-K_{\mathscr{F}}\cdot \ell$, where $\ell$ is a general curve from the family $H$,
we will use the following observations.
\begin{lemma}\label{lemma:bound_on_pseudo_index_bis}
Let $X$ be a complex projective manifold, and $\mathscr{F}$ a codimension one foliation on $X$.
Let $C\subset X$ be a curve not contained in the singular locus of $\mathscr{F}$,
and denote by $g$ its geometric genus.
If $C$ is not tangent to $\mathscr{F}$, then
$-K_\mathscr{F}\cdot C \le -K_X\cdot C + 2g-2$.
\end{lemma}
\begin{proof}
Set $n:=\dim(X)$, and let $\omega\in H^0\big(X,\Omega^1_{X}\otimes \det(N_\mathscr{F})\big)$
be a $1$-form
defining $\mathscr{F}$, as in \ref{q-forms}.
Consider the normalization morphism $f:\tilde C\to C\subset X$. The
pullback of $\omega$ to $\tilde C$ yields a nonzero $1$-form
$\tilde \omega \in H^0\big(\tilde{C},\Omega^1_{\tilde{C}}\otimes f^*\det(N_\mathscr{F})\big)$.
Thus
$\deg\big(\Omega^1_{\tilde{C}}\otimes f^*\det(N_\mathscr{F})\big) \ge 0$,
proving the lemma.
\end{proof}
\begin{lemma}\label{lemma:bound_on_pseudo_index}
Let $X$ be a uniruled complex projective manifold, and $\mathscr{F}$ a foliation on $X$.
Let $\ell\subset X$ be a general member of a minimal dominating family of rational curves on $X$.
If $\ell$ is not tangent to $\mathscr{F}$, then
$-K_\mathscr{F}\cdot \ell \le -K_X\cdot \ell -2$.
\end{lemma}
\begin{proof}
Set $n:=\dim(X)$.
Consider the normalization morphism $f:\p^1\to \ell\subset X$.
By \cite[IV.2.9]{kollar96},
$f^*T_X\cong \mathscr{O}_{\p^1}(2)\oplus \mathscr{O}_{\p^1}(1)^{\oplus d}\oplus
\mathscr{O}_{\p^1}^{\oplus (n-d-1)}$, where $0\le d=-K_X\cdot \ell-2\le n-1$.
Write $f^*\mathscr{F}\cong \oplus_{i=1}^{r_{\mathscr{F}}}\mathscr{O}_{\p^1}(a_i)$.
Since $\ell$ is general, $f^*\mathscr{F}$ is a subbundle of $f^*T_X$, and
since $\ell$ is not tangent to $\mathscr{F}$,
the inclusion $f^*\mathscr{F} \hookrightarrow f^*T_X$ induces an inclusion
$$
\oplus_{i=1}^{r_{\mathscr{F}}}\mathscr{O}_{\p^1}(a_i)\cong f^*\mathscr{F} \hookrightarrow f^*T_X/T_{\p^1}\cong \mathscr{O}_{\p^1}(1)^{\oplus d}\oplus \mathscr{O}_{\p^1}^{\oplus (n-d-1)}.
$$
Thus $a_i\le 1$ for $1\le i\le r_{\mathscr{F}}$, and
$$
-K_\mathscr{F}\cdot \ell \ = \\ \sum_{i=1}^{r_{\mathscr{F}}}a_i \ \le \\ d \ = \\ -K_X\cdot \ell-2.
$$
This completes the proof of the lemma.
\end{proof}
\begin{defn}\label{log_leaf}
Let $\mathscr{F}$ be an algebraically integrable foliation on a complex projective manifold $X$.
Let $i:\tilde F\to X$ be the normalization of the closure of a general leaf of $\mathscr{F}$.
There is an effective divisor $\tilde \Delta$ on $\tilde F$ such that
$K_{\tilde F} + \tilde \Delta \sim i^*K_{\mathscr{F}} $ (\cite[Definition 3.4]{fano_fols}).
The pair $( \tilde F, \tilde \Delta)$ is called a \emph{general log leaf} of $\mathscr{F}$.
\end{defn}
In \cite{fano_fols}, we applied the notions of singularities of pairs,
developed in the context of the minimal model program, to the log leaf $( \tilde F, \tilde \Delta)$ .
The case when $( \tilde F, \tilde \Delta)$ is \emph{log canonical} is specially interesting.
We refer to \cite[section 2.3]{kollar_mori} for the definition of log canonical pairs.
Here we only remark that if $\tilde F$ is smooth and $\tilde \Delta$ is a reduced simple normal crossing divisor, then
$(\tilde F,\tilde \Delta)$ is log canonical.
\begin{prop}[{\cite[Proposition 5.3]{fano_fols} }]\label{prop:common_pt}
Let $\mathscr{F}$ be an algebraically integrable Fano foliation on a complex projective manifold $X$.
Suppose that the general log leaf of $\mathscr{F}$ is log canonical.
Then there is a common point contained in the closure of a general leaf of $\mathscr{F}$.
\end{prop}
\subsection{Fano Foliations with large index on $\p^n$ and $Q^n$}\label{fols_in_p^n} \label{fols_in_Q^n} \
\smallskip
Jouanolou's classification of codimension $1$ foliations on $\p^n$ of degree $0$ and $1$
has been generalized to arbitrary rank in \cite{cerveau_deserti} and \cite{lpt3fold}, respectively.
The \emph{degree} $\deg(\mathscr{F})$ of a foliation $\mathscr{F}$ on $\p^n$ is defined as the degree
of the locus of tangency of $\mathscr{F}$ with a general linear subspace $\p^{n-r_{\mathscr{F}}}\subset \p^n$.
By \ref{q-forms}, a codimension $q$ foliation on $\p^n$ of degree $d$ is given by a twisted $q$-form
$\omega\in H^0\big(\p^n,\Omega^{q}_{\p^n}(q+d+1)\big)$.
One easily checks that
$$
\deg(\mathscr{F})=\deg(K_{\mathscr{F}})+r_{\mathscr{F}}.
$$
\begin{say}[{ \cite[Th\'eor\`eme 3.8]{cerveau_deserti}}] \label{cerveau_deserti}
A codimension $q$ foliation of degree $0$ on $\p^n$
is induced by a linear projection $\p^n \dashrightarrow \p^q$.
\end{say}
\begin{say}[{\cite[Theorem 6.2]{lpt3fold}}] \label{lpt3fold}
A codimension $q$ foliation $\mathscr{F}$ of degree $1$ on $\p^n$
satisfies one of the following conditions.
\begin{itemize}
\item $\mathscr{F}$ is induced by a dominant rational map $\p^n\dashrightarrow \p(1^{q},2)$,
defined by $q$ linear forms $L_1,\ldots,L_q$
and one quadratic form $Q$; or
\item $\mathscr{F}$ is the linear pullback of a foliation on $\p^{q+1}$ induced by a global
holomorphic vector field.
\end{itemize}
In the first case, $\mathscr{F}$ is induced by the $q$-form on $\mathbb{C}^{n+1}$
\begin{multline*}
\Omega = \sum_{i=1}^q(-1)^{i+1}L_idL_1\wedge\cdots\wedge \widehat{dL_i}\wedge\cdots \wedge dL_q\wedge dQ +(-1)^{q}2QdL_1\wedge\cdots\wedge dL_q\\
= (-1)^q \Big(\sum_{i=q+1}^{n+1}L_j\frac{\partial Q}{\partial L_i}\Big)dL_1\wedge\cdots\wedge dL_q\\
+\sum_{i=1}^{q}\sum_{j=q+1}^{n+1}(-1)^{i+1}L_i\frac{\partial Q}{\partial L_j}
dL_1\wedge\cdots\wedge \widehat{dL_i}\wedge\cdots \wedge dL_q\wedge dL_j,
\end{multline*}
where
$L_{q+1},\ldots,L_{n+1}$ are linear forms such that
$L_{1},\ldots,L_{n+1}$ are linearly independent.
The singular locus of $\mathscr{F}$ is the union of the quadric
$\{L_1=\cdots=L_q=Q=0\}\cong Q^{n-q-1}$ and the linear subspace
$\{\frac{\partial Q}{\partial L_{q+1}}=\cdots =\frac{\partial Q}{\partial L_{n+1}}=0\}$.
In the second, case the singular locus of $\mathscr{F}$ is the union of linear
subspaces of codimension at least $2$ containing the center $\p^{n-q-2}$ of the projection.
\end{say}
\begin{say}[{\cite[Proposition 3.17]{fano_fols_2}}]\label{lemma:fols_in_Q^n}
A codimension $q$ del Pezzo foliation on a smooth quadric hypersurface $Q^n \subset \p^{n+1}$
is induced by the restriction of a linear projection $\p^{n+1} \dashrightarrow \p^{q}$.
\end{say}
\section{Codimension $1$ Mukai foliations on Fano manifolds with $\rho=1$}\label{section:rho=1}
\subsection{Fano manifolds of high index}\label{Fano_manifolds_high_index} \
\smallskip
A \emph{Fano manifold} $X$ is a complex projective manifold whose anti-canonical class $-K_X$
is ample.
The index $\iota_{X}$ of $X$ is the largest integer dividing $-K_{X}$ in $\textup{Pic}(X)$.
By Kobayachi-Ochiai's theorem (\cite{kobayashi_ochiai}),
$\iota_X\le n+1$, equality holds if and only if $X\cong \p^n$, and $\iota_X= n$ if and only if
$X\cong Q^n\subset \p^{n+1}$.
Fano manifolds with $\iota_X = \dim X-1$ were classified by Fujita in \cite{fujita1}, \cite{fujita2} and \cite{fujita3}.
Those with Picard number $1$ are isomorphic to one of the following.
\begin{enumerate}
\item A cubic hypersurface in $\p^{n+1}$.
\item An intersection of two hyperquadrics in $\p^{n+2}$.
\item A linear section of the Grassmannian $G(2,5)\subset\p^9$ under the Pl\"ucker embedding.
\item A hypersurface of degree $4$ in the weighted projective space $\p(2,1,\ldots,1)$.
\item A hypersurface of degree $6$ in the weighted projective space $\p(3,2,1,\ldots,1)$.
\end{enumerate}
A Fano manifold $X$ such that $(\dim X-1)$ divides $\iota_X$ is called a \emph{del Pezzo} manifold.
In this case, either $X\cong \p^3$, or $\iota_X = \dim X-1$.
Fano manifolds with $\iota_X = \dim X-2$ are called \emph{Mukai} manifolds.
Their classification was first announced in \cite{Mukai89}.
We do not include it here.
Instead, we refer to \cite[Theorem 7]{AC_Harris} for the full list of Mukai manifolds with Picard number $1$, and
state below the property that we need.
This property can be checked directly for each Mukai manifold in the list.
\begin{rem}\label{Mukai_manifolds_covered_by_lines}
Let $X$ be a Mukai manifold with $\rho(X)=1$ and $\dim X\ge 4$. Denote by
$\mathscr{L}$ the ample generator of $\textup{Pic}(X)$.
Then $X$ is covered by rational curves having degree $1$ with respect to $\mathscr{L}$.
\end{rem}
If a complex projective manifold $X$ with $\rho(X)=1$ admits a Fano foliation $\mathscr{F}$, then
$X$ is a Fano manifold with index $\iota_X>\iota_{\mathscr{F}}$ by Theorem~\ref{Thm:KX-KF_not_nef}.
When $\iota_{\mathscr{F}}$ is high, we can improve this bound.
\begin{lemma}\label{lemma:bound_on_index2}
Let $X$ be an $n$-dimensional Fano manifold with Picard number $1$, $n\ge 4$,
and $\mathscr{F}$ a Fano foliation on $X$.
Suppose that $\iota_{\mathscr{F}}\ge n-3$.
Then $\iota_{X}\ge \iota_{\mathscr{F}}+2$.
\end{lemma}
\begin{proof}
By Theorem \ref{Thm:ADK}, $\iota_{\mathscr{F}}\le r_{\mathscr{F}}$, and equality holds only if $X\cong \p^n$.
If $\iota_{\mathscr{F}}= r_{\mathscr{F}}-1=n-2$, then either $X\cong \p^n$ or $X\cong Q^n\subset \p^{n+1}$ by Theorem~\ref{Thm:codim1_dP}.
In all these cases we have $\iota_{X}\ge \iota_{\mathscr{F}}+2$.
So we may assume from now on that $\iota_{\mathscr{F}}= n-3$ and $r_{\mathscr{F}}\in\{n-2, n-1\}$.
Let $\mathscr{L}$ be the ample generator of $\textup{Pic}(X)$.
Let $H$ be a minimal dominating family of rational curves on $X$, and
$\ell$ a general curve parametrized by $H$.
By \cite{CMSB}, $-K_X\cdot \ell =n+1$ if and only if $X\cong \p^n$.
So we may assume that $-K_X\cdot \ell \le n$.
Set $\lambda :={\mathscr{L}}\cdot \ell$.
By \cite[Proposition 2]{hwang98}, $\ell$ is not tangent to $\mathscr{F}$. Hence,
by Lemma \ref{lemma:bound_on_pseudo_index},
$$\lambda (n-3) = -K_\mathscr{F}\cdot \ell \le -K_X\cdot \ell - 2 \le n-2.$$
If $\lambda >1$, then $\lambda=2$, $n=4$ and $\iota_{X}=2$,
contradicting Remark~\ref{Mukai_manifolds_covered_by_lines}.
So we conclude that $\lambda =1$, and thus
$$\iota_{X} = -K_X\cdot \ell \ge -K_\mathscr{F}\cdot \ell +2 = \iota_{\mathscr{F}}+2.$$
This completes the proof of the lemma.
\end{proof}
\subsection{Del Pezzo foliations of codimension $2$} \label{section:del_pezzo_codim2} \
\smallskip
When proving Theorem~\ref{main_thm_rho=1}, we distinguish the cases when the
codimension $1$ Mukai foliation $\mathscr{F}$ is semi-stable or not.
When $\mathscr{F}$ is not semi-stable, we will show that it contains
a codimension $2$ del Pezzo subfoliation.
The aim of this subsection is to provide a classification of these.
\begin{thm}\label{thm:codim2_delPezzo}
Let $X$ be an $n$-dimensional Fano manifold with $\rho(X)=1$, $n\ge 4$,
and $\mathscr{G}$ a codimension $2$ del Pezzo foliation on $X$.
Then the pair $(X,\mathscr{G})$ satisfies one of the following conditions.
\begin{enumerate}
\item $X\cong \p^n$ and $\mathscr{G}$ is the pullback under a linear projection of a foliation on $\p^{3}$ induced by a global
vector field.
\item $X\cong \p^n$ and $\mathscr{G}$ is induced by a rational map $\p^n\dashrightarrow \p(2,1,1)$ defined by one quadratic form and two
linear forms.
\item $X\cong Q^n\subset \p^{n+1}$ and $\mathscr{G}$ is induced by the restriction of a linear projection $\p^{n+1}\dashrightarrow \p^2$.
\end{enumerate}
\end{thm}
Throughout this subsection, we will use the following notation.
\smallskip
\noindent {\bf Notation.} Let $X$ be an $n$-dimensional Fano manifold with $\rho(X)=1$
and $\mathscr{L}$ an ample line bundle on $X$ such that $\textup{Pic}(X)=\mathbb{Z}[\mathscr{L}]$.
Given a sheaf of $\mathscr{O}_X$-modules $\mathscr{E}$ on $X$ and an integer $m$, we denote by
$\mathscr{E}(m)$ the twisted sheaf $\mathscr{E}\otimes \mathscr{L}^{\otimes m}$.
\smallskip
Under the assumptions of Theorem~\ref{thm:codim2_delPezzo},
$\mathscr{G}$ is defined by a nonzero section $\omega\in H^0\big(X, \Omega_X^2(\iota_X - n+3)\big)$
as in \ref{q-forms}.
In order to compute these cohomology groups, we will use the knowledge of
several cohomology groups of special Fano manifolds, which we gather below.
\begin{say}[Bott's formulae]\label{bott}
Let $p,q$ and $k$ be integers, with $p$ and $q$
nonnegative. Then
\begin{equation*}
h^p\big(\p^n,\Omega_{\p^n}^q(k)\big) =
\begin{cases}
\binom{k+n-q}{k}\binom{k-1}{q} & \text{for } p=0, 0\le q\le n \text{ and } k>q,\\
1 & \text{for } k=0 \text{ and } 0\le p=q\le n,\\
\binom{-k+q}{-k}\binom{-k-1}{n-q} & \text{for } p=n, 0\le q\le n \text{ and } k<q-n,\\
0 & \text{otherwise.}
\end{cases}
\end{equation*}
\end{say}
\begin{say}[{\cite[Satz 8.11]{flenner81}}]\label{flenner}
Let $X$ be a smooth $n$-dimensional complete intersection in a weighted projective space. Then
\begin{enumerate}
\item $h^q(X,\Omega_{X}^q) =1 $ for $0\le q\le n$, $q\neq \frac{n}{2}$.
\item $h^p\big(X,\Omega_{X}^q(t)\big) = 0$ in the following cases
\begin{itemize}
\item $0<p<n$, $p+q\neq n$ and either $p\neq q$ or $t\neq 0$;
\item $p+q > n$ and $t>q-p$;
\item $p+q < n$ and $t<q-p$.
\end{itemize}
\end{enumerate}
\end{say}
\begin{say} \label{vanishing_ADK} \label{vanishing_Q}
Let $X$ be an $n$-dimensional Fano manifold with $\rho(X)=1$.
By \cite[Theorem 1.1]{adk08}, $h^0\big(X, \Omega_X^q(\iota_X - n+q)\big)=0$
unless $X\cong \p^n$, or $X\cong Q^n$ and $q=n$.
In particular for a smooth hyperquadric $Q=Q^n\subset \p^{n+1}$, $n\ge 3$,
$h^0\big(Q, \Omega_Q^2(2)\big)=0$.
\end{say}
\begin{say}[{\cite[Lemma 4.5]{fano_fols}}] \label{vanishing_AD}
Let $X\subset \p^{n+1}$ be a smooth hypersurface of degree $d\ge 3$.
Suppose that $q\ge 1$ and $t\le q \le n-2$.
Then $h^0\big(X, \Omega_X^q(t)\big)=0$.
\end{say}
\begin{say}\label{restriction_exact_seqs}
Let $Y$ be an $n$-dimensional Fano manifold with $\rho(Y)=1$, and $X\in \big|\mathscr{O}_Y(d) \big|$ a smooth divisor.
The following exact sequences will be used to relate foliations on $X$ with foliations on $Y$.
\begin{equation}\label{restriction2}
0 \ \to \ \Omega_Y^{q}(t-d) \ \to \ \Omega_Y^{q}(t) \ \to \ \Omega_Y^{q}(t)|_X \ \to \ 0, \ \text{ and }
\end{equation}
\begin{equation}\label{restriction1}
0 \ \to \ \Omega_X^{q-1}(t-d) \ \to \ \Omega_Y^{q}(t)|_X \ \to \ \Omega_X^{q}(t) \ \to \ 0.
\end{equation}
By \cite[Lemma 1.2]{PW_Stability},
if $h^p(Y,\Omega_{Y}^{q-1}) \neq 0$ and $p+q-1 < n$, then
the map in cohomology induced by the exact sequence \eqref{restriction1} (with $t=d$)
$$
H^p(X,\Omega_{X}^{q-1}) \ \to \ H^p\big(X,\Omega_{Y}^{q}(d)_{|X})
$$
is nonzero.
\end{say}
\begin{say} \label{WPS}
Let $a_0,\ldots, a_n$ be positive integers such that $\gcd(a_0,\ldots, \hat a_i, \ldots a_n)=1$ for every $i\in\{0, \ldots, n\}$.
Denote by $S=S(a_0,\ldots, a_n)$ the polynomial ring $\c[x_0,\ldots, x_n]$ graded by $\deg x_i=a_i$, and
by $\p:=\p(a_0,\ldots, a_n)$ the weighted projective space $\textup{Proj} \big(S(a_0,\ldots, a_n)\big)$.
For each $t\in\mathbb{Z}$, let $\mathscr{O}_{\p}(t)$ be the $\mathscr{O}_{\p}$-module associated to the graded $S$-module $S(t)$.
Consider the sheaves of $\mathscr{O}_{\p}$-modules
$\overline{\Omega}^{q}_{\p}(t)$ defined in \cite[2.1.5] {dolgachev} for $q, t\in\mathbb{Z}$, $q\ge 0$.
If $U\subset \p$ denotes the smooth locus of $\p$, and $\mathscr{O}_{U}(t)$ is the line bundle obtained by restricting
$\mathscr{O}_{\p}(t)$ to $U$, then ${\overline{\Omega}^{q}_{\p}(t)}_{|U}=\Omega^q_{U}\otimes \mathscr{O}_{U}(t)$.
The cohomology groups $H^p\big(\p, \overline{\Omega}^{q}_{\p}(t)\big)$ are described in \cite[2.3.2] {dolgachev}.
We will need the following:
\begin{itemize}
\item $h^0\big(\p, \overline{\Omega}^{q}_{\p}(t)\big)=\sum_{i=0}^q \Big((-1)^{i+q} \sum_{\#J=i}\dim_{\c}\big(S_{t-a_J}\big)\Big)$,
where $J\subset \{0, \ldots, n\}$ and $a_J:=\sum_{i\in J}a_i$.
\item $h^p\big(\p, \overline{\Omega}^{q}_{\p}(t)\big)=0$ if $p\not\in \{0, q,n\}$.
\end{itemize}
Now suppose that $\p$ has only isolated singularities, let $d>0$ be such that $\mathscr{O}_{\p}(d)$ is a line bundle generated by
global sections, and $X\in \big|\mathscr{O}_{\p}(d)\big|$ a smooth hypersurface.
We will use the cohomology groups $H^p\big(\p, \overline{\Omega}^{q}_{\p}(t)\big)$ to compute some cohomology groups
$H^p\big(X,\Omega_{X}^q(t)\big)$.
Note that $X$ is contained in the smooth locus of $\p$, so we have an exact sequence as in \eqref{restriction1}:
\begin{equation}\label{restriction1_P}
0 \ \to \ \Omega_X^{q-1}(t-d) \ \to \ \overline{\Omega}^{q}_{\p}(t)|_X \ \to \ \Omega_X^{q}(t) \ \to \ 0.
\end{equation}
Tensoring the sequence
$$
0 \ \to \ \mathscr{O}_{\p}(-d) \ \to \ \mathscr{O}_{\p} \ \to \ \mathscr{O}_{X} \ \to \ 0.
$$
with the sheaf $\overline{\Omega}^{q}_{\p}(t)$, and noting that $\overline{\Omega}^{q}_{\p}(t)\otimes \mathscr{O}_{\p}(-d) \cong \overline{\Omega}^{q}_{\p}(t-d)$,
we get an exact sequence as in \eqref{restriction2}:
\begin{equation}\label{restriction2_P}
0 \ \to \ \overline{\Omega}^{q}_{\p}(t-d) \ \to \ \overline{\Omega}^{q}_{\p}(t) \ \to \ \overline{\Omega}^{q}_{\p}(t)|_X \ \to \ 0.
\end{equation}
\end{say}
\medskip
\begin{proof}[{Proof of Theorem~\ref{thm:codim2_delPezzo}}]
By Lemma~\ref{lemma:bound_on_index2}, $\iota_X \ge n - 1$.
Recall the classification of Fano manifolds of high index discussed in Subsection~\ref{Fano_manifolds_high_index}.
We will go through the manifolds in that list, and determine all codimension $2$ del Pezzo foliations on them.
\medskip
Suppose first that $X\cong \p^n$.
Then $\mathscr{G}$ is a codimension $2$ foliation of degree $1$ on $\p^n$.
Such foliations are described in \ref{lpt3fold}.
\medskip
Suppose that $X$ is a smooth hyperquadric $Q=Q^n\subset \p^{n+1}$.
Codimension $2$ del Pezzo foliation on $Q^n$
are described in \ref{lemma:fols_in_Q^n}.
\medskip
From now on we suppose that $\iota_X = n - 1$.
We consider the 5 possibilities for $X$ described in Subsection~\ref{Fano_manifolds_high_index}.
If we show that $h^0\big(X, \Omega_X^2(2)\big)=0$, then it follows from
\ref{q-forms} that $X$ does not admit del Pezzo foliations of codimension $2$.
\
\noindent {\bf $\bullet$ $X$ is a cubic hypersurface in $\p^{n+1}$}. The vanishing
$h^0\big(X, \Omega_X^2(2)\big)=0$ follows from \ref{vanishing_AD} above.
\
\noindent {\bf $\bullet$ $X$ is the intersection of two hyperquadrics in $\p^{n+2}$}.
Let $Q$ and $Q'$ be smooth hyperquadrics in $\p^{n+2}$ such that $X=Q\cap Q'$.
Consider the exact sequences of \ref{restriction_exact_seqs} for $Y=Q$, $d=2$, $q=2$ and $t=2$.
They induce maps of cohomology groups:
$$
H^0\big(Q, \Omega_Q^2(2)\big)\ \to \ H^0\big(X, \Omega_Q^2(2)_{|X}\big)\ \to \ H^0\big(X, \Omega_X^2(2)\big).
$$
We will show that the composed map $H^0\big(Q, \Omega_Q^2(2)\big)\to H^0\big(X, \Omega_X^2(2)\big)$
is surjective.
Since $h^0\big(Q, \Omega_Q^2(2)\big)=0$ by \ref{vanishing_Q}, the required vanishing
$h^0\big(X, \Omega_X^2(2)\big)=0$ will follow.
Surjectivity of the first map $H^0\big(Q, \Omega_Q^2(2)\big) \to H^0\big(X, \Omega_Q^2(2)_{|X}\big)$ follows from
the vanishing of $H^1\big(Q, \Omega_Q^2\big)$ granted by \ref{flenner}.
To prove surjectivity of the second map,
we consider the long exact sequence in cohomology associated to the sequence \eqref{restriction1}.
By \ref{flenner}, $H^1\big(Q, \Omega_Q^1\big)\cong \mathbb{C}$.
So, as we noted in \ref{restriction_exact_seqs} above,
the map $H^1(X,\Omega_{X}^{1}) \to H^1\big(X,\Omega_{Q}^{2}(2)_{|X})$
is nonzero.
Since $H^1\big(X, \Omega_X^1\big)\cong \mathbb{C}$ by \ref{flenner}, we conclude that the map
$H^1(X,\Omega_{X}^{1}) \to H^1\big(X,\Omega_{Q}^{2}(2)_{|X})$ is injective, and thus the map
$H^0\big(X, \Omega_Q^2(2)_{|X}\big) \to H^0\big(X, \Omega_X^2(2)\big)$ is surjective.
\
\noindent {\bf $\bullet$ $X$ is a linear section of the Grassmannian $G(2,5)\subset\p^9$ of codimension $c\le 2$}.
We will show that $X$ does not admit del Pezzo foliations of codimension 2.
By \ref{restricting_fols} and \ref{vanishing_ADK}, it is enough to prove this in the case $c=2$.
By \cite[Theorem 10.26]{fujita2}, $X$ can be described as follows. There is a plane $\p^2 \cong P \subset X$
such that the blow-up $f : Y \to X$ of $X$ along $P$ admits a morphism
$g : Y \to \p^4$. Moreover, $g$ is the blow-up of $\p^4$ along a rational normal curve $C$ of degree 3 contained in an hyperplane $H \subset \p^4$.
Denote by $E$ and $F$ the exceptional loci of $f$ and $g$, respectively.
Then $q(E)=H$, $f^*\mathscr{O}_X(1)\cong g^*\mathscr{O}_{\p^4}(2)\otimes \mathscr{O}_Y(-F)$, and
$g^*\mathscr{O}_{\p^4}(1)\cong \mathscr{O}_Y(E+F)$.
Suppose that $X$ admits a codimension $2$ del Pezzo foliation
$\mathscr{G}$, which is defined by a twisted $2$-form
$\omega\in H^0\big(X, \Omega_X^2(2)\big)$.
Then $\omega$ induces a twisted $2$-form
$\alpha \in H^0\big(Y, \Omega_Y^2\otimes f^*\mathscr{O}_X(2)\big)
\cong H^0\big(Y, \Omega_Y^2\otimes g^*\mathscr{O}_{\p^4}(4)\otimes \mathscr{O}_Y(- 2 F)\big)$.
The restriction of $\alpha$ to $Y \setminus F$ induces a twisted $2$-form
$\tilde \alpha \in H^0\big(\p^4, \Omega_{\p^4}^2(4)\big)$
vanishing along $C$.
Denote by $\tilde \mathscr{G}$ the foliation on $\p^4$ induced by $\tilde \alpha$.
There are two possibilities:
\begin{itemize}
\item Either $\tilde \alpha$ vanishes along $H$,
and hence $\tilde \mathscr{G}$ is a degree $0$ foliation on $\p^4$; or
\item $\tilde \mathscr{G}$ is a degree $1$ foliation on $\p^4$
containing $C$ in its singular locus.
\end{itemize}
In the first case,
$\alpha$ vanishes along $E$, and thus
$\alpha \in H^0\big(Y, \Omega_Y^2\otimes f^*\mathscr{O}_X(2)\otimes\mathscr{O}_Y(-E)\big)
\cong H^0\big(Y, \Omega_Y^2\otimes g^*\mathscr{O}_{\p^4}(3)\otimes \mathscr{O}_Y(- F)\big)$.
Therefore $C$ must be tangent to $\tilde{\mathscr{G}}$, which is impossible
since $\tilde \mathscr{G}$ is induced by a linear projection $\p^4\dashrightarrow \p^2$.
To see that the second case cannot occur either, recall the
description of the two types of codimension $2$ degree $1$ foliations on $\p^4$
from \ref{lpt3fold}.
In all these foliations, any irreducible component
of the singular locus is either a linear subspace of dimension at most $2$, or a conic. This proves the claim.
\
\noindent {\bf $\bullet$ $X$ is a hypersurface of degree $4$ in the weighted projective space $\p(2,1,\ldots,1)$
or a hypersurface of degree $6$ in the weighted projective space $\p(3,2,1,\ldots,1)$
}.
Consider the exact sequences of \ref{WPS} for $\p:= \p(2,1,\ldots,1)$, $d=4$, $q=2$ and $t=2$
(respectively $\p:= \p(3,2,1,\ldots,1)$, $d=6$, $q=2$ and $t=2$).
They induce maps of cohomology groups:
$$
H^0\big(\p, \overline{\Omega}_{\p}^{2}(2)\big)\ \to \ H^0\big(X, \overline{\Omega}_{\p}^{2}(2)_{|X}\big)\ \to \ H^0\big(X, \Omega_X^2(2)\big).
$$
We will show that the composed map $H^0\big(\p, \overline{\Omega}_{\p}^{2}(2)\big)\to H^0\big(X, \Omega_X^2(2)\big)$
is surjective.
Since $h^0\big(\p, \overline{\Omega}_{\p}^{2}(2)\big)=0$ by Dolgachev's formulae described in \ref{WPS}, the required vanishing
$h^0\big(X, \Omega_X^2(2)\big)=0$ follows.
Surjectivity of the first map $H^0\big(\p, \overline{\Omega}_{\p}^{2}(2)\big) \to H^0\big(X, \overline{\Omega}_{\p}^{2}(2)_{|X}\big)$ follows from
the vanishing of $H^1\big(\p, \overline{\Omega}_{\p}^{2}(2-d)\big)$, granted by Dolgachev's formulae.
Surjectivity of the second map $H^0\big(X, \overline{\Omega}_{\p}^{2}(2)_{|X}\big) \to H^0\big(X, \Omega_X^2(2)\big)$ follows from
the vanishing of $H^1\big(X, \Omega_X^1(2-d)\big)$, granted by \ref{flenner}.
\end{proof}
\subsection{Proof of Theorem~\ref{main_thm_rho=1}} \label{proof_main_thm_rho=1} \
Let $\mathscr{F}$ be a codimension $1$ Mukai foliation on an $n$-dimensional complex projective manifold
$X\not\cong \p^n$
with $\rho(X)=1$, $n\ge 4$.
We will consider two cases, according to whether or not $\mathscr{F}$ is semi-stable.
\begin{lemma}[{\cite[Proposition 7.5]{fano_fols}}] \label{lemma:subfoliation}
Let $\mathscr{F}$ be a Fano foliation on a Fano manifold $X$ with $\rho(X)=1$.
Then one of the following holds.
\begin{enumerate}
\item Either $\mathscr{F}$ is semi-stable; or
\item there exits a Fano subfoliation $\mathscr{G}\subsetneq \mathscr{F}$
such that $\iota_{\mathscr{G}}\ge \iota_{\mathscr{F}}$.
\end{enumerate}
\end{lemma}
Suppose first that $\mathscr{F}$ is semi-stable, and denote by $\mathscr{L}$ the ample generator of $\textup{Pic}(X)$.
In this case, we will use the following result to classify the possible pairs $(X,\mathscr{F})$.
\begin{lemma}[{\cite[Proposition 3.5]{lpt3fold}}] \label{lemma:pencil}
Let $X$ be a Fano manifold with $\rho(X)=1$, and
$\mathscr{F}$ a semi-stable codimension $1$ Fano foliation on $X$.
Then $\mathscr{F}$ is induced by a dominant rational map of the form
$$
\varphi=(s_1^{\otimes m_1}:s_2^{\otimes m_2}):X\dashrightarrow \p^1,
$$
where $m_1$ and $m_2$ are relatively prime positive integers, and $s_1$ and $s_2$ are sections of line bundles
$\mathscr{L}_1$ and $\mathscr{L}_2$ such that $\mathscr{L}_1^{\otimes m_1}\cong \mathscr{L}_2^{\otimes m_2}$, and
$\mathscr{L}_1\otimes \mathscr{L}_2\cong \mathscr{O}_X(-K_X+K_{\mathscr{F}})$.
\end{lemma}
In our setting, Lemma~\ref{lemma:pencil} says that
there are positive integers $\lambda$ and $a\ge b$ such that $a$ and $b$ relatively prime, $\lambda(a+b)=\iota_X-\iota_{\mathscr{F}}$,
and $\mathscr{F}$ is induced by a pencil of hypersurfaces generated by $b\cdot F$ and $a\cdot G$,
where $F\in \big|\mathscr{L}^{\otimes \lambda a}\big|$ and $G\in \big|\mathscr{L}^{\otimes \lambda b}\big|$.
If $X\cong Q^n\subset \p^{n+1}$, then $a=2$ and $b=1$. Thus $\mathscr{F}$
is cut out by a pencil of hyperquadrics of $\p^{n+1}$ containing a double hyperplane.
If $X$ is a del Pezzo manifold, then $a=b=1$, and $\mathscr{F}$
is induced by a pencil in $|\mathscr{L}|$.
\medskip
From now on, we assume that $\mathscr{F}$ is not semi-stable.
By Lemma~\ref{lemma:subfoliation}, there exits a Fano subfoliation $\mathscr{G}\subsetneq \mathscr{F}$
such that $\iota_{\mathscr{G}}\ge n-3$.
By Theorem~\ref{Thm:ADK}, $\iota_{\mathscr{G}}= n-3$, and $r_{\mathscr{G}} = n-2$,
\textit{i.e.}, $\mathscr{G}$ is a
codimension $2$ del Pezzo foliation on $X$.
By Theorem~\ref{thm:codim2_delPezzo}, $X\cong Q^n\subset \p^{n+1}$,
and $\mathscr{G}$ is induced by the restriction to $Q^n$ of a
linear projection $\varphi: \p^{n+1}\dashrightarrow \p^{2}$.
By \eqref{K_pullback_fol}, $\mathscr{F}$ is the pullback via $\varphi_{|Q^n}$ of
a foliation on $\p^2$ induced by a global vector field.
This completes the proof of Theorem~\ref{main_thm_rho=1}.
\qed
\section{Codimension $1$ Mukai foliations on manifolds with $\rho > 1$}\label{section:rho>1}
In this section we prove Theorem~\ref{main_thm_rho>1}.
Our setup is the following.
\begin{assumptions}\label{assumptions}
Let $X$ be an $n$-dimensional complex projective manifold with $\rho(X) > 1$,
and $\mathscr{F}$ a codimension $1$ Mukai foliation on $X$ ($n\ge 4$).
Let $L$ be an ample divisor on $X$ such that $-K_{\mathscr{F}} \sim (n-3)L$,
and set $\mathscr{L}:=\O_{X}(L)$.
\end{assumptions}
Under Assumptions~\ref{assumptions},
Theorem~\ref{Thm:KX-KF_not_nef} implies $K_X+(n-3)L$ is not nef.
Smooth polarized varieties $(X,L)$ satisfying this condition have been classified.
We explain this classification in Subsection~\ref{subsection:adjunction},
and then use it in the following subsections to prove Theorem~\ref{main_thm_rho>1}.
\subsection{Adjunction Theory}\label{subsection:adjunction}
\
We will need the following classification of Fano manifolds with large index with respect to the dimension.
For $n\ge 5$, the list follows from \cite{wis}. The classification for $n=4$ can be found in \cite[Table 12.7]{IP}.
\begin{thm}\label{Thm:Classification_Mukai}
Let $X$ be an
$n$-dimensional Fano manifold
with Picard number $\rho(X)>1$, $n\ge 4$.
Let $\mathscr{L}$ be an ample line bundle such that $\mathscr{O}_X(-K_X)\cong \mathscr{L}^{\otimes \iota_X}$.
\begin{itemize}
\item If $\iota_X = n-1$, then $n=4$ and $(X,\mathscr{L})\cong \big(\p^2\times \p^2, \O_{\p^2}(1)\boxtimes\O_{\p^2}(1)\big)$.
\item If $\iota_X = n-2$, then $n\in \{4,5,6\}$.
\begin{enumerate}
\item If $n=6$, then $(X,\mathscr{L})\cong \big(\p^3\times \p^3, \O_{\p^3}(1)\boxtimes\O_{\p^3}(1)\big)$.
\item If $n=5$, then one of the following holds.
\begin{enumerate}
\item $(X,\mathscr{L})\cong \big(\p^2\times Q^3, \O_{\p^2}(1)\boxtimes\O_{Q^3}(1)\big)$.
\item $(X,\mathscr{L})\cong \Big(\p_{\p^3}\big(T_{\p^3}\big), \O_{\p(T_{\p^3})}(1)\Big)$.
\item $(X,\mathscr{L})\cong \Big(\p_{\p^3}\big(\O_{\p^3}(2)\oplus \O_{\p^3}(1)^{\oplus 2}\big),
\O_{\p(\O_{\p^3}(2)\oplus \O_{\p^3}(1)^{\oplus 2})}(1)\Big)$.
\end{enumerate}
\item If $n=4$, then $X$ is isomorphic to one of the following.
\begin{enumerate}
\item $\p^1\times Y$, where $Y$ is a Fano $3$-fold with index $2$, or $Y\cong \p^3$.
\item A double cover of $\p^2\times\p^2$ whose branch locus is a divisor of bidegree $(2,2)$.
\item A divisor of $\p^2\times\p^3$ of bidegree $(1,2)$.
\item An intersection of two divisors of bidegree $(1,1)$ on $\p^3\times\p^3$.
\item A divisor of $\p^2\times Q^3$ of bidegree $(1,1)$.
\item The blow-up of $Q^4$ along a conic $C$ which is not contained in a plane in $Q^4$.
\item $\p_{\p^3}(\mathscr{E})$, where $\mathscr{E}$ is the null-correlation bundle on $\p^3$.
\item The blow-up of $Q^4$ along a line $\ell$.
\item $\p_{Q^3}\big(\mathscr{O}_{Q^3}(-1)\oplus\mathscr{O}_{Q^3}\big)$.
\item $\p_{\p^3}\big(\mathscr{O}_{\p^3}(-1)\oplus\mathscr{O}_{\p^3}(1)\big)$.
\end{enumerate}
\end{enumerate}
\end{itemize}
\end{thm}
\begin{say}[Nef values]\label{nef_value}
Let $X$ be $\textup{\textbf{Q}}$-factorial terminal $n$-dimensional projective variety, and $L$ an
ample $\textup{\textbf{Q}}$-divisor on $X$.
The \emph{nef value} of $L$ is defined as
$$
\tau(L)\ := \ \min\big\{t\ge 0\ \big| \ K_X+tL \text{ is nef } \big\}.
$$
It is a rational number by the rationality theorem (\cite[Theorem 3.5]{kollar_mori}).
By the basepoint free theorem (\cite[Theorem 3.7.3]{kollar_mori}), for $m$ sufficiently large and divisible, the linear system $\big|m\big(K_X+\tau(L)L\big)\big|$
defines a morphism $\varphi_L:X\to X'$ with connected fibers onto a normal variety.
We refer to $\varphi_L$ as the \emph{nef value morphism} of the polarized variety $(X,L)$.
\end{say}
The next theorem summarizes the classification of smooth polarized varieties $(X,L)$ such that
$K_X+(n-3)L$ is not nef, i.e., $\tau(L)>n-3$.
\begin{thm}\label{tironi}
Let $(X,L)$ be an $n$-dimensional smooth polarized variety, with $\rho(X)>1$ and $n\ge 4$.
Set $\mathscr{L}:=\O_{X}(L)$.
Suppose that $\tau(L)>n-3$.
Then $\tau(L)\in \{n-2, n-1, n\}$, unless $\big(n,\tau(L)\big)\in \big\{(5,\frac{5}{2}),(4,\frac{3}{2}),(4,\frac{4}{3}) \big\}$.
\begin{enumerate}
\item Suppose that $\tau(L)=n$.
Then $\varphi_L$ makes $X$ a $\p^{n-1}$-bundle over a smooth curve $C$, and for a general fiber $F\cong \p^{n-1}$ of
$\varphi_L$, $\mathscr{L}_{|F}\cong \mathscr{O}_{\p^{n-1}}(1)$.
\item Suppose that $\tau(L)=n-1$. Then $(X,\mathscr{L})$ satisfies one of the following conditions.
\begin{enumerate}
\item $(X,\mathscr{L})\cong \big(\p^2\times \p^2, \O_{\p^2}(1)\boxtimes\O_{\p^2}(1)\big)$.
\item $\varphi_L$ makes $X$ a quadric bundle over a smooth curve $C$, and for a general fiber $F\cong Q^{n-1}$ of
$\varphi_L$, $\mathscr{L}_{|F}\cong \mathscr{O}_{Q^{n-1}}(1)$.
\item $\varphi_L$ makes $X$ a $\p^{n-2}$-bundle over a smooth surface $S$, and for a general fiber $F\cong \p^{n-2}$ of
$\varphi_L$, $\mathscr{L}_{|F}\cong \mathscr{O}_{\p^{n-2}}(1)$.
\item $\varphi_L$ is the blowup of a smooth projective variety at finitely many points,
and for any component $E\cong \p^{n-1}$ of the exceptional locus of $\phi_L$, $\mathscr{L}_{|E}\cong \mathscr{O}_{\p^{n-1}}(1)$.
\end{enumerate}
\item Suppose that $\tau(L)=n-2$. Then $(X,\mathscr{L})$ satisfies one of the following conditions.
\begin{enumerate}
\item $-K_X \sim (n-2)L$, and $(X,\mathscr{L})$ is as in Theorem~\ref{Thm:Classification_Mukai}.
\item $\varphi_L$ makes $X$ a generic del Pezzo fibration over a smooth curve $C$, and, for a general fiber $F$ of
$\varphi_L$, either $\textup{Pic}(F)=\mathbb{Z}[\mathscr{L}_{|F}]$, or
$(F,\mathscr{L}_{|F})\cong \big(\p^2\times \p^2, \O_{\p^2}(1)\boxtimes\O_{\p^2}(1)\big)$.
\item $\varphi_L$ makes $X$ a generic quadric bundle over a normal surface $S$, and, for a general fiber $F\cong Q^{n-2}$ of
$\varphi_L$, $\mathscr{L}_{|F}\cong \mathscr{O}_{Q^{n-2}}(1)$.
\item $\varphi_L$ makes $X$ a generic $\p^{n-3}$-bundle over a normal $3$-fold $Y$, and, for a general fiber $F\cong \p^{n-3}$ of
$\varphi_L$, $\mathscr{L}_{|F}\cong \mathscr{O}_{\p^{n-3}}(1)$.
\item $\varphi_L:X\to X'$ is the composition of finitely many disjoint divisorial contractions. In particular, $X'$ is $\textup{\textbf{Q}}$-factorial and
terminal. (See Proposition~\ref{prop:BS} below.)
\end{enumerate}
\item Suppose that $n=5$ and $\tau(L)=\frac{5}{2}$.
Then $\varphi_L$ makes $X$ a $\p^{4}$-bundle over a smooth curve $C$ and, for a general fiber $F\cong \p^{4}$ of
$\varphi_L$, $\mathscr{L}_{|F}\cong \mathscr{O}_{\p^{4}}(2)$.
\item Suppose that $n=4$ and $\tau(L)=\frac{3}{2}$. Then $(X,\mathscr{L})$ satisfies one of the following conditions.
\begin{enumerate}
\item $(X,\mathscr{L})\cong \big(\p^2\times \p^2, \O_{\p^2}(2 )\boxtimes\O_{\p^2}(2)\big)$.
\item $\varphi_L$ makes $X$ a generic quadric bundle over a smooth curve $C$, and for a general fiber $F\cong Q^{3}$ of
$\varphi_L$, $\mathscr{L}_{|F}\cong \mathscr{O}_{Q^{3}}(2)$.
\item $\varphi_L$ makes $X$ a generic $\p^{2}$-bundle over a normal surface $S$, and for a general fiber $F\cong \p^{2}$
of $\varphi_L$, $\mathscr{L}_{|F}\cong \mathscr{O}_{\p^{2}}(2)$.
\end{enumerate}
\item Suppose that $n=4$ and $\tau(L)=\frac{4}{3}$. Then $\varphi_L$ makes $X$ a $\p^{3}$-bundle over a smooth curve $C$,
and for a general fiber $F\cong \p^{3}$ of
$\varphi_L$, $\mathscr{L}_{|F}\cong \mathscr{O}_{\p^{3}}(3)$.
\end{enumerate}
\end{thm}
\begin{proof}
The main references for the proof of Theorem~\ref{tironi} are \cite[Chapter 7]{beltrametti_sommese} and
\cite{andreatta_wisniewski}.
By \cite[Proposition 7.2.2, Theorems 7.2.3 and 7.2.4]{beltrametti_sommese},
either $\tau(L)=n$ and $(X,L)$ is as in (1) above, or $\tau(L)\le n-1$.
Suppose that $\tau(L)\le n-1$.
By \cite[Theorems 7.3.2 and 7.3.4]{beltrametti_sommese}, one of the following holds.
\begin{itemize}
\item[(2a)] $-K_X \sim (n-1)L$, and hence $(X,\mathscr{L})\cong \big(\p^2\times \p^2, \O_{\p^2}(1)\boxtimes\O_{\p^2}(1)\big)$ by Theorem~\ref{Thm:Classification_Mukai}.
\item[(2b)] $\varphi_L$ makes $X$ a generic quadric bundle over a smooth curve $C$, and for a general fiber $F\cong Q^{n-1}$ of
$\varphi_L$, $\mathscr{L}_{|F}\cong \mathscr{O}_{Q^{n-1}}(1)$.
\item[(2c)] $\varphi_L$ makes $X$ a generic $\p^{n-2}$-bundle over a normal surface $S$, and for a general fiber $F\cong \p^{n-2}$ of
$\varphi_L$, $\mathscr{L}_{|F}\cong \mathscr{O}_{\p^{n-2}}(1)$.
\item[(2d)] $\varphi_L$ is the blowup of a smooth projective variety at finitely many points,
and for any component $E\cong \p^{n-1}$ of the exceptional locus of $\phi_L$, $\mathscr{L}_{|E}\cong \mathscr{O}_{\p^{n-1}}(1)$.
\item[(e)] $\tau(L)\le n-2$.
\end{itemize}
In case (2b), it follows from \cite[Theorem 5.1]{andreatta_wisniewski} that $X$ is in fact a quadric bundle over $C$.
In case (2c), it follows from \cite[Theorem 5.1]{andreatta_wisniewski} that $S$ is smooth and $X$ is in fact a $\p^{n-2}$-bundle over $S$.
If $\tau(L)= n-2$, the classification under (3) follows from \cite[Theorem 7.5.3]{beltrametti_sommese}.
If $\tau(L)< n-2$, then, by \cite[Theorems 7.7.2, 7.7.3, 7.7.5 and 7.7.8]{beltrametti_sommese}
$\big(n,\tau(L)\big)\in \big\{(5,\frac{5}{2}),(4,\frac{3}{2}),(4,\frac{4}{3}) \big\}$
and $(X,L)$ is as in (4--6) above.
\end{proof}
We will also need the following result.
\begin{prop}[{\cite[Theorems 7.5.3, 7.5.6, 7.7.2, 7.7.3, 7.7.5 and 7.7.8]{beltrametti_sommese}}]\label{prop:BS}
Let $(X,L)$ be an $n$-dimensional smooth polarized variety, $n\ge 4$.
Suppose that $\tau(L)=n-2$, and the nef value morphism $\varphi_L:X\to X'$ is birational.
Then $\varphi_L$ is the composition of finitely many disjoint divisorial contractions $\varphi_i:X\to X_{i}$,
with exceptional divisor $E_i$, of the following types:
\begin{itemize}
\item $\varphi_i:X\to X_{i}$ is the blowup of a smooth curve $C_i\subset X_{i}$.
In this case $X_{i}$ is smooth and the restriction of $\mathscr{L}$ to a fiber $F\cong \p^{n-2}$ of $(\varphi_i)_{|E_i}:E_i\to C_i$
satisfies $\mathscr{L}_{|F}\cong \mathscr{O}_{\p^{n-2}}(1)$.
\item $\big(E_i, \mathscr{N}_{E_i/X}, \mathscr{L}_{|E_i}\big)\cong \big(\p^{n-1}, \mathscr{O}_{\p^{n-1}}(-2), \mathscr{O}_{\p^{n-1}}(1)\big)$.
In this case $X_i$ is $2$-factorial. In even dimension it is Gorenstein.
\item $\big(E_i, \mathscr{N}_{E_i/X}, \mathscr{L}_{|E_i}\big)\cong \big(Q^{n-1}, \mathscr{O}_{Q^{n-1}}(-1), \mathscr{O}_{Q^{n-1}}(1)\big)$.
In this case $X_i$ is singular and factorial.
\end{itemize}
Set $L':=(\varphi_L)_*(L)$.
Then $K_{X'}+(n-3)L'$ is nef except in the following cases.
\begin{enumerate}
\item $n=6$ and $\big(X',\mathscr{O}_{X'}(L')\big) \cong \big(\p^6, \mathscr{O}_{\p^6}(2)\big)$.
\item $n=5$ and one of the following holds.
\begin{enumerate}
\item $\big(X',\mathscr{O}_{X'}(L')\big) \cong \big(Q^5, \mathscr{O}_{Q^5}(2)\big)$.
\item $X'$ is a $\p^4$-bundle over a smooth curve, and the restriction of $\mathscr{O}_{X'}(L')$ to a general fiber is $\mathscr{O}_{\p^4}(2)$.
\item $\big(X,\mathscr{O}_{X}(L)\big) \cong \Big(\p_{\p^4}\big(\mathscr{O}_{\p^4}(3)\oplus \mathscr{O}_{\p^4}(1)\big), \mathscr{O}_{\p}(1)\Big)$.
\end{enumerate}
\item $n=4$ and one of the following holds.
\begin{enumerate}
\item $\big(X',\mathscr{O}_{X'}(L')\big) \cong \big(\p^4, \mathscr{O}_{\p^4}(3)\big)$.
\item $X'$ is a Gorenstein del Pezzo $4$-fold and $3L'\sim_{\textup{\textbf{Q}}}-2K_{X'}$.
\item $\varphi_{L'}$ makes $X'$ a generic quadric bundle over a smooth curve $C$, and for a general fiber $F\cong Q^{3}$ of
$\varphi_{L'}$, $\mathscr{O}_F\big(L'_{|F}\big)\cong \mathscr{O}_{Q^{3}}(2)$.
\item $\varphi_{L'}$ makes $X'$ a generic $\p^{2}$-bundle over a normal surface $S$, and for a general fiber $F\cong \p^{2}$
of $\varphi_{L'}$, $\mathscr{O}_F\big(L'_{|F}\big)\cong \mathscr{O}_{\p^{2}}(2)$.
\item $\big(X',\mathscr{O}_{X'}(L')\big) \cong \big(Q^4, \mathscr{O}_{Q^4}(3)\big)$.
\item $\varphi_L:X\to X'$ factors through $\tilde X$, the blowup of $\p^4$ along a cubic surface contained in
a hyperplane $E\subset \p^4$. Denote by $\tilde E\subset \tilde X$ the strict transform of $E$,
and by $\tilde L$ the push-forward of $L$ to $\tilde X$.
Then $N_{\tilde E/ \tilde X}\cong \mathscr{O}_{\p^{3}}(-2)$, $\mathscr{O}_{\tilde E}\big(\tilde L_{|\tilde E}\big)\cong \mathscr{O}_{\p^{3}}(1)$,
and only $\tilde E$ is contracted by $\tilde X\to X'$.
\item $\varphi_L:X\to X'$ factors through $\tilde X$, a conic bundle over $\p^3$.
Denote by $\tilde L$ the push-forward of $L$ to $\tilde X$.
The morphism $\tilde X\to X'$ only contracts a subvariety $\tilde E\cong \p^3$ such that
$N_{\tilde E/ \tilde X}\cong \mathscr{O}_{\p^{3}}(-2)$ and $\mathscr{O}_{\tilde E}\big(\tilde L_{|\tilde E}\big)\cong \mathscr{O}_{\p^{3}}(1)$.
\item $\varphi_{L'}$ makes $X'$ a $\p^{3}$-bundle over a smooth curve $C$,
and for a general fiber $F\cong \p^{3}$ of
$\varphi_{L'}$, $\mathscr{O}_F\big(L'_{|F}\big)\cong \mathscr{O}_{\p^{3}}(3)$.
\item $\big(X',\mathscr{O}_{X'}(L')\big) \cong \big(\p^4, \mathscr{O}_{\p^4}(4)\big)$.
\item $X'\subset \p^{10}$ is a cone over $\big(\p^3, \mathscr{O}_{\p^3}(2)\big)$ and $L'\sim_{\textup{\textbf{Q}}} 2 H$, where $H$
denotes a hyperplane section in $\p^{10}$.
\end{enumerate}
\end{enumerate}
\end{prop}
\begin{rem}
In \cite[Theorems 7.5.3]{beltrametti_sommese}, the description of the first type of divisorial contraction is as follows:
$X_{i}$ is smooth, and $\varphi_i:X\to X_{i}$ contracts a smooth divisor $E_i\subset X$ onto a smooth curve $C_i\subset X_{i}$.
By \cite[Theorem 2]{luo}, $\varphi_i$ is a smooth blowup.
\end{rem}
\subsection{Codimension $1$ Mukai foliations on projective space bundles over curves}\label{subsection:P-bdles/curves}
\
In this subsection, we work under Assumptions~\ref{assumptions}, supposing moreover that $\tau(L)=n$, and thus
$\varphi_L$ makes $X$ a $\p^{n-1}$-bundle over a smooth curve $C$.
We start with the following observation, which is a special case of Proposition \ref{prop:common_pt}.
\begin{prop}\label{prop:Fano_fol_not_fibration}
Let $\mathscr{F}$ be a codimension 1 Fano foliation on a smooth projective variety $X$.
Then $\mathscr{F}$ is not the relative tangent sheaf of any surjective morphism $\pi:X\to C$ onto a smooth curve.
\end{prop}
As a consequence of Proposition~\ref{prop:Fano_fol_not_fibration}, $\mathscr{H}:=T_{X/C}\cap \mathscr{F}$
is a codimension $2$ foliation on $X$.
The restriction of $\mathscr{H}$ to the general fiber $F\cong \p^{n-1}$ of $\varphi_L$ inherits positivity of $\mathscr{F}$,
which allows us to describe it explicitly.
In order to do so, we recall the description of families of degree $0$ foliations on
projective spaces from \cite[7.8]{fano_fols}.
\begin{say}[{Families of degree $0$ foliations on $\p^m$}]\label{V_in_E}
Let $Y$ be a positive dimensional smooth projective variety, and
$\mathscr{E}$ a locally free sheaf of rank $m+1\ge 2$ on $Y$. Set $X:=\p_Y(\mathscr{E})$,
denote by $\mathscr{O}_X(1)$ the tautological line bundle on $X$,
by $\pi:X\to Y$ the natural projection, and by $F\cong \p^m$ a general fiber of $\pi$.
Let $\mathscr{H}\subsetneq T_{X/Y}$ be a foliation of rank $r\le m-1$ on $X$, and suppose that
$\mathscr{H}|_{F}\cong \mathscr{O}_{\p^m}(1)^{\oplus r}\subset T_{\p^m}$.
Let $\mathscr{V}^*$ be the saturation of $\pi_*\big(\mathscr{H}\otimes\mathscr{O}_X(-1)\big)$ in
$\mathscr{E}^*\cong \pi_*\big(T_{X/Y}\otimes\mathscr{O}_X(-1)\big)$,
and set $\mathscr{V}:=(\mathscr{V}^*)^*$.
Then $\mathscr{H}\cong(\pi^*\mathscr{V}^*)\otimes \mathscr{O}_X(1)$.
In particular, $\det (\mathscr{H})\cong \pi^*\det( \mathscr{V}^*)\otimes \mathscr{O}_X(r)$.
The description of $\mathscr{H}|_{F}$ as the relative tangent sheaf of a linear projection $\p^m\dashrightarrow \p^{m-r}$
globalizes as follows.
Let $\mathscr{K}$ be the (rank $m+1-r$) kernel of the dual map $\mathscr{E}\to \mathscr{V}$.
Then there exists an open subset $Y^{\circ}\subset Y$,
with $\textup{codim}_Y(Y\setminus Y^{\circ})\ge 2$, over which we have an exact sequence
of vector bundles
$$
0\ \to \ \mathscr{K}|_{Y^{\circ}} \ \to \ \mathscr{E}|_{Y^{\circ}} \ \to \ \mathscr{V}|_{Y^{\circ}} \ \to \ 0.
$$
This induces a relative linear projection $\varphi: \p_{Y^{\circ}}(\mathscr{E}|_{Y^{\circ}}) \dashrightarrow \p_{Y^{\circ}}(\mathscr{K}|_{Y^{\circ}})=:Z$,
which restricts to a smooth morphism $\varphi^{\circ}:X^{\circ}\to Z$, where $X^{\circ}\subset X$ is an open subset
with $\textup{codim}_X(X\setminus X^{\circ})\ge 2$.
The restriction of $\mathscr{H}$ to $X^{\circ}$ is precisely $T_{X^{\circ}/Z}$.
\end{say}
\begin{prop}\label{lemma:P-bdle_over_C}
Let $X$, $\mathscr{F}$, $L$ and $\mathscr{L}$ be as in Assumptions~\ref{assumptions}.
Suppose that $\tau(L)=n$, and thus $\varphi_L$ makes $X$ a $\p^{n-1}$-bundle over a smooth curve $C$.
Set $\mathscr{E}:=(\varphi_L)_*\mathscr{L}$, so that $X\cong \p_C(\mathscr{E})$.
Then one of the following holds.
\begin{enumerate}
\item $C\cong \p^1$, $\mathscr{F}$ is algebraically integrable, and its
restriction to a general fiber is induced by a pencil of hyperquadrics in $\p^{n-1}$ containing a double hyperplane.
\item There exist
\begin{itemize}
\item an exact sequence
$
0\ \to \ \mathscr{K} \ \to \ \mathscr{E} \ \to \ \mathscr{V} \ \to \ 0
$
of vector bundles on $C$, with $\textup{rank}(\mathscr{K})=3$;
\item a rank 2 foliation $\mathscr{G}$ on $\p_C(\mathscr{K})$, generically transverse to the natural projection $p:\p_C(\mathscr{K})\to C$,
satisfying $\det(\mathscr{G}) \cong p^*\big(\det(\mathscr{V})\big)$ and $r_{\mathscr{G}}^a\ge 1$;
\end{itemize}
such that $\mathscr{F}$ is the pullback of $\mathscr{G}$ via the induced relative linear projection $\p_C(\mathscr{E})\dashrightarrow \p_C(\mathscr{K})$.
In this case, $r_{\mathscr{F}}^a\ge r_{\mathscr{F}}-1$.
\item There exist
\begin{itemize}
\item an exact sequence
$
0\ \to \ \mathscr{K} \ \to \ \mathscr{E} \ \to \ \mathscr{V} \ \to \ 0
$
of vector bundles on $C$, with $\textup{rank}(\mathscr{K})=2$;
\item a foliation by curves $\mathscr{G}$ on $\p_C(\mathscr{K})$, generically transverse to the natural projection $p:\p_C(\mathscr{K})\to C$,
and satisfying $\mathscr{G} \cong p^*\big(\det(\mathscr{V})\big)\otimes \mathscr{O}_{\p(\mathscr{K})}(-1)$;
\end{itemize}
such that $\mathscr{F}$ is the pullback of $\mathscr{G}$ via the induced relative linear projection $\p_C(\mathscr{E})\dashrightarrow \p_C(\mathscr{K})$.
In this case, $r_{\mathscr{F}}^a\ge r_{\mathscr{F}}-1$.
\end{enumerate}
\end{prop}
\begin{proof}
By Proposition~\ref{prop:Fano_fol_not_fibration}, $\mathscr{F}\neq T_{X/C}$.
So $\mathscr{H}:=\mathscr{F}\cap T_{X/C}$ is a codimension $2$ foliation on $X$.
Set $\mathscr{Q}:=(\mathscr{F}/\mathscr{H})^{**}$. It is an invertible subsheaf of $(\varphi_L)^*T_{C}$,
and we have
\begin{equation}\label{eq1_P-bdle_over_C}
\det(\mathscr{H})\ \cong \ \det(\mathscr{F})\otimes \mathscr{Q}^*.
\end{equation}
We want to describe the codimension $1$ foliation $\mathscr{H}_F$
obtained by restricting $\mathscr{H}$ to a general fiber $F\cong \p^{n-1}$ of $\varphi_L$.
By \ref{restricting_fols}, there exists a non-negative integer $b$ such that
$$
-K_{\mathscr{H}_F} \ = \ (n-3+b) H,
$$
where $H$ denotes a hyperplane in $F\cong \p^{n-1}$.
By Theorem~\ref{Thm:ADK}, we must have $b\in\{0,1\}$.
\medskip
First we suppose that $b=0$, i.e., $\mathscr{H}_F$ is a degree $1$ foliation on $\p^{n-1}$.
Then $\mathscr{Q}|_F\cong \mathscr{O}_{\p^{n-1}}$, and thus $\mathscr{Q}\cong (\varphi_L)^*\mathscr{C}$ for some line bundle
$\mathscr{C}\subset T_C$ on $C$.
Recall that there are two types codimension $1$ degree $1$ foliations on $\p^{n-1}$:
\begin{itemize}
\item[(i)] either $\mathscr{H}_{F}$ is induced by pencil of hyperquadrics containing a double hyperplane, or
\item[(ii)] $\mathscr{H}_{F}$ is the linear pullback of a foliation on $\p^{2}$ induced by a global
holomorphic vector field.
\end{itemize}
\medskip
Suppose that we are in case (i).
Then $\mathscr{H}$ is algebraically integrable, and its general log leaf is $(Q^{n-2}, H)$, where $Q^{n-2}\subset F\cong \p^{n-1}$
is an irreducible (possibly singular) hyperquadric, and $H$ is a hyperplane section.
Note that $(Q^{n-2}, H)$ is log canonical, unless $Q^{n-2}$ is a cone over a conic curve
and $H$ is a tangent hyperplane through the $(n-4)$-dimensional vertex.
The latter situation falls under case (ii), treated below. So we may assume that the
general log leaf of $\mathscr{H}$ is log canonical.
By Proposition~\ref{prop:common_pt}, $\det(\mathscr{H})$ cannot be ample.
By \eqref{eq1_P-bdle_over_C}, we must have $\deg(\mathscr{C})>0$, and hence $C\cong \p^1$.
Next we show that $\mathscr{F}$ is algebraically integrable.
It then follows that we are in case (1) in the statement of Proposition~\ref{lemma:P-bdle_over_C}.
Since $\mathscr{H}$ is algebraically integrable,
there is a smooth surface $S$ with a generic $\p^1$-bundle structure $p:S\to \p^1$, and a rational
map $\psi:X\dashrightarrow S$ over $\p^1$ inducing $\mathscr{H}$.
By \ref{pullback_foliations}, $\mathscr{F}$ is the pullback via $\psi$ of a rank $1$ foliation $\mathscr{G}$ on $S$.
Moreover, there is an inclusion $p^*\mathscr{C}\subset \mathscr{G}$.
It follows from Theorem~\ref{thm:BM} that the leaves of $\mathscr{G}$ are algebraic, and so are the leaves of $\mathscr{F}$.
\medskip
Suppose that we are in case (ii). Then there exists a codimension $3$ foliation $\mathscr{W}\subset \mathscr{H}$
whose restriction to $F\cong \p^{n-1}$ is a degree $0$ foliation on $\p^{n-1}$.
By \ref{V_in_E}, there exists an exact sequence of vector bundles on $C$
$$
0\ \to \ \mathscr{K} \ \to \ \mathscr{E} \ \to \ \mathscr{V} \ \to \ 0
$$
with $\textup{rank}(\mathscr{K})=3$, such that $\mathscr{W}\cong \big((\varphi_L)^*\mathscr{V}^*\big)\otimes \mathscr{L}$ is the tangent sheaf to the relative linear projection
$\varphi: X\cong \p_C(\mathscr{E})\dashrightarrow \p_C(\mathscr{K})$.
Denote by $p: \p_C(\mathscr{K})\to C$ the natural projection.
By \eqref{K_pullback_fol},
there is a codimension $1$ foliation $\mathscr{G}$ on $\p_C(\mathscr{K})$ such that $\mathscr{F}$ is the pullback of $\mathscr{G}$ via $\varphi$, and
$\det(\mathscr{G}) \ \cong \ p^*\big(\det(\mathscr{V})\big)$.
Note that $\det(\mathscr{V})$ is an ample line bundle on $C$.
Thus, applying Theorem~\ref{thm:BM} to a suitable destabilizing subsheaf of $\mathscr{G}$, we conclude that $r_{\mathscr{G}}^a\ge 1$.
We are in case (2) in the statement of Proposition~\ref{lemma:P-bdle_over_C}.
\medskip
From now on we assume that $b=1$, i.e., $\mathscr{H}_F$ is a degree $0$ foliation on $\p^{n-1}$.
By \ref{V_in_E}, there exists an exact sequence of vector bundles on $C$
$$
0\ \to \ \mathscr{K} \ \to \ \mathscr{E} \ \to \ \mathscr{V} \ \to \ 0
$$
with $\textup{rank}(\mathscr{K})=2$, such that $\mathscr{H}\cong \big((\varphi_L)^*\mathscr{V}^*\big)\otimes \mathscr{L}$ is the tangent sheaf to the relative linear projection
$\varphi: X\cong \p_C(\mathscr{E})\dashrightarrow \p_C(\mathscr{K})$.
Denote by $p: \p_C(\mathscr{K})\to C$ the natural projection.
By \eqref{K_pullback_fol},
there is a foliation by curves on $\p_C(\mathscr{K})$
$$
\mathscr{G} \ \cong \ p^*\big(\det(\mathscr{V})\big)\otimes \mathscr{O}_{\p(\mathscr{K})}(-1)\ \hookrightarrow \ T_{\p(\mathscr{K})}
$$
such that $\mathscr{F}$ is the pullback of $\mathscr{G}$ via $\varphi$.
We are in case (3) in the statement of Proposition~\ref{lemma:P-bdle_over_C}.
\end{proof}
Next we describe the codimension $1$ foliations on $\p_C(\mathscr{K})$ that appear in Proposition~\ref{lemma:P-bdle_over_C}(2).
\begin{prop}\label{proposition:P-bdle_over_curve_(2)}
Let
$\mathscr{K}$ be a rank $3$ vector bundle on a smooth complete curve $C$, and set $Y:= \p_C(\mathscr{K})$,
with natural projection $p:Y\to C$.
Let $\mathscr{G}$ be a rank 2 foliation on $Y$, generically transverse to
$p:Y\to C$, and satisfying $\det(\mathscr{G}) \cong p^*\mathscr{A}$ for some ample line bundle $\mathscr{A}$ on $C$.
Then one of the following holds.
\begin{enumerate}
\item There exist
\begin{itemize}
\item an exact sequence
$
0\ \to \ \mathscr{K}_1 \ \to \ \mathscr{K} \ \to \ \mathscr{B} \ \to \ 0
$
of vector bundles on $C$, with $\textup{rank}(\mathscr{K}_1)=2$;
\item a rank 1 foliation $\mathscr{N}$ on $\p_C(\mathscr{K}_1)$, generically transverse to the natural projection $p_1:\p_C(\mathscr{K}_1)\to C$,
and satisfying $\mathscr{N} \cong p_1^*(\mathscr{B})\otimes \mathscr{O}_{\p(\mathscr{K}_1)}(-1)$;
\end{itemize}
such that $\mathscr{G}$ is the pullback of $\mathscr{N}$ via the induced relative linear projection $Y=\p_C(\mathscr{K})\dashrightarrow \p_C(\mathscr{K}_1)$.
\item There exist
\begin{itemize}
\item a $\p^1$-bundle $q:S\to C$;
\item a rational map $\psi: Y\dashrightarrow S$ over $C$ whose restriction to a general fiber $F\cong \p^2$ of $p$ is given
by a pencil of conics containing a double line;
\item a rank 1 foliation $\mathscr{N}$ on $S$, generically transverse to $q:S\to C$,
and satisfying $\mathscr{N} \cong q^*\big(T_C(-B)\big)$ for some effective divisor $B$ on $C$;
\end{itemize}
such that $\mathscr{G}$ is the pullback of $\mathscr{N}$ via $\psi$. Moreover, the set of critical values
of $\psi$ is invariant by $\mathscr{N}$.
\item $C\cong \p^1$, and there exist
\begin{itemize}
\item a rational map $\psi: Y\dashrightarrow \p^2$, which restricts to an isomorphism on all but possibly one
fiber of $p:Y\to C$;
\item a rank 1 foliation $\mathscr{N}$ on $\p^2$ induced by a global vector field;
\end{itemize}
such that $\mathscr{G}$ is the pullback of $\mathscr{N}$ via $\psi$.
\end{enumerate}
\end{prop}
\begin{proof}
Consider the rank 1 foliation $\mathscr{C}:=\mathscr{G} \cap T_{Y/C}\subsetneq T_{\p_C(\mathscr{K})}$.
It induces a rank 1 foliation $\mathscr{C}_F $ on a general fiber $F\cong \p^2$ of $p$.
By \ref{restricting_fols}, there exists a non-negative integer $b$ such that
$$
-K_{\mathscr{C}_F} \ = \ b H,
$$
where $H$ denotes a hyperplane in $F\cong \p^{2}$.
By Thereom~\ref{Thm:ADK}, we must have $b\in\{0,1\}$.
\medskip
First we suppose that $b=1$, i.e., $\mathscr{C}_F$ is a degree $0$ foliation on $\p^2$.
The same argument used in the last paragraph of the proof of Proposition \ref{lemma:P-bdle_over_C} shows that
$(\mathscr{K},\mathscr{G})$ satisfies condition (1) in the statement of Proposition \ref{proposition:P-bdle_over_curve_(2)}.
\medskip
From now on we assume that $b=0$, i.e., $\mathscr{C}_F$ is a degree $1$ foliation on $\p^2$.
It follows that $\mathscr{C}_{|F} \cong \mathscr{O}_{\p^2}$ and
there exists an effective divisor $B$ on $C$ such that $\mathscr{C}\cong p^*\big(\mathscr{A}\otimes \mathscr{O}_C(K_C+B)\big)$.
We will distinguish two cases, depending on whether or not $\mathscr{C}$ is algebraically integrable.
\medskip
Suppose first that $\mathscr{C}$ is algebraically integrable.
Then $\mathscr{C}_F$ is induced by a pencil of conics containing a double line.
We will show that $\mathscr{G}$ satisfies condition (2) in the statement of Proposition \ref{proposition:P-bdle_over_curve_(2)}.
Let $S$ be the space of leaves of $\mathscr{C}$.
Then $S$ comes with a natural morphism onto $C$, whose general fiber
parametrizes a pencil of conics in $F\cong \p^2$.
We conclude that
$S\to C$ is a $\p^1$-bundle.
So $\mathscr{C}$ is induced by
a rational map $\psi: Y\dashrightarrow S$ over $C$,
and the restriction of $\psi$ to a general fiber $F\cong \p^2$ of $p$, $\psi_{|F}:F\cong\p^2\to \p^1$, is
given by a pencil of conics containing a double line $2\ell_F$.
Let $R\subset Y$ be the closure of the union of the lines $\ell_F$ when $F$ runs through general fibers of $p:Y\to C$.
Then $p_{|R}:R\to C$ is a $\p^1$-bundle.
Next we show that $R$ is the singular locus of $\psi$.
Let $F'\cong\p^2$ be a special fiber of $p$ such that $\mathscr{C}_{|F'}\hookrightarrow T_{F'}$ vanishes in codimension one.
Then the foliation $\mathscr{C}_{F'}$ on $F'$ induced by $\mathscr{C}$ is a degree $0$ foliation on $\p^2$, and
the cycle in $S$ corresponding to the leaf $\ell$ of $\mathscr{C}_{F'}$ is $R\cap F'+\ell$.
We conclude that $R$ is the singular locus of $\psi$.
By \ref{pullback_foliations},
there is a foliation by curves $\mathscr{N}$ on $S$ such that $\mathscr{G}$ is the pullback of $\mathscr{N}$ via $\psi$.
If $\psi(R)$ is not invariant by $\mathscr{N}$, then
$\psi^*\mathscr{N}\cong \mathscr{O}_{\p_C(\mathscr{K})}(R)\otimes p^*\mathscr{O}_C(-K_C-B)=
\mathscr{O}_{\p_C(\mathscr{K})}(R)\otimes \psi^*\big(q^*\mathscr{O}_C(-K_C-B)\big)$
by \eqref{pullback_fol}. Thus
$\mathscr{O}_{\p_C(\mathscr{K})}(R)\cong \psi^*\big(\mathscr{N}\otimes q^*\mathscr{O}_C(-K_C-B)\big)$, yielding a contradiction.
Therefore $\psi(R)$ is invariant by $\mathscr{N}$. Moreover,
$\psi^*\mathscr{N}\cong p^*\mathscr{O}_C(-K_C-B)$.
\medskip
Suppose from now on that $\mathscr{C}$ is not algebraically integrable, and hence neither is $\mathscr{G}$.
We will show that $\mathscr{G}$ satisfies condition (3) in the statement of Proposition \ref{proposition:P-bdle_over_curve_(2)}.
Let $\mathscr{L}$ be a very ample line bundle on $Y$.
By \cite[Proposition 7.5]{fano_fols}, there exists an algebraically integrable subfoliation by curves
$\mathscr{M} \subset \mathscr{G}$, $\mathscr{M}\not\subset T_{Y/C}$, such that
$\mathscr{M} \cdot \mathscr{L}^2 \ge \det(\mathscr{G}) \cdot \mathscr{L}^2 \ge 1$.
Moreover the general leaf of $\mathscr{M}$ is a rational curve.
Since $\mathscr{M}\not\subset T_{Y/C}$, the general leaf of $\mathscr{M}$ dominates $C$, and
we conclude that $C\cong \p^1$.
Next we show that $\mathscr{M}\cong p^*\mathscr{O}_{\p^1}(c)$, with $c\in\{1,2\}$.
Write $\mathscr{M}\cong \mathscr{O}_{\p_C(\mathscr{K})}(a)\otimes p^*\mathscr{O}_{\p^1}(c)$ for some integers $a$ and $c$.
First note that, for a general line $\ell \subset F$,
$\mathscr{C}_{|\ell}\subset \mathscr{G}_{|\ell}$ is a subbundle.
Since $\mathscr{C}_{|\ell}\cong \det(\mathscr{G})_{|\ell}\cong \mathscr{O}_\ell$, we must have
$\mathscr{G}_{\ell}\cong \mathscr{O}_\ell\oplus \mathscr{O}_\ell$. This implies that $a \le 0$.
Now observe that, since $\mathscr{G}$ is not algebraically integrable, $\mathscr{M}$ does not depend on the choice
of $\mathscr{L}$.
Therefore
$\mathscr{M} \cdot (\mathscr{O}_{Y}(kF)\otimes \mathscr{L})^2
\ge \det(\mathscr{G})\cdot (\mathscr{O}_{Y}(kF)\otimes \mathscr{L})^2 >0$ for all $k\ge 1$.
Thus $\mathscr{M} \cdot F \cdot \mathscr{L} \ge 0$, and hence $a \ge 0$.
We conclude that $a=0$ and $\mathscr{M}\cong p^*\mathscr{O}_{\p^1}(c)$.
Since $\mathscr{M} \cdot \mathscr{L}^2 \ge 1$, we have $c \ge 1$.
Since $\mathscr{M}\subset p^*T_{\p^1}$, we conclude that $c\in\{1,2\}$.
If $c=2$, then $\mathscr{M}$ yields a flat connection on $p$. Hence,
$\mathscr{K}\cong \mathscr{O}_{\p^1}(d)\oplus\mathscr{O}_{\p^1}(d)\oplus \mathscr{O}_{\p^1}(d)$ for some integer $d$, and
$\mathscr{M}$ is induced by the projection $\psi: Y\cong \p^1\times \p^2 \to \p^2$.
Now suppose that $c=1$.
We may assume that $\mathscr{K}$ is of the form
$\mathscr{K}\cong \mathscr{O}_{\p^1}\oplus\mathscr{O}_{\p^1}(-a_1)\oplus \mathscr{O}_{\p^1}(-a_2)$ for integers $a_2\ge a_1 \ge 0$.
Let $\tilde C \subset Y$ be the closure of a general leaf of $\mathscr{M}$.
We will show that $\tilde C$ is a section of $p$.
Suppose to the contrary that
$\tilde C$ has degree $\ge 2$ over $\p^1$. By \cite[Lemme 1.2 and Corollaire 1.3]{druel04},
$\tilde C$ has degree $2$ over $\p^1$, and $\mathscr{M}$ is regular in a neighbourhood of $\tilde C$.
In particular, we have
$\mathscr{N}_{\tilde C/Y}\cong \mathscr{O}_{\tilde C}\oplus\mathscr{O}_{\tilde C}$.
Write $\tilde C\sim 2 \sigma +kf$, where $\sigma$ is the section of $p$ corresponding to
the surjection $\mathscr{K}\twoheadrightarrow \mathscr{O}_{\p^1}(-a_2)$, and $f$ is a line on a fiber of $p$.
Let $E\subset Y$ be the divisor corresponding to the surjection
$\mathscr{K}\twoheadrightarrow \mathscr{O}_{\p^1}(-a_1)\oplus \mathscr{O}_{\p^1}(-a_2)$, so that
$\mathscr{O}_Y(E)\cong \mathscr{O}_{\p_C(\mathscr{K})}(1)$.
Since the deformations of $\tilde C$ sweep out a dense open subset of $Y$, we must have
\begin{equation} \label{E.tildeC}
E \ \cdot \ \tilde C \ = \ -2a_2 \ + \ k \ \ge \ 0.
\end{equation}
On the other hand, since $\mathscr{N}_{\tilde C/Y}\cong \mathscr{O}_{\tilde C}\oplus\mathscr{O}_{\tilde C}$, we have
\begin{equation} \label{deg_NC}
0 \ = \ \deg(\mathscr{N}_{\tilde C/Y}) \ = \ \deg\big((T_Y)_{|\tilde C}\big) \ - \ 2 \ = \ 2 \ + \ 2a_1\ - \ 4a_2 \ + \ 3k.
\end{equation}
Equations \eqref{E.tildeC} and \eqref{deg_NC} together yield a contradiction,
proving that $\tilde C$ is a section of $p$.
The map $\mathscr{M}\cong p^*\mathscr{O}_{\p^1}(1) \to p^*T_{\p^1}$ vanishes exactly along one fiber $F_0$ of
$p$. This implies that $\mathscr{M} \subset T_Y$ restricts to a regular foliation (with algebraic leaves) over
$Y\setminus F_0$. This foliation is induced by a smooth morphism
$\psi: Y\setminus F_0 \to \p^2$, which restricts to an isomorphism on all fibers $F\neq F_0$ of $p:Y\to \p^1$.
In either case, by \eqref{pullback_foliations}, $\mathscr{G}$ is the pullback via $\psi$ of a rank $1$ foliation
$\mathscr{N}$ on $\p^2$.
A straightforward computation shows that $\mathscr{N}\cong \mathscr{O}_{\p^2}$. This completes the proof of the proposition.
\end{proof}
We construct examples of foliations described in Proposition~\ref{proposition:P-bdle_over_curve_(2)}(2).
\begin{exmp}Let $C$ be a smooth complete curve, $\mathscr{A}$ an ample line bundle on $C$, and $P\in C$.
Set $\mathscr{L}:=\mathscr{A}\otimes\mathscr{O}_C(K_C+P)$,
$\mathscr{K}:=\mathscr{L}^{\otimes 2}\oplus\mathscr{L}\oplus\mathscr{O}_C$, and $\mathscr{W}:=\mathscr{L}^{\otimes 4}\oplus\mathscr{L}^{\otimes 2}$.
Suppose that $\deg(\mathscr{L})\neq 0$.
Let $s$ be a local frame for $\mathscr{L}$.
It induces local frames $(k_1,k_2,k_3)$ and $(w_1,w_2)$ for $\mathscr{K}$ and $\mathscr{W}$, respectively.
We view $\mathscr{W}$ as a subbundle of $Sym^2\mathscr{K}$ by mapping
$w_1$ to $k_1\otimes k_1$, and $w_2$ to $k_2\otimes k_2-k_1\otimes k_3$. This gives rise to a rational map
$\psi : \p_C(\mathscr{K}) \dashrightarrow \p_C(\mathscr{W})$ such that $\psi^*\mathscr{O}_{\p_C(\mathscr{W})}(1)\cong \mathscr{O}_{\p_C(\mathscr{K})}(2)$.
Note that the set $\sigma$ of critical values of $\psi$ is the section of $q$ corresponding to
$\mathscr{W}=\mathscr{L}^{\otimes 4}\oplus\mathscr{L}^{\otimes 2} \twoheadrightarrow \mathscr{L}^{\otimes 2}$.
Denote by $q : \p_C(\mathscr{W})\to C$ the natural morphism.
By \cite[Lemma 9.5]{fano_fols}, the inclusion $\mathscr{N}:=q^*\big(T_C(-P)\big)\hookrightarrow p^*T_C$
lifts to an inclusion $\iota : \mathscr{N}\hookrightarrow T_{\p_C(\mathscr{W})}$.
We claim that the cokernel of $\iota$ is torsion-free, and thus it defines a foliation on $\p_C(\mathscr{W})$.
Indeed, if $T_{\p_C(\mathscr{W})}/\mathscr{N}$ is not torsion-free, then we get an inclusion
$\mathscr{N}\subset T_{\p_C(\mathscr{W})}\otimes q^*\mathscr{O}_C(-P)$ (see \cite[Lemma 9.7]{fano_fols}).
Thus $p^*T_C\cong \mathscr{N}\otimes q^*\mathscr{O}_C(P) \subset T_{\p_C(\mathscr{W})}$, and
the natural exact sequence
$$
0\ \to \ T_{\p_C(\mathscr{W})/C} \ \to \ T_{\p_C(\mathscr{W})} \ \to \ p^*T_C \ \to \ 0
$$
splits. This implies that $\mathscr{K}$ admits a flat projective connection, which is absurd.
This proves the claim.
An easy compuation shows that
$\deg(\omega_\sigma\otimes\mathscr{N}_{|\sigma})=-\deg(\mathscr{A})-\deg(\mathscr{L})<0$, and thus $\sigma$ is invariant under
$\mathscr{N}$.
Now set $\mathscr{G}:=\psi^{-1}(\mathscr{N})$. Then $\mathscr{G}$ is a rank 2 foliation on $\p_C(\mathscr{K})$, generically transverse to the natural projection
$p:\p_C(\mathscr{K})\to C$, and satisfies $\det(\mathscr{G}) \cong p^*\mathscr{A}$.
\end{exmp}
Next we construct examples of foliations described in Proposition~\ref{lemma:P-bdle_over_C}(3).
\begin{exmp}
Let $C$ be a smooth complete curve and $\mathscr{V}$ an ample vector bundle of rank $n-2$ on $C$.
Let $\mathscr{K}_0$ be a vector bundle of rank $2$ on $C$, and suppose that $\mathscr{K}_0$
does not admit a flat projective connection.
Choose a sufficiently ample line bundle $\mathscr{A}$ on $C$ such that the following conditions hold:
\begin{enumerate}
\item $\mathscr{K}:=\mathscr{K}_0\otimes \mathscr{A}$ is an ample vector bundle;
\item there is a nowhere vanishing section $\alpha\in H^0\big(C, T_C\otimes \det(\mathscr{V}^*)\otimes \mathscr{K}_0\otimes \mathscr{A}\big)$; and
\item $h^1\big(C, \det(\mathscr{V}^*)\otimes Sym^3(\mathscr{K}_0)\otimes \det(\mathscr{K}_0^*)\otimes \mathscr{A}\big)=0$.
\end{enumerate}
Set $S:= \p_C(\mathscr{K})$, denote by $p:S\to C$ the natural projection, and by $\mathscr{O}_S(1)$ the tautological line bundle.
The section $\alpha$ from condition (2) yields an inclusion
$$
\mathscr{G} \ := \ p^*\big(\det(\mathscr{V})\big)\otimes \mathscr{O}_S(-1) \ \hookrightarrow \ p^*T_C,
$$
which does not vanish identically on any fiber of $p$.
Notice that $\mathscr{G}\otimes \mathscr{O}_S(B) \cong p^*T_C$
for some section $B$ of $p$.
By Lemma~\ref{lemma:lifting} below, condition (3) implies that the inclusion
$\mathscr{G}\hookrightarrow p^*T_C$ can be lifted to an inclusion
$$
\iota: \ \mathscr{G} \ \hookrightarrow \ T_S.
$$
We claim that the cokernel of $\iota$ is torsion-free, and thus it defines a foliation on $S$.
Indeed, if $T_S/\mathscr{G}$ is not torsion-free, then we get an inclusion
$\mathscr{G}\subset T_S(-B)$ (see \cite[Lemma 9.7]{fano_fols}).
Thus $p^*T_C \cong \mathscr{G}\otimes \mathscr{O}_S(B) \subset T_S$, and
the natural exact sequence
$$
0\ \to \ T_{S/C} \ \to \ T_S \ \to \ p^*T_C \ \to \ 0
$$
splits. This implies that $\mathscr{K}$ admits a flat projective connection, contradicting our assumption.
This proves the claim.
Now set $\mathscr{E}:= \mathscr{V}\oplus \mathscr{K}$, $X:=\p_C(\mathscr{E})$,
denote by $\pi:X\to C$ the natural projection, and by $\mathscr{O}_X(1)$ the tautological line bundle.
Condition (1) above implies that $\mathscr{O}_X(1)$ is an ample line bundle on $X$.
The natural quotient $\mathscr{E}\to \mathscr{V}$ defines a relative linear projection
$\varphi:X\dashrightarrow S$.
Let $\mathscr{F}$ be the codimension $1$ foliation on $X$ obtained as pullback of $\mathscr{G}$ via $\varphi$.
Recall that $T_{X/S}\cong \pi^*\det(\mathscr{V}^*)\otimes \mathscr{O}_X(1)$, and thus, by \eqref{K_pullback_fol},
$$
\det(\mathscr{F})\ \cong \ \mathscr{O}_X(n-3),
$$
i.e., $\mathscr{F}$ is a codimension $1$ Mukai foliation on $X$.
\end{exmp}
\begin{lemma}\label{lemma:lifting}
Let $\mathscr{K}$ be a vector bundle of rank $2$ on a smooth projective curve $C$,
$p:S=\p(\mathscr{K})\to C$ the corresponding ruled surface, and $\mathscr{O}_S(1)$ the tautological line bundle.
Let $\mathscr{B}$ be a line bundle on $C$ such that there is an inclusion
$j : \ p^*\mathscr{B} \otimes \mathscr{O}_S(-1)\hookrightarrow p^*T_C$.
If $h^1\big(C, \mathscr{B}\otimes Sym^3(\mathscr{K})\otimes \det(\mathscr{K}^*)\big)=0$, then
$j$ can be lifted to an inclusion $p^*\mathscr{B} \otimes \mathscr{O}_S(-1)\hookrightarrow T_S$.
\end{lemma}
\begin{proof}
Let $e$ be the class in $H^1(S,T_{S/C}\otimes p^*\omega_C)$
corresponding to the exact sequence
$$
0 \ \to \ T_{S/C} \ \to \ T_S \ \to \ p^*T_C\ \to \ 0 \ .
$$
An inclusion of line bundles $j: \mathscr{G}\hookrightarrow p^*T_C$ extends to
an inclusion $\mathscr{G} \hookrightarrow T_S$ if and only if the induced section
$j^*e\in H^1(S,T_{S/C}\otimes \mathscr{G}^{-1})$ vanishes identically.
Setting $\mathscr{G}:=p^*\mathscr{B} \otimes \mathscr{O}_S(-1)$, we get
$$
H^1(S,T_{S/C}\otimes \mathscr{G}^{-1}) \ =\ H^1\Big(C, p_*\big(T_{S/C} \otimes \mathscr{O}_S(1)\big)\otimes \mathscr{B}\Big).
$$
Since $T_{S/C}\cong p^*\big(\det(\mathscr{K}^*))\otimes \mathscr{O}_S(2)$, this gives
$$
H^1(S,T_{S/C}\otimes \mathscr{G}^{-1}) \ =\ H^1\big(C, p_*\mathscr{O}_S(3)\otimes \det(\mathscr{K}^*) \otimes \mathscr{B}\big) \ =\
H^1\big(C, Sym^3(\mathscr{K})\otimes \det(\mathscr{K}^*) \otimes \mathscr{B}\big).
$$
The latter vanishes by assumption, and thus $j: p^*\mathscr{B} \otimes \mathscr{O}_S(-1)\hookrightarrow p^*T_C$ extends to
an inclusion $p^*\mathscr{B} \otimes \mathscr{O}_S(-1) \hookrightarrow T_S$.
\end{proof}
\subsection{Codimension $1$ Mukai foliations on quadric bundles over curves}\label{subsection:Q-bdles/curves}
\
In this subsection, we work under Assumptions~\ref{assumptions}, supposing moreover that $\tau(L)=n-1$ and
$\varphi_L$ makes $X$ a quadric bundle over a smooth curve $C$. This is case (2b) of Theorem~\ref{tironi}.
We start with two useful observations.
\begin{rem} \label{rem:sing_Q-bdle_over_C}
Let $\phi:X\to C$ be a quadric bundle over a smooth curve, with $X$ smooth.
An easy computation shows that
the (finitely many) singular fibers of $\pi$ have only isolated singularities.
\end{rem}
\begin{lemma}\label{lemma:line_bundle_base}
Let $T$ be a complex variety, and $\phi : X \to T$ a flat projective morphism whose fibers are all irreducible and reduced. Let $\mathscr{Q}$ be a line bundle on $X$ such that $\mathscr{Q}_{|F}\cong \mathscr{O}_F$ for a general fiber $F$ of $\phi$. Then there exists a line bundle $\mathscr{M}$ on $T$ such that $\mathscr{Q}\cong \phi^*\mathscr{M}$.
\end{lemma}
\begin{proof}
Let $t\in T$ be any point, and denote by $X_t$ the corresponding fiber of $\phi$. By the semicontinuity theorem,
$h^0(X_t,\mathscr{Q}_{|X_t}) \ge 1$, and $h^0(X_t,\mathscr{Q}^*_{|X_t}) \ge 1$.
It follows that $\mathscr{Q}_{|X_t}\cong \mathscr{O}_{X_t}$,
since $X_t$ is irreducible and reduced.
By \cite[Corollary III.12.9]{hartshorne77},
$\mathscr{M}:=\phi_*\mathscr{Q}$ is a line bundle on $T$, and the evaluation map $\phi^*\mathscr{M}=\phi^*\phi_*\mathscr{Q}\to \mathscr{Q}$ is an isomorphism.
\end{proof}
\begin{prop}\label{proposition:Q-bdle_over_C}
Let $X$, $\mathscr{F}$ and $L$ be as in Assumptions~\ref{assumptions}.
Suppose that $\tau(L)=n-1$, and $\varphi_L$ makes $X$ a quadric bundle over a smooth curve $C$.
Then $C \cong \p^1$, and there exist
\begin{itemize}
\item an exact sequence of vector bundles on $\p^1$
$$
0\ \to \ \mathscr{K} \ \to \ \mathscr{E} \ \to \ \mathscr{V} \ \to \ 0,
$$
with $\textup{rank}(\mathscr{E})=n+1$, $\textup{rank}(\mathscr{K})=2$, and natural projections $\pi:\p_{\p^1}(\mathscr{E})\to \p^1$
and $q:\p_{\p^1}(\mathscr{K})\to \p^1$;
\item an integer $b$ and a foliation by rational curves $\mathscr{G}\cong q^*\big(\det(\mathscr{V})\otimes \mathscr{O}_{\p^1}(b)\big)$ on $\p_{\p^1}(\mathscr{K})$;
\end{itemize}
such that $X\in \big|\mathscr{O}_{\p_{\p^1}(\mathscr{E})}(2)\otimes \pi^*\mathscr{O}(b)\big|$, and
$\mathscr{F}$ is the pullback of $\mathscr{G}$ via the restriction to $X$ of the relative linear projection $\p_{\p^1}(\mathscr{E}) \dashrightarrow \p_{\p^1}(\mathscr{K})$.
Moreover, one of the following holds.
\begin{enumerate}
\item $(\mathscr{E},\mathscr{K})\cong (\mathscr{O}_{\p^1}(a)^{\oplus 2}\oplus\mathscr{O}_{\p^1}^{\oplus 3},\mathscr{O}_{\p^1}(a)^{\oplus 2})$
for some integer $a\ge 1$, and $b=2$.
\item $(\mathscr{E},\mathscr{K})\cong (\mathscr{O}_{\p^1}(a)^{\oplus 2}\oplus\mathscr{O}_{\p^1}^{\oplus 2}\oplus \mathscr{O}_{\p^1}(1),\mathscr{O}_{\p^1}(a)^{\oplus 2})$
for some integer $a\ge 1$, and $b=1$.
\item $(\mathscr{E},\mathscr{K})\cong (\mathscr{O}_{\p^1}(a)^{\oplus 2}\oplus\mathscr{O}_{\p^1}\oplus \mathscr{O}_{\p^1}(1)^{\oplus 2},\mathscr{O}_{\p^1}(a)^{\oplus 2})$ for some integer $a\ge 1$, and $b=0$.
\item $\mathscr{K}\cong \mathscr{O}_{\p^1}(a)^{\oplus 2}$ for some integer $a$, and
$\mathscr{E}$ is an ample vector bundle of rank $5$ or $6$ with $\deg(\mathscr{E})=2+2a-b$.
\item $\mathscr{K}\cong \mathscr{O}_{\p^1}(a)\oplus \mathscr{O}_{\p^1}(c)$ for distinct integers $a$ and $c$, and
$\mathscr{E}$ is an ample vector bundle of rank $5$ or $6$ with $\deg(\mathscr{E})=1+a+c-b$.
\end{enumerate}
In particular, $n\in\{4,5\}$ and $\mathscr{F}$ is algebraically integrable.
Conversely, given $\mathscr{K}$, $\mathscr{E}$ and $b$ satisfying any of the conditions (1-5),
and a smooth member $X\in \big|\mathscr{O}_{\p_{\p^1}(\mathscr{E})}(2)\otimes \pi^*\mathscr{O}_{\p^1}(b)\big|$,
there exists a codimension one Mukai foliation on $X$ as described above.
\end{prop}
\begin{proof}
Denote by $F\cong Q^{n-1}\subset \p^n$ a general (smooth) fiber of $\varphi_L$, and recall
from Theorem~\ref{tironi}(2b) that $\mathscr{L}_{|F}\cong \mathscr{O}_{Q^{n-1}}(1)$.
Set $\mathscr{E}:=(\varphi_L)_*\mathscr{L}$, and
denote by $\pi:\p_{C}(\mathscr{E})\to C$ the natural projection.
Then $X$ is a divisor of relative degree $2$ on $\p_C(\mathscr{E})$, i.e.,
$X\in \big|\mathscr{O}_{\p_C(\mathscr{E})}(2)\otimes \pi^*\mathscr{B}\big|$ for some line bundle $\mathscr{B}$ on $C$.
By Proposition~\ref{prop:Fano_fol_not_fibration}, $\mathscr{F}\neq T_{X/C}$.
So $\mathscr{H}:=\mathscr{F}\cap T_{X/C}$ is a codimension $2$ foliation on $X$.
Set $\mathscr{Q}:=(\mathscr{F}/\mathscr{H})^{**}$. It is an invertible subsheaf of $(\varphi_L)^*T_{C}$,
and $\det(\mathscr{H})\ \cong \ \det(\mathscr{F})\otimes \mathscr{Q}^*$.
Denote by $\mathscr{H}_F$ the codimension $1$ foliation on $F\cong Q^{n-1}$
obtained by restriction of $\mathscr{H}$.
By \ref{restricting_fols}, there exists a non-negative integer $d$ such that
$-K_{\mathscr{H}_F} \ = \ (n-3+d) H$,
where $H$ denotes a hyperplane section of $Q^{n-1}\subset \p^n$.
By Thereom~\ref{Thm:ADK}, we must have $d=0$. Hence,
\begin{itemize}
\item $\mathscr{H}_{F}$ is induced by a pencil of hyperplane sections
on $Q^{n-1}\subset \p^n$ by Theorem \ref{Thm:codim1_dP}. In paticular
the general log leaf of $\mathscr{H}$ is log canonical.
\item $\det(\mathscr{H})_{|F}\ \cong \ \det(\mathscr{F})_{|F}$, and thus $\mathscr{Q}\cong (\varphi_L)^*\mathscr{M}$ for some line bundle $\mathscr{M}\subset T_C$
by Lemma~\ref{lemma:line_bundle_base}.
\end{itemize}
By Proposition~\ref{prop:common_pt}, $\det(\mathscr{H})$ is not ample.
Since $\det(\mathscr{H})\ \cong \ \det(\mathscr{F})\otimes (\varphi_L)^*(\mathscr{M}^*)$ and $\det(\mathscr{F})$ is ample,
the line bundle $\mathscr{M}$ has positive degree.
Hence $C\cong \p^1$, $\mathscr{B}\cong \mathscr{O}_{\p^1}(b)$ for some $b\in \mathbb{Z}$,
and $\deg(\mathscr{M})\in \{1,2\}$.
The linear span of $\textup{Sing}(\mathscr{H}_{F})$ in $\p^n$ is the base locus of the pencil of hyperplanes in
$\p^n$ inducing $\mathscr{H}_{F}$ on $F\cong Q^{n-1}\subset \p^n$.
So $\mathscr{H}$ is the restriction to $X$ of a foliation $\tilde \mathscr{H}$ on $\p_{\p^1}(\mathscr{E})$ whose restriction to a general fiber of $\pi$ is a
degree zero foliation on $\p^n$.
By \eqref{V_in_E}, there is a sequence of vector bundles on $\p^1$,
\begin{equation}\label{exact_sequence}
0\ \to \ \mathscr{K} \ \to \ \mathscr{E} \ \to \ \mathscr{V} \ \to \ 0,
\end{equation}
with $\textup{rank}(\mathscr{K})=2$ and natural projection $q:\p_{\p^1}(\mathscr{K})\to \p^1$,
such that $\tilde \mathscr{H}$ is induced by the relative linear projection $\tilde \psi: \p_{\p^1}(\mathscr{E}) \dashrightarrow \p_{\p^1}(\mathscr{K})$.
So $\mathscr{H}$ is induced by the restriction $\psi= \tilde \psi_{|X}:X\dashrightarrow \p_{\p^1}(\mathscr{K})$.
By Remark~\ref{rem:sing_Q-bdle_over_C}, there is an open subset $X^{\circ}\subset X$ with $\textup{codim}_X(X\setminus X^{\circ})\ge 2$
such that $\psi^{\circ}=\psi_{|X^{\circ}}:X^{\circ}\to \p_{\p^1}(\mathscr{K})$ is a smooth morphism with connected fibers.
In particular, $\mathscr{H}\cong T_{X/\p(\mathscr{K})}$, where $T_{X/\p(\mathscr{K})}$ denotes the saturation of $T_{X^\circ/\p(\mathscr{K})}$ in $T_X$.
By \ref{pullback_foliations}, $\mathscr{F}$ is the pullback via $\psi$ of a rank $1$ foliation $\mathscr{G}$ on $\p_{\p^1}(\mathscr{K})$.
By \eqref{K_pullback_fol}, $\mathscr{G} \cong q^*\mathscr{M}$ and
$$
\mathscr{L}^{\otimes n-3}\cong \det(\mathscr{F})\cong \det(T_{X/\p(\mathscr{K})})\otimes (\varphi_L)^*\mathscr{M}.
$$
Since $\deg(\mathscr{M})>0$, the leaves of $\mathscr{G}$ are rational curves by Theorem~\ref{thm:BM}.
A straightforward computation gives $\mathscr{M}\cong \det (\mathscr{V})\otimes \mathscr{O}_{\p^1}(b)$, and so
\begin{equation}\label{degreeV}
\deg(\mathscr{M})=\deg(\mathscr{V})+b\in \{1,2\}.
\end{equation}
If $\deg(\mathscr{M})=2$, i.e., if $\mathscr{M}\cong T_C$, then $q^*\mathscr{M} \subset T_{\p_{\p^1}(\mathscr{K})}$
yields a flat connection on $q:\p_{\p^1}(\mathscr{K})\to \p^1$. Hence,
$\p_{\p^1}(\mathscr{K})\cong \p^1\times \p^1$, and $\mathscr{G}$
is induced by the projection to $\p^1$ transversal to $q$.
In this case, $\mathscr{K}\cong\mathscr{O}_{\p^1}(a)^{\oplus 2}$ for some integer $a$.
If $\deg(\mathscr{M})=1$, then $\mathscr{K}\cong \mathscr{O}_{\p^1}(a)\oplus \mathscr{O}_{\p^1}(c)$ for distinct integers $a$ and $c$.
This can be seen from the explicit description of the Atiyah classes in the proof of
\cite[Theorem 9.6]{fano_fols}.
By \cite[Theorem 4.13]{campana04}, the vector bundle
$(\varphi_L)_*\mathscr{O}_{X}(K_{X/\p^1}+m L)\cong S^{m-n+1}\mathscr{E}\otimes \det(\mathscr{E})\otimes \mathscr{O}_{\p^1}(b)$
is nef for all $m\ge n-1$. Therefore $\mathscr{E}$
is a nef vector bundle on $\p^1$, and we write $\mathscr{E}\cong\mathscr{O}_{\p^1}(a_1)\oplus\cdots\oplus\mathscr{O}_{\p^1}(a_{n+1})$,
with $0 \le a_1\le \cdots \le a_{n+1}$.
\medskip
First suppose that $\mathscr{E}$ is not ample.
Let $r\in \{1, \ldots, n+1\}$ be the largest positive integer
such that $a_1=\cdots =a_{r}=0$.
Let $\varpi : \p_{\p^1}(\mathscr{E}) \to \p\Big(H^0\big(\p_{\p^1}(\mathscr{E}),\mathscr{O}_{\p_{\p^1}(\mathscr{E})}(1)\big)^*\Big)$ be the morphism induced by the complete
linear system $\big|\mathscr{O}_{\p_{\p^1}(\mathscr{E})}(1)\big|$.
If $r=n+1$, then $\p_{\p^1}(\mathscr{E})\cong \p^1 \times \p^{n}$, and $\varpi$ is induced by the projection morphism
$\p^1 \times \p^{n} \to \p^{n}$.
If $r \le n$, then $\varpi$ is a birational morphism, and its restriction to the exceptional
locus $\textup{Exc}(\varpi)=\p_{\p^1}\big(\mathscr{O}_{\p^1}(a_1)\oplus\cdots\oplus\mathscr{O}_{\p^1}(a_{r})\big)\cong \p^1 \times \p^{r-1}$
corresponds to the projection $\p^1 \times \p^{r-1} \to \p^{r-1}$.
Thus, if $Z\subset \p_{\p^1}(\mathscr{E})$ is any closed subset, then
${\mathscr{O}_{\p_{\p^1}(\mathscr{E})}(1)}_{|Z}$ is ample if and only if
$Z$ does not contain any fiber of
$\textup{Exc}(\varpi)\cong\p^1 \times \p^{r-1} \to \p^{r-1}$.
Since ${\mathscr{O}_{\p_{\p^1}(\mathscr{E})}(1)}_{|X}\cong \mathscr{L}$ is ample,
$X$ does not contain any fiber of
$\textup{Exc}(\varpi)\cong\p^1 \times \p^{r-1} \to \p^{r-1}$.
Since $\big(\mathscr{O}_{\p_{\p^1}(\mathscr{E})}(2)\otimes \pi^*\mathscr{O}_{\p^1}(b)\big)_{|\textup{Exc}(\varpi)}
\cong \mathscr{O}_{\p^1}(b) \boxtimes \mathscr{O}_{\p^{r-1}}(2)$ and
$X\in \big|\mathscr{O}_{\p_{\p^1}(\mathscr{E})}(2)\otimes \pi^*\mathscr{O}_{\p^1}(b)\big|$, we must have $b \ge 0$ and
\begin{equation}\label{rnb}
h^0\big(\p^1,\mathscr{O}_{\p^1}(b)\big)= b+1 \ge r.
\end{equation}
If $\deg(\mathscr{V})=0$, then $\mathscr{V}\cong \mathscr{O}_{\p^1}^{\oplus (n-1)}$ and the exact sequence \eqref{exact_sequence} splits.
This implies that $r \ge n-1$. On the other hand, by \eqref{degreeV} and \eqref{rnb}, $r\le b+1\le 3$.
Thus $n=4$, $r=3$, $b=2$, and $\deg(\mathscr{M})=2$.
This is case (1) described in the statement of Proposition~\ref{proposition:Q-bdle_over_C}.
If $\deg(\mathscr{V}) \ge 1$, then $b\in\{0,1\}$ by \eqref{degreeV}.
Suppose that $b=1$. Then $\deg(\mathscr{V})=1$ by \eqref{degreeV}.
Thus $\mathscr{V}\cong \mathscr{O}_{\p^1}^{\oplus (n-2)}\oplus \mathscr{O}_{\p^1}(1)$, and
the exact sequence \eqref{exact_sequence} splits. By \eqref{rnb}, we have $n-2 \le 2$.
This implies $n=4$, and $\deg(\mathscr{M})=2$.
This is case (2) described in the statement of Proposition~\ref{proposition:Q-bdle_over_C}.
Suppose that $b=0$. Then we must have $r=1$ by \eqref{rnb}.
By \eqref{degreeV}, we have $\deg(\mathscr{V})\le 2$.
On the other hand, $\deg(\mathscr{V})\ge a_1 + \cdots + a_{n-1} \ge n-2$. Thus
$n=4$, $\mathscr{V}\cong \mathscr{O}_{\p^1}\oplus \mathscr{O}_{\p^1}(1)^{2}$, the exact sequence \eqref{exact_sequence} splits,
and $\deg(\mathscr{M})=2$.
This is case (3) described in the statement of Proposition~\ref{proposition:Q-bdle_over_C}.
\medskip
Suppose from now on that $\mathscr{E}$ is an ample vector bundle on $\p^1$.
Since
$\deg(\mathscr{V})\ge a_1 + \cdots + a_{n-1}$, we have
$b \le 2 - (a_1 + \cdots + a_{n-1})$ by \eqref{degreeV}.
We claim that $n\in\{4,5\}$.
Suppose to the contrary that $n\ge 6$.
Then
$h^0\big(\p^1,\mathscr{O}_{\p^1}(a_i+a_j+b)\big)=0$
if either $1 \le i,j \le n-2$, or
$1 \le i\le n-2$ and $j=n-1$. This implies that
in suitable homogeneous coordinates $(x_1:\cdots:x_{n+1})$,
$F\cong Q^{n-1}\subset \p^n$ is given by equation
$$
c_{n-1}x_{n-1}^2 + x_{n}l_{n}(x_1,\ldots,x_{n+1}) + x_{n+1}l_{n+1}(x_1,\ldots,x_{n+1})=0,
$$
where $c_{n-1}\in\mathbb{C}$, and $l_n$ and $l_{n+1}$ are linear forms.
This contradicts the fact that $F$ is smooth, and proves the claim.
So we are in one of cases (4) and (5) of Proposition~\ref{proposition:Q-bdle_over_C},
depending on whether $\deg(\mathscr{M})$ is $2$ or $1$, respectively.
\medskip
Now we proceed to prove the converse statement.
Let $\mathscr{K}$, $\mathscr{E}$ and $b$ satisfy one of the conditions (1-5) in the statement of Proposition~\ref{proposition:Q-bdle_over_C}, and
$X\in \big|\mathscr{O}_{\p_{\p^1}(\mathscr{E})}(2)\otimes \pi^*\mathscr{O}_{\p^1}(b)\big|$ a smooth member.
Then, one easily checks that $\mathscr{O}_{\p_{\p^1}(\mathscr{E})}(1)_{|X}$
is an ample line bundle on $X$.
We shall construct a codimension 1 Mukai foliation $\mathscr{F}$ on $X$
such that $\mathscr{O}_X(-K_{\mathscr{F}})\cong \mathscr{O}_{\p_{\p^1}(\mathscr{E})}(1)_{|X}$.
First, let $\mathscr{V}$ be a vector bundle of rank $n-1$ on $\p^1$
fitting into an exact sequence of vector bundles
$$
0 \ \to \ \mathscr{K} \ \to \ \mathscr{E} \ \to \ \mathscr{V}\ \to 0,
$$
and consider the induced rational map
$\tilde{\psi}:\p_{\p^1}(\mathscr{E}) \dashrightarrow \p_{\p^1}(\mathscr{K})$.
Let
$\psi:=\psi_{|X} : X \dashrightarrow \p_{\p^1}(\mathscr{K})$ be the restriction of $\tilde{\psi}$ to $X$, and let
$q:\p_{\p^1}(\mathscr{K})\to \p^1$ be the
natural projection.
By Remark~\ref{rem:sing_Q-bdle_over_C}, there is an open subset $X^{\circ}\subset X$ with $\textup{codim}_X(X\setminus X^{\circ})\ge 2$
such that $\psi^{\circ}=\psi_{|X^{\circ}}:X^{\circ}\to \p_{\p^1}(\mathscr{K})$ is a smooth morphism with connected fibers.
Set $\mathscr{M}:=\det(\mathscr{V})\otimes\mathscr{O}_{\p^1}(b)\subset T_{\p^1}$.
This inclusion lifts
to an inclusion of vector bundles $q^*\mathscr{M}\subset T_{\p(\mathscr{K})}$.
Let $\mathscr{F}$ be the pullback via $\psi$ of the foliation defined by $q^*\mathscr{M}$ in $\p_{\p^1}(\mathscr{K})$.
One computes that $\mathscr{O}_X(-K_{\mathscr{F}})\cong \mathscr{O}_{\p_{\p^1}(\mathscr{E})}(1)_{|X}$.
\end{proof}
\begin{exmp}
Set $\mathscr{E}=\mathscr{O}_{\p^1}(a_1)\oplus\cdots\oplus\mathscr{O}_{\p^1}(a_6)$,
with $1=a_1=a_2\le a_3\le a_4 \le a_5=a_6=a$, and $a_3+a_4 \le a+1$.
Set $\mathscr{K}=\mathscr{O}_{\p^1}(a)^{\oplus 2}$, and
$b=-(a_3+a_4)$. Then $\mathscr{E}$ and $b$ satisfy condition (4) in Proposition \ref{proposition:Q-bdle_over_C}.
Let $\lambda_{4,4} \in H^0\big(\p^1,\mathscr{O}_{\p^1}(2a_4+b)\big)$,
$\lambda_{2,5}\in H^0\big(\p^1,\mathscr{O}_{\p^1}(a_2+a_5+b)\big)$
and $\lambda_{1,6} \in H^0\big(\p^1,\mathscr{O}_{\p^1}(a_1+a_6+b)\big)$
be general sections. Then
$\lambda_{4,4}+\lambda_{2,5}+\lambda_{1,6}
\in
H^0\big(\p^1,S^2\mathscr{E}\otimes \mathscr{O}_{\p^1}(b)\big)
\cong H^0\big(\p_{\p^1}(\mathscr{E}),\mathscr{O}_{\p_{\p^1}(\mathscr{E})}(2)\otimes \pi^*\mathscr{O}(b)\big)$
defines a smooth hypersurface $X\subset \p_{\p^1}(\mathscr{E})$.
\end{exmp}
\subsection{Codimension $1$ Mukai foliations on projective space bundles over surfaces}\label{subsection:P-bdles/surfaces}
\
In this subsection, we work under Assumptions~\ref{assumptions}, supposing moreover that $\tau(L)=n-1$ and
$\varphi_L$ makes $X$ a $\p^{n-2}$-bundle over a smooth surface $S$. This is case (2c) of Theorem~\ref{tironi}.
We start with some easy observations.
\begin{lemma}\label{lemma:hirzebruch}
Let $C$ be a smooth proper curve of genus $g\ge 0$,
and $p: S \to C$ a ruled surface.
Suppose that $-K_S\sim A+B$ where $A$ is an ample divisor, and $B$ is effective.
Then $g=0$. If moreover $B$ is nonzero and supported on fibers of $p$, then
$S\cong\p^1\times\p^1$.
\end{lemma}
\begin{proof}
Let $\mathscr{E}$ be a normalized vector bundle
on $C$ such that $S\cong \p_C(\mathscr{E})$, and set $e:=-\deg(\mathscr{E})$,
as in \cite[Notation V.2.8.1]{hartshorne77}.
Denote by $f$ a general fiber, and by
$C_0$ a minimal section of $p: S \to C$.
Let $a$ and $b$ be integers such that
$A\sim aC_0+bf$. We have $-K_S\sim 2C_0+(2-2g+e)f$, and hence
$a\in\{1,2\}$, and $B\sim (2-a)C_0+(e-2g+2-b)f$.
Since $B$ is effective, $e-2g+2-b \ge 0$.
We claim that $e\ge 0$. Suppose to the contrary that $e<0$.
Then $b>\frac{1}{2}ae$ by \cite[Proposition V.2.21]{hartshorne77}. Thus
$2g-2\le e-b <e-\frac{1}{2}ae<0$,
and $g=0$. But this contradicts \cite[Theorem V.2.12]{hartshorne77}, proving the claim.
By \cite[Proposition V.2.20]{hartshorne77},
we must have $b\ge ae+1$, and thus $-2 \le 2g-2\le e-b \le (1-a)e-1<0$.
This implies $g=0$, and $b-e\in\{1,2\}$. If moreover $a=2$ and $B\neq 0$,
then $e=0$, completing the proof of the lemma.
\end{proof}
\begin{lemma}\label{lemma:big_anticanonical}
Let $S$ be a smooth projective surface such that
$-K_S\sim A+B$ where $A$ is ample, and $B\neq 0$ is effective.
Then
either $S\cong \p^2$, or $S$ is a Hirzebruch surface.
\end{lemma}
\begin{proof}
It is enough to show that either $S$ is minimal, or $S \cong \mathbb{F}_1$.
Suppose otherwise, and let $c : S \to T$ be a proper birational morphism onto a ruled surface $q:T\to C$.
Set $A_T:=c_*A$, and $B_T:=c_*B$. Then $A_T$ is ample, $B_T$ is effective, and $-K_{T}\sim A_T+B_T$.
By Lemma \ref{lemma:hirzebruch}, $C \cong \p^1$ and $T \cong \mathbb{F}_e$ for some $e \ge 0$.
Let $p : S \to C$ be the induced morphism, and denote by $f$ a fiber of $p$ or $q$.
Since $c$ is not an isomorphism by assumption, $A \cdot f \ge 2$.
On the other hand, $-K_S\cdot f=2$, and thus $A \cdot f = -K_S\cdot f-B\cdot f\le 2$. Hence
$A \cdot f = A_T\cdot f =2$, and $B\cdot f= B_T\cdot f = 0$.
These equalities, together with the fact that $A_T$ is ample and $-K_{T}\sim A_T+B_T$,
imply that one of the following holds:
\begin{enumerate}
\item $T \cong \mathbb{F}_1$ and $B_T=0$; or
\item $T\cong \p^1\times \p^1$ and either $B_T=0$ or $B=f$.
\end{enumerate}
Suppose that $B_T=0$. Intersecting $-K_S$ with the (disjoint) curves $E_i$'s contracted by $c$ gives that
$A=c^*A_T - \sum E_i=-K_S$. But this forces $B=0$, contrary to our assumptions.
So we must have $T\cong \p^1\times \p^1$ and $B=f$.
Intersecting $-K_S$ with the (disjoint) curves $E_i$'s contracted by $c$ gives that
$A=c^*A_T - \sum E_i$ and $B=f$.
Let $\ell\subset T$ be a fiber of the projection $T \cong \p^1\times \p^1\to \p^1$
transversal to $q:T\to \p^1$. Then $A_T\cdot \ell= 1$.
We choose $\ell$ to contain the image of some $E_i$, and
let $\tilde \ell\subset S$ be its strict transform.
Then $A\cdot \tilde \ell\le 0$, contrary to our assumptions.
We conclude that either $S$ is minimal, or $S \cong \mathbb{F}_1$.
\end{proof}
\begin{prop}\label{proposition:P-bdle_over_S}
Let $X$, $\mathscr{F}$ and $L$ be as in Assumptions~\ref{assumptions}.
Suppose that $\tau(L)=n-1$, and $\varphi_L$ makes $X$ a $\p^{n-2}$-bundle over a smooth surface $S$.
Then there exist
\begin{itemize}
\item an exact sequence of sheaves of $\mathscr{O}_S$-modules
$$
0\ \to \ \mathscr{K} \ \to \ \mathscr{E} \ \to \ \mathscr{Q} \ \to \ 0,
$$
where $\mathscr{K}$, $\mathscr{E}$, and $\mathscr{V}:=\mathscr{Q}^{**}$ are vector bundles on $S$, $\mathscr{E}$ is ample of rank $n-1$, and
$\textup{rank}(\mathscr{K})=2$;
\item a codimension $1$ foliation $\mathscr{G}$ on $\p_S(\mathscr{K})$, generically transverse to the natural projection $q:\p_{S}(\mathscr{K})\to S$,
satisfying $\det(\mathscr{G})\cong q^*\det(\mathscr{V})$ and $r_{\mathscr{G}}^a\ge 1$;
\end{itemize}
such that $X\cong \p_{S}(\mathscr{E})$, and
$\mathscr{F}$ is the pullback of $\mathscr{G}$ via the induced relative linear projection $\p_C(\mathscr{E})\dashrightarrow \p_C(\mathscr{K})$.
In this case, $r_{\mathscr{F}}^a\ge r_{\mathscr{F}}-1$.
Moreover, one of the following holds.
\begin{enumerate}
\item $S\cong\p^2$, $\det(\mathscr{V})\cong\mathscr{O}_{\p^2}(i)$ for some $i\in \{1,2,3\}$, and $4\le n\le 3+i$.
\item $S$ is a del Pezzo surface $\not\cong\p^2$, $\det(\mathscr{V})\cong \mathscr{O}_S(-K_S)$ , and $4\le n\le 5$.
\item $S\cong \p^1\times \p^1$, $\det(\mathscr{V})$ is a line bundle of type $(1,1)$, $(2,1)$ or $(1,2)$, and $n=4$.
\item $S\cong \mathbb{F}_e$ for some integer $e \ge 1$, $\det(\mathscr{V})\cong\mathscr{O}_{\mathbb{F}_e}(C_0+(e+i)f)$,
where $i\in\{1,2\}$, $C_0$ is the minimal section of the natural morphism $\mathbb{F}_e\to\p^1$,
$f$ is a general fiber, and $n=4$.
\end{enumerate}
Conversely, given $S$, $\mathscr{K}$, $\mathscr{E}$, $\mathscr{V}$ as above, and a codimension one foliation $\mathscr{G}\subset T_{\p_{S}(\mathscr{K})}$
satisfying $\det(\mathscr{G})\cong q^*\det(\mathscr{V})$,
the pullback of $\mathscr{G}$ via the relative linear projection $X\cong \p_{S}(\mathscr{E}) \dashrightarrow \p_{S}(\mathscr{K})$ is a codimension one Mukai foliation on $X$.
\end{prop}
\begin{proof}
Denote by $F\cong \p^{n-2}$ a general fiber of $\varphi_L$, and recall from Theorem~\ref{tironi}(2c) that $\mathscr{L}_{|F}\cong \mathscr{O}_{\p^{n-2}}(1)$.
Set $\mathscr{E}:=(\varphi_L)_*\mathscr{L}$. Then $\mathscr{E}$ is an ample vector bundle of rank $n-1$, and $X\cong \p_S(\mathscr{E})$.
We claim that $T_{X/S}\not \subseteq \mathscr{F}$. Indeed, if $T_{X/S}\subseteq \mathscr{F}$,
then $\mathscr{F}$ would be the pullback via $\varphi_L$ of a foliation on $S$, and hence
${\mathscr{L}^{\otimes n-3}}_{|F}\cong \det(\mathscr{F})_{|F}\cong \det(T_{X/S})_{|F}\cong {\mathscr{L}^{\otimes n-1}}_{|F}$
by \ref{pullback_foliations}, which is absurd.
So $\mathscr{H}:=\mathscr{F}\cap T_{X/S}$ is a codimension $3$ foliation on $X$. Denote by $\mathscr{H}_F$ the codimension $1$ foliation on $F\cong \p^{n-2}$
obtained by restriction of $\mathscr{H}$.
By \ref{restricting_fols}, there exists a non-negative integer $c$ such that
$-K_{\mathscr{H}_F} \ = \ (n-3+c) H$,
where $H$ denotes a hyperplane in $\p^{n-2}$.
By Thereom~\ref{Thm:ADK}, $c=0$, and
$\mathscr{H}_{F}$ is a degree zero foliation on $F\cong \p^{n-2}$.
Let $\mathscr{K}$ and $\mathscr{V}$ be as defined in \ref{V_in_E}.
By \cite[Corollary 1.4]{hartshorne80} and \cite[Remark 2.3]{fano_fols}, $\mathscr{K}$ and $\mathscr{V}$ are vector bundles on $S$, and there is an exact sequence
$$
0\ \to \ \mathscr{K} \ \to \ \mathscr{E} \ \to \ \mathscr{Q} \ \to \ 0
$$
with $\mathscr{Q}^{**}\cong\mathscr{V}$.
The foliation $\mathscr{H}$ is induced by the relative linear projection $\psi: X\cong\p_{S}(\mathscr{E}) \dashrightarrow \p_{S}(\mathscr{K})$.
So, by \ref{pullback_foliations}, $\mathscr{F}$ is the pullback via $\psi$ of a rank $2$ foliation $\mathscr{G}$ on $\p_{S}(\mathscr{K})$.
There is an open subset $X^{\circ}\subset X$ with $\textup{codim}_X(X\setminus X^{\circ})\ge 2$
such that $\psi^{\circ}=\psi_{|X^{\circ}}:X^{\circ}\to \p_{S}(\mathscr{K})$ is a smooth morphism with connected fibers.
So, by \eqref{K_pullback_fol},
$$
\mathscr{L}^{\otimes n-3}\cong \det(\mathscr{F})\cong \det(T_{X/\p_{S}(\mathscr{K})})\otimes \varphi^*\det(\mathscr{G}),
$$
where $T_{X/\p_{S}(\mathscr{K})}$ denotes the saturation of $T_{X^\circ/\p_{S}(\mathscr{K})}$ in $T_X$.
A straightforward computation gives $\det(\mathscr{G})\cong q^*\det (\mathscr{V})$, where $q:\p_{S}(\mathscr{K})\to S$
denotes the natural projection.
By Lemma \ref{lemma:almost_quotient_of_ample_is_ample}, $\det(\mathscr{V})$ is an ample line bundle on $S$.
Thus, applying Theorem~\ref{thm:BM} to a suitable destabilizing subsheaf of $\mathscr{G}$, we conclude that $r_{\mathscr{G}}^a\ge 1$.
The natural morphism $\mathscr{G} \to q^*T_{S}$ is injective since
$T_{\p_{S}(\mathscr{K})/S}\not\subseteq \mathscr{G}$. Let $q^*B$ be the divisor of zeroes of
the induced map $q^*\det(\mathscr{V})\cong\det(\mathscr{G}) \to q^*\det(T_{S})$.
\medskip
Suppose first that $B=0$. Then
$\det(\mathscr{V})\cong \mathscr{O}_S(-K_S)$, and hence $S$ is a del Pezzo surface.
If $S\cong\p^2$, then $\det(\mathscr{V})\cong\mathscr{O}_{\p^2}(3)$.
Since the restriction of $\mathscr{V}$ to a general line on $\p^2$ is an ample vector bundle,
$\textup{rank}(\mathscr{V}) \le 3$, and hence $4\le n\le 6$.
Suppose that $S\not \cong\p^2$, and let $\ell \subset S$ be a general free rational curve of minimal anticanonical degree.
Then $\det(\mathscr{V})\cdot \ell = -K_S \cdot \ell = 2$.
Since $\mathscr{V}_{|\ell}$ is an ample vector bundle, $\textup{rank}(\mathscr{V}) \le 2$, and hence $4\le n\le 5$.
\medskip
Suppose now that $B\neq 0$. By Lemma \ref{lemma:big_anticanonical},
either $S\cong \p^2$, or $S$ is a Hirzebruch surface.
If $S\cong\p^2$, then $\det(\mathscr{V})\cong\mathscr{O}_{\p^2}(i)$, with $i\in \{1,2\}$.
As above, we see that $\textup{rank}(\mathscr{V}) \le i$, and hence $4\le n\le 3+i$.
If $S\cong \mathbb{F}_e$ for some $e\ge 0$, then
a straightforward computation gives that
either $\det(\mathscr{V})\cong\mathscr{O}_{\mathbb{F}_e}(C_0+(e+i)f)$, with $i\in\{1,2\}$,
or $e=0$ and $\det(\mathscr{V})\cong\mathscr{O}_{\mathbb{F}_e}(2C_0+f)$.
In any case, $\det(\mathscr{V})\cdot \ell=1$ for a suitable free rational curve $\ell\subset \mathbb{F}_e$.
Since $\mathscr{V}_{|\ell}$ is an ample vector bundle, $\textup{rank}(\mathscr{V}) =1$, and hence $n=4$.
\medskip
Conversely, given $S$, $\mathscr{K}$, $\mathscr{E}$, $\mathscr{V}$ satisfying one the conditions in the statement of Proposition
\ref{proposition:P-bdle_over_S}, and a codimension one foliation $\mathscr{G}\subset T_{\p_{S}(\mathscr{K})}$
satisfying $\det(\mathscr{G})\cong q^*\det(\mathscr{V})$, a straightforward computation shows that
the pullback of $\mathscr{G}$ via the relative linear projection $X\cong \p_{S}(\mathscr{E}) \dashrightarrow \p_{S}(\mathscr{K})$ is a codimension one foliation on $X$
with determinant $\mathscr{O}_{\p_S(\mathscr{E})}(n-3)$.
\end{proof}
\begin{lemma}\label{lemma:almost_quotient_of_ample_is_ample}
Let $S$ be a smooth projective surface, $W\subset S$ a closed subscheme with $\textup{codim}_S W \ge 2$,
$\mathscr{E}$ an ample vector bundle on $S$,
and $\mathscr{V}$ a vector bundle on $S$ such that there exists a surjective morphism of sheaves of
$\mathscr{O}_S$-modules
$\mathscr{E} \twoheadrightarrow \mathscr{I}_W\mathscr{V}$. Then $\det(\mathscr{V})$ is an ample line bundle.
\end{lemma}
\begin{proof}
Let $r$ be the rank of $\mathscr{V}$.
The $r$-th wedge product of the morphism
$\mathscr{E} \twoheadrightarrow \mathscr{I}_W\mathscr{V}$ gives rise to
a surjective morphism $\wedge^r\mathscr{E}_{|S \setminus \textup{Supp}(W)} \twoheadrightarrow
\det(\mathscr{V})_{|S \setminus \textup{Supp}(W)}$. It follows that $\det(\mathscr{V}) \cdot C \ge 1$ for any curve $C \subset S$.
To conclude that $\det(\mathscr{V})$ is ample, it is enough to show that $h^0(S,\det(\mathscr{V})^{\otimes m})\ge 1$
for some integer $m\ge 1$
by the Nakai-Moishezon criterion.
Set $Y:=\p_S(\wedge^r\mathscr{E})$. Denote by $\mathscr{O}_Y(1)$ the tautological line bundle on $Y$, and
by $q : Y \to S$ the natural projection.
Let $T\subset Y$ be the closure of the section of
$q_{|S\setminus \textup{Supp}(W)}$ corresponding to
$\wedge^r\mathscr{E}_{|S \setminus \textup{Supp}(W)} \twoheadrightarrow \det(\mathscr{V})_{|S \setminus \textup{Supp}(W)}$.
Then $\mathscr{O}_Y(1)_{|T}\cong (q_{|T})^*\det(\mathscr{V})\otimes\mathscr{O}_T(E)$ for some divisor $E \subset T$
with $\textup{Supp}(E)\subset T \cap q^{-1}(\textup{Supp}(W))$.
Since $\mathscr{O}_Y(1)_{|T}$ is an ample line bundle, we must have
$$h^0(S\setminus \textup{Supp}(W),{\det(\mathscr{V})^{\otimes m}}_{|S\setminus \textup{Supp}(W)}) \ge
h^0(T\setminus \textup{Supp}(E),{\mathscr{O}_Y(m)}_{|T\setminus \textup{Supp}(E)})
\ge h^0(T,{\mathscr{O}_Y(m)}_{|T})
\ge 1$$ for some $m\ge 1$,
and hence
$h^0(S,\det(\mathscr{V})^{\otimes m})\ge 1$.
\end{proof}
Our next goal is to classify pairs $(\mathscr{K},\mathscr{G})$ that appear in Proposition~\ref{proposition:P-bdle_over_S}.
When $\det(\mathscr{G})\cong q^*\mathscr{O}_S(-K_S)$, the situation is easily described as follows.
This includes the cases described in Proposition~\ref{proposition:P-bdle_over_S}(1, $i=3$) and (2).
\begin{rem}\label{rem:flat_connection}
Let $Z$ be a simply connected smooth projective variety, and let $\mathscr{K}$ be a rank $2$ vector bundle on $Z$.
Set $Y:=\p_Z(\mathscr{K})$, with natural projection $q : Y \to Z$. Denote by $\mathscr{O}_{Y}(1)$ the tautological line bundle on $Y$. Let $\mathscr{G} \subset T_Y$ be a codimension one foliation on $Y$ such that
$\mathscr{G}\cong q^*\mathscr{O}_Z(-K_Z)$.
Then $\mathscr{G} \subset T_Y$ induces a flat connection on $q$. Thus $\mathscr{K}\cong\mathscr{M}\oplus \mathscr{M}$ for some line bundle $\mathscr{M}$ on $Z$,
and $\mathscr{G}$ is induced by the natural morphism $\p_Z(\mathscr{K})\cong Z\times\p^1\to \p^1$.
\end{rem}
Suppose now that $S\cong \p^2$ or $\mathbb{F}_e$, and $\det(\mathscr{G})\not \cong q^*\mathscr{O}_S(-K_S)$.
We will describe $\mathscr{K}$ and $\mathscr{G}$ that appear in Proposition~\ref{proposition:P-bdle_over_S}
by restricting them to special rational curves on $S$.
Our analysis will rely on the following result.
\begin{lemma}\label{lemma:foliation_surface}
Let $m \ge 0$ be an integer, and consider the ruled surface $q : \mathbb{F}_m \to \p^1$.
Let $\mathscr{C}\cong q^*\mathscr{O}_{\p^1}(a)$ be a foliation by curves on $\mathbb{F}_m$ with $a> 0$.
Then $a\in\{1,2\}$, and one of the following holds.
\begin{enumerate}
\item If $a=2$, then $m=0$, and $\mathscr{C}$ is induced by the projection $\mathbb{F}_0 \cong \p^1\times\p^1 \to \p^1$ transversal to $q$.
\item If $a=1$, then $m\ge 1$, and $\mathscr{C}$
is induced by a pencil containing $C_0+m f_0$,
where $C_0$ denotes the minimal section and $f_0$ a fiber of $q : \mathbb{F}_m \to \p^1$.
\end{enumerate}
\end{lemma}
\begin{proof}
Notice that $\mathscr{C}\neq T_{\mathbb{F}_m/\p^1}$, thus the natural map
$q^*\mathscr{O}_{\p^1}(a)\cong \mathscr{C} \to q^*T_{\p^1}\cong q^*\mathscr{O}_{\p^1}(2)$ is nonzero,
and hence $a\in\{1,2\}$.
If $a=2$, then, as in Remark~\ref{rem:flat_connection},
$\mathscr{C}$ yields a flat connexion on $q$, $m=0$, and $\mathscr{C}$ is induced by the projection
$\mathbb{F}_0 \cong \p^1\times\p^1 \to \p^1$ transversal to $q$, proving (1).
From now on we assume that $a=1$. Then we must have $m\ge 1$,
since the map $\mathscr{C} \to T_{\mathbb{F}_m}$ does not vanish in codimension one.
Denote by $C_0$ be a minimal section, and by $f$ a fiber of $q : \mathbb{F}_m \to \p^1$.
By Theorem \ref{thm:BM}, $\mathscr{C}$ is algebraically integrable and its leaves are rational curves.
So it is induced by a rational map with irreducible general fibers $\pi : \mathbb{F}_m \dashrightarrow \p^1$.
Let $C$ be the closure of a general leaf of $\mathscr{C}$.
As in the proof of Proposition~\ref{proposition:P-bdle_over_curve_(2)},
one shows that $C$ is a section of $q$, and $\mathscr{C}$ is not regular along $C$.
Write $C \sim C_0+b f$ with $b \ge m$ (see \cite[Proposition V.2.20]{hartshorne77}).
The foliation $\mathscr{C}$ is induced by a pencil $\Pi$ of members of $|\mathscr{O}_{\mathbb{F}_m}(C)|$.
Observe that the space of reducible members of $|\mathscr{O}_{\mathbb{F}_m}(C)|$ is a codimension one linear subspace.
Therefore, $\Pi$ has a unique reducible member.
Let $f_0$ be the divisor of zeroes of
$q^*\mathscr{O}_{\p^1}(1)\cong \mathscr{C} \to q^*T_{\p^1}\cong q^*\mathscr{O}_{\p^1}(2)$. It is a fiber of $q$.
Note that $\mathscr{C}$ induces a flat connection on $q$ over $\mathbb{F}_m \setminus f_0$. In particular,
$\mathscr{C}$ is regular over $\mathbb{F}_m \setminus f_0$.
Let $R(\pi)$ be the ramification divisor of $\pi$, and notice that $R(\pi)$ is supported on $f_0$.
A straightforward computation gives $R(\pi)\equiv (2b-(m+1))f$.
Let $C_1+kf_0$ the reducible member of $\Pi$ ($k\ge 1$), where $C_1$ is irreducible.
Write $C_1 \sim C_0+b_1f$.
Then
$k=2b-(m+1)+1$, and $b_1+k=b$.
Thus $b-k=m-b \le 0$, and $b-k=b_1 \ge 0$.
Hence $b_1=0$, and $k=b=m$. This proves (2).
\end{proof}
\begin{prop}\label{proposition:P-bdle_over_S_hirzebruch}
Let $\mathscr{K}$ be a rank $2$ vector bundle on a ruled surface $p : \mathbb{F}_e \to \p^1$, $e \ge 0$.
Set $Y:=\p_{\mathbb{F}_e}(\mathscr{K})$,
with natural projection $q : Y \to \mathbb{F}_e$, and tautological line bundle $\mathscr{O}_{Y}(1)$.
Let $\mathscr{G}\subset T_Y$ be a codimension one foliation on $Y$ with $\det(\mathscr{G})\cong q^*\mathscr{A}$
for some ample line bundle $\mathscr{A}$ on $\mathbb{F}_e$. Then one of the following holds.
\begin{enumerate}
\item $e\in\{0,1\}$ and there exists a line bundle $\mathscr{B}$ on $\mathbb{F}_e$ such that
\begin{itemize}
\item $\mathscr{K}\cong\mathscr{B}\oplus \mathscr{B}$, and
\item $\mathscr{G}$ is induced by the natural morphism $Y\cong \mathbb{F}_e\times\p^1\to \p^1$
and thus $\det(\mathscr{G})\cong q^*\mathscr{O}_{\mathbb{F}_e}(-K_{\mathbb{F}_e})$.
\end{itemize}
\item There exist a line bundle $\mathscr{B}$ on $\mathbb{F}_e$, integers $s \ge 1$ and $t\ge 0$,
a minimal section $C_0$ and a fiber $f$ of $p : \mathbb{F}_e \to \p^1$ such that
\begin{itemize}
\item $\mathscr{K}\cong\mathscr{B}\otimes \big(\mathscr{O}_{\mathbb{F}_e}\oplus\mathscr{O}_{\mathbb{F}_e}(sC_0+tf)\big)$,
\item $\mathscr{G}$ is induced by a pencil in $|\mathscr{O}_{Y}(1)\otimes q^*\mathscr{B}^*|$
containing $\Sigma+q^*(sC_0+tf)$, where $\Sigma$ is the section of $q : Y \to \mathbb{F}_e$
corresponding to the surjection
$\mathscr{O}_{\mathbb{F}_e}\oplus\mathscr{O}_{\mathbb{F}_e}(sC_0+tf)\twoheadrightarrow \mathscr{O}_{\mathbb{F}_e}$,
\item $\det(\mathscr{G})\cong q^*\mathscr{O}_{\mathbb{F}_e}(C_0+(e+2)f)$ if $t=0$, and
$\det(\mathscr{G})\cong q^*\mathscr{O}_{\mathbb{F}_e}(C_0+(e+1)f)$ if $t>0$.
\end{itemize}
\item $e\in\{0,1\}$, there exist a line bundle $\mathscr{B}$ on $\mathbb{F}_e$, an integer $s \ge 1$, and
an irreducible divisor $B\sim C_0+f$ on $\mathbb{F}_e$, where $C_0$ is a minimal section and
$f$ a fiber of $p : \mathbb{F}_e \to \p^1$, such that
\begin{itemize}
\item $\mathscr{K}\cong\mathscr{B}\otimes \big(\mathscr{O}_{\mathbb{F}_e}\oplus\mathscr{O}_{\mathbb{F}_e}(s(C_0+f))\big)$,
\item $\mathscr{G}$ is induced by a pencil in $|\mathscr{O}_{Y}(1)\otimes q^*\mathscr{B}^*|$
containing $\Sigma+sq^*B$, where $\Sigma$ is the section of $q : Y \to \mathbb{F}_e$
corresponding to the surjection
$\mathscr{O}_{\mathbb{F}_e}\oplus\mathscr{O}_{\mathbb{F}_e}(s(C_0+f))\twoheadrightarrow \mathscr{O}_{\mathbb{F}_e}$,
\item $\det(\mathscr{G})\cong q^*\mathscr{O}_{\mathbb{F}_e}(C_0+(e+1)f)$.
\end{itemize}
\item There exist a line bundle $\mathscr{B}$ on $\mathbb{F}_e$, integers $s,t \ge 1$,
a minimal section $C_0$ and a fiber $f$ of $p : \mathbb{F}_e \to \p^1$ such that
\begin{itemize}
\item $\mathscr{K}\cong\mathscr{B}\otimes \big(\mathscr{O}_{\mathbb{F}_e}(sC_0) \oplus \mathscr{O}_{\mathbb{F}_e}(t f)\big)$,
\item $\mathscr{G}$ is induced by a pencil in $|\mathscr{O}_{Y}(1)\otimes q^*\mathscr{B}^*|$ generated by
$\Sigma+sq^*C_0$ and $\Sigma'+tq^*f$, where
$\Sigma$ and $\Sigma'$ are sections of $q : Y \to \mathbb{F}_e$ corresponding to the surjections
$\mathscr{O}_{\mathbb{F}_e}(sC_0) \oplus \mathscr{O}_{\mathbb{F}_e}(t f) \twoheadrightarrow \mathscr{O}_{\mathbb{F}_e}(t f)$, and
$\mathscr{O}_{\mathbb{F}_e}(sC_0) \oplus \mathscr{O}_{\mathbb{F}_e}(t f) \twoheadrightarrow \mathscr{O}_{\mathbb{F}_e}(s C_0)$, respectively,
\item $\det(\mathscr{G})\cong q^*\mathscr{O}_{\mathbb{F}_e}(C_0+(e+1)f)$.
\end{itemize}
\item There exist a line bundle $\mathscr{B}$ on $\mathbb{F}_e$, integers $s \ge 1$ and $\lambda\ge 0$,
a minimal section $C_0$ and a fiber $f$ of $p : \mathbb{F}_e \to \p^1$ such that
\begin{itemize}
\item $\mathscr{K}$ fits into an exact sequence
$$
0\ \to \ \mathscr{O}_{\mathbb{F}_e}\oplus \mathscr{O}_{\mathbb{F}_e}(sC_0) \ \to \
\mathscr{K}\otimes\mathscr{B}^*
\ \to \
\mathscr{O}_{f}(-\lambda)
\to \ 0,
$$
\item $\mathscr{G}$ is induced by a pencil in $|\mathscr{O}_{Y}(1)\otimes q^*\mathscr{B}^*|$ generated by
$\Sigma+sq^*C_0$ and $\Sigma'$, where
$\Sigma$ is the zero locus of the section of
$\mathscr{O}_{Y}(1)\otimes q^*(\mathscr{B}^*\otimes \mathscr{O}_{\mathbb{F}_e}(-sC_0))$
corresponding to
$\mathscr{O}_{\mathbb{F}_e}(sC_0) \to \mathscr{O}_{\mathbb{F}_e}\oplus \mathscr{O}_{\mathbb{F}_e}(sC_0) \to \mathscr{K}\otimes\mathscr{B}^*$, and
$\Sigma'$ corresponds to
$\mathscr{O}_{\mathbb{F}_e} \to \mathscr{O}_{\mathbb{F}_e}\oplus \mathscr{O}_{\mathbb{F}_e}(sC_0) \to \mathscr{K}\otimes\mathscr{B}^*$,
\item $\det(\mathscr{G})\cong q^*\mathscr{O}_{\mathbb{F}_e}(C_0+(e+1)f)$.
\end{itemize}
\item There exist a line bundle $\mathscr{B}$ on $\mathbb{F}_e$, an integer $t\ge 0$,
a minimal section $C_0$ and a fiber $f$ of $p : \mathbb{F}_e \to \p^1$,
and a local complete intersection subscheme $\Lambda \subset \mathbb{F}_e$ of codimension $2$, with
$h^0(\mathbb{F}_e,\mathscr{I}_\Lambda\mathscr{O}_{\mathbb{F}_e}(C_0))\ge 1$, such that
\begin{itemize}
\item $\mathscr{K}$ fits into an exact sequence
$$
0 \ \to \ \mathscr{O}_{\mathbb{F}_e}(tf) \ \to \
\mathscr{K}\otimes\mathscr{B}^* \ \to \ \mathscr{I}_\Lambda\mathscr{O}_{\mathbb{F}_e}(C_0) \ \to \ 0,
$$
\item $\mathscr{G}$ is induced by a pencil in $|\mathscr{O}_{Y}(1)\otimes q^*\mathscr{B}^*|$
containing $\Sigma+tq^*f$, where $\Sigma$ is the zero locus of the section of
$\mathscr{O}_{Y}(1)\otimes q^*(\mathscr{B}^*\otimes\mathscr{O}_{\mathbb{F}_e}(-tf))$
corresponding to
$\mathscr{O}_{\mathbb{F}_e}(tf) \to \mathscr{K}\otimes\mathscr{B}^*$,
\item $\det(\mathscr{G})\cong q^*\mathscr{O}_{\mathbb{F}_e}(C_0+(e+1)f)$.
\end{itemize}
\item There exist a line bundle $\mathscr{B}$ on $\mathbb{F}_e$,
a minimal section $C_0$ and a fiber $f$ of $p : \mathbb{F}_e \to \p^1$,
and a local complete intersection subscheme $\Lambda \subset \mathbb{F}_e$ of codimension $2$
such that
\begin{itemize}
\item there exists a curve $C \sim C_0+f$ with $\Lambda \subset C$ and,
for any proper subcurve $C' \subsetneq C$, $\Lambda\not\subset C'$,
\item $\mathscr{K}$ fits into an exact sequence
$$
0 \ \to \ \mathscr{O}_{\mathbb{F}_e} \ \to \
\mathscr{K}\otimes\mathscr{B}^* \ \to \ \mathscr{I}_\Lambda\mathscr{O}_{\mathbb{F}_e}(C_0+ f) \ \to \ 0,
$$
\item $\mathscr{G}$ is induced by a pencil of irreducible members of $|\mathscr{O}_{Y}(1)\otimes q^*\mathscr{B}^*|$
containing $\Sigma$, the zero locus of the section of
$\mathscr{O}_{Y}(1)\otimes q^*\mathscr{B}^*$ corresponding to
$\mathscr{O}_{\mathbb{F}_e} \to \mathscr{K}\otimes\mathscr{B}^*$,
\item $\det(\mathscr{G})\cong q^*\mathscr{O}_{\mathbb{F}_e}(C_0+(e+1)f)$.
\end{itemize}
\end{enumerate}
\end{prop}
\begin{proof}
To ease notation, set $S:=\mathbb{F}_e$.
Denote by $C_0$ a minimal section, and by $f$ a general fiber of $p : S \to \p^1$.
Denote by $F\cong \p^1$ a general fiber of $q:Y\to S$.
Given any curve $C\subset S$, we set $Y_C:=q^{-1}(C)$, and denote by
$q_C : Y_C \to C$ the restriction of $q$ to $Y_C$.
We claim that $T_{Y/S}\not \subseteq \mathscr{G}$.
Indeed, if $T_{Y/S}\subseteq \mathscr{G}$,
then $\mathscr{G}$ would be the pullback via $q$ of a foliation on $S$, and hence
$\mathscr{O}_{\p^1}\cong \det(\mathscr{G})_{|F}\cong (T_{Y/S})_{|F}\cong \mathscr{O}_{\p^1}(2)$
by \ref{pullback_foliations}, which is absurd.
Therefore, the natural map $T_Y \to q^*T_{S}$
induces an injective morphism of sheaves $\mathscr{G} \to q^*T_{S}$.
Let $q^*B$ be the divisor of zeroes of
the induced map $q^*\det(\mathscr{A})\cong \det(\mathscr{G}) \to q^*\det(T_{S})$.
Suppose that $B=0$. Then
$\mathscr{A}\cong \mathscr{O}_S(-K_S)$ is ample, and hence $e\in\{0,1\}$. Moreover,
$\mathscr{G} \subset T_Y$ induces a flat connection on $q$. Thus $\mathscr{K}\cong\mathscr{B}\oplus \mathscr{B}$ for some line bundle $\mathscr{B}$ on $S$,
and $\mathscr{G}$ is induced by the natural morphism $Y\cong S\times\p^1\to \p^1$.
This is case (1) in the statement of Proposition~\ref{proposition:P-bdle_over_S_hirzebruch}.
Suppose from now on that $B\neq 0$.
A straightforward computation shows that one of the following holds
(up to possibly exchanging $C_0$ and $f$ when $e=0$).
\begin{itemize}
\item $\mathscr{A}\cong\mathscr{O}_{S}(C_0+(e+1)f)$ and $B\sim C_0+f$, or
\item $\mathscr{A}\cong\mathscr{O}_{S}(C_0+(e+2)f)$ and $B\sim C_0$.
\end{itemize}
In either case, $B$ contains a unique irreducible component dominating $\p^1$.
We denote this irreducible component by $B_1$, and set $B_2:=B-B_1$.
Let $C\cong\p^1$ be a general member of $|\mathscr{O}_S(C_0+ef)|$.
Since $T_{Y/S}\not \subseteq \mathscr{G}$, $\mathscr{G}$ induces foliations by curves
$\mathscr{C}_f\subset {T_{Y_f}}$ and $\mathscr{C}_C\subset {T_{Y_C}}$ on $Y_f$ and $Y_C$, respectively.
By \ref{restricting_fols}, there exist effective divisors
$D_f$ on $Y_f$ and $D_C$ on $Y_C$ such that
\begin{equation}\label{equ:canonical_bundle}
\mathscr{O}_{Y_f}(-K_{\mathscr{C}_f})\cong (q^*\mathscr{A})_{|Y_f}
\otimes\mathscr{O}_{Y_f}(D_f) \text{, and }
\mathscr{O}_{Y_C}(-K_{\mathscr{C}_C})\cong q^*\big(\mathscr{A}\otimes\mathscr{O}_{S}(-C_0-ef)\big)_{|Y_C}
\otimes\mathscr{O}_{Y_C}(D_C).
\end{equation}
\noindent {\bf Claim.}
\begin{itemize}
\item[(a)] $\mathscr{C}_f\cong q_f^*\mathscr{O}_{\p^1}(1)$, $Y_f\cong \mathbb{F}_m$ with $m\ge 1$,
and $\mathscr{C}_f$ is induced by a pencil containing $\sigma_0+m \ell_0$,
where $\sigma_0$ denotes the minimal section and $\ell_0$ a fiber of $q_f : Y_f \to \p^1$.
\item[(b)] If $B\sim C_0+f$ (and so $C \cap \textup{Supp}(B) \neq \emptyset$),
then $\mathscr{C}_C\cong q_C^*\mathscr{O}_{\p^1}(1)$, $Y_C\cong \mathbb{F}_m$ with $m\ge 1$,
and $\mathscr{C}_C$ is induced by a pencil containing $\sigma_0+m \ell_0$,
where $\sigma_0$ denotes the minimal section and $\ell_0$ a fiber of $q_C : Y_C \to \p^1$.
\item[(c)] If $B\sim C_0$ (and so $C \cap \textup{Supp}(B) = \emptyset$), then $\mathscr{C}_C\cong q_C^*\mathscr{O}_{\p^1}(2)$,
$Y_C\cong C\times \p^1$, and $\mathscr{C}_C$ is induced by the projection morphism $C\times \p^1 \to \p^1$.
\end{itemize}
On the open subset $Y\setminus q^{-1}(\textup{Supp}(B))$, $\mathscr{G}$ is regular and induces a flat connection.
Therefore $\mathscr{G}_{|Y_f}$ intersects $T_{Y_f/f}$ transversely over $f\setminus \{f\cap \textup{Supp}(B)\}$,
and thus the support of $D_f$ is contained in $q^{-1}(f\cap \textup{Supp}(B))$.
It follows from \eqref{equ:canonical_bundle} that $\mathscr{C}_f\cong q_f^*\mathscr{O}_{\p^1}(k)$ for some
positive integer $k$.
Since the natural map $\mathscr{C}_f \to q_f^*T_f$ is injective, we must have $k\in\{1,2\}$.
The same argument shows that $\mathscr{C}_C\cong q_C^*\mathscr{O}_{\p^1}(l)$, with $l\in\{1,2\}$.
If $\mathscr{C}_f\cong q_f^*\mathscr{O}_{\p^1}(2)$, then $\mathscr{C}_f$ is a regular foliation, $Y_f\cong f\times \p^1$, and $\mathscr{C}_f$
is induced by the projection morphism $f\times \p^1 \to \p^1$.
On the other hand, if $b$ is a general point in $\textup{Supp}(B)$,
then $q^{-1}(b)$ is tangent to $\mathscr{G}$,
while $\mathscr{G}$ is regular at a general point of $q^{-1}(b)$.
Since $f$ is assumed to be general, $f\cap \textup{Supp}(B)$ is a general point $b\in \textup{Supp}(B)$, and we
conclude from this observation that $Y_f$ must be a leaf of $\mathscr{G}$, which is absurd.
Therefore we must have $\mathscr{C}_f\cong q_f^*\mathscr{O}_{\p^1}(1)$ and $D_f=0$.
Analogously, we prove that if $C \cap \textup{Supp}(B) \neq \emptyset$, then $\mathscr{C}_C\cong q_C^*\mathscr{O}_{\p^1}(1)$
and $D_C=0$.
The description of $(Y_f,\mathscr{C}_f)$ and $(Y_C,\mathscr{C}_C)$ in this case follow from Lemma~\ref{lemma:foliation_surface}.
Finally, if $B\sim C_0$, then
$D_C=0$, and $\mathscr{C}_C$ induces a flat connection on $q_C$.
Therefore $\mathscr{C}_C\cong q_C^*\mathscr{O}_{\p^1}(2)$,
$Y_C\cong C\times \p^1$, and $\mathscr{C}_C$
is induced by the projection morphism $C\times \p^1 \to \p^1$.
This proves the claim.
\smallskip
Next we show that $\mathscr{G}$ has algebraic leaves,
and that a general leaf has relative degree $1$ over $S$.
From the claim, we know that the general leaves of $\mathscr{C}_f$ and $\mathscr{C}_C$ are sections of
$q_f:Y_f\to f$ and $q_C:Y_C\to C$, respectively.
Let $F_C$ be a general leaf of $\mathscr{C}_C$ mapping onto $C$.
For a general fiber $f$ of $p:S\to \p^1$, $Y_f$ meets $F_C$
in a single point, and there is a unique leaf $F_f$ of $\mathscr{C}_f$ through this point.
We let $\Sigma$ be the closure of the union of the $F_f$'s obtained in this way, as
$f$ varies through general fibers of $p:S\to \p^1$.
It is a general leaf of $\mathscr{G}$, and has relative degree $1$ over $S$.
Since $\mathscr{G}$ is algebraically integrable, we can consider the rational first integral
for $\mathscr{G}$, $\pi : Y \dashrightarrow \tilde W$, as described in \ref{notation:family_leaves}.
Since $Y$ is a rational variety, $\tilde W\cong\p^1$. So
$\mathscr{G}$ is induced by a pencil $\Pi$ in the linear system $|\mathscr{O}_Y(1)\otimes q^*\mathscr{M}|$
for some line bundle $\mathscr{M}$ on $S$.
Notice that
$\pi_f:=\pi_{|Y_f} : Y_f \dashrightarrow \p^1$ and
$\pi_C:=\pi_{|Y_C} : Y_C \dashrightarrow \p^1$ are rational first integrals for
$\mathscr{C}_f$ and $\mathscr{C}_C$, respectively, and $\mathscr{C}_f$ and $\mathscr{C}_C$ are
induced by the restricted pencils $\Pi_{|Y_f}$ and $\Pi_{|Y_C}$, respectively.
Our next task is to determine the line bundle $\mathscr{M}$.
From claim (a--c), there are integers $a,b,s,t$, with $s\ge 1$ and $t\ge 0$
such that $\mathscr{K}_{|f}\cong\mathscr{O}_{\p^1}(a)\oplus\mathscr{O}_{\p^1}(a+s)$, and
$\mathscr{K}_{|C}\cong\mathscr{O}_{\p^1}(b)\oplus\mathscr{O}_{\p^1}(b+t)$.
Moreover, $\mathscr{M}_{|f}\cong \mathscr{O}_{\p^1}(-a)$ and $\mathscr{M}_{|C}\cong \mathscr{O}_{\p^1}(-b)$.
This implies that $\mathscr{M}\cong\mathscr{O}_S(-aC_0-b f)$.
Any member of $\Pi$ can be written as $\Sigma+uq^*B_1+vq^*B_2$, where
$\Sigma$ is irreducible and has relative degree $1$ over $S$,
and $u,v\ge 0$ are integers.
In particular, the ramification divisor $R(\pi)$ of $\pi$ must be of the form
$R(\pi)=cq^*B_1+d q^*B_2$, with $c,d \ge 0$ integers.
We have
$$N_\mathscr{G}\cong \big(\pi^*\Omega^1_{\p^1}\otimes \mathscr{O}_Y(R(\pi))\big)^*\cong \mathscr{O}_Y(2)\otimes q^*\mathscr{M}^{\otimes 2}\otimes \mathscr{O}_Y(-R(\pi)).$$
On the other hand,
\begin{multline*}
$$
N_\mathscr{G}\cong\mathscr{O}_{Y}(-K_{Y})\otimes \mathscr{O}_{Y}(K_\mathscr{G})\cong \mathscr{O}_{Y}(2)\otimes q^*(\det(\mathscr{K}^*)\otimes
\mathscr{O}_S(-K_S)\otimes\mathscr{A}^*) \\
\cong \mathscr{O}_{Y}(2)\otimes q^*(\det(\mathscr{K}^*)\otimes
\mathscr{O}_S(B)),
$$
\end{multline*}
and hence
$$\mathscr{O}_{Y}(R(\pi)) \cong q^*(\mathscr{M}^{\otimes 2}\otimes \det(\mathscr{K})\otimes
\mathscr{O}_S(-B))\cong q^*(\mathscr{O}_S(s C_0+ t f)\otimes
\mathscr{O}_S(-B)).$$
It follows from claim (b--c) that, if $B\sim C_0$ then
$t=d=0$, and if $B\not\sim C_0$, then $c=s-1$ and $d=t-1$.
Notice also that, if $s \ge 2$, then the pencil $\Pi$ contains a member of the form
$\Sigma+sq^*B_1+vq^*B_2$, where $\Sigma$ is irreducible and has relative degree $1$ over $S$,
and $v\ge 0$.
Similarly, if $t \ge 2$, then the pencil $\Pi$ contains a member of the form
$\Sigma+uq^*B_1+tq^*B_2$, where $\Sigma$ is irreducible and has relative degree $1$ over $S$,
and $u \ge 0$.
\medskip
\noindent {\bf Case 1:} Suppose that $\Pi$ contains a member of the form
$\Sigma+uq^*B_1+vq^*B_2$, where $\Sigma$ is irreducible and has relative degree $1$ over $S$, $B_2\neq 0$,
and $u,v >0$.
Up to replacing $C_0$ and $f$ with linearly equivalent curves on $S$, we may write
$B=C_0+f$.
It follows from claim (a--b) that $u=s$ and $v=t$.
Moreover $\Sigma\cap Y_f$ and $\Sigma \cap Y_C$
are the minimal sections of $q_f:Y_f\to f$ and $q_C:Y_C\to C$, respectively.
If $\Sigma'$ is the closure of a general leaf of $\mathscr{G}$, then
$\Sigma\cap \Sigma'\cap Y_f = \emptyset=\Sigma\cap \Sigma'\cap Y_C$.
This implies that $\Sigma\cap \Sigma' \cap q^{-1}(b)=\emptyset$ for a general point $b \in\textup{Supp}(B)$.
One can find an open subset $V \subset S$,
with $\textup{codim}_S(S\setminus V)\ge 2$, such that $\Sigma\cap q^{-1}(V)$
and $\Sigma'\cap q^{-1}(V)$ are sections of $q_{|q^{-1}(V)}$, and
$\Sigma\cap\Sigma'\cap q^{-1}(V)=\emptyset$.
Therefore, there are line bundles $\mathscr{B}_1$ and $\mathscr{B}_2$ on $S$ such that
$\mathscr{K} \cong \mathscr{B}_1\oplus \mathscr{B}_2$, and $\Sigma$ corresponds to the surjection
$\mathscr{B}_1\oplus\mathscr{B}_2\twoheadrightarrow \mathscr{B}_1$.
From the description of $\mathscr{K}_{|f}$ and $\mathscr{K}_{|C}$ above, we see that
${\mathscr{B}_1}_{|f}\cong\mathscr{O}_{\p^1}(a)$,
${\mathscr{B}_1}_{|C}\cong\mathscr{O}_{\p^1}(b)$,
${\mathscr{B}_2}_{|f}\cong\mathscr{O}_{\p^1}(a+s)$, and
${\mathscr{B}_2}_{|C}\cong\mathscr{O}_{\p^1}(b+t)$.
Thus $\mathscr{B}_1\cong\mathscr{O}_S(aC_0+b f)$, and $\mathscr{B}_2\cong\mathscr{O}_S((a+s)C_0+(b+t) f)$.
We are in case (2) in the statement of Proposition~\ref{proposition:P-bdle_over_S_hirzebruch}, with $t\ge 1$.
\medskip
\noindent {\bf Case 2:} Suppose that $\Pi$ contains a member of the form
$\Sigma+uq^*B_1$, where $\Sigma$ is irreducible and has relative degree $1$ over $S$,
and $u>0$.
It follows from claim (a) that $u=s$.
Next we prove that any other reducible divisor of $\Pi$ must be of the form
$\Sigma'+tq^*B_2$, where $\Sigma'$ is irreducible and has relative degree $1$ over $S$.
In particular, if there exists such divisor in $\Pi$, we must have $B_2\neq 0$ and $t>0$.
Indeed, let $D=\Sigma'+iq^*B_1+jq^*B_2$ be a reducible member of $\Pi$,
where $\Sigma'$ is irreducible and has relative degree $1$ over $S$.
If $i>0$, then it follows from the claim that $i=s$ and $\Sigma = \Sigma'$.
Since $D\sim \Sigma+sq^*B_1$, we must have $D= \Sigma+sq^*B_1$.
If $i=0$, then we must have $B_2\neq 0$ and $j>0$.
It follows from claim (b) that $j=t$, and so $D=\Sigma'+tq^*B_2$.
We consider three cases.
\noindent {\bf Case 2.1:} Suppose that $B_2=0$, and $B=B_1\sim C_0+f$.
Then we must have $e\in\{0,1\}$.
Let $\Sigma'$ is the closure of a general leaf of $\mathscr{G}$.
It follows from claim (a) that $\Sigma\cap Y_f$ is the minimal section of $q_f$, and
$\Sigma\cap \Sigma'\cap Y_f = \emptyset$.
Proceeding as in case 1, we show that we must be in case (3)
in the statement of Proposition~\ref{proposition:P-bdle_over_S_hirzebruch}.
\noindent {\bf Case 2.2:} Suppose $B_2=0$, and $B=B_1\sim C_0$.
Up to replacing $C_0$ with a linearly equivalent curve on $S$, we may write
$B=C_0$.
Since $\Sigma$ is irreducible and has relative degree $1$ over $S$,
it contains only finitely many fibers of $q$, and corresponds to
a surjective morphism of sheaves $\mathscr{K} \twoheadrightarrow \mathscr{I}_{\Lambda}\mathscr{S}$, where $\mathscr{S}$ is a line bundle on $S$, and
$\Lambda \subset S$ is a closed subscheme with $\textup{codim}_S\Lambda \ge 2$.
Denote by $\mathscr{T}$ the kernel of this morphism.
Then $\mathscr{T}$ is a line bundle on $S$, and $\mathscr{T} \otimes \mathscr{S}\cong \det(\mathscr{K})\cong \mathscr{O}_S((2a+s)C_0+2bf)$.
Since $\Sigma+sq^*C_0 \in \Pi\subset |\mathscr{O}_Y(1)\otimes q^*\mathscr{O}_S(-aC_0-b f)|$,
we must have $\mathscr{T} \otimes \mathscr{O}_S(-aC_0-b f)\cong \mathscr{O}_S(s C_0)$, and
$\mathscr{S} \otimes \mathscr{O}_S(-aC_0-b f) \cong \mathscr{O}_S$.
Let $\Sigma'$ is the closure of a general leaf of $\mathscr{G}$, and
$\sigma' \in H^0(Y,\mathscr{O}_Y(1)\otimes q^*\mathscr{O}_S(-aC_0-b f))$ a nonzero section vanishing along
$\Sigma'$.
We claim that
$\sigma'$ is mapped to a nonzero element in $H^0(S,\mathscr{I}_{\Lambda}\mathscr{S}\otimes\mathscr{O}_S(-aC_0-b f))\cong H^0(S,\mathscr{I}_{\Lambda})$
under the natural morphism
$H^0(S,\mathscr{K}\otimes \mathscr{O}_S(-aC_0-b f)) \to
H^0(S,\mathscr{I}_{\Lambda}\mathscr{S}\otimes\mathscr{O}_S(-aC_0-b f))$.
Indeed, if $0\neq \sigma\in H^0\big(Y,\mathscr{O}_Y(1)\otimes q^*\mathscr{O}_S(-aC_0-b f))\big)$ comes from
$H^0\big(S,\mathscr{T}\otimes \mathscr{O}_S(-aC_0-b f))\big)
\subset H^0\big(S,\mathscr{K}\otimes \mathscr{O}_S(-aC_0-b f))\big) \cong
H^0\big(Y,\mathscr{O}_Y(1)\otimes q^*\mathscr{O}_S(-aC_0-b f))\big)$, then its zero locus on $Y$ must be reducible, yielding a contradiction.
We conclude that $\Lambda=\emptyset$, and
the exact sequence
$0 \to \mathscr{T} \to \mathscr{K} \to \mathscr{S} \to 0$ splits.
So we are in case (2) in the statement of Proposition~\ref{proposition:P-bdle_over_S_hirzebruch}, with
$s>0$ and $t=0$.
\noindent {\bf Case 2.3:} Suppose that $B_2\neq 0$.
Up to replacing $C_0$ and $f$ with linearly equivalent curves on $S$, we may write
$B=C_0+f$.
As in case 2.2, $\Sigma$ corresponds to
a surjective morphism of sheaves $\mathscr{K} \twoheadrightarrow \mathscr{I}_{\Lambda}\mathscr{S}$, where $\mathscr{S}$ is a line bundle on $S$, and
$\Lambda \subset S$ is a closed subscheme with $\textup{codim}_S\Lambda \ge 2$.
Denote by $\mathscr{T}$ the kernel of this morphism.
Then $\mathscr{T}$ is a line bundle on $S$, and $\mathscr{T} \otimes \mathscr{S}\cong \det(\mathscr{K})\cong \mathscr{O}_S((2a+s)C_0+(2b+t)f)
\cong \mathscr{O}_S(sC_0+tf)\otimes\mathscr{O}_S(2aC_0+2b f)$.
Since $\Sigma+sq^*C_0 \in |\mathscr{O}_Y(1)\otimes q^*\mathscr{O}_S(-aC_0-b f)|$, we must have
$\mathscr{T} \otimes \mathscr{O}_S(-aC_0-b f)\cong \mathscr{O}_S(s C_0)$, and
$\mathscr{S} \otimes \mathscr{O}_S(-aC_0-b f) \cong \mathscr{O}_S(t f)$.
Thus, we have an exact sequence
$$
0 \ \to \ \mathscr{O}_S(sC_0) \ \to \
\mathscr{K}\otimes\mathscr{O}_S(-aC_0-b f) \ \to \ \mathscr{I}_{\Lambda}\mathscr{O}_S(t f) \ \to \ 0,
$$
where $\mathscr{O}_S(sC_0) \to \mathscr{K}\otimes\mathscr{O}_S(-aC_0-b f)$ is the map corresponding to
$\Sigma+sq^*C_0\in \Pi$.
\smallskip
Suppose that $\Sigma+sq^*C_0$ is the only reducible member of
$\Pi$. Then we must have
$t=1$.
Let $\Sigma'$ is the closure of a general leaf of $\mathscr{G}$, and
$\sigma' \in H^0(Y,\mathscr{O}_Y(1)\otimes q^*\mathscr{O}_S(-aC_0-b f))$ a nonzero section vanishing along
$\Sigma'$.
As in case 2.2, we see that
$\sigma'$ is mapped to a nonzero element $\bar \sigma'\in H^0(S,\mathscr{I}_{\Lambda}\mathscr{S}\otimes\mathscr{O}_S(-aC_0-b f))\cong H^0(S,\mathscr{I}_{\Lambda}\mathscr{O}_S( f))$
under the natural morphism
$H^0(S,\mathscr{K}\otimes \mathscr{O}_S(-aC_0-b f)) \to H^0(S,\mathscr{I}_{\Lambda}\mathscr{S}\otimes\mathscr{O}_S(-aC_0-b f))$.
Let $f'\sim f$ be the divisor of zeros of $\bar \sigma'$.
Then $\Lambda \subset f'$.
Since
$\Sigma \cap \Sigma' \cap q^{-1}(b)=\emptyset$ for any point
$b \in C_0\setminus f$, we must have
$f=f'$.
We obtain an exact sequence
$$
0 \ \to \ \mathscr{O}_S\oplus \mathscr{O}_S(sC_0) \ \to \
\mathscr{K}\otimes\mathscr{O}_S(-aC_0-b f) \ \to \ \mathscr{O}_{f}(-\Lambda) \ \to \ 0,
$$
where $\mathscr{O}_S \to \mathscr{K}\otimes\mathscr{O}_S(-aC_0-b f)$ is the map given by $\sigma'$.
We are in case (5) in the statement of Proposition~\ref{proposition:P-bdle_over_S_hirzebruch}
\smallskip
Now suppose that $\Pi$ contains a second reducible divisor.
We have seen above that it must be of the form
$\Sigma'+tq^*f$, where $\Sigma'$ is irreducible and has relative degree $1$ over $S$.
As before, it gives rise to an exact sequence
$$
0 \ \to \ \mathscr{O}_S(t f) \ \to \
\mathscr{K}\otimes\mathscr{O}_S(-aC_0-b f) \ \to \ \mathscr{I}_{\Lambda'}\mathscr{O}_S(s C_0) \ \to \ 0,
$$
where
$\Lambda' \subset S$ is a closed subscheme with $\textup{codim}_S\Lambda' \ge 2$.
Notice that $\Sigma\neq \Sigma'$, and so the induced morphism
$\mathscr{O}_S(sC_0) \oplus \mathscr{O}_S(t f) \to \mathscr{K}\otimes\mathscr{O}_S(-aC_0-b f)$ is injective.
Since $\det(\mathscr{O}_S(sC_0) \oplus \mathscr{O}_S(t f))\cong\det(\mathscr{K}\otimes\mathscr{O}_S(-aC_0-b f))$,
it is in fact an isomorphism.
We are in case (4) in the statement of Proposition~\ref{proposition:P-bdle_over_S_hirzebruch}.
\medskip
\noindent {\bf Case 3:} Suppose that $\Pi$ contains a member of the form
$\Sigma+vq^*B_2$, where $\Sigma$ is irreducible and has relative degree $1$ over $S$, $B_2\neq 0$,
and $v >0$.
Up to replacing $C_0$ and $f$ with linearly equivalent curves on $S$, we may write
$B=C_0+f$.
It follows from claim (b) that $v=t$.
As in case (2), we see that any other reducible divisor of $\Pi$ must be of the form
$\Sigma'+sq^*C_0$, where $\Sigma'$ is irreducible and has relative degree $1$ over $S$.
If there exists such divisor, we are in case 2.3 above.
So we may assume that $\Sigma+tq^*f$ is the only reducible member of $\Pi$.
This implies that $s=1$, and $\Sigma$ gives rise to an exact sequence
$$
0 \ \to \ \mathscr{O}_S(tf) \ \to \
\mathscr{K}\otimes\mathscr{O}_S(-aC_0-b f) \ \to \ \mathscr{I}_{\Lambda}\mathscr{O}_S(C_0) \ \to \ 0,
$$
where $\Lambda \subset S$ is a closed subscheme with $\textup{codim}_S\Lambda \ge 2$.
If $\Lambda=\emptyset$, then the sequence splits since
$h^1(S,\mathscr{O}_S(-C_0+tf))=0$, and
we are in case (4) in the statement of Proposition~\ref{proposition:P-bdle_over_S_hirzebruch}, with $s=1$.
If $\Lambda\neq\emptyset$, then $\Lambda$ is a local complete intersection subscheme, and we are in case
(6) in the statement of Proposition~\ref{proposition:P-bdle_over_S_hirzebruch}, with $t \ge 1$.
\medskip
\noindent {\bf Case 4:} Suppose that all members of $\Pi$ are irreducible.
Then $s=1$ and $t\le 1$.
Let $\Sigma'$ be the closure of a general leaf of $\mathscr{G}$.
It gives rise to an exact sequence
$$
0 \ \to \ \mathscr{O}_S \ \to \
\mathscr{K}\otimes\mathscr{O}_S(-aC_0-b f) \ \to \ \mathscr{I}_{\Lambda}\mathscr{O}_S(C_0+tf) \ \to \ 0,
$$
where
$\Lambda \subset S$ is a closed subscheme with $\textup{codim}_S\Lambda \ge 2$.
We claim that $\Lambda\neq\emptyset$.
Indeed, if $\Lambda=\emptyset$, then the sequence splits since $h^1(S,\mathscr{O}_S(-C_0-f))=0$.
But this implies that $\Pi$ contains a reducible member, contrary to our assumptions.
Hence $\textup{codim}_S\Lambda = 2$, and $\Lambda$ is a local complete intersection subscheme.
If $t=0$, then we are in case
(6) in the statement of Proposition~\ref{proposition:P-bdle_over_S_hirzebruch}.
Suppose from now on that $t=1$.
Then we must have $h^0(S,\mathscr{I}_{\Lambda}\mathscr{O}_{S}(C_0+f))\ge 1$.
We will show that, there exists a curve $C \sim C_0+f$ with $\Lambda \subset C$
such that, for any proper subcurve $C' \subsetneq C$, $\Lambda\not\subset C'$.
Suppose to the contrary that any curve $C \sim C_0+f$ with $\Lambda \subset C$
can be written as $C=C_1\cup f_1$ with $C_1\sim C_0$, $f_1\sim f$,
and either $\Lambda \subset C_1$, or $\Lambda \subset f_1$.
This implies that the set of reducible members of $|\mathscr{O}_Y(1)\otimes q^*\mathscr{O}_S(-aC_0-b f)|$ has
codimension 1 since
$h^0(S,\mathscr{O}_{S}(-f))=0=h^0(S,\mathscr{O}_{S}(-C_0))$. This contradicts the fact that all members of $\Pi$ are irreducible.
We are in case (7) in the statement of Proposition~\ref{proposition:P-bdle_over_S_hirzebruch}.
\end{proof}
Next we investigate whether all the 7 cases described in Proposition~\ref{proposition:P-bdle_over_S_hirzebruch} in fact occur.
\begin{say}[{Proposition~\ref{proposition:P-bdle_over_S_hirzebruch}(1--4)}]
Let $\mathscr{K}$ be a rank $2$ vector bundle on a ruled surface $p : \mathbb{F}_e\to\p^1$, $e \ge 0$.
Set $Y:=\p_{\mathbb{F}_e}(\mathscr{K})$,
with natural projection $q : Y \to \mathbb{F}_e$, and denote
by $\mathscr{O}_{Y}(1)$ the tautological line bundle on $Y$.
Suppose that $\mathscr{K}$ satisfies one of the conditions (1-4) in the statement of
Proposition~\ref{proposition:P-bdle_over_S_hirzebruch}.
Then the pencil $\Pi$ described in the statement yields a codimension one foliation
$\mathscr{G}$ on $Y$ with $\det(\mathscr{G})\cong q^*\mathscr{A}$ where
$\mathscr{A}$ is an ample line bundle.
\end{say}
\begin{say}[{Proposition~\ref{proposition:P-bdle_over_S_hirzebruch}(5)}]
Consider the ruled surface $p : \mathbb{F}_e\to\p^1$, with $e \ge 0$,
denote by $C_0$ a minimal section, and by $f$ a fiber of $p$.
Let $s, \lambda \ge 0$ be integers, and suppose that $\mathscr{K}$ is a coherent sheaf on $\mathbb{F}_e$
fitting into an exact sequence
\begin{equation}\label{eq:sequence1}
0 \ \to \ \mathscr{K} \ \to \ \mathscr{O}_{\mathbb{F}_e}(f)\oplus \mathscr{O}_{\mathbb{F}_e}(sC_0+f)
\ \to \ \mathscr{O}_{f_0}(s+\lambda+1) \ \to \ 0.
\end{equation}
By \cite[Proposition 5.2.2]{HuyLehn}, $\mathscr{K}$ is a rank $2$ vector bundle on $\mathbb{F}_e$.
Since $\det(\mathscr{K})\cong \mathscr{O}_{\mathbb{F}_e}(sC_0+f)$, we have $\mathscr{K}^*\cong \mathscr{K}\otimes \mathscr{O}_{\mathbb{F}_e}(-sC_0-f)$.
Dualizing sequence~\eqref{eq:sequence1}, and twisting it with $\mathscr{O}_{\mathbb{F}_e}(sC_0+f)$
yields
\begin{equation}\label{eq:sequence2}
0\ \to \ \mathscr{O}_{\mathbb{F}_e}\oplus \mathscr{O}_{\mathbb{F}_e}(sC_0) \ \to \
\mathscr{K}
\ \to \
\mathscr{O}_{f_0}(-\lambda)
\to \ 0
\end{equation}
Conversely, dualizing sequence~\eqref{eq:sequence2}, and twisting it with $\mathscr{O}_{\mathbb{F}_e}(sC_0+f)$
yields sequence~\eqref{eq:sequence1}.
Set $Y:=\p_{\mathbb{F}_e}(\mathscr{K})$, with natural projection $q : Y \to \mathbb{F}_e$
and tautological line bundle $\mathscr{O}_{Y}(1)$.
Let $\Sigma$ be the zero locus of the section of
$\mathscr{O}_{Y}(1)\otimes q^*\mathscr{O}_{\mathbb{F}_e}(-sC_0)$
corresponding to the map
$\mathscr{O}_{\mathbb{F}_e}(sC_0) \to \mathscr{O}_{\mathbb{F}_e}\oplus \mathscr{O}_{\mathbb{F}_e}(sC_0) \to \mathscr{K}$
induced by \eqref{eq:sequence2}.
Similarly, let
$\Sigma'$ be the
zero locus of the section of
$\mathscr{O}_{Y}(1)$
corresponding to the map
$\mathscr{O}_{\mathbb{F}_e} \to \mathscr{O}_{\mathbb{F}_e}\oplus \mathscr{O}_{\mathbb{F}_e}(sC_0) \to \mathscr{K}$
induced by \eqref{eq:sequence2}.
Let $\Pi$ be the pencil in $|\mathscr{O}_Y(1)|$ generated by $\Sigma+sq^*C_0$ and $\Sigma'$.
If $\Sigma+sq^*C_0$ is the only reducible member of $\Pi$, then this pencil
induces a foliation $\mathscr{G}$ on $Y$ as in Proposition \ref{proposition:P-bdle_over_S_hirzebruch}(5).
So we investigate this condition.
Suppose that there exists another reducible divisor in $\Pi$, and write it as $\Sigma''+q^*D\neq \Sigma+sq^*C_0$,
where $D\sim uC_0+vf$ is a nonzero effective divisor on $\mathbb{F}_e$.
By restricting $\Pi$ to $Y_f=q^{-1}(f)$, we see that $u=0$ and $v>0$.
Thus
$h^0(\mathbb{F}_e,\mathscr{K}\otimes\mathscr{O}_{\mathbb{F}_e}(-vf_0))\ge 1$. On the other hand,
$h^0(\mathbb{F}_e,\mathscr{O}_{\mathbb{F}_e}(-vf_0)\oplus \mathscr{O}_{\mathbb{F}_e}(sC_0-vf_0))=0$.
This implies $\lambda = 0$, and thus
$\mathscr{K} \cong \mathscr{O}_{\mathbb{F}_e}(f)\oplus \mathscr{O}_{\mathbb{F}_e}(sC_0)$.
\end{say}
\begin{say}[{Proposition~\ref{proposition:P-bdle_over_S_hirzebruch}(6)}]
\label{rem:existence}
Consider the ruled surface $p : \mathbb{F}_e\to\p^1$, with $e \ge 0$,
denote by $C_0$ a minimal section, and by $f$ a fiber of $p$.
Let $\Lambda \subset \mathbb{F}_e$ be a local complete intersection subscheme of codimension $2$, and $t \ge 0$ an integer.
By \cite[Theorem 5.1.1]{HuyLehn}, there exists a vector bundle $\mathscr{K}$ on $\mathbb{F}_e$ fitting into an exact sequence
$$
0 \ \to \ \mathscr{O}_{\mathbb{F}_e}(tf) \ \to \
\mathscr{K}\ \to \ \mathscr{I}_\Lambda\mathscr{O}_{\mathbb{F}_e}(C_0) \ \to \ 0.
$$
Set $Y:=\p_{\mathbb{F}_e}(\mathscr{K})$, with natural projection $q : Y \to \mathbb{F}_e$ and tautological line bundle $\mathscr{O}_{Y}(1)$.
Note that the map
$H^0(\mathbb{F}_e,\mathscr{K})
\to H^0(\mathbb{F}_e,\mathscr{I}_\Lambda\mathscr{O}_{\mathbb{F}_e}(C_0))$
is surjective since
$H^1(\mathbb{F}_e,\mathscr{O}_{\mathbb{F}_e}(tf))$ vanishes.
Suppose moreover that
$h^0(\mathbb{F}_e,\mathscr{I}_\Lambda\mathscr{O}_{\mathbb{F}_e}(C_0))\ge 1$.
Let $s'\in H^0(Y,\mathscr{O}_Y(1))$
be a section mapping to a nonzero section in
$H^0(\mathbb{F}_e,\mathscr{I}_\Lambda\mathscr{O}_{\mathbb{F}_e}(C_0))$, and denote by $\Sigma'$ its zero locus.
We claim that $\Sigma'$ is irreducible.
Indeed, if $\Sigma'$ is reducible, then, up to replacing $C_0$ by a linearly equivalent curve, we see that
$s'$ must vanish along $q^{-1}(C_0)$.
Therefore,
$h^0(\mathbb{F}_e,\mathscr{K}\otimes\mathscr{O}_{\mathbb{F}_e}(-C_0))\ge 1$,
and hence
$h^0(\mathbb{F}_e,\mathscr{O}_{\mathbb{F}_e}(tf-C_0))\ge 1$, which is absurd.
This shows that $\Sigma'$ is irreducible.
Let $\Sigma$ be the zero locus of the section of
$\mathscr{O}_{Y}(1)\otimes q^*\mathscr{O}_{\mathbb{F}_e}(-tf)$
corresponding to $\mathscr{O}_{\mathbb{F}_e}(tf) \to \mathscr{K}$.
Then the pencil $\Pi\subset |\mathscr{O}_Y(1)|$ generated by $\Sigma+tq^*f_0$ and $\Sigma'$
induces a foliation $\mathscr{G}$ on $Y$ as in Proposition \ref{proposition:P-bdle_over_S_hirzebruch}(6).
\end{say}
\begin{say}[{Proposition~\ref{proposition:P-bdle_over_S_hirzebruch}(7)}]
Consider the ruled surface $p : \mathbb{F}_e\to\p^1$, with $e \ge 0$,
denote by $C_0$ a minimal section, and by $f$ a fiber of $p$.
Let $\Lambda \subset \mathbb{F}_e$ be a local complete intersection subscheme of codimension $2$.
By \cite[Theorem 5.1.1]{HuyLehn}, there exists a vector bundle $\mathscr{K}$ on $\mathbb{F}_e$ fitting into an exact sequence
$$
0 \ \to \ \mathscr{O}_{\mathbb{F}_e} \ \to \
\mathscr{K}\ \to \ \mathscr{I}_\Lambda\mathscr{O}_{\mathbb{F}_e}(C_0+f) \ \to \ 0.
$$
Set $Y:=\p_{\mathbb{F}_e}(\mathscr{K})$, with natural projection $q : Y \to \mathbb{F}_e$ and tautological line bundle $\mathscr{O}_{Y}(1)$.
Note that the map
$H^0(\mathbb{F}_e,\mathscr{K})
\to H^0(\mathbb{F}_e,\mathscr{I}_\Lambda\mathscr{O}_{\mathbb{F}_e}(C_0+f))$
is surjective since
$H^1(\mathbb{F}_e,\mathscr{O}_{\mathbb{F}_e})$ vanishes.
We assume moreover that $h^0(\mathbb{F}_e,\mathscr{I}_\Lambda\mathscr{O}_{\mathbb{F}_e}(C_0+f))\ge 1$,
and that there exists a curve $C \sim C_0+f$ with $\Lambda \subset C$ and
such that, for any proper subcurve $C' \subsetneq C$, $\Lambda\not\subset C'$.
Let $\Sigma$ be the zero locus of the section of $\mathscr{O}_{Y}(1)$
corresponding to the map
$\mathscr{O}_{\mathbb{F}_e} \to \mathscr{K}$.
Then $\Sigma$ is a section of $q$ over
$q^{-1}(\textup{Supp}(\Lambda))$, and hence it is irreducible.
Let $\Sigma'$ be the zero locus of the section $s'\in H^0(\mathbb{F}_e,\mathscr{O}_Y(1))$
that lifts the section of
$H^0(\mathbb{F}_e,\mathscr{I}_\Lambda\mathscr{O}_{\mathbb{F}_e}(C_0+f))$ whose divisor of zeroes
is $C$.
We claim that $\Sigma'$ is irreducible. Suppose otherwise. Then $s'$ must vanish along a subcurve
$C'$ of $C$. This implies that
$h^0(S,\mathscr{I}_\Lambda\mathscr{O}_{\mathbb{F}_e}(C_0+f-C'))\neq 0$.
Therefore,
$C'$ is a proper subcurve of $C$,
and $\Lambda\subset C''$ where $C''$ is such that $C=C'\cup C''$,
contrary to our assumptions.
This shows that $\Sigma'$ is irreducible.
The pencil $\Pi\subset |\mathscr{O}_Y(1)|$
generated by $\Sigma$ and $\Sigma'$
has only irreducible members, and induces
a foliation $\mathscr{G}$ on $Y$ as in Proposition \ref{proposition:P-bdle_over_S_hirzebruch}(7).
\end{say}
Now we go back to the problem of describing
$\mathscr{K}$ and $\mathscr{G}$ that appear in Proposition~\ref{proposition:P-bdle_over_S}.
It remains to consider the case when $S\cong \p^2$, and
$\det(\mathscr{G})\cong q^*\mathscr{O}_{\p^2}(a)$ with $a\in\{1,2\}$.
Proposition \ref{proposition:P-bdle_over_S_plane} below addresses the case $a=2$,
while Proposition \ref{proposition:P-bdle_over_S_plane_2} addresses the case $a=1$.
\begin{prop}\label{proposition:P-bdle_over_S_plane}
Let
$\mathscr{K}$ be a rank $2$ vector bundle on $\p^2$. Set $Y:=\p_{\p^2}(\mathscr{K})$, with natural projection $q : Y \to \p^2$
and tautological line bundle $\mathscr{O}_{Y}(1)$.
Let $\mathscr{G}\subset T_Y$ be a codimension one foliation on $Y$ with $\det(\mathscr{G})\cong q^*\mathscr{O}_{\p^2}(2)$.
Then one of the following holds.
\begin{enumerate}
\item There exist integers
$a$ and $s$, with $s \ge 1$, such that $\mathscr{K}\cong \mathscr{O}_{\p^2}(a)\oplus\mathscr{O}_{\p^2}(a+s)$, and
$\mathscr{G}$ is induced by a pencil in $|\mathscr{O}_{Y}(1)\otimes q^*\mathscr{O}_{\p^2}(-a)|$
containing a divisor of the form $\Sigma+sq^*\ell_0$, where $\Sigma$ is the section of
$q$ corresponding to the map $\mathscr{O}_{\p^2}(a)\oplus\mathscr{O}_{\p^2}(a+s)\twoheadrightarrow \mathscr{O}_{\p^2}(a)$,
and $\ell_0\subset \p^2$ is a line.
\item There exist an integer $a$ and a local complete intersection subscheme
$\Lambda \subset \p^2$ of codimension $2$ such that
$h^0(\p^2,\mathscr{I}_\Lambda\mathscr{O}_{\p^2}(1))\ge 1$, $\mathscr{K}$ fits into
an exact sequence
$$
0 \ \to \ \mathscr{O}_{\p^2} \ \to \
\mathscr{K}\otimes\mathscr{O}_{\p^2}(-a) \ \to \ \mathscr{I}_\Lambda\mathscr{O}_{\p^2}(1) \ \to \ 0,
$$
and $\mathscr{G}$ is induced by a pencil of irreducible members of $|\mathscr{O}_{Y}(1)\otimes q^*\mathscr{O}_{\p^2}(-a)|$
containing the zero locus of the section of
$\mathscr{O}_{Y}(1)\otimes q^*\mathscr{O}_{\p^2}(-a)$
corresponding to the map
$\mathscr{O}_{\p^2} \to \mathscr{K}\otimes\mathscr{O}_{\p^2}(-a)$.
\end{enumerate}
\end{prop}
\begin{proof}
The proof is very similar to that of Proposition \ref{proposition:P-bdle_over_S_hirzebruch},
and so we leave some easy details to the reader.
To ease notation, set $S:=\p^2$, and write $\mathscr{O}_S(1)$ for $\mathscr{O}_{\p^2}(1)$.
Since $T_{Y/S}\not \subseteq \mathscr{G}$, the natural map $T_Y \to q^*T_{S}$
induces an injective morphism of sheaves $\mathscr{G} \to q^*T_{S}$.
There is a line $\ell_0 \subset \p^2$ such that the divisor of zeroes of
the induced map $q^*\mathscr{O}_{\p^2}(2)\cong \det(\mathscr{G}) \to q^*\det(T_{S})$ is
$q^*\ell_0$. Note that $\mathscr{G}$ induces a flat connection
on $q : Y \to S$ over $q^{-1}(\textup{Supp}(\ell_0))$.
Let $\ell \subset \p^2$ be a general line, set
$Y_\ell:=q^{-1}(\ell)$ and $q_\ell := q_{|Y_\ell} : Y_\ell \to \ell$.
Since $T_{Y/S}\not \subseteq \mathscr{G}$, $\mathscr{G}$ induces a foliation by curves $\mathscr{C}_\ell\subset T_{Y_\ell}$.
As in the claim in the proof of Proposition \ref{proposition:P-bdle_over_S_hirzebruch}, one checks that
$\mathscr{C}_\ell\cong q_\ell^*\mathscr{O}_{\p^1}(1)$,
$Y_\ell\cong \mathbb{F}_s$ with $s\ge 1$,
and $\mathscr{C}_\ell$ is induced by a pencil containing $\sigma_0+s f$,
where $\sigma_0$ denotes the minimal section, and $f$ a fiber of $q_\ell : Y_\ell \to \ell$.
One then shows that $\mathscr{G}$ has algebraic leaves, and it is induced by a pencil $\Pi\subset |\mathscr{O}_Y(1)\otimes q^*\mathscr{O}_S(-a)|$,
where $a$ is such that $\mathscr{K}_{|\ell}\cong\mathscr{O}_{\p^1}(a)\oplus\mathscr{O}_{\p^1}(a+s)$.
Any member of $\Pi$ is of the form $\Sigma+uq^*{\ell_0}$, where
$\Sigma$ is irreducible and has relative degree $1$ over $S$,
and $u\ge 0$ is an integer.
In particular, the ramification divisor of the rational first integral
for $\mathscr{G}$, $\pi : Y \dashrightarrow \p^1$,
must be of the form $R(\pi)=cq^*\ell_0$, $c \ge 0$.
An easy computation shows that $c=s-1$.
In particular, if $s \ge 2$, then $\Pi$ contains a member of the form
$\Sigma+sq^*\ell_0$, where
$\Sigma$ is irreducible and has relative degree $1$ over $S$.
\medskip
\noindent {\bf Case 1:}
Suppose that $\Pi$ contains a member of the form $\Sigma+uq^*\ell_0$, where
$\Sigma$ is irreducible and has relative degree $1$ over $S$,
and $u\ge 0$ is an integer.
It follows from the description of $\mathscr{C}_\ell$ above that $u=s$, and
$\Sigma\cap Y_\ell$ is the minimal section of $q_\ell:Y_\ell\cong \mathbb{F}_s\to \ell$.
If $\Sigma'$ is the closure of a general leaf of $\mathscr{G}$, then
$\Sigma\cap \Sigma'\cap Y_\ell= \emptyset$.
This implies that $\Sigma\cap \Sigma' \cap q^{-1}(b)=\emptyset$ for a general point $b \in\ell_0$.
One can find an open subset $V \subset S$,
with $\textup{codim}_S(S\setminus V)\ge 2$, such that $\Sigma\cap q^{-1}(V)$
and $\Sigma'\cap q^{-1}(V)$ are sections of $q_{|q^{-1}(V)}$, and
$\Sigma\cap\Sigma'\cap q^{-1}(V)=\emptyset$.
Therefore, there are line bundles $\mathscr{B}_1$ and $\mathscr{B}_2$ on $S$ such that
$\mathscr{K} \cong \mathscr{B}_1\oplus \mathscr{B}_2$, and $\Sigma$ corresponds to the surjection
$\mathscr{B}_1\oplus\mathscr{B}_2\twoheadrightarrow \mathscr{B}_1$.
From the description of $\mathscr{K}_{|\ell}$, we see that
$\mathscr{B}_1\cong\mathscr{O}_S(a)$, and $\mathscr{B}_2\cong\mathscr{O}_S(a+s)$.
This is case (1) in the statement of Proposition~\ref{proposition:P-bdle_over_S_plane}.
\medskip
\noindent {\bf Case 2:}
Suppose then that all members of $\Pi$ are irreducible.
In particular, we must have $s=1$.
Then the section $\Sigma$ gives rise to an exact sequence
$$
0 \ \to \ \mathscr{O}_S \ \to \
\mathscr{K}\otimes\mathscr{M} \ \to \ \mathscr{I}_{\Lambda}\mathscr{O}_S(1) \ \to \ 0,
$$
where
$\Lambda \subset S$ is a closed subscheme with $\textup{codim}_S\Lambda \ge 2$.
If $\Lambda=\emptyset$, then the sequence splits since
$h^1(S,\mathscr{O}_S(-1))=0$. But then $\Pi$ contains a reducible member, a contradiction.
Thus $\Lambda\neq\emptyset$, and $\Lambda$ is a local complete intersection subscheme.
This is case (2) in the statement of Proposition~\ref{proposition:P-bdle_over_S_plane}.
\end{proof}
Next we give examples of foliations of the type described in Proposition \ref{proposition:P-bdle_over_S_plane}(2).
\begin{say}Let $\Lambda \subset \p^2$ be a local complete intersection subscheme of codimension 2.
By \cite[Theorem 5.1.1]{HuyLehn}, there exists a vector bundle $\mathscr{K}$ on $\p^2$ fitting into an exact sequence
$$
0 \ \to \ \mathscr{O}_{\p^2} \ \to \
\mathscr{K}\ \to \ \mathscr{I}_\Lambda\mathscr{O}_{\p^2}(1) \ \to \ 0.
$$
Set $Y:=\p_{\mathbb{F}_e}(\mathscr{K})$, with natural projection $q : Y \to \p^2$ and tautological line bundle $\mathscr{O}_{Y}(1)$.
Note that the map
$H^0(\p^2,\mathscr{K})
\to H^0(\p^2,\mathscr{I}_\Lambda\mathscr{O}_{\p^2}(1))$
is surjective since
$H^1(\p^2,\mathscr{O}_{\p^2})$ vanishes.
We assume moreover that $h^0(\p^2,\mathscr{I}_\Lambda\mathscr{O}_{\p^2}(1))\ge 1$,
and let $\ell_0$ be a line in $\p^2$ such that $\Lambda \subset \ell_0$.
Let $\Sigma$ be the zero locus of the section of
$\mathscr{O}_{Y}(1)$
corresponding to
$\mathscr{O}_{\p^2} \to \mathscr{K}$.
Then $\Sigma$ is a section of $q$ over
$q^{-1}(\textup{Supp}(\Lambda))$, and hence it is irreducible.
Let $s'\in H^0(\p^2,\mathscr{O}_Y(1))$
be a section lifting the section
$H^0(\p^2,\mathscr{I}_\Lambda\mathscr{O}_{\p^2}(1))$ corresponding
to $\ell_0$, and denote by $\Sigma'$ its zero locus.
We claim that $\Sigma'$ is irreducible.
Indeed, if $\Sigma'$ is reducible, then $s'$ must vanish along $\ell_0$.
On the other hand, $h^0(\p^2,\mathscr{O}_{\p^2}(-1))=h^0(\p^2,\mathscr{I}_\Lambda)=0$, and hence
$h^0(\p^2,\mathscr{K}\otimes\mathscr{O}_{\p^2}(-1))=0$, yielding a contradiction.
Therefore, the pencil $\Pi\subset |\mathscr{O}_Y(1)|$
generated by $\Sigma$ and $\Sigma'$ induces
a foliation $\mathscr{G}$ on $Y$ as in Proposition \ref{proposition:P-bdle_over_S_plane}(2).
\end{say}
\begin{exmp}
Set $Y:=\p_{\p^2}(T_{\p^2})$, denote by $q : Y \to \p^2$ the natural projection, and by $\mathscr{O}_Y(1)$ the
tautological line bundle.
Let $p : Y \to \p^2$ be the morphism
induced by the linear system $|\mathscr{O}_Y(1)\otimes q^*\mathscr{O}_{\p^2}(-1)|$.
If $\mathscr{C}\subset T_{\p^2}$ is a degree zero foliation, then $\mathscr{G}:= p^{-1}(\mathscr{C})$
is a codimension one foliation on $Y$ with
$\det(\mathscr{G})\cong q^*\mathscr{O}_{\p^2}(2)$ as in
Proposition \ref{proposition:P-bdle_over_S_plane}(2).
Let $F$ be the closure of a leaf of $\mathscr{G}$.
Then $q_{|F}: F\to \p^2$
is the blow up of $\p^2$ at a point on the line
$q(p^{-1}(\textup{Sing}(\mathscr{C})))$.
\end{exmp}
\begin{prop}\label{proposition:P-bdle_over_S_plane_2}
Let
$\mathscr{K}$ be a rank $2$ vector bundle on $\p^2$. Set $Y:=\p_{\p^2}(\mathscr{K})$, with natural projection $q : Y \to \p^2$
and tautological line bundle $\mathscr{O}_{Y}(1)$.
Let $\mathscr{G}$ be a codimension one foliation on $Y$ with $\det(\mathscr{G})\cong q^*\mathscr{O}_{\p^2}(1)$,
and suppose that $\mathscr{G}$ is not algebraically integrable.
Then there exist
\begin{itemize}
\item a rational map $\psi: Y\dashrightarrow S=\p_\ell(\mathscr{K}_{|\ell})$, where $\ell \subset \p^2$ is a general line,
giving rise to a foliation by rational curves $\mathscr{M}\cong q^*\mathscr{O}_{\p^2}(1)$ on $Y$, which lifts a degree zero foliation on $\p^2$;
\item a rank $1$ foliation $\mathscr{N}$ on $S$ induced by a global vector field;
\end{itemize}
such that $\mathscr{G}$ is the pullback of $\mathscr{N}$ via $\psi$.
\end{prop}
\begin{proof}
First note that $T_{Y/\p^2}\not \subseteq \mathscr{G}$.
Indeed, if $T_{Y/\p^2}\subseteq \mathscr{G}$,
then $\mathscr{G}$ would be the pullback via $q$ of a foliation on $\p^2$.
Denote by $f\cong\p^1$ a general fiber of $q : Y \to \p^2$.
Then, by \ref{pullback_foliations},
$\mathscr{O}_{\p^1}\cong \det(\mathscr{G})_{|f}\cong (T_{Y/S})_{|f}\cong \mathscr{O}_{\p^1}(2)$
which is absurd.
Let $\mathscr{A}$ be a very ample line bundle on $Y$.
Since $\mathscr{G}$ is not algebraically integrable, by \cite[Proposition 7.5]{fano_fols}, there exists an algebraically integrable subfoliation by curves
$\mathscr{M} \subset \mathscr{G}$ such that
$\mathscr{M} \cdot \mathscr{A}^2 \ge \det(\mathscr{G}) \cdot \mathscr{A}^2 \ge 1$.
Moreover, $\mathscr{M}$ does not depend on the choice
of the very ample line bundle $\mathscr{A}$ on $Y$.
Therefore
$\mathscr{M} \cdot (q^*\mathscr{O}_{\p^2}(k)\otimes\mathscr{A})^2
\ge \det(\mathscr{G})\cdot (q^*\mathscr{O}_{\p^2}(k)\otimes\mathscr{A})^2 >0$ for all $k\ge 1$, and
hence $\mathscr{M} \cdot f \ge 0$.
So we can write $\mathscr{M}\cong \mathscr{O}_Y(a)\otimes q^*\mathscr{O}_{\p^2}(b)$ for some integers $a$ and $b$, with $a\ge 0$.
Since $T_{Y/\p^2}\not \subseteq \mathscr{G}$, there exists an injection of sheaves
$\mathscr{M} \to q^*T_{\p^2}$, and hence we must have $a=0$.
On the other hand, since $\mathscr{M} \cdot \mathscr{A}^2 \ge 1$, we must have $b \ge 1$.
Since the map $\mathscr{M} \to q^*T_{\p^2}$ induces a nonzero map $\mathscr{O}_{\p^2}(b) \to T_{\p^2}$, we conclude that
$b = 1$ by Bott's formulas. So $\mathscr{M}\cong q^*\mathscr{O}_{\p^2}(1)$.
Let $p \in \p^2$ be the singular locus of the degree zero foliation $\mathscr{O}_{\p^2}(1) \subset T_{\p^2}$
induced by the map $\mathscr{M} \to q^*T_{\p^2}$.
Then $\mathscr{M} \subset T_Y$ is a regular foliation (with algebraic leaves) over $q^{-1}(\p^2\setminus \{p\})$.
By Lemma \ref{lemma:foliation_surface}, a general leaf of $\mathscr{M}$ maps isomorphically to a line in $\p^2$ through the point $p$.
This implies that the space of leaves of $\mathscr{M}$ can be naturally identified with
$S=\p_\ell(\mathscr{K}_{|\ell})$, where $\ell \subset \p^2$ is a general line.
Moreover, the natural morphism
$q^{-1}(\p^2\setminus \{p\}) \to S$ is smooth. Hence, by \eqref{pullback_foliations}, $\mathscr{G}$ is the pullback of a rank $1$ foliation
$\mathscr{N}\cong \mathscr{O}_S$ on $S$.
\end{proof}
\begin{rem}
Let $\mathscr{E}$ be a vector bundle on a smooth complex variety $Z$.
Set $Y:=\p_{Z}(\mathscr{E})$,
with natural projection $q : Y \to Z$.
Let $\mathscr{W}$ be a line bundle on $Z$, and $V\in H^0(Z,T_Z\otimes\mathscr{W})$
a twisted vector field on $Z$.
By \cite[Proposition 1.1]{carrell_lieberman} and \cite[Lemma 9.5]{fano_fols},
the map $\mathscr{W}^* \subset T_Z$ induced by $V$ lifts to a
map $q^*\mathscr{W}^* \to T_{Y}$ if and only if
$\mathscr{E}$ is $V$-equivariant, i.e., if there exists a $\mathbb C$-linear map
$\tilde V: \mathscr{E}\to \mathscr{W}\otimes\mathscr{E}$ lifting the derivation $V:\mathscr{O}_T\to \mathscr{W}$.
\end{rem}
\begin{exmp}
Set $Y:=\p_{\p^2}(\mathscr{O}_{\p^2}\oplus \mathscr{O}_{\p^2}(1))$,
with natural projection $q : Y \to \p^2$ and tautological line bundle $\mathscr{O}_Y(1)$.
Let $p : Y \to \p^3$ be the morphism
induced by the linear system $|\mathscr{O}_Y(1)|$. It is the blow-up of
$\p^3$ at a point $x$.
Let $y \in \p^3\setminus \{x\}$, and denote by $\varpi: \p^3 \dashrightarrow \p^2$
the linear projection from $y$.
Let $\mathscr{C}\cong \mathscr{O}_{\p^2}\subset T_{\p^2}$ be a
degree one foliation on $\p^2$, singular at the point $\varpi(x)$,
and let $\mathscr{G}\subset T_Y$ be the pullback of $\mathscr{C}$ via $\varpi\circ p$.
It is a codimension one foliation on $Y$.
An easy computation shows that $\det(\mathscr{G})\cong q^*\mathscr{O}_{\p^2}(1)$.
The rational map
$\varpi: \p^3 \dashrightarrow \p^2$ induces a foliation by curves
$\mathscr{M}\cong q^*\mathscr{O}_{\p^2}(1)$ on $Y$, which lifts
the degree zero foliation on $\p^2$
given by the linear projection from the point
$q(p^{-1}(y))\in \p^2$.
The space $T$ of leaves of $\mathscr{M}$ can be naturally identified with $S=\mathbb{F}_1$,
and $\mathscr{G}$ is the pullback via the induced rational map
$Y \dashrightarrow S$ of a foliation induced by a global vector field on $S$.
\end{exmp}
\subsection{{Proof of Theorem~\ref{main_thm_rho>1}}} \label{subsection:proof_main_rho>0}
\
Let $X$, $\mathscr{F}$ and $L$ be as in Assumptions~\ref{assumptions}.
By Theorem~\ref{Thm:KX-KF_not_nef}, $K_X+(n-3)L$ is not nef, i.e.,
$\tau(L)>n-3$.
By Theorem~\ref{tironi}, $\tau(L)\in \{n-2, n-1, n\}$, unless
$\big(n,\tau(L)\big)\in \big\{(5,\frac{5}{2}),(4,\frac{3}{2}),(4,\frac{4}{3}) \big\}$.
\medskip
\noindent {\bf Step 1: } We show that $\tau(L)\ge n-2$.
\medskip
Suppose to the contrary that $\big(n,\tau(L)\big)\in \big\{(5,\frac{5}{2}),(4,\frac{3}{2}),(4,\frac{4}{3}) \big\}$
(Theorem~\ref{tironi}(4--6)).
In cases (4), (5b) and (6) described in Theorem~\ref{tironi},
$\varphi_L$ makes $X$ a fibration over a smooth curve $C$.
Denote by $F$ the general fiber of $\varphi_L$, which is either a projective space or a quadric.
By Proposition~\ref{prop:Fano_fol_not_fibration}, $\mathscr{F}\neq T_{X/C}$.
Therefore, if $\ell\subset X$ is a general line on $F$, then $\ell$ is not tangent to $\mathscr{F}$.
By Lemma \ref{lemma:bound_on_pseudo_index},
$$
(n-3)L\cdot \ell \ = \ -K_\mathscr{F} \cdot\ell \ \le \ -K_F \cdot\ell -2.
$$
One easily checks that this inequality is violated for those $(X,L)$
in Theorem~\ref{tironi}(4), (5b) and (6), yielding a contradiction.
It remains to consider cases (5a) and (5c) described in Theorem~\ref{tironi}.
In both cases, $X$ admits a morphism $\pi:X\to S$ onto a normal surface with general fiber
$F\cong \p^{2}$, and $\mathscr{L}_{|F}\cong \mathscr{O}_{\p^2}(2)$.
Let $\ell\subset X$ be a general line on $F\cong \p^{2}$.
Then $L\cdot \ell = 2$.
It follows from Lemma \ref{lemma:bound_on_pseudo_index} that
$\ell$ is tangent to $\mathscr{F}$.
By \ref{pullback_foliations}, $\mathscr{F}$ is the pullback via $\pi$
of a foliation by curves $\mathscr{G}$ on $S$.
Thus
$$
L_{|F} \ = \ (-K_{\mathscr{F}})_{|F} \ = \ (-K_{X/S})_{|F},
$$
which is a contradiction.
We conclude that $\tau(L)\ge n-2$.
\medskip
\noindent {\bf Step 2:} We show that $\tau(L)\ge n-1$.
\medskip
Suppose to the contrary that $\tau(L)= n-2$.
Then either
\begin{itemize}
\item $(X,L)$ is as in Theorem~\ref{tironi}(3a--d); or
\item $\varphi_L: X\to X'$ is birational.
\end{itemize}
Suppose that $(X,L)$ is one of the pairs described in Theorem~\ref{tironi}(3a--d) and Theorem~\ref{Thm:Classification_Mukai}.
Then $X$ admits a morphism $\pi:X\to Y$ onto a normal
variety of dimension $d$, $1\le d\le 3$, with general fiber
$F$ a Fano manifold of dimension $n-d$, index
$\iota_F=n-2$, and
\begin{equation}\label{-KF=(n-2)L}
-K_F\ = \ (n-2)L_{|F}.
\end{equation}
Since $\iota_F\ge \dim(F)-1$, $F$ is covered by rational curves of $L$-degree $1$.
So we can apply Lemma \ref{lemma:bound_on_pseudo_index}, and conclude that $F$
is tangent to $\mathscr{F}$.
By \ref{pullback_foliations}, $\mathscr{F}$ is the pullback via $\pi$
of a codimension $1$ foliation $\mathscr{G}$ on $Y$. So
$$
(n-3) L_{|F} \ = \ (-K_{\mathscr{F}})_{|F} \ = \ (-K_{X/Y})_{|F},
$$
which contradicts \eqref{-KF=(n-2)L}.
\medskip
Suppose now that $\varphi_L: X\to X'$ is birational.
By Proposition~\ref{prop:BS},
$\varphi_L$ is the composition of finitely many disjoint divisorial contractions
$\varphi_i:X\to X_{i}$ of the following types:
\begin{itemize}
\item[(E)] $\varphi_i:X\to X_{i}$ is the blowup of a smooth curve $C_i\subset X_{i}$,
with exceptional divisor $E_i$.
In this case $X_{i}$ is smooth, and the restriction of $\mathscr{L}$ to a fiber of $(\varphi_i)_{|E_i}:E_i\to C_i$
is isomorphic to $\mathscr{O}_{\p^{n-2}}(1)$.
\item[(F)] $\varphi_i:X\to X_{i}$ contracts a divisor $F_i\cong \p^{n-1}$ to a singular point, and
$$
\big(F_i, \mathscr{N}_{F_i/X}, \mathscr{L}_{|F_i}\big)\cong \big(\p^{n-1}, \mathscr{O}_{\p^{n-1}}(-2), \mathscr{O}_{\p^{n-1}}(1)\big).
$$
In this case $X_i$ is $2$-factorial. In even dimension it is Gorenstein.
\item[(G)] $\varphi_i:X\to X_{i}$ contracts a divisor $G_i\cong Q^{n-1}$ to a singular point, and
$$
\big(G_i, \mathscr{N}_{G_i/X}, \mathscr{L}_{|G_i}\big)\cong \big(Q^{n-1}, \mathscr{O}_{Q^{n-1}}(-1), \mathscr{O}_{Q^{n-1}}(1)\big).
$$
In this case $X_i$ factorial.
\end{itemize}
In particular $X'$ is $\textup{\textbf{Q}}$-factorial and terminal.
Set $L':=(\varphi_L)_*(L)$.
The Mukai foliation $\mathscr{F}$ induces a foliation $\mathscr{F}'$ on $X'$ such that $-K_{\mathscr{F}'} \sim (n-3)L'$.
We claim that $K_{X'}+(n-3)L'$ is not pseudo-effective.
To prove this, let
$\Delta\sim_{\mathbb{Q}} (n-3)L$ be an effective $\mathbb{Q}$-divisor on $X$ such that $(X,\Delta)$ is klt, and set
$\Delta':=(\varphi_L)_*(\Delta)\sim_{\mathbb{Q}} (n-3)L'$.
Since $-(K_X+\Delta)$ is $\phi_L$-ample,
$(X',\Delta')$ is also klt.
Suppose that $K_{X'}+(n-3)L'\sim_{\mathbb{Q}} K_{X'}+\Delta'$ is pseudo-effective.
Under these assumptions, \cite[Theorem 2.11]{codim_1_del_pezzo_fols} states that
for any integral divisor $D$ on $X'$ such that $D\sim_{\mathbb{Q}} K_{X'}+\Delta'$,
$h^0\big(X,\Omega_{X'}^{1}[\otimes]\mathscr{O}_{X'}(-D)\big)=0$.
On the other hand, by \ref{q-forms},
$\mathscr{F}'$ gives rise to a nonzero global section
$\omega\in H^0\big(X,\Omega_{X'}^{1}[\otimes]\mathscr{O}_{X'}\big(-(K_{X'}-(n-3)L')\big)\big)$,
yielding a contradiction and proving the claim.
In particular, $K_{X'}+(n-3)L'$ is not nef, and Proposition~\ref{prop:BS}
implies that one of the following holds.
\begin{enumerate}
\item $n=6$ and $\big(X',\mathscr{O}_{X'}(L')\big) \cong \big(\p^6, \mathscr{O}_{\p^6}(2)\big)$.
\item $n=5$ and one of the following holds.
\begin{enumerate}
\item $\big(X',\mathscr{O}_{X'}(L')\big) \cong \big(Q^5, \mathscr{O}_{Q^5}(2)\big)$.
\item $X'$ is a $\p^4$-bundle over a smooth curve, and the restriction of $\mathscr{O}_{X'}(L')$ to a general fiber is $\mathscr{O}_{\p^4}(2)$.
\item $\big(X,\mathscr{O}_{X}(L)\big) \cong \Big(\p_{\p^4}\big(\mathscr{O}_{\p^4}(3)\oplus \mathscr{O}_{\p^4}(1)\big), \mathscr{O}_{\p}(1)\Big)$.
\end{enumerate}
\item $n=4$ and one of the following holds.
\begin{enumerate}
\item $\big(X',\mathscr{O}_{X'}(L')\big) \cong \big(\p^4, \mathscr{O}_{\p^4}(3)\big)$.
\item $X'$ is a Gorenstein del Pezzo $4$-fold and $3L'\sim_{\textup{\textbf{Q}}} -2K_{X'}$.
\item $\varphi_{L'}$ makes $X'$ a generic quadric bundle over a smooth curve $C$, and for a general fiber $F\cong Q^{3}$ of
$\varphi_{L'}$, $\mathscr{O}_F\big(L'_{|F}\big)\cong \mathscr{O}_{Q^{3}}(2)$.
\item $\varphi_{L'}$ makes $X'$ a generic $\p^{2}$-bundle over a normal surface $S$, and for a general fiber $F\cong \p^{2}$
of $\varphi_{L'}$, $\mathscr{O}_F\big(L'_{|F}\big)\cong \mathscr{O}_{\p^{2}}(2)$.
\item $\big(X',\mathscr{O}_{X'}(L')\big) \cong \big(Q^4, \mathscr{O}_{Q^4}(3)\big)$.
\item $\varphi_L:X\to X'$ factors through $\tilde X$, the blowup of $\p^4$ along a cubic surface $S$ contained in
a hyperplane.
The exceptional locus of the contraction $\tilde X\to X'$ is the strict transform of the hyperplane of $\p^4$ containing $S$,
and it is of type (F) above.
\item $\varphi_L:X\to X'$ factors through $\tilde X$, a conic bundle over $\p^3$.
The exceptional locus of the contraction $\tilde X\to X'$ consists of a single prime divisor of type (F) above.
\item $\varphi_{L'}$ makes $X'$ a $\p^{3}$-bundle over a smooth curve $C$,
and for a general fiber $F\cong \p^{3}$ of
$\varphi_{L'}$, $\mathscr{O}_F\big(L'_{|F}\big)\cong \mathscr{O}_{\p^{3}}(3)$.
\item $\big(X',\mathscr{O}_{X'}(L')\big) \cong \big(\p^4, \mathscr{O}_{\p^4}(4)\big)$.
\item $X'\subset \p^{10}$ is a cone over $\big(\p^3, \mathscr{O}_{\p^3}(2)\big)$ and $L'\sim_{\textup{\textbf{Q}}}2 H$, where $H$
denotes a hyperplane section in $\p^{10}$.
\end{enumerate}
\end{enumerate}
If $X'$ is a Fano manifold with $\rho(X')=1$, then $\mathscr{F}'$ is a Fano foliation with $-K_{\mathscr{F}'} \sim (n-3)L'$.
By Theorem~\ref{Thm:ADK}, $\iota_{\mathscr{F}'}\le n-1$, and equality holds only if $X'\cong \p^n$,
in which case $\mathscr{F}'$ is induced by a pencil of hyperplanes.
As a consequence, $(X',L')$ cannot be as in (1), (2a), (3e) and (3i).
Suppose that $\big(X',\mathscr{O}(L')\big) \cong \big(\p^4, \mathscr{O}_{\p^4}(3)\big)$, i.e., $(X,L)$ is as in (3a).
Then $\mathscr{F}'$ is induced by a pencil of hyperplanes in $\p^4$.
Denote by $H\cong \p^2$ the base locus of this pencil.
Since $X'\cong \p^4$ is smooth, by Proposition~\ref{prop:BS}, $\varphi_L: X\to \p^4$ is the blowup of finitely many
disjoint smooth curves $C_i\subset \p^4$, $1\le i\le k$.
Denote by $E_i\subset X$ the exceptional divisor over $C_i$,
and by $F_i\cong \p^{n-2}$ a fiber of $(\varphi_L)_{|E_i}$.
Let
$$
\omega\in H^0\big(\p^4,\Omega^1_{\p^4}\otimes \mathscr{O}_{\p^4}(2)\big)
$$
be the $1$-form
defining $\mathscr{F}'$.
An easy computation shows that $(\varphi_L)^*\omega$ vanishes along $E_i$ (with multiplicity exactly $2$) if and only if $C_i\subset H$.
So $(\varphi_L)^*\omega$ induces a section that does not vanish in codimension $1$
$$
\omega_{\mathscr{F}}\ \in \ H^0\big(X,\Omega^1_X\otimes (\varphi_L)^*(\mathscr{O}_{\p^4}(2)) \otimes \mathscr{O}_X(-\sum_{i=1}^{k}\epsilon_iE_i)\big),
$$
where $\epsilon_i=2$ if $C_i\subset H$, and $\epsilon_i=0$ otherwise.
This is precisely the $1$-form defining $\mathscr{F}$.
Hence,
$$
N_\mathscr{F}\cong \mathscr{O}_X(-K_X+K_{\mathscr{F}})\cong (\varphi_L)^*(\mathscr{O}_{\p^4}(2)) \otimes \mathscr{O}_X(-\sum_{i=1}^{k}\epsilon_iE_i),
$$
and thus
$$
\mathscr{O}_{\p^{n-2}}(1) \cong \mathscr{O}_X(-K_X+K_{\mathscr{F}})_{|F_i}\cong \mathscr{O}_{\p^{n-2}}(\epsilon_i),
$$
yielding a contradiction.
We conclude that $(X',L')$ cannot be as in (3a).
\medskip
Next we consider the cases in which $X'$ admits a morphism $\pi:X'\to C$ onto a smooth curve, with general fiber
$F$ isomorphic to either $\p^{n-1}$ or $Q^{n-1}$
(these are cases (2b), (3c) and (3h)).
By Proposition~\ref{prop:Fano_fol_not_fibration}, $\mathscr{F}$ is not the relative tangent to the
composed morphism $\varphi_L\circ \pi:X\to C$.
Hence, $\mathscr{F}'\neq T_{X'/C}$, and a general line $\ell\subset F$ is not tangent to $\mathscr{F}'$.
By Lemma \ref{lemma:bound_on_pseudo_index_bis},
$$
(n-3)L'\cdot \ell \ = \ -K_{\mathscr{F}'} \cdot\ell \ \le \ -K_F \cdot\ell -2.
$$
One easily checks that this inequality is violated for those $(X',L')$
in cases (2b), (3c) and (3h).
\medskip
Next we show that $(X,L)$ cannot be as in (2c).
Suppose to the contrary that
$$
\big(X,\mathscr{O}_X(L)\big) \cong \Big(\p_{\p^4}\big(\mathscr{O}_{\p^4}(3)\oplus \mathscr{O}_{\p^4}(1)\big), \mathscr{O}_{\p}(1)\Big).
$$
Let $\ell\subset X$ be a general fiber of the natural projection $\pi:X=\p_{\p^4}\big(\mathscr{O}_{\p^4}(3)\oplus \mathscr{O}_{\p^4}(1)\big)\to \p^4$.
Since $-K_{\mathscr{F}} \cdot \ell=2$, $\ell$ is tangent to $\mathscr{F}$ by Lemma \ref{lemma:bound_on_pseudo_index}.
By \ref{pullback_foliations}, $\mathscr{F}$ is the pullback via $\pi$ of a codimension $1$ foliation $\mathscr{G}$ on $\p^4$.
By \eqref{K_pullback_fol}, $\det(\mathscr{G})\cong \mathscr{O}_{\p^4}(4)$, which is impossible
by Theorem~\ref{Thm:ADK}.
\medskip
We show that $(X',L')$ cannot be as in (3b).
Suppose to the contrary that $X'$ is a Gorenstein del Pezzo $4$-fold and $3L'\sim_{\textup{\textbf{Q}}} -2K_{X'}$.
Then there is an ample Cartier divisor $H'$ on $X'$ such that $L'\sim_{\textup{\textbf{Q}}} 2H'$ and $-K_{X'}\sim 3H'$.
Notice that $X'$ has isolated singularities.
Let $Y \in |H'|$ be a general member.
We claim that $Y$ is a smooth $3$-fold. Suppose first
that $(H')^4 \ge 2$. Then $|H'|$ is basepoint free by \cite[Corollary 1.5]{fujita90}, and hence $Y$ is smooth by Bertini's Theorem. Suppose now that $(H')^4 = 1$. By \cite[Theorem 4.2]{fujita_classification} (see also
\cite[6.3 and 6.4]{fujita_classification}), $\dim(\textup{Bs}(H'))\le 0$. Thus, if
$H'_1,\ldots,H'_4$ are general members of $|H'|$, then
$H'_1,\ldots, H'_4$ meets properly in a (possibly empty) finite set of points, and
$(H')^4=\deg(H'_1\cap\cdots\cap H'_4)$ (see \cite[Example 2.4.8]{fulton}).
Since $(H')^4=1$, $H'_1\cap\cdots\cap H'_4$ is a reduced point $x$,
$X'$ is smooth at $x$, and the local equations of
$H'_1,\ldots, H'_4$ at $x$ form a regular sequence in $\mathscr{O}_{X',x}$. In particular,
$H'_i$ is smooth at $x$ for all $i\in\{1,\ldots,4\}$.
By Bertini's Theorem again, we conclude that $Y$ is smooth.
Set $H_Y:={H'}_{|Y}$, and denote by $\mathscr{H}$ the codimension one foliation on $Y$ induced by $\mathscr{F}'$.
By the adjunction formula, $-K_Y=2H_Y$, and hence $Y$ is a del Pezzo threefold.
By \ref{restricting_fols}, there exists a non-negative integer $b$ such that
$
-K_{\mathscr{H}} \ = \ (1+b) H_Y.
$
By Theorem~\ref{Thm:ADK}, we must have $b\in\{0,1\}$, and hence $\mathscr{H}$ is a Fano foliation.
It follows from Thereom \ref{Thm:codim1_dP} and the classification of
del Pezzo manifolds that
$Y \cong \p^3$, $b=0$, and $(H_Y)^3=(H')^4=8$.
Therefore, $H'$ is very ample by \cite[6.2.3]{fujita_classification}, so that
we can apply \cite[Theorem 3]{fujita82a} to conclude that one of the following holds.
\begin{itemize}
\item $X'$ is a cubic hypersurface in $\p^5$.
\item $X'$ is a complete intersection of two quadric hypersurfaces in $\p^6$.
\item $X'$ is a cone over a Gorenstein del Pezzo $3$-fold.
\item $\dim(\textup{Sing}(X'))\ge 1$.
\end{itemize}
In the first two cases, $(H')^4=3$ and $(H')^4=4$, respectively. Since $X'$ has isolated singularities, we conclude that
$X'$ must be a cone over $\big(\p^3,\mathscr{O}_{\p^3}(2)\big)$.
Denote by $\pi:X\to \p^3$ the induced map, and by $\ell$ a general fiber of $\pi$.
One computes that $L \cdot \ell =1$.
By Lemma \ref{lemma:bound_on_pseudo_index}, $\ell$ is tangent to $\mathscr{F}$.
So, by \ref{pullback_foliations}, $\mathscr{F}$ is the pullback via $\pi$
of a codimension $1$ foliation $\mathscr{G}$ on $\p^3$. Thus
$$
L_{|\ell} \ = \ (-K_{\mathscr{F}})_{|\ell} \ = \ (-K_{X/\p^3})_{|\ell} \ = \ -K_\ell,
$$
which is absurd. This shows that $(X',L')$ cannot be as in (3b).
\medskip
We show that $(X',L')$ cannot be as in (3d).
Suppose to the contrary that $\varphi_{L'}$ makes $X'$ a generic $\p^{2}$-bundle over a normal surface $S$,
and for a general fiber $F\cong \p^{2}$ of $\varphi_{L'}$, $\mathscr{O}_F\big(L'_{|F}\big)\cong \mathscr{O}_{\p^{2}}(2)$.
Lemma \ref{lemma:bound_on_pseudo_index_bis} implies that $F$ is tangent to $\mathscr{F}'$.
By \ref{pullback_foliations}, $\mathscr{F}'$ is the pullback via $\varphi_{L'}$
of foliation by curves $\mathscr{G}$ on $S$, and thus
$$
L'_{|F} \ = \ (-K_{\mathscr{F}'})_{|F} \ = \ (-K_{X'/S})_{|F},
$$
which is a contradiction.
\medskip
In cases (3f), (3g) and (3j), $\varphi_L:X\to X'$ factors through a factorial $4$-fold $\tilde X$.
Denote by $\tilde L$ the push-forward of $L$ to $\tilde X$.
The Mukai foliation $\mathscr{F}$ induces a foliation $\tilde \mathscr{F}$ on $\tilde X$ such that $-K_{\tilde \mathscr{F}} \sim \tilde L$.
In case (3f), $\tilde X$ is the blowup of $\p^4$ along a cubic surface $S$ contained in
a hyperplane $F\subset \p^4$.
Denote by $\tilde F\subset \tilde X$ the strict transform of $F$,
so that $N_{\tilde F/ \tilde X}\cong \mathscr{O}_{\p^{3}}(-2)$ and $\mathscr{O}_{\tilde F}\big(\tilde L_{|\tilde F}\big)\cong \mathscr{O}_{\p^{3}}(1)$.
We will reach a contradiction by exhibiting a family $H$ of rational curves on $\tilde X$ such that
\begin{enumerate}
\item the general member of $H$ is a curve tangent to $\tilde \mathscr{F}$; and
\item two general points of $\tilde X$ can be connected by a chain of curves from $H$
avoiding the singular locus of $\tilde \mathscr{F}$.
\end{enumerate}
We take $H$ to be the family of strict transforms of lines in $\p^4$ meeting $S$ and not contained in $F\cong \p^3$.
It is a minimal dominating family of rational curves on $\tilde X$ satisfying condition (2) above.
Let $\ell\subset \tilde X$ be a general member of $H$.
One computes that $-K_{\tilde X}\cdot \ell =4$ and $-K_{\tilde \mathscr{F}}\cdot \ell = \tilde L \cdot \ell \ge 3$.
For the latter, notice that $\ell \equiv \ell_1+ 2\ell_2$, where $\ell_1\subset \tilde F$ is a line under the isomorphism
$\tilde F\cong\p^3$, and $\ell_2$ is a nontrivial fiber of the blowup $\tilde X\to \p^4$.
Condition (1) above then follows from Lemma \ref{lemma:bound_on_pseudo_index}.
We conclude that $(X',L')$ cannot be as in (3f).
In case (3g), $\tilde X$ is a conic bundle over $\p^3$.
Moreover, there is a divisor $F\subset X$ mapping isomorphically onto its image by $X\to \tilde X$,
and such that $(F, N_{F/ X})\cong \big(\p^3, \mathscr{O}_{\p^{3}}(-2)\big)$.
Denote by $\pi:X\to \tilde X\to \p^3$ the composite map, and by $\ell$ a general fiber of $\pi$.
By Lemma \ref{lemma:bound_on_pseudo_index}, $\ell$ is tangent to $\mathscr{F}$.
So, by \ref{pullback_foliations}, $\mathscr{F}$ is the pullback via $\pi$
of a codimension $1$ foliation $\mathscr{G}$ on $\p^3$.
Let $C\subset \p^3$ be a general line, and set $S:=\pi^{-1}(C)$.
Note that $S$ is smooth, $\pi_C:=\pi_{|C}:S\to C$ is a conic bundle, and the foliation on $S$ induced by
$\mathscr{F}$ is precisely $T_{S/C}$.
Hence, by \ref{restricting_fols}, there is an effective divisor $B$ on $S$ such that
\begin{equation}\label{eq13g}
-K_S \ = L_{|S} \ + \ B.
\end{equation}
On the other hand, by \ref{pullback_foliations},
\begin{equation}\label{eq23g}
L_{|S} \ = -K_{S/C} \ + \ (\pi_C)^*c_1\big(\mathscr{G}_{|C}\big).
\end{equation}
Equations \eqref{eq13g} and \eqref{eq23g} together imply that $B=(\pi_C)^*B_C$ for some
effective divisor $B_C$ on $C$, and thus $-K_S$ is ample.
We will reach a contradiction by exhibiting a curve $\sigma \subset S$ such that
$-K_S\cdot \sigma \le 0$.
We take $\sigma:= F\cap S$.
Using that $\mathscr{O}_{F}(F) \cong \mathscr{O}_{\p^{3}}(-2)$, the adjunction formula implies that $-K_S\cdot \sigma \le 0$.
We conclude that $(X',L')$ cannot be as in (3g).
In case (3j), $\tilde X=\p_{\p^3}(\mathscr{O}_{\p^3}\oplus \mathscr{O}_{\p^3}(-2))$.
Denote by $\pi:X\to \tilde X\to \p^3$ the composite map, and by $\ell$ a general fiber of $\pi$.
One computes that $L \cdot \ell =1$.
By Lemma \ref{lemma:bound_on_pseudo_index}, $\ell$ is tangent to $\mathscr{F}$.
So, by \ref{pullback_foliations}, $\mathscr{F}$ is the pullback via $\pi$
of a codimension $1$ foliation $\mathscr{G}$ on $\p^3$. Thus
$$
L_{|\ell} \ = \ (-K_{\mathscr{F}})_{|\ell} \ = \ (-K_{X/\p^3})_{|\ell} \ = \ -K_\ell,
$$
which is a contradiction.
We conclude that $\tau(L)\ge n-1$.
\medskip
\noindent {\bf Step 3: } We show that if $\tau(L)= n-1$, then one of the following conditions hold.
\begin{enumerate}
\item[(i)] $X$ admits a structure of quadric bundle over a smooth curve. In this case, by Proposition~\ref{proposition:Q-bdle_over_C},
$X$ and $\mathscr{F}$ are as described in Theorem~\ref{main_thm_rho>1}(3).
\item[(ii)] $X$ admits a structure of $\p^{n-2}$-bundle over a smooth surface. In this case, by section \ref{subsection:P-bdles/surfaces},
$X$ and $\mathscr{F}$ are as described in Theorem~\ref{main_thm_rho>1}(4).
\item[(iii)] $n=5$, $\varphi_L: X\to \p^5$ is the blowup of one point $P\in \p^5$, and $\mathscr{F}$ is induced by a pencil
of hyperplanes in $\p^5$ containing $P$ in its base locus. This gives Theorem~\ref{main_thm_rho>1}(5).
\item[(iv)] $n=4$, $\varphi_L: X\to \p^4$ is the blowup of at most $8$ points in general position on a plane $\p^2 \cong S\subset \p^4$,
and $\mathscr{F}$ is induced by the pencil of hyperplanes in $\p^4$ with base locus $S$. This gives Theorem~\ref{main_thm_rho>1}(6).
\item[(v)] $n=4$, $\varphi_L: X\to Q^4$ is the blowup of at most $7$ points in general position on a
codimension $2$ linear section $Q^2 \cong S\subset Q^4$,
and $\mathscr{F}$ is induced by the pencil of hyperplanes sections of $Q^4\subset \p^5$ with base locus $S$.
This gives Theorem~\ref{main_thm_rho>1}(7).
\end{enumerate}
\medskip
Suppose that $\tau(L)= n-1$.
By Theorem~\ref{tironi}, one of the following holds.
\begin{itemize}
\item $X$ admits a structure of quadric bundle over a smooth curve. This is case (i).
\item $X$ admits a structure of $\p^{n-2}$-bundle over a smooth surface. This is case (ii).
\item $\varphi_L: X\to X'$ is the blowup of a smooth projective variety at finitely many points $P_1, \ldots, P_k\in X'$.
\end{itemize}
Suppose that we are in the latter case, and
denote the exceptional prime divisors of $\varphi_L$ by $E_i$, $1\le i\le k$.
Set $L':=(\varphi_L)_*(L)$. It is an ample divisor on $X'$ and
\begin{equation}\label{L+E=L'}
L \ +\ \sum_{i=1}^k E_i \ = \ (\varphi_L)^*L'.
\end{equation}
The Mukai foliation $\mathscr{F}$ induces a Fano foliation $\mathscr{F}'$ on $X'$ such that $-K_{\mathscr{F}'} \sim (n-3)L'$.
By Theorem~\ref{Thm:KX-KF_not_nef}, $K_{X'}+(n-3)L'$ is not nef, i.e., $\tau(L') > n-3$.
On the other hand, since $K_{X'}+(n-1)L'= (\varphi_L)_*\big(K_{X}+(n-1)L\big)$ is ample, $\tau(L') < n-1$.
It follows from Theorem~\ref{tironi}, together with steps 1 and 2 above, that $\rho(X')=1$.
Let $H'$ be the ample generator of $\textup{Pic}(X')$, and
write $L'\sim \lambda H'$, $\lambda\ge 1$.
Then $-K_{X'}= \iota_{X'}H'$, $-K_{\mathscr{F}'}= \lambda (n-3)H'$ and
\begin{equation} \label{lambda1}
\tau(L') =\frac{\iota_{X'}}{\lambda} < n-1.
\end{equation}
By Lemma~\ref{lemma:bound_on_index2},
\begin{equation} \label{lambda2}
\iota_{X'} \ge \lambda (n-3) +2.
\end{equation}
Inequalities \eqref{lambda1} and \eqref{lambda2} together yield that $\lambda \ge 2$.
On the other hand, by Theorem~\ref{Thm:ADK}, $\iota_{\mathscr{F}'}=\lambda (n-3)\le n-1$.
Thus $(n,\lambda)\in \big\{(5,2),(4,3),(4,2)\big\}$.
Let $\omega' \in H^0\big(X', \Omega_{X'}^1\big( -K_{X'}-(n-3)L' \big)\big)$
be the twisted $1$-form defining $\mathscr{F}'$.
The induced twisted $1$-form $(\varphi_L)^*\omega' \in H^0\big(X, \Omega_X^1\big( (\varphi_L)^*(-K_{X'}-(n-3)L') \big)\big)$
saturates to give the twisted $1$-form defining $\mathscr{F}$,
$\omega_{\mathscr{F}} \in H^0\big(X, \Omega_{X}^1\big( -K_{X}-(n-3)L \big)\big)$.
Using \eqref{L+E=L'}, one computes that
$$
-K_{X}-(n-3)L= (\varphi_L)^*\big( -K_{X'}-(n-3)L' \big) + 2 \sum_{i=1}^k E_i.
$$
Thus $(\varphi_L)^*\omega'$ must vanish along each $E_i$ with multiplicity exactly $2$.
\medskip
Suppose that $(n,\lambda)=(5,2)$. Then $\iota_{\mathscr{F}'}=\textup{rank}(\mathscr{F}')$ and, by Theorem~\ref{Thm:ADK},
$X'\cong \p^5$, and $\mathscr{F}'$ is
a foliation induced by a pencil of hyperplanes in $\p^5$.
We claim that
$\varphi_L: X\to \p^5$ is the blowup of only one point.
Indeed, if $\varphi_L: X\to \p^5$ blows up at least 2 points $P$ and $Q$, let $\ell$ be a line connecting $P$ and $Q$, and
$\tilde \ell\subset X$ its strict transform.
We get a contradiction by intersecting \eqref{L+E=L'} with $\tilde \ell$, and conclude that
$\varphi_L: X\to \p^5$ is the blowup of a single point $P\in \p^5$, with exceptional divisor $E$.
Moreover, $(\varphi_L)^*\omega'$ vanishes along $E$ with multiplicity exactly $2$.
A local computation shows that this happens precisely when $P$ is in the base locus of
the pencil of hyperplanes defining $\mathscr{F}'$.
\medskip
Suppose that $(n,\lambda)=(4,3)$. Then $\iota_{\mathscr{F}'}=\textup{rank}(\mathscr{F}')$ and, by Theorem~\ref{Thm:ADK},
$X'\cong \p^4$, and $\mathscr{F}'$ is a foliation induced by a pencil of hyperplanes in $\p^4$.
Moreover, $(\varphi_L)^*\omega'$ vanishes along each $E_i$ with multiplicity exactly $2$.
A local computation shows that this happens precisely when the points $P_i$ all lie in the base locus of
the pencil of hyperplanes defining $\mathscr{F}'$, which is a codimension $2$ linear subspace .
Since $L = (\varphi_L)^*L' - \sum_{i=1}^k E_i$ is ample, we must have $k\le 8$
and the $P_i$'s are in general position by
Lemma~\ref{number_of_pts} below.
\medskip
Suppose that $(n,\lambda)=(4,2)$.
Then $\mathscr{F}'$ is a codimension $1$ del Pezzo foliation on $X'$.
By Theorem~\ref{Thm:codim1_dP},
either $X'\cong\p^4$ and $\mathscr{F}'$ is a degree $1$ foliation, or
$X'\cong Q^4\subset \p^{5}$ and $\mathscr{F}'$ is induced by a pencil of hyperplane sections.
Suppose first that $X'\cong\p^4$ and $\mathscr{F}'$ is a degree $1$ foliation.
The same argument used in the case $(n,\lambda)=(5,2)$ shows that
$\varphi_L: X\to \p^4$ is the blowup of only one point $P\in \p^4$, with exceptional divisor $E$.
Moreover, $(\varphi_L)^*\omega'$ vanishes along $E$ with multiplicity exactly $2$.
A local computation shows that this cannot happen.
Finally suppose that $X'\cong Q^4\subset \p^{5}$ and $\mathscr{F}'$ is induced by a pencil of hyperplane sections.
Moreover, $(\varphi_L)^*\omega'$ vanishes along each $E_i$ with multiplicity exactly $2$.
A local computation shows that this happens precisely when the points $P_i$ all lie in the base locus of
the pencil defining $\mathscr{F}'$, which is a codimension $2$ linear section of $Q^4$.
Since $L = (\varphi_L)^*L' - \sum_{i=1}^k E_i$ is ample, we must have $k\le 7$
and the $P_i$'s are in general position by
Lemma~\ref{number_of_pts} below.
\medskip
\noindent {\bf Step 4: } Finally suppose that $\tau(L)= n$.
By Theorem~\ref{tironi}, $X$ admits a structure of $\p^{n-1}$-bundle over a smooth curve.
In this case, by Proposition~\ref{lemma:P-bdle_over_C},
$X$ and $\mathscr{F}$ are as described in Theorem~\ref{main_thm_rho>1}(1) or (2).
\qed
\
\begin{lemma}\label{number_of_pts}
\begin{enumerate}
\item Let $\pi:X\to \p^4$ be the blowup of finitely many points $P_1, \ldots, P_k$ contained in a codimension 2 linear subspace $S\cong \p^2$,
and denote by $E_i$ the exceptional divisor over $P_i$.
Then the line bundle $\pi^*\mathscr{O}_{\p^4}(3) \otimes \mathscr{O}_X(- \sum_{i=1}^k E_i)$ is ample if and only if
$k\le 8$ and the $P_i$'s are in general position in $\p^2$.
\item Let $\pi:X\to Q^4$ be the blowup of a smooth quadric at finitely many points $P_1, \ldots, P_k$
contained in a codimension 2 linear section $S$,
and denote by $E_i$ the exceptional divisor over $P_i$.
Then the line bundle $\pi^*\mathscr{O}_{Q^4}(2) \otimes \mathscr{O}_X(- \sum_{i=1}^k E_i)$ is ample if and only if
$k\le 7$ and the $P_i$'s are in general position in $S$.
\end{enumerate}
\end{lemma}
\begin{proof}
Under the assumptions of (1) above, write
$L= \pi^*3H - \sum_{i=1}^k E_i$, where $H$ is a hyperplane in $\p^4$.
The divisor $L$ is ample if and only if $|mL|$ separates points in $X$ for $m\gg 1$.
Notice that $|L|$ always separates points outside the strict transform $\tilde S$ of the plane
$S\subset \p^4$.
Moreover, for $m\ge 0$, any global section of $\mathscr{O}_{\tilde S}\big(mL_{|\tilde S}\big)$ extends to a
global section
of $\mathscr{O}_X(mL)$.
Hence, $L$ is ample if and only if $L_{|\tilde S}$ is ample.
Now notice that
$$
L_{|\tilde S}\ = \ p^*(-K_S)\ -\ \sum_{i=1}^k {E_i}_{|\tilde S} \ = \ -K_{\tilde S},
$$
where $p=\pi_{|\tilde S}:\tilde S\to S\cong \p^2$ is the blowup of $\p^2$ at $P_1, \ldots, P_k$.
Therefore, $L$ is ample if and only if $-K_{\tilde S}$ is ample, i.e.,
$k\le 8$ and the $P_i$'s are in general position in $\p^2$.
Now we proceed to prove (2). Let $\tilde S$ be the strict transform of the (possibly singular)
irreducible quadric surface $S\subset Q^4$.
Write as above $L= \pi^*2H - \sum_{i=1}^k E_i$, where $H\subset Q^4$ is a hyperplane section.
Notice that $|L|$ always separates points outside $\tilde S$.
Moreover, any global section of $\mathscr{O}_{\tilde S}\big(L_{|\tilde S}\big)$ extends to a
global section of $\mathscr{O}_X(L)$.
Suppose that $L$ is ample, so that $L_{|\tilde{S}}=-K_{\tilde{S}}$ is ample as well. Then $k\le 7$ and the $P_i$'s are in general position in $S$.
Conversely, suppose that $k\le 7$, and the $P_i$'s are in general position in $S$.
Then $\dim(\textup{Bs}(-K_{\tilde{S}}))\le 0$, and hence
$\dim(\textup{Bs}(L))\le 0$.
We conclude that
$L$ is ample by Zariski's Theorem (\cite[Remark 2.1.32]{lazarsfeld1}).
\end{proof}
\bibliographystyle{amsalpha}
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\section{Introduction}
The structure functions are important objects in QCD. For example, the nucleon structure functions control the cross section of
deep inelastic scattering and they are related to the parton densities inside the initial hadrons.
The nucleon structure functions however are defined by nucleon matrix elements of the electro-magnetic currents,
so they cannot be calculated in the perturbative method.
We can only calculate energy scale evolution by perturbation theory.
AdS/CFT correspondence relates $\mathcal{N}=4$ super Yang-Mills theory at large 't Hooft coupling in four dimensions
to weakly-coupled string theory in $AdS_5 \times S^5$ \cite{maldacena}.
The authors of \cite{PS} have studied dual gravity description of nucleon structure functions and have calculated nucleon structure functions.
This is called the hard-wall model. In this model, the AdS space has a cut-off at infrared region and conformal invariance is broken.
This cut-off scale corresponds to the infrared mass scale of the gauge theory. At small coupling, probe photons scatter off
partons inside hadrons. But at large 't Hooft coupling, the situation is completely different; probe photons scatter off entire
hadrons and do not destroy hadronic states. Hadronic states are dual to (massless) string states in AdS space and
we can calculate structure functions from supergravity interactions.
Usually, the leading-twist operators
have the leading contribution to OPE of currents correlators at small coupling. But at large 't Hooft coupling, these operators have
large anomalous dimensions \cite{GKP2} and do not dominate in OPE.
In this article, we consider the virtual photon structure functions and study their property from bulk dynamics in AdS space.
The photon structure functions are defined by the absorptive part of the four quark currents correlator.
Naively, they can be obtained by replacing the initial hadronic states in hadronic tensors by the target photon states.
So their properties are similar to nucleon structure functions; we can apply standard OPE technique and calculate the anomalous dimensions
of twist-two operators and coefficient functions perturbatively. A crucial difference of the virtual photon
structure functions from the nucleon structure functions
is that one can determine $x$-behavior of the virtual photon structure functions perturbatively.
But from theoretical view point, it is still interesting to consider QCD objects at strong coupling which
are observed at small coupling \cite{HM}.
\section{Review of photon structure functions}
We review photon structure functions briefly \cite{UW}.
We take the Lorentz metric $\eta_{\mu \nu }=\rm{diag}(-1,1,1,1) $.
For definiteness, we consider virtual photon scattering on $e^+ e^- \rightarrow e^+ e^- + \rm{hadrons} $ (fig.\ref{feyn}).
Four dimensional momenta of two virtual photons are $q^{\mu}$ and $p^{\mu}$ $(p^2 \le q^2)$.
We call the photon whose momentum is $q^{\mu} $ $(p^{\mu})$ the probe (target) photon.
If the invariant mass of the target photon is close to on-shell $p^2 \simeq 0$,
then vector meson dominance is realized. This process is rather well described by vector meson-photon coupling
than by photon-photon scattering. Dual gravity descriptions of such interactions are studied in \cite{gomez}, \cite{PRSW}.
If the invariant mass of the target photon is far off-shell $ \Lambda^2_{QCD} \ll p^2 $, this process is described
by virtual photon-virtual photon scattering.
\begin{figure}
\begin{center}
\includegraphics[width=7cm,clip]{feynman1.eps}
\caption{$e^- + e^+ \to e^- + e^+ + X$. $X$ represents the final hadron states.
Wavy lines denote virtual photons which couple to quark currents.
$q^2$ $(p^2)$ is invariant mass squared of the probe (target) photon. }
\label{feyn}
\end{center}
\end{figure}
We define the tensor $T^{\mu \nu \alpha \beta}$ as
\begin{eqnarray}
T^{ \mu \nu \alpha \beta} (p,q)=i \int d^4x d^4y d^4z e^{i q\cdot x } e^{i p\cdot (y-z) }
\langle 0 |T( J^{\alpha}(y) J^{\mu}(x) J^{\nu}(0) J^{\beta}(z) ) | 0 \rangle ,
\end{eqnarray}
where the $J^{\mu}$ are quark currents which couple to the photons.
The structure tensor of virtual photons is defined by the absorptive part of $T^{\mu \nu \alpha \beta}$.
\begin{eqnarray}
W^{ \mu \nu \alpha \beta} (p,q)&=&\frac{1}{\pi} {\rm{Im}} T^{\mu \nu \alpha \beta} (p,q) .
\end{eqnarray}
This tensor has eight independent components.
If the initial photon states are unpolarized,
we average the target photon helicities and we obtain
\begin{eqnarray} \label{structure}
W^{\mu \nu } (p,q)&=&\frac{1}{2} \sum_{\lambda} {\epsilon^{(\lambda)}_{\alpha}}^{*} (p)
W^{ \mu \nu \alpha \beta} (p,q) \epsilon^{(\lambda)}_{\beta} (p) \nonumber \\
&=&\frac{1}{2} \eta_{\alpha \beta} W^{ \mu \nu \alpha \beta} (p,q) .
\end{eqnarray}
$W^{\mu \nu }$ can be decomposed into two structure functions $F_1(x,q^2,p^2)$ and $F_{2}(x,q^2,p^2)$ as
\begin{eqnarray}
W^{\mu \nu } (p,q)= ({\eta}^{\mu \nu}-\frac{q^{\mu} q^{\nu}}{q^2}) F_{1}(x,q^2,p^2)
+(p^{\mu}+\frac{q^{\mu}}{2x})(p^{\nu}+\frac{q^{\nu}}{2x}) \frac{4x}{q^2}F_2 (x,q^2,p^2) ,
\end{eqnarray}
where $x=\frac{-q^2}{2p\cdot q}$. $x$ take $0 \le x \le 1/(1+\frac{p^2}{q^2})$ .
The structure functions $F_{i}(x,q^2,p^2)$ are called the real photon structure functions in the region $p^2 \simeq 0 $
and are called the virtual photon structure functions in the region $\Lambda^2_{QCD} \ll p^2 $.
In order to apply OPE for the product of the quark currents which couple to probe photons, the condition $p^2 \ll q^2$ is required.
This means that the distance between quark currents correlators which couple to the probe photons is much shorter than the others.
So in the region $\Lambda^2_{QCD} \ll p^2 \ll q^2$, we can calculate the virtual photon structure functions perturbatively at small
coupling.
Inserting the final hadronic states $\sum_{X} | X \rangle \langle X |=1$, we can represent the structure tensor $W^{\mu \nu}$ as
\begin{eqnarray} \label{tens}
W^{\mu \nu} (p,q)&=&\frac{1}{2}
\langle 0 |\tilde{J}^{\alpha}(-p) \tilde{J}^{\mu}(q) | X \rangle \langle X |J^{\nu}(0) \tilde{J}_{\alpha}(p) | 0\rangle \nonumber \\
&=&\frac{1}{2} (2\pi)^4 \sum_{X} \delta^{(4)}(p+q-P_{X})
\langle 0 |\tilde{J}^{\alpha}(-p) {J}^{\mu}(0) | X \rangle \langle X |J^{\nu}(0) \tilde{J}_{\alpha}(p) | 0\rangle , \nonumber \\
\end{eqnarray}
where we denote $P_{X}$ the momentum of the final state hadron $X$ and $\tilde{J}^{\mu}$ is the Fourier transformation of ${J}^{\mu}$.
\section{Virtual photon structure functions in AdS/QCD}
In this section, we calculate the virtual photon structure functions from supergravity interaction.
Scattering process is caused by gauged $U(1)$ R-symmetry currents $J^{\mu}$.
We treat the process to the lowest order in $U(1)$ coupling constant and to all orders in strong coupling constant.
Indices $m, n, \cdots $ denote the $\rm{AdS}_{5}$ space and $\mu, \nu, \cdots $ denote four dimensional Minkowski space.
Indices $m, n, \cdots $ are raised with the curved metric $g^{m n}$ and $\mu, \nu, \cdots $ are raised with $\eta^{\mu \nu}$.
We define $q^2 = \eta_{\mu \nu} q^{\mu} q^{\nu}$ and $q=\sqrt{q^2}$.
In the hard-wall model, the AdS space has a infrared cut-off at $r_0=\Lambda R^2$ and
$\Lambda$ corresponds to the infrared mass scale of the gauge theory.
The metric of $AdS_5 \times W$ is
\begin{eqnarray}
ds^2=\frac{r^2}{R^2} \eta_{\mu \nu} dy^{\mu} dy^{\nu}+ \frac{R^2}{r^2} d^2 r + R^2 ds^2_{W}
\end{eqnarray}
where $R=(4\pi g_2 N)^{1/4} \alpha'$ is the AdS radius and $W$ is the five-dimensional internal space which has
a $U(1)$-isometry.
The final state hadron is dual to a string state in ten-dimensional space \cite{PS}.
We consider this string state is given by the dilaton.
The dilaton wave function $\Phi_X$ is given by
\begin{eqnarray} \label{dila}
\Phi_X=c_X {P_X}^{1/2} \Lambda^{1/2} \frac{e^{-i P_X \cdot y}}{r^2} J_{\Delta-2} (P_X R^2/r) Y(\Omega),
\end{eqnarray}
where $c_X$ is a dimensionless constant. $J_{\Delta-2}$ is a Bessel function and $\Delta$ is related to
the dilaton mass $M^2=\Delta(\Delta-4)/R^2$ in $AdS_5$, and
$Y(\Omega)$ is the wave function of dilaton in $W$ direction which is normalized
\begin{eqnarray}
\int d^5\Omega \sqrt{g_{W}} |Y(\Omega)|^2 =1 .
\end{eqnarray}
The Kaluza-Klein gauge field $A_m$ couples to the R-current at AdS boundary \cite{GKP1}, \cite{witten2}.
The boundary condition of gauge field is imposed by
\begin{eqnarray}
\lim_{r \to \infty} A_{\mu} (y, r) =\epsilon_{\mu} (q) e^{i q\cdot y}
\end{eqnarray}
where $\epsilon_{\mu}$ denotes the polarization vector of the gauge field in four dimensions.
The bulk gauge field satisfies the Maxwell equation in the $AdS_5$ space. We take Lorentz gauge in five dimensions,
then the nonnormalizable mode of gauge field are
\begin{eqnarray}
A_{\mu} (y, r)&=&\epsilon_{\mu} (q) e^{i q\cdot y} \frac{qR^2}{r} K_{1} (q R^2/r), \nonumber \\
A_{r} (y, r)&=&i (\epsilon (q) \cdot q) e^{i q\cdot y} \frac{R^4}{r^3} K_{0} (q R^2/r) ,
\end{eqnarray}
where $K_0, K_1$ are modified Bessel functions of second type.
Field strength tensors of $A_{\mu}$ and $A_{r}$ are
\begin{eqnarray}
F_{\mu \nu}(q)&=&i[ q_{\mu} \epsilon_{\nu}(q)- \epsilon_{\mu}(q) q_{\nu}] \frac{qR^2}{r} K_1 (qR^2/r) e^{iq \cdot y}, \nonumber \\
F_{\mu r}(q)&=&[ \epsilon_{\mu}(q) q^2 - q_{\mu} (\epsilon(q) \cdot q)] \frac{R^4}{r^3} K_0 (qR^2/r) e^{iq \cdot y} . \label{field}
\end{eqnarray}
The insertion of R-symmetry currents excites the metric perturbation $\delta g_{m a}=A_{m}(y,r) v_a(\Omega)$ with
Killing vector $v_a(\Omega)$ associated to a U(1)-isometry of $W$.
The correlation function of two R-symmetry currents and a scalar corresponds to the supergravity interaction \cite{freedman}
\begin{eqnarray}
\epsilon_{\mu} (q)\epsilon_{\nu} (p) \langle X |\tilde{J}^{\mu}(q) \tilde{J}^{\nu}(p) | 0\rangle
\sim \int d^{10}x \sqrt{-g} \Phi_X F_{mn} F^{mn} v^{a} v_{a} . \label{int}
\end{eqnarray}
From eq.(\ref{field}), eq.(\ref{int}) becomes
\begin{eqnarray} \label{corelator}
\int d^{10}x \sqrt{-g} \Phi_X F_{mn}(q) F^{mn}(p) v^{a} v_{a} &=&(2\pi)^4 \delta^{(4)}(p+q-P_X) c_X C s^{1/4} \Lambda^{1/2} \nonumber \\
&& \times \int dr \frac{r}{R^3}(\frac{R^4}{r^4} F_{\mu \nu} (q) F^{\mu \nu} (p)+2 F_{\mu r} (q) F^{\mu}_{r}(p) ) \nonumber \\
\end{eqnarray}
where $C=\int \sqrt{g_{W}} Y(\Omega) v^{a} v_{a}$. This factor does not affect $x$-behavior of $F_1(x, q^2, p^2)$ and $F_{2}(x, q^2, p^2)$, so we
do not write it explicitly, then
\begin{eqnarray}
\epsilon_{\mu} (q)\epsilon_{\nu} (p) \langle X |J^{\mu}(0) \tilde{J}^{\nu}(p) | 0\rangle
&=& c_X C s^{1/4} \Lambda^{1/2}
\int dr \frac{r}{R^3}(\frac{R^4}{r^4} F_{\mu \nu} (q) F^{\mu \nu} (p)+2 F_{\mu r} (q) F^{\mu}_{r}(p) ) \nonumber \\
&=&c_X C s^{1/4} \Lambda^{1/2} \Bigr\{ -2 \Bigr[(q \cdot p)(\epsilon(q) \cdot \epsilon(p))
-(q\cdot \epsilon(p)) (p\cdot \epsilon(q)) \Bigr] \nonumber \\
&&
\times q p A(x, q^2, p^2) + 2\Bigr[(\epsilon(q) \cdot \epsilon(p)) q^2 p^2 \nonumber \\
&&-(p\cdot \epsilon(q)) (p\cdot \epsilon(p)) q^2 -(q\cdot \epsilon(q)) (q\cdot \epsilon(p)) p^2 \nonumber \\
&&+(q \cdot p) (q \cdot \epsilon(q)) (p \cdot \epsilon(p)) \Bigr] B(x,q^2,p^2) \Bigr\} . \label{aaa}
\end{eqnarray}
In the above equation, $A(x, q^2, p^2)$ and $B(x, q^2, p^2)$ are defined by
\begin{eqnarray} \label{AB}
A(x,q^2,p^2)=\int_{0}^{1/ \Lambda} dr \frac{R^5}{r^5} K_1(qR^2/r) K_1(pR^2/r) J_{\Delta-2} (s^{1/2} R^2/r) , \nonumber \\
B(x,q^2,p^2)=\int_{0}^{1/ \Lambda} dr \frac{R^5}{r^5} K_0(qR^2/r) K_0(pR^2/r) J_{\Delta-2} (s^{1/2} R^2/r) ,
\end{eqnarray}
where $s=-(p+q)^2=q^2(1/x-1-p^2/q^2)$.
We have to sum up the final-state hadrons to get the structure tensor $W^{\mu \nu}$.
In the hard-wall model, the space has a cut-off at $r_0=\Lambda R^2$,
so the zeros of the Bessel function eq.(\ref{dila}) are at $M_n=n\pi \Lambda$ and the on-shell condition of
the dilaton becomes
\begin{eqnarray} \label{zero}
\sum_{n} \delta(M^2_n-s) \sim (\partial M^2_n/\partial n)^{-1} \sim (2\pi s^{1/2} \Lambda)^{-1} .
\end{eqnarray}
Using eq.(\ref{aaa}) and eq.(\ref{zero}) and taking average for the polarization
vectors of the target photon, we get the structure tensor eq.(\ref{tens})
\begin{eqnarray}
W^{\mu \nu } (p,q)&=&\frac{|c_X|^2 |C|^2}{\pi} \Bigr[ \left({\eta}^{\mu \nu}-\frac{q^{\mu} q^{\nu}}{q^2} \right) p^2 q^6
\Bigr(\frac{A(x,q^2,p^2)}{2x}+\frac{p}{q}B(x,q^2,p^2) \Bigr)^2 \nonumber \\
&&+\left( p^{\mu}+\frac{q^{\mu}}{2x} \right) \left(p^{\nu}+\frac{q^{\nu}}{2x} \right) p^2 q^4 \Bigr( A(x,q^2,p^2)^2-B(x,q^2,p^2)^2 \Bigr) \Bigr].
\end{eqnarray}
Hence we have the following structure functions
\begin{eqnarray} \label{str}
F_1 (x,q^2,p^2) &=&\frac{|c_X|^2 |C|^2}{\pi} p^2 q^6
\Bigr(\frac{A(x,q^2,p^2)}{2x}+\frac{p}{q}B(x,q^2,p^2) \Bigr)^2, \\
F_2 (x,q^2,p^2)&=&\frac{|c_X|^2 |C|^2}{4\pi x}p^2 q^6 \Bigr( A(x,q^2,p^2)^2-B(x,q^2,p^2)^2 \Bigr) .
\end{eqnarray}
We stress that the condition $p^2 \ll q^2$ is not used in the supergravity calculation of the structure functions. This is quite different
from pQCD analysis in which the condition $p^2 \ll q^2$ is necessary for applying OPE to the product of quark currents.
We analyze $F_i (x,q^2,p^2), (i=1,2) $ in the kinematical region $p^2 \ll q^2$. $K_i(qR^2/r)$ damps exponentially in small $r $ region.
So the region which contributes to integrals in $A(x,q^2,p^2)$ and $B(x,q^2,p^2)$ is $ qR^2 \le r$.
To begin with, we evaluate $F_1 (x,q^2,p^2)$.
We show later that the dominant contribution to $F_i (x,q^2,p^2) $ comes from $A(x,q^2,p^2)$ and
we can neglect the second term in eq.(\ref{str}) for $p^2 \ll q^2$.
When $p^2 \ll q^2$ is satisfied, we can also use the leading behavior $K_1(pR^2/r) \simeq r/pR^2$.
Then, with an approximation $1/ \Lambda \sim \infty$, this integrals can be performed and we get
\begin{eqnarray}
A(x,q^2,p^2) &\simeq& \int_{0}^{\infty} dr \frac{R^3}{p r^4} K_1(qR^2/r) J_{\Delta-2} (s^{1/2} R^2/r) \nonumber \\
&=&\frac{2}{p q^3 R^3} \left(\frac{1}{x} - 1 -\frac{p^2}{q^2}\right)^{\frac{\Delta}{2}-1}
\frac{\Gamma(\frac{\Delta+2}{2})\Gamma(\frac{\Delta}{2})}{\Gamma(\Delta-1)}
\quad _{2}F_{1}(\frac{\Delta+2}{2},\frac{\Delta}{2};\Delta-1;1-\frac{1}{x} +\frac{p^2}{q^2}) \nonumber \\
&=&\frac{2}{p q^3 R^3} x^2 \Bigr[1 - (1+\frac{p^2}{q^2})x \Bigr]^{\frac{\Delta}{2}-1} \Bigr(1-\frac{p^2}{q^2}x \Bigr)^{1-\frac{\Delta}{2}}
\frac{\Gamma(\frac{\Delta+2}{2})\Gamma(\frac{\Delta}{2})}{\Gamma(\Delta-1)} \nonumber \\
&& \times {}_2F_{1}(\frac{\Delta+2}{2},\frac{\Delta-2}{2};\Delta-1;\frac{1-(1+\frac{p^2}{q^2})x}{1-\frac{p^2}{q^2}x }) \nonumber \\
\end{eqnarray}
where $ {}_2F_{1} (a,b;c;z)$ is a hypergeometric function.
From the second line to the third line we use the identity
$ {}_2F_{1} (a,b;c;z)=(1-z)^{-a} {}_2F_{1} (a,c-b;c;z/(z-1))$.
Then $F_{1}$ becomes
\begin{eqnarray} \label{f1}
F_1 (x,q^2,p^2) &\simeq&\frac{|c_X|^2 |C|^2}{4\pi x^2} p^2 q^6 A(x,q^2,p^2)^2 \nonumber \\
&\simeq&\frac{|c_X|^2 |C|^2 }{\pi R^6} \frac{\Gamma(\frac{\Delta+2}{2})^2 \Gamma(\frac{\Delta}{2})^2}{\Gamma(\Delta-1)^2}
x^2 \Bigr[1 - (1+\frac{p^2}{q^2})x \Bigr]^{\frac{\Delta}{2}-1} \Bigr(1-\frac{p^2}{q^2}x \Bigr)^{1-\frac{\Delta}{2}}
\nonumber \\
&& \times {}_2F_{1}(\frac{\Delta+2}{2},\frac{\Delta-2}{2};\Delta-1;\frac{1-(1+\frac{p^2}{q^2})x}{1-\frac{p^2}{q^2}x }). \nonumber \\
\end{eqnarray}
One can see that $F_1 (x,q^2,p^2)$ increases with $x$ in the small $x$-region and decreases near $x_{\rm{max}} =1/(1+\frac{p^2}{q^2}) $.
The structure functions should vanish at $x_{\rm{max}}$ and actually eq.(\ref{f1}) satisfies this requirement.
From eq.(\ref{f1}), we have found that $F_1(x,q^2,p^2) $ scales as $F_1(x)$ in the limit $p^2/q^2 \to 0$.
This is a crucial difference between the photon structure functions and the nucleon structure functions in gravity calculation.
If we define $\Delta_{i}$ as the conformal dimension of the dilaton or the dilatino which is dual to the initial state hadron,
the momentum dependence of the nucleon structure functions always behave $(\Lambda/q)^{2\Delta_i-2}$ \cite{PS}.
Therefore the nucleon structure functions do not have the scaling property in general $\Delta_{i}$.
The nucleon structure functions have the scaling property when the conformal dimension is special value $\Delta_{i}=1$ \cite{PRSW}.
Next, we evaluate $F_{2}$.
We transform $w=qR^2/r$ in eq.(\ref{AB}), $B(x,q^2,p^2) $ can be written as
\begin{eqnarray}
\frac{1}{R^3 q^4}\int_{0}^{\infty} dw w^3 K_0(wp/q) K_0(w) J_{\Delta-2} (ws^{1/2}/q).
\end{eqnarray}
When the condition $p^2 \ll q^2 $ is satisfied, we can use the asymptotic form $K_0(wp/q) \simeq \log (2q/wp)$.
Then
\begin{eqnarray}
B(x,q^2,p^2)&=&\frac{1}{R^3 q^4} \lim_{\epsilon \to 0}\int_{0}^{\infty}
dw w^{3} \frac{(2q/wp)^{\epsilon}-1}{\epsilon} K_0(w) J_{\Delta-2} (ws^{1/2}/q). \nonumber \\
&\simeq& \frac{2^2\log(q/p)}{R^3 q^4} \frac{\Gamma(\frac{\Delta+2}{2})^2}{ \Gamma(\Delta-1)}x^2(1-x)^{\frac{\Delta}{2}-1} \nonumber \\
&&\times {}_2F_{1}(\frac{\Delta+2}{2},\frac{\Delta}{2}-2;\Delta-1;1-x) +O(q^{-4}),
\end{eqnarray}
where we used $\log x= \lim_{\epsilon \to 0}(x^{\epsilon}-1)/\epsilon$.
Therefore $B(x,q^2,p^2)^2$ is a subleading contribution to $F_{2}$.
From the expression for $F_2$, we calculate the longitudinal structure function defined as $F_{L}= F_{2}-xF_{1}$.
\begin{eqnarray}
F_L (x,q^2,p^2)&\simeq&\frac{|c_X|^2 |C|^2}{4\pi x}p^2 q^6 B(x,q^2,p^2)^2 \nonumber \\
&\simeq&\frac{2^2|c_X|^2 |C|^2}{\pi } \frac{\Gamma(\frac{\Delta+2}{2})^4}{ \Gamma(\Delta-1)^2} \frac{p^2}{q^2} \log^2(q/p) \nonumber \\
&& \times x^3(1-x)^{\Delta-2} {}_2F_{1}(\frac{\Delta+2}{2},\frac{\Delta}{2}-2;\Delta-1;1-x)^2 .
\end{eqnarray}
From the above equation, in the limit $p^2 /q^2 \to 0$,
one can see that the Callan-Gross relation $F_{L}=0$ holds in gravity calculation.
In QCD, the Callan-Gross relation implies the partons involved in the scattering process are spin 1/2 particles.
This relation is weakly broken by higher order corrections in real QCD.
In our calculation, the Callan-Gross relation is weakly broken by the effect of gauge field in radial direction.
\section{Summary and discussions}
In this article, we have calculated the virtual photon structure functions in the hard-wall model.
The structure functions $F_i (x,q^2,p^2)$ $(i=1,2)$ we obtained possess the following properties which we have in QCD.
1) $F_i$'s increase at small $x$-region, decrease at large $x$-region and vanish at $x=x_{\rm{max}}$.
2) $F_{L}$ is small compared to $F_{i}$'s. It goes to zero when the limit $p^2 \ll q^2$ as $q^2 \to \infty $.
It seems peculiar that the Callan-Gross relation holds in gravity calculation, although there exists no spin 1/2 particles in our case.
But as mentioned in \cite{gomez}, the interaction $\Phi F^2$ in eq.(\ref{int}) is
the low-energy effective action describing a higgs decay into two gammas through a heavy quark triangle
loop \cite{shifman}. Since the heavy quark is a spin 1/2 particle , this might be a reason why the Callan-Gross relation holds in our calculation.
We simply comment some extension of the virtual photon structure functions at strong coupling.
In the hard-wall model, mass square of dilatons and radial numbers exhibit quadratic relation, but actually relations mass square
between radial number is linear (Regge trajectory). A background which reproduces Regge trajectory is
proposed in \cite{karch}. This is called the soft-wall model. Deep inelastic scattering in the soft-wall model
is studied and the leading order results in soft wall model is same as the one in the hard-wall model \cite{BBNB1}.
We treat the virtual photon structure functions in not so small $x$-region.
At small $x$-region, we have to take into account excited string states\cite{PS}, \cite{BPST}.
Then, we cannot use the insertion of dilaton states and have to treat four currents correlator directly.
The analysis would become much more complicated.
It is interesting to investigate the virtual photon structure functions in such situations.
\section*{Acknowledgments}
We are grateful to Y.Kitadono, T.Matsuo, K.Sugiyama and T.Uematsu for useful discussions.
We also thanks T.Uematsu for careful reading of the manuscript and helpful comments.
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train/arxiv
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BkiUcv45qoTAk-0Y1neC
| 5
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\section{Introduction}
\label{sec1}
There has been a tremendous activity in the field
of pyrochlore ice materials~\cite{PhysRevLett.98.157204,Gingras2014,BalentsSavary,PhysRevLett.105.047201,Savary12,Sungbin2012,SavaryPRB,PhysRevLett.87.067203,PhysRevB.65.054410,PhysRevLett.87.047205,Gingras2014,Ross2009,Huang2014,PhysRevB.94.205107,PhysRevLett.108.247210,PhysRevB.95.041106,PhysRevB.95.094422,Chen2015,SavaryPRB,PhysRevLett.109.017201,Yasui2002,PhysRevB.64.224416,PhysRevB.90.214430,Chang2012,Kimura2012,RevModPhys.82.53,Lhotel2014,Chang2014,Yasui2003,Ross11,Shannon12,Goswami2016,PhysRevB.95.094407,PhysRevLett.118.107206,PhysRevB.92.054432,PhysRevLett.113.197202,fu2017fingerprints,PhysRevB.86.075154,PhysRevLett.115.267208,PhysRevLett.109.097205,PhysRevLett.107.207207,PhysRevLett.115.097202,Shannon2017,GangChen2017,PhysRevLett.118.087203}.
Besides the early efforts in classical spin ice and dipolar spin ice where
quantum effects are negligible~\cite{Gingras2001,PhysRevLett.87.067203,Castelnovo12008},
a recent motivation of this exciting area is to
search for the three-dimensional U(1) quantum spin liquid (QSL)~\cite{Hermele04}
in the pyrochlore quantum spin ice systems where quantum effects
are significant~\cite{PhysRevLett.98.157204,Savary12,Sungbin2012,PhysRevLett.105.047201}.
The existence of this exotic quantum phase of
matter has been firmly established by the theoretical studies of the relevant
and even realistic spin models on the pyrochlore lattice~\cite{Hermele04,Savary12,Shannon12,Sungbin2012,Huang2014,Gingras2014,BalentsSavary,
PhysRevLett.100.047208,PhysRevLett.115.077202,PhysRevLett.115.037202}.
The experimental confirmation of this interesting
phase of matter, however, is still open. Even if this phase may have already
existed in some candidate materials since the original proposal
in Tb$_2$Ti$_2$O$_7$~\cite{PhysRevLett.98.157204} and Yb$_2$Ti$_2$O$_7$~\cite{Ross11,Savary12},
the firm identification of this exotic phase requires the strong mutual feedback between
the experimental progress and the theoretical development
that provides and clarifies unique and clear physical observables for the
experiments.
The pyrochlore spin ice U(1) QSL is described by the emergent compact
U(1) lattice gauge theory with deconfined and fractionalized
excitations~\cite{Hermele04,Savary12}. There are three elementary excitations,
namely, spinon, ``magnetic monopole'', and gauge photon in this U(1) QSL.
Here the nomenclature for the excitations follows from the original work
by Hermele, Fisher and Balents~\cite{Hermele04}
(see Table.~\ref{tab1}). To confirm the realization of the U(1) QSL,
one would need at least observe one such emergent and exotic excitation.
Inelastic neutron scattering, that is a spectroscopic measurement,
is likely to provide much richer information than any other experimental
probes for the pyrochlore spin ice systems~\cite{Ross11}. It is thus of
great importance to understand how the neutron moments are coupled to
the microscopic degrees of freedom and how the inelastic neutron
scattering (INS) results are related to the emergent and exotic
properties of the pyrochlore ice U(1) QSL. It is this purpose that
motivates our work in this paper.
\begin{table}[b]
\begin{tabular}{p{4cm} p{4cm}}
\hline\hline
Excitations (notation 1) & Excitations (notation 2)
\\
Spinon & Magnetic monopole
\\
``Magnetic monopole'' & Electric monopole
\\
Gauge photon & Gauge photon \\
\hline \hline
\end{tabular}
\caption{Two different but equivalent notations for the excitations
in the pyrochlore ice U(1) QSL. The notation 1 was introduced
in Ref.~\onlinecite{Hermele04} and is adopted in this paper.
The notation 2 can be found in Ref.~\onlinecite{PhysRevX.6.011034},
and the magnetic monopole in this notation has a classical analogue
that is a defect tetrahedron with either ``3-in 1-out'' or ``1-in 3-out''
spin configurations~\cite{Castelnovo12008}.}
\label{tab1}
\end{table}
We mainly deal with the non-Kramers doublets in most parts of this paper.
The non-Kramers doublets on the pyrochlore system have been discussed
by several previous works. In particular, the generic spin model
was introduced and studied in Refs.~\onlinecite{PhysRevLett.105.047201,PhysRevB.83.094411,Sungbin2012},
and more recently, the random strain effect was discussed for Pr$^{3+}$ ions
in Pr$_2$Zr$_2$O$_7$ in Refs.~\onlinecite{PhysRevLett.118.087203,PhysRevLett.118.107206}.
In Ref.~\onlinecite{PhysRevB.94.205107}, we have pointed out the magnetic transition
out of U(1) QSL should be a confinement transition by a simple symmetry analysis.
For a non-Kramers doublet~\cite{PhysRevLett.105.047201,PhysRevB.83.094411}
that is described by a pseudospin-1/2 operator $\boldsymbol{S}$,
the time reversal symmetry, $\mathcal{T}$, acts rather
peculiarly such that~\cite{Sungbin2012,PhysRevB.94.205107},
\begin{eqnarray}
\mathcal{T}: && \quad\quad S^{x,y} \rightarrow S^{x,y},
\quad
S^z \rightarrow - S^z.
\end{eqnarray}
This property means the neutron moments would merely pick up the $S^z$
component and naturally measure the $S^z$ correlation. By examining the
connection with the emergent variables such as gauge fields and
matter fields, we point out that, the $S^z$ correlation should detect
the gauge photons as well as the ``magnetic monopoles''.
The ``magnetic monopole'' is the topological defect of the emergent
vector gauge potential in the compact U(1) quantum electrodynamics
and has no classical analogue. Even though the spinon and the
``magnetic monopole'' can be interchanged by the electromagnetic
duality of the lattice gauge theory, the ``magnetic monopole'' might
be more close in spirit to the {\sl Dirac's magnetic monopole}~\cite{Dirac}
from the original definition and theory of the pyrochlore U(1) QSL~\cite{Hermele04}.
The existence of the ``magnetic monopole'' is one of the key properties
of the compact U(1) lattice gauge theory~\cite{Fradkinbook} and
the pyrochlore ice U(1) QSL~\cite{Hermele04}, and it is of great
importance to demonstrate that the ``magnetic monopole'' is
a {\sl real physical entity} rather than any artificial or
fictitious excitation.
So far, there were only limited studies of ``monopole'' physics in the U(1)
QSL of the pyrochlore ice context~\cite{Hermele04,PhysRevB.94.205107,PhysRevB.95.134439}.
We here realize that the ``magnetic monopole'' could manifest itself as
the ``monopole'' continuum in the INS result on the non-Kramers doublet
pyrochlore spin ice systems. Our renewed understanding of the INS
measurement for non-Kramers doublets is further extended to the
Kramers doublets and the quantum Monte carlo
simulation, and henceforth provides a new insight for the experimental
observation and the numerical simulation. Moreover, the ``magnetic monopole''
experiences a background $\pi$ flux as the ``magnetic monopole'' hops
around the perimeter on the elementary plaquette of the dual diamond lattice.
We then point out that the background $\pi$ flux immediately modulates
the spectral structure of the ``monopole'' continuum by enhancing the
spectral periodicity. This is an unique experimental signature for the
``monopole'' continuum in the INS measurement. More generally, this
is an example of translation symmetry fractionalization in topologically
ordered phases~\cite{PhysRevB.90.121102,Hermele04,WenPSG}.
Combining with the prior work on the translation symmetry
fractionalization of the spinons~\cite{GangChen2017},
we establish a general classification for the pyrochlore ice U(1)
QSLs based on the translation symmetry and list their relevant
spectral properties.
The following part of the paper is organized as follows. In Sec.~\ref{sec2},
we introduce the microscopic model for the non-Kramers doublets, and
explain the application of several effective models. In Sec.~\ref{sec3},
we point out the presence of the ``monopole'' dynamics
in the spin correlation function from the INS measurements.
In Sec.~\ref{sec4}, we establish the spectral structure of the
``monopole'' continuum. In Sec.~\ref{sec5}, we carry out the ``monopole''
mean field theory and explicitly compute the ``monopole'' dynamics.
Finally in Sec.~\ref{sec6}, we give a {\sl broad} discussion about the
spectral properties of non-Kramers doublet and Kramers doublet spin ice materials
and present a classification of the U(1) QSLs based on the translation symmetry
fractionalization patterns of the ``magnetic monopoles'' and the spinons.
\section{Model for non-Kramers doublets and the low-energy field theory}
\label{sec2}
Due to the peculiar property of the non-Kramers doublets under the time reversal
symmetry, the generic spin model, that describes the interaction between these
doublets on the pyrochlore lattice, is actually simpler than the usual Kramers
doublets and is given by~\cite{PhysRevLett.105.047201,PhysRevB.83.094411,Sungbin2012}
\begin{eqnarray}
H &=& \sum_{\langle ij \rangle}
{J_{zz}^{} S^z_i S^z_j
- J_{\pm}^{} (S^+_i S^-_j + h.c.)}
\nonumber \\
&+& {J_{\pm\pm}^{} (\gamma_{ij}^{} S^+_i S^+_j + h.c.)}
+ {\text{dipolar interaction}},
\end{eqnarray}
where ${S^{\pm}_i \equiv S^{x}_i \pm i S^y_i}$ and $\gamma_{ij}$ is the
bond-dependent phase variable that arises from the spin-orbit-entangled
nature of the non-Kramers doublet. The dipolar interaction includes the
further neighbor interactions between the $S^z$ components since only
$S^z$ is time reversally odd and contributes to the dipole moment.
It has been shown in Ref.~\onlinecite{Sungbin2012} that, in the perturbative
Ising limit with ${|{J_{\pm}}| \ll J_{zz}}$ and ${|J_{\pm\pm}| \ll J_{zz}}$,
the system realizes the U(1) QSL. Moreover, it was demonstrated that
the U(1) QSL is more robust on the frustrated side~\cite{Sungbin2012}
with ${J_{\pm} < 0}$ and along the axis of $J_{\pm\pm}$.
Throughout the paper, we deliver our theory through the non-Kramers doublet system.
Only in the Sec.~\ref{sec6}, we extend our theory to the Kramers doublet system.
\subsection{Effective theories}
Our purpose is not to understand the energetics of the relevant
microscopic spin model. We assume that the U(1) QSL has been realized
in the system and try to understand its manifestation in the physical
observables. For the U(1) QSL, we can then start from the
ring exchange model that is obtained from the perturbative treatment
of the $J_{\pm}$ and $J_{\pm\pm}$ interactions in the Ising limit.
With the mapping ${S^z_i = E_{{\boldsymbol r}{\boldsymbol r}'}
+ \frac{1}{2}}, S^{\pm}_i = e^{\pm i A_{{\boldsymbol r} {\boldsymbol r}'}}$
and ${[E_{{\boldsymbol r}{\boldsymbol r}'}, A_{{\boldsymbol r} {\boldsymbol r}'}]=i}$,
one obtains the U(1) lattice gauge theory on the diamond lattice formed by the
tetrahedral centers of the pyrochlore lattice~\cite{Hermele04,Savary12}.
In this lattice gauge theory,
the spinon excitations that violate the ice rule have been traced out
in the perturbative treatment, and thus, the effective model
captures the physics below the spinon gap. The lattice gauge theory
Hamiltonian is given as~\cite{Hermele04}
\begin{eqnarray}
H_{\text{LGT}} = - K \sum_{\hexagon} \cos (curl A)
+ \sum_{\langle {\boldsymbol r}{\boldsymbol r}' \rangle }
\frac{U}{2} (E_{{\boldsymbol r}{\boldsymbol r}'}
- \frac{\eta_{\boldsymbol r}}{2} )^2,
\end{eqnarray}
where ``${\boldsymbol r}, {\boldsymbol r}'$'' stand for the diamond lattice
sites, ${\eta_{\boldsymbol r} = \pm 1}$ for the two sublattices of the
diamond lattice, and
$E_{{\boldsymbol r}{\boldsymbol r}'} = -E_{{\boldsymbol r}'{\boldsymbol r}},
{A_{{\boldsymbol r}{\boldsymbol r}'} = -A_{{\boldsymbol r}'{\boldsymbol r}}}$.
Here, $curl A$ is defined as
\begin{eqnarray}
curl A \equiv{ \sum_{{\boldsymbol r} {\boldsymbol r}' \in
\hexagon} \!\!\!\!\!\!\!\!\!\!\!\! \circlearrowleft} \quad A_{{\boldsymbol r} {\boldsymbol r}'},
\end{eqnarray}
and thus corresponds to the magnetic field $B$ through the hexagon center.
The magnetic coupling $K$ is of the order of the ring exchange coupling
in the perturbation theory, and the electric field term is introduced to
enforce the spin-1/2 Hilbert space. If one focuses on the low-energy and
long-distance physics, one can further coarsen grain and obtain the
continuous Maxwell field theory with~\cite{Hermele04}
\begin{eqnarray}
H_{\text{Maxwell}} \simeq \frac{\mathcal{K}}{2} B^2 + \frac{\mathcal{U}}{2} E^2,
\end{eqnarray}
where $\mathcal{K}$ and $\mathcal{U}$ are coarse-grained magnetic and electric couplings.
\begin{figure*}[t]
\centering
\includegraphics[width=0.9\textwidth]{fig1.pdf}
\caption{(Color online.)
(a) The diamond lattice (in thin line) and its dual diamond lattice (in thick line).
The physical spin is located in the mid of the link on the diamond lattice.
The spinons (``monopoles'') hop on the diamond (dual diamond) lattice.
The colored dots correspond to the tetrahedral centers of the pyrochlore lattice.
(b) Every buckled hexagon on the dual diamond lattice traps a ``$\pi$''
background dual U(1) flux that is experienced by the ``monopole'' hopping.
``I'' and ``II'' refer to the two sublattices of the dual diamond lattice.
In (c) and (d), the background flux trapped in the (dashed) parallelogram
is identical to the flux in the (colored) buckled hexagon.
}
\label{fig1}
\end{figure*}
\subsection{Photon in low-energy theory}
Based on the mapping from the microscopic spin degrees of
freedom to the emergent field variables in the lattice gauge theory,
one could establish the connection between the spin correlation function
with the emergent degrees of freedom. For the non-Kramers doublet,
the INS measurement would merely pick up the $S^z$ correlator and
thus measure the correlation function of the emergent electric field.
It was then shown, within the low-energy Maxwell field theory,
that the spin correlation corresponds to the electric field
correlator~\cite{Hermele04,Savary12,PhysRevB.86.075154},
\begin{eqnarray}
\langle {E}^{\mu}_{-{\boldsymbol q},-\omega} E^{\nu}_{{\boldsymbol q},\omega} \rangle
\sim
[\delta_{\mu\nu}- \frac{q^{\mu}q^{\nu}}{q^2}] \omega \delta (\omega - v|{\boldsymbol q}|),
\label{eqmax}
\end{eqnarray}
where $v$ is the speed of the photon mode. Apart from the angular dependence,
the spectral weight of the photon mode is suppressed~\cite{Savary12,PhysRevB.86.075154}
as the energy transfer $\omega \rightarrow 0$.
\section{The loop current of ``magnetic monopoles''}
\label{sec3}
The well-known result of the photon modes in the INS measurement
was obtained by considering the low-energy field theory that describes
the long-distance quantum fluctuation within the spin ice manifold.
The actual spin dynamics, that is captured by the $S^z$
correlation in the INS measurement, operates in a broad energy scale
up to the exchange energy and certainly contains more information
than just the photon mode from the low-energy Maxwell field
theory. What is the other information hidden behind?
To address this question, we have to leave the low-energy Maxwell
field theory and include the gapped matters into our consideration.
The gapped matters are spinons and ``magnetic monopoles''.
The spinons are sources and sinks of the emergent $E$ field
and live on the diamond lattice sites or the tetrahedral centers.
These spinon are excitations out of the spin ice manifold
and are created by the $S^{x}$ or $S^y$ operator. For the
non-Kramers' doublet systems, the neutron scattering does not
allow such spin-flipping processes. So we turn to the
``magnetic monopoles''. The ``magnetic monopole'' is the
source or the sink of the emergent $B$ field and is
the excitation within the spin ice manifold.
Since the ``magnetic monopole'' is located on the dual
diamond lattice site (see Fig.~\ref{fig1}), to make the ``magnetic monopole''
explicit, one needs to do a duality transformation on the
lattice gauge Hamiltonian $H_{\text{LGT}}$~\cite{Hermele04,Bergman2006,PhysRevB.94.205107}.
This standard procedure~\cite{Hermele04,Bergman2006,PhysRevB.94.205107}
yields the following dual theory
\begin{eqnarray}
H_{\text{dual}} & = & -t \sum_{\langle \boldsymbol{\mathsf R} \boldsymbol{\mathsf R}' \rangle}
e^{-i 2\pi \alpha_{\boldsymbol{\mathsf R}\boldsymbol{\mathsf R}'}}
\Phi^{\dagger}_{\boldsymbol{\mathsf R}} \Phi^{\phantom\dagger }_{\boldsymbol{\mathsf R}'}
-\mu\sum_{\boldsymbol{\mathsf R}}
\Phi^{\dagger}_{\boldsymbol{\mathsf R}} \Phi^{\phantom\dagger}_{\boldsymbol{\mathsf R}}
\nonumber \\
&& + \frac{U}{2} \sum_{\hexagon^{\ast}} (curl \alpha - \frac{\eta_{\boldsymbol r}}{2})^2
- K \sum_{\langle \boldsymbol{\mathsf R}\boldsymbol{\mathsf R}'\rangle }
\cos B_{\boldsymbol{\mathsf R}\boldsymbol{\mathsf R}'}^{}
\nonumber \\
&& + \cdots,
\end{eqnarray}
where ${\Phi}^{\dagger}_{\boldsymbol{\mathsf R}}$
(${\Phi}^{}_{\boldsymbol{\mathsf R}}$) creates (annihilates)
the ``magnetic monopole'' at the dual diamond lattice site
${\boldsymbol{\mathsf R}}$, ``$\hexagon^{\ast}$'' is the hexagon
on the dual diamond lattice, ``$t$'' is the ``monopole'' hopping,
and ``$\cdots$'' refers to the ``monopole''
interaction. Here $\alpha$ is the dual U(1) gauge field that lives
on the links of the dual diamond lattice, and $curl \, \alpha$ is
defined as
\begin{eqnarray}
curl \alpha \equiv { \sum_{{\boldsymbol {\mathsf R}} {\boldsymbol {\mathsf R}}' \in
\hexagon^{\ast}} \!\! \!\!\! \!\! \!\! \!\! \!\! \circlearrowleft}
\quad
\alpha_{{\boldsymbol {\mathsf R}} {\boldsymbol{\mathsf R}}'}^{}
\end{eqnarray}
and is simply the electric field going through the center of the hexagon plaquette
on the dual diamond lattice. This dual model describes the coupling between the
``magnetic monopoles'' and the fluctuating dual U(1) gauge fields,
and is the starting point to explore the dynamics of the ``magnetic monopoles''.
For our purpose to capture the generic {\sl spectral structure} of
the ``monopole'' dynamics, we here keep only the nearest-neighbor
``monopole'' hopping.
Since the neutron picks up the $S^z$ component for non-Kramers doublets,
we want to find what kind of ``monopole'' operators in the dual theory
correspond to the $S^z$ component. Since this is a gauge theory, only
gauge invariant quantity is physical according to Elitzur's theorem~\cite{PhysRevD.12.3978}.
It has been shown from the Maxwell's equations in the early
studies of critical theories for the ``magnetic monopole''
condensation transition~\cite{Bergman2006,PhysRevB.94.205107,PhysRevB.71.125102},
that the ``magnetic monopole'' current on a closed hexagon
loop of the dual diamond lattice induces the electric field
through the center of the loop (see Fig.~\ref{fig1}), {\sl i.e.}
\begin{eqnarray}
{\sum_{{{\boldsymbol{\mathsf R}}{\boldsymbol{\mathsf R}}' } \in \hexagon^{\ast}} \!\!\!\!
\!\!\!\!\!\!\!\!\!
\circlearrowleft}
\, \,\, {J}_{{\boldsymbol{\mathsf R}}{\boldsymbol{\mathsf R}}' } = E \sim S^z ,
\end{eqnarray}
where ${J}_{{\boldsymbol{\mathsf R}}{\boldsymbol{\mathsf R}}'}$
is the ``monopole'' current between the nearest neighbors with
\begin{eqnarray}
{J}_{{\boldsymbol{\mathsf R}}{\boldsymbol{\mathsf R}}' }^{}
\equiv i [ {\Phi}^{\dagger}_{\boldsymbol{\mathsf R}}
{\Phi}^{\phantom\dagger}_{\boldsymbol{\mathsf R}'}
e^{- i 2\pi \alpha_{{\boldsymbol{\mathsf R}}{\boldsymbol{\mathsf R}}' }}
- h.c. ].
\end{eqnarray}
How do we understand the above relation? First, we emphasize that
this relation is applicable beyond the early studies of identifying
the proximate {\sl static} $S^z$ Ising order through the ``monopole''
condensation, and holds even for the {\sl dynamical property
inside the U(1) QSL phase}. Second, there is no contradiction between
this relation with Eq.~(\ref{eqmax}) that is a coarse-grained low-energy
and long-distance result. This relation here includes the short distance
and finite energy dynamics of the ``magnetic monopoles''.
From this relation, we conclude that the $S^z$ correlation
contains the contribution of the ``monopole'' current correlator.
The above analysis does not provide the information about the spectral
weight of the ``monopole'' continuum in the $S^z$ correlation.
It was pointed out that increasing further neighbor $S^z$-$S^z$
interaction could drive a quantum phase transition from the U(1) QSL
to the Ising order via the ``monopole'' condensation~\cite{PhysRevB.94.205107}.
We thus think that the systems with extended $S^z$ coupling may have
more visible ``monopole'' continuum in the INS result.
\section{The spectral structure of the ``monopole'' continuum}
\label{sec4}
We realize that the physical spin operator, $S^z$, creates one
``monopole''-``anti-monopole'' pair. The dynamic spin structure
factor of the non-Kramers doublet would contain a broad ``monopole''
continuum due to this ``fractionalization'' of the spin into the
two ``monopoles''. Here we are interested in the generic and unique
spectral structure rather than some specific details that can be
used to uniquely identify the ``monopole'' continuum in the INS
results.
The ``magnetic monopole'' hops on the dual diamond lattice and
experiences the dual U(1) gauge flux. The background gauge flux
thus {\sl modulates} the ``monopole'' dynamics. Due to the electric
field offset, ${\eta_{\boldsymbol r}/2}$, that originates fundamentally
from the effective spin-1/2 nature of the local moment, there
exists a background gauge flux on each hexagon plaquette
of the dual diamond lattice with~\cite{PhysRevB.94.205107}
\begin{eqnarray}
2\pi \langle curl \, \alpha \rangle = {\pi \eta_{\boldsymbol r} \equiv \pi}
\quad (\text{mod}\, 2\pi).
\end{eqnarray}
To see the effect of the background dual gauge flux, we introduce
the translation operator for the ``magnetic monopole'', $T^{m}_{\mu}$,
that translates the ``monopole'' by a basis lattice vector
${\boldsymbol {\mathsf a}}_{\mu}$ of the dual diamond lattice,
where ${\mu=1,2,3}$, and
${{\boldsymbol {\mathsf a}}_{1}= \frac{1}{2}(011)},
{{\boldsymbol{\mathsf a}}_{2}=\frac{1}{2}(101)},
{{\boldsymbol{\mathsf a}}_{3}=\frac{1}{2}(110)}$.
We use the cubic coordinate system and set the lattice
constant to unity throughout the paper.
As the ``magnetic monopole'' hops successively
through the parallelogram defined by
$T^m_{\mu} T^m_{\nu} (T^m_{\mu})^{-1} (T^m_{\nu})^{-1}$ with
${\mu \neq \nu}$, the ``monopole'' experiences an identical
Aharonov-Bohm flux as the background flux trapped in the hexagon
plaquette of the dual diamond lattice (see Fig.~\ref{fig1}).
This is because of the lattice geometry of the diamond lattice.
Thus, we have the following algebraic relation
\begin{eqnarray}
T^m_{\mu} T^m_{\nu} (T^m_{\mu})^{-1} (T^m_{\nu})^{-1} = e^{i \pi} = -1.
\label{anticomm}
\end{eqnarray}
This algebraic relation means the lattice translation symmetry is realized
{\sl projectively} for the ``magnetic monopoles''.
The translation symmetry fractionalization for the ``magnetic monopole''
is intimately connected to the spectral periodicity of the ``monopole continuum''
~\cite{PhysRevB.90.121102,WenPSG,Wen2002175}.
To demonstrate the enhanced spectral periodicity of the ``monopole''
continuum, we introduce a 2-``monopole'' scattering state
${|\text{A} \rangle \equiv |{\boldsymbol {\mathsf q}}_{\text A};
{\mathsf z}_{\text A} \rangle}$, where ${\boldsymbol {\mathsf q}}_{\text A}$
is the total crystal momentum of this state and ${\mathsf z}_{\text A}$
represents the remaining quantum number that specifies the state~\cite{PhysRevB.90.121102}.
The translation symmetry fractionalization acts on the individual ``monopole'',
such that
\begin{eqnarray}
T_{\mu} |\text{A} \rangle \equiv T_{\mu}^m (1) T_{\mu}^m (2) |\text{A} \rangle ,
\end{eqnarray}
where $T_{\mu}$ is the translation operator for the system, and
``1'' and ``2'' refer to the two ``monopoles'' of this state. By translating
one ``monopole'' by the basis lattice vector ${\boldsymbol{\mathsf a}}_{\mu}$,
we obtain another three 2-``monopole'' scattering states,
\begin{eqnarray}
|\text{B} \rangle & = & T^m_1 (1) |\text{A} \rangle,\\
|\text{C} \rangle & = & T^m_2 (1) |\text{A} \rangle,\\
|\text{D} \rangle & = & T^m_3 (1) |\text{A} \rangle .
\end{eqnarray}
It is ready to compare the translation eigenvalues of these
four states by making use of Eq.~(\ref{anticomm}) and obtain the following
relations for the crystal momentum of these states,
\begin{eqnarray}
{\boldsymbol {\mathsf q}}_{\text B} & = & {\boldsymbol {\mathsf q}}_{\text A} + 2\pi (100) , \\
{\boldsymbol {\mathsf q}}_{\text C} & = & {\boldsymbol {\mathsf q}}_{\text A} + 2\pi (010) , \\
{\boldsymbol {\mathsf q}}_{\text D} & = & {\boldsymbol {\mathsf q}}_{\text A} + 2\pi (001).
\end{eqnarray}
Since these scattering states have the same energy, we thus conclude that
the ``monopole continuum'' of the two ``monopole'' excitations
have the following enlarged spectral periodicity such that
\begin{eqnarray}
{\mathsf L}_{m} ({\boldsymbol {\mathsf q}} ) & = &
{\mathsf L}_m ({\boldsymbol {\mathsf q}} + 2\pi (100) )
\nonumber \\
& = & {\mathsf L}_m ({\boldsymbol {\mathsf q}} + 2\pi (010) )
\nonumber \\
& = & {\mathsf L}_m ({\boldsymbol {\mathsf q}} + 2\pi (001) ) ,
\end{eqnarray}
where ${\mathsf L}_{m} ({\boldsymbol {\mathsf q}} )$ is the lower
excitation edge
of the ``monopole'' continuum for a given momentum ${\boldsymbol {\mathsf q}}$
because there is a finite energy cost to excite two ``monopoles''.
This enhanced spectral periodicity also appears in the upper excitation
edges of the ``monopole'' continuum.
There is no symmetry breaking nor any static magnetic order in the system,
but the spectral periodicity is enhanced. The spectrum is
invariant if one translates the spectrum by $2\pi (100)$, $2\pi (010)$,
or $2\pi (001)$. This is very different from the conventional case
where the spectral periodicity is given by
the reciprocal lattice vectors, $2\pi (\bar{1}11)$,
$2\pi (1\bar{1}1)$ and $2\pi (11\bar{1})$, for the FCC bravais lattice.
Therefore, the spectral periodicity enhancement
with a fold Brillouin zone is a strong indication of
the fractionalization in the system.
\section{The ``monopole'' mean-field theory and the continuum}
\label{sec5}
To explicitly compute the ``monopole'' dynamics and demonstrate the
spectral periodicity enhancement, we carry out the mean-field
approximation for the ``monopole''-gauge coupling. To capture
the $\pi$ background flux, we set the dual gauge potential as~\cite{Sungbin2012,PhysRevB.94.205107}
\begin{eqnarray}
2\pi \langle \alpha^{}_{{\boldsymbol{\mathsf R}},{\boldsymbol{\mathsf R}}
+ {\boldsymbol{\mathsf e}}_{\mu}} \rangle
= {\mathsf{\xi}}_{\mu} ({\boldsymbol{\mathsf Q}} \cdot {\boldsymbol{\mathsf R}}),
\end{eqnarray}
where ${\boldsymbol {\mathsf R}} \in$ I sublattice of the dual diamond lattice, and
${{\boldsymbol{\mathsf R}} + {\boldsymbol{\mathsf e}}_{\mu} }\in$ II sublattice of
the dual diamond lattice with ${\boldsymbol{\mathsf e}}_{\mu}$ (${\mu=0,1,2,3}$)
the nearest-neighbor vectors connecting two sublattices.
Here $\boldsymbol{\mathsf{e}}_{\mathsf 0} = \frac{1}{4}(111),
\boldsymbol{\mathsf e}_{\mathsf 1} = \frac{1}{4}(1\bar{1}\bar{1}),
\boldsymbol{\mathsf e}_{\mathsf 2} = \frac{1}{4}(\bar{1}1\bar{1}),
\boldsymbol{\mathsf e}_{\mathsf 3} = \frac{1}{4}(\bar{1}\bar{1}1)$,
${(\xi_{\mathsf 0},\xi_{\mathsf 1},\xi_{\mathsf 2},\xi_{\mathsf 3} )
=(0,1,1,0)}$ and ${\boldsymbol{\mathsf Q}} = 2\pi(100)$.
Under this above gauge fixing, we have the ``monopole'' mean-field Hamiltonian,
\begin{eqnarray}
{H_{\text{MFT}} =
-t \sum_{\langle \boldsymbol{\mathsf R} \boldsymbol{\mathsf R}' \rangle}
e^{-i 2\pi \langle \alpha_{\boldsymbol{\mathsf R}\boldsymbol{\mathsf R}'} \rangle }
\Phi^{\dagger}_{\boldsymbol{\mathsf R}} \Phi^{\phantom\dagger }_{\boldsymbol{\mathsf R}'}
-\mu\sum_{\boldsymbol{\mathsf R}}
\Phi^{\dagger}_{\boldsymbol{\mathsf R}} \Phi^{\phantom\dagger}_{\boldsymbol{\mathsf R}} } ,
\end{eqnarray}
where the ``monopole'' spectrum is found to be
\begin{eqnarray}
\Omega^+_{\pm} ({\boldsymbol {\mathsf q}}) &=& +t [4\pm 2 (3+
{\mathsf C}_{\mathsf x} {\mathsf C}_{\mathsf y} -
{\mathsf C}_{\mathsf x} {\mathsf C}_{\mathsf z} +
{\mathsf C}_{\mathsf y} {\mathsf C}_{\mathsf z})^{\frac{1}{2}}]^{\frac{1}{2}}- \mu ,
\nonumber
\\
\Omega^-_{\pm} ({\boldsymbol {\mathsf q}}) &=& -t [4\pm 2 (3+
{\mathsf C}_{\mathsf x} {\mathsf C}_{\mathsf y} -
{\mathsf C}_{\mathsf x} {\mathsf C}_{\mathsf z} +
{\mathsf C}_{\mathsf y} {\mathsf C}_{\mathsf z})^{\frac{1}{2}}]^{\frac{1}{2}}- \mu ,
\nonumber
\end{eqnarray}
where ${{\mathsf C}_{\mu} = \cos {\mathsf q}_{\mu} }$ (${\mu = {\mathsf{x,y,z}}}$).
There are four ``monopole'' bands: two arise from the two sublattices of
the dual diamond lattice, and two arise from the gauge fixing that doubles
the unit cell.
As we point out in Sec.~\ref{sec4}, the ``monopole'' continuum
is contained in the ``monopole'' current correlation.
Here we are interested in
the spectral structure of the upper and lower excitation edges
of the ``monopole'' continuum. From the momentum and the energy
conservation, we have for the two ``monopoles''
\begin{eqnarray}
{\boldsymbol{\mathsf q}} & = & {\boldsymbol{\mathsf q}}_{\mathsf 1} +
{\boldsymbol{\mathsf q}}_{\mathsf 2} + {\boldsymbol{\mathsf Q}}, \\
{\mathsf E} & = & \Omega^{i_1}_{j_1} ({\boldsymbol{\mathsf q}}_{\mathsf 1}) +
\Omega^{i_2}_{j_2} ({\boldsymbol{\mathsf q}}_{\mathsf 2}),
\end{eqnarray}
where ${\boldsymbol{\mathsf q}}$ and ${\mathsf E}$ are the momentum and energy
transfer of the neutrons, ${\boldsymbol{\mathsf q}}_{\mathsf 1}$ and
${\boldsymbol{\mathsf q}}_{\mathsf 2}$ are the crystal momenta of the
two ``monopoles'', and the offset ${\boldsymbol{\mathsf Q}}$ arises
from the dual gauge link that is present in the ``monopole'' current.
The minimum (maximum) of the energy ${\mathsf E}$
is obtained when ${i_1 = i_2 = -}$ and ${j_1=j_2=+}$
(${i_1 = i_2 = +}$ and ${j_1=j_2=+}$).
In Fig.~\ref{fig2}, we depict the upper and lower excitation edges
of the ``monopole'' continuum for a specific choice of
``monopole'' hopping and chemical potential. Clearly,
the spectral periodicity is enhanced in both plots.
\begin{figure}
\includegraphics[width=5.8cm]{fig2.pdf}
\caption{(Color online.) (a) The upper excitation edge of the ``monopole'' continuum.
(b) The lower excitation edge of the ``monopole'' continuum.
For both figures, we set ${\mu = -3t}$, and the $\Gamma$ points
are the Brillouin zone centers. The important information of the plot
is not the dispersion itself, instead is the enhanced spectral
periodicity as if the Brillouin zone is folded. Here
${\Gamma_0\Gamma_1} = 2\pi (\bar{1}11)$
and ${\Gamma_0\Gamma_2} = 2\pi(1\bar{1}1)$ are the
reciprocal lattice vectors. }
\label{fig2}
\end{figure}
\section{Discussion}
\label{sec6}
\subsection{Non-Kramers doublets}
We discuss the application of our results to
various pyrochlore ice systems. We begin with
the non-Kramers doublets. The continuous excitations
have actually been observed from the INS measurements
on Pr$_2$Zr$_2$O$_7$, Tb$_2$Ti$_2$O$_7$ and Pr$_2$Hf$_2$O$_7$~\cite{Romain2017,PhysRevLett.118.107206,0953-8984-24-5-052201}.
In particular, in the INS result for Pr$_2$Hf$_2$O$_7$~\cite{Romain2017},
besides the very low-energy features that seem to resemble the suppressed
spectral intensity of the photon mode, there exists a broad excitation
continuum extending to higher energies.
This continuum may be attributed to the random strain effect that has
already been suggested to Pr$_2$Zr$_2$O$_7$~\cite{Baker156,PhysRevLett.118.107206,PhysRevLett.118.087203}.
Nevertheless, the random strain effect was also suggested to create
quantum entanglement and induce U(1) QSL phase in non-Kramers doublet
systems~\cite{PhysRevLett.118.087203}. Therefore, if the underlying
systems realize the U(1) QSL, according to our theory,
these mysterious continuous excitations may at least contain the
contribution from the two-``monopole'' continuum that is predicted
in this work.
How does one verify the above claim of the ``monopole'' continuum
in the INS measurement? We here propose a scheme to exclude the presence
of the spinon continuum in the INS result by conducting a thermal
transport measurement. Spinons are higher energy excitations,
and their contribution to thermal conductivity should appear
at higher temperatures~\cite{Matsuda}. If one observes that
the energy scale of the continuum in the INS measurement is clearly lower
than the temperature scale where the spinons contribute to the thermal
conductivity, one could then conclude the presence of the spinon excitation
in the thermal conductivity results and the absence of the spinon excitation
in the continuum of the INS results. The direct measurement would
be the confirmation of the enhanced spectral periodicity of the
``monopole'' continnum in the momentum space. This may be difficult
as the low-energy photon excitation is also present in the low-energy
INS data. Thus, the higher energy part of the ``monopole'' continnum may
provide more useful information. It is certainly very exciting if all
the three excitations, spinon, ``magnetic monopole'', and gauge photon
are confirmed by a combination of the INS and the thermal transport
measurements.
For the ``monopoles continuum'', probably the most positive side
in this identification of ``monopole continuum'' is that
weak external magnetic field can be used to manipulate the ``monopole''
continuum. With weak magnetic fields, the U(1) QSL will not be destroyed,
and the ``magnetic monopole'' remains to be a valid description of the
excitation of the system. However, the external magnetic field, that
only couples linearly to the $S^z$ components, polarizes $S^z$ slightly
and thereby modifies the background dual U(1) gauge flux
that is experienced by the ``monopole''. As a result,
the ``monopole'' band would probably develop a Hofstadter band~\cite{PhysRevB.14.2239},
and the spectral structure of the ``monopole'' continuum is modified.
How this ``monopole'' continuum is modulated depends on the orientation
and the amplitude of the external magnetic fields. The detailed behavior
of the ``monopole'' continuum in the weak field will be explored in future works.
\begin{table}[t]
\begin{tabular}{lcc}
\hline\hline
Properties & U(1)$_{0,\pi}$ QSL & U(1)$_{\pi,\pi}$ QSL \\
spinon flux & $0$ & $\pi$ \\
``monopole'' flux & $\pi$ & $\pi$ \\
spinon continuum & not enhanced & enhanced \\
``monopole'' continuum & enhanced & enhanced
\\
\hline\hline
\end{tabular}
\caption{A classification of distinct U(1) QSLs from the symmetry
classification patterns of the spinons and the ``magnetic monopoles''.
The first subindex refers to the flux that is experienced by
the spinon hopping around the hexagon plaquette on the diamond
lattice (see the second row), while the second subindex refers to
the flux that is experienced by the ``monopole'' hopping around
the hexagon plaquette on the dual diamond lattice (see the third row).
In the table, ``enhanced'' and ``not enhanced'' refer
to the spectral periodicity of the related excitation continuum. }
\label{tab2}
\end{table}
\subsection{Kramers doublets and numerical simulation}
As for the usual Kramers doublets~\cite{Ross11,Savary12,Gingras2014},
all the three components of the local moments
are odd under the time reversal symmetry, and the neutron spin would couple
to all of them. Therefore, the INS results on the U(1) QSL with the usual Kramers
doublets would also detect the spin flipping events out of the spin ice manifold
and measure the spinon continuum in addition to the gauge photon and the
``monopole'' continuum. As we have already pointed out in the previous
sections, the visibility of the ``monopole'' continuum in the INS data depends
on how much weight of the ``monopole'' continuum, and may vary for different materials.
If the neutron energy transfer is located within the ``monopole''
continuum, the spectral periodicity would experience an enhancement.
If the neutron energy transfer is located in the spinon continuum,
the spectral periodicity is enhanced (not enhanced)
if the spinon experiences a background ${\pi}$ ($0$)
flux on the diamond lattice~\cite{GangChen2017}.
The U(1) QSL has been explored by quantum Monte carlo simulation,
and the photon mode was identified in the $S^z$ correlation
function~\cite{PhysRevLett.115.037202,PhysRevLett.115.077202,PhysRevLett.100.047208}.
It might be of interest to introduce further $S^z$ interactions to
possibly enhance and manifest the ``monopole'' continuum
in the $S^z$ correlation~\cite{PhysRevB.94.205107}.
\subsection{A classification of the U(1) QSLs}
Finally, let us remark on the translation symmetry fractionalization patterns
for the U(1) QSLs. In this work, we have focused on the ``magnetic monopole''
excitation and found that the ``magnetic monopole'' experiences a background
dual U(1) flux on the dual diamond lattice. In the previous work~\cite{GangChen2017},
we studied the spectral periodicity
and the translation symmetry fractionalization for the spinon excitation.
The combination of the ``magnetic monopole'' and the spinon symmetry fractionalization
patterns results in a classification of the distinct symmetry enriched U(1)
QSLs in Table~\ref{tab2}. Like the classification scheme that was developed for
the two-dimensional $\mathbb{Z}_2$ QSLs and applied to the $\mathbb{Z}_2$
toric code model~\cite{PhysRevB.87.104406},
one could use the result in Table~\ref{tab2}
to further establish the translation symmetry fractionalization for the (fermionic)
dyon that is a bound state of the spinon and the ``monopole''.
Our classification not only helps improve the understanding
of the crystal symmetry fractionalization in the U(1) QSLs, but also
provides unique and detectable experimental signatures for the U(1) QSLs.
\section{Acknowledgments}
We acknowledege Nic Shannon and Mike Hermele for useful discussion, Chenjie Wang
for various related and unrelated philosophical conversations, Zhong Wang for a
comment, and one anonymous referee for a comment that improves this work. We acknowledge
Michel Gingras for the invitation to the ``International Workshop on Quantum Spin Ice''
at Perimeter Institute for Theoretical Physics where this work is carried out and
finalized. This work is supported by the ministry of science and technology of
China with the Grant No.2016YFA0301001, the start-up fund for original research
and the first-class university construction fund of Fudan University, and the
thousand-youth-talent program of China.
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\section{Introduction}
Assume $\Om\subset \mathbb R^3$ is a $C^3$-- smooth bounded domain
and $Q_T=\Om\times (0,T)$. In this paper we investigate the boundary
regularity of solutions to the principal system of
magnetohydrodynamics (the MHD equations):
\begin{equation}
\left.
\begin{array}c
\partial_t v + (v\cdot \nabla )v - \Delta v + \nabla
p = \rot H \times H
\\
\div v =0
\end{array} \right\}
\quad\mbox{in } Q_T, \label{MHD_NSE}
\end{equation}
\begin{equation}
\left.
\begin{array}c
\partial_t H +\rot \rot H = \rot (v\times H)
\\ \div H =0
\end{array} \right\} \quad \mbox{in }Q_T.
\label{MHD_Magnetic}
\end{equation}
Here unknowns
are the velocity field $v:Q_T\to \mathbb R^3$, pressure
$p:Q_T\to \mathbb R$, and the magnetic field $H:Q_T\to \mathbb
R^3$. We impose on $v$ and $H$ the boundary conditions:
\begin{equation}
v|_{\cd\Om\times(0,T)}=0, \quad H_{\nu}|_{\cd\Om\times(0,T)}=0, \quad (\rot H)_{\tau}|_{\cd\Om\times(0,T)}=0,
\label{BC}
\end{equation}
Here by $\nu$ we denote the outer normal to $\cd\Om$ and $H_{\nu}= H\cdot \nu$, $(\rot H)_\tau=\rot H- \nu(\rot H\cdot \nu)$.
These conditions correspond to the case of liquid flowing in the area bounded by ideal conductor.
\begin{definition}
\label{sws_def}
Assume $\Ga\subset \cd\Om$.
The functions $(v,H,p)$ are called a {\it boundary suitable
weak solution } to the system (\ref{MHD_NSE}), (\ref{MHD_Magnetic})
near $\Ga_T\equiv \Ga\times (0,T)$ if
\begin{itemize}
\item[1)] \ $v\in L_{2,\infty}(Q_T)\cap W^{1,0}_2(Q_T)\cap W^{2,1}_{\frac 98, \frac 32}(Q_T)$,
$H \in L_{2,\infty}(Q_T)\cap W^{1,0}_2(Q_T)$,
\item[2)] \ $p\in L_{\frac 32}(Q_T)\cap W^{1,0}_{\frac 98, \frac 32}(Q_T)$,
\item[3)] \ $\div v=0$, \ $\div H=0$ \ a.e. in \ $Q_T$,
\item[4)] \ $v|_{\cd \Om}=0$, \ $H_\nu|_{\cd\Om}=0$ \ in the sense of traces,
\item[5)] \ for any $w\in L_2(\Om)$ the functions
$$
t\mapsto \int\limits_{\Om} v(x,t)\cdot w(x)~dx \qquad\mbox{and}\qquad t\mapsto \int\limits_{\Om} H(x,t)\cdot w(x)~dx
$$
are continuous,
\item[6)] \ $(v,H)$ satisfy the following integral identities: for any $t\in [0,T]$
$$
\begin{array}c
\int\limits_{\Om} ~v(x,t)\cdot \eta(x,t)~dx -\int\limits_{\Om} ~v_0(x)\cdot \eta(x,0)~dx \ + \\ + \
\int\limits_0^t \int\limits_\Om ~ \Big( - v\cdot \cd_t \eta + (\nabla v - v\otimes v + H \otimes H) : \nabla \eta - (p+\frac 12 |H|^2)\div \eta \Big)~dxdt \ = \ 0,
\label{isl1}
\end{array}
$$
for all \ $\eta \in W^{1,1}_{\frac 52}(Q_t)$ \ such that \ $\eta|_{\cd\Om\times (0,t)}=0$,
$$
\begin{array}c
\int\limits_{\Om}~ H(x,t)\cdot \psi (x,t)~dx \ - \ \int\limits_{\Om}~ H_0(x)\cdot \psi (x,0)~dx \ + \\ + \
\int\limits_0^t \int\limits_\Om ~ \Big( - H\cdot \cd_t \psi + \rot H\cdot \rot \psi - (v\times H) \cdot \rot \psi \Big)~dxdt \ = 0,
\label{isl2}
\end{array}
$$
for all \ $\psi \in W^{1,1}_{\frac 52}(Q_t)$ \ such that \ $\psi_\nu |_{\cd\Om\times (0,t)}=0$.
\item[7)] For every $z_0=(x_0,t_0)\in \Ga_T$
such that $ \Om_R(x_0)\times (t_0-R^2, t_0)\subset Q_T$ where $\Om(x_0,R) \equiv \Om \cap B(x_0,R)$ and for any $\zeta\in C_0^\infty(B_R(x_0)\times(t_0-R^2,t_0])$
such that $\left.\frac{\cd\zeta}{\cd \nu}\right|_{\cd\Om}=0$ the following ``local energy inequality near $\Ga_T$'' holds:
\begin{equation}
\begin{array}c
\sup\limits_{t\in (t_0-R^2,t_0)}\int\limits_{\Om_R(x_0)} \zeta \Big( |v|^2+|H|^2\Big)~dx \ + \\
+ \ 2 \int\limits_{t_0-R^2}^{t_0}\int\limits_{\Om_R(x_0)} \zeta \Big( |\nabla v|^2+|\rot H|^2\Big)~dxdt \ \le \\
\le\ \int\limits_{t_0-R^2}^{t_0}\int\limits_{\Om_R(x_0)} \Big(| v|^2 +|H|^2\Big) (\cd_t \zeta + \Delta \zeta )~dxdt \ + \\ + \
\int\limits_{t_0-R^2}^{t_0}\int\limits_{\Om_R(x_0)} \Big ( | v|^2+ 2\bar p \Big ) v\cdot\nabla \zeta~dxdt \ + \\
- \ 2 \int\limits_{t_0-R^2}^{t_0}\int\limits_{\Om_R(x_0)} (H\otimes H): \nabla^2 \zeta ~dxdt \ + \\ + \
2 \int\limits_{t_0-R^2}^{t_0}\int\limits_{\Om_R(x_0)} (v\times H)(\nabla \zeta\times H)~dxdt
\end{array}
\label{LEI}
\end{equation}
\end{itemize}
\end{definition}
\noindent
Here $L_{s,l}(Q_T)$ is the anisotropic Lebesgue space equipped with the norm
$$
\|f\|_{L_{s,l}(Q_T)}:=
\Big(\int_0^T\Big(\int_\Om |f(x,t)|^s~dx\Big)^{l/s}dt\Big)^{1/l} ,
$$
and we use the following notation for the functional spaces:
$$
\gathered
W^{1,0}_{s,l}(Q_T)\equiv L_l(0,T; W^1_s(\Om))= \{ \ u\in L_{s,l}(Q_T): ~\nabla u \in L_{s,l}(Q_T) \ \},
\\
W^{2,1}_{s,l}(Q_T) = \{ \ u\in W^{1,0}_{s,l}(Q_T): ~\nabla^2 u, \ \cd_t u \in L_{s,l}(Q_T) \ \},
\\
\overset{\circ}{W}{^1_s}(\Om)=\{ \ u\in W^1_s(\Om):~ u|_{\cd\Om}=0 \ \},
\endgathered
$$
and the following notation for the norms:
$$
\gathered
\| u \|_{W^{1,0}_{s,l}(Q_T)}= \| u \|_{L_{s,l}(Q_T)}+ \|\nabla u\|_{L_{s,l}(Q_T)}, \\
\| u \|_{W^{2,1}_{s,l}(Q_T)}= \| u \|_{W^{1,0}_{s,l}(Q_T)}+ \| \nabla^2 u \|_{L_{s,l}(Q_T)}+\|\cd_t u\|_{L_{s,l}(Q_T)}, \\
\endgathered
$$
\begin{theorem} \label{Existense_Suitable}
For any sufficiently smooth divergent-free \ $v_0$, $H_0$ satisfying (\ref{BC}) there exists at least one boundary suitable weak solution near $\cd\Om\times (0,T)$ which satisfies the initial conditions :
$$\| v(\cdot, t) -v_0(\cdot)\|_{L_2(\Om)}\to 0, \quad \| H(\cdot, t) -H_0(\cdot)\|_{L_2(\Om)}\to 0 \quad\mbox{as} \quad t\to+0,$$
and additionally satisfies the global energy inequality
$$
\gathered
\| v\|_{L_{2,\infty}(Q_T)}+ \| H\|_{L_{2,\infty}(Q_T)} + \| \nabla v\|_{L_{2}(Q_T)} + \| \rot H\|_{L_{2}(Q_T)} \le \\ \le \| v_0\|_{L_2(\Om)}+ \| H_0\|_{L_2(\Om)}
\endgathered
$$
\end{theorem}
The global existence of weak solutions to the MHD equations (\ref{MHD_NSE}) --- (\ref{BC}) was established originally in \cite{LadSol}.
The proof of Theorem \ref{Existense_Suitable} can be found in \cite{VyaShi}.
\noindent
{\bf Notations} In this paper we will use the following notations
$$B(x_0,R) = \l\{ x \in \R^3 : |x-x_0| < R \r\},$$
$$\Bdva(x'_0,R) = \l\{ x' \in \R^3 : |x'-x'_0| < R \r\},$$
$$\Bdva(R) = \Bdva(0,R), \quad \Bdva = \Bdva(1)$$
$$\B_0(R) = \l\{ x \in B(0,R) : x_3 > 0 \r\}, \quad \Q_0(R) = \B_0(R) \times (-R^2,0)$$
\section{Main Results}
\setcounter{equation}{0}
\noindent
{\bf Main condition on $\d \Om$}. {\it There exist positive numbers $\mu$ and $R_0$ depending only on $\d \Om$ such that
for each point $x_0 \in \d \Om$ we can choose a Cartesian coordinate system $\{y_i\}^3_{i=1}$ associated to the origin $x_0$, and some function $\ph_{x_0} \in C^3(\Bdva(R_0))$ such that
$$\Om(x_0,R_0) \equiv \Om \cap B(x_0,R_0) = \{ y \in B(R_0) : y_3 > \ph_{x_0}(y_1,y_2) \},$$
and
\begin{equation}
\vphi_{x_0}(0) = 0, \quad \gr \vphi_{x_0} (0) = 0, \quad \| \vphi_{x_0} \|_{W^2_{\infty}} \leq \mu.
\label{phi_cond}
\end{equation}
}
The main results of the present paper are the following theorems on boundary regularity of suitable weak solutions of MHD system
\begin{theorem}
\label{CKN_theorem_Omega}
Assume that $\d\Om$ satisfies Main Condition. Then for any $K>0$ there exists $\ep_0(K)>0$ with the following property.
Assume $(v,H,p)$ is a boundary suitable weak solution in $Q_T$ and $z_0=(x_0,t_0)\in \cd\Om\times (0,T)$.
If
\begin{equation}
\limsup\limits_{r\to 0}\Big(~\frac 1r \int\limits_{t_0-r^2}^{t_0} |\nabla H|^2~dxdt ~\Big)^{1/2}\ < \ K
\end{equation}
and
\begin{equation}
\limsup\limits_{r\to 0} \Big(~\frac 1r \int\limits_{t_0-r^2}^{t_0} |\nabla v|^2~dxdt~\Big)^{1/2} \ < \ \ep_0,
\end{equation}
then the functions $v$ and $H$ are H\" older continuous in some neighborhood of $z_0$.
\end{theorem}
\begin{theorem}\label{Partial_Regularity}
Assume that $\d\Om$ satisfies Main Condition and $(v,H,p)$ is a boundary suitable weak solution in $Q_T$. Then there exists a closed set $\Sigma\subset \d\Om \times (0,T]$ such that
for any $z_0\in (\Ga\setminus \Sigma)\times (0,T]$ the functions $(v,H)$ are H\" older continuous in some neighborhood of $z_0$,
\begin{equation}
\mathcal P^1(\Sigma) \ = \ 0,
\label{Sigma_estimate}
\end{equation}
where $\mathcal P^1(\Sigma)$ is the one-dimensional parabolic Hausdorff measure of $\Sigma $.
\end{theorem}
Our Theorem \ref{CKN_theorem_Omega} presents for the MHD equations a result which is a boundary analogue of the famous Caffarelli--Kohn--Nirenberg (CKN)
theorem for the Navier-Stokes system, see \cite{CKN}, see also \cite{Lin}.
The boundary regularity of solutions to the Navier-Stokes equations was originally investigated by G.~Seregin in \cite{Seregin_JMFM} and \cite{Seregin_Aa} in the case of a plane part of the boundary
and by G.~Seregin, T.~Shilkin, and V.~Solonnikov in \cite{SSS} in the case of a curved boundary.
The internal partial regularity of solutions to the MHD system was originally proved by C.~He and Z.~Xin in \cite{China}, see also \cite{Vya}, \cite{Vya1}. The local regularity near the plan
part of the boundary for MHD equations was investigated in \cite{VyaShi} (see also \cite{Vya2011}). In the case of boundary regularity due to boundary conditions on the derivatives of $H$
there will be some problems if try to directly generalize the approach used for Navier-Stokes equations. To solve this problem we will consider the equation \eqref{MHD_Magnetic} as a linear heat equation on $H$
and this gives us some additional estimates. This idea was originally proposed in \cite{China} to obtain regularity theorems with just boundedness conditions instead of smallness on magnetic component the internal case.
In the present paper corresponding estimates are proved in the sections \ref{Heat_equation_estimate_section} and \ref{H_estimate_section}.
Note that using the methods of our paper one can prove various $\ep$--regularity conditions involving various scale--invariant functionals
(such it was done for the plane part of the boundary in \cite{Vya2011}, see also \cite{Mih}).
In the present paper we concentrate on the condition of the theorem \ref{CKN_theorem_Omega} as this condition provides the optimal estimate of the Hausdorff measure of the singular set $\Sigma$
in Theorem \ref{Partial_Regularity}. Note that we also have \eqref{Sigma_estimate} in the internal case, so combining these two estimates we will obtain \eqref{Sigma_estimate} for the set of singular points
in any bounded domain $\Omega$ with $C^3$ smooth boundary.
Our paper is organized as follows:
in Section \ref{Symmetry_section} using symmetries of \eqref{MHD_NSE}-\eqref{MHD_Magnetic} we present more convenient statement of Theorem \ref{CKN_theorem_Omega}.
In Section \ref{Perturbed_Stokes_section} we describe coercive estimates for solutions of the Stokes equations near the boundary.
Section \ref{Main_criterion_section} contains the proof of the Decay Lemma and the sketch of the proof of Theorem \ref{Fixed_r}.
Sections \ref{Heat_equation_estimate_section} and \ref{H_estimate_section} is concerned with the estimate of some Morrey functional for weak solutions to the heat equation near the boundary.
These estimates together with the estimates of the scale invariant energy functionals obtained in Section \ref{EnergyEstimates_Section} turn to be crucial for the prove of theorem \ref{CKN_theorem_Omega}
presented in section \ref{Final_section}.
\section{Symmetry group of MHD system and new statement of the main results}
\label{Symmetry_section}
\setcounter{equation}{0}
The solutions of MHD system (\ref{MHD_NSE}), (\ref{MHD_Magnetic}) have the same set of symmetries as the Navier-Stokes equations i.e.
they are invariant under translations, rotations and scaling
\begin{equation}
\begin{aligned}
& v_R(y,s) = R v(R y, R^2 s),\\
& H_{R}(y,s) = R H(R y, R^2 s),\\
& p_{R}(y,s) = R^2 p(R y, R^2 s).
\label{scaling}
\end{aligned}
\end{equation}
So we can consider that in the statement of theorem \ref{CKN_theorem_Omega} $z_0 = 0$ and the boundary of the domain is described by function $\ph$ satisfying \eqref{phi_cond}.
Also it will be convenient to consider the function $\ph$ as a part of the problem data
and deal with the "local version" of suitable weak solution in parabolic cylinder
$$\Q(R) \equiv (-R^2,0) \times \B(R),$$
where
$$\B(R) \equiv \{ x \in B(R) : x_3 > \ph (x_1,x_2) \}.$$
\begin{definition}
Let $R>0$ and $\vphi \in C^2(\Bdva(R))$ satisfies \eqref{phi_cond}.
The functions $(v,H,p,\vphi)$ are called a boundary suitable
weak solution to the system (\ref{MHD_NSE}), (\ref{MHD_Magnetic})
in $\Q(R)$ if there is a domain $\Om$ such that $\Gamma = \l\{ x_3 = \ph(x_1,x_2) \r\} \subseteq \d \Om$ and functions $v$, $p$, $H$ can be extended up to
suitable weak solution near $\Gamma$.
\end{definition}
Then theorem \ref{CKN_theorem_Omega} can be formulated by following way
\begin{theorem}
\label{CKN_theorem}
For any $K>0$ there exists $\ep_0(K)>0$ with the following property.
Assume $(v,H,p,R)$ is a boundary suitable weak solution in $\Q(R)$ for some $R>0$.
If
\begin{equation}
\limsup\limits_{r\to 0}\Big(~\frac 1r \int\limits_{\Q(r)} |\nabla H|^2~dxdt ~\Big)^{1/2}\ < \ K
\label{ep-regularity-1}
\end{equation}
and
\begin{equation}
\limsup\limits_{r\to 0} \Big(~\frac 1r \int\limits_{\Q(r)} |\nabla v|^2~dxdt~\Big)^{1/2} \ < \ \ep_0,
\label{ep-regularity-2}
\end{equation}
then there exists $\rho_*>0$ such that the functions $v$ and $H$ are H\" older continuous on the closure of $ Q^+(\rho_*)$.
\end{theorem}
To prove this theorem we will generalize the approach introduced in \cite{VyaShi} in the case of the plane part of the boundary. The first step is the following theorem
\begin{theorem}\label{Fixed_r}
There exists an absolute constant $\ep_*>0$ with the following property. Assume $(v,H,p,\vphi)$ is a boundary suitable weak solution in $\Q(R)$
If there exists $0 < r_0 <R$ such
$$
\frac 1{r_0^2} \int\limits_{Q^+(r_0)} \Big(~ |v|^3+|H|^3+|p|^{\frac 32}~\Big) dxdt \ < \ \ep_*
$$
and
\begin{equation}
\| \ph \|_{C^2(\Bdva(r_0))} < \ep_*,
\label{ssl1}
\end{equation}
then the functions $v$ and $H$ are H\" older continuous on the closure of $ Q^+(\frac {r_0}2)$.
\end{theorem}
Note, that \eqref{ssl1} is just the condition on the smallness of $r_0$. Indeed
if $(v,p,H,\vphi)$ are the suitable weak solution in $\Q(R)$, then if we apply the scaling transformations \eqref{scaling},
then $(v_R,p_R,H_R,\ph_R)$ where
$$
\vphi_R = \frac1{R}\vphi
$$
will be the solution in $\Q(1)$ and from Taylor formula we have
$$ \| \frac{\vphi}{R} \|_{C^2 (\Bdva)} \leq R \| \vphi \|_{C^2}.$$
\section{Estimates for perturbed Stokes system}
\label{Perturbed_Stokes_section}
\setcounter{equation}{0}
In this section we describe coercive estimates for linearisation of the \eqref{MHD_NSE}. We start from the Stokes problem in $\Q$ with
some $\ph \in C^3(\Bdva)$
\begin{equation}
\begin{aligned}
& \d_t u - \Delta u + \nabla p = f\\
& \nabla \cdot u = 0 \\
& u|_{x_3=\ph(x_1,x_2)} = 0,
\end{aligned}
\qquad \text{ in } \Q.
\label{pssl1}
\end{equation}
and define new coordinates $\{ y_i \}_{i=1}^3$ connected with the original ones via formula
\begin{equation}
x = e (y) \equiv \begin{pmatrix} y_1\\ y_2 \\ y_3 + \vphi(y_1,y_2) \end{pmatrix}.
\label{change_variables}
\end{equation}
Denote by $\L$ the Jacobi matrix of the map $x = e(y)$ i.e.
\begin{equation}
\L =
\begin{pmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
\vphi_{,1} & \vphi_{,2} & 1
\end{pmatrix},
\label{Jmatrix}
\end{equation}
and by $\tilde \nabla_\ph$ and $\tilde \Delta_\ph $ the following differential operators with variable coefficients:
\begin{equation}
\gathered
(\tilde \nabla_\ph p)_{i} = p_{,k}{e_{k,i}}, \\ (\tilde \Delta_\ph v)_i = v_{i,jl}e_{j,k}e_{l,k} +v_{i,j}e_{j,kk}, \\
\tilde\nabla_\ph \cdot v = v_{i,k}{e_{k,i}}
\endgathered
\label{Hat operators-1}
\end{equation}
Also we have the relation
\begin{equation}
\rot\nolimits_x H = \L \rot\nolimits_y (\L^T \tH),
\label{rot_tr}
\end{equation}
here we have used that $\det \L = 1$.
Then the system \eqref{pssl1} transforms into so called perturbed Stokes system
\begin{equation}
\begin{aligned}
& \d_t u - \tilde\Delta_{\ph} u + \tilde\nabla_{\ph} p = f\\
& \tilde\nabla_{\ph} \cdot u = 0 \\
& u|_{y_3=0} = 0.
\end{aligned}
\label{zerodiv_Stokes}
\end{equation}
Without loss of generality we can consider \eqref{zerodiv_Stokes} in $\Q_0$.
We recall that the function $\vphi$ satisfy the following relations
\begin{equation}
\vphi (0) = 0, \qquad \nabla \vphi (0) = 0, \qquad \| \vphi \|_{W^2_{\infty}} < \mu,
\label{phi_conditions}
\end{equation}
and we can think, that the constant $\mu$ is sufficiently small.
For our further arguments we will need two lemmas about the solutions of this problem.
We start from consideration of initial-boundary problem for perturbed Stokes system \eqref{zerodiv_Stokes} with homogenous data:
\begin{equation}
u|_{\d \B} = 0, \quad u|_{t=-1} = 0
\label{nonzerodiv_Stokes_BC}
\end{equation}
\begin{lemma}
Assume that $s,l \in (1,\infty)$. There is $\mu_* \in (0,\frac1{100})$ if $\vphi$ satisfies \eqref{phi_conditions} for some $\mu < \mu_*$,
then there is a pair of functions $(u,p)$ such that
$$ u \in W^{2,1}_{s,l}(\Q), \qquad \nabla p \in L_{s,l}(\Q),$$
$(v,p)$ satisfy \eqref{zerodiv_Stokes} with initial-boundary conditions \eqref{nonzerodiv_Stokes_BC} and the following estimate holds
\begin{equation}
\| v \|_{W^{2,1}_{s,l}(\Q)} + \| \nabla p \|_{L_{s,l}(\Q)} \leq C_* \| f \|_{L_{s,l}(\Q)}.
\label{linest1}
\end{equation}
here $C_*$ is an absolute constant depending only on the dimension.
\label{nonzerodivStokes_Lemma}
\end{lemma}
The second lemma is the coercive estimate for the solution of \eqref{zerodiv_Stokes}.
\begin{lemma}
Assume that $s,m,l \in (1,\infty)$, $m \geq s$. Assume that $\vphi \in C^3(\Bdva)$ satisfies \eqref{phi_conditions} for some $\mu < \mu_*$.
Then for every functions $u \in W^{2,1}_{s,l}(\Q)$, $\nabla p \in L_{s,l}(\Q)$ and $f \in L_{m,l}(\Q)$ satisfying \eqref{zerodiv_Stokes} we have
$u \in W^{2,1}_{m,l}(\Q(\frac12))$, $\nabla p \in L_{m,l}(\Q(\frac(1/2))$ and the following estimate holds
\begin{equation}
\gathered
\| v \|_{W^{2,1}_{m,l}(\Q(\frac12))} + \| \nabla p \|_{L_{m,l}(\Q(\frac12))} \leq\\
\leq C \l( \| f \|_{L_{s,l}(\Q)} + \|\nabla v \|_{L_{s,l}(\Q)} + \| p - p_0 \|_{L_{s,l}(\Q)} \r)
\endgathered
\label{linest2}
\end{equation}
for some absolute constant $C$ and arbitrary function $p_0 = p_0 (t)$, $p \in L_l(-1,0)$.
\label{zerodivStokes_Lemma}
\end{lemma}
\section{The main criterion of $\ep$-regularity}
\label{Main_criterion_section}
\setcounter{equation}{0}
In this section we will prove the theorem \ref{Fixed_r}. Let $(v,p,H,\ph)$ are the suitable weak solution in $\Q$, then we will use the following notations
$$
\gathered
Y_\tau (v) \ = \l(\frac{1}{|Q(\tau)|} \int\limits_{Q^+(\tau)} ~|v|^3~dxdt\ \r)^{1/3}, \\
\tilde Y_\tau (H) \ = \l(\frac{1}{|Q(\tau)|} \int\limits_{Q^+(\tau)} ~|H - b_R(H)|^3~dxdt\ \r)^{1/3}, \\
\hat Y_\tau (p)\ = \
\tau ~\l(\frac{1}{|Q(\tau)|}\int\limits_{Q^+(\tau)} ~|p-[p]_{B^+(\tau)}|^{3/2}~dxdt\ \r)^{2/3},\\
Y_\tau (v,p,H) = Y_\tau(v) + \tilde Y_\tau (H) + \hat Y_\tau (p),\\
b_\tau(H) = \L \begin{pmatrix} (h_1)_\tau \\ (h_2)_\tau \\ 0 \end{pmatrix},
\endgathered
$$
here $h = \L (H \circ e)$, where $e$ and $\L$ defined by \eqref{change_variables} and \eqref{Jmatrix},
$$
(h_i)_\tau = \frac1{|\Q(\tau)|} \intl_{e(\Q(\tau))} h_i(z)\,dz
$$
and
$$
[p]_{\B(\tau)}(t) = \frac1{|\B(\tau)|} \intl_{\B(\tau)} p(x,t)\,dt.
$$
Note, that $\tilde Y_\tau(H)$ is equivalent to norm $H$ in Morrey space
with taking into account boundary conditions. It can be easily seen if we replace $H$ by $\L h$ and use, that matrix $\L$ is close to
the identity one.
We start our considerations from the modification of local energy inequality.
\begin{lemma}
Assume $(v,H,p,\ph)$ is a boundary suitable weak solution satisfied the MHD equations in $\Q(R)$.
Let $\zeta\in C_0^\infty(B\times (-R^2,0])$ be a cut-off function such that $\frac{\d\zeta}{\d \nu}|_{x_3 = \ph(x_1,x_2)}=0$.
Assume $b\in \mathbb R^3$ is an arbitrary constant vector of the form $b=(b_1, b_2, 0)$. Then the following inequality holds
\begin{equation}
\gathered
\intl_{\B(R)} \zeta \l( |v|^2 + | \bar{H}|^2 \r)\,dx \\
+ 2 \intl_{\B(R)\times (t_0,t)} \zeta\l( |\nabla v| + |\rot \bar{H}|^2\r)\,dz\\
\leq \intl_{\B(R)\times (t_0,t)} (\d_t \zeta + \lap \zeta) (|v|^2 + |\bar{H}|^2)\,dz\\
+\intl_{\B(R)\times (t_0,t)} \Big ( | v|^2+ 2\bar p \Big ) v\cdot\nabla \zeta\,dz \\
- \ 2 \intl_{\B(R)\times (t_0,t)} (H\otimes \bar{H}): \nabla^2 \zeta \,dz \ + \\ + \
2 \intl_{\B(R)\times (t_0,t)} (v\times H)(\nabla \zeta\times \bar{H})\,dz\\
- \intl_{\B(R)\times (t_0,t)} \l[ |\rot \L b|^2 \zeta + \nabla \zeta \cdot \nabla (|\L b|^2) - (v\times H) \cdot \rot(\L b) \zeta\r]\,dz,
\endgathered
\label{mlei1}
\end{equation}
where $\bar{H} = H - \L b$.
\end{lemma}
{\bf Proof.}
We use \eqref{LEI} and transform the remaining terms. Via integration by parts formula
\begin{equation}
\gathered
\intl_{\B(R)\times (t_0,t)} (\d_t \zeta + \lap \zeta) |\L b|^2\,dz \\
= \intl_{\B(R)} \zeta |\L b|^2 \,dx -
\intl_{\B(R)\times (t_0,t)} \nabla \zeta \cdot \nabla (|\L b|^2)\,dz
\endgathered
\label{mlei2}
\end{equation}
Also we have
\begin{equation}
\gathered
\intl_{\B(R)\times (t_0,t)} \d_t \zeta H \cdot \L b \,dz = \intl_{\B(R)} \zeta H \cdot \L b \,dx +\\
+ \intl_{\B(R)\times (t_0,t)} (-\d_t H \cdot \L b)\,dz.
\endgathered
\label{mlei3}
\end{equation}
Now we consider two terms: one from left hand side another from right hand side of \eqref{mlei1}
\begin{equation}
\gathered
\intl_{\B(R)\times (t_0,t)} \lap \zeta H \cdot \L b\,dz -
\intl_{\B(R)\times (t_0,t)} \rot H \cdot \rot (\L b)\,dz =\\
= - \intl_{\B(R)\times (t_0,t)} \l[ \rot H \cdot \l(\nabla \zeta \times (\L b) \r) + \rot H \cdot \rot (\L b) \zeta \r.\\
\l.- H \otimes (\L b) : \nabla^2 \zeta\r]\,dz\\
-\intl_{\B(R)\times (t_0,t)} H \otimes (\L b) : \nabla^2 \zeta\,dz=\\
= -\intl_{\B(R)\times (t_0,t)} H \otimes (\L b) : \nabla^2 \zeta\,dz
- \intl_{\B(R)\times (t_0,t)} \rot \rot H \cdot (\L b \zeta)\,dz.
\endgathered
\label{mlei4}
\end{equation}
The last terms of \eqref{mlei3} and \eqref{mlei4} can be modified using equation \eqref{MHD_Magnetic}
\begin{equation}
\gathered
\intl_{\B(R)\times (t_0,t)} (\d_t H \cdot \L b + \rot \rot H \cdot (\L b \zeta))\,dz=\\
= \intl_{\B(R)\times (t_0,t)} \rot (v \times H) \cdot (\L b\zeta)\,dz=\\
= \intl_{\B(R)\times (t_0,t)} (v \times H) \cdot \rot (\L b) \zeta -\\
- \intl_{\B(R)\times (t_0,t)} (v \times H) \cdot (\nabla \zeta \times (\L b))\,dz.
\endgathered
\label{mlei5}
\end{equation}
Combining \eqref{mlei2}-\eqref{mlei5} we obtain the statement of lemma.
\qed
\begin{lemma} \label{Decay estimate theorem}
There exists an absolute constant $\ep_*>0$ such that for any $M>0$ there exists $C_*=C_*(M)$ with the following properties.
For any boundary suitable weak solution $(v,H,p,\vphi)$ of the MHD system (\ref{MHD_NSE}), (\ref{MHD_Magnetic}) near the boundary in $\Q(1)$
the following implication holds:
if
$$
Y_1(v,H,p) + \| \vphi \|_{C^2} \ < \ \ep_0,
$$
and
$$
b_R(H) \le \ M
$$
then
\begin{equation}
Y_\tau (v,H,p) \ \le \ C_* ~\tau^{1/3}~Y_1 (v,H,p)
\label{Decay_Estimate}
\end{equation}
\end{lemma}
{\bf Proof.}
Arguing by contradiction we assume there exists a sequence of numbers $\ep_m\to 0$,
and a sequence of boundary suitable weak solutions $(v^m,H^m, p^m, \vphi^m)$ such that
$$
Y_1(v^m,H^m,p^m) + \| \vphi^m \|_{C^2} \ < \ \ep_m \ \to \ 0,
$$
and
$$
Y_\tau (v^m, H^m, p^m)\ \ge \ C_* \tau^{1/3} \delta_m
$$
here $\delta_m = Y_1(v^m,H^m,p^m)$.
Let us introduce functions
$$
\gathered
u^m(y, s) \ = \ \frac{1}{\delta_m}~ v^m \circ e_m, \\
q^m(y, s) \ = \ \frac{1}{\delta_m}~ \Big(p^m(x, t) - [p^m]_{B^+}(t)\Big) \circ e_m, \\
h^m(y, s) \ = \ \frac{1}{\delta_m}~ \Big(H^m(x, t) - b_1(H^m)\Big) \circ e^m,\\
\endgathered
$$
here $e^m$ denotes the map \eqref{change_variables} corresponding to $\vphi^m$.
Then
\begin{equation}
Y_1(u^m,h^m,q^m)\ = \ 1,
\qquad
Y_\tau (u^m, h^m, q^m )\ge C_* \tau^{1/3}
\label{Contradiction_asumptions}
\end{equation}
and $(u^m, h^m, q^m)$ satisfy the following equations in $\mathcal D'(Q^+)$
\begin{equation}
\gathered
\partial_t u^m + \delta_m (u^m\cdot \tilde\nabla_{\vphi^m} )u^m - \tilde\Delta_{\vphi^m} u^m + \tilde\nabla_{\vphi^m} q^m =\\
= \L_m \rot \L_m^{-T} h^m \times ( \delta_m h^m + b_1(H^m))
\\
\L_m^{-T} \cdot \nabla u^m =0
\endgathered \label{MHD_NS_m}
\end{equation}
\begin{equation}
\gathered
\partial_t h^m - \tilde\Delta_{\vphi^m} h^m = \L_m \rot \L_m^{T} \big(u^m\times (\delta_m h^m+ (H^m))\big)
\\ \L_m^{-T} \cdot \nabla h^m =0
\endgathered
\label{MHD_Magnetic_m}
\end{equation}
The conditions (\ref{Contradiction_asumptions}) imply in particular the boundedness
\begin{equation}
\sup\limits_m~\Big(\| u^m\|_{L_3(Q^+)} + \| h^m \|_{L_3(Q^+)} + \| q^m \|_{L_{\frac 32}(Q^+)}\Big) \ < \ +\infty
\label{Basic_boundedness}
\end{equation}
From the local energy inequality near the boundary and the relation obtained from (\ref{MHD_Magnetic})
multiplied by the test function $\psi=\zeta (H^m)$ we obtain
\begin{equation}
\gathered
\| u^m \|_{L_{2,\infty}(Q^+_0(\frac 9{10}))} + \| h^m \|_{L_{2,\infty}(Q^+_0(\frac 9{10}))} \ + \\
+ \ \| u^m \|_{W^{1,0}_{2}(Q^+_0(\frac 9{10}))} + \| h^m \|_{W^{1,0}_{2}(Q^+_0(\frac 9{10}))} \ \le C.
\endgathered
\label{Gradient boundedness}
\end{equation}
From the equations (\ref{MHD_NS_m}), (\ref{MHD_Magnetic_m}) we also obtain the estimate
$$
\| \cd_t u^m \|_{L_{\frac 53}(-1, 0; W^{-1}_{\frac 53} (B^+_0)) } + \| \cd_t h^m \|_{L_{\frac 53}(-1, 0; W^{-1}_{\frac 53} (B^+_0)) }\ \le \ C.
$$
Hence we can extract subsequences
\begin{equation}
\begin{array}c
u^m \ \rightharpoonup \ u \quad \mbox{in} \quad L_3(Q^+_0), \\
h^m \ \rightharpoonup \ h \quad \mbox{in} \quad L_3(Q^+_0), \\
q^m \ \rightharpoonup \ q \quad \mbox{in} \quad L_{\frac 32}(Q^+_0), \\
\end{array}
\label{Weak_Convergence0}
\end{equation}
\begin{equation}
\begin{array}c
u^m \ \rightharpoonup \ u \quad \mbox{in} \quad W^{1,0}_2(Q^+_0(\frac 9{10})), \\
h^m \ \rightharpoonup \ h \quad \mbox{in} \quad W^{1,0}_2(Q^+_0(\frac 9{10})),
\end{array}
\label{Weak_Convergence}
\end{equation}
\begin{equation}
\begin{array}c
u^m \ \to \ u \quad \mbox{in} \quad L_3 (Q^+_0( \frac 9{10})), \\
h^m \ \to \ h \quad \mbox{in} \quad L_3(Q^+_0( \frac 9{10})), \\
\vphi^m \to 0 \quad \mbox{in} \quad C^2(\Bdva),\\
b_R(H^m)\ \to \ a \quad \mbox{in} \quad C^2,\qquad \ \qquad
\end{array}
\label{Strong_Convergence}
\end{equation}
here $a \in \R^3$ is the constant vector.
Passing to the limit in the equations (\ref{MHD_NS_m}), (\ref{MHD_Magnetic_m}) we obtain
\begin{equation}
\gathered
\cd_t u - \Delta u + \nabla q = \rot h \times a \qquad \mbox{in}\quad Q^+_0,
\\
\div u =0 \qquad \mbox{in}\quad Q^+_0,
\\
u|_{y_3=0}=0,
\endgathered
\label{LinProb1}
\end{equation}
\begin{equation}
\gathered
\cd_t h - \Delta h = \rot (u\times a) \qquad \mbox{in}\quad Q^+_0,
\\
\div h =0 \qquad \mbox{in}\quad Q^+_0,
\\
\begin{array}c
h_3|_{y_3=0}=0, \qquad \frac{\cd h_1}{\cd y_3}\big|_{y_3=0}\ =\ \frac{\cd h_2}{\cd y_3}\big|_{y_3=0} \ = \ 0.
\end{array}
\endgathered
\label{LinProb2}
\end{equation}
For the solution to the linear problem (\ref{LinProb1}) --- (\ref{LinProb2}) by a standard way (see \cite{VyaShi} Theorem 4.1) we obtain
\begin{equation}
Y_\tau(u) + \tilde Y_\tau(h) \ \le \ C(M)~ \tau^{1/3} ~Y_{1} (u, h, q)
\label{Y_u_h}
\end{equation}
Moreover from the second relation in \eqref{Contradiction_asumptions} we have
\begin{equation}
\liminf_{m \to \infty} Y_\tau (u^m,p^m,h^m) \geq c \tau^{\frac13}.
\label{lmc1}
\end{equation}
On the other hand we will show that
\begin{equation}
\limsup_{m \to \infty} Y_\tau (u^m,p^m,h^m) \leq c_* \tau^{\frac13}
\label{lmc2}
\end{equation}
for some constante $c_*$ Taking in \eqref{lmc1} a constant $c > c_*$ we obtain a contradiction. This
contradiction will prove the theorem.
From (\ref{Strong_Convergence}) we conclude
$$
\lim\limits_{m\to +\infty} Y_\tau(u^m) = Y_\tau(u), \qquad \lim\limits_{m\to +\infty} \tilde Y_\tau(h^m) = \tilde Y_\tau(h)
$$
and hence
\begin{equation}
\limsup\limits_{m\to \infty} Y_\tau (u^m, h^m , q^m ) \ \le \ Y_\tau(u) + \tilde Y_\tau(h) + \limsup\limits_{m\to \infty} \hat Y_\tau (q^m).
\label{Y estimate}
\end{equation}
Then to prove \eqref{lmc2} it is sufficient to show that
\begin{equation}
\limsup_{m \to \infty} \hat Y_\tau (q^m) \leq c(M) \tau^{\frac13}.
\label{lcm3}
\end{equation}
For this purpose we decompose $(u^m, q^m)$ and $(u,q)$ as
$$
\gathered
u^m = u^m_1 + u^m_2, \qquad q^m = q^m_1+q^m_2, \\
u=u_1+u_2,\ \qquad\quad q = q_1+q_2,
\endgathered
$$
where $(u^m_1, q^m_1)\in W^{2,1}_{\frac 98, \frac 32}(\Q_0)\times W^{1,0}_{\frac 98, \frac 32}(\Q_0)$
are determined as a solutions of the following initial boundary-value problems in $\Q_0$:
$$
\gathered
\cd_t u^m_1 - \tilde \Delta_{\vphi_m} u^m_1 + \tilde\nabla_{\vphi_m} q_1^m = f^m \qquad \mbox{in}\quad \Q_0, \\
\tilde\nabla_{\vphi_m} \cdot u^m_1 =0 \qquad \mbox{in}\quad \Q_0, \\
u^m_1|_{t=-1} =0, \qquad u^m_1|_{y_3=0}=0,
\endgathered
$$
where $f^m$ is defined by the expression $\L_m \rot \L_m^{-T} h^m \times ( \delta_m h^m + b_1(H^m)) -\delta_m (u^m\cdot \tilde\nabla_{\vphi^m} )u^m$
on the set $\Q_0(\frac 9{10})$
and extended by zero onto the whole $\Q_0$. Similarly, $(u_1, q_1)$ are determined by the relations
\begin{equation}
\gathered
\cd_t u_1 - \Delta u_1 + \nabla q_1 = f \qquad \mbox{in}\quad \Q_0, \\
\div u_1 =0 \qquad \mbox{in}\quad \Q_0, \\
u_1|_{t=-1} =0, \qquad u_1|_{y_3=0}=0,
\endgathered
\label{Global_Linear_Problem}
\end{equation}
where $f$ determined by the expression $\rot h\times a $ on the set $\Q_0(\frac 9{10})$ and extended by zero onto the whole $\Q_0$.
As functions $u^m_1-u_1$, $q^m_1-q_1$ are the solution of the first initial boundary-value problem in $\Q_0$
with the right-hand side $f^m-f$ and zero initial and boundary conditions from lemma \ref{nonzerodivStokes_Lemma}, we obtain the estimate
\begin{equation}
\gathered
\| u^m_1 \|_{W^{2,1}_{\frac 98, \frac 32}(\Q_0)} +
\| \nabla q^m_1 \|_{L_{\frac 98, \frac 32}(\Q_0) } \ \le C \| f^m\|_{L_{\frac 98, \frac 32}(\Q_0(\frac 9{10}))}
\\
\| u^m_1-u_1 \|_{W^{2,1}_{\frac 98, \frac 32}(\Q_0)} + \| \nabla q^m_1 - \nabla q_1 \|_{L_{\frac 98, \frac 32}(\Q_0) } \
\le C \| f^m-f\|_{L_{\frac 98, \frac 32}(\Q_0(\frac 9{10}))}
\endgathered
\label{u_1 convergence}
\end{equation}
Note that
\begin{equation}
\gathered
\| f^m\|_{L_{\frac 98, \frac 32}(\Q_0(\frac 9{10}))} \ \le \ C(M)
\\
\| f^m - f\|_{L_{\frac 98, \frac 32}(\Q_0(\frac 9{10}))} \ \to \ 0, \quad \mbox{as} \quad m\to \infty.
\endgathered
\label{f^m boundedness}
\end{equation}
So, taking into account the imbedding $W^{1,0}_{\frac 98, \frac 32}(\Q_0(\frac 9{10})) \hookrightarrow L_{ \frac 32}(\Q_0(\frac 9{10}))$ we can conclude that
$$
\begin{array}c
q^m_1\to q_1\quad \mbox{in} \quad L_{\frac 32}(\Q_0(\frac 9{10}))
\end{array}
$$
and hence for any $\tau \in (0, \frac 9{10})$
$$
\lim\limits_{m\to \infty } Y_\tau (q^m_1 ) \ = \ Y_\tau (q_1).
$$
On the other hand, $(u_1, q_1)$ is a solution of the linear Stokes problem in $\Q_0$. Hence from lemma \ref{zerodivStokes_Lemma} we conclude
$$
Y_\tau (q_1) \ \le \ C(M)~\tau^{1/3}~ Y_{\frac 9{10}} (q_1)
$$
We need to estimate $Y_{\frac 9{10}} (q_1)$.
From imbedding theorem $L_{\frac 32}(B^+_0(\frac{9}{10})) \hookrightarrow W^1_{\frac 98}(B^+_0(\frac{9}{10}))$ we conclude
$$
Y_{\frac 9{10}} (q_1)\ \le \ C ~\| \nabla q_1\|_{L_{\frac 98,\frac 32}(B^+_0(\frac{9}{10}))}
$$
For the solution $(u_1, q_1)$ of the initial-boundary value problem (\ref{Global_Linear_Problem}) we have the estimate
$$
\| u_1\|_{W^{2,1}_{\frac 98, \frac 32}(\Q_0(\frac 9{10}))} + \| \nabla q_1\|_{L_{\frac 98, \frac 32}(\Q_0(\frac 9{10}))} \
\le \ C(M)~ \| \nabla h\|_{L_{\frac 98, \frac 32}(\Q_0(\frac 9{10}))}
$$
Using H\" older inequality \ $\| \nabla h\|_{L_{\frac 98, \frac 32}(\Q_0(\frac 9{10}))} \ \le \ C~\| \nabla h\|_{L_2(\Q_0(\frac 9{10}))}$
and taking into account the weak convergence (\ref{Weak_Convergence}) from which we conclude
$$
\| \nabla h\|_{L_2(\Q_0(\frac 9{10}))} \ \le \ \liminf\limits_{m\to \infty} \| \nabla h^m \|_{L_2(\Q_0(\frac 9{10}))},
$$
and using (\ref{Gradient boundedness}) we obtain
$$
Y_{\frac 9{10}} (q_1) \ \le \ C(M).
$$
Now we consider functions $(u^m_2, q^m_2)$ determined by relations
\begin{equation}
\gathered
u^m_2:=u^m-u^m_1, \qquad q^m_2:= q^m -q^m_1.
\endgathered
\label{u_2 definition}
\end{equation}
These functions satisfy the homogeneous Stokes problems in $\Q_0(\frac{9}{10})$:
$$
\begin{array}c
\cd_t u^m_2 - \Delta u^m_2 + \nabla q_2^m = 0 \qquad \mbox{in}\quad \Q_0(\frac{9}{10}), \\
\div u^m_2 =0 \qquad \mbox{in}\quad \Q_0(\frac{9}{10}), \\
u^m_2|_{x_3=0}=0,
\end{array}
$$
$$
\begin{array}c
\cd_t u_2 - \Delta u_2 + \nabla q_2 = 0 \qquad \mbox{in}\quad \Q_0(\frac{9}{10}), \\
\div u_2 =0 \qquad \mbox{in}\quad \Q_0(\frac{9}{10}), \\
u_2|_{x_3=0}=0.
\end{array}
$$
Then
$$
\begin{array}c
\| u^m_2\|_{W^{2,1}_{9, \frac 32}(\Q_0(\frac 4{5}))}+ \| \nabla q^m_2\|_{L_{9, \frac 32}(\Q_0(\frac 4{5}))} \ \le \ C ~\Big(
\| u^m_2\|_{L_3(\Q_0(\frac 9{10}))} + \| q^m_2\|_{L_{\frac 32}(\Q_0(\frac 9{10}))}\Big)
\end{array}
$$
Note that due to (\ref{u_2 definition}), (\ref{Basic_boundedness}) and the first inequalities in (\ref{u_1 convergence}), (\ref{f^m boundedness})
we have the estimate
$$
\gathered
\| u^m_2\|_{L_3(\Q_0(\frac 9{10}))} + \| q^m_2\|_{L_{\frac 32}(\Q_0(\frac 9{10}))} \ \le \\
\le \
\| u^m\|_{L_3(\Q_0(\frac 9{10}))} + \| q^m\|_{L_{\frac 32}(\Q_0(\frac 9{10}))} \ + \
\| u^m_1\|_{L_3(\Q_0(\frac 9{10}))} + \| q^m_1\|_{L_{\frac 32}(\Q_0(\frac 9{10}))} \ \le \\ \le \ C(M)
\endgathered
$$
On the other hand, from the H\" older inequality and lemma \ref{zerodivStokes_Lemma} we obtain for any $\tau \in (0, \frac 45)$
$$
\gathered
\hat Y_{\tau}(q^m_2)= \tau \Big( \frac1{|\Q_0(\tau)|} \int\limits_{\Q_0(\tau)} |q_2^m- [q_2^m]_{\Q_0(\tau)}|^{\frac 32}~dx dt\Big)^{\frac 23} \ \le \\
C \tau^2 ~ \Big( \frac1{|\Q_0(\tau)|} \int\limits_{\Q_0(\tau)} |\nabla q_2^m |^{\frac 32}~dx dt\Big)^{\frac 23} \\
\le \ C~\tau^{\frac 76} ~\| \nabla q^m_2 \|_{L_{9, \frac 32}(\Q_0(\frac 45))} \ \le \ C(M)~\tau^{\frac 76}
\endgathered
$$
Summarizing all previous estimates we arrive at
$$
\gathered
\limsup\limits_{m\to \infty} \hat Y_\tau (q^m) \ \le \ \lim\limits_{m\to \infty} \hat Y_\tau (q^m_1) + \limsup\limits_{m\to \infty} \hat Y_\tau (q^m_2) \
\le \ C(M) ~ \tau^{\frac 13}
\endgathered
$$
which gives us a contradiction with \eqref{lmc1}.
\qed
Iterating \eqref{Decay_Estimate} and using scaling argument it is easy to obtain the following lemma (see \cite{ESS} and \cite{SSS} for details).
\begin{lemma}
There exists an absolute constant $\ep_{**}>0$ such that for any $M>0$ and $\beta \in (0,1/3)$ there exists $\tau \in (0,1/2)$ with the following properties.
For any boundary suitable weak solution $(v,H,p,\vphi)$ of the MHD system (\ref{MHD_NSE}), (\ref{MHD_Magnetic}) near the boundary in $\Q(1)$
the following implication holds:
if
$$
Y_1(v,H,p) + \| \vphi \|_{C^2} \ < \ \ep_{**},
$$
and
$$
b_R(H) \le \ M
$$
then
\begin{equation}
Y_{\tau^k} (v,H,p) \ \le \tau^{\beta k}~Y_1 (v,H,p)
\end{equation}
\end{lemma}
Theorem \ref{CKN_theorem} follows from this lemma in the standard way
by scaling arguments, and combination of boundary estimates with the internal estimates obtained in \cite{Vya}.
See details in \cite{Seregin_JMFM}, \cite{Seregin_Aa}, \cite{SSS}, \cite{Seregin_Handbook}.
\section{Estimates of solutions of the heat equation with homogeneous boundary data}
\label{Heat_equation_estimate_section}
\setcounter{equation}{0}
In this section we will obtain some estimates for $L_2$-norms of solutions of homogeneous initial and boundary problem for the heat equation in half-ball.
Namely, we consider the following problem
\begin{equation}
\gathered
\cd_t h - \lap h = f \quad \text{ in } \Q_0(R),\\
h_3|_{x_3=0} = 0, \quad \frac{\cd h_i}{\cd x_3}|_{x_3=0} = 0 \quad \text{in } \Qdva(R) \quad i=1,2,\\
h|_{t=-R^2} = 0,
\endgathered
\label{rhsel1}
\end{equation}
here $h: \Q_0(R) \to \R^3$ is an unknown function.
The main result of this section is the following theorem
\begin{theorem}
Let $f \in L_{\frac 32,1}( \Q_0(R) )$, and $h$ is the solution of \eqref{rhsel1}. Then the following estimate holds
\begin{equation}
\| h \|_{L_2( \Q_0(R) )} \leq c R^{\frac 12} \| f \|_{L_{\frac 32, 1}}( \Q_0(R) ).
\label{lh1}
\end{equation}
\label{lhTh1}
\end{theorem}
We note, that conditions for $h$ on a plain part of a boundary allow us to extend this function
into whole $B$ by the following way: components $h_1$ and $h_2$ will be extended as even functions and component $h_3$ as odd function. The right hand side
can be extended by the same manner. We also put $f \equiv 0$ in $\R^3 \backslash \B(R)$.
So it is sufficient to prove the theorem for the solution of the following Cauchy problem for the heat equation.
\begin{equation}
\gathered
\d_t h - \Delta{h} = f \quad \text{ in } \quad \Pi_R, \\
h|_{t= -R^2} = 0,
\endgathered
\label{Heat_eq}
\end{equation}
here $\Pi_R = \R^3 \times [-R^2,0)$.
To prove this theorem we will need the Young inequality for convolutions (se. \cite{Hardy}, \cite{Stein}). Namely, let
$$ g(x) = \intl_{\R^n} K(x-y) f(y)\, dy,$$
then for arbitrary $1 \leq p \leq q \leq \infty$, the following estimate holds
\begin{equation}
\| g \|_q \leq \| K \|_l \| f \|_p, \quad \text{ here } \quad 1 - \frac1p + \frac 1q = \frac 1l.
\label{convYung}
\end{equation}
In particular we will use an inequality
\begin{equation}
\| g \|_2 \leq \| K \|_{\frac65} \| f \|_{\frac 32}.
\label{lh3}
\end{equation}
\begin{lemma}
Let $f \in L_1(-R^2,0)$, for some $R > 0$ and
\begin{equation}
g(t) = \intl_{-R^2}^t \frac{ f(\tau)\, d\tau}{ ( t - \tau )^{\frac14}},
\label{lh3}
\end{equation}
then
\begin{equation}
\| g \|_{L_2(-R^2,0)} \leq c R^{\frac12} \| f \|_{L_1(-R^2,0)}
\label{lh4}
\end{equation}
\end{lemma}
{\bf Proof.} In the case $R = 1$ inequality \eqref{lh4} is the corollary of \eqref{convYung} with $p = 1$ and $q = l = 2$.
In general case we make scaling transformations.
Namely, let consider the functions
$$f^*(s) = f (R^2 s), \quad g^*(s) = g(R^2 s), \quad \hat{g}(s) = \intl_{-R^2}^s \frac{ f^*(\sigma)\, d\sigma}{(s - \sigma)^{\frac14}}.$$
Changing variables under the integral, we have
$$
\gathered
g^*(s) = R^{\frac32} \hat{g}(s), \\
\|f^* \|_{L_1(-1,0)} = R^{-2} \|f \|_{L_1(-R^2,0)}, \quad \| g \|_{L_2(-R^2,0)} = R \| g^* \|_{L_2(-1,0)}.
\endgathered
$$
And then we get
$$
\gathered
\| g \|_{L_2(-R^2,0)} = R \| g^* \|_{L_2(-1,0)} = R^{\frac52} \| \hat{g} \|_{L_2(-1,0)} \leq \\
\leq c R^{\frac52} \| f^* \|_{L_1(-1,0)} = c R^{\frac12} \| f \|_{L_1(-R^2,0)}.
\endgathered
$$
\qed
{\bf Proof of theorem \ref{lhTh1}.} The solution of \eqref{Heat_eq} can be found as follows
$$ h(t,x) = \intl_{-R^2}^t \intl_{\R^3} \frac{ e^{-\frac{|x-y|^2}{4(t-\tau)}}}{(4 \pi (t - \tau))^{\frac32}} f(\tau,y) \,dy d\tau.$$
We fix $t$, look on excretion under the integral by time as function with values in Banach space $L_2(\R^3)$ and use
inequality to the norm of its integral
\begin{equation}
\| h (t,\cdot) \|_{L_2(\R^3)} \leq
\intl_{-R^2}^t \l\| \intl_{\R^3} \frac{ e^{-\frac{|x-y|^2}{4(t-\tau)}}}{(4 \pi (t - \tau))^{\frac32}} f(\tau,y) \,dy\r\|_{L_2(\R^3)} \,d\tau.
\label{lh5}
\end{equation}
By direct computations we find
\begin{equation}
\l( \intl_{\R^3} e^{-\frac{z^2}{4(t-\tau)} \cdot \frac65} \,dz \r)^{\frac56} = c (t - \tau )^{\frac54}.
\label{lh6}
\end{equation}
Then from \eqref{lh3}, \eqref{lh5} and \eqref{lh6} we obtain
$$\| h(t,\cdot) \|_{L_2(\R^3)} \leq c \intl_0^t \frac{\| f(\tau,\cdot) \|_{L_{\frac32}(\R^3)}}{(t - \tau)^{\frac14}} \,d\tau.$$
Next form \eqref{lh4}
$$ \|h \|_{L_2(\Pi_R)} \leq c R^{\frac12} \| f \|_{L_{\frac32 , 1}(\Pi_R)}.$$
\qed
\section{Estimates for the magnetic component}
\label{H_estimate_section}
\setcounter{equation}{0}
In this section we will get estimates for magnetic component of the suitable weak solution in $\Q(R)$.
We recall, that $H$ satisfy the following integral identity
\begin{equation}
\intl_{\Q(R)} ~ \Big( - H\cdot \cd_t \psi + \rot H\cdot \rot \psi - (v\times H) \cdot \rot \psi \Big)~dxdt \ = 0,
\label{lhes1}
\end{equation}
for all \ $\psi \in C^{\infty}(\Q(R))$ \ such that \ $\psi_\nu |_{\cd\Om\times (-R^2,0)}=0$ and $\psi(-R^2,x) = \psi(0,x) = 0$.
Also we have $H_\nu |_{\Gamma_R \times (-R^2,0)}=0$ and $\div H = 0$.
Note, that without loss of generality we can assume, that \eqref{lhes1} holds only for test functions with $\div \psi = 0$.
\begin{theorem}
Assume that \eqref{lhes1} holds for some function $v \in \WO$.
Then there exist absolute positive constants $\ep_1$, $\al$ and $c$ such that for any $\ep\in (0,\ep_1)$ and any $K>0$ if
\begin{equation}
\sup\limits_{r\in (0,1)} E(r) < \ep, \quad \| \vphi \|_{C^2} < \ep \quad \text{and} \quad \sup\limits_{r\in (0,1)} E_*(r) < K
\label{Assumptions}
\end{equation}
then for any $0<r< R \leq 1$
\begin{equation}
F_2(r) \ \le \ c\left(\frac{r}{R}\right)^{2} F_2(R) + c \ep (F_2(R) + K + 1).
\label{F_2_l1}
\end{equation}
\label{Bound_F_2}
\end{theorem}
{\bf Proof.} We proceed to the coordinates \eqref{change_variables}.
Then \eqref{lhes1} transforms to
$$
\intl_{-R^2}^0 \l[ -(\tH,\d_t\tpsi) + ( \L \rot \L^T \tH , \L \rot \L^T \tpsi) - ( \tv \times \tH , \L \rot \L^T \tpsi) \r]\,dt = 0,
$$
here $(\cdot , \cdot)$ is $L_2$ inner product.
Next we introduce new functions $h = \L^{-1} \tH$ and $\eta = \L^{-1} \tpsi$. Then the last identity can be written as follows
\begin{equation}
\intl_{-R^2}^0 \l[ -(h,\d_t \A \eta) + (\A \rot \A h,\rot \A \eta) - (\tv \times \tH, \L \rot \A \eta) \r]\,dt = 0
\label{lhes2}
\end{equation}
here $\A = \L^T \L$.
Note, that $\nabla_y \cdot h = \nabla_y \cdot \L^{-1} \tH = \L^{-T} \nabla_y \cdot \tH = \nabla_x \cdot H \circ e = 0$.
Also $H \cdot \nu \circ e = \L h \cdot \tilde{\nu} = h \cdot \L^{T} \tilde{\nu} = -h_3$. Similar identities holds for function $\eta$.
As the result we can consider function $h$ as the generalized solution of parabolic system which corresponds to holding the identity \eqref{lhes2}
for arbitrary function $\eta \in C^{\infty}$, such that
\begin{equation}
\div \eta = 0 \quad \text{ and } \quad \eta_3|_{y_3 = 0} = 0.
\label{lhes5}
\end{equation}
Then to estimate $L_2$-norm of $H$ it will sufficient to obtain
inequality for $\| h \|_2$. To do this we decompose it into three
parts
\begin{equation}
h = \hi + \hii + \hiii.
\label{lhes7}
\end{equation}
Here $\hi$ is the solution of the following initial-boundary problem
\begin{equation}
\begin{aligned}
&\d_t \hi - \lap \hi = \A \rot \l( \L^T ( \tv \times \tH) \r)\\
& \hi_3|_{y_3 = 0} = 0, \quad \frac{\d \hi_1}{\d y_3} |_{y_3 = 0} = 0, \quad \frac{\d \hi_2}{\d y_3} |_{y_3 = 0} = 0,\\
&\hi|_{t = -R^2} = 0.
\end{aligned}
\label{lhes3}
\end{equation}
From \eqref{rhsel1} we have
$$ \| \hi \|_{2,R} \leq c R^{\frac12} \| \A \rot \l( \L^T ( \tv \times \tH) \r) \|_{\frac32,1}
\leq c(\vphi) R^{\frac12} ( \| |v||\nabla H| + |\nabla v||H| \|_{\frac32,1}).$$
Next we use H\"older inequality and the embedding theorem
\begin{equation}
F_2(R,\hi) \leq c(\vphi) \l( E(R)E_*(R) + E(R)F_2(R) \r).
\label{lhes4}
\end{equation}
Boundary conditions for function $\hi$ imply that the following identity
\begin{equation}
\intl_{-R^2}^0 \l[ -(\hi,\d_t \eta) + (\rot \hi,\rot \eta) - (\tv \times \tH, \L \rot \A \eta) \r]\,dt = 0
\label{lhes6}
\end{equation}
for every function $\eta$ satisfying \eqref{lhes5}.
As \eqref{lhes4} and \eqref{lhes6} are stored at the replacing $\hi$ to the solenoidal component of its Weil decomposition
without loss of generality we can assume, that $\div \hi = 0$.
The second component of \eqref{lhes7} is the solution of the following problem
\begin{equation}
\gathered
\intl_{-R^2}^0 \l[ -(\hii,\d_t \eta) + (\rot \hii, \rot \eta)\r]\,dt =\\
\intl_{-R^2}^0 \l[ (h, \d_t( \A - I)) + (\rot h, \rot \eta) - (\A \rot \A h, \rot \A \eta) \r]\,dt
\endgathered
\label{lhes8}
\end{equation}
for arbitrary function $\eta$ satisfying \eqref{lhes5} with the initial and boundary conditions
$$
\hii_3|_{y_3 = 0} = 0, \quad \hii|_{t = -R^2} = 0.
$$
To obtain the estimate for $\hii$ we consider the dual problem
\begin{equation}
\begin{aligned}
& \d_t \eta + \lap \eta = -\hii \\
& \eta_3|_{y_3 = 0} = 0, \quad \frac{\d \eta_1}{\d y_3} |_{y_3 = 0} = 0, \quad \frac{\d \eta_2}{\d y_3} |_{y_3 = 0} = 0,\\
&\eta|_{t = 0} = 0.
\end{aligned}
\label{lhes9}
\end{equation}
For the right hand side of \eqref{lhes8} we have the following identity
\begin{equation}
\gathered
(\rot h , \rot \eta) - (\A \rot \A h, \rot \A \eta) = \\
( \rot h , \rot (I - \A) \eta) + (\rot (I - \A)h, \rot \A \eta) + ((I - \A) \rot \A h, \rot \A \eta).
\endgathered
\label{lhes10}
\end{equation}
Substituting into \eqref{lhes8} the solution of \eqref{lhes9} we have
$$
\| \hii \|_2 = \intl_{-R^2}^0 \l[ (h, \d_t( \A - I)) + (\rot h, \rot \eta) - (\A \rot \A h, \rot \A \eta) \r]\,dt
$$
Note that the matrix $\A$ is close the identity, so from \eqref{lhes10} and coercive estimates for \eqref{lhes8} we obtain
\begin{equation}
F_2(R, \hii) \leq c \| \vphi \|_{C^2} \l( F_2(R) + E_*(R) \r).
\label{lhes11}
\end{equation}
The third component of \eqref{lhes7} satisfy to the homogenous boundary problem for the heat equation
\begin{equation}
\begin{aligned}
& \d_t \hiii - \lap \hiii = 0 \\
& \hiii_3|_{y_3 = 0} = 0, \quad \frac{\d \hiii_1}{\d y_3} |_{y_3 = 0} = 0, \quad \frac{\d \hiii_2}{\d y_3} |_{y_3 = 0} = 0.
\end{aligned}
\label{lhes12}
\end{equation}
Extending $\hiii$ into whole cylinder and using mean value theorem we have
\begin{equation}
\gathered
F_2(r,\hiii) \leq c \l(\frac{r}{R} \r)^2 F_2 (R, \hiii) \\
\leq c \l(\frac{r}{R} \r)^2 \l( F_2 (R, h) + F_2(R,\hi + \hii) \r).
\endgathered
\label{lhes13}
\end{equation}
Combining \eqref{lhes4}, \eqref{lhes11} and \eqref{lhes13} we obtain the statement of the theorem.
\qed
\section{Estimates of Energy Functionals}
\label{EnergyEstimates_Section}
\setcounter{equation}{0}
Now we define few more functionals.
Note that all these functionals are invariant with respect to the natural scaling of the MHD system.
For $r\le 1$, $q\in [1,\frac{10}3]$, $s\in [1,\frac 98]$ and $(v,p,H,\vphi)$ suitable weak solution to the MHD system in $\Q(R)$ $0 < r < R <1$
we introduce the following quantities:
$$
\begin{array}c
A( r) \equiv \Big( \frac 1{r}
\sup\limits_{t\in (-r^2, 0)} \int\limits_{B^+(r)} |v|^{2}~dy
\Big)^{1/2}, \\
A_*( r) \equiv \Big( \frac 1{r}
\sup\limits_{t\in (-r^2, 0)} \int\limits_{B^+(r)} |H |^{2}~dy
\Big)^{1/2},
\\
C_q( r) \equiv \Big( \frac 1{r^{5-q}} \int\limits_{Q^+(r)} |v|^q~dydt
\Big)^{1/q},
\\ D(r) \equiv \Big( \frac 1{r^2} \int\limits_{Q^+(r)}
|p - [p]_{B^+(r)}|^{3/2}~dydt \Big)^{2/3},
\\
D_s(r) = R^{\frac 53 -\frac 3s} \Big( \int\limits_{-r^2}^{0} \Big(
\int\limits_{B^+(r)} |\nabla p|^{s}~dy \Big)^{\frac 1s \cdot
\frac 32}~dt \Big)^{2/3},
\end{array}
$$
$$
C(r) = C_3(r), \qquad F(r)= F_3(r), \qquad D_*(r)= D_{\frac{36}{35}}(r).
$$
\noindent
First we formulate the set of results following from the general theory of functions:
\begin{lemma}
\label{Interpolation}
Let $R > 0$, $\vphi \in C^2(\Bdva(R))$, $v$, $H\in W^{1,0}_2(Q^+(R))$ and $p\in W^{1,0}_{\frac 98, \frac 32}(Q^+(R))$ are arbitrary functions.
Assume $v|_{x_3= \vphi(x_1,x_2)}=0$. Then for any $0 < r <R$ the following inequalities hold:
\begin{equation}\label{C_3}
C(r) \ \le \ ~A^{\frac 12}(r)E^{\frac 12}(r),
\qquad
F(r)\ \le \ A_*^{\frac 12}(r)[ E_*^{\frac 12}(r) + F_2^{\frac 12}(r)]
\end{equation}
\begin{equation}
D(r)\ \le \ c D_1(r), \qquad D_1(r)\ \le \ c D_s(r), \qquad \forall s>1.
\label{D}
\end{equation}
\end{lemma}
The proof of this lemma follows from interpolation inequalities and imbedding theorems.
Proof of the similar inequalities for the Navier-Stokes system can be found in \cite{LS}.
\begin{lemma}
\label{EnergyEstimate_Lemma}
Assume $(v,p,H,\vphi)$ is a boundary suitable weak solution to the MHD equations in $Q^+$.
Then for any $r\in (0,1)$ the following inequality holds
\begin{equation}
\gathered
A(r/2) + A_*(r/2) + E(r/2) + E_*(r/2) \ \le \\ \le \ c~ \Big( C_2(r)+ F_2(r) + C^{\frac 12}(r)D^{\frac 12}(r) + C^{\frac 32}(r)\Big) \ + \\
+ \ c~ \Big( C^{\frac 12}(r)A_*^{\frac 12}(r)E_*^{\frac 12}(r) + F^{\frac 12}(r)A_*^{\frac 12}(r)E^{\frac 12}(r) \Big)
\label{Lo_En_Iq}
\endgathered
\end{equation}
\end{lemma}
\noindent
{\bf Proof.} Estimate (\ref{Lo_En_Iq}) follows from (\ref{LEI}) in a standard way. We just explain the specific estimates of the terms
$$
I_1:=\int\limits_{Q^+(r)} ~ |H|^2 (v \cdot \nabla \zeta) ~dxdt \quad \mbox{and} \quad I_2:=\int\limits_{Q^+(r)} ~ (v \cdot H) (H\cdot \nabla \zeta) ~dxdt.
$$
$I_1$ we transform in the following way
$$
\begin{array}c
I_1\ = \ \int\limits_{Q^+(r)} ~ \Big( |H|^2 - [|H|^2]_{B^+(r)} \Big) (v \cdot \nabla \zeta) ~dxdt
\end{array}
$$
Applying the H\" older inequality we obtain
$$
|I_1| \ \le \ \frac cr ~\int\limits_{-r^2}^0 \left\| |H|^2 - [|H|^2]_{B^+(r)}\right\|_{L_{\frac 32}(B^+(r))} \| v\|_{L_{3}(B^+(r))}~dt
$$
Applying the inequality $\| f-[f]_{B^+(r)}\|_{L_{\frac 32}(B^+(r))} \le c \| \nabla f \|_{L_1(B^+(r))}$, we arrive at
$$
\gathered
|I_1| \ \le \ \frac cr ~\int\limits_{-r^2}^0 \| \nabla |H|^2 \|_{L_{1}(B^+(r))} \| v\|_{L_{3}(B^+(r))}~dt \ \le \\ \le \
\frac cr ~\int\limits_{-r^2}^0 \| H\|_{L_2(B^+(r))} \| \nabla H \|_{L_{2}(B^+(r))} \| v\|_{L_{3}(B^+(r))}~dt \ \le \\ \le \
\frac c{r^{2/3}} ~\| H\|_{L_{2,\infty}(Q^+(r))} \| \nabla H \|_{L_2(Q^+(R))} \| v \|_{L_3(Q^+(r))}\ \le \ cr~ A_*(r) E_*(r) C(r)
\endgathered
$$
For $I_2$ we obtain relations
$$
I_2 \ = \ \int\limits_{Q^+(r)} ~ \Big ( (v \cdot H)- [v \cdot H]_{B^+(r)}\Big) (H\cdot \nabla \zeta) ~dxdt
$$
Hence
$$
\gathered
|I_2| \ \le \ \frac cr ~\int\limits_{-r^2}^0 \left\| (v \cdot H)- [v \cdot H]_{B^+(r)} \right\|_{L_{2}(B^+(r))} \| H\|_{L_{2}(B^+(r))}~dt \ \le \\
\le \ \frac cr ~ \| H\|_{L_{2,\infty}(Q^+(r))} \int\limits_{-r^2}^0 \left\| \nabla (v \cdot H) \right\|_{L_{\frac 65}(B^+(r))} ~dt \ \le \
\frac cr ~ \| H\|_{L_{2,\infty}(Q^+(r))} \ \times \\ \times \ \int\limits_{-r^2}^0 \Big( \| \nabla v \|_{L_2(B^+(r))} \| H \|_{L_3(B^+(r))}
+ \| \nabla H \|_{L_2(B^+(r))} \| v \|_{L_3(B^+(r))}\Big)~dt \ \le \\
\le \ \frac c{r^{2/3}} ~\| H\|_{L_{2,\infty}(Q^+(r))} \Big( \| \nabla v \|_{L_2(Q^+(r))} \| H \|_{L_3(Q^+(r))}
+ \| \nabla H \|_{L_2(Q^+(r))} \| v \|_{L_3(Q^+(r))}\Big) \endgathered
$$
So, we obtain
$$
|I_2| \ \le \ cr~ A_*(r) ~\Big(~E(r) F(r) + E_*(r) C(r)~\Big)
$$
\qed
\begin{lemma}
\label{PresureEstimate_Lemma}
Assume $(v,p,H,\vphi)$ is a boundary suitable weak solution to the MHD equations in $Q^+$ and $\| \vphi \|_{C^2(\Bdva)} < \mu < \frac{\mu_*}2$
where $\mu_*$ is the constant defined in lemma \ref{nonzerodivStokes_Lemma}.
Then for any $r\in (0,1)$ and $\theta \in (0,\frac14)$ the following inequality holds
\begin{equation}
\gathered
D_*(\theta r) \ \le \ c ~\theta^{\frac 43} ~\Big( D_*(r) + E(r) \Big) \ + \\
+ \ c(\theta) ~\Big( A^{\frac 23}(r) E^{\frac 43}(r) + A^{\frac 56}_*(r) F^{\frac 16}(r) E_*(r) \Big)
\endgathered
\label{D_*}
\end{equation}
\end{lemma}
\noindent
{\bf Proof.}
To obtain (\ref{D_*}) we apply the method developed in \cite{Seregin_JMFM}, \cite{Seregin_ZNS271}, see also \cite{SSS}.
Let $e(y)$ is the map defined by \eqref{change_variables}.
We fix $r\in (0,1]$ and $\theta\in (0,\frac 14)$ and without loss of generality we can assume, that
$e^{-1}(\Q(\theta r) \subset \Q_0(2\theta r) \subset \Q_0(r/2) \subset e^{-1}(\Q(r))$.
Then we decompose $v$ and $p$ as
$$
v\ = \ \hat v + \check v,\qquad p \ = \ \hat p +\check p,
$$
where $(\hat v, \hat p)$ is a solution of the perturbed Stokes initial boundary value problem in a half-space
$$
\gathered
\left\{\begin{array}c \cd_t \hat v - \tilde\Delta_{\vphi} \hat v+\tilde\nabla_{\vphi} \hat p \ = \L \rot (\L^{-T} \tH) \times \tH-(\tv \cdot \tilde\nabla_{\vphi} ) \tv , \\
\tilde\nabla_{\vphi} \hat v =0 \end{array}\right. \qquad \mbox{in}\quad \Q_0(\frac{r}2), \\
\hat v|_{t=0}=0, \qquad \hat v|_{y_3=0}=0,
\endgathered
$$
and $(\check v, \check p)$ is a solution of the homogeneous perturbed Stokes system in $Q_0^+(\frac{r}2)$:
$$
\gathered
\left\{\begin{array}c \cd_t \check v - \tilde\Delta_{\vphi} \check v +\tilde\nabla_{\vphi} \check p\ = \ 0, \\
\tilde\nabla_{\vphi} \check v =0 \end{array}\right. \qquad \mbox{in}\quad Q^+_0(\frac{r}2), \\
\check v|_{y_3=0}=0.
\endgathered
$$
For $\nabla \hat p$ and $\nabla \check p$ from lemmas \ref{nonzerodivStokes_Lemma} and \ref{zerodivStokes_Lemma} we have the following estimates.
$$
\gathered
\| \nabla \hat p\|_{L_{\frac{36}{35},\frac 32}(Q^+_0(\frac{r}2))} \ + \ \frac 1r \| \nabla \hat v \|_{L_{\frac{36}{35},\frac 32}(Q^+_0(\frac{r}2))} \ \le \\
\le \ c~\Big( ~\| H\times \rot H\|_{L_{\frac{36}{35}, \frac 32}(Q^+(r))} \ + \ \| (v\cdot \nabla)v \|_{L_{\frac{36}{35}, \frac 32}(Q^+(r))}~\Big),
\endgathered
$$
$$
\| \nabla \check p\|_{L_{\frac{36}{35},\frac 32}(Q^+_0(\theta r))} \
\le \ c~\theta^{\frac{31}{12}} ~\Big( ~\frac 1r \| \nabla \check v \|_{L_{\frac{36}{35},\frac 32}(Q_0^+(\frac{r}2))}
\ + \ \| \nabla \check p \|_{L_{\frac{36}{35},\frac 32}(Q_0^+(\frac{r}2))}~\Big).
$$
From the H\" older inequality we obtain
$$
\gathered
\| H\times \rot H\|_{L_{\frac{36}{35}, \frac 32}(Q^+(r))} \
\le \ c ~r^{\frac 29} ~\| H \|_{L_{2,\infty}(Q^+(r))}^{\frac 56} \| \nabla H\|_{L_2(Q^+(r))} \| H \|_{L_3(Q^+(r))}^{\frac 16}
\endgathered
$$
$$
\gathered
\| (v\cdot \nabla)v \|_{L_{\frac{36}{35}, \frac 32}(Q^+(r))} \ \le \ c~r^{\frac 14}~ \| (v\cdot \nabla )v \|_{L_{\frac{9}{8}, \frac 32}(Q^+(r))} \ \le \\
\le \ c~r^{\frac 14}~
\| v\|_{L_{2,\infty}(Q^+(r))}^{\frac 23} \| \nabla v\|_{L_2(Q^+(r))}^{\frac 43}
\endgathered
$$
Representing $\check v = v-\hat v$, $\check p=p-\hat p$ and gathering all above estimates for $\hat p$ and $\hat v$ we obtain
$$
\gathered
D_*( \theta r) \ \le \ c~\theta^{\frac 43} ~\Big( ~D_*(r) + E(r) + A^{\frac 23}(r) E^{\frac 43}(r) + A_*^{\frac 56}(r) E_*(r) F^{\frac 16}(r) ~\Big) \ + \\
+ \ c(\theta)~ \Big( ~A^{\frac 23}(r) E^{\frac 43}(r) + A_*^{\frac 56}(r) E_*(r) F^{\frac 16}(r) ~\Big)
\endgathered
$$
\qed
\section{CKN condition and Partial Regularity of Solutions}
\label{Final_section}
\setcounter{equation}{0}
In this section we present the proofs of Theorems \ref{CKN_theorem} and \ref{Partial_Regularity}.
We start from proof of the modified version of \eqref{F_2_l1}.
\begin{lemma}
For any $K>0$ there exists a constants $c(K)>0$ and $\ep_2>0$ such that for any $\ep\in (0,\ep_2]$ and
any boundary suitable weak solution $(v,H,p,\vphi)$ of the MHD system in $Q^+$ if
\begin{equation}
\sup\limits_{r\in (0,1)} E(r)\le \ep, \quad \| \vphi \|_{C^2(\Bdva)} < \ep, \quad \sup\limits_{r\in (0,1)} E_*(r) \ \le \ K,
\label{Statement_of_Theorem_E_E1}
\end{equation}
then for some $\al > 0$ and any $0<r< R \leq 1$
\begin{equation}
F_2(r) \ \le \ c\left(\frac{r}{R}\right)^{\al} F_2(R) + c(K) \ep_2.
\label{F_2}
\end{equation}
\label{ckn_lemma1}
\end{lemma}
\noindent
{\bf Proof.} We will use standard iteration technic. Let $R > 0$ and $\theta \in (0,1/2)$. We fix $\ep_1$ from theorem \ref{Bound_F_2}.
Then from \eqref{F_2_l1} we have
\begin{equation}
F_2(\theta R) \leq c_1 (\theta^2 + \ep_1) F_2(R) + c \ep_1 (K+1).
\label{cknl1}
\end{equation}
Next we choose $\theta$ and $\ep_2 < \ep_1$ such that
$$ c_1 \theta^2 \leq \frac14, \quad c_1 \ep_2 \leq \frac14.$$
Then from \eqref{cknl1} we obtain
$$ F_2(\theta R) \leq \frac12 F_2(R) + c \ep_2 (K+1).$$
Next we will iterate the last inequality
\begin{equation}
\gathered
F_2 (\theta^k R) \leq \frac12 F_2(\theta^{k-1} R) + c \ep_2 (K+1) \leq\\
\leq \frac14 F_2(\theta^{k-2} R) + \l( 1 + \frac12 \r) c \ep_2 (K+1)\leq\\
\leq \frac1{2^k} F_2(R) + c \ep_2 (K+1).
\endgathered
\label{cknl2}
\end{equation}
Finally we put $\al = \log_{\theta} \frac12$ and chose $k>0$ such that $ \theta^{k+1} R \leq r \leq \theta^k R$.
Then from \eqref{cknl2} we obtain
$$
\gathered
F_2(r) \leq c F_2(\theta^k R) \leq c \frac{1}{2^k} F_2(R) + c(K) \ep_2 \leq\\
\leq c \theta^{k\al} F_2(R) + c(K) \ep_2 \leq c \theta^{-\al} \l( \frac{\theta^{k+1} R}{R} \r)^{\al} F_2(R) + c(K) \ep_2 \leq\\
\leq c \l( \frac{r}{R} \r)^{\al} F_2(R) + c(K) \ep_2.
\endgathered
$$
\qed
\begin{lemma} \label{Boundedness_A_A_D}
Denote by \ $\mathcal E(r)$ \ the following functional
$$
\mathcal E (r) \ = \ A(r)+ A_*(r) + D_*(r),
$$
and let $\ep_2>0$ be the absolute constant defined in lemma \ref{ckn_lemma1}.
For any $K>0$ there exists a constant $c(K)>0$ such that for any $\ep\in (0,\ep_2]$ and
any boundary suitable weak solution $(v,H,p,\vphi)$ of the MHD system in $Q^+$ if
\begin{equation}
\sup\limits_{r\in (0,1)} E(r)\le \ep, \quad \| \vphi \|_{C^2(\Bdva)} < \ep, \quad \sup\limits_{r\in (0,1)} E_*(r) \ \le \ K,
\label{Statement_of_Theorem_E_E1}
\end{equation}
and
\begin{equation}
F_2(1) \ \le \ M,
\label{M}
\end{equation}
then for any $0< r < R \le 1$
\begin{equation}
\mathcal E(r) \ \le \ c \left(\frac{r}{R}\right)^\be\mathcal E(R) \ + \ c(K)(1+R^\al M).
\label{Estimate_varE}
\end{equation}
where $\be>0$ is some absolute constant.
\end{lemma}
\noindent
{\bf Proof.} Without loss of generality we can assume $K\ge 1$. Then from (\ref{F_2}) we obtain
$$
F_2(R)\le cr^\al M + c \ep_2 C(K).
$$
From this inequality and (\ref{C_3}) we obtain
\begin{equation}
C(R) \ \le \ c~ \mathcal E^{\frac 12} (R) \ep_1^{\frac 12}, \qquad F(R)\
\le \ c~ \mathcal E^{\frac 12} (R)\Big(C(K) + R^{\frac \al 2}M^{\frac 12} \Big)
\label{C}
\end{equation}
Assume $r\in (0,1)$ and $\theta \in (0,\frac 12)$.
From (\ref{Lo_En_Iq}) with the help of (\ref{D}) and the Young inequality we obtain
$$
\gathered
\mathcal E(\theta R) \ \le \ c~\Big( F_2(2\theta R) + D_*(2\theta R)\Big) \ +
\\ + \
c(\theta)\Big( C_2(R) + C(R) + C^{\frac 32}(R) + C^{\frac 12}(R)A_*^{\frac 12}(R)E_*^{\frac 12}(R) + F^{\frac 12}(R)A_*^{\frac 12}(R)E^{\frac 12}(R)\Big)
\endgathered
$$
Taking into account (\ref{C}) and (\ref{Statement_of_Theorem_E_E1}) we obtain
\begin{equation}
\gathered
\mathcal E(\theta R) \ \le \ c~\Big( F_2(2\theta R) + D_*(2\theta R)\Big) \ +
\\ + \
c(\theta)\l(\mathcal E^{\frac 12} (R) \ep_1^{\frac 12} + \mathcal E^{\frac 34} (R) \ep_2^{\frac 34} + \ep_2^{\frac 14}\mathcal E^{\frac 34}(R)K^{\frac 12} + \r.\\
\l.+ ( C(K) + R^{\frac \al 4}M^{\frac 14})
\mathcal E^{\frac 34}(R)\ep_1^{\frac 12}\r)
\endgathered
\label{Similar to this}
\end{equation}
Applying the Young inequality $ab\le \ep a^p+C_\ep b^{p'}$ we obtain
$$
\gathered
\mathcal E(\theta R) \le \frac 14 \mathcal E(R) + c\Big( F_2(2\theta R) + D_*(2\theta R)\Big) + c(\theta )c(K) + c(\theta) R^\al M.
\endgathered
$$
From (\ref{F_2}) and (\ref{D_*}) we obtain
$$
\gathered
F_2(2\theta R) + D_*(2\theta R) \le c \theta^\al \Big(F_2(R) + D_*(R) \Big) + C(K,\theta) \ep_2 + \\ +
c(\theta) ~\Big( A^{\frac 23}(R) E^{\frac 43}(R) + A^{\frac 56}_*(R) F^{\frac 16}(R) E_*(R) \Big)
\endgathered
$$
Taking into account (\ref{C}) and the obvious inequality $F_2(R)\le A_*(R)$ we arrive at
$$
\gathered
F_2(2\theta R) + D_*(2\theta R) \ \le \ c \theta^\al \mathcal E(R) + c(K,\theta) +
\\ + c(\theta) \Big(\mathcal E^{\frac 23}(R) \ep_2^{\frac 43} +
\mathcal E^{\frac {11}{12}}(R) ( C(K)+ R^{\frac \al{12}} M^{\frac 1{12}}) K\Big)
\endgathered
$$
Applying the Young inequality we get
$$
\gathered
F_2(2\theta R) + D_*(2\theta R) \ \le \ \Big(\frac 14 + c \theta^\al \Big) \mathcal E(R) + C(\theta,K) (1+
R^\al M)
\endgathered
$$
Gathering the estimates we obtain
$$
\gathered
\mathcal E(\theta R) \le \l(\frac 14 + c \theta^\al \r) \mathcal E(R) + C(\theta,K) (1+ R^\al M).
\endgathered
$$
Fixing $\theta\in (0,\frac 12)$ so that
$$
c \theta^\al \leq \frac 14
$$
Hence
$$
\gathered
\mathcal E(\theta R) \le \frac 12\mathcal E(R) + C(K) (1+
R^\al M).
\endgathered
$$
Next with the help of technic used in the proof of lemma \ref{ckn_lemma1} we obtain (\ref{Estimate_varE}).
\qed
\begin{lemma} \label{Estimate_A_A_D} Assume all conditions of Theorem \ref{Boundedness_A_A_D} hold and fix $R_0\in (0,1)$ so that
\begin{equation}
R_0^\al M \ \le \ 1.
\label{rho_0}
\end{equation}
Then for any $0<r<R\le R_0 $ the following estimates hold:
\begin{equation}
\gathered
A(r) + A_*(r) \le c\left(\frac{r}{R} \right)^{\ga}
\Big( A( R ) + A_*( R) \Big) + \ep^{\frac 14}D(R) + G(K, \ep)
\endgathered
\label{Statement_of_Theorem_A_A}
\end{equation}
\begin{equation}
\gathered
D(r) \ \le \ c\left( \frac{r}{R}\right)^{\ga} D(R)\ + \ c(K) \Big(A^{\frac {11}{12}}(R)+A_*^{\frac {11}{12}}(R)\Big) + G(K, \ep)
\endgathered
\label{Cor}
\end{equation}
where $\ga>0$ is some absolute constant and $G$ is a continuous function possessing the following property:
\begin{equation}
\gathered
\mbox{for any fixed}\quad K >0 \quad G(K, \ep)\to 0 \quad \mbox{as}\quad\ep\to 0.
\endgathered
\label{Property of G}
\end{equation}
\end{lemma}
\noindent
{\bf Proof.} From (\ref{C_3}) taking into account (\ref{rho_0}) we obtain
\begin{equation}
C(r) \ \le A^{\frac 12}(r)\ep^{\frac 12}, \qquad F(r) \ \le \ A_*^{\frac 12}(r)C(K)
\label{F_K}
\end{equation}
Take arbitrary $r\in (0,R_0)$ and $\theta\in (0,\frac 12)$.
Denote by \ $\mathcal E_*(R)$ \ the following functional
$$
\mathcal E_* (R) \ = \ A(R)+ A_*(R),
$$
Then from (\ref{Lo_En_Iq}) similar to (\ref{Similar to this}) using (\ref{F_K}) we derive
$$
\gathered
\mathcal E_*(\theta R) \ \le \ F_2(2\theta R) \ + \ C^{\frac 12}(2\theta R)D^{\frac 12}(2\theta R) \ +
\\
+\ c(\theta)\Big( \mathcal E_*^{\frac 12}(R)\ep^{\frac 12} + \mathcal E_*^{\frac 34}(R)\ep^{\frac 34} +
\mathcal E_*^{\frac 34}(R) K^{\frac 12}\ep^{\frac 14} + \mathcal E_*^{\frac 34}(R) C(K) \ep^{\frac 12}\Big)
\endgathered
$$
Applying the Young inequality and using (\ref{D}) we obtain
\begin{equation}
\gathered
\mathcal E_*(\theta R) \ \le \ \frac 18 ~\mathcal E_*(R) \ + \ c(\theta)G(K,\ep) \ +
\\
+ \ F_2(2\theta R) \ + \ C^{\frac 12}(2\theta R)D^{\frac 12}_*(2\theta R)
\endgathered
\label{1}
\end{equation}
From (\ref{F_2}) we conclude
\begin{equation}
F(2\theta R) \ \le c\theta^\al \mathcal E_*(R) \ + \ G(K, \ep).
\label{2}
\end{equation}
From (\ref{D_*}) for $R\le R_0$ with the help of (\ref{F_K}) and the Young inequality we obtain
\begin{equation}
\gathered
D_*(2\theta R) \ \le \ c ~\theta^{\be} D_*(R) \ + \ C(\theta,K)\mathcal E^{\frac {11}{12}}_*(R) + c(\theta)G(K,\ep)
\endgathered
\label{DDD}
\end{equation}
Hence from (\ref{F_K}) we obtain
$$
\gathered
C^{\frac 12}(2\theta R)D^{\frac 12}(2\theta R) \ \le \ c(\theta) \mathcal E_*^{\frac 14}(R)\ep^{\frac 14} D_*^{\frac 12}(R) \ + \\
+ \ C(\theta,K) \ep^{\frac 14} \mathcal E^{\frac {17}{24}}_*(R) + c(\theta)G(K,\ep)
\endgathered
$$
Applying the Young inequality we arrive at
\begin{equation}
\gathered
C^{\frac 12}(2\theta R)D^{\frac 12}(2\theta R) \ \le \ \frac 18 \mathcal E_*(R) + \frac 12 \ep^{\frac 14} D_*(R) + \ c(\theta)G(K, \ep)
\endgathered
\label{3}
\end{equation}
Gathering estimates (\ref{1}) --- (\ref{3}) we obtain the inequality
$$
\gathered
\mathcal E_*(\theta R) \ \le \ \Big(\frac 14 + c\theta^\ga \Big) ~\mathcal E_*(R) + \frac 12 \ep^{\frac 14} D_*(R) + c(\theta)G(K,\ep)
\endgathered
$$
Choosing $\theta\in (0,\frac 12)$ so that
$$
\frac 14 + c\theta^\al \ = \ \frac 12
$$
we obtain
$$
\gathered
\mathcal E_*(\theta R) \ \le \ \frac 12 ~\mathcal E_*(R) + \frac 12 \ep^{\frac 14} D_*(R) + c(\theta)G(K,\ep)
\endgathered
$$
Iterating this inequality we obtain (\ref{Statement_of_Theorem_A_A}).
Choosing in (\ref{DDD}) $\theta\in (0,\frac 12)$ so that
$$
c\theta^\be \ = \ \frac 12
$$
and iteration the inequality we derive (\ref{Cor}).
\qed
\begin{theorem} \label{CKN_condition} For any $K>0$ there exists a constant $\ep_0(K)>0$ such that if the condition
(\ref{Statement_of_Theorem_E_E1}) holds with $\ep\le \ep_0$, then
there exists $\rho_*\in (0,1)$ such that
$$
\begin{array}c
\Big( C(\rho_*) + F(\rho_*) + D(\rho_*)\Big) \ < \ \ep_*^{\frac 13},
\end{array}
$$
where the constant $\ep_*>0$ is defined in Theorem \ref{Fixed_r}.
\end{theorem}
\noindent
{\bf Proof.} From (\ref{Estimate_varE}) we obtain
$$
\limsup\limits_{r\to 0} D_*(r) \le c(K).
$$
From (\ref{Statement_of_Theorem_A_A}) we derive
$$
\gathered
\limsup\limits_{r\to 0} \Big(A(r)+A_*(r)\Big) \ \le
\ \ep^{\frac 14} \limsup\limits_{\rho\to 0} D(\rho) + G(K, \ep ) \ \le \\ \le \ \ep^{\frac 14} c(K) + G(K, \ep).
\endgathered
$$
From (\ref{Cor}) we obtain
$$
\gathered
\limsup\limits_{r\to 0} D_*(r) \ \le \ c(K)\limsup\limits_{\rho\to 0} \Big(A^{\frac {11}{12}}(\rho)+ A^{\frac {11}{12}}_*(\rho)\Big) + G(K, \ep ) \ \le \\
\le \ c(K) \Big(\ep^{\frac 14} c(K) + G(K, \ep)\Big)^{\frac {11}{12}} + G(K, \ep).
\endgathered
$$
From (\ref{C_3}) we conclude
$$
\gathered
\limsup\limits_{r\to 0} \Big( C(r)+F(r)\Big) \ \le (\ep^{\frac 12} + C(K)) \limsup\limits_{r\to 0} \Big(A(r)+A_*(r)\Big) \ \le \\ \le \
(\ep^{\frac 12} + C(K))\Big( \ep^{\frac 14} c(K) + G(K, \ep)\Big)^{\frac 12}.
\endgathered
$$
Taking into account (\ref{Property of G}) for any $K>0$ we can find $\ep_0(K)>0$ such that for any $\ep\in (0,\ep_0)$
$$
c(K) \Big(\ep^{\frac 14} c(K) + G(K, \ep)\Big)^{\frac {11}{12}}
+ G(K, \ep) \ < \ \frac{\ep_*^{\frac 13}}{2}
$$
and
$$
(\ep^{\frac 12} + K^{\frac 12})\Big( \ep^{\frac 14} c(K) + G(K, \ep)\Big)^{\frac 12} \ < \ \frac{\ep_*^{\frac 13}}{2}.
$$
Hence for $\ep\in (0,\ep_0)$
$$
\limsup\limits_{r\to 0} \Big( C(r)+F(r)+ D_*(r)\Big) \ < \ \ep_*^{\frac 13}.
$$
\qed
|
train/arxiv
|
BkiUbQ45jDKDyE8-Kwp5
| 5
| 1
|
\section{Introduction}\label{sec1}
The model selection step is a fundamental task in statistical modelling and its implementation typically depends upon the objective of the exercise. In the time series framework the focus is on either forecasting future values or describing/controlling the process that has generated the data (DGP). A good model selection criterion must feature a good ability to identify the model with the ``best'' fit to future values, in a specified sense. In particular, in the parametric time series framework, we can identify two main properties. The first one is consistency, i.e., the ability to select the true DGP with probability one as the sample size diverges. This assumes that a true model exists and is among the set of candidate models. If either the set of candidate models does not contain the true DGP, or, for some reason, a true model cannot be postulated, then a selection criterion should be asymptotically efficient, for instance, in the mean square sense, i.e. it minimizes the mean squared prediction error as the sample size diverges. Starting from the seminal work of Akaike, \cite{AKA1973} a plethora of model selection criteria has been proposed. These include Akaike's AIC \cite{AKA1973, AKA1974}, Schwarz's Bayesian Information Criterion (BIC) \cite{SCH1978}, and Rissanen's Minimum Description Length (MDL) \cite{RIS1978}. Such criteria paved the way for various extensions dealing with different unsolved issues. For instance, the AIC is efficient but not consistent (i.e. it leads to select overfitting models), whereas the BIC is consistent but not efficient, see \cite{HSU2019} for a discussion.
\par
A recent development for model selection in possibly misspecified parametric time series models in the fixed-dimensionality setting is given by the Misspecification-Resistant Information Criterion (hereafter $\mric$) \cite{HSU2019}. Fixed-dimensionality means that the number of observations increases to infinity while the number of 'true' parameters is finite. In this respect, the MRIC provides a solution to the original research question of Akaike: it enjoys both consistency, in case the true model is included as a candidate, and asymptotic efficiency when a true model either cannot be assumed or is not included. Moreover, when the number of variables in the model grows with the sample size, the $\mric$ can achieve asymptotic efficiency, without the need for additional criteria. Finally, in the high-dimensional setting, the $\mric$ can be used together with appropriate model selection criteria to identify the best predictive models. The $\mric$ is based upon the additive decomposition of the mean squared prediction error in a term that depends upon the misspecification level and a term that measures the sampling variability of the predictor. The idea is to select the model with smallest variability among those that minimize the misspecification index.
\par
The appealing properties of the $\mric$ make it an ideal tool for omnibus time series model selection but, to date, only the univariate response case has been studied \cite{HSU2019}. In this work we extend the $\mric$ to multivariate time series with a single regressor as to obtain the vectorial MRIC (hereafter $\vmric$). As it will be clear, such an extension does not easily derive from the univariate case since it requires dealing with the dependence structure within the components of the vector of forecasting error and hence relies upon random matrix theory. Such multivariate extension can be used in all those models where many time series depend upon a single regressor, like for instance, in econometrics, where many interest rates depend upon a single macroeconomic indicator, such as inflation. Other possible applications include dimension reduction and hedging, which is intimately connected to the problem of model selection \cite{BES16}.
\par
The rest of the paper is organized as follows: in Section~\ref{sec:notation} we introduce the notation and in Section~\ref{subsec:MRIC} summarize the available results for the univariate case; in Section~\ref{sec:VMIRC} we extend the MRIC approach to multivariate time series with a single regressor. In particular, in Section~\ref{subsec:MSPE_dec} we obtain the asymptotic decomposition of the Mean Squared Prediction Error (hereafter MSPE) matrix into two parts: the first one is linked to the goodness of fit of the model and the second one depends upon the prediction variance. In Section~\ref{subsec:consistency} we present the $\vmric$ and derive a consistent estimator for it, whereas in Section~\ref{subsec:eff}, we prove the asymptotic efficiency of the $\vmric$. Section~\ref{sec:example} presents an example to assess the effect of misspecification in the $\vmric$ framework. All the proofs are detailed in Section~\ref{sec:proof}. \ref{appendx} contains auxiliary technical lemmas.
\section{Notation and preliminaries}\label{sec:notation}
For each $t$, let $\{\bx_t\}$ and $\{\by_t\}$, with $\bx_t=(x_{t,1},\dots,x_{t,m})^\top$ and $\by_t=(y_{t,1},\dots,y_{t,w})^\top$, be two weakly
stationary stochastic processes defined over the probability space $\left(\Omega, \mathcal{F}, \mathbb{P}\right)$. When $m=1$ ($w=1$, respectively) we write $x_t$ ($y_t$). Given a vector $\mathbf{v}$ and a matrix $\mathbf{M}$, we use $\norm{\mathbf{v}}$ and $\norm{\mathbf{M}}$ to refer to the $\mathcal{L}_2$ vectorial norm and the matrix norm induced by the Euclidean norm, respectively. We write $o(1)$ ($o_p(1)$) to indicate a sequence that converges (in probability) to zero and $O(1)$ ($O_p(1)$) to indicate a sequence that is bounded (in probability). Moreover, let $\{c_n\}$ be a sequence of scalar random variables whereas $\{\mathbf{v}_n\}$ and $\{\mathbf{M}_n\}$ are sequences of random vectors and random matrices, respectively. We adopt the following notation: $\mathbf{v}_n =o_p(c_n)$ if $\norm{\mathbf{v}_n}/c_n=o_p(1)$; $\mathbf{v}_n=O_p(c_n)$, if $\norm{\mathbf{v}_n}/c_n=O_p(1)$,
$\mathbf{M}_n =o_p(c_n)$ if $\norm{\mathbf{M}_n}/c_n=o_p(1)$; $\mathbf{M}_n = O_p(c_n)$ if $\norm{\mathbf{M}_n}/c_n=O_p(1)$.
For further details on matrix algebra see \cite{SEB2008, HOR2013, ODE2018}, for multivariate time series see \cite{REI1993, LUT2005, TSA2013}, and for asymptotic tools for vector and matrices, see \cite{JIA2010}
\par
Let $\{(\bx_t,\by_t),t \in \{1,\dots,n\}\}$ be the observed sample, and divide the interval $[1,n]$ into the \emph{training set} $[1,N]$ and the \emph{test set} $[N+1,N+h]$, with $h$ being the forecasting horizon. Note that $\bx_t$ can contain both endogenous and exogenous variables, therefore, Model~(\ref{eqn:for_mod}) encompasses many different models including, inter alia, VAR and VARX models. We denote $\bar{\bx}=n^{-1}\sum_{t=1}^{n}\bx_t$ and $\bar{\by}=n^{-1}\sum_{t=1}^{n}\by_t$, i.e. the two sample means. Without loss of generality assume $\E[\bx_t]=\E[\by_t]=\boldsymbol 0$. In order to forecast $\by_{n+h}$, $h\geq 1$, we adopt the following $h$-step ahead forecasting Model:
\begin{equation}\label{eqn:for_mod}
\by_{t+h}= \mathbf{B}_h \bx_t+\beps_t^{(h)},
\end{equation}
\par\noindent
where $\mathbf{B}_h$ is a $(w\times m)$ matrix and $\beps_t^{(h)}$ is the vector containing the $w$ $h$-step ahead forecast errors; as before, if $w=1$ we write $\eps_t^{(h)}$.
\begin{remark}
Since the model can possibly be misspecified, the prediction error vector $\beps_t^{(h)}$ can be serially correlated, and also correlated with $\bx_s$, $s\neq t$. Moreover, the multivariate framework differs from \cite{HSU2019} in different key aspects. For instance, $(i)$ the components of the error vector can be cross-correlated, and $(ii)$ $\bx_t \beps_t^{(h)}$ and $\bx_k \beps_k^{(h)}$, for $t\neq k$, can also be both serially and cross correlated.
\end{remark}
Define
\begin{equation}\label{eqn:def_R}
\hbR=N^{-1} \sum_{t=1}^{N}\bx_t\bx_t^\top\qquad\text{ and }\qquad \bR =\E[\bx_1 \bx_1^\top].
\end{equation}
Then, the ordinary least squares estimator (hereafter OLS) of $\mathbf{B}_h$ results:
\begin{equation}\label{eqn:ols}
\hat{\mathbf{B}}_n(h)= \hbR^{-1}\left(N^{-1}\sum_{t=1}^{N} \bx_t \by_{t+h}^\top\right).
\end{equation}
When $w=1$, $\mathbf{R}$ and $\mathbf{B}$ become $R$ and $\boldsymbol\beta$, respectively. The prediction of $\by_{n+h}$, $h\geq 1$, is given by
\begin{equation}\label{eqn:for_y}
\hat{\by}_{n+h}=\hat{\mathbf{B}}_n(h)\bx_n
\end{equation}
and the corresponding Mean Squared Prediction Error matrix is
\begin{equation}\label{eqn:MSPE}
\bmspe_h=\E\left[(\by_{n+h}-\hat{\by}_{n+h})(\by_{n+h}-\hat{\by}_{n+h})^\top\right].
\end{equation}
\subsection{The MRIC for parametric univariate time series models}\label{subsec:MRIC}
In \cite{HSU2019}, the authors focused on the case $w=1$ and $m\geq1$. Under appropriate conditions, they obtained the following asymptotic decomposition of $\mspe$:
\begin{align}
\mspe_h &= \E\left[(y_{n+h}-\hat{y}_{n+h})^2\right] = \mi_h+n^{-1}(\vi_h+o(1)),\label{eqn:MSPE_dec}\\
\text{with}\quad\mi_h &= \E\left[\left(\eps_n^{(h)}\right)^2\right],\quad
\vi_h = \trace{\mathbf{R}^{-1}\mathbf{C}_{h,0}} +2\sum_{s=1}^{h-1}\trace{\mathbf{R}^{-1}\mathbf{C}_{h,s}},\nonumber
\end{align}
\noindent
where $\mathbf{C}_{h,s} = \E\left[\bx_1\bx_{1+s}^\top \eps_1^{(h)} \eps_{1+s}^{(h)}\right]$, $s\geq0$, is the cross-covariance matrix between the regressors and the $h$-step ahead prediction error at lag $s$.
\begin{remark}
The first part of Eq.~(\ref{eqn:MSPE_dec}) is the Misspecification Index ($\mi$), linked to the goodness-of-fit of the model and coincides with the $h$-step ahead prediction error variance. The second component is the Variability Index ($\vi$), which depends upon the variance of the $h$-step ahead predictor, $\hat{y}_{n+h}= \hat{\boldsymbol{\beta}}_n^\top (h) \mathbf{x}_n$, and is also linked to the bias of the estimator of $\boldsymbol{\beta}_h$.
\end{remark}
\noindent
Based upon the above decomposition, the $\mric$ is defined as follows:
\begin{align}
\mric_h &= \hat{\mi}_h+\frac{\alpha_n}{n}\hat{\vi}_h,\label{eqn:MRIC_uni}
\end{align}
with $\hat{\mi}_h$ and $\hat{\vi}_h$ being the estimators of $\mi_h$ and $\vi_h$ respectively, i.e.:
\[
\hat{\mi}_h = N^{-1}\sum_{t=1}^{N}\left(\hat{\eps}_t^{(h)}\right)^2,\quad
\hat{\vi}_h = \trace{\hat R^{-1}\hat{\mathbf{C}}_{h,0}} + 2 \sum_{s=1}^{h-1} \trace{\hat R^{-1} \hat{\mathbf{C}}_{h,s}},
\]
\noindent
where $\hat{\mathbf{C}}_{h,s} = (N-s)^{-1} \sum_{t=1}^{N-s}\bx_t\bx_{t+s}^\top \hat{\eps}_t^{(h)}\hat{\eps}_{t+s}^{(h)}$ and
$\hat{\eps}_t^{(h)}=y_{t+h}-\hat{\bBeta}_n(h)\bx_t$ is the estimated forecast error; $\alpha_n$ is a penalization term sequence such that, as $n$ increases:
\begin{align}\label{eqn:penalty}
\frac{\alpha_n}{\sqrt{n}}\to+\infty\qquad\text{ and }\qquad \frac{\alpha_n}{n}\to0.
\end{align}
\noindent
It is shown that $\hat{\mi}_h$ and $\hat{\vi}_h$ are consistent estimators of ${\mi}_h$ and ${\vi}_h$, moreover the asymptotic efficiency of the $\mric$ is proved. By minimizing this criterion, the model which minimizes $\vi$ among those with minimum $\mi$ is selected. Among other features, the $\mric$ is particularly helpful in situations where competing models present the same goodness-of-fit and the same number of parameters.
\begin{remark}
The type of penalty considered in \cite{HSU2019} is similar to that used in \cite[p. 230]{SHI1989} for the correctly specified case.
\end{remark}
\section{A multivariate extension of the MRIC framework}\label{sec:VMIRC}
In this section we extend the $\mric$ approach to the case where the response is a multivariate time series ($w\geq2$) and the predictor is univariate ($m=1$), for a generic $h$-step ahead forecast.
\noindent
Hence, Model~(\ref{eqn:for_mod}) reduces to $\by_{t+h}=\bBeta_h x_t+ \beps_t^{(h)}$, namely:
\begin{equation}\label{eqn:for_mod_case1}
\begin{cases}
y_{t+h,1}= \beta_{h,1}x_t+\eps_{t,1}^{(h)} \\
y_{t+h,2}= \beta_{h,2}x_t+\eps_{t,2}^{(h)}\\
\vdots\\
y_{t+h,w}= \beta_{h,w}x_t+\eps_{t,w}^{(h)}
\end{cases}
\end{equation}
\subsection{Asymptotic decomposition of the MSPE matrix}\label{subsec:MSPE_dec}
We extend the asymptotic representation of the $\mspe_h$ defined in (\ref{eqn:MSPE_dec}) which is the key step to derive the $\vmric$ in this multivariate framework. We rely upon the following assumptions, which are the natural multivariate extensions of those in \cite{HSU2019}.
\begin{assumptions}
\begin{align*}
(\text{C}1) & \quad \exists \ q_1 > 5, 0<K_1<\infty : \ \text{for any } 1 \leq n_1 < n_2 \leq n,\\
& \E\left[
\left\lvert \left( n_2-n_1+1 \right)^{-1/2}
\sum_{t=n_1}^{n_2} x_{t}^2 - \E\left[ x_{t}^2 \right]
\right\rvert^{q_1}
\right]\leq K_1.\\
(\text{C}2) &\quad 1. \ \mathbf{C}_{h,s} = \ \E\left[ \boldsymbol{\eps}_t^{(h)} x_t \left(\boldsymbol{\eps}_{t+s}^{(h)} x_{t+s}\right)^\top \right] \perp t,\\
&\quad 2. \ \E\left[ x_1 x_n \eps_{1,i}^{(h)} \eps_{n,j}^{(h)} \right] = \ o(n^{-1}) \ \forall \ i,j \in\{ 1, \dots, w\}.\\
(\text{C}3) & \quad 1. \sup_{-\infty<t<\infty} \E\left[ | x_t |^{10} \right] < \infty,\\
& \quad 2. \sup_{-\infty<t<\infty} \E\left[ \norm{\boldsymbol{\eps}_{t}^{(h)}}^{6} \right] < \infty.\\
(\text{C}4) & \quad \exists \ 0 < K_2 < \infty : \ \text{for} \ 1\leq n_1 < n_2 \leq n,
\quad \E\left[\norm{\left(n_2-n_1+1\right)^{-\frac{1}{2}} \sum_{t=n_1}^{n_2} \boldsymbol{\eps}_t^{(h)} x_t }^5 \right] < K_2.\\
(\text{C}5) & \quad \text{For any } q>0, \ \E\left[ \left|\hat{R}^{-1} \right|^q \right] = O(1).\\
(\text{C}6) & \quad \exists \mathcal{F}_t \subseteq \mathcal{F}, \mathcal{F}_t \ \text{an increasing sequence of } \sigma\text{-fields such that: }\\
& \quad 1. \
x_t \ \text{is} \ \mathcal{F}_t\text{-measurable}\\
& \quad 2. \ \sup_{-\infty<t<\infty} \E\left[ \left|\E\left[ x_t^2 \mid \mathcal{F}_{t-k}\right] - R\right|^3\right] = \ o(1), \text{ as } k \rightarrow\infty,\\
& \quad 3. \ \sup_{-\infty<t<\infty} \E\left[ \norm{\E\left[ \boldsymbol{\eps}_{t}^{(h)} x_t \mid \mathcal{F}_{t-k}\right]}^3\right] = \ o(1), \text{ as } k \rightarrow\infty.
\end{align*}
\end{assumptions}
\begin{theorem}\label{thm:MSPE_dec_case1}
Under the regularity conditions (C1) -- (C6), the asymptotic expression of the $\bmspe_h$ defined in (\ref{eqn:MSPE}) results
\begin{align}\label{eqn:MSPE_dec_case1}
&N \left\{ \E\left[ \left( \by_{n+h} - \hat{\by}_{n+h} \right) \left( \by_{n+h} - \hat{\by}_{n+h} \right)^\top - \E\left[
\beps_{n}^{(h)} \beps_{n}^{(h)^\top} \right] \right] \right\}\\
& = R^{-1} \E\left[ \left(\beps_{1}^{(h)} x_1\right)\left(\beps_{1}^{(h)} x_1\right)^\top \right]\nonumber\\
&+ R^{-1} \E\left[\sum_{s=1}^{h-1} \left\{\left( \beps_1^{(h)} x_1\right) \left(\beps_{s+1}^{(h)} x_{s+1}\right)^\top + \left(\beps_{s+1}^{(h)} x_{s+1}\right) \left( \beps_1^{(h)} x_1 \right)^\top\right\} \right] + o(1).\nonumber
\end{align}
\end{theorem}
\noindent
\subsection{VMRIC and its consistent estimation}\label{subsec:consistency}
In this section we introduce the $\vmric$. Let $\{\alpha_n\}$ be the penalization term sequence defined as in Eq.~(\ref{eqn:penalty}).
\begin{align}
\vmric_h &= \norm{{\bmi}_h} + \norm{ \frac{\alpha_n}{n} {\bvi}_h} \label{eqn:VMRIC}\\
\text{where } \bmi_h &= \E\left[\left(\beps_t^{(h)}\beps_t^{(h)\top} \right)\right],\quad
\bvi_h = R^{-1} \left( \bC_{h,0} + \sum_{s=1}^{h-1} \left( \bC_{h,s} + \bC_{h,s}^\top \right) \right),\nonumber\\
\bC_{h,s} &= \E\left[\left(x_t\beps_t^{(h)}\right)\left(x_t\beps_t^{(h)}\right)^\top\right].\nonumber
\end{align}
The $\vmric$ can be estimated via the method of moments as to obtain:
\begin{align}
\hat\vmric_h &\equiv \norm{\hat{\bmi}_h} + \norm{ \frac{\alpha_n}{n} \hat{\bvi}_h \label{eqn:VMRIC_hat}},\\
\text{where } \hat{\bmi}_h &= N^{-1}\sum_{t=1}^{N}\left(\hat{\beps}_t \hat{\beps}_t^\top \right),\quad
\hat{\bvi}_h = \hat{R}^{-1} \left[ \hbC_{h,0} + \sum_{s=1}^{h-1} \left( \hbC_{h,s} + \hbC_{h,s}^\top \right) \right],\nonumber
\end{align}
and $\hbC_{h,s} = (N-s)^{-1} \sum_{t=1}^{N-s} x_t x_{t+s} \hat{\beps}_t \hat{\beps}_{t+s}^\top$, and $\hat{\beps}_t=\by_{t+h}-\hat{\bBeta}_n(h) x_t$ is the estimated forecast error vector.
\par
\noindent In Theorem~\ref{thm:MOME_case1} we prove that $\hat{\bmi}_h$ and $\hat{\bvi}_h$ are consistent estimators of ${\bmi}_h$ and ${\bvi}_h$, respectively. Theorem~\ref{thm:MOME_case1} relies upon the following assumptions, that are less restrictive with respect to (C1) -- (C6). For further discussions on the assumptions see \cite[][Remark~1--3, p. 1073]{HSU2019}.
\begin{assumptions}
For each $0\leq s \leq h-1$, we assume the following:
\begin{align*}
(\text{A}1) & \quad n^{-1} \sum_{t=1}^{n} \left( \beps_t^{(h)} \beps_t^{(h)\top}\right) = \E\left[ \beps_1^{(h)} \beps_1^{(h)\top} \right] + O_p \left(n^{-1/2}\right)\\%, \text{and any } 1\leq i,j \leq m,\\
(\text{A}2)
&\quad n^{-1} \sum_{t=1}^{n} \left(x_t \beps_t^{(h)}\right) \left(x_{t+s}\beps_{t+s}^{(h)}\right)^\top = \boldsymbol{C}_{h,s} + o_p(1),\\
(\text{A}3)
& \quad n ^{-1/2} \sum_{t=1}^{n} x_t \beps_t^{(h)} = O_p(1).\\
(\text{A}4)
& \quad n ^{-1} \sum_{t=1}^{n} x_t^{2} = R + o_p(1),\\
(\text{A}5)
& \quad \sup_{-\infty < t < \infty} \E \left[\left\|\beps_t^{(h)}\right\|^4\right] +
\sup_{-\infty<t<\infty} \E\left[\abs {x_t}^4 \right] < \infty.
\end{align*}
\end{assumptions}
\begin{theorem}\label{thm:MOME_case1}
If Assumptions (A1) -- (A5) hold, then for the case $w\geq2$, and $m=1$ we obtain:
\begin{align*}
\hat{\bmi}_h&= \bmi_h +\, O_p (n^{-1/2}),\\
\hat{\bvi}_h&= \bvi_h +\, o_p (1).
\end{align*}
\end{theorem}
\subsection{Asymptotic efficiency}\label{subsec:eff}
In this section we prove the asymptotic efficiency of the $\vmric$ in the fixed dimensionality framework. To this end, let $\mathcal{M}$ be the set of $K$ candidate models; each model is indicated either by $\ell$ or $\kappa$, $1\leq \ell,\kappa\leq K$. Define the subsets $M_1$ and $M_2$ as follows:
\begin{equation}
M_1 = \left\{
\kappa: 1 \leq \kappa \leq K,
\norm{\bmi_h(\kappa)} = \min_{1\leq \ell \leq K}
\norm{\bmi_h(\ell)}
\right\}
\end{equation}
\begin{equation}
M_2 = \left\{
\kappa: \kappa \in M_1,
\norm{\bvi_h(\kappa)} = \min_{\ell \in M_1}
\norm{\bvi_h(\ell)}
\right\}.
\end{equation}
In short, for a given forecast horizon $h$, $M_1$ contains the models with the minimum $\bmi_h$ whereas in $M_2$ we are minimizing $\bvi_h$ among the candidates models in $M_1$. The definition of efficiency used in our framework is the same as that of \cite{HSU2019}:
\begin{definition}\label{def:eff}
Given a sample of size $n$, a model selection criterion is said to be asymptotically efficient if it selects the model $\hat{\ell}_h$ such that
$$\lim_{n\to\infty}\Pr \left( \hat\ell_h \in M_2\right) = 1.$$
\end{definition}
\begin{remark}
Alternative definitions of asymptotic efficiency for model selection are available. For instance, in the framework of linear stationary processes, \cite{SHI1980} defines the Mean Efficiency when a criterion attains asymptotically a lower bound for the sum of squared prediction errors. Also, the notion of Approximate Efficiency is given in \cite{SHI1984}. In \cite{LI1987}, a criterion that depends upon the ratio between loss functions is introduced. This latter definition is similar to the Loss Efficiency proposed in \cite{SHA1997}.
\end{remark}
The $\vmric$ selects the model with the smallest variability index among those that achieve the best goodness of fit. Hence, the selected model $\hat\ell_h$ is such that:
\begin{equation}\label{eq:vmric}
\vmric_h \left( \hat{\ell}_h\right) \equiv
\min_{1 \leq \ell \leq K}
\norm{ \hat{\boldsymbol{\mi}}_h (\ell) } +
\min_{ \ell \in M_1}
\norm{ \frac{C_n}{n} \hat{\boldsymbol{\vi}}_h (\ell)}.
\end{equation}
In the next Theorem we show that the $\vmric$ is an asymptotic efficient model selection criterion in the sense of Definition~\ref{def:eff}.
\begin{theorem}\label{thm:EFF_case1}
Assume that for each $ 1 \leq \ell \leq K$, $0\leq s \leq h-1$, Theorem~\ref{thm:MOME_case1} holds and let $\hat\ell_h$ be the model selected by the $\vmric$. Then we have that:
\begin{align*}
\label{eq:theo3}
\lim_{n\rightarrow \infty} \Pr \left( \hat{\ell}_h \in M_2\right) &= 1,
\end{align*}
namely, the $\vmric$ is asymptotically efficient in the sense of Definition~\ref{def:eff}.
\end{theorem}
\section{Example: a misspecified bivariate AR(2) model}\label{sec:example}
The aim of this section is twofold. First, we assess the goodness of the theoretical derivations and the finite sample behaviour of the method of moments estimator for the $\vmric$. Second, we show that in presence of misspecification the $\vmric$ leads to selecting the best predictive model (i.e. is asymptotically efficient) whereas both the $\aic$ and the $\bic$ fail to do so. In order to achieve the goals we consider a bivariate AR(2) DGP and use two misspecified predictive models for it: in Model 1 there is one omitted lagged predictor, whereas Model 2 uses only one non-informative predictor. We derive theoretically the Mean Square Prediction Error matrix and the $\vmric$ for both models and these show that Model 1 is a better predictive model over Model 2. Based on this, we assess the ability of the $\vmric$, and of the multivariate versions of the $\aic$ and $\bic$ to select the best model (Model 1) in finite samples and for different parameterizations.
\par
We start by providing the definition of misspecification. Consider an increasing sequence of $\sigma$-fields, $\left\{ \mathcal{G}_t \right\}$ such that $\sigma\left( \bx_s, s\leq t \right) \subseteq \mathcal{G}_t \subseteq \mathcal{F} $, where $\left\{ \bx_t \right\}$ is an $m$-dimensional weakly stationary process defined over the probability space $\left(\Omega, \mathcal{F}, \mathbb{P} \right)$.
\begin{definition}\label{def:miss}
The $h$-step ahead forecasting model:
\begin{equation}
\by_{t+h} = \bBeta_h^\top \bx_t + \beps_t^{(h)},
\end{equation}
is correctly specified with respect to an increasing sequence of $\sigma$-fields, $\left\{ \mathcal{G}_t \right\}$ if
\begin{equation}
E\left[\by_{t+h} \mid \mathcal{G}_t \right] = \bBeta_h^\top \bx_t \ \text{a.s.}, \ \forall \ -\infty < t < \infty.
\end{equation}
Otherwise, it is misspecified.
\end{definition}
\begin{remark}\label{re:miss}
The presence of misspecification implies that: $E\left[ \beps_t^{(h)} x_t \right] = \boldsymbol{0}$, while it is possible to have
$E\left[ \beps_t^{(h)} x_s \right] \neq \boldsymbol{0}$, for $s\neq t$, i.e. we have null simultaneous correlation and non-null cross correlation between the forecasting error vector and the regressor.
\end{remark}
Consider the following DGP :
\begin{equation}\label{eq:DGP}
\by_{t+1} = \boldsymbol{a} w_t + \beps_{t+1},
\end{equation}
where $\boldsymbol{a}\neq \boldsymbol{0}$, $\left\{ \beps_t \right\}$ is a sequence of independent and identically distributed (hereafter i.i.d.) bivariate random vectors with $E\left[ \beps_1 \right] = \boldsymbol{0} $, $E\left[ \beps_1 \beps_1^{\top} \right] >\boldsymbol{0} $ and $w_t$ is the following scalar AR(2) process:
\begin{equation}\label{eq:AR2}
w_t = \phi_1 w_{t-1} + \phi_2 w_{t-2} + \delta_t,
\end{equation}
where $\phi_1 \phi_2 \neq 0$ , $\left\{ \delta_t \right\}$ a sequence of i.i.d. random variables independent of $\left\{ \beps_t \right\}$ such that
$$E\left[ \delta_1 \right] = 0\quad\text{and}\quad E\left[ \delta_1^2 \right] = 1 - \phi_2^2 - \left\{ \phi_1^2 \frac{1+\phi_2}{1-\phi_2} \right\}.$$
Hence, we obtain $E\left[ w_t^2 \right] \equiv \gamma_w(0) = 1$, where $\gamma_w(j) = E\left[ w_t w_{t+j} \right]$ is the $j$-th lag autocovariance of $w$. \par
We consider the correctly specified $2$-step ahead forecasting model:
\begin{align}\label{eqn:fcst_mod_corr}
\by_{t+2} &= \boldsymbol{a} w_{t+1} + \beps_{t+1}, \text{ which leads to } \nonumber\\
\by_{t+2} &= \boldsymbol{a} \phi_1 w_t + \boldsymbol{a} \phi_2 w_{t-1} + \beps_t^{*(2)},
\end{align}
where $\beps_t^{*(2)} = \beps_{t+2} + \boldsymbol{a} \delta_{t+1}$. It can be easily proved that $E\left[ \beps_t^{*(2)} w_{t-j} \right] = \boldsymbol{0}$ for $j \geq 0$.
\par
Now, consider the following \textit{misspecified} model, Model 1:
\begin{align*}
\by_{t+2}& = \bBeta w_t + \beps_t^{(2)}, \quad \text{with}\quad\bBeta = \frac{E\left[ \by_{t+2} w_t \right]}{V\left[ w_T \right]} = \boldsymbol{a} \left( \phi_1 + \frac{\phi_1 \phi_2}{1- \phi_2} \right).
\end{align*}
The forecasting error results:
\begin{equation}\label{eqn:fcst_error_miss}
\beps_t^{(2)} = \beps_t^{*(2)} - \boldsymbol{a} \phi_2 \left[ \frac{\phi_1}{1-\phi_2} w_t - w_{t-1} \right].
\end{equation}
\par\noindent
\begin{remark}
In presence of misspecification $E\left[ \beps_{t}^{(2)} w_{t} \right] = \boldsymbol{0}$, whereas $E\left[ \beps_{t}^{(2)} w_{t-j} \right] \neq \boldsymbol{0}$ for $j\neq0$. We show that this occurs in our case:
\begin{align}
E\left[ \beps_{t}^{(2)} w_{t-j} \right]& = -\boldsymbol{a} \frac{\phi_2}{1-\phi_2} \left\{ \phi_1 E\left[ w_t w_{t-j} \right] - \left( 1- \phi_2 \right) E\left[ w_{t-1} w_{t-j} \right] \right\}\nonumber\\
& = -\boldsymbol{a} \frac{\phi_2}{1-\phi_2} \left\{\gamma_w(j+1) - \gamma_w (j-1)\right\},\nonumber
\end{align}
which is zero if $j=0$, otherwise this is generally not the case.
\end{remark}
We compute the theoretical value of the $\vmric$ by using Eq.~(\ref{eqn:VMRIC}).
After some routine algebra, we get:
\begin{equation}\label{eqn:example_mi2}
\bmi = E\left[
\beps_{n}^{(2)}
\beps_{n}^{(2)^\top}
\right] = \boldsymbol{\sigma}^2_\eps +
\boldsymbol{a} \boldsymbol{a}^\top
\left[
\sigma^2_{\delta}
+ \phi_2^2
\left(
1 - \gamma_w^2 (1)
\right)
\right],
\end{equation}
which highlights how the variance-covariance matrix of the $2$-step ahead forecast vector is equal to the {DGP}'s variance-covariance plus a bias term that depends upon the misspecification considered.
\par
Now we focus on the variability index $\bvi$. We get
\begin{align}\label{eqn:example_c20}
\bC_{2,0} = \boldsymbol{\sigma}^2_{\beps} +
\boldsymbol{a} \boldsymbol{a}^\top \left\{
\sigma^2_{\delta}
+ \phi^2_2 \left(
\gamma_w(1)^2 E\left[w_t^4\right]- 2 \gamma_w(1)E\left[w_t^3 w_{t-1}\right] +
E\left[w_t^2 w_{t-1}^2\right]
\right)
\right\}
\end{align}
and
\begin{align}\label{eqn:example_c21}
\bC_{2,1} =
\boldsymbol{a} \boldsymbol{a}^\top \gamma_w(1)\left(
b_1 E\left[w_{t-1}^3 w_{t-2}\right] +
b_2 E\left[w_{t-1} w_{t-2}^3 \right] +
b_3 E\left[w_{t-1}^2 w_{t-2}^2\right]
\right),
\end{align}
where
\begin{align*}
b_1= 2\phi_1\phi_2\gamma_w(1) - \phi_2,\qquad
b_2= -\phi_2^2, \qquad
b_3= \phi_2 \left(\phi_2\gamma_w(1) - 2\phi_1 + \gamma_w(1)^{-1} \right).
\end{align*}
Following Eq.~(\ref{eqn:VMRIC}), the results from Eq.~(\ref{eqn:example_mi2}), (\ref{eqn:example_c20}), and (\ref{eqn:example_c21}), deliver the $\vmric$ for this case.
\par
Now we consider a second misspecified model, Model 2:
\begin{equation}
\by_{t+2} = \bRho z_t + \bEta_t^{(2)},
\end{equation}
where $z_t$ is a weakly stationary linear AR(1) process independent of $w_t$:
\begin{equation}\label{eq:AR1}
z_t = \psi_1 z_{t-1} + \upsilon_t
\end{equation}
with $\psi_1 \in (-1, 1)$, and $\left\{ \upsilon_t \right\}$ is a sequence of i.i.d. random variables independent of both the error terms $\left\{ \delta_t\right\}$ and $\left\{\beps_{t}\right\}$ such that $E[\upsilon_t] = 0$ and $E[\upsilon_t^2] = 1 - \psi_1^2$, delivering $E[z_t] = 0$ and $E[z_t^2]=1$. Thus, $z_t$ is uncorrelated with both $w_t$ and $\by_t$, therefore $\bRho=\boldsymbol{0}$. The forecasting error in this case results $\bEta_t^{(2)} = \boldsymbol{a} w_{t+1} + \beps_{t+2}$. Following similar arguments as above we obtain ${\bmi}$ and ${\bvi}$ for Model 2:
\begin{align}
{\bmi} &= \boldsymbol{\sigma}^2_{\beps} + \boldsymbol{a} \boldsymbol{a}^\top \\
{\bvi} &= \boldsymbol{\sigma}^2_{\beps} + \boldsymbol{a} \boldsymbol{a}^\top (1 + 2 \psi_1 \gamma_w(1))
\end{align}
\par
As mentioned above, Model 1 is misspecified since it omits the lagged predictor $w_{t-1}$, while Model 2 only includes the non-informative predictor $z_t$.
\subsection{Finite sample performance}
First, we compare the above theoretical derivations with their sample counterpart. We consider three different parameterizations, presented in Table~\ref{tab:1}. Also, $\alpha_n = n^\alpha$ with $\alpha=0.85$. Note that, in order for Eq.~(\ref{eqn:penalty}) to hold, $\alpha$ must range in $(0, 1)$. Further experiments showed that results are fairly robust if reasonable values of $\alpha$ are selected. For an empirical method to determine it, see \cite[Section 5]{hsu2019_supp}. We take the following variance/covariance matrix for the innovations:
$$E[\beps_t\beps_t^{\top}] =
\left[\begin{array}{cc}
1 & 0.5 \\
0.5 & 1
\end{array}\right].$$
\par
We compute both the {$\vmric$} for Model 1 and Model 2, and estimate the $\hat{\vmric}$ and $\hat{\vmric}$ on a large sample of $n=10^6$ observations. The results are shown in Table~\ref{tab:2} for the two models, where the theoretical $\vmric$ (rows 1 and 3) is compared with the estimated one (rows 2 and 4). The results seem to confirm the consistency of the estimator shown in Eq.~(\ref{eqn:VMRIC_hat}). Clearly, the $\vmric$ of Model 1 is consistently smaller than that of Model 2 and indicates its superior predictive capability.
\begin{table}[]
\centering \caption{Parameters' combinations for the DGP of Eq.~(\ref{eq:DGP}), (\ref{eq:AR2}), and (\ref{eq:AR1}).}\label{tab:1}
\begin{tabular}{crrrrr}
Case & $\phi_1$ & $\phi_2$ & $a_1$ & $a_2$ & $\psi_1$ \\
\cmidrule(lr){2-6}
1 & 0.4 & -0.75 & 1.50 & -2.00 & 0.80 \\
2 & -0.4 & -0.45 & -0.75 & 1.25 & -0.65 \\
3 & 0.3 & -0.80 & 1.00 & 0.50 & -0.75 \\
\cmidrule(lr){2-6}
\end{tabular}
\end{table}
\begin{table}[]
\centering \caption{Theoretical and estimated {$\vmric$} of Models 1 and 2, for the three parameterizations of Table~\ref{tab:1}, computed on a data set of $n=10^6$ observations.}
\label{tab:2}
\begin{tabular}{ccccc}
& \multicolumn{2}{c}{Model 1} & \multicolumn{2}{c}{Model 2} \\
\cmidrule(lr){2-3}\cmidrule(lr){4-5}
Case & $\vmric$ & $\hat{\vmric}$ & $\vmric$ & $\hat{\vmric}$ \\
\cmidrule(lr){2-3}\cmidrule(lr){4-5}
1 & 6.671 & 6.636 & 7.914 & 7.902 \\
2 & 2.777 & 2.768 & 3.164 & 3.168 \\
3 & 2.801 & 2.784 & 2.994 & 2.993 \\
\cmidrule(lr){2-3}\cmidrule(lr){4-5}
\end{tabular}
\end{table}
The finite sample behaviour of the method of moments estimator of the $\vmric$ can be further appreciated in Table~\ref{tab:3} where we show their bias and Mean Squared Error (MSE).
The results are based upon 1000 Monte Carlo replications and seem to indicate a rate of convergence of the order of $n^{-1}$.
\par
In Table~\ref{tab:4}, we show the percentages of correct model selection by the $\vmric$, compared with the multivariate version of the $\aic$ and $\bic$ for the three parameterizations of Table~\ref{tab:1}. For a sample size of $n=100$, both the $\aic$ and $\bic$ select the best predictive model in about 50\% of the cases and relying upon them is tantamount to tossing a fair coin. In such a case, the $\vmric$ selects the correct model in about $80\%$ of the cases and reaches $100\%$ for $n=1000$. On the contrary, for Case 3, both the $\aic$ and $\bic$ cannot go above $64\%$ for a sample size as large as $n=10000$ observations and this is a general indication of their lack of asymptotic efficiency.
\begin{table}[]
\centering
\caption{Bias and Mean-Squared Error (MSE) for the (method of moments) estimator of the $\vmric$ for the three parameterizations, $\alpha=0.85$ and different sample size $n$. The results are based upon $1000$ Monte Carlo replications.}
\label{tab:3}
\begin{tabular}{rcccccc}
& \multicolumn{2}{c}{Case 1} & \multicolumn{2}{c}{Case 2} & \multicolumn{2}{c}{Case 3} \\
\cmidrule(lr){2-3}\cmidrule(lr){4-5} \cmidrule(lr){6-7}
$n$ & Bias & MSE & Bias & MSE & Bias & MSE \\
\cmidrule(lr){2-3}\cmidrule(lr){4-5} \cmidrule(lr){6-7}
100 & 0.227 & 1.137 & 0.063 & 0.306 & 0.030 & 0.182 \\
250 & 0.117 & 0.455 & 0.022 & 0.107 & 0.032 & 0.076 \\
500 & 0.061 & 0.225 & 0.015 & 0.048 & 0.004 & 0.032 \\
1000 & 0.019 & 0.109 & 0.010 & 0.023 & 0.002 & 0.015 \\
2500 & 0.008 & 0.044 & 0.001 & 0.009 & 0.001 & 0.006 \\
5000 & 0.009 & 0.023 & 0.001 & 0.004 & 0.003 & 0.003 \\
10000 & 0.001 & 0.012 & 0.003 & 0.002 & 0.001 & 0.002 \\
15000 & 0.004 & 0.008 & 0.001 & 0.001 & 0.002 & 0.001 \\
30000 & 0.002 & 0.004 & 0.001 & 0.001 & 0.001 & 0.001 \\
\cmidrule(lr){2-3}\cmidrule(lr){4-5} \cmidrule(lr){6-7}
\end{tabular}
\end{table}
\begin{table}[]
\centering
\caption{Percentages of correctly selected models by the three information criteria for the three parameterizations and varying sample size $n$.}\label{tab:4}
\small
\begin{tabular}{rccccccccc}
&\multicolumn{3}{c}{Case 1} & \multicolumn{3}{c}{Case 2} & \multicolumn{3}{c}{Case 3} \\
\cmidrule(lr){2-4}\cmidrule(lr){5-7} \cmidrule(lr){8-10}
$n$ & VMRIC & AIC & BIC & VMRIC & AIC & BIC &VMRIC & AIC & BIC \\
\cmidrule(lr){2-4}\cmidrule(lr){5-7} \cmidrule(lr){8-10}
100 & 85.9 & 52.5 & 52.5 & 84.6 & 56.2 & 56.2 & 72.1 & 49.0 & 49.0 \\
1000 & 99.9 & 65.6 & 65.6 & 99.9 & 73.7 & 73.7 & 97.0 & 56.8 & 56.8 \\
10000 & 100 & 88.0 & 88.0 & 100 & 97.8 & 97.8 & 100 & 63.8 & 63.8 \\
\cmidrule(lr){2-4}\cmidrule(lr){5-7} \cmidrule(lr){8-10}
\end{tabular}
\end{table}
\section{Proofs}\label{sec:proof}
In this section we detail the proofs of the three theorems. Hereafter all the derivations hold for any fixed $h\geq 1$; for the sake of presentation we write $ \beps_{t}$ instead of $ \beps_{t}^{(h)}$.
Remember that $\{l_n\}$ indicates an increasing sequence of positive integers such that:
\begin{equation}\label{eqn:ln}
l_n \rightarrow \infty,\qquad \dfrac{l_n}{\sqrt{n}}=o\left(1\right)
\end{equation}
and define $a = n-l_n-h$ and $b=n-l_n-h+1$.
\subsection{Proof of Theorem~\ref{thm:MSPE_dec_case1}}\label{sec:A1}
The proof of Theorem~\ref{thm:MSPE_dec_case1} relies upon four propositions.
\begin{proposition}\label{prop:thm1_1}
Under assumptions of Theorem~\ref{thm:MSPE_dec_case1}, it holds that:
\begin{equation}
N(\mathrm{I}) = (\mathrm{III}) + o(1),\label{eqn:thm1_1}
\end{equation}
where
\begin{align*}
(\mathrm{I}) =- \E\left[x_n \hat{R}^{-1} \left( \boldsymbol{\hat{\Sigma}} \boldsymbol{\eps}_{n}^\top + \boldsymbol{\eps}_{n} \boldsymbol{\hat{\Sigma}}^\top \right) \right],\quad
(\mathrm{III})= \ - \E\left[x_n R^{-1} \left( \boldsymbol{\hat{\Sigma}}_A \boldsymbol{\eps}_{n}^\top + \boldsymbol{\eps}_{n} \boldsymbol{\hat{\Sigma}}_A^\top\right) \right],
\end{align*}
with $\boldsymbol{\hat{\Sigma}} = \left( N^{-1} \sum_{t=1}^{N}x_t \boldsymbol{\eps}_{t} \right)$ and $\boldsymbol{\hat{\Sigma}}_A = \sum_{t=1}^{N}\boldsymbol{\varepsilon_{t}} x_t$.
\end{proposition}
\begin{proof}
Let $\boldsymbol{A}_1 = \sum_{t=1}^{N} \left(\beps_t x_t\right) \beps_n^\top$ and note that
\begin{align}
\label{I-IIIdiv} \norm{(\mathrm{I})-(\mathrm{III}) }
&= \norm{\E\left[x_n \left(\hat{R}^{-1}-R^{-1}\right)\left(\boldsymbol{A}_1+\boldsymbol{A}_1^\top\right)\right] }.
\end{align}
By using standard properties of the norm, (\ref{eqn:thm1_1}) follows upon proving that
\begin{equation}\label{eqn:thm1_1main}
\norm{ \E\left[x_n \left(\hat{R}^{-1}-R^{-1}\right)\boldsymbol{A}_1^\top\right]}=o(1).
\end{equation}
Let
\begin{equation}\label{eqn:R_tilde}
\tilde{R} = \left(n-l_n\right)^{-1} \sum_{t=1}^{n-l_n} x_t^2.
\end{equation}
By adding and subtracting $ \beps_n x_n \left(\tilde{R}^{-1} \left[\sum_{t=1}^{N}\left(\beps_t x_t\right)\right]^\top \right)$, we have
\begin{align}
\E\left[x_n \left(\hat{R}^{-1}-R^{-1}\right)\boldsymbol{A}_1^\top\right]&=\E\left[\beps_n x_n \left(\hat{R}^{-1}-\tilde{R}^{-1}\right) \sum_{t=1}^{N} \beps_t^\top x_t\right]\nonumber\\
&+ \E\left[\beps_n x_n \left(\tilde{R}^{-1}-R^{-1}\right) \sum_{t=1}^{N} \beps_t^\top x_t\right]\nonumber\\
&= \E\left[\beps_n x_n \left(\hat{R}^{-1}-\tilde{R}^{-1}\right) \left(\sum_{t=1}^{N} \beps_t x_t\right)^\top\right]\label{eqn:1}\\
&+ \E\left[\beps_n x_n \left(\tilde{R}^{-1}-R^{-1}\right) \left(\sum_{t=b}^{N} \beps_t x_t\right)^\top\right]\label{eqn:2}\\
&+ \E\left[\beps_n x_n \left(\tilde{R}^{-1}-R^{-1}\right) \left(\sum_{t=1}^{a} \beps_t x_t\right)^\top\right]\label{eqn:3}.
\end{align}
We show below that the norms of (\ref{eqn:1}), (\ref{eqn:2}) and (\ref{eqn:3}) are asymptotically negligible. Focus on the first one: by combining conditions (C3), (C4), Lemma \ref{lma:technical}, and H\"older's inequality, it follows that $\norm{(\ref{eqn:1})}$ is bounded by
\begin{align*}
\E \left[\norm{\beps_n x_n \left(\hat{R}^{-1}-\tilde{R}^{-1}\right) \left(\sum_{t=1}^{N} \beps_t x_t\right)^\top}\right]
&\leq \E\left[\norm{\beps_n}^6\right]^{\frac{1}{6}} \E\left[\left\lvert x_n \right\rvert^{6} \right]^{\frac{1}{6}} \E\left[\left\lvert \hat{R}^{-1}-\tilde{R}^{-1} \right\rvert^{3} \right]^{\frac{1}{3}}\\
& \times \E\left[\norm{ N^{\frac{1}{2}}N^{-\frac{1}{2}}\sum_{t=1}^{N} \beps_t x_t } ^{3}\right]^{\frac{1}{3}} = O\left(\frac{l_n}{n^{1/2}}\right),
\end{align*}
which converges to zero due to the definition of $l_n$ in (\ref{eqn:ln}). Similarly, we have that $\norm{(\ref{eqn:2})}$ is bounded by
\begin{align*}
\E\left[\norm{\beps_n}^6\right]^\frac{1}{6} \E\left[\left\lvert x_n\right\rvert^6\right]^\frac{1}{6} \E\left[\left\lvert\tilde{R}^{-1}-R^{-1}\right\rvert^3\right]^\frac{1}{3} \E\left[\norm{\left(\left(N-b+1\right)^{\frac{1}{2}}\left(N-b+1\right)^{-\frac{1}{2}}\sum_{t=b}^{N} \beps_t x_t\right)^\top}^3\right]^\frac{1}{3}.
\end{align*}
which is an $O\left(n^{-1/2}l_n\right)$ thereby vanishing asymptotically. Lastly, Condition (C6), Lemma \ref{lma:technical}, and H\"older's inequality imply that $\norm{(\ref{eqn:2})}$ is bounded by
\begin{align*}
\E\left[\norm{\E\left[ \beps_t x_t \mid \mathcal{F}_{t-l_n}\right]}^3\right]^\frac{1}{3} \E\left[\left\lvert\tilde{R}^{-1}-R^{-1}\right\rvert^{3}\right]^\frac{1}{3} \E\left[\norm{a^{\frac{1}{2}} a^{-\frac{1}{2}}\sum_{t=1}^{a} \beps_t^\top x_t}^3\right]^\frac{1}{3}= o\left(1\right)
\end{align*}
and this completes the proof.
\end{proof}
\begin{proposition}\label{prop:thm1_2}
Under assumptions of Theorem~\ref{thm:MSPE_dec_case1}, it holds that:
\begin{equation}\label{eqn:thm1_2}
N(\mathrm{II}) = (\mathrm{IV}) + o(1),
\end{equation}
where
\begin{align*}
(\mathrm{II}) = \E\left[\hat{R}^{-1} \boldsymbol{\hat{\Sigma}}x_n x_n \boldsymbol{\hat{\Sigma}}^\top \hat{R}^{-1}\right],\quad
(\mathrm{IV}) = \ \E\left[ \boldsymbol{\hat{\Sigma}}_B R^{-1} \boldsymbol{\hat{\Sigma}}_B^\top \right],
\end{align*}
with $\boldsymbol{\hat{\Sigma}}$ being defined in Proposition~\ref{prop:thm1_1} and $\boldsymbol{\hat{\Sigma}}_B = N^{-\frac{1}{2}} \sum_{t=1}^{N}\boldsymbol{\varepsilon_{t}} x_t.$
\end{proposition}
\begin{proof}
Let $\boldsymbol{M}_1 = x_n \left(\hat{R}^{-1} - R^{-1}\right) \boldsymbol{\hat{\Sigma}}_B$ and $\boldsymbol{M}_2 = x_n R^{-1} \boldsymbol{\hat{\Sigma}}_B$. Since
\begin{align*}
N \left(\mathrm{II}\right)= \E\left[\left(\boldsymbol{M}_1+ \boldsymbol{M}_2\right) \left(\boldsymbol{M}_1+ \boldsymbol{M}_2\right)^\top\right]
&= \E\left[\boldsymbol{M}_1 \boldsymbol{M}_1^\top\right] + \E\left[\boldsymbol{M}_2\boldsymbol{M}_2^\top\right]\\
& + \E\left[\boldsymbol{M}_1 \boldsymbol{M}_2^\top\right] + \E\left[\boldsymbol{M}_2 \boldsymbol{M}_1^\top\right]
\end{align*}
the proof of (\ref{eqn:thm1_2}) reduces to show that the following conditions hold:
\begin{align}
\norm{\E\left[ \boldsymbol{M}_1 \boldsymbol{M}_1^\top \right]} &= o\left(1\right),\label{prop:cond1}\\
\norm{\E\left[ \boldsymbol{M}_1 \boldsymbol{M}_2^\top \right]} &= o\left(1\right),\label{prop:cond2} \\
\norm{\E\left[ \boldsymbol{M}_2 \boldsymbol{M}_2^\top \right]- (\mathrm{IV})}&= o\left(1\right).\label{prop:cond3}
\end{align}
Conditions (\ref{prop:cond1}) and (\ref{prop:cond2}) readily follow from Assumptions (C3) and (C4), Lemma \ref{lma:technical}, the non singularity of $R$ and H\"older's inequality:
\begin{align*}
\E\left[\norm{\boldsymbol{M}_1 \boldsymbol{M}_1^\top }\right] &= \E\left[\norm{x_n^2 \left(\hat{R}^{-1} - R^{-1}\right)^2 \boldsymbol{\hat{\Sigma}}_B \boldsymbol{\hat{\Sigma}}_B^\top }\right]
\leq \left(\E\left[\left\lvert x_n \right\rvert^{10}\right]\right)^\frac{1}{5} \left(\E\left[\left\lvert\hat{R}^{-1}-R^{-1}\right\rvert^5\right]\right)^\frac{2}{5}\\ &\times\left(\E\left[\norm{\boldsymbol{\hat{\Sigma}}_B}^5\right]\right)^\frac{2}{5}= o\left(1\right);\\
\E\left[ \norm{\boldsymbol{M}_1\boldsymbol{M}_2^\top} \right] &= \E\left[\norm{x_n^2 \left(\hat{R}^{-1} - R^{-1}\right) R^{-1} \boldsymbol{\hat{\Sigma}}_B \boldsymbol{\hat{\Sigma}}_B^\top }\right]
\leq \left(\E\left[\left\lvert x_n\right\rvert^{10}\right]\right)^\frac{1}{5} \left(\E\left[\left\lvert\hat{R}^{-1}-R^{-1}\right\rvert ^5\right]\right)^\frac{1}{5}\\
&\times \left(\E\left[\left\lvert R^{-1} \right\rvert^5\right]\right)^\frac{1}{5} \left(\E\left[\norm{\boldsymbol{\hat{\Sigma}}_B}^5\right]\right)^\frac{2}{5}= o\left(1\right).
\end{align*}
As concerns (\ref{prop:cond3}), decompose the vector $\boldsymbol{\hat{\Sigma}}_B$ as follows:
\begin{align*}
\boldsymbol{\hat{\Sigma}}_B = N^{-\frac{1}{2}} \sum_{t=1}^{N}\boldsymbol{\varepsilon_{t}} x_t = \boldsymbol{u} + \boldsymbol{w}, \quad
\text{ with } \quad \boldsymbol{u} = N^{-\frac{1}{2}} \sum_{t=1}^{a}\boldsymbol{\varepsilon_{t}} x_t \quad\text{and}\quad \boldsymbol{w} = N^{-\frac{1}{2}} \sum_{t=b}^{N}\boldsymbol{\varepsilon_{t}} x_t.
\end{align*}
Hence, we have that
\begin{align*}
\E\left[\boldsymbol{M}_2\boldsymbol{M}_2^\top\right]- (\mathrm{IV}) &= \E\left[\boldsymbol{u} R^{-1} x_n x_n R^{-1} \boldsymbol{u}^\top\right] - \E\left[\boldsymbol{u} R^{-1} R R^{-1} \boldsymbol{u}^\top\right]\\
& + \E\left[\boldsymbol{u} R^{-1} x_n x_n R^{-1} \boldsymbol{w}^\top\right] - \E\left[\boldsymbol{u} R^{-1} R R^{-1} \boldsymbol{w}^\top\right]\\
& + \E\left[\boldsymbol{w} R^{-1} x_n x_n R^{-1} \boldsymbol{u}^\top\right] - \E\left[\boldsymbol{w} R^{-1} R R^{-1} \boldsymbol{u}^\top\right]\\
& + \E\left[\boldsymbol{w} R^{-1} x_n x_n R^{-1} \boldsymbol{w}^\top\right] - \E\left[\boldsymbol{w} R^{-1} R R^{-1} \boldsymbol{w}^\top\right].
\end{align*}
The law of iterated expectations implies that:
\begin{align}
&\norm{\E\left[\boldsymbol{M}_2 \boldsymbol{M}_2^\top\right]-(\mathrm{IV})}\nonumber\\
&\leq \norm{\E\left[\boldsymbol{u} R^{-1} \left(\E\left[x_n^2 \mid \mathcal{F}_{n-l_n}\right]-R\right) R^{-1} \boldsymbol{u}^\top\right]}\label{norm1}\\
&+ \norm{\E\left[\boldsymbol{u} R^{-1} \left(\E\left[x_n^2 \mid \mathcal{F}_{n-l_n}\right]-R\right) R^{-1} \boldsymbol{w}^\top\right]}\label{norm2}\\
&+ \norm{\E\left[\boldsymbol{w} R^{-1} \left(\E\left[x_n^2 \mid \mathcal{F}_{n-l_n}\right]-R\right) R^{-1} \boldsymbol{u}^\top\right]}\label{norm3}\\
&+ \norm{\E\left[\boldsymbol{w} R^{-1} \left(\E\left[x_n^2 \mid \mathcal{F}_{n-l_n}\right]-R\right) R^{-1} \boldsymbol{w}^\top\right]}\label{norm4}.
\end{align}
By using arguments previously developed, it is easy to see that, under Assumptions (C4) and (C6), (\ref{norm1}) -- (\ref{norm4}) asymptotically vanish. Therefore, conditions (\ref{prop:cond1}) -- (\ref{prop:cond3}) are fulfilled and the proof is completed.
\end{proof}
\begin{proposition}\label{prop:thm1_3}
Under assumptions of Theorem~\ref{thm:MSPE_dec_case1}, it holds that:
\begin{equation}\label{eqn:prop3}
(\mathrm{III}) = - (D) + o\left(1\right),
\end{equation}
where
\begin{align*}
(D) &= \E\left[ R^{-1} \left[\sum_{j=h}^{N-1} \left\{\left( \beps_1 x_1\right) \left(\beps_{j+1} x_{j+1}\right)^\top + \left(\beps_{j+1} x_{j+1}\right) \left( \beps_1 x_1\right)^\top\right\} \right] \right]
\end{align*}
\end{proposition}
\begin{proof}
The result readily follows upon noting that, under Assumption (C2) and the weakly stationarity of the process $\{x_t\}$, it holds that:
\begin{align*}
(\mathrm{III}) & = - \sum_{t=1}^{N} \E\left[R^{-1} \left\{ \left(\beps_t x_t\right) \left(\beps_n x_n\right)^\top + \left(\beps_n x_n\right) \left(\beps_t x_t\right)^\top\right\} \right]\\
& = - \ \sum_{j=h}^{n-1} \E\left[R^{-1} \left\{ \left(\beps_1 x_1\right) \left(\beps_{j+1} x_{j+1}\right)^\top + \left(\beps_{j+1} x_{j+1}\right) \left(\beps_1 x_1\right)^\top\right\} \right]\\
&= - \E\left[ R^{-1} \left(\sum_{j=h}^{N-1} \left\{\left( \beps_1 x_1\right) \left(\beps_{j+1} x_{j+1}\right)^\top + \left(\beps_{j+1} x_{j+1}\right) \left( \beps_1 x_1\right)^\top\right\} \right) \right] +o(1).
\end{align*}
\end{proof}
\begin{proposition}\label{prop:thm1_4}
Under assumptions of Theorem~\ref{thm:MSPE_dec_case1}, it holds that:
\begin{equation}\label{eqn:prop4}
(\mathrm{IV}) = (1) + \left(Q\right) + \left(D\right) + o(1),
\end{equation}
where
\begin{align*}
(1) & = N^{-1} \E\left[ R^{-1} \left\{\sum_{t=1}^{N} \left(\beps_t x_t\right)\left(\beps_t x_t\right)^\top \right\} \right], \\
\left(Q\right) &= \E\left[ R^{-1} \left[\sum_{s=1}^{h-1} \left\{\left( \beps_1 x_1\right) \left(\beps_{s+1} x_{s+1}\right)^\top + \left(\beps_{s+1} x_{s+1}\right) \left( \beps_1 x_1 \right)^\top\right\} \right] \right]\\
\left(D\right) &= \E\left[ R^{-1} \left[\sum_{j=h}^{N-1} \left\{\left( \beps_1 x_1\right) \left(\beps_{j+1} x_{j+1}\right)^\top + \left(\beps_{j+1} x_{j+1}\right) \left( \beps_1 x_1\right)^\top\right\} \right] \right]
\end{align*}
\end{proposition}
\begin{proof}
Let
\begin{equation*}
(2) = N^{-1} \E\left[ R^{-1} \left\{\sum_{j=1}^{N-1} \sum_{k=j+1}^{N} \left(\beps_j x_j\right) \left(\beps_k x_k\right)^\top\right\} \right],
\end{equation*}
and note that $(\mathrm{IV})-(1) = (2) + (2)^\top$. Moreover
\begin{align}
(2) & = N^{-1} \E\left[ R^{-1} \left\{\sum_{j=1}^{N-1} \left( N-j \right) \left( \beps_1 x_1\right) \left(\beps_{j+1} x_{j+1}\right)^\top \right\} \right] \nonumber\\
& = \E\left[ R^{-1} \left\{\sum_{j=1}^{N-1} \left( \beps_1 x_1\right) \left(\beps_{j+1} x_{j+1}\right)^\top \right\} \right] \label{eqn:a} \\
& - N^{-1} \E\left[ R^{-1} \left\{\sum_{j=1}^{N-1} j\left( \beps_1 x_1\right) \left(\beps_{j+1} x_{j+1}\right)^\top \right\} \right].\label{eqn:b}
\end{align}
Assumptions (C2) implies that (\ref{eqn:b}) is $o(1)$. Since (\ref{eqn:a}) can be written as
\begin{equation*}
\E\left[ R^{-1} \left\{\sum_{s=1}^{h-1} \left( \beps_1 x_1\right) \left(\beps_{s+1} x_{s+1}\right)^\top \right\} \right] + \E\left[ R^{-1} \left\{\sum_{j=h}^{N-1} \left( \beps_1 x_1\right) \left(\beps_{j+1} x_{j+1}\right)^\top \right\} \right],
\end{equation*}
then $(\ref{eqn:a})+(\ref{eqn:a})^\top=(Q)+(D)$ and this completes the proof.
\end{proof}
\subsubsection*{Proof of Theorem~\ref{thm:MSPE_dec_case1}}
We prove that:
\begin{align}
&N \left\{ \E\left[ \left( \by_{n+h} - \hat{\by}_{n+h} \right) \left( \by_{n+h} - \hat{\by}_{n+h} \right)^\top - \E\left[
\beps_{n}^{(h)} \beps_{n}^{(h)^\top} \right] \right] \right\}\nonumber\\
& = R^{-1} \E\left[ \left(\beps_{1}^{(h)} x_1\right)\left(\beps_{1}^{(h)} x_1\right)^\top \right] \label{eq:main1} \\
&+ R^{-1} \E\left[ \sum_{s=1}^{h-1} \left\{\left( \beps_1^{(h)} x_1\right) \left(\beps_{s+1}^{(h)} x_{s+1}\right)^\top + \left(\beps_{s+1}^{(h)} x_{s+1}\right) \left( \beps_1^{(h)} x_1 \right)^\top\right\} \right] \label{eq:main2}\\
&+ o(1). \nonumber
\end{align}
Since $$\left( \boldsymbol{\hat{\beta}}-\bBeta\right) = \hat{R}^{-1} \left( N^{-1} \sum_{t=1}^{N} x_t \boldsymbol{y}_{t+h} \right) - \bBeta = \hat{R}^{-1} \left(N^{-1} \sum_{t=1}^{N} x_t \beps_t \right),$$
routine algebra implies that:
\begin{equation}
\E\left[ \left(\by_{n+h} - \mathbf{\hat{y}}_{n+h}\right) \left(\by_{n+h} - \mathbf{\hat{y}}_{n+h}\right)^\top \right]
- \E\left[\beps_n \beps_n^\top\right] =
(\mathrm{I})+(\mathrm{II}).
\end{equation}
By applying Propositions~\ref{prop:thm1_1} -- Propositions~\ref{prop:thm1_4}, we have:
\begin{align*}
N \left\{ \E\left[ \left( \by_{n+h} - \hat{\by}_{n+h} \right) \left( \by_{n+h} - \hat{\by}_{n+h} \right)^\top - \E\left[
\beps_{n}^{(h)} \beps_{n}^{(h)^\top} \right] \right] \right\}= N(\mathrm{I})+N(\mathrm{II})\\
=(\mathrm{III})+(\mathrm{IV})+o(1)
=(1)+(Q)+o(1).
\end{align*}
The proof is completed upon noting that $(1)=(\ref{eq:main1})$ and $(Q)=(\ref{eq:main2})$.
\subsection{Proof of Theorem~\ref{thm:MOME_case1}}\label{sec:A2}
We start proving that
\begin{equation}\label{mi_hat}
\hat{\bmi}_h= \bmi_h + O_p (n^{-1/2}).
\end{equation}
Note that
\begin{align*}
\hat{\bmi}_h=N^{-1} \left(\sum_{t=1}^{N} \beps_{t} \beps_{t}^\top\right)- \left(N^{-1}\sum_{t=1}^{N} x_t \beps_{t}\right) \hat{R}^{-1}\left(N^{-1}\sum_{s=1}^{N} x_s \beps_{s}\right)^\top
\end{align*}
hence, it holds that $\hat{\bmi}_h - \bmi_h$ equals
\begin{align}
& N^{-1} \left\{\sum_{t=1}^{N} \left( \beps_{t} \beps_{t}^\top - \E\left[ \beps_{1} \beps_{1}^\top\right] \right) \right\}\label{eq:mi_A}\\
& -\left(N^{-1}\sum_{t=1}^{N} x_t \beps_{t}\right)\hat{R}^{-1} \left(N^{-1}\sum_{t=1}^{N} x_t \beps_{t}\right)^\top.\label{eq:mi_B}
\end{align}
Assumption (A1) implies that $(\ref{eq:mi_A})=O_p(n^{-1/2})$ whereas, by combining Assumptions (A3) and (A4) with the non-singularity of $R$ and H\"older's inequality, it can be shown that $(\ref{eq:mi_B})=O_p(n^{-1})$ and hence the proof of (\ref{mi_hat}) is complete.
\par\noindent
Next, we prove that
\begin{equation*}
\hat{\bvi}_h= \bvi_h + o_p (1).
\end{equation*}
It suffices to show that
\begin{equation}\label{Cs_hat}
\hat{\bC}_{h,s} = \bC_{h,s}+ o_p (1).
\end{equation}
It holds that $\hat{\bC}_{h,s}$ is equal to
\begin{align}
&\phantom{-}\left( N-s\right)^{-1} \sum_{t=1}^{N-s} \left( x_t \beps_{t}^{\top}\right)^{\top} \left( x_{t+s} \beps_{t+s}^{\top}\right)\label{Cs_A}\\
& - \left( N-s\right)^{-1} \sum_{t=1}^{N-s} x_t^2 x_{t+s} \left( \hat{\bBeta}_n(h) - \bBeta_h \right) \beps_{t+s}^{\top}\label{Cs_B}\\
& - \left( N-s\right)^{-1} \sum_{t=1}^{N-s} x_t x_{t+s}^2 \beps_{t} \left( \hat{\bBeta}_n(h) - \bBeta_h \right)^{\top}\label{Cs_C}\\
& + \left( N-s\right)^{-1} \sum_{t=1}^{N-s} x_t^2 x_{t+s}^2 \left( \hat{\bBeta}_n(h) - \bBeta_h \right)\left( \hat{\bBeta}_n(h) - \bBeta_h \right)^{\top}.\label{Cs_D}
\end{align}
We prove that (\ref{Cs_B}) is $o_p(1)$ componentwise. To this end consider:
$$\E\left[\left( N-s\right)^{-1}\abs*{\sum_{t=1}^{N-s} x_t^2 x_{t+s} \eps_{t+s,i}}\right],$$
with $\eps_{t+s,i}$ being the $i$-th component of the vector $\beps_{t+s}$. The triangular inequality and H\"older's inequality imply that:
\begin{align*}
\E\left[\left( N-s\right)^{-1}\abs*{\sum_{t=1}^{N-s} x_t^2 x_{t+s} \eps_{t+s,i}}\right]
&\leq \left( N-s\right)^{-1}\sum_{t=1}^{N-s}\E\left[\abs*{x_t^2 x_{t+s} \eps_{t+s,i}}\right]\\
&\leq (N-s)^{-1}\sum_{t=1}^{N-s}\left\{\left(\E\left[x_t^4\right]\right)^{1/2}\left( \E\left[\abs*{x_{t+s}\varepsilon_{t+s,i}}^2 \right]\right)^{1/2}\right\}.
\end{align*}
Since $ \hat{\bBeta}_n(h) - \bBeta_h = \hat{R}^{-1} \left( N^{-1} \sum_{j=1}^{N} x_j \beps_{j,h} \right)$, by combining Assumptions (A3), (A4) and (A5) with Chebyshev's inequality we obtain that (\ref{Cs_B}) is $o_p(1)$. Similarly, we can verify that (\ref{Cs_C}) and (\ref{Cs_D}) are $o_p(1)$. Lastly, Condition (A2) implies that $(\ref{Cs_A}) = \bC_{h,s}+ o_p (1)$, hence (\ref{Cs_hat}) is verified and the whole proof is complete.
\subsection{Proof of Theorem~\ref{thm:EFF_case1}}\label{sec:A3}
By Theorem \ref{thm:MOME_case1} the $\vmric_h$ defined in (\ref{eq:vmric}) can be written as:
\begin{equation}\label{eq:theo3_proof_1}
\vmric_h \left( \hat{\ell}_h\right) = \min_{1 \leq \ell \leq K} \norm{ \bmi_h + O_p (n^{-1/2}) } + \min_{ \ell \in M_1} \norm{\frac{\alpha_n}{n}\bvi_h + o_p \left(\frac{\alpha_n}{n}\right)}.
\end{equation}
Therefore,
\begin{equation}
\lim\limits_{n\rightarrow \infty} \vmric_h \left( \hat{\ell}_h\right) = \min_{1 \leq \ell \leq K}
\norm{\bmi_h}
\end{equation}
and hence
\begin{equation}
\lim\limits_{n\rightarrow +\infty} \Pr \left( \hat{\ell}_h \in M_1 \right) = 1.
\end{equation}
Now, consider two models $\ell_1$ and $\ell_2$ in the candidates set $J_{\ell_1}, J_{\ell_2} \in M_1$ such that $\vi_h(\ell_1)\neq \vi_h(\ell_2)$. We show that
\begin{equation}\label{eqn:eff_main}
\lim_{n\rightarrow \infty}\Pr \left[ \sign{\vmric_h(\ell_1) - \vmric_h(\ell_2)} = \sign{\norm{\boldsymbol{\vi}_h(\ell_1)}- \norm{\boldsymbol{\vi}_h(\ell_2)}}\right] = 1.
\end{equation}
By defining $\bmi^*_h$ to be the minimum value of $\bmi_h$ over the family of candidate models, we have:
\begin{align*}
\vmric_h (\ell_1) &= \norm{ \bmi_h^* + O_p (n^{-1/2}) } + \norm{ \frac{\alpha_n}{n}\bvi_h(\ell_1) + o_p\left(\frac{\alpha_n}{n}\right)},\\
\vmric_h (\ell_2) &= \norm{ \bmi_h^* + O_p (n^{-1/2}) } + \norm{ \frac{\alpha_n}{n} \bvi_h(\ell_2) + o_p\left(\frac{\alpha_n}{n}\right)}.
\end{align*}
Therefore, for sufficiently large $n$, it holds that:
\begin{equation*}
\vmric_h (\ell_1) - \vmric_h (\ell_2) = \left\|\frac{\alpha_n}{n}\right\| \left( \norm{ \bvi_h(\ell_1)} - \norm{ \bvi_h(\ell_2)}\right).
\end{equation*}
Thus
\begin{equation*}
\sign{ \vmric_h(\ell_1) - \vmric_h(\ell_2)} = \sign{ \norm{\bvi_h(\ell_1)} - \norm{\bvi_h(\ell_2)}},
\end{equation*}
and (\ref{eqn:eff_main}) is verified and implies that
\begin{equation}
\lim_{n\rightarrow \infty} \Pr\left(\hat{\ell}_h \in M_2\right) = 1.
\end{equation}
This completes the proof.
\section*{Acknowledgments}
Greta Goracci acknowledges the support of Libera Università di Bolzano, Grant WW201L (ESAMD).
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\section{Introduction}
\vspace{-1em}
\noindent
While its specific nature is still unclear, the presence of Dark Matter (DM) in our universe has been corroborated by several experimental evidences, and its relic density has been measured very precisely by the Planck~\cite{Aghanim:2018eyx}.
While cold DM with mass in the GeV to TeV range has been used to explain the large structure of the universe, warm DM such as sterile neutrinos can better account for other cosmological observations, such as the missing satellite galaxies, the cusp-core problem of inner DM density profiles and the too-big-to fail problem.
Recent measurements in the neutrino sector, such as the solar and atmospheric neutrino mass difference~\cite{Tanabashi:2018oca}, also call for physics Beyond the Standard Model (BSM).
A common theoretically well justified extension of the SM consists in the inclusion of right-handed neutrinos, which can generate small neutrino masses through various realisations of the see-saw mechanism.
When this mechanism is realised at one loop, the neutrino and DM sectors become connected, leading to interesting phenomenological signatures.
BSM constructions which explore such possibility are called Minimal Models, and several examples can be found in the literature.
Recently these models have also been classified according to their neutrino masses generating loop topologies~\cite{Restrepo:2013aga}.
Here we consider a variant of the SLIM (Scalar as Light as MeV) model~\cite{Boehm:2006mi,Farzan:2009ji} with a light sterile right-handed neutrino playing the role of DM candidate with mass in the MeV region, and were the correct neutrino masses are generated radiatively~\cite{Fiaschi:2019evv}.
\section{The SLIM model with MeV neutrino DM}
\vspace{-1em}
In this version of the SLIM model, the particle content of the SM is augmented by an extra complex scalar singlet field ($\rho$), a complex scalar doublet ($\eta$) and by two generations of Majorana right-handed neutrinos ($N_i$, $i$ = 1,2).
The new fields are stabilised by an extra global $U(1)$ symmetry, which gets softly broken to $Z_2$.
The Lagrangian with the Higgs ($\Phi$) potential and the new terms is
\vspace{-0.5em}
\begin{eqnarray}
\mathcal{L} &=& - m_1^2 \Phi^\dagger \Phi - m_2^2 \eta^\dagger \eta - m_3^2 \rho^*\rho - \frac{1}{2} m_4^2 \left(\rho^2+ (\rho^*)^2 \right) - \mu (\eta^\dagger \Phi \rho + h.c.) - \frac{1}{2}m_{N_i} \overline{N^c_i}N_i \nonumber\\
&& - \frac{1}{2}\lambda_1 (\Phi^\dagger\Phi)^2 - \frac{1}{2}\lambda_2 (\eta^\dagger\eta)^2 - \frac{1}{2}\lambda_3 (\rho^*\rho)^2 - \lambda_4 (\eta^\dagger\eta)(\Phi^\dagger\Phi)- \lambda_5 (\eta^\dagger\Phi)(\Phi^\dagger\eta) \nonumber\\
&& - \lambda_6 (\rho^*\rho)(\Phi^\dagger\Phi)- \lambda_7 (\rho^*\rho)(\eta^\dagger\eta) - \left(\lambda_8 \right)_{ij} (\overline{N^c_i} \eta^\dagger L_j + h.c.).
\end{eqnarray}
\noindent
The parameter $m_4$ is assumed to be small as it is responsible for the soft breaking of $U(1)$ into $Z_2$.
The new scalars acquire mass after electroweak symmetry breaking.
The charged components will have mass
\begin{equation}
m_{\eta^\pm}^2 = m_2^2 + \frac{1}{2}\lambda_4 v^2
\end{equation}
\noindent
while the neutral components of the singlet and doublet mix and their mass matrix reads
\begin{equation}
M_{R,I}^2 = \begin{pmatrix}
m_2^2 + (\lambda_4 + \lambda_5) \frac{v^2}{2} & \mu \frac{v}{\sqrt{2}} \\
\mu \frac{v}{\sqrt{2}} & m_3^2 + \lambda_6 \frac{v^2}{2} \pm m_4^2
\end{pmatrix} =:
\begin{pmatrix}A & \!\!\!\!B \\ B & \ C_{R,I} \end{pmatrix}
\end{equation}
\noindent
with the positive and negative signs associated to the real and imaginary components respectively.
The mass splitting between the two is thus small, and the masses of their eigenstates is
\begin{equation}
m_{R,I}^2 = \frac{1}{2}\left(A+C_{R,I} \pm \sqrt{(A-C_{R,I})^2 +4B^2}\right).
\end{equation}
\noindent
We parametrise their mass difference by introducing the parameter $\epsilon$
\vspace{-0.5em}
\begin{equation}
AC_{R,I}-B^2=:\epsilon(A+C_{R,I})
\end{equation}
\noindent
By tuning this small parameter we obtain two MeV eigenvalues, while the other two remain as heavy as the charged scalars.
\section{Experimental constrains}
\vspace{-1em}
The large parameter space of the model is strongly narrowed by experimental constrains.
The masses of the new scalars, thus the parameters $m_{2,3}$, and the couplings $\lambda_{4,5,6}$ are constrained by ATLAS~\cite{Aaboud:2019rtt} and CMS~\cite{Sirunyan:2018owy} limits on the Higgs branching ratio into invisible decays.
The masses of the charged scalars are excluded below 98.5 GeV by LEP measurements~\cite{Abbiendi:2003yd} and their couplings to the Higgs are restricted below a certain threshold by measurements on the Higgs to two photon branching ratio~\cite{Sirunyan:2018koj}.
The choice $\lambda_{4,5} \simeq$ 0.1 satisfies the above constrains, and at the same time provides with a relatively large electron recoil cross section.
To provide for a valid solution for the cosmological problems cited above, the masses of the singlet and of the doublet neutral eigenstates shall be close.
For a better control on the mass splitting, we fix $\lambda_6$ = 2.3, while we vary the parameter $\epsilon$ between 10$^{-5}$ and 60 GeV$^2$.
Moreover, in order not to erase primordial DM fluctuation, for MeV DM and weak DM self-interactions, we need a small mass difference between the DM particle and the lightest scalar.
For this purpose we vary the ratio of the right handed neutrino masses over the lighter scalars ones between 0.1 and 0.98.
Finally, the couplings $\lambda_{2,3,7}$ have a small impact on the phenomenology, and they are fixed $\simeq$ 0.1 - 0.2.
Because of the residual $Z_2$ symmetry, no tree-level see-saw mechanism is allowed.
The neutrino masses are generated at one-loop through the interaction of the new neutral scalar fields and the right-handed neutrinos.
In particular, similarly to the scotogenic model, the neutrino masses arise from the small mass splitting between the real and imaginary scalars, and they are directly proportional to the coupling $\lambda_8$.
We adopt the very convenient Casas-Ibarra parametrisation~\cite{Casas:2001sr}, which allows to take as input the experimentally measured values of the neutrino mass differences and mixing angles of the PMNS matrix, together with the dark particles masses, and it returns the associated value of $\lambda_8$.
With this method the parameters of the neutrino sector are automatically satisfied for any point in the parameter space of the model.
\section{Numerical Results}
\vspace{-1em}
We obtained the following numerical results scanning over appropriate regions of the model's parameter space.
Fig.~\ref{fig:Relic_density} on the left shows the right-handed neutrino DM relic density that is obtained for the parameter space points with certain electrons to electron neutrinos coupling $\lambda_8^{e1}$. The color scheme shows the respective value of the DM mass.
The measured relic density is represented by the blue horizontal line, and it can be generated by various appropriate combinations of couplings and DM masses.
In particular, as the coupling rises, the DM mass increases from the MeV to the GeV region.
In the figure on the right is shown the projection of the model's parameter space points leading to the correct relic density (and satisfying the other experimental constrains discussed above) in the plane of DM-electron coupling and DM mass.
Again, larger couplings, thus more efficient DM annihilation processes, are in order to reduce the DM abundance for heavier DM.
\begin{figure}
\centering
\includegraphics[width=0.46\textwidth]{Relic_density.pdf}{(a)}
\includegraphics[width=0.46\textwidth]{Neutrino_coupling.pdf}{(b)}
\caption{(a) The relic density $\Omega h^2$ in the plane right-handed neutrino DM mass versus the absolute value of its coupling to electrons and electron neutrinos. The relic density measured by Planck~\cite{Aghanim:2018eyx}
is shown as a blue horizontal band, and the DM mass is given on a colour scale.
(b) The projection of the points satisfying all constrains, including correct relic density, in the plane DM-electron coupling versus DM mass.}
\label{fig:Relic_density}
\end{figure}
The right-handed neutrinos are thus connected to charged leptons through sizeable coupling $\lambda_8$, and in this context Lepton Flavour Violating (LFV) processes occur at one-loop.
The experimentally most sensitive LFV process being the flavour-changing neutral current $\mu \rightarrow e\gamma$, sets strong bounds on the $\lambda_8^{ei}$ and $\lambda_8^{\mu i}$ couplings, which in turn were connected to the parameters of the neutrino section through the Casas-Ibarra parametrisation.
The BR for this process is shown in Fig.~\ref{fig:Lepton_flavour_violation_Electron_recoil_XS}(a) as function of $\lambda_8^{e1}$ in the $x$-axis and of $\lambda_8^{\mu 1}$ in the colour scale.
The black solid and dashed lines represent the current~\cite{TheMEG:2016wtm} and expected future~\cite{Renga:2018fpd} limits imposed by the MEG experiment, and as visible they already exclude large part of the model's parameter space, forcing $\lambda_8^{e1}$ below 6 $\times$ 10$^{-3}$ and $\lambda_8^{\mu 1}$ below 10$^{-2}$.
Future constrains will lower these threshold further by roughly a factor four.
The experimental limits on other LFV processes carry similar but somewhat slightly weaker constrains on the same couplings.
\begin{figure}
\centering
\includegraphics[width=0.46\textwidth]{Lepton_flavour_violation.pdf}{(a)}
\includegraphics[width=0.46\textwidth]{Electron_recoil_XS.pdf}{(b)}
\caption{(a) Branching ratio of the lepton flavour violating process
$\mu \rightarrow e \gamma$ versus the DM-electron coupling $\lambda_8^{e1}$.
The DM-muon coupling $\lambda^{\mu 1}_8$ is shown on a colour scale.
Solid and dashed black lines represent the current \cite{TheMEG:2016wtm} and expected future~\cite{Renga:2018fpd} limits imposed by the MEG experiment.
(b) Scattering cross section of right-handed neutrino DM off electrons versus the DM mass. The DM-electron coupling $\lambda_8^{e1}$ is shown on a colour scale.}
\label{fig:Lepton_flavour_violation_Electron_recoil_XS}
\end{figure}
\section{Sensitivity on electron recoil}
\vspace{-1em}
The relatively large couplings between the DM right-handed neutrinos and charged leptons make the SLIM model an ideal candidate to explore the sensitivity of direct detection experiment in their novel electron-recoil analysis.
A generalisation of the usual scattering over nuclei analysis has been proposed by the XENON collaboration~\cite{Aprile:2016wwo} to lower their detector energy threshold down to few KeV by abandoning the scintillation light requirement S1 and using only the charge signal S2.
The disadvantage of this method is that fiducialisation is limited, since without S1 the event depth $z$ cannot be accurately estimated, yielding a potentially increased background.
In an optimal configuration, the XENON100 experiment could became sensitive to DM masses down to 600 MeV with a lowest cross section of 6$\times$10$^{-35}$ cm$^2$ for a DM mass of 2 GeV and axial-vector couplings.
Very recently XENON1T published a S2-only analysis~\cite{Aprile:2019xxb} with an electron energy threshold of 0.4 keV, which would be sensitive to light DM scattering off electrons down to masses of 100 to 20 MeV.
In Fig.~\ref{fig:Lepton_flavour_violation_Electron_recoil_XS}(b), we show the DM-electron scattering cross section in the SLIM model for the parameter space points satisfying all the collider, LFV and astrophysical constrains, as function of the DM mass and with colour scheme reflecting the size of the $\lambda_8^{e1}$ coupling.
Even for the largest possible coupling ($\lambda_8^{e1} \geq$ 0.1), the cross section is at most of the order of 10$^{-46}$ cm$^2$, while when further imposing the LFV constrains it is bounded below 10$^{-52}$ cm$^2$ as visible in the plot.
The strong suppression is due to the large mass of the mediator $\eta^{\pm}$, which enters with the fourth power in the calculation of the electron recoil cross section.
\section{Conclusions}
\vspace{-1em}
We have analysed a variant of the SLIM model, predicting MeV right-handed neutrino DM where the particles in the dark sector are responsible for the small neutrino masses which are generated at one-loop.
We have shown that collider, astrophysical and LFV constrains drastically restrict the allowed parameter space of the model, where however various viable points can be found.
The leptophilic nature of the model allows for potential discovery in the XENON1T direct detection experiment exploiting the recently proposed analysis of electron recoil signals by the collaboration.
We have verified that once all experimental constrains are taken into account, the size of the DM-electron cross sections of the viable points are several orders of magnitude below the sensitivity of the experiment.
\section*{Acknowledgements}
\vspace{-1em}
\noindent
We thank J.\ Alvey for useful discussions.
This work has been supported by the BMBF under contract 05H18PMCC1 and the
DFG through the Research Training Group 2149 ``Strong and weak interactions
-- from hadrons to dark matter''.
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train/arxiv
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\section{Introduction}
\label{sec:introduction}
Antiferromagnetically interacting spins are sensitive to both quantum mechanics and frustration~\cite{Lhuillier,Indrani}. Therefore, the quantum antiferromagnets with frustration can realize interesting non-magnetic ground states such as the dimerized singlet (valence bond) states~\cite{MG, SS, Kageyama}, plaquette singlets~\cite{CaV4O9, CaV4O9_Troyer}, spin-liquids~\cite{Kagome_Material, Mila,kappa_salt} etc., apart from having an antiferromagnetically ordered ground state (say, N\'eel type). In the present study, we are concerned with those systems in which the ground state has a dimer order, either spontaneous or given. There is an ever-increasing number of quantum antiferromagnetic (AF) materials which exhibit, or seem to exhibit, dimerization physics at low temperatures~\cite{Kageyama, TlCuCl3, TlCuCl3Nature, BaCuSi2O6,CuFeGeO,Sr3Cr2O8}. Such systems are typically characterized by an energy gap to spin excitations, thereby showing sharp drop in the magnetic susceptibility as the temperature is lowered below a certain temperature characteristic of the interaction between spins.
We are presently interested in generic theoretical questions concerning the instability of a dimer singlet ground state towards AF ordering as competing interactions in a system are varied. For a spin-1/2 system, such investigations at the simplest level can be conveniently carried out by doing a mean-field triplon analysis with respect to a dimer ground state. A {\em triplon} is a triplet excitation residing on a bond (dimer), and dispersing from one bond to another under the exchange interactions present in the system. An energy gap in the triplon dispersion implies a stable dimer phase, while gaplessness signifies an AF order of some kind (that depends upon the dispersion). The underlying formulation is facilitated by what is called the {\em bond-operator} representation of the spin operators~\cite{sach-bhatt,chub}. This approach has been successfully applied to many different spin-1/2 systems~\cite{gopalan_rice,rkbk,rkdkbk}. Subsequent to this, the bond-operator method has also been developed for spin-1 dimer problems~\cite{spin1_bo1,spin1_bo2,spin1_bo3, spin1_bo4}. Moreover, the triplon analysis has also been suitably extended to the square-plaquette problems (spin-1/2 case)~\cite{plaquette1}. However, no such formulation exists for a general spin-$S$ dimer problem. In the present work, we precisely set out to achieve this objective. That is, to derive the bond-operator representation for spin-$S$ operators (in Sec.~\ref{sec:boS}), and to do the mean-field triplon analysis for some model systems of interest (in Secs.~\ref{sec:coupledCD} and~\ref{sec:honeycomb}).
Obviously, it is impractical to be working with all the $(2S+1)^2$ states of a spin-S dimer. We therefore restrict our analysis to the subspace of singlet and triplet states only. Apart from simplifying our labor, which it does, it is enough for a primary discussion of the problem. Here, we adopt a simple working philosophy that, for an antiferromagnetic spin-S dimer problem, the triplet excitations are the principal cause of instability (if it occurs) of a singlet phase, as the higher spin excitations are further up in energy and hence irrelevant for an effective low energy description. It is implicit in our discussion that a given system only has exchange interactions. The treatment, however, will have to be extended to include quintet or higher total-spin states, if the single-ion anisotropy effects are present. While it is a real concern, presently we focus only on developing the triplon analysis for dimerized spin-$S$ quantum antiferromagnets. As an interesting byproduct of this exercise, we have also developed a nice and simple method for evaluating the total-spin eigenstates for a pair of spin-$S$ (angular momentum addition) using Schwinger boson representation. Our method resembles that of Schwinger's, but it is different in actual details of the procedure that generates the compound spin eigenstates.~\cite{Schwinger,Mattis} For the benefit of readers, it is presented in detail in the appendix.
Of the two spin-$S$ quantum antiferromagnets, that we apply this mean-field triplon analysis to, the first one in Sec.~\ref{sec:coupledCD} is a coupled columnar-dimers model on square lattice. Different variations of this model for the spin-1/2 (and spin-1) case have been studied extensively for investigating the quantum phase transition from dimer to AF ordered phases~\cite{SinghGelfandHuse, AffleckGelfandSingh, Matsumoto, Ivanov, Imada, Richter_SpinDependence,Wenzel_Janke}. Here, we investigate this transition as a function of $S$, the size of spin. Interestingly, we find that the dimer singlet phase (a quantum mechanical phase) survives even in the so-called classical limit ($S\rightarrow\infty$), under suitable conditions (such as strong frustration). For example, we find the columnar-dimer phase to be stable and present in the limit $S\rightarrow\infty$ in a small range of coupling around $J_2/J_1=0.5$ for the $J_1$-$J_2$ model. While other kinds of gapped phases (such as the Haldane phase) are likely to arise in different regions of the phase diagram (especially for $S\ge 1$)~\cite{Matsumoto}, we can not study them here within a bond-operator mean-field theory. However, the conclusion drawn from the present calculation about the $S\rightarrow\infty$ limit, which is certainly valid in the vicinity of strongly dimerized limit, suggests a generic possibility of this kind, and asks for a rethinking of the classical limit in quantum antiferromagnets.
The second one is a model on honeycomb lattice, recently constructed and shown by the present author to have an exact triply degenerate dimer ground state for a certain value of the interaction parameter, for any spin. In Ref.~\onlinecite{rkdkbk}, we have already presented the mean-field triplon analysis results for the spin-1/2 case of this model. However, we did not know then, how to do it for $S>1/2$. This, in fact, was our original motivation behind developing the triplon analysis for spin-$S$ dimer problems. From the mean-field triplon analysis of this model in Sec.~\ref{sec:honeycomb}, for $S\ge1$, we find the dimer phase giving way to the N\'eel ordered AF phase as soon as one moves away from the point of exact dimer ground state. It is unlike the spin-1/2 case where the dimer phase was found to survive over a finite range of coupling.
In Sec.~\ref{sec:extend}, we present two straightforward extensions of the spin-S bond-operator representation. First, we extend it to include the quintet states. Next, we work out a similar representation on a square plaquette, in the restricted space of a plaquette singlet and certain low-lying triplets. Finally, we conclude with a summary.
\section{Bond-operator representation for spin-$S$ operators}
\label{sec:boS}
Consider the Heisenberg exchange interaction, ${\bf S}_1\cdot{\bf S}_2$, on a bond. The eigenstates, $\{|j,m_j\rangle \}$, of this problem are such that, ${\bf S}_1\cdot{\bf S}_2|j,m_j\rangle=[-S(S+1)+\frac{1}{2}j(j+1)]|j,m^{ }_j\rangle$, where $j=0,1,\dots,2S$ is the total-spin quantum number of the two spins. For a given $j$, the eigenvalues $m_j$ of operator $({\bf S}_1+{\bf S}_2)_z$ are given by $m^{ }_j=-j,-j+1,\dots, j$. Therefore, the bond eigenstate for a given $j$ is ($2j+1$)-fold degenerate. It is a singlet for $j=0$, triplet for $j=1$, quintet for $j=2$, and so on. We denote the singlet state as $|s\rangle$, and the triplets as $|t^{ }_{m^{ }_1}\rangle $ where $m^{ }_1=0,\pm 1$. The quintet states are denoted as $|q^{ }_{m^{ }_2}\rangle$ where $m^{ }_2=0,\pm 1,\pm 2$. The eigenstates for $j > 2$ may in general be denoted as $|h_{j,m_j}\rangle$. Below we define the bosonic creation operators, $ \hat{s}^\dag, \hat{t}^\dag_{m^{ }_1}$, $ \hat{q}^\dag_{m^{ }_2}$ and $ \hat{h}^\dag_{j,m_j}$, corresponding to the bond eigenstates. These operators are called bond-operators.
\begin{subequations}
\begin{eqnarray}
|s\rangle &:=& \hat{s}^\dag|0\rangle \label{eq:sbop}\\
|t^{ }_{m^{ }_1}\rangle &:=& \hat{t}^\dag_{m^{ }_1}|0\rangle \label{eq:tbop}\\
|q^{ }_{m^{ }_2}\rangle &:=& \hat{q}^\dag_{m^{ }_2} |0\rangle \label{eq:qbop}\\
|h^{ }_{j,m_j}\rangle &:=& \hat{h}^\dag_{j,m_j}|0\rangle \label{eq:hbop}
\end{eqnarray}
\end{subequations}
Here, $|0\rangle$ denotes the vacuum of the bosonic Fock space. The completeness of the bond eigenstates implies the following physical constraint on the bond-operators.
\begin{equation}
\hat{s}^\dag\hat{s} + \hat{t}^\dag_{m^{ }_1}\hat{t}^{ }_{m^{ }_1}+\hat{q}^\dag_{m^{ }_2}\hat{q}^{ }_{m^{ }_2}+\hat{h}^\dag_{j,m_j}\hat{h}^{ }_{j,m_j}=1
\label{eq:constraint}
\end{equation}
Here, the repeated indices are summed over.
In terms of the bond-operators, the exchange Hamiltonian on the bond can be written as:
\begin{eqnarray}
& J{\bf S}_1\cdot{\bf S}_2 = \nonumber \\
& -JS(S+1)\left[\hat{s}^\dag\hat{s}+\hat{t}^\dag_{m^{ }_1}\hat{t}^{ }_{m^{ }_1}+\hat{q}^\dag_{m^{ }_2}\hat{q}^{ }_{m^{ }_2} +\hat{h}^\dag_{j,m_j}\hat{h}^{ }_{j,m_j}\right] \nonumber \\
& +J\left[\hat{t}^\dag_{m^{ }_1}\hat{t}^{ }_{m^{ }_1} +3\hat{q}^\dag_{m^{ }_2}\hat{q}^{ }_{m^{ }_2}+\frac{1}{2} j(j+1) \hat{h}^\dag_{j,m_j}\hat{h}^{ }_{j,m_j}\right]. \label{eq:bondH}
\end{eqnarray}
Furthermore, to describe the interaction between the spins of different bonds in the bosonic Fock space, we must know the spin operators in terms of the bond-operators. Below we develop the bond-operator representation for spins, which is a generalization of the well-known bond-operator representation for spin-1/2 operators to the case of arbitrary spin-$S$.
In order to construct the bond-operator representation for the spins, we first find out the explicit forms of all the eigenstates on a bond. In the appendix to this paper, we have worked out an elegant procedure to write down the total-spin eigenstates for a pair of arbitrary spins. Following this approach, we can write the singlet wavefunction on a bond as:
\begin{equation}
|s\rangle = \frac{1}{\sqrt{2S+1}}\sum_{m=0}^{2S} (-)^m |S-m,-S+m\rangle. \label{eq:singlet_state}
\end{equation}
Here, the state, $|S-m,-S+m\rangle$, denotes a product-state, $|S,S-m\rangle \otimes|S,-S+m\rangle$, of the two spins of a bond. For the derivation of Eq.~(\ref{eq:singlet_state}), refer to Proposition 1 in the appendix, and also see Fig.~\ref{fig:singlet_triplets}. We can similarly write the three triplet states as:
\begin{subequations}
\begin{eqnarray}
|t_1\rangle &=& \frac{1}{\sqrt{{\tt N}_t}}\sum_{m=0}^{2S-1}(-)^m \sqrt{(2S-m)(m+1)} \times \nonumber \\
&& \hspace{2cm} |S-m,-S+m+1\rangle \label{eq:triplet_plus1}\\
|t_0\rangle &=& \sqrt{\frac{2}{{\tt N}_t}}\sum_{m=0}^{2S}(-)^m (S-m) |S-m,-S+m \rangle \label{eq:triplet_0} \\
|t_{\bar{1}}\rangle &=& \frac{1}{\sqrt{{\tt N}_t}}\sum_{m=0}^{2S-1}(-)^m\sqrt{(2S-m)(m+1)} \times \nonumber \\
&& \hspace{2cm} |S-m-1,-S+m\rangle \label{eq:triplet_minus1}
\end{eqnarray}
\end{subequations}
where $\bar{1}$ in the above equation denotes $m^{ }_1=-1$ (we will sometime denote negative integers as integers with a bar), and the normalization, ${\tt N}_t=2S(S+1)(2S+1)/3$.
Refer to Eqs.~(\ref{eq:1plus1}) to (\ref{eq:1minus1}) and Fig.~\ref{fig:singlet_triplets} for the derivation of the triplet states. Below we also write the quintet states for $m^{ }_2=0$, $1$ and $2$ (refer to Fig.~\ref{fig:quintets} and related discussion for details).
\begin{subequations}
\begin{eqnarray}
|q^{ }_2\rangle &=& \frac{1}{\sqrt{{\tt N}_q}}\sum_{m=0}^{2S-2}(-)^m \sqrt{(2S-m)(2S-m-1)} \nonumber \\
&& \times \sqrt{(m+1)(m+2)} \, |S-m,-S+m+2\rangle \label{eq:quintet_plus2}\\
|q^{ }_1\rangle &=& \frac{1}{\sqrt{{\tt N}_q}}\sum_{m=0}^{2S-1}(-)^m (2S-2m-1)\sqrt{2S-m} \nonumber\\
&& \hspace{1cm} \times \sqrt{m+1} \, |S-m,-S+m+1 \rangle
\label{eq:quintet_plus1} \\
|q^{ }_{0}\rangle &=& \sqrt{\frac{2}{3{\tt N}_q}}\sum_{m=0}^{2S}(-)^m [3(S-m)^2-S(S+1)] \nonumber \\
&& \hspace{2cm} \times |S-m,-S+m\rangle \label{eq:quintet_0}
\end{eqnarray}
\end{subequations}
Here, ${\tt N}_q=2S(S+1)(2S-1)(2S+1)(2S+3)/15$. The state $|q^{ }_{\bar{1}} \rangle$ can be generated from $|q^{ }_1\rangle$ by changing $|S-m,-S+m+1\rangle $ to $|S-m-1,-S+m\rangle$ (that is, ${\bf S}_{1z} \leftrightarrow -{\bf S}_{2z}$). Similarly, for $|q^{ }_{\bar{2}}\rangle$. We can also evaluate the states for higher $j$ values following the Propositions 5 and $5^*$ in the appendix.
As emphasized earlier, the present discussion will be restricted to the subspace of singlet and triplet states only. By computing the matrix elements of ${\bf S}_1$ and ${\bf S}_2$ in this restricted subspace, we derive the following bond-operator representation for spin-$S$ operators.
\begin{subequations}
\begin{eqnarray}
{\bf S}_{1\alpha} &\approx& \sqrt{\frac{S(S+1)}{3}}\left(\hat{s}^\dag\hat{t}^{ }_\alpha + \hat{t}^\dag_\alpha\hat{s}\right)-\frac{i}{2}\epsilon_{\alpha\beta\gamma}\hat{t}_\beta^\dag\hat{t}^{ }_\gamma \label{eq:bo_s1}\\
{\bf S}_{2\alpha} &\approx& -\sqrt{\frac{S(S+1)}{3}}\left(\hat{s}^\dag\hat{t}^{ }_\alpha + \hat{t}^\dag_\alpha\hat{s}\right)-\frac{i}{2}\epsilon_{\alpha\beta\gamma}\hat{t}_\beta^\dag\hat{t}^{ }_\gamma
\label{eq:bo_s2}
\end{eqnarray}
\end{subequations}
Here, $\alpha=x,y,z$ (for three components of the spin operators), and the same for $\beta$ and $\gamma$. The $\epsilon_{\alpha\beta\gamma}$ denotes the totally antisymmetric tensor. Moreover, $\hat{t}^\dag_x = \frac{1}{\sqrt{2}}(\hat{t}^\dag_{\bar{1}}-\hat{t}^\dag_1)$, $\hat{t}^\dag_y = \frac{i}{\sqrt{2}}(\hat{t}^\dag_{\bar{1}}+\hat{t}^\dag_1)$ and $\hat{t}^\dag_z= \hat{t}^\dag_0$. Since it is convenient to write the bond-operator representation, Eqs.~(\ref{eq:bo_s1}) and (\ref{eq:bo_s2}), in terms of $\hat{t}^\dag_\alpha$, we also write the constraint, Eq.~(\ref{eq:constraint}), and the bond Hamiltonian, Eq.~(\ref{eq:bondH}), using the same, replacing $\hat{t}^\dag_{m^{ }_1}\hat{t}^{ }_{m^{ }_1}$ by $\hat{t}^\dag_\alpha \hat{t}^{ }_\alpha$.
While our representation is valid for arbitrary $S$, it is obviously approximate for $S\ge 1 $, because it is constructed in a restricted subspace, ignoring the contributions from $j\ge 2$ states. However, it is exact for $S=1/2$, and correctly reproduces the known representation~\cite{sach-bhatt}. As briefly discussed in the introduction, for doing a simple stability analysis of the dimer phase of a spin-$S$ quantum antiferromagnet, it would suffice to know the dynamics of triplet excitations, except when it may be necessary to consider higher spin states.
In the following sections, we do mean-field triplon analysis of two different spin-$S$ models using this bond-operator representation. The first one is a model of coupled columnar-dimers on square lattice. It reduces to many different models of interest such as $J_1$-$J_2$ model, the coupled ladders and so on. The second model is defined on the honeycomb lattice. It admits an exact dimer ground state for arbitrary $S$, and is expected to undergo a transition to the N\'eel ordered phase away from the exactly solvable case.
\section{Coupled dimers on square lattice}
\label{sec:coupledCD}
\begin{figure}[t]
\centering
\includegraphics[width=7cm]{CoupledDimers2
\caption{Quantum spin-$S$ coupled dimer model. As in Eq.~\ref{eq:model1}, the exchange interactions are: thick bonds=$J$, thin horizontal lines=$\eta_x J$, not-so-thin vertical lines=$\eta_y J$, and thin dashed lines=$\eta^\prime J$, where $J$, $\eta_x$, $ \eta_y$ and $\eta^\prime>0$. Also shown are the primitive translations, and the spin labeling on a dimer.}
\label{fig:model1}
\end{figure}
Consider a spin-$S$ quantum antiferromagnet of interacting dimers on square lattice as shown in Fig.~\ref{fig:model1}. The arrangement of dimers is taken to be columnar because it occurs in the disordered ground state of the $J_1$-$J_2$ model, which is a special case of the present model.
However, one may also consider other arrangements, if one wants. The Hamiltonian of this model is written below.
\begin{eqnarray}
H^{ }_{\tt I} &=& J \sum_{{\bf r}} \Big[ {\bf S}_{{\bf r},1} \cdot {\bf S}_{{\bf r},2} + \eta_x {\bf S}_{{\bf r},2} \cdot {\bf S}_{{\bf r}+2a{\hat{x}},1} \nonumber \\
&& + \eta_y \left( {\bf S}_{{\bf r},1} \cdot {\bf S}_{{\bf r}+a{\hat{y}},1}+ {\bf S}_{{\bf r},2} \cdot {\bf S}_{{\bf r}+a{\hat{y}},2}\right) \nonumber\\
&& +\eta^\prime\sum_{\underline{\delta}}^{\pm a{\hat{y}}}\left( {\bf S}_{{\bf r},1} \cdot {\bf S}_{{\bf r}+\underline{\delta},2}+{\bf S}_{{\bf r},2} \cdot {\bf S}_{{\bf r}+2a{\hat{x}}+\underline{\delta},1} \right) \Big]
\label{eq:model1}
\end{eqnarray}
Here, $1$ and $2$ denote the two spins of a dimer; ${\bf r}$ denotes the position of a dimer, and is summed over all dimers; $J$ is the intra-dimer antiferromagnetic exchange; various inter-dimer spin-exchange interactions (relative to $J$) are given by $0\le \eta_x, \eta_y, \eta^\prime\le 1$. Refer to Fig.~\ref{fig:model1} for clarification. Physically, the single-ion anisotropy is also expected to be present for $S \ge 1$ (except when the ground state of the ion is an orbital singlet). However, presently we work without such anisotropy effects. Our objective here is to apply the bond-operator representation of the previous section to study the stability of a dimer phase in a simple model quantum antiferromagnet, and not study any particular physical system. The model given by $H_{\tt I}$ reduces to the following simpler models of interest for certain special choices of the interactions.
\begin{enumerate}
\item $\eta^\prime=0$. It presents an {\em unfrustrated} model which interpolates between coupled dimer chains and coupled ladders, passing through a square lattice model at $\eta_x=\eta_y$. For $\eta_x\simeq 0$, it describes a set of decoupled (or weakly coupled) two-leg ladders, which is physically the case of a family of ladder compounds Na$_2$T$_2$(C$_2$O$_4$)$_3$(H$_2$O)$_2$, where T$=$Ni, Co, Fe and Mn in the increasing order of spin from $S=1$ to $5/2$. As noted earlier, one must also consider the single-ion anisotropy in real materials, which has been ignored presently. A realistic calculation for this family of ladder compounds will be discussed elsewhere. For $\eta_y=0$, it describes a set of decoupled dimerized spin chains.
\item $\eta^\prime \neq 0$ and $\eta_x=\eta_y=\eta$. It is a frustrated model of coupled columnar dimers on square lattice which reduces to the well-known $J_1$-$J_2$ model when $\eta=1$.
\end{enumerate}
Below the two cases are studied within a triplon mean-field theory of $H_{\tt I}$. This is however not an exhaustive discussion of the problem, as it does not address other kinds of gapped phases that may arise for larger values of $S$. In any case, let us see what we learn from this simple stability analysis of the dimer phase.
\subsection{Mean-field triplon analysis}
\label{subsec:mft}
Imagine a special limit of $H_{\tt I}$ in which all couplings, except the intra-dimer exchange $J$, were zero. Then, the spins on each thick bond (dimer) in Fig.~\ref{fig:model1} would form an exact singlet in the ground state. Moreover, the elementary excitation in this case would correspond to creating a triplet on it. Since a triplet state on a bond costs an extra energy $J$, there is an energy gap to such excitations. Besides, these triplet excitations are localized because of the absence of inter-dimer interactions in this special case of independent dimers. This limit presents an idealized version of what is called a {\em spin-gapped} dimer phase in quantum antiferromagnets. In general, the inter-dimer spin interactions are non-zero, and the triplet excitations disperse, thereby lowering the spin-gap. A dispersing bond-triplet is often called a {\em triplon}. As long as the triplon spin-gap is non-zero, the dimer phase is stable against these excitations, and such a ground state will have zero magnetic moment. For some values of the competing interactions in a problem, the spin-gap may however close. This marks the onset of a quantum phase transition from the gapped dimer phase to a gapless ordered AF phase. We study such quantum phase transitions in $H_{\tt I}$ within a simple mean-field theory using bond operators.
The key steps of a mean-field triplon analysis are the following. First, identify a configuration of the singlet forming dimers as expected in the ground state. In the present case, by construction, the preferred dimers are the tick bonds in Fig.~\ref{fig:model1}. Using the bond-operator representation for the spins on each dimer, rewrite the spin Hamiltonian in terms of the bond operators, including the constraint by means of a Lagrange multiplier. Now replace the singlet bond-operators, $\hat{s}$ and $\hat{s}^\dag$, on every dimer by a mean-field, $\bar{s}$. This results in a model of interacting triplons (on a mean-field singlet background given by $\bar{s}$). To make the problem tractable, ignore the trilpon-triplon interaction (similar to the spin-wave analysis). The last two steps essentially amount to writing the spins on a dimer as: \( {\bf S}_{1\alpha} =-{\bf S}_{2\alpha} \approx \bar{s} \sqrt{S(S+1)/3} (\hat{t}^\dag_\alpha +\hat{t}^{ }_\alpha) \), and replacing $\hat{s}^\dag\hat{s}$ by $\bar{s}^2$ in Eqs.~(\ref{eq:constraint}) and (\ref{eq:bondH}). As an additional simplification, we satisfy the bond-operator constraint only globally. This gives us a bilinear Hamiltonian of triplons which can be studied fairly straightforwardly. In the present formulation, the quintets and higher bond eignestates for $S\ge 1$ are dispersion-less higher energy excitations, and will play no role in determining the ground state properties.
Applying the above prescription to $H_{\tt I}$ gives the following mean-field triplon Hamiltonian.
\begin{eqnarray}
H^{ }_{{\tt I},mf} = N_d\left[J-JS(S+1)-\frac{5}{2}\lambda +\bar{s}^2 (\lambda-J)\right] + \nonumber\\ \frac{1}{2}\sum_{{\bf k},\alpha}\Bigg\{[\lambda-\bar{s}^2 S(S+1) \xi_{{\bf k}} ]\left(\hat{t}^\dag_{{\bf k}\alpha}\hat{t}^{ }_{{\bf k}\alpha} + \hat{t}^{ }_{-{\bf k}\alpha}\hat{t}^\dag_{-{\bf k}\alpha}\right) \nonumber \\
- \bar{s}^2 S(S+1)\xi_{{\bf k}} \left(\hat{t}^\dag_{{\bf k}\alpha}\hat{t}^\dag_{-{\bf k}\alpha} + \hat{t}^{ }_{-{\bf k}\alpha}\hat{t}^{ }_{{\bf k}\alpha}\right) \Bigg\} +\nonumber \\
\sum_{{\bf k}}\sum^{j=2S}_{j\ge 2}\sum_{m_j}\left[\lambda+\frac{J}{2}j(j+1)-J\right] \hat{h}^\dag_{{\bf k},j,m_j} \hat{h}^{ }_{{\bf k},j,m_j} \label{eq:H1mf}
\end{eqnarray}
Here, $N_d$ is the number of dimers, and $\xi_{{\bf k}} = 2J\epsilon_{\bf k}/3$.
Moreover, \(\epsilon_{\bf k} = \eta_x\cos{(2k_xa)}+2(\eta^\prime-\eta_y)\cos{(k_ya)}+2\eta^\prime\cos{(2k_xa)}\cos{(k_ya)}\).
The triplon part of the mean-field Hamiltonian, $H^{ }_{{\tt I},mf}$, can be diagonalized using Bogoliubov transformation. Define the triplon quasi-particle operators, $\hat{\gamma}^{ }_{{\bf k}\alpha}$, such that \( \hat{t}^{ }_{{\bf k}\alpha} = \cosh{\theta_{{\bf k}\alpha}} \hat{\gamma}^{ }_{{\bf k}\alpha} +\sinh{\theta_{{\bf k}\alpha}}\hat{\gamma}^\dag_{-{\bf k}\alpha}\), and demand that the triplon terms in Eq.~(\ref{eq:H1mf}) be diagonal in $\hat{\gamma}^{ }_{{\bf k}\alpha}$. This is achieved for \(\tanh{2\theta_{{\bf k}\alpha}} =\bar{s}^2 S(S+1) \xi_{{\bf k}}/[\lambda-\bar{s}^2 S(S+1) \xi_{{\bf k}}]\), giving the following diagonal mean-field Hamiltonian.
\begin{eqnarray}
H^{ }_{{\tt I},mf} = e_0 N_d + \sum_{{\bf k},\alpha}E_{{\bf k}}\left(\hat{\gamma}^\dag_{{\bf k}\alpha}\hat{\gamma}^{ }_{{\bf k}\alpha} +\frac{1}{2}\right)+\nonumber \\
\sum_{{\bf k}}\sum^{j=2S}_{j\ge 2}\sum_{m_j}\left[\lambda+\frac{J}{2}j(j+1)-J\right] \hat{h}^\dag_{{\bf k},j,m_j} \hat{h}^{ }_{{\bf k},j,m_j} \label{eq:diagonalH1mf}
\end{eqnarray}
Here, $e_0 = J-JS(S+1)-(5\lambda/2) +\bar{s}^2(\lambda-J)$, and $E_{{\bf k}} = \sqrt{\lambda[\lambda-2\bar{s}^2 S(S+1)\xi_{{\bf k}}]}$ is the triplon dispersion. The ground state in this mean-field theory is given by the vacuum of the triplon quasi-particles, and of the excitations for $j\ge 2$. The ground state energy per dimer, $e_g$, of the $H^{ }_{{\tt I},mf}$ is given by
\begin{equation}
e_g[\lambda,\bar{s}^2] = e_0 + \frac{3}{2 N_d} \sum_{{\bf k}} E_{{\bf k}}. \label{eq:eg1}
\end{equation}
Minimizing $e_g$ with respect to $\lambda$ and $\bar{s}^2$
gives the following mean-field equations.
\begin{subequations}
\begin{eqnarray}
\bar{s}^2 &=& \frac{5}{2} - \frac{3}{2N_d}\sum_{{\bf k}}\frac{\lambda- \bar{s}^2 S(S+1)\xi_{{\bf k}}}{E_{{\bf k}}} \label{eq:H1sbar2}\\
\lambda &=& J+\frac{3\lambda S(S+1)}{2N_d}\sum_{{\bf k}} \frac{\xi_{{\bf k}}}{E_{{\bf k}}} \label{eq:H1lambda}
\end{eqnarray}
\end{subequations}
The physical solution corresponds to solving these equations self-consistently for $\bar{s}^2$ and $\lambda$. This we will do separately for different cases of the model. One can calculate the spin-gap, and also the magnetic moment in the ordered phase, by solving these equations in the entire parameter space. The staggered magnetic moment in the ordered AF phase is given by $M_s=2\bar{s}\sqrt{S(S+1)n_c/3}$, where $n_c$ is the triplon condensate density in the gapless phase~\cite{rkbk}. Presently, we only compute the phase boundaries between the dimer and the magnetically ordered phases. This is done by tracing the closing of the triplon gap. The wavevector ${\bf Q}$, at which the bottom of the dispersion touches zero ($E_{\bf Q} =0$), decides the magnetic order in the AF phase. The vanishing triplon gap also fixes $\lambda$ as \(\lambda^*=2\bar{s}^2 S(S+1) \xi_{{\bf Q}}$. After a few steps of algebra on Eqs.~(\ref{eq:H1sbar2}) and (\ref{eq:H1lambda}), we get the following equation for the phase boundary between the columnar dimer phase and the ${\bf Q}$-ordered antiferromagnetic phase in the space of coupling parameters.
\begin{equation}
\epsilon_{{\bf Q}}\left[5-\frac{3}{N_d}\sum_{{\bf k}}\sqrt{\frac{\epsilon_{\bf Q}}{\epsilon_{\bf Q} - \epsilon_{\bf k}}} \right] = \frac{1.5}{S(S+1)}
\label{eq:H1QPBoundary}
\end{equation}
In the above equation, the spin appears as $S(S+1)$ only on the right-hand-side of the equality and all the couplings are in the expression on the left-hand-side. It implies that the phase boundaries for different spins will collapse onto a single boundary surface in the space of couplings rescaled by a factor of $S(S+1)$ [for example, $\eta_x S(S+1)$ as so on]. Moreover, we can access the so-called ``classical'' limit by making the right-hand-side of the equality in Eq.~(\ref{eq:H1QPBoundary}) zero (that is, $S\rightarrow \infty$). Below we will see that even in the classical limit, one finds a finite region of phase diagram in which the {\em quantum mechanical} singlet dimer phase survives! This seems to happens when the frustration is high, or when the problem is {\em sufficiently} quasi one dimensional.
\subsection{Calculations and discussion}
In all our calculations, $J=1$ sets the unit of energy. Below we discuss two special cases of $H_{\tt I}$. In the first case, we set $\eta^\prime=0$. This is a model of coupled two-leg ladders and coupled dimerized chains interpolating between one another. The second case is for $\eta^\prime\ne 0$, but $\eta_x=\eta_y=\eta$. For $\eta=1$, it is the $J_1$-$J_2$ model.
\subsubsection{$\eta^\prime=0$}
\begin{figure}[t]
\centering
\includegraphics[width=7cm]{Case1_etaXY}
\includegraphics[width=7cm]{Case1_etatildeXY}
\caption{The mean-field quantum phase diagram of $H_{\tt I}$ for case-1 ($\eta^\prime=0$). The {\em top panel} shows the phase boundaries between the columnar-dimer and N\'eel ordered phases for different spins. For a given $S$, the region between the axes and the phase boundary is the spin-gapped dimer phase, and on the other side of the boundary is the N\'eel ordered phase. Note the quantum phase boundary in the ``classical'' limit ($S=\infty$). Surprisingly, the gapped dimer phase survives even in the classical limit for sufficiently weak $\eta_x$ (coupled two-leg ladders) or $\eta_y$ (coupled chains). In the plane of rescaled couplings ({\em bottom panel}), the quantum phase boundaries for different spins collapse onto a single line.}
\label{fig:QPD_H1_Case1}
\end{figure}
We find ${\bf Q}=(0,\pi/a)$ in this case. Thus, it is a case of quantum phase transition from the columnar dimer to N\'eel ordered phase. We compute the phase boundaries in the $\eta_x$-$\eta_y$ plane for different values of $S$ by solving Eq.~(\ref{eq:H1QPBoundary}). Here, $\epsilon_{{\bf Q}}=\eta_x+2\eta_y$. The mean-field quantum phase diagram is presented in Fig.~\ref{fig:QPD_H1_Case1}.
As discussed earlier, the phase boundaries for different spins collapse onto a single line in the plane of rescaled parameters, $\eta_x\,S(S+1)$ and $\eta_y\,S(S+1)$. Fig.~\ref{fig:QPD_H1_Case1} should in principle be complemented with other calculations for a correct picture in the strongly anisotropic weak dimerization cases (for example, to have a Haldane phase for higher spins, and the like). But presently, we discuss a few interesting things about this mean-field phase diagram.
First about the model corresponding to $\eta_x=\eta_y=\eta$, a popular fully two-dimensional case on a dimerized square lattice. For a finite value of $S$, there occurs a quantum phase transition from the dimer to N\'eel ordered phase at some non-zero value of $\eta=\eta*$. We find that $\eta^* = \frac{0.466}{S(S+1)}$. In the limit $S\rightarrow\infty$, $\eta^*$ goes to zero however. That is, in the classical limit of this case, the ground state is N\'eel ordered even for an infinitesimally small inter-dimer coupling $\eta$. This conforms to the usual expectations in the classical limit. Moreover, the spin dependence of critical $\eta^*\sim \frac{1}{S(S+1)}$ is a simple analytical confirmation of a suggestion from the numerical studies of a similar model.~\cite{Richter_SpinDependence} Besides the qualitative agreement, the mean-field calculation overestimates the dimer phase in the present case. For the spin-1/2 case, $\eta^*=0.523$ from quantum Monte Carlo simulations~\cite{Matsumoto,Wenzel_Janke}, 0.535 from spin-wave analysis~\cite{Imada} and 0.54 from dimer series expansion~\cite{SinghGelfandHuse} as compared to 0.62 from the present calculation.
More importantly, we want to take note of the behavior in the extremely large $S$ limit of the general case (that is, $\eta_x\ne \eta_y$). Look at the phase boundary for $S=\infty$, in the top panel of Fig.~\ref{fig:QPD_H1_Case1}, given by the lines: $\eta_y= 0.0222 \eta_x $ and $\eta_y= 11.288 \eta_x$. Here, we find two disjoint regions (one bounded by the lines: $\eta_y=0$ and $\eta_y= 0.0222 \eta_x $, and the other by $\eta_y= 11.288 \eta_x$ and $\eta_x=0$) of the dimer phase (which is a quantum mechanical state with zero magnetic moment) existing even in this classical limit. This is a striking {\em deviation} from the normally expected behavior in the limit $S\rightarrow\infty$. The two regions can be viewed as corresponding to the quasi-1d cases of the coupled {\em dimerized} chains and the coupled two-leg ladders, respectively. It seems that the strong spatial anisotropy in the (dimerized) lattice helps the dimer state to survive even when the spins are very large. Below we will see the same behavior also recurring in the highly frustrated situations of a fully two-dimensional case (the present case of $\eta^\prime=0$ is not frustrated, but $\eta^\prime\ne 0$ in the following subsection is). While a case like $\eta_x\sim 1$ and $\eta_y\ll 1$ is known to be more sensitive towards N\'eel ordering than what the present calculation suggests ~\cite{AffleckGelfandSingh,Matsumoto,Sandvik}, we believe the existence of a dimer phase for $S\rightarrow\infty$ is very likely to come true in more accurate numerical calculations for the strongly dimerized cases ($\eta_x$, $\eta_y \ll 1$).
\subsubsection{$\eta^\prime\ne 0$ {\rm and} $\eta_x=\eta_y=\eta$}
\begin{figure}[t]
\centering
\includegraphics[width=7cm]{Case2_etaetaprime}
\includegraphics[width=7cm]{Case2_tilde_etaetaprime}
\caption{The mean-field quantum phase diagram of $H_{\tt I}$ for case-2 ($\eta^\prime \ne 0$ and $\eta_x=\eta_y=\eta$). Here, $\eta=1$ corresponds to the $J_1$-$J_2$ model. {\em Top panel}. For a given $S$, the region above the corresponding upper phase boundary, and bounded by the axes, is the collinear phase, while that below the lower phase boundary is the N\'eel phase. In between the two transition lines lies the columnar dimer phase. For $S=\infty$, the phase boundaries are given by the equations, $\eta^\prime = 0.408\eta$ and $\eta^\prime=0.525\eta$.
{\em Bottom panel}. The phase boundaries of for different spins collapse onto two lines for two different transitions.}
\label{fig:QPD_H1_Case2}
\end{figure}
As noted above, this is the case of a frustrated two-dimensional model on a dimerized square lattice. In this case, there are two different choices of ${\bf Q}$. For weaker $\eta^\prime$, ${\bf Q}=(0,\pi/a)$, and for stronger $\eta^\prime$, it is ${\bf Q}=(0,0)$. While the former corresponds to having N\'eel order in the ground state, the latter gives collinear order (in which the magnetic moments are parallel, for the spins along the $y$-direction in Fig.~\ref{fig:model1}, and anti-parallel along the $x$-direction). From Eq.~(\ref{eq:H1QPBoundary}), we calculate the quantum phase diagram shown in Fig.~\ref{fig:QPD_H1_Case2}. Here, $\epsilon_Q=3\eta-4\eta^\prime$ for ${\bf Q}=(0,\pi/a)$, and $4\eta^\prime-\eta$ for ${\bf Q}=(0,0)$. For $\eta^\prime=0$, the quantum critical point for different spins is given by $\eta^* = \frac{0.466}{S(S+1)}$ (same as in the previous case), and it is $\eta^{\prime *}=\frac{0.272}{S(S+1)}$ for $\eta=0$. The phase boundaries for different spins collapse to a single line for the dimer to N\'eel transition, and similarly for the dimer to collinear transition.
The dimer phase survives again in the limit $S\rightarrow\infty$ when the frustration is strong. To discuss this point, consider $\eta=1$ case. It corresponds to the $J_1$-$J_2$ model. In the present notation, $J_1=J$ and $J_2=\eta^\prime J$. In the classical version of this model, $\eta^\prime=0.5$ is the transition point between the N\'eel and collinear ordered ground states~\cite{j1j2_Chandra_Doucot}. It is also the point of infinitely degenerate classical ground state manifold, and hence of very high frustration. In the quantum case, for spin-1/2 specifically, it is known from many numerical studies that there exists a quantum disordered spin-gapped state (most likely a columnar dimer state) in a small range of $.4\lesssim \eta^\prime \lesssim .6$ around the $0.5$ point~\cite{vbc1,vbc2,vbc4,vbc5}. This is about $.19\lesssim \eta^\prime \lesssim .61$ from the triplon mean-field calculation~\cite{sach-bhatt,rkbk}. Below and above this range, one finds the N\'eel and collinear ordered ground state, respectively, as in the classical case.
Interestingly, even when $S$ is arbitrarily large, we find the dimer phase to be stable in a small window of $\eta^\prime$ around 0.5. For $S=\infty$ , this range is $0.41 \lesssim \eta^\prime\lesssim 0.53$ at $\eta=1$. Away from $\eta=1$, the region of dimer phase is bounded by the lines, $\eta^\prime = 0.408\eta$ and $\eta^\prime=0.525\eta$. Furthermore, it shrinks smoothly as one moves towards $\eta=0$. The ``quantum region'' of the phase diagram manages to survive in the classical limit seemingly because of the strong frustration. While there may be concerns about the bond-operator mean-field theory overestimating the dimer region (as it does for spin-1/2 case), a sufficient amount of frustration may always help a quantum state. Hence, we have a reasonable qualitative finding which needs to investigated further. Besides, it should be asked afresh, "is $S\rightarrow\infty$ necessarily classical?".
\section{A Model on Honeycomb lattice}
\label{sec:honeycomb}
We now investigate a quantum spin-$S$ model on honeycomb lattice given by the following Hamiltonian~\cite{rkdkbk}.
\begin{eqnarray}
H^{ }_{\tt II} &=&J\sum_{\langle {\bf r}, {\bf r}^\prime \rangle} {\bf S}_{\bf r}\d{\bf S}_{{\bf r}^\prime} + \nonumber \\
&& \frac{K}{8} \sum_{\setlength{\unitlength}{0.4cm}
\begin{picture}(1,1)
\put(.0,-.1){\includegraphics[width=.6cm, angle=90]{honey_little}}
\end{picture}
} \left[{\bf S}_{12}^2{\bf S}_{34}^2{\bf S}_{56}^2 +{\bf S}_{23}^2{\bf S}_{45}^2{\bf S}_{61}^2\right] \label{eq:model2}
\end{eqnarray}
Here, $J$ is the nearest neighbor Heisenberg interaction, and $K$ denotes the strength of a multiple-spin-exchange interaction generated by the product of pairwise total-spins of three pairs of neighboring spins on a hexagonal plaquette. The six spins on a hexagonal plaquette are labeled as 1 to 6 (see Fig.~\ref{fig:model2}). In the second term of $H_{\tt II}$, ${\bf S}_{ij}^2=({\bf S}_i+{\bf S}_j)^2$. The interaction parameters $J$ and $K$ are taken to be positive.
\begin{figure}[htbp]
\centering
\includegraphics[width=7cm]{HoneycombModel}
\caption{The model of Eq.~\ref{eq:model2}. The lines indicate the nearest neighbor Heisenberg exchange, $J$. The multiple spin-exchange proportional to $K$ is represented by a hexagon itself, with six spins labeled as 1 to 6. This multiple spin-exchange interaction is present on every hexagonal plaquette of the honeycomb. Moreover, $J$, $K>0$.}
\label{fig:model2}
\end{figure}
An important feature of this model is that it has an exact triply degenerate dimerized singlet ground state for $J=0$ for any value of $S$. It presents an example of spontaneous dimerization in the ground state as $H_{\tt II}$ itself has no preferred dimer order (unlike $H_{\tt I}$ in the previous section). One of these dimer states corresponds to forming a singlet on every vertical nearest-neighbor bond of the honeycomb lattice in Fig.~\ref{fig:model2}. The other two states are generated from the first one by making $\pm 2\pi/3$ rotation of the lattice. For $J>0$, and for spin-1/2, we had earlier performed a triplon mean-field calculation to investigate the transition from dimer to N\'eel order in the ground state~\cite{rkdkbk}. A similar analysis was desired for higher spins, but could not be done at that point due to the lack of spin-$S$ bond-operator representation. It was our {\em original} motivation for developing the bond-operator mean-field theory for arbitrary spins. Having achieved this objective in Sec.~\ref{sec:boS}, we can now do a triplon mean-field theory for $H_{\tt II}$, exactly in the same way as done for $H_{\tt I}$, by taking the configuration of vertical dimers on honeycomb lattice as a reference state.
The mean-field triplon Hamiltonian in this case also looks the same as in Eq.~(\ref{eq:H1mf}). The diagnoalized mean-field Hamiltonian for $H_{\tt II}$ can therefore be written as:
\begin{eqnarray}
H_{{\tt II},mf} &=& N_d e_0 + \sum_{{\bf k},\alpha}\sqrt{\lambda(\lambda-2\bar{s}^2\xi_{\bf k})} \left(\hat{\gamma}^\dag_{{\bf k}\alpha}\hat{\gamma}^{ }_{{\bf k}\alpha} +\frac{1}{2}\right) \nonumber\\
&& + \mbox{(localized higher $j$ dimer states)},
\end{eqnarray}
where $e_0 = -\frac{5}{2}\lambda+\bar{s}^2(\lambda-J-2K[S(S+1)]^2) + J[1-S(S+1)]+2K[S(S+1)]^2$, and $\xi_{\bf k} = \frac{2}{3}S(S+1)\{J+2K[S(S+1)]^2(1-\bar{s}^2)\}\epsilon_k$. Here, $\epsilon_{\bf k} = \cos{2\pi k_2}+\cos{2\pi(k_1-k_2)}$, and ${\bf k}$ is defined as ${\bf k}=2\pi(k_1{\bf G}_1+k_2{\bf G}_2)$ where ${\bf G}_1$ and ${\bf G}_2$ are the primitive reciprocal lattice vectors, and $ k_1, k_2 \in [0,1]$. The self-consistent equations for this problem are given by
\begin{equation}
\bar{s}^2 = \frac{5}{2}-\frac{3}{2N_d}\sum_{\bf k}\frac{\lambda-\bar{s}^2\xi_{\bf k}}{\sqrt{\lambda(\lambda-2\bar{s}^2\xi_{\bf k})}}
\end{equation}
and
\begin{equation}
\lambda=J+2K[S(S+1)]^2+\frac{3}{2N_d}\sum_{\bf k}\frac{\xi_{\bf k}+\bar{s}^2\frac{\partial\xi_{\bf k}}{\partial\bar{s}^2}}{\sqrt{\lambda(\lambda-2\bar{s}^2\xi_{\bf k})}}.
\end{equation}
The closing of the triplon gap, which marks the instability of the dimer phase to a magnetically ordered phase, fixes $\lambda$ as $\lambda^*=2\bar{s}^2\xi_{\bf Q}$, where ${\bf Q}=(0,0)$ in the present case. At this quantum critical point, we get
\begin{subequations}
\begin{eqnarray}
\bar{s}^2 &=& \frac{5}{2}-\frac{3}{2N_d}\sum_k\frac{1-\frac{\epsilon_{\bf k}}{4}}{\sqrt{1-\frac{\epsilon_{\bf k}}{2}}} \simeq 0.817,~\mbox{and} \\
\frac{J}{K} &=& 2[S(S+1)]^2\left\{-1+2\bar{s}^2\frac{u-\frac{4}{3}\bar{s}^2}{\frac{1}{S(S+1)}+u-\frac{8}{3}\bar{s}^2}\right\} \nonumber\\
&\simeq&2[S(S+1)]^2\left[ \frac{0.71}{1.524-\frac{1}{S(S+1)}}-1\right] \equiv \zeta^*_{S} \label{eq:JbyK}
\end{eqnarray}
\end{subequations}
where $u=\frac{1}{N_d}\sum_{\bf k}\frac{\epsilon_{\bf k}}{\sqrt{1-\frac{\epsilon_{\bf k}}{2}}} \simeq 0.655$. Equation~(\ref{eq:JbyK}) is a closed form expression for the critical $J/K$, denoted as $\zeta^*_S$, as a function of $S$. For the dimer order to become unstable to N\'eel order in this mean-field theory, $J/K$ must be greater than $\zeta^*_S$ for a given $S$. We find $\zeta^*_{1/2} \simeq 3.067$, which agrees with our earlier calculation for the spin-1/2 case of this model. Next we find $\zeta^*_{1}\simeq -2.454$, $\zeta^*_{3/2}\simeq -12.247$, and so on. For positive $J$ and $K$, we therefore conclude that the mean-field triplon calculation predicts a N\'eel ordered ground state for $H_{\tt II}$ for $S\ge 1$ for any non-zero value of $J/K$.
We know for sure that the dimer ground state is exact for $J=0$. For the spin-1/2 case, the triplon analysis predicts that the dimer state will give way to the N\'eel state only when $J/K$ is sufficiently strong. However, for spin-1 and higher, it seems to happen for arbitrarily small $J$. At this point, it is important to note the following. While deriving the mean-field triplon Hamiltonian for $H_{\tt II}$, following the steps outlined in the previous section, we end up having {\em no contributions} from the six-spin terms of the form $({\bf S}_1\cdot{\bf S}_2) ({\bf S}_3\cdot{\bf S}_4) ({\bf S}_5\cdot{\bf S}_6)$. For $S=1/2$, it does not seem to affect the dimer phase for small $J$ as the mean-field theory suggests. We have some evidence of this from a numerical calculation in our earlier work on this model~\cite{rkdkbk}. However, it is not clear as to how the absence of contribution from the six-spin terms in the present mean-field calculation will affect the case of spin-1 and higher. May be, in a {\em renormalized} triplon analysis of $H_{\tt II}$, one gets the dimer phase over a small but finite range of $J/K$ for $S\ge 1$. One such calculation is done by writing $({\bf S}_1\cdot{\bf S}_2) ({\bf S}_3\cdot{\bf S}_4) ({\bf S}_5\cdot{\bf S}_6)$ as $-[S(S+1)]^2\bar{s}^2\chi \left[ {\bf S}_3\cdot{\bf S}_4+{\bf S}_5\cdot{\bf S}_6-2\chi S(S+1)\right]$, and similarly for $({\bf S}_2\cdot{\bf S}_3) ({\bf S}_4\cdot{\bf S}_5) ({\bf S}_6\cdot{\bf S}_1)$. Here, the expectation values $\langle{\bf S}_3\cdot{\bf S}_4\rangle$, $\langle{\bf S}_5\cdot{\bf S}_6\rangle$, $\langle{\bf S}_6\cdot{\bf S}_1\rangle$ and $\langle{\bf S}_2\cdot{\bf S}_3\rangle$ are all taken to be equal to $S(S+1) \chi$. It gives the following critical value of $J/K$.
\begin{equation}
\frac{J}{K} \simeq 2[S(S+1)]^2\left[ \frac{0.856}{1.524-\frac{1}{S(S+1)}}-1\right] \equiv \zeta^*_{S}
\end{equation}
It is similar to Eq.~(\ref{eq:JbyK}), except the numerator inside the square-brackets in different, which only slightly increases the value of $\zeta^*_{S}$. But the qualitative conclusion remains the same. That is, the N\'eel order sets in for arbitrarily small $J$ for $S\ge 1$. Well, this is the result from triplon mean-field calculation. Alternative calculations are needed to resolve this conclusively.
\section{\label{sec:extend} Extensions of the representation}
Below we present two immediate extensions of the bond-operator representation derived in Sec.~\ref{sec:boS}. First, we go beyond singlet and triplets to include quintet states on a bond. In the second case, we derive a similar representation on square plaquette in terms of the plaquette bosons, which turns out to be an easy extension of Eqs.~(\ref{eq:bo_s1}) and (\ref{eq:bo_s2}) to a plaquette problem.
\subsection{\label{sec:including_quintets} Including quintets on a bond}
The bond-operator representation derived in Sec.~\ref{sec:boS} is in the subspace of singlet and triplet states only.
With some labor, we can extend this to include the quintets, knowing how to systematically construct the bond-eigenstates (see Appendix). Including higher states is possible, but it requires even more effort, and will not be considered presently. The bond operator representation including quintets is written below.
\begin{eqnarray}
S^z_{1,2} &\approx& \pm\sqrt{\frac{{\tt N}_t}{2{\tt N}_s}} \left( \hat{s}^\dag\hat{t}^{ }_0 + \hat{t}^\dag_0 \hat{s} \right) \pm\sqrt{\frac{{\tt N}_q}{{\tt N}_t}}\bigg[\frac{1}{\sqrt{3}}\left(\hat{t}^\dag_0\hat{q}^{ }_0 + \hat{q}^\dag_0\hat{t}^{ }_0\right) \nonumber \\
& +& \frac{1}{2}\left(\hat{t}^\dag_1\hat{q}^{ }_1 + \hat{q}^\dag_1\hat{t}^{ }_1+ \hat{t}^\dag_{\bar{1}} \hat{q}^{ }_{\bar{1}} + \hat{q}^\dag_{\bar{1}}\hat{t}^{ }_{\bar{1}}\right)\bigg] + \frac{1}{2}\Big(\hat{t}^\dag_1\hat{t}^{ }_1 - \hat{t}^\dag_{\bar{1}}\hat{t}^{ }_{\bar{1}} \nonumber \\
& +& \hat{q}^\dag_1\hat{q}^{ }_1 - \hat{q}^\dag_{\bar{1}}\hat{q}^{ }_{\bar{1}}\Big) + \left(\hat{q}^\dag_2\hat{q}^{ }_2 - \hat{q}^\dag_{\bar{2}}\hat{q}^{ }_{\bar{2}}\right) \label{eq:bo1_quintet}
\end{eqnarray}
\begin{eqnarray}
S^+_{1,2} & \approx & \pm\sqrt{\frac{{\tt N}_t}{{\tt N}_s}} \left( \hat{s}^\dag\hat{t}^{ }_{\bar{1}} - \hat{t}^\dag_1 \hat{s} \right) \pm\sqrt{\frac{{\tt N}_q}{{\tt N}_t}}\bigg[ \left(\hat{t}^\dag_{\bar{1}}\hat{q}^{ }_{\bar{2}} - \hat{q}^\dag_2\hat{t}^{ }_1\right) \nonumber \\
& & + \frac{1}{\sqrt{2}}\left(\hat{t}^\dag_0\hat{q}^{ }_{\bar{1}} - \hat{q}^\dag_1\hat{t}^{ }_0\right)+\frac{1}{\sqrt{6}}\left(\hat{t}^\dag_1 \hat{q}^{ }_0 - \hat{q}^\dag_0\hat{t}^{ }_{\bar{1}}\right)\bigg]\nonumber \\
& & + \frac{1}{\sqrt{2}}\left(\hat{t}^\dag_1\hat{t}^{ }_0 + \hat{t}^\dag_0\hat{t}^{ }_{\bar{1}}\right) + \sqrt{\frac{3}{2}}\left(\hat{q}^\dag_1\hat{q}^{ }_0 + \hat{q}^\dag_0\hat{q}^{ }_{\bar{1}}\right) \nonumber \\
&& + \left(\hat{q}^\dag_2\hat{q}^{ }_1 + \hat{q}^\dag_{\bar{1}}\hat{q}^{ }_{\bar{2}}\right) \label{eq:bo2_quintet}
\end{eqnarray}
Here, ${\tt N}_s = 2S+1$, ${\tt N}_t= 2S(S+1){\tt N}_s/3$ and ${\tt N}_q= {\tt N}_t(2S-1)(2S+3)/5$, are the normalization constants for the singlet, triplet and quitet states respectively. Moreover, in the notation $\pm$, the `$+$' corresponds to ${\bf S}_1$ and `$-$' to ${\bf S}_2$.
Equations~(\ref{eq:bo1_quintet}) and (\ref{eq:bo2_quintet}) are exact for spin-1 case~\cite{spin1_bo1}, and reduce to the representation for spin-1/2 operators by dropping the terms involving quintets. Note that the coefficients of the terms mixing singlet with triplets and triplets with quintets scale as $S$ for large $S$. While the strengths of different mixing terms grow similarly as $S$ grows large, the hierarchy of mixing suggests that for a spin-gapped phase in a system of exchange-interacting quatum spins, the triplon analysis is a minimal reasonable thing to do. It is because the condensation of ``quintons'' is facilitated only by that of the triplons. In a gapped phase where triplons have not condensed, it is {\em unlikely} that the quintons will condense. Therefore, it seems okay to ignore the quintet states to first approximation. It will not be the same however if we take into account the single anisotropy effects like $(S^z_{1,2})^2$. In this case, the singlet state will directly mix with quintets, and therefore, it will be better to work with Eqs.~(\ref{eq:bo1_quintet}) and (\ref{eq:bo2_quintet}) instead of Eqs.~(\ref{eq:bo_s1}) and (\ref{eq:bo_s2}). This combined ``tiplon-quinton'' analysis will be useful in investigating the influence of single-ion anisotropy on the stability of a dimer phase, and on its existence in the limit $S\rightarrow\infty$.
\subsection{\label{sec:plaquette} Representation on a square plaquette}
Consider a spin-$S$ problem on a single square plaquette given by the Hamiltonian: $H_{sp} = J({\bf S}_1\cdot{\bf S}_2+{\bf S}_2\cdot{\bf S}_3+{\bf S}_3\cdot{\bf S}_4+{\bf S}_4\cdot{\bf S}_1)+J^\prime ({\bf S}_1\cdot{\bf S}_3+{\bf S}_2\cdot{\bf S}_4)$, where $J$ is the exchange interaction along the edges of the square and $J^\prime$ is the interaction along the diagonals. The subscript $sp$ stands for square plaquette. This problem can be solved by rewriting it as:
\(H_{sp}=\frac{J}{2}{\bf S}^2_{tot} - \frac{(J-J^\prime)}{2}({\bf S}^2_{13} +{\bf S}^2_{24})- 2J^\prime S(S+1) \), where ${\bf S}_{tot} ={\bf S}_1+{\bf S}_2+{\bf S}_3+{\bf S}_4 $ is the total spin of the plaquette, and ${\bf S}_{13}={\bf S}_1+{\bf S}_3$ and ${\bf S}_{24}={\bf S}_2+{\bf S}_4$ are the total spins on the two diagonals. The eigenstates of this problem are completely specified by three quantum numbers: the total spin of the plaquette, $j$, and the two diagonal spins, $j_{13}$ and $j_{24}$, with eigenvalues, $E_{sp}(j,j_{13},j_{24}) = \frac{J}{2}j(j+1) -\frac{(J-J^\prime)}{2}[j_{13}(j_{13}+1) + j_{24}(j_{24}+1)] -2J^\prime S(S+1)$. Given that we are interested in antiferromagnetic interactions ($J$, $J^\prime >0$), let us figure out the possible ground states, and derive a bosonic representation for spin operators, considering only the lowest energy excitations.
Since $J>0$, for a given $j_{13}$ and $j_{24}$, the $E_{sp}$ would be lowest for the smallest value of $j$. Moreover, when $J>J^\prime$, the ground state of $H_{sp}$ is given by: $j_{13}=j_{24}=2S$ and $j=0$, and for $J<J^\prime$, it corresponds to $j_{13}=j_{24}=0$. The latter is a case of dimer ground state, in which the two diagonal bonds separately become singlet, and $j$ is trivially zero. The elementary excitations in this case would just correspond to making a diagonal bond a triplet. In short, for $J<J^\prime$, the bond-operator representation of Eqs.~(\ref{eq:bo_s1}) and (\ref{eq:bo_s2}) is applicable as it is. However, for $J>J^\prime$, the ground state is a true plaquette-singlet, involving all four spins. Therefore, we must separately find out a representation of the spin operators in terms of this plaquette-singlet and the corresponding plaquette-triplet excitations of the elementary kind.
For $J>J^\prime$, the ground state lies in the sector given by $j_{13}=j_{24}=2S$. The plaquette states in this sector, for different values of $j$, are the compound eigenstates of two spins of size $2S$. That is, in Eqs.~(\ref{eq:singlet_state}) and (\ref{eq:triplet_plus1}-\ref{eq:triplet_minus1}), replace $S$ by $2S$. This immediately suggests that the diagonal spins, ${\bf S}_{13}$ and ${\bf S}_{24}$, are represented by Eqs.~(\ref{eq:bo_s1}) and (\ref{eq:bo_s2}) with $S$ written as $2S$, where $\hat{s}$ and $\hat{t}_\alpha$ operators are now the bosons corresponding to the plaquette singlet, $|j=0;j_{13}=2S,j_{24}=2S\rangle$, and triplet states, $|j=1;j_{13}=2S,j_{24}=2S \rangle$, respectively. In order to find the representation for individual spins, ${\bf S}_1$ and ${\bf S}_3$, we must also find ${\bf S}_1-{\bf S}_3$ in terms of the plaquette bosons, and do similarly for ${\bf S}_2$ and ${\bf S}_4$. Since we consider only those states given by $j_{13}=j_{24}=2S$, the operators ${\bf S}_1-{\bf S}_3$ and ${\bf S}_2-{\bf S}_4$ would be null operators in this restricted subspace because they change the values of $j_{13}$ and $j_{24}$. Hence, their matrix elements in the subspace of the ground state singlet and the lowest triplets are zero. It leads to the following representation of the spin operators on the plaquette.
\begin{subequations}
\begin{eqnarray}
{\bf S}_{1\alpha} &=& {\bf S}_{3\alpha} \nonumber \\
&\approx& \sqrt{\frac{S(2S+1)}{6}}\left(\hat{s}^\dag\hat{t}^{ }_\alpha + \hat{t}^\dag_\alpha\hat{s}\right)-\frac{i}{4}\epsilon_{\alpha\beta\gamma}\hat{t}_\beta^\dag\hat{t}^{ }_\gamma \label{eq:bo_s13}\\
{\bf S}_{2\alpha} &=& {\bf S}_{4\alpha} \nonumber \\
&\approx& -\sqrt{\frac{S(2S+1)}{6}}\left(\hat{s}^\dag\hat{t}^{ }_\alpha + \hat{t}^\dag_\alpha\hat{s}\right)-\frac{i}{4}\epsilon_{\alpha\beta\gamma}\hat{t}_\beta^\dag\hat{t}^{ }_\gamma
\label{eq:bo_s24}
\end{eqnarray}
\end{subequations}
The above equations are written in the standard notation, except that the bosons are now defined on a square plaquette. It correctly reproduces the representation for spin-1/2 case~\cite{plaquette1}. While it is an approximate representation, it provides a simple framework for discussing the low energy physics of a (coupled) plaquette problem for $J$ sufficiently stronger than $J^\prime$ (and other couplings in a given problem). However, when $J^\prime$ is strong enough, the states from other sectors begin to compete. For example, on a single plaquette for $J^\prime > (1-\frac{1}{4S})J$, the singlet state for $j_{13}=j_{24}=2S-1$ becomes lower in energy than the triplets in the sector containing ground state. This renders the above representation insufficient for an effective low energy description. It is, in any case, a useful representation, if considered within limits.
\section{Conclusion}
\label{sec:summary}
To summarize, we have derived the bond-operator representation for spin-S dimer problems, and also worked out a similar representation on a square plaquette. Using this bond-operator representation, we have done the mean-field triplon analysis of two model quantum antiferromagnets: 1) a coupled columnar dimers model on square lattice, and 2) a model on honeycomb lattice with spontaneous dimer order in the ground state. Through this mean-field calculation, we have studied the quantum phase transition from the dimer to AF ordered phases as a function of spin. A notable outcome of this analysis is that one finds the dimer phase, which is a quantum mechanical phase, to exist even in the limit $S\rightarrow\infty$, under the conditions of strong frustration (or spatial anisotropy with strong dimerization). It suggests that the limit $S\rightarrow\infty$ is not necessarily ``classical'', as there may not exist any ground state with non-zero classical magnetic moments for a system of (frustrated) quantum spins. Such quantum ground states are known to exist for arbitrarily large spins in specially constructed models, such as the Shastry-Sutherland model~\cite{SS} or the exactly solvable case of the model on honeycomb lattice in Sec.~\ref{sec:honeycomb}. However, we believe this behavior of having quantum states in the so-called classical limit to occur more generically. The present observations offer an interesting view on the classical limit of frustrated quantum antiferromagnets, which further needs to be investigated carefully.
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BkiUdRI4ukPiEebgxIPJ
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\section{Introduction}
The study of the flow around building-like obstacles has been extensively addressed in the literature~\cite{Hunt1978,Oke1988,Zajic2011} due to its implications in urban-environment phenomena, i.e. pollutant dispersion, air quality, and heat propagation. The very high levels of air pollution to which the vast majority of the urban population is exposed are undoubtedly related to a myriad of health issues~\cite{Heaviside2016}. The hunt is for predictive models capable of accurately reproducing the pollutant and thermal distributions within urban environments. Some of these models have already been introduced by the European Union (EU)~\cite{EUUrbWorld}. However, their inability to provide the spatio-temporal accuracy required to model pollutant dispersion through urban environments forces researchers to improve those methods to ensure urban sustainability. As an example, to establish a proper action plan to alleviate the associated adverse consequences, several studies~\cite{Hunt1978,Oke1988,Zajic2011, Becker2002, zhu2017, Bourgeois2012} have focused their efforts on analyzing the spatio-temporal structures of the flow. The main point is to identify the three-dimensional flow regions responsible for the pollutant dispersion within a given urban geometry. Therefore, the objective of this study is to apply recently-developed tools from system dynamics, notably higher-order dynamic mode decomposition (HODMD), to turbulent flows within urban environments to understand how different city configurations influence the mechanisms leading the flow dynamics.
The large number of spatio-temporal features present in the high-dimensional nonlinear system of a turbulent flow complicates the analysis. Nonetheless, the fact that physical-flow features are shared across a wide variety of flows suggests that they may be used to describe the dynamics of such a flow. One of the first studies aiming at identifying the flow structures around a wall-mounted obstacle was performed by Hunt et al.~\cite{Hunt1978}. Using flow-visualization techniques, they proved the absence of a closed surface, i.e. a separation bubble or cavity, in the wake of the obstacle, due to the interaction of four different vortical structures. They discovered two large vortices on both sides of the obstacle, as well as a roof vortex at the cube top. The well-known arch vortex is formed on the leeward side of the obstacle due to the separation region appearing downstream of the block, which causes the structures above to move closer. Together with the horseshoe vortex, these vortical structures are the consequence of the interaction of the outer flow with the urban-canopy layer. Understanding their underlying physics is essential for developing strategies to reduce pollution-dispersion and perform pedestrian-comfort assessments. Based on this idea, the arch vortex has been extensively analyzed in various configurations. For instance, Becker et al.~\cite{Becker2002} experimentally studied the flow structures around three-dimensional rectangular blocks in a suburban boundary layer. They analyzed the arc-shaped vortices formed on the leeward side of the obstacle for different angle-of-incidence (AOI) values using laser-Doppler-anemometry (LDA) measurements together with oil-film visualizations.
They reported the dislocation of one of the vortex legs until $\text{AOI}=60^{\circ}$, which causes the vortex leg to be displaced to the top of the obstacle. Furthermore, Zhu et al.~\cite{zhu2017} used particle-image-velocimetry (PIV) measurements to analyze the three-dimensional wake patterns of the flow through a wall-mounted cylinder. They concentrated on the formation process of the arch and tip vortices and documented their apparition in both the time-averaged and the instantaneous fields. Therefore, the arch vortex is a prominent structure in flows through surface-mounted obstacles responsible for the flow interaction on the leeward side of the block.
Vinuesa et al.~\cite{Vinuesa2015} also aimed to investigate the influence of the inflow conditions on such coherent structures around a wall-mounted square cylinder using direct numerical simulation (DNS). While the sharp edges of the obstacle maintain the separation point regardless of inflow conditions, the resultant structures and topology of the wake downstream in both laminar- and turbulent- inflow scenarios varies considerably. The influence of the growing boundary layers on both sides of the obstacle, in particular, causes a difference in behavior in both wakes, since the turbulent one is slightly wider in the time-averaged field. In addition, Vinuesa et al.~\cite{Vinuesa2015} noted that the horseshoe vortex was modified by streamwise variations in the turbulent-inflow simulation. This emphasizes the importance of having well-established inflow conditions in both experimental and numerical analyses to ensure flow consistency.
The apparent complexity of urban-based environments, on the other hand, leads to more intricate physics due to the interaction of flow structures around individual buildings. Oke~\cite{Oke1988} provided an analysis of the resultant flow regimes as a function of the geometrical parameters that define an urban model. Interestingly, the author discovered that the street width was the critical parameter in establishing the flow regimes~\cite{Oke1988}: in the case of narrow streets, the flow above the canopy can barely reach down to the street (skimming flow), and only one vortex can be seen between the obstacles; gradually broader streets lead to the wake-interference regime first and then to the isolated-roughness flow, which exhibits much more contact with the flow above the roofline. Meinders~\cite{Meinders1998} further examined this classification by analyzing the interaction of flow patterns around wall-mounted rectangular obstacles with different spacing ratios in the streamwise direction. The separated shear layer from the first obstacle reattached on the windward side edge of the downstream obstacle for the lowest separation, resulting in an inter-obstacle area with an arc-shaped vortex confined by the side flow~\cite{Meinders1998}. With larger separation ratios, flow reattachment occurs in the region between the obstacles, from which a second horseshoe vortex emerges on the windward side of the downstream obstacle. This results in similar flow patterns for both obstacles, but with lower intensity in the downstream block due to the flow disruption of the upstream block~\cite{Meinders1998}.
A wide range of criteria to identify these vortical structures has been developed. Monnier et al.~\cite{Monnier2018} aimed at identifying the main flow patterns present in the wind-tunnel flow around the geometry of the Mock Urban Setting Test (MUST) experiment using different criteria. They started evaluating the vorticity components, namely the wall-normal and spanwise mean components, $\hat{\omega}_y$ and $\hat{\omega}_z$ respectively, to identify the location of the arch vortex. Using the modulus of the spatially-averaged vorticity vector allowed them to define a local threshold to properly characterize the influence of the AOI on this vortical structure, extracting similar conclusions to those of Becker et al.~\cite{Becker2002}. They also employed some popular methods for vortex identification, based on the second invariant of the velocity gradient tensor, i.e. the Q-criterion~\cite{Hunt1978} and the $\lambda_2$ criterion~\cite{jeong_hussain_1995}. However, they improved the identification of large-scale vortical structures using the normalized angular momentum technique $\Gamma_1$, introduced by Sousa~\cite{Sousa2002} to locate the center of vortical structures downstream of a single cuboid obstacle. This method allowed the authors to describe the relationship between the arch vortex and high-turbulence areas. Monnier et al.~\cite{Monnier2018} concluded that the arch vortex is located in between high-turbulence areas. They consist of two regions of significant streamwise velocity fluctuations on both sides of the obstacles due to the separation of the shear layer and a high spanwise velocity fluctuating region along the windward face of the downstream obstacle. This experimental study led to relevant conclusions in analyzing coherent structures in a more realistic urban model.
Here, we focus on Oke's classification~\cite{Oke1988} to extract through data-driven procedures the key dominant patterns present in the three-dimensional instantaneous fields of the flow through urban environments with different separation ratios. In this regard, the objective is to characterize the main flow processes responsible for the formation and destruction of such vortical structures, thus gaining further understanding of the dynamical processes governing urban flows. To this end, we employ different techniques related to modal decompositions. First, we use proper-orthogonal decomposition (POD)~\cite{Lumley1967} to identify those spatial modes energetically more relevant to the system as well as their associated time coefficients. We compare them with the results obtained using a recently-developed higher-order variant of dynamic-mode decomposition (DMD)~\cite{Schmid2010}, named HODMD~\cite{LeClainche2017b}. Via this novel nonlinear dynamic mode decomposition approach, we can analyze the dynamics of a highly complex turbulent flow~\cite{LeClainche2017,LeClainche2017b,LeClainche2017b,LeClainche2017c,LECLAINCHE2017d,LeClainche2018b}, cleaning noisy artifacts and small amplitude modes from data. Recently, Amor et al.~\cite{Amor2020} showed the potential of HODMD to understand the complicated physics of the wake in a wall-mounted square cylinder ~\cite{LeClainche2017b}. Like DMD, HODMD decomposes spatio-temporal data into a group of modes oscillating in time and space, representing the leading flow dynamics.
Following the introduction to urban flows, the present work provides a general overview of the performed numerical simulations in \S~\ref{sec: Numerical simulations}. A summary of the mathematical concepts behind the modal-decomposition techniques used to characterize the flow structures over the numerical simulation data is addressed during \S~\ref{sec: Methodology}. The mechanisms driving the flow dynamics within urban environments, which are the result of the application of different data-driven tools, are further investigated in \S~\ref{sec: Results}. Finally, a summary of the main conclusions of the project is provided in \S~\ref{sec: Conclusions}, and the justification of the selected modes is performed in Appendix \ref{Appendix: Calibration} through the calibration process of the methods.
A review of the formation and destruction mechanisms of arch vortices in urban flows has already been addressed in a companion paper~\cite{PRL}. Here, a detailed analysis of the modal-decomposition techniques, which shed light on the mechanisms driving the flow dynamics, will be addressed with an overview of the high-order numerical simulations carried out to perform the present analysis.
\section{Numerical simulations} \label{sec: Numerical simulations}
\begin{figure}[b]
\centering
\includegraphics[width=\textwidth]{NumericalDomainPRF.pdf}
\caption{Schematic representation of the numerical domain, where $L_y=3h$ and $L_z=4h$. The flow is from left to right. (Top) and (bottom) show side and top views, respectively.}
\label{fig: numerical domain}
\end{figure}
The high-order spectral-element code Nek5000~\cite{Nek5000} was used to solve the incompressible Navier--Stokes equations governing the flow in the cases under consideration. Based on the spectral-element method (SEM) of Patera~\cite{PATERA1984}, Nek5000 exhibits both geometrical flexibility and the accuracy of the high-order spectral methods~\cite{Vinuesa2015,Hoyas2006,SIMENS2009, NEGI2018,Noorani2016,VINUESA2021,tanarro_vinuesa_schlatter_2020}. Due to the flow complexity in urban environments, high-order methods need to be used to resolve all the relevant flow structures properly. In this database, we use a \textit{well-resolved} large-eddy simulation (LES), the resolution of which is close to that of a direct numerical simulation (DNS)~\cite{NEGI2018}. This code has been extensively used for high-fidelity simulations of complex turbulent flows, see Refs.~\cite{Noorani2016, VINUESA2021, tanarro_vinuesa_schlatter_2020}. Additional details on the numerical scheme and employed resolution can be found in Ref.~\cite{torres2021thesis}.
The geometrical domain is composed of two wall-mounted obstacles, as depicted in Fig.~\ref{fig: numerical domain}. The size of the computational box dimension varies according to the separation of the obstacles. While the wall-normal and spanwise directions remain the same for the three cases, the streamwise length changes proportionally to the separation $\ell$, which modifies the computational cost of the numerical simulation associated with each case. The obstacles are then defined by the height $h$, length $w_b$ and width $b$. All dimensions are normalized with the height of the obstacle $h$. The velocity field is given by $\bm{v}(x,y,z,t)$, where $x, y,$ and $z$ are the streamwise, wall-normal, and spanwise directions, respectively, and $t$ is time. Every velocity is normalized with the free stream velocity. The components of the velocity are $\bm{v}=(u,v,w)$, which denote the streamwise, wall-normal, and spanwise components, respectively. Using Reynolds decomposition, $\bm{v}$ is defined as $\bm{v} = V + \tilde{\bm{v}}$, where $V=\overline{\bm{v}}$ is the average in time and $\bm{\tilde{v}}$ is the turbulent fluctuation. Primes are reserved for intensities $\bm{v'}=\overline{\tilde{\bm{v}}^2}^{1/2}$.
As inflow condition, a numerically-tripped~\cite{VINUESA201886, Vinuesa2017} laminar Blasius profile allows the flow for undergoing a rapid transition to turbulence without needing to accelerate the flow before reaching the obstacles~\cite{schlatter2012}. This numerical tripping consists of a weak wall-normal volume that is randomly added in the forcing terms of the incompressible Navier--Stokes equations to create flow disturbances, thus inducing turbulence. The inflow is located at $x/h = -10$, the tripping force is applied at $x/h=-9$, allowing the boundary layer for developing in the region upstream the obstacles, i.e. $-8\leq x/h \leq -1$. In this region, both $z$-averaged friction and momentum-thickness Reynolds numbers, i.e. ${Re}_\tau$ and ${Re}_\theta$ respectively, increase in the streamwise direction, reaching turbulent conditions for $x/h\geq -2$. The adverse pressure gradient induced by the obstacles leads to an increase of the Rota-Clause pressure-gradient parameter and a decrease in the skin-friction coefficient. The stabilized outflow condition developed by Dong et al.~\cite{DONG2014} is used as an outflow condition. At the upper part of the domain, a combination of outflow and Dirichlet conditions is used to simulate an open-air urban environment: a zero-stress condition is applied in the wall-normal direction, and a Dirichlet condition in the other two directions~\cite{Nek5000}. Finally, periodicity is applied in the spanwise direction. Wall condition is applied to the bottom plane of the domain and the surfaces of the obstacles.
We consider a spectral-element mesh with an eight-point Gauss--Lobatto--Legendre (GLL) quadrature in each element. The mesh is refined in the near-obstacle area in order to increase resolution, which has a direct impact on flow statistics. Following the criteria of Negi et al.~\cite{NEGI2018}, the mesh here employed satisfies all the resolution criteria to be considered a \textit{well-resolved} LES.
The analyzed database has the temporal parameters gathered in Table \ref{tab: Numerical Simulation}. Note that this information is critical for the analysis of the spatio-temporal structures of the flow since they define the system's dynamical behavior, which is closely related to the time span and time step of the snapshots to be analyzed. All the introduced parameters are expressed in convective time units, i.e. a ratio between a characteristic length and a velocity. In the present case, time is obtained from the freestream velocity $U_\infty$ and the height of the obstacle, $h$.
\begin{table}
\caption{\label{tab: Numerical Simulation}%
Temporal parameters of the datasets of the numerical simulations carried out for the three flow regimes identified by Oke~\cite{Oke1988}. The parameter $\ell/h$ refers to the separation-to-height ratio between the obstacles in the streamwise direction, $T_i$ and $T_f$ are the initial and final times of the database and $\Delta T$ is the total database timespan. Furthermore, $\Delta t$ represents the time interval between snapshots and $N_f$ is the number of fields.
}
\begin{ruledtabular}
\begin{tabular}{lccccccr}
\textrm{Flow regime}& $\ell /h$ & $T_i$ & $T_f$& $\Delta T$& $\Delta t$ & $N_f$ & Color code\\
\colrule
Skimming flow (SF) & 1 & 38.65 & 117.38 & 78.73 & 0.35 & 225 & \multicolumn{1}{c}{\textcolor{red}{\rule{0.05\textwidth}{.05cm}}} \\
Wake interference (WI) & 2 & 80.40 & 144.60 & 64.20 & 0.3 & 215 & \multicolumn{1}{c}{\textcolor{blue}{\rule{0.05\textwidth}{.05cm}}} \\
Isolated roughness (IR) & 4 & 83.30 & 145.60 & 62.30 & 0.7 & 90 & \multicolumn{1}{c}{\textcolor{black}{\rule{0.05\textwidth}{.05cm}}}
\end{tabular}
\end{ruledtabular}
\end{table}
\section{Methodology for modal decomposition} \label{sec: Methodology}
\subsection{Proper-orthogonal decomposition (POD)}
The proper orthogonal decomposition (POD) is a modal-decomposition technique, introduced in the field of fluid mechanics by Lumley~\cite{Lumley1967}, which aims at extracting coherent patterns from a given flow field. Thus, the objective of the POD algorithm is to decompose a set of data of a given field variable into a minimal number of modes (basis functions) that capture as much energy as possible. This process implies that POD modes are optimal in minimizing the mean-square error between the signal and its reconstructed representation. For instance, if the field variable to be examined is the velocity, the modes representing such variable are optimal to capture the kinetic energy of the flow field. This low-dimensional latent space provided by the POD modes is attractive for interpreting the most energetic and dominant patterns within a given flow field. Let us consider a vector field $\bm{q}\left(\bm{\xi},t\right)$, which may represent e.g. the velocity or the vorticity field depending on a spatial vector $\bm{\xi}$ and time. In fluid-flow applications, subtracting the temporal mean $\bm{\bar{q}}\left(\bm{\xi}\right)$ allows for the analysis of the unsteady component of the field variable:
\begin{equation}
\bm x(t) = \bm{q}\left(\bm{\xi},t\right) - \bm{\bar{q}}\left(\bm{\xi}\right),\quad\quad t = t_1,t_2,\dots,t_k
\end{equation}
\noindent where $\bm x(t)$ represents the fluctuating component of the vector data with its temporal mean removed. This representation emphasizes the idea that the data vector $\bm x(t)$ is being considered as a collection of snapshots at different time instants $t_k$. If the $m$ snapshots are then stacked into a matrix from, we obtain the so-called snapshot matrix $\bm X$:
\begin{equation}
\bm X = \left[\bm x(t_1), \bm x(t_2),\dots,\bm x(t_m)\right] \in \mathbb{R}^{J\times K},
\end{equation}
\noindent where $J$ represents the number of points in $x$, $y$ and $z$. The objective of the POD analysis is to find the optimal basis to represent the given set of data $\bm x(t)$. This can be solved finding the eigenvectors $\bm{\Phi}_j$ and the eigenvalues $\lambda_j$ from:
\begin{equation} \label{eq: POD basic eq}
\bm C \bm{\Phi}_j = \lambda_j \bm{\Phi}_j, \quad\quad \bm{\Phi}_j \in \mathbb{R}^{J}, \quad\quad \lambda_1\geq\dots\geq\lambda_N\geq0,
\end{equation}
\noindent where $\bm C$ states for the covariance matrix of the input data, defined as
\begin{equation}
\bm C = \sum_{i=1}^{K} \bm x\left(t_i\right)\bm x^\text{T}\left(t_i\right) = \bm X \bm X ^\text{T} \in \mathbb{R}^{J\times J}.
\end{equation}
The size of this matrix depends on the spatial degrees of freedom of the problem. In the case of fluid flows, this value is usually large since it equals the number of grid points times the variables to be considered. The POD modes are derived from the eigenvectors of Eq.~(\ref{eq: POD basic eq}, with the eigenvalues reflecting how well each eigenvector $\bm{\Phi}_j$ represents the original data in the $\ell_2$-sense. This enables a hierarchy of modes in terms of captured energy, which improves understanding of the most prominent patterns, e.g. in a specific flow field.
Another method for computing the POD algorithm is based on the singular-value decomposition (SVD)~\cite{Sirovich1987}, which can be applied directly on the snapshot matrix $\bm X$ to obtain the left $\bm{\Phi}$ and right $\bm \Psi$ singular vectors as
\begin{equation}
\bm X = \bm\Phi \bm\Sigma \bm\Psi ^\text{T},
\end{equation}
\noindent where $\bm\Phi \in \mathbb{R}^{J\times J}$, $\bm\Psi \in \mathbb{R}^{K\times K}$ and $\bm\Sigma \in \mathbb{R}^{J\times K}$. The matrix $\bm \Sigma$ contains the singular values $\left(\sigma_1,\sigma_2,\dots,\sigma_N\right)$ along its diagonal, which relates to the eigenvalues as $\sigma_j^2 = \lambda_j$. Moreover, the left and right singular vectors correspond to the eigenvectors of matrices $\bm X \bm X ^\text{T}$ and $\bm X ^\text{T}\bm X$, respectively. Therefore, the SVD can be seen as a rectangular-matrix decomposition technique capable of computing the POD modes.
\subsection{Higher order dynamic mode decomposition (HODMD)} \label{sec: HODMD methodology}
Aiming at identifying the spatio-temporal coherent patterns present in high-dimensional flow data, Schmid~\cite{Schmid2010} developed a data-driven tool which retrieved the spatially-correlated structures with similar behavior in time. This methodology, known as dynamic-mode decomposition (DMD), provides not only a reduction in dimension concerning a reduced set of modes which best reproduce the input flow-field, but also a model for the interaction of those modes in time.
The method decomposes the vector field data $\bm v\left(\textbf{x},t\right)$ as an expansion of $M$ Fourier-type modes:
\begin{equation} \label{eq: Fourier modes}
\bm v\left(\textbf{x},t\right) \simeq \sum_{m=1}^M a_m {\bm u}_m\left(\textbf{x}\right)e^{\left(\delta_m+i{\omega}_m\right)t_k},
\end{equation}
\noindent for $k=1,...,K$, where ${\bm u}_m$ represents the DMD modes weighted by an amplitude $a_m$, ${\omega}_m$, their associated frequencies and $\delta_m$, their associated growth rates, which symbolize the temporal growth or decay of the ${\bm u}_m$ modes in time.
The standard DMD algorithm assumes a linear relationship of two consecutive snapshot matrices using the linear Koopman operator \textbf{R}. To this end, a general snapshot matrix $\textbf{V}_{k_1}^{k_2}$ can be defined for $k_1<k_2$ so that its columns represent the snapshots varying equidistantly between $k_1$ and $k_2$, namely:
\begin{equation}
\textbf{V}_{k_1}^{k_2} = \left[{\bm v}_{k_1}, {\bm v}_{k_1+1}, ..., {\bm v}_{k_2}\right].
\end{equation}
Therefore, using the previous nomenclature, the standard DMD can be defined based on the Koopman operator as
\begin{equation}
\textbf{V}_{2}^{K} \simeq \textbf{R} \textbf{V}_{1}^{K-1}
\end{equation}
\noindent where $\textbf{V}_{2}^{K}$ and $\textbf{V}_{1}^{K-1}$ represent here the second to last snapshots and the first to the second last snapshots of the data matrix, respectively. Recalling Eq.~(\ref{eq: Fourier modes}), this equation might be seen as the simplest equation exhibiting such behavior~\cite{Vega2020Book}. The Koopman matrix \textbf{R}, which is independent of $k$, contains the dynamical information of the system. Recently, Le Clainche \& Vega~\cite{Vega2020Book} extended the DMD method for the analysis of various types of flows, e.g. turbulent, multi-scale or transitional flows and noisy experimental data. Based on Takens' delayed-embedded theorem~\cite{Vega2020Book}, the higher order dynamic mode decomposition (HODMD) relates $d$ time-delayed snapshots using higher-order Koopman assumption defined as
\begin{equation} \label{eq: HO Koopman General}
\textbf{V}_{d+1}^{K} \simeq \textbf{R}_1 \textbf{V}_{1}^{K-d} + \textbf{R}_2 \textbf{V}_{2}^{K-\left(d-1\right)} + ... + \textbf{R}_d \textbf{V}_{d}^{K-1},
\end{equation}
which relates each flow field with the $d$ subsequent fields. The HODMD algorithm can be encompassed into three main steps.
\subsubsection{Step 1: Dimension reduction} \label{sec: DMD Step 1}
First of all, the SVD technique is employed to reduce spatial redundancy and filter out noise caused by numerical or experimental errors. The truncated SVD allows for the reduction of the original snapshot data into a series of linearly independent vectors of dimension $N$ (where $N<J$ is the spatial complexity), based on a certain tolerance $\varepsilon_{\text{SVD}}$:
\begin{equation} \label{eq: SVD decomposition DMD}
\textbf{V}_1^K \simeq \textbf{W}\mathbf{\Sigma} \textbf{T}^\text{T},
\end{equation}
\noindent where $\mathbf{\Sigma}$ includes the singular values $\sigma_1,...,\sigma_N$ and $\textbf{W}^\text{T}\textbf{W}=\textbf{T}^\text{T}\textbf{T}={\bm I}$ are $N\times N$ unitary matrices. Note that the parameter $\varepsilon_{\text{SVD}}$ is tunable based on previous information of the simulation or experimental data, e.g. if the noise level of the snapshots is known in advance, then $\varepsilon_{\text{SVD}}$ may be set to be comparable to that level (see details in Ref.~\cite{LECLAINCHE2017d}). Above all, this parameter determines the number $N$ of SVD retained modes as
\begin{equation} \label{eq: Dimension reduction Step 1}
\frac{\sigma_{N+1}}{\sigma_1} \leq \varepsilon_{\text{SVD}}.
\end{equation}
Following the definition in Eq.~(\ref{eq: SVD decomposition DMD}), the reduced snapshot matrix $\hat{\textbf{T}}$ can be defined as
\begin{equation}
\textbf{V}_1^K \simeq \textbf{W}\mathbf{\Sigma} \textbf{T}^\text{T} \equiv \textbf{W} \hat{\textbf{T}}_1^K.
\end{equation}
The dimension of this reduced snapshot matrix is $N\times K$.
\subsubsection{Step 2: The DMD-d algorithm} \label{sec: DMD Step 2}
The higher-order Koopman assumption, defined in Eq.~(\ref{eq: HO Koopman General}), is now applied to the reduced snapshot matrix as
\begin{equation}
\hat{\textbf{V}}_{d+1}^{K} \simeq \hat{\textbf{R}}_1 \hat{\textbf{V}}_{1}^{K-d} + \hat{\textbf{R}}_2 \hat{\textbf{V}}_{2}^{K-\left(d-1\right)} + ... + \hat{\textbf{R}}_d \hat{\textbf{V}}_{d}^{K-1},
\end{equation}
\noindent where $\hat{\textbf{R}}_k=\textbf{W}^\text{T}{\textbf{R}}_k\textbf{W}$ is used for $k=1,...,d$. The above equation may be cast in a more generic form by incorporating the modified snapshot matrix $\tilde{\textbf{V}}_1^{k-d+1}$ and the modified Koopman matrix $\tilde{\textbf{R}}$ as
\begin{equation} \label{eq: general koopman matrix}
\tilde{\textbf{V}}_2^{K-d+1} = \tilde{\textbf{R}} \tilde{\textbf{V}}_1^{K-d},
\end{equation}
\noindent where the many Koopman operators $\hat{\bm{R}}_1,\dots,\hat{\bm{R}}_K$ are then combined into a single matrix after some computations, from which the eigenvalue problem can be solved to obtain the DMD modes, frequencies and growth rates defining the DMD expansion of Eq.~(\ref{eq: Fourier modes}). Sorted in decreasing order of the mode amplitudes, this expansion is further reduced by removing the modes such that:
\begin{equation}
a_m/a_1 < \epsilon_\text{DMD},
\end{equation}
\noindent for $m = 1,\dots,M$, where $\epsilon_\text{DMD}$ represents a parameter tunable by the user. The number of retained modes, $M$, represents the spectral complexity of the analysis. This complexity, together with the spatial one, determines the performance of the HODMD algorithm, which reduces to the standard DMD when $d=1$. In complex fluid flows, the spatial complexity is usually smaller than the spectral one, $N<M$, where the standard DMD fails, thus requiring the use of the DMD-d algorithm. Furthermore, using the tunable parameters $\epsilon_\text{SVD}$ and $\epsilon_\text{DMD}$, enables for retaining only the large scales of the input data, which is particularly interesting for complex turbulent flows involving a large number of scales.
\subsection{Spatio-temporal Koopman decomposition (STKD)} \label{sec: STKD methodology}
Spatio-temporal Koopman decomposition (STKD) is an extension of HODMD introduced to identify spatio-temporal structures as an expansion of traveling and standing waves driving the flow dynamics both in the streamwise and spanwise directions. For the streamwise direction, the spatio-temporal modes $\mathbf{u}_{mn_1}$ and growth rates $\nu_{mn_1}$ are defined in the following modal expansion, which reconstruct the original flow field analyzed as
\begin{equation}
\bm{v}(x_j,y,z,t_{k})\simeq
\sum_{m,n_1=1}^{M,N_1} a_{mn_1}\widehat{\mathbf{u}}_{mn_1}(y,z)e^{(\delta_m+i \omega_m)t_k+(\nu_{mn_1} + i \alpha_{mn_1})x_j},\label{ab002}
\end{equation}
\noindent for $k=1,\ldots,K$ and $j=1,\ldots,J$. It must be emphasized that the spatio-temporal expansion is useful when the data exhibit exponential/oscillatory behavior in both the x coordinate and time. In this case, it is interesting to compare the expansion \ref{ab002} with the purely temporal
expansion \ref{eq: Fourier modes}. This expansion can be easily obtained by simply applying HODMD to the DMD modes in Eq.~(\ref{eq: Fourier modes}), resulting in the following DMD expansion:
\begin{equation}
\mathbf{u}_m(x_j,y,z)\simeq
\sum_{n_1=1}^{N_1} a_{n_1}\widehat{\mathbf{u}}_{mn_1}(y,z)e^{(\nu_{mn_1} + i \alpha_{mn_1})x_j},\label{ab003}
\end{equation}
\noindent for $j=1,\ldots,J$. Eq.~(\ref{ab002}) is obtained by combining this solution with Eq.~(\ref{eq: Fourier modes}), where the spatio-temporal amplitudes are defined as $a_{mn_1}=a_m a_{n_1}$. In a similar way, it is possible to obtain spatio-temporal expansions defined along the spanwise direction as
\begin{equation}
\bm{v}(x,y,z_r,t_{k})\simeq
\sum_{m,n_1=1}^{M,N_1} a_{mn_1}\bar{\mathbf{u}}_{mn_1}(x,y)e^{(\delta_m+i \omega_m)t_k+(\lambda_{mn_1} + i \beta_{mn_1})z_r},\label{ab004}
\end{equation}
\noindent for $k=1,\ldots,K$ and $r=1,\ldots,R$, where $\lambda_{mn_1}$ and $\beta_{mn_1}$ are the growth rates and wavenumbers related with the spanwise direction. Using this expansion it is also possible to describe the analyzed data as a group of traveling waves, the phase velocity of which is defined as $c_{mn_1}=\omega_m/\beta_{mn_1}$. A more detailed description of the method can be found in Refs.~\cite{Clainche2018,LeClainche2018b}.
\section{Mean-flow structures} \label{sec: flow structures}
\begin{figure}[b]
\centering
\begin{subfigure}[b]{0.32\textwidth}
\centering
\includegraphics[width=\textwidth]{ArchVortexSF.png}
\caption{Skimming flow}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
\centering
\includegraphics[width=\textwidth]{ArchVortexWI.png}
\caption{Wake interference}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
\centering
\includegraphics[width=\textwidth]{ArchVortexIR.png}
\caption{Isolated roughness}
\end{subfigure}
\caption{Main vortical structures formed around two wall-mounted obstacles with different separation ratios: (\textbf{a}) $h/\ell=1$, (\textbf{b}) $0.5$ and (\textbf{c}) $0.25$, visualized by means of streamlines. Note the arch on the leeward side of the obstacles. The arrow indicates flow direction.}
\label{fig: Arch Vortex}
\end{figure}
A large number of studies have focused on the flow around a square wall-mounted cylinder of different aspect ratios, which is interesting due to the myriad of physical phenomena occurring at the same time: a recirculation bubble formed on the windward side of the cylinder induces an adverse pressure gradient that thickens the incoming boundary layer, which then produces a shear layer around the obstacle. Simultaneously, a horseshoe vortex progressively gets wider around the two sides of the cylinder, which accelerates the flow close to the obstacle due to the favorable pressure gradient induced by the geometry. A separated wake is then formed downstream the obstacle with a self-sustained oscillation process and a downward motion from the top of the obstacle, which is responsible for the widening of the wake~\cite{Vinuesa2015}. This configuration has been extensively analyzed both experimentally~\cite{Luo2003,Luo2007} and numerically~\cite{Sohankar1999,Saha2003}. In such a fashion, a two-dimensional approach allows for studying the turbulent features in the von Kármán vortex street occurring in the wake in a more organized fashion than in a finite-length wall-mounted cylinder~~\cite{Oertel1990,zdravkovich1997}.
However, the more complex flow encountered in wall-mounted cylinders is characteristic of important technological applications such as pollutant dispersion in urban environments and impact on pedestrian comfort~\cite{Oke1988}. The flow around a wall-mounted square cylinder of finite length is highly three-dimensional. The topology of the flow consists of free-end downwash flow, spanwise shear flow and upwash flow from the wall, which relate to the tip, base and spanwise vortices. Due to the interaction between these three components, the near wake is then characterized by an arch-shaped structure downstream the obstacle~\cite{wang_zhou_2009}. The formation of the arch vortex, which consists of two spanwise vortical legs, one on each side of the cylinder, and their connection or bridge near the free end, is closely related to the symmetric shedding modes, which induce an arch-type structure even on the instantaneous field~\cite{zhu2017}. This section aims at providing an overview of the main vortical structures present in the flow around two wall-mounted obstacles using the mean-flow streamlines, which are obtained from the database discussed in \S~\ref{sec: Numerical simulations}.
Hunt et al.~\cite{Hunt1978} performed an experimental analysis in order to examine the general pattern of the streamlines of the flow around a single surface-mounted bluff obstacle. They concluded that a closed mean streamline surface did not exist in the wake of the obstacle due to the interaction of 4 different vortical structures: the horseshoe vortex formed around the obstacle, the roof vortex and the vortices of the obstacles sides, both having a strong interaction with the wake, which yield to the so-called arch vortex downstream the obstacle. Therefore, the characterization of these vortices allows the hypothesis of no-closed surface in the wake to be established. The interaction between these structures emerges from the trailing vortex pattern that starts at the upper corners of the cube. This pair of counter-rotating vortices are known to extend farther downstream the obstacle, forming a dipole or a quadrupole structure depending on the obstacle aspect ratio. Wang and Zhou~\cite{wang_zhou_2009} found that above a critical falling in between $h/b=3$ and $5$ ($h$ and $b$ being the height and width, respectively) the streamwise vorticity distribution was of a quadrupole type and below this critical value, the distribution was of a dipole type. Burgeois et al.~\cite{Bourgeois2011} reviewed the salient flow features using the experimental results from an open-test-section suction wind-tunnel. They identified the vortex cores by means of the $\lambda_2$-criterion~\cite{jeong_hussain_1995} and they confirmed the existence of a single pair of streamwise vortices in the wake (for $h/b$ below the critical value).
Meinders~\cite{Meinders1998,Meinders2002} extended the work on flow structures around wall-mounted cubes by analyzing the interaction between the obstacles when more than one cuboid was introduced. Using oil-film visualizations, Meinders experimentally analyzed the influence of the separation distance between the obstacles, i.e. $\ell$, on the flow around around an in-line tandem disposition of two cubes. It was proved that the separation variance only led to a substantial modification of the mean flow patterns. In Fig.~\ref{fig: Arch Vortex}, the streamline time-averaged flow patterns for the three flow regimes are depicted. These flow regimes are the skimming flow (SF), the wake interference (WI) and the isolated roughness (IR)~\cite{Oke1988}. The main conclusions here addressed are in good agreement with the experimental results of Meinders~\cite{Meinders1998}. For the lowest separation $\ell/h=1$, i.e. the skimming-flow regime, the separated shear layer reattaches on the downstream side edge of the downstream obstacle, which leads to an inter-obstacle region characterized by an arc-shaped vortex, confined by the flow on the sides and the separated shear layer. In addition to this, a horseshoe vortex emerges upstream of the windward face of the leading cube and it is deflected downstream along the sides due to the presence of the downstream obstacle. The wake-interference regime, with a separation ratio $\ell/h = 2$, exhibits similar flow patterns to those of the skimming flow. However, in this case, the intermediate arch vortex does not span the whole region in between the obstacles: the separation is large enough for the flow on the sides to interact slightly with the inter-obstacle region. For the isolated-roughness regime, i.e. $\ell/h=4$, the flow eventually reattaches within the inter-obstacle region: the shear layer detaches from the sides and top edges of the upstream obstacle and breaches the inter-obstacle spacing before reattaching on the lower wall. Because of that flow reattachment, a second horseshoe vortex emerges in front of the downstream cube. In the three flow regimes, a second arch vortex is formed on the leeward side of the downstream obstacle, albeit with a lower intensity. This is mainly due to the flow disruption of the upstream obstacle, which induces a different turbulent-intensity level upstream of the second one~\cite{Oke1988,Meinders2002,Vinuesa2015}.
\section{Spatio-temporal structures: understanding urban flows} \label{sec: Results}
\begin{figure}[b]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{SingularValues_UY25_All_grid.png}
\end{subfigure}
\hspace{0.04\textwidth}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{SingularValues_UY25_sum.png}
\end{subfigure}
\caption{Left: Singular-value distribution normalized with its maximum value $\sigma_1$ of the POD modes corresponding to the complete set of velocity components (streamwise, wall-normal and spanwise) of the ({red}) skimming-flow, ({blue}) wake-interference and ({black}) isolated-roughness regime. Right: Cumulative singular-value spectrum normalized with the cumulative sum of the eigenvalues $\sum_{i=1}^{i=M} \sigma_m$, where $m$ indicates the number of modes. Note that the number of retained modes coincides with the number of columns of the snapshot matrix, i.e. $225$, $215$ and $90$, respectively.}
\label{fig: Singular Values}
\end{figure}
Data-driven modal decompositions are powerful techniques to extract the energetically-important features from a given flowfield. Having identified the main flow structures present in the time-averaged fields in \S~\ref{sec: flow structures}, the analysis over the instantaneous fields will allow for the characterization of the main mechanisms driving the flow dynamics. In this section, we first analyze the results obtained using POD for modal decomposition of the urban database discussed in \S~\ref{sec: Numerical simulations}. In Fig.~\ref{fig: Singular Values}, we show the singular-value distribution normalized with its maximum value $\sigma_m/\sigma_1$, where $m$ represents the mode number, and the cumulative energy for the POD modes corresponding to the three reference cases. The objective is to identify energy gaps that make some modes energetically more relevant than others. However, this distribution is highly dependent on the input data, where the number of fields varies from case to case, see \S~\ref{sec: Numerical simulations}. Despite this, an energy gap between the second and third modes for all regimes reveals that the first two modes contain the more relevant information of the flow. The energy of the most dominant POD mode for the wall-normal component, in particular, represents roughly $45\%$ of the maximum energy value of the other velocity components. This means that the influence of the wall-normal velocity fluctuations is lower than that of the other velocity components. This is consistent with the findings of Monnier et al.~\cite{Monnier2018}, who discovered that for a zero-incidence angle, the streamwise and spanwise fluctuating components are more significant than the wall-normal component. On the basis of the above, the wall-normal component will be omitted from the analysis and only the streamwise and spanwise components will be further studied for the three flow regimes.
\begin{figure}
\centering
\begin{subfigure}[b]{\textwidth}
\centering
\adjincludegraphics[width=0.95\textwidth,trim={{.05\width} {.025\width} {.05\width} {.02\width}}]{Modes_U_Y25_all_SF.png}
\caption{Skimming flow}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}
\centering
\adjincludegraphics[width=0.95\textwidth,trim={{.05\width} {.025\width} {.05\width} {.02\width}}]{Modes_U_Y25_all_WI.png}
\caption{Wake interference}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}
\centering
\adjincludegraphics[width=0.95\textwidth,trim={{.05\width} {.025\width} {.05\width} {.02\width}}]{Modes_U_Y25_all_IR.png}
\caption{Isolated roughness}
\end{subfigure}
\caption{POD orthogonal basis of the streamwise velocity fields at $y/h=0.25$ for the different flow regimes. For each regime, from the upper left to the lower right, first to tenth modes are sequentially presented. Contours of the velocity of the modes are normalized with the $L_\infty$-norm and vary between $-1$ (blue) and $+1$ (red).}
\label{fig: POD modes streamwise}
\end{figure}
Starting with the streamwise velocity field, Fig.~\ref{fig: POD modes streamwise} depicts the orthogonal POD basis for the modes corresponding to the three flow regimes. First of all, regarding the first two modes, it could be noted their apparent similarity, dominated by fluctuating regions on both sides of the blocks, mainly due to the interaction of the wake region with the shear layer on both sides. These regions match with the high-turbulent-kinetic-energy (TKE) regions of the streamwise component identified by Monnier et al.~\cite{Monnier2018} for an array of building-like blocks. However, the main differences among regimes depend on the position of the secondary structures, which are associated with the downstream block. For instance, while these fluctuating regions span the zone in between the obstacles for the skimming flow and the wake interference cases, in the isolated roughness, they are only located on the immediate leeward side of the upstream block. In such a fashion, increasing the separation of the obstacles does not yield more fluctuating regions, at least for the more energetic modes, as it occurs in a vortex-shedding case. The structures observed in the first two modes are remarkably similar to those associated with the high-frequency vortex-breaker modes. Higher modes are more focused on the wake as well as the creation of these vortical structures. Particularly, for all cases, the third and fourth modes are clearly impacted by certain major fluctuating areas on the wake. This pattern is quite similar to that of the time-averaged field, therefore these results suggest that these modes can be associated with the formation process of such structures.
\begin{figure}
\centering
\begin{subfigure}[b]{\textwidth}
\centering
\adjincludegraphics[width=0.95\textwidth,trim={{.05\width} {.025\width} {.05\width} {.02\width}}]{Modes_W_Y25_all_SF.png}
\caption{Skimming flow}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}
\centering
\adjincludegraphics[width=0.95\textwidth,trim={{.05\width} {.025\width} {.05\width} {.02\width}}]{Modes_W_Y25_all_WI.png}
\caption{Wake interference}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}
\centering
\adjincludegraphics[width=0.95\textwidth,trim={{.05\width} {.025\width} {.05\width} {.02\width}}]{Modes_W_Y25_all_IR.png}
\caption{Isolated roughness}
\end{subfigure}
\caption{POD orthogonal basis of the spanwise velocity fields at $y
/h=0.25$ for the different flow regimes. For each regime, from the upper left to the lower right, first to tenth modes are sequentially presented. Contours of the velocity of the modes are normalized with the $L_\infty$-norm and vary between $-1$ (blue) and $+1$ (red).}
\label{fig: POD modes spanwise}
\end{figure}
After these first modes, the similarity among the many cases begins to diverge, each with its own set of properties. In the isolated-roughness regime, modes 5, 6, 7 and 10 exhibit flow structures around the downstream obstacle independently, a fact that suggests that the flow behaves independently around both obstacles. This is the only regime providing such isolated structures. The difference between the skimming-flow and the wake-interference regimes, on the other hand, is based on the intersection of the fluctuating areas in the wake-interference regime owing to the increasing spacing between the obstacles, as shown in modes 4 and 6.
In terms of the spanwise component, Fig.~\ref{fig: POD modes spanwise} depicts the set of POD orthogonal modes for all flow regimes. On the basis of the singular-value distribution of Fig.~\ref{fig: Singular Values}, the first two modes will be firstly analyzed. Unlike the streamwise-component scenario, increasing the distance between the obstacles increases the number of high-intensity flow structures of spanwise fluctuating regions, although the width of these structures decreases. These results could be connected to the interplay of lateral flow within the canopy, which suggests that the arch vortex formed on the leeward side of the upstream obstacle is shattered. Moving on to the modes that are related to the creation of such vortical structures, it is worth noting that modes 6 and 7 are not as uniform as the previous ones; the two separated and symmetric areas may suggest that these areas are connected to the formation of each of the legs of the arch vortex. The spanwise-velocity component, like the streamwise one, reveals some isolated features exclusively for the downstream obstacle of the isolated-roughness regime. As a result of the loss of energy in the flow after the first obstacle, the method identifies two types of flow structures, placed upstream and downstream of the obstacles.
Each of these POD modes are associated with temporal coefficients, and a quantitative analysis can be performed on them with the aim of characterizing their dominant frequencies. The fast-Fourier-transform (FFT) technique will be used to generate, from the temporal data, the power spectrum revealing the most prominent frequencies of each mode.
\begin{figure}
\centering
\adjincludegraphics[width=0.95\textwidth,trim={{.05\width} {.035\width} {.05\width} 0}]{FFT_All.png}
\caption{Power spectrum of FFT scaled with the Strouhal number $St$ and applied to the temporal coefficients of the POD modes: ({red}) skimming-flow, ({blue}) wake-interference and ({black}) isolated-roughness regimes. As in Figures~\ref{fig: POD modes streamwise} and \ref{fig: POD modes spanwise}, modes from 1 to 10 are shown from top-left to bottom-right.}
\label{fig: FFT POD All}
\end{figure}
Fig.~\ref{fig: FFT POD All} depicts the power spectrum for the first ten POD modes of all flow regimes. Low- and high-frequency modes within the frequency range $\left[0,2\right]$ are distinguished for the three flow regimes. Frequency values are found to match in all cases, with the isolated-roughness case yielding clearer results due to a higher power-spectral density in the peak frequency. The first two modes are dominated by a peak frequency in the range ${\omega}_m = \left[1-1.2 \right]$. Modes $3 - 6$ are seen to be of low frequency, which suggests that they could be connected with the creation of significant vortical structures such as the arch vortex. Higher modes present flow structures which might result from the combination of the previous modes and if even higher modes $\left(m > 10\right)$ were studied, certain high-frequency phenomena would be captured due to the smaller turbulent-flow scales associated with them.\\
Because of the large number of frequencies and spatio-temporal structures present in turbulent flows, identifying flow patterns is challenging, and therefore it is critical to test the robustness of the results. To that end, the previous results will be compared to those obtained with a highly efficient tool for the analysis of complex flows, i.e. the higher-order DMD. The modified Koopman operator employed in the HODMD algorithm yields a solution fulfilled by all sub-groups of data (snapshots) simultaneously evaluated and capable of capturing large-scale and large-amplitude features from the highly-varied frequencies seen in the flow~\cite{LeClainche2020}. Therefore, the HODMD algorithm (also known as DMD-d) is used for this purpose with varied tolerances and values of the $d$ parameter.
\begin{figure}[b]
\centering
\includegraphics[width=\textwidth]{DMDsolutionTogether.png}
\caption{DMD-d modes. Amplitude scaled with its maximum value ($\hat{a}_m=a_m/a_0$) versus frequency ${\omega}_m$ computed for ({red}) skimming flow, ({blue}) wake interference and ({black}) isolated roughness. The modes here represented are the result of a calibration process of the user parameters, from where $\epsilon_\text{SVD} = \epsilon_\text{DMD} = 10^{-3}$, $d=20$ for SF ($K=225$ snapshots) and WI ($K=215$ snapshots) and $d=10$ for IR ($K=90$ snapshots) have been selected as reference.}
\label{fig: DMD spectrum}
\end{figure}
In Appendix \ref{Appendix: Calibration} we provide a detailed overview of the calibration process of the identification method of the modes. In particular, Fig.~\ref{fig: DMD spectrum} shows the frequency versus amplitude of the different modes computed using DMD-d for the three flow regimes. Although this technique can be used to analyze datasets with snapshots that are not equally spaced in time, the intricacy of the flow described here would make modeling much more challenging. As a result, the DMD-d analysis can be conducted considerably more efficiently if the data specifications in Table \ref{tab: Numerical Simulation} are used. The very large number of modes calculated with each variation requires the use of well-established criteria to identify the most robust modes that best characterize the entire system~\cite{LeClainche2020}. The dominant mode, i.e. the one with the highest amplitude, is located between the frequencies $\omega_m=1$ and $\omega_m=1.2$, while the rest of the modes are subharmonics and harmonics of it. This mode will referred to as mode A. In addition, another relevant mode is the one with the lowest frequency ($\omega_m = 0.1$), called as mode B, since it is the first mode to appear in the spectrum and the periodicity of the main physics is led by its frequency. Hence, studying in detail the two modes selected, will provide a general idea of the main dynamics driving the flow. Details about the remaining modes are presented below in \S~\ref{sec: Interaction}. It is worth noting that for the SF case, the method identifies two large-amplitude modes with similar frequency values, a fact that does not occur for the other two flow regimes. However, we have selected the higher amplitude mode as mode B in this case. The lower-frequency mode, mode A, has been selected comparing the three different test cases satisfying the criterion $|f_{mi}-f_{mj}|<\epsilon$, where $f_{mi}$ and $f_{mj}$ represent the frequency value in two different test cases and $\epsilon$ is a tolerance set by the user. Besides, the modal-assurance criterion~\cite{Pastor2012,Mendez2021} has been used to conclude that the mode selected in the three cases as modes A and B represent the same flow physics. Note as well that the relative error made in the calculations remains fenced in the set of tolerances used. The goal of this study, however, is to identify the largest-amplitude modes in order to provide a broad description of the fundamental patterns driving the flow, rather than to build any accurate reduced-order models based on the physical knowledge of the flow. As a result, the relative error will not be examined further in this study.
Fig.~\ref{fig: HODMD main modes} shows a three-dimensional view of the main DMD modes presented in Fig.~\ref{fig: DMD spectrum} as a function of the separation ratio between the obstacles, aiming at providing a general overview of the main flow structures related to each mode. Fig.~\ref{fig: HODMD main modes} (left) corresponds to the vortex-generator low-frequency mode, mode A, whereas the vortex-breaker high-frequency mode, mode B, is depicted in Fig.~\ref{fig: HODMD main modes} (right). Indeed, their structures are linked to the first two highest-in-energy POD modes. These results are further discussed in the following sections.
\begin{figure}
\centering
\begin{subfigure}[b]{0.483\textwidth}
\centering
\includegraphics[width=\textwidth]{Vortex-generator.pdf}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{Vortex-breaker.pdf}
\end{subfigure}
\caption{Three-dimensional iso-surfaces of the ({left}) streamwise, ({middle}) wall-normal and ({right}) spanwise velocities of the vortex-generator and vortex-breaker modes shown in Fig.~\ref{fig: DMD spectrum}. The left and right frames refer to vortex-generating and vortex-breaking modes, respectively. In each panel, the flow moves from left to right. Velocity values are normalized using the $L_\infty$-norm. The iso-values employed are given by $a\,U_\text{max}$ (blue) and $b\,U_\text{min}$ (red): for the vortex-generator mode, (left) streamwise velocity is represented with $a=0.6$ and $b=0.7$, (middle) wall-normal velocity is depicted using $a=0.6$ and $b=0.5$ and (right) spanwise velocity is represented with $a=b=0.5$; for the vortex-breaker mode, $a=b=0.4$ for all velocity components.}
\label{fig: HODMD main modes}
\end{figure}
\subsection{Generation process of the main vortices}
In the first case, the vortex-generator mode presents some characteristic three-dimensional structures which relate to the formation of the arch and horseshoe vortices in a low-frequency fashion.
First, with regard to the streamwise velocity component, a dome-like structure prevails in the intermediate section of the obstacles. Increasing the distance between the obstacles does not result in a reallocation of such a flow pattern, which remains in the same place throughout all flow regimes. This indicates that when the flow reaches this region in the WI and IR cases, it interacts with the canopy, which does not happen in the SF regime, where the flow in between the obstacles remains trapped, i.e. the spherical structure covers the same region as the separated zone between the obstacles. Note as well that the dome feature in the IR case is complemented by a structure on the wake which extends through both lateral sides up to the leeward side of the downstream obstacle. These conclusions were already extracted from the analysis of the time-averaged fields in the streamwise direction at the symmetry plane $z/h=0$, where the increased separation between the buildings yielded a higher interaction of the flow within this zone.
On the other hand, although not presenting relevant flow structures in the constant$-y$ planes analyzed in Fig.~\ref{fig: HODMD harmonic modes}, the wall-normal velocity component of mode A exhibits a three-dimensional pattern on the upper windward side of the upstream obstacle, which is shared among all the flow regimes. At this location, the flow experiences a high-velocity region due to the impact of the flow over the edge and the shear layer on the upper part of the obstacle. This region is then followed by another fluctuating part in the wall-normal direction in between the obstacles. Albeit to a lesser extent, the downstream obstacle in the IR case also exhibits a similar flow structure.
Finally, similar to the wall-normal component, the flow encounters a high-spanwise velocity gradient over this zone owing to the effect of the flow over the edges of the first obstacle, which tends to deviate the flow towards the outside regions of the domain. Furthermore, the second obstacle has comparable structures on its windward lateral edges as a result of the flow reattachment that occurs in the IR case. On the other hand, due to the slight interaction of the flow in the spanwise direction within this region, several fluctuating zones develop in between the obstacles. This characteristic distinguishes this generating mode from the breaker modes, which exhibit substantial spanwise variations between the obstacles. This result suggests that these structures could be connected with the destruction of the vortical structures in this region. To conclude, it can be state that the flow features that dominate in the vortex-generator mode are located close to the first obstacle rather than on the wake, emphasizing the idea of being a formation-type mode.
\begin{figure}
\centering
\begin{subfigure}[b]{\textwidth}
\centering
\adjincludegraphics[width=0.95\textwidth,trim={{.05\width} {.025\width} {.05\width} {.02\width}}]{DMDmodes_SF_All_UW.png}
\caption{Skimming flow}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}
\centering
\adjincludegraphics[width=0.95\textwidth,trim={{.05\width} {.025\width} {.05\width} {.02\width}}]{DMDmodes_WI_All_UW.png}
\caption{Wake interference}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}
\centering
\adjincludegraphics[width=0.95\textwidth,trim={{.05\width} {.025\width} {.05\width} {.02\width}}]{DMDmodes_IR_All_UW.png}
\caption{Isolated roughness}
\end{subfigure}
\caption{DMD modes of the streamwise and spanwise velocity fields with selected frequencies, represented at $y/h=0.25$ for the different flow regimes. Contours of the velocity of the modes are normalized with the $L_\infty$-norm and vary between $-1$ (blue) and $+1$ (red). The bold-face frequencies represent the vortex-generator and the vortex-breaker modes, modes A and B, respectively. The rest of the modes are the result of the interaction between the above-described modes and are known as harmonic modes.}
\label{fig: HODMD harmonic modes}
\end{figure}
\subsection{Breaking process of the main vortices}
The three-dimensional structures of the vortex-breaker mode are illustrated in Fig.~\ref{fig: HODMD main modes} (right). This mode is closely connected with the wake as opposed to what was observed in the previous case. For all of the flow regimes investigated here, the spanwise-fluctuating areas are shown to occupy the whole intermediate zone between the obstacles. As a result, increasing the distance between the obstacles leads to a greater number of fluctuation areas in the spanwise direction: in the IR case, up to three alternating structures may be seen, whereas only one can be seen in the SF regime. Conversely, the streamwise component of the present mode does not exhibit the same behavior; the structures on the leeward side of the upstream obstacle remain unchanged for all three flow regimes. However, another oscillating zone arises connected to the downstream obstacle, the position of which is modified among the various regimes. Finally, in terms of velocity in the wall-normal direction, as the separation increases, the flow interacts inside the canopy with considerably more significance, resulting in larger flow structures for both the WI and IR cases.
Apart from being quite similar to those of the first two POD modes, these structures are consistent with the results of Monnier et al.~\cite{Monnier2018}, with strong streamwise fluctuations on both lateral sides and a high turbulent spanwise region near to the windward side of the downstream obstacle. The arch vortex is known to exist between these regions, and these structures will be related to the process of breaking rather than creation, owing to its location on the wake.
\subsection{Interaction of vortex-generator and vortex-breaker modes\label{sec: Interaction}}
Finally, the three-dimensional structures of the most significant DMD modes can be compared to those of the various modes identified by the algorithm. Specifically, the following lines will be dedicated to the classification of the modes in vortex-generator or breaking-vortex modes based on resemblance with the prior patterns. This allows for selecting the more robust modes identified by HODMD using a varied set of parameters. To complement the previous results related to modes A and B, Fig.~\ref{fig: HODMD harmonic modes} shows a contour representation of the DMD modes presented in Fig.~\ref{fig: DMD spectrum}. By general inspection, one could see the structures previously discussed for the main HODMD modes, which are highlighted in bold. From these structures, a limit frequency can be established such that greater frequency values result in flow patterns more focused on the wake and thus being of breaking-type.
For the SF case, the mode ${{\omega_m}} = 0.37$ still exhibits some flow trapped in between the obstacles and high spanwise fluctuations on the lateral edges of the first obstacle. Even though the flow in this region appears to be modified by the slight interaction with the surrounding flow, this mode can be thought of as a vortex-production mode with a different production mechanism. Higher-frequency modes (${\omega_m}>0.85$) are characterized as breaker modes since both the streamwise and spanwise components share the same flow features as the main one ($\omega_{m}=1.22$). Note as well that higher-frequency modes exhibit smaller turbulent scales. This highlights the association of low-frequency modes with large flow scales (dominant patterns) and high-frequency modes with smaller turbulent structures. A similar conclusion can be extracted for the IR regime, where the threshold value is set for the mode with frequency ${\omega_m}=0.57$, which shows some flow structures around the upstream obstacle combined with particular features on the wake. Hence, this mode may be regarded as a transitory mode between the vortex-generator and vortex-breaker modes. Finally, regarding the WI case, apart from the vortex-generator mode (${\omega_m}=0.13$), the lowest-frequency mode (${\omega_m}=0.45$) exhibits a flow pattern similar to that of the vortex-breaker mode. Therefore, in this situation, the threshold value should be set lower than this frequency, resulting in all modes fulfilling ${\omega_m}>0.45$ becoming of breaking-type.\\
\begin{figure}
\centering
\begin{subfigure}[b]{\textwidth}
\centering
\includegraphics[width=\textwidth]{Streamlines_Generator_Ar.png}
\caption{Vortex-generator mode}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}
\centering
\includegraphics[width=\textwidth]{Streamlines_Breaking.png}
\caption{Vortex-breaker mode}
\end{subfigure}
\caption{Main flow patterns of the vortex-generator and vortex-breaker modes shown in Fig.~\ref{fig: DMD spectrum} visualized by means of streamlines for the three flow regimes. The arrows indicate the direction of the flow in each panel. Note the arch-shaped structure on the leeward side of the upstream obstacle and the helicoidal flow structures in between the obstacles.}
\label{fig: HODMD streamlines}
\end{figure}
The dominant flow structures of the above-described modes can be elucidated by means of streamlines. To that end, using this visualization technique, it is possible to relate the previous three-dimensional structures with the vortex-generating and vortex-breaking processes. Fig.~\ref{fig: HODMD streamlines} depicts the streamlines flow patterns of the vortex-generator and vortex-breaker modes for the three flow regimes. While the first mode resembles the arch-vortex structure, the second presents a helicoidal tunnel-shaped flow pattern in the region in between the obstacles, owing to the increased correlation in the spanwise direction. The location of these structures perfectly matches the gaps in between the velocity-fluctuating regions in the streamwise and spanwise directions. Therefore, since these regions define the location of such patterns, the number of structures is modified from case to case: up to three structures are observed in between the buildings for the IR regime. Consequently, the interaction of the flow within this region results in a mixing procedure, leading to the breaking process of the vortical structures.
Knowledge of the mechanisms of generation and destruction of relevant vortical structures within urban flows provides sufficient information to be able to perform studies of pollutant dispersion within urban environments, so that ground-level concentrations significantly higher than those occurring in the absence of the building can be avoided. Regions of strong recirculation have been demonstrated in the literature to increase the concentration of scalars~\cite{zhu2017}. In this sense, for pollutants emitted at street level, the vortex-breaker mode could provide, at a high frequency, a higher interaction with the surrounding clean air, while for the arc-generating mode, the flow between buildings could be hardly influenced by the flow outside. Therefore, B--type modes could be connected to the promotion of the pollutant dispersion within cities, whereas the generation of A--type modes should be minimized owing to their low interaction with the surrounding atmosphere. Another important aspect closely related with the pollutant-dispersion aspect is the direction of the fluctuations. Regarding vortex-breaker modes, the tunnel-shaped structures, mainly influenced by the arrow-shaped spanwise fluctuations, would disperse rapidly those pollutant emitted at the street level towards the atmosphere. However, the streamwise fluctuations, which are known to appear close to the building sides, produce an increased concentration of pollutants within the city.\\
\begin{figure}
\centering
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{Spectre_STKD_X.png}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{Spectre_STKD_Z.png}
\end{subfigure}
\caption{Spectra of the STKD modes for the (left) streamwise and (right) spanwise directions. Amplitude scaled with the maximum value ($\hat{a}_m=a_m/a_0$) versus wavenumber, ${\alpha}_m$ for the $x$-analysis and ${\beta}_m$ for the $z$-analysis, computed for ({red}) skimming flow, ({blue}) wake interference and ({black}) isolated roughness. The filled points represent the spatial modes obtained from the generator temporal modes while the empty markers represent the ones obtained from the breakers. The modes here represented are the result of a calibration process of the user parameters, from where $\epsilon_\text{SVD} = \epsilon_\text{DMD} = 10^{-4}$, $d=5$ for every case.}
\label{fig: STKD spectrum}
\end{figure}
After discussing the different generation and breaking mechanisms of the main vortical structures associated with the temporal modes obtained by HODMD, the STKD algorithm is applied to these modes in order to obtain the spatio-temporal modes, with the aim of understanding the mentioned mechanisms and how they link with the physics of the problem.
As already shown in \S~\ref{sec: STKD methodology}, mode expansion can be applied to different spatial directions. In this work, we analyze both the streamwise and spanwise directions. Once the STKD is applied to the temporal HODMD modes, we obtain a spectrum of the spatio-temporal modes as can be seen in the Fig.~\ref{fig: STKD spectrum} for both directions, where the modes obtained for the $x$-direction will be called X modes and the ones obtained in the $z$-direction will be the Z modes. As it is shown in this figure, the dominant wave-number is $\alpha_m=0.6$ for the X modes, and $\beta_m=2.2$ for the Z modes. In general, the spatial modes obtained from the A modes have lower amplitude than the ones obtained from the B modes. After studying the results obtained from the spatio-temporal modes, it can be observed that the X modes are connected to the mechanisms of the breaking process and the Z modes showed the results of these mechanisms.
Fig.~\ref{fig: STKD generators} shows the different structures appearing in the STKD analysis when applied to generator modes. The interaction zone for all the displayed modes is near and between the buildings, hence, the shedding of the arch vortex still have not started. This fact is also backed by the fact that the mean flow, represented by white-transparent structures, appears enveloping the structures of the dominant mode for each case preventing it from breaking. Many interesting phenomena can be extracted from these modes. Also, a phase shift appears in the streaks formed in the streamwise component of the Z mode between the real and imaginary part (the imaginary part is not shown for the sake of brevity), suggesting that these structures are traveling waves along the spanwise direction. It can also be seen that this structures move along the streamwise direction. For the Z modes, the streaks that appear in the streamwise component of the velocity envelope both buildings for the SF and WI regimes while these streaks are divided, creating a new structure in the second building. This phenomenon occurs since the IR regime has enough distance between the buildings for the turbulent flow to adapt. Something similar happens for the wall-normal component, where the cap-like structure starts to appear again in the second building.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{STKD_generators.png}
\caption{Three-dimensional iso-surfaces of the different spatio-temporal modes obtained from the generator modes. (Top), (middle) and (bottom) show the SF, WI and IR cases respectively, while each column represents a different mode: the first column shows the streamwise velocity of the temporal generator mode, the second and third columns display the streamwise and wall-normal velocity of the dominant Z mode with a wavenumber of $\beta_m=2.2$, and the last column represents the streamwise velocity of the dominant X mode with a wavenumber of $\alpha_m=0.6$. Red and blue denote positive and negative velocities respectively, and the white-transparent structures represent the mean-flow of the spatio-temporal modes with zero wavenumber.}
\label{fig: STKD generators}
\end{figure}
Regarding Fig.~\ref{fig: STKD breakers}, where the vortex-breaker modes are shown, the structures of the Z modes continue appearing between the buildings rather than on the sides of the buildings, which is what was expected. The reason behind this phenomena lies in the strong streamwise influence present in the flow, making the breaking mechanisms to appear near the arch vortex. Still, there is a big difference in the interaction of the dominant mode with the mean flow, since in the generator case it wraps the structures while in the breakers it cuts them. As explained previously, the Z modes show the results of the breaking mechanisms shown on the X modes. The X modes display large streamwise structures on the sides of the buildings (except in the IR case) that indicate the destruction of the main vortices and generating the turbulent wake. The results of this mechanism are shown in the Z modes, where a large cluster appears between the buildings after the arch vortex, suggesting an association with the shedding of the arch vortex.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{STKD_breakers.png}
\caption{Three-dimentional iso-surfaces of the different spatio-temporal modes obtained from the breaker modes. (Top), (middle) and (bottom) show the SF, WI and IR cases respectively, while each column represents a different mode: the first column shows the streamwise velocity of the temporal generator mode, the second and third columns display the streamwise and wall-normal velocity of the dominant Z mode with a wavenumber of $\beta_m=2.2$, and the last column represents the streamwise velocity of the dominant X mode with a wavenumber of $\alpha_m=0.6$. Red and blue denote positive and negative velocities respectively, and the white-transparent structures represent the mean-flow of the spatio-temporal modes with zero wavenumber.}
\label{fig: STKD breakers}
\end{figure}
\newpage
\section{Summary and conclusions} \label{sec: Conclusions}
A simplified urban environment model consisting of an array of two buildings with variable spacing ratios was examined using high-fidelity simulations. These simulations were carried out to provide a complete physical description of the fundamental mechanisms controlling the dynamics in different urban streets. The aim was to provide an insightful analysis of the physics of the flow within environments that simulate different types of urban areas. The growing expansion of cities boosts the search for physical models capable of reproducing the pollutant and thermal distributions within cities. Here, the three-dimensional flow patterns responsible for pollutant dispersion have been characterized. Isosurfaces and contour slices were used to demonstrate the complicated flow behavior of the modes identified by the POD and HODMD algorithms.
The results show that the flow behavior can be split into low- and high-frequency phenomena, each with significant consequences related to the formation and destruction of vortical structures such as the arch or horseshoe vortices. The low- and high-frequency modes are named vortex-generator modes since their associated structures have been related to the mechanism triggering the formation of the arch vortex formed on the leeward side of both buildings. These structures are particularly noticeable on the windward side of the upstream obstacle for the wall-normal and spanwise velocity components and the leeward side for the streamwise one, which defines the location and shape of the arch vortex. Furthermore, this location is kept constant among the different flow regimes, which highlights the idea that the process of formation of the arch vortex does not strongly depend on the separation between the obstacles.
HODMD, on the other hand, identifies a high-frequency mode that correlates with the largest-amplitude mode in all cases. \sergio{no see entiende bien} Because of the streamline flow patterns in the intermediate section of the obstacles, they are referred to as vortex-breaker modes. The large amplitude of these modes emphasizes their importance in this sort of urban flow. Indeed, their structures are linked to the first two highest-in-energy POD modes. Furthermore, as the separation increases, the flow becomes more correlated in the spanwise direction, owing to the more significant interaction of the flow with the wake layer inside the canopy. This effect yields to fluctuating velocity regions occupying the whole section between the obstacles, thus being associated with the destruction of the vortical structures rather than their formation.
Besides, the vortex-breaker modes will be responsible for the emergence of high-TKE regions on both sides of the obstacles and on the windward side of the downstream obstacle. Interestingly, these results are consistent with the wind-tunnel results of Monnier et al.~\cite{Monnier2018}, performed on a scaled and slightly more complex urban environment. Therefore, the conclusions of the present work can be extrapolated to more realistic urban environments by considering that turbulence levels from one street to another are expected to decrease significantly. It is also interesting to note that the results provided by both POD and HODMD show that the wall-normal velocity component does not significantly influence the more prominent structures as the streamwise and the spanwise components do.
Regarding the results obtained from the STKD analysis, the Z modes display the results of the mechanisms obtained in the X modes about the generation and destruction of the coherent structures. When analyzing the temporal generator modes, the X and Z modes show a phase shift between the real and imaginary parts. Consequently, traveling waves appear in each direction. In addition, the symmetry is conserved, and the influence area shows that the structures are still unbroken. On the other hand, when the STKD analysis is applied to the temporal breaker modes, the X modes show the main structures are broken, and the structures causing this destruction are shaped as large streaks in the streamwise direction. Meanwhile, the Z modes show the result of these mechanisms.
To conclude, from an environmental point of view, urban areas with highly-separated buildings, namely the isolated-roughness regime, would exhibit much more interaction with clean air sources, thus enabling the rapid propagation of those pollutants emitted at the street level. However, power plants, commonly located close to urban centers, are also responsible for pollution issues within cities. In those cases, owing to the low interaction of the flow above the urban canopy with the streets, it would be convenient to decrease the separation between buildings, i.e. the skimming-flow regime. In such a fashion, the air at street level would also be in contact with clean sources of air through the arch-vortex legs.
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\section{Introduction:}
Existence of dark matter (DM) particle is supported by many gravitational phenomena in astrophysics and cosmology but the nature of the dark matter
is still to be discovered. These gravitational evidences motivate the physicist to know about the dark matter (DM) particle candidates and their
interactions with ordinary matter if any. None of the running experiments are able to probe the DM candidates hence different theoretical assumptions or
models are floating in scientific community to describe their nature. DM candidate are assume to be a non-baryonic, weakly interacting and stable.
Further these DM candidates
are assumed to be weakly interacting and stable.
On the basis of DM mass and their velocities, DM candidates can be broadly classified into two categories; hot and cold dark matter.
All relativistic and super-relativistic DM particles are called hot DM whereas all non-relativistic DM particles are known as cold DM. Most part of the
DM is cold because they move with non-relativistic speed and they are capable of forming small structure galaxies rather than massive galaxies.
While hot DM can not form small structure of galaxies by the free streaming process. As we observe
small structure galaxies hence it is assumed that DM is a mixture of small amount of hot DM with predominant cold DM.
Different DM physics models needs different experimental environment for the detection of DM.
The experimental techniques proposed for dark matter detection can be broadly classified into three categories.
\begin{itemize}
\item \textbf{Direct detection:} In the direct detection technique we suggest that the DM candidates can be detected by the measurement of recoil
energy of nucleons but this detection technique becomes less sensitive for the detection of sub-GeV dark matter because in this mass range of DM
the recoil energy of nucleons
decreases below the detector threshold value
of the detector.
\item \textbf{Indirect detection:} In indirect detection, the standard model (SM) particles produced via annihilation of DM particles are studied.
\item \textbf{Collider technique:} Beam collider technique looks for the missing transverse energy in a event, for the DM detection.
\end{itemize}
In this work, we have focused on the MeV to GeV mass range of DM because a lot of work have been done for mass range 1 GeV to 10 TeV [WIMP
(weakly interactive massive particles) as a DM candidate] at direct detection experiments but these experiments have estimated null results for WIMP
\cite{79, 80, 81, 82, 83}.
Hence sub-GeV mass DM particles can be a potential candidates of DM family and
the sub-GeV DM candidates can be explored by direct detection, in neutrino laboratories.
To improve the sensitivity of DM candidates fixed target approach is considered in direct detection experiment but
one major drawback of this approach is the presence of huge neutrino
background. These particles (neutrino) can mimic the DM signatures hence for the detection of DM neutrino background reduction becomes necessary.
To mitigate the neutrino background in our DM detection work,
we have considered DUNE experiment running in beam dump mode.
Fir the first time, neutrino experiment in the beam dump mode was carried by MiniBooNE in 2014 \cite{69,8}.
The fixed target experiment in beam dump mode opens the door for the study of DM and hidden sector physics via different DM portals.
A large number of DM studies \cite{69, 66, 67, 68, 70, 71, 72, e} have focused on the kinetic mixing scenario, in which DM of hidden sector
couples with the SM particles via kinetic mixing parameter $(\epsilon)$; which kinetically mixes the dark photon $(\gamma_{D})$ mediator of hidden sector
and ordinary photon. This is one of the
possible way of interaction of DM with SM particles but there are other large number of scenarios of DM interaction with SM particles.
These scenarios needs to be probed for imposing better constraints on DM parameter space.
In one of this scenario, DM candidates of hidden sector interacts with the SM particles via a new massive vector gauge boson mediator
$(V_{B})$
of new baryonic gauge group $U(1)_{B}$. This vector gauge boson $V_{B}$
dominantly couples with the quarks and is called \enquote{leptophobic dark matter}.
In our work, using fixed target experiment in beam dump mode we have estimated the sensitivity of DUNE (Deep underground neutrino) experiment \cite{1}
for capturing the signatures of leptophobic dark matter. A highly energetic proton beam of energy 120 GeV is used to produce a
boosted dark matter beam through vector boson mediator $V_{B}$. Here three modes of DM production are taken into consideration in our analysis.
These modes are pseudosmesons decay,
bremsstrahlung and parton-level production channels. The DM signatures are captured in
near DUNE detector by looking at DM-nucleon elastic scattering.
\vspace{-3mm}
\section{Leptophobic Dark Matter model:}
The accurate measurements of CMB (Cosmic Microwave Background) provides an important information regarding the DM beyond its gravitational interactions.
In large class of models where DM is considered to be thermal relic, its cosmological abundance is calculated by correct thermal relic density of DM
($\sim$ 22\% of the energy density of the universe) which can be obtained by the measurement of s-wave annihilation cross-section
$\langle\sigma_{s}\rangle \sim 3\times10^{-26} cm^{3}/s$ during freeze-out. In our studies we are interested in DM candidates whose masses are below
a few GeV, hence bounds
imposed by the direct detection will be very mild. The most stringent constraints on these DM candidates can be imposed by the bounds on energy injection
around redshift
$z \sim 100 - 1000$, coming from the CMB.
The s-wave annihilation of DM into charged SM particles, in particular rules out DM masses below 10 GeV but by adding p-wave process into
annihilation cross-section of DM, the DM of masses less than 10 GeV can be produced.
For the correct reproduction of DM annihilation cross-section with only s-wave process, the most viable annihilation mode of DM is, its annihilation
into neutrino like states.
In leptophobic DM model, the DM can be a viable thermal relic DM candidate only if the DM annihilation occurs into the light baryonic neutrinos via
s-wave process.
In this annihilation process because of weak interaction of baryonic neutrinos with matter they are unable to ionize the hydrogen and helium gases
hence the problem of energy injection around redshift $z \sim 100 - 1000$ are
completely ignore, during and after the recombination.
In this scenario the required annihilation cross-section of thermal relic DM (of $\sim pb$ order) can be achieved.
The required value of annihilation cross-section imposes constraints on the on-shell production mode and
off-shell production mode of DM candidates. In off-shell production mode ($m_{V_{B}} < m_{\chi}$; $m_{V_{B}}$ is the mass of vector boson
mediator $V_{B}$ and
$m_{\chi}$ is the DM mass), $\chi\chi^{\dagger} \to V_{B}^{*} \to \nu_{b}\bar{\nu_{b}}$
annihilation process can achieve thermal relic DM annihilation cross-section $\langle\sigma v\rangle \sim 1 pb$ by imposing
$\alpha_{B}^{2} \sim 10^{-11}(m_{\chi}/100 MeV)^{2}$ bound. Whereas for the on-shell production mode ($m_{V_{B}} > m_{\chi}$), baryonic
fine structure constant $\alpha_{B}$ would require slightly larger value.
In our work we have considered, on-shell and off-shell production modes for the production of DM.
In both
production modes baryonic constant $G_{B} = 4\pi\alpha_{B}/m_{V_{B}}^{2}$ would necessarily be greater than the weak Fermi constant $G_{F}$,
$G_{B} \sim (10^{2} - 10^{3}) \times G_{F}$ \cite{A}.
A benchmark model for sub-GeV leptophobic scalar DM candidates which are charged under a new baryonic gauge group $U(1)_{B}$,
couples with the SM particles through a new vector boson mediator $V_{B}$ which dominantly interacts with quarks.
We have used generation independent coupling of $V_{B}$ with quarks to make model simple as it avoids the tree level flavor
changing neutral current interactions. In this consideration, to allow the renormalizable Yukawa couplings of quarks to the Higgs boson of SM,
coupling of $V_{B}$ with right and left handed quarks should be the same.
Our considered model, due to the addition of new gauge group $U(1)_{B}$ to the standard model gauge group suffers from gauge anomalies and can be
considered as
non-renormalizable effective field theory with a cutoff $\Lambda_{UV}$ \cite{q}. New states, either at or below this cutoff must be introduced
for the theory to remain consistent. The simplest choice for addition of new states are new chiral fermion which help in the cancellation
of the anomaly.
Here in this work
our focus is on GeV scale phenomenology hence the exact details of the UV completion can be ignored and we can focus our attention on low energy effective
field theory of a local $U(1)_{B}$ symmetry under which the DM is charged.
The Lagrangian of leptophobic scalar DM for low energy effective field theory can be expressed as,
\begin{equation}
\mathcal{L}_{DM} = |D_{\mu}\chi|^{2} - m_{\chi}^{2}|\chi|^{2} - \frac{1}{4}(F_{B}^{\mu\nu})^{2} + \frac{1}{2}m_{V_{B}}^{2}(F_{B}^{\mu})^{2} - \frac{\epsilon}{2}F_{B}^{\mu\nu}
F_{\mu\nu} + g_{B}F_{B}^{\mu}J_{B\mu} + ......
\end{equation}
where $F_{\mu\nu} = [\partial_{\mu}F_{\nu} - \partial_{\nu}F_{\mu}]$ is the field of SM,
$F_{B}^{\mu\nu} = [\partial^{\mu}V_{B}^{\nu} - \partial^{\nu}V_{B}^{\mu}]$ is field strength of $U(1)_{B}$,
$D = \partial - ig_{B}q_{B}V_{B}$, $g_{B}$ is baryonic gauge coupling and $q_{B}$ is baryonic charge of $U(1)_{B}$. The $J_{B}^{\mu}$
represents sum of baryonic current over all quarks species i.e. $J_{B}^{\mu} = \frac{1}{3}\sum_{i}\bar{q_{i}}\gamma^{\mu}q_{i}$. Above Lagrangian includes
baryonic coupling ($g_{B}$) scenario as well as kinetic mixing ($\epsilon$) interaction scenario.
Two different scenarios at a time are possible and may give some interesting results,
but for most cases either the baryonic coupling scenario or the kinetic mixing scenario will dominate.
Therefore for leptophobic DM scenario we have set the value of kinetic mixing parameter $\epsilon \to 0$ as we want to check constraint on DM
parameter space in the presence of baryonic coupling.
\section{Production of Leptophobic Dark Matter:}
We have focused on three production channels of leptophobic dark matter for DUNE experiment. The considered production modes are as follows,
\begin{itemize}
\item Pseudoscalar mesons decay
\item Bremsstrahlung process
\item Parton-level production
\end{itemize}
\subsection{Pseudoscalar Mesons Decay}
This production channel dominates over all production modes of dark matter at lower masses of vector boson mediator $V_{B}$. In this mode
vector boson mediator $V_{B}$ are produced via radiative decay of pseudoscalar mesons $\varrho = \pi^{0}, \eta$ \cite{66}\cite{67}.
These secondary mesons are produced from the primary interaction of $p(p)$ and $p(n)$.
\begin{equation}
p + p(n) \to X + \pi^{0}/\eta \to X + \gamma + V_{B} \to X + \gamma + \chi + \chi^{\dagger}
\label{1}
\end{equation}
The pseudoscalar mesons coupling with vector boson $V_{B}$ takes place under the gauged Wess-Zumino-Witten (WZW) lagrangian \cite{b,c,d}.
If the mass of secondary meson is greater than the mass of the vector boson mediator $V_{B}$ i.e. $m_{V_{B}} < m_{\varrho}$
or if the mass of
DM particle produced is such that $2m_{\chi} < m_{V_{B}} < m_{\varrho}$ then $V_{B}$ will be produced on-shell and further it will
decay into a pair of DM candidates.
Branching ratio of pseudoscalar mesons decay to dark matter particles is calculated using narrow width approximation \cite{67} which is equal
to the product of
mesons decay to $V_{B}$ and decay of $V_{B}$ to dark matter.
\begin{equation}
Br(\varrho \to \gamma\chi\chi^{\dagger}) = Br(\varrho \to \gamma V_{B}) Br(V_{B} \to \chi\chi^{\dagger}) \\
\label{2}
\end{equation}
\begin{equation}
Br(\varrho \to \gamma V_{B}) = 2\left(c_{\varrho}\frac{g_{B}}{g} - \epsilon\right)^{2} \left(1 - \frac{m_{V_{B}}^{2}}{m_{\varrho}^{2}}\right)^{3}
Br(\varrho \to \gamma \gamma)
\label{3}
\end{equation}
Where $g_{B}$ is baryonic coupling constant, $g$ is electromagnetic coupling constant, $m_{\varrho}$ is mass of the pseudoscalar mesons and
the value of $c_{\varrho}$ is different for different mesons i.e. $c_{\pi^{0}} = 1$ and $c_{\eta} \approx 0.11$ \cite{B}.
We have used the BMPT (Beryllium Material Proton Target) distribution fits \cite{54} in rejection sampling
to simulate the momentum and angular distribution of mesons.
For the detailed calculation check the reference \cite{a}.
\subsection{Bremsstrahlung Process}
For intermediate masses of vector boson mediator $V_{B}$, resonant vector mesons $R = \omega$ decay by the bremsstrahlung process
dominates over other decay processes.
Here the mass of $V_{B}$ is close to the mass of resonant vector mesons.
\begin{equation}
p + p(n) \to p + p(n) + V_{B} \to .... + \chi + \chi^{\dagger}
\label{4}
\end{equation}
In this process vector mesons mixes with vector boson mediator $V_{B}$.
This process generates a nearly collimated beam of vector boson $V_{B}$. The four momentum of incident proton of mass $m_{p}$ is
$q = (E_{p}, 0, 0, Q)$ where $E_{p} = Q + \frac{m_{p}^{2}}{2Q}$. The four momentum of outgoing vector boson mediator of mass $m_{V_{B}}$ is
$q_{V_{B}} = (E_{V_{B}}, q_{\perp}\cos(\phi),
q_{\perp}\sin(\phi), Q.z)$
where $E_{V_{B}} = Q.z + \frac{q_{\perp}^{2} + m_{V_{D}}^{2}}{2Q.z}$, $Q.z = q_{\parallel}$ and z is a fraction of proton beam momentum
carried away by outgoing vector boson $V_{B}$ in the direction of proton beam. Here $q_{\perp}$ and $q_{\parallel}$ are transverse and longitudinal
components
of the momenta of $V_{B}$ .\\
By Weizs$\ddot{a}$cker-Williams approximation the rate of production of vector boson $V_{B}$ production per proton is as follows \cite{56,57,e},\\
\begin{equation}
\frac{d^{2}N_{V_{B}}}{dz dq_{\perp}^{2}} =
\frac{\sigma_{pA}[2m_{p}(E_{p} - E_{V_{B}})]}{\sigma_{pA}(2m_{p}E_{p})}F^{2}_{1, N}(q^{2})f_{yx}(z, q_{\perp}^{2})
\label{14}
\end{equation}
\\
here $\sigma_{pA} = f(A)\sigma_{pp}$, $f(A)$ is a function of atomic number $A$ and $f_{yx}(z, q_{\perp}^{2})$
is a splitting weight-function of photon which relates before and after differential scattering cross section \cite{56},\\
$f_{yx}(z, q_{\perp}^{2}) = \frac{\alpha_{B}}{2\pi H}\left[\frac{1 + (1 - z)^{2}}{z} - 2z(1-z)\left
(\frac{2m_{p}^{2} + m_{V_{B}}^{2}}{H} - z^{2}\frac{2m_{p}^{4}}{H^{2}}\right)
+ 2z(1 - z)(z + (1 - z)^{2})
\frac{m_{p}^{2}m_{V_{B}}^{2}}{H^{2}} + 2z(1 - z)^{2}\frac{m_{V_{B}}^{4}}{H^{2}} \right]$
\\
here $H = q_{\perp}^{2} + (1 - z)m_{V_{B}}^{2} + z^{2}m_{p}^{2}$.
Since radiative $V_{B}$ has time-like momentum and
time-like form factor $F_{1,N}(q^{2})$ expresses off-shell mixing of vector bosons with vector mesons in appropriate kinematic region.
Baryonic vector portal considers both; proton form factor $F_{1,p}(q^{2})$ and neutron form factor $F_{1,n}(q^{2})$ \cite{58}.
These incorporates only isoscalar Breit-Wigner components \cite{58} $\omega$-like in the spacelike regime
and this form factor is not completely resolved for $\omega$.
Above 1 GeV, the form factors suppresses the rate of production of virtual bosonic mediator hence above this energy direct parton level
production dominates over other channels.
To calculate the dark photon production rate, equation [\ref{14}] must be integrated over $p_{\perp}$ and $z$ in a range that satisfies
some kinematic conditions \cite{56} expressed as,
\begin{equation}
E_{p}, E_{V_{B}}, E_{P} - E_{V_{B}} \gg m_{p}, m_{V_{B}}, \left|{q_{\perp}}\right|
\end{equation}
A range $z \in [0.2,0.8]$ and $\left|{p_{\perp}}\right| = 0.4$ for DUNE is selected as is satisfies the above kinematic conditions.
\subsection{Parton-Level Production of Dark Matter}
Above 1 GeV of vector boson mediator $V_{B}$ mass, this channel becomes significant. This process works under the narrow width approximation via
$q\bar{q} \to V_{B}$ and can be written as,
\begin{equation}
p + p(n) \to X + V_{B} \to X + \chi\chi^{\dagger}
\label{5}
\end{equation}
Dark matter pair production cross-section at parton-level can be expressed as,
\begin{equation}
\sigma(pp(n) \to X + V_{B} \to X + \chi\chi^{\dagger}) = \sigma(pp(n) \to V_{B}) Br(V_{B} \to \chi\chi^{\dagger})
\label{6}
\end{equation}
where $\sigma(pp(n) \to V_{B})$ cross-section for the production of vector boson $V_{B}$ and can be written as,
\begin{equation}
\sigma(pp(n) \to V_{B}) =
\frac{\pi}{3m_{V_{B}}^{2}}
\sum_{q}\left(\frac{g_{B}}{3} - \epsilon gQ_{q}\right)^{2} \int_{\zeta}^{1} \frac{dx}{x} \tau \left[f_{q/p}(x)f_{\overline{q}/p(n)}(\frac{\zeta}{x})
+ f_{\overline{q}/p}(x)f_{q/p(n)}(\frac{\zeta}{x})\right]
\label{7}
\end{equation}
where $\zeta = m_{V_{B}}^{2}/s$, $\sqrt{s}$ is the hadron-level
center of mass energy and $Q_{q}$ is quark charge in the unit of positron electric charge.
To calculate the cross-section of the DM production, we have used CTEQ6.6 PDFs \cite{60} and have set $Q = m_{V_{B}}$ which is allowed to vary from
$m_{V_{B}}/2$ to $2m_{V_{B}}$.
In above equation $f_{q/p(n)}(x)$ is the parton distribution function (PDF) which gives the probability of extraction of
quarks and gluons with longitudinal momentum fraction $x$ from a proton (neutron).
Details of the cross-section calculation are discussed in the references \cite{68, e, a}.
\begin{figure}[H]
\centering
\includegraphics[width=0.8\linewidth]{Events.eps}
\caption{Dark matter-nucleon scattering event plot with the variation of mediator vector boson mass for all distinct channels.
Here $m_{\chi} = 0.01$ GeV, $\epsilon$ = $0$, $\alpha_{B} = 10^{-6}$ and POT = $1.1 \times 10^{21}$.}
\label{fig2}
\end{figure}
\section{Scattering Cross-Section of Leptophobic Dark Matter:}
In the considered leptophobic dark matter model, dark matter dominantly couples with the quarks whereas it do not couple with the leptons.
In our work we have focused on the
neutral current elastic scattering of dark matter with nucleons present in the DUNE near detector. The differential cross-section of
neutral current DM-nucleon
elastic scattering which is similar to the neutrino-nucleon neutral current scattering \cite{f, g} can be expressed as,
\begin{equation}
\frac{d\sigma_{\chi N\to \chi N}}{dE_{\chi}} = \alpha_{B}q_{B}^{2} \times
\frac{\tilde{F}_{1,N}^{2}(Q^{2})A(E,E_{\chi}) + \tilde{F}_{2,N}^{2}(Q^{2})B(E,E_{\chi}) + \tilde{F}_{1,N}(Q^{2})\tilde{F}_{2,N}(Q^{2})C(E,E_{\chi})}
{(m_{\gamma_{D}}^{2} + 2m_{N}(E - E_{\chi}))^{2}(E^{2} - m_{\chi}^{2})}
\label{8}
\end{equation}
where $E$ and $E_{\chi}$ represents energy of the incoming and outgoing DM, $m_{N}$ is the mass of nucleons $(N = p, n)$
and $Q^{2} = 2m_{N}(E - E_{\chi})$
is the momentum transfer. The $\tilde{F}_{1,N}^{2}(Q^{2})$ and $\tilde{F}_{2,N}^{2}(Q^{2})$ are monopole and dipole form factors \cite{a} and the
value of kinematic functions for complex scalar DM are listed as,
$ A = 2m_{N}EE_{\chi} - m_{\chi}^{2}(E - E_{\chi});$ \\
$ B = \frac{1}{4}(E - E_{\chi})[(E + E_{\chi})^{2} - 2m_{N}(E - E_{\chi}) -4m_{\chi}^{2}];$ \\
$ C = -(E - E_{\chi})(m_{N}(E - E_{\chi}) + 2m_{\chi}^{2})$.
\section{Signal Rates:}
To simulate the DM event rates its essential to incorporate the relevant cuts of detector geometry and energy resolution
in the simulation tool. In this work the detector limitations are taken from the reference \cite{e}.
The expression for the DM events produced by the pseudoscalar mesons and vector mesons can be stated as,
\begin{equation}
N_{\chi N \to \chi N} = n_{A}\epsilon_{eff} \sum_{M=\pi^{0},\eta,\omega}\left[N_{M}Br(M \to V_{B} + ...)Br(V_{B}
\to \chi\chi^{\dagger}) \times \left(\frac{1}{N_{\chi M}} \sum_{i} L_{i} \sigma_{\chi N, i}\right)\right]
\label{10}
\end{equation}
where $n_{A}$ is atomic number density of detector material, $\epsilon_{eff}$ is detector efficiency, $N_{M}$ is total number of mesons produced in the
target, $N_{\chi M}$ is total number of DM trajectories produced by relevant production channels, $L_{i}$ is the length of DM trajectory in the detector.
The DM-nucleon elastic scattering cross-section $\sigma_{\chi N}$ is defined as,
\begin{equation}
\sigma_{\chi N}(E) = \int_{E_{\chi}^{min}}^{E_{\chi}^{max}} dE_{\chi} \sum_{N=p,n} f_{N}
\frac{d\sigma_{\chi,N}}{dE_{\chi}}
\label{11}
\end{equation}
where $E$ represents the energy of incoming dark matter, $E_{\chi}$ represents the energy of outgoing dark matter and $E_{\chi}^{min/max}$ is minimum
and maximum energy of outgoing DM which is calculated by the relevant
experimental cuts that are derived by the experimental data of nucleon recoil momentum $q$ ($q$ = $\sqrt{2m_{N}(E - E_{\chi})}$).
In baryonic vector portal we take $f_{p,n}$ = A (A is atomic number) for the elastic or quasi-elastic scattering of DM. For parton-level production
channel we have substituted $N_{V_{B}}$ (total number of produced vector bosons) in place of $N_{M}Br(M \to V_{B} + ...)$ in above equation [\ref{10}].
Total DM events are evaluated by adding the DM events produced via three different channels considered in this work.
\section{Constraints on the Leptophobic Dark Matter:}
In this paper we are checking the sensitivity of DUNE fixed target experiment in beam dump mode for leptophobic DM.
Several essential constraints that needs to be considered for checking the DUNE sensitivity for leptophobic DM are listed below in brief.
\begin{itemize}
\item \textbf{Direct Detection:} Direct detection experiments probe the cross-section of DM-nucleon elastic scattering.
The CRESST-II \cite{80}
experiment provides best limit on the recoil energy of nucleons. The CRESST-II experiment can explore the sensitivity of DM masses
below 0.5 GeV with detection threshold of nuclear recoil 307 eV. The DM-nucleon scattering cross-section for baryonic current can be expressed as,
\begin{equation}
\sigma_{\chi N} \sim \frac{16\pi\alpha_{B}^{2}\mu_{\chi,N}^{2}}{m_{V_{B}}^{4}}
\end{equation}
where, $\mu_{\chi,N}$ is the reduced mass of DM and nucleons.
\item \textbf{Constraints on masses of the vector boson mediator $V_{B}$:}
The CDF (Collider Detector at Fermilab) collaboration imposes a rigorous constraints on monojet,
$pp \to $jet + missing energy. The limit imposed on the quarks coupling $g_{u} < 0.026$ and $g_{d} < 0.04$ are largely independent
of vector boson mass $m_{V_{B}} \textless 10 GeV$ \cite{h, i}.
\item \textbf{$\pi^{0} \to \gamma + invisible$:} The Brookhaven alternating gradient synchrotron \cite{j} imposes limit on the branching ratio
$Br(\pi^{0} \to \gamma V_{B}) < 5 \times 10^{-4}$.
\item \textbf{$K^{+} \to \pi^{+}\nu\bar{\nu}$:} The $K$-decay imposes a limits on the branching ratio of $Br(K^{+} \to \pi^{+} V_{B}) < 10^{-6}$ for
$m_{V_{B}}$ = 1.8 MeV and $Br(K^{+} \to \pi^{+} V_{B}) < 7 \times 10^{-7}$ for $m_{V_{B}}$ = 100 MeV \cite{k, l}.
\item \textbf{$J/\psi \to invisible$:}
BES (Beijing Spectrometer) collaboration imposes a constraint on the branching ratio
$Br(J/\psi \to invisible) < 7\times10^{-4}$ for larger value of $m_{V_{B}}$ \cite{o}.
\item \textbf{Angular Dependence in Neutron Scattering:}
The constraints imposed by neutron scattering on the baryonic fine structure constant $\alpha_{B}$
($\alpha_{B} = \frac{g_{B}^{2}}{4\pi}$)
which couples the DM candidates and SM particles for mediator vector boson mass $m_{V_{B}} > 1$ MeV is expressed below \cite{m, n},
\begin{equation}
\alpha_{B} < 3.4 \times 10^{-11} \left(\frac{m_{V_{B}}}{MeV}\right)^{4}
\end{equation}
\end{itemize}
\iffalse
\begin{figure}[H]
\centering
\begin{minipage}[b]{0.48\textwidth}
\includegraphics[width=\textwidth]{sens.png}
\captionsetup{width=0.9\textwidth}
\caption{The contour sensitivity plot for leptophobic dark matter signatures from distinct experiments in the parameter space of $\alpha_{B} - m_{V_{B}}$
\cite{a}.}
\label{sens}
\end{minipage}
\begin{minipage}[b]{0.48\textwidth}
\includegraphics[width=\textwidth]{yield.png}
\captionsetup{width=0.9\textwidth}
\caption{The contour sensitivity plot for leptophobic dark matter signatures from distinct experiments in the parameter space of $\alpha_{B} - m_{\chi}$
\cite{61}.}
\label{yield}
\end{minipage}
\end{figure}
\fi
Existing constraints on the leptophobic DM model are shown in figure [\ref{fig1}] and [\ref{fig3}]. The plot [\ref{fig1}] shows the constraints on
the $U(1)_{B}$ model in the $\alpha_{B} - m_{V_{B}}$ parameter space while the plot [\ref{fig3}] shows the constraints on the $U(1)_{B}$ model in
the $\alpha_{B} - m_{\chi}$ parameter space for 10 MeV DM mass and $\epsilon = 0$. The other constraints plotted in figure [\ref{fig1}] and [\ref{fig3}]
are taken from
$K^{+} \to \pi^{+}\nu\bar{\nu}$,
$\pi^{0} \to \gamma + invisible$, Monojet(CDF), Neutron Scattering and $J/\psi \to invisible$ experiments.
\section{Simulation and Results:}
We have used BdNMC (Beam dump Neutrino Monte Carlo) \cite{61} simulation tool to compute the sensitivity of DUNE detector for leptophobic DM. This
simulation tool works on the direct detection technique to probe the sensitivity of the considered experiment running in beam dump mode.
The $120 GeV$ proton beam of DUNE experiment produces charged and neutral mesons after hitting the beryllium target.
These mesons are allowed to propagate through the decay pipe which is made up of steel. The neutral mesons quickly decays in the decay pipe while charged
mesons propagate further and get absorbed in the decay pipe before they decay.
The neutral mesons decay into vector mediator $V_{B}$ which further decays into a pair of DM candidates in the center of mass frame of $V_{B}$.
The total DM particles produced via distinct production channels
(visit section[3]) reaches the DUNE near detector which is a cylinder of length $5$ m, diameter $5$ m and $1$ ton ($Ar:CH_{4}::90:10$) fiducial mass
\cite{63}.
Using BdNMC simulation tool the DM event rate are estimated by imposing
production distribution function $f(p, \theta)$, specific geometric cuts \cite{63} and recoil energy cuts on nucleons
$E_{R} \in [0.1,2]$ in equation \ref{10}.
The trajectory of each DM particle which intersects the fiducial mass
of near detector are recorded in the form of the energy of DM candidates.
In our work, the sensitivity of fixed target DUNE experiment in beam dump mode is explored for leptophobic DM at DUNE near detector.
In the considered DM model,
DM candidates couples
with the SM particles via vector boson mediator $V_{B}$ of a new baryonic gauge group $U(1)_{B}$ with coupling strength $\alpha_{B}$.
\begin{figure}[H]
\centering
\includegraphics[width=1.0\linewidth]{Newleptosens.png}
\caption{The contour sensitivity plot for light dark matter signatures at DUNE experiment in the parameter space of $\alpha_{B} - m_{V_{B}}$.
These dark matter are produced from distinct channels by using 120 GeV proton beam. Here we have considered $m_{\chi} = 0.01$ GeV,
$\epsilon$ = $0$ and POT = $1.1 \times 10^{21}$. In above plot,the gray regions are excluded by existing constraints, while the yellow
contours indicate 1, 10 and 1000 events.}
\label{fig1}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=1.0\linewidth]{Newleptoyield.png}
\caption{The contour sensitivity plot for light dark matter signatures at DUNE experiment in the parameter space of $\alpha_{B} - m_{\chi}$.
These dark matter are produced from distinct channels by using 120 GeV proton beam. Here we have considered $m_{V_{B}} = 3m_{\chi}$ GeV,
$\epsilon$ = $0$ and POT = $1.1 \times 10^{21}$. In above plot,the gray regions are excluded by existing constraints, while the yellow
contours indicate 1, 10 and 1000 events.}
\label{fig3}
\end{figure}
\iffalse
\begin{figure}[H]
\centering
\begin{subfigure}{.51\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{leptosens.png}
\captionsetup{width=0.9\textwidth}
\caption{The contour sensitivity plot for light dark matter signatures at DUNE in the parameter space of $\alpha_{B} - m_{V_{B}}$.
These dark matter are produced from distinct channels by using 120 GeV proton beam. Here we have considered $m_{\chi} = 0.01$ GeV,
$\epsilon$ = $0$ and POT = $1.1 \times 10^{21}$. In above plot, blue, green and red contour lines corresponds
to 10, 100 and 1000 events.}
\end{subfigure}%
\begin{subfigure}{.51\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{leptoyield.png}
\captionsetup{width=0.9\textwidth}
\caption{The contour sensitivity plot for light dark matter signatures at DUNE in the parameter space of $\alpha_{B} - m_{\chi}$.
These dark matter are produced from distinct channels by using 120 GeV proton beam. Here we have considered $m_{V_{B}} = 3m_{\chi}$ GeV,
$\epsilon$ = $0$ and POT = $1.1 \times 10^{21}$. In above plot, blue, green and red contour lines corresponds
to 10, 100 and 1000 events.}
\end{subfigure}
\caption{The sensitivity plot of DUNE in $\alpha_{B} - m_{V_{B}}$ (left panel) and $\alpha_{B} - m_{\chi}$ (right panel) parameter space.}
\label{fig1}
\end{figure}
\fi
In the figure [\ref{fig2}] we have explored the nature of DM-nucleon scattering event with varying mass of the vector boson mediator
$m_{V_{B}}$ for a fix value
of DM mass i.e. 10 MeV. From this piece of work we observe that meson decay channel is the dominant channel for the production of DM candidates till
600 MeV mass of $V_{B}$.
For intermediate mass range of $V_{B}$ i.e. from 40 MeV to 1 GeV the DM particles are dominantly produced via resonance decay channel
whereas parton-level DM production channel is active for complete range of
$m_{V_{B}}$ but for $m_{V_{B}} > 1$ GeV the DM production is mainly due to this channel only.
In the figure [\ref{fig1}] we have presented the sensitivity of DUNE for leptophobic DM scenario in $m_{V_{B}}-\alpha_{B}$ parameter space
where $m_{V_{B}}$ is allowed to vary between 0.03 GeV to 2 GeV and baryonic fine structure constant $\alpha_{B}$ varies from $10^{-9}$ to
$10^{-3}$.
This study is performed for the DM mass 10 MeV and kinetic mixing parameter $\epsilon$ = 0.
The DM-nucleon elastic scattering event contours for 1, 10 and 1000 events are shown
in the figure [\ref{fig1}] along with other experimental constraints.
This plot shows that DUNE experiment can prob DM for baryonic fine
structure coupling below the existing bound on the
coupling value $\alpha_{B} \simeq 10^{-6}$.
This lower coupling values are present
for mass of vector boson mediator $V_{B}$ less than 200 MeV.
The figure [\ref{fig3}] shows the contour plot of DM-nucleon elastic scattering events in $m_{\chi}-\alpha_{B}$ plane
where the variation of $m_{\chi}$ is from 0.01 GeV to 1 GeV and $\alpha_{D}$ is varied from $10^{-9}$ to $10^{-3}$.
This study is performed for
kinetic mixing parameter $\epsilon$ = 0 and vector boson mass $m_{V_{B}} = 3m_{\chi}$.
The three contours of figure [\ref{fig3}] represents the 1, 10 and 1000 events in the parameter space of $m_{\chi}-\alpha_{B}$.
In this study we observe that the vector boson mediator $V_{B}$ shows weak coupling (below the existing limit of $\alpha_{B} = 10^{-6}$)
for DM masses less than 250 MeV.
In all three plots we notice a sharp peak around $m_{V_{B}} = 3m_{\chi} \sim m_{\omega} \sim 800$ MeV which can be attributed to the resonance
production via bremsstrahlung process at this point.
\section{Conclusion:}
In our work DUNE simulation study is performed for leptophobic light DM signatures via via DM-nucleon elastic scattering in the parameter space of
$m_{V_{B}}-\alpha_{B}$ and $m_{\chi}-\alpha_{B}$.
The study is based on the benchmark DM model in which DM candidates dominantly interacts with quarks via vector boson mediator $V_{B}$
of baryonic gauge group
$U(1)_{B}$. The analysis of figure [\ref{fig1}] and [\ref{fig3}] illuminates the DUNE experiment sensitivity for
leptophobic light DM imposes constraints on the coupling value $\alpha_{B}$, which is lower than the present constraint value of the coupling
$\alpha_{B} \simeq 10^{-6}$.
These lower coupling value can be explored by DUNE for vector boson mediator mass $m_{V_{B}} < 200$ MeV. Therefore DUNE experiment in beam dump mode
will be able to provide the new results for leptophobic DM.
These results will help us to understand the nature of DM and its interactions.
\begin{acknowledgement}
We are thankful to Prof. Raj Gandhi for his valuable help at all stages of work and for his comments on the manuscript.
\end{acknowledgement}\vspace{-10mm}
|
train/arxiv
|
BkiUdRk4eIZjyluAAvLg
| 5
| 1
|
\section{Proof Sketches}
Here we include the proof sketches mentioned before.
\subsection{For Implication Completeness}
\vspace{-1em}
\label{completeSketch}
\input{sections/proofs/completeSketch.tex}
\vspace{-1em}
\subsection{For Notions of other Frameworks}
\vspace{-1em}
\label{OtherFrameworksProof}
\begin{figure*}[h!]
\center
\includegraphics[width=0.95\textwidth]{images/hierarchy_author_notations}
\caption{\mbox{Our hierarchy with the mapping of the other works (Bohli's, \textcolor{red}{AnoA}, \textcolor{blue}{Hevia's}, \textcolor{yellow}{Gelernter's framework}, \textcolor{cyan}{Loopix's \ac{ACN}} and \textcolor{green}{new notions})}}
\label{fig:hierarchyold}
\end{figure*}
\input{sections/proofs/otherFrameworksSketch.tex}
\vspace{-1.5em}
\subsection{For Loopix's Notions 1}
\vspace{-1em}
\label{Loopix1Sketch}
\input{sections/proofs/Loopix1Sketch.tex}
\vspace{-1.5em}
\subsection{For Loopix's Notions 2}
\vspace{-1em}
\label{Loopix2Sketch}
\input{sections/proofs/Loopix2Sketch.tex}
\fi
\iflong
\input{sections/appendix/HierarchyAndTables.tex}
\fi
\input{sections/appendix/graphicOtherFrameworks.tex}
\end{document}
\section{Discussion} \label{discussion}
\vspace{-0.2cm}
In this section, we present the lessons learned while creating our framework.
\inlineheading{Learning about privacy goals}
The need for formal definitions is emphasized by the mapping of Loopix's privacy goals to notions as example that less formal alternatives leave room for interpretation. Further, a result like our hierarchy would be much harder to achieve without formal definitions.
These definitions allow us to point out the relation of privacy and confidentiality ($M\overline{O}-|M|$). The way we ordered the notions in the hierarchy allows easy identification the notions implying $M\overline{O}-|M|$ (the middle of the upper part). Note that any privacy notion implying $M\overline{O}-|M|$ can be broken by distinguishing the message's content. Further, nearly all those notions also imply $M\overline{O}$ and hence, all such notions can be broken by learning the message length.
Our formal definitions also enabled the comparison of existing frameworks. Excluding differences in the adversarial model, quantifications and restrictions that do not apply to all \acp{ACN}, we observe that equivalent definitions are often defined independently by the authors of the analytical frameworks. For this reason, we included the notions of the other frameworks in our hierarchy in Figure \ref{fig:hierarchyold} of Appendix \ref{hierarchyFrameworks}.
$\overline{O}$, $S\overline{O}-P$, $SM\overline{L}-P$, $R\overline{O}\{SM\overline{L}\}$ and $SM\overline{L}$ are defined (under different names) in multiple works; $S\overline{O}$ is even defined in all works.
Although previous work includes equivalent definitions, we realized that some notions are still missing. For example, we added weak notions like $(SM)\overline{L}$, $(RM)\overline{L}$ and $(SR)\overline{L}$ because they match our understanding of anonymity. Our understanding was confirmed by the analysis of Loopix' goals. Further, we defined all analogous notions for all communication parties involved (senders and receivers) as real-world application define which party is more vulnerable. For the concrete applications we refer the reader to Section \ref{sec:notionExamples}.
Consequently, we present a broad selection of privacy notions. We are aware that understanding them all in detail might be a challenging task, so we want to provide some intuitions and preferences, based on what we know and conjecture. We expect the lower part of the hierarchy to be more important for $\acp{ACN}$ as \cite{gelernter13limits} already includes an inefficiency result for $S\overline{O}$ and thus for all notions implying $S\overline{O}$ \iflong(see Figure \ref{fig:hierarchyInefficient} of the Appendix)\fi. As a first guess, we think $S\overline{O}$, if higher overhead is manageable, $SF\overline{L}$, $SM\overline{L}$, $(SM)\overline{L}$ (and receiver counterparts), $M\overline{O}-|M|$ and $(SR)\overline{L}$ are the most popular notions for \acp{ACN}.
Further, we want to add some results concerning two well-known systems to ease intuition. \cite{backes17anoa}'s analysis of Tor results in a small, but non-negligible probability to break $S\overline{O}$ and thus Tor does not achieve $S\overline{O}$ with our strict definition. Classical DC-Nets, on the other hand, do achieve at least $S\overline{O}-P$ \cite{gelernter13limits}.
\iflong
We present our selection of notions also graphically in Figure \ref{fig:hierarchyAnnotated} of the Appendix.
\fi
\inlineheading{Correcting Inconsistencies}
While the above similarities most likely stem from the influence of prior informal work on privacy goals, attempts to provide concrete mappings have led to contradictions.
The AnoA framework maps its notions to their interpretation of Pfitzmann and Hansen's terminology.
Pfitzmann and Hansen match their terminology to the notions of Hevia's framework.
This means that, notions of AnoA and Hevia's framework are indirectly mapped.
However, those notions are not equivalent.
While AnoA's sender anonymity and Hevia's sender unlinkability are both mapped to Pfitzmann and Hansen's sender anonymity, they differ: In Hevia's sender unlinkability the number of times every sender sends can leak to the adversary, but in AnoA's sender anonymity it cannot.
We belive that AnoA's sender anonymity should be called sender unobservability, which is also our name for the corresponding notion.
This follows the naming proposal of Pfitzmann and Hansen and their mapping to Hevia.
It is also more suitable because AnoA's sender anonymity can be broken by learning whether a certain sender is active, i.e. sends a real message, in the system ($u \in U_b$).
In order to achieve this notion, all senders have to be unobservable.
To verify this, we looked at how the notions of AnoA have been used.
For example in \cite{chaum16cmix} the protocol model contains an environment that lets all senders send randomly.
Hence, $U_b$ is hidden by the use of this environment.
We consider that the information that is allowed to be disclosed should instead be part of the notion and not modified by an environment.
Only then are the notions able to represent what information is protected by the protocol.
Another lesson learned by comparing privacy notions is the power of names, because they introduce intuitions.
The fact that Hevia's strong sender anonymity is equivalent to Bohli's receiver weak unlinkability seems counter-intuitive, since a sender notion is translated to a receiver notion. This might also be the reason for the incorrect mapping in \cite{bohli11relations}.
However, Bohli's receiver weak unlinkability is named this way because receivers are the ``interesting'' users, whose communication is restricted.
It does not restrict senders in any way and hence should be, in most cases, easier to break according to some information about the sender. This is why we and Hevia have classified it as a sender notion.
An analogous argument explains why Bohli's receiver weak anonymity $R/WA$ implies the restricted case of Bohli's sender strong anonymity $S/SA^\circ$.
\iflong
\inlineheading{Use and Limits}
Because there is no restriction in the use of protocol queries, the only restriction to what can be analyzed is what is modeled in the protocol model.
So, if the protocol model includes e.g. insider corruption and active behavior of the insider, like delaying or modifying of messages, those functionality can be used via protocol queries.
The same applies to timing; if the protocol model specifies that it expects protocol queries telling it, that $x$ seconds passed and the adversary gets meaningful answers after this protocol messages and only an empty answer after batch queries (because they are only processed after some time passed),
attacks on this can be analyzed.
However, it needs to be specified in the protocol model.
This model also defines the exact meaning of a batch query, whether messages of one batch are sent at the same time or in a sequence without interruption and specifies whether a synchronous or asynchronous communication model is used.
Defining the protocol model with the strongest adversary imaginable and restricting it later on with adversary classes is a way to limit the work, when analyzing against different adversary models.
We decided not to increase the complexity of the framework further by adding interfaces for dimensions of adversary models to the protocol model, i.e. adding more dedicated query types instead of the versatile protocol query. So far, our decisions for query types are driven by the related work. Differentiating between the different possible use cases of protocol queries and defining a set of adversary classes defining typical adversaries based on this is future work. One of them should allow to limit the amount of corrupted users compared to the amount or honest users or specify n-1 attacks.
Although we presented all notions that we deemed important, there might still be use cases that are not covered. With our properties as building blocks, we conjecture that it is easy to add the needed properties and use them in combination with ours. Further, for adding new notions to the hierarchy, our proofs can be used as templates.
\else
\inlineheading{Long Version \cite{longVersion}} Besides giving more technical details, we focus on making our results easier to apply for practitioners
in the long version of thus paper by presenting an analysis framework, along with a how-to-use section.
The extended version includes different parts of the adversary model, like user corruption and limiting the number of adversarial users, and discusses how typical attacks, like n-1, intersection and active attacks (e.g. delaying or dropping messages), apply to our framework.
To further simplify the proofs practitioners have to make, it allows privacy goals to be quantified by using multiple challenges or multiple challenge rows and includes results on how the limited case of challenge rows generalizes to more, such that only the limited case needs to be proven. Further, as we are aware that our strict definition of achieving a notion might not work for some practical cases, we point out the relaxed definition that allows for a non-negligible distinguishing probability.
However, none of those extensions limits or contradicts the results regarding the hierarchy of privacy notions built from observable properties that we presented here, as they work independently.
\fi
\iflong
Another possible extension is including and extending more results of the existing frameworks.
Such results are Bohli's closed hierarchy or Gelernter's inefficiency result.
Further, in Hevia's framework techniques to achieve a stronger \ac{ACN} are included.
For instance, given an \ac{ACN} achieving $SM\overline{L}$ adding a certain cover traffic creates an \ac{ACN} achieving $S\overline{O}$.
Those techniques hide certain information that is allowed to leak in the weaker but not in the stronger notion.
The proof of our Theorem \ref{the:noImpl} already includes some information that is allowed to leak in the weaker but not in the stronger notion.
Hence, it is a good starting point for finding more such techniques that help understanding and constructing \acp{ACN} better.
We make the conjecture that adapting the proofs of all these results for our framework is possible.
However, at the moment we leave proving of these results as future work.
\fi
\section{Protected Properties}
\label{aspects}
We define properties to specify which information is allowed to be disclosed to the adversary, and which must be protected to achieve a privacy notion, as mentioned in Section \ref{background}.
\begin{table}
\center
\resizebox{0.48\textwidth}{!}{%
\begin{tabular}{ c p{3.5cm} p{4.8cm} }
Symbol &Description&Translation to Game\\ \hline
$|M|$&Message Length& Messages in the two scenarios always have the same length.\\
$E_{S}$&Everything but Senders& Everything except the senders is identical in both scenarios.\\
$E_{R}/ E_{M}$&Everything but Receivers/Messages& Analogous\\
$E_{SM}$&Everything but Senders and Messages& Everything except the senders and messages is identical in both scenarios.\\
$E_{RM}/ E_{SR}$&Analogous& Analogous\\
\something &Something is sent& In every communication something must be sent ($\Diamond$ not allowed).\\
$\aleph$ &Nothing&Nothing will be checked; always true.\\
$U/U'$& Active Senders/Receivers& Who sends/receives is equal for both scenarios.\\
$Q/Q'$& Sender/Receiver Frequencies&Which sender/receiver sends/receives how often is equal for both scenarios.\\
$|U|/|U'|$& Number of Senders/ Receivers& How many senders/receivers communicate is equal for both scenarios.\\
$P/P'$& Message Partitioning per Sender/Receiver& Which messages are sent/received from the same sender/receiver is equal for both scenarios.\\
$H/H'$& Sender/Receiver Frequency Histograms&How many senders/receivers send/receive how often is equal for both scenarios.\\
\end{tabular}}
\caption{Simple properties: Information about communications that may be required to remain private}
\label{tab:information}
\end{table}
\vspace{-0.3cm}
\subsection{Simple Properties}
\vspace{-0.3cm}
\todo[inline]{"We distinguish between simple and complex properties, where simple properties describe information that intuitively can be observed when communicating and that are easily modeled with the game." -- what does it mean that information can intuitively be observed when communicating?the caption of Figure 1 is a bit vague as well. Your adversary observes whatever the protocol outputs, right? So what does it mean that this information "can be observed but may be required to remain private"?
CKu: Tried to change this, but not sure if I made it better or worse...}
We distinguish between simple and complex properties, where simple properties describe an intuitive selection of observable information of communications that is easily modeled with the game. We summarize their informal meaning in Table \ref{tab:information}.
Assume an \ac{ACN} aims to hide the sender but discloses message lengths to observers. For this case, we specify the property ($|M|$) that the message length must not differ in both scenarios, as this information must not help the adversary to distinguish which scenario the challenger chose to play.
Next, we might want an \ac{ACN} to protect the identity of a sender, as well as any information about who sent a message, but deliberately disclose which messages are received by which receiver,
who the receivers are, and potentially other auxiliary information.
We hence specify a property ($E_{S}$) where only the senders differ in both scenarios\footnote{$E$ symbolizes that only this property may vary in the two submitted scenarios and everything else remains equal.}, to ensure that the adversary in our game can only win by identifying senders.
The same, of course, has to be defined for protection of the receivers ($E_{R}$) and of the messages ($E_{M}$).
Further, we might want the \ac{ACN} to protect senders and in addition also the messages, leaving the receiver and auxiliary information to be disclosed to the adversary.
This is achieved by specifying a property where only senders and messages differ in the two scenarios and everything else remains equal ($E_{SM}$).
Again, the same is specified for receivers and messages ($E_{RM}$) or senders and receivers ($E_{SR}$).
Lastly, \ac{ACN}s might allow the adversary to learn whether a real message is sent or even how many messages are sent. Therefore, we specify a property (\something) that requires real communications in both scenarios, i.e. it never happens that nothing is sent in one scenario but something is sent in the other. We ensure this by not allowing the empty communication ($\diamond$).
However, a very ambitious privacy goal might even require that the adversary learns no information about the communication at all ($\aleph$). In this case, we allow any two scenarios and check nothing.
\inlineheading{Formalizing those Simple Properties}
In the following definition all simple properties mentioned so far are formally defined.
Therefore, we use $\top$ as symbol for the statement that is always true.
\begin{definition}[Properties $|M|$, $E_{S}$, $E_{SM}$, \something, $\aleph$]\label{def:properties}
Let the checked batches be $\underline{r_0},\underline{r_1}$, that include the communications \mbox{${r_0}_j \in \{ (u_{0_j},u'_{0_j},m_{0_j},aux_{0_j}), \diamond\}$} and \mbox{ ${r_1}_j \in \{ (u_{1_j},u'_{1_j},m_{1_j},aux_{1_j}), \diamond\}$} with $ j \in \{1, \dots l\} $. We say the following properties are met, iff for all $j \in \{1, \dots l\}$:
\begin{align*}
|M|& : |{m_0}_j|=|{m_1}_j| \\[0.5em]
E_{S}&: {r_1}_j =(\mathbf{u_{1_j}},u'_{0_j},m_{0_j},aux_{0_j}) \\
E_{R}&: {r_1}_j =(u_{0_j},\mathbf{u'_{1_j}},m_{0_j},aux_{0_j}) \\
E_{M} &: {r_1}_j =(u_{0_j},u'_{0_j},\mathbf{m_{1_j}},aux_{0_j}) \\[0.5 em]
E_{SM}&: {r_1}_j =(\mathbf{u_{1_j}},u'_{0_j},\mathbf{m_{1_j}},aux_{0_j}) \\
E_{RM}&: {r_1}_j =(u_{0_j},\mathbf{u'_{1_j}},\mathbf{m_{1_j}},aux_{0_j}) \\
E_{SR} &: {r_1}_j =(\mathbf{u_{1_j}},\mathbf{u'_{1_j}},m_{0_j},aux_{0_j}) \\[0.5 em]
\text{\something} &: \Diamond \not \in \underline{r}_0 \land \Diamond \not \in \underline{r}_1\\
\text{$\aleph$}&: \top
\end{align*}
\label{SimpleProp1}
\end{definition}
\inlineheading{More Simple Properties: Active Users, Frequencies}
The properties of Definition \ref{SimpleProp1} are important to formalize privacy, but are by itself not sufficient.
Take the \ac{ACN} Tor as an example: While the set of active senders is trivially known to their ISPs and the guard nodes, we still require that the senders are unlinkable with the messages they are sending (and their receivers).
Also the message count a party is sending or receiving is an important information that we cannot formalize so far.
To formalize these properties, we use sets that capture which user sent which messages in a certain period, i.e. a batch of communications (and similarly sets to capture which user received which messages). Note that we use primes for the corresponding sets and properties of the receivers.
\begin{definition}[Sender-Message Linking]\label{def:senderMessageSet}
We define the sender-message linkings for scenario $b$ ($L'_{b_i}$ the receiver-message linkings analogous) as:
\begin{align*}
L_{b_i}:= &\{(u,\{m_1,...,m_h\}) \bigm| u \text { sent messages }m_1, \dots , m_h\\
&\text{ in batch }i\}\text{.}\\
\end{align*}
\end{definition}
The sets from Definition \ref{def:senderMessageSet} easily allow to identify who an active sender in this batch was and how often each of those sent something:
\begin{definition}[Active Sender Set, Frequency Set]\label{UbQb}
Let the current batch be the $k$-th one.
For $ b \in \{0,1\}$ $U_b,Q_b$ ($U_b',Q_b'$ for $L_b'$) are defined as:
\begin{align*}
U_b &:= \{ u \bigm| (u,M) \in L_{b_k}\}\hfill\\
Q_b &:= \{ (u,n) \bigm| (u,M) \in L_{b_k}\land |M|=n\}\\
\end{align*}
\end{definition}
Recall that we currently define properties for ACNs that allow the adversary to learn which senders are active at different times, or the number of messages they send during some periods while they hide some other properties (e.g. which messages they have sent).
Hence with the respective sets for active users and user frequencies defined, we only need to request that they are equal in both scenarios:
\begin{definition}[Properties $U$, $Q$, $|U|$]
We say that the properties $U,Q,|U|$ ($U',Q', |U'|$ analogous) are met, iff:
\[U: U_0=U_1 \hspace{1,5em} Q: Q_0=Q_1 \hspace{1,5em} |U|: |U_0|=|U_1| \]
\end{definition}
\inlineheading{More Simple Properties: Message Partitions, Histograms}
Other interesting properties are which messages stem from the same sender or how many senders sent how many messages. If the adversary knows which messages are sent from the same sender, e.g. because of a pseudonym, she might be able to combine information from all of them and identify the sender. If she knows how many senders sent how many messages, she knows the sender activity and hence can make conclusions about the nature of the senders.
As before, we introduce auxiliary variables to formally define these two properties.
We use $M_{b,I}$ to denote the collection of messages that has been sent by the same sender (e.g. linked by a shared pseudonym) in a set of batches, and $M_{b,I,n}$ to denote the union of all these sets of the cardinality $n$.
The equality of the properties in the two scenarios has to pertain throughout all comparable batches in the scenarios. Otherwise the inequality would help the adversary to distinguish the scenarios without learning the protected information e.g. identifying the sender.
\begin{definition}[Multi-Batch Message Linkings]\label{def:messageSets} Let the current batch be the $k$-th, \mbox{$\mathcal{K}:=\{1, \dots, k\}$}, $\mathcal{P}(\mathcal{K})$ the power set of $\mathcal{K}$ and $\mathcal{U}$ the set of all possible senders ($\mathcal{U'}$ receivers).
For $b \in \{0,1\}$ and $I \in \mathcal{P}(\mathcal{K})$: We define ($M'_{b,I}, M'_{b,I,n}$ for $L'_{b_i}$)
\begin{itemize}
\item the multi-batch-message-sender-linking:\\ \mbox{$M_{b,I} := \cup_{u\in \mathcal{U}} \{ \cup_{i \in I}\{M| (u,M)\in L_{b_i}\}\}$} and
\item the cardinality restricted multi-batch-message-sender-linking: $M_{b,I,n}:=\{ M \in M_{b,I} \bigm| |M|=n \}$.
\end{itemize}
\end{definition}
As before we define auxiliary variables capturing the information that we want to be equal in both scenarios: We define ordered sets specifying which messages are sent from the same user for any set of batches (Message Partition $P_b$) and how many users sent how many messages for any set of batches (Histogram $H_b$). Therefore, we use a slightly unusual notation: For any set Z, we use $(Z_i)_{i\in\{1,\dots, k\}}$ to denote the sequence $(Z_1, Z_2, \dots, Z_k)$ and $\overrightarrow{\mathcal{P}}(Z)$ to denote a sorted sequence of the elements of the power set\footnote{For brevity we use $\in$ to iterate through a sequence.} of $Z$.
\begin{definition}[Message partitions, Histograms]\label{def:properties}
Consider the $k$-th batch, \mbox{$\mathcal{K}:=\{1, \dots, k\}$}.
For $ b \in \{0,1\}$ $P_b, H_b$ ($P_b',H_b'$ analogous) are defined as:
\allowdisplaybreaks
\begin{align*}
P_b &:= (M_{b,I})_{ I \in \overrightarrow{\mathcal{P}}(\mathcal{K})}\\
H_b &:= (\{(n, i) \bigm| i= |M_{b,I,n}|\})_{ I \in \overrightarrow{\mathcal{P}}(\mathcal{K})}\\
\end{align*}
\vspace{-0.5cm}
Further, we say that properties $P,H$ ($P',H' $ analogous) are met, iff:
\begin{align*}
P&: P_0=P_1 \hspace{3em} H: H_0=H_1\\
\end{align*}
\end{definition}
\subsection{Complex Properties}
\label{sec:ComplexProperties}
\vspace{-0.3cm}
So far we defined various properties to protect senders, messages, receivers, their activity, frequency and grouping of messages.
However, this does not suffice to formalize several relevant privacy goals, and we hence introduce complex properties.
\inlineheading{Learning Sender and Receiver}
Consider that one aims to hide which sender is communicating with which receiver.
This goal is already used in early \acp{ACN} like the classical Mix-Nets~\cite{chaum81untraceable} and also in Tor~\cite{dingledine04tor}. Therefore, we want the adversary to win the game only if she identifies both: sender and receiver of the same communication.
An intuitive solution may be to model this goal by allowing the adversary to pick different senders and receivers ($E_{SR}$) in both scenarios (see Fig. \ref{fig:senderReceiver} (a) for an example).
This, however, does not actually model the privacy goal: by identifying only the sender or only the receiver of the communication, the game adversary could tell which scenario was chosen by the challenger.
We hence have to extend the simple properties and introduce scenario \emph{instances} to model dependencies.
\inlineheadingTwo{Scenario instances}
We now require the adversary to give alternative instances for both scenarios (Fig. \ref{fig:senderReceiver} (b)). The challenger chooses the scenario of the challenge bit, which is picked randomly for every game, and the instance according to the instance bit, which is picked randomly for every challenge.
Formally, we replace step 2- 5 of the game with the following steps:
\begin{description}
\item[2.] $\mathcal{A}$ sends a batch query, containing $\underline{r}^0_{0}$, $\underline{r}^1_{0}$, $\underline{r}^0_{1}$and $\underline{r}^1_{1}$ to $Ch$.
\item[3.] $Ch$ checks if the query is valid according to the analyzed notion $X$.
\item[4.] If the query is valid and $Ch$ has not already picked an instance bit $a$ for this challenge, $Ch$ picks $a \in \{0,1\}$ randomly and independent of $b$. Then it inputs the batch corresponding to $b$ and $a$ to $\Pi$.
\item[5.] $\Pi$'s output $\Pi(\underline{r}^a_{b})$ is forwarded to $\mathcal{A}$.
\end{description}
This allows us to model the goal that the adversary is not allowed to learn the sender and receiver: We allow the adversary to pick two sender-receiver pairs, which she uses as instances for the first scenario.
The mixed sender-receiver pairs then have to be provided as instances for the second scenario (see Fig. \ref{fig:senderReceiver} (b)).
We thus force the game adversary to provide alternative assignments for each scenario.
This way she cannot abuse the model to win the game by identifying only the sender or the receiver.
We call this property \textit{Random Sender Receiver} $R_{SR}$.
This complex property still is not sufficient to model the situation in, for example, Tor:
The adversary can distinguish the scenarios without learning who sent to whom, just by learning which senders and which receivers are active.
We hence further restrict the adversary to pick instances where both senders and both receivers are active by defining property \textit{Mix Sender Receiver} $M_{SR}$.
Here, the adversary picks two instances for $b=0$ where her chosen sender-receiver pairs communicate and two for $b=1$ where the mixed sender-receiver pairs communicate.
The two instances simply swap the order in which the pairs communicate (Fig. \ref{fig:senderReceiver} (c)).
This way, we force the adversary to provide alternative assignments for each scenario where both suspected senders and both suspected receivers are active.
This combination prevents the adversary to win the game without learning the information that the real system is actually supposed to protect, i.e. the sender-receiver pair. We call this property \textit{Mixed Sender Receiver} $M_{SR}$.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.48\textwidth]{images/complexProp.pdf}
\caption{Examples showing the general structure of communications that differ in both scenarios: a) Naive, but incorrect b) Random Sender Receiver $R_{SR}$ c) Mixed Sender Receiver $M_{SR}$}
\label{fig:senderReceiver}
\end{figure}
\inlineheading{Defining Complex Properties}
For an easier formal definition of the complex properties, we introduce \emph{challenge rows}. A challenge row is a pair of communications with the same index that differ in the two scenarios (e.g. ${r}_{0_j}, {r}_{1_j}$ with index $j$).
For complex properties the challenger only checks the differences of the challenge rows in the two scenarios.
\begin{definition}[Properties $R_{SR}$, $M_{SR}$]\label{def:complexProperties}
Let the given batches be $\underline{r}_b^a$ for instances $a \in \{0,1\}$ and scenarios $b \in \{0,1\}$,
$\mathsf{CR}$ the set of challenge row indexes,
$(u^a_0, {u'}^{\:a}_0)$ for both instances $a\in \{0,1\}$ be the sender-receiver-pairs of the first challenge row of the first scenario ($b=0$) \iflong in this challenge \fi.
Random Sender Receiver $R_{SR}$, Mixed Sender Receiver $M_{SR}$ ($R_{SM}, R_{RM}, M_{SM}, M_{RM}$ analogous) are met, iff:
\allowdisplaybreaks
\begin{align*}
R_{SR}:\quad{r_{0}^a}_{cr}&=(\mathbf{u^{a}_0}, \mathbf{{u'}^{\:a}_0}, m^1_{0_{cr}}, aux^1_{0_{cr}})~\land \\
{r_{1}^a}_{cr}&=(\mathbf{u^{a}_0}, \mathbf{ {u'}^{\:1-a}_0}, m^1_{0_{cr}},aux^1_{0_{cr}})\\[0.2em]
&\forall cr \in \mathsf{CR} , a \in \{0,1\}&
\end{align*}
\begin{align*}
M_{SR}:\quad&{r_0^a}_{cr}=(\mathbf{u^{a}_0}, \mathbf{ {u'}^{\:a}_0},m^1_{0_{cr}},aux^1_{0_{cr}})~\land \\
&{r_0^a}_{cr+1}= (\mathbf{u^{1-a}_0}, \mathbf{ {u'}^{\:1-a}_0}, m^1_{0_{cr}},aux^1_{0_{cr}})~\land \\[0.2em]
&{r_1^a}_{cr}=(\mathbf{u^{a}_0}, \mathbf{ {u'}^{\:1-a}_0},m^1_{0_{cr}},aux^1_{0_{cr}})~\land \\
&{r_1^a}_{cr+1}=(\mathbf{u^{1-a}_0},\mathbf{ {u'}^{\:a}_0}, m^1_{0_{cr}},aux^1_{0_{cr}})\\[0.2em]
&\text{for every second } cr \in \mathsf{CR} , a \in \{0,1\}&
\end{align*}
\end{definition}
\inlineheading{Linking message senders}
A last common privacy goal that still cannot be covered is the unlinkability of senders over a pair of messages (Twice Sender Unlinkability}%/ Double Sender Unobservability).
Assume a real world adversary that can determine that the sender of two messages has been the same entity.
If subsequently she discovers the identity of the sender of one of the messages through a side channel, she can also link the second message to the same individual.
\inlineheadingTwo{Stages}
To model this goal, we need two scenarios (1) both messages are sent by the same sender, and (2) every message is sent by different senders.
Further, the adversary shall pick the messages, for which she wants to decide whether they are sent from the same individual, and which other messages are sent between those two messages.
Therefore, we add the concept of \emph{stages} and ensure that only one sender sends in the challenge rows of stage 1 and in stage 2 either the same sender continues sending ($b=0$) or another sender sends those messages ($b=1$). This behavior is specified as the property \emph{Twice Sender} $T_S$.
\begin{definition}[Property $T_S$]\label{def:complexProperties}
Let the given batches be $\underline{r}_b^a$ for instances $a \in \{0,1\}$ and scenarios $b \in \{0,1\}$, $x$ the current stage,
$\mathsf{CR}$ the set of challenge row indexes,
$(u^a_0, {u'}^a_0)$ for both instances $a\in \{0,1\}$ be the sender-receiver-pairs of the first challenge row of the first scenario ($b=0$) \iflong in this challenge \fi in stage 1 and $(\tilde{u}^a_0,\tilde{u}'^a_0)$ the same pairs in stage 2.
Twice Sender $T_S$ is met, iff ($T_R$ analogous):
\begin{align*}
T_{S}:&\quad x=stage1~\land\\
&\qquad{r_0^a}_{cr}=(\mathbf{u^a_0}, {u'}^{\:0}_0,m^1_{0_{cr}},aux^1_{0_{cr}})~\land \\
&\qquad{r_1^a}_{cr}=(\mathbf{u^a_0}, {u'}^{\:0}_0,m^1_{0_{cr}},aux^1_{0_{cr}})\\[0.2em]
\mathbf{\lor} &\quad x=stage2~\land\\
&\qquad{r_0^a}_{cr}= (\mathbf{u^{a}_0}, \tilde{u}'^{\:0}_0, m^1_{0_{cr}},aux^1_{0_{cr}})~\land \\
&\qquad{r_1^a}_{cr}= (\mathbf{u^{1-a}_0}, \tilde{u}'^{\:0}_0, m^1_{0_{cr}},aux^1_{0_{cr}})\\
&\quad\forall cr \in \mathsf{CR} , a \in \{0,1\}&
\end{align*}
\end{definition}
Hence, we need to facilitate distinct stages for notions with the complex properties $T_S$ or $T_R$. Precisely, in step 2 of the game the adversary is additionally allowed to switch the stages.
Note that the above definition can easily be extended to having more stages and hence, more than two messages for which the adversary needs to decide whether they have originated at the same sender.
This set of properties allows us to specify all privacy goals that have been suggested in literature as privacy notions and additionally all that we deem important.
It is of course difficult to claim completeness, as future \acp{ACN} may define diverging privacy goals and novel observable properties (or: side-channels) may be discovered.
\iflong
\vspace{-0.3cm}
\subsubsection{Sessions}\label{sec:sessions}
\vspace{-0.3cm}
Some \ac{ACN} protocols, like e.g. Tor, use sessions. Sessions encapsulate sequences of communications from the same sender to the same receiver by using the same session identifier for them. In our formalization, the adversary can pick this identifier as part of the auxiliary information.
However, in reality the adversary often does not have this capacity. Hence, also in the formalization she should not be able to distinguish the scenarios just because she can identify that two messages are part of the same session or because the session identifier is leaked.
So, to model sessions, the auxiliary information of a communication is set to be the session ID ($sess$): $aux=sess$.
In most communications, $sess$ is a number. Only for the session notions, we require special session IDs that correspond to the current challenge $\Psi$ in all challenge rows: $Ch\Psi$. In this way, they have to be the same in both scenarios and a concrete $sess$ is only used in one stage of one challenge.
%
The session identifier that is handed to the \ac{ACN} protocol model is a random number that is generated by the challenger when a new $sess$ is seen. Hence, neither leaking (it is a random number) nor linking session identifiers (it will be picked new for every challenge and stage) will give the attacker an advantage.
We formalize this in the following definition, where we also use `$\_$' to declare that this part of a tuple can be any value.\footnote{E.g.
$\exists(u,m,\_) \in r$ will be true iff $\exists u': \exists(u,m,u') \in r$.}
\begin{definition}[Sessions]\label{def:sessions}
Let $x$ be the stage and $u^a_0,u^a_1,u'^a_0,u'^a_1$ be the senders and receivers of the first challenge row of this challenge $\Psi$ and stage in instance $a \in \{0,1\}$.
Property $sess$ is met, iff for all $a\in \{0,1\}$:
{\footnotesize
\begin{align*}
sess&: \forall (r^a_0,r^a_1)\in \mathsf{CR}(\underline{r}^a_0,\underline{r}^a_1):(r^a_0,r^a_1)=\\
&\quad (u^a_0,u'^a_0,\_,(x, Ch\Psi) ),(u^a_1,u'^a_1,\_,(x,Ch\Psi))
\end{align*}
}
\end{definition}
\vspace{-0.3cm}
As not all protocols use sessions, we allow to add sessions as an option to the notion $X$ abbreviated by $\manySess{X}$
\iflong
\subsection{Corruption} \label{sec:advCap}
Some adversary capabilities like user corruption imply additional checks our challenger has to do. In reality for most ACNs the privacy goal can be broken for corrupted users, e.g. a corrupted sender has no unobservability. Hence, we include this adversary capability into our definition for the notion. As all properties are independent from the adversary capability, we add adversary capabilities as an option, that can be more or less restricted as shown in Table \ref{tab:corruption options}.
\paragraph{Corruption}
We allow the adversary to corrupt users in some notions.
This has implications on the challenger, when a corrupt query or a batch query arrives.
\begin{table} [t]
\center
\resizebox{0.35\textwidth}{!}{%
\begin{tabular}{ l p{7cm}}
Symbol &Description\\ \hline
$X$&Adaptive corruption is allowed.\\%&Bohli\\
$\static{X}$ & Only static corruption of users is allowed.\\%& AnoA, Gelernter
$\noCorr{X}$ & No corruption of users is allowed.\\ \hlin
$X$&Corrupted users not restricted.\\
$\corrNoComm{X}$&Corrupted users are not allowed to be chosen as senders or receivers.\\
$\corrOnlyPartnerSender{X}$&Corrupted users are not allowed to be senders.\\
$\corrOnlyPartnerReceiver{X}$&Corrupted users are not allowed to be receivers.\\
$\corrStandard{X}$&Corrupted users send/receive identical messages in both scenarios.\\
\end{tabular}}
\caption{Options for corruption and for corrupted communication}
\label{tab:corruption options}
\end{table}
\inlineheadingTwo{Check on corrupt queries}
This check depends on whether the user corruption is adaptive, static, or not allowed at all.
The default case for notion $X$ is adaptive corruption, i.e.
the adversary can corrupt honest users at any time.
With static corruption $\static{X}$, the adversary has to corrupt a set of users before she sends her first batch.
The third option, $\noCorr{X}$, is that no user corruption is allowed.
We denote the set of corrupted users as $\hat{U}$.
\begin{definition}[Corruption: Check on Corrupt Query]\label{def:corruptionCorrupt}
Let $\hat{U}$ be the set of already corrupted users, $u$ the user in the corrupt query and $subsequent$ be true iff at least one batch query happened.
The following properties are met, iff:
{\footnotesize
\begin{align*}
{corr}_{static}&: subsequent \implies u \in \hat{U} \\
{corr}_{no}&: \perp \quad \quad {corr}_{adaptive}: \top\\
\end{align*}
}
\end{definition}
\vspace{-2em}
\inlineheadingTwo{Check on batch queries}
As it would not make sense to protect the privacy of corrupted users, we need to assure that the adversary cannot distinguish the scenario because the behavior of corrupted users differs. This is done by assuring equal behavior $corr$ or banning such users from communicating $\corrNoComm{corr},\corrOnlyPartnerSender{corr},\corrOnlyPartnerReceiver{corr}$.
\begin{definition}[Corruption: Check on Batch Query]\label{def:corruption}
The following properties are met, iff for all $a \in \{0,1\}$:
{\footnotesize
\begin{align*}
\corrNoComm{corr}&: \forall (u,u',m,aux) \in \underline{r}^a_0\cup \underline{r}^a_1: S \not \in \hat{U} \land R \not \in \hat{U} \\
\corrOnlyPartnerSender{corr}&: \forall (u,u',m,aux) \in \underline{r}^a_0\cup \underline{r}^a_1: u \not \in \hat{U} \\
\corrOnlyPartnerReceiver{corr}&: \forall (u,u',m,aux) \in \underline{r}^a_0\cup \underline{r}^a_1: u' \not \in \hat{U}\\
corr&: \forall \hat{u} \in \hat{U}:r^a_{0_i}=(\hat{u},\_,m,\_) \implies r^a_{1_i}=(\hat{u},\_,m,\_)\\
&\quad \quad \quad \quad \land r^a_{0_i}=(\_,\hat{u},m,\_) \implies r^a_{1_i}=(\_,\hat{u},m,\_) \\
\end{align*}
}
\end{definition}
\vspace{-0.5cm}
Of course, other adversarial capabilities can be adjusted with other parts of the model (like the corruption of other parts of the \ac{ACN} with protocol queries).
\subsection{Quantification}\label{sec:challenges}
The aspects introduced in this section are based on the game and do not translate into the real world easily. However, they are useful for the analysis, since they quantify privacy.
\inlineheading{Challenge Complexity}
\todo[inline]{Why do we really want to do this?}
To quantify how different the scenarios are allowed to be, we add the concept of \emph{challenge complexity}. Challenge complexity is measured in
\textit{Challenge rows}, the pairs of communications that differ in the two scenarios as defined earlier. $c$ is the maximal allowed number of challenge rows in the game. Additionally, we abbreviate the maximal allowed numbers of challenge rows per challenge to at most $\#cr$ as option to a notion $X$ with $X_{\mathsf{CR}_{\#cr}}$.
\example{Consider Alice using a protocol, that achieves $S\overline{O}$ for one challenge row ($S\overline{O}_{\mathsf{CR}_{1}}$), but not for two($S\overline{O}_{\mathsf{CR}_{2}}$). This means in the case that Alice only communicates once, the adversary is not able to distinguish Alice from any other potential sender Charlie. However, if Alice communicates twice the regime might distinguish her existence from the existence of some other user.\todo{intersection attacks?}}
\begin{definition}[Challenge Complexity]\label{def:challengeComplexity}
Let $\#\mathsf{CR}$ be the number of challenge rows in this challenge so far,
We say that the following property is met, iff:
\allowdisplaybreaks
\begin{align*}
\mathsf{CR}_{\#cr}:& \#\mathsf{CR} \leq \#cr
\end{align*}
\end{definition}
\inlineheading{Challenges Cardinality}
So far, we defined all aspects for one challenge. We now bound the number of challenges to $n$, as the adversary potentially gains more information the more challenges are played. While challenge complexity defines a bound on the total number of differing rows within a single challenge, cardinality bounds the total number of challenges.
Communications belonging to a challenge are identified by the challenge number $\Psi$, which has to be between 1 and $n$ to be valid. The challenge number is a part of the auxiliary information of the communication and is only used by the challenger, not by the protocol model.
This dimension of quantification can be useful for analysis, since for certain assumptions it is known, how the privacy of the $n$-challenge-case can be bounded in the privacy of the single-challenge-case (see Section \ref{advClass} for more information).
\fi
\fi
\section{Privacy Notions}
\label{notions}
Given the properties above, we can now set out to actually express intuitive privacy goals as formal privacy notions.
We start by specifying sender unobservability as an example first, to give a general definition of our privacy notions after.
Recall the first game we defined in Section \ref{background}, which corresponds to sender unobservability ($S\overline{O}$ = S(ender) $\lnot$ O(bservability)). There, in both scenarios something has to be sent, i.e. we need to specify that sending nothing is not allowed: \something. Further, both scenarios can only differ in the senders, i.e. we also need the property that everything but the senders is equal: $E_{S}$. Hence, we define sender unobservability as $S\overline{O} :=$\something$ \land E_{S}$.
\footnote{Technically $E_{S}$ already includes \something. However, to make the differences to other notions more clear, we decide to mention both in the definition.}
We define all other notions in the same way:
\begin{definition}[Notions]
Privacy notions are defined as boolean expression of the properties
according to Table~\ref{NotionsDefinition}.
\end{definition}
\begin{table}[htb]
\center
\resizebox{0.4\textwidth}{!}{%
\begin{tabular}{ l l }
Notion&\iflong Aspects\else Properties \fi \\ \hline
$(SR)\overline{L}$ &\something $ \land E_{SR} \land M_{SR}$ \\
$(SR)\overline{O}$ &\something$\land E_{SR} \land R_{SR}$ \\
$M\overline{O}$ &\something $ \land E_{M}$ \\
$M\overline{O}-|M|$ &\something $ \land E_{M} \land |M| $ \\
$M\overline{O}[M\overline{L}]$& \something$\land Q \land Q'$\\
$\overline{O}$& \something \\
$C\overline{O}$& $\aleph$\\ \hline
$S\overline{O}$& \something $\land E_{S}$\\
$S\overline{O}-|U| $& \something$\land E_{S} \land |U|$\\
$S\overline{O}-H$ &\something$\land E_{S} \land H$\\
$S\overline{O}-P $&\something $\land E_{S} \land P$\\
$SF\overline{L}$ &\something$\land E_{S} \land U$ \\
$SF\overline{L}-H$ &\something $\land E_{S} \land U \land H$\\
$SF\overline{L}-P $&\something $\land E_{S} \land U \land P$\\
$SM\overline{L}$ &\something $\land E_{S} \land Q$\\
$SM\overline{L}-P$ &\something $\land E_{S} \land Q \land P$\\
$(2S)\overline{L}$ &\something$\land E_{S} \land T_S $ \\
$R\overline{O}$ \ etc.&analogous\\ \hline
\mbox{$S\overline{O}[M\overline{O}]$}&\something $ \land E_{SM} $ \\
\mbox{$S\overline{O}[M\overline{O}-|M|]$}&\something $ \land E_{SM} \land |M| $ \\
$(SM)\overline{O}$ &\something $ \land E_{SM} \land R_{SM}$ \\
$(SM)\overline{L}$&\something $ \land E_{SM} \land M_{SM} $ \\
\mbox{$R\overline{O}[M\overline{O}-|M|]$} \ etc.&analogous \\ \hline
$\sgame{X'}$& Properties of $X'$, remove $E_{R}$ \\
\multicolumn{2}{c}{for $X' \in \{R\overline{O},$ $ R\overline{O}-|U'|,$ $R\overline{O}-H',$ $R\overline{O}-P',$ $RF\overline{L},$} \\
\multicolumn{2}{c}{$RF\overline{L}-H',$ $RF\overline{L}-P',$ $RM\overline{L},$ $RM\overline{L}-P'\}$}\\
$\rgame{X}$ &analogous \\
\iflong
\hline
$\challengeRows{X}$& Properties of $ X \land \mathsf{CR}_c$\\%) $\lor E$ \\
$\manySess{X}$&Properties of $X \land sess$\\
$\corrStandard{X}$, $\corrNoComm{X}$ \ etc.& $\land corr$, $\land \corrNoComm{corr}$ \ etc.\\%\footnote{for all corruption options as defined in Table~\ref{tab:corruption options}} \\
\fi
\end{tabular}}
\caption{Definition of the notions\iflong for all corruption options as defined in Table~\ref{tab:corruption options}\fi. A description of simple properties was given in Table~\ref{tab:information}. }
\label{NotionsDefinition}
\end{table}
\iflong
Modeling the notions as a game, the respective challenger will check all aspects of the adversary's queries.
A complete description of the challenger can be found in Appendix \ref{sec:challenger}.
Further, an example of how the definitions of aspects can be represented by using a challenge specific state, which the challenger maintains, is shown in Algorithm \ref{Challenger} and \ref{calcNewState} in Appendix \ref{pseudocode}.
\else
Modeling the notions as a game, the respective challenger will check all aspects of the adversary's queries.
A complete description of the challenger can be found in Appendix \ref{sec:challenger}.
\fi
\section{On the Choice of Notions}
\label{sec:choiceNotions}
The space of possible combinations of properties, and hence of conceivable privacy notions, naturally is large.
Due to that, we verified our selection of privacy goals by finding exemplary use cases.
Additionally, we approve the choice and the applicability of our definition by analyzing the privacy goals of Loopix, an \ac{ACN} that was recently published.
We additionally verify that our privacy notions include those of previous publications that suggest frameworks based on indistinguishability games, and we provide a complete mapping in Section \ref{sec:ComparingFrameworks}.
\iflong
\else
\fi
\subsection{Exemplary Use Cases for the Notions}
\label{sec:notionExamples}
We illustrate our notions by continuing the example of an activist group trying to communicate in a repressive regime, although our notions are applicable in general.
Recall the general idea of an indistinguishability game from the examples in Section \ref{background}:
To prove that an \ac{ACN} hides certain properties, whatever is allowed to be learned in the actual \ac{ACN} must not help a game adversary to win, such that she is forced to win the game solely based on properties that are required to remain hidden.
Therefore, the information allowed to be disclosed cannot be used in the game and hence has to be kept identical in both scenarios.
Before giving examples, we want to order the notions. Hence, we group them semantically.
Our resulting clusters are shown as gray boxes in Figure \ref{fig:hierarchyColored}.
Horizontally, we categorize notions that focus on receiver or sender protection (Receiver Privacy Notions or Sender Privacy Notions, respectively) or treat both with the same level of importance (Impartial Notions).
Inside those categories we use clusters concerning the general leakage type: Both-side Unobservability means that neither senders, nor receivers or messages should be leaked.
Both-side Message Unlinkability means that it should neither be possible to link senders nor receivers to messages. In Sender Observability, the sender of every communication can be known, but not the message she sends or to whom she sends (Receiver and Message Observability analogous). In Sender-Message Linkability, who sends which message can be known to the adversary (Receiver-Message and Sender-Receiver Linkability analogous).
\iflong
\begin{table} [thb]
\center
\resizebox{0.4\textwidth}{!}{%
\begin{tabular}{ c p{6cm} }
Usage&Explanation\\ \hline
$D \in \{S,R,M\}$& Dimension $\in \{$Sender, Receiver, Message$\}$\\
Dimension $D$ not mentioned& Dimension can leak \\
Dimension $D$ mentioned &Protection focused on this dimension exists\\ \hline
$D \overline{O}$& not even the participating items regarding D leak,(e.g. $S\overline{O}$: not even $U$ leaks)\\
$DF \overline{L}$& participating items regarding D can leak, but not which exists how often (e.g. $SF\overline{L}$: $U$leaks, but not $Q$)\\
$DM \overline{L}$& participating items regarding D and how often they exist can leak ( e.g. $SM\overline{L}$: $U,Q$ leaks)\\ \hline
$X -Prop, $& like X but additionally Prop can leak\\
$Prop \in \{|U|,H,P,|U'|, H',P', |M| \}$&\\ \hline
$(D_1 D_2)\overline{O}$& uses $R_{D_1 D_2}$; participating items regarding $D_1,D_2$ do not leak, (e.g. $(SR)\overline{O}$: $R_{SR}$)\\
$(D_1 D_2)\overline{L}$& uses $M_{D_1 D_2}$; participating items regarding $D_1,D_2$ can leak, (e.g. $(SR)\overline{L}$: $M_{SR}$)\\
$(2D)\overline{L}$& uses $T_{D}$; it can leak whether two participating item regarding $D$ are the same, (e.g. $(2S)\overline{L}$: $T_{S}$)\\ \hline
$\overline{O}$&short for $S \overline{O} R\overline{O} M\overline{O} $\\
$M\overline{O}[M\overline{L}]$& short for $ M\overline{O}(SM\overline{L}, RM\overline{L})$\\
$S\overline{O}\{X\}$& short for $S\overline{O} M\overline{O} X$\\
$D_1 X_1[ D_2 X_2]$& $D_1$ is dominating dimension, usually $D_1$ has more freedom, i.e. $X_2$ is a weaker restriction than $X_1$ \\ \hline
$C\overline{O}$& nothing can leak (not even the existence of any communication)\\
\end{tabular}}
\caption{Naming Scheme}
\label{Tab:NamingScheme}
\end{table}
We first want to explain our naming scheme, which we also summarize in Table \ref{Tab:NamingScheme}. Our notions consider three dimensions: senders, messages and receivers.
Each notion restricts amount of leakage on each of those dimensions. However, if one dimension can be leaked to the adversary, we do not mention this type.
We use $\overline{O}$, short for unobservability, whenever the set of such existing items of this dimension cannot be leaked to the adversary.
E.g. $S\overline{O}$ cannot be achieved if the set of senders $U$ is leaked.
We use some form of $\overline{L}$, short for unlinkability, if $U$ can be leaked, but some other property related to the item cannot.
E.g. we use $SF\overline{L}$ if the frequency $Q$ cannot be leaked and $SM\overline{L}$, if $Q$ can be leaked, but not the sender-message relation.
With a ``$- Prop$'' we signal that the property $Prop$ can additionally leak to the adversary. We distinguish those properties from $U$ and $Q$ used before as they give another leakage dimension (as illustrated later in the hierarchy).
Further, we use $(SR)\overline{O}$ to symbolize that if not only one set, but both sets of senders and receivers ($U$ and $U'$) are learned the notion is broken. Analogously, in $(SR)\overline{L}$ both sets can be learned but the linking between sender and receiver cannot.
For the last missing complex property, we use $(2S)\overline{L}$ to symbolize that two senders have to be linked to be the same identity to break this notion.
For readability we also add some abbreviations:
We use $\overline{O}= S \overline{O} R\overline{O} M\overline{O} $ to symbolize unobservability on all three types and we summarize the remaining types in $ M\overline{O}(SM\overline{L}, RM\overline{L})$ to $M\overline{O}[M\overline{L}]$. $C\overline{O}$ symbolizes the notion in which nothing is allowed to leak.
Further, we use curly brackets to symbolize that the message cannot be leaked $ S\overline{O}\{X\}= S\overline{O} M\overline{O} X$ and we also try to put the in our understanding non dominating part of the notion in brackets $S\overline{O} M\overline{O}= S\overline{O}[M\overline{O}]$.
\else
Table \ref{Tab:NamingScheme} of Appendix \ref{app:summaryNamingScheme} summarizes our naming scheme.
\fi
\vspace{-0.3cm}
\subsubsection{Impartial Privacy Notions}
\vspace{-0.2cm}
These notions treat senders and receivers equally.
\inlineheading{Message Observability}
Messages' content can be learned in notions of this group, because messages are not considered confidential. Because the adversary in reality can learn the content, we have to avoid that she trivially wins the game by choosing different content.
Hence, such notions use the property that the scenarios are identical except for the senders and receivers ($E_{SR}$) to ensure that the messages are equal in both scenarios.
\example{One person of the activist group is already well known and communication with that person leads to persecution of Alice.}
Alice needs a protocol that hides whether a certain sender and receiver communicate with each other; cf. Section \ref{sec:ComplexProperties} motivation of the complex property $M_{SR}$.
The resulting notion is \emph{Sender-Receiver Pair Unlinkability \ ($(SR)\overline{L}$)}.
\exampleCont{Only few people participate in the protocol. Then, already using the protocol to receive (send) something, when the well known activist is acting as sender (receiver) threatens persecution. }
Alice needs a protocol that hides whether a certain sender and receiver actively participate at the same time or not; cf. Section \ref{sec:ComplexProperties} motivation of the complex property $R_{SR}$.
The resulting notion is \emph{Sender-Receiver Unobservability\ ($(SR)\overline{O}$)}.
\inlineheading{Sender-Receiver Linkability (Message Confidentiality)}
Senders and receivers can be learned in notions of this group, because they are not considered private.
Hence, such notions include the property that the scenarios are identical, except for the messages ($E_{M}$) to ensure that the sender-receiver pairs are equal in both scenarios.
\example{Alice wants to announce her next demonstration. (1) Alice does not want the regime to learn the content of her message and block this event. (2) Further, she is afraid that the length of her messages could make her suspicious, e.g. because activists tend to send messages of characteristic length.}
In (1) Alice needs a protocol that hides the content of the messages. However, the adversary is allowed to learn all other attributes,
especially also the length of the message.
Modeling this situation, the scenarios may differ solely in the message
content, all other attributes have to be identical in both scenarios, as
they may not help the adversary distinguish between them.
Beyond the above described $E_{M}$, we thus also have to request that the
length of the messages $|M|$ is identical in both scenarios.
The resulting notion is \emph{Message Unobservability leaking Message Length\ ($M\overline{O}-|M|$)}\footnote{We stick to our naming scheme here, although we would commonly call this confidentiality.}.
In the second case (2), the protocol is required to hide the length of the message.
The length of the messages thus may differ in the two scenarios, as the
protocol will need to hide this attribute. Hence, we remove the restriction that the message length has to be equal in both scenarios $|M|$ from the above notion and end up with \emph{Message Unobservability\ $M\overline{O}$}.
\iflong
\inlineheading{Both-Side Message Unlinkability}
Notions of this group are broken if the sender-message or receiver-message relation leaks.
\example{The activists know that their sending and receiving frequencies are similar to regime supporters and that using an \ac{ACN} is in general not forbidden, but nothing else. Even if the message's content and length ($M\overline{O}$) and the sender-receiver relationship ($(SR)\overline{L}$) is hidden, the regime might be able to distinguish uncritical from critical communications, e.g. whether two activists communicate ``Today'' or innocent users an innocent message. In this case, the regime might learn that currently many critical communications take place and improves its measures against the activists.}
In this case, the activists want a protocol that hides the communications, i.e. relations of sender, message and receiver. However, as using the protocol is not forbidden and their sending frequencies are ordinary, the adversary can learn which users are active sender or receiver and how often they sent and receive. Modeling this, the users need to have the same sending and receiving frequencies in both scenarios $Q,Q'$, since it can be learned. However, everything else needs to be protected and hence, can be chosen by the adversary. This corresponds to the notion \emph{\heviaULLong \ ($M\overline{O}[M\overline{L}]$)}.
\fi
\inlineheading{Both-Side Unobservability}
Even the activity of a certain sender or receiver is hidden in notions of this group.
\exampleCont{It is a risk for the activists, if the regime can distinguish between two leading activists exchanging the message ``today'' and two loyal regime supporters exchanging the message ``tomorrow''.}
In this case, Alice wants to disclose nothing about senders, receivers, messages or their combination . However, the adversary can learn the total number of communications happening in the \ac{ACN}. Modeling this, we need to assure that for every communication in the first scenario exists one in the second. We do this prohibiting the use of the empty communication with property \something. This results in the notion \emph{Unobservability \ ($\overline{O}$)}.
\example{The regime knows that a demonstration is close, if the total number of communications transmitted over this protocol increases. Then it prepares to block the upcoming event.}
To circumvent this, Alice needs a protocol that additionally hides the total number of communications. Modeling this, we need to allow the adversary to pick any two scenarios. Especially, also using the empty communication $\Diamond$ is allowed. This is represented in the property that nothing has to be equal in the two scenarios: $\aleph$\ and results in the notion \emph{Communication Unobservability \ ($C\overline{O}$)}.
Note, that this is the only notion, where the existence of a communication is hidden. All other notions include \something\ and hence do not allow for the use of the empty communication.
\subsubsection{Sender (and Receiver) Privacy Notions}
\vspace{-0.2cm}
These notions allow a greater freedom in picking the senders (resp.
receivers: analogous notions are defined for receivers.).
\inlineheading{Receiver-Message Linkability}
The receiver-message relation can be disclosed in notions of this group.
Hence, such notions include the property that the scenarios are identical except for the senders ($E_{S}$) to ensure the receiver-message relations are equal in both scenarios.
In \emph{Sender-Message Unlinkability \ ($SM\overline{L}$)} additionally the total number of communications and how often each user sends can be learned.
However, especially who sends which message is hidden.
In \emph{Sender-Frequency Unlinkability } \emph{($SF\overline{L}$)} additionally the set of users and the total number of communications can be disclosed.
However, especially how often a certain user sends is hidden, since it can vary between the two scenarios.
In \emph{Sender Unobservability \ ($S\overline{O}$)}, the total number of communications can additionally be disclosed.
However, especially the set of active senders $U_b$ is hidden.
If a notion further includes the following abbreviations, the following information can be disclosed as well:
\begin{itemize}
\item \emph{with User Number Leak} ($-|U|$): the number of senders that send something in the scenario
\item \emph{with Histogram Leak} ($-H$): the histogram of how many senders send how often
\item \emph{with Pseudonym Leak} ($-P$): which messages are sent from the same user
\end{itemize} \vspace*{-\baselineskip}
\exampleSpec{Alice is only persecuted when the regime can link a message with compromising content to her {\normalfont -- she needs a protocol that at least provides $SM\overline{L}-P$.}
However, since such a protocol does not hide the message content, the combination of all the messages she sent, might lead to her identification. {\normalfont Opting for a protocol that additionally hides the message combination ($P$), i.e. provides $SM\overline{L}$, can protect her from this thread.} \\
Further, assuming most users sent compromising content, and Alice's message volume is high, the regime might easily suspect her to be the origin of some compromising messages even if she is careful that the combination of her messages is not reidentifying her {\normalfont - she needs a protocol that does not disclose her sending frequencies ($Q$) although the combination of her messages ($P$) might be learned, i.e. achieving $SF\overline{L}-P$.}
However, Alice might fear disclosing the combination of her messages {\normalfont - then she needs a protocol achieving at least $SF\overline{L}-H$, which hides the frequencies ($Q$) and the message combination ($P$), but discloses the sending histogram, i.e. how many people sent how many messages ($H$).}
However, if multiple activist groups use the \ac{ACN} actively at different time periods, disclosing the sending histogram $H$ might identify how many activist groups exist and to which events they respond with a more active communication {\normalfont - to prevent this she needs a protocol that hides the frequencies $Q$ and the histogram $H$, i.e. provides $SF\overline{L}$.} \\
Further, not only sending a certain content, but also being an active sender (i.e. being in $U$) is prosecuted {\normalfont she might want to pick a protocol with at least $S\overline{O}-P$.
Again if she is afraid that leaking $P$ or $H$ together with the expected external knowledge of the regime leads to identifying her, she picks the corresponding stronger notion.}
If the regime knows that mostly activists send in the \ac{ACN} and learns that the number of active senders is high, it blocks the \ac{ACN}.
{\normalfont In this case at least $S\overline{O}$ should be picked to hide the number of senders ($|U|$).}}
\example{ For the next protest, Alice sends two messages: (1) a location, and (2) a time. If the regime learns that both messages are from the same sender, they will block the place at this time even if they do not know who sent the messages.}
Alice then needs a protocol that hides whether two communications have the same sender or not. We already explained how to model this with complex property $T_{S}$ in Section \ref{sec:ComplexProperties}.
The resulting notion is Twice Sender Unlinkability}%/ Double Sender Unobservability ($(2S)\overline{L}$).
\iflong
\inlineheading{Receiver Observability}
In notions of this group the receiver of each communication can be learned.
Hence, such notions include the property that the scenarios are equal except for the senders and messages ($E_{SM}$) to ensure that they are equal in both scenarios.
\example{Consider not only sending real messages is persecuted, but also the message content or any combination of senders and message contents is exploited by the regime. If the regime e.g. can distinguish activist Alice sending ``today'' from regime supporter Charlie sending ``see u'', it might have learned an information the activists would rather keep from the regime. Further, either (1) the activists know that many messages of a certain length are sent or (2) they are not sure that many messages of a certain length are sent.
In case (1), Alice needs a \ac{ACN}, that hides the sender activity, the message content and their combination. However, the adversary can especially learn the message length. Modeling this, beyond the above described $E_{SM}$, the message lengths have to be equal $|M|$. This results in the notion \emph{Sender Unobservability with Message Unobservability leaking Message Length} ($S\overline{O}[M\overline{O}-|M|]$).
Note that in $S\overline{O}[M\overline{O}-|M|]$ the properties of $M\overline{O}-|M|$ are included and further the senders are allowed to differ in the two scenarios.
The second case (2) requires a protocol that additionally hides the message length. Hence, in modeling it we remove the property that the message lengths are equal $|M|$ from the above notion. This results in \emph{Sender Unobservability with Message Unobservability} ($S\overline{O}[M\overline{O}]$).
\example{ Alice's demonstration is only at risk if the regime can link a message with a certain content to her as a sender with a non negligible probability.}
Then at least \emph{Sender-Message Pair Unlinkability\ ($(SM)\overline{L}$)}, which is defined analogous to $(SR)\overline{L}$ is needed.
\exampleCont{However, $(SM)\overline{L}$ only allows Alice to claim that not she, but Charlie sent a critical message $m_a$ and the regime cannot know or guess better. Now assume that Dave is also communicating, then the regime might be able to distinguish Alice sending $m_a$, Charlie $m_c$ and Dave $m_d$ from Alice sending $m_d$, Charlie $m_a$ and Dave $m_c$. In this case, it might not even matter that Alice can claim that Charlie possibly sent her message. The fact that when comparing all three communications that possibly happened, Alice is more likely to have sent the critical message $m_a$ means a risk for her.}
To circumvent this problem Alice needs a protocol that not only hides the difference between single pairs of users, but any number of users. Modeling this, instead of the complex property $M_{SM}$, we need to restrict that the active senders' sending frequencies are equal.
\example{In another situation our activists already are prosecuted for being a sender while a message with critical content is sent. }
In this case at least \emph{Sender-Message Pair Unobservability\ ($(SM)\overline{O}$)}, which is defined analogous to $(SR)\overline{O}$ is needed.
Analogous notions are defined for receivers.
\inlineheading{Both-Side Unlinkability}
As explained with the example before in the case that Alice does not want any information about senders, receivers and messages or their combination to leak, she would use $\overline{O}$. However, the privacy in this scenario can be tuned down, if she assumes that the regime does not have certain external knowledge or that the users are accordingly careful. As explained for the Sender Notions with Receiver-Message Linkability before, in this case we might decide to allow $U', |U'|,Q',H',P'$ to leak.
If a notion $X \in \{R\overline{O}, R\overline{O}-|U'|, R\overline{O}-H', R\overline{O}-P',RF\overline{L},RF\overline{L}-H',RF\overline{L}-P',RM\overline{L},RM\overline{L}-P'\}$ is extended to \emph{Sender Unobservability by X} \emph{($ \sgame{X}$)}, the leaking of the sender-message relation is removed.
This is done by removing $E_{R}$.
Since the attacker now has a greater degree of freedom in choosing the senders and is (if at all) only restricted in how she chooses the receivers and messages, this is a special strong kind of Sender Unobservability.
Analogous notions are defined for receivers.\footnote{Note that $\sgame{R\overline{O}}=\rgame{S\overline{O}}=\overline{O}$.}
\subsubsection{Options}
Options, which can be combined arbitrarily to all so far explained notions, are sessions, corruption and challenge complexity. Their relation to the real world resp. to easier analysis can be found in Sections \ref{sec:sessions}, \ref{sec:advCap} and \ref{sec:challenges}.
\else
Due to the page limits the examples for the remaining notions can be found in Appendix \ref{app:examples}.
\fi
\subsection{Model/The simple anonymity game}
\subsection{Analyzing Loopix's Privacy Goals}
\label{application}
\label{sec:loopix}
\vspace{-0.2cm}
To check if we include currently-used privacy goals, we decide on a current \ac{ACN} that has defined its goals based on an existing analytical framework and which has already been analyzed: the Loopix anonymity system \cite{piotrowska17loopix}. In this section, we show that the privacy goals of Loopix map to notions we have defined (although the naming differs).
Loopix aims for Sender-Receiver Third-Party Unlinkability, Sender online Unobservability and Receiver Unobservability.
\inlineheading{Sender-Receiver Third-Party Unlinkability}
Sender-Receiver Third-Party Unlinkability means that an adversary cannot distinguish scenarios where two receivers are switched:
\begin{quote}
``The senders and receivers should be unlinkable by any unauthorized party. Thus, we consider an adversary that wants to infer whether two users are communicating. We define \emph{sender-receiver third party unlinkability} as the inability of the adversary to distinguish whether $\{S_1\rightarrow R_1, S_2 \rightarrow R_2\}$ or $\{S_1 \rightarrow R_2, S_2 \rightarrow R_1\}$ for any concurrently online honest senders $S_1,S_2$ and honest receivers $R_1,R_2$ of the adversary's choice.'' \cite{piotrowska17loopix}
\end{quote}
The definition in Loopix allows the two scenarios to be distinguished by learning the first receiver.
We interpret the notion such that it is only broken if the adversary learns a sender-receiver-pair, which we assume is what is meant in \cite{piotrowska17loopix}. This means that the sender and receiver of a communication must be learned and is exactly the goal that motivated our introduction of complex properties: $(SR)\overline{L}$.
\inlineheading{Unobservability}
In sender online unobservability the adversary cannot distinguish whether an adversary-chosen sender communicates ($\{S\rightarrow\}$) or not \mbox{($\{S \centernot \rightarrow\}$)}:
\begin{quote}
``Whether or not senders are communicating should be hidden from an unauthorized third party. We define \emph{sender online unobservability} as the inability of an adversary to decide whether a specific sender $S$ is communicating with any receiver $\{S \rightarrow\}$ or not $\{S \centernot \rightarrow\}$, for any concurrently online honest sender $S$ of the adversary's choice.'' \cite{piotrowska17loopix}
\end{quote}
Receiver unobservability is defined analogously.
\iflong
\begin{table}
\center
\resizebox{0.3\textwidth}{!}{%
\begin{tabular}{ c c c }
Notion&Name&Aspects\\ \hline
$LS\overline{O}$ &Loopix's Sender Unobservavility& $ E_{\Diamond}$\\
$LR\overline{O}$ & Loopix's Receiver Unobservability&$E_{\Diamond}$\\% \lor E$ \\
$S\overline{O}'$ &Restricted Sender Unobservability& $\centernot \rightarrow \land E_{S}$ \\%\& delete $\centernot \rightarrow$\\
$R\overline{O}'$ &Restricted Receiver Unobservability& $\centernot \rightarrow' \land E_{R}$ \\
\end{tabular}}
\caption{Definition of the Loopix notions}
\label{NotionsLoopix}
\end{table}
\fi
Those definitions are open to interpretation.
On the one hand, $\{S \centernot \rightarrow\}$ can mean that there is no corresponding communication in the other scenario.
\iflong
This corresponds to our $\Diamond$ and the definition of $LS\overline{O}$ and $LR\overline{O}$ according to Table \ref{NotionsLoopix}.
\else
This corresponds to our $\Diamond$ and the definition of $LS\overline{O}$ and $LR\overline{O}$ in Appendix \ref{Loopix1Sketch}.
\fi
When a sender is not sending in one of the two scenarios, this means that there will be a receiver receiving in the other, but not in this scenario.
Hence, $LS\overline{O}$ can be broken by learning about receivers and the two notions are equal.
\iflong
These notions are equivalent to $C\overline{O}$:
\begin{theorem}
\label{loopixUO}
It holds that
\[(c, \epsilon, \delta)-C\overline{O} \Rightarrow (c, \epsilon, \delta)- LS\overline{O}_{CR_1}\text{.}\]
\[(c, \epsilon, \delta)-C\overline{O} \Leftarrow (2c, \epsilon, \delta)- LS\overline{O}_{CR_1}\text{.}\]
\end{theorem}
\begin{proofsketch}
$C\overline{O} \implies LS\overline{O}$ by definition.
For $LS\overline{O} \implies C\overline{O}$ we use the following argumentation: Given an attack on $C\overline{O}$, we can construct an attack on $LS\overline{O}$ with the same success. Assume a protocol has $LS\overline{O}$, but not $C\overline{O}$. Because it does not achieve $C\overline{O}$, there exists a successful attack on $C\overline{O}$. However, this implies that there exists a successful attack on $LS\overline{O}$ (we even know how to construct it). This contradicts that the protocol has $LS\overline{O}$.
We construct an successful attack on $LS\overline{O}$ by creating two new batches $(r_0, \Diamond )$ and $( \Diamond , r_1)$ for every challenge row $(r_0,r_1)$ in the successful attack on $C\overline{O}$.
\end{proofsketch}
\else
These notions are equivalent to $C\overline{O}$ (see Appendix \ref{Loopix1Sketch}).
\fi
On the other hand, $\{S \centernot \rightarrow\}$ can mean that sender $u$ does not send anything in this challenge.
In this case, the receivers can experience the same behavior in both scenarios and the notions differ.
\iflong
We formulate these notions as $S\overline{O}'$ and $R\overline{O}'$ according to Table \ref{NotionsLoopix}.
Therefore, we need a new property that some sender/receiver is not participating in any communication in the second scenario:
\begin{definition}[Property $\centernot\rightarrow$]
Let $u$ be the sender of the first scenario in the first challenge row of this challenge.
We say that $\centernot \rightarrow$ is fulfilled iff for all $j:$ $ u_{1_j}\neq u$.
(Property $\centernot \rightarrow'$ is defined analogously for receivers.)
\end{definition}
\begin{theorem}
\label{loopixUO2}
It holds that
\begin{align*} (c, \epsilon, \delta)-S\overline{O} &\Rightarrow (c, \epsilon, \delta)- S\overline{O}' \text{ and}\\
(c, 0, 2\delta)-S\overline{O} &\Leftarrow (c, 0, \delta)- S\overline{O}' \text{.}
\end{align*}
\end{theorem}
\begin{proofsketch}
Analogously to Theorem \ref{loopixUO}.
$\Rightarrow$: Every attack on $S\overline{O}'$ is by definition a valid attack on $S\overline{O}$.
$\Leftarrow$: Given an attack $\mathcal{A}$ on $(c, 0, 2\delta)-S\overline{O}$, where both scenarios of a challenge use all users (otherwise it would be a valid attack on $S\overline{O}'$).
Let $(r_{2_1}, \dots, r_{2_l})$ be the same batch as the second of $\mathcal{A}$ except that whenever one of the two senders of the first challenge row from the original scenarios is used, it is replaced with an arbitrary other sender (that was not used in the first challenge row of the original scenarios).
Let
$P(0|2)$ be the probability that $\mathcal{A}$ outputs 0, when the new batches are run; $P(0|0)$ when the first scenario of $\mathcal{A}$ is run and $P(0|1)$ when the second is run.
In the worst case for the attacker $P(0|2)=\frac{P(0|0)+P(0|1)}{2}$ (otherwise we would replace the scenario $b$ where $|P(0|2)-P(0|b)|$ is minimal with the new one and get better parameters in the following calculation).
Since $\mathcal{A}$ is an attack on $(c, 0, 2\delta)-S\overline{O}$, $P(0|0)> P(0|1) + 2\delta$.
Transposing and inserting the worst case for $P(0|2)$ leads to: $ P(0|0) > 2P(0|2)- P(0|0)+ 2 \delta \iff P(0|0)> P(0|2) + \delta $.
Hence, using $\mathcal{A}$ with the adapted scenario is an attack on $(c, 0, \delta)- S\overline{O}'$\footnote{An analogous argumentation works for $(c, \epsilon- \ln(0.5), \delta)-S\overline{O} \Leftarrow (c, \epsilon, \delta)- S\overline{O}'$.}.
\end{proofsketch}
\else
We formulate this notion and argue its equivalence to $S\overline{O}$ (with a change in parameters) in Appendix \ref{Loopix2Sketch}.
\fi
This is equivalent to AnoA's sender anonymity $\alpha_{SA}$. Analogously, Loopix's corresponding receiver notion is equivalent to $R\overline{O}$, which is even weaker than AnoA's receiver anonymity.
\inlineheading{Remark}
We do not claim that the Loopix system achieves or does not achieve any of these notions,
since we based our analysis on the definitions of their goals, which were not sufficient to unambiguously derive the corresponding notions.
\section{How to Use}\label{howToUse}
The framework described above offers the opportunity to thoroughly analyze \acp{ACN}. To perform such an analysis, we advice a top-down approach as follows.
\begin{enumerate}
\item In case the \ac{ACN} under analysis can be instantiated to protect against different adversaries, fix those parameters.
\item Extract capabilities of the adversary and general protocol properties from the \ac{ACN} description:
Specify the allowed \emph{user corruption}.
Is it none, static, adaptive? See Table~\ref{tab:corruption options}.
Are \emph{sessions} (channels) constructed that link messages from the same sender? See Section~\ref{sec:sessions}.
Extract all other capabilities to include them in the protocol model.
\item Simplify the \ac{ACN} protocol in a protocol model:
Generate a simplified protocol (\emph{ideal functionality}) without cryptography by assuming secure communication.
Show indistinguishability between this ideal functionality and the real-world protocol using a \emph{simulation based proof}.
Previous work~\cite{backes_provably_2012} can guide the modeling step. See Section \ref{advClass} (UC-realizability) for how the result of the simplified protocol can be transferred to the real-world protocol.
\item Extract properties based upon the input to the adversary from the ideal functionality:
Start with \emph{simple properties}, see Table~\ref{tab:information}.
What does the adversary learn from the protocol execution?
Continue with \emph{complex properties}.
See Section~\ref{sec:ComplexProperties}.
\item After mapping all properties from the protocol and adversary model to our framework, a privacy notions must be selected.
Either the description of the \ac{ACN} already specifies (in-)formally which privacy goal should be achieved, or the \ac{ACN} under analysis should be shown to achieve a certain notion.
See Table~\ref{NotionsDefinition} for an overview of our defined notions.
\item As it is easier to show that a certain notion is not fulfilled compared to show that it is fulfilled, we propose to start with the strongest notions extracted this way.
A notion is not fulfilled if the functionality (and thus the protocol) leaks a property to the adversary that he is not allowed to learn for the given notion.
If it is not obvious that a notion is not fulfilled, check if the notion can be proven for the protocol model.
The related work of Gelernter~\cite{gelernter13limits} and Backes~\cite{backes17anoa} serve as examples for such proofs.
\end{enumerate}
If the proof cannot be constructed or $\delta=1$, a weaker notion can be selected for analysis.
It might also help to consider the case of a limited number of challenge rows (see Section~\ref{sec:challenges}) and limit the adversary by using a single-challenge reducable adversary class (see Section~\ref{advClass} Adversary Class).
In case the proof goes through and yields $\epsilon = 0$ and a negligible $\delta$, the protocol was shown to achieve the selected notion as per Definition~\ref{def:achieve}. If $\epsilon > 0$ or a non negligible $\delta < 1$, the protocol achieves the selected notion as per Definition~\ref{def:achieveEpsilon}.
If the protocol supports different adversaries, the steps described above can be repeated. This typically leads to adjusting the ideal functionality or adding different adversary classes (see Section~\ref{advClass}) and thus fulfilling different properties of our framework. Analysis results under a variation of \ac{ACN} parameters may achieve different notions in our hierarchy (Figure~\ref{fig:hierarchyColored} and Figure~\ref{HierarchyExtended}). Based on our established relations between notions, analysis results can be compared for various parameters or parameter ranges, as well as against results of other \acp{ACN}.
\section{Overview}
\label{background}
\vspace{-0.5cm}
We start with an example of a use case and the corresponding implicit privacy goal, to then introduce the idea of the related indistinguishability game.
We show how such a game works and what it means for a protocol to be secure according to this goal.
Furthermore, by adopting the game we sketch how privacy goals can be formalized as notions and provide an intuition for the relations between different goals.
\example{Alice is a citizen of a repressive regime and engaged with a resistance group. Despite the regime's sanctions on distributing critical content, Alice wants to publish her latest critical findings.}
A vanilla encryption scheme would reduce Alice's potential audience and thus does not solve her problem. Hence, she needs to hide the link between the message and herself as the sender. We call this goal sender-message unlinkability.\footnote{Usually this is called sender anonymity. However, since the term sender anonymity is overloaded and sometimes also used with a slightly different meaning, we refer to it as sender-message unlinkability, as the message should not be linkable to the sender.}
\inlineheading{First attempt}
We start by presenting an easy game, that at first glance looks like the correct formalization for the goal of the example, but turns out to model an even stronger goal.
For Alice's safety, the regime should not suspect her of being the sender of a compromising message, otherwise she risks persecution. Thus, we need to show for the applied protection measure, that compared to any other sender of this message, it is not more probable that Alice is the sender. We analyze the worst case: in a group of users, let Charlie be a user for whom the probability of being the sender differs most from Alice's probability. If even these two are too close to distinguish, Alice is safe, since all other probabilities are closer. Hence, the regime cannot even exclude a single user from its suspects.
We abstract this idea into a game\footnote{Similar to indistinguishability games in cryptology \cite{goldwasser84probabilistic}.}, where the adversary aims to distinguish two ``worlds'' or scenarios.
These may only differ in the properties the protocol is required to protect, but within these restrictions the adversary can choose freely, especially the worst case that is easiest for her to distinguish (e.g. in one scenario Alice sends the message, in the other Charlie).
Fig. \ref{fig:game1} shows such a game.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.4\textwidth]{images/sampleGame.pdf}
\caption{Steps of the sample game: \textbf{1)}~adversary picks two scenarios; \textbf{2)}~challenger checks if scenarios only differ in senders; \textbf{3)}~based on random bit $b$ the challenger inputs a scenario into the \ac{ACN}; \textbf{4)}~adversary observes execution; \textbf{5)}~adversary outputs `guess' as to which scenario was executed}
\label{fig:game1}
\end{figure}
What the adversary can observe in step 4 depends on her capabilities and area of control. A weak adversary may only receive a message from somewhere, or discover it on a bulletin board. However, a stronger adversary could e.g. also observe the activity on the Internet uplinks of some parties.
The adversary wins the game if she guesses the correct scenario. If she can devise a strategy that allows her to win the game repeatedly with a probability higher than random guessing, she must have learned some information that is supposed to be protected, here the sender (e.g. that Alice is more probable the sender of the message than Charlie), since everything else was identical in both scenarios. Hence, we say that, if the adversary can find such a strategy, we do not consider the analyzed protocol secure regarding the respective privacy goal.
\inlineheading{Why this is too strong}
As argued, a protocol achieving this goal would help Alice in her use case. However, if an adversary learns who is sending any message with real information (i.e. no random bits/dummy traffic), she can distinguish both scenarios and wins the game.
As an example, consider the following two scenarios: (1) Alice and Bob send messages (2) Charlie and Dave send messages.
If the adversary can learn the active senders, she can distinguish the scenarios and win the game.
However, if she only learns the set of active senders, she may still not know who of the two active senders in the played scenario actually sent the regime-critical content.
Thus, a protocol hiding the information of who sent a message within a set of active senders is good enough for the given example.
Yet, it is considered insecure regarding the above game, since an adversary can learn the active senders.
Hence, the game defines a goal stronger than the required sender-message unlinkability.
As the \ac{ACN} in this case needs to hide the sending activity (the adversary does not know if a certain possible sender was active or not), we call the goal that is actually modeled sender unobservability.
\inlineheading{Correcting the formalization}
However, we can adjust the game of Fig. \ref{fig:game1} to model sender-message unlinkability. We desire that the only information about the communications that differs between the scenarios is who is sending which message. Thus, we allow the adversary to pick scenarios that differ in the senders, but not in the activity of the senders, i.e. the number of messages each active sender sends.
This means, we change what the adversary is allowed to submit in step 1 and what the challenger checks in step 2. So, if the adversary now wants to use Alice and Charlie, she has to use both in both scenarios, e.g. (1) Alice sends the critical message, Charlie a benign message and (2) Charlie sends the critical message, Alice the benign message.
Hence, given an \ac{ACN} where this game cannot be won, the adversary is not able to distinguish whether Alice or another active user sent the regime-critical message. The adversary might learn, e.g. that someone sent a regime-critical message and the identities of all active senders (here that Alice and Charlie are active senders). However, since none of this is sanctioned in the above example, Alice is safe, and we say such an \ac{ACN} provides sender-message unlinkability.
\inlineheading{Lessons learned}
Depending on the formalized privacy goal (e.g. sender unobservability) the scenarios are allowed to differ in certain properties of the communications (e.g. the active senders) as we have illustrated in two exemple games.
Following the standard in cryptology, we use the term \emph{privacy notion}, to describe such a formalized privacy goal that defines properties to be hidden from the adversary.
Further, the games used to prove the privacy notions only differ in how scenarios can be chosen by the adversary and hence what is checked by the challenger.
This also holds for all other privacy notions; they all define certain properties of the communication to be private and other properties that can leak to the adversary.
Therefore, their respective games are structurally identical and can be abstracted to define one general game, whose instantiations represent notions.
We explain and define this general game in Section \ref{sec:GeneralGame}.
We then define the properties (e.g. that the set of active senders can change) in Section \ref{aspects} and build notions (e.g. for sender unobservability) upon them in Section \ref{notions}.
Additionally, we already presented the intuition that sender unobservability is stronger than sender-message unlinkability. This is not only true for this example, in fact we prove: every protocol achieving sender unobservability also achieves sender-message unlinkability. Intuitively, if whether Alice is an active sender or not is hidden, whether she sent a certain message or not is also hidden. We will prove relations between our privacy notions in Section \ref{sec:hierarchy} and show that the presented relations (depicted in Figure \ref{fig:hierarchyColored}) are complete. Before that, we argue our choice of notions in Section \ref{sec:choiceNotions}.
\section{Our Game model}
\label{sec:GeneralGame}
Our goal in formalizing the notions as a game is to analyze a given ACN protocol w.r.t. to a notion, i.e. the game is a tool to investigate if an adversary can distinguish two self-chosen, notion-compliant scenarios. Scenarios are sequences of communications.
A \emph{communication} is described by its sender, receiver, message and auxiliary information (e.g. session identifiers) or the empty communication, signaling that nobody wants to communicate at this point.
Some protocols might restrict the information flow to the adversary to only happen at specific points in the execution of the protocol, e.g. because a component of the \ac{ACN} processes a batch of communications before it outputs statistics about them.
Therefore, we introduce \emph{batches} as a sequence of communications, which is processed as a unit before the adversary observes anything\footnote{We use the word batch to designate a bunch of communications. Besides this similarity, it is not related to batch mixes.}.
When this is not needed, batches can always be replaced with single communications.
As explained in Section \ref{background}, we do not need to define a complete new game for every privacy goal, since notions only vary in the difference between the alternative scenarios chosen by the adversary. Hence, for a given ACN and notion, our general game is simply instantiated with a model of the \ac{ACN}, which we call the protocol model, and the notion.
The protocol model accepts a sequence of communications as input. Similar to the real implementations the outputs of the protocol model are the observations the real adversary can make.
Note, the adversaries in the game and the real world have the same capabilities\footnote{A stronger game adversary also implies that the protocol is safer in the real world.}, but differ in their aims: while the real world adversary aims to find out something about the users of the system, the game adversary merely aims to distinguish the two scenarios she has constructed herself.
In the simplest version of the game, the adversary constructs two scenarios, which are just two batches of communications and sends them to the challenger. The challenger checks that the batches are compliant with the notion. If so, the challenger tosses a fair coin to randomly decide which of the two batches it executes with the protocol model. The protocol model's output is returned to the game adversary. Based on this information, the game adversary makes a guess about the outcome of the coin toss.
We extend this simple version of the game, to allow the game adversary to send multiple times two batches to the challenger. However, the challenger performs a single coin flip and sticks to this scenario for this game, i.e. it always selects the batches corresponding to the initial coin flip. This allows analyzing for adversaries, that are able to base their next actions in the attack on the observations they made previously.
\iflong
Further, we allow for user (a sender or receiver) corruption, i.e. the adversary learns the user's momentary internal state, by sending corrupt queries to the challenger. Note that although the adversary decides on all the communications that happen in the alternative scenarios, she does not learn secret keys or randomness unless the user is corrupted. This allows to add several options for different corruption models to the privacy goals
\fi
\iflong
To model all possible attacks,
\else
To unfetter our general game from the concrete adversary model,
\fi we allow the adversary to send protocol queries. This is only a theoretical formalization to reflect what information the adversary gets and what influence she can exercise. These protocol query messages are sent to the protocol model without any changes by the challenger. The protocol model covers the adversary to ensure that everything the real world adversary can do is possible in the game with some query message. For example, protocol query messages can be used to add or remove nodes from the \ac{ACN} by sending the appropriate message.
As introduced in Section \ref{background}, we say that an adversary has an advantage in winning the game, if she guesses the challenger-selected scenario correctly with a higher probability than random guessing.
A protocol achieves a certain privacy goal, if an adversary has at most negligible advantages in winning the game.
\subsection*{Formalization}
In this subsection, we formalize the game model to conform to the above explanation.
We use $\Pi$ to denote the analyzed \textit{\ac{ACN} protocol model}, $Ch$ for the challenger and $\mathcal{A}$ for the adversary, which is a probabilistic polynomial time algorithm.
Additionally, we use $X$ as a placeholder for the specific notion, e.g. sender unobservability, if we explain or define something for all the notions.
A \textit{communication} $r$ in $\Pi$ is represented by a tuple $(u,u',m,aux)$ with a sender $u$, a receiver $u'$, a message $m$, and auxiliary information $aux$ (e.g. session identifiers).
Further, we use $\Diamond$ instead of the communication tuple $(u,u',m,aux)$ to represent that no communication occurs.
Communications are clustered into \textit{batches} $\underline{r}_b=(r_{b_1},\dots, r_{b_l})$, with $r_{b_i}$ being the $i$-th communication of batch $\underline{r}_b$. Note that we use $\underline{r}$ (underlined) to identify batches and $r$ (no underline) for single communications. Batches in turn are clustered into \textit{scenarios}; the first scenario is $(\underline{r}_{0_1}, \dots,\underline{r}_{0_k} )$. A \textit{challenge} is defined as the tuple of two scenarios $\left ( (\underline{r}_{0_1}, \dots,\underline{r}_{0_k} ), (\underline{r}_{1_1}, \dots,\underline{r}_{1_k} ) \right )$.
All symbols used so far and those introduced later are summarized in \mbox{Tables~\ref{tab:allNotions}~--~\ref{tab:allSymbols}} in Appendix~\ref{sec:allSymbols}.
\inlineheading{Simple Game}
\begin{enumerate}
\item $Ch$ randomly picks challenge bit $b$.
\item $\mathcal{A}$ sends a batch query, containing $\underline{r}_{0}$ and $\underline{r}_{1}$, to $Ch$.
\item $Ch$ checks if the query is valid, i.e. both batches differ only in information that is supposed to be protected according to the analyzed notion $X$.
\item If the query is valid, $Ch$ inputs the batch corresponding to $b$ to $\Pi$.
\item $\Pi$'s output $\Pi(\underline{r}_{b})$ is handed to $\mathcal{A}$.
\item After processing the information, $\mathcal{A}$ outputs her guess $g$ for $b$.
\end{enumerate}
\inlineheading{Extensions}
As explained above, there are useful extensions we make to the simple game:
\begin{description}
\item[\emph{Multiple Batches}] Steps 2-5 can be repeated.
\iflong
\item[\emph{User corruption}] Instead of Step 2, $\mathcal{A}$ can also decide to issue a corrupt query specifying a user $u$ and receive $u$'s internal state as output. This might change $\Pi$'s state, lead to different behavior of $\Pi$ in following queries and yield a higher advantage in guessing than before.
\fi
\item[\emph{Other parts of the adversary model}] Instead of Step 2, $\mathcal{A}$ can also decide to issue a protocol query, containing an input specific to $\Pi$ and receive $\Pi$'s output to it (e.g. the internal state of a router that is corrupted in this moment). This might change $\Pi$'s state.
\end{description}
\inlineheading{Achieving notion $X$}
Intuitively, a protocol \(\Pi\) achieves a notion \(X\) if any possible adversary has at most negligible advantage in winning the game.
To formalize the informal understanding of $\Pi$ achieving goal $X$, we need the following denotation. \iflong ${\text{Pr}[g= \langle \mathcal{A} \bigm| Ch(\Pi, X,c,b)\rangle ]}$ \else ${\text{Pr}[g= \langle \mathcal{A} \bigm| Ch(\Pi, X,b)\rangle ] }$ \fi describes the probability that $\mathcal{A}$ \iflong (with at most $c$ challenge rows, i.e. communications differing in the scenarios) \fi outputs $g$, when $Ch$ is instantiated with $\Pi$ and $X$ and the challenge bit was chosen to be $b$. With this probability, achieving a notion translates to Definition \ref{def:achieve}.
\begin{definition}[Achieving a notion $X$]\label{def:achieve}
An \ac{ACN} Protocol $\Pi$ achieves $X$, iff for all probabilistic polynomial time(PPT) algorithms $\mathcal{A}$ there exists a negligible $\delta$ such that
\small
\[ \bigm| \text{Pr}[0= \langle \mathcal{A} \mid Ch(\Pi, X,\iflong c, \fi 0)\rangle ] -
\text{Pr}[0= \langle \mathcal{A} \mid Ch(\Pi, X,\iflong c, \fi 1)\rangle]\bigm|\leq \delta \text{.} \]
\end{definition}
\iflong
We use a variable \(\delta\), which is referred to as negligible, as an abbreviation when we actually mean a function \(\delta(\kappa)\) that is negligible in a security parameter \(\kappa\).
\fi
\iflong
\paragraph{Equivalence to Other Definitions}
Notice, that this definition is equivalent to
\begin{align*}
\text{(1) Pr}[0= \langle \mathcal{A} \bigm| Ch(\Pi, X,c,0)\rangle ] &\leq \\
\text{Pr}[0= \langle \mathcal{A} \bigm| Ch(\Pi, X, c,1)\rangle]&+ \delta \text{.}
\end{align*}
and
\begin{align*}
\text{(2) Pr}[1= \langle \mathcal{A} \bigm| Ch(\Pi, X,c,1)\rangle ] &\leq \\
\text{Pr}[1= \langle \mathcal{A} \bigm| Ch(\Pi, X, c,0)\rangle]&+ \delta \text{.}
\end{align*}
(1): $|Pr[0\mid 0]- Pr[0 \mid 1]|\leq \delta $ for all $\mathcal{A}$ $\iff (Pr[0\mid 0]- Pr[0 \mid 1]\leq \delta $ for all $\mathcal{A}$) $\land $ $( Pr[0 \mid 1] - Pr[0\mid 0]\leq \delta $ for all $\mathcal{A}$). To every attack $\mathcal{A}$ with $Pr[0 \mid 1] - Pr[0\mid 0]> \delta$, we can construct $\mathcal{A'}$ with $Pr[0\mid 0]- Pr[0 \mid 1]> \delta$. Since the definition requires the inequality to hold for all attacks, this is enough to prove that (1) implies the original, the other way is trivial.
This is how we construct it: Given attack $\mathcal{A}$, we construct $\mathcal{A'}$ by changing the batches of the first with the second scenario. Hence, $\text{Pr}[0= \langle \mathcal{A} \bigm| Ch(\Pi, X,c,0)\rangle ] =\text{Pr}[0= \langle \mathcal{A'} \bigm| Ch(\Pi, X,c,1)\rangle ] $ and $\text{Pr}[0= \langle \mathcal{A} \bigm| Ch(\Pi, X,c,1)\rangle ] =\text{Pr}[0= \langle \mathcal{A'} \bigm| Ch(\Pi, X,c,0)\rangle ] $.
(2): To every attack $\mathcal{A}$ breaking (1), we can construct one with the same probabilities in (2). Given attacker $\mathcal{A}$, we construct $\mathcal{A'}$ as the one that changes the batches of the first with the second scenario and inverts the output of $\mathcal{A}$. Hence, $\text{Pr}[0= \langle \mathcal{A} \bigm| Ch(\Pi, X,c,0)\rangle ] =\text{Pr}[1= \langle \mathcal{A'} \bigm| Ch(\Pi, X,c,1)\rangle ] $ and $\text{Pr}[1= \langle \mathcal{A} \bigm| Ch(\Pi, X, c,0)\rangle]=\text{Pr}[0= \langle \mathcal{A'} \bigm| Ch(\Pi, X, c,1)\rangle]$. Since we can reverse this operations by applying them again, we can also translate in the other direction.
\paragraph{Differential Privacy based Definition}
For some use cases, e.g. if the court of your jurisdiction requires that the sender of a critical content can be identified with a minimal probability of a certain threshold e.g. 70\%, a non-negligible $\delta$ might be sufficient. Hence, we allow to specify the parameter of $\delta$ and extend it with the allowed number of challenge rows $c$ to finally include the well-known concept of differential privacy as AnoA does in the following definition:
\begin{definition}[Achieving $(c,\epsilon, \delta)-X$]\label{def:achieveEpsilon}
An \ac{ACN} protocol $\Pi$ is $(c,\epsilon, \delta) -X$ with $c>0$, $\epsilon \geq 0$ and $0 \leq \delta \leq 1$, iff for all PPT algorithms $\mathcal{A}$:
\begin{align*}
\text{Pr}[0= \langle \mathcal{A} \bigm| Ch(\Pi, X,c,0)\rangle ] &\leq \\
e^{\epsilon} \text{Pr}[0= \langle \mathcal{A} \bigm| Ch(\Pi, X, c,1)\rangle]&+ \delta\text{.}
\end{align*}
\end{definition}
Notice that $\epsilon$ describes how close the probabilities of guessing right and wrong have to be. This can be interpreted as the quality of privacy for this notion. While $\delta$ describes the probability with which the $\epsilon$-quality can be violated. Hence, every \ac{ACN} protocol will achieve $(0,1)-X$ for any notion $X$, but this result does not guarantee anything, since with probability $\delta =1$ the $\epsilon$-quality is not met.
The first variant can be expressed in terms of the second as $\Pi$ achieves $X$, iff $\Pi$ is $(c,0,\delta)-X$ for a negligible $\delta$ and any $c\geq 0$.
\fi
\section{Options for Notions}
\label{sec:options}
Additionally to the properties, we define options. Options can be added to any notion and allow for a more precise mapping of real world protocols, aspects of the adversary model, or easier analysis by quantification.
\vspace{-0.3cm}
\subsection{Protocol-dependent: Sessions}\label{sec:sessions}
\vspace{-0.3cm}
Some \ac{ACN} protocols, like e.g. Tor, use sessions. Sessions encapsulate sequences of communications from the same sender to the same receiver by using the same session identifier for them. In reality, the adversary might be able to observe the session identifiers but (in most cases) not to link them to a specific user.
To model sessions, we therefore set the auxiliary information of a communication to the session ID ($sess$): $aux=sess$.
However, as the adversary can choose this auxiliary information, we need to ensure that the scenarios cannot be distinguished just because the session identifier is observed.
Hence, we definine $sess$ to be a number in most communications. Only for the session notions, we require special session IDs that correspond to the current challenge $\Psi$ and stage $x$ in all challenge rows: $(x,Ch\Psi)$. In this way, they have to be the same in both scenarios and a concrete $sess$ is only used in one stage of one challenge.
The session identifier that is handed to the \ac{ACN} protocol model is a random number that is generated by the challenger when a new $sess$ is seen. Hence, neither leaking (it is a random number) nor linking session identifiers (it will be picked new and statistically independent for every challenge and stage) will give the attacker an advantage.
We formalize this in the following definition, where we also use `$\_$' to declare that this part of a tuple can be any value.\footnote{E.g.
$\exists(u,m,\_) \in r$ will be true iff $\exists u': \exists(u,m,u') \in r$.}
\begin{definition}[Sessions]\label{def:sessions}
Let $x$ be the stage and $u^a_0,u^a_1,u'^a_0,u'^a_1$ be the senders and receivers of the first challenge row of this challenge $\Psi$ and stage in instance $a \in \{0,1\}$.
Property $sess$ is met, iff for all $a\in \{0,1\}$:
{\footnotesize
\begin{align*}
sess&: \forall (r^a_0,r^a_1)\in \mathsf{CR}(\underline{r}^a_0,\underline{r}^a_1):(r^a_0,r^a_1)=\\
&\quad (u^a_0,u'^a_0,\_,(x, Ch\Psi) ),(u^a_1,u'^a_1,\_,(x,Ch\Psi))
\end{align*}
}
\end{definition}
\vspace{-0.3cm}
As not all protocols use sessions, we allow to add sessions as an option to the notion $X$ abbreviated by $\manySess{X}$.
\subsection{Adversary Model: Corruption} \label{sec:advCap}
Some adversary capabilities like user corruption imply additional checks our challenger has to do. As all properties are independent from corruption, we add corruption as an option, that can be more or less restricted as shown in Table \ref{tab:corruption options}.
The different corruption options have implications on the challenger, when a corrupt query or a batch query arrives.
\begin{table} [t]
\center
\resizebox{0.35\textwidth}{!}{%
\begin{tabular}{ l p{7cm}}
Symbol &Description\\ \hline
$X$&Adaptive corruption is allowed.\\%&Bohli\\
$\static{X}$ & Only static corruption of users is allowed.\\%& AnoA, Gelernter
$\noCorr{X}$ & No corruption of users is allowed.\\ \hlin
$X$&Corrupted users not restricted.\\
$\corrNoComm{X}$&Corrupted users are not allowed to be chosen as senders or receivers.\\
$\corrOnlyPartnerSender{X}$&Corrupted users are not allowed to be senders.\\
$\corrOnlyPartnerReceiver{X}$&Corrupted users are not allowed to be receivers.\\
$\corrStandard{X}$&Corrupted users send/receive identical messages in both scenarios.\\
\end{tabular}}
\caption{Options for corruption and for corrupted communication}
\label{tab:corruption options}
\end{table}
\inlineheadingTwo{Check on corrupt queries}
This check depends on whether the user corruption is adaptive, static, or not allowed at all.
The default case for notion $X$ is adaptive corruption, i.e.
the adversary can corrupt honest users at any time.
With static corruption $\static{X}$, the adversary has to corrupt a set of users before she sends her first batch.
The third option, $\noCorr{X}$, is that no user corruption is allowed.
We denote the set of corrupted users as $\hat{U}$.
\begin{definition}[Corruption: Check on Corrupt Query]\label{def:corruptionCorrupt}
Let $\hat{U}$ be the set of already corrupted users, $u$ the user in the corrupt query and the bit $\mathrm{subsequent}$ be true iff at least one batch query happened.
The following properties are met, iff:
{\footnotesize
\begin{align*}
{corr}_{static}&: \mathrm{subsequent} \implies u \in \hat{U} \\
{corr}_{no}&: \perp \quad \quad {corr}_{adaptive}: \top\\
\end{align*}
}
\end{definition}
\vspace{-2em}
\inlineheadingTwo{Check on batch queries}
In reality for most ACNs the privacy goal can be broken for corrupted users, e.g. a corrupted sender has no unobservability.
Therefore, we need to assure that the adversary cannot distinguish the scenario because the behavior of corrupted users differs. This is done by assuring equal behavior $\corrStandard{corr}$ or banning such users from communicating $\corrNoComm{corr},\corrOnlyPartnerSender{corr},\corrOnlyPartnerReceiver{corr}$.
\begin{definition}[Corruption: Check on Batch Query]\label{def:corruption}
The following properties are met, iff for all $a \in \{0,1\}$:
{\footnotesize
\begin{align*}
\corrNoComm{corr}&: \forall (u,u',m,aux) \in \underline{r}^a_0\cup \underline{r}^a_1: u \not \in \hat{U} \land u' \not \in \hat{U} \\
\corrOnlyPartnerSender{corr}&: \forall (u,u',m,aux) \in \underline{r}^a_0\cup \underline{r}^a_1: u \not \in \hat{U} \\
\corrOnlyPartnerReceiver{corr}&: \forall (u,u',m,aux) \in \underline{r}^a_0\cup \underline{r}^a_1: u' \not \in \hat{U}\\
\corrStandard{corr}&: \forall \hat{u} \in \hat{U}:r^a_{0_i}=(\hat{u},\_,m,\_) \implies r^a_{1_i}=(\hat{u},\_,m,\_)\\
&\quad \quad \quad \quad \land r^a_{0_i}=(\_,\hat{u},m,\_) \implies r^a_{1_i}=(\_,\hat{u},m,\_) \\
\end{align*}
}
\end{definition}
\vspace{-0.5cm}
Of course user corruption is not the only important part of an adversary model. Other adversarial capabilities can be adjusted with other parts of our framework (like the corruption of other parts of the \ac{ACN} with protocol queries).
\subsection{Easier Analysis: Quantification}\label{sec:challenges}
For an easier analysis, we allow the quantification of notions in the options. This way a reduced number of challenge rows (challenge complexity) or of challenges (challenge cardinality) can be required. The next section includes information on how results with low challenge cardinality imply results for higher challenge cardinalities.
\inlineheading{Challenge Complexity}
\example{Consider Alice using a protocol, that achieves $S\overline{O}$ for one challenge row ($S\overline{O}_{\mathsf{CR}_{1}}$), but not for two ($S\overline{O}_{\mathsf{CR}_{2}}$). This means in the case that Alice only communicates once, the adversary is not able to distinguish Alice from any other potential sender Charlie. However, if Alice communicates twice the regime might distinguish her existence from the existence of some other user, e.g. by using an intersection attack.}
To quantify how different the scenarios can be, we add the concept of \emph{challenge complexity}. Challenge complexity is measured in \textit{Challenge rows}, the pairs of communications that differ in the two scenarios as defined earlier. $c$ is the maximal allowed number of challenge rows in the game. Additionally, we add the maximal allowed numbers of challenge rows per challenge $\#cr$ as option to a notion $X$ with $X_{\mathsf{CR}_{\#cr}}$.
\begin{definition}[Challenge Complexity]\label{def:challengeComplexity}
Let $\#\mathsf{CR}$ be the number of challenge rows in this challenge so far,
We say that the following property is met, iff:
\allowdisplaybreaks
\begin{align*}
\mathsf{CR}_{\#cr}:& \#\mathsf{CR} \leq \#cr
\end{align*}
\end{definition}
\begin{table}[htb]
\center
\resizebox{0.35\textwidth}{!}{%
\begin{tabular}{ l l }
Notion including option& Definition \\ \hline
$\manySess{X}$&Properties of $X \land sess$\\
$\corrStandard{X}$, $\corrNoComm{X}$ \ etc.& $\land \corrStandard{corr}$, $\land \corrNoComm{corr}$ \ etc.\\
$\challengeRows{X}$& Properties of $ X \land \mathsf{CR}_{\#cr}$\\%) $\lor E$ \\
\end{tabular}}
\caption{Definition of notions including the options; for all notions $X$ }
\label{NotionsDefinition}
\end{table}
\inlineheading{Challenges Cardinality}
So far, our definitions focused on one challenge. We now bound the number of challenges to $n$, as the adversary potentially gains more information the more challenges are played. While challenge complexity defines a bound on the total number of differing rows within a single challenge, cardinality bounds the total number of challenges.
Communications belonging to a challenge are identified by the challenge number $\Psi$, which has to be between 1 and $n$ to be valid. The challenge number is a part of the auxiliary information of the communication and is only used by the challenger, not by the protocol model.
This dimension of quantification can be useful for analysis, since for certain assumptions the privacy of the $n$-challenge-case can be bounded in the privacy of the single-challenge-case as we will discuss in the next section.
\section{Introduction}
\label{intro}
With our frequent internet usage of, e.g., social networks, instant messaging, and web browsing, we constantly reveal personal data. Content encryption can reduce the footprint, but metadata (e.g. correspondents' identities) still leaks.
To protect metadata from state and industrial surveillance, a broad variety of \acp{ACN} has emerged; one of the most deployed is Tor \cite{dingledine04tor}, but also others, e.g. I2P \cite{zantout11i2p} or Freenet \cite{clarke01freenet}, are readily available. Additionally, many conceptual systems, like Mix-Nets~\cite{chaum81untraceable}, DC-Nets~\cite{chaum88dining}, Loopix \cite{piotrowska17loopix} and Crowds \cite{reiter98crowds} have been published.
The published \acp{ACN} address a variety of privacy goals. However, many definitions of privacy goals are ad hoc and created for a particular use case. We believe that a solid foundation for future analysis is still missing. This hinders the understanding and comparison of different privacy goals and, as a result, comparison and improvement of \acp{ACN}.
In general, comparing privacy goals is difficult since their formalization is often incompatible and their naming confusing.
This has contributed to a situation where existing informal comparisons disagree: e.g., Sender Unlinkablity of Hevia and Micciancio's framework~\cite{hevia08indistinguishability} and Sender Anonymity of AnoA~\cite{backes17anoa} are both claimed to be equivalent to Sender Anonymity of Pfitzmann and Hansen's terminology~\cite{pfitzmann10terminology}, but significantly differ in the protection they actually provide. These naming issues further complicate understanding of privacy goals and hence analysis of \acp{ACN}.
To allow rigorous analysis, i.e. provable privacy, of \acp{ACN}, their goals need to be unambiguously defined. Similar to the notions of semantic security (like CPA, CCA1, CCA2 \cite{goos_relations_1998}) for confidentiality, privacy goals can be formally defined as indistinguishability games.
We call such formally defined privacy goals \emph{privacy notions}. Further, notions need to be compared according to their strength:
achieving the stronger notion implies the weaker one. Comparison of notions, and of the \acp{ACN} achieving them, is otherwise impossible.
To understand the ramifications of privacy goals, we aim at setting all notions into mutual relationships. This means for every pair of notions it must be clear if one is stronger or weaker than the other, or if they have no direct relationship. Such a comparison has already been made for the notions of semantic security~\cite{goos_relations_1998}.
\iflong
Further, all the assumptions of different existing analysis frameworks, e.g. regarding corruption or specific protocol parts like sessions, have to be unified in one framework to find a common basis for the comparison.
\fi
\iflong
In this work, we introduce such a unified framework.
\else
In this work, we tackle the formal definition and comparison of privacy goals.
\fi
To achieve this, we build on the foundations of existing analytical frameworks~\cite{backes17anoa,hevia08indistinguishability,gelernter13limits,bohli11relations}.
With their preparatory work, we are able to present basic building blocks of privacy notions: observable properties of a communication, that (depending on the notion) must either be protected, i.e. kept private, by the protocol, or are permitted to be learned by the adversary. Defining our notions based on the idea of properties simplifies comparison.
Further, we map practitioners' intuitions to their underlying formal model, justify our choice of notions with exemple use cases for each, and make a sanity check to see that the privacy goals of a current \ac{ACN} (Loopix~\cite{piotrowska17loopix}) are covered.
\iflong
As a next step, we include assumptions of existing analysis frameworks by defining them similarly as building blocks that can be combined to any notion. Finally, we argue how the notions and assumptions of existing works map to ours.
\else
Additionally, for all formalized goals of existing analysis frameworks~\cite{backes17anoa,hevia08indistinguishability,gelernter13limits,bohli11relations} we reason to which notions they correspond if they are broken down to the general observable properties and interpreted for \acp{ACN}.
This means that we focus on general privacy goals and do not present aspects regarding the adversary model, infrequently-used observable information, or the quantification of privacy goals. However, those aspects are compatible with our formalization and have not been ignored; they are presented in the long version of this paper~\cite{longVersion}.
\fi
We compare all identified privacy notions and present a complete proven hierarchy. As a consequence of our comparison, we are able to rectify mapping inconsistencies of previous work and show how privacy notions and data confidentiality interact.
Furthermore, the proofs for building the hierarchy include templates in order to compare and add new privacy notions to the established hierarchy, if necessary.
\iflong
As we added the assumptions, our resulting framework captures all the assumptions and notions of the AnoA \cite{backes17anoa}, Hevia and Miccianchio's \cite{hevia08indistinguishability}, Gelernter and Herzberg's \cite{gelernter13limits} frameworks, captures most and adapts some of Bohli and Pashalidis's framework \cite{bohli11relations} and adds missing ones.
We capture the assumptions and notions of the other frameworks by demonstrating equivalences between their and our corresponding notion.
This removes the constraints of co-existing frameworks and allows to use all options when analyzing an \ac{ACN}. To make our work more accessible, we included a how-to-use section and intuitions, recommendations and limits of this work in the discussion.
\fi
\noindent
In summary, our main contributions are:
\begin{itemize}
\iflong
\item a holistic framework for analyzing \acp{ACN}, capturing more notions and assumptions than each existing framework,
\fi
\item the mapping of practitioners' intuitions to game-based proofs,
\item the definition of building blocks for privacy notions,
\item the selection and unified definition of notions,
\item a complete hierarchy of privacy notions, which simplifies comparison of \acp{ACN}, \iflong \else and \fi
\item the resolution of inconsistencies and revision of mistakes in previous (frame)works%
\iflong
\item the definition of building blocks for assumptions compatible to our notions and
\item a guide to use the framework and an example of mapping the goals of an \ac{ACN} into our hierarchy.
\else .
\fi
\end{itemize}
\iflong
\else
\fi
\inlineheading{Outline}
Section \ref{background} contains an introductory example and gives an overview of our paper.
In Section \ref{sec:GeneralGame}, we introduce the underlying model and indistinguishability games.
In Section \ref{aspects}, we introduce the basic building blocks of privacy notions: properties.
In Section \ref{notions}, we define the privacy notions.
In Section \ref{sec:choiceNotions}, we argue our choice of notions.
\iflong
In Section \ref{sec:options}, we introduce further assumptions, that can be combined with our notions as options.
In Section \ref{sec:adversary}, we explain how results regarding restricted adversaries carry over to our work.
In Section \ref{sec:mappingpapers}, we state the relation of our notions to the other existing analytical frameworks.
\fi
In Section \ref{sec:hierarchy}, we present the relations between the notions.
\iflong
In Section \ref{howToUse}, we explain how to use the framework for analysis.
\fi
In Section \ref{discussion}, we discuss our results.
In Section \ref{conclusion}, we conclude our paper and give an outlook.
\section{Protected Properties}
\label{aspects}
We define properties to specify which information about the communication is allowed to be disclosed to the adversary, and which must be protected to achieve a privacy notion, as mentioned in Section \ref{background}.
We distinguish between simple and complex properties. Simple properties can be defined with the basic game model already introduced, while complex properties require some extensions to the basic model.
\begin{table}
\center
\resizebox{0.48\textwidth}{!}{%
\begin{tabular}{ c p{3.5cm} p{4.8cm} }
Symbol &Description&Translation to Game\\ \hline
$|M|$&Message Length& Messages in the two scenarios always have the same length.\\
$E_{S}$&Everything but Senders& Everything except the senders is identical in both scenarios.\\
$E_{R}/ E_{M}$&Everything but Receivers/Messages& Analogous\\
$E_{SM}$&Everything but Senders and Messages& Everything except the senders and messages is identical in both scenarios.\\
$E_{RM}/ E_{SR}$&Analogous& Analogous\\
\something &Something is sent& In every communication something must be sent ($\Diamond$ not allowed).\\
$\aleph$ &Nothing&Nothing will be checked; always true.\\
$U/U'$& Active Senders/Receivers& Who sends/receives is equal for both scenarios.\\
$Q/Q'$& Sender/Receiver Frequencies&Which sender/receiver sends/receives how often is equal for both scenarios.\\
$|U|/|U'|$& Number of Senders/ Receivers& How many senders/receivers communicate is equal for both scenarios.\\
$P/P'$& Message Partitioning per Sender/Receiver& Which messages are sent/received from the same sender/receiver is equal for both scenarios.\\
$H/H'$& Sender/Receiver Frequency Histograms&How many senders/receivers send/receive how often is equal for both scenarios.\\
\end{tabular}}
\caption{Simple properties; information about communications that may be required to remain private}
\label{tab:information}
\end{table}
\vspace{-0.3cm}
\subsection{Simple Properties}
\vspace{-0.3cm}
We summarize the informal meaning of all simple properties in Table \ref{tab:information} and introduce them in this section.
Assume an \ac{ACN} aims to hide the sender but discloses message lengths to observers. For this case, we specify the property ($|M|$) that the message length must not differ between the two scenarios, as this information must not help the adversary to distinguish which scenario the challenger chose to play.
Next, we might want an \ac{ACN} to protect the identity of a sender, as well as any information about who sent a message, but deliberately disclose which messages are received by which receiver,
who the receivers are, and potentially other auxiliary information.
We hence specify a property ($E_{S}$) where only the senders differ between the two scenarios\footnote{$E$ symbolizes that only this property may vary in the two submitted scenarios and everything else remains equal.}, to ensure that the adversary in our game can only win by identifying senders.
In case the protection of the receiver identities or messages is required, the same can be defined for receivers ($E_{R}$) or messages ($E_{M}$).
Further, we might want the \ac{ACN} to protect senders and also the messages; leaving the receiver and auxiliary information to be disclosed to the adversary.
This is achieved by specifying a property where only senders and messages differ between the two scenarios and everything else remains equal ($E_{SM}$).
Again, the same can be specified for receivers and messages ($E_{RM}$) or senders and receivers ($E_{SR}$).
Lastly, \ac{ACN}s might allow the adversary to learn whether a real message is sent or even how many messages are sent. We specify a property (\something) that requires real communications in both scenarios, i.e. it never happens that nothing is sent in one scenario but something is sent in the other. We ensure this by not allowing the empty communication ($\diamond$).
However, a very ambitious privacy goal might even require that the adversary learns no information about the communication at all ($\aleph$). In this case, we allow any two scenarios and check nothing.
\inlineheading{Formalizing those Simple Properties}
In the following definition all simple properties mentioned so far are formally defined.
Therefore, we use $\top$ as symbol for the statement that is always true.
\begin{definition}[Properties $|M|$, $E_{S}$, $E_{SM}$, \something, $\aleph$]\label{def:properties}
Let the checked batches be $\underline{r_0},\underline{r_1}$, which include the communications \mbox{${r_0}_j \in \{ (u_{0_j},u'_{0_j},m_{0_j},aux_{0_j}), \diamond\}$} and \mbox{ ${r_1}_j \in \{ (u_{1_j},u'_{1_j},m_{1_j},aux_{1_j}), \diamond\}$} with $ j \in \{1, \dots l\} $. We say the following properties are met, iff for all $j \in \{1, \dots l\}$:
\begin{align*}
|M|& : |{m_0}_j|=|{m_1}_j| \\[0.5em]
E_{S}&: {r_1}_j =(\mathbf{u_{1_j}},u'_{0_j},m_{0_j},aux_{0_j}) \\
E_{R}&: {r_1}_j =(u_{0_j},\mathbf{u'_{1_j}},m_{0_j},aux_{0_j}) \\
E_{M} &: {r_1}_j =(u_{0_j},u'_{0_j},\mathbf{m_{1_j}},aux_{0_j}) \\[0.5 em]
E_{SM}&: {r_1}_j =(\mathbf{u_{1_j}},u'_{0_j},\mathbf{m_{1_j}},aux_{0_j}) \\
E_{RM}&: {r_1}_j =(u_{0_j},\mathbf{u'_{1_j}},\mathbf{m_{1_j}},aux_{0_j}) \\
E_{SR} &: {r_1}_j =(\mathbf{u_{1_j}},\mathbf{u'_{1_j}},m_{0_j},aux_{0_j}) \\[0.5 em]
\text{\something} &: \Diamond \not \in \underline{r}_0 \land \Diamond \not \in \underline{r}_1\\
\text{$\aleph$}&: \top
\end{align*}
\label{SimpleProp1}
\end{definition}
\inlineheading{More Simple Properties: Active Users, Frequencies}
The properties of Definition \ref{SimpleProp1} are important to formalize privacy, but are by themselves not sufficient.
Take the \ac{ACN} Tor as an example: While the set of active senders is trivially known to their ISPs and the guard nodes, we still require that the senders are unlinkable with the messages they are sending (and their receivers).
Similarly, the sending (receiving) frequency of a party may be important and is not formalized yet.
To formalize these properties, we use sets that capture which user sent which messages in a certain period, i.e. a batch of communications (and similarly sets to capture which user received which messages). Note that we use primes ($'$) for the corresponding sets and properties of the receivers.
\begin{definition}[Sender-Message Linking]\label{def:senderMessageSet}
We define the sender-message linkings for scenario $b$ ($L'_{b_i}$ the receiver-message linkings are analogous) as:
\begin{align*}
L_{b_i}:= &\{(u,\{m_1,...,m_h\}) \bigm| u \text { sent messages }m_1, \dots , m_h\\
&\text{ in batch }i\}\text{.}\\
\end{align*}
\end{definition}
The sets from Definition \ref{def:senderMessageSet} allow easy identification of who an active sender in this batch was and how often each sent something:
\begin{definition}[Active Sender Set, Frequency Set]\label{UbQb}
Let the current batch be the $k$-th one.
For $ b \in \{0,1\}$ $U_b,Q_b$ ($U_b',Q_b'$ for $L_b'$) are defined as:
\begin{align*}
U_b &:= \{ u \bigm| (u,M) \in L_{b_k}\}\hfill\\
Q_b &:= \{ (u,n) \bigm| (u,M) \in L_{b_k}\land |M|=n\}\\
\end{align*}
\end{definition}
Recall that we currently define properties for ACNs that allow the adversary to learn which senders are active at different times, or the number of messages they send during some periods, while hiding some other properties (e.g. which messages they have sent).
Hence, with the respective sets for active users and user frequencies defined, we need only to request that they are equal in both scenarios:
\begin{definition}[Properties $U$, $Q$, $|U|$]
We say that the properties $U,Q,|U|$ ($U',Q', |U'|$ analogous) are met, iff:
\[U: U_0=U_1 \hspace{1,5em} Q: Q_0=Q_1 \hspace{1,5em} |U|: |U_0|=|U_1| \]
\end{definition}
\inlineheading{More Simple Properties: Message Partitions, Histograms}
Other interesting properties are which messages came from a given sender and how many senders sent how many messages. If the adversary knows which messages are sent from the same sender, e.g. because of a pseudonym, she might be able to combine information from them all to identify the sender. If she knows how many senders sent how many messages, she knows the sender activity and hence can make conclusions about the nature of the senders.
As before, we introduce auxiliary variables to formally define these two properties.
We use $M_{b,I}$ to denote the collection of messages that has been sent by the same sender (e.g. linked by a shared pseudonym) in a set of batches, and $M_{b,I,n}$ to denote the union of all these sets of cardinality $n$.
The equality of the properties in the two scenarios must pertain throughout all comparable batches in the scenarios. If this were not true, the inequality would help the adversary to distinguish the scenarios without learning the protected information e.g. identifying the sender.
\begin{definition}[Multi-Batch-Message Linkings]\label{def:messageSets} Let the current batch be the $k$-th, \mbox{$\mathcal{K}:=\{1, \dots, k\}$}, $\mathcal{P}(\mathcal{K})$ the power set of $\mathcal{K}$ and $\mathcal{U}$ the set of all possible senders ($\mathcal{U'}$ receivers).
For $b \in \{0,1\}$ and $I \in \mathcal{P}(\mathcal{K})$: We define ($M'_{b,I}, M'_{b,I,n}$ for $L'_{b_i}$)
\begin{itemize}
\item the multi-batch-message-sender linking:\\ \mbox{$M_{b,I} := \cup_{u\in \mathcal{U}} \{ \cup_{i \in I}\{M| (u,M)\in L_{b_i}\}\}$} and
\item the cardinality restricted multi-batch-message-sender linking: $M_{b,I,n}:=\{ M \in M_{b,I} \bigm| |M|=n \}$.
\end{itemize}
\end{definition}
As before, we define auxiliary variables capturing the information that we want to be equal in both scenarios: We define ordered sets specifying which messages are sent from the same user for any set of batches (Message Partition $P_b$) and how many users sent how many messages for any set of batches (Histogram $H_b$). Therefore, we use a slightly unusual notation: For any set Z, we use $(Z_i)_{i\in\{1,\dots, k\}}$ to denote the sequence $(Z_1, Z_2, \dots, Z_k)$ and $\overrightarrow{\mathcal{P}}(Z)$ to denote a sorted sequence of the elements of the power set\footnote{For brevity we use $\in$ to iterate through a sequence.} of $Z$.
\begin{definition}[Message partitions, Histograms]\label{def:properties}
Consider the $k$-th batch, \mbox{$\mathcal{K}:=\{1, \dots, k\}$}.
For $ b \in \{0,1\}$ $P_b, H_b$ ($P_b',H_b'$ analogous) are defined as:
\allowdisplaybreaks
\begin{align*}
P_b &:= (M_{b,I})_{ I \in \overrightarrow{\mathcal{P}}(\mathcal{K})}\\
H_b &:= (\{(n, i) \bigm| i= |M_{b,I,n}|\})_{ I \in \overrightarrow{\mathcal{P}}(\mathcal{K})}\\
\end{align*}
\vspace{-0.5cm}
Further, we say that properties $P,H$ ($P',H' $ analogous) are met, iff:
\begin{align*}
P&: P_0=P_1 \hspace{3em} H: H_0=H_1\\
\end{align*}
\end{definition}
\subsection{Complex Properties}
\label{sec:ComplexProperties}
\vspace{-0.3cm}
So far, we have defined various properties to protect senders, messages, receivers, their activity, frequency and the grouping of messages.
However, this is not sufficient to formalize several relevant privacy goals, and we must hence introduce complex properties.
\inlineheading{Learning Sender and Receiver}
Consider that one aims to hide which sender is communicating with which receiver.
Early \acp{ACN} like classical Mix-Nets~\cite{chaum81untraceable}, and also Tor~\cite{dingledine04tor}, already used this goal. Therefore, we want the adversary to win the game only if she identifies both: sender and receiver of the same communication.
An intuitive solution may be to model this goal by allowing the adversary to pick different senders and receivers ($E_{SR}$) in both scenarios (see Fig. \ref{fig:senderReceiver} (a) for an example).
This, however, does not actually model the privacy goal: by identifying only the sender or only the receiver of the communication, the game adversary could tell which scenario was chosen by the challenger.
We hence must extend the simple properties and introduce scenario \emph{instances} to model dependencies.
\inlineheadingTwo{Scenario instances}
We now require the adversary to give alternative instances for both scenarios (Fig. \ref{fig:senderReceiver} (b)). The challenger chooses the scenario according to the challenge bit, which is picked randomly for every game, and the instance according to the instance bit, which is picked randomly for every challenge.
Formally, we replace steps 2--5 of the game with the following steps:
\begin{description}
\item[2.] $\mathcal{A}$ sends a batch query, containing $\underline{r}^0_{0}$, $\underline{r}^1_{0}$, $\underline{r}^0_{1}$and $\underline{r}^1_{1}$ to $Ch$.
\item[3.] $Ch$ checks if the query is valid according to the analyzed notion $X$.
\item[4.] If the query is valid and $Ch$ has not already picked an instance bit $a$ for this challenge, $Ch$ picks $a \in \{0,1\}$ randomly and independent of $b$. Then it inputs the batch corresponding to $b$ and $a$ to $\Pi$.
\item[5.] $\Pi$'s output $\Pi(\underline{r}^a_{b})$ is forwarded to $\mathcal{A}$.
\end{description}
This allows us to model the goal that the adversary is not allowed to learn the sender and receiver: We allow the adversary to pick two sender-receiver pairs, which she uses as instances for the first scenario.
The mixed sender-receiver pairs must then be provided as instances for the second scenario (see Fig. \ref{fig:senderReceiver} (b)).
We thus force the game adversary to provide alternative assignments for each scenario.
This way she cannot abuse the model to win the game by identifying only the sender or the receiver.
We call this property \textit{Random Sender Receiver} $R_{SR}$.
This complex property is still not sufficient to model the situation in, for example, Tor:
The adversary can distinguish the scenarios without learning who sent to whom, just by learning which senders and which receivers are active.
Hence, we further restrict the adversary picking instances where both senders and both receivers are active by defining the property \textit{Mix Sender Receiver} $M_{SR}$.
Here, the adversary picks two instances for $b=0$ where her chosen sender-receiver pairs communicate, and two for $b=1$ where the mixed sender-receiver pairs communicate.
The two instances simply swap the order in which the pairs communicate (Fig. \ref{fig:senderReceiver} (c)).
This way, we force the adversary to provide alternative assignments for each scenario where both suspected senders and both suspected receivers are active.
This combination prevents the adversary from winning the game without learning the information that the real system is actually supposed to protect, i.e. the sender-receiver pair.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.48\textwidth]{images/complexProp.pdf}
\caption{Examples showing the general structure of communications that differ in both scenarios: a) Naive, but incorrect b) Random Sender Receiver $R_{SR}$ c) Mixed Sender Receiver $M_{SR}$}
\label{fig:senderReceiver}
\end{figure}
\inlineheading{Defining Complex Properties}
To simplify the formal definition of complex properties, we introduce \emph{challenge rows}. A challenge row is a pair of communications with the same index that differ in the two scenarios (e.g. ${r}_{0_j}, {r}_{1_j}$ with index $j$).
For complex properties, the challenger only checks the differences of the challenge rows in the two scenarios.
\begin{definition}[Properties $R_{SR}$, $M_{SR}$]\label{def:complexProperties}
Let the given batches be $\underline{r}_b^a$ for instances $a \in \{0,1\}$ and scenarios $b \in \{0,1\}$,
$\mathsf{CR}$ the set of challenge row indexes,
$(u^a_0, {u'}^{\:a}_0)$ for both instances $a\in \{0,1\}$ be the sender-receiver-pairs of the first challenge row of the first scenario ($b=0$)\iflong in this challenge \fi.
Random Sender Receiver $R_{SR}$, Mixed Sender Receiver $M_{SR}$ ($R_{SM}, R_{RM}, M_{SM}, M_{RM}$ analogous) are met, iff:
\allowdisplaybreaks
\begin{align*}
R_{SR}:\quad{r_{0}^a}_{cr}&=(\mathbf{u^{a}_0}, \mathbf{{u'}^{\:a}_0}, m^1_{0_{cr}}, aux^1_{0_{cr}})~\land \\
{r_{1}^a}_{cr}&=(\mathbf{u^{a}_0}, \mathbf{ {u'}^{\:1-a}_0}, m^1_{0_{cr}},aux^1_{0_{cr}})\\[0.2em]
&\forall cr \in \mathsf{CR} , a \in \{0,1\}&
\end{align*}
\begin{align*}
M_{SR}:\quad&{r_0^a}_{cr}=(\mathbf{u^{a}_0}, \mathbf{ {u'}^{\:a}_0},m^1_{0_{cr}},aux^1_{0_{cr}})~\land \\
&{r_0^a}_{cr+1}= (\mathbf{u^{1-a}_0}, \mathbf{ {u'}^{\:1-a}_0}, m^1_{0_{cr}},aux^1_{0_{cr}})~\land \\[0.2em]
&{r_1^a}_{cr}=(\mathbf{u^{a}_0}, \mathbf{ {u'}^{\:1-a}_0},m^1_{0_{cr}},aux^1_{0_{cr}})~\land \\
&{r_1^a}_{cr+1}=(\mathbf{u^{1-a}_0},\mathbf{ {u'}^{\:a}_0}, m^1_{0_{cr}},aux^1_{0_{cr}})\\[0.2em]
&\text{for every second } cr \in \mathsf{CR} , a \in \{0,1\}&
\end{align*}
\end{definition}
\inlineheading{Linking message senders}
A final common privacy goal that still cannot be covered is the unlinkability of senders over a pair of messages (Twice Sender Unlinkability}%/ Double Sender Unobservability).
Assume a real world adversary that can determine that the sender of two messages is the same entity.
If subsequently she discovers the identity of the sender of one of the messages through a side channel, she can also link the second message to the same individual.
\inlineheadingTwo{Stages}
To model this goal, we need two scenarios (1) both messages are sent by the same sender, and (2) each message is sent by a different sender.
Further, the adversary picks the messages for which she wants to decide whether they are sent from the same individual, and which other messages are sent between those two messages.
Therefore, we add the concept of \emph{stages} and ensure that only one sender sends in the challenge rows of stage 1, and in stage 2 either the same sender continues sending ($b=0$) or another sender sends those messages ($b=1$). This behavior is specified as the property \emph{Twice Sender} $T_S$.
\begin{definition}[Property $T_S$]\label{def:complexProperties}
Let the given batches be $\underline{r}_b^a$ for instances $a \in \{0,1\}$ and scenarios $b \in \{0,1\}$, $x$ the current stage,
$\mathsf{CR}$ the set of challenge row indexes,
$(u^a_0, {u'}^a_0)$ for both instances $a\in \{0,1\}$ be the sender-receiver-pairs of the first challenge row of the first scenario ($b=0$) \iflong in this challenge \fi in stage 1 and $(\tilde{u}^a_0,\tilde{u}'^a_0)$ the same pairs in stage 2.
Twice Sender $T_S$ is met, iff ($T_R$ analogous):
\begin{align*}
T_{S}:&\quad x=stage1~\land\\
&\qquad{r_0^a}_{cr}=(\mathbf{u^a_0}, {u'}^{\:0}_0,m^1_{0_{cr}},aux^1_{0_{cr}})~\land \\
&\qquad{r_1^a}_{cr}=(\mathbf{u^a_0}, {u'}^{\:0}_0,m^1_{0_{cr}},aux^1_{0_{cr}})\\[0.2em]
\mathbf{\lor} &\quad x=stage2~\land\\
&\qquad{r_0^a}_{cr}= (\mathbf{u^{a}_0}, \tilde{u}'^{\:0}_0, m^1_{0_{cr}},aux^1_{0_{cr}})~\land \\
&\qquad{r_1^a}_{cr}= (\mathbf{u^{1-a}_0}, \tilde{u}'^{\:0}_0, m^1_{0_{cr}},aux^1_{0_{cr}})\\
&\quad\forall cr \in \mathsf{CR} , a \in \{0,1\}&
\end{align*}
\end{definition}
Hence, we need to facilitate distinct stages for notions with the complex properties $T_S$ or $T_R$. Precisely, in step 2 of the game, the adversary is additionally allowed to switch the stages.
Note that the above definition can easily be extended to having more stages and hence, more than two messages for which the adversary needs to decide whether they have originated at the same sender.
This set of properties allows us to specify all privacy goals that have been suggested in literature as privacy notions and additionally all that we consider important.
It is of course difficult to claim completeness, as future \acp{ACN} may define diverging privacy goals and novel observable properties (or side-channels) may be discovered.
\section{Relations to Prior Work}
\label{sec:mappingpapers}
In this Section, we introduce existing frameworks and point out to which of our notions their notions corresponds.
\else
\subsection{Relation to Existing Analysis Frameworks}
\label{sec:mappingpapers}
In this section, we briefly introduce the existing frameworks based on indistinguishability games.
\fi
\label{sec:ComparingFrameworks}
\iflong
We argue that our framework includes all their assumptions and notions relevant for \acp{ACN} and thus provides a combined basis for an analysis of \acp{ACN}.
\else
We argue that our summary of notions includes all their notions\footnote{Where necessary, we have interpreted them for \acp{ACN} and broken them down to the general observable information.} and therefore allows a comparison along this dimension
\fi
\label{advClass}
\iflong
For each framework, we first quickly give an idea why the properties and options match the notions of it and focus on how the concepts (like batches) relate later on. The resulting mapping is shown in Table \ref{mapping} and reasoned below.
\else
The resulting mapping is shown in Table \ref{mapping} of Appendix~\ref{app:NotionMapping}. Since the mapping of our properties to the notions of the other frameworks is obvious in most cases, we reason the remaining cases and concepts here and refer to the long version of this paper \cite{longVersion} for the complete verification.
\fi
\iflong
\begin{table} [b!]
\center
\resizebox{0.4\textwidth}{!}
\begin{tabular}{ c c c }
Framework & Notion & Equivalent to \\ \hline
AnoA& $\alpha_{SA}$&$ {\manySess{\static{\corrStandard{S\overline{O}}}}}_{CR_1}$\\
&$\alpha_{RA}$&${\manySess{\static{\corrStandard{R\overline{O}[M\overline{O}-|M|]}}}}_{CR_1}$\\
&$ \alpha_{REL}$&${\manySess{\static{\corrStandard{(SR)\overline{O}}}}}_{CR_2}$\\
&$ \alpha_{UL}$&${\manySess{\static{\corrStandard{(2S)\overline{L}}}}}_{CR_2}$\\
& $\alpha_{sSA}$&$\manySess{\static{\corrStandard{S\overline{O}}}}$\\
&$\alpha_{sRA}$&$\manySess{\static{\corrStandard{R\overline{O}[M\overline{O}-|M|]}}}$\\
&$ \alpha_{sREL}$\footnotemark&$\manySess{\static{\corrStandard{(SR)\overline{O}}}}$\\
&$ \alpha_{sUL}$\footnotemark&$\manySess{\static{\corrStandard{(2S)\overline{L}}}}$\\ \hline
Bohli's&$S/SA=R/SA$&$\overline{O}$\\
&$ R/SUP$& $S\overline{O}\{R\overline{O}-|U'|\}$\\
&$ R/WUP$& $S\overline{O}\{R\overline{O}-H'\}$\\
& $ R/PS$&$S\overline{O}\{R\overline{O}-P'\}$\\
&$ R/SUU$& $S\overline{O}\{RF\overline{L}\}$\\
&$ R/WUU$&$S\overline{O}\{RF\overline{L}-H'\}$\\
&$ R/AN$&$S\overline{O}\{RF\overline{L}-P'\}$\\
&$ R/WU$&$S\overline{O}\{RM\overline{L}\}$\\
& $ R/WA$&$S\overline{O}\{RM\overline{L}-P'\}$\\
&$ S/SA^\circ$& $S\overline{O}$\\
&$ S/SUP^\circ$&$S\overline{O}-|U|$\\
&$ S/WUP^\circ$&$S\overline{O}-H$\\
&$S/PS^\circ$&$S\overline{O}-P$\\
&$S/SUU^\circ$&$SF\overline{L}$\\
& $S/WUU^\circ$&$SF\overline{L}-H$\\
&$S/AN^\circ$&$SF\overline{L}-P$\\
&$S/WU^\circ$&$SM\overline{L}$\\
&$S/WA^\circ$&$SM\overline{L}-P$\\
&$S/X, R/X^\circ$&analogous\\
&$X^+$&$\corrStandard{\langle X\rangle}$\\
& $X^*$&$\corrStandard{\langle X^\circ\rangle}$\\ \hline
Hevia's&$UO$&$\noCorr{C\overline{O}}$, $k=1$\\
&$SRA$&$\noCorr{\overline{O}}$, $k=1$\\
&$SA^*$&$\noCorr{S\overline{O}\{RM\overline{L}\}}$, $k=1$\\
&$SA$&$\noCorr{S\overline{O}}$, $k=1$\\
&$UL$&$\noCorr{M\overline{O}[M\overline{L}]}$, $k=1$\\
&$SUL$&$\noCorr{SM\overline{L}}$, $k=1$\\
&$RA^*, RUL, RA$&analogous\\ \hline
Gelernter's&$R^{H, \tau}_{SA}$&$\noCorr{{{R^{H, \tau}_{SA}}} } \iff \noCorr{S\overline{O}-P}$, $k=1$\\
&$R^{H, \tau}_{SUL}$&$\noCorr{{{R^{H, \tau}_{S\overline{L}}}}} \iff \noCorr{SM\overline{L}-P}$, $k=1$\\
&$R_X$& analogous Hevia: $\langle X\rangle$ \\
&$R^H_X$& analogous Hevia: $\corrNoComm{\langle X\rangle}$\\
&$\hat{R}^H_X$& analogous Hevia $\corrOnlyPartnerSender{\langle X\rangle}$\\
\end{tabular}}
\caption{Equivalences, $\langle X \rangle$ equivalence of $X$ used}
\label{mapping}
\end{table}
\addtocounter{footnote}{-1}
\footnotetext{Under the assumption that in all cases $m_0$ is communicated like in $\alpha_{REL}$ of \cite{backes17anoa} and in $\alpha_{SREL}$ of one older AnoA version \cite{backes14anoa}.}
\stepcounter{footnote}
\footnotetext{Under the assumption that the receiver in stage 2 can be another than in stage 1 like in $\alpha_{UL}$ of \cite{backes17anoa}.}
\fi
\paragraph{AnoA Framework}
AnoA \cite{backes17anoa} builds its privacy notions on $(\epsilon, \delta)$~differential privacy and compares them to their interpretation of the terminology paper of Pfitzmann and Hansen \cite{pfitzmann10terminology}.
\iflong
AnoA's $\alpha_{SA}$ allows only one sender to change, the same is achieved with the combination of $E_{S}$ and $CR_1$. In AnoA's $\alpha_{RA}$ also the messages can differ, but have to have the same length, which we account for with using $E_{RM}$ and $|M|$. AnoA's $\alpha_{REL}$ will either end in one of the given sender-receiver combinations been chosen ($b=0$) or one of the mixed cases ($b=1$). This is exact the same result as $R_{SR}$ generates. For AnoA's $\alpha_{UL}$ either the same sender is used in both stages or each of the senders is used in one of the stages. This behavior is achieved by our property $T_S$. Although AnoA checks that the message length of the communication of both scenarios is equal, only the first message is used in any possible return result of $\alpha_{UL}$. Hence, not checking the length and requiring the messages to be the same as we do in $T_S$ is neither weaker nor stronger.
\else
\fi
\iflong
Our model differs from AnoA's model in the batch queries, the adaptive corruption, the arbitrary sessions and the use of notions instead of anonymity functions.
\else
Conceptually our model differs from AnoA's model in the definition of achieving a notion, batch queries, and the use of notions instead of anonymity functions.
\fi
\iflong
Instead of \emph{batch queries} AnoA distinguishes between input, i.e.
communications that are equal for both scenarios, and challenge queries, i.e.
challenge rows.
Input queries are always valid in AnoA.
They are also valid in our model, because all the privacy aspects used for our notions equivalent to AnoA's hold true for identical batches without $\Diamond$ and $\Diamond$ is not allowed in the equivalent notions.
\else
AnoA's definition of achieving a notion can be easily included (see Appendix \ref{app:epsilonDef}), if needed.
\fi
\iflong
In AnoA's single-message anonymity functions only a limited number of challenge queries, i.e. challenge rows, is allowed per challenge.
We ensure this restriction with $CR_{\#cr}$.
\fi
In AnoA, the adversary gets information after every communication.
This is equivalent to multiple batches of size one in our case.
\iflong
We assume that for the analyzed protocol a protocol model can be created, which reveals the same or less information when it is invoked on a sequence of communications at once instead of being invoked for every single communication.
Our notions, which match the AnoA notions, allow for batches of size one.
So, our batch concept neither strengthens nor weakens the adversary.
\fi
\iflong
AnoA's \emph{corruption} is static, does not protect corrupted users\footnote{Although AnoA does not explicitly state this, we understand the analysis and notions of AnoA this way, as scenarios differing in the messages corrupted users send/receive could be trivially distinguished.} and AnoA includes restrictions on \emph{sessions}.
Hence, AnoA's notions translate to ours with the static corruption $\static{X}$, the corrupted communication have to be equal in both scenarios $\corrStandard{X}$ and the session option of our model $\manySess{X}$.
\fi
AnoA's challenger does not only check properties, but modifies the batches with the \emph{anonymity functions}. However, the modification results in one of at most four batches. We require those four batches (as combination of scenario and instances) as input from the adversary, because it is more intuitive that all possible scenarios stem from the adversary. This neither increases nor reduces the information the adversary learns, since she knows the challenger algorithm.
\vspace{-0.5cm}
\paragraph{Bohli's Framework}
Bohli and Pashalidis \cite{bohli11relations} build a hierarchy of application-independent privacy notions based on what they define as ``interesting properties'', that the adversary is or is not allowed to learn.
Additionally, they compare their notions to Hevia's, which we introduce next, and find equivalences.
\iflong
It is easy to see, that our definitions of $U,Q, H$ ($P$ is not easy and hence, explained more detailed below) match the ones of Bohli's properties (who sent, how often any sender sends and how many senders sent how often) although we do not use a function that links every output message with the sender(/receiver), but the sender-messages-sets(/receiver-messages-sets).
Bohli and Pashalidis additionally define the restriction of picking their communications equal except for the user (depending on the current notion sender or receiver) $\circ$. This is the same as allowing only the senders resp. receivers to differ ($E_{S}$ resp. $E_{R}$).
Conceptually, our model differs from Bohli's model in the concept of challenges, the advantage definition, the order of outputs, and the allowed behavior of corrupted users.
\else
\fi
\iflong
Bohli's notions can be understood as one \emph{challenge} ($n=1$) with arbitrarily many challenge rows (any $c$).
Further, it does not use a multiplicative term in its \emph{advantage} ($\epsilon=0$).
Then $\delta$ equals the advantage, which has to be 0 to unconditionally provide a privacy notion or negligible to computationally provide this notion.
\else
\fi
\iflong
\else
To achieve the mapping, we need to interpret one property of Bohli's framework for \acp{ACN}. Our message partitionings ($P,P'$) group the messages by their sender/receiver. However, Bohli's corresponding linking relation groups the indexes of the outputs of the analyzed system. Since messages are usually the interesting output elements, the adversary tries to link in \acp{ACN}; we consider this as a suitable mapping when analyzing \acp{ACN}.
\fi
\iflong
Bohli's framework assumes that the protocol outputs information as an information vector, where each entry belongs exactly to one communication.
The adversary's goal in Bohli's framework is to link the index number of the output vector with the sender or receiver of the corresponding communication.
All except one of their properties can be determined given the batches of both scenarios.
However, the linking relation property that partitions the index numbers of the output vector by user (sender or receiver depending on the notion), can only be calculated once the output order is known.
Since our notions shall be independent from the analyzed protocol, the challenger cannot know the protocol and the way the output order is determined.
Running the protocol on both scenarios might falsely result in differing output orders for non-deterministic protocols.
Thus, we adapt the linking relation for \acp{ACN} to be computable based on the batches.
The interesting output elements the adversary tries to link in \acp{ACN} are messages.
Hence, here the linking relation partitions the set of all messages into the sets of messages sent/received by the same user, which can be calculated based on the batches.
This adaption is more restrictive for an adversary, since the partition of output numbers can be equal for both scenarios even though the sent messages are not.
However, if the adversary is able to link the output number to the message, she can calculate our new linking relations based on Bohli's.
\fi
\iflong
Further, Bohli's framework allows for notions, where the \emph{behavior of corrupted users} differs in the two scenarios.
This means privacy of corrupted users is provided, i.e.
the adversary wins if she can observe the behavior of corrupted users.
Those notions are the ones without the option $\corrStandard{X}$.
To match our batch query, Bohli's input queries, which include communications of both scenarios, have to be combined with a nextBatch query, which signals to hand all previous inputs to the protocol.
\fi
\vspace{-0.5cm}
\paragraph{Hevia's Framework}
Hevia and Micciancio \cite{hevia08indistinguishability} define scenarios based on message matrices.
Those message matrices specify who sends what message to whom.
Notions restrict different communication properties like the number or set of sent/received messages per fixed user, or the number of total messages.
Further, they construct a hierarchy of their notions and give optimal \ac{ACN} protocol transformations that, when applied, lead from weaker to stronger notions.
\iflong
Mapping of the properties follows mainly from Bohli's and the equivalences between Bohli and Hevia (including the one we correct in the following paragraph). Besides this, only Hevia's Unobservability ($UO$), where the matrices can be picked arbitrary, is new. However, this corresponds to our $\aleph$ \ property, that always returns TRUE and allows any arbitrary scenarios.
Our model differs from Hevia's, since ours considers the order of communications, allows adaptive attacks and corruption.
Our game allows to consider the \emph{order of communications}.
\else
In contrast, our model considers the order of communications.
\fi
Analyzing protocol models that ignore the order will lead to identical results. However, protocol models that consider the order do not achieve a notion -- although they would in Hevia's framework, if an attack based on the order exists.
\iflong \footnote{
Creating an adapted version left a degree of freedom.
Our choice of adaptation corresponds with the interpretation of Hevia's framework that was used, but not made explicit in Bohli's framework.}\fi
\iflong
Most of Hevia's notions are already shown to match Bohli's with only one batch ($k=1$) and no corruption ($\noCorr{X}$) \cite{bohli11relations}.
\else
Most of Hevia's notions are already shown to match Bohli's with only one batch \cite{bohli11relations}.
\fi
However, we have to correct two mappings:
in \cite{bohli11relations} Hevia's strong sender anonymity (${SA^*}$), which requires the number of messages a receiver receives to be the same in both scenarios was mistakenly matched to Bohli's sender weak unlinkability ($S/WU^+$), in which every sender sends the same number of messages in both scenarios.
\iflong
The needed restriction is realized in Bohli's $R/WU^+$ instead.
The proof is analogous to Lemma 4.3 in \cite{bohli11relations}.
\else
Hence, the sender and receiver restrictions become confused and it needs to be mapped to Bohli's receiver weak unlinkability ($R/WU^+$) instead.
\fi
The same reasoning leads to Bohli's sender weak unlinkability ($S/WU^+$) as the mapping for Hevia's strong receiver anonymity (${RA^*}$).
\paragraph{Gelernter's Framework}
Gelernter and Herzberg \cite{gelernter13limits} extend Hevia's framework to include corrupted participants.
Additionally, they show that under this strong adversary an \ac{ACN} protocol achieving the strongest notions exists.
However, they prove that any \ac{ACN} protocol with this strength has to be inefficient, i.e.
the message overhead is at least linear in the number of honest senders.
Further, they introduce relaxed privacy notions that can be efficiently achieved.
\iflong
The notions of Gelernter's framework build on Hevia's and add corruption, which is covered in our corruption options.
Only the relaxed notions $R^{H, \tau}_{SA}$ and $R^{H, \tau}_{SUL}$ are not solely a corruption restriction.
\else
The notions of Gelernter's framework build on Hevia's and add corruption, which we do not discuss in this work, but include in the long version of this paper \cite{longVersion}. However, the relaxed notions are not solely an extension regarding corruption.
\fi
\iflong
We define new notions as ${{R^{H, \tau}_{SA}}} $=\something $\land G$ and ${{R^{H, \tau}_{S\overline{L}}}}$= \something $\land Q \land G$ that are equivalent to some of the already introduced notions to make the mapping to the Gelernter's notions obvious.
They use a new property $G$, in which scenarios are only allowed to differ in the sender names.
\begin{definition}[Property $G$]
Let $\mathcal{U}$ be the set of all senders, $s_{b_i}= \{(u,\{m_1, \dots ,m_h\})\bigm| u $ send message $m_1, \dots , m_h$ in batch $i\}$ the sender-messages sets for scenario $b\in \{0,1\}$.
We say that $G$ is met, iff a permutation $perm$ on $\mathcal{U}$ exists such that for all $ (u,M)\in s_{0_k}: (\text{perm}(u),M) \in s_{1_k}$.
\end{definition}
Note that Gelernter's relaxed notions (indistinguishability between permuted scenarios) is described by our property $G$, the need for the existence of such a permutation.
\begin{theorem}It holds that
\begin{align*}
(c, \epsilon, \delta)-{{R^{H, \tau}_{SA}}} &\iff (c, \epsilon, \delta)-S\overline{O}-P,\\
(c, \epsilon, \delta)-{{R^{H, \tau}_{S\overline{L}}}} &\iff (c, \epsilon, \delta)-SM\overline{L}-P\text{.}
\end{align*}
\end{theorem}
\begin{proofsketch}
Analogous to Theorem \ref{loopixUO}.
${{R^{H, \tau}_{SA}}} \Rightarrow S\overline{O}-P$: Every attack on $S\overline{O}-P$ is valid against ${{R^{H, \tau}_{SA}}} $: Since $P$ is fulfilled, for every sender $u_0$ in the first scenario, there exists a sender $\tilde{u}_0$ in the second scenario sending the same messages.
Hence, the permutation between senders of the first and second scenario exists.
${{R^{H, \tau}_{SA}}} \Leftarrow S\overline{O}-P$: Every attack on ${{R^{H, \tau}_{SA}}} $ is valid against $S\overline{O}-P$: Since there exists a permutation between the senders of the first and second scenario sending the same messages, the partitions of messages sent by the same sender are equal in both scenarios, i.e.
$P$ is fulfilled.
${{R^{H, \tau}_{S\overline{L}}}} \iff SM\overline{L}-P:$
$Q$ is required in both notions by definition.
Arguing that $P$ resp.
$G$ is fulfilled given the other attack is analogous to ${{R^{H, \tau}_{SA}}} \iff S\overline{O}-P$.
\end{proofsketch}
\else
In Appendix \ref{OtherFrameworksProof} we formalize them and shown to be equivalent to two previously defined notions.
\fi
\section{Privacy Notions}
\label{notions}
Given the properties above, we can now set out to express intuitive privacy goals as formal privacy notions.
We start by specifying sender unobservability as an example leading to a general definition of our privacy notions.
Recall the first game we defined in Section \ref{background}, which corresponds to sender unobservability ($S\overline{O}$ = S(ender) $\lnot$ O(bservability)). There, in both scenarios something has to be sent, i.e. we need to specify that sending nothing is not allowed: \something. Further, both scenarios can only differ in the senders, i.e. we also need the property that everything but the senders is equal: $E_{S}$. Hence, we define sender unobservability as $S\overline{O} :=$\something$ \land E_{S}$.
\footnote{Technically $E_{S}$ already includes \something. However, to make the differences to other notions more clear, we decide to mention both in the definition.}
We define all other notions in the same way:
\begin{definition}[Notions]
Privacy notions are defined as a boolean expression of the properties
according to Table~\ref{NotionsDefinition}.
\end{definition}
\begin{table}[htb]
\center
\resizebox{0.4\textwidth}{!}{%
\begin{tabular}{ l l }
Notion&Properties \\ \hline
$(SR)\overline{L}$ &\something $ \land E_{SR} \land M_{SR}$ \\
$(SR)\overline{O}$ &\something$\land E_{SR} \land R_{SR}$ \\
$M\overline{O}$ &\something $ \land E_{M}$ \\
$M\overline{O}-|M|$ &\something $ \land E_{M} \land |M| $ \\
$M\overline{O}[M\overline{L}]$& \something$\land Q \land Q'$\\
$\overline{O}$& \something \\
$C\overline{O}$& $\aleph$\\ \hline
$S\overline{O}$& \something $\land E_{S}$\\
$S\overline{O}-|U| $& \something$\land E_{S} \land |U|$\\
$S\overline{O}-H$ &\something$\land E_{S} \land H$\\
$S\overline{O}-P $&\something $\land E_{S} \land P$\\
$SF\overline{L}$ &\something$\land E_{S} \land U$ \\
$SF\overline{L}-H$ &\something $\land E_{S} \land U \land H$\\
$SF\overline{L}-P $&\something $\land E_{S} \land U \land P$\\
$SM\overline{L}$ &\something $\land E_{S} \land Q$\\
$SM\overline{L}-P$ &\something $\land E_{S} \land Q \land P$\\
$(2S)\overline{L}$ &\something$\land E_{S} \land T_S $ \\
$R\overline{O}$ \ etc.&analogous\\ \hline
\mbox{$S\overline{O}[M\overline{O}]$}&\something $ \land E_{SM} $ \\
\mbox{$S\overline{O}[M\overline{O}-|M|]$}&\something $ \land E_{SM} \land |M| $ \\
$(SM)\overline{O}$ &\something $ \land E_{SM} \land R_{SM}$ \\
$(SM)\overline{L}$&\something $ \land E_{SM} \land M_{SM} $ \\
\mbox{$R\overline{O}[M\overline{O}-|M|]$} \ etc.&analogous \\ \hline
$\sgame{X'}$& Properties of $X'$, remove $E_{R}$ \\
\multicolumn{2}{c}{for $X' \in \{R\overline{O},$ $ R\overline{O}-|U'|,$ $R\overline{O}-H',$ $R\overline{O}-P',$ $RF\overline{L},$} \\
\multicolumn{2}{c}{$RF\overline{L}-H',$ $RF\overline{L}-P',$ $RM\overline{L},$ $RM\overline{L}-P'\}$}\\
$\rgame{X}$ &analogous \\
\end{tabular}}
\caption{Definition of the notions. A description of simple properties was given in Table~\ref{tab:information}. }
\label{NotionsDefinition}
\end{table}
\iflong
Modeling the notions as a game, the respective challenger verifies all properties (and the later introduced options) of the adversary's queries.
A complete description of the challenger can be found in Appendix \ref{sec:challenger}.
Further, an example of how the definitions can be represented by using a challenge specific state, which the challenger maintains, is shown in Algorithms \ref{Challenger} and \ref{calcNewState} in Appendix \ref{pseudocode}.
\else
Modeling the notions as a game, the respective challenger will check all aspects of the adversary's queries.
A complete description of the challenger can be found in Appendix \ref{sec:challenger}.
\fi
\section{Hierarchy}
\label{sec:hierarchy}
\vspace{-1em}
\iflong
\begin{figure*}[h!]
\center
\includegraphics[width=0.95\textwidth]{images/hierarchy_structured.pdf}
\caption{Our new hierarchy of privacy notions divided into sender, receiver and impartial notions and clustered by leakage type}\label{fig:hierarchyColored}
\end{figure*}
\else
\begin{figure*}[h!]
\center
\includegraphics[width=0.95\textwidth]{images/hierarchy_structured_less_colors.pdf}
\caption{Our hierarchy of privacy notions divided into sender, receiver and impartial notions and clustered by leakage type. Table~\ref{NotionsDefinition} provides definitions for the presented notions based on properties. Table~\ref{tab:allNotions} gives an overview on all properties. For a summary of the naming scheme, see Table \ref{Tab:NamingScheme} of Appendix \ref{app:summaryNamingScheme}. } \label{fig:hierarchyColored}
\end{figure*}
\fi
\iflong
\begin{figure}[thb]
\begin{center}
\resizebox{0.25\textwidth}{!}{%
\begin{tikzpicture}[>=triangle 45,font=\sffamily]
\node (X) at (0,0) {$\corrStandard{X}$};
\node (XSpecial) [above =0.3cm of X] {$X \iff X_{CR_c}$};
\node (corrNoComm)[below = 0.6cm of X] {$\corrNoComm{X}$};
\node (invis)[below = 0.1cm of X] {};
\node(corrSender)[left = 0.5cm of X] {$\corrOnlyPartnerSender{X}$};
\node(corrRec)[right = 0.5cm of X] {$\corrOnlyPartnerReceiver{X}$};
\node (static)[right = 0.5cm of corrRec] {$\static{X}$};
\node (noCorr)[right = 1.5cm of corrNoComm] {$\noCorr{X}$};
\node (manySess) [ left=0.5cm of corrSender] {$\manySess{X}$};
\draw [semithick,->] (XSpecial) -- (static);
\draw [semithick,->] (XSpecial) -- (X);
\draw [semithick,->] (XSpecial) -- (corrSender);
\draw [semithick,->] (XSpecial) -- (corrRec);
\draw [semithick,->] (corrSender) -- (corrNoComm);
\draw [semithick,->] (corrRec) -- (corrNoComm);
\draw [semithick,->] (X) -- (corrNoComm);
\draw [semithick,->] (static) -- (noCorr);
\draw [semithick,->] (corrNoComm) -- (noCorr);
\draw [semithick,->] (XSpecial) -- (manySess);
\end{tikzpicture}}
\caption{Additional implications for corruption and sessions} \label{HierarchyExtended}
\end{center}
\end{figure}
\fi
Next, we want to compare all notions and establish their hierarchy. To do this, for any pair of notions we analyze which one is stronger than, i.e. implies, the other. This means, any \ac{ACN} achieving the stronger notion also achieves the weaker (implied) one. Our result is shown in Figure~\ref{fig:hierarchyColored}, where all arrow types represent implications, and is proven as Theorem \ref{the:impl} below.
Further, obvious implications between every notion $\sgame{X}$, $\rgame{X}$ and $X$ exist, since $\sgame{X}$ only adds more possibilities to distinguish the scenarios. However, to avoid clutter we do not show them in Figure \ref{fig:hierarchyColored}.
\iflong
To ease understanding the hierarchy for the first time, we added Appendix \ref{sec:HierarchyAndTables} where it is plotted together with the most important symbol tables. Further, the same hierarchy exists between notions with the same session, corruption and quantification options.
Further, we add a small hierarchy for the options that holds by definition in Figure \ref{HierarchyExtended}.
\fi
\begin{theorem}
\label{the:impl}
The implications shown in Figure \ref{fig:hierarchyColored} hold.
\end{theorem}
\iflong
\input{sections/proofs/hierarchyProof.tex}
So far we have proven that implications between notions exist. Further, we assure that the hierarchy is complete, i.e. that there exist no more implications between the notions of the hierarchy:
\else
\input{sections/proofs/hierarchySketch.tex}
\fi
\begin{theorem}
\label{the:noImpl}
For all notions $X_1$ and $X_2$ of our hierarchy, where $X_1 \implies X_2$ is not proven or implied by transitivity, there exists an \ac{ACN} protocol achieving $X_1$, but not $X_2$.
\end{theorem}
\iflong
\input{sections/proofs/completeProof.tex}
\else
\begin{proofsketch}
We construct the protocol in the following way: Given a protocol $\Pi$ that achieves $X'_1$ ($X_1$ itself or a notion that implies $X_1$), let protocol $\Pi'$ run $\Pi$ and additionally output some information $I$. We argue that learning $I$ does not lead to any advantage in distinguishing the scenarios for $X_1$. Hence, $\Pi'$ achieves $X_1$. We give an attack against $X_2$ where learning $I$ allows the scenarios to be distinguished. Hence, $\Pi'$ does not achieve $X_2$.
Further, we use the knowledge that $\implies$ is transitive\footnote{If $X_1 \implies X_2$ and $X_1 \centernot \implies X_3$, it follows that $X_2 \centernot \implies X_3$.}.
Some concrete cases are shown in Appendix \ref{completeSketch}. We provide the complete list of proofs in the long version of this paper \cite{longVersion}.
\end{proofsketch}
\fi
\section{Conclusion and Future Work}
\label{conclusion}
\iflong
We have presented a framework of privacy notions for sharper analysis of \acp{ACN} that, to the best of our knowledge, includes more notions and assumptions than all existing frameworks based on indistinguishability games. To achieve this, we
\else
We
\fi
expressed privacy goals formally as privacy notions. We first presented their basic building blocks: properties.
Those properties cover the observable information of communications, which is either required to remain private or allowed to be learned by an adversary, depending on the goal.
\iflong
Furthermore, we checked the sanity of the notions by finding exemplary use cases and by providing a mapping of the privacy goals of a current ACN to them.
\else
We formally specified privacy goals from \acp{ACN} and sorted them into a proven hierarchy, according to their strength.
\fi
\iflong
Our framework allows to compare and understand the differences in privacy goals. We proved the relations between our notions. This means that, for every pair of notions, we know which one is stronger than the other or if they are separate. This way, we resolved inconsistencies between the existing frameworks and built the basis to understand the strengths and weaknesses of \acp{ACN} better, which helps building improved \acp{ACN}.
Further, it creates a unified basis for the analysis and comparison of \acp{ACN}.
\else
This means, for every pair of notions, we know which one is the stronger; or if they do not imply each other.
As a result, we resolved inconsistencies between existing analytical frameworks and built the foundations to understand the strengths and weaknesses of \acp{ACN} better, which helps analyzing and building improved \acp{ACN}.
\fi
\inlineheading{Future Work}
\iflong
Although our framework allows to analyze all types of attacks with the versatile protocol queries, the protocol model must support those attacks without systematic guidance by interfaces of the framework.
Restrictions of such attacks thus cannot be expressed formally as part of the notion and hence are not easily represented.
In future work we want to introduce more dedicated queries to also formalize other attack dimensions and based on this adversary classes for typical attackers.
\fi
As we mentioned in the discussion, providing more intuitions and understanding the significance of notions is necessary. Therefore, analogous to the analysis of Loopix's privacy goals, more current ACNs can be analyzed to understand which parts of the hierarchy they cover. This can also identify gaps in research; privacy goals for which \acp{ACN} are currently missing. Further, a survey of goals in greater depth would be useful to identify the most important notions in the hierarchy and to provide intuitions and thus ease deciding on the correct notions for practitioners.
Additionally, such a survey helps to understand the relationships between currently-employed privacy enhancing technologies.
Finally, this understanding and the knowledge about how notions are related and differ can be used to define general techniques that strengthen \acp{ACN}.
Beyond that, an investigation of the applicability of our \iflong framework \else notions and hierarchy \fi to other areas, like e.g. anonymous payment channels, would be interesting.
\section*{Acknowledgements}
We would like to thank the anonymous reviewers for their helpful comments and feedback. Our work was partially funded by the German Research Foundation (DFG) within the Research Training Group GRK 1907, the German Federal Ministry of Education and Research (BMBF) within the EXPLOIDS project grant no. 16KIS0523 and European Union's Horizon 2020 project SAINT grant no. 740829.
\section{On the Choice of Notions}
\label{sec:choiceNotions}
The space of possible combinations of properties, and hence of conceivable privacy notions, is naturally large.
Due to this, we verify our selection of privacy goals by finding exemple use cases.
Additionally, we demonstrate the choice and the applicability of our definition by analyzing the privacy goals of Loopix, an \ac{ACN} that was recently published.
We additionally verify that our privacy notions include those of previous publications that suggest frameworks based on indistinguishability games, and provide a complete mapping in Section \ref{sec:ComparingFrameworks}.
\subsection{Example Use Cases for the Notions}
\label{sec:notionExamples}
We illustrate our notions by continuing the example of an activist group trying to communicate in a repressive regime, although our notions are generally applicable.
Recall the general idea of an indistinguishability game from the examples in Section \ref{background}:
To prove that an \ac{ACN} hides certain properties, whatever is allowed to be learned in the actual \ac{ACN} must not help a game adversary to win. This way, she is forced to win the game solely based on those properties that are required to remain hidden.
Therefore, the information allowed to be disclosed cannot be used in the game and hence must be kept identical in both scenarios.
Before giving examples, we need to order the notions. We chose to group them semantically.
Our resulting clusters are shown as gray boxes in Figure \ref{fig:hierarchyColored}.
Horizontally, we categorize notions that focus on receiver or sender protection (Receiver Privacy Notions or Sender Privacy Notions, respectively) or treat both with the same level of importance (Impartial Notions).
Inside those categories, we use clusters concerning the general leakage type: Both-side Unobservability means that neither senders, nor receivers or messages should be leaked.
Both-side Message Unlinkability means that it should be possible to link neither senders nor receivers to messages. In Sender Observability, the sender of every communication can be known, but not the message she sends or to whom she sends (Receiver and Message Observability analogous). In Sender-Message Linkability, who sends which message can be known to the adversary (Receiver-Message and Sender-Receiver Linkability analogous).
\iflong
\begin{table} [thb]
\center
\resizebox{0.45\textwidth}{!}{%
\begin{tabular}{ c p{6cm} }
Usage&Explanation\\ \hline
$D \in \{S,R,M\}$& Dimension $\in \{$Sender, Receiver, Message$\}$\\
Dimension $D$ not mentioned& Dimension can leak \\
Dimension $D$ mentioned &Protection focused on this dimension exists\\ \hline
$D \overline{O}$& not even the participating items regarding D leak,(e.g. $S\overline{O}$: not even $U$ leaks)\\
$DF \overline{L}$& participating items regarding D can leak, but not which exists how often (e.g. $SF\overline{L}$: $U$ leaks, but not $Q$)\\
$DM \overline{L}$& participating items regarding D and how often they exist can leak ( e.g. $SM\overline{L}$: $U,Q$ leaks)\\ \hline
$X -Prop, $& like X but additionally Prop can leak\\
$Prop \in \{|U|,H,P,|U'|, H',P', |M| \}$&\\ \hline
$(D_1 D_2)\overline{O}$& uses $R_{D_1 D_2}$; participating items regarding $D_1,D_2$ do not leak, (e.g. $(SR)\overline{O}$: $R_{SR}$)\\
$(D_1 D_2)\overline{L}$& uses $M_{D_1 D_2}$; participating items regarding $D_1,D_2$ can leak, (e.g. $(SR)\overline{L}$: $M_{SR}$)\\
$(2D)\overline{L}$& uses $T_{D}$; it can leak whether two participating item regarding $D$ are the same, (e.g. $(2S)\overline{L}$: $T_{S}$)\\ \hline
$\overline{O}$&short for $S \overline{O} R\overline{O} M\overline{O} $\\
$M\overline{O}[M\overline{L}]$& short for $ M\overline{O}(SM\overline{L}, RM\overline{L})$\\
$S\overline{O}\{X\}$& short for $S\overline{O} M\overline{O} X$\\
$D_1 X_1[ D_2 X_2]$& $D_1$ is dominating dimension, usually $D_1$ has more freedom, i.e. $X_2$ is a weaker restriction than $X_1$ \\ \hline
$C\overline{O}$& nothing can leak (not even the existence of any communication)\\
\end{tabular}}
\caption{Naming Scheme}
\label{Tab:NamingScheme1}
\end{table}
We also want to explain our naming scheme, which we summarize in Table \ref{Tab:NamingScheme1}. Our notions consider three dimensions: senders, messages and receivers.
Each notion restricts the amount of leakage on each of those dimensions. However, only dimensions that are to be protected are part of the notion name.
We use $\overline{O}$, short for unobservability, whenever the set of such existing items of this dimension cannot be leaked to the adversary.
E.g. $S\overline{O}$ cannot be achieved if the set of senders $U$ is leaked.
Notions carrying $\overline{L}$, short for unlinkability, can leak $U$ (for sender related notions), but not some other property related to the item.
E.g. we use $SF\overline{L}$ if the frequency $Q$ cannot be leaked and $SM\overline{L}$, if $Q$ can be leaked, but not the sender-message relation.
With a ``$-Prop$'' we signal that the property $Prop$ can additionally leak to the adversary. We distinguish those properties from $U$ and $Q$ used before as they give another leakage dimension (as illustrated later in the hierarchy).
Further, we use parentheses as in $(SR)\overline{O}$ to symbolize that if not only one set, but both sets of senders and receivers ($U$ and $U'$) are learned the notion is broken. Analogously, in $(SR)\overline{L}$ both sets can be learned but the linking between sender and receiver cannot.
For the last missing complex property, we use $(2S)\overline{L}$ to symbolize that two senders have to be linked to be the same identity to break this notion.
For readability we add some abbreviations:
We use $\overline{O}= S \overline{O} R\overline{O} M\overline{O} $ to symbolize unobservability on all three types and we summarize the remaining types in $ M\overline{O}(SM\overline{L}, RM\overline{L})$ to $M\overline{O}[M\overline{L}]$. $C\overline{O}$ symbolizes the notion in which nothing is allowed to leak.
Further, we use curly brackets to symbolize that the message cannot be leaked $ S\overline{O}\{X\}= S\overline{O} M\overline{O} X$ and we put the (in our understanding) non dominating part of the notion in brackets $S\overline{O} M\overline{O}= S\overline{O}[M\overline{O}]$.
\else
Table \ref{Tab:NamingScheme} of Appendix \ref{app:summaryNamingScheme} summarizes our naming scheme.
\fi
\vspace{-0.3cm}
\subsubsection{Impartial Privacy Notions}
\vspace{-0.2cm}
These notions treat senders and receivers equally.
\inlineheading{Message Observability}
The content of messages can be learned in notions of this group, as messages are not considered confidential. Because the real world adversary can learn the content, we must prevent her from winning the game trivially by choosing different content.
Hence, such notions use the property that the scenarios are identical except for the senders and receivers ($E_{SR}$) to ensure that the messages are equal in both scenarios.
\example{An activist of the group is already well-known and communication with that person leads to persecution of Alice.}
Alice needs a protocol that hides whether a certain sender and receiver communicate with each other; cf. Section \ref{sec:ComplexProperties} motivation of the complex property $M_{SR}$.
The resulting notion is \emph{Sender-Receiver Pair Unlinkability \ ($(SR)\overline{L}$)}.
\exampleCont{Only few people participate in the protocol. Then, just using the protocol to receive (send) something, when the well known activist is acting as sender (receiver) threatens persecution. }
Alice needs a protocol that hides whether a certain sender and receiver actively participate at the same time or not; cf. Section \ref{sec:ComplexProperties} motivation of the complex property $R_{SR}$.
The resulting notion is \emph{Sender-Receiver Unobservability\ ($(SR)\overline{O}$)}.
\inlineheading{Sender-Receiver Linkability (Message Confidentiality)}
Senders and receivers can be learned in notions of this group, because they are not considered private.
Hence, such notions include the property that the scenarios are identical, except for the messages ($E_{M}$) to ensure that the sender-receiver pairs are equal in both scenarios.
\example{Alice wants to announce her next demonstration. (1) Alice does not want the regime to learn the content of her message and block this event. (2) Further, she is afraid that the length of her messages could put her under suspicion, e.g. because activists tend to send messages of a characteristic length.}
In (1) Alice needs a protocol that hides the content of the messages. However, the adversary is allowed to learn all other attributes,
in particular the length of the message.
Modeling this situation, the scenarios may differ solely in the message
content; all other attributes must be identical in both scenarios, as
they may not help the adversary distinguish between them.
Beyond the above-described $E_{M}$, we must thus also request that the
length of the messages $|M|$ is identical in both scenarios.
The resulting notion is \emph{Message Unobservability leaking Message Length\ ($M\overline{O}-|M|$)}\footnote{We stick to our naming scheme here, although we would commonly call this confidentiality.}.
In the second case (2), the protocol is required to hide the length of the message.
The length of the messages thus may differ in the two scenarios, as the
protocol will need to hide this attribute. Hence, we remove the restriction that the message length $|M|$ has to be equal in both scenarios from the above notion and end up with \emph{Message Unobservability\ $M\overline{O}$}.
\iflong
\inlineheading{Both-Side Message Unlinkability}
Notions of this group are broken if the sender-message or receiver-message relation is revealed.
\example{The activists know that their sending and receiving frequencies are similar to regime supporters' and that using an \ac{ACN} is in general not forbidden, but nothing else. Even if the content and length of the message ($M\overline{O}$) and the sender-receiver relationship ($(SR)\overline{L}$) is hidden, the regime might be able to distinguish uncritical from critical communications, e.g. whether two activists communicate ``Today'' or innocent users an innocent message. In this case, the regime might learn that currently many critical communications take place and improves its measures against the activists.}
In this case, the activists want a protocol that hides the communications, i.e. relations of sender, message and receiver. However, as using the protocol is not forbidden and their sending frequencies are ordinary, the adversary can learn which users are active senders or receivers and how often they sent and receive. Modeling this, the users need to have the same sending and receiving frequencies in both scenarios $Q,Q'$, since it can be learned. However, everything else needs to be protected and hence, can be chosen by the adversary. This corresponds to the notion \emph{\heviaULLong \ ($M\overline{O}[M\overline{L}]$)}.
\fi
\inlineheading{Both-Side Unobservability}
Even the activity of a certain sender or receiver is hidden in notions of this group.
\exampleCont{It is a risk for the activists, if the regime can distinguish between two leading activists exchanging the message ``today'' and two loyal regime supporters exchanging the message ``tomorrow''.}
In this case, Alice wants to disclose nothing about senders, receivers, messages or their combination. However, the adversary can learn the total number of communications happening in the \ac{ACN}. Modeling this, we need to assure that for every communication in the first scenario, there exists one in the second. We achieve this by prohibiting the use of the empty communication with property \something. This results in the notion \emph{Unobservability \ ($\overline{O}$)}.
\example{The regime knows that a demonstration is close, if the total number of communications transmitted over this protocol increases. It then prepares to block the upcoming event.}
To circumvent this, Alice needs a protocol that additionally hides the total number of communications. Modeling this, we need to allow the adversary to pick any two scenarios. Particularly, use of the empty communication $\Diamond$ is allowed. This is represented in the property that nothing needs to be equal in the two scenarios, $\aleph$\ , and results in the notion \emph{Communication Unobservability \ ($C\overline{O}$)}.
Note that this is the only notion where the existence of a communication is hidden. All other notions include \something\ and hence do not allow for the use of the empty communication.
\subsubsection{Sender (and Receiver) Privacy Notions}
\vspace{-0.2cm}
These notions allow a greater freedom in picking the senders (or
receivers: analogous notions are defined for receivers.).
\inlineheading{Receiver-Message Linkability}
The receiver-message relation can be disclosed in notions of this group.
Hence, such notions include the property that the scenarios are identical except for the senders ($E_{S}$) to ensure the receiver-message relations are equal in both scenarios.
In \emph{Sender-Message Unlinkability \ ($SM\overline{L}$)} the total number of communications and how often each user sends can be additionally learned.
However, who sends which message is hidden.
In \emph{Sender-Frequency Unlinkability } \emph{($SF\overline{L}$)} the set of users and the total number of communications can be additionally disclosed.
However, how often a certain user sends is hidden, since it can vary between the two scenarios.
In \emph{Sender Unobservability \ ($S\overline{O}$)}, the total number of communications can additionally be disclosed.
However, especially the set of active senders $U_b$ is hidden.
If a notion further includes the following abbreviations, the following information can be disclosed as well:
\begin{itemize}
\item \emph{with User Number Leak} ($-|U|$): the number of senders that send something in the scenario
\item \emph{with Histogram Leak} ($-H$): the histogram of how many senders send how often
\item \emph{with Pseudonym Leak} ($-P$): which messages are sent from the same user
\end{itemize} \vspace*{-\baselineskip}
\exampleSpec{Alice is only persecuted when the regime can link a message with compromising content to her {\normalfont -- she needs a protocol that at least provides $SM\overline{L}-P$.}
However, since such a protocol does not hide the message content, the combination of all the messages she sent might lead to her identification. {\normalfont Opting for a protocol that additionally hides the message combination ($P$), i.e. provides $SM\overline{L}$, can protect her from this threat.} \\
Further, assuming most users send compromising content, and Alice's message volume is high, the regime might easily suspect her to be the origin of some compromising messages even if she is careful that the combination of her messages does not reidentify her {\normalfont -- she needs a protocol that does not disclose her sending frequencies ($Q$) although the combination of her messages ($P$) might be learned, i.e. achieving $SF\overline{L}-P$.}
However, Alice might fear disclosing the combination of her messages {\normalfont - then she needs a protocol achieving at least $SF\overline{L}-H$, which hides the frequencies ($Q$) and the message combination ($P$), but discloses the sending histogram, i.e. how many people sent how many messages ($H$).}
However, if multiple activist groups use the \ac{ACN} actively at different time periods, disclosing the sending histogram $H$ might identify how many activist groups exist and to which events they respond by more active communication {\normalfont -- to prevent this she needs a protocol that hides the frequencies $Q$ and the histogram $H$, i.e. provides $SF\overline{L}$.} \\
Further, not only sending a certain content, but also being an active sender (i.e. being in $U$) is prosecuted {\normalfont she might want to pick a protocol with at least $S\overline{O}-P$.
Again if she is afraid that leaking $P$ or $H$ together with the expected external knowledge of the regime would lead to her identification, she picks the corresponding stronger notion.}
If the regime knows that senders in the \ac{ACN} are activists and learns that the number of active senders is high, it blocks the \ac{ACN}.
{\normalfont In this case at least $S\overline{O}$ should be picked to hide the number of senders ($|U|$).}}
\example{ For the next protest, Alice sends two messages: (1) a location, and (2) a time. If the regime learns that both messages are from the same sender, they will block the place at this time even if they do not know who sent the messages.}
Alice then needs a protocol that hides whether two communications have the same sender or not. We already explained how to model this with complex property $T_{S}$ in Section \ref{sec:ComplexProperties}.
The resulting notion is Twice Sender Unlinkability}%/ Double Sender Unobservability ($(2S)\overline{L}$).
\iflong
\inlineheading{Receiver Observability}
In notions of this group the receiver of each communication can be learned.
Hence, such notions include the property that the scenarios are equal except for the senders and messages ($E_{SM}$) to ensure that they are equal in both scenarios.
\example{Consider not only sending real messages is persecuted, but also the message content or any combination of senders and message contents is exploited by the regime. If the regime e.g. can distinguish activist Alice sending ``today'' from regime supporter Charlie sending ``see u'', it might have learned an information the activists would rather keep from the regime. Further, either (1) the activists know that many messages of a certain length are sent or (2) they are not sure that many messages of a certain length are sent.
In case (1), Alice needs a \ac{ACN}, that hides the sender activity, the message content and their combination. However, the adversary can especially learn the message length. Modeling this, beyond the above described $E_{SM}$, the message lengths have to be equal $|M|$. This results in the notion \emph{Sender Unobservability with Message Unobservability leaking Message Length} ($S\overline{O}[M\overline{O}-|M|]$).
Note that in $S\overline{O}[M\overline{O}-|M|]$ the properties of $M\overline{O}-|M|$ are included and further the senders are allowed to differ in the two scenarios.
The second case (2) requires a protocol that additionally hides the message length. Hence, in modeling it we remove the property that the message lengths are equal $|M|$ from the above notion. This results in \emph{Sender Unobservability with Message Unobservability} ($S\overline{O}[M\overline{O}]$).
\example{ Alice's demonstration is only at risk if the regime can link a message with a certain content to her as a sender with a non negligible probability.}
Then at least \emph{Sender-Message Pair Unlinkability\ ($(SM)\overline{L}$)}, which is defined analogous to $(SR)\overline{L}$ is needed.
\exampleCont{However, $(SM)\overline{L}$ only allows Alice to claim that not she, but Charlie sent a critical message $m_a$ and the regime cannot know or guess better. Now assume that Dave is also communicating, then the regime might be able to distinguish Alice sending $m_a$, Charlie $m_c$ and Dave $m_d$ from Alice sending $m_d$, Charlie $m_a$ and Dave $m_c$. In this case, it might not even matter that Alice can claim that Charlie possibly sent her message. The fact that when comparing all three communications that possibly happened, Alice is more likely to have sent the critical message $m_a$ means a risk for her.}
To circumvent this problem Alice needs a protocol that not only hides the difference between single pairs of users, but any number of users. Modeling this, instead of the complex property $M_{SM}$, we need to restrict that the active senders' sending frequencies are equal, i.e. $SM\overline{L}$.
\example{In another situation our activists already are prosecuted for being a sender while a message with critical content is sent. }
In this case at least \emph{Sender-Message Pair Unobservability\ ($(SM)\overline{O}$)}, which is defined analogous to $(SR)\overline{O}$ is needed.
Analogous notions are defined for receivers.
\inlineheading{Sender Privacy Notions: Both-Side Message Unlinkability}
As explained with the example before in the case that Alice does not want any information about senders, receivers and messages or their combination to leak, she would use $\overline{O}$. However, the privacy in this example can be tuned down, if she assumes that the regime does not have certain external knowledge or that the users are accordingly careful. As explained for the Sender Notions with Receiver-Message Linkability before, in this case we might decide to allow $U', |U'|,Q',H',P'$ to leak.
If a notion $X \in \{R\overline{O}, R\overline{O}-|U'|, R\overline{O}-H', R\overline{O}-P',RF\overline{L},RF\overline{L}-H',RF\overline{L}-P',RM\overline{L},RM\overline{L}-P'\}$ is extended to \emph{Sender Unobservability by X} \emph{($ \sgame{X}$)}, the leaking of the sender-message relation is removed.
This is done by removing $E_{R}$.
Since the attacker now has a greater degree of freedom in choosing the senders and is (if at all) only restricted in how she chooses the receivers and messages, this is a special strong kind of Sender Unobservability.
Analogous notions are defined for receivers.\footnote{Note that $\sgame{R\overline{O}}=\rgame{S\overline{O}}=\overline{O}$.}
\else
Due to page limits the examples for the remaining notions can be found in Appendix \ref{app:examples}.
\fi
\section{Capturing Different Adversaries}
\label{sec:adversary}
The adversary model assumed in the protocol model can be further restricted by adding adversary classes, that filter the information from the adversary to the challenger and vice versa. Potentially many such adversary classes can be defined.
\inlineheadingTwo{Adversary Classes}
AnoA introduces adversary classes, i.e.
a PPT algorithm that can modify and filter all in- and outputs from/to the adversary. Adversary classes $\mathcal{C}$ can be included into our framework in exactly the same way: to wrap the adversary $\mathcal{A}$. Instead of sending the queries to $Ch$, $\mathcal{A}$ sends the queries to $\mathcal{C}$, which can modify and filter them before sending them to $Ch$. Similarly, the answers from $Ch$ are sent to $\mathcal{C}$ and possibly modified and filtered before sent further to $\mathcal{A}$. Adversary classes that fulfill reliability, alpha-renaming and simulatability (see \cite{backes17anoa} for definitions) are called single-challenge reducible. For such adversary classes it holds that every protocol $\Pi$ that is $(c, \epsilon,\delta)$-X for $\mathcal{C}(\mathcal{A})$, is also $(n \cdot c, n\cdot \epsilon,n\cdot \epsilon^{n \epsilon}\delta)$-X for $\mathcal{C}(\mathcal{A})$. Even though our framework extends AnoA's in multiple ways, the proof for multi-challenge generalization of AnoA is independent from those extensions and still applies to our framework.
\input{sections/proofs/SingleChallenge.tex}
Note that since the adversary class $\mathcal{C}$ is only a PPT algorithm, $\mathcal{C}(\mathcal{A})$ still is a PPT algorithm and hence, a possible adversary against X when analyzed without an adversary class. So, while adversary classes can help to restrict the capabilities of the adversary, results shown in the case without an adversary class carry over.
\inlineheadingTwo{Using UC-Realizability}
AnoA shows that, if a protocol $\Pi$ UC-realizes an ideal functionality $\mathcal{F}$, which achieves $(c,\epsilon, \delta)-X$, $\Pi$ also achieves ($c,\epsilon, \delta+\delta'$)-X for a negligible $\delta'$. As the proof is based on the $(\epsilon, \delta)$- differential privacy definition of achieving a notion and independent from our extensions to the AnoA framework, this result still holds.
\begin{proofsketch}
AnoA's proof assumes that $\Pi$ does not achieve ($\epsilon, \delta+\delta'$)-X. Hence, there must be an attack $\mathcal{A}$ distinguishing the scenarios. With this, they build a PPT distinguisher $\mathcal{D}$ that uses $\mathcal{A}$ to distinguish $\Pi$ from $\mathcal{F}$. Since, even with our extensions $\mathcal{A}$ still is a PPT algorithm, that can be used to build distinguisher $\mathcal{D}$ and the inequalities that have to be true are the same (since the definition of achieving ($\epsilon, \delta$)-X is the same as being $(\epsilon, \delta)$-differentially private. The combination of $\Pi$ not being ($\epsilon, \delta+\delta'$)-X and $\mathcal{F}$ being $(\epsilon, \delta)-X$ results in the same contradiction as in AnoA's proof.
\end{proofsketch}
\subsection{For CR Option}
\label{CRProofSketch}
\vspace{-1em}
\begin{theorem}
For all $X$:
\begin{enumerate}
\item $(c,n,\epsilon, \delta)-X \Rightarrow (c,n,\epsilon,\delta)X_{CR_{\#cr}}$.
\item $(c,n,\epsilon, \delta)-X \Leftarrow (c,n',\epsilon,\delta)X_{CR_{\#cr}}$ with $n \leq n'$
\end{enumerate}
\end{theorem}
\begin{proofsketch}
1.) follows from definition.
2.) Given an attack $\mathcal{A}_2$, we construct attack $\mathcal{A}_1$.
Let $\bar{n}$ be the number of previous challenges used in $\mathcal{A}_1$ so far.
For every batch query $bq$ with $n''$ challenge rows replace the challenge number of the $1$st $,\dots, {\#cr}$'st challenge row with $\bar{n}+1$ in $\mathcal{A}_1$. Continue for the next $\#cr$ challenge rows with the next higher challenge number. Repeat till all challenge rows have a new number.
Use all other queries as $\mathcal{A}_2$ does, give the answers to $\mathcal{A}_2$ and output whatever $\mathcal{A}_2$ outputs.
We show that $\mathcal{A}_1$ is valid against $X_{CR_{\#cr}}$:
The attack construction assures, that at most $\#cr$ challenge row are used in every challenge ($CR_{\#cr}$), all other aspects of $X$ are fullfilled in $\mathcal{A}_2$, too.
Since $\mathcal{A}_1$ perfectly simulates the given attack $\mathcal{A}_2$, it has the same success.
\end{proofsketch}
\fi
\section{Additional Tables and Lists}
\vspace{-2.7em}
\iflong
\else
\label{app:summaryNamingScheme}
\begin{table} [thb]
\center
\resizebox{0.48\textwidth}{!}{%
\begin{tabular}{ c p{6cm} }
Usage&Explanation\\ \hline
$D \in \{S,R,M\}$& Dimension $\in \{$Sender, Receiver, Message$\}$\\
Dimension $D$ not mentioned& Dimension can leak \\
Dimension $D$ mentioned &Protection focused on this dimension exists\\ \hline
$D \overline{O}$& not even the active participating items regarding D leak,(e.g. $S\overline{O}$: not even $U$ leaks)\\
$DF \overline{L}$& active participating items regarding D can leak, but not which exists how often (e.g. $SF\overline{L}$: $U$leaks, but not $Q$)\\
$DM \overline{L}$& active participating items regarding D and how often they exist can leak ( e.g. $SM\overline{L}$: $U,Q$ leaks)\\ \hline
$X -Prop, $& like X but additionally Prop can leak\\
$Prop \in \{|U|,H,P,|U'|, H',P', |M| \}$&\\ \hline
$(D_1 D_2)\overline{O}$& uses $R_{D_1 D_2}$; active participating items regarding $D_1,D_2$ do not leak, (e.g. $(SR)\overline{O}$: $R_{SR}$)\\
$(D_1 D_2)\overline{L}$& uses $M_{D_1 D_2}$; active participating items regarding $D_1,D_2$ can leak, (e.g. $(SR)\overline{L}$: $M_{SR}$)\\
$(2D)\overline{L}$& uses $T_{D}$; one active participating item regarding $D$ has to be identified twice, (e.g. $(2S)\overline{L}$: $T_{S}$)\\ \hline
$\overline{O}$&short for $S \overline{O} R\overline{O} M\overline{O} $\\
$M\overline{O}[M\overline{L}]$& short for $ M\overline{O}(SM\overline{L}, RM\overline{L})$\\
$S\overline{O}\{X\}$& short for $S\overline{O} M\overline{O} X$\\
$D_1 X_1[ D_2 X_2]$& $D_1$ is dominating dimension, usually $D_1$ has more freedom, i.e. $X_2$ is a weaker restriction than $X_1$ \\ \hline
$C\overline{O}$& nothing can leak (not even the existence of any communication)\\
\end{tabular}}
\caption{Naming Scheme}
\label{Tab:NamingScheme}
\end{table}
\label{app:NotionMapping}
\iflong
\begin{table} [b!]
\center
\resizebox{0.4\textwidth}{!}
\begin{tabular}{ c c c }
Framework & Notion & Equivalent to \\ \hline
AnoA& $\alpha_{SA}$&$ {\manySess{\static{\corrStandard{S\overline{O}}}}}_{CR_1}$\\
&$\alpha_{RA}$&${\manySess{\static{\corrStandard{R\overline{O}[M\overline{O}-|M|]}}}}_{CR_1}$\\
&$ \alpha_{REL}$&${\manySess{\static{\corrStandard{(SR)\overline{O}}}}}_{CR_2}$\\
&$ \alpha_{UL}$&${\manySess{\static{\corrStandard{(2S)\overline{L}}}}}_{CR_2}$\\
& $\alpha_{sSA}$&$\manySess{\static{\corrStandard{S\overline{O}}}}$\\
&$\alpha_{sRA}$&$\manySess{\static{\corrStandard{R\overline{O}[M\overline{O}-|M|]}}}$\\
&$ \alpha_{sREL}$\footnotemark&$\manySess{\static{\corrStandard{(SR)\overline{O}}}}$\\
&$ \alpha_{sUL}$\footnotemark&$\manySess{\static{\corrStandard{(2S)\overline{L}}}}$\\ \hline
Bohli's&$S/SA=R/SA$&$\overline{O}$\\
&$ R/SUP$& $S\overline{O}\{R\overline{O}-|U'|\}$\\
&$ R/WUP$& $S\overline{O}\{R\overline{O}-H'\}$\\
& $ R/PS$&$S\overline{O}\{R\overline{O}-P'\}$\\
&$ R/SUU$& $S\overline{O}\{RF\overline{L}\}$\\
&$ R/WUU$&$S\overline{O}\{RF\overline{L}-H'\}$\\
&$ R/AN$&$S\overline{O}\{RF\overline{L}-P'\}$\\
&$ R/WU$&$S\overline{O}\{RM\overline{L}\}$\\
& $ R/WA$&$S\overline{O}\{RM\overline{L}-P'\}$\\
&$ S/SA^\circ$& $S\overline{O}$\\
&$ S/SUP^\circ$&$S\overline{O}-|U|$\\
&$ S/WUP^\circ$&$S\overline{O}-H$\\
&$S/PS^\circ$&$S\overline{O}-P$\\
&$S/SUU^\circ$&$SF\overline{L}$\\
& $S/WUU^\circ$&$SF\overline{L}-H$\\
&$S/AN^\circ$&$SF\overline{L}-P$\\
&$S/WU^\circ$&$SM\overline{L}$\\
&$S/WA^\circ$&$SM\overline{L}-P$\\
&$S/X, R/X^\circ$&analogous\\
&$X^+$&$\corrStandard{\langle X\rangle}$\\
& $X^*$&$\corrStandard{\langle X^\circ\rangle}$\\ \hline
Hevia's&$UO$&$\noCorr{C\overline{O}}$, $k=1$\\
&$SRA$&$\noCorr{\overline{O}}$, $k=1$\\
&$SA^*$&$\noCorr{S\overline{O}\{RM\overline{L}\}}$, $k=1$\\
&$SA$&$\noCorr{S\overline{O}}$, $k=1$\\
&$UL$&$\noCorr{M\overline{O}[M\overline{L}]}$, $k=1$\\
&$SUL$&$\noCorr{SM\overline{L}}$, $k=1$\\
&$RA^*, RUL, RA$&analogous\\ \hline
Gelernter's&$R^{H, \tau}_{SA}$&$\noCorr{{{R^{H, \tau}_{SA}}} } \iff \noCorr{S\overline{O}-P}$, $k=1$\\
&$R^{H, \tau}_{SUL}$&$\noCorr{{{R^{H, \tau}_{S\overline{L}}}}} \iff \noCorr{SM\overline{L}-P}$, $k=1$\\
&$R_X$& analogous Hevia: $\langle X\rangle$ \\
&$R^H_X$& analogous Hevia: $\corrNoComm{\langle X\rangle}$\\
&$\hat{R}^H_X$& analogous Hevia $\corrOnlyPartnerSender{\langle X\rangle}$\\
\end{tabular}}
\caption{Equivalences, $\langle X \rangle$ equivalence of $X$ used}
\label{mapping}
\end{table}
\addtocounter{footnote}{-1}
\footnotetext{Under the assumption that in all cases $m_0$ is communicated like in $\alpha_{REL}$ of \cite{backes17anoa} and in $\alpha_{SREL}$ of one older AnoA version \cite{backes14anoa}.}
\stepcounter{footnote}
\footnotetext{Under the assumption that the receiver in stage 2 can be another than in stage 1 like in $\alpha_{UL}$ of \cite{backes17anoa}.}
\else
\vspace{-1cm}
\begin{table} [h!]
\center
\resizebox{0.35\textwidth}{!}
\begin{tabular}{ c c c }
Framework & Notion & Equivalent to \\ \hline
AnoA& $\alpha_{SA}$&$ {S\overline{O}}$\\
&$\alpha_{RA}$&${R\overline{O}[M\overline{O}-|M|]}$\\
&$ \alpha_{REL}$&${(SR)\overline{O}}$\\
&$ \alpha_{UL}$&${(2S)\overline{L}}$\\ \hline
Bohli's&$S/SA=R/SA$&$\overline{O}$\\
&$ R/SUP$& $S\overline{O}\{R\overline{O}-|U'|\}$\\
&$ R/WUP$& $S\overline{O}\{R\overline{O}-H'\}$\\
& $ R/PS$&$S\overline{O}\{R\overline{O}-P'\}$\\
&$ R/SUU$& $S\overline{O}\{RF\overline{L}\}$\\
&$ R/WUU$&$S\overline{O}\{RF\overline{L}-H'\}$\\
&$ R/AN$&$S\overline{O}\{RF\overline{L}-P'\}$\\
&$ R/WU$&$S\overline{O}\{RM\overline{L}\}$\\
& $ R/WA$&$S\overline{O}\{RM\overline{L}-P'\}$\\
&$ S/SA^\circ$& $S\overline{O}$\\
&$ S/SUP^\circ$&$S\overline{O}-|U|$\\
&$ S/WUP^\circ$&$S\overline{O}-H$\\
&$S/PS^\circ$&$S\overline{O}-P$\\
&$S/SUU^\circ$&$SF\overline{L}$\\
& $S/WUU^\circ$&$SF\overline{L}-H$\\
&$S/AN^\circ$&$SF\overline{L}-P$\\
&$S/WU^\circ$&$SM\overline{L}$\\
&$S/WA^\circ$&$SM\overline{L}-P$\\
&$S/X, R/X^\circ$&analogous\\ \hline
Hevia's&$UO$&${C\overline{O}}$\\
&$SRA$&${\overline{O}}$\\
&$SA^*$&${S\overline{O}\{RM\overline{L}\}}$\\
&$SA$&${S\overline{O}}$\\
&$UL$&${M\overline{O}[M\overline{L}]}$\\
&$SUL$&${SM\overline{L}}$\\
&$RA^*, RUL, RA$&analogous\\ \hline
Gelernter's&$R^{H, \tau}_{SA}$&${{{R^{H, \tau}_{SA}}} } \iff {S\overline{O}-P}$\\
&$R^{H, \tau}_{SUL}$&${{{R^{H, \tau}_{S\overline{L}}}}} \iff {SM\overline{L}-P}$\\
&$R_X$& analogous Hevia: $\langle X\rangle$ \\
\end{tabular}}
\caption{Equivalences, $\langle X \rangle$ equivalence of $X$ used}
\label{mapping}
\end{table}
\fi
\fi
\section{Relations to Other Frameworks}
\label{hierarchyFrameworks}
\section{Achieving $(\epsilon, \delta)$-X}
\label{app:epsilonDef}
For some use cases, e.g. if the court of your jurisdiction requires that the sender of a critical content can be identified with a minimal probability of a certain threshold e.g. 70\%, a non-negligible $\delta$ is suitable. Hence, we allow to specify the parameter of $\delta$ and include the well-known concept of differential privacy \cite{dwork14algorithmic} as AnoA does in the following Definition:
\begin{definition}[Achieving $(\epsilon, \delta)-X$]\label{def:achieveEpsilon}
An \ac{ACN} protocol $\Pi$ is $(\epsilon, \delta)$ -$X$ with $\epsilon \geq 0$ and $0 \leq \delta \leq 1$, iff for all PPT algorithms $\mathcal{A}$:
\begin{align*}
\text{Pr}[0= \langle \mathcal{A} \bigm| Ch(\Pi, X,0)\rangle ] &\leq \\
e^{\epsilon} \text{Pr}[0= \langle \mathcal{A} \bigm| Ch(\Pi, X,1)\rangle]&+ \delta\text{.}
\end{align*}
\end{definition}
Note that $\epsilon$ describes how close the probabilities of guessing right and wrong have to be. This can be interpreted as the quality of privacy for this notion. While $\delta$ describes the probability with which the $\epsilon$-quality can be violated. Hence, every \ac{ACN} protocol will achieve $(0,1)-X$ for any notion $X$, but this result does not guarantee anything, since with probability $\delta =1$ the $\epsilon$-quality is not met.
Note $\Pi$ is $(0,\delta)-X$ for a negligible $\delta$ is equivalent to the first definition of $\Pi$ achieves $X$.
\section{Challenger in Pseudocode}
\label{challengerAppendix}
\begin{algorithm}[h!]
\scriptsize
$\hat{U}=\emptyset$\\
$\text{stage}=1$\\
\uponMes{(Batch, $\underline{r}^0_0, \underline{r}^1_0, \underline{r}^0_1,\underline{r}^1_1, \Psi$)}{
\If {$\Psi \notin \{1,\dots, n\}$}
{output $\perp$}
\uIf{$\Psi \in T$}
{Retrieve $s:= s_\Psi$}
\Else
$s:= initializeState$ \\
\uIf{X uses only simple properties}
{$a\gets 0$}
\Else{
$a\gets ^R \{0,1\}$}
add $\Psi$ to $T$}
\If{$\lnot Validate(r)$}
{output $\perp$}
Compute $c_t=c_t+|CR(\underline{r}^0_0, \underline{r}^1_0, \underline{r}^0_1,\underline{r}^1_1)|$\\
\If{$c_t>c$}{output $\perp$}
$s'= calculateNewState( stage, s,\underline{r}^0_0,\underline{r}^1_0, \underline{r}^0_1,\underline{r}^1_1))$\\
\uIf {$checkFor(X, \Psi, s',\hat{U}, \underline{r}_0= (\underline{r}^0_0,\underline{r}^1_0), \underline{r}_1= (\underline{r}^0_1,\underline{r}^1_1)) $}
{$(\underline{r}, s_\Psi) \gets (\underline{r}^a_b,s'$)}
\Else{ output $\perp$}
Store $s_\Psi$\\
RunProtocol ($\underline{r}$) }
\uponMes{(Protocol, x)}{
\If{$x$ allowed}{Send $x$ to $\Pi$ }
}
\uponMes{(Corrupt, u)}{
\If{$X$= $\noCorr{X'}$ ($X' \in$ Privacy notions)}{
output $\perp$}
\If{$X$= $\static{X'}$ ($X' \in$ Privacy notions) and a batch query occurred before and $u\notin \hat{U}$}
{output $\perp$}
$\hat{U}$= $\hat{U}\cup \{u\}$ \\
Send internal state of $u$ to $\mathcal{A}$
}
\uponMes{(SwitchStage)}{
\If {$\lnot X$ includes $T_S$ or $T_R$}
{output $\perp$}
$\text{stage}=2$
}
\Validate{($\underline{r}^0_0=(S^0_{0_i},R^0_{0_i},m^0_{0_i},aux^0_{0_i})_{i \in \{1,\dots, l\}}, \underline{r}^1_0, \underline{r}^0_1,\underline{r}^1_1$)}{
\For{$r =(S,R,m,aux) \in \{\underline{r}^{a'}_{b'} \mid a',b' \in \{0,1\}\}$}
{\If {$\lnot Validate(S,R,m)$}
{output FALSE}
}
output TRUE
}
\RunProtocol{ ($\underline{r}=(S_i,R_i,m_i,aux_i)_{i \in \{1,\dots, l\}}$)}{
\For {$r_i \in \underline{r}$}{
\If{$aux_i=(session_i, ID_i)$}{
\uIf{$\not \exists y: (\text{session},y, ID_i)\in S_i$}{
$y' \gets \{0,1\}^k$\\
Store ($\text{session},y',ID$) in $S_i$
}
\Else{
$y':=y$ from $(\text{session},y, ID_i)\in S_i$ \\
}
Run $\Pi$ on $r_i$ with session ID $y'$\\ Forward responses sent by $\Pi$ to $\mathcal{A}$
}
\Else
{Run $\Pi$ on $r_i$\\ Forward responses sent by $\Pi$ to $\mathcal{A}$}
}}
\caption{Challenger $Ch(\Pi, X, c, n, b)$}
\label{Challenger}
\end{algorithm}
\FloatBarrier
\section{Notions in Pseudocode}
\label{pseudocode}
CalcNewState always calculates the states for all user roles (senders and receivers). This is for improved readability. It would be sufficient to calculate the parts of the state needed for the current notion.
\begin{algorithm} [h!]
\footnotesize
\initializeState{}{
$s=(1,1,(\tilde{s},\tilde{s},\tilde{s},\tilde{s},\tilde{r},\tilde{r},\tilde{r},\tilde{r}), 0, \emptyset, \emptyset)$\\
return $s$}
\calcNewState{$(newStage, s=(stage, session, users, cr, $ $s_{sender}=(L^0_{0_i},L^1_{0_i}, L^0_{1_i},L^1_{1_i})_{i \in \{1, \dots, k-1\}}$, $s_{rec}=(L'^0_{0_i},L'^1_{0_i}, L'^0_{1_i},L'^1_{1_i})_{i \in \{1, \dots, k-1\}}),$ $\underline{r}^0_0=(S^0_{0_i}, R^0_{0_i}, m^0_{0_i}, aux^0_{0_i})_{i \in \{1,..,l\}},$ $\underline{r}^1_0=(S^1_{0_i}, R^1_{0_i}, m^1_{0_i}, aux^1_{0_i})_{i \in \{1,..,l\}},$ $ \underline{r}^0_1=(S^0_{1_i}, R^0_{1_i}, m^0_{1_i}, aux^0_{1_i})_{i \in \{1,..,l\}})$, $ \underline{r}^1_1=(S^1_{1_i}, R^1_{1_i}, m^1_{1_i}, aux^1_{1_i})_{i \in \{1,..,l\}})$}{
\For {$a \in \{0,1\}$}{
\For {$b \in \{0,1\}$}{
$L^a_{b_k}=\{(u,M)\bigm|$ $M = \cup_{j:S^a_{b_j}=u} m^a_{b_i}\}$\\
$L'^a_{b_k}=\{(u,M)\bigm|$ $ M = \cup_{j:R^a_{b_j}=u} m^a_{b_i}\}$\\}}
{$cr=cr+|CR(\underline{r}^0_0,\underline{r}^1_0,\underline{r}^0_1,\underline{r}^1_1)|$}\\
\If{users=$(\tilde{s},\tilde{s},\tilde{s},\tilde{s},\tilde{r},\tilde{r},\tilde{r},\tilde{r})\land cr>0$}{
$((S^0_0,R^0_0,\_,\_),$ $(S^1_0,R^1_0,\_,\_),(S^0_1,R^0_,\_,\_),$ $(S^1_1,R^1_1,\_,\_), \dots)= CR(\underline{r}^0_0,\underline{r}^1_0, \underline{r}^0_1,\underline{r}^1_1)$\\
users= ($S^0_0,S^1_0,S^0_1,S^1_1,R^0_0,R^1_0,R^0_1,R^1_1$)
}
\If{users= ($S^0_0,S^1_0,S^0_1,S^1_1,R^0_0,R^1_0,R^0_1,R^1_1)$ $\land \exists (r^0_0,r^1_0,r^0_1, r^1_1)\in CR(\underline{r}^0_0,\underline{r}^1_0, \underline{r}^0_1,\underline{r}^1_1): (r^0_0,r^1_0,r^0_1, r^1_1)\neq ((S^0_0,R^0_0,\_,\_,\_),(S^1_0,R^1_0,\_,\_,\_),$ $(S^0_1,R^0_1,\_,\_,\_),(S^1_1,R^1_1,\_,\_,\_))$}
{session=$\perp$}
$\text{stage}=\text{newStage}$\\
output $s$
}
\caption{State Management}
\label{calcNewState}
\end{algorithm}
For the simple properties $checkFor$ uses $s^0_{b_k}$ from $s_{sender}$ resp. $s'^0_{b_k}$ from $s_{rec}$ to calculate $U_b,Q_b,P_b$ and $H_b$ and compares them like in Definition \ref{def:properties}. For the complex properties the senders and receivers of the first challenge row are stored in the $users$-part and the current stage in the $stage$-part of s. With this complex properties are computed as stated in Definition \ref{def:complexProperties}. Further, for the sessions-aspect the $session$-part of the state is set to $\perp$ if another sender-receiver-pair is used. With this and the $users-$ and $stage$-information the Definition \ref{def:sessions} can be checked. For the corruption it gets all the required information direct as input and can check it like defined in Definition \ref{def:corruption}. For the challenge complexity the number of challenge rows of this challenge is counted in the $cr$-part of the state and hence, Definition \ref{def:challengeComplexity} can be calculated.
\section{Challenger}
\label{sec:challenger}
\iflong
\vspace{-0.5cm}
This section describes the queries to the challenger $Ch(\Pi, X,c, n,b)$.
Pseudocode of our challenger is shown in Algorithm \ref{Challenger} of Appendix \ref{challengerAppendix}.
\else
This section describes the queries to the challenger $Ch(\Pi, X,b)$.
\fi
\inlineheading{Batch Query}
The batches $\underline{r}_0,\underline{r}_1$ that the adversary chooses for the two scenarios are represented
in batch queries.
\iflong
When the challenger receives a batch query, it will first check if their challenge number $\Psi$ is valid, i.e. $\Psi \in \{1, \dots, n\}$.
Further, the challenger will validate the communications that would be input to $\Pi$ for $b=0$ and $b=1$ as explained below.
\else
When the challenger receives a batch query, it will validate the communications that would be input to $\Pi$ for $b=0$ and $b=1$ as explained below.
\fi
\iflong If the game is not aborted so far, the challenger \iflong will retrieve or create the current state $s$ of the challenge $\Psi$, which stores \else uses stored \fi information to calculate the \iflong aspects. \else properties. \fi \fi
\iflong
Afterwards it checks if the allowed total number of challenge rows $c$ is met.
\fi
If all criteria are met so far, it checks that the \iflong aspects \else properties \fi of the privacy notion $X$ are met by using \iflong the current state of the challenge $s$ \else stored information about the past batches and \fi \iflong, the set of corrupted users $\hat{U}$,\fi the instances for both scenarios $\underline{r_0}^a, \underline{r^a}_1, a \in \{0,1\}$. Finally, it runs the instance belonging to the challenge bit $b$ of this game and the for this challenge randomly chosen instance bit $a$, if the \iflong aspects \else properties \fi are matched.
Otherwise, it returns $\perp$ and aborts the experiment.
Running the scenario in the \ac{ACN} protocol will return information that is forwarded to the adversary\iflong (or adversary class)\fi.
This information is what an adversary is assumed to be able to observe.
\iflong
\inlineheading{Corrupt Query}
Corrupt queries represent adaptive, momentary corruption of users (senders or receivers).
If the corrupt query is valid, the challenger forwards it to the \ac{ACN} protocol.
The \ac{ACN} protocol returns the current state of the user to the challenger, who forwards it to the adversary.
Active attacks based on corruption are realized with protocol queries if the protocol model allows for them.
\fi
\inlineheading{Protocol Query}
Protocol queries allow the adversary e.g. to compromise parts of the network\iflong(not the users)\fi, set parameters of the \ac{ACN} protocol or use other functionalities modeled in the protocol model, like e.g. active attacks.
The meaning and validity of those queries is specific to the analyzed \ac{ACN} protocol.
\inlineheading{Switch Stage Query}
If this query occurs and it is allowed, i.e. the notion contains a relevant property, the stage is changed from 1 to 2.
\inlineheading{Validate Communications}
If the analyzed \ac{ACN} protocol specifies restrictions of senders and receiver-message pairs, their validity is checked by this function.
\iflong
\inlineheading{Run Protocol}
\iflong
Run protocol first creates a new random session identifier if there is not already one for this session identifier of the adversary chosen $\text{session}$ with the extension $ID$.
This is done to ensure that the \ac{ACN} protocol is not broken only because the session identifier is leaked.
Afterwards it passes the communications to the \ac{ACN} protocol formalization.
\else
Run protocol passes the chosen communications to the \ac{ACN} protocol formalization.
\fi
\fi
\inlineheading{Remark to simple properties and instances}
In case the notion only uses simple properties, the challenger will pick $a=0$ and check the properties for
$\underline{r_1}_j = \underline{r_1^0}_j $ and $\underline{r_0}_j = \underline{r_0^0}_j$.
In case the notion uses a combination of simple and complex properties, the challenger will check the simple properties for any pair $\underline{r_1}_j = \underline{r_1^a}_j $ and $\underline{r_0}_j = \underline{r_0^{a'}}_j$ resulting by any $a, a' \in \{0,1\}$.
\section{Remaining Examples}
\label{app:examples}
\inlineheading{Impartial Notions: Both-Side Message Unlinkability}
Notions of this group are broken if the sender-message or receiver-message relation is revealed.
\example{The activists know that their sending and receiving frequencies are similar to regime supporters' and that using an \ac{ACN} is in general not forbidden, but nothing else. Even if the content and length of the message ($M\overline{O}$) and the sender-receiver relationship ($(SR)\overline{L}$) is hidden, the regime might be able to distinguish uncritical from critical communications, e.g. whether two activists communicate ``Today'' or innocent users an innocent message. In this case, the regime might learn that currently many critical communications take place and improves its measures against the activists.}
In this case, the activists want a protocol that hides the communications, i.e. relations of sender, message and receiver. However, as using the protocol is not forbidden and their sending frequencies are ordinary, the adversary can learn which users are active senders or receivers and how often they sent and receive. Modeling this, the users need to have the same sending and receiving frequencies in both scenarios $Q,Q'$, since it can be learned. However, everything else needs to be protected and hence, can be chosen by the adversary. This corresponds to the notion \emph{\heviaULLong \ ($M\overline{O}[M\overline{L}]$)}.
\inlineheading{Sender Privacy Notions: Receiver Observability}
In notions of this group the receiver of each communication can be learned.
Hence, such notions include the property that the scenarios are equal except for the senders and messages ($E_{SM}$) to ensure that they are equal in both scenarios.
\example{Consider not only sending real messages is persecuted, but also the message content or any combination of senders and message contents is exploited by the regime. If the regime e.g. can distinguish activist Alice sending ``today'' from regime supporter Charlie sending ``see u'', it might have learned an information the activists would rather keep from the regime. Further, either (1) the activists know that many messages of a certain length are sent or (2) they are not sure that many messages of a certain length are sent.
In case (1), Alice needs a \ac{ACN}, that hides the sender activity, the message content and their combination. However, the adversary can especially learn the message length. Modeling this, beyond the above described $E_{SM}$, the message lengths have to be equal $|M|$. This results in the notion \emph{Sender Unobservability with Message Unobservability leaking Message Length} ($S\overline{O}[M\overline{O}-|M|]$).
Note that in $S\overline{O}[M\overline{O}-|M|]$ the properties of $M\overline{O}-|M|$ are included and further the senders are allowed to differ in the two scenarios.
The second case (2) requires a protocol that additionally hides the message length. Hence, in modeling it we remove the property that the message lengths are equal $|M|$ from the above notion. This results in \emph{Sender Unobservability with Message Unobservability} ($S\overline{O}[M\overline{O}]$).
\example{ Alice's demonstration is only at risk if the regime can link a message with a certain content to her as a sender with a non negligible probability.}
Then at least \emph{Sender-Message Pair Unlinkability\ ($(SM)\overline{L}$)}, which is defined analogous to $(SR)\overline{L}$ is needed.
\exampleCont{However, $(SM)\overline{L}$ only allows Alice to claim that not she, but Charlie sent a critical message $m_a$ and the regime cannot know or guess better. Now assume that Dave is also communicating, then the regime might be able to distinguish Alice sending $m_a$, Charlie $m_c$ and Dave $m_d$ from Alice sending $m_d$, Charlie $m_a$ and Dave $m_c$. In this case, it might not even matter that Alice can claim that Charlie possibly sent her message. The fact that when comparing all three communications that possibly happened, Alice is more likely to have sent the critical message $m_a$ means a risk for her.}
To circumvent this problem Alice needs a protocol that not only hides the difference between single pairs of users, but any number of users. Modeling this, instead of the complex property $M_{SM}$, we need to restrict that the active senders' sending frequencies are equal, i.e. $SM\overline{L}$.
\example{In another situation our activists already are prosecuted for being a sender while a message with critical content is sent. }
In this case at least \emph{Sender-Message Pair Unobservability\ ($(SM)\overline{O}$)}, which is defined analogous to $(SR)\overline{O}$ is needed.
Analogous notions are defined for receivers.
\inlineheading{Sender Privacy Notions: Both-Side Message Unlinkability}
As explained with the example before in the case that Alice does not want any information about senders, receivers and messages or their combination to leak, she would use $\overline{O}$. However, the privacy in this example can be tuned down, if she assumes that the regime does not have certain external knowledge or that the users are accordingly careful. As explained for the Sender Notions with Receiver-Message Linkability before, in this case we might decide to allow $U', |U'|,Q',H',P'$ to leak.
If a notion $X \in \{R\overline{O}, R\overline{O}-|U'|, R\overline{O}-H', R\overline{O}-P',RF\overline{L},RF\overline{L}-H',RF\overline{L}-P',RM\overline{L},RM\overline{L}-P'\}$ is extended to \emph{Sender Unobservability by X} \emph{($ \sgame{X}$)}, the leaking of the sender-message relation is removed.
This is done by removing $E_{R}$.
Since the attacker now has a greater degree of freedom in choosing the senders and is (if at all) only restricted in how she chooses the receivers and messages, this is a special strong kind of Sender Unobservability.
Analogous notions are defined for receivers.\footnote{Note that $\sgame{R\overline{O}}=\rgame{S\overline{O}}=\overline{O}$.}
\section{Hierarchy And Tables}
\label{sec:HierarchyAndTables}
On the next page the hierarchy can be found combined with the symbol tables (Fig. \ref{fig:hierarchyColored} and Tables of \ref{tab:combinationOfTables}). Further, Fig. \ref{fig:hierarchyInefficient} and Fig. \ref{fig:hierarchyAnnotated} highlight special parts of the hierarchy and Fig. \ref{fig:hierarchyold} presents the mapping of the notions of the other frameworks to ours.
\begin{figure*}[h!]
\center
\includegraphics[width=0.8\textwidth]{images/hierarchy_structured.pdf}
\caption{Our new hierarchy of privacy notions divided into sender, receiver and impartial notions and clustered by leakage type.}\label{fig:hierarchyColored}
\end{figure*}
\begin{table*}
\centering
\begin{subtable}[t]{0.3\linewidth}
\resizebox{\linewidth}{!}{%
\begin{tabular}{ p{2.5cm} p{5.32cm} }
Usage&Explanation\\% \hline
$D \in \{S,R,M\}$& Dimension $\in \{$Sender, Receiver, Message$\}$\\
Dimension $D$\linebreak \emph{not} mentioned& Dimension can leak \\
Dimension $D$\linebreak mentioned &Protection focused on this dimension exists\\ \hline
$D \overline{O}$& not even the participating items regarding D leak,(e.g. $S\overline{O}$: not even $U$ leaks)\\
$DF \overline{L}$& participating items regarding D can leak, but not which exists how often (e.g. $SF\overline{L}$: $U$leaks, but not $Q$)\\
$DM \overline{L}$& participating items regarding D and how often they exist can leak ( e.g. $SM\overline{L}$: $U,Q$ leaks)\\ \hline
$X -Prop, $& like X but additionally Prop can leak\\
$Prop \in \{|U|,H,\allowbreak P,|U'|, H',P', |M| \}$&\\ \hline
$(D_1 D_2)\overline{O}$& uses $R_{D_1 D_2}$; participating items regarding $D_1,D_2$ do not leak, (e.g. $(SR)\overline{O}$: $R_{SR}$)\\
$(D_1 D_2)\overline{L}$& uses $M_{D_1 D_2}$; participating items regarding $D_1,D_2$ can leak, (e.g. $(SR)\overline{L}$: $M_{SR}$)\\
$(2D)\overline{L}$& uses $T_{D}$; it can leak whether two participating item regarding $D$ are the same, (e.g. $(2S)\overline{L}$: $T_{S}$)\\ \hline
$\overline{O}$&short for $S \overline{O} R\overline{O} M\overline{O} $\\
$M\overline{O}[M\overline{L}]$& short for $ M\overline{O}(SM\overline{L}, RM\overline{L})$\\
$S\overline{O}\{X\}$& short for $S\overline{O} M\overline{O} X$\\
$D_1 X_1[ D_2 X_2]$& $D_1$ is dominating dimension, usually $D_1$ has more freedom, i.e. $X_2$ is a weaker restriction than $X_1$ \\ \hline
$C\overline{O}$& nothing can leak (not even the existence of any communication)\\
\end{tabular}}
\subcaption{Naming Scheme}
\label{Tab:NamingScheme}
\end{subtable}
\hfill
\begin{subtable}[t]{0.2295\linewidth}
\resizebox{\linewidth}{!}{%
\begin{tabular}{ p{2cm} p{3.33cm} }
Notion&Properties \\% \hline
$(SR)\overline{L}$ &\something $ \land E_{SR} \land M_{SR}$ \\
$(SR)\overline{O}$ &\something$\land E_{SR} \land R_{SR}$ \\
$M\overline{O}$ &\something $ \land E_{M}$ \\
$M\overline{O}-|M|$ &\something $ \land E_{M} \land |M| $ \\
$M\overline{O}[M\overline{L}]$& \something$\land Q \land Q'$\\
$\overline{O}$& \something \\
$C\overline{O}$& $\aleph$\\ \hline
$S\overline{O}$& \something $\land E_{S}$\\
$S\overline{O}-|U| $& \something$\land E_{S} \land |U|$\\
$S\overline{O}-H$ &\something$\land E_{S} \land H$\\
$S\overline{O}-P $&\something $\land E_{S} \land P$\\
$SF\overline{L}$ &\something$\land E_{S} \land U$ \\
$SF\overline{L}-H$ &\something $\land E_{S} \land U \land H$\\
$SF\overline{L}-P $&\something $\land E_{S} \land U \land P$\\
$SM\overline{L}$ &\something $\land E_{S} \land Q$\\
$SM\overline{L}-P$ &\something $\land E_{S} \land Q \land P$\\
$(2S)\overline{L}$ &\something$\land E_{S} \land T_S $ \\
$R\overline{O}$ \ etc.&analogous\\ \hline
\mbox{$S\overline{O}[M\overline{O}]$}&\something $ \land E_{SM} $ \\
\mbox{$S\overline{O}[M\overline{O}-|M|]$}&\something $ \land E_{SM} \land |M| $ \\
$(SM)\overline{O}$ &\something $ \land E_{SM} \land R_{SM}$ \\
$(SM)\overline{L}$&\something $ \land E_{SM} \land M_{SM} $ \\
\mbox{$R\overline{O}[M\overline{O}-|M|]$} \ etc.&analogous \\ \hline
$\sgame{X'}$& Properties of $X'$, remove $E_{R}$ \\
\multicolumn{2}{p{4.5cm}}{for $X' \in \{R\overline{O},$ $ R\overline{O}-|U'|,$ $R\overline{O}-H',$ $R\overline{O}-P',$ $RF\overline{L},RF\overline{L}-H',$ $RF\overline{L}-P',$ $RM\overline{L},$ $RM\overline{L}-P'\}$}\\
$\rgame{X'}$ &analogous, remove $E_{S}$ \\
\end{tabular}}
\subcaption{Definition of the notions\iflong ~for all corruption options as defined in Table~\ref{tab:corruption options}\fi. A description of simple properties was given in Table~\ref{tab:information}. }
\label{NotionsDefinition2}
\end{subtable}
\hfill
\begin{subtable}[t]{0.4495\linewidth}
\resizebox{\linewidth}{!}{%
\begin{tabular}{ p{1.5cm} p{9cm}
Symbol &Description\\ \hline
$U/U'$& Who sends/receives is equal for both scenarios.\\
$Q/Q'$&Which sender/receiver sends/receives how often is equal for both scenarios.\\
$H/H'$& How many senders/receivers send/receive how often is equal for both scenarios.\\
$P/P'$& Which messages are sent/received from the same sender/receiver is equal for both scenarios.\\
$|U|/|U'|$& How many senders/receivers communicate is equal for both scenarios.\\
$|M|$& Messages in the two scenarios always have the same length.\\
$E_{S}$& Everything but the senders is identical in both scenarios.\\
$E_{R}, E_{M}$& analogous\\
$E_{SM}$&Everything but the senders and messages is identical in both scenarios.\\
$E_{RM}, E_{SR}$& analogous\\
$\aleph$¬hing will be checked; always true\\
$E_{\Diamond}$&If something is sent in both scenarios, the communication is the same.\\
\something & In every communication something must be sent.\\ \hline
$R_{SR}$& Adversary picks two sender-receiver-pairs. One of the senders and one of the receivers is chosen randomly.
For b=0 one of the adversary chosen sender-receiver pairs is drawn.
For b=1 the sender is paired with the receiver of the other pair.\\
$R_{SM}, R_{RM}$& analogous\\ \hline
$T_S$&Adversary picks two senders. The other sender might send the second time (stage 2).
For b=0 the same sender sends in both stages, for b=1 each sender sends in one of the stages.\\
$T_R$& analogous \\ \hline
$M_{SR}$&Adversary picks two sender-receiver-pairs. Sender-receiver-pairs might be mixed.
For b=0 both adversary chosen sender-receiver-pairs communicate.
For b=1 both mixed sender-receiver-pairs communicate.\\
$M_{SM},M_{RM}$&analogous\\
\end{tabular}}
\subcaption{Properties}
\label{tab:allNotions2}
\end{subtable}
\caption{Tables for our naming scheme~(\subref{Tab:NamingScheme}), notion definitions~(\subref{NotionsDefinition2}) and property definitions~(\subref{tab:allNotions2})}
\label{tab:combinationOfTables}
\end{table*}
\begin{figure*}
\center
\includegraphics[width=0.8\textwidth]{images/hierarchy_highlight_inefficient.pdf}
\caption{Protocols for notions highlighted are inefficient due to a result by Gelernter~\cite{gelernter13limits}.}\label{fig:hierarchyInefficient}
\end{figure*}
\begin{figure*}
\center
\includegraphics[width=0.8\textwidth]{images/hierarchy_highlight_important.pdf}
\caption{Depicted notions are a first guess on which notions might be important based on informal and formal usage in the related work.}\label{fig:hierarchyAnnotated}
\end{figure*}
|
train/arxiv
|
BkiUeSHxK02iP1lCY2ub
| 5
| 1
|
\section{\label{intro}Introduction}
BPM is used in many accelerator facilities to control beam orbit. There exists various types of BPMs, such as button BPM for electron accelerators and proton linac, linear cut BPM for proton and heavy ion synchrotrons, and stripline BPM for short bunch observation. However, bunched beam is essential for a BPM to induce a significant signal. For most synchrotrons, the injected beam is always stored as a coasting beam after many revolutions due to debunching. So, a RF cavity is indispensable to capture coasting beam to a bunched one for BPM observation. Meanwhile, in order to accelerate beam to a high energy, the coasting beam has to be captured with a high capture efficiency to perform a high acceleration efficiency. In general, capture process can be considered as adiabatic and nonadiabatic. Adiabatic capture states that parameters in the synchrotron Hamiltonian change slowly so that the particle orbit is a torus of constant Hamiltonian value\cite{Lee1}, and verse vice for nonadiabatic capture. To obtain high capture efficiency, adiabatic capture should be chosen firstly, however, which leads to a long capture time that is not suitable for rapid cycling synchrotron. So, in some sense, a trade-off has to be made between high capture efficiency and short capture time. In this article, two kinds of capture curves corresponding to adiabatic and nonadiabatic capture respectively are investigated and a comparison is discussed.
\begin{figure}
\includegraphics[width=8cm]{linear_cut.eps
\caption{\label{fig1} Schematic of a typical linear cut BPM.}
\end{figure}
BPM is of great importance for accelerator commission and operation, which consists of two pairs of plates inserted in the beam pipe, and horizontal and vertical beam center are determined by the induced signal on the plates. While many kinds of BPMs there exist, we limit our analysis to capacitive linear cut BPM, or shoe box BPM, which is widely used in proton and heavy ion synchrotrons such as HIMM. Although we considered only linear cut BPM here, the signal treatment methods we used can be applied to any types of capacitive BPMs. The schematic of a typical linear cut BPM is shown in Fig.\ref{fig1}. Such type of BPM can induce a precise linear dependence signal with respect to beam displacement, that is to say, having a good linearity. In this article, BPM signal is estimated for different input impendence, and an example is taken to deepen the understand of BPM signal.
\section{The synchrotron mapping equation and the stability of algorithm}
In the synchrotron phase space coordinates $(\phi,\Delta E)$, the mapping equations can be written as\cite{Lee1}
\begin{equation}
\label{eq1}
\left\{
\begin{aligned}
\Delta E_{n+1} & = \Delta E_{n}+qV(sin\phi_n-sin\phi_s) \\
\phi_{n+1} & = \phi_{n}+\frac{2\pi h\eta}{\beta^2E}\Delta E_{n+1}
\end{aligned}
\right.
\end{equation}
in which $\Delta E=E-E_s$, $\phi$ and $E$ are respectively the synchronous phase angle and the total energy for an off-momentum particle. $E_s$ refers to the total energy for an on-momentum particle. $q$ is the particle charge, $V$ is the RF voltage amplitude, $h$ is the harmonic number, $\eta=\frac{1}{\gamma^2}-\frac{1}{\gamma^2_T}$ is the phase slip factor, $\gamma=\frac{1}{\sqrt{1-\beta^2}}$ is the relativistic factor and $\gamma_T$ is the transition energy factor.
To check whether the algorithm is stable or not, the Jacobian for variables transformation plays an important role. The phase space area from $(\Delta E_{n}, \phi_{n})$ to $(\Delta E_{n+1}, \phi_{n+1})$ can be formulated as
\begin{equation}
\label{eq2}
\iint d\Delta E_{n+1}d\phi_{n+1}=\iint J(\Delta E,\phi)d\Delta E_{n}d\phi_{n}
\end{equation}
where the Jacobian is
\begin{equation}
\label{eq3}
\begin{aligned}
J(\Delta E,\phi)&=
\left|
\begin{array}{ccc}
\frac{\partial \Delta E_{n+1}}{\partial \Delta E_{n}} & \frac{\partial \Delta E_{n+1}}{\partial\phi_{n}} \\
\frac{\partial \phi_{n+1}}{\partial \Delta E_{n}} & \frac{\partial \phi_{n+1}}{\partial\phi_{n}}
\end{array}
\right|\\
&=
\left|
\begin{array}{ccc}
1 & qVcos\phi_{n} \\
0 & 1
\end{array}
\right|\\
&=1
\end{aligned}
\end{equation}
Obviously, the mapping from $(\Delta E_{n}, \phi_{n})$ to $(\Delta E_{n+1}, \phi_{n+1})$ preserves the phase space area. The mapping algorithm is therefore stable. Because of the simplicity and stability of the mapping equations, they are frequently used in particle tracking calculations.
\section{Synchrotron Hamiltonian}
Although Eq.\ref{eq1} can track the evolution of particles numerically, it is more transparent to work with the Hamiltonian for a qualitative understanding of bunches of particles\cite{Weng1}. Here, we introduce the phase angle $\phi$ canonically conjugate to $\frac{\Delta E}{\omega_0}$, so that the phase space has the dimensions of action (energy times time), where $\omega_0$ is the angular revolution frequency of the on-momentum particle. In the conjugate phase space coordinates $(\phi,\frac{\Delta E}{\omega_0})$, the differential synchrotron equations of motion are
\begin{equation}
\label{eq4}
\left\{
\begin{aligned}
\frac{d}{dt}\left(\frac{\Delta E}{\omega_0}\right) & = \frac{1}{2\pi}qV(sin\phi-sin\phi_s) \\
\frac{d}{dt}\phi & = \frac{\omega_0^2 h\eta}{\beta^2E}\frac{\Delta E}{\omega_0}
\end{aligned}
\right.
\end{equation}
If the Hamiltonian doesn't contain time explicitly, or Hamiltonian is adiabatic, we can say
\begin{equation}
\label{eq5}
\frac{dH}{dt}=\frac{\partial H}{\partial t} \equiv 0
\end{equation}
So, Hamiltonian can be a integral as follows
\begin{equation}
\label{eq6}
H\left(\phi,\frac{\Delta E}{\omega_0}\right)=\int_{\phi_s}^\phi\frac{\partial H}{\partial \phi}d\phi+\int_{0}^{\frac{\Delta E}{\omega_0}}\frac{\partial H}{\partial \frac{\Delta E}{\omega_0}}d\left(\frac{\Delta E}{\omega_0}\right)
\end{equation}
Combining Eq.\ref{eq4} and the canonical equation
\begin{equation}
\label{eq7}
\left\{
\begin{aligned}
\frac{\partial H}{\partial \frac{\Delta E}{\omega_0}} & = \frac{d}{dt}\phi\\
-\frac{\partial H}{\partial \phi} & = \frac{d}{dt}\left(\frac{\Delta E}{\omega_0}\right)
\end{aligned}
\right.
\end{equation}
we obtain
\begin{equation}
\label{eq8}
\begin{aligned}
H\left(\phi,\frac{\Delta E}{\omega_0}\right)&=\frac{\omega_0^2h\eta}{2\beta^2E}\left(\frac{\Delta E}{\omega_0}\right)^2\\
&+\frac{qV}{2\pi}\left[cos\phi-cos\phi_s+(\phi-\phi_s)sin\phi_s\right]
\end{aligned}
\end{equation}
Through simple analysis, we can easily know that the Hamiltonian has two fixed point $(0,\phi_s)$ and $(0,\pi-\phi_s)$ corresponding to coordinates $(\frac{\Delta E}{\omega_0}, \phi)$. $(0,\phi_s)$ is the stable fixed point, or centre point, and $(0,\pi-\phi_s)$ is the unstable fixed point, or saddle point. The Hamiltonian torus passes through saddle point is called the separatrix, which defines the longitudinal phase space area, or bucket area. Substituting saddle point $(0,\pi-\phi_s)$ to Eq.\ref{eq8}, the Hamiltonian value of the separatrix can be written as
\begin{equation}
\label{eq9}
H_{sx}=\frac{qV}{2\pi}\left[-2cos\phi_s+(\pi-2\phi_s)sin\phi_s\right]
\end{equation}
Thus, the phase space trajectory of the separatrix is $H=H_{sx}$, or
\begin{equation}
\label{eq10}
\Delta E_{sx}^2+\frac{qV\beta^2E}{\pi h\eta}\left[cos\phi+cos\phi_s-(\pi-\phi-\phi_s)sin\phi_s\right]=0
\end{equation}
Bucket height or maximum energy deviation is
\begin{equation}
\label{eq11}
\begin{aligned}
\Delta E_{max}&=\Delta E_{sx}(\phi_s)\\
&=\sqrt{-\frac{qV\beta^2E}{\pi h\eta}\left[2cos\phi_s-(\pi-2\phi_s)sin\phi_s\right]}
\end{aligned}
\end{equation}
Bucket area, namely the phase space area enclosed by the separatrix, is
\begin{equation}
\label{eq12}
\begin{aligned}
A_B&=\iint d\Delta E_{sx}d\phi=\oint \Delta E_{sx}(\phi)d\phi\\
&=16\sqrt{\frac{qV\beta^2E}{2\pi h|\eta|}}\alpha_b(\phi_s),\alpha_b(\phi_s)\approx \frac{1-sin\phi_s}{1+sin\phi_s}
\end{aligned}
\end{equation}
In the process of capture, $\phi_s=0$, so
\begin{equation}
\label{eq13}
\Delta E_{max}=2\sqrt{\frac{qV\beta^2E}{2\pi h|\eta|}}
\end{equation}
and
\begin{equation}
\label{eq14}
A_B=16\sqrt{\frac{qV\beta^2E}{2\pi h|\eta|}}
\end{equation}
Obviously, the phase space area and bucket height for a stationary bucket have the following relation,
\begin{equation}
\label{eq15}
A_B=8\Delta E_{max}
\end{equation}
\section{Adiabatic capture voltage curve}
\begin{figure}[!htb]
\includegraphics[width=8cm]{flow_chart.eps}
\caption{\label{fig2} Simulation flow chart. }
\end{figure}
To capture coasting beam (it is true for most sychrotrons) to a bucket without loss, the bucket area after capture finish should be chosen so that it is large enough compared to initial beam area. Assume that the energy deviation of the injected beam is $\Delta E$, the phase space area of the coasting beam in one RF period is
\begin{equation}
\label{eq16}
A_0=4\pi\Delta E
\end{equation}
Let $A_B=FA_0$, in which $F>1$ represents the phase space dilution factor. Thus, final capture voltage can be calculated via $A_B=FA_0$, that is
\begin{equation}
\label{eq17}
V_1=\frac{\pi^3h|\eta|}{8q\beta^2E}F^2\Delta E^2
\end{equation}
Usually, we select $F=\frac{3}{2}$\cite{LiuWei}.
To preserve emittance as a constant as much as possible in the process of capture, according to numerous experience, the initial bucket area should be much smaller than the coasting beam area\cite{NG1}. Suppose the bucket height equals initial energy deviation, that is $\Delta E_{max}=\Delta E$. Hence, the corresponding voltage is
\begin{equation}
\label{eq18}
V=\frac{\pi h|\eta|}{2q\beta^2E}\Delta E^2.
\end{equation}
So, initial capture voltage should obey $V_0<V$. Let $V_0=F_0V$, in which $F_0<1$ represents the phase space conservation factor. Thus, the initial capture voltage is
\begin{equation}
\label{eq19}
V_0=\frac{\pi h|\eta|}{2q\beta^2E}F_0\Delta E^2
\end{equation}
In principle, the smaller the $F_0$, the more conservation the emittance, however, much smaller $F_0$ leads to a much longer capture time. Therefore, $F_0$ should be reasonable in some sense, i.e. 0.2.
Adiabatic capture criterion\cite{Ng2} states that the relative change in bucket height or bucket area should be much slower than the synchrotron frequency $\omega_s/2\pi$, or
\begin{equation}
\label{eq20}
\omega_s\gg \frac{1}{A}\frac{dA}{dt}
\end{equation}
Since bucket area $A$ and synchrotron angular frequency $\omega_s$ are both proportional $\sqrt{V}$, so we can write down
\begin{equation}
\label{eq21}
A=k_1\sqrt{V}, \omega_s=k_2\sqrt{V}
\end{equation}
Substituting Eq. \ref{eq21} to Eq. \ref{eq20}, we have
\begin{equation}
\label{eq22}
\frac{1}{2V^{3/2}}\frac{dV}{dt}\ll k
\end{equation}
After integral,
\begin{equation}
\label{eq23}
V(0)^{-1/2}-V(t)^{-1/2}\ll kt
\end{equation}
Let $V(0)=V_0$, and then initial synchrotron angular frequency can be denoted as $\omega_{s0}=k\sqrt{V_0}$. So, above equation can be written as
\begin{equation}
\label{eq24}
\frac{\omega_{s0}t}{1-\sqrt{V_0/V(t)}}\gg 1
\end{equation}
Defining adiabatic parameter $n_{ad}=\frac{\omega_{s0}t}{1-\sqrt{V_0/V(t)}}$, thus adiabatic condition becomes $n_{ad}\gg 1$. Hence, adiabatic capture voltage is
\begin{equation}
\label{eq25}
V(t)=\frac{V_0}{\left(1-\omega_{s0}t/n_{ad}\right)^2}
\end{equation}
Assume the capture time and voltage change from $(V_0,0)$ to $(V_1,t_1)$ in the process of adiabatic capture,
where $V_0,V_1,t_1$ are initial capture voltage, final capture voltage and capture time respectively, so
\begin{equation}
\label{eq26}
V_1=\frac{V_0}{\left(1-\omega_{s0}t_1/n_{ad}\right)^2}
\end{equation}
Substituting Eq. \ref{eq26} to Eq. \ref{eq25}, finally, RF capture voltage curve can be formulated as
\begin{equation}
\label{eq27}
V(t)=\frac{V_0}{\left[1-\left(1-\sqrt{V_0/V_1}\right)\frac{t}{t_1}\right]^2}
\end{equation}
If $V_0,V_1,n_{ad}$ are given, capture time can be easily calculated as follows,
\begin{equation}
\label{eq28}
t_1=\frac{n_{ad}\left(1-\sqrt{V_0/V_1}\right)}{\omega_{s0}}
\end{equation}
where $\omega_{s0}=\omega_{0}\sqrt{\frac{h|\eta|qV_0}{2\pi\beta^2E}}$, and $\omega_{0}$ is the angular revolution frequency.
\section{Longitudinal RF capture simulation}
\begin{figure*}
\centering
\includegraphics[width=16cm,height=21cm]{adiabatic.eps}
\caption{\label{fig3}Plot of longitudinal phase space evolution after adiabatic capture with different adiabaticity under adiabatic voltage curve. The initial and final capture voltage are fixed at $19.9V$ and $553V$ and the initial bunch area is $27.494 MeVrad$. (a) is coasting beam with uniform distribution and energy deviation is $2.188MeV$. (b) is adiabatic capture with adiabatic parameter $n_{ad}=5$ corresponding to capture time $5.472ms$ or $16579$ turns. (c) is adiabatic capture with adiabatic parameter $n_{ad}=30$ corresponding to capture time $32.834ms$ or $99478$ turns. (d) is coasting beam with Gaussian distribution and RMS energy deviation is $0.729MeV$. (e) and (f) are adiabatic capture with the same settings with (b) and (c) respectively. }
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=16cm,height=21cm]{nonadiabatic.eps}
\caption{\label{fig4} Plot of longitudinal phase space evolution after nonadiabatic capture under Kang's voltage curve. The initial and final capture voltage are fixed at $19.9V$ and $553V$ and the initial bunch area is $27.494 MeVrad$. (a) is coasting beam with uniform distribution and energy deviation is $2.188MeV$. (b) is nonadiabatic capture with capture time $5.472ms$ or $16579$ turns. (c) is nonadiabatic capture with capture time $32.834ms$ or $99478$ turns. (d) is coasting beam with Gaussian distribution and RMS energy deviation is $0.729MeV$. (e) and (f) are nonadiabatic capture with the same settings with (b) and (c) respectively. }
\end{figure*}
The simulation flow chart is illustrated in Fig. \ref{fig2}. Initial conditions used in this simulation: particle $^{12}C^{6+}$, kinetic energy $E_k=200MeV/u$, fractional momentum deviation $\frac{\Delta p}{p}=\pm 5.0\times 10^{-4}$ or energy deviation $2.188MeV$, harmonic number $1$, transition energy $1.741$, phase space dilution factor $F=1.5$, phase space conservation factor $F_0=0.2$, the number of macro-particles $2000$. The initial coasting beam area is $A_0=4\pi\Delta E=4\pi\times 2.188 MeVrad=27.494MeVrad$. Considering phase space conservation factor and phase space dilution factor, the initial capture voltage and final capture voltage are $19.9V$ and $553V$, and the corresponding bucket area or bucket height is $7.827MeVrad$ or $0.978MeV$ and $41.241MeVrad$ or $5.155MeV$, respectively. The initial bucket area is small enough with respect to initial coasting beam area, and meanwhile, the final bucket area is large enough with respect to initial coasting beam area, which guarantees high capture efficiency.
Simulation of adiabatic capture with different adiabatic parameters is illustrated in Fig.\ref{fig3}. In the case of uniform distribution, for $n_{ad}=5$, the corresponding capture time is $5.472ms$ or $16579$ turns, and the final bunch length is $0.673rad$, which is large compared to the case of $n_{ad}=30$, i.e. $0.497rad$. And meanwhile, the phase space area is also large. So, small adiabatic parameter leads to short capture time, and therefore the bad capture. In addition, it is apparent that Gaussian distribution performs a better capture compared to uniform distribution, because Gaussian distribution has a dense beam core.
As a comparison, we introduce another kind of capture voltage curve, namely Kang's curve or nonadiabatic curve, which is employed by Kang Xiaojian to study adiabatic capture of a linac beam into buckets of the CIS booster of the IUCF Electron Cooler\cite{Kang}. The RF voltage used has the following form
\begin{equation}
\label{eq29}
V(t)=\left[3\left(\frac{t}{t_1}\right)^2-2\left(\frac{t}{t_1}\right)^3\right](V_1-V_0)+V_0
\end{equation}
The simulation result for nonadiabatic capture is shown in Fig.\ref{fig4}. The simulation parameters applied are identical to adiabatic capture, and also the phase space evolution is similar to adiabatic situation. However, the capture is much worse under the identical capture time compared to adiabatic capture, which performs a more loose phase space distribution and longer bunch length. The reason why this happens is that the much more rapid increase in RF voltage in nonadiabatic curve leads to a heavy filament of bunch at the beginning of the capture, which is shown in Fig.\ref{fig5}. Therefore, slow increase of RF voltage, especially, in the beginning of capture, is essential for adiabaticity.
From Fig.\ref{fig6}, we can see that, with the increase of capture time, the capture becomes more and more better. And with very long capture time or extremely slow capture, the four kinds of results become roughly similar, which indicates that the capture process reaches the adiabatic limit for either one case. Careful analysis reveals that Gaussian distribution with adiabatic capture is the best one. Meanwhile, to reach similar capture result, adiabatic capture takes less time than nonadiabatic capture. Finally, whatever which one case, there is no beam loss, because the final bucket area is large enough and the initial energy deviation is small enough.
\begin{figure}[!htb]
\includegraphics[width=8cm]{v_t.eps}
\caption{\label{fig5} Two kinds of capture voltage curves. }
\end{figure}
\begin{figure}[!htb]
\includegraphics[width=8cm]{tcap_phase.eps}
\caption{\label{fig6} Bunch length or phase distribution with capture time. }
\end{figure}
\begin{figure}[!htb]
\includegraphics[width=8cm]{circuit.eps}
\caption{\label{fig7} Equivalent circuit for a typical capacitive BPM. }
\end{figure}
\section{BPM signal estimation}
As described in introduction, BPM is a non-destructive diagnostics tool used most frequently at linacs, transfer line, and synchrotrons. The equivalent circuit\cite{Forck2011} for a typical capacitive BPM is shown in Fig.\ref{fig7}. Where $I_{im}(t)$ is the image current driven by the image charge on the plates, $U_{im}(t)$ is the induced voltage that is the signal we can observe directly, $C$ is the equivalent capacity of the BPM electrode against the beam pipe, and $R$ is the input resistor of amplifier.
For such a circuit, the impedance can be written easily as
\begin{equation}
\label{eq30}
Z=\frac{R}{1+j\omega RC}
\end{equation}
Hence, the voltage drop at the impedance $Z$ is
\begin{equation}
\label{eq31}
U_{im}(t)=ZI_{im}(t)
\end{equation}
The image current can be expressed as a function of beam current,
\begin{equation}
\label{eq32}
\begin{aligned}
I_{im}(t)\equiv \frac{dQ_{im}(t)}{dt}
&=k\frac{dQ_{beam}(t)}{dt}\\
&=K \frac{dI_{beam}(t)}{dt}\\
&=K j\omega I_{beam}(t)
\end{aligned}
\end{equation}
in which, $k$ and $K$ are constant, $I_{beam}(t)=I_0e^{j\omega t}$. So, time domain signal is
\begin{equation}
\label{eq33}
U_{im}(t)=K \frac{j\omega RC}{1+j\omega RC}I_{beam}(t)=Z(\omega)I_{beam}(t)
\end{equation}
Because $Z(\omega)$ is a frequency domain value, we have to transform $I_{beam}(t)$ to frequency domain, i.e. via FFT, which yields
\begin{equation}
\label{eq34}
U_{im}(\omega)=Z(\omega)I_{beam}(\omega)
\end{equation}
After inverse FFT, the time domain voltage signal becomes back to $U_{im}(t)$.
In the high frequency range, or $f\gg f_{cut}$, we have
\begin{equation}
\label{eq35}
Z(\omega)\propto \frac{j\omega/\omega_{cut}}{1+j\omega/\omega_{cut}}\approx 1
\end{equation}
Thus,
\begin{equation}
\label{eq36}
U_{im}(t)\propto I_{beam}(t)
\end{equation}
Therefore, voltage signal is the direct image of beam current in the range of high frequency.
However, in the low frequency range, or $f\ll f_{cut}$, we have
\begin{equation}
\label{eq37}
Z(\omega)\propto \frac{j\omega/\omega_{cut}}{1+j\omega/\omega_{cut}}\approx j\frac{\omega}{\omega_{cut}}
\end{equation}
Thus,
\begin{equation}
\label{eq38}
U_{im}(t)\propto \frac{dI_{beam}(t)}{dt}
\end{equation}
Therefore, voltage signal is the derivative of beam current in the range of low frequency.
\begin{figure*}
\centering
\includegraphics[width=15cm]{bpmSignalSimulated.eps}
\caption{\label{fig8}Simulated BPM signal for various bunch length with high and low input impedance. Figure (a), (b) and (c) correspond to input impedance of $50\Omega$, and bunch length is $1ns$, $10ns$ and $100ns$, respectively. Figure (d), (e) and (f) correspond to input impedance of $1\Omega$, and bunch length is $10\mu s$, $100\mu s$ and $1ms$, respectively. The solid blue curve with circle is the beam current distribution. }
\end{figure*}
The cut-off frequency for Fig.\ref{fig7} described circuit is $f_{cut}=\frac{1}{2\pi RC}$. For the case of so-called linear cut BPM or shoe box BPM used most often at synchrotrons, the typical value of the capacitance is about $C\approx 100pF$. Therefore, the cut-off frequency for low input impedance, i.e. $50\Omega$, is $f_{cut}=\frac{1}{2\pi\times 50 \times 10^{-10}}=31.831MHz$, and for high input impedance, i.e. $1M\Omega$, is $f_{cut}=\frac{1}{2\pi\times 10^6 \times 10^{-10}}=1.592KHz$. For a Gaussian current distribution in time domain with a width of $\sigma_t$, after FFT, it is still a Gaussian distribution, however, with a width of $\sigma_f=\frac{1}{2\pi\sigma_t}$ instead. In the case of low input impedance, if the bunch length is $\sigma_t=1ns$, and hence $f=159.155MHz$, the BPM voltage signal is a direct image of beam structure since the frequency is much larger than cut-off frequency. The calculated signal shape is shown in Fig.\ref{fig8}(a). If the bunch length is $\sigma_t=100ns$ or frequency $f=1.592MHz$, the signal is a derivative of beam structure since the frequency is much smaller than cut-off frequency, as shown in Fig.\ref{fig8}(c). And if the bunch length is $\sigma_t=10ns$ or frequency $f=15.915MHz$, the signal is a intermediate state since the frequency is comparable to cut-off frequency, which is shown in Fig.\ref{fig8}(b). For the case of longer bunch, the simulation results are shown in Fig.\ref{fig8}(d),(e) and (f), which is corresponding to high impedance case, i.e. $1M\Omega$ or $f_{cut}=1.592KHz$, and similar results are achieved. According to the simulation, we can conclude that the signal is always a derivative of beam current for a very long bunch, i.e. synchrotrons, and a direct image of beam current for a very short bunch, i.e. electron machine.
In the previous capture simulation, the smallest bunch length is $\sigma_{\phi}=0.473rad$ or $\sigma_t=\frac{\sigma_{\phi}}{2\pi f_0}=\frac{0.473}{2\pi\times 10^6}=75.3ns$ corresponding to revolution frequency $1MHz$, which is similar to the case of $\sigma_t=100ns$ with input impedance of $50\Omega$. The longest bunch length is $\sigma_{\phi}=0.771rad$ or $\sigma_t=123ns$, which is also similar to the case of $\sigma_t=100ns$. However, if the input impedance is $1M\Omega$, the signal is the direct image of beam current, which is similar to the case of $\sigma_t=10\mu s$. For HIMM, the BPM has a low input impedance $50\Omega$, so the signal should be a derivative of beam current. In the next section we will present the BPM signal measured at HIMM.
\section{HIMM BPM signal}
Heavy Ion Medical Machine (HIMM)\cite{Yang} is a new compact accelerator facility dedicated to carbon cancer therapy, which having been constructed in Wuwei and Lanzhou city. Eight BPMs are installed along the synchrotron, which are mainly used to control beam orbit.
The BPM is linear cut type with a low input impedance $50\Omega$, and the capacity is $100pF$. The raw analog signal is sent to NI 5734 sample card that has a sample rate of $120MS/s$ with 16 bit, and a attenuator is adapted to avoid signal saturation. A Labview based program is developed to acquire turn-by-turn data. For such machine, the bunch length is about $100ns$ according to simulation, so the signal shape is the same as Fig.\ref{fig8}(c). The measured BPM signal is shown in Fig.\ref{fig9}. Obviously, an expected agreement is presented.
\begin{figure}
\includegraphics[width=8cm]{bpm_signal_measured.eps}
\caption{\label{fig9} Measured BPM signal of HIMM. }
\end{figure}
\section{Conclusion}
In this paper, the longitudinal RF capture theory is reviewed in details and a simulation code is developed. The simulation of capture with various distribution and capture time is performed. A comparison between adiabatic and nonadiabatic capture is discussed, and adiabatic capture with Gaussian distribution is the best choice for capture process. Meanwhile, we can conclude that capture time is essential for the whole capture process and the longer the capture time the better the capture. In addition, BPM signal with different input impedance is simulated and compared to measured one from HIMM, and a good agreement is achieved. Hence, this method to simulate BPM signal is absolutely feasible.
\section*{References}
|
train/arxiv
|
BkiUdCTxK02iP4Y2ujgQ
| 5
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\section{Bottom$\rightarrow$up approach to spacetime}
In this section we will discuss mainly spinless particles to establish some
concepts in a familiar setting. A similar analysis applies to particles with
spin\footnote{A detailed discussion of the spinning particle and the related
generalized twistors, including higher dimensions and supersymmetry, will be
given in a future paper \cite{twistorsSpin}.}, therefore when we discuss
twistors in four dimensions, $d=4$ in section-\ref{twistd4}, we include the
spin in some of the discussion. Generalizations of the twistors to higher
dimensions, and D-branes, will be presented in the following sections. Other
generalizations, including spinning particles, supersymmetry, compactified
internal spaces, will be given elsewhere.
\subsection{Phase space}
A massless and spinless relativistic particle in $\left( d-1\right) $ space
dimensions is described by its position-momentum phase space coordinates
$\left( \vec{x}^{i}\left( t\right) ,\vec{p}^{i}\left( t\right) \right)
,$ $i=1,2,\cdots,\left( d-1\right) $ while its time development is governed
by the Hamiltonian $H=\left\vert \vec{p}\right\vert =\sqrt{\vec{p}^{2}}.$
Hence its action is
\begin{equation}
S=\int dt\left( \partial_{t}\vec{x}\cdot\vec{p}-\left\vert \vec{p}\right\vert
\right) . \label{noncov}%
\end{equation}
For a spinning particle we include spin degrees of freedom in an enlarged
phase space. Of course, a massless particle is a relativistically invariant
system, and this is verified by the fact that this action is invariant under
Lorentz transformations SO$\left( d-1,1\right) $. However, this symmetry is
only partially manifest in this action: rotation symmetry SO$\left(
d-1\right) $ is evident while the boost symmetry is hidden. To make the
Lorentz symmetry fully manifest one must introduce a gauge symmetry together
with extra matter gauge degrees of freedom and a gauge field. This process is
a first step in the bottom$\rightarrow$up approach that helps us discover a
deeper point of view of symmetries and their connection to spacetime.
The well known bottom-up approach in this example is to introduce the
worldline reparametrization gauge symmetry and then use the larger phase space
$\left( x^{\mu}\left( \tau\right) ,p^{\mu}\left( \tau\right) \right) $
with $\mu=0,1,2,\cdots,\left( d-1\right) .$ The action in the first order
formalism is
\begin{equation}
S\left( x,p\right) =\int d\tau\left( \partial_{\tau}x^{\mu}p_{\mu}-\frac
{1}{2}ep_{\mu}p_{\nu}\eta^{\mu\nu}\right) , \label{cov}%
\end{equation}
where $\eta_{\mu\nu}$ is the Minkowski metric and $e\left( \tau\right) $ is
the gauge field coupled to the generator of gauge transformations $p^{2}/2 $.
The gauge transformations
\begin{equation}
\delta_{\varepsilon}e=\partial_{\tau}\varepsilon\left( \tau\right)
,\;\;\delta_{\varepsilon}x^{\mu}=\varepsilon\left( \tau\right) p^{\mu
},\;\;\delta_{\varepsilon}p_{\mu}=0,
\end{equation}
transform the Lagrangian into an ignorable total derivative $\delta
_{\varepsilon}S=\int d\tau\partial_{\tau}\left( \varepsilon p^{2}/2\right)
=0.$ The action (\ref{cov}) has an evident global Lorentz symmetry due to the
fact that all terms are Lorentz dot products. Noether's theorem gives the
conserved charges $L^{\mu\nu}=x^{\mu}p^{\nu}-x^{\nu}p^{\mu}.$ These are gauge
invariant $\delta_{\varepsilon}L^{\mu\nu}=0$, and hence they are physical
observables. So the $L^{\mu\nu}$ remain conserved, and act as the generators
of symmetry of the gauge invariant action $S\left( x,p\right) ,$ even if
some arbitrary gauge is fixed.
The relation between Eqs.(\ref{noncov},\ref{cov}) is obtained in a fixed
gauge. This is the reverse process, namely it is part of the top$\rightarrow
$bottom approach that will be discussed in the next section. The equation of
motion with respect to the gauge field $e$ requires that the gauge generator
vanishes $p^{2}$=0, implying that the physical sector (massless particle on
mass shell) must be gauge invariant. The gauge symmetry can be fixed by taking
$x^{0}\left( \tau\right) =t\left( \tau\right) =\tau,$ and the constraint
can be solved for the canonical conjugate to $x^{0}=\tau,$ namely $p^{0}%
=\pm\left\vert \vec{p}\right\vert .$ The remaining phase space $\left(
\vec{x},\vec{p}\right) $ provides a parametrization of the gauge invariant
sector. Taking the positive root $p^{0}=+\left\vert \vec{p}\right\vert $, we
derive the non-covariant action Eq.(\ref{noncov}) for the massless particle as
the gauge fixed form of the gauge invariant Eq.(\ref{cov}). Similarly, the
gauge fixed form of the gauge invariant $L^{\mu\nu},$ given by $L^{ij}%
=x^{i}p^{j}-x^{j}p^{i},$ $L^{0i}=\tau p^{i}-x^{i}\left\vert \vec{p}\right\vert
,$ are the generators of the non-linearly realized hidden global SO$\left(
d-1,1\right) $ symmetry of the gauge fixed Lagrangian in Eq.(\ref{noncov}).
These generate the Lorentz transformations of phase space $\left( \vec
{x}\left( \tau\right) ,\vec{p}\left( \tau\right) \right) $ at any $\tau$
through the Poisson brackets $\delta_{\omega}\vec{x}\left( \tau\right)
=\frac{1}{2}\omega_{\mu\nu}\left\{ L^{\mu\nu},\vec{x}\right\} \left(
\tau\right) ,$ and $\delta_{\omega}\vec{p}\left( \tau\right) =\frac{1}%
{2}\omega_{\mu\nu}\left\{ L^{\mu\nu},\vec{p}\right\} \left( \tau\right) $
where $\tau$ is treated like a parameter. It can be checked that under these
transformations the non-covariant looking action in Eq.(\ref{noncov}) is
Lorentz invariant (dropping an ignorable total derivative $\delta S=\int
d\tau~\partial_{\tau}\left( f\left( \tau\right) \right) =0$).
The covariant formulation in Eq.(\ref{cov}) provides a greater flexibility to
analyze the system from a broader and more fundamental perspective. For
example, one may choose other gauges besides the timelike gauge $x^{0}\left(
\tau\right) =\tau$ that relates Eqs.(\ref{noncov}) and (\ref{cov}). In
particular the lightcone gauge $x^{+}\left( \tau\right) =\tau$, in which the
constraint $p^{2}=0$ is solved for the canonical conjugate $p^{-}=p_{\perp
}^{2}/2p^{+},$ has certain advantages in computation.
One may also analyze the system covariantly. For example in covariant
quantization one may apply the constraints on the physical states to derive
the Klein-Gordon equation, and from it the Klein-Gordon free field theory%
\begin{align}
p^{2}|\varphi\rangle & =0~\text{gauge invariant }\Leftrightarrow\text{
physical states.}\\
\langle x|p^{2}|\varphi\rangle & =0=-\partial^{\mu}\partial_{\mu}%
\varphi\left( x\right) \;\rightarrow S_{KG}=\int d^{4}x~\partial^{\mu
}\varphi^{\ast}\partial_{\mu}\varphi. \label{kg}%
\end{align}
The covariant formulation in Eq.(\ref{cov}) is one of the stations in the
bottom$\rightarrow$up approach toward a deeper point of view of symmetries and spacetime.
We are not done yet with the hidden symmetries of the non-covariant action in
Eq.(\ref{noncov}). This system has the larger symmetry SO$\left( d,2\right)
,$ namely conformal symmetry which is a general symmetry of massless systems.
This symmetry persists as a hidden symmetry in the covariant action in
Eq.(\ref{cov}) and in the Klein-Gordon action in Eq.(\ref{kg}). It has been
known that the SO$\left( d,2\right) $ symmetry can be made manifest in two
ways: one is twistors \cite{penrose}\cite{penrose2} (in $d=4$), and the other
is two time physics (2T-physics) \cite{2treviews}\cite{2tHandAdS} in any $d$.
Actually these are related to each other by gauge transformations in the
2T-physics formalism \cite{2ttwistor}\cite{twistorBP1} in the twistor gauge as
we will discuss in the rest of the paper in more detail. Either way, the route
to making the SO$\left( d,2\right) $ symmetry manifest involves introducing
a gauge symmetry, extra matter gauge degrees of freedom, and gauge fields.
\subsection{Twistor space in d=4\label{twistd4}}
The twistor formalism in $d=4$ \cite{penrose}\cite{penrose2} starts from a
different description of the massless particle. Instead of phase space degrees
of freedom $x^{\mu},p^{\mu}$ that are SO$\left( 3,1\right) $ vectors, it
introduces SO$\left( 3,1\right) $=SL$\left( 2,C\right) $ spinor degrees of
freedom $Z_{A}=\left(
\genfrac{}{}{0pt}{}{\mu^{\dot{\alpha}}}{\lambda_{\alpha}}%
\right) ,$ $A=1,2,3,4,$ constructed from SL$\left( 2,C\right) $ doublet
spinors $\mu^{\dot{\alpha}}$, $\lambda_{\alpha},$ each described by two
complex degrees of freedom $\alpha,\dot{\alpha}=1,2.$ The quartet $Z_{A}$ is
the spinor representation $\mathbf{4}$ of the conformal group SO$\left(
4,2\right) =$SU$\left( 2,2\right) $, while its conjugate $\bar{Z}%
^{A}=\left( Z^{\dagger}C\right) ^{A}=\left( \bar{\lambda}_{\dot{\alpha}%
}~\bar{\mu}^{\alpha}\right) ,$with the SU$\left( 2,2\right) $ metric
\begin{equation}
C=\sigma_{1}\times1, \label{C}%
\end{equation}
is the anti-quartet $\mathbf{\bar{4}}$ that corresponds to the second spinor
representation of SO$\left( 4,2\right) .$ An over-bar such as $\bar{\lambda
}_{\dot{\alpha}}$ means complex conjugate of $\lambda_{\alpha}.$ The spinor is
subject to a SU$\left( 2,2\right) $ invariant helicity constraint $Z_{A}%
\bar{Z}^{A}=\mu^{\dot{\alpha}}\bar{\lambda}_{\dot{\alpha}}+\lambda_{\alpha
}\bar{\mu}^{\alpha}=2h,$ where $h$ is the helicity of the particle. The
helicity constraint is the generator of a U$\left( 1\right) $ gauge symmetry
that acts on the twistor through the local phase transformation $Z_{A}\left(
\tau\right) \rightarrow Z_{A}^{\prime}\left( \tau\right) =e^{i\Lambda
\left( \tau\right) }Z_{A}\left( \tau\right) .$ The gauge invariant action
that describes the dynamics of twistors in four dimensions is%
\begin{equation}
S\left( Z\right) =\int d\tau\left( i\bar{Z}^{A}DZ_{A}-2hV\right)
,\;\;DZ_{A}\equiv\frac{\partial Z_{A}}{\partial\tau}-iVZ_{A}. \label{action}%
\end{equation}
Here the 1-form $Vd\tau$ is a U$\left( 1\right) $ gauge field on the
worldline, $DZ_{A}$ is the gauge covariant derivative that satisfies
$\delta_{\Lambda}\left( DZ_{A}\right) =i\Lambda\left( \tau\right) \left(
DZ_{A}\right) $ for $\delta_{\Lambda}V=\partial\Lambda/\partial\tau$ and
$\delta_{\Lambda}Z_{A}=i\Lambda\left( \tau\right) Z_{A}$. Note that the term
$2hV$ (absent in previous literature) is gauge invariant since it transforms
as a total derivative under the gauge transformation$.$ The reason for
requiring the U$\left( 1\right) $ gauge symmetry is the fact that the
overall phase of the $Z_{A}$ is unphysical and drops out in the relation
between phase space and twistors, as in Eq.(\ref{penrose}). Furthermore, the
equation of motion with respect to $V$ imposes the constraint $Z_{A}\bar
{Z}^{A}-2h=0,$ which is interpreted as the helicity constraint. Taking into
account that $\left( Z_{A}\bar{Z}^{A}-2h\right) $ is the generator of the
U$\left( 1\right) $ gauge transformations, the meaning of the vanishing
generator (or helicity constraint) is that only the U$\left( 1\right) $
gauge invariant sector of twistor space is physical.
To establish the equivalence between the massless spinless particle in
Eqs.(\ref{noncov},\ref{cov}) in $d=4,$ and twistors with vanishing helicity
$h=0,$ we must choose a U$\left( 1\right) $ gauge for $Z_{A}$ and solve the
constraint $Z_{A}\bar{Z}^{A}=\mu^{\dot{\alpha}}\bar{\lambda}_{\dot{\alpha}%
}+\lambda_{\alpha}\bar{\mu}^{\alpha}=0.$ First we count degrees of freedom.
The twistor $Z_{A}$ has 4 complex , or 8 real, degrees of freedom. Gauge
fixing the U$\left( 1\right) $ symmetry and solving the helicity constraint
removes 2 real degrees of freedom, leaving behind 6 real degrees of freedom,
which is the same number as the phase space degrees of freedom $\left(
\vec{x},\vec{p}\right) .$
More explicitly, Penrose has provided the transformation between twistors and
the phase space of spinless massless particles as follows%
\begin{equation}
\mu^{\dot{\alpha}}=-ix^{\dot{\alpha}\beta}\lambda_{\beta},\;\lambda_{\alpha
}\bar{\lambda}_{\dot{\beta}}=p_{\alpha\dot{\beta}}, \label{penrose}%
\end{equation}
where the $2\times2$ Hermitian matrices $x^{\dot{\alpha}\beta},$
$p_{\alpha\dot{\beta}}$ are expanded in terms of the Pauli matrices%
\begin{equation}
x^{\dot{\alpha}\beta}\equiv\frac{1}{\sqrt{2}}x^{\mu}\left( \bar{\sigma}_{\mu
}\right) ^{\dot{\alpha}\beta},\;p_{\alpha\dot{\beta}}\equiv\frac{1}{\sqrt{2}%
}p^{\mu}\left( \sigma_{\mu}\right) _{\alpha\dot{\beta}};\;\sigma_{\mu}%
\equiv\left( 1,\vec{\sigma}\right) ,\;\bar{\sigma}_{\mu}\equiv\left(
-1,\vec{\sigma}\right) .
\end{equation}
Here $\lambda_{\alpha}$ can be gauge fixed by choosing a phase, and the
helicity constraint is explicitly solved since
\begin{equation}
\bar{Z}^{A}Z_{A}=\left( \bar{\lambda}_{\dot{\alpha}}~\bar{\mu}^{\alpha
}\right) \left(
\genfrac{}{}{0pt}{}{\mu^{\dot{\alpha}}}{\lambda_{\alpha}}%
\right) =\bar{\lambda}_{\dot{\alpha}}\mu^{\dot{\alpha}}+\bar{\mu}^{\alpha
}\lambda_{\alpha}=-i\bar{\lambda}_{\dot{\alpha}}x^{\dot{\alpha}\beta}%
\lambda_{\beta}+i\bar{\lambda}_{\dot{\beta}}x^{\dot{\beta}\alpha}%
\lambda_{\alpha}=0. \label{tw1}%
\end{equation}
Furthermore, since $\lambda_{\alpha}\bar{\lambda}_{\dot{\beta}}$ is a
2$\times2$ matrix of rank one, its determinant vanishes. Then the
parametrization $p_{\alpha\dot{\beta}}=\lambda_{\alpha}\bar{\lambda}%
_{\dot{\beta}}$ of the momentum $p^{\mu},$ with $\sqrt{2}p^{0}=Tr\left(
p\right) =\bar{\lambda}\lambda,$ insures automatically that $p^{0}>0$ and
that the mass shell condition is satisfied $\det\left( \lambda\bar{\lambda
}\right) =\det\left( p\right) =p^{\mu}p_{\mu}=0.$ Furthermore, by inserting
the twistor transform in Eq.(\ref{penrose}) into the twistor action $S\left(
Z\right) $ we recover the massless particle action Eq.(\ref{cov}) on which
$p^{2}=0$ is already imposed by the form of $p^{\mu}$
\begin{align}
S\left( Z\right) & =\int d\tau\left[ \bar{\lambda}_{\dot{\alpha}}%
\frac{\partial}{\partial\tau}\left( x^{\dot{\alpha}\beta}\lambda_{\beta
}\right) -\bar{\lambda}_{\dot{\alpha}}x^{\dot{\alpha}\beta}\frac
{\partial\lambda_{\beta}}{\partial\tau}+0\right] =\int d\tau\frac{\partial
x^{\dot{\alpha}\beta}}{\partial\tau}\lambda_{\beta}\bar{\lambda}_{\dot{\alpha
}}\\
& =\frac{1}{2}\int d\tau\frac{\partial x_{\mu}}{\partial\tau}p_{\nu}Tr\left(
\bar{\sigma}^{\mu}\sigma^{\nu}\right) =\int d\tau\frac{\partial x_{\mu}%
}{\partial\tau}p_{\mu},\;\text{(with }p^{2}=0\text{)}%
\end{align}
Finally, fixing the $\tau$ reparametrization symmetry $x^{0}=\tau$ shows that
$S\left( Z\right) $ gives the original massless particle action
Eq.(\ref{noncov}). From this we conclude that the canonical structure of
twistors determined by $S\left( Z\right) ,$ namely $\left[ Z_{A},\bar
{Z}^{B}\right] =\delta_{A}^{~B},$ is equivalent to the canonical structure in
phase space determined by $S\left( x,p\right) ,$ namely $\left[ x^{\mu
},p_{\nu}\right] =i\delta_{~\nu}^{\mu},$ in the gauge invariant sectors
determined respectively by the constraints $\bar{Z}Z=0$ and $p^{2}=0$.
The analog of covariant quantization described in Eq.(\ref{kg}) can also be
done in twistor space. At the quantum level the Hermitian ordered product is
applied on physical states $\frac{1}{2}(Z_{A}\bar{Z}^{A}+\bar{Z}^{A}%
Z_{A})|\psi\rangle=2h|\psi\rangle$. Wavefunctions in twistor space are
obtained as $\psi\left( Z\right) =\langle Z|\psi\rangle$ where the operators
$Z_{A}$ are diagonalized on the states labelled as $\langle Z|$. Using the
fact that the canonical conjugate acts as $\bar{Z}^{A}\psi\left( Z\right)
=\langle Z|\bar{Z}^{A}|\psi\rangle=-\frac{\partial}{\partial Z_{A}}\psi\left(
Z\right) ,$ the physical state condition in $Z$ space $\frac{1}{2}\langle
Z|(Z_{A}\bar{Z}^{A}+\bar{Z}^{A}Z_{A})|\psi\rangle=2h\langle Z|\psi\rangle$
produces Penrose's homogeneity constraint, $Z_{A}\frac{\partial}{\partial
Z_{A}}\psi\left( Z\right) =\left( -2h-2\right) \psi\left( Z\right) ,$
known to correctly describe the quantum wavefunction $\psi\left( Z\right) $
of a particle of helicity $h.$
As the analog of the Klein-Gordon field theory of Eq.(\ref{kg}), we propose an
action in twistor space field theory that takes into account at once both the
positive and negative helicities
\begin{equation}
S_{h}\left( \psi\right) =\int d^{4}Z~\psi^{\ast}\left( Z_{A}\frac{\partial
}{\partial Z_{A}}+2h+2\right) \psi. \label{Sh}%
\end{equation}
The minimal action principle yields the homogeneity constraints. Indeed, the
equations of motion derived from $S_{h}\left( \psi\right) $ for $\psi
,\psi^{\ast}$ show that $\psi\left( Z\right) $ is the helicity $+h$
wavefunction, while $\psi^{\ast}\left( Z\right) $ is the helicity $-h$
wavefunction, so that together they describe a CPT invariant free field
theory. Therefore $S_{h}\left( \psi\right) $ is the twistor equivalent of
the Klein-Gordon, Dirac, Maxwell and higher spinning particle free field
actions in four flat dimensions. The free field theory $S_{h}\left(
\psi\right) $ is evidently invariant under conformal transformations
SU$\left( 2,2\right) $ by taking $\psi\left( Z\right) $ to transform like
a scalar while $Z_{A}$ transforms like the fundamental representation of
SU$\left( 2,2\right) .$ Our field theory proposal for an action principle
$S_{h}\left( \psi\right) $ for any spinning particle seems to be new in the literature.
What do we learn from the twistor approach? Perhaps, as Penrose would suggest,
twistors may be more basic than spacetime? Without a conclusive answer to that
question so far, it is nevertheless evident through the twistor program
\cite{penrose},\cite{penrose2}, and the recent twistor superstring
\cite{witten}-\cite{2tstringtwistors} that led to computational advances in
Super Yang Mills theory \cite{witten2}\cite{cachazo}, that twistor space is a
useful space, as an alternative to space-time, to discuss the physics of
massless systems.
But there is more to be said about twistors and spacetime. In addition to what
has traditionally been known about twistors, it has recently been shown
\cite{twistorBP1} that the \textit{same} twistor $Z_{A}$ in Eq.(\ref{action})
which is known to describe the on-shell massless particle, also describes a
variety of other on-shell dynamical systems. In particular the twistor
transform of Eq.(\ref{penrose}) has been generalized so that the \textit{same}
twistor also gives the d=4 particle worldline actions for the massive
relativistic particle, the particle on AdS$_{4}$ or AdS$_{3}\times$S$^{1}$ or
AdS$_{2}\times$S$^{2},$ the particle on $R\times S^{3},$ the nonrelativistic
free particle in 3 space dimensions, the nonrelativistic hydrogen atom in 3
space dimensions, and a related family of other particle systems. For example
the twistor transform for the massive relativistic particle is
\cite{twistorBP1}%
\begin{equation}
\mu^{\dot{\alpha}}=-ix^{\dot{\alpha}\beta}\lambda_{\beta}\frac{2a}%
{1+a},\;\;\lambda_{\alpha}\bar{\lambda}_{\dot{\beta}}=\frac{1+a\mathbf{\;}%
}{2a\mathbf{\;}}p_{\alpha\dot{\beta}}+\frac{m^{2}}{2(x\cdot p)a}x_{\alpha
\dot{\beta}}, \label{twistormassive}%
\end{equation}
where $a\equiv\sqrt{1+\frac{m^{2}x^{2}}{\left( x\cdot p\right) ^{2}}}$.
Using this transform instead of Eq.(\ref{penrose}) we find that the action
$S\left( Z\right) $ in Eq.(\ref{action}) reduces to the action for the
massive particle $S=\int dt\left( \partial_{t}\vec{x}\cdot\vec{p}-\sqrt
{\vec{p}^{2}+m^{2}}\right) $ instead of the massless particle of
Eq.(\ref{noncov}). The mass parameter emerges as a modulus in relating the
twistor components $\left( \mu^{\dot{\alpha}},\lambda_{\alpha}\right) $ to
phase space $\left( x^{\mu},p^{\mu}\right) $ in a different way than
Eq.(\ref{penrose}). As seen in \cite{twistorBP1} the mass parameter can also
be thought of as the value of an extra momentum component in $4+2$ dimensions.
Similarly in the case of the twistor transform for the H-atom, a combination
of mass and the Coulomb coupling constant is a modulus, and so on for other
moduli in more general cases. We see that certain mass parameters, certain
curvature or other spacetime metric parameters, and certain coupling constants
emerge as moduli in the generalized twistor transform.
The results in \cite{twistorBP1} imply that twistor space is a unifying space
for various dynamics, with different Hamiltonians, which must be related to
one another through a web of dualities. This raises deeper questions on the
meaning of space and time, and accentuates the feeling that twistor space may
be even more fundamental than was thought of before: namely, from the point of
view of twistor space, spacetime and dynamics are emergent concepts, as
explicitly shown in the examples in ref.\cite{twistorBP1}. There seems to be a
deeper meaning for twistors in the context of unification that goes beyond the
originally envisaged role for twistors\footnote{Promoting these results to the
quantum level we must expect another remarkable result that the wave equations
that follow from our proposed action $S_{h}\left( \psi\right) $ in
Eq.(\ref{Sh}) must also correctly describe all the other cases unified by the
same twistor as given in \cite{twistorBP1}. This point will not be further
discussed in this paper and will be taken up in a future publication.}.
Actually ref.\cite{twistorBP1} provides a connection between the more general
twistor properties just outlined and the concept of 2T-physics. This relation
will be explained through the top$\rightarrow$down approach in the next
section. Here we will briefly describe the relevant properties of 2T-physics
that unify various particle dynamics in 1T-physics, and thus promote the
notion of spacetime to a higher level.
\subsection{2T-physics}
2T-physics can be viewed as a unification approach for one-time physics
(1T-physics) systems through higher dimensions. It is distinctly different
than Kaluza-Klein theory because there are no Kaluza-Klein towers of states,
but instead there is a family of 1T systems with duality type relationships
among them.
A particle interacting with various backgrounds in $\left( d-1\right) +1$
dimensions (e.g. electromagnetism, gravity, high spin fields, any potential,
etc.), usually described in a worldline formalism in 1T-physics, can be
equivalently described in 2T-physics. The 2T theory is in $d+2$ dimensions,
but has enough gauge symmetry to compensate for the extra $1+1$ dimensions, so
that the physical (gauge invariant) degrees of freedom are equivalent to those
encountered in 1T-physics.
One of the strikingly surprising aspects of 2T-physics is that a given $d+2$
dimensional 2T theory descends, through gauge fixing, down to a family of
holographic 1T images in $\left( d-1\right) +1$ dimensions. Each image fully
captures the gauge invariant physical content of a unique parent 2T theory,
but from the point of view of 1T-physics each image appears as a different
1T-dynamical system. The members of such a family naturally must obey
duality-type relationships among them and share many common properties. In
particular they share the same overall global symmetry in $d+2$ dimensions
that becomes hidden and non-linear when acting on the fewer $\left(
d-1\right) +1$ dimensions in 1T-physics. Thus 2T-physics unifies many 1T
systems into a family that corresponds to a given 2T-physics parent in $d+2$ dimensions.
The essential ingredient in 2T-physics is the basic gauge symmetry Sp(2,R)
acting on phase space $X^{M},P_{M}$ in $d+2$ dimensions. The two timelike
directions is not an input, but is one of the outputs of the Sp$\left(
2,R\right) $ gauge symmetry. A consequence of this gauge symmetry is that
position and momentum become indistinguishable at any instant, so the symmetry
is of fundamental significance. The transformation of $X^{M},P_{M}$ is
generally a nonlinear map that can be explicitly given in the presence of
background fields \cite{2tbacgrounds}, but in the absence of backgrounds the
transformation reduces to a linear doublet action of Sp$\left( 2,R\right) $
on $\left( X^{M},P^{M}\right) $ for each $M$ \cite{2treviews}. The physical
phase space is the subspace that is gauge invariant under Sp$\left(
2,R\right) .$ Since Sp$\left( 2,R\right) $ has 3 generators, to reach the
physical space we must choose 3 gauges and solve 3 constraints. So, the gauge
invariant subspace of $d+2$ dimensional phase space $X^{M},P_{M}$ is a phase
space with six fewer degrees of freedom in $\left( d-1\right) $
\textit{space} dimensions $\left( x^{i},p_{i}\right) ,$ $i=1,2,\cdots\left(
d-1\right) .$
In some cases it is more convenient not to fully use the three Sp$\left(
2,R\right) $ gauge symmetry parameters and work with an intermediate space in
$\left( d-1\right) +1$ dimensions $\left( x^{\mu},p_{\mu}\right) ,$ that
includes time. This space can be further reduced to $d-1$ space dimensions
$\left( x^{i},p_{i}\right) $ by a remaining one-parameter gauge symmetry.
There are many possible ways to embed the $\left( d-1\right) +1$ or $\left(
d-1\right) $ phase space in $d+2$ phase space, and this is done by making
Sp(2,R) gauge choices. In the resulting gauge fixed 1T system, time,
Hamiltonian, and in general curved spacetime, are emergent concepts. The
Hamiltonian, and therefore the dynamics as tracked by the emergent time, may
look quite different in one gauge versus another gauge in terms of the
remaining gauge fixed degrees of freedom. In this way, a unique 2T-physics
action gives rise to many 1T-physics systems.
The general 2T theory for a particle moving in any background field has been
constructed \cite{2tbacgrounds}. For a spinless particle it takes the form%
\begin{equation}
S=\int d\tau~\left( \dot{X}^{M}P_{M}-\frac{1}{2}A^{ij}Q_{ij}\left(
X,P\right) \right) ,
\end{equation}
where the symmetric $A^{ij}\left( \tau\right) $ $,$ $i,j=1,2,$ is the
Sp$\left( 2,R\right) $ gauge field, and the three Sp$\left( 2,R\right) $
generators $Q_{ij}\left( X\left( \tau\right) ,P\left( \tau\right)
\right) , $ which generally depend on background fields that are functions of
$\left( X\left( \tau\right) ,P\left( \tau\right) \right) $, are required
to form an Sp$\left( 2,R\right) $ algebra. The background fields must
satisfy certain conditions to comply with the Sp$\left( 2,R\right) $
requirement. An infinite number of solutions to the requirement can be
constructed \cite{2tbacgrounds}. So any 1T particle worldline theory, with any
backgrounds, can be obtained as a gauge fixed version of some 2T particle
worldline theory.
The 1T systems discussed in \cite{twistorBP1}, and alluded to in connection
with twistors above, are obtained by considering the simplest version of
2T-physics without any background fields. The 2T action for a
\textquotedblleft free\textquotedblright\ 2T particle is \cite{2treviews}
\begin{equation}
S_{2T}\left( X,P\right) =\frac{1}{2}\int d\tau~D_{\tau}X_{i}^{M}X_{j}%
^{N}\eta_{MN}\varepsilon^{ij}=\int d\tau~\left( \dot{X}^{M}P^{N}-\frac{1}%
{2}A^{ij}X_{i}^{M}X_{j}^{N}\right) \eta_{MN}. \label{2Taction}%
\end{equation}
Here $X_{i}^{M}=\left( X^{M}~P^{M}\right) ,$ $i=1,2,$ is a doublet under
Sp$\left( 2,R\right) $ for every $M,$ the structure $D_{\tau}X_{i}%
^{M}=\partial_{\tau}X_{i}^{M}-A_{i}^{~j}X_{j}^{M}$ is the Sp(2,R) gauge
covariant derivative, Sp(2,R) indices are raised and lowered with the
antisymmetric Sp$\left( 2,R\right) $ metric $\varepsilon^{ij},$ and in the
last expression an irrelevant total derivative $-\left( 1/2\right)
\partial_{\tau}\left( X\cdot P\right) $ is dropped from the action. This
action describes a particle that obeys the Sp$(2,R)$ gauge symmetry, so its
momentum and position are locally indistinguishable due to the gauge symmetry.
The $\left( X^{M},P^{M}\right) $ satisfy the Sp$\left( 2,R\right) $
constraints
\begin{equation}
Q_{ij}=X_{i}\cdot X_{j}=0:\;X\cdot X=P\cdot P=X\cdot P=0,
\label{2Tconstraints}%
\end{equation}
that follow from the equations of motion for $A^{ij}$. The vanishing of the
gauge symmetry generators $Q_{ij}=0$ implies that the physical phase space is
the subspace that is Sp$\left( 2,R\right) $ gauge invariant. These
constraints have non-trivial solutions only if the metric $\eta_{MN}$ has two
timelike dimensions. So when position and momentum are locally
indistinguishable, to have a non-trivial system, two timelike dimensions are
necessary as a consequence of the Sp$\left( 2,R\right) $ gauge symmetry.
Thus the $\left( X^{M},P^{M}\right) $ in Eq.(\ref{2Taction}) are SO$\left(
d,2\right) $ vectors, labelled by $M=0^{\prime},1^{\prime},\mu$ or
$M=\pm^{\prime},\mu,$ and $\mu=0,1,\cdots,\left( d-1\right) $ or $\mu
=\pm,1,\cdots,\left( d-2\right) ,$ with lightcone type definitions of
$X^{\pm^{\prime}}=\frac{1}{\sqrt{2}}\left( X^{0^{\prime}}\pm X^{1^{\prime}%
}\right) $ and $X^{\pm}=\frac{1}{\sqrt{2}}\left( X^{0}\pm X^{3}\right) .$
The SO$\left( d,2\right) $ metric $\eta^{MN}$ is given by
\begin{align}
ds^{2} & =dX^{M}dX^{N}\eta_{MN}=-2dX^{+^{\prime}}dX^{-^{\prime}}+dX^{\mu
}dX^{\nu}\eta_{\mu\nu}\\
& =-\left( dX^{0^{\prime}}\right) ^{2}+\left( dX^{1^{\prime}}\right)
^{2}-\left( dX^{0}\right) ^{2}+\left( dX^{1}\right) ^{2}+\left(
dX_{\perp}\right) ^{2}\\
& =-2dX^{+^{\prime}}dX^{-^{\prime}}-2dX^{+}dX^{-}+\left( dX_{\perp}\right)
^{2}.
\end{align}
where the notation $X_{\perp}$ indicates SO$\left( d-2\right) $ vectors. So
the target phase space $X^{M},P_{M}$ is flat in $d+2$ dimension, and hence the
system in Eq.(\ref{2Taction}) has an SO$(d,2)$ global symmetry. The conserved
generators of SO$\left( d,2\right) $
\begin{equation}
L^{MN}=X^{M}P^{N}-X^{N}P^{M},\;\partial_{\tau}L^{MN}=0,
\end{equation}
commute with the SO$\left( d,2\right) $ invariant Sp$\left( 2,R\right) $
generators $X\cdot X$, $P\cdot P$, $X\cdot P$.
The Sp$\left( 2,R\right) $ local symmetry can be gauge fixed by choosing
three gauges and solving three constraints, but to keep some of the subgroups
of SO$\left( d,2\right) $ as evident symmetries it is more convenient to
choose two gauges and solve two constraints.
The SO$\left( d-1,1\right) $ covariant massless particle emerges if we
choose the two gauges, $X^{+^{\prime}}\left( \tau\right) =1$ and
$P^{+^{\prime}}\left( \tau\right) =0$, and solve the two constraints
$X^{2}=X\cdot P=0$ to obtain the $\left( d-1\right) +1$ dimensional phase
space $\left( x^{\mu},p_{\mu}\right) $ embedded in $\left( d+2\right) $
dimensions
\begin{align}
X^{M} & =\left( \overset{+^{\prime}}{1},\;\overset{-^{\prime}}{x^{2}%
/2}~,\;\overset{\mu}{x^{\mu}}\right) ,\label{massless1}\\
P^{M} & =\left( ~0~,~x\cdot p~,\;~p^{\mu}\right) . \label{massless2}%
\end{align}
The remaining constraint, $P^{2}=-2P^{+^{\prime}}P^{-^{\prime}}+P^{\mu}P_{\mu
}=p^{2}=0,$ which is the third Sp$\left( 2,R\right) $ generator, remains to
be imposed on the physical sector. In this gauge the 2T-physics action in
Eq.(\ref{2Taction}) reduces to the covariant massless particle action in
Eq.(\ref{cov}). Furthermore, the Sp$\left( 2,R\right) $ gauge invariant
$L^{MN}=X^{M}P^{N}-X^{N}P^{M}$ take the following nonlinear form
\begin{equation}
L^{\mu\nu}=x^{\mu}p^{\nu}-x^{\nu}p^{\mu},\;L^{+^{\prime}-^{\prime}}=x\cdot
p,\;L^{+^{\prime}\mu}=p^{\mu},\;L^{-^{\prime}\mu}=\frac{x^{2}}{2}p^{\mu
}-x^{\mu}x\cdot p. \label{conf0}%
\end{equation}
These are recognized as the generators of SO$\left( d,2\right) $ conformal
transformations of the $\left( d-1\right) +1$ dimensional phase space at the
classical level. Thus the conformal symmetry of the massless system is now
understood as the Lorentz symmetry in $d+2$ dimensions.
Having established the higher symmetrical version of the theory for the
massless particle as in Eq.(\ref{2Taction}) we reach a deeper level of
understanding of the symmetries as well as the presence of the $d+2$ nature of
the underlying spacetime. Furthermore we learn that the higher symmetrical
parent theory can be gauge fixed in many ways that produce not only the
massless particle system Eq.(\ref{noncov}) we started from, but also an
assortment of other particle dynamical systems, as discussed before
\cite{2treviews}\cite{2tHandAdS}\cite{twistorBP1}.
To emphasize this point we give also the massive relativistic particle gauge
by fixing two gauges and solving the constraints $X^{2}=X\cdot P=0$ explicitly
as follows%
\begin{align}
X^{M} & =\left( \overset{+^{\prime}}{\frac{1+a\mathbf{\;}}{2a\mathbf{\;}}%
},\;\overset{-^{\prime}}{\;\frac{x^{2}a\mathbf{\;}}{1+a\mathbf{\;}}%
}~,~~\overset{\mu}{x^{\mu}}\right) ,\;a\equiv\sqrt{1+\frac{m^{2}x^{2}%
}{\left( x\cdot p\right) ^{2}}}\label{massive2x}\\
P^{M} & =\left( \frac{-m^{2}}{2(x\cdot p)a},\;\;\left( x\cdot p\right)
a\;\mathbf{,\;\;}p^{\mu}\right) ,\;P^{2}=p^{2}+m^{2}=0. \label{massive2p}%
\end{align}
In this gauge the 2T action reduces to the relativistic massive particle
action
\begin{equation}
S=\int d\tau~\left( \dot{X}^{M}P^{N}-\frac{1}{2}A^{ij}X_{i}^{M}X_{j}%
^{N}\right) \eta_{MN}=\int d\tau\left( \dot{x}^{\mu}p_{\mu}-\frac{1}%
{2}A^{22}\left( p^{2}+m^{2}\right) \right) .
\end{equation}
A little recognized fact is that this action is invariant under SO$\left(
d,2\right) $. This SO$\left( d,2\right) $ does not have the form of
conformal transformations of Eq.(\ref{conf0}), but is a deformed version of
it, including the mass parameter. Its generators are obtained by inserting the
massive particle gauge into the gauge invariant $L^{MN}=X^{M}P^{N}-X^{N}P^{M}$%
\begin{align}
L^{\mu\nu} & =x^{\mu}p^{\nu}-x^{\nu}p^{\mu},\text{ \ \ \ }L^{+^{\prime
}-^{\prime}}=\left( x\cdot p\right) a,\label{LMNmassive1}\\
L^{+^{\prime}\mu} & =\frac{1+a\mathbf{\;}}{2a\mathbf{\;}}p^{\mu}+\frac
{m^{2}}{2\left( x\cdot p\right) a}x^{\mu}\label{LMNmassive2}\\
L^{-^{\prime}\mu} & =\frac{x^{2}a\mathbf{\;}}{1+a\mathbf{\;}}p^{\mu}-\left(
x\cdot p\right) ax^{\mu} \label{LMNmassive3}%
\end{align}
It can be checked explicitly that the massive particle action above is
invariant under the SO$\left( d,2\right) $ transformations generated by the
Poisson brackets $\delta x^{\mu}=\frac{1}{2}\omega_{MN}\left\{ L^{MN},x^{\mu
}\right\} $ and $\delta p^{\mu}=\frac{1}{2}\omega_{MN}\left\{ L^{MN},p^{\mu
}\right\} ,$ up to a reparametrization of $A^{22}$ by a scale and an
irrelevant total derivative.
Since both the massive and massless particles give bases for the same
representation of SO$\left( d,2\right) $, we must expect a duality
transformation between them. Of course this transformation must be an
Sp$\left( 2,R\right) =$SL$\left( 2,R\right) $ local gauge transformation
$\left(
\genfrac{}{}{0pt}{}{\alpha}{\gamma}%
\genfrac{}{}{0pt}{}{\beta}{\delta}%
\right) \left( \tau\right) $ with unit determinant $\alpha\delta
-\beta\gamma=1,$ that transforms the doublets $\left(
\genfrac{}{}{0pt}{}{X^{M}}{P^{M}}%
\right) \left( \tau\right) $ from Eqs.(\ref{massive2x},\ref{massive2p}) to
Eqs.(\ref{massless1},\ref{massless2}). The $\alpha,\beta,\gamma,\delta$ are
fixed by focussing on the doublets labelled by $M=+^{\prime}$
\begin{equation}
\left(
\begin{array}
[c]{c}%
\left( \frac{1+a\mathbf{\;}}{2a\mathbf{\;}}\right) \\
\left( \frac{-m^{2}}{2(x\cdot p)a}\right)
\end{array}
\right) =\left(
\begin{array}
[c]{cc}%
\left( \frac{1+a\mathbf{\;}}{2a\mathbf{\;}}\right) & 0\\
\left( \frac{-m^{2}}{2(x\cdot p)a}\right) & \left( \frac{2a\mathbf{\;}%
}{1+a\mathbf{\;}}\right)
\end{array}
\right) \left(
\begin{array}
[c]{c}%
1\\
0
\end{array}
\right) .
\end{equation}
Applying the inverse of this transformation on the doublets labelled by
$M=\mu$ gives the massless particle phase space (re-labelled by $\left(
\tilde{x}^{\mu},\tilde{p}^{\mu}\right) $ below) in terms of the massive
particle phase space (labelled by $\left( x^{\mu},p^{\mu}\right) $)%
\begin{equation}
\left(
\begin{array}
[c]{cc}%
\left( \frac{2a\mathbf{\;}}{1+a\mathbf{\;}}\right) & 0\\
\left( \frac{m^{2}}{2(x\cdot p)a}\right) & \left( \frac{1+a\mathbf{\;}%
}{2a\mathbf{\;}}\right)
\end{array}
\right) \left(
\begin{array}
[c]{c}%
x^{\mu}\\
p^{\mu}%
\end{array}
\right) =\left(
\begin{array}
[c]{c}%
\frac{2a\mathbf{\;}}{1+a\mathbf{\;}}x^{\mu}\\
\frac{1+a\mathbf{\;}}{2a\mathbf{\;}}p^{\mu}+\frac{m^{2}}{2(x\cdot p)a}x^{\mu}%
\end{array}
\right) \equiv\left(
\begin{array}
[c]{c}%
\tilde{x}^{\mu}\\
\tilde{p}^{\mu}%
\end{array}
\right)
\end{equation}
This duality transformation is a canonical transformation $\left\{ \tilde
{x}^{\mu},\tilde{p}^{\nu}\right\} =\eta^{\mu\nu}=\left\{ x^{\mu},p^{\nu
}\right\} .$ Also note that the time coordinate $\tilde{x}^{0}$ is different
than the time coordinate $x^{0},$ and so are the corresponding Hamiltonians
for the massless particle $\tilde{p}^{0}=\sqrt{\tilde{p}^{i}\tilde{p}^{i}}$
versus the massive particle $p^{0}=\sqrt{p^{i}p^{i}+m^{2}}.$
The same reasoning applies among all gauge choices of the 2T theory in
Eq.(\ref{2Taction}). All resulting 1T dynamical systems are holographic images
of the same parent theory. The global symmetry SO$\left( d,2\right) $ of the
2T-physics action is shared in the same singleton\footnote{At the classical
level all Casimir eigenvalues vanish, but at the quantum level, due to
ordering of factors the Casimir eigenvalues are non-zero (see
Eq.(\ref{casimirs})) and correspond to the singleton representation.}
representation by all the emergent lower dimensional theories obtained by
different forms of gauge fixing. These include special cases of particles that
are massive or massless, relativistic and nonrelativistic, in flat or curved
spaces, free or interacting. This is an established fact in previous work on
2T-physics \cite{2treviews}\cite{2tHandAdS}, and it came into new focus by
displaying the explicit twistor$/$phase space transforms given in
\cite{twistorBP1}.
As seen above, the descendants of the $d+2$ dimensional 2T-physics action are
1T-physics dynamical systems that are dual to each other. Therefore we must
expect that they all have the same twistor representation modulo twistor gauge
transformations. This will be derived through the top$\rightarrow$down
approach in the next section.
It must be emphasized that as a by product of the top$\rightarrow$down
approach certain physical parameters, such as mass, parameters of spacetime
metric, and some coupling constants appear as moduli in the holographic image
while descending from $d+2$ dimensional phase space to $\left( d-1\right)
+1$ dimensions or to twistors. Explicit examples of these have appeared in
\cite{twistorBP1}.
\section{Top$\rightarrow$down approach}
The 2T-physics action (\ref{2Taction}) and the twistor action (\ref{action})
in four dimensions are related to one another and can both be obtained as
gauge choices from the same theory in the 2T-physics formalism. To demonstrate
this fact and setup a general formalism for deriving the twistor transform in
any dimension, with or without supersymmetry, we discuss a unified theory that
defines the top$\rightarrow$down approach. This formalism was introduced in
\cite{2ttwistor} and developed further in the context of the twistor
superstring \cite{2tsuperstring}\cite{2tstringtwistors}. In this section we
begin without supersymmetry or compactified dimensions. These will be
introduced later.
In the case of $d=4$ the generalized twistor transform was applied explicitly
to specific cases in \cite{twistorBP1}, but the derivation of the general
formula was relegated to the present paper. In this section we will derive the
general twistor transform between twistor space $Z$ in $d$-dimensions and the
$d+2$ dimensional phase space $X^{M},P_{M}$ or $d$ dimensional phase space
$x^{\mu},p_{\mu}.$ We will show how it works explicitly in $d=3,4,5,6$ and
higher dimensions.
\subsection{SO$\left( d,2\right) $ local and SO$\left( d,2\right) $ global
symmetry\label{ggroup}}
In addition to the phase space SO$\left( d,2\right) $ vectors $\left(
X^{M},P^{M}\right) ,$ we introduce a group element $g\left( \tau\right)
\in$SO$\left( d,2\right) $ in the \textit{spinor} representation. It is
given by
\begin{equation}
g\left( \tau\right) =\exp\left( \frac{i}{2}S^{MN}\omega_{MN}\left(
\tau\right) \right) =\exp\left( \frac{1}{4}\Gamma^{MN}\omega_{MN}\left(
\tau\right) \right) \label{g}%
\end{equation}
The 2T particle action with Sp$\left( 2,R\right) $ \& SO$\left( d,2\right)
$ local and SO$\left( d,2\right) $ global symmetry is%
\begin{equation}
S_{2T}\left( X,P,g\right) =\int d\tau\left[ \frac{1}{2}\varepsilon
^{ij}\partial_{\tau}X_{i}\cdot X_{j}-\frac{1}{2}A^{ij}X_{i}\cdot X_{j}%
+\frac{4}{s_{d}}Tr\left( ig^{-1}\partial_{\tau}gL\right) \right] ,\;\;
\label{S2T}%
\end{equation}
where the trace is in spinor space\footnote{The trace in spinor space gives
the dimension of the spinor $Tr\left( 1\right) =s_{d}$ and $Tr\left(
\Gamma^{M}\bar{\Gamma}^{N}\right) =s_{d}\eta^{MN}.$ For even dimensions
$s_{d}=2^{d/2}$ for the Weyl spinor of SO$\left( d,2\right) ,$ and the
$\bar{\Gamma}^{M},\Gamma^{M}$ are the gamma matrices in the bases of the two
different spinor representations$.$The correctly normalized generators of
SO$\left( d,2\right) $ in the spinor representation are $S^{MN}=\frac{1}%
{2i}\Gamma^{MN}$, where the even-dimension gamma matrices satisfy $\Gamma
^{M}\bar{\Gamma}^{N}+\Gamma^{N}\bar{\Gamma}^{M}=2\eta^{MN},$ while
$\Gamma^{MN}=\frac{1}{2}\left( \Gamma^{M}\bar{\Gamma}^{N}-\Gamma^{N}%
\bar{\Gamma}^{M}\right) $, $\Gamma^{MNK}=\frac{1}{3!}\left( \Gamma^{M}%
\bar{\Gamma}^{N}\Gamma^{K}\mp\text{permutations}\right) $, etc. There exists
a metric $C$ of SO$\left( d,2\right) $ in the spinor representation such
that when combined with hermitian conjugation it gives $C^{-1}\left(
\Gamma^{M}\right) ^{\dagger}C=-\bar{\Gamma}^{M}$ and $C^{-1}\left(
\Gamma^{MN}\right) ^{\dagger}C=-\Gamma^{MN}.$ So the inverse $g^{-1}$ is
obtained by combining hermitian and $C$-conjugation $g^{-1}=C^{-1}\left(
g\right) ^{\dagger}C\equiv\bar{g}.$ In odd number of dimensions the
even-dimension gamma matrices above are combined to a larger matrix
$\hat{\Gamma}^{M}=\left(
\genfrac{}{}{0pt}{}{0}{\Gamma^{M}}%
\genfrac{}{}{0pt}{}{\bar{\Gamma}^{M}}{0}%
\right) $ for $M=0^{\prime},1^{\prime},0,1,\cdots,\left( d-2\right) $ and
add one more matrix for the additional last dimension $\hat{\Gamma}%
^{d-1}=\left(
\genfrac{}{}{0pt}{}{1}{0}%
\genfrac{}{}{0pt}{}{0}{-1}%
\right) $. The text is written as if $d$ is even; for odd dimensions we
replace everywhere $\hat{\Gamma}^{M}$ for both $\Gamma^{M}$ and $\bar{\Gamma
}^{M}.$ \label{ginverse}} and the matrix $L$ is given by%
\begin{equation}
L\equiv\frac{1}{4i}\Gamma_{MN}L^{MN}=\frac{1}{4i}\left( \Gamma\cdot
X~\bar{\Gamma}\cdot P-\Gamma\cdot P~\bar{\Gamma}\cdot X\right) . \label{L}%
\end{equation}
The first two terms of the action $S_{2T}\left( X,P,g\right) $ are the same
as Eq.(\ref{2Taction}), hence these terms are invariant under Sp$\left(
2,R\right) $ which acts on $X_{i}^{M}=\left( X^{M},P^{M}\right) $ as a
doublet for every $M,$ and on $A^{ij}$ as the gauge field$.$ Furthermore, by
taking $g\left( \tau\right) $ as a singlet while noting that $L^{MN}%
=\varepsilon^{ij}X_{i}^{M}X_{j}^{N}=X^{M}P^{N}-X^{N}P^{M}$ is Sp$\left(
2,R\right) $ gauge invariant$,$ we see that the full action is gauge
invariant under Sp$\left( 2,R\right) .$ The action can be rewritten in the
form%
\begin{equation}
S_{2T}\left( X,P,g\right) =\int d\tau\left\{ \frac{1}{2s_{d}}%
\varepsilon^{ij}Tr\left[ \partial_{\tau}\left( gX_{i}\cdot\Gamma
g^{-1}\right) \left( gX_{j}\cdot\bar{\Gamma}g^{-1}\right) \right]
-\frac{1}{2}A^{ij}X_{i}\cdot X_{j}\right\} .
\end{equation}
When both $X_{i}^{M}$ and $g\left( \tau\right) $ are transformed under local
Lorentz transformations as $\delta_{R}X_{i}^{M}=\varepsilon_{R}^{MN}\left(
\tau\right) X_{iN}$ and $\delta_{R}g=-\frac{1}{4}\left( g\Gamma_{MN}\right)
\varepsilon_{R}^{MN}\left( \tau\right) ,$ the structures $\left(
gX_{j}\cdot\Gamma g^{-1}\right) $ and $X_{i}\cdot X_{j}$ are gauge invariant
under $\delta_{R}$. Therefore the Lagrangian has a gauge symmetry with local
SO$\left( d,2\right) _{R}$ parameters $\varepsilon_{R}^{MN}\left(
\tau\right) $ when $g$ is transformed on the right side. In addition, there
is a global symmetry under SO$\left( d,2\right) _{L}$ when $g\left(
\tau\right) $ is transformed from the left side as $\delta_{L}g=\frac{1}%
{4}\varepsilon_{L}^{MN}\left( \Gamma_{MN}g\right) $, with $\tau$ independent
parameters $\varepsilon_{L}^{MN}$.
Using Noether's theorem we construct the conserved charge $J^{MN}\left(
\tau\right) $ of the global left side symmetry SO$\left( d,2\right) _{L}.$
We find $J^{MN}\sim iTr\left( \Gamma^{MN}gLg^{-1}\right) ,$ but we prefer to
write it in spinor space in the form
\begin{equation}
J_{A}^{~B}=\left( gLg^{-1}\right) _{A}^{~B}=J^{MN}\left( \tau\right)
\left( \frac{1}{4i}\Gamma_{MN}\right) _{A}^{~B}. \label{J}%
\end{equation}
Note that the matrix $J_{A}^{~B}$ must have the same form as the matrix
$L_{A}^{~B}$ of Eq.(\ref{L}), i.e. $J=\left( \frac{1}{4i}\Gamma_{MN}\right)
J^{MN}\left( \tau\right) ,$ since $gLg^{-1}=\frac{1}{4i}\left( g\Gamma
_{MN}g^{-1}\right) L^{MN}$ is a Lorentz transformation of the gamma matrices
that mixes them among themselves. By using the equations of motion for
$\left( X,P,g\right) $ one can show that these charges are conserved
$\partial_{\tau}J_{A}^{~B}\left( \tau\right) =0.$
As seen from the form of $J$ in Eq.(\ref{J}), it is gauge invariant under the
local SO$\left( d,2\right) _{R}$ as well as the local Sp$\left( 2,R\right)
$ transformations. Therefore the SO$\left( d,2\right) _{L}$ charges
$J_{A}^{~B}$ are physical observables that classify the physical states under
SO$\left( d,2\right) _{L}$ representations. In particular the Casimir
operators of these representations are given by $C_{n}=\frac{1}{s_{d}%
}tr\left( \left( 2J\right) ^{n}\right) .$ With this in mind we study the
properties of $J.$ In particular the square of the matrix $J,$ given by
$\left( J^{2}\right) _{A}^{~B}=\left( gLg^{-1}gLg^{-1}\right) _{A}%
^{~~B}=\left( gL^{2}g^{-1}\right) _{A}^{~~B},$ contains important
information about the physical states as we will see below. To proceed from
here we will outline the rest of the computation of $J^{2}$ at the classical
and quantum levels.
If the square of the matrix $L^{2}$ is computed at the classical level, i.e.
not caring about the orders of generators $L_{MN},$ then one finds that
$\left( L^{2}\right) _{A}^{~B}$ is proportional to the identity matrix
$\delta_{A}^{B},$ $\left( L^{2}\right) =\left( \frac{1}{4i}\Gamma
_{MN}L^{MN}\right) ^{2}=\frac{1}{8}L^{MN}L_{MN}~1.$ Furthermore by computing,
still at the classical level $\frac{1}{2}L^{MN}L_{MN}=X^{2}P^{2}-\left(
X\cdot P\right) ^{2},$ and imposing the classical constraints $X^{2}%
=P^{2}=\left( X\cdot P\right) =0,$ one finds that $L^{2}=0$ in the space of
gauge invariants of the classical theory. Then this implies also $J^{2}=0$ in
the space of gauge invariants of the classical theory. By taking higher powers
of $J,$ we find $J^{n}=0$ for all positive integers $n\geq2.$ Therefore all
Casimir eigenvalues are zero $C_{n}=0$ for all the classical physical
configurations of phase space. This is a special non-trivial representation of
the non-compact group SO$\left( d,2\right) _{L}$, and all classical gauge
invariants, which are functions of $L^{MN},$ can be classified as irreducible
multiplets of SO$\left( d,2\right) _{L}$.
We now consider the quantum theory. All the physical (gauge invariant) states
must fall into irreducible representations of the global symmetry SO$\left(
d,2\right) _{L}.$ In the quantum theory the $L_{MN}$ form the Lie algebra of
SO$\left( d,2\right) ,$ therefore if the square of the matrix $L$ is
computed at the quantum level, by taking into account the orders of the
operators $L^{MN},$ one finds%
\begin{equation}
L^{2}=\left( \frac{1}{4i}\Gamma_{MN}L^{MN}\right) ^{2}=-\frac{d}{2}\left(
\frac{1}{4i}\Gamma_{MN}L^{MN}\right) +\frac{1}{8}L^{MN}L_{MN}~1. \label{L2}%
\end{equation}
In this computation we used the properties of gamma matrices%
\[
\Gamma_{MN}\Gamma_{RS}=\Gamma_{MNRS}+\left( \eta_{NR}\Gamma_{MS}-\eta
_{MR}\Gamma_{NS}-\eta_{NS}\Gamma_{MR}+\eta_{MS}\Gamma_{NR}\right) +\left(
\eta_{NR}\eta_{MS}-\eta_{MR}\eta_{NS}\right) .
\]
The term $\Gamma_{MNRS}L^{MN}L^{RS}$ vanishes for $L^{MN}=X^{[M}P^{N]}$ due to
a clash between symmetry/antisymmetry. The term \textquotedblleft$\eta
_{NR}\Gamma_{MS}\cdots$\textquotedblright\ turns into a commutator, and after
using the SO$\left( d,2\right) $ Lie algebra for $\left[ L^{MN}%
,L^{RS}\right] $ it produces the linear term proportional to $d/2$ in
Eq.(\ref{L2}). The term \textquotedblleft$\eta_{NR}\eta_{MS}\cdots
$\textquotedblright\ produces the last term in Eq.(\ref{L2}). Furthermore the
Casimir $\frac{1}{2}L^{MN}L_{MN}$ does not vanish at the quantum level. As
shown in \cite{2treviews}, in the Sp$\left( 2,R\right) $ gauge invariant
physical sector of phase space one finds that it has the fixed value $\frac
{1}{2}L^{MN}L_{MN}=1-d^{2}/4$ rather than zero. Hence, in the physical sector
of the quantum theory the matrix $J_{A}^{~B}$ satisfies the following algebra%
\begin{equation}
\left( J^{2}\right) _{A}^{~B}=-\frac{d}{2}~J_{A}^{~B}+\frac{1}{8}\left(
1-\frac{d^{2}}{4}\right) ~\delta_{A}^{~B},\;\text{on physical states.}
\label{singleton}%
\end{equation}
We can compute the higher powers $J^{n}$ on physical states by repeatedly
using this relation,
\begin{equation}
\left( J^{n}\right) _{A}^{~B}=\alpha_{n}~J_{A}^{~B}+\beta_{n}~\delta
_{A}^{~B},
\end{equation}
and then compute the Casimir eigenvalues\footnote{Note that in the literature
one may find that the definition of the cubic and higher Casimir eigenvalues
are given as a linear combination of our $C_{n}.$} $C_{n}=\frac{1}{s_{d}%
}Tr\left( \left( 2J\right) ^{n}\right) =2^{n}\beta_{n}$. Evidently the
$C_{n}$ will end up having fixed values determined by the dimension $d$ of
SO$\left( d,2\right) _{R}.$ In particular,
\begin{equation}
C_{2}=1-\frac{d^{2}}{4},\;\;C_{3}=d\left( 1-\frac{d^{2}}{4}\right)
,\;C_{4}=\left( 1-\frac{d^{2}}{4}\right) \left( 1+\frac{3d^{2}}{4}\right)
,\;\text{etc.} \label{casimirs}%
\end{equation}
Therefore, at the quantum level we have identified a special unitary
representation that classifies all physical states of the theory. This is the
singleton representation of SO$\left( d,2\right) $ for any $d$. Our approach
shows that the singleton is more fully characterized by the constraints
satisfied by the charges in Eq.(\ref{singleton}). We will see in the next
section that these constraints will be satisfied explicitly at the quantum
level by constructing $J_{A}^{~B}$ in terms of twistors.
\subsection{Twistor gauge and the general twistor transform
\label{sttransform}}
There are different ways of choosing gauges to express the theory given by
$S_{2T}\left( X,P,g\right) $ in terms of the physical sector. One extreme in
gauge space is to eliminate $g$ completely, while another extreme is to
eliminate $\left( X,P\right) $ completely. When $g$ is eliminated we obtain
the phase space description, and when $\left( X,P\right) $ is eliminated we
obtain the twistor description.
Since SO$\left( d,2\right) _{R}$ is a local symmetry that acts on $g\left(
\tau\right) $ from the right, $g\rightarrow g^{\prime}=gg_{R},$ and $g_{R}$
has exactly the same number of degrees of freedom as $g,$ one can gauge fix
the extended classical theory by choosing the gauge $g\left( \tau\right) =1.
$ In that case the theory described by Eq.(\ref{L}) in terms of $\left(
X,P,g\right) $ reduces to the theory described by only $\left( X,P\right) $
in Eq.(\ref{2Taction})%
\begin{equation}
S_{2T}\left( X,P,g\right) \overset{g=1}{=}S_{2T}\left( X,P\right)
,\;\;J_{A}^{~B}=L_{A}^{~B}. \label{JL}%
\end{equation}
In this gauge the conserved charge becomes $J=L,$ so that the global symmetry
SO$\left( d,2\right) _{L}$ becomes the SO$\left( d,2\right) $ global
symmetry of the $\left( X,P\right) $ theory. This reflects the fact that to
maintain the gauge $g=1,$ a transformation of $g$ from the left must be
compensated by a transformation from the right, therefore SO$\left(
d,2\right) _{L}$ and SO$\left( d,2\right) _{R}$ become identified.
In the $g=1$ gauge there still remains the Sp$\left( 2,R\right) $ gauge
symmetry. If one fixes this gauge as in Eqs.(\ref{massless1},\ref{massless2})
then we see that the original SO$\left( d,2\right) _{L}$ is interpreted, in
this gauge, as the conformal symmetry of the relativistic massless particle
given in Eqs.(\ref{conf0}). But if one fixes Sp$\left( 2,R\right) $ as in
Eqs.(\ref{massive2x}-\ref{massive2p}) then the original SO$\left( d,2\right)
_{L}$ is interpreted as the hidden SO$\left( d,2\right) $ of the massive
particle given in Eqs.(\ref{LMNmassive1}-\ref{LMNmassive3}). So, the original
SO$\left( d,2\right) _{L}$ applied on $g$ can take on many possible physical
interpretations as the hidden symmetry of various dynamical particle phase
spaces that arise from Sp$\left( 2,R\right) $ gauge choices. Recall that for
all cases the conserved SO$\left( d,2\right) $ charges are just the physical
charges $J_{A}^{~B}=\left( g^{-1}Lg\right) _{A}^{~B}$ whose classical and
quantum properties were already computed in a gauge invariant way in the
previous section.
To obtain the twistor description of the system we eliminate $\left(
X^{M},P^{M}\right) $ completely and keep only $g$ as discussed in
\cite{2ttwistor}. This is done by using the Sp$\left( 2,R\right) $ and the
SO$\left( d,2\right) _{R}$ local symmetries to completely fix $X^{M},P^{M}$
to the convenient form $X^{+^{\prime}}=1$ and $P^{+}=1,$ while all other
components vanish%
\begin{equation}
X^{M}=(\overset{+^{\prime}}{{1}},\overset{-^{\prime}}{{0}},\overset{+}{{0}%
},\overset{-}{{0}},\overset{i}{{0}}),\;P^{M}=(\overset{+^{\prime}}{{0}%
},\overset{-^{\prime}}{{0}},\overset{+}{{1}},\overset{-}{{0}},\overset{i}{{0}%
}),\;i=1,\cdots,\left( d-2\right) . \label{twistgauge1}%
\end{equation}
These $X^{M},P^{M}$ already satisfy the constraints $X^{2}=P^{2}=X\cdot P=0$.
In this gauge the only non-vanishing component of $L^{MN}$ is $L^{+^{\prime}%
+}=1$, so that
\begin{equation}
L_{fixed}=\frac{-2}{4i}\Gamma^{-^{\prime}-}L^{+^{\prime}+}=\frac{i}{2}%
\Gamma^{-^{\prime}-}\equiv\Gamma. \label{Lfixed}%
\end{equation}
Hence the physical content of the theory is now described only in terms of $g$
and the fixed matrix $\Gamma$ embedded in the Lie algebra of SO$\left(
d,2\right) .$
The matrix $\Gamma$ has very few non-zero entries as seen by choosing a
convenient form of gamma matrices\footnote{An explicit form of SO$\left(
d,2\right) $ gamma matrices that we find convenient in even dimensions, is
given by $\Gamma^{0}=1\times1$,\ $\Gamma^{i}=\sigma_{3}\times\gamma^{i}%
$,\ $\Gamma^{\pm^{\prime}}=-i\sqrt{2}\sigma^{\pm}\times1$ (note $\Gamma
^{0^{\prime}}=-i\sigma_{1}\times1$ and $\Gamma^{1^{\prime}}=\sigma_{2}\times
1$), \ where $\gamma^{i}$ are the SO$\left( d-1\right) $ gamma matrices. The
$\bar{\Gamma}^{M}$ are the same as the $\Gamma^{M}$ for $M=\pm^{\prime},i,$
but for $M=0$ we have $\bar{\Gamma}^{0}=-\Gamma^{0}=-1\times1.$ From these we
construct the traceless $\Gamma^{+^{\prime}-^{\prime}}=\left(
\genfrac{}{}{0pt}{}{-1}{0}%
\genfrac{}{}{0pt}{}{0}{1}%
\right) $,\ $\Gamma^{+^{\prime}\mu}=i\sqrt{2}\left(
\genfrac{}{}{0pt}{}{0}{0}%
\genfrac{}{}{0pt}{}{\bar{\gamma}^{\mu}}{0}%
\right) $,\ $\Gamma^{-^{\prime}\mu}=-i\sqrt{2}\left(
\genfrac{}{}{0pt}{}{0}{\gamma^{\mu}}%
\genfrac{}{}{0pt}{}{0}{0}%
\right) $,\ $\Gamma^{\mu\nu}=\left(
\genfrac{}{}{0pt}{}{\bar{\gamma}^{\mu\nu}}{0}%
\genfrac{}{}{0pt}{}{0}{\gamma^{\mu\nu}}%
\right) $, with $\gamma_{\mu}=\left( 1,\gamma^{i}\right) $ and $\bar
{\gamma}_{\mu}=\left( -1,\gamma^{i}\right) .$ Then $\frac{1}{2}\Gamma
_{MN}J^{MN}=-\Gamma^{+^{\prime}-^{\prime}}J^{+^{\prime}-^{\prime}}$+~
$\frac{1}{2}J_{\mu\nu}\Gamma^{\mu\nu}-$ $\Gamma_{~\mu}^{+^{\prime}%
}J^{-^{\prime}\mu}-$ $\Gamma_{~\mu}^{-^{\prime}}J^{+^{\prime}\mu}$ takes the
matrix form given in Eq.(\ref{GL}). We can further write $\gamma^{1}=\tau
^{1}\times1,$ $\gamma^{2}=\tau^{2}\times1$ and $\gamma^{r}=\tau^{3}\times
\rho^{r},$ where the $\rho^{r}$ are the gamma matrices for SO$\left(
d-3\right) $. These gamma matrices are consistent with the metric
$C=\sigma_{1}\times1\times c$ of Eq.(\ref{C}), and footnote (\ref{ginverse}),
provided $c^{-1}\left( \rho^{r}\right) ^{\dagger}c=\rho^{r}.$ It is possible
to choose hermitian $\rho^{r}$ with $c=1$ for SO$\left( d-3\right) .$ If one
works in a basis with $c\neq1,$ then hermitian conjugation of of SO$\left(
d-3\right) $ spinors (which occur e.g. in $\bar{\lambda}$ of Eq.(\ref{zbar}))
must be supplemented by multiplying with $c,$ as in $\bar{\lambda}%
\equiv\lambda^{\dagger}\left( 1\times c\right) .$ \label{gamms}} for
SO$\left( d,2\right) $. Then, up to similarity transformations, $\Gamma$ can
be brought to the form\footnote{The gamma matrices $\Gamma^{M}$ of footnote
(\ref{gamms}) can be redefined differently for the left or right sides of $g$
up to similarity transformations. Thus, for the right side of $g$ we apply a
similarity transformation so that $\gamma^{1}=\tau^{3}\times1$, etc., to
obtain $\gamma^{-}=\left( \gamma^{0}-\gamma^{1}\right) /\sqrt{2}$ in the
form given in Eq.(\ref{g--}).}
\begin{equation}
\Gamma=\frac{i}{2}\Gamma^{-^{\prime}-}=\frac{1}{\sqrt{2}}\left(
\genfrac{}{}{0pt}{}{0}{\gamma^{-}}%
\genfrac{}{}{0pt}{}{0}{0}%
\right) =\left(
\genfrac{}{}{0pt}{}{\genfrac{}{}{0pt}{}{0}{0}\genfrac{}{}{0pt}{}{0}{0}%
}{\genfrac{}{}{0pt}{}{0}{0}\genfrac{}{}{0pt}{}{0}{1}}%
\genfrac{}{}{0pt}{}{\genfrac{}{}{0pt}{}{0}{0}\genfrac{}{}{0pt}{}{0}{0}%
}{\genfrac{}{}{0pt}{}{0}{0}\genfrac{}{}{0pt}{}{0}{0}}%
\right) . \label{g--}%
\end{equation}
The identity matrix $1,$ and the $0$'s, are $\frac{s_{d}}{4}\times\frac{s_{d}%
}{4}$ square block matrices embedded in the spinor representation of
SO$\left( d,2\right) .$ Then the gauge invariant 2T action in Eq.(\ref{S2T}%
), and the gauge invariant SO$\left( d,2\right) _{L}$ charges in
Eq.(\ref{J}), take the twistor form similar to Eq.(\ref{action})
\begin{align}
S_{2T}\left( X,P,g\right) & =\frac{4}{s_{d}}\int d\tau~Tr\left(
i\partial_{\tau}g\Gamma g^{-1}\right) =\frac{4}{s_{d}}\int d\tau
~i\partial_{\tau}Z_{A}^{~a}\bar{Z}_{a}^{~A}\equiv S_{twistor},\label{Stw}\\
\left( J\right) _{A}^{B} & =\left( g\Gamma g^{-1}\right) _{A}%
^{B}=\left( Z_{A}^{~a}\bar{Z}_{a}^{~B}-\frac{1}{s_{d}}tr\left( Z\bar
{Z}\right) \delta_{A}^{B}\right) ,\; \label{Jtw}%
\end{align}
The $Z_{A}^{~a},\bar{Z}_{a}^{~B}$ are the twistors that already obey the
constraints $\left( \bar{Z}Z\right) _{a}^{~b}=0$ in Eq.(\ref{ZZconstraint})
below, so $\frac{4}{s_{d}}\int d\tau~i\partial_{\tau}Z_{A}^{~a}\bar{Z}%
_{a}^{~A}$ is the full twistor action (for an equivalent gauge invariant
action that also produces the constraints, see Eq.(\ref{stwistor})). Due to
the form of $\Gamma$ it is useful to think of $g$ as written in the form of
$\frac{s_{d}}{4}\times\frac{s_{d}}{4}$ square blocks. Then $Z_{A}^{~a}$ with
$A=1,2,\cdots,s_{d}$ and $a=1,2,\cdots,\frac{s_{d}}{4}$ emerges as the
rectangular matrix that corresponds to the last block of columns of the matrix
$g,$ and similarly $\bar{Z}_{a}^{~A}$ corresponds to the second block of rows
of $g^{-1}.$ Since $g^{-1}=C^{-1}g^{\dagger}C,$ we find that $\bar{Z}%
=c^{-1}Z^{\dagger}C,$ where $C=\sigma_{1}\times1\times c$ is given in footnote
(\ref{gamms}). Furthermore, as part of $g,g^{-1},$ the $Z_{A}^{~a},\bar{Z}%
_{a}^{~B}$ must satisfy the constraint $\bar{Z}_{a}^{~A}Z_{A}^{~b}=0$ since
the product $\bar{Z}_{a}^{~A}Z_{A}^{~b}$ contributes to an off-diagonal block
of the matrix $1$ in $g^{-1}g=1,\;$
\begin{equation}
g^{-1}g=1\;\rightarrow\bar{Z}_{a}^{~A}Z_{A}^{~b}=0. \label{ZZconstraint}%
\end{equation}
A constraint such as this one must be viewed as the generator of a gauge
symmetry that operates on the $a$ index (the columns) of the twistor
$Z_{A}^{~a}.$
Let us do some counting of degrees of freedom. To describe the particle in $d
$ dimensions we only need $2\left( d-1\right) $ physical degrees of freedom
corresponding to phase space $\left( \vec{x},\vec{p}\right) .$ This counting
applies no matter if the particle is massless or massive, relativistic or not
relativistic, in flat space or curved space, interacting or not interacting.
Our twistors are expected to apply to all these cases, so we must have the
same number of physical parameters in the twistors given above. Any extra
parameters in $Z_{A}^{~a}$ beyond $2\left( d-1\right) $ must be either gauge
degrees of freedom of the twistor, or there must be additional relations among
the $Z_{A}^{~a}$. The $s_{d}\times s_{d}$ matrix $g$ (with $s_{d}=2^{d/2}$ for
even $d$) is constructed from $\frac{1}{2}\left( d+2\right) \left(
d+1\right) $ group parameters $\omega_{MN}$ as in Eq.(\ref{g}), but only
$2\left( d-1\right) $ of those parameters, corresponding to a coset,
contribute in Eqs.(\ref{Stw},\ref{Jtw}) due to the form of $\Gamma$ as will be
explained in section (\ref{coset}). So for sufficiently large $d,$ we expect
to find many relations among the $s_{d}^{2}/4=2^{2d-2}$ entries in the
rectangular matrix $Z_{A}^{~a}\left( g\right) .$
In $d=3,$ with $s_{3}/4=1$ columns, the $Z_{A}$ form the fundamental
representation of Sp$\left( 4,R\right) =$SO$\left( 3,2\right) .$ The
single $Z_{A}$ has just 4 real components that automatically satisfy the
constraint $\bar{Z}^{~A}Z_{A}=0.$ This number of degrees of freedom precisely
matches the expected number 4 of physical degrees of freedom, $2\left(
d-1\right) =2\left( 3-1\right) =4,$ for $d=3.$ So there are no extra
relations among the 4 real twistor entries in $Z_{A}.$
In $d=4$ dimensions, with s$_{4}/4=1$ columns, the $Z_{A}$ form the
fundamental representation of SU$\left( 2,2\right) =$SO$\left( 4,2\right)
.$ The single $Z_{A}$ has 4 complex components or 8 real parameters that must
satisfy the U$\left( 1\right) $ gauge constraint $\bar{Z}^{~A}Z_{A}=0$ of
Eq.(\ref{ZZconstraint}). The U$\left( 1\right) $ gauge symmetry together
with the constraint remove 2 real parameters. So the $Z_{A}$ contains $8-2=6$
physical degrees of freedom, which is just the correct number $2\left(
4-1\right) =6$ in $d=4,$ as discussed in section (\ref{twistd4}). So there
are no extra relations among the 4 complex twistor entries in $Z_{A}.$
For $d=5,6,$ with $s_{5}/4=s_{6}/4=2$ columns, the $Z_{A}^{~a}$ is a doublet
under an SU$\left( 2\right) $ gauge symmetry for $d=6,$ and SU$\left(
2\right) \times$U$\left( 1\right) $ for $d=5$. Beyond the SU$\left(
2\right) $ or SU$\left( 2\right) \times$U$\left( 1\right) $ gauge freedom
there seems to be further relations, but these amount to a simple
pseudo-reality condition on $Z_{A}^{~a},$ consistent with the transformation
rules of $Z_{A}^{~a}$ under Spin$\left( 6,2\right) \times$SU$\left(
2\right) $. The pseudo-reality condition emerges from the pseudo-reality of
the spinor representation of SO$\left( 6,2\right) $ or SO$\left(
5,2\right) . $ So, again there are no complicated relations among the entries
of $Z_{A}^{~a}$ for $d=5,6$ as seen by the following simple counting. The
number of real entries in the pseudo-real $Z_{A}^{~a}$ is $s_{6}^{2}%
/8=s_{5}^{2}/8=8^{2}/4=16.$ The 3 SU$\left( 2\right) $ gauge conditions
together with the 3 constraints remove 6 parameters, leaving $16-6=10$, which
is precisely the correct number of physical degrees of freedom for $d=6,$ i.e.
$2\left( 6-1\right) =10.$ Similarly, for $d=5$ the extra U$\left( 1\right)
$ and its constraint removes two more real parameters and this matches the
correct number $2\left( 5-1\right) =8.$
For higher dimensions there are gauge symmetries among the columns but there
also are further complicated relations among the $Z_{A}^{~a}$. It turns out
that in all cases $d\geq3$ just the first column of $Z_{A}^{~a}$ (i.e. $a=1$)
already contains all the parameters that describe the physical degrees of
freedom, but it is still useful to deal with the full $Z_{A}^{~a}$ since all
components can be conveniently given explicitly in terms of gamma matrices, as
seen in Eqs.(\ref{mu},\ref{lambd}) below.
We can construct explicitly the $Z_{A}^{~a},\bar{Z}_{a}^{~A}$ that satisfy all
of the relations discussed above at the classical level. This will give the
twistor transform we are after. The key is the gauge invariant $J_{A}^{B}.$ We
identify its two different forms in the two different gauges, one in terms of
phase space and the other in terms of twistors, as given in Eqs.(\ref{JL}%
,\ref{Jtw})
\begin{equation}
\frac{1}{4i}\Gamma_{MN}L^{MN}\;\overset{g=1}{=}\;J\;\overset{X,P\sim0}%
{=}\;Z\bar{Z},\;\text{with }\left( \bar{Z}Z\right) _{a}^{~b}=0,~Tr\left(
Z\bar{Z}\right) =0.\;\; \label{basicrelation}%
\end{equation}
Of course, the $\left( X,P,g=1\right) $ on the left side of the equation are
gauge transformations of the ($X^{M}=\delta_{+^{\prime}}^{M}$, $P^{M}%
=\frac{s_{d}}{4}\delta_{+}^{M}$, and $Z\left( g\right) $) on the right side.
So this equation must contain the twistor transform. More explicitly we write
$Z$ in terms of its components
\begin{equation}
Z_{A}^{~a}=\left(
\begin{array}
[c]{c}%
\mu\\
\lambda
\end{array}
\right) ,\;\bar{Z}_{a}^{~A}=\left(
\begin{array}
[c]{cc}%
\bar{\lambda} & \bar{\mu}%
\end{array}
\right) ,\;\bar{Z}_{a}^{~A}Z_{A}^{~b}=\left( \bar{\lambda}\mu+\bar{\mu
}\lambda\right) _{a}^{~b}=0, \label{zbar}%
\end{equation}
where $\mu,\lambda$ are $\frac{s_{d}}{2}\times\frac{s_{d}}{4}$ rectangular
matrices. Then we use the gamma matrices, and the definition of $\bar{\lambda
},\bar{\mu}$ in footnote (\ref{gamms}), to write the basic relation
(\ref{basicrelation}) between phase space and twistors more explicitly as%
\begin{equation}
\frac{1}{2i}\left(
\begin{array}
[c]{cc}%
L^{+^{\prime}-^{\prime}}+\frac{1}{2}L_{\mu\nu}\bar{\gamma}^{\mu\nu} &
-i\sqrt{{2}}L^{-^{\prime}\mu}~\bar{\gamma}_{\mu}\\
i\sqrt{{2}}L^{+^{\prime}\mu}~\gamma_{\mu} & -L^{+^{\prime}-^{\prime}}+\frac
{1}{2}L_{\mu\nu}\gamma^{\mu\nu}%
\end{array}
\right) =J=\left(
\begin{array}
[c]{cc}%
\mu\bar{\lambda} & \mu\bar{\mu}\\
\lambda\bar{\lambda} & \lambda\bar{\mu}%
\end{array}
\right) . \label{GL}%
\end{equation}
Comparing the lower off diagonal blocks we learn part of the twistor transform%
\begin{equation}
\lambda\bar{\lambda}=\frac{1}{\sqrt{2}}L^{+^{\prime}\mu}~\gamma_{\mu}=\frac
{1}{\sqrt{2}}\left( X^{+^{\prime}}P^{\mu}-P^{+^{\prime}}X^{\mu}\right)
\gamma_{\mu}. \label{lambda}%
\end{equation}
We should also note the twistor relations that follow from the other three
blocks. In three or four dimensions a single doublet $\lambda$ satisfies this
equation automatically. In higher dimensions a single column $\lambda$ cannot
do it automatically, instead we have $\frac{s_{d}}{4}$ columns in $\lambda$
with certain relations among them. Thanks to the relations among columns, that
can be expressed in terms of gamma matrices as in Eqs.(\ref{lambd}) below, the
equation above will be satisfied.
Next we consider $\left( \bar{\Gamma}\cdot X\right) J$ and use the different
gauge fixed forms of the gauge invariant $J$ to show that it vanishes as
follows%
\begin{align}
\left( \bar{\Gamma}\cdot X\right) J & =\left( \bar{\Gamma}\cdot X\right)
L=\frac{1}{4i}\left( \bar{\Gamma}\cdot X\right) \left( \Gamma\cdot
X~\bar{\Gamma}\cdot P-\Gamma\cdot P~\bar{\Gamma}\cdot X\right) \nonumber\\
& =\frac{1}{2i}X\cdot X\left( \bar{\Gamma}\cdot P\right) -\frac{1}%
{4i}X\cdot P~\left( \bar{\Gamma}\cdot X\right) =0\text{.}%
\end{align}
The last zero is because $X\cdot P=P\cdot P=0$ in the Sp$\left( 2,R\right) $
gauge invariant physical sector. Similarly we show also $\left( \bar{\Gamma
}\cdot P\right) J=0.$ Therefore
\begin{equation}
\left( \bar{\Gamma}\cdot P\right) J=0,\;\left( \bar{\Gamma}\cdot X\right)
J=0.
\end{equation}
Hence, every column of $J$ is a simultaneous null eigenstate of the matrices
$\left( \bar{\Gamma}\cdot X\right) $ and $\left( \bar{\Gamma}\cdot
P\right) . $
Furthermore, because $J$ can be written in the form $J=Z\bar{Z},$ it must be
that the $s_{d}\times\frac{s_{d}}{4}$ matrix $Z$ is a collection of these null
eigenstates, so it is possible to write $Z$ as a linear combination of
$\frac{s_{d}}{4}$ columns of $L=J$ as follows
\begin{equation}
Z_{A}^{~a}=\left(
\genfrac{}{}{0pt}{}{\mu}{\lambda}%
\right) =L\left(
\genfrac{}{}{0pt}{}{\genfrac{}{}{0pt}{}{\alpha_{1}}{\alpha_{2}}%
}{\genfrac{}{}{0pt}{}{\alpha_{3}}{\alpha_{4}}}%
\right) ,\text{ } \label{ZJa}%
\end{equation}
where $\alpha_{i}$ are $\frac{s_{d}}{4}\times\frac{s_{d}}{4}$ matrices that
will be determined below by requiring $J=L=Z\bar{Z}$. So $Z$ must also satisfy
the null conditions
\begin{equation}
\left( \bar{\Gamma}\cdot P\right) Z=0,\text{ }\;\left( \bar{\Gamma}\cdot
X\right) Z=0\;\text{and }\;LZ=0. \label{null}%
\end{equation}
The last null relation $LZ=0$ is satisfied automatically if the first two are
satisfied. This property is sufficient to determine $Z_{A}^{~a}$ up to gauge
transformations as follows. The explicit matrices are obtained by using the
gamma matrices in footnote (\ref{gamms})
\begin{equation}
\left( \bar{\Gamma}\cdot X\right) =\left(
\begin{array}
[c]{cc}%
X_{\mu}\gamma^{\mu} & -i\sqrt{2}X^{-^{\prime}}\\
-i\sqrt{2}X^{+^{\prime}} & -X_{\mu}\bar{\gamma}^{\mu}%
\end{array}
\right) ,\;\;\left( \bar{\Gamma}\cdot P\right) =\left(
\begin{array}
[c]{cc}%
P_{\mu}\gamma^{\mu} & -i\sqrt{2}P^{-^{\prime}}\\
-i\sqrt{2}P^{+^{\prime}} & -P_{\mu}\bar{\gamma}^{\mu}%
\end{array}
\right)
\end{equation}
Then the zero eigenvalue conditions $\left( \bar{\Gamma}\cdot X\right)
Z=0=\left( \bar{\Gamma}\cdot P\right) Z$ are solved by%
\begin{equation}
\mu=-i\frac{X_{\mu}\bar{\gamma}^{\mu}}{\sqrt{2}X^{+^{\prime}}}\lambda
=-i\frac{P_{\mu}\bar{\gamma}^{\mu}}{\sqrt{2}P^{+^{\prime}}}\lambda
\;,\;\;\lambda=i\frac{X_{\mu}\gamma^{\mu}}{\sqrt{2}X^{-^{\prime}}}\mu
=i\frac{P_{\mu}\gamma^{\mu}}{\sqrt{2}P^{-^{\prime}}}\mu. \label{mulambda}%
\end{equation}
To show that these expressions are consistent with each other, note that the
second set is obtained by inverting the first set as long as $X^{2}%
=P^{2}=X\cdot P=0$ are satisfied in the physical sector. For example, multiply
both sides of the equation $\mu=-i\frac{X_{\mu}\bar{\gamma}^{\mu}}{\sqrt
{2}X^{+^{\prime}}}\lambda$ by $i\frac{X_{\mu}\gamma^{\mu}}{\sqrt
{2}X^{-^{\prime}}},$ then use $\bar{\gamma}^{\mu}\gamma^{\nu}+\bar{\gamma
}^{\nu}\gamma^{\mu}=2\eta^{\mu\nu}$ and $X^{2}=X^{\mu}X_{\mu}-2X^{+^{\prime}%
}X^{-^{\prime}}=0,$ to obtain $\;\lambda=i\frac{X_{\mu}\gamma^{\mu}}{\sqrt
{2}X^{-^{\prime}}}\mu.$ So we can concentrate on the consistency of the first
set only. The difference between the two expressions for $\mu$ must vanish,
this implies that $\lambda$ should satisfy the Dirac-like equation%
\begin{equation}
\left( X^{+^{\prime}}P^{\mu}-P^{+^{\prime}}X^{\mu}\right) \bar{\gamma}_{\mu
}\lambda=L^{+^{\prime}\mu}~\gamma_{\mu}\lambda=0. \label{lambdaeq}%
\end{equation}
This is a consistent equation provided the vector $L^{+^{\prime}\mu}~$is null
\begin{equation}
L^{+^{\prime}\mu}L_{~\mu}^{+^{\prime}}=\left( X^{+^{\prime}}P^{\mu
}-P^{+^{\prime}}X^{\mu}\right) ^{2}=0.
\end{equation}
This is indeed correct in the physical sector that satisfies $X^{2}%
=P^{2}=X\cdot P=0.$ Then we can solve for $\lambda$ by writing%
\begin{equation}
\lambda=\left( X^{+^{\prime}}P^{\mu}-P^{+^{\prime}}X^{\mu}\right)
\gamma_{\mu}\left(
\genfrac{}{}{0pt}{}{\alpha}{\beta}%
\right) .
\end{equation}
This satisfies the $\lambda$ equation (\ref{lambdaeq}) automatically for any
$\frac{s_{d}}{4}\times\frac{s_{d}}{4}$ matrices $\alpha,\beta$. These in turn
are determined, up to $\frac{s_{d}}{4}\times\frac{s_{d}}{4}$ gauge
transformations, by satisfying the $\frac{s_{d}}{4}\times\frac{s_{d}}{4}$
matrix equation (\ref{lambda}).
In this way we have seen that all of the forms given in Eq.(\ref{mulambda})
are consistent with each other and determine $Z_{A}^{~a}$ up to a gauge
transformation. Thus the null conditions in Eq.(\ref{null}) is all that is
needed to determine $Z_{A}^{~a}$ up to a gauge transformation, but in turn
these followed from the basic relation in Eq.(\ref{GL}).
The following subset of our relations resemble the twistor transform in four
dimensions, but $\mu,\lambda$ are $\frac{s_{d}}{2}\times\frac{s_{d}}{4}$
matrices that must obey all the relations above
\begin{equation}
\mu=-i\frac{X_{\mu}\bar{\gamma}^{\mu}}{\sqrt{2}X^{+^{\prime}}}\lambda
,\;\text{and ~}\lambda\bar{\lambda}=\frac{1}{\sqrt{2}}\left( X^{+^{\prime}%
}P^{\mu}-P^{+^{\prime}}X^{\mu}\right) \gamma_{\mu}. \label{tttransf}%
\end{equation}
Indeed we can check directly that by inserting only these relations into the
right hand side of Eqs.(\ref{GL},\ref{basicrelation}), we derive the
SO$\left( d,2\right) $ generators in terms of phase space $L^{MN}=X^{M}%
P^{N}-X^{N}P^{N}$ that appear on the left side of those equations. Furthermore
by inserting only these relations into the twistor action we derive the phase
space action that determines the canonical structure%
\begin{align}
\frac{4}{s_{d}}\int d\tau~i\partial_{\tau}Z_{A}^{~a}\bar{Z}_{a}^{~A} &
=i\frac{4}{s_{d}}\int d\tau~Tr\left( \partial_{\tau}\mu\bar{\lambda}%
+\partial_{\tau}\lambda\bar{\mu}\right) =\frac{4}{s_{d}}\int d\tau~Tr\left(
\partial_{\tau}\left( \frac{X_{\mu}\bar{\gamma}^{\mu}}{\sqrt{2}X^{+^{\prime}%
}}\right) \lambda\bar{\lambda}\right) \\
& =\frac{4}{s_{d}}\int d\tau~\frac{1}{\sqrt{2}}\left( X^{+^{\prime}}P^{\mu
}-P^{+^{\prime}}X^{\mu}\right) Tr\left( \partial_{\tau}\left( \frac{X_{\mu
}\bar{\gamma}^{\mu}}{\sqrt{2}X^{+^{\prime}}}\right) \gamma_{\mu}\right) \\
& =\int d\tau~\left( X^{+^{\prime}}P_{\mu}-P^{+^{\prime}}X_{\mu}\right)
\partial_{\tau}\left( \frac{X^{\mu}}{X^{+^{\prime}}}\right) \\
& =\int d\tau~\left( P_{\mu}-\frac{P^{+^{\prime}}}{X^{+^{\prime}}}X_{\mu
}\right) \left( \partial_{\tau}X^{\mu}-\frac{\partial_{\tau}X^{+^{\prime}}%
}{X^{+^{\prime}}}X^{\mu}\right) \\
& =\int d\tau~\left( \partial_{\tau}X^{\mu}P_{\mu}-\partial_{\tau
}X^{+^{\prime}}P^{-^{\prime}}-\partial_{\tau}X^{-^{\prime}}P^{+^{\prime}%
}\right) =\int d\tau~\partial_{\tau}X^{M}P_{M}.
\end{align}
The last line follows when the constraints $X^{2}=P^{2}=X\cdot P=0$ are
satisfied in the physical sector. This shows the consistency of our twistor
transform of Eq.(\ref{tttransf}) for spinless particles in all dimensions.
It is also interesting to give an explicit formula for both $\mu$ and
$\lambda$ in terms of the Sp$\left( 2,R\right) $ gauge invariant $L^{MN}.$
This is already obtained through the relation between $Z$ and $L$ given in
Eq.(\ref{ZJa}). It turns out that it is sufficient to take only one of the
$\alpha_{i}$ to be nonzero. So we will take $\alpha_{2}=\alpha_{3}=\alpha
_{4}=0$ and determine $\alpha_{1}\neq0$ from the relation $J=L=Z\bar{Z}$. The
other possibilities are gauge equivalent. The equivalence is guaranteed by the
Sp$\left( 2,R\right) $ gauge invariance conditions $X^{2}=P^{2}=X\cdot P=0$.
This gives%
\begin{equation}
Z_{A}^{~a}=\left(
\genfrac{}{}{0pt}{}{\mu}{\lambda}%
\right) =L\left(
\genfrac{}{}{0pt}{}{\genfrac{}{}{0pt}{}{\alpha_{1}}{0}%
}{\genfrac{}{}{0pt}{}{0}{0}}%
\right)
\end{equation}
We write out the $L=J$ in Eq.(\ref{GL}) more explicitly in terms of
$\frac{s_{d}}{4}\times\frac{s_{d}}{4}$ blocks by using the gamma matrices in
footnote (\ref{gamms}), and we obtain
\begin{align}
\mu & =\left(
\begin{array}
[c]{c}%
L^{+^{\prime}-^{\prime}}+L^{r0}\rho_{r}+iL^{12}+\frac{1}{2}L_{rs}\rho^{rs}\\
-L^{01}-iL^{02}-\left( L^{r1}+iL^{r2}\right) \rho_{r}%
\end{array}
\right) hu,\;\;\label{mu}\\
\lambda & =\left(
\begin{array}
[c]{c}%
L^{+^{\prime}0}+L^{+^{\prime}r}~\rho_{r}\\
L^{+^{\prime}1}+iL^{+^{\prime}2}%
\end{array}
\right) i\sqrt{2}hu.\; \label{lambd}%
\end{align}
Here $\alpha_{1}=hu$ is written as a product of a unitary matrix $u$ and a
Hermitian matrix $h.$ The matrix $u$ is an arbitrary $\frac{s_{d}}{4}%
\times\frac{s_{d}}{4}$ unitary matrix that belongs to the gauge group that
acts on the twistors. The generator of this gauge group is the constraint
$\left( \bar{Z}Z\right) _{a}^{~b}=0.$ The matrix $h$ is determined by
insuring $\lambda\bar{\lambda}=\frac{1}{\sqrt{2}}L^{+^{\prime}\mu}\gamma_{\mu
}$ that was established in Eq.(\ref{lambda}) and is given by
\begin{equation}
h=2^{-3/4}\left( L^{+^{\prime}0}+L^{+^{\prime}r}~\rho_{r}\right) ^{-1/2}.
\label{h}%
\end{equation}
To summarize, in Eqs.(\ref{mu}-\ref{h}) we have given a more fundamental form
of the twistor transform between $Z_{A}^{~a}$ and the $\left( d+2\right) $
dimensional phase space $\left( X^{M},P^{M}\right) .$ The formulas are
Sp$\left( 2,R\right) $ gauge invariant since $\left( X,P\right) $ appear
only in the form $L^{MN}=X^{M}P^{N}-X^{N}P^{M}.$ These transforms are
consistent with Eqs.(\ref{tttransf},\ref{mulambda},\ref{null}) which somewhat
resemble the more traditional form of the twistor transform.
Note that $Z\left( L\right) $ written in terms of $L^{MN}$ depends only on
$2\left( d-1\right) $ independent combinations of the $L^{MN},$ since the
$L^{MN}$ obey the constraints $L^{MN}L_{NK}=0$ that follow from $X^{2}%
=P^{2}=X\cdot P=0$ in the physical sector. From these we derive the explicit
relations%
\[
\left( L^{+^{\prime}-^{\prime}}\right) ^{2}=-L^{+^{\prime}\mu}L_{~\mu
}^{-^{\prime}},\;L^{\mu\nu}=\frac{L^{+^{\prime}\mu}L^{-^{\prime}\nu
}-L^{+^{\prime}\nu}L^{-^{\prime}\mu}}{L^{+^{\prime}-^{\prime}}},\;L^{+^{\prime
}\mu}L_{~\mu}^{+^{\prime}}=0=L^{-^{\prime}\mu}L_{~\mu}^{-^{\prime}}.
\]
So, all $L^{MN}$ are then written only in terms of the $2d$ vector components
$L^{\pm^{\prime}\mu},$ but those are lightlike vectors and therefore contain
only $2\left( d-1\right) $ independent degrees of freedom.
Both the 2T particle and the corresponding twistors are SO$\left( d,2\right)
$ covariant descriptions, the first is in terms of vectors $X_{i}^{M}=\left(
X^{M},P^{M}\right) $ and the second is in terms of spinors $Z_{A}^{~a}$ of
SO$\left( d,2\right) $. This covariance is achieved by having gauge
symmetries in both versions, in the first case the gauge symmetry is applied
on the $i$ index of $X_{i}^{M}$ and in the second case the gauge symmetry is
applied on the $a$ index of $Z_{A}^{~a}.$
To account for the gauge symmetry and the constraint (\ref{ZZconstraint}) we
may derive them from a twistor action principle. This is done by rewriting the
twistor action in Eq.(\ref{Stw}) in a form similar to Eq.(\ref{action})
\begin{equation}
S_{twistor}=\frac{4}{s_{d}}\int d\tau~Tr\left[ \left( i\bar{Z}DZ\right)
-h_{d}V\right] ,\;\;\left( DZ_{A}\right) ^{a}\equiv\frac{\partial
Z_{A}^{~a}}{\partial\tau}-iZ_{A}^{~b}V_{b}^{~a}. \label{stwistor}%
\end{equation}
The equation of motion of the $\frac{s_{d}}{4}\times\frac{s_{d}}{4}$ matrix
gauge field $V_{b}^{~a}$ generates the constraint $\left( \bar{Z}Z\right)
_{a}^{~b}-h_{d}\delta_{a}^{~b}=0$ where $h_{d}$ is chosen to make the matrix
traceless or with trace depending on the number of dimensions $d$ (see e.g.
the counting of degrees of freedom for $d=5,6$ following
Eq.(\ref{ZZconstraint})). Equivalently, the matrix $V$ itself can be taken as
traceless or with trace depending on the dimension, and this will result in
the same constraint. Once this constraint is satisfied this action reduces to
the previous one in Eq.(\ref{Stw}). For $d=3,4,5,6$ this is the full action
principle in terms of twistors since there are no other conditions (other than
reality or pseudo-reality in some dimensions) as discussed following
Eq.(\ref{ZZconstraint}). However, for $d\geq7$ some more conditions on a
general complex or (pseudo)real $Z$ are required to make it satisfy also the
basic relation Eq.(\ref{basicrelation}) or its equivalent null conditions in
Eq.(\ref{null}). Although we have given the full twistor transform for any
dimension, we have so far given the full action principle only for $d\leq6.$
Up to now we have described the twistor transform for the \textquotedblleft
free\textquotedblright\ 2T particle in $d+2$ dimensions. But from here it is
an easy step to obtain the twistor transform for an assortment of non-trivial
particle dynamics in 1T-physics. Having established the transform between
twistors and the Sp$\left( 2,R\right) $ doublets $\left( X^{M}%
,P^{M}\right) $ in $\left( d+2\right) $ dimensions, we can now make
Sp$\left( 2,R\right) $ gauge choices to produce various dynamical systems in
$\left( d-1\right) +1$ dimensions, in the physical sector that satisfies
$X^{2}=P^{2}=X\cdot P=0$. Examples that occur in this paper are the massless
particle in $d$ dimensions of Eq.(\ref{massless1},\ref{massless2}), or the
massive particle in $d$ dimensions of Eq.(\ref{massive2x},\ref{massive2p}).
Other Sp$\left( 2,R\right) $ gauge choices that include interacting and
curved background cases are found in \cite{2treviews}\cite{2tHandAdS}. By
inserting the gauge choice for $\left( X^{M},P^{M}\right) $ into
Eqs.(\ref{tttransf},\ref{mulambda}) we obtain the corresponding twistors in
$d$ dimensions, such as the twistors for the massless particle of
Eq.(\ref{penrose}), or the massive particle in of Eq.(\ref{twistormassive}).
For more examples see \cite{twistorBP1} where the computations for the twistor
transform were done explicitly in $d=4,$ but the same explicit formulas also
apply in $d$ dimensions by inserting the corresponding gamma matrices in $d$
dimensions, as given in the expressions above.
\subsection{Geometry: twistors as the coset SO$\left( d,2\right)
/$H$_{\Gamma}$ \label{coset}}
A geometric view of twistors in $d$ dimensions can also be given in the form
of a coset as follows. The starting point for twistors was the twistor gauge
of 2T-physics in Eq.(\ref{Stw}) which involved the group element $g\left(
\tau\right) $ in the spinor representation of SO$\left( d,2\right) $ and
the special matrix $\Gamma$ in Eq.(\ref{g--}). The action and its SO$\left(
d,2\right) $ global symmetry charge (on the left side of $g)$ have the form%
\begin{equation}
S\left( g\right) =\frac{4}{s_{d}}\int d\tau~Tr\left( ig^{-1}\partial_{\tau
}g\Gamma\right) ,\;\;\left( J\right) _{A}^{B}=\left( g\Gamma
g^{-1}\right) _{A}^{B}.\; \label{Sg}%
\end{equation}
The action is like a sigma model, but it is linear instead of being quadratic
in the Cartan connection $ig^{-1}\partial_{\tau}g,$ and has the special
insertion $\Gamma$ on the right side of $g.$ The insertion $\Gamma$ determines
important properties of this action. The equation of motion for $g $ is
$\left[ \Gamma,g^{-1}\partial_{\tau}g\right] =0.$ Using this one can show
that the global current is conserved $\partial_{\tau}\left( g\Gamma
g^{-1}\right) =0,$ as expected from Noether's theorem.
We recall that the current $J_{A}^{~B}$ is gauge invariant and contains all
the physical information of the theory as seen in the previous sections. In
particular the current satisfies $J^{2}=\left( g\Gamma g^{-1}\right) \left(
g\Gamma g^{-1}\right) =\left( g\Gamma^{2}g^{-1}\right) =0$ at the classical
level, which is consistent with the covariant approach in section
(\ref{ggroup}). This property of the current captures all the essential
aspects of the physical sector at the classical level.
Now, let us determine the independent degrees of freedom that contribute to
the current. We will find that there are precisely 2$\left( d-1\right) $
degrees of freedom, precisely equal to the number of physical degrees of
freedom. Since $g^{-1}\Gamma g$ is a SO$\left( d,2\right) $ transformation
applied on a generator $\Gamma$ in the algebra of SO$\left( d,2\right) ,$ we
can eliminate from $g\left( \tau\right) $ the subgroup H$_{\Gamma}$ that
leaves $\Gamma$ invariant, and keep only the coset degrees of freedom in
SO$\left( d,2\right) /$H$_{\Gamma}.$ To do this, we can decompose $g\left(
\tau\right) =T_{\Gamma}\left( \tau\right) H_{\Gamma}\left( \tau\right) $
and write $g\Gamma g^{-1}=T_{\Gamma}\Gamma T_{\Gamma}^{-1}$ since by
definition $H_{\Gamma}\Gamma H_{\Gamma}^{-1}=\Gamma.$ Here $H_{\Gamma}%
=\exp\left( h_{\Gamma}\right) $ and $T_{\Gamma}=\exp\left( t_{\Gamma
}\right) ,$ where $h_{\Gamma}$ ($t_{\Gamma}$) is a linear combination of all
the SO$\left( d,2\right) $ generators $\Gamma^{MN}$ that commute (do not
commute) with $\Gamma,$ i.e. $\left[ h_{\Gamma},\Gamma\right] =0$ and
$\left[ t_{\Gamma},\Gamma\right] \neq0.$
To characterize the sets of generators $\left( h_{\Gamma},t_{\Gamma}\right)
$ consider the decomposition of SO$\left( d,2\right) $ with respect to the
SO$\left( d-2\right) \times$SO$\left( 2,2\right) $ subgroup. We can write
$J^{MN}=J^{ij}\oplus J^{\mu\nu}\oplus J^{\mu i}$ where $i=1,2,\cdots,\left(
d-2\right) $ spans the SO$\left( d-2\right) $ basis and $\mu=+^{\prime
},-^{\prime},+,-$ (or $0^{\prime},0,1^{\prime},1$) spans the SO$\left(
2,2\right) $ basis. Furthermore we decompose SO$\left( 2,2\right)
=$SL$\left( 2,R\right) _{+}\times$SL$\left( 2,R\right) _{-}$ and note that
each $\mu$ index is in the $\left( \frac{1}{2},\frac{1}{2}\right) $
representation of SL$\left( 2,R\right) _{+}\times$SL$\left( 2,R\right)
_{-}. $ The SL$\left( 2,R\right) _{+}\times$SL$\left( 2,R\right) _{-}$
generators can be identified explicitly as
\begin{align}
\text{SL}\left( 2,R\right) _{+} & :\;\frac{1}{2}\left( J^{+^{\prime
}-^{\prime}}+J^{+-}\right) ,~J^{+^{\prime}+},~J^{-^{\prime}-}\\
\text{SL}\left( 2,R\right) _{-} & :\;\frac{1}{2}\left( J^{+^{\prime
}-^{\prime}}-J^{+-}\right) ,~J^{+^{\prime}-},~J^{-^{\prime}+}%
\end{align}
From the general SO$\left( d,2\right) $ commutation rules
\begin{equation}
\left[ J^{MN},J^{KL}\right] =i\left[ \left( J^{ML}\eta^{NK}-\left(
M\leftrightarrow N\right) \right) -\left( K\leftrightarrow L\right)
\right] ,
\end{equation}
with $\eta^{+^{\prime}-^{\prime}}=\eta^{+-}=-1$ and $\eta^{ij}=\delta^{ij},$
it is easy to verify that these indeed form the SL$\left( 2,R\right)
_{+}\times$SL$\left( 2,R\right) _{-}$ algebra. Under commutation with the
generator $\frac{1}{2}\left( J^{+^{\prime}-^{\prime}}+J^{+-}\right) $ each
SO$\left( d,2\right) $ generator has a definite charge $0,\pm\frac{1}{2}%
,\pm1.$ We list the generators according to those charges as follows%
\begin{equation}%
\begin{tabular}
[c]{ccccc}%
$-1$ & $-\frac{1}{2}$ & $0$ & $+\frac{1}{2}$ & $+1$\\
$J^{-^{\prime}-}$ & $\left(
\begin{array}
[c]{c}%
J^{-i}\\
J^{-^{\prime}i}%
\end{array}
\right) $ & $\left.
\begin{array}
[c]{c}%
J^{ij},\;\frac{J^{+^{\prime}-^{\prime}}+J^{+-}}{2}\\
\left( J^{-^{\prime}+},\frac{J^{+^{\prime}-^{\prime}}-J^{+-}}{2}%
,J^{+^{\prime}-}\right)
\end{array}
\right. $ & $\left(
\begin{array}
[c]{c}%
J^{+^{\prime}i}\\
J^{+i}%
\end{array}
\right) $ & $J^{+^{\prime}+}$%
\end{tabular}
\end{equation}
These charges are conserved additively in the general commutation rules
$\left[ J^{MN},J^{KL}\right] =\cdots$ given above. Furthermore the charge
$\pm\frac{1}{2}$ generators form doublets under SL$\left( 2,R\right) _{-}$
and vectors under SO$\left( d-2\right) $ as indicated, while the charge
$\pm1$ generators are singlets under both. From this structure of the
commutation rules $\left[ J^{MN},J^{KL}\right] =\cdots$ we easily see that
the generators that commute or do not commute with $\Gamma\sim\Gamma
^{-^{\prime}-}$ (or $J^{-^{\prime}-}$) are%
\begin{align}
h_{\Gamma} & :\;J^{-^{\prime}-},~\left(
\begin{array}
[c]{c}%
J^{-i}\\
J^{-^{\prime}i}%
\end{array}
\right) ,\left.
\begin{array}
[c]{c}%
J^{ij}\;\\
\left( J^{-^{\prime}+},\frac{J^{+^{\prime}-^{\prime}}-J^{+-}}{2}%
,J^{+^{\prime}-}\right)
\end{array}
\right. \\
t_{\Gamma} & :\;\frac{1}{2}\left( J^{+^{\prime}-^{\prime}}+J^{+-}\right)
,\left(
\begin{array}
[c]{c}%
J^{+^{\prime}i}\\
J^{+i}%
\end{array}
\right) ,J^{+^{\prime}+}%
\end{align}
We note that each set forms a subalgebra $\left[ h_{\Gamma},h_{\Gamma
}\right] \sim h_{\Gamma},\;\left[ t_{\Gamma},t_{\Gamma}\right] \sim
t_{\Gamma}$ while $\left[ h_{\Gamma},t_{\Gamma}\right] \sim h_{\Gamma
}+t_{\Gamma}.$ In particular, within $t_{\Gamma}$ the generator $\frac{1}%
{2}\left( J^{+^{\prime}-^{\prime}}+J^{+-}\right) $ forms a U$\left(
1\right) $ subgroup and classifies the others according to their charges
above, while the remaining coset $T_{\Gamma}/U\left( 1\right) $ forms an
algebra similar to the Heisenberg algebra, $\left[ J^{+^{\prime}i}%
,J^{+j}\right] =i\delta^{ij}J^{+^{\prime}+},$ since $J^{+^{\prime}+}$
commutes with both $J^{+^{\prime}i},J^{+j}.$ From this we conclude that the
general $T_{\Gamma}=\exp\left( t_{\Gamma}\right) $ can be parametrized as
follows%
\begin{equation}
T_{\Gamma}=\exp\left( t_{\Gamma}\right) =\exp\left( \frac{1}{2}\left(
\Gamma^{+^{\prime}-^{\prime}}+\Gamma^{+-}\right) \omega\left( \tau\right)
\right) \exp\left( \Gamma^{+^{\prime}+}z\left( \tau\right) \right)
\exp\left( \Gamma^{+i}k_{i}\left( \tau\right) +\Gamma^{+^{\prime}i}%
q_{i}\left( \tau\right) \right)
\end{equation}
The number of parameters in this coset is precisely $2\left( d-1\right) ,$
which is the same as the number of physical degrees of freedom. If we insert
the explicit set of gamma matrices $\Gamma^{MN}$ given in footnote
(\ref{gamms}) into this expression, we can write $T_{\Gamma}$ in the form of
$\frac{s_{d}}{4}\times\frac{s_{d}}{4}$ similar to $\Gamma$. Then, as seen from
Eq.(\ref{Stw}), the first block of columns that forms an $s_{d}\times
\frac{s_{d}}{4}$ rectangular matrix is the twistor $Z_{A}^{~a}\left(
t_{\Gamma}\right) $ now written in terms of only the $2\left( d-1\right) $
parameters of the coset $t_{\Gamma}\in$SO$\left( d,2\right) /$H$_{\Gamma}$.
This result is of course in agreement with the form of the Sp$\left(
2,R\right) $ invariant $Z_{A}^{~a}\left( L\right) $ of Eqs.(\ref{mu}%
,\ref{lambd}), which can also be written only in terms of $2\left(
d-1\right) $ parameters as detailed in Sec.(\ref{2dm1}). We now understand
that there is a close relationship with the geometric interpretation as a coset.
\section{D-branes and twistors}
In the discussion following the twistor action in Eq.(\ref{stwistor}) we
explained that more conditions are needed on $Z$ in order to obtain the proper
twistor that is equivalent to the phase space of a particle when $d\geq7.$
What if those conditions are never imposed? What would then be the content of
$Z_{A}^{~a}$ ? We find that the extra degrees of freedom can be interpreted as
collective coordinates of D-branes.
To see this let us consider the properties of the rectangular matrix
$Z_{A}^{~a}$ that follow from the action in Eq.(\ref{stwistor}). The gauge
group acts on the right side and there is a global symmetry with conserved
charges $J$ that acts on the left side$.$ These properties are summarized by
the equations%
\begin{equation}
\bar{Z}Z=0,\;J=Z\bar{Z}-trace.
\end{equation}
Thus, the global current $J$ is a $s_{d}\times s_{d}$ matrix in the
fundamental representation of the global group $G$ that can be expanded in a
complete set of SO$\left( d,2\right) $ gamma matrices as follows
\begin{equation}
J=Z\bar{Z}=J_{0}+\Gamma^{M}J_{M}+\frac{1}{2}\Gamma^{MN}J_{MN}+\frac{1}%
{3!}\Gamma^{MNK}J_{MNK}+\cdots\label{ggg}%
\end{equation}
There are as many terms $\Gamma^{M_{1}\cdots M_{n}}$ as necessary to span all
the generators of some group $G$ whose fundamental representation has the same
dimension $s_{d}$ of the SO$\left( d,2\right) $ spinor. If the group is
constrained to be SO$\left( d,2\right) $ there is only the term proportional
to $\Gamma^{MN}$ in the expansion (\ref{ggg}), and then the group element $g$
is constructed by exponentiating the generators as in Eq.(\ref{g}). But if the
group is more general, then the exponent in Eq.(\ref{g}) contains all the
terms that appear in Eq.(\ref{ggg}).
Thus, with a more general $g$ represented as a $s_{d}\times s_{d}$ matrix, a
more general twistor $Z$ would emerge, with more degrees of freedom than the
particle phase space. This is achieved with an action that is written in the
form of Eq.(\ref{Sg}), which in turn is obtained by gauge fixing from the
2T-physics action in (\ref{S2T}), but by taking $g$ to be a group element not
just in SO$\left( d,2\right) ,$ but an element in the smallest group $G$
that contains SO$\left( d,2\right) $ in the spinor representation. The
parent 2T-physics theory in Eq.(\ref{S2T}) has an interaction term of the form
of $\frac{4}{s_{d}}Tr\left( ig^{-1}\partial_{\tau}gL\right) $ which is
unchanged. But we emphasize that now $L$ is proportional to only $\Gamma
^{MN},$ while $ig^{-1}\partial_{\tau}g$ has all the terms in Eq.(\ref{ggg}),
so $L$ couples to only the SO$\left( d,2\right) $ subgroup of $G.$ Then the
action in Eq.(\ref{S2T}) still has local Sp$\left( 2,R\right) $ and
SO$\left( d,2\right) $ symmetries, but now it has global $G$ symmetry
instead of only global SO$\left( d,2\right) $ on the left side of $g.$
This generalization of the group element $g$ allows $Z$ to contain the extra
degrees of freedom. We emphasize that the spinor representation of SO$\left(
d,2\right) ,$ whose dimension is $s_{d},$ must correspond to the fundamental
representation of $G.$ This requirement determines $G$ as we see in Table 1 below.
We already know that for $d=3,4,6$ the groups $G=$Sp$\left( 4,R\right) $,
SU$\left( 2,2\right) $, Spin$\left( 6,2\right) $ respectively are exactly
equal to SO$\left( d,2\right) $ in the spinor representation. Therefore for
these cases there are no other terms in Eq.(\ref{ggg}) other than $\Gamma
^{MN},$ provided some (pseudo)reality conditions are imposed as given
following Table 1 below. Extra terms usually appear for $d\geq7.$
As an example of what terms appear, consider SO$\left( 7,2\right) $ for
$d=7$. The spinor representation has dimension 16. The smallest group with a
$16$ dimensional fundamental representation is SO$^{\ast}\left( 16\right) $,
where the $^{\ast}$ indicates the appropriate analytic continuation that
contains SO$\left( 7,2\right) $ as a subgroup. The number of generators of
SO$^{\ast}\left( 16\right) $ is $\frac{16\cdot15}{2}=\allowbreak120.$ The
number of generators represented by the 16$\times16$ gamma matrices is
$\Gamma^{M}\rightarrow9$, $\Gamma^{MN}\rightarrow\frac{9\cdot8}{2}=36$,
$\Gamma^{MNK}\rightarrow\frac{9\cdot8\cdot7}{1\cdot2\cdot3}=\allowbreak84$,
$\Gamma^{MNKL}\rightarrow\frac{9\cdot8\cdot7\cdot6}{1\cdot2\cdot3\cdot
4}=\allowbreak126.$ We see that the 120 generators of SO$^{\ast}\left(
16\right) $ are represented by $\frac{1}{2}\Gamma^{MN}L_{MN}+\frac{1}%
{3!}\Gamma^{MNK}L_{MNK}.$ Therefore the more general twistor in $d=7$ has the
expansion%
\begin{equation}
d=7:\;J=Z\bar{Z}=\frac{1}{2}L^{MN}\Gamma_{MN}+\frac{1}{3!}L^{MNK}\Gamma_{MNK}.
\end{equation}
If we impose additional conditions on $Z$ as in the previous sections, then we
eliminate the term $L_{MNK}=0,$ and the remaining $L_{MN}$ necessarily
satisfies all the conditions of section (\ref{sttransform}) since they all
followed from $\bar{Z}Z=0$ that was imposed by the gauge symmetry H.
However, if we do not impose the conditions of section (\ref{sttransform})
then we can interpret the degrees of freedom $L_{MNK}$ as D-brane degrees of
freedom. To see this, consider the smallest extended super-\textit{conformal}
algebra that contains spin$\left( 7,2\right) \subset$SO$^{\ast}\left(
16\right) $. This is OSp$\left( 16|2\right) .$ Its two supercharges satisfy
$\left\{ Q_{A}^{i},Q_{B}^{j}\right\} =\varepsilon^{ij}\left[ \frac{1}%
{2}\left( \Gamma^{MN}\right) _{AB}~L_{MN}+\frac{1}{3!}\left( \Gamma
^{MNK}\right) _{AB}~L_{MNK}\right] +q^{ij}C_{AB},$ with $i=1,2$ labelling
the Sp$\left( 2\right) ,$ and $C_{AB},q^{ij}$ both symmetric. The usual
$d=7$ Poincar\'{e} super-algebra is a subalgebra obtained from the above by
decomposing the SO(7,2)$\rightarrow$SO(6,1)$\times$SO(1,1) for spinors (16
=8$_{+}$+8$_{-}$ ) as $A=\alpha_{+}\oplus\alpha_{-}$ and vectors as
$M=\pm^{\prime},\mu,$ and keeping only the operators $Q_{\alpha+}^{i}$ and all
the $L^{MN},L^{MNK}$ with a single $+^{\prime}$ as follows%
\begin{equation}
\left\{ Q_{\alpha+}^{1},Q_{\beta+}^{2}\right\} =L^{+^{\prime}\mu}\left(
\Gamma_{+^{\prime}\mu}\right) _{\alpha\beta}+\frac{1}{2}L^{+^{\prime}\mu\nu
}\left( \Gamma_{+^{\prime}\mu\nu}\right) _{\alpha\beta}. \label{Dbrane}%
\end{equation}
In the massless particle gauge, $L^{+^{\prime}\mu}$ is the momentum $p^{\mu}$
and then $L^{+^{\prime}\mu\nu}$ are the \textit{commuting} D2-brane charges in
$d=7$ dimensions (like generalized momenta). The other components of $L_{MN},$
and $L_{MNK}$ are functions of phase space, including the particle as well as
D-brane canonical degrees of freedom, and do not generally commute among
themselves\footnote{The commutation rules of the $L^{M_{1}\cdots M_{n}}$ are
isomorphic to the commutation rules of the $\Gamma^{M_{1}\cdots M_{n}}.$ From
this we see that the D-brane charges $L^{+^{\prime}\mu\nu}$ commute among
themselves as well as with the momenta $p^{\mu}=L^{+^{\prime}\mu
}.\label{Dcharges}$}.
As another example consider SO$\left( 8,2\right) $ for $d=8$. The two spinor
representations are 16,$\overline{16}$. The smallest groups with $16$
dimensional fundamental representations are SO$^{\ast}\left( 16\right) $,
Sp$^{\ast}\left( 16\right) $, SU$^{\ast}\left( 16\right) .$ To decide
which is the smallest one that contains SO$\left( 8,2\right) $ as a subgroup
we analyze the number of generators represented by the gamma matrices. The
number of generators of SO$^{\ast}\left( 16\right) $ is $\frac{16\cdot15}%
{2}=\allowbreak120,$ for Sp$^{\ast}\left( 16\right) $ is $\frac{16\cdot
17}{2}=\allowbreak136,$ and for SU$^{\ast}\left( 16\right) $ is $\left(
16\right) ^{2}-1=255.$ The number of generators represented by the gamma
matrices is $\Gamma^{M}\rightarrow10$, $\Gamma^{MN}\rightarrow\frac{10\cdot
9}{2}=\allowbreak45$, $\Gamma^{MNK}\rightarrow\frac{10\cdot9\cdot8}%
{1\cdot2\cdot3}=\allowbreak120$, $\Gamma^{MNKL}\rightarrow\frac{10\cdot
9\cdot8\cdot7}{1\cdot2\cdot3\cdot4}=210$, $\Gamma_{+}^{MNKLR}\rightarrow
\frac{1}{2}\frac{10\cdot9\cdot8\cdot7\cdot6}{1\cdot2\cdot3\cdot4\cdot
5}=\allowbreak126,$ where the last one is self dual (hence the extra factor of
$\frac{1}{2}$). The 45 generators of SO$\left( 8,2\right) $ represented by
$\Gamma^{MN}$ must be included as one of the criteria in choosing the smallest
$G$. Then we see that there is no combination of gamma matrices that can be
used to construct SO$^{\ast}\left( 16\right) $ and Sp$^{\ast}\left(
16\right) $ and therefore we must take SU$^{\ast}\left( 16\right) $ as the
smallest group that contains spin$\left( 8,2\right) .$ The smallest
superconformal algebra is SU$\left( 16|1\right) .$ The 255 generators of
SU$^{\ast}\left( 16\right) $ that appear in $\left\{ Q_{A},\bar{Q}%
^{B}\right\} $ are then represented by $\frac{1}{2}\Gamma^{MN}L_{MN}+\frac
{1}{4!}\Gamma^{MNKL}L_{MNKL}.$ The extra $L_{MNKL}$ lead to the D-brane
degrees of freedom. To see the content of D-brane \textit{commuting}%
$^{\ref{Dcharges}}$ charges we must decompose SO$\left( 8,2\right) $ to
SO$\left( 7,1\right) $ by $M=\pm^{\prime},\mu$ and identify the D-brane
commuting charges as the D3-brane $Z^{+^{\prime}\mu\nu\lambda}$ in $8$ dimensions.%
\begin{gather*}%
\begin{tabular}
[c]{|l|l|l|l|l|l|l|}\hline
d & Spin$\left( d,2\right) $ & {\small spinor} & G & G$_{\text{super}%
}\left( N\right) $ & {\small generators of G in Spin}$\left( d,2\right)
${\small basis} & $%
\genfrac{}{}{0pt}{}{\text{contained}}{\text{in product}}%
$\\\hline
3 & Spin$\left( 3,2\right) $ & 4 & Sp$\left( 4,R\right) $ & OSp$\left(
N|4\right) $ & $\Gamma^{MN}$ $_{10}$ & $\left( {\small 4\times4}\right)
_{s}$\\\hline
4 & Spin$\left( 4,2\right) $ & 4$,\bar{4}$ & SU$\left( 2,2\right) $ &
SU$\left( 2,2|N\right) $ & $\Gamma^{MN}$ $_{15}$ & ${\small 4\times\bar{4}}%
$\\\hline
5 & Spin$\left( 5,2\right) $ & 8$_{+}$ & $%
\genfrac{}{}{0pt}{}{\text{spin}^{\ast}\text{(7)~}}{\text{SO}^{\ast}\left(
8\right) ~~}%
$ & $%
\genfrac{}{}{0pt}{}{\text{F(4)~~~~~~~~~~}}{\text{OSp}\left( 8|2N\right) ~}%
$ & $%
\genfrac{}{}{0pt}{}{\Gamma^{MN}\text{~}_{21}\text{~~~~~~~~~~~~~~}}{\Gamma
^{MN}~_{21}~\oplus~~\Gamma^{M}~_{7}}%
$ & $\left( {\small 8\times8}\right) _{a}$\\\hline
6 & Spin$\left( 6,2\right) $ & 8$_{+}$ & SO$^{\ast}\left( 8\right) $ &
OSp$\left( 8|2N\right) $ & $\Gamma^{MN}$ $_{28}$ & $\left( {\small 8\times
8}\right) _{a}$\\\hline
7 & Spin$\left( 7,2\right) $ & 16 & SO$^{\ast}\left( 16\right) $ &
OSp$\left( 16|2N\right) $ & $\Gamma^{MN}$ $_{36}$ $\oplus$ $\Gamma^{MNK}$
$_{84}$ & $\left( 1{\small 6\times16}\right) _{a}$\\\hline
8 & Spin$\left( 8,2\right) $ & {\small 16}$,\overline{{\small 16}}$ &
SU$^{\ast}\left( 16\right) $ & SU$\left( 16|N\right) $ & $\Gamma^{MN}$
$_{45}$ $\oplus$ $\Gamma^{MNKL}$ $_{210}$ & ${\small 16\times~}\overline
{{\small 16}}$\\\hline
9 & Spin$\left( 9,2\right) $ & 32 & Sp$^{\ast}\left( 32\right) $ &
OSp$\left( N|32\right) $ & $\Gamma^{MN}$ $_{55}$ $\oplus$ $\Gamma^{M}$
$_{11}$ $\oplus$ $\Gamma^{M_{1}\cdots M_{5}}$ $_{462}$ & $\left(
{\small 32\times32}\right) _{s}$\\\hline
10 & Spin$\left( 10,2\right) $ & 32$_{+}$ & Sp$^{\ast}\left( 32\right) $ &
OSp$\left( N|32\right) $ & $\Gamma^{MN}$ $_{66}$ $\oplus$ $\Gamma_{+}%
^{M_{1}\cdots M_{6}}$ $_{462}$ & $\left( {\small 32\times32}\right) _{s}%
$\\\hline
11 & Spin$\left( 11,2\right) $ & 64 & Sp$^{\ast}\left( 64\right) $ &
OSp$\left( N|64\right) $ & $\Gamma^{MN}$ $_{78}$ $\oplus$ $\Gamma^{MNK}$
$_{286}$ $\oplus$ $\Gamma^{M_{1}\cdots M_{6}}$ $_{1716}$ & $\left(
{\small 64\times64}\right) _{s}$\\\hline
12 & Spin$\left( 12,2\right) $ & {\small 64,}$\overline{{\small 64}}$ &
SU$^{\ast}\left( 64\right) $ & SU$\left( 64|N\right) $ & $\Gamma^{MN}$
$_{91}$ $\oplus$ $\Gamma^{MNKL}$ $_{1001}$ $\oplus$ $\Gamma^{M_{1}\cdots
M_{6}}$ $_{3003}$ & ${\small 64\times~}\overline{{\small 64}}$\\\hline
\end{tabular}
\\
\text{{\small Table 1:} {\small Smallest} {\small groups} }{\small G}\text{
{\small and supergroups} }{\small G}_{\text{super}}\text{ {\small that contain
Spin}}\left( {\small d,2}\right) {\small ,}\text{ {\small and D-branes}.}%
\end{gather*}
In Table-1 we give a list of the smallest groups $G$ that contain spin$\left(
d,2\right) $ for $3\leq d\leq12.$ We also include the smallest supergroup
$G_{\text{super}}$ that contains $G.$ We list the gamma matrix representation
of the generators of $G$, and their numbers as subscripts, as represented by
antisymmetrized products of gamma matrices $\Gamma^{M_{1}\cdots M_{n}}%
\equiv\frac{1}{n!}\left( \Gamma^{M_{1}}\bar{\Gamma}^{M_{2}}\Gamma^{M_{3}%
}\cdots\Gamma^{M_{n}}\mp\text{permutations}\right) $ in dimension $d+2$
labelled by $M.$ The last column gives information on whether the gamma
matrices occur in the symmetric or antisymmetric products of the spinors of
SO$\left( d,2\right) $, when both spinor indices $A,B$ are lowered or raised
in the form $\left( \Gamma^{M_{1}\cdots M_{n}}\right) _{AB}$ by using the
metric $C$ in spinor space. The phase space of the D-branes correspond to the
extra generators beyond $\Gamma^{MN}$ as explained in the examples above. In
the case of $d=5,$ one option is to keep the D0-brane associated with
$\Gamma^{M}\rightarrow\Gamma^{+^{\prime}},$ another option is to remove it
with the extra U$\left( 1\right) $ gauge symmetry as discussed in the
counting done in section (\ref{sttransform}). The D-brane does not occur for
the supergroup F$\left( 4\right) $ that can be used for the $d=5$
superparticle as described below.
Groups that are larger than the listed $G$ may be considered in our scheme in
every dimension (e.g. SU$\left( 8\right) $ instead of SO$\left( 8\right) $
in $d=6$, etc.). In that case the number of generators $\Gamma^{M_{1}\cdots
M_{n}}$ increases compared to the ones listed in the table for each $d.$
Furthermore the corresponding D-brane degrees of freedom also get included in
the model.
When $Z_{A}^{~a}$ is obtained from the group element $g$ through the relation
$Z\bar{Z}=g\Gamma g^{-1},$ with the group $G$ listed in the table above, then
$Z$ is real or pseudo-real when the group is SO or Sp and it is complex when
the group is SU. Given those properties, in general the quadratic $\left(
Z\bar{Z}\right) _{A}^{~~B}$ contains just the gamma matrices listed above
which correspond to the generators of the group $G.$ If the (pseudo)reality
properties associated with $G$ are not obeyed by $Z$ then more D-brane terms
will appear generally in the expansion of $Z\bar{Z}$ as in Eq.(\ref{ggg}) as
compared to those on the table.
The case of $d=11$ is particularly interesting since it relates to M-theory as
follows. The corresponding twistors are spinors of Spin$(11,2)$ that are 64
dimensional. The smallest group is Sp$^{\ast}\left( 64\right) $ whose
generators are represented by $\Gamma^{MN}$ $\left( 78\right) $ +
$\Gamma^{MNK}$ $\left( 286\right) $ + $\Gamma^{M_{1}\cdots M_{6}}$ $\left(
1716\right) .$ To identify the commuting charges of D-branes we decompose
$M=\pm^{\prime},\mu$ and keep all the generators with a single $+^{\prime},$
as follows $L^{+^{\prime}\mu}$ $\oplus$ $L^{+^{\prime}\mu\nu}$ $\oplus$
$L^{+^{\prime}\mu_{1}\cdots\mu_{5}}.$ Here $L^{+^{\prime}\mu}$ is the momentum
in 11 dimensions and $L^{+^{\prime}\mu\nu},L^{+^{\prime}\mu_{1}\cdots\mu_{5}}$
are the D2-brane and D5-brane \textit{commuting}$^{\ref{Dcharges}}$ charges respectively.
Let us mention that the discussion above with the group $G$ can be directly
generalized to the supergroup $G_{\text{super}}\left( N\right) $ listed in
Table 1 (with some limits on $N$ as discussed below) by following
\cite{2ttwistor}. The 2T-physics action is still of the same form as $S\left(
X,P,g\right) $ of Eq.(\ref{S2T}), but now we have a supergroup element $g$
and a supertrace coupling $\frac{4}{s_{d}}Tr\left( ig^{-1}\partial_{\tau
}gL\right) ,$ and the matrix $L$ is of the form $L=\frac{1}{4i}\left(
\genfrac{}{}{0pt}{}{\Gamma^{MN}}{0}%
\genfrac{}{}{0pt}{}{0}{0}%
\right) L_{MN}$ where the gamma matrices couple to the bosonic subgroup $G$
as above. This 2T superparticle action reduces to the standard massless
superparticle action in the particle gauge for dimensions $d=3,4,5,6$
\cite{2ttwistor} with $N$ supersymmetries. It can also be gauge fixed to the
twistor gauge to give supertwistors that are equivalent to the super phase
space in those dimensions \cite{2tsuperstring}\cite{2tstringtwistors}. One can
go beyond those gauge choices and obtain a twistor description of many other
dual superparticle theories that give the super generalizations of the ones
studied recently in \cite{twistorBP1}.
The supergroup can be enlarged to have more fermionic generators, but keeping
$G$ as a bosonic subgroup. For example, for $d=4$ we may take SU$\left(
2,2|N\right) $ instead of the smallest $N=1$ shown in the table. For physical
purposes the total number of real fermionic generators cannot exceed 64 (32
ordinary supercharges and 32 conformal supercharges). For example, for $d=4$
we can go as far as $N=8$, or $G_{\text{super}}=$SU$\left( 2,2|8\right) $
which has 64 real fermionic parameters. Similarly, for $d=10$ we may take
OSp$\left( 1|32\right) $ or OSp$\left( 2|32\right) .$ In the more general
cases the coupling $L$ can be of the form of the previous paragraph
$L=\frac{1}{4i}\left(
\genfrac{}{}{0pt}{}{\Gamma^{MN}}{0}%
\genfrac{}{}{0pt}{}{0}{0}%
\right) L_{MN}.$
The model can also be generalized by adding $d^{\prime}$ more dimensions
$\left( X^{I},P^{I}\right) $ in addition to the $d+2$ dimensions $\left(
X^{M},P^{M}\right) $, but keeping the same $g\in G_{\text{super}}.$ The
generalized action has the form \cite{2tAdSs}\cite{2tsuperstring}%
\cite{2tstringtwistors}\cite{chinaLect}%
\begin{equation}
S_{2T}\left( \hat{X},\hat{P},g\right) =\int d\tau\left[ \frac{1}%
{2}\varepsilon^{ij}\partial_{\tau}\hat{X}_{i}\cdot\hat{X}_{j}-\frac{1}%
{2}A^{ij}\hat{X}_{i}\cdot\hat{X}_{j}+\frac{4}{s_{d}}Str\left( ig^{-1}%
\partial_{\tau}g\hat{L}\right) \right]
\end{equation}
where $\hat{X}^{\hat{M}}=\left( X^{M},X^{I}\right) ,$ $\hat{P}^{\hat{M}%
}=\left( P^{M},P^{I}\right) ,$ and we now take the more general coupling
$\hat{L}=\frac{1}{4i}\left(
\genfrac{}{}{0pt}{}{\Gamma^{MN}}{0}%
\genfrac{}{}{0pt}{}{0}{0}%
\right) L_{MN}+\frac{\alpha}{4i}\left(
\genfrac{}{}{0pt}{}{0}{0}%
\genfrac{}{}{0pt}{}{0}{\Gamma^{IJ}}%
\right) L_{IJ}$. The fixed parameter $\alpha=\frac{s_{d}}{s_{d^{\prime}}}$ is
determined by local bosonic and fermionic symmetries in this action, to be the
ratio of the spinor dimensions of SO$\left( d+2\right) $ and SO$\left(
d^{\prime}\right) .$ In this latter scheme we obtain interesting cases, such
as supertwistors with some compactified subspaces, without D-branes. For
example supertwistors for AdS$_{4}\times$S$^{7},$ AdS$_{5}\times$S$^{5},$
AdS$_{7}\times$S$^{4}$ with a of total 10 or 11 dimensions emerge
\cite{2tsuperstring}\cite{2tstringtwistors}\cite{chinaLect} by using
supergroups with only 32 real fermions, namely $G_{\text{super}}=$OSp$\left(
8|4^{\ast}\right) $, SU$\left( 2,2|4\right) $, OSp$\left( 8^{\ast
}|4\right) $ respectively.
This analysis taken to the maximum allowed number of supersymmetries and the
maximum number of dimensions leads to $d=11$ with OSp$\left( 1|64\right) $
as the hidden global supersymmetry for M-theory, and suggests that the
extended supertwistors may well play a role in a nice description of M-theory.
This can be studied through the toy M-model \cite{2ttoyM} that has the action
$S\left( X,P,g\right) $ of Eq.(\ref{S2T}) with the group OSp$\left(
1|64\right) ,$ and includes the D2 and D5-branes. The supergroup OSp$\left(
1|64\right) $ is motivated by other considerations as well \cite{Stheory}%
\cite{liftM}\cite{west}. In other approaches to twistors in D=11
\cite{azcarraga2}\cite{azcarraga3}, twistors in the fundamental representation
of OSp$(1|64)$ were used in \cite{azcarraga3} for the formulation of a
superstring action in an extended D=11 superspace.
The twistor action (\ref{stwistor}) is a gauge fixed form of the 2T-physics
action $S\left( X,P,g\right) $ of Eq.(\ref{S2T}) for a general $g$. We can
play the game of gauge fixing the local symmetries Sp$\left( 2,R\right) $
and SO$\left( d,2\right) $ of $S\left( X,P,g\right) $ in many possible
ways and derive a multitude of 1T-physics systems with a rich web of dualities
among them. D-branes are included in this web of dualities. Then twistors can
be shown to unify many dual theories including D-branes. It would be
interesting to pursue this line of reasoning in more detail.
\section{Discussion}
In this paper we gave the general twistor transform that maps twistor space to
phase space in $d$ dimensions with one time. The general transform can be
specialized to a variety of special dynamical particle systems that include
particles with or without mass, relativistic or nonrelativistic, in flat or
curved spaces, interacting or non-interacting. Thus, the scope of our formulas
is much larger than the traditional twistor transform. The special cases of
phase space described by the same twistor are those that can be derived by
gauge fixing the parent unifying theory in 2T-physics. Thus, either the
twistor description or the vector SO$\left( d,2\right) $ description in
2T-physics provide a unification of those 1T-physics systems and establishes a
duality relationship between them.
To our knowledge this is the first time that twistors have been successfully
defined generally in $d$ dimensions. We have insured that our twistor
transform is fully equivalent to particle phase space for all dimensions. If
we specialize to the phase space of massless particles only, then our result
agrees with twistors that were previously defined for $d\leq6$ in another
approach \cite{cederwal}. Even for $d\leq6$ our twistors for the phase spaces
other than the massless particle are all new structures. For a rather
different approach to twistors for massive particles, which uses double the
number of twistors compared to our formulas and only in $d=4$, see
\cite{penrose2},\cite{perjes}-\cite{azcarraga}.
Beyond twistors for particles, we have also defined twistors for a phase space
that includes also D-brane degrees of freedom. Including the D-branes may lead
to some interesting applications of twistors, in particular for M-theory. The
twistor action principle in Eq.(\ref{Stw}) applies generally to twistors
including D-branes for a generally complex $Z$ in every dimension. If
(pseudo)reality conditions are imposed on $Z$ as mentioned after the table in
the previous section, then for $d\leq6$ we obtain only the particle phase
space out of the twistor. In $d\geq7$ there are automatically D-branes even
with the (pseudo) reality conditions. However, if extra constraints are
applied on $Z,$ as detailed in section (\ref{sttransform}), then again the
degrees of freedom in $Z$ are thinned down to only the particle phase space
without D-branes in every dimension.
Quantization of twistors for any spin in four dimensions was discussed in
section (\ref{twistd4}). Here we suggested a free field theory in twistor
space that describes any spinning particle. This could also lead to some
interesting applications of twistors in field theory with interactions in four dimensions.
Generalizations of our results in many directions are possible. Some of these
are already briefly described in recent papers \cite{2tsuperstring}%
\cite{2tstringtwistors}, such as twistors for spinning particles,
supertwistors in various dimensions, including compactified dimensions, and
supertwistors for supersymmetric AdS$_{5}\times$S$^{5},$ AdS$_{4}\times$%
S$^{7},$ AdS$_{7}\times$S$^{4}.$ We plan to give details of those structures
in future publications.
It must be emphasized that in all cases the underlying theory is anchored in
2T-physics, and therefore by gauge fixing the Sp$\left( 2,R\right) $ gauge
symmetry these twistors describe not only massless systems, but much more, as
discussed in this paper and \cite{twistorBP1} with examples. Hence the
twistors play a role in some kind of unification of 1T-physics systems via
dualities, or via higher dimensions with 2T, but in a way that is distinctly
different than the Kaluza-Klein scheme, since there are no Kaluza-Klein
excitations, but instead there is a web of dualities.
In this context it is also interesting that some parameters such as mass,
moduli of some metrics, and some coupling constants for interactions, emerge
from the higher dimensions as moduli while holographically projecting from
$d+2$ dimensions down to $d$ dimensions. Furthermore concepts such as time and
Hamiltonian in 1T-physics are derived concepts that emerge either from
2T-physics and its gauge choices, or from the details of the twistor transform
to 1T-physics systems.
\bigskip{\Large Acknowledgments}\textbf{\bigskip}
I. Bars was supported by the US Department of Energy under grant No.
DE-FG03-84ER40168; M. Pic\'{o}n was supported by the Spanish Ministerio de
Educaci\'{o}n y Ciencia through the grant FIS2005-02761 and EU FEDER funds,
the Generalitat Valenciana and by the EU network MRTN-CT-2004-005104
\textquotedblleft Constituents, Fundamental Forces and Symmetries of the
Universe\textquotedblright. M. Pic\'{o}n wishes to thank the Spanish
Ministerio de Educaci\'{o}n y Ciencia for his FPU research grant, and the USC
Department of Physics and Astronomy for kind hospitality.
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BkiUcMbxK0iCl7DT7ga4
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\section{Introduction}\label{sec1}
Neutron stars are mysterious compact objects where strong gravitational and electromagnetic fields emerge. These objects usually manifest as pulsars, emitting energetic electromagnetic signals detected at very precise intervals \citep{Camenzind}.
Even though they were discovered more than fifty years ago, many ingredients of these astrophysical systems are still poorly known: the composition of the compact object itself \citep{Weber2005}, the composition and structure of their magnetosphere \citep{Petri2016}, and the generation of their strong magnetic fields \citep{Duncan1992, Dieters1998}.
Concerning the magnetosphere, three fundamental models can be considered: the simplest one consists of a naked star, without any kind of plasma in its neighborhood; the opposite one in which we have the compact object fully immersed in a plasma, and finally, an intermediate model that admits the existence of a magnetosphere partially filled with electrons and positrons \citep{Petri2016}
On the other hand, a magnetic field strength on the surface of neutron stars of the order of Schwinger's critical field $B_c= 4.41 \times 10^{13}$~G (e.g. \citep{Ciolfi2014}), and beyond ( e.g. \citep{Dieters1998}), imposes to study the behavior of matter at extreme conditions.
On the observational side, the information we can get from pulsars is based on the measurement of their pulses arrival times. During their propagation from the source to Earth, photons can experience a variety of time delays. In particular, a most known time delay is the dispersion of photons when interacting with the electrons in the interstellar medium \citep{Pushkarev2010},\citep{Wang2004}, \citep{Bosnjakl2012}, \citep{Waxman1996}. This type of time delay mainly depends on the electron column density and the distance to the source, and causes that pulses at lower frequencies are delayed with respect to those emitted at higher frequencies (the delay is inversely proportional to the square of the photon frequency; see e.g.~\citep{2004hpa..book.....L}).
Therefore, the knowledge of this type of time delay probes the interstellar medium properties rather than the source site ones.
We are here interested in a photon time delay process occurring in the vicinity of the source, and unrelated with the photon interaction with matter. The propagation of light in vacuum is modified by various external agents: electromagnetic fields, temperature, geometric boundary configurations, gravitational background and non-trivial topologies. In particular, the problem of light propagation in electron-positron vacuum in the presence of a magnetic field is similar to the dispersion of light in an anisotropic medium, where the external field axis sets the anisotropic direction. Therefore, the photon dispersion relation is corrected by adding the polarization tensor $\Pi(k_{\perp},k_{\parallel}, B,\omega)$. This takes into account the indirect interaction with the magnetic field through the virtual electron-positron pairs \citep{PerezRojasH.Shabad1978} and depend on components of the wave vector, the photon frequency and the external magnetic field.
In this paper, we study photon time delay which might occur in the pulsar magnetosphere. We use the simplest approximation to describe it, which consists in considering it as a magnetized vacuum.
We start by solving the photon dispersion equation considering the radiative corrections given by the magnetized photon self-energy. Then, we compute the phase velocity and the photon time delay.
The propagation of photons is considered perpendicular to the magnetic field ($k \perp B)$ since, for parallel propagation, photons behave like in absence of the magnetic field, namely with no deviation from the light-cone.
From a physical aspect, we estimate the time delay of photons in the region of the pulsar magnetosphere, modeling it as an electromagnetic vacuum.
From a mathematical point of view, our calculation is more robust (and elegant) than others \citep{PerezRojas2014}, since our analytic expressions for the solution of the dispersion equation are presented in term of A-hypergeometric functions \citep{Sturmfels2000}.
The paper is organized as follows. In section \ref{sec2}, we solve the dispersion equations considering the radiative corrections given by the photon self-energy in presence of magnetic field. We devote section \ref{sec3} to discuss the phase velocity and time delay of photons traveling in a magnetized vacuum. The dependence with the radial coordinate of the photon delay is studied assuming a dipole configuration for the magnetic field in the magnetosphere. Finally, we present in section~\ref{sec4} the conclusions of our work.
\section{Propagation of photon in magnetized vacuum} \label{sec2}
In this section, we study the propagation of photons perpendicular to the constant and uniform external magnetic field in vacuum\footnote{We use natural units $\hbar=c=1$.}.
It is well known that photons in vacuum obey the dispersion equation
\begin{equation}\label{lightcone}
k^2_{\perp}+k^2_{\parallel}-\omega^2=0,
\end{equation}
that implies that photons travel at the speed of light.
The effect of the presence of the magnetic field on the dispersion relation, Eq.~(\ref{lightcone}), can be included through radiative corrections to the photon self-energy.
The modified dispersion equations ~\citep{Sha1984} are
\begin{equation}
k^2=\kappa^{(i)}(\omega,k_{\parallel},k_{\perp},b),
\end{equation}
where $b$ is the magnetic field normalized to the Schwinger's field, $b=B/B_c$, and $\kappa^{(i)}$ are the eigenvalues of the photon self-energy given in the appendix.
In what follows, we consider photon propagation perpendicular to the magnetic field ($\vec{k}\perp \vec{B}$). Three modes appear: one longitudinal mode $i=1$, that is not physical and two transverse ones $i=2,3$. The threshold of pair creation of second and third modes are $\omega=2m_e$ and $\omega=m_e+\sqrt{m_e^2+2eB}$, respectively \citep{PerezRojas2014}.
In our study, we consider only the second mode, which is more relevant in the region of transparency. The corrections of dispersion relations become relevant close to the thresholds, and the second mode threshold is independent on the magnetic field, being much lower than the threshold for the third mode, for the considered values of the magnetic field.
For a large range of frequencies, the solution of the dispersion equation corresponds to relatively small deviations from the light-cone, $k^2\ll e B$, except for values of $\omega^2-k^2_{\parallel}$ extremely close to the vacuum threshold for pair creation \citep{PerezRojas2014}. As it is shown in the appendix, in this case, we can write the photon self-energy eigenvalues as polynomials in $k^2$:
\begin{equation}
\kappa_{i}=\sum_{l=0}^{\infty}\chi_{il}(k^2)^l.
\end{equation}
If we truncate the first four terms of the power series, for $j=3$ we obtain a cubic equation in $k^2$. This equation has been solved in \citet{PerezRojas2014} using Cardano formulas for polynomials of third degree. However, numerical calculations with quadratic and cubic roots are thorny, so in this work we solve it with the aid of hypergeometric functions \citep{Sturmfels2000}:
\begin{equation}
k^2=-\sum_{j_2,...,j_n=0}^{\infty}\dfrac{(-1)^{j_1}j_1!}{(j_0+1)!j_2!...j_n!}\dfrac{\chi_{0}^{j_0+1} \chi_{i2}^{j_2} ...\chi_{in}^{j_n} }{(\chi_{i1}-1)^{j_1+1}},
\end{equation}
where $ j_0=j_2+2j_3+...+(n-1)j_n $, $j_1=2j_2+3j_3+...+nj_n $, and integral expressions of $\chi_{i1}$ are written in the appendix.
The solution of the dispersion equation is shown in Fig.~\ref{fig:lightcone} for selected values of the magnetic field strength. The figure shows that, when the magnetic field increases, the deviation from the light-cone is higher. Besides, for any value of the magnetic field, a threshold exists $\omega=2m_{e}$, above which the photons have a high probability to decay in electron-positron pairs \citep{PerezRojas2009}.
\begin{figure}[t]
\centering
\includegraphics[width=.6\linewidth]{lightcone_mode2.eps}
\caption{Dispersion relation for selected values of the magnetic field strength. The orange line corresponds to the propagation of light for $B=0$ (light-cone) and the gray line marks the first threshold of pair creation. We recall that $B_{c}=4.41 \times 10^{13}$~G and $m_{e}= 0.511$~MeV.}
\label{fig:lightcone}
\end{figure}
For our purposes, we are only interested in the study of the region of transparency ($ 0<\omega<2m_e$), which is the region of momentum space where the photon self-energy and its frequency have real values.
\section{Phase velocity and time delay in magnetized vacuum} \label{sec3}
In this section, we calculate the phase velocity and the photon time delay taking advantage of the previous calculations.
The photon phase velocity takes the form
\begin{eqnarray}
v_{ph}(\omega,B)&=&\dfrac{\omega}{k_{\perp}}\nonumber\\
&=&\left (1-\dfrac{1}{\omega^2}\sum_{j_2,j_3=0}^{\infty}\dfrac{(-1)^{j_1}j_1!}{(j_0+1)!j_2!j_3!}\dfrac{\chi_{i0}^{j_0+1} \chi_{i2}^{j_2} \chi_{i3}^{j_3}}{(\chi_{i1}-1)^{j_1+1}} \right )^{-1/2}.
\end{eqnarray}
Figure~\ref{figure2} shows the photon phase velocity as a function of magnetic field for fixed values of the frequencies.
\begin{figure}[h!]
\centering
\includegraphics[width=.6\linewidth]{velocityphase_field_mode2.eps}
\caption{ Phase velocity in function of the magnetic field for different frequencies }
\label{figure2}
\end{figure}
We can see that photons of higher energies have lower phase velocity than the lower energy ones, hence the latter suffer a longer time delay. Besides, we can appreciate that in the limit of low frequency (black solid line), the phase velocity decreases linearly with the external magnetic field strength.
\subsection{Photon time delay in magnetosphere}
To calculate the photons time delay when crossing the magnetosphere (magnetized vacuum), for different energies, we consider a magnetic dipole configuration:
\begin{equation}\label{dipolar}
B(r)=B_{0} \left (\dfrac{r_{0}}{r}\right )^3,
\end{equation}
where $B_{0}$ and $r_{0}$ are, respectively, the surface magnetic field and radius of the neutron star.
We consider for $B_{0}$ values from $10^{12}$~G all the way up to $10^{15}$~G, covering the range of (theoretically) estimated fields from radio pulsars to soft gamma repeaters and anomalous X-ray pulsars (``magnetars'') \citep[e.g.][]{Duncan1992,Dieters1998}.
Using Eq.~(\ref{dipolar}), we can compute the time delay of the radiation crossing the magnetosphere of the pulsars given by the expression
\begin{equation}
\tau=\int_{r_o}^ r \frac{dr}{v_{ph}(\omega,B(r))}.
\end{equation}
Figure~\ref{Velocidaddefase} shows the phase velocity as a function of the distance traveled by the photons for different values of frequencies and two different values of $B_{0}$. It can be seen how, as expected, the phase velocity tends to the speed of light as the magnetic field decreases.
\begin{figure}[h!]
\centering
\includegraphics[width=.6\linewidth]{velocityphase_distance_field_10_range_1_2.eps}
\includegraphics[width=.6\linewidth]{velocityphase_distance_field_100_range_1_4.eps}
\caption{Phase velocity as a function of the distance traveled by the photons for different frequencies, top panel ($B_{0}= 10 B_{c}$) and bottom panel ($B_{0}= 100 B_{c}$) }
\label{Velocidaddefase}
\end{figure}
Figure~\ref{timedelay} shows the time delay of photons as a function of the distance, for two fixed values of the surface magnetic field $B_{0}$. The time delay is of the order of nanosecond for different values of frequencies. In spite of its shortness, it already shows that the solely presence of the magnetic field is sufficient to cause a time delay that grows with the photon frequency.
We would like to stress that more complex configurations of the magnetosphere, including for instance the magnetized electron-positron plasma, might have relevant effects. A study in this direction is currently in progress.
\begin{figure}[h!]
\centering
\includegraphics[width=.6\linewidth]{timedelay_field_10_range_1_4.eps}
\includegraphics[width=.6\linewidth]{timedelay_field_100_range_1_4.eps}
\caption{Time delay of photons for fixed values of frequencies as a function of the distance traveled, top panel ($B_{o}= 10 B_{c}$) and bottom panel ($B_{o}= 100 B_{c}$) }
\label{timedelay}
\end{figure}
\section{Conclusions}\label{sec4}
We solved the dispersion equation for photons propagating perpendicular to a constant and uniform magnetic field.
The analytical and approximate expressions for the phase velocity, valid in a wide range of the characteristic parameters, have been calculated. The photon phase velocity depends on the magnetic field and the photon energy.
In the limit of low frequencies, the phase velocity tends to have a linear behavior with the magnetic field.
The photon time delay was calculated starting from a simple model of the magnetic field configuration in the neutron star magnetosphere. We found that differences between the photon time delay of $\gamma$-radiation $\lesssim 1~ MeV$ is of the order of nanosecond. This difference might be due to the fact that more energetic photons interact stronger with virtual electron-positron pairs, being closer to the threshold ($\omega=2 m_{e}$)
In this work, we have made a first attempt to study the photon time delay considering a simple model of the magnetosphere. An improvement of our study should include, for instance, a more realistic model of pulsar magnetosphere and the magnetized electron-positron plasma, as well as other possible geometrical configurations of the magnetic field that surrounds the neutron star.
A relevant result of our work is that, contrary to the traditional time delay of photons in the interstellar medium, in the present quantum electrodynamical process the more energetic photons are delayed with respect to the lower energetic ones. An essential pending task is to understand the physical reason at the core of this theoretical result.
|
train/arxiv
|
BkiUd3s5qU2Ap6C-CoSd
| 5
| 1
|
\section{Introduction}
The success of homogeneous Big-Bang Nucleosynthesis (BBN) is one of the
lynchpins of cosmology. Using the baryon-to-photon ratio, $\eta$, inferred from
measurements of the cosmic microwave background (CMB) radiation \cite{wmap7},
homogeneous BBN predicts successfully the astrophysical abundances of
Deuterium, \he3 and \he4~\cite{cfo1,cfo2,bbn2,fs,cyburt,cfo5}. On the other hand, there are issues with the
abundances of \li7 \cite{cfo5} and potentially with \li6 \cite{rvo}.
In particular, the predicted abundance of \li7 is
considerably larger than the range suggested by
observations~\cite{spite,rbn,rbofn,liglob,liglob2,asp06,hos,aoki,sbordone,newer}~\footnote{A
globular cluster star with a \li7 abundance comparable to the BBN prediction has recently been observed~\cite{LiNa}:
this value may be due to production by a previous generation of stars.}, and there
are suggestions that the astrophysical value of the \li6 abundance may be
much higher than predicted by homogeneous BBN \cite{asp06}. However, one should note that
the line asymmetries which have been interpreted as \li6 could be the result of
convective processes affecting \li7 \cite{cayrel}.
We and others have investigated previously whether the late decays of massive particles,
such as the gravitino in the constrained minimal supersymmetric extension of the
Standard Model (the CMSSM) with a neutralino as the lightest supersymmetric particle (LSP),
could improve significantly the \li6 and \li7
abundances predicted by homogeneous BBN~\cite{Lindley:1984bg,Ellis:1984er,Lindley:1986wt,Scherrer:1987rr,Reno,Dimopoulos:1988ue,ellis,Kawasaki:1994af,kawmoroi,holtmann,karsten,kkm,kohri,cefo,jed,Jedamzik:2004ip,kkm2,EOV,kmy,Jed1,Jed2,Jedamzik06,stef,jp,ceflos}. We did not find a solution to the \li6
problem, but we did find a region of supersymmetric parameter space where
gravitino decays might alleviate or even solve the \li7 problem~\cite{ceflos,ceflos2}. On the other hand
the \li7 problem might have a more banal solution, such as the existence of a
suitable carbon, boron or beryllium resonance~\cite{resonance}.
In addition to decays, the late-time annihilations of cold dark matter may also affect the abundances of the light elements
\cite{Reno,earlylate,hisano,Jedamzik:2004ip,jed}.
In particular, these annihilations may have a significant
effect on the abundance of \li6 \cite{Jedamzik:2004ip}. There it was argued that \li6 production may occur
if the $s$-wave annihilation cross-section is sufficiently large, and it was assumed that the relic density
of the annihilating dark matter particles is controlled largely by the $s$-wave part of the cross-section.
However, in supersymmetric models where the LSP is a neutralino, such as the CMSSM,
the relic density is in fact largely determined by the
$p$-wave part of the cross-section, which by the time of BBN is essentially ineffective. Therefore,
a re-analysis of the suggestion of~\cite{Jedamzik:2004ip} in the context of the CMSSM and related
models is timely, and is the subject of this paper.
In this paper we study the possible effects on the cosmological light-element
abundances of residual late-time annihilations of neutralino LSPs during or after BBN~\cite{earlylate,jed,Jedamzik:2004ip}.\footnote
{
Recent papers have also considered the BBN consequences of
WIMP models having residual annihilations increased by
Sommerfeld or Briet-Wigner enhancements \cite{enhance}.
In the CMSSM model, neither effect occurs,
due to the lack of a light boson and of
extreme degeneracy in the funnel, respectively.
}
We find negligible effects on the abundances of Deuterium, \he3, \he4 and \li7 predicted
by homogeneous BBN, but potentially a large enhancement in the predicted abundance of \li6,
as suggested in~\cite{Jedamzik:2004ip}.
The physics of this effect is the following. It is well understood that the famous $A = 5$ gap
in the spectrum of stable nuclei impedes the production of heavier nuclei in BBN.
The dominant mechanism for making \li6 in annihilating-particle scenarios
is initiated by $p$ and $n$ spallation of \he4. This yields many $A = 3$ nuclei with only a tiny
reduction in \he4 abundance. The tritium and \he3 nuclei are produced with
large, nonthermal energies, and subsequently slow down due to
ionization losses,
but have some probability of inducing $t (\alpha, n)\li6$ or $\he3 ( \alpha, p) \li6$
reactions first. In this way, an amount of \li6 may be produced that is large relative
to the standard homogeneous BBN abundance, without making large
amounts of extra Deuterium and $A = 3$ or reducing the \he4 abundance, and leaving the \li7
abundance unaffected.
In the CMSSM~\cite{cmssm1}, it is assumed that all supersymmetry-breaking gaugino masses have a common value
$m_{1/2}$ at some grand unification scale before renormalization, and likewise all the soft
supersymmetry-breaking scalar masses are assumed to have a common $m_0$. The other
parameters of this model are the (supposedly) universal trilinear parameter $A_0$ (taken here to be $A_0 = 0$),
and the ratio of Higgs vacuum expectation values, $\tan \beta$. In addition, one must specify the
sign of the Higgs mixing mass, $\mu$, which is generally taken to be positive in the CMSSM so as to
improve compatibility with measurements of $g_\mu - 2$ and $b \to s \gamma$ decays.
As is well known, in the CMSSM there are strips in the $(m_{1/2}, m_0)$ planes
for fixed $A_0$ and $\tan \beta$
along which the relic density of the
neutralino LSP, $\chi$, lies within the range favoured by WMAP and other
astrophysical observations~\cite{cmssmwmap}. At relatively low values of $m_0$ there is generically
a coannihilation strip close the boundary where the LSP would become charged
that extends, for large $\tan \beta$, into a funnel at large $m_{1/2}$ where the LSPs annihilate rapidly
via direct-channel heavy Higgs resonances.
At relatively high values of $m_0$, there is a focus-point strip close to the boundary of
consistent electroweak symmetry breaking. In light of present experimental constraints from
the LHC and elsewhere, plausible values of $\tan \beta$ range between $\sim 10$ and $\sim 55$~\cite{LHC}.
Representative $(m_{1/2}, m_0)$ planes for these values of $\tan \beta$ are shown in Fig.~\ref{fig:planes},
and are discussed below in more detail.
In this paper, we explore the effects on cosmological light-element abundances of residual
late-time $\chi \chi$ annihilations for CMSSM parameters along the coannihilation/funnel
and focus-point strips for these reference values of $\tan \beta$, and also remark on additional
possibilities in one- and two-parameter generalizations of the CMSSM with non-universal soft supersymmetry-breaking contributions to
Higgs masses (the NUHM1 and NUHM2)~\cite{nuhm1,nuhm12,nuhm2}.
\begin{figure}[htb]
\begin{center}
\epsfig{file=cmssm10s,width=0.475\textwidth}
\epsfig{file=cmssm55anns,width=0.48\textwidth}
\caption{
\it Left: The CMSSM $(m_{1/2}, m_0)$ plane for $A_0 = 0$ and $\tan \beta = 10$,
and Right: the corresponding plane for $\tan \beta =55$, both with $\mu > 0$,
displaying contours of the \li6 abundance including the effects of late-time
$\chi \chi$ annihilations. Contours of the \li6 abundance
are coloured light blue, and the WMAP-compatible~\protect\cite{wmap7} strips of
parameter space are shaded dark blue. The brown shaded region at large $m_{1/2}$ and
small $m_0$ is excluded because the LSP would be charged, and in the pink
shaded region at small $m_{1/2}$ and large $m_0$ there would be no consistent
electroweak vacuum. Also shown are the exclusion by LEP searches
for the Higgs boson (red dash-dotted line) and charginos (black dashed line),
and by LHC searches for sparticles (purple solid line)~\protect\cite{LHC,CMS}.
The green shaded region is excluded by $b \to s \gamma$, and the paler pink region is favoured by
$g_\mu - 2$ at the 1- (2-)$\sigma$ level, as indicated by the dashed (solid) black lines.}
\label{fig:planes}
\end{center}
\end{figure}
We find that such late annihilations have no significant effect on the cosmological
abundances of Deuterium, \he3, \he4 and \li7 in any of the CMSSM scenarios studied.
However, they may enhance the \li6 abundance by up to two orders of magnitude: in some
instances, we find \li6/H $\sim 10^{-12}$, compared to the value $\sim 10^{-14}$ found in
standard homogeneous BBN \cite{bbnli6,van}. This possibility arises at low $m_{1/2}$ along the
focus-point strips for $\tan \beta = 10$ and 55, and also along the coannihilation strip
for $\tan \beta = 55$. The values of \li6/H along these strips decrease to $\sim 10^{-13}$
at large $m_{1/2}$, which is typical also of values in the funnel region for $\tan \beta = 55$.
On the other hand, we find no substantial enhancement of \li6/H along the
coannihilation strip for $\tan \beta = 10$ for values of $m_{1/2}$ consistent with the LHC
constraints~\cite{LHC,CMS}, though values as large as $\sim 10^{-13}$ might have been reached at
lower $m_{1/2}$. The possibility of a large \li6 enhancement is extended in the NUHM1
to a large range of $m_0$ with $\tan \beta = 10$ and low $m_{1/2}$.
\section{The Cosmological Lithium Problems}
Cosmologically, the dominant Lithium isotope is \li7, whose abundance is commonly
inferred from observations of low-metallicity halo dwarf stars. These indicate a plateau of
Lithium versus metallicity~\cite{spite}, with
\begin{equation}
\left( \frac{\li7}{\rm H} \right)_{\rm halo*} \; = \; (1.23^{+0.34}_{-0.16}) \times 10^{-10} ,
\label{Li7Hhalo}
\end{equation}
whereas observations of globular clusters~\cite{liglob} yield somewhat higher values:
\begin{equation}
\left( \frac{\li7}{\rm H} \right)_{\rm GC} \; = \; (2.35 \pm 0.05) \times 10^{-10} .
\label{Li7HGC}
\end{equation}
For comparison, the standard BBN result for \li7/H is $(5.12^{+0.71}_{-0.62}) \times 10^{- 10}$~\cite{cfo5},
and the difference between this and (\ref{Li7Hhalo}, \ref{Li7HGC}) constitutes the
cosmological \li7 problem. As already mentioned, this might be resolved by new
physics beyond the Standard Model such as late decays of massive gravitinos~\cite{ceflos2},
or by some undocumented Standard Model effect such as a suitable carbon, boron or
beryllium resonance~\cite{resonance}. The cosmological \li7 problem is not our focus in this paper.
\li6 has been observed in some halo stars \cite{li6obs} with [Fe/H] $\sim$ -2, and
with an isotopic ratio that is
\begin{equation}
\left( \frac{\li6}{\li7} \right)_{\rm halo*} \; \sim \; 0.05 .
\label{Li6Li7}
\end{equation}
These observations are consistent with the results of Galactic cosmic-ray (GCR) nucleosynthesis \cite{gcr,van,fo}, though see below for results
at lower metallicity.
This confirms that most of the Lithium is in the form of \li7, leaving unscathed the cosmological
\li7 problem.
However, a recent paper has reported the presence of a similar isotopic abundance in halo stars
over a broad range of metallicities that extends to significantly lower
values ([Fe/H] $\sim$ -1 to -3) \cite{li6obs,asp06}.
The inferred plateau \li6/H ratio $\sim (6$ to $25) \times 10^{-12}$
is about 1000 times higher than the \li6/H ratio predicted by standard homogeneous BBN~\cite{bbnli6,van},
namely $\li6/{\rm H} \sim 10^{-14}$. The isotopic ratio (\ref{Li6Li7})
cannot be explained by conventional GCR nucleosynthesis,
at the lowest metallicities:
this is the cosmological \li6 problem. The reliability of the \li6 plateau at very
low metallicity has been questioned \cite{cayrel},
so the \li6 problem should be taken with a grain of salt. But in any
case, these exiting if controversial results demonstrate that
\li6 abundances at levels $\li6/{\rm H} \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} {\rm few} \times 10^{-12}$
are at or near the reach of present observational techniques.
Thus the current observational situation is evolving, but
without question is interesting:
at the very least, the present results serve as upper limits to
primordial \li6,
and impose bounds on nonstandard BBN.
At most, current data may already point to a primordial
\li6 problem which would {\em demand} new BBN physics,
and probe its details.
Our focus in this paper is to determine the \li6 production
and its observational implications in the context of
some of the most popular supersymmetric dark matter scenarios.
It has been proposed that some decaying-particle scenario might produce \li6 at the plateau level
with some destruction of \li7~\cite{Jed2,cefos,grant,jed3,cumber,bailly},
offering the possibility of solving both Lithium problems simultaneously.
However, we note that solving the \li6 problem would use up only a small fraction of the \li7 whose
destruction would be needed to solve the \li7 problem, leading one to consider separate solutions
for the two Lithium problems. It is also possible that the \li6 problem might be explained
by nucleosynthesis due to cosmological cosmic rays produced at the epoch of structure formation~\cite{rvo,ccr}.
We have previously demonstrated that late-decaying massive gravitinos
might resolve the \li7 problem within the CMSSM framework~\cite{ceflos2}. Here we show that the \li6 problem
might, independently and in parallel, have at least a partial supersymmetric solution,
via the late annihilations of neutralino LSPs.
\section{Residual Late-Time Neutralino Annihilations}
Assuming that the lightest neutralino $\chi$ is the LSP, and that R-parity is conserved,
the relic neutralino density is essentially fixed at a freeze-out temperature $T_f \sim m_\chi/20$.
At lower temperatures, the local density of neutralinos, $n_\chi$, decreases as the universe expands
(presumably) adiabatically, and subsequent annihilations have very little effect on the dark matter
density, but may have important effects on the light-element abundances~\cite{Reno,earlylate,Jedamzik:2004ip,jed}.
The rate per volume of annihilation {\em events} is
\begin{equation}
q_{\rm ann} = \frac{1}{2} n_\chi^2 \avg{\sigma v}_{\rm ann} ,
\label{qrate}
\end{equation}
and so the annihilation event rate {\em per $\chi$} is
\begin{equation}
\Gamma_{\rm ann} = \frac{q_{\rm ann}}{n_{\chi}}
= \frac{1}{2}\avg{\sigma v}_{\rm ann} n_\chi = \frac{1}{2}\avg{\sigma v}_{\rm ann} Y_\chi n_{\rm b} ,
\end{equation}
and thus the annihilation event rate {\em per baryon} is
\begin{equation}
\label{eq:decaysperb}
\frac{q_{\rm ann}}{n_{\rm b}}
= \Gamma_{\rm ann} Y_\chi
= \frac{1}{2}\avg{\sigma v}_{\rm ann} Y_\chi^2 n_{\rm b} ,
\end{equation}
where the $\chi$ abundance is
\begin{equation}
Y_\chi = \frac{n_\chi}{n_{\rm b}}
= \frac{m_{\rm b}}{m_\chi} \frac{\Omega_\chi}{\Omega_{\rm b}} \, .
\end{equation}
At the temperatures of interest here,
$T_{\rm BBN} \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 1 \ {\rm MeV} \ll m_{\chi}$,
the annihilation rate coefficient $\avg{\sigma v}_{\rm ann}$ is very well approximated
as a constant, the value of which depends on the specific underlying
supersymmetry model. In (\ref{qrate}), we are interested in only the $s$-wave part of the
cross-section whereas a combination of $s$- and (mainly) $p$-wave cross-sections is constrained by the requirement of
reproducing the present dark matter density
within errors.
The annihilations inject nonthermal
Standard Model particles, including both
electromagnetic as well as hadronic species.
For electromagnetic products we need only track the
total energy injected per annihilation.
For nonthermal hadrons (nucleons)
$h = n, p$, we calculate the spectrum
$Q_h^{\rm ann}(\epsilon)$ of annihilation products,
normalized such that
$\int Q_h^{\rm ann} (\epsilon) \ d\epsilon = B_h$,
the expected number of $h$ created per annihilation.
Then the injection/source rate of $h$ due to annihilations,
per unit volume, per unit time, and per unit kinetic energy $\epsilon$, is
\begin{equation}
\frac{d{\cal N}_{h,\rm inj}^{\rm ann}}{dV \, dt \, d\epsilon}
= q_{\rm ann} Q_h^{\rm ann}(\epsilon) \ .
\end{equation}
These particles then lose energy as they propagate
in the cosmic plasma. The propagated spectrum of nonthermal
particles must be calculated, and this
produces the nonthermal reactions on ambient thermal light nuclides
that perturb BBN.
The effect of nonthermal particle injection in BBN has
been well-studied in the case of decays of some unstable
particle $X$. Much of the physics
carries over here, once one makes the appropriate substitution
of abundances $n_X \rightarrow n_\chi$ and
of annihilation rate for decay rate:
$\Gamma_X = \tau_X^{-1} \longrightarrow \Gamma_{\rm ann}$.
After injection, the nonthermal particle propagation remains the
same, and we treat this as in \cite{ceflos}.
We also adopt the same set of nonthermal BBN reactions
which we include in the same manner, making the
appropriate substitution of annihilations for decays.
\subsection{Order-of-Magnitude Calculation}
Before turning to our numerical results,
we first present an order-of-magnitude calculation that illustrates
the basic physics in play, and also serves as a check on our numerical results.
The total number of annihilation events
per baryon occurring after a given time $t_i$ is
the time integral of eq.~(\ref{eq:decaysperb})
\begin{eqnarray}
{\cal N}_{\rm ann} & = & \int_{t_i} \Gamma_{\rm ann} \ Y_\chi \ dt
\sim Y_\chi^2 \avg{\sigma v}_{\rm ann} n_{\rm b}(t_i) \ t_i \\
& = & 5 \times 10^{-9} \ {\rm events/baryon} \
\pfrac{\avg{\sigma v}_{\rm ann}}{10^{-26} \ \rm cm^3 \, s^{-1}} \
\pfrac{300 \rm GeV}{m_\chi}^2 \ \ .
\end{eqnarray}
Our fiducial values correspond to $t_i \sim 100 \ \rm sec$
and $T_i \sim 100 \ \rm keV$,
since this marks the epoch when the \he4
abundance becomes large.
Given this number of annihilations per baryon, we now need
the branching for \li6 production per annihilation.
As discussed earlier, nonthermal particles from annihilations or decays produce \li6
as a secondary by-product of \he4 spallation:
\begin{eqnarray}
\nonumber
p_{\rm nonthermal} \he4 & \rightarrow & \iso{H}{3}_{\rm nonthermal} + \cdots \\
& & \iso{H}{3}_{\rm nonthermal} + \he4 \rightarrow \li6 + n \, ,
\end{eqnarray}
and similarly with nonthermal $\he3$; nonthermal D also contributes but
is subdominant.
As discussed for the late-decay case in~\cite{ceflos}, each late annihilation produces a mass-3
abundance increment $\Delta Y({}^{3}A)$ which is given in the thin target limit by
\begin{equation}
\Delta Y({}^{3}A) \sim {\cal N}_{\rm ann} B_N
\frac{\sigma(N\alpha \rightarrow {}^{3}A+\cdots)}
{\sigma(N\alpha \rightarrow {\rm inelastic})} ,
\end{equation}
where $B_N \sim 0.4$ is the number of nucleons per annihilation.
Typically this increases the mass-3 abundance by an amount
$\Delta Y({}^3A) \sim 10^{-9} \ll Y_{\rm BBN}({}^3A) \sim 10^{-5}$, i.e.,
much smaller than the standard primordial abundance,
and thus we do not expect substantial perturbations to mass-3 nuclides, or to D,
which has similar cross sections, or to \he4.
The energetic $A=3$ particles are slowed
in the cosmic plasma by ionization and related losses,
with a range $R_3 = \int (dE/dX)^{-1} dE$, where
$dE/dX$ is the loss rate per
thickness $dX = \rho_{\rm b} dx$ in $[\rm g/cm^2]$.
Hence the stopping length is $R_3/\rho_{\rm b}$.
The fraction of mass-3 nuclides which produce \li6
before stopping is this stopping length
divided by the mean free path for \li6 production, namely:
\begin{equation}
f({}^{3}A \rightarrow \li6) \sim n_\alpha \sigma({}^{3}{A}\alpha \rightarrow \li6) \frac{R_3}{\rho_{\rm b}}
\sim Y_\alpha \frac{\sigma({}^{3}{A}\alpha \rightarrow \li6) R_3}{m_{\rm b}}
\sim 7 \times 10^{-4} \ \ .
\end{equation}
Collecting these results, the residual late-time annihilation
contribution to the \li6 abundance per baryon is
\begin{eqnarray}
\Delta Y(\li6) & = & \Delta Y({}^{3}A)
\ f({}^{3}A \rightarrow \li6) \\
& \sim & B_N Y_\chi^2 \avg{\sigma v}_{\rm ann} Y_\alpha
\frac{\sigma(N\alpha \rightarrow {}^{3}A+\cdots)}
{\sigma(N\alpha \rightarrow {\rm inelastic})} \
\frac{\sigma({}^{3}{A}\alpha \rightarrow \li6) R_3}{m_{\rm b}} \
n_{\rm b}(t_i) \ t_i \\
& = & 7 \times 10^{-13} \,
\pfrac{\avg{\sigma v}_{\rm ann}}{10^{-26} \ \rm cm^3 \, s^{-1}} \,
\pfrac{300 \ \rm GeV}{m_\chi}^2 \ \ .
\label{FoM}
\end{eqnarray}
The numerical results given above are evaluated for $t_i = 100 \ \rm sec$,
and we also take $R_3 = 1 \, \rm g/cm^2$ and
$\sigma({}^{3}{A}\alpha \rightarrow \li6) = 30 \ \rm mb$.
This formula gives the scaling
$\Delta Y(\li6) \propto B_N \avg{\sigma v}_{\rm ann} (\Omega_\chi/m_\chi)^2$
which we verify with our full numerical results,
and the normalization agrees to within a factor $\sim 2$.
This agreement lends confidence in our code and our
understanding of the physics.
\subsection{Numerical Results}
We turn now to our full numerical results. In order to establish the context for our
subsequent analysis of the possible annihilation effects along the strips in
CMSSM parameter space that are compatible with WMAP and other constraints
on the present-day dark matter density, we first discuss the full CMSSM
$(m_{1/2}, m_0)$ planes shown in Fig.~\ref{fig:planes}. The light blue lines
are contours of the \li6 abundance, and the relic density is WMAP-compatible~\cite{wmap7}
along the dark blue strips, assuming that the lightest neutralino $\chi$ is the LSP
and is stable, as in $R$-conserving models. There would be no consistent
electroweak vacuum in the pink shaded region at small $m_{1/2}$ and large $m_0$,
the lighter ${\tilde \tau}$ would be the LSP in the brown shaded region, and the
green shaded region is excluded by $b \to s \gamma$ decay~\footnote{According to
conventional Big-Bang cosmology and in the absence of $R$ violation,
the LSP $\chi$ would be overdense in the regions between the WMAP strips. It would be
underdense in the regions between these strips and the pink and brown shaded regions.}. Regions to the left of
the red dash-dotted (black dashed) (purple) line are excluded by searches
for the Higgs boson at LEP (charginos) (LHC searches for sparticles)~\cite{LHC,CMS}.
In the paler pink region the supersymmetric contribution remedies the discrepancy between the experimental measurement of
$g_\mu - 2$ and theoretical calculation within the Standard Model using low-energy $e^+ e^-$ data,
with 1- (2-)$\sigma$ consistency being indicated by the dashed (solid) black lines.
We see in the left panel of Fig.~\ref{fig:planes} showing the $(m_{1/2}, m_0)$ plane for $\tan \beta = 10$ that
most of the lower (coannihilation) WMAP strip has \li6/H $< 10^{-13}$, whereas the upper
(focus-point) strip may have \li6/H as large as $10^{-12}$. There is a region where \li6/H
seems able to exceed $10^{-11}$, but this is well inside the region between the WMAP strips,
where the relic $\chi$ is overdense according to conventional Big-Bang cosmology.
In the right panel of Fig.~\ref{fig:planes} for $\tan \beta = 55$,
we see that \li6/H $\sim 10^{-13}$ along the coannihilation strip and in the funnel region at large
$m_{1/2}$ and $m_0$ where the relic density is brought into the WMAP range by rapid
annihilations through direct-channel $H/A$ resonances, though somewhat larger values of
\li6/H are possible at small $m_{1/2}$. Along the focus-point strip, we see that values of
\li6/H $\sim 10^{-12}$ are also possible at small $m_{1/2}$, falling to $\sim 10^{-13}$
at large $m_{1/2}$. The range \li6/H $\sim 10^{-11}$ is never attained for $\tan \beta =55$,
even in the overdense region of the $(m_{1/2}, m_0)$ plane.
We now focus on the WMAP strips in Fig.~\ref{fig:planes}. The left panel of
Fig.~\ref{fig:ratefom} displays the figure-of-merit combination
$\avg{\sigma v}_{\rm ann} (\Omega_\chi h^2/m_\chi)^2$ as a function of $m_{1/2}$ along the WMAP strips in the
CMSSM for $\tan \beta = 10$ and 55.
We see that $\avg{\sigma v}_{\rm ann} (\Omega_\chi h^2/m_\chi)^2$ along the coannihilation
strip for $\tan \beta = 10$ is much smaller than along the other strips. This can be
understood from the fact that along this strip several coannihilation processes
involving sleptons contribute to reducing the relic $\chi$ density into the WMAP range,
and that their relative contributions become more important as $m_{1/2}$ increases.
In addition, along this strip the $s$-wave cross-section relevant during BBN
is significantly smaller than the $p$-wave cross-section that dominates during freeze-out.
These coannihilation processes are less important along the corresponding strip for
$\tan \beta = 55$, and unimportant in the funnel region and along the focus-point
strips, where the $s$-wave cross-section becomes
comparable to the total cross section.
Hence, along these strips $\avg{\sigma v}_{\rm ann}$ must be larger, in
order to bring the relic density down into the WMAP range unaided.
\begin{figure}[htb!]
\begin{center}
\epsfig{file=sigv_ochi_mchiks,width=0.4775\textwidth}
\epsfig{file=br_m12ks,width=0.37\textwidth}
\caption{
\it Left: The figure of merit for the late-time annihilation rate, namely the velocity-averaged
$\chi \chi$ annihilation cross section divided by the square of the neutralino mass,
$\avg{\sigma v}_{\rm ann} (\Omega_\chi h^2/m_\chi)^2$, along the WMAP strips in the
coannihilation, focus-point and funnel regions for $\tan \beta = 10, 55$,
$A_0 = 0$ and $\mu > 0$, as functions of $m_{1/2}$. We see that the figure of merit
along the $\tan \beta = 10$ coannihilation strip is much smaller
than along the other strips, and that all decrease rapidly as $m_{1/2}$ increases.
Right: The numbers $B_{p,n}$ of protons (solid or dotted lines) and neutrons (dashed or dash-dotted lines) produced per $\chi \chi$ annihilation
event, as calculated using {\tt PYTHIA}, along the WMAP strips in the
coannihilation, focus-point and funnel regions for $\tan \beta = 10, 55$,
$A_0 = 0$ and $\mu > 0$, as functions of $m_{1/2}$. We see that in general
the numbers increase significantly as $m_{1/2}$ increases.}
\label{fig:ratefom}
\end{center}
\end{figure}
The $\chi \chi$ annihilations feed many different particle species into the
cosmological background, initially with nonthermal spectra that we model using {\tt PYTHIA}~\cite{pythia}.
The only species that survive long enough to interact significantly with background nuclei are
protons and neutrons (and their antiparticles) and photons. The former are far
more important for the nuclear reactions of interest here, so we focus on their
numbers and spectra. The right panel of Fig.~\ref{fig:ratefom} displays the numbers of protons (solid or dotted lines) and neutrons (dashed or dash-dotted lines) produced per annihilation event, again along the WMAP strips
for $\tan \beta = 10, 55$ discussed previously. We see that in general the numbers of
protons and neutrons increase significantly as $m_{1/2}$ increases, with some bumps
as new annihilation thresholds are crossed.
Fig.~3 of~\cite{EOSgamma} displays the most important branching fractions for
final states in $\chi \chi$ annihilations as functions of $m_{1/2}$ along the
WMAP strips for $\tan \beta = 10$ and 55, which include the
final states $\tau^+ \tau^-$, $b {\bar b} $, $W^+ W^-$, $t {\bar t} $, $hZ$ and $ZZ$.
Of these, the $\tau^+ \tau^-$ final state clearly yields no baryons, while the
numbers of baryons yielded by the final states $W^+ W^-$, $hZ$ and $ZZ$
are all independent of the annihilation centre-of-mass energy $2 m_\chi$. Only the
$b {\bar b}$ and $t {\bar t}$ final states yield numbers of baryons that increase
with the annihilation centre-of-mass energy.
Fig.~\ref{fig:baryons} displays the spectra of protons (upper panel) and neutrons
(lower panel) for the $W^+ W^-$, $hZ$ and $ZZ$ final states for $m_\chi = 250$~GeV, and for the
$b {\bar b} $ final state for $ m_\chi = 100, 250$~GeV, all calculated using
{\tt PYTHIA}.
We display the number of protons or neutrons per unit of the parameter
$x \equiv \sqrt{E_i^2-m_i^2}/m_\chi$, where $i=p,n$.
The proton and neutron spectra are almost identical. They differ in the small $x$ region primarily because of
the difference of $m_p$ and $m_n$. We also see that the
$W^+ W^-$, $hZ$ and $ZZ$ final states yield rather similar spectra, with
the spectrum from the $hZ$ final state rising slightly higher. The spectra
of baryons from $b {\bar b}$ final states rise from being lower at
$ m_\chi = 100$~GeV to being higher at $ m_\chi = 250$~GeV.
\begin{figure}[htb!]
\begin{center}
\epsfig{file=p_sp_x,width=0.64\textwidth} \\[1cm]
\epsfig{file=n_sp_x,width=0.64\textwidth}
\caption{
\it The spectra of protons (upper panel) and neutrons (lower panel) injected by $\chi \chi$ annihilations
into the $Z h$, $W^+ W^-$, $ZZ$ and $b \bar{b}$ (for $m_\chi$=100 and $250$ GeV)
final states, as calculated using {\tt PYTHIA}.
}
\label{fig:baryons}
\end{center}
\end{figure}
Together with Fig.~3 of~\cite{EOSgamma}, Fig.~\ref{fig:baryons} enables us to
understand the salient features of the baryon production rates shown in the
right panel of Fig.~\ref{fig:ratefom}. The large branching fraction for $\tau^+ \tau^-$
suppresses baryon injection along the coannihilation strip for $\tan \beta = 10$,
particularly for small $m_{1/2}$ but less so for large $m_{1/2}$ where the $W^+ W^-$
branching fraction grows.
Following their injection into the primordial plasma, some of the nucleons
cause spallation of \he4, yielding $A = 3$ nuclei as discussed above. These
are produced with large, nonthermal energies and subsequently thermalize,
but may previously induce $t (\alpha, n)\li6$ or $\he3 ( \alpha, p) \li6$ reactions.
Fig.~\ref{fig:li6results} displays the enhancement of the cosmological \li6 abundance
that we find along the WMAP strips discussed above. The homogeneous BBN
value $\sim 10^{-14}$ is attained at large $m_{1/2}$ along the WMAP
coannihilation strip for $\tan \beta = 10$, but much larger values are possible
along the other WMAP strips~\footnote{The region of enhanced \li6 along the WMAP coannihilation strip for $\tan \beta = 10$ with $m_{1/2} < 400$~GeV
is now excluded by the unsuccessful LHC searches for supersymmetry~\cite{LHC,CMS}.}, where
we find \li6/H $\sim 10^{-13}$ at large $m_{1/2}$ to $10^{-12}$ at small $m_{1/2}$.
We have found that the enhancement of \li6 scales very closely with the combination
$B_N \avg{\sigma v}_{\rm ann} / m_\chi^2$, as was to be expected.
\begin{figure}[htb!]
\begin{center}
\epsfig{file=Li6_m12ks,width=0.75\textwidth}
\caption{
\it The enhancement of the cosmological \li6 abundance as a function of
$m_{1/2}$ along the WMAP strips discussed in the text. The standard homogeneous
BBN value $\sim 10^{-14}$ is attained at large $m_{1/2}$ along the WMAP
coannihilation strip for $\tan \beta = 10$.}
\label{fig:li6results}
\end{center}
\end{figure}
We recall that the enhancement of \li6/H that would be required for consistency
with (\ref{Li6Li7}) is by a factor $\sim 1000$, rather than the factor of up to
$\sim 100$ that we find here. However,
as we have noted there remains a question as to whether
or not the plateau ratio of 0.05 should be attributed to \li6.
The abundance of \li6 we find here is potentially observable
and would in fact be seen as a plateau extending to low metallicities.
Optimistically, we could envision \li6 observations playing a role in
discerning between supersymmetric models. In any case,
we regard the enhancement we find as
already an interesting contribution to the analysis of the \li6 problem.
\subsection{Exploration of Non-Universal Higgs Models}
It is quite possible that some modifications of the CMSSM
might yield even greater enhancements of the \li6 abundance.
To be successful in this respect, it is apparent from (\ref{FoM}) that
such a model would require a relatively large annihilation
cross section $\langle \sigma v \rangle_{\rm ann}$ combined with a small value of $m_\chi$, as in the
focus-point region of the CMSSM. There, the relatively large value of $\langle \sigma v \rangle_{\rm ann}$
is made possible by the admixture of a Higgsino component in the $\chi$, and along this strip the low value
of $m_\chi$ is consistent with the LHC and other constraints~\cite{LHC}.
In an initial probe of other possibilities for a large enhancement of the \li6
abundance, we have explored the NUHM1 model, in which the
soft supersymmetry-breaking contributions to the Higgs masses have a
common value that differs from $m_0$. It is known that in this model the
Higgsino component in the LSP $\chi$ may be enhanced
at values of $m_{1/2}$ and $m_0$ away from the focus-point region,
thanks to a level-crossing transition at particular values of $\mu/m_{1/2}$ \cite{nuhm12}. In the CMSSM,
the value of $\mu$ is generally fixed by applying the conditions for a consistent
electroweak vacuum. However, in the NUHM1 the value of $\mu/m_{1/2}$ can be adjusted by
varying the degree of non-universality in the soft supersymmetry-breaking Higgs
masses, enabling a WMAP-compatible relic density to be found in models with values of $(m_{1/2}, m_0)$
different from those allowed in the CMSSM.
We have explored the conditions under which such transition regions in the NUHM1 may yield an
enhancement of \li6/H similar to, or (possibly) greater than the value $\sim 10^{-12}$ attainable in the
CMSSM in the focus-point region. To this end, we have studied over a dozen NUHM1 parameter
planes. In no case did we find enhancements of \li6/H significantly larger than in the CMSSM (and
this is also the case in some planes we explored in the NUHM2, in which both Higgs soft supersymmetry-breaking
masses are treated as free, non-universal parameters).
Fig.~\ref{fig:NUHM1} shows results in a couple of
selected NUHM1 parameter planes. The left panel shows an $(m_{1/2}, m_0)$ plane for
$\tan \beta = 10, A_0 = 0$ and fixed $\mu = 250$~GeV. In this case, there is a near-vertical
WMAP-compatible strip in a transition region at $m_{1/2} \sim 400$~GeV.
This transition strip is compatible with the LEP Higgs constraint,
and the upper part of the strip above $m_0 \sim 700$~GeV is compatible with the constraints
imposed by LHC searches for sparticles. We see that the \li6/H ratio is remarkably
constant at $\sim 5 \times 10^{-13}$ along this strip. It would be possible to increase
\li6/H to $\sim 10^{-12}$ by choosing $\mu$ somewhat smaller, in which case the
WMAP-compatible strip would be at smaller $m_{1/2}$. In that case, the LHC
would enforce a stronger lower limit on $m_0$, closer to the CMSSM focus-point strip.
On the other hand, larger values of $\mu$ yield small values of the \li6 abundance, and we find
no increase in the \li6 abundance for larger $\tan \beta$.
\begin{figure}[htb]
\begin{center}
\epsfig{file=Mvm0mu8_10_250anns,width=0.475\textwidth}
\epsfig{file=m0mu8_20_500anns,width=0.48\textwidth}
\caption{
\it Left: The NUHM1 $(m_{1/2}, m_0)$ plane for $\mu = 250$~GeV, $A_0 = 0$ and $\tan \beta = 10$,
and Right: the NUHM1 $(\mu, m_0)$ plane for $m_{1/2} = 500$~GeV, $A_0 = 0$ and $\tan \beta = 20$,
displaying contours of the \li6 abundance including the effects of late-time
$\chi \chi$ annihilations. Contours of the \li6 abundance
are coloured light blue, and the WMAP-compatible strips of
parameter space are shaded dark blue. The other shaded regions and lines have the same
meanings as in Fig.~\protect\ref{fig:planes}.}
\label{fig:NUHM1}
\end{center}
\end{figure}
The right panel of Fig.~\ref{fig:NUHM1} displays a $(\mu, m_{0})$ plane in the NUHM1 for
$\tan \beta = 20, A_0 = 0$ and fixed $m_{1/2} = 500$~GeV, at the lower end of the range allowed
by the LHC and other data for $m_0 < 1000$~GeV. In this case,
there are near-vertical WMAP-compatible strips in transition region at $|\mu| \sim 300$~GeV,
where \li6/H approaches $10^{-12}$. There is also a WMAP-compatible strip near
$\mu \sim 1000$~GeV that parallels the
region without a consistent electroweak vacuum (here caused by $m_A^2 < 0$),
where \li6/H is again somewhat below
$10^{-12}$. Connecting these two regions is a co-annihilation segment at $m_0 \sim 100-200$~GeV where the \li6 abundance is relatively small.
We have explored several other NUHM1 $(\mu, m_0)$ planes, finding that
increasing $m_{1/2}$ decreases the attainable value of \li6/H. We have also explored
several other projections of the NUHM1 and NUHM2, including $(m_A, m_{1/2})$,
$(m_A, m_0)$, $(\mu, m_A)$ and $(m_1, m_2)$ planes, without finding values of \li6/H above $10^{-12}$.
\section{Summary and Conclusions}
We have demonstrated in this paper that in both the CMSSM and the NUHM1
it is possible that late neutralino LSP annihilations may enhance significantly
the cosmological \li6 abundance, without affecting significantly the BBN
abundances of the other light element Deuterium, \he3, \he4 and \li7~\cite{Jedamzik:2004ip}.
This enhancement may be up to two orders of magnitude, yielding \li6/H $\sim 10^{-12}$
compared to the BBN value $\sim 10^{-14}$.
As we have shown, this enhancement occurs typically when the neutralino LSP
is relatively light and has a large annihilation cross section, as occurs when the LSP
contains a strong Higgsino admixture. This phenomenon appears, in particular,
in the focus-point region of the CMSSM and in transition regions of the NUHM1.
While interesting for the debates on the astrophysical Lithium abundances,
this enhancement falls short of resolving by itself the cosmological \li6 problem.
Further work could include a more exhaustive study of other supersymmetric
models, to see whether they could reconcile a larger enhancement with the
available theoretical, phenomenological, experimental and cosmological
constraints. Alternatively, is it possible that the height of the \li6 plateau may
receive contributions from other sources such as an early generation of stars,
or might the height of the \li6 plateau be over-estimated, perhaps because of
convective processes involving \li7 \cite{cayrel}? It is clearly
desirable to pin down more definitively the magnitude of the \li6 problem
by establishing more solidly the existence and height of the inferred \li6
plateau in halo stars. However, it already seems that a substantial
enhancement of the standard homogeneous BBN prediction for \li6/H
might be an interesting signature of supersymmetric models.
Finally, our work illustrates in detail
the more general point that \li6 production should
play a role in--and thus probe--{\em any} WIMP
dark matter scenario involving hadronic annihilation products \cite{karsten}.
Specifically, we have seen that \li6 production
is essentially guaranteed provided there are
nonthermal nucleons injected with kinetic energies $\mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}} {\rm few}$ MeV.
We also find that the level of \li6 abundance due to
residual annihilations is model-dependent,
in our
case spanning a range from 100 times the standard yield
down to an unobservable perturbation to this level.
The lessons for WIMP modelers would seem to be that
\li6 observations already provide important constraints
which one must test against, and that a confirmed detection
of primordial \li6--particularly if it is above the standard level--will
likely shed light on the details of the nonstandard physics which
produced it.
\section*{Acknowledgments}
We would like to thank R.V. Wagoner for interesting discussions on this problem.
The work of J.E. was supported partly by the London
Centre for Terauniverse Studies (LCTS), using funding from the European
Research Council
via the Advanced Investigator Grant 267352. This Grant also supported visits by K.A.O. and
V.C.S. to the CERN TH Division, which they thank for its hospitality. The work of F.L. and K.A.O. was supported in part
by DOE grant DE--FG02--94ER--40823 at the University of Minnesota.
The work of V.C.S. was supported by Marie Curie International
Reintegration grant SUSYDM-PHEN, MIRG-CT-2007-203189.
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\section{Introduction}\label{sec1}
Accurate diagnosis is a key target of diagnostic decision-making. Before a medical diagnostic test is routinely applied in practice, it is important to evaluate its performance in discriminating between diseased and non-diseased subjects. The most well-known summary accuracy measures are the $\text{AUC}$ (Area Under the receiver operating characteristic Curve) and the Youden index \citep{youden1950}; other summary indexes can be found in \citet[][Section~4.3.3]{pepe2003}. {These well-known summary measures at times gloss over important differences between diseased and non-diseased subjects. To see when this might happen let} $Y_D \sim F_D$ and $Y_{\ensuremath{\bar{D}}} \sim F_{\bar{D}}$ denote the test results for diseased and non-diseased subjects. Formally, the AUC consists of $P(Y_D > Y_{\ensuremath{\bar{D}}})$ and it is typically argued that $\text{AUC} = 0.5$ for a test {that does} no better than chance {in discriminating between diseased and non-diseased individuals}, while $\text{AUC} = 1$ for a test that perfectly distinguishes between diseased and non-diseased subjects. While AUC is widely used in practice, Figure~\ref{motivation} illustrates a setting under which the AUC is known to perform poorly. {Regarding this setting, \cite{lee1996} make the following comment}:
\begin{quote}
``For the two populations of the diseased and the non-diseased [...] the marker perfectly separates the two. Therefore, any clinician (or epidemiologist) will have no trouble in choosing a decision rule for the marker, that is, high and low cutoff points. Nevertheless, adopting the AUC as the measure of overall performance leads one to conclude that the marker is not better than flipping a
fair coin (its AUC is 0.5) ''
\end{quote}
\begin{figure}
\centering
\includegraphics[scale = 0.45]{Figures/motivation.pdf}
\caption{\small The separation trap: Perfect discrimination but $\text{AUC} = 0.5$. Details on the truncated normal densities used to construct this example can be found in Example~\ref{trap}; the black and grey lines respectively denote the densities of the biomarkers of the diseased and non-diseased subjects.}
\label{motivation}
\end{figure}
Throughout, we will refer to the situation in Figure~\ref{motivation} as the `separation trap,' since one has perfect discrimination but $\text{AUC} = 0.5$.
As can be seen from Figure~\ref{motivation}, even though the populations of diseased and non-diseased subjects are perfectly separated, half of the diseased-subjects are predicted to have a test result higher than the non-diseased subjects, and thus $\text{AUC} = 0.5$.
{But beyond the AUC, the Youden index (YI) also falls into the separation trap. To see this recall that $\protect{\text{YI}= \max_{c \in \mathbb{R}}\,\{F_{\bar{D}}(c)-F_{D}(c)\}}$, with $\text{YI}=0$ corresponding to complete overlap ($F_{\bar{D}}(c)=F_{D}(c)$), and it is often argued that $\text{YI}=1$ when the distributions are `completely separated'. It is straightforward to show that in the example in Figure~1, it holds that $\text{YI} = 1/2$, while the distributions of the markers for diseased and non-diseased subjects are completely separated---thus confirming that the Youden index would fall into the separation trap. The optimal cutoff region yielded through the Youden index} is
\begin{eqnarray*}\label{yi}
\arg \max_{c \in \mathbb{R}}\,\{F_{\bar{D}}(c)-F_{D}(c)\} = [2, 4].
\end{eqnarray*}
Interestingly, however, the more sensible cutoff region $[-4, -2] \cup [2, 4]$ could have been obtained by adjusting the definition of Youden index to consider the absolute value of the difference between distribution functions, but even this modified index would be equal to 1/2.
A main goal of this article is {to propose} new diagnostic accuracy measures {that accommodate the separation trap and that} can be used as companions, or possibly as alternatives, to existing diagnostic accuracy measures. {Conceptually, our summary measures can be motivated by first considering a geometric interpretation of covariance and Pearson correlation}. By recalling the well-known fact that for zero-mean finite-variance random variables $X$ and $Y$, the covariance can be interpreted as an inner product between random variables \citep{williams1991}, it follows that Pearson correlation
\begin{equation}
\label{eq:pearson}
\rho = \frac{\text{cov}(X, Y)}{\text{sd}(X) \text{sd}(Y)} = \cos(X \angle Y),
\end{equation}
can be interpreted as a cosine of the angle between $X$ and $Y$. This simple geometric interpretation is handy for understanding some basic properties of the Pearson correlation, including the fact that just like a cosine, $\rho$ is between -1 and 1, and that orthogonality in such context corresponds to the case where {there is no agreement between $X$ and $Y$,}
that is, $\text{cov}(X, Y) = 0$. The summary measures we develop, introduced in Section~2, are constructed along the same lines, but are based on evaluating the level of agreement between densities (of the biomarkers of diseased and non-diseased subjects) instead of focusing on random variables---as Pearson correlation does.
A covariate-specific version of our main summary index is here devised, which can be used to assess the discrimination performance of a diagnostic test, conditionally on the value of a covariate. Nonparametric Bayesian estimators for all proposed indexes are developed and the numerical performance of a specific implementation is evaluated in detail through a simulation study. Using Bayesian nonparametric (BNP) inference in a medical diagnostic setting is not unprecedented \citep{erkanli2006, gu2008, inaciodecarvalho2013, rodriguez2014, inaciodecarvalho2015, branscum2015, johnson2015, inaciodecarvalho2016}, {and doing so provides a great deal of flexibility particularly in the dependent case (i.e., a covariate is present). An additional computational advantage of our covariate-specific summary measure with respect to that of \cite{inaciodecarvalho2013} is that it avoids the need of computing conditional quantiles over a grid of covariates---a task which requires a substantial computational investment. More importantly, as we elaborate below, our summary measures do not fall into the separation trap depicted in Figure~\ref{motivation}.}
The article is organized as follows. In the next section we introduce the proposed measures along with the corresponding inference tools. In Section~3 we conduct a simulation study. Section~4 offers an illustration of our methods in a prostate cancer diagnosis case study. Proofs are included in the online supplementary materials.
\section{Geometric measures of diagnostic test accuracy}
\subsection{Angle-based summary measures of diagnostic test accuracy}\label{unconditional}
Our summary measures are built on similar construction principles as Pearson correlation, but instead of looking at the angle between random variables as in \eqref{eq:pearson}, we work directly with the density of the biomarker outcome for diseased and non-diseased subjects, which we denote as $f_{D}$ and $f_{\ensuremath{\bar{D}}}$, respectively. Thus, in place of the covariance inner product we use $\langle f_D, f_{\ensuremath{\bar{D}}}\rangle = \int_{-\infty}^{\infty} f_D(y) f_{\ensuremath{\bar{D}}}(y)\, \mathrm{d} y$, and in place of the standard deviation (sd) norms, we use $\|f_D\| = \{\int_{-\infty}^{\infty} f_D^2(y)\, \mathrm{d} y\}^{1/2} < \infty$, and $\|f_{\ensuremath{\bar{D}}}\| = \{\int_{-\infty}^{\infty} f_{\ensuremath{\bar{D}}}^2(y) \, \mathrm{d} y\}^{1/2} < \infty$. The starting point for the construction of our measure is given by the following standardized inner product:
\begin{equation}
\bar{\kappa} = \frac{\langle f_D, f_{\ensuremath{\bar{D}}}\rangle}{\|f_D\| \|f_{\ensuremath{\bar{D}}}\|}.
\label{kappabar}
\end{equation}
For a medical test with perfect discriminatory ability we would have $\bar{\kappa} = 0$, as $f_D$ would be orthogonal to $f_{\ensuremath{\bar{D}}}$. The higher the value of $\bar{\kappa}$, the lower the discriminatory ability of the corresponding biomarker. Indeed, similar to the Pearson correlation, our measure can be interpreted as an angle between $f_D$ and $f_{\ensuremath{\bar{D}}}$. However, since $f_D \geqslant 0$ and $f_{\ensuremath{\bar{D}}} \geqslant 0$ it follows that $\langle f_D, f_{\ensuremath{\bar{D}}}\rangle \geqslant 0$, and
thus the angle between $f_D$ and $f_{\ensuremath{\bar{D}}}$ can only be acute or right, that is $f_D \angle f_{\ensuremath{\bar{D}}}$ is in $[0, \pi/2]$, and thus $\bar{\kappa}$ is in $[0, 1]$. Orthogonality between the biomarker outcome for diseased and non-diseased subjects corresponds to a diagnostic test that perfectly discriminates between diseased and non-diseased subjects.
{However natural the $\bar{\kappa}$ in \eqref{kappabar} may appear,} in practice one may want to avoid the division by the norms---which in theory could lead to more unstable estimates and constrain us to work with square-integrable densities---while retaining the main ingredients of the construction above. Since $f_D$ and $f_{\ensuremath{\bar{D}}}$ are valid densities, it follows that $\|\sqrt{f_D}\| = \|\sqrt{f_D}\| = 1$, and thus we define our summary measure as
\begin{equation}
\kappa = \frac{\langle \sqrt{f_D}, \sqrt{f_{\ensuremath{\bar{D}}}} \rangle}
{\|\sqrt{f_{D}}\| \|\sqrt{f_{\ensuremath{\bar{D}}}}\|}
= \langle \sqrt{f_D}, \sqrt{f_{\ensuremath{\bar{D}}}} \rangle
= \int_{-\infty}^{+\infty} \sqrt{f_D(y)} \sqrt{f_{\ensuremath{\bar{D}}}(y)} \, \mathrm{d} y.
\label{kappa}
\end{equation}
Some comments are in order. Similar to \eqref{kappabar}, orthogonality between the biomarker outcome for diseased and non-diseased subjects corresponds to the case where the diagnostic test is perfect, that is $\kappa = 0$ for a perfect test---which perfectly discriminates diseased subjects from non-diseased subjects---and $\kappa = 1$ for a useless test---for which $f_D = f_{\ensuremath{\bar{D}}}$.
Interestingly, the measure $\kappa$ in \eqref{kappa} is known in mathematical statistics under the name of Hellinger affinity \citep{vaart1998}, but we are unaware of applications of the concept in medical diagnostic statistics. In our context, $\kappa$ can be interpreted as a measure of the level of agreement between the densities of the biomarker outcomes for diseased and non-diseased subjects, or equivalently, as a measure of {the highest possible} diagnostic test accuracy {for a test based on the biomarker}.
\begin{example}[Binormal affinity]\normalfont \label{binormal}
Suppose $f_D(y) = \phi(y \mid \mu_D, \sigma^2_D)$ and $f_{\ensuremath{\bar{D}}}(y) = \phi(y \mid \mu_{\ensuremath{\bar{D}}}, \sigma^2_{\ensuremath{\bar{D}}})$. As stated in Table~\ref{parametric}:
\begin{equation}
\kappa = \sqrt{\frac{2 \sigma_D \sigma_{\ensuremath{\bar{D}}}}{\sigma_D^2 + \sigma_{\ensuremath{\bar{D}}}^2}} \exp\bigg\{-\frac{1}{4}
\frac{(\mu_D - \mu_{\ensuremath{\bar{D}}})^2}{\sigma_D^2 + \sigma_{\ensuremath{\bar{D}}}^2}\bigg\}.
\label{bincomp}
\end{equation}
As expected, for a useless test---that is $\mu_D = \mu_{\ensuremath{\bar{D}}}$ and $\sigma_{D} = \sigma_{\ensuremath{\bar{D}}}$---it holds that $\kappa = 1$. For fixed $\sigma_D > 0$ and $\sigma_{\ensuremath{\bar{D}}} > 0$ it follows that as $\mu_{D} \to \infty$ and $\mu_{\ensuremath{\bar{D}}} \to -\infty$, that is as populations become more separated, then $\kappa \to 0$. Indeed, as it can be seen from Figure~\ref{comp} the more separated the populations---that is the more orthogonal they are---the closer $\kappa$ gets to zero. Notice also that, in this setting, the larger the $\text{AUC}$ the lower $\kappa$. However, this needs not always be the case, and there are actually situations for which $\text{AUC}$ and $\kappa$ may recommend different decisions, as will be seen in Examples~\ref{trap} and \ref{binormal2}.
\end{example}
\begin{example}[Separation trap]\normalfont \label{trap}
Let's revisit the setting from Figure~\ref{motivation}. The exact setup is \begin{equation*}
\begin{cases}
f_{D}(y) = 1/2 \phi_{\textsc{t}}(y \mid -6, -4, -5, 1/3^2) +
1/2 \phi_{\textsc{t}}(y \mid 4, 6, 5, 1/3^2), \\
f_{\ensuremath{\bar{D}}}(y) = \phi_{\textsc{t}}(y \mid -2, 2, 0, 1/4^2).
\end{cases}
\end{equation*}
Here $\phi_{\textsc{t}}(y \mid a, b, \mu, \sigma^2)$ is the density of a truncated normal with lower bound $a$ and upper bound $b$. In this case it holds that
\begin{equation*}
\begin{split}
\kappa & = \int_{-\infty}^{\infty} \sqrt{f_D(y)} \sqrt{f_{\ensuremath{\bar{D}}}(y)} \, \mathrm{d} y \\
& = \int_{-6}^{-4} \sqrt{f_D(y)} \sqrt{f_{\ensuremath{\bar{D}}}(y)} \, \mathrm{d} y
+ \int_{-2}^{2} \sqrt{f_D(y)} \sqrt{f_{\ensuremath{\bar{D}}}(y)} \, \mathrm{d} y +
\int_{4}^{6} \sqrt{f_D(y)} \sqrt{f_{\ensuremath{\bar{D}}}(y)} \, \mathrm{d} y = 0.
\end{split}
\end{equation*}
Thus, $\kappa$ claims that both populations are perfectly separated---and so it would not fall into the separation trap.
\end{example}
\begin{figure} \centering
\begin{minipage}{0.275\linewidth}\footnotesize \centering
$\hspace{-1cm}\kappa = 1$ \\ {$\hspace{-1.4cm} (\text{AUC} = 0.5)$}
\end{minipage}
\begin{minipage}{0.275\linewidth}\footnotesize \centering
$\hspace{0cm}\kappa = 0.61$ \\ {\hspace{0cm} $(\text{AUC}~= 0.92)$}
\end{minipage}
\begin{minipage}{0.275\linewidth}\footnotesize \centering
$\hspace{1.8cm}\kappa = 0.32$ \quad \\ {\hspace{1.5cm} $(\text{AUC}~= 0.98)$}
\end{minipage}
\begin{minipage}{0.275\linewidth} \vspace{-.5cm}
\includegraphics[scale = 0.25]{Figures/binormal1.pdf}
\end{minipage}\hspace{0.7cm}
\begin{minipage}{0.275\linewidth} \vspace{-.5cm}
\includegraphics[scale = 0.25]{Figures/binormal2.pdf}
\end{minipage}\hspace{0.7cm}
\begin{minipage}{0.275\linewidth} \vspace{-.5cm}
\includegraphics[scale = 0.25]{Figures/binormal3.pdf}
\end{minipage} \\
\begin{minipage}{0.275\linewidth}\footnotesize
~~~~~~~~~~~~~~~~~~~(a)
\end{minipage}\hspace{0.4cm}
\begin{minipage}{0.275\linewidth}\footnotesize \centering
~~~(b)
\end{minipage}\hspace{0.7cm}
\begin{minipage}{0.275\linewidth}\footnotesize \centering
~~~(c)
\end{minipage}
\caption{\small Affinity for Binormal model from Example~\ref{binormal}; the black and grey lines respectively denote the densities of the biomarkers of the diseased and non-diseased subjects; the configurations of parameters are as follows: a) $(\mu_D, \sigma_D) = (\mu_{\ensuremath{\bar{D}}}, \sigma_{\ensuremath{\bar{D}}}) = (0, 1)$; b) $(\mu_D, \sigma_D) = (2, 1)$ and $(\mu_{\ensuremath{\bar{D}}}, \sigma_{\ensuremath{\bar{D}}}) = (0, 1)$; c) $(\mu_D, \sigma_D) = (3, 1)$ and $(\mu_{\ensuremath{\bar{D}}}, \sigma_{\ensuremath{\bar{D}}}) = (0, 1)$.}
\label{comp}
\end{figure}
Table~\ref{parametric} contains two other examples of affinity for parametric models. For completeness we include derivations of these expressions in the supplementary materials.
\begin{table}\centering
\caption{\small Affinity ($\kappa$) for bibeta, biexponential, and binormal models; here, $\lambda_D$ and $\lambda_{\ensuremath{\bar{D}}}$ are the rate parameters of the corresponding exponential distributions} \footnotesize \vspace{.3cm}
\begin{tabular}{ll} \hline
Model & Affinity\\
\hline
Bibeta & $\frac{B((a_D +a_{\ensuremath{\bar{D}}}) / 2, (b_D + b_{\ensuremath{\bar{D}}}) / 2)}{\{B(a_D, b_D)B(a_{\ensuremath{\bar{D}}}, b_{\ensuremath{\bar{D}}})\}^{1/2}}$\\[.3cm]
Biexponential & $\frac{2(\lambda_D \lambda_{\ensuremath{\bar{D}}})^{1/2}}{\lambda_D + \lambda_{\ensuremath{\bar{D}}}}$\\[.3cm]
Binormal & $\sqrt{\frac{2 \sigma_D \sigma_{\ensuremath{\bar{D}}}}{\sigma_D^2 + \sigma_{\ensuremath{\bar{D}}}^2}} \exp\bigg\{-\frac{1}{4}
\frac{(\mu_D - \mu_{\ensuremath{\bar{D}}})^2}{\sigma_D^2 + \sigma_{\ensuremath{\bar{D}}}^2}\bigg\}$\\
\hline
\end{tabular}
\label{parametric}
\end{table}
\subsection{Properties and covariate-specific affinity}
The following proposition documents two elementary properties associated with our measure of diagnostic test accuracy.
\begin{proposition}\label{prop}
Affinity, as defined in \eqref{kappa}, obeys the following properties:
\begin{enumerate}
\item $\kappa \in [0, 1]$.
\item $\kappa$ is invariant to monotone increasing data transformations.
\end{enumerate}
\end{proposition}
{A proof of Proposition~\ref{prop} can be found in the online supplementary materials.} Interestingly, just like affinity, the $\text{AUC}$ is also invariant to monotone increasing data transformations \citep{pepe2003}. Affinity is also invariant to whether we work with a test for which larger values of the biomarker are more indicative of disease, or the other way around; this is an obvious consequence of the fact that $\langle \sqrt{f_D}, \sqrt{f_{\ensuremath{\bar{D}}}} \rangle = \langle \sqrt{f_{\ensuremath{\bar{D}}}}, \sqrt{f_{D}} \rangle$. Thus, for instance, Binormal affinity in \eqref{bincomp} is the same, regardless of whether $\mu_D > \mu_{\ensuremath{\bar{D}}}$ or vice versa. For the lack of better terminology, below we refer to an \textit{upper-tailed diagnostic test} as one for which larger values of the biomarker are more indicative of disease, and to a \textit{lower-tailed diagnostic test} as to one where larger values of the biomarker are less indicative of disease. {Another parallel to the AUC is the fact that $\kappa$ can be regarded as an area under a curve, with the curve of interest being
\begin{equation*}
c(y) = \sqrt{f_D(y) f_{\ensuremath{\bar{D}}}(y)}.
\end{equation*}
Another interesting aspect is that $\kappa$ can also be regarded as an average squared likelihood ratio, in the sense that}
\begin{equation*}
\kappa = \int_{-\infty}^{\infty} \sqrt{\frac{f_D(y)}{f_{\ensuremath{\bar{D}}}(y)}} f_{\ensuremath{\bar{D}}}(y) \, \mathrm{d} y
= E_{\ensuremath{\bar{D}}}\bigg(\sqrt{\frac{f_D(Y_{\ensuremath{\bar{D}}})}{f_{\ensuremath{\bar{D}}}(Y_{\ensuremath{\bar{D}}})}} \bigg).
\end{equation*}
If covariates are available the question arises {of how} to
conduct a covariate-specific analysis for measuring diagnostic test accuracy using affinity. A natural extension of \eqref{kappa} to the conditional setting is
\begin{equation}
\kappa(x) = \langle \sqrt{f_{D \mid x}}, \sqrt{f_{\ensuremath{\bar{D}} \mid x}}\rangle =
\int_{-\infty}^{+\infty} \sqrt{f_D(y \mid x)} \sqrt{f_{\ensuremath{\bar{D}}}(y \mid x)} \, \mathrm{d} y,
\label{kpred}
\end{equation}
where $x \in \mathcal{X} \subseteq \mathbb{R}^p$ is a covariate, $f_{D \mid x} = f_D(\, \cdot \mid x)$, and $f_{\ensuremath{\bar{D}} \mid x} = f_{\ensuremath{\bar{D}}}(\, \cdot \mid x)$. Below we refer to $\kappa(x)$ as the covariate-specific affinity. As with $\kappa$, it holds that $\kappa(x) \in [0, 1]$, and that $\kappa(x)$ is invariant to monotone increasing data transformations.
\begin{figure} \hspace{2cm}
\begin{minipage}{0.45\linewidth}\hspace{.4cm}
\includegraphics[scale = 0.3]{Figures/cov.pdf}
\end{minipage}
\begin{minipage}{0.45\linewidth}\hspace{-.4cm}
\includegraphics[scale = 0.3]{Figures/cov4.pdf}
\end{minipage}
\begin{minipage}{0.45\linewidth}\centering \footnotesize
~~~~~~~~~~~~~~~~~~~~~~(a)
\end{minipage}
\begin{minipage}{0.45\linewidth}\centering \footnotesize
~~~~~~~~~~~~(b)
\end{minipage}
\caption{\small a) Covariate-specific affinity (solid line) for binormal model from Example~\ref{binormal2}, and corresponding covariate-specific $\text{AUC}$
(dotted line); b) Density of the diseased (black line) and non-diseased (grey line) subjects, for $x = 4$.}
\label{cov}
\end{figure}
\begin{example}[Binormal covariate-specific affinity]\normalfont \label{binormal2}
Extending Example~\ref{binormal}, suppose that $f_D(y \mid x) = \phi(y \mid \mu_D(x), \sigma^2_D(x))$ and $f_{\ensuremath{\bar{D}}}(y) = \phi(y \mid \mu_{\ensuremath{\bar{D}}}(x), \sigma^2_{\ensuremath{\bar{D}}}(x))$. It then follows that
\begin{equation*}
\kappa(x) = \sqrt{\frac{2 \sigma_D(x) \sigma_{\ensuremath{\bar{D}}}(x)}{\sigma_D^2(x) + \sigma_{\ensuremath{\bar{D}}}^2(x)}} \exp\bigg\{-\frac{1}{4}
\frac{\{\mu_D(x) - \mu_{\ensuremath{\bar{D}}}(x)\}^2}{\sigma_D^2(x) + \sigma_{\ensuremath{\bar{D}}}^2(x)}\bigg\}.
\end{equation*}
In particular, for $\mu_D(x) = x$ and $\mu_{\ensuremath{\bar{D}}}(x) = x - 3$, and $\sigma_D(x) = 1$ and $\sigma_{\ensuremath{\bar{D}}}(x) = 1 + x^2$, we obtain the covariate-specific affinity plotted in Figure~\ref{cov}(a). As it can be observed from Figure~\ref{cov}, for values of the predictor between 0 and approximately 1.2, both $\kappa$ and $\text{AUC}$ agree that the quality of the test deteriorates as $x$ increases ($\kappa$ increases and $\text{AUC}$ decreases). {As $x$ increases beyond 1.2, each measure suggests a different conclusion as to how the test accuracy changes with $x$.} To understand the reason for this, we analyze in further detail the case of $x = 4$, whose corresponding densities are plotted in Figure~\ref{cov}(b). In the case $x = 4$ we have an $\text{AUC} = 0.504$ whereas $\kappa = 0.34$. Thus, on the one hand the $\text{AUC} = 0.504$ suggests that the test is quite poor, whereas the value of $\kappa = 0.34$ suggests that it could be satisfactory, though far from excellent. The intuition underlying this lack of agreement is as follows: $\kappa$ is taking into account that around 95\% of the mass for the test values for diseased subjects will be on the $[0, 8]$ interval, whereas around 95\% of the mass for the test values of non-diseased subjects will be on the $[-30, 38]$ interval.
\end{example}
\subsection{Nonparametric Bayesian inference for affinity and covariate-specific affinity}\label{NonparInf}
In this section we discuss Bayesian nonparametric estimators for affinity, as defined in \eqref{kappa}, and covariate-specific affinity, as defined in \eqref{kpred}. {Let $\{Y_{D, i}\}_{i = 1}^{n_D}$ and $\{Y_{\ensuremath{\bar{D}}, i}\}_{i = 1}^{n_{\ensuremath{\bar{D}}}}$ be random samples from $F_{D}$ and $F_{\ensuremath{\bar{D}}}$}. We propose to estimate $\kappa$ in \eqref{kappa} by modeling each conditional density $f_D$ and $f_{\ensuremath{\bar{D}}}$ as an infinite mixture model of the type
\begin{equation}
f(y) = \int_{\Theta} K(y \mid \theta) \, G(\mathrm{d} \theta),
\label{mixgen}
\end{equation}
where $K$ is a kernel and $G$ is a random mixing measure. The corresponding induced prior is
\begin{equation}
\kappa = \int_{-\infty}^{+\infty} \bigg\{\int_{\Theta} K(y \mid \theta) \, G_D(\mathrm{d} \theta)\bigg\}^{1/2} \bigg\{\int_{\Theta} K(y \mid \theta) \, G_{\ensuremath{\bar{D}}}(\mathrm{d} \theta)\bigg\}^{1/2} \, \mathrm{d} y.
\label{plugin}
\end{equation}
A natural approach is to consider each $G$ as a Dirichlet process \citep{ferguson1973}, and to rely on normal kernels, in which case \eqref{mixgen} becomes a so-called Dirichlet process mixture of normal kernels,
\begin{equation}
f(y) = \int_{\mathbb{R} \times (0, \infty)} \phi(y\mid \mu, \sigma) \,
G(\mathrm{d} \mu, \mathrm{d} \sigma) = \sum_{h=1}^\infty \pi_h \phi(y\mid \mu_h, \sigma_h), \quad G \sim \text{DP}(\alpha, G_0).
\label{mixDPM}
\end{equation}
Here $\alpha > 0$ is the so-called precision parameter, $G_0$ is the centering distribution function, or baseline measure, and we use the notation $G \sim \text{DP}(\alpha, G_0)$ to represent that $G$ follows a Dirichlet process (DP).
A celebrated representation of the DP is the so-called stick-breaking construction \citep{sethuraman1994}. According to this representation a random distribution function $G$ follows a DP if it admits a representation of the type
\begin{equation}
G = \sum_{h=1}^{\infty}\pi_h \delta_{\theta_h} , \quad \theta_h \overset{\text{iid}}{\sim} G_0,
\label{stick_break}
\end{equation}
where $\pi_1 = V_1$, and
$\pi_h = V_h\prod_{k < h}(1 - V_k)$, with $V_h \iid \text{Beta}(1,\alpha)$, for $h = 2, \dots$. The $\theta_h$ are known as atoms, the $\pi_h$ as mixing weights, and the $V_h$ are the so-called stick-breaking weights.
{For regression data, $\{(x_i, Y_{D, i})\}_{i = 1}^{n_D}$ and $\{(x_i, Y_{\ensuremath{\bar{D}}, i})\}_{i = 1}^{n_{\ensuremath{\bar{D}}}}$, we} propose to estimate $\kappa(x)$ in \eqref{kappa} by modeling each density $f_D$ and $f_{\ensuremath{\bar{D}}}$ as an infinite mixture model of regressions
\begin{equation}
f(y \mid x) = \int_{\Theta} K(y \mid \theta) \, G_{x}(\mathrm{d} \theta),
\label{mixgenreg}
\end{equation}
where $K$ is a kernel and $G_{x}$ is a covariate-specific random mixing measure. The corresponding induced prior is
\begin{equation}
\kappa(x) = \int_{-\infty}^{+\infty} \bigg\{\int_{\Theta} K(y \mid \theta) \, G_{D, x}(\mathrm{d} \theta)\bigg\}^{1/2} \bigg\{\int_{\Theta} K(y \mid \theta) \, G_{\ensuremath{\bar{D}}, x}(\mathrm{d} \theta)\bigg\}^{1/2} \, \mathrm{d} y.
\label{plugin}
\end{equation}
A natural approach is to consider each $G_{x}$ as a dependent Dirichlet process (DDP) \citep{maceachern2000}, and to rely on normal kernels in which case \eqref{mixgenreg} becomes an infinite mixture of regression models,
\begin{equation}
f(y \mid x) = \int_{\mathbb{R} \times (0, \infty)} \phi(y\mid \mu, \sigma) \,
G_{x}(\mathrm{d} \mu, \mathrm{d} \sigma).
\label{mixDP2}
\end{equation}
Because of the support properties in Theorem~4 of \cite{barrientos2012}, we consider a `single-weights' DDP \citep{deiorio2004, deiorio2009}
\begin{equation}\label{ddp}
G_{x} = \sum_{h=1}^{\infty}\pi_{h}\delta_{\theta_{x, h}}.
\end{equation}
The random support locations $\theta_{x,h}$ are, for $h = 1,2,\ldots$ independent and identically distributed realizations from a stochastic process over the covariate space $\mathcal{X}$ and the weights $\{\pi_h\}_{h = 1}^{\infty}$ match those from a standard DP; in this specific version of \eqref{ddp} we obtain
\begin{equation}
f(y \mid x) = \sum_{h=1}^\infty \pi_h \phi(y\mid \mu_h(x), \sigma_h).
\label{cond_ddp}
\end{equation}
To achieve a reasonable tradeoff between flexibility and parsimony, in practice we choose to model $\mu(x)$ as a linear model, that is, $\mu_h(x) = x^{{ \mathrm{\scriptscriptstyle T} }} \beta_h$, where $x^{{ \mathrm{\scriptscriptstyle T} }}$ corresponds {to the cubic B-spline basis evaluated at the predictor. Finally, {to facilitate prior specification we suggest} \textit{standardizing} the {biomarkers} (i.e., $Z_{Di} =(Y_{Di} - \bar{Y}_{D})/s_{D}$ and $z_{\bar{D}j}=(Y_{\bar{D}j}- \bar{Y}_{\bar{D}})/s_{\bar{D}}$) {and rescaling the covariate (i.e., $\min\{ x_{\bar{D}}, x_{D}\}=-1$ and $\max\{x_{\bar{D}}, x_{D}\}=1$)}. Having estimated the densities on the standardized data, the location-scale adjustment may be applied to easily convert to densities for the untransformed data.
{We now present a specific embodiment of our model.} Let $x_{\bar{D}i}^{{ \mathrm{\scriptscriptstyle T} }}$ represent the cubic B-spline representation of $x_{\bar{D}i}$, with $x_{\bar{D}i}$ having been rescaled to lie in $[-1,1]$. The assumptions for the non-diseased population in the conditional case are that
\begin{align}\label{likelihood}
f_{\bar{D}}(Z_{\bar{D}i} \mid x_{\bar{D}i}) & =
\int {\phi(Z_{\bar{D}i} \mid x_{\bar{D}i}^{{ \mathrm{\scriptscriptstyle T} }} \beta, \sigma^2)} \,\mathrm{d} G_{\bar{D}}(\beta, \sigma^2) \\
G_{\bar{D}}( \beta, \sigma^2) \mid G_{\bar{D}0}(\beta, \sigma^2) & \sim \text{DP}(1, G_{\bar{D}0}( \beta, \sigma^2)) \\
G_{\bar{D}0}( \beta, \sigma^2) & \equiv \mbox{N}(\beta_{\bar{D}0}, \Sigma_{\bar{D}0}) \times \mbox{IG}(\mbox{shape}=1, \mbox{rate}=50)\\
\beta_{\bar{D}0} & \sim \mbox{N}(0, I) \\
\Sigma_{\bar{D}0} & \sim \mbox{IWish}(\nu=1, S= I),
\end{align}
where IG and IWish respectively denote the inverse Gamma and inverse Wishart distributions. Two aspects of this specification are particularly noteworthy. First, it is assumed that the number and locations of all knots are known, although this could be relaxed. Second, the prior on the within-cluster variance (i.e., $\sigma^2$) was chosen to favor variances much less than one. The justification for this is immediate when recognizing that the likelihood is on standardized data with a marginal sample variance of one; the within-cluster variance ought to be substantially smaller than the marginal variance. The assumptions are analogous for the diseased population; the only difference is the substitution of $D$ for $\bar{D}$. To apply the model specification without conditioning on any covariate, we can simply substitute
{$x_i^{{ \mathrm{\scriptscriptstyle T} }}=1$.}
\subsection{Theoretical properties on induced priors}\label{theory}
This section includes theoretical properties on the induced priors for the summary measures introduced in Section~\ref{unconditional}. {Although in practice we model the densities from which $\kappa$ is estimated with a Dirichlet process mixture, as in {\eqref{mixDPM}} and \eqref{cond_ddp}, below we document theoretical results which apply more generally to \eqref{mixgen} and \eqref{mixgenreg} and only require that the mixing distribution has a full weak support, which includes the Dirichlet process as a particular case. In what follows, we assume the same setting as in \cite{lijoi2004}, namely:
\begin{enumerate}
\item[A$_1$)] The random mixing distribution(s) has (have) full weak support.
\item[A$_2$)] $\int_{-\infty}^{+\infty} K(y \mid \theta) \, \mathrm{d} y = 1$, for $\theta \in \Theta$.
\item[A$_3$)] $\theta \mapsto K(y \mid \theta)$ is bounded, continuous, and $\mathbb{B}_{\Theta}$-measurable for $y \in \mathbb{R}$.
\item[A$_4$)] The family of mappings $\{\theta \mapsto K(y \mid \theta): y \in C\}$, is uniformly equicontinuous, for every compact $C \subset \mathbb{R}$.
\end{enumerate}
Here, A$_1$ is a condition on the support of the mixing, whereas A$_2$--A$_4$ are regularity conditions on the kernel. Under these conditions, it can be shown that $f(y)$ in \eqref{mixgen} has full Hellinger support \citep{lijoi2004}. As a consequence, the following result holds.
\begin{theorem}\label{thm1} \normalfont
Suppose A$_1$--A$_4$ and let $(\Omega, \mathcal{A}, P)$ be the probability space associated with the infinite mixture model in \eqref{mixgen}, which induces $\kappa = \int \sqrt{f_D(y)} \sqrt{f_{\ensuremath{\bar{D}}}(y)} \, \mathrm{d} y$. Let $\kappa^\omega$ be a realization of the $\kappa$ index under \eqref{mixgen}. Then, for every $\varepsilon > 0$, it holds that $P\{\omega \in \Omega: |\kappa^\omega - \kappa| < \varepsilon\} > 0$.
\end{theorem}
Under the same conditions as above, it can be shown that $f(y \mid x)$ in \eqref{mixgenreg} has full Hellinger support \citep{barrientos2012}. Thus, the following analogous result to Theorem~\ref{thm1} holds for the covariate-specific version of our summary measure as defined in \eqref{kpred}.
\begin{theorem}\label{thm2} \normalfont
Suppose A$_1$--A$_4$ and let $(\Omega, \mathcal{A}, P)$ be the probability space associated with the infinite mixture of regression models in \eqref{mixgenreg}, which induces $\kappa(x) = \int \sqrt{f_D(y \mid x)} \sqrt{f_{\ensuremath{\bar{D}}}(y \mid x)} \, \mathrm{d} y$. Let $\kappa^\omega(x)$ be a trajectory of covariate-specific affinity $\kappa(x)$ under \eqref{mixgenreg}. Then, for $x_1,\ldots,x_n \in \mathcal{X}$, for every positive integer $n$ and $\varepsilon>0$, it holds that $P\{\omega \in \Omega: |\kappa^\omega(x_i) - \kappa(x_i)| < \varepsilon,~i=1,\ldots,n\} > 0$.
\end{theorem}
Proofs of Theorems~\ref{thm1} and~\ref{thm2} can be found in the online supplementary materials.
\begin{figure} \centering
\includegraphics[scale = 0.77]{Figures/SimSetU2.pdf}
\caption[Distributions from Unconditional Setting 2]{
Densities for the second unconditional simulation study
setting in Table~\ref{simset}; the black and grey lines respectively denote the densities of the biomarkers of the diseased and non-diseased subjects.}
\label{simsetu2}
\end{figure}
\section{Simulation study}
\subsection{Data generating processes}\label{dgp}
The simulation settings are summarized in Table~\ref{simset}. The simulation employed pairs of biomarker distributions that were either conditional on a single uniformly distributed covariate or were unconditional. In the unconditional settings, each distribution was either normal or a mixture of normals, and the means and standard deviations were systematically altered so that a range of $\kappa$ and $\text{AUC}$ values were considered. In the conditional settings, the conditional distributions were likewise normal or a mixture of normals, and the covariate's effect on the mean and standard deviation were altered to have varying levels of complexity. In terms of the conditional setting, we consider the same scenarios as in \cite{inaciodecarvalho2013}.
\begin{table}
\caption{\small Simulation study settings}
\begin{center}
\resizebox{\textwidth}{!}{%
\begin{tabular}{lllc}
\hline
{Scenario} & {Non-Diseased} ($f_{\bar{D}}$) & {Diseased} ($f_{D}$) & Notes$^\dagger$ \\ \midrule
Unconditional \#1 & $\phi(.4, .8)$ & $\phi(\mu_D, \sigma_D)$ & 1) \\ [0.25 cm]
Unconditional \#2 & $.7\phi(.1,.2) + .3\phi(3.1, .2)$ & $.7\phi(\mu_{1D}, \sigma_D) + .3\phi(\mu_{2D}, \sigma_D)$ & 2) \\ [0.25 cm]
Conditional \#1 & $\phi(.5+x_{\bar{D}}, 1.5)$ & $\phi(2+4x_D, 2)$ & 3) \\ [0.25 cm]
Conditional \#2 & $\phi(\sin(\pi (x_{\bar{D}}+1)), .5)$ & $\phi( .5+x_D^2, 1)$ & 3) \\ [0.25 cm]
\multirow{2}{*}{Conditional \#3} & $\phi(\sin(\pi x_{\bar{D}}), \sqrt{.2+.5\exp(x_{\bar{D}})})$ & $(1+\exp(-x))^{-1}\phi( x_D, .5) + (1+\exp(x))^{-1}\phi( x_D^3, 1)$ & 3) \\ [0.1 cm]
\bottomrule
\end{tabular}}
{\footnotesize $^\dagger$Each unconditional setting has nine distinct pairs of means and variances where for 1): $\mu_D$ in $\{.8, 1.6, 3.2\}$ and $\sigma_D$ in $\{.8, 1.2, 1.6 \}$ and 2): $\sigma_D = .2c$ and $(\mu_{1D}, \mu_{2D})$ in $\{(.2c, 3.2c), (1.1c, 4.1c), (2c, 5c) \}$, with $c$ in $\{.6, 1, 1.6\}$. For the conditional setting we have 3): $x_{\bar{D}}, x_{D} \iid \mbox{Unif}(-1,1)$.}
\end{center}
\label{simset}
\end{table}
\begin{figure} \hspace{2cm}
\begin{minipage}{0.45\linewidth} \hspace{-.5cm}
\includegraphics[scale = 0.4]{Figures/Uncond1SimResultsFinal.pdf}
\end{minipage}\hspace{0.7cm}
\begin{minipage}{0.45\linewidth} \hspace{-.5cm}
\includegraphics[scale = 0.4]{Figures/Uncond2SimResultsFinal.pdf}
\end{minipage} \\
\begin{minipage}{0.45\linewidth} \footnotesize
\hspace{4.2cm}(a)
\end{minipage}\hspace{0.7cm}
\begin{minipage}{0.45\linewidth} \footnotesize
\hspace{4.2cm}(b)
\end{minipage}
\caption{\small $\text{AUC}$ and $\kappa$ estimates (average across 100 simulations) along with true values in the unconditional scenarios of the simulation study (Table~\ref{simset}): a) the first unconditional setting (normals); b) the second unconditional setting (mixtures of normals).}
\label{fig:u1u2results}
\end{figure}
Figure~\ref{simsetu2} depicts the density pairs from the second unconditional {setting}; the plots for the {remaining} scenarios are included in the supplementary materials. Of particular note is the pattern of possibilities for $\kappa$ and $\text{AUC}$ when the biomarker densities are mixtures of normals. In particular, the middle plot in Figure~\ref{simsetu2} displays a situation where $\kappa$ is particularly adept at identifying the distinctiveness of the diseased and non-diseased populations {as can be seen by the very small $\kappa$ value}. However, our convention that $\text{AUC}$ be computed assuming the diagnostic test will be one-sided forces the $\text{AUC}$ to be lower than might be expected given the distinctiveness of the populations. While it is certainly possible to entertain more flexible regions at which the diagnostic test would be considered to have a positive result, this would require another nontrivial step before $\text{AUC}$ could even be calculated, whereas such a step is not needed to calculate $\kappa$.
\subsection{Monte Carlo simulation study}\label{MonteCarlo}
For each setting in Table~\ref{simset}, we generated 100 data sets from $f_D$ and from $f_{\bar{D}}$. The sample sizes were varied at $n_D=n_{\bar{D}}=150$, 500, or 2000 to provide some sense of how reliably $\kappa$ and $\text{AUC}$ were estimated in moderate to large samples. {In implementing the model, detailed in \eqref{likelihood}, no additional knots for the cubic B-splines were included; this lets us ascertain the covariate-dependent model's flexibility in the absence of extra knots.} {Additionally, because the covariate values were simulated from the Unif(-1,1) distribution, we did not rescale the covariate prior to computing the B-spline representation.} {For each synthetic data set $\kappa$ was estimated by collecting 300 MCMC iterates after a burn in of 2000 and thinning of 40. Algorithm~8 of \cite{neal2000} was employed to collect the iterates.}
{The results from the unconditional settings are summarized in} Figure~\ref{fig:u1u2results}, which depicts the Monte Carlo average (across 100 simulations) of the estimated values for $\kappa$ and $\text{AUC}$, along with the actual values. In part a), which was characterized by each population having a normal distribution, $\kappa$ and $\text{AUC}$ were both estimated with little bias. Not surprisingly, the bias is reduced by having larger sample sizes. In part b), which was characterized by each population having a mixture of normals distribution, the same pattern was exhibited.
\begin{figure}[H]\centering
\includegraphics[scale = 0.525]{Figures/CondSimSummaryPlotFinal.pdf}
\caption{\small Estimated covariate-specific affinity, $\kappa(x)$, and covariate-specific AUC, $\text{AUC}$$(x)$, across the 100 simulated data sets for the conditional settings described in Table~\ref{simset}. The bands represent the pointwise empirical 2.5th and 97.5th percentiles of the 100 point estimates, while the dark grey lines represent the average of the 100 estimates.}
\label{fig:condplot}
\end{figure}
The results from the conditional settings are summarized in Figure~\ref{fig:condplot}. For each of the 100 simulated data sets, the conditional means for $\kappa$ and $\text{AUC}$ were estimated at values of $x$ ranging from -1 to 1. The pointwise averages of the 100 estimated means are plotted in this figure, as well as the 2.5th and 97.5th empirical percentiles of these estimated means. This gives some sense for how variable the estimates are (primarily attributable to differences between the 100 simulated data sets).
{Point estimates of $\text{AUC}(x)$ and $\kappa(x)$ were quite successful in estimating the corresponding true values. Predictably, the estimates exhibited less variability as more data were available. Recall that a strength of $\kappa(x)$ is that it is not susceptible to the separation trap, nor does it require us to distinguish between upper- and lower-tailed diagnostic tests. This distinction for $\text{AUC}(x)$ explains why the AUC is sometimes estimated to be well below 0.5. Given these advantages of $\kappa(x)$ over $\text{AUC}(x)$, it is even more notable that $\kappa(x)$ can be reliably estimated. An important collateral suggestion of the simulation is that the model is quite flexible even if the cubic B-spline basis does not include additional knots, though of course knots may be added if desired.}
\section{Revisiting a prostate cancer diagnosis study}
{We now turn our attention to an application that has been regularly employed} {to demonstrate diagnostic test accuracy that is covariate-dependent.}
\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.5]{Figures/UncondPSA_AllObsFinal.pdf}\\ \vspace{-4.2cm}
\includegraphics[scale=0.7]{Figures/hist.pdf} \\ \vspace{-3.3cm}
\caption{\small Top: DPM-based estimated densities along with AUC and $\kappa$ values when age is not considered. The black and grey lines respectively denote the densities of the biomarkers of the diseased and non-diseased subjects. Bottom: Overlapping histograms.}
\label{Figure:UnconditionalCancer}
\end{center}
\end{figure}
\subsection{Study data and preliminary considerations}
The data were gathered from the Beta-Carotene and Retinol Efficacy Trial (CARET)---a lung cancer prevention trial, conducted at the Fred Hutchinson Cancer Research Center. During this study longitudinal measurements of two Prostate Specific Antigen (PSA)-based biomarkers were collected for 71 prostate cancer cases and 70 controls. The biomarker measurements were taken on males between 46 and 80 years old. The number of repeated measures per subject ranged from one to nine, with $n = 683$ total observations. Further details on this study can be found, for instance, in \cite{etzioni1999} and \cite{pepe2003}. To make our inferences directly comparable with those of \cite{rodriguez2014}---who consider a Gaussian process prior-based model for $\text{AUC}(x)$---we follow the latter authors and ignore the longitudinal nature of the data; however, for reference, we also include in the supplementary materials the results from restricting analysis to each subject's last available observation. A test based on total PSA concentration (Biomarker~1, ng/ml) was assumed to have a positive test result if the measurement was sufficiently large. Conversely, a test based on the free-to-total PSA ratio (Biomarker~2, f/t) was assumed to have a positive test result if the measurement was sufficiently small.
The direction of the tendency is of no consequence in estimating $\kappa$. In estimating AUC, however, we must consider the direction of the diagnostic test, that is whether larger values of the biomarker are more indicative of disease or the other way around. A main goal below will be on illustrating how the proposed methods can be used to assess which biomarker might screen better for prostate cancer.
\subsection{PSA-based analysis}\label{uncondData}
{We first fit the unconditional model (i.e., sans covariate so that
{$x_i^{{ \mathrm{\scriptscriptstyle T} }}=1$}) detailed in \eqref{likelihood} by collecting 1\,800 MCMC iterates after a burn in of 20\,000 and thinning of 100.} To visualize differences between the biomarkers we provide Figure~\ref{Figure:UnconditionalCancer}. For each biomarker, the estimated density among cases and controls are superimposed. It is readily apparent that there are differences between cases and controls, and that the direction of the differences depends on which biomarker we consider. Both univariate summaries, $\kappa$ and AUC, signal a preference for the first biomarker as a screening mechanism. {The 95\% credible interval of $\kappa$ associated with Biomarker~1 is (0.71, 0.77) and for $\text{AUC}$ is (0.80, 0.86), while for Biomarker~2 the interval for $\kappa$ is (0.83, 0.90) and for $\text{AUC}$ (0.70, 0.77) respectively. }
On the one hand, total PSA (Biomarker~1) looks like a reasonable biomarker since $\text{AUC} = 0.83$, but on the other hand $\kappa$ seems to be putting into question that evidence ($\kappa = 0.74$). {The overlapping histograms in Figure~\ref{Figure:UnconditionalCancer} shed some light on the reasons underlying the lack of consensus between $\text{AUC}$ and $\kappa$. Despite the fact that the AUC is moderately large in both cases, there is a considerable overlap between the distributions of the biomarkers for diseased and non-diseased subjects. And interestingly, the modes of both distributions are not that far apart. But perhaps this should not be regarded surprising since the discrimination power of PSA has been called into question and often regarded as controversial \citep{prensner2012, harvard2017}.}
\subsection{PSA-based analysis with age-adjustment}\label{condData}
It is well known that PSA levels may be age-dependent---for both diseased and non-diseased subjects---since both benign prostate conditions and prostate cancer become more common with age. With this in mind, we obtained conditional density estimates for each biomarker in each population to estimate $\kappa(\mbox{age})$ and $\text{AUC}(\mbox{age})$ {by fitting the conditional model and collecting 1\,800 MCMC iterates after a burn in of 20\,000 and thinning of 100 and using the same specifications as before.} In model fitting, the patients' ages were first rescaled from the interval $[46.75, 80.83]$ to the interval $[-1,1]$, and, following numerical evidence from \citet[][Section~3]{inaciodecarvalho2017}, we elected to not to include any additional knots in the cubic B-splines.
{Figure~\ref{condPSA} displays the posterior mean and pointwise 95\% credible intervals for $\kappa$ and AUC as a function of age. Notice first that for Biomarker~1 our estimated $\text{AUC}(\mbox{age})$ is very similar to that found in Figure~4 of \cite{rodriguez2014}, with the largest discriminatory power occurring when an individual is in their late 50s. Regarding comparisons with $\kappa$, generally speaking Biomarker~1 exhibits less affinity than Biomarker~2 between the distributions of those with and without a prostate cancer diagnosis. This suggests that a diagnostic test based on Biomarker~1 would be preferred to a test based on Biomarker~2. The first biomarker's affinity appears to be sensitive to the subject's age. In line with the findings in the previous section, AUC seems to indicate that PSA is a reasonably good diagnostic test, while $\kappa$ seems to be more pessimistic regarding the test's ability ($\kappa \approx 0.6$); a similar conclusion holds for Biomarker 2. In addition, $\kappa$ more clearly identifies the difference in screening ability of the two biomarkers for males aged 55 to 70. Furthermore, it is invariant to whether the diagnostic test is assumed to be lower- or upper-tailed.}
\begin{figure} \hspace{0.8cm}
\begin{minipage}{0.275\linewidth} \hspace{-1.2cm}
\includegraphics[scale = 0.4]{Figures/CondPSA_Kappas_AllObsFinal.pdf}
\end{minipage}
\begin{minipage}{0.275\linewidth} \hspace{1cm}
\includegraphics[scale = 0.4]{Figures/CondPSA_UpperTailAUCs_AllObsFinal.pdf}
\end{minipage} \hspace{1cm}
\begin{minipage}{0.275\linewidth} \hspace{1cm}
\includegraphics[scale = 0.4]{Figures/CondPSA_AUCs_AllObsFinal.pdf}
\end{minipage} \\
\begin{minipage}{0.275\linewidth} \centering \footnotesize
~~~~~~~(a)
\end{minipage}\hspace{0.7cm}
\begin{minipage}{0.275\linewidth} \centering \footnotesize
~~~~~~~~~~~~~~(b)
\end{minipage}\hspace{0.7cm}
\begin{minipage}{0.275\linewidth} \centering \footnotesize
~~~~~~~~~~~~~~~~~~~~~~(c)
\end{minipage}
\caption{\small Means and 95\% pointwise credible intervals for the age-adjusted affinity and $\text{AUC}$ of two biomarkers in cases and controls. a) is the age-adjusted affinity; b) is the age-adjusted $\text{AUC}$ if both biomarkers have upper-tailed diagnostic tests; c) is the age-adjusted $\text{AUC}$ if the second biomarker diagnostic test is lower-tailed. In each panel, the black and grey lines respectively denote the first and second biomarkers.}
\label{condPSA}
\end{figure}
Finally, both analyses suggest Biomarker~1 is a better alternative than Biomarker~2 in screening older males with lung cancer for prostate cancer. This latter conclusion is supported even more emphatically by $\kappa(\mbox{age})$ than by $\text{AUC}(\mbox{age})$, as seen in Figure~\ref{condPSA}.
\section{Discussion}
In this paper we show how Hellinger affinity can be used as a natural summary measure of medical diagnostic accuracy. The summary measure has several desirable properties that motivate its use as a supplement, if not competitor, to other existing summaries such as $\text{AUC}$ and the Youden index. Affinity shares some of the properties of the AUC---such as invariance to monotone increasing transformations---, but it does not fall into the separation trap, {whereas both the AUC and the Youden index would}. Indeed, a principal advantage of $\kappa$ is that it is readily calculated and interpreted without assuming anything about the biomarker threshold(s) that demarcate positive and negative test diagnoses. This can be especially beneficial if, for instance, a biomarker's distribution when the disease is present favors both atypically low and atypically high values. Affinity-based measures can be framed into the same geometrical principles as Pearson correlation, and they focus on the overlap between $f_D$ and $f_{\ensuremath{\bar{D}}}$ rather than on always presuming that larger values of {a biomarker} are more indicative of disease. Nonparametric Bayes estimators for affinity and covariate-specific affinity are discussed, and theoretical properties of the corresponding priors have been derived. While it could be natural to fit parametric models such as the ones in Table~\ref{parametric}, the added flexibility of the proposed inferences allows us to model diagnostic test accuracy in a way that offers flexibility and robustness against misspecification.
While not explored here, our summary measure has the potential to be applied to the more general setting where $p > 1$ biomarkers per subject are available. Indeed, if $f_D(y)$ and $f_{\ensuremath{\bar{D}}}(y)$ denote the joint distributions of the $p$ biomarkers, for diseased and non-diseased subjects, similarly to \eqref{kappa} one can define
\begin{equation*}
\kappa = \langle \sqrt{f_D}, \sqrt{f_{\ensuremath{\bar{D}}}} \rangle
= \int_{\mathbb{R}^p} \sqrt{f_D(y)} \sqrt{f_{\ensuremath{\bar{D}}}(y)} \, \mathrm{d} y.
\end{equation*}
{However, estimation of $\kappa$ would become more challenging in the multivariate case than in the univariate case presented in this article.} Future work could also entail nonparametric Bayesian inference for a covariate-specific version of the so-called overlap coefficient \citep{wang2017} which can be defined as
\begin{equation*}
\mbox{OVL}(x) = \int_{-\infty}^{\infty} \min\{f_{D}(y \mid x), f_{\ensuremath{\bar{D}}}(y \mid x) \}\, \mathrm{d} y.
\label{OVL}
\end{equation*}
The index in \eqref{OVL} would quantify the proportion of overlap area between $f_{D}(y \mid x)$ and $f_{\ensuremath{\bar{D}}}(y \mid x)$, and thus it could be used as a companion to $\text{AUC}(x)$ and $\kappa(x)$. Finally, another direction which we may revisit in future work rests on the study of $\kappa(x)$ and $\mbox{OVL}(x)$ on settings where a gold standard test is unavailable.
\section*{Funding}
This work was partially supported by FCT (Funda\c c\~ao para a Ci\^encia e a Tecnologia, Portugal), through the project UID/MAT/00006/2013.
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\section{Introduction}
In this paper we apply some constructions from the theory of braided groups and
braided geometry\cite{Ma:introm} to obtain a new construction for matrix
solutions of the celebrated Quantum Yang-Baxter Equations (QYBE). Equivalently,
we provide a new and canonical class of representations of the Artin braid
group. The importance of such representations or R-matrices has been very
clearly established in the last few years and is one of the primary motivations
behind the celebrated quantum groups $U_q(g)$\cite{Dri}\cite{Jim:dif}.
Representations lead ultimately to link invariants and families of
representations to 3-manifold invariants.
By contrast to the theory of quantum groups, our construction is based on what
we believe to be a more primitive object, called a {\em braided Lie
algebra}\cite{Ma:lie}. The famous quantum groups $U_q(g)$ have
finite-dimensional braided-Lie algebras associated to them and one can work
with them instead of the quantum group. In this case our canonical braiding
reproduces the braiding associated to the quantum double\cite{Dri}.
More generally however, our notion of braided Lie algebra also includes as a
special case the notion of a rack, see e.g\cite{FenRou:rac}. In this case we
recover the rack braiding. Super-Lie algebras, super-racks and other much more
esoteric objects are also included in the theory.
The axioms of a braided-Lie algebra are recalled in the Preliminaries. They are
Lie algebra-like objects living in a braided tensor category with braiding
$\Psi$. Ordinary Lie algebras and ordinary racks are defined with $\Psi$ given
by the usual transposition. Their super-versions are defined with $\Psi=\pm 1$
according to a ${\Bbb Z}_2$-grading.
Unlike previous attempts to go beyond supersymmetry, we need not assume that
$\Psi^2={\rm id}$.
In Section~3 we give a parallel theorem for Hopf algebras, super-Hopf algebras
and more generally, for braided-Hopf algebras\cite{Ma:bg}. The latter are Hopf
algebras living in our braided tensor category with background braiding $\Psi$.
This theory is more general because not every braided-Hopf algebra is the
enveloping algebra of a braided-Lie algebra, but in the case that it is, we
recover the results of Section~2. The example of a finite group and its
canonical braiding fits comfortably into either setting. We will also give some
more novel examples in Section~4, including one based on the anyonic line where
$\Psi$ is given by a root of unity. Many other important algebras of interest
in the theory of $q$-deformations are not naturally Hopf algebras but rather
braided ones.
Finally, we show in the Appendix how the extended Artin braid relations for
braids in the complement of the unknot in $S^3$ can be represented equally well
using the same techniques. We assume that we are given a cocommutative
representation of a braided-Hopf algebra in the sense that it is central in the
braided representation ring of the braided-Hopf algebra\cite{Ma:tra}.
Representations of such extended braid relations have been used by
knot theorists in \cite{LamPrz:hom} and elsewhere. In short, the techniques
which we use here are of quite wide applicability and this appendix
demonstrates one more instance of them.
Although we will not go as far as constructing knot and three-manifold
invariants from our canonical braiding, knot theory nevertheless enters in a
fundamental way. This is because we will be working throughout in a background
braided category with braiding $\Psi$. Usually one uses quantum groups etc to
construct such a braided category and hence to obtain knot-invariants: we
proceed in exactly the reverse direction by assuming that $\Psi$ is given and
doing all our proofs by drawing braids and tangles. These techniques and the
formulation of a large number of geometrical constructions of planes, lines,
matrices, groups, differential operators etc., is the topic of braided geometry
as developed over 30-40 papers by the author in the last few years. We refer to
\cite{Ma:introp}\cite{Ma:introm} for reviews and to
\cite{Ma:bg}\cite{Ma:tra}\cite{Ma:bos}\cite{Ma:skl}\cite{Ma:lin}\cite{Ma:fre}
for some of the basic theory.
\subsection*{Preliminaries} Here we recall very briefly the definition of
braided or quasitensor categories and the diagrammatic notation for them.
Firstly, a monoidal category consists of a category $\CC$ equipped with a
functor
$\mathop{\otimes}:\CC\times \CC \to \CC$ and functorial isomorphisms
$\Phi_{V,W,Z}:V\mathop{\otimes} (W\mathop{\otimes} Z)\to
(V\mathop{\otimes} W)\mathop{\otimes} Z$ for all objects $V,W,Z$, and a unit
object $\und 1$ with
functorial isomorphisms $l_V:V\to \und 1\mathop{\otimes} V,r_V:V\to
V\mathop{\otimes} \und 1$ for all objects $V$.
The $\Phi$ should obey a well-known pentagon coherence
identity while the $l$ and $r$ obey triangle identities of
compatibility with $\Phi$\cite{Mac:cat}. We assume such a monoidal category and
suppress writing
$\Phi,l,r$ explicitly. A monoidal category also has an opposite tensor product
$\mathop{\otimes}^{\rm op}:\CC\times\CC\to \CC$ defined in the obvious way.
A braided monoidal or quasitensor category $(\CC,\Psi)$ is a monoidal category
$\CC$ equipped further with a natural transformation $\Psi:\mathop{\otimes}^{\rm op}\to
\mathop{\otimes}$ called the {\em braiding} or quasisymmetry and subject to two hexagon
coherence identities. Explicitly, this means a collection of functorial
isomorphisms
$\Psi_{V,W}:V\mathop{\otimes} W\to W\mathop{\otimes} V$ for any two objects and such that
\eqn{psi-hex}{ \Psi_{V,W\mathop{\otimes} Z}=\Psi_{V,Z}\circ\Psi_{V,W},\quad\Psi_{V\mathop{\otimes}
W,Z}=\Psi_{V,Z}\circ\Psi_{W,Z}.}
One can deduce also that $\Psi_{V,\und 1}={\rm id}=\Psi_{\und 1,V}$ for all $V$. If
$\Psi^2={\rm id}$ then one of the
hexagons is superfluous and we have an ordinary symmetric monoidal category or
tensor
category. Braided monoidal categories were formally introduced in
\cite{JoyStr:bra}, while being known also to specialists in the representation
theory of quantum groups\cite[Sec. 7]{Ma:qua}.
Crucial for us is the following diagrammatic notation for working with
algebraic objects in braided categories. Firstly, we write all morphisms
pointing downwards (say) and in the case of the braiding morphism, we use the
shorthand
\eqn{Psi-bra}{\epsfbox{psinoarrow.eps}}
This distinguishes between $\Psi$ and $\Psi^{-1}$, while the hexagons
(\ref{psi-hex}) appear as
\eqn{Psi-hex-bra}{\epsfbox{smallhex.eps}}
The doubled lines refer to the composite objects $V\mathop{\otimes} W$ and $W\mathop{\otimes} Z$ in a
convenient extension of the notation. The coherence theorem for braided
categories says then that if two series of morphisms
built from $\Psi,\Phi$ correspond to the same braid then they compose to the
same morphism. The proof is just the same as Mac Lane's proof in the symmetric
case with the action of the symmetric group replaced by that of the Artin braid
group.
Finally, we take this notation further by writing any other
morphisms as nodes on a string connecting the inputs down to the outputs.
Functoriality of the braiding then says that morphisms $\phi:V\to Z$, $W\to Z$,
etc. can
be pulled through braid crossings,
\eqn{Psi-funct}{\epsfbox{funcnode.eps}}
Similarly for $\Psi^{-1}$ with inverse braid crossings. The simplest example is
with $\CC={\rm SuperVec}$, the category of ${\Bbb Z}_2$-graded vector spaces and
braiding
\eqn{supertran}{\Psi(v\mathop{\otimes} w)=(-1)^{|v|
|w|}w\mathop{\otimes} v}
where $|\ |$ denotes the degree of a homogeneous element. Of course, this
example is not truly braided since $\Psi^2={\rm id}$.
We recall also the celebrated Yang-Baxter equations or Artin braid relations.
Thus, a Yang-Baxter operator is a morphism $\check{R }:V\mathop{\otimes} V\to
V\mathop{\otimes} V$ such that
\eqn{YBop}{{\check{R}}_{23}\circ {\check{R}}_{12}\circ {\check{R
}}_{23}={\check{R }}_{12}\circ {\check{R }}_{23}
\circ {\check{R }}_{12}}
where the suffices refer to the copy of $V$ in $V\mathop{\otimes} V\mathop{\otimes} V$. If $V$ is an
ordinary vector space and everything is linear then we can write $\check{R
}=PR$ where $P:V\mathop{\otimes} V\to V\mathop{\otimes} V$ is the permutation operator. Then the
corresponding equation for $R$ is
\eqn{QYBE}{ R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}}
which is the so-called {\em quantum Yang-Baxter equation} (QYBE). The matrices
of such operators are called in physics `R-matrices'.
There is a close relation between R-matrices and braided categories for which
the objects are built on vector spaces. Obviously, if $\Psi$ is a braiding then
$\Psi_{V,V}$ is an invertible Yang-Baxter operator and hence when $V$ is a
finite-dimensional vector space we have an associated invertible R-matrix.
Conversely, any invertible R-matrix defines a braiding on the monoidal category
generated by $V$.
Note that the general theory of Sections~2,3 works in any braided monoidal
category. In this case we use the word `operator' etc here a bit loosely. On
the other hand, our examples in Section~4 are in a $k$-linear setting where $k$
is a field, and then our operators are indeed linear maps.
\section{Canonical braiding of a braided-Lie algebra}
We have introduced in \cite{Ma:lie} the notion of a braided-Lie algebra or
Lie-algebra-like object in a braided monoidal category $(\CC,\Psi)$ as
$(\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}},\Delta,{\epsilon},[\ ,\ ])$ where $(\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}},\Delta,{\epsilon})$ is a coalgebra in the
category and $[\ , \ ]:\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}\mathop{\otimes}\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}\to \hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}\mathop{\otimes}\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}$ is the {\em braided Lie
bracket} and is required to obey
\ceqn{Lie}{ \epsfbox{Lie1.eps}\quad\qquad\epsfbox{Lie2.eps}\\
\epsfbox{Lie3.eps}}
We use here the diagrammatic notation described in the preliminaries. A
coalgebra in the category is defined in just the same way as an algebra, but
with arrows reversed. Thus, $\Delta:\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}\to \hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}\mathop{\otimes} \hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}$, the comultiplication,
is coassociative in an obvious sense and ${\epsilon}:\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}\to \und 1$ is a counit for
it in the obvious sense. Explicitly,
\eqn{Liecoalg}{\epsfbox{Liecoalg.eps}}
The condition (L1) is called the {\em braided-Jacobi} identity axiom, (L2) the
{\em braided-cocommutativity} axiom and (L3) the {\em coalgebra-compatibility
axiom}. We refer to \cite{Ma:lie} for the justification and full explanation of
these axioms. Suffice it to say that in a truly braided category the naive
notions of $\Psi$-anticommutativity and $\Psi$-Jacobi identity are not
appropriate and one needs a genuinely new idea. The new idea in \cite{Ma:lie}
is to allow ourselves a more general coalgebra $\Delta$ instead of the
primitive coalgebra structure $\Delta\xi=\xi\mathop{\otimes} 1+1\mathop{\otimes}\xi$ on $k\oplus g$
which is implicitly assumed in the theory of Lie algebras.
The basic theory of braided-Lie algebras has also been developed in
\cite{Ma:lie}. This includes such things as (in the Abelian category case) a
braided-enveloping bialgebra $U(\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}})$, braided-Killing forms and
braided-Casimirs etc. To this theory we want to add now the following theorem
announced in \cite{Ma:mex}.
\begin{theorem} Let $(\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}},\Delta,{\epsilon},[\ ,\ ])$ be a braided-Lie algebra. Then
\[ \check{\bf R }=\ {{}\atop \epsfbox{Liebraid.eps}}=([\ ,\
]\mathop{\otimes}{\rm id})\circ({\rm id}\mathop{\otimes}\Psi)\circ(\Delta\mathop{\otimes}{\rm id}):\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}\mathop{\otimes}\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}\to \hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}\mathop{\otimes}
\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}\]
is a Yang-Baxter operator.
\end{theorem}
\begin{figure}
\[\epsfbox{Liebraproof.eps}\]
\caption{Proof of Theorem~2.1}
\end{figure}
\goodbreak\noindent{\bf Proof\quad} We do this diagrammatically in Figure~1, using the notation explained in
the preliminaries. The vertices are $\Delta=\epsfbox{deltafrag.eps}$ and $[\ ,\
]=\epsfbox{prodfrag.eps}$ throughout.
The first expression is the right-hand side of (\ref{YBop}) for $\check{\bf R
}$ as stated. The first equality is (L3). The second equality is
coassociativity (\ref{Liecoalg}) and functoriality to put the diagram in a form
suitable for (L2), which is the third equality. The fourth equality is
coassociativity (\ref{Liecoalg}) again. The fifth then uses our braided-Jacobi
identity axiom (L1). The sixth is coassociativity once more and finally we use
functoriality to slide the diagram into the final form, which is the left hand
side of (\ref{YBop}) for $\check{\bf R }$. {\ $\lform$}\bigskip
Moreover, it is evident from its diagrammatic definition in \cite{Ma:lie} that
the braided enveloping algebra $U(\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}})$ is generated by $1$ and
$\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}$ with the relations
\eqn{bracom}{ \cdot\circ\check{\bf R }=\cdot}
of braided commutativity.
\section{Canonical braiding of a braided-Hopf algebra}
In this section, we further generalise the result of the last section to
associate to any braided-Hopf algebra
at all a canonical Yang-Baxter operator. Braided-Hopf algebras were introduced
by the author in \cite{Ma:bg}\cite{Ma:exa}\cite{Ma:eul}\cite{Ma:bra} as a
generalisation to braided categories of the usual notion of
Hopf algebra or super-Hopf algebra. Briefly, a braided-Hopf algebra means
$(B,\Delta,{\epsilon},S)$ where firstly $B$ is a unital algebra in a braided monoidal
category. This means it comes equipped with product and unit morphisms $B\mathop{\otimes}
B\to B$ and $\und1\to B$ respectively, obeying the obvious axioms of
associativity and unity. Secondly, $\Delta:B\to B\und\mathop{\otimes} B$ and
${\epsilon}:B\to\und 1$ form a coalgebra as already encountered in Section~2. We
require further that $\Delta$ is an algebra homomorphism where $B\und\mathop{\otimes} B$
is the braided tensor product algebra (as also introduced by the author). This
forms a braided-bialgebra or bialgebra in a braided category. Finally, we
require an antipode $S:B\to B$ obeying axioms similar to those for quantum
groups or Hopf algebras, but as a morphism in our category. In the diagrammatic
notation with
$\Delta=\epsfbox{deltafrag.eps}$ and $\cdot=\epsfbox{prodfrag.eps}$, our axioms
read
\ceqn{hopfax}{ \epsfbox{bialg-ax.eps}\\ \epsfbox{ant-ax.eps}}
The general theory of braided-Hopf algebras has also been developed by now in a
diagrammatic form\cite{Ma:tra}\cite{Ma:bos}\cite{Ma:introp}. One has left and
right dual Hopf algebras (when $B$ has a dual object), regular actions and
coactions, braided-adjoint actions and coactions, cross products etc. We use
the braided-adjoint action
now.
\begin{theorem} Let $(B,\Delta,{\epsilon},S)$ be a braided-Hopf algebra and ${\rm Ad}$ the
braided-adjoint action. Then
\[ \check{\bf R }=\ {{}\atop
\epsfbox{hopfbraid.eps}}
=({\rm Ad}\mathop{\otimes}{\rm id})\circ({\rm id}\mathop{\otimes}\Psi)\circ(\Delta\mathop{\otimes}{\rm id}):B\mathop{\otimes} B\to B\mathop{\otimes} B\]
is a Yang-Baxter operator.
\end{theorem}
\begin{figure}
\[\epsfbox{hbraproof.eps}\]
\caption{Proof of Theorem~3.1}
\end{figure}
\goodbreak\noindent{\bf Proof\quad} This is given in diagrammatic form in Figure~2. Some of the
$\epsfbox{prodfrag.eps}$ vertices are the braided-adjoint action
${\rm Ad}$\cite{Ma:exa}\cite{Ma:lie} and the rest are the product in $B$. The
$\epsfbox{deltafrag.eps}$ are the coproduct throughout. The first expression is
the right hand side of (\ref{YBop}) for $\check{\bf R }$ as stated. The first
equality
uses that ${\rm Ad}$ is indeed an action of $B$ on $B$. The second equality
substitutes the form of ${\rm Ad}$ in terms of the braided-Hopf algebra structure,
as shown in the definition of $\check{\bf R }$.
The third equality uses the bialgebra axiom that $\Delta$ is an algebra
homomorphism to the braided tensor product as on the left in (\ref{hopfax}). We
also adopt the convention that repeated applications of $\Delta$ can be
represented by multiple branches. Likewise for multiple products. This
convention expresses coassociativity and associativity respectively. The
fourth equality is the lemma proven in \cite{Ma:tra} that $S$ is a
braided-anti-algebra homomorphism in the sense
$S\circ\cdot=\cdot\circ\Psi\circ(S\mathop{\otimes} S)$. The fifth equality recognises a
loop involving the antipode and cancels it according to the left-hand antipode
axiom shown in (\ref{hopfax}). We also recognise the remaining antipode $S$ as
part of an application of ${\rm Ad}$. The sixth equality is coassociativity and
functoriality to push this ${\rm Ad}$ down to the bottom of the expression. Finally,
we use again that ${\rm Ad}$ is an action to obtain the left hand side of
(\ref{YBop}) for $\check{\bf R }$. {\ $\lform$}\bigskip
Moreover, it is obvious from coassociativity, associativity and the axioms for
the antipode that $B$ itself is braided-commutative in the sense
\[ \epsfbox{hbracom.eps}\]
Let us note also that the axioms of a braided-Hopf algebra in (\ref{hopfax})
are symmetric under the operations of left-right reflection and braid crossing
reversal, up-down reflection and braid crossing reversal, and rotation by 180
degrees. As explained in \cite{Ma:introm}, it means that the diagrammatic
method always gives three theorems for the price of one. Applying these
symmetries to Figure~2 and its associated lemmas gives
\[\check{\bf R }=\ {{}\atop \epsfbox{hbravar3.eps}}\qquad\quad \check{\bf R }=\
{{}\atop \epsfbox{hbravar1.eps}}\qquad\quad \check{\bf R }=\ {{}\atop
\epsfbox{hbravar2.eps}}\]
as three other Yang-Baxter operators. $B$ is also braided-commutative with
respect to the first and braided-cocommutative with respect to the second and
third.
\section{Examples}
In this section we describe various examples and special cases of the general
constructions above. Throughout this
section we work over a field $k$ of characteristic zero. The same results apply
more generally with appropriate care. As far as I know, only the case in
subsection~4.4 was known before from the theory of racks and its generalisation
in subsection~4.5 from Drinfeld's quantum double construction for Hopf algebras
and later from \cite{Wor:sol}. The rest appear to be a product of our
construction, as announced recently in the conference proceedings\cite{Ma:mex}.
\subsection{Ordinary Lie Algebras}
Note that an ordinary Lie algebra obeys these axioms if one
puts $[1,\xi]=\xi$, $[\xi,1]=0$, $[1,1]=1$ and
\eqn{ordLie}{\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}=k\oplus g,\quad \Delta 1=1\mathop{\otimes} 1,\ {\epsilon} 1=1,\quad
\Delta\xi=\xi\mathop{\otimes}
1+1\mathop{\otimes}\xi,\ {\epsilon}\xi=0,\quad\forall \xi\in g.}
So this structure $\Delta,{\epsilon}$ is implicit for an ordinary Lie algebra but we
never think about it because it has this standard form. This was our motivation
in
\cite{Ma:lie}.
\begin{propos} Let $V=k\oplus g$ and define the linear map
\cmath{ \check{\bf R }(1\mathop{\otimes} 1)=1\mathop{\otimes} 1,\quad {\check{\bf R }} (1\mathop{\otimes}
\xi)=\xi\mathop{\otimes} 1,\quad
{\check{\bf R }} (\xi\mathop{\otimes} 1)=1\mathop{\otimes}\xi\\
{\check{\bf R }} (\xi\mathop{\otimes}\eta)=\eta\mathop{\otimes}\xi+[\xi,\eta]\mathop{\otimes}
1,\qquad\qquad\quad \forall
\xi,\eta\in g.}
Then ${\check{\bf R }} $ is a braiding {\em iff} $[\ ,\ ]:g\mathop{\otimes} g\to g$ obeys
the Jacobi
identity. It has minimal polynomial
\eqn{minpoly}{({\check{\bf R }} ^2-{\rm id})({\check{\bf R }} +{\rm id})=0}
{\em iff} $[\ ,\ ]$ is non-zero and antisymmetric.
\end{propos}
\goodbreak\noindent{\bf Proof\quad} The forward direction is a special case of Theorem~2.1 where we view
$\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}=k\oplus g$ as a braided-Lie algebra with trivial braiding $\Psi$ as
explained above. On the other hand, at least in this setting, one can compute
it explicitly and see that it is reversible. Thus
\align{{\check{\bf R }}_{23}\circ {\check{\bf R }}_{12}\circ {\check{\bf R
}}_{23}(\xi\mathop{\otimes}\eta\mathop{\otimes}\zeta)&&\kern -1.7em
=\zeta\mathop{\otimes}\eta\mathop{\otimes}\xi+\zeta\mathop{\otimes}[\xi,\eta]\mathop{\otimes} 1+[\xi,\zeta]\mathop{\otimes}\eta\mathop{\otimes}
1\\
&&\quad +[\eta,\zeta]\mathop{\otimes} 1\mathop{\otimes}\xi+[\xi,[\eta,\zeta]]\mathop{\otimes} 1\mathop{\otimes} 1\\
{\check{\bf R }}_{12}\circ {\check{\bf R }}_{23}
\circ {\check{\bf R }}_{12}(\xi\mathop{\otimes}\eta\mathop{\otimes}\zeta)&&\kern -1.7em
=\zeta\mathop{\otimes}\eta\mathop{\otimes} \xi+[\xi,\zeta]\mathop{\otimes}\eta\mathop{\otimes} 1+[\eta,\zeta]\mathop{\otimes}
1\mathop{\otimes}\xi\\
&&\quad +[\eta,[\xi,\zeta]]\mathop{\otimes} 1\mathop{\otimes} 1 +\zeta\mathop{\otimes} [\xi,\eta]\mathop{\otimes}
1+[[\xi,\eta],\zeta]\mathop{\otimes} 1\mathop{\otimes} 1}
so that the only condition is the Jacobi identity in the form that $[\xi,\ ]$
acts like a Lie derivation. The braid relations involving the basis element $1$
are all empty. Secondly, we compute
\[ (\check{\bf R }^2-{\rm id})\circ (\check{\bf R
}+{\rm id})(\xi\mathop{\otimes}\eta)=([\xi,\eta]+[\eta,\xi])\mathop{\otimes} 1+1\mathop{\otimes}
([\xi,\eta]+[\eta,\xi])\]
so this vanishes {\em iff} the bracket is antisymmetric. Finally, we compute
\cmath{ (\check{\bf R }\pm{\rm id})(\xi\mathop{\otimes}\eta)=\eta\mathop{\otimes}\xi+[\xi,\eta]\mathop{\otimes} 1\pm
\xi\mathop{\otimes} \eta\\
(\check{\bf R }^2-{\rm id})(\xi\mathop{\otimes}\eta)=[\eta,\xi]\mathop{\otimes} 1+1\mathop{\otimes}[\xi,\eta]\\
(\check{\bf R }+{\rm id})^2(\xi\mathop{\otimes}\eta)=2\xi\mathop{\otimes}\eta+[\eta,\xi]\mathop{\otimes}
1+1\mathop{\otimes}[\xi,\eta]+2\eta\mathop{\otimes}\xi+2[\xi,\eta]\mathop{\otimes} 1}
which are all non-zero for some $\xi,\eta$ if $[\ ,\ ]$ is non-zero. Hence in
this case (\ref{minpoly}) is the minimum polynomial. Conversely, if this is the
minimum-polynomial then
in each case there exist $\xi,\eta$ such that the expression is non-zero. In
particular, $(\check{\bf R }^2-{\rm id})\ne 0$ implies that $[\xi,\eta ]\ne 0$ for
some $\xi,\eta$. {\ $\lform$}\bigskip .
This says that the definition of a Lie algebra
is mathematically completely equivalent to looking for a braiding of a certain
form. We say accordingly that a Yang-Baxter operator obeying (\ref{minpoly}) is
of {\em Lie type}.
We next discuss the braided-enveloping algebra. To avoid confusion here we
denote the basis element of $k$ in $k\oplus g$ by
$\lambda$ rather than $1$ as above. So our starting point is that that $\Psi$
trivial (usual transposition) and
\cmath{ \Delta \lambda=\lambda\mathop{\otimes}\lambda, \quad \Delta
\xi=\xi\mathop{\otimes}\lambda+\lambda\mathop{\otimes}\xi,\quad {\epsilon}\lambda=1,\quad {\epsilon}\xi=0\\
{}[\lambda,\lambda]=\lambda,\quad [\lambda,\xi]=\xi,\quad [\xi,\lambda]=0,\quad
\forall\xi\in g}
extends a usual Lie algebra $g$ to a braided Lie algebra $\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}=k\oplus g$.
The braided enveloping algebra in this case is $U(\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}})=\widetilde{U(g)}$, a
homogenised form of the enveloping algebra $U(g)$. It is an ordinary bialgebra
since the braiding $\Psi$ is trivial. On the other hand we know from the
definition of $U(\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}})$ that it is braided-commutative in the sense
(\ref{bracom}) relative to $\check{\bf R }$. Moreover, everything descends to
the quotient $\lambda=1$ so we recover the usual enveloping algebra too as
braided-commutative in this sense, a fact which is in any case evident from the
form of $\Psi$.
Note also that if we allow $\lambda^{-1}$ then this form of $\widetilde{U(g)}$
has $\xi\lambda^{-1}$ primitive and $\lambda$ group-like, and forms a Hopf
algebra. It is only for this part of its structure that antisymmetry of the Lie
bracket of $g$ is needed. For a general braided-Lie algebra I do not know any
very natural notion of antisymmetry for the bracket.
Finally, we give the concrete matrix version in which we choose a basis
$g=\{x_i\}$ for $i=1,2,\cdots ,n-1$ and $k=\{x_0\}$ for $V=k\oplus g$. Then
\eqn{g-R}{{\bf R}=\pmatrix{1&0&0&0\cr 0&I&0 & c\cr 0&0&I&0\cr 0&0&0&I}}
where $I$ are identity matrices and $c^i{}_{jk}$ are the structure constants of
the bracket $[\ ,\ ]$ on $g$. The basis for $V\mathop{\otimes} V$ used here is $\{x_0\mathop{\otimes}
x_0,x_0\mathop{\otimes}
x_j,x_i\mathop{\otimes} x_0,x_i\mathop{\otimes} x_j\}$. Explicitly,
\[ {\bf R}^0{}_i{}^k{}_j=c^k{}_{ij},\ {\bf
R}^i{}_j{}^k{}_l=\delta^i{}_j\delta^k{}_l,\ {\bf
R}^0{}_0{}^i{}_j=\delta^i{}_j={\bf R}^i{}_j{}^0{}_0,\ {\bf R}^0{}_0{}^0{}_0=1\]
and zero for the rest. This obeys the QYBE and has minimal polynomial of Lie
type {\em iff} $c$ defines a non-zero Lie algebra. The quantum R-plane or
Zamolodchikov algebra
\[ x_ix_j=x_bx_a{\bf R}^a{}_i{}^b{}_j\]
for this R-matrix recovers the homogenised enveloping algebra above in our
basis. The indices here range $0,\cdots,n-1$ and summation of the repeated
indices is understood.
Finally, we note that homogenised Lie algebras have recently been studied in
\cite{LeBSmi:hom}\cite{LeBBer:spa} as examples of a kind of non-commutative
geometry based on `projective line modules'. It would be interesting to try
connect this with the braided-geometrical picture developed above.
\subsection{Super-Lie Algebras}
A super-Lie algebra is a ${\Bbb Z}_2$-graded or `super' vector space $g$ with a
degree-preserving map $[\ ,\ ]:g\mathop{\otimes} g\to g$ obeying the axiom of
graded-antisymmetry and the graded-Jacobi identity:
\[ [\xi,\eta]=(-1)^{|\xi||\eta|+1}[\xi,\eta],\quad
[[\xi,\eta],\zeta]+[\eta,[\xi,\zeta]](-1)^{|\xi||\eta|}=[\xi,[\eta,\zeta]]\]
on homogeneous elements $\xi,\eta,\zeta\in g$. One can view any super-Lie
algebra as a braided-Lie algebra $\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}=k\oplus g$ in the category of super
vector spaces with braiding given by super-transposition (\ref{supertran}) and
the remaining structure
as in subsection~4.1 in (\ref{ordLie}). The $k$ part of $\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}$ is given degree
zero.
\begin{propos} Let $g$ be a ${\Bbb Z}_2$-graded vector space and $V=k\oplus g$ with
$k$ given degree zero, and $[\ ,\ ]:g\mathop{\otimes} g\to g$ a degree-preserving linear
map. Then
\cmath{ \check{\bf R }(1\mathop{\otimes} 1)=1\mathop{\otimes} 1,\quad {\check{\bf R }} (1\mathop{\otimes}
\xi)=\xi\mathop{\otimes} 1,\quad
{\check{\bf R }} (\xi\mathop{\otimes} 1)=1\mathop{\otimes}\xi\\
{\check{\bf R }} (\xi\mathop{\otimes}\eta)=(-1)^{|\xi||\eta|}\eta\mathop{\otimes}\xi+[\xi,\eta]\mathop{\otimes}
1,\qquad\qquad\quad \forall
\xi,\eta\in g.}
obeys the braid relations {\em iff} $[\ ,\ ]:g\mathop{\otimes} g\to g$ obeys the
graded-Jacobi
identity. Moreover, it has minimal polynomial (\ref{minpoly}) {\em iff} $[\ ,\
]$ is graded-antisymmetric and non-zero.
\end{propos}
\goodbreak\noindent{\bf Proof\quad} That $\check{\bf R }$ obeys the braid relations follows from Theorem~2.1
in the category of ${\Bbb Z}_2$-graded vector spaces where $V$ is viewed as a
braided-Lie algebra in this category as explained. Conversely, an explicit
computation along the same lines as the proof of Proposition~4.1 gives that the
braid relations force $[\ ,\ ]$ to obey the graded Jacobi identity. Similarly
for the minimal polynomial by explicit computation. {\ $\lform$}\bigskip
The braided-enveloping algebra $U(\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}})$ in this case is a homogenised
super-bialgebra version of the enveloping super-Hopf algebra $U(g)$. It is the
Zamolodchikov or quantum plane algebra for the matrix ${\bf R}$ corresponding
to $\check{\bf R }$ in this case.
This generalisation of the preceding subsection is immediate because the
category is not truly braided, i.e. one has $\Psi^2={\rm id}$ and hence all the
properties familiar in the category of vector spaces. The same applies if we
work in any symmetric monoidal category of vector spaces with $\Psi^2={\rm id}$ and
the same form of coproduct $\Delta$ and $[\ ,\ ]$ on an object $\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}=k\mathop{\otimes} g$.
The analogue of Proposition~4.2 recovers the obvious axioms of a $\Psi$-Lie
algebra as studied, for example, in \cite{Gur:yan}. We still find
(\ref{minpoly}) for $\check{\bf R}$ even for this more general case.
\subsection{Matrix braided-Lie algebras}
The data we need is a matrix solution $R\in M_n\mathop{\otimes} M_n$ of the QYBE
which is bi-invertible. The `second inverse' $\widetilde R$ which we suppose
here is
characterised by
\[\widetilde{R}^i{}_a{}^b{}_l
R^a{}_j{}^k{}_b=\delta^i{}_j\delta^k{}_l
=R^i{}_a{}^b{}_l\widetilde{R}^a{}_j{}^k{}_b.\]
We assume summation of repeated indices throughout this section. These
$R,\widetilde{R}$ generate a braided monoidal category $\CC$ and this has an
associated braided group ${\rm Aut}(\CC)$\cite{Ma:bra}\cite{Ma:bg} which has in
turn, a braided-Lie algebra $\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}$. Explicitly\cite{Ma:skl}\cite{Ma:lie},
\eqn{L(R)-coalg}{\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}=k^{n^2}=\{u^i{}_j\},\quad \Delta u^i{}_j=u^i{}_k\mathop{\otimes}
u^k{}_j,\quad {\epsilon}
u^i{}_j=\delta^i{}_j.}
\eqn{B(R)-stat}{ \Psi(u_J\mathop{\otimes} u_L)= u_K\mathop{\otimes} u_I \Psi^I{}_J{}^K{}_L;\quad
\Psi^I{}_J{}^K{}_L=R^{j_0}{}_a{}^d{}_{k_0}
R^{-1}{}^a{}_{i_0}{}^{k_1}{}_b
R^{i_1}{}_c{}^b{}_{l_1} {\widetilde R}^c{}_{j_1}{}^{l_0}{}_d}
\eqn{L(R)}{{}[u_I,u_J]=u_K
c^K{}_{IJ};\qquad c^K{}_{IJ}=\widetilde{R}^{a}{}_{i_1}{}^{j_0}{}_b
R^{-1}{}^b{}_{k_0}{}^{i_0}{}_c R^{k_1}{}_e{}^c{}_d R^d{}_{a}{}^{e}{}_{j_1} }
where we write $I=(i_0,i_1)$ etc as multi-indices. We changed conventions here
from \cite{Ma:lie} to lower indices for the $\{u_I\}$. There is also a nice
compact notation used in physics where
subscripts refer to the positions in a matrix tensor product (as in the QYBE
above). In this notation,
\cmath{ \Delta {\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}={\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}\mathop{\otimes}{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p},\ {\epsilon}{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}={\rm id}\\
\Psi(R_{12}^{-1}{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_1\mathop{\otimes} R_{12}{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_2)={\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_2 R_{12}^{-1}{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_1
R_{12},\quad R_{21}[{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_1,R_{12}{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_2]={\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_2 R_{21}R_{12}.}
\begin{corol} Let $R\in M_n\mathop{\otimes} M_n$ be a bi-invertible solution of the QYBE
and $\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}(R)$ its associated matrix braided-Lie algebra. Then the associated
canonical braiding from Theorem~2.1 is
\[ {\check{\bf R }} (u_J\mathop{\otimes} u_L)= u_K\mathop{\otimes} u_I {\bf R}^I{}_J{}^K{}_L;\quad
{\bf R}^I{}_J{}^K{}_L=R^{-1}{}^{d}{}_{k_0}{}^{j_0}{}_{a}
R^{k_1}{}_{b}{}^{a}{}_{i_0}R^{i_1}{}_c{}^b{}_{l_1} {\widetilde
R}^c{}_{j_1}{}^{l_0}{}_d\]
and ${\bf R}\in M_{n^2}\mathop{\otimes} M_{n^2}$ necessarily obeys the QYBE.
\end{corol}
\goodbreak\noindent{\bf Proof\quad} We compute the canonical braiding for the matrix braided-Lie algebra
above. In fact, the necessary computation was done already in the proof of
\cite[Prop. 5.2]{Ma:lie} in the course of computing the relations of $U(\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}})$.
We need only the matrix form of $\Delta$ and the formulae for
$\Psi=\epsfbox{braid.eps},[\ ,\ ]=\epsfbox{prodfrag.eps}$ in
(\ref{L(R)-coalg})--(\ref{L(R)}). Hence from Theorem~2.1 we conclude that
$\check{\bf R }$ obeys the QYBE too. {\ $\lform$}\bigskip
The braided enveloping algebra $U(\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}})$ for this class was computed and
identified in \cite[Prop. 5.2]{Ma:lie} as the braided-bialgebra of $B(R)$ of
`braided matrices' as introduced in \cite{Ma:exa}. This is the associative
algebra generated by $1$ and ${\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}=\{u^i{}_j\}$
with the braided-commutativity relations (\ref{bracom}) which are now
\eqn{B(R)}{u_Ju_L= u_Ku_I {\bf R}^I{}_J{}^K{}_L,\quad {\rm i.e.}\quad
R_{21}{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_1R_{12}{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_2= {\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_2 R_{21} {\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_1 R_{12}}
where the second puts two of the $R$'s to the left and uses the matrix
notation. The coproduct $\Delta {\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}={\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}\mathop{\otimes} {\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}$ extends as an algebra
homomorphism $B(R)\to B(R)\und\mathop{\otimes} B(R)$ to the braided tensor product algebra
determined by $\Psi$, i.e. according to the axiom on the left in
(\ref{hopfax}). Note that the motivation in \cite{Ma:exa} for $B(R)$ was as a
braided-version of quantum or super matrices, with braid statistics $\Psi$,
i.e. the generators are to be regarded as, by definition, the
braided-commutative ring of co-ordinate functions
on a braided space. Hence it is remarkable that this $B(R)$ is also the
enveloping algebra of a braided-Lie algebra. We obtain in the corollary a new
and conceptual proof that the matrix $\bf R$ that describes its relations
indeed obeys the QYBE.
This class of examples generalises those of Section~4.1 for ordinary Lie
algebras and Section~4.2 for super-Lie algebras, as well as including the case
of Lie algebras defined in an obvious way relative to any background braiding
where $\Psi^2={\rm id}$. The way to obtain these from the notion of braided-Lie
algebras is explained in \cite{Ma:lie}. One uses $\bar\chi={{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}-{\rm id}\over
\hbar}$
along with $1$ as generators of $U(\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}})$ in place of $1$ and ${\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}$, where $R$
is parametrised in such a way that $R_{21}R=O(\hbar)$. The standard R-matrices
associated to semisimple Lie algebras $g$ in \cite{FRT:lie} then give
deformations as braided-bialgebras of their homogenised enveloping algebras
$\widetilde U(g)$ from this point if view.
Unfortunately, the simplest non-trivial example of a matrix braided-Lie algebra
has to be four-dimensional. We mention the standard one from \cite{Ma:lie},
namely the braided-Lie algebra $\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}=gl_{2,q}$ with basis
\[ gl_{2,q}=\{\pmatrix{a& b\cr c& d}\},\quad t=q^{-1}a+qd,\quad x={b+c\over
2},\quad y={b-c\over 2i},\quad z=d-a\]
if we work over ${\Bbb C}$. The braided-Lie bracket is obtained from (\ref{L(R)})
with the standard $sl_2$ R-matrix, and given explicitly in \cite[Example
5.5]{Ma:lie}. The braided-enveloping algebra here in terms of the $\chi$
variables is a deformation of $U(gl_2)$ or from another point of view, of
$\widetilde{U(sl_2)}$.
The canonical braiding $\check{\bf R }$ from Theorem~2.1 for this example is
the braided-commutativity relations for the algebra of $2\times 2$ braided
matrices\cite{Ma:exa} and with the braided-determinant
\[ {\rm BDET}\pmatrix{a& b\cr c& d}={q^2\over(q^2+1)^2}t^2-q^2x^2-q^2 y^2-
{(q^4+1)q^2\over 2(q^2+1)^2}z^2+\left({q^2-1\over q^2+1}\right)^2{q\over 2}
tz\]
it can be viewed as a braided $q$-deformation of the algebra of functions on
Minkowski space with its Lorentzian metric\cite{Ma:mec}\cite{Mey:new}. These
$t,x,y,z$ are the non-commutative spacetime co-ordinates. The algebra here also
agrees with the proposal for $q$-Minkowski space based on spinors in the
approach \cite{CWSSW:lor}\cite{OSWZ:def}.
We note that the FRT bialgebra $A({\bf R})$ associated to this canonical
braiding has a Hopf algebra quotient $SO_q(1,3)$, the $q$-Lorentz group in the
interpretation above. On the other hand, this is also closely related to the
dual of the quantum double of $U_q(su_2)$. These points are described in detail
elsewhere.
The braided-enveloping algebra here of $2\times 2$ braided matrices is also
isomorphic to a degenerate form of the 4-dimensional Sklyanin algebra as shown
in \cite{Ma:skl}, so the latter has the R-matrix form (\ref{B(R)}). More
recently, some remarkable homological properties of braided-matrix algebras
have been found in \cite{LeB:hom}.
\note{\align{&&[h,x_+]=(q^{-2}+1)q^{-2}x_+=-q^{-2}[x_+,h]\\
&&[h,x_-]=-(q^{-2}+1)x_-=-q^{2}[x_-,h]\\
&&[x_+,x_-]= q^{-2}h=-[x_-,x_+]\\
&&[h,h]=(q^{-4}-1)h, \quad [\lambda,\cases{h\cr x_+\cr x_-}]=(q^3+q^{-3}\over
q+q^{-1})\cases{h\cr x_+\cr x_-}}
and zero for the rest ***. The braiding $\Psi=\epsfbox{braid.eps}$ is
\cmath{ \Psi(\lambda\mathop{\otimes} x)=x\mathop{\otimes}\lambda,\quad \Psi(x\mathop{\otimes}\gamma)=\gamma\mathop{\otimes}
x,\ \forall x\in\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}},\quad\Psi(h\mathop{\otimes} x_-)=x_-\mathop{\otimes} h, \quad \Psi(x_+\mathop{\otimes}
h)=h\mathop{\otimes} x_+ \\
\Psi(h\mathop{\otimes} h)=h\mathop{\otimes} h +(1-1^2)x_-\mathop{\otimes} x_+ ,\quad \Psi(x_-\mathop{\otimes}
x_-)=q^2x_-\mathop{\otimes} x_- ,\quad \Psi(x_+\mathop{\otimes} x_+)=q^2x_+\mathop{\otimes} x_+ ,\quad
\Psi(x_+\mathop{\otimes} x_-)=q^{-2}x_-\mathop{\otimes} x_+ \\
\Psi(h\mathop{\otimes} x_+)=x_+\mathop{\otimes} h +h\mathop{\otimes} x_+ (q^2-q^{-2}) ,\quad \Psi(x_-\mathop{\otimes}
h)=h\mathop{\otimes} x_- +x_-\mathop{\otimes} h (q^2-q^{-2})\\
\Psi(x_-\mathop{\otimes} x_+)=q^{-2}x_+\mathop{\otimes} x_- +(1+q^2)(1-q^{-2})^2x_-\mathop{\otimes} x_+
-(1-q^{-2})h\mathop{\otimes} h. }
One says that $\lambda$ is `bosonic' in the sense that its braiding with all
elements is trivial. Finally, the coalgebra structure on $\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}$ is ${\epsilon} h={\epsilon}
x_\pm=0$, ${\epsilon}\lambda=1$ and
\cmath{\Delta h=q^2(h\mathop{\otimes} 1+1\mathop{\otimes} h)+(q^2-1)\left(x_-\mathop{\otimes} x_+-x_+\mathop{\otimes}
x_-+h\mathop{\otimes}\lambda+\lambda\mathop{\otimes} h+{(1-q^{-4})\over(1+q^{-2})^2}h\mathop{\otimes} h\right)\\
\Delta \gamma=\gamma\mathop{\otimes} 1+1\mathop{\otimes} \gamma +(q^2-1)\left(q^{-2}x_-\mathop{\otimes}
x_++x_+\mathop{\otimes} x_-+{h\mathop{\otimes} h+\gamma\mathop{\otimes} \gamma\over 1+q^{-2}}\right)\\
\Delta x_-=q^2(x_-\mathop{\otimes} 1+1\mathop{\otimes} x_-)+(q^2-1)\left(x_-\mathop{\otimes}\lambda+\lambda\mathop{\otimes}
x_-+{h\mathop{\otimes} x_--q^{-2}x_-\mathop{\otimes} h\over (q^{-2}+1)}\right)\\
\Delta x_+=q^2(x_+\mathop{\otimes} 1+1\mathop{\otimes} x_+)+(q^2-1)\left(x_+\mathop{\otimes}\lambda+\lambda\mathop{\otimes}
x_+ \mathop{\otimes} h -{q^{-2}\over (q^{-2}+1)}h\mathop{\otimes} x_+\right)}}
\note{h'b=bh', \quad c'h=hc'\\
h'h=hh'+(1-1^2)bc',\quad b'b=q^2bb',\quad c'c=q^2cc',\quad c'b=q^{-2}bc'\\
h'c=ch'+hc'(q^2-q^{-2}) ,\quad b'h=hb'+bh'(q^2-q^{-2})\\
b'c=q^{-2}cb'+(1+q^2)(1-q^{-2})^2bc'-(1-q^{-2})hh'
\cmath{\Delta h=h\mathop{\otimes} 1+1\mathop{\otimes} h+(q^2-1)\left(x_-\mathop{\otimes} x_+-x_+\mathop{\otimes}
x_-+{h\mathop{\otimes}\gamma+\gamma\mathop{\otimes} h\over
1+q^{-2}}+{(1-q^{-4})\over(1+q^{-2})^2}h\mathop{\otimes} h\right)\\
\\
\Delta x_-=x_-\mathop{\otimes} 1+1\mathop{\otimes} x_-+{(q^2-1)\over
(q^{-2}+1)}\left(x_-\mathop{\otimes}\gamma+\gamma\mathop{\otimes} x_-+h\mathop{\otimes} x_--q^{-2}x_-\mathop{\otimes}
h\right)\\
\Delta x_+=x_+\mathop{\otimes} 1+1\mathop{\otimes} x_++{(q^2-1)\over
(q^{-2}+1)}\left(x_+\mathop{\otimes}\gamma+\gamma\mathop{\otimes} x_+ \mathop{\otimes} h -q^{-2}h\mathop{\otimes}
x_+\right)}}
\subsection{Finite Groups and Racks}
A {\em rack} is a set $X$ and a map $X\times X\to X$ denoted $x\times y\mapsto
{}^xy$ obeying the `rack-identity'
\[ {}^{({}^{\scriptstyle x}y)}({}^x z)={}^x({}^yz),\qquad \forall x,y,z\in X.\]
One usually adds to this that the map ${}^x(\ )$ is bijective for each $x$, but
we do insist on this here. One may also have conventions in which the notation
is $y^x$ rather than ${}^x y$. Such objects have a long history and some
applications in algebraic topology\cite{FenRou:rac}. It is easy to see that
every rack provides an example of a braided-Lie algebra if we take as our
category $\CC$ as the category of sets, with tensor product provided by the
direct product of sets, and with the usual permutation map as $\Psi$. This is a
symmetric monoidal category rather than a truly braided one. We just take
\[ [x,y]={}^xy,\quad \Delta x=x\times x\]
and note that the axiom (L1) in (\ref{Lie}) then becomes the rack identity
above, while the others are empty. At the level of sets the braiding from
Theorem~2.1 is
\[ \check{\bf R }(x\times y)=[x,y]\times x\]
and recovers the braiding associated to a rack in \cite{FenRou:rac}.
For a $k$-linear setting over a field we let $\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}=kX$, the vector space with
basis $X$, and the above definitions extended $k$-linearly. So $\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}}$ is the
coalgebra with basis $X$ and all basis elements grouplike. Then we have a
(trivially braided) braided-Lie algebra in the category of vector spaces and
the canonical braiding from Theorem~2.1 is just
\[ \check{\bf R }(x\mathop{\otimes} y)=[x,y]\mathop{\otimes} x.\]
The braided-enveloping algebra $U(\hbox{{$\cal L$}}} \def\CH{\hbox{{$\cal H$}})$ consists of the algebra generated by
elements of $X$ modulo the relations $xy=[x,y]x$ for all $x,y$. This is the
bialgebra generated by the rack monoid. Here the rack monoid is the free monoid
generated by symbols from $x,y$ modulo such relations, and is also a classical
construction for racks.
Our examples of matrix braided-Lie algebras in subsection~4.3 are a deformation
of a mixture of some Lie-algebra like elements as in subsection~4.1 and some
rack-like elements. In the $gl_{2,q}$ example mentioned there, the rack-like
element is proportional to the `time' direction $t$ and the Lie-algebra like
elements are the `space' directions $x,y,z$.
We see that even when the braiding is trivial, the notion of a braided-Lie
algebra in (\ref{Lie}) is still useful. Moreover, it is more general than a
rack because we are free to specify a more general coproduct $X\to X\times X$
than the diagonal map, as long as we obey (L1)--(L3) in the category of sets.
The simplest way to obey (L2) is for $\Delta$ to be cocommutative.
The classic example of a rack is a group with the rack operation
$[x,y]=xyx^{-1}$. The braiding $\check{\bf R }$ in the $k$-linear setting is
then the braiding associated to the quantum double Hopf algebra
$D(X)$\cite{Dri}. The latter is defined for $X$ finite but the rack point of
view is more general and works for any group. The associated 3-manifold
invariants in this case are well-known, see for example \cite{FreYet:bra}. On
the other hand, we have super-racks, etc. just as easily as examples of
braided-Lie algebras, and their associated braidings may prove more
interesting.
\subsection{Ordinary Hopf Algebras}
If $H$ is any Hopf algebra then it acts on itself by the Hopf-algebra adjoint
action ${\rm Ad}_h(g)=\sum h{}_{\scriptscriptstyle(1)} g S h{}_{\scriptscriptstyle(2)}$ where we use the notation $\Delta h=\sum
h{}_{\scriptscriptstyle(1)}\mathop{\otimes} h{}_{\scriptscriptstyle(2)}$ of \cite{Swe:hop} for the coproduct. Theorem~3.1 reduces for
ordinary Hopf algebras (with trivial braiding) to
\eqn{brahopf}{{\check{\bf R }} (h\mathop{\otimes} g)=\sum {\rm Ad}_{h{}_{\scriptscriptstyle(1)}}(g)\mathop{\otimes}
h{}_{\scriptscriptstyle(2)},\qquad\forall h,g\in H.}
One can easily verify in a couple of lines that $\check{\bf R }$ obeys the
braid relations. This was perhaps first explicitly remarked in \cite{Wor:sol}.
The case of $H$ a group algebra clearly reduces us to the rack braiding in
subsection~4.4.
Once again, this braiding can be viewed as originating in Drinfeld's quantum
double construction $D(H)$\cite{Dri}, this time applied to a general
finite-dimensional Hopf algebra $H$. Drinfeld introduced $D(H)$ as a
quasitriangular Hopf algebra defined by generators and relations in a basis.
Here the quasitriangular structure $\hbox{{$\cal R$}}} \def\CD{\hbox{{$\cal D$}}\in D(H)\mathop{\otimes} D(H)$ obeys Drinfeld's
axioms which are such as to ensure that its image in any representation obeys
the QYBE. We introduced a form of this in \cite{Ma:phy} built explicitly on the
vector space $H^*\mathop{\otimes} H$ with product
\[ (a\mathop{\otimes} h)(b\mathop{\otimes} g)=\sum \<Sh{}_{\scriptscriptstyle(1)},b{}_{\scriptscriptstyle(1)}\>b{}_{\scriptscriptstyle(2)} a\mathop{\otimes} h{}_{\scriptscriptstyle(2)} g \<
h{}_{\scriptscriptstyle(3)},b{}_{\scriptscriptstyle(3)}\>,\qquad\forall a,b\in H^*,\ h,g\in H\]
and tensor product unit and coalgebra. In writing this we switch also to
conventions with $H$ and $H^{*\rm op}$ (the opposite algebra) as sub-hopf
algebras, rather than Drinfeld's original conventions with $H,H^{*\rm cop}$
(the opposite coalgebra). Let $\{e_a\}$ be a basis of $H$ and $\{f^a\}$ a dual
basis then
\[ \hbox{{$\cal R$}}} \def\CD{\hbox{{$\cal D$}}=\sum_a(f^a\mathop{\otimes} 1)\mathop{\otimes} (1\mathop{\otimes} e_a)\]
is Drinfeld's quasitriangular structure in these conventions.
\begin{propos} $D(H)$ acts on $H$ by
\[ (1\mathop{\otimes} h){\triangleright}}\def\ra{{\triangleleft} g=\sum h{}_{\scriptscriptstyle(1)} g Sh{}_{\scriptscriptstyle(2)},\quad (a\mathop{\otimes} 1){\triangleright}}\def\ra{{\triangleleft} g=\sum \<a, h{}_{\scriptscriptstyle(1)}\>h{}_{\scriptscriptstyle(2)}\]
and the associated braiding is (\ref{brahopf}).
\end{propos}
\goodbreak\noindent{\bf Proof\quad} It is easy to see that this defines an action of $D(H)$ (this is
modelled on quantum mechanics and could be called the `Schroedinger
representation' of the quantum double). Then the action of $\hbox{{$\cal R$}}} \def\CD{\hbox{{$\cal D$}}$ is $\hbox{{$\cal R$}}} \def\CD{\hbox{{$\cal D$}}{\triangleright}}\def\ra{{\triangleleft}
(h\mathop{\otimes} g)=\sum_a (f^a\mathop{\otimes} 1){\triangleright}}\def\ra{{\triangleleft} h\mathop{\otimes} (1\mathop{\otimes} e_a){\triangleright}}\def\ra{{\triangleleft} g=\sum h{}_{\scriptscriptstyle(2)}\mathop{\otimes}
(1\mathop{\otimes} h{}_{\scriptscriptstyle(1)}){\triangleright}}\def\ra{{\triangleleft} g=\sum h{}_{\scriptscriptstyle(2)}\mathop{\otimes} {\rm Ad}_{h{}_{\scriptscriptstyle(1)}}(g)$ giving exactly (\ref{brahopf})
for the corresponding $\check{\bf R }$. {\ $\lform$}\bigskip
Thus the braiding from Theorem~3.1 does not give anything genuinely new for an
ordinary Hopf algebra. It is slightly more general than the braiding coming
from the quantum double in that it does not require $H$ to be finite
dimensional, but this issue too can be dealt with in other ways\cite{Dri}. On
the other hand, it is still a useful observation, as is the fact which is
obvious from (\ref{brahopf}) that $\cdot\circ \check{\bf R }=\cdot$ holds in
$H$. See for example \cite{AndDev:ext} where such an observation recently
proved very useful for some Hopf algebraic constructions.
\subsection{Super Hopf Algebras}
To obtain something new from Theorem~3.1 we can consider Hopf algebras in
categories other than the usual one of vector spaces. The simplest setting is
that of ${\Bbb Z}_2$-graded or super Hopf algebras. These are defined in the obvious
way with all maps degree-preserving, where the degree of a term in a tensor
product is the sum of the degrees in a homogeneous decomposition. Plenty of
super-Hopf algebras are known, not least in algebraic
topology\cite{MilMor:str}.
The braiding in this case is
\[ \check{\bf R }(h\mathop{\otimes} g)=\sum {\rm Ad}_{h{}_{\scriptscriptstyle(1)}}(g)\mathop{\otimes} (-1)^{|h{}_{\scriptscriptstyle(2)}||g|} h{}_{\scriptscriptstyle(2)};\quad
{\rm Ad}_{h}(g)=\sum h{}_{\scriptscriptstyle(1)} g Sh{}_{\scriptscriptstyle(2)} (-1)^{|h{}_{\scriptscriptstyle(2)}||g|}\]
where it is assumed that all tensor product elements are decomposed
homogeneously.
\subsection{The Braided Line $k[x]$}
The previous subsection is still not a truly braided example of Theorem~3.1.
Truly braided-Hopf algebras were first introduced and studied by the author
through a number of papers. In this subsection we compute Theorem~3.1 for the
simplest of these\cite{Ma:any}, where the braiding is still a factor but not
necessarily $\pm 1$ as it was in subsection~4.6.
The braided-line $k[x]$ as an algebra is nothing other than the polynomials in
one variable. However, we regard it as an algebra in the category of
${\Bbb Z}$-graded vector spaces. As such, it has a braiding
\eqn{anybraid}{ \Psi(x^m\mathop{\otimes} x^n)=q^{nm}x^n\mathop{\otimes} x^m}
where $q\ne 0,1$ is a fixed but otherwise arbitrary element of $k$. The ideas
here are from \cite{Ma:any}. We define $\Delta x=x\mathop{\otimes} 1+1\mathop{\otimes} x$, ${\epsilon} x=0$
and extend to products as a braided-Hopf algebra according to (\ref{hopfax}).
It is easy to see that
\eqn{anydelta}{ \Delta x^m=\sum_{r=0}^m [{m\atop r};q] x^r\mathop{\otimes} x^{m-r},\quad
[{m\atop r};q]={[m;q]!\over [r;q]![m-r;q]!},\quad [m;q]={q^m-1\over q-1}.}
The $q$-integers and $q$-binomial coefficients here are
well-known\cite{And:ser} but we use them in a novel way as defining a
braided-Hopf algebra structure\cite{Ma:fre}. Using the lemma that $S$ is a
braided-antialgebra map we have also that
\eqn{anyS}{ S x^m=q^{m(m-1)\over 2}(-x)^m}
to complete the braided-Hopf algebra structure. Finally, we note that this
braided-Hopf algebra approach to $q$-analysis derives the standard
$q$-derivative
\eqn{anydif}{ ({\partial}_q f)(x)={f(qx)-f(x)\over x(q-1)},\quad {\partial}_q
x^m=[m;q]x^{m-1}}
as infinitesimal translations\cite{Ma:fre}, a point of view which generalises
at once to $n$-dimensional quantum plane algebras.
\begin{lemma} The braided adjoint action of $k[x]$ on itself as in Theorem~3.1
is
\[ {\rm Ad}_f(g)(x)=f(x^2(1-q){\partial}_q)g(x),\qquad \forall f,g\in k[x]\]
\end{lemma}
\goodbreak\noindent{\bf Proof\quad} We know from the general theory of braided-Hopf algebras\cite{Ma:introp}
that the braided-adjoint action is an action and also that it acts on itself as
a braided module-algebra. The latter condition means in the present case that
it acts as a braided-derivation
\[ {\rm Ad}_x(fg)={\rm Ad}_x(f)g+\cdot\circ \Psi({\rm Ad}_x\mathop{\otimes}
f)g={\rm Ad}_x(f)g+L_q(f){\rm Ad}_x(g);\quad L_q(f)(x)=f(qx).\]
Since ${\rm Ad}_x(x)=xx-qxx=(1-q)x^2$ we deduce from this $q$-derivation property of
the adjoint action that ${\rm Ad}_x(x^n)=x^{n+1}(1-q^n)=x^2(1-q){\partial}_q x^n$. Since
${\rm Ad}_x$ is an action, we deduce the result stated. Explicitly, this has on
monomials the form
\[ {\rm Ad}_{x^m}(x^n)=(1-q^n)\cdots (1-q^{n+m-1})x^{n+m}=(1-q)^m{[n+m-1;q]!\over
[n-1;q]!}x^{n+m}.\]
{\ $\lform$}\bigskip
We note in passing that our derivation here depends strongly on the properties
of ${\rm Ad}$ proven in \cite{Ma:lin}\cite{Ma:exa} using the same novel diagrammatic
techniques as in Figure~2. If we try to compute it directly from
(\ref{anydelta})--(\ref{anyS}) on monomials then we derive the novel
$q$-identity
\[ \sum_{r=0}^m [{m\atop r};q] q^{r(r-1)\over 2} (-1)^r
q^{rn}=(1-q)^m{[n+m-1;q]!\over [n-1;q]!}.\]
This is the content of the lemma from the point of view of $q$-analysis.
\begin{corol} The braiding $\check{\bf R }:k[x]\mathop{\otimes} k[x]\to k[x]\mathop{\otimes} k[x]$
obtained from Theorem~3.1 is
\[ \check{\bf R }(x^m\mathop{\otimes} x^n)=\sum_{r=0}^m [{m\atop r};q] q^{n(m-r)}(1-q)^r
{[n+r-1;q]!\over [n-1;q]!} x^{n+r}\mathop{\otimes} x^{m-r}.\]
\end{corol}
\goodbreak\noindent{\bf Proof\quad} We compute from Theorem~3.1 in our case. Thus from the formula above for
$\Delta$ we have
\[ \check{\bf R }(x^m\mathop{\otimes} x^n)=\sum_{r=0}^m [{m\atop r};q] {\rm Ad}_{x^r}(x^n)\mathop{\otimes}
x^{m-r} q^{n(m-r)}\]
and putting in the form of ${\rm Ad}$ computed in Lemma~4.5 gives the result stated.
{\ $\lform$}\bigskip
We now provide a braided-geometrical picture of this $\check{\bf R }$ as an
operator on polynomials in two variables. Thus we distinguish the two copies of
$k[x]$ in both the input and output, so that $\check{\bf R }:k[y]\mathop{\otimes} k[x]\to
k[x]\mathop{\otimes} k[y]$ say. Next we consider these variables to be non-commuting with
the quantum-plane relations, i.e.
\[ \check{\bf R }:k[x,y;q]\to k[x,y;q],\quad k[x,y;q]={k\<x,y\>\over yx-qxy}.\]
This is a purely notational device because this algebra has a basis
$\{y^nx^m\}$ so as a linear space can be identified with $k[y]\mathop{\otimes} k[x]$, but
also has a basis $\{x^my^n\}$ and so can be identified with the linear space
$k[x]\mathop{\otimes} k[y]$ as well.
\begin{propos} $\check{\bf R }$ in the form $k[x,y;q]\to k[x,y;q]$ is the
operator
\[ \check{\bf R }=e_q^{x^2(1-q){\partial}_{q,x}|{\partial}_{q,y}}=\sum_{m=0}^\infty
{(x^2(1-q){\partial}_{q,x})^m{\partial}_{q,y}^m\over [m;q]!}\]
where the $|$ denotes that the $q$-exponential is to be understood as ordered
in the form shown and ${\partial}_{q,x}$ and ${\partial}_{q,y}$ are as in (\ref{anydif})
acting on $x,y$ respectively.
\end{propos}
\goodbreak\noindent{\bf Proof\quad} Note that the expression is a well-defined operator since on any
polynomial the power-series always terminates. The $q$-exponential is the
standard one except that we have adopted the ordering convention stated. This
is such that we have
\eqn{anyTay}{ e_q^{x{\partial}_{q,y}}f(y)=(\Delta f)(x,y)=f(x+y)}
where we recall that $\Delta$ is an algebra homomorphism to the braided tensor
product algebra which we identify as $k[x]\und\mathop{\otimes} k[y]=k[x,y;q]$ for the
braiding (\ref{anybraid}). This (\ref{anyTay}) describes a `braided-Taylors
theorem' as explained in \cite{Ma:fre}, where it is also generalised to
$n$-dimensions. Applying the braided-adjoint representation
$x\mapsto{\rm Ad}_x=(1-q)x^2{\partial}_{q,x}$ to both sides of (\ref{anyTay}) allows us to
recompute $\check{\bf R }$ from Theorem~3.1 as
\[ \check{\bf R
}(f(y)g(x))=f(x^2(1-q){\partial}_{q,x}+y)g(x)
=e_q^{x^2(1-q){\partial}_{q,x}|{\partial}_{q,y}}f(y)g(x)\]
which is the form stated. Here $f,g$ are arbitrary polynomials and a general
polynomial in $x,y$ can be written as a linear combination of such products.
{\ $\lform$}\bigskip
In this form, our Yang-Baxter operator $\check{\bf R }$ has some similarities
with the quasitriangular structure $\hbox{{$\cal R$}}} \def\CD{\hbox{{$\cal D$}}$ of $U_q(sl_2)$. To see this we note
that classically one can embed $sl_2$ inside the Witt algebra of `vector
fields' on $k[x]$ by
\[ L_0=x{d\over dx},\quad L_1=x^2{d\over dx},\quad L_{-1}={d\over dx}\]
so $\check{\bf R }$ resembles the factor $e_q^{(1-q)L_{1}\mathop{\otimes} L_{-1}}$
ocurring in the formula for $\hbox{{$\cal R$}}} \def\CD{\hbox{{$\cal D$}}$ in \cite{KirRes:rep}, cf\cite{Dri} and
elsewhere. On the other hand, there is no Gaussian factor $q^{L_0\mathop{\otimes} L_0}$ as
to be found there.
If one works over ${\Bbb C}[[\hbar]]$ rather than over a field and sets
$q=e^{\hbar}$ then $\check{\bf R }$ has a an expansion
\[ \check{\bf R }=P\circ({\rm id}+\hbar r+O(\hbar^2));\quad r=x{d\over dx}\mathop{\otimes}
x{d\over dx}-{d\over dx}\mathop{\otimes} x^2{d\over dx}\]
which $r$ is necessarily an operator realisation $k[x]\mathop{\otimes} k[x]\to k[x]\mathop{\otimes}
k[x]$ of the Classical Yang-Baxter equation (CYBE). Here $P$ is permutation. To
obtain this formula one can work from Corollary~4.6 or else from
Proposition~4.7 provided one remembers the contribution (the first term in $r$)
coming from the fact that the output of $\check{\bf R }$ has to be viewed in
$k[x]\mathop{\otimes} k[y]$ while its input is viewed in $k[y]\mathop{\otimes} k[x]$. In this case
$r$ is indeed the image in the Witt algebra of the Drinfeld-Jimbo
solution\cite{Dri}\cite{Jim:dif} of the CYBE on $sl_2$ when represented as
vector fields on $k[x]$. The connection with theory of quantum groups is
provided by the `bosonisation theorem' introduced in \cite{Ma:bos}. In the
present case this turns constructions on $k[x]$ into equivalent ones on the
Hopf algebra $U_q(b_+)$ in \cite{Dri}.
On the other hand, we have obtained this $\check{\bf R }$ starting from nothing
other than $k[x]$ regarded as a braided-Hopf algebra (the braided-line) and
Theorem~3.1. Moreover, we can suppose that $q\in k^*$ is a root of unity and
proceed with the same calculations. More precisely, we can repeat the above
calculations for the braided-Hopf algebra $U_n(k)=k[x]/x^n$ introduced in
\cite{Ma:any}, where $q$ is a primitive $n$'th root of $1$. This `anyonic line'
braided-Hopf algebra is $n$-dimensional and hence $\check{\bf R }$ corresponds
to a matrix solution of the QYBE in $M_n\mathop{\otimes} M_n$. An elementary computation
gives for example,
\begin{example} The braiding from Theorem~3.1 applied to the anyonic line for
$n=3$ has minimal polynomial
\[ (\check{\bf R }^2-1)^2(\check{\bf R }-q)=0\]
and corresponding matrix solution of the QYBE
\[ {\bf R}=\pmatrix{1&0&0 &0&0&0 &0&0&0\cr
0&1&0 &0&0&0 &0&0&0\cr
0&0&1 &0&1-q&0 &0&0&0\cr
0&0&0 &1&0&0 &0&0&0\cr
0&0&0 &0&q&0 &0&0&0\cr
0&0&0 &0&0&q^2 &0&$q$-1&0\cr
0&0&0 &0&0&0 &1&0&0\cr
0&0&0 &0&0&0 &0&q^2&0\cr
0&0&0 &0&0&0 &0&0&q}\]
where $q^3=1$ is a primitive root and the basis is $\{1\mathop{\otimes} 1,1\mathop{\otimes}
x,\cdots,x^2\mathop{\otimes} x^2\}$.
\end{example}
This braided-line which we have studied above is only the very simplest example
of a braided-Hopf algebra. The next simplest is probably the quantum plane
$k[x,y;q]$ this time regarded itself as a braided-Hopf algebra with braiding
provided by the standard $sl_2$ R-matrix corresponding to the Jones knot
polynomial\cite{Ma:poi}. Explicitly,
\cmath{\Psi(x\mathop{\otimes} x)=q^2 x\mathop{\otimes} x,\ \Psi(x\mathop{\otimes} y)=q y\mathop{\otimes} x,\ \Psi(y\mathop{\otimes}
y)=q^2 y\mathop{\otimes} y\\
\Psi(y\mathop{\otimes} x)=q x\mathop{\otimes} y+(q^2-1)y\mathop{\otimes} x.}
Using this, one obtains the braided adjoint action on generators as
\[ {\rm Ad}_x(x)=x^2(1-q^2),\ {\rm Ad}_x(y)=(1-q^2)xy,\ {\rm Ad}_y(x)=yx(1-q^2),\
{\rm Ad}_x(y)=y^2(1-q^2).\]
Extending this to higher products along the lines above, one obtains the
braided-vector fields for the braided-adjoint action as
\[ {\rm Ad}_x=(1-q^2)x(x{\partial}_{q,x}+y{\partial}_{q,y}),\quad
{\rm Ad}_y=(1-q^2)y(x{\partial}_{q,x}+y{\partial}_{q,y})\]
where ${\partial}_{q,x}$ and ${\partial}_{q,y}$ are the partial derivatives on the quantum
plane\cite{WesZum:cov} in the form obtained by an infinitesimal translation in
\cite{Ma:fre}.
{}From this one can compute the braiding $\check{\bf R }$ from Theorem~3.1 as
represented by $q$-deformed vector fields on a quantum plane. We have not
found an explicit exponential formula for it along the lines of
Proposition~4.7.
This class of examples generalises further to any quantum plane algebra of the
R-matrix type. The necessary braided-Hopf algebra structure, braided
differential calculus and R-binomial theorem are in \cite{Ma:fre}. For example,
the $q$-Minkowski space example with generators $t,x,y,z$ fits into this
setting with additive braided-Hopf algebra structure found in \cite{Mey:new}.
In these cases the appropriate $q$-exponential map needed for the analogue of
Proposition~4.7 is not yet known.
In a different direction, we can take the free braided-Hopf algebra
$k\<x_1,\cdots,x_n\>$ with braiding determined by $R$ as a generalisation of
the example $k[x]$ above. In this case the relevant exponential map is provided
in \cite{Ma:fre}.
|
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\section{Introduction}
Despite huge efforts of researchers and industry put into identifying vulnerable
software,
many software systems still suffer from various security weaknesses.
The concept of a code property graph (CPG) \cite{yamaguchi2014modeling} has been
introduced to simplify the identification of vulnerabilities and bugs in the
source code of programs. A CPG is a super-graph covering properties of an
abstract syntax tree (AST), a control flow graph and data flow graphs, among
others, thus containing all information relevant for a security analysis. The
CPG enables its user to identify vulnerabilities or bugs by performing reusable
graph queries. This perk led to a widespread adaption of the technique with
several implementations \cite{xiaomeng2018cpgva,yamaguchi2014modeling,Graft,Plume,Joern,CPG,banse2021cloud,weiss2022languageindependent}.
Even if the graphs mimic the source code with a minimal loss of information, the
graph provides an abstraction of the actual code. This abstraction is
suitable to support a language-agnostic analysis of software.
Unfortunately, the implementations are limited with respect to the
supported programming languages since each language requires a separate
translation.
As compilers suffer from a similar problem, the use of intermediate
representations (IR) has become popular. The IR abstracts from the programming
language but, in many cases, still contains a significant amount of high-level
information such as the types of variables which is lost in the compiled binary
and can barely be recovered \cite{mantovani2022the}.
The lack of such information can worsen the analysis results. A very
popular IR is LLVM-IR \cite{llvmir} which is part of the LLVM
project. Numerous compiler frontends exist to translate programming languages to
LLVM-IR. E.g., clang \cite{clang} translates the
languages C, C++ and Objective-C to LLVM-IR and has been extended by Apple to
support Swift \cite{swiftClang}. Other frontends exist to support a wide range
of programming languages (e.g., Rust).
While LLVM-IR was designed for compilers, it is also frequently used by
binary lifters. E.g., RetDec \cite{RetDec}, McSema
\cite{mcsema}, llvm-mctoll \cite{mctoll} and reopt \cite{reopt} can
lift a binary to LLVM-IR. Since some lifters
support multiple architectures and types of binary files, this avoids to
implement the translation for different flavors of assembly code or application
binary interfaces.
Recent research \cite{liu2022sp} showed that binary lifting is meanwhile a
stable technique and its produced output is suitable for a security analysis of
the program. Another use-case for analyzing LLVM-IR is to consider the effects
of compiler optimizations during the analysis. As an example, recent research
showed that side channel vulnerabilities introduced by the compiler are still a
major concern of developers of cryptographic libraries \cite{jancar2022sp} and
that the source code and the final binary files can differ significantly
\cite{balakrishnan2010wysinwyx}. Since the LLVM-IR can be emitted after the
optimizations, it can already contain the vulnerabilities or bugs which stem
from the compiler and is therefore an interesting analysis target.
In this paper, we present an approach how to overcome shortcomings of existing
CPG tools by enabling the analysis of LLVM-IR in a code property graph. This
bridges the gap between the analysis of source code written in higher-level
programming languages and the analysis of programs (or dependencies) that may
only exist in binary form. While supporting LLVM-IR in a CPG seems to be
straightforward, several challenges arise from the static single assignment
(SSA) form, the exception handling routine, instructions which do not exist in
high-level programming languages and significantly different syntactic
representations of some concepts in LLVM-IR and other languages. Contrary to
prior work, we do not require to run any LLVM passes beforehand, which helps us
to keep the graph smaller. At the same time, we aim to retrieve as much
high-level information as possible and map the code to high-level concepts
whenever possible. Rather than handling LLVM-IR-specific instructions, e.g.,
\textit{cmpxchg}\footnote{The \textit{cmpxchg} instruction compares a given
argument against a value stored in a memory address. If they are equal, a new
value, specified in a second argument is stored in memory. This is similar to
\texttt{if(*addr == arg0) {*addr = arg1;}} in C/C++.}, only as a generic
function call, we translate the concepts into existing CPG node types that
represent the behavior of a higher-level programming language. This allows us to
re-use existing concepts in queries, such as if-statements or pointer
referencing. Overall, integrating LLVM-IR into a CPG allows to support more
programming languages, to analyze binary files and to validate that a compiler
did not introduce new bugs. It enables that existing analyses queries for source
code can be applied to the LLVM-IR without any modifications. In summary, our
contributions are as follows:
\begin{itemize}
\item We are the first to present a mapping of all LLVM-IR instructions to
existing CPG nodes with full compatibility to the existing structure.
This ensures that existing analyses are fully compatible with the
representation.
\item We show how we can keep the size of the CPG minimal.
\item We are the first to include LLVM-IR's exception handling routines in a CPG.
\item We extended the open source project \textit{cpg}
\cite{weiss2022languageindependent,CPG} to support our concepts.
\end{itemize}
\section{Background}
\label{sec:background}
\subsection{The Code Property Graph}
The \textit{cpg} project \cite{weiss2022languageindependent,CPG} enables a
graph-based representation of
source code of different programming languages. To date, the focus lies in
Java and C/C++ but experimental support for Python, Go and TypeScript is also
available. The goal of the project is to provide a language-agnostic
representation of the source code. This enables a security expert to identify
vulnerabilities or bugs.
Furthermore, the \textit{cpg} library comprises a way to store the graph in
neo4j\footnote{\url{https://neo4j.com/}}, and makes the graph
accessible via a command line interface.
For some cases, the library can also evaluate the value
which can be held by a node. All this allows a security expert to write custom
queries either to the graph database or the in-memory representation of the CPG.
The \textit{cpg} library is designed in a way to allow reusing these
queries among all supported programming languages.
To fulfill this goal, the \textit{cpg} library implements a thorough class hierarchy which
accounts for various types of statements and expressions. The CPG encodes
information such as the class hierarchy of the code under analysis, the control
flow graph, and the call graph in a single graph. The current design
mainly targets object-oriented programming languages. To deal with a possible
lack of some code fragments or errors in the code, the library is resilient to
incomplete, non-compilable and to a certain extent even incorrect code.
\subsection{The LLVM Intermediate Representation}
\noindent\textbf{The Instructions.}~
The LLVM-IR is used as IR of the LLVM project. Its main
purpose lies in providing an abstraction of code
to ease the optimization and analysis of the program in a language-
and architecture-independent way. The LLVM-IR holds values in global variables
(prefixed with \texttt{@}) and local variables (prefixed with \texttt{\%})
both of which can be named or unnamed. The LLVM-IR follows the static single
assignment (SSA) form. Hence, every variable can be written to exactly
once. This limitation does not affect global variables as they are represented
as memory locations and are accessed via store or load operations.
Overall,
the LLVM-IR differentiates between 65
instructions. Of these, 13 are arithmetic operations, 6 are bitwise
operations, and 13 instructions cast types. The remaining instructions call
functions, handle exceptions, load from or store to memory,
manipulate aggregated types or jump to other program locations.
The instructions can be enhanced with metadata to note the calling
convention, optimizations or desired properties of functions and parameters,
among others.
Besides the basic instructions, LLVM-IR contains numerous so-called
``intrinsics''. Those are functions which model certain standard library
functionality, or model frequent actions which have to be represented
differently on different architectures. Some intrinsics repeat or refine basic
instructions, others insert functionality such as the automated memory
management in Objective-C.
The LLVM-IR supports a simple type system and differentiates between a set of
primitive types
and aggregated types such as
structs, arrays and vectors. Additionally, LLVM-IR has a type for
labels (i.e., jump targets), metadata and a so-called token which is used by
certain instructions to transport information. Overall, the type system
resembles C rather than object-oriented programming languages. In fact,
object-oriented concepts are handled by the respective language
frontend in LLVM. The frontend translates the object-oriented properties to
concepts such as VTables for overriding methods, and method
name mangling to support overloaded functions. In the case of
Objective-C, it uses the dynamic dispatching strategy. Other languages make use
of similar concepts.
\noindent\textbf{Accessing LLVM-IR.}~
The LLVM project offers a C++ and a C API to parse LLVM-IR and
LLVm bitcode files.
As the CPG project is mainly implemented in
Java, access to the API has to be provided via the Java Native Interface (JNI).
We use the open source project
javacpp-presets\footnote{\url{https://github.com/bytedeco/javacpp-presets/tree/master/llvm/src/gen/java/org/bytedeco/llvm}}
which provides access to the C API via JNI. Unfortunately, the C API has a flat
type hierarchy in its functions to access the LLVM-IR's AST, thus making the parsing
of instructions and the extraction of their elements more error-prone if not
parsed correctly\footnote{Typically, an incorrect API call leads to a segfault.}
However, as our evaluation
in Section \ref{sec:eval} shows, our implementation works in a stable way.
\section{Related Work}
\label{sec:related_work}
\noindent\textbf{Code Property Graphs.}~
Researchers and industry proposed multiple use cases and implementations of CPGs
and analysis tools \cite{xiaomeng2018cpgva,click1995a,yamaguchi2014modeling,Graft,Plume,Joern,CPG,banse2021cloud,weiss2022languageindependent,schuette2019lios}. All of these tools differ in
their support for programming languages.
Closest to our work is the tool \texttt{llvm2cpg} \cite{llvm2cpg} which uses
Joern \cite{Joern} as graph representation.
The respective CPG represents most instructions as function calls and
does not try to infer any of the high-level information. Furthermore, it uses
the \textit{reg2mem} LLVM pass to address the $\varphi$ instruction of LLVM-IR,
which significantly increases the code base. This results in additional
instructions present in the graph and thus slows down the analysis
and makes it more error-prone.
liOS \cite{schuette2019lios} constructs
a CPG holding assembly instructions and the function
bodies lifted to LLVM-IR to analyze iOS apps. The graph model cannot be used to
represent source code. Furthermore, liOS does not
specifically address LLVM-IR instructions since the analyses mainly operate on
assembly code.
Plume \cite{Plume} and Graft \cite{Graft,keirsgieter2020graft} only support
Java bytecode, a different low-level language. Plume
builds the graph incrementally to analyze data flows and has been merged
into Joern in a revised version. Graft follows a similar goal.
Other tools \cite{click1995a,yamaguchi2014modeling,Joern,CPG,weiss2022languageindependent} analyze source code and differ in their level of
abstractions and supported languages. Some tools extend CPGs for specific use
cases, e.g., analyzing cloud apps \cite{banse2021cloud} or finding
vulnerabilities with deep learning \cite{xiaomeng2018cpgva}.
\noindent\textbf{Graph-based Security Analysis.}~
Various other works investigated in the usage of other graph-based
representations of the source code to identify bugs or vulnerabilities
\cite{urma2015source,yamaguchi2012generalized,yamaguchi2015automatic} or
similar code fragments\cite{gascon2013structural,baxter1998clone},
traverse the graph \cite{Rodriguez2015} or improve the analysis
\cite{lam2005context}. These works aim to provide a rich basis for analyzing the
graphs. Many of the proposed techniques operate on other graph structures (e.g.
the AST). However, the CPG combines a multitude of information and includes the
respective relations, thus making the required information available for the
analysis. Hence, these approaches can still be applied to the CPG
\noindent\textbf{Static Analysis of Multiple Programming Languages.}~
Other works target the analysis of multiple programming languages
\cite{caracciolo2014pangea,flores2015cross,flores2011towards,angerer2014variability,mushtaq2017multilingual,mayer2012cross}.
Some of the frameworks rely on language-agnostic ASTs
\cite{schiewe2022advancing,zugner2021language} or aim to provide a
common pattern for the AST of multiple languages \cite{rakic2013language,strein2006cross}. However, ASTs are cannot be used to find all kinds of bugs
as they do not contain the required information \cite{yamaguchi2014modeling}.
Teixeira et al. \cite{teixeira2021multi} even propose to translate source code
to a custom language.
Furthermore, various intermediate representations (IRs) have been proposed
either for compilers (e.g., LLVM \cite{lattner2004llvm}, GIMPLE \cite{gimple},
HIR \cite{hir} or CIL \cite{ecmaecma}), or specifically targeting code analysis
(e.g. VEX IR \cite{nethercote2007valgrind,vex}, jimple \cite{vallee1998jimple},
BIL \cite{brumley2011bap}, REIL \cite{dullien2009reil}, ESIL \cite{esil}, DBA
\cite{bardin2011bincoa,david2016binsec} or RASCAL \cite{klint2009rascal}).
Since the IRs are often tailored to a specific use case or language, they
differ in the information available in the instructions and their abstractions.
Many of the IRs are integrated in analysis toolchains whose analyses are often
specific to a use case and cannot easily be ported to other tools. Therefore,
integrating such IRs in an abstract analysis platform like the CPG can enable
further generalized security analysis.
Numerous tools \cite{fbinfer,sonarqube,checkmarx,appscreener,codacy,codeql,avgustinov2016ql,de2007keynote,codechecker,coverity,deepsource,lgtm,wala} can analyze multiple programming
languages.
However, they can often barely share the analyses between the
languages.
The CPG representation allows reusing analyses across languages.
\section{Mapping LLVM-IR to CPG nodes}
\label{sec:mapping}
We aim to include LLVM-IR in the CPG while reusing only the existing node types
and representing LLVM-specific constructs similar to their equivalents in
languages which are already supported by the CPG.
We also want to keep the number of nodes minimal.
In this section,
we present how we represent 1) arithmetic and logical instructions, 2) access
to aggregate types, 3) the $\varphi$ instruction with a minimal increase of
nodes, and 4) LLVM-IR's exception handling routine.
\subsection{Basic Instructions}
Many instructions are known from other programming languages. We can coarsely
differentiate between arithmetic and logical operations, operations which
enforce specific interpretations of types, and operations which are composed of
numerous steps but are often performed atomically on the CPU. In this section,
we explain how we include those respective instructions in the CPG.
Almost all programming languages have a common subset of instructions or
operations. This includes arithmetic, bitwise and logic operations, or
comparisons which we map to their representation in high-level
languages (\texttt{+}, \texttt{-}, \texttt{*}, \texttt{/}, \texttt{\%},
\texttt{$\hat{ }$} , \texttt{\&}, \texttt{|}, \texttt{<<}, \texttt{>>},
\texttt{<}, \texttt{<=}, etc.).
Other instructions like jumps, calls, return instructions are modeled with
their representation in C code. For if- and switch/case-statements,
the branches or cases contain a simple goto statement. Later, a CPG pass
removes such indirections whenever possible to reduce the size of the graph.
For some instructions, LLVM-IR can enforce a specific interpretation of the
types of the arguments. E.g., the instructions \texttt{udiv},
\texttt{sdiv} and \texttt{fdiv} represent a division and are mapped to the
binary operator \texttt{/}. However, they interpret the values as unsigned
(\texttt{udiv}), signed (\texttt{sdiv}) or as floating point value
(\texttt{fdiv}). In the CPG, we add typecasts to the arguments to enforce the
correct interpretation.
In addition,
some comparators of floating point values
check if a number is ordered or not (i.e., if it is \texttt{NAN}).
We split these comparisons into a check if
the number is ordered and then the actual comparison. E.g., the
comparators \texttt{ult} and \texttt{olt}
compare two floating point values and are mapped to the \texttt{<} operator.
However, the \texttt{ult} comparison checks if a value \texttt{a} is unordered
or less than value \texttt{b} and thus is modeled
as the statement \texttt{std::isunordered(a)||a<b}.
Similarly, we model the \texttt{olt} comparison with
\texttt{!std::isunordered(a)\&\&!std::isunordered(b)\&\&a<b}.
Some of LLVM's instructions like \texttt{atomicrmw} and \texttt{cmpxchg} are
known from assembly code rather than high-level languages and perform multiple
operations atomically.
The \texttt{cmpxchg} instruction loads a value from memory and
replaces it with an operand if the value equals another operand. In the CPG,
we model this by a block of statements holding the comparison, an if statement and
the assignment in the then-branch.
We annotate the block to keep the information that all this is
performed atomically. Similarly, we model \texttt{atomicrmw} as a block of
statements performing a load, an optional comparison and if-statement and an
assignment to a variable.
By modeling these instructions with a representation similar to
source code, we simplify subsequent analyses. In contrast, prior work
\cite{llvm2cpg} models these instructions as a call to custom functions.
\subsection{Handling Aggregate Types}
High-level languages provide syntactic means to access elements of complex types
like arrays, structs or objects. In LLVM-IR, arrays and structs are still
present and their values can be accessed by special instructions. For arrays
which are represented as a vector, the instructions \texttt{extractelement} and
\texttt{insertelement} provide access to the elements.
\begin{figure}[t]
\centering
\includegraphics[width=0.78\columnwidth]{figures/insertvalue.png}
\caption{The graph representing the insertvalue instruction. We can see the
literal struct which is generated as well as the access to the field.}
\label{fig:insertvalue}
\end{figure}
Both instructions are represented as an \texttt{ArraySubscriptionExpression} in
the CPG, one being the left-hand side of the assignment and one the right-hand
side. Note that \texttt{insertelement} returns the modified vector and does not
modify the existing one. For all other aggregated types, the instructions
\texttt{getelementptr}, \texttt{extractvalue}, and \texttt{insertvalue} model
the access to the element either by the index inside an array or by the position
of a field inside a structure. The code \texttt{\%b = insertvalue {i32, i8} \%a,
i8 7, 1} shows how the second element of the variable \texttt{a} is set to
\texttt{7}.
We model the instruction as a copy of \texttt{a} to the variable
\texttt{b} and an assignment of the value \texttt{7} to the accessed
\texttt{field\_1}.
Figure \ref{fig:insertvalue} shows the resulting graph with the initialization
of \texttt{b} on the bottom right, and the access to the field on the left.
The example uses an interesting concept of LLVM-IR: a
so-called literal structure, a struct whose layout is
defined in the instruction. For such structs, we generate a type which is
identified by the types of its fields. Hence, all literal structs
with the same list of fields are regarded as the same type.
In our example, the struct is named \texttt{literal\_i32\_i8} and has the
fields \texttt{field\_0} of type \texttt{i32} and \texttt{field\_1} of type
\texttt{i8}.
The top left of Figure \ref{fig:insertvalue} shows the declaration of the
type.
While the instructions \texttt{insertvalue} or \texttt{extractvalue}
read or write values from memory, it is sometimes desirable to retrieve a pointer
to an element of a structure. For this case, the instruction
\texttt{getelementptr} computes a memory address without accessing
memory. Listing \ref{lst:getelementptr} illustrates the usage of this
instruction on a named struct. Listing \ref{lst:cGetelementptr}, in turn, shows
the same code written as C.
Figure \ref{fig:getelementptr} shows the definition of the named struct and the
connections between the fields for the graph retrieved from LLVM-IR.
The result is remarkably similar to the graph in Figure \ref{fig:cGetelementptr}
which represents the C code.
This similarity lets us reuse existing
analyses for the graphs retrieved from LLVM-IR and shows that the graphs
are structurally identical.
In fact, the relations between variables and fields could be better resolved
which can lead to improved analysis results.
\begin{minipage}{\linewidth}
\begin{lstlisting}[caption={The instruction getelementptr for a named struct},label={lst:getelementptr}]
define i32* @foo
i64 1, i32 2, i32 1, i64 5, i64 13
ret i32*
}
\end{lstlisting}
\end{minipage}
\begin{minipage}{\linewidth}
\begin{lstlisting}[caption={The C code for the example in Listing \ref{lst:getelementptr}},label={lst:cGetelementptr}]
struct RT { char A; int B[10][20]; char C; };
struct ST { int X; double Y; struct RT Z; };
int *foo(struct ST *s) {return &s[1].Z.B[5][13];}
\end{lstlisting}
\end{minipage}
\begin{figure}[htb!]
\centering
\begin{subfigure}{\textwidth}
\includegraphics[width=\columnwidth]{figures/graph_getelementptr_comparison_llvm.png}
\caption{The graph representing the getelementptr instruction.}
\label{fig:getelementptr}
\end{subfigure}
\begin{subfigure}{\textwidth}
\includegraphics[width=\columnwidth]{figures/graph_getelementptr_comparison_c.png}
\caption{The graph representing the C code.}
\label{fig:cGetelementptr}
\end{subfigure}
\caption{Comparison of the LLVM code using getelementptr and the respective
C code. The graph contains structs (light pink) and their fields (light
brown), the access to fields (dark pink), access to elements in arrays
(brown) and the return instruction (purple). The green nodes are constant
values, the yellow node is the method's argument. The structure of both
graphs is nearly identical.}
\end{figure}
\subsection{The $\varphi$-Instruction}
The SSA form enforces that each variable is assigned exactly once in LLVM-IR.
However, in some cases, it is required to assign a value multiple times.
A frequent example is a loop counter which is set before executing the
loop and is updated on each iteration.
To allow such behavior without duplicating code and without storing the values
in memory, the $\varphi$-instruction is used.
It assigns the target variable one of the inputs based on the
previously executed basic block (BB).
As most programming languages do not have such an instruction, there is no
fitting node to represent this in the CPG.
To address this issue, prior work \cite{llvm2cpg} relied on the LLVM
reg2mem pass\footnote{\url{https://llvm.org/doxygen/Reg2Mem_8cpp_source.html}}
which translates the instruction to multiple load
and store operations. However, this pass also transforms the access to other
variables and thus significantly increases the size of the resulting CPG.
As this reduces the scalability of subsequent analyses, we avoid this
LLVM pass.
We collect all
$\varphi$-instructions during the translation. Finally, we parse the
instructions to identify the predecessor BBs and
add an assignment to the target variable at the end of the BB. To
keep the CPG clean, we further insert a declaration of the variable at the
beginning of the function containing the $\varphi$-instruction and all
BB
\footnote{For all other variables, the statement of the assignment performs the
declaration.}. This, however, breaks the SSA form. The snippet in Listing
\ref{lst:phi} contains the $\varphi$-instruction while Listing \ref{lst:phiAfter} shows the
function's model in the CPG.
\begin{figure}[tb]
\noindent\begin{minipage}{.45\textwidth}
\begin{lstlisting}[caption={Code snippet using the $\varphi$-instruction},label={lst:phi}]
define i32 @main(i32
br i1
label
BB1:
br label
BB2:
br label
BB3:
[
ret i32
}
\end{lstlisting}
\end{minipage}
\hfill
\begin{minipage}{0.45\textwidth}
\begin{lstlisting}[caption={Snippet using the $\varphi$-instruction as modeled in the CPG},label={lst:phiAfter}]
define i32 @main(i32
; VariableDeclaration of
br i1
label
BB1:
br label
BB2:
br label
BB3:
ret i32
}
\end{lstlisting}
\end{minipage}
\end{figure}
\subsection{Exception handling}
LLVM-IR offers a rich system for exception handling. The CPG
represents exception handling routines with try-catch
statements. To make the LLVM-IR fit into this pattern, we need to identify which
instructions form a try-block and which ones a catch-block.
Concerning the try-block, we represent the \texttt{invoke} instruction
as a try-block surrounding a function call and a goto-statement.
For the catch-blocks, however, such a straightforward model is not possible.
In LLVM, the \texttt{catchswitch} instruction selects a matching
\texttt{catchpad} based on the signature of the catchpad-instruction of a basic
block. The catchpad contains the code of the catch-block and is ended by a
\texttt{catchret} instruction. However, the matching and signature cannot easily
be transferred to a high-level name. Therefore, we model this construct as a
catch-block which catches all exceptions and contains if-statements representing
the signature matching. If none of them matches, the exception is thrown again.
The remaining constructs such as the \texttt{cleanuppad} and its
\texttt{cleanupret} instruction are not modeled specifically.
Another way to
mark a catch-block is the \texttt{landingpad}-instruction which, again, filters
for the right object to catch. Once more, the matching is specific to the
programming language and thus, modelling this is left to future work. If we
cannot translate the instructions to concepts supported by the CPG, we model
them as special functions similar to the LLVM intrinsics.
\section{LLVM-Specific CPG Passes}
\label{sec:pass}
During the translation of the LLVM instructions to CPG nodes, the frontend
generates various instructions which later turn out to be unnecessary and thus
can be removed. This clean-up phase takes place in a pass over the CPG nodes.
First, none of the conditional jumps and switch/case-statements incorporates a
meaningful body of statements. Instead, they are modeled as goto
statements to another basic block. The pass identifies all basic blocks which
have only a single predecessor and replaces
the respective goto-statement with the basic block. Note that we do not perform
this transformation if multiple predecessors exist because it would
unnecessarily increase the number of nodes in the graph.
Second, the pass removes the instructions which serve as intermediate
steps during the generation of catch-blocks and propagates the caught exception
to the final throw statement if none of the
catchpad instructions matches.
As we explicitly aim to handle lifted or decompiled code, a second pass can
remove method stubs, i.e., methods whose only purpose is to call a library
method. The main purpose of this pass is to simplify subsequent analyses.
\section{Experimental Evaluation}
\label{sec:eval}
To reuse the same analyses for the graphs constructed from source code as well
as the ones containing LLVM-IR, we carefully designed the translation in a way
to mimic the concepts used in source code as closely as possible. In this
section, we first show a case study which advocates that we can reuse queries that
aim to identify security concerns in source code to query LLVM-IR. Second, we
test the implementation against the Rust standard library to
show the applicability of the approach to large-scale projects.
All measurements were performed on a Ubuntu 20.04 running on an Intel i5-6200U
CPU and 20 GB of RAM.
\begin{table}[tb]
\centering
\caption{Results for detecting misuse of cryptographic libraries.}
\label{tab:ssl_res}
\begin{tabular}{|l|c|c|c|c|}\hline
& \textbf{Analysis time [ms]} & \textbf{\# Nodes} & \textbf{\# Functions} & \textbf{Problem found} \\\hline
\multicolumn{5}{|c|}{\cellcolor{gray}Source Code} \\\hline
Original file & 171 & 328 & 38 & Yes \\\hline
\multicolumn{5}{|c|}{\cellcolor{gray}macOS M1 using XCode} \\\hline
Compiled ll & 1091 & 5279 & 151 & Yes \\\hline
Lifted ll & 256 & 1743 & 76 & Yes \\\hline
Decompiled & 179 & 971 & 149 & No \\\hline
\multicolumn{5}{|c|}{\cellcolor{gray}Ubuntu x86-64 clang} \\\hline
Compiled ll & 163 & 1371 & 57 & Yes \\\hline
Lifted ll & 127 & 911 & 48 & Yes \\\hline
Decompiled & 80 & 594 & 101 & Yes \\\hline
\multicolumn{5}{|c|}{\cellcolor{gray}Ubuntu x86-64 g++} \\\hline
Lifted ll & 242 & 1702 & 89 & Yes \\\hline
Decompiled & 148 & 1137 & 200 & Yes \\\hline
\multicolumn{5}{|c|}{\cellcolor{gray}Linux AArch64 (cross compiled)} \\\hline
Lifted ll & 250 & 1891 & 93 & Yes \\\hline
Decompiled & 158 & 1176 & 209 & Yes \\\hline
\multicolumn{5}{|c|}{\cellcolor{gray}Linux arm 32 bit (cross compiled)} \\\hline
Lifted ll & 132 & 1123 & 51 & Yes \\\hline
Decompiled & 71 & 626 & 102 & Yes \\\hline
\end{tabular}
\end{table}
\subsection{Case Study: Cryptographic Misuse}
\label{sec:crypto}
This case study is driven by the anticipated usages of the CPG on LLVM-IR.
First, it should enable a security analysis of the LLVM-IR without the need to
rewrite existing analyses. Second, it should be scalable by introducing a
minimal number of nodes.
The toolchain should be able to operate on LLVM-IR emitted during
the compilation of a program (subsequently, we call this ``compiled LLVM-IR'')
or when lifting a binary (we call this ``lifted LLVM-IR'').
To show that these properties are fulfilled, we
1) compare the sizes of graphs retrieved from compilers and lifters,
2) compare the runtime of the analysis,
and 3) show that the weakness can be identified with the same analysis in
all samples.
We implemented a TLS-client in C++ which uses the \texttt{openssl} library. It
accepts the insecure hashing algorithm MD5 as one of the options.
First, we tested the toolchain against the original cpp file, which
identified the respective issue. Next, we used XCode on macOS with the M1 chip
and clang on Ubuntu running on a x64 CPU to emit the LLVM-IR which can be
retrieved during compilation.
As LLVM-IR also serves as target LLVM-IR for many lifters, we lifted binaries
of the test file which had been compiled on the Mac and on Ubuntu with various
compilers.
We use RetDec \cite{RetDec} to lift the binaries to LLVM-IR and also
decompiled them to a C-style file\footnote{We compiled a custom version
of RetDec to update the disassembler and support the \texttt{endbr64}
instruction which had not been supported at the time of the experiments.}.
Table \ref{tab:ssl_res} summarizes
the analysis time, how many nodes and functions are included
in the graph and if the problem could be found successfully. We discuss the
observations
in the following paragraphs.
\noindent\textbf{Size of the graphs.}~
One of our goals is to keep the sizes of the graph small. Therefore, we compare
the size of the graphs retrieved from compiled and lifted LLVM-IR
and when decompiling a binary file.
One observation is the significant increase in functions contained in the
LLVM-IR compared to the original C file.
This can be explained by stubs introduced
by the compiler. Note, however,
that RetDec seems to remove
some of the functions which have been introduced during compilation. This
reduction facilitates and speeds up a subsequent security analysis on the
resulting graph.
Not only does RetDec reduce the number of functions contained
in the binary but it also reduces the number of nodes compared to compiled
LLVM-IR. This observation is in-line with recent research which found that some
lifters, including RetDec, can reduce the complexity of the code
represented by LLVM-IR as well as the number of elements it contains
\cite{liu2022sp} while keeping the main functionality of the code available.
The authors further observed that RetDec's output is not suitable for
recompiling in most cases.
However, as the CPG
library aims to handle incomplete, non-compilable and to a certain extent even
incorrect code, this limitation should not affect the representation and further
analysis.
Compared to the lifted LLVM-IR, the decompiled C files
contain more functions but less nodes. This is
explained by the possibility to summarize multiple LLVM-IR instructions in a
single C statement.
Overall, the reduction of nodes can be explained by RetDec's passes which aim to
eliminate unnecessary code.
\noindent\textbf{Runtime of the analysis.}~
We ran the translation to the CPG and the bug detection query 100 times for each
of the files and report the average runtimes in Table \ref{tab:ssl_res}.
First, it is interesting to note that the analysis time of the decompiled files
is comparable to the one of the original cpp-file. The reduced number of nodes
explains the speedup in some cases.
The overall analysis time for the LLVM-IR files is
ranging between 0.74
to 11.1 times the time of the original file. It is notable that the graphs
retrieved from the LLVM-IR files contain 2.8 to 16.1 times the amount of nodes
of the original file and still the runtime improved.
\noindent\textbf{Identification of weaknesses.}~ To detect the misconfiguration
in the test file, we implemented a query to identify the arguments of calls to
the function \texttt{SSL\_CTX\_set\_cipher\_list}. To implement this analysis,
we use the constant propagation implemented in the analysis module included in
the CPG
library\footnote{\url{https://github.com/Fraunhofer-AISEC/cpg/tree/master/cpg-analysis}}.
With the query, we are able to identify the flaw in the original C file and in
the compiled and lifted LLVM-IR files.
However, when
decompiling the binary compiled on macOS using the M1 chip,
we failed to identify the misuse. We manually investigated the case and found
that the CDT library\footnote{\url{https://www.eclipse.org/cdt/}} which the CPG
library uses for parsing the C file fails to identify the name of a field
correctly. Therefore, the data flow between the field and the method call is not
resolved.
\noindent\textbf{Stability of the translation.}~
All samples could be represented in the CPG without crashes. However, the
LLVM-IR retrieved during compilation of a program contains a much richer
semantics and uses various different instructions.
This results in warnings, some of which show that nested instructions
are not yet handled. The other ones indicate that a different scoping for
variables in a try-catch block is expected because LLVM-IR's scoping differs
to other languages.
\begin{table}[tb]
\centering
\caption{Performance when analyzing Rust libraries.}
\label{tab:rust}
\resizebox{\textwidth}{!}{
\begin{tabular}{|l|l|c|c|c|c|c|}\hline
\# & \textbf{Filename} & \textbf{LoC} & \textbf{\# Nodes} & \textbf{\# Functions} & \textbf{\# Errors} & \textbf{Analysis time [ms]} \\\hline
1 & addr2line & 879 & 2327 & 29 & 9 & 3641 \\\hline
2 & adler & 488 & 1707 & 25 & 2 & 507 \\\hline
3 & alloc & 4925 & 13482 & 253 & 91 & 6505 \\\hline
4 & cfg\_if & 9 & 1 & 0 & 0 & 23 \\\hline
5 & compiler\_builtins & 9990 & 34304 & 338 & 0 & 23670 \\\hline
6 & core & 80193 & 263729 & 3608 & 1879 & 2872096 \\\hline
7 & gimli & 23702 & 72845 & 411 & 43 & 112269 \\\hline
8 & hashbrown & 276 & 529 & 26 & 0 & 193 \\\hline
9 & libc & 1477 & 3619 & 130 & 0 & 646 \\\hline
10 & memchr & 11063 & 40602 & 257 & 108 & 32639 \\\hline
11 & miniz\_oxide & 15760 & 54868 & 294 & 166 & 79863 \\\hline
12 & object & 14174 & 50060 & 277 & 5 & 47806 \\\hline
13 & panic\_abort & 71 & 87 & 9 & 0 & 124 \\\hline
14 & panic\_unwind & 927 & 2619 & 67 & 25 & 610 \\\hline
15 & proc\_macro & 92115 & 244010 & 5488 & 2570 & 15260350 \\\hline
16 & rustc\_demangle & 14669 & 44069 & 437 & 309 & 43281 \\\hline
17 & rustc\_std\_workspace\_alloc & 9 & 1 & 0 & 0 & 107 \\\hline
18 & rustc\_std\_workspace\_core & 9 & 1 & 0 & 0 & 102 \\\hline
19 & std & 157377 & 468223 & 5923 & 2629 & 9303378 \\\hline
20 & std\_detect & 558 & 1921 & 15 & 0 & 659 \\\hline
21 & unwind & 106 & 230 & 2 & 0 & 273 \\\hline
\end{tabular}}
\end{table}
\subsection{Application to the Rust Runtime}
To assess the applicability to real-world programs, we retrieved the LLVM-IR
from the standard and core libraries of Rust. We chose Rust since it is
not yet supported by the CPG implementation and provides the option to compile
to LLVM-IR. Overall, the test set includes 21 distinct LLVM files which are
listed in Table \ref{tab:rust} together with their size and the results. We
report the time it took to translate the file (including various CPG passes) as
well as the number of nodes which could not be parsed accurately. For the
latter, we need to extend the LLVM-specific translation to include more cases of
``nested'' LLVM expressions.
\begin{figure}[tb]
\centering
\resizebox{0.9\textwidth}{!}{
\begin{tikzpicture}
\begin{axis}[
xlabel={Nodes/LoC},
ylabel={ProblemNodes/Nodes [\%]},
]
\addplot[color=blue, mark=*, only marks]
coordinates {(2.647326507,0.3867641) (3.49795082,0.1171646)
(2.7374619,0.674974) (0.11111111,0) (3.433833934,0) (3.288678563,0.71247379)
(3.073369336,0.05902944) (1.91166667,0) (2.450236967,0)
(3.670071409,0.265996749) (3.481472081,0.30254428811)
(3.531818823,0.00998801438) (1.225352113, 0) (2.825242718,0.95456281023)
(2.648971394,1.05323552313) (3.0042266,0.70117316027) (0.11111111,0)
(0.11111111,0) (2.97516791,0.56148459174) (3.44265233,0) (2.169811321,0)};
\end{axis}
\begin{axis}[
xlabel={LoC},
ylabel={ProblemNodes/Nodes [\%]},
xticklabel pos=right,
xlabel near ticks,
legend pos=outer north east
]
\addplot[color=blue, mark=*, only marks] coordinates {(1,0)};
\addlegendentry{Nodes/LoC vs. ProblemNodes/Nodes};
\addplot[color=red, mark=o, only marks]
coordinates {(879,0.3867641) (488,0.1171646)
(4925,0.674974) (9,0) (9990,0) (80193,0.71247379)
(23702,0.05902944) (276,0) (1477,0)
(11063,0.265996749) (15760,0.30254428811)
(14174,0.00998801438) (71, 0) (927,0.95456281023)
(92115,1.05323552313) (14669,0.70117316027) (9,0)
(9,0) (157377,0.56148459174) (558,0) (106,0)};
\addlegendentry{LoC vs. ProblemNodes/Nodes};
\end{axis}
\end{tikzpicture}}
\caption{Relation between lines of code, nodes in the CPG and the fraction
of ProblemNodes. For non-trivial samples, the error-rates are randomly
distributed.}
\label{fig:errors}
\end{figure}
\noindent\textbf{Stability.}~
We want to assess the maturity level of the translation step
against a large and unknown codebase consisting of a total of $428,777$ lines of
LLVM-IR. To measure this, the graph includes specific nodes, called
\texttt{ProblemNode}, for each expression which could not be parsed correctly.
While we handle all types of instructions, some arguments of the instructions
can be computed in line by type casts, or simple arithmetic operations, among
others. Overall, we could observe $7,836$ of such \texttt{ProblemNode}s, which
accounts for $0.60\%$ of all $1,299,234$ nodes. This result is encouraging and
indicates that the current implementation is already capable of handling the vast
majority of all combinations of statements\footnote{We will manually
investigate the ProblemNodes to parse the statements in the future.}.
The fraction of nodes which cannot be handled differs significantly among the
samples and ranges between $0\%$ to $1.05\%$. Larger files are more likely to
lead to an error during the translation. In addition, it is possible that the
varying amount of complexity of the code could trigger more errors. To validate
this, we set the average number of CPG nodes per line of code as complexity of
the LLVM instructions. Among the samples, this ratio ranges between $1.22$ and
$3.67$.
We plot this relation in Figure \ref{fig:errors}. Neither of the graphs
gives a strong indication for this idea since the error rates seem to be
randomly distributed for all non-trivial samples. Neither the size nor the
complexity of the samples lead to a conceptual limitation. Instead, some samples
use unsoppurted expressions more frequently which can easily be addressed in the
implementation.
\begin{figure}[tb]
\centering
\resizebox{0.6\textwidth}{!}{
\begin{tikzpicture}
\begin{axis}[
xlabel={\# Nodes in the CPG},
ylabel={Analysis time [s]},
ymode=log,
legend pos=south east
]
\addplot[color=blue, mark=square]
coordinates {(1,0.023) (1,0.102) (1,0.107) (87,0.124) (230,0.273)
(529,0.193) (1707,0.507) (1921,0.659) (2327,3.641) (2619,0.610) (3619,0.646)
(13482,6.505) (34304,23.670) (40602,32.639) (44069,43.281) (50060,47.806)
(54868,79.863) (72845,112.269) (244010,15260.350) (263729,2872.096)
(468223,9303.378)};
\addlegendentry{Including CPG passes};
\addplot[color=red, mark=o]
coordinates {(1,0.012) (1,0.181) (1,0.519) (87,0.137) (230,0.22)
(529,0.129) (1707,0.885) (1921,0.349) (2327,5.108) (2619,0.296) (3619,0.404)
(13482,2.057) (34304,5.203) (40602,3.828) (44069,7.850) (50060,3.846)
(54868,10.354) (72845,24.883) (244010,248.362) (263729,132.879)
(468223,515.532)};
\addlegendentry{Only LLVM-related CPG pass};
\end{axis}
\end{tikzpicture}}
\caption{Analysis time vs. \# Nodes. Note the logarithmic y scale.}
\label{fig:at_nodes}
\end{figure}
\noindent\textbf{Scalability.}~
Another goal is to assess the scalability of the implementation on
real-world software with many lines of code. Two factors can impact the analysis
time: The lines of code and the number of nodes in the graph. According to Table
\ref{tab:rust}, an increase of LoC leads to more nodes in the graph in most
cases. Figure~\ref{fig:at_nodes} plots the time of the analysis (i.e., the
translation to the CPG and all CPG passes but the
\texttt{ControlFlowSensitiveDFGPass}) for the number of nodes. With the
exception of one sample, the analysis time seems to grow linearly depending on
the number of nodes in the graph. Interestingly, when we only consider the
analysis time of the LLVM-specific translation and pass of the CPG, the outlier
is no longer present. This shows that the LLVM-related translation and CPG pass
do scale well even for larger samples but that some of the other CPG passes seem
to perform poorly in the presence of a specific combination of nodes.
\begin{figure}[h!]
\centering
\begin{minipage}{.5\textwidth}
\centering
\begin{tikzpicture}
\begin{axis}[
xbar=0pt,
width = \textwidth,
axis y line*=left,
axis x line=bottom,
height = 300pt,
enlarge y limits=0.025,
bar width=5pt,
xmajorgrids = true,
ylabel = {File},
xlabel = {Analysis time [ms]},
ytick = data,
xmode=log,
scaled y ticks = true,
axis line style={-},
legend columns=2,
legend cell align=left,
legend style={
at={(0.5,-0.11)},
anchor=north,
column sep=1ex,
nodes={scale=0.75, transform shape}
},
nodes near coords,
nodes near coords style={font=\tiny},
nodes near coords align=horizontal,
point meta=rawx
]
\addplot+[xbar]
coordinates {(3641, 1) (507, 2) (6505, 3) (0023, 4) (23670, 5) (2872096, 6)
(112269, 7) (0193, 8) (0646, 9) (32639, 10) (79863, 11) (47806, 12) (0124, 13)
(0610, 14) (15260350, 15) (43281, 16) (0107, 17) (0102, 18)
(9303378, 19) (0659, 20) (0273, 21)};
\addplot+[xbar]
coordinates {(3847, 1) (1118, 2) (13347, 3) (0011, 4) (108124, 5)
(41934277, 6) (635146, 7) (2477, 8) (1687, 9) (184776, 10) (476777, 11)
(64749, 12) (0051, 13) (0762, 14) (19795079, 15) (182274, 16) (0034, 17)
(0020, 18) (30598095, 19) (0382, 20) (0067, 21)};
\legend{Our approach, reg2mem}
\end{axis}
\end{tikzpicture}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\begin{tikzpicture}
\begin{axis}[
xbar=0pt,
width=\textwidth,
axis x line=bottom,
height=300pt,
bar width=5pt,
enlarge y limits=0.025,
xmajorgrids=true,
xlabel={\#Nodes},
ylabel={File},
ytick=data,
axis line style={-},
nodes near coords,
node near coords style={font=\tiny},
nodes near coords align=horizontal,
point meta=rawx,
legend columns=2,
legend cell align=left,
legend style={
at={(0.5,-0.11)},
anchor=north,
column sep=1ex,
nodes={scale=0.75, transform shape}
},
]
\addplot+[xbar, style=black!60!green, fill=black!40!green]
coordinates {(2327, 1) (1707, 2) (13482, 4) (1, 4) (34304, 5) (263729, 6)
(72845, 7) (529, 8) (3619, 9) (40602, 10) (54868, 11) (50060, 12)
(87, 13) (2619, 14) (244010, 15) (44069, 16) (1, 17) (1, 18)
(468223, 19) (1921, 20) (230, 21)};
\addplot+[xbar, style=black!40!orange, fill=black!20!orange]
coordinates {(3695, 1) (2952, 2) (21858, 3) (1, 4) (59673, 5) (400775, 6)
(102504, 7) (687, 8) (4111, 9) (70026, 10) (102923, 11) (56010, 12)
(103, 13) (4288, 14) (385270, 15) (78289, 16) (1, 17) (1, 18)
(770152, 19) (2554, 20) (369, 21)};
\legend{Our approach, reg2mem}
\end{axis}
%
\end{tikzpicture}
\end{minipage}
\caption{Performance comparison of our approach and prior work. The analysis time
and the number of nodes are typically much smaller with our improvements.}
\label{fig:comparison}
\end{figure}
\noindent\textbf{Comparison to prior work.}~
To compare our approach to prior work which relied on LLVM's reg2mem pass to
remove $\varphi$ nodes, we ran our toolchain but first executed the respective
pass.
As Figure \ref{fig:comparison} shows, our approach leads to a significant
reduction of nodes and time required to generate the graph.
\section{Discussion}
\label{sec:discussion}
Our evaluation suggests that our translation and CPG model can
unify source code and low-level representations such as
LLVM-IR in a single graph representation. This increases the reusability of
analyses and queries on the graph.
We found that the LLVM-IR retrieved from binary lifters is significantly easier
to handle by the graph. This is due to the fact that most lifters tend to use
rather conservative steps for their translation. This results in the LLVM-IR being
closer to assembly code with comparably simple types of instructions. The LLVM-IR
retrieved during the compilation, in contrast, features numerous
highly specialized instructions which typically make the translation more
difficult. Furthermore, the graphs retrieved from lifted binaries
are typically smaller than the ones which can be retrieved when the LLVM-IR is
retrieved during the compilation. This makes it an interesting application since
it simplifies and speeds up the analysis. Last, we found that the graph of the
decompiled binary is only marginally smaller than the one holding the lifted
LLVM-IR instructions. This small advantage will, however, not outweigh the
error-prone and time-consuming decompliation step in most scenarios which is
required to retrieve the code.
\noindent\textbf{Validity of the Results.}~
The main threat to the validity of the findings is the set of test samples. In
particular, as we could see in Section \ref{sec:crypto}, the compiler has a
significant impact on the generated LLVM-IR and the resulting complexity which needs
to be handled by our toolchain. Hence, testing the toolchain against different
compilers and configurations might lead to different results. To address this
potential issue, we used XCode on macOS and clang on Ubuntu,
and we also generated the LLVM-IR with Rust's crates build system.
Furthermore, we used a binary lifter to showcase a possible application to such a
scenario.
\noindent\textbf{Limitations.}~
As our evaluation against the Rust standard library showed, a small amount of
instructions could not be parsed correctly. This is explained by the possibility
of LLVM-IR to hold sub-statements for the arguments. While we do handle the
concepts and operators (e.g., casts), their
potential usage in a specific sub-statement needs to be added to
the translation step. To identify all possible combinations, a more extensive
testing is required.
\noindent\textbf{Future Work and Research Directions.}~
The resulting graph can be used as an entry point for further research to better
include specifics of certain platforms.
One example is the analysis of the LLVM-IR emitted by XCode for apps written in
Apple's programming languages Swift or Objective-C. Their calling
conventions differ significantly from other programming languages. As an
example, Objective-C makes use of a dynamic dispatching routine which requires
extensive tracing of a method's arguments to recover type information and the
method name as a string \cite{schuette2019lios,egele2011pios}. This
information is present in the CPG but has to be combined to identify
the calls. Similarly, it is necessary to model Swift's calling conventions and
memory model since it differs significantly from the one of C++
\cite{tiganov2020swan,kraus2018the}. However, to date, the differences are
not fully explored. Future work should identify differences and
integrate this knowledge into the CPG.
Furthermore, software written in C or C++ can rely on macros which are used
similar to function calls in the source code and represented as such but are
replaced with their specific implementation in LLVM-IR. This discrepancy needs
to be addressed appropriately to better analyze such programs. In the current
stage, addressing such inconsistencies between source code and the binary is
left to manual efforts of the user of the cpg library. Additional efforts
are necessary to reduce these manual efforts and ease the usability of the
analysis toolchain.
Last, adapting the solution to the analysis of closed-source software is
promising.
Recent research \cite{liu2022sp} showed that lifting is a stable technique for many
applications. However, lifted or decompiled binaries still
suffer from a lack of information which are crucial for a security analysis
\cite{mantovani2022the}. Hence, further research should
study which gaps still exist to apply existing tools to lifted binaries.
\noindent\textbf{Generalizability.}~
Since the SSA form is also used by other IRs (e.g. Shimple \cite{shimple}, WALA
\cite{wala}, SIL \cite{sil}), some of the challenges generalize to those
IRs.
Hence, the concepts presented in this paper can be reused to add further code
representations using the SSA form to the CPG.
Furthermore, some parts of our concept could be ported to other projects
which suffer from similar issues. However, the applicability and impact depend
on the projects' data models.
\section{Conclusion}
\label{sec:conclusion}
We showed how we extended an open source CPG implementation to handle
LLVM-IR. While the majority of instructions can easily be mapped to the high-level
equivalents, the $\varphi$ instruction and the LLVM exception handling
instructions impose challenges to the translation. However, we could transform
the program to the CPG representation with a reasonable increase in nodes
while prior work suffered from huge performance penalties.
The similarity between the resulting graph and the one of the code fractions in
high-level languages allows to reuse existing analyses detecting security
weaknesses or bugs. Our evaluation suggests that the approach scales to larger
projects. Future work is necessary to include characteristics of some
programming languages (e.g. Swift), to add analyses for further use cases,
and to study the gaps of binary lifting.
\textbf{Acknowledgements.}~
This work was partially funded by the Horizon 2020 project MEDINA, grant agreement ID 952633.
\bibliographystyle{splncs04}
\urlstyle{tt}
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\section{Introduction}
We investigate the global existence of smooth solutions to the following
two-dimensional isentropic compressible Navier-Stokes equations describing the motion of viscous barotropic compressible flows:
\be\label{CNS}
\left\{\begin{aligned}
&\d_t \rho + \dive (\rho u)= 0,\qquad (t,x,y)\in\R^+\times\TT\times\R,\\
&\d_t(\rho u) + \dive(\rho u \otimes u) - \mu \Delta u - \nabla\left(\mu' \dive u\right) + \nabla p(\rho) = 0,
\end{aligned}\right.
\ee
where $\rho(t,x,y) \in \R_+$ and $u(t,x,y)\in \R^2$ designate the the density and the velocity field of the fluid respectively.
Here the shear viscosity $\mu$ and bulk viscosity $\mu'$ are assumed to be constant with $\mu>0$ and $ \mu+\mu' >0$. The pressure function $p$
is a smooth non-decreasing function of the density $\rho.$ For simplicity, we just consider power law pressure with
\be\label{pressure}
p(\rho) = a \rho^\g, \quad a>0, \quad \g \geq 1.
\ee
The typical range of the adiabatic constant is $1<\g\leq \frac{5}{3}$ where the maximum value $\g = \frac{5}{3}$ is related to monatomic gases, the intermedius value $\g = \frac{7}{5}$ is related to diatomic gases including air, and lower values close to $1$ is related to polyatomic gases at high temperature. In this paper, we allow any $\g \in [1, 2]$. One may check \cite{Lions-C} for more background of the system \eqref{CNS}.
\smallskip
The global existence of smooth solution to the multi-dimensional compressible Navier-Stokes equations is one of the most important
problems in the theory which describes the motion of the viscous compressible fluids. In \cite{MN83}, Matsumura and Nishida proved the global existence of solutions to the full compressible Navier-Stokes equations when the initial data is close to a constant equilibrium state in $H^3(\R^n).$ Danchin \cite{Dan00} worked out similar result yet with initial data in certain optimal function spaces. Huang, Li and Xin \cite{HLX12} established the global existence and uniqueness of classical solutions to the Cauchy problem of the barotropic compressible Navier-Stokes equations in three space dimensions with smooth initial data which are of small energy but possibly large oscillations. Concerning general smooth data without size restriction and with initial density being away from vacuum, the first well-posedness result is due to Va\u{\i}gant and Kazhikhov \cite{VK95} where the authors proved the global unique solution to the two-dimensional compressible Navier-Stokes system \eqref{CNS} in $\R^+\times\TT^2$ with the bulk viscosity $\mu'=b\rho^\b$ and $b>0, \b>3.$ Lately this result was improved by Huang and Li \cite{HL16} for $\b>\frac43.$ For further results in higher dimensional setting, we refer to \cite{Valli83, CK03, FNP} and the references therein.
\smallskip
On the other hand, Chemin and Gallagher \cite{CG10} proved the global existence of smooth solution to three-dimensional incompressible Navier-Stokes equations $(NS)$ with initial data which is slowly varying in one direction. This type of result was extended by Chemin and the second author \cite{CZ6}
to the three-dimensional incompressible inhomogeneous Navier-Stokes equations and by Liu and the second author \cite{LZ4} for $(NS)$ with unidirectional derivative of the initial velocity being sufficiently small in some critical functional space.
Motivated by \cite{CG10,CZ6,LZ4}, we are looking for global solutions of \eqref{CNS} that are periodic in $x$ variable, and have slow variation in $y$ variable.
This means that we implement the system \eqref{CNS} with initial data of the form
\be\label{ini-data}
\rho(0,x,y) = [\varsigma_0]_\e(x, y), \quad u(0,x,y) = \big([w_0]_\e(x, y), [\frak{w}_0]_\e(x, y) \big),
\ee
where $(x,y)\in \TT \times \R$ with $\TT $ being the torus $ \R/\Z$. Here and in all that follows, we shall always denote
$$
[f]_\e(x,y) \eqdefa f (x, \e y).
$$
We aim at showing the global well-posedness of \eqref{CNS}--\eqref{ini-data} without assuming any size restriction on $(\varsigma_0,w_0,\frak{w}_0)$
provided that $\e$ is sufficiently small.
We remark that data of the form \eqref{ini-data} are neither of small perturbation of some constant equilibrium state, say $(1,0),$ nor of small energy. Hence in particular, our result gives examples indicating that 2D compressible Navier-Stokes equations with constant viscosity might
be globally well-posed even for general initial data with initial density being away from vacuum.
\smallskip
Formally with initial data given by \eqref{ini-data}, we may seek solution of \eqref{CNS} as follows
\ba\label{vr-v}
\rho(t,x,y) = [{\xi}]_\e(t,x,y), \quad u(t,x,y) =[v]_\e(t,x,y),
\nn \ea
then it follows from \eqref{CNS} that
\be\label{CNS-e}
\left\{\begin{aligned}
&\d_t \xi + \divee (\xi v)= 0,\\
&\d_t(\xi v) + \divee(\xi v \otimes v) - \mu \Delta_\e v - \mu' \nabla_\e \divee v + \nabla_\e p(\xi) = 0,
\end{aligned}\right.
\ee
with initial data
\ba\label{ini-data-v}
\xi(0,x,y) = \varsigma_0(x,y), \quad v(0,x,y) = \big(w_0(x, y),\frak{ w}_{0}(x,y) \big).
\nn \ea
Here
\be\label{nabla-e}
\nabla_\e \eqdefa \bp\d_x\\ \e \d_y \ep, \quad \divee \eqdefa \nabla_\e \cdot \andf \Delta_\e \eqdefa \d_x^2 + \e^2 \d_y^2.
\ee
Formally passing $\e \to 0$ in \eqref{CNS-e} leads to a limit system of the form with a parameter $y:$
\be\label{CNS-limit}
\left\{\begin{aligned}
&\d_t \eta + \d_x ( \eta w)= 0,\qquad (t,x,y)\in\R^+\times\TT\times\R,\\
& \eta (\d_t w + w \d_x w) - \nu \d_x^2 w - \d_x p( \eta ) = 0 \with \nu\eqdefa \mu + \mu',
\end{aligned}\right.
\ee
and
\be\label{tw}
\eta ( \d_t\frak{w} + w \d_x \frak{w}) - \mu \d_x^2 \frak{w} = 0, \qquad (t,x,y)\in\R^+\times\TT\times\R.
\ee
Accordingly, the initial data are
\be\label{CNS-limit-initial}
\eta (0,x,y) = \varsigma_0(x,y), \quad w(0,x,y) = w_{0}(x,y), \quad \frak{w}(0,x,y) = \frak{w}_{0}(x,y).
\ee
We observe that the equations of $(\eta ,w)$ and the equation of $\frak{w}$ are decoupled.
Furthermore,
$( \eta ,w)$ satisfies one dimensional compressible Navier-Stokes equations with a parameter $y.$ We shall study in detail about this system in Section \ref{sec:1dNS}.
Before proceeding, we assume that
\ba\label{ass-ini-1}
&\underline\varsigma_0 \leq \varsigma_0 \leq \bar \varsigma_0, \quad (\d_{y}^{j}(\varsigma_0-1), \d_{y}^{j}w_{0}, \d_{y}^{j}\frak{w}_{0})\in (L_{\rmv}^{2} \cap L_{\rmv}^{\infty})(H_{\h}^{5})\with\\
&A_{k}(y) \eqdefa \sum_{j=0}^{2} \|\d_{y}^{j}(\varsigma_0-1, w_{0}, \frak{w}_{0})(\cdot,y)\|_{H^k_\h}^2 \andf \bar A_{k} \eqdefa \sup_{y\in \R} A_{k} (y), \quad 0\leq k\leq 5,
\ea
for some positive constants $\underline\varsigma_0 $ and $\bar \varsigma_0$, and for $j = 0,1,2$. Here the subscript $\h$ (resp. ${\rmv}$) denotes the norm on $\TT_{x}$ (resp. $\R_{y}$). We assume moreover that
\ba\label{ini-1}
\int_{\TT} \varsigma_0(x,y)\,\dx = 1,\quad \int_{\TT} (\varsigma_0 w_0)(x,y)\,\dx = \int_{\TT} (\varsigma_0 \frak{w}_0)(x,y)\,\dx = 0, \quad \forall\, y\in \R.
\ea
Our first result is concerned with the large time exponential decay for the solutions to the equations \eqref{CNS-limit} and \eqref{tw}
with initial data \eqref{CNS-limit-initial}.
\begin{theorem}\label{thm1}
{\sl Let $(\varsigma_{0}, w_{0}, \frak{w}_0)$ satisfy \eqref{ass-ini-1} and \eqref{ini-1}. Then for each $y\in \R$,
the system \eqref{CNS-limit}--\eqref{CNS-limit-initial} has a unique global-in-time strong solution $(\eta, w, \frak{w}) \in C([0,\infty); H^2(\TT))$ so that
\beq\label{thm1-2}
\underline \eta \leq \eta(t,x,y) \leq \bar \eta, \quad \mbox{for some positive constants $\underline \eta$ and $\bar \eta$.}
\eeq
Moreover,
there exist positive constants $C$ and $\a$ solely depending on $(a, \g, \nu, \bar \varsigma_0, \underline{\varsigma}_0, \bar A_{4})$ such that
for all $ t\in \R_+$ and $ y\in \R$
\ba\label{thm1-0}
&\Bigl( \| (\eta -1)(t,\cdot,y) \|_{H^{4}_\h} + \| (w, \frak{w})(t,\cdot,y)\|_{H^{5}_\h} + \| (\eta_{t}, w_{t}, \frak{w}_{t})(t,\cdot,y)\|_{H^3_\h}\\
&\ + \| (\eta_{tt}, w_{tt}, \frak{w}_{tt})(t,\cdot,y)\|_{H^1_\h} \Bigr) + \sum_{j=1}^{2} \Bigl( \|\d_{y}^{j} \eta \|_{H^{3-j}_\h} + \|\d_{y}^{j} (w, \frak{w})(t,\cdot,y)\|_{H^{4-j}_\h}\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad +\| \d_{y}^{j} (\eta_{t},w_{t}, \frak{w}_{t})(t,\cdot,y)\|_{H^{2-j}_\h}\Bigr) \leq C A_{5}^{\frac{1}{2}}(y) e^{-\a t}.
\ea
}\end{theorem}
We point out that to prove Theorem \ref{thm1},
it is crucial to establish the related estimates for $(\eta, w)$ in \eqref{thm1-0}.
Indeed with thus obtained estimates for $(\eta, w)$,
those estimates for $\frak{w}$ follow immediately. The proof of Theorem \ref{thm1} will be
presented from Section \ref{sec:1dNS} to Section \ref{Sect6}.
The main result of this paper states as follows:
\begin{theorem}\label{thm2}
{\sl Let $(\varsigma_{0}, w_{0}, \frak{w}_0)$ satisfy \eqref{ass-ini-1} and \eqref{ini-1}. Then
\eqref{CNS}--\eqref{ini-data} has a unique global-in-time strong solution $(\rho_\e, u_\e) \in C([0,\infty); H^2(\TT\times \R)).$
Furthermore, let $(\eta, w,\frak{w})$ be the global solution to (\ref{CNS-limit}--\ref{CNS-limit-initial})
obtained in Theorem \ref{thm1}, we denote
\be\label{thm2-1}
\vr_\e \eqdefa \rho_\e - [\eta]_\e \andf R_\e \eqdefa u_\e - \left([w]_\e, [\frak{w}]_\e\right)^{\rm T},
\ee
and the energy functional
\ba\label{thm2-2}
E_\e(T) \eqdefa & \sup_{0 < t < T} \int_{\TT\times \R} \left( |R_\e|^{2} + |\vr_\e|^{2} + |\nabla R_\e|^{2} + |\fD_{t}R_\e|^{2} + |\nabla \o_\e|^{2}\right)\,\dx\,\dy \\
& + \int_{0}^{T}\int_{\TT\times \R} \left( |\nabla R_\e|^{2} + |\fD_{t}R_\e|^{2} + |\nabla \o_\e|^{2} + \ |\nabla \fD_{t}R_\e |^{2} \right)\,\dx\,\dy\,\dt,\\
\th_\e(T)\eqdefa & \sup_{0 < t < T} \|\vr_\e(t)\|_{L^{\infty}(\TT\times \R)},
\ea
where $\o_\e \eqdefa curl R_\e = \d_{y} R_\e^{1} - \d_{x} R_\e^{2}$, $\fD_{t}R_\e \eqdefa \d_{t}R_\e + u_\e \cdot \nabla R_\e$. Then there exists a constant $C$ solely depending on $(a, \g, \mu, \mu', \bar \varsigma_0, \underline{\varsigma}_0, \|A_{5}\|_{L^{1}\cap L^{\infty}(\R_{y})})$ such that
\ba\label{thm2-3}
E_\e(\infty) + \th_\e^{2}(\infty) \leq C \e.
\ea
}
\end{theorem}
We remark that the main idea used to prove Theorem \ref{thm2} is to approximate 2D compressible Navier-Stokes equations with a slow variable via 1D compressible Navier-Stokes equations with a parameter. This is inspired by the study in \cite{CG10,CZ6} where 3D incompressible Navier-Stokes equations with a slow variable can be well approximated by 2D Navier-Stokes equations which is globally well-posed.
\smallskip
Let us end this section with some notations that we shall use throughout this paper.
\noindent{\bf Notations:} In the whole paper,
we designate $\OO\eqdefa\TT\times \R,$ $ Q_{T} \eqdefa (0,T)\times \OO$ and
$Q_{\infty} \eqdefa (0,+\infty)\times \OO.
$
We shall always denote $C$ to be a uniform constant
which may vary from line to line, and $a\lesssim b$ means that $a\leq Cb$, and $a\thicksim b$
means that both $a\leq Cb$ and $b\leq Ca$ hold.
And we use subscript $\h$ (resp. ${\rmv}$) to denote the norm taken
on $\TT_{x}$ (resp. $\R_{y}$).
\section{Ideas and structure of the proof}\label{sec2}
In this section, we shall sketch the main ideas of the proof to Theorems \ref{thm1} and \ref{thm2}.
For each fixed $y\in \R$, the global existence and uniqueness of strong solutions to 1D compressible Navier-Stokes equations \eqref{CNS-limit} is well known (see
for instance \cite{Hoff87,KS77, Solo76, Solo80, SZ02}).
Although the results in the above references focus on domains either being the whole line $\R$ or being a bounded interval,
the well-posedness results can be easily modified to the torus $\TT$. More precisely, under the assumptions that
\be\label{ini-data-1d-ass}
0<\underline\varsigma_0 \leq \varsigma_0 \leq \bar \varsigma_0 <\infty, \quad \varsigma_0-1 \in H^1(\TT), \quad w _0\in H^1(\TT),
\ee
\eqref{CNS-limit}--\eqref{CNS-limit-initial} has a unique global solution $( \eta , w )$ so that for each $T>0$
\ba\label{sl-1d-0}
&0<\underline\varsigma(T) \leq \eta \leq \bar \varsigma(T)<\infty, \quad \eta-1 \in C([0,T];H^1(\TT)),\quad \eta _t\in C([0,T];L^2(\TT)),\\
& w \in C([0,T]; H^1(\TT)),\quad w \in L^2(0,T;H^2(\TT)), \andf w _t \in L^2(0,T;L^2(\TT)).
\ea
In general, the related estimates in \eqref{sl-1d-0} depend on the time interval $[0,T]$. We shall show below that
such estimates hold uniformly in time. Actually, we shall show exponential decay-in-time estimates of the solutions. In the case when spatial domain is a bounded interval and
with homogeneous
boundary condition for $w,$ Stra\v{s}kraba and Zlotnik \cite{SZ02} established exponential decay-in-time estimate for the $H^1$ norms of $ \eta - 1$ and $w$. It is not obvious that the same exponential decay estimate holds for the case of torus uniformly in $y\in \R$. One difficulty lies in that, unlike the case in bounded domain with homogeneous boundary condition for $w,$ we can not have a similar version of Poincar\'e inequality on torus $\TT$.
Another difficulty is due to the presence of the parameter $y\in \R.$ We will have to show that the corresponding decay estimates are uniform with respect to $y\in \R$ and $(\d_y^j \eta ,\d_y^j w ),$ $j=1,2,$ share the same decay estimate. To achieve this, we need to show the density function $\eta$ admits a uniform finite upper bound and a uniform positive lower bound with respect to $y\in \R$. In particular, in order to derive the uniform strictly positive lower bound, one needs to show that as time goes to infinity, the kinetic energy goes to zero and the integral of the pressure goes to a strictly positive number with a speed independent of $y\in \R$. The result in \cite{SZ02}
can not be directly applied here.
Instead, we shall first establish the uniform upper bound of the density by using the idea in \cite{SZ02}. Then we find that this is enough to derive the exponential decay for the basic energy, see Proposition \ref{prop-decay-exp-L2}. Since $(\d_y^j \eta ,\d_y^j w ),$ $j=1,2,$ does not fulfill a typical 1D compressible Navier-Stokes system, there arise new difficulties in this procedure of deriving the exponential decay estimates for $(\d_y^j \eta,\d_y^j w),$ $j=1,2$. A further difficulty comes from the Gagliardo-Nirenberg interpolation inequality in $\OO$. Since $\OO = \TT \times \R$ which is essentially bounded in $x$ variable, thus the classical Gagliardo-Nirenberg interpolation inequality reads ($p>2$):
$$
\|f\|_{L^{p}(\OO)} \leq C \| f \|_{L^{2}(\OO)}^{\frac 2p} \| f \|_{H^{1}(\OO)}^{1-\frac{2}{p}} \leq C \| f \|_{L^{2}(\OO)} + C \| f \|_{L^{2}(\OO)}^{\frac 2p} \| \nabla f \|_{L^{2}(\OO)}^{1-\frac{2}{p}} , \quad \forall f \in H^{1}(\OO).
$$
However, it happens to us that we can control the integral of $\| \nabla f \|_{L^{2}(\OO)}^{2}$ in time variable over $\R^+$ for some $f$, but not for the quantity $\| f \|_{L^{2}(\OO)}^{2},$ see for instance \eqref{vflux-2} and \eqref{vflux-4} on the effective viscous flux in the proof of Lemma \ref{prop-nablaR-6}. So we need a modified version of Gagliardo-Nirenberg interpolation inequality in $\OO$ which always involves the term $\| \nabla f \|_{L^{2}(\OO)},$ which is Lemma \ref{lem-GN} below.
\medskip
In Section \ref{sec:1dNS}, we shall present the detailed decay-in-time estimates for solutions of \eqref{CNS-limit}--\eqref{CNS-limit-initial}.
\begin{prop}\label{S2prop1}
{\sl Under the assumptions of Theorem \ref{thm1}, \eqref{CNS-limit}--\eqref{CNS-limit-initial} has a unique global solution $(\eta,w)$ and there exist positive constants $\underline{\eta}, \bar{\eta},$ $C$ and $\a$ solely depending on $(a, \g, \nu, \bar \varsigma_0, \underline{\varsigma}_0, \bar A_{4})$ such that \eqref{thm1-2} and
\ba\label{S2eq1}
\| (\eta -1)(t,\cdot,y) \|_{H^{4}_\h} + \|w(t,\cdot,y)\|_{H^{5}_\h} &+ \sum_{i=1}^2\bigl\| (\d_{t}^i\eta, \d_{t}^iw)(t,\cdot,y)\bigr\|_{H^{5-2i}_\h}\leq C A_{5}^{\frac{1}{2}}(y) e^{-\a t}.
\ea
hold for any $ t\in \R_+$ and $ y\in \R.$
}
\end{prop}
To avoid notational complexity, we shall denote $E(y)\in L^1(\R)\cap L^{\infty}(\R)$ to a universal function, which is determined by the initial conditions. The positive constants $C$ and $\a$ are also determined by the initial conditions. From now on, we shall not point out the precise dependence of $E(y)$ and the constants $C, \alpha,$ and we neglect the dependence on the $y$ variable.
\begin{prop}\label{S2prop2}
{\sl Under the assumptions of Theorem \ref{thm1}, there exist positive constants $C$ and $\a$ so that
the global solution $(\eta,w)$ of \eqref{CNS-limit}--\eqref{CNS-limit-initial} verifies
\ba\label{decay-exp-all-y-2}
\| \eta_y(t) \|_{H^2_\h} + \| w_{y}(t) \|_{H^3_\h} + \|(\eta _{yt},w_{yt})(t)\|_{H^1_\h} \leq C E^{\frac 12}(y) e^{-\a t}, \quad \forall \, t\in \R^{+}, \ y\in \R.
\ea
}
\end{prop}
\begin{prop}\label{S2prop3}
{\sl Under the assumptions of Theorem \ref{thm1}, there exist positive constants $C$ and $\a$ so that the global solution $(\eta,w)$ of \eqref{CNS-limit}--\eqref{CNS-limit-initial} fulfills
\ba\label{decay-exp-all-yy-2}
\| \eta_{yy}(t) \|_{H^1_\h} + \| w_{yy}(t) \|_{H^2_\h} + \|(\eta_{yyt},w_{yyt})(t)\|_{L^2_\h} \leq
E^{\frac 12}(y) e^{-\a t}, \quad \forall \, t\in \R^{+}, \ y\in \R.
\ea
}
\end{prop}
\begin{prop}\label{S2prop4}
{\sl Under the assumptions of Theorem \ref{thm1}, \eqref{tw}--\eqref{CNS-limit-initial} has a unique global solution
$\frak{w}$ and there exists positive constants $C$ and $\a$ so that for all $t\in \R^{+}$ and
$y\in \R$
\ba\label{S2eq4}
\sum_{j=1}^{2} \bigl(\|\d_t^j\frak{w}(t)\|_{H^{5-2j}_\h} +\|\d_{y}^{j} \frak{w}(t)\|_{H^{4-j}_\h} +\| \d_{y}^{j} \frak{w}_{t}(t)\|_{H^{2-j}_\h}\bigr)
\leq C E^{\frac 12}(y) e^{-\a t} \quad \forall \, t\in \R^{+}, \ y\in \R.
\ea
}
\end{prop}
The proof of Proposition \ref{S2prop2} will be presented in Section \ref{sec:1dNS-y}. While we shall outline the proof of Propositions \ref{S2prop3} and \ref{S2prop4} in Sections \ref{sec:1dNS-yy}
and \ref{Sect6} respectively. By combing Propositions \ref{S2prop1}-\ref{S2prop4}, we conclude the proof of Theorem \ref{thm1}.
Let $(\eta,w,\frak{w})$ be the unique global solution of \eqref{CNS-limit}-\eqref{CNS-limit-initial}, which has been
constructed in Theorem \ref{thm1}. We define
\be\label{zeta-W-def}
\rho^{\rm a}_\e \eqdefa [\eta]_\e, \quad u^{\rm a}_\e \eqdefa ([w]_\e, [\frak{w}]_\e)^{\rm T}.
\ee
Then in view of \eqref{CNS-limit} and \eqref{tw}, we have
\be\label{CNS-1d-W}
\left\{\begin{aligned}
&\d_t \rho^{\rm a}_\e + \dive (\rho^{\rm a}_\e u^{\rm a}_\e ) = \e [(\eta \frak{w})_{y}]_\e,\\
&\rho^{\rm a}_\e(\d_t u^{\rm a}_\e + u^{\rm a}_\e \cdot \nabla u^{\rm a}_\e ) - \mu \Delta u^{\rm a}_\e- \mu' \nabla \dive u^{\rm a}_\e + \nabla p(\rho^{\rm a}_\e) = - G_\e,
\end{aligned}\right.
\ee
where $G_\e = (G_{1,\e}, G_{2,\e})^{\rm T}$ is given by
\ba\label{def-G12}
& G_{1,\e}\eqdefa - \e [\eta\frak{w}w_{y} ]_\e + \mu\e^{2 }[w_{yy}]_\e + \mu' \e [\frak{w}_{xy}]_\e, \\
& G_{2,\e}\eqdefa - \e [\eta\frak{w}\frak{w}_{y} ]_\e + \nu \e^{2} [\frak{w}_{yy} ]_\e + \mu' \e [w_{xy}]_\e - \e [p(\eta)_{y}]_\e.
\ea
Then by virtue of \eqref{thm1-0}, we deduce that
\ba\label{def-G12-est}
\|\d_{t} G_{\e}\|_{L^{2}(\Omega)} \leq C \e^{\frac{1}{2}}e^{-\a t}, \quad \| G_{\e}\|_{L^{2}(\Omega)} \leq C \e^{\frac{1}{2}}e^{-\a t}, \quad \| G_{\e}\|_{L^{\infty}(\Omega)} \leq C \e e^{-\a t}.
\ea
On the other hand, with initial data given by \eqref{ini-data} and $(\varsigma_0, w_0,\frak{w}_0)$ satisfying \eqref{ass-ini-1} and \eqref{ini-1},
there exists a positive time $T^\ast_\e>0$ so that \eqref{CNS}--\eqref{ini-data} has a unique solution $( \rho_\e , u_\e )$ on
$[0,T^\ast_\e)$ and for any $T<T^\ast_\e,$
\ba\label{sl-1d-00}
&0<\underline\rho(T) \leq \rho_\e \leq \bar \rho(T)<\infty, \quad \rho_\e \in C([0,T];H^2(\OO)),\quad \d_t\rho_\e \in C([0,T];H^1(\OO)),\\
& u_\e \in C([0,T]; H^2(\OO))\cap L^2(0,T;H^3(\OO)), \quad \d_tu_\e \in C([0,T];L^2(\OO)) \cap L^2(0,T;H^1(\OO)).
\ea
One may check \cite{Nash62, Serrin59} for details.
Let $T^\ast_\e$ be the maximal existence time of $(\rho_\e, u_\e)$ so that \eqref{sl-1d-00} holds. We are going to prove that
$T^\ast_\e=\infty$ for $\e$ being sufficiently small. In order to do so, we define the remaining term
\be\label{error}
\vr_\e \eqdefa \rho_\e - \rho^{\rm a}_\e \andf R_\e \eqdefa u_\e - u^{\rm a}_\e .
\ee
Recall that $\frak{D}_t\eqdefa \d_t + u_\e \cdot \nabla.$ Then it follows from \eqref{CNS} and \eqref{CNS-1d-W} that
\be\label{CNS-error-new}
\left\{\begin{aligned
&\d_t \vr_\e + \dive (\rho_\e R_\e) + \dive (\vr_\e u^{\rm a}_\e) = - \e [(\eta \frak{w})_{y}]_\e,\\
&\rho_\e \frak{D}_t R_\e- \mu \Delta R_\e- \mu' \nabla\dive R_\e + \nabla \big(p(\rho_\e) - p(\rho^{\rm a}_\e) \big) \\
&\qquad+ \rho_\e R_\e\cdot \nabla u^{\rm a}_\e + \vr_\e (\d_{t}u^{\rm a}_\e + u^{\rm a}_\e \cdot \nabla u^{\rm a}_\e) = G_\e,\\
&\vr_\e(0,x,y) = 0, \quad R_\e(0,x,y) = 0.
\end{aligned}\right.
\ee
We define
\ba\label{a-priori-r}
T^\star_\e\eqdefa \sup\big\{ T<T^\ast_\e; \ \theta_\e(T) \eqdefa \sup_{0<t<T}\| \vr_\e \|_{L^{\infty}(\OO)} \leq \frac{1}{2} \min\left\{1,\underline\eta\right\} \big\}.
\ea
It is easy to observe that for all $t< T^\star_\e$:
\ba\label{upper-lower-rho-0}
0<{\underline \eta}/{ 2 } \leq \rho_\e \leq \bar \eta + {\underline \eta}/{ 2 } < \infty.
\ea
We shall first derive the basic energy estimate for $(\vr_\e,R_\e)$ for $t < T^\star_\e.$
A standard way to perform this estimate is to test the momentum type equation in \eqref{CNS-error-new} by $R_\e$ and using integration by parts. A tricky term to handle is the one related to the pressure:
\be\label{tricky-p-diveR}
\int_{\Omega}\big( p(\rho_\e) - p(\rho_\e^{\rm a}) \big) \dive R_\e\,\dx\,\dy.
\ee
Similar term $p(\rho_{\e}) \dive u_{\e}$ appears in the renormalized formulation to the continuity equation of the
compressible Navier-Stokes equations:
$$
\d_{t}P(\rho_{\e}) + \dive (P(\rho_{\e}) u_{\e}) + (\rho_{\e} P'(\rho_{\e}) - P(\rho_{\e}))\dive u_{\e} = 0,
$$
where $P(\rho)$ is called the pressure potential, which is determined by
$$
\rho P'(\rho) - P(\rho) = p(\rho).
$$
As a result, it comes out
$$
\int_{\Omega} p(\rho_{\e}) \dive u_{\e} \,\dx\,\dy= -\frac{\rm d}{\dt} \int_{\Omega} P(\rho_{\e})\,\dx\,\dy.
$$
In particular, for our case $p(\rho) = a \rho^{\g}$ with $\g\geq 1$, the corresponding pressure potential is
\be\label{pre-pot}
P(\rho)\eqdefa \frac{1}{\g-1} \rho^{\g} \ \mbox{if $\g>1$}, \quad P(\rho) \eqdefa a (\rho \log \rho + 1) \ \mbox{if $\g = 1$}.
\ee
However, it is not clear at a first glance how to use such an argument to deal with the term \eqref{tricky-p-diveR}. Here we shall employ the well-known {\rm relative entropy inequality} to derive the basic energy estimate for $\vr_\e$ and $R_\e$.
Relative entropy inequality is widely used in the study of uniqueness and stability for Navier--Stokes equations and some related models, see \cite{FJN12,Germain10} for compressible Navier--Stokes equations, \cite{FNS14} for compressible Navier--Stokes--Fourier equations, and \cite{Lu-Zhang18} for a compressible Oldroyd model. The result states as follows:
\begin{prop}\label{prop-rela}
{\sl Let $(\rho_\e, u_\e)$ be the local-in-time strong solution of \eqref{CNS}-\eqref{ini-data} satisfying \eqref{sl-1d-00}. For each pair $(\tilde \rho, \tilde u)$ satisfying the same regularity assumption as that of $(\rho_\e, u_\e)$ listed in \eqref{sl-1d-00}, we define the {\em relative energy functional}
\ba\label{def-rel-fuc}
\calE_{1}\bigl((\rho_\e, u_\e) | (\tilde \rho, \tilde u)\bigr) (t)\eqdefa \int_{\Omega} \Bigl(\frac{1}{2} \rho_\e |u_\e-\tilde u|^{2} + P(\rho_\e) - P(\tilde \rho) - P'(\tilde \rho) (\rho_\e - \tilde \rho)\Bigr)\,\dx\,\dy,
\ea
where the pressure potential $P$ is defined by \eqref{pre-pot}. Then for any $t\in (0,T^\ast_\e)$, the following \emph{relative entropy equality} holds
\ba\label{ineq-entropy}
\calE_1\bigl((\rho_\e,u_\e) | (\tilde\rho,\tilde u)\bigr)(t)
+ \int_0^t \int_\Omega \bigl(\mu \left| \nabla (u_\e-\tilde u) \right|^2 +& \mu' |\dive (u_\e-\tilde u) |^2\bigr)\,\dx\,\dy \,\dt' \\
=& \calE_1\bigl((\rho_\e,u_\e)|_{t=0} | (\tilde\rho,\tilde u)|_{t=0}\bigr) + \int_0^t \calR_\e (t') \,\dt',
\ea
where
\ba\label{R-def}
\calR_\e&(t) \eqdefa \int_\Omega \rho_\e\frak{D}_t \tilde u\cdot (\tilde u - u_\e)\,\dx\,\dy
+ \int_\Omega \bigl(\mu \nabla \tilde u :\nabla (\tilde u - u_\e) + \mu' \dive \tilde u \,\dive (\tilde u-u_\e)\bigr)\,\dx\,\dy\\
& + \int_\Omega\bigl( (\tilde \rho - \rho_\e) \d_t P'(\tilde \rho_\e) + (\tilde \rho \tilde u -\rho_\e u_\e) \cdot \nabla P'(\tilde \rho)\bigr)\,\dx\,\dy - \int_\Omega \dive \tilde u (p(\rho_\e ) - p(\tilde \rho))\,\dx\,\dy.
\ea
}
\end{prop}
There are also similar relative entropy inequalities related to finite energy weak solutions of different models: for instance, Theorem 2.4 of \cite{FJN12} for the compressible Navier--Stokes equations and Proposition 5.3 of \cite{Lu-Zhang18} for a compressible Oldroyd-B model. The proof of Proposition \ref{prop-rela} follows the same line as the argument in \cite{FJN12} or \cite{Lu-Zhang18}, and we skip the details here.
Thanks to Proposition \ref{prop-rela}, we shall prove in Section \ref{Sect7.1} that
\begin{prop}\label{prop-energy-basic}
{\sl Let $T_\e^\star$ be given by \eqref{a-priori-r}. Then for all $ t < T^\star_\e$, one has
\ba\label{energy-bas}
\int_{\Omega} \bigl(|R_\e|^{2} + \vr_\e^{2}\bigr)(t) \,\dx\,\dy + \int_{0}^{t} \int_{\Omega} |\nabla R_\e|^{2}\,\dx\,\dy\,\dt' \leq C \e,
\ea
where the positive constants $C$ is independent of $T^\star_{\e}$.
}
\end{prop}
In Section \ref{Sect7.4}, we shall prove the energy estimate for the derivatives of $R_\e.$
\begin{prop}\label{energy}
{\sl Let the energy functional $E_\e(T)$ be given by \eqref{thm2-2}. Then for all $T \leq T^\star_\e,$ one has
\be\label{E(T)-1}
E_\e(T) \leq C \e + C\int_{0}^{T} \int_{\OO} \bigl(|\nabla R_\e|^{3} + |\nabla R_\e|^{4}\bigr)\,\dx\,\dy \,\dt.
\ee}
\end{prop}
In order to close the energy estimate, \eqref{E(T)-1}, we need to handle the estimates of the cubic and quadratic terms:
$$
\int_{0}^{t} \int_{\OO} |\nabla R_\e|^{3}\,\dx\,\dy \,\dt' \andf \int_{0}^{t} \int_{\OO} |\nabla R_\e|^{4}\,\dx\,\dy \,\dt'.
$$
Here we introduce the following refined Gagliardo-Nirenberg interpolation inequality in $\OO$:
\begin{lem}\label{lem-GN}
{\sl Let $2<p<\infty$, there exists a constant $C$ depending solely on $p$ such that for all $f\in H^{1}(\OO)$ there holds
\ba\label{GN-O}
\|f\|_{L^{p}(\OO)} \leq C \Bigl( \| f \|_{L^{2}(\OO)}^{\frac 2p} \|\nabla f \|_{L^{2}(\OO)}^{1-\frac{2}{p}} + \| f \|_{L^{2}(\OO)}^{\frac{1}{2} + \frac{1}{p}} \|\nabla f \|_{L^{2}(\OO)}^{\frac{1}{2} - \frac{1}{p}} \Bigr).
\ea
}
\end{lem}
The proof of Lemma \ref{lem-GN} will be presented in Appendix \ref{appb}.
\medskip
Motivated by \cite{Hoff95, Lions-C}, we define the effective viscous flux $\frak{F}$ for the system \eqref{CNS-error-new} as follows
\be\label{visflux-def}
\fF_\e\eqdefa \nu \dive R_\e -\big(p(\rho_\e) - p(\rho_\e^{\rm a}) \big).
\ee
\begin{lem}\label{lem-p-p}
{\sl For $T\leq T^\star_\e,$ one has \be\label{p-p-0}
\int_{0}^{T}\int_{\OO} | \rho_\e-\rho^{\rm a}_\e |^{6}\,\dx\,\dy \,\dt \leq C \e^{3} + C \e^{2} E_{\e}(T) + C \int_0^T\int_{\OO} |\fF_\e|^{6}\,\dx\,\dy \,\dt.
\ee
}
\end{lem}
\begin{proof}
In view of the continuity equations of \eqref{CNS} and \eqref{CNS-error-new}, we write
\ba
\fD_{t} (\log \rho_\e - \log \rho^{\rm a}_\e ) + R_\e \cdot \nabla \log \rho^{\rm a}_\e + \dive R_\e = - \e (\rho_\e^{\rm a})^{-1} [(\eta \frak{w})_{y}]_\e,
\nn \ea
which together with \eqref{visflux-def} implies
\ba
\fD_{t} (\log \rho_\e - \log \rho_\e^{\rm a} )
= - R_\e \cdot \nabla \log \rho_\e^{\rm a} - \nu^{-1} \bigl(\fF_\e+ \big(p(\rho_\e) - p(\rho_\e^{\rm a})\big)\bigr) - \e (\rho_\e^{\rm a})^{-1} [(\eta \frak{w})_{y}]_\e.
\nn \ea
Multiplying the above equation by $6 (\log \rho_\e - \log \rho_\e^{\rm a} )^{5} $ and then applying Young's inequality gives
\begin{align*}
& \fD_{t} (\log \rho_\e - \log \rho_\e^{\rm a} )^{6} + 6 \nu^{-1} \big(p(\rho_\e) - p(\rho_\e^{\rm a})\big) (\log \rho_\e - \log \rho_\e^{\rm a} )^{5} \\
& =- 5 \left( R_\e\cdot \nabla \log \rho_\e^{\rm a} +\nu^{-1} \fF_\e + \e (\rho_\e^{\rm a})^{-1} [(\eta \frak{w})_{y}]_\e\right) (\log \rho_\e - \log \rho_\e^{\rm a} )^{5}\\
& \leq C_{\de} \left(|R_\e \cdot \nabla \log \rho_\e^{\rm a}|^{6} + |\fF_\e|^{6} + | \e (\rho_\e^{\rm a})^{-1} [(\eta \frak{w})_{y}]_\e |^{6}\right) + \de |\log \rho_\e - \log \rho_\e^{\rm a} |^{6}.
\end{align*}
Notice that for $t < T^\star_\e,$ there holds \eqref{upper-lower-rho-0} so that
\ba
6 \nu^{-1} \big(p(\rho_\e) - p(\rho_\e^{\rm a})\big) (\log \rho_\e - \log \rho_\e^{\rm a} )^{5} \geq 2 C \de |\log \rho_\e - \log \rho_\e^{\rm a} |^{6} \geq \de |\log \rho_\e - \log \rho_\e^{\rm a} |^{6} + \de (\rho_\e - \rho_\e^{\rm a})^{6},
\nn \ea
for some small positive constant $\de$ and some $C\geq 2$. We thus obtain
\ba\label{p-p-7}
\fD_{t} (\log \rho_\e - \log \rho_\e^{\rm a} )^{6} + \de (\rho_\e - \rho_\e^{\rm a})^{6} \leq C_{\de}\left( | R_\e \cdot \nabla \log \rho_\e^{\rm a}|^{6} + |\fF_\e|^{6} + | \e (\rho_\e^{\rm a})^{-1} [(\eta \frak{w})_{y}]_\e |^{6}\right).
\ea
Let us introduce the particle trajectory $X_\e(t,x,y)$ of $u_{\e}$ via
\be\label{flow-def}
\frac{\rm d}{\dt} X_\e(t,x,y) = u_\e(t, X_\e(t,x,y)); \quad X_\e(0,x,y) = (x,y).
\ee
Then we deduce from \eqref{p-p-7} that
\ba\label{p-p-8}
& \frac{\rm d}{\dt} \bigl(\log \rho_\e - \log \rho_\e^{\rm a} \bigr)^{6} (t, X_\e(t,x,y)) + \de \bigl(\rho_\e - \rho_\e^{\rm a}\bigr)^{6} (t, X_\e(t,x,y)) \\
& \leq C_{\de}\left( | R_\e \cdot \nabla \log \rho_\e^{\rm a}|^{6} + |\fF_\e|^{6} + \e^6 | (\rho_\e^{\rm a})^{-1} [(\eta \frak{w})_{y}]_\e |^{6}\right) (t, X_\e(t,x,y)) .
\ea
Let $T\leq T_{\e}^{\star}$. Integrating the above inequality over $(0,T)\times\Omega$ yields
\ba\label{p-p-11}
&\int_{\OO}\bigl(\log \rho_\e - \log \rho_\e^{\rm a} \bigr)^{6} (T, X_\e(T,x,y))\,\dx\,\dy + \delta\int_{0}^{T} \int_{\OO} (\rho_\e - \rho_\e^{\rm a})^{6} (t, X_\e(t,x,y))\dx\dy\dt \\
&\leq C \int_{0}^{T} \int_{\OO} \bigl(| R_\e \cdot \nabla \log \rho_\e^{\rm a}|^{6}(t, X_\e(t,x,y))+|\fF_\e|^{6}(t, X_\e(t,x,y))\bigr) \dx\dy\dt\\
&
+ C \e^6 \int_{0}^{T} \int_{\OO} |(\rho_\e^{\rm a})^{-1} [(\eta \frak{w})_{y}]_\e |^{6} (t, X_\e(t,x,y))\,\dx\dy\dt.
\ea
Recall \eqref{upper-lower-rho-0} and Lemma 3.2 of \cite{Hoff95} that there exists $C$ depending only on the positive lower bound and upper bound of $\rho_{\e}$ such that for any nonnegative smooth integrable function $g,$ there holds for all $t<T_{\e}^{\star}$,
\ba
C^{-1} \int_{\OO} g(t,x,y) \dx\dy \leq \int_{\OO} g(t,X_\e(t,x,y)) \dx\dy \leq C\int_{\OO} g(t,x,y) \dx\dy.
\nn \ea
We thus deduce from \eqref{thm1-2} and \eqref{p-p-11} that
\ba\label{p-p-12}
&\int_{0}^{T} \int_{\OO} (\rho_\e - \rho_\e^{\rm a})^{6} (t, x,y)\dx\dy\dt \leq C \int_{0}^{T} \int_{\OO} | R_\e \cdot \nabla \log \rho_\e^{\rm a}|^{6}(t, x,y) \dx\dy\dt\\
&\quad + C \int_{0}^{T} \int_{\OO} |\fF_\e|^{6}(t, x,y)\dx\dy\dt+ C \e^6 \int_{0}^{T} \int_{\OO} | (\rho_\e^{\rm a})^{-1} [(\eta \frak{w})_{y}]_\e |^{6} (t, x,y)\,\dx\dy\dt.
\ea
It follows from Theorem \ref{thm1}, Proposition \ref{prop-energy-basic} and Lemma \ref{lem-GN} that
\begin{align*}
& \int_{0}^{T} \int_{\OO} | R_\e \cdot \nabla \log \rho_\e^{\rm a}|^{6}(t, x,y) \,\dx\,\dy\,\dt\\
& \leq C \int_{0}^{T}\int_{\OO} |R_\e|^{6} e^{-\a t}\,\dx\,\dy\,\dt\\
& \leq C \int_{0}^{T}e^{-\a t} \big(\|R_\e\|_{L^{2}(\OO)}^{2} \|\nabla R_\e\|_{L^{2}(\OO)}^{4} + \|R_\e\|_{L^{2}(\OO)}^{4} \|\nabla R_\e\|_{L^{2}(\OO)}^{2} \big)\dt \\
& \leq C \sup_{0<t<T} \big(\|R_\e\|_{L^{2}(\OO)}^{2} \|\nabla R_\e\|_{L^{2}(\OO)}^{2} + \|R_\e\|_{L^{2}(\OO)}^{4} \big) \int_{0}^{T} e^{-\a t} \|\nabla R_\e\|_{L^{2}(\OO)}^{2}\,\dt\\
& \leq C \e^{2}(E_{\e}(T) + \e),
\end{align*}
and
\begin{align*}
\e^6\int_{0}^{T} \int_{\OO} | (\rho_\e^{\rm a})^{-1} [(\eta \frak{w})_{y}]_\e |^{6} (t, x,y)\,\dx\,\dy\,\dt
& \leq \underline \eta^{-1} \e^{6} \int_{0}^{T}\int_{\OO}\e^{-1} |(\eta \frak{w})_{y}|^{6} \,\dx\,\dy\,\dt\\
& \leq C \e^{5} \int_{0}^{T}e^{-\a t} \dt \leq C \e^{5}.
\end{align*}
By inserting the above estimates into \eqref{p-p-12}, we conclude the proof of \eqref{p-p-0}.
\end{proof}
\begin{lem}\label{prop-nablaR-6}
{\sl For $T\leq T^\star_\e,$ there holds
\ba\label{nablaR-6-0}
\int_{0}^{T} \int_{\OO} |\nabla R_\e|^{6} \,\dx\,\dy\,\dt \leq C\e^{3} + C \e^{2} E_{\e}(T) + C \e E^{2}_\e(T).
\ea
}
\end{lem}
\begin{proof}
Recall that $ \o_\e=\d_yR^1_\e-\d_xR^2_\e,$ we observe that
$$
\Delta R_\e = \nabla \dive R_\e + \nabla^{\perp} \o_\e, \quad \nabla^{\perp} \eqdefa \bp \d_{y} \\ - \d_{x}\ep,
$$
which together with \eqref{visflux-def}
implies
\ba
\nu \Delta R_\e = \nabla \fF_\e + \nabla \big(p(\rho_\e) - p(\rho_\e^{\rm a}) \big) + \nu \nabla^{\perp} \o_\e.
\nn \ea
Then we get, by using standard elliptic estimates that
\ba\label{cubic-0}
\|\nabla R_\e\|_{L^{6}(\OO)} \leq C \big(\|\o_\e\|_{L^{6}(\OO)} + \|\fF_\e\|_{L^{6}(\OO)} + \|p(\rho_\e) - p(\rho_\e^{\rm a}) \|_{L^{6}(\OO)} \big).
\ea
Next let us estimate term by term on the right-hand side of above equation.
By applying Lemma \ref{lem-GN}, we obtain
\ba
\|\o_\e\|_{L^{6}(\OO)} \leq C \Bigl( \| \o_{\e} \|_{L^{2}(\OO)}^{\frac 13} \|\nabla \o_{\e} \|_{L^{2}(\OO)}^{\frac 23} + \| \o_{\e} \|_{L^{2}(\OO)}^{\frac 23} \|\nabla \o_{\e} \|_{L^{2}(\OO)}^{\frac 13} \Bigr).
\nn \ea
Together with \eqref{thm2-2} and \eqref{energy-bas}, we infer
\ba\label{vorticity-new-0}
\int_{0}^{T}\int_{\OO} |\o_\e|^{6}\,\dx\,\dy\,\dt & \leq C \int_{0}^{T} \left( \| \o_{\e} \|_{L^{2}(\OO)}^{ 2} \|\nabla \o_{\e} \|_{L^{2}(\OO)}^{ 4} + \| \o_{\e} \|_{L^{2}(\OO)}^{ 4} \|\nabla \o_{\e} \|_{L^{2}(\OO)}^{ 2} \right)\,\dt \\
& \leq C \sup_{0 < t < T} \big(\|\nabla \o_{\e}\|_{L^{2}}^{4} + \|\o_\e\|_{L^{2}}^{2} \|\nabla \o_\e\|_{L^{2}}^{2} \big)\int_{0}^{T}\|\nabla R_\e\|_{L^{2}(\OO)}^{2}\,\dt \\
& \leq C \e E^{2}_\e(T).
\ea
While in view of \eqref{energy-bas} and \eqref{visflux-def}, we have
\be\label{vflux-2}
\| \fF_\e \|_{L^{2}(\OO)} \leq \nu \| \nabla R_\e \|_{L^{2}(\OO)} + C \| \rho_\e - \rho_\e^{\rm a} \|_{L^{2}(\OO)} \leq C \big(\| \nabla R_\e \|_{L^{2}(\OO)} + \e^{\frac 12} \big).
\ee
By \eqref{visflux-def} and the $R_\e$ equation of \eqref{CNS-error-new}, we write
\b
\Delta \fF_\e & = \dive \big( \mu \Delta R_\e + \mu' \nabla \dive R_\e - \nabla \big(p(\rho_\e ) - p(\rho_\e ^{\rm a}) \big) \big) \\
& = \dive \left(\rho_\e \fD_{t}R_\e + \rho_\e R_\e\cdot \nabla u^{\rm a}_\e + \vr_\e (\d_{t} u^{\rm a}_\e + u^{\rm a}_\e \cdot \nabla u_\e^{\rm a}) - G_\e\right),\nonumber
\ea
from which, we infer
\ba\label{vflux-4}
\|\nabla \fF_\e\|_{L^{2}(\OO)}
& \leq C\big( \| \fD_{t}R_\e \|_{L^{2}(\OO)} + \|R_\e\| _{L^{2}(\OO)} \|\nabla u_\e^{\rm a}\|_{L^{\infty}(\OO)} \\
&\qquad+ \|\vr_\e\|_{L^{2}(\OO)} \bigl\|\d_{t} u_\e^{\rm a} + u_\e^{\rm a} \cdot \nabla u_\e^{\rm a}\bigr\|_{L^{\infty}(\OO)} + \|G_\e\|_{L^{2}(\OO)}\big) \\
& \leq C \| \fD_{t}R_\e \|_{L^{2}(\OO)} + C e^{-\a t} \bigl( \|R_\e\| _{L^{2}(\OO)} + \|\vr_\e\|_{L^{2}(\OO)} + \e^{\frac 12} \bigr)\\
& \leq C \bigl(\| \fD_{t}R_\e \|_{L^{2}(\OO)} + \e^{\frac 12} e^{-\a t}\bigr).
\ea
Thanks to \eqref{vflux-2} and \eqref{vflux-4}, we get, by applying Lemma \ref{lem-GN}, that
\ba\label{vflux-0}
\int_{0}^{T}\int_{\OO} |\fF_\e|^{6}\,\dx\,\dy\,\dt
& \leq C \int_{0}^{T}\big(\|\fF_\e\|_{L^{2}(\OO)}^{2} \|\nabla \fF_\e\|_{L^{2}(\OO)}^{4} + \|\fF_\e\|_{L^{2}(\OO)}^{4} \|\nabla \fF_\e\|_{L^{2}(\OO)}^{2} \big)\,\dt\\
& \leq C \int_{0}^{T} \big(\| \nabla R_\e \|_{L^{2}(\OO)}^{2} + \e \big) \big(\| \fD_{t} R_\e \|_{L^{2}(\OO)}^{2} + \e e^{-\a t}\big)\\
&\qquad\qquad\times\big(\| \nabla R_\e \|_{L^{2}(\OO)}^{2} + \e+ \| \fD_{t} R_\e \|_{L^{2}(\OO)}^{2} + \e e^{-\a t}\big) \,\dt\\
&\leq C \e^{3} + C \e E^{2}_\e(T).
\ea
By inserting the estimates \eqref{p-p-0}, \eqref{vorticity-new-0} and \eqref{vflux-0} into \eqref{cubic-0}, we achieve \eqref{nablaR-6-0}.
This completes the proof of Lemma \ref{prop-nablaR-6}.
\end{proof}
Now we are ready to give the following lemma in order to close the energy estimates:
\begin{lem}\label{prop-nablaR-3-4}
{\sl For $T\leq T^\star_\e,$ there holds
\ba\label{nablaR-4-0}
&\int_{0}^{T} \int_{\OO} |\nabla R_\e|^{4} \,\dx\,\dy\,\dt \leq C\e^{2} + C \e E_\e(T),\\
&\int_{0}^{T} \int_{\OO} |\nabla R_\e|^{3}\,\dx\,\dy\,\dt \leq C\e^{\frac 32} + C \e E^{\frac 12}_{\e}(T).
\ea
}
\end{lem}
\begin{proof}
Indeed it follows from Proposition \ref{prop-energy-basic}, Lemma \ref{prop-nablaR-6} and H\"older's inequality that
\ba
&\|\nabla R_\e\|_{L^{4}((0,T)\times \OO)}^{4} \leq \|\nabla R_\e\|_{L^{2}((0,T)\times \OO)} \|\nabla R_\e\|_{L^{6}((0,T)\times \OO)}^{3} \leq C \e^{\frac 12} \big(\e^{\frac 32} + \e^{\frac 12} E_{\e}(T)\big),\\
&\|\nabla R_\e\|_{L^{3}((0,T)\times \OO)}^{3} \leq \|\nabla R_\e\|_{L^{2}((0,T)\times \OO)}^{\frac 32} \|\nabla R_\e\|_{L^{6}((0,T)\times \OO)}^{\frac 32} \leq C \e^{\frac 34} \big(\e^{\frac 34} + \e^{\frac 14} E^{\frac 12}_{\e}(T)\big).
\nonumber
\ea
Then \eqref{nablaR-4-0} follow immediately.
\end{proof}
Next let us turn to the {estimate of $\th_\e(T).$}
\begin{lem}\label{prop-thT}
{\sl For $T\leq T^\star_\e,$ there holds
\ba\label{thT-0}
\th_\e^{2}(T) \leq C \e + C E_\e(T).
\ea
}
\end{lem}
\begin{proof}
Notice that for all $T\leq T^\star_{\e},$ $0< \underline \eta/2 \leq \rho_\e , \rho_\e ^{\rm a} \leq 3\bar \eta/2,$ we deduce from \eqref{p-p-8} that
\begin{align*}
\vr_\e^{6} (T, X_\e(T,x,y)) & \leq C\bigl(\log \rho_\e - \log \rho_\e ^{\rm a} \bigr)^{6} (T, X_\e(T,x,y))\\
& \leq C\int_{0}^{T} \Big( | R_\e \cdot \nabla \log \rho_\e ^{\rm a} |^{6} + |\fF_\e|^{6} + |\e [(\eta \frak{w})_{y}]_\e |^{6}\Big) (t, X_\e(t,x,y))\,\dt,
\end{align*}
from which, we infer
\ba\label{thT-7}
\| \vr_\e (T)\|_{L^{\infty}(\OO)}^{6} & \leq C\int_{0}^{T} \big( \| R_\e \|_{L^{\infty}(\OO)}^{6} \|(\rho_\e ^{\rm a})^{-1} \nabla \rho_\e ^{\rm a} \|_{L^{\infty}(\OO)}^{6} + \|\fF_\e\|_{L^{\infty}(\OO)}^{6} + \e^{6} \| (\eta \frak{w})_{y} \|_{L^{\infty}(\OO)}^{6} \big)\dt\\
& \leq C\int_{0}^{T } \big(e^{-\a t}\bigl( \| R_\e \|_{L^{\infty}(\OO)}^{6} + \e^{6}\bigr)
+ \|\fF_\e\|_{L^{\infty}(\OO)}^{6} \big)\,\dt.
\ea
It follows from Sobolev embedding theorem, Proposition \ref{prop-energy-basic} and Lemma \ref{prop-nablaR-6} that
\ba\label{EstR}
\int_{0}^{T} e^{-\a t} \|R_\e\|_{L^{\infty}(\OO)}^{6}\,\dt & \leq C\int_{0}^{T} e^{-\a t} \bigl(\|R_\e\|_{L^{2}(\OO)}^{6}
+ \|\nabla R_\e\|_{L^{6}(\OO)}^{6}\bigr)\,\dt\\
& \leq C\e^{3} + C \e E^{2}_\e(T).
\ea
Concerning the term related to the effective viscous flux, we get, by using Sobolev embedding inequality, that
\be\label{F-embed-Linfty}
\|\fF_\e\|_{L^{\infty}(\OO)} \leq C \big( \|\fF_\e\|_{L^{6}(\OO)} + \|\nabla \fF_\e\|_{L^{3}(\OO)}\big).
\ee
The estimate related to $\|\fF_\e\|_{L^{6}(\OO)}$ is given in \eqref{vflux-0}. While for $\|\nabla \fF_\e\|_{L^{3}(\OO)}$, it follows from a similar derivation of \eqref{vflux-4} that
\begin{align*}
\|\nabla \fF_\e\|_{L^{3}(\OO)} & \leq C\Bigl( \| \fD_{t}R_\e \|_{L^{3}} + \|R_\e\| _{L^{3}}
\|\nabla u^{\rm a}_\e\|_{L^{\infty}} + \|\vr_\e \|_{L^{3}} \bigl\|\d_{t} u^{\rm a}_\e + u_\e^{\rm a}\cdot \nabla u_\e^{\rm a}\bigr\|_{L^{\infty}} + \|G_\e\|_{L^{3}} \Bigr) \\
& \leq C \| \fD_{t}R_\e \|_{L^{3}(\OO)} + Ce^{-\a t} \bigl(\|R_\e\| _{L^{3}(\OO)} + \|\vr_\e \|_{L^{3}(\OO)} + \e^{\frac 23}\bigr),
\end{align*}
{where we used the estimate of $G_{\e}$ in \eqref{def-G12-est}.} By using the Gagliardo-Nirenberg interpolation inequality and \eqref{thm2-2}, we find
\ba
\int_{0}^{T} \| \fD_{t}R_\e \|_{L^{3}(\OO)}^{6}\,\dt & \leq C \int_{0}^{T}\| \fD_{t}R_\e \|_{L^{2}(\OO)}^{4} \| \fD_{t}R_\e \|_{H^{1}(\OO)}^{2}\,\dt \\
& \leq C \sup_{0 < t < T} \| \fD_{t}R_\e(t) \|_{L^{2}}^{4} \int_{0}^{T} \| \fD_{t}R_\e \|_{H^{1}(\OO)}^{2}\,\dt \leq C E^{3}_{\e}(T).
\nonumber\ea
Similarly, we have
\begin{align*}
\int_{0}^{T} e^{-\a t} \|R_\e\| _{L^{3}(\OO)}^{6}\,\dt & \leq C \int_{0}^{T} e^{-\a t} \|R_\e\| _{L^{2}(\OO)}^{4} \| R_\e\|_{H^{1}(\OO)}^{2}\,\dt \\
& \leq C \sup_{0 < t < T} \| R_\e(t) \|_{L^{2}(\OO)}^{4} \int_{0}^{T} e^{-\a t} \| R_\e \|_{H^{1}(\OO)}^{2}\,\dt \leq C\e^{3}.
\end{align*}
Finally, in view of \eqref{a-priori-r}, one has
\ba
\|\vr_\e (t)\|_{L^{3}(\OO)} \leq \|\vr_\e (t)\|_{L^{2}(\OO)}^{\frac 23}\|\vr_\e (t)\|_{L^{\infty}(\OO)}^{\frac 13} \leq C \e^{\frac 13} \th_\e^{\frac 13}(t),
\nn \ea
so that
\ba
\int_{0}^{T} e^{-\a t} \|\vr_\e (t)\| _{L^{3}(\OO)}^{6}\,\dt \leq C \e^{2}\int_{0}^{T} e^{-\a t} \th_{\e}^{2}(t)\,\dt \leq C \e^{2} \th_\e^{2}(T).
\nn \ea
As a result,
\ba
\int_{0}^{T} \|\nabla \fF_\e(t)\|_{L^{3}(\OO)}^6\,\dt \leq C \big( E_\e^{3}(T)+\e^{2} \th_\e^{2}(T)+\e^{3}\big),
\nn \ea
which together with \eqref{vflux-0} and \eqref{F-embed-Linfty} ensures that
\ba\label{thT-9}
\int_{0}^{T}\|\fF_\e(t)\|_{L^{\infty}(\OO)}^6\,\dt & \leq C\int_{0}^{T}\bigl(\|\fF_\e\|_{L^{6}(\OO)}^6 + \|\nabla \fF_\e\|_{L^{3}(\OO)}^6\bigr)\,\dt
\\
&\leq C \big( E_\e^{3}(T)+\e^{2} \th_\e^{2}(T)+\e^{3}\big).
\ea
By inserting the estimates \eqref{EstR} and \eqref{thT-9} into \eqref{thT-7}, we arrive at
\ba
\th_\e^{6}(T)\leq C \big( E_\e^{3}(T)+\e^{2} \th_\e^{2}(T)+\e^{3}\big) \leq C \e^{3} + C E_\e^{3}(T) + \frac{\th_\e^{6}(T)}{2},
\nn \ea
which leads to \eqref{thT-0}.
\end{proof}
Now we are in a position to complete the proof of Theorem \ref{thm2}.
\begin{proof}[Proof of Theorem \ref{thm2}]
We first deduce from \eqref{E(T)-1} and \eqref{nablaR-4-0} that
\ba
E_\e(T) \leq C \e + C \e^{\frac 12} E_\e(T)\quad \mbox{for}\ T\leq T_\e^\star,
\nn \ea
which implies that for $\e \leq \e_{1}$ small such that that $C\e_{1}^{\frac12} = \frac{1}{2},$ there holds
\be\label{E(T)-f}
E_\e(T) \leq C \e \quad \mbox{for}\ T\leq T_\e^\star.
\ee
This together with Lemma \ref{prop-thT} ensures that
\be\label{E-th-T-1}
\th_\e^{2}(T) \leq C \e \quad \mbox{for}\ T\leq T_\e^\star.
\ee
In particular, if we take $\e \leq \e_{2}$ to be small that $C\e_{2} = \frac{1}{6} \min\left\{1,\underline\eta\right\}^2,$
\eqref{E-th-T-1} contradicts with the definition of $T_\e^\star$ given by \eqref{a-priori-r}. As a consequence,
we deduce that $T_\e^\star=T_\e^\ast$.
It remains to show that the life-span $T_\e^\ast= \infty$. Indeed we have shown that the estimate \eqref{upper-lower-rho-0} holds
for all $t < T_\e^\ast$ and $\e\leq \e_0\eqdefa \min\{ \e_{1}, \e_{2}\}$, then regularity criteria for smooth solutions of compressible Navier-Stokes equations
(see for instance \cite{HLX11, SWZ11, SZ11}) ensures that $T_\e^\ast= \infty.$
This completes the proof of Theorem \ref{thm2}.
\end{proof}
\section{1D compressible Navier-Stokes equations with a parameter}\label{sec:1dNS}
In this section, we investigate the 1D compressible Navier-Stokes equations \eqref{CNS-limit} with a parameter $y.$
We assume that $(\eta,w)$ is a global smooth solution of \eqref{CNS-limit} determined at the beginning of Section \ref{sec2}.
For simplicity, we shall always neglect $y$ variable and
denote $D_t \eqdefa \d_t + w \d_x$ to be the material derivative in the rest of this section.
\subsection{Conservation of mass, momentum, and energy}
Integrating \eqref{CNS-limit} in $x$ over $\TT$ leads to the conservations of the mass and of the momentum:
\beq\label{csv-m-m}
\frac{\rm d}{\dt}\int_{\TT} \eta (t,x,y)\,\dx = 0,\andf \frac{\rm d}{\dt}\int_{\TT} ( \eta w )(t,x,y)\,\dx = 0,
\eeq
which together with \eqref{ini-1} ensures that
\ba\label{csv-m-m-1}
\int_{\TT} \eta (t,x,y)\,\dx = 1,\quad \int_{\TT} ( \eta w )(t,x,y)\,\dx = 0, \quad \forall\, t\in\R^{+}, \ y\in \R.
\ea
\medskip
Recall the conservation of energy:
\begin{lem}\label{prop-basic-energy}
{\sl There holds
\ba\label{csv-eng}
\frac{\rm d}{\dt}&E_0(t) + \nu\int_{\TT} w _x^2\,\dx = 0 \with E_0(t) \eqdefa \int_{\TT} \bigl(\frac{1}{2} \eta w ^2 + P( \eta )\bigr) \,\dx, \andf\\
& P(\eta) \eqdefa \frac{a}{\g-1} \eta^\g \quad \mbox{if $\g >1$} \andf P(\eta) \eqdefa a (\eta \log \eta + 1) \quad \mbox{if $\g = 1$}.
\ea
}
\end{lem}
It follows from \eqref{ass-ini-1} that
\be\label{ini-2}
\begin{split}
\frak{E}_{00}(y) & \eqdefa \int_{\TT} \bigl(\frac{1}{2} \varsigma_0 w _0^2 + (\varsigma_0-1)^2\bigr)(x,y) \,\dx \in (L^1 \cap L^{\infty})(\R)\with \bar{\frak{E}}_{00} \eqdefa \sup_{y\in \R} \frak{E}_{00}(y) ,\\
E_{00}(y) &\eqdefa \int_{\TT} \bigl(\frac{1}{2} \varsigma_0 w _0^2 + P(\varsigma_0)\bigr)(x,y) \,\dx \in L^{\infty}(\R)
\with \bar E_{00} \eqdefa \sup_{y\in \R} E_{00}(y).
\end{split}
\ee
Then for all $t\in \R^{+}, \ y\in \R$, there holds
\ba\label{csv-eng-1}
E_0(t) + \nu \int_0^t \int_{\TT} w _x^2\,\dx\,\dt' \leq E_{00}(y).
\ea
\subsection{Upper bound for the density function}
We observe from \eqref{sl-1d-0} that the density function $\eta$ admits an upper bound depending on time.
We shall derive here a time uniform upper bound for $ \eta.$
We first recall from \cite{SZ02} that for $ u \in L^1(\TT)$ and for all $x\in\TT$
\be\label{I-def}
I(u)(x) \eqdefa \int_0^x u(x')\,\dx', \quad \tilde I(u) \eqdefa I(u) - \langle I(u) \rangle \andf
\langle u \rangle \eqdefa \int_{\TT} u (x)\,\dx.
\ee
It is easy to observe that $I(u) \in C(\TT) \cap W^{1,1}(\TT)$ with
\be\label{I-pt1}
\|I(u)\|_{W^{1,1}_\h\cap L^\infty_\h} \leq \|u\|_{L^1_\h}.
\ee
Moreover, for each $u\in W^{1,1}(\TT) \cap C(\TT)$, standard density argument implies
\ba\label{I-pt2}
I(u_x)(x) = \int_0^x u_x \,\dx' = u(x) - u(0) \andf \tilde I(u_x)(x) = u(x) - \langle u \rangle.
\ea
\begin{prop}\label{prop-upperbd-density}
{\sl For all $(t,x,y) \in\R^{+} \times \TT \times \R$, there holds
\be\label{upperb-vtr}
\eta (t,x,y) \leq \max\left\{ \varsigma_0, \bar \varsigma_1 \right\} \exp\bigl(4 \bar E_{00}^{\frac 12}\bigr)\eqdefa \bar{ \eta },
\ee
where $\bar\varsigma_0$ is given in \eqref{ini-data-1d-ass} and $\bar \varsigma_1$ will be determined by \eqref{def-bar-vr1} below.
}
\end{prop}
\begin{proof}
We rewrite the momentum equation in \eqref{CNS-limit} as
\ba
( \eta w)_t + ( \eta w ^2)_x - \nu w _{xx} + p( \eta )_x = 0.
\nn \ea
Applying the operator $\tilde I$ to the above equation gives
\be\label{tI-mm-1}
\tilde I\big(( \eta w)_t\big) + \eta w ^2 - \nu w _{x} + p( \eta ) - \langle \eta w ^2 - \nu w _{x} + p( \eta ) \rangle = 0.
\ee
We first compute
\begin{align*}
\tilde I\big(( \eta w)_t\big) = \d_t \tilde I( \eta w) = D_t \tilde I( \eta w ) - w \d_x \tilde I( \eta w ) = D_t \tilde I( \eta w ) - \eta w ^2.
\end{align*}
While it follows from the transport equation of \eqref{CNS-limit} that
\ba\label{tI-mm-3}
- w _x = \frac{- \eta w _x}{ \eta } = \frac{ \eta _t + w \eta _x}{ \eta } = D_t \log \eta .
\nn \ea
By inserting the above equalities into \eqref{tI-mm-1} and
the fact that $\langle w_x \rangle = 0$, we obtain
\be\label{tI-mm-4}
D_t \tilde I( \eta w ) + \nu D_t \log \eta + p( \eta ) - \langle \eta w ^2 + p( \eta ) \rangle = 0.
\ee
Let ${X}(t,x,y)$ be the trajectory of $w,$ which is determined by
\be\label{flow-defw}
\frac{\rm d}{\dt} X_w(t,x,y) = w(t, X(t,x,y)); \quad X_w(0,x,y) = x.
\ee
Let $\frak{y}(t,x,y) = \log \eta(t,X_w(t,x,y),y) $. Then in view of \eqref{tI-mm-4}, we write
\be\label{tI-mm-6}
\nu \frac{\rm d}{\dt} \frak{y} = g(\frak{y}) + \frac{\rm d}{\dt}\frak{b}
\ee
with
$$
g(\frak{y}) = - p(\eta) + \langle \eta w ^2 + p( \eta ) \rangle, \quad \frak{b} = - \tilde I( \eta w ).
$$
Before proceeding, we recall the following lemma:
\begin{lem}[Lemma 1.3 of \cite{SZ03}]\label{lem-uperb}
{\sl Suppose $g\in C(\R)$ and $\frak{y}, b\in W^{1,1}(0,T)$ for all $T>0$. Suppose that $\frak{y}$ verifies
$$
\frac{\rm d}{\dt} \frak{y} = g(\frak{y}) + \frac{\rm d}{\dt} \frak{b} \quad \mbox{on} \ \R^{+}, \quad \frak{y}(0) = \frak{y}_0.
$$
If $g(+\infty) = -\infty$ and there exist non-negative constants $N_0,N_1\geq 0$ so that
for any $0\leq t_1<t_2 <\infty,$
$$
\frak{b}(t_2) - \frak{b}(t_1) \leq N_0 + N_1(t_2 - t_1).
$$
Then one has
$$
\frak{y}(t) \leq \bar{\frak{y}} <\infty, \quad \forall \, t\in \R^{+} \with
\bar{\frak{y}} \eqdefa \max\{ \frak{y}_0, \frak{y}_1 \} + N_0,
$$
where $\frak{y}_1$ is such that $g(\frak{y}) \leq -N_1$ for all $\frak{y}\geq \frak{y}_1$.
}
\end{lem}
Now we would like to apply Lemma \ref{lem-uperb} to derive an upper bound of $\eta$ via \eqref{tI-mm-6}. Firstly, it follows from
\eqref{csv-eng-1} and \eqref{ini-2} that
$$
g(\frak{y}) = - p(e^{\frak{y}}) + \int_{\TT} \left(\eta w ^2 + p( \eta )\right) \,\dx
\leq -p(e^{\frak{y}}) + \g \bar E_{00} \to -\infty, \quad \mbox{as} \ \frak{y}\to +\infty.
$$
While in view of \eqref{I-def}, we have for all $t\in \R^{+}$,
\be\label{I(rv)}
|\ti ( \eta w )|_{L^\infty_\h} \leq 2\| \eta w \|_{L^1_\h} \leq 2 \| \eta \|_{L^1_\h}^{\frac 12}\| \eta w ^2\|_{L^1_\h}^{\frac 12}\leq 2 \bar E_{00}^{\frac 12},
\ee
which implies
$$
{ |\tilde I( \eta w )(t_{2}) -\tilde I( \eta w )(t_{1}) |\leq 4 E_{00}^{\frac 12}, \ \forall \, t_1, t_2 \in [0,\infty).}
$$
On the other hand, we observe that
\be\label{def-bar-vr1}
- p(\bar\varsigma_1) + \g \bar E_{00} = 0 \Rightarrow
\bar \varsigma_1\eqdefa \Big(\frac{ \g \bar E_{00} }{a}\Big)^{\frac{1}{\g}}.
\ee
Hence we get, by applying Lemma \ref{lem-uperb}, that
\be\label{upperb-vtr0}
\frak{y} \leq \max\{ \log \bar \varsigma_0, \log \bar \varsigma_1 \} + 4 E_{00}^{\frac 12},
\ee
where $\bar \varsigma_0$ is given in \eqref{ini-data-1d-ass}. This leads to \eqref{upperb-vtr}.
\end{proof}
\subsection{Decay estimates of $L^2$ norms}
With the upper bound of $ \eta$ obtained in \eqref{upperb-vtr}, similarly to Lemma 2.1 of \cite{ZZ20}, we have
\begin{lem}\label{prop-kinetic-tx}
{\sl Under the assumptions of \eqref{ini-1}, one has
\ba
\int_{\TT} \eta w ^2 \, \dx \leq \bar{ \eta }^2 \int_{\TT} w _x^2 \, \dx.
\nn \ea}
\end{lem}
\begin{proof}
Under the assumptions of \eqref{ini-1}, one has \eqref{csv-m-m-1} for all $t>0.$ Then we deduce from
\eqref{upperb-vtr} and Poincar\'e's inequality that
\ba
\int_{\TT} \eta w ^2 \, \dx & = \int_{\TT} \eta w ^2 \, \dx - \langle \eta w \rangle^2\\
& = \frac12\int_{\TT} \int_{\TT} \eta (x) \eta (x') | w (x) - w (x')|^2\,\dx \,\dx'\\
& \leq\frac{\bar{ \eta }^2}2 \int_{\TT} \int_{\TT} | w (x) - w (x')|^2\,\dx \,\dx'\\
& = \bar{ \eta }^2 \int_{\TT} | w - \langle w \rangle|^2 \,\dx \leq \bar{ \eta }^2 \int_{\TT} w _x^2 \,\dx.
\nn\ea
\end{proof}
By combining Lemma \ref{prop-kinetic-tx} with the energy estimate \eqref{csv-eng-1}, we achieve
\be\label{tvr-tv2-tx}
\int_0^\infty \int_{\TT} \eta w ^2 \,\dx\,\dt \leq \bar{ \eta }^2 \int_0^\infty \int_{\TT} w _x^2 \,\dx\,\dt \leq \bar{ \eta }^2 \nu^{-1} E_{00}.
\ee
\begin{prop}\label{prop-decay-exp-L2}
{\sl There exist positive constants $\a>0$ and $C$ which depend on $(a, \g, \nu, \bar \varsigma_0, \bar E_{00})$ such that
\ba\label{decay-exp-L2-4}
\| \eta ^{\frac{1}{2}} w(t) \|_{L^2_\h} + \|(\eta -1)(t)\|_{L^2_\h} \leq C E_{00}^{\frac{1}{2}}(y) e^{-\a t}, \quad \forall \, t\in \R^{+}, \ y\in \R.
\ea
}
\end{prop}
\begin{proof} By
combining \eqref{csv-m-m} with \eqref{csv-eng}, we obtain
\ba\label{csv-L2-1}
\frac{\rm d}{\dt} \int_{\TT} \Bigl(\frac{1}{2} \eta w ^2 + \big(P( \eta ) - P(1) - P'(1)( \eta - 1)\big)\Bigr) \,\dx
+ \nu\int_{\TT} w _x^2\,\dx = 0.
\ea
While we get, by using Taylor's formula, that
\be\label{P-1-0}
P( \eta ) - P(1) - P'(1)( \eta - 1) = P''(\hat \eta) ( \eta -1)^2 = \frac{p'(\hat \eta)}{\hat \eta} ( \eta -1)^2 = a \g \hat \eta^{\g-2} ( \eta -1)^2,
\nn\ee
for some $\hat\eta$ between $ \eta $ and $1$. Due to the upper bound $0\leq \eta \leq \bar{ \eta } <\infty$, we have
$$0\leq \hat \eta \leq \max\{1,\bar{ \eta }\} \leq 1 + \bar{ \eta }.$$
So that if $\g\in [1,2]$, we have
\ba\label{P-1-1}
P( \eta ) - P(1) - P'(1)( \eta - 1) \geq a \g (1+\bar{ \eta })^{\g-2} ( \eta -1)^2.
\ea
While for $\g\in [1,2]$, it is easy to observe that
\ba\label{P-1-2}
P( \eta ) - P(1) - P'(1)( \eta - 1) \leq C(\g) a ( \eta - 1)^2,
\ea
where the constant $C$ is solely determined by $\g$. It follows from \eqref{P-1-1} and \eqref{P-1-2} that $P( \eta ) - P(1) - P'(1)( \eta - 1)$ behaves like $( \eta -1)^2$.
Notice that $I( \eta - 1)|_{x=0}=I( \eta - 1)|_{x=1}=0.$ By multiplying the momentum equation of \eqref{CNS-limit} by $ I( \eta - 1)$
and using integration by parts, one has
\beq\label{csv-L2-2}
\int_{\TT} p( \eta ) ( \eta -1) \,\dx = \int_{\TT} ( \eta w )_t I( \eta -1)\,\dx - \int_{\TT} ( \eta w ^2) ( \eta -1) \,\dx + \nu \int_{\TT} w _x ( \eta -1) \,\dx.
\eeq
We first compute
\be\label{csv-L2-3}
\int_{\TT} ( \eta w )_t I( \eta -1)\,\dx = \frac{\rm d}{\dt} \int_{\TT} ( \eta w ) I( \eta -1)\,\dx - \int_{\TT} ( \eta w ) \d_tI( \eta -1) \,\dx.
\nn\ee
By virtue of \eqref{csv-m-m-1} and Lemma \ref{prop-kinetic-tx}, one has
\ba
\bigl| \int_{\TT} ( \eta w )\d_t I( \eta -1) \,\dx \bigr| & = \bigl| \int_{\TT} ( \eta w ) I(( \eta w )_x)\,\dx \bigr|\\
& = \bigl| \int_{\TT} ( \eta w )( \eta w (t,x) - \eta w (t,0)) \,\dx \bigr| \\
& = \bigl| \int_{\TT} ( \eta w )^2 \,\dx \bigr| \leq \bar{ \eta }\int_{\TT} \eta w ^2 \,\dx \leq \bar{ \eta }^3 \int_{\TT} w _x^2 \,\dx.
\nn\ea
Similarly, we have
\be\label{csv-L2-5}
\bigl| \int_{\TT} ( \eta w ^2) ( \eta -1) \,\dx \bigr| \leq (1+\bar{ \eta }) \int_{\TT} \eta w ^2 \,\dx \leq (1+\bar{ \eta }) \bar{ \eta }^2 \int_{\TT} w _x^2 \,\dx.
\ee
While applying Young's inequality yields
\be\label{csv-L2-6}
\nu \bigl| \int_{\TT} w _x ( \eta -1) \,\dx \bigr| \leq \de \int_{\TT} | \eta -1|^2 \,\dx \ + \de^{-1} \nu^2 \int_{\TT} w _x^2 \,\dx.
\nn\ee
By inserting the above estimates into \eqref{csv-L2-3}, we find
\ba\label{csv-L2-7}
\int_{\TT} p( \eta ) ( \eta -1) \,\dx &- \frac{\rm d}{\dt} \int_{\TT} ( \eta w ) I( \eta -1)\,\dx \\
&\leq \de \int_{\TT} | \eta -1|^2 \,\dx + \big(\bar{ \eta }^2+\bar{ \eta }^3 + \de^{-1} \nu^2\big)\int_{\TT} w _x^2 \,\dx.
\ea
Notice that as long as $\g \geq 1$, one has
\be\label{csv-L2-8}
\big(p(r) - p(1)\big) (r-1) - a (r-1)^2 = a r (r^{\g-1}-1)(r-1) \geq 0, \quad \forall\, r\geq 0.
\nn\ee
As a result, it comes out
\ba\label{csv-L2-9}
\int_{\TT} p( \eta ) ( \eta -1) \,\dx = \int_{\TT} \big(p( \eta ) - p(1)\big) ( \eta -1) \,\dx \geq a \int_{\TT} ( \eta -1)^2 \,\dx.
\nn \ea
Then choosing $\de = \frac{a}{2}$ in \eqref{csv-L2-7} yields
\ba\label{csv-L2-10}
\frac{a}{2}\int_{\TT} ( \eta -1)^2 \,\dx - \frac{\rm d}{\dt} \int_{\TT} ( \eta w ) I( \eta -1)\,\dx \leq \big(\bar{ \eta }^2+\bar{ \eta }^3 + 2 a^{-1} \nu^2\big)\int_{\TT} w _x^2 \,\dx.
\ea
Multiplying \eqref{csv-L2-1} by a constant $A_1$ and summing up the resulting inequality with \eqref{csv-L2-10}, we get
\ba\label{csv-L2-11}
& \frac{\rm d}{\dt}\int_{\TT} \Bigl(\frac{A_1}{2} \eta w ^2 + A_1 \big(P( \eta ) - P(1) - P'(1)( \eta - 1)\big) - ( \eta w ) I( \eta -1) \Bigr)\,\dx \\
&\quad + A_1 \nu \int_{\TT} w _x^2\,\dx + \frac{a}{2}\int_{\TT} ( \eta -1)^2 \,\dx \leq \big(\bar{ \eta }^2+\bar{ \eta }^3 + 2 a^{-1} \nu^2\big)\int_{\TT} w _x^2 \,\dx.
\nonumber\ea
Taking $A_1$ large enough so that
\ba\label{csv-L2-13}
A_1 \geq 4, \quad A_1 a \g (1+\bar{ \eta })^{\g-2} \geq 2, \quad A_1 \nu \geq \bar{ \eta }^2+\bar{ \eta }^3 + 2 a^{-1} \nu^2 + 1,
\ea
we obtain
\ba\label{decay-exp-L2-1}
\frac{\rm d}{\dt}\int_{\TT} \Bigl(\frac{A_1}{2} \eta w ^2 + A_1 \big(P( \eta ) - P(1) - P'(1)( \eta - 1)\big) &- ( \eta w ) I( \eta -1)\Bigr) \,\dx \\
&+ \int_{\TT} w _x^2\,\dx + \frac{a}{2}\int_{\TT} ( \eta -1)^2 \,\dx \leq 0.
\ea
Furthermore, due to
\ba
\int_{\TT} \bigl|( \eta w ) I( \eta -1)\bigr| \,\dx \leq \int_{\TT} | \eta w | \,\dx \int_{\TT} | \eta -1| \,\dx \leq \int_{\TT} \eta w ^2 \,\dx + \int_{\TT} | \eta -1|^2 \,\dx,
\nn \ea
and taking into account \eqref{P-1-1} and \eqref{P-1-2}, we have
\ba\label{decay-exp-L2-2}
\int_{\TT} \Bigl( \frac{A_1}{2} \eta w ^2 + A_1 \big(P( \eta ) - P(1) &- P'(1)( \eta - 1)\big) - ( \eta w ) I( \eta -1)\Bigr) \,\dx \\
&\geq \int_{\TT} \Bigl(\frac{A_1}{4} \eta w ^2 + \frac{A_1}{2}a \g (1+\bar{ \eta })^{\g-2} ( \eta - 1)^2 \Bigr) \,\dx,
\ea
and
\ba\label{decay-exp-L2-3}
\int_{\TT} \Bigl(\frac{A_1}{2} \eta w ^2 + A_1 \big(P( \eta ) - P(1)& - P'(1)( \eta - 1)\big) - ( \eta w ) I( \eta -1)\Bigr) \,\dx \\
&\leq \int_{\TT} \Bigl(\big(\frac{A_1}{2} + 1 \big) \eta w ^2 + \big(A_1 a C(\g) +1 \big) ( \eta - 1)^2\Bigr) \,\dx.
\ea
Then \eqref{decay-exp-L2-4} follows from \eqref{decay-exp-L2-1}--\eqref{decay-exp-L2-3} and Lemma \ref{prop-kinetic-tx}.
\end{proof}
\subsection{Lower bound of the density function}
\begin{prop}\label{prop-lowerbd-density}
{\sl There exists a positive constant $\underline \eta $ which depends on $(a, \g, \nu, \bar \varsigma_0 , \underline \varsigma_0, \bar E_{00})$ so that
\be\label{lowerb-vtr}
\eta (t,x,y) \geq \underline \eta , \quad \forall \, (t,x,y)\in \R^{+}\times \TT\times \R.
\ee
}
\end{prop}
\begin{proof}
Let
$
\frak{y}_1 \eqdefa \log \frac{1}{ \eta } = - \log \eta .
$
Then it follows from \eqref{tI-mm-4} that
\ba\label{density-lower1}
D_t\big(\nu \frak{y}_1 - \tilde I( \eta w )\big) = p( \eta ) - \langle \eta w ^2 + p( \eta ) \rangle,
\nn \ea
from which, we deduce that for each $t\in\R^{+}$,
\ba\label{density-lower2}
\max_{x\in \TT} \big(\nu \frak{y}_1 - \tilde I( \eta w )\big)(t,\cdot) &\leq \max_{x\in \TT} \big(\nu \frak{y}_1 - \tilde I( \eta w )\big)(0,\cdot) + \int_0^t \max_{x\in \TT}\big(p( \eta ) + \langle \eta w ^2 + p( \eta ) \rangle\big) \dt'\\
& \leq - \nu \log\underline \varsigma_0 + E_{00}^{\frac{1}{2}} + (a \bar{ \eta }^\g + \g E_{00}) t.
\nonumber
\ea
This implies
\ba
\frak{y}_1 (t) \leq -\log\underline \varsigma_0 + 2 \nu^{-1}E_{00}^{\frac{1}{2}} + \nu^{-1} (a \bar{ \eta }^\g + \g E_{00}) t,
\nn \ea
which is equivalent to
\ba\label{density-lower4}
\eta (t) &\geq \underline \varsigma_0 \exp\big(- 2 \nu^{-1}\bar E_{00}^{\frac{1}{2}} - \nu^{-1} (a \bar{ \eta }^\g + \g \bar E_{00}) t \big) \eqdefa
\underline{ \eta }_1(t).
\ea
To obtain a lower bound of the density function for large time, we
define
$$
\frak{y}_2 = \exp(\l(\nu\frak{y}_1 - \tilde I( \eta w ))), \quad \l >0
$$
which solves
\ba\label{density-lower5}
D_t \frak{y}_2 & = \l \exp(\l(\nu\frak{y}_1 - \tilde I( \eta w ))) \big(p( \eta ) - \langle \eta w ^2 + p( \eta ) \rangle\big)\\
& = - \l \frak{y}_2 \langle \eta w ^2 + p( \eta ) \rangle + \l \eta ^{-\l\nu} a \eta ^\g \exp( - \l \tilde I( \eta w )).
\nonumber\ea
Taking $\l$ so that $\l \nu = \g$ gives rise to
\ba\label{density-lower6}
D_t \frak{y}_2 + \nu^{-1} \g \frak{y}_2 \langle \eta w ^2 + p( \eta ) \rangle = a \nu^{-1} \g \exp( - \l \tilde I( \eta w ) ).
\ea
Yet we observe from Proposition \ref{prop-decay-exp-L2} that
\ba\label{density-lower6.5}
\langle p( \eta ) \rangle & = \int_{\TT} p( \eta ) \, \dx
\geq p(1) - \int_{\TT} |p( \eta ) - p(1)| \, \dx \\
& \geq p(1) - p'(\bar{ \eta }) \int_{\TT} | \eta - 1| \, \dx \geq a - a \g \bar{ \eta }^{\g - 1} \big(C e^{-\a t}\big)^{\frac{1}{2}}.
\ea
Let $T_1$ be such that
\ba\label{T1-def}
\g \bar{ \eta }^{\g - 1} \big(C e^{-\a T_1}\big)^{\frac{1}{2}} = \frac{1}{2}.
\ea
Then for all $t\geq T_1$, there holds
$$
\langle \eta w ^2 + p( \eta ) \rangle \geq \langle p( \eta ) \rangle \geq \frac{a}{2},
$$
so that we deduce from \eqref{density-lower6} that for $t\geq T_1$,
\ba\label{density-lower7}
\frak{y}_2 (t) \leq e^{- \tilde a t}\frak{y}_2(T_1) + \int_{T_1}^t e^{- \tilde a (t-t')} a \nu^{-1} \g \exp( - \l \tilde I( \eta w ) )(t')\,\dt',
\ea
where $\tilde a\eqdefa \frac{a \nu^{-1} \g}{2}>0$. Together with \eqref{I(rv)}, \eqref{density-lower7} ensures that
for $t\geq T_1$,
\ba
\frak{y}_2 (t) \leq e^{- \tilde a t}\frak{y}_2 (T_1) + a\left(\tilde a \nu\right)^{-1} \g e^{2\l E_{00}^{\frac 12}},
\nn \ea
from
which, we infer
\ba\label{density-lower10}
\eta ^{-\lambda\g} (t) \leq e^{- \tilde a t} \eta ^{-\lambda\g} (T_1) e^{2 \l E_{00}^{\frac 12}} +
a\left(\tilde a \nu\right)^{-1} \g e^{4 \l E_{00}^{\frac 12}},
\nn \ea
that is,
for $t\geq T_1$ there holds
\ba\label{density-lower12}
\eta (t) \geq \underline{ \eta }_2(T_1) \eqdefa \Big( e^{- \tilde a T_1}\underline{ \eta }_1^{-\lambda\g} (T_1) e^{2 \l \bar E_{00}^{\frac 12}}
+ a \left(\tilde a\nu\right)^{-1} \g e^{4 \l \bar E_{00}^{\frac 12}}\Big)^{-\frac{1}{\lambda\g}}.
\ea
Combining \eqref{density-lower4} and \eqref{density-lower12}, we deduce that \eqref{lowerb-vtr} holds with
\ba\label{density-lower13}
\underline{ \eta } \eqdefa \min\left\{\underline{ \eta }_1(T_1),\underline{ \eta }_2(T_1) \right\}.
\ea
This completes the proof of the proposition.
\end{proof}
\subsection{Decay estimate of $\| w _x(t)\|_{L^2_\h}$}
It follows from \eqref{ass-ini-1} that
\be\label{ini-3}
E_{10} (y) \eqdefa \|\varsigma_0-1\|_{H^{1}_\h}^2 + \| w _0\|_{H^{1}_\h}^2 \in ( L^{1} \cap L^{\infty})(\R)
\andf \bar E_{10} \eqdefa \sup_{y\in \R}E_{10} (y) <\infty.
\ee
We start with the following two lemmas:
\begin{lem}\label{prop-est-rv4}
{\sl There holds for all $t\in \R^{+}$
\be\label{est-rv4}
\frac{\rm d}{\dt}\int_{\TT} \eta w ^4 \,\dx + 6 \nu \int_{\TT} | w |^2 | w _x|^2\,\dx \leq 24a \nu^{-1}\bar{ \eta }^{2\g-1} \int_{\TT} w _x^2\,\dx.
\ee}
\end{lem}
\begin{proof}
Multiplying $4 w ^3$ to the momentum equation of \eqref{CNS-limit} and integrating the resulting equation over $\TT$ gives
\ba\label{est-rv4-1}
\frac{\rm d}{\dt}\int_{\TT} \eta w ^4 \,\dx + 12 \nu \int_{\TT} | w |^2 | w _x|^2\,\dx & = 12 \int_{\TT} p( \eta ) w ^2 w _x\,\dx \\
& \leq 6\nu \int_{\TT} | w |^2 | w _x|^2\,\dx + 24a \nu^{-1} \int_{\TT} \eta ^{2\g} w ^2\,\dx \\
& \leq 6\nu \int_{\TT} | w |^2 | w _x|^2\,\dx + 24 a\nu^{-1} \bar{ \eta }^{2\g-1} \int_{\TT} w _x^2\,\dx,
\nonumber\ea
where we used Lemma \ref{prop-kinetic-tx} in the last step. And \eqref{est-rv4} follows.
\end{proof}
\begin{lem}\label{prop-est-vx}
{\sl There exist positive constants $B_1$ and $B_2$ solely depending on $(a, \g, \nu , \bar\varsigma_0, \bar E_{00})$, so that for all $t\in \R^{+}$
\ba\label{est-vx}
\frac{\rm d}{\dt} \int_{\TT}\Bigl( \frac\nu2 w _x^2 + \frac{a^2}{2\nu} \big( \eta ^{2\g} & - 1 - 2\g ( \eta -1)\big) - \big(p( \eta ) - p(1)\big) w _{x}\Bigr) \,\dx + \frac{1}{2} \int_{\TT} \eta w _t^2 \,\dx \\
& \qquad\quad \leq 5 \bar{ \eta }\int_{\TT} | w |^2 | w _x |^2 \,\dx + B_{1} \int_{\TT} w _x^2 \,\dx + B_2 \int_{\TT} ( \eta - 1)^2 \,\dx.
\ea}
\end{lem}
\begin{proof}
Multiplying $ w _t$ to the momentum equation of \eqref{CNS-limit} and integrating the resulting equation over $\TT$ gives
\be\label{est-vx-1}
\int_{\TT} ( \eta w )_t w _t \,\dx + \int_{\TT} ( \eta w ^2)_x w _t \,\dx + \frac\nu2 \frac{\rm d}{\dt} \int_{\TT} w _x^2 \,\dx + \int_{\TT} p( \eta )_x w _t \,\dx = 0.
\ee
We now handle term by term above. Firstly, it follows from the continuity equation of \eqref{CNS-limit} that
\begin{align*}\label{est-vx-2}
\int_{\TT} ( \eta w )_t w _t \,\dx
& = \int_{\TT} \eta w _t^2 \,\dx - \int_{\TT} ( \eta w ^2)_x w _t \,\dx + \int_{\TT} \eta w w _x w _t \,\dx.
\end{align*}
This implies
\ba\label{est-vx-3}
\int_{\TT} ( \eta w )_t w _t \,\dx + \int_{\TT} ( \eta w ^2)_x w _t \,\dx =& \int_{\TT} \eta w _t^2 \,\dx + \int_{\TT} \eta w w _x w _t \,\dx\\
\geq & \frac{3}{4} \int_{\TT} \eta w _t^2 \,\dx - 4 \bar{ \eta }\int_{\TT} | w |^2 | w _x |^2 \,\dx.
\ea
It is rather complicated to estimate the term related to the pressure in \eqref{est-vx-1}. We compute
\beq\label{est-vx-4}
\int_{\TT} p( \eta )_x w _t \,\dx
= - \frac{\rm d}{\dt} \int_{\TT} \big(p( \eta ) - p(1)\big) w _{x} \,\dx + \int_{\TT} p( \eta )_t w _{x} \,\dx.
\eeq
For the last term of \eqref{est-vx-4}, we decompose it as
\begin{align*}\label{est-vx-5}
\int_{\TT} p( \eta )_t w _{x} \,\dx & = \nu^{-1}\int_{\TT} p( \eta )_t \big(\nu w _{x} - p( \eta )\big) \,\dx + \nu^{-1}\int_{\TT} p( \eta )_t p( \eta ) \,\dx \\
& = \nu^{-1}\int_{\TT} p( \eta ) w \big(\nu w _{xx} - p( \eta )_x\big) \,\dx - a (\g-1)\nu^{-1}\int_{\TT} \eta ^\g w _x \big(\nu w _{x} - p( \eta )\big) \,\dx \\
& \quad + \frac{a^2}{2\nu}\frac{\rm d}{\dt}\int_{\TT} \big( \eta ^{2\g} - 1 - 2\g ( \eta -1)\big)\,\dx.
\end{align*}
Observing that
\ba
\nu^{-1}\int_{\TT} p( \eta ) w \big(\nu w _{xx} - p( \eta )_x\big) \,\dx= \nu^{-1}\int_{\TT} p( \eta ) w \eta \big( w _t + w w _x)\big) \,\dx.
\nn \ea
Applying Young's inequality yields
\ba
\nu^{-1}\bigl|\int_{\TT} p( \eta ) w \eta w _t \,\dx \bigr|
& \leq \frac{1}{4}\int_{\TT} \eta w _t^2 \,\dx + 4 \nu^{-2} a \bbeta^{2\g+1} \int_{\TT} w _x^2 \,\dx,\\
\nu^{-1}\bigl| \int_{\TT} p( \eta ) w \eta w w _x \,\dx \bigr|
& \leq a^2\nu^{-2} \bbeta^{2\g+2} \int_{\TT} w _x^2 \,\dx + \bbeta \int_{\TT} | w |^2 | w _x|^2 \,\dx,
\nonumber
\ea
where we used Lemma \ref{prop-kinetic-tx} in the second inequality.
We finally compute
\ba
a (\g-1)\bigl| \int_{\TT} \eta ^\g w _x^2 \,\dx \bigr| \leq a (\g-1) \bbeta^\g \int_{\TT} w _x^2 \,\dx,
\nn \ea
and
\begin{align*}
\nu^{-1}a (\g-1)\bigl| \int_{\TT} \eta ^\g w _x p( \eta ) \,\dx\bigr|
& = \nu^{-1} a (\g-1) \bigl| \int_{\TT} ( \eta ^{2\g} -1) w _x \,\dx \bigr| \\
& \leq \nu^{-1} a (\g-1) \Big( \int_{\TT} ( \eta ^{2\g} -1)^2 \,\dx + \int_{\TT} w _x^2 \,\dx\Big) \\
& \leq \nu^{-1} a (\g-1) \Big( \bigl(2\g \bbeta^{2\g-1}\bigr)^2 \int_{\TT} ( \eta -1)^2 \,\dx + \int_{\TT} w _x^2 \,\dx\Big).
\end{align*}
By inserting the above estimates into \eqref{est-vx-4}, we achieve
\beq\label{est-vx-6}
\begin{split}
\int_{\TT} p( \eta )_x w _t \,\dx\geq & \frac{\rm d}{\dt} \int_{\TT} \Bigl( \frac{a^2}{2\nu} \big( \eta ^{2\g} - 1 - 2\g ( \eta -1)\big)
- \big(p( \eta ) - p(1)\big) w _{x}\Bigr) \,\dx\\
&-\Bigl(\bar{ \eta }\int_{\TT} | w |^2 | w _x |^2 \,\dx+\frac14\int_{\TT}\eta w_t^2\,\dx
+B_{1} \int_{\TT} w _x^2 \,\dx + B_2 \int_{\TT} ( \eta - 1)^2 \,\dx\Bigr),
\end{split}
\eeq
where
\ba
&B_1\eqdefa 4 \nu^{-2} a \bbeta^{2\g+1} + \nu^{-2} a^2 \bbeta^{2\g+2} + a (\g-1) \bigl(\bbeta^\g + \nu^{-1}\bigr)
\andf B_2\eqdefa 4a \g^2(\g-1) \nu^{-1 } \bbeta^{2(2\g-1)}.
\nn \ea
By substituting \eqref{est-vx-3} and \eqref{est-vx-6} into \eqref{est-vx-1}, we achieve \eqref{est-vx}.
\end{proof}
By multiplying \eqref{est-rv4} by a large enough positive constant $A_2,$ which
depends on $(\nu, \bbeta),$ and summing up the resulting inequality
with \eqref{est-vx}, we obtain
\begin{corollary}
{\sl Let $A_2$ and $B_3$ be determined by
\ba\label{A2-B3}
6 A_2 \nu = 5\bbeta + 1, \quad B_3 = B_1 + 24 A_2 \nu^{-1} a \bar{ \eta }^{\g+1}.
\nn \ea There holds
\ba\label{est-vx-rv4}
& \frac{\rm d}{\dt} \int_{\TT} \Bigl(A_2 \eta w ^4 + \nu w _x^2 + \frac{a^2}{2\nu} \big( \eta ^{2\g} - 1 - 2\g ( \eta -1)\big) - \big(p( \eta ) - p(1)\big) w _{x} \Bigr)\,\dx \\
& \qquad\quad + \frac{1}{2} \int_{\TT} \eta w _t^2 \,\dx + \int_{\TT} | w |^2 | w _x |^2 \,\dx \leq B_{3} \int_{\TT} w _x^2 \,\dx + B_2 \int_{\TT} ( \eta - 1)^2 \,\dx.
\ea
}
\end{corollary}
\begin{prop}\label{prop-decay-exp-H1}
{\sl There exist positive constants $\a$ and $C$ solely depending on $(a, \g, \nu, \bar \varsigma_0, \underline \varsigma_{0}, \bar E_{00})$ such that
\ba\label{decay-exp-H1-2}
\| w(t)\|_{H^1_\h} + \| w(t) \|_{C^{0,\frac{1}{2}}_\h} \leq C E_{10}^{\frac{1}{2}}(y) e^{-\a t}, \quad \forall \, t\in \R^{+}, \ y\in \R.
\ea
}
\end{prop}
\begin{proof}
Thanks to \eqref{decay-exp-L2-1} and \eqref{est-vx-rv4}, we deduce that
there exists a large enough positive constant $A_3$ depending only on $(a, \g, \nu , \bar\varsigma_0, \bar E_{00})$ so that
\ba\label{decay-exp-H1-1}
& \frac{\rm d}{\dt} F_2(t)
+ \int_{\TT} \bigl( w _x^2 + ( \eta -1)^2 + | w |^2 | w _x |^2 + \frac{1}{2} \eta w _t^2\bigr) \,\dx \leq 0,
\ea
where
\ba\label{decay-exp-H1-3}
F_{2}(t) \eqdefa \int_{\TT} \Bigl(&\frac{A_1 A_3}{2} \eta w ^2 + A_1 A_3 \big(P( \eta ) - P(1) - P'(1)( \eta - 1)\big) - A_3 ( \eta w ) I( \eta -1) \\
&+ A_2 \eta w ^4 + \nu w _x^2 + \frac{a^2}{2\nu} \big( \eta ^{2\g} - 1 - 2\g ( \eta -1)\big) - \big(p( \eta ) - p(1)\big) w _{x} \Bigr) \,\dx.
\ea
Observing that
\begin{align*}\label{decay-exp-H1-4}
\bigl|\int_{\TT} \big(p( \eta ) - p(1)\big) w _{x}\,\dx \bigr| & \leq (2 \nu)^{-1}\int_{\TT} |p( \eta ) - p(1)|^2 \,\dx + \frac{\nu}{2}\int_{\TT} | w _{x}|^2\,\dx \\
&\leq (2 \nu)^{-1}p'(\bbeta)^2 \int_{\TT} | \eta - 1|^2 \,\dx + \frac{\nu}{2}\int_{\TT} | w _{x}|^2\,\dx.
\end{align*}
By choosing $A_3$ sufficiently large, we have
\ba\label{decay-exp-H1-5}
\int_{\TT} \left(\eta w ^2 + ( \eta -1)^2 + \eta w ^4 + w _x^2\right)
\,\dx \leq F_{2} \leq C \int_{\TT} \left( \eta w ^2 + ( \eta -1)^2 + \eta w ^4 + w _x^2\right) \,\dx,
\ea
where $C$ solely depends on $(a, \g, \nu , \bar\varsigma_0, \bar E_{00})$. Then \eqref{decay-exp-H1-2} follows from \eqref{decay-exp-H1-1},
\eqref{decay-exp-H1-5} and \eqref{decay-exp-L2-4}.
\end{proof}
We remark that up to now, the strictly lower bound of $\eta$ in \eqref{lowerb-vtr} is not really needed.
Indeed let us recall
\begin{lem}[Lemma 3.2 in \cite{F-book}]\label{lem-Poincare}
{\sl Let $\Omega\subset \R^d$ be a bounded domain with $d\geq 1.$ Let $\rho$ be a non-negative function satisfying
$$
\int_\Omega \rho \,\dx \geq M >0 \andf \int_\Omega \rho^q \,\dx \leq E_0 <\infty,
$$
for some $q>1.$ Then for each $u\in H^1(\Omega)$, there holds
$$
\|u\|_{L^2(\Omega)}^2 \leq C(M, E_0) \left(\|\nabla u\|_{L^2(\Omega)}^2 + \Big(\int_\Omega \rho |u|\,\dx \Big)^2\right).
$$
}
\end{lem}
\begin{remark}
By integrating \eqref{decay-exp-H1-1} over $[0,t],$ we obtain for all $t\in \R^{+}$
\ba\label{decay-exp-H1-6}
\int_{\TT} \bigl(\eta w ^2 + ( \eta -1)^2 + \eta w ^4 + w _x^2\bigr)(t) \,\dx
&+ \int_0^t \int_{\TT} \bigl(( \eta -1)^2 + | w |^2 | w _x |^2 + \frac{1}{2} \eta w _t^2\bigr) \,\dx\,\dt' \\
& \leq C \int_{\TT} \bigl(\eta _0 w _0^2 + ( \eta _0 -1)^2 + \eta _0 w _0^4 + w _{0,x}^2\bigr) \,\dx < \infty,
\nonumber
\ea
from which and Lemma \ref{lem-Poincare}, we infer
$$
\int_{\TT} |w|^{2} \, \dx\leq \int_{\TT} \bigl(\eta w ^2 + w _x^2\bigr) \,\dx \in L^{\infty}(\R_{+}).
$$
Then along the same lines of proof of \eqref{decay-exp-H1-2} , yet without using the positive lower bound of $\eta$,
we have
$$
\bigl\|\bigl( \eta ^{\frac{1}{2}} w, \eta -1, \eta ^{\frac{1}{2}} w ^2, w _x\bigr)\bigr\|_{L^2_\h} + \| w \|_{L^\infty_\h} \leq C E_{10}^{\frac{1}{2}}(y) e^{-\a t}, \quad \forall \, t\in \R^{+}, \ y\in \R,
$$
where $\a>0$ solely depending on $(a, \g, \nu, \bar \varsigma_0)$ and $C>0$ solely depending on $(a, \g, \nu, \bar \varsigma_0, \bar E_{00})$.
\end{remark}
\subsection{Decay estimate of $\| \eta _x(t)\|_{L^2_x}$}
From this subsection on,
we need to use the lower bound of $ \eta $ obtained in Proposition \ref{prop-lowerbd-density}.
\begin{prop}\label{prop-decay-exp-rx}
{\sl
There exist positive constants $C$ and $\a$ depending solely on $(a, \g, \nu, \bar \varsigma_0, \bar E_{00}, \underline{\varsigma}_0)$ so that
\be\label{decay-exp-rx}
\| \eta_x(t)\|_{L^2_\h} \leq C E_{10}^{\frac{1}{2}} e^{-\a t}, \quad \forall\, t\in \R^{+}.
\ee
}
\end{prop}
\begin{proof} Let $\zeta \eqdefa \eta ^{-1}.$ By multiplying $- \eta ^{-2}$ to the density equation
of \eqref{CNS-limit}, we get
\ba\label{decay-exp-rx-1}
\zeta_t + w \zeta_x - \zeta w _x = 0.
\nn \ea
Applying $\d_x$ to the above equation gives
\be\label{decay-exp-rx-2}
\zeta_{tx} + w \zeta_{xx} - \zeta w _{xx} = 0 \Longrightarrow \zeta w _{xx} = \zeta_{tx} + w \zeta_{xx}.
\ee
While we rewrite the momentum equation of \eqref{CNS-limit} as
\ba\label{decay-exp-rx-4}
w _t + w w _x - \nu (\zeta_{tx} + w \zeta_{xx}) - \zeta^{-1} p'( \eta )\zeta_x = 0.
\nn \ea
Multiplying the above equation by $\eta$ yeilds
\be\label{decay-exp-rx-6}
\eta \big(( w - \nu \zeta_x)_t + w ( w - \nu \zeta_x)_x \big) - \eta ^2 p'( \eta )\zeta_x = 0.
\ee
By multiplying $( w - \nu \zeta_x)$ to \eqref{decay-exp-rx-6} and integrating the resulting equation over $\TT,$ we find
\be\label{decay-exp-rx-7}
\frac{1}{2}\frac{\rm d}{\dt }\int_{\TT} \eta ( w - \nu \zeta_x)^2\,\dx - \int_{\TT} \eta ^2 p'( \eta )\zeta_x ( w -\nu\zeta_x)\,\dx = 0.
\ee
We compute
\begin{align*}\label{decay-exp-rx-8}
- \int_{\TT} \eta ^2 p'( \eta )\zeta_x ( w -\nu\zeta_x)\,\dx
& =\nu a \g \int_{\TT} \eta ^{\g+1} \zeta_x^2 \,\dx + \int_{\TT} p( \eta )_x w \,\dx.
\end{align*}
In view of Proposition \ref{prop-lowerbd-density}, we have $ \eta \geq \underline{ \eta }>0,$ so that
\ba\label{decay-exp-rx-9}
\nu a \g \int_{\TT} \eta ^{\g+1} \zeta_x^2 \,\dx \geq \nu a \g \underline{ \eta }^{\g+1} \int_{\TT} \zeta_x^2 \,\dx.
\nn \ea
We observe that
\begin{align*}
\bigl|\int_{\TT} p( \eta )_x w \,\dx\bigl| & = \bigl|\int_{\TT} p( \eta ) w _x \,\dx\bigl| = \bigl|\int_{\TT} (p( \eta )-p(1)) w _x \,\dx\bigl| \\
& \leq \int_{\TT} (p( \eta )-p(1))^2\,\dx + \int_{\TT} w _x^2 \,\dx \\
& \leq a^2 \g^2 \bbeta^{2\g-2} \int_{\TT} ( \eta -1)^2\,\dx + \int_{\TT} w _x^2 \,\dx.
\end{align*}
By substituting the above estimates into \eqref{decay-exp-rx-7}, we achieve
\be\label{decay-exp-rx-0}
\frac{1}{2}\frac{\rm d}{\dt }\int_{\TT} \eta ( w - \nu \zeta_x)^2\,\dx + \nu a \g \underline{ \eta }^{\g+1} \int_{\TT} \zeta_x^2 \,\dx \leq a^2 \g^2 \bbeta^{2\g-2} \int_{\TT} ( \eta -1)^2\,\dx + \int_{\TT} w _x^2 \,\dx.
\ee
Let $A_4$ be a large enough positive constant so that
\be\label{A4}
A_4 \geq 4 + \nu \andf \frac{a}{2}A_4 \geq \a^2 \g^2 \bbeta^{2\g-2} + 1.
\ee
Then by multiplying \eqref{decay-exp-L2-1} by $A_4$ and summing up the resulting inequality with \eqref{decay-exp-rx-0}, we obtain
\ba\label{decay-exp-rx-11}
\frac{\rm d}{\dt} F_3(t) + \int_{\TT} \bigl( w _x^2 + ( \eta -1)^2 +& \nu a \g \underline{ \eta }^{\g+1} \zeta_x^2\bigr) \,\dx \leq 0
\with\\
F_3(t) \eqdefa \int_{\TT} \Bigl(\frac{A_1A_4}{2} \eta w ^2 +& A_1 A_4 \big(P( \eta ) - P(1) - P'(1)( \eta - 1)\big)\\
&- A_4( \eta w ) I( \eta -1) + \frac{1}{2} \eta ( w - \nu \zeta_x)^2\Bigr)\,\dx.
\ea
Notice that
\ba
2 a^2 + (a-b)^2 = a^2 + 2 (a-\frac{b}{2})^2 + \frac{b^2}{2} \geq a^2 + \frac{b^2}{2},\nn
\ea
by choosing $A_4$ sufficiently large, we find
\ba\label{decay-exp-rx-13}
\int_{\TT} \bigl(\eta w ^2 + ( \eta - 1)^2 + \frac{\nu \underline{ \eta }}{4} \zeta_x^2\bigr)\,\dx
\leq F_3(t) \leq C \int_{\TT} \bigl(\eta w ^2 + ( \eta - 1)^2 + \frac{\nu}{2} \zeta_x^2\bigr)\,\dx,
\ea
where $C$ depends solely on $(a,\g, \nu , \bar\varsigma_0, \bar E_{00})$.
It follows from \eqref{decay-exp-rx-11}--\eqref{decay-exp-rx-13} that
\ba\label{decay-exp-rx-14}
\|\zeta_x(t)\|_{L^2_\h} \leq CE_{10}^{\frac12} \e^{-\a t}, \quad \forall\, t\in \R^{+},
\nn \ea
with $\a$ and $C$ satisfying the assumptions in the proposition.
Then \eqref{decay-exp-rx} follows from the fact that
$$
\| \eta _x(t)\|_{L^2_\h} = \| \eta ^{2} \zeta_x(t) \|_{L^2_\h} \leq \bbeta^2 \|\zeta_x(t)\|_{L^2_\h}.
$$
This completes the proof of Proposition \ref{prop-decay-exp-rx}.
\end{proof}
\subsection{Decay estimates of $H^2$ norms}
We first deduce from \eqref{ass-ini-1} that
\be\label{ini-4}
E_{20} (y) \eqdefa \|\varsigma_0-1\|_{H^{2}_\h}^2 + \| w _0\|_{H^{2}_\h}^2 \in (L^1 \cap L^{\infty})(\R)\andf
\bar E_{20} \eqdefa \sup_{y\in \R}E_{20} (y) <\infty.
\ee
\begin{lem}\label{prop-vxx-L2}
{\sl We have
\be\label{vxx-L2-1}
\int_0^\infty \int_{\TT} w _{xx}^2\,\dx\,\dt \leq C E_{10},
\ee
where $C$ solely depends on $(a, \g, \nu, \bar \varsigma_0, \bar E_{00}, \underline{\varsigma}_0)$.}
\end{lem}
\begin{proof} Indeed by multiplying $ \eta ^{-1} w _{xx}$ to the momentum equation of \eqref{CNS-limit} and integrating
the resulting equation over $\TT,$ we find
\ba\label{vxx-L2}
\frac{\rm d}{\dt} \int_{\TT} w _x^2 \, \dx + \frac{\nu}{\bbeta} \int_{\TT} w _{xx}^2\,\dx \leq \frac{4 \bbeta}{\nu}\Big(\int_{\TT} w ^2 w _x^2 \,\dx + a^2 \g^2 \int_{\TT} \eta ^{2\g - 4} \eta _x^2\,\dx \Big).
\ea
Yet by virtue of \eqref{decay-exp-H1-2} and \eqref{decay-exp-rx}, we have
\ba
& \int_{\TT} w ^2 w _x^2(t) \,\dx \leq \| w(t) \|_{L^\infty_\h}^2 \| w _x(t)\|_{L^2_\h}^2 \leq C E_{10} e^{-\a t}, \\
&\int_{\TT} \eta ^{2\g - 4} \eta _x^2(t)\,\dx \leq \underline{ \eta }^{2\g - 4} \| \eta _x(t)\|_{L^2_\h}^2 \leq C E_{10} e^{-\a t},
\nn\ea
where $C$ solely depends on $(a, \g, \nu, \bar \varsigma_0, \bar E_{00}, \underline{\varsigma}_0)$.
Then integrating \eqref{vxx-L2} over $\R^+$ leads to \eqref{vxx-L2-1}.
\end{proof}
\begin{prop}\label{prop-Dtv-L2}
{\sl Let $D_t\eqdefa\d_t + w \d_x$ be the material derivative. Then there exist positive constants $C$ and $\a$ depending
on $(a, \g, \nu, \bar \varsigma_0, \underline{\varsigma}_0,\bar E_{10})$ so that
\be\label{Dtv-L2-1}
\int_{\TT} \eta |D_t w |^2 \, \dx\leq C E_{20} (y) e^{-\a t} \andf \int_0^\infty \int_{\TT} |(D_t w )_x|^2\,\dx\,\dt \leq C E_{20}(y).
\ee
}
\end{prop}
\begin{proof}
Applying the material derivative $D_t$ to the momentum equation of \eqref{CNS-limit} gives
\ba\label{Dtv-L3}
\eta D_t^2 w + D_t \eta D_t w - \nu D_t w _{xx} + D_t p( \eta )_x = 0.
\nn \ea
We compute
\ba
D_t \eta D_t w = - \eta w _x D_t w = - w _x (\nu w _{xx} - p( \eta )_x) = - \frac{\nu}{2}( w _x^2)_x + w _x p( \eta )_x,
\nn \ea
and
\ba
- \nu D_t w _{xx} = -\nu ( w _{txx} + w w _{xxx}) = -\nu (D_t w )_{xx} + \frac{3\nu}{2}( w _x^2)_x.
\nn \ea
As a result, it comes out
\ba\label{Dtv-L7}
\eta D_t^2 w -\nu (D_t w )_{xx} +\nu ( w _x^2)_x + \big( w p( \eta )_x \big)_x + p( \eta )_{tx} = 0.
\ea
Multiplying \eqref{Dtv-L7} by $D_t w $ and integrating the resulting equation over $\TT$ yields
\ba\label{Dtv-L8}
\frac{1}{2}& \frac{\rm d}{\dt} \int_{\TT} \eta |D_t w |^2 \, \dx + \nu \int_{\TT} |(D_t w )_x|^2\,\dx \\
& = \nu\int_{\TT} w _x^2 (D_t w )_x\,\dx + \int_{\TT} w p( \eta )_x (D_t w )_x\,\dx + \int_{\TT} p( \eta )_t (D_t w )_x\,\dx\\
& \leq \frac{\nu}{2}\int_{\TT} |(D_t w )_x|^2\,\dx + 8 \nu^{-1} \int_{\TT} w _x^4\,\dx + C \int_{\TT} (\eta _x^2 + w _x^2)\,\dx,
\ea
where $C$ solely depend on $(a, \g, \nu, \bar \varsigma_0, \bar E_{00}, \underline{\varsigma}_0)$, and we used the uniform boundedness of $\| \eta \|_{L^\infty}$ and $\| w \|_{L^\infty}$.
While by applying Sobolev embedding theorem and H\"older inequality, one has
\ba\label{Dtv-L9}
\| w _x\|_{L^4_\h}^4 \leq \| w _x\|_{L^2_\h}^2 \| w _x\|_{L^\infty_\h}^2 \leq C \| w _x\|_{L^2_\h}^2 \| w _x\|_{H^1_\h}^2 \leq C e^{- 2\a t} \big(\| w _x\|_{L^2_\h}^2 + \| w _{xx}\|_{L^2_{\h}}^2\big).
\nn \ea
Then we deduce from \eqref{Dtv-L8} that there exists $C$ solely depending on $(a, \g, \nu, \bar \varsigma_0, \bar E_{00}, \underline{\varsigma}_0,\bar E_{10})$ so that
\ba\label{Dtv-L2}
\frac{\rm d}{\dt} \int_{\TT} \eta |D_t w |^2 \, \dx + \nu \int_{\TT} |(D_t w )_x|^2\,\dx \leq C \int_{\TT} \bigl( w _x^2 + w _{xx}^2 + \eta _x^2\bigr)\,\dx.
\ea
By multiplying \eqref{vxx-L2} by a sufficiently large constant $A_5$ and summing up the resulting inequality with \eqref{Dtv-L2}, we get
\ba\label{vxx-Dtv-L2}
\frac{\rm d}{\dt} \int_{\TT} \bigl(A_5 w _x^2 + \eta |D_t w |^2\bigr) \, \dx + \int_{\TT} w _{xx}^2\,\dx + \nu \int_{\TT} |(D_t w )_x|^2\,\dx \leq C \int_{\TT} \bigl(w _x^2 + \eta _x^2\bigr)\,\dx.
\ea
Observing that
\ba\label{vxx-Dtv-L2-2}
\int_{\TT} \eta |D_t w |^2 \, \dx \leq C \int_{\TT} \bigl(\eta w _t^2 + \eta w _x^2\bigr) \, \dx.
\nn \ea
Then by virtue of \eqref{decay-exp-H1-1} and \eqref{decay-exp-rx-11}, we can find a large enough constant $A_6$ such that the quantity
\ba\label{def-F4}
F_4(t)\eqdefa A_6 F_3(t) + \int_{\TT} \bigl( A_5 w _x^2 + \eta |D_t w |^2 \bigr)\, \dx
\ea
satisfies
\begin{align*}\label{vxx-Dtv-L2-4}
\int_{\TT} \bigl(\eta w ^2 + ( \eta - 1)^2 + & \eta _x^2 + \eta w _x^2 + w _x^2 + \eta |D_t w |^2\bigr)\,\dx \leq F_4(t) \\
& \leq C \int_{\TT} \bigl( \eta w ^2 + ( \eta - 1)^2 + \eta _x^2 + \eta w _x^2 + \eta |D_t w |^2\bigr)\,\dx,
\end{align*}
and
\ba\label{vxx-Dtv-L2-5}
\frac{\rm d}{\dt}F_4(t) + \int_{\TT} \bigl( w _x^2 + ( \eta -1)^2 + | w |^2 | w _x |^2 + \eta w _t^2 + \eta _x^2 + w_{xx}^2\bigr)\,\dx \leq 0.
\nn \ea
Here $F_3(t)$ is defined in \eqref{decay-exp-rx-11} and $C$ solely depends on $(a, \g, \nu, \bar \varsigma_0, \bar E_{00}, \underline{\varsigma}_0,\bar E_{10})$. Then there exists $\a>0$ solely depending on $(a, \g, \nu, \bar \varsigma_0, \bar E_{00}, \underline{\varsigma}_0)$ such that
\ba\label{def-F4-est}
F_4(t)\leq C E_{20} e^{-\a t}.
\nn \ea
And \eqref{Dtv-L2-1} follows.
\end{proof}
\subsection{Proof of Proposition \ref{S2prop1}} \label{sec-decay-all}
With the estimates obtained in the previous sections, we shall prove Proposition \ref{S2prop1}
by induction method and along the same line as that of Propositions \ref{prop-upperbd-density},
\ref{prop-decay-exp-L2}, \ref{prop-decay-exp-H1}, \ref{prop-decay-exp-rx}
and \ref{prop-Dtv-L2}. Since it involves only technicalities, we postpone the proof in Appendix \ref{appa}.
\section{Decay estimates of $(\eta_{y},w_{y})$}\label{sec:1dNS-y}
In this section, we investigate the decay in time estimates of $(\eta _y, w _y)$. We first get, by applying $\d_y$ to \eqref{CNS-limit}, that
\be\label{CNS-1d-dy}
\left\{\begin{aligned}
& \eta _{yt} + ( \eta w )_{yx}= 0,\\
&( \eta w )_{yt} + ( \eta w ^2)_{yx} - \nu w _{yxx} + p( \eta )_{yx} = 0.
\end{aligned}\right.
\ee
Integrating \eqref{CNS-1d-dy} with respect to $x$ over $\TT$ gives
\ba\label{csv-m-m-dy}
\frac{\rm d}{\dt}\int_{\TT} \eta _y \,\dx = 0 \andf \frac{\rm d}{\dt}\int_{\TT} ( \eta w )_y \,\dx = 0.
\nn \ea
It follows from \eqref{ini-1} that
\ba\label{ini-1-dy}
\int_{\TT} \varsigma_{0y}\,\dx = 0 \andf \int_{\TT} (\varsigma_0 w _0)_y \,\dx = 0.
\nn \ea
This implies
\ba\label{csv-m-m-1-dy}
\int_{\TT} \eta _y \,\dx = 0 \andf \int_{\TT} ( \eta w )_y\,\dx = 0, \quad \forall \, t\in\R_{+}.
\ea
\subsection{Decay estimates of $L^2$ norms}
Without loss of generality, we may assume that
\be\label{pressure-2}
p'(1) = 1.
\ee
Note that this assumption \eqref{pressure-2} can always hold after a suitable normalization.
In view of \eqref{ass-ini-1}, one has
\be\label{ini-6}
E_{00}^{(1)} \eqdefa \|(\d_y \varsigma_{0},\d_y w _{0})\|_{L^2_\h}^2 \in (L^{1} \cap L^{\infty})(\R) \andf \bar E_{00}^{(1)} \eqdefa \sup_{y\in \R} E_{00}^{(1)} <\infty.
\ee
Throughout this subsection, $A, \a$ and $C$ are positive numbers solely depending on $(a, \g, \nu, \bar \varsigma_0, \bar E_{10}, \underline{\varsigma}_0),$ which may differ from line to line.
\begin{lem}\label{energy-basic-y}
{\sl For all $t\in \R^{+}$,
one has
\ba\label{energy-0-0}
\int_{\TT} \bigl( \eta w _y^2 + \eta _y^2\bigr) \,\dx + \nu \int_0^t \int_{\TT}| w _{yx}|^2 \,\dx \leq C E_{00}^{(1)} .
\ea}
\end{lem}
\begin{proof}
Taking $L^2(\TT)$ inner product of $\eqref{CNS-1d-dy}_2$ with $ w _y$ gives
\ba\label{energy-0-1}
\int_{\TT}( \eta w )_{yt} w _y \,\dx + \int_{\TT}( \eta w ^2)_{yx} w _y \,\dx + \nu \int_{\TT}| w _{yx}|^2 \,\dx + \int_{\TT} p( \eta )_{yx} w _y \,\dx = 0.
\ea
Next we handle term by term above.
For the first term in \eqref{energy-0-1}, we have
\begin{align*}
\int_{\TT}( \eta w )_{yt} w _y \,\dx & = \int_{\TT}( \eta w _y + \eta _y w )_{t} w _{y} \,\dx\\
& = \frac{1}{2}\frac{\rm d}{\dt} \int_{\TT} \eta w _y^2 \,\dx + \frac{1}{2} \int_{\TT} \eta _t w _y^2 \,\dx+ \int_{\TT}( \eta w )_{y} ( w w _{y})_x \,\dx + \int_{\TT} w _t \eta _y w _y \,\dx.
\end{align*}
For the second term in \eqref{energy-0-1}, we have
\ba\label{energy-0-5}
\int_{\TT}( \eta w ^2)_{yx} w _y \,\dx = - \int_{\TT}( \eta w ^2)_{y} w _{yx} \,\dx = - \int_{\TT}(2 \eta w w _y + w ^2 \eta _y) w _{yx} \,\dx .
\nn \ea
And
for the last term in \eqref{energy-0-1}, one has
\ba\label{energy-0-6}
\int_{\TT} p( \eta )_{yx} w _y \,\dx = - \int_{\TT} p( \eta )_{y} w _{yx} \,\dx = - \int_{\TT} p'( \eta ) \eta _{y} w _{yx} \,\dx.
\nn \ea
By substituting the above equalities into \eqref{energy-0-1}, we achieve
\ba\label{energy-0-7}
\frac{1}{2}& \frac{\rm d}{\dt} \int_{\TT} \eta w _y^2 \,\dx + \nu \int_{\TT}| w _{yx}|^2 \,\dx \\
& \quad = - \frac{1}{2} \int_{\TT} \eta _t w _y^2 \,\dx - \int_{\TT}( w w _x \eta _y w _y + \eta w _x w _y^2 ) \,\dx\\
& \qquad - \int_{\TT} w _t \eta _y w _y \,\dx + \int_{\TT} \eta w w _y w _{yx} \,\dx + \int_{\TT} p'( \eta ) \eta _{y} w _{yx} \,\dx.
\ea
On the other hand, by
taking $L^2(\TT)$ inner product of $\eqref{CNS-1d-dy}_1$ with $ \eta _y,$ we find
\ba\label{energy-0-9}
\frac{1}{2} \frac{\rm d}{\dt} \int_{\TT} \eta _y^2 \,\dx & = - \int_{\TT}( \eta w )_{yx} \eta _y \,\dx \\
& = - \int_{\TT} w \eta _{yx} \eta _y \,\dx - \int_{\TT} ( w _x \eta _{y}^2 + \eta _{x} w _y \eta _y) \,\dx - \int_{\TT} \eta w _{yx} \eta _y \,\dx.
\ea
Summing up \eqref{energy-0-7} and \eqref{energy-0-9} gives rise to
\begin{align*}
&\frac{1}{2} \frac{\rm d}{\dt} \int_{\TT} \bigl(\eta w _y^2 + \eta _y^2\bigr) \,\dx + \nu \int_{\TT}| w _{yx}|^2 \,\dx \\
& \quad = -\int_{\TT}\big( \frac{1}{2} \eta _t + \eta w _x \big) w _y^2 \,\dx -
\int_{\TT}\big( w w _x + w _t + \eta _x \big) \eta _y w _y \,\dx \\
& \qquad+ \frac{1}{2} \int_{\TT} w _x \eta _{y}^2 \,\dx + \int_{\TT}{( p'( \eta ) - \eta )} \eta _y w _{yx} \,\dx + \int_{\TT} \eta w w _y w _{yx} \,\dx,
\end{align*}
from which, \eqref{pressure-2}, \eqref{decay-exp-all-1} and \eqref{decay-exp-all-2}, we deduce
\ba\label{energy-0}
\frac{\rm d}{\dt} \int_{\TT} \bigl(\eta w _y^2 + \eta _y^2\bigr) \,\dx + \nu \int_{\TT} w _{yx}^2 \,\dx \leq C e^{-\a t}
\int_{\TT} \bigl( \eta w _y^2 + \eta _y^2\bigr) \,\dx.
\ea
Applying Gronwall's inequality leads to
\eqref{energy-0-0}.
\end{proof}
\begin{prop}\label{prop-decay-L2-y}
{\sl We have
\ba\label{decay-L2-y-2}
\int_{\TT} \bigl( \eta w _y^2 + \eta _y^2\bigr)(t) \,\dx \leq C E_{00}^{(1)}(y) e^{-\a t} \andf
\int_0^\infty \int_{\TT} \bigl(w _{yx}^2 + \eta _y^2\bigr) \,\dx \,\dt \leq C E_{00}^{(1)}(y).
\ea}
\end{prop}
\begin{proof} Recall \eqref{I-pt2}, we get, by
multiplying the momentum equation of \eqref{CNS-1d-dy} by $I( \eta _y)$ and integrating the resulting equality over $\TT,$ that
\ba\label{decay-L2-y-3}
\int_{\TT} p( \eta )_{y} \eta _y \,\dx = \int_{\TT} ( \eta w )_{yt} I( \eta _y)\,\dx - \int_{\TT} ( \eta w ^2)_{y} \eta _y\,\dx + \nu\int_{\TT} w _{yx} \eta _y \,\dx .
\ea
It is easy to observe that
\begin{align*}
- \int_{\TT} &( \eta w ^2)_{y} \eta _y \,\dx = - \int_{\TT} w ^2 \eta _y^2 \,\dx - \int_{\TT} 2 \eta w w _{y} \eta _y \,\dx,\\
&\nu\int_{\TT} w _{yx} \eta _y \,\dx \leq C\int_{\TT} w _{yx}^2 \,\dx + \de \int_{\TT} \eta _y^2\,\dx,
\end{align*}
and
\ba\label{decay-L2-y-6}
\int_{\TT} ( \eta w )_{yt} I( \eta _y)\,\dx = \frac{\rm d}{\dt}\int_{\TT} ( \eta w )_{y} I( \eta _y)\,\dx - \int_{\TT} ( \eta w )_{y} I( \eta _{yt})\,\dx.
\nn \ea
By virtue of \eqref{csv-m-m-1-dy}, one has
\begin{align*}
- \int_{\TT} ( \eta w )_{y} I( \eta _{yt})\,\dx & = \int_{\TT} ( \eta w )_{y} I\left(( \eta w )_{yx}\right)\,\dx \\
& = \int_{\TT} ( \eta w )_{y} (( \eta w )_{y} - ( \eta w )(t,0,y))\,\dx = \int_{\TT} ( \eta w )_{y}^2 \,\dx.
\end{align*}
By inserting the above estimates into \eqref{decay-L2-y-3}, we obtain
\begin{align*}
\int_{\TT} p'( \eta ) \eta _y^2 \,\dx\leq \frac{\rm d}{\dt}\int_{\TT} ( \eta w )_{y} I( \eta _y)\,\dx+ \int_{\TT} \eta ^2 w _y^2 \,\dx
+ C\int_{\TT} w _{yx}^2 \,\dx + \de \int_{\TT} \eta _y^2\,\dx.
\end{align*}
Choosing $\de = \frac{p'(\underline{\eta})}{2}$ in the above inequality gives rise to
\ba\label{decay-L2-y-8}
\frac{p'(\underline{\eta})}{2}\int_{\TT} \eta _y^2 \,\dx - \frac{\rm d}{\dt}\int_{\TT} ( \eta w )_{y} I( \eta _y)\,\dx \leq C \int_{\TT} w _{yx}^2 \,\dx + \int_{\TT} \eta ^2 w _y^2 \,\dx.
\ea
Notice from Lemma \ref{prop-kinetic-tx} and \eqref{csv-m-m-1-dy} that
\ba\label{decay-L2-y-9}
\int_{\TT} \eta w _y^2 - \langle \eta w _y \rangle^2 \,\dx = \int_{\TT} \eta ( w _y - \langle \eta w _y \rangle)^2 \,\dx \leq \bbeta^2 \int_{\TT} w _{yx}^2 \,\dx,
\nn \ea from which and
\eqref{csv-m-m-1-dy}, we infer
\ba\label{decay-L2-y-10}
\int_{\TT} \eta w _y^2
& \leq \bbeta^2 \int_{\TT} w _{yx}^2 \,\dx + \langle \eta _y w \rangle^2 \\
& \leq \bbeta^2 \int_{\TT} w _{yx}^2 \,\dx + \int_{\TT} \eta _y^2 w ^2 \,\dx.
\ea
Hence thanks to \eqref{decay-exp-H1-2}, we deduce from \eqref{decay-L2-y-8} that
\ba\label{decay-L2-y-8a}
\frac{p'(\underline{\eta})}{2}\int_{\TT} \eta _y^2 \,\dx - \frac{\rm d}{\dt}\int_{\TT} ( \eta w )_{y} I( \eta _y)\,\dx \leq C \int_{\TT} w _{yx}^2 \,\dx + Ce^{-\alpha t} \int_{\TT} \eta_y^2 \,\dx.
\ea
Let $A$ be a sufficiently large constant, we denote
\be\label{F1-1-def}
F_1^{(1)}\eqdefa \int_{\TT} \bigl(A ( \eta w _y^2 + \eta _y^2) - ( \eta w )_y I( \eta _y) \bigr)\,\dx.
\ee
Thanks to Lemma \ref{energy-basic-y}, we get, by multiplying \eqref{energy-0} by $A$ and summing up the resulting inequality with \eqref{decay-L2-y-8a}, that
\ba\label{decay-L2-y-1}
\frac{\rm d}{\dt} F_1^{(1)}(t) + \int_{\TT}\bigl( w _{yx}^2 + \frac{ p'(\underline{\eta})}{2} \eta _y^2 \bigr)\,\dx \leq C E_{00}^{(1)} e^{-\a t}.
\ea
Due to
\ba \label{decay-L2-y-12}
\bigl|\int_{\TT}( \eta w )_y I( \eta _y) \,\dx\bigr| \leq \| \eta _y\|_{L^1_\h} \bigl(\| w \eta _y\|_{L^1_\h} + \| \eta w _y\|_{L^1_\h}\bigr)
\leq (1 + \| w \|_{L^\infty}) \int_{\TT} ( \eta w _y^2 + \eta _y^2)\,\dx,
\nn \ea
we deduce from \eqref{decay-L2-y-10} that
\ba\label{F1-1-pt}
\int_{\TT} \bigl(\eta w _y^2 + \eta _y^2\bigr) \,\dx \leq F_1^{(1)}(t) \leq C \int_{\TT}
\bigl(\eta w _y^2 + \eta _y^2\bigr) \,\dx \leq C \int_{\TT} \bigl(w _{yx}^2 + \eta _y^2\bigr) \,\dx,
\nn \ea
which together with \eqref{decay-L2-y-1} ensures \eqref{decay-L2-y-2}.
\end{proof}
\subsection{Decay estimates of $H^1$ norms}
It follows from \eqref{ass-ini-1} that
\be\label{ini-7}
E_{10}^{(1)} \eqdefa \|\d_y \varsigma_{0} \|_{H^1_\h}^2 + \|\d_y w _{0}\|_{H^1_\h}^2 \in (L^{1} \cap L^{\infty}) (\R) \andf
\bar E_{10}^{(1)} \eqdefa \sup_{y\in \R} E_{10}^{(1)} <\infty.
\ee
Throughout this subsection, $A, \a, C$ are positive constants solely depending on $(a, \g, \nu, \bar \varsigma_0, \bar E_{10}, \underline{\varsigma}_0, \bar E_{00}^{(1)}),$ which may differ from line to line.
\begin{lem}\label{prop-H1-w-y}
{\sl For all $t\in \R^{+}$, there holds
\ba\label{decay-H1-y-0}
\frac{\rm d}{\dt} \int_{\TT} w_{yx}^2\,\dx + \nu \int_{\TT} \eta^{-1} w_{yxx}^2 \,\dx \leq C e^{-\a t}
\int_{\TT}\bigl( w_{yx}^2 + \eta_y^2\bigr) \,\dx + C \int_{\TT} \eta_{yx}^2 \,\dx.
\ea
}
\end{lem}
\begin{proof}
We rewrite the equation $\eqref{CNS-1d-dy}_2$ as
\beq\label{mm-y-1}
\eta (w_{yt} + w w_{yx} + w_y w_x) + \eta_y (w_t + w w_x) - \nu w _{yxx} + p( \eta )_{yx} = 0.
\eeq
Multiplying the above equation by $\eta^{-1} w_{yxx}$ and integrating the resulting equation over $\TT$ yields
\ba\label{decay-H1-y-1}
\frac{1}{2}& \frac{\rm d}{\dt} \int_{\TT} w_{yx}^2\,\dx + \nu \int_{\TT} \eta^{-1} w_{yxx}^2 \,\dx \\
&\quad = \int_{\TT} \big(\eta (w w_{yx} + w_y w_x) + \eta_y (w_t + w w_x) + p(\eta)_{yx}\big) \eta^{-1} w_{yxx} \,\dx.
\ea
Applying Young's inequality and using \eqref{decay-exp-all-2} gives
\begin{align*}
& \int_{\TT} \big(\eta (w w_{yx} + w_y w_x) + \eta_y (w_t + w w_x) + p(\eta)_{yx}\big) \eta^{-1} w_{yxx} \,\dx \\
& \quad \leq \frac{\nu}{2} \int_{\TT} \eta^{-1} w_{yxx}^2 \,\dx + C e^{-\a t} \int_{\TT}\left( w_{yx}^2 + w_y^2 + \eta_y^2 \right)\,\dx + C \int_{\TT} |p(\eta)_{yx}|^2 \,\dx.
\end{align*}
Notice that
\ba\label{decay-H1-y-3}
\int_{\TT} |p(\eta)_{yx}|^2 \,\dx = \int_{\TT} |p'(\eta)\eta_{yx} + p''(\eta) \eta_y \eta_x|^2 \,\dx.
\nn \ea
As a consequence, thanks to \eqref{decay-L2-y-10}, we thus deduce \eqref{decay-H1-y-0} from \eqref{decay-H1-y-1}.
\end{proof}
\begin{lem}\label{prop-H1-eta-y}
{\sl For all $t\in \R^{+}$, there holds
\ba\label{decay-H1-eta-y-1}
\frac{\rm d}{\dt} \int_{\TT} \eta(w_y - \nu (\eta^{-1})_{yx})^2 \,\dx + \nu \int_{\TT} p'(\eta) \eta^{-2} \eta_{yx}^2 \,\dx
\leq C \int_{\TT}\bigl( w_{yx}^2 + \eta_y^2\bigr) \,\dx.
\ea
}
\end{lem}
\begin{proof}
We rewrite the equation $\eqref{CNS-1d-dy}_2$ as
\beq\label{mm-y-2}
D_t w_y + w_x w_y + \eta^{-1} \eta_y D_t w - \nu \eta^{-1} w _{yxx} + \eta^{-1} p( \eta )_{yx} = 0.
\eeq
Let $\zeta \eqdefa \eta^{-1}$. Then we deduce from equation $\eqref{CNS-1d-dy}_1$ that
\ba\label{mass-y-2}
\zeta w_{yxx} = (\zeta_{tx} + w \zeta_{xx})_y - \zeta_y w_{xx} = D_t \zeta_{yx} + w_y \zeta_{xx} - \zeta_y w_{xx} .
\nn \ea
Therefore, we obtain
\ba\label{decay-H1-eta-y-2}
D_t (w_y - \nu \zeta_{yx}) + w_x w_y + \eta^{-1} \eta_y D_t w - \nu(w_y \zeta_{xx} - \zeta_y w_{xx}) + \eta^{-1} p( \eta )_{yx} = 0.
\nn \ea
By taking $L^2$ inner product of the above equation with $\eta (w_y - \nu \zeta_{yx}),$ we find
\ba\label{decay-H1-eta-y-4}
\frac{1}{2}&\frac{\rm d}{\dt} \int_{\TT} \eta (w_y - \nu \zeta_{yx})^2\,\dx + \int_{\TT} p( \eta )_{yx}(w_y - \nu \zeta_{yx}) \,\dx \\
& + \int_{\TT} \big( \eta w_x w_y + \eta_y D_t w - \nu \eta (w_y \zeta_{xx} - \zeta_y w_{xx}) \big) (w_y - \nu \zeta_{yx}) \,\dx = 0.
\ea
Notice that
\begin{align*}
\int_{\TT} p( \eta )_{yx}(w_y - \nu \zeta_{yx}) \,\dx & = \int_{\TT} (p'(\eta)\eta_{yx} + p''(\eta) \eta_y \eta_x) (w_y - \nu (2 \eta^{-3} \eta_x \eta_y - \eta^{-2} \eta_{yx})) \,\dx \\
& = \nu \int_{\TT} p'(\eta) \eta^{-2} \eta_{yx}^2 \,\dx +\int_{\TT} p'(\eta)\eta_{yx} (w_y - 2 \nu \eta^{-3} \eta_x \eta_y ) \,\dx \\
& \quad + \int_{\TT} p''(\eta) \eta_y \eta_x \bigl(w_y - 2 \nu \eta^{-3} \eta_x \eta_y +\nu \eta^{-2} \eta_{yx}\bigr)) \,\dx.
\end{align*}
Hence thanks to \eqref{decay-exp-all-3}, we deduce from \eqref{decay-H1-eta-y-4} that
\ba\label{decay-H1-eta-y-6}
&\frac{1}{2}\frac{\rm d}{\dt} \int_{\TT} \eta (w_y - \nu \zeta_{yx})^2\,\dx + \nu \int_{\TT} p'(\eta) \eta^{-2} \eta_{yx}^2 \,\dx \leq \de \int_{\TT} \eta_{yx}^2 \,\dx + C \int_{\TT} \bigl(w_y^2 + \eta_y^2\bigr) \,\dx .
\ea
Choosing $\de>0$ small so that
$
\de \leq \frac{\nu}{2} p'(\underline \eta) {\underline \eta}^{-2},
$
and using \eqref{decay-L2-y-10}, we conclude the proof of \eqref{decay-H1-eta-y-1}.
\end{proof}
\begin{prop}\label{prop-decay-H1-y}
{\sl For all $t\in \R^{+}$, there holds
\ba\label{decay-H1-eta-w-y-1}
\int_{\TT} \left(w_{yx}^2 + \eta_{yx}^2\right)(t) \,\dx \leq C E_{10}^{(1)}(y) e^{-\a t}, \quad \int_0^\infty \int_{\TT} w _{yxx}^2 \,\dx \,\dt \leq C E_{10}^{(1)}(y),
\ea
and
\be\label{decay-H1-eta-w-y-t-2}
\int_0^\infty\int_{\TT} \eta w_{yt}^2\,\dx\,\dt \leq C E_{10}^{(1)}(y).
\ee }
\end{prop}
\begin{proof} Thanks to Lemmas \ref{prop-H1-w-y} and \ref{prop-H1-eta-y}, we deduce \eqref{decay-H1-eta-w-y-1}
by similar arguments using in the proof of Proposition \ref{prop-decay-exp-rx}.
While we get, by taking $L^2(\TT)$ inner product of \eqref{mm-y-1} with $w_{yt},$ that
\ba\label{decay-H1-eta-w-y-t-3}
\int_{\TT} \eta w_{yt}^2 \,\dx + \nu \frac{\rm d}{\dt} \int_{\TT} w_{yx}^2 \,\dx = - \int_{\TT}\big(w w_{yx} + w_y w_x + \eta_y (w_t + w w_x) + p( \eta )_{yx} \big) w_{yt}\,\dx,
\nn \ea
which implies
\ba\label{decay-H1-eta-w-y-t-1}
\nu\frac{\rm d}{\dt} \int_{\TT} w_{yx}^2\,\dx + \frac{1}{2} \int_{\TT} \eta w_{yt}^2\,\dx \leq C \int_{\TT}\bigl( w_{yx}^2 + w_y^2 + \eta_y^2 + \eta_{yx}^2\bigr)\,\dx.
\ea
By integrating \eqref{decay-H1-eta-w-y-t-1} over $[0,t]$ and using \eqref{decay-H1-eta-w-y-1},
we obtain \eqref{decay-H1-eta-w-y-t-2}.
\end{proof}
\subsection{Decay estimate of $\|w_y\|_{H^{2}}$}
In view of \eqref{ass-ini-1}, we have
\be\label{ini-8}
E_{20}^{(1)} \eqdefa \|\d_y \varsigma_{0}\|_{H^2_\h}^2+ \|\d_y w _{0}\|_{H^2_\h}^2 \in (L^{1}\cap L^{\infty}) (\R)
\andf \bar E_{20}^{(1)} \eqdefa \sup_{y\in \R} E_{20}^{(1)} <\infty.
\ee
Throughout this subsection, $A, \a, C$ are positive numbers solely depending on $(a, \g, \nu, \bar \varsigma_0, \bar E_{20}, \underline{\varsigma}_0, \bar E_{10}^{(1)}),$ which may change from line to line.
\begin{prop}\label{prop-Dtv-y-L2}
{\sl Let $D_t\eqdefa\d_t + w \d_x$ be the material derivative. Then for all $t \in \R^{+}$, there holds
\be\label{Dtv-L2-y-1}
\int_{\TT} \eta |D_t w_y |^2 \, \dx\leq C E_{20}^{(1)} (y) e^{-\a t} \andf \int_0^\infty \int_{\TT} |(D_t w_y )_x|^2\,\dx\,\dt \leq C E_{20}^{(1)} (y) ,
\ee and
\be\label{vxx-L2-y-1}
\int_{\TT} \bigl(|\eta_{yt}|^2 + |w_{yt}|^2 + |w_{yxx}|^2\bigr) \, \dx\leq C E_{20}^{(1)} (y)e^{-\a t}.
\ee
}
\end{prop}
\begin{proof}
Applying $D_t$ to $\eta\times$\eqref{mm-y-2} gives
\ba\label{Dtv-L2-y-3}
\eta D_t^2 w_y + D_t \eta D_t w_y + D_t (\eta w_x w_y) + D_t (\eta_y D_t w ) - \nu D_t w _{yxx} + D_t p( \eta )_{yx} = 0.
\ea
It is easy to observe that
\begin{align*
D_t \eta D_t w_y & = - \eta w _x D_t w_y = w_x (\eta w_x w_y + \eta_y D_t w - \nu w _{yxx} + p( \eta )_{yx} ),
\\
D_t (\eta w_x w_y) &= D_t (\eta w_x) w_y + \eta w_x D_t w_y,
\\
D_t (\eta_y D_t w ) & = -(\eta_y w_x + \eta_x w_y + \eta w_{yx}) D_t w + \eta_y D_t^2 w,
\\
- \nu D_t w _{yxx} &
= -\nu (D_t w_y )_{xx} + \nu( 2 w_x w_{yxx} + w_{xx} w_{yx}),
\end{align*}
and
\ba
D_t p( \eta )_{yx} = p( \eta )_{tyx} + w p( \eta )_{yxx}.
\nn \ea
Then by taking $L^2(\TT)$ inner product of \eqref{Dtv-L2-y-3} with $D_t w_y$ and using the fact
$$
w_x p( \eta )_{yx} + w p( \eta )_{yxx} = (w p(\eta)_{yx})_x,
$$ we find
\begin{align*}
\frac{1}{2}& \frac{\rm d}{\dt} \int_{\TT} \eta |D_t w_y |^2 \, \dx + \nu \int_{\TT} |(D_t w_y )_x|^2\,\dx \\
& \leq \frac{\nu}{2} \int_{\TT} |(D_t w_y )_x|^2\,\dx + C \int_{\TT} \bigl( w_y^2 + w_{yx}^2 + w _{yxx}^2 + \eta_y^2 +\eta_{yx}^2\bigr)\,\dx.
\end{align*}
This implies
\ba\label{Dtv-L2-y}
\frac{\rm d}{\dt} \int_{\TT} \eta |D_t w_y |^2 \, \dx
+ \nu \int_{\TT} |(D_t w_y )_x|^2\,\dx \leq C \int_{\TT} \bigl(w_y^2 + w_{yx}^2 + w _{yxx}^2 + \eta_y^2 +\eta_{yx}^2\bigr)\,\dx,
\ea
from which,
Propositions \ref{prop-decay-L2-y} and \ref{prop-decay-H1-y}, Lemmas \ref{prop-H1-w-y} and \ref{prop-H1-eta-y},
we deduce \eqref{Dtv-L2-y-1} by the similar argument used in the proof of Proposition \ref{prop-decay-exp-rx}.
Thanks to \eqref{Dtv-L2-y-1}, we conclude \eqref{vxx-L2-y-1} by using the mass and momentum equations \eqref{CNS-1d-dy}.
\end{proof}
\subsection{Proof of Proposition \ref{S2prop2}} \label{sec-decay-y-all}
\begin{proof}[Proof of Proposition \ref{S2prop2}] By summarizing Propositions \ref{prop-decay-L2-y}, \ref{prop-decay-H1-y} and \ref{prop-Dtv-y-L2},
we deduce that there exist two positive constants $C$ and $\a$ solely depends on $(a, \g, \nu, \bar \varsigma_0, \bar E_{20}, \underline{\varsigma}_0, \bar E_{10}^{(1)})$ such that
\ba\label{decay-exp-all-y-1}
\| \eta_y \|_{H^1_\h} + \| w_{y} \|_{H^2_\h} + \| \eta _{yt}\|_{L^2_\h} + \|w_{yt}\|_{L^2_\h} \leq C \left( E_{20}^{(1)}(y)\right)^{\frac 12} e^{-\a t}, \quad \forall \, t\in \R^{+}.
\ea
In what follows, we shall
follow the same strategy as that of the proof of Proposition \ref{S2prop1}.
We first get, by applying $\d_x$ to \eqref{CNS-1d-dy}, that
\be\label{CNS-1d-dy-dx}
\left\{\begin{aligned}
& \eta _{yxt} + ( \eta w )_{yxx}= 0,\\
&( \eta w )_{yxt} + ( \eta w ^2)_{yxx} - \nu w _{yxxx} + p( \eta )_{yxx} = 0.
\end{aligned}\right.
\ee
We can also rewrite $\eqref{CNS-1d-dy-dx}_{2}$ as
\beq\label{CNS-1d-dy-dx-1}
\begin{split}
D_{t} w_{yx} &+ (w_{y}w_{xx} + 2 w_{x}w_{yx}) + \eta^{-1} \eta_{y} (D_{t} w_{x} + w_{x}^{2}) \\
&+ \eta^{-1} \eta_{x} (D_{t} w_{y} + w_{x}w_{y}) + \eta^{-1} \eta_{yx} D_{t}w
- \nu \eta^{-1} w_{yxxx} + \eta^{-1} p(\eta)_{yxx} = 0.
\end{split}
\eeq
We split the proof of the remaining estimates in \eqref{decay-exp-all-y-2} into the following steps:
\medskip
\noindent{\bf Step 1.} {Decay estimates for $\eta_{yxx}.$}
\medskip
Recalling that $\zeta \eqdefa \eta^{-1},$
we deduce from $\eqref{CNS-1d-dy-dx}_{1}$ that
\be\label{CNS-1d-dy-dx-2}
\zeta w_{yxxx} = D_{t}\zeta_{yxx} - \zeta_{yx} w_{xx} - \zeta_{x} w_{yxx} - \zeta_{y} w_{xxx} + \zeta_{xxx} w_{y} + \zeta_{yxx} w_{x} + \zeta_{xx} w_{yx}.
\ee
Plugging \eqref{CNS-1d-dy-dx-2} into \eqref{CNS-1d-dy-dx-1} gives rise to
\begin{align*
&D_{t} (w_{yx} - \nu \zeta_{yxx}) + w_{x}(w_{yx} - \nu\zeta_{yxx}) + (w_{y}w_{xx} + w_{x}w_{yx}) \\
& \quad + \eta^{-1} \eta_{y} (D_{t} w_{x} + w_{x}^{2}) + \eta^{-1} \eta_{x} (D_{t} w_{y} + w_{x}w_{y}) + \eta^{-1} \eta_{yx} D_{t}w \\
& \quad - \nu \bigl(- \zeta_{yx} w_{xx} - \zeta_{x} w_{yxx} - \zeta_{y} w_{xxx} + \zeta_{xxx} w_{y} + \zeta_{xx} w_{yx}\bigr)+ \eta^{-1} p(\eta)_{yxx} = 0.
\end{align*}
By taking $L^2(\TT)$ inner product of the above equation with $\eta (w_{yx} - \nu \zeta_{yxx}) $ and using the decay estimates we have derived in the previous sections, we obtain
\ba\label{CNS-1d-dy-dx-4}
&\frac{1}{2}\frac{\rm d}{\dt} \int_{\TT} \eta (w_{yx} - \nu \zeta_{yxx})^{2} \,\dx + \int_{\TT} (w_{yx} - \nu \zeta_{yxx}) p(\eta)_{yxx} \,\dx\\
& \quad \leq C \left(E_{20}(y)\right)^{\frac 12} e^{-\a t} \int_{\TT} \eta (w_{yx} - \nu \zeta_{yxx})^{2} \,\dx + \de \int_{\TT} \eta (w_{yx} - \nu \zeta_{yxx})^{2} \,\dx \\
& \qquad + C \de^{-1} \left( E_{30}(y) + E_{20}^{(1)}(y) \right) e^{-\a t}.
\ea
To handle the term related to the pressure, we write
\ba\label{CNS-1d-dy-dx-5}
p(\eta)_{yxx} & = p'(\eta)\eta_{yxx} +p''(\eta) \eta_{y}\eta_{xx} + p''(\eta) 2 \eta_{x}\eta_{yx} + p'''(\eta)\eta_{y} \eta_{x}^{2}\\
&= p'(\eta) (-\zeta^{-2} \zeta_{yxx} + 2 \zeta^{-3} \zeta_{y} \zeta_{xx} + 4 \zeta^{-3} \zeta_{x}\zeta_{yx} - 6 \zeta^{-4} \zeta_{y}\zeta_{x}^{2}) \\
& \quad + p''(\eta) \eta_{y}\eta_{xx} + p''(\eta) 2 \eta_{x}\eta_{yx} + p'''(\eta)\eta_{y} \eta_{x}^{2}\\
& = p'(\eta) \eta^{2} \nu^{-1} (w_{yx} - \nu\zeta_{yxx}) - p'(\eta) \eta^{2} \nu^{-1} w_{yx} + p'(\eta) \eta^{3} (2 \zeta_{y} \zeta_{xx} + 4 \zeta_{x}\zeta_{yx} - 6 \eta^{2} \zeta_{y}\zeta_{x}^{2}) \\
& \quad + p''(\eta) \eta_{y}\eta_{xx} + p''(\eta) 2 \eta_{x}\eta_{yx} + p'''(\eta)\eta_{y} \eta_{x}^{2}.
\ea
Notice that
$$
p'(\eta)\eta \geq p'(\underline \eta) \underline \eta >0,
$$
by choosing $\de$ sufficiently small, we deduce from \eqref{CNS-1d-dy-dx-4} and \eqref{CNS-1d-dy-dx-5} that
\begin{align*
&\frac{\rm d}{\dt} \int_{\TT} \eta (w_{yx} - \nu \zeta_{yxx})^{2}\,\dx
+ \nu^{-1} p'(\underline \eta) \underline \eta \int_{\TT} \eta (w_{yx} - \nu \zeta_{yxx})^{2} \,\dx\\
& \leq C e^{-\a t} \int_{\TT} \eta (w_{yx} - \nu \zeta_{yxx})^{2} \,\dx + C \left( E_{30}(y) + E_{20}^{(1)}(y) \right) e^{-\a t}.
\end{align*}
Applying Gronwall's inequality gives
\ba\label{CNS-1d-dy-dx-7}
\int_{\TT} \eta (w_{yx} - \nu \zeta_{yxx})^{2} \,\dx \leq C \left( E_{30}(y) + E_{20}^{(1)}(y) \right) e^{-\a t},
\nn \ea
from which and \eqref{decay-exp-all-y-1}, we infer
\ba\label{CNS-1d-dy-dx-8}
\int_{\TT} \zeta_{yxx}^{2}\,\dx \leq C \left( E_{30}(y) + E_{20}^{(1)}(y) \right) e^{-\a t}.
\ea
This leads to
\ba\label{CNS-1d-dy-dx-9}
\int_{\TT} \eta_{yxx}^{2}\,\dx \leq C \left( E_{30}(y) + E_{20}^{(1)}(y) \right) e^{-\a t}.
\ea
\medskip
\noindent{\bf Step 2.} {Decay estimates for $D_{t} w_{yx}.$}
\medskip
We first rewrite $\eqref{CNS-1d-dy-dx}_{2}$ as
\beq\label{Dt-wys}\begin{split}
&\eta D_{t} w_{yx} - \nu w_{yxxx} + \frak{g} + p(\eta)_{yxx} = 0,\\
\frak{g} \eqdefa \eta (w_{y}w_{xx} &+ 2 w_{x}w_{yx}) + \eta_{y} (D_{t} w_{x} + w_{x}^{2}) + \eta_{x} (D_{t} w_{y} + w_{x}w_{y}) + \eta_{yx} D_{t}w.
\end{split}
\eeq
By taking $L^2(\TT)$ inner product of \eqref{Dt-wys} with $\eta^{-1} w_{yxxx},$ we obtain
\ba\label{Dt-wys-2}
\int_{\TT} D_{t} w_{yx} w_{yxxx}\,\dx-\nu \int_{\TT} \eta^{-1} |w_{yxxx}|^{2}\,\dx+\int_{\TT} \bigl(\frak{g}
+ p(\eta)_{yxx} \bigr) \eta^{-1} w_{yxxx}\,\dx=0.
\ea
It is easy to observe that
\begin{align*}
\int_{\TT} D_{t} w_{yx} w_{yxxx}\,\dx & = \int_{\TT} \d_{t} w_{yx} w_{yxxx}\,\dx + \int_{\TT} w w_{yxx} w_{yxxx}\,\dx\\
& = -\frac{1}{2} \frac{\rm d}{\dt} \int_{\TT} |w_{yxx}|^{2} - \frac{1}{2}\int_{\TT} w_{x} (w_{yxx})^{2}.
\end{align*}
While it follows from \eqref{CNS-1d-Dtwn-26}, \eqref{decay-exp-all-y-1} and \eqref{CNS-1d-dy-dx-9} that
\ba
\int_{\TT} \bigl(\frak{g} + p(\eta)_{yxx} \bigr) \eta^{-1} w_{yxxx}\,\dx \leq C \de^{-1} E(y) e^{-\a t} + \de \int_{\TT} |w_{yxxx}|^{2}\,\dx.
\nn \ea
By inserting the above estimates into \eqref{Dt-wys-2} and choosing $\de$ suitably small, we achieve
\ba\label{Dt-wys-1}
\frac{\rm d}{\dt} \int_{\TT} |w_{yxx}|^{2}\,\dx + \nu \bar\eta^{-1} \int_{\TT} |w_{yxxx}|^{2}\,\dx \leq C E(y) e^{-\a t}.
\ea
While by taking $L^2(\TT)$ inner product of \eqref{Dt-wys} with $ D_{t} w_{yx},$ we find
\ba\label{Dt-wys-7}
\int_{\TT} \eta |D_{t}w_{yx}|^{2}\,\dx-\nu\int_{\TT}D_t w_{yxxx}D_{t} w_{yx}\,\dx +\int_{\TT}\bigl( D_t\frak{g} +D_t p(\eta)_{yxx}\bigr)D_{t} w_{yx}\,\dx = 0.
\ea
Observing that
\begin{align*
- \nu\int_{\TT} w_{yxxx} D_{t} w_{yx}\,\dx & = - \nu \int_{\TT} w_{yxxx} \d_{t} w_{yx} \,\dx - \nu \int_{\TT} w_{yxxx} w w_{yxx}\,\dx \\
& = \frac\nu2\frac{\rm d}{\dt} \int_{\TT} |w_{yxx}|^{2}\,\dx + \frac{\nu}{2} \int_{\TT} (w_{yxx})^{2} w_{x}\,\dx\\
& \geq \frac\nu2\frac{\rm d}{\dt} \int_{\TT} |w_{yxx}|^{2} \,\dx- C E(y) e^{-\a t}.
\end{align*}
By using the estimates \eqref{CNS-1d-Dtwn-26}, \eqref{decay-exp-all-y-1} and \eqref{CNS-1d-dy-dx-9}, we find
\ba
\int_{\TT} \bigl(\frak{g} + p(\eta)_{yxx} \bigr) D_{t} w_{yx}\,\dx \leq C \de^{-1} E(y) e^{-\a t} + \de \int_{\TT} | D_{t} w_{yx}|^{2}\,\dx.
\nn \ea
By substituting the above estimates into \eqref{Dt-wys-7}
and taking $\de$ to be suitably small, we achieve
\ba\label{Dt-wys-5}
\nu\frac{\rm d}{\dt} \int_{\TT} |w_{yxx}|^{2}\,\dx + \int_{\TT} \eta |D_{t}w_{yx}|^{2}\,\dx \leq C E(y) e^{-\a t}.
\ea
On the other hand, we get, by applying $D_{t}$ to \eqref{Dt-wys} and then taking $L^2(\TT)$ inner product of the resulting
equation with $D_{t} w_{yx},$ that
\ba\label{Dt-wys-6}
\int_{\TT}\bigl(D_t(\eta D_{t} w_{yx})- \nu D_t w_{yxxx} + D_t\frak{g} +D_t p(\eta)_{yxx}\bigr)D_{t} w_{yx}\,\dx = 0.
\ea
Next we handle term by term above. We first observe that
\begin{align*
\int_{\TT}D_{t}( \eta D_{t} w_{yx} ) D_{t} w_{yx}\,\dx & = \int_{\TT} (D_{t} \eta) D_{t} w_{yx} D_{t} w_{yx}\,\dx
+ \int_{\TT}\eta (D_{t} D_{t} w_{yx})D_{t} w_{yx}\,\dx \\
& = \frac{1}{2} \frac{\rm d}{\dt} \int_{\TT}\eta |D_{t} w_{yx}|^{2}\,\dx + \frac12\int_{\TT} (D_{t} \eta) |D_{t} w_{yx}|^{2}\,\dx \\
& \geq \frac{1}{2}\frac{\rm d}{\dt} \int_{\TT}\eta |D_{t} w_{yx}|^{2}\,\dx - C \int_{\TT} |D_{t} w_{yx}|^{2}.
\end{align*}
Notice that
\ba\label{Dt-wys-12}
D_{t} w_{yxxx} = (D_{t} w_{yx})_{xx} - w_{xx} w_{yxx} - 2 w_{x} w_{yxxx},
\nn \ea
we have
\begin{align*
- \nu \int_{\TT} D_{t} w_{yxxx} D_{t} w_{yx}\,\dx & = -\nu \int_{\TT}(D_{t} w_{yx})_{xx} D_{t} w_{yx}\,\dx
+ \nu \int_{\TT} \bigl(w_{xx} w_{yxx} + 2 w_{x} w_{yxxx}\bigr)D_{t} w_{yx}\,\dx \\
& \geq \nu \int_{\TT} |(D_{t} w_{yx})_{x}|^{2}\,\dx - C \int_{\TT} \bigl(|D_{t} w_{yx}|^{2} + |w_{yxxx}|^{2}\bigr)\,\dx.
\end{align*}
We get, by applying the estimates \eqref{CNS-1d-Dtwn-26}, \eqref{decay-exp-all-y-1} and \eqref{CNS-1d-dy-dx-9}, that
\ba\label{Dt-wys-14}
\int_{\TT} (D_{t} \bigl(\frak{g}-\eta_{x} D_{t} w_{y} \bigr) D_{t} w_{yx}\,\dx \leq C E(y) e^{-\a t} + C \int_{\TT} | D_{t} w_{yx}|^{2} \,\dx.
\nn \ea
In view of \eqref{mm-y-2}, we write
\ba\label{Dt-wys-15}
D_{t} (\eta_{x} D_{t} w_{y}) = D_{t} \eta_{x} D_{t} w_{y} - \eta_{x} D_{t} \big( w_x w_y + \eta^{-1} \eta_y D_t w - \nu \eta^{-1} w _{yxx} + \eta^{-1} p( \eta )_{yx} \big).
\nn \ea
It suffices to deal with the highest order derivative term $\eta_{x}\eta^{-1} D_{t} w_{yxx}$ above. Other terms can be estimated similarly, even easier.
Observing that
\ba\label{Dt-wys-16}
D_{t} w_{yxx} = (D_{t} w_{yx})_{x} - w_{x} w_{yxx},
\nn \ea
we get
\ba\label{Dt-wys-17}
\int_{\TT} \eta_{x}\eta^{-1} (D_{t} w_{yxx}) D_{t} w_{yx}\,\dx \leq C E(y) e^{-\a t} + C \int_{\TT} | D_{t} w_{yx}|^{2}\,\dx
+ \frac{\nu}{4} \int_{\TT} |(D_{t} w_{yx} )_{x} |^{2}\,\dx.
\nn \ea
For the term related to the pressure in \eqref{Dt-wys-6}, we have
\begin{align*}\label{Dt-wys-18}
&\int_{\TT} (D_{t} p(\eta)_{yxx}) D_{t} w_{yx}\,\dx = \int_{\TT} \bigl(\d_{t} p(\eta)_{yxx} + w p(\eta)_{yxxx}\bigr) D_{t} w_{yx}\,\dx \\
& = - \int_{\TT} \bigl((\d_{t} p(\eta)_{yx} + w p(\eta)_{yxx}) (D_{t} w_{yx} )_{x} - w_{x} p(\eta)_{yxx} D_{t} w_{yx}\bigr)\,\dx\\
& \leq C \int_{\TT} |\d_{t} p(\eta)_{yx} + w p(\eta)_{yxx}|^{2}\,\dx + \frac{\nu}{4} \int_{\TT} |(D_{t} w_{yx} )_{x} |^{2}\,\dx
+ \int_{\TT} \bigl(| w_{x} p(\eta)_{yxx}|^{2} + | D_{t} w_{yx}|^{2}\bigr)\,\dx\\
& \leq C E(y) e^{-\a t} + C \int_{\TT} | D_{t} w_{yx}|^{2}\,\dx + \frac{\nu}{4} \int_{\TT} |(D_{t} w_{yx} )_{x} |^{2}\,\dx.
\end{align*}
By substituting the above estimate into \eqref{Dt-wys-6}, we arrive at
\ba\label{Dt-wys-9}
\frac{\rm d}{\dt} \int_{\TT} \eta |D_{t} w_{yx}|^{2}\,\dx + \nu \int_{\TT} |(D_{t} w_{yx})_{x}|^{2} \,\dx\leq C E(y) e^{-\a t} + C \int_{\TT} \bigl(|w_{yxxx}|^{2} + |D_{t}w_{yx}|^{2}\bigr)\,\dx.
\ea
By virtue of \eqref{Dt-wys-1}, \eqref{Dt-wys-5} and \eqref{Dt-wys-9}, we can find a large enough positive constant $A>0$ so that
\ba\label{Dt-wys-19}
\frac{\rm d}{\dt} &\int_{\TT}\bigl( A \bigl(|w_{yxx}|^2+|w_{yxx}|^{2}\bigr) + \eta |D_{t} w_{yx}|^{2}\bigr)\,\dx\\
&+ \int_{\TT} \bigl(|w_{yxxx}|^{2} + |D_{t}w_{yx}|^{2} + \nu |(D_{t} w_{yx})_{x}|^{2}\bigr)\,\dx \leq C E(y) e^{-\a t}.
\ea
Together with \eqref{decay-exp-all-y-1}, we deduce from \eqref{Dt-wys-19} that
\ba\label{Dt-wys-20}
\|D_{t} w_{yx}\|_{L^{2}_{h}}^{2} \leq C E(y) e^{-\a t} .
\ea
Together with \eqref{decay-exp-all-y-1}, this implies
\ba\label{Dt-wys-21}
\|w_{yxxx}\|_{L^{2}_{h}}^{2} \leq C E(y) e^{-\a t} .
\ea
By summarizing the estimates \eqref{decay-exp-all-y-1}, \eqref{CNS-1d-dy-dx-9}, \eqref{Dt-wys-20} and \eqref{Dt-wys-21}, we conclude the proof of \eqref{decay-exp-all-y-2}.
\end{proof}
\section{Decay estimates of $(\eta_{yy}, w_{yy})$}\label{sec:1dNS-yy}
In this section, we investigate the decay in time estimates of $(\eta _y, w _y)$. We first get,
by applying $\d_y^{2}$ to \eqref{CNS-limit}, that
\be\label{CNS-1d-dyy}
\left\{\begin{aligned}
& \eta _{yyt} + ( \eta w )_{yyx}= 0,\\
&( \eta w )_{yyt} + ( \eta w ^2)_{yyx} - \nu w _{yyxx} + p( \eta )_{yyx} = 0.
\end{aligned}\right.
\ee
Integrating \eqref{CNS-1d-dyy} with respect to $x$ over $\TT$ gives
\ba\label{csv-m-m-dyy}
\frac{\rm d}{\dt}\int_{\TT} \eta_{yy} \,\dx = 0 \andf \frac{\rm d}{\dt}\int_{\TT} ( \eta w )_{yy} \,\dx = 0.
\nn \ea
This implies
\ba\label{csv-m-m-1-dyy}
\int_{\TT} \eta_{yy} \,\dx = 0 \andf \int_{\TT} ( \eta w )_{yy}\,\dx = 0, \quad \forall \, t\in\R_{+}.
\ea
\subsection{Decay estimates of $L^{2}$ norms}
Introduce
\be\label{ini-6-dyy}
E_{00}^{(2)} (y)\eqdefa \|(\d_y^{2} \varsigma_{0},\d_y^{2} w _{0})(y)\|_{L^2_\h}^2 \in (L^{1} \cap L^{\infty})(\R) \andf \bar E_{00}^{(2)} \eqdefa \sup_{y\in \R} E_{00}^{(2)} <\infty.
\ee
Throughout this subsection, $A, \a$ and $C$ are positive numbers solely depending on $(a, \g, \nu, \bar \varsigma_0, \bar E_{20}, \bar E_{10}^{(1)}, \underline{\varsigma}_0),$ which may differ from line to line. We first give the basic energy estimates:
\begin{lem}\label{energy-basic-yy}
{\sl For all $t\in \R^{+}$,
one has
\ba\label{energy-0-0-yy}
\int_{\TT} \bigl( \eta w_{yy}^2 + \eta_{yy}^2\bigr) \,\dx + \nu \int_0^t \int_{\TT}| w _{yyx}|^2 \,\dx \leq C \left( E_{10}^{(1)} (y) + E_{20}(y)\right).
\ea}
\end{lem}
\begin{proof}
Firstly, we rewrite $\eqref{CNS-1d-dyy}_{2}$ as
\ba\label{CNS-1d-dyy-1}
\eta D_t w_{yy} + \eta (2 w_{y} w_{yx} + w_{yy}w_{x})+ 2 \eta_{y} (D_{t} w_{y} + w_{y} w_{x}) + \eta_{yy} D_{t} w - \nu w _{yyxx} + p( \eta )_{yyx} = 0.
\ea
By using the decay estimates we have derived in the previous sections and testing \eqref{CNS-1d-dyy-1} by $ w_{yy}$, we obtain
\ba\label{energy-0-1-dyy}
\frac{1}{2} & \frac{\rm d}{\dt} \int_{\TT} \eta w_{yy}^2 \,\dx + \nu \int_{\TT}| w _{yyx}|^2 \,\dx \\
& \quad \leq \frac{\nu}{8} \int_{\TT}| w _{yyx}|^2 \,\dx + C e^{-\a t} \int_{\TT} \bigl( \eta w_{yy}^2 + \eta_{yy}^2\bigr) \,\dx + \int_{\TT} p'( \eta ) \eta_{yy} w _{yyx} + C E_{10}^{(1)} (y) e^{-\a t}.
\ea
On the other hand, by taking $L^2(\TT)$ inner product of $\eqref{CNS-1d-dyy}_1$ with $ \eta_{yy},$ we find
\ba\label{energy-0-2-dyy}
\frac{1}{2}&\frac{\rm d}{\dt} \int_{\TT} \eta_{yy}^2 \,\dx = - \int_{\TT}( \eta w )_{yyx} \eta_{yy} \,\dx \\
& = - \int_{\TT} \bigl( \eta w_{yyx} + \eta_{yy} w_{x} + \eta_{yyx} w + 2 \eta_{yx} w_{y} + 2 \eta_{y} w_{yx} + \eta w_{yyx} + \eta_{x} w_{yy}\bigr) \eta_{yy}\,\dx \\
& = - \int_{\TT} \eta w_{yyx} \eta_{yy}\,\dx - \int_{\TT} \bigl(\frac{1}{2}\eta_{yy} w_{x} + 2 \eta_{yx} w_{y} + 2 \eta_{y} w_{yx} + \eta w_{yyx} + \eta_{x} w_{yy}\bigr) \eta_{yy}\,\dx.
\ea
By summing up \eqref{energy-0-1-dyy} with \eqref{energy-0-2-dyy}, we achieve
\begin{align*}
&\frac{\rm d}{\dt} \int_{\TT} \frac{1}{2} \bigl(\eta w_{yy}^2 + \eta_{yy}^2\bigr) \,\dx + \nu \int_{\TT}| w _{yyx}|^2 \,\dx \\
& \quad \leq \frac{\nu}{4} \int_{\TT}| w _{yyx}|^2 \,\dx + C e^{-\a t} \int_{\TT} \bigl( \eta w_{yy}^2 + \eta_{yy}^2\bigr) \,\dx + \int_{\TT} (p'( \eta ) -\eta) \eta_{yy} w _{yyx}\,\dx + C E_{10}^{(1)} (y) e^{-\a t},
\end{align*}
from which and the decay of $p'(\eta) - \eta$ obtained in \eqref{S2eq1}, we infer
\ba\label{energy-0-3-dyy}
&\frac{\rm d}{\dt} \int_{\TT} \bigl(\eta w_{yy}^2 + \eta_{yy}^2\bigr) \,\dx + \nu \int_{\TT}| w _{yyx}|^2 \,\dx \\
& \quad \leq C e^{-\a t} \int_{\TT} \bigl( \eta w_{yy}^2 + \eta_{yy}^2\bigr) \,\dx + C E_{10}^{(1)} (y) e^{-\a t}.
\ea
Applying Gronwall's inequality leads to \eqref{energy-0-0-yy}.
\end{proof}
\begin{prop}\label{prop-decay-L2-yy}
{\sl For all $t\in \R_{+}$,
\ba\label{decay-L2-yy}
\int_{\TT} \bigl( \eta w_{yy}^2 + \eta_{yy}^2\bigr) (t)\,\dx \leq C \left( E_{10}^{(1)} (y) + E_{20}(y)\right) e^{-\a t}.
\ea}
\end{prop}
\begin{proof} We first get, by multiplying the momentum equation of \eqref{CNS-1d-dyy} by $I( \eta_{yy})$ and integrating the resulting equality over $\TT,$ that
\ba\label{decay-L2-yy-1}
\int_{\TT} p( \eta )_{yy} \eta_{yy} \,\dx = \int_{\TT} ( \eta w )_{yyt} I( \eta_{yy})\,\dx - \int_{\TT} ( \eta w^2)_{yy} \eta_{yy}\,\dx + \nu \int_{\TT}w_{yyx} \eta_{yy} \,\dx .
\ea
We now handle term by term above. For the term on the left-hand side of \eqref{decay-L2-yy-1}, we have
\begin{align*
\int_{\TT} p( \eta )_{yy} \eta_{yy} \,\dx & = \int_{\TT} p'( \eta ) (\eta_{yy})^{2} \,\dx + \int_{\TT} p''( \eta ) (\eta_{y})^{2} \eta_{yy} \,\dx\\
& \geq \int_{\TT} p'( \eta ) (\eta_{yy})^{2} \,\dx - C \de E_{10}^{(1)} (y) e^{-\a t} \int_{\TT} (\eta_{yy})^{2} \,\dx - C \de^{-1} E_{10}^{(1)} (y) e^{-\a t}.
\end{align*}
We then compute the first term on the right-hand side of \eqref{decay-L2-yy-1} as follows
\begin{align*
\int_{\TT} ( \eta w )_{yyt} I( \eta_{yy})\,\dx & = \frac{\rm d}{\dt}\int_{\TT} ( \eta w )_{yy} I( \eta_{yy})\,\dx - \int_{\TT} ( \eta w )_{yy} I( \eta _{yyt})\,\dx\\
& = \frac{\rm d}{\dt}\int_{\TT} ( \eta w )_{yy} I( \eta_{yy})\,\dx + \int_{\TT} ( \eta w )_{yy} I\left(( \eta w )_{yyx}\right)\,\dx \\
& = \frac{\rm d}{\dt}\int_{\TT} ( \eta w )_{yy} I( \eta_{yy})\,\dx + \int_{\TT} ( \eta w )_{yy} (( \eta w )_{yy} (t,x,y) - ( \eta w )(t,0,y))\,\dx \\
& = \frac{\rm d}{\dt}\int_{\TT} ( \eta w )_{yy} I( \eta_{yy})\,\dx + \int_{\TT} ( ( \eta w )_{yy} )^{2}\,\dx\\
& = \frac{\rm d}{\dt}\int_{\TT} ( \eta w )_{yy} I( \eta_{yy})\,\dx + \int_{\TT} (\eta_{yy} w + 2 \eta_{y} w_{y} + \eta w_{yy})^{2}\,\dx \\
& \leq \frac{\rm d}{\dt}\int_{\TT} ( \eta w )_{yy} I( \eta_{yy})\,\dx + C \left( E_{10}^{(1)} (y) + E_{20}(y)\right) e^{-\a t}+ C \int_{\TT} \eta w_{yy}^{2}\,\dx,
\end{align*}
where we used $\eqref{CNS-1d-dyy}_{1}$, \eqref{csv-m-m-1-dyy}, \eqref{energy-0-0-yy}, and the decay estimates we have obtained in the previous sections.
Similarly, we have
\begin{align*
- \int_{\TT} ( \eta w^2)_{yy} \eta_{yy}\,\dx & = - \int_{\TT} ( w^{2} \eta_{yy} + 2 \eta_{y} w w_{y} + \eta (w w_{yy} + w_{y}^{2})) \eta_{yy}\,\dx \\
& \leq C \left( E_{10}^{(1)} (y) + E_{20}(y)\right) e^{-\a t}+ C \int_{\TT} \eta w_{yy}^{2}\,\dx.
\end{align*}
For the last term in \eqref{decay-L2-yy-1}, we find
\ba\label{decay-L2-yy-40}
\nu \int_{\TT}w_{yyx} \eta_{yy} \,\dx \leq \de \nu \int_{\TT} \eta_{yy}^{2}\,\dx + \de^{-1} \nu \int_{\TT} w_{yyx}^{2}\,\dx.
\nn \ea
Notice that $p'(\eta) \geq p'(\underline \eta) >0$, by substituting the above estimates into \eqref{decay-L2-yy-1}
and taking $\de$ to be suitably small, we achieve
\ba\label{decay-L2-yy-5}
\frac{p'(\underline{\eta})}{2} \int_{\TT} \eta_{yy}^2 \,\dx\leq \frac{\rm d}{\dt}\int_{\TT} ( \eta w )_{yy} I( \eta_{yy})\,\dx + C \left( E_{10}^{(1)} (y) + E_{20}(y)\right) e^{-\a t}+ C \int_{\TT} \bigl(\eta w_{yy}^{2}+ w _{yyx}^2\bigr) \,\dx.
\ea
While it follows from a similar proof of Lemma \ref{prop-kinetic-tx} that
\ba\label{decay-L2-yy-6}
\int_{\TT} \bigl( \eta w_{yy}^2 - \langle \eta w_{yy} \rangle^2\bigr) \,\dx \leq \bbeta^2 \int_{\TT} w _{yyx}^2 \,\dx.
\nn \ea
In view of \eqref{csv-m-m-1-dyy}, it holds that
$$
\langle \eta w_{yy} \rangle = - \langle w \eta_{yy} \rangle - \langle 2 \eta_{y} w_{y} \rangle.
$$
As a result, we infer
\ba\label{decay-L2-yy-7}
\int_{\TT} \eta w_{yy}^2\,\dx
& \leq \bbeta^2 \int_{\TT} w_{yyx}^2 \,\dx + \langle \eta w_{yy} \rangle^2 \\
& \leq \bbeta^2 \int_{\TT} w _{yyx}^2 \,\dx + 8 \int_{\TT} \bigl( \eta_{yy}^2 w^2 + \eta_{y}^{2} w_{y}^{2} \bigr) \,\dx\\
& \leq C \left( E_{10}^{(1)} (y) + E_{20}(y)\right) e^{-\a t} + C\int_{\TT} w _{yyx}^2 \,\dx.
\ea
Thanks to \eqref{decay-L2-yy-5} and \eqref{decay-L2-yy-7}, we deduce that
\ba\label{decay-L2-yy-8}
\frac{p'(\underline{\eta})}{2}\int_{\TT} \eta_{yy}^2 \,\dx - \frac{\rm d}{\dt}\int_{\TT} ( \eta w )_{yy} I( \eta_{yy})\,\dx \leq C \left( E_{10}^{(1)} (y) + E_{20}(y)\right) e^{-\a t} + C\int_{\TT} w _{yyx}^2 \,\dx.
\ea
Let $A$ be a sufficiently large positive constant, we denote
\be\label{F2-1-def}
F_1^{(2)}\eqdefa \int_{\TT} \bigl(A ( \eta w_{yy}^2 + \eta_{yy}^2) - ( \eta w )_{yy} I( \eta_{yy}) \bigr)\,\dx \andf
F_2^{(2)}\eqdefa F_1^{(1)} + F_1^{(2)},
\ee
where $F_1^{(1)} $ is given in \eqref{F1-1-def}.
Then by virtue of \eqref{energy-0-3-dyy} and \eqref{decay-L2-yy-8}, we get
\ba\label{decay-L2-yy-9}
\frac{\rm d}{\dt} F_1^{(2)}(t) + \int_{\TT}\bigl( w _{yyx}^2 + \frac{ p'(\underline{\eta})}{2} \eta_{yy}^2 \bigr)\,\dx \leq C \left( E_{10}^{(1)} (y) + E_{20}(y)\right) e^{-\a t} .
\ea
Notice that
\ba\label{decay-L2-yy-10}
\left| \int_{\TT} ( \eta w )_{yy} I ( \eta_{yy})\,\dx \right| & \leq \| ( \eta w )_{yy} \|_{L^1_\h} \| \eta_{yy} \|_{L^1_\h} \\
& \leq \| \eta_{yy} w + 2 \eta_{y} w_{y} + \eta w_{yy} \|_{L^1_\h} \| \eta_{yy} \|_{L^1_\h} \\
& \leq C \int_{\TT}\bigl( \eta w_{yy}^2 + \eta_{yy}^2 + \eta_{y}^{2} + w_{y}^{2}\bigr)\,\dx.
\ea
Then by \eqref{decay-L2-y-1} and \eqref{decay-L2-yy-9}, we infer
\ba\label{decay-L2-yy-11}
\frac{\rm d}{\dt} F_2^{(2)}(t) + \int_{\TT}\bigl( w_{yx}^{2} + w _{yyx}^2 + \frac{ p'(\underline{\eta})}{2} (\eta_{y}^{2} + \eta_{yy}^2 )\bigr)\,\dx \leq C \left( E_{10}^{(1)} (y) + E_{20}(y)\right) e^{-\a t} .
\ea
On the other hand, in view of \eqref{decay-L2-y-10}, \eqref{decay-L2-yy-7} and \eqref{decay-L2-yy-10}, by choosing $A$ suitably large, we deduce that
\ba\label{decay-L2-yy-12}
\int_{\TT} \bigl(\eta w _y^2 + \eta _y^2 + \eta w_{yy}^2 + \eta_{yy}^2 \bigr) \,\dx \leq F_2^{(2)}(t) & \leq C \int_{\TT}
\bigl(\eta w _y^2 + \eta _y^2 + \eta w_{yy}^2 + \eta_{yy}^2\bigr) \,\dx \\
& \leq C \int_{\TT} \bigl(w_{xy}^2 + \eta_y^2 + w_{yyx}^2 + \eta_{yy}^2\bigr) \,\dx.
\ea
Our desired estimate \eqref{decay-L2-yy} follows from \eqref{decay-L2-yy-11} and \eqref{decay-L2-yy-12}.
\end{proof}
\subsection{Proof of Proposition \ref{S2prop3}}
\begin{proof}[Proof of Proposition \ref{S2prop3}] We divide the proof into the following two steps:
\medskip
\noindent{\bf Step 1.} Decay estimate of $\eta_{yyx}.$
\medskip
The idea to derive the decay estimate of $\eta_{yyx}$ is similar as that of $\eta_{yxx}$ in Section \ref{sec-decay-y-all}.
In what follows, we just outline its derivation. In order to do so, wfirst e rewrite $\eqref{CNS-1d-dyy}_{2}$ as
\ba\label{decay-H1-yy-1}
D_t w_{yy} + (2 w_{y} w_{yx} + w_{yy}w_{x}) +\eta^{-1}\bigl( 2 \eta_{y} (D_{t} w_{y} + w_{y} w_{x}) +
\eta_{yy} D_{t} w - \nu w _{yyxx} + p( \eta )_{yyx}\bigr) = 0.
\ea
While it follows from $\eqref{CNS-1d-dyy}_{1}$ that
\ba\label{decay-H1-yy-2}
\zeta w_{yyxx} = D_{t}\zeta_{yyx} - \zeta_{yy} w_{xx} - \zeta_{y} w_{yxx} - \zeta_{y} w_{yxx} + \zeta_{yxx} w_{y} + \zeta_{yxx} w_{y} + \zeta_{xx} w_{yy}.
\ea
Plugging \eqref{decay-H1-yy-2} into \eqref{decay-H1-yy-1} gives
\ba\label{decay-H1-yy-3}
D_t (w_{yy} -\nu \zeta_{yyx}) + \eta^{-1} p( \eta )_{yyx} + \frak{g}_1 = 0,
\ea
with
\begin{align*
\frak{g}_1 =& (2 w_{y} w_{yx} + w_{yy}w_{x}) + 2 \eta^{-1}\eta_{y} (D_{t} w_{y} + w_{y} w_{x}) + \eta^{-1} \eta_{yy} D_{t} w\\
&- \nu (- \zeta_{yy} w_{xx} - \zeta_{y} w_{yxx} - \zeta_{y} w_{yxx} + \zeta_{yxx} w_{y} + \zeta_{yxx} w_{y} + \zeta_{xx} w_{yy}).
\end{align*}
Then we deduce from Propositions \ref{S2prop1}, \ref{S2prop2} and \ref{prop-decay-L2-yy} that
\ba\label{decay-H1-yy-5}
\|\frak{g}_2\|_{L^2_{h}}^{2} \leq C E(y) e^{-\a t}.
\ea
By taking $L^2(\TT)$ inner product of \eqref{decay-H1-yy-3} with $\eta (w_{yy} -\nu \zeta_{yyx})$ and using
\eqref{decay-H1-yy-5}, we find
\begin{align*
\frac 12 \frac{\rm d}{\dt}\int_{\TT}\eta(w_{yy} -\nu \zeta_{yyx})^{2}\,\dx &-\int_{\TT}p'(\eta) \zeta^{-2} \zeta_{yxx} (w_{yy} -\nu \zeta_{yyx})\,\dx\\
&\leq C \de^{-1}E(y) e^{-\a t} + \de \int_{\TT} \eta(w_{yy} -\nu \zeta_{yyx})^{2}\,\dx,
\end{align*}
from which, we infer
\ba\label{decay-H1-yy-7}
\frac 12 \frac{\rm d}{\dt}\int_{\TT} \eta(w_{yy} -\nu \zeta_{yyx})^{2}\,\dx
&+ \nu^{-1}\int_{\TT} p'(\eta) \eta^{2}(w_{yy} -\nu \zeta_{yyx})^{2}\,\dx \\
&\leq C \de^{-1}E(y) e^{-\a t} + \de \int_{\TT} \eta(w_{yy} -\nu \zeta_{yyx})^{2}\,\dx.
\ea
Taking $\de = (2\nu)^{-1}p'(\underline{\eta})\underline{\eta}$ in \eqref{decay-H1-yy-7} gives
\ba\label{decay-H1-yy-8}
\frac{\rm d}{\dt} \int_{\TT} \eta(w_{yy} -\nu \zeta_{yyx})^{2}\,\dx + \nu^{-1}p'(\underline{\eta})\underline{\eta} \int_{\TT} \eta (w_{yy} -\nu \zeta_{yyx})^{2}\,\dx \leq C E(y) e^{-\a t},
\nn \ea
from which, we infer
\ba\label{decay-H1-yy-9}
\int_{\TT} \eta(w_{yy} -\nu \zeta_{yyx})^{2}\,\dx \leq C E(y) e^{-\a t},
\ea
and
\ba\label{decay-H1-yy-10}
\int_{\TT} |\zeta_{yyx}|^{2}\,\dx \leq C E(y) e^{-\a t} \andf \int_{\TT} |\eta_{yyx}|^{2}\,\dx \leq C E(y) e^{-\a t}.
\ea
\medskip
\noindent{\bf Step 2.} {Decay estimates of $D_{t} w_{yy}.$}
\medskip
The main idea to derive the decay estimates of $D_{t} w_{yy}$ is analogues to that of $D_{t} w_{yx}$
in Section {sec-decay-y-all}. We shall outline its proof below.
Observing that
\begin{align*}\label{Dt-wyy-2}
\int_{\TT} D_{t} w_{yy} w_{yyxx} & = \int_{\TT} \d_{t} w_{yy} w_{yyxx}\,\dx + \int_{\TT} w w_{yyx} w_{yyxx}\,\dx\\
& = - \frac{1}{2}\frac{\rm d}{\dt} \int_{\TT} |w_{yyx}|^{2} \,\dx- \frac{1}{2} \int_{\TT} w_{x} (w_{yyx})^{2}\,\dx.
\end{align*}
Then by taking $L^2(\TT)$ inner product of \eqref{CNS-1d-dyy-1} with $\eta^{-1} w_{yyxx},$ we deduce that
\ba\label{Dt-wyy-1}
\frac{\rm d}{\dt} \int_{\TT} |w_{yyx}|^{2}\,\dx + \nu \bar\eta^{-1} \int_{\TT} |w_{yyxx}|^{2}\,\dx \leq C E(y) e^{-\a t}.
\ea
While due to
\begin{align*
- \nu\int_{\TT} w_{yyxx} D_{t} w_{yy}\,\dx & = - \nu \int_{\TT} w_{yyxx} \d_{t} w_{yy}\,\dx
- \nu \int_{\TT} w_{yyxx} w w_{yyx}\,\dx \\
& =\frac\nu2 \frac{\rm d}{\dt} \int_{\TT} |w_{yyx}|^{2} + \frac{\nu}{2} \int_{\TT} (w_{yyx})^{2} w_{x}\\
& \geq \frac\nu2\frac{\rm d}{\dt} \int_{\TT} |w_{yyx}|^{2}\,\dx - C E(y) e^{-\a t}\int_{\TT} |w_{yyx}|^{2}\,\dx,
\end{align*}
we get, by taking $L^2(\TT)$ inner product of \eqref{CNS-1d-dyy-1} with $ D_{t} w_{yy},$ that
\ba\label{Dt-wyy-5}
\nu\frac{\rm d}{\dt} \int_{\TT} |w_{yyx}|^{2}\,\dx + \int_{\TT} \eta |D_{t}w_{yy}|^{2}\,\dx \leq C E(y) e^{-\a t} + C E(y) e^{-\a t}\int_{\TT} |w_{yyx}|^{2}\,\dx.
\ea
Notice that
\begin{align*
\int_{\TT}D_{t} (\eta D_{t} w_{yy} ) D_{t} w_{yy}\,\dx & = \int_{\TT} (D_{t} \eta) D_{t} w_{yy} D_{t} w_{yy}\,\dx
+ \int_{\TT}\eta (D_{t}^2 w_{yy})D_{t} w_{yy} \,\dx\\
& = \frac{1}{2}\frac{\rm d}{\dt} \int_{\TT}\eta |D_{t} w_{yy}|^{2} +\frac12 \int_{\TT} (D_{t} \eta) |D_{t} w_{yy}|^{2} \\
& \geq \frac{1}{2}\frac{\rm d}{\dt} \int_{\TT}\eta |D_{t} w_{yy}|^{2} - C \int_{\TT} |D_{t} w_{yy}|^{2}\,\dx
\end{align*}
and
\begin{align*
- \nu \int_{\TT} (D_{t} w_{yyxx}) D_{t} w_{yy}\,\dx & = -\nu\int_{\TT} (D_{t} w_{yy})_{xx} D_{t} w_{yy}\,\dx
+ \int_{\TT} \nu (w_{xx} w_{yyx} + 2 w_{x} w_{yyxx})D_{t} w_{yy}\,\dx \\
& \geq \nu \int_{\TT} |(D_{t} w_{yy})_{x}|^{2}\,\dx - C \int_{\TT} \bigl(|D_{t} w_{yy}|^{2} + |w_{yyxx}|^{2}\bigr)\,\dx,
\end{align*}
we get, by applying $D_{t}$ to \eqref{CNS-1d-dyy-1} and then taking $L^2(\TT)$ inner product of the resulting equation with $D_{t} w_{yy},$
that
\ba\label{Dt-wyy-9}
\frac{\rm d}{\dt} \int_{\TT} \eta |D_{t} w_{yy}|^{2}\,\dx + \nu \int_{\TT} |(D_{t} w_{yy})_{x}|^{2}\,\dx \leq C E(y) e^{-\a t} + C \int_{\TT} \bigl(|w_{yyxx}|^{2} + |D_{t}w_{yy}|^{2} + |w_{yyx}|^{2}\bigr)\,\dx.
\ea
By virtue of \eqref{decay-L2-yy-9}--\eqref{decay-L2-yy-12}, we deduce from \eqref{Dt-wyy-1}, \eqref{Dt-wyy-5} and \eqref{Dt-wyy-9} that
\ba\label{Dt-wyy-20}
\|D_{t} w_{yy}\|_{L^{2}_{h}}^{2} \leq C E(y) e^{-\a t},
\ea
from which and \eqref{CNS-1d-dyy}, we infer
\ba\label{Dt-wyy-21}
\|w_{yyxx}\|_{L^{2}_{h}}^{2} \leq C E(y) e^{-\a t} .
\ea
By summarizing the estimates \eqref{decay-L2-yy}, \eqref{decay-H1-yy-10}, \eqref{Dt-wyy-20} and
\eqref{Dt-wyy-21}, we conclude the proof of \eqref{decay-exp-all-yy-2}.
\end{proof}
\section{Decay estimates of $\frak{w}$}\label{Sect6}
With the decay estimates for $(\eta, w)$ obtained in the previous sections,
we are going to derive the same exponential decay estimates for $\frak{w}$.
Notice that the equation for $\frak{w},$ \eqref{tw}, is of standard parabolic type, we shall only
present the main decay in time estimates of $\frak{w}$ and skip the derivation of the related estimates
concerning $(\d_y\frak{w},\d_y^2\frak{w}).$
The main results state as follows
\begin{prop}\label{prop-tw-L2}
{\sl For all $t\in\R^{+}$, one has
\begin{subequations} \label{Streq19}
\begin{gather}
\label{tw-L2}
\int_{\TT} |\frak{w}|^{2}\,\dx \leq \underline \eta^{-1} e^{- \mu\bar \eta^{-2} t } \int_{\TT}\eta_{0} |\frak{w}_{0}|^{2}\,\dx,\\
\label{tw-H1}
\int_{\TT}\bigl( |\frak{w}|^{2} + |\frak{w}_{x}|^2\bigr)\,\dx \leq CE(y) e^{- \a t },\\
\label{tw-H2}
\int_{\TT}\left( |\frak{w}|^{2} + |\frak{w}_{x} | + | D_{t} \frak{w} |^{2} + |\d_{xx} \frak{w}|^{2}\right)\,\dx \leq C E(y)e^{- \a t }.
\end{gather}
\end{subequations}}
\end{prop}
\begin{proof} 1) We first get, by
taking $L^{2}(\TT)$ inner product of \eqref{tw} with $\frak{w}$ and using integrating by parts, that
\ba\label{tw-L2-0}
\frac{1}{2}\frac{\rm d}{\dt}\int_{\TT} \eta |\frak{w}|^{2} \,\dx + \mu\int_{\TT} |\frak{w}_{x}|^{2}\,\dx = 0.
\ea
While observing from the density equation of \eqref{CNS-limit} and \eqref{tw} that
\ba\label{tw-new-1}
(\eta \frak{w})_{t} + (\eta w \frak{w})_{x} - \mu \d_x^2 \frak{w} = 0.
\nn \ea
Integrating the above equation over $\TT$ gives
\ba\label{tw-L2-1}
\frac{\rm d}{\dt} \int_{\TT}(\eta \frak{w})\,\dx = 0,
\nn \ea
which together with \eqref{ini-1} ensures that
\be\label{tw-L2-2}
\int_{\TT}(\eta \frak{w})\,\dx = \int_{\TT} \eta_{0} \frak{w}_{0} \,\dx= 0.
\ee
Then we get, by using a similar proof of Lemma \ref{prop-kinetic-tx}, that
\be\label{tw-L2-3}
\int_{\TT}\eta |\frak{w}|^{2}\,\dx \leq \bar \eta^{2} \int_{\TT} | \frak{w}_{x} |^{2}\,\dx.
\ee
By virtue of \eqref{tw-L2-3}, we deduce from \eqref{tw-L2-0} that
\ba
\int_{\TT}\eta |\frak{w}|^{2}\,\dx \leq e^{\mu\bar \eta^{-2} t } \int_{\TT}\eta_{0} |\frak{w}_{0}|^{2}\,\dx,
\nn \ea which together with \eqref{lowerb-vtr} ensures \eqref{tw-L2}.
\medskip
\noindent 2) By taking $L^2$ inner product of \eqref{tw} with $D_{t} \frak{w}$ and using integrating by
parts, we obtain
\ba\label{tw-H1-0}
\frac{1}{2}\int_{\TT} \eta |D_{t}\frak{w}|^{2} \,\dx + \frac{\mu}{2}\frac{\rm d}{\dt}\int_{\TT} |\frak{w}_{x}|^{2} \,\dx \leq \frac{\mu}{2} \|w_{x}\|_{L^{\infty}(\OO)} \int_{\TT} |\frak{w}_{x}|^{2} \,\dx.
\ea
By multiplying \eqref{tw-L2-0} by $\frak{A}_{1}\eqdefa 1 + \|w_{x}\|_{L^{\infty}([0,\infty)\times\OO)}$ and summing up the
resulting inequality with \eqref{tw-H1-0}, we get
\ba
\int_{\TT}\eta |D_{t}\frak{w}|^{2} \,\dx + \frac{\rm d}{\dt} \int_{\TT}\bigl({\frak{A}_{1}}\eta | \frak{w}|^{2} + {\mu} |\frak{w}_{x}|^{2}\bigr) \,\dx + \frak{A}_{1}\int_{\TT} \mu |\frak{w}_{x}|^{2}\,\dx \leq 0.
\nn \ea
Due to $D_t\frak{w}=\frac\mu\eta\d_x^2 \frak{w},$ \eqref{tw-H1} follows.
\medskip
\noindent 3)
Applying $D_{t}$ to \eqref{tw} gives
\ba\label{tw-Dtw}
D_{t} (\eta D_{t} \frak{w}) - \mu D_{t} \d_x^2 \frak{w} = 0.
\nn \ea
We observe that
\ba\label{tw-Dtw-1}
D_{t} (\eta D_{t} \frak{w}) = (-\eta \d_{x} w) D_{t} \frak{w} + \eta D_{t}^{2} \frak{w},
\nn \ea
and
\ba
D_{t} \d_x^2 \frak{w}
= \d_{x}^{2} (D_{t} \frak{w}) - \d_{x} ( \d_{x} w \d_{x} \frak{w}).
\nn \ea
As a result, it comes out
\ba\label{tw-Dtw-3}
\eta D_{t}^{2} \frak{w} - \mu \d_{x}^{2} (D_{t} \frak{w}) = (\eta \d_{x} w) D_{t} \frak{w} - \mu \d_{x} ( \d_{x} w \d_{x} \frak{w}).
\ea
By taking $L^2(\TT)$ inner product of \eqref{tw-Dtw-3} with $D_{t} \frak{w}$, we find
\ba\label{tw-Dtw-4}
\frac{1}{2} &\frac{\rm d}{\dt} \int_{\TT} \eta |D_{t} \frak{w}|^{2} \,\dx + \mu\int_{\TT} | \d_{x} (D_{t} \frak{w})|^{2}\,\dx \\
& = \int_{\TT} (\eta \d_{x} w) |D_{t} \frak{w}|^{2}\,\dx + \int_{\TT} \mu (\d_{x} w \d_{x} \frak{w}) \d_{x} D_{t} \frak{w}\,\dx\\
& \leq \| \d_{x} w \|_{L^{\infty}(\OO)} \int_{\TT} \eta |D_{t} \frak{w}|^{2}\,\dx + \frac{\mu}{2} \|\d_{x} w \|_{L^{\infty}(\OO)}^{2} \int_{\TT} |\d_{x} \frak{w}|^{2}\,\dx + \frac{\mu}{2}\int_{\TT} | \d_{x} (D_{t} \frak{w})|^{2}\,\dx.
\ea
Multiplying \eqref{tw-H1-0} by $\frak{A}_{2} \eqdefa 2 + 2 \|\d_{x} w \|_{L^{\infty}([0,\infty)\times\OO)}^{2}$ and summing up the resulting inequality with \eqref{tw-Dtw-4} yields
\ba\label{tw-Dtw-5}
\frac{\rm d}{\dt} \int_{\TT}\bigl( \frak{A}_{1}\frak{A}_{2} \eta | \frak{w}|^{2} + \mu A_{2} |\frak{w}_{x}|^{2} + \eta |D_{t} \frak{w}|^{2}\bigr) \,\dx + \int_{\TT} \bigl(\mu |\frak{w}_{x}|^{2} + \eta |D_{t}\frak{w}|^{2} + \mu | \d_{x} (D_{t} \frak{w})|^{2}\bigr)\,\dx \leq 0.
\ea
Then by using a similar proof of Lemma \ref{prop-kinetic-tx} and Gronwall's inequality, we find
\be\label{tw-Dtw-6}
\int_{\TT} \bigl(|\frak{w}|^{2} + |\frak{w}_{x} | + | D_{t} \frak{w} |^{2}\bigr)\,\dx \leq C E(y) e^{- \a t }.
\ee
Observing that $\mu\d_{xx} \frak{w} = \eta D_{t} \frak{w}, $ we conclude the proof of \eqref{tw-H2}.
This completes the proof of Proposition \ref{prop-tw-L2}.
\end{proof}
Let us now outline the proof of Proposition \ref{S2prop4}.
\begin{proof}[Proof of Proposition \ref{S2prop4}]
Along the same line to proof of Proposition \ref{prop-tw-L2} and through the induction method as what we used in
the Appendix \ref{appa}, we deduce that
\ba\label{tw-Hk-0}
\| \frak{w}(t)\|_{H^5_\h} + \|\frak{w}_{t}(t)\|_{H^3_\h} + \| \frak{w}_{tt}(t) \|_{H^1_\h} \leq CE(y) e^{-\a t}.
\ea
The decay estimates related to $y$-derivatives of $\frak{w}$ in \eqref{S2eq4} can be derived along the same line. We omit the details here.
\end{proof}
\section{Energy estimates for the perturbed equations}\label{sec-error}
The purpose of this section is to present the proof of Propositions \ref{prop-energy-basic} and \ref{energy}.
For simplicity, we shall neglect the subscript $\e$ in the rest of this section.
\subsection{Basic energy estimate}\label{Sect7.1}
In this subsection, we shall derive a basic energy estimate for all $t < T^\star.$ We first deduce from Proposition \ref{prop-rela}
that
\begin{lem}\label{prop-rela-w}
{\sl Let $(\rho, u)$ and $(\rho^{\rm a}, u^{\rm a})$ be respectively given by \eqref{zeta-W-def} and \eqref{sl-1d-00}. Then one has
\ba\label{ineq-entropy-1}
\calE_1\bigl((\rho,u) | (\rho^{\rm a}, u^{\rm a} )\bigr)(t)
+ \int_0^t \int_\Omega \bigl(\mu \left| \nabla (u-u^{\rm a}) \right|^2 + \mu' |\dive (u-u^{\rm a}) |^2\bigr)\,\dx\,\dy \,\dt' =\int_0^t \calR (t') \,\dt',
\ea
where for $G$ given by \eqref{def-G12},
\ba\label{R-def-1}
\calR (t) & \eqdefa \int_\Omega \Bigl((\rho^{\rm a})^{-1} (\rho -\rho^{\rm a}) ( \mu \Delta u^{\rm a} + \mu' \nabla \dive u^{\rm a}
)+ \rho (u-u^{\rm a} )\cdot \nabla u^{\rm a}\\
&\qquad+\rho (\rho^{\rm a})^{-1} G\Bigr)\cdot (u^{\rm a} - u)\,\dx\,\dy + \e\int_{\OO}(\rho^{\rm a}-\rho) (\rho^{\rm a})^{-1} p'(\rho^{\rm a}) [(\eta \frak{w})_{y}]_\e \,\dx \, \dy \\
& \quad - \int_\Omega \dive u^{\rm a} \bigl(p(\rho ) - p(\rho^{\rm a}) - p'(\rho^{\rm a})(\rho - \rho^{\rm a})\bigr)\,\dx\,\dy.
\ea
}
\end{lem}
\begin{proof} Since $(\rho, u)$ and $(\rho^{\rm a}, u^{\rm a})$ have the same initial data, we get, by
applying Proposition \ref{prop-rela-w}, that \eqref{ineq-entropy-1} holds with
\ba\label{R-def2}
\calR(t) \eqdefa \int_\Omega \Bigl(&\rho \frak{D}_t u^{\rm a} \cdot (u^{\rm a} - u)+ \mu \nabla u^{\rm a}:\nabla (u^{\rm a} - u)
+ \mu' \dive u^{\rm a} \,\dive (u^{\rm a} - u)\\
& + (\rho^{\rm a} - \rho) \d_t P'(\eta) + (\rho^{\rm a} u^{\rm a}-\rho u) \cdot \nabla P'(\rho^{\rm a})- \dive u^{\rm a} (p(\rho ) - p(\rho^{\rm a}))\Bigr)\,\dx\,\dy.
\ea
It follows from the $u^{\rm a}$ equation of \eqref{CNS-1d-W} that
\begin{align*}
\calR_{1}(t) & \eqdefa \int_\Omega\rho \bigl( \d_t u^{\rm a} + u^{\rm a}\cdot \nabla u^{\rm a} + (u-u^{\rm a})\cdot \nabla u^{\rm a}\bigr)
\cdot ( u^{\rm a} - u)\,\dx\,\dy\\
& = \int_\Omega \rho\Bigl((\rho^{\rm a})^{-1} \bigl( \mu \Delta u^{\rm a} + \mu' \nabla \dive u^{\rm a}
- \nabla p( \rho^{\rm a} ) + G\bigr)+ (u-u^{\rm a})\cdot \nabla u^{\rm a}\Bigr) ( u^{\rm a} - u)\,\dx\,\dy.
\end{align*}
By using integration by parts and the fact that $P''(s) = s^{-1} p'(s)$, we find
\ba
\calR_{1}(t)
& = \int_\Omega \bigl((\rho^{\rm a})^{-1} (\rho -\rho^{\rm a}) \bigl( \mu \Delta u^{\rm a} + \mu' \nabla \dive u^{\rm a} \bigr)
- \rho \nabla P'(\rho^{\rm a})+\rho (\rho^{\rm a})^{-1} G\bigr)\cdot ( u^{\rm a}- u)\,\dx\,\dy \\
&\quad - \int_\Omega\bigl( \mu \nabla u^{\rm a} :\nabla ( u^{\rm a} - u) + \mu' \dive u^{\rm a}
\,\dive ( u^{\rm a} - u)\bigr)\,\dx\,\dy \\
& \quad + \int_\Omega \rho (u- u^{\rm a})\cdot \nabla u^{\rm a} \cdot ( u^{\rm a} - u)\,\dx\,\dy.
\nn\ea
Plugging the above equality into \eqref{R-def2} gives
\ba\label{R-def2-2}
\calR(t) & = \int_\Omega \Bigl((\rho^{\rm a})^{-1} (\rho -\rho^{\rm a}) \bigl( \mu \Delta u^{\rm a} + \mu' \nabla \dive u^{\rm a} \bigr)
+\rho (\rho^{\rm a})^{-1} G + (u-u^{\rm a})\cdot \nabla u^{\rm a}\\
&\qquad- \rho \nabla P'(\rho^{\rm a})\Bigr)
\cdot (u^{\rm a}- u)\,\dx\,\dy- \int_\Omega \dive u^{\rm a} (p(\rho ) - p(\rho^{\rm a}))\,\dx\,\dy\\
&\quad+ \int_\Omega \bigl((\rho^{\rm a} - \rho) \d_t P'(\eta) + (\rho^{\rm a} u^{\rm a} -\rho u) \cdot \nabla P'(\rho^{\rm a})\bigr) \,\dx\,\dy.
\ea
Notice that
$P''(s) = s^{-1} p'(s)$ and the renormalized equation
\be
\d_t P'(\rho^{\rm a}) + \dive(u^{\rm a} P'(\rho^{\rm a})) + (P''(\rho^{\rm a})\rho^{\rm a} - P'(\rho^{\rm a})) \dive u^{\rm a} = \e P''(\rho^{\rm a}) [(\eta \frak{w})_{y}]_\e.
\ee
We get, by using the continuity equation $\eqref{CNS-1d-W}_{1}$, that
\ba\label{R-def2-3}
& - \rho \nabla P'(\rho^{\rm a}) \cdot ( u^{\rm a} - u) + (\rho^{\rm a} - \rho) \d_t P'(\eta) + (\rho^{\rm a} u^{\rm a} -\rho u) \cdot \nabla P'(\rho^{\rm a})\\
&= (\rho^{\rm a} - \rho) \left( \d_t P'(\rho^{\rm a}) + u^{\rm a} \cdot \nabla P'(\rho^{\rm a})\right) \\
&= (\rho^{\rm a}-\rho) \left(\left( \d_t P'(\rho^{\rm a}) + \dive(u^{\rm a}P'(\rho^{\rm a})) + (P''(\rho^{\rm a})\rho^{\rm a} - P'(\rho^{\rm a})) \dive u^{\rm a} \right) - P''(\rho^{\rm a})\rho^{\rm a} \dive u^{\rm a}\right) \\
& = \e (\rho^{\rm a}-\rho) (\rho^{\rm a})^{-1} p'(\rho^{\rm a}) [(\eta \frak{w})_{y}]_\e
- (\rho^{\rm a} - \rho) p'(\rho^{\rm a}) \dive u^{\rm a}.
\ea
Then \eqref{R-def-1} follows by inserting \eqref{R-def2-3} into \eqref{R-def2-2}.
\end{proof}
We now present the proof of Proposition \ref{prop-energy-basic}.
\begin{proof}[Proof of Proposition \ref{prop-energy-basic}]
We first get, by applying \eqref{upper-lower-rho-0} and Taylor's expansion, that for $t < T^\star,$
\ba\label{rela-entr-lower}
\calE_1(t)\eqdefa
\calE_1\bigl((\rho,u) | (\rho^{\rm a}, u^{\rm a})\bigr)(t) & = \int_{\Omega} \Bigl(\frac{1}{2} \rho |u- u^{\rm a}|^{2} + P(\rho) - P(\rho^{\rm a}) - P'(\rho^{\rm a}) (\rho - \rho^{\rm a})\Bigr)\,\dx\,\dy\\
& \geq \int_{\OO} \bigl({\underline \eta}/{4} | R|^{2} + \g (2\bar \eta)^{\g-2} | \vr |^{2}\bigr)\,\dx\,\dy.
\ea
While it follows from Lemma \ref{prop-rela-w} that
\ba\label{energy-bas1}
\calE_1(t)+ \int_0^t \int_\Omega\bigl( \mu \left| \nabla R \right|^2 + \mu' |\dive R|^2\bigr)\,\dx\,\dy \,\dt' \leq\int_0^t \calR (t') \,\dt'.
\ea
According to \eqref{R-def-1}, we decompose $\calR$ as
$
\calR (t) \eqdefa \sum_{j=1}^{5} \calR_{j}(t).
$
We first deduce from Theorem \ref{thm1} that
\begin{align*}
\calR_{1} (t) & \eqdefa \int_\Omega (\rho^{\rm a})^{-1} (\rho -\rho^{\rm a}) ( \mu \Delta u^{\rm a}
+ \mu' \nabla \dive u^{\rm a} )\cdot (u^{\rm a} - u)\,\dx\,\dy\\
& \leq \|(\rho^{\rm a})^{-1}\|_{L^{\infty}(\OO)} \| (\mu \Delta u^{\rm a} + \mu' \nabla \dive u^{\rm a})\|_{L^{\infty}(\OO)} \|\vr\|_{L^{2}(\OO)} \|R\|_{L^{2}(\OO)} \\
& \leq C e^{-\a t} \bigl( \|\vr\|_{L^{2}(\OO)}^{2} + \|R\|_{L^{2}(\OO)}^{2} \bigr).
\end{align*}
For the second term in \eqref{R-def-1}, we have
\ba
\calR_{2} (t) \eqdefa \int_\Omega \rho (\rho^{\rm a})^{-1} G \cdot ( u^{\rm a}- u)\,\dx\,\dy
\leq \|\rho\|_{L^{\infty}(\OO)} \|(\rho^{\rm a})^{-1}\|_{L^{\infty}(\OO)} \|G\|_{L^{2}(\OO)} \|R\|_{L^{2}(\OO)},
\nn \ea
which together with \eqref{def-G12-est} and Theorem \ref{thm1} ensures that
\ba
\calR_{2} (t) \leq C e^{-\a t} \e^{\frac 12} \|R\|_{L^{2}(\OO)}.
\nn \ea
For the third term in \eqref{R-def-1}, we get, by applying Theorem \ref{thm1}, that
\begin{align*}
\calR_{3} (t) & \eqdefa \int_\Omega \rho (u-u^{\rm a})\cdot \nabla u^{\rm a} \cdot (u^{\rm a} - u)\,\dx\,\dy\\
& \leq \|\nabla u^{\rm a}\|_{L^{\infty}(\OO)} \int_\Omega \rho |R|^{2} \,\dx\,\dy \leq C e^{-\a t} \calE_{1}(t).
\end{align*}
Along the same line and thanks to \eqref{pre-pot}, we have
\begin{align*}
\calR_{4} (t) & \eqdefa - \int_\Omega \dive u^{\rm a} \bigl(p(\rho ) - p(\rho^{\rm a}) - p'(\rho^{\rm a})(\rho - \rho^{\rm a})\bigr)\,\dx\,\dy \\
&\leq (\g-1) \| \dive u^{\rm a} \|_{L^{\infty}(\OO)} \int_\Omega \bigl(P(\rho ) - P(\rho^{\rm a}) - P'(\rho^{\rm a})(\rho - \rho^{\rm a})\bigr)\,\dx\,\dy \leq C e^{-\a t} \calE_{1}(t),
\end{align*}
and
\begin{align*}
& \calR_{5}(t) \eqdefa \e \int_{\OO}(\rho^{\rm a}-\rho) (\rho^{\rm a})^{-1} p'(\rho^{\rm a}) [(\eta \frak{w})_{y}]_\e \,\dx \, \dy \\
& \leq \e \|(\rho^{\rm a})^{-1} p'(\rho^{\rm a}) \|_{L^{\infty}(\OO)} \|[(\eta \frak{w})_{y}]_\e\|_{L^{2}(\OO)} \|\vr\|_{L^{2}(\OO)} \leq C \e^{\frac{1}{2}} e^{-\a t} \|\vr\|_{L^{2}(\OO)}.
\end{align*}
By inserting the above estimates into \eqref{energy-bas1} and using \eqref{rela-entr-lower},
we deduce that
\ba
\calE_{1}(t) + \int_0^t \int_\Omega \mu \left| \nabla R \right|^2 \,\dx\,\dy \,\dt'\leq
C \int_{0}^{t} e^{-\a t'}\bigl( \e^{\frac 12} \calE_{1}(t')^{1/2} + \calE_{1}(t')\bigr)\,\dt'.
\nn \ea
Applying Gornwall's inequality gives rise to
\ba
\calE_{1}(t) + \int_0^t \int_\Omega \mu \left| \nabla R \right|^2 \,\dx\,\dy \,\dt'\leq C \e,
\nn \ea
which together with \eqref{rela-entr-lower} ensures \eqref{energy-bas}. This completes the proof of Proposition \ref{prop-energy-basic}.
\end{proof}
\subsection{Estimates of $\nabla R$}
\begin{lem}\label{prop-DtR}
{\sl For each $t < T^\star$, there holds
\ba\label{DtR-0}
\int_{\OO} |\nabla R|^{2}\,\dx\,\dy + \int_{0}^{t} \int_{\OO} |\frak{D}_{t}R|^{2}\,\dx\,\dy
\leq C \int_{0}^{t}\int_{\OO} |\nabla R|^{3}\,\dx\,\dy + C \e.
\ea}
\end{lem}
\begin{proof} We first get, by taking $L^2$ inner product of the $R$ equation of \eqref{CNS-error-new} with $\frak{D}_tR$, that
\ba\label{DtR-1}
\int_{\OO} \rho |\fD_{t} R|^{2}\,\dx\,\dy - \int_{\OO} \bigl(\mu \Delta R+ \mu' \nabla\dive R\bigr)\cdot \fD_tR \,\dx\,\dy
+\int_{\OO} \nabla \big(p(\rho) - p(\rho^{\rm a}) \big)\cdot \fD_tR \,\dx\,\dy \\
+ \int_{\OO}\Bigl(\rho R\cdot \nabla u^{\rm a} + \vr \bigl(\d_{t}u^{\rm a} + u^{\rm a} \cdot \nabla u^{\rm a}\bigr)- G\Bigr) \cdot \fD_tR \,\dx\,\dy = 0.
\ea
Let us now handle term by term above. By using integration by parts, we find
\ba
- \int_{\OO} \Delta R \cdot \fD_{t} R\,\dx\,\dy
= \frac{1}{2}\frac{\rm d}{\dt} \int_{\OO} |\nabla R|^{2}\,\dx\,\dy
+ \int_{\OO} \nabla R: \bigl( \nabla u \nabla R + (u \cdot \nabla)\nabla R\bigr)\,\dx\,\dy.
\nn \ea
Due to $u=u^{\rm a}+R,$ one has
\begin{align*}
\int_{\OO}\nabla R: ( \nabla u \nabla R)\,\dx\,\dy & = \int_{\OO} \nabla R: \bigl( \nabla u^{\rm a} \nabla R\bigr)\,\dx\,\dy
+ \int_{\OO} \nabla R: ( \nabla R \nabla R) \,\dx\,\dy \\
& \leq \|\nabla u^{\rm a}\|_{L^{\infty}} \int_{\OO} |\nabla R|^{2}\,\dx\,\dy + \int_{\OO}|\nabla R|^{3}\,\dx\,\dy,
\end{align*}
and
\begin{align*}
\int_{\OO} \nabla R: ( (u \cdot \nabla)\nabla R) \,\dx\,\dy
& = - \frac{1}{2}\int_{\OO} (\dive u^{\rm a}) |\nabla R|^{2} \,\dx\,\dy
- \,\frac{1}{2} \int_{\OO} (\dive R) |\nabla R|^{2}\,\dx\,\dy.
\end{align*}
This together with Theorem \ref{thm1} ensures that
\ba\label{DtR-3-1}
- \int_{\OO} \Delta R \cdot \fD_{t} R\,\dx\,\dy
\geq \frac{1}{2}\frac{\rm d}{\dt} \int_{\OO} |\nabla R|^{2}\,\dx\,\dy -C e^{-\alpha t} \int_{\OO} |\nabla R|^{2}\,\dx\,\dy
-\int_{\OO}|\nabla R|^{3}\,\dx\,\dy.
\ea
Exactly along the same line, one has
\ba\label{DtR-3-2}
-\int_{\OO} \nabla \dive R \cdot \fD_{t} R\,\dx\,\dy\geq &\frac{1 }{2}\frac{\rm d}{\dt} \int_{\OO} |\dive R|^{2}\,\dx\,\dy\\
& -C e^{-\a t} \int_{\OO} |\nabla R|^{2}\,\dx\,\dy -\int_{\OO} |\nabla R|^{3}\,\dx\,\dy .
\ea
It is a little trickier to deal with the pressure term in \eqref{DtR-1}. Indeed we first observe that
\ba\label{DtR-4-1}
\int_{\OO}& \nabla (p(\rho) - p(\rho^{\rm a})) \cdot \fD_{t}R\,\dx\,\dy
= - \frac{\rm d}{\dt} \int_{\OO} (p(\rho) - p(\rho^{\rm a})) \dive R\,\dx\,\dy\\
&+ \int_{\OO} \d_{t}(p(\rho) - p(\rho^{\rm a})) \dive R \,\dx\,\dy
- \int_{\OO} (p(\rho) - p(\rho^{\rm a})) \dive(u\cdot \nabla R)\,\dx\,\dy.
\ea
It follows from the continuity equations of \eqref{CNS} and \eqref{CNS-1d-W} that
\begin{align*}
\d_{t}(p(\rho) - p(\rho^{\rm a}))
& = - \dive\big(p(\rho) u - p(\rho^{\rm a}) u^{\rm a}\big) - a (\g-1) \big( \rho^{\g} \dive u - (\rho^{\rm a})^{\g} \dive u^{\rm a}\big)
-\e p'(\rho^{\rm a}) [(\eta \frak{w})_{y}]_\e
\end{align*}
so that
\begin{align*}
&\int_{\OO} \d_{t}(p(\rho) - p(\rho^{\rm a})) \dive R\,\dx\,\dy = \int_{\OO} \big(p(\rho) u - p(\rho^{\rm a}) u^{\rm a}\big) \cdot \nabla \dive R\,\dx\,\dy \\
&- a (\g-1)\int_{\OO} \big( \rho^{\g} \dive u - (\rho^{\rm a})^{\g} \dive u^{\rm a}\big) \dive R\,\dx\,\dy
- \e \int_{\OO} p'(\rho^{\rm a}) [(\eta \frak{w})_{y}] \dive R\,\dx\,\dy.
\end{align*}
While we observe that
\begin{align*}
& - \int_{\OO} (p(\rho) - p(\rho^{\rm a})) \dive(u\cdot \nabla R)\,\dx\,\dy = - \int_{\OO} \bigl(p(\rho) u- p(\rho^{\rm a}) u^{\rm a}\bigr)\cdot \nabla \dive R \,\dx\,\dy \\
& \qquad\qquad + \int_{\OO} p(\rho^{\rm a}) R \cdot \nabla \dive R \,\dx\,\dy
- \int_{\OO} (p(\rho) - p(\rho^{\rm a})) (\nabla u^{\rm a}+\nabla R): \nabla R\,\dx\,\dy.
\end{align*}
As a result, it comes out
\ba\label{DtR-4-5}
& \int_{\OO} \nabla (p(\rho) - p(\rho^{\rm a})) \cdot \fD_{t} R \,\dx\,\dy
= - \frac{\rm d}{\dt} \int_{\OO} (p(\rho) - p(\rho^{\rm a})) \dive R\,\dx\,\dy\\
&\quad- a (\g-1)\int_{\OO} \big( \rho^{\g} \dive u - (\rho^{\rm a})^{\g} \dive u^{\rm a}\big) \dive R\,\dx\,\dy
- \e \int_{\OO} p'(\rho^{\rm a}) [(\eta \frak{w})_{y}]_\e \dive R\,\dx\,\dy\\
&\quad + \int_{\OO} p(\rho^{\rm a}) R \cdot \nabla \dive R\,\dx\,\dy - \int_{\OO} (p(\rho) - p(\rho^{\rm a})) (\nabla u^{\rm a}+\nabla R): \nabla R\,\dx\,\dy.
\ea
Observe that
\begin{align*}
&\int_{\OO} \big( \rho^{\g} \dive u - (\rho^{\rm a})^{\g} \dive u^{\rm a}\big) \dive R \,\dx\,\dy\\
& = \int_{\OO} \big( \rho^{\g} \dive R + (\rho^{\g}- (\rho^{\rm a})^{\g})\dive u^{\rm a} \big) \dive R \,\dx\,\dy\\
& \leq \bar\rho^{\g}\int_{\OO} |\dive R |^{2}\,\dx\,\dy + \|\dive u^{\rm a}\|_{L^{\infty}(\OO)}
\int_{\OO}\bigl( (\rho^{\g}- (\rho^{\rm a})^{\g})^{2} + |\dive R|^{2}\bigr)\,\dx\,\dy,
\end{align*}
and
\ba
\e \int_{\OO} p'(\rho^{\rm a}) [(\eta \frak{w})_{y}]_\e \dive R\,\dx\,\dy
\leq \e^{\frac 12} \|p'(\rho^{\rm a})\|_{L^{\infty}(\OO)} \| (\eta \frak{w}_{y} )\|_{L^{2}(\OO)} \|\dive R\|_{L^{2}(\OO)},
\nn \ea
and
\begin{align*}
& \int_{\OO} p(\rho^{\rm a}) R \cdot \nabla \dive R\,\dx\,\dy - \int_{\OO} (p(\rho) - p(\rho^{\rm a})) (\nabla u^{\rm a}+\nabla R): \nabla R\,\dx\,\dy \\
& =- \int_{\OO} \Bigl( (\nabla p(\rho^{\rm a}) ) \cdot R \cdot \nabla R + (p(\rho) - p(\rho^{\rm a})) \nabla u^{\rm a}: \nabla R
+ p(\rho) \nabla R: \nabla R\Bigr)\,\dx\,\dy \\
& \leq \|\nabla p(\rho^{\rm a})\|_{L^\infty} \| R\|_{L^2}^2 + \|\nabla u^{\rm a}\|_{L^\infty}
\| (\rho-\rho^{\rm a})\|_{L^2}^{2} + C \|\nabla R\|_{L^2}^{2}.
\end{align*}
By substituting the above estimates into \eqref{DtR-4-5} and using Theorem \ref{thm1}, we find
\ba\label{DtR-4-5a}
\int_{\OO} \nabla (p(\rho) - p(\rho^{\rm a})) \cdot \fD_{t} R \,\dx\,\dy
\leq & - \frac{\rm d}{\dt} \int_{\OO} (p(\rho) - p(\rho^{\rm a})) \dive R\,\dx\,\dy\\
&+Ce^{-\alpha t}\bigl(\|\vr\|_{L^2}^2+\| R\|_{L^2}^2 +\|\nabla R\|_{L^2}^2\bigr)+ C \|\nabla R\|_{L^2}^{2}.
\ea
We further observe that
\begin{align*}
\int_{\OO} \rho R\cdot \nabla u^{\rm a} \cdot \fD_{t} R\,\dx\,\dy &
\leq 2 \|\nabla u^{\rm a} (t)\|_{L^{\infty}}^{2}\int_{\OO} \rho |R|^{2}\,\dx\,\dy + \frac{1}{12}\int_{\OO} \rho (\fD_{t} R)^{2}\,\dx\,\dy,\\
\int_{\OO} \vr \bigl(\d_{t} u^{\rm a} + u^{\rm a} \cdot \nabla u^{\rm a}\bigr) \cdot \fD_{t} R\,\dx\,\dy
& \leq 2 \bigl\| \bigl(\d_{t} u^{\rm a} + u^{\rm a} \cdot \nabla u^{\rm a}\bigr) \bigr\|_{L^{\infty}}^{2}
\int_{\OO} \rho^{-1} \vr^{2}\,\dx\,\dy + \frac{1}{12}\int_{\OO} \rho (\fD_{t} R)^{2}\,\dx\,\dy,\\
\int_{\OO} G \cdot \fD_{t}R\,\dx\,\dy & \leq 2\int_{\OO} \rho^{-1} |G|^{2}\,\dx\,\dy
+ \frac{1}{12}\int_{\OO} \rho (\fD_{t} R)^{2}\,\dx\,\dy.
\end{align*}
This together with \eqref{def-G12-est} and
\eqref{energy-bas} that
\ba\label{DtR-6-1}
\int_{\OO}\Bigl(\rho R\cdot \nabla u^{\rm a} + \vr \bigl(\d_{t}u^{\rm a} + u^{\rm a} \cdot \nabla u^{\rm a}\bigr)- G\Bigr)
\cdot \fD_tR \,\dx\,\dy\leq C e^{-\a t} \e + \frac{1}{4}\int_{\OO} \rho (\fD_{t} R)^{2}\,\dx\,\dy.
\ea
By inserting the estimates \eqref{DtR-3-1}, \eqref{DtR-3-2}, \eqref{DtR-4-5a} and \eqref{DtR-6-1} into \eqref{DtR-1}
and
and integrating the resulting inequality over $[0,t]$, we achieve
\ba\label{DtR-7-1}
&\int_{\OO} \bigl(\mu |\nabla R|^{2} + \mu'| \dive R|^{2}\bigr) \,\dx\,\dy + \int_{0}^{t} \int_{\OO} \rho |\fD_{t}R|^{2} \,\dx\,\dy\,\dt'\\
& \leq C \e + \int_{\OO} (p(\rho) - p(\rho^{\rm a})) \dive R \,\dx\,\dy+ 4C\int_{0}^{t}\int_{\OO} |\nabla R|^{3} \,\dx\,\dy\,\dt' + C \int_{0}^{t}\int_{\OO} |\nabla R|^{2}\,\dx\,\dy\,\dt'.
\ea
Notice that
\begin{align*}
\int_{\OO} (p(\rho) - p(\rho^{\rm a})) \dive R\,\dx\,\dy & \leq \mu^{-1}\int_{\OO} |p(\rho) - p(\rho^{\rm a})|^{2}\,\dx\,\dy + \frac{\mu}{4} \int_{\OO} |\nabla R|^{2}\,\dx\,\dy \\
& \leq C\mu^{-1}\|\vr\|_{L^2}^{2} + \frac{\mu}{4} \|\nabla R\|_{L^2}^{2},
\end{align*}
from which and \eqref{energy-bas}, we deduce \eqref{DtR-0} from \eqref{DtR-7-1}.
This completes the proof of the lemma.
\end{proof}
\subsection{Estimates of $\fD_{t} R$}
\begin{lem}\label{prop-DtR-new}
{\sl For each $t < T^\star$, one has
\ba\label{DtR-0-new}
\int_{\OO} |D_{t}R|^{2}\,\dx\,\dy + \int_{0}^{t} \int_{\OO} |\nabla \fD_{t} R|^{2}\,\dx\,\dy\,\dt' \leq C \e
+ C \int_{0}^{t} \int_{\OO}|\nabla R|^{4}\,\dx\,\dy\,\dt'.
\ea}
\end{lem}
\begin{proof}
By applying $\fD_{t}$ to the $R$ equation of \eqref{CNS-error-new} and taking $L^2$ inner product of the resulting
equation with $\fD_tR,$ we find
\ba\label{DtR-1-new}
\int_{\OO} \Bigl(&\fD_{t} (\rho \fD_{t} R) - \mu \fD_{t} \Delta R - \mu' \fD_{t} \nabla\dive R + \fD_{t} \nabla \big(p(\rho) - p(\rho^{\rm a}) \big) \\
&+ \fD_{t} \big(\rho R\cdot \nabla u^{\rm a} \big) + \fD_{t} \big( \vr (\d_{t} u^{\rm a} + u^{\rm a} \cdot \nabla u^{\rm a})\big)\Bigr)
\cdot \fD_{t} R\,\dx\,\dy = \int_{\OO} \fD_{t} G\cdot \fD_{t} R\,\dx\,\dy.
\ea
Let us now handle term by term above. Notice that
\ba
\fD_{t} (\rho D_{t} R) =\rho \fD_{t}^{2} R + (\fD_{t} \rho) \fD_{t}R.
\nn \ea
Firstly,
\ba\label{DtR-2-2-new}
\int_{\OO} \rho \fD_{t}^{2} R \cdot \fD_{t} R \,\dx\,\dy= \frac{1}{2}\frac{\rm d}{\dt}\int_{\OO} \rho |\fD_{t}R|^{2}\,\dx\,\dy.
\ea
By virtue of \eqref{CNS}, we write
\ba\label{DtR-2-3-new}
\fD_{t} \rho \fD_{t}R & = \dive u (-\rho \fD_{t}R) \\
& = \dive u \big( - \mu \Delta R- \mu' \nabla\dive R + \nabla \big(p(\rho) - p(\rho^{\rm a}) \big) + \rho R \cdot \nabla u^{\rm a} + \vr (\d_{t} u^{\rm a} + u^{\rm a} \cdot \nabla u^{\rm a}) - G \big).
\ea
We shall postpone the estimate of the above terms below.
For $i=1,2,$ we write
\begin{align*}
-\mu \fD_{t}\Delta R_{i}
& = - \mu \Delta \d_{t} R_{i} - \mu \dive\big( (u \cdot \nabla) \nabla R_{i} \big) + \mu \nabla u_{j} \cdot \d_{j}\nabla R_{i} \\
& = - \mu \Delta \fD_{t} R_{i} + \mu \dive\big( \nabla u_{j} \cdot \d_{j} R_{i} \big) + \mu \nabla u_{j} \cdot \d_{j}\nabla R_{i}.
\end{align*}
The last term above can be written as
\begin{align*}
\mu \nabla u_{j} \cdot \d_{j}\nabla R_{i} = \mu \d_{k} u_{j} \d_{j} \d_{k} R_{i}
&= \mu \d_{j} \big( \d_{k} u_{j} \d_{k} R_{i}\big) - \mu \d_{k}\d_{j} u_{j} \d_{k} R_{i} \\
& = \mu \d_{j} \big( \d_{k} u_{j} \d_{k} R_{i}\big) - \mu \d_{k}\big(\d_{j} u_{j} \d_{k} R_{i}\big) + \mu \dive u \Delta R_{i}.
\end{align*}
We observe that the last term above cancels with the first term on the right-side of \eqref{DtR-2-3-new},
so that there holds
\begin{align*}
& -\int_{\OO} \big( \dive u \Delta R + \fD_{t}\Delta R\big) \cdot \fD_{t} R\,\dx\,\dy\\
& = - \int_{\OO}\Delta D_{t} R \cdot D_{t} R\,\dx\,\dy
+\int_{\OO} \Big( \d_k\big( \d_{k} u_{j} \d_{j} R \big) + \d_{j} \big( \d_{k} u_{j} \d_{k} R\big) - \d_{k}\big(\d_{j} u_{j} \d_{k} R\big) \Big)\cdot D_{t} R_i\,\dx\,\dy.
\end{align*}
We get, by using integration by parts, that
\ba
-\int_{\OO} \Delta \fD_{t} R \cdot \fD_{t} R \,\dx\,\dy = \int_{\OO} |\nabla \fD_{t} R |^{2}\,\dx\,\dy,
\nn \ea
and
\begin{align*}
& \mu\int_{\OO} \Big( \d_k\big( \d_{k} u_{j} \d_{j} R \big) + \d_{j} \big( \d_{k} u_{j} \d_{k} R\big) - \d_{k}\big(\d_{j} u_{j} \d_{k} R\big) \Big)\cdot \fD_{t} R\,\dx\,\dy \\
& \leq 12\mu \int_{\OO} | \nabla u |^{2} |\nabla R|^{2}\,\dx\,\dy + \frac{\mu}{8} \int_{\OO} |\nabla \fD_{t} R |^{2}\,\dx\,\dy \\
& \leq 12\mu \int_{\OO} \bigl(\| \nabla u^{\rm a} \|_{L^\infty}^{2} + |\nabla R|^{2}\bigr) |\nabla R|^{2}\,\dx\,\dy
+ \frac{\mu}{8} \|\nabla \fD_{t} R \|_{L^2}^{2}.
\end{align*}
Thus, we get, by applying Theorem \ref{thm1}, that
\ba\label{DtR-3-6-new}
-\mu\int_{\OO} \big( \dive u \Delta R &+ \fD_{t}\Delta R\big) \cdot \fD_{t} R\,\dx\,\dy\\
&\geq \frac{7 }{8}\mu \|\nabla \fD_{t} R \|_{L^2}^{2} - C e^{-\a t} \|\nabla R\|_{L^2}^{2} - 12\mu \int_{\OO} |\nabla R|^{4}\,\dx\,\dy.
\ea
Similar argument leads to
\ba\label{DtR-4-1-new}
- \mu'\int_{\OO} \big(\dive u ( \nabla \dive R) &+ \fD_{t}\nabla \dive R) \cdot \fD_{t} R\,\dx\,\dy\\
& \geq \frac{7}{8} \mu' \|\dive \fD_{t} R \|_{L^2}^{2} - C e^{-\a t} \|\nabla R\|_{L^2}^{2} - 12\mu' \int_{\OO} |\nabla R|^{4}\,\dx\,\dy.
\ea
To handle the pressure related terms, we write
\begin{align*}
& ( \dive u) \nabla(p(\rho) - p(\rho^{\rm a})) + (u \cdot \nabla )\nabla(p(\rho) - p(\rho^{\rm a})) \\
& = \d_{j} \big(u_{j} \nabla (p(\rho) - p(\rho^{\rm a}))\big) = \nabla \dive \big(u (p(\rho) - p(\rho^{\rm a}))\big) - \d_{j} \big( \nabla u_{j} (p(\rho) - p(\rho^{\rm a}))\big).
\end{align*}
In view of the continuity equations of \eqref{CNS} and \eqref{CNS-1d-W}, one has
\begin{align*}
\d_{t}(p(\rho) - p(\rho^{\rm a}))
=& - \dive\big( u (p(\rho)- p(\rho^{\rm a})) + p(\rho^{\rm a}) R \big)\\
&- a (\g-1) \big( \rho^{\g} \dive u - (\rho^{\rm a})^{\g} \dive u^{\rm a}\big) -\e p'(\rho^{\rm a}) [(\eta \frak{w})_{y}]_\e.
\end{align*}
As a consequence, we deduce that
\begin{align*}
( \dive u) \nabla(p(\rho) - p(\rho^{\rm a})) + \fD_{t} \nabla \big(p(\rho) & - p(\rho^{\rm a}) \big)
= - \d_{j} \big( \nabla u_{j} (p(\rho) - p(\rho^{\rm a}))\big) - \nabla \dive (p(\rho^{\rm a}) R)\\
&- a (\g-1) \nabla\big( \rho^{\g} \dive u - (\rho^{\rm a})^{\g} \dive u^{\rm a}\big) - \e \nabla \big( p'(\rho^{\rm a})[(\eta \frak{w})_{y}]_\e \big),
\end{align*}
from which and Theorem \ref{thm1}, we deduce that
\ba\label{DtR-5-5-new}
&\int_{\OO} \Big(( \dive u) \nabla(p(\rho) - p(\rho^{\rm a})) + \fD_{t} \nabla \big(p(\rho) - p(\rho^{\rm a}) \big)\Big) \cdot \fD_{t}R \,\dx\,\dy \\
& = \int_{\OO} (p(\rho) - p(\rho^{\rm a}))(\nabla u^{\rm a} + \nabla R) : \nabla \fD_{t} R \,\dx\,\dy
+ \int_{\OO} \Bigl((p(\rho^{\rm a}) \dive R + \nabla p(\rho^{\rm a}) \cdot R) \\
& \qquad +a (\g-1) \big( \rho^{\g} \dive R + (\rho^{\g}- (\rho^{\rm a})^{\g} ) \dive u^{\rm a}\big)
+ \e p'(\rho^{\rm a}) [(\eta \frak{w})_{y}]\Bigr) \dive \fD_{t}R\,\dx\,\dy \\
& \leq C e^{-\a t} \bigl(\e+\|\vr|_{L^2}^{2} + \|R\|_{L^2}^{2}\bigr) + C \|\nabla R\|_{L^2}^{2} + \frac{\mu}{2} \|\nabla \fD_{t} R \|_{L^2}^{2}.
\ea
On the other hand, thanks to the continuity equation of \eqref{CNS-limit}, we infer
\begin{align*}
& \int_{\OO} \big( \fD_{t} (\rho R \cdot \nabla u^{\rm a}) + (\dive u) \rho R\cdot \nabla u^{\rm a}\big) \cdot \fD_{t} R\,\dx\,\dy \\
& = \int_{\OO} \big( \rho (\fD_{t} R) \cdot \nabla u^{\rm a}+ \rho R \cdot D_{t} \nabla u^{\rm a}\big) \cdot \fD_{t} R\,\dx\,\dy \\
& \leq \bigl(\| \nabla u^{\rm a}\|_{L^{\infty}} + \| \d_{t} \nabla u^{\rm a} \|_{L^{\infty}}
+ \| u^{\rm a} \nabla^{2} u^{\rm a}\|_{L^{\infty}}+ \| \nabla^{2} u^{\rm a} \|_{L^{\infty}} \bigr) \\
&\qquad\times\int_{\OO} \rho \left( |\fD_{t} R|^{2} + |R|^{2}\right)\,\dx\,\dy + \| \nabla^{2} u^{\rm a} \|_{L^{\infty}} \int_{\OO} \rho |R|^{4}\,\dx\,\dy.
\end{align*}
It follows from the Gagliardo-Nirenberg interpolation inequality and \eqref{energy-bas} that
\ba
\| R \|_{L^{4}(\OO)}^{4} \leq C \| R \|_{L^{2}(\OO)}^{2} \| R\|_{H^{1}(\OO)}^{2} \leq C\e^{2} + C \e \|\nabla R\|_{L^{2}(\OO)}^{2}.
\nn \ea
As a result, we get, by applying Theorem \ref{thm1}, that
\ba\label{DtR-6-4-new}
\int_{\OO} \big( \fD_{t} (\rho R \cdot \nabla u^{\rm a}) &+ (\dive u) \rho R\cdot \nabla u^{\rm a}\big) \cdot \fD_{t} R\,\dx\,\dy\\
&\leq C e^{-\a t}\Bigl(\e \bigl(1+ \|\nabla R\|_{L^2}^{2}\bigr) + \int_{\OO} \rho |D_{t} R|^{2}\,\dx\,\dy\Bigr).
\ea
In view of the $\vr$ equation of \eqref{CNS-error-new}, we write
\begin{align*}
& \fD_{t} \big( \vr \bigl(\d_{t} u^{\rm a} + u^{\rm a} \cdot \nabla u^{\rm a})\bigr)
+ \dive u \big(\vr \bigl(\d_{t} u^{\rm a} + u^{\rm a} \cdot \nabla u^{\rm a}) \bigr) \\
&= \bigl(\fD_{t}\vr + \vr \dive u\bigr) \bigl(\d_{t} u^{\rm a} + u^{\rm a} \cdot \nabla u^{\rm a})\bigr)
+ \vr \fD_{t} \bigl(\d_{t} u^{\rm a} + u^{\rm a} \cdot \nabla u^{\rm a})\bigr) \\
& = - \big( \dive(\rho^{\rm a} R) + \e [(\eta \frak{w})_{y}]_\e\big) \bigl(\d_{t} u^{\rm a} + u^{\rm a} \cdot \nabla u^{\rm a})\bigr)
+ \vr \fD_{t} \bigl(\d_{t} u^{\rm a} + u^{\rm a} \cdot \nabla u^{\rm a})\bigr) .
\end{align*}
It is easy to observe that
\begin{align*}
&\int_{\OO} \big( \dive(\rho^{\rm a} R) + \e [(\eta \frak{w})_{y}]_\e\big)\bigl(\d_{t} u^{\rm a} + u^{\rm a} \cdot \nabla u^{\rm a})\bigr) \cdot \fD_{t} R \,\dx\,\dy\\
&\leq \bigl\|\bigl(\d_{t} u^{\rm a} + u^{\rm a} \cdot \nabla u^{\rm a})\bigr)\bigr\|_{L^{\infty}}
\bigl\| \rho^{\rm a} \dive R + R\cdot \nabla \rho^{\rm a} + \e [(\eta \frak{w})_{y}]_\e\bigr\|_{L^{2}}\| \fD_{t} R\|_{L^{2}}\\
& \leq C e^{-\a t} \bigl(\|R\|_{L^{2}} + \|\nabla R\|_{L^{2}} + \e^{\frac 12}\bigr)\| \fD_{t} R\|_{L^{2}} ,
\end{align*}
and
\begin{align*}
& \int_{\OO} \vr \d_{t}\bigl(\d_{t} u^{\rm a} + u^{\rm a} \cdot \nabla u^{\rm a})\bigr)\cdot \nabla u^{\rm a})\bigr) \cdot \fD_{t} R \,\dx\,\dy\leq { \bigl\|\d_{t}\bigl(\d_{t} u^{\rm a} + u^{\rm a} \cdot \nabla u^{\rm a})\bigr)\bigr\|_{L^{\infty}}} \| \vr\|_{L^{2}}\| \fD_{t} R\|_{L^{2}},
\end{align*}
and
\begin{align*}
& \int_{\OO} \vr u \cdot \nabla \bigl(\d_{t} u^{\rm a} + u^{\rm a} \cdot \nabla u^{\rm a})\bigr) \cdot \fD_{t} R\,\dx\,\dy \\
&\leq \| \fD_{t} R\|_{L^{2}}\big( \bigl\|u^{\rm a}\cdot \nabla \bigl(\d_{t} u^{\rm a} + u^{\rm a} \cdot \nabla u^{\rm a})\bigr) \bigr\|_{L^{\infty}} \| \vr\|_{L^{2}}+ \|\vr\|_{L^{\infty}} \bigl\|\bigl(\d_{t} u^{\rm a} + u^{\rm a} \cdot \nabla u^{\rm a})\bigr)\bigr\|_{L^{\infty}}
\| R \|_{L^{2}}\big)\\
&\leq C e^{-\a t} \e^{\frac 12} \| D_{t} R\|_{L^{2}(\OO)}.
\end{align*}
Therefore, we obtain
\ba\label{DtR-9-5-new}
\int_{\OO}
\Bigl(\fD_{t} \big( \vr \bigl(\d_{t} u^{\rm a} + u^{\rm a} \cdot \nabla u^{\rm a})\bigr)
+ \dive u \big(\vr \bigl(\d_{t} u^{\rm a} &+ u^{\rm a} \cdot \nabla u^{\rm a}) \bigr)\Bigr)\cdot \fD_{t} R \,\dx\,\dy\\
&\leq C e^{-\a t} \e + C e^{-\a t}\| \fD_{t} R\|^{2}_{L^{2}(\OO)}.
\ea
We remark that it is exactly the term ${ \bigl\|\d_{t}\bigl(\d_{t} u^{\rm a} + u^{\rm a} \cdot \nabla u^{\rm a})\bigr)\bigr\|_{L^{\infty}}}$ that needs the highest regularity of the approximate solution. It follows from \eqref{thm1-0} that ${\bigl\|\d_{t}\bigl(\d_{t} u^{\rm a} + u^{\rm a} \cdot \nabla u^{\rm a})\bigr)\bigr\|_{L^{\infty}}} \leq C. $
Finally, thanks to \eqref{def-G12-est}, we compute
\ba\label{DtR-7-2-new}
\int_{\OO} &\big( \d_{t} G + \dive (u G) \big) \cdot \fD_{t}R\,\dx\,\dy\\
&\leq \|\d_{t} G\|_{L^{2}} \| \fD_{t} R\|_{L^{2}} + \| G\|_{L^{\infty}} (\|u^{\rm a}\|_{L^{2}} + \|R\|_{L^{2}})\| \nabla \fD_{t} R\|_{L^{2}} \\
& \leq C \e^{\frac{1}{2}} e^{-\a t}\| \fD_{t} R\|_{L^{2}} + C \e^{\frac 12} e^{-\a t} \| \nabla \fD_{t} R\|_{L^{2}} \\
& \leq C \e e^{-\a t} + C e^{-\a t}\| \fD_{t} R\|_{L^{2}}^{2} + \frac{1}{8}\| \nabla \fD_{t} R\|_{L^{2}}^2.
\ea
By inserting the estimates \eqref{DtR-2-2-new} and (\ref{DtR-3-6-new}--\ref{DtR-7-2-new}) into \eqref{DtR-1-new} and then
integrating the resulting inequality over $[0,t],$ we arrive at
\ba\label{DtR-8-1-new}
\|\fD_{t}R\|_{L^2}^{2} + \int_{0}^{t} \|\nabla \fD_{t} R(t')\|^{2}\,\dt' \leq C \e +
C \int_{0}^{t} |\nabla R|^{4}\,\dx\,\dy
+ C \int_{0}^{t} e^{-\a t'} \|D_{t}R(t')\|^{2}\,\dt'.
\ea
Applying Gronwall's inequality leads to \eqref{DtR-0-new}.
This finishes the proof of Lemma \ref{prop-DtR-new}.
\end{proof}
\subsection{Estimates for the vorticity and proof of Proposition \ref{energy}} \label{Sect7.4}
For each vector valued function $v = (v_{1},v_{2})^{\rm T}:\R^{2} \to \R^{2}$, we define its vorticity $\curl v$ as follows:
$$
\curl v \eqdefa \nabla^{\perp} \cdot v = \d_{y} v_{1} - \d_{x} v_{2}, \quad \nabla^{\perp} \eqdefa \bp \d_{y} \\ - \d_{x}\ep.
$$
In particular, we denote $\omega \eqdefa \curl R$.
Applying $\curl$ to $\eqref{CNS-error-new}_{2}$ gives
\ba\label{vor-1}
\curl (\rho \fD_{t} R ) - \mu \Delta \o + \curl \big( \rho R\cdot \nabla u^{\rm a} \big) + \curl \big(\vr (\d_{t} u^{\rm a}
+ u^{\rm a} \cdot \nabla u^{\rm a} ) \big) = \curl G.
\ea
\begin{lem}\label{prop-vorticity}
{\sl Let $\o\eqdefa \curl R$. Then there holds
\ba\label{vor-4}
\int_{\OO} |\nabla \o|^{2}\,\dx\,\dy \leq C \e + C\int_{0}^{t}\int_{\OO} |\nabla R|^{4}\,\dx\,\dy,
\ea
and
\ba\label{vor-5}
\int_{0}^{t} \int_{\OO} |\nabla \o|^{2}\,\dx\,\dy\,\dt' \leq C \e + C \int_{0}^{t} \int_{\OO} |\nabla R|^{3}\,\dx\,\dy\,\dt'.
\ea}
\end{lem}
\begin{proof} We first get, by taking $L^2$ inner product of \eqref{vor-1} with $\o$ and using integration by parts, that
\begin{align*}
\mu \int_{\OO} |\nabla \o|^{2} \,\dx\,\dy=\int_{\OO}\Bigl(&- G \cdot \nabla^{\perp} \o + \rho \fD_{t} R \cdot \nabla^{\perp} \o
\\
& + \rho R\cdot \nabla u^{\rm a} \cdot \nabla^{\perp} \o+ \vr \bigl(\d_{t} u^{\rm a} + u^{\rm a} \cdot \nabla u^{\rm a}\bigr) \cdot \nabla^{\perp} \o\Bigr)\,\dx\,\dy.
\end{align*}
Applying Young's inequality gives
\begin{align*}
\mu \int_{\OO} |\nabla \o|^{2} \,\dx\,\dy \leq \frac{4}{\mu} \int_{\OO} \Big(& |G|^{2} + \rho^{2} |\fD_{t} R|^{2} + |\nabla u^{\rm a}| ^{2} \rho^{2} | R|^{2} \\
&+ \vr^{2} \bigl| \d_{t} u^{\rm a} + u^{\rm a} \cdot \nabla u^{\rm a}\bigr|^{2} \Big)\,\dx\,\dy
+ \frac{\mu}{2} \int_{\OO} |\nabla \o|^{2}\,\dx\,\dy,
\end{align*}
which together with Proposition \ref{prop-energy-basic}, Lemmas \ref{prop-DtR} and \ref{prop-DtR-new}, and Theorem
\ref{thm1} ensures \eqref{vor-4} and \eqref{vor-5}.
\end{proof}
By summarizing Proposition \ref{prop-energy-basic}, Lemmas \ref{prop-DtR}, \ref{prop-DtR-new} and \ref{prop-vorticity}, we conclude the proof of
Proposition \ref{energy}.
|
train/arxiv
|
BkiUbOE4eIXgu1jpCouR
| 5
| 1
|
\section{Introduction}
Light field imaging has recently emerged as a promising technology able to discriminate and capture light rays along different directions \cite{Ng2005,Wilburn2005}.
This rich visual description of the scene enables the creation of immersive experience in AR/VR applications and facilitates the integration of computer-generated graphics for post-production editing.
Together with proper computational algorithms, this technology is expected to impact the field of digital photography, by enabling post-capture re-focusing, depth of field extension, or 3D scene models estimation.
However, light field imaging systems trade-off spatial
resolution with angular information in the light field. Rigs of cameras capture views with a high spatial resolution but in general with limited angular sampling to reduce costs \cite{Wilburn2005}. On the other hand, plenoptic cameras
use an array of microlenses placed in front of the sensor to capture multiple low-resolution (LR) views in one 2D sensor image \cite{Ng2005}.
This is a way to cost-effectively capture multiple views with a high angular sampling, but at the expense of reducing the
spatial resolution by orders of magnitude compared to the raw sensor image.
To tackle this problem, various methods have been developed,
which are aimed to achieve better spatial and angular resolution trade-off from a plenoptic camera. These methods go from the use of coded aperture techniques, using e.g. a programmable non-refractive mask
placed at the aperture as in \cite{Liang2008}, or optically
coded projections as in \cite{Xu2012}, to light field super-resolution methods \cite{Levin2008,Bishop2012,Mitra2012,Wanner2014,Farrugia2017,Yoon2015,Yoon2017,Rossi2017}.
While research in light field super-resolution is at its infancy, research in the related field of single-image super-resolution (SISR) is quite mature with methods based on very deep convolutional neural networks achieving state-of-the-art performances \cite{Kim2016,Kim2016b,Mao2016,Lim2017,Bae2017}.
This paper presents a framework which allows to leverage state-of-the-art 2D image super-resolution techniques to light field super-resolution.
The energy of the light field is first compacted, to capture the coherent information, and to then apply 2D single-image super-resolution to restore the whole light field.
To do so, we first align each view to the centre view using optical flows.
The alignment plays an important role since it removes the disparities across the views which can be easily recovered by inverting the alignment process.
This aligned light field is then decomposed using singular value decomposition (SVD) where the eigenvectors of the SVD capture dominant variations (or eigenimages) of the different views.
It will be shown in Section \ref{sec:lf_decomposition} that aligning the light field allows to put more information within the \textit{principal basis} (\textit{a.k.a}. dominant eigenvector) which captures the coherent information within the light field.
We then apply a state-of-the-art SISR algorithm to restore the \textit{principal basis}.
The information restored in the \textit{principal basis} is then propagated to all the other views in a consistent manner.
Inverse warping is then applied to restore the original disparities in the light field.
The results in Section \ref{sec:results} show that the proposed method achieves sharper light field images with results superior than existing methods for applications such as digital refocusing.
Supplementary material attached to this paper also show that the restored light fields are angularly coherent and that it is able to restore real-world plenoptic light fields. It is also shown that the method manages to restore light fields containing non-Lambertian surfaces\footnote{While a .ppsx file is included as supplementary material and uploaded on ScholarOne, the reviewers can watch the video at \url{https://youtu.be/HHmUZSP7HU4}}.
The main contributions of this paper are as follows:
\begin{itemize}
\item We present a framework that enables to leverage SISR methods to restore the \textit{principal basis} capturing the coherent information across the entire light field.
\item Based on this framework, we describe a light field super-resolution method that yield sharper light field images with results superior than existing methods for applications such as digital refocusing.
\item The proposed framework allows to inherit the benefits of the SISR methods employed to restore the \textit{principal basis} and we are therefore presenting the first light field super-resolution algorithm which uses only one model to cater for different magnification factors.
\end{itemize}
The remainder of this paper is organized as follows. Work related to the method described in this paper is provided in Section \ref{sec:related_work} while the light field energy compaction method is explained in Section \ref{sec:lf_decomposition}.
The proposed \textit{principal basis} VDSR (PB-VDSR) is described in Seciton \ref{sec:principal_basis_SR} while the experimental results are delived in the following section.
Section \ref{sec:conclusion} concludes with the final remarks.
\section{Related Work}
\label{sec:related_work}
This section gives a brief overview of work related to the key concepts of the proposed spatial light field super-resolution approach and the light field super-resolution methods that are found in literature.
\subsection{Single Image Super-Resolution}
\label{sec:sisr_related_work}
Single-image super-resolution is an ill-posed inverse problem with infinite possible solutions.
These methods use priors to derive a more plausible solution that satisfies a predefined assumption.
These priors are either hand-crafted, such as total variation or Bayesian models, or data driven that are learned using machine learning methods.
Pixel-based methods have been proposed in \cite{HeSiu2011}, \cite{ZhangGao2012} where each pixel in the high-resolution (HR) image is inferred via statistical learning. To improve spatial coherency, patch-based approaches, referred to as example-based methods, have been proposed.
Freeman \emph{et. al}. \cite{Freeman2002} presented the first single-image example-based super-resolution algorithm that used a coupled dictionary to learn a mapping between LR and HR patches.
More advanced methods based on manifold learning \cite{Chang2004,Gao2012,ferreira:hal-01388955} and sparse coding \cite{Yang2012,Timofte2015} were investigated to regularize the problem and were found to provide sharper images.
Other approaches \cite{Glasner2009,Freedman2011,Yang2013,bevilacqua:hal-01088753} utilized image self-similarities to avoid using dictionaries constructed using external images.
Deep neural networks have contributed to a drastic improvement in the field of single-image super-resolution.
Dong \emph{et. al}. \cite{Dong2014} were the first to use a rather shallow convolutional neural network (SRCNN).
Residual learning was introduced in \cite{Kim2016,Kim2016b,Bae2017} for training deeper network architectures and achieved state-of-the-art performance. The authors in \cite{Mao2016} pose the general image restoration problem with encoder-decoder networks and systematic skip connections.
This architecture was later on extended in \cite{Lim2017} where the authors expanded the model size and removed unnecessary modules in the convolutional residual networks.
\subsection{Light Field Super-Resolution}
\label{sec:lf_SR_related_work}
Early light field super-resolution approaches pose the problem as one of recovering the high-resolution views from multiple low-resolution images with unknown non-integer translation misalignment.
The authors in \cite{Levin2008,Bishop2012} proposed a two-step approach where they first estimate a depth map and then formulate the super-resolution problem either as a simple linear problem \cite{Levin2008} or as a Beyesian inference problem \cite{Bishop2012} assuming an image formation model with Lambertian reflectance priors and depth-dependent blurring kernels.
A patch-based technique was proposed in \cite{Mitra2012} where high-resolution 4D patches are estimated using a linear minimum mean square error (LMMSE) estimator assuming a disparity-dependent Gaussian Mixture Model (GMM) for the patch structure.
A variational optimization framework was proposed in \cite{Wanner2014} to spatially super-resolve the light field given their estimated depth maps and to increase the angular resolution.
Example-based light field super-resolution methods have been recently proposed. These methods use machine learning to learn a mapping between low- and high-resolution light fields.
In \cite{Farrugia2017}, the authors show that a 3D patch-volume resides on a low-dimensional subspace and propose to learn a projection between low- and high-resolution subspaces of patch-volumes using ridge-regression.
Deep learning techniques for light field super-resolution have been first proposed in \cite{Yoon2015} where 4-tuples of neighbouring views are stacked into groups and restored using SRCNN \cite{Dong2014}.
The spatially restored light field is then fed into a second CNN that up-scales the angular resolution.
The same authors have later proposed to restore each view independently using SRCNN in \cite{Yoon2017} showing superior performance over their original method.
More recently, graph based light field super-resolution algorithm was presented in \cite{Rossi2017} that enforces the optimization to preserve the light field structure.
A shallow neural network was proposed in \cite{Gul2018} to restore light fields captured by a plenoptic camera.
However, this method is only suitable to achieve a magnification factor of $\times 2$ and needs to train a CNN for every angular view.
Very recently, a multi-scale fusion scheme was used to accumulate contextual information from multiple scales while Recurrent Convolutional Neural Networks (BRCNN) is used to model the spatial relation between adjacent views and restore the light field.
A hybrid light field super-resolution method was proposed in \cite{Wang2017} where a high-resolution camera was coupled with a plenoptic camera.
The authors in \cite{Wang2017b} describe an acquisition device formed by eight low-resolution side cameras arranged around a central high-quality camera.
Iterative patch- and depth-based synthesis (iPADS) is then used to reconstruct a light field with the spatial resolution of the SLR camera and an increased number of views.
While the methods in \cite{Yoon2015, Yoon2017,Gul2018,Wang2018} use deep learning to super-resolve the light field, our method is considerably different.
The novelty of our approach is that the proposed framework allows to use SISR techniques for light field super-resolution.
The deep learning SR method used are not retrained on light fields and use models that are trained on natural images.
Moreover, our framework inherits the benefits of the SISR algorithm used.
In our study, we used VDSR \cite{Kim2016} which allows us to use a very deep super-resolution method that adopts one single model to cater for different magnification factors.
\subsection{Light Field Edit Propagation}
\label{sec:lf_edit_propagation}
Light field edit propagation involves the restoration of the centre view followed by the propagation of the restored information to all the other views.
The authors in \cite{Seitz1998} described an approach using a 3D voxel-based model of the scene with an associated radiance function to propagate pixel edits and illumination changes in a consistent manner from one view to the other views of the light field.
The authors in \cite{Jarabo2011} extend the 2D image stroke-based edit propagation method of \cite{An2008} to light fields, where they reduce the complexity by propagating the edits in a downscaled version of the light field.
In \cite{Ao2015}, a method based on a reparameterization of the light field is proposed to better preserve coherence of the edits in the angular domain.
However, these methods deal with simple stroke-based editing and are not suitable to propagate complex edits, such as inpainting or super-resolution, to all the other views.
A patch-based depth-layer-aware image synthesis algorithm was adopted in \cite{Zhang2017} to propagate the edits from the centre view to all the other views.
The authors in \cite{Frigo2017} use tensor driven diffusion to propagate information from the centre view along the Epipolar Plane Image (EPI) structure of the light field.
These methods were used to propagate either simple edits, recolorization or inpainting from the center view to all the other views.
However, up to the knowledge of the authors, such approaches were never been considered for light field super-resolution.
\section{Light Field Energy Compaction}
\label{sec:lf_decomposition}
Lets consider an input light field $I(x,y,s,t)$ represented with the two plane parametrization proposed in \cite{Levoy1996,Gortler1996}, where $(x,y)$ and $(s,t)$ represent spatial and angular coordinates respectively.
The light field can be seen as a 2D array of images, where each image $\mathbf{I}_{s,t}$ captures the scene from a viewpoint defined by angular coordinates $(s,t)$.
One can use single image super-resolution technique to restore every angular view independently.
However, these methods do not exploit the geometrical structure of the light field \cite{Liang2015} and are not guaranteed to provide angularly coherent solutions \cite{Farrugia2017}.
On the other hand, several light field super-resolution techniques have been proposed that either exploit the disparity/depth information \cite{Levin2008,Bishop2012,Mitra2012,Wanner2014} or else use learning based methods \cite{Farrugia2017,Yoon2015,Yoon2017,Gul2018,Wang2018} to improve the quality of the light field.
However, these algorithms do not benefit from the recent advances in single image super-resolution where very deep Convolutional Neural Networks are achieving outstanding performances \cite{Dong2014,Kim2016,Kim2016b,Lim2017,Mao2016}.
A light field consists of a very large volume of high-dimensional data.
Nevertheless, it exhibits redundancies in all four dimensions since every view captures the same scene from a slightly different viewpoint.
Early work in the field of light field compression used 3D/4D wavelet transforms to decompose the light field into a number of sub--bands \cite{Peter99,Lalonde1999,Chang2006}, where each sub--band gives information at different spatial and angular frequencies.
Figure \ref{fig:svd_decomposition2}(a) shows the the first six orthogonal basis when decomposing the light field using SVD.
It can be seen that while most of the energy resides in the \textit{principal basis} $\mathbf{B}_0$, there is still a lot of high frequency detail in the other basis.
Moreover, the \textit{principal basis} $\mathbf{B}_0$, which captures the average energy in the scene is blurred.
This is attributed to variations in disparities across the views which result in high-frequency angular details that are not captured by the \textit{principal basis}.
\begin{figure*}[htb]
\begin{minipage}[b]{0.5\linewidth}
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.3\linewidth]{imgs/lf_decomposition/original/basis/boardgames_1.jpg} & \includegraphics[width=0.3\linewidth]{imgs/lf_decomposition/original/basis/boardgames_2.jpg} & \includegraphics[width=0.3\linewidth]{imgs/lf_decomposition/original/basis/boardgames_3.jpg} \\
$\mathbf{B}_0: 7.66$ & $\mathbf{B}_1: 5.45$ & $\mathbf{B}_2: 5.50$ \\
\includegraphics[width=0.3\linewidth]{imgs/lf_decomposition/original/basis/boardgames_4.jpg} & \includegraphics[width=0.3\linewidth]{imgs/lf_decomposition/original/basis/boardgames_5.jpg} & \includegraphics[width=0.3\linewidth]{imgs/lf_decomposition/original/basis/boardgames_6.jpg} \\
$\mathbf{B}_3: 5.09$ & $\mathbf{B}_4: 5.64$ & $\mathbf{B}_5: 4.72$ \\
\end{tabular}
\end{minipage}
\begin{minipage}[b]{0.5\linewidth}
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.3\linewidth]{imgs/lf_decomposition/original/basis/antinous_1.jpg} & \includegraphics[width=0.3\linewidth]{imgs/lf_decomposition/original/basis/antinous_2.jpg} & \includegraphics[width=0.3\linewidth]{imgs/lf_decomposition/original/basis/antinous_3.jpg} \\
$\mathbf{B}_0: 7.56$ & $\mathbf{B}_1: 5.76$ & $\mathbf{B}_2: 5.71$ \\
\includegraphics[width=0.3\linewidth]{imgs/lf_decomposition/original/basis/antinous_4.jpg} & \includegraphics[width=0.3\linewidth]{imgs/lf_decomposition/original/basis/antinous_5.jpg} & \includegraphics[width=0.3\linewidth]{imgs/lf_decomposition/original/basis/antinous_6.jpg} \\
$\mathbf{B}_3: 4.95$ & $\mathbf{B}_4: 4.90$ & $\mathbf{B}_5: 5.06$ \\
\end{tabular}
\end{minipage}
\centerline{(a) First six-basis computed using SVD.}\\
\begin{minipage}[b]{0.48\linewidth}
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.3\linewidth]{imgs/lf_decomposition/aligned/basis/boardgames_1.jpg} & \includegraphics[width=0.3\linewidth]{imgs/lf_decomposition/aligned/basis/boardgames_2.jpg} & \includegraphics[width=0.3\linewidth]{imgs/lf_decomposition/aligned/basis/boardgames_3.jpg} \\
$\mathbf{B}_0: 7.55$ & $\mathbf{B}_1: 4.17$ & $\mathbf{B}_2: 3.75$ \\
\includegraphics[width=0.3\linewidth]{imgs/lf_decomposition/aligned/basis/boardgames_4.jpg} & \includegraphics[width=0.3\linewidth]{imgs/lf_decomposition/aligned/basis/boardgames_5.jpg} & \includegraphics[width=0.3\linewidth]{imgs/lf_decomposition/aligned/basis/boardgames_6.jpg} \\
$\mathbf{B}_3: 3.21$ & $\mathbf{B}_4: 3.48$ & $\mathbf{B}_5: 3.67$ \\
\end{tabular}
\end{minipage}
\begin{minipage}[b]{0.48\linewidth}
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.3\linewidth]{imgs/lf_decomposition/aligned/basis/antinous_1.jpg} & \includegraphics[width=0.3\linewidth]{imgs/lf_decomposition/aligned/basis/antinous_2.jpg} & \includegraphics[width=0.3\linewidth]{imgs/lf_decomposition/aligned/basis/antinous_3.jpg} \\
$\mathbf{B}_0: 7.29$ & $\mathbf{B}_1: 4.88$ & $\mathbf{B}_2: 4.71$ \\
\includegraphics[width=0.3\linewidth]{imgs/lf_decomposition/aligned/basis/antinous_4.jpg} & \includegraphics[width=0.3\linewidth]{imgs/lf_decomposition/aligned/basis/antinous_5.jpg} & \includegraphics[width=0.3\linewidth]{imgs/lf_decomposition/aligned/basis/antinous_6.jpg} \\
$\mathbf{B}_3: 3.94$ & $\mathbf{B}_4: 4.30$ & $\mathbf{B}_5: 4.11$ \\
\end{tabular}
\end{minipage}
\centerline{(b) First six-basis computed using SVD of the aligned light field.}\\
\caption{Comparing the information contained in each basis when computing (a) SVD decomposition and (b) Aligned SVD decomposition for the (left) Boardgames and (right) Antinous synthetic light fields.
The entropy measure for each basis is given below the corresponding image.}
\label{fig:svd_decomposition2}
\end{figure*}
The authors in \cite{Jiang2017} tried to reduce the energy within the high-frequency basis by jointly aligning the angular views and estimating a low--rank approximation (LRA) of the light field.
This approach has shown very promising results in the field of light field compression.
In the same spirit, the RASL algorithm \cite{Peng2010} was used to find the homographies that globally align a batch of linearly correlated images.
Both methods find an optimal set of homographies such that the matrix of aligned images can be decomposed in a low--rank matrix of aligned images, with the latter constraining the error matrix to be sparse.
However, as it can be seen in Figure \ref{fig:subjective_Eval1}, while both RASL and HLRA methods manage to globally align the angular views, the resulting mean view, that is computed by averaging all the views, are still blurred indicating that the views are not well aligned.
\begin{figure*}[ht]
\centering
\begin{tabular}{cccccc}
\centering
\footnotesize{{No Align}} & \footnotesize{{RASL}} \cite{Peng2010} & \footnotesize{HLRA \cite{Jiang2017}} & \footnotesize{SIFT Flow \cite{Liu2011}} & \footnotesize{CPM \cite{Hu2016}} & \footnotesize{SPM-BP \cite{Li2015}} \\
\includegraphics[width=2.5cm]{imgs/mean_subAps/Bee_2/no_align.png} &
\includegraphics[width=2.5cm]{imgs/mean_subAps/Bee_2/rasl.png} &
\includegraphics[width=2.5cm]{imgs/mean_subAps/Bee_2/hlrma.png}
&
\includegraphics[width=2.5cm]{imgs/mean_subAps/Bee_2/siftflow.png}
&
\includegraphics[width=2.5cm]{imgs/mean_subAps/Bee_2/cpm.png}
&
\includegraphics[width=2.5cm]{imgs/mean_subAps/Bee_2/spm_bp.png}
\\
\footnotesize{99.092} & \footnotesize{98.261} & \footnotesize{99.080} & \footnotesize{19.164} & \footnotesize{51.073} & \footnotesize{71.652} \\
\includegraphics[width=2.5cm]{imgs/mean_subAps/Chess/no_align.png} &
\includegraphics[width=2.5cm]{imgs/mean_subAps/Chess/rasl.png} &
\includegraphics[width=2.5cm]{imgs/mean_subAps/Chess/hlrma.png}
&
\includegraphics[width=2.5cm]{imgs/mean_subAps/Chess/siftflow.png}
&
\includegraphics[width=2.5cm]{imgs/mean_subAps/Chess/cpm.png}
&
\includegraphics[width=2.5cm]{imgs/mean_subAps/Chess/spm_bp.png}
\\
\footnotesize{102.625} & \footnotesize{100.529} & \footnotesize{102.478} & \footnotesize{3.595} & \footnotesize{23.978} & \footnotesize{4.638} \\
\includegraphics[width=2.5cm]{imgs/mean_subAps/Railway_Lines_1/no_align.png} &
\includegraphics[width=2.5cm]{imgs/mean_subAps/Railway_Lines_1/rasl.png} &
\includegraphics[width=2.5cm]{imgs/mean_subAps/Railway_Lines_1/hlrma.png}
&
\includegraphics[width=2.5cm]{imgs/mean_subAps/Railway_Lines_1/siftflow.png}
&
\includegraphics[width=2.5cm]{imgs/mean_subAps/Railway_Lines_1/cpm.png}
&
\includegraphics[width=2.5cm]{imgs/mean_subAps/Railway_Lines_1/spm_bp.png}
\\
\footnotesize{76.039} & \footnotesize{75.150} & \footnotesize{75.944} & \footnotesize{12.973} & \footnotesize{18.815} & \footnotesize{15.100} \\
\end{tabular}
\caption{Cropped regions of the mean view when using different disparity compensation methods. Underneath each image we provide the average variance across the $n$ angular views which was used in \cite{Farrugia2017} to characterize the performance of the alignment algorithm, where smaller values indicate better alignment. }
\label{fig:subjective_Eval1}
\end{figure*}
In the sequel, we consider $\mathbf{I}_{s,t}$ to represent different views, where $(s,t)$ define the angular coordinates.
This notation will be further simplified as $\mathbf{I}_i$ with a bijection between $(s,t)$ and $i$.
The complete light field can hence be represented by a matrix $\mathbf{I} \in \mathbb{R}^{m,n} $:
\begin{equation}
\mathbf{I} = [vec(\mathbf{I}_{1}) \quad | \quad vec(\mathbf{I}_{2}) \quad | \quad \cdots \quad | \quad vec(\mathbf{I}_{n})]
\label{eq:lf_matrix}
\end{equation}
\noindent with $vec(\mathbf{I}_i)$ being the vectorized representation of the $i$-th angular view, $m$ represents the number of pixels in each view $(m = X \times Y)$ and $n$ is the number of views in the light field $(n = P \times Q)$, where $P$ and $Q$ represent the number of vertical and horizontal angular views respectively.
We then formulate the light field decomposition problem as that of finding a set of orthogonal basis $\mathbf{B}$ that is able to capture most of the information contained in the light field.
This can be achieved by minimizing the following optimization problem
\begin{equation}
\label{eq:lf_decomposition}
\underset{\mathbf{u},\mathbf{v},\mathbf{B},\mathbf{C}}{
min}_{}{ || \Gamma_{\mathbf{u},\mathbf{v}} \left( \mathbf{I}\right) - \mathbf{B}\mathbf{C}||^2_2 }
\end{equation}
\noindent where $\mathbf{u} \in \mathbb{R}^{m,n}$ and $\mathbf{v} \in \mathbb{R}^{m,n}$ are flow vectors that specify the displacement of each pixel needed to align each view with the centre view, $\mathbf{B} \in \mathbb{R}^{m,n}$ represents the basis matrix, $\mathbf{C} \in \mathbb{R}^{n,n}$ is the combination weight matrix and $\Gamma_{\mathbf{u},\mathbf{v}}(\cdot)$ is the forward warping operator.
This optimization problem is computationally intractable.
Instead, we decompose this problem in two sub--problems: \romannumeral 1) use an optical flow estimation technique to find the flow vectors $\mathbf{u}$ and $\mathbf{v}$ that best align each view with the centre view and \romannumeral 2) decompose the aligned light field into a set of basis $\mathbf{B}$ and coefficient matrix $\mathbf{C}$ using SVD.
The results in Figure \ref{fig:subjective_Eval1} clearly show that the mean views are much sharper when aligning the light field using optical flows.
Moreover, optical flows significantly reduce the variance across the angular views, with the SIFT flow method \cite{Liu2011} achieving the best performance. It reduces the mean variance across views by a factor of nine, and thus we will use it to align the views.
Reducing the total variance across the views (as shown in Figure \ref{fig:subjective_Eval1}) allows to compact more information in the low-frequency basis.
The solution of the first sub-problem gives the flow-vectors $\mathbf{u}$ and $\mathbf{v}$ which are used to align the light field using forward warping \emph{i.e}. $\tilde{\mathbf{I}} = \Gamma_{\mathbf{u},\mathbf{v}} \left( \mathbf{I}\right)$.
The aligned light field $\tilde{\mathbf{I}} = \mathbf{U} \boldsymbol{\Sigma} \mathbf{V}^T$ is then decomposed using SVD, where $\mathbf{U}$ and $\mathbf{V}$ are unitary matrices and $\boldsymbol{\Sigma}$ is a diagonal matrix containing the singular values.
The basis matrix is computed as $\mathbf{B} = \mathbf{U} \boldsymbol{\Sigma}$ while the coefficient matrix is given by $\mathbf{C}= \mathbf{V}^T$.
Figure \ref{fig:svd_decomposition2}(b) shows the first six orthogonal basis using our proposed light field decomposition method.
It can be seen that the \textit{principal basis} $\mathbf{B}_0$ is much sharper indicating that it captures more information from the light field.
Moreover, the energy in the higher-frequency basis is significantly reduced as indicated by the significant drop in entropy when using our proposed light field decomposition method.
The light field can then be easily reconstructed using $\tilde{\mathbf{I}} = \mathbf{B} \mathbf{C}$ without losing any information since $\mathbf{B}$ is orthogonal and full-rank.
In the sequel, the decompositon of the aligned light field will be referred to as A-SVD.
\section{Principal Basis Super-Resolution}
\label{sec:principal_basis_SR}
Let $\mathbf{I}^H$ and $\mathbf{I}^L$ denote the high- and low-resolution light fields. The super-resolution problem can be formulated in Banach space as
\begin{equation}
\mathbf{I}^L = \downarrow_\alpha \mathbf{I}^H + \boldsymbol{\eta}
\label{eq:acquision_model}
\end{equation}
\noindent where $\boldsymbol{\eta}$ is an additive noise matrix and $\downarrow_\alpha$ is a downsampling operator applied on each angular view with a scale--factor $\alpha$.
Figure \ref{fig:pb_vdsr_block_diagram} illustrates a block diagram of the proposed light field super-resolution algorithm where for simplicity a $3 \times 3$ matrix of angular views is shown.
The A-SVD algorithm, described in section \ref{sec:lf_decomposition}, is applied on the low-resolution light field $\mathbf{I}^L$ to decompose the light field into a set of orthogonal basis $\mathbf{B} \in \mathbb{R}^{m,n}$ and coefficient matrix $\mathbf{C} \in \mathbb{R}^{n,n}$.
As shown in more detail in Section \ref{sec:lf_decomposition}, the A-SVD algorithm is able to capture more information in the \textit{principal basis} $\mathbf{B}_0$.
\begin{figure*}[ht]
\centering
\def\svgwidth{\linewidth}
\input{block_diagram.pdf_tex}
\caption{The proposed light field super-resolution algorithm that takes a $3 \times 3$ matrix of low-resolution views as input, denoted by $\mathbf{I}^L$, to estimate the high-resolution light field $\mathbf{I}^H$.}
\label{fig:pb_vdsr_block_diagram}
\end{figure*}
Driven by the observation that the \textit{principal basis} $\mathbf{B}_0$ is a natural image that captures most of the information in the light field, we pose the problem of light field super-resolution as that of restoring the resolution of the \textit{principal basis}.
The higher order basis $\mathbf{B}_j$, $j \in [1,n-1]$, that capture the discrepancies across the views in terms of occlusions and illumination, are not modified.
Any single SISR method can be used to restore the low resolution \textit{principal basis} $\mathbf{B}_0$ and to estimate the high resolution \textit{principal basis} $\hat{\mathbf{B}}_0$.
The high frequency basis are simply approximated using $\hat{\mathbf{B}}_j = \mathbf{B}_j \text{ for } j \in [1,n-1]$.
The restored aligned light field is then reconstructed using a simple matrix multiplication which is then inverse-warped to restore the original disparities \textit{i.e}. $\tilde{\mathbf{I}}^H = \Gamma^{-1}_{\mathbf{u},\mathbf{v}} (\hat{\mathbf{B}} \mathbf{C})$, where $\Gamma^{-1}_{\mathbf{u},\mathbf{v}}(\cdot)$ stands for the inverse warping operator.
While the theoretical and implementation details of A-SVD were provided in Section \ref{sec:lf_decomposition}, the following sub-sections will deal with the implementation detail of the SISR and LF Reconstruction modules.
\subsection{SISR Module}
\label{sec:restore_pb}
In this work we consider some of the most promising SISR methods found in literature to restore the \textit{principal basis} and their performance is summarized in Table \ref{tbl:SISR_performance}.
In essence we consider the first deep-learning based super-resolution method SRCNN \cite{Dong2014}, the very deep convolutional neural network (VDSR) which uses residual learning with 20 convolutional layers \cite{Kim2016} and the Lab402 method which was ranked third in the recent NTIRE workshop challenge.
The network models of these methods were not retrained on light field data and therefore this experiment evaluates the generalization abilities of these methods.
These results demonstrate that while both VDSR and Lab402 manage to outperform SRCNN, the VDSR method is able to achieve the best performance in terms of both PSNR and SSIM quality measures.
This indicates that while other methods can be used to restore the \textit{principal basis}, the VDSR algorithm achieves the best performance and will therefore be considered in the experimental results in Section \ref{sec:results}.
Given that our method uses VDSR to restore the \textit{principal basis} we named our method PB-VDSR.
It is important to mention here that unlike SRCNN, VDSR uses a single network model to cater for different magnification factors and PB-VDSR inherits this property.
\begin{table}[htb]
\caption{Quality analysis (PSNR with SSIM in parenthesis) using different single-image super-resolution algorithms to restore the \textit{principal basis} $\mathbf{B}_0$ at a magnification factor $\times 3$.}
\label{tbl:SISR_performance}
\begin{center}
\begin{tabular}{|l|c|c|c|}
\hline
\bf{Light Field} & \bf{SRCNN} \cite{Dong2014} & \bf{VDSR} \cite{Kim2016} & \bf{Lab402} \cite{Bae2017} \\
\hline
Antinous & 33.32 (0.954) & \bf{35.74} (\bf{0.978}) & 33.81 (0.977) \\
Boardgames & 23.68 (0.835) & \bf{24.65} (\bf{0.865}) & 23.92 (0.859)\\
Greek & 30.78 (0.935) & \bf{33.55} (\bf{0.966}) & 31.70 (0.961) \\
Medieval 2 & 30.32 (0.952) & \bf{32.10} (\bf{0.962}) & 31.74 (\bf{0.962})\\
Origami & 25.32 (0.951) & 28.89 (\bf{0.973}) & \bf{28.97} (\bf{0.973})\\
Books & 29.73 (0.966) & \bf{30.78} (0.974) & 29.86 (0.970))\\
Friends 2 & 29.31 (0.935) & \bf{31.13} (\bf{0.944}) & 30.79 (\bf{0.944}) \\
Game Board & 31.75 (0.972) & \bf{32.12} (\bf{0.976}) & 31.54 (0.974) \\
Graffiti & 28.56 (0.870) & 29.90 (0.880) & \bf{29.91} (\bf{0.883}) \\
Parc du Luxembourg & 28.49 (0.926) & \bf{29.08} (\bf{0.935}) & 28.39 (0.928) \\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Light Field Reconstruction Module}
\label{sec:lf_reconstruct}
The aligned high resolution light field can be estimated by multiplying the restored basis $\hat{\mathbf{B}}$ and weight matrix $\mathbf{C}$ \textit{i.e}. $\hat{\mathbf{I}}^H = \hat{\mathbf{B}} \mathbf{C}$.
The views of $\hat{\mathbf{I}}^H$ are aligned with the center view.
Forward warping can be used to recover the original disparities of the restored views.
However, as can be seen in the first column of Figure \ref{fig:inpainting}, forward warping is not able to restore all pixels and results in a number of cracks and holes.
Another approach is to use inverse warping and use neighbouring pixels to estimate the missing information.
However, as can be seen in the second column of Figure \ref{fig:inpainting}, missing pixels due to occlusion are not well correlated with the neighbouring pixels and result in inaccurate estimates.
\begin{figure}[htb]
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.3\linewidth]{imgs/inpaint/LegoKnightForwardWarping.png} &
\includegraphics[width=0.3\linewidth]{imgs/inpaint/LegoKnightInverseWarping.png} &
\includegraphics[width=0.3\linewidth]{imgs/inpaint/LegoKnightProposedWarping.png} \\
\includegraphics[width=0.3\linewidth]{imgs/inpaint/BoxesForwardWarping.png} &
\includegraphics[width=0.3\linewidth]{imgs/inpaint/BoxesInverseWarping.png} &
\includegraphics[width=0.3\linewidth]{imgs/inpaint/BoxesProposedWarping.png} \\
\footnotesize{Forward Warping} & \footnotesize{Inverse Warping} & \footnotesize{Proposed Inpainting}\\
\end{tabular}
\caption{Inpainting the cracks marked in green}
\label{fig:inpainting}
\end{figure}
In this work we observe that the pixels warped using forward warping are very accurate.
Instead of interpolating the missing pixels, in this work we simply copy the collocated pixels from the low-resolution light field to replace the missing pixels.
The light field reconstructed in this work is depicted as the third column of Figure \ref{fig:inpainting} where it can be seen that the recovered pixels are more accurate than those estimated using inverse warping.
\subsection{Edit Propagation Methods}
\label{sec:edit_prop_compare}
This work is related to the Light Field Edit Propagation methods described in Section \ref{sec:lf_edit_propagation} which allow the user to edit the center view and propagate the edits to all the other views.
Figure \ref{fig:edit_propagation_analysis_} illustrates the PSNR measure at each view and compares the proposed method, which we call PB-VDSR, against the edit propagation method that will be described next.
The edit propagation considered applies VDSR to restore the center view and then propagate the information to the other views using forward warping.
The missing pixels due to occlusions are estimated using collocated pixels from the low-resolution light field as described in the previous subsection.
It can be seen that edit propagation achieves larger PSNR for the center view (view 41). However its performance degrades significantly when propagating the information to all the other views.
This can be explained since the edit propagation ignores the variations across the views caused by illumination and occlusions.
On the other hand, PB-VDSR restores the \textit{principal basis} that captures the angular consistent information in the light field while the variations caused by illumination and occlusion are preserved in the higher frequency basis.
This implies that PB-VDSR propagates the high angular frequency information in $\hat{\mathbf{I}}^H$.
Moreover, the higher PSNR achieved at the center view by the edit propagation is obtained using the VDSR network trained to restore natural images like the center view and not the \textit{principal basis} which captures the dominant information in the light field.
This result suggests that the performance of the algorithm can be further improved by retraining the VDSR neural network to specifically restore \textit{principal basis} rather than considering it as a generic image.
However, retraining the SISR is not in scope of this paper since the objective here is to show that SISR can be extended using our framework to restore light fields.
\begin{figure*}[ht]
\centering
\setlength\tabcolsep{1.5pt}
\begin{tabular}{cc}
\centering
\includegraphics[width=0.4\linewidth]{imgs/edit_prop_analysis/Bee_2.png} &
\includegraphics[width=0.4\linewidth]{imgs/edit_prop_analysis/Bikes.png} \\
Bee 2 (INRIA) & Bikes (EPFL) \\
\end{tabular}
\caption{PSNR analysis at each view comparing the proposed PB-VDSR to an edit propagation method. The view index indicates the index of the view when scanned using raster scan ordering.}
\label{fig:edit_propagation_analysis_}
\end{figure*}
\section{Experimental Results}
\label{sec:results}
The experiments conducted in this paper use real-world light fields from the EPFL \cite{Rerabek2016}, INRIA\footnote{INRIA dataset: https://www.irisa.fr/temics/demos/IllumDatasetLF/index.html}, and Stanford\footnote{Stanford dataset: http://lightfield.stanford.edu/} datasets.
Both EPFL and INRIA are light fields that are captured by a plenoptic camera and therefore have low angular disparities while the Stanford dataset is captured using a Gantry which have larger angular disparities.
While the angular views of the EPFL and Stanford datasets are available, the light fields in the INRIA dataset were decoded using the method in \cite{Dansereau2013}.
In all our experiments we consider a $9 \times 9$ matrix of angular views.
For computational purposes, the high-resolution views of the Stanford dataset were down-sampled such that the lowest dimension is set to 400 pixels.
The high-resolution images of the other datasets were kept unchanged \textit{i.e}. $625 \times 434$.
We compare the performance of our proposed PB-VDSR method against some of the best performing methods in the field of light field super-resolution, namely the CNN based light field super-resolution algorithm (LF-SRCNN) \cite{Yoon2017}, the linear subspace projection based method (BM-PCARR) \cite{Farrugia2017} and the Graph-based light field super resolution (GRAPH) \cite{Rossi2017}.
It must be mentioned that while the BM+PCARR and LF-SRCNN were retrained on 98 light fields that were not considered in the evaluation phase, the network model adopted by VDSR was not retrained on light fields and we used the original model adopted for single image super-resolution.
Moreover, PB-VDSR inherits the benefits of VDSR and adopts one single model to cater for different magnification factors.
The other light field super-resolution methods described in the related work section were not considered since they either were reported to achieve performance inferior to the methods considered here \cite{Yoon2017,Farrugia2017,Rossi2017} or the code was not made publicly available by the authors at the time of writing the paper.
The MATLAB code of the proposed method will be made available online upon publication\footnote{LF-Editing Repository: https://github.com/rrfarr/LF-Editing}.
\begin{table*}[ht]
\caption{Quality analysis (PSNR with SSIM in parenthesis) using different light field super-resolution algorithms when considering a magnification factor of $\times 3$.}
\label{tbl:SR_psnr_analysis_x3}
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|}
\hline
\bf{Light Field} & \bf{Bicubic} & \bf{BM-PCARR} & \bf{LF-SRCNN} & \bf{GRAPH} & \bf{PB-VDSR} \\
\hline
Bikes & 27.55 (0.87) & 28.78 (0.89) & 28.73 (0.88) & 29.22 (0.90) & 29.87 (0.90)\\
Bench in Paris & 22.43 (0.79) & 23.33 (0.83) & 23.07 (0.82) & 23.25 (0.83) & 23.48 (0.83)\\
Friends 1 & 31.17 (0.90) & 32.17 (0.92) & 32.17 (0.92) & 32.17 (0.92) & 33.16 (0.92)\\
Sphynx & 27.65 (0.77) & 28.73 (0.81) & 28.45 (0.80) & 28.88 (0.81) & 28.76 (0.80)\\
Bee 2 & 31.02 (0.91) & 32.03 (0.91) & 32.24 (0.92) & 32.74 (0.93) & 32.62 (0.92)\\
Duck & 23.35 (0.84) & 24.22 (0.86) & 24.19 (0.87) & 24.43 (0.88) & 24.50 (0.88)\\
Fruits & 28.74 (0.85) & 30.21 (0.89) & 29.87 (0.88) & 30.91 (0.91) & 30.20 (0.89)\\
Rose & 34.05 (0.90) & 35.30 (0.92) & 35.00 (0.91) & 36.19 (0.94) & 34.98 (0.91)\\
Mini & 27.30 (0.77) & 28.23 (0.79) & 28.03 (0.79) & 28.31 (0.81) & 28.55 (0.80)\\
Chess & 30.04 (0.92) & 31.02 (0.93) & 30.88 (0.93) & 31.69 (0.94) & 31.61 (0.94)\\
Bunny & 32.91 (0.94) & 34.31 (0.94) & 34.14 (0.94) & 35.31 (0.96) & 35.64 (0.95)\\
Lego Bulldozer & 26.21 (0.86) & 27.05 (0.87) & 27.10 (0.87) & 28.27 (0.90) & 28.15 (0.89)\\
Lego Truck & 30.26 (0.89) & 31.18 (0.91) & 30.99 (0.91) & 31.62 (0.92) & 31.39 (0.92)\\
Lego Knights & 27.28 (0.86) & 28.15 (0.88) & 28.24 (0.87) & 28.62 (0.90) & 29.01 (0.90)\\
\hline
\hline
\bf{Overall} & \bf{ 28.57(0.86)} & \bf{29.62 (0.88)} & \bf{29.51 (0.88)} & \bf{30.12 (0.90)} & \bf{30.14 (0.89)}\\
\hline
\end{tabular}
\end{center}
\end{table*}
\begin{table*}[ht]
\caption{Quality analysis (PSNR with SSIM in parenthesis) using different light field super-resolution algorithms when considering a magnification factor of $\times 4$.}
\label{tbl:SR_psnr_analysis_x4}
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|}
\hline
\bf{Light Field} & \bf{Bicubic} & \bf{BM-PCARR} & \bf{LF-SRCNN} & \bf{GRAPH} & \bf{PB-VDSR} \\
\hline
Bikes & 25.33 (0.80) & 26.42 (0.82) & 26.28 (0.82) & 26.62 (0.84) & 27.85 (0.82)\\
Bench in Paris & 21.00 (0.72) & 21.74 (0.75) & 21.50 (0.75) & 21.57 (0.75) & 21.73 (0.75)\\
Friends 1 & 29.15 (0.86) & 30.14 (0.88) & 30.10 (0.88) & 30.08 (0.88) & 30.89 (0.88)\\
Sphynx & 25.88 (0.70) & 26.89 (0.74) & 26.62 (0.72) & 26.86 (0.74) & 26.79 (0.72)\\
Bee 2 & 28.72 (0.86) & 29.85 (0.87) & 29.85 (0.88) & 30.25 (0.89) & 30.27 (0.88)\\
Duck & 21.62 (0.76) & 22.29 (0.79) & 22.25 (0.79) & 22.44 (0.81) & 22.50 (0.80)\\
Fruits & 26.60 (0.78) & 27.82 (0.82) & 27.53 (0.80) & 28.28 (0.84) & 27.28 (0.80)\\
Rose & 31.86 (0.84) & 33.05 (0.87) & 32.57 (0.85) & 33.42 (0.88) & 32.12 (0.84)\\
Mini & 25.71 (0.70) & 26.40 (0.72) & 26.30 (0.71) & 26.45 (0.73) & 26.73 (0.72)\\
Chess & 28.03 (0.87) & 28.90 (0.88) & 28.77 (0.88) & 29.31 (0.90) & 28.65 (0.88)\\
Bunny & 30.47 (0.90) & 31.80 (0.91) & 31.57 (0.91) & 32.30 (0.92) & 32.33 (0.91)\\
Lego Bulldozer & 24.29 (0.79) & 25.00 (0.80) & 25.02 (0.81) & 25.85 (0.84) & 25.13 (0.81)\\
Lego Truck & 28.55 (0.85) & 29.35 (0.87) & 29.15 (0.86) & 29.56 (0.87) & 29.09 (0.86)\\
Lego Knights & 25.20 (0.79) & 26.13 (0.81) & 26.84 (0.81) & 26.67 (0.84) & 25.89 (0.81)\\
\hline
\hline
\bf{Overall} & \bf{26.60 (0.80)} & \bf{27.60 (0.82)} & \bf{27.45 (0.82)}& \bf{27.83 (0.84)} & \bf{27.66 (0.82)}\\
\hline
\end{tabular}
\end{center}
\end{table*}
The results in Table \ref{tbl:SR_psnr_analysis_x3} and Table \ref{tbl:SR_psnr_analysis_x4} compare these light field super-resolution methods in terms of both PSNR and SSIM for magnification factors of $\times3$ and $\times4$ respectively.
It can be seen that our proposed method outperforms both BM-PCARR and LF-SRCNN and it is competitive to the GRAPH light field super-resolution method when considering both PSNR and SSIM objective quality metrics.
Moreover, it can be seen in Figure \ref{fig:SR_subjective_Eval} that our method is able to restore central views that are much sharper and of higher quality (see bee in first row, text on the bicycle in second row, eyes of the duck in forth row and edges of the chess board in the sixth row of Figure \ref{fig:SR_subjective_Eval}) compared to the three leading light field super-resolution methods found in literature.
One can also notice that the other methods provide aliasing (see bicycle rim in second row and teeth of the second female on the left in the fifth row of Figure \ref{fig:SR_subjective_Eval}) and ghosting artifacts (see rabbit ears in third row in Figure \ref{fig:SR_subjective_Eval}).
It can also be seen that our proposed method yields sharper results on non-Lambertian surfaces as can be seen on the Mini light field (bottom row in Figure \ref{fig:SR_subjective_Eval}).
\begin{figure*}[ht]
\centering
\setlength\tabcolsep{1.5pt}
\begin{tabular}{cccc}
\centering
\footnotesize{BM+PCARR} & \footnotesize{LF-SRCNN} &\footnotesize{GRAPH}& \footnotesize{PB-VDSR}
\\
\includegraphics[width=4.3cm]{imgs/results/bee_2/bm_pca_rr.png} &
\includegraphics[width=4.3cm]{imgs/results/bee_2/lf_srcnn.png} &
\includegraphics[width=4.3cm]{imgs/results/bee_2/graph_sr.png} &
\includegraphics[width=4.3cm]{imgs/results/bee_2/pb_vdsr.png}
\\
\includegraphics[width=4.3cm]{imgs/results/bikes/bm_pca_rr.png} &
\includegraphics[width=4.3cm]{imgs/results/bikes/lf_srcnn.png} &
\includegraphics[width=4.3cm]{imgs/results/bikes/graph_sr.png} &
\includegraphics[width=4.3cm]{imgs/results/bikes/pb_vdsr.png}
\\
\includegraphics[width=4.3cm]{imgs/results/bunny/bm_pca_rr.png} &
\includegraphics[width=4.3cm]{imgs/results/bunny/lf_srcnn.png} &
\includegraphics[width=4.3cm]{imgs/results/bunny/graph_sr.png} &
\includegraphics[width=4.3cm]{imgs/results/bunny/pb_vdsr.png}
\\
\includegraphics[width=4.3cm]{imgs/results/duck/bm_pca_rr.png} &
\includegraphics[width=4.3cm]{imgs/results/duck/lf_srcnn.png} &
\includegraphics[width=4.3cm]{imgs/results/duck/graph_sr.png} &
\includegraphics[width=4.3cm]{imgs/results/duck/pb_vdsr.png}
\\
\includegraphics[width=4.3cm]{imgs/results/friends_1/bm_pca_rr.png} &
\includegraphics[width=4.3cm]{imgs/results/friends_1/lf_srcnn.png} &
\includegraphics[width=4.3cm]{imgs/results/friends_1/graph_sr.png} &
\includegraphics[width=4.3cm]{imgs/results/friends_1/pb_vdsr.png}
\\
\includegraphics[width=4.3cm]{imgs/results/chess/bm_pca_rr.png} &
\includegraphics[width=4.3cm]{imgs/results/chess/lf_srcnn.png} &
\includegraphics[width=4.3cm]{imgs/results/chess/graph_sr.png} &
\includegraphics[width=4.3cm]{imgs/results/chess/pb_vdsr.png}
\\
\includegraphics[width=4.3cm]{imgs/results/mini/bm_pca_rr.png} &
\includegraphics[width=4.3cm]{imgs/results/mini/lf_srcnn.png} &
\includegraphics[width=4.3cm]{imgs/results/mini/graph_sr.png} &
\includegraphics[width=4.3cm]{imgs/results/mini/pb_vdsr.png}
\\
\end{tabular}
\caption{Restored center view of light fields using different light field super-resolution algorithms. These are best viewed in color and by zooming on the views.}
\label{fig:SR_subjective_Eval}
\end{figure*}
\begin{figure*}[ht]
\centering
\def\svgwidth{0.95\linewidth}
\input{bulldozer-06.pdf_tex}
\\
\centerline{(a) Refocus of Lego Bulldozer at a slope of -0.6.}
\def\svgwidth{0.95\linewidth}
\input{chess-01.pdf_tex}
\\
\centerline{(b) Refocus of Chess at a slope of +0.1.}
\def\svgwidth{0.95\linewidth}
\input{duck-00.pdf_tex}
\\
\centerline{(c) Refocus of Duck at a slope of +0.0.}
\caption{Refocusing of different light field at different depths.}
\label{fig:refocus}
\end{figure*}
One important feature of a light field is that it enables to digitally refocus the image after production.
The quality of the refocused image depends on the quality of the light field and of its coherence across all the views.
The results in Figure \ref{fig:refocus} shows a number of refocused images obtained from light fields restored using the GRAPH \cite{Rossi2017} and PB-VDSR where the images were refocused using the Light Field Toolbox \cite{Dansereau2015}.
These results clearly show that the refocused images computed on light fields restored using PB-VDSR are sharper and of better quality.
Moreover, the supplementary multimedia files show a pseudovideo of different light fields reconstructed using PB-VDSR where it is evident that the proposed method manages to restore light fields of higher quality and better angularly coherence compared to those obtained using the GRAPH method, even when considering non-Lambertian surfaces as the Tarot Cards and Crystal Ball light field where the latter fails.
Moreover, these result also show the restoration of real-world applications when super-resolving the plenoptic image from $625 \times 434$ to $1875 \times 1302$.
The complexity of PB-VDSR is mainly affected by the computation of the optical flows used to align the light field (SIFT Flow in our case), of the SVD decomposition used to decompose the aligned light field, of the single image super resolution method used to restore the \textit{principal basis} (VDSR in our case) and of the matrix multiplication that is used to propagate the restored information in the principle basis to all the other views.
The Sift Flow is used to align all the $n$ views to the center view and is reported in \cite{Liu2011} to have a time complexity of the order $O(nm\log (\sqrt{m}))$, where $m$ represents the number of pixels in each view.
The SVD decomposition and the matrix multiplication incur a time complexity of the order $O(n^2m)$ each.
Moreover, the feed-forward part of VDSR which is used during evaluation has a fixed depth and width and its complexity is mainly dependent on the resolution of the \textit{principal basis}.
This implies that the VDSR algorithm has a time complexity of the order $O(m)$.
This complexity analysis concludes that the proposed method has a time complexity that is mainly dependent on the resolution and number of views in the light field.
This contrasts with the GRAPH method presented in \cite{Rossi2017} whose time complexity is proportional to $\alpha^4$ where $\alpha$ is the magnification factor.
A quantitative assessment of the complexity of different light field super-resolution methods considered in this work is summarized in Table \ref{tbl:complexity}.
These methods were implemented using MATLAB with code provided by the authors and tested on an Intel Core(TM)i7 with a Windows 10 64-bit Operating System, 32-GByte RAM and a Titan GTX1080Ti GPU. The LF-SRCNN has the smallest time complexity. However, it registered the worst performance in terms of quality (see Tables \ref{tbl:SR_psnr_analysis_x3} and \ref{tbl:SR_psnr_analysis_x4}).
Our proposed method achieved the second lowest complexity which is clearly independent on the target magnification factor.
On the other hand, the complexity of GRAPH is orders of magnitudes larger than our method and its complexity increases exponentially with increasing magnification factors.
\begin{table}[ht]
\caption{Processing time of different light field super-resolution algorithms at different magnification factors.}
\label{tbl:complexity}
\begin{center}
\begin{tabular}{|l|c|c|c|}
\hline
\bf{Algorithm} & $\times2$ & $\times3$ & $\times4$ \\
\hline
BM-PCARR & 22 min. & 23 min. & 23 min.\\
LF-SRCNN & 33 sec. & 33 sec. & 33 sec. \\
GRAPH & 4 hrs. & 7 hrs.& 1 day \\
PB-VDSR & 9 min. & 9 min. & 9 min. \\
\hline
\end{tabular}
\end{center}
\end{table}
\section{Comments and Conclusion}
\label{sec:conclusion}
This paper has proposed a simple framework allowing to apply state-of-the-art SISR methods for light field super-resolution while preserving light field geometrical constraints.
The problem is decomposed into two sub--problems where we first align each view to the center view using optical flows and we then decompose the aligned light field using SVD.
Experimental results show that the \textit{principal basis} captures the coherent information in the light field and is a natural image that can be restored using state-of-the-art SISR methods.
We also demonstrate that the information restored in the \textit{principal basis} can be propagated in a consistent manner to all the other views.
Experimental results show that the use of the VDSR SISR technique in the proposed framework manages to restore light fields that are sharper and coherent across the angular views, compared to existing light field super-resolution methods.
Moreover, results in the supplementary material show that the restored light field is able to restore reflections on non-Lambertian surfaces.
The proposed framework can be extended to other light field image processing applications such as inpainting and recolouring where one can edit the \textit{principal basis} using state-of-the-art 2D image processing methods, and then propagate the restored information to all the other views as it was done here.
\bibliographystyle{IEEEtran}
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\section{Introduction}
\label{sec:intro}
Directional data are ubiquitous in oceanography with wave directions as an example, in meteorology where wind directions are directional data of interest, and in biology where protein backbone structures are directional data researchers study. These exemplify directional data of dimension no higher than three. Other examples of low dimensional direction data include migratory movements of animals, and measurements on a periodic scale, such as weekdays and hours. Directional data of higher dimensions arise in bioinformatics and hydrogeology, among many other fields of research. For example, gene expression data associated with a large number of genes for each experimental unit are often standardized to preserve directional characteristics when studying the fluctuation of gene expressions over cell cycles \citep{dortet2008model}. By transforming the original gene expression data on a high dimensional Euclidean space to a unit hypersphere, one ignores absolute expression levels and can obtain better clustering of genes that are functionally related \citep{banerjee2005clustering}. Another example of directional data with dimension usually higher than three is compositional data \citep{pawlowsky2006compositional, alenazi2021review}. For instance, microbiome data are often summarized as the composition of bacterial taxa so that one can focus on the microbial relative abundances as opposed to absolute abundances in microbiome analysis \citep{shi2016regression}. A compositional data point is a vector with non-negative components that sum to one, hence a component-wise square-root transformation of this vector yields a vector on a unit hypersphere \citep{scealy2011regression}.
Each of the above examples of directional data can be viewed as realizations of a random variable supported on a unit-radius $d$-dimensional spherical space defined by $\mathbb{S}^{d-1} = \lbrace \by\in \mathbb{R}^d : \|\by\| = 1 \rbrace$, for $d\geq 2$, where $\|\by\|$ is the $L_2$-norm of $\by$. \citet{lee2010circular} provided a brief survey of statistical methods for analyzing circular data, i.e., directional data with $d=2$. Two general strategies for constructing a circular distribution are highlighted in this review paper: one uses a ``wrapped" circular version of a random variable supported on $\mathbb{R}$ to formulate a circular distribution; the other deduces a circular distribution via projecting a univariate random variable on $\mathbb{R}$ or a bivariate random variable on $\mathbb{R}^2$ onto the circle. Both strategies have been generalized and used to formulate directional distributions on $\mathbb{S}^{d-1}$ for $d>2$. With the Gaussian distribution playing an important role in statistics, it is not surprising that directional distributions originating from a Gaussian distribution have been most studied and adopted in practice, including the so-called wrapped normal distribution and projected normal distribution, with more attention on the latter in recent literature. In particular, \citet{presnell1998projected} used a projected multivariate normal distribution to construct a regression model for a circular response and linear predictors, and employed the maximum likelihood method to infer unknown parameters. \citet{wang2013directional} incorporated projected normal distributions to develop Bayesian
hierarchical models for analyzing circular data.
\citet{hernandez2017general} proposed Bayesian inferential method for directional data of arbitrary dimension, again modelled by projected normal distributions.
Projected normal distributions are also referred to as angular Gaussian distributions. Different angular Gaussian distributions are created by imposing different constraints on the parameter space associated with a multivariate Gaussian distribution in order to resolve the non-identifiability issue that arises when the support of a random variable changes from a Euclidean space to a spherical space. \citet{paine2018elliptically} imposed constraints on the mean vector and variance-covariance matrix of a Gaussian distribution so that the resultant angular Gaussian distribution is identifiable and, more interestingly, elliptically symmetric. The authors thus coined their proposed distribution as the elliptically symmetric angular Gaussian distribution, ESAG for short. \citet{paine2020spherical} further developed regression models for directional data assuming an ESAG distribution for the response given covariates. Both works on ESAG focus on directional data with $d\le 3$. More recently, \citet{scealy2019scaled} proposed a new directional distribution, called scaled von Mises-Fisher distribution, using grouped transformations of the von Mises-Fisher distribution to achieve elliptical symmetry. The authors used this new distribution to model archeomagnetic data that can be converted to directional data with $d=3$. The feature of elliptical symmetry of a distribution makes capturing certain anisotropic pattern of directional data possible. An added benefit of ESAG is that the normalization constant in its probability density function is much easier to compute compared to many existing directional distributions, such as the Kent distribution \citep{kent1982fisher}. This makes maximum likelihood estimation under the ESAG model for directional data more straightforward.
To incorporate the constraints imposed on the mean vector and variance-covariance matrix of a Gaussian distribution when formulating ESAG, \citet{paine2018elliptically} designed a parameterization of ESAG when $d=3$, which allows one to bypass the complicated problem of optimization with constraints when finding the maximum likelihood estimators of the induced parameters. But their parameterization cannot be easily generalized to cases with $d>3$. This limits the use of ESAG in applications where directional data of higher dimension are observed. The first contribution of our study presented in this paper is a novel parameterization of ESAG that yields a mathematically sophisticated model for directional data of arbitrary dimension. This new parameterization of ESAG for $d\ge 3$ is presented in Section~\ref{sec:parameterization}.
Under the new parameterization, maximum likelihood estimation translates to a routine numerical problem of optimization without constraints, as we describe in Section~\ref{sec:mle}. A legitimate concern in any parametric modelling is potential violations of certain model assumptions in a given application. To address this concern, we propose model diagnostics methods that exploit directional residuals in Section~\ref{sec:verify}, which constitutes a second major contribution of our study. Operating characteristics of the proposed model diagnostics methods are demonstrated in simulation study in Section~\ref{sec:sim}. In Section~\ref{sec:realdata}, we entertain data from hydrogeological research, where we fit ESAG to compositional data from different geographic locations. Section~\ref{sec:conc} summarizes the contributions of the study and outlines the follow-up research agenda.
\section{The ESAG distribution}
\label{sec:parameterization}
\subsection{Constraints on parameters}
\label{sec:constraints}
Let $\bX$ be a $d$-dimensional Gaussian variable with mean $\bmu$ and variance-covariance $\bV$, i.e., $\bX\sim N_d (\bmu, \bV)$. Then the normalized variable, $\bY=\bX/\|\bX\|$, follows an angular Gaussian distribution, $\mbox{AG}(\bmu, \bV)$, supported on $\mathbb{S}^{d-1}$. Parameters in $\bmu$ and $\bV$ associated with $\mbox{AG}(\bmu, \bV)$ are not identifiable because $\bX/\|\bX\|$ and $c\bX/\|c\bX\|$ are equal for $c>0$, and thus they follow the same angular distribution, even though $\bX$ and $c\bX$ have different mean or/and variance-covariace when $c\ne 1$. To construct an identifiable angular Gaussian distribution, \citet{paine2018elliptically} impose the following two sets of constraints on $\bmu$ and $\bV$, where $\mbox{det}(\cdot)$ refers to the determinant of a matrix,
\begin{align}
\bV\bmu & = \bmu, \label{eq:constraint1}\\ \mbox{det}(\bV) & =1,
\label{eq:constraint2}
\end{align}
leading to the ESAG distribution, with the probability density function given by \begin{equation}
f(\by|\bmu,\bV)=\frac{(2\pi)^{-(d-1)/2}}{ (\by^{ \mathrm{\scriptscriptstyle T} } \bV^{-1} \by)^{d/2}}\exp \left[ \frac{1}{2}\left\{\frac{(\by^{ \mathrm{\scriptscriptstyle T} }\bmu)^2}{\by^{ \mathrm{\scriptscriptstyle T} } \bV^{-1}\by} -\bmu^{ \mathrm{\scriptscriptstyle T} } \bmu\right\}\right] M_{d-1}\left\{\frac{\by^{ \mathrm{\scriptscriptstyle T} }\bmu}{(\by^{ \mathrm{\scriptscriptstyle T} }\bV^{-1}\by)^{1/2}}\right\},
\label{eq:pdf}
\end{equation}
where $M_{d-1}(t)=(2\pi)^{-1/2}\int_{0}^{\infty}x^{d-1} \exp\{ -(x-t)^2/2 \} dx$. Henceforth, we say that $\bY$ follows a $(d-1)$-dimensional ESAG, or $\bY\sim \mbox{ESAG}_{d-1}(\bmu, \bV)$, if $\bY$ follows a distribution specified by the density in (\ref{eq:pdf}) with constraints in (\ref{eq:constraint1}) and (\ref{eq:constraint2}).
Figure~\ref{fig:ESAGdata} presents four random samples scattering on 3-dimensional spheres, generated from $\mbox{ESAG}_2(\bmu, \, \bV)$ with the following parameters specifications, where $\bone_d$ is a vector of $d$ ones and $\bI_d$ is $d$-dimensional identity matrix:
\vspace{0.5cm}
\noindent\begin{minipage}{.5\linewidth}
\begin{itemize}
\item[(a)]$\bmu\ =\ 2\times \bone_3, \, \ \ \bV\ =\ \bI_3$;
\item[(c)]$\bmu\ =\ 2\times \bone_3, \\ \bV = \begin{bmatrix}
1.57 & -0.08 & -0.50 \\
-0.08 & 0.74 & 0.34 \\
-0.50 & 0.34 & 1.16
\end{bmatrix}$;
\end{itemize}
\end{minipage}%
\begin{minipage}{.5\linewidth}
\begin{itemize}
\item[(b)]$\bmu\ =\ 4\times \bone_3, \,\ \ \bV\ =\ \bI_3$;
\item[(d)]$\bmu\ =\ 2\times \bone_3, \\ \bV =
\begin{bmatrix}
0.74 & -0.08 & 0.34 \\
-0.08 & 1.57 & -0.50 \\
0.34 & -0.50 & 1.16
\end{bmatrix}$.
\end{itemize}
\end{minipage}
\vspace{0.5cm}
\begin{figure}[h]
\centering
\subfigure[]{\includegraphics[width=1.5in]{1.jpg}}
\subfigure[]{\includegraphics[width=1.5in]{2.jpg}}\\
\subfigure[]{\includegraphics[width=1.5in]{3.jpg}}
\subfigure[]{\includegraphics[width=1.5in]{4.jpg}}
\caption{\label{fig:ESAGdata}Four random samples from $\mbox{ESAG}_2(\bmu, \, \bV)$ with $\bmu$ and $\bV$ specified by (a)--(d) in Section~\ref{sec:constraints}.}
\end{figure}
\noindent Comparing the four data clouds depicted in Figure~\ref{fig:ESAGdata}, one can see that a larger $\|\bmu\|$ leads to less variability in a random sample (e.g., contrasting (a) with (b)); and $\bV$ also influences the orientation of the data cloud (e.g., comparing (a), (c), and (d)).
Because the dimension of the parameter space associated with $N_d(\bmu, \bV)$ is $d(d+3)/2$, and there are $d+1$ constraints imposed by (\ref{eq:constraint1}) and (\ref{eq:constraint2}), there are at most $p=(d-1)(d+2)/2$ identifiable parameters for $\mbox{ESAG}_{d-1}(\bmu, \bV)$. Let $\bOmega$ be the $p\times 1$ parameter vector that specifies $\mbox{ESAG}_{d-1}(\bmu, \bV)$. To facilitate likelihood-based inference, it is desirable to formulate $\bOmega$ so that the parameter space is $\mathbb{R}^p$. For this purpose, we define $\bOmega=(\bmu^{ \mathrm{\scriptscriptstyle T} }, \bgamma^{ \mathrm{\scriptscriptstyle T} })^{ \mathrm{\scriptscriptstyle T} }$, where, clearly, $\bmu\in \mathbb{R}^d$, and thus $\bgamma\in \mathbb{R}^{(d-2)(d+1)/2}$ includes parameters needed to specify $\bV$ that satisfies (\ref{eq:constraint1}) and (\ref{eq:constraint2}) after $\bmu$ is given.
The parameterization leading to $\bgamma$ starts from the spectral decomposition of $\bV$,
\begin{align}
\bV = \sum_{j=1}^{d} \lambda_j \bv_j\bv_j^{ \mathrm{\scriptscriptstyle T} }, \label{eq:Vdecom}
\end{align}
where $\lambda_1,...,\lambda_d\in (0,\, +\infty)\triangleq\mathbb{R}_+$ are eigenvalues of $\bV$, and $\bv_1,...,\bv_d$ are the corresponding orthonormal eigenvectors. According to (\ref{eq:constraint1}), one of the eigenvalues of $\bV$ is equal to 1, with $\bmu$ being the corresponding (non-zero) eigenvector. Without loss of generality, we set $\lambda_d = 1$ and $\bv_d = \bmu/\|\bmu\|$. It follows that $\lambda_1 = 1/\prod_{j=2}^{d-1}\lambda_j$ since $\mbox{det}(\bV)=\prod_{j=1}^d \lambda_j=1$ by (\ref{eq:constraint2}). To this end, once $\bmu$ is given, one needs to formulate $\bgamma$ so that it can be mapped to $ \lambda_2,...,\lambda_{d-1}$ and $\bv_1,...,\bv_{d-1}$, through which $\bV$ is determined via (\ref{eq:Vdecom}).
In what follows, we present the derivations leading to such mapping in two steps.
\subsection{Constructing eigenvectors}
\label{sec:basis}
We first define an orthonormal basis of $\mathbb{R}^d$ as a function of $\bmu=(\mu_1, \ldots, \mu_d)^{ \mathrm{\scriptscriptstyle T} }$, denoted by $(\tilde \bv_1, \ldots, \tilde \bv_d)$, with $\tilde \bv_j=\bu_j/\|\bu_j\|$, for $j=1, \ldots, d$, and
\begin{equation}
\label{eq:vtilde}
\begin{aligned}
\bu_j & =
\left\{
\begin{array}{ll}
(-\mu_2,\, \mu_1,\,0,...,0)^{ \mathrm{\scriptscriptstyle T} }, & \mbox{for $j=1$,}\\
(\mu_1\mu_{j+1},\, ...,\, \mu_j\mu_{j+1},\,- \sum_{k=1}^{j} \mu_k^2, \, 0, \ldots, 0)^{ \mathrm{\scriptscriptstyle T} }, & \mbox{for $j=2,...,d-1$,} \\
\bmu & \mbox{for $j=d$}.
\end{array}
\right.
\end{aligned}
\end{equation}
If (\ref{eq:vtilde}) yields $\bu_j=\bzero_d$, for $j\in \{1, \ldots, d-1\}$, then we set $\bu_j=\be_j$, i.e., the unit vector with 1 at the $j$-th entry.
By (\ref{eq:vtilde}), $\tilde \bv_d=\bv_d$. We next relate $\{\tilde \bv_j\}_{j=1}^{d-1}$ to $\{\bv_j\}_{j=1}^{d-1}$ via a $(d-1)$-dimensional rotation matrix $\mathcal{R}_{d-1}$ formulated following the strategy proposed by \citet{murnaghan1952element}, which \citet{scealy2011regression} exploited to parameterize the Kent distribution for modeling compositional data. Following this strategy, for $d>3$, we write $\mathcal{R}_{d-1}$ as a product of $(d-2)(d-1)/2$ plane rotation matrices that are functions of longitude angles, $ \theta_1,..,\theta_{d-2}\in [-\pi , \pi )$, and latitude angles, $\phi_1,..,\phi_{(d-2)(d-3)/2}\in [0,\pi]$. Here, a $(d-1)$-dimensional plane rotation matrix $R_{jk}^*(\cdot)$ comes from replacing the $(j,j)$, $(j,k)$, $(k,j)$, and $(k,k)$ entries of $\bI_{d-1}$ by $\cos(\cdot)$, $-\sin(\cdot)$, $\sin(\cdot)$, and $\cos(\cdot)$, respectively. More specifically, we define $(\bv_1,...,\bv_{d-1})=(\tilde{\bv}_1,...,\tilde{\bv}_{d-1})\mathcal{R}_{d-1}$, where
\begin{equation}
\mathcal{R}_{d-1} =\left[\prod_{m=1}^{d-3}\left\{ R_{12}^*(\theta_{d-m-1})\prod_{j=1}^{d-m-2}R_{j+1,j+2}^*(\phi_{1-j+(d-m-1)(d-m-2)/2}) \right\} \right]R_{12}^*(\theta_1).
\label{eq:rotation}
\end{equation}
The rotation matrix in (\ref{eq:rotation}) depends on $(d-2)(d-1)/2$ angles that we refer to as orientation parameters in the sequel. Putting the orientation parameters along with the eigenvaules, we have the collection of parameters needed to specify $\bV$ after $\bmu$ is given in $(\lambda_2, \ldots, \lambda_{d-1}, \, \theta_1, \ldots, \theta_{d-2}, \, \phi_1, \ldots, \phi_{(d-2)(d-3)/2})$. We next turn to defining $\bgamma \in \mathbb{R}^{(d-2)(d+1)/2}$ that can be mapped to this collection of parameters via groups of transformations.
\subsection{Grouped spherical transformations}
\label{sec:grouping}
Following setting $\lambda_d=1$, we now let $\lambda_1 \le ...\le \lambda_{d-1}$, and write $\lambda_j=(r_{j-1}+1)\lambda_{j-1}$, where $r_{j-1}\ge 0$, for $j=2, \ldots, d-1$. Because $\prod_{j=1}^{d-1} \lambda_j=1$ by (\ref{eq:constraint2}), the first $d-1$ eigenvalues can be expressed in terms of $r_1, \ldots, r_{d-2}$ as follows,
\begin{equation}
\lambda_1 = \left\{ \prod_{j=1}^{d-2} (r_j+1)^{d-(j+1)}\right\}^{-1/(d-1)} \textrm{ and }
\lambda_j = \lambda_1\prod_{k=1}^{j-1}(r_k+1), \mbox{ for $j=2,...,d-1$.}
\label{eq:lambdas}
\end{equation}
We call $r_1, \ldots, r_{d-2}$ radial parameters for a reason to become clear momentarily. In what follows, we define transformations mapping $\bgamma$ to radial and orientation parameters in $\widetilde\bOmega=(r_1, \ldots, r_{d-2}, \, \theta_1, \ldots, \theta_{d-2}, \, \phi_1, \ldots, \phi_{(d-2)(d-3)/2})^{ \mathrm{\scriptscriptstyle T} }$ after partitioning these parameters into $d-2$ groups motivated by the following observations.
As the dimension of $\bY$ increases from $k$ to $k+1$, where $k \ge 3$, we need one additional radial parameter to account for the additional eigenvalue of $\bV$, along with, by (\ref{eq:rotation}), one additional longitude angle and $k-2$ additional latitude angles, yielding a total of $k$ additional parameters needed to specify $\bV$ when one increases the dimension of $\bY$ by one from $k$. This collection of additional parameters can be viewed as the parameters needed to specify a $(k-1)$-sphere under a spherical coordinate system \citep{moon1988spherical}, in terms of both parameter counts and parameter interpretations. A spherical coordinate system for characterizing $(k-1)$-spheres of arbitrary radius consists of one radial coordinate ranging over $[0, \, +\infty)$, where a radial parameter falls, one angular coordinate ranging over $[-\pi, \, \pi)$, which a longitude angle is within, and another $k-2$ angular coordinates, each ranging over $[0, \, \pi]$, which a latitude angle belongs to. These $k$ radial and orientation parameters can then link to $k$ parameters in $\mathbb{R}^k$ using the connection between the spherical coordinate system in the $(k-1)$-dimensional spherical space and the Cartesian coordinate system in the $k$-dimensional Euclidean space \citep{blumenson1960derivation}. This is the connection that relates $\bgamma$ to $\widetilde\bOmega$ after partitioning $\widetilde\bOmega$ in a way that we demonstrate in a concrete example next.
Suppose that $\bY\sim \mbox{ESAG}_4(\bmu, \, \bV)$ and thus $d=5$. After $\bmu$ is specified, we need radial and orientation parameters in $\widetilde\bOmega=(r_1, r_2, r_3, \, \theta_1, \theta_2, \theta_3, \, \phi_1, \phi_2, \phi_3)^{ \mathrm{\scriptscriptstyle T} }$ to specify $\bV$. We divide $\widetilde\bOmega$ into $3(=d-2)$ groups of parameters as follows:
\begin{itemize}
\item $(r_1, \, \theta_1)$, which are the only radial and orientation parameters needed to specify an ESAG resulting from normalizing a bivariate Gaussian random variable, i.e., $k=2$;
\item $(r_2, \, \theta_2, \, \phi_1)$, which includes the three additional radial and orientation parameters as we move from a 2-dimensional ESAG to a 3-dimensional ESAG, i.e., the dimension of the random variable changes from $k=3$ to $k+1=4$;
\item $(r_3, \, \theta_3, \, \phi_2, \, \phi_3)$, which contains the four additional radial and orientation parameters as the dimension of the ESAG random variable increases from $k=4$ to $k+1=5$.
\end{itemize}
In general, for $d\ge 3$, we divide $(d-2)(d+1)/2$ parameters in $\widetilde\bOmega$ into $d-2$ groups, with the first group being $(r_1, \,\theta_1)$, and, for $j=2,...,d-2$, the $j$-th group being $(r_j, \, \theta_j, \, \tilde \bphi_j)$, where
$\tilde\bphi_j = (\tilde\phi_{j,1},...,\tilde\phi_{j,j-1})^{ \mathrm{\scriptscriptstyle T} }$. In other words, $\tilde \phi_{j,k}$, for $j=2, \ldots, d-2$ and $k=1, \ldots, j-1$, are the original latitude angles assigned to the $j$-th group. We then formulate each group of parameters in $\widetilde\bOmega$ using a group of new parameters in the corresponding Euclidean space by invoking the connection between a spherical coordinate system and the corresponding Cartesian coordinate system \citep{blumenson1960derivation}.
To adapt to the grouping for $\widetilde\bOmega$, we also define $\bgamma$ as $d-2$ groups of parameters, $\bgamma = (\tilde\bgamma_1^{ \mathrm{\scriptscriptstyle T} },...,\tilde\bgamma_{d-2}^{ \mathrm{\scriptscriptstyle T} })^{ \mathrm{\scriptscriptstyle T} }$, where
$\tilde\bgamma_j = (\gamma_{j,1},...,\gamma_{j,j+1})^{ \mathrm{\scriptscriptstyle T} }\in \mathbb{R}^{j+1}$, for $ j=1,...,d-2$.
Then the transformations that map $\bgamma$ to $\widetilde\bOmega$ are given by $
r_1 = \|\tilde \bgamma_1\|, \, \,
\theta_1 = \mbox{atan2}(\gamma_{1,2},\gamma_{1,1}),
$ and, for $j=2, \ldots, d-2$, $r_j = \|\tilde\bgamma_{j}\|$,
\begin{equation*}
\begin{aligned}
\theta_j & = \begin{cases}
0, & \text {if $\gamma_{j,j}^2+\gamma_{j,j+1}^2 = 0$,}\\
\displaystyle{\mbox{arccos}\frac{\gamma_{j,j}}{\sqrt{\gamma_{j,j}^2+\gamma_{j,j+1}^2}}}, & \text{if $\gamma_{j,j+1} \ge 0$ and $\gamma_{j,j}^2+\gamma_{j,j+1}^2 \neq 0$,}\\
\displaystyle{-\mbox{arccos}\frac{\gamma_{j,j}}{\sqrt{\gamma_{j,j}^2+\gamma_{j,j+1}^2}}}, & \text{if $\gamma_{i,i+1} < 0$}, \end{cases}\\
\tilde{\phi}_{j,k} & = \begin{cases}
0, & \text{if $\sum_{\ell=k}^{j+1} \gamma_{j,\ell}^2 = 0$, for $k=1, ..., j-1$ },\\
\displaystyle{\mbox{arccos} \frac{\gamma_{j,k} }{\sqrt{\sum_{\ell=k}^{j+1} \gamma_{j,\ell}^2 }}}, & \text{otherwise, for $k=1, ..., j-1$.}
\end{cases}
\end{aligned}
\end{equation*}
This completes the parameterization of $\mbox{ESAG}_{d-1}(\bmu, \, \bV)$ so that all identifiable parameters in $\bOmega=(\bmu^{ \mathrm{\scriptscriptstyle T} }, \, \bgamma^{ \mathrm{\scriptscriptstyle T} })^{ \mathrm{\scriptscriptstyle T} }$ range over the entire real line. Having the parameter space being $\mathbb{R}^p$ greatly simplifies the implementation of maximum likelihood estimation for $\bOmega$.
\section{Maximum likelihood estimation}
\label{sec:mle}
Using the parameterization of ESAG developed in Section~\ref{sec:parameterization}, one can easily derive the likelihood function of a sample from ESAG, following which one can maximize the logarithm of it with respect to $\bOmega$ over $\mathbb{R}^p$ to obtain the maximum likelihood estimator (MLE) of $\bOmega$. Straightforward as it appears, some cautions should be given in this likelihood-based inference procedure, in part due to the nature of $\bgamma$.
\subsection{Interpretations of parameters}
\label{sec:interp}
According to Section~\ref{sec:grouping}, $\bgamma=\bzero$ implies $\lambda_j=1$, for $j=1,...,d$, and thus $\bV=\bI_d$, leading to an isotropic hyperspherical distribution \citep{mardia2014statistics}. If $\bY \sim \mbox{ESAG}_{d-1}(\bmu,\, \bI_d)$, then, for any orthogonal matrix $\bP$ such that $\bP\bmu=\bmu$, we have $\bP\bY\sim \mbox{ESAG}_{d-1}(\bmu,\, \bI_d)$, i.e., $\bP\bY=\bY$ in distribution, or, $\bP\bY\stackrel{\mathcal{L}} {=}\bY$ in short. In addition, if $\tilde \bgamma_j=\bzero$, then $r_j=0$, and thus $\lambda_{j+1}=(r_j+1)\lambda_j=\lambda_j$, in which case we say that the distribution is isotropic in the subspace spanned by $\lbrace \bv_j,\bv_{j+1}\rbrace$, or partially isotropic. That is, given any orthogonal matrix $\bP$ such that $\bP\bmu =\bmu$ and $\bP\bv_k=\bv_k$, for $k\ne j, j+1$, we have $\bP\bY\stackrel{\mathcal{L}}{=}\bY$. Practically speaking, this means that rotating data from an isotropic (a partially isotropic) ESAG via certain orthogonal matrix that rotates the mean direction to itself (and rotates certain eigenvectors of $\bV$ to themselves) does not change the distribution of the data. From the modelling point of view, any level of isotropy of ESAG implies a reduced model. Hence, testing whether or not a data set can be modelled by a reduced, thus more parsimonious, ESAG amounts to testing hypotheses regarding parameters in $\bgamma$. For example, testing $\bV=\bI_d$ is equivalent to testing $\bgamma=\bzero$.
A note of caution one should bear in mind when obtaining the MLE of $\bOmega$ is that,
even though the mapping from $\bgamma$ to $(\lambda_2,\, \ldots, \, \lambda_{d-1}, \, \bv_1^{ \mathrm{\scriptscriptstyle T} }, \, \ldots, \, \bv_{d-1}^{ \mathrm{\scriptscriptstyle T} })$ is bijective, the mapping from the latter to the former is not a bijection because, as one can see in (\ref{eq:Vdecom}), if $\bv_j$ is an eigenvector of $\bV$ corresponding to the eigenvalue $\lambda_j$, then so is $-\bv_j$. This suggests that there exist $\bgamma\ne \bgamma'$ yet both $\bgamma$ and $\bgamma'$ map to the same $\bV$ given $\bmu$. When this happens, we say that $\bgamma$ and $\bgamma'$ are equivalent. We show in Appendix A that, if $\bgamma$ and $\bgamma'$ are equivalent, then $\|\tilde\bgamma_j\| = \|\tilde\bgamma_j'\|$, for $j=1,...,d-2$, which in turn suggests that the interpretations of $\bgamma$ and $\bgamma'$ relevant to isotropy of ESAG are the same.
A theoretical implication of the existence of equivalent $\bgamma$ and $\bgamma'$ is that, although one cannot claim consistency of the MLE of $\bgamma$ (since the MLE may consistently estimate $\bgamma$ or $\bgamma'$), the consistency of the MLE of $\bV$ is guaranteed by the invariance property of MLE \citep[Theorem 7.2.10,][]{casella2021statistical}. A numerical implication of this is that maximum likelihood estimation of $\bOmega$ tends to be very forgiving in terms of the starting value for $\bOmega$, especially when the focal point of inference lies in $\bmu$ and $\bV$. We provide empirical evidence of these implications in a simulation experiment next.
\subsection{Empirical evidence}
\label{eq:mlesim}
Using the proposed parameterization, we generate a random sample of size $n=1000$ from $\mbox{ESAG}_3(\bmu, \bV)$, where $\bmu=(2,\,-2,\,-1,\,-3)^{ \mathrm{\scriptscriptstyle T} }$, and $\bV$ is determined via $\bmu$ and $\bgamma=(\gamma_{1,1}, \, \gamma_{1,2}, \, \gamma_{2,1}, \, \gamma_{2,2}, \, \gamma_{2,3})^{ \mathrm{\scriptscriptstyle T} }=(-2,\,5,\, 3, \,5, \,-8)^{ \mathrm{\scriptscriptstyle T} }$.
We then maximize the log-likelihood function of this random sample to find the MLE of $\bOmega$, denoted by $\hat \bOmega$, using two different starting values of $\bOmega$: one coincides with the truth, the other is given by $\bmu_0 = \bone_4$ and $\bgamma_0 = \bzero$. This produces two estimates of $\bOmega$. We repeat this experiment 100 times. In all 100 Monte Carlo replicates, we employ the Broyden-Fletcher-Goldfarb-Shanno algorithm \citep{fletcher2013practical} to find a maximizer of the log-likelihood function. In fact, we find that most commonly used optimization algorithms work well in maximizing the objective function despite the choice of starting values, partly thanks to the fact that transformations involved in the parameterization derivations in Section~\ref{sec:parameterization} are mostly smooth and simple enough.
Figure~\ref{fig:mle} presents graphical summaries of 100 realizations of a subset of $\hat\bOmega=(\hat \bmu^{ \mathrm{\scriptscriptstyle T} }, \hat \bgamma^{ \mathrm{\scriptscriptstyle T} })^{ \mathrm{\scriptscriptstyle T} }$, $(\hat \mu_2, \, \hat \gamma_{1,1}, \, \hat \gamma_{2,1})$, corresponding to each choice of starting value. In particular, for each parameter, a kernel density estimate based on 100 realizations of its MLE is depicted in Figure~\ref{fig:mle}. The top panels of Figure~\ref{fig:mle}, which present results from using the truth of $\bOmega$ to start the optimization algorithm, provide empirical evidence suggesting that the usual asymptotic properties of an MLE, including consistency and asymptotic normality, are expected to hold for $\hat\bOmega$ when one uses a starting value in a neighborhood of the truth. The bottom panels of Figure~\ref{fig:mle}, which show results from using a starting value that has little resemblance with the truth, indicate that $\hat \bmu$ still behaves like a regular MLE that is consistent and asymptotically normally distributed, but $\hat \bgamma$ appears to follow a bimodal distribution. The two modes of the distribution of $\hat \bgamma$ are expected to be the true value of $\bgamma$ and another value $\bgamma'$ that is equivalent to $\bgamma$.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=4.8in]{MLE_density.png}
\end{center}
\caption{\label{fig:mle}Estimated distributions of estimators for selected parameters in $\bOmega$ based on 100 realizations of each parameter estimator when the true parameter values are used as the starting value (upper panels) and when $\bmu_0$ and $\bgamma_0$ not equal to the truth are used as starting values (lower panels) in search for a maximizer of the log-likelihood. Vertical lines mark the true values of the corresponding parameters.}
\end{figure}
Despite the potential bimodality of $\hat \bgamma$ when a less carefully chosen starting value of $\bOmega$ is used to find $\hat \bOmega$, the resultant estimate of $\bV$, $\hat \bV$, is similar, if not identical, to the estimate one obtains when using the truth as the starting value. Figure~\ref{fig:V_consistency} shows boxplots of the Frobenius norm of $\bV-\hat \bV$ corresponding to 100 realizations of $\hat \bV$ resulting from each choice of the starting value. From there one can see that $\hat \bV$ is virtually unaffected by the choice of starting values. Although the robustness of $\hat \bmu$ and $\hat \bV$ to the choice of starting value is reassuring, one should not treat $\hat \bgamma$ as a conventional MLE due to its behavior observed in Figure~\ref{fig:mle}. Consequently, the usual Fisher information matrix or the sandwich variance does not serve well for estimating the variance of $\hat\bOmega$. We thus recommend use of bootstrap for the uncertainty assessment of $\hat \bOmega$.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=4in]{V_consistency.pdf}
\end{center}
\caption{\label{fig:V_consistency}Boxplots of the Frobenius norm of $\bV-\hat \bV$ as sample size $n$ varies when the true parameter values are used as the starting value (in the left panel) and when $\bmu_0$ and $\bgamma_0$ not equal to the truth are used as starting values (in the right panel) in search for a maximizer of the log-likelihood.}
\end{figure}
\section{Model diagnostics}
\label{sec:verify}
Even though the ESAG family accommodates certain anisotropic feature of a distribution and thus offers some flexibility in modelling, it remains fully parametric and thus is subject to model misspecification in a given application. In this section, we develop residual-based model diagnostics tools that data analysts can use to assess whether or not an ESAG distribution provides adequate fit for their directional data, either as a marginal distribution, or a conditional distribution of the directional response given covariates $\bW$ as in a regression setting.
\subsection{Residuals}
\label{sec:resid}
Denote by $\{\bY_i\}_{i=1}^n$ the observed directional data of size $n$, where $\bY_1, \ldots, \bY_n$ are independent with $\bY_i\sim \mbox{ESAG}_{d-1}(\bmu_i, \, \bV_i)$, for $i=1, \ldots, n$. The subscript $i$ attached to the mean and variance-covariance can be dropped if one aims to assess the goodness of fit (GOF) for the observed data using an ESAG as the marginal distribution. Otherwise the subscript implies covariate-dependent model parameters in ESAG as in a regression model for $\bY$.
In a non-regression or regression setting, after one obtains the MLE of all unknown parameters in the model, one has the MLEs $\hat \bmu_i$ and $\hat \bV_i$, following which a prediction can be made by $\hat{\bY}_i = \hat{\bmu}_i/\|\hat{\bmu}_i\|$, for $i=1, \ldots, n$. Similar to a directional residual defined in \citet{jupp1988residuals}, we define residuals as
\begin{equation}
\hat \br_i = \left(\bI_d-\hat \bY_i \hat \bY_i^{ \mathrm{\scriptscriptstyle T} }\right)\bY_i, \mbox{ for $i=1, \ldots, n$}.\label{eq:ourresid}
\end{equation}
In (\ref{eq:ourresid}), $\hat{\bY}_i\hat{\bY}^{ \mathrm{\scriptscriptstyle T} }_i$ can be viewed as the projection onto the space spanned by $\hat{\bmu}_i$, and thus $\bI_d-\hat \bY_i \hat \bY_i^{ \mathrm{\scriptscriptstyle T} }$ is the projection onto the space orthogonal to the space spanned by $\hat{\bmu}_i$. Equivalently, by the orthogonality of eigenvectors of $\hat \bV_i$, $\bI_d-\hat \bY_i \hat \bY_i^{ \mathrm{\scriptscriptstyle T} }$ is the projection onto the space spanned by the $d-1$ eigenvectors of $\hat \bV_i$ that are orthogonal to $\hat{\bmu}_i$, denote by $\lbrace \hat{\bv}_{i,j} \rbrace_{j=1}^{d-1}$. Hence (\ref{eq:ourresid}) can be re-expressed as $\hat \br_i=\hat \bP_{-d} \hat \bP_{-d}^{ \mathrm{\scriptscriptstyle T} } \bY_i$, where $\hat{\bP}_{-d}=[\hat{\bv}_{i,1} \mid ...\mid \hat{\bv}_{i, d-1}]$, that is, $\hat{\bP}_{-d}$ is the $d\times (d-1)$ matrix with the $j$-th column being $\hat{\bv}_{i,j}$, for $j=1,\ldots, d-1$. The potential dependence $\hat{\bP}_{-d}$ on covariates via the subscript $i$ is suppressed for simplicity.
For model diagnostic purposes, we use the following quadratic form of residuals,
\begin{equation}
\hat Q_i = \hat{\br}_i^{ \mathrm{\scriptscriptstyle T} } \hat{\bV}_i^{-1} \hat{\br}_i, \mbox{ for $i=1, \ldots, n$}.
\label{eq:sqr}
\end{equation}
Note that $\hat \br_i=\hat \bP_{-d} \hat \bP_{-d}^{ \mathrm{\scriptscriptstyle T} } \bY_i$ converges to $\br_i=\bP_{-d} \bP_{-d}^{ \mathrm{\scriptscriptstyle T} } \bY_i$ in distribution, where $\bP_{-d}$ results from excluding the $d$-th column of the $d \times d$ matrix $\bP=[\bv_1 \mid ... \mid \bv_{d-1} \mid \bv_{d}]$, and $\bP_{-d}\bP_{-d}^{ \mathrm{\scriptscriptstyle T} } = \bI_d - \bmu_i\bmu_i^{ \mathrm{\scriptscriptstyle T} }/\|\bmu_i\|^2$. Additionally, $\hat{\bV}_i$ converges to $\bV_i$ in probability as $n\to \infty$. Thus, (\ref{eq:sqr}) converges to $ Q_i=\br_i^{ \mathrm{\scriptscriptstyle T} } \bV^{-1}_i \br_i$ in distribution as $n\to \infty$. In what follows, we investigate the distribution of $Q_i$ to gain insight on the asymptotic distribution of (\ref{eq:sqr}). The subscript $i$ as the data point index is suppressed in this investigation.
For $\bY \sim \mbox{ESAG}_{d-1}(\bmu, \, \bV)$, the random variable can be expressed as $\bY = \bX/\|\bX\| = (\bV^{1/2}\bZ+\bmu)/\|\bX\|$, where $\bZ \sim N_d(\bzero, \, \bI_d)$. Hence, $
\br = \bP_{-d}\bP_{-d}^{ \mathrm{\scriptscriptstyle T} }\bY = {\bP_{-d}} \bP_{-d}^{ \mathrm{\scriptscriptstyle T} }\bV^{1/2}\bZ/\|\bX\|$, following which
we show in Appendix B that
\begin{align}
Q = \br^{ \mathrm{\scriptscriptstyle T} } \bV^{-1} \br =\frac{\|\bU_{-d}\|^2}{\|\bX\|^2}, \label{eq:Q2}
\end{align}
where $\bU_{-d}$ results from replacing the $d$-th entry of $\bU=\bP^{ \mathrm{\scriptscriptstyle T} }\bZ$ with zero. Since $\bP$ is an orthogonal matrix, $\bU=\bP^{ \mathrm{\scriptscriptstyle T} }\bZ \sim N_d(\bzero, \, \bI_d)$, and thus $\|\bU_{-d}\|^2 \sim \chi_{d-1}^2 $.
Now we see that $Q$ relates to the quotient of norms of Gaussian vectors, the distribution of which was studied in \citet{miller1964distributions}, following which one can derive the distribution of $Q$ analytically. One then can see that $Q$ is not a pivotal quantity and its distribution is not of a form familiar or easy enough for direct use for model diagnosis. We next construct a transformation of $Q$ aiming at attaining an approximate pivotal quantity for the purpose of model diagnostics.
\subsection{Graphical model diagnostic}
\label{sec:graphdiag}
Diagnostics methods proposed by \citet{paine2020spherical} and \citet{scealy2019scaled} build upon the finding that, if $\bY=(Y_1, \ldots, Y_d)^{ \mathrm{\scriptscriptstyle T} } \sim \mbox{ESAG}_{d-1}(\bmu, \, \bV)$, then $\|\bmu\|(Y_1, \ldots, Y_{d-1})^{ \mathrm{\scriptscriptstyle T} }$ converges in distribution to $N_{d-1}(\bzero, \, \sum_{j=1}^{d-1} \lambda_j^{-1} \bv_j \bv_j^{ \mathrm{\scriptscriptstyle T} })$ as $\|\bmu\|\to \infty$ \citep{paine2018elliptically}. Following this finding, one also has that $T_0= \|\bmu\|^2 Q =(\|\bmu\|^2/\|\bX\|^2) \|\bU_{-d}\|^2$
converges in distribution to $\chi_{d-1}^2$ for ESAG, and thus is a pivot in limit as $\|\bmu\|\to \infty$ (instead of $n\to \infty$). One may thus assess adequacy of a posited ESAG model for a data set by checking if $\{\hat T_{0,i}\}_{i=1}^n=\{\|\hat \bmu_i\|^2\hat Q_i\}_{i=1}^n$ approximately come from $\chi^2_{d-1}$. As seen in Figure~\ref{fig:ESAGdata}, a larger $\|\bmu\|$ implies that the distribution has a higher concentration and thus less variability in data. This diagnostic strategy based on $T_0$ is thus intuitively well motivated since, with $\|\bmu\|$ large, $\|\bmu\|^2/\|\bX\|^2$ is expected to be close to one, making $T_0$ close to $\|\bU_{-d}\|^2\sim \chi_{d-1}^2$. However, empirical evidence from our extensive simulation study suggest that a practically unreasonably large $\|\bmu\|$ is needed to make $\chi_{d-1}^2$ a reasonably good approximation of the distribution of $T_0$. Consequently, this strategy based on $T_0$ is of little practical value since data observed in most applications can rarely have low enough variability to make this approximation satisfactory.
Motivated by the fact that $E(\|\bX\|^2)=\|\bmu\|^2+\sum_{j=1}^d \lambda_j$ \citep[Theorem 5.2.1,][]{rencher2008linear}, we propose the following random quantity for diagnostics purposes,
\begin{align}
T_1 & = \left(\|\bmu\|^2 + \sum_{j=1}^{d}\lambda_j\right) Q, \label{eq:newteststat2}
\end{align}
which follows $\chi^2_{d-1}$ approximately when $\|\bmu\|$ is large, with the approximation improves much faster than that for $T_0$ as $\|\bmu\|$ increases, and thus is more like a pivot than $T_0$ is. Figure~\ref{fig:T0T1T2} presents kernel density estimates of the distributions of $T_0$ and $T_1$ based on random samples of these random quantities, each of size 500, generated based on Monte Carlo replicates from $\mbox{ESAG}_3(\bmu, \, \bV)$. More specifically, we set $\|\bmu\|=4.24$, which is not large enough to make the $\chi^2$- approximation for $T_0$ satisfactory, and $\sum_{j=1}^{d}\lambda_j=11.1$. As one can see in this figure, the variability of $T_0$ is way too low to make $\chi^2_{d-1}$ approximate its distribution well, and $T_1$ greatly improves over $T_0$ in its proximity to $\chi^2_{d-1}$. In general, $T_1$ only requires a moderate $\|\bmu\|$ to make the $\chi^2$-approximation practically useful.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=3in]{Residual_density.pdf}
\end{center}
\caption{Kernel density estimates of $T_0$ (dashed line) and $T_1$ (dotted line) comparing with the density of $\chi^2_3$ (solid line). \label{fig:T0T1T2}}
\end{figure}
Following maximum likelihood estimation of all unknown parameters, one can exploit an empirical version of $T_1$, $\{\hat T_{1,i}\}_{i=1}^n$, where $\hat T_{1,i}=(\|\hat\bmu_i\|^2 + \sum_{j=1}^{d}\hat\lambda_{i,j}) \hat Q_i$, for $i=1, \ldots, n$, and check if $\{\hat T_{1,i}\}_{i=1}^n$ can be reasonably well modeled by $\chi_{d-1}^2$. It can be a graphical check via a quantile-quantile (QQ) plot, for example, to see if there exists any clear signal of this sample deviating from $\chi_{d-1}^2$. Such graphical check is easy to implement following parameter estimation, and can provide visual warning signs when ESAG is a grossly inadequate model for the observed data $\{\bY_i\}_{i=1}^n$. Certainly, in a given application, the quality of $\chi^2$-approximation for $T_1$ is unknown with its true distribution yet to be estimated. We next propose a bootstrap procedure to facilitate a quantitative test for model misspecification, which leads to another graphical diagnostic tool as a byproduct that does not rely on a $\chi^2$-approximation for $T_1$.
\subsection{Goodness of fit test}
\label{sec:GOF}
Consider testing the null hypothesis that $\bY$ follows an ESAG. Although $T_1$ defined in (\ref{eq:newteststat2}) approximately follows $\chi^2_{d-1}$ under the null hypothesis, a testing procedure based on $T_1$ that does not acknowledge its exact null distribution can lead to misleading conclusion, e.g., an inflated Type I error for the test. Instead of estimating the exact null distribution of $T_1$, we use a random sample of $T_1$ induced from an ESAG as a reference sample, and quantify the dissimilarity between this reference sample and the observed empirical version of $T_1$, $\{\hat T_{1,i}\}_{i=1}^n$. One may use a nonparametric test for testing if two data sets come from the same distribution, such as the Kolmogorov–Smirnov (KS) test \citep{chakravarti1967handbook} and the Cram\'er-von Mises test \citep{anderson1962distribution}, to compare $\{\hat T_{1,i}\}_{i=1}^n$ and the reference sample induced from an ESAG. We employ the KS test in all presented simulation study in this article. A smaller $p$-value from the test indicates a larger distance between the underlying distribution of $\{\hat T_{1,i}\}_{i=1}^n$ and that of the reference sample, with the latter approximately representing what one expects for $T_1$ under the null hypothesis. Here, the ultimate test statistic for testing the null hypothesis is a $p$-value from the KS test. Denote this test statistic as $\mbox{KS}_p$. Even when data are from an ESAG, it is analytically unclear what $\mbox{KS}_p$ should be because the ESAG from which the reference sample is induced is not exactly the true ESAG (as to be seen next). We thus use parametric bootstrap to estimate the null distribution of $\mbox{KS}_p$ to obtain an approximate $p$-value to compare with a preset nominal level, such as 0.05, according to which we conclude to reject or fail to reject the null at the chosen nominal level. The following presents a detailed algorithm for this hypothesis testing procedure.
\begin{algorithm}[h!]\label{alg:gof}
\footnotesize
\caption{\footnotesize Goodness-of-Fit Test Procedure }\label{alg:gof test}
\begin{algorithmic}[1]
\Procedure{Compare observed empirical version of $T_1$ with a reference sample}{}
\State Given data $\{\bY_i\}_{i=1}^n$ for a non-regression setting or $\lbrace(\bY_i,\bW_i)\rbrace_{i=1}^n$ for a regression setting, find the MLE $\hat{\bmu}_i$ and $\hat{\bgamma}_i$, for $i=1, \ldots, n$, assuming an ESAG model for $\bY_i$ or $\bY_i$ conditioning on $\bW_i$.
\State Compute $\hat \bV_i$ and $\lbrace \hat{\lambda}_{i,j}\rbrace_{j=1}^{d-1}$ based on $\hat{\bmu}_i$ and $\hat{\bgamma}_i$, for $i=1, \ldots, n$.
\State Compute $\hat T_{1,i}= (\|\hat{\bmu}_i\|^2+\sum_{j=1}^{d-1}\hat{\lambda}_{i,j})\hat Q_i$, for $i = 1 ,\ldots, n$.
\State Generate $\{\tilde\bY_i\}_{i=1}^n$, where $\tilde \bY_i \sim \mbox{ESAG}(\hat{\bmu}_i,\hat{\bV}_i)$, for $i = 1, ..., n$.
\State Compute $\tilde T_{1,i}= (\|\hat{\bmu}_i\|^2+\sum_{j=1}^{d-1}\hat{\lambda}_{i,j})\tilde Q_i$, where $\tilde Q_i =\tilde r_i ^{ \mathrm{\scriptscriptstyle T} } \hat \bV_i^{-1}\tilde r_i$ and $\tilde r_i=\hat \bP_{-d}\hat \bP_{-d}^{ \mathrm{\scriptscriptstyle T} } \tilde \bY_i$, for $i = 1 ,... , n $.
\State Use the KS test to test if $\lbrace \hat T_{1,i}\rbrace_{i=1}^n$ and $\lbrace \tilde T_{1,i}\rbrace_{i=1}^n$ arise from the same distribution. Denote by $\mbox{KS}_p$ the resultant $p$-value of the KS test.
\EndProcedure
\Procedure{Bootstrap procedure to estimate the null distribution of $\mbox{KS}_p$}{}
\State Set $B$ = number of bootstraps
\State Initiate $s = 0$
\For{$b$ in $1,...,B$}
\State Generate the $b$-th bootstrap sample $\{\bY_i^{(b)}\}_{i=1}^n$, where $\bY_i^{(b)}\sim\mbox{ESAG}(\hat{\bmu}_i, \, \hat{\bV}_i)$ for $i = 1, ..., n$.
\State Repeat steps 2--7 using data $\{\bY_i^{(b)}\}_{i=1}^n$ for a non-regression setting or $\lbrace(\bY_i^{(b)}, \bW_i)\rbrace_{i=1}^n$ for a regression setting. Denote the $p$-value of the KS test as $\mbox{KS}_p^{(b)}$.
\State \textbf{if} $\mbox{KS}_p^{(b)} < \mbox{KS}_p$ \textbf{then} $s = s + 1$
\EndFor
\State Define an estimated $p$-value for this GOF test as $s/B$. \EndProcedure
\end{algorithmic}
\end{algorithm}
Several remarks are in order for this algorithm. First, in Step 5, $\mbox{ESAG}(\hat \bmu_i, \, \hat\bV_i)$, from which we induce a data point $\tilde T_{1,i}$ in the reference sample $\{\tilde T_{1,i}\}_{i=1}^n$, can be viewed as the member of the ESAG family that is closest to the distribution that characterizes the true data generating process producing $\bY_i$, where the closeness between two distributions is quantified by the Kullback-Leibler divergence \citep{white1982maximum}. Hence, $\tilde \bY_i$ generated from $\mbox{ESAG}(\hat \bmu_i, \, \hat\bV_i)$ at this step is expected to resemble $\bY_i$ if the null hypothesis is true, with $\hat \bmu_i$ and $\hat \bV_i$ consistently estimating $\bmu_i$ and $\bV_i$, respectively. Second, in Step 6, $\tilde T_{1,i}$ is constructed in a way that closely mimics $T_1$ instead of $\hat T_{1,i}$. In particular, just like $T_1$ where all population parameters are used in its construction, such as $\bmu$, $\{\lambda_j\}_{j=1}^d$, as well as $\bV$ and $\bP_{-d}$ that $Q$ depends on, computing $\tilde T_{1,i}$ (following steps 2--5) requires no parameter estimation although it depends on $\hat \bmu_i$, $\{\hat\lambda_{i,j}\}_{j=1}^d$, $\hat\bV_i$ and $\hat\bP_{-d}$, which are viewed as population parameters associated with $\tilde \bY_i$. One may certainly construct in Step 6 a random quantity closely mimicking $\hat T_{1,i}$ instead, but that would involve another round of parameters estimation based on $\{\tilde \bY_i\}_{i=1}^n$ and thus is computationally unattractive. Third, we acknowledge that, even under the null hypothesis, $\mbox{ESAG}(\hat \bmu_i, \, \hat \bV_i)$ is not the true distribution of $\bY_i$, with MLEs in place of the true model parameters. Hence, even when the null hypothesis is true, $\{\hat T_{1,i} \}_{i=1}^n$ do not come from the same distribution as that of the reference sample $\{\tilde T_{1,i}\}_{i=1}^n$, but the two distributions are expected to be closer than when the null hypothesis is severely violated. The bootstrap procedure is designed to estimate the null distribution of the distance between these two distributions that is quantified by $\mbox{KS}_p$, with a smaller value of $\mbox{KS}_p$ indicating a larger distance and thus stronger evidence against the null. As to be seen in the upcoming simulation study, this bootstrap procedure is capable of approximating the null distribution of $\mbox{KS}_p$ well enough to yield an empirical size of the test matching closely with any given nominal level.
In the absence of model misspecification, the distribution of $\{\tilde T_{1,i}\}_{i=1}^n$ approximates the distribution of $T_1$, with the accuracy of the approximation depends less on $\|\bmu\|$ than the $\chi^2$-approximation does. Therefore, a more reliable graphical diagnostic device than the aforementioned QQ plot using $\chi^2_{d-1}$ as a reference distribution is a QQ plot based on $\{\hat T_{1,i}\}_{i=1}^n$ and $\{\tilde T_{1,i}\}_{i=1}^n$, as we demonstrate in the upcoming empirical study.
\section{Simulation study}
\label{sec:sim}
\subsection{Design of simulation}
To demonstrate operating characteristics of the diagnostics methods proposed in Section~\ref{sec:verify}, we apply them to data $\{\bY_i\}_{i=1}^n$ generated according to four data generating processes specified as follows:
\begin{itemize}
\item[(M1)]An ESAG model, $\mbox{ESAG}_3(\bmu, \bV)$, with $\bmu=(2, \, -2, \, 3, \,-3)^{ \mathrm{\scriptscriptstyle T} }$ and $\bV$ defined via $\bmu$ and $\bgamma=(2,\,3, \,5, \,8, \,2)^{ \mathrm{\scriptscriptstyle T} }$.
\item[(M2)]A mixture of ESAG and angular Cauchy, with a mixing proportion of $1-\alpha$ on $\mbox{ESAG}_3(\bmu, \bV)$ specified in (M1), where a random vector from an angular Cauchy is generated by normalizing a random vector from a multivariate Cauchy with mean $\bmu$. This creates a scenario where $(1-\alpha)\times 100\%$ of the data arise from EAG but the rest of the data deviate from ESAG, where $\alpha\in \{0.05, \, 0.1, \, 0.2\}$.
\item[(M3)]An angular Gaussian distribution, $\mbox{AG}(\bmu, \, \tilde\bV)$, where $\det(\tilde \bV)=\alpha \ne 1$, which creates a scenario where the constraint in (\ref{eq:constraint2}) is violated. More specifically, when formulating (M1), one has the eigenvalues $\{\lambda_j\}_{j=1}^{d-1}$ and the corresponding eigenvectors $\{\bv_j\}_{j=1}^{d-1}$ of $\bV$, besides $\lambda_d=1$ and $\bv_d=\bmu/\|\bmu\|$. Using these quantities from (M1), we define $\tilde \bV=\sum_{j=1}^d \tilde\lambda_j\bv_j\bv_j^{ \mathrm{\scriptscriptstyle T} }$, where $\tilde \lambda_j=\alpha^{1/(d-1)}\lambda_j$, for $j=1, \ldots, d-1$, and $\tilde \lambda_d=1$, with $\alpha\in \{0.05, 0.1, 5, 10\}$. Because $\tilde \bV\bmu=\bmu$, the constraint in (\ref{eq:constraint1}) for ESAG is satisfied for this angular Gaussian distribution.
\item[(M4)]Similar to (M3) but $\tilde \lambda_j=\alpha^{-1/(d-1)}\lambda_j$, for $j=1, \ldots, d-1$, and $\tilde \lambda_d=\alpha\in \{0.1, 0.5, 2.5, 5\}$. This leads to $\tilde \bV \bmu=\alpha \bmu$ and thus violates constraint (\ref{eq:constraint1}). Because now $\mbox{det}(\tilde\bV)=1$, the constraint in (\ref{eq:constraint2}) for ESAG is satisfied for this angular Gaussian distribution.
\end{itemize}
We generate random samples of size $n\in \{250, 500, 1000\}$ following each data generating process. The proportions of data sets across 300 Monte Carlo replicates for which the GOF test rejects the null hypothesis at various significance levels are recorded for each simulation setting. This rejection rate estimates the size of the test under (M1), and sheds light on how sensitive the proposed diagnostic methods are to various forms and severity of deviations from ESAG exhibited in (M2)--(M4). We set $B=200$ in the bootstrap algorithm.
\subsection{Simulation results}
Under (M1), Figure~\ref{fig:M1} shows the rejection rate versus the nominal level when the null hypothesis stating that $\bY\sim \mbox{ESAG}$ is true. This figure suggests that the null distribution of the test statistic $\mbox{KS}_p$ is approximated well enough over a wide range of nominal levels based on merely $B=200$ bootstrap samples, especially at the lower tail so that the size of the test is close to a low nominal level such as 0.05.
\begin{figure}[h]
\begin{center}
\includegraphics[width=3in]{Test_size.pdf}
\caption{\label{fig:M1}Rejection rates of the GOF test versus nominal levels under (M1) when $n=250$ (dashed line), 500 (dotted line), and 1000 (dash-dotted line). The solid line is the $45^\circ$ reference line.}
\end{center}
\end{figure}
\begin{table}
\caption{\label{table:M2toM5}Rejection rates of the GOF test under (M2)--(M4) at nominal level 0.05}
\centering
\begin{tabular}{*{14}{c}}
\hline
$n$ & \multicolumn{3}{c}{(M2)} && \multicolumn{4}{c}{(M3)} && \multicolumn{4}{c}{(M4)} \\
\hline
$\{\alpha\}$ & 0.05 & 0.1 & 0.2 && 0.05 & 0.1 & 5 & 10 && 0.1 & 0.5 & 2.5 & 5\\
\cline{2-4} \cline{6-9} \cline{11-14}
250 & 0.10 & 0.27 & 0.65 && 0.27 & 0.17 & 0.14 & 0.17 && 0.47 & 0.16 & 0.75 & 1.00\\
500 & 0.17 & 0.42 & 0.89 && 0.38 & 0.30 & 0.22 & 0.33 && 0.73 & 0.26 & 0.98 & 1.00 \\
1000 & 0.26 & 0.69 & 0.99 && 0.60 & 0.46 & 0.30 & 0.52 && 0.96 & 0.42 & 1.00 & 1.00 \\
\hline
\end{tabular}
\end{table}
Table~\ref{table:M2toM5} presents rejection rates of the GOF test at nominal level 0.05 under the remaining three data generating processes (M2)--(M4). Under (M2), when $\alpha\times 100\%$ of the observed data are not from ESAG, the power of the test steadily increases as $\alpha$ increases. A larger sample size also boosts the power of detecting violation of the null. Under (M3), when data are from $\mbox{AG}(\bmu, \, \tilde \bV)$ that does not satisfy constraint (\ref{eq:constraint2}) due to $\det(\tilde \bV)=\alpha(\ne 1)$, one can see from Table~\ref{table:M2toM5} that, depending on the severity of the violation of (\ref{eq:constraint2}) that is controlled by the deviation of $\alpha$ from 1, the proposed test has a moderate power to detect this particular violation of ESAG, with a higher power at a larger sample size. Under (M4), when data are from $\mbox{AG}(\bmu,\tilde{\bV})$ with constraint (\ref{eq:constraint1}) violated due to $\tilde\bV\bmu=\alpha\bmu$, one can see from Table~\ref{table:M2toM5} that, as $\alpha$ deviates from 1 from either directions, the proposed test possesses moderate to high power to detect violation of the null hypothesis, with the power increasing quickly as $n$ grows larger. This can also serve as evidence for that, between the two constraints of ESAG in (\ref{eq:constraint1}) and (\ref{eq:constraint2}), violating the first constraint leads to an angular Gaussian deviating from ESAG more.
Besides the quantitative GOF test that performs satisfactorily according to the above empirical evidence, one can also inspect the QQ plot based on $\{\hat T_{1,i}\}_{i=1}^n$ and the bootstrap sample $\{\tilde T_{1,i}\}_{i=1}^n$ to graphically check ESAG assumptions. Figure~\ref{fig:M1M2M3M4} shows a collection of such plots based on a randomly chosen Monte Carlo replicate from each of the four considered data generating processes. As evidenced in Figure~\ref{fig:M1M2M3M4}, violation of the ESAG assumptions as designed in (M2)--(M4) causes a QQ plot deviating from a straight-line pattern, a pattern more or less observed in the absence of model misspecification as in (M1). To create such QQ plots does not require the full $B$-round bootstrap procedure in the above algorithm, and provides a convenient graphical check on the goodness of fit.
\begin{figure}[h]
\begin{center}
\includegraphics[width=2.3in]{M1.pdf}
\includegraphics[width=2.3in]{M2_20percent.pdf}\\
\includegraphics[width=2.3in]{M3_0.05.pdf}
\includegraphics[width=2.3in]{M4_2.5.pdf}
\end{center}
\caption{ \label{fig:M1M2M3M4}QQ plots based on $\{\hat T_{1,i}\}_{i=1}^n$ and the bootstrap sample $\{\tilde T_{1,i}\}_{i=1}^n$ under (M1) (top-left panel), (M2) with $\alpha=0.2$ (top-right panel), (M3) with $\alpha=0.05$ (bottom-left panel), and (M4) with $\alpha=2.5$ (bottom-right panel), respectively. Solid lines are $45^\circ$ reference lines.}
\end{figure}
\section{Application to hydrochemical data}
\label{sec:realdata}
In this section, we analyze the hydrochemical data containing 14 molarities measured monthly at different stations along the Llobregat River and its tributaries in northeastern Spain between the summer of 1997 and the spring of 1999 \citep{otero2005relative}. The complete data are available in the \texttt{R} package, \texttt{compositions} \citep{van2008compositions}. For illustration purposes, we focus on the compositional data recording relative abundance of two major ions, $\mbox{K}^+$ and $\mbox{Na}^+$, and two minor ions, $\mbox{Ca}^{2+}$ and $\mbox{Mg}^{2+}$. Taking the square-root transformation of the compostional data gives directional data with $d=4$. The four considered ions are mostly from potash mine tailing, which is one of the major sources of anthropogenic pollution in the Llobregat Basin \citep{soler2002sulfur}.
We first assume that the composition of $(\mbox{K}^+, \mbox{ Na}^+, \mbox{ Ca}^{2+}, \mbox{ Mg}^{2+})$ in tributaries of Anoia, one of the two main tributaries of the Llobregat River, follows an ESAG distribution. Using 67 records collected from stations placed along tributaries of Anoia, we obtain the estimated mean and variance-covariance of the compositional vector given by $$\hat{\bmu}_{\hbox {\tiny A}} = \begin{bmatrix}
1.99 \\
5.74 \\
7.95 \\
4.59
\end{bmatrix},
\hspace{0.5cm}
\hat{\bV}_{\hbox {\tiny A}}= \begin{bmatrix}
0.93 & 1.15 & -0.76 & -0.09 \\
1.15 & 2.77 & -1.41 & -0.27 \\
-0.76 & -1.41 & 1.99 & 0.38 \\
-0.09 & -0.27 & 0.38 & 0.73
\end{bmatrix}.$$ The GOF test yields an estimated $p$-value of 0.66, suggesting that the estimated ESAG distribution may provide an adequate fit for the data. The QQ plot in Figure~\ref{fig:modelfit} (see the left panel) may indicate some disagreement in the upper tail when it comes to the distribution of $\hat T_1$ and its bootstrap counterpart induced from an ESAG distribution, but otherwise mostly resemble each other in distribution. Transforming the estimated mean $\hat{\bmu}_{\hbox {\tiny A}}$ back to the composition of four considered ions, we estimate the mean composition of $(\mbox{K}^+, \mbox{ Na}^+, \mbox{ Ca}^{2+}, \mbox{ Mg}^{2+})$ to be (0.03, 0.27, 0.52, 0.18).
We repeat the above exercise for another compositional data of size 43 collected from stations placed along tributaries of the lower Llobregat course, and find the estimated mean vector and variance-covariance matrix to be
$$\hat{\bmu}_{\hbox {\tiny L}} = \begin{bmatrix}
3.27 \\
8.56 \\
9.01 \\
5.78
\end{bmatrix},
\hspace{0.5cm}
\hat{\bV}_{\hbox {\tiny L}}=
\begin{bmatrix}
0.63 & 1.50 & -0.71 & -0.90 \\
1.50 & 5.36 & -2.66 & -3.17 \\
-0.71 & -2.66 & 2.43 & 2.10 \\
-0.90 & -3.17 & 2.10 & 2.91
\end{bmatrix}.$$
The estimated $p$-value from the GOF test is 0.55 in this case. This, along with the QQ plot in Figure~\ref{fig:modelfit} (see the middle panel), also implies that the inferred ESAG distribution fits the data reasonably well. According to the estimated mean direction $\hat{\bmu}_{\hbox {\tiny L}}$, the estimated the mean composition of $(\mbox{K}^+, \mbox{ Na}^+, \mbox{ Ca}^{2+}, \mbox{ Mg}^{2+})$ is (0.05, 0.37, 0.41, 0.17), which shares some similarity with the estimated mean composition associated with Anoia tributaries in that $\mbox{Ca}^+$ and $\mbox{Na}^+$ are the two dominating components among the four, and $\mbox{K}^+$ is the minority. The two estimated variance-covariance matrices, $\hat{\bV}_{\hbox {\tiny A}}$ and $\hat{\bV}_{\hbox {\tiny L}}$, also share some implications in common: the two major ions, $\mbox{K}^+$ and $\mbox{Na}^+$, are positively correlated, so are the two minor ions, $\mbox{Ca}^{2+}$ and $\mbox{Mg}^{2+}$; but a major ion is negatively correlated with a minor ion in composition.
Diagonal entries of $\hat{\bV}_{\hbox {\tiny A}}$ and $\hat{\bV}_{\hbox {\tiny L}}$ should not be interpreted or compared here in the same way as if data were not directional because the variability of ESAG($\bmu$, $\bV$) depends on both $\bmu$ and $\bV$. For the compositional vector as a whole, with $\|\hat \bmu_{\hbox {\tiny A}}\|\approx 11.00<\|\hat \bmu_{\hbox {\tiny L}}\|\approx 14.10$, we have data evidence suggesting that the compositional data from Anoia tributaries are less concentrated around its mean direction, and thus more variable, than those from tributaries of the lower Llobregat course. When zooming in on one component at a time in the compositional vector, one can compare variability between two ESAG distributions base on $\bV/\|\bmu\|^2$. For instance, even though $\hat \bV_{\hbox {\tiny A}}[3,3]=1.99<\hat \bV_{\hbox {\tiny L}}[3,3]=2.43$, we would not jump to the conclusion that the composition of $\mbox{Ca}^{2+}$ is less variable in Anoia tributaries than that in the other set of locations. Instead, because $\hat \bV_{\hbox {\tiny A}}[3,3]/\|\hat \bmu_{\hbox {\tiny A}}^2\|= 0.18>\hat\bV_{\hbox {\tiny L}}[3,3]/\|\hat \bmu_{\hbox {\tiny L}}^2\|= 0.17$, we conclude that the composition of $\mbox{Ca}^{2+}$ is similar in variability between the two sets of locations, but tributaries of Anoia may be subject to slightly higher variability in this regard. This conclusion is also consistent with the comparison of the sample standard deviation of the composition of $\mbox{Ca}^{2+}$ between the two data sets.
Moreover, estimates for the other set of parameters of ESAG arising in the new parameterization, $\bgamma$, also provide statistically interesting insights on the underlying distributions. Denote by $\hat\bgamma_{\hbox {\tiny A}}$ the estimate based on data from Anoia tributaries, and by $\hat\bgamma_{\hbox {\tiny L}}$ the estimate based on data from tributaries of the lower Llobregat course. We find that $\|\hat\bgamma_{\hbox {\tiny A}}\|=6.24 <\|\hat\bgamma_{\hbox {\tiny L}}\|=17.03$, indicating that neither of the two ESAG distributions is isotropic, with the second ESAG deviating from isotropy further. To check partial isotropy, we look into the estimated eigenvalues associated with $\hat \bV_{\hbox {\tiny A}}$ and $\hat \bV_{\hbox {\tiny L}}$. With one eigenvalue fixed at 1, the three estimated eigenvalues associated with $\hat \bV_{\hbox {\tiny A}}$ are 0.37 (0.05), 0.62 (0.10), and 4.44 (0.64), with the estimated standard errors in parentheses obtained based on 300 bootstrap data sets, each of the same size as the raw data sampled from the raw data with replacement. Similarly, we have the three estimated eigenvalues associated with $\hat \bV_{\hbox {\tiny L}}$ given by 0.19 (0.04), 0.54 (0.29), and 9.61 (1.84). Taking the estimated standard errors into consideration, with the large discrepancy between the estimated (and fixed) eigenvalues, neither of the two data sets provides sufficient evidence indicating partial isotropy.
Lastly, we fit the ESAG model to the 110 records that combine the above two data sets and obtain an estimated $p$-value of 0.02 from the GOF test, with the corresponding QQ plot clearly deviating from a straight line (see the right panel in Figure~\ref{fig:modelfit}). We thus conclude that an ESAG distribution is inadequate for modeling the data that mix compositional data from Anoia tributaries and those from tributaries of the lower Llobregat course. This lack of fit is not surprising because Anoia mostly passes through vineyards and industrialized zones, whereas the Llobergat lower course also flows through densely populated areas with high demands of water besides agricultural and industrial areas. This explains the vastly different patterns and sources of anthorpogenic and geological pollution between Anoia and the lower Llobregat course \citep{gonzalez2012presence}, which create substantial heterogeneity in the mixed compositional data that an ESAG model is unlikely to capture.
\begin{figure}[h]
\begin{center}
\includegraphics[width=1.56in]{At.pdf}
\includegraphics[width=1.56in]{LLt.pdf}
\includegraphics[width=1.56in]{At+LLt.pdf}
\end{center}
\caption{QQ plots from the GOF test applied to compositional data from tributaries of Anoia (left panel), those from tributaries of the lower Llobregat course (middle panel), and the data that combine the previous two data sets (right panel). \label{fig:modelfit}}
\end{figure}
\section{Discussion}
\label{sec:conc}
Given the wide range of applications where directional data are of scientific interest and typically of dimension higher than three, an important first step towards sound statistical analysis of such data is the formulation of a directional distribution of arbitrary dimension. We adopt the initial formulation of the ESAG distribution proposed by \citet{paine2018elliptically}, and take it to the next level via a sequence of reparameterizations leading to a distribution family indexed by parameters ranging over the entire real space. The resultant parametric family for directional data avoids pitfalls that many existing directional distributions suffer so that, unlike the Kent distribution for instance, there is no hard-to-compute normalization constant in the density function, and it is easy to simulate data from an ESAG of any dimension. More importantly, the proposed parameterization of ESAG lends itself to straightforward maximum likelihood inference procedures that are numerically stable and less dependent on ``good" starting values for parameter estimation. New parameters introduced along the way of reparameterization have statistically meaningful interpretations, which facilitate formulating hypothesis testing where one compares a reduced ESAG model, such as an isotropic or a partially isotropic model, with a saturated ESAG model. In summary, the proposed ESAG family of arbitrary dimension sets the stage for carrying out a full range of likelihood-based inference for directional data, including parameter estimation, uncertainty assessment, and hypothesis testing.
To ease the concerns of model misspecification when assuming a parametric family in a given application, we develop graphical and quantitative diagnostics methods that utilize directional residuals. Maximum likelihood estimation and the proposed diagnostics methods for ESAG can be easily implemented using the R code developed and maintained by the first author that is available upon request.
An immediate follow-up step is to consider regression models for directional data, which is well motivated by the lack of fit of a marginal ESAG distribution for the mixed compositional data entertained in Section~\ref{sec:realdata}. We conjecture that, conditioning on covariates relating to geological features of considered tributaries and covariates reflecting human activities developed in regions these tributaries running through, the mixed compositioinal data can be better modelled by an ESAG distribution with covariate-dependent $\bmu$ and $\bgamma$. With $\bmu$ and $\bgamma$ ranging over the entire real space of adequate dimensions, the proposed ESAG family prepares itself well for regression analysis of directional data without using complicated link functions to introduce dependence of model parameters on covariates $\bW$. For example, one may consider a fully parametric regression model as simple as $\bY|\bW\sim \mbox{ESAG}(\bmu(\bW), \, \bV(\bW))$, where $\bmu(\bW)$ is a linear function of covariates $\bW$, and $\bV(\bW)$ is determined by $\bmu(\bW)$ and $\bgamma(\bW)$, with the latter also a linear function of covariates. More flexible dependence structures of $\bmu$ and $\bgamma$ on covariates are also worthy of consideration in the follow-up research along the line of regression analysis. Once we enter the realm of regression models, the dimension of the parameter space grows more quickly as $d$ increases than before considering regression analysis for directional data. Upon completion of the study presented in this article, we have embarked on the exciting journey of developing scalable inference procedures suitable for settings with high dimensional parameter space following the strategies of frequentist penalized maximum likelihood estimation and Bayesian shrinkage estimation via hierarchical modeling.
\par
\section*{Appendix A: Implication of $\bgamma$ and $\bgamma'$ being equivalent}
Under the proposed parameterization of $\mbox{ESAG}_{d-1}(\bmu, \bV)$, $\bV$ is determined by $\bgamma$ after $\bmu$ is specified. We thus write $\bV$ as $\bV(\bgamma)$ in this appendix, and view quantities related to $\bV$ as functions of $\bgamma$, such as the eigenvalues of $\bV$ and the radial parameters in (\ref{eq:lambdas}). If $\bgamma$ and $\bgamma'$ are equivalent, then $\bV(\bgamma)=\bV(\bgamma')$, and thus $\bV(\bgamma)$ and $\bV(\bgamma')$ share the same eigenvalues. By (\ref{eq:lambdas}), $\{\lambda_j(\bgamma)=\lambda_j(\bgamma')\}_{j=1}^{d-1}$ implies that $\{r_j(\bgamma)=r_j(\bgamma')\}_{j=1}^{d-2}$ . Lastly, from Section~\ref{sec:grouping}, $r_j=\|\tilde \bgamma_j\|$, for $j=1, \ldots, d-2$. Therefore, if $\bgamma$ and $\bgamma'$ are equivalent, $\|\tilde \bgamma_j\|=r_j(\bgamma)=r_j(\bgamma')=\|\tilde \bgamma_j'\|$, for $j=1, \ldots, d-2$.
\setcounter{equation}{0}
\defB.\arabic{equation}{B.\arabic{equation}}
\section*{Appendix B: Proof of equation (\ref{eq:Q2})}
By the spectral decomposition theorem, $\bV^{\alpha}=\bP\bD^{\alpha}\bP^{ \mathrm{\scriptscriptstyle T} }$, where $\bD^{\alpha} = \mbox{diag}(\lambda_1^\alpha,...,\lambda_d^\alpha)$ and $\bP=[\bv_1 \mid ... \mid \bv_{d}]$. Using this decomposition with $\alpha=-1$ and 1/2, we have
\begin{align*}
Q & = \br^{ \mathrm{\scriptscriptstyle T} } \bV^{-1} \br \\
& = \frac{\bZ^{ \mathrm{\scriptscriptstyle T} }}{\|\bX\|} \bV^{1/2}\bP_{-d}\bP_{-d}^{ \mathrm{\scriptscriptstyle T} } \times \bV^{-1} \times \bP_{-d}\bP_{-d}^{ \mathrm{\scriptscriptstyle T} } \bV^{1/2}\frac{\bZ}{\|\bX\|} \\
& = \frac{\bZ^{ \mathrm{\scriptscriptstyle T} }}{\|\bX\|^2} \bP\bD^{1/2} \bP^{ \mathrm{\scriptscriptstyle T} }\bP_{-d}\bP_{-d}^{ \mathrm{\scriptscriptstyle T} } \times \bP\bD^{-1} \bP^{ \mathrm{\scriptscriptstyle T} } \times \bP_{-d}\bP_{-d}^{ \mathrm{\scriptscriptstyle T} } \bP\bD^{1/2} \bP^{ \mathrm{\scriptscriptstyle T} } \bZ,
\end{align*}
where $$
\bP^{ \mathrm{\scriptscriptstyle T} }\bP_{-d} =
\begin{bmatrix}
\bP_{-d}^{ \mathrm{\scriptscriptstyle T} } \\
\bv_d^{ \mathrm{\scriptscriptstyle T} }
\end{bmatrix}
\bP_{-d}=
\begin{bmatrix}
\bI_{d-1} \\
\bzero^{ \mathrm{\scriptscriptstyle T} }
\end{bmatrix},
$$
and thus
$$\bP^{ \mathrm{\scriptscriptstyle T} }\bP_{-d} \bP_{-d}^{ \mathrm{\scriptscriptstyle T} } \bP=
\begin{bmatrix}
\bI_{d-1} & \bzero \\
\bzero^{ \mathrm{\scriptscriptstyle T} } & 0
\end{bmatrix}\triangleq \tilde \bI_d.$$
It follows that
\begin{align*}
Q & =\frac{1}{\|\bX\|^2} \bZ^{ \mathrm{\scriptscriptstyle T} } \bP \bD^{1/2} \tilde \bI_d \bD^{-1} \tilde \bI_d \bD^{1/2} \bP^{ \mathrm{\scriptscriptstyle T} }\bZ \\
& =
\frac{1}{\|\bX\|^2} \bZ^{ \mathrm{\scriptscriptstyle T} }\bP \tilde \bI_d \bD^{1/2}\bD^{-1}\bD^{1/2} \tilde \bI_d \bP^{ \mathrm{\scriptscriptstyle T} } \bZ\\
& = \frac{1}{\|\bX\|^2} \bU^{ \mathrm{\scriptscriptstyle T} } \tilde \bI_d \tilde \bI_d\bU, \mbox{ where $\bU=\bP^{ \mathrm{\scriptscriptstyle T} }\bZ$,}\\
& = \frac{1}{\|\bX\|^2} \bU_{-d}^{ \mathrm{\scriptscriptstyle T} } \bU_{-d}, \mbox{ where $\bU_{-d}= \tilde \bI_d \bU$,}
\end{align*}
which gives (\ref{eq:Q2}).
\bibliographystyle{apalike}
|
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BkiUc9nxK7Ehm79fcgcE
| 5
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\section{Introduction}
There are many well known exact solutions in general relativity (see, i.e.\
\cite{KrStMaHe80}). To give physical
interpretation of any solution to Einstein's equation, we must know not only the
metric satisfying equations of general relativity but the global structure of
space-time. By this we mean a pair $({\mathbb M},g)$, where ${\mathbb M}$ is the
four-dimensional space-time manifold and $g$ is the metric on ${\mathbb M}$ such, that
manifold ${\mathbb M}$ is maximally extended along geodesics: any geodesic line on ${\mathbb M}$
either can be continued to infinite value of the canonical parameter in both
directions, or it ends up at a singular point, where one of the geometric
invariants becomes infinite. The famous example is the
Kruskal--Szekeres extension \cite{Kruska60,Szeker60} of the Schwarzschild
solutions. In this case, the space-time ${\mathbb M}$ is globally the topological
product of a sphere (spherical symmetry) with the two-dimensional Lorentzian
surface depicted by the well known Carter--Penrose diagram. The knowledge of
this global structure of space-time allows one to introduce the notion of black
and white holes.
The famous Reissner--Nordstr\"om solution \cite{Reissn16,Nordst18}, which is the
spherically symmetric solution of Einstein's equations with electromagnetic
field, is also known globally. There are three types of Carter--Penrose
diagrams: the Reissner-Nordstr\"om black hole, extremal black hole and naked
singularity. The type of the Carter--Penrose diagram depends on the relation
between mass and charge parameters. The spherically symmetric exact solution of
Einstein's equations with electromagnetic field and cosmological constant is
known locally but not analyzed in full detail globally. In this paper, in
particular, we give complete classification of global spherically symmetric
solutions of Einstein's equations with electromagnetic field and cosmological
constant, which depends on relations between three parameters: mass, charge, and
cosmological constant. We show that there are 16 different Carter--Penrose
diagrams in the spherically symmetric case.
In fact, more general classification is given. We do not assume that
solutions have any symmetry from the very beginning. Instead, we require the
space-time to be the warped product of two surfaces: ${\mathbb M}={\mathbb U}\times{\mathbb V}$, where
${\mathbb U}$ and ${\mathbb V}$ are two two-dimensional surfaces with Lorentzian and Euclidean
signature metrics, respectively. As the consequence of the equations of motion,
at least one of the surfaces must be of constant curvature. In this paper, we
consider the cases when (i) both surfaces ${\mathbb U}$ and ${\mathbb V}$ are of constant
curvature and when (ii) only surface ${\mathbb V}$ is of constant curvature. In the
latter case, there are three possibilities: ${\mathbb V}$ is the sphere ${\mathbb S}^2$ (the
spherical ${\mathbb S}{\mathbb O}(3)$ symmetry), the Euclidean plane (the Poincare
${\mathbb I}{\mathbb S}{\mathbb O}(2)$ symmetry), and the two-sheeted hyperboloid ${\mathbb H}^2$ (the
Lorentzian ${\mathbb S}{\mathbb O}(1,2)$ symmetry). We see that the symmetry of solutions is not
assumed from the beginning but arise as the consequence of the equations of
motions. This effect is called ``spontaneous symmetry emergence''. We classify
all global solutions by drawing their Carter--Penrose diagrams for surface
${\mathbb U}$ depending on relations between mass, charge, and cosmological constant.
Totally, there are 4 different Carter--Penrose diagrams in case (i) and 33
globally different solutions in case (ii).
Moreover, we prove that there is the additional forth Killing vector field in
each case. This is a generalization of Birkhoff's theorem stating that any
spherically symmetric solution of vacuum Einstein's equations must be static.
The existence of extra Killing vector field is proved for ${\mathbb S}{\mathbb O}(3)$,
${\mathbb I}{\mathbb S}{\mathbb O}(2)$, and ${\mathbb S}{\mathbb O}(1,2)$ symmetry groups.
This paper follows the classification of global warped product solutions of
general relativity with cosmological constant (without electromagnetic field)
given in \cite{KaKlKu99}. The Carter--Penrose diagrams are constructed using the
conformal block method described in \cite{Katana00A}.
As in \cite{KaKlKu99}, we assume that space-time ${\mathbb M}$ is the warped product
of two surfaces: ${\mathbb M}={\mathbb U}\times{\mathbb V}$, where ${\mathbb U}$ and ${\mathbb V}$ are surfaces with
Lorentzian and Euclidean signature metrics, respectively. Local coordinates on
${\mathbb M}$ are denoted by $x^i$, $i=0,1,2,3$, and coordinates on the surfaces by
Greek letters from the beginning and middle of the alphabet:
\begin{equation*}
(x^\alpha)\in{\mathbb U},\quad\alpha=0,1,\qquad(y^\mu)\in{\mathbb V},\quad\mu=2,3.
\end{equation*}
That is $(x^i):=(x^\alpha,y^\mu)$. Geometrical notions on four-dimensional
space-time are marked by the hat to distinguish them from notions on surfaces
${\mathbb U}$ and ${\mathbb V}$, which appear more often.
We do not assume any symmetry of solutions from the very beginning.
Four-dimensional metric of the warped product of two surfaces has block diagonal
form by definition:
\begin{equation} \label{egdtrf}
\widehat g_{ij}=\begin{pmatrix} k(y)g_{\alpha\beta}(x) & 0 \\ 0 & m(x)h_{\mu\nu}(y)
\end{pmatrix},
\end{equation}
where $g_{\alpha\beta}(x)$ and $h_{\mu\nu}(y)$ are some metrics on surfaces ${\mathbb U}$ and
${\mathbb V}$, respectively, $k(y)$ and $m(x)$ are scalar (dilaton) fields on
${\mathbb V}$ and ${\mathbb U}$.
The Ricci tensor components for metric (\ref{egdtrf}) are
\begin{equation} \label{eritab}
\begin{split}
\widehat R_{\alpha\beta}&=R_{\alpha\beta}+\frac{\nabla_\alpha\nabla_\beta m}m
-\frac{\nabla_\alpha m\nabla_\beta m}{2m^2}+\frac{g_{\alpha\beta}\nabla^2 k}{2m}
\\
\widehat R_{\alpha\mu}&=\widehat R_{\mu\alpha}=-\frac{\nabla_\alpha m\nabla_\mu k}{2mk}
\\
\widehat R_{\mu\nu}&=R_{\mu\nu}+\frac{\nabla_\mu\nabla_\nu k}k
-\frac{\nabla_\mu k\nabla_\nu k}{2k^2}+\frac{h_{\mu\nu}\nabla^2 m}{2k},
\end{split}
\end{equation}
where, for brevity, we introduce notation
\begin{equation} \label{edalan}
\nabla^2 m:=g^{\alpha\beta}\nabla_\alpha\nabla_\beta m,\qquad
\nabla^2 k:=h^{\mu\nu}\nabla_\mu\nabla_\nu k.
\end{equation}
Here and in what follows symbol $\nabla$ denotes covariant derivative with the
corresponding Christoffel's symbols. The four-dimensional scalar curvature is
\begin{equation} \label{escacu}
\widehat R=\frac1kR^g+2\frac{\nabla^2 m}{km}-\frac{(\nabla m)^2}{2km^2}
+\frac1mR^h+2\frac{\nabla^2 k}{km}-\frac{(\nabla k)^2}{2k^2m},
\end{equation}
where
\begin{equation} \label{egrsqn}
(\nabla m)^2:=g^{\alpha\beta}\partial_\alpha m\partial_\beta m,\qquad
(\nabla k)^2:=h^{\mu\nu}\partial_\mu k\partial_\nu k.
\end{equation}
Scalar curvatures of surfaces ${\mathbb U}$ and ${\mathbb V}$ are denoted by $R^g$ and $R^h$,
respectively.
\section{Solution for electromagnetic field}
We assume that electromagnetic field is minimally coupled to gravity. Then the
action takes the form
\begin{equation} \label{ubcftg}
S=\int\!dx\sqrt{|\widehat g|}\left(\widehat R-2\Lambda-\frac14\widehat F^2\right),
\end{equation}
where $\widehat R$ is the scalar curvature for metric $\widehat g_{ij}$,
$\widehat g:=\det\widehat g_{ij}$, $\Lambda$ is a cosmological constant, and
$\widehat F^2$ is the square of electromagnetic field strength:
\begin{equation*}
\widehat F^2:=\widehat F_{ij}\widehat F^{ij},\qquad
\widehat F_{ij}:=\partial_i \widehat A_j-\partial_j\widehat A_i.
\end{equation*}
Here, $\widehat A_i$ are components of electromagnetic field potential. For
brevity, gravitational and electromagnetic coupling constants are set to unity.
Variation of action (\ref{ubcftg}) with respect to metric yields
four-dimensional Einstein's equations:
\begin{equation} \label{ubsvgr}
\widehat R_{ij}-\frac12\widehat g_{ij}\widehat R+\widehat g_{ij}\Lambda
=-\frac12 \widehat T_{{\textsc{e}}{\textsc{m}} ij},
\end{equation}
where
\begin{equation} \label{ubnhgy}
\widehat T_{{\textsc{e}}{\textsc{m}} ij}:=-\widehat F_{ik}\widehat F_j{}^k
+\frac14\widehat g_{ij}\widehat F^2
\end{equation}
is the electromagnetic field energy-momentum tensor. Variation of
the action with respect to electromagnetic field yields Maxwell's equations:
\begin{equation} \label{uncmhy}
\partial_j\big(\sqrt{|\widehat g|}\widehat F^{ji}\big),
\end{equation}
where
\begin{equation*}
\widehat g=k^2m^2gh,\qquad g:=\det g_{\alpha\beta},\qquad h:=\det h_{\mu\nu}.
\end{equation*}
To simplify the problem, we assume that the four-dimensional electromagnetic
potential consists of two parts:
\begin{equation*}
\widehat A_i=\big(A_\alpha(x),A_\mu(y)\big),
\end{equation*}
where $A_\alpha(x)$ and $A_\mu(y)$ are two-dimensional electromagnetic potentials
on surfaces ${\mathbb U}$ and ${\mathbb V}$, respectively. Then the electromagnetic field
strength becomes block diagonal:
\begin{equation} \label{unfhgt}
\widehat F_{ij}=\begin{pmatrix}F_{\alpha\beta} & 0 \\ 0 & F_{\mu\nu} \end{pmatrix},
\end{equation}
where
\begin{equation*}
F_{\alpha\beta}(x):=\partial_\alpha A_\beta-\partial_\beta A_\alpha,\qquad
F_{\mu\nu}(y):=\partial_\mu A_\nu-\partial_\nu A_\mu
\end{equation*}
are strength components for two-dimensional electromagnetic potentials.
In what follows, the raising of Greek indices from the beginning and middle
of the Greek alphabet is performed by using the inverse metrics $g^{\alpha\beta}$ and
$h^{\mu\nu}$. Therefore
\begin{equation*}
\widehat F^{\alpha\beta}=\frac1{k^2}F^{\alpha\beta},\qquad
\widehat F^{\mu\nu}=\frac1{m^2}F^{\mu\nu},
\end{equation*}
where $k(y)$ and $m(x)$ are dilaton fields entering four-dimensional metric
(\ref{egdtrf}). The square of four-dimensional electromagnetic field strength is
\begin{equation*}
\widehat F^2=\frac1{k^2}F_{\alpha\beta}F^{\alpha\beta}+\frac1{m^2}F_{\mu\nu}F^{\mu\nu}.
\end{equation*}
In the case under consideration, Maxwell's Eqs.~(\ref{uncmhy}) for $i=\alpha$ lead
to equality
\begin{equation*}
\frac1{|k|}\sqrt{|h|}\partial_\beta\left(|m|\sqrt{|g|}F^{\beta\alpha}\right)=0.
\end{equation*}
A general solution to these equations has the form
\begin{equation} \label{uncbfh}
|m|\sqrt{|g|}F^{\alpha\beta}=2\hat\varepsilon^{\alpha\beta}Q,\qquad Q={\sf\,const},
\end{equation}
where $\hat\varepsilon^{\alpha\beta}$ is the totally antisymmetric second rank tensor
density. The factor 2 is introduced in the right hand side of general solution
for simplification of subsequent formulae. This solution is rewritten as
\begin{equation} \label{ubvcfr}
F^{\alpha\beta}=\frac{2Q}{|m|}\varepsilon^{\alpha\beta},
\end{equation}
where $\varepsilon^{\alpha\beta}:=\hat\varepsilon^{\alpha\beta}/\sqrt{|g|}$ is now the totally
antisymmetric second rank tensor.
If $i=\mu$, then Maxwell's Eqs.~(\ref{uncmhy}) yield the equality
\begin{equation*}
\frac1{|m|}\sqrt{|g|}\partial_\mu\left(|k|\sqrt h F^{\mu\nu}\right)=0.
\end{equation*}
Its general solution is
\begin{equation} \label{unbgtm}
F^{\mu\nu}=\frac{2P}{|k|}\varepsilon^{\mu\nu},\qquad P={\sf\,const}.
\end{equation}
Now the four-dimensional electromagnetic energy-momentum tensor (\ref{ubnhgy})
is easily calculated. It is block diagonal:
\begin{equation} \label{uvbxfp}
\widehat T_{ij}=\begin{pmatrix}
\widehat T_{\alpha\beta} & 0 \\ 0 & \widehat T_{\mu\nu} \end{pmatrix},
\end{equation}
where
\begin{equation*}
\widehat T_{\alpha\beta}=\frac{2g_{\alpha\beta}}{km^2}(Q^2+P^2),\qquad
\widehat T_{\mu\nu}=-\frac{2h_{\mu\nu}}{k^2m}(Q^2+P^2).
\end{equation*}
Note that we do not need the electromagnetic potentials $A_\alpha$ and $A_\mu$ for
the calculation of the energy-momentum tensor. It is sufficient to know
strengthes (\ref{ubvcfr}) and (\ref{unbgtm}).
Now we have to solve Einstein's Eqs.~(\ref{ubsvgr}) with right hand side
(\ref{uvbxfp}). Since energy-momentum tensor depends only on the sum $Q^2+P^2$,
we set $P=0$ to simplify formulae. In the final answer, this constant is easily
reconstructed by substitution $Q^2\mapsto Q^2+P^2$.
In what follows, we consider only the case $Q\ne0$, because the case $Q=0$ was
considered in \cite{KaKlKu99} in full detail.
\section{Einstein's equations}
The right hand side of Einstein's Eqs.~(\ref{ubsvgr}) is defined by general
solution of Maxwell's equations, which leads to electromagnetic energy-momentum
tensor (\ref{uvbxfp}). The trace of Einstein's equations can be easily solved
with respect to the scalar curvature:
\begin{equation*}
\widehat R=4\Lambda,
\end{equation*}
which does not depend of the electromagnetic field, because the trace of the
electromagnetic field energy-momentum tensor equals zero. After elimination of
the scalar curvature, Einstein's equations are simplified:
\begin{equation} \label{ubbdgt}
\widehat R_{ij}-\widehat g_{ij}\Lambda=-\frac12\widehat T_{{\textsc{e}}{\textsc{m}} ij}.
\end{equation}
For indices values $(ij)=(\alpha,\beta)$, $(\mu\nu)$, and $(\alpha,\mu)$, these
equations yield the following system of equations:
\begin{align} \label{ubvcfl}
R_{\alpha\beta}+\frac{\nabla_\alpha\nabla_\beta m}m-\frac{\nabla_\alpha m\nabla_\beta m}{2m^2}
+g_{\alpha\beta}\left(\frac{\nabla^2k}{2m}-k\Lambda+\frac{Q^2}{m^2k}\right)=&0,
\\ \label{ubncjh}
R_{\mu\nu}+\frac{\nabla_\mu\nabla_\nu k}k-\frac{\nabla_\mu k\nabla_\nu k}{2k^2}
+h_{\mu\nu}\left(\frac{\nabla^2 m}{2k}-m\Lambda-\frac{Q^2}{k^2m}\right)=&0,
\\
-\frac{\nabla_\alpha m\nabla_\mu k}{2mk}=&0,
\end{align}
where $R_{\alpha\beta}$ and $R_{\mu\nu}$ are Ricci tensors for two-dimensional
metrics $g_{\alpha\beta}$ and $h_{\mu\nu}$, respectively, $\nabla_\alpha$ and $\nabla_\mu$ are
two-dimensional covariant derivatives with Christoffel's symbols on surfaces
${\mathbb U}$ and ${\mathbb V}$, $\nabla^2:=g^{\alpha\beta}\nabla_\alpha\nabla_\beta$ or
$\nabla^2:=h^{\mu\nu}\nabla_\mu\nabla_\nu$, which is clear from the context. Sure, the
equalities $\nabla_\alpha m=\partial_\alpha m$ and $\nabla_\mu k=\partial_\mu k$ hold. But we keep
the symbol of covariant derivative for uniformity.
For subsequent analysis of Einstein's equations, we extract the traces and
traceless parts from Eqs.~\ (\ref{ubvcfl}) and (\ref{ubncjh}). Then the full
system of Einstein's equations takes the form
\begin{align} \label{unncbg}
\nabla_\alpha\nabla_\beta m-\frac{\nabla_\alpha m\nabla_\beta m}{2m}-\frac12\left(\nabla^2 m
-\frac{(\nabla m)^2}{2m}\right)=&0,
\\ \label{ubmsdi}
\nabla_\mu\nabla_\nu k-\frac{\nabla_\mu k\nabla_\nu k}{2k}-\frac12\left(\nabla^2 k
-\frac{(\nabla k)^2}{2k}\right)=&0,
\\ \label{undbyt}
R^g+\frac{\nabla^2 m}m-\frac{(\nabla m)^2}{2m^2}+\frac{\nabla^2 k}m-2k\Lambda
+\frac{2Q^2}{m^2k}=&0,
\\ \label{undhtt}
R^h+\frac{\nabla^2 k}k-\frac{(\nabla k)^2}{2k^2}+\frac{\nabla^2 m}k-2m\Lambda
-\frac{2Q^2}{k^2m}=&0,
\\ \label{ubvfds}
\nabla_\alpha m\nabla_\beta k=&0,
\end{align}
where $(\nabla m)^2:=g^{\alpha\beta}\nabla_\alpha m\nabla_\beta m$,
$(\nabla k)^2:=g^{\mu\nu}\nabla_\mu k\nabla_\nu k$, $R^g$ and $R^h$ are scalar curvatures
of two-dimensional surfaces ${\mathbb U}$ and ${\mathbb V}$ for metrics $g$ and $h$,
respectively. In the above formulae, we used equalities
$R_{\alpha\beta}=\frac12g_{\alpha\beta}R^g$ and $R_{\mu\nu}=\frac12h_{\mu\nu}R^h$ valid in
two dimensions.
The last Eq.~(\ref{ubvfds}), which corresponds to mixed values of indices
$(ij)=(\alpha\mu)$ in Einstein's equations results in strong restrictions on
solutions. Namely, as in the case without electromagnetic field, there are only
three cases:
\begin{equation} \label{ecasek}
\begin{array}{lrr}
{\sf A}: & \qquad k={\sf\,const}\ne0, & \qquad m={\sf\,const}\ne0, \\
{\sf B}: & k={\sf\,const}\ne0, & \nabla_\alpha m\ne0, \\
{\sf C}: & \nabla_\mu k\ne0, & m={\sf\,const}\ne0.
\end{array}
\end{equation}
We shall see in what follows, that this leads to ``spontaneous symmetry
emergence''.
Now we consider the first two cases in detail.
\section{Product of constant curvature surfaces}
The most symmetric solutions of Einstein's equations with electromagnetic field
in the form of the product of two constant curvature surfaces arise in case
{\sf A} (\ref{ecasek}), when both dilaton fields are constant. If $k$ and $m$
are constant, then Eqs.~(\ref{unncbg}) and (\ref{ubmsdi}) are identically
satisfied, and Eqs.~(\ref{undbyt}) and (\ref{undhtt}) take the form
\begin{equation} \label{ubbcnd}
R^q=2k\Lambda-\frac{2Q^2}{m^2k}=-2K^g,\qquad R^h=2m\Lambda+\frac{2Q^2}{k^2m}
=-2K^h,
\end{equation}
where
\begin{equation*}
K^g:=-k\left(\Lambda-\frac{Q^2}{k^2m^2}\right),\qquad
K^h:=-m\left(\Lambda+\frac{Q^2}{k^2m^2}\right)
\end{equation*}
are Gaussian curvatures of surfaces ${\mathbb U}$ and ${\mathbb V}$, respectively. It means that
both surfaces are of constant curvature in case {\sf A}. The metric on each
surface is invariant under three-dimensional transformation group.
In stereographic coordinates on both surfaces, the metric of four-dimensional
space-time takes the form
\begin{equation} \label{ubvxgh}
\begin{split}
ds^2=&k g_{\alpha\beta}dx^\alpha dx^\beta+mh_{\mu\nu}dy^\mu dy^\nu=
\\[6pt]
=&k\frac {dt^2-dx^2}{\big[1+\frac{K^g}4(t^2-x^2)\big]^2}
+m\frac{dy^2+dz^2}{\big[1+\frac{K^h}4(y^2+z^2)\big]^2},
\end{split}
\end{equation}
where $(x^\alpha):=(t,x)$ and $(y^\mu):=(y,z)$.
We can put $k=\pm1$ and $m=\pm1$ by rescaling coordinates. One has also to
redefine the constant of integration $Q^2/(k^2m^2)\mapsto Q^2$. We choose $k=1$
and $m=-1$ for the metric signature to be $(+---)$. Then the Gaussian curvatures
are
\begin{equation} \label{umnkiu}
K^g=Q^2-\Lambda,\qquad K^h=Q^2+\Lambda.
\end{equation}
There are four qualitatively different cases for topologically inequivalent
global solutions depending on relations between cosmological constant and
charge:
\begin{equation} \label{ubsndi}
\begin{aligned}
\Lambda<-Q^2: & \qquad K^g>0, & K^h<0, & \qquad {\mathbb M}={\mathbb L}^2\times{\mathbb H}^2,
\\
\Lambda=-Q^2: & \qquad K^g>0, & K^h=0, & \qquad {\mathbb M}={\mathbb L}^2\times{\mathbb R}^2,
\\
-Q^2<\Lambda<Q^2: & \qquad K^g>0, & K^h>0, & \qquad {\mathbb M}={\mathbb L}^2\times{\mathbb S}^2,
\\
\Lambda=Q^2: & \qquad K^g=0, & K^h>0, & \qquad {\mathbb M}={\mathbb R}^{1,1}\times{\mathbb S}^2,
\\
\Lambda>Q^2: & \qquad K^g<0, & K^h>0, & \qquad {\mathbb M}={\mathbb L}^2\times{\mathbb S}^2,
\end{aligned}
\end{equation}
where ${\mathbb L}^2$ is the one sheet hyperboloid (more precisely, its universal
covering) embedded in three-dimensional Minkowskian space ${\mathbb R}^{1,2}$,
${\mathbb H}^2$ is the Lobachevsky plane (the upper sheet of two-sheeted hyperboloid
embedded in ${\mathbb R}^{1,2}$), and ${\mathbb S}^2$ is the two-dimensional sphere. From
topological point of view the third and fifth cases in Eq.~(\ref{ubsndi})
coincide. Therefor there are only four topologically inequivalent global
solutions of Einstein's equations in the form of direct product of two constant
curvature surfaces. Note that for $Q=0$, there are only three topologically
inequivalent solutions \cite{KaKlKu99}.
All solutions have exactly six Killing vector fields and belong to type $D$ in
Petrov's classification.
The cases of other signatures of four-dimensional metric for $k=\pm1$ and
$m=\pm1$ are analysed similarly. Qualitative properties of global solutions are
the same.
We see that symmetry properties in this case are not imposed from the very
beginning but arise as the result of solution of equations of motion. This
effect is called ``spontaneous symmetry emergence''.
\section{Solutions with spatial symmetry}
The dilaton field $k$ is constant in second case {\sf B} (\ref{ecasek}).
Without loss of generality, we put $k=1$. Then Einstein's equations
(\ref{unncbg})--(\ref{ubvfds}) take the form
\begin{align} \label{ubvhhj}
\nabla_\alpha\nabla_\beta m-\frac{\nabla_\alpha m\nabla_\beta m}{2m}
-\frac12g_{\alpha\beta}\left[\nabla^2 m-\frac{(\nabla m)^2}{2m}\right]=&0,
\\ \label{undmwi}
R^h+\nabla^2 m-2m\Lambda-\frac{2Q^2}m=&0,
\\ \label{usneoo}
R^g+\frac{\nabla^2 m}m-\frac{(\nabla m)^2}{2m^2}-2\Lambda+\frac{2Q^2}{m^2}=&0.
\end{align}
Consider Eq.~(\ref{undmwi}). The scalar curvature $R^h$ depends on coordinates
$y$ on surface ${\mathbb V}$, whereas all other terms depend on coordinates $x$ on
surface ${\mathbb U}$. For this equation to be fulfilled, it is necessary that
equation $R^h={\sf\,const}$ holds. It means that surface ${\mathbb V}$ must be of constant
curvature as the consequence of Einstein equations. Therefor the
four-dimensional metric of space-time has at least three independent Killing
vector fields. So, there is spontaneous symmetry emergence.
Let us put $R^h:=-2K^h={\sf\,const}$. Then Eq.~(\ref{undmwi}) is
\begin{equation} \label{unnvbh}
\nabla^2m-2m\Lambda-2K^h-\frac{2Q^2}m=0.
\end{equation}
Excluding the case {\sf A} considered in the previous section, we proceed
further assuming $\nabla_\alpha m\ne0$ on the whole ${\mathbb U}$.
\begin{prop} \label{pkfgju}
Equation (\ref{unnvbh}) is the first integral of Eqs.~(\ref{ubvhhj}) and
(\ref{usneoo}).
\end{prop}
\begin{proof}
Differentiate Eq.~(\ref{unnvbh}) and use the equality
\begin{equation*}
[\nabla_\alpha,\nabla_\beta]A_\gamma=-R^g_{\alpha\beta\gamma}{}^\delta A_\delta,
\end{equation*}
valid for any covector field $A_\alpha$, to change the order of
derivatives in the first term:
\begin{multline*}
\nabla_\alpha(\ref{unnvbh})=\frac{\nabla^\beta m\nabla_\alpha m\nabla_\beta m}{2m}
+\frac{\nabla_\alpha m\nabla^2m}{2m}-\frac{\nabla_\alpha m(\nabla m)^2}{2m^2}+\\
\\
+\frac12\nabla_\alpha\left(\nabla^2 m-\frac{(\nabla m)^2}{2m}\right)
+\frac12\nabla_\alpha m R^g-2\nabla_\alpha m\Lambda+\nabla_\alpha m\frac{2Q^2}{m^2}.
\end{multline*}
Now exclude derivatives $\nabla^\beta m\nabla_\alpha m$ and $\nabla^2 m$ using
Eqs.~(\ref{ubvhhj}) and (\ref{undmwi}) in the first and fourth terms on the
right hand side. After rearranging terms, the sum of the first and fourth terms
takes the form
\begin{equation*}
\nabla_\alpha m\left(\frac{(\nabla m)^2}{4m^2}+\Lambda-\frac{Q^2}{m^2}\right).
\end{equation*}
Taking all terms together, we obtain
\begin{equation} \label{uncbvg}
\nabla_\alpha(\ref{unnvbh})=\frac12\nabla_\alpha m(\ref{usneoo}).
\end{equation}
Since $\nabla_\alpha m\ne0$, it implies the statement of the proposition.
\end{proof}
The proof of the proposition implies that it is sufficient to solve
Eqs.~(\ref{ubvhhj}) and (\ref{unnvbh}), Eq.~(\ref{usneoo}) being satisfied
automatically.
To solve Eqs.~(\ref{ubvhhj}) and (\ref{unnvbh}) explicitly, we fix the
conformal gauge for metric $g_{\alpha\beta}$ on Lorentzian surface ${\mathbb U}$:
\begin{equation} \label{endhft}
g_{\alpha\beta}dx^\alpha dx^\beta=\Phi d\xi d\eta,
\end{equation}
where $\Phi(\xi,\eta)\ne0$ is the conformal factor depending on light cone
coordinates $\xi:=\tau+\sigma$, $\eta:=\tau-\sigma$ on ${\mathbb U}$. The respective four
dimensional metric is
\begin{equation} \label{unjhsy}
ds^2=\Phi d\xi d\eta+md\Omega,
\end{equation}
where $d\Omega$ is the metric on the Riemannian surface of constant curvature
${\mathbb V}={\mathbb S}^2$, ${\mathbb R}^2$, or ${\mathbb H}^2$. The sign of the conformal factor $\Phi$ is not
fixed for the present.
For $\Phi>0$ and $m<0$ the signature of metric (\ref{unjhsy}) is $(+---)$. If we
change the sign of $m$, the signature of the metric becomes $(+-++)$. The same
transformation of the signature can be achieved by changing the overall sign of
the metric $\widehat g_{ij}\mapsto-\widehat g_{ij}$, and interchanging the
first two coordinates, $\tau\leftrightarrow\sigma$. Einstein's equations with
cosmological constant and electromagnetic field (\ref{ubbdgt}) are not invariant
with respect to these transformations with simultaneous changing the sign of the
cosmological constant, because the right hand side changes its sign. Therefor,
for $\Phi>0$, we have to consider two cases:
\begin{equation*}
m<0\quad\Leftrightarrow\quad{\sf\,sign\,}\widehat g_{ij}=(+---)\qquad\text{и}\qquad
m>0\quad\Leftrightarrow\quad{\sf\,sign\,}\widehat g_{ij}=(-+++).
\end{equation*}
This is the difference for Einstein's equations without electromagnetic field
considered in \cite{KaKlKu99}.
\subsubsection{Metric signature $(+---)$}
For $\Phi>0$ and $m<0$, we introduce convenient parameterization
\begin{equation} \label{ubndoi}
m:=-q^2,\qquad q(\xi,\eta)>0.
\end{equation}
Afterwards, we obtain the full system of equations:
\begin{align} \label{egcpga}
-\partial^2_{\xi\xi}q+\frac{\partial_\xi\Phi\partial_\xi q}\Phi&=0,
\\ \label{egcygb}
-\partial^2_{\eta\eta}q+\frac{\partial_\eta\Phi\partial_\eta q}\Phi&=0,
\\ \label{egcugc}
-2\frac{\partial^2_{\xi\eta}q^2}\Phi-K^h+\Lambda q^2+\frac{Q^2}{q^2}&=0.
\end{align}
The first two equations which do not depend on the electromagnetic field imply
the following assertion.
\begin{prop}
If $\partial_\xi q\partial_\eta q>0$, then the function $q(\tau)$ depends only on
timelike coordinate $\tau:=\frac12(\xi+\eta)$. If $\partial_\xi q\partial_\eta q<0$, then
the function $q(\sigma)$ depends only on spacelike coordinate
$\sigma:=\frac12(\xi-\eta)$. And the following equality holds
\begin{equation} \label{unbcgt}
|\Phi|=|q'|,
\end{equation}
where prime denotes differentiation on the argument (either $\tau$, or $\sigma$).
\end{prop}
This proposition provides a general solution to equations (\ref{egcpga}) and
(\ref{egcygb}) up to conformal transformations. This statement is proved in
\cite{KaKlKu99,Katana13B}.
Thus, we can always choose coordinates in such a way that $q$ and $\Phi$
depend simultaneously on timelike or spacelike coordinate
\begin{equation} \label{eindvj}
\zeta:=\frac12(\xi\pm\eta)=:\begin{cases} \tau,\qquad \partial_\xi q\,\partial_\eta q>0,
\\ \sigma,\qquad \partial_\xi q\,\partial_\eta q<0. \end{cases}
\end{equation}
It means that two-dimensional metric (\ref{endhft}) and consequently
four-dimensional metric (\ref{unjhsy}) have the Killing vector $\partial_\sigma$ or
$\partial_\tau$, as the consequence of equations (\ref{egcpga}) and (\ref{egcygb}).
We call these solutions homogeneous and static, respectively, though it is
related to the fixed coordinate system. The existence of additional Killing
vector is the generalization of Birkhoff's theorem \cite{Birkho23} stating that
arbitrary spherically symmetric solution of vacuum Einstein's equations must be
static. (This statement was previously published in \cite{Jebsen21}.)
The generalization includes the addition of electromagnetic field, and, in
addition, the existence of extra Killing vector is proved not only for
spherically symmetric solution $(K^h=1)$, but also for solutions invariant with
respect to ${\mathbb I}{\mathbb S}{\mathbb O}(2)$ $(K^h=0)$ and ${\mathbb S}{\mathbb O}(1,2)$ $(K^h=-1)$ transformation
groups.
We are left to solve equation (\ref{egcugc}). In static, $q=q(\sigma)$, and
homogeneous, $q=q(\tau)$, cases, equation (\ref{egcugc}) takes the form
\begin{align} \label{ufijhg}
(q^2)''&=~~2\left(K^h-\Lambda q^2-\frac{Q^2}{q^2}\right)\Phi, & q&=q(\sigma),
\\ \label{usedrf}
(q^2)''&=-2\left(K^h-\Lambda q^2-\frac{Q^2}{q^2}\right)\Phi, & q&=q(\tau).
\end{align}
To integrate the derived equations, one has to express $\Phi$ through $q$ using
equation (\ref{unbcgt}) and removing the modulus sign.
We consider the static case $q=q(\sigma)$, $\Phi>0$ and $q'>0$ in detail. Then
Eq.~(\ref{ufijhg}) together with Eq.~(\ref{unbcgt}) reduces to
\begin{equation*}
(q^2)''=2\left(K^h-\Lambda q^2-\frac{Q^2}{q^2}\right)q'.
\end{equation*}
It can be easily integrated:
\begin{equation*}
(q^2)'=2\left(K^hq-\frac{\Lambda q^3}3-2M+\frac{Q^2}q\right),
\end{equation*}
where $M={\sf\,const}$ is an integration constant, which coincides with mass in the
Schwarzschild solution. Differentiating the left hand side and dividing it by
$2q>0$, we obtain equation
\begin{equation*}
q'=K^h-\frac{2M}q+\frac{Q^2}{q^2}-\frac{\Lambda q^2}3.
\end{equation*}
Since $q'=\Phi$ in the case under consideration, it implies expression for the
conformal factor through variable $q$:
\begin{equation} \label{ucohjy}
\Phi(q)=K^h-\frac{2M}q+\frac{Q^2}{q^2}-\frac{\Lambda q^2}3.
\end{equation}
If $q=q(\sigma)$, $\Phi>0$ and $q'<0$, then the similar integration yields
\begin{equation*}
q'=-\Phi(q),
\end{equation*}
where the same conformal factor (\ref{ucohjy}) stands in the right hand side.
This case can be united with the previous one by re-writing equation for $q$ in
the form
\begin{equation} \label{ueqsra}
|q'|=\Phi(q),\qquad q=q(\sigma),\quad \Phi>0.
\end{equation}
The modulus sign in the left hand side means that if $q(\sigma)$ is a solution, then
the function $q(-\sigma)$ is also the solution.
The static case for $\Phi<0$ is integrated in the same way:
\begin{equation} \label{uftrye}
|q'|=-\Phi(q),\qquad q=q(\sigma),\quad \Phi<0.
\end{equation}
If solution is homogeneous, $q=q(\tau)$ and $\Phi>0$, $q'>0$, then integration
of Eq.~(\ref{usedrf}) yields the equality
\begin{equation*}
q'=-\left(K^h-\frac{2M}q+\frac{Q^2}{q^2}-\frac{\Lambda q^2}3\right).
\end{equation*}
That is the conformal factor must be identified with the right hand side
\begin{equation} \label{unewsd}
\hat\Phi=-\left(K^h-\frac{2M}q+\frac{Q^2}{q^2}-\frac{\Lambda q^2}3\right).
\end{equation}
We denote the expression for the conformal factor through $q$ by hat because in
homogeneous case it differs by the sign. Thus, homogeneous solutions of
Einstein's equations can be written in the form
\begin{alignat}{3} \label{ufredl}
|q'|&=~~\hat\Phi(q),\qquad & q&=q(\tau),\quad && \hat\Phi>0.
\\ \label{usdeax}
|q'|&=-\hat\Phi(q),\qquad & q&=q(\tau),\quad && \hat\Phi<0.
\end{alignat}
If the conformal factor is negative, then the signature of the metric is
$(-+--)$. In this case, we return to the previous signature $(+---)$ after
substitution $\tau\leftrightarrow\sigma$. This transformation allows us to unite
static and homogeneous solutions by taking the modulus of the conformal factor
in the expression for metric (\ref{unjhsy}). Then a general solution of vacuum
Einstein's equations with electromagnetic field (\ref{ubsvgr}) in the
corresponding coordinate system takes the form
\begin{equation} \label{qnhtsd}
ds^2=|\Phi|(d\tau^2-d\sigma^2)-q^2d\Omega,
\end{equation}
where the conformal factor $\Phi$ is given by Eq.~(\ref{ucohjy}). Here the
variable $q$ depends on $\sigma$ (static local solution) or $\tau$
(homogeneous local solution) through the differential equation
\begin{equation} \label{qdtres}
\left|\frac{dq}{d\zeta}\right|=\pm\Phi(q),
\end{equation}
where the sign rule holds:
\begin{equation} \label{esignk}
\begin{array}{ccl}
\Phi>0: & \quad \zeta=\sigma, &\quad \text{the sign $+$ (static local solution)},\\
\Phi<0: & \quad \zeta=\tau,&\quad \text{the sign $-$ (homogeneous local solution)}.
\end{array}
\end{equation}
Thus the four-dimensional Einstein's equations imply that there is the metric
with one Killing vector field on surface ${\mathbb U}$ which was considered in full
detail in \cite{Katana00A}. Now we can construct global solutions (maximally
extended along geodesics) of vacuum Einstein's equations using the conformal
block method. The number of singularities and zeroes of conformal factor
(\ref{ucohjy}) depends on relations between constants $K$, $M$, $Q$, and $\Lambda$.
Therefor there are many qualitatively different global solutions, which are
considered in next sections.
Conformal factor (\ref{ucohjy}) has one singularity: the second order pole at
$q=0$. Therefor according to the rules formulated in \cite{Katana00A,Katana13B}
every global solution correspond to one of the intervals $(-\infty,0)$ or
$(0,\infty)$. The form of conformal factor (\ref{ucohjy}) implies that
these global solutions are obtained one from the other by the transformation
$M\mapsto-M$. Hence, without loss of generality, we describe global solutions
corresponding to both intervals but positive values of $M$.
Because conformal factor $\Phi(q)$ is a smooth function for $q\ne0$, all
arising Lorentzian surfaces ${\mathbb U}$ and metrics on them are smooth.
To conclude the section we compute geometrical invariants which show that
obtained solution of Einstein's equations are nontrivial. First, we compute
the scalar curvature $R^g$ of the surface ${\mathbb U}$. Equations (\ref{undmwi}) and
(\ref{usneoo}) imply
\begin{equation*}
R^g=-\frac{2K^h}m+\frac{(\nabla m)^2}{2m^2}-\frac{4Q^2}{m^2}
=\frac{2K^h}{q^2}+\frac{2(\nabla q)^2}{q^2}-\frac{4Q^2}{q^4}.
\end{equation*}
Since
\begin{equation*}
(\nabla q)^2=\frac1\Phi\eta^{\alpha\beta}\partial_\alpha q\partial_\beta q
=-\frac{q'{}^2}\Phi
\end{equation*}
both for static and homogeneous solutions, the final expression is
\begin{equation} \label{ubjdgh}
R^g=\frac{2\Lambda}3+\frac{4M}{q^3}-\frac{6Q^2}{q^4}.
\end{equation}
It does not depend on Gaussian curvature $K^h$ of Riemannian surface ${\mathbb V}$ and
is singular for $q=0$ if $M\ne0$ and/or $Q\ne0$.
\subsubsection{Metric signature $(-+++)$}
If $m>0$, then the signature of the metric is opposite $(-+++)$, and we
introduce parameterization
\begin{equation*}
m:=q^2,\qquad q>0,
\end{equation*}
instead of Eq.~(\ref{ubndoi}). Performing the same calculation as in the previous
section, we obtain the first order equation for $q$:
\begin{equation} \label{uncmxy}
\left|\frac{dq}{d\zeta}\right|=\pm\Phi(q),
\end{equation}
where $M$ is an integration constant and
\begin{equation} \label{ubbchd}
\Phi(q):=\left(K^h-\frac{2M}q-\frac{Q^2}{q^2}+\frac{\Lambda q^2}3\right).
\end{equation}
Here we must take into account that for getting the signature $(-+++)$ we have
to make interchanging $\tau\leftrightarrow\sigma$. We see that for drawing the
Carter--Penrose diagram one has to simply make replacement $Q^2\mapsto-Q^2$ and
$\Lambda\mapsto-\Lambda$ as compared to signature $(+---)$.
Now we describe all spatially symmetric global solution of Einstein's equations
with electromagnetic field which are defined by zeroes and their types of the
conformal factor $\Phi(q)$.
\subsection{Spherically symmetric solutions $K^h=1$ \label{sphers}}
In the considered case, global spherically symmetric solutions, that is pairs
$({\mathbb M},\widehat g)$, have the form ${\mathbb M}={\mathbb U}\times{\mathbb S}^2$, where ${\mathbb U}$ is the
maximally extended Lorentzian surface which is depicted by the Carter--Penrose
diagram. Four-dimensional metric on ${\mathbb M}$ has the form (\ref{qnhtsd}), where
$d\Omega$ is the metric on sphere ${\mathbb S}^2$ for signature $(+---)$. If the signature
is opposite $(-+++)$, then we have to replace $Q^2\mapsto-Q^2$ and $\Lambda\to-\Lambda$
in the conformal
factor and change the sign of $d\Omega$ in metric (\ref{qnhtsd}). Due to the
existence of one Killing vector on Lorentzian surface ${\mathbb U}$, we are able to
classify all global solutions. To construct Carter--Penrose diagrams, we use the
conformal block method described in \cite{Katana00A} (see also,
\cite{Katana13B}). First, we consider
solutions of signature $(+---)$, and then with signature $(-+++)$.
\subsubsection{Metric signature $(+---)$}
If the metric signature is $(+---)$, then the conformal factor is
\begin{equation} \label{uvcfre}
\Phi(q)=1-\frac{2M}q+\frac{Q^2}{q^2}-\frac{\Lambda q^2}3=:
\frac{\varphi(q)+3Q^2}{3q^2},
\end{equation}
where we introduced the auxiliary function
\begin{equation} \label{uvxgij}
\varphi(q):=-\Lambda q^4+3q^2-6Mq
\end{equation}
which is needed for further analysis. The case $Q=0$ was analyzed in
\cite{KaKlKu99}. Therefor, we classify solutions for $Q\ne0$. Without loss of
generality, we consider the case $Q>0$, because only $Q^2$ enters the conformal
factor.
Conformal factor (\ref{uvcfre}) has the second order pole $Q^2/q^2$ at zero and
the following asymptotic at infinity
\begin{equation*}
\Phi\approx1-\frac{\Lambda q^2}3,\qquad q\to\infty.
\end{equation*}
If cosmological constant is equal to zero, then metric is asymptotically flat.
For $\Lambda>0$ and $\Lambda<0$, we have asymptotically de Sitter and anti-de Sitter
spacetime, respectively.
A global solution corresponds to one of the intervals $q\in(0,\infty)$ or
$q\in(-\infty,0)$ and $M>0$, because the curvature has singularity
(\ref{ubjdgh}) at zero, and space-time is not extendable through this point.
Roots of conformal factor (\ref{uvcfre}) correspond to horizons of space-time,
and Carter--Penrose diagrams are defined by the number and type of zeroes of the
conformal factor \cite{Katana00A}. Thus we have to analyse the number and type
of zeroes of conformal factor (\ref{uvcfre}) for all possible values of
constants $\Lambda$, $M\ge0$, and $Q>0$.
Note that conformal factor (\ref{uvcfre}) is invariant with respect to
transformation
\begin{equation*}
M\to-M,\qquad q\to-q.
\end{equation*}
Therefor, instead of constructing global solutions on the interval
$q\in(0,\infty)$ for all values of $M$, we restrict ourselves only for
nonnegative $M\ge0$, but on two intervals $q\in(-\infty,0)$ and $(0,\infty)$.
This simplifies the analysis of the conformal factor.
We start with the simplest and well known case $\Lambda=0$.
\subsubsection{Metric signature $(+---)$. The case $\Lambda=0$.}
If cosmological constant vanishes, then zeroes of conformal factor
(\ref{uvcfre}) are defined by the quadratic equation
\begin{equation} \label{umdopi}
q^2-2Mq+Q^2=0,
\end{equation}
which has two roots:
\begin{equation} \label{unbvgt}
q_\pm=M\pm\sqrt{M^2-Q^2}.
\end{equation}
{\bf The Reissner--Nordstr\"om solution.}
For $Q<M$, there are two positive simple roots. This solution is called the
Reissner--Nordstr\"om solution \cite{Reissn16,Nordst18} and depicted by the
Carter--Penrose diagram S1 shown in Fig.\ref{fcarpenspherEM}. It was also found
by H.~Weyl \cite{Weyl17}. The solution has two horizons at $q_-$ and $q_+$ and
naked timelike singularity at $q=0$. The conformal factor tends to unity at
infinity, and, consequently, the Reissner--Nordstr\"om solution is
asymptotically flat. Arrows on the diagram show directions in which the solution
can be periodically extended in time. Instead of periodic extension, there is
the possibility to identify the opposite horizons. The singularity at $q=0$ is
timelike, and an observer can approach it as close as he likes in conformal
blocks I or III, and then enter universe III or I by going through conformal
block IV. Therefor, the Reissner--Nordstr\"om solution does not describe a
black hole.
\begin{figure}[p
\hfill\includegraphics[width=0.9\textwidth]{fcarpenspherEM}
\hfill {}
\centering\caption{The Carter--Penrose diagrams for spherically symmetric
solutions of Einstein's equations with electromagnetic field. Diagrams S1--S11
and S12--S16 correspond to metrics of signature $(+---)$ and $(-+++)$,
respectively.}
\label{fcarpenspherEM}
\end{figure
{\bf Extremal black hole.}
For $Q=M$, the conformal factor is
\begin{equation*}
\Phi=\frac{(q-M)^2}{q^2}.
\end{equation*}
It has one positive root of second order at $q=M$. The corresponding
Carter--Penrose diagram is shown in Fig.\ref{fcarpenspherEM}, S4. It is called
extremal black hole, though there is no any black hole since the singularity is
timelike and horizon surrounding the singularity is absent. There is also
space-reflected diagram.
{\bf Naked singularity.}
For $Q>M$, horizons are absent, and we have naked singularity shown in
Fig.\ref{fcarpenspherEM}, S5. There is also space-reflected diagram.
\subsubsection{Metric signature $(+---)$. The case $\Lambda>0$.}
For positive cosmological constant, zeroes of the conformal factor are defined
by the fourth order equation
\begin{equation} \label{ubbcgt}
\varphi(q)+3Q^2=0,
\end{equation}
where function $\varphi(q)$ is given by the fourth order polynomial (\ref{uvxgij}).
To draw Carter--Penrose diagrams, we do not need to know exact position of
zeroes. We have to know only their existence and type. Therefor, we analyze
function $\varphi(q)$ qualitatively and then move its graphic up, which corresponds
to increasing value of $Q^2$.
First, we differentiate function (\ref{ubbcgt}):
\begin{equation} \label{unmjqj}
\begin{split}
\varphi'(q)=&-4\Lambda q^3+6q-6M,
\\
\varphi''(q)=&-12\Lambda q^2+6=-6(2\Lambda q^2-1).
\end{split}
\end{equation}
The asymptotics of function $\varphi(q)$ ($\Lambda>0$) and its derivatives for $q=0$
and $q\to\infty$ are easily found:
\begin{align} \nonumber
\varphi(0)=&~~0, & \varphi(q\to\infty)\approx&-\Lambda q^4,
\\ \label{edswhy}
\varphi'(0)=&-6M, & \varphi'(q\to\infty)\approx&-4\Lambda q^3,
\\ \nonumber
\varphi''(0)=&~~6, & \varphi''(q\to\infty)\approx&-12\Lambda q^2.
\end{align}
Zeroes of function $\varphi(q)+3Q^2$ require more work. As we see later, their
number does not exceed three. To find the types of zeroes, we have to know local
extrema of function $\varphi(q)$, which become zeroes of order two or three after
shifting on $3Q^2$.
Local extrema of function $\varphi$ are defined by cubic equation (the solution is
given, i.e.\ in \cite{KorKor68})
\begin{equation} \label{uvskju}
q^3-\frac3{2\Lambda} q+\frac{3M}{2\Lambda}=0.
\end{equation}
There are three qualitatively distinct cases depending on the value of constant
\begin{equation} \label{ebgtdj}
\Upsilon:=-\frac1{8\Lambda^3}+\frac{9M^2}{16\Lambda^2}.
\end{equation}
Namely,
\begin{align*}
\Upsilon>0\quad\Leftrightarrow\quad |M|>&\frac13\sqrt{\frac2\Lambda} \qquad
\text
{--\quad \parbox[c]{0.5\textwidth}{one real and two complex conjugate roots,}}
\\[6pt]
\Upsilon=0\quad\Leftrightarrow\quad |M|=&\frac13\sqrt{\frac2\Lambda} \qquad
\text{--\quad \parbox[c]{0.5\textwidth}{three real roots \\
(at least two roots coincide),}}
\\[6pt]
\Upsilon<0\quad\Leftrightarrow\quad |M|<&\frac13\sqrt{\frac2\Lambda} \qquad
\text{--\quad three different real roots.}
\end{align*}
We start with the simplest case $\Upsilon=0$. This equality implies restriction
on ``mass'':
\begin{equation} \label{edbgdt}
\Upsilon=0\qquad\Leftrightarrow\qquad M=\frac13\sqrt{\frac2\Lambda}.
\end{equation}
Moreover, roots of Eq.~(\ref{uvskju}) take the simple form:
\begin{equation} \label{unbhsk}
M=\frac13\sqrt{\frac2\Lambda}:\qquad q_1=-\sqrt{\frac2\Lambda}, \qquad
q_{2,3}=\frac12\sqrt{\frac2\Lambda},
\end{equation}
As we see, there are one simple negative root and one positive root of second
order for positive ``mass'' (\ref{edbgdt}).
If inequality $\Upsilon<0$ holds, then real roots of cubic equation
(\ref{uvskju}) are (see, i.e., \cite{KorKor68})
\begin{equation} \label{ubcndt}
q_3=\sqrt{\frac2\Lambda}\cos\frac\alpha3,\qquad
q_{2,1}=-\sqrt{\frac2\Lambda}\cos\left(\frac\alpha3\pm\frac\pi3\right),
\end{equation}
where
\begin{equation*}
\cos\alpha:=-3M\sqrt{\frac\Lambda2}.
\end{equation*}
Since we consider only nonnegative $M$, then
$\alpha\in\big[\frac\pi2,\frac{3\pi}2\big]$. It implies existence of one negative
root $q_1$ and two positive: $q_2$ and $q_3$. We enumerate the zeroes in
Eq.~(\ref{ubcndt}) in such a way, that, in the limit
\begin{equation*}
M\to\frac13\sqrt{\frac2\Lambda},
\end{equation*}
they take values (\ref{unbhsk}).
If $\Upsilon>0$, then we have only one negative root $q_1$. Its exact
position can be written but it is not needed.
Figure \ref{fgremone}, {\em a}, shows qualitative behavior of function $\varphi(q)$
for $\Lambda>0$ and different values of $M\ge0$.
\begin{figure}[hbt
\hfill\includegraphics[width=.9\textwidth]{fgremone}
\hfill {}
\centering\caption{Auxiliary function $\varphi(q)$ for $\Lambda>0$ {\em (a)} and
conformal factor $\Phi(q)$ for $Q=0$ {\em (b)}. The curves correspond to the
following values of the constant: (1) $\Upsilon>0$, (2) $\Upsilon=0$, (3)
$\Upsilon<0$, and (4) $\Upsilon=-\frac1{8\Lambda^3}$. Local extrema for curve 3
on the left picture are located at points $q_1$, $q_2$, and $q_3$. For curve
2, local maximum and minimum coincide, that is $q_2=q_3$, and are denoted by the
fat point. For curve 1, there is only one maximum for negative $q$. Curve 4 on
the left is symmetric with respect to substitution $q\mapsto-q$, has local
minimum at $q=0$, and two maxima at points $q=\pm\sqrt{\frac3{2\Lambda}}$.}
\label{fgremone}
\end{figure
Now, to construct all global solutions which exist in the theory for signature
$(+---)$, we have to analyse zeroes of conformal factor $\Phi(q)$, qualitative
behavior of which for $Q=0$ is shown in Fig.\ref{fgremone}, {\it b}. Zeroes of
the conformal factor and their type coincide with that of function
$\varphi(q)+3Q^2$. Therefor, we have to shift up curves 1--4 in Fig.\ref{fgremone},
{\it a}, on $3Q^2$ to analyse its qualitative behavior. The number and type of
zeroes depend on curves 1--4 and on the value of the shift $3Q^2$. All possible
Carter--Penrose diagrams are drawn in Fig.\ref{fcarpenspherEM}.
The conformal factor depicted by curve 4 in Fig.\ref{fgremone}, {\it b}, does
not have zero at $q=0$. It corresponds to de Sitter space, and is degenerate at
this presentation of the problem ($M=0$, $Q=0$), which is not considered here
because of the assumption $Q>0$.
For qualitative description of behavior of the conformal factor, we introduce
notation:
\begin{equation} \label{ubvsik}
\varphi_1:=\varphi(q_1),\qquad\varphi_2:=\varphi(q_2),\qquad\varphi_3:=\varphi(q_3),
\end{equation}
where $\varphi_1$ is the maximum, $\varphi_2$ is local minimum, and $\varphi_3$ is local
maximum of the auxiliary function $\varphi(q)$. One can easily verify, that, for
$\Lambda>0$ and $q<0$, the maximum is positive: $\varphi_1>0$. On positive half line
$q>0$, the local minimum is always negative: $\varphi_2<0$, and local maximum
$\varphi_3$ can take negative as well as positive values:
\begin{align*}
0<M<\frac1{\sqrt\Lambda},& &&\varphi_3>0,
\\
M=\frac1{\sqrt\Lambda}, & &&\varphi_3=0,
\\
M>\frac1{\sqrt\Lambda}, & &&\varphi_3<0.
\end{align*}
When Eq.~(\ref{edbgdt}) holds, local minimum and maximum coincide: $q_2=q_3$.
Now we list all possibilities in the considered case.
{\bf Three horizons.} Under condition
\begin{equation} \label{ubcvdr}
-\varphi_3<3Q^2<-\varphi_2,
\end{equation}
the conformal factor has three simple zeroes on positive half line. The
corresponding Carter--Penrose diagram of surface ${\mathbb U}$ is given by
S2 in Fig.\ref{fcarpenspherEM}. Here we have two timelike naked singularities.
Arrows show that this diagram can be either periodically continued in space- and
timelike directions, or opposite horizons can be identified. If we identify
horizons in one direction, them topologically the surface ${\mathbb U}$ is a cylinder.
If identification is performed in both directions, then it is a torus.
{\bf One simple horizon and timelike singularity.}
The conformal factor has one simple zero on positive half line under the
following conditions:
\begin{equation} \label{ubvgfg}
\begin{aligned}
&\Lambda>0,\qquad& \Upsilon<0,& & {}\qquad&M=0, & {}\qquad&{} & &Q\ne0,
\\
&\Lambda>0,& \Upsilon<0,& & &M>0, & &\varphi_3<0, & {}\qquad 3&Q^2<-\varphi_3,
\\
&\Lambda>0,& \Upsilon<0,& & &M>0, & &\forall\,\varphi_3, & 3&Q^2>-\varphi_2,
\\
&\Lambda>0,& \Upsilon=0,& & &M>0, & & & 3&Q^2\ne-\varphi_2,
\\
&\Lambda>0,& \Upsilon>0,& & &M>0, & & & &Q\ne0,
\\
&\Lambda>0,& \forall\,\Upsilon,& & &M<0, & &\forall\,\varphi_3, & &Q\ne0.
\end{aligned}
\end{equation}
This global solution is depicted by the Carter--Penrose diagram S7. It has
timelike singularity.
{\bf Triple horizon.} Under conditions:
\begin{equation} \label{uvbxfr}
\Lambda>0,\qquad\Upsilon=0,\qquad M>0,\qquad 3Q^2=-\varphi_2.
\end{equation}
local maximum and minimum of auxiliary function $\varphi(q)$ coincide:
$q_2=q_3$, and the conformal factor has zero of third order at point $q_2$
(triple horizon). This case is depicted by diagram S6. It coincides with
diagram S7, but there is one important difference: the saddle point $q_2$ in the
center of the diagram is geodesically complete.
This diagram is interesting from physical standpoint. Consider a spacelike
section of this diagram. If the section does not go through the saddle point,
which is located in the center of the diagram, then it is an interval of finite
length with singular ends where two-dimensional curvature becomes infinite. If
the space section goes through the saddle point then it is the union of two
half-infinite intervals, because the central point in the center of the diagram
is the space infinity. If we introduce now global evolution parameter $T$, for
instance, vertical line on the diagram, then topology of space sections change
during evolution: for some value of $T$, there are two half-infinite intervals
instead of one finite interval. This example shows that changing topology of
space in time can occur already at the classical level. This type of diagram
appeared first in two-dimensional gravity with torsion \cite{Katana93A}.
{\bf Two horizons with double local minima.} Under conditions:
\begin{equation} \label{ubvcgt}
\Lambda>0,\qquad\Upsilon<0,\qquad M>0,\qquad 3Q^2=-\varphi_2,
\end{equation}
the conformal factor has one zero of second order at point $q_2$ and one simple
simple zero at some point lying to the right from $q_2$. This solution is
depicted by Carter--Penrose diagram S8 with two timelike singularities, which
can be periodically extended in timelike direction.
{\bf Two horizons with double local maximum.} Under conditions:
\begin{equation} \label{ubvcgk}
\Lambda>0,\qquad\Upsilon<0,\qquad M>0,\qquad \varphi_3<0,\qquad 3Q^2=-\varphi_3,
\end{equation}
the conformal factor has one double zero at $q_3$ and one simple zero at some
point lying to the left from $q_2$. This solution corresponds to Carter--Penrose
diagram S3 with two timelike singularities, which can be periodically extended
in spacelike direction.
\subsubsection{Metric signature $(+---)$. The case $\Lambda<0$.}
For negative cosmological constant, the conformal factor have the same form and
asymptotics remain the same (\ref{edswhy}). Equation (\ref{uvskju}) and
constant (\ref{ebgtdj}), defining the roots, do not change. We see that values
of constant $\Upsilon$ are positive for all $\Lambda$ and $M$. Consequently,
Eq.~(\ref{uvskju}) has only one nonnegative real root. Moreover, now branches of
auxiliary function $\varphi(q)$ are directed upwards as shown in Fig.\ref{fvfnelm},
and three new Carter--Penrose diagrams appear in the spherically symmetric case.
\begin{figure}[hbt
\hfill\includegraphics[width=.9\textwidth]{fvfnelm}
\hfill {}
\centering\caption{Auxiliary function $\varphi(q)$ {\em (a)} and conformal factor
$\Phi(q)$ for $\Lambda<0$ and $Q=0$ {\em (b)}. The curves correspond to the
following values of the constant: (1) $\Upsilon>-\frac1{8\Lambda^3}$ and (2)
$\Upsilon=-\frac1{8\Lambda^3}$ ($M=0$). On the left picture, the only minimum of
curve 1 is located at point $q_4$. Curve 2 on the left is invariant with respect
to the map $q\mapsto-q$ and has minimum at $q=0$.}
\label{fvfnelm}
\end{figure
The conformal factor depicted by curve 2 in Fig.\ref{fvfnelm}, {\it b}, has zero
at point $q=0$. It corresponds to anti-de Sitter space and is the degenerate
case in the problem under consideration ($M=0$, $Q=0$).
Now we list all possibilities for negative cosmological constant.
{\bf Timelike singularity.} Under conditions:
\begin{equation} \label{ubvgfl}
\begin{aligned}
&\Lambda<0,\qquad& {}\qquad&M>0, & {}\qquad&{} & &3Q^2>-\varphi_4,
\\
&\Lambda<0,& &M\le0, & &\varphi_3<0, & {}\qquad 3&Q\ne0,
\end{aligned}
\end{equation}
the conformal factor does not have zeroes, and, consequently, horizons are
absent. In this case, the Carter--Penrose diagram has the lens form S9 in
Fig.\ref{fcarpenspherEM}. There is also space-reflected diagram.
{\bf Naked singularity.} Under conditions:
\begin{equation} \label{ubndht}
\Lambda<0,\qquad M>0,\qquad 3Q^2=-\varphi_4,
\end{equation}
the conformal factor has one positive root of second order at the minimum of
the auxiliary function at $q_4$. In this case, the Carter--Penrose diagram is
S10 in Fig.\ref{fcarpenspherEM}. In contrast to the naked singularity S4, the
right complete infinity $q=\infty$ is timelike. It is due to asymptotic of the
conformal factor at infinity, because space-time is asymptotically
anti-de Sitter for $\Lambda<0$. There is also space-reflected diagram.
{\bf Timelike singularity and two horizons.} Under conditions:
\begin{equation} \label{ubvcfj}
\Lambda<0,\qquad M>0,\qquad 3Q^2<-\varphi_4,
\end{equation}
the conformal factor has two zeroes. In this case, the Carter--Penrose diagram
is given by S11 in Fig.\ref{fcarpenspherEM}. This solution can either be
periodically extended in timelike direction or opposite horizons can be
identified. In contrast to diagram S1, space infinities are timelike, which is
due to asymptotic at infinity.
Thus we classified all spherically symmetric global solutions of Einstein's
equations with electromagnetic field for metric signature $(+---)$. We see, that
all solutions of signature $(+---)$ contain timelike singularity.
Totally, we get 11 topologically inequivalent solutions S1--S11. It is possible
to give more subtle classification taking into account existence of
degenerate and oscillating geodesics. The latter appears, if the conformal
factor has local extremum inside one of the conformal blocks. This
classification was given for global solutions of two-dimensional gravity with
torsion \cite{Katana93A}.
\subsubsection{Metric signature $(-+++)$}
If the signature is opposite, the conformal factor has the form (\ref{uvcfre})
but with the replacement $Q^2\mapsto-Q^2$. It means that auxiliary function
$\varphi(q)$ in Figs.~\ref{fgremone} and \ref{fvfnelm}, {\it a}, remains the same,
but we have to move it downwards instead of upwards. There are 5 new
Carter--Penrose diagrams.
We start with the simplest case.
\subsubsection{Metric signature $(-+++)$. The case $\Lambda=0$.}
In the considered case, zeroes of the conformal factor are defined by quadratic
equation
\begin{equation*}
q^2-2Mq-Q^2=0,
\end{equation*}
which has two roots:
\begin{equation*}
q_\pm=M\pm\sqrt{M^2+Q^2}.
\end{equation*}
It implies inequalities $q_+>0$ and $q_-<0$ for $Q>0$. Therefor, there is one
simple horizon for any $M$. Consequently, the Carter--Penrose diagram has
exactly the same form as for Schwarzschild black hole S12 in
Fig.\ref{fcarpenspherEM}.
\subsubsection{Metric signature $(-+++)$. The case $\Lambda>0$.}
Auxiliary function $\varphi(q)$ is the same (\ref{uvxgij}), but it has to be moved
on $3Q^2$ downwards. For positive cosmological constant, the qualitative
behavior of the auxiliary function is shown in Fig.\ref{fgremone}, {\em a}.
{\bf Spacelike singularity.} Under conditions:
\begin{equation} \label{ubvgfj}
\begin{aligned}
&\Lambda>0,\qquad& &\Upsilon<0,& {}\qquad&M>0, & {}\qquad\varphi_3\le0,&{} &&Q\ne0,
\\
&\Lambda>0,& &\Upsilon<0,& &M>0, & \varphi_3>0,& & {}\qquad 3&Q^2>\varphi_3,
\\
&\Lambda>0,& &\forall\Upsilon,& &M<0, & \varphi_3>0,& & {}\qquad 3&Q^2>\varphi_1,
\end{aligned}
\end{equation}
the conformal factor does not have roots. In this case, there is spacelike
singularity without horizons. Its Carter--Penrose diagram is S14 in
Fig.~\ref{fcarpenspherEM}. There is also time-reflected diagram.
{\bf Spacelike singularity with two horizons.} Under conditions:
\begin{equation} \label{ubvgff}
\begin{aligned}
&\Lambda>0,\qquad& \Upsilon<0,&{}\qquad&M>0,&\qquad\varphi_3>0,& & 0<3Q^2<\varphi_3,
\\
&\Lambda>0,& \forall\,\Upsilon,& &M<0, & & {}\qquad &0<3Q^2<\varphi_1,
\end{aligned}
\end{equation}
the conformal factor has two simple zeroes, and, consequently, two horizons.
Moreover the singularity at $q=0$ is spacelike. This solution is depicted by
diagram S13 in Fig.~\ref{fcarpenspherEM}. It can be either periodically extended
in spacelike direction, or we can identify the opposite horizons. This solution
describes white and black holes, which are periodically located in spacelike
directions. Moreover, if an observer is located in the domain IV, he has the
opportunity either to live forever, or to fall on one of two black holes.
{\bf Spacelike singularity with one double horizon.} Under conditions:
\begin{equation} \label{ubvgfk}
\begin{aligned}
&\Lambda>0,\qquad& \Upsilon<0,&{}\qquad&M>0,&\qquad\varphi_3>0,& & 3Q^2=\varphi_3,
\\
&\Lambda>0,& \forall\,\Upsilon,& &M<0, & & {}\qquad &3Q^2=\varphi_1,
\end{aligned}
\end{equation}
the conformal factor has one double zero, and the singularity is spacelike. This
global solution is given by the Carter--Penrose diagram S15 in
Fig.~\ref{fcarpenspherEM}, which can be periodically extended in spacelike
direction. This solution describes the collection of black and white (after
time reflection) holes. As in the previous case, an observer in domain II has
the choice either to live forever or to fall on one of two black holes. There is
also time-reflected diagram.
\subsubsection{Metric signature $(-+++)$. The case $\Lambda<0$.}
For negative cosmological constant and signature $(-+++)$, the auxiliary
function has previous form and is shown in Fig.\ref{fvfnelm}. To find zeroes,
its graphic must be moved downwards. Thus for all values of parameters:
\begin{equation} \label{unbhju}
\Lambda<0,\qquad\forall\,\Upsilon,\qquad\forall M,\qquad Q\ne0,
\end{equation}
it has one simple zero. This global solution is given by diagram S16 in
Fig.~\ref{fcarpenspherEM}. In this case we have asymptotically anti-de Sitter
black hole. Note, that, for positive mass, the space-time has degenerate and
oscillating geodesics, because local minimum exists for $q>0$. For $M<0$ these
geodesics are absent.
Thus, for metric signature $(-+++)$, there are only 5 topologically different
global solutions S12--S16. All singularities in this case are spacelike and
correspond either to black or white holes.
\subsection{Planar solutions $K^h=0$ \label{sdbhft}}
If Gaussian curvature of surface ${\mathbb V}$ equals to zero, then it is either the
Euclidean plane ${\mathbb R}^2$, or a cylinder, or a torus (after factorization).
Thus, there is spontaneous ${\mathbb I}{\mathbb S}{\mathbb O}(2)$ symmetry arising if the surface
${\mathbb V}$ is Euclidean plane ${\mathbb R}^2$. That is, the space-time metric becomes
invariant with respect to ${\mathbb I}{\mathbb S}{\mathbb O}(2)$ transformation group on the equations
of motion. In Schwarzschild coordinates $(\zeta,q,y,z)$, it is written in the form
(for $m=-q^2<0$, corresponding to signature $(+---)$):
\begin{equation} \label{uvxcfd}
ds^2=\Phi(q)d\zeta^2-\frac{dq^2}{\Phi(q)}-q^2d\Omega_{\textsc{p}},
\end{equation}
where
\begin{equation} \label{ubcvdf}
\Phi(q)=-\frac{2M}q+\frac{Q^2}{q^2}-\frac{\Lambda q^2}3,\qquad
d\Omega_{\textsc{p}}:=dy^2+dz^2.
\end{equation}
To draw Carter--Penrose diagrams for Lorentzian surface ${\mathbb U}$, we have to
analyse zeroes and asymptotics of conformal factor $\Phi(q)$. For $Q\ne0$,
we have the second order pole $Q^2/q^2$ at zero and asymptotic at infinity
\begin{equation*}
\Phi\approx-\frac{\Lambda q^2}3,\qquad q\to\infty
\end{equation*}
On intervals $(0,\infty)$ and $(-\infty,0)$, the conformal factor is smooth,
and, consequently, every global solution corresponding to one of these
intervals is smooth.
As for spherically symmetric solutions, we consider positive $M$ on both
intervals due to the symmetry transformation $(M,q)\mapsto(-M,-q)$.
We start with the simplest case.
\subsubsection{Metric signature $(+---)$. The case $\Lambda=0$.}
The conformal factor is
\begin{equation} \label{ubjsuu}
\Phi(q)=\frac{Q^2-2Mq}{q^2}.
\end{equation}
It has obviously one simple zero
\begin{equation*}
q=\frac{Q^2}{2M}.
\end{equation*}
Moreover, there are only two cases.
{\bf Timelike singularity and one horizon.}
Under conditions:
\begin{equation} \label{ubcvdi}
\Lambda=0,\qquad M>0,
\end{equation}
the conformal factor has one simple positive zero. The corresponding
Carter--Penrose diagram is P1 in Fig.~\ref{fcarpenpl}. This diagram has the
same form as the Schwarzschild black hole S12 but turned over on $90^\circ$.
\begin{figure}[hbt
\hfill\includegraphics[width=.3\textwidth]{fcarpenpl}
\hfill {}
\centering\caption{The Carter--Penrose diagram for planar solution for
$\Lambda=0$ and $M>0$.}
\label{fcarpenpl}
\end{figure
{\bf Naked singularity.}
Under conditions:
\begin{equation} \label{ubvcgf}
\Lambda=0,\qquad M\le0,
\end{equation}
positive roots of the conformal factor are absent, and we have naked singularity
S5 in Fig.~\ref{fcarpenspherEM}.
\qed
To find zeroes for nonzero cosmological constant $\Lambda\ne0$, we introduce
auxiliary function $\phi(q)$ representing the conformal factor for signature
$(+---)$ in the form
\begin{equation} \label{ubvuiu}
\Phi(q)=:\frac{\phi(q)+3Q^2}{3q^2},
\end{equation}
where
\begin{equation} \label{uvxbsf}
\phi(q):=-6Mq-\Lambda q^4.
\end{equation}
For the opposite signature, ${\sf\,sign\,}\widehat g=(-+++)$, it is needed to make
replacement $Q^2\mapsto-Q^2$. We see that on intervals $(0,\infty)$ and
$(-\infty,0)$ the number and type of zeroes of the conformal factor coincide
with zeroes of numerator $\phi(q)+3Q^2$. It means that auxiliary function must
be shifted either downwards (signature $(+---)$), or upwards (signature
$(-+++)$).
Auxiliary function (\ref{uvxbsf}) has two real roots:
\begin{equation*}
q=0,\qquad q=\sqrt[3]{-\frac{6M}\Lambda},
\end{equation*}
and two complex conjugate roots which do not interest us. Qualitative behavior
of the auxiliary function and corresponding conformal factor are shown in
Fig.~\ref{fvplanar}. Position of extrema of the auxiliary function is defined by
the equality
\begin{equation*}
\phi'(q)=-6M-4\Lambda q^3=0\qquad\Rightarrow\qquad q=\sqrt[3]{-\frac{3M}{2\Lambda}}.
\end{equation*}
We denote them by $q_5$ and $q_6$ for $\Lambda>0$ and $\Lambda<0$, respectively
(see.\ Fig.~\ref{fvplanar}, {\it a}).
\begin{figure}[hbt
\hfill\includegraphics[width=.9\textwidth]{fvplanar}
\hfill {}
\centering\caption{Auxiliary function $\phi(q)$ {\em (a)} and conformal factor
$\Phi(q)$ {\em (b)} при $M>0$. Maximum and minimum of the auxiliary function
are located ap points $q_5$ and $q_6$ for $\Lambda>0$ and $\Lambda<0$, respectively.}
\label{fvplanar}
\end{figure
The maximal and minimal values of the auxiliary function are denoted by
\begin{equation*}
\phi_{5,6}:=\phi(q_{5,6})=\frac92M\sqrt[3]{\frac{3M}{2\Lambda}}.
\end{equation*}
It is clear, that $\phi_5>0$ for $\Lambda>0$ and $\phi_6<0$ for $\Lambda<0$.
Detailed analysis show that Carter--Penrose diagrams for all planar solutions
for $\Lambda\ne0$ were already met in the spherically symmetric case. Therefor,
to save space, we give classification of all planar solutions in table
\ref{tplanar}.
\begin{table}[htb
\begin{center}
\begin{tabular}{|c|c|c|c|l|} \hline
$+---$ & $\Lambda>0$ & $\forall M$ & $Q\ne0$ & S7 \\ \hline
$+---$ & $\Lambda<0$ & $M>0$ & $0<3Q^2<-\phi_6$ & S11 \\ \hline
$+---$ & $\Lambda<0$ & $M>0$ & $3Q^2=-\phi_6$ & S10 \\ \hline
$+---$ & $\Lambda<0$ & $M>0$ & $3Q^2>-\phi_6$ & S9 \\ \hline
$+---$ & $\Lambda<0$ & $M\le0$ & $Q\ne0$ & S9 \\ \hline
$-+++$ & $\Lambda>0$ & $M\ge0$ & $Q\ne0$ & S14 \\ \hline
$-+++$ & $\Lambda>0$ & $M<0$ & $0<3Q^2<\phi_5$ & S13 \\ \hline
$-+++$ & $\Lambda>0$ & $M<0$ & $3Q^2=\phi_5$ & S15 \\ \hline
$-+++$ & $\Lambda>0$ & $M<0$ & $3Q^2>\phi_5$ & S14 \\ \hline
$-+++$ & $\Lambda<0$ & $\forall M$ & $Q\ne0$ & S16 \\ \hline
\end{tabular}
\caption{\label{tplanar} Classification of global planar solutions
for $\Lambda\ne0$.}
\end{center}
\end{table
Note, that diagrams S7, S9, S10 and S11 differ from diagrams S16, S14, S15 and
S13 by the turn on $90^\circ$ degrees, respectively.
\section{Hyperbolic global solutions \label{sbcvdg}}
If Gaussian curvature of surface ${\mathbb V}$ is negative, $K^h=-1$, then the surface
is two-sheeted hyperboloid ${\mathbb H}^2$, more precisely, the upper sheet of
two-sheeted hyperboloid (the Lobachevsky plane). It is the universal covering
surface for closed Riemannian surfaces of genus two and higher. If ${\mathbb V}={\mathbb H}^2$,
then the isometry group is the Lorentz group ${\mathbb S}{\mathbb O}(1,2)$. In this case, the
metric in Schwarzschild coordinates $(\zeta,q,\theta,\varphi)$ for signature $(+---)$
has the form
\begin{equation} \label{uvxbif}
ds^2=\Phi(q)d\zeta^2-\frac{dq^2}{\Phi(q)}-q^2d\Omega_{\textsc{h}},
\end{equation}
where
\begin{equation*}
\Phi(q)=-1-\frac{2M}q+\frac{Q^2}{q^2}-\frac{\Lambda q^2}3,\qquad
d\Omega_{\textsc{h}}:=d\theta^2+{\sf\,sh\,}^2\theta d\varphi^2.
\end{equation*}
The conformal factor for this metric differs from that in the spherically
symmetric case (\ref{uvcfre}) by the transformation
\begin{equation} \label{unvbfg}
\Phi\mapsto-\Phi,\qquad M\mapsto-M,\qquad Q^2\mapsto-Q^2,\qquad\Lambda\mapsto-\Lambda.
\end{equation}
In addition, transformation $Q^2\mapsto-Q^2$ corresponds to signature change of
the metric, $(+---)\mapsto(-+++)$. Since we have already described global
spherically symmetric solutions for all values of $M,Q^2$ and $\Lambda$, all
hyperbolic solutions are obtained from spherically symmetric ones by simple
rotation of Carter--Penrose diagrams by $90^\circ$, which corresponds to
transformation $\Phi\mapsto-\Phi$. In this way we get 16 additional
Carter--Penrose diagrams.
\section{Conclusion}
We assumed that four-dimensional space-time is the warped product of two
surfaces, ${\mathbb M}={\mathbb U}\times{\mathbb V}$, and find a general solution of Einstein's
equations with cosmological constant and electromagnetic field. These solutions
are well known locally and partly globally. We give classification of all global
solutions in the case when surface ${\mathbb V}$ is of constant curvature. Totally,
there are 37 topologically different global solutions.
These solutions in case {\sf B} have four Killing vector fields, three
of them corresponding to symmetry of the metric on constant curvature surface
${\mathbb V}$. They are generators of isometry groups ${\mathbb S}{\mathbb O}(3)$, ${\mathbb I}{\mathbb S}{\mathbb O}(2)$, and
${\mathbb S}{\mathbb O}(1,2)$ in cases when surface ${\mathbb V}$ is a sphere ${\mathbb S}^2$, Euclidean plane
${\mathbb R}^2$, and two-sheeted hyperboloid ${\mathbb H}^2$, respectively. The fourth Killing
vector
generalizes Birkhoff's theorem. In all cases, there is ``spontaneous symmetry
emergence'' because the existence of Killing vector fields was not assumed at
the beginning, and their appearance is the consequence of Einstein's equations.
Most probably, part of the constructed solutions are not satisfactory from
physical point of view. For example, for given signs in the Lagrangian and
signature of the metric $(-+++)$, the Carter--Penrose diagram for charged
black hole coincide with the Schwarzschild solution. However, the quadratic form
of momenta in the canonical Hamiltonian for physical degrees of freedom is not
positive definite (ghosts appearance), and this solution have to be discarded as
unphysical. Nevertheless, the given classification of global solutions of
Einstein's equations in the form of warped product of two surfaces is important,
because we must know what is to be discarded.
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\section{Introduction}
The electromagnetic flares associated with tidal disruption events (TDEs), in which a star is broken apart by the differential gravity of a supermassive black hole \citep[SMBH;][]{Hills75, Lidskii79, Rees1988, Evans1989}, offer a direct and promising route to constrain the mass and spin of otherwise dormant SMBHs. TDE candidates were first found in soft X-ray wavelengths \citep{Bade+96, Komossa+04}, where emission is often but not always quasi-thermal \citep{Saxton+17}. TDEs have also been discovered in thermal optical/UV radiation \citep{Gezari+06, van+11}; optical surveys dominate the current rate of TDE detection, finding $\mathcal{O}(10)$ events per year \citep{vanVelzen+20}. In principle, emission at all of these wavelengths can be used to constrain the handful of underlying event parameters, such as SMBH mass $M_\bullet$ and spin $a_\bullet$.
TDEs have two advantages over traditional techniques for measuring $M_\bullet$ and $a_\bullet$. First, in contrast to other methods, TDEs probe lower mass SMBHs, and possibly even intermediate mass black holes (IMBHs). Loss cone modeling predicts that the volumetric rate of TDEs is dominated by the lowest mass dwarf galaxies with a high black hole (BH) occupation fraction \citep{WangMerritt04, StoneMetzger16}. A volume-complete TDE sample would therefore measure the low end of the SMBH mass function. The gravitational influence radius of IMBHs is usually too compact for stellar dynamical mass measurements\footnote{IMBH candidates can be found dynamically \citep[e.g., ][]{Nguyen+19}, but only in the very nearest galaxies.}, and the faintness and short light-crossing times of AGN in dwarf galaxies have limited reverberation mapping mass measurements to a handful of systems \citep[see, e.g., ][for a review]{Reines+13,Greene+20}.
The second advantage TDEs bring to the black hole census is their unique ability to measure $a_\bullet$ in {\it quiescent} galactic nuclei. Far from the horizon, $a_\bullet$ is a higher order correction to the gravitational potential, challenging to observe even for resolved S-star orbits in the Milky Way \citep{Merritt+10}, and impossible for unresolved stellar orbits around extragalactic SMBHs. Iron K-$\alpha$ spectroscopy allows spin measurements in bright AGN \citep{Reynolds20}, which, however, may not be representative of the SMBH population \citep{BertiVolonteri08}.
The masses of the black holes causing TDEs are typically inferred from galaxy scaling relations, such as the $M_\bullet-\sigma$ or $M_\bullet-M_{\rm bulge}$ correlations \citep{Wevers+17, Wevers+19}, an indirect approach that is not well-calibrated for IMBHs. It is therefore desirable to measure $M_\bullet$ directly, from the light curves or spectra of the flares themselves. Multiple models exist for constraining $M_\bullet$ from a TDE optical/UV light curve \citep{Guillochon+14, Mockler+19, Ryu+20}, but challenges for this approach include the unknown power source \citep{Loeb+1997, Piran+15, MetzgerStone16} and three-dimensional geometry \citep{Guillochon+14, Shiokawa+15, Dai+18} of the optical/UV photosphere. Likewise, it is not clear how (or even whether) $a_\bullet$ will affect the optical/UV emission\footnote{One possibility, not yet quantified in light curve modeling, is the sub-leading role spin plays in setting the self-intersection radii of eccentric debris streams \citep{Wevers+17}.}. Neither scaling relations nor optical/UV light curve fitting have produced constraints on $a_\bullet$.
These limitations motivated us to use X-ray continuum fitting to determine $M_\bullet$ and $a_\bullet$ in TDEs. To do so, we extended stationary general relativistic "slim disk" accretion models from stellar-mass black holes to SMBHs for the first time \citep[][hereafter W20]{Wen+20}. These slim disk models extend standard thin disk accretion theory \citep{ShakuraSunyaev1973,NovikovThorne1973} to accretion rates
comparable to or larger than approximately ten per cent of the Eddington limit, where sub-Keplerian gas motion and advective heat losses can no longer be neglected \citep{Abramowicz+1988}. We ray-traced the trajectories of photons from the image plane to the disk surface, including gravitational redshift, Doppler, and lensing effects self-consistently.
In W20, we applied these general relativistic slim disk models to two well-studied SMBH TDEs: ASASSN-14li \citep{Holoien+16a} and ASASSN-15oi \citep{Holoien+16b}, placing strong constraints on $M_\bullet$ for both flares and on $a_\bullet$ for ASASSN-14li. In this paper, we apply our models to X-ray observations of 3XMM~J215022.4$-$055108, hereafter ``J2150.'' The J2150 flare is a luminous X-ray outburst in a small optical source
adjacent to the large, barred lenticular galaxy 6dFGS~gJ215022.2$-$055059 at $z=0.055$ \citep[][hereafter L18]{Lin+18}. J2150, one of the most compelling IMBH TDE candidates to date, was first detected by L18 in the XMM-{\it Newton}\, X-ray source catalog. If the position is not a chance association, the host is a star cluster
of mass $\sim 10^7 M_\odot$ and a half-light radius of about 27 pc, offset by $\approx 12$ kpc from the lenticular's center \citep[][hereafter L20]{lin+20}. L18 fit standard thin disk accretion models to this flare and estimated an IMBH mass of $5\times 10^4 M_\odot \lesssim M_\bullet \lesssim 1\times 10^5 M_\odot$. Their work hints that $a_\bullet$ is large, but they only consider two possible values for it in their fits. In this paper, we re-reduce and reanalyze existing multi-epoch X-ray observations and simultaneously fit the continuum with our relativistic slim disk model to constrain $M_\bullet$ and $a_\bullet$.
In \S \ref{sec:method}, we summarize our theoretical model for quasi-thermal X-ray emission from TDE accretion disks. In \S \ref{sec:data}, we review the X-ray observations of J2150 and our data reduction methods. In \S \ref{sec:results}, we present our X-ray spectral fits, the resulting constraints on the ($M_\bullet, a_\bullet$) plane, sources of uncertainty, and implications for particle physics. We summarize in \S \ref{sec:conclusions}. Throughout, we assume a flat cosmology with $H_0 = 69.6~\rm{km}~\rm{s}^{-1}\rm{Mpc}^{-1}$, $\Omega_M=0.29$ and $\Omega_\Lambda=0.71$.
\section{METHODOLOGY}
\label{sec:method}
In this work, we follow the general procedures of W20. We use the general relativistic stationary slim disk to model the dynamic TDE accretion disk, a color-modified, multi-color black body model \citep{DE18} to calculate the local X-ray emission, and a geodesic ray-tracing code \citep{JP11}, which includes gravitational redshift, Doppler, and lensing effects self-consistently, to calculate the synthetic X-ray spectrum.
For high spin BHs, the radiative efficiency $\eta$ can be up to 0.42, which would allow the radiation to take away a significant part of the angular momentum \citep{Abramowicz+1996}. Here, we update our code to account for the angular momentum loss by radiation in the disk equations. For the reader's convenience, we write the underlying slim disk equations in the Appendix \ref{app:slimdisk} and explore the importance of angular momentum loss by radiation. Neglecting this angular momentum loss produces changes in the local effective temperature that are always $<10\%$, and usually $<5\%$ (see Fig. 8 in Appendix \ref{app:slimdisk}). The effect is maximized for black holes with spins near the Thorne limit ($a_\bullet = 0.998$; \citealt{Thorne74}) and sub-Eddington accretion rates.
A fully circularized TDE debris stream would form an accretion disk with an initial radius $R_{\rm c}=\frac{2R_{\rm t}}{\beta}$, where the tidal disruption radius is
\begin{align}
R_{\rm t} =& R_\star \left( \frac{M_\bullet}{M_\star} \right)^{1/3}\\ \notag \approx& 1.0 \times 10^3 R_{\rm g} \left(\frac{M_\bullet}{10^4 M_\odot} \right)^{-2/3} \left(\frac{M_\star}{M_\odot} \right)^{-1/3} \left(\frac{R_\star}{R_\odot} \right),
\end{align}
to within factors of order two \citep{Guillochon13}. Here $\beta = \frac{R_{\rm t}}{R_{\rm p}}$ with $R_{\rm p}$ the periastron radius of the star's orbit, $R_{\rm g} = GM_\bullet / c^2$ is the gravitational radius, and $M_\star$ and $R_\star$ are the mass and radius, respectively, of the disrupted star.
As twice the tidal radius, in an IMBH system, is about one thousand gravitational radii away from the IMBH (and thus too cold to produce significant X-ray flux), we set the outer edge of the disk to $\le$ 600 gravitational radii. The error on the flux caused by the choice of outer edge is $< 1\%$ (see appendix \ref{rout_rin}). However, the choice speeds up our calculations by at least 4 times. When we do the ray tracing, we cut off the disk at the innermost stable circular orbital (ISCO), due to a singularity inside the ISCO when calculating the spectral hardening factor. The error on the flux caused by the different choices of inner edge is less than $2\%$ (see appendix \ref{rout_rin}). For an accretion disk around a high-spin BH, the error is less than $0.5\%$ (see Appendix \ref{rout_rin}).
We do not assume any prior on the disk accretion rate from TDE gas fallback hydrodynamic simulations; instead, we treat the disk accretion rate as a free parameter for the fit in each epoch of data. The free parameters for the slim disk model are $M_\bullet$, $a_\bullet$, $\dot m_i$, and $\theta$. Here, the subscript index $i$ denotes the $i$-th observational epoch and $\theta$ is the inclination angle of the accretion disk with respect to our line of sight. The X-ray spectrum is subject to circumnuclear and interstellar extinction, so we introduce the extinction parameter $N_{\rm H,i}$.
The three largest assumptions in our model, as applied to TDEs, are that (1) the dynamic inner disk can be approximated by a time series of stationary accretion disks, (2) the inner accretion disk is axisymmetric, and (3) the inner disk is always aligned with the BH equatorial plane. The first assumption is true at all times, because, at each timestep, the accretion rate across the disk is nearly the same. At early times, this is because the viscous timescale is much shorter than the mass fallback timescale.
At late times, this is because the decreased importance of mass fallback has caused the disk to settle into a self-similar spreading state, where mass accretion rates are almost constant in the inner disk, the source of the X-ray emission fit with our model.
The latter two assumptions are questionable in the earliest phases of TDEs, but are likely better for later observational epochs. Initially, the gas from the disrupted star returns to the BH on highly eccentric trajectories, which dissipate excess kinetic energy in shocks, thereby circularizing into an accretion disk. The efficiency and progression of this circularization process is currently unknown for realistic TDE parameters, and if the inner disk remains significantly non-axisymmetric, it will bias the results of our continuum fitting\footnote{Note that the slim disk models from W20 account self-consistently for sub-Keplerian fluid motion, one aspect of incompletely circularized accretion flows, but not for the large-scale apsidal misalignment characteristic of a globally eccentric flow.}. However, we note that for IMBHs, the characteristic circularization radius $R_{\rm c} \sim 1000 R_{\rm g}$ is about two orders of magnitude larger than the radii that emit most of the observed X-ray radiation, and matter that has reached scales of $\sim R_{\rm g}$ has thus dissipated about two orders of magnitude in orbital energy through shocks or magnetized turbulence. Even if early-time epochs in TDE disks feature globally eccentric structures, it seems reasonable to assume -- especially in the case of IMBHs -- that the X-ray emitting inner annuli will have mostly circularized.
Likewise, orbital dynamics of the loss cone suggest that disrupted stars approach BHs from a quasi-isotropic distribution of directions, implying that TDE disks should be born with a substantial tilt \citep{StoneLoeb12} that is not accounted for in our models. The most immediate observational consequence of disk tilt will be a softening of the X-ray spectrum (as the tilted analogue of the ISCO sits further out than the ISCO itself); there may also be more complex lensing effects related to the geometry of null geodesics in the axisymmetric Kerr spacetime. This tilt will decay over time due to the onset of the Bardeen-Petterson effect \citep{StoneLoeb12}, inter-annulus torques in a globally precessing thick disk \citep{Franchini+16}, and torques from returning debris streams impacting a misaligned, precessing accretion flow \citep{XiangGruess+16, ZanazziLai19}. Because this effect may potentially bias the results of early-time observations, we will perform two different multi-epoch fits: one that uses all observational epochs, and one which only employs the late-time epochs, when we can be more confident that the inner disk has aligned itself (this alternate fit focused on late epochs may also be more trustworthy with regards to the assumption of axisymmetry).
\section{Data Reduction}
\label{sec:data}
We will include in our analysis pointed observations obtained with both the XMM-{\it Newton}\, \citep{Jansen+01} and the {\it Chandra}\, \citep{weisskopf+02} satellites. Table~\ref{tab:xmmchan} lists some properties of these observations. Note that all these observations were also used by L20 but our analysis differs from theirs in important points. First, we employ Poisson statistics in all of our spectral fits \citep{Cash1979}. As shown in the work of \citet{Kaastra17}, the use of $\chi^2$ statistics in spectral fits even for 20-30 counts per spectral bin will bias the results of the fit. Second, we correct for the influence of an interloping nearby source on the XMM-{\it Newton}\, spectral fits (see \ref{sec:chan} below).
\subsection{{\it Chandra}\,data}
\label{sec:chan}
Two {\it Chandra}\, observations of J2150 exist (see Table~\ref{tab:xmmchan} for more information). We used {\sc ciao} version 4.12 for our {\it Chandra}\, data analysis (\citealt{Fruscione+06}). During the first observation obtained in 2006, the source was observed serendipitously on the ACIS-I CCD array and at a large off-axis angle where the {\it Chandra}\, point spread function is degraded with respect to that on-axis. We extracted the source and background spectrum using circular regions with a radius of 7.7 and 60\arcsec, respectively.
During the on-axis {\it Chandra}\, observation of 2016 with the ACIS-S3 CCD, in addition to J2150, another source was clearly detected at $\alpha$: 21:50:22.2, $\delta$: 05:50:58.7 (J2000), a location consistent with the nucleus of the nearby, barred lenticular galaxy (6dFGS~gJ215022.2$-$055059; see L18 and L20; see our Fig.~\ref{J2150chan}). This location is 11.2\arcsec\, away from the position of J2150. As this source position is close enough, given the size of the XMM-{\it Newton}\, point spread function, to influence the XMM-{\it Newton}\, TDE spectra, we assessed its flux and spectral shape to treat it as background in the spectral fitting of the XMM-{\it Newton}\, data (assuming its flux and spectrum are constant).
We extracted the spectrum of both this interloper and J2150 using a circular region with a radius of 1.5\arcsec. We used a source-free, nearby circular region with a 1\arcmin\, radius on the same detector to estimate the background spectrum. For all our fits, we first fit the background spectrum separately with two power law models. Next, to correct the TDE spectrum for the background contribution, the best model-fit values for the background model are kept fixed during the fit of the TDE spectrum.
\subsection{XMM-{\it Newton}\,data}
\label{sec:xmm}
We run the {\sc SAS} v18 (20190531) tools under the HEASOFT {\sl ftools} software version 6.26.1 to extract the spectra of J2150 and filter both the EPIC pn as well as the MOS detector data. All the observations are done with the pn and the MOS detectors in Prime Full Window mode, providing a time resolution of 73.4~ms and 2.6~s for the pn and MOS detectors, respectively. We filtered the pn and MOS data for periods of enhanced background radiation, where we require that the 10--12 keV detection rate of pattern 0 events is $<$~0.4 counts s$^{-1}$ for the pn and that the $>$~10 keV detection rate of pattern 0 events is $<$~0.35 counts s$^{-1}$, for both MOS detectors. The effective exposure time for each observation after filtering is given in Table~\ref{tab:xmmchan}. To investigate if pile-up is important, we use the {\sc SAS} command {\sc epatplot} to compare the observed and expected number of single and double event pattern as a function of photon energy. We conclude that pile-up is not important.
\begin{figure}
\plotone{image-chandra-on-axis2.jpeg}
\caption{{\it Chandra}\, 0.1--7 keV image from observation ID 17862 of the field around J2150 showing the nearby interloper that contaminates the XMM-{\it Newton}\, spectra of J2150. Note that the circle centered on the J2150 position has a radius of 1 pn pixel (i.e., 4.1\arcsec). In our spectral fits of the XMM-{\it Newton}\, data we treated this interloper as an additional contribution to the background, assuming its flux and spectral shape did not vary over time.
}
\label{J2150chan}
\end{figure}
We extracted the spectrum of J2150 using a circular aperture of 30\arcsec\, radius centered on the optical position of J2150 for the pn and the both MOS detectors, except for the pn data of observation ID 0603590101, where we used a radius of 20\arcsec\, to avoid bad pixels falling into the source extraction region. To extract the background spectrum, we used a circular region on the same CCD as close to the position of J2150 as possible with a radius of 75\arcsec, except for the pn observations of observation ID 0603590101 and 0823360101, where we had to use a circle with radius of 45\arcsec\, to avoid bad pixels falling into the extraction region.
\begin{deluxetable*}{lrcc}
\tablecaption{XMM--{$\it Newton$} (top part of the Table) and {\it Chandra}\, (bottom) observations of 3XMM~J215022.4$-$055108\, used in this paper.}
\tablewidth{0pt}
\tablehead{
\colhead{Observing } & \colhead{Start date \& time} & \colhead{Exp time } & \colhead{Counts$^{\dagger}$} \\
\colhead{ID } & \colhead{[UTC]} & \colhead{pn/MOS1/MOS2 [ks]} & \colhead{pn/MOS1/MOS2}
}
\startdata
0404190101 & 2006-05-05 12:24:35 & 22.2/49.6/49.6 & 2605/3559/3000 \\
0603590101 & 2009-06-07 07:53:31 & 40.8/68.3/67.9 & 3724/1687/1629 \\
0823360101 & 2018-05-24 08:28:11 & 40.3/57.8/57.8 & 1179/327/362 \\
\hline
6791 & 2006-09-28 20:49:55 & 100.6 & 4801 \\
17862 & 2016-09-14 07:31:09 & 77.1 & 152
\label{tab:xmmchan}
\enddata
\tablecomments{$^{\dagger}$ Counts detected after filtering on photon energy between 0.3--10 keV for the pn, MOS1, MOS2 detectors and 0.3--7 keV for the {\it Chandra}\, ACIS-I and -S detectors for observation IDs 6791 and 17862, respectively. }
\end{deluxetable*}
\section{Results and Discussion}
\label{sec:results}
Throughout this paper, we fit the spectra by using {\sc XSPEC} version 12.11.1 \citep{Arnaud1996} applying Poisson statistics (\citealt{Cash1979}; {\sc C-stat} in {\sc XSPEC}). We quote all the parameter errors at a $1\sigma$ $(68.3\%)$ confidence level (CL), using the method $\rm{Statistic} = \rm{Statistic_{best-fit}} + \Delta\,C$ \citep{Arnaud1996} and assuming
$\Delta Cstat = 1.0$ and $\Delta Cstat = 2.3$ for single and two parameter models, respectively \citep{Mao+18}. We explore the statistical properties of the fitting results further in Appendix~\ref{cstatisitc}.
\subsection{Model Fitting}
We fitted the {\it Chandra}\, observation ID 17862 spectrum of the interloper
and found it to be well-fit ($Cstat/\nu=42.22/38$) with a power law with index 1.7.
We add this power law as an additional background component to our XMM-{\it Newton}\, spectral fits of J2150.
The quasi-thermal slim disk model of the previous section forms the basis for our X-ray spectral fitting and parameter estimation for J2150. We fit the multi-epoch spectra of J2150 by combining our slim disk model with two absorption parameters, $N_{\rm H}$, one fixed at $2.6\times 10^{20}\,\rm{cm}^{-2}$ at redshift $z=0$ (L18) to account for Galactic absorption, the other allowed to float using a redshift fixed at $z=0.055$ to describe the effect of any extinction in the host, TDE, and perhaps the nearby, barred lenticular (although TDE may be in front of it). The general absorption models {\sc phabs} and {\sc zphabs} in XSPEC \citep{Arnaud1996} are added as multiplication models to our slim disk model to account for these components.
Using Cash statistics \citep{Cash1979}, we need to fit the source and background together. As explained above, the background spectrum in the {\it Chandra}\, observations is well-fit using two power law models. The XMM-{\it Newton}\, background can be well fitted with two power laws plus two Gaussian emission lines (arising from the satellite, and there is only one Gaussian emission line for the pn spectra). If we would ignore the two emission lines at about 1.5 and 1.8 keV, the results would be biased favoring high spin values. During the fit of J2150, we fix the respective background models at their best-fit values from the background-only fit to accelerate the calculation. The final fit-function is a combination of the background fit-function plus our absorbed slim disk model. The complete fit-function in XSPEC
is {\sc po+po+po+agauss+agauss+phabs( zphabs(slimdisk))}, with the first {\sc po} accounting for the interloper contamination. However, for the {\it Chandra} ~spectra, which are neither affected by the interloper nor by the background Gaussian emission lines, we fix the normalization of the interloper power law and the two Gaussian emission lines to 0.
Following the procedure of W20, we fit the five spectra simultaneously, by fixing ($M_\bullet$, $a_\bullet$) at the grid value, while allowing all five accretion rates ($\dot m_i$), all five absorption parameters ($N_{\rm H,i}$), and the one inclination ($\theta$) to float. In order to evaluate the significance of each ($M_\bullet$, $a_\bullet$) pair, we minimize $Cstat$ for each ($M_\bullet, a_\bullet$) in our ($M_\bullet$, $a_\bullet$) grid. The parameter priors are listed in Table 2.
In the initial fit, we use a spectral hardening factor $f_{\rm c}$ as calculated by \citep{DE18} (see W20 for more detail; we call this the fiducial treatment of $f_{\rm c}$, with more details in Appendix \ref{f-fc}). In all the fitting, we allow for the possibility that the absorption local to the TDE changes with time by keeping the host$+$TDE absorption component $N_{\rm{H,i}}$ as a free parameter, while the absorption from Milky Way has been fixed at $2.6\times 10^{20} ~\rm{cm}^{-2}$ (L18).
\begin{deluxetable}{ccccccc}
\tablecaption{Parameter Priors for Fitting X-Ray Spectra.}
\tablewidth{0pt}
\tablehead{
\colhead{$M_\bullet^a$} & \colhead{$a_\bullet^a$} & \colhead{$\dot m_i^b$} & \colhead{$\theta$} & \colhead{$N_{\rm H, i}$} & \\
\colhead{$[10^4 M_\odot]$} & \colhead{} & \colhead{$[\rm Edd]$} & \colhead{$[^{\circ}]$} & \colhead{$[10^{21}{\rm cm}^{-2}]$} &
}
\startdata
[0.15, 10] & [-0.9, 1.0) & [0.05, 100] & [5, 90] & (0, 1] & \\
\enddata
\label{tab:prior}
\tablecomments{$^a$For individual epoch fits, $M_\bullet$ and $a_\bullet$ are discretely sampled at each grid point across the given ranges.~$^b$We use linear interpolation to estimate spectra for accretion rate values $\dot{m_i}$ between discrete grid points.
The accretion rate
is calculated by assuming $\eta=0.1$ and listed in dimensionless Eddington units. }
\end{deluxetable}
In preliminary initial fits where we did not require $M_\bullet$ and $a_\bullet$ to be the same across all five epochs, we found that the last three of the five epochs can be well fitted and constrain the $M_\bullet$ and $a_\bullet$ to values that are mutually consistent within errors. However, the first two epochs only find their best fit at a high accretion rate and an extremely high $a_\bullet$. The constraint on $a_\bullet$ derived using the first two epochs is $> 3\sigma$ away from the value derived using the last three epochs. This inconsistency stems from the fact that, except for extremely high values for $a_\bullet$, the disk is not bright enough to fit the first two epochs under the initial assumptions.
This result suggests that the source was accreting at a highly super-Eddington rate during the first two epochs, as is theoretically predicted for main sequence-IMBH disruptions \citep[e.g.,~][]{Rees1988, Chen+18}. For such a highly super-Eddington accretion rate, the X-ray luminosity of the slim disk is virtually insensitive to the actual super-Eddington accretion rate value, but it does depend on the choice of $f_{\rm c}$ (W20). We provide more information on the role of $f_{\rm c}$ in this regime in Appendix \ref{f-fc}, but note here that the fiducial prescription of \citet{DE18} is only tailored for sub-Eddington accretion disks.
There are two possible reasons for why the disk is not bright enough during the first two epochs: (1) the fiducial $f_{\rm c}$ value we assume is too low; (2) the redshift $z$ to J2150 is overestimated. With these options in mind, we refit the data with two different models: Model 1, where we adopt the same fiducial $f_{\rm c}$ prescription but allow it to float for highly super-Eddington (i.e., the first two) epochs; Model 2, where we keep the fiducial $f_{\rm c}$ prescription, but allow the redshift of the source $z$ to float.
\begin{figure*}
\plotone{J215sp.pdf}
\caption{Simultaneous slim disk fits to three XMM--{\it Newton} and two {\it Chandra}\ spectra for J2150 from early (Epoch 1) to late (Epoch 5) times over a twelve-year period. The spectra are from the XMM--{\it Newton} PN (green line), MOS1 (red line) and MOS2 (blue line) detectors, which are most sensitive over the 0.3--10 keV range, and the {\it Chandra}\, ACIS-I and -S detectors, which are most sensitive over the 0.3--7 keV range. All spectra include the background spectral contribution. The data are binned so that there is at least one X-ray photon per energy bin. These two features of the data allow us to employ Cash statistics \citep{Cash1979} in fitting the spectra. To avoid the plot becoming too crowded we show the Model~1 fit to only the MOS1 data for the top three panels, excluding the pn and MOS2 data. Each panel shows the best-fit slim disk (dashed line), the interloper power law (dot dashed line), the background (power law + power law + agauss + agauss) (dotted line), and the combined model (solid line). The interloper power law index and normalization are fixed to the same values for all the XMM-{\it Newton}\, epochs derived from the fit to the interloper spectrum in the {\it Chandra}\, data from observation ID 17862. The background spectral parameters are also fixed to the best-fit values derived from fits to the background spectra separately (see text). The accretion rate is allowed to float between the epochs. The sub-plot panels with Y-axis label ``Error" denote $\rm{(model - data)/ \sigma}$ for each spectral energy bin.
We provide the best-fit results in Table~\ref{J2150p}. }
\label{J2150sp}
\end{figure*}
\begin{deluxetable*}{cccccccccccccc}
\tablecaption{The best-fit parameters to J2150's five epochs of XMM-{\it Newton}\, and {\it Chandra}\, spectroscopic data derived using three different models.}
\tablewidth{0pt}
\tablehead{
\colhead{ }&& \colhead{XMM 1} & & \colhead{Chandra 1} & & \colhead{XMM 2} & & \colhead{Chandra 2} & & \colhead{XMM 3} \\
\colhead{Date } && \colhead{2006-05-05} & & \colhead{2006-09-28} & & \colhead{2009-06-07} & & \colhead{2016-09-14} & & \colhead{2018-05-24}
}
\startdata
Model 1: flexible $f_{\rm c}$\\
$N_{\rm H}$ $[10^{20}{\rm cm}^{-2}]$ && $3.6\pm1.9$ && $0^{+0.7}$ && $3.8\pm0.5$ && $0^{+0.7}$ &&$1.3\pm0.9$\\
$f_{\rm c}$ && $2.4_{-0.06}^{+0}$ && $=f_{\rm c1}$ && - && - &&-\\
$\theta$ $ [{^\circ}]$ && $5.0_{-0}^{+10}$ && $=\theta_1$ && $=\theta_1$ &&$=\theta_1$ &&$=\theta_1$ &&\\
$\dot m^a$ ${\rm [Edd]}$ && $56\pm20$ && $5.1\pm1.0$ && $1.8\pm0.1$ && $0.35\pm0.02$ && $0.31\pm0.02$\\
$M_\bullet$ $[10^4{\rm M}_\odot]$ &&$1.75$&&$=M_{\bullet,1}$&&$=M_{\bullet,1}$&&$=M_{\bullet,1}$&&$=M_{\bullet,1}$\\
$a_\bullet$ &&$0.8$&&$=a_{\bullet, 1}$&&$=a_{\bullet, 1}$&&$=a_{\bullet, 1}$&&$=a_{\bullet, 1}$\\
$Cstat/\nu$ &&$298.94/270$ && $224.07/180$ && $271.17/283$ && $60.23/58$ &&$272.16/240$\\
\hline
Model 2: free $z$\\
$N_{\rm H}$ $[10^{20}{\rm cm}^{-2}]$ && $1.6\pm1.4$ && $0.3^{+1.0}_{-0.3}$ && $2.1\pm0.4$ && $0^{+0.7}$ &&$1.5\pm0.9$\\
$z$ && $0.017\pm0.004$ && $=z_1$ && $=z_1$ && $=z_1$ &&$=z_1$\\
$\theta$ $ [{^\circ}]$ && $61\pm26$ && $=\theta_1$ && $=\theta_1$ &&$=\theta_1$ &&$=\theta_1$ \\
$\dot m^a$ ${\rm [Edd]}$ && $37\pm28$ && $10.0\pm1.9$ && $2.9\pm0.2$ && $0.72\pm0.05$ && $0.64\pm0.04$\\
$M_\bullet$ $[10^4{\rm M}_\odot]$ &&$0.3$&&$=M_{\bullet,1}$&&$=M_{\bullet,1}$&&$=M_{\bullet,1}$&&$=M_{\bullet,1}$\\
$a_\bullet$ &&$-0.7$&&$=a_{\bullet, 1}$&&$=a_{\bullet, 1}$&&$=a_{\bullet, 1}$&&$=a_{\bullet, 1}$\\
$Cstat/\nu$ &&$301.20/270$ && $227.31/180$ && $270.43/283$ && $59.55/58$ &&$269.68/240$\\
\hline
Model 3: diskbb\\
$N_{\rm H}$ $[10^{20}{\rm cm}^{-2}]$ && $0.0^{+0.2}$ && $0^{+0.001}$ &&$2.1\pm 0.9$ && $0.0^{+7.2}$ &&$17.2\pm 5.2$ \\
$T_{\rm disk}[{\rm keV}]$ &&$0.261\pm0.003$ && $0.262\pm0.003$ &&$0.215\pm0.005$ && $0.13\pm0.01$ &&$0.103\pm0.008$\\
$N_{\rm disk}^b$ && $14.4\pm 0.8$ && $9.6\pm 0.7$ &&$10.3\pm 1.5$ && $23\pm 9$ &&$271\pm 170$ \\
$Cstat/\nu$ && $302.77/273$ && $236.71/179$ &&$270.4/282$ && $58.11/57$ && $254.87/239$ \\
\enddata
\tablecomments{For the slim disk model, the errors on the parameters are calculated keeping $M_\bullet$ and $a_\bullet$ fixed at their best-fit values. The total $Cstat/\nu$ for the combined five-epoch fit is $1126.58/1031=1.093$ and $1128.09/1031=1.094$, for Model 1 and Model 2, respectively. For the {\sc diskbb} model, the total $Cstat/\nu$ is $1122.86/1030=1.088$. The total $Cstat$ of the slim disk fit is close to the expected $Cstat=1124.1\pm1.4$ (\citealt{Kaastra17}), indicating a good fit to the data. $^{\rm a}$ Accretion rate (in dimensionless Eddington units) is calculated by assuming a radiative efficiency of $\eta=0.1$. $^{\rm b}N_{\rm disk}$ is defined as $(R_{\rm in}/D_{10})^2\cos\theta$, where $R_{\rm in}$ is the inner disk radius in km, $D_{10}$ is the distance to the source in units of 10 kpc, and $\theta$ is the inclination angle of the disk.
In the fit to the XMM-{\it Newton}\, spectra, we add a power-law component, $\Gamma=1.7$ and $A_{\rm pl}=1.1\times10^{-6}$ $\rm {photons ~s^{-1}~cm^{-2}~keV^{-1}}$, to account for the contamination from the interloper which is the nuclear source of the nearby brighter barred lenticular galaxy. The Milky Way $N_{\rm H}$ absorption is fixed at $2.6\times10^{20}$ ${\rm cm}^{-2}$ (L18). The $N_{\rm H}$ shown in the Table is for the TDE, its host system, and possibly the nearby, interloping lenticular galaxy at $z=0.055$.}
\label{J2150p}
\end{deluxetable*}
\begin{figure*}[ht!]
\gridline{ \fig{FCj2150.pdf}{0.5\textwidth}{Model 1: $f_{\rm c1}\in [2.0, 2.4]$ is allowed to float.}
\fig{redshift.pdf}{0.5\textwidth}{Model 2: $\rm{z}$ is allowed to float. }
}
\caption{Black hole mass $M_\bullet$ and spin $a_\bullet$ for our slim disk Model 1 and 2 (see text). We calculate the $\Delta Cstat$ across a model grid in the $(M_\bullet, a_\bullet)$ plane (grid points are indicated by vertices of the black lines) and then fill in the color contours by linear interpolation. The left panel shows one choice for modeling the spectral hardening factor $f_{\rm c}$, where it is fixed with a fiducial, theory-driven value \citep{DE18} for the sub-Eddington epochs for which it is calculated, but allowed to float for super-Eddington epochs (i.e., Epochs 1 and 2). The host redshift is taken to be that of the nearby, barred lenticular galaxy at $z=0.055$. The right panel shows the result of treating $z$ as a free parameter, but using the theoretical \citet{DE18} treatment of $f_{\rm c}$ at all epochs. The total $Cstat$ for the two models are 1124.36 and 1125.99, respectively.
The left panel constrains $M_\bullet$ and $a_\bullet$ to $1.75^{+0.45}_{-0.05}\times 10^4 M_\odot$ and $0.8^{+0.12}_{-0.02}$, respectively, within 1$\sigma$.
The right panel constrains
$M_\bullet$ and $a_\bullet$ to be $3.0^{+4.0}_{-0.5}\times 10^3 M_\odot$ and $-0.7^{+1.0}_{-0.1}$, respectively.
For the best fit $M_\bullet=3.0\times 10^3 M_\odot$ and $a_\bullet=-0.7$ in the right panel, we obtain a redshift of
$z=0.017 \pm 0.004$ ($1\sigma$ CL).
These two panels together show that regardless of uncertainties on $z$ and $f_{\rm c}$, this BH is an IMBH with mass less than $2.2\times 10^4 M_\odot$. The constraint on $M_\bullet$ is lower than that predicted by L18. Our spin constraint is the first of its kind.
}
\label{J2150}
\end{figure*}
For Model 1, we show the best fit results in
Fig.~\ref{J2150sp} and the left panel of Fig.~\ref{J2150} and the first section of Table~\ref{J2150p}. Here, $f_{\rm c1}$ is a free parameter with a flat prior between $2.0-2.4$. This prior is only applied to the first two epochs. For the last three epochs, $f_{\rm c}$ is calculated as in our fiducial method following the prescription provided in \citet{DE18}. The spectra, for each of the five epochs are well-fit with $Cstat/\nu$ $< 1.3$. The total $Cstat$ for the five epochs together is 1126.58, which is close to that expected ($Cstat=1124.1\pm1.4$; calculated following \citealt{Kaastra17}), indicating a good fit to the data. The mass accretion rate we derive to explain the Epoch 1 spectrum is highly super-Eddington even when using the high $f_{\rm c}$ value, while it is more mildly super-Eddington at Epoch 2.
For Model 2, we list the best fit results in
the second section of Table~\ref{J2150p} and show the constraint on $M_\bullet$ and $a_\bullet$ in the right panel of Figure~\ref{J2150}. Here, the redshift is a free parameter (the redshift for our {\sc slimdisk} model and that of {\sc zphabs} are required to be the same). The best fit value of ($M_\bullet, a_\bullet)$ is $(3\times10^3M_\odot,-0.7$). The total $Cstat=1125.99$, similar to that of Model 1. The fit of Model 2 prefers a lower redshift, a lower $M_\bullet$, and a lower $a_\bullet$ than for Model 1. In other words, the observed spectrum can either be described by a softer intrinsic source at lower redshift or by a harder intrinsic source at higher redshift. A lower $M_\bullet$ implies a higher disk temperature, which is counteracted by the larger inner disk radius caused by the retrograde spin.
For comparison with the work of L18, we also fit the spectra with a simple {\sc diskbb} model. The fitted temperatures for the last four epochs are lower than in L18. This difference may come from the different statistics we employ, and/or from our different $N_{\rm{H}}$ treatment (they tie the $N_{\rm{H}}$ of four epochs together, while we let them float for each epoch).
Our {\sc diskbb} fit yields a lower fitted $Cstat=1122.86$ with one more free parameter than the slim disk model fit. The better fit for the slim disk model can be explained by its improved fit to Epochs 1, 2, and 5. The best fit of the {\sc diskbb} model finds a larger absorption $N_{\rm {H}}=17.2\times10^{20}~ \rm{cm}^{-2}$ (10 times bigger than that of slim disk model) for Epoch 5. Even with stronger priors, e.g.,~requiring the same $M_\bullet$, the same $a_\bullet$ and the same $\theta$ for all epochs, the slim disk yields a better fit for the two early epochs, with $\Delta Cstat > 10$ lower than for the {\sc diskbb} model. The slim disk also finds a decreasing mass accretion rate, which is consistent with the expectation that the mass accretion rate in TDEs should decrease after rising to a peak.
\subsection{Event Parameters}
Fig.~\ref{J2150} shows the constraints on $M_\bullet$ and $a_\bullet$ for Models 1 and 2. Here, we assume $\Delta Cstat = 2.3$ and $\Delta Cstat = 6.2$ correspond to $1\sigma$ and $2\sigma$ confidence levels (CLs), respectively. For Model 1, we constrain $M_\bullet$ and $a_\bullet$ to $1.75^{+0.45}_{-0.05}\times 10^4$ $M_\odot$ and $0.8^{+0.12}_{-0.02}$, respectively.
For Model 2, we
constrain $M_\bullet$ and $a_\bullet$ to $3.0^{+4.0}_{-0.5} \times 10^3 M_\odot$ and $-0.7^{+1.0}_{-0.1}$, respectively.
Model 2
rules out $z > 0.055$ at $>3\sigma$.
For the best fit $M_\bullet = 3\times10^3 M_\odot $ and $a_\bullet =-0.7$ from Model 2, the best fit $z$ is $0.017 \pm 0.004$ at $1\sigma$ CL.
The NASA Extragalactic Database (NED)\footnote{The NASA/IPAC Extragalactic Database (NED) is funded by the National Aeronautics and Space Administration and operated by the California Institute of Technology.} lists two faint galaxies projected within $\sim$8$^\prime$ of the TDE host and at photometrically-estimated
redshifts of $z=0.015$ and 0.019, implying offsets of $\sim$200 kpc. Three other faint galaxies within $\sim$8$^\prime$ of the host lie at $z = 0.027$-0.028. It is possible that the star cluster hosting the TDE is associated with any of these galaxies, although none are as intrinsically bright or as near to the TDE host as the barred lenticular at $z=0.055$.
For both Model 1 and 2, we find a lower $M_\bullet$ value than L18, who used a thin disk model.
From the results in Fig.~\ref{J2150}, as well as the additional analyses in Appendix A and Fig.~\ref{fiduclai-fc}, we see that the L18 identification of the host as an IMBH is robust to both a range of choices regarding disk modeling (e.g.,~slim vs.~thin disk) and prescriptions for spectral hardening. Although other TDE IMBH candidates have been identified in the past \citep[e.g.,][]{Maksym+14}, J2150 has the richest set of observational data and appears generally inconsistent with an SMBH origin.
If we do not assume anything about the redshift of the host, $a_\bullet$ is largely unconstrained: at a 2$\sigma$ CL, the Model 2 fit for $|a_\bullet|$ is consistent with both $0$ and $1$. However, if we make the reasonable assumption that the host of J2150 is located at $z=0.055$, the redshift of the adjacent, barred lenticular galaxy, then $a_\bullet=0.8^{+0.12}_{-0.02}$. IMBH spins have not been measured before, which makes J2150 a valuable object for testing different theories of IMBH formation and growth. Different IMBH formation scenarios predict different IMBH spins at birth, some of which are compatible with the values inferred from Model 1 here. For example, supermassive stars may form in runaway collisions of main sequence stars and then collapse due to subsequent general relativistic instability. Simulations of this collapse process find high, but considerably sub-extremal spins; for example, \citet{ShibataShapiro02} find IMBH birth spins $a_\bullet \sim 0.75$, while \citet{Reisswig+13} find $a_\bullet \approx 0.9$ under substantially different collapse evolution.
Other IMBH formation/growth scenarios would be less compatible with the range of $a_\bullet$ we infer from Model 1. For example, IMBHs may form with masses $M_\bullet \sim 10^2~M_\odot$ as the remnants of Pop~III stars in the early Universe \citep{MadauRees01, Greif+11} and then grow to larger sizes via gas accretion. If the incoming gas maintains a fixed orientation for timescales much longer than the Salpeter time, then the accreting IMBH seed will become nearly extremal in spin \citep{Thorne74}; conversely, if the angular momentum of the incoming gas randomizes its direction on timescales much shorter than the Salpeter time (the so-called ``chaotic accretion regime'' of \citealt{KingPringle06}), then the IMBH seed spins down to a spin in the range of $a_\bullet\sim 0.1-0.3$, depending on black hole mass and the details of disk realignment, with fluctuations between accretion episodes of $\Delta a_\bullet \pm 0.2$ \citep{King+08}. In a cosmological context, growth of massive black holes through the chaotic accretion mode can produce even lower mean spin values \citep{BertiVolonteri08}.
This second latter option includes growth through stellar tidal disruptions. Recent cosmological simulations show that the high-redshift growth of IMBHs up to BH masses of $\sim 5 \times 10^5$ M$_\odot$ might indeed be dominated by TDEs (e.g., \citealt{pfister2020}). These outcomes are incompatible with the spin inference in Model 1, so if that measurement is correct, it rules out accretion- or TDE-driven growth of an IMBH seed with initial $M_\bullet \ll 10^4 M_\odot$, unless unusual conditions are met. For example, one could fine-tune accretion-driven growth models to result in $a_\bullet \approx 0.8$ if the last $e$-fold of growth saw a large-scale accretion disk reverse its angular momentum over a timescale comparable to the Salpeter time. Likewise, the estimated spin could be attained if the last e-fold of growth was driven by tidal disruption from a disk of stars with aspect ratio $\sim 0.1$. We note also that $a_\bullet \approx 0.8$ is compatible with the last e-fold of growth happening in a comparable-mass IMBH merger.
\begin{figure}[ht!]
\plotone{mfj215.pdf}
\caption{The evolution of unabsorbed X-ray flux (top panel) and mass accretion rate (bottom panel). The error on the best fit flux is estimated by fixing the best fit inclination and absorption parameters, but varying only the accretion rate within the $1\sigma$ CL. Here, we set the peak date (arbitrarily) as 500 days prior to the first epoch of the X-ray observations. The red dashed lines represent the best fit power law to the last three epochs of the flux with $\chi^2/\nu=0.39/1$ ($ t^{-1.77\pm0.08}$) and to all five epochs of accretion rate with $\chi^2/\nu=10.78/3$. The disk accretion rate decays as $\propto t^{-1.53\pm0.06}$, steeper than that of $t^{-1.1}$ ($\chi^2/\nu=42.92/3$) in the TDE ASSASN-14li (W20) and marginally consistent with $t^{-5/3}$, indicating no significant
circularization or viscous delay. The late-time X-ray flux approximately traces the accretion rate.}
\label{mfJ2150}
\end{figure}
Fig.~\ref{mfJ2150} shows the evolution of the unabsorbed X-ray flux and the disk accretion rate. Here and in the remainder of this section, we only consider the case of Model 1. The error on the measured flux is calculated by fixing the inclination and absorption, but adopting the lower and upper limit of $\dot m$ at $1\sigma$ CL. We plot the evolution of the X-ray flux for the best fit pair ($M_\bullet$, $a_\bullet$) and the evolution of accretion rate for the pairs ($M_\bullet$, $a_\bullet$) that fall within the $1\sigma$ contour in the left panel of Fig.~\ref{J2150}. The late time evolution of X-ray flux traces the decay of the accretion rate and is close to $t^{-5/3}$. This behavior is inconsistent with the prediction of an exponential decay from W20, because here we have an IMBH \citep{LodatoRossi11}. For IMBHs, the X-ray spectrum peaks at $\sim 0.8$ keV (see Fig.~\ref{J2150sp}), in contrast to the SMBH TDEs in W20, for which the 0.3 $-$ 7 keV band we observe is far down the Wien tail of the accretion disk.
The accretion rate decays roughly as $t^{-1.53\pm0.06}$, which is also close to a $t^{-5/3}$ decay rate expected for late-time mass fallback. As a result, there should not be a significant viscous or circularization delay for this TDE during the epochs we observe. Indeed, the viscous timescale for an IMBH TDE disk is roughly $T_{\rm{vis}}=\alpha^{-1}\Omega^{-1}(2R_t)(H/2R_t)^{-2}\sim 0.1^{-1}\times 0.0036\times 0.3^{-2}~\rm{days}\sim 4.0 ~\rm{days}$, shorter than the gas fallback time scale $\sim 8.7 ~\rm{days}$ for $\beta=2$ ($\sim 7.8 ~\rm{days}$ for $\beta=1$).
The mass accreted by the IMBH during the five epochs is $(6.7\pm0.4)\times 10^{-3} ~M_\odot$, $(5.9\pm0.4)\times 10^{-3} ~M_\odot$ and $(4.8\pm0.4)\times 10^{-3} ~M_\odot$ for the three $(M_\bullet/M_\odot, a_\bullet)$ pairs $(1.75\times 10^4, 0.8)$, $(1.75\times 10^4, 0.85)$ and $(2 \times 10^4, 0.9)$, respectively. However, enhanced optical emission was seen from the TDE host roughly one year prior to the first X-ray observation (L18), suggesting that there may have been an earlier phase of significant accretion. If we assume that the mass accretion peaked 500 days prior to the first epoch of the X-ray observations,
then the mass accreted from peak to Epoch 5 is $0.09\pm0.02 ~M_\odot$, $0.08\pm0.02 ~M_\odot$ and $0.05\pm0.02 ~M_\odot$, respectively (for the different mass-spin pairs given above). As a result, the flare is consistent with a full disruption if it maintained or exceeded its Epoch 1 luminosity for 1-2 years prior to the first XMM-{\it Newton}\, observation, but would be consistent with a partial disruption (or the loss of most of the dynamically bound stellar debris in an outflow, as in \citealt{MetzgerStone16}) if the accretion rate rose quickly prior to the start of X-ray observations.
\subsection{Particle Physics Implications}
\begin{figure}[ht!]
\plotone{darkphoton.pdf}
\caption{Exclusion regions in the hypothetical masses of ultralight bosons, including both Proca vector bosons (mass $M_{\rm V}$) and scalar axion-like particles (mass $M_{\rm ALP}$), with our J2150 constraints (green). Both panels use the formalism of \citet{Cardoso+18} to show mass ranges of elementary particles excluded by astrophysical spin measurements of different black holes; in all cases the x-axis denotes the instability BH timescale, while the y-axis shows the particle mass. The upper panel shows the excluded mass of the Proca vector bosons, while the lower panel shows the excluded axion-like particle mass. The contours denote the excluded masses for a given $M_\bullet$, $a_\bullet$ and instability time $\tau_I$ (this can be interpreted as the astrophysical spin-up time for the most recent e-fold of growth of the BH in question; it cannot be less than the Salpeter time $\sim 4 \times 10^7$ yr for Eddington-limited accretion). The green contours show the exclusion regions based on our mass and spin measurements for J2150. The cyan and yellow contours denote the constraint of the ALP and Proca mass from the stellar BH system Cygnus~X-1 ($M_\bullet=14.80 M_\odot$, $a_\bullet=0.97$; \citealt{orosz2011}, \citealt{Parker+15}) and the SMBH system NGC~4051 ($M_\bullet=1.91 \times 10^6 M_\odot$, $a_\bullet=0.99$; \citealt{denney2009}, \citealt{patrick2012}).
The dark, light and lighter contours in the lower panel denote the cases of low-order instability modes (modenumber $m=$1, 2 and 3, respectively). This figure shows that measurement of rapid spin in IMBHs, as we have performed for J2150, can exclude the existence of ultralight bosons at novel mass scales.}
\label{fig:ALP_V}
\end{figure}
If our Model 1 assumptions (source location at $z=0.055$; super-Eddington disk spectra can be modeled as a spectrally hardened multi-color disk blackbody) are correct, then the IMBH powering J2150 is rapidly spinning, with $0.78 < a_\bullet < 0.92$ at a 1$\sigma$ CL. This represents the first spin measurement of an IMBH, and the high spin measured carries notable implications for particle physics. In particular, rapidly rotating Kerr BHs are well known to exhibit a superradiant scattering instability in which spindown is triggered by interactions with ultralight bosons \citep{Bardeen+72, PressTeukolsky72}; the spin kinetic energy and angular momentum of the BH is converted into a bound cloud of elementary particles, saturating only after an order unity fraction of both have been transferred into the cloud \citep{EastPretorius17}. This spindown instability is only efficient if bosons exist with mass $m$ such that $\frac{GM_\bullet m}{c\hbar} \sim 1$ (here $\hbar$ is the reduced Planck constant). The timescale for linear growth of the instability grows exponentially if $\frac{GM_\bullet m}{c\hbar} \gg 1$ or $\ll 1$.
In practice, measurements of large $a_\bullet$ values in astrophysical BHs can be used to rule out roughly one order-of-magnitude in ultralight particle mass $m$ \citep[e.g.][]{Cardoso+18}. Figure 5 shows the excluded particle masses (colored regions) derived from the black hole masses and spins of Cygnus X-1 (blue), J2150 (green), and NGC 4051 (yellow). The black hole instability time $\tau_{\rm I}$ on the x-axis is calculated from Eq.~2.13 and Eq.~2.18 of \cite{Cardoso+18} for a given particle mass, $M_\bullet$, and $a_\bullet$. This is roughly equivalent to a "spindown time," in that an isolated black hole will lose an order unity fraction of its spin over the timescale $\tau_{\rm I}$. When this spindown/instability timescale is much shorter than any plausible spinup timescale (which, for Eddington-limited accretion, would be of order the Salpeter time), an observed $(M_\bullet, a_\bullet)$ combination can be said to exclude a given boson mass. The stellar-mass BH system Cygnus X-1 excludes massive scalar fields (e.g., axion-like particles) and massive vector fields (e.g., dark photons) with $m\sim 10^{-12}$ eV, whereas the SMBH system NGC 4051 excludes those particles at $m\sim 10^{-17}$ eV. Spin measurements of larger SMBHs have been used in the past to exclude even smaller ultralight bosons \citep{Cardoso+18}. Critically, the mass and spin constraints that we derive here from J2150 (Model 1), an IMBH, exclude a new, intermediate particle mass range around $10^{-15}$ eV. For each astrophysical system, the range of excluded particle masses widens as instability timescale increases.
To our knowledge, there have been no astrophysical spin measurements of IMBHs prior to our results; therefore, this is the first superradiance constraint on ultra-light boson masses. As such, it complements existing laboratory experiments, such as the CASPEr project \citep{Garcon+18}, which has already placed bounds on the existence of low-mass bosons in the $\sim 10^{-16}$-$10^{-13}$ eV range \citep{Garcon+19}. In contrast to laboratory experiments, however, superradiance constraints depend primarily on the mass of the boson and do not require any significant interactions with baryonic matter. Yet if the bosonic self-interaction is too strong, bound clouds formed through superradiance can self-annihilate \citep{YoshinoKodama12}, greatly reducing the degree of black hole spindown and potentially creating an observable gravitational wave signal \citep{Arvanitaki+15}. We do not consider the possibility of boson self-interaction and its effect on spindown rates here, except to note that if the boson self-interaction is too strong, superradiance constraints are weakened. This scenario is treated in detail in \citep{Mathur+20}.
\section{Conclusions}
\label{sec:conclusions}
We have fit our general relativistic slim disk accretion model to the unusual TDE J2150. Our approach is similar to our earlier work in W20, although here we improve our model to account for angular momentum lost by radiation. We use a Kerr metric ray-tracing code to simultaneously fit
five epochs of X-ray continuum spectra in J2150.
We explore fits with different priors to test the uncertainties
in our model, including the assumed
disk outer and inner radius (Appendix E), spectral hardening parameterization (Appendix B), and TDE host redshift (Appendix C). We find that:
\begin{enumerate}
\item Regardless of our choice of priors, we identify the central engine of the accretion disk to be an {\it intermediate-mass} black hole where
$M_\bullet$ is less than
2.2$\times 10^4 ~M_\odot $ at $1\sigma$.
\item
If we assume the TDE host is associated with the adjacent, barred lenticular galaxy at $z=0.055$ (``Model 1''), we
achieve a good fit across 12 years of observations and two orders of magnitude in disk accretion rate. We
constrain the black hole mass $M_\bullet$ and spin $a_\bullet$ to be $1.75^{+0.45}_{-0.05}\times 10^4$ $M_\odot$ and $0.8^{+0.12}_{-0.02}$, respectively, at $1\sigma$.
This high, but significantly sub-extremal, spin suggests that, if the IMBH has grown significantly since formation, it has acquired its last e-fold in mass in a way incompatible with both the ``standard'' and ``chaotic'' gas accretion limits, which predict spins that are too high and too low, respectively \citep{BertiVolonteri08}.
The spin
$a_\bullet$ depends sensitively on
the unconfirmed
redshift of J2150's host system.
Measuring that redshift would
eliminate a major systematic uncertainty on the spin measurement.
\item If our Model 1 is correct, we have discovered a rapidly spinning IMBH, the first measurement of its kind. The IMBH in J2150 would thus also represent the first ``superradiant scattering'' constraint on ultralight elementary bosons with masses $\sim 10^{-15}$ to $\sim 10^{-16}$ eV. The existence of ultralight scalar (e.g.,~axion-like particles) or vector (e.g.,~dark photons) bosons in these mass ranges, respectively, can be ruled out due to the failure of the IMBH powering J2150 to spin down under the effects of superradiant scattering (although this conclusion is weakened for bosons with sufficiently strong self-interaction cross-sections).
\item
The flare is consistent with a full disruption if it maintained its Epoch 1 luminosity for 1-2 years prior to the first XMM-{\it Newton}\, observation in May 2006, as is suggested by the 2005 identification of an optical outburst (L18). If, however, the accretion rate rose quickly prior to the start of X-ray observations, then the accreted mass is very low ($\sim 10^{-3}$ to $10^{-2}M_\odot$) and requires either a partial disruption or the loss of most of the dynamically bound mass.
\end{enumerate}
In the near future, the X-ray satellites {\it SRG/eROSITA}, {\it Einstein Probe}, and possibly {\it Theseus} will together likely discover hundreds of new soft X-ray TDEs \citep{Khabibullin+14, Yuan+15, Jonker+19}. Targeted X-ray followup of TDEs found in optical surveys such as ZTF may find additional X-ray bright TDEs. Our analysis of J2150 demonstrates that if high-quality, multi-epoch X-ray spectra can be acquired for some of these TDEs, for instance through pointed XMM-{\it Newton}, {\it Chandra}, {\it SRG/eROSITA} or {\it Athena} observations, then it will be possible to map out the IMBH mass function and potentially constrain IMBH spins as well.
\section*{Acknowledgements}
We thank Dacheng Lin for his help in providing X-ray spectra, offering useful advice on X-ray fitting in the early stages of this work and useful comments on the draft. We thank the anonymous referee for their helpful comments. We thank B.~Metzger, C.~Miller, A.~Loeb, and E. Kara for their guidance and suggestions. SW and AIZ thank Steward Observatory and the UA Department of Astronomy for post-doctoral support for SW.
NCS received support from the Israel Science Foundation (Individual Research Grant 2565/19), and thanks Lam Hui for informative conversations.
Our calculations were carried out at UA on the El Gato and Ocelote supercomputers, which are supported by the National Science Foundation under Grant No.~1228509. Our work here is partly based on observations obtained with XMM-Newton, an ESA science mission
with instruments and contributions directly funded by ESA Member States and NASA.
Our research has made use of data obtained from the Chandra Data Archive and software provided by the Chandra X-ray Center (CXC) in the application package CIAO.
\begin{appendix}
\begin{deluxetable*}{cccccccccccccc}
\tablecaption{Testing error estimation of parameters for C-statistic}
\tablewidth{0pt}
\tablehead{
\colhead{Spectrum }&&\colhead{counts }&&\colhead{$kT_{\chi^2}$ [keV] }&&\colhead{$1\sigma$ width$_{\chi^2}$ }&&\colhead{$kT_{Cstat}$ [keV]}&&\colhead{$1\sigma$ width$_{Cstat}$ }&&}
\startdata
$N_1=0.01$ && 452413 && $0.30045^{+0.00039}_{-0.00039}$ && 0.00078 && $0.30058^{+0.00039}_{-0.00039}$ &&0.00078\\
$N_2=0.001$ && 45067 && $0.30114^{+0.00126}_{-0.00124}$ && 0.00250 &&$0.30206^{+0.00125}_{-0.00125}$ &&0.00250 &&\\
$N_3=0.0001$ && 4411 && $0.28644^{+0.00365}_{-0.00359}$ && 0.00724 && $0.29650^{+0.00390}_{-0.00381}$ && 0.00771\\
Chandra 6791 && 4801 && $0.25021_{-0.00303}^{+0.00307}$ && 0.00610 && $0.26146^{+0.00330}_{-0.00318}$&& 0.00648 \\
Chandra 17862 && 152 && $0.12916_{-0.00704}^{+0.00758}$ && 0.01462 &&$0.13426_{-0.00618}^{+0.00665}$ && 0.01283 \\
\enddata
\label{stat}
\tablecomments{While the {\it Chandra}\, data have a background component that we modeled separately (see text), the simulated data has no background component. Unlike the first three lines, the temperature of the blackbody fit to the {\it Chandra}\, data decreases with time as the source evolves.}
\end{deluxetable*}
\section{Statistical analysis}
\label{cstatisitc}
\cite{Kaastra17} shows that the C-statistic can be used for
assessing the goodness of fit of a spectral model
and that it is
preferred for X-ray spectra, as the $\chi^2$ statistic gives biased results even for 20-30 counts per spectral bin. However, when estimating the error on the fit parameters for the C-statistic, the assumption is often made that the C-statistic converges to the $\chi^2$ statistic without confirming that there are sufficient number of counts per spectral bin and/or a sufficient number of spectral data bins to justify the Central Limit Theorem, which underlies the assumption that one can use the $\chi^2$-like statistical distribution of C-statistic values. In this work, we also assume that the C-statistic converges to the $\chi^2$ statistic, i.e., the $1\sigma$ errors on the fit parameters are determined by using $\Delta Cstat =1.0$ for single parameter and $\Delta Cstat = 2.3$ for two parameters. Here, we check if the C-statistic indeed converges to the $\chi^2$ statistic when estimating the uncertainty regions on the best-fit parameters in this paper.
We first generate three mock spectra using the XMM-{\it Newton} ~pn response file and the {\sc XSPEC} {\sc bb} (blackbody) model, and then fit these spectra employing either the C-statistic or the $\chi^2$ statistic. The three spectra have the same temperature (kT=0.3 keV) and exposure time (1~ks), but a different normalization, e.g., 0.01, 0.001 and 0.0001 ($N_1$ through $N_3$ in Table~\ref{stat}). We only fit the spectra with the {\sc bb} model over the 0.3-2.0 keV band. The total number of X-ray photons is 452413, 45067, and 4411 in the 0.3-2.0 keV band for the three normalisations listed above, respectively. There are 342 spectral data bins for all three cases. $kT_{\chi^2}$ is determined by $\Delta \chi^2= 1$, while $kT_{Cstat}$ is determined by $\Delta Cstat= 1$ with the {\sc steppar} command in {\sc XSPEC}.
We also extend our analysis to the two {\it Chandra} ~spectra of J2150; see Table~\ref{stat} for the number of X-ray photons in those spectra. We fit the two spectra separately using the {\sc bb} model attenuated by absorption. The model parameters used to describe the background are fixed at their best-fit value from the background-only spectral fits (see main text). We use the {\sc steppar} command to determine the $1\sigma$ CL on the best-fit value of kT at $\Delta \rm{statistic} = 1$ for both the C-statistic and $\chi^2$ statistic. Here, $N_{\rm H}$ is fixed at the best-fit value.
As we can see from Table~\ref{stat}, the value of the $1\sigma$ error is consistent when calculated using the $\chi^2$ and C-statistic. This result indicates that
using $\Delta Cstat=1$ to
estimate the $1\sigma$ CL uncertainties on the best-fit value (single parameter) is justified and that the C-statistic converges to the $\chi^2$ statistic when estimating errors.
As the total number of detected X-ray photons decreases, using the $\chi^2$ statistic in the fit starts to bias the fit result, which is in line with the results of \cite{Kaastra17}.
\section{fiducial {\lowercase{$f_{\rm c}$}} treatment over different epochs}
\label{f-fc}
\begin{deluxetable*}{cccccccccccccc}
\tablecaption{Results from fitting five epochs of J2150 with our fiducial $f_{\rm c}$ slim disk model.}
\tablewidth{0pt}
\tablehead{
\colhead{ }&& \colhead{XMM 1} && \colhead{Chandra 1} && \colhead{XMM 2} && \colhead{Chandra 2} && \colhead{XMM 3} \\
\colhead{Date} && \colhead{2006-05-05} && \colhead{2006-09-28} && \colhead{2009-06-07} && \colhead{2016-09-14} && \colhead{2018-05-24}
}
\startdata
Separate fitting \\
$N_{\rm H}$ $[10^{20}{\rm cm}^{-2}]$ && $2.4\pm2.4$ && $0.4\pm0.7$ && $6.2\pm2.3$ && $0^{+1.1}$ && $2.4\pm1.1$ \\
$\theta$ $[{^\circ}]$ && $5.0^{+17}_{-0}$ && $49.7\pm3.2$ && $5.0_{-0}^{+25}$ && $=\theta_3$ && $=\theta_3$ \\
$\dot m$ ${\rm [Edd]}$ && $8.9^{+18.8}_{-1.2}$ && $100_{-53}^{+0}$ && $14.9\pm8.0$ && $1.3\pm0.1$ && $1.2\pm0.7$ \\
$M_\bullet$ $[10^4{\rm M}_{\odot}]$ && $2.0$ && $1.5$ && $1.0$ && $=M_{\bullet,3}$ && $=M_{\bullet,3}$ \\
$a_\bullet$ && $0.9995$ && $0.9995$ && $-0.3$ && $=a_{\bullet, 3}$ && $=a_{\bullet, 3}$ \\
$Cstat/\nu$ && $297.78/270$ && $211.93/177$ && $273.64/283$ && $58.71/58$ && $268.79/240$ \\
\hline
Combining fitting \\
$N_{\rm H}$ $[10^{20}{\rm cm}^{-2}]$ && $2.7\pm0.6$ && $0.7\pm0.7$ && $1.0\pm0.4$ && $0^{+0.4}$ && $0^{+0.4}$ \\
$\theta$ $[{^\circ}]$ && $5.0^{+10}_{-0}$ && $=\theta_1$ && $=\theta_1$ && $=\theta_1$ && $=\theta_1$ \\
$\dot m$ ${\rm [Edd]}$ && $6.4\pm1.5$ && $2.0\pm0.2$ && $0.58\pm0.02$ && $0.135\pm0.007$ && $0.120\pm0.005$ \\
$M_\bullet$ $[10^4{\rm M}_\odot]$ && $2.25$ && $=M_{\bullet,1}$ && $=M_{\bullet,1}$ && $=M_{\bullet,1}$ && $=M_{\bullet,1}$ \\
$a_\bullet$ && $0.9995$ && $=a_{\bullet, 1}$ && $=a_{\bullet, 1}$ && $=a_{\bullet, 1}$ && $=a_{\bullet, 1}$ \\
$Cstat/\nu$ &&$300.01/271$ && $234.09/180$ && $274.91/283$ && $66.33/58$ &&$281.92/240$ \\
\enddata
\tablecomments{We adopt the same fit function as in Table~\ref{J2150p}, e.g., {\sc po + po + po + agauss + agauss + phabs(zphabs(slimdisk))}. Here, $f_{\rm c}$ is calculated by the fiducial method (see W20 and Appendix~\ref{f-fc}). The Milky Way $N_{\rm H}$ absorption is fixed at $2.6\times10^{20}$ ${\rm cm}^{-2}$ (L18). $N_{\rm H}$ shown in the table is thus associated with the TDE itself, its host star cluster, and possibly the nearby, barred lenticular galaxy at $z=0.055$. The error on the parameters are calculated with fixed $M_\bullet$ and $a_\bullet$.}
\label{J2150p2}
\end{deluxetable*}
\begin{figure*}[ht!]
\gridline{ \fig{ep1.pdf}{0.5\textwidth}{Epoch 1}
\fig{ep2.pdf}{0.497\textwidth}{Epoch 2 }
}
\gridline{ \fig{ep35.pdf}{0.495\textwidth}{Epoch 3-5}
\fig{MAj215.pdf}{0.5\textwidth}{Combine }
}
\caption{For our fiducial $f_{\rm c}$, the effects of using different observed epochs to constrain $M_\bullet$ and $a_\bullet$ are shown. We calculate the $\Delta Cstat$ on a grid in the $(M_\bullet, a_\bullet)$ plane and then fill the intermediate parameter space by linear interpolation. The top left panel, top right panel, and the lower left panel show the results from Epoch 1, Epoch 2, and Epoch 3-5, while the lower right panel shows the result of the combining all five epochs. The best fit parameters of each panel are listed in Table~\ref{J2150p2}. The first two epochs produce constraints on $M_\bullet$ and $a_\bullet$ that are inconsistent with the later three epochs. The combined constraint on $M_\bullet$ and $a_\bullet$ is driven by the first two epochs. This inconsistency may arise from the underestimation of $f_{\rm c}$ for the early two super-Eddington spectra, Epochs 1 and 2.
}
\label{fiduclai-fc}
\end{figure*}
Early studies \citep{ST93,ST95} showed that the local X-ray flux at each annulus will be higher than the corresponding blackbody flux, due to electron scattering and the temperature gradient in the atmosphere. The local X-ray emission can be approximated by a color-corrected blackbody \citep{ST95},
\begin{equation}
\label{Iv}
I(\nu)=\frac{2h\nu^3c^{-2}f_{\rm c}^{-4}}{\exp(h\nu/k_{\rm B} f_{\rm c}T)-1}.
\end{equation}
Here, $h$ and $k_{\rm B}$ are the Planck constant and the Boltzmann constant, respectively; $f_{\rm c}$ is the spectral hardening factor. For a sub-Eddington disk, $f_{\rm c}$ is about 1.7 and insensitive to disk parameters \citep{ST95}. Later studies showed that $f_{\rm c}$ may increase with accretion rate \citep{GD04,DBHT05}. \citet{DE18} estimated $f_{\rm c}$ for a non-spinning SMBH,
\begin{eqnarray}
\nonumber
f_{\rm c}=1.74+&&1.06(\log_{10} T-7)-0.14[\log_{10} Q-7] \\
&&-0.07\{ log_{10}[\Sigma/2]-5 \}.
\label{fc}
\end{eqnarray}
Here, $Q$ and $\Sigma$ are the strength of vertical gravity and surface density at each annulus of the disk, respectively. This $f_{\rm c}$ estimate holds for accretion rates between 0.01 to 1 Eddington units. In the super-Eddington regime, $f_{\rm c}$ would not increase to infinity as accretion increases and would instead saturate \citep{Davis2006} at about 2.4 for a SMBH accretion disk. In this paper, we take $f_{\rm c}$ from Eq.~\ref{fc} as our fiducial $f_{\rm c}$ treatment. As this fiducial $f_{\rm c}$ may not work well for a highly super-Eddington accretion disk, we parameterize $f_{\rm c}$ and set a flat prior of (2.0, 2.4) in that case.
In this section, we examine further the results of fitting the spectra with our fiducial $f_{\rm c}$ treatment and the problems that arise from this assumption. We first fit the spectral epochs separately, constraining the corresponding $M_\bullet$ and $a_\bullet$. We divide the five spectra into three groups, with Epoch 1 and Epoch 2 as two separate groups and Epoch 3-5 as the third group. We break up our analysis in this manner, because it is unclear whether the fiducial $f_{\rm c}$ prescription of \citet{DE18} can be successfully extrapolated beyond the sub-Eddington regime in which it was derived, and applied to strongly super-Eddington accretion rates such as those in Epochs 1 and 2. We also perform a simultaneous fit to all five epochs with the fiducial \citet{DE18} prescriptions.
Figure \ref{fiduclai-fc} shows the fitting results, and the corresponding best fit parameters are listed in Table~\ref{J2150p2}. Epoch 1 and Epoch 2 yield very narrow contours in the ($M_\bullet$, $a_\bullet$) plane. The best-fit $a_\bullet$ is pushed to an extremely high value beyond theoretically predicted saturation spins \citep[e.g.][]{Thorne74}, $\approx 0.9995$. Epoch 3-5 yields contours in ($M_\bullet$, $a_\bullet$) similar to ASSASN-14li (W20), but with even less constraint on $a_\bullet$. Our separate analysis of these three groups with the fiducial \citet{DE18} $f_{\rm c}$ prescription have consistent constraints on $M_\bullet$, but not on $a_\bullet$. The $1\sigma$ contours of Epoch 1 and Epoch 2 are at least $3\sigma$ away from those of Epoch 3-5.
The combined Epoch 1-5 ($M_\bullet$, $a_\bullet$) constraints are driven by the first two epochs. $Cstat$ increases very quickly as spin decreases, indicating that the disk of a low spin BH is not bright enough to fit the early observations, which may be caused by an underestimated $f_{\rm c}$ in the fiducial model. From the separate fits, we see that both two early epochs are indeed in a highly super-Eddington phase (for $a_\bullet=0.9995$, the radiation efficiency is $\eta\approx 0.36$). In W20, we showed that X-ray flux would be nearly constant in the highly super-Eddington regime. As a result, the X-ray luminosity of the disk is insensitive to accretion rate, but sensitive to the choice of $f_{\rm c}$. The
unusual behavior of the $M_\bullet$ and $a_\bullet$ contours in the combined fitting may arise from the underestimation of $f_{\rm c}$ for a low spin disk. Motivated by (1) the lack of theoretical calculations for super-Eddington $f_{\rm c}$ values and (2) the incompatibility between the Epoch 1/Epoch 2 and Epoch 3-5 $a_\bullet$ constraints under \citet{DE18} $f_{\rm c}$ prescriptions, we allow $f_{\rm c}$ to float as a free fit parameter for the first two epochs.
\section{effects of redshift uncertainty}
\label{app:simulation}
\begin{figure}[ht!]
\plotone{fake2.pdf}
\plotone{fake.pdf}
\caption{Constraints on $M_\bullet$ and $a_\bullet$ from varying the TDE redshift assumption. We generate two mock spectra with the pn respond file with slim disk model. The value of parameters are $M_\bullet=2 \times 10^4 ~M_\odot$, $a_\bullet=0.9$, $N_{\rm{H}}=3.0\times10^{20} ~\rm{cm}^{-1}$, $\theta=10^\circ$, $z=0.055$, $\dot m_1 = 1$, and $\dot m_2 = 0.3$. The upper panel fits the spectra with $z=0.055$, while the lower panel allows $z$ to float. The best $Cstats$ for these two models are 819.08 and 818.94, respectively, slightly preferring a smaller $M_\bullet$, a lower $a_\bullet$, and a lower $z$.
}
\label{simulation}
\end{figure}
In the section, we explore the effect of the unknown TDE redshift on our ($M_\bullet$, $a_\bullet$) constraints. We first generate two mock spectra with the XMM pn response file using our slim disk Model 1, and then refit the spectra using different settings of $z$. We generate the spectra with parameters $M_\bullet=2 \times 10^4 M_\odot$, $a_\bullet=0.9$, $N_{\rm{H}}=3.0\times10^{20} \rm{cm}^{-2}$, $\theta=10^\circ$, $z=0.055$, $\dot m_1 = 1$ Edd, and $\dot m_2 = 0.3$ Edd. The exposure time for both spectra is 200,000 seconds, and the number of counts in the spectra are 39,927 and 11,389 counts in band 0.3 -- 7.0 keV, respectively.
Fig.~\ref{simulation} shows the results of these fits. In the upper panel, we fit the spectra by fixing $z=0.055$. Within the $1\sigma$ contour is the pair value ($M_\bullet=2 \times 10^4 ~M_\odot$, $a_\bullet=0.9$), which we have used to generate the spectra. From the contours, we see that $M_\bullet$ is degenerate with $a_\bullet$ (W20). This degeneracy arises from the fact that either smaller $M_\bullet$ or higher $a_\bullet$ can produce a hotter disk.
For the lower panel, we use the same fit function as the upper panel, although in addition we treat the redshift $z$ as a free floating parameter in the fit. The best fit $Cstat$ is 818.94, very close to that of 819.08 from the upper panel fit. Therefore, allowing $z$ to float does not improve the fit significantly. However, the $M_\bullet$ and $a_\bullet$ contours become bigger, e.g., more models with smaller $M_\bullet$ and lower $a_\bullet$ can describe the spectra well. We also find that $z$ can be smaller than 0.055; e.g., the best fit is $z=0.033$ for $M_\bullet=5 \times 10^3 M_\odot$ and $a_\bullet=-0.9$. A lower $z$ is always associated with a lower $M_\bullet$, as a lower $z$ makes the expected spectra brighter by reducing the luminosity distance and redshifting the spectra less, while a lower $M_\bullet$ requires the disk to be hotter, increasing the flux as well as blueshifting the spectra.
Therefore, a smaller $M_\bullet$ and a lower $z$ may arise from the degeneracy of the $z$, $M_\bullet$ and $a_\bullet$ parameters.
\section{Stationary slim disk model}
\label{app:slimdisk}
We adopt the same procedure as in W20 to reduce the relativistic slim disk equations. The equations of state, vertical hydrostatic equilibrium \citep{Abramowicz+1997}, and mass conservation are the same as that in W20. For brevity, the aforementioned equations are not detailed in this appendix, and we refer the reader to W20 for more detail. Here, we only write the three equations:
(1) angular momentum conservation equation \citep{Abramowicz+1996},
\begin{equation}
\frac{\dot{M}}{2\pi}\frac{d {\cal L}}{{\rm d}r}-\frac{{\rm d}}{{\rm d}r}\left(\frac{\alpha A^{1/2}\Delta^{1/2}\gamma P}{r}\right)=Q^{\rm rad}{\cal L};
\label{L}
\end{equation}
(2) radial momentum conservation equation,
\begin{equation}
\frac{V^2}{1-V^2}\frac{{\rm d}\ln V}{{\rm d} r}=\frac{{\cal A}}{r}-\frac{P}{\Sigma}\frac{{\rm d}\ln P}{{\rm d} r};
\label{dvdr}
\end{equation}
(3) energy conservation equation,
\begin{equation}
Q^{\rm adv}=-\alpha P\frac{A\gamma^2}{r^3}\frac{{\rm d}\Omega}{{\rm d}r }- \frac{64\sigma T_{\rm c}^4}{3\Sigma\kappa}.
\label{Qadv}
\end{equation}
All the parameters are defined the same as in W20. These equations can be simplified to:
\begin{eqnarray}
\label{2D1}
&&\frac{{\rm d}\ln V}{{\rm d} r}=\frac{N}{D}=\frac{A_3B_2-B_3A_2}{A_1B_2-A_2B_1}, \\
\label{2D2}
&&\frac{{\rm d}\ln T_{\rm c}}{{\rm d}r}=\frac{B_3}{B_2}-\frac{B_1N}{B_2D},\\
&&c_3\frac{{\rm d}\ln \cal L}{{\rm d} r}=c_4-c_1\frac{{\rm d}\ln V}{{\rm d} r}-c_2\frac{{\rm d} \ln T_{\rm c}}{{\rm d} r},
\end{eqnarray}
where
\begin{eqnarray}
\nonumber
&&A_1=a_1c_3-a_3c_1, A_2=a_2c_3-a_3c_2, A_3=a_4c_3-a_3c_4,\\ \nonumber
&&B_1=b_1c_3-b_3c_1, B_2=b_2c_3-b_3c_2, B_3=b_4c_3-b_3c_4.
\end{eqnarray}
Here $a_i$, $b_i$ and $c_i$ (i=1,2,3,4) are function of $T_{\rm c}$, $V$, $\cal{L}$ and r. They can be written as:
\begin{eqnarray}
\nonumber
&&a_1=P_1+\frac{V^2 \Sigma}{(1-V^2)P}, ~~a_2=P_2, \\ \nonumber
&&a_3=P_3, ~~a_4=\frac{{\cal A} \Sigma}{rP}-P_4,\\ \nonumber
&&b_1=\frac{4-3\beta_{\rm{p}}}{1-V^2}-\frac{2\pi\alpha A\lambda^2\Sigma}{\dot Mr^2}O_1,\\ \nonumber
&&b_2=12-10.5\beta_{\rm{p}}, ~~b_3=-\frac{2\pi\alpha A\lambda^2\Sigma}{\dot Mr^2}O_2,\\
\nonumber
&&b_4=\frac{2\pi\alpha A\lambda^2\Sigma}{\dot Mr^2}O_3+\frac{2\pi rQ^{rad}\Sigma}{\dot M P}-\frac{4-3\beta_{\rm{p}}}{2}\frac{{\rm d} \ln\Delta}{{\rm d}r}, \\
\nonumber
&&c_1=-\lambda_1-P_1, ~~c_2=-P_2, \\ \nonumber
&&c_3=\frac{\dot Mr\cal L}{2\pi\alpha \sqrt{A\Delta}\lambda P}-P_3-\lambda_2,\\ \nonumber
&&c_4=\frac{Q^{rad}r^2\cal L }{\alpha \sqrt{A\Delta}\lambda P}+P_4+\lambda_3+\frac{{\rm d} \ln(\sqrt{A\Delta}/r)}{{\rm d}r},\\
\nonumber
&&\lambda_1=\frac{V^2}{(1-V^2)^2\lambda^2},~\lambda_2=\frac{{\cal L}^2r^2}{A\lambda^2},\\\nonumber
&&\lambda_3=\lambda_2\left(\frac{1}{r}-\frac{\rm{d}\ln A}{2\rm{d}r}\right),\\
\nonumber
&&O_1=-(\Omega-\omega)\lambda_1,~O_2=(\Omega-\omega)(1-\lambda_2),\\\nonumber
&&O_3=\frac{{\rm d}\omega}{{\rm d}r}+(\Omega-\omega)(\frac{3}{r}+0.5\frac{\rm{d}\ln \Delta}{{\rm d}r}-\frac{3}{2}\frac{\rm{d}\ln A}{{\rm d}r}-\lambda_3), \\
\nonumber
&&P_1=-\frac{3\beta_{\rm p}-1}{1+\beta_{\rm p}}\frac{1}{1-V^2}-\frac{\beta_{\rm p}-1}{1+\beta_{\rm p}}\frac{a^2u_t(u_t+\omega{\cal L})\lambda_1}{r^4\cal G },\\ \nonumber
&&P_2=\frac{8-6\beta_{\rm p}}{1+\beta_{\rm p}}, \\
\nonumber
&&P_3=\frac{\beta_{\rm p}-1}{1+\beta_{\rm p}}\left(2-a^2\frac{u_t^2-1 -u_t((u_t+\omega{\cal L})\lambda_2-\omega{\cal L})}{r^4\cal G}\right),\\
\nonumber
&&P_4=-\frac{1}{2}\frac{3\beta_{\rm p}-1}{1+\beta_{\rm p}}\frac{{\rm d}\ln \Delta}{{\rm d}r}-\frac{\beta_{\rm p}-1}{1+\beta_{\rm p}}\frac{4}{r} -\frac{\beta_{\rm p}-1}{1+\beta_{\rm p}}\frac{a^2u_t}{r^4\cal G}\times\\ \nonumber
&&\left((u_t+\omega{\cal L})(\frac{{\rm d}\ln\Delta}{2{\rm d}r}-\frac{{\rm d}\ln A}{2{\rm d}r}+\frac{1+r\lambda_3}{r})-\frac{{\rm d}\omega}{{\rm d}r}{\cal L}\right).
\end{eqnarray}
We estimate the initial conditions by assuming the Novikov-Thorne disk ($\Omega=\Omega_k^+$, $Q^{\rm adv}=0$ and $Q^{rad}{\cal L}=0$). As a result, angular momentum conservation and energy conservation equations can be rewritten as,
\begin{eqnarray}
&&\frac{\dot{M}}{2\pi}({\cal L}-{\cal L}_{\rm in})=\frac{A^{1/2}\Delta^{1/2}\gamma}{r}\alpha P,\\
&&\alpha P\frac{A\gamma^2}{r^3}\frac{{\rm d}\Omega}{{\rm d}r }=\alpha P\frac{A\gamma^2}{r^3}\frac{{\rm d}\Omega_k^+}{{\rm d}r }=- \frac{64\sigma T_{\rm c}^4}{3\Sigma\kappa}.
\end{eqnarray}
Combined with other equations, one can solve $V$, $T_{\rm c}$ and $\cal {L}$ for a given r. Here, ${\cal L}_{in}$ is the integration constant, which denotes the angular momentum component at the disk inner edge. The free parameters are $M$, $a$, $\dot M$ and $\alpha$. ${\cal L}_{in}$ is the eigenvalue of the problem, which must be chosen properly to ensure that $N=0$ and $D=0$ at the sonic point. We use the shooting technique to narrow ${\cal L}_{in}$ (W20). The ${\cal L}_{in}$ estimation is updated iteratively, until $\Delta {\cal L}_{in}/{\cal L}_{in}$ is less than $10^{-6}$. We iteratively integrate the equations to radius near the sonic point with the latest ${\cal L}_{in}$ estimate, then take a large step ahead and continue to solve the equations to near-horizon distances. Our solutions are insensitive to the initial conditions.
\begin{figure}[ht!]
\plotone{AM-test.pdf}
\caption{The comparison of effective temperature $T(R)$ between disk models that account for angular momentum lost in radiation ($T_2$) and those that do not ($T_1$). In both panels, we plot the temperature difference against dimensionless radius $R$. The figure show that the angular momentum loss effect is strongest for high spin and low accretion rate disks. For $a_\bullet<0.8$, the effect is weak, and the error is $< 5\%$.
}
\label{AM}
\end{figure}
Figure \ref{AM} shows the comparison of effective temperature between this disk model and the one in W20. We consider the case of $M_\bullet=10^4~M_\odot$. The upper panel shows the temperature differences for different accretion rate for $a_\bullet=0.998$. The differences grow for lower accretion rate, because the radiative efficiency $\eta$ is bigger for low $\dot m$, where advection cooling is unimportant. The angular momentum removes by radiation is larger for a lower accretion rate disk. As a result, the lower accretion rate disk becomes dimmer. The lower panel shows the temperature differences with disk radius for different $a_\bullet$. Here we fix $\dot m= 0.1$. The differences become bigger as $a_\bullet$ increases, again because $\eta$ becomes bigger as $a_\bullet$ increases. As a result, the radiation removes more angular momentum, making the disk dimmer. As we can see from both panels, for cases of high spin and low accretion, the effective temperature becomes dimmer by less than $10\%$.
\section{choice of outer and inner radius}
\label{rout_rin}
\begin{figure}[ht!]
\gridline{\fig{Rout-test.pdf}{0.45\textwidth}{}
}
\caption{The effects on the spectrum of different choices of outer disk edge. The error is calculated as $E=\frac{F_{r}-F_{800}}{F_{800}}$ ($F_r$ denotes the flux of the disk with $r_{out}=r$) for a given frequency, and is $< 1.0\%$ when $r_{out}>600 ~\rm{R_g}$. For lower accretion rates, the error would be smaller due to a cooler disk.
}
\label{rout}
\end{figure}
\begin{figure}[ht!]
\plotone{ISCO-test.pdf}
\caption{The effects on the spectrum of different choices of inner disk edge for different $a_\bullet$ and $\dot m$ values. Here, two kinds of inner edge are considered: event horizon and ISCO. The error is calculated as $E=\frac{F_{EH}-F_{ISCO}}{F_{EH}}$ for a given frequency.
We set $M_\bullet =10^4 M_\odot$ for both panels. For the upper panel, $\dot m = 10$ Edd, while for the lower panel $a_\bullet = -0.9$. This figure shows that the error is $< 2\%$ over a range of disk inner edges.
}
\label{rin}
\end{figure}
In this section, we test the effects of different choices of disk outer and inner radius. Most of the X-ray photons are emitted in the inner disk (within 30~$R_g$, see W20). However, for an IMBH disk, the effective temperature at several hundred gravitational radii can be as high as several $\times 10^5 ~\rm{K}$. As a result, it could impact the low energy part of the 0.3--7 keV spectrum.
Fig.~\ref{rout} shows the effects of different choices for the outer radius. We consider a disk with parameters $M_\bullet=10^4M_\odot$, $a_\bullet=0.998$, and $\dot m=10$ Edd. For this disk, two times the tidal radius is $2r_t=2000 ~\rm{R_g}$.
We fix the disk inclination at $45^\circ$, and calculate the spectrum for different outer disk radii. The relative error on the flux is $< 1 \%$ when $r_{out}>600 ~\rm{R_g}$. A lower $M_\bullet$, a lower $a_\bullet$ and a lower $\dot m$ produce a lower disk temperature in the outer disk region, moving the spectrum to a lower energy and making the relative error smaller.
Therefore, in the main paper we fix the outer radius at 600 $~\rm{R_g}$, even if the disk outer radius is bigger than 600$~\rm{R_g}$. The main effect is to accelerate the calculation, as the error introduced this way is negligible (Fig.~\ref{rout}).
Fig.~\ref{rin} shows the effect of the difference between using the ISCO or the event horizon as the inner disk radius for purposes of ray tracing. For all the disks considered here, we fix the inclination at $\theta=45^\circ$ and the BH mass at $M_\bullet=10^4M_\odot$. For high accretion rate disks, the slim disk assumptions would push the inner edge inside the ISCO. As a result, our choice to use the ISCO as the inner disk edge \citep{Sadowski09} could impact the spectrum and therefore the best-fit parameters. In Fig.~\ref{rin} we compare the results obtained if we set the boundary condition of the slim disk at close to the event horizon. In the main text, we cut off the disk at the ISCO when ray tracing, due to a singularity in the $f_c$ prescription. Instead, in this analysis when $f_c$ runs into this singularity, we reset it to 1. We study the slim disk model with different $a_\bullet$ and $\dot m$. For the upper panel, we fix the accretion rate at $\dot m=10$ Edd. The negative spin disk yields the biggest error, because the ISCO is relatively far away from the BH, and for a high accretion rate disk, the inner edge can be pushed to near the event horizon even for such a retrograde spinning disk. In the lower panel, we fix $a_\bullet=-0.9$. There is little difference between the spectra when choosing the ISCO or the event horizon as the inner edge for a low accretion rate disk. This is because for low accretion rate disks, the disk terminates at ISCO, and extending the disk inward would not affect the spectrum too much. For both panels, the error caused by choosing ISCO as inner radius is always $< 2\%$. For a lower accretion rate disk, the error is lower. As a result, we set the inner edge of the disk at ISCO in the main paper.
\end{appendix}
|
train/arxiv
|
BkiUb1DxK2li-DeXy421
| 5
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|
\section{Introduction and summary}
An important direction is to extend checks of AdS/CFT
correspondence to subleading orders in $1/N$ expansion on the gauge theory side
or higher genus corrections on the dual string theory side.
One of the simplest observables to consider is the expectation value $\langle \mathcal W \rangle $ of
$1\ov2$-BPS circular Wilson loop for which the exact in $N$
expressions are available in both $SU(N)$ $\mathcal N=4$ SYM
\cite{Erickson:2000af,Drukker:2000rr,Pestun:2007rz,Zarembo:2016bbk} and $U(N)_{k}\times U(N)_{-k}$ ABJM \cite{Kapustin:2009kz,Marino:2009jd,Drukker:2010nc} theories.
It was recently observed in \cite{Giombi:2020mhz} that the expressions for
$\langle \mathcal W\rangle$ expanded first in $1/N$ and then
in large 't Hooft coupling $\l$ have a universal form when written in terms of the corresponding string coupling $g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} $ and
string tension $T= {\RR^2\over 2 \pi \alpha'}$ defined as
\cite{Maldacena:1997re,Aharony:2008ug}
\ba \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){1.1}
&{\rm SYM}:\qquad g_{s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s}} = \frac{g^2_{\rm YM}}{4\pi}=\frac{\l}{4\pi N} \ , \qquad
\l = g^2_{\rm YM} N \ , \qquad
= {{\sqrt{\l}}\ \over 2 \pi} \ , \\
& {\rm ABJM}:\qquad g_{s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s}} = { \sqrt \pi\,(2 \l)^{5/4}\over N} \ , \qquad
\l = {N \over k} \ , \qquad \ \ \ \ \ T = {\sqrt{2 \l} \over 2} \ . \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){1.2}
\ea
Explicitly,\footnote{\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){f0} Following \cite{Giombi:2020mhz} in this paper we define ${\cal W}=\tr [ P \exp (...)] $ without the
$1/N$ prefactor.}
\ba
\nonumber
\langle \mathcal W \rangle =& \ e^{2\pi\,T} \sum^\infty_{p=0}\text{c}_{p}\Big(\frac{g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} }{\sqrt T}\Big)^{2p-1}\,\Big[1+\mathcal O(T^{-1})\Big]\\
= &\ e^{2\pi\,T}\, {\sqrt T \over g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s}} \Big\{ \text{c}_{0} \big[ 1 + {\cal O}} \def \bG {\bar \G (T^{-1}) \big]
+ \text{c}_{1} { g^2_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} \over T } \big[ 1 + {\cal O}} \def \bG {\bar \G (T^{-1}) \big]
+ \text{c}_{2} \Big({g^2_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} \over T } \Big)^2 \big[ 1 + {\cal O}} \def \bG {\bar \G (T^{-1}) \big] + ... \Big\} \ . \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){1.3}
\ea
Indeed, this is what one finds by expanding (in large $N$ and then in large $\l$)
the exact result in, e.g., SYM theory \cite{Drukker:2000rr}\footnote{
Here $L^{(k)}_n$ is the generalized Laguerre polynomial.
This expression is for the $U(N)$ group while for $SU(N)$ one gets an additional factor
$e^{-\frac{\lambda}{8N^2}}$ so that $ \langle \mathcal W\rangle$ is 1 for $N=1$.}
\ba
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){1.4}
\langle \mathcal W\rangle &= e^{\frac{\lambda}{8N}}\,L_{N-1}^{(1)}\big(-\frac{\lambda}{4N}\big)
= N\,e^{\sqrt\lambda}\sum_{p=0}^{\infty}\frac{\sqrt 2}{96^{p}\sqrt\pi p!}\frac{\lambda^{\frac{6p-3}{4}}}{N^{2p}}
\Big[1+\mathcal O\big(\frac{1}{\sqrt\lambda}\big)\Big]\ .
\ea
The universal structure of \rf{1.3} is a manifestation of the fact that the two gauge theories are expected to be dual to similar
superstring theories in ${\rm AdS}_5 \times S^5\ $ and ${\rm AdS}_4 \times CP^3$ where $\langle \mathcal W \rangle$ should be given by a string path integral over surfaces
ending on circle at the boundary of AdS. $e^{2\pi\,T} $ in \rf{1.3}
is the semiclassical factor corresponding to AdS$_2$ minimal surface
\cite{Berenstein:1998ij,Drukker:1999zq,Drukker:2000ep}.
The expansion is done first in small string coupling $g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} $ (i.e. large $N$ for fixed $T\sim {\sqrt{\l}}\ $)
and then in $1/T$ at each order in $g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} $. The power of string coupling is
the Euler number $\chi = 1- 2p$ of a disc with $p=0,1,2, ...$ handles.
A non-trivial feature of \rf{1.3} is that the leading power of the inverse string tension $1/T$
at each order in $g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} $ is precisely
$-\ha \chi = p-\ha$, i.e. it is correlated with the power of $g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} $.
A string theory explanation of this fact was suggested in \cite{Giombi:2020mhz} by showing that dependence of the string partition function
on the AdS$_n$ radius $\RR$ (and thus on the string tension) is controlled by the Euler number of the surface.
Another remarkable fact about the SYM result \rf{1.4} is that the leading large $T$ terms in \rf{1.3} exponentiate \cite{Drukker:2000rr} (according to \rf{1.4}, the coefficients $\text{c}_{p} $ in \rf{1.3} are given by
$\text{c}_{p} = \frac{1}{2\pi p!}\big(\frac{\pi}{12}\big)^{p}$)
\iffa
\footnote{\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){f1} One may try
understand this exponentiation as a result of resummation of thin far separated handles;
an attempt \cite{Drukker:2000rr} to explain the negative power of $T$ in the $ {g_{\text{s}}^{2}}/{T}$ argument in the exponent in \rf{1.5} based on supergravity approximation as in \cite{Berenstein:1998ij}
does not appear to work directly requiring subtle cancellations presumably implied by supersymmetry
(see Appendix \ref{app:KK} below).}
\fi
\ba
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){1.5}
\langle \mathcal W\rangle =
{ 1 \over 2 \pi} \frac{\sqrt T}{g_{\text{s}}}\, e^{2\pi\,T \ +\ \frac{\pi}{12}\frac{g_{\text{s}}^{2}}{T} }\,\Big[1+\mathcal O(T^{-1})\Big]\ .
\ea
Surprisingly, in the ABJM theory the coefficient of the first subleading correction is the same $\pi \over 12$ as in the SYM case
\cite{Drukker:2010nc}
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){1.6}
\langle \mathcal W \rangle = \Big(\frac{N}{4\pi\,\lambda}+\frac{\pi\,\lambda}{6N}+\cdots\Big)\,e^{\pi\,\sqrt{2\lambda}} =
{1 \over \sqrt{2\pi}} \frac{\sqrt T}{g_{\text{s}}}\,e^{2\pi\,T}
\Big[1+\frac{\pi}{12}\,\frac{g_{\text{s}}^{2}}{T}+{\cal O}} \def \bG {\bar \G(g^4_s)\Big]\ .
\end{equation}
However, the coefficients of higher order terms (that can be found from \cite{Klemm:2012ii})
turn out to be different than in the SYM case \rf{1.4}, i.e. here the exponentiation
does not happen.\footnote{This may be surprising given that such an exponentiation
may be expected in the large tension (``thin handle'') approximation on the string theory side \cite{Drukker:2000rr,Giombi:2020mhz}
and the fact that the
dual string theories in ${\rm AdS}_5 \times S^5\ $ and AdS$_4\times CP^3$ are similar.}
Instead, we will find (see Appendix ~\ref{ABJM:simple:genus}) that in the ABJM case the leading strong-coupling terms in
\rf{1.3} can be resummed as
\begin{equation} \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){1.7}
\langle \mathcal W \rangle = {1 \over 2 \sin [ \sqrt\frac{\pi}{2}\,\frac{g_{\text{s}}}{\sqrt T}] } \, e^{2\pi\,T} \Big[1+\mathcal O(T^{-1})\Big] \ , \end{equation}
where $\sqrt\frac{\pi}{2}\,\frac{g_{\text{s}}}{\sqrt T}= 2\pi\,\frac{\lambda}{N}= {2 \pi \over k }$
(see \rf{1.2}).
\
Our aim below will be to extract similar predictions about the structure of
small $g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s}$, large $T$ string theory
corrections and their possible resummation
for other closely related observables for which the exact gauge theory results can be found
from matrix model representations following from localization
(in some cases generalizing partial results in the literature).
Namely, we shall consider correlators of $\ha$-BPS Wilson loop with chiral primary operators (CPO)
and also correlators of several coincident Wilson loops (mostly in the SYM theory).
Like in the case of $ \langle \mathcal W \rangle$ in \rf{1.3} we
will observe certain universal patterns in their expansion in small $g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} $ and large $T$
that should be related to supersymmetry of these observables.
This may hopefully aid future investigations on the dual string theory side.
Let us summarize our main results.
\subsection{Correlators of $\frac{1}{2}$-BPS Wilson loop with chiral primary operators}
In section \ref{2} we shall consider the SYM correlator of a circular Wilson loop with a chiral primary
operator $\mathcal O_{J} = \tr \varphi^{J}$.\footnote{
As is well known, in $\mathcal N=4$ SYM one can construct Maldacena-Wilson loops with various amounts of supersymmetry
\cite{Zarembo:2002an}, \textit{e.g.} the $\frac{1}{4}$-BPS circular loop \cite{Drukker:2006ga}
and $\frac{1}{8}$-BPS loops \cite{Drukker:2007yx,Drukker:2007dw,Drukker:2007qr}.
Correlators of these loops
and local operators were considered in
\cite{Semenoff:2001xp,Pestun:2002mr,Semenoff:2006am}.
Correlators of $\frac{1}{8}$-BPS circular loop and various chiral primaries have been computed
by localization in
\cite{Giombi:2009ds,Giombi:2012ep,Bassetto:2009rt,Bassetto:2009ms,Bonini:2014vta}.
Correlators involving Wilson loops in higher representations were discussed in
\cite{Giombi:2006de,Gomis:2008qa}.
In the planar limit at strong coupling the results were successfully compared with AdS/CFT predictions
\cite{Berenstein:1998ij,Giombi:2006de,Gomis:2008qa}
~Beyond the planar limit and for $J>3$
the definition of the $\mathcal N=4$ SYM BPS operators dual to
single-particle string (supergravity) states requires the addition to $\tr \varphi^J$ of multi-trace terms
(see \cite{Aprile:2020uxk} and references therein). This happens not change the qualitative structure of the
$1/N$ expansions discussed below. }
The correlator $\langle \mathcal W\,\mathcal O_{J}\rangle $
was originally discussed in \cite{Berenstein:1998ij} at the
leading order in strong coupling in connection with the Wilson loop
OPE expansion.
In the planar limit this correlator was computed exactly in $\l$ in \cite{Semenoff:2001xp}:
$\langle \mathcal W\,\mathcal O_{J}\rangle \sim I_J ({\sqrt{\l}}\ ) $ ($I_J$ is the Bessel function).
We have extended the computation to non-planar corrections; expanded in small $g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s}$ and then in large $T$
as in \rf{1.3} the result reads\footnote{Here we ignore
the R-symmetry factor $Y$ depending on the choice of the CPO and the scalar coupling in ${\cal W}$
\cite{Semenoff:2001xp}
and the factor of dependence on the operator insertion point
which is fixed by conformal invariance
(see section 2).}
\ba
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){1.8}
\frac{\vev{\mathcal W\,\mathcal O_{J}}}{\vev{\mathcal W}} = c_{J}\,T\,\Big\{1 &+\frac{a_{0}^{(1)}(J)}{T}+\frac{a_{0}^{(2)}(J)}{T^{2}}+\cdots +\frac{g_{\text{s}}^{2}}{T^{2}}\,\Big[a_{1}^{(0)}(J)+\frac{a_{1}^{(1)}(J)}{T}+\cdots\Big] \notag \\ &
+\frac{g_{\text{s}}^{4}}{T^{4}} \,\Big[a_{2}^{(0)}(J)+\frac{a_{2}^{(1)}(J)}{T}+\cdots\Big]+{\cal O}} \def \bG {\bar \G\Big( \frac{g_{\text{s}}^{6}}{T^{6}}\Big)\Big\}\ ,
\ea
where dots stand for terms subleading in $1/T$.
On the string theory side, the overall factor of $T$ should come from the semiclassical value of the vertex
operator dual to ${\cal O}} \def \bG {\bar \G_J$ evaluated on the AdS$_2$ minimal surface.
The coefficient $c_J$ is fixed by normalization of $\mathcal O_{J}$ and
$a_{i}^{(j)}(J)$ are polynomials in $J$, cf. \rf{2.28}
(for example,
$ a_{0}^{(1)}= - {1 \over 4 \pi} (J^2-1)$ as in \cite{Semenoff:2001xp}).
Compared to the series in $\frac{g_{\text{s}}^2}{T} \sim \frac{\lambda^{3/2}}{N^2}$ in \rf{1.3} here the
natural expansion parameter turns out to be $x\equiv \frac{g_{\text{s}}^2}{T^2} \sim \frac{\lambda^{3}}{N^2}$.
Remarkably, it is possible to explicitly sum up all leading large $T$ terms in \rf{1.8}
as
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){1.9}
\frac{\vev{\mathcal W\,\mathcal O_{J}}}{\vev{\mathcal W}} = c_{J}\,T\,F_{J}\Big(\frac{g_{\text{s}}^{2}}{T^{2}}\Big) \, \Big[ 1 + {\cal O}} \def \bG {\bar \G(T^{-1})\Big]\
,\qquad\qquad F_{J}(x) = \frac{2}{J\sqrt x}\sinh\Big(J\,\text{arcsinh}\frac{\sqrt x}{2}\Big).
\end{equation}
Here $F_J$ is a finite polynomial for odd $J$ and $\sqrt{1+\tfrac{1}{4} {g_{\text{s}}^{2}\over T^2}} $ times a polynomial for even $J$.
For example, in the $J=2$ case
one finds simply
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){1.10}
\frac{\vev{\mathcal W\,\mathcal O_{2}}}{\vev{\mathcal W}} = \pi\,\sqrt{T^{2}+\tfrac{1}{4} g_{\text{s}}^{2}}\,\Big[1+\mathcal O\Big(T^{-1}\Big)\Big].
\end{equation}
The same expression applies also to the correlator of $\mathcal W$ with the
dimension 4 dilaton operator $\mathcal O_{\rm dil}$ which is a supersymmetry descendant of
${\cal O}} \def \bG {\bar \G_2$.\footnote{\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){f4} For higher $J$, the generalized dilaton operator $\mathcal O_{{\rm dil}, J'}$ with non-zero $R$-charge $J'$
and dimension $\Delta= 4 + J'$ is a supersymmetry descendant of $ \mathcal O_{J}$
with $J= 2 + J'$ and thus ${\vev{\mathcal W\,\mathcal O_{{\rm dil}, J-2}}}/{\vev{\mathcal W}}$ is the same as \rf{1.8}.}
For any $g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} $ and $T$ the expectation value $\frac{\vev{\mathcal W\,\mathcal O_{\rm dil }}}{\vev{\mathcal W}} $
can be found directly from $\vev{\mathcal W}$ in \rf{1.3},\rf{1.4} by differentiating over $\l$
so that using \rf{1.1} we have (see \cite{Giombi:2020mhz} and refs. there)
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){1.11}
\frac{\vev{\mathcal W\,\mathcal O_{2}}}{\vev{\mathcal W}} = \frac{\vev{\mathcal W\,\mathcal O_{\rm dil}}}{\vev{\mathcal W}}
= \l { \partial \over \partial \l}
\log \langle \mathcal W \rangle= \Big( \ha T { \partial \over \partial T} + g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} {\partial \over \partial g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} } \Big) \log \langle \mathcal W \rangle \ .\end{equation}
The small $g_{\text{s}}$, large $T$ expansion of $\log\vev{\mathcal W}$ following from (\ref{1.4}) is found to be
\ba
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){x1}
\log\vev{\mathcal W} = 2\pi T &- \ha \log\Big(4\pi^2 { g^2_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} \over T}
\Big) -\frac{3}{16\pi T}+\cdots+\frac{\pi}{12}\frac{g_{\text{s}}^{2}}{T}\,\Big(
1-\frac{3}{4\pi T}+\cdots\Big)\notag \\ &
-\frac{\pi}{320}\frac{g_{\text{s}}^{4}}{T^{3}}\,\Big(1-\frac{1945}{768\pi T}+\cdots\Big)+\mathcal O\Big({g_{\text{s}}^{6}\over T^5}\Big)\ ,
\ea
and therefore
\ba
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){x2} \Big( \ha T { \partial \over \partial T} + g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} {\partial \over \partial g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} } \Big) \log \langle \mathcal W \rangle
= \pi\,T\,\Big[1 & -\frac{3}{4\pi T}+\cdots+\frac{g_{\text{s}}^{2}}{8\,T^{2}}\Big(1-\frac{1}{2\pi T}+\cdots\Big)\notag \\ & -\frac{g_{\text{s}}^{4}}{128\,T^{4}}
\Big(1-\frac{389}{192\pi T}+\cdots\Big)
+ \mathcal O\Big({g_{\text{s}}^{6}\over T^6}\Big)
\Big] \ ,
\ea
in agreement with \rf{1.8}.
The reason why the large $T$ expansion \rf{1.8}
has a different structure than \rf{1.3} and thus also why the resummed expressions in \rf{1.5} and in \rf{1.10}
are not directly related by \rf{1.11} is that subleading in $1/T$ terms at each order in $g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} $ in $ \langle \mathcal W \rangle$ in \rf{1.3} contribute to
$\log \langle \mathcal W \rangle$
and, as a result, reorganize its large $T$ expansion (see \rf{2.39}--\rf{2.41} for details).
There is still an interesting connection between the resummed expressions for $\langle \mathcal W \rangle$ in \rf{1.5} and
the correlator ${\vev{\mathcal W\,\mathcal O_{2}}}$
in \rf{1.9},\rf{1.10}: both can be given a ``D3-brane'' interpretation \cite{Drukker:2005kx,Giombi:2006de}.
To recall,
for a circular Wilson loop in $k$-symmetric $SU(N)$ representation in the limit of
large $k,\, N$ and $\l$ with $\varkappa= { k {\sqrt{\l}}\ \over 4 N} = {k\, g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} \over 2\, T}=$fixed one expects that $ \langle \mathcal W \rangle$
should be given by $\exp ( - S_{\rm D3})$ where $S_{\rm D3}$ is the D3-brane
action on the corresponding classical solution \cite{Drukker:2005kx}.
For $1 \ll k \ll N $ this should apply also to the Wilson loop
in the $k$-fundamental representation
described by a minimal surface ending on a multiply wrapped circle; here one finds \cite{Drukker:2005kx}: \ $ S_{\rm D3} =N f(\varkappa) =
- k {\sqrt{\l}}\ - { k^3 \l^{3/2} \over 96 N^2} + {\cal O}} \def \bG {\bar \G ({ k^5 \l^{5/2} \over N^4}) $.
Extrapolating this to the
$k=1$ case corresponds to the resummation of the expansion in
\rf{1.3},\rf{1.4} for fixed $ { {\sqrt{\l}}\ \over N} \sim {g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} \over T}$
(i.e.
when $g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} \sim T $ formally are both large, cf. \cite{Drukker:2005kx,Okuyama:2006ir}).
Then
$ S_{\rm D3} = - 2 \pi T - { \pi \over 12 }\, { g^2_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} \over T} + {\cal O}} \def \bG {\bar \G( { g^4_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} \over T^3} )$,
reproducing the exponential factor in \rf{1.5}.
Similar D3-brane interpretation is possible also in the case of the correlator ${\vev{\mathcal W\,\mathcal O_{J}}}$
\cite{Giombi:2006de}.\footnote{We thank S. Giombi for pointing this out to us.}
Indeed, the resummed expression \rf{1.9} is in perfect agreement with the result found in \cite{Giombi:2006de}
in the fixed $\varkappa={k\, g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} \over 2\, T}$ limit (both from the derivative of the D3-brane action over the corresponding graviton source and
from the matrix model in the case of $k$-fundamental representation) after formally interpolating
to the $k=1$ case.
In section \ref{3}
we shall also consider the correlator $\vev{\mathcal W\,\mathcal O_{J_1}\,\mathcal O_{J_2}}$
with two chiral primary operators.
In the two special cases
(a) $J_1=J_2$ and (b) $J_1=2, J_2=2J$
it is possible to
reduce their computation
to correlators in the Gaussian 1-matrix model.
The structure of the resulting $1/N$ strong coupling expansion
is found to be similar to \rf{1.8}
\ba
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){1.14}
&\frac{\vev{\mathcal W\,\mathcal O_{J_1}\,\mathcal O_{J_2}}}{\vev{\mathcal W}} = d_{J_1J_2}\,T^2\,\Big\{1+\frac{b_{0}^{(1)}(J_1,J_2)}{T}+\frac{b_{0}^{(2)}(J_1,J_2)}{T^{2}}+\cdots \\
&\quad+\frac{g_{\text{s}}^{2}}{T^{2}}\,\Big[b_{1}^{(0)}(J_1,J_2)+\frac{b_{1}^{(1)}(J_1,J_2)}{T}+\cdots\Big]
+\frac{g_{\text{s}}^{4}}{T^{4}}\,\Big[b_{2}^{(0)}(J_1,J_2)+\frac{b_{2}^{(1)}(J_1,J_2)}{T}+\cdots\Big]+ {\cal O}} \def \bG {\bar \G\Big( { g^6_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} \over T^6}\Big)\Big\},\nonumber
\ea
where $b_{i}^{(j)}$ are polynomials in $J_1,J_2$ (see \rf{3.19},\rf{332},\rf{3322}).
\subsection{Correlators of coincident Wilson loops}
\def {\rm T} {{\rm T}}
Another class of tractable examples that we shall consider in section \ref{4}
are the expectation values of coincident circular Wilson loops $\vev{\mathcal W^{n}}$ in SYM theory.\footnote{
Correlators of separated loops were considered in
\cite{Gross:1998gk,Zarembo:1999bu,Correa:2018lyl,Correa:2018pfn};
supersymmetric configurations with oppositely oriented loops were discussed in
\cite{Dorn:2018srz,Giombi:2009ms}; for various
matrix model calculations, see \cite{Sysoeva:2018xig,CanazasGaray:2018cpk,Okuyama:2018aij,CanazasGaray:2019mgq,Muck:2019hnz}.
}
\iffa
Notice also that the multi-trace $\mathcal W^{n} = (\tr U)^{n}$ is completely different from the $n$-wound simple loop, \textit{i.e.} $\tr(U^{n})$. The multiply wound case
is obtained from the standard simple loop by the rescaling $\lambda\to n^{2}\,\lambda$. This means $T\to n\,T$ and $g_{\text{s}}\to n^{2}\,g_{\text{s}}$, so instead of (\ref{1.5})
we get $W_{1}\to W_{1}\,n^{-3/2}$ (including also the rescaled AdS area) and $H\to n^{3}H$.
\fi
The $n=2$ case
in the planar limit was discussed, in particular, in \cite{Drukker:2000rr,Arutyunov:2001hs}.
Extending calculation to subleading orders in $1/N$
in large $\l$ limit and rewriting the resulting expansion in terms of $g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} $ and $T$ as in \rf{1.3}
we have found that
\ba
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){1.15}
\frac{\vev{\mathcal W^{2}}}{\vev{\mathcal W}^{2}} = &
\sum_{p=0}^{\infty}\frac{\pi^p }{(2p-1)!!}\Big( {g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} \over T}\Big)^p\Big[1 +\mathcal O(T^{-1})\Big] \nonumber\\
= &1+e^{\frac{\xi}{2}}\sqrt\frac{\pi\,\xi}{2}\,\text{erf}\Big(
\sqrt\frac{\xi}{2}\Big)+\mathcal O(T^{-1}),
\ \ \ \ \ \qquad
\xi \equiv \pi\frac{g^2_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s}}{T}\ .
\ea
The analogous expression for $n=3$ is
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){1.16}
\frac{\vev{\mathcal W^{3}}}{\vev{\mathcal W}^{3}} = 1+3e^{\frac{\xi}{2}}\sqrt\frac{\pi\,\xi}{2}\,\text{erf}\Big(
\sqrt\frac{\xi}{2}\Big)
+\frac{4\pi}{3\sqrt 3}\,\xi\,e^{2\xi}\,\Big[1-12\,{\rm T}\Big(\sqrt{3\xi}, \frac{1}{\sqrt 3}\Big)\Big]+\mathcal O(T^{-1}),
\end{equation}
where ${\rm T}(h,a)$
is the Owen T-function (see \rf{ttt}).
For general $n$ we found similar expansion
(see \rf{443} and Appendix \ref{anew})
\ba
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){1.17}
\frac{\vev{\mathcal W^{n}}}{\vev{\mathcal W}^{n}}=
1&+\frac{n\,(n-1)}{2}\,\xi+\frac{\,n(n-1)(3n-5)\,(n+2)}{24}\,\xi^{2}\notag \\ &+\frac{\,n(n-1)(15n^{4}+30n^{3}-75n^{2}-610 n+1064)}{720}\,\xi^{3}+... \ .
\ea
Thus like for $\vev{{\cal W}}$ in \rf{1.3} here we get again series in $\xi= \pi \frac{g^2_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s}}{T} $
while for the correlators
with chiral primary operators \rf{1.8},\rf{1.14}
the expansion was in powers of $ x= {g^2_s \over T^2}$.
In section \ref{s4.3} we shall derive a similar expansion
for the correlator $\mathcal W^{(1,-1)}$ of coincident Wilson loops in fundamental and anti-fundamental representations. It turns out that in contrast to \rf{1.15},\rf{1.17}
$\mathcal W^{(1,-1)}$ has trivial connected part, i.e.
to all orders in $\xi$ (and to leading order in $1/T$)
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){1.18}
\langle{\mathcal W^{(1,-1)}}\rangle = \vev{\mathcal W}^{2}\,\Big[1+\mathcal O(T^{-1})\Big]\simeq \vev{\mathcal W}^{2}\ .
\end{equation}
It would be interesting to explain this fact
from the string theory point of view.
\subsection{Comments on correlators in ABJM}
Obtaining the above results in the ${\cal N}=4$ SYM theory case is facilitated by a relative simplicity of the
associated Gaussian matrix model.
In the ABJM theory the computations of similar correlators involving
$\frac{1}{2}$-BPS circular Wilson loop \cite{Drukker:2009hy}
are substantially more involved.
The structure of the strong-coupling expansion of the correlators
with chiral primary operators
is expected to be similar to \rf{1.8}.\footnote{For a discussion of single-trace
CPO in ABJM theory see, e.g., \cite{Aharony:2008ug,Papathanasiou:2009zm}.}
This is suggested by the observation \cite{Giombi:2020mhz} that
the expansion \rf{1.3} of $ \vev{\mathcal W}$ looks the same in the SYM and ABJM theories
and that, in particular, for the dilaton operator the correlators
$\vev{\mathcal W\,\mathcal O_{\rm dil }}$ and $ \vev{\mathcal W}$ should be
again related as in the last equality in \rf{1.11}.
\iffa
\footnote{Note that in view of \rf{1.2}
here $\l { \partial \over \partial \l} = \ha T { \partial \over \partial T} + {5 \over 4} g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} {\partial \over \partial g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} }
$ so that the insertion of the dilaton operator on the ABJM theory side does not appear to
correspond to applying $\l { \partial \over \partial \l}$.}
\fi
Indeed, the dilaton vertex operator
has the same structure in both ${\rm AdS}_5 \times S^5\ $ and AdS$_{4}\times CP^{3}$ string theories
and thus the derivative over the zero-momentum dilaton should be related to the string partition function
in the same way as in ${\rm AdS}_5 \times S^5\ $ case in \cite{Giombi:2020mhz}, i.e. as in \rf{1.11}.
Below in Appendix \ref{5} we shall discuss the computation of correlators of coincident Wilson loops
$\vev{{\mathcal W}^{n}}$ in ABJM theory.
In particular, for the $n=2,3$
we will find
\ba
\vev{\mathcal W^{2}} = \vev{\mathcal W}^{2}
\,\Big[1+0\times \frac{g_{\text{s}}^{2}}{T}+0\times \Big(\frac{g_{\text{s}}^{2}}{T}\Big)^{2}+\cdots
\Big], \qquad
\vev{\mathcal W^{3}} =\vev{\mathcal W}^{3}
\, \Big[1+0\times \frac{g_{\text{s}}^{2}}{T}+0\times \Big(\frac{g_{\text{s}}^{2}}{T}\Big)^{2}+\cdots
\Big], \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){1.19}
\ea
suggesting the conjecture that, up to subleading $1/T$ terms at each order in the genus expansion, here
the connected part of the correlator $\vev{\mathcal W^{n}} $ vanishes,
i.e. $\vev{\mathcal W^{n}} \simeq \vev{\mathcal W}^{n}$.
This is in contrast to the non-trivial relation \rf{1.17} found for $\vev{\mathcal W^{n}} $ in the SYM case
(but is similar to the behaviour of $ \langle{\mathcal W^{(1,-1)}}\rangle $ in \rf{1.18}).
\subsection{Structure of the paper}
In section \ref{2} we compute the $1/N$ expansion \rf{1.8} of
the SYM correlator $\vev{\mathcal W\, \mathcal O_{J}}$ of the $\frac{1}{2}$-BPS Wilson loop with a chiral primary
operator starting with its matrix model representation implied by localization.
In section \ref{3} we repeat the same analysis for the correlator $\vev{\mathcal W\, \mathcal O_{J_{1}}(x_1) \mathcal O_{J_{2}}(x_2) }$
assuming a special (supersymmetric) choice of insertion points $x_1$ and $x_2$
that allows a matrix model calculation confirming
that its strong coupling expansion has the form \rf{1.14}.
In section \ref{4} we consider correlators of
coincident BPS Wilson loops. We establish the structure of the expansions in \rf{1.15},\rf{1.16} and prove
their exact form by exploiting the Toda integrability
structure of the underlying Gaussian matrix model.
In section \ref{s4.3} we consider the correlator of Wilson loops in the fundamental and in the anti-fundamental representation
where special features are expected due supersymmetry.
Indeed, in this case one finds \rf{1.8}, i.e. there are no leading order corrections to $\mathcal W^{(1,-1)}$
beyond those in $\vev{\mathcal W^{2}}$.
In Appendix \ref{app:KK} we discuss an attempt \cite{Drukker:2000rr} to explain the negative power of $T$ in the $ {g_{\text{s}}^{2}}/{T}$ term in \rf{1.3} by
assuming that for large $T$ one can use supergravity approximation as in \cite{Berenstein:1998ij}. As we explain,
this
argument may work only if there are non-trivial cancellations of the dominant large $T$ terms
that should be implied by supersymmetry.
Appendix \ref{app:B} contains some technical details of the $1/N$ expansion
of $\vev{\mathcal W}$. In
Appendix \ref{sec:semi} we consider the $1/N$ expansion of the
correlator $\vev{{\cal W}\, \mathcal O_{J}}$ in the
string semiclassical limit $J\sim {\sqrt{\l}}\ \gg 1$.
In Appendix \ref{anew} we work out the $1/N$ expansion of $\vev{\mathcal W^n}$ deriving the expansion \rf{1.17}.
The Appendices \ref{ABJM:simple:genus} and \ref{5} are devoted to the
correlators $\vev{\mathcal W^n}$ of $\frac{1}{2}$-BPS circular Wilson loop in the ABJM theory.
In Appendix \ref{ABJM:simple:genus} we comment on the single Wilson loop case
case by reviewing the known matrix model results
pointing out that here the expansion has again the same structure as in the SYM case in \rf{1.3}
and deriving the representation \rf{1.7}.
Appendix \ref{5} discusses correlators of
$n=2,3$ coincident Wilson loops where we
use the topological expansion of the algebraic curve characterizing the ABJM matrix model
to first derive the exact expressions valid for all couplings, and then expand at strong coupling
demonstrating the validity of \rf{1.19}.
\iffa
\item The relation (\ref{1.17}) shows that exponentiation does not hold for $n\neq 1$, at least not in terms of a simple linear
function of $g_{\text{s}}^{2}/T$. Instead ``simple factorization'' holds for (\ref{1.18}), where it is just inherited from squaring $\vev{\mathcal W}$.
Is there an explanation in terms of the dual minimal surfaces and/or supersymmetry ?
\item Apart from ``simple exponentiation'', a separate issue is the presence of ``irreducible'' corrections in $\vev{\mathcal W^{n}}/\vev{\mathcal W}^{n}$.
In this regard, what happens in ABJM ? What is the high genera extension of (\ref{1.21}) ? Is it true that in that case we have at all orders the relation
$\vev{\mathcal W^{2}}/\vev{\mathcal W}^{2} = 1+\mathcal O(T^{-1})$ ? And if yes, why ?
\fi
\section
Expansion of
$ \vev{\mathcal W\, \mathcal O_J} $
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){2} }
In this section we will compute the $1/N$ expansion of the ${\cal N}=4$ SYM correlator of $\frac{1}{2}$-BPS circular Wilson loop
with chiral primary operators
$\mathcal O_{J}$.
As was shown in \cite{Semenoff:2001xp},
in the leading planar approximation the
expression for this correlator
is proportional to the Bessel function, $\langle \mathcal W\,\mathcal O_{J}\rangle \sim I_J ({\sqrt{\l}}\ )$. This result
was
obtained by summing all planar rainbow Feynman graphs under the assumption that radiative corrections from planar graphs with internal vertices
cancel to all orders in perturbation theory. This result was later confirmed in the framework of
supersymmetric localization where $\vev{\mathcal W\,\mathcal O_{J}}$
was computed using a suitable hermitian 2-matrix model \cite{Giombi:2009ds}.
Below we shall first obtain the finite $N$ localization result for this correlator using
a simplified equivalent 1-matrix model suggested by similar computations in the
$\mathcal N=2$ superconformal models \cite{Fucito:2015ofa,Billo:2018oog}. We shall then derive the
$1/N$ expansion of this correlator (up to the $1/N^{6}$ order)
with the coefficients
being
$J$-dependent combinations of Bessel functions of $ \sqrt \l$.
Finally, we will extract the leading large $\l$ behaviour of these coefficients.
In general, the $\frac{1}{2}$-BPS Wilson loop depends \cite{Maldacena:1998im} on a unit 6-vector
$n_i$
defining the coupling to the SYM scalars $n_{i}\Phi_{i}$.
The chiral primary operator
may be chosen as $\mathcal O_{J} = \tr\big(u_{i}\Phi_{i}(x)\big)^{J}$
where $u$ is a complex null 6-vector $u^{2}_i=0$.
The dependence of the correlator $\vev{\mathcal W\,\mathcal O_{J}}$
on $n$ and $u$ factorizes \cite{Semenoff:2001xp}, i.e. is contained only in the overall
factor $Y(n, u) = (n_i u_i)^{J}$.
We shall choose the 6-vector $n_i$ in ${\cal W}$ along the 1-direction and the
vector $u_i$ to be non-zero only in (1,2) directions, so that
\ba &\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){1w}
\mathcal W = \tr P \exp\Big\{g_{_{\rm YM}}\int_{C} d\sigma\,[i\,A_{\mu}(x)\,\dot x^{\mu}(\sigma)+\, R\,\Phi_{1}(x)]\Big\},
\\ \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){2w}
&\mathcal O_{J} = \tr \big[\varphi (x)\big]^J \ ,\qquad \qquad \varphi = \tfrac{1}{\sqrt 2}(\Phi_{1}+i\Phi_{2}),
\ea
where $C$ is a circle of radius $R$ (that can be set to 1 as we assume below).
Then $ Y(n, u) = 2^{-J/2}$; we will not explicitly indicate this factor in $\vev{\mathcal W\, \mathcal O_J(x)}$
as it can be absorbed into normalization of $\mathcal O_{J}$ discussed below.
Note also that $\varphi=\varphi^{I}T^{I}$ where $T^{I}$ are $U(N)$ generators.
Let us also assume that the unit-radius circular loop in 4-space $(x_1,x_2,x_3,x_4) $
lies in the $(x_1,x_2)$ plane (with the center at the origin)
and define the ``transverse distance''
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){2.2}
d_{\perp}(x) = \tfrac{1}{2}\big[(r^{2}+h^{2}-1)^{2}+4h^{2}\big]^{1/2},\qquad \qquad
r^{2}\equiv x_{1}^{2}+x_{2}^{2}, \qquad h^{2}\equiv x_{3}^{2}+x_{4}^{2}\ .
\end{equation}
Conformal symmetry implies that \cite{Berenstein:1998ij,Alday:2011pf,Giombi:2018hsx}
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){2.3}
\vev{\mathcal W\, \mathcal O_J(x)} = \frac{1 }{\big[2d_{\perp}(x) \big]^{J}} \vev{\mathcal W\,\mathcal O_{J}(0)} \ .
\end{equation}
In what follows we shall thus assume that $\vev{\mathcal W\, \mathcal O_J}$ stands
for the $x$-independent part of \rf{2.3}, i.e. its value at $x=0$.
\subsection{Matrix model formulation \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){s2.1}}
\def g_{_{\rm YM}} {g_{_{\rm YM}}}
We we will use a hermitian 1-matrix model formulation
that computes expectation values like \rf{2.3} using
a Gaussian hermitian 1-matrix model with the variable $a=a^{I}T^{I}$ as
\footnote{
This is the approach that can be used for chiral correlators in $\mathcal N=2$ superconformal theories \cite{Fucito:2015ofa,Billo:2018oog}.
Other equivalent approaches are available in the $\mathcal N=4$ case, such
as the complex matrix model formulation \cite{Kristjansen:2002bb}
or the associated normal matrix model version applicable for suitable chiral observables \cite{Okuyama:2006jc}.
Nevertheless, our choice will be more convenient for most of our purposes.}
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){2.5}
\vev{\mathcal O} = \int Da\,\mathcal O(a)\,e^{-\tr a^{2}}\ ,\ \qquad \qquad Da = \prod_{I=1}^{N^{2}}\frac{da^{I}}{\sqrt{2\pi}} \ .
\end{equation}
The explicit map from the gauge theory operator to the matrix model one is
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){2.6}
\mathcal O_{J} = \tr \big[\varphi(0)\big]^{J} \
{\longrightarrow}\ \Big(\frac{g^2}{8\pi^{2}}\Big)^{J/2}\,\mathsf{O}_{J},\qquad\qquad \mathsf{O}_{J} = :\tr a^{J}:\ , \qquad \qquad g\equiv g_{_{\rm YM}} \ ,
\end{equation}
where the coupling factor comes from the scaling needed to have the simple normalization in the
exponent $e^{-\tr a^{2}}$ in \rf{2.5}.
Normal ordering in $\mathsf{O}_{J}$ (\ref{2.6}) amounts to subtraction of all self-contractions.
It is required as the correlators involving the standard (flat 4-space) chiral
operator do not have self-contractions (there is no $\varphi\varphi$ propagator) so that
on the matrix model side (derived from gauge theory formulated on 4-sphere)
these self-contractions should be explicitly removed (see, e.g., \cite{Gerchkovitz:2016gxx}).\footnote{\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){x3}
Subtraction of self-contractions is in turn equivalent to the requirement of orthogonality
to all lower dimensional operators \cite{Billo:2017glv}. Denoting by $\{\Omega_{\alpha}\}$ the (single or multi-trace) operators with dimension
strictly less than $\dim \mathcal O$, one has
\begin{equation}
\nonumber
:\mathcal {O}: = \mathcal {O}-\sum_{\alpha, \beta}\vev{\mathcal O\,\Omega_{\alpha}}\,(C^{-1})_{\alpha\beta}\,\Omega_{\beta},\qquad C_{\alpha\beta}=\vev{\Omega_{\alpha}\Omega_{\beta}}.
\end{equation}
}
The matrix model counterpart of the BPS Wilson loop operator is simply (with no $1/N$ normalization)
\begin{equation}\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){266}
\mathcal W \
{\longrightarrow} \ \tr \, e^{\, \frac{g}{\sqrt 2}\,a} \ .
\end{equation}
The correlator
in (\ref{2.5}) is computed by Wick contractions
with the free propagator $\vev{a^{I}a^{J}} = \delta^{IJ}$.
In the following, we will need generic multi-trace correlation functions of the form
\begin{equation}
t_{\bm n} = \vev{\tr a^{n_{1}}\cdots \tr a^{n_{\ell}}}.
\end{equation}
They may be computed by repeated application of the $U(N)$ fusion/fission identities
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){2.10}
\tr (T^{I}\, A \, T^{I} \, B) = \frac{1}{2}\tr A\, \tr B\ ,\qquad\qquad \tr(T^{I}A)\,\tr(T^{I}B) = \frac{1}{2}\tr(AB)\ ,
\end{equation}
leading to the recursion relations \cite{Billo:2017glv}
\ba
t_{n} &= \frac{1}{2}\sum_{m=0}^{n-2}t_{m,n-m-2}\ ,\qquad\qquad
t_{n, n'} = \frac{1}{2}\sum_{m=0}^{n-2}t_{m,n-m-2,n'}+\frac{n'}{2}t_{n+n'-2}\ ,\qquad \textit{etc.}
\ea
\subsection{Differential relations \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){s2.2}}
Using the methods of \cite{Billo:2018oog} we can compute the one-point correlation
functions \rf{2.3} in presence of the Wilson loop.
Remarkably,
they can be found directly from
the knowledge of $\vev{\mathcal W}$ since it is possible to show that for all $J$ one has
$\vev{\mathcal W\,\mathsf{O}_{J}} = \mathscr{D}_{J}(g, \partial_{g})\,\vev{\mathcal W}$,
where $\mathscr D_{J}$ is a linear differential operator of order $J-1$.
This follows from the matrix model
representation of $\vev{\mathcal W\,\mathsf{O}_{J}}$ and is ultimately related to the supersymmetry.
For example,
the $J=2$ CPO correlator is related to the correlator with the dilaton operator and the latter may be found by differentiation over
the coupling $g$ or $\l= g^2 N $ as in \rf{1.11}.
Explicitly, in the $J=2$ case one finds
\ba
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){2.12}
\vev{\mathcal W\,\mathsf{O}_{2}} = \vev{\mathcal W\,:\tr a^{2}:} &= \sum_{k=0}^{\infty}\frac{g^{k}}{2^{\frac{k}{2}}\,k!}\,\Big(t_{k,2}-\frac{N^{2}}{2}t_{k}\Big)
= \sum_{k=0}^{\infty}\frac{g^{k}}{2^{\frac{k}{2}}\,k!}\,\Big(\frac{k+N^{2}}{2}t_{k}-\frac{N^{2}}{2}t_{k}\Big)\notag \\
&= \frac{1}{2}\sum_{k=0}^{\infty}\frac{g^{k}}{2^{\frac{k}{2}}\,k!}\,k\,t_{k} = \frac{1}{2}g \,\partial_{g}\vev{\mathcal W} \ .
\ea
The $J=4$ case is slightly more complicated
\ba
\vev{\mathcal W\,\mathsf{O}_{4}} = \vev{\mathcal W\,:\tr a^{4}:} &= \sum_{k=0}^{\infty}\frac{g^{k}}{2^{\frac{k}{2}}\,k!}\tr a^{k}\,\Big(\tr a^{4}-(\tr a)^{2}-2N\,\tr a^{2}+\frac{N^{3}}{2}+\frac{N}{4}\Big)\notag \\
&= \sum_{k=0}^{\infty}\frac{g^{k}}{2^{\frac{k}{2}}\,k!}\Big(t_{k,4}-t_{k,1,1}-2N\,t_{k,2}+(\frac{N^{3}}{2}+\frac{N}{4})\,t_{k}\Big) \ ,
\ea
where we used the explicit form of $:\tr a^{4}:$ obtained by resolving the mixing with dimension $<4$ operators. From the relations (\ref{2.10}), we find (doing Wick contractions)
\ba
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){2.14}
t_{k,2} &= \frac{N^{2}+k}{2}t_{k}, \qquad\qquad t_{k,1,1} = \frac{N}{2}t_{k}+\frac{k(k-1)}{4}t_{k-2},\notag \\
t_{k,4} &= N\,t_{k,2}+\frac{1}{2}\,t_{k,1,1}+\frac{k}{2}\,t_{k+2} = \frac{k}{2}t_{k+2}+\frac{N(2N^{2}+1+2k)}{4}t_{k}+\frac{k(k-1)}{8}t_{k-2}.
\ea
Hence,
\ba
\vev{\mathcal W\,\mathsf{O}_{4}} &= \sum_{k=0}^{\infty}\frac{g^{k}}{2^{\frac{k}{2}}\,k!}\Big(\frac{k}{2}t_{k+2}-\frac{k N}{2}t_{k}-\frac{k(k-1)}{8}t_{k-2}\Big)=
\sum_{k=0}^{\infty}\frac{g^{k}}{2^{\frac{k}{2}}\,k!}\Big(-\frac{g^{2}}{16}-\frac{kN}{2}+\frac{k(k-1)(k-2)}{g^{2}}\Big)\, t_{k}
\notag \\
&= \Big(-\frac{g^{2}}{16}-\frac{N}{2}\,g\,\partial_{g}+g\,\partial_{g}^{3}\Big)\,\vev{\mathcal W}. \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){213}
\ea
A completely similar calculation for $\mathsf{O}_{6}$ and $\mathsf{O}_{8}$ gives
\ba
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){2.16}
\vev{\mathcal W\,\mathsf{O}_{6}} &= \Big[
2g\partial_{g}^{5}+\frac{3}{8}(N^{2}+1)\,g\partial_{g}-\frac{3}{4}(g\partial_{g})^{2}-2N\, g\partial_{g}^{3}+\frac{3N}{32}g^{2}
\Big]\,\vev{\mathcal W},
\\
\vev{\mathcal W\,\mathsf{O}_{8}} =& \Big[
4g\partial_{g}^{7}
-6N g\partial_{g}^{5}
-\frac{1}{64} g^2 (1+6 N^2)
+\frac{5}{32} g^3 \partial_{g}
-\frac{1}{8} N (17+2 N^2) g\partial_{g}
\notag \\ &
+\frac{15}{8} N (g\partial_{g})^{2}
+\frac{5 (4+N^2))}{2} g\partial_{g}^{3}
-\frac{15}{4 g} \partial_{g}(g^{3}\partial_{g}^{3})
\Big]\,\vev{\mathcal W}. \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){217}
\ea
The above differential relations \rf{2.12},\rf{213},\rf{2.16} written in terms of $\lambda=N\,g^{2}$ read
\ba
\vev{\mathcal W\,\mathsf{O}_{2}} =& \lambda\partial_{\lambda}\vev{\mathcal W}, \qquad \qquad
\vev{\mathcal W\,\mathsf{O}_{4}} = \Big(8N\lambda^{2}\partial^{3}_{\lambda}+12N\lambda\partial^{2}_{\lambda}-N\lambda\partial_{\lambda}
-\frac{\lambda}{16N}\Big)\,\vev{\mathcal W}, \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){216} \\
\vev{\mathcal W\,\mathsf{O}_{6}} = &\Big[
64N^{2}\lambda^{3}\partial_{\lambda}^{5}+320N^{2}\lambda^{2}\partial^{4}_{\lambda}+16N^{2}(\lambda-15)\,\lambda\,\partial_{\lambda}^{3}-3\lambda(8N^{2}+\lambda)\partial_{\lambda}^{2}\notag \\ &
\qquad +\frac{3}{4}(N^{2}-3)\lambda
\partial_{\lambda}+\frac{3\lambda}{32}
\Big]\,\vev{\mathcal W}\ .
\ea
Similar representations are found for higher even $J$ and also for odd $J$, e.g.,
\ba
& \vev{\mathcal W\,\mathsf{O}_{1}} = \frac{1}{2}\sqrt{\textstyle \frac{\lambda}{2N}} \, \vev{\mathcal W} \ ,
\qquad \qquad \vev{\mathcal W\,\mathsf{O}_{3}} = -\frac{1}{4}\sqrt{\tfrac{\lambda }{2N}}\,N\big(1-8\partial_\l-16\partial_\l^{2}\big) \vev{\mathcal W} \ ,\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){2188} \\
& \vev{\mathcal W\,\mathsf{O}_{5}} = \frac{1}{8}\sqrt{\tfrac{\lambda }{2N}}\Big[ N^{2}-1-6 (\lambda
+4 N^2) \partial_{\l}-48 (-4+\lambda ) N^2 \partial_{\l}^{2}+768 \lambda
N^2 \partial_{\l}^{3}+256 \lambda ^2 N^2 \partial_{\l}^{4}\Big]\vev{\mathcal W} , \notag \\
& \vev{\mathcal W\,\mathsf{O}_{7}} = \frac{1}{32N}\sqrt{\textstyle \frac{\lambda}{2N}}\Big[
\lambda -2 N^4 +16 N^2 (-5+3 \lambda +6 N^2) \partial_{\l}+32 N^2 (-35 \lambda -60 N^2+6 \lambda N^2) \partial_{\l}^{2}\notag \\
& -640 N^2 (
\lambda ^2-24 N^2+12 \lambda N^2) \partial_{\l}^{3}-2560
(-36+\lambda ) \lambda N^4 \partial_{\l}^{4}+61440 \lambda ^2 N^4 \partial_{\l}^{5}+8192 \lambda ^3 N^4 \partial_{\l}^{6}
\Big] \vev{\mathcal W}. \notag \ea
\def \textstile {\textstile }
\subsection{$1/N$ and strong coupling expansion \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){s2.3}}
From the large $N$ expansion of $\vev{\mathcal W}$ in (\ref{B.2}) we can then compute the
corresponding expansion of the ratios $\vev{\mathcal W\,\mathsf{O}_{J}}/\vev{\mathcal W}$.
The strong coupling regime we are interested is defined by
first expanding in large $N$ for fixed $\lambda$ and then expanding the coefficient of each $1/N$ term
at large $\lambda$.
We find for $J=2,4,6$ (the expressions for $J=8$ and odd $J$ are similar)
\ba
& \Big. \frac{\vev{\mathcal W\,\mathsf{O}_{2}}}{\vev{\mathcal W}}\Big|_{N\gg 1,\, \lambda \gg 1} = \frac{\sqrt\lambda}{2}\,\Big[
1-\frac{3}{2\,\sqrt\lambda}+\cdots
+\frac{1}{N^{2}}\Big(
\frac{\lambda }{32}-\frac{\sqrt{\lambda
}}{32}+\cdots\Big)\notag \\
& \qquad \qquad +\frac{1}{N^{4}}\Big(
-\frac{\lambda ^2}{2048}+\frac{\lambda ^{3/2}}{512}+\cdots\Big)
+\frac{1}{N^{6}}\Big(
\frac{\lambda ^3}{65536}-\frac{\lambda ^{5/2}}{8192}+\cdots\Big)
+\mathcal O\Big(\frac{1}{N^{8}}\Big)
\Big], \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){218}
\\
& \Big. \frac{\vev{\mathcal W\,\mathsf{O}_{4}}}{\vev{\mathcal W}}\Big|_{N\gg 1,\, \lambda \gg 1} = \frac{N\,\sqrt\lambda}{2}\Big[
1-\frac{15}{2\sqrt\lambda}+\cdots
+\frac{1}{N^{2}}\,\Big(\frac{5\lambda}{32}-\frac{15\sqrt\lambda}{32}+\cdots\Big)\notag \\
& \qquad\qquad +\frac{1}{N^{4}}\,\Big(\frac{7\lambda^{2}}{2048}-\frac{\lambda^{3/2}}{512}+\cdots\Big)
+\frac{1}{N^{6}}\,\Big(-\frac{3\lambda^{3}}{65536}+\frac{\lambda^{5/2}}{8192}+\cdots\Big)
+\mathcal O\Big(\frac{1}{N^{8}}\Big)
\Big], \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){219}
\\
& \Big. \frac{\vev{\mathcal W\,\mathsf{O}_{6}}}{\vev{\mathcal W}} \Big|_{N\gg 1,\, \lambda \gg 1} = \frac{3N^{2}\,\sqrt\lambda}{8}\Big[
1-\frac{35}{2\sqrt\lambda}+\cdots
+\frac{1}{N^{2}}\,\Big(\frac{35\lambda}{96}-\frac{105\sqrt\lambda}{32}+\cdots\Big)\notag \\
& \qquad \qquad +\frac{1}{N^{4}}\,\Big(\frac{63\lambda^{2}}{2048}-\frac{63\lambda^{3/2}}{512}+\cdots\Big)
+\frac{1}{N^{6}}\,\Big(\frac{33\lambda^{3}}{65536}-\frac{\lambda^{5/2}}{8192}+\cdots\Big)
+\mathcal O\Big(\frac{1}{N^{8}}\Big)
\Big] \ . \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){220}
\ea
The leading (planar) terms in the square brackets are $1-\frac{J^{2}-1}{2\sqrt\lambda}+\cdots$ in agreement with the
expansion of $I_{J}(\sqrt\lambda)$ in
\cite{Semenoff:2001xp}.\footnote{This planar $1/N^{0}$ part in the square brackets has the full $\lambda$
dependence given
by the Bessel function ratio $I_{J}(\sqrt\lambda)/I_{1}(\sqrt\lambda)$ up to a power of $\lambda$ fixed by the choice of normalization
of the operator.
Notice that we are considering the $U(N)$ gauge theory.
Beyond the planar level, results in the $SU(N)$ gauge theory differ in the subleading terms in the $1/N$ expansion
due to an additional factor $\exp\big(-\frac{\lambda}{8N^{2}}\big)$, and also
due to $1/N$ modifications in the fusion/fission relations for the $SU(N)$ generators compared to (\ref{2.10}).}
To determine the higher order $J$-dependent terms in the expansion of $\vev{\mathcal W\,\mathsf{O}_{J}}$ up to some fixed order in $1/N$ it is
convenient to use the
representation derived in \cite{Okuyama:2006jc}
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){222}
\vev{\mathcal W\, \mathsf{O}_{J}} = \frac{2^{1-J/2}N^{1+J/2}}{\sqrt\lambda}\,e^{\frac{\lambda}{8N}}\oint\frac{dw}{2\pi i}
w^{J}\,e^{\frac{\sqrt\lambda}{2}w}\,\Big(1+\frac{\sqrt\lambda}{2N w}\Big)^{N}\,
\Big[\Big(1+\frac{\sqrt\lambda}{2N w}\Big)^{J}-1\Big].
\end{equation}
Expanding at large $N$ gives
\begin{equation}
\vev{\mathcal W\, \mathsf{O}_{J}} = J\,(N/2)^{J/2}\,\oint\frac{dw}{2\pi i}
w^{J-1}\exp\Big(\frac{\sqrt\lambda}{2w}+\frac{w\sqrt\lambda}{2}\Big)\,\Big[
1+\frac{2(J-1)w\sqrt\lambda+\lambda(w^{2}-1)}{8w^{2}N}+\cdots\Big].
\end{equation}
Using the identity
\begin{equation}
I_{J}(\sqrt\lambda) = \oint\frac{dw}{2\pi i}
w^{J-1}\exp\Big(\frac{\sqrt\lambda}{2w}+\frac{w\sqrt\lambda}{2}\Big),
\end{equation}
this gives
\begin{equation}
\vev{\mathcal W\, \mathsf{O}_{J}} = J\,(N/2)^{J/2}\Big[
I_{J}+\frac{1}{N^{2}}\Big(
\frac{(J+1)(J-2)}{96}\lambda I_{J}+\frac{\lambda+2(J^{2}-1)(J-2)}{96}\sqrt\lambda I_{J-1}
\Big)+\cdots
\Big].
\end{equation}
Extending this procedure to determine all terms up to order $1/N^{6}$, we find
\ba
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){2.24}
\vev{\mathcal W\,\mathsf{O}_{J}} &= (N/2)^{J/2}\,
J\,\Big[A^{(0)}_{J}(\lambda)+\frac{1}{N^{2}}\,A^{(1)}_{J}(\lambda)+\frac{1}{N^{4}}\,A^{(2)}_{J}(\lambda) +\frac{1}{N^{6}}\,A^{(3)}_{J}(\lambda) + \cdots\Big],
\ea
where $A_{J}$ are expressed in terms of the modified Bessel functions $I_{n}\equiv I_{n}(\sqrt\lambda)$
\ba
& A^{(0)}_{J}(\lambda) = I_{J},\qquad \qquad
A^{(1)}_{J}(\lambda) =
\frac{(J+1)(J-2)}{96}\,\lambda\,I_{J}+\frac{\lambda+2(J^{2}-1)(J-2)}{96}\sqrt\lambda\,I_{J-1},\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){225} \\
& A^{(2)}_{J}(\lambda) =
\frac{(J-3) (-40-18 J+19 J^2+3 J^3)}{92160}\,\lambda^{2}\,I_{J} \notag \\ & \qquad\qquad\qquad
+\Big[\frac{(J-3) (J-2) (-16-8 J+9 J^2+3 J^3)}{23040}+\frac{-12-5 J+5 J^2}{46080}\,\lambda\Big]\,\lambda^{3/2}\,I_{J-1} \notag \\ & \
\qquad\qquad\qquad
+\Big[\frac{(J-4) (J-3) (J-2) (J+1) (-6-J+3 J^2)}{23040}+\frac{1}{18432}\,\lambda^{2}\Big]\,\lambda\,I_{J-2}, \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){226} \\
& A^{(3)}_{J}(\lambda) = \frac{4800+8248 J-1254 J^{2}-1891 J^{3}+447 J^{4}+27 J^{5}-9 J^{6}}{92897280}\,\lambda^{3}\,I_{J} \notag \\ & \quad
+\Big[\frac{1}{8064}+\frac{828+196 J-413 J^{2}+21 J^{4}}{61931520}\,\lambda\Big]\,\lambda^{5/2}\,I_{J-1} \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){224}\\ \quad
&+\Big[\frac{(J-5) (J-4) (J-3) (288+228 J-184 J^2-125 J^3+24 J^4+9
J^5)}{15482880}\notag \\ &
+\frac{(J+1) (-8592+2788 J+1604 J^2-733 J^3+36 J^4+9 J^5)}{61931520}\,\lambda-\frac{14+5 J-5 J^2}{8847360}\,\lambda^{2}\Big]\,\lambda^{2}\,I_{J-2}\notag \\ &
+\Big[\frac{(J-6) (J-5) (J-4) (J-3) (J+1) (80+34 J-57 J^2-18 J^3+9
J^4)}{23224320}
+\frac{\lambda^{3}}{5308416}\Big]\,\lambda^{3/2}\,I_{J-3} \nonumber
\ea
It is then straightforward to expand the coefficients $A^{(n)}_J (\lambda)$ at large $\l$ for any $J\ge 2$.\footnote{
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){fo14} The expressions \rf{222} and \rf{225}--\rf{224} apply for $J\ge 2$.
The case of $J=0$ is trivial
while for $J=1$ we get $\vev{\mathcal W\,\mathsf{O}_{1}} = \frac{g}{2\sqrt 2}\,\vev{\mathcal W}$
by contraction of $\tr a$ with $\tr \exp(\frac{g}{\sqrt 2}a)$. The $J\ge 2$
restriction can be understood at the planar level by noting that the recursion relation
leading to the $I_{J}$ term is based on a recursion over the number of
scalar propagator endpoints and this has a regular structure only for $J\ge 2$ (\textit{cf.} section 2.2 in
\cite{Semenoff:2001xp}).}
\subsectio
{String theory interpretation}
Let us now rewrite the above expansions
in terms of the string coupling and tension in \rf{1.1},
setting $N = \pi\,T^{2}/g_{\text{s}}$ and $\lambda=(2\pi T)^{2}$.
Let us also choose a particular normalization of the chiral primary operator.
One possibility could be to impose as in \cite{Semenoff:2001xp} the condition that the two-point function
should be unit-normalized.
However, this choice does not appear to be natural in the string theory context.\footnote{For example, the
string dilaton vertex has a factor of $T\sim \sqrt\lambda$ and no $g_{\text{s}}\sim 1/N$ factors (see, e.g.,
\cite{Giombi:2020mhz}).
Its gauge theory counterpart is the SYM Lagrangian
$\frac{1}{g_{\rm YM}^2} \tr F^2_{mn}+...$ and its 2-point function scales as $N^2$.}
Below we shall assume that the operators $\mathcal O_{J} $
that should correspond to the string vertex operators should be normalized relative to the matrix model operator
$\mathsf{O}_{J}$ in \rf{2.6} as\footnote{Below we shall use the label $\mathcal O_{J} $ for the CPO as in \rf{2.6}
even though its normalization will be different.}
\begin{equation}\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){2255}
\mathcal O_{J} = \Big({g_{\text{s}}\over T^2} \Big)^{\frac{J}{2}-1}\,\mathsf{O}_{J} = \Big({\pi \over N} \Big)^{\frac{J}{2}-1}\, \mathsf{O}_{J} \ .
\end{equation}
Since at strong coupling the correlators in \rf{218}--\rf{220} scale as $ N^{{J\over 2}-1} \sqrt \l $
we will then have at the leading planar order
\begin{equation}\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){2311}
\frac{\vev{\mathcal W\, \mathcal O_{J}}}{\vev{\mathcal W}}\ \sim \ \sqrt \l \, N^{{J\over 2}-1} \, \Big({\pi \over N} \Big)^{\frac{J}{2}-1} \sim T \ ,
\end{equation}
in agreement with the canonical normalization of the corresponding
string vertex operator.
Including subleading corrections and using (\ref{B.2}), we then obtain from (\ref{2.24}) the following
expression for general value of $J$
\ba
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){2.28}
\Big. \frac{\vev{\mathcal W\,\mathcal{O}_{J}}}{\vev{\mathcal W}}\Big|_{g_{\text{s}}\ll 1, \, T\gg 1} = &
c_{J}\,T
\,\Big\{
1 -\frac{J^{2}-1}{4\pi\,T}
+\cdots
+\frac{J^{2}-1}{24}\,\frac{g_{\text{s}}^{2}}{T^{2}}\,\Big[
1-\frac{J^{2}-4J+6}{4\pi\,T}
+\cdots
\Big]\notag \\ &
+\frac{J^{2}-1}{1920}\,\frac{g_{\text{s}}^{4}}{T^{4}}\,\Big[
J^{2}-9-\frac{J^{4}-8J^{3}+16J^{2}+32J-120}{4\pi\,T}
+\cdots
\Big]\\
&+\frac{J^{2}-1}{322560}\,\frac{g_{\text{s}}^{6}}{T^{6}}\,\Big[
(J^{2}-9)(J^{2}-25)\notag \\ &
-\frac{5040-768 J-944 J^2+240 J^3+22 J^4-12 J^5+J^6}{4 \pi T}
+\cdots
\Big]
+\mathcal O\Big(\frac{g_{\text{s}}^{8}}{T^{8}}\Big)\Big\},\notag
\ea
where dots stand for terms subleading at large $T$ and the value of the overall coefficient
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){2.29}
c_{J} = J\,\Big(\frac{\pi}{2}\Big)^{J/2}
\end{equation}
reflects our choice of normalization of $\mathcal O_{J}$ in \rf{2255}.
\subsubsection{Resummation of leading strong coupling
terms}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){s2.4.1}
Separating the leading $(g_{\text{s}}/T)^{2n}$ terms in the brackets in (\ref{2.28}) we get
\ba
& \frac{\vev{\mathcal W\,\mathcal{O}_{J}}}{\vev{\mathcal W}} =
c_{J}\,T\,
\Big[ \,F_{J}\Big(\frac{g_{\text{s}}^{2}}{T^{2}}\Big)\, +\cdots\Big],\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){229} \\ \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){2.30}
F_{J}(x) &= 1+\frac{J^{2}-1}{24}\,x+\frac{(J^{2}-1)(J^{2}-9)}{1920}\,x^{2}+
\frac{(J^{2}-1)(J^{2}-9)(J^{2}-25)}{322560}\,x^{3}+\cdots\ ,
\ea
where
dots in \rf{229} stand for the terms which are subleading in $1/T$ at each order in $g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s}$.
Thus, formally, keeping only $F_J$ part of \rf{229}
is the same as keeping only the terms that are non-vanishing
at $T\to\infty$ for fixed $x= \frac{g_{\text{s}}^2}{T^2}= \frac{\l}{4 N^2 }$.
The pattern of the leading coefficients
in \rf{2.30} suggests the all-order conjecture
\begin{equation}\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){231}
F_{J}(x) = \sum_{n=0}^{\infty}\frac{\prod_{k=1}^{n}\big[J^{2}-(2k-1)^{2}\big]}{4^{n}(2n+1)!}\,x^{n} = \frac{2}{J\,\sqrt x}\sinh\Big(J\,\text{arcsinh}\frac{\sqrt x}{2}\Big).
\end{equation}
As was mentioned in section 1.1 in the Introduction, this resummed expression
agrees with the semiclassical D3-brane calculation in \cite{Giombi:2006de}
generalizing the computation of $\vev{\mathcal W}$ in \cite{Drukker:2005kx} to the case of
correlators with chiral primary operators.
To explain the reason this agreement, let us recall that the semiclassical D3-brane probe
description applies to the
expectation value of the circular Wilson loop in the $k$-symmetric representation
and in the limit where $\varkappa= \frac{k{\sqrt{\l}}\ }{4N}$ is fixed for large $\lambda$ and $N$.
Remarkably, at large $N$ and large $\lambda$ the result for the $k$-symmetric Wilson loop
is the same as for the simpler $k$-fundamental Wilson loop
\cite{Okuyama:2006jc,Hartnoll:2006is,Kawamoto:2008gp}
for which the dependence on $k$ is obtained from the $k=1$ case by simply
rescaling $\lambda\to k^{2}\lambda$.
Hence, in the above large $N,\l$ limit with fixed $\frac{{\sqrt{\l}}\ }{N}\sim \frac{g_{\text{s}}}{T}$
the semiclassical D3-brane description should also reproduce the
result for the Wilson loop in the
fundamental ($k=1$) representation,
but this limit is equivalent to the one we considered
when we neglected the subleading $1/T$ terms in the full expansion (\ref{2.28}).
This leading
contribution \rf{229},\rf{231} may be obtained also by directly from the matrix model saddle point at fixed
$\frac{{\sqrt{\l}}\ }{N}$ \cite{Giombi:2006de}.
For odd $J$
the function $F_{J}(x)$ in \rf{231} reduces to a polynomial in $x$, while for even $J$
the series expansion in $x$ does not truncate -- in this case
$F_{J}(x)$ turns out to be $\sqrt{1+x/4}$ times a polynomial in $x$.
Indeed, from the definition of the Chebyshev polynomials
\begin{equation}
{\rm T}_{n}(\cos\theta) = \cos(n\,\theta),\qquad {\rm U}_{n}(\cos\theta)\sin\theta = \sin\big((n+1)\theta\big)\ ,
\end{equation}
we obtain
\begin{equation}
\sinh(J\,\text{arcsinh}\, t) = \begin{cases}
i\,(-1)^{\frac{J}{2}}\,\sqrt{1+t^{2}}\,{\rm U}_{J-1}(i\,t), & \quad J\ \text{even} \\
i\,(-1)^{\frac{J+1}{2}}\,{\rm T}_{J}(i\,t), & \quad J\ \text{odd}
\end{cases}
\end{equation}
For even $J$, the overall factor $\sqrt{1+t^{2}}=(1+x/4)^{1/2} = (1+\frac{g_{\text{s}}^{2}}{4T^{2}})^{1/2}$ (that has an imaginary branch point)
shows that the $\frac{g_{\text{s}}}{T}$ expansion has a finite radius of
convergence. Explicitly, one finds for $F_J$ in \rf{231}
\ba
F_{2}(x) &= \frac{1}{2}\sqrt{4+x}\ , \qquad
F_{3}(x) = 1+\frac{x}{3}\ , \qquad
F_{4}(x) = \frac{1}{4}(2+x)\,\sqrt{4+x}\ ,\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){235} \\
F_{5}(x) &= 1+x+\frac{x^{2}}{5}\ ,\qquad F_{6}(x) = \frac{1}{6}(1+x)(3+x)\,\sqrt{4+x}\ .
\ea
Starting with the resummed expression \rf{229},\rf{231}
we can formally consider the limit when the parameter $x= \frac{g_{\text{s}}^2}{T^2}$ that was fixed in the resummation is
now taken to be
large. Using that
$F_{J}(x) \stackrel{x\to\infty}{=} {J}^{-1}\,(\sqrt x\,)^{{J-1}}+\cdots, $ and \rf{2.29}
we then get\footnote{The limit $ g_{\text{s}}/T \gg 1$ assumed here is of course formal as in the original expansion we assumed that both
$g_{\text{s}}$ and $1/T$ are small.}
\begin{equation}
\frac{\vev{\mathcal W\,\mathcal{O}_{J}}}{\vev{\mathcal W}} \simeq
\Big(\frac{\pi}{2}\Big)^{J/2} T\,\Big(\frac{g_{\text{s}}}{T}\Big)^{J-1}+\cdots \ .
\end{equation}
One can also consider the limit of large $J$. The result depends on the assumption about growth of $J$ relative to $T$.
To be able to ignore the $1/T$ corrections in square brackets in (\ref{2.28}) and thus
use the resummed expression in \rf{229},\rf{231}
$J^2 $ should grow slower than $T$ (i.e. $J \ll \lambda^{1/4}$).
Then $c_J F_J(x) \sim \exp ( J \text{arcsinh}\frac{\sqrt x}{2})$.
Another interesting limit corresponds to the semiclassical large charge
expansion in the dual string theory when $J\sim T \gg 1$.
In this case the $1/T$ corrections in (\ref{2.28})
are not negligible and \rf{229},\rf{231} cannot be used. This limit will be discussed in Appendix \ref{sec:semi} below.
\subsubsection{Comparison of expansions of $\vev{\mathcal W}$ and $\vev{\mathcal W\,\mathcal O_{2}}/ \vev{\mathcal W}$}
Let us recall that
the chiral primary operator $\mathcal O_{J}\sim \tr \varphi^{J}$ with dimension $\Delta=J$
belongs to the same short supermultiplet as the R-charge generalization of the dilaton operator $\mathcal O_{{\rm dil}, J'}\sim \tr (\varphi^{J'} F^2_{mn} ) + ...$ of dimension $\Delta = 4 + J'$ with $J'= J-2$.
The standard $\Delta =4$ dilaton operator is the supersymmetry descendant of the
$J=2$ CPO and thus their correlators with the
BPS Wilson loop should be directly related.
Indeed, like the dilaton correlator,
the CPO correlator can be obtained from $ \vev{\mathcal W}$ by the differentiation over the coupling using
(\ref{2.12}),\rf{216},\rf{2255} (cf. \rf{1.11})\footnote{For general $J$, the supersymmetry relation between the Wilson loop
correlators with
CPO $\mathcal O_{J}$
and with the dilaton operator $\mathcal O_{{\rm dil}, J-2}$
imply that $\vev{\mathcal W\, \mathcal O_{{\rm dil}, J-2} }$ can also be obtained from $\vev{{\cal W}}$ by the differential
relations like \rf{216}--\rf{2188}.}
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){2.37}
\frac{\vev{\mathcal W\,\mathcal O_{2}}}{\vev{\mathcal W}}= \lambda\,\partial_{\lambda}\log \vev{\mathcal W} \ .
\end{equation}
According to \rf{229},\rf{235},\rf{2.29} the result of the resummation of the
strong coupling expansion for the $J=2$ case is simply
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){2.38}
\frac{\vev{\mathcal W\,\mathcal{O}_{2}}}{\vev{\mathcal W}} \simeq c_{2}\,T\,\sqrt{1+\frac{g_{\text{s}}^{2}}{4T^{2}}} = \pi\,\sqrt{T^{2}+\frac{1}{4}g_{\text{s}}^{2}} \ ,
\end{equation}
or, in gauge theory notation, $\frac{1}{2}\,\sqrt{\lambda+\frac{\lambda^{2}}{16\pi^{2}\,N^{2}}}$.
The leading strong coupling term here agrees with \rf{2.37} since $\vev{\mathcal W} \sim e^{{\sqrt{\l}}\ }$ and thus
$\lambda\partial_{\lambda}\log\vev{\mathcal W} = \frac{1}{2}\sqrt\lambda+\cdots$.
However,
the resummed expression \rf{1.5} for $\vev{\mathcal W} $ does not lead to \rf{2.38} if substituted into \rf{2.37}.
As already discussed in the Introduction, the reason why the two resummations are not directly related is that
subleading in $1/T$ terms in $\vev{\mathcal W} $ cannot be in general ignored in $\log \vev{\mathcal W} $ in \rf{2.37} (see \rf{x1}--\rf{x2}).
In more detail,
the structure of the expansion of $\vev{\mathcal W}$ is
\ba
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){2.39}
\vev{\mathcal W} &= e^{2\pi T}\Big(\frac{g_{\text{s}}^{2}}{T}\Big)^{-1/2}\,\sum_{p=0}^{\infty}\frac{1}{2\pi p!}\Big(\frac{\pi}{12}\Big)^{p}
\Big(\frac{g_{\text{s}}^{2}}{T}\Big)^{p}\,
\Big(1+\frac{a_{p}^{(1)}}{T}+\frac{a_{p}^{(2)}}{T^{2}}+\cdots\Big),
\ea
where the values of the coefficients $a_{p}^{(n)}$ may be extracted from (\ref{1.4})
\cite{Drukker:2000rr}. Then including the subleading terms we have
\ba
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){2.41}
\frac{\vev{\mathcal W\,\mathcal O_{2}}}{\vev{\mathcal W}} = \lambda\,\partial_{\lambda}\log \vev{\mathcal W}
= \pi\,T\,\Big[
1 &-\frac{3}{4\pi T}-\frac{a_{0}^{(1)}}{2\pi T^{2}}+\cdots+\frac{g_{\text{s}}^{2}}{T}\,\Big(\frac{1}{8T}
+\frac{a_{1}^{(1)}-a_{0}^{(1)}}{12\,T^{2}}+\cdots\Big)\notag \\ &
+\Big(\frac{g_{\text{s}}^{2}}{T}\Big)^{2}\,\Big(
\,\frac{5 \pi \big(a_{0}^{(1)}-2a_{1}^{(1)}+a_{2}^{(1)}\big)}{576 T^{2}}+\cdots\Big)+\cdots\,
\Big].
\ea
The resummation of $\vev{\mathcal W}$ leading to the $g^2_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s}\over T$ exponent in
(\ref{1.5}) amounts to dropping
all subleading $a_{p}^{(n)}$ corrections in \rf{2.39} but they actually contribute to the leading order terms in
\rf{2.41} starting with the order $\big(\frac{g_{\text{s}}^{2}}{T}\big)^{2}$.
Using that $a_{p}^{(1)} = -\frac{3(12p^{2}+8p+5)}{80\pi}$
one finds indeed the agreement with the result of the direct computation of the order $\frac{g_{\text{s}}^{4}}{T^4}$
term in the $J=2$ CPO correlator in the brackets in \rf{218},\rf{2.28} which corresponds to the
$g^4_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s}$ term in the expansion of the square root in \rf{2.38}.
\iffa
\footnote{
Of course, from the exact expression of $\vev{\mathcal W}$ we can improve with respect to (\ref{1.5}).
From instance, one has, \textit{cf.} eq.(B.7) of ,
$a_{p}^{(0)} = -\frac{3(12p^{2}+8p+5)}{80\pi}$,
and plugging this into (\ref{2.39}) gives
\ba
\notag
\frac{\vev{\mathcal W\,\mathcal O_{2}}}{\vev{\mathcal W}} &= c_{2}T\,\Big[
1-\frac{3}{4\pi T}+\frac{3}{32\pi^{2}T^{2}}+\cdots+\frac{g_{\text{s}}^{2}}{T}\,\Big(\frac{1}{8T}
-\frac{1}{16\pi T^{3}}+\cdots\Big)
+\Big(\frac{g_{\text{s}}^{2}}{T}\Big)^{2}\,\Big(-\frac{1}{128T^{4}}+\cdots\Big)+\cdots\,
\Big],
\ea
where the omitted terms depend on $a_{p}^{(1)}$. The predicted terms are again in agreement with (\ref{2.28}).
}
\fi
Similar remarks apply to higher $J$ cases of the resummed expression for the
correlator \rf{229},\rf{231} (understood using analytic continuation in $J$)
when applying the differential relations like \rf{2.16}--\rf{2188}
and comparing to the resummed expression for $\vev{\mathcal W}$.
\def {\rm J} {{\rm J}}
\section{Expansion of $\vev{\mathcal W\,\mathcal O_{J_{1}}\,\mathcal O_{J_{2}}}$
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){3}}
One may also consider a correlation function of a circular Wilson loop with two scalar
chiral primary
operators at generic positions $x_1$, $x_2$.
Such correlator is fixed by conformal invariance up to a function of $N$ and $\lambda$
and two scalar combinations u and v
of the positions invariant under the conformal transformations preserving the circle \cite{Buchbinder:2012vr}.
Explicitly, for
$\vev{\mathcal W\,\mathcal O_{1}(x_{1})\,\mathcal O_{2}(x_{2})}$
where $\mathcal O_{1}$ and $\mathcal O_{2}$ are scalar primary
operators of dimensions $\Delta_{1}$,$\Delta_{2}$
at points $x_{1}, x_{2}\in \mathbb R^{4}$ and $\mathcal W$ is the circular $\frac{1}{2}$-BPS loop
of unit radius the conformal symmetry implies that\footnote{One can conformally map $\mathbb R^{4}\to AdS_{2}\times S^{2}$ so that the circle is mapped to the boundary
of $AdS_{2}$. Then $ \vev{\mathcal W\,\mathcal O_{1}(x_{1})\,\mathcal O_{2}(x_{2})}/\vev{{\cal W}}$
is invariant under the 6 isometries of $AdS_{2}\times S_{2}$ (corresponding to
6 conformal transformations that preserve the circle in $\mathbb R^{4}$). It is expressed in terms of two functions
(u and v) of the $AdS_{2}$
and $S^{2}$ geodesic distances between the operators (see \cite{Buchbinder:2012vr} for details).}
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){3.1}
\frac{\vev{\mathcal W\,\mathcal O_{1}(x_{1})\,\mathcal O_{2}(x_{2})}}{\vev{\mathcal W}} = \frac{{\cal F}({\rm u}, {\rm v}; N, \lambda)}{d_{\perp}^{\Delta_{1}}(x_1)\, d_{\perp}^{\Delta_{2}}(x_2) }\ ,
\end{equation}
where $d_{\perp}(x) $ for a point $x\in \mathbb R^{4}$ was
defined in (\ref{2.2}).
Fixing particular values of $x_1$,$x_2$ and thus of
u and v one may then study the $1/N$ expansion of the resulting function.
It turns out that for special supersymmetric configurations
correlators of certain BPS Wilson loops with local operators
may be computed to all orders by localization by reducing them to
correlators in a multi-matrix model \cite{Giombi:2012ep}.
Examples include
special $\frac{1}{8}$-BPS Wilson loop which is a contour on a 2-sphere $S^{2}\subset \mathbb R^{4}$.
In the general $1\over 8$-BPS case, one considers \cite{Giombi:2012ep} the operators
${\mathcal O }_J (x) = \tr \big[ x_n \Phi_n(x) + i \Phi_4(x)\big]^J $ (for $x^2_n=1$, \ $n=1,2,3$)
and the Wilson loop for a contour on $S^{2}\subset \mathbb R^{4}$
with the scalar coupling being $\int \epsilon_{nkl} \Phi_n x_k d x_l $ (cf. \rf{1w}).
The special $1\over 2$-BPS case we are interested in here corresponds to
placing the operators
at the poles of the 2-sphere and the unit-circle Wilson loop at its
equator.
This results in the following choice of $x_1$ and $x_2$
\begin{equation}
x_{1}=(0,0,1,0)\ , \ \quad \ x_{2}=(0,0,-1,0)\ , \ \ \qquad
{\rm u}=-{\rm v}=1\ , \ \ \qquad \
d_{\perp}(x_1)=d_{\perp}(x_2)=\tfrac{1}{2} \ . \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){ddd}
\end{equation}
Then the correlator in \rf{3.1} becomes explicitly
\begin{equation} \vev{\mathcal W\, \tr\big[(\Phi_3 + i \Phi_4)^{J_1}\big] \, \tr\big[ (-\Phi_3 + i \Phi_4)^{J_2} \big] } \ , \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){333} \end{equation}
and the Wilson loop scalar coupling becomes the same as in \rf{1w} with $\Phi_1\to \Phi_3$ (and $R=1$).
For general $x_1,x_2$ the correlator \rf{333} has the structure \rf{3.1} but its value can be
computed by localization at specific positions in \rf{ddd}.
In detail, it can be computed using a 3-matrix model with the following
action depending on the hermitian matrices $X_{1}$, $X_{2}$, $X_{3}$ \cite{Giombi:2012ep}\footnote{
We specialize
the expression in \cite{Giombi:2012ep}
to the case of \rf{ddd}.}
\ba\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){32}
S &= \frac{8\pi^{2}}{g^{2}}\tr\Big(X_{1}^{2}-\frac{1}{4\pi^{2}} X_{2}^{2}+X_{3}^{2}-\frac{i}{\pi}X_{1}X_{2}+\frac{i}{\pi}X_{2}X_{3}\Big),\qquad \ \ g\equiv g_{_{\rm YM}}.
\ea
The connected part of the correlator \rf{3.1} is related to a particular matrix model
correlator which admits the following $1/N$ expansion
\begin{equation}\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){34}
\vev{\tr X_{1}^{J_{1}}\tr e^{X_{2}}\tr X_{3}^{J_{2}}}_{\rm conn} \equiv Q_{J_{1},J_{2}}(\lambda; N) = \frac{Q_{J_{1},J_{2}}^{(1)}(\lambda)}{N}+ \frac{Q_{J_{1},J_{2}}^{(2)}(\lambda)}{N^{2}}+\cdots\ .
\end{equation}
For the coefficient $Q_{J_{1},J_{2}}^{(1)}(\lambda) $ of the leading planar contribution one finds \cite{Giombi:2012ep}
\ba
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){3.5}
&Q_{J_{1},J_{2}}^{(1)}(\lambda) = J_{1}J_{2}\,\Big(\frac{i{\sqrt{\l}}\ }{4\pi}\Big)^{J_{1}} \Big(-\frac{i{\sqrt{\l}}\ }{4\pi}\Big)^{J_{2}}
\Big[\sum_{k=1}^{\min(J_{1},J_{2})}(J_{1}+J_{2}-2k)\,I_{J_{1}+J_{2}-2k}({\sqrt{\l}}\ )\notag \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
+\sum_{k=1}^{\infty}(J_{1}+J_{2}+2k-2)\,I_{J_{1}+J_{2}+2k-2}({\sqrt{\l}}\ )\Big].
\ea
The 3-matrix model representation \rf{32},\rf{34}
can be translated into a Gaussian 1-matrix model one similar to the one
considering in the previous section (cf. \rf{2.5},\rf{2.6}). Indeed,
after the change of variables
\begin{equation}
X_{1}\to A+i C\ ,\qquad X_{2}\to 2\pi C\ ,\qquad X_{3}\to B-i C\ ,
\end{equation}
the correlator in \rf{34} becomes
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){3.12}
\vev{\tr (A+i C)^{J_1} \tr(B-i C)^{J_2} \tr e^{2\pi C} },
\end{equation}
computed in the matrix model with the
decoupled Gaussian action $S\sim A^{2}+B^{2}+C^{2}$.
Integrating out the $A$ and $B$ matrices amounts to subtracting from $\tr (A+i C)^{J_1}$ and $\tr(B-iC)^{J_2}$ their self contractions, resulting in
the normal ordering
discussed in section \ref{s2.1}.\footnote{The
relation between the 2-matrix model and the 1-matrix model with explicit normal ordering
follows also from the equivalence between the 2-matrix model and the complex matrix model of \cite{Kristjansen:2002bb}
(see, for instance, Appendix C of \cite{Okuyama:2006jc}).}
We then end up with the following correlator in the 1-matrix model for $C$
\begin{equation}\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){38}
\vev{:\tr C^{J_{1}}:\ :\tr C^{J_{2}}:\ \tr e^{2\pi C}}\ .
\end{equation}
Below we shall consider two examples of the
correlators \rf{3.1}. The first has $J_{1}=J_{2}=J$ and the second $J_{1}=2$ and $J_{2}=2J$ ($J$ is integer).
We shall use them to
illustrate the general features of the strong coupling limit of the coefficients of the $1/N$ expansion of
(\ref{3.1}).
\subsection{$J_{1}=J_{2}=J$ \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){s3.1}}
In this case the explicit form of the relation between
the matrix model correlator and the function of $\l,N$ in
\rf{3.1},\rf{ddd} is
\ba\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){71}
Q_{J}(\lambda; N) =\vev{\tr X_{1}^{J}\tr e^{X_{2}}\tr X_{3}^{J}}_{\rm conn} &=
\Big(\frac{\lambda}{8\pi^{2}N}\Big)^{J}\,\Big[\vev{ \mathcal W\,\mathsf O_{J}\mathsf O_{J}}-\vev{\mathcal W}\, \vev{\mathsf O_{J}\mathsf O_{J}}\Big]\ ,
\notag \\
&= \frac{\pi^{2}}{N^{2}}\Big(\frac{\lambda}{8\pi^{3}}\Big)^{J}\,\Big[\vev{ \mathcal W\,\mathcal O_{J}\mathcal O_{J}}-\vev{\mathcal W}\, \vev{\mathcal O_{J}\mathcal O_{J}}\Big],
\ea
where $\mathsf O_{J}$ are the matrix model operators
the notation
of section \ref{s2.1} (cf. \rf{2.6}), i.e. $\mathsf O_{J} = :\tr a^ J: $ after
renaming $C\to a$.
We used that $\vev{\mathsf O_{J}}=0$.\footnote{Recall that for any 3 operators
$\vev{O_1 O_2 O_3 }_{\rm conn} = \vev{O_1 O_2 O_3 }-\vev{O_1}\vev{O_2 O_3 }-\vev{O_2}\vev{O_1O_3}-\vev{O_3}\vev{O_1O_2}+2\vev{O_1}\vev{O_2}\vev{O_3}$.}
The operators $\mathcal O_{J}$ in \rf{71} are assumed to be
normalized as in (\ref{2255}).
Let us consider explicitly the $J=2$ case when
\ba\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){316}
\mathsf{O}_{2}^{2} &= \mathsf{O}_{2,2}+2\mathsf{O}_{2}+\frac{N^{2}}{2}, \qquad \vev{\mathsf{O}_{2}^{2}} = \frac{N^{2}}{2}, \qquad \mathsf{O}_{J} = :\tr a^ J: \ , \quad \mathsf{O}_{2,2} = :(\tr a^{2})^{2}:\ , \\
\vev{\,\mathcal W\,\mathsf{O}_{2}^{2}} &= \vev{\mathcal W\,\mathsf{O}_{2,2}}+2\,\vev{\mathcal W\,\mathsf{O}_{2}}+\frac{N^{2}}{2}\vev{\mathcal W}
= \lambda^{2}\partial_{\lambda}^{2}\vev{\mathcal W}+2\,\lambda\partial_{\lambda}\vev{\mathcal W}+\frac{N^{2}}{2}\vev{\mathcal W}\ .
\ea
Here we used the relation
$ \vev{\mathcal W\,:(\tr a^{2})^{2}:} = \frac{1}{4}(g^{2}\partial_{g}^{2}-g \partial_{g})\,\vev{\mathcal W} = \lambda^{2}\partial_{\lambda}^{2}\vev{\mathcal W} $
that may be proved using the same method as in section \ref{s2.2}.
As a result, we get the following differential relation for the $J=2$ case of \rf{71}
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){3.16}
Q_{2}(\lambda; N) = \Big(\frac{\lambda}{8\pi^{2}N}\Big)^{2}\,(\lambda^{2}\partial^{2}_{\lambda}+2\lambda\partial_{\lambda})\, \vev{\mathcal W}\ .
\end{equation}
Using the $1/N$ expansion of $\vev{{\cal W}}$ in (\ref{B.2}) we find
\begin{equation}
Q_{2}(\lambda; N) = \frac{1}{N}\,\Big(\frac{\lambda}{16\pi^{2}}\Big)^{2}\,\Big[2\sqrt\lambda\,I_{1}(\sqrt\lambda)
+\frac{1}{N^{2}}\frac{\lambda^{3/2}}{48}\Big(\sqrt\lambda\,I_{0}(\sqrt\lambda)+4I_{1}(\sqrt\lambda)\Big)+\mathcal O\Big(\frac{1}{N^{4}}\Big)\Big],
\end{equation}
Similar calculation can be repeated for higher $J$ and leads to
\ba
Q_{J}(\lambda;N ) =& { 1\over N} \Big[ Q_{J}^{(1)}(\lambda) +{1 \over N^2} Q_{J}^{(2)}(\lambda) + {\cal O}} \def \bG {\bar \G\big({1\over N^4}\big)\Big] \ , \qquad \qquad
Q_{J}^{(1)}(\lambda) = \frac{J^{2}}{2}\,\Big(\frac{\lambda}{16\pi^{2}}\Big)^{J}\,\sqrt\lambda\,I_{1}(\sqrt\lambda), \notag \\
Q_{J}^{(2)}(\lambda) =&\frac{J^{2}}{192}\,\Big(\frac{\lambda}{16\pi^{2}}\Big)^{J}\,\sqrt\lambda\,\Big\{
\big[4 (J^{2}-1)(J-2)+\lambda\big]\,\sqrt\lambda\,I_{0}(\sqrt\lambda)\notag \\ & \qquad \qquad \qquad \qquad \qquad
+\big[4(J^{2}-1)(J-2)^{2}+2(J^{2}-2)\,\lambda\big]\,I_{1}(\sqrt\lambda)\Big\}.
\ea
Dividing \rf{71} over ${\vev{\mathcal W}} $ leads to (cf. \rf{3.1})
\ba
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){3.19}
{\vev{ \mathcal W\,\mathcal O_{J}\mathcal O_{J}}\over \vev{{\cal W}}} -\vev{\mathcal O_{J}\mathcal O_{J}}
& = N^{2}\pi^{J-2}\,\Big[
\frac{J^{2}\lambda}{4N^{2}}+\frac{J^{2}(J^{2}-1)}{N^{4}}\,\Big(
\frac{\,\lambda^{2}}{192}+\frac{(J-2)\,\lambda^{3/2}}{96}+ ...
\Big)
+\mathcal O\Big(\frac{1}{N^{6}}\Big)
\Big]
\notag \\
& =
\pi^{J}\, J^2\,T^{2}\,\Big[1+\frac{J^{2}-1}{12}\frac{g_{\text{s}}^{2}}{T^{2}}\,\Big(1+\frac{J-2}{\pi T}+\cdots\Big)+\cdots\Big].
\ea
\subsection{$J_{1}=2$, $J_{2}=2J$ \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){s3.2}}
\iffa
In this case the specialization of (\ref{3.5}) is
\ba
&Q_{J_{1},J_{2}}^{(1)}(\lambda) = 2 (2J)\,\Big(\frac{i\sqrt\lambda}{4\pi}\Big)^{2}\Big(-\frac{i\sqrt\lambda}{4\pi}\Big)^{2J}\times\notag \\
&\times\Big[\sum_{k=1}^{2}(2J+2-2k)\,I_{2J+2-2k}(\sqrt\lambda)+\sum_{k=1}^{\infty}(2J+2k)\,I_{2J+2k}(\sqrt\lambda)\Big] \notag \\
&= 4J\,(-1)^{J+1}\Big(\frac{\lambda}{16\pi^{2}}\Big)^{J+1}\,\Big[\frac{\sqrt\lambda}{2}I_{1}(\sqrt\lambda)-\sum_{k=1}^{J-2}2k\,I_{2k}(\sqrt\lambda)\Big]
\ea
The first cases are $J=2, 3$ for which the planar results read
\ba
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){3.21}
Q_{2,4}^{(1)}(\lambda) &= -4\,\Big(\frac{\lambda}{16\pi^{2}}\Big)^{3}\,\sqrt\lambda\,I_{1}(\sqrt\lambda), \notag \\
Q_{2,6}^{(1)}(\lambda) &= 12\,\Big(\frac{\lambda}{16\pi^{2}}\Big)^{4}\,\Big[\frac{\sqrt\lambda}{2}I_{1}(\sqrt\lambda)-2\,I_{2}(\sqrt\lambda)\Big].
\ea
\fi
In this case
the 1-matrix model representations for the correlators \rf{34} with $J=2,3$ are\footnote{The sign is $i^{J_{1}}(-i)^{J_{2}}$ from (\ref{3.12}).}
\ba
&Q_{2,4}(\lambda; N) = -\Big(\frac{\lambda}{8\pi^{2}N}\Big)^{3}\,\vev{\mathcal W\,\mathsf{O}_{2}\,\mathsf{O}_{4}} = -\frac{\pi^{2}}{N^{2}}\,\Big(\frac{\lambda}{8\pi^{3}}\Big)^{3}\,\vev{\mathcal W\,\mathcal {O}_{2}\,\mathcal {O}_{4}}, \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){339} \\
& Q_{2,6}(\lambda; N) = \Big(\frac{\lambda}{8\pi^{2}N}\Big)^{4}\,\vev{\mathcal W\,\mathsf{O}_{2}\,\mathsf{O}_{6}} = \frac{\pi^{2}}{N^{2}}\Big(\frac{\lambda}{8\pi^{3}}\Big)^{4}\,\vev{\mathcal W\,\mathcal {O}_{2}\,\mathcal {O}_{6}}, \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){319}
\\
&\mathsf{O}_{4} = :\tr a^{4}: =\textstyle \tr a^{4}-2N\,\tr a^{2}-(\tr a)^{2}+\frac{1}{4}N(1+2N^{2})\ , \notag \\
&\mathsf{O}_{6} = :\tr a^{6}: =\textstyle \tr a^{6}-3N\,\tr a^{4}+\frac{15}{4}\,(N^{2}+1)\,\tr a^{2}-\frac{5}{8}N^{2}(2+N^{2})\notag \\ & \qquad\qquad\qquad\quad\textstyle
+\frac{15}{4}N\,(\tr a)^{2}-3 \tr a\,\tr a^{3}-\frac{3}{2}(\tr a^{2})^{2}.
\ea
The {exact} differential relations for \rf{339},\rf{319} are found to be
\ba
\vev{\mathcal W\,\mathsf{O}_{2}\,\mathsf{O_{4}}} &= \Big[
-\frac{\lambda}{16 N}-\frac{\lambda (\lambda +16 N^2)}{16 N}\,\partial_{\lambda}
-(-36+\lambda ) \lambda N \,\partial^{2}_{\lambda}+44
\lambda ^2 N\,\partial_{\lambda}^{3}+8 \lambda ^3 N\,\partial_{\lambda}^{4}
\Big]\,\vev{\mathcal W}\ , \notag \\
\vev{\mathcal W\,\mathsf{O}_{2}\,\mathsf{O_{6}}} &= \Big[
\frac{3}{32} (-80+\lambda ) \lambda \partial_{\lambda}+\frac{3}{4} \lambda
(-17 \lambda -80 N^2+\lambda N^2) \partial_{\lambda}^{2}-\lambda (3 \lambda
^2-960 N^2+80 \lambda N^2) \partial_{\lambda}^{3}\notag \\ & \qquad
-16 (-115+\lambda )
\lambda ^2 N^2 \partial_{\lambda}^{4}+704 \lambda ^3 N^2 \partial_{\lambda}^{5}+
64 \lambda ^4 N^2 \partial_{\lambda}^{6}
\Big]\,\vev{\mathcal W}\ .
\ea
As a result, using \rf{B.2} we get
\ba
&Q_{2,4}(\lambda; N) = -\frac{4}{N}\Big(\frac{\lambda}{16\pi^{2}}\Big)^{3}\,\Big\{
\sqrt{\lambda } I_1(\sqrt{\lambda })+\frac{1}{N^{2}}\Big[\frac{1}{96} \lambda (24+
\lambda ) I_0(\sqrt{\lambda })+\frac{1}{6} \lambda ^{3/2}
I_1(\sqrt{\lambda })\Big]\notag \\ &
\qquad +\frac{1}{N^{4}}\,\Big[\frac{\lambda ^2 (32+13 \lambda )
I_0(\sqrt{\lambda })}{7680}+\frac{\lambda ^{3/2} (-768+672 \lambda +5
\lambda ^2) I_1(\sqrt{\lambda })}{92160}\Big]+\mathcal O\Big(\frac{1}{N^{6}}\Big)
\Big\}, \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){323} \\
&Q_{2,6}(\lambda; N) = \frac{12}{N}\Big(\frac{\lambda}{16\pi^{2}}\Big)^{4}\,\Big\{
-2I_{0}(\sqrt\lambda)+\frac{(8+\lambda)I_{1}(\sqrt\lambda)}{2\sqrt\lambda}
+\frac{1}{N^{2}}\Big[\frac{ \lambda (24+\lambda ) I_ 0(\sqrt{\lambda}) }{192}
\notag \\ & \qquad +\frac{\sqrt{\lambda } (-3+2 \lambda ) I_ 1(\sqrt{\lambda
})}{12} \Big]
+\frac{1}{N^{4}}\Big[\frac{\lambda ^2 (160+7 \lambda ) I_ 0(\sqrt{\lambda})}{3840}
+\frac{\lambda ^{5/2} (3408+5 \lambda ) I_ 1(\sqrt{\lambda})}{184320}\Big]
+\mathcal O\Big(\frac{1}{N^{6}}\Big)
\Big\}.\nonumber
\ea
Taking the ratio of \rf{319} and $\vev{\mathcal W}$ in \rf{B.2}
and expanding at strong coupling gives\footnote{The absence of $1/T$ corrections at leading planar order in \rf{332} is due to cancellation of the planar $I_1({\sqrt{\l}}\ )$ term in
$Q_{2,4}(\lambda; N)$ in \rf{323} and in $\vev{{\cal W}}$ in \rf{B.2}.}
\ba
\frac{\vev{\mathcal W\,\mathcal {O}_{2}\,\mathcal {O}_{4}}}{\vev{\mathcal W}} = &
\pi^3 T^2 \,\Big[1+
\frac{3}{4}\frac{g_{\text{s}}^{2}}{T^{2}}\Big(1+\frac{2}{3\pi T}+\cdots\Big)
+\frac{1}{8}\frac{g_{\text{s}}^{4}}{T^{4}}\,\Big(1+\frac{1}{2\pi T}+\cdots\Big)\notag \\ & \qquad\quad \quad
+ \frac{g_{\text{s}}^{6}}{T^{6}}\Big( -\frac{1}{256\pi T}+\cdots\Big)+
\cdots
\Big], \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){332} \\
\frac{\vev{\mathcal W\,\mathcal O_{2}\,\mathcal O_{6}}}{\vev{\mathcal W}} = &
\frac{3\pi^4}{4}\,T^{2} \,\Big[
1-\frac{2}{\pi T}+\cdots
+\frac{19}{12}\frac{g_{\text{s}}^{2}}{T^{2}}\,\Big(1+\frac{3}{19\pi T}+\cdots\Big)
+\frac{2}{3}\frac{g_{\text{s}}^{4}}{T^{4}}\,\Big(1+\frac{143}{128\pi T}+\cdots\Big)\notag \\ &
\qquad \qquad\qquad\qquad \qquad +\frac{1}{12}\frac{g_{\text{s}}^{6}}{T^{6}}\,\Big(1+\frac{119}{128\pi T}+\cdots\Big)
+\cdots
\Big] .\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){3322}
\ea
Similar expansions may be found for other values of $J$.
Thus from in \rf{3.19} and \rf{332},\rf{3322} we conclude that
the small $g_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s}$, large $T$ expansion of the correlators \rf{3.1} goes
in powers of $\frac{g_{\text{s}}^2}{T^2}$, up to subleading $1/T$ corrections.
This is the same pattern as was found in the case of $\frac{\vev{\mathcal W\,\mathcal O_{J}}}{\vev{\mathcal W}}$
in \rf{2.28}.
\section{Correlators of coincident circular Wilson loops \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4}}
As was mentioned in the Introduction, we can also study the
$1/N$ expansion for other observables, like correlators of several circular
Wilson loops $\vev{\mathcal W^n}$.
Such correlators were previously discussed in particular in the planar approximation
in the $n=2$ case with two circular loops in parallel planes separated by some distance;
at strong coupling one finds a transitional behaviour \cite{Gross:1998gk} at certain critical distance
when the associated minimal surface reduces to independent
surfaces attached to separate loops \cite{Zarembo:1999bu,Correa:2018lyl,Correa:2018pfn}.
Here we will consider the limiting case
when the loops have the same radii and are coincident. In this case the correlator $\vev{\mathcal W^n}$
can be found exactly using the matrix model methods \cite{Drukker:2000rr,Okuyama:2018aij,Sysoeva:2018xig}.
\iffa \footnote{
The $n=2$ example was used to check
the consistency of the localization approach with weak-coupling perturbation theory \cite{Arutyunov:2001hs}.}
\fi
Our aim below will be to work out
the large $N$, large $\lambda$ expansion of such
correlators.
\subsection{$\vev{\mathcal W^2}$ for loops in fundamental representation}
The coincident Wilson loops may be considered in generic representations
(see, e.g., \cite{Sysoeva:2018xig,Aguilera-Damia:2017znn}).
Let us consider the case of two loops in the fundamental representation.\footnote{Let us note that a discussion of similar correlator
in planar limit at strong coupling (i.e. using semiclassical string theory)
was in section 6 of \cite{Giombi:2009ms} where the
coincident $1\ov4$-BPS ``latitudes'' were considered; the present example of
$1\ov2$-BPS circular loops is a special case.}
The relevant $1/N$ expansions may be written in terms of matrix model correlators as
\ba
\vev{ \mathcal W } &= \vev{\tr\exp\Big(\sqrt{\tfrac{\lambda}{2N}}\, a\Big)} =
\sum_{n=0}^{\infty}\frac{1}{(2n)!}\Big(\frac{\lambda}{2N}\Big)^{n}\vev{\tr a^{2n}}, \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){411} \\
\vev{ \mathcal W^{2} } &=
\sum_{n,m=0}^{\infty}\frac{1}{n!\,m!}\Big(\frac{\lambda}{2N}\Big)^{\frac{n+m}{2}}\vev{\tr a^{n}\,\tr a^{m}}\ . \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.1}
\ea
The expression for \rf{411} is given by (\ref{1.4}). A similar exact result for \rf{4.1}
was found in \cite{Drukker:2000rr,Kawamoto:2008gp,Okuyama:2018aij} (here $L^{(i)}_j$ are the generalized
Laguerre polynomials and $L_i= L^{(0)}_i$)
\ba
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.2}
\langle \mathcal W^{2}\rangle
= &
e^{\frac{\lambda}{2N}}\,L_{N-1}^{(1)}\Big(-\frac{\lambda}{N}\Big)
2e^{\frac{\lambda}{4N}}\,\sum_{i=0}^{N-1}\sum_{j=0}^{i-1}\Big\{
L_{i}\Big(-\frac{\lambda}{4N}\Big)\,L_{j}\Big(-\frac{\lambda}{4N}\Big) \notag \\ & \qquad \qquad\qquad \qquad \qquad \qquad \qquad \qquad \qquad
-\frac{j!}{i!}\Big(\frac{\lambda}{4N}\Big)^{i-j}\,\Big[L_{j}^{(i-j)}\Big(-\frac{\lambda}{4N}\Big)\Big]^{2}
\Big\}.
\ea
This expression can be checked by directly evaluating $\langle \mathcal W^{2}\rangle$ at weak coupling and finite $N$
using the Gaussian matrix model, which gives
\ba
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.3}
\vev{\mathcal W^{2}} =& N^2 \Big[
1+\tfrac{1}{4}\textstyle \big(1+\frac{1}{N^2}\big) \lambda +\tfrac{1}{192}
\big(5+\frac{19}{N^2}\big) \lambda ^2+\tfrac{(24+65 N^2+7 N^4) }{4608 N^4}\lambda
^3 \notag \\
&\qquad +\tfrac{(554+385 N^2+21 N^4) }{368640
N^4}\lambda ^4 + \tfrac{(320+1239 N^2+350 N^4+11 N^6) }{7372800 N^6}\lambda ^5 +{\cal O}} \def \bG {\bar \G(\l^6)\Big] \ .
\ea
While \rf{4.2} is exact, it is non-trivial to extract the exact $\l$ dependence
of its coefficients in the $1/N$ expansion so some indirect approach may be required.
The first non-planar contribution to the $1/N$ expansion of \rf{4.2} was computed exactly
in $\lambda$
in \cite{Akemann:2001st} (and was checked in \cite{Arutyunov:2001hs} by the standard weak coupling perturbation theory)
\ba
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.4}
\vev{\mathcal W^{2}} &= N^2 \Big\{ \frac{4}{\lambda}\big[I_{1}(\sqrt\lambda)\big]^{2}+\frac{\sqrt\lambda}{2N^{2}}\Big[
I_{0}(\sqrt\lambda)\,I_{1}(\sqrt\lambda)+\frac{1}{6}\,I_{1}(\sqrt\lambda)\,I_{2}(\sqrt\lambda)
\Big]+\mathcal O\Big({1\over N^{4}}\Big)\Big\}.
\ea
Expanding \rf{4.4} at large $\lambda$ gives (cf. \rf{1.5})
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.5}
\vev{\mathcal W^{2}} = N^{2}\,e^{2\sqrt\lambda}\,
\Big(\frac{2}{\pi}\lambda^{-3/2}+\frac{7}{24\pi N^2}+\cdots\Big) =
W_{1}^{2}\, \Big(1+\frac{7\pi}{6}\,\frac{g_{\text{s}}^{2}}{T}+\cdots\cdots\Big) \ , \qquad
W_1= \frac{\sqrt T}{{2\pi}g_{\text{s}}}\,e^{2\pi\,T} \ .
\end{equation}
In general, writing the $1/N$ expansion as
\begin{equation}\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){47}
\vev{\mathcal W^{2}} = \sum_{p=0}^{\infty}\frac{1}{N^{2p-2}}\, \vev{\mathcal W^{2}}_{p}\ ,
\end{equation}
the above previously known expressions \rf{4.4} for the
$p=0,1$ terms may be written in terms of the $_{1}F_{2}$ hypergeometric function as
\ba
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.7}
\vev{\mathcal W^{2}}_{0} &= {}_1F_2\Big(\tfrac{3}{2};2,3;\lambda \Big), \qquad \qquad
\vev{\mathcal W^{2}}_{1} &= \tfrac{1}{4} \lambda \, {}_1F_2\Big(\tfrac{3}{2};2,3;\lambda \Big)+\tfrac{7}{192}
\lambda ^2 \, {}_1F_2\Big(\tfrac{5}{2};3,4;\lambda \Big).
\ea
Extending the weak-coupling expansion (\ref{4.3}) up to $\mathcal O(\lambda^{17})$ order
one can come up with
similar results for the $p=2,3,4$ terms in \rf{47}
\ba
\vev{\mathcal W^{2}}_{2} = &
\tfrac{1}{192} \lambda ^3 \, _1F_2\Big(\tfrac{5}{2};4,5;\lambda \Big)+\tfrac{157
}{184320}\lambda ^4 \, _1F_2\Big(\tfrac{7}{2};5,6;\lambda \Big)+\tfrac{679
}{22118400}\lambda ^5 \, _1F_2\Big(\tfrac{9}{2};6,7;\lambda \Big)\notag \\ &
+\tfrac{37}{141557760} \lambda ^6 \, _1F_2\Big(\tfrac{11}{2};7,8;\lambda \Big),\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.9} \\
\vev{\mathcal W^{2}}_{3} =& \tfrac{1}{23040}\lambda ^5 \, _1F_2\Big(\tfrac{7}{2};6,7;\lambda
\Big)+\tfrac{7883}{1238630400} \lambda ^6 \, _1F_2\Big(\tfrac{9}{2};7,8;\lambda
\Big)+\tfrac{7073}{26011238400} \lambda ^7 \,
_1F_2\Big(\tfrac{11}{2};8,9;\lambda \Big)\notag \\ &
+\tfrac{176671}{39953262182400} \lambda
^8 \, _1F_2\Big(\tfrac{13}{2};9,10;\lambda \Big)+\tfrac{38753
}{1369826131968000}\lambda ^9 \, _1F_2\Big(\tfrac{15}{2};10,11;\lambda
\Big)\notag \\ &
+\tfrac{11531}{197254963003392000} \lambda ^{10} \,
_1F_2\Big(\tfrac{17}{2};11,12;\lambda \Big). \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){49}
\ea
These expressions can be written also in terms of Bessel functions; for \rf{4.9} one finds (cf. \rf{4.4})
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.10}
\vev{\mathcal W^{2}}_{2} =\tfrac{37 \lambda ^2}{2304} \big[I_0(\sqrt{\lambda })\big]^2 -\tfrac{\sqrt\lambda(24 +131 \lambda )
}{2880}I_0(\sqrt{\lambda }) I_1(\sqrt{\lambda
}) +\tfrac{(192+332 \lambda +185 \lambda ^2) }{11520}\big[I_1(\sqrt{\lambda
})\big]^2 .
\end{equation}
This agrees with the result in \cite{Okuyama:2018aij} found using
the topological recursion.
From the point of view of computational
efficiency, our procedure based on the hypergeometric representation of the connected
part of the $\vev{\mathcal W^{2}}$ correlator has an advantage that it can be
easily coded and extended to higher order terms in $1/N$ expansion in \rf{47}.
Continuing to order $p=6$ in \rf{47},
expanding for large $\lambda$ and dropping subleading $1/T$ terms
we get the following generalization of
(\ref{4.5})
\ba
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.11}
\vev{\mathcal W^{2}} &\simeq W_{1}^{2}\, \Big(1+\tfrac{7}{6}\,\xi+\tfrac{37}{72}\xi^{2}
+\tfrac{887}{6480}\,\xi^{3}+\tfrac{28379}{1088640}\,\xi^{4}+\tfrac{5045}{1306368}\,\xi^{5}+
\tfrac{1210793}{2586608640}\,\xi^{6}
+\cdots\cdots\Big), \\
\frac{\vev{\mathcal W^{2}}}{\vev{\mathcal W}^{2}} &\simeq 1+\xi+\frac{\xi^{2}}{3}+\frac{\xi^{3}}{15}
+\frac{\xi^{4}}{105}+\frac{\xi^{5}}{945}+\frac{\xi^{6}}{10395}+\cdots\ , \qquad \qquad \xi \equiv \frac{\pig_{\text{s}}^{2}}{T} \ .
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){413}
\ea
This suggests a natural all-order conjecture for the resummed
leading-order strong-coupling terms (cf. \rf{1.5},\rf{1.9})
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.13}
\frac{\vev{\mathcal W^{2}}}{\vev{\mathcal W}^{2}} \simeq
1 + \sum_{p=1}^{\infty}\frac{\xi^{p}}{(2p-1)!!} = 1+e^{\frac{\xi}{2}}\sqrt\frac{\pi\,\xi}{2}\,\text{erf}\Big(
\sqrt\frac{\xi}{2}\Big).
\end{equation}
We prove (\ref{4.13}) using the Toda integrability structure of the
underlying Gaussian matrix model in the next subsection.
Let us note that one can easily find also the correlation function
of ${\cal W}^2 $ with $J=2$ chiral primary operator.
Indeed, the insertion of $\mathcal {O}_{2}$ is equivalent to $\lambda\partial_{\lambda}$ in presence of
any power of $\mathcal W$ in the correlator (cf. \rf{2.12},\rf{216}).
Then
from (\ref{4.4})
one finds
\begin{equation}\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){415}
\frac{\vev{\mathcal W^{2}\,\mathcal {O}_{2}}}{\vev{\mathcal W^{2}}} = \lambda\partial_{\lambda }\log {\vev{\mathcal W^{2}}} = 2\pi\,T\,\Big[
1-\frac{3}{4\pi T}+\cdots+\frac{7}{8}\frac{g_{\text{s}}^{2}}{T^{2}}\,\Big(1+\frac{1}{14\pi T}+\cdots\Big)\Big] \ ,
\end{equation}
which has a similar structure to the one of the previously found correlator in \rf{2.28}
\begin{equation}\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){416}
\frac{\vev{\mathcal W\,\mathcal {O}_{2}}}{\vev{\mathcal W}} = \pi\,T\,\Big[
1-\frac{3}{4\pi T}+\cdots+\frac{1}{8}\frac{g_{\text{s}}^{2}}{T^{2}}\,\Big(1-\frac{1}{2\pi T}+\cdots\Big)\Big].
\end{equation}
\subsection{Resummation of the $g_{\text{s}}^{2}/T$ expansion using Toda integrability structure}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){s4.2}
In the Gaussian matrix model case, the Toda integrability structure
\cite{Gerasimov:1990is,Morozov:1994hh,Morozov:1995pb,Mironov:2005qn} is a useful alternative to the topological recursion. Let us now show how to use it
to prove the relation (\ref{4.13})
to all orders in $\xi =\pi g_{\text{s}}^{2}/T$. From (\ref{4.1}) it follows that we need to find
the exponential generating functions (here $x,y$ are free parameters)
\ba
e_{N}(x) &= \vev{\tr e^{x a}} = \sum_{n=0}^{\infty}\frac{x^{n}}{n!}\,\vev{\tr a^{n}}\,, \quad
e_{N}(x, y) = \vev{\tr e^{x a}\, \tr e^{y a}}_{\rm conn} = \sum_{n,m=0}^{\infty}\frac{x^{n}\,y^{m}}{n!\, m!}\,\vev{\tr a^{n}\, \tr a^{m}}_{\rm conn}\,\nonumber
\\
\vev{\mathcal W} &= e_{N}\Big(\sqrt{\tfrac{\lambda}{2N}}\Big),\qquad\qquad \qquad
\vev{\mathcal W^{2}}_{\rm conn} = e_{N}\Big(\sqrt{\tfrac{\lambda}{2N}}, \sqrt{\tfrac{\lambda}{2N}}\Big).\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){417}
\ea
The Toda hierarchy analysis of \cite{Morozov:2009uy} shows that\footnote{
Note that our normalization of $a$ is different by $\sqrt 2$ from the one in \cite{Morozov:2009uy}.}
\ba
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.18}
& e_{N+1}(x)+e_{N-1}(x) = 2e_{N}(x)+\frac{x^{2}}{2N}e_{N}(x), \\
& e_{N+1}(x,y)+e_{N-1}(x,y) = 2e_{N}(x,y)+\frac{(x+y)^{2}}{2N}e_{N}(x,y)-\frac{x^{2}y^{2}}{4N^{2}}e_{N}(x)e_{N}(y)\ .\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){418}
\ea
The first recursion is solved by
\begin{equation}
e_{N}(x) = e^{\frac{x^{2}}{4}}\,L_{N-1}^{(1)}\big(-\tfrac{1}{2}x^{2}\big)\ ,
\end{equation}
reproducing the expression for $ \vev{\mathcal W}$ in \rf{1.4}.
\subsubsection{$1/N$ expansion from Toda recursion and proof of (\ref{4.13}) \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.2.1}}
Using \rf{4.18},\rf{418} one can generate the $1/N$ expansion of $\vev{\mathcal W^{2}}$.
Let us first show how this is done for $\vev{\mathcal W}$.
At large $N$ we have
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.20}
\vev{\mathcal W} \equiv w(N, \lambda) = e_{N}\Big(\sqrt{\tfrac{\lambda}{2N}}\Big) =N\, w_{0}(\lambda)+\frac{1}{N}\,w_{1}(\lambda)+\frac{1}{N^{3}}\,w_{2}(\lambda) + \cdots.
\end{equation}
The first recursion (\ref{4.18}) gives
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.21}
w\Big(N+1, \lambda\frac{N+1}{N}\Big)+w\Big(N-1,\lambda\frac{N-1}{N}\Big) = 2\,\Big(1+\frac{\lambda}{8\,N^{2}}\Big)\,w(N, \lambda).
\end{equation}
Making the following ansatz for the large $\l$ expansion of the $1/N$ coefficients as
(dots stand for subleading terms at large $\l$)
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.22}
w(N,\lambda) =e^{\sqrt\lambda} \sum_{n=0}^{\infty} C_{n}\,(\lambda^{-3/4}N)^{1-2n}\,+...\ , \qquad
C_{0} = \frac{1}{48\sqrt{2\pi}} \
\end{equation}
and plugging it in the recursion (\ref{4.21}) gives
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.23}
w(N,\lambda) = e^{\sqrt\lambda}\,F(\lambda^{-3/4}\,N)+...\ .
\end{equation}
Setting $z=\lambda^{-3/4}\,N$ and taking $N\to \infty$ gives
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.24}
F'(z)+\Big(1-\frac{1}{48z^{2}}\Big)\,F(z)=0\quad\to\quad F(z) = C_{0}\,z\,e^{\frac{1}{96z^{2}}} \ .
\end{equation}
Thus it reproduces the
resummed expression in (\ref{1.5}). The derivation of the ``D3-brane'' limit in this approach is
presented for completeness in Appendix~\ref{app:Dbrane-Toda}.
For the case of $\vev{\mathcal W^{2}} $ in \rf{417} we define similarly
\ba
\vev{\mathcal W^{2}}_{\rm conn} & \equiv \sigma(N,\lambda) = e_{N}\Big(\sqrt{\textstyle \frac{\lambda}{2N}}, \sqrt{\textstyle \frac{\lambda}{2N}}\Big) = N^{2}\,\sigma_{0}(\lambda)+\sigma_{1}(\lambda)+\frac{1}{N^{2}}\sigma_{2}(\lambda)+\cdots, \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){426}
\\
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.26}
\vev{\mathcal W^{2}} &= \sigma(N,\lambda)+\big[w(N,\lambda)\big]^{2} = e^{2\sqrt\lambda}\,\sum_{n=0}^{\infty} S_{n}\,(\lambda^{-3/4}N)^{2-2n}+ ... \ .
\ea
Eq. (\ref{4.11}) gives the ``initial data'' values
$S_{0}, ...,S_6 =\big\{ \tfrac{2}{\pi },\ \tfrac{7}{24 \pi },\ \tfrac{37}{2304 \pi },\ \tfrac{887}{1658880 \pi}, \ \tfrac{28379}{2229534720 \pi }\big\}$.
The recursion relation in (\ref{418}) reads
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.28}
\sigma\Big(N+1, \lambda\frac{N+1}{N}\Big) +\sigma\Big(N-1, \lambda\frac{N-1}{N}\Big) = \Big(2+\frac{\lambda}{N^{2}}\Big)\, \sigma(N,\lambda)-\frac{\lambda^{2}}{16N^{4}}\, \big[w(N,\lambda)\big]^{2}.
\end{equation}
Making, like in (\ref{4.23}), the strong-coupling ansatz (cf. (\ref{4.24}))
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.29}
\sigma(N,\lambda) = e^{2\sqrt\lambda}\,G(\lambda^{-3/4}N)+... \ ,
\end{equation}
and taking large $N$ limit this gives the differential equation for $G(z)$, $z= \lambda^{-3/4}N$
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.30}
G'(z)+\frac{1-6z^{2}}{6x^{3}}\,G(z)+\frac{1}{38864\,\pi\,z}\,e^{\frac{1}{48z^{2}}}=0\ .
\end{equation}
Its general solution is
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.31}
G(z) = c\,z\,e^{\frac{1}{12z^{2}}}+\frac{1}{18432\,\sqrt\pi}\,z\,e^{\frac{1}{12z^{2}}}\,\text{erf}\Big(\frac{1}{4z}\Big) \ ,
\end{equation}
where the integration constant $c$ should be set to zero to match the leading terms in \rf{4.11}.
As a result, we find from \rf{4.24} and \rf{4.31}\footnote{As in similar relations above, here
``$\simeq$'' stands again for the procedure of first making the $1/N$ expansion
and then keeping the leading large $\lambda$ term at each order in $1/N$. We will understand this notation in the rest of the paper.}
\begin{equation}
\vev{\mathcal W^{2}} \simeq e^{2\sqrt\lambda}(G+F^{2}) = \vev{\mathcal W}^{2}\,\Big[1+\sqrt\frac{\pi}{2}\,e^{\xi/2}\,\sqrt\xi\,\text{erf}\Big(\sqrt\frac{\xi}{2}\Big)\Big] \ , \ \ \ \ \qquad \xi= {1\over 8 z^2} = \pi {g^2_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s}\over T} \ . \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){432}
\end{equation}
This proves our conjecture in \rf{4.13}.
\subsubsection{Case of $\vev{{\cal W}^3}$}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){s4.2.2}
Similar approach can be applied also for higher correlators $\vev{{\cal W}^n}$.
For $n=3$ we need the generating functions with 3 arguments
\ba
e_{N}(x, y, z) &= \vev{\tr e^{x a}\, \tr e^{y a}\, \tr e^{z a}}_{\rm conn},\qquad
\vev{\mathcal W^{3}}_{\rm conn} =e_{N}\Big(\sqrt{\textstyle \frac{\lambda}{2N}}, \sqrt{\textstyle \frac{\lambda}{2N}}, \sqrt{\textstyle \frac{\lambda}{2N}}\Big) = \,t(N,\lambda).\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){433}
\ea
The Toda recursion relation here reads
\ba
& e_{N+1}(x,y,z)+e_{N-1}(x,y,z) = 2e_{N}(x,y,z)+\frac{(x+y+z)^{2}}{2N}e_{N}(x,y,z)
-\frac{(x+y)^{2}\,z^{2}}{4N^{2}}e_{N}(x,y)e_{N}(z)\notag \\
&\qquad
-\frac{(x+z)^{2}y^{2}}{4N^{2}}e_{N}(x,z)e_{N}(y)
-\frac{(y+z)^{2}x^{2}}{4N^{2}}e_{N}(y,z)e_{N}(x)
+\frac{x^{2}y^{2}z^{2}}{4N^{3}}e_{N}(x)\,e_{N}(y)\,e_{N}(z).
\ea
Writing it in terms of the functions $t$ in \rf{433}, $w$ in \rf{4.20} and $\sigma$ in \rf{426} we get
\ba
t\Big(N+1,\lambda\frac{N+1}{N}\Big)+t\Big(N-1,\lambda\frac{N-1}{N}\Big) = &\Big( 2+\frac{9\lambda}{4N^{2}}\Big)\,t(N,\lambda)
-\frac{3\lambda^{2}}{4N^{4}}\,w(N,\lambda)\,\sigma(N,\lambda)\notag \\
&+\frac{\lambda^{3}}{32N^{6}}\,\big[w(N,\lambda)\big]^{3}\ .
\ea
Making an ansatz
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.37}
t(N,\lambda) = e^{3\sqrt\lambda}\,U(\lambda^{-3/4}N)+...
\end{equation}
and using \rf{4.23},\rf{4.24} and \rf{4.29},\rf{4.31} gives\footnote{The fact that the resulting differential equation is
1st order and separable holds for any $\vev{\mathcal W^{n}}$ due to the universal finite difference form of
the Toda recursion.}
\begin{equation}
U'(z)-\frac{16z^{2}-9}{16z^{3}}\,U(z)+\frac{1}{884736\sqrt 2\,\pi z}e^{\frac{3}{32z^{2}}}\text{erf}\Big(\frac{1}{4z}\Big)=0,
\end{equation}
Solving for $U(x)$ and using that \begin{equation}
\vev{\mathcal W^{3}} = e^{3\sqrt\lambda}\Big[t+3(G+F^{2})F-2F^{3}\Big],
\end{equation}
gives the analog of \rf{4.13},\rf{432} ($\xi=\pig_{\text{s}}^{2}/T$)
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.40}
\frac{\vev{\mathcal W^{3}}}{\vev{\mathcal W}^{3}} \simeq 1+3\,\sqrt\frac{\pi}{2}\,e^{\xi/2}\,\sqrt\xi\,\text{erf}\Big(\sqrt\frac{\xi}{2}\Big)
-\frac{4\pi}{3\sqrt 3}\,\xi\,e^{2\xi}\,\Big[-1+12\,{\rm T}\Big(\sqrt{3\xi}, \frac{1}{\sqrt 3}\Big)\Big],
\end{equation}
where ${\rm T}(h,a)$ is the Owen T-function
\begin{equation}\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){ttt}
{\rm T}(h,a) = \frac{1}{2\pi}\,\int_{0}^{a}dx\,\frac{e^{-\frac{h^{2}}{2}\,(1+x^{2})}}{1+x^{2}}
= \frac{\text{arctan}(a)}{2\pi}-\frac{1}{2\pi}\sum_{n=0}^{\infty}\frac{(-1)^{n}a^{2n+1}}{2n+1}\Big(1-e^{-\frac{h^{2}}{2}}\sum_{m=0}^{n}\frac{h^{2m}}{2^{m}m!}\Big).
\end{equation}
Explicitly, the first few terms in the expansion of \rf{4.40} in powers of $\xi$ are thus (cf. \rf{413})
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.43}
\frac{\vev{\mathcal W^{3}}}{\vev{\mathcal W}^{3}} \simeq 1+3\, \xi +5\, \xi ^2+\tfrac{73}{15}\,\xi ^3
+\tfrac{113}{35}\, \xi ^4+\tfrac{508}{315}\, \xi ^5+\tfrac{33521}{51975}\, \xi ^6
+\cdots.
\end{equation}
Similar expansion can be found for all $n$; as we show in Appendix \ref{anew}, we have
\ba
{\vev{\mathcal W^{n}}\over \vev{\mathcal W}^{n}} = 1 &+\frac{n\,(n-1)}{2}\,\xi+\frac{n\,(n-1)\, (3n-5)\,(n+2)}{24}\,\xi^{2}\notag \\
&+\frac{n\,(n-1)\, (15n^{4}+30n^{3}-75n^{2}-610 n+1064)}{720}\,\xi^{3}+
\mathcal O(\xi^{4}).\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){443}
\ea
\subsection{Correlator of loops in fundamental and anti-fundamental representations}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){s4.3}
Let us consider now a correlator of one Wilson loop in $k$-fundamental and another in $k$-anti-fundamental representation of $U(N)$. In the matrix model description it is given by (cf. \rf{266})
\begin{equation}
\vev{\mathcal W^{(k,-k)}}\equiv \vev{ {\cal W}^{(k)} \, {\cal W}^{(-k)} } = \vev{ \tr U^{k}\,\tr U^{-k} } \ , \qquad \qquad
U= e^{\, \frac{g}{\sqrt 2}\,a} \ . \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){41}
\end{equation}
We will focus on the $k=1$ case as (like in the case of $k$-fundamental -- $k$-fundamental correlator
discussed above)
the dependence on $k$ can be recovered by the rescaling $g\to k g $ or
$\lambda\to k^{2}\lambda$.
Instead of $\vev{{\cal W}^2}$ in \rf{4.2} here one finds
\cite{Okuyama:2018aij}\footnote{The peculiar first term in the r.h.s. of (\ref{42}) is due to would-be term in $\vev{\mathcal W^{(k, k')}}$ proportional to
a certain Laguerre $L_j^{(i)}\big(-(k+k')^{2}\frac{\lambda}{4N}\big)$
contribution
that happens to be $\lambda$ independent for $k+k'=0$.}
\ba\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){42}
\langle \mathcal W^{(1,-1)}\rangle = N + \Big[e^{\frac{\lambda}{8N}}\,L_{N-1}^{(1)}\Big(-\frac{\lambda}{4N}\Big)\Big]^{2
-e^{\frac{\lambda}{4N}}\,\sum_{i=0}^{N-1}\sum_{j=0}^{N-1}
(-1)^{i-j}\,\frac{j!}{i!}\Big(\frac{\lambda}{4N}\Big)^{i-j}\,\Big[L_{j}^{(i-j)}\Big(-\frac{\lambda}{4N}\Big)\Big]^{2}.
\ea
Its weak coupling expansion reads (cf. \rf{4.3})
\ba
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.46}
\vev{\mathcal W^{(1,-1)}} = N^2 + (N^2-1)\Big[
\tfrac{1}{4} \lambda +\tfrac{5 }{192 } \lambda
^2+\tfrac{7 }{4608}\lambda ^3
+\tfrac{(7N^2 +2) }{122880 N^2} \lambda ^4
+\tfrac{11 (N^2+1)}{7372800
N^2} \lambda ^5
+\cdots\Big].
\ea
The first two terms in the $1/N$ expansion are as in \rf{4.4},\rf{47}:
\ba
\vev{\mathcal W^{(1,-1)}} &= \sum^\infty_{p=0} {1\over N^{2p-2}}\, \vev{\mathcal W^{(1,-1)}}_{p} \ , \qquad \qquad
\vev{\mathcal W^{(1,-1)}}_{0} = \frac{4}{\lambda}\,\big[I_{1}(\sqrt\lambda)\big]^{2}, \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){777} \\
\vev{\mathcal W^{(1,-1)}}_{1} &= -\frac{1}{2} \,\lambda\big[ I_0(\sqrt{\lambda })\big]^2+\frac{7}{12}
\sqrt{\lambda }\, I_0(\sqrt{\lambda }) I_1(\sqrt{\lambda })+\frac{1}{6}
(-1+3 \lambda ) \big[I_1(\sqrt{\lambda })\big]^2.\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){447}
\ea
Expanding these first two terms at large $\lambda$ gives the analog of \rf{4.5},\rf{4.11}
\begin{equation}
\vev{\mathcal W^{(1,-1)}} = W_{1}^{2}\,\Big(1+\frac{\xi}{6} +\cdots\Big)\ , \qquad \qquad
W_1= \frac{\sqrt T}{{2\pi}g_{\text{s}}}\,e^{2\pi\,T} \ , \qquad \xi = \pi { g^2_s} \def \l {\lambda} \def \sql {\sqrt{\l}}\def \RR {{\rm R}} \def \str {{\rm s} \over T} \ . \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){448}
\end{equation}
To make an efficient ansatz for higher order terms it is useful to use as in \rf{4.7} the
representation in terms of the $_1F_2$ hypergeometric function
\ba
\vev{\mathcal W^{(1,-1)}}_{0} &= {}_1F_2\Big(\tfrac{3}{2};2,3;\lambda \Big),
\qquad\nonumber \\
\vev{\mathcal W^{(1,-1)}}_{1}
& = -\tfrac{1}{4} \lambda \, _1F_2\Big(\tfrac{1}{2};2,3;\lambda
\Big)-\tfrac{1}{192} \lambda ^2 \, _1F_2\Big(\tfrac{3}{2};3,4;\lambda
\Big)+\tfrac{1}{2304}\lambda ^3 \, _1F_2\Big(\tfrac{5}{2};4,5;\lambda \Big).
\ea
By some trial and error it is then possible to determine the higher genus contributions, e.g.,\footnote{As we mentioned previously,
this is an efficient procedure equivalent to the rigorous analysis based on the topological recursion \cite{Eynard:2004mh,Eynard:2008we}.}
\ba
\vev{\mathcal W^{(1,-1)}}_{2} =& -\tfrac{\lambda ^4}{61440} \, _1F_2\Big(\tfrac{5}{2};5,6;\lambda
\Big)-\tfrac{\lambda ^5}{7372800} \, _1F_2\Big(\tfrac{7}{2};6,7;\lambda
\Big)+\tfrac{\lambda ^6}{78643200} \, _1F_2\Big(\tfrac{9}{2};7,8;\lambda
\Big)\notag \\ &
+\tfrac{\lambda ^7}{7927234560} \, _1F_2\Big(\tfrac{11}{2};8,9;\lambda
\Big), \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){451}\\
\vev{\mathcal W^{(1,-1)}}_{3} = &-\tfrac{13 \lambda ^6}{1238630400} \, _1F_2\Big(\tfrac{7}{2};7,8;\lambda \Big)-
\tfrac{71 \lambda ^7}{208089907200} \, _1F_2\Big(\tfrac{9}{2};8,9;\lambda
\Big)\notag \\ &
+\tfrac{389 \lambda ^8}{119859786547200} \,
_1F_2\Big(\tfrac{11}{2};9,10;\lambda \Big)+\tfrac{4499
\lambda ^9}{28766348771328000} \, _1F_2\Big(\tfrac{13}{2};10,11;\lambda \Big)\notag \\ &
+
\tfrac{169 \lambda ^{10}}{140896402145280000} \, _1F_2\Big(\tfrac{15}{2};11,12;\lambda
\Big)+\tfrac{13 \lambda ^{11}}{5207531023289548800} \,
_1F_2\Big(\tfrac{17}{2};12,13;\lambda \Big). \label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){452}
\ea
Their weak-coupling expansions
\ba
\vev{\mathcal W^{(1,-1)}}_{2} &= -\tfrac{\lambda ^4}{61440}-\tfrac{11 \lambda ^5}{7372800}-\tfrac{13
\lambda ^6}{235929600}-\tfrac{13 \lambda ^7}{15854469120}+\tfrac{187
\lambda ^8}{22830435532800}+\cdots, \\
\vev{\mathcal W^{(1,-1)}}_{3} &= -\tfrac{13 \lambda ^6}{1238630400}-\tfrac{83 \lambda
^7}{83235962880}-\tfrac{289 \lambda ^8}{7491236659200} -\tfrac{12331 \lambda ^9}{17259809262796800}
+\cdots,
\ea
agree with the large $N$ expansion of (\ref{4.46}) (as we checked up to $\mathcal O(\lambda^{30})$).
Converting the hypergeometric functions into Bessel functions gives
(cf. \rf{4.10})
\ba
\vev{\mathcal W^{(1,-1)}}_{2} = \tfrac{1}{11520} \Big[ -55 \lambda ^2 \big[I_0(\sqrt{\lambda })\big]^2&-4 \sqrt{\lambda }
(24+11 \lambda ) I_0(\sqrt{\lambda }) I_1(\sqrt{\lambda })\nonumber\\ & +(192+332
\lambda +65 \lambda ^2) \big[I_1(\sqrt{\lambda })\big]^2\Big].
\ea
Expanding at large $\lambda$, we obtain higher order terms in \rf{448} (cf. \rf{4.11})
and observe that they exponentiate
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.56}
\vev{\mathcal W^{(1,-1)}} \simeq W_{1}^{2}\Big(
1+\tfrac{1 }{6}\xi +\tfrac{1}{72}\xi^2+\tfrac{1}{1296}\xi ^3+\tfrac{1}{31104}\xi ^4+\tfrac{1}{933120}\xi ^5+\tfrac{1}{33592320}\xi ^6+\cdots
\Big) \simeq W_{1}^{2}\, e^{\xi\ov6} \ .
\end{equation}
Comparing this with the sum of the leading strong coupling terms in $\vev{{\cal W}}$ given by $e^{{\xi\over 12}}$ in \rf{1.5}
we conclude that in contrast to the nontrivial result for $\vev{W^2}$ in \rf{4.13}
here one finds a simple factorization relation (valid again up to subleading terms in $1/T$)
\begin{equation}
\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){4.57}
\vev{\mathcal W^{(1,-1)}} \simeq \vev{\mathcal W}^{2} \ .
\end{equation}
Like \rf{4.13} this can be proved to all orders
in $\xi$ using the Toda recursion relations (cf. section \ref{4.2.1}). To this aim,
let us define as in \rf{417},\rf{4.26}
\begin{equation}\label}\def\footnote{\footnote}\newcommand{\rf}[1]{(\ref{#1}){458}
\vev{\mathcal W^{(1,-1)}}_{\rm conn}= e_{N}\Big(\sqrt{\textstyle \frac{\lambda}{2N}}, -\sqrt{\textstyle \frac{\lambda}{2N}}\Big) \equiv \overline\sigma(N,\lambda) = N^{2}\,\overline\sigma_{0}(\lambda)+\overline\sigma_{1}(\lambda)+\frac{1}{N^{2}}\overline\sigma_{2}(\lambda)+\cdots\ .
\end{equation}
The second recursion relation in (\ref{4.18}) reads (cf. \rf{4.28},\rf{4.22})
\ba
& \overline\sigma\Big(N+1, \lambda\frac{N+1}{N}\Big) +\overline\sigma\Big(N-1, \lambda\frac{N-1}{N}\Big) = 2\, \overline\sigma(N,\lambda)-\frac{\lambda^{2}}{16N^{4}}\, \big[w(N,\lambda)\big]^{2}, \notag \\
& w(N,\lambda) = \sqrt\frac{2}{\pi}\,e^{\sqrt\lambda}\,\sum_{n=0}^{\infty} \frac{1}{96^{n}\,n!}\,(\lambda^{-3/4}N)^{1-2n}+\cdots\ .
\ea
Making an ansatz as in \rf{4.29}
\begin{equation}
\overline\sigma(N,\lambda) = e^{2\sqrt\lambda}\,\overline G(z)+\cdots \ , \qquad \qquad z=\lambda^{-3/4}N \ ,
\end{equation}
we find, expanding in large $N$
\begin{equation}
z^{-4/3}N^{-2/3}\overline G(z)+ z^{-8/3} N^{-4/3}\Big[\tfrac{1}{73728\pi}
z^{2}e^{\frac{1}{48z^{2}}}+\tfrac{1}{12} \big((1-6z^{2})\overline G(z)+6z^{3}\overline G'(z)\big)\Big]+\mathcal O(N^{-2}) = 0.
\end{equation}
In contrast to the differential equation in \rf{4.30} here at leading order
in large $N$ we get simply the constraint
\begin{equation}
\overline G(x)=0\ ,
\end{equation}
implying the vanishing of the connected part \rf{458} of $\vev{\mathcal W^{(1,-1)}}$ and thus proving \rf{4.57}.
\section*{Acknowledgments}
We are grateful to Simone Giombi for a collaboration at an early stage of this project and many useful remarks and suggestions.
We also thank Nadav Drukker, Marcos Mari\~{n}o, Albrecht Klemm, Francesco Galvagno, and Marco Billo' for useful communications and
discussions on various aspects of this work. M.B. acknowledges the support of the INFN grant GSS (Gauge Theories, Strings and Supergravity).
A.A.T. acknowledges the support of the
STFC grants ST/P000762/1 and ST/T000791/1.
|
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BkiUcOvxK6EuM_Ubp4_s
| 5
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\section{Introduction}
It is well known that the usual heat equation
$$\ro c{\d T\over\d t}=\kappa\triangle T+Q\eqno(1.1)$$
has the defect that heat can propagate instantaneously through space. In (1.1) $T$ is temperature, $t$ is time, $\triangle$ is the Laplace operator and $Q$ describes the heat generation. The constant $\kappa$ is the thermal conductivity, $\ro$ the mass density and $c$ the specific heat. Still the equation is widely used to simulate heat transfer problems ([1] and references given there). The reason for this causality defect is that the heat equation is first order in time. A second defect related to this is the restriction of the initial value problem. In the Cauchy problem for (1.1) only the temperature $T$ at time $t=0$ can be specified. But in experiments also the temporal derivative $\d T/\d t$ at $t=0$ must be adjusted to the experimental situation. This is possible in the hyperbolic heat transfer equation which is second order in time.
It is our aim to compare the two equations in an analytically solvable case of some practical interest. In the next section the 1D ablation problem in infinite space is solved for the heat equation and in sect.3 for the hyperbolic heat transfer equation. In sect.4 some special functions are discussed which appear in the solutions. In sect.5 we study the problem on a finite spherical volume. This is important because it turns out that the infinite
problem cannot be viewed as the limit of the finite problem. So for real applications only this finite problem is relevant.
\section{The 1D ablation problem according to the bio-heat equation}
Let us consider a spherical electrode of radius $r_0$ in an infinite medium with electrical conductivity $\si$. Assuming the second dispersive electrode at infinity with potential $V=0$ the potential in the medium is given by the simple solution of Laplace's equation
$$V(r)={V_0r_0\over r}\eqno(2.1)$$
where $V_0$ is the applied potential on the ablation electrode. The corresponding electric field strength is
$$E_r=-{\d V\over\d r}={V_0r_o\over r^2}\eqno(2.2)$$
and the heat generation
$$Q=\si{(V_0r_0)^2\over r^4}.\eqno(2.3)$$
With this constant heating we want to calculate the transient temperature $T(t,r)$ as solution of equation (1.1) which becomes
$${\d T\over\d t}={a\over r^2}{\d\over\d r}\B(r^2{\d T\over\d r}\B)+{\beta\over r^4},\quad a={\kappa\over\ro c},$$
where $\beta$ is the constant appearing in (2.3) divided by $\ro c$ and $a$ the thermal diffusivity.
We write the solution as
$$T(t,r)=T_1(r)+T_2(t,r),\eqno(2.4)$$
where $T_1(r)$ is the steady state solution satisfying
$${d\over dr}\B(r^2{dT_1\over dr}\B)=-{\beta\over ar^2}.\eqno(2.5)$$
Assuming a temperature $T_\infty$ at infinity we get
$$T_1(r)=T_\infty+{C_1\over r}-{b\over 2r^2}.\quad b={\beta\over a}\eqno(2.6)$$
The integration constant $C_1$ is fixed by the assumption of no heat flux at the electrode
$${dT_1\over dr}\vert_{r_0}=0,\eqno(2.7)$$
which is reasonable for a small electrode. This gives the following steady state solution
$$T_1(r)=T_\infty+{b\over r_0 r}-{b\over 2r^2}.\eqno(2.8)$$
The maximal temperature is found at the electrode, of course
$$T_1(r_0)=T_\infty+{b\over 2r_0^2}.\eqno(2.9)$$
The remaining homogeneous equation for $T_2$ is solved by separation of variables
$$T_2(t,r)=T_3(t)T_4(r).\eqno(2.10)$$
Then we have
$${1\over a}{\dot T_3\over T_3}={T_4''\over T_4}+{2\over r}{T_4'\over T_4}={\rm const.}=-\,k^2$$
where the dot means time derivative and the prime radial derivative. This yields
$$T_3(t)=C_3e^{-\,k^2at}\eqno(2.11)$$
and
$$r^2T_4''+2rT_4'+\,k^2r^2T_4=0.\eqno(2.12)$$
This last equation is a special case of Bessel's equation with the solution
$$T_4(r)={1\over r}\B(C_4\sin\,k r+C_5\cos\,k r\B).\eqno(2.13)$$
Now the general solution of our problem is given by
$$T(t,r)=T_\infty+{b\over r_0 r}-{b\over 2r^2}+{1\over r}\int\limits_0^\infty d\,k\,e^{-\,k^2at}\B(f_1(\,k)\sin\,k r+$$
$$+f_2(\,k)\cos\,k r\B).\eqno(2.14)$$
The unknown functions $f_1$ and $f_2$ in (2.14) are determined by the initial condition
$$T(0,r)=T_\infty,\eqno(2.15)$$
$${b\over 2r^2}-{b\over r_0r}={1\over r}\int\limits_0^\infty[f_1(\,k)\sin\,k r+f_2(\,k)\cos\,k r]d\,k.\eqno(2.16)$$
Here the l.h.side is only defined for $r>r_0$. To use the theorems on the {\it real} Fourier transform we continue the functions to $0<r<r_0$. The even function $b/2r^2$ is continued by the constant $b/2r_0^2$. The inverse Fourier transform then yields
$$f_1(\,k)={b\over\pi}\int\limits_{r_0}^\infty{\sin\,k r\over r}dr+{b\over\pi r_0^2}\int\limits_0^{r_0}r\sin\,k r\,dr=$$
$$={b\over 2}-{b\over\pi}{\rm Si}(\,k r_0)+{b\over\pi\,k^2r_0^2}(\sin\,k r_0-\,k r_0\cos\,k r_0)=$$
$$={b\over 2}+O(\,k r_0).\eqno(2.17)$$
Here {\rm Si} is the sine integral. For small electrode radius $r_0$ this gives the following contribution to (2.14)
$$\int\limits_0^\infty d\,k e^{-\,k^2at}f_1(\,k)\sin\,k r={b\over 4\sqrt{at}}\int\limits_0^\infty dy\,e^{-y^2/4}\sin{ry\over 2\sqrt{at}}=$$
$$={b\over 2\sqrt{at}}D_+\B({r\over 2\sqrt{at}}\B).\eqno(2.18)$$
Here $D_+(z)$ is the Dawson function (see Wikipedia and references given there).
The function $f_2(\,k)$ in (2.16) must clearly degenerate to a delta-distribution
$$f_2(\,k)=-{2b\over r_0}\delta(\,k).\eqno(2.19)$$
This contribution then cancels the $1/r$-term in (2.14) which is necessary for $t=0$. Then the final result is
$$T(t,r)=T_\infty-{b\over 2r^2}+{b\over 2\sqrt{at}}D_+\B({r\over 2\sqrt{at}}\B)+O(r_0).\eqno(2.20)$$
The Dawson function has the following asymptotic expansion
$$D_+(x)={2\over 2x}+{1\over 4x^3}+{3\over 8x^5}+\ldots$$
for $x\gg 1$. Using this in (2.20), the first term cancels the negative contribution so that the result for small times seems to be correct. It shows the expected rise of the temperature above $T_\infty$. However the Dawson function has a maximum at $x=0.924...$, $D(x)=0.541...$ and for smaller $x$ it decreases to 0. As a consequence for large $t$ the temperature (2.20) falls below $T_\infty$ which is completely wrong ! The reason for this disaster is simple: The heat equation is first order in time. Therefore only one condition, namely the initial condition at $t=0$ is at our disposal. The hyperbolic heat equation in the next section is second order in time, then we have two free constants of integration, so that we can get the right behavior of the solution at $t=0$ {\it and} $t=\infty$.
\section{Approach to steady state in the hyperbolic heat equation}
According to Cattaneo [2] and Vernotte [3] a better description of heat transfer is obtained by assuming a time delay $\tau$ between heat flux $q$ and temperature gradient
$$q(t+\tau,x)=-\kappa \nabla T(t.x).\eqno(3.1)$$
Expanding up to first order in $\tau$ we have
$$q+\tau{\d q\over\d t}=-\kappa\nabla T$$
and
$$\nabla q=-\tau{\d\over \d t}\nabla q-\nabla \kappa\nabla t$$
which is substituted into the energy conservation equation
$$-\nabla q(t,x)+Q(t,x)=\ro c{\d T(t,x)\over\d t}.\eqno(3.2)$$
For constant thermal conductivity $k$ this gives the hyperbolic heat equation
$${\d^2 T\over\d t^2}+{1\over\tau}{\d T\over\d t}-{\kappa\over\tau\ro c}\triangle T={1\over\ro c}{\d Q\over\d t}+{Q\over\tau\ro c}.\eqno(3.3)$$
For our 1D ablation problem this equation assumes the following form
$${\kappa\over\tau\ro c}{1\over r^2}{\d\over\d r}\B(r^2{\d\over\d r}T\B)+{d\over r^4}={1\over\tau}{\d T\over\d t}+{\d^2T\over\d t^2}.\eqno(3.4)$$
With another boundary condition this problem has been studied in [4]. These authors use the method of Laplace transform which becomes very complicated. By the method of the previous section we get the solution in much simpler form. We do not include a switching of the heat generation $Q$ by means of a Heaviside step function as in [4]. This would give an additional singular term $u_1(r)\delta(t)$. Then we have a so-called generalized Cauchy problem in the sense of distributions, which has been treated by Vladimirov [7]. The $\delta$-term then fixes the initial condition at $t=0$ as
$${\d T(t,r)\over\d t}\B\vert_{t=0}=u_1(r).$$
But in [4] the simple initial condition
$${\d T(t,r)\over\d t}\B\vert_{t=0}=0\eqno(3.5)$$
was used, this is a certain inconsistency. We consider the classical Cauchy problem where we have two initial conditions at $t=0$ for free.
As before we write the solution in the form (2.4) where the steady state solution $T_1(r)$ satisfies the equation
$${d\over dr}\B(r^2{dT_1\over dr}\B)=-{\beta\over ar^2}.\eqno(3.6)$$
We have the same steady state solution (2.8)
$$T_1(r)=T_\infty+{b\over r_0 r}-{b\over 2r^2}.\eqno(3.7)$$
But the homogeneous equation now reads
$${1\over r^2}\B(2r{\d T_2\over\d r}+r^2{\d ^2T_2\over\d r^2}\B)={1\over a}{\d T_2\over\d t}+\eps{\d ^2T_2\over\d t^2}\eqno(3.8)$$
with
$$a={\ro c\over\kappa},\quad \eps=a\tau.\eqno(3.9)$$
For $\eps=0$ we are back at the parabolic heat equation.
Again equation (3.8) is solved by separating the variables
$$T_2(t,r)=T_3(t)T_4(r)\eqno(3.10)$$
which yields
$${1\over a}{\dot T_3\over T_3}+\eps{\ddot T_3\over T_3}={T_4''\over T_4}+{2\over r}{T_4'\over T_4}=-\,k^2.\eqno(3.11)$$
The equation for $T_4$ is the same as before (2.12), but for $T_3$ we now have the second order equation
$$\eps\ddot T_3+{1\over a}\dot T_3=-\,k^2T_3.\eqno(3.12)$$
It has two exponential fundamental solutions
$$T_\pm (t)=C_\pm e^{\omega_\pm t}\eqno(3.13)$$
where $\omega_\pm$ are the two solutions of the quadratic equation
$$\eps\omega^2+{1\over a}\omega+\,k^2=0.$$
We have two negative roots
$$\omega_\pm=-{1\over 2a\eps}\pm\sqrt{{1\over 4a^2\eps^2}-{\,k^2\over\eps}}=-{1\over 2\tau}\pm\sqrt{{1\over 4\tau^2}-{a\,k^2\over\tau}},\eqno(3.14)$$
Then the general solution with the same boundary condition (2.7) as in the last section is given by
$$T(t,r)=T_\infty+{b\over r_0r}-{b\over 2r^2}+{1\over r}\int\limits_0^\infty d\,k e^{\omega_+t}\B[f_1(\,k)\sin \,k r+$$
$$+f_2(\,k)\cos\,k r\B]+{1\over r}\int\limits_0^\infty d\,k e^{\omega_-t}\B[g_1(\,k)\sin \,k r
+g_2(\,k)\cos\,k r\B].\eqno(3.15)$$
To satisfy the initial condition $T=T_\infty$ we must again compensate the second term $\beta/r_0r$ by some contribution from the integrals for $\,k =0$. For $\,k=0$ we have
$$\omega_+=0,\quad \omega_-=-{1\over\tau}.\eqno(3.16)$$
In the first case with $f_2(k)\sim\delta(\,k)$ we are in the same situation as in the last section and get no approach to steady state. So we take $f_2=0$. But now we can choose
$$g_2(\,k)=-{b\over r_0}\delta(\,k)\eqno(3.17)$$
and have the desired compensation for $t=0$. But for $t\to\infty$ this term goes to 0 because $\omega_-$ (3.16) gives an exponential fall off $\sim\exp -t/\tau$. That means we obtain the correct steady state as far as the $1/r$ term is concerned.
Regarding the $1/r^2$ term we must determine $f_1$ and $g_1$ such that
$${b\over r}=\int\limits_0^\infty d\,k\,[f_1(\,k)+g_1(\,k)]\sin\,k r.\eqno(3.18)$$
As before (2.17) this gives
$$f_1(\,k)+g_1(\,k)={b\over 2}+O(\,k r_0).\eqno(3.19)$$
To determine $f_1$ and $g_1$ separately we need a second initial condition. Preliminary experiments show that the above condition (3.5) is physically correct, so we assume it. The condition (3.5) implies
$$\omega_+f_1+\omega_- g_1=0$$
so that
$$f_1(\,k)={b\over 2}{\omega_-\over\omega_--\omega_+},\quad
g_1(\,k)={b\over 2}{\omega_+\over\omega_+-\omega_-}.\eqno(3.20)$$
Inserting the roots (3.14) we obtain the following final result
$$T(t,r)=T_\infty+{b\over r_0r}\B(1-e^{-t/\tau}\B)-{b\over 2r^2}+$$
$$+{b\over 4r}\int\limits_0^\infty d\,k\B[e^{\omega_+t}\B(1+(1-4a\tau\,k^2)^{-1/2}\B)+e^{\omega_-t}\B(1-(1-4a\tau\,k^2)^{-1/2}\B)\B]\sin\,k r.\eqno(3.21)$$
However, we note that this total solution does not satisfy the initial condition (3.5) of vanishing temporal derivative. We shall return to this point in sect.5.
The integral in (3.21) must be split at
$$\,k={1\over 2\sqrt{a\tau}}=\,k_0\eqno(3.22)$$
because the roots (3.14) become complex for $k>k_0$. For small $\tau$ only the integral
$$I_1=\int\limits_0^{\,k_0}d\,k\B(1+(1-4a\tau\,k^2)^{-1/2}\B)e^{\omega_+t}\sin\,k r\eqno(3.23)$$
is important. Extending the upper limit to infinity and expanding the square root we get
$$I_1=\int\limits_0^\infty d\,k(2+2a\tau\,k^2)e^{-at\,k^2}\sin\,k r=$$
$$=2(1-\tau\d_t)\int\limits_0^\infty d\,k\,e^{-at\,k^2}\sin\,k r.\eqno(3.24)$$
This gives the Dawson function $D_+$ (2.18) again, up to a correction $O(\tau)$
$$I_1=2(1-\tau\d_t){1\over\sqrt{at}}D_+\B({r\over 2\sqrt{at}}\B).\eqno(3.25)$$
So for small thermal relaxation time $\tau$ we recover the term in the solution (2.20) of the usual heat equation.
For large $\tau$ the integral over $[\,k_0,\infty]$ gives the leading contribution. Since we have two complex conjugate roots (3.14) we obtain a real part
$$I_2=\int\limits_{k_0}^\infty=2{\rm Re}\int\limits_{k_0}^\infty d\,k\B(1-i(4a\tau\,k^2-1)^{-1/2}\B)e^{-{t\over 2\tau}(1-
\sqrt{4a\tau\,k^2-1}}\sin\,k r=$$
$$=2e^{-t/2\tau}\int\limits_{\,k_0}^\infty d\,k\,\B[\cos\B({t\over 2\tau}\sqrt{4a\tau\,k^2-1}\B)+{\sin ({t\over 2\tau}\sqrt{4a\tau\,k^2-1})\over\sqrt{4a\tau\,k^2-1}}\B]\sin\,k r.\eqno(3.26)$$
Here the periodic time dependence indicates the appearance of thermal waves [4] which, however, are damped with a time constant $2\tau$. This damping is essential for the approach to steady state The integral (3.26) is investigated in the next section.
\section{Some special functions}
According to (3.21-23) we must calculate the integrals
$$I^\pm_1=\int\limits_0^{k_0}dk\B(1\pm(1-4a\tau k^2)^{-1/2}\B)e^{\omega_\pm t}\sin kr.\eqno(4.1)$$
With the new integration variable
$$x=2\sqrt{a\tau}k\eqno(4.2)$$
we get the dimensionless form
$$I_1^\pm={1\over 2\sqrt{a\tau}}\int\limits_0^1 dx\B(1\pm{1\over\sqrt{1-x^2}}\B)e^{-{t\over 2\tau}(1\mp\sqrt{1-x^2})}\sin\B({rx\over 2\sqrt{a\tau}}
\B)=$$
$$={1\over 2\sqrt{a\tau}}I_1^\pm(s,u)\eqno(4.3)$$
where
$$s={t\over 2\tau},\quad u={r\over 2\sqrt{a\tau}}\eqno(4.4)$$
and
$$I_1^\pm(s,u)=\int\limits_0^1 dx\B(1\pm{1\over\sqrt{1-x^2}}\B)e^{-s\pm s\sqrt{1-x^2}}\sin ux=$$
$$=\pm e^{-s}(\d_s+1)S_1(\pm s,u).\eqno(4.5)$$
The remaining integral
$$S_1(s,u)=\int\limits_0^1 dx\,e^{s\sqrt{1-x^2}}{\sin ux\over\sqrt{1-x^2}}\eqno(4.6)$$
can be easily calculated by numerical integration, together with its derivatives. But it seems not possible to write it in terms of known special functions.
With the substitution $y=\sqrt{1-x^2}$ we get the form
$$S_1(s,u)=\int\limits_0^1 dy\,{\sin(u\sqrt{1-x^2})\over\sqrt{1-x^2}}e^{sy}\eqno(4.7)$$
and
$${\d S_1\over\d u}=\int\limits_0^1 dy\,s^{sy}\cos(u\sqrt{1-y^2}).\eqno(4.8)$$
On the other hand
$${\d S_1\over\d s}=\int\limits_0^1 dx\,e^{s\sqrt{1-x^2}}\sin ux=\int\limits_0^1 dy\,{y\over\sqrt{1-y^2}}\sin(u\sqrt{1-y^2})e^{sy}=$$
$$=-{1\over u}\int\limits_0^1{d\over dy}(\cos u\sqrt{1-y^2})e^{sy}dy\eqno(4.9)$$
allows partial integration which brings us back to (4.8), so that we get the following differential equation for $S_1$:
$$u{\d S_1\over\d s}=s{\d S_1\over\\d u}+\cos u-e^s.\eqno(4.10)$$
For the approach to steady state we need a bound of $S_1(s,u)$ for large $s$. Such a bound is obtained by means of the confluent hypergeometric function [5]
$$\vert S_1(s,u)\vert<\int\limits_0^1{e^{sx}\over\sqrt{1-x}}dx=2 M(1,{3\over 2},s).\eqno(4.11)$$
Using the asymptotic behavior of $M(a,b,s)$ [5] we get for positive $s>0$
$$\vert S_1(s,u)\vert <\sqrt{\pi}e^s s^{-1/2}(1+O(s^{-1})).\eqno(4.12)$$
The exponential factor is cancelled in (4.5) so that we find a slow approach to steady state with $s^{-1/2}$. The other term with negative $s$ behaves
better
$$\vert S_1(-s,u)\vert < 2M(1.{3\over 2},-s)<{1\over s}+O(s^{-2}).\eqno(4.13)$$
This leads to an exponential decrease in (4.5).
To calculate the thermal wave integral (3.26) we introduce the second special function
$$S_2(s,u)=\int\limits_1^\infty{dx\over\sqrt{x^2-1}}\sin(s\sqrt{x^2-1})\sin ux.\eqno(4.14)$$
This is obtained from (3.26) with the substitution (4.2) again. With the new integration variable $y=\sqrt{x^2-1}$ we get a Fourier - sine integral
$$S_2(s,u)=\int\limits_0^\infty{dy\over\sqrt{y^2+1}}\sin(u\sqrt{y^2+1})\sin sy.\eqno(4.15)$$
The same integral with two cosine or one sine and one cosine function can be expressed by Bessel functions [6], but the integral (4.15) cannot. This might indicate that it is a new special function. As it stands the integral is not well suited for numerical integration. We get a better form by using the Euler substitution
$$\sqrt{y^2+1}=yx+1\eqno(4.16)$$
in
$$J_2^\pm=\int\limits_0^\infty{dy\over\sqrt{y^2+1}}\cos (u\sqrt{y^2+1}\pm sy).\eqno(4.17)$$
This follows from (4.15) by simple trigonometric formulas. With the substitution (4.16) we find
$$J_2^\pm(s,u)=2\int\limits_0^1 dx\,{1\over 1-x^2}\cos\B({ux^2\pm 2sx+u\over 1-x^2}\B)\eqno(4.18)$$
which can be easily calculated by numerical integration. For the special case $u=s$ we obtain
$$J_2^\pm(s,s)=2\int\limits_0^1{dx\over 1-x^2}\cos\B(s{1+x\over 1-x}\B)=$$
$$=\int\limits_1^\infty dz\,{\cos sz\over z}=-{\rm Ci}(s),\eqno(4.19)$$
which is the cosine-integral [5]
To obtain a bound for $J_2^\pm$ we use partial integration again:
$$J_2^\pm=\int\limits_0^\infty{dy\over uy\pm s\sqrt{y^2+1}}{d\over dy}\sin(u\sqrt{y^2+1\pm sy})=$$
$$=\mp{\sin u\over s}+\int\limits_0^\infty{\sin(u\sqrt{y^2+1}\pm sy)\over (uy\pm s\sqrt{y^2+1})^2}\B(u\pm{sy\over\sqrt{y^2+1}}\B)\, dy.\eqno(4.20)$$
This decreases as $1/s$ for fixed $u$ or $r$.
\section{Finite spherical geometry}
In the results of the previous sections the zero-mode $k=0$ (3.16) has played an important role. This mode only appears in the infinite system. To be physically relevant we must check whether the infinite system can be considered as a limit of a large finite system. For this purpose let us assume a large spherical boundary of radius $r_1$ which is kept at a constant temperature $T_{01}$, $r_0$ is the radius of the electrode as before. We require the two boundary conditions
$${\d T\over\d r}(t,r_0)=0,\quad T(t,r_1)=T_{01}.\eqno(5.1)$$
The steady state solution satisfying these conditions is now given by
$$T_1(r)=T_{01}-{b\over r_0r_1}+{b\over 2r_1^2}+{b\over r_0r}-{b\over 2r^2}.\eqno(5.2)$$
At the electrode we have the higher temperature
$$T_1(r_0)=T_{01}+{b\over 2}\B({1\over r_0}-{1\over r_1}\B)^2.$$
The remaining time dependent solution $T_2(t,r)$ is again factorized $=T_3(t)T_4(r)$ (2.10) where $T_4$ is the solution of
$$r^2T_4''+2rT_4'=-k^2r^2T_4\eqno(5.3)$$
with the boundary conditions
$${\d T_4\over\d r}(r_0)=0,\quad T_4(r_1)=0.\eqno(5.4)$$
We transform the equation (5.3) into a selfadjoint form with the substitution
$$y(r)=rT_4(r)\eqno(5.5)$$
yielding
$$-y''=k^2 y\eqno(5.6)$$
and the boundary conditions
$$y(r_1)=0,\quad r_0y'(r_0)-y(r_0)=0.\eqno(5.7)$$
This is a simple standard Sturm-Liouville eigenvalue problem [8] in the Hilbert space $L^2([r_0,r_1])$. It has a discrete spectrum of eigenvalues $k_n$ in contrast to the infinite problem in the previous sections. The number $n=0,1,2,\ldots$ gives the number of zeros of the eigenfunctions $y_n$.
The first boundary condition (5.7) is immediately satisfied by
$$y_n(r)=\sin k_n(r_1-r)\eqno(5.8)$$
and the second condition gives the transcendental equation
$$\tan k_n(r_1-r_0)=-k_nr_0.\eqno(5.9)$$
This equation can easily be discussed graphically. It seems as if $k=0$ were the lowest eigenvalue, but a glance to (5.8) shows that this is not the case because $y_n=0$. To get the eigenvalues analytically we put
$$k_n(r_1-r_0)=(2n+1){\pi\over 2}+\delta.\eqno(5.10)$$
Then (5.9) leads to
$$\tan k_n(r_1-r_0)=-\cot\delta=-{1\over\delta}+{\delta\over 3}+\ldots =-(2n+1){\pi\over 2}{r_0\over r_1-r_0}$$
which for large $n$ gives
$$\delta={2(r_1-r_0)\over (2n+1)\pi r_0}$$
so that
$$k_n={(2n+1)\pi\over 2(r_1-r_0)}+{2\over (2n+1)\pi r_0}+O\B({1\over n^2}\B).\eqno(5.11)$$
In the infinite volume limit $r_1\to\infty$ the first term goes to 0, but the second term does not. There remains a finite gap between $k=0$ and the lowest eigenvalue $k_0$. That means the zero-mode of the previous sections is exceptional and not physical. The reason is that the boundary condition (5.7) at $r_0$ is not fulfilled for all $t$ in the infinite problem.
For the selfadjoint eigenvalue problem we have expansion and completeness theorems [8]. The eigenfunctions $y_n$ for different $n$ are orthogonal and complete in $L^2([r_0,r_1])$. To normalize them we compute
$$\int\limits_{r_0}^{r_1}\sin^2k_n(r-r_1)dr={r_1-r_0\over 2}+O\B({1\over n^2}\B)$$
so that
$$\fii_n(r)=\B(\sqrt{{2\over r_1-r_0}}+O({1\over n^2})\B)\sin k_n(r_1-r)\eqno(5.12)$$
is a complete orthonormal system. The general solution of the finite ablation problem for the hyperbolic heat equation is now given by
$$T(t,r)=T_{01}-{b\over r_0r_1}+{b\over 2r_1^2}+{b\over r_0r}-{b\over 2r^2}+$$
$$+{1\over r}\sum_{n=0}^\infty\B[a_n\fii_n(r)e^{\omega_n^+t}+b_n\fii_n(r)e^{\omega_n^-t}\B].\eqno(5.13)$$
Here the two roots (3.14) appear again with $k=k_n$. For the parabolic equation we have only the terms with $\omega_n^+=-ak_n^2$. To satisfy the initial condition
$$T(0,r)=T_{01}\eqno(5.14)$$
we must expand the function
$${b\over 2r}-{b\over r_0}+r\B({b\over r_0r_1}-{b\over 2r_1^2}\B)=\sum_n(a_n+b_n)\fii_n(r)$$
into a Fourier series. To do so we need the $L^2$-scalar products of $\fii_n$ with the functions 1, $r$, $1/r$ which can easily be calculated. The Fourier coefficients $a_n+b_n$ are of the order $1/n$ which gives a slow convergence of the series. For large $t$ the exponential factors in (5.13) give a rapid convergence.
For the hyperbolic equation we again require the second initial condition
$${\d T(t,r)\over\d t}\B\vert_{t=0}=0\eqno(5.15)$$
which yields
$$b_n=-{\omega_n^+\over\omega_n^-}a_n.\eqno(5.16)$$
In the parabolic case this condition is violated. This seems to be in contradiction to experiments. The reason for condition (5.15) is the finite propagation speed of the heat which follows from the characteristics of the hyperbolic equation (3.8) [7] [9]. The characteristics are given by
$${dr\over dt}=\pm{1\over\sqrt{\eps}}=\pm\sqrt{{\kappa\over\ro c\tau}}.\eqno(5.17)$$
\section{Conclusions}
The approach to steady state for large times is determined by the exponential term $\exp(\omega t)$ with frequency $\omega$ closest to 0 in (5.13). Leaving aside unrealistically large $\tau$, this is given by
$$\omega_0^+=-ak_0^2\approx -a\B({\pi\over 2(r_1-r_0)}+{2\over\pi r_0}\B)$$
where $\tau$ has cancelled. Therefore, in contrast to the infinite geometry in sect.2-4, both equations show the same approach to steady state in the finite system.
For small times the solutions of the two equations differ considerably. One reason for this is the different initial condition (5.15). A second interesting difference has been observed by Lopez Molina et al.[4]. The temperature $T$ at fixed radius $r$ shows ``cuspidal-type'' singularities as a function of time $t$. These are discontinuities in the derivative $\d_t T(t,r)$. The origin of this phenomenon is the change of branch in the characteristic frequencies $\omega_n^\pm$ which is connected with the appearance of thermal waves. Indeed, according to (3.26) we must consider
$$F(t)={\rm Re}\B(e^{\sqrt{1-x^2}t}\B)=\cases{e^{\sqrt{1-x^2}t}&if $x<1$\cr \cos(\sqrt{x^2-1}t)&if $x>1.$\cr}$$
This function is continuous at $t=0$, but $\d_t F(t)$ makes a jump. The theory of characteristics [9] implies that such a discontinuity travels through the medium with the velocity (5.17).
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BkiUd_3xK0wg09lKDHOV
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\section{Introduction}
\label{sec:introduction}
A human can skim through a minute-long video in a few seconds, and still grasp its underlying story~\cite{szelag2004individual}.
This extreme efficiency in temporal information processing raises a question. Can a neural model achieve such efficiency in recognizing minutes-long activities in videos?
Related works propose different CNN models with efficiency in mind~\cite{howard2017mobilenets,zhang2018shufflenet,zoph2018learning}.
However, such models~\cite{kopuklu2019resource} address only short-range actions, as in Kinetics~\cite{kay2017kinetics}, UCF-101~\cite{soomro2012ucf101}, or HMDB~\cite{kuehne2011hmdb}.
On average, these actions take ten seconds or less, where recognizing only a few frames would suffice~\cite{schindler2008action}.
However, this paper focuses on long-range activities, as in Charades~\cite{sigurdsson2016hollywood}, Breakfast~\cite{kuehne2014language} or MultiThumos~\cite{yeung2018every}.
These activities can take up to a few minutes to unfold.
Current methods fully process the entire video of long-range activity to successfully recognize it~\cite{wang2018non,carreira2017quo}.
As a result, the major computational bottleneck of such methods is the sheer number of video frames to be processed.
\begin{figure}[!t]
\begin{center}
\includegraphics[trim=2mm 0mm 1mm 0mm,width=1.0\linewidth]{1-1-v1}
\end{center}
\vspace*{-5mm}
\caption{Top, short-range action ``disc throw" in an untrimmed video.
Based on each segment, you can tell whether is it relevant (green) to the action or not (red).
But in long-range activities, middle and bottom, the importance of each segment is conditioned on the video context.
The segment ``get food from cupboard" is relevant to ``cook food" but not to ``washing dishes".}
\label{fig:1-1}
\vspace*{-5mm}
\end{figure}
Another solution is frame sampling~\cite{yeung2016end}.
The recently proposed SCSampler~\cite{korbar2019scsampler} achieves efficiency by sampling the most salient segments from an untrimmed video of short-range action.
The sampling is conditioned on only the segment level, which is plausible for short-range actions in trimmed videos, such as Kinetics~\cite{kay2017kinetics} or untrimmed videos, such as Sports-1M~\cite{karpathy2014large}.
The reason is that, on the segment level, one can easily tell if the segment is relevant to the action or it is just background, see figure 1.
So, segment-level classification probabilities would suffice for sampling~\cite{korbar2019scsampler}.
In contrast, long-range activities are known for being diverse and complex~\cite{hussein2019timeception,ye2015eventnet}.
Thus, the importance of one segment to a certain activity is not self-described but rather depends on the context, \textit{i.e.} the long-range activity itself.
That is to say, while a segment is relevant to one activity, it is not relevant to another.
So, sampling conditioned only on the segment level is not the most optimal choice for long-range activities.
To address the limitations of the previous methods, we propose TimeGate, a two-stage neural network for the efficient recognition of long-range activities without compromising the performance.
Different from previous sampling methods, such as SCSampler, TimeGate solves two problems.
\textit{i.} Conditional selection: when selecting segments from the long-range activity, TimeGate is conditioned on both the segment- and context-level features.
Context-conditioning better suited for long-range activities than only the segment-conditioning of SCSampler.
\textit{ii.} Differentiable gating: the selection mechanism of TimeGate is differentiable, so it is trained end-to-end with modern 2D and 3D CNNs~\cite{carreira2017quo,he2016deep}, resulting in a better performance.
Our novelties are:
\textit{i.} Gating module for the conditional sampling of segments in videos.
Our gating is more suited to long-range activities than other methods, such as SCSampler.
The reason is that the sampling is conditioned on the segment- and context-level features.
\textit{ii.} The proposed gating module is differentiable, which enables end-to-end training with existing CNNs.
\textit{iii.} The proposed model, TimeGate, reduces the computational cost of existing CNNs in recognizing long-range activities.
In end-to-end training, the cost is reduced even further.
Finally, we conduct experiments and report the results on three datasets for long-range activity recognition: Charades~\cite{sigurdsson2016hollywood}, Breakfast~\cite{kuehne2014language} and MultiThumos~\cite{yeung2018every}.
\section{Related Work}
\label{sec:related_work}
\partitle{Long-range Activities.}
Short-range actions, such as Kinetics~\cite{kay2017kinetics} and UCF-101~\cite{soomro2012ucf101}, have an average length of 10 seconds or less.
Practically, they can be classified with CNNs using as little as ten frames per video~\cite{wang2016temporal}, and in some cases, even one frame would suffice~\cite{schindler2008action}.
Therefore, building efficient CNNs is a plausible choice to reduce the computational cost of recognizing short-range actions.
However, in long-range activities, such as Charades~\cite{sigurdsson2016hollywood} and Breakfast~\cite{kuehne2014language}, the activity can take up to five minutes to unfold.
Thus, requiring as many as a thousand frames~\cite{hussein2019timeception,hussein2019videograph,hussein2020pic,hussein2017unified} to be correctly classified.
As such, analyzing all the frames using efficient CNNs is still computationally expensive.
Nevertheless, having a mechanism to select the most relevant frames can boost efficiency~\cite{bhardwaj2019efficient}.
Therefore, this paper focuses on reducing the number of video frames needed for activity recognition.
Though, our work is orthogonal to prior work of efficient CNNs for action recognition.
\vspace*{5pt}
\partitle{Efficient Architectures.}
CNNs are the go-to solution when it comes to video classification.
Thus, one prospective of reducing the computation of video recognition is to build efficient CNNs.
Methods for pruning less important weights~\cite{hassibi1993optimal,han2015learning} or filters~\cite{li2016pruning} were previously proposed.
Careful design choices result in very efficient 2D CNNs such as MobileNet~\cite{howard2019searching} and ShuffleNet~\cite{zhang2018shufflenet}.
These 2D CNNs are extended to their 3D counterparts, such as ShuffleNet3D and MobileNet3D~\cite{kopuklu2019resource}, to learn spatio-temporal concepts for video classification.
Neural architecture search~\cite{zoph2016neural} is used to find the lightweight NasNet-Mobile~\cite{zoph2018learning}.
While efficient architectures are successful in the case of short-range actions, they are not the most viable solution for long-range activities.
The reason is that these activities span up to a few minutes.
Naively processing the video in its entirety undermines the computation saved by these efficient CNNs.
In other words, in the case of long-range activities, the computational bottleneck is the sheer number of video segments needed to be processed.
\vspace*{5pt}
\partitle{Conditional Computing.}
Another solution to reduce the computation is to dynamically route the computational graph of a neural network.
The assumption is that not all input signals require the same amount of computation -- some are complicated while others are seemingly easy.
Thanks to categorical reparameterization~\cite{jang2016categorical}, it becomes possible to discretize a continuous distribution, and effectively learn binary gating.
In~\cite{veit2018convolutional}, a dynamical graph is built by gating the layers of a typical CNN.
While in~\cite{chen2019you,bejnordi2019batch}, the gating is achieved on the level of convolutional channels.
In the same vein, GaterNet~\cite{chen2019you} proposes a separate gating network to learn binary gates for the backbone network.
Rather than gating the network layers, this paper focuses on gating the video frames themselves, to realize the efficiency in recognizing long-range activities.
In all cases, our paper benefits from prior work of differentiable gating~\cite{jang2016categorical}.
\vspace*{5pt}
\partitle{Sampling of Video Segments.}
Several works discuss frame sampling for short-range videos.
In~\cite{bhardwaj2019efficient}, a student-teacher model for trimmed video classification is presented.
With reinforcement learning in~\cite{yeung2016end}, an agent predicts the next move.
Most recently, SCSampler~\cite{korbar2019scsampler} proposes a method for sampling salient segments in the untrimmed videos of Sports-1M~\cite{karpathy2014large}.
Conditioned on only the segment, it predicts a score for how salient this segment is to the action.
Conversely, in long-range activities, the importance of each segment is conditioned on not only the segment but also its context.
Thus, SCSampler is less suited for such activities.
This paper presents TimeGate, a novel selection method tailored for these activities.
\begin{figure*}[!ht]
\begin{center}
\includegraphics[trim=0mm 5mm 0mm 5mm,width=0.8\linewidth]{3-1-v2}
\end{center}
\caption{Overview of the proposed model, TimeGate, with two stages.
The first stage is the timestep selector, left.
Based on a lightweight CNN, \textit{LightNet}, the model learns to select the most relevant timesteps for classifying the video.
This selection is conditioned on both the features of timestep and its context.
The second stage is the video classifier, right.
In which, only the selected timesteps (\textcolor{ForestGreen}{\makebox[0pt][l]{$\square$}\raisebox{.15ex}{\hspace{0.1em}$\boldsymbol{\checkmark}$}}) are considered, while the unselected timesteps (\textcolor{BrickRed}{\CrossedBox}) are completely ignored.
In this stage, a heavyweight CNN, \textit{HeavyNet} is used for feature representation of only the selected timesteps, followed by MLP for classification.}
\label{fig:3-1}
\end{figure*}
\section{Method}
\label{sec:method}
\subsection{TimeGate}
\partitle{Model Overview.}
TimeGate consists of two stages: timestep selector and video classifier, see figure~\ref{fig:3-1}.
The first stage is the selector, which consists of a lightweight CNN, \textit{LightNet}, followed by a novel gating module, see figure~\ref{fig:3-2}.
Its purpose is to select the most relevant timesteps from a minutes-long video.
The second stage is the classifier.
Its purpose is to learn deep and discriminatory video-level representations for maximum classification accuracy.
Thus, it resides on top of a heavyweight CNN, \textit{HeavyNet}, followed by a Multi-Layer Perceptron (MLP) for classification.
Only the timesteps chosen by the first stage, the timestep selector, are considered by the second stage, the video classifier.
\vspace*{5pt}
\partitle{Timestep Selector.}
The selector takes as an input a uniformly sampled $T$ frames from a long-range video $v = \{ f_i \; | \; i \in [1, ..., T] \}$.
All the frames are represented as convolutional features $ X = \{ x_i \; | \; i \in [1, ..., T] \}$, $ X \in \mathbb{R}^{ T {\mkern-2mu\times\mkern-2mu} C {\mkern-2mu\times\mkern-2mu} 1 {\mkern-2mu\times\mkern-2mu} 1}$, where $C$ is the number of channels.
The objective of the selector is to choose only a few of these features.
In other words, we want to select only the timesteps that are most representative of the activity in the video, where each timestep is represented as a feature $x_i$.
Our hypothesis is that, a lightweight feature representation using an efficient CNN, \textit{LightNet}, would suffice for the selection.
Thus, the features $X$ are obtained from the last convolutional layer of the LightNet, and average-pooled globally over space, so the spatial dimensions of $X$ are $1 {\mkern-2mu\times\mkern-2mu} 1$.
\vspace*{5pt}
\partitle{Concept Kernels.}
The next step is to take binary decision of considering or discarding the timesteps.
But how to decide if a timestep feature $x_i$ is relevant or not?
Conceptually speaking, a long-range activity consists of few yet dominant and discriminative visual evidences, based on which, the video can be recognized~\cite{hussein2019timeception}.
Take for example ``making pancake''.
One can easily discriminate it by observing the evidences ``pancake'', ``eggs'', ``pan'', and ``stove''.
These evidences can be thought of as \textit{latent} concepts.
To represent them, we learn a set of concept kernels $ K =\{k_1, k_2, ...k_N\}, K \in \mathbb{R} ^ {N \times C}$, where $N$ is the number of kernels, and $C$ is the kernel dimension.
$K$ are randomly initialized and are part of the network parameters.
They are learned during the training of the selector.
Our concept kernels $K$ are reminiscent of the centroids in ActionVlad~\cite{girdhar2017actionvlad}.
\vspace*{5pt}
\partitle{Gating Module.}
The purpose of the gating module is to select the video timesteps, see figure~\ref{fig:3-2}, top.
The first step is to measure how relevant each timestep feature $x_i$ is to all of the concept kernels $K$ using an inner product $\odot$.
The result is the similarity vector $s_i = K^{\top} \odot x_i$, $s_i \in \mathbb{R}^{N {\mkern-2mu\times\mkern-2mu} 1}$.
Our understanding is that the vector $s_i$ encodes how relevant a timestep is to each of $N$ concept kernels.
Then, based on this similarity vector $s_i$, we want to take a binary decision of considering or discarding the timestep feature $x_i$.
Therefore, we model the similarity vector $s_i$ using a two-layer MLP $f_\theta(\cdot)$.
The output layer of the MLP has a single neuron, denoted as $\alpha_i = f_{\theta}(s_i)$, $a_i \in \mathbb{R}^{1}$.
Intuitively, $\alpha_i$ is the gating decision corresponding to the timestep feature $x_i$.
Since $\alpha_i$ is a continuous variable, we cannot make a binary gating decision.
Thus, we make use of~\cite{jang2016categorical} to discretize $\alpha_i$ to binary gating variable $\hat{\alpha}_i$.
More formally, following the gating mechanism of~\cite{bejnordi2019batch}, we add gumbel noise $G$ to $\alpha_i$ and follow with \texttt{sigmoid} activation, thus $\hat{\alpha_i} = \texttt{sigmoid}(\alpha_i + G)$ .Then, we apply binary thresholding using the threshold value $\delta=0.5$.
So, we arrive at the binary gating value $\hat{\alpha}_i =\mathbb{I}_{(\delta > 0.5)} (\delta) $, $ \hat{\alpha}_i \in \{0, 1\}$, see figure~\ref{fig:3-2}, top.
Finally, for gating the $i$-th timestep, we multiply its feature $x_i$ with the binary value, resulting in the gated feature $ \hat{x_i} = x_i \cdot \hat{\alpha}_i$, $\hat{x_i} \in \mathbb{R}^{C {\mkern-2mu\times\mkern-2mu} 1 {\mkern-2mu\times\mkern-2mu} 1}$.
\begin{figure*}[t]
\minipage{0.24\textwidth}
\includegraphics[trim=3mm 8mm 3mm 5mm,width=\linewidth]{3-3-1}
\begin{center}
(a) ReLU
\end{center}
\endminipage\hfill
\minipage{0.24\textwidth}
\includegraphics[trim=3mm 8mm 3mm 5mm,width=\linewidth]{3-3-2}
\begin{center}
(b) Sigmoid
\end{center}
\endminipage\hfill
\minipage{0.24\textwidth}
\includegraphics[trim=3mm 8mm 3mm 5mm,width=\linewidth]{3-3-3}
\begin{center}
(c) Clipped sigmoid
\end{center}
\endminipage\hfill
\minipage{0.24\textwidth}
\includegraphics[trim=3mm 8mm 3mm 5mm,width=\linewidth]{3-3-4}
\begin{center}
(d) Step Function
\end{center}
\endminipage
\vspace*{-2mm}
\label{fig:3-3}
\caption{
In training, we use \texttt{gated-sigmoid} to activate the gating value $\alpha_i$ and to select the timesteps.
\texttt{gated-sigmoid} has some desirable properties.
\textit{i}. Unlike \texttt{ReLU}, having upper bound does not allow a timestep feature to dominate others.
\textit{ii}. Different from \texttt{sigmoid}, being clipped allows the network to discard insignificant timesteps, \textit{i.e.} those with gating values $\alpha_i < 0.5$.
In test, we replace the \texttt{gated-sigmoid} with \texttt{step function} for binary gating of timesteps.}
\vspace*{-5mm}
\end{figure*}
\vspace*{5pt}
\partitle{Gating Activation.}
A problem with using binary thresholding for gating, as in~\cite{bejnordi2019batch}, is that during training, the classifier does not know out of the gated timestep features, which is more relevant than the other.
Each $x_i$ is multiplied by a binary value $\hat{\alpha}_i \in \{0, 1\}$.
As a remedy, we propose \texttt{clipped-sigmoid} activation to replace the \texttt{sigmoid} activation used in~\cite{bejnordi2019batch}.
We find that this simply modified activation \texttt{clipped-sigmoid} is better suited for timestep gating due to three desirable properties, see figure 3.
\textit{i.} Being a relaxed version of the \texttt{step function} makes it differentiable.
\textit{ii.} Retaining the \texttt{sigmoid} value above the threshold means that the classifier gets the chance to know, out of the selected timesteps, which is relatively more important than the other.
\textit{iii.} Conversely to \texttt{ReLU}, the activation \texttt{clipped-sigmoid} is upper-bounded by one, thus preventing a single timestep feature $x_i$ from dominating the others by being multiplied by unbounded gating value $\hat{\alpha}_i$.
\begin{figure}[ht]
\begin{center}
\includegraphics[trim=0mm 5mm 0mm 2mm,width=0.7\linewidth]{3-2-v1}
\end{center}
\caption{Bottom, the timestep selector learns concept kernels $K$ to represent the most representative visual evidence.
Top, the gating module learns to select only a timestep feature $x_i$ according to its importance to the current video.}
\label{fig:3-2}
\end{figure}
\vspace*{5pt}
\partitle{Context-Conditional Gating.}
Up till now, the selector learns to gate each timestep regardless of its context, \textit{i.e.} the other timesteps in the video.
To achieve context-conditional gating, where both the timestep and its context affect the gating decision, we opt for a temporal modeling layer, self-attention~\cite{wang2018non}, before the gating module, See figure~\ref{fig:3-2}, bottom.
This layer learns to correlate each timestep $x_i$ with all the others in the video $\{ x_1, ..., x_T\}$ before gating.
\vspace*{5pt}
\partitle{Sparse Selection.}
The last component of the selector is to enforce sparsity on timestep selection, \textit{i.e.} choose as few timesteps as possible, yet retain the classification accuracy.
Simply put, the selector can cheat by predicting gating values just higher than the threshold $\alpha > \delta$, $\delta=0.5$, resulting in all gates opened and all timesteps selected.
The selector has a natural tendency to such a behaviour, as the only loss used so far is that of the classification.
And the more timesteps used by the classifier, the better the classification accuracy.
To prevent such a behaviour, we apply $L_0$ regularization~\cite{bejnordi2019batch,louizos2017learning} to all the gating values $\{ \hat{\alpha}_i \; | \; i \in [1, ..., T] \} $ to enforce sparsity on the selected timesteps.
We note that the sparsity regularization is necessary for a properly functioning gating mechanism.
\vspace*{5pt}
\partitle{Video Classifier.}
The assumption of TimeGate is that having efficiently selected the most crucial timesteps from the video using the LightNet and the selector, one can opt for a much more powerful HeavyNet to effectively classify the video.
Thus, the second stage of TimeGate is the video classifier, see figure~\ref{fig:3-1}, left.
This classifier takes as input only the subset $T'$ of timesteps chosen by the selector, $T^{\prime} \subset T, \; T^{\prime} \ll T$.
Each timestep is represented as the feature of last convolutional layer of the HeavyNet.
The video-level features are denoted as $Y = \{ y_i \; | \, i \in [1, ..., T^{\prime}] \}$, $Y \in \mathbb{R}^{T^{\prime} {\mkern-2mu\times\mkern-2mu} C^{\prime} {\mkern-2mu\times\mkern-2mu} H {\mkern-2mu\times\mkern-2mu} W}$, where $C^{\prime}$ is the number of channels, $T'$ is the number of selected timesteps, and $H, W$ are the spatial dimensions.
After the last convolutional layer, the video level features $Y$ are max-pooled over the spatial dimension and fed-forwarded to a two-layer MLP for classification. We follow~\cite{wang2018non} and max-pool the temporal dimension before the MLP \texttt{logits}.
\begin{figure*}[t]
\minipage{0.32\textwidth}
\includegraphics[trim=6mm 8mm 6mm 0mm,width=\linewidth]{4-1-1-v3}
\begin{center}
(a) I3D
\end{center}
\endminipage\hfill
\minipage{0.32\textwidth}
\includegraphics[trim=6mm 8mm 6mm 0mm,width=\linewidth]{4-1-2-v3}
\begin{center}
(b) ShuffleNet3D
\end{center}
\endminipage\hfill
\minipage{0.32\textwidth}
\includegraphics[trim=6mm 8mm 6mm 0mm,width=\linewidth]{4-1-3-v3}
\begin{center}
(c) ResNet2D
\end{center}
\endminipage
\vspace*{-2mm}
\caption{Our stand-alone timestep selector helps improving the performance and reduces the computation of off-the-shelf CNN classifiers -- be it 2D/3D heavyweight CNN or even lightweight 3D CNN.
More over, if TimeGate is trained end-to-end, the selector learns a better gating to the benefit of the classifier. So, the performance is improved even further.}
\label{fig:4-1}
\vspace*{-4mm}
\end{figure*}
\subsection{TimeGate Implementation}
\partitle{Backbone Choices.}
LightNet and HeavyNet are the backbone CNNs used by TimeGate.
Our choice for the LightNet is MobileNet-V3~\cite{sandler2018mobilenetv2}.
As for the HeavyNet, we explore three choices.
A powerful 3D CNN I3D~\cite{carreira2017quo}, an efficient 3D CNN ShuffleNet3D-V2~\cite{kopuklu2019resource}, and a powerful 2D CNN ResNet2D-50~\cite{he2016deep}.
Before training TimeGate on a specific dataset, the backbone CNNs are pre-trained on the dataset at hand.
We use the same training procedures specified by the authors of these CNNs.
\vspace*{5pt}
\partitle{Timestep Alignment.}
When the HeavyNet is a 3D CNN, the $i$-th timestep feature $y_i$ is obtained from processing the $i$-th video segment $s_i$ of $M$ successive frames $s_i = \{ f_{j}, ...., f_{j+M} \}$.
For I3D, $M=8$, and for ShuffleNet3D, $M=16$.
But since the LightNet of the selector is a 2D CNN, how can we align the timestep of the selector, with that of the classifier?
Simply put, for the aforementioned HeavyNet feature $y_i$, the aligned LightNet feature $x_i$ has to be obtained from the middle frame of the video segment $s_i$. More formally, the frame $f_{j + \lceil M/2 \rceil }$.
\vspace*{5pt}
\partitle{Model Training.}
TimeGate is trained with batch size 32 and for 100 epochs.
We use Adam with learning rate $1e\text{-}3$ and epsilon $1e\text{-}4$.
We use PyTorch and TensorFlow for our implementation.
As for the number of concept kernels $N$, we found that $N=128$ is a good choice for all the experiments, similar to~\cite{hussein2019videograph}.
As for the gating module, see figure~\ref{fig:3-2}, during the training phase, we use gumbel noise and \texttt{clipped-sigmoid} to get the activated gating value $\hat{\alpha}_i$.
In the test phase, we don't use gumbel noise, and we replace \texttt{clipped-sigmoid} with \texttt{step-function}, to get the binary gating value $\hat{\alpha}_i =\mathbb{I}_{(\delta > 0.5)} (\delta) $.
That means alpha is binarized $ \hat{\alpha}_i \in \{0, 1\}$ with thresholding $\delta = 0.5$.
\section{Experiments}
\label{sec:experiments}
\subsection{Datasets}
\label{subsec:4-1}
\partitle{Charades}
is a widely used benchmark for human action recognition.
It is a diverse dataset with 157 action classes in total.
The task is mult-label recognition, where each video is assigned to one or more action class.
It is divided into 8k, 1.2k and 2k videos for training, validation and test splits, respectively, covering 67 hours.
On average, each video spans 30 seconds, and is labeled with 6 and 9 actions for training and test splits, respectively.
Thus, Charades meets the criteria of long-range activities.
We use Mean Average Precision (mAP) for evaluation.
\vspace*{5pt}
\partitle{Breakfast}
is a benchmark for long-range activities, depicting cooking activities.
Overall, it contains 1712 videos, divided into 1357 and 335 for training and testing, respectively.
The task is video recognition into 10 classes of making different breakfasts.
Added to the video-level annotation, we are given temporal annotations of 48 unit-actions.
In our experiments, we only use the video-level annotation, and we do not use the temporal annotation of the unit-actions.
The videos are long-range, with the average length of 2.3 minutes per video.
Which makes it ideal for testing the efficiency of recognizing long-range activities.
The evaluation method is the classification accuracy.
\vspace*{5pt}
\partitle{MultiThumos}
is a benchmark for long-range videos, depicting sports activities.
It consists of 413 videos, divided into 200 and 213 for training and testing, respectively.
Each video has multi-labels, where the total number of action classes across the dataset is 65.
The average length is 3.5 minutes per video.
The original task of this dataset~\cite{yeung2018every} is the temporal segmentation of these short-range actions.
Recently, it is repurposed by~\cite{hussein2019timeception} into multi-label classification of long-range videos.
We adopt the same experimental setup of~\cite{hussein2019timeception}.
That is to say, each long-range video is classified into multi-labels, and the mAP is used for evaluation.
\vspace*{5pt}
\partitle{Ablation Studies.}
We use Breakfast as the primary dataset for the ablation experiments and studies.
These experiments highlight our contributions, as follows.
\textit{i.} In \textsection~\ref{subsec:4-3}, we discuss to what extend the end-to-end training of TimeGate helps.
\textit{ii}. In \textsection~\ref{subsec:4-4}, we show how context-conditional gating is more important than frame-conditional.
\textit{iii.} In \textsection~\ref{subsec:4-2},~\ref{subsec:4-5}, we demonstrate the improvements of TimeGate over the current CNN classifiers, in terms of accuracy and efficiency.
\subsection{Stand-alone Timestep Selector}
\label{subsec:4-2}
One might raise an important question
-- will a timestep selector based on LightNet features $X$ benefit a classifier based on HeavyNet features $Y$, given the differences between the feature spaces of LightNet and HeavyNet $C \neq C^{\prime} $?
To answer this question, we construct an experiment of two steps on Breakfast.
The first step is training a stand-alone selector, where we choose MobileNet for both LightNet and HeavyNet.
During training, we randomly sample $T=32$ timesteps from each video.
Since MobileNet is a 2D CNN, a timestep here is practically a video frame.
With the help of the $L_0$ regularization, the selector achieves sparse selection of timesteps, by as little as $T^{\prime}=16$ without degrading the classification performance.
The second step is testing how will the selector benefit off-the-shelf CNN classifiers: I3D, ShuffleNet3D and ResNet2D.
Then, we measure their performance using different time scales.
More formally, from each test video, we sample $T^{\prime}$ timesteps , $T^{\prime} \in \{1, 2, 4, 8, 16\}$, and we use different sampling methods: random, uniform, and timestep selector.
During testing, the output of the timestep selector is a per-timestep binary value $\hat{\alpha}_i \in \{0, 1\}$ of whether to consider or discard the $i$-th timestep.
So, if $T$ timesteps are processed by the selector, it is able to choose a subset $T'$ timesteps and discard the others, where $T' \subset T, T' \ll T$.
And to evaluate the benefit of the selector, the off-the-self classifier then uses only $T'$.
\begin{table}[!ht]
\centering
\renewcommand{\arraystretch}{1.0}
\setlength\tabcolsep{3.4pt}
\begin{tabular}{lccccccc}
\specialrule{0.3mm}{.0em}{.3em}
\multirow{2}{*}{Baseline} & \multicolumn{7}{c}{Accuracy (\%) @ Timesteps} \\
\cmidrule(lr){2-8}
& 4 & 8 & 16 & 32 & 64 & 128 & 256 \\
\midrule
R2D & 61.0 & 67.1 & 67.3 & 71.0 & 72.9 & 74.3 & 73.8 \\
R2D+\textbf{TG} & 63.9 & 68.2 & 70.2 & 73.3 & 74.3 & \textbf{76.4} & 74.3 \\
\midrule
S3D & 46.3 & 60.8 & 63.4 & 67.2 & 67.3 & 65.8 & 66.3 \\
S3D+\textbf{TG} & 54.4 & 65.1 & 66.2 & \textbf{69.8} & 69.7 & 66.7 & 67.8 \\
\midrule
I3D & 66.8 & 74.3 & 82.8 & 84.7 & 85.7 & 86.5 & 85.4 \\
I3D+SCS~\cite{korbar2019scsampler} & 61.4 & 74.7 & 81.8 & 84.4 & 84.4 & 85.4 & 84.6 \\
I3D+\textbf{TG} & 69.5 & 77.9 & 85.2 & 85.9 & 86.7 & \textbf{88.1} & 86.5 \\
\specialrule{0.3mm}{.0em}{.0em}
\end{tabular}
\caption{The stand-alone selector of our model TimeGate (\textbf{TG}) benefits off-the-shelf CNN classifiers.
The benefit is consistent for various classifiers: I3D, ShuffleNet3D (S3D), and ResNet2D (R2D).}
\label{tbl:4-1}
\vspace*{-5pt}
\end{table}
As reported in table~\ref{tbl:4-1}, and shown in figure~\ref{fig:4-1}, we observe that the stand-alone selector improves the accuracy of off-the-shelf classifiers.
The reason is that the selector, based on LightNet, is able to select the most relevant timesteps from the video.
Also, we notice that the improvements are consistent for three different classifiers: I3D, ResNet2D and ShuffleNet3D.
\subsection{End-to-End TimeGate}
\label{subsec:4-3}
Having experimented with the stand-alone selector, we pose another question.
Is it possible to train TimeGate end-to-end, given that the selector and the classifier are based on two different CNNs, with two different feature spaces, $C \neq C^{\prime}$?.
Our experiments show that indeed, in end-to-end training, the gating module learns a better selection to the benefit of the classifier.
The outcome is improvement in performance over the stand-alone selection, as reported in table~\ref{tbl:4-4}.
We conclude that in end-to-end, the gating module learns to determine the importance of the $i$-th heavyweight feature $y_i$ based on the corresponding lightweight feature $x_i$.
\begin{table}[!ht]
\centering
\renewcommand{\arraystretch}{1.0}
\setlength\tabcolsep{3.0pt}
\begin{tabular}{lccccccc}
\specialrule{0.3mm}{.0em}{.3em}
\multirow{2}{*}{Baseline} & \multicolumn{7}{c}{Accuracy (\%) @ Timesteps} \\
\cmidrule(lr){2-8}
& 4 & 8 & 16 & 32 & 64 & 128 & 256 \\
\midrule
SCSampler~\cite{korbar2019scsampler} & 61.4 & 74.7 & 81.8 & 84.4 & 84.4 & 85.4 & 84.6 \\
TimeGate \textbf{SA} & 69.5 & 77.9 & 85.2 & 85.9 & 86.7 & 88.1 & 86.5 \\
TimeGate \textbf{ETE} & 74.4 & 78.1 & 82.9 & 86.7 & 87.4 & \textbf{89.3} & 86.1 \\
\specialrule{0.3mm}{.0em}{.0em}
\end{tabular}
\caption{Our stand-alone (\textbf{SA}) selector benefits off-the-shelf CNN classifiers.
End-to-end (\textbf{ETE}) training is even better.}
\label{tbl:4-4}
\end{table}
In addition, figure~\ref{fig:4-5} shows the average ratio of selected timesteps for each activity class of Breakfast dataset.
The ratios of the stand-alone (red) is changed when it is trained end-to-end with different HeavyNet: ResNet2D, (blue), I3D (yellow), and ShuffleNet3D (blue).
We observe that these ratios have similar trends when the HeavyNet is 3D CNN, regardless of which 3D CNN is used.
Between yellow and blue, there is a similar trend in 8 out of 10 activities.
However, these ratios tend to vary between 2D and 3D as HeavyNet -- only 3 out of 10 actions tend to have similar trends, see green and yellow.
From this experiment, we conclude that the gating module, depending on LightNet features, learns to select better timesteps to the benefit of the HeavyNet classifier.
\begin{figure}[!ht]
\begin{center}
\includegraphics[trim=5mm 18mm 5mm 2mm,width=1.0\linewidth]{4-5}
\end{center}
\caption{
The ratios of selected timesteps for the activity classes of Breakfast.
Note the change in these ratios from stand-alone selector (red) to end-to-end training with the HeavyNets: ResNet2D (green) I3D (yellow) and ShuffleNet3D (blue).}
\label{fig:4-5}
\end{figure}
\begin{table*}[!ht]
\centering
\renewcommand{\arraystretch}{1.0}
\setlength\tabcolsep{8.0pt}
\begin{tabular}{lcccccc}
\specialrule{0.3mm}{.0em}{.3em}
\multirow{2}{*}{} & \multicolumn{2}{c}{Timesteps} & \multicolumn{2}{c}{FLOPS (G) } & \multirow{2}{*}{Total FLOPS $\downarrow$} & \multirow{2}{*}{Acc. (\%) $\uparrow$} \\
\cmidrule(lr){2-3}
\cmidrule(lr){4-5}
& LightNet & HeavyNet & LightNet+Gating & HeavyNet & & \\
\midrule
R2D & --- & 64 & --- & 246.6 & 246.6 & 72.9 \\
S3D+SCSampler~\cite{korbar2019scsampler} & 128 & 16 & 7.5 & 61.7 & 69.2 & 68.6 \\
R2D+TimeGate & 128 & 16 & 7.8 & 61.7 & 69.5 & 70.2 \\
\midrule
S3D & --- & 64 & --- & 61.8 & 61.8 & 67.3 \\
S3D+SCSampler~\cite{korbar2019scsampler} & 128 & 16 & 7.5 & 17.3 & 24.8 & 64.1 \\
S3D+TimeGate & 128 & 16 & 7.8 & 17.3 & 25.1 & 66.2 \\
\midrule
I3D & --- & 64 & --- & 830.7 & 830.7 & 85.7 \\
I3D+SCSampler~\cite{korbar2019scsampler} & 128 & 16 & 7.5 & 207.8 & 215.3 & 81.8 \\
I3D+TimeGate & 128 & 16 & 7.8 & 207.8 & 215.6 & 85.2 \\
\specialrule{0.3mm}{.0em}{.0em}
\end{tabular}
\caption{Breakdown of the computational cost of TimeGate \textit{v.s.} SCSampler.
Three choices of HeavyNet: ResNet2D (R2D), ShuffleNet3D (S3D) and I3D.
The computational cost of LightNet and the gating module is marginal compared to that of the HeavyNet.
TimeGate reduces the cost by almost half.
Our selector improves over SCSampler.}
\label{tbl:4-3}
\end{table*}
\subsection{Context-Conditional Gating}
\label{subsec:4-4}
When selecting the timesteps of long-range activities, TimeGate is conditioned on both the segment and its context.
This context-conditioning is an important novelty of TimeGate.
Also, this property is desired for long-range activities, because the importance of a certain segment is not always self-described, but rather depends on the context.
\begin{table}[!ht]
\centering
\renewcommand{\arraystretch}{1.0}
\setlength\tabcolsep{3.0pt}
\begin{tabular}{lccccccc}
\specialrule{0.3mm}{.0em}{.3em}
\multirow{2}{*}{Baseline} & \multicolumn{7}{c}{Accuracy (\%) @ Timesteps} \\
\cmidrule(lr){2-8}
& 4 & 8 & 16 & 32 & 64 & 128 & 256 \\
\midrule
SCSampler~\cite{korbar2019scsampler} & 61.4 & 74.7 & 81.8 & 84.4 & 84.4 & 85.4 & 84.6 \\
TG Frame & 69.2 & 73.8 & 80.7 & 81.5 & 83.9 & 83.1 & 83.6 \\
TG Context & 69.5 & 77.9 & 85.2 & 85.9 & 86.7 & \textbf{88.1} & 86.5 \\
\specialrule{0.3mm}{.0em}{.0em}
\end{tabular}
\caption{TimeGate (TG) is better when the gating module is conditioned on both the frame-level and the context-level.
More over, TimeGate outperforms SCSampler in long-range activities.}
\label{tbl:4-2}
\end{table}
To validate this assumption, we design the following experiment.
We devise a baseline model of our timestep selector, that does not have a temporal layer before the gating module.
Thus, in this baseline, the gating is frame-conditioned.
Also, we include SCSampler~\cite{korbar2019scsampler} in this comparison.
We use I3D for the HeavyNet and we use MobileNet as the backbone CNN for both our timestep selector and SCSampler.
As reported in table~\ref{tbl:4-2}, we observe a drop in the performance when using the frame-conditioned TimeGate.
The reason is that, for long-range activities, its important for the selector to pay attention to the context of the video segment before sampling it.
\begin{figure}[!ht]
\begin{center}
\includegraphics[trim=3mm 13mm 5mm 5mm,width=1.0\linewidth]{4-6-v3}
\end{center}
\caption{In both the frame-conditioned TimeGate and SCSampler, the ratio of the selected timesteps have small variance across the activity classes of Breakfast.
In contrast, in context-conditioned TimeGate, the ratio is highly dependent on the activity, which means context-conditional gating is archived.}
\label{fig:4-6}
\end{figure}
We report another analysis in figure~\ref{fig:4-6}.
On the left, we show the ratio of selected timesteps for each activity class of Breakfast.
The frame-conditioned gating (dark blue) tends to select similar ratios regardless of the category, so does the SCSampler (light blue).
In contrast, we see more diverse ratios for the context-conditioned gating.
Figure~\ref{fig:4-6}, right, shows the ratio variances.
The much higher variance for context-conditional TimeGate means that it is more dependent on the activity class than the case of SCSampler or frame-conditional TimeGate.
\subsection{Computation-Performance Tradeoff}
\label{subsec:4-5}
When it comes to the recognition of long-range activities, the golden rule is the more timesteps the better the accuracy, and the heavier the computation.
But given the huge redundancies of the visual evidences in these timesteps, there is a tradeoff between accuracy and computation.
In this experiment, we explore what is the effect of such a tradeoff on TimeGate, and we compare against SCSampler.
Figure~\ref{fig:4-3} shows this tradeoff using I3D as the video classifier.
We notice that both TimeGate and SCSampler can dramatically reduce the cost of I3D.
However, TimeGate outperforms SCSampler.
\begin{figure}[!ht]
\begin{center}
\includegraphics[trim=10mm 12mm -10mm 5mm,width=0.8\linewidth]{4-3-v2}
\end{center}
\caption{TimeGate (\textbf{TG}) is better than SCSampler (\textbf{SCS}) in reducing computational cost of I3D.}
\label{fig:4-3}
\end{figure}
In table~\ref{tbl:4-3}, we report the exact computational budget of TimeGate \textit{v.s.} SCSampler and I3D.
We notice that with, carefully selected 16 timesteps out of 128, TimeGate is able to match the performance of off-the-shelf CNNs which use 64 uniformly sampled timesteps.
Also, we notice the computational cost of selecting these timesteps is marginal to the cost of the CNN classifier itself.
For example, to select 8 out of 128 Timesteps, TimeGate spends 7.5 G-FLOPS, while to classify only one timestep using I3D, 3.9 G-FLOPS are needed.
\subsection{Experiments on Charades}
\label{subsec:4-6}
In this experiment, we test how TimeGate would fair against off-the-shelf CNN for recognizing the multi-label action videos of Charades.
This dataset is different from Breakfast in two ways.
First, the videos are mid-range with average length of 0.5 minutes, compared to 2 minutes of Breakfast.
Second, it is multi-label classification, but breakfast is single-label classification.
So, it is more challenging to select unrelated timesteps from the videos of Charades than Breakfast.
Most of the timesteps are already relevant to recognizing the mid-range videos of Charades.
Still, TimeGate outperforms I3D at different time scales, see figure~\ref{fig:4-4} and table~\ref{tbl:4-5}.
\begin{table}[!ht]
\centering
\renewcommand{\arraystretch}{1.0}
\setlength\tabcolsep{2.8pt}
\begin{tabular}{lccccccc}
\specialrule{0.3mm}{.0em}{.3em}
\multirow{2}{*}{Baseline} & \multicolumn{7}{c}{mAP (\%) @ Timesteps} \\
\cmidrule(lr){2-8}
& 4 & 8 & 16 & 32 & 64 & 128 & 256 \\
\midrule
I3D & 20.4 & 22.3 & 26.8 & 28.3 & 30.1 & 30.9 & 31.5 \\
I3D + TimeGate & 21.6 & 24.7 & 27.9 & 29.7 & 30.8 & 32.4 & \textbf{33.1} \\
\specialrule{0.3mm}{.0em}{.0em}
\end{tabular}
\caption{TimeGate improves the performance of the backbone CNNs (\textit{i.e.} I3D) on the challenging task of multi-label classification of Charades.}
\label{tbl:4-5}
\end{table}
Worth mentioning that TimeGate consistently improves the efficiency of HeavyNet CNNs other than I3D.
For example, if TimeGate uses 3D-ResNet-101~\cite{wang2018non} as the HeavyNet, we achieve 36.2\% using 256 timesteps compared to 35.5\% achieved by~\cite{wang2018non} using dense sampling of 1024 timesteps.
In other words, TimeGate retains the performance of 3D-ResNet-101 using only 25\% of the computation.
The reason is that, when TimeGate selects the most relevant segments from each video, it improves the signal-to-noise ratio.
In analogy, ~\cite{korbar2019scsampler} concluded that when the CNN video classifier considers the unrelated video segments, the accuracy degrades.
\begin{figure}[!ht]
\begin{center}
\includegraphics[trim=5mm 10mm -5mm 5mm,width=0.9\linewidth]{4-4-v2}
\end{center}
\caption{TimeGate improves the performance of the off-the-shelf I3D for recognizing the actions of Charades.}
\label{fig:4-4}
\end{figure}
\begin{figure*}[!ht]
\begin{center}
\includegraphics[trim=0mm 2mm 0mm 5mm,width=1.0\linewidth]{4-8-v3.pdf}
\end{center}
\caption{Top, frames corresponding to the selected timesteps by TimeGate. Bottom, are those discarded by TimeGate.
The shown figures are for three activities: ``making sandwich", ``preparing coffee", and ``making pancake".
The general observation is that TimeGate tends to discard the segments with little discriminative visual evidences.}
\label{fig:4-9}
\end{figure*}
\subsection{Experiments on MultiThumos}
\label{subsec:4-7}
Our final experiment is to use TimeGate in classifying the long-range activities of MultiThumos.
This dataset is particularly challenging, as each video is multi-labeled.
Nevertheless, we observe that TimeGate is able to retain the performance of the HeavyNet (I3D) with much reduced computation, see table~\ref{tbl:4-9}.
In addition, it outperforms SCSampler in reducing the computational cost.
Worth mentioning that TimeGate achieves 75.11\% mAP using 256 timesteps compared to 74.79\% mAP achieved by~\cite{hussein2019timeception} using dense-sampling of 1024 timesteps.
In other words, TimeGate retains the performance of~\cite{hussein2019timeception} with almost 25\% of the computational cost.
\begin{table}[!ht]
\centering
\renewcommand{\arraystretch}{1.0}
\setlength\tabcolsep{2.8pt}
\scalebox{0.85}{
\begin{tabular}{lccccccc}
\specialrule{0.3mm}{.0em}{.3em}
\multirow{2}{*}{Baseline} & \multicolumn{7}{c}{mAP (\%) @ Timesteps} \\
\cmidrule(lr){2-8}
& 4 & 8 & 16 & 32 & 64 & 128 & 256 \\
\midrule
I3D & 41.85 & 45.02 & 52.75 & 58.41 & 64.74 & 67.19 & 69.32 \\
I3D + SCSampler & 43.51 & 47.68 & 54.14 & 60.87 & 67.23 & 69.83 & 72.46 \\
I3D + TimeGate & 45.38 & 50.02 & 57.63 & 63.34 & 69.07 & 73.20 & \textbf{75.11} \\
\specialrule{0.3mm}{.0em}{.0em}
\end{tabular}}
\vspace*{5pt}
\caption{TimeGate improves the performance of I3D when classifying the long-range activities of MultiThumos.
Also, it outperforms SCSampler.}
\label{tbl:4-9}
\end{table}
\subsection{Qualitative Results}
\label{subsec:4-8}
\partitle{Examples of Gated Timesteps.}
In figure~\ref{fig:4-9}, we show a few visual examples of the timesteps selected, top, and discarded, bottom, by the gating module.
We consider three activities: ``making sandwich", ``preparing coffee", and ``making pancake".
The general observation is that TimeGate tends to discard the segments with little discriminative visual evidences.
\vspace*{5pt}
\partitle{Distribution of Gating Values.}
One might ask the question, how evenly distributed are the timesteps selected by TimeGate?
To answer this question, we uniformly sample $T=128$ timesteps from each test video.
Then, we predict the gating value $\alpha_i$ for each timestep.
After that, for all the videos of the same activity class, we average their gating values.
Next, we normalize these values between zero and one, and visualize them in figure~\ref{fig:4-7}.
Our observation is that, some activities are simple and usually happen in the middle of the video, such as ``preparing tea", or ``making coffee".
Others are complex and occupy the entire video, such as ``fried egg" or ``making sandwich".
\begin{figure}[!ht]
\begin{center}
\includegraphics[trim=2mm 14mm 2mm 4mm,width=1.0\linewidth]{4-7-v2}
\end{center}
\caption{Distribution of the gating values across time for each activity of Breakfast.
In simple activities, such as ``making coffee", most of the selected segments happen in the middle of the video.
This means the middle of the video is much more relevant than the other parts.
While in complex activities, such as ``making sandwich", the selected segments tend to distribute across the entire video.
This means that almost the entire video contains relevant and important segments.}
\label{fig:4-7}
\end{figure}
\section{Conclusion}
\label{sec:conclusions}
In this paper, we proposed TimeGate, a neural model for efficient recognition of long-range activities in videos.
Our approach for realizing the efficiency is sampling the most relevant segments from the activity video.
We highlighted the differences between sampling for short-range actions \textit{v.s.} long-range activities.
We also stated the limitations of existing works, such as SCSampler.
TimeGate overcomes these limitations using three contributions.
First, a differentiable gating module for timestep selection.
Second, the selection that is conditioned on both the timestep and its context.
Third, TimeGate, an end-to-end neural model to retain the performance of existing CNN classifiers at a fraction of the computational budget.
We experimented on three benchmarks and compared against related works.
TimeGate consistently outperforms competing methods on all three benchmarks and reduces the computation of I3D by 50\% while maintaining the classification accuracy. On MultiThumos, TimeGate sets a new state-of-the-art mAP of 75.11\% compared to 74.79\% mAP of Timeception~\cite{hussein2019timeception} while consuming only 25\% of the computation cost.
Our empirical evaluations and results demonstrate the efficiency of TimeGate in recognizing long-range activities.
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\section{Introduction}
The oriented chromatic number of an oriented graph is a natural generalization of the usual chromatic number. The chromatic number of an undirected graph $G=(V,E)$, denoted $\chi(G)$, is the smallest $k$ for which there is a function $c:V \to [k]$ such that for every $uv \in E,\ c(u) \ne c(v)$. The chromatic number of $G$ can equivalently be defined as the smallest $k$ for which there is a graph homomorphism from $G$ to a complete graph $K_k$. The oriented chromatic number of an oriented graph $\vec{G}$, denoted ${\chi_o}(\vec{G})$, is the smallest $k$ for which there is a homomorphism (of directed graphs) from $\vec{G}$ to some tournament of order $k$. An equivalent formulation is that ${\chi_o}(\vec{G})$ is the least $k$ for which there is a partition of $V$ into $k$ sets, called colour classes, so that each colour class contains no arcs and so that between every pair of colour classes, all arcs have the same direction. The notion of such colourings of oriented graphs was first introduced by Courcelle~\cite{bC94} in a series of papers studying graph properties described in monadic second order logic.
While ${\chi_o}(\vec{G})$ is always bounded from below by the usual chromatic number of the underlying graph, these parameters can, in general, be far. It was noted by Kostochka, Sopena, and Zhu~\cite{KSZ97} that there is an orientation of the complete bipartite graph $K_{n, n}$ with oriented chromatic number $2n$. Oriented graphs, $\vec{G}$, that have ${\chi_o}(\vec{G})$ equal to the number of vertices in $\vec{G}$ are sometimes called \emph{oriented cliques} and are precisely the oriented graphs with diameter at most $2$. The problem of determining the minimum number of arcs in a directed graph with diameter at most $2$ was proposed by Erd\H{o}s, S\'{o}s, and R\'{e}nyi~\cite{ERS66} (see also~\cite{sZ70, DM87}) and F\"{u}redi, Horak, Pareek, and Zhu~\cite{FHPZ98} showed that this minimum is $(1+o(1))n \log n$ (see also~\cite{KLSS99}). The maximum order of outerplanar and planar oriented cliques was studied by Klostermeyer and MacGillivray~\cite{kMac04b} and Sen~\cite{sS12}.
Another closely-related concept is the \emph{acyclic chromatic number} of an oriented graph, denoted $\chi_a(\vec{G})$, which is the smallest number of colours needs to properly colour the vertices of an oriented graph so that every pair of colour classes induce an acyclic graph. Raspaud and Sopena~\cite{RS94} showed that ${\chi_o}(\vec{G})$ can be bounded in terms of $\chi_a(\vec{G})$ and Kostochka, Sopena and Zhu~\cite{KSZ97} gave bounds on $\chi_a(\vec{G})$ in terms of ${\chi_o}(\vec{G})$.
There has been significant prior investigation into ${\chi_o}(\vec{G})$ focusing on extremal results for graphs with bounded degree properties. In a recent series of papers~\cite{Du20, DMS19}, Duffy, with others, prove that the oriented chromatic number of connected cubic graphs is at most eight. Oriented graphs with maximum degree $4$ have an oriented chromatic number at most $69$ (see~\cite{DMS19}) although it is believed that this upper bound can be improved. In~\cite{KSZ97}, Kostochka, Sopena, and Zhu consider deterministic bounds on ${\chi_o}(\vec{G})$ in terms of its maximum degree. They show that for any digraph $\vec{G}$ with maximum degree $d$, ${\chi_o}(\vec{G}) \le 2d^22^d$. They also prove that when $n$ is sufficiently large, for every $d$-regular digraph $G$ on $n$ vertices, there is an orientation $\vec{G}$ with ${\chi_o}(\vec{G}) \ge \lceil 2^{d/2} \rceil$.
In this paper, we consider the question of bounding the oriented chromatic number of a random directed graph, extending results on the chromatic number of undirected random graphs. The random graph $\mathcal{G}(n,p)$ was introduced in 1960 by Erd\H{o}s and R\'enyi. In this model, each possible edge of a graph on $n$ vertices is chosen independently with probability $p$. This model marks the beginning of the field of random graph theory, the study of random graph models and their properties. We direct the reader to the books~\cite{AS16, Bo01, JLR00} for results on random graphs and their connections to other fields. In addition to $\mathcal{G}(n,p)$, we consider the model $\mathcal{G}(n,d)$ in which a $d$-regular graph on $n$ vertices is selected uniformly at random.
The two models of random directed graphs that we consider in this paper are $\vec{\mathcal{G}}(n,p)$ and $\vec{\mathcal{G}}(n,d)$. In each, after selecting an appropriate undirected graph according to $\mathcal{G}(n, p)$ or $\mathcal{G}(n, d)$, respectively, each edge is given an orientation uniformly at random. In particular, $\vec{\mathcal{G}}(n,p)$ and $\vec{\mathcal{G}}(n,d)$ contain no directed loops or two-cycles as they are orientations of simple graphs.
The question of the chromatic number of a random graph was mentioned by Erd\H{o}s and R\'enyi~\cite{ER60} and investigated as early as 1974 by Grimmet and McDiarmid~\cite{GM75}, but the topic has received considerable attention since 1987 following the breakthrough result of Shamir and Spencer~\cite{SS87} in which martingales were used to show $\chi(\mathcal{G}(n,p))$ is concentrated in a small set of values with probability tending to $1$ as $n$ tends to infinity (which we refer to as \emph{asymptotically almost surely} or a.a.s.). Bollob\'as~\cite{Bo88} and later \L uczak~\cite{tL91a} prove that for $pn \to \infty$, a.a.s. $\chi(\mathcal{G}(n,p))$ is $(\frac{1}{2}+o(1))\log(1/(1-p))\frac{n}{\log n}$. \L uczak also proved~\cite{tL91b} that when $p = d/n$, so that on average each vertex has degree $d$, there is a value $k_d$ such that the chromatic number of $G(n,p)$ is a.a.s. $k_d$ or $k_d+1$. In 2005, Achlioptas and Naor~\cite{AN05} determined $k_d$ to be the smallest integer such that $d < 2k_d\log k_d$. Building on that work, Achloptas and Moore~\cite{AM04} showed that the $\chi(\mathcal{G}(n,d))$ is a.a.s. $k_d, k_d+1$, or $k_d+2$ for the same $k_d$. Kemkes et al.~\cite{KPW10} improved that result to $k_d$ or $k_d+1$, and Coja-Oghlan et al.~\cite{CEH13} proved that the chromatic number is concentrated at $k_d$ for sufficiently large $d$.
Directed random graphs have received significantly less attention than undirected models. The aforementioned Kostochka, Sopena, and Zhu bound, ${\chi_o}(\vec{G}) \le 2d^22^d$ for digraphs with maximum degree $d$, also serves as an upper bound for ${\chi_o}(\vec{\mathcal{G}}(n,d))$. Chebolu and Frieze considered Hamilton cycles in random lifts of directed graphs~\cite{CF08}. Bensmail, Duffy, and Sen~\cite{BDS17} noted that a.a.s. ${\chi_o}(\vec{\mathcal{G}}(n,p=\frac{1}{2})) = n$. (We expand on this observation in Proposition~\ref{prop:gnp_large_p}.) For $d=1$ or $2$, $\vec{\mathcal{G}}(n,d)$ can be readily calculated exactly, and we do so in Section~\ref{ssec:vsparse_vdense}. Unlike in any known undirected case, ${\chi_o}(\vec{\mathcal{G}}(n,2))$ is not concentrated in a single value; instead, a.a.s. ${\chi_o}(\vec{\mathcal{G}}(n,2)) \in \{4,5\}$ and each value occurs with positive probability. We believe this difference suggests the oriented chromatic number of directed graphs deserves further study.
\textbf{Our contribution.} We extend the literature on undirected graph models to the directed case by analyzing the chromatic number of the random graph models $\vec{\mathcal{G}}(n,p=\frac{d}{n})$ and $\vec{\mathcal{G}}(n,d)$.
A `doubly regular' tournament $T$ is a tournament in which every vertex has
out-degree $\frac{n-1}{2}$ and for every pair of vertices $v,w$ there are
$\frac{n-3}{4}$ vertices $u$ such that $v \to u$ and $w \to u$.
Now set
\begin{equation}\label{eq:uk}
u_k = \frac{2\log k}{\log k - \log(\frac{k-1}{2})} < \frac{2}{\log 2}\log k
\end{equation}
and
\begin{equation}\label{eq:lk}
\ell_k = \frac{2(k-1)^3}{k(k+1)(k-2)}\log(k-1) \ge 2\left(1-\frac{2}{k+1}\right)\log(k-1).
\end{equation}
The main results of this paper are
\begin{theorem}\label{thm:gnp_main}
Let $d > 0$ be a real number and let $k_1, k_2 \in \mathbb{Z}$ such that there exists a doubly-regular tournament of order $k_2$ and $d \in (u_{k_1-1}, \ell_{k_2}]$. Then a.a.s. ${\chi_o}(\vec{\mathcal{G}}(n,p=\frac{d}{n})) \in [k_1,3k_2+11]$.
\end{theorem}
and
\begin{theorem}\label{thm:gnd_main}
Let $d\ge 2$ be an integer and let $k_1, k_2 \in \mathbb{Z}$ such that there exists a doubly-regular tournament of order $k_2$ and $d \in (u_{k_1-1}, \ell_{k_2}]$. Then a.a.s. ${\chi_o}(\vec{\mathcal{G}}(n,d)) \in [k_1,3k_2+11]$.
\end{theorem}
We also use Theorems~\ref{thm:gnp_main} and \ref{thm:gnd_main} to prove the following corollary:
\begin{corollary}\label{cor:main_cor}
Let $d > 1$ be given. Then a.a.s.
\[ {\chi_o}(\vec{\mathcal{G}}(n,p=\tfrac{d}{n})) \in (2^{d/2}, 6e^{d/2}+6d+17]. \]
Furthermore, if $d$ is an integer, a.a.s.
\[ {\chi_o}(\vec{\mathcal{G}}(n,d)) \in (2^{d/2}, 6e^{d/2}+6d+17] \]
as well.
\end{corollary}
Corollary~\ref{cor:main_cor} allows us to contrast our results with the undirected case. For all $d$, we know $\mathcal{G}(n,p=\frac{d}{n})$ and $\mathcal{G}(n,d)$ are concentrated in at most two values. The gap in our directed result is exponential in $d$. While the bounds of Corollary~\ref{cor:main_cor} may be improvable, the case $d=2$ suggests the oriented chromatic number may not be concentrated in as small a set of values as the unoriented cases.
The lower bound on ${\chi_o}$ given by Kostochka, Sopena and Zhu~\cite{KSZ97} in fact showed that for $d$ fixed and $n$ large, every graph $G$ with $n$ vertices and average degree $d$ has some orientation with ${\chi_o}(\vec{G}) > 2^{d/2} -1$, roughly corresponding to both lower bounds on ${\chi_o}$ in Corollary~\ref{cor:main_cor}. Neither result follows directly from the other, but the underlying proofs are similar as the lower bound for a worst-case orientation of a fixed graph roughly comes from counting proper oriented colourings.
As part of our argument, we maximize a specific function over doubly stochastic matrices. This is common in investigations of the chromatic number of regular graphs; see~\cite{AM04, KPW10, NP21}, each of which rely on a result of Achlioptas and Naor~\cite{AN05}. However, unlike previous investigations, we need to optimize a function not covered by the original theorem of Achlioptas and Naor. We therefore extended their result in the following proposition which we believe may be of independent interest.
\begin{restatable}{proposition}{optimization} \label{prop:generalized_AN}
Let $k \geq 3$, let $G = (V, E)$ be a connected regular graph on vertex set $[k]^2$, and let $\lambda \in \mathbb{R}$ be the second-largest eigenvalue of the adjacency matrix for $G$. Then, for every $d < \frac{4|E|}{k^3 \lambda} \cdot \frac{(k-1)}{(k-2)} \log(k-1)$, the function $\gamma_d$ defined on doubly-stochastic matrices by
\[
\gamma_d(\mbf A) = -\frac{1}{k} \sum_{v \in V(G)} a_v \log a_v + \frac{d}{2}\log\left(\frac{2\sum_{uv \in E} a_u a_v}{k^2}\right)
\]
has a unique maximum value at $\mbf A = \frac{1}{k} J_k$ of $\gamma_d(\frac{1}{k}J_k) = \log k + \frac{d}{2}\log\left(\frac{2|E|}{k^4}\right)$.
\end{restatable}
\textbf{Outline of the argument.} It will be convenient to work with slightly different random graph models. Let $\vec{\mathcal{M}}(n,m)$ be a random multigraph model on graphs of order $n$ generated by selecting one of the $n^2$ ordered pairs of vertices (chosen with replacement) and adding a directed edge (or loop) from the first vertex to the second and then repeating that process to add a total of $m$ arcs. The configuration model $C(n,d)$, introduced by Bollob\'as~\cite{Bo80}, is a random $d$-regular multigraph model in which a random perfect matching is chosen for a set of $nd$ vertices partitioned into $n$ groups of $d$ points. By contracting each part of the partition to a vertex and adding edges between two vertices if their corresponding parts held adjacent vertices (or a loop if two vertices of the same part were adjacent), one generates a $d$-regular multigraph with loops on $n$ vertices. By orienting each edge before contraction we get the oriented configuration model $\vec{\mathcal{C}}(n,d)$.
In Section~\ref{sec:optimization} we prove
Proposition~\ref{prop:generalized_AN}. We develop some results regarding
doubly-regular tournaments in Section~\ref{sec:tour_prods}. In
Section~\ref{sec:lb} we use first moment arguments to establish lower bounds on
${\chi_o}(\vec{\mathcal{M}}(n,m))$ and ${\chi_o}(\vec{\mathcal{C}}(n,d))$. The upper
bounds rely on more delicate second moment arguments. First, we narrow our
consideration to certain nice colourings, simplifying the second
moment calculations. We conduct the second moment calculations in
Section~\ref{sec:ub} which give a value $k$ for which there exist
$k$-oriented-colourings of $\vec{\mathcal{M}}(n,m)$ and $\vec{\mathcal{C}}(n,d)$
with positive probability. In Section~\ref{sec:window} we prove that a.a.s. the
values of ${\chi_o}(\vec{\mathcal{M}}(n,m))$ and ${\chi_o}(\vec{\mathcal{C}}(n,d))$
fall in a window of size linear in the number of colours. We prove
Theorems~\ref{thm:gnp_main} and \ref{thm:gnd_main} in Section~\ref{sec:main} by
relating $\vec{\mathcal{G}}(n,p)$ to $\vec{\mathcal{M}}(n,m)$ and
$\vec{\mathcal{G}}(n,d)$ to $\vec{\mathcal{C}}(n,d)$ before concluding with
additional thoughts in Section~\ref{sec:further_disc}.
\textbf{Notation and conventions.} As previously mentioned, we say that a family of events dependent on $n$ happens asymptotically almost surely, abbreviated a.a.s., if the probability of that event approaches 1 as $n$ approaches infinity. For any integers $m, n$, $I_n$ is an $n \times n$ square identity matrix with entries $1$ along the main diagonal and zero elsewhere, $J_{m,n}$ is an $m \times n$ matrix with every entry $1$, and $0_{m,n}$ is an $m \times n$ matrix with every entry 0. If $m=n$, we write $J_n$ or $0_n$. We take $0^0 = 1$ and $0 \log 0 = 0$. There are a few occasions where a $k \times k$ matrix is treated as interchangeable with a vector of length $k^2$ in a tensor space. Such instances will be highlighted when it may cause confusion. We frequently use Stirling's formula $n! = \xi(n)(n/e)^n$ where $\xi(n)$ is a function satisfying $1 \le \xi(n)$ and $\lim_{n \to \infty} \xi(n) = \sqrt{2\pi n}$.
\section{Optimization Over Doubly Stochastic Matrices} \label{sec:optimization}
In this section we prove Proposition~\ref{prop:generalized_AN} which describes how to maximize certain functions over doubly stochastic matrices. This optimization result generalizes that given by Achlioptas and Naor~\cite{AN05} in their proof of the concentration of the usual chromatic number in random graphs.
For every $k \geq 3$ and constant $c$, Achlioptas and Naor define a function $g_c$ on $k \times k$ doubly-stochastic matrices $\mbf A = (a_{ij})$ by
\begin{equation}\label{eq:AN-function}
g_c(\mbf A) = - \frac{1}{k} \sum_{i = 1}^k \sum_{j = 1}^k a_{ij}\log a_{ij} + c \log\left(1 - \frac{2}{k} + \frac{1}{k^2}\sum_{i = 1}^k\sum_{j = 1}^k a_{ij}^2\right)
\end{equation}
and show that for $c \leq \frac{(k-1)^3}{k(k-2)}\log(k-1)$, $g_c$ is maximized in its domain by the matrix $\frac{1}{k} J_k$ that has all entries equal to $\frac{1}{k}$. The first sum in the expression for $g_c$ is viewed as the `entropy part' of the function, while the second term is viewed as a function of the 2-norm of $\mbf A$: $||\mbf A||_2^2 = \sum_{i = 1}^k \sum_{j = 1}^k a_{ij}^2$. In their paper, Achlioptas and Naor, in fact, give a much more general result, replacing the first sum with one of the form $\frac{1}{k} \sum_{i = 1}^k \sum_{j = 1}^k h(a_{ij})$ where $h$ satisfies a number of conditions involving its derivatives.
We use their techniques to show a generalization involving the second term,
replacing the 2-norm with an expression involving the `Lagrangian' of a regular
graph, a function over its edges. The proof uses the key tools developed by Achlioptas and Naor in their
optimization result, which are stated here for reference with some of their
notation.
For any $k > 1$, define a function $f: [1/k, 1] \to \mathbb{R}$ as follows. For any $r \in [1/k, 1]$, set
\[
x(r) = \frac{1 + \sqrt{(k-1)(kr-1)}}{k}, \qquad \text{and} \qquad y(r) = \frac{1-x(r)}{k-1} = \frac{k-1 -\sqrt{(k-1)(kr-1)}}{k(k-1)}
\]
defined to be the unique values $x, y \geq 0$ satisfying $x + (k-1)y = 1$ and $x^2 + (k-1)y^2 = r$. Set
\begin{equation}\label{eq:single-row-opt-fn}
f(r) = -x(r)\log x(r) - (k-1) y(r)\log y(r).
\end{equation}
\begin{theorem}[see {\cite[Theorem 9]{AN05}}]\label{thm:AN-entropy-ub}
Let $k > 1$ and $1 \leq \rho \leq k$. For every row stochastic matrix $\mbf A$ satisfying $||\mbf A||_2^2 = \rho$,
\begin{equation}
-\sum_{i = 1}^k \sum_{j = 1}^k a_{ij} \log a_{ij}
\leq \max\left\{ m \log k + (k-m) f\left(\frac{k\rho-m}{k(k-m)}\right) :\ 0 \leq m \leq \frac{k(k-\rho)}{k-1} \right\}.
\end{equation}
\end{theorem}
Note that, in their proof, Achlioptas and Noar~\cite{AN05} show that the function $-\sum_{i = 1}^k\sum_{j = 1}^k a_{ij}\log a_{ij}$, subject to the condition $||\mbf A||_2^2 = \rho$, achieves a unique maximum that is of the form given in Theorem~\ref{thm:AN-entropy-ub}, but the corresponding value of $m$, determined by $k$ and $\rho$, is not determined.
\begin{lemma}[see {\cite[Proof of Theorem 7]{AN05}}]\label{lem:eta-min}
For any $k \geq 3$, define a function $\eta: [0, 1-1/k] \to \mathbb{R}$ by
\[
\eta(x) = \begin{cases}
k/2 &\text{if $x = 0$}\\
\frac{f\left(\frac{1}{k}\right) - f\left(\frac{1}{k} + x\right)}{x} &\text{if $x > 0$}
\end{cases}.
\]
Then, $\eta$ achieves its unique global minimum at $x = \frac{(k-2)^2}{k(k-1)}$ with a value of $\eta\left( \frac{(k-2)^2}{k(k-1)}\right) = \frac{k-1}{k-2} \log(k-1)$.
\end{lemma}
In~\cite{AN05}, the uniqueness of the global minimum in Lemma~\ref{lem:eta-min} is not stated, but is shown in the proof.
\begin{definition}
For any graph $G = (V, E)$, define the \emph{Lagrangian of $G$} to be a function $\mathcal{L}_G$ defined on stochastic vectors in $[0, 1]^V$ by
\[
\mathcal{L}_G(\mbf a) = \sum_{uv \in E} a_u a_v.
\]
\end{definition}
Lagrangians of graphs were studied by Motzkin and Straus~\cite{MS65} who showed that $\mathcal{L}_G$ is maximized by vectors whose support is a maximum clique in $G$, with equal entries on those coordinates.
We are now ready to prove Proposition~\ref{prop:generalized_AN}, which we restate here:
\optimization*
\begin{proof}
Let $\mbf A$ be a doubly-stochastic matrix. In this proof, $\mbf A$ is treated as a vector in the tensor product space $\mathbb{R}^k \otimes \mathbb{R}^k$. Similarly, the constant $1$ matrix $J_k$ is treated as interchangeable with the vector $1_k \otimes 1_k$.
Let $M$ be the adjacency matrix for $G$. Then,
\begin{align}
2 \mathcal{L}_G(\mbf A)
&=2 \sum_{uv \in E} a_u \cdot a_v \notag\\
&=\sum_{u \in V} \sum_{v \in N(u)} a_u \cdot a_v \notag\\
&= {\mbf A}^T M {\mbf A}.
\end{align}
Let $r$ be the degree of vertices in $G$. Since $G$ is connected and $r$-regular, the largest eigenvalue for $M$ is $r$, of multiplicity 1, spanned by the vector $J_k$. The second-largest eigenvalue is $\lambda < r$. Since $\mbf A$ is doubly-stochastic, $J_k^T \bullet \mbf A = k$ and so $J_k^T M \mbf A = r J_k^T \mbf A = r k$. Thus,
\begin{align}
\left({\mbf A} - \frac{1}{k}J_k\right)^T M \left({\mbf A} - \frac{1}{k} J_k\right)
&= {\mbf A}^T M \mbf A - \left(\frac{1}{k}J_k\right)^T M \mbf A - {\mbf A}^T M \left(\frac{1}{k}J_k\right) + \frac{1}{k^2} J_k^T M J_k \notag \\
&={\mbf A}^T M \mbf A - 2r + \frac{1}{k^2} \cdot 2|E| \notag\\
&={\mbf A}^T M \mbf A - 2\cdot \frac{2|E|}{k^2} + \frac{2|E|}{k^2} \notag\\
&={\mbf A}^T M \mbf A - \frac{2|E|}{k^2}. \label{eq:lagrangian-AminusJ}
\end{align}
Note that since $\mbf A$ is double-stochastic, $(\mbf A - \frac{1}{k}J_k) \bullet J_k = 0$ and so
\begin{equation}\label{eq:norm}
|| \mbf A - \tfrac{1}{k} J_k ||_2^2 = ||\mbf A||_2^2 - 2 \mbf A \bullet \left(\frac{1}{k} J_k \right) + \left(\frac{1}{k} J_k \right) \bullet \left(\frac{1}{k} J_k \right) = ||\mbf A||_2^2 - \mbf A \bullet \left(\frac{1}{k} J_k\right) = ||\mbf A||_2^2 - 1.
\end{equation}
By the Rayleigh-Ritz theorem from linear algebra, since the vector $\mbf A - \frac{1}{k} J_k$ is orthogonal to the eigenspace of the largest eigenvector for $M$,
\begin{equation}\label{eq:RR-2nd-eval}
\left({\mbf A} - \frac{1}{k}J_k\right)^T M \left({\mbf A} - \frac{1}{k} J_k \right) \leq \lambda ||\mbf A - \tfrac{1}{k}J_k||_2^2.
\end{equation}
Thus,
\begin{align}
2 \mathcal{L}_G({\mbf A}) &= {\mbf A}^T M \mbf A \notag\\
&=\left({\mbf A} - \tfrac{1}{k}J_k\right)^T M ({\mbf A} - \tfrac{1}{k} J_k) + \frac{2|E|}{k^2} &&\text{(by eqn.~\eqref{eq:lagrangian-AminusJ})} \notag\\
&\leq \lambda ||\mbf A - \frac{1}{k}J_k||_2^2 + \frac{2|E|}{k^2} &&\text{(by eqn.~\eqref{eq:RR-2nd-eval})} \notag\\
&= \lambda\left(||\mbf A||_2^2 - 1\right) + \frac{2|E|}{k^2} &&\text{(by eqn.~\eqref{eq:norm})} \label{eq:ub-lagrangian}
\end{align}
Furthermore,
\[
2\mathcal{L}_G\left(\tfrac{1}{k}J_k\right) = \frac{2|E|}{k^2} = \lambda\left(||\tfrac{1}{k}J_k||_2^2-1\right) + \frac{2|E|}{k^2}.
\]
Thus, if the function $\nu_d$ defined by on doubly-stochastic $k \times k$ matrices by
\begin{equation}\label{eq:new-fn-optimization}
\nu_d(\mbf A) = -\frac{1}{k} \sum_{v \in V(G)} a_v \log a_v + \frac{d}{2}\log\left(\frac{\lambda}{k^2} \left(||\mbf A||_2^2 - 1 \right) + \frac{2|E|}{k^4}\right)
\end{equation}
is maximized at $\mbf A = \frac{1}{k} J_k$ for a given value of $d$, then the same is true for the function $\gamma_d$. From here, the optimization tools developed in~\cite{AN05} are used to optimize the sum in the function $\nu_d$, subject to a fixed value of $||\mbf A||_2^2$.
In order to show that for all doubly-stochastic $\mbf A$, $\nu_d(\mbf A) \leq \nu_d \left(\frac{1}{k} J_k \right)$, fix the value of $||\mbf A||_2^2 = \rho \in [1, k]$. By Theorem~\ref{thm:AN-entropy-ub}, it suffices to show that for every choice of $\rho \in [1, k]$ and $m \in \left[0, \frac{k(k - \rho)}{k-1}\right]$,
\begin{equation*}
\frac{m \log k}{k} + \frac{(k-m)}{k} f\left(\frac{k\rho-m}{k(k-m)}\right) + \frac{d}{2}\log\left(\frac{\lambda (\rho-1)}{k^2} + \frac{2|E|}{k^4}\right)
\leq \nu_d\left(\tfrac{1}{k} J_k \right)
=\log k + \frac{d}{2}\log\left(\frac{2|E|}{k^4}\right).
\end{equation*}
Rearranging gives the condition
\begin{align*}
\frac{d}{2} \log\left(1 + \frac{\lambda(\rho - 1)}{2|E|/k^2}\right) &\leq \left(1 - \frac{m}{k}\right)\left[\log k - f\left(\frac{k\rho-m}{k(k-m)}\right) \right]\\
&=\left(1 - \frac{m}{k}\right)\left[f\left(\frac{1}{k}\right) - f\left(\frac{k\rho-m}{k(k-m)}\right) \right].
\end{align*}
Using the inequality $\log(1+x) \leq x$, it thus suffices to show that for every $\rho \in [1, k]$ and $m \in \left[0, \frac{k(k - \rho)}{k-1}\right]$,
\[
d \leq \frac{2 \left(1 - \frac{m}{k}\right)}{\frac{\lambda(\rho - 1)}{2|E|/k^2}}\left[f\left(\frac{1}{k}\right) - f\left(\frac{k\rho-m}{k(k-m)}\right) \right]
\]
Consider now the right hand side of the above equation.
\begin{align}
\frac{2 \left(1 - \frac{m}{k}\right)}{\frac{\lambda(\rho - 1)}{2|E|/k^2}}\left[f\left(\frac{1}{k}\right) - f\left(\frac{k\rho-m}{k(k-m)}\right) \right]
&=\frac{2 \left(1 - \frac{m}{k}\right)}{\frac{\lambda(\rho - 1)}{2|E|/k^2}}\left[f\left(\frac{1}{k}\right) - f\left(\frac{1}{k} + \frac{k\rho-k}{k(k-m)}\right) \right] \notag\\
&=\frac{4|E| \left(1 - \frac{m}{k}\right)}{k^2\lambda(\rho - 1)}\left[f\left(\frac{1}{k}\right) - f\left(\frac{1}{k} + \frac{\rho-1}{k(1-m/k)}\right) \right] \notag\\
&= \frac{4|E|}{k^3\lambda} \left[\frac{f\left(\frac{1}{k}\right) - f\left(\frac{1}{k} + \frac{\rho-1}{k(1-m/k)}\right) }{\frac{\rho-1}{k(1-m/k)}}\right] \notag\\
&=\frac{4|E|}{k^3\lambda} \eta\left(\frac{\rho-1}{k(1-m/k)}\right) \notag\\
&\geq \frac{4|E|}{k^3\lambda} \cdot \frac{k-1}{k-2} \log(k-1). \label{eq:dbound-general}
\end{align}
The final inequality in Equation~\eqref{eq:dbound-general} follows from Lemma~\ref{lem:eta-min} since $0 \leq \frac{\rho-1}{1-m/k} \leq \frac{\rho - 1}{(\rho-1)/(k-1)} = k-1$ for $0 \leq m \leq \frac{k(k-\rho)}{k-1}$ and hence $0 \leq \frac{\rho-1}{k(1-m/k)} \leq 1-\frac{1}{k}$ is in the domain of $\eta$.
Thus, $d \leq \frac{4|E|}{k^3\lambda} \cdot \frac{k-1}{k-2} \log(k-1)$ suffices to guarantee that $\nu_d$ is maximized at $\mbf A = \frac{1}{k} J_k$ and hence that the same holds for the function $\gamma_d$.
Since the maximum in Theorem~\ref{thm:AN-entropy-ub} and the minimum in Lemma~\ref{lem:eta-min} are unique, then $d > \frac{4|E|}{k^3\lambda} \cdot \frac{k-1}{k-2} \log(k-1)$, the function $\nu_d$ attains a unique maximum at $\frac{1}{k}J_k$, and hence the same is true for the function $\gamma_d$.
\end{proof}
To see that Proposition~\ref{prop:generalized_AN} generalizes the optimization result for the function in Equation~\eqref{eq:AN-function}, consider the graph $G = K_k \otimes K_k$, the graph on vertex set $[k]^2$ with $(u,v) \sim (a, b)$ if{f} $u \neq a$ and $v \neq b$. The graph $G$ is a connected $(k-1)^2$ regular graph. Since the adjacency matrix of $G$ is $(J_k - I_k) \otimes (J_k - I_k)$ it has as eigenvalues $(k-1)^2$, $1$, and $-(k-1)$. The eigenspace for $-(k-1)$ is the set of all vectors of the form $1_k \otimes \mbf v$ or $\mbf v \otimes 1_k$, where $\mbf v \bullet 1_k = 0$. For any such eigenvector and any doubly-stochastic matrix $\mbf A$, $(1_k \otimes \mbf v) \bullet \mbf A = (\mbf v \otimes 1_k) \bullet A = 0$ and so, decomposing $\mbf A$ into the basis of eigenvectors for the adjacency matrix of $G$ gives
\[
2 \mathcal{L}_G(\mbf A) = (k-1)^2\cdot ||\tfrac{1}{k} J_k||_2^2 + 1 \cdot ||\mbf A - \tfrac{1}{k}J_k||_2^2 = (k-1)^2 + ||\mbf A||_2^2 - 1 = k^2 - 2k + ||\mbf A||_2^2.
\]
Since $G$ has $\frac{k^2(k-1)^2}{2}$ edges, Proposition~\ref{prop:generalized_AN} says that for $k \geq 3$, if
\[
c = \frac{d}{2} \leq \frac{2|E|}{k^3 \cdot 1} \cdot \frac{(k-1)}{(k-2)} \log(k-1) = \frac{(k-1)^3}{k(k-2)} \log(k-1),
\]
then the function
\[
\gamma_d(\mbf A) = -\frac{1}{k} \sum_{v \in V(G)} a_v \log a_v + \frac{d}{2}\log\left(\frac{2\mathcal{L}_G(\mbf A)}{k^2}\right) = -\frac{1}{k} \sum_{v \in V(G)} a_v \log a_v + \frac{d}{2}\log\left(\frac{k^2 - 2k + ||\mbf A||_2^2}{k^2}\right)
\]
is maximized at $\mbf A = \frac{1}{k}J_k$.
For the result here, we shall take $G$ to be the Kronecker product of a doubly-regular tournament with itself, as defined in Section~\ref{sec:tour_prods}.
\section{Properties of Doubly Regular Tournaments} \label{sec:tour_prods}
In this section we develop several results regarding Kronecker products of tournaments with nice properties.
Doubly-regular tournaments were introduced by Brown and Reid~\cite{RB72}.
\begin{definition}
A tournament $T$ is said to be \emph{doubly-regular} if{f} there are integers $k, \ell$ so that for every $v \in V(T)$, $\deg^+(v) = k$ and for every pair of vertices $v, w$, $|N^+(v) \cap N^+(w)| = \ell$.
\end{definition}
Brown and Reid~\cite{RB72} note that if $T$ is a doubly-regular tournament of order $n$, then $n \equiv 3 \pmod{4}$, $k = \frac{n-1}{2}$ and $\ell = \frac{n-3}{4}$. They further relate the existence of a tournament of order $n$ to the existence of a `skew Hadamard matrix' of order $n+1$. A recent survey on the existence of skew-Hadamard matrices was given by Koukouvinos and Stylianou~\cite{KS08}.
A special case of a doubly regular tournament are the `Paley tournaments' (see, eg~\cite{Hall86}). For each prime $q \equiv 3 \mod 4$, the \emph{Paley} tournament $T_q$ is defined on vertex set $\mathbb{F}_q$ by directing the edge $\{u,v\}$ from $u$ to $v$ if{f} $(v-u)$ is a square modulo $q$ and otherwise directing it from $v$ to $u$. Standard number theoretic arguments give that exactly one of $v-u, u-v \in (\mathbb{F}_q^\times)^2$, so the direction is well-defined, and as
\[ |(\mathbb{F}_q^\times)^2| = \frac{q-1}{2} = \frac{1}{2}|\mathbb{F}_q^\times| \]
we see that $T_q$ satisfies $d^+(v) = d^-(v) = \frac{q-1}{2}$ for every $v \in V(T_q)$.
As shown in~\cite{Hall86}, for each pair of distinct vertices $u,v \in V(T_q)$, there are exactly $\frac{k-3}{4}$ vertices $w$ that satisfy $u \to w$ and $v \to w$.
\begin{lemma} \label{lem:T_props}
Let $T$ be doubly-regular tournament with $k$ vertices.
Then if $u \to v$, the number of oriented triangles containing the edge $u$--$v$
paths of length two in each possible orientation is given in the table below:
\begin{center}
$\begin{array}{c|c|c|c|c}
\text{adjacency} &
\vcenteredhbox{\begin{tikzpicture}
\node[circle,draw,label=u,minimum size=2mm,inner sep=0] (u) at (-0.5,0.866) {};
\node[circle,draw,label=v,minimum size=2mm,inner sep=0] (v) at (0.5,0.866) {};
\node[circle,draw,label=below:w,anchor=south,minimum size=2mm,inner sep=0] (w) at (0,0) {};
\draw[->] (u) -> (v);
\draw[->] (u) -> (w);
\draw[->] (v) -> (w);
\end{tikzpicture}} &
\vcenteredhbox{\begin{tikzpicture}
\node[circle,draw,label=u,minimum size=2mm,inner sep=0] (u) at (-0.5,0.866) {};
\node[circle,draw,label=v,minimum size=2mm,inner sep=0] (v) at (0.5,0.866) {};
\node[circle,draw,label=below:w,anchor=south,minimum size=2mm,inner sep=0] (w) at (0,0) {};
\draw[->] (u) -> (v);
\draw[->] (u) -> (w);
\draw[->] (w) -> (v);
\end{tikzpicture}} &
\vcenteredhbox{\begin{tikzpicture}
\node[circle,draw,label=u,minimum size=2mm,inner sep=0] (u) at (-0.5,0.866) {};
\node[circle,draw,label=v,minimum size=2mm,inner sep=0] (v) at (0.5,0.866) {};
\node[circle,draw,label=below:w,anchor=south,minimum size=2mm,inner sep=0] (w) at (0,0) {};
\draw[->] (u) -> (v);
\draw[->] (w) -> (u);
\draw[->] (v) -> (w);
\end{tikzpicture}} &
\vcenteredhbox{\begin{tikzpicture}
\node[circle,draw,label=u,minimum size=2mm,inner sep=0] (u) at (-0.5,0.866) {};
\node[circle,draw,label=v,minimum size=2mm,inner sep=0] (v) at (0.5,0.866) {};
\node[circle,draw,label=below:w,anchor=south,minimum size=2mm,inner sep=0] (w) at (0,0) {};
\draw[->] (u) -> (v);
\draw[->] (w) -> (u);
\draw[->] (w) -> (v);
\end{tikzpicture}}\\
\hline
&&&&\\
\# w & \displaystyle\frac{k-3}{4} & \displaystyle\frac{k-3}{4} & \displaystyle\frac{k+1}{4} & \displaystyle\frac{k-3}{4}
\end{array}$
\end{center}
\end{lemma}
\begin{proof}
As noted above, the first value is shown in~\cite{Hall86}. For the second count, there are $d^{+}(u) - 1 = \frac{k-3}{2}$ out-neighbours of $u$ other than $v$. Of those, $v$ is directed towards $\frac{k-3}{4}$ of them, leaving $\frac{k-3}{2}-\frac{k-3}{4} = \frac{k-3}{4}$ of them satisfying $u \to w \to v$. The third and fourth counts follow from similar arguments.
\end{proof}
Recall that the signed adjacency matrix of an oriented graph $D$ on $n$ vertices, $\{v_1, v_2, \ldots, v_n\}$, is an $n \times n$ matrix $A$ with
\[
A_{i,j} = \begin{cases}
$1$ &\text{if } v_i \to v_j \in E(G)\\
$-1$ &\text{if } v_j \to v_i \in E(G), \text{ and}\\
$0$ &\text{otherwise}.
\end{cases}
\]
\begin{lemma}\label{lem:eval-dr-tourn}
If $T$ is a doubly-regular tournament on $k$ vertices with signed adjacency matrix $M$, then $M$ is diagonalizable over $\mathbb{C}$ and has eigenvalues:
\begin{itemize}
\item $0$, with the eigenspace equal to the span of $\mathbf{1}$.
\item $i\sqrt{k}$ with multiplicity $\frac{k-1}{2}$, and
\item $-i\sqrt{k}$ with multiplicity $\frac{k-1}{2}$
\end{itemize}
\end{lemma}
\begin{proof}
Since the signed adjacency matrix satisfies $M^T = -M$, it is skew symmetric and real skew symmetric matrices are diagonalizable over $\mathbb{C}$.
For the eigenvalues of $M$, consider the product, $M^2$. If $M = (m_{ij})_{k \times k}$, the $(i,j)$ entry of $M^2$ is $\sum_{\ell = 1}^k m_{i \ell} m_{\ell j}$. If $i= j$, then $\sum_{\ell = 1}^k m_{i \ell} m_{\ell i} = \sum_{\ell = 1}^k -m_{i \ell}^2 = -(k-1)$. If $i \neq j$, then for any $\ell \notin \{i, j\}$, $m_{i \ell} m_{\ell j}$ is equal to $1$ if $(i, \ell, j)$ is a directed path (in either direction) and $-1$ otherwise. By Lemma~\ref{lem:T_props}, this is
\[
\sum_{\ell = 1}^k m_{i \ell} m_{\ell j} = -\frac{(k-3)}{4} + \frac{(k-3)}{4} + \frac{(k+1)}{4} - \frac{(k-3)}{4} = 1.
\]
Thus $M^2 = J_k - kI_k$, which has eigenvalues $0$ with multiplicity $1$ and eigenvector $\mathbf{1}$ and $-k$ with multiplicity $k-1$. Thus, the eigenvalues for $M$ are $0$ with multiplicity $1$ and eigenvector $\mathbf{1}$ and complex eigenvalues $i\sqrt{k}$ and $-i\sqrt{k}$. Since $tr(M) = 0$, the eigenvalues $i\sqrt{k}$ and $-i\sqrt{k}$ occur with equal multiplicity.
\end{proof}
\begin{definition}\label{def:kroenecker_tournaments}
For any tournament $T$, define the Kronecker product of $T$ with itself, $T^{\otimes 2}$, as the graph with vertex set $V(T)^2$ and edges $(u,v) \sim (x,y)$ if{f} either $u \to x$ and $v \to y$ or $x \to u$ and $y \to v$. Note that while $T$ is a directed graph, we have defined $T^{\otimes 2}$ to be undirected. This is more convenient for our calculations, but it should be noted there is a natural orientation of $T^{\otimes 2}$ where $(u,v) \to (x,y)$ if $u \to x$ and $v \to y$.
\end{definition}
\begin{proposition}
Let $T$ be a regular tournament of order $k \geq 5$. Then the Kroenecker product graph $T^{\otimes 2}$ is connected.
\end{proposition}
\begin{proof}
First, consider two non-adjacent vertices $(u, v), (x, y) \in V(T^{\otimes 2})$ with $u \neq x$ and $v \neq y$. Without loss of generality, assume that, in $T$, $u \to x$ and $y \to v$. Define the following:
\begin{align*}
a_{++} &= |N^+(u) \cap N^+(x)|\\
a_{+-} &= |N^+(u) \cap N^-(x)|\\
a_{-+} &= |N^-(u) \cap N^+(x)|\\
a_{--} &= |N^-(u) \cap N^-(x)|
\end{align*}
Since $T$ is regular and $u \to x$, then $a_{++} + a_{+-} = \frac{k-3}{2}$, $a_{-+} + a_{--} = \frac{k-1}{2}$, $a_{+-} + a_{--} = \frac{k-3}{2}$, and $a_{++} + a_{-+} = \frac{k-1}{2}$. Similarly, define
\begin{align*}
b_{++} &= |N^+(v) \cap N^+(y)|\\
b_{+-} &= |N^+(v) \cap N^-(y)|\\
b_{-+} &= |N^-(v) \cap N^+(y)|\\
b_{--} &= |N^-(v) \cap N^-(y)|
\end{align*}
Since $y \to v$, then $b_{++} + b_{+-} = \frac{k-1}{2}$, $b_{-+} + b_{--} = \frac{k-3}{2}$, $b_{++} + b_{-+} = \frac{k-3}{2}$, and $b_{+-} + b_{--} = \frac{k-1}{2}$.
If there are any $i, j \in \{+, -\}$ with $a_{ij} > 0$ and $b_{ij} > 0$, then there are vertices $w, z$ (not necessarily distinct) so that the direction of $(u, w)$ matches $(v, z)$ and the direction of $(x, w)$ matches $(y, z)$. Thus, $(u, v), (w, z), (x, y)$ is a path in $T^{\otimes}$. To see that such pair always exists, suppose that one does not. If $a_{++} \neq 0$, then $b_{++} = 0$ and so $b_{+-} = \frac{k-1}{2}$ and $b_{-+} = \frac{k-3}{2}$ which implies that $b_{--} = 0$. The fact that $b_{+-}, b_{-+} > 0$ would imply that $a_{+-} = a_{-+} = 0$. However, this is impossible since $a_{-+} + a_{--} = \frac{k-1}{2} \neq \frac{k-3}{2} = a_{+-} + a_{--}$.
Next, consider the case of vertices $(u, u)$ and $(u, x)$. Note that the vertices $\{(v, v) \mid v \in V(T)\}$ form a clique in $T^{\otimes 2}$. Let $z \in V\setminus \{u, x\}$. Then, $(u, u)$ is adjacent to $(z, z)$ and by the previous case, there is a path between $(z, z)$ and $(u,x)$.
For the final case, consider vertices $(u, v), (u, x)$ with $v \neq x, v \neq u$, and $x \neq u$. By the previous case, there is a path between $(u, v)$ and $(u, u)$ and a path between $(u, x)$ and $(u, u)$. Thus, $(u, v)$ and $(u, x)$ are in the same component.
Since the graph $T^{\otimes 2}$ is isomorphic to itself under the operation of interchanging the order of pairs, this covers all cases and shows that every pair of vertices in $T^{\otimes 2}$ are joined by a path.
\end{proof}
\begin{remark} \label{rem:k=3}
Note that the result does not hold for $k = 3$ since the unique regular tournament on $3$ vertices is the directed $3$-cycle, whose Kroenecker square is a disjoint union of three copies of $K_3$.
Throughout the paper, it is regularly assumed that $k > 3$ and part of the reason for this is that when the product graph $G$ is a disjoint union of cliques, the optimization result from Proposition~\ref{prop:generalized_AN} holds without any assumptions on $d$ in terms of $k$. Indeed, suppose that $G$ is a graph on vertex set with $[k]^2$ and $G \cong k K_k$, a disjoint union of $k$ cliques with the property that the vertices of each clique form a transversal in $[k]\times [k]$. Fix $s_1, s_2, \ldots, s_k$ with $\sum_{i = 1}^k s_i = k$ and restrict to doubly-stochastic matrices with the property that the sum of entries on vertices, $V_i$, corresponding to the $i$-th copy of $K_k$ is exactly $s_i$. Among such matrices,
\begin{align*}
\mathcal{L}_G(\mbf A) &= \sum_{i = 1}^k \sum_{v, w \in V_i} a_v a_w\\
&= \frac{1}{2}\sum_{i = 1}^k \left(\left(\sum_{v \in V_i} a_v\right)^2 - \sum_{v \in V_i} a_v^2 \right)\\
&\leq \frac{1}{2} \sum_{i = 1}^k \left(s_i^2 - \frac{s_i^2}{k}\right) &&\text{(by convexity)}\\
&= \frac{(k-1)}{2k}\sum_{i = 1}^k s_i^2,
\end{align*}
Then, for any $d$,
\[
\gamma_d(\mbf A) \leq -\frac{1}{k}\sum_{v \in V} a_v \log a_v + \frac{d}{2}\log\left( \frac{(k-1)}{2k}\sum_{i = 1}^k s_i^2\right).
\]
Because both functions $-x \log x$ and $\log x^2$ are concave down, the upper bound for $\gamma_d(\mbf A)$ is maximized when $s_1 = s_1 = \cdots = s_k =1$, which gives precisely the value of $\gamma_d(\frac{1}{k}J_k)$.
In particular, this means that the optimization result holds without condition on $d$ for oriented $3$-colourings with the directed $3$-cycle. However, for the expected number of such $3$-colourings to be bounded away from $0$, $d \leq 2$ is required and this case is covered in Section~\ref{ssec:vsparse_vdense} using structural results of underlying graphs and their possible colourings.
\end{remark}
\begin{proposition}\label{prop:tourn_prod_eigenvectors}
Let $T$ be a doubly-regular tournament on $k$ vertices with signed adjacency
matrix $M$. Then the adjacency matrix for the graph $T^{\otimes 2}$ is
\[
\frac{1}{2}\left(M \otimes M + (J_k - I_k) \otimes (J_k - I_k)\right)
\]
with eigenvalues and eigenspaces
\begin{itemize}
\item $\frac{(k-1)^2}{2}$ with multiplicity $1$ and eigenspace spanned by $\mathbf{1} \otimes \mathbf{1}$,
\item $\frac{-(k-1)}{2}$ with multiplicity $\frac{(k-1)(k+3)}{2}$ and eigenspace consisting of
\begin{itemize}
\item a subspace of dimension $2(k-1)$ spanned by $\{\mathbf{u} \otimes \mathbf{1}, \mathbf{1} \otimes \mathbf{u} \mid \mathbf{u} \neq \mathbf{0},\ \mathbf{u} \bullet \mathbf{1} = 0\}$,
\item a subspace of dimension $\frac{(k-1)^2}{4}$ spanned by
\[
\{\mathbf{u} \otimes \mathbf{v} \mid \mathbf{u}, \mathbf{v} \text{ are eigenvectors for $M$ for the eigenvalue $i \sqrt{k}$}\},
\]
and
\item a subspace of dimension $\frac{(k-1)^2}{4}$ spanned by
\[
\{\mathbf{u} \otimes \mathbf{v} \mid \mathbf{u}, \mathbf{v} \text{ are eigenvectors for $M$ for the eigenvalue $-i \sqrt{k}$}\}.
\]
\end{itemize}
\item $\frac{(k+1)}{2}$ with multiplicity $\frac{(k-1)^2}{2}$ spanned by
\begin{multline*}
\{\mathbf{u} \otimes \mathbf{v}, \mathbf{v} \otimes \mathbf{u} \mid \mathbf{u} \text{ an eigenvector for $M$ for the eigenvalue $i\sqrt{k}$ and } \\
\mathbf{v} \text{ an eigenvector for $M$ for the eigenvalue $-i \sqrt{k}$}\}.
\end{multline*}
\end{itemize}
\end{proposition}
\begin{proof}
To see that the adjacency matrix for $T^{\otimes 2}$ is $\frac{1}{2} \left(M
\otimes M + (J_k - I_k) \otimes (J_k - I_k)\right)$, note that the entry of
$M \otimes M$ at $((u, v), (x, y))$ is
\begin{itemize}
\item $0$ if either $u = x$ or $v = y$,
\item $+1$ if either $u \to x$ and $v \to y$ in $T$ or else $x \to u$ and $y \to v$,
\item $-1$ otherwise.
\end{itemize}
Since the $((u, v), (x, y))$ entry of $(J_k - I_k) \otimes (J_k - I_k)$ is $1$ if{f} $u \neq x$ and $v \neq y$, and $0$ otherwise, then the $((u, v), (x, y))$ entry of $\frac{1}{2}\left(M \otimes M + (J_k - I_k) \otimes (J_k - I_k)\right)$ is $1$ if{f} either $u \to x$ and $v \to y$ or else $x \to u$ and $y \to v$ in $T$, and $0$ otherwise, exactly matching the adjacency conditions for $T^{\otimes 2}$.
The rest of the Proposition follows from Lemma~\ref{lem:eval-dr-tourn} and the fact that $(J_k - I_k)$ is diagonalizable with eigenvalues $(k-1)$ (with multiplicity $1$ and eigenspace spanned by $\mathbf{1}$) and $-1$ (with multiplicity $(k-1)$). Thus, there is an orthogonal basis for $\mathbb{C}^k$ $\{\mathbf{v}_1 = \mathbf{1}, \mathbf{v}_2, \mathbf{v}_3, \ldots, \mathbf{v}_k\}$ whose elements are all eigenvectors for both $M$ (corresponding to eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_k$) and $(J_k - I_k)$ (corresponding to eigenvalues $\alpha_1, \alpha_2, \ldots, \alpha_k$). Therefore, $\{\mathbf{v}_i \otimes \mathbf{v}_j \mid 1 \leq i, j \leq k\}$ is a basis for $\mathbb{C}^k \otimes \mathbb{C}^k$ consisting of eigenvectors for $\frac{1}{2}\left(M \otimes M + (J_k - I_k) \otimes (J_k - I_k)\right)$.
\end{proof}
Note that is it also possible to use Lemma~\ref{lem:T_props} to show that if $T$ is a doubly regular tournament on $k$ vertices, then $T^{\otimes 2}$ is a strongly regular graph with the property that every vertex has $\frac{(k-1)^2}{2}$ neighbours, every pair of adjacent vertices has $\frac{(k-1)(k-3)}{4} + 1$ common neighbours, and every non-adjacent pair of vertices have $\frac{(k-1)(k-3)}{4}$ common neighbours. From this, the eigenvalues of $T^{\otimes 2}$ and their multiplicities follow from standard results in algebraic graph theory on strongly regular graphs. In Section~\ref{ssec:second_moment}, the precise structure of the eigenspaces of the adjacency matrix of $T^{\otimes 2}$ are used and so the more precise result in Proposition~\ref{prop:tourn_prod_eigenvectors} is given.
It is conjectured that for every $n$, there are skew-Hadamard matrices of order $4n$ (see~\cite{jW71, KS08}). By results of Brown and Reid~\cite{RB72}, this would correspond to the existence of doubly-regular tournaments of order $k$, for every $k \equiv 3 \pmod{4}$. According to Koukouvinos and Stylianou~\cite{KS08}, the smallest un-resolved case is the existence of a skew-Hadamard matrix of order 276, corresponding to the existence of a doubly regular tournament of order $275$. Brown and Reid note that if all of the odd prime factors of $k+1$ are $3\pmod{4}$ and occur to an odd power, then there exist doubly-regular tournaments of order $k$.
Using results about the existence of skew-Hadamard matrices due to Williamson~\cite{jW44}, Brown and Reid~\cite{RB72} note that for any $t_0, t_1, \ldots, t_r$ and primes $p_1, p_2, \ldots, p_r$ with the property that for every $i$, $p_i^{t_i} \equiv 3 \pmod{4}$, there exist doubly regular tournaments of order $n$ when $n = 2^{t_0}(p_1^{t_1} + 1)(p_2^{t_2} + 1) \cdots (p_r^{t_r} + 1) - 1$.
\begin{corollary}\label{cor:opt_doubly_reg_tour}
Let $k > 3$ be such that there exist doubly regular tournaments of order $k$, fix such a tournament $T$ and set $G = T^{\otimes 2}$. For every $d \leq \frac{2 (k-1)^3}{k(k+1)(k-2)}\log(k-1)$, the function $\gamma_d$ defined on doubly-stochastic matrices by
\[
\gamma_d(\mbf A) = -\frac{1}{k} \sum_{v \in V(G)} a_v \log a_v + \frac{d}{2}\log\left(\frac{2\mathcal{L}_G(\mbf A)}{k^2}\right)
\]
has a unique maximum value at $\mbf A = \frac{1}{k} J_k$ of $\gamma_d(\frac{1}{k} J_k) = \log k + \frac{d}{2} \log\left(\frac{(k-1)^2}{2k^2}\right)$.
\end{corollary}
The proof of Corollary~\ref{cor:opt_doubly_reg_tour} follows immediately from Propositions~\ref{prop:generalized_AN} and \ref{prop:tourn_prod_eigenvectors}.
In order to give some explicit bounds on $\chi_o(\vec{\mathcal{G}}(n,p=\frac{d}{n}))$ and $\chi_o(\vec{\mathcal{G}}(n,d))$ in terms of $d$ in Corollary~\ref{cor:main_cor}, it is useful to have some quantitative bounds on the gaps between orders of doubly-regular tournaments. In particular, there is a doubly-regular tournament of every order that is prime and equivalent to $3 \pmod{4}$. Erd\H{o}s~\cite{pE35} showed that for $n$ sufficiently large, there are always primes equivalent to each of $1 \pmod{4}$ and $3 \pmod{4}$ in the interval $[n, 2n]$, extending Bertrand's postulate. The precise bounds given in the paper by Ramar\'{e} and Rumely~\cite{RR96} can be used to show that this property holds for all $n \geq 2$. Since Paley tournaments are doubly-regular, this gives the following result.
\begin{lemma}\label{lem:bertrand_postulate_dr_tourn}
For every $n \geq 2$, there exists $k$ with $n \leq k \leq 2n$ so that there exists a doubly-regular tournament of order $k$.
\end{lemma}
Very precise results about the number of primes in a given arithmetic progression are given by Bennett, Martin, O'Bryant, and Rechnitzer~\cite{BMOBR18} and give a smaller interval in which one can guarantee a prime of a given residue class.
\section{Lower Bounds} \label{sec:lb}
In this section we prove lower bounds on ${\chi_o}(\vec{\mathcal{M}}(n,m))$ and ${\chi_o}(\vec{\mathcal{C}}(n,d))$ using first moment arguments.
\begin{proposition}\label{prop:gnm_lb}
Let $c > 0$ be a real number and $k$ be an integer such that $c > \frac{1}{2}u_k$. Then a.a.s. there are no proper oriented $k$-colourings of $\vec{\mathcal{M}}(n,m=cn)$.
\end{proposition}
\begin{proof}
Let $X$ be a random variable counting the number of proper oriented $k$-colourings of $\vec{\mathcal{M}}(n,m=cn)$. Fix a colouring of $n$ vertices and a tournament on $k$ vertices. Let $a_i$ be the proportion of vertices that received colour $i$. Then the probability that a directed edge connects two vertices of colour $i$ is $a_i^2$, so the probability that it connects different colours is $1 - \sum_{i=1}^k a_i^2$ and the probability it connects two different colours and its orientation matches the tournament is $\frac{1}{2}(1-\sum_{i=1}^k a_i^2)$. By convexity, since the colour proportions must add to one, this probability is at most $\frac{1}{2}(1-\frac{1}{k})$. As each edge is added independently, the probability that the resulting multigraph is properly coloured is at most $(\frac{1}{2}(1-\frac{1}{k}))^m$. Therefore
\[ \mathbf{E} X \le 2^{\binom{k}{2}} k^n\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right)^m = 2^{\binom{k}{2}} \left(k\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right)^c\right)^n. \]
Noting
\[ c > \frac{1}{2}u_k \implies c\log\left(\frac{2k}{k-1}\right) > \log k \implies 1 > k\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right)^c, \]
we see $\mathbf{E} X = o(1)$ and thus a.a.s. there are no proper $k$-oriented-colourings of $\vec{\mathcal{M}}(n,m=cn)$.
\end{proof}
\begin{proposition}\label{prop:gnd_lb}
Let $d \ge 2$ and $k$ be integers such that $d > u_k$. Then a.a.s. there are no proper oriented $k$-colourings of $\vec{\mathcal{C}}(n,d)$.
\end{proposition}
\begin{proof}
Let $X$ be the number of oriented $k$-colourings of a random $d$-regular oriented multigraph $\vec{\mathcal{C}}(n,d)$. We calculate $\mathbf{E} X$ by summing over pairs $(\vec{C},{\chi_o})$, where $\vec{C} \in \vec{\mathcal{C}}(n,d)$ is a configuration with edges given orientations independently and uniformly at random and ${\chi_o}$ is an oriented colouring that respects $\vec{C}$, then dividing by the total number of oriented configurations.
Starting with the denominator, each of the $|\vec{\mathcal{C}}(n,d)|$ many graphs
of the configuration model has $\frac{dn}{2}$ edges, each with one of two
orientations, giving
\[|\vec{\mathcal{C}}(n,d)| \cdot 2^{\frac{1}{2}dn} = \frac{(dn)!}{(\frac{1}{2}dn)!2^{\frac{1}{2}dn}} \cdot 2^{\frac{1}{2}dn}= \frac{(dn)!}{(\frac{1}{2}dn)!}\]
total oriented configurations.
Now we turn to the numerator. For each $i \in [k]$, let $a_i \in
\frac{1}{n}\mathbb{Z}$ be the proportion of the vertices of $\vec{C}$ that
${\chi_o}$ stipulates receive colour $i$. We have
\begin{equation}\label{eq:acons}
a_i \ge 0 \ \forall i \in [k] \quad \text{and} \quad \sum_{i \in [k]} a_i = 1.
\end{equation}
Set ${\bf a} = (a_i)_{i \in [k]}$ and note that ${\bf a} \in \frac{1}{n}\mathbb{Z}^k$.
To be a proper oriented $k$-colouring, all edges between colour classes
must have the same orientation which may be chosen arbitarily. Fix a $k$-colouring of
$K_k$ and choose any of the $2^{\binom{k}{2}}$ possible orientations uniformly
at random. Then orient any edges between vertices of colour $i$ and $i'$
according to that orientation.
Having fixed the direction of all the edges, we now need only specify their
endpoints. To generate edges in the configuration model, we replace each vertex
with a collection of $d$ vertices and select a perfect matching on the resulting
$dn$ vertices. The larger collection of vertices inherits a colouring from the
colouring of the base vertices. For each pair of distinct colours $i,i'$, let
$b_{i,i'}\in\frac{1}{n}\mathbb{Z}$ denote the proportion of the vertices that
${\chi_o}$ colours with colour $i$ and are paired with a vertex of colour $i'$. (As ${\chi_o}$ is a
proper colouring there are no edges from a vertex of colour $i$ to another
vertex of colour $i$.) Let $K$ be the set $\{(i,i') \in [k]^2 \mid i \ne i'\}$. We have
\begin{align}
b_{i,i'} \ge 0 \quad &\quad \forall (i,i') \in K \notag\\
b_{i,i'} = b_{i',i} \quad &\quad \forall (i,i') \in K \notag\\
\sum_{i \ne i'} b_{i,i'} = a_{i'} \quad &\quad \forall i \in [k] \notag\\
\sum_{i' \ne i} b_{i,i'} = a_i \quad & \quad \forall i' \in [k] \label{eq:bcons}
\end{align}
Set ${\bf b} = (b_{i,i'})_{(i,i') \in K}$ and note that ${\bf b} \in \frac{1}{n}\mathbb{Z}^{k(k-1)}$.
To generate a pair $(\vec{C},{\chi_o})$, we first fix ${\bf a}$ and ${\bf b}$. We then colour the $n$
vertices in accordance with ${\bf a}$, which can be done in
\[ \frac{n!}{\prod\limits_{i\in[k]} (a_in)!} \]
many ways. To add edges in accordance with ${\bf b}$ we first select the
appropriate numbers of vertices in each colour class. The configuration model
blows up the $a_in$ vertices of colour $i$ to $a_idn$ vertices, so this
selection can be made in
\[ \prod_{i \in [k]} \frac{(a_idn)!}{\prod\limits_{i' \ne i} (b_{i,i'}dn)!}\]
many ways. Note that each pair of distinct colours $\{i,i'\}$ contributes two terms to the
denominator of this product. We then select a perfect matching between the
vertices of each pair of colours. For a given pair of coulours $(i,i')$, we only
chose one matching, so there are
\[ \prod_{i < i'} (b_{i,i'}dn)! \]
many such selections. Putting these steps together, we see there are
\[ \frac{n!}{\prod_{i \in k} (a_in)!} \prod_{i\in k} \frac{(a_i dn)!}{\prod_{i'
\ne i} (b_{i,i'}dn)!} \prod_{i < i'} (b_{i,i'}dn)! = n!\prod_{i
\in k} \frac{(a_idn)!}{(a_in)!} \prod_{i < i'} \frac{1}{(b_{i,i'}dn)!} \]
many colourings that satisfy ${\bf a}$ and ${\bf b}$. Finally, colour the vertices of $K_k$ with $[k]$ and choose a random orientation of the edges, then orient the edges of $\vec{P}$ in accordance with that colouring; there are $2^{\binom{k}{2}}$ such orientations of $K_k$ from which to choose.
Altogether, keeping in mind (\ref{eq:acons}) and (\ref{eq:bcons}),
\begin{equation} \label{eq:sum_half_bs}
\sum_{i < i'} b_{i,i'} = \frac{1}{2} \sum_{i < i'} 2b_{i,i'} = \frac{1}{2}
\sum_{i < i'} (b_{i,i'} + b_{i',i}) = \frac{1}{2} \sum_{i \in [k]} \sum_{i' \ne i}
b_{i,i'} = \frac{1}{2} \sum_{i \in [k]} a_i = \frac{1}{2},
\end{equation}
and applying Stirling's approximation $x! \sim \sqrt{2\pi x}(x/e)^x$, we get
\begin{align*}
\mathbf{E} X &= \frac{\sum_{{\bf a}, {\bf b}} 2^{\binom{k}{2}} \cdot n! \prod_{i \in k} \frac{(a_idn)!}{(a_in)!} \prod_{i < i'}
\frac{1}{(b_{i,i'}dn)!}}{\frac{(dn)}{(\frac{dn}{2})!}}\\
&\sim \frac{\sum_{{\bf a}, {\bf b}} 2^{\binom{k}{2}} \cdot \sqrt{\pi n} (n/e)^n \prod_{i \in k} \frac{\sqrt{d}(a_idn/e)^{a_idn}}{(a_in/e)^{a_in}} \prod_{i < i'} \frac{1}{\sqrt{2\pi b_{i,i'}dn}(b_{i,i'}dn/e)^{b_{i,i'}dn}}}{\frac{(dn/e)^{dn}}{(dn/(2e))^{dn/2}}}\\
&=\frac{\sum_{{\bf a}, {\bf b}} 2^{\binom{k}{2}} \cdot \sqrt{\pi n} \prod_{i \in k} \sqrt{d}(a_i)^{a_i(d-1)n} \prod_{i < i'} \frac{1}{\sqrt{2\pi b_{i,i'}dn}(b_{i,i'})^{b_{i,i'}dn}}}{2^{dn/2}}\\
&= \sum_{{\bf a}, {\bf b}} \frac{2^{\binom{k}{2}} \cdot \sqrt{\pi n} \cdot d^{k/2}}{\prod_{i < i'} \sqrt{2\pi b_{i,i'}dn}} \left(\frac{1}{2}\right)^{dn/2}\prod_{i \in [k]}a_i^{a_i(d-1)n} \prod_{i < i'} \frac{1}{b_{i,i'}^{b_{i,i'}dn}}\\
&= \sum_{{\bf a}, {\bf b}} \poly(n) \prod_{i < i'} \left(\frac{(a_ia_{i'})^{d-1}}{2(b_{i,i'})^d}\right)^{b_{i,i'}n}\\
&= \sum_{{\bf a}, {\bf b}} \poly(n) e^{nf({\bf a}, {\bf b})}
\end{align*}
where
\begin{equation}\label{eq:fab}
f({\bf a},{\bf b}) = \sum_{i < i'} b_{i,i'}
\log\left(\frac{(a_ia_{i'})^{d-1}}{2(b_{i,i'})^d}\right)
\end{equation}
and $\poly(n)$ is a function bounded above by a function that is polynomial in $n$.
Let ${\bf \hat{a}} = (\frac{1}{k})_{i \in [k]}$ be a vector representing an equal
distribution of colours. Define ${\bf \hat{b}} = (\frac{1}{k(k-1)})_{(i,i') \in K}$.
Due to Molloy and Reed~\cite{Mo92}, the number of configurations respecting a fixed ${\bf a}$ is maximized when ${\bf a} =
{\bf \hat{a}}$. Furthermore, we claim that if ${\bf a} = {\bf \hat{a}}$, then $f({\bf \hat{a}},{\bf b})$ is maximized at ${\bf b} = {\bf \hat{b}}$.
Note that
\[ f({\bf \hat{a}}, {\bf b}) = \sum_{i < i'} -d b_{i,i'} \log(b_{i,i'}) -(d-1)\log k -
\frac{d}{2}\log\left(2\right). \]
We maximize $f({\bf \hat{a}}, {\bf b})$ subject to the relaxed constraints
\begin{equation}\label{eq:brelaxedcons}
b_{i,i'} \ge 0 \ \forall (i,i') \in K \quad \text{and} \quad \sum_{(i,i') \in K} b_{i,i'} = 1.
\end{equation}
First we note that the boundary is closed but does not contain a maximum: if,
say, $b_{1,2} = 0$ and $b_{2,1} > 0$, then setting $b'_{1,2} = b'_{2,1} =
\frac{1}{2}b_{2,1}$ increases $f$ as
\[ -b_{2,1} \log (b_{2,1}) < -b_{2,1} \log (\tfrac{1}{2}b_{2,1}) \]
where we take the convention $0 \log 0 = 0$.
Using the method of Lagrange multipliers, we see
\[ \frac{\partial}{\partial b_{i,i'}}[-d b_{i,i'}\log(b_{i,i'}) ] =
-d(1+\log(b_{i,i'})) = \lambda \]
for each $i \ne i'$. Then
\[ b_{i,i'} = \exp\left(-1-\frac{\lambda}{d}\right) \]
and thus
\[ 1 = \sum_{\substack{i,i'\in[k]\\i\ne i'}} b_{i,i'} =
k(k-1)\exp(-1-\frac{\lambda}{d}) \]
or
\[ \lambda = -d\left(1+\log\left(\frac{1}{k(k-1)}\right)\right) \]
which gives
\[ b_{i,i'} = \frac{1}{k(k-1)} \]
for each $i \ne i'$.
Note that the constraints in Equation~\eqref{eq:brelaxedcons} are a relaxation of the
constraints in Equation~\eqref{eq:bcons}. Therefore
\[ \max_{{\bf b} \text{ s.t. } (\ref{eq:bcons})} f({\bf \hat{a}},{\bf b}) \le \max_{{\bf b} \text{
s.t. } (\ref{eq:brelaxedcons})} f({\bf \hat{a}},{\bf b}) = f({\bf \hat{a}}, {\bf \hat{b}}) \]
which gives
\[ \exp(nf({\bf \hat{a}},{\bf \hat{b}})) = \left(k\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right)^{\frac{d}{2}}\right)^n. \]
Note that each of the $\frac{dn}{2}$ edges of $G$ is adjacent to two vertices of
different colours and determining those colours for each edge specifies ${\bf b}$.
Thus there are no more than
\[ \left(\frac{dn}{2}\right)^{k(k-1)} = O\left(n^{k(k-1)}\right) \]
possible assignments for ${\bf b}$, so
\[ \sum_{{\bf b} \text{ s.t.} (\ref{eq:bcons})} \exp(nf({\bf \hat{a}},{\bf b})) =
O\left(n^{k(k-1)}\right)\left(k\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right)^{\frac{d}{2}}\right)^n. \]
Similarly, there are at most $n^k$ assignments of colours to vertices, each of
which determines an assignment to ${\bf a}$, so
\[ \sum_{{\bf a} \text{ s.t. } (\ref{eq:acons})} \sum_{{\bf b} \text{ s.t. }
(\ref{eq:bcons})} \exp(nf({\bf a}, {\bf b})) = O(n^k) \sum_{{\bf b} \text{ s.t. }
(\ref{eq:bcons})} \exp(nf({\bf \hat{a}}, {\bf b})) =
O\left(n^{k^2}\right)\left(k\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right)^{\frac{d}{2}}\right)^n. \]
Now recalling that $\poly(n) = O(n^m)$ for some $m$,
\[ \mathbf{E} X = \sum_{{\bf a} \text{ s. t. } (\ref{eq:acons})} \sum_{{\bf b} \text{ s.t. } (\ref{eq:bcons})} \poly(n)e^{nf({\bf a},{\bf b})} = O(n^{k^2+m})\left(k\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right)^{\frac{d}{2}}\right)^n. \]
Note that
\[ d > u_k \implies d\log\left(\frac{2k}{k-1}\right) > 2\log k \implies 1 >
k\left(\frac{1}{2}\left(1 - \frac{1}{k}\right)\right)^{d/2}. \]
Therefore when $d > u_k$, $\mathbf{E} X = o(1)$ and there are a.a.s. no $k$-colourings of $\vec{\mathcal{C}}(n,d)$.
\end{proof}
\section{Upper Bounds} \label{sec:ub}
In this section we use the second moment method to get upper bounds for
${\chi_o}(\vec{\mathcal{M}}(n,m))$ and ${\chi_o}(\vec{\mathcal{C}}(n,d))$ by finding
values of $k$ for which oriented colourings are likely to exist. To simplify the
calculations, we narrow our consideration to particularly well-behaved oriented
colourings. If $T$ is a tournament on $k$ vertices, then a $T$-colouring of
$\vec{C}$ is a colouring with a homomorphism onto $T$. We call a $k$-colouring
(oriented or otherwise) \textit{equitable} if there are $\frac{n}{k}$ vertices
of each colour. Note that this condition requires that $n$ is divisible by $k$.
We make that assumption throughout this section and discuss how to modify these
arguments for $n$ not a multiple of $k$ in Section~\ref{ssec:divisibility}.
\subsection{$\vec{\mathcal{M}}(n,m)$}
Let $c > 0 $ be real and $k \ge 3$ be an integer such that there exists a doubly regular tournament of order $k$, denoted $T_k$. Let $Y$ be the number of equitable $T_k$ oriented colourings of $\vec{\mathcal{M}}(n,m=cn)$. The first moment is straightforward to calculate.
\begin{lemma} \label{lem:gnm_first_mom}
\[ \mathbf{E} Y \sim k^{k/2}(2\pi n)^{-(k-1)/2}\left(k\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right)^c\right)^n. \]
\end{lemma}
\begin{proof}
As we are only counting equitable colourings, the probability an arc chosen at random is correct is precisely $\frac{1}{2}(1-\frac{1}{k})$. Therefore the probability a colouring is correct is $(\frac{1}{2}(1-\frac{1}{k}))^m$ and
\begin{align*}
\mathbf{E} Y &= \frac{n!}{((n/k)!)^k}\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right)^m\\
&\sim \frac{\sqrt{2 \pi n}}{(\sqrt{2 \pi n/k})^k} \cdot \frac{(n/e)^n}{((n/ek)^{n/k})^k}\left(\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right)^c\right)^n\\
&= k^{k/2}(2\pi n)^{-(k-1)/2}\left(k\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right)^c\right)^n.
\end{align*}
\end{proof}
The second moment will require summing over the possible overlap matrices of
pairs of colourings. To accurately estimate the sum of this lattice, we use the
following Laplace summation technique of P\'erez and the second author~\cite{NP21} based on
work by Greenhill, Janson, and Ruc\'inski~\cite{GJR10}:
\begin{theorem}[{\cite[Proposition 3.4]{NP21}}]\label{thm:laplace_summation_gamma}
Suppose the following:
\begin{enumerate}[(i)]
\item $\Gamma=(V_\Gamma, E_\Gamma)$ is a non-empty bipartite multigraph with at least one cycle.
\item $D$ is the unsigned incidence matrix of $\Gamma$ .
\item $\tau(\Gamma)$ is the number of maximal forests in $\Gamma$.
\item $\mathbb V = \Ker(D) \subseteq \mathbb{R}^{|E_\Gamma|}$ is a vector space of dimension $r$.
\item $\mbf y\in\mathbb{R}^{|V_\Gamma|}$ such that
\begin{equation}
D\mbf x = \mbf y
\label{eq:Dxy}
\end{equation}
is a consistent linear system.
\item $K \subset \mathbb{R}^{|E_\Gamma|}$ is a compact convex set with non-empty interior $K^\circ$.
\item $\phi : K \to \mathbb{R}$ is a continuous function and the maximum of $\phi$ in $K$ subject to~\eqref{eq:Dxy} is attained at a unique maximizer $\mbf{\hat x} \in K^\circ$.
\item $\phi$ is twice continuously differentiable in a neighbourhood of $\mbf{\hat x}$ and $H$ is its Hessian matrix at $\mbf{\hat x}$.
\item $\psi : K_1 \to \mathbb{R}$ is a continuous function on some neighbourhood $K_1 \subseteq K$ of $\mbf{\hat x}$ with $\psi(\mbf{\hat x}) > 0$.
\item For each positive integer $n$,
\[
\mathbb X_n = \left\{ \mbf x\in K \cap \frac{1}{n} \mathbb{Z}^{|E_\Gamma|} : D\mbf x = \mbf y \right\}
\]
is non-empty, and there is a positive real number $b_n$ and a function $T_n : \mathbb X_n \to \mathbb{R}$ such that, as $n \to \infty$,
\[ T_n(\mbf x) = O( b_n e^{n\phi(\mbf x)+o(n)}),\quad \mbf x \in \mathbb X_n \]
and
\[ T_n(\mbf x) = b_n (\psi(\mbf x) + o(1)) e^{n\phi(\mbf x)}, \quad \mbf x \in \mathbb X_n \cap K_1,\]
uniformly for $\mbf x$ in the indicated sets.
\end{enumerate}
Then provided $\det(-H|_{\mathbb V} )\neq 0$, as $n \to \infty$,
\[
\sum_{\mbf x \in \mathbb X_n} T_n(\mbf x) \sim \frac{\psi(\mbf{\hat x})}{\tau(\Gamma)^{1/2} \det(-H|_{\mathbb V})^{1/2}} (2\pi n)^{r/2} b_n e^{n\phi(\mbf{\hat x})}.
\]
\end{theorem}
Using Theorem~\ref{thm:laplace_summation_gamma} and
Proposition~\ref{prop:generalized_AN}, we give the following asymptotic value for the
expected value of the square of the number of oriented colourings.
\begin{lemma}\label{lem:gnm_second_mom}
For any integer $k > 3$ such that there exists a doubly regular tournament $T_k$ of order $k$ and $0 < c < \frac{1}{2}\ell_k$,
\[ \mathbf{E} Y^2 \sim \left(\frac{(k-1)^4}{((k-1)^2-2c)^2-4c^2k^2}\right)^{(k-1)^2/4} (\mathbf{E} Y)^2.\]
\end{lemma}
\begin{proof}
For any oriented graph $\vec{G}$ with $n$ vertices, let $h_1, h_2: V(\vec{G})
\to V(T_k)$ be any two equitable oriented colourings with the tournament $T_k$.
For every $i, j \in [k]$, let $na_{i, j}$ be the number of vertices of $\vec{G}$
that receive colour $i$ with colouring $h_1$ and that receive colour $j$ with
colouring $h_2$. Let $\mbf A=(a_{ij})_{i,j=1}^k$ be the \emph{overlap matrix}
for the pair of colourings. As the two colourings are both equitable,
\begin{align}
\forall (i,j) \in [k]^2, \qquad a_{ij} &\ge 0 \notag\\
\forall i \in [k], \qquad \sum_{j=1}^k a_{ij} &= \frac{1}{k} \notag\\
\forall j \in [k], \qquad \sum_{i=1}^k a_{ij} &= \frac{1}{k} \label{eq:gnm_Acons}
\end{align}
Note that the overlap matrix can be prescribed first and then the arcs in the random oriented graph chosen.
Under the two colourings $h_1, h_2$, every vertex of $\vec{G}$ receives a pair
of colours in $[k]^2$. Let $x, y \in V(\vec{G})$ be two vertices and set $u =
(h_1(x), h_2(x))$, $v = (h_1(y), h_2(y))$. There can be an edge in $\vec{G}$
between $x$ and $y$ only in the case that there are arcs in $T_k$ between
$h_1(x)$ and $h_1(y)$ and between $h_2(x)$ and $h_2(y)$ and that the direction
of the edges matches: either $h_1(x) \to h_1(y)$ and $h_2(x) \to h_2(y)$ or else
$h_1(y) \to h_1(x)$ and $h_2(y) \to h_2(x)$. Thus, in terms of the Kroenecker
product of tournaments (Definition~\ref{def:kroenecker_tournaments}), arcs in
$\vec{G}$ between $x$ and $y$ can occur exactly when $u v \in E(T_k^{\otimes
2})$ and the direction of the arc is prescribed by the tournament $T_k$.
Thus, given an overlap matrix $\mbf A = (a_{ij})_{i, j=1}^k$, the probability an edge in a random graph $\vec{G}$ is coloured correctly is $\sum\limits_{uv \in E(T^{\otimes 2})}a_ua_v$, so
\begin{align}
\mathbf{E} Y^2 &= \sum_{\mbf A \text{ s.t. } \eqref{eq:gnm_Acons}} \frac{n!}{\prod_{v \in V} (na_v)!} \left(\sum_{uv \in E(T^{\otimes 2})}a_u a_v\right)^m \label{eq:mnm_count}\\
&=\sum_{\mbf A \text{ s.t. } \eqref{eq:gnm_Acons}} \frac{\xi(n) (n/e)^n}{\prod_{v \in V} \xi(na_v) (na_v/e)^{na_v}} \left(\sum_{uv \in E(T^{\otimes 2})}a_u a_v\right)^{cn} \notag\\
&=\sum_{\mbf A \text{ s.t. } \eqref{eq:gnm_Acons}} \xi(n) \left(\prod_{v \in V} \xi(na_v) \right)^{-1} \left(\prod_{v \in V}a_v^{-a_v} \left(\sum_{uv \in E(T^{\otimes 2})}a_u a_v\right)^c \right)^n \notag\\
&=\sum_{\mbf A \text{ s.t. } \eqref{eq:gnm_Acons}} p(n,\mbf A) e^{nf(\mbf A)} \label{eq:exp_sum}
\end{align}
where
\begin{equation}
p(n,\mbf A) = \xi(n) \left(\prod_{v \in V} \xi(na_v) \right)^{-1}
\end{equation}
and
\begin{equation}
f(\mbf A) = -\sum_{v \in V} a_v \log a_v + c \log\left(\sum_{uv \in E(T^{\otimes 2})}a_u a_v\right)
\end{equation}
We seek to apply Theorem~\ref{thm:laplace_summation_gamma} to the function $f$. Assign to each constraint of Equation~\eqref{eq:gnm_Acons} of the form $\sum_{j=1}^k a_{ij} = \frac{1}{k}$ a vertex $w_{1,i}$ and to each constraint of the form $\sum_{i=1}^k a_{ij} = \frac{1}{k}$ a vertex $w_{2,j}$. Let $V_\Gamma = \{w_{1,i}, w_{2,j}\}_{i,j=1}^k$, $E_\Gamma = \{\{w_{1,i}, w_{2,j}\} \mid i,j \in [k]\}$, and $\Gamma = (V_\Gamma, E_\Gamma)$. Then $\Gamma \cong K_{k,k}$, and as $k > 3$, $\Gamma$ contains a cycle. The equality constraints in Equation~\eqref{eq:gnm_Acons} are equivalent to $D\mbf x = \frac{1}{k} \mbf{1}_{2k}$ where $D$ is the unsigned incidence matrix of $\Gamma$. Onodera~\cite{On73} proved that the number of spanning trees of $K_{m,n}$ is $m^{n-1}n^{m-1}$, so $\tau(\Gamma) = k^{2k-2}$. As $\Gamma$ is bipartite and connected, standard results (see e.g. \cite[Theorem 8.2.1]{GR01}) give $\dim D = |V_\Gamma|-1 = 2k-1$ and therefore
\[ r = \dim \mathbb{V} = k^2 - \dim D = k^2-2k+1 = (k-1)^2. \]
Setting $\mbf y = \frac{1}{k} \mbf{1}_{2k}$ we see $D \mbf x = \mbf y$ is a consistent linear system; $\mbf x = \frac{1}{k^2} \mbf{1}_{k^2}$ is a solution. Set
\[ K = \{ \mbf x \in \mathbb{R}^{k^2} \mid 0 \le a_v \le \tfrac{1}{k} \} \quad \text{and} \quad K_1 = \{ \mbf x \in \mathbb{R}^{k^2} \mid \tfrac{0.9}{k^2} \le a_v \le \tfrac{1.1}{k^2}\}. \]
Then $K$ is a compact convex set with nonempty interior, $K_1 \subseteq K$, and $K_1$ contains the point $\mbf{\hat{x}} = \frac{1}{k^2}J_k$. Set $\alpha_v = ka_v$, let $\mbf x = (\alpha_v)_{v \in V}$, and define $\phi(\frac{1}{k} \mbf x) = f(\mbf{A})$ so that
\[ \phi(\mbf x) = -\frac{1}{k}\sum_{v \in V} \alpha_v \log \alpha_v + c\log\left(\frac{2\sum_{uv \in E} \alpha_u \alpha_v}{k^2}\right)+\log k - c \log 2. \]
Then by Corollary~\ref{cor:opt_doubly_reg_tour}, for $2c \le \frac{2(k-1)^3}{k (k+1)(k-2)} \log(k-1) = \ell_k$, $\phi$, which is continuous on $K^\circ$, is uniquely maximized at $\mbf{\hat{x}} \in K_1$. We have
\[ \frac{\partial \phi}{\partial a_u} = -\log a_u + 1 +c \cdot \frac{\sum_{w \sim u} a_w}{\sum_{xy \in E} a_xa_y} \]
and therefore
\[ \frac{\partial^2 \phi}{\partial a_u\partial a_v} = \begin{cases} \displaystyle -\frac{1}{a_u} - c\left(\frac{\sum_{w \sim u}a_w}{\sum_{xy \in E} a_xa_y} \right)^2 & v = u\\ \displaystyle -c\cdot \frac{(\sum_{w \sim u}a_w)(\sum_{w \sim v}a_w)}{\left(\sum_{xy \in E}a_xa_y\right)^2} & v \not\sim u \\ \displaystyle c\left(\frac{1}{\sum_{xy \in E}a_xa_y} - \frac{(\sum_{w \sim u}a_w)(\sum_{w \sim v}a_v)}{\left(\sum_{xy \in E}a_xa_y\right)^2}\right) & v \sim u \end{cases}. \]
Letting $B$ denote the adjacency matrix for the graph $T^{\otimes 2}$. This gives
\begin{equation}\label{eq:Mnm-hessian}
H|_{\mbf{\hat{x}}} = -\left(k^2I_{k^2}+4cJ_{k^2}-\frac{4ck^2}{(k-1)^2}B\right).
\end{equation}
Then for any matrix $U$ whose columns form a basis for $\ker(D)$,
\begin{equation}\label{eq:Hessian-det}
\det(-H|_{\mathbb{V}}) = \frac{\det(U^T(-H)U)}{\det(U^TU)}.
\end{equation}
We give an explicit expression for the matrix $U$ and the corresponding determinant of the Hessian over the vector space $\ker(D)$. Letting $e_i$ denote the $i$-th unit vector in $\mathbb{R}^k$ and using the natural correspondence between $\mathbb{R}^{k^2}$ and $\mathbb{R}^k \otimes \mathbb{R}^k$, the following is a basis for $\ker(D)$:
\[
\mathcal{B} = \left\{ (e_1 - e_i) \otimes (e_1 - e_j) :\ 2 \leq i, j \leq k \right\}.
\]
Since the set $\{(e_1 - e_i) :\ 2 \leq i \leq k \}$ is linearly independent over $\mathbb{R}^k$, then $\mathcal{B}$ is a set of $(k-1)^2$ linearly independent vectors in $\mathbb{R}^k \otimes \mathbb{R}^k$. With the appropriate indexing, these are naturally elements in $\ker(D)$. Recall that $D$ is the incidence matrix of the graph $K_{k, k}$. Label the edges of $K_{k, k}$ in a natural way with the elements of the set $[k]^2$. Then, the vector $(e_1 - e_i) \otimes (e_1 - e_j)$ corresponds to the vector in $\mathbb{R}^{k^2}$ whose entries corresponding to the edges $(1, 1)$ and $(i, j)$ are $1$, entries corresponding to the edges $(i, 1)$ and $(1, j)$ are $-1$ and all other entries are $0$.
Define the $k \times (k-1)$ matrix
\[
V = \left[\begin{array}{c}
\mbf{1}_{k-1}^T \\ \hline
-I_{k-1}
\end{array}\right].
\]
Then the matrix $U = V \otimes V$ has columns that form a basis for $\ker(D)$. Note that
\begin{equation}\label{eq:vtransposev}
V^T V = \mbf{1}_{k-1} \mbf{1}_{k-1}^T + (-1)^2 I_{k-1} = J_{k-1} + I_{k-1}
\end{equation}
and since every column of $V$ has entries summing to $0$, $J_k V = 0_{k \times (k-1)}$.
With this in mind, consider the Hessian matrix in Equation~\eqref{eq:Mnm-hessian}. Recall that $M$ is the signed adjacency matrix of the doubly-regular tournament $T_k$.
\begin{align}
-H
&=k^2I_{k^2}+4cJ_{k^2}-\frac{4ck^2}{(k-1)^2}B \notag \\
&=k^2I_{k^2}+4cJ_{k^2} - \frac{4ck^2}{(k-1)^2} \cdot \frac{1}{2}\left(M \otimes M + (J_k - I_k) \otimes (J_k - I_k)\right) \notag\\
&=k^2 \left(1 - \frac{2c}{(k-1)^2}\right) I_k \otimes I_k + 4cJ_k \otimes J_k \notag \\
& \qquad - \frac{2ck^2}{(k-1)^2}\left(M \otimes M - J_k \otimes I_k - I_k \otimes J_k + J_k \otimes J_k\right) \notag
\end{align}
Thus, considering the matrix expression in the numerator of Equation~\eqref{eq:Hessian-det}, and using the fact that $J_k V = 0_{k \times (k-1)}$ and hence $V^T J_k = 0_{(k-1) \times k}$,
\begin{align*}
U^T(-H)U
&=(V^T \otimes V)(-H) (V \otimes V)\\
&= k^2 \left(1 - \frac{2c}{(k-1)^2}\right) (V^T V \otimes V^T V) - \frac{2ck^2}{(k-1)^2}\left((V^T \otimes V^T) (M \otimes M) (V \otimes V)\right)\\
\end{align*}
By a determinant theorem of Sylvester,
\begin{multline}\label{eq:Sylvester-det-swap}
\det(U^T(-H)U)
=\det\left(k^2 \left(1 - \frac{2c}{(k-1)^2}\right) (V^T V \otimes V^T V)\right)\cdot\\
\det\left(I_{k}^2 - \frac{2c/(k-1)^2}{1 - 2c/(k-1)^2} \cdot (M \otimes M) (V \otimes V) (V^TV \otimes V^T V)^{-1} (V^T \otimes V^T)\right)
\end{multline}
Note that $V^TV = J_{k-1} + I_{k-1}$ and hence $(V^T V)^{-1} = \frac{1}{k}\left(k I_{k-1} - J_{k-1}\right)$. Thus,
\[
(V^TV \otimes V^T V)^{-1} = (V^TV)^{-1} \otimes (V^TV)^{-1} = \frac{1}{k^2}\left(k I_{k-1} - J_{k-1}\right) \otimes \left(k I_{k-1} - J_{k-1}\right).
\]
Since
\begin{align*}
MV(k I_{k-1} - J_{k-1}) V^T
&=M \left(k V V^T - VJ_{k-1}V^T\right)\\
&=M\left(k \begin{bmatrix} \begin{tabular}{c|c} $(k-1)$ &$-1_{k-1}^T$\\ \hline $-1_{k-1}$ &$I_{k-1}$ \end{tabular}\end{bmatrix} - \begin{bmatrix} \begin{tabular}{c|c} $(k-1)^2$ &$-(k-1)1_{k-1}^T$\\ \hline $-(k-1)1_{k-1}$ &$J_{k-1}$ \end{tabular}\end{bmatrix}\right)\\
&=M(kI_k - J_k)\\
&=kM,
\end{align*}
then
\[
(M \otimes M) (V \otimes V) (V^TV \otimes V^T V)^{-1} (V^T \otimes V^T) = \frac{1}{k^2} (kM) \otimes (kM) = M \otimes M.
\]
Therefore, using Equation~\eqref{eq:Sylvester-det-swap} and the eigenvalues for $M$ given in Lemma~\ref{lem:eval-dr-tourn},
\begin{align*}
\det(U^T(-H)U)
&=\det\left(k^2 \left(1 - \frac{2c}{(k-1)^2}\right) (V^T V \otimes V^T V)\right) \det\left(I_{k}^2 - \frac{2c/(k-1)^2}{1 - 2c/(k-1)^2} \cdot (M \otimes M)\right)\\
&=\det\left(k^2 \left(\frac{(k-1)^2 - 2c}{(k-1)^2}\right) (V^T V \otimes V^T V)\right) \det\left(I_{k}^2 - \frac{2c}{(k-1)^2 - 2c} \cdot (M \otimes M)\right)\\
&=\left(k^2 \left( \frac{(k-1)^2- 2c}{(k-1)^2}\right)\right)^{(k-1)^2} \det(U^TU) \cdot (1 - 0)^{2k-1} \cdot\\
&\qquad \left(1 - k \frac{2c}{(k-1)^2 - 2c}\right)^{(k-1)^2/2} \cdot \left(1 + k \frac{2c}{(k-1)^2 - 2c}\right)^{(k-1)^2/2}\\
&=\left(\frac{k^2}{(k-1)^2}\right)^{(k-1)^2} \det(U^TU) \cdot \left(((k-1)^2 - 2c)^2 - 4k^2c^2\right)^{(k-1)^2/2}
\end{align*}
Thus, from Equation~\eqref{eq:Hessian-det},
\begin{align*}
\det(-H|_{\mathbb{V}})
&= \left(\frac{k^2}{(k-1)^2}\right)^{(k-1)^2} \det(U^TU) \cdot \left(((k-1)^2 - 2c)^2 - 4k^2c^2\right)^{(k-1)^2/2}/ \det(U^TU)\\
&= \left(\frac{k^2}{(k-1)^2}\right)^{(k-1)^2} \cdot \left(((k-1)^2 - 2c)^2 - 4k^2c^2\right)^{(k-1)^2/2}.
\end{align*}
Note that for $k > 1$, $\det(-H|_{\mathbb{V}})$ has zeros at $k = (c+1) \pm \sqrt{c^2+4c}$, so for $c < \frac{1}{2}\ell_k = o(k), \det(-H|_{\mathbb{V}}) > 0$.
Let $\psi = \prod_{v\in V} a_v^{-1/2}$. Then $\psi$ is continuous on $K_1$ and $\psi(\mbf{\hat{x}}) = k^{k^2} > 0$. Set $b_n = (2 \pi n)^{-(k^2-1)/2}$ and $T_n(\mbf x) = p(n,\mbf{x})e^{nf(\mbf{x})}$. Then as $1 \le \xi(n) \sim \sqrt{2\pi n}$,
\[ p(n,\mbf{x}) = \xi(n)\left(\prod_{v \in V} \xi(na_v)\right)^{-1} = O(\sqrt{n}) \]
so
\[ p(n, \mbf{x})/b_n = O(\sqrt{n}/b_n) = e^{o(n)} \]
and
\[ T_n(\mbf x) = b_n(p(n,\mbf{x})/c_b)e^{nf(\mbf x)} = O(b_ne^{n\phi(\mbf{\hat{x}})+o(n)}). \]
Furthermore, for $\mbf{x} \in K_1$,
\[ p(n, \mbf x) = b_n(\psi(\mbf x)+o(1)). \]
Therefore Theorem~\ref{thm:laplace_summation_gamma} gives
\begin{align*}
\mathbf{E} Y^2 = \sum_{\mbf x \in \mathbb{X}_n} T_n(\mbf x) &\sim \frac{\psi(\mbf{\hat{x}})}{\tau(\Gamma)^{1/2}\det(-H|_{\mathbb{V}})^{1/2}}(2\pi n)^{r/2}c_ne^{n\phi(\mbf{\hat{x}})}\\
&=\frac{k^{k^2}(2\pi n)^{(k-1)^2/2}(2\pi n)^{-(k^2-1)/2}}{k^{(k-1)}(\frac{k^2}{(k-1)^2})^{(k-1)^2/2}(((k-1)^2-2c)^2-4c^2k^2)^{(k-1)^2/4}}e^{2n\log(k(\frac{1}{2}(1-\frac{1}{k}))^c)}\\
&= \left(\frac{(k-1)^4}{((k-1)^2-2c)^2-4c^2k^2}\right)^{(k-1)^2/4} k^k(2\pi n)^{-(k-1)}\left(k\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right)^c\right)^{2n}\\
&\sim \left(\frac{(k-1)^4}{((k-1)^2-2c)^2-4c^2k^2}\right)^{(k-1)^2/4} (\mathbf{E} Y)^2
\end{align*}
\end{proof}
\begin{proposition} \label{prop:gnm_ub}
For $k > 3$ an integer such that there exists a doubly regular tournament of order $k$ and $0 < c < \frac{1}{2}\ell_k$, with positive probability ${\chi_o}(\vec{\mathcal{M}}(n,m=cn)) \le k$.
\end{proposition}
\begin{proof}
By the Paley-Zygmund inequality and Lemmas~\ref{lem:gnm_first_mom} and \ref{lem:gnm_second_mom},
\[ \mathbf{Pr}[Y > 0] \ge \frac{(\mathbf{E} Y)^2}{\mathbf{E}(Y^2)} \sim \left(\frac{((k-1)^2-2c)^2-4c^2k^2}{(k-1)^4}\right)^{(k-1)^2/4} > 0. \]
\end{proof}
Ideally, the ratio $\frac{(\mathbf{E} Y)^2}{\mathbf{E}(Y^2)}$ would tend to $1$. This would
show that a.a.s. $k$ is an upper bound for ${\chi_o}(\vec{\mathcal{M}}(n,m=cn))$.
In many investigations into the chromatic number of random graphs, however, this
ratio does not tend towards $1$. When considering the undirected question
$\chi(\mathcal{G}(n,m=cn))$, Achlioptas and Naor~\cite{AN05} circumvent this
issue using a sharp threshold result of Achlioptas and Friedgut~\cite{AF99}.
There is no reason to believe an equivalent result holds in the directed case.
As we will see in Section~\ref{ssec:vsparse_vdense}, there are values for $d$ for
which $\vec{\mathcal{G}}(n,p=\frac{d}{n})$ and $\vec{\mathcal{G}}(n,d)$ are
concentrated in more than one value. Instead, in Section~\ref{sec:window} we
adapt an argument of {\L}uczak~\cite{tL91b}, using the fact that $k$ is an upper
bound with positive probability to find a weaker bound that holds with high
probability.
\subsection{$\vec{\mathcal{C}}(n,d)$}
In this section we use a second moment argument to get an upper bound on ${\chi_o}(\vec{\mathcal{C}}(n,d))$. Fix a doubly regular tournament on $k$ vertices $T_k$ and let $Y$ count the number of equitable doubly regular $T_k$-colourings of $\vec{\mathcal{C}}(n,d)$.
We start with the first moment by adapting the following result of Kemkes et al.~\cite{KPW10}:
\begin{proposition}[{\cite[Proposition 2(a)]{KPW10}}]
Fix integers $d,k \ge 3$. Let $Z$ be the number of equitable $k$-colourings of a random $d$-regular multigraph $\mathcal{C}(n,d)$ (where $n$ is restricted to the set of multiples of $k$). Then
\[ \mathbf{E} Z \sim k^{k/2} \left(\frac{k-1}{2\pi(k-2)}\right)^{(k-1)/2}n^{-(k-1)/2}k^n\left(1-\frac{1}{k}\right)^{dn/2}. \]
\end{proposition}
\begin{corollary}\label{cor:first_mom}
Fix integers $d,k \ge 3$ and a tournament $T_k$ on $k$ vertices (not necessarily doubly regular). Then
\[ \mathbf{E} Y \sim k^{k/2} \left(\frac{k-1}{2\pi(k-2)}\right)^{(k-1)/2}n^{-(k-1)/2}k^n\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right)^{dn/2}. \]
\end{corollary}
\begin{proof}
Each proper (unoriented) $k$-colouring of $\mathcal{C}(n,d)$ has exactly one orientation that adheres to $T_k$, so there are the same number of proper $k$-oriented-colourings of $\vec{\mathcal{C}}(n,d)$ that are coloured by $T_k$ as there are unoriented colourings. However, each $C \in \mathcal{C}(n,d)$ has $2^{dn/2}$ orientations. Thus we have
\begin{align*}
|\vec{\mathcal{C}}(n,d)| \mathbf{E} Y &= |\{(\vec{C}, \chi_o) : \vec{C} \sim \vec{\mathcal{C}}(n,d) \text{ and $\chi_o$ properly $T_k$-colours $\vec{C}$}\}|\\
&= |\{(C, \chi) : C \sim \mathcal{C}(n,d) \text{ and $\chi$ properly $k$-colours $C$}\}|\\
&= |\mathcal{C}(n,d)| \mathbf{E} Z
\end{align*}
so
\[ \mathbf{E} Y = \frac{|\mathcal{C}(n,d)|}{|\vec{\mathcal{C}}(n,d)|} \mathbf{E} Z = \frac{1}{2^{dn/2}} \mathbf{E} Z. \]
\end{proof}
In order to prove Proposition~\ref{prop:gnd_ub}, we also need an estimate for the second moment $\mathbf{E} Y^2$. This will require a substantial amount more effort. First, in Section~\ref{ssec:counting} we give a counting argument similar to that in Section~\ref{sec:lb}. Then we use Proposition~\ref{prop:generalized_AN} to optimize the exponential part of resulting equation in Section~\ref{ssec:optimization}. Finally, we complete the second moment argument and prove Proposition~\ref{prop:gnd_ub} in Section~\ref{ssec:second_moment}.
\subsubsection{Counting Argument} \label{ssec:counting}
To calculate $\mathbf{E}(Y^2)$, we count ordered triples $(\vec{C},h_1,h_2)$ such that $h_1$ and $h_2$ are balanced proper $T_k$-colourings of $\vec{C}\in \vec{\mathcal{C}}(n,d)$. Let $A = (a_{ij})_{\substack{1 \le i \le k \\ 1 \le j \le k}}$ be the overlap matrix of $h_1$ and $h_2$; that is, $a_{ij}$ is the proportion of vertices $v$ such that $h_1(v) = i$ and $h_2(v) = j$. Note that $a_{ij} \ge 0$ for each $i,j \in [k]$ and, as $h_1$ and $h_2$ are balanced,
\begin{equation} \label{eq:Acons}
\forall i \in [k] \quad \sum_{j=1}^k a_{ij} = \frac{1}{k} \qquad \text{and} \qquad \forall j \in [k] \quad \sum_{i=1}^k a_{ij} = \frac{1}{k}.
\end{equation}
We introduce an (undirected) auxiliary graph $\Gamma$. The vertex set of $\Gamma$ is $[k]^2$, so there is $v \in V(\Gamma)$ corresponding to each $a_{ij}$, and we may, when convenient, refer to $a_{ij}$ as $a_v$. The edge set of $\Gamma$ is defined by
\[ E(\Gamma) = \bigl\{ \{u,v\} \mid u = (i,j), v = (i',j'), \ \text{and} \ i \to i', j \to j' \in A(T_k)\bigr\}. \]
Note that if we partition edges in $\vec{C}$ by the pairs of colours their vertices receive in $h_1$ and $h_2$, the set of equivalence classes is exactly the edges of $\Gamma$: in order for an edge to be directed from a vertex receiving colour $i$ in $h_1$ and colour $j$ in $h_2$ towards a vertex receiving colour $i'$ in $h_1$ and $j'$ in $h_2$, we require that $i \to i'$ and $j \to j'$ in $T_k$. Note also that as $T_k$ is doubly regular, $\Gamma$ is regular of degree $\frac{(k-1)^2}{2}$.
Then let $B = (b_{uv})_{u=(i,j), v = (i',j')}$ record the proportion of edges between a vertex $u$ such that $h_1(u) = i$ and $h_2(u) = j$ and a vertex $v$ such that $h_1(v) = i'$ and $h_2(v) = j'$. Then for any tournament $T_k$, $B$ satisfies
\begin{align}\label{eq:Bcons}
\forall u,v & \quad b_{uv} \ge 0 \notag\\
\forall u,v &\quad \text{$b_{uv} = 0$ unless $i \to i'$ and $j \to j'$ (or $i' \to i$ and $j' \to j$) in $T_k$} \notag\\
\forall u,v & \quad b_{uv} = b_{vu} \notag\\
\forall v = (i,j) & \quad \sum\limits_{u \sim v} b_{uv} = a_{ij}.
\end{align}
To choose a $\vec{C} \in \vec{\mathcal{C}}(n,d)$ that respects $h_1$ and $h_2$, first select vertices to receive each pair of colours, which we do in one of
\[ \frac{n!}{\prod_{i,j} (na_{ij})!} \]
ways. Then, for each pair of colour classes, select the vertices that will be connected by an edge, which we do in
\[ \prod_{i,j} \frac{(dna_{ij})!}{\prod_{u\sim v=(i,j)} (dnb_{uv})!} \]
many ways. Finally, match the chosen vertices of each edge type in one of
\[ \prod_{uv \in E} (dnb_{uv})! \]
ways. Noting that
\[ \frac{\prod_{uv \in E} (dnb_{uv})!}{\prod_{v \in V} \prod_{u \sim v} (dnb_{uv})!} = \frac{1}{\prod_{uv \in E} (dnb_{uv})!} ,\]
repeating this process for each choice of $A$ and $B$ that satisfy (\ref{eq:Acons}) and (\ref{eq:Bcons}), and dividing by $|\vec{\mathcal{C}}(n,d)|$ gives
\begin{align}
\mathbf{E}(Y^2) &= \frac{1}{2^{dn/2}(dn-1)!!} \sum_{A \text{ s.t. } \eqref{eq:Acons}} \frac{n!}{\prod_{ij} (na_{ij})!} \sum_{B \text{ s.t. } \eqref{eq:Bcons}} \prod_{i,j} \frac{(dna_{ij})!}{\prod_{u\sim v=(i,j)} (dnb_{uv})!} \prod_{uv \in E} (dnb_{uv})! \notag\\
&= \sum_{A \text{ s.t.} \eqref{eq:Acons}, \ B \text{ s.t.} \eqref{eq:Bcons}} \frac{n!}{\prod_{ij} (na_{ij})!} \cdot \frac{\prod_{ij} (dna_{ij})!}{(dn)!} \cdot \frac{(\frac{dn}{2})!}{\prod_{uv \in E} (dnb_{uv})!} \notag\\
&= \sum_{A \text{ s.t.} \eqref{eq:Acons}, \ B \text{ s.t.} \eqref{eq:Bcons}} \frac{\xi(n)(\frac{n}{e})^n}{\prod_{ij} \xi(na_{ij})(\frac{na_{ij}}{e})^{na_{ij}}} \cdot \frac{\prod_{ij} \xi(dna_{ij})(\frac{dna_{ij}}{e})^{dna_{ij}}}{\xi(dn)(\frac{dn}{e})^{dn}} \cdot \frac{\xi(\frac{dn}{2})(\frac{dn}{2e})^{\frac{dn}{2}}}{\prod_{uv \in E} \xi(dnb_{uv})(\frac{dnb_{uv}}{e})^{dnb_{uv}}} \notag\\
&= \sum_{A \text{ s.t.} \eqref{eq:Acons}, \ B \text{ s.t.} \eqref{eq:Bcons}} p(A,B,n) e^{nf(A,B)} \label{eq:second_moment}
\end{align}
where
\begin{equation} \label{eq:p_def}
p(A,B,n) = \frac{\xi(n)\xi(\frac{dn}{2})\prod\limits_{v \in
V}\xi(dna_v)}{\xi(dn)\prod\limits_{v \in V}\xi(na_v) \prod\limits_{uv
\in E}\xi(dnb_{uv})}
\end{equation}
and
\begin{equation}\label{eq:f1}
f(A,B) = (d-1)\sum_{v \in V} a_v \log a_v - d\sum_{uv \in E} b_{uv} \log
(2b_{uv}).
\end{equation}
\subsubsection{Optimization} \label{ssec:optimization}
In order to estimate $\mathbf{E} Y^2$, we will use a Laplace summation technique. As we are summing over a polynomial number of exponential terms, all but $o(1)$ of the weight of the sum comes from terms with maximum exponential contribution. In this section we use Proposition~\ref{prop:generalized_AN} to determine that maximum value.
\begin{lemma} \label{lem:second_mom_opt}
Fix an integer $k$ such that there exists a doubly regular tournament $T_k$ of
order $k$ and let $d < \ell_k$. Let $A = (a_{ij})_{i,j=1}^k$ be a $k \times k$
real positive matrix satisfying Equation~\eqref{eq:Acons} and let $B = (b_{uv})_{u=(i,j),
v = (i',j')}$ be a $k^2 \times k^2$ real positive matrix satisfying
Equation~\eqref{eq:Bcons}. Define $f(A,B)$ as in Equation~\eqref{eq:f1}. Then $f(A,B)$ is uniquely
maximized at $f(\hat{A},\hat{B}) = \log k^2 + d\log(\frac{1}{2}(1-\frac{1}{k}))$
where $\hat{A} = \frac{1}{k^2}J_{k,k}$ and $\hat{B} = (\hat{b}_{uv})_{u=(i,j), v
= (i',j')}$ where
\[ \hat{b}_{uv} = \begin{cases} \frac{2}{k^2(k-1)^2} & u\sim v \\ 0 & u \not\sim v \end{cases}. \]
\end{lemma}
\begin{proof}
We start by finding a more convenient expression of $f$. Note that using Equation~\eqref{eq:Bcons},
\begin{align}
(d-1)\sum_{v \in V} a_v\log a_v - d\sum_{uv \in V} &b_{uv}\log(2b_{uv}) = -\sum_{v \in V} a_v \log a_v +d\left(\sum_{v \in V} a_v \log a_v + \sum_{uv \in E} b_{uv} \log\left(\frac{1}{2b_{uv}}\right) \right) \notag\\
&= -\sum_{v \in V} a_v \log a_v +d\left(\sum_{v \in V} \left(\sum_{u \sim v} b_{uv}\right) \log a_v + \sum_{uv \in E} b_{uv} \log\left(\frac{1}{2b_{uv}}\right) \right) \notag\\
&= -\sum_{v \in V} a_v \log a_v +d\sum_{uv \in E} b_{uv} \left(\log a_u + \log a_v + \log\left(\frac{1}{2b_{uv}}\right) \right) \notag\\
&= -\sum_{v \in V} a_v \log a_v +d\sum_{uv \in E} b_{uv} \log\left(\frac{a_ua_v}{2b_{uv}}\right) \label{eq:f2}
\end{align}
Using an idea from~\cite{NP21}, we now maximize $f$ maintaining the constraints from Equation~\eqref{eq:Acons} but relaxing those in Equation~\eqref{eq:Bcons} to $b_{uv} \ge 0$ and
\begin{equation} \label{eq:Brelaxedcons}
\sum_{u, v \in V} b_{uv} = 1.
\end{equation}
In view of this relaxation, we can regard $B$ as an arbitrary probability distribution. We define another probability distribution given by
\[
b^*_{uv} = \begin{cases} \frac{a_{u}a_{v}}{2\sum_{xy \in E} a_{x}a_{y}} & u\sim v, \\ 0 & u \not\sim v\end{cases},
\]
and we write $B^*=(b^*_{uv})_{u,v \in V}$. Then
\[
\sum_{uv \in E} b_{uv}\log\left( \frac{a_{u}a_{v}}{2b_{uv}}\right) = \frac{1}{2}\log \left(\sum_{xy \in E} a_xa_y\right) - D_{KL}(B \| B^*),
\]
where $D_{KL}(B \| B^*) = \sum_{uv} b_{uv}\log\left( \frac{b_{uv}}{b^*_{uv}}\right)$ is the Kullback-Leibler divergence from $B$ to $B^*$. By Gibb's inequality, $D_{KL}(B \| B^*)\ge0$ with equality if and only if $B = B^*$. As a result,
\begin{equation}\label{eq:maxgamma}
\max_{B\text{ s.t.~\eqref{eq:Brelaxedcons}}} f(A, B) = -\sum_{v\in V} a_v \log a_v + \frac{d}{2}\log\left(\sum_{xy \in E} a_xa_y\right) =: \tilde{f}_d(A),
\end{equation}
with one unique maximizer at $B = B^*$. Note that if $A=\hat A$ then $B^*=\hat B$.
Next we seek to apply Corollary~\ref{cor:opt_doubly_reg_tour}, but our matrix $A$ is not currently doubly stochastic. We reparameterize, setting $\mbf \alpha = kA$. Then
\[ \tilde{f}_d(A) = \tilde{f}_d(\tfrac{1}{k}\mbf \alpha) = \log k-\frac{1}{k} \sum_{v \in V} \alpha_v \log \alpha_v + \frac{d}{2} \log\left(\frac{2\sum_{xy \in E} \alpha_x\alpha_y}{k^2}\right) - \frac{d}{2}\log2. \]
Then as $\log k-\frac{d}{2}\log2$ is constant with respect to $\mbf \alpha$, Corollary~\ref{cor:opt_doubly_reg_tour} gives
\begin{equation} \label{eq:gamma_ub}
\tilde{f}_d(A) \le \log k +\log k + \frac{d}{2}\log\left(\frac{2|E|}{k^4}\right) -\frac{d}{2}\log2 = \log k^2 + d\log\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right)
\end{equation}
whenever $d \leq \frac{2 (k-1)^3}{k(k+1)(k-2)}\log(k-1)$. Now, using Equations \eqref{eq:maxgamma} and \eqref{eq:gamma_ub}, we have
\begin{align*}
f(A,B) \le \max_{A \text{ s.t.} \eqref{eq:Acons}, \ B \text{ s.t.} \eqref{eq:Bcons}} f(A,B) &\le \max_{A \text{ s.t.} \eqref{eq:Acons}, \ B \text{ s.t.} \eqref{eq:Brelaxedcons}} f(A,B)\\
&\le \max_{A \text{ s.t.} \eqref{eq:Acons}} \tilde{f}_d(A) = \log k^2 + d\log\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right) = f(\hat{A},\hat{B})
\end{align*}
\end{proof}
\subsubsection{Second Moment Argument} \label{ssec:second_moment}
We now return to Equation~\eqref{eq:second_moment}. In order to estimate the
summation, we again use a version of Laplace summation. This time, however, the
constraints in Equations \eqref{eq:Acons} and \eqref{eq:Bcons} cannot be expressed as the signed incidence matrix of an auxiliary graph. Instead, we use the following theorem proved by Greenhill et al. in \cite{GJR10}.
\begin{theorem}[{\cite[Theorem 2.3]{GJR10}}]\label{thm:laplacian_summation}
Suppose the following:
\begin{enumerate}[(i)]
\item $\Lambda \subset \mathbb{R}^N$ is a lattice with rank $1\le r \le N$.
\item $\mathbb V \subseteq \mathbb{R}^N$ is the $r$-dimensional subspace spanned by $\Lambda$.
\item $\mathbb W = \mathbb V + \mbf w$ is an affine subspace parallel to $\mathbb V$, for some $\mbf w \in \mathbb{R}^N$.
\item $K \subset \mathbb{R}^N$ is a compact convex set with non-empty interior $K^\circ$.
\item $\phi : K \to \mathbb{R}$ is a continuous function and the restriction of $\phi$ to $K \cap \mathbb W$ has a unique maximum at some point $\mbf z_0 \in K^\circ \cap \mathbb W$.
\item $\phi$ is twice continuously differentiable in a neighbourhood of $\mbf z_0$ and $H := D^2\phi(\mbf z_0)$ is its Hessian at $\mbf z_0$.
\item $\psi : K_1 \to \mathbb{R}$ is a continuous function on some neighbourhood $K_1 \subseteq K$ of $\mbf z_0$ with $\psi(\mbf z_0) > 0$.
\item For each positive integer $n$ there is a vector $\ell_n \in \mathbb{R}^N$ with $\ell_n/n \in \mathbb W$,
\item For each positive integer $n$ there is a positive real number $b_n$ and a function $a_n : (\Lambda + \ell_n) \cap nK \to \mathbb{R}$ such that, as $n \to \infty$,
\[ a_n(\ell) = O(b_n e^{n\phi(\ell/n)+o(n)}),\quad \ell \in (\Lambda +\ell_n) \cap nK \]
and
\[ a_n(\ell) = b_n(\psi(\ell/n) + o(1)) e^{n\phi(\ell/n)}, \quad \ell \in (\Lambda +\ell_n) \cap nK_1,\]
uniformly for $\ell$ in the indicated sets.
\end{enumerate}
Then provided $\det(-H|_{\mathbb V} )\neq 0$, as $n \to \infty$,
\[ \sum_{\ell\in(\Lambda+\ell_n)\cap nK}a_n(\ell) \sim \frac{(2\pi)^{r/2}\psi(\mbf z_0)}{\det(\Lambda)\det(-H|_{\mathbb V})^{1/2}}b_nn^{r/2}e^{n\phi(\mbf z_0)}. \]
\end{theorem}
We also reference this additional results from that paper:
\begin{lemma}[{\cite[Lemma 6.2]{GJR10}}] \label{lem:det_lattice}
Let $0 \le m \le N$. Let $x_1, \ldots, x_m$ be linearly independent vectors in $\mathbb{Z}^N$. Let $V$ be the subspace of $\mathbb{R}^N$ spanned by $x_1, \ldots, x_m$ and let $V^{\perp}$ be its orthogonal complement; thus
\[ V^{\perp} = \{y \in \mathbb{R}^N : \langle y,x_i \rangle = 0 \ \text{for} \ i=1,\ldots,m\}. \]
Let $\mathcal{L}$ and $\mathcal{L}^{\perp}$ be the lattices $V \cap \mathbb{Z}^N$ and $V^{\perp} \cap \mathbb{Z}^N$, and let $\mathcal{L}_0$ be the lattice spanned by $x_1, \ldots, x_m$ (i.e., the set $\{\sum_{i=1}^m n_ix_i : n_ \in \mathbb{Z}\}$ of integer combinations). Then $\mathcal{L}^{\perp}$ has rank $N - m$ and
\[ \det(\mathcal{L}^{\perp}) = \det(\mathcal{L}) = \det(\mathcal{L}_0)/q, \]
where $q$ is the order of the finite group $\mathcal{L}/\mathcal{L}_0$. Explicitly, $q$ is the number of solutions $(t_1, \ldots, t_m)$ in $(\mathbb{R}/\mathbb{Z})^m$ (or $(\mathbb{Q}/\mathbb{Z})^m$) of the system
\[ \sum_i x_{ij}t_i \equiv 0 (\text{mod }1), \quad j=1, \ldots, N, \]
where $x_i = (x_{ij})_{j=1}^N$ for $i = 1,\ldots, m$.
\end{lemma}
To each restriction in Equation~\eqref{eq:Acons} of the form $\sum_{j = 1}^k a_{ij} = \frac{1}{k}$ we associate the vector $r_i \in \mathbb{R}^{k^2+\frac{1}{4}k^2(k-1)^2}$ with value $1$ at index $j+ki$ if the variable $a_{ij}$ appears in the constraint and $0$ at all other indices. To each restriction in Equation~\eqref{eq:Acons} of the form $\sum_{i=1}^k a_{ij} = \frac{1}{k}$ we associate the vector $c_j \in \mathbb{R}^{k^2+\frac{1}{4}k^2(k-1)^2}$ in the same manner. Finally, to each restriction in Equation~\eqref{eq:Bcons} of the form $\sum_{u \sim v} b_{uv} = a_{ij}$ we associate a vector $e_{ij} \in \mathbb{R}^{k^2+\frac{1}{4}k^2(k-1)^2}$ with entry $-1$ at index $j+ki$ and, given some fixed ordering of the edges, index $1$ at position $k^2+\ell$ if the $\ell$th edge is $\{u,v\}$ for some $b_{uv}$ in the constraint.
Define the $(2k+k^2) \times (k^2+\frac{1}{4}k^2(k-1)^2)$ matrix $D$ by setting the first $k$ rows equal to $r_i$, the next $k$ rows equal to $c_j$, and the last $k^2$ rows equal to $e_{ij}$. Let $\mbf y \in \mathbb{R}^{2k+k^2}$ be the
vector with first $2k$ entries $\frac{1}{k}$ and last $k^2$ entries $0$.
\begin{lemma}\label{lem:cons_space}
If $\mbf x \in \mathbb{R}^{k^2+\frac{1}{4}k^2(k-1)^2}$ satisfies $D\mbf x = \mbf
y$ then assigning (as vectors) $A = (x_i)_{i=1}^{k^2}$ and $B =
(x_i)_{i=k^2+1}^{k^2+\frac{1}{4}k^2(k-1)^2}$ gives an assignment to $A$ and $B$ that
satisfies Equations \eqref{eq:Acons} and \eqref{eq:Bcons}, respectively, with the exception that entries may not be positive.
\end{lemma}
\begin{proof}
By construction,
\[ r_i \cdot \mbf x = \sum_{j=1}^k a_{ij} \quad \text{and} \quad c_j \cdot \mbf x = \sum_{i=1}^k a_{ij} \]
so as the first $2k$ entries of $\mbf y$ are $\frac{1}{k}$, $D\mbf x = \mbf y$ enforces the equality constraints in Equation~\eqref{eq:Acons}. For the latter $k^2$ rows,
\[ e_{ij} \cdot \mbf x = -a_{ij} + \sum_{u \sim v} b_{uv} \]
and thus as $\mbf y$ has $0$s in the corresponding rows, $D\mbf x = \mbf y$ also enforces the last constraints in Equation~\eqref{eq:Bcons}. Note that each edge appears only as a single variable in $D$, and thus we may set $b_{uv} = b_{vu}$ to satisfy the third constraint, and that nonedges do not appear in the constraints, so we may set them to $0$ to satisfy the second constraint.
\end{proof}
\begin{lemma}\label{lem:rankD}
The collection $ \{r_i\}_{i=2}^k \cup \{c_j\}_{j=1}^{k} \cup \{e_{ij}\}_{i,j=1}^k$ is linearly independent, and furthermore
\[ \rank(D) = k^2+2k-1. \]
\end{lemma}
\begin{proof}
As each vector contains a nonzero entry in a unique column, the collection
$\{e_{ij}\}$ is linearly independent. We claim that the collection $R =
\{r_i\}_{i=2}^k \cup \{c_j\}_{j=1}^{k}$ is linearly independent. Consider any
linear combination of $R$ that sums to $\vec{0}$. Note that $a_{1j}$ occurs in
$c_j$ but does not occur in any $r_i$ as $r_1 \notin R$. Thus each $c_j$ must
have a coefficient of $0$. But each $r_i$ has unique entries (there is no
$a_{ij}$ that occurs in two different row constraints because the value of $i$
would differ) so these must also have coefficients $0$. Now $R \cup \{e_{ij}\}$
is also linearly independent: given any linear combination that sums to $0$, the
coefficients of the $\{e_{ij}\}$ must be $0$ because $\{e_{ij}\}$ is linearly
independent and $R$ does not contain any entries past index $k^2$ while each
$\{e_{ij}\}$ does, and then because $R$ is linearly independent its coefficients
must also be zero. Finally, as each $a_{ij}$ occurs in exactly $r_i$ and $c_j$,
\[ r_1 = \sum_{j=1}^k c_j - \sum_{i=2}^{k} r_i \]
so no larger collection of rows of $D$ is linearly independent and $\rank(D) = k^2+2k-1$.
\end{proof}
\begin{lemma} \label{lem:det_l_nought}
Let $\hat{D}$ be the matrix that results from removing the row corresponding to $r_1$ from $D$. Then
\[ \det(\hat{D}^T\hat{D}) = \frac{(k-1)^{2k}(k(k-2))^{2k-2}((k^2-k+4)(k^2-3k+4))^{\frac{1}{2}(k-1)^2}}{2^{k^2-1}}.\]
\end{lemma}
\begin{proof}
We begin by considering the structure of $\hat{D}$. We claim that $\hat{D}$ is a block matrix
\[ \hat{D} = \begin{bmatrix} U_k & 0_{2k,|E|} \\ -I_{k^2} & E_\Gamma \end{bmatrix} \]
where $E_\Gamma$ is the $k^2 \times \frac{1}{4}k^2(k-1)^2$ incidence matrix of our auxiliary graph $\Gamma$ and $U_k$ is itself a block matrix, defined by
\[ U_k = \begin{bmatrix} 0_{k-1,k} & \mathbf{1}_k \otimes I_{k-1} \\ I_k & I_k \otimes \mathbf{1}_{k-1} \end{bmatrix}. \]
The $0_{2k,|E|}$ block occurs in $\hat{D}$ because the constraints in
Equation~\eqref{eq:Acons} only contain entries in the first $k^2$ columns. The
negative identity block comes from the $-a_{ij}$ terms that arise when
rearranging the constraints in Equation~\eqref{eq:Bcons}. The incidence matrix
of $\Gamma$ arises because the latter $\frac{1}{4}k^2(k-1)^2 = |E(\Gamma)|$
columns of $\hat{D}$ contain exactly two non-zero entries corresponding to the
two constraints in Equation~\eqref{eq:Bcons} in which the corresponding $b_{uv}$
appears. To understand the structure of $U_k$, note that each $r_i$ vector
contains $k$ consecutive 1 entries while each $c_j$ vector contains $k$ entries
with value 1, each separated by $k-1$ consecutive 0 entries, with the first 1 entry occurring at
index $j$. As we removed $r_1$ from $D$ to get $\hat{D}$, the first block of
$\hat{D}$ contains no 1 entries.
Now set $G = \hat{D}^T\hat{D}$ and note that $G$ is also a block matrix with blocks
\[ G = \begin{bmatrix} U_kU_k^T & -U_k \\ -U_k^T & I_{k^2} + E_\Gamma E_\Gamma^T\end{bmatrix} \]
It is well-known from spectral graph theory that $E_\Gamma E_\Gamma^T = A_{\Gamma} + D_\Gamma$ where $A_\Gamma$ is the adjacency matrix of $\Gamma$ and $D_\Gamma$ is the degree matrix, a diagonal matrix with $d_{ii} = d_\Gamma(i)$ and zero entries elsewhere. Straightforward calculation gives
\[ U_kU_k^T = \begin{bmatrix} kI_{k-1} & J_{k-1,k} \\ J_{k,k-1} & kI_k \end{bmatrix}. \]
To determine $\det(G)$, we once again use Sylvester's block matrix determinant formula:
\[ \det\left(\begin{bmatrix} A & B \\ C & D \end{bmatrix}\right) = \det(A)\det(D-CA^{-1}B). \]
This requires several calculations. First,
\[ \det(U_kU_k^T) = \det(kI_{k-1})\det(kI_{k}-J_{k,k-1}(kI_{k-1})^{-1}J_{k-1,k}) = k^{k-1}\det(kI_{k} - J_{k}). \]
The eigenvalues of $-J_{k}$ are $-k$ with multiplicity 1 and $0$ with multiplicity $k-1$. Shifting these by $k$ gives $\det(kI_{k}-J_{k,k}) = k^{k-1}$ and therefore
\begin{equation}\label{eq:detF}
\det(U_kU_k^T) = k^{2k-2}.
\end{equation}
One can verify that
\[ (U_kU_k^T)^{-1} = \frac{1}{k^2} \begin{bmatrix} kJ_{k-1} + I_{k-1} & -kJ_{k-1,k} \\ -kJ_{k,k-1} & (k-1)J_k + kI_k \end{bmatrix} \]
and then, using standard results on tensor products and an iron will, one can calculate
\[ (-U_k^T)(U_kU_k^T)^{-1}(-U_k) = \frac{1}{k^2}(kI_k \otimes J_{k} + kJ_{k} \otimes I_k - J_{k^2}). \]
Thus
\[ I_{k^2}+E_\Gamma E_\Gamma^T - (-U_k^T)(U_kU_k^T)^{-1}(-U_k) = I_{k^2} + A_\Gamma + D_{\Gamma} - \frac{1}{k^2}(kI_k \otimes J_{k} + kJ_{k} \otimes I_k - J_{k^2}). \]
We now investigate the eigenvalues and eigenspaces of the matrices $A_\Gamma, D_{\Gamma}$, and $kI_k \otimes J_{k} + kJ_{k} \otimes I_k - J_{k^2}$.
As $\Gamma$ is regular, $D_{\Gamma} = \frac{(k-1)^2}{2}I_{k^2}$.
By Proposition~\ref{prop:tourn_prod_eigenvectors}, the eigenvalues and eigenspaces of $A_\Gamma$ are
\[ \begin{array}{c|c}
\text{e-value} & \text{e-space}\\
\hline
\frac{(k-1)^2}{2} & \mathbf{1}_k \otimes \mathbf{1}_k = \mathbf{1}_{k^2}\\
-\frac{(k-1)}{2} & \Span(\{\mathbf{u} \otimes \mathbf{1} \mid \mathbf{u} \in \mathbf{1}_k^\perp\})\\
-\frac{(k-1)}{2} & \Span(\{\mathbf{1} \otimes \mathbf{u} \mid \mathbf{u} \in \mathbf{1}_k^\perp\})\\
-\frac{(k-1)}{2} & \mathbb{S}_1\\
-\frac{(k-1)}{2} & \mathbb{S}_2\\
\frac{(k+1)}{2} & \mathbb{S}_3\\
\end{array}\]
where the $\mathbb{S}_i$ are subspaces of $\Span(\{\mathbf{u} \otimes \mathbf{v} \mid \mathbf{u}, \mathbf{v} \in \mathbf{1}_k^\perp\})$ defined carefully in Proposition~\ref{prop:tourn_prod_eigenvectors}.
Finally, for any constant $\alpha$, $J_{\alpha}$ has two eigenvalues, $\alpha$ with multiplicity one corresponding to the eigenspace spanned by $\mathbf{1}_\alpha$ and $0$ with multiplicity $\alpha-1$ corresponding to the eigenspace $\mathbf{1}_k^\perp$. Therefore the eigenvalues and eigenspaces of $kI_k \otimes J_{k}$ are
\[ \begin{array}{c|c}
\text{e-value} & \text{e-space}\\
\hline
k^2 & \mathbf{1}_k \otimes \mathbf{1}_k = \mathbf{1}_{k^2}\\
k^2 & \Span(\{\mathbf{u} \otimes \mathbf{1} \mid \mathbf{u} \in \mathbf{1}_k^\perp\})\\
0 & \Span(\{\mathbf{1} \otimes \mathbf{u} \mid \mathbf{u} \in \mathbf{1}_k^\perp\})\\
0 & \Span(\{\mathbf{u} \otimes \mathbf{v} \mid \mathbf{u}, \mathbf{v} \in \mathbf{1}_k^\perp\})
\end{array}\]
while the eigenvalues of $kJ_{k} \otimes I_k$ are
\[ \begin{array}{c|c}
\text{e-value} & \text{e-space}\\
\hline
k^2 & \mathbf{1}_k \otimes \mathbf{1}_k = \mathbf{1}_{k^2}\\
0 & \Span(\{\mathbf{u} \otimes \mathbf{1} \mid \mathbf{u} \in \mathbf{1}_k^\perp\})\\
k^2 & \Span(\{\mathbf{1} \otimes \mathbf{u} \mid \mathbf{u} \in \mathbf{1}_k^\perp\})\\
0 & \Span(\{\mathbf{u} \otimes \mathbf{v} \mid \mathbf{u}, \mathbf{v} \in \mathbf{1}_k^\perp\})
\end{array}\]
Therefore the eigenvalues and eigenspaces of $-\frac{1}{k^2}(kI_k \otimes J_{k} + kJ_{k} \otimes I_k - J_{k^2})$ are
\[ \begin{array}{c|c}
\text{e-value} & \text{e-space}\\
\hline
-1 & \mathbf{1}_k \otimes \mathbf{1}_k = \mathbf{1}_{k^2}\\
-1 & \Span(\{\mathbf{u} \otimes \mathbf{1} \mid \mathbf{u} \in \mathbf{1}_k^\perp\})\\
-1 & \Span(\{\mathbf{1} \otimes \mathbf{u} \mid \mathbf{u} \in \mathbf{1}_k^\perp\})\\
0 & \Span(\{\mathbf{u} \otimes \mathbf{v} \mid \mathbf{u}, \mathbf{v} \in \mathbf{1}_k^\perp\})
\end{array}\]
Fortunately, the eigenspaces of $A_\Gamma$ and $kI_k \otimes J_{k} + kJ_{k} \otimes I_k - J_{k^2}$ align, so, shifting by $1+\frac{(k-1)^2}{2}$ to account for $I_{k^2}+D_\Gamma = (1+\frac{(k-1)^2}{2})I_{k^2}$, we see that the eigenvalues of $I_{k^2} + A_\Gamma + D_{\Gamma} - \tfrac{1}{k^2}(kI_k \otimes J_{k} + kJ_{k} \otimes I_k - J_{k^2})$ are
\[ \begin{array}{c|c}
\text{e-value} & \text{multiplicity}\\
\hline
1+\frac{(k-1)^2}{2}+\frac{(k-1)^2}{2}-1 = (k-1)^2& 1\\
1+\frac{(k-1)^2}{2}-\frac{(k-1)}{2}-1 = \frac{1}{2}(k-1)(k-2)& 2(k-1)\\
1+\frac{(k-1)^2}{2}-\frac{(k-1)}{2} = \frac{1}{2}(k^2-3k+4)& \frac{(k-1)^2}{2}\\
1+\frac{(k-1)^2}{2}+\frac{(k+1)}{2} = \frac{1}{2}(k^2-k+4)& \frac{(k-1)^2}{2}
\end{array}\]
and therefore
\begin{align*}
\det\bigl(I_{k^2} + A_\Gamma + D_{\Gamma} - &\tfrac{1}{k^2}(kI_k \otimes J_{k} + kJ_{k} \otimes I_k - J_{k^2})\bigr)\\
&= \frac{(k-1)^{2k}(k-2)^{2(k-1)}\bigl((k^2-3k+4)(k^2-k+4)\bigr)^{\frac{1}{2}(k-1)^2}}{2^{k^2-1}}
\end{align*}
which, along with Equation~\eqref{eq:detF}, gives
\[ \det(G) = \frac{(k-1)^{2k}(k(k-2))^{2k-2}((k^2-k+4)(k^2-3k+4))^{\frac{1}{2}(k-1)^2}}{2^{k^2-1}}. \]
\end{proof}
\begin{proposition}
There is a value $C_{d,k}$ depending on $d$ and $k$ such that
\[ \mathbf{E} Y^2 \sim C_{d,k} (\mathbf{E} Y)^2. \]
\end{proposition}
\begin{proof}
We use Theorem~\ref{thm:laplacian_summation}. Let $\Lambda = \ker(D) \cap
\mathbb{Z}^{k^2+\frac{1}{4}k^2(k-1)^2}$ and $\mathbb{V} = \ker(D)$. Then
$\Lambda$ and $\mathbb{V}$ satisfy (i) and (ii).
Let $\mbf w \in \mathbb{R}^{k^2+\frac{1}{4}k^2(k-1)^2}$ be defined by $w_i =
\frac{1}{k^2}$ for $1 \le i \le k^2$ and $w_i = \frac{2}{k^2(k-1)^2}$ for $k^2+1
\le i \le k^2+\frac{1}{4}k^2(k-1)^2$ and define $\mathbb{W} = \mathbb{V} + \mbf
w$, satisfying (iii). Note that for any $\mbf x \in \mathbb{W}$,
\[ D\mbf x = D\mbf x + D(\mbf w - \mbf x) = D \mbf w = \mbf y \]
as $\mbf w - \mbf x \in \ker(D)$.
Define
\[ K = \{ \mbf x \in \mathbb{R}^{k^2+\frac{1}{4}k^2(k-1)^2} : 0 \le x_i \le
\frac{1}{k} \} \]
which satisfies (iv).
Let $\phi = f$. By Lemma~\ref{lem:cons_space}, the restriction of $\phi$ to $K
\cap \mathbb{W}$ ensures $A$ satisfies Equation~\eqref{eq:Acons} and $B$ satisfies
Equation~\eqref{eq:Bcons} and thus Lemma~\ref{lem:second_mom_opt} assures $\phi$ has a unique maximum at $z_0 =
(\hat{A},\hat{B}) \in K^\circ \cap \mathbb{W}$.
Using Equation~\eqref{eq:f1} it is easy to see that $\phi$ is twice differentiable and
furthermore that
\begin{align*}
\frac{d^2\phi}{da_uda_v} &= \begin{cases} \frac{d-1}{a_u} & a_u = a_v \\ 0 & a_u
\ne a_v \end{cases},\\
\frac{d^2\phi}{da_udb_{xy}} &= 0, \qquad \text{and}\\
\frac{d^2\phi}{db_{uv}db_{xy}} &= \begin{cases} -\frac{d}{b_{uv}} & b_{uv} =
b_{xy} \\ 0 & b_{uv} \ne b_{xy} \end{cases}.
\end{align*}
and therefore $H$ is a diagonal matrix with $k^2$ entries $(d-1)k^2$ and
$\frac{1}{4}k^2(k-1)^2$ entries $-\frac{1}{2}dk^2(k-1)^2$. We can write
\[ H = -\tfrac{1}{2}dk^2(k-1)^2I_{k^2+\frac{1}{4}k^2(k-1)^2} +
\left(\bigl((d-1)k^2+\tfrac{1}{2}dk^2(k-1)^2\bigr)I_{k^2} \otimes
E_{\frac{1}{4}(k-1)^2}\right) \]
where $E_{\frac{1}{4}(k-1)^2}$ is a matrix with $e_{11} = 1$ and zeroes
elsewhere. Thus $H$ satisfies (vi).
Define $K_1 \subseteq K$ by those $\mbf x \in
\mathbb{R}^{k^2+\frac{1}{4}k^2(k-1)^2}$ such that
\[ \frac{0.9}{k^2} \le x_i \le \frac{1.1}{k^2}, 1 \le i \le k^2 \]
and
\[ \frac{1.9}{k^2(k-1)^2} \le x_i \le \frac{2.1}{k^2(k-1)^2}, k^2+1 \le i \le
k^2 + \frac{1}{4}k^2(k-1)^2. \]
Let $\psi = \prod_{uv \in E} \frac{1}{\sqrt{b_{uv}}}$. Then $K_1$ and $\psi$
satisfy (vii).
For each $n$, define $\ell_n \in \mathbb{R}^{k^2+\frac{1}{4}k^2(k-1)^2}$ to have
first $k^2$ entries $\frac{n}{k^2}$ and remaining entries
$\frac{2n}{k^2(k-1)^2}$, satisfying (viii).
Finally, set
\[ b_n = \frac{1}{\sqrt{2}} \cdot (\sqrt{2\pi n})^{1-\frac{k^2(k-1)^2}{4}}
(\sqrt{d})^{k^2-\frac{k^2(k-1)^2}{4}} \]
and $a_n(\ell) = p(A,B,n)e^{nf(A,B)}$ where $A$ is the first $k^2$ elements of
$\ell/n$ and $B$ is the remaining elements of $\ell/n$. Then as $p(A,B,n) =
O(n^{\frac{1}{2}(k^2+2)})$, for any $\ell \in (\Lambda + \ell_n) \cap nK$,
\[ a_n(\ell) = p(A,B,n)e^{nf(A,B)} =
b_n\left(\frac{p(A,B,n)}{b_n}\right)e^{n\phi(\ell/n)} =
O(b_ne^{n\phi(\ell/n)+o(n)}) \]
and for any $\ell \in (\Lambda + \ell_n) \cap nK_1$,
\[ a_n(\ell) = p(A,B,n)e^{nf(A,B)} = b_n(\psi(\ell/n)+o(1))e^{n\phi(\ell/n)}, \]
both uniformly.
We need to determine $\rank(\Lambda)$ and $\det(\Lambda)$. We do so using Lemma~\ref{lem:det_lattice}. As we proved in Lemma~\ref{lem:rankD}, the $k^2+2k-1$ rows of $D$ are linearly independent. Our $\mathbb{V} = V^{\perp}$ in Lemma~\ref{lem:det_lattice} and thus
\[ \rank(\Lambda) = \rank(\mathcal{L}^\perp) = N-m = k^2+\frac{1}{4}k^2(k-1)^2 - (k^2+2k-1) = \frac{1}{4}k^2(k-1)^2-2k+1. \]
We claim the order of $\mathcal{L}/\mathcal{L}_0$ is $2$. Consider a solution $(t_1, \ldots, t_m)$. It will be
convenient to rename these vectors $t_{r_2}, \ldots, t_{r_k}$ corresponding to
the rows $r_2, \ldots r_k$, $t_{c_1}, \ldots, t_{c_k}$ corresponding to the rows
$c_1, \ldots, c_k$, and $t_{e_{11}}, \ldots, t_{e_{kk}}$ corresponding to the
rows $e_{11}, \ldots, e_{kk}$. The first $k$ columns of $\hat{D}$ require
$t_{c_j} - t_{e_{1j}} \equiv 0 (\text{mod }1)$. The next $(k-1)k$ columns
require $t_{r_i} + t_{c_j} - t_{e_{ij}} \equiv 0 (\text{mod }1)$. The remaining
$|E|$ columns require $t_{e_u} + t_{e_v} \equiv 0 (\text{mod }1)$ whenever $u
\sim v$. As $\Gamma$ is connected and contains an odd cycle, we may either set
$t_{e_{ij}} = 0$ for each $i,j$ or $t_{e_{ij}} = \frac{1}{2}$. Whichever choice
we make determine $t_{c_j}$ for all $j$, due to the first $k$ columns, which in
turn determine $t_{r_i}$. Therefore once we make the choice to set each
$t_{e_{ij}}$ to $0$ or $\frac{1}{2}$, the other variables are determined,
meaning $q=2$. Thus
\[ \det(\Lambda) = \det(\mathcal{L}^\perp) = \frac{1}{2}\det(\mathcal{L}_0). \]
By Lemma 2.1 in \cite{GJR10},
\[ \det(\mathcal{L}_0) = \sqrt{\det(\hat{D}\hat{D}^T)} \]
and thus by Lemma~\ref{lem:det_l_nought}
\[ \det(\Lambda) = \frac{(k-1)^{k}(k(k-2))^{k-1}((k^2-k+4)(k^2-3k+4))^{\frac{1}{4}(k-1)^2}}{\sqrt{2}^{k^2+1}}. \]
Unfortunately, we were not able to determine $\det(-H|_{\mathbb{V}})$. We
consider the implications of finding an exact value in
Section~\ref{ssec:second_mom_const}.
For now, we note that since the function $f(A, B)$ achieves a unique global maximum in the lattice, there is some constant, $h_{d, k} > 0$, depending on $d$ and $k$ so that
\[
\sum_{\ell \in (\Lambda +\ell_n) \cap nK}a_n(\ell) \leq h_{d, k} (2\pi n)^{r/2} \psi(z_0) b_n e^{n\phi(z_0)}
\]
Noting that
\[\psi(z_0) = \prod_{uv \in E} \frac{1}{\sqrt{\frac{2}{k^2(k-1)^2}}} = \left(\frac{k(k-1)}{\sqrt{2}}\right)^{\frac{1}{4}k^2(k-1)^2}\]
and
\[ \phi(z_0) = -\sum_{v \in V} \frac{1}{k^2} \log \frac{1}{k^2} + d\sum_{uv \in E} \frac{2}{k^2(k-1)^2}
\log\left(\frac{\frac{1}{k^2}\cdot\frac{1}{k^2}}{2\cdot\frac{2}{k^2(k-1)^2}}\right) = \log k^2 + d\log \left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right),\]
we have
\begin{align}
\mathbf{E} Y^2 &= \sum_{A \text{ s.t.} \eqref{eq:Acons}, \ B \text{ s.t.}
\eqref{eq:Bcons}} p(A,B,n) e^{nf(A,B)} \notag\\
&= \sum_{\ell \in (\Lambda +\ell_n) \cap nK}a_n(\ell) \notag\\
&\leq h_{d, k} (2\pi n)^{r/2} \psi(z_0) b_n e^{n\phi(z_0)} \notag \\
&= C_{d, k} \cdot k^k\left(\frac{k-1}{2\pi(k-2)}\right)^{(k-1)}n^{-(k-1)}
k^{2n}\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right)^{dn} \notag\\
&= C_{d,k} (\mathbf{E} Y)^2 \label{eq:PZ_constant}
\end{align}
where $C_{d, k}$ is some positive constant depending on $d$ and $k$.
\end{proof}
We conclude this section by proving the following proposition.
\begin{proposition} \label{prop:gnd_ub}
For $k \ge 3$ such that a doubly regular tournament exists on $k$ vertices and $d < \ell_k$, with positive probability ${\chi_o}(\vec{\mathcal{C}}(n,d)) \le k$.
\end{proposition}
\begin{proof}
By the Paley-Zygmund inequality,
\[ \mathbf{Pr}[Y > 0] > \frac{(\mathbf{E} Y)^2}{\mathbf{E}(Y^2)} \sim \frac{(\mathbf{E} Y)^2}{C_{d,k}(\mathbf{E} Y)^2} = \frac{1}{C_{d,k}} > 0 \]
and thus with positive probability there exists a $T_k$-colouring of $\vec{\mathcal{C}}(n,d)$ and therefore ${\chi_o}(\vec{\mathcal{C}}(n,d)) \le k$.
\end{proof}
\section{Range of Possible Values of the Oriented Chromatic Number} \label{sec:window}
In this section, we adapt an argument of {\L}uczak~\cite{tL91b} who proved that for every $p=\frac{d}{n}$ with $d < n^{1/6-\varepsilon}$, the chromatic number of $\mathcal{G}(n,p)$ is concentrated in an interval of length two. (We note that our adaptation is closely based on that of Achlioptas and Moore~\cite{AM04}.) We prove a weaker result for $\vec{\mathcal{C}}(n,d)$ and $\vec{\mathcal{M}}(n,m=cn)$, showing that a.a.s. their chromatic numbers are concentrated in an interval of length linear in the number of colours.
We start with a result regarding the oriented chromatic number of sparse digraphs.
\begin{lemma}[{\cite[Theorem 1]{BKNRS99}}]\label{lem:eleven_colour}
Let $\vec{G}$ be directed graph with the property that any subgraph of $\vec{G}$ has average degree strictly smaller than three. Then $\chi_o(\vec{G}) \le 11$.
\end{lemma}
Now we argue that a.a.s. subgraphs of $\vec{\mathcal{C}}(n,d)$ of sub-linear size are sparse.
\begin{lemma}\label{lem:gnd_subgraph_colourable}
For any $0 < \varepsilon < \frac{1}{2}$ and constant $d$, with high probability every subgraph of $\vec{C} \sim \vec{\mathcal{C}}(n,d)$ induced by $s \le e^{-5}d^{-3}n^{1-\varepsilon}$ vertices contains at most $(\frac{3}{2}-\varepsilon)s$ edges.
\end{lemma}
It should be noted that it is possible to prove a stronger version of Lemma~\ref{lem:gnd_subgraph_colourable} where $d$ is allowed to grow bounded by a function of $n$ rather than being constant; however, this simpler version meets our current needs.
\begin{proof}
Let $\vec{C} \sim \vec{\mathcal{C}}(n,d)$ and let $S \subseteq V(\vec{C})$ such that $|S| = s$. Let $v \in S$ and $(u,v) \in E(\vec{C})$. The probability $u \in S$ is at most $\frac{s}{n}$. Thus
\[ \mathbf{Pr}[\text{$S$ has at least $k$ edges}] \le \binom{ds}{k} \left(\frac{s}{n}\right)^k \]
and
\[ \mathbf{Pr}[\text{There exists a set $S$ with at least $k$ edges}] \le \binom{n}{s}\binom{ds}{k} \left(\frac{s}{n}\right)^k. \]
Using $k = \frac{3}{2}-\varepsilon, 1 \le s \le e^{-5}d^{-3}n^{1-\varepsilon}$, and $0 < \varepsilon < \frac{1}{2}$ we get
\[ \binom{n}{s}\binom{ds}{k} \left(\frac{s}{n}\right)^k < \frac{de^2}{n^{\varepsilon(\frac{1}{2}-\varepsilon)}} = o(1) \]
as $\varepsilon(\frac{1}{2}-\varepsilon) > 0$ for $0 < \varepsilon <
\frac{1}{2}$. Therefore the probability such a set $S$ exists approaches zero as $n$ approaches infinity.
\end{proof}
Next we appeal to a martingale result of Wormald on undirected configurations.
\begin{theorem}[{\cite[Theorem 2.19]{Wo99}}]\label{thm:original_martingale}
Given an undirected configuration $C$, we define a \emph{switching} in $C$ to be the replacement of two edges $\{v_1, v_2\}$, $\{v_3, v_4\}$ by the edges $\{v_1, v_3\}$ and $\{v_2,v_4\}$ or by the edges $\{v_1,v_4\}$ and $\{v_3,v_2\}$. Let $X_n$ be a random variable defined on $\mathcal{C}(n,d)$ such that for any configurations $C, C'$ that differ by a switching
\[ |X_n(C)-X_n(C')| \le b \]
for some constant $b > 0$. Then for every $t > 0$,
\[ \mathbf{Pr}[X_n \le \mathbf{E}[X_n] - t] \le \exp\left(\frac{-t^2}{dnb^2}\right) \quad \text{and} \quad \mathbf{Pr}[X_n \ge \mathbf{E}[X_n] + t] \le \exp\left(\frac{-t^2}{dnb^2}\right). \]
\end{theorem}
As long as the value $b$ does not depend on the orientations chosen for the new edges, we can extend Wormald's result to directed configurations by ``forgetting'' the orientations, applying the result, and then reorienting the edges, including assigning new orientations to the new edges.
Our next step is to prove that as soon as there is some small probability that $k$ colours are sufficient to colour $\vec{C} \sim \vec{\mathcal{C}}(n,d)$, we can almost surely $k$-colour all but a small number of vertices.
\begin{lemma}[{\cite[Lemma 2]{AM04}}]\label{lem:gnd_mostly_colourable}
For a given function $\omega(n)$, let $k = k(n, d, \omega(n))$ be the smallest $k$ such that
\[ \mathbf{Pr}[{\chi_o}(\vec{\mathcal{C}}(n,d)) \le k] \ge 1/\omega(n) . \]
With probability greater than $1-1/\omega(n)$, all but $8\sqrt{nd\log \omega(n)}$ vertices of $\vec{\mathcal{C}}(n,d)$ can be properly oriented-coloured using $k$ colours.
\end{lemma}
\begin{proof}
For $\vec{C}\sim\vec{\mathcal{C}}(n,d)$, let $Y_j(\vec{C})$ be the minimal size of a set of vertices $S$ such that $\vec{C}-S$ is $j$-oriented-colourable. For any $j$ and $\vec{C}$, switching two arcs of $\vec{C}$ can affect $Y_j(\vec{C})$ by at most 4 as, in the worst case, we can add all four vertices to $S$ and still $j$-oriented-colour $\vec{C}-S$. By Theorem~\ref{thm:original_martingale}, with $\mu_j = \mathbf{E}[Y_j(\vec{\mathcal{C}}(n,d))]$,
\[ \mathbf{Pr}[Y_j \le \mu_j - \lambda \sqrt{n}] < e^{-\frac{\lambda^2}{16d}} \quad \text{and} \quad \mathbf{Pr}[Y_j \ge \mu_j + \lambda \sqrt{n}] < e^{-\frac{\lambda^2}{16d}}. \]
Choose $\lambda = \lambda(n) = 4\sqrt{d\log \omega(n)}$ so that $e^{-\frac{\lambda^2}{16d}} = \frac{1}{\omega(n)}$. Then, recalling the definition of $k$,
\[ \mathbf{Pr}[Y_k \le \mu_k - \lambda \sqrt{n}] < \frac{1}{\omega(n)} \le \mathbf{Pr}[{\chi_o}(\vec{C}) \le k] = \mathbf{Pr}[Y_k = 0]. \]
If $\mu_k \ge \lambda \sqrt{n}$, then the event $Y_k = 0$ implies the event $Y_k \le \mu_k-\lambda \sqrt{n}$; since we have $\mathbf{Pr}[Y_k \le \mu_k-\lambda \sqrt{n}] < \mathbf{Pr}[Y_k = 0]$, this must not be the case and instead we must have $\mu_k < \lambda\sqrt{n}$. Therefore
\[ \frac{1}{\omega(n)} = e^{-\frac{\lambda^2}{16d}} > \mathbf{Pr}[Y_k \ge \mu_k + \lambda \sqrt{n}] \ge \mathbf{Pr}[Y_k \ge 2\lambda\sqrt{n}] \]
which completes the proof.
\end{proof}
Finally, we prove that the oriented chromatic number of $\vec{\mathcal{C}}(n,d)$ is concentrated in an interval of length linear in the number of colours.
\begin{proposition} \label{prop:gnd_window}
For every integer $d$, there exists an integer $k = k(d)$ such that a.a.s.
\[ {\chi_o}(\vec{\mathcal{C}}(n,d)) \in [k,3k+11]. \]
\end{proposition}
\begin{proof}
Let $d$ be given and let $k = k(d,n)$ be the smallest integer for which the probability that $\vec{\mathcal{C}}(n,d)$ is $k$-colourable is at least $1/\log \log n$. Fix $\vec{C} \sim \vec{\mathcal{C}}(n,d)$. By Lemma~\ref{lem:gnd_mostly_colourable}, with high probability there exists a set of vertices $S$ such that $\vec{C}-S$ can be $k$-oriented-coloured and $|S| < 8\sqrt{nd \log \log \log n} < \sqrt{nd} \log n \equiv s_0$. From $S$, we will construct an increasing sequence of sets of vertices $\{U_i\}$ as follows. $U_0 = S$; for $i \ge 0, U_{i+1} = U_i \cup W$, where either $W = \{w_1,w_2\}$ for some $w_1,w_2 \notin U_i$ which are adjacent and each of which has some neighbour in $U_i$ or $W = \{w\}$ for some $w \notin U_i$ for which there are $u_1,u_2 \in U_i$ such that $(u_1,w)$ and $(w,u_2)$ are arcs in $\vec{C}$. The construction ends, with $U_t$, when no such pair exists.
Define $N = \{v \in \vec{G} : v \notin U_t \ \text{and} \ \exists u \in U_t \ \text{such that} \ u\sim v\}$
to be the neighbourhood of $U_t$ in the rest of the graph. Note that $N$ is an independent set; if $v_1, v_2 \in N$ and $v_1 \sim v_2$, then we could have continued in the construction with $U_{t+1} = U_t\cup\{v_1,v_2\}$. Furthermore, $N = N_I \cup N_O$ where each edge is directed from $U_t$ into $N_I$ and each edge is directed from $N_O$ out to $U_t$; if some $v \in N$ satisfies neither of these constraints, then there are $u_1, u_2 \in U_t$ such that $(u_1, v)$ and $(v,u_2)$ are arcs of $\vec{C}$ and we could have continued the construction with $U_{t+1} = U_t \cup \{v\}$.
We claim we can properly $11$-oriented-colour $U_t$ with high probability. Set $\varepsilon = 0.1$ and let $n$ be large enough so that
\[ \frac{s_0(1+2\log n)}{s_0(1+2\log n)+2} > 1-\frac{\varepsilon}{3}, \quad \frac{3\log n}{1+2\log n} > \frac{3}{2}- \frac{\varepsilon}{2}, \]
and that $n^{1/2-\varepsilon} > 3e^{5}d^{7/2}(\log n)^2$. It suffices to show that $|U_t| < s_0(1+2\log n)$: then
\[ s_0(1+2\log n) = \sqrt{nd}\log n(1+2 \log n) \le 3\sqrt{nd}(\log n)^2 \le e^{-5}d^{-3}n^{1-\varepsilon} \]
so Lemma~\ref{lem:gnd_subgraph_colourable} ensures the condition of Lemma~\ref{lem:eleven_colour}.
Assume for contradiction that $|U_t| \ge s_0(1+2\log n)$ and let $i^\ast$ be the first step at which $|U_{i^\ast}| \ge s_0(1+2\log n)$. Because $|U_0| = s_0$, at least $2s_0\log n$ vertices have been added. At step $i^\ast$ we added at most two vertices to $U_{i^\ast-1}$, and therefore $|U_{i^\ast}| \le s_0(1+2\log n)+2$. Note that in steps in which two vertices are added to $U_i$, three edges are added, and in steps in which one vertex is added, we add two edges to $U_i$. Thus each vertex added to $U_{i^\ast}$ also added at least $\frac{3}{2}$ edges to $U_{i^\ast}$, from which we conclude $U_{i^\ast}$ contains at least $\frac{3}{2}(2s_0\log n) = 3s_0\log n$ edges. But then
\[ e(U_{i^\ast}) = 3s_0\log n = (s_0 (1+2\log n)+2)\cdot \frac{s_0(1+2\log n)}{s_0(1+2\log n)+2} \cdot \frac{3\log n}{1+2\log n} > |U_{i^\ast}|(1-\tfrac{\varepsilon}{3})(\tfrac{3}{2}-\tfrac{\varepsilon}{2}) > |U_{i^\ast}|(\tfrac{3}{2}-\varepsilon). \]
This contradicts Lemma~\ref{lem:gnd_subgraph_colourable} as
\[ |U_{i^\ast}| \le s_0(1+2\log n)+2 = \sqrt{nd}\log n(1+2 \log n)+2 \le 3\sqrt{nd}(\log n)^2 \le e^{-5}d^{-3}n^{1-\varepsilon}. \]
Now recall that $\vec{C}-S$ is $k$-oriented-colourable, say by ${\chi_o}:\vec{C}-S \to [k]$. We colour $\vec{C}$ as follows. First, for $v \in \vec{C} - U_t - N$, colour $v$ with $({\chi_o}(v),0)$. As $N \subseteq \vec{C}-U_t \subseteq \vec{C}-S$, each vertex in $N$ is also coloured by ${\chi_o}$. Colour each $v \in N_I$ with colour $({\chi_o}(v),1)$ and each $v \in N_O$ with colour $({\chi_o}(v),2)$. Finally, use the eleven unique colours $(1,4), \ldots, (11,4)$ to colour $U_t$. We claim this is a proper oriented $(3k+11)$-colouring of $\vec{C}$. First, no adjacent vertices share a colour: $U_t$ is properly coloured by Lemma~\ref{lem:eleven_colour}, $\vec{C}-U_t$ is properly $k$-oriented-coloured as $\vec{C}-S$ was, and the vertices in $U_t$ are coloured with colours distinct from those in $\vec{C}-U_t$. The only other issue would be a directed path $v_1 \to v_2 \to v_3$ such that $v_1$ and $v_3$ receive the same colour. Consider any directed path $v_1 \to v_2 \to v_3$. If $v_1, v_3 \notin U_t$ then in particular they are not in $S$ and thus ${\chi_o}(v_1) \ne {\chi_o}(v_3)$. If exactly one of $v_1, v_3 \in U_t$ then one is coloured with second coordinate 4 and the other is not. Finally, if $v_1, v_3 \in U_t$ then $v_2 \in U_t$ as well because $U_t$ only has edges directed internally and towards $N_I$, but $N_I$ has no edges directed back towards $U_t$. Then $v_1$ and $v_3$ receive different colours as $U_t$ was properly coloured.
\end{proof}
A version of Proposition~\ref{prop:gnd_window} holds for $\vec{\mathcal{M}}(n,m)$ holds for $m=cn, c > 0$ as well.
\begin{proposition} \label{prop:gnm_window}
For every $c > 0$, there exists an integer $k = k(c)$ such that a.a.s.
\[ {\chi_o}(\vec{\mathcal{M}}(n,m=cn)) \in [k,3k+11]. \]
\end{proposition}
The proof of Proposition~\ref{prop:gnm_window} is nearly identical to that of Proposition~\ref{prop:gnd_window}, so rather than repeat each of the intermediate results, we summarize the small changes in the proofs. In Lemma~\ref{lem:gnd_subgraph_colourable}, the probability that a ``bad'' subset exists is $\binom{n}{s}\binom{m}{k}(\binom{s}{2}/\binom{n}{2})^k$. This probability is still less than $(de^2)/n^{\varepsilon(\frac{1}{2}-\varepsilon)}$, so the corresponding result holds. Instead of appealing to Theorem~\ref{thm:original_martingale}, we can use Azuma's inequality directly by exposing the edges of the graph one at a time. This has the effect of changing the size of the uncoloured set in Lemma~\ref{lem:gnd_mostly_colourable} by a constant multiple. That difference disappears in the step of the proof of Proposition~\ref{prop:gnd_window} in which we set $s_0 = \sqrt{nd} \log n$.
\section{Proofs of Main Theorems} \label{sec:main}
\begin{lemma}\label{lem:gnm_simple}
For any constant $c > 0$ and $m \leq cn + n^{2/3}$, the probability that $\vec{\mathcal{M}}(n,m)$ is a simple oriented graph (without loops or multiple edges) is at least $\exp(-5c(c+1)) > 0$.
\end{lemma}
\begin{proof}
The probability that $\vec{M} \sim \vec{\mathcal{M}}(n,m)$ is an orientation of a simple graph is
\begin{align*}
\frac{1}{n^{2m}}\prod_{j=0}^{m-1} n(n-1)-2j
&\geq \left(\frac{n^2 - 2(m+n)}{n^2}\right)^m\\
&= \left(1 - \frac{2(m+n)}{n^2}\right)^m\\
&\geq \exp\left(-\frac{4(m+n)}{n^2} \cdot m \right) &\text{(for $n$ large enough)}\\
&\geq \exp\left(-\frac{4((cn+n^{2/3}+n)(cn+n^{2/3})}{n^2}\right)\\
&=\exp\left( - 4c^2 - \frac{8c}{n^{1/3}} - \frac{4}{n^{2/3}} -4c -\frac{4}{n^{1/3}}\right)\\
&\geq \exp\left(-5c(c+1)\right). &\text{(for $n$ large enough)}
\end{align*}
\end{proof}
\begin{corollary}\label{cor:gnm_mnm_equiv}
For any constant $c > 0$, any event $A$ in both $\vec{\mathcal{G}}(n,p)$ and $\vec{\mathcal{M}}(n,m)$, and any $m \le cn + n^{2/3}$,
\[ \mathbf{Pr}_{\vec{G} \sim \vec{\mathcal{G}}(n,p=\frac{d}{n})}[A \mid e(\vec{G}) = m] \le e^{5c(c+1)}\mathbf{Pr}_{\vec{M}\sim\vec{\mathcal{M}}(n,m)}[A]. \]
\end{corollary}
\begin{proof}
The key idea is that $\vec{G} \sim \vec{\mathcal{G}}(n,p)$ conditioned on having $m$ edges and $\vec{M} \sim \vec{\mathcal{M}}(n,m)$ conditioned on being simple are both uniform over the collection of graphs of order $n$ with $m$ edges. Thus
\begin{align*}
\mathbf{Pr}_{\vec{G} \sim \vec{\mathcal{G}}(n,p=\frac{d}{n})}[A \mid e(\vec{G}) = m] &= \mathbf{Pr}_{\vec{M} \sim \vec{\mathcal{M}}(n,m)}[A \mid \text{$\vec{M}$ is simple}]\\
&= \frac{\mathbf{Pr}_{\vec{M} \sim \vec{\mathcal{M}}(n,m)}[A \text{ and $\vec{M}$ is simple}]}{\mathbf{Pr}_{\vec{M} \sim \vec{\mathcal{M}}(n,m)}[\text{$\vec{M}$ is simple}]}\\
&\le e^{5c(c+1)}\mathbf{Pr}_{\vec{M} \sim \vec{\mathcal{M}}(n,m)}[A] & \text{Lemma~\ref{lem:gnm_simple}}.
\end{align*}
\end{proof}
\begin{lemma}\label{lem:edge_conc}
For any constant $d > 0$, the number of edges of $\vec{G} \sim \vec{\mathcal{G}}(n,p=\frac{d}{n})$ satisfies
\[
\mathbf{Pr}_{\vec{G} \sim \vec{\mathcal{G}}(n,p=\frac{d}{n})}[|e(\vec{G}) - dn/2| \geq n^{2/3}] \leq 2\exp\left(-\frac{n^{1/3}}{4d}\right) = o(1).
\]
\end{lemma}
\begin{proof}
Set $c = d/2$. By standard concentration results,
for $\vec{G} \sim \vec{\mathcal{G}}(n,p=\frac{d}{n})$, since $\mathbf{E}(e(\vec{G})) = \binom{n}{2}p = \frac{d(n-1)}{2} = c(n-1)$,
\begin{align*}
\mathbf{Pr}_{\vec{G} \sim \vec{\mathcal{G}}(n,p=\frac{d}{n})}[|e(\vec{G}) - cn| \geq n^{2/3}]
&\leq \mathbf{Pr}_{\vec{G} \sim \vec{\mathcal{G}}(n,p=\frac{d}{n})}[|e(\vec{G})) - \mathbf{E}(e(\vec{G}))| \geq n^{2/3}-c]\\
&\leq 2 \exp\left(-\frac{(n^{2/3}-c)^2}{2(\binom{n}{2}\frac{d}{n} +(n^{2/3}-c)/3)}\right)\\
&\leq 2 \exp\left(- \frac{n^{4/3}/2}{d(n-1) + n^{2/3}}\right)\\
&\leq 2 \exp\left(-\frac{n^{1/3}}{4d}\right) &&\text{(for $n$ large enough)}\\
&=o(1).
\end{align*}
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:gnp_main}]
Let $d > 1$ be given and let $k_1$ be an integer satisfying $d > u_{(k_1-1)}$. Set $c = \frac{d}{2}$. By Corolloary~\ref{cor:gnm_mnm_equiv}, letting $A$ be the event that the chromatic number is at most $k_1-1$, for any $m \le cn+n^{2/3}$,
\[
\mathbf{Pr}_{\vec{G} \sim \vec{\mathcal{G}}(n,p=\frac{d}{n})}[{\chi_o}(\vec{G}) \leq k_1-1 \mid e(\vec{G}) = m] \le e^{5c(c-1)}\mathbf{Pr}_{\vec{M} \sim \vec{\mathcal{M}}(n, m)}[{\chi_o}(\vec{M}) \leq k_1-1].
\]
Thus, by Lemma~\ref{lem:edge_conc}, the law of total probability, and noting that the probability that $\vec{M} \sim \vec{\mathcal{M}}_{n, m}$ is simple decreases as $m$ increases, with $\vec{\mathcal{G}} = \vec{\mathcal{G}}(n,p=\frac{d}{n})$,
\begin{align*}
\mathbf{Pr}_{\vec{G} \sim \vec{\mathcal{G}}}[{\chi_o}(\vec{G}) \leq k_1-1] &= \mathbf{Pr}_{\vec{G} \sim \vec{\mathcal{G}}}[{\chi_o}(\vec{G}) \leq k_1-1 \text{ and } |e(\vec{G}) - cn| \le n^{2/3}] \quad + \\
&\qquad \qquad \mathbf{Pr}_{\vec{G} \sim \vec{\mathcal{G}}}[{\chi_o}(\vec{G}) \leq k_1-1 \text{ and } |e(\vec{G}) - cn| > n^{2/3}]\\
&\leq \sum_{m=\lceil cn-n^{2/3}\rceil}^{\lfloor cn+n^{2/3}\rfloor}\mathbf{Pr}_{\vec{G} \sim \vec{\mathcal{G}}}[{\chi_o}(\vec{G}) \leq k_1-1 \mid e(\vec{G}) = m]\mathbf{Pr}_{\vec{G} \sim \vec{\mathcal{G}}}[e(\vec{G}) = m] \quad +\\
&\qquad\qquad \mathbf{Pr}_{\vec{G} \sim \vec{\mathcal{G}}}[|e(\vec{G}) - cn| > n^{2/3}] \\
&\le e^{5c(c+1)}\mathbf{Pr}_{\vec{M} \sim \vec{\mathcal{M}}(n, cn-n^{2/3})}[{\chi_o}(\vec{M}) \leq k_1-1]\sum_{m=\lceil cn-n^{2/3}\rceil}^{\lfloor cn+n^{2/3}\rfloor}\mathbf{Pr}_{\vec{G} \sim \vec{\mathcal{G}}}[e(\vec{G}) = m] + o(1)\\
&\le e^{5c(c+1)}\mathbf{Pr}_{\vec{M} \sim \vec{\mathcal{M}}(n, cn-n^{2/3})}[{\chi_o}(\vec{M}) \leq k_1-1] + o(1)
\end{align*}
Pick $c'$ satisfying $c > c' > \frac{1}{2}u_{(k_1-1)}$ and let $n$ be large enough that $cn-n^{2/3} \ge c'n$. Then by Proposition~\ref{prop:gnm_lb}, noting the probability that an oriented $(k_1-1)$-colouring exists is decreasing in the number of edges,
\[ e^{5c(c+1)}\mathbf{Pr}_{\vec{M} \sim \vec{\mathcal{M}}(n, cn-n^{2/3})}[{\chi_o}(\vec{M}) \leq k_1-1] \le e^{5c(c+1)}\mathbf{Pr}_{\vec{M} \sim \vec{\mathcal{M}}(n,m=c'n)}[{\chi_o}(\vec{M}) \leq k_1-1] = o(1) \]
and therefore a.a.s. ${\chi_o}(\vec{\mathcal{G}}(n,p=\frac{d}{n})) > k_1-1$.
Now let $k_2 \ge 3$ such that $d < \ell_{k_2}$ and there exists a doubly regular tournament of order $k_2$. Set $c = d/2$, let $c'$ be such that $c < c' < \frac{1}{2}\ell_{k_2}$ and let $n$ be large enough that $cn + n^{2/3} \leq c' n$. Using Corollary~\ref{cor:gnm_mnm_equiv} where $A$ is the event that the chromatic number is greater than $3k_2+11$, for any $m \le c'n+n^{2/3}$,
\[
\mathbf{Pr}_{\vec{G} \sim \vec{\mathcal{G}}(n,p=\frac{d}{n})}[{\chi_o}(\vec{G}) > 3k_2+11 \mid e(\vec{G}) = m] \le e^{5c'(c'+1)}\mathbf{Pr}_{\vec{M} \sim \vec{\mathcal{M}}_{n, m}}[{\chi_o}(\vec{M}) > 3k_2+11].
\]
Thus, using Lemma~\ref{lem:edge_conc}, that the probability the oriented chromatic number is at least $3k_2+11$ is increasing in the number of edges, and the law of total probability, with $\vec{\mathcal{G}} = \vec{\mathcal{G}}(n,p=\frac{d}{n})$,
\begin{align*}
\mathbf{Pr}_{\vec{G} \sim \vec{\mathcal{G}}}[{\chi_o}(\vec{G}) > 3k_2+11] &= \mathbf{Pr}_{\vec{G} \sim \vec{\mathcal{G}}}[{\chi_o}(\vec{G}) > 3k_2+11 \text{ and } e(\vec{G}) > c'n] \quad + \\
&\qquad\qquad \sum_{m=0}^{\lfloor c'n\rfloor}\mathbf{Pr}_{\vec{G} \sim \vec{\mathcal{G}}}[{\chi_o}(\vec{G}) > 3k_2+11 \text{ and } e(\vec{G}) = m]\\
&\leq \mathbf{Pr}_{\vec{G} \sim \vec{\mathcal{G}}}[e(\vec{G}) > c'n \geq cn + n^{2/3}] \quad +\\
&\qquad \qquad e^{5c'(c'+1)}\mathbf{Pr}_{\vec{M} \sim \vec{\mathcal{M}}(n,m=c'n)}[{\chi_o}(\vec{M}) > 3k_2+11]\sum_{m=0}^{\lfloor c'n\rfloor}\mathbf{Pr}_{\vec{G} \sim \vec{\mathcal{G}}}[e(\vec{G}) = m]\\
&\le o(1) + e^{5c'(c'+1)}\mathbf{Pr}_{\vec{M} \sim \vec{\mathcal{M}}(n,m=c'n)}[{\chi_o}(\vec{M}) > 3k_2+11].
\end{align*}
Then by Proposition~\ref{prop:gnm_ub}, as $1 < c' < \frac{1}{2}\ell_{k_2}$, $\vec{M}\sim\vec{\mathcal{M}}(n,m=c'n)$ has a proper oriented $k_2$-colouring with positive probability. Proposition~\ref{prop:gnm_window} then guarantees $\mathbf{Pr}_{\vec{M}\sim\vec{\mathcal{M}}(n,m=c'n)}[{\chi_o}(\vec{M}) > 3k_2+11] = o(1)$. We conclude a.a.s.
\[ {\chi_o}(\vec{\mathcal{G}}(n,p=\tfrac{d}{n})) \in [k_1,3k_2+11]. \]
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:gnd_main}]
It is well-known (see, for example, \cite{Bo80}) that the (undirected) configuration model produces a simple graph with probability tending toward $\exp(-\frac{1}{2}(d-1)-\frac{1}{4}(d-1)^2) =: \kappa > 0$. Orienting an undirected simple graph produces a simple oriented graph, so with positive probability $\vec{C} \sim \vec{\mathcal{C}}(n,d)$ is a simple digraph. Furthermore, these simple digraphs are uniformly distributed as each simple graph corresponds to the same number of configurations. Therefore, if $A$ is any event in both $\vec{\mathcal{G}}(n,d)$ and $\vec{\mathcal{C}}(n,d)$, then
\begin{align*}
\mathbf{Pr}_{\vec{G}\sim\vec{\mathcal{G}}(n,d)}[A] &= \mathbf{Pr}_{\vec{C}\sim\vec{\mathcal{C}}(n,d)}[A \mid \text{$\vec{C}$ is simple}]\\
&= \frac{\mathbf{Pr}_{\vec{C}\sim\vec{\mathcal{C}}(n,d)}[A \text{ and $\vec{C}$ is simple}]}{\mathbf{Pr}_{\vec{C}\sim\vec{\mathcal{C}}(n,d)}[\text{$\vec{C}$ is simple}]}\\
&\le \kappa^{-1}\mathbf{Pr}_{\vec{C}\sim\vec{\mathcal{C}}(n,d)}[A]
\end{align*}
For $k_1 \ge 2$ and $d \ge u_{(k_1-1)}$, Proposition~\ref{prop:gnd_lb} assures $\mathbf{Pr}_{\vec{C}\sim\vec{\mathcal{C}}(n,d)}[{\chi_o}(\vec{C}) \le k_1-1] = o(1)$ and therefore
\[ \mathbf{Pr}_{\vec{G}\sim\vec{\mathcal{G}}(n,d)}[{\chi_o}(\vec{G}) \le k_1-1] \le \kappa^{-1}\cdot o(1) = o(1).\]
On the other hand, if there is a $k_2 \ge 3$ such that $d < \ell_{k_2}$ and there exists a doubly regular tournament of order $k_2$, then Proposition~\ref{prop:gnd_ub} gives ${\chi_o}(\vec{\mathcal{C}}(n,d)) \le k_2$ with positive probability. Then Proposition~\ref{prop:gnd_window} guarantees $\mathbf{Pr}_{\vec{C} \sim \vec{\mathcal{C}}(n,d)}[{\chi_o}(\vec{C}) > 3k_2+11] = o(1)$, so
\[ \mathbf{Pr}_{\vec{G}\sim\vec{\mathcal{G}}(n,d)}[{\chi_o}(\vec{G}) > 3k_2+11] \le \kappa^{-1} \cdot o(1) = o(1).\]
We conclude a.a.s.
\[ {\chi_o}(\vec{\mathcal{G}}(n,d)) \in [k_1,3k_2+11]. \]
\end{proof}
\begin{proof}[Proof of Corollary~\ref{cor:main_cor}]
Let $d > 1$ be given and set $k = 2^{d/2}$. Then
\[ u_{k} < \frac{2}{\log 2} \log(k) = d \]
Setting $k' = k^{1/\log 2}+\frac{2}{\log 2}\log k + 1 = e^{d/2}+d+1$,
\[ d < 2\left(1-\frac{2}{k'+1}\right)\log(k'-1) < \ell_{k'}. \]
Thus we can apply Theorems~\ref{thm:gnp_main} and \ref{thm:gnd_main} for any $k'' \ge k'$ such that there exists a doubly regular tournament of order $k''$. Lemma~\ref{lem:bertrand_postulate_dr_tourn} guarantees the smallest $k''$ for which this holds is at most $2k'$, and thus
\[ {\chi_o}(\vec{\mathcal{G}}(n,p=\tfrac{d}{n})), {\chi_o}(\vec{\mathcal{G}}(n,d)) \in (k,3(2k')+11] = (2^{d/2}, 6e^{d/2}+6d+17]. \]
\end{proof}
\section{Further Discussion} \label{sec:further_disc}
\subsection{Very sparse or very dense graphs}\label{ssec:vsparse_vdense}
Throughout the majority of this paper, we have focused on the case of oriented graphs with edge density $\frac{d}{2}$, mainly for $d$ fixed and large. However, in the case that $d \leq 2$ it is possible to use the properties of random graphs and constraints on oriented colourings with a small number of colours to say something more precise than what is given by the second moment argument. What makes these cases more tractable for analysis is that there are, up to isomorphism, only two different tournaments on $3$ vertices ($C_3$ and $T_3$ in Figure~\ref{fig:small-tourn}) and there are characterizations of when an oriented unicyclic graph can be properly coloured by either one.
\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}
[decoration={markings, mark=at position 0.7 with {\arrow{>}}}]
\tikzstyle{vertex}=[circle, fill=black, minimum size=5pt,inner sep=0pt]
\foreach \x/\l in {1/right, 2/above, 3/left}
{
\node[vertex, label = \l:{$\x$}] at ({120*\x-150}:0.75cm) (\x) {};
}
\draw[postaction={decorate}] (1) -- (2);
\draw[postaction={decorate}] (2) -- (3);
\draw[postaction={decorate}] (3) -- (1);
\node at (0, -1) {$C_3$};
\end{tikzpicture}
\hspace*{0.5in}
\begin{tikzpicture}
[decoration={markings, mark=at position 0.7 with {\arrow{>}}}]
\tikzstyle{vertex}=[circle, fill=black, minimum size=5pt,inner sep=0pt]
\foreach \x/\l in {1/right, 2/above, 3/left}
{
\node[vertex, label = \l:{$\x$}] at ({120*\x-150}:0.75cm) (\x) {};
}
\draw[postaction={decorate}] (1) -- (2);
\draw[postaction={decorate}] (2) -- (3);
\draw[postaction={decorate}] (1) -- (3);
\node at (0, -1) {$T_3$};
\end{tikzpicture}
\hspace*{0.5in}
\begin{tikzpicture}
[decoration={markings, mark=at position 0.7 with {\arrow{>}}}]
\tikzstyle{vertex}=[circle, fill=black, minimum size=5pt,inner sep=0pt]
\node[vertex, label = above:{$1$}] at (1, 1) (1) {};
\node[vertex, label = above:{$2$}] at (0, 1) (2) {};
\node[vertex, label = below:{$3$}] at (0, 0) (3) {};
\node[vertex, label = below:{$4$}] at (1, 0) (4) {};
\draw[postaction={decorate}] (1) -- (2);
\draw[postaction={decorate}] (2) -- (3);
\draw[postaction={decorate}] (3) -- (4);
\draw[postaction={decorate}] (4) -- (1);
\draw[postaction={decorate}] (3) -- (1);
\draw[postaction={decorate}] (2) -- (4);
\node at (0.5, -1) {$T_4$};
\end{tikzpicture}
\hspace*{0.5in}
\begin{tikzpicture}
[decoration={markings, mark=at position 0.7 with {\arrow{>}}}]
\tikzstyle{vertex}=[circle, fill=black, minimum size=5pt,inner sep=0pt]
\foreach \x/\l in {1/right, 2/above, 3/left, 4/below, 5/below}
{
\node[vertex, label = \l:{$\x$}] at ({72*\x-54}:1cm) (\x) {};
}
\draw[postaction={decorate}] (1) -- (2);
\draw[postaction={decorate}] (2) -- (3);
\draw[postaction={decorate}] (3) -- (4);
\draw[postaction={decorate}] (4) -- (5);
\draw[postaction={decorate}] (5) -- (1);
\draw[postaction={decorate}] (1) -- (3);
\draw[postaction={decorate}] (3) -- (5);
\draw[postaction={decorate}] (5) -- (2);
\draw[postaction={decorate}] (2) -- (4);
\draw[postaction={decorate}] (4) -- (1);
\node at (0, -1.5) {$T_5$};
\end{tikzpicture}
\end{center}
\caption{Small tournaments for colouring oriented $2$-regular graphs}\label{fig:small-tourn}
\end{figure}
In his survey, Sopena~\cite{eS16} gave a complete characterization of the
oriented chromatic number of oriented cycles, showing that for an oriented cycle
$\vec{C}$, the possible values of $\chi_o(\vec{C})$ are $2, 3, 4$, or $5$,
exactly characterizing each case. The proofs can be modified to give the
corresponding results for here for $2$-regular oriented graphs that are not
necessarily connected. The main ideas of the proofs are that both of the
tournaments $T_4$ and $T_5$ in Figure~\ref{fig:small-tourn} contain directed
$3$-cycles, directed $4$-cycles, and the transitive tournament on $3$ vertices.
This can be used to show that every directed cycle can be properly coloured with
either one, except in the case of the directed $5$-cycle that can only be
properly $T_5$-coloured. Since both tournaments $T_4$ and $T_5$ have the
property that every vertex has non-zero in-degree and out-degree, the rest
follows by induction on the number of source or sink vertices, with a few
special cases to check. Since both tournaments $T_4$ and $T_5$ have the
property that every vertex has both in-neighbours and out-neighbours, any proper
colouring of the cycle in an orientation of a unicyclic graph can be extended to
a colouring of the remaining vertices.
\begin{proposition}[see Sopena~{\cite[Proposition 10]{eS16}}]\label{prop:4or5-colouring}
If $\vec{G}$ is an orientation of a graph in which each component is a tree or unicyclic, then $\chi_o(\vec{G}) \leq 5$ and $\vec{G}$ can be properly $5$-coloured by the tournament $T_5$ in Figure~\ref{fig:small-tourn}. Moreover, if no component of $\vec{G}$ contains a directed $5$-cycle, then $\chi_o(\vec{G}) \leq 4$ and $\vec{G}$ can be properly $4$-coloured by the tournament $T_4$ in Figure~\ref{fig:small-tourn}.
\end{proposition}
Within the proof, Sopena~\cite[Proposition 10]{eS16} notes the following straightforward characterization of the $2$-regular oriented graphs that have oriented $3$-colourings. Up to isomorphism, there are only two different tournaments on $3$ vertices: the directed $3$-cycle ($C_3$) and the transitive tournament on $3$ vertices ($T_3$). Let $\vec{G}$ be an oriented cycle with vertices labelled around the cycle $u_1, u_2, \ldots, u_k$. Call an arc a \emph{forward arc} if it is directed $u_i \to u_{i+1}$ (indices computed modulo $k$) and a \emph{backward arc} if it is directed $u_{i+1} \to u_i$. Then $\vec{G}$ has a proper $C_3$-colouring if{f} the difference between the number of forward arcs and the number of backward arcs is divisible by $3$. The graph $\vec{G}$ has a proper $T_3$-colouring if{f} there are no 3 consecutive arcs in the same direction. Neither of these properties depends on the choice of starting vertex for the cycle's labelling, nor the direction of the labelling. Using these properties it is possible to show that with high probability, some component of a large $2$-regular oriented graph does not satisfy either property.
To see that a proper $C_3$-colouring is not likely, one can show, by induction, that for any $k \geq 3$, in a random orientation of a $k$-cycle, with probability at most $1/2$, the difference between the number of forward arcs and backward arcs is divisible by $3$. One can check that in a random $2$-regular graph on $n$ vertices, with high probability, there are at least $\log n$ components. Thus, the probability that a random orientation of a random $2$-regular graph is $C_3$-colourable is $o(1)$.
We now consider the property of being $T_3$-colourable, which is equivalent to not having three consecutive arcs in the same direction. The number of orientations of a $k$-cycle with this property is the same as the number of cyclic sequences of $0$s and $1$s of length $k$ that start with a $1$ and have no substring $000$ or $111$. This is bounded above by the number of binary strings (not cyclic) of length $k$ that start with a $1$ and have no three consecutive $0$s nor three consecutive $1$s. By checking cases for how such a string can begin, one can give a linear recurrence for this number and show that, if $\phi = \frac{1+ \sqrt{5}}{2}$, the number of such strings is bounded above by $\phi^k$. Since the sum of all cycle lengths in a $2$-regular graph on $n$ vertices is $n$, this shows that the probability that $\vec{G} \sim \vec{\mathcal{G}}_{n, 2}$ is properly $T_3$-colourable is at most $\left(\frac{\phi}{2}\right)^n = o(1)$.
Thus, with high probability, for $\vec{G} \sim \vec{\mathcal{G}}(n, 2)$, $\chi_o(\vec{G}) > 3$.
Every oriented $2$-regular graph has a proper oriented $5$ colouring and Proposition~\ref{prop:4or5-colouring} shows that whether $5$ colours are needed depends on the presence of a directed $5$-cycle. Since the number of $5$-cycles in the undirected graph $G_{n, 2}$ is asymptotically distributed as a Poisson random variable with mean $\frac{1}{2 \cdot 5}$, the number of directed $5$-cycles is a asymptotically distributed as a Poisson random variable with mean $\frac{1}{2 \cdot 5} \cdot \frac{1}{2^4}$. This gives the following complete result for random $2$-regular oriented graphs.
\begin{theorem}
For $\vec{G} \sim \vec{\mathcal{G}}(n, 2)$, with high probability, $\chi_o(\vec{G}) \in \{4, 5\}$ with
\begin{align*}
\mathbf{Pr}(\chi_o(\vec{G}) = 4) &= e^{-1/160} (1+o(1))\\
\mathbf{Pr}(\chi_o(\vec{G}) = 5) &= \left(1 - e^{-1/160}\right) (1+o(1)).
\end{align*}
\end{theorem}
A similar analysis can be applied to the random binomial graph $\vec{\mathcal{G}}(n, p)$ in the case that $p < 1/n$ to show that in that case, with high probability, the oriented chromatic number takes at one of $3$ possible values, each with positive probability.
\begin{theorem}\label{thm:gnd_small_d}
For $d$ constant with $0 < d < 1$ and $\vec{G} \sim \vec{\mathcal{G}}(n, p=\frac{d}{n})$, with high probability, $\chi_o(\vec{G}) \in \{3, 4, 5\}$ with
\begin{align*}
\mathbf{Pr}(\chi_o(\vec{G})) = 3) &> (1-d)^{1/2} \exp\left( \frac{d}{2} + \frac{d^2}{4}\right)(1+o(1)),\\
\mathbf{Pr}(\chi_o(\vec{G})) = 4) & > \frac{d^5}{32}e^{-d^5/10},\\
\mathbf{Pr}(\chi_o(\vec{G})) = 5) & = (1-\exp(-\frac{d^5}{160}))(1+o(1))
\end{align*}
\end{theorem}
\begin{proof}
For $p = d/n$, with $d < 1$, Erd\H{o}s and R\'{e}nyi~\cite{ER60} showed that with high probability, for $G \sim \mathcal{G}(n, p)$, every component of $G$ is either a tree or unicyclic. By Proposition~\ref{prop:4or5-colouring}, then with high probability $\chi_o(\vec{G}) \leq 5$ and $\chi_o(\vec{G}) = 5$ if and only if some component of $\vec{G}$ contains a directed $5$-cycle. Again, due to results of Erd\H{o}s and R\'{e}nyi~\cite{ER60}, the number of $5$-cycles in $\vec{G}$ is asymptotically a Poisson random variable with mean $\frac{d^5}{10}$. Thus, the probability that there is at least one directed $5$-cycle in $\vec{G}$ is asymptotically $(1-\exp(-\frac{d^5}{10}\cdot\frac{1}{16}))$. This is precisely the probability that $\vec{G}$ has oriented chromatic number equal to $5$.
By a similar argument, with probability asymptotically, $\frac{d^5}{10}e^{-d^5/10} \cdot \frac{5}{16}$, there is exactly one $5$-cycle in $\vec{G}$ and it is oriented isomorphically to the cycle in Figure~\ref{fig:5cycle4cols}. If this is the case, then $\chi_o(\vec{G}) = 4$.
\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}
[decoration={markings, mark=at position 0.7 with {\arrow{>}}}]
\tikzstyle{vertex}=[circle, fill=black, minimum size=5pt,inner sep=0pt]
\foreach \x in {0, 1, 2, 3, 4}
{
\node[vertex] at ({18+72*\x}:1cm) (\x) {};
}
\draw[postaction={decorate}] (1) -- (0);
\draw[postaction={decorate}] (1) -- (2);
\draw[postaction={decorate}] (2) -- (3);
\draw[postaction={decorate}] (4) -- (3);
\draw[postaction={decorate}] (0) -- (4);
\end{tikzpicture}
\end{center}
\caption{Oriented $5$-cycle with $\chi_o(C) = 4$}\label{fig:5cycle4cols}
\end{figure}
On the other hand, based on results of Tak\'{a}cs~\cite{jT88} or Janson~\cite{SJ87}, the probability that $G \sim \mathcal{G}(n, p=\frac{d}{n})$ is a forest is asymptotically $(1-d)^{1/2} e^{d/2+d^2/4}$. If $G$ is a forest, then any orientation of $G$ can be properly $3$-coloured by the directed $3$-cycle. On the other hand, to see that $\chi_o(\vec{G}) > 2$, note that by straightforward second moment arguments, with high probability, $\vec{G}$ contains a directed path of length $2$, which has no proper oriented $2$-colouring.
\end{proof}
It was noted by Bensmail, Duffy, and Sen~\cite{BDS17} that looking at a uniform random oriented graph on $n$ vertices, that is $\vec{G} \sim \vec{\mathcal{G}}(n, 1/2)$, with high probability, every pair of vertices are joined by a directed path of length $2$ (in one direction or the other) and hence none can be coloured with the same colour in any oriented colouring. Klostermeyer and MacGillivray~\cite{KMac04} called such graphs \emph{oriented cliques} and noted that the condition of having weak diameter $2$ is equivalent to having oriented chromatic number equal to the number of vertices. Using a standard first moment argument shows that in a large range of values for $p$, $\vec{G} \sim \vec{\mathcal{G}}(n, p)$ is an oriented clique.
\begin{proposition}\label{prop:gnp_large_p}
For any function $\omega(n) \to \infty$, if $p = p(n)$ is such that $p^2 n = 4\log n + \omega(n)$, then with high probability, for $\vec{G} \sim \vec{\mathcal{G}}(n, p)$, $\chi_o(\vec{G}) = n$.
\end{proposition}
\begin{proof}
Since there are two different directed paths of length $2$ between two given vertices, via a third. Therefore, the expected number of pairs of vertices in $\vec{G}$ not joined by a directed path of length $2$ is
\[
\binom{n}{2}(1 - p^2/2)^{n-2} \leq \frac{n^2}{2} \exp\left(-(n-2)\frac{p^2}{2}\right) \leq \frac{e}{2}\exp\left(2 \log n - \frac{np^2}{2} \right)
\]
and the result follows.
\end{proof}
\subsection{Precise value of the second moment} \label{ssec:second_mom_const}
To improve the results of Achlioptas and Moore~\cite{AM04} from a collection of
three possible values for $\chi(\mathcal{G}(n,d))$ to two values, Kemkes at
al.~\cite{KPW10} used the small subgraph conditioning method of Robinson and
Wormald (see~\cite[Chapter 9]{JLR00} and \cite{Wo99}). The small subgraph
conditioning method explains that $\mathbf{E} Y^2$ exceeds $(\mathbf{E} Y)^2$ because the
presence or absence of small cycles artificially inflates the variance. In cases
where the small subgraph conditioning method is useful, conditioning on the
presence of these cycles ``luckily and yet mysteriously'' accounts for the
difference between $\mathbf{E} Y^2$ and $(\mathbf{E} Y)^2$. The method requires
accurate estimates for the first and second moments as well as certain joint
moments.
In Section~\ref{ssec:second_moment}, we were unable to calculate
$\det(-H|_{\mathbb{V}})$ exactly. With an exact value
for $\det(-H|_{\mathbb{V}})$, we could have gotten an exact value for $C_{d,k}$
in Equation~\eqref{eq:PZ_constant}. Though we have not done so carefully, it seems likely
that there is an extension of the small subgraph conditioning method to directed
graphs. With such an extension and an exact value for $C_{d,k}$,
one could use Proposition 2 of Kemkes et al.~\cite{KPW10} to calculate
the required joint factorials and replace ``with positive probability'' in
Proposition~\ref{prop:gnd_ub} with ``almost surely.'' This argument would supersede
Section~\ref{sec:window} and allow for the marginally stronger result
\[ {\chi_o}(\mathcal{G}(n,d)) \in (2^{d/2}, e^{d/2}+d+1]. \]
Extending the small subgraph conditioning method to directed graphs would also
open up the possibility of improving the bounds on ${\chi_o}(\mathcal{G}(n,m))$.
In addition, one would need to calculate the joint factorial moments of small
cycles in $\mathcal{M}(n,m)$. We chose not to pursue this approach because the
it only offers a scalar improvement, a marginal difference compared to the
exponential gap between the upper and lower bounds.
\subsection{Upper bounds when $k$ does not divide $n$} \label{ssec:divisibility}
In Section~\ref{sec:ub} we calculated upper bounds for ${\chi_o}(\vec{\mathcal{M}}(n,m=cn))$ and ${\chi_o}(\vec{\mathcal{C}}(n,d))$ under the assumption that $n$ was divisible by $k$. In this section we discuss how to remove that condition using ideas of~\cite{NP21}.
Suppose $n = qk+r$ where $r \in [0,k-1]$. We extend the concept of equitable colourings by requiring every colour classes 1 through $r$ to contain $q+1$ vertices and colour classes $r+1$ through $k$ to contain $q$ vertices. (Note that for $r=0$ this coincides with our original definition of equitable.) The strategy to adapt Propositions~\ref{prop:gnm_ub} and \ref{prop:gnd_ub} for $r > 0$ is to show that fixing these $r = O(1)$ vertices does not affect the ratio $(\mathbf{E} Y)^2/(\mathbf{E} Y^2)$. To get a feeling for the argument, we prove the following proposition.
\begin{proposition} \label{prop:proportional}
Let $k$ be an integer such that there exists a doubly regular tournament of order $k$ and let $c > 0$. Fix a doubly regular tournament $T_k$. For $n = qk+r, r \in [0,k-1]$, let $Y_n$ count the number of equitable oriented $T_k$-colourings of $\vec{\mathcal{M}}(n,m=cn)$. Let $n' = n-r$. Then
\[ \mathbf{E} Y_n^2 \sim \left(k\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right)^c\right)^{2r} \mathbf{E} Y_{n'}^2. \]
\end{proposition}
Before proving Proposition~\ref{prop:proportional}, we introduce a few ideas of P\'erez and the second author. Recall that in the proof of Lemma~\ref{lem:gnm_second_mom}, we defined the set $\mathbb{X}_n$ which contained, for each $n$, the valid overlap matrices on $n$ vertices. We proved that the exponential contribution to each term of the sum was maximized at $\mbf{\hat{x}}$, the vector with every entry $\frac{n}{k^2}$. As seen in Equation~\eqref{eq:exp_sum}, the second moment is a sum containing polynomially many exponential terms, and thus the terms close to this exponential maximum contribute all but a negligible portion of the sum. This fact is captured in the following result:
\begin{proposition}[see {\cite[Proposition 3.5]{NP21}}] \label{prop:central_sum}
Suppose the same set of conditions hold as in Theorem~\ref{thm:laplace_summation_gamma}. In addition, suppose there is $N > 0$ such that $\psi(x) \le n^N$ for each $\mbf x \in K_1$. For any $\gamma > 0$, define $\mathbb{Y}_n(\gamma) \subseteq \mathbb{X}_n$ as
\[ \mathbb{Y}_n(\gamma) = \left\{ \mbf x \in \mathbb{X}_n : ||\mbf{\hat{x}} - \mbf x||_{\infty} < \gamma \frac{\log n}{\sqrt{n}} \right\}. \]
Then
\[ \sum_{\mbf x \in \mathbb{Y}_n(\gamma)} T_n(\mbf x) \sim \sum_{\mbf x \in \mathbb{X}_n} T_n(\mbf x). \]
\end{proposition}
We now prove Proposition~\ref{prop:proportional}, following the proof of Proposition 7.1 in~\cite{NP21}.
\begin{proof}[Proof of Proposition~\ref{prop:proportional}]
Given $\mbf A \in \mathbb{Y}_n(1)$, define
\[ \mbf{A}^\ast = \left(\frac{1}{n'}(na_{ij}-\delta_{ij})\right)_{i,j \in [k]} \]
where
\[ \delta_{ij} = \begin{cases} 1 & i =j \le r\\ 0 & \text{else} \end{cases}. \]
Then $\mbf{A}^\ast$ defines the overlap matrix of a pair of equitable oriented $T_k$-colourings of $\vec{\mathcal{M}}(m,cn)$ as we have removed the $r$ vertices of prescribed colours 1 through $r$. As $\mbf A \in \mathbb{Y}_n(1)$, we know $\mbf{A}^\ast$ is non-negative.
Recall from Equation~\eqref{eq:mnm_count} that the number of oriented $T_k$-colourings respecting $\mbf A$ is
\[ \frac{n!}{\prod_{v \in V} (na_v)!}\left(\sum_{uv \in E(T^{\otimes 2})}a_ua_v \right)^m =: \ell(\mbf A). \]
We have
\[ \frac{\ell(\mbf A)}{\ell(\mbf{A}^\ast)} = \frac{n_{(r)}}{n^r} \cdot \frac{1}{\prod_{i=1}^r a_{ii}} \cdot \left(\frac{n-r}{n}\right)^{2cr}\cdot \frac{(\sum_{uv \in E}a_ua_v)^{cn}}{(\sum_{uv \in E}(a_ua_v - \frac{1}{n}(a_u\delta_v + a_v\delta_u) + \frac{1}{n^2}\delta_u\delta_v))^{c(n-r)}}. \]
Note that $\delta_u\delta_v = 1$ if and only if $u = (i,i), v = (j,j)$, and $i\ne j \le r$. Therefore
\[ \sum_{uv \in E(T^{\otimes 2})} \delta_u \delta_v = \binom{r}{2}. \]
Furthermore,
\[ 0 \le \sum_{uv \in E(T^{\otimes 2})} (a_u\delta_v + a_v\delta_u) \le \sum_{uv \in E(T^{\otimes 2})} (a_u + a_v) = \frac{(k-1)^2}{2} \sum_{v \in V(T^{\otimes 2})} a_v = \frac{(k-1)^2}{2} \]
as each $v \in T^{\otimes 2}$ has degree $\frac{(k-1)^2}{2}$. Therefore, keeping in mind $\mbf A \in \mathbb{Y}_n(1)$, we have
\[ \frac{\ell(\mbf A)}{\ell(\mbf{A}^\ast)} \ge \frac{n_{(r)}}{n^r} \cdot \left(\frac{n-r}{n}\right)^{2cr} \cdot \frac{(\sum_{uv \in E}(\frac{1}{k^2}-\frac{|log n}{\sqrt{n}})^2)^{cn}}{(\frac{1}{k^2}+\frac{\log n}{\sqrt{n}})^r(\sum_{uv \in E}(\frac{1}{k^2}+\frac{\log n}{\sqrt{n}})^2+\frac{1}{n^2}\binom{r}{2})^{c(n-r)}} \to \left(k\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right)^c\right)^{2r} \]
and
\[ \frac{\ell(\mbf A)}{\ell(\mbf{A}^\ast)} \le \frac{n_{(r)}}{n^r} \cdot \left(\frac{n-r}{n}\right)^{2cr} \cdot \frac{(\sum_{uv \in E}(\frac{1}{k^2}+\frac{|log n}{\sqrt{n}})^2)^{cn}}{(\frac{1}{k^2}-\frac{\log n}{\sqrt{n}})^r(\sum_{uv \in E}(\frac{1}{k^2}-\frac{\log n}{\sqrt{n}})^2-\frac{(k-1)^2}{2n})^{c(n-r)}} \to \left(k\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right)^c\right)^{2r} \]
from which we conclude
\[ \frac{\ell(\mbf A)}{\ell(\mbf{A}^\ast)} \to \left(k\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right)^c\right)^{2r}. \]
Let $\mathcal{A} = \{ \mbf{A}^\ast \mid \mbf A \in \mathbb{Y}_n(1) \}$. Then if there is $\gamma > 0$ such that $\mathbb{Y}_n(\gamma) \subseteq \mathcal{A}$, Proposition~\ref{prop:central_sum} gives
\begin{align*}
\left(k\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right)^c\right)^{2r} \mathbf{E} Y_{n'}^2 &\sim \sum_{\mbf A \in \mathbb{Y}_{n'}(\gamma)} \left(k\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right)^c\right)^{2r} p(n,\mbf A) \exp(n f(\mbf A))\\
& \le \sum_{\mbf A \in \mathcal{A}} \left(k\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right)^c\right)^{2r} p(n,\mbf A) \exp(n f(\mbf A)) \sim \mathbf{E} Y_n^2\\
&\le \sum_{\mbf A \in \mathbb{X}_{n'}} \left(k\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right)^c\right)^{2r} p(n,\mbf A) \exp(n f(\mbf A))\\
&= \left(k\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right)^c\right)^{2r} \mathbf{E} Y_{n'}^2,
\end{align*}
completing the proof of the proposition.
Thus it suffices to show for some $\gamma >0$ and all $\mbf{\tilde{A}} \in \mathbb{Y}_{n'}(\gamma)$ there is $\mbf A \in \mathbb{Y}_n(1)$ such that $\mbf{A}^\ast = \mbf{\tilde{A}}$. Each $\tilde{a}_v \in \mbf{\tilde{A}}$ satisfies
\[ \frac{1}{k^2} - \gamma\frac{\log n'}{\sqrt{n'}} \le \tilde{a}_v \le \frac{1}{k^2} + \gamma\frac{\log n'}{\sqrt{n'}} \]
and we want
\[ \frac{1}{n'}(na_v-\delta_v) = \tilde{a}_v \]
where $a_v$ must satisfy
\[ \frac{1}{k^2} - \frac{\log n}{\sqrt{n}} \le a_v \le \frac{1}{k^2} + \frac{\log n}{\sqrt{n}}. \]
Thus it suffices to pick $\gamma$ so that
\[ \frac{1}{k^2}-\frac{\log n}{\sqrt{n}} \le \frac{n'}{n}\left(\frac{1}{k^2}-\gamma\frac{\log n'}{\sqrt{n'}}\right) \]
and
\[ \frac{1}{n}\left(n'\left(\frac{1}{k^2}+\gamma \frac{\log n'}{\sqrt{n'}}\right)+1\right) \le \frac{1}{k^2}+\frac{\log n}{\sqrt{n}}. \]
These are identical conditions to those of~\cite{NP21}, where the authors show $\gamma=0.4$ suffices.
\end{proof}
To complete the argument that ${\chi_o}(\vec{\mathcal{M}}(n,m=cn)) \le k$ with positive probability, one should prove that
\[ \mathbf{E} Y_n \sim \left(k\left(\frac{1}{2}\left(1-\frac{1}{k}\right)\right)^c\right)^{r} \mathbf{E} Y_{n'} \]
in a similar (but easier) manner; then
\[ \mathbf{Pr}[Y_n > 0] \ge \frac{(\mathbf{E} Y_n)^2}{\mathbf{E}(Y_n^2)} \sim \frac{(\mathbf{E} Y_{n'})^2}{\mathbf{E}(Y_{n'}^2)} \sim \frac{(\mathbf{E} Y)^2}{\mathbf{E}(Y^2)} \sim \left(\frac{((k-1)^2-2c)^2-4c^2k^2}{(k-1)^4}\right)^{(k-1)^2/4} > 0. \]
In proving the upper bound for ${\chi_o}(\vec{\mathcal{C}}(n,d))$, we used the Laplace Summation technique in~\cite{GJR10}, rather than that in~\cite{NP21}; however, the latter is an extension of the former and thus a version of Proposition~\ref{prop:central_sum} holds under the conditions of Theorem~\ref{thm:laplacian_summation} as well. Then one should prove a version of Proposition~\ref{prop:proportional} for the first and second moments of $\vec{\mathcal{C}}(n,d)$.
\subsection{Colouring with tournaments that are not doubly-regular}
The exponential gap between the bounds in Theorem~\ref{thm:gnd_main} comes from the different constants in $u_k$ and $\ell_k$; $u_k$ behaves like $\frac{2}{\log 2} \log k$ while $\ell_k$ grows as $2\log k$. While the argument for the lower bound is straightforward, one might reasonably ask if a better constant for $\ell_k$ could be found by counting equitable colourings of a different tournament, hoping to improve Corollary~\ref{cor:opt_doubly_reg_tour}. In the calculations of the second moment for the number of equitable oriented colourings in Lemma~\ref{lem:gnm_second_mom} and Section~\ref{ssec:counting}, the initial computation of the second moment would apply equally well to colouring by \emph{any} tournament on $k$ vertices, not just a doubly-regular tournament. Similarly, the function, $\gamma_d$, defined in Proposition~\ref{prop:generalized_AN} can be defined with the Lagrangian part given for any graph $G$. In the case where a tournament $T$ is not regular, that is, not all vertices have the same in-degree or out-degree, the product graph $G = T^{\otimes 2}$ will not be regular.
If $G$ is not regular, then subject to the condition that the matrix $\mbf A$ be doubly-stochastic, the point at $\mbf A = \frac{1}{k}J_k$ is not a critical point for the function $-\frac{1}{k} \sum_{v \in V(G)} a_v \log a_v + \frac{d}{2}\log\left(\frac{2\sum_{uv \in E} a_u a_v}{k^2}\right)$ and so cannot be the location of the maximum value. This would mean that the maximum value is strictly greater and so the exponential part of the second moment of the number of equitable $T$-colourings has a larger base that the square of the expected value and a standard second moment method argument would not yield a positive probability of finding an equitable $T$-colouring.
Among the regular tournaments, one might seek a tournament whose Kronecker square has a smaller second-largest eigenvalue to improve the bound on $d$ obtained from Proposition~\ref{prop:generalized_AN}. However, one can show that the signed adjacency matrix of any regular tournament of order $k$ has an eigenvalue $\lambda$ with $|\lambda| \ge \sqrt{k}$ and use this fact to prove that the second-largest eigenvalue has size at least $\frac{1}{2}(1+\lambda^2) \ge \frac{k+1}{2}$, precisely the value of the second-largest eigenvalue of the Kronecker square of a doubly regular tournament. The fact that the signed adjacency matrix of a regular tournament of order $k$ has an eigenvalue of modulus at least $\sqrt{k}$ follows from a result by Brauer and Gentry~\cite{BG72} on the unsigned adjacency matrices of regular tournaments, but one can also prove it directly by showing that if $M$ is the signed adjacency matrix of a regular tournament of order $k$ and $\mathbf{v}$ is any column of $M$, then $||M \mathbf{v}||_2/||\mathbf{v}||_2 \geq \sqrt{k}$. From this, one can show that the largest eigenvalue of $T^{\otimes 2}$ is exactly $\frac{(k-1)^2}{2}$ and the second-largest eigenvalue is at least $\frac{k+1}{2}$. Thus, for the approach of finding oriented colourings using the second-moment method, doubly-regular tournaments will give the best results.
\section*{Acknowledgements}
The authors would like to thank Stephen Kirkland for help with references on the algebraic properties of tournaments and Siddarth Sankaran for assistance with reference for number theoretic results. They also thank Xavier P\'erez for offering his expertise on the history of problems on random graphs of bounded degree and Cris Moore for helpful clarifications about the results for undirected regular graphs.
\bibliographystyle{abbrv}
\nocite{*}
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\section{Introduction}
Could quark and lepton masses be related at TeV scales? Not long ago, one of us
explored this possibility in the framework of the so-called \textit{early
quark-lepton mass unification}~\cite{Hung2005}. The idea was to combine two TeV
scale scenarios, namely one of the two petite unification models
$\mathrm{PUT}_1=\mathrm{SU}(4)_{\mathrm {PS}} \otimes \mathrm{SU}(2)_{L} \otimes \mathrm{SU}(2)_{R}
\otimes \mathrm{SU}(2)_{H}$, and TeV scale large extra dimensions~\cite{LED,Arkani1998}.
The Petite Unification Theories (PUT's)~\cite{Hung1982,Buras2003} are
quark-lepton unification models, which occur at TeV scales and have the gauge
group structure $G = G_{S}(g_S) \otimes G_{W}(g_W)$. Both PUT models propose
unusually charged heavy quarks and leptons, in addition to the fermion content
of the Standard Model (SM).
The model in Ref.~\cite{Hung2005} made use of the mechanism of wave function
overlap along the large extra dimension~\cite{Arkani1998,Antoniadis1998}, which
was originally employed to justify the smallness of Dirac neutrino
mass~\cite{Arkani2000,Hung2003a,SNM}. The mechanism connects the strengths of
the couplings in the mass terms of the fermions in four dimensions, as
\textit{effective} Yukawa couplings, to the magnitudes of wave function overlaps
between the corresponding left- and right-handed fermionic zero modes along the
large extra dimension~\cite{Arkani2000,Hung2003a}.
In this framework, therefore, the shapes of the wave functions of left- and
right-handed fermions plus distances between those wave functions in the extra
dimension determine the strengths of the mass terms in four dimensions.
The geometry of the fermionic zero modes along the extra dimension was
systematically set in Ref.~\cite{Hung2005} by breaking the symmetries of the
model in the extra dimension down to that of the Standard Model, which was the
approach originally suggested in Ref.~\cite{Hung2003a}.
As a result, Ref.~\cite{Hung2005} obtained \textit{early quark-lepton mass
unification}, within which the four-dimensional (4D) Yukawa couplings of the
chiral fermions of the model related to each other and a light Dirac neutrino
was made possible.
The present work intends to build a model based on the marriage of the other
petite unification model, ${\mathrm {PUT}}_2=\mathrm{SU}(4)_{\mathrm {PS}} \otimes
\mathrm{SU}(3)_L \otimes \mathrm{SU}(3)_{H}$, and the physics of large extra dimension in the
context of ``brane world" picture, in order to explore its implications. Similar
to the work in Ref.~\cite{Hung2005}, we make use of the idea of wave function
overlaps along the extra dimension and set the geometry of the zero modes by
symmetry breakings.
Historically, questions on quark-lepton mass relation were addressed in a
quark-lepton unification scenario, e.g., Grand Unified Theories
(GUT's)~\cite{GUT}. A well-known example of this is the equality of
$\tau$-lepton and bottom-quark masses~\cite{Buras1978} at $M_{GUT}$ in $\mathrm{SU}(5)$
scenario. A TeV scale quark-lepton mass relation differs from a GUT one in the
amount of ``running\footnote{including both coupling constants and masses.}" one
needs to be concerned about if one attempts to explore the implications at lower
energies, say $M_Z$.
On another front, the present work assumes a Dirac neutrino, which will turn out
light in a direct correlation with the masses of heavy unconventional fermions.
Such connection between a light Dirac neutrino and TeV-scale physics is in
contrast with the traditional seesaw mechanism~\cite{Seesaw}, where its scale is
limited perhaps only by Planck mass. Very recently, however, a TeV scale
scenario for seesaw mechanism~\cite{Hung2007} has been put forward, which
broadens the implications on TeV-scale physics to both Dirac and Majorana light
neutrinos. Of course, the final word on the nature of neutrino, whether it is a
Majorana or Dirac particle, must come from experiment, in particular those
regarding lepton number violation.
The outline of the paper is as follows. First, we go over the idea of petite
unification theories briefly followed by a review on the group structure and the
particle content of $\mathrm{PUT}_2$ scenario. Then, we present a five
dimensional model based on $\mathrm{PUT}_2$ scenario plus a short review on the
wave function overlap mechanism. Afterward, we set the geometry of the zero mode
wave functions of chiral fermions by systematic symmetry breakings in the extra
dimension. In subsequent sections, we move toward the computation of chiral
fermion mass scales by relating them to the magnitudes of applicable overlaps in
the extra dimension. A numerical analysis concludes the mass scale computation,
which substantiates the notion of early quark-lepton mass unification. Then, we
examine the validity of our model by computing the electroweak oblique parameter
$S$ and the lifetimes of heavy chiral fermions.
\section{Petite unification of quarks and leptons}\label{sec:PUT}
Petite unification models \cite{Hung1982} were built around the idea of unifying
quarks and leptons at an energy scale not too much higher than the electroweak
scale. They have the gauge group structure of $G = G_{S}(g_S) \otimes
G_{W}(g_W)$ with two independent couplings $g_S$ and $g_W$, which must contain
the SM fields. The first PUT model was constructed based on the knowledge of the
low-energy $\sin^{2} \theta_W$ value and known fermion representations at the
time. With the $\mathrm{SU}(4)_{\mathrm{PS}}$ group of Pati and Salam \cite{Pati1974}
chosen for $G_S$ and the constraint from the experimental value of
$\sin^{2}\theta_{W}(M_{Z}^{2})$, known at the time, the gauge group
$\mathrm{PUT}_0=\mathrm{SU}(4)_{\mathrm {PS}} \otimes [\mathrm{SU}(2)]^4$ with unification scale
of several hundreds of TeV emerged and was proposed in Ref.~\cite{Hung1982}.
Later precise measurements of $\sin^{2}\theta_{W}(M_{Z}^{2})$ plus renewed
interest in TeV scale physics, however, resulted in a thorough re-examination of
the PUT idea \cite{Buras2003}, yielding three favorable PUT models:
$\mathrm{PUT}_0$ and $\mathrm{PUT}_{1,2}$, where
\begin{equation}
{\mathrm {PUT}}_1=\mathrm{SU}(4)_{\mathrm {PS}} \otimes \mathrm{SU}(2)_{L} \otimes
\mathrm{SU}(2)_{R} \otimes \mathrm{SU}(2)_{H} ,
\end {equation}
and
\begin{equation}
{\mathrm {PUT}}_2=\mathrm{SU}(4)_{\mathrm {PS}} \otimes \mathrm{SU}(3)_L \otimes \mathrm{SU}(3)_{H} .
\end {equation}
The new measured value of $\sin^{2}\theta_{W}(M_{Z}^{2})$, which was higher than
its old value, lowered the unification scale down to a few-TeV region. This
lower scale rules out $\mathrm{PUT}_0$ scenario due to problems with the decay
rate of $K_L \rightarrow \mu e$ at tree level. The remaining two models,
$\mathrm{PUT}_{1}$ and $\mathrm{PUT}_{2}$, however, are found to naturally avoid
the violation of the upper bound on the $K_L \rightarrow \mu e$ rate at tree
level. The SM gauge group with three couplings, $\mathrm{SU}(3)_c (g_3)\otimes \mathrm{SU}(2)_L
(g_2) \otimes \mathrm{U}(1)_Y (g_1)$, is assumed to be embedded into the PUT
groups with two couplings. The symmetry breaking scheme of PUT scenarios is
given by\footnote{The gauge symmetry breakdown of PUT scenarios down to that of
the SM with an additional discrete $\mathcal{Z}$ symmetry and its implications
on monopoles is discussed in Ref.~\cite{Zubkov2007}.}
\begin{subequations}\label{equ:sb}
\begin{equation}
G \stackrel{\textstyle M}{\longrightarrow} G_1
\stackrel{\textstyle \tilde{M}}{\longrightarrow} G_2
\stackrel{\textstyle M_Z}{\longrightarrow} \mathrm{SU}(3)_c \otimes \mathrm{U}(1)_{EM} ,
\end{equation}
where
\begin{equation}
G_1 = \mathrm{SU}(3)_{c}(g_3) \otimes \mathrm{U}(1)_S(\tilde{g}_S) \otimes
G_{W}(g_W) \, ,
\end {equation}
and
\begin{equation}
G_2 = \mathrm{SU}(3)_{c}(g_3) \otimes \mathrm{SU}(2)_{L}(g_2) \otimes
\mathrm{U}(1)_{Y}(g^\prime)\ ,
\end{equation}
\end{subequations}
with $M_Z < \tilde{M} \leq M$. The two PUT scenarios have three new generations
of unconventional quarks and leptons, in addition to the three standard
generations of quarks and leptons. The magnitude of the charges of these new
particles can reach up to $4/3$ (for ``quarks'') and 2 (for ``leptons''). The
horizontal groups $\mathrm{SU}(2)_H$ and $\mathrm{SU}(3)_H$ connect the standard fermions to the
unconventional ones, as well as the gauge bosons of
$\mathrm{SU}(4)_{\mathrm{PS}}/\left[\mathrm{SU}(3)_c \otimes \mathrm{U}(1)_S \right]$.
In both PUT models the $\mathrm{SU}(4)_{\mathrm {PS}}$ quartets contain either
``unconventional quark and the SM lepton'' or ``SM quark and unconventional
lepton.'' As a result, there is no tree-level transition between ordinary quarks
and leptons mediated by the $\mathrm{SU}(4)_{\mathrm{PS}}/\left[\mathrm{SU}(3)_c \otimes
\mathrm{U}(1)_S \right]$ gauge bosons. This important property prevents rare
decays such as $K_L \rightarrow \mu e$ from acquiring large rates, since it can
only occur through one-loop processes which can be made small enough to comply
with the experimental bound.
Another property of PUT scenarios is the existence of new contributions to
flavor changing neutral current (FCNC) processes, involving standard quarks and
leptons, which are mediated by the horizontal $\mathrm{SU}(2)_H$ and $\mathrm{SU}(3)_H$ weak
gauge bosons and the new unconventional quarks and leptons. Nonetheless, they
appear at one-loop level and can be made consistent with the existing
experimental bounds. A thorough analysis of $\mathrm{PUT}_1$ was carried out by
the authors of Ref.~\cite{Buras2004}.
\section{\texorpdfstring{$\text{PUT}_2$}{PUT2} model}\label{sec:PUT2}
In this scenario the weak gauge group is $G_W=\mathrm{SU}(3)_{L} \otimes \mathrm{SU}(3)_{H}$,
where the SM's $\mathrm{SU}(2)_L$ is the subgroup of its $\mathrm{SU}(3)_L$. The gauge symmetry
breaking of $\mathrm{PUT}_2$ follows the scheme given in Eqs.~(\ref{equ:sb}).
Within such symmetry breaking, the strong $\mathrm{U}(1)_S$ group corresponds to
the unbroken diagonal generator of $\mathrm{SU}(4)_{\mathrm{PS}}$, i.e., $\hat Y_S$. The
weak hypercharge $\mathrm{U}(1)_Y$ group emerges from $\mathrm{U}(1)_S$ and
$G_W$ breaking, whose generator $\hat{Y}_W$ can be written as $\hat Y_W = C_S
\hat T_{15\mathrm{PS}} + C_L \hat T_{8L} + C_{1H} \hat T_{8H} + C_{2H} \hat
T_{3H}$ where $\hat T$'s are the diagonal generators of $\tilde{G}_S$,
$\mathrm{SU}(3)_L$ and $\mathrm{SU}(3)_H$ symmetries. The SM's $\hat T_{3L}$ generator is simply
the third generator of $\mathrm{SU}(3)_L$, which goes into the unbroken $\mathrm{SU}(2)_L$
subgroup. Note that this is all in the ``unlocked standard model'' picture of
Ref.~\cite{Hung1982}, where the generators of $\mathrm{SU}(2)_L$ are the unbroken
generators of $G_W$. The $C_i$ coefficients in $\hat{Y}_W$ define the embedment
of the SM's weak hypercharge group $\mathrm{U}(1)_Y$ into $G_1$.
The two symmetry breaking scales $M$ and $\tilde M$ were determined in
Ref.~\cite{Buras2003} by renormalization group (RG) evolution combined with the
very precise experimental value of $\sin ^2 \theta _W \left( {M_Z^2 } \right)$.
The values could differ by up to an order of magnitude, roughly $3 \leqslant M
\leqslant 10{\text{ TeV}}$ and $0.8 \leqslant \tilde M \leqslant 3{\text{
TeV}}$.
The charge operator in PUT scenarios is defined as $\hat Q = \hat Q_W + C_S
\hat T_{15\mathrm{PS}}$, where $\hat Q_W$ is the weak charge given by $\hat Q_W
= \hat T_{3L} + C_L \hat T_{8L} + C_{1H} \hat T_{8H} + C_{2H} \hat T_{3H}$.
The weak charge $Q_W $, as shown in Ref.~\cite{Hung1982}, is related to $\sin ^2
\theta _W^0 $ defining the charge distribution of the relevant representations
of PUT scenarios. For $\mathrm{PUT}_2$ model, $C_S^2 = {8 \mathord{\left/
{\vphantom {8 3}} \right. \kern-\nulldelimiterspace} 3}$ and the important group
theoretical factor $\sin ^2 \theta _W^0 $ is given by $\sin ^2 \theta _W^0 = {1
\mathord{\left/ {\vphantom {1 {\left( {1 + C_W^2 } \right)}}} \right.
\kern-\nulldelimiterspace} {\left( {1 + C_W^2 } \right)}} = {3 \mathord{\left/
{\vphantom {3 8}} \right. \kern-\nulldelimiterspace} 8}$ , where $C_W^2 = C_L^2
+ C_{1H}^2 + C_{2H}^2 = {5 \mathord{\left/ {\vphantom {5 3}} \right.
\kern-\nulldelimiterspace} 3}$.
For the model in question, the fermion representations, which together are
anomaly-free, are $(4,3,\bar{3})$ and $(4,\bar{3},3)$. The charge distribution
of the fermion content of $(4,3,\bar{3})$ representation is
\begin{equation}\label{equ:charge1}
\mathcal{Q}_1 = \Biggl( {\left[ {\left( {\frac{1}
{3},\frac{4}
{3},\frac{4}
{3}} \right),\left( { - 1,0,0} \right)} \right],\left[ {\left( { - \frac{2}
{3},\frac{1}
{3},\frac{1}
{3}} \right),\left( { - 2, - 1, - 1} \right)} \right],\left[ {\left( { -
\frac{2}
{3},\frac{1}
{3},\frac{1}
{3}} \right),\left( { - 2, - 1, - 1} \right)} \right]} \Biggr) \, ,
\end{equation}
Similarly, for $(4,\bar{3},3)$ the charge distribution is given by
\begin{equation}\label{equ:charge2}
\mathcal{Q}_2 = \Biggl( {\left[ {\left( {\frac{1}
{3}, - \frac{2}
{3}, - \frac{2}
{3}} \right),\left( { - 1, - 2, - 2} \right)} \right],\left[ {\left( {\frac{4}
{3},\frac{1}
{3},\frac{1}
{3}} \right),\left( {0, - 1, - 1} \right)} \right],\left[ {\left( {\frac{4}
{3},\frac{1}
{3},\frac{1}
{3}} \right),\left( {0, - 1, - 1} \right)} \right]} \Biggr) \, ,
\end{equation}
In terms of $\mathrm{SU}(2)_{L}$ doublets and singlets, one can write $(4,3,\bar{3})$ as
\begin{equation}\label{equ:psi1}
\Psi _{1,L} = \Biggl( {\bigg[ {\Big( {\bm{\psi}^{Q*} ,D^c } \Big),\Big( {\psi
^l ,\nu ^c } \Big)} \bigg],\bigg[ {\Big( {\bm{\psi} ^{q*} ,d^c } \Big),\Big(
{\bm{\psi}^{L*} ,l_{d}^c } \Big)} \bigg],\bigg[ {\Big( {\tilde {\bm{\psi}} ^{q*}
,\tilde{\mathtt{d}}^{*} } \Big),\Big( {\tilde{\bm{\psi}}^{L*}
,\tilde{\mathtt{l}}^{*} } \Big)} \bigg]} \Biggr)_L ,
\end{equation}
and $(4,\bar{3},3)$ as
\begin{equation}\label{equ:psi2}
\Psi _{2,L} = \Biggl( {\bigg[ {\Big( {\tilde {\bm{\psi}} ^{q,c} ,u^c }
\Big),\Big( {\tilde{\bm{\psi}}^{L,c} ,l_{u}^c } \Big)} \bigg],\bigg[ {\Big(
{\tilde{\bm{\psi}}^{Q,c} ,U^c } \Big),\Big( {\tilde \psi ^{l,c*} ,l^{c*} }
\Big)} \bigg],\bigg[ {\Big( {\tilde{\bm{\psi}}^{Q*} ,\tilde{\mathtt{d}}^{c} }
\Big),\Big( {\tilde \psi ^l ,\tilde{\mathtt{l}}^{c} } \Big)} \bigg]} \Bigg)_L .
\end{equation}
Before we identify the $\mathrm{SU}(2)_{L}$ doublets and singlets appearing in
Eqs.~(\ref{equ:psi1} and \ref{equ:psi2}), let us first point out that in
Eqs.~(\ref{equ:psi1} and \ref{equ:psi2}) the right-handed fields are written in
terms of the left-handed charge conjugates; so that the whole representation is
left handed, e.g., $\nu_L^c$ or $u_L^c$. Besides, to match the charge
distributions of Eqs.~(\ref{equ:charge1} and \ref{equ:charge2}), some
$\mathrm{SU}(2)_{L}$ doublets, in Eqs.~(\ref{equ:psi1} and \ref{equ:psi2}), appear in
italic-boldface typeset. To explain this notation, consider an arbitrary doublet
\begin {equation}
\psi _{L,R} = \left( {\begin{array}{*{20}c}
{\psi _u } \\
{\psi _d } \\
\end{array} } \right)_{L,R} ,
\end {equation}
then $\bm{\psi} _{L,R}$, the rotated doublet in $\mathrm{SU}(2)$ space by $\pi$ about
the second axis, is defined as
\begin {equation}\label{equ:cpsi}
\bm{\psi} _{L,R} \equiv i\tau _2 \psi _{L,R} = \left( {\begin{array}{*{20}c}
{\psi _d } \\
{ - \psi _u } \\
\end{array} } \right)_{L,R}.
\end{equation}
The $\mathrm{SU}(2)_L$ doublets and singlets present in $(4,3,\bar{3})$ are\footnote{As
a convention, the fields presented by tilded letters are vector-like (i.e., not
chiral).}
\begin{subequations}\label{rep1}
\begin{equation}
\psi^{q}_{L} = \left(
\begin{array}{c}
u(2/3)\\
d(-1/3)
\end{array}
\right)_L \,; \,\, d^{c}_L(1/3)=C \bar{d}^{\,T}_{R}\, ,
\end{equation}
\begin{equation}
\psi^{l}_{L} = \left(
\begin{array}{c}
\nu(0) \\
l(-1)
\end{array}
\right)_L \, ; \,\, \nu^{c}_L= C \bar{\nu}_{R}^{T} \, ,
\end{equation}
\begin{equation}
\psi^Q_{L} = \left(
\begin{array}{c}
U(-1/3)\\
D(-4/3)
\end{array}
\right)_L \,; \,\, D^{c}_{L}(4/3) = C \bar{D}^{T}_{R} \, ,
\end{equation}
\begin{equation}
\psi^L_{L} = \left(
\begin{array}{c}
l_{u}(2)\\
l_{d}(1)
\end{array}
\right)_L \,; \,\, l^{c}_{d,L}(-1)= C \bar{l}^{\,T}_{d,R}\, ,
\end{equation}
\begin{equation}
\tilde{\psi}^L_{L} = \left(
\begin{array}{c}
\tilde{l}_{u}(2)\\
\tilde{l}_{d}(1)
\end{array}
\right)_L \, ; \,\, \tilde{\mathtt{l}}_L(+1) \, ,
\end{equation}
\begin{equation}
\tilde{\psi}^{q}_{L} = \left(
\begin{array}{c}
\tilde{u}(2/3)\\
\tilde{d}(-1/3)
\end{array}
\right)_L \, ; \,\, \tilde{\mathtt{d}}_L(-1/3) \, .
\end{equation}
\end{subequations}
In the above list, one notices normal quarks and leptons, and those with unusual
electric charges. On the other hand, the $\mathrm{SU}(2)$ doublets and singlets of
$(4,\bar{3},3)$ are
\begin{subequations}\label{rep2}
\begin{equation}
\tilde{\psi}^{l}_{L,R} = \left(
\begin{array}{c}
\tilde{\nu}(0)\\
\tilde{l}(-1)
\end{array}
\right)_{L,R} \, ; \,\,\, l^{c}_{L}(+1) = C \bar{l}^{\,T}_{R} \, ,
\end{equation}
\begin{equation}
\tilde{\mathtt{l}}_R (+1)\ \, ;\ \, u_L^c \left( {{{ - 2} \mathord{\left/
{\vphantom {{ - 2} 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)=C \bar{u}^{T}_{R} \,,
\end{equation}
\begin{equation}
\tilde{\psi}^{q}_{R} = \left(
\begin{array}{c}
\tilde{u}(+2/3)\\
\tilde{d}(-1/3)
\end{array}
\right)_R\ \, ; \,\, \tilde{\mathtt{d}} _R (-1/3) \, ,
\end{equation}
\begin{equation}
\tilde{\psi}^L_{R} = \left(
\begin{array}{c}
\tilde{l}_{u}(2)\\
\tilde{l}_{d}(1)
\end{array}
\right)_{R}\ \, ; \,\,
l^{c}_{u,L}(-2)= C \bar{l}^{\,T}_{u,R} \, ,
\end{equation}
\begin{equation}
\tilde{\psi}^Q_{L,R} = \left(
\begin{array}{c}
\tilde{U}(-1/3)\\
\tilde{D} (-4/3)
\end{array}
\right)_{L,R}\ \,; \;\, U^{c}_{L}(1/3) = C
\bar{U}^{T}_{R} \, .
\end{equation}
\end{subequations}
One notices two types of families with SM transformation property in both
$\Psi_1$ and $\Psi_2$. This means left-handed doublets and right-handed singlets
for each family. One family includes SM quarks and leptons (normal fermions) and
the other contains unconventional quarks and leptons, i.e., those with unusual
charges. These unconventional particles are $\psi^Q_L$, $D^c_L$, $U^c_L$, and
$\psi^L_{L}$, $l^c_{d,L}$, $l^c_{u,L}$. The normal and unconventional quarks and
leptons will receive mass through their couplings with the SM Higgs field.
In addition, the fermion content of $\mathrm{PUT}_2$ includes two vector-like
$\mathrm{SU}(2)_L$ doublets of quarks and leptons $(\tilde\psi^q,\tilde\psi^l)_{L,R}$
and $(\tilde{\psi}^Q,\tilde{\psi}^L)_{L,R}$, with normal and unusual charges,
and two vector-like $\mathrm{SU}(2)_L$ singlets $\tilde{\mathtt{l}}_{L,R}$ and
$\tilde{\mathtt{d}}_{L,R}$. These vector-like particles can obtain large bare
masses as mentioned in Ref.~\cite{Buras2003}.
Let us write the two representations in terms of quartets and triplets of the
corresponding gauge symmetry groups. For $\Psi _{1L}$, we have the following
multiplets:
\begin{itemize}
\item
$\mathrm{SU}(4)_{\mathrm{PS}}$ quartets
\begin{subequations}\label{equ:quartets1}
\begin{equation}
\begin{array}{*{20}c}
{\left( {\begin{array}{*{20}c}
{D^* _L \left( {{4 \mathord{\left/
{\vphantom {4 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{\nu _L \left( 0 \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{-U^* _L \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{l_L \left( { - 1} \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{D_L^c \left( {{4 \mathord{\left/
{\vphantom {4 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{\nu _L^c \left( 0 \right)} \\
\end{array} } \right)} \,, \\
\end{array}
\end{equation}
\begin{equation}
\begin{array}{*{20}c}
{\left( {\begin{array}{*{20}c}
{d^* _L \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{l^* _{d,L} \left( { - 1} \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{-u^* _L \left( {{{ - 2} \mathord{\left/
{\vphantom {{ - 2} 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{-l^* _{u,L} \left( { - 2} \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{d_L^c \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{l_{d,L}^c \left( { - 1} \right)} \\
\end{array} } \right)}\,, \\
\end{array}
\end{equation}
\begin{equation}
\begin{array}{*{20}c}
{\left( {\begin{array}{*{20}c}
{\tilde{d}^* _L \left( {{{ - 2} \mathord{\left/
{\vphantom {{ - 2} 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{\tilde{l}^* _{d,L} \left( { - 1} \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{-\tilde{u}^* _L \left( {{{ - 2} \mathord{\left/
{\vphantom {{ - 2} 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{-\tilde{l}^* _{u,L} \left( { - 2} \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{\tilde{\mathtt{d}}_L^{*} \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} \right)} \\
{\tilde{\mathtt{l}}_L^{*} \left( { - 1} \right)} \\
\end{array} } \right)}\,, \\
\end{array}
\end{equation}
\end{subequations}
\item
$\mathrm{SU}(3)_L$ triplets
\begin{subequations}\label{equ:tripletsL1}
\begin{equation}
\begin{array}{*{20}c}
{\left( {\begin{array}{*{20}c}
{D^* _L \left( {{4 \mathord{\left/
{\vphantom {4 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{-U^* _L \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{D_L^c \left( {{4 \mathord{\left/
{\vphantom {4 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{\nu _L \left( 0 \right)} \\
{l_L \left( { - 1} \right)} \\
{\nu _L^c \left( 0 \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{d^* _L \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{-u^* _L \left( {{{ - 2} \mathord{\left/
{\vphantom {{ - 2} 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{d_L^c \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
\end{array} } \right)}\,, \\
\end{array}
\end{equation}
\begin{equation}\label{equ:tripletsL12}
\begin{array}{*{20}c}
{\left( {\begin{array}{*{20}c}
{l^* _{d,L} \left( { - 1} \right)} \\
{-l^* _{u,L} \left( { - 2} \right)} \\
{l_{d,L}^c \left( { - 1} \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{\tilde{d}^* _L \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{-\tilde{u}^* _L \left( {{{ - 2} \mathord{\left/
{\vphantom {{ - 2} 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{\tilde{\mathtt{d}}_L^{*} \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{\tilde{l}^* _{d,L} \left( { - 1} \right)} \\
{-\tilde{l}^* _{u,L} \left( { - 2} \right)} \\
{\tilde{\mathtt{l}}_L^{*} \left( { - 1} \right)} \\
\end{array} } \right)}\,, \\
\end{array}
\end{equation}
\end{subequations}
\item
$\mathrm{SU}(3)_H$ antitriplets
\begin{subequations}\label{equ:atripletsH1}
\begin{equation}
\begin{array}{*{20}c}
{\left( {\begin{array}{*{20}c}
{d^* _L \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{D^* _L \left( {{4 \mathord{\left/
{\vphantom {4 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{\tilde{d}^* _L \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{-u^* _L \left( {{{ - 2} \mathord{\left/
{\vphantom {{ - 2} 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{-U^* _L \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{-\tilde{u}^* _L \left( {{{ - 2} \mathord{\left/
{\vphantom {{ - 2} 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{d_L^c \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{D_L^c \left( {{4 \mathord{\left/
{\vphantom {4 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{\tilde{\mathtt{d}}_L^{*} \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
\end{array} } \right)}\,, \\
\end{array}
\end{equation}
\begin{equation}
\begin{array}{*{20}c}
{\left( {\begin{array}{*{20}c}
{l^* _{d,L} \left( { - 1} \right)} \\
{\nu _L \left( 0 \right)} \\
{\tilde{l}^* _{d,L} \left( { - 1} \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{-l^* _{u,L} \left( { - 2} \right)} \\
{l_L \left( { - 1} \right)} \\
{-\tilde{l}^* _{u,L} \left( { - 2} \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{l_{d,L}^c \left( { - 1} \right)} \\
{\nu _L^c \left( 0 \right)} \\
{\tilde{\mathtt{l}}_L^{*} \left( { - 1} \right)} \\
\end{array} } \right)}\,, \\
\end{array}
\end{equation}
\end{subequations}
\end{itemize}
For $\Psi _{2L} $, on the other hand, the corresponding multiplets are:
\begin{itemize}
\item
$\mathrm{SU}(4)_{\mathrm{PS}}$ quartets
\begin{subequations}\label{equ:quartets2}
\begin{equation}
\begin{array}{*{20}c}
{\left( {\begin{array}{*{20}c}
{\tilde d_L^{c} \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{\tilde{l}_{d,L}^{c} \left( { - 1} \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{-\tilde u_L^{c} \left( {{{ - 2} \mathord{\left/
{\vphantom {{ - 2} 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{-\tilde{l}_{u,L}^{c} \left( { - 2} \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{u_L^c \left( {{{ - 2} \mathord{\left/
{\vphantom {{ - 2} 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{l_{u,L}^c \left( { - 2} \right)} \\
\end{array} } \right)} \,, \\
\end{array}
\end{equation}
\begin{equation}
\begin{array}{*{20}c}
{\left( {\begin{array}{*{20}c}
{\tilde{D}_L^{c} \left( {{4 \mathord{\left/
{\vphantom {4 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{\tilde \nu _L^{c*} \left( 0 \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{-\tilde{U}_L^{c} \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{\tilde l_L^{\,c*} \left( { - 1} \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{U_L^c \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{l_L^{c*} \left(-1 \right)} \\
\end{array} } \right)} \,, \\
\end{array}
\end{equation}
\begin{equation}
\begin{array}{*{20}c}
{\left( {\begin{array}{*{20}c}
{\tilde{D}_L^{*} \left( {{4 \mathord{\left/
{\vphantom {4 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{\tilde \nu _L \left( 0 \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{-\tilde{U}_L^{*} \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{\tilde l_L \left( { - 1} \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{\tilde{\mathtt{d}}_L^{c} \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{\tilde{\mathtt{l}}_L^{c} \left( { - 1} \right)} \\
\end{array} } \right)} \,, \\
\end{array}
\end{equation}
\end{subequations}
\item
$\mathrm{SU}(3)_L$ antitriplets
\begin{subequations}\label{equ:atripletsL2}
\begin{equation}
\begin{array}{*{20}c}
{\left( {\begin{array}{*{20}c}
{-\tilde u_L^{c} \left( {{{ - 2} \mathord{\left/
{\vphantom {{ - 2} 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{\tilde d_L^{c} \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{u_L^c \left( {{{ - 2} \mathord{\left/
{\vphantom {{ - 2} 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{-\tilde{l}_{u,L}^{c} \left( { - 2} \right)} \\
{\tilde{l}_{d,L}^{c} \left( { - 1} \right)} \\
{l_{u,L}^c \left( { - 2} \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{-\tilde{U}_L^{c} \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{\tilde{D}_L^{c} \left( {{4 \mathord{\left/
{\vphantom {4 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{U_L^c \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
\end{array} } \right)} \,, \\
\end{array}
\end{equation}
\begin{equation}\label{equ:atripletsL22}
\begin{array}{*{20}c}
{\left( {\begin{array}{*{20}c}
{\tilde l_L^{c*} \left( { - 1} \right)} \\
{\tilde \nu _L^{c*} \left( 0 \right)} \\
{l_L^{c*} \left( { - 1} \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{-\tilde{U}_L^{*} \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{\tilde{D}_L^{*} \left( {{4 \mathord{\left/
{\vphantom {4 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{\tilde{\mathtt{d}}_L^{c} \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{\tilde l_L \left( { - 1} \right)} \\
{\tilde \nu _L \left( 0 \right)} \\
{\tilde{\mathtt{l}}_L^{c} \left( { - 1} \right)} \\
\end{array} } \right)} \,, \\
\end{array}
\end{equation}
\end{subequations}
\item
$\mathrm{SU}(3)_H$ triplets
\begin{subequations}\label{equ:tripletsH2}
\begin{equation}
\begin{array}{*{20}c}
{\left( {\begin{array}{*{20}c}
{\tilde{D}_L^{c} \left( {{4 \mathord{\left/
{\vphantom {4 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{\tilde d_L^{c} \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{\tilde{D}_L^{*} \left( {{4 \mathord{\left/
{\vphantom {4 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{-\tilde{U}_L^{c} \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{-\tilde u_L^{c} \left( {{{ - 2} \mathord{\left/
{\vphantom {{ - 2} 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{-\tilde{U}_L^{*} \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{U_L^c \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{u_L^c \left( {{{ - 2} \mathord{\left/
{\vphantom {{ - 2} 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
{\tilde{\mathtt{d}}_L^{c} \left( {{1 \mathord{\left/
{\vphantom {1 3}} \right.
\kern-\nulldelimiterspace} 3}} \right)} \\
\end{array} } \right)} \,, \\
\end{array}
\end{equation}
\begin{equation}
\begin{array}{*{20}c}
{\left( {\begin{array}{*{20}c}
{\tilde \nu _L^{c*} \left( 0 \right)} \\
{\tilde{l}_{d,L}^{c} \left( { - 1} \right)} \\
{\tilde \nu _L \left( 0 \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{\tilde l_L^{c*} \left( { - 1} \right)} \\
{-\tilde{l}_{u,L}^{c} \left( { - 2} \right)} \\
{\tilde l_L \left( { - 1} \right)} \\
\end{array} } \right)} & ; & {\left( {\begin{array}{*{20}c}
{l_L^{c*} \left( { - 1} \right)} \\
{l_{u,L}^c \left( { - 2} \right)} \\
{\tilde{\mathtt{l}}_L^{c} \left( { - 1} \right)} \\
\end{array} } \right)} \,, \\
\end{array}
\end{equation}
\end{subequations}
\end{itemize}
Before we end this section, it is worth mentioning that all left-handed SM-type
fermions are in $\Psi _1 $. Plus, four of the corresponding right-handed fields
are in $\Psi _1 $ (i.e., $d_L^c$, $D_L^c$, $l_{d,L}^c$, $\nu _L^c$) and the
other four in $\Psi _2 $ (i.e., $u_L^c$, $U_L^c$, $l_L^{c*}$, $l_{u,L}^c$). The
right-handed fields, in both representations, are the third components of the
$\mathrm{SU}\left( 3 \right)_L $ triplets.
\section{Early quark-lepton unification in five dimensions}
Generalization to five-dimensional (5D) space is simply done by introducing an
extra spatial dimension, $y$. It is well known that 5D fermions are of Dirac
type and not chiral. As we would like the SM-type fermion content of our five
dimensional model to mimic the \textit{chiral} spectrum of the 4D SM-type
fermions; we compactify the extra dimension on an ${{\mathrm{S}_1 }
\mathord{\left/ {\vphantom {{S_1 } {\mathbb{Z}_2 }}} \right.
\kern-\nulldelimiterspace} {\mathbb{Z}_2 }}$ orbifold with a TeV-scale size.
That means the size of the extra dimension for our model is about the inverse of
the partial unification scale ($M \sim 3.3 - 10$ TeV).
In the ``brane world" picture, however, such chiral fermions are assumed to be
trapped onto a three-dimensional (3D) sub-manifold (``brane'' or ``domain wall''
\cite{Rubakov1983}) as zero modes. The localization of fermions into brane is
achievable by coupling the fermionic field to a background scalar field with a
kink solution.
In addition to localization, the shapes of zero-mode wave functions are to be
set. For doing that, we follow the idea in Ref.~\cite{Hung2003a} for which a
short review is given here.
In Ref.~\cite{Hung2003a} a 5D left-right symmetric model was considered. After
localizing the right-handed fermions of a given doublet at the same point, the
$\mathrm{SU}(2)_R$ symmetry was spontaneously broken along the extra dimension via the
kink solution of a triplet scalar field. The outcome of such symmetry breaking
is significant in the sense that one element of the right-handed doublet obtains
a narrow, while the other element acquires a broad wave function along the extra
dimension. With left handed doublet localized at some other point along the
extra dimension, two very different left-right overlaps are resulted. An
exaggerated depiction of such overlaps is shown in Fig.~\ref{fig:lroverlaps} for
a leptonic doublet, $\nu$ and $l$. Fermionic Dirac mass terms involve left- and
right-handed fields and when the extra dimension is integrated out, the Yukawa
coupling in 4D space will be proportional to the corresponding left-right
overlaps in the extra dimension. The spirit of the work presented in
Ref.~\cite{Hung2003a} is that when zero-mode wave functions of the right-handed
fields overlap with the left-handed wave function (common for both $\nu$ and
$l$) there will be a large difference between the effective Yukawa couplings of
neutrino and charged lepton.
The objective in our 5D model is to localize the SM-type fermions of our model
on 3D slices and break the relevant symmetries along the extra dimension, which
in turn define the geometry of zero modes and ultimately will determine the
effective Yukawa couplings in the 4D theory.
\begin{figure*}
\includegraphics{Adibzadeh_Overlaps}
\caption{\label{fig:lroverlaps} Schematic depiction of left-right overlaps for
neutrino and charged lepton in the extra dimension: $\xi^{\nu.l}_R$, zero-mode
wave function for right-handed neutrino and charged lepton; $\xi_L$, zero-mode
wave function for left-handed leptonic doublet.}
\end{figure*}
The localization and symmetry breakings along the extra dimension involve Yukawa
couplings, e.g., in the form $f\bar \Psi _1 \Phi \Psi _1 + f\bar \Psi _2 \Phi
\Psi _2 $, where $\Psi _1 $ and $\Psi _2 $ couple to the same scalar field with
the same coupling constant to localize at the same point or shift position with
the same amount. This suggests an $\mathrm{SU}(2)_G$ global symmetry among $\Psi _1 $
and $\Psi _2 $ in the extra-dimensional Yukawa sector. For the Yukawa sector in
the extra dimension, therefore, the symmetry group of the theory can be written
as the product of global and gauge groups, $G=G_{\text{gauge}} \otimes
G_{\text{global}}$. Although the $\mathrm{SU}(2)_G$ global symmetry is limited to the
extra-dimensional Yukawa sector, there is an analogous, however implicit, global
$\mathrm{SU}(2)$ symmetry among $\Psi_1$ and $\Psi_2$ if only the strong
$\mathrm{SU}(4)_{\mathrm{PS}}$ quartets are looked at, i.e., weak group neglected. The
fact that the weak group representations of $\Psi_1$ and $\Psi_2$ differ means
that such extensive symmetry is explicitly broken by $G_W$.
The fermion representation of the model for couplings with scalar fields in the
extra dimension can be written as
\begin {equation}
\Psi \left( {x^\mu ,y} \right) = \left( {\begin{array}{*{20}c}
{\begin{array}{*{20}c}
{4,3,\bar 3} \\
{4,\bar 3,3} \\
\end{array} } & ; & 2 \\
\end{array} } \right) = \left( {\begin{array}{*{20}c}
{\Psi _1 \left( {x^\mu ,y} \right)} \\
{\Psi _2 \left( {x^\mu ,y} \right)} \\
\end{array} } \right),
\end {equation}
where we used the notation $\left( {\begin{array}{*{20}c}{\mathrm{Gauge}} & ; &
{\mathrm{Global}} \\ \end{array} } \right)$ to articulate the multiplet
structure of $\Psi$ with respect to the gauge and global groups.
To find out the appropriate group representations of the background scalar
fields, needed for localization and symmetry breakings, we should examine the
bilinear form of $\Psi$ under $\mathrm{SU}(4)_{\mathrm {PS}} \otimes \mathrm{SU}(3)_L \otimes
\mathrm{SU}(3)_{H} \otimes \mathrm{SU}(2)_{G}$, explicitly
\begin{equation}\label{equ:bilinear}
\bar \Psi \left( {x,y} \right)\Psi \left( {x,y} \right) = \left(
{\begin{array}{*{20}c}
{15 \oplus 1,8 \oplus 1,8 \oplus 1} & ; & {3 \oplus 1} \\
\end{array} } \right).
\end{equation}
From Eq.~(\ref{equ:bilinear}), one can pick suitable scalar fields to
\begin{enumerate}
\item
Localize the right-handed and left-handed fermions in the extra dimension at
different locations,
\item
Give different profiles to up and down sectors of the right-handed fermions,
\item
Differentiate between normal and unconventional fermions, also quarks and
leptons.
\end{enumerate}
In the following sections, we shall carry out these tasks one by one.
\subsection{Localization of fermions}\label{sec:localization}
To localize the SM-type fermions as chiral zero modes, we first note that we
wrote the fermion representations of $\mathrm{PUT}_2$ as left-handed multiplets
(see section~\ref{sec:PUT2}). Therefore, by choosing a chiral ${{\mathrm{S}_1 }
\mathord{\left/ {\vphantom {{S_1 } {\mathbb{Z}_2 }}} \right.
\kern-\nulldelimiterspace} {\mathbb{Z}_2 }}$ orbifold and positive couplings in
the localization process \cite{Georgi2001}, we can assign zero modes to all
left-handed fields of the representation including the right handed fields which
are written as charge conjugates. This way, the relevant 5D Dirac spinors
transform as left-handed SM fermions. Symbolically, the 5D representation of the
model as chiral zero mode can be imagined as
\begin {equation}
\Psi_L \left( {x ,y} \right) = \left( {\begin{array}{*{20}c}
{\Psi _{1L} \left( x\right) \xi_1 \left( y \right)} \\
{\Psi _{2L} \left( x \right) \xi_2 \left( y \right)} \\
\end{array} } \right),
\end {equation}
where $\xi$'s take on the appropriate zero-mode wave functions for each
constituent field of the multiplet. Throughout this work and for clarity, we
denote the zero-mode wave functions of the left- and right-handed fields with
subscripts $L$ and $R$, respectively.
For localization, consider a singlet scalar field $\Phi _S = \left(
{\begin{array}{*{20}c}{1,1,1} & ; & 1 \\ \end{array} } \right)$. The gauge- and
global-invariant Yukawa coupling of such scalar field with fermions looks like
\begin{equation}\label{equ:singcoup}
\mathcal{L}_S = f_S \bar \Psi \Phi _S \Psi = f_S \left( {\bar \Psi _1 \Phi _S
\Psi _1 + \bar \Psi _2 \Phi _S \Psi _2 } \right),
\end{equation}
where $f_S > 0$. To localize at some non-zero point, $y \ne0$, let the kink
solution of $\Phi _S$ be in the form
\begin{equation}
\left\langle {\Phi _S } \right\rangle = h_S \left( y \right) +v_S \, .
\end{equation}
The equation of motion for the zero-mode wave functions of the left- and
right-handed SM-type fermions is then given by
\begin{equation}\label{equ:localized}
\partial _y \xi _{L,R} + \left[ {f_S h_S \left( y \right) + f_S v_S } \right]
\xi _{L,R} = 0 \, .
\end{equation}
However, if one wants to have left-right overlaps between the zero modes, one
needs to separate the zero-mode wave functions of the left- and right-handed
fields along the extra dimension. This can be done by moving the left- and
right- handed zero-modes asymmetrically. To do this, we need to couple fermions
to a background scalar field that would only acquire a minimum energy solution
and \textit{not} a kink solution. We introduce a scalar field $\Phi _O = \left(
{\begin{array}{*{20}c} {1,8 ,1} & ; & 1 \\ \end{array} } \right)$, whose
coupling with fermions takes the form
\begin{equation}\label{equ:Octcoup}
\mathcal{L}_O = - f_O \bar \Psi \Phi _O \Psi = - f_O \left( \bar \Psi_1 \Phi _O
\Psi_1 + \bar \Psi_2 \Phi _O \Psi_2 \right),
\end{equation}
where $f_O > 0$. The minimum energy solution of $\Phi _O$ (which leaves
$\mathrm{SU}(2)_L$ unbroken) for such asymmetrical shift can simply take on the eighth
direction of $\mathrm{SU}(3)_L$, i.e.,
\begin{equation}\label{equ:octVEV}
\left\langle {\Phi _O } \right\rangle = \delta \left( {\begin{array}{*{20}c}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & { - 2} \\
\end{array} } \right),
\end{equation}
where $\delta$ is the vacuum expectation value (VEV) of $\Phi _O$. The coupling
in Eq.~(\ref{equ:Octcoup}), when $\Phi _O$ develops VEV, shifts the position of
the left- and right-handed zero modes along the extra dimension differently,
which is obvious from their equations of motion
\begin{subequations}\label{equ:lrheq0}
\begin{equation}\label{equ:lheq0}
\partial _y \xi _L \left( y \right) + \left[ {f_S h_S \left( y \right) + f_S
v_S - f_O \delta } \right]\xi _L \left( y \right) = 0,
\end{equation}
\begin{equation}\label{equ:rheq0}
\partial _y \xi _R \left( y \right) + \left[ {f_S h_S \left( y \right) + f_S
v_S + 2f_O \delta} \right] \xi _R \left( y \right) = 0.
\end{equation}
\end{subequations}
with $\delta \ne 0$. The possibility of $\delta = 0$ will be discussed later.
We remind ourselves that the left-handed zero-mode wave functions $\xi _L$, are
$\mathrm{SU}(2)_L$ doublets, while the right-handed zero-mode wave functions $\xi _R$,
are just singlets.
\subsection{Distinguishing the up and down sectors of the right-handed
fermions}\label{sec:uddis}
Since one sector of the right-handed SM-type fields are in $\Psi_1$ and the
other in $\Psi_2$, distinguishing these two sectors along the extra dimension
demands a coupling which differentiates between them in the extra dimension.
Looking at Eq.~(\ref{equ:bilinear}), we consider two $\mathrm{SU}(2)_G$ triplet fields
$\Phi _T = \left( {\begin{array}{*{20}c} {1,1 ,1} & ; & 3 \\ \end{array} }
\right)$ and $\Phi' _T = \left( {\begin{array}{*{20}c} {1,8,1} & ; & 3 \\
\end{array} } \right)$, for an asymmetrical profile changing. The Yukawa
couplings with fermions would be
\begin{equation}\label{equ:Gcoupling}
\mathcal{L}_T = f_T \bar \Psi \Phi _T \Psi + f'_T \bar \Psi \Phi' _T \Psi,
\end{equation}
where $f_T , f'_T > 0$. To alter the shapes of the right-handed zero-mode wave
functions, these two triplet fields must attain kink solutions, they are
\begin{equation}
\left\langle {\Phi _{T} } \right\rangle = h_T \left( y \right) \left(
{\begin{array}{*{20}c}
1 & 0 \\
0 & { - 1} \\
\end{array} } \right),
\end{equation}
and
\begin{equation}
\left\langle {\Phi' _{T} } \right\rangle = h'_T \left( y \right) \left(
{\begin{array}{*{20}c}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & { - 2} \\
\end{array} } \right) \otimes \left( {\begin{array}{*{20}c}
1 & 0 \\
0 & { - 1} \\
\end{array} } \right),
\end{equation}
where $h_T \left( y \right)$ and $h'_T \left( y \right)$ are the kink solutions
of $\Phi _T$ and $\Phi' _T$, respectively.
The equations of motion for the zero-mode wave functions now read
\begin{subequations}\label{equ:eomTT}
\begin{equation}\label{equ:eomTTl0}
\partial _y \xi _L \left( y \right) + \left[ {f_S h_S \left( y \right) + f_T h_T
\left( y \right) + f'_T h'_T \left( y \right) + f_S v_S - f_O \delta}
\right]\xi _L \left( y \right) = 0 \, ,
\end{equation}
\begin{equation}\label{equ:eomTTru0}
\partial _y \xi _R^{up} \left( y \right) + \left[ {f_S h_S \left( y \right) +
\left({ f_T h_T \left( y \right) -2f'_T h'_T \left( y \right) } \right) + f_S
v_S - 2f_O \delta }\right]\xi _R^{up} \left( y \right) = 0 \, ,
\end{equation}
\begin{equation}\label{equ:eomTTrd0}
\partial _y \xi _R^{up} \left( y \right) + \left[ {f_S h_S \left( y \right) -
\left({ f_T h_T \left( y \right) -2f'_T h'_T \left( y \right) } \right) + f_S
v_S - 2f_O \delta }\right]\xi _R^{up} \left( y \right) = 0 \, ,
\end{equation}
\end{subequations}
where $\xi _R^{up}$ and $\xi _R^{down}$ refer to the right-handed zero-mode wave
functions of $\Psi_1$ (i.e., those of $d$, $D$, $l_{d}$, and $\nu$) and $\Psi_2$
(i.e., those of $u$, $U$, $l_{u}$, and $l$), respectively. The doublet $\xi _L$
still refers to both normal and unconventional left-handed fermion zero-mode
wave functions, which means the left-handed fermions of $\Psi_1$.
It can be seen, from Eqs.~(\ref{equ:eomTTru0} and \ref{equ:eomTTrd0}) that the
profiles of the right-handed zero-mode wave functions of $\Psi_1$ and $\Psi_2$,
which we denote by $\xi _R^{up}$ and $\xi _R^{down}$, are now different: a broad
wave function for $\xi _R^{down}$ and a narrow wave function for $\xi _R^{up}$.
This disparity between the profiles of the two sectors of right-handed zero
modes may become more clear in section~\ref{sec:separation}. Let us define
\begin{subequations}\label{equ:hs}
\begin{equation}
h_{sym} \left( y \right) \equiv f_S h_S \left( y \right) + \left({ f_T h_T
\left( y \right) - 2f'_T h'_T \left( y \right)} \right) \, ,
\end{equation}
\begin{equation}
h_{asym} \left( y \right) \equiv f_S h_S \left( y \right) - \left({ f_T h_T
\left( y \right) - 2f'_T h'_T \left( y \right)} \right) \, ,
\end{equation}
\end{subequations}
for future compactness of equations.
\subsection{Distinguishing normal and unconventional fermions, quarks and
leptons}\label{sec:undis}
As the geometry of the zero-mode wave functions in the extra dimension
determines the overlaps and therefore the effective Yukawa couplings, one would
like to differentiate between the zero-mode wave functions of normal and
unconventional fermions, also between those of quarks and leptons. Since these
fermions are mixed by groups $\mathrm{SU}(4)_\mathrm{PS}$ and $\mathrm{SU}(3)_H$, breaking those
symmetries along the extra dimension seems plausible.
The desired symmetry breaking can be achieved by four scalar fields, which only
develop VEV's and not kink solutions. The scalar fields are $\Sigma = \left(
{\begin{array}{*{20}c} {15,8,1} & ; & 1 \\ \end{array} } \right)$, $\Sigma'
= \left({\begin{array}{*{20}c} {15,1,1} & ; & 1 \\ \end{array} } \right)$,
$\Omega = \left( {\begin{array}{*{20}c} {15,8,8} & ; & 1 \\ \end{array} }
\right)$, and $\Omega' = \left( {\begin{array}{*{20}c} {15,1,8} & ; & 1 \\
\end{array} } \right)$, with Yukawa couplings in the form
\begin{equation}
\mathcal{L}_Y = \bar \Psi \left( {f_{\Sigma } \Sigma + f_{\Sigma' } \Sigma' +
f_{\Omega } \Omega + f_{\Omega' } \Omega' } \right)\Psi ,
\end{equation}
where $f_{\Sigma }, \, f_{\Sigma'}, \, f_{\Omega}, \, f_{\Omega'} > 0$. The
minimum energy solutions of these fields are taken as
\begin{subequations}\label{equ:sigomegvev}
\begin{equation}
\left\langle {\Sigma } \right\rangle = \sigma \left( {\begin{array}{*{20}c}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & { - 3} \\
\end{array} } \right) \otimes \left( {\begin{array}{*{20}c}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & { - 2} \\
\end{array} } \right),
\end{equation}
\begin{equation}
\left\langle {\Sigma'} \right\rangle = \sigma' \left( {\begin{array}{*{20}c}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & { - 3} \\
\end{array} } \right),
\end{equation}
\begin{equation}
\left\langle {\Omega} \right\rangle = \left( {\begin{array}{*{20}c}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & { - 3} \\
\end{array} } \right) \otimes \left( {\begin{array}{*{20}c}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & { - 2} \\
\end{array} } \right) \otimes \left( {\begin{array}{*{20}c}
{\omega} & 0 & 0 \\
0 & { - \omega} & 0 \\
0 & 0 & 0 \\
\end{array} } \right),
\end{equation}
\begin{equation}
\left\langle {\Omega' } \right\rangle = \left( {\begin{array}{*{20}c}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & { - 3} \\
\end{array} } \right) \otimes \left( {\begin{array}{*{20}c}
{\omega'} & 0 & 0 \\
0 & { - \omega'} & 0 \\
0 & 0 & 0 \\
\end{array} } \right).
\end{equation}
\end{subequations}
Similar to $\Phi_O$'s role in section \ref{sec:localization}, the role of these
scalar fields is to shift the positions of the zero-mode wave functions of
normal and unconventional fermions, even those of quarks and leptons along the
extra dimension. That means different left-right separations for each one of
those classes, which would indicate different overlaps and therefore effective
Yukawa couplings.
Let us start with the left-handed zero-mode wave functions. Their equation of
motion, Eq.~(\ref{equ:eomTTl0}), now splits into four different equations
\begin{subequations}\label{equ:lheom0}
\begin{equation}
\partial _y \xi _L^q \left( y \right) + \left[ {f_S h_S \left( y \right) + f_T
h_T \left( y \right) + f'_T h'_T \left( y \right) + f_S v_S - f_O \delta +
X_L^q} \right]\xi _L^q \left( y \right) = 0 \, ,
\end{equation}
\begin{equation}
\partial _y \xi _L^Q \left( y \right) + \left[ {f_S h_S \left( y \right) + f_T
h_T \left( y \right) + f'_T h'_T \left( y \right) + f_S v_S - f_O \delta +
X_L^Q} \right]\xi _L^Q \left( y \right) = 0 \, ,
\end{equation}
\begin{equation}
\partial _y \xi _L^l \left( y \right) + \left[ {f_S h_S \left( y \right) + f_T
h_T \left( y \right) + f'_T h'_T \left( y \right) + f_S v_S - f_O \delta -
3X_L^q} \right]\xi _L^l \left( y \right) = 0 \, ,
\end{equation}
\begin{equation}
\partial _y \xi _L^L \left( y \right) + \left[ {f_S h_S \left( y \right) + f_T
h_T \left( y \right) + f'_T h'_T \left( y \right) + f_S v_S - f_O \delta - 3
X_L^Q} \right]\xi _L^L \left( y \right) = 0 \, ,
\end{equation}
\end{subequations}
where
\begin{subequations}\label{equ:xls}
\begin{equation}
X_L^q = f_{\Sigma} \sigma + f_{\Sigma'} \sigma' + f_{\Omega} \omega +
f_{\Omega'} \omega' \, ,
\end{equation}
\begin{equation}
X_L^Q = f_{\Sigma } \sigma + f_{\Sigma'} \sigma' - f_{\Omega} \omega -
f_{\Omega'} \omega' \, ,
\end{equation}
\end{subequations}
In Eqs.~(\ref{equ:lheom0}), the superscripts $q$, $l$, $Q$, and $L$, correspond
to normal quark, normal lepton, unconventional quark and unconventional lepton,
respectively.
The two equations of motion for right-handed zero-mode wave functions,
Eqs.~(\ref{equ:eomTTru0} and \ref{equ:eomTTrd0}), also split into eight
equations for those of quarks and leptons, unconventional and normal. For $\xi
_R^{up}$, we obtain
\begin{subequations}\label{equ:rheom0}
\begin{equation}
\partial _y \xi _R^{q,up} \left( y \right) + \left[ {h_{sym} \left( y \right) +
f_S v_S + 2f_O \delta + X_R^q} \right]\xi _R^q \left( y \right) = 0 \, ,
\end{equation}
\begin{equation}
\partial _y \xi _R^{Q,up} \left( y \right) + \left[ {h_{sym} \left( y \right) +
f_S v_S + 2f_O \delta + X_R^Q} \right]\xi _R^Q \left( y \right) = 0 \, ,
\end{equation}
\begin{equation}
\partial _y \xi _R^{l,up} \left( y \right) + \left[ {h_{sym} \left( y \right) +
f_S v_S + 2f_O \delta -3X_R^q} \right]\xi _R^l \left( y \right) = 0 \, ,
\end{equation}
\begin{equation}
\partial _y \xi _R^{L,up} \left( y \right) + \left[ {h_{sym} \left( y \right) +
f_S v_S + 2f_O \delta -3X_R^Q} \right]\xi _R^L \left( y \right) = 0 \, ,
\end{equation}
\end{subequations}
where
\begin{subequations}\label{equ:xrs}
\begin{equation}
X_R^q = -2 f_{\Sigma } \sigma + f_{\Sigma'} \sigma' -2 f_{\Omega} \omega +
f_{\Omega'} \omega' \, ,
\end{equation}
\begin{equation}
X_R^Q = -2 f_{\Sigma} \sigma + f_{\Sigma'} \sigma' +2 f_{\Omega} \omega -
f_{\Omega'} \omega' \, ,
\end{equation}
\end{subequations}
while for $\xi _R^{down}$ the same equations are valid with $h_{sym} \left( y
\right) \rightarrow h_{asym} \left( y \right)$ of Eqs.~(\ref{equ:hs}). From
Eqs.~(\ref{equ:lheom0} and \ref{equ:rheom0}), it is clear that due to strong and
horizontal symmetry breaking, each type of left- and right-handed zero mode is
localized at different point in the extra dimension. Therefore, the left-right
separations which determine the overlaps would be different for each type, as we
desired.
Although Eqs.~(\ref{equ:lheom0} and \ref{equ:rheom0}) seem to suggest that the
displacements due to strong and horizontal symmetry breakings are expressed in
terms of four parameters $X_L^q$, $X_L^Q$, $X_R^q$, and $X_R^Q$, there are only
two independent parameters involved. For example, since
\begin{subequations}\label{equ:reduction}
\begin{equation}
X_L^q + X_L^Q = 2f_{\Sigma} \sigma + 2f_{\Sigma'} \sigma' \, ,
\end{equation}
\begin{equation}
X_R^q + X_R^Q = -4f_{\Sigma} \sigma + 2f_{\Sigma'} \sigma' \, ,
\end{equation}
\end{subequations}
once one fixes the two coupling constants and vacua on the right hand side of
Eqs.~(\ref{equ:reduction}), the $X$'s can be expressed in terms of each other.
That means two of these $X$'s are indeed arbitrary and can be viewed as
references for the other two.
Hence, let us set $X_R^q=X_R^Q=0$ and let $X_L^q$, $X_L^Q$ be the two
independent parameters of strong and horizontal symmetry breakings, they become
\begin{subequations}\label{equ:xlsn}
\begin{equation}
X_L^q = 3f_{\Sigma} \sigma + 3f_{\Omega} \omega \, ,
\end{equation}
\begin{equation}
X_L^Q = 3f_{\Sigma} \sigma - 3f_{\Omega} \omega \, .
\end{equation}
\end{subequations}
At this stage and to differentiate the normal fermions from the unconventional
ones, we demand the important phenomenological constraint
\begin{equation}\label{equ:const6}
f_{\Sigma} \sigma = f_{\Omega} \omega \, .
\end{equation}
This assumption separates the zero-mode wave functions of normal and
unconventional SM-type fermions in a fashion that results in stronger left-right
overlaps for unconventional fermions and consequently higher mass scales. That
is what we expect, since the unconventional fermions have not been
experimentally detected yet. With the constraint of Eq.~(\ref{equ:const6}), the
two independent distances $X_L^q$ and $X_L^Q$ become
\begin{subequations}\label{equ:XLf}
\begin{equation}
X_L^q = 6f_\Omega \omega \, ,
\end{equation}
\begin{equation}
X_L^Q = 0 \, ,
\end{equation}
\end{subequations}
Therefore, the zero-mode wave functions of left-handed SM-type fermions satisfy
\begin{subequations}\label{equ:lheomj}
\begin{equation}
\partial _y \xi _L^q \left( y \right) + \left[ {f_S h_S \left( y \right) + f_T
h_T \left( y \right) + f'_T h'_T \left( y \right) + f_S v_S - f_O \delta +
6f_\Omega \omega } \right]\xi _L^q \left( y \right) = 0 \, ,
\end{equation}
\begin{equation}\label{equ:QL0}
\partial _y \xi _L^Q \left( y \right) + \left[ {f_S h_S \left( y \right) + f_T
h_T \left( y \right) + f'_T h'_T \left( y \right) + f_S v_S - f_O \delta }
\right]\xi _L^Q \left( y \right) = 0 \, ,
\end{equation}
\begin{equation}
\partial _y \xi _L^l \left( y \right) + \left[ {f_S h_S \left( y \right) + f_T
h_T \left( y \right) + f'_T h'_T \left( y \right) + f_S v_S - f_O \delta -
18f_\Omega \omega } \right]\xi _L^l \left( y \right) = 0 \, ,
\end{equation}
\begin{equation}\label{equ:LL0}
\partial _y \xi _L^L \left( y \right) + \left[ {f_S h_S \left( y \right) + f_T
h_T \left( y \right) + f'_T h'_T \left( y \right) + f_S v_S - f_O \delta }
\right]\xi _L^L \left( y \right) = 0 \, ,
\end{equation}
\end{subequations}
Looking at Eqs.~(\ref{equ:QL0} and \ref{equ:LL0}), one notices that $\xi
_L^Q=\xi _L^L$. On the other hand, the zero-mode wave functions for the
right-handed SM-type fermions still obey Eqs.~(\ref{equ:eomTTru0} and
\ref{equ:eomTTrd0}).
\subsection{Simplification of numerical algorithm}
So far, we have localized SM-type fermions at different points and given
different shapes to the right-handed zero mode wave functions by symmetry
breakings along the extra dimension. The equations of motion for left- and
right-handed zero mode wave functions can be simplified considerably, for
numerical ease, however without affecting the values of left-right overlaps.
To begin with, let us assume
\begin{equation}\label{equ:const1}
f_S v_S = - 2 f_O\delta \, ,
\end{equation}
which preserves the distance between the left- and right-handed zero modes,
however places the right-handed zero modes at the origin. As the distance and
profiles of the zero modes are the only important factors in determining the
overlaps, such assumption only simplifies numerical procedure.
On the other hand, in analogy with the idea presented in
Refs.~\cite{Hung2003a,Hung2005}, where the difference in profiles for the up and
down sectors of the right-handed zero-mode wave functions is sufficient to
describe the sizes of corresponding overlaps, we may also consider
\begin{equation}\label{equ:const2}
f'_T h'_T \left( y \right) = - f_T h_T \left( y \right) \, ,
\end{equation}
which simplifies the left-handed zero-mode wave functions. The dissimilar
(narrow and broad) profiles of the right-handed zero-mode wave functions $\xi
_R^{up}$ and $\xi _R^{down}$ remain in place regardless of the condition of
Eq.~(\ref{equ:const2}) and since that difference in shapes is what matters (see
Fig.~\ref{fig:lroverlaps}), the numerical value of left-right overlaps will not
change. With these simplifications, the equations of motion for zero-mode wave
functions read
\begin{subequations}\label{equ:lheom}
\begin{equation}
\partial _y \xi _L^q \left( y \right) + \left[ {f_S h_S \left( y \right) - 3f_O
\delta + 6f_\Omega \omega } \right]\xi _L^q \left( y \right) = 0 \, ,
\end{equation}
\begin{equation}\label{equ:QL}
\partial _y \xi _L^Q \left( y \right) + \left[ {f_S h_S \left( y \right) - 3f_O
\delta } \right]\xi _L^Q \left( y \right) = 0 \, ,
\end{equation}
\begin{equation}
\partial _y \xi _L^l \left( y \right) + \left[ {f_S h_S \left( y \right) - 3f_O
\delta - 18f_\Omega \omega } \right]\xi _L^l \left( y \right) = 0 \, ,
\end{equation}
\begin{equation}\label{equ:LL}
\partial _y \xi _L^L \left( y \right) + \left[ {f_S h_S \left( y \right) - 3f_O
\delta } \right]\xi _L^L \left( y \right) = 0 \, ,
\end{equation}
\end{subequations}
and
\begin{subequations}\label{equ:rheom}
\begin{equation}
\partial _y \xi _R^{up} \left( y \right) + \left[ {f_S h_S \left( y \right) +
3f_T h_T \left( y \right) } \right] \xi _R^{up} \left( y \right) = 0 \, ,
\end{equation}
\begin{equation}
\partial _y \xi _R^{down} \left( y \right) + \left[ {f_S h_S \left( y \right) -
3f_T h_T \left( y \right) } \right] \xi _R^{down} \left( y \right) = 0 \, .
\end{equation}
\end{subequations}
In mass scale calculations, we find out that the distances between localized
left-handed and right-handed zero-mode wave functions along the extra dimension
are needed. Estimating those separations is the subject of the next section.
\subsection{Left-right separations along the extra
dimension}\label{sec:separation}
The localization process of SM-type fermions involved scalar fields, with
classical kink solutions. The kink solutions, however, yet to be specified. In
order to estimate the left-right separations, we give a Gaussian shape to zero
mode wave functions. Let us consider a linear approximation for the kink
solutions, explicitly
\begin{subequations}\label{equ:linapp}
\begin{equation}
h_S \left( y \right) \approx \mu _S^2 y \, ,
\end{equation}
\begin{equation}
h_T \left( y \right) \approx \mu _T^2 y \, .
\end{equation}
\end{subequations}
In this linear approximation, the equations of motion for the right-handed
zero-mode wave functions, Eqs.~(\ref{equ:rheom}), become
\begin{subequations}\label{equ:aprheom1}
\begin{equation}
\partial _y \xi _R^{up} \left( y \right) + \left( {f_S \mu _S^2 + 3f_T \mu _T^2}
\right)y\xi _R^{up} \left( y \right) = 0 \, ,
\end{equation}
\begin{equation}
\partial _y \xi _R^{down} \left( y \right) + \left( {f_S \mu _S^2 - 3f_T \mu
_T^2} \right) y \xi _R^{down} \left( y \right) = 0 \, ,
\end{equation}
\end{subequations}
The Gaussians defined by Eqs.~(\ref{equ:aprheom1}) are clearly localized at
$y=0$, meaning
\begin{equation}\label{equ:rhlocs}
y_R^{up} =y_R^{down}= 0 \,,
\end{equation}
where $y_R^{up}$ corresponds to the location of the right-handed zero modes of
$\Psi_1$ (i.e., those of $d$, $D$, $l_{d}$, and $\nu$) and $y_R^{down}$ refers
to the location of the right-handed zero modes of $\Psi_2$ (i.e., those of $u$,
$U$, $l_{u}$, and $l$). From Eqs.~(\ref{equ:aprheom1}), one clearly sees the
shape notion of narrow $\xi _R^{up}$ and wide $\xi _R^{down}$. The locations of
the left-handed zero-mode wave functions, on the other hand, can be determined
from their differential equations, Eqs.~(\ref{equ:lheom}). Those equations, in
the linear approximation scheme, now read
\begin{subequations}\label{equ:aplheom}
\begin{equation}
\partial _y \xi _L^q \left( y \right) + \left( {f_S \mu _{S}^2 y - 3f_O \delta
+ 6f_\Omega \omega} \right)\xi _L^q \left( y \right) = 0 \, ,
\end{equation}
\begin{equation}
\partial _y \xi _L^Q \left( y \right) + \left( {f_S \mu _{S}^2 y - 3f_O \delta }
\right)\xi _L^Q \left( y \right) = 0 \, ,
\end{equation}
\begin{equation}\partial _y \xi _L^l \left( y \right) + \left( {f_S \mu _{S}^2 y
- 3f_O \delta - 18f_\Omega \omega} \right)\xi _L^l \left( y \right) = 0 \, ,
\end{equation}
\begin{equation}
\partial _y \xi _L^L \left( y \right) + \left( {f_S \mu _{S}^2 y - 3f_O \delta }
\right)\xi _L^L \left( y \right) = 0 \, .
\end{equation}
\end{subequations}
For our future convenience, let us define
\begin{equation}\label{equ:w}
w \equiv \frac{3f_O \delta }{f_\Omega \omega} \, .
\end{equation}
With this definition, The locations of localized left-handed zero-mode wave
functions can be written as
\begin{subequations}\label{equ:lhloc}
\begin{align}
y_L^q &= \frac{f_\Omega \omega }{f_S \mu _{S}^2}\left( {w - 6} \right)\, ,\\
y_L^Q &= \frac{f_\Omega \omega }{f_S \mu _{S}^2} w \, ,\\
y_L^l &= \frac{f_\Omega \omega }{f_S \mu _{S}^2} \left( {w + 18} \right) \, ,\\
y_L^L &= \frac{f_\Omega \omega }{f_S \mu _{S}^2} w\, .
\end{align}
\end{subequations}
The superscripts on $y$'s in Eqs.~(\ref{equ:lhloc}) have the same meanings
explained in sections \ref{sec:uddis} and \ref{sec:undis} for $\xi$'s. Each
location given in Eqs.~(\ref{equ:lhloc}) is applicable to both components of the
left-handed zero-mode wave function doublet to which it refers. Since the mass
terms involve left- and right-handed fields, the relevant wave function
separations are those between the left- and right-handed ones. Using the
locations we already found, those left-right separations can be computed easily.
They are
\begin{itemize}
\item
For normal quarks
\begin{equation}\label{equ:qseps}
\left| {\Delta y^{q} } \right| = \left| {y_R - y_L^q } \right| = \left|
{\frac{f_\Omega \omega }{f_S \mu _{S}^2}\left( {6 - w} \right)} \right|,
\end{equation}
\item
For normal leptons
\begin{equation}\label{equ:lseps}
\left| {\Delta y^{l} } \right| = \left| {y_R - y_L^l } \right| = \left|
{\frac{f_\Omega \omega }{f_S \mu _{S}^2} \left( {w + 18} \right)} \right|,
\end{equation}
\item
For unconventional quarks and leptons
\begin{equation}\label{equ:QLseps}
\left| {\Delta y^{L} } \right| = \left| {\Delta y^{Q} } \right| = \left| {y_R -
y_L^L } \right| = \left| {y_R - y_L^Q } \right| = \left| {\frac{f_\Omega \omega
}{f_S \mu _{S}^2} w} \right|.
\end{equation}
\end{itemize}
In these left-right separations, $\Delta y$'s refer to both up and down sectors
of each flavor doublet and $y_R = y_R^{up} = y_R^{down}$. The identical
left-right separations of unconventional quarks and leptons,
Eq.~(\ref{equ:QLseps}), imply similar mass scales. Obviously, the magnitude of
such mass scale can be large and remains to be explored. Alternatively, and with
the help of Eqs.~(\ref{equ:qseps} and \ref{equ:lseps}), the relation between the
wave function separations of quarks and leptons can be found, i.e.,
\begin{equation}\label{equ:lqsep}
\left| {\Delta y^l } \right| = 3\left| {\frac{{{w \mathord{\left/
{\vphantom {w {18}}} \right.
\kern-\nulldelimiterspace} {18}} + 1}}
{{1 - {w \mathord{\left/
{\vphantom {w 6}} \right.
\kern-\nulldelimiterspace} 6}}}} \right|\left| {\Delta y^q } \right|
\end{equation}
We can also find relationship between the left-right separations of
unconventional fermions and ordinary quarks, i.e.,
\begin{equation}\label{equ:unqsep}
\left| {\Delta y^{Q/L} } \right| = \left|{\frac{w}{6 - w}}\right| \left|{\Delta
y^{q} } \right| = \frac{1}{4}\left( {\frac{\left| {\Delta y^{l} }
\right|}{\left| {\Delta y^{q} } \right|} - 3} \right)\left| {\Delta y^{q} }
\right|.
\end{equation}
Since the left-right separations are determinant factors in mass scale
computations, Eq.~(\ref{equ:lqsep}) implies relationship between the mass scales
of ordinary quarks and leptons in one generation, as fixing one would restrain
the other.
This can also be extended to unconventional fermions, as Eq.~({\ref
{equ:unqsep}) relates the left-right separations of unconventional fermions to
those of ordinary fermions. Thus the masses of unconventional fermions cannot
just be heavy enough to escape detection; they must yield meaningful masses for
ordinary fermions, as the known physics is concerned.
We have then arrived at a point where the masses of unconventional fermions not
only should comply with the existent experimental check on ordinary quarks and
charged lepton's masses but they could in principle restrain the mass scales for
the neutrino sector of ordinary leptons, as their left-right separations in the
extra dimension restrain the left-right separations of ordinary leptons
including those of neutrinos.
Let us now discuss the possibility of having $\delta = 0$, which we left aside
in section~\ref{sec:localization}. Obviously, $\delta = 0$ corresponds to $w =
0$, which would mean $\left| {\Delta y^{l} } \right| = 3\left| {\Delta y^{q} }
\right|$ and $ \left| {\Delta y^{Q/L} } \right| = 0$. Phenomenologically, we
prefer $\delta \ne 0$ for the reason that will be clear when we give numerical
results for the mass scales. With a minimum at $w = 0$, Eq.~(\ref{equ:lqsep})
can be also written as $\left| {\Delta y^l } \right| \ge 3\left| {\Delta y^q }
\right|$, which clearly indicates that lepton's wave function overlaps can be
potentially weaker than those of quarks. This seems plausible knowing the
profound differences between the mass scales of quarks and leptons.
\section{Return to four dimensions: The mass scales}
Speaking of SM-type fermion mass terms and mass scales implies that the gauge
symmetry is reduced to that of the SM and is going to break further down by the
SM Higgs vacuum.
A complete analysis of the gauge symmetry breakdown of the model is rather
lengthy and is not consistent with the flow of the paper at this point. However,
the necessary scalar fields for the gauge symmetry breakdown and the mixing of
charged gauge bosons are crucial to our analyses in sections~\ref{sec:EWP} and
\ref{sec:decays}. For that reason and completeness, a detailed gauge symmetry
breakdown is given in Appendix~\ref{app:gsb}.
As we only concentrate on the mass scales, we therefore will not discuss issues
such as the fermion mixings in the mass matrix~\cite{Split}. We follow the mass
scale calculations with some rough numerical analysis.
\subsection{Effective Yukawa couplings and the mass scales}
Dirac mass terms for chiral fermions involve couplings of left-handed and
right-handed fields with a Higgs field, which acquires VEV and breaks the SM
symmetry as well. The minimal SM symmetry breakdown of our model can occur
through a Higgs multiplet transforming as $\Theta = \left( 1,8,8 \right)$. The
decomposition of $\Theta$'s $\mathrm{SU}(3)_L$ octet in terms of $\mathrm{SU}(2)_L \otimes
\mathrm{U}(1)_Y$ multiplets or quantum numbers
\begin{equation}
\left[ 8 \right]_{\mathrm{SU}\left( 3 \right)_L} = \left( 3,0 \right) \oplus \left( 2,1
\right) \oplus \left( 2,-1 \right) \oplus \left( 1,0 \right),
\end{equation}
shows that $\Theta$ indeed possesses a SM Higgs field, which we denote by
$H=\left( {2,1} \right)$. Thus, $\Theta$ can break the SM symmetry and give mass
to chiral fermions by developing a VEV in $H$. The Yukawa couplings between the
left- and right-handed SM-type fermions can be written in the form
\begin{equation}\label{equ:mcouplings}
\mathcal{L}_{\mathrm{mass}} = \kappa _1 \Psi_1^T \Theta C \Psi_2^\ast + \kappa
_2 \Psi_1^T \tilde \Theta C \Psi_1^\ast + h.c.\, .
\end{equation}
In the above couplings, $\kappa_1$, $\kappa_2$ can be different in general,
$\tilde \Theta= i \hat \lambda_{2L} \Theta^*$, and $C = i \gamma^{2}
\gamma^{0}$. The mass terms in Eq.~(\ref{equ:mcouplings}) seem compact but they
can be expanded very easily. For example, they yield
\begin{equation}\label{equ:qmass}
\mathcal{L}_{\mathrm{mass}}^{q} =\kappa _1 \frac{v}{{\sqrt 2 }}\bar u_L u_R -
\kappa _2 \frac{v}{{\sqrt 2 }}\bar d_L d_R + h.c. \;,
\end{equation}
for normal quarks. These mass expressions have been worked out for transparency
in Appendix~\ref{app:gsb}. Similar expressions for other SM-type fermions can be
obtained easily.
We assume a delocalized Higgs field along the extra dimension and use its lowest
KK mode, which entirely depends on 4D coordinates. This means that the zero mode
of the Higgs field is independent of $y$, and can be written as, e.g., $H^0
\left( {x,y} \right) = K\phi \left( x \right)$. The zero mode of the SM Higgs
field $\phi$, then obtains VEV in the usual form $\bigl( \begin{smallmatrix} 0
\\ {{v \mathord{\left/ {\vphantom {v {\sqrt 2 }}} \right.
\kern-\nulldelimiterspace} {\sqrt 2 }}} \end{smallmatrix} \bigl)$.
The mass terms involve Yukawa couplings determining the magnitude of each mass
term. In our model and in four dimensional space, those couplings can be viewed
as ``effective'' Yukawa couplings whose strengths are determined by the geometry
of the zero-mode wave functions in the extra dimension. The reduction to 4D
space is simply done by integrating the extra dimension out, and that is how the
couplings in mass terms become ``effective'' 4D Yukawa couplings.
Mass scales can be computed from the mass terms in Eq.~(\ref{equ:mcouplings}).
To proceed, we define dimensionless couplings
\begin{equation}
g_{Y1,2} = \kappa_{1,2} K \, .
\end{equation}
The relationship between the mass scales and the mass matrix is given by
\begin{equation}
\mathcal{M} = \Lambda M \, ,
\end{equation}
where $M$ is a dimensionless matrix, whose form depends on the model for fermion
masses. In our case, we may write explicitly
\begin{equation}
\mathcal{M}_{u,d,\nu ,l,U,D,l_{u} ,l_{d} } = \Lambda _{u,d,\nu ,l,U,D,l_{u}
,l_{d} } M_{u,d,\nu ,l,U,D,l_{u} ,l_{d} } \; ,
\end{equation}
where $\Lambda _{u,d,\nu ,l,U,D,l_{u} ,l_{d} }$ are the mass scales of interest
and the subscripts refer to the SM-type fermions of the theory. The mass scales
in 4D space are proportional to the 4D effective Yukawa couplings, which in turn
are proportional to the overlaps between the relevant left- and right-handed
zero-mode wave functions in the extra dimension. Therefore, they simply are
\begin{align}\label{equ:mscales}
\Lambda _u &= \frac{v}{\sqrt 2} g_{Y1} \int_0^L {dy\xi _L^q } \left( y
\right)\xi _R^{down} \left( y \right)
\qquad ; &
\Lambda _d &= \frac{v}{\sqrt 2} g_{Y2} \int_0^L {dy\xi _L^q } \left( y
\right)\xi _R^{up} \left( y \right),
\\
\Lambda _\nu &= \frac{v}{\sqrt 2} g_{Y2} \int_0^L {dy\xi _L^l } \left( y
\right)\xi _R^{up} \left( y \right)
\qquad \; \; \; ; &
\Lambda _l &= \frac{v}{\sqrt 2} g_{Y1} \int_0^L {dy\xi _L^l } \left( y
\right)\xi _R^{down} \left( y \right),
\\
\Lambda _U &= \frac{v}{\sqrt 2} g_{Y1} \int_0^L {dy\xi _L^Q } \left( y
\right)\xi _R^{down} \left( y \right)
\qquad ; &
\Lambda _D &= \frac{v}{\sqrt 2} g_{Y2} \int_0^L {dy\xi _L^Q } \left( y
\right)\xi _R^{up} \left( y \right),
\\
\Lambda _{l_{u}} &= \frac{v}{\sqrt 2} g_{Y1} \int_0^L {dy\xi _L^L } \left( y
\right)\xi _R^{down} \left( y \right)
\qquad ; &
\Lambda _{l_{d}} &= \frac{v}{\sqrt 2} g_{Y2} \int_0^L {dy\xi _L^L } \left( y
\right)\xi _R^{up} \left( y \right).
\end{align}
Note that the left-handed $\xi _L $'s appearing in Eqs.~(\ref{equ:mscales}) are
no longer doublets, but the relevant components of those doublets. The fact is
that the geometry of each flavor component is the same as that assigned to the
corresponding doublet. As we are only concerned with the geometry of zero-mode
wave functions, we do not introduce new notation for the flavor components, as
if they were the relevant doublets. There are two possibilities that one can
explore: $g_{Y1} = g_{Y2} $ and $g_{Y1} \ne g_{Y2} $. The relationships
between the mass scales may depend on those choices.
\begin{enumerate}
\item
$g_{Y1} = g_{Y2}$ : One can write all sorts of ratios, which would only depend
on wave function overlaps. For example, we can write ratios relating mass scales
of two sectors of one family, or ratios involving mass scales from different
families. Some of those ratios are
\begin{subequations}\label{equ:ratios1}
\begin{equation}\label{equ:du}
\frac{{\Lambda _d }}
{{\Lambda _u }} = \frac{{\int_0^L {dy\xi _L^q } \left( y \right)\xi _R^{up}
\left( y \right)}}
{{\int_0^L {dy\xi _L^q } \left( y \right)\xi _R^{down} \left( y \right)}} \, ,
\end{equation}
\begin{equation}\label{equ:nl}
\frac{{\Lambda _\nu }}
{{\Lambda _l }} = \frac{{\int_0^L {dy\xi _L^l } \left( y \right)\xi _R^{up}
\left( y \right)}}
{{\int_0^L {dy\xi _L^l } \left( y \right)\xi _R^{down} \left( y \right)}} \, ,
\end{equation}
\begin{equation}\label{equ:DU}
\frac{{\Lambda _D }}
{{\Lambda _U }} = \frac{{\int_0^L {dy\xi _L^Q } \left( y \right)\xi _R^{up}
\left( y \right)}}
{{\int_0^L {dy\xi _L^Q } \left( y \right)\xi _R^{down} \left( y \right)}} \, ,
\end{equation}
\begin{equation}\label{equ:nu}
\frac{{\Lambda _\nu }}
{{\Lambda _u }} = \frac{{\int_0^L {dy\xi _L^l } \left( y \right)\xi _R^{up}
\left( y \right)}}
{{\int_0^L {dy\xi _L^q } \left( y \right)\xi _R^{down} \left( y \right)}} \, ,
\end{equation}
\begin{equation}\label{equ:lu}
\frac{{\Lambda _l }}
{{\Lambda _u }} = \frac{{\int_0^L {dy\xi _L^l } \left( y \right)\xi _R^{down}
\left( y \right)}}
{{\int_0^L {dy\xi _L^q } \left( y \right)\xi _R^{down} \left( y \right)}} \, ,
\end{equation}
\begin{equation}\label{equ:Du}
\frac{{\Lambda _D }}
{{\Lambda _u }} = \frac{{\int_0^L {dy\xi _L^Q } \left( y \right)\xi _R^{up}
\left( y \right)}}
{{\int_0^L {dy\xi _L^q } \left( y \right)\xi _R^{down} \left( y \right)}} \, ,
\end{equation}
\begin{equation}\label{equ:Ud}
\frac{{\Lambda _U }}
{{\Lambda _d }} = \frac{{\int_0^L {dy\xi _L^Q } \left( y \right)\xi _R^{down}
\left( y \right)}}
{{\int_0^L {dy\xi _L^q } \left( y \right)\xi _R^{up} \left( y \right)}}\, ,
\end{equation}
\begin{equation}\label{equ:nU}
\frac{{\Lambda _\nu }}
{{\Lambda _U }} = \frac{{\int_0^L {dy\xi _L^l } \left( y \right)\xi _R^{up}
\left( y \right)}}
{{\int_0^L {dy\xi _L^Q } \left( y \right)\xi _R^{down} \left( y \right)}} \, ,
\end{equation}
\begin{equation}\label{equ:lD}
\frac{{\Lambda _l }}
{{\Lambda _D }} = \frac{{\int_0^L {dy\xi _L^l } \left( y \right)\xi _R^{down}
\left( y \right)}}
{{\int_0^L {dy\xi _L^Q } \left( y \right)\xi _R^{up} \left( y \right)}} \, .
\end{equation}
\end{subequations}
One notices that the ratios involving unconventional leptons are identical to
those of unconventional quarks, since they share the same left-handed wave
functions and the same separations.
\item
$g_{Y1} \ne g_{Y2}$:
In this case, we may still find some ratios, depending only on wave function
overlaps. They are
\begin{subequations}\label{equ:ratios2}
\begin{equation}\label{equ:lu2}
\frac{{\Lambda _l }}
{{\Lambda _u }} = \frac{{\int_0^L {dy\xi _L^l } \left( y \right)\xi _R^{down}
\left( y \right)}}
{{\int_0^L {dy\xi _L^q } \left( y \right)\xi _R^{down} \left( y \right)}} \, ,
\end{equation}
\begin{equation}
\frac{{\Lambda _\nu }}
{{\Lambda _d }} = \frac{{\int_0^L {dy\xi _L^l } \left( y \right)\xi _R^{up}
\left( y \right)}}
{{\int_0^L {dy\xi _L^q } \left( y \right)\xi _R^{up} \left( y \right)}} \, ,
\end{equation}
\begin{equation}
\frac{{\Lambda _U }}
{{\Lambda _u }} = \frac{{\int_0^L {dy\xi _L^Q } \left( y \right)\xi _R^{down}
\left( y \right)}}
{{\int_0^L {dy\xi _L^q } \left( y \right)\xi _R^{down} \left( y \right)}} \, ,
\end{equation}
\begin{equation}
\frac{{\Lambda _D }}
{{\Lambda _d }} = \frac{{\int_0^L {dy\xi _L^Q } \left( y \right)\xi _R^{up}
\left( y \right)}}
{{\int_0^L {dy\xi _L^q } \left( y \right)\xi _R^{up} \left( y \right)}} \, ,
\end{equation}
\begin{equation}
\frac{{\Lambda _l }}
{{\Lambda _U }} = \frac{{\int_0^L {dy\xi _L^l } \left( y \right)\xi _R^{down}
\left( y \right)}}
{{\int_0^L {dy\xi _L^Q } \left( y \right)\xi _R^{down} \left( y \right)}} \, ,
\end{equation}
\begin{equation}
\frac{{\Lambda _\nu }}
{{\Lambda _D }} = \frac{{\int_0^L {dy\xi _L^l } \left( y \right)\xi _R^{up}
\left( y \right)}}
{{\int_0^L {dy\xi _L^Q } \left( y \right)\xi _R^{up} \left( y \right)}} \, .
\end{equation}
\end{subequations}
\end{enumerate}
So far, we have been able to find relationships between the mass scales of the
fermions of interest. However, there exist parameters in these relations, which
need to be determined in order to give numerical results. In the next section,
we attempt to express mass scales in terms of the mass scales of up- and
down-sectors of ordinary quarks and charged-sector of ordinary leptons by fixing
some of the parameters and deriving others.
\subsection{Numerical analysis}
To obtain numerical values for the mass scales of neutrino and unconventional
fermions, we first need to specify the analytical expressions for the zero mode
wave functions involved in the overlap integrals. To start, let us consider the
general case of $w \ne 0$.
For the left- and right-handed zero-mode wave functions, we employ the same
expressions as those used in Refs.~\cite{Hung2003a, Hung2005, Georgi2001}. The
left-handed zero mode wave functions are
\begin{equation}\label{equ:lhexp}
\xi _L^i \left( y \right) = N^i_L \exp \Big[ { - C_S \ln \Big( {\cosh \left(
{\mu _S \left( {y - y_i } \right)} \right)} \Big)} \Big],
\end{equation}
where $i=q,l,Q,L$, and $N_L $'s are normalization factors, $C_{S} = f_{S}
\sqrt{{2 \mathord{\left/ {\vphantom {2 {\lambda _{S} }}} \right.
\kern-\nulldelimiterspace} {\lambda _{S} }}}$, and $y_i$'s are the positions of
the left-handed zero modes along the extra dimension. The right-handed zero mode
wave functions, which are slightly more complicated, are expressed in the form
\begin{subequations}\label{equ:rhexp}
\begin{equation}
\xi _R^{up} \left( y \right) = N_R^{up} \exp \Big[ { - \Big( {C_S \ln \left(
{\cosh \mu _S y} \right) + C_T \ln \left( {\cosh \mu _T y} \right)} \Big)}
\Big],
\end{equation}
and
\begin{equation}
\xi _R^{down} \left( y \right) = N_R^{down} \exp \Big[ { - \Big( {C_S \ln \left(
{\cosh \mu _S y} \right) - C_T \ln \left( {\cosh \mu _T y} \right)} \Big)}
\Big],
\end{equation}
\end{subequations}
where $N_R $'s are normalization factors and $C_T = 3f_T \sqrt{{2
\mathord{\left/ {\vphantom {2 {\lambda _{T} }}} \right.
\kern-\nulldelimiterspace} {\lambda _{T} }}}$. Note that $C_S $ and $C_T $
contain factors from both the Yukawa coupling with fermions $f_{S,T} $, and the
scalar field self-interaction $\lambda _{S,T} $.
To calculate the mass scales of interest, we note that the mass scale ratios of
ordinary quarks and leptons may be estimated from the experimental values for
mass eigenstates (we shall elucidate this issue momentarily). Therefore, we can
use the \textit{estimated} value of ${{\Lambda _d } \mathord{\left/ {\vphantom
{{\Lambda _d } {\Lambda _u }}} \right. \kern-\nulldelimiterspace} {\Lambda _u
}}$ in conjunction with the relevant mass scale ratio of Eq.~(\ref{equ:du}) to
obtain the quark left-right separation, $\Delta y^q $. In addition, we may use
the \textit{estimated} value of ${{\Lambda _l } \mathord{\left/ {\vphantom
{{\Lambda _l } {\Lambda _d }}} \right. \kern-\nulldelimiterspace} {\Lambda _d
}}$ and the ratio in Eq.~(\ref{equ:lu}) to find the lepton left-right
separation, $\Delta y^l $, which in turn can determine the mass scale of Dirac
neutrino, say using Eqs.~(\ref{equ:nl},\ref{equ:nu}). Once $\Delta y^q $ and
$\Delta y^l $ are known, we can find the unconventional fermion left-right
separation, using Eq.~(\ref{equ:unqsep}), in the linear approximation scheme of
section~\ref{sec:separation}. Consequently, we can estimate unconventional
fermion mass scales, say using Eqs.~(\ref{equ:Du},\ref{equ:Ud},\ref{equ:lD}).
The outlined numerical method makes use of the ratios of
Eqs.~(\ref{equ:ratios1}), which are obtained assuming that $g_{Y1} = g_{Y2}$. It
turns out that $g_{Y1} \ne g_{Y2}$ case gives the same mass scales, however with
a bit different numerical approach. We shall explain this at the closing of this
section.
To evaluate the mass scales, we need to fix some of the parameters in zero-mode
wave function expressions, Eqs.~(\ref{equ:lhexp} and \ref{equ:rhexp}), and vary
some. Since the difference between $C_S \ln \left( {\cosh \mu _S y} \right)$ and
$C_T \ln \left( {\cosh \mu _T y} \right)$, in Eqs.~(\ref{equ:rhexp}), is what
matters, we choose $C_S = C_T = 1$, set $\mu _S = 1$ (in some units) and let
$\mu _T $ vary. Therefore, for a given $\mu _T $ we may find the quark and
lepton left-right separations that satisfy the phenomenological constraints and
use those separations to estimate the Dirac neutrino and unconventional fermion
mass scales. Technically speaking, varying $\mu_T$ means varying the width of
the right-handed zero-mode wave functions; therefore we look for
width-separation combinations that would satisfy the estimated mass scale
ratios.
To estimate the phenomenological constraints on the mass scale ratios, we need
to make an assumption concerning the nature of mass matrices of up- and
down-quark sectors and charged-lepton sector of ordinary fermions. The mass
matrix $\mathcal{M}$, is related to the mass scale $\Lambda$, through the
expression
\begin{equation}
\mathcal{M} = \Lambda M \, ,
\end{equation}
where $M$ is a dimensionless matrix. Obviously, mass scale $\Lambda$ is a common
factor in the mass matrix and $M$, which determines the flavor mixings and
masses, is to be specified by the model describing the mass issues. We shall not
engage in discussing mass matrices here, as the subject itself is rich and well
beyond the scope of this work. Nevertheless, to relate the mass scales of
up-quark, down-quark and charged-lepton sectors of ordinary fermions to
experimentally measured mass eigenvalues, a general case could be considered,
where the relevant mass scales lie within two bounds, namely
\begin{equation}\label{equ:scalebounds}
\begin{gathered}
\frac{{m_t }}
{3} \leqslant \Lambda _u \leqslant m_t , \hfill \\
\frac{{m_b }}
{3} \leqslant \Lambda _d \leqslant m_b , \hfill \\
\frac{{m_\tau }}
{3} \leqslant \Lambda _l \leqslant m_\tau , \hfill \\
\end{gathered}
\end{equation}
where $m_t$, $m_b$, and $m_{\tau}$ are the largest eigenvalues of up-quark,
down-quark and charged-lepton mass matrices, respectively. The lower bounds
correspond to pure democratic mass matrices \cite{Dem}, which are impractical
since they cannot replicate proper mass spectrum and CKM matrix. The upper
bounds, on the other hand, refer to ``highly hierarchical'' mass
matrices\footnote{There have been a lot of works done on hierarchical mass
matrices, which span from phenomenological to superstring theory inspired
models. See Ref.~\cite{Fritzsch2000} for a mini review and references therein.},
where the largest eigenvalues are approximately equal to the mass scales, i.e.,
\begin{equation}\label{equ:eigenmass}
\begin{gathered}
\Lambda_u \approx m_t \,, \hfill \\
\Lambda_d \approx m_b \, , \hfill \\
\Lambda_l \approx m_{\tau} \, . \hfill \\
\end{gathered}
\end{equation}
To carry out the mass scale calculations, we consider this highly hierarchical
scheme. We will come back to Eqs.~(\ref{equ:scalebounds}) and the mass scales
within the two bounds, which do not correspond to pure democratic mass matrices.
\begin{table}
\caption{\label{tab:parameters} Values of $\mu_T$, $\Delta y^l$, $\Delta y^q$,
and $\Delta y^{L,Q}$ that give meaningfull results in accordance with the
phenomenological constraints of Eqs.~(\ref{equ:expconst}). Each set of values is
labeled with a roman letter.}
\begin{ruledtabular}
\begin{tabular}{ccccc}
& $\mu_T$ & $\left|{\Delta y^l}\right|$& $\left|{\Delta
y^q}\right|$& $\left|{\Delta y^{Q/L}}\right|$ \\
\hline
a & 0.81 & 31.360 & 6.940 & 2.635 \\
b & 0.80 & 30.200 & 7.000 & 2.300 \\
c & 0.79 & 29.170 & 7.070 & 1.990 \\
d & 0.75 & 24.715 & 7.530 & 0.531 \\
e & 0.73 & 24.115 & 7.690 & 0.211 \\
f & 0.70 & 23.285 & 7.815 & 0.040 \\
\end{tabular}
\end{ruledtabular}
\end{table}
We employ the masses of top and bottom quarks and tau lepton at $M_Z$ for $m_t$,
$m_b$, and $m_{\tau}$, and to simplify our numerical computations ignore any
running between $M_Z$ and the early unification scale. That seems plausible as
the early unification scale is not much higher than $M_Z$, meaning that there
would not be much of a ``running.'' We use
\begin{equation}
\begin{gathered}
m_t \left( {M_Z } \right) = 181{\text{ GeV,}} \hfill \\
m_b \left( {M_Z } \right) = 3{\text{ GeV,}} \hfill \\
m_\tau \left( {M_Z } \right) = 1.747{\text{ GeV}}{\text{.}} \hfill \\
\end{gathered}
\end{equation}
Therefore, the phenomenological constraints on the mass scale ratios can be
written as
\begin{subequations}\label{equ:expconst}
\begin{equation}\label{equ:duratio}
\frac{{\Lambda _d }}
{{\Lambda _u }} \approx \frac{{m_b \left( {M_Z } \right)}}
{{m_t \left( {M_Z } \right)}} \approx 0.0166,
\end{equation}
and
\begin{equation}\label{equ:luratio}
\frac{{\Lambda _l }}
{{\Lambda _u }} \approx \frac{{m_\tau \left( {M_Z } \right)}}
{{m_t \left( {M_Z } \right)}} \approx 0.00965.
\end{equation}
\end{subequations}
With the mass scale ratios of Eqs.~(\ref{equ:expconst}), the left-right
separations of normal quarks and leptons are at grab, which then lead us to the
left-right separation for unconventional fermions and finally the mass scales
for neutrino and unconventional fermions.
It turns out that there are a few width-separation combinations that satisfy the
phenomenological constraints. Consequently, there will be a few sets of mass
scales for neutrino and unconventional fermions which in turn imply a
relationship between the masses. The possible values of $\mu_T$, $\Delta y^l$,
$\Delta y^q$, and $\Delta y^{L,Q}$, which satisfy the phenomenological
conditions are listed in Table~\ref{tab:parameters} for completeness. The
left-right separations of Table~\ref{tab:parameters} demonstrate a hierarchy in
the form $\Delta y^l > \Delta y^q > \Delta y^{L,Q}$, which means a hierarchy in
overlaps where the largest is that of unconventional fermions and the smallest
belongs to ordinary leptons.
With the values of Table~\ref{tab:parameters}, the left- and right-handed zero
mode wave functions are specified and finally the mass scales of Dirac neutrino
and unconventional fermions for each allowed case can be determined. Those mass
scales are listed in Table~\ref{tab:massscales} for each allowed set of
parameters.
\begin{table}
\caption{\label{tab:massscales} Predicted mass scales for Dirac neutrino
$\Lambda_\nu$, and unconventional quarks $\Lambda_U$, $\Lambda_D$, and leptons
$\Lambda_{l_{u}}$, $\Lambda_{l_{d}}$. Note that $\Lambda_U \approx
\Lambda_{l_{u}}$ and $\Lambda_D \approx \Lambda_{l_{d}}$ and that each label in
the left column refers to the corresponding set of parameters in
Table~\ref{tab:parameters}.}
\begin{ruledtabular}
\begin{tabular}{cccc}
&$\Lambda_\nu \,\, \text{(eV)} \approx$ & $\Lambda_U \,\, \text{(GeV)} \approx$
& $\Lambda_D \,\, \text{(GeV)} \approx$ \\
\hline
a & 0.065 & 406 & 181 \\
b & 0.23 & 456 & 252 \\
c & 0.67 & 513 & 336 \\
d & 23 & 802 & 791 \\
e & 87 & 988 & 1053 \\
f & 486 & 1321 & 1435 \\
\end{tabular}
\end{ruledtabular}
\end{table}
Looking at Table~\ref{tab:massscales}, it is obvious that the mass scales of
neutrino and unconventional fermions increase monotonically together. One can
argue that there is a correlation between the masses of neutrino and those of
unconventional fermions, such that the mass of one can set a bound on the mass
of the other. For instance, we could start with a mass scale for unconventional
fermions and find the corresponding left-right separation, which together with
quark left-right separation would determine that of lepton and therefore the
mass scale of neutrino. Such relationship can also be seen, however naively, by
looking at Eq.~(\ref{equ:nU}) where the mass scales of one sector can set a
bound on another.
The neutrino oscillation data provide mass differences between the neutrinos of
different families. The most recent data~\cite{PDBook} on neutrino mass
differences indicate $\Delta m_{21}^2 = \left( {8.0_{ - 0.3}^{ + 0.4} } \right)
\times 10^{ - 5} {\text{ eV}}^2 $ and $\left| {\Delta m_{32}^2 } \right| =
1.9\text{ to }3.0 \times 10^{ - 3} {\text{ eV}}^2 $.
The neutrino mass scales of of Table~\ref{tab:massscales} increase with those of
unconventional fermions. One could see two distinct possibilities by looking at
$\Delta m_{21}^2$ and $\left| {\Delta m_{32}^2 } \right|$, namely:
\begin{enumerate}
\item
For the lightest unconventional fermions, i.e., mass scales not smaller than 180
GeV, the neutrino sector is very light, about 0.065 eV. That corresponds to
either quasi-degenerate or hierarchical mass matrix for neutrinos.
\item
For heavier unconventional fermions, i.e., mass scales between 250 and 500 GeV,
the neutrino sector is light, ranging between 0.2 and 0.7 eV. In this case the
neutrino mass matrix ought to be quasi-degenerate in order to satisfy the
neutrino oscillation data.
\end{enumerate}
The mass scales in Table~\ref{tab:massscales} are similar to those obtained in
Ref.~\cite{Hung2005} based on $\mathrm{PUT}_1$ scenario. This similarity is
mainly due to the common strong $\mathrm{SU}(4)_{\mathrm{PS}}$ group. The breaking of
this symmetry in the extra dimension yields similar relations between the
left-right distances of quarks and leptons. Nevertheless, the actual masses for
unconventional fermions in each scenario can be different in principal, as the
mass matrices can be different.
In obtaining the mass scales of Table~\ref{tab:massscales}, we assumed highly
hierarchical mass matrices for up-quark, down-quark and charged-lepton sectors
of ordinary fermions, which resulted in phenomenological conditions of
Eqs.~(\ref {equ:expconst}). However, between the two bounds defined by
Eqs.~(\ref{equ:scalebounds}), the corresponding mass matrices are no longer
purely democratic \cite{NonDem}. There are models (e.g., those in
Refs.~\cite{HungSeco}) where the mass matrices deviate enough from pure
democratic case that can generate suitable mass spectrum and CKM matrix. In
those models, the mass scales of interest can be taken nearly as low as half of
the largest eigenvalues. In such regime, we end up with mass scales at least
half of those given in Table~\ref{tab:massscales}, which make the degeneracy of
neutrino sector for heavier (250~-~500 GeV) unconventional fermions seem less
reflective.
Let us talk about the possible mass scales that can be computed when $w=0$.
Although such case simplifies the relation between the left-right separations of
ordinary quarks and leptons, it yields ordinary charged lepton with mass scale
in order of 11 GeV. That alone is sufficient to dismiss the $w=0$ case, as
Ref.~\cite{Hung2005} also suggests.
Now that we know $w=0$ leads to unphysical mass scales, we may explain the
numerical method for computing mass scales if we were to use the ratios in
Eqs.~(\ref{equ:ratios2}) when $g_{Y1} \ne g_{Y2}$, which would only make sense
if $w\ne0$. Similar to $g_{Y1} = g_{Y2}$ case, the ratios of
Eqs.~(\ref{equ:ratios2}) should also comply with the corresponding
\textit{estimated} ratios. That, nevertheless, requires adjusting the left-right
separations of quarks and leptons accordingly (e.g., to fix the ratio in
Eq.~(\ref{equ:lu2})), which means varying one more parameter and that is $w$.
Once $\Delta y^l$ and $\Delta y^q$ are known, $\Delta y^{Q/L}$ can be estimated
and then the mass scales of interest can be evaluated.
A few remarks are in order here. If the unconventional fermions are very heavy,
the neutrinos are quasi-degenerate. That would imply that the mixing angles in
PMNS matrix~\cite{PMNS} will be mainly determined by the angles of the charged
lepton sector. If the unconventional fermions are lighter, it would imply that
the mixing angles could come from both charged lepton and neutrino sectors,
since the neutrino sector could also be hierarchical in this case.
\section{Constraints from precision electroweak measurements} \label{sec:EWP}
The oblique corrections to the SM are best presented in terms of the so-called
electroweak oblique parameters $S$, $T$, and $U$ \cite{Peskin}. They are
primarily defined for sorts of new physics that have no or insignificant direct
couplings to the SM particle content and have mass scales larger than $M_Z$.
Of these parameters, $U$ plays a relatively minor role and is not linked to any
precision measurement but that of $M_W$. The other two, however, are strongly
correlated and important in limiting the type of new physics that could couple
to the SM. To give a conceptual sense, $S$ measures the momentum dependence of
the vacuum polarization and $T$ measures the custodial isospin violation.
The new physics corrections to oblique parameters in our model come from the
SM-type unconventional fermions, and the scalars of the theory, since the
vector-like fermions decouple for large vector-like masses (\textit{decoupling
theorem}).
The experimental values of oblique parameters refer to the allowed contributions
from new physics with respect to the SM reference point. The latest experimental
values of oblique parameters are \cite{PDBook}
\begin{subequations}\label{equ:expoblique}
\begin{equation}\label{equ:expS}
S = - 0.13 \pm 0.10\left( { - 0.08} \right),
\end{equation}
\begin{equation}\label{equ:expT}
T = - 0.13 \pm 0.11\left( { - 0.09} \right),
\end{equation}
\begin{equation}\label{equ:expU}
U = 0.20 \pm 0.12\left( { + 0.01} \right),
\end{equation}
\end{subequations}
where the central values assume the SM Higgs mass $M_H = 117$ GeV, and the
values in parentheses show the change for $M_H = 300$ GeV.
The custodial isospin symmetry constraint presented by $T$, forbids too much of
a difference between the masses of $U$, $l_{u}$ and $D$, $l_{d}$ respectively.
This will have implications on the decay modes of unconventional fermions, as it
constrains the phase space for decays such as $U \to D + W_L$ or $D \to U + W_L$
happening in real $W_L$'s.
The $S$ parameter, on the other hand, can be estimated for our model. For
fermionic contribution to $S$, we note that the mass scales give the maximum
masses for unconventional fermions and therefore their maximum contributions to
$S$. The total $S$ from one extra generation of fermions can be estimated that
way, i.e.,
\begin{equation}\label{equ:Sfermions}
S = \frac{1}
{{6\pi }}\left[ {16x_U + 32x_D + 2\ln \frac{{x_U }}
{{x_D }} + \left( {4x_U - 1} \right)G\left( {x_U } \right) + \left( {8x_D + 1}
\right)G\left( {x_D } \right)} \right],
\end{equation}
with
\[
x_{U} = \frac{{M_U }}
{{M_Z }} \approx \frac{{M_{l_{u} } }}
{{M_Z }} \,\,\,{\text{ and }}\,\,\,x_{D} = \frac{{M_D }}
{{M_Z }} \approx \frac{{M_{l_{d} } }}
{{M_Z }}\,\,.
\]
This expression obviously depends only on the masses of unconventional fermions
for which we use the calculated mass scales. The derivation of
Eq.~(\ref{equ:Sfermions}) is given in Appendix~\ref{app:Scalc}, for
completeness. The minimum and maximum fermionic contributions to the electroweak
$S$ parameter, obtained using the computed mass scales as maximum masses, are
given in Table~\ref{tab:Sferm}. The $S$ values in Table~\ref{tab:Sferm} indicate
$0.391\geq S \geq 0.195$, form one generation of unconventional fermions.
\begin{table}
\caption{\label{tab:Sferm} Minimum and maximum values of electroweak $S$
parameter from one generation of unconventional fermions for given masses.}
\begin{ruledtabular}
\begin{tabular}{ccc}
$M_U$(GeV) & $M_D$(GeV) & $S$ \\
\hline
406 & 181 & 0.391 \\
1321 & 1435 & 0.195 \\
\end{tabular}
\end{ruledtabular}
\end{table}
If we assume generational mass degeneracy among the three generations of
unconventional fermions, the total fermionic correction to $S$ can reach up to
three times those values.
Correction to electroweak $S$ parameter from scalar fields, generally, takes
negative sign~\cite{Georgi1991,Dugan1991}. The scalar contributions to $S$ come
from the Higgs multiplets responsible for the gauge symmetry breakdown (see
Appendix~\ref{app:gsb}).
The gauge symmetry breaking of our model involves many scalar multiplets.
However, only those with $\mathrm{SU}(2)_L$ quantum number can contribute to $S$. Those
are $\Phi _L = \left( {1,8,1} \right)$, $\Phi _H^{\left( 2 \right)} = \left(
{4,3,3} \right)$, and $\Theta = \left( {1,8,8} \right)$. In terms of their
$\mathrm{SU}(2)_L$ multiplets, the scalar fields which carry $\mathrm{SU}(2)_L$ quantum number
consist of 9 triplets and 18 doublets. The computations of scalar corrections to
$S$, in the paradigm of Ref.~~\cite{Dugan1991}, are given in detail in
Appendix~\ref{app:Scalc}. The $S$ parameter due to an $\mathrm{SU}(2)_L$ doublet with
mass $m$ and mass splitting parameter $m' \geq 0$ is
\begin{equation}\label{equ:Sdoublet}
S_{\mathrm{doublet}} = \frac{1}
{\pi }\int_0^1 {dx{\text{ }}x\left( {1 - x} \right)\ln \left( {x + \zeta \left(
{1 - x} \right)} \right)} \, ,
\end{equation}
where $\zeta = {{\left( {1 - {{3\beta ^2 } \mathord{\left/ {\vphantom {{3\beta
^2 } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)} \mathord{\left/
{\vphantom {{\left( {1 - {{3\beta ^2 } \mathord{\left/ {\vphantom {{3\beta ^2 }
2}} \right. \kern-\nulldelimiterspace} 2}} \right)} {\left( {1 + {{\beta ^2 }
\mathord{\left/ {\vphantom {{\beta ^2 } 2}} \right. \kern-\nulldelimiterspace}
2}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 + {{\beta ^2 }
\mathord{\left/ {\vphantom {{\beta ^2 } 2}} \right. \kern-\nulldelimiterspace}
2}} \right)}}$ with $\beta = {{m'} \mathord{\left/ {\vphantom {{m'} m}} \right.
\kern-\nulldelimiterspace} m}$. For an $\mathrm{SU}(2)_L$ triplet with mass $m$ and mass
splitting parameter $m' \geq 0$, contribution to $S$ is
\begin{equation}\label{equ:Striplet}
S_{\mathrm{triplet}} = \frac{2}
{{9\pi }}\left\{ {\frac{1}
{3}\ln \zeta + 8\int_0^1 {dx{\text{ }}x\left( {1 - x} \right)\ln \left( {x +
\zeta \left( {1 - x} \right)} \right)} } \right\},
\end{equation}
where $\zeta = {{\left( {1 - 2\beta ^2 } \right)} \mathord{\left/ {\vphantom
{{\left( {1 - 2\beta ^2 } \right)} {\left( {1 + \beta ^2 } \right)}}} \right.
\kern-\nulldelimiterspace} {\left( {1 + \beta ^2 } \right)}}$ with $\beta =
{{m'} \mathord{\left/ {\vphantom {{m'} m}} \right. \kern-\nulldelimiterspace}
m}$. The integrals in Eqs.~(\ref{equ:Sdoublet} and \ref{equ:Striplet}) can be
easily computed, which yield $S$ parameters that depend only on $\beta$ for each
scalar multiplet. Figure~\ref{fig:S} shows the dependence of those $S$
parameters on $\beta$.
\begin{figure*}
\includegraphics{Adibzadeh_Sgraph}
\caption{\label{fig:S} Electroweak $S$ parameter from left-handed doublet and
triplet scalars: thick solid line, left-handed triplet; thin solid line,
left-handed doublet.}
\end{figure*}
Since the $S$ parameters in Fig.~\ref{fig:S} are bounded from above, at
$\beta=0$, the contribution from the scalars of the theory is net negative. The
total contribution from the scalar fields to $S$, however, is a sum of all
scalar contributions. As the $\beta$ parameter for each scalar multiplet is not
known, we may only speak of the bounds the total scalar contribution should lie
within, namely
\begin{equation}\label{equ:Ss}
0 \geqslant S_{\mathrm{scalars}} \geqslant - 3.13 \,\, .
\end{equation}
To be inclusive, we may also consider the contribution from a heavy SM model
Higgs\footnote{By heavy SM Higgs, we mean heavier than the 300 GeV Higgs for
which the experimental \protect $S$ value is provided in Eq.~(\ref{equ:expS}).}
to $S$, which can be positive depending on its mass. This contribution, however,
is relatively small even for an exotic Higgs with 1 TeV mass, where it can reach
up to 0.06. With such contribution, if heavy SM Higgs exists at all, it would
not change the maximum negative $S$ provided by other scalars, significantly. On
the other hand, the fermionic contribution to $S$ from three generations of
unconventional fermions is within
\begin{equation}\label{equ:Sf}
1.17 \geqslant S_{\mathrm{fermions}} \geqslant 0.58 \,\, .
\end{equation}
The positive contribution from unconventional fermions obviously violates the
experimental bounds. Nonetheless, the negative scalar contribution to $S$ has
the potential to bring the total $S$ in agreement with the experimental
constraint on new physics, given in Eq.~(\ref{equ:expS}). Therefore, the notion
of three extra generations of heavy fermions can, in principal, be accommodated
within the model.
\section{The decay of unconventional fermions}\label{sec:decays}
Although unconventional fermions can have transitions involving ordinary
fermions via $\mathrm{SU}(4)_{\mathrm{PS}}$ and $\mathrm{SU}(3)_{H}$ mediated processes, the
lightest unconventional fermion cannot decay into light ordinary fermions, even
through weak channels. That poses an alarming danger: a stable unconventional
fermion. There are stringent constraints on heavy stable fermions (quarks or
leptons) from cosmology (e.g., nucleosynthesis) and earth-based experiments.
Those constraints are discussed in length in Ref.~\cite{Frampton2000} and the
references therein.
Fortunately, the decay of the lightest unconventional fermion is possible via
the mixing among the charged gauge bosons. Such mixing is possible through
$\Theta$'s VEV, which mixes the charged gauge bosons $W_{\mu H}^ \pm $ and
$W_{\mu L}^ \pm $, since it carries $\mathrm{SU}(2)_L$ and $\mathrm{SU}(2)_H$ quantum numbers.
This mixing has been discussed in Appendix~\ref{app:gsb}, where its rather long
expressions are given for completeness.
The mixing between the charged gauge bosons corresponds to an equivalent mixing
among the relevant currents as well. If we denote gauge eigenstates of charged
bosons by $W_{\mu H}^ \pm $ and $W_{\mu L}^ \pm $, and mass eigenstates by
$\tilde W_{\mu H}^ \pm $ and $\tilde W_{\mu L}^ \pm $, we may write (to first
approximation)
\begin{subequations}
\begin{equation}
\tilde W_{\mu L}^ \pm = W_{\mu L}^ \pm + \mathcal{O}\left({\frac{{v^2}}
{{v_H^{'2}}}} \right) W_{\mu H}^ \pm \, ,
\end{equation}
\begin{equation}
\tilde W_{\mu H}^ \pm = W_{\mu H}^ \pm + \mathcal{O}\left({\frac{{v^2}}
{{v_H^{'2}}}} \right) W_{\mu L}^ \pm \, ,
\end{equation}
\end{subequations}
where $v$ and $ v'_H $ are the VEV's of the SM Higgs $H$, and the horizontal
breaking Higgs $\Phi _H^{\left( 2 \right)}$, respectively. Let us also denote
currents coupled to gauge eigenstates by $J_{\mu H}$ and $J_{\mu L}$, and those
coupled to mass eigenstates by $\tilde J_{\mu H}$ and $\tilde J_{\mu L}$. The
same matrix that connects the gauge and mass eigenstate charged bosons relates
the corresponding currents to each other as well. Therefore, we have
\begin{subequations}
\begin{align}
\tilde J_{\mu L} &= J_{\mu L} + \mathcal{O}\left({\frac{{v^2}}{{ v_H^{'2}}}}
\right) J_{\mu H} \, ,\\
\tilde J_{\mu H} &= J_{\mu H} + \mathcal{O}\left({\frac{{v^2}}{{ v_H^{'2}}}}
\right) J_{\mu L} \, .
\end{align}
\end{subequations}
The consequent interaction terms are then given by
\begin{subequations}\label{equ:ints}
\begin{equation}\label{equ:Lint}
\mathcal{L}_L = g_W \tilde J_L^\mu \tilde W_{\mu L} = g_W \left[ {J_L^\mu +
\mathcal{O}\left( {\frac{{v^2 }}
{{v_H^{'2} }}} \right)J_H^\mu } \right]\tilde W_{\mu L} ,
\end{equation}
and
\begin{equation}\label{equ:Hint}
\mathcal{L}_H = g_W \tilde J_H^\mu \tilde W_{\mu H} = g_W \left[ {J_H^\mu +
\mathcal{O}\left( {\frac{{v^2 }}
{{v_H^{'2} }}} \right)J_L^\mu } \right]\tilde W_{\mu H} .
\end{equation}
\end{subequations}
The interaction term of Eq.~(\ref{equ:Lint}) portrays how the lightest
unconventional fermion, in $J_{\mu H}$, can couple to $\tilde W_{\mu L}^ \pm $
and therefore decay into ordinary fermions. The decay mechanism falls within one
of the two possibilities:
\begin{enumerate}
\item
The mass of the lightest unconventional fermion is large enough to decay into a
real $\tilde W_{ L}$ and a regular fermion, according to $g_W \mathcal{O}\left(
{{{v^2 } \mathord{\left/ {\vphantom {{v^2 } {v_H^{'2} }}} \right.
\kern-\nulldelimiterspace} {v_H^{'2} }}} \right)J_H^\mu \tilde W_{\mu L}$.
Although the $\mathcal{O}\left( {{{v^2 } \mathord{\left/ {\vphantom {{v^2 }
{v_H^{'2} }}} \right. \kern-\nulldelimiterspace} {v_H^{'2} }}} \right)$ factor
is small, the corresponding decay rate can be sizeable, since the unconventional
fermion decays into a real $\tilde W_{ L}$.
\item
The mass of the lightest unconventional fermion is not large enough to decay
into a real $\tilde W_{ L}$ and a regular fermion. In that case, $\tilde W_{ L}$
would be virtual and the interaction involves a $\tilde W_{ L}$ propagator,
i.e.,
\[
\mathcal{L}_{\mathrm{int}} \sim g_W^2 \mathcal{O}\left( {\frac{{v^2 }}
{{v_H^{'2} }}} \right)J_L^{\mu \dagger} \frac{{1}}
{{p^2 - M_{\tilde W_ L}^2 }} J_{\mu H} .
\]
\end{enumerate}
The fermions appearing in $J_{\mu L}$ and $J_{\mu H}$ should be expressed in
terms of mass eigenstates. This means new mixing angels, which are totally
different from the known CKM matrix elements. Thus, the computation of the
lifetimes of unconventional fermions involves unknown mixing angels; one could
only have a rational estimate for.
To estimate the lifetime and decay length of the lightest unconventional
fermion, which is the long-lived one, we note that the dominant decay is that
into a real $\tilde W_L$. For illustration purposes, let us assume that the
lightest unconventional quark and lepton are $D$ and $l_{d} $, respectively.
Their dominant decay modes would be
\begin{subequations}
\begin{equation}
D\left( -{\frac{4}{3}} \right) \to d^j \left( -{\frac{1}{3}} \right) + \tilde
W_L^ - ,
\end{equation}
\begin{equation}
l_{d} \left( 1 \right) \to \nu ^k \left( 0 \right) + \tilde W_L^ + ,
\end{equation}
\end{subequations}
where $j = d,s,b$, and $k=e,\mu,\tau$, and all fermions ought to be mass
eigenstates. The decay widths can be found easily. For $m_{d^j ,\nu ^k } \ll
m_{D,l_{d} } $
, the decay widths are simply given by
\begin{subequations}\label{equ:widths1}
\begin{equation}
\Gamma _D = \frac{{G_F m_D^3 }}
{{8\pi \sqrt 2 }}\left| {\mathcal{O}\left( {\frac{{v^2 }}
{{v_H^{'2} }}} \right)} \right|^2 \left| {V_Q } \right|^2 \left( {1 +
\frac{{2M_{\tilde W_L }^2 }}
{{m_D^2 }}} \right)\left( {1 - \frac{{M_{\tilde W_L }^2 }}
{{m_D^2 }}} \right)^2 ,
\end{equation}
\begin{equation}
\Gamma _{l_{d} } = \frac{{G_F m_{l_{d} }^3 }}
{{8\pi \sqrt 2 }}\left| {\mathcal{O}\left( {\frac{{v^2 }}
{{v_H^{'2} }}} \right)} \right|^2 \left| {V_{L} } \right|^2 \left( {1 +
\frac{{2M_{\tilde W_L }^2 }}
{{m_{l_{d} }^2 }}} \right)\left( {1 - \frac{{M_{\tilde W_L }^2 }}
{{m_{l_{d} }^2 }}} \right)^2 .
\end{equation}
\end{subequations}
The factors $V_Q$ and $V_{L}$ are the relevant elements of matrices
$V_Q=\mathrm{U}_D^{-1}\mathrm{U}_U$ and
$V_{L}=\mathrm{U}_{l_{d}}^{-1}\mathrm{U}_{l_{u}}$, which describe the mixings
among the unconventional quarks and leptons, respectively. To be precise,
$\mathrm{U}_U$, $\mathrm{U}_D$ and $\mathrm{U}_{l_{u}}$, $\mathrm{U}_{l_{d}}$
are matrices which diagonalize the up-, down-sector of unconventional quarks and
leptons, respectively. To obtain an estimate, let us take a rational value for
the masses of $D$ and $l_{d} $, namely $m_D \approx m_{l_{d}} \approx 250$ GeV
and apply realistic assumption $\mathcal{O}\left( {{{v^2 } \mathord{\left/
{\vphantom {{v^2 } {v_H^{'2} }}} \right. \kern-\nulldelimiterspace} {v_H^{'2}
}}} \right) \approx 10^{ - 2} $. Then typical lifetimes for the lightest
unconventional fermions can be estimated
\begin{subequations}\label{equ:LT}
\begin{equation}
\tau _D \approx 1.3 \times 10^{ - 21} \left| {V_Q } \right|^{ - 2} \,\,
\mathrm{s} \, ,
\end{equation}
\begin{equation}
\tau _{l_{d} } \approx 1.3 \times 10^{ - 21} \left| {V_{L} } \right|^{ - 2}
\,\, \mathrm{s} \, .
\end{equation}
\end{subequations}
These lifetimes are obviously short, which indicate that unconventional fermions
decay fast and therefore pose no cosmological problems, unless the mixing
factors are peculiarly small. A typical decay length for the lightest
unconventional quark and lepton can also be estimated from the lifetimes of
Eqs.~(\ref{equ:LT}), they are
\begin{subequations}\label{equ:DL}
\begin{equation}
l _D \approx 400 \left| {V_Q } \right|^{ - 2} \,\, \mathrm{fm} \, ,
\end{equation}
\begin{equation}
l _{l_{d} } \approx 400 \left| {V_{L} } \right|^{ - 2} \,\, \mathrm{fm} \, ,
\end{equation}
\end{subequations}
which also depend on the mixings $V_Q$ and $V_{L}$. To summarize, we showed that
the lightest of unconventional quarks or leptons will not be stable and can
decay through the mixing among the horizontal and left-handed charged gauge
bosons. Very short lifetimes (for reasonably small mixings) are possible for the
longest-lived unconventional fermions, which alleviate cosmological concerns on
heavy stable fermions.
\section{Summary}
We examined the idea of \textit{early quark-lepton mass unification} through one
of petite unification models $\mathrm{PUT}_2=\mathrm{SU}(4)_{\mathrm {PS}} \otimes
\mathrm{SU}(3)_L \otimes \mathrm{SU}(3)_{H}$.
For this we embedded $\mathrm{PUT}_2$ into a 5D model, in brane world picture.
The petite unification scenario calls for new chiral fermions with
unconventional charges. The philosophy was that the sizes of 4D couplings in
chiral fermion mass terms are controlled by the left-right overlaps between the
corresponding localized fermions along the extra dimension. The magnitudes of
these overlaps are set by the geometry of the localized zero modes in the extra
dimension. We chose to establish such geometry by reducing the symmetry of the
model to that of the SM. This way the quark-lepton unification structure
translates into the geometry of the localized zero modes and yields a
\textit{quark-lepton mass unification structure}. This idea therefore sets the
symmetry breakings along the extra dimension in motion systematically.
As a result, the effective Yukawa couplings of quarks and leptons even those of
unconventional fermions and the SM fermions relate to each other. A numerical
estimation of mass scales showed that the unconventional fermion mass scales can
set bounds on the mass scales of Dirac neutrino and vice versa. For example,
light unconventional fermions (up to 500 GeV mass scales) set 1 eV bound on
neutrino mass scale, which imply a near-degenerate mass matrix for neutrino
sector. On the other hand, lighter unconventional fermions (as light as 180 GeV)
yield light (less than 0.1 eV) neutrino sector, which corresponds to a
hierarchical or near-degenerate mass matrix for neutrinos.
The mass scales obtained in this model are similar to those of $\mathrm{PUT}_1$.
The strong $\mathrm{SU}(4)_{\mathrm {PS}}$ is the common group of PUT scenarios. The
unconventional quarks and leptons are connected to their normal siblings through
the quartets of this group in both scenarios. Breaking $\mathrm{SU}(4)_{\mathrm {PS}}$
along the extra dimension results in similar relations between the left-right
separations of quarks and leptons and that translates into similar mass scales
for both models.
We computed the contributions of extra heavy fermions and scalars of the model
to the electroweak oblique parameter S, and showed that the extra generations of
heavy fermions may not violate the experimental bounds on new physics, in
principle.
The issue of the decay of the lightest unconventional fermion was also
discussed. We showed that the lightest unconventional fermion is indeed
unstable, as it decays to ordinary fermions through the mixing of charged gauge
bosons. The estimated lifetimes for the lightest unconventional quark and lepton
also appeared to be small enough to comply with the cosmological bounds on
stable heavy fermions.
In addition, we discussed the gauge symmetry breaking of the model in length,
where the mixings of neutral and charged gauge bosons of the model were
explained explicitly.
\begin{acknowledgments}
This work was supported, in part, by the U.S. Department of Energy under grant
No. DE-A505-89ER40518.
\end{acknowledgments}
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\section{Successes of WIMP models}
The most successful dark matter model is the cold dark matter (CDM) WIMP. These are nonrelativistic particles which interact gravitationally at long distances and via weak interactions at short distances. While experimental searches for their weak interactions have yielded no convincing signals \cite{lux}, the consequences of their gravitational interactions are regularly verified.
Perhaps the most famous of these confirmations comes from the bullet cluster: Two clusters of galaxies have passed through each other, leaving most of their luminous matter in the form of a hot plasma at the collision point. Via gravitational lensing it was shown in Ref.~\cite{bullet} that most of the matter continued past the collision point, separating from the plasma. The fact that the dark matter separated from the visible matter means that it has its own inertia. This observation eliminates the most elegant modified gravity explanations of dark matter. In addition, the fact that the dark matter halos of the clusters passed through each other yields a strong upper bound on the dark matter self-interaction cross section \cite{bulletlim}. It has been claimed \cite{nobec} that this observation rules out a large class of models in which, like the model presented below, dark matter consists of classical field theory solutions such as the scalar condensate model of Ref.~\cite{bec}.
Once structure formation begins, CDM WIMPs are nonrelativistic and only interact gravitationally. Based on numerical simulations it was claimed in Ref.~\cite{nfw} that, as a result of these two facts, the shape of a pure dark matter halo assumes a universal form which is well-approximated by the NFW profile. Observations of galaxy clusters confirm that the NFW profile indeed provides an excellent approximation to the dark matter density profile \cite{cluster}.
The CMB power spectrum observed up to multipoles of about $l=2000$ \cite{planck} is just what would arise from an initial power law spectrum of adiabatic fluctuations in a mixture of baryonic matter, photons and a nonrelativistic fluid in a universe with a cosmological constant and massless neutrinos. Around $l=2000$, combining CMB data with other data, the interpretation becomes more controversial \cite{spt} and above $l=3000$ the primordial fluctuations have been erased by Silk damping \cite{silk}. Therefore CMB observations demonstrate that up to redshifts of about $10^4$ and at angular scales down to about $l=2000$ dark matter behaves as a perfect fluid with an equation of state of about $w=0$.
The above evidence for dark matter is subject to various assumptions and unknown systematics. For example, since the power spectrum of primordial perturbations is entirely unknown there is no way to estimate the error caused by its deviation from a power law. While the bullet cluster seems to be a convincing demonstration of the existence of particulate dark matter, it is worth noting that in the case of the merger Abell 520 the dark matter coincides with the plasma. If the dark matter was stopped by a self-interaction, the cross section for this process would need to be appreciably larger than the upper bound established using the bullet cluster \cite{bulletcattivo}. Similarly the dark matter profile of a galaxy cluster depends upon its unknown formation history. We have chosen not to discuss type Ia supernovae as the standardizability of these candles is an empirical observation, still not supported by a calculation, and so again the errors cannot be estimated. Indeed, assuming $\Lambda$CDM the cosmic distance ladder based on standardizable candles \cite{hst,hst2} is now in about 3$\sigma$ disagreement with CMB observations \cite{planck}.
However one cosmological probe is independent of any known systematics or cosmological assumptions, the baryonic acoustic oscillation peak (BAO) \cite{bao1}. This is the location of an isolated peak in the position space two point function of the matter distribution. As the matter moves nonrelativistically, the location of this peak in comoving coordinates is time independent. Thus, while the size itself depends on the cosmological model, once the size is measured at any redshift, it will be the same in comoving coordinates at all redshifts. Furthermore, the location of the peak is isotropic in the sense that the peak of the matter distribution in the radial and angular directions at the same redshift agree. Thus at each redshift one can measure two quantities, the angular and radial BAO size. The knowledge that each scale is the same in comoving coordinates then allows one to robustly determine the expansion of the universe as a function of time. Using Einstein's equations this rate of expansion yields the pressure and density of the universe as a function of time. If one further assumes that the universe consists of various components whose stress energy tensors are separately conserved and that dark matter corresponds to a nonrelativistic fluid $(w=0)$ then one obtains the average dark matter density at each time.
The dark matter density so obtained, going back to about $z=2.3$ \cite{baolyman}, is consistent with that obtained from the CMB \cite{planck} corresponding to redshifts going back to $10^4$. This implies that the quantity of dark matter in a sufficiently large comoving volume should have been roughly constant since redshift $z=10^4$. While this result is inconsistent with modified gravity models of dark matter, it is consistent with models of stable particles or other such compact objects.
\section{Challenges faced by WIMPs} \label{challsez}
The successes of WIMPs have one thing in common. They are all at large distance scales, comoving scales of Mpc. What about short distance predictions?
One robust prediction of CDM WIMPs is that the Milky Way should have at least of order $10^4$ satellite dark matter halos \cite{klypin,moore}. Only about 25 such satellites have so far been observed, in particular the lightest ones seem to be missing. In itself this is not a contradiction, current lensing probes are not yet sufficiently sensitive to observe these halos. It may be that, since the lightest halos lead to shallow gravitational potential wells, the primordial gas in these wells was blown away by ultraviolet radiation from reionization \cite{uv1,uv2} or cosmic ray pressure \cite{cr} before it could condense into stars, or that the supernovas of the first generation of stars blew out of all the gas so that no further generations formed \cite{snfeedback}. Whether these mechanisms really would have suppressed star formation in these light halos depends on many unknowns, such as the ultraviolet and cosmic ray fluxes in the distant past and the efficiency with which supernovae can transfer their energy to the gas. Even if these mechanisms may evade star formation in small satellites and so explain the nondetection of light halos, there also appear to be missing heavy halos \cite{toobig} for which such mechanisms are of no avail \cite{garrison}. One can eliminate these heavy halos from simulations by reducing the mass of the Milky Way \cite{mwleggera,mwleggera2} but this leads to tension with theoretical models of the mass distribution of the Milky Way~\cite{mcmillan}. Also such a low Milky Way mass would imply that the dwarf galaxy Leo I \cite{leounbound} and the Magellanic clouds \cite{magunbound} are gravitationally unbound, which makes their observed proximity to the Milky Way unlikely.
Pure dark matter simulations yield a second prediction, that the density profile of a dark matter halo diverges at small radii \cite{nfw}. Such divergent halos are said to be cusped. Recent simulations \cite{aquarius} suggest that the power law divergence be replaced by a more complicated Einasto form \cite{einasto}, but in practice the diminishing divergence occurs at such small radii that it will not have any effect on the discussion below. Observed halos do not appear cusped, but they are not pure dark matter, they also contain stars and often gas. Could the baryonic physics of stars eliminate the cusps? A number of simulations have indicated that in principle the outflow of gas from a supernova may turn a cusped profile into a cored profile \cite{sn1,sn2}. However this transformation only works in the supernovae is able to efficiently transfer its energy, which requires these models to use a very high threshold gas density for star formation \cite{troppogas}. There has been considerable debate concerning whether such a threshold is reasonable.
While it may be that baryonic physics can transform the cusps into cores in some dark matter halos, clearly such a transformation is impossible if the baryon density is insufficient. The main proponents of this mechanism \cite{gov2012} find, as summarized in their Eq.~(1), that it can totally remove cusps in galaxies with more than $10^9\ M_\odot$ of stars whereas the central density diverges at least as $1/r$ if the mass is beneath $4\times 10^ 6\ M_\odot$. Thus smaller galaxies, such as the majority of the dwarf spheroidal galaxies in our local group, will unavoidably have cusped dark matter halos if dark matter is made of WIMPs. Are these cusps in contradiction with observations? These light galaxies are not rotating, so their rotation curves can not be used to determine their dark matter halo profiles. The stars are dispersion supported, so in principle the halo profiles can be determined from the Jeans equation \cite{jeans}. However, as a result of a degeneracy \cite{jeansdegen} this is not sufficient to determine the halo density. Instead, if a few basic assumptions are included \cite{wolf}, the Jeans equation yields the amount of mass within the half-light radius \cite{walker} of each stellar population \cite{pops}. Applying this method to galaxies with multiple stellar populations one can obtain the densities at distinct radii. This has been done for the Fornax \cite{pops} and Sculptor \cite{sculptor} dwarfs and in both cases the result is consistent with a cored profile and not a cusped profile, although the stellar mass of Sculptor is only $2\times 10^ 6\ M_\odot$ and so the results of \cite{gov2012} suggest a cusped profile. Furthermore, assuming it not to be recently accreted, the existence of old substructure in the Fornax \cite{fornax}, Ursa Minor \cite{ursaminor} and Sextans \cite{sextans} dwarfs is inconsistent with a cusped profile, as the resulting tidal force would both delocalize the substructure and also pull it towards the center of the halo.
Recently WIMP models have faced new challenges arising from the concentration in phase space of dwarf satellites of the Milky Way \cite{milkydisco} and Andromeda \cite{andromedadisco} as well as the planes of Ref.~\cite{duedischi} and the filamentary structure of \cite{filament} and the abundance of pairs in Refs.~\cite{fattahi,paii}. In CDM WIMP simulations such concentrations do not appear, although perhaps they may be caused by recent mergers \cite{belreview}.
\section{Proposal: giant monopoles}
The standard model contains no dark matter candidates which are consistent with the tight bounds on MACHOs. As a result, any explanation of dark matter must introduce either new and exotic couplings or else new fields. We choose the later path, introducing a dark SU(2) gauge field, a scalar Higgs field which transforms in the adjoint representation of this dark SU(2) and also an undetermined number of fundamental fermions. These fields are coupled via the same gauge-invariant and renormalizable terms that appear in the standard model. These fields all inhabit a dark sector, they are in addition to the usual standard model fields. They will interact with the standard model fields gravitationally, although renormalizability and gauge invariance also allow the Higgs portal interaction of Ref.~\cite{kiasmonopole}.
This dark sector admits a stable classical field theory solution, the 't Hooft-Polyakov monopole. While it is known that a small fraction of dark matter consists of active, massive neutrinos, following Ref.~\cite{bjarke}, we will propose that the rest of the dark matter in our universe consists of these monopoles. Each monopole, in a steady state and up to a discrete choice that we will discuss below, is completely characterized by a single integer, its charge $Q$. We will propose that each dark matter halo consists of a {\it single} monopole of some charge $Q$, thus recovering the observational fact that galaxies tend to inhabit one parameter families. This observation has consistently posed a challenge to WIMP models which naturally yield two parameters, corresponding intuitively to a mass and a temperature. Note that, since halos are extremely large, so are the monopoles. The monopoles will have diameters measured in parsecs and masses measured in solar masses, in contrast with the more familiar and smaller GUT monopoles. In the rest of this talk we investigate the consequences of this proposal.
\section{Giant monopoles at large scales}
Does the giant monopole model reproduce the successes of WIMPs at large scales? At scales much larger than the monopole, the monopoles can be treated as point particles. As we will discuss later, at long distances only their gravitational interactions will be significant. Therefore their behavior will be indistinguishable from WIMPs. In particular, the profiles of galaxy clusters will be the same as will the relative BAO scales.
The overall BAO scale is determined by the speed of sound of the primordial plasma, which is unchanged in our scenario, and by the age of the universe at recombination. What is the age of the universe at recombination in this model? First of all we can calculate the time when the monopoles form using the usual Kibble mechanism. The monopoles form when the scalar Higgs field unfreezes, which occurs when the Hubble length grows beyond $r_1$, the inverse tachyonic mass of the Higgs field. The scale $r_1$ is just the radius of the core of a dark matter halo which is about 1 kpc, so monopoles form when the Hubble radius is about one inverse kpc. This is before matter-radiation equality, so before the monopoles have formed the dark matter, in either WIMP or monopole models, is an insignificant part of the total energy budget of the universe. At matter-radiation equality, when the contribution of dark matter to the energy density of the universe becomes relevant, the monopoles have already formed and so, as the monopoles are stable, it is reasonable that the mass in each comoving volume is then fixed as in a WIMP cosmology. In this case the energy density of monopoles before recombination is the same as that of WIMPs in the standard cosmological model, and so recombination happens at the same moment. This implies that the overall scale of the BAO feature will be the same as in WIMP cosmologies, in agreement with observations. Note that, had the cores of galaxies today been larger by a factor of ten, the monopoles would have formed too late and this model would have been falsified.
What about the CMB? As the monopoles have formed before matter-radiation equality, they may contribute the evolution of the power spectrum. Once they are fully formed, they will have the correct nonrelativistic equation of state $w=0$, although at early times $w$ begins at $-1$. Were they a fluid? Evolving back the density of dark matter today one finds about $10^2$ monopoles in each $l=2000$ volume \cite{bjarke}, just enough for the fluid approximation at these multipoles.
The monopoles are particles with inertia and so, as is observed in the bullet cluster, will tend to continue straight through a collision unless scattered. The bullet cluster bound on the cross section \cite{bulletlim} of about 1 cm${}^2$/g, for a $10^{12} M_\odot$ halo, corresponds to the cross section of a black disk of radius equal to about 5 kpc. While we have not yet computed the cross sections for our solutions, the dark force interactions will be stronger than the gravitational interactions by only about 5 orders of magnitude and so it is plausible that this bound is satisfied.
\section{Giant monopoles at small scales}
We have seen that it is plausible that giant monopoles share the same large scale successes as WIMPs, although this is contingent on details of their formation before recombination. The real strength of the monopole models is at small scales, where they effortlessly solve the challenges described in Sec.~\ref{challsez}. First of all, the missing satellite problem. The missing light satellite problem is essentially the observation that satellite galaxies are observed to have a minimum mass, in contradiction with the universal expectation from WIMP simulations. On the other hand, Dirac has shown that monopole solutions always have a minimum mass, corresponding to the $Q=1$ monopole. Below we will fix the parameters of our theory such that the $Q=1$ monopole has the observed minimum mass, and thus the missing halo problem will be solved.
What about the cusp problem? The profile of a 't Hooft-Polyakov monopole in the presence of gravity, at least in the case $Q=1$, is well known. There is a central core, of radius $r_1$ with a reasonably constant density and a nearly zero Higgs and gauge field. Thus the core problem is automatically solved. At the radius $r_1$ the Higgs field turns on, as $r_1$ is proportional to the inverse Higgs mass, breaking the SU(2) gauge symmetry to U(1). Let $r_2$ be the inverse W boson mass, where the W boson is not the standard model W boson but the gauge field which has been Higgsed. Then, between $r_1$ and $r_2$ the gauge fields are essentially zero but the Higgs field enjoys a topologically nontrivial winding which gives this region a density proportional to $1/r^2$. This automatically yields the observed flat rotation curves. At higher radii the gauge fields also become nontrivial.
And the phase space correlations? The dark forces in these models are more attractive than gravity in some regimes, and so one expects more binding and so more correlations. Whether this increase agrees with observations will be studied in the near future.
What are the parameters of the model? Dwarf spheroidal galaxies (dSphs) are among the purest concentrations of dark matter in the universe. By using Refs.~\cite{walker,wolf} to fix to their masses inside of their half-light radii and assigning each dSph a charge $Q$ we have fit the Higgs VEV to be $v\sim 10^{14}$ GeV, the quartic Higgs selfcoupling to be $\lambda\sim 10^{-96}$ and the gauge coupling to be less than $10^{-48}$. The value of $v$ is tantalizing, the inputs were astrophysical scales and the output is the leptogenesis scale.
\section{Challenge}
Unfortunately these monopoles repel and so the charge $Q>1$ halos are unstable. This may rule out our model. Then again, protons repel but visible matter is mostly made of protons, as the repulsion at small distances is canceled by neutrons and at large distances is screened by electrons. The monopoles only repel at long distances. So what are the analogs of the electrons? Electrons carry the opposite charge from protons but cannot annihilate with protons as they carry a flavor quantum number and the lightest state for a decay product, the neutron, is too massive for the decay to be kinematically allowed. Similarly such a flavor quantum number for the monopoles is an automatic consequence of our fermionic couplings \cite{jr}. The masses of the various flavors of monopoles can be adjusted by choosing the Yukawa couplings. We propose to include light antimonopoles of a different flavor which screen the long distance repulsion of our monopoles. If such a screening cannot be made to work, our proposal will be excluded.
\ack
Many of the results presented in this talk were obtained in collaboration with Sven Bjarke Gudnason. JE is supported by the Chinese Academy of Sciences Fellowship for
Young International Scientists grant number 2010Y2JA01.
\section*{References}
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\section{Power Counting and decuplet resonances in B$\chi$PT}
Unlike in the meson sector, in B$\chi$PT the power counting (PC) is violated by the presence of $M_N$ as a heavy scale and the baryonic loop diagrams do not fulfill the chiral order prescribed by their topology~\cite{Gasser:1987rb}. A crucial observation follows from noticing that this naive PC arises from considering the nucleons in a non-relativistic expansion, which eventually can be implemented from the outset using heavy-baryon (HB)$\chi$PT~\cite{Jenkins:1990jv}. The HB is an elegant formalism with a neat PC, but it alters the analytic structure of the baryon propagators. This has been argued to be the reason behind the problematic convergence showed by the HB expansion in some observables, motivating the study and emergence of Lorentz covariant approaches. An important development in this sense comes from realizing that the genuine non-analytical chiral corrections in a covariant formalism satisfy the PC and are identical to those obtained in the HB expansion. The PC-breaking pieces, on the other hand, are analytic and can be eventually absorbed into the local counterterms or low energy constants (LECs) in some (renormalization) prescription~\cite{becher,scherer1}. There are two main covariant approaches, the infrared (IR)~\cite{becher} and the extended-on-mass-shell (EOMS) scheme~\cite{scherer1}. The former one, however, introduces unphysical cuts that can disrupt the chiral expansion. The EOMS scheme, on the other hand, is nothing else than conventional dimensional regularization in which the finite parts of the counter-terms are adjusted to cancel the PC-breaking pieces. In this way, it recovers the PC at the same time as it does not change the analytic structure dictated by $S$-matrix theory.
Another difficulty in B$\chi$PT is the treatment of the lowest-lying decuplet resonances. In the conventional approach, these resonances and other heavier degrees of freedom are integrated out and accounted for by the LECs. This is a valid prescription as long as the energies probed in the theory are well below the mass gap of these states with respect to the ground state octet baryons, or $M_N$. However, in case of the decuplet resonances like the $\Delta(1232)$, the mass gap is only of $\delta\sim M_\Delta -M_N\sim 300$~MeV and, moreover, this resonance couples strongly to the $\pi N$ system. In a $SU(3)$-B$\chi$PT approach, the size of the perturbative parameter, $\sim M_K/\Lambda_{\chi SB}$ is even larger than this typical scale $\delta/\Lambda_{\chi SB}$. Therefore, it becomes necessary to properly take the $\Delta(1232)$ and other decuplet resonances into account as explicit degrees of freedom. In order to do so, one needs to define a suitable PC for the new scale $\delta$~\cite{Hemmert:1997ye,Pascalutsa:2002pi}, and also to tackle the {\it consistency} problem that afflicts interacting spin-3/2 theories (see Ref.~\cite{Pascalutsa:2006up} and references therein). Once these two issues are solved, one can apply any of the formalisms to cure the power counting problem and explicitly calculate their contributions to low-energy baryon structure.
\section{Experimental determinations of the sigma terms}
\subsection{The $\pi N$ scattering amplitude and $\sigma_{\pi N}$}
The elastic scattering of pions and nucleons probes their scalar vertices, and this is formalized in the nucleon case by the Cheng-Dashen theorem connecting $\sigma_{\pi N}$ to the isospin-even scalar scattering amplitude at the kinematical point $(s=m_N^2,\,t=2M_\pi^2)$, which lies in the unphysical region of the process. The traditional extrapolation is done using an energy-dependent parameterization of the data in partial waves (PW) supplemented by dispersion relations that impose strong analyticity and unitarity constraints onto the scattering amplitude at low energies. There are two ``classic'' determinations of $\sigma_{\pi N}$, the one based on the Karlsruhe-Helsinki (KH) group, $\sigma_{\pi N}\simeq 45(8)$~MeV~\cite{KA85,sigmatermupdate}, and the other performed by the George-Washington (GW) group, $\sigma_{\pi N}=64(7)$~MeV~\cite{WI08,Pavan:2001wz}. Although the difference between these two determinations is not too large, it leads to radically different interpretations of the strangeness content in the nucleon, as we will see below. Besides, a substantial reduction of the $\sim30$ MeV uncertainty involved by these two determinations would increase the significance of the constraints set on the parameter space of models from the experimental bounds on the dark-matter nucleon cross sections.
In order to understand some of the systematic effects, one would wish to complement the dispersive treatments with B$\chi$PT. Ideally, one would even dream of performing a completely model-independent analysis of the scattering data
leading to a systematic study of the subthreshold region, $\sigma_{\pi N}$ and all other related quantities without any further input. However these studies have faced important difficulties. The classical works of Fettes et {\it al.} in HB at $\mathcal{O}(p^3)$~\cite{fettes3} and $\mathcal{O}(p^4)$~\cite{fettes4} were able to reproduce the $S$- and $P$-wave phase shifts in the threshold region, but they didn't succeed to give a realistic description of the subthreshold region and, consequently, they overestimated the value of the sigma-term. They concluded that an order-by-order improvement in the extrapolation onto the subthreshold region was far from obvious~\cite{fettes4}. The inclusion of the $\Delta$ as explicit degree of freedom in the small-scale-expansion (SSE) ($\delta\sim \mathcal{O}(p)$) allowed to stretch significantly the reproduction of the phase-shifts to larger energies~\cite{fettesD}. However, large correlations among the LECs were reported, with values depending much on the PW analysis used as input. As a result a stable value of $\sigma_{\pi N}$ and extrapolation to the subthreshold region could be only achieved using the Olsson dispersive sum rules~\cite{Olsson}. These problems were corroborated by the $\mathcal{O}(p^4)$ calculation without explicit $\Delta$'s done in the IR scheme~\cite{beche2}. In this case, though, an inverse approach was followed. The subthreshold description was investigated and the extension into the physical region was then attempted, without success.
The situation has recently improved with a novel chiral analysis of the $\pi N$ scattering amplitude introducing two main innovations over previous work. In the first place, a fully covariant approach in the EOMS scheme is employed~\cite{Alarcon:2011zs,Alarcon:2012kn,Chen:2012nx}. This proves to be not only important in the extrapolation onto the subthreshold region (in comparison with HB) but also in extending the framework to higher energies (as the IR scheme becomes sensitive to its unphysical cuts). The second essential ingredient is the inclusion of the $\Delta(1232)$ as an explicit degree of freedom in the $\delta$-counting~\cite{Pascalutsa:2002pi}, which exploits the hierarchy $M_\pi<\delta<\Lambda_{\chi SB}$ by counting $\delta$ as $\mathcal{O}(p^{1/2})$. This analysis was performed up to $\mathcal{O}(p^3)$ in this counting, implying that the only explicit $\Delta$ contributions are those stemming from the Born-Term diagrams. The strategy followed in this work was to determine the different LECs with the $S$- and $P$-wave phase shifts provided by the KH, GW and Matsinos'~\cite{EM02} groups and then discuss the resulting phenomenology.
\begin{figure}[t]
\centering
\includegraphics[width=10cm]{PlotGW.eps}
\caption{Chiral analysis of the GW phase shifts (blue points) up to $\mathcal{O}(p^3)$ in the EOMS scheme without $\Delta$ (dashed green line) and with $\Delta$ (solid red line) in the $\delta$-counting.\label{Fig:piNPSs}}
\end{figure}
The quality of the corresponding fits to the GW PW analysis is shown in Fig.~\ref{Fig:piNPSs} where the phase shifts are perfectly reproduced up to energies of $W=\sqrt{s}\simeq1.2$ GeV. For the sake of comparison, we also include the result obtained without the explicit $\Delta$ contribution, which achieves a good description of the phase shifts only up to energies slightly above threshold.\footnote{The inclusion of the $\Delta$ explicitly up to $\mathcal{O}(p^3)$ in the $\delta$-counting introduces 3 new LECs through the Born-term diagram. However, one of these parameters can be fixed with the $\Delta(1232)$-resonance width and the other two can be shown to be redundant~\cite{Long:2010kt,Alarcon:2012kn}.} In fact, a comparison between the contributions at different orders shows that only in the former case a good convergence is obtained up to $\mathcal{O}(p^3)$ in all the low-energy region.
\begin{table}
\centering
\caption{Physical observables obtained from the $\mathcal{O}(p^3)$ $\pi N$ scattering amplitude in the EOMS renormalization scheme fitted to different PW analyses. We show the $\pi N \Delta$ coupling $h_A$, the $\pi N$ coupling and corresponding Goldberger-Treiman discrepancy $\Delta_{GT}$, the scattering lengths, which are given in units of 10$^{-2}$ $M_\pi^{-1}$ and the pion-nucleon sigma term, which is given in units of MeV~\label{Table:Observables}}
\begin{tabular}{|cccccccc|}
\hline
&$\chi^2_{\rm d.o.f.}$&$h_A$&$g_{\pi N}$&$\Delta_{GT}$ [\%] &$a_{0+}^+$ &$a_{0+}^-$&$\sigma_{\pi N}$ \\
\hline
KH~\cite{KA85}&0.75&3.02(4)&13.51(10)&4.9(8)&$-1.2(8)$&8.7(2)&43(5)\\
GW~\cite{WI08}&0.23&2.87(4)&13.15(10)&2.1(8)&$-0.4(7)$&8.2(2)&59(4)\\
EM~\cite{EM02}&0.11&2.99(2)&13.12(5)&1.9(4)&0.2(3)&7.7(1)&59(2)\\
\hline
\end{tabular}
\end{table}
Once the LECs are determined, one can predict and study different $\pi N$ scattering observables or to investigate the extrapolation onto the subthreshold region. In Table~\ref{Table:Observables} we present the results for some selected observables, that can be checked to be in good agreement with those reported by the respective PW analyses. Then, one can see that a B$\chi$PT analysis of the phase shifts ratifies the discrepancy between different PW analyses, in particular regarding the value of $\sigma_{\pi N}$. A comparison of the values of some of these observables with the alternative determinations that can be obtained from other sources like pionic atoms or $NN$-scattering favors the GW solution. The KH solution gives rise to a value for $h_A$ that is not compatible with the $\Delta(1232)$ width and to a value for $g_{\pi N}$ that leads to a significant violation of the GT relation. As for our study of the EM PW analysis, we found a value for the isovector scattering length that is too small as compared with the accurate values obtained from pion-atoms data~\cite{EM02}. However, the most important observable in the discussion of $\sigma_{\pi N}$ concerns the scalar-isoscalar scattering length. While the KH result is compatible with the old negative values, it is not with the more recent determinations from modern pionic-atom data, $a_{0+}^+=-0.1(1)$ $10^{-2} M_\pi^{-1}$~\cite{Baru:2010xn}.\footnote{See Ref.~\cite{Alarcon:2012kn} for details.} These are, on the other hand, compatible with the scattering data extractions from the GW and EM solutions. Finally, notice that the Matsinos and GW analyses lead to the same $\sigma_{\pi N}$. This is relevant because these are two completely different analyses that incorporate the new high quality data collected over the 2 last decades, whereas the KH group stopped updating theirs in the mid 80's. With these considerations, we reported the value~\cite{Alarcon:2011zs}
\begin{equation}
\sigma_{\pi N}=59(7) {\rm MeV},\label{Eq:SpiNValue}
\end{equation}
where the error includes a theoretical uncertainty estimated with the explicit calculation of higher-order graphs added in quadrature with the one given by the dispersion of the values in the average of the GW and EM results.
\begin{table
\begin{center}
\begin{tabular}{|c||c|c|}
\hline
& B$\chi$PT & Dispersive \\
\hline
$d_{00}^+$ ($M_\pi^{-1}$) & $-1.48(15)$ & $-1.46$ \\
$d_{01}^+$ ($M_\pi^{-3}$) & $1.21(10)$ & $1.14$ \\
$d_{10}^+$ ($M_\pi^{-3}$) & 0.99(14) & 1.14(2) \\
$d_{02}^+$ ($M_\pi^{-5}$) & 0.004(6) & $0.036$ \\
$b_{00}^+$ ($M_\pi^{-3}$) & -5.1(1.7) & $-3.54(6)$ \\
$d_{00}^-$ ($M_\pi^{-2}$) & 1.63(9) & 1.53(2) \\
$d_{01}^-$ ($M_\pi^{-4}$) & -0.112(25) & $-0.134(5)$ \\
$d_{10}^-$ ($M_\pi^{-4}$) & -0.18(5) & $-0.167(5)$ \\
$b_{00}^-$ ($M_\pi^{-2}$) & 9.63(30) & 10.36(10) \\
\hline
\end{tabular}
{\caption[pilf]{Results for different subthreshold coefficients obtained from the fits to the KH analysis and in B$\chi$PT in the EOMS scheme and with explicit $\Delta$ contributions up to $\mathcal{O}(p^3)$ in the $\delta$-counting. The results obtained using dispersive techniques are included for the sake of comparison.
\label{subthreshold-results}}}
\end{center}
\end{table}
The success of this modern calculation to provide a reliable determination of $\sigma_{\pi N}$ from phase shifts can be understood by analyzing the scattering amplitude in the subthreshold region, where it comes usually characterized by the so-called subthreshold coefficients stemming from a kinematic expansion about the point $s=u=m_N^2+M_\pi^2$ and $t=0$. In Table~\ref{subthreshold-results} we show the values of these quantities obtained after fitting the LECs to the KH phase shifts compared to those given by dispersive techniques~\cite{Alarcon:2012kn}. As it can be seen, in this approach to B$\chi$PT, the long-sought connection between the physical and subthreshold regions is accomplished. In particular, $d_{00}^+$ and $d_{01}^+$ correspond to the leading orders of the expansion in $t$ of the Born-subtracted scalar-isoscalar amplitude and so they are essential to understand the extrapolation to the Cheng-Dashen point~\cite{sigmatermupdate} and determination of $\sigma_{\pi N}$. For the details on the discrepancy on $d_{02}^+$ and its meaning we refer the reader to Ref.~\cite{Alarcon:2012kn}.
\subsection{The baryon-octet mass splittings and the strangeness content of the nucleon}
The contribution of the strange quark to the nucleon mass can be related with $\sigma_{\pi N}$ and the $SU(3)_F$-breaking of the baryon masses in the octet. Namely, one can re-express the pion-nucleon sigma term as~\cite{Gasser:1980sb}
\begin{equation}
\sigma_{\pi N}=\frac{\sigma_0}{1-y}, \label{Eq:sigmapiN-sigma0-y}
\end{equation}
where $y$ is the so-called ``strangeness content'' of the nucleon,
\begin{equation}
y=\frac{2\langle N|\bar{s}s|N\rangle}{\langle N|\bar{u}u+\bar{d}d|N\rangle}=\frac{2\hat{m}\sigma_{s}}{m_s\sigma_{\pi N}}=1-\frac{\sigma_0}{\sigma_{\pi N}},\label{Eq:Defy}
\end{equation}
and
\begin{equation}
\sigma_0=\hat{m}\langle N|\bar{u}u+\bar{d}d-2\bar{s}s|N\rangle.\label{Eq:Defsigma0}
\end{equation}
Thus, $\sigma_0$ can be understood as the value of the pion-nucleon sigma term in case that the strange-quark contribution to the nucleon wave function is null (Zweig rule). The interest of $\sigma_0$ lies on the fact that it can be calculated in $SU(3)_F$ B$\chi$PT up to $\mathcal{O}(p^3)$ using the experimental baryon-octet mass splittings. Subsequently, using Eq.~\ref{Eq:sigmapiN-sigma0-y}, one can obtain the strangeness content of the nucleon from a given experimental determination of $\sigma_{\pi N}$.
At LO, $\sigma_0=\hat{m}/(m_s-\hat{m})\left(M_\Xi+M_\Sigma-2M_N\right)\simeq27$ MeV. The NLO or $\mathcal{O}(p^3)$ corrections were first calculated by Gasser in 1982 using an early version of B$\chi$PT regularized with phenomenological form factors, giving $\sigma_0=35(5)$ MeV. This result was later bolstered by an $\mathcal{O}(p^4)$ HB calculation~\cite{Borasoy:1996bx} which obtained $\sigma_0=36(7)$. However, at $\mathcal{O}(p^4)$ several new unknown LECs contribute and they had to be determined in this calculation using model estimates, or even the value $\sigma_{\pi N}=45$ MeV as input. Besides, it is known that the HB approach suffers from a problematic convergence in $SU(3)$-flavor applications~\cite{Geng:2008mf} (for a recent review see~\cite{Geng:2013xn}). Nevertheless, and despite the possible problems in these numerical determinations, they have settled in the field, becoming an important source of distrust in ``relatively large'' values of $\sigma_{\pi N}$. Indeed, taking the Gasser result on $\sigma_0$ with the value of $\sigma_{\pi N}=64(8)$ MeV extracted from $\pi N$ scattering data by the GW group, one obtains a strangeness contribution to $M_N$ of about 300 MeV, a scenario that is considered implausible.
\begin{table*}[h]
\centering
\caption{Values of $\sigma_0$ for different B$\chi$PT approaches. \label{Table:sigma0}}
\begin{tabular}{c|c|cc|cc|}
\cline{2-6}
& &\multicolumn{2}{|c|}{ \raisebox{-1ex}[0.pt]{Octet $\mathcal{O}(p^3)$}}&\multicolumn{2}{|c|}{\raisebox{-1ex}[0.pt]{Octet+Decuplet $\mathcal{O}(p^3)$}} \\
& \raisebox{2ex}[10.pt]{Tree level $\mathcal{O}(p^2)$}& HB & Covariant& HB-SSE& Covariant\\
\hline
\multicolumn{1}{|c|}{$\sigma_0$ [MeV]}&27&58(23)&46(8)&89(23)&58(8)\\
\hline
\end{tabular}
\end{table*}
These calculations were recently revisited in the context of B$\chi$PT framed in a covariant framework within the EOMS scheme and considering explicitly the contributions of the decuplet~\cite{Alarcon:2012nr}, which were neglected in previous works. The results of this analysis are summarized in Table~\ref{Table:sigma0}, where we present those corresponding to the EOMS and HB approaches, with and without decuplet contributions. The errors are obtained by the explicit computation of $\mathcal{O}(p^4)$ diagrams stemming from the $SU(3)$-breaking of the baryon masses in the $\mathcal{O}(p^3)$ diagrams. As we can see, the corrections to the LO result on $\sigma_0$ studied are large. The HB expansion has severe problems of convergence in the description of the sigma terms at ${\cal O}(p^3)$. Focusing in the following on the covariant results, we see that considering only the octet contributions the result is reasonably close to the classical result of Gasser~\cite{Gasser:1980sb}, whereas the new contributions given by the decuplet baryons are not negligible producing a $\sim$10 MeV rise on $\sigma_0$. In summary, this implies that a pion-nucleon sigma term of $\sim 60$ MeV is not at odds with a small strangeness content in the nucleon. In fact, plugging the result for $\sigma_{\pi N}$ from $\pi N$-scattering reported in the previous section, we obtain
\begin{equation}
y=0.02(13).
\end{equation}
\section{Determinations from lattice QCD}
A theoretical determination of the sigma terms is accessible through the LQCD simulations. There are two possible strategies. The most straightforward one consists of directly evaluating the matrix elements~(\ref{Eq:STsDefinition}) in the lattice. However this is computationally very expensive due to the evaluation of the contributions of the current coupled to disconnected quark lines, which are expected to play an important role in the numerical determinations. The second and most widely used strategy is based on the Hellmann-Feynman theorem which relates the sigma-terms to the quark-mass derivatives of the nucleon mass.
Therefore, one can obtain $\sigma_{\pi N}$ and $\sigma_s$ by interpolating the physical nucleon mass with determinations of $M_N$ at different values of quark masses. One needs enough accurate determinations close to the physical point and the main difficulty lies in assessing the systematic effects originating from a specific choice of interpolators. In this sense, it is natural to use B$\chi$PT to perform these studies. Interpolators based on $SU(2)$ HB$\chi$PT up to $\mathcal{O}(p^3)$ and $\mathcal{O}(p^4)$ have become standard in the extrapolations of $M_N$ and determinations of $\sigma_{\pi N}$ performed by the lattice collaborations.
Two main difficulties have been encountered in the development of this program based on B$\chi$PT. First, the extension of the formalism into a $SU(3)$ setup, describing the quark-mass dependence of the masses of all the octet (and decuplet) baryons and giving access to $\sigma_s$, has been troubled by the problematic convergence of the HB approach in this sector of the theory. Only after the application of approaches with cut-off regularization prescriptions~\cite{Young:2009zb} or in the covariant formalisms ~\cite{MartinCamalich:2010fp,Geng:2011wq,Semke:2011ez,Semke:2012gs,Ren:2012aj,Ren:2013dzt}, it has been possible to perform reliable $SU(3)_F$-B$\chi$PT extrapolations. Second, the systematic effects given by the decuplet degrees of freedom in the extrapolation of the baryon masses and on the value of the sigma-terms remains unclear. While the effect of $\Delta$ pieces on $\sigma_{\pi N}$ at $\mathcal{O}(p^3)$ in a $SU(2)$ calculation has been claimed to be negligible~\cite{Procura:2006bj}, a more thorough calculation of these effects up to $\mathcal{O}(p^4)$ contradicted this statement~\cite{WalkerLoud:2008bp}.
\begin{figure}[t]
\centering
\includegraphics[width=7cm]{ChiralExtrap1.eps}
\caption{Extrapolation of the PACS-CS results~\cite{Aoki:2008sm} on the lowest-lying baryon masses within the
covariant formulation of $SU(3)_F$-B$\chi$PT up to $\mathcal{O}(p^3)$~\cite{MartinCamalich:2010fp}.\label{fig_extrapolationp3}}
\end{figure}
In order to address these two issues, we report on the results of the extrapolation of the octet (and decuplet) baryon masses obtained in $SU(3)$-B$\chi$PT in the EOMS scheme. In contrast with the results obtained using the HB expansion, it has been found that a good description of the LQCD results can be achieved within the Lorentz covariant approach to $SU(3)$-B$\chi$PT up to $\mathcal{O}(p^4)$ and considering the explicit inclusion of decuplet degrees of freedom~\cite{MartinCamalich:2010fp,Geng:2011wq,Ren:2012aj,Ren:2013dzt}. Moreover, an order-by-order improvement in the description of the lattice results on the octet baryon masses was found~\cite{MartinCamalich:2010fp,Ren:2013dzt}. Similar efforts in self-consistent formalisms up this accuracy have been reported by Semke and Lutz in~\cite{Semke:2011ez,Semke:2012gs} and also by the latter author in this conference.
In Fig.~\ref{fig_extrapolationp3}, we show the quark mass dependence and extrapolation of the lowest-lying baryon masses in Lorentz covariant B$\chi$PT up to $\mathcal{O}(p^3)$ for the case of the analysis of the PACS-CS results~\cite{Aoki:2008sm}. The strategy followed was to fit the 4 (3) LECs appearing at this order for the octet (decuplet) baryons using the results of different LQCD collaborations. As it can be seen from the figure, the lattice results are well reproduced and the extrapolation to the physical point agrees with the experimental values within errors~\footnote{Notice that in these fits the experimental baryon masses are not included in the fit, so the results obtained at the physical point are a prediction of the extrapolation.}. The improvement obtained at this order in covariant $SU(3)$-B$\chi$PT in comparison with the description provided by the Gell-Mann-Okubo approach at $\mathcal{O}(p^2)$, stresses the relevance of the leading chiral non-analytical terms in the understanding of the nucleon mass from quark masses as light as those reached by PACS-CS~\cite{Aoki:2008sm} ($M_\pi\simeq156$ MeV), whereas the comparison with HB~\cite{MartinCamalich:2010fp} highlights the fact that the relativistic corrections greatly improve the description of the LQCD results on the baryon masses at heavier quark-masses.
\begin{table}[t]
\renewcommand{\arraystretch}{1.3}
\setlength{\tabcolsep}{0.3cm}
\centering
\caption{Predictions on the $\sigma_{\pi N}$ and $\sigma_{sN}$ terms (in MeV) of the baryon-octet in covariant $SU(3)_F$-B$\chi$PT by fitting the LECs to the PACS-CS~\cite{Aoki:2008sm} or LHPC~\cite{WalkerLoud:2008bp} results. The errors are only statistical. \label{Table:ResSigmasB}}
\begin{tabular}{c|cc|cc|}
\cline{2-5}
&\multicolumn{2}{|c|}{PACS-CS}&\multicolumn{2}{|c|}{LHPC}\\
\cline{2-5}
\multicolumn{1}{c|}{}&No Dec.&Dec.&No Dec.&Dec.\\
\hline
\multicolumn{1}{|c|}{$\sigma_{\pi N}$}&46(2)&59(2)&43(2)&61(2)\\
\multicolumn{1}{|c|}{$\sigma_{sN}$}&28(23)&$-7$(23)&6(20)&$-4$(20)\\
\hline
\end{tabular}
\end{table}
As a result of the determinations of the LECs from the fits, one can predict the nucleon sigma terms. In Table ~\ref{Table:ResSigmasB} we present the results on $\sigma_{\pi N}$ and $\sigma_{sN}$ after fitting the LECs to the PACS-CS and LHPC results. We also present the results that are obtained in fits with (Dec.) and without (No Dec.) the inclusion of decuplet resonances to discuss the systematic effects stemming from the treatment of these contributions. As we can see from this table, the results on $\sigma_{\pi N}$ in either case are very consistent with the analysis of the experimental data described in the previous section in the case of approximate fulfillment of the Zweig rule. This confirms, in a highly non-trivial fashion, the conclusions at $\mathcal{O}(p^3)$ in $SU(3)$-B$\chi$PT in the EOMS scheme derived exclusively from experimental data. In particular, it confirms that a scenario with a $\sigma_{\pi N}\simeq 60$ MeV can not be ruled out on the grounds of a small strangeness content of the nucleon at this order and that an irreducible uncertainty of about $15$ MeV originates from the omission of the decuplet.
\begin{figure}[t]
\centering
\includegraphics[width=14cm]{ChiralExtrap2.eps}
\caption{Extrapolation of the PACS-CS results~\cite{Aoki:2008sm} on the lowest-lying baryon masses within the
covariant formulation of $SU(3)_F$-B$\chi$PT up to $\mathcal{O}(p^4)$ and without decuplet degrees of freedom~\cite{Ren:2012aj}. \label{fig_extrapolationp4}}
\end{figure}
In order to settle the question of the strangeness $\sigma_s$ it is clear that one needs calculations at $\mathcal{O}(p^4)$. However, at this order a staggering amount of 15 new unknown LECs enter the calculation and determining their values in a statistically sound fashion becomes a challenge. In fact, it seems impossible to constraint their values resorting to experimental data only and results from LQCD calculations have to be massively used. Nevertheless, first steps in this direction have been given and stable fits to global LQCD results on the baryon masses have been obtained. In particular, in the works by Semke and Lutz~\cite{Semke:2011ez,Semke:2012gs}, reported also in this conference, the usual chiral expansion is supplemented with another one in $1/N_c$ which allows to uncover hierarchies among the LECs and to reduce their total number. More general fits taking into account all the 19 LECs and also finite volume corrections have been presented in the EOMS scheme without~\cite{Ren:2012aj} and with the decuplet degrees of freedom~\cite{Ren:2013dzt} and using a total of 11 configurations at different quark masses and volumes (each of which contains four points for the $N$, $\Lambda$, $\Sigma$ and the $\Xi$). The resulting good description of the quark mass dependence of the lowest lying octet baryons in this approach is illustrated in Fig.~\ref{fig_extrapolationp4} where the results of these fits is plotted against PACS-CS~\cite{Aoki:2008sm}, LHPC~\cite{WalkerLoud:2008bp}, HSC~\cite{Lin:2008pr} and UKQCD~\cite{Bietenholz:2011qq} results. The NPLQCD~\cite{Beane:2011pc} results are also used but not plotted and the BMW results~\cite{Durr:2011mp} are not included in the analysis. As for the sigma terms, the situation is at the moment unclear. In this calculation the values $\sigma_{\pi N}=43(1)(6)$ MeV and $\sigma_{s N}=126(24)(54)$ MeV are reported~\cite{Ren:2012aj}, whilst in the calculation at the same order by Semke {\it et al.} the values $\sigma_{\pi N}=32(2)$ MeV and $\sigma_{s N}=22(20)$ MeV are given~\cite{Semke:2012gs} (see M.~Lutz's talk in this conference). Therefore, further efforts are required to understand these inconsistencies and to assess the convergence of the chiral expansion of these observables. Agreement with the results obtained with cut-off regularization~\cite{Young:2009zb} in the context of dimensional regularized approaches shall also be pursued.
\section{Conclusions}
We have reviewed the status and potential of the modern approaches to B$\chi$PT by showing different recent determinations of the sigma terms. Besides being very important properties of the nucleon, they can be determined from different perspectives, based on $SU(2)$ or $SU(3)$ approaches along one direction but also using either experimental data or LQCD results along an orthogonal one. We have seen how the situation in our understanding of the $\pi N$ scattering data in a chiral framework has been greatly improved thanks to the application of modern Lorentz covariant techniques and dealing rigorously with the contributions of the $\Delta(1232)$. Although at the moment the resulting phenomenology still depends on the PW used as input, this progress offers the possibility of extracting the $\pi N$ scattering observables, and in particular $\sigma_{\pi N}$, in B$\chi$PT using directly the scattering cross-section data. As for the LQCD determinations, there has been much progress both in the quality of the LQCD results as well as on the accuracy of the B$\chi$PT calculations. Nevertheless, further work is needed to settle this issue from $\chi$PT perspective. On one hand, it would be desirable to revisit the lattice world data on $M_N$ using a $SU(2)$ framework to determine $\sigma_{\pi N}$. On the other, more data and, ideally, calculations and extrapolations on other observables in $SU(3)$ seem necessary to understand better the strangeness content of the nucleon at $\mathcal{O}(p^4)$.
\section{Acknowledgments}
I would like to thank the organizers for inviting me to this extremely interesting meeting. Also I would like to thank my collaborators J. M. Alarc\'on, L. S. Geng, J. Meng, J. A. Oller, X. L. Ren, H. Toki and M. J. Vicente Vacas who have contributed to the work presented in this talk. This work is funded by the Science Technology and Facilities Council (STFC) under grant ST/J000477/1, the Spanish Government and FEDER funds under contract FIS2011-28853-C02-01 and the grants FPA2010-17806 and Fundaci\'on S\'eneca 11871/PI/09.
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\section{Introduction}\label{Sec:Introduction}
The optical Hall effect (OHE) is a phenomenon in which the optical response of a conductive material is altered by the presence of an externally applied magnetic field.\cite{SchubertJOSAA20_2003} This effect can be measured with generalized ellipsometry at oblique angles of incidence and at terahertz (THz) frequencies. Previously, the THz-OHE has been proven as a viable non-contact method to obtain the free charge carrier properties of semiconductor heterostructures using high-field superconducting magnets.\cite{KuehneRSI85_2014,HofmannAPL101_2012,HofmannTSF519_2011,HofmannAPL98_2011,SchocheAPL98_2011,Hofmannpssa205_2008,HofmannRSI77_2006} This approach allows the extraction of a sample's carrier concentration, mobility, and effective mass parameters by using a THz-transparent substrate as a Fabry-P\'{e}rot cavity to resonantly enhance the THz-OHE signal. Recently, it has been shown these properties can be conveniently obtained with permanent magnets.\cite{KnightOL40_2015,armakavicius2017cavity,KuehneIEEETHz2017,armakavicius2016properties} Using low-field permanent magnets to provide the external field significantly decreases the magnitude of the THz-OHE signal. However, one can compensate for this by exploiting an externally-coupled Fabry-P\'{e}rot cavity to further enhance the signal. In Ref.~\onlinecite{KnightOL40_2015}, different external cavity thickness values are achieved by simply stacking multiple layers of adhesive spacers between the sample and magnet. This method is useful because it is straight-forward and low cost, but only large increments of cavity thickness can be produced. The cavity-tuning optical stage described in this work improves on this previous approach and is capable of finely tuning the cavity thickness thus providing a new measurement dimension. As an example of the cavity-tuning stage described here a sample-permanent magnet arrangement is placed inside a gas flow cell to improve sensitivity to small variations of free charge carrier parameters under varying gas flow conditions. This experiment highlights the advantage of the small footprint of this enhancement technique.
In this work, we discuss the concept of THz-OHE signal enhancement due to an externally-coupled cavity. An optical model is used to choose desirable measurement parameters, such as angle of incidence, frequency, and external cavity thickness. Details of the instrument design and data acquisition are explained. Experimental and model-calculated data are presented and compared with data for the case of no cavity-enhancement. It is demonstrated that the cavity-enhancement technique allows extraction of the free charge carrier properties of a two-dimensional electron gas (2DEG) at THz frequencies.
\section{Method}\label{Sec:Method}
\subsection{Optical Hall effect}\label{subsec:OHE}
We refer to the OHE as a physical phenomenon that describes the occurrence of magnetic-field-induced dielectric displacement at optical wavelengths, transverse and longitudinal to the incident electric field, and analogous to the static electrical Hall effect.\cite{SchubertJOSAA20_2003,SchubertJOSAOHE2016} We have previously described data acquisition and analysis approaches for the method of the OHE in the mid-infrared, far-infrared, and THz spectral regions.\cite{SchubertJOSAA20_2003,HofmannRSI77_2006,KuehneRSI85_2014,KuehneIEEETHz2017}
\subsection{Optical Hall effect model for thin film layer stacks}\label{subsec:OHEmodel}
The OHE can be calculated by use of appropriate physical models. The models contain two portions, one describes a given material's dielectric function under an external magnetic field, the second portion describes the wave propagation within a given layer stack. We have provided a recent review on this topic in Ref.~\onlinecite{SchubertJOSAOHE2016}. Briefly, in the THz spectral range the dielectric function can be approximated by a static contribution due to phonon excitations and higher energy electronic band-to-band transitions. Contributions due to free charge carriers can be well described by the Drude quasi-free electron model.\cite{Drude04,ANDP:ANDP19003060312} In the presence of a static external magnetic field, an extension of the Drude model predicts magneto-optic anisotropy\cite{DrudeAP1892} and which is the cause of the classical OHE in conductive materials. (For quantum effects see, for example, Refs.~\onlinecite{KuehnePRL111_2013,bouhafs2016decoupling,bouhafs2017multi})
\subsection{Tunable cavity-enhanced optical Hall effect}\label{subsec:TunablecavityenhancedopticalHalleffect}
\begin{figure*}[pbt]
\includegraphics[keepaspectratio=true,width=\linewidth]{Figure1.jpg}
\caption{Principle of the tunable cavity-enhanced frequency-domain THz-OHE method, here applied to characterize a two-dimensional electron gas (2DEG). The 2DEG of interest may be part of a multiple layer stack with differently, low-conducting constituents, for example, directly at the interface of a substrate (Fig.~\ref{fig:Sample-cavity_schematic}(a-1)), or ontop of the layer structure (Fig.~\ref{fig:Sample-cavity_schematic}(a-2)), or within (Fig.~\ref{fig:Sample-cavity_schematic}(a-3)). The differently, low-conducting constituents (layers) should themselves be sufficiently THz transparent. The principle configuration requires a THz-transparent substrate. If the substrate has a finite thickness ($d_{\mathrm{sub}}$), incident plane wave electric field ($E_i$) components are retro-reflected and pass the layer structure multiple times, where the first 2 orders are shown here for brevity only ($E_{r,1}, E_{r,2}$). A mirror placed at the opposite side of the substrate (Fig.~\ref{fig:Sample-cavity_schematic}b) can be used to control the frequencies of constructive interference maxima by $d_{\mathrm{gap}}$, where fractions of plane wave components reflected off the mirror exit the sample, and only the first order of those are shown ($E_{r,3}$). If the mirror surface is distanced to the backside of the substrate by $d_{\mathrm{gap}}$, the frequencies of constructive interference maxima can be tuned by $d_{\mathrm{gap}}$ (Fig.~\ref{fig:Sample-cavity_schematic}(c)). The angle of incidence is $\Phi_a$. Fig.~\ref{fig:Sample-cavity_schematic}(d) depicts the situation when the mirror is removed ($d_{\mathrm{gap}}\rightarrow \infty$). Fig.~\ref{fig:Sample-cavity_schematic}(e) depicts the case when the substrate is optically infinite ($d_{\mathrm{sub}}\rightarrow \infty$), and against which the enhancement of the tunable cavity-enhanced OHE is to be referenced. The magnetic field \textbf{B} direction is not relevant for the enhancement. Here, all examples are discussed with direction of \textbf{B} perpendicular to the cavity interfaces. Drawing not to scale.
\label{fig:Sample-cavity_schematic}}
\end{figure*}
The principle of the enhancement of the OHE in a layer stack is the constructive superposition of the magneto-optically polarization converted electromagnetic field components from multiple passages through the layer stack in the presence of an external magnetic field. Each time of passage, the electromagnetic field components undergo an additional polarization rotation caused by the magneto-optic anisotropy created by the response of the free charge carriers under the influence of the Lorentz force and within the conductive layer(s). At an interference maximum, the sensitivity to the portion of reflected light that has undergone polarization discriminating reflection or transmission optical intensity measurement is greatly enhanced towards the causes of the magneto-optic anisotropy.\cite{KnightOL40_2015} The magnitude of the enhancement can be significant and which depends on the free charge carrier properties in a given sample configuration. Examples are discussed in this work.
The principle of the tunable cavity-enhanced OHE method is demonstrated in Fig.~\ref{fig:Sample-cavity_schematic}. All configurations require the layer stack to be supported by a THz-transparent substrate. The substrate must have a flat and polished backside whose surface is parallel to the front of the substrate carrying the layer stack. The thickness of the substrate $d_{\mathrm{sub}}$ should be such that spectrally neighboring Fabry-P\'{e}rot interference maxima and minima within the substrate can be sufficiently resolved with a given spectroscopic setup. Plane wave electric field ($E_i$) components incident under an angle $\Phi_a$ then pass the sample layer structure multiple times due to multiple internal reflections within the substrate cavity. The frequencies of such maxima are controlled by the angle of incidence, the substrate thickness and the substrate index of refraction. In principle, the substrate thickness is adjustable by depositing the sample layer stack onto different substrates. This, however, requires multiple fabrication steps. If a second cavity is created by the introduction of a mirror, placed at distance $d_{\mathrm{gap}}$, the portion of electromagnetic waves lost at the backside of the substrate is fed back into the substrate, and introduces additional fractions of plane wave components passing the layer stack. The two Fabry-P\'{e}rot cavities (substrate, gap) couple, and produce coupled Fabry-P\'{e}rot resonances. The frequencies of the coupled interference maxima can then be tuned by $d_{\mathrm{gap}}$, for any given but fixed $d_{\mathrm{sub}}$. Then, the magneto-optic signal enhancement occuring at interference maxima, limited to certain frequencies for a given $d_{\mathrm{sub}}$ without external cavity, can be tuned spectrally. Thereby, a new magneto-optic spectroscopy method is created where in addition to frequency, the external cavity is tuned by changing its thickness, $d_{\mathrm{gap}}$.
Also shown in Fig.~\ref{fig:Sample-cavity_schematic} are the limiting cases, when the external cavity is zero ($d_{\mathrm{gap}}\rightarrow 0$, Fig.~\ref{fig:Sample-cavity_schematic}(b)), infinite ($d_{\mathrm{gap}}\rightarrow \infty$, Fig.~\ref{fig:Sample-cavity_schematic}(d)), and when both cavity and substrate are infinite ($d_{\mathrm{sub}}\rightarrow \infty$, Fig.~\ref{fig:Sample-cavity_schematic}(e)). The case $d_{\mathrm{gap}}\rightarrow 0$ requires deposition of a metal layer onto the backside of the substrate. The case $d_{\mathrm{gap}}\rightarrow \infty$ occurs when the substrate is THz-transparent and has parallel interfaces. The case $d_{\mathrm{sub}}\rightarrow \infty$ occurs when the layer stack is deposited onto a non-transparent substrate or when the backside of a transparent substrate is not parallel, for example, if the substrate consists of a wedge or a prism.
\subsection{Mueller matrix spectroscopic ellipsometry}\label{subsec:MMSE}
The generalized ellipsometry concept\cite{Azzam72}, its spectroscopic extension,\cite{SchubertJOSAA13_1996} and the Mueller matrix formalism\cite{JarrendahlMuellerJANews2011} are employed in this work. The Mueller matrix connects Stokes vector\cite{JonesJOSA1947} components of electromagnetic waves before and after interaction with the sample upon reflection or transmission. For the use of the Mueller matrix concept in spectroscopic generalized ellipsometry we refer the reader to recent reviews (see, e.g., Refs.~\onlinecite{SchubertIRSEBook_2004,Fujiwara_2007}). For use of the Mueller matrix formalism in the OHE\cite{footnoteSeanRSI2017A}, and in particular, for data format definition we refer to Ref.~\onlinecite{KuehneRSI85_2014}.
\subsection{Data analysis}\label{subsec:dataanalysis}
Non-linear parameter regression analysis methods are used for data analysis. The experimental data are compared with calculated OHE data. The calculated data are obtained with appropriate physical models and model parameters. Parameters are varied until a best-match is obtained minimizing an appropriately weighted error sum. The error sum takes into account the systematic uncertainties determined during the measurement for each experimental data value. Best-match model parameter uncertainties are obtained from the covariance matrix using the 90$\%$ confidence interval.\cite{KuehneRSI85_2014}
\section{Instrument}\label{Sec:Instrument}
The tunable cavity and sample must be placed within a THz spectroscopic ellipsometer system, and subjected to an external magnetic field. Ellipsometer, sample stage, and magnetic field designs are discussed in this section.
\subsection{Terahertz frequency-domain ellipsometer}\label{subs:ecTHzOHE}
Two THz frequency-domain ellipsometer instruments are used in this work. Both instruments operate in the rotating-analyzer configuration which enables acquisition of the upper left 3$\times$3 block of the 4$\times$4 Mueller matrix. The frequency-domain source is a backward wave oscillator (BWO) with GaAs Schottky diode frequency multipliers. Technical details are described in Ref.~\onlinecite{KuehneRSI85_2014} and Ref.~\onlinecite{KuehneIEEETHz2017}.
\subsection{Tunable cavity stage}\label{subsec:tunablecavitystage}
\begin{figure}
\includegraphics[keepaspectratio=true,width=\linewidth, clip, trim=0cm 0cm 0cm 0cm ]{Figure2.jpg}
\caption{Technical schematic of the sample holder with cavity-tuning adjustment controls. The instrument is comprised of the spring-loaded sample tip-tilt plate (STP), the sample (SA), the stepper motor (SM), the mirror housing plate (MHP), the adjustment screw (AS), and the permanent magnet (PM). The external cavity distance (sample-magnet air gap) is labeled $d_{\text{gap}}$. The plate containing the AS is removable which allows the user to flip the permanent magnet to the opposite pole-face and redo experiments with opposite field direction without disturbing the sample alignment.} \label{fig:optical_stage}
\end{figure}
\paragraph{Use of permanent magnet:} A principle design of the sample holder for the tunable cavity-enhanced THz-OHE is shown in Fig.~\ref{fig:optical_stage}. In this design, the permanent magnet serves both as mirror as well as for providing the external magnetic field. The mirror properties of the magnet surface must be characterized by THz spectroscopic ellipsometry measurements at multiple angles of incidence prior to its use in the sample stage. The permanent magnet (PM) sits flush inside a hole in the mirror housing plate (MHP). The micrometer adjustment screw (AS) rests inside a brass bushing in the back plate. The rounded tip of the AS is made of ferromagnetic material which attracts the magnet providing synchronous PM-AS movement. The stepper motor (SM) is attached to the back of the setup and is connected to the AS by a flexible bellows shaft coupler. The flexible coupler allows a $d_{\text{gap}}$ range of approximately 0~$\mu$m to 600~$\mu$m. The stepper motor is operated by a commercially available motor controller (Thor Labs Inc.), which uses LabVIEW programming. The minimum $d_{\text{gap}}$ increment is 1.6~$\mu$m, corresponding to one step of the motor. The back plate in Fig.~\ref{fig:optical_stage} is removable and allows the user to flip the magnet to the opposite pole-face and redo experiments without disturbing the sample alignment.
\paragraph{Use of fixed cavity spacer adjustments:} For simplifying the tunable cavity-enhanced sample stage, non-magnetic adhesive spacers can be placed between the sample and the mirror surface dispensing with the need for the stepper motor in Fig.~\ref{fig:optical_stage}. This option is suitable for \textit{in-situ} measurements when limited space is available. However, no tuning of the cavity after sample mounting can be performed.
\paragraph{Use of external electromagnet:} For use of the sample holder with an external electromagnet, the permanent magnet can be replaced by a non-magnetic insert with a THz mirror at the front towards $d_{\mathrm{gap}}$ and the sample backside. The normal reflectance properties of the mirror can be evaluated by performing THz spectroscopic ellipsometry measurements at multiple angles of incidence prior to its use. The external magentic field can be provided by electromagnets, for example, by placing the stage within a Helmholtz coil arrangement.
\section{Data Acquisition and Analysis}\label{Sec:Dataacquisitionandanalysis}
\subsection{Data acquisition}\label{subsec:dataaquisition}
\paragraph{Magnetic field calibration:} The permanent magnet mounted in the sample holder is a high-grade neodymium (N42) magnet. With the use of a permanent magnet the change in magnetic field strength at the sample surface upon variation of $d_{\text{gap}}$ can be substantial. Hence, it is necessary to implement the magnetic field as a function of distance in the optical model. Using a commercially available Hall probe (Lakeshore), the magnetic field is measured at multiple $d_{\text{gap}}$ values. For our instrument, within approximately 1~mm of the magnet surface the field is approximately linear and can be approximated using by:
\begin{equation}
\pm B = \pm[0.55 - (5.1\times10^{-5})\times (d_{\text{gap}}+d_{\text{sub}})]~\text{[T]},
\end{equation}
where the plus and minus sign refers to the two respective pole orientations of the magnet. The parameters $d_{\text{gap}}$ and $d_{\text{sub}}$ are in units of micrometers.
\paragraph{Mirror calibration:} Separate ellipsometry experiments are performed in the mid-infrared spectral range to determine the optical properties of the metallic permanent magnet surface as mirror. Data analysis is performed using the classical Drude model parameters for static resistivity of $\rho = (9.53\pm 0.04) \times 10^{-5}~\Omega$cm and the average-collision time $\tau = (1.43\pm0.08) \times 10^{-16}~$s. These parameters are used here to model-calculate the optical reflectance of the magnet surface for the model analysis in the THz spectral range for the cavity-enhanced measurements. The magnet surface behaves as an ideal metallic ``Drude'' mirror characterized by metal electron carrier scattering time and resistivity, and no magneto-optic polarization coupling occurs because the metal electron effective mass is too large and the mean scattering time is too short in order for the free charge carriers to respond to the external magnetic field producing measurable magnetooptic birefringence.
\paragraph{Mirror-to-ellipsometer alignment:} The mirror surface is aligned first and then the sample is mounted and aligned. The mirror surface is aligned to the ellipsometer's coordinate system by use of a laser diode mounted such that the laser diode beam is parallel to the plane of incidence, perpendicular to the sample surface, and coincides with the center of the THz beam at the sample surface. A gap value $d_{\text{gap}}$ is selected in the middle of the range of values anticipated for experiments. To align the mirror, the alignment laser diode beam is reflected off the mirror surface and the mirror is adjusted until the beam reflects back into the laser aperture. The adjustment is performed by moving the entire stage relative to the ellipsometer system.
\paragraph{Sample-to-mirror alignment:} Once the mirror is aligned, the sample is mounted to the sample tip-tilt plate STP (Fig.~\ref{fig:optical_stage}). STP serves as an adjustable frame to mount the sample. The sample can be mounted via adhesive, for example, or mechanical clamps. STP contains three micrometer screws secured against the MHP by springs, creating a tip-tilt ability. This is necessary to ensure that the sample surface is also aligned to the ellipsometer's coordinate system. The sample surface is aligned using the same alignment laser as for the mirror.
\begin{figure}
\includegraphics[keepaspectratio=true,width=\linewidth, clip, trim=0cm 0cm 0cm 0cm ]{Figure3.pdf}
\caption{Flow chart describing the data acquisition process of the cavity-tuning optical stage and ellipsometer.
\label{fig:flow_chart}}
\end{figure}
\paragraph{Ellipsometry data acquisition:} After mounting the sample stage into the ellipsometer system, data are acquired in a selected spectral range, for selected angles of incidence $\Phi_a$, and gap distance $d_{\mathrm{gap}}$. Figure~\ref{fig:flow_chart} depicts a flow chart describing the data acquisition process. First, the sample-mirror air gap distance $d_{\text{gap}}$ is set. Next, the frequency-domain source frequency is set. Then the polarizer angle is set and the intensity at the detector is recorded. The process is repeated for all polarizer settings as described in Ref.~\onlinecite{KuehneRSI85_2014}, and a Fourier transform of the signal is performed to determine the Fourier coefficients, which are stored and then subject to a regression analysis. As a result, the elements of the upper 3$\times$3 block of the Mueller matrix are obtained. The procedure is repeated for different settings of gap distance, frequency, or angle of incidence, for example. The acquisition process can be repeated with the magnetic field direction reversed, for example, by reverting the magnet direction, or by reverting the currents in external electromagnetic coils. The experiment can also be repeated with a mirror without a magnet for acquisition of field-free ellipsometry data.
\subsection{Data analysis}\label{subsec:dataanalysis}
Data measured by tunable cavity-enhanced THz-OHE are analyzed using model calculations and numerical regression procedures. Multiple data acquisition modes are available, and which will be discussed by examples further below.
\paragraph{Cavity-enhanced data at tunable gap thickness:} Data obtained as a function of gap thickness are compared with calculated data.
\paragraph{Cavity-enhanced data at tunable frequency:} Data obtained at a fixed gap and/or substrate thickness but as a function of frequency are compared with calculated data.
\paragraph{Cavity-enhanced data at tunable frequency and tunable gap thickness:} Data over a 2-dimensional parameter set can be obtained tuning both gap thickness and frequency and are compared with calculated data.
\paragraph{Cavity-enhanced data at magnetic field reversal:} Field-reversal OHE data obtained at opposing magnetic field directions, $\Delta M_{ij} = M_{ij}(B) - M_{ij}(-B)$, are taken and the difference data is compared with calculated data.
\section{Results and Discussion}\label{Sec:Resultsanddiscussion}
Here we discuss two sample systems as examples for the application of the tunable cavity-enhanced OHE. Both samples contain 2DEGs. The characterization of their free charge carrier properties is demonstrated. One sample is comprised of a transistor device structure for a high electron mobility transistor (HEMT) based on group-III nitride semiconductor layer structures. The second sample is an epitaxial graphene sample grown on a silicon carbide substrate.
\subsection{Two-dimensional electron gas characterization in a HEMT device structure}\label{subsec:HEMT}
\subsubsection{Sample structure}\label{sec:samplestructure}
\paragraph{Growth:} The sample investigated is an AlInN/AlN/GaN HEMT structure grown using an AIXTRON 200/4 RF-S metal-organic vapor phase epitaxy system. The HEMT structure consists of a bottom 2~$\mu$m thick undoped GaN buffer layer, a 1~nm thick AlN spacer layer, followed by a 12.3~nm thick Al$_{\text{0.82}}$In$_{\text{0.18}}$N top layer.\cite{DarakchievaJAP08,schoeche2014_phd-dissertation} The substrate is single-side polished \textit{c}-plane sapphire with a nominal thickness of 350~$\mu$m.
\paragraph{Optical sample structure:} All sample constituents are optically uniaxial and the layer interfaces are plane parallel. In a separate experiment, the HEMT structure was investigated using a commercial (J.A. Woollam Co. Inc.) mid-infrared (MIR) ellipsometer from 300-1200~cm$^{-1}$ at $\Phi_a$~=~60$^{\circ}$ and 70$^{\circ}$ at room temperature in order to determine phonon mode parameters of the AlInN top layer. No distinct phonon features are seen in the THz measurements. However, the MIR analysis is used to help determine the dielectric function of the HEMT structure constituents in the THz spectral range. Phonon parameters for the substrate, GaN buffer layer, and AlN spacer layer are taken from Ref. \onlinecite{SchubertPRB61_2000}, \onlinecite{SchocheAPL98_2011}, and \onlinecite{SchocheAPL103_2013}, respectively. The thickness of the AlInN and AlN layers are found by growth rate calculations and not varied in the analysis. The best-match model layer thickness for the GaN layer is $(2.11\pm0.01)~\mu$m. For the AlInN top layer, the best-match model frequency and broadening parameters for the one-mode type E1- and A1-symmetry are
$\omega_{\text{TO},\perp}=(625.4\pm0.8)~\text{cm}^{-1}$, $\omega_{\text{LO},\perp}=(877.8)~\text{cm}^{-1}$, $\gamma_{\perp}=(40.8\pm1.5)~\text{cm}^{-1}$,
$\omega_{\text{TO},\parallel}=(610)~\text{cm}^{-1}$, $\omega_{\text{LO},\parallel}=(847.8\pm0.4)~\text{cm}^{-1}$, $\gamma_{\parallel}=(11.3\pm0.4)~\text{cm}^{-1}$
which are in good agreement with previous works\cite{KasicPSSC_2003}. Note, certain phonon parameters are functionalized according to Ref. \onlinecite{KasicPSSC_2003}, and were not varied in the analysis. In order to obtain an excellent match between experimental and model-calculated THz-OHE data, a low-mobility electron channel was included in the AlInN top layer. This same low-mobility channel was also included in our previous model analysis for the same HEMT structure.\cite{schoeche2014_phd-dissertation,KnightOL40_2015} A mobility value of $\mu =50~\text{cm}^{2}/\text{Vs}$ for a similar HEMT structure is adopted for this sample, and an effective mass parameter of 0.3~$m_{0}$ is taken from density function calculations in Ref.~\onlinecite{tsai2019structural}. The volume density value for the low-mobility channel was previously determined to be $N = 1.02 \times 10^{20}~\text{cm}^{-3}$. This value is not varied in our analysis.
\paragraph{Previous OHE characterization:} In Ref.~\onlinecite{schoeche2014_phd-dissertation} we reported field-reversal high-field OHE measurements on the same HEMT sample without external cavity. The high field measurements were performed in a cryogenic superconducting magnet setup. In Ref.~\onlinecite{KnightOL40_2015} we reported field-reversal cavity-enhanced OHE measurements using a permanent magnet and various adhesive spacers (discrete settings for $d_{\mathrm{gap}}$) on the same sample. The results reported in this work are in excellent agreement with those reported previously. All THz-OHE results further compare well with Hall effect and C-V measurements done on similar samples.\cite{GonschorekJAP103_2008,GonschorekAPL89_2006}
\subsubsection{Single-frequency tunable-cavity measurements}
\begin{figure}
\includegraphics[keepaspectratio=true,width=\linewidth, clip, trim=0cm 0cm 0cm 0cm ]{Figure4.pdf}
\caption{Experimental (green symbols) and best-match model calculated data (red solid lines) field-reversal cavity-enhanced OHE data as a function of the external cavity thickness $d_{\mathrm{gap}}$, at two frequencies for a HEMT layer structure on sapphire. Closed and open triangles represent $\Delta M_{\text{13}}$ and $\Delta M_{\text{31}}$ respectively. Closed and open squares represent $\Delta M_{\text{23}}$ and $\Delta M_{\text{32}}$ respectively. The blue solid lines are model-calculated data for the case of no cavity-enhancement ($d_{\text{gap}}=\infty$, $d_{\text{sub}}=\infty$). All data is obtained at angle of incidence $\Phi_a = 45^{\circ}$ and at room temperature.
\label{fig:line_scans}}
\end{figure}
Figure~\ref{fig:line_scans} shows experimental and best-match model data for field-reversal cavity-enhanced OHE data as a function of $d_{\text{gap}}$ for the HEMT sample. The experiment was performed at two different, fixed frequencies of $\nu$~=~860~GHz and 880~GHz for a $d_{\text{gap}}$ range of 120~$\mu$m to 520~$\mu$m in increments of 3~$\mu$m. The experimental data for both frequencies are analyzed simultaneously. The layer stack optical model for the best-match model calculation is AlInN/AlN/GaN/sapphire substrate/external cavity/mirror (magnet surface). The external magnetic field is oriented normal to the sample surface. All off-block diagonal Mueller matrix elements are zero for the HEMT sample structure without external magnetic field. To begin with, the solid blue lines in Figs.~\ref{fig:line_scans}(a,b) are model calculated data for the same HEMT structure in the absence of the cavity enhancement (Fig.~\ref{fig:Sample-cavity_schematic}(e), $d_{\text{sub}} = \infty$, $d_{\text{gap}} = \infty$), where we assumed that the field at the layer stack is $B=\pm 0.55$~T. Specifically, $\Delta M_{\text{13}}=\Delta M_{\text{31}}=-0.0004$ and $\Delta M_{\text{23}}=\Delta M_{\text{32}}=0.004$. Data are below our current instrumental uncertainty limit for the individual Mueller matrix elements of $\delta M_{ij} \approx \pm 0.01$. Hence, the 2DEG within the HEMT layer structure would not be detectable.
In the cavity-enhanced mode, however, large off-diagonal Mueller matrix elements appear, far above the current instrumental uncertainty level, upon variation of the gap thickness $d_{\text{gap}}$. Features in Figs.~\ref{fig:line_scans}(a,b) are due to Fabry-P\'{e}rot interference enhanced cross-polarized field components after reflection at the layer stack. Minima and maxima occur as a function of gap thickness. The cross polarization is produced only by the free charge carrier gas within the HEMT structure under the influence of the Lorentz force. The potential to use the variation of gap thickness as a new parameter variation measurement configuration is obvious. In particular, for this sample, and for the scans shown in Figs.~\ref{fig:line_scans}(a,b), compared with the blue lines (no cavity enhancement) cavity thickness parameters can be adjusted where the OHE signal enhancement reaches -0.124 at $\nu$=860 GHz in Fig.~\ref{fig:line_scans}(a) for $\Delta M_{\text{13,31}}$ and 0.088 for $\nu$=880 GHz in Fig.~\ref{fig:line_scans}(b) for $\Delta M_{\text{23,32}}$. The best-match model 2DEG sheet density, mobility, and effective mass parameters obtained from the OHE data in Figs.~\ref{fig:line_scans}(a,b) are $N_\text{s} = (1.23\pm0.13) \times 10^{13}~\text{cm}^{-2}$, $\mu = (1245\pm64)~\text{cm}^{2}/\text{Vs}$, $m^{\ast} = (0.272\pm0.013) m_{0}$, respectively. The results are in excellent agreement with electrical measurements and with our previous THz-OHE experiments.\cite{KnightOL40_2015}
\subsubsection{Tunable-frequency tunable-cavity measurements}
\begin{figure}
\includegraphics[keepaspectratio=true,width=\linewidth, clip, trim=0cm 0cm 0cm 0cm ]{Figure5.jpg}
\caption{False-color three-dimensional surface rendering of model-calculated cavity-enhanced field-reversal THz-OHE data for an AlInN/AlN/GaN HEMT structure grown on a sapphire substrate as functions of frequency $\nu$, external cavity distance $d_{\text{gap}}$, and angle of incidence $\Phi_a$. Data for $\Delta M_{13,31} = M_{13,31}(+B) - M_{13,31}(-B)$ are shown as example. Values within the range of -0.02 to 0.02 are omitted for clarity. The green horizontal plane at $\Phi_a = 45^{\circ}$ indicates the instrumental settings for the angle of incidence in this work. Model parameters given in text. Note that the model calculated plot for $\Delta M_{23,32}$ is similar in appearance to $\Delta M_{13,31}$ and is excluded here for brevity.
\label{fig:M13-31_3D}}
\end{figure}
\begin{figure*}
\includegraphics[keepaspectratio=true,width=\linewidth, clip, trim=0cm 0.2cm 0cm 0.2cm ]{Figure6.pdf}
\caption{(a) False-color two-dimensional surface rendering of experimental (left two columns) and model-calculated (right column) cavity-enhanced field-reversal THz-OHE data for an AlInN/AlN/GaN HEMT structure grown on a sapphire substrate as functions of frequency $\nu$ and external cavity distance $d_{\text{gap}}$. The angle of incidence is $\Phi_a = 45^{\circ}$. Data for $\Delta M_{13,31} = M_{13,31}(+B) - M_{13,31}(-B)$ are shown as example. Parameter details given in text. (b) Same as in (a) for fixed cavity thickness $d_{\text{gap}}$~=~420~$\mu$m. The solid blue line indicates model calculated data when $d_{\text{sub}}=\infty$ and $d_{\text{gap}}=\infty$. Solid green symbols indicate $\Delta M_{\text{13}}$ and $\Delta M_{\text{23}}$, and open symbols indicate $\Delta M_{\text{31}}$ and $\Delta M_{\text{32}}$. The blue solid lines are model-calculated data for the case of no cavity-enhancement ($d_{\text{gap}}=\infty$, $d_{\text{sub}}=\infty$). All data taken at room temperature. \label{fig:OHE_data}}
\end{figure*}
False-color rendering of model calculated data of cavity-enhanced field-reversal THz-OHE data versus frequency, gap thickness, and angle of incidence are shown in Fig.~\ref{fig:M13-31_3D}. The color type indicates positive or negative values, the color intensity indicates the magnitude of the OHE data. The three-dimensional rendering is insightful as it indicates distinct regions within which the data rapidly switches signs, and regions within which data takes very large values. All three parameters, frequency, gap thickness, and angle of incidence influence the OHE data, and proper selection may result in strong OHE data, while poor choices may result in disappearance of the OHE data. The horizontal plane indicated at 45$^{\circ}$ identifies the angle of incidence at which experiments are performed in this work. False-color rendering of experimental and model-calculated data at angle of incidence of $45^{\circ}$ are shown in Fig.~\ref{fig:OHE_data}(a) as a function of frequency and gap thickness. An excellent agreement between both experiment and model calculation is obtained. Fig.~\ref{fig:OHE_data}(b) shows data at a fixed cavity thickness. Data are similar to those in Figs.~\ref{fig:line_scans}(a,b), except now the frequency is tuned. Figs.~\ref{fig:OHE_data}(a,b) identify frequency and gap regions where the OHE data is very small. The blue solid lines are identical to those in Figs.~\ref{fig:line_scans}(a,b) for the case of no cavity-enhancement. All experimental data in Fig.~\ref{fig:OHE_data} is analyzed simultaneously and the resulting best-model sheet density, mobility, and effective mass parameters for the 2DEG are $N_\text{s} = (1.22\pm0.12) \times 10^{13}~\text{cm}^{-2}$, $\mu = (1262\pm59)~\text{cm}^{2}/\text{Vs}$, $m^{\ast} = (0.268\pm0.012) m_{0}$. The results are identical within the error bars to those obtained from the single frequency gap thickness scans as well as to our previous OHE investigation reports.\cite{KuehneRSI85_2014,KnightOL40_2015}
The central results obtained from this section are (i) the demonstration of the strong enhancement obtained by use of multiple interference through substrate and external cavities, and (ii) the obvious appearance from Fig.~\ref{fig:M13-31_3D} and the support from our experiments that OHE measurements can be performed both as a function of frequency and gap thickness.
\subsection{Environmental gas doping characterization in epitaxial graphene}\label{subsec:gasdoping}
In this section we demonstrate the use of the cavity-enhanced OHE method for detecting changes in the properties of a 2DEG upon exposure to various external gas compositions. The purpose of this section is to demonstrate the use of this method when transient physical changes to a sample limit the time durations during which spectroscopic scanning measurements can be performed.
\subsubsection{Sample structure}
\paragraph{Growth:} The sample studied here is graphene epitaxially grown on Si-face (0001) 4H-SiC by high-T sublimation in Ar atmosphere.\cite{knight2017situ} Reflectivity and low-energy electron microscopy mapping, and scan lines verify the primarily one monolayer coverage across the 10$\times$10 mm sample surface.
\paragraph{Optical sample structure:} The sample is optically modeled by considering the graphene monolayer as a 1~nm highly conductive thin film on top of the SiC substrate as described in Ref.~\onlinecite{knight2017situ}. All sample constituents have plane parallel interfaces. No free charge carriers are detected in the SiC substrate. Due to the ultrathin layer thickness of the graphene, ellipsometry data cannot differentiate between the thickness and the dielectric function of the layer. Instead, a new parameter emerges, the sheet free charge carrier density, This parameter takes a constant ratio with the assumed layer thickness, and hence can be determined independently and accurately (Further details are discussed in Refs.~\onlinecite{knight2017situ, SchubertJOSAOHE2016}).
\paragraph{\textit{In-situ} gas cell design:} The schematics of the \textit{in-situ} gas flow cell used in this work is shown in Fig.~\ref{fig:gas-cell-experiments}(a). The THz ellipsometer instrument is schematically indicated by source, polarizer, analyzer, and detector at angle of incidence $\Phi_a$~=~45$^{\circ}$.\cite{KuehneIEEETHz2017} The cell is equipped with a humidity and temperature sensor, gas inlets, and gas outlets. The side walls of the flow cell are made from Delrin, and the cover and base portions are made from acrylic. THz-transparent windows are produced from homopolymer polypropylene. The thickness of the transparent sheets is 0.27~mm. Normal ambient gas is pushed through the cell using a vacuum pump (Linicon). Nitrogen and helium flow was provided by additional purge lines. The background pressure in the cell was 1~atm throughout the experiment. The flow rate was 0.5~liters/minute.
\paragraph{Fixed cavity-enhancement settings:} The sample consists of a 2DEG (graphene) at the surface of a THz-transparent substrate ($d_{\mathrm{sub}}$~=~355~$\mu$m). The substrate is placed with its backside using adhesive spacers onto the permanent magnet (Fig.~\ref{fig:gas-cell-experiments}(a)). The sample is mounted with the neodymium (N42) magnet into the gas cell. The gap thickness $d_{\text{gap}}$ was thereby fixed at 100~$\mu$m. OHE data acquisition is identical to the procedure in Fig.~\ref{fig:flow_chart} with fixed cavity thickness.
\subsubsection{Cavity-enhanced optical Hall effect simulations}
\begin{figure}
\includegraphics[keepaspectratio=true,width=\linewidth, clip, trim=0cm 0cm 0cm 0cm ]{Figure7.jpg}
\caption{False-color three dimensional surface rendering of model-calculated cavity-enhanced single-field THz-OHE data for an epitaxial graphene layer on SiC as a function of frequency $\nu$, external cavity distance $d_{\text{gap}}$, and angle of incidence $\Phi_a$. Data for $M_{23}(+B)$ are shown only (0.02 to 0 (red) and -0.04 to -0.17 (blue)). Data with values between -0.04 and 0 are omitted for clarity. The black sphere at the intersection of the three black lines illustrates the point where \textit{in-situ} gas cell data was taken for the gas flow experiments on the sample shown on this work. For model calculations the following parameters are used: $N_\text{s} = 8.69\times10^{11}~\text{cm}^{-2}$ and $\mu = 2550~\text{cm}^{2}/\text{Vs}$. The effective mass parameter of $m^{\ast}$~=~0.019~$m_0$ is calculated as a function of $N_\text{s}$ as in Ref.~\onlinecite{novoselov2005two}.}
\label{fig:graphene_m23_32_4d}
\end{figure}
Figure~\ref{fig:graphene_m23_32_4d} shows model-calculated THz-OHE data for $M_{23}$ as function of $\nu$, $d_{\text{gap}}$, and $\Phi_a$ for epitaxial graphene grown on SiC. Figure~\ref{fig:graphene_m23_32_4d} can be used as a guide to find optimal values for $\nu$, $d_{\text{gap}}$, and $\Phi_a$ to perform a THz-OHE measurement. The black sphere and three intersecting lines illustrates the point chosen to perform the \textit{in-situ} THz-OHE measurement. Only a single set of measurement parameters is chosen to minimize time between measurements in order to resolve sharp dynamic changes in the Mueller matrix data during the gas flow experiment. For practical reasons, $d_{\text{gap}} = 100~\mu$m and $\Phi_a = 45^{\circ}$ were chosen. Therefore, $\nu$ = 428~GHz was selected for the \textit{in-situ} measurement. Unlike Fig.~\ref{fig:M13-31_3D}, Fig.~\ref{fig:graphene_m23_32_4d} is not THz-OHE difference data since the gas flow experiment is only performed using the north pole-face of the permanent magnet.
\subsubsection{\textit{In-situ} tunable-frequency single-cavity measurements}
Figures~\ref{fig:gas-cell-experiments}(c,d) depict experimental and best-match model data for single-field-orientation cavity-enhanced OHE data as a function of frequency for the graphene sample. These measurements were performed at two different points during the gas exposure experiment: the first spectral measurement was after three hours of exposure to helium, and the subsequent spectral measurement after two hours of exposure to ambient air (labeled `T$_{1}$' and `T$_{2}$' in Fig.~\ref{fig:gas-cell-experiments}(b), respectively). Indicated in Figs.~\ref{fig:gas-cell-experiments}(c,d) are also the THz-OHE data without cavity enhancement (blue lines: $d_{\text{gap,sub}} = \infty$). Comparing the cavity-enhanced data to the case of no cavity-enhancement indicates large changes in the Mueller matrix data are entirely due to interference enhancement in the substrate and external cavity. The off-block-diagonal element $M_{23}$ is selected to show the OHE signature enhancement.
\begin{figure*}
\includegraphics[keepaspectratio=true,width=\linewidth, clip, trim=0cm 0cm 0cm 0cm ]{Figure8.pdf}
\caption{\textit{In-situ} cavity-enhanced THz-OHE gas flow experiment and results. Contents of Figs.~\ref{fig:gas-cell-experiments}(a,b) are adapted from Ref.~\onlinecite{knight2017situ} which is licensed under a Creative Commons Attribution 4.0 International License\cite{creative_commons_license_url}. Gas cell schematic (a) for gas exposure experiment on epitaxial graphene grown on SiC. In panel (b), \textit{in-situ} experimental (open triangles) and best-match model data (solid red line) at single frequency ($\nu$ = 428 GHz) for two selected Mueller matrix elements are shown. Normalized $M_{\text{12}}$ (c) and $M_{\text{23}}$ (d) spectra are shown at two different times during the gas exposure experiment (labeled `T$_{1}$' and `T$_{2}$') for the cases with the cavity-enhancement effect ($d_{\text{gap}} = 100~\mu$m and $d_{\text{sub}} = 355~\mu$m) and without ($d_{\text{gap,sub}} = \infty$). In panels (c) and (d) experimental data is shown as open and closed squares, and model-calculated data are solid lines. Note, the experiment is performed only on one side of the permanent magnet (north pole-face) and thus Mueller matrix difference data is not obtained.
\label{fig:gas-cell-experiments}}
\end{figure*}
\subsubsection{\textit{In-situ} time-dependent single-frequency single-cavity measurements}
Figure~\ref{fig:gas-cell-experiments}(b) shows the \textit{in-situ} cavity-enhanced THz-OHE data taken at a single frequency, $\nu$ = 428 GHz, for $M_{12}$ and $M_{23}$. Analyzing the data allows the extraction of the graphene's free charge carrier properties $N_\text{s}$, $\mu$, and charge carrier type as a function of time. The carrier type is determined to be \textit{n}-type during each gas phase. It is found that $N_\text{s}$ increases with helium and nitrogen exposure, and decreases with air exposure. An inverse relationship is observed for $\mu$ and $N_\text{s}$ throughout the gas flow experiment. The lowest $N_\text{s}$ occurred at the end of the second air phase where $N_\text{s} = (8.40\pm0.72) \times 10^{11}~\text{cm}^{-2}$ and $\mu = (2595\pm217)~\text{cm}^{2}/\text{Vs}$. The highest $N_\text{s}$ occurred at the end of the helium phase where $N_\text{s} = (2.31\pm0.29) \times 10^{12}~\text{cm}^{-2}$ and $\mu = (1961\pm250)~\text{cm}^{2}/\text{Vs}$. Further details on the \textit{in-situ} THz-OHE gas exposure experiment can be found in our previous publication.\cite{knight2017situ} Sensitivity to the free charge carrier properties as a function of gas flow is entirely dependent on the cavity-enhancement effect, as demonstrated in Figs.~\ref{fig:gas-cell-experiments}(c,d). The variations in free charge carrier properties upon exposure with He and Air, indicated by the blue lines in Figs.~\ref{fig:gas-cell-experiments}(c,d), would not have been detectable with the same instrument without the external cavity stage since the changes in Mueller matrix elements are below the detection limit.
\section{Conclusion}
We demonstrated a tunable cavity-enhanced THz frequency-domain OHE technique to extract the free charge carrier properties of 2DEG layers situated on top of THz-transparent substrates. A HEMT structure grown on sapphire and epitaxial graphene grown on SiC are studied as examples. For the HEMT structure sample, the OHE signatures are enhanced by tuning an externally coupled Fabry-P\'{e}rot cavity via stepper motor. Data measured as a function of external cavity size and frequency are analyzed to obtain the carrier concentration, mobility, and effective mass parameters of the 2DEG located within the HEMT structure. For the epitaxial graphene on SiC sample, an external cavity of fixed size is used to enhance the OHE signal during a gas flow experiment. This enhancement effect allows the extraction of the graphene's carrier concentration and mobility as a function of time throughout the experiment. In our experiments, the Fabry-P\'{e}rot cavity-enhancement is made possible by the THz-transparent substrates as well as the external cavity (air gap) between the sample's backside and the reflective metal surface. The magnetic field necessary for the OHE experiments is provided by a permanent magnet; for which the metallic coating also provides the reflective surface for the external cavity. Our enhancement technique can be expanded upon by using superconducting magnets to measure samples with much lower free charge carrier contributions. In general, our technique is a powerful method for materials characterization, and can be used to study even more complex sample structures.
\section{Acknowledgments}
We thank Dr.~Craig~M.~Herzinger for helpful discussions. Prof. Nikolas Grandjean is gratefully acknowledged for providing the AlInN/GaN HEMT structure. We thank Dr. Chamseddine Bouhafs and Dr. Vallery Stanishev for growing the graphene sample, and Prof. Rositsa Yakimova for providing access to her sublimation facility for graphene growth. This work was supported in part by the National Science Foundation under award DMR 1808715, by Air Force Office of Scientific Research under award FA9550-18-1-0360, by the Knut and Alice Wallenbergs Foundation supported grant 'Wide-bandgap semiconductors for next generation quantum components', by the Swedish Agency for Innovation Systems under the Competence Center Program Grant No. 2016-05190 and 2014-04712, Swedish Foundation for Strategic Research (RFI14-055, EM16-0024), Swedish Research Council (2016-00889), Swedish Government Strategic Research Area in Materials Science on Functional Materials at Link\"oping University (Faculty Grant SFO Mat LiU No. 2009 00971), and the J.A. Woollam Foundation.
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\section{Proof of Lemma \ref{submodular reduction}}\label{appendix-submodular-reduction}
\begin{proof}
Assume that all weights $w_i$ are integers. We reduce our weighted problem with monotone submodular utility function $u$ to the unweighted one as follows.
\begin{itemize}
\item For each item $i\in[n]$ we consider new $w_i$ items of unit weight.
Denote them as $i_j$ for $j\in[w_i]$ and call $i$ to be the type of the
unit $i_j$.
\item We make the new valuation function $\nu$ depends only on the amounts of
unit items of each type.
\item Let a set $S$ contains $a_i$ units of each type $i$. Independently
for each type pick at random in the set $\calR$ with probability
$\frac{a_i}{w_i}$ weighted item $i$. Define $\nu(S)=E(u(\calR))$.
\end{itemize}
Therefore
$$\nu(S)=\frac{1}{w_1\cdot\ldots\cdot w_n}\sum_{\pi}u\left(S\cdot \pi\right)$$
where $\pi$ is a sampling of units one for each type (there are $w_1\cdot\ldots\cdot w_n$ variants for $\pi$); $S\cdot \pi$ is a vector of types at which $\pi$ hits $S$.
Using this formula it is not hard to verify monotonicity and submodularity of $\nu$. Indeed, e.g. to verify submodularity one only need to check that the marginal contribution of any unit is smaller for a large set, i.e. for $S\subset T$ and $i_j\notin T$ verify inequality $\nu(S\cap\{i_j\})-\nu(S)\ge\nu(T\cap\{i_j\})-\nu(T)$, which is pretty straightforward.
For any $T\subseteq [n]$ if we consider a set of units $S=\{i_k|i\in T, 1\le k\le w_i\}$, then according to the definition $\nu(S)=u(T)$. Hence, optimal solution to the unit weights problem is large or equal than the optimal solution to the original problem.
To conclude the proof it is only left to show that the greedy scheme for the unit weights gives us the same result as our fractional greedy scheme for an integer weights. Note that once we have taken a unit of type $i$ we will proceed to take units of type $i$ until exhaust it completely (we brake ties in favor to the last type we have picked). Indeed, let $i_k, i_{k+1}\notin S$ then
\begin{eqnarray*}
\nu(S\cup\{i_k\})-\nu(S) &=& \nu(S\cup\{i_{k+1}\})-\nu(S) \\
&=& \frac{1}{w_1\cdot\ldots\cdot w_n}\sum_{\{\pi|i_{k+1}\in\pi\}} u\left(S\cup\{i_{k+1}\}\cdot\pi\right)-u\left(S\cdot\pi\right) \\
&=& \frac{1}{w_1\cdot\ldots\cdot w_n}\sum_{\{\pi|i_{k+1}\in\pi\}} u\left(S\cup\{i_k,i_{k+1}\}\cdot\pi\right)-u\left(S\cup \{i_k\}\cdot\pi\right) \\
&=& \nu(S\cup\{i_k,i_{k+1}\})-\nu(S\cup\{i_k\})
\end{eqnarray*}
Therefore, marginal contribution of the type $i$ does not decrease if we include in the solution units of type $i$. On the other hand, because $\nu$ is submodular, marginal contribution of any other type can not increase. So we will take unit $i_{k+1}$ right after $i_k$.
Assume we already have picked set $S$ and now are picking the first unit of a type $i$. Hence, $S$ comprises all units of a type set $T$. Then we have
\[\nu\left(S\cup\{i_1\}\right)-\nu\left(S\right)=
\frac{1}{\prod_{k=1}^{n}w_k}\sum_{\{\pi|i_{1}\in\pi\}} u\left(S\cup\{i_1\}\cdot\pi\right)-u\left(S\cdot\pi\right)
=\frac{\prod_{k\neq i}w_k}{\prod_{k=1}^{n}w_k}m_{T}(i)=\frac{m_T(i)}{w_i}\]
Thus $i=argmax_{i\notin T}\frac{m_T(i)}{w_i}$ which coincides with the rule of our fractional greedy scheme.
In case of real weights the same approach can be applied but in more tedious way.
\end{proof}
\section{Mechanisms for Knapsack}\label{appendix-knapsack}
In this section, we describe our deterministic and randomized mechanisms for knapsack, yielding a proof for Theorem~\ref{theorem-knapsack-mechanism}.
\subsection{Deterministic Mechanism}
We consider the following greedy strategy studied by Singer~\cite{PS10}.
\begin{center}
\small{}\tt{} \fbox{
\parbox{4.0in}{
\hspace{0.05in} \\[-0.05in] $\gre(A)$
\begin{enumerate}
\item Order all items in $A$ s.t. $\frac{v_1}{c_1}\ge \frac{v_2}{c_2}\ge \cdots \ge \frac{v_{|A|}}{c_{|A|}}$
\item Let $k=1$ and $S=\emptyset$
\item While $k\le |A|$ and $c_k\le B\cdot \frac{v_k}{\sum_{i\in S\cup\{k\}}v_i}$
\begin{itemize}
\item $S \leftarrow S\cup \{k\}$
\item $k\leftarrow k+1$
\end{itemize}
\item Return winning set $S$
\end{enumerate}
}}
\end{center}
It is shown that the above greedy strategy is monotone (and therefore truthful).
Actually, it has the following remarkable property: any $i\in S$ cannot control the output set given that $i$
is guaranteed to be a winner. That is, if the winning sets are $S$ and $S'$ when $i$ bids $c_i$ and $c'_i$,
respectively, where $i\in S\cap S'$, then $S=S'$. Otherwise, consider the item $i_0\notin S\cap S'$ with the smallest index;
assume without loss of generality that $i_0\in S\setminus S'$. Let $T = \{j\in S\cap S'~|~j<i_0, j\neq i\}$ be the winning
items in $S\cap S'\setminus \{i\}$ before $i_0$. Then
$c_{i_0}\le B\cdot \frac{v_{i_0}}{\sum_{j\in S}v_j} \le B\cdot \frac{v_{i_0}}{\sum_{j\in T}v_j + v_i+v_{i_0}}$, which implies that
$i_0$ should be a winner in $S'$ as well, a contradiction.
Given the greedy strategy described above, our mechanism for
knapsack is as follows (where $fopt(A)$ denotes the value of the
optimal fractional solution; for knapsack it can be computed in polynomial time).
\begin{center}
\small{}\tt{} \fbox{
\parbox{4.0in}{
\hspace{0.05in} \\[-0.05in] \Mknapsack
\begin{enumerate}
\item Let $A=\{i~|~c_i\le B\}$ and $i^*\in \arg\max_{i\in A} v_i$
\item If $(1+\sqrt{2})\cdot v_{i^*} \ge fopt(A\setminus \{i^*\})$, return $i^*$
\item Otherwise, return $S= \gre(A)$
\end{enumerate}
}}
\end{center}
\begin{theorem}\label{th_M_knapsack}
\Mknapsack\ is a $2+\sqrt{2}$ approximation budget feasible truthful
mechanism for knapsack.
\end{theorem}
\begin{proof}
The proof consists of each property stated in the claim.
\begin{itemize}
\item {\em Truthfulness.} We analyze monotonicity of the mechanism according to the condition of Step~2 and 3, respectively. If $i^*$ wins in Step~2 (note that the fractional optimal value computed in Step~2 is independent of the bid of $i^*$), then $i^*$ still wins if he decreases his bid.
If the condition in Step~2 fails and the mechanism runs Step~3, for any $i\in S$, the subset $S$ remains the same if $i$ decreases his bid. Note that if $i\neq i^*$, when $i$ decreases his bid, the value of the fractional optimal solution computed in Step~2 will not decrease. Hence $i$ is still a winner, which implies that the mechanism is monotone.
\item {\em Individual rationality and budget feasibility.} If $i^*$ wins in Step~2, his payment is the threshold bid $B$. Otherwise, assume that all buyers in $A$ are ordered by $1,2,\ldots,n$; let $S=\{1,\ldots,k\}$. Note that it is possible that $i^*\in S$. For any $i\in S$, let $q_i$ be the maximum value that $i$ can bid such that the fractional optimal value on instance $A\setminus \{i^*\}$ is still larger than $v_{i^*}$. Note that $c_i\le q_i$.
The payment to any winner $i\in S\setminus \{i^*\}$ is
$p_i=\min\left\{v_i\cdot \frac{c_{k+1}}{v_{k+1}},B\cdot \frac{v_i}{\sum_{j\in S}v_j}, q_i\right\}$,
and $p_{i^*}=\min\left\{v_{i^*}\cdot \frac{c_{k+1}}{v_{k+1}},B\cdot \frac{v_{i^*}}{\sum_{j\in S}v_j}\right\}$ if $i^*\in S$.
It can be seen that the mechanism is individually rational. Further,
$\sum_{i\in S}p_i \le \sum_{i\in S} B\cdot \frac{v_i}{\sum_{j\in S}v_j} = B$, which implies that the mechanism is budget feasible.
\item {\em Approximation.} Assume that all buyers in $A$ are ordered by $1,2,\ldots,n$, and $T=\{1,\ldots,k\}$ is the subset returned by $\gre(A)$. Let $\ell$ be the maximal item for which $\sum_{i=1,\ldots,\ell}c_i\le B$. Let $c'_{\ell+1}=B-\sum_{i=1,\ldots,\ell}c_i$ and $v'_{\ell+1}=v_{\ell+1}\cdot \frac{c'_{\ell+1}}{c_{\ell+1}}$. Hence, the optimal fractional solution is \[fopt(A)=\sum_{i=1}^{\ell}v_i + v'_{\ell+1}\]
For any $j=k+1,\ldots,\ell$, we have $\frac{c_j}{v_j}\ge \frac{c_{k+1}}{v_{k+1}} > \frac{1}{v_{k+1}}\cdot B\cdot \frac{v_{k+1}}{\sum_{i=1}^{k+1}v_i}$, where the last inequality follows from the fact that the greedy strategy stops at item $k+1$. Hence, $c_j>B\cdot \frac{v_j}{\sum_{i=1}^{k+1}v_i}$. Same analysis shows $c'_{\ell+1}>B\cdot \frac{v'_{\ell+1}}{\sum_{i=1}^{k+1}v_i}$. Therefore,
$B\cdot \frac{\sum_{j=k+1}^{\ell}v_j+v'_{\ell+1}}{\sum_{i=1}^{k+1}v_i}<\sum_{j=k+1}^{\ell}c_j+c'_{\ell+1}<B$,
which implies that $\sum_{i=1}^{k}v_i > \sum_{j=k+2}^{\ell}v_j+v'_{\ell+1}$. Hence,
\[fopt(A)=\sum_{i=1}^{\ell}v_i + v'_{\ell+1} < 2\sum_{i\in S}v_i + v_{i^*}\]
A basic observation of the mechanism is that
\[fopt(A)-v_{i^*} \le fopt(A\setminus\{i^*\})\le fopt(A)\]
Hence, if the condition in Step~2 holds and the mechanism outputs $i^*$, we have
\[ fopt(A) \le fopt(A\setminus \{i^*\}) + v_{i^*}\le (2+\sqrt{2})\cdot v_{i^*}\]
If the condition in Step~3 fails and the mechanism outputs $S$ in Step~4, we have
\[(1+\sqrt{2})\cdot v_{i^*} < fopt(A\setminus\{i^*\})\le fopt(A) < 2\sum_{i\in S}v_i + v_{i^*}\]
which implies that $v_{i^*} < \sqrt{2}\cdot \sum_{i\in S}v_i$. Hence,
\[opt\le fopt(A)=\sum_{i=1,\ldots,\ell}v_i + v'_{\ell+1} < 2\sum_{i\in S}v_i + v_{i^*}\le (2+\sqrt{2})\cdot \sum_{i\in S}v_i.\]
Therefore, the mechanism is $(2+\sqrt{2})$ approximation.
\end{itemize}
\end{proof}
\subsection{Randomized Mechanism}
Our randomized mechanism for knapsack is as follows.
\begin{center}
\small{}\tt{} \fbox{
\parbox{3.5in}{
\hspace{0.05in} \\[-0.05in] \RMknapsack
\begin{enumerate}
\item Let $A=\{i~|~c_i\le B\}$ and $i^*\in \arg\max_{i\in A} v_i$
\item With probability $\frac{1}{3}$, return $i^*$
\item With probability $\frac{2}{3}$, return $\gre(A)$
\end{enumerate}
}}
\end{center}
\begin{theorem}\label{theorem-random-knapsack}
\label{th_RM_knapsack}
\RMknapsack\ is a $3$ approximation universal truthful budget feasible
mechanism for knapsack.
\end{theorem}
\begin{proof}
Since both mechanisms in Step~2 and 3 are budget feasible and truthful, it is left only to prove approximation ratio.
Using the same notation and argument in the proof of Theorem~\ref{th_M_knapsack}, assume that all buyers in $A$ are ordered by $1,2,\ldots,n$, and $T=\{1,\ldots,k\}$ is the subset returned by $\gre(A)$. Let $\ell$ be the maximal item for which $\sum_{i=1,\ldots,\ell}c_i\le B$. Let $c'_{\ell+1}=B-\sum_{i=1,\ldots,\ell}c_i$ and $v'_{\ell+1}=c'_{\ell+1}\cdot \frac{v_{\ell+1}}{c_{\ell+1}}$. Hence, the optimal fractional solution is \[fopt(A)=\sum_{i=1}^{\ell}v_i + v'_{\ell+1}\]
and
\[fopt(A)=\sum_{i=1}^{\ell}v_i + v'_{\ell+1} < v_{i^*} + 2\sum_{i\in S}v_i.\]
The excepted value of \RMknapsack\ is therefore
\[\frac{1}{3}v_{i^*} + \frac{2}{3}\sum_{i\in S}v_i = \frac{1}{3}\Big(v_{i^*} + 2\sum_{i\in S}v_i\Big) > \frac{1}{3} opt \]
which completes the proof.
\end{proof}
\section{Knapsack with Heterogeneous Items}\label{appendix-heterogeneous}
In this section we analyze heterogeneous knapsack problem and $\greH$, which leads to a proof of Theorem~\ref{theorem-star-knapsack-mechanism}.
\subsection{Optimal Fractional Solution}
We start our study again on fractional solutions to the optimization problem. First we have to define what is a fractional relaxation for heterogeneous knapsack or more precisely what is a feasible fractional solution.
\begin{defi}
A feasible solution for heterogeneous knapsack is an $n$-tuple of real numbers $(\alpha_1,\ldots,\alpha_n)\in [0,1]^n$ satisfying $\sum_{i=1}^{n}\alpha_i c_i \le B$ and $\sum_{i\in t^{-}_j}\alpha_i \le 1$ for any $j\in [m]$. An optimal fractional solution is a feasible solution that maximizes $\sum_{i=1}^{n}\alpha_i v_i$.
\end{defi}
We have the following observation on optimal solution.
\begin{lemma}
\label{lemma_HK}
For a given budget $B$ we can pick an optimal fractional solution $f_{OPT}$ such that
\begin{itemize}
\item there are at most two nonzero amounts of items of any type in $f_{OPT}$. \item there is exactly one item of any
type in $f_{OPT}$ except maybe only for one type.
\end{itemize}
\end{lemma}
\begin{proof}
Consider any optimal solution $f^{'}_{OPT}$. Fix the price $p_j$ spent on the particular type $j$ in it. We can use only two items of type $j$ in order to provide the maximum value for the price $p_j$. Indeed, if one draws all items of type $j$ in the plain with $x$-coordinate corresponding to the cost and $y$-coordinate corresponding to the value of an item together with the point $(0,0)$, then the condition $\sum_{i\in t^{-}_j}\alpha_i \le 1$ will describe a point in the convex hull of the drawn set.
Thus we can take $f_{OPT}$ with at most two items of a type and derive the first part of the lemma.
One can derive the second part of the lemma by changing $p_{j_1}$ and $p_{j_2}$ in $f_{OPT}$ such that $p_{j_1}+p_{j_2}$ remains constant. Indeed,
appealing to the picture again, we consider two convex polygons $P_1$ and $P_2$ for the types $j_1$ and $j_2$. If both prices $p_{j_1}$ and $p_{j_2}$ get strictly inside the corresponding sides of those polygons, then by stirring $p_{j_1}$ and $p_{j_2}$ in $f_{OPT}$ with keeping $p_{j_1}+p_{j_2}$ constant we can get to a vertex of $P_1$ or $P_2$ and do not decrease the total value.
\end{proof}
The following algorithm computes an optimal fractional solution for heterogeneous knapsack. (For convenience we add an item numbered by $0$ of a new type with cost $0$ and value $0$; this does not affect any optimal solution.)
\begin{center}
\small{}\tt{} \fbox{
\parbox{6.0in}{
\hspace{0.05in} \\[-0.05in] \fHK
\begin{enumerate}
\item For each type $j\in[m]$, (partially) order items of type $j$ as follows:
\begin{itemize}
\item let $last = 0$, $tg = 0$ and $A_j=\emptyset$
\item while $v(last)< \max\limits_{i\in t^{-}_j}v(i)$
\begin{itemize}
\item let $k = \arg\max_{i\in t^{-}_j} \frac{v(i)-v(last)}{|c(i)-c(last)|}$ and add $k$ to $A_j$
\item define $tg_k = \frac{v(k)-v(last)}{|c(k)-c(last)|}$
\item let $last = k$
\end{itemize}
\end{itemize}
\item Comprise all $A_j$ into one big set $A$ and order all items s.t. $tg_1\ge \cdots\ge tg_{|A|}$
\item Let $last[j]=0$ for each $j\in[m]$, $\alpha_i=0$ for each $i\in[n]$ and $k=1$
\item While $k\le|A|$ and $c_k + \sum_{i=1}^{k-1}\alpha_{i}\cdot c_i\le B$
\begin{itemize}
\item let $\alpha_{last[t_k]}\leftarrow 0$
\item let $last[t_k]\leftarrow k$, $\alpha_k\leftarrow 1$
\item let $k\leftarrow k+1$
\end{itemize}
\item If $k\le|A|$, then let $\alpha_k = \frac{B-\sum_{i=1}^{k-1}\alpha_{i}c_i}{c_k}$ and $\alpha_{last[t_k]} = 1-\alpha_k$
\item Return vector $(\alpha_i)_{i\in [n]}$
\end{enumerate}
}}
\end{center}
\begin{figure}
\begin{center}
\include{fig2}
\end{center}
\caption{Convex hull}
\label{figure:star}
\end{figure}
\begin{theorem}
\fHK\ computes an optimal fractional solution for heterogeneous knapsack.
\end{theorem}
\begin{proof}~
\begin{itemize}
\item If we draw every item $i\in t^{-}_j\cup\{0\}$ as a point $(c_i,v_i)$ in
the plain (see fig.~\ref{figure:star}), then all picked items in $A_j$ will correspond to the part of
convex hull's vertices of the drawn set from $(0,0)$ to the item with
maximal value. Computed value of $tg$ will correspond then to the
tangent of the side of the convex hull with the right end at the given
item.
\item As in the proof of lemma \ref{lemma_HK} one can find the optimal value,
that we can get for a type $j$ at the price $c$, by taking the
$y$-coordinate of the point on a side of convex hull with $c$ at
$x$-coordinate. Thus for the optimal fractional
solution we only need items from $A=\cup_j A_j$.
\item Taking everything above into account we can reduce the heterogeneous
knapsack to the basic knapsack problem. Fix a type $j$ and construct the
instance of the reduced problem $\tilde{K}_j$ as follows. For each item
$k\in A_j$ assign the cost $\tilde{c}_k:=c_k-c(last[t_k])$ and the value
$\tilde{v}_k:=v_k-v(last[t_k])$. It is easy to see that optimal
solution to basic knapsack problem $\tilde{K}_j$ gives the same value as
the solution to the original heterogeneous problem restricted to the
items of type $j$ for any given budget. Hence the optimal fractional
solution to basic knapsack problem $\cup_j\tilde{K}_j$ has the same
value as the optimal fractional solution to the original problem.
\item Now it easy to check that our algorithm at stages $2-5$ computes the
optimal fractional solution to the reduced knapsack problem and
thus finds the optimal fractional solution to our original problem.
\end{itemize}
\end{proof}
\subsection{Greedy Strategy with Deletions}
We consider the following greedy strategy mechanism.
\begin{center}
\small{}\tt{} \fbox{
\parbox{5.0in}{
\hspace{0.05in} \\[-0.05in] $\greH$
\begin{enumerate}
\item Take the same ordered set $A$ as in Step~2 of \fHK
\item Let $k=1$, $S=\emptyset$, and $last[j]=0$ for $j\in[m]$
\item While $k\le |A|$ and $c(k)-c(last[t_k])\le B\cdot \frac{v(k)-v(last[t_k])}{v(k)-v(last[t_k])+\sum_{i\in S}v(i)}$
\begin{itemize}
\item let $S \leftarrow (S\setminus\{last[t_k]\})\cup \{k\}$
\item let $last[t_k] = k$
\item let $k\leftarrow k+1$
\end{itemize}
\item Return winning set $S$
\end{enumerate}
}}
\end{center}
Recall the notation in the algorithm \fHK, $tg_k=\frac{v(k)-v(last[t_k])}{|c(k)-c(last[t_k])|}$, where $last[t_k]$ is the last item of type $t_k$ in $A$ at the moment when we are about to add $k$ into $A$. Define $S_k=(S\setminus\{last[t_k])\cup\{k\}$.
Then the second condition in Step~3 of $\greH$ can be rewritten as $$tg_k\ge \frac{v(S_k)}{B}$$
We next analyze the mechanism \greH. Let us denote by $\calM_b$ the run of mechanism $\greH$ on bid $b$ (with the corresponding ordered set $A_b$, the last item of each type $last_b[t_k]$ and marginal tangent $tg_k(\calM_b)$).
\begin{claim}
$\greH$ is monotone (and therefore truthful).
\end{claim}
\begin{proof}
We will show that any losing item cannot bid more and become a winner.
Assume otherwise that item $j$ loses with bid $c_j$ but wins with bid $b_j>c_j$, given that all others bid $c_i$, $i\neq j$.
Note that when $j$ changes his bid, it will only affect the convex hull of items in $t_j^-\cup \{0\}$. The following observations can be verified easily (see fig.~\ref{figure:star}):
\begin{enumerate}
\item Values $v(S)$ of the set of winners and $v(last[t_k])$ for each type $t_k$, taking dynamically in the process of the mechanism, keep increasing.
\item Value $tg_j$ decreases when $j$ increases its bid (since point $(b_j,v_j)$ is on the right hand side of point $(c_j,v_j)$).
\item Ordered set $A_b\setminus t^{-}_j$ is the same as ordered set $A_c\setminus t^{-}_j$
\end{enumerate}
By considering the convex hull for $t^{-}_j$, one can easily see that
if $j$ was not getting at any moment in the winning set $S$ in $\calM_c$ it also will never get in the winning set in $\calM_b$.
Let us explain why for $j$ increasing its bid can not help to remain in the winning set if for the current cost $c_j$ it has been dropped off.
Note that in the new ordered set $A_b$, there can be new items of the same type as $j$ (e.g. $last_{c}[j]$ can be different from $last_{b}[j]$), but nevertheless $tg_j(\calM_b)\le tg_j(\calM_c)$. Let $j'\in t^{-}_j$ be the item that substitutes $j$ in $\calM_c$, then $tg_{j'}(\calM_c)\le tg_{j'}(\calM_b)$ (note that $j'$ necessarily appears in $A_b$). Let $k$ be an item at which $\calM_b$ has stopped, i.e. the first item that we have not taken in the winning set. Assume $k$ stands in $A_b$ not further than $j'$.
Consider two cases.
\begin{enumerate}
\item Let $t_k\neq t_j$. Then
\begin{itemize}
\item $tg_{k}(\calM_c)=tg_{k}(\calM_b)$
\item $v(S_{j'}(\calM_c))\ge v(S_k(\calM_c))$, as $j'$ stands later than $k$ in $A_c$
\item $v(S_k(\calM_c))=v(S_k(\calM_b))$, since in both $S_k(\calM_b)$
and $S_k(\calM_c)$ for $t_j$ type we have taken $j$ as well as for
each other type we have taken the same item.
\end{itemize}
\item $t_k=t_j$. Then
\begin{itemize}
\item $tg_{j'}(\calM_c)\le tg_{j'}(\calM_b)\le tg_{k}(\calM_b)$
\item $v(S_{j'}(\calM_c))\ge v(S_k(\calM_b))$. The last equality
holds true, because for each type the value of the item in
$S_{j'}(\calM_c)$ is greater or equal than value of the
corresponding item in $S_{k}(\calM_b)$.
\end{itemize}
\end{enumerate}
In both cases we can write
$$tg_k(\calM_b)\ge tg_{j'}(\calM_b)\ge tg_{j'}(\calM_c)\ge\frac{v(S_{j'}(\calM_c))}{B}\ge\frac{v(S_{k}(\calM_b))}{B}$$
Thus we have to take $k$ in $\calM_b$ to the winning set. Hence we arrive at a contradiction. Hence we have taken $j'$ to the winning set in $\calM_b$ and therefore exclude $j$.
\end{proof}
Unfortunately, in contrast to knapsack case this scheme does not possess the following property: any $i\in S$ cannot control the output set given that $i$ is guaranteed to be a winner.
\begin{claim}
\label{bound_payment_hk}
Let $S$ be the winning set of $\greH$ on cost vector $c$. Then no item $j\in S$ can be remained a winner with bid $b_j$ satisfying
$$b_j > (v(j)-v(last_c[t_j]))\cdot\frac{B}{V(S)}+c(last_c[t_j])$$
\end{claim}
\begin{proof}
Assume to the contrary that there exist such $j$ and bid $b_j$. We can write
$$tg_j(\calM_b)=\frac{v(j)-v(last_b[t_j])}{b_j-c(last_b[t_j])}\le \frac{v(j)-v(last_c[t_j])}{b_j-c(last_c[t_j])}<\frac{v(S)}{B}$$
Consider the ordered set $A_c$ and let $k$ be the last item we have taken in the winning set in $\calM_c$. Now consider any item $i\in[1,k]$ where $t_j\neq t_i$. We have $\frac{v(S)}{B}\le tg_k(\calM_c)\le tg_i(\calM_c)=tg_i(\calM_b)$. By the assumption that $j$ is in the winning set in $\calM_b$ and $tg_j(\calM_b)<\frac{v(S)}{B}\le tg_i(\calM_b)$, we get that $S_j(\calM_b)$ contains an item $i'$ with $t_i=t_{i'}$ and $v(i')\ge v(i)$. Since $j$ is in $S$ and in $S_j(\calM_b)$ we get $v(S_j(\calM_b))\ge v(S)$. Hence
$$\frac{v(S)}{B}> tg_j(\calM_b)\ge\frac{v(S_j(\calM_b))}{B}\ge\frac{v(S)}{B}$$
which gives a contradiction.
\end{proof}
\begin{claim}
\label{budget_feasible_hk}
Greedy scheme $\greH$ is budget feasible.
\end{claim}
\begin{proof}
Let $S$ be a winning set for $\calM$. By Claim~\ref{bound_payment_hk}, we have an upper bound on the payment $p_j$ to each item $j\in S$, i.e.,
$$p_j\le (v(j)-v(last_c[t_j]))\cdot\frac{B}{V(S)}+c(last_c[t_j])$$
Let $0 = i_0, i_1,\ldots ,i_r, i_{r+1} =j$ be the items of type $t_j$ that have appeared in the winning set. We have $tg_{i_\ell}\ge\frac{v(S)}{B}$ for each $\ell=1,\ldots,r$. Hence $$c(i_\ell)-c(i_{\ell-1})\le (v(i_\ell)-v(i_{\ell-1}))\frac{B}{v(S)}$$
Now if we sum up the above inequalities on $c(i_l)-c(i_{l-1})$ for all $\ell=1,\ldots,r$ and plug it in the bound on $p_j$, we get
$$p_j\le\frac{B}{v(S)}\sum_{\ell=1}^{r+1}v(i_\ell)-v(i_{\ell-1})=v(j)\frac{B}{v(S)}$$
Therefore, $\sum_{j\in S}p_j\le B$, which concludes the proof.
\end{proof}
\subsection{Mechanisms}
Given the greedy strategy described above, our mechanism for heterogeneous knapsack is as follows.
\begin{center}
\small{}\tt{} \fbox{
\parbox{4.0in}{
\hspace{0.05in} \\[-0.05in] \MHknapsack
\begin{enumerate}
\item Let $A=\{i~|~c_i\le B\}$ and $i^*\in \arg\max_{i\in A} v_i$
\item If $(1+\sqrt{2})\cdot v_{i^*} \ge \fHK(A\setminus \{i^*\})$, return $i^*$
\item Otherwise, return $S= \greH$
\end{enumerate}
}}
\end{center}
\begin{theorem}
\MHknapsack\ is a $2+\sqrt{2}$ approximation budget feasible truthful
mechanism for heterogeneous knapsack.
\end{theorem}
\begin{proof}
The proof consists of each property stated in the claim.
\begin{itemize}
\item {\em Truthfulness.} The same proof as for knapsack also works here.
\item {\em Individual rationality and budget feasibility.} If $i^*$ wins in Step~2, his payment is the threshold bid $B$. Otherwise, payment to each item has an upper bound from the payment rule in $\greH$ and thus according to the claim \ref{budget_feasible_hk} final total payment will be below given budget $B$.
\item {\em Approximation.} Return back to the algorithm for optimal fractional heterogeneous knapsack. Consider the stage where we add item $k$ to a set $A_j$, let us define $\tilde{v}(k)=v(k)-v(last[t_k])$ and $\tilde{c}(k)=c(k)-c(last[t_k])$ to be modified value and cost of item $k$. Let us consider fractional knapsack $\tilde{FK}$ problem for those modified costs and values for all items in $A$. It turns out that for any budget this new problem $\tilde{FK}$ has the same answer as initial heterogeneous knapsack $HK$. Note that our greedy scheme $\gre$ for modified costs and values and our greedy scheme $\greH$ for original heterogeneous knapsack also give the same answer.
Thus applying the part {\em approximation} of claim \ref{th_M_knapsack} to the modified problem we obtain desired bound.
\end{itemize}
\end{proof}
We can also have the following randomized mechanism with approximation ratio of $3$ (its proof is similar to Theorem~\ref{theorem-random-knapsack}).
\begin{center}
\small{}\tt{} \fbox{
\parbox{3.8in}{
\hspace{0.05in} \\[-0.05in] \RMHknapsack
\begin{enumerate}
\item Let $A=\{i~|~c_i\le B\}$ and $i^*\in \arg\max_{i\in A} v_i$
\item With probability $\frac{1}{3}$, return $i^*$
\item With probability $\frac{2}{3}$, return $S= \greH$
\end{enumerate}
}}
\end{center}
\begin{theorem}
\label{th_RHM_knapsack}
\RMHknapsack\ is a $3$ approximation universal truthful budget feasible
mechanism for heterogeneous knapsack.
\end{theorem}
\section{Introduction}
It is well-known that a mechanism may have to pay a large amount to enforce incentive compatibility (i.e., truthfulness). For example, the seminal VCG mechanism may have unbounded payment (compared to the shortest path) in path auctions~\cite{AT02}.
The negative effect of truthfulness on payments leads to a broad study of frugal mechanism design, i.e., how should one minimize his payment to get a desired output with incentive agents? While a class of results have been established~\cite{AT02,talwar03,KKT05,KSM10,CEG10}, in practice, one cannot expect a negative overhead for a few perspectives, e.g., budget or resource limit.
Recently, Singer~\cite{PS10} consider mechanism design problems from a converse angle and initiate the study of truthful mechanism design with a sharp budget constraint: the total payment of a mechanism is upper bounded by a given value $B$. Formally, in a marketplace each agent/item has a {\em privately} known incurred cost $c_i$. For any given subset $S$ of agents, there is a {\em publicly} known valuation $v(S)$, meaning the social welfare derived from $S$. A mechanism selects a subset $S$ of agents and decides a payment $p_i$ to each $i\in S$. Agents bid strategically on their costs and would like to maximize their utility $p_i-c_i$. The objective is to design truthful budget feasible mechanisms with outputs approximately close to socially optimal solution. In other words, it studies the ``price of being truthful" in a budget constraint framework\footnote{Note that if we do not consider truthful mechanism design, the problem is purely an optimization question with an extra capacity (i.e., budget) constraint, which has been well-studied in, e.g.,~\cite{NW81,Svi04,KG05,FMV07,LMN09}, in the framework of submodularity. It is well-known that a simple greedy algorithm gives the best possible approximation ratio $1-1/e$~\cite{NW81}. When agents are weighted (corresponding to costs in our setting), the simple greedy algorithm may have unbounded approximation ratio~\cite{KMN99}; a variant of the greedy by picking the maximum of the original greedy and the agent with the largest value provides the same approximation ratio $1-1/e$~\cite{KG05}.}.
Although budget is a realistic condition that appears almost everywhere in daily life, it has not received much attention until very recently~\cite{DLN08,BGG10,CHM10,PS10}. In the framework of worst case analysis, most results are negative~\cite{DLN08}. The introduction of budget adds another dimension to mechanism design; it further limits the searching space, especially given the (already) strong restriction of truthfulness. Designing budget feasible mechanisms even requires us to bound the threshold payment of each individual, which, not surprisingly, is tricky to analyze.
While the problem in general does not admit any budget feasible mechanism\footnote{For example, one with budget $B=1$ would like to purchase a path from $s$ to $t$ in a network $\{(s,v),(v,t)\}$ where each edge has incurred cost 0. In any truthful mechanism that guarantees to buy the path (i.e., outputs the socially optimum solution), one has to pay each edge the threshold value $B$, leading to a total payment $2B$ which exceeds the given budget.}, Singer~\cite{PS10} studied an important class of valuation functions, i.e., monotone submodular functions. He gives a randomized truthful mechanism with constant approximation ratio 112 for any monotone submodular functions, and deterministic mechanisms for special cases including knapsack (ratio 6) and coverage. Further, he shows that no deterministic truthful mechanism can obtain an approximation ratio better than 2 even for knapsack.
\subsection{Our Results}
In this paper, we improve upper and lower bounds of budget feasible mechanisms for general submodular functions and knapsack, summarized in the following table.
\small
\begin{table*}[t]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\hline
& \multicolumn{4}{c|}{Submodular functions} & \multicolumn{4}{c|}{Knapsack} \\
\cline{2-9}
& \multicolumn{2}{c|}{deterministic} & \multicolumn{2}{c|}{randomized} & \multicolumn{2}{c|}{deterministic} & \multicolumn{2}{c|}{randomized} \\
\cline{2-9}
& upper & lower & upper & lower & upper & lower & upper & lower \\ \hline
Singer~\cite{PS10} & $-$ & 2 & 112 & $-$ & $6$ & $2$ & $-$ & $-$ \\ \hline
Our results & $8.34^*$ & $1+\sqrt{2}$ & 7.91 & 2 & $2+\sqrt{2}$ & $1+\sqrt{2}$ & 3 & 2 \\
\hline
\end{tabular}
\\[.1in] *It may require exponential running time for general submodular functions.
\end{center}
\end{table*}
\normalsize
In truthful mechanism design, if there is no restriction on total payment, it is sufficient to focus on designing monotone allocations --- the payment to each individual winner is the unique threshold to maintain the winning status~\cite{myerson}. With a sharp budget constraint, in addition to monotone allocation, we also have to upper bound the sum of threshold payments. For submodular functions, the natural greedy algorithm is a good candidate for designing budget feasible mechanisms due to its nice monotonicity and small approximation ratio. However, the threshold payment to each winner can be very complicated because an agent can bid differently to get other ranking positions in the greedy algorithm, which results in different ways of computing the marginal contributions for the rest agents, and therefore, unpredictably change the set of winners.
Singer~\cite{PS10} bound the threshold of each winner by considering all possible ranking positions for his bids and taking the maximum of the thresholds of all these positions. In Section~3, we give a clean and tight analysis for the upper bound on threshold payment by applying the combinatorial structure of submodular functions (Lemma~\ref{lemma-pay-upper}). These upper bounds on payments suggest appropriate parameters in our randomized mechanism, which roughly speaking, selects the greedy algorithm or the agent with the largest value at a certain probability.
A difficulty of deriving deterministic mechanisms is related to the agent $i^*$ with the largest value $v(i^*)$ --- due to its (possibly large) cost greedy may not include it, which could result in a solution with an arbitrarily bad ratio. However, we cannot simply compare greedy with $v(i^*)$ because this breaks monotonicity as $i^*$ is able to manage the greedy solution by his bid (this is exactly where randomization helps). To get around of this issue, we drop $i^*$ out of the market and compare $v(i^*)$ with remaining agents in an appropriate way --- now $i^*$ is completely independent of the rest of the market and cannot affect its output --- this gives our deterministic mechanisms for submodular functions and knapsack with small approximation ratios (note that we still need to be careful about the agents in the remaining market as they are still able to manage their bids to beat $v(i^*)$).
On the other hand, it is interesting to explore limitations of budget feasible mechanisms. Singer gives a simple lower bound of $2$ on the approximation ratio and proposes that exploring the lower bounds that are dictated by budget feasibility is ``perhaps the most interesting question"~\cite{PS10}. In Section~4, we prove a stronger lower bound $1+\sqrt{2}$ for deterministic mechanisms. In most lower bounds proofs for truthful mechanisms, a number of related instances are constructed and one shows that a truthful mechanism cannot do well for all of them~\cite{CKV07,KV07,NR01,PT09}. (For example, in Singer's proof, three instances are constructed.) Our lower bound proof uses a slightly different approach: We first establish a property of a truthful mechanism for all instances provided that the mechanism has a good approximation ratio
(Lemma~\ref{lemma-knapsack-lower}), then we conclude that this property is inconsistent with the budget feasibility condition for a carefully constructed instance. Furthermore, we show a lower bound of $2$ for universally randomized budget feasible mechanisms. Both our lower bounds are independent of computational assumptions and hold for instances with small number of agents.
While submodular functions admit a good approximation budget feasible mechanism, its generalization seems to be a very difficult task and there may not be any good approximation mechanisms for instances like path and spanning tree~\cite{PS10}. In Section~5, we take a first step of this generalization by considering an extended knapsack problem with heterogeneous items, where items are of different types and we are only allowed to pick one item from each type.
Here we cannot apply the same greedy mechanism for the original knapsack as it may not even generate a feasible solution; and its approximation ratio can be arbitrarily bad if we only take the first agent from each type.
To construct a truthful mechanism with good approximation, we employ a greedy strategy with {\em deletions} --- in the process of the greedy, we either add a new item whose type has not been considered yet or replace an existing item with the new one of same type.
Although there are deletions, the greedy algorithm is still monotone (but its proof is much more involved), based on which we have similar approximation mechanisms for heterogeneous knapsack. We believe that the greedy strategy with deletions can be extended to a number of interesting non-submodular settings to derive budget feasible mechanisms.
\section{Preliminaries}
In a marketplace there are $n$ agents (or items), denoted by $A=\{1,\ldots,n\}$. Each agent $i$ has a privately known incurred {\em cost} $c_i$ (or denoted by $c(i)$). For any given subset $S\subseteq A$ of agents, there is a publicly known valuation $v(S)$, meaning the social welfare derived from $S$. We assume $v(\emptyset)=0$ and $v(S)\le v(T)$ for any $S\subset T\subseteq A$. We say the valuation function {\em submodular} if $v(S)+v(T)\ge v(S\cap T)+ v(S\cup T)$ for any $S,T\subseteq A$.
Upon receiving a {\em bid} cost $b_i$ from each agent, a mechanism decides an {\em allocation} $S\subseteq A$ as winners and a {\em payment} $p_i$ to each $i\in A$. We assume that the mechanism has no positive transfer (i.e., $p_i=0$ if $i\notin S$) and individually rational (i.e., $p_i\ge b_i$ if $i\in S$). Agents bid strategically on their costs and would like to maximize their utilities, which is $p_i-c_i$ if $i$ is a winner and $0$ otherwise. We say a mechanism is {\em truthful} if it is of best interests for all agents to bid their true costs. For randomized mechanisms, we consider universal truthfulness in this paper (i.e., a randomized mechanism takes a distribution over deterministic truthful mechanisms).
Since our setting is in single parameter domain as each agent has one private cost, it is well-known~\cite{myerson} that a mechanism is truthful if and only if its allocation rule is monotone (i.e., a winner keeps winning if he unilaterally decreases his bid) and the payment to each winner is his threshold bid to win. Therefore, we will only focus on designing monotone allocations and do not specify the payment to each winner explicitly.
A mechanism is said to be {\em budget feasible} if $\sum_i p_i \le B$, where $B$ is a given sharp budget constraint.
Assume without loss of generality that $c_i\le B$ for any agent $i\in A$, since otherwise he will never win in any (randomized) budget feasible truthful mechanism. Our objective is to design truthful budget feasible mechanisms with outputs approximately close to the social optimum. That is, we want to minimize the {\em approximation ratio} of a mechanism, which is defined as $\max_I \frac{\mathcal{M}(I)}{opt(I)}$, where $\mathcal{M}(I)$ is the (expected) value of mechanism $\mathcal{M}$ on instance $I$ and $opt(I)$ is the optimal value of the integer program: $\max_{_{S\subseteq A}}v(S)$ subjected to $c(S)\le B$.
\section{Budget Feasible Mechanisms}
For any given submodular function, denote the marginal contribution of an item $i$ with respect to set $S$ by $m_S(i)=v(S\cup\{i\})-v(S)$.
We assume that agents are sorted according to their increasing marginal
contributions relative to cost, recursively defined by: $i + 1 = \arg\max_{j \in [n]} \frac{m_{S_i}(j)}{c_j}$, where $S_i =\{1,\ldots,i\}$ and $S_0=\emptyset$. To simplify notations we will denote this order by $[n]$ and write $m_i$ instead of $m_{S_{i-1}}(i)$. This sorting, in the presence of submodularity, implies that
\[ \frac{m_1}{c_1} \geq \frac{m_2}{c_2} \geq \cdots \geq \frac{m_n}{c_n}.\]
Notice that $v(S_k) = \sum_{i\leq k} m_i$ for all $k\in [n]$.
The following greedy scheme is the core of our mechanism (where the parameters denote the set of agents $A$ and available budget $B/2$).
\begin{center}
\small{}\tt{} \fbox{
\parbox{3.0in}{
\hspace{0.05in} \\[-0.05in] $\Gmodular(A,B/2)$
\begin{enumerate}
\item Let $k=1$ and $S=\emptyset$
\item While $k\le |A|$ and $c_k\le \frac{B}{2} \cdot \frac{m_k}{\sum_{i\in S\cup\{k\}}m_i}$
\begin{itemize}
\item $S \leftarrow S\cup \{k\}$
\item $k\leftarrow k+1$
\end{itemize}
\item Return winning set $S$
\end{enumerate}
}}
\end{center}
Our mechanism for general submodular functions is as follows.
\begin{center}
\small{}\tt{} \fbox{
\parbox{4.0in}{
\hspace{0.05in} \\[-0.05in] \RMmodular
\begin{enumerate}
\item Let $A=\{i~|~c_i\le B\}$ and $i^*\in \arg\max_{i\in A} v(i)$
\item with probability $\frac{2}{5}$, return $i^*$
\item with probability $\frac{3}{5}$, return $\Gmodular(A,B/2)$
\end{enumerate}
}}
\end{center}
In the above mechanism, if it returns $i^*$, the payment to $i^*$ is $B$; if it returns $\Gmodular(A,B/2)$, the payment is more complicated and is given in~\cite{PS10}. Actually,
we do not need this explicit payment formula to prove our result.
\begin{theorem}\label{theorem-mechanism-SM}
\RMmodular \ is a budget feasible universally truthful mechanism for submodular valuation function with approximation ratio $\frac{5e}{e-1} (\approx 7.91)$.
\end{theorem}
\subsection{Analysis of \RMmodular}
In this subsection we analyze \RMmodular\ in terms of three respects: truthfulness, budget feasibility and approximation. They together yield a proof for Theorem~\ref{theorem-mechanism-SM}.
\subsubsection{Universal Truthfulness}
Our mechanism is a simple random combination of two mechanisms. To prove that the \RMmodular \ is universally truthful, it suffices to prove that these two mechanisms are truthful, i.e., the allocation rule is monotone.
The scheme where we simply return $i^*$ is obviously truthful. Also it is easy to see that prior step when we throw away the agents having the cost greater than $B$ does not affect truthfulness. The greedy scheme $\Gmodular(A,B/2)$ is monotone as well, since any item out of a winning set can not increase its bid and become a winner.
\subsubsection{Budget Feasibility}
While truthfulness is quite straightforward, the budget feasibility part turns out to be quite tricky.
The difficulties arise when we compute the payment to each item. Indeed, it can happen that an item changes its bid (while still remaining in the winning set) to force the mechanism to change its output. In other words, an item can control the output of the mechanism. Fortunately, in such a case no item can reduce the valuation of the output too much. That enables us to write an upper bound on the bid of each item in case of submodularity; summing up these bounds yields budget feasibility.
If the mechanism returns $i^*$, his payment is $B$ and it is clearly budget feasible. It remains to prove budget feasibility for $\Gmodular(A,B/2)$. A similar but weaker result has been proven in~\cite{PS10} using the characterization of payments and arguing that the total payment is not larger than $B$. Here we directly show that the payment to any item $i$ in the winning set $S$ is bounded above by $\frac{m_i}{v(S)}\cdot B$; then the total payment will be bounded by $B$. Before doing that, we first prove a useful lemma.
\begin{lemma}
\label{submodular average}
Let $S\subset T\subseteq [n]$ and $t_0=\arg\max_{t\in T\setminus S}\frac{m_S(t)}{c(t)}$. Then $$\frac{v(T)-v(S)}{c(T)-c(S)}\le\frac{m_S(t_0)}{c(t_0)}.$$
\end{lemma}
\begin{proof}
Assume for contradiction that the lemma does not hold, then for all $t\in T\setminus S$, we have
$\frac{v(T)-v(S)}{c(T)-c(S)}> \frac{m_S(t)}{c(t)}.$
Then add all inequalities together, we have
$$\frac{v(T)-v(S)}{c(T)-c(S)}> \frac{\sum_{t\in T\setminus S} m_S(t)}{\sum_{t\in T\setminus S} c(t)} = \frac{\sum_{t\in T\setminus S} m_S(t)}{c(T)-c(S)}.$$
This implies that $v(T)-v(S) >\sum_{t\in T\setminus S} m_S(t)$, which contradicts the submodularity.
\end{proof}
Let $1,\ldots, k$ be the order of items in which we add them to the winning set.
Let $\emptyset=S_0\subset S_1\subset\ldots\subset S_k\subseteq [n]$ be the sequence of winning sets that we pick at each step by applying our mechanism. Thus we have $S_j=[j]$.
Now, since $v$ is sumbodular, we can write the following chain of inequalities (note that marginal contribution is smaller for larger sets).
$$\frac{m_{S_0}(1)}{c_1}\ge\frac{m_{S_1}(2)}{c_2}\ge\ldots\ge\frac{m_{S_{k-1}}(k)}{c_k}\ge\frac{2 v(S_k)}{B}.$$
The following is our main lemma.
\begin{lemma}\label{lemma-pay-upper}
There is no $j\in S=\Gmodular(A,B/2)$ such that it can bid more than $m_{S_{j-1}}(j)\frac{B}{v(S_k)}$ and still get into a winning set. Thus the payment to $j$ is upper bounded by $m_{S_{j-1}}(j)\frac{B}{v(S_k)}$.
\end{lemma}
\begin{proof}
Assume that there is $j\in[k]$ such that it can bid $b_j>m_{S_{j-1}}(j)\frac{B}{v(S_k)}$ and still wins. We will use notation $b$ instead of $c$ to emphasize that we consider a new scenario where $j$ has increased its bid to $b_j$ and others remain the same.
Note that $\frac{m_{S_0}(1)}{c_1}\ge\frac{m_{S_1}(2)}{c_2}\ge\ldots\ge\frac{m_{S_{j-1}}(j)}{c_j}\ge\frac{m_{S_{j-1}}(j)}{b_j}.$ Thus $S_{j-1}$ still get into the winning set.
For bid vector $b$, denote by $S$ the set we have chosen right before $j$ is included into the winning set. Thus we have
\begin{eqnarray}
j &=& \arg\max_{i\in[n]\setminus S}\frac{m_S(i)}{b_i}, \\
\frac{m_S(j)}{b_j} &\ge& \frac{2 v(S\cup\{j\})}{B}.
\end{eqnarray}
We may assume that $S_k\cup S\supsetneq S\cup\{j\}$. Indeed, otherwise $S\cup\{j\}=S_k\cup S$ and $$\frac{m_{S_{j-1}}(j)}{b_j}\ge\frac{m_{S}(j)}{b_j}\ge\frac{2 v(S\cup\{j\})}{B}\ge \frac{2 v(S_k)}{B}\ge\frac{v(S_k)}{B}.$$
Thus $b_j\le m_{S_{j-1}}\frac{B}{v(S_k)}$ and we get a contradiction.
Let $R=S_k\setminus S$. Applying equation~(1) and Lemma~\ref{submodular average} to $S_k\cup S$ and $S\cup \{j\}$, we know that for some $r_0\in R\setminus\{j\}$,
$$\frac{v(S_k\cup S)-v(S\cup\{j\})}{b(S_k\cup S)-b(S\cup\{j\})}\le\frac{m_{S\cup\{j\}}(r_0)}{b(r_0)}\le\frac{m_S(j)}{b_j}.$$
On the other hand we know that $b_j> m_{S_{j-1}}(j)\frac{B}{v(S_k)}$. Hence,
$\frac{m_S(j)}{b_j}< \frac{m_S(j)}{m_{S_{j-1}}(j)}\frac{v(S_k)}{B}<\frac{v(S_k)}{B}.$
Combining these inequalities, we get $$\frac{v(S_k\cup S)-v(S\cup\{j\})}{b(S_k\cup S)-b(S\cup\{j\})}<\frac{v(S_k)}{B}.$$
We have $b(S_k\cup S)-b(S\cup\{j\})=b(R\setminus\{j\})=c(R\setminus\{j\})\le c(S_k).$
Recall that $\frac{m_{S_{i-1}}(i)}{c_i}\ge \frac{2 \emph{v}(S_k)}{B}$ for $i\in[k]$. Thus $c_i\le m_{S_{i-1}}(i)\frac{B}{2 v(S_k)}$ and
$c(S_k)=\sum_{i=1}^{k}c(i)\le \frac{B}{2}.$
We get
$$\frac{v(S_k)-v(S\cup\{j\})}{B/2}\le\frac{v(S_k)-v(S\cup\{j\})}{c(S_k)}\le\frac{v(S_k\cup S)-v(S\cup\{j\})}{b(S_k\cup S)-b(S\cup\{j\})}<\frac{v(S_k)}{B}.$$
Thus, $v(S_k)<2v(S\cup \{j\}).$
Recalling inequality~(2) on $\frac{m_S(j)}{b_j}$, we derive
$$\frac{m_{S_{j-1}}(j)}{b_j}\ge\frac{m_S(j)}{b_j}\ge\frac{2 v(S\cup\{j\})}{B}>\frac{v(S_k)}{B}.$$
Hence, we arrive at the contradiction with $b_j > m_{S_{j-1}}(j)\frac{B}{v(S_k)}$.
\end{proof}
\subsubsection{Approximation Ratio}
Before analyzing the performance of our mechanism, we consider a simple greedy algorithm (without considering bidding strategies): order items according to their marginal contributions and add as many items as possible (i.e., it stops when we cannot add the next item as the sum of $c_i$ otherwise will be bigger than $B$).
Moreover we can consider the fractional variant of that, i.e., for the remaining budget we take a portion of the item at which we have stopped.
Let $\ell$ be the maximal index for which $\sum_{i=1,\ldots,\ell}c_i\le B$. Let $c'_{\ell+1}=B-\sum_{i=1,\ldots,\ell}c_i$ and $m'_{\ell+1}=m_{\ell+1}\cdot \frac{c'_{\ell+1}}{c_{\ell+1}}$. Hence, the fractional greedy solution is defined to be
\[fgre(A)\triangleq \sum_{i=1}^{\ell}m_i + m'_{\ell+1}.\]
It is well-known that the greedy algorithm is a $1-1/e$ approximation to maximization of monotone submodular functions with a cardinality constraint~\cite{NW81}. Also there was shown that the simple greedy algorithm has unbounded approximation ratio in case of weighted items with a capacity constraint. Nevertheless, a variant of greedy was suggested in \cite{KG05} which gives the same $1-1/e$ approximation to the weighted case. Next we present the following lemma, which is fundamental to our analysis, establishing the same approximation ratio for the simple greedy algorithm with fractional solution. (The proof is deferred to Appendix~\ref{appendix-submodular-reduction}.)
\begin{lemma}\label{submodular reduction}
Fractional greedy solution has approximation ratio $1-1/e$ for the weighted submodular maximization problem. That is,
\[fgre(A)\ge (1-1/e)\cdot opt(A),\]
where $opt(A)$ is the value of the optimal integral solution for the given instance $A$.
\end{lemma}
Now we are ready to analyze the approximation ratio of \RMmodular. Let $T=\{1,\ldots,k\}$ be the subset returned by $\Gmodular(A,\frac{B}{2})$.
For any $j=k+1,\ldots,\ell$, we have $\frac{c_j}{m_j}\ge \frac{c_{k+1}}{m_{k+1}} > \frac{B}{2 \sum_{i=1}^{k+1}m_i}$, where the last inequality follows from the fact that the greedy strategy stops at item $k+1$. Hence, $c_j>B\cdot \frac{m_j}{2 \sum_{i=1}^{k+1}m_i}$. Same analysis shows that $c'_{\ell+1}>B\cdot \frac{m'_{\ell+1}}{2 \sum_{i=1}^{k+1}m_i}$. Therefore,
$$B\cdot \frac{\sum_{j=k+1}^{\ell}m_j+m'_{\ell+1}}{2\sum_{i=1}^{k+1}m_i}<\sum_{j=k+1}^{\ell}c_j+c'_{\ell+1}\le B.$$
Which implies that $2 \sum_{i=1}^{k+1}m_i > \sum_{j=k+1}^{\ell}m_j+m'_{\ell+1}$ and $m_{k+1}+ 2 \sum_{i=1}^{k}m_i > \sum_{j=k+2}^{\ell}m_j+m'_{\ell+1}$. Hence,
\[fgre(A)=\sum_{i=1}^{\ell}m_i + m'_{\ell+1} = \sum_{i=1}^{k+1}m_i + \sum_{j=k+2}^{\ell}m_j+m'_{\ell+1} < 3\sum_{i\in S}m_i + 2 m_{k+1} \leq 3\sum_{i\in S}m_i + 2 v(i^*).\]
Together with Lemma~\ref{submodular reduction}, we can bound the optimal solution as
\begin{equation}\label{eqn:opt-upper}
opt(A)\le\frac{e}{e-1} \Big( 3 \Gmodular(A,B/2) + 2 v(i^*) \Big).
\end{equation}
Therefore, the expected value of our randomized mechanism is
$\frac{3}{5} \Gmodular(A,B/2) + \frac{2}{5} v(i^*)\ge \frac{e-1}{5e}opt.$
\subsection{Derandomization}
In this section, we provide a deterministic truthful mechanism which is budget feasible and has constant approximation ratio (where $opt(A\setminus \{i^*\},B)$ denotes the value of the optimal solution for the weighted submodular maximization problem given instance $A\setminus \{i^*\}$ with budget $B$).
\begin{center}
\small{}\tt{} \fbox{
\parbox{4.0in}{
\hspace{0.05in} \\[-0.05in] \Mmodular
\begin{enumerate}
\item Let $A=\{i~|~c_i\le B\}$ and $i^*\in \arg\max_{i\in A} v(i)$
\item If $\frac{1 + 4 e + \sqrt{1 + 24 e^2}}{2 (e-1)}\cdot v(i^*) \ge opt(A\setminus \{i^*\},B)$,\footnote{} return $i^*$
\item Otherwise, return $\Gmodular(A,B/2)$
\end{enumerate}
}}
\end{center}
\footnotetext[3]{Our deterministic mechanism works in general not in polynomial time because of the hardness of computing an optimal solution for submodular maximization problems. However, we may substitute it by the optimum of the fractional problem; therefore for special problems like knapsack (discussed in the following subsection), we can get a polynomial time deterministic mechanism.
Note however that we cannot replace it by the simple greedy solution as it breaks monotonicity.
Our mechanism suggests a natural question on the power of computation in (budget feasible) mechanism design at the price of being truthful~\cite{PSS08,Dob}. In particular, can an (exponential runtime) mechanism beat the lower bound of all polynomial time mechanisms? We leave this as future work.}
\begin{theorem}
\Mmodular \ is a budget feasible truthful mechanism for submodular functions with approximation ratio $\frac{6 e -1 + \sqrt{1 + 24 e^2}}{2 (e-1)} (\approx 8.34)$.
\end{theorem}
\begin{proof}
Note that the bid of $i^*$ is independent to the value of $opt(A\setminus \{i^*\},B)$. Therefore, the mechanism is truthful (a detailed similar argument is given in the proof of Theorem~\ref{th_M_knapsack} in Appendix~\ref{appendix-knapsack}). Budget feasibility follows from Lemma~\ref{lemma-pay-upper} and the observation that Step~2 only gives additional upper bounds on the thresholds of winners from $\Gmodular(A,B/2)$.
It remains to prove the approximation ratio. Let $x=\frac{1 + 4 e + \sqrt{1 + 24 e^2}}{2 (e-1)}(\approx 7.34)$.
We observe that $opt(A,B) - v(i^*) \le opt(A\setminus\{i^*\},B) \le opt(A,B).$
If the condition in Step~2 holds and the mechanism outputs $i^*$, then
\[opt(A,B) \le opt(A\setminus\{i^*\},B)+v(i^*) \le (x+1)\cdot v(i^*).\]
Otherwise, the condition in Step~2 fails and the mechanism outputs $\Gmodular(A,B/2)$ in Step~3.
Recall that in formula~(\ref{eqn:opt-upper}),
$opt(A,B)\leq \frac{e}{e-1} \Big( 3 \Gmodular(A,B/2) + 2 v(i^*) \Big).$
We have
\[ x \cdot v(i^*) < opt(A\setminus \{i^*\},B)\le opt(A,B) \leq \frac{e}{e-1} \Big( 3 \Gmodular(A,B/2) + 2 v(i^*) \Big). \]
This implies that
$ v(i^*) \leq \frac{3 e}{x(e-1)-2e} \Gmodular(A,B/2).$
Hence,
\[ opt \leq \frac{e}{e-1} \Big( 3 \Gmodular(A,B/2) + 2 v(i^*) \Big) \le \frac{e}{e-1} \left( 3 + \frac{6 e}{x(e-1)-2e} \right)\cdot \Gmodular(A,B/2).\]
Simple calculations show that
$1+x= \frac{6 e -1 + \sqrt{1 + 24 e^2}}{2 (e-1)}=\frac{e}{e-1} \left( 3 + \frac{6 e}{x(e-1)-2e} \right).$
Therefore, we have $opt \le (x+1)\cdot \Gmodular(A,B/2)$ in the both cases, which concludes the proof of the claim with approximation ratio
$\frac{e}{e-1} \left( 3 + \frac{6 e}{x(e-1)-2e} \right) (\approx 8.34)$.
\end{proof}
\subsection{Improved Mechanisms for Knapsack}
In this subsection, we consider a special model of submodular functions where the valuations of agents are additive, i.e., $v(S)=\sum_{i\in S}v_i$ for $S\subseteq [n]$. This leads to an instance of the Knapsack problem, where items correspond to agents and the size of the knapsack corresponds to budget $B$. Singer~\cite{PS10} give a 6-approximation deterministic mechanism. By applying approaches from the previous subsections, we have the following results (proofs are deferred to Appendix~\ref{appendix-knapsack}).
\begin{theorem}\label{theorem-knapsack-mechanism}
There are $2+\sqrt{2}$ approximation deterministic and 3 approximation randomized polynomial truthful budget feasible mechanisms for knapsack.
\end{theorem}
\section{Lower Bounds}
In this section we focus on lower bounds for the approximation ratio of truthful budget feasible mechanisms for knapsack. Note that the same lower bounds can be applied to the general submodular functions as well. In~\cite{PS10}, a lower bound of $2$ is obtained by the following argument: Consider the case with two items, both of unit value
(the value of two items together is 2). If their costs are $(B-\epsilon, B-\epsilon)$, at least one item should win,
otherwise the approximation ratio is infinite. Without loss of generality, we can assume that the first item wins, then its payment is
at least $B-\epsilon$. Now consider another profile $(\epsilon, B-\epsilon)$, the first item should also win and get payment at least
$B-\epsilon$ by truthfulness. Then the second item could not win because of the budget constraint and individual rationality. Therefore, the mechanism
can only archieve value $1$ for that instance while the optimal solution is $2$. This gives us the lower bound of $2$.
We improve the deterministic lower bound to $1+\sqrt{2}$ by a more involved proof. We also adduce a lower bound of $2$ for
universally randomized truthful mechanisms. All our lower bounds are unconditional, which implies that we do not impose any complexity assumption and constraints of the running time on the mechanism. Our lower bounds relys only on truthfulness and budget feasibility.
\subsection{Deterministic Lower Bound}
\begin{theorem}\label{theorem-knapsack-lower}
There is no truthful budget feasible mechanism that
can achieve an approximation ratio better than $1+\sqrt{2}$, even if
there are only three items.
\end{theorem}
Assume otherwise that there is a budget feasible truthful mechanism
that can achieve a ratio better than $1+\sqrt{2}$. We consider the
following scenario: budget $B=1$, and values $v_1=\sqrt{2}$,
$v_2=v_3=1$. Then the mechanism on a scenario has the following two
properties: (i) if all items are winners in the optimal solution,
the mechanism must output at least two items; and (ii) if $\{1,2\}$ or $\{1,3\}$ is the optimal solution, the mechanism cannot output either $\{2\}$ or $\{3\}$ (i.e., a single item with
unit value).
For any item $i$, let function $p_i(c_j,c_k)$ be the payment offered
to item $i$ given that the bids of the other two items are $c_j$ and
$c_k$. That is, $p_i(c_j,c_k)$ is the threshold bid of $i$ to be a winner.
\begin{lemma}\label{lemma-knapsack-lower}
For any $c_3>0.5$ and any domain $(a,b)\subset (0,1-c_3)$, there is
$c_2\in (a,b)$ such that $p_1(c_2,c_3) < 1-c_2$.
\end{lemma}
\begin{proof}
Assume otherwise that there are $c_3>0.5$ and domain $(a,b)\subset
(0,1-c_3)$ such that for any $c_2\in (a,b)$, $p_1(c_2,c_3) \ge
1-c_2$. Let $c_1=1-c_3-b$, then $c_1+c_2+c_3<1=B$, which implies
that the mechanism has to output at least two items. Since
$c_1=1-c_3-b<1-c_2\le p_1(c_2,c_3)$, item 1 is a winner. Further,
$p_1(c_2,c_3) \ge 1-c_2 > 0.5$, which together with budget feasibility imply that item 3 cannot be a winner. Therefore, item 2 must be a
winner with payment $p_2(c_1,c_3) = c_2$ due to individual
rationality and budget feasibility. The same analysis still holds if
the true cost of item 2 becomes $c'_2=\frac{c_2+b}{2}$, i.e., item 2
is still a winner with payment $c'_2$. Thus for the sample $(c_1,c_2,c_3)$ the payment $p_2(c_1,c_3) \ge c'_2 > c_2$, a contradiction.
\end{proof}
Since item 2 and 3 are identical, the above lemma still holds if we
switch item 2 and 3 in the claim. We are now ready to prove
Theorem~\ref{theorem-knapsack-lower}.
\bigskip \noindent {\em Proof of
Theorem~\ref{theorem-knapsack-lower}.} Define $c_3=0.7$ and
$(a,b)=(0.2,0.3)$. Note that $c_3$ and $(a,b)$ satisfy the condition
of Lemma~\ref{lemma-knapsack-lower}. Hence, there is $c\in
(0.2,0.3)$ such that $p_1(c,0.7)<1-c$. Define
$p_1(c,0.7)= 1-c-x, \ \textup{where} \ x>0.$
Symmetrically, define $c_2=0.7$ and $(a',b')=(c,\min\{0.3,c+x\})$.
Again by Lemma~\ref{lemma-knapsack-lower}, there is $d\in (a',b')$
such that $p_1(0.7,d)<1-d$. Define
$p_1(0.7,d)=1-d-y, \ \textup{where} \ y>0.$
Pick $c_1=1-d-\epsilon$, where $\epsilon>0$ is sufficiently small so
that $c_1\in (1-c-x,1-c)\cap (1-d-y,1-d)$. Note that since $d\in
(c,c+x)$, $c_1$ is well-defined.
Consider a true cost vector $(c_1,c,0.7)$. Since
$p_1(c,0.7)=1-c-x<c_1$, item 1 cannot be a winner. Since
$c_1+c=1-d-\epsilon+c<1$, the optimal solution has value at least
$v_1+v_2=1+\sqrt{2}$; therefore the mechanism has to output both
item 2 and 3. Hence, $p_3(c_1,c)\ge c_3 = 0.7.$
Similarly, consider true cost vector $(c_1,0.7,d)$; we have
$p_2(c_1,d)\ge c_2=0.7.$
Finally, consider cost vector $(c_1,c,d)$. By the above two
inequalities, both items 2 and 3 are the winners; this contradicts the
budget feasibility. \hfill $\square$
\subsection{Randomized Lower Bound}
\begin{theorem}\label{theorem-knapsack-lower-random}
There is no randomized (universally) truthful budget feasible mechanism that
can achieve an approximation ratio better than $2$, even in case of
two items.
\end{theorem}
\begin{proof}
We use Yao's min-max principle, which is a typical tool used to prove lower bounds. By the principle, we need to design
a distribution of instances and argue that any deterministic budget feasible mechanism cannot get an expected approximation ratio which is better than
$2$.
All the instances contain two items both with value $1$. Their costs $(c_1,c_2)$ are drawn from the following distribution (see Fig.~\ref{figure:square} in Appendix):
\begin{enumerate}
\item $(\frac{kB}{n}, \frac{(n-k)B}{n})$ with probability $\frac{1-\epsilon}{n-1}$, where $k=1,2,\ldots, n-1$,
\item $(\frac{iB}{n}, \frac{jB}{n})$ with probability $\frac{2\epsilon}{(n-1)(n-2)}$, where $i,j\in\{1,\ldots,n-1\}$ and $i+j>n$,
\end{enumerate}
where $1>\epsilon >0$ and $n$ is a larger integer.
We first claim that for any deterministic truthful budget feasible mechanism with finite expected approximation ratio, there is at most one instance, for which both items win in the mechanism.
Assume for contradiction that there are at least two such instances.
Note that for the second distribution $(\frac{iB}{n}, \frac{jB}{n})$, where $i+j>n$, it cannot be the case that both items win
due to the budget constraint. Hence, the two instances must be of the first type; denote them as $(\frac{k_1B}{n}, \frac{(n-k_1)B}{n})$
and $(\frac{k_2 B}{n}, \frac{(n-k_2 )B}{n})$, where $k_1 > k_2$. Consider then the instance $(\frac{k_1B}{n}, \frac{(n-k_2 )B}{n})$ .
Since $k_1+ n-k_2 > n$, this is the instance of the second type in our distribution. Therefore it has nonzero probability (see fig.~\ref{figure:square}). The mechanism has finite approximation ratio, thus it should have finite approximation ratio on the instance $(\frac{k_1B}{n}, \frac{(n-k_2 )B}{n})$ as well. As a result, it cannot be the case that
both items loss. We assume that item 1 wins (the proof for the other case is similar); the payment to him is at least $\frac{k_1B}{n}$ due to
individual rationality. Then consider the original instance $(\frac{k_2 B}{n}, \frac{(n-k_2 )B}{n})$; item 1 should also win and get a threshold payment, which is equal to or greater than $\frac{k_1B}{n}$. Therefore the payment to second item because of the budget constraint is at most $B-\frac{k_1B}{n}= \frac{(n-k_1 )B}{n}$.
Since $\frac{(n-k_1)B}{n}<\frac{(n-k_2)B}{n}$, we arrive at a contradiction with either individual rationality or assumption that both items won in the instance $(\frac{k_2 B}{n}, \frac{(n-k_2)B}{n})$.
On the other hand, for all instances $(\frac{kB}{n}, \frac{(n-k)B}{n})$, both items win in the optimal solution with value 2.
Hence, the expected approximation ratio of any deterministic truthful budget feasible mechanism is at least
$\frac{1-\epsilon}{n-1}\cdot 1 + (n-2) \cdot \frac{1-\epsilon}{n-1}\cdot 2+ \epsilon \cdot 1=2-\epsilon -\frac{1-\epsilon}{n-1}.$
The ratio approaches to $2$ when $\epsilon \rightarrow 0$ and $n\rightarrow \infty$. This completes the proof.
\end{proof}
\section{Beyond Submodularity}
A natural generalization of knapsack is to consider heterogeneous items.
That is, we are given $m$ different types of items and each item has a (private) cost $c_i$ and a (public) value $v_i$,
as well as an indicator $t_i\in [m]$ standing for the type of item $i$. The goal is to pick items of different types
(i.e., one cannot pick more than one item of the same type) to maximize total value given a budget constraint $B$.
The knapsack problem studied in the last section is therefore a special case of the heterogeneous problem when all items are of different types.
However, we cannot simply apply the mechanisms for knapsack here because of heterogeneous items. (Notice however that the lower bounds established in the last section still work.)
The main difference of this problem with knapsack or general submodular functions is that here not every subset is a feasible solution. A straightforward greedy could end up with a very poor solution: Consider a situation that every type contains one very small item (both $v_i$ and $c_i$ are very small) but with large value cost ratio $\frac{v_i}{c_i}$; greedy will take all these small items first and therefore not be able to take more since each type already has one item. The overall value of this greedy solution can be arbitrarily bad compared to the optimal solution.
To construct a truthful mechanism for heterogeneous knapsack, we employ a greedy strategy with {\em deletions}. The main idea is that at every time making a greedy move, we consider two possible changes: (i) add a new item whose type has not been considered before, and (ii) replace an existing item with the new one of same type. Among all the possible choices (of two types), we greedily select items with highest value cost ratio: In the case of adding a new item, its value cost ratio is defined as usually $\frac{v_i}{c_i}$. For the replacement case where we replace $i$ with $j$, its marginal value is $v_j-v_i$ and marginal cost is $c_j-c_i$, and hence its value cost ratio is defined to be $\frac{v_j-v_i}{c_j-c_i}$.
As before, now we assume that all the items are ordered according to their appearances in the greedy algorithm (note that some items never appear in the algorithm and we simply ignore them). The following greedy strategy is similar to what we did for the knapsack problem. In Appendix~\ref{appendix-heterogeneous}, we prove that it is monotone (therefore truthful) and budget feasible.
(Here for notation simplicity, we assume that we already take an item with $c=0$ and $v=0$ for each type, thus every greedy step can be viewed as a replacement.)
\begin{center}
\small{}\tt{} \fbox{
\parbox{4.5in}{
\hspace{0.05in} \\[-0.05in] $\greH$
\begin{enumerate}
\item Let $k=1$, $S=\emptyset$, and $last[j]=0$ for $j\in[m]$
\item While $k\le |A|$ and $c(k)-c(last[t_k])\le B\cdot \frac{v(k)-v(last[t_k])}{v(k)-v(last[t_k])+\sum_{i\in S}v(i)}$
\begin{itemize}
\item let $S \leftarrow (S\setminus\{last[t_k]\})\cup \{k\}$
\item let $last[t_k] = k$
\item let $k\leftarrow k+1$
\end{itemize}
\item Return winning set $S$
\end{enumerate}
}}
\end{center}
By applying the above $\greH$, we have the following claim for heterogeneous knapsack. (Details can be found in Appendix~\ref{appendix-heterogeneous}.)
\begin{theorem}\label{theorem-star-knapsack-mechanism}
There are $2+\sqrt{2}$ approximation deterministic and 3 approximation randomized polynomial truthful budget feasible mechanisms for knapsack with heterogeneous items.
\end{theorem}
Finally, we comment that greedy is typically the first choice when we consider designing truthful mechanisms because it usually has a nice monotone property. However, when we allow cancelations in the greedy process, its monotonicity may fail. In the heterogeneous knapsack problem, fortunately $\greH$ is still monotone (although its proof is much more involved) and therefore we are able to apply it to design truthful mechanisms with good approximation ratios. Our idea sheds light on the possibility of exploring budget feasible mechanisms in larger domains beyond submodularity.
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\section{Introduction} \vspace{-2mm}
Neutral $B$-$\bar B$-mixing is a phenomenologically important quantity because it allows us to determine CKM matrix elements. Within the Standard Model the dominant contribution is given by box diagrams with top quarks as shown in Figure \ref{fig:MixingDiagrams}. Experimentally, the observable quantity is the mass difference (also named oscillation frequency) $\Delta m_q$, where the subscript $q$ labels the light quark content ($d$ or $s$) of the $B$ meson. The mass difference is parametrized as \cite{Buras:1990fn}
\begin{align}
\Delta m_q = \frac{G_F^2m^2_W}{6\pi^2} \eta_B S_0 m_{B_q}{f_{B_q}^2B_{B_q}} \abs{V_{tq}^*V_{tb}}^2.
\end{align}
where $m_{B_q}$ is the mass of the $B_q$-meson, and $V_{tq}^*$ and $V_{tb}$ denote CKM matrix-elements. The Inami-Lim function, $S_0$ \cite{Inami:1980fz}, and the QCD coefficient, $\eta_B$ \cite{Buras:1990fn}, can be computed perturbatively, while $f_q^2 B_{B_q}$ is the non-perturbative input: the decay constant $f_q$ and the $B$-meson bag parameter $B_{B_q}$.
We define $SU(3)$ breaking ratios as the ratio of a quantity for the $B_s$-meson over the same quantity for the $B_d$ -meson. In particular we are interested in
\begin{align}
\xi &= \frac{f_{B_s}\sqrt{B_{B_s}}}{f_{B_d}\sqrt{B_{B_d}}}
\end{align}
and we also consider the ratio of the decay constants ${f_{B_s}}/{f_{B_d}}$. Computing $\xi$ non-perturbatively allows one with additional experimental input to extract the ratio of CKM matrix elements
\begin{align}
\frac{\Delta m_s}{\Delta m_d} = \frac{m_{B_s}}{m_{B_d}}\,{\xi^2} \, \frac{\abs{V_{ts}}^2}{\abs{V_{td}}^2}.
\end{align}
The phenomenological importance of $B$-$\bar B$ mixing is given by the fact that the ratio ${\abs{V_{ts}}^2}/{\abs{V_{td}}^2}$ constrains the apex of the unitarity triangle \cite{Antonelli:2009ws}. Experimentally, $\Delta m_d$ and $\Delta m_s$ are measured to better than a percent \cite{Amsler:2008zzb,Abazov:2006dm,Abulencia:2006mq,Abulencia:2006ze}, whereas we know $\xi$ only to about $3\%$. Hence in order to get a stronger constraint on the apex of the unitarity triangle and consequently on new physics beyond the Standard Model, we need to improve the determination of $\xi$.
\begin{figure}[b]
\begin{center}
\vspace{-3mm}
\includegraphics[width=0.8\textwidth]{images/Mixing}
\caption{The dominant contribution to $B$-$\bar B$-mixing.}
\end{center}
\label{fig:MixingDiagrams}
\end{figure}
The project presented here aims at this task by demonstrating the viability of our method on $16^3\times 32 \times 16$ dynamical domain-wall fermion ensembles generated by the RBC and UKQCD collaborations \cite{Allton:2007hx}. Our central values agree with the ones published in the literature, however our errors are large. A future follow-up project will hopefully lead to results with comparable uncertainties to those obtained by Gamiz et al.~\cite{Gamiz:2009ku} and Evans et al.~\cite{ToddEvans:2008}. Details of the present computation will soon be published in \cite{ThePaper}.
\section{Actions and parameters} \label{sec:Method}\vspace{-2mm}
In order to accomplish the computation of $B$-$\bar B$-mixing we choose domain-wall fermions \cite{Kaplan:1992bt,Shamir:1993zy} for the light quarks ($u$, $d$, $s$), the Iwasaki gauge action \cite{Iwasaki:1983ck}, and a static action with link-smearing for the $b$-quarks \cite{Eichten:1989kb}.
Domain-wall fermions are formulated in five dimensions and posses an approximate chiral symmetry. Left-handed modes are bound to a 4-d brane at $s=0$, while right-handed modes are bound to a 4-d brane at $s=L_s-1$. The overlap between both is exponentionally suppressed. Besides the approximate chiral symmetry, domain-wall fermions are advantageous because the renormalization is simplified due to reduced operator mixing. The choice of the Iwasaki gauge action is motivated by the fact that it improves chiral symmetry and reduces the residual quark mass in combination with domain-wall sea quarks \cite{Aoki:2002vt} over e.g.~the Wilson gauge action \cite{Wilson:1974sk}.
Our simulations use $16^3 \times 32 \times 16$ dynamical 2+1 flavor domain-wall lattices \cite{Allton:2007hx} in which the up and down sea quarks are degenerate and the strange sea quark is slightly heavier than its physical value \cite{Allton:2008pn}. The domain-wall height is set to $M_5 = 1.8$ and we have $\beta = 2.13$. We estimate the inverse lattice spacing to be $a^{-1} = 1.729(28)$ GeV and the residual quark mass to be $a m_\text{res}=0.00315$ \cite{Allton:2008pn}. Hence $a$ is approximately $0.11$ fm and we have a (1.8 fm)$^3$ box. In Table \ref{tab:SimParameters} we list the values of the light ($m_l$) and heavy ($m_h$) sea quark masses, the values of the valence quark masses ($m_x$), the mass of the pions, and the number of configurations used in our simulations.
\begin{table}[bt]
\begin{center}
\begin{tabular}{cccccc}\toprule
& & & &\multicolumn{2}{c}{\# configs.}\\
$a m_l$ & $a m_h$ & $a m_x$ & $m_\pi$ (MeV)& APE & HYP \\ \midrule
0.01 & 0.04 & 0.01, 0.0359 & 400 & 298& 300\\
0.02 & 0.04 & 0.02, 0.0359 & 530 & 298& 300\\
0.03 & 0.04 & 0.03, 0.0359 & 630 & 298& 300\\ \bottomrule
\end{tabular}
\caption{The light ($m_l$) and heavy ($m_h$) sea quark masses, the valence quark masses ($m_x$), the mass of the pions ($m_\pi$) and the number of configurations used in our simulations.}
\label{tab:SimParameters}
\end{center}\vspace{-5mm}
\end{table}
For the $b$-quarks we use a static action which is based on the original lattice formulation by Eichten and Hill \cite{Eichten:1989kb} but improved by link-smearing. This formulation corresponds to an effective action in the limit of infinitely heavy $b$-quarks, i.e.~corrections of order $1/m_b$ are neglected. Advantages of this static action are that the static quark propagator is given as product of gauge links, the propagator is $O(a)$ improved and has a simple continuum limit. We also enhance the signal-to-noise ratio by link-smearing in comparison to the original formulation \cite{DellaMorte:2003mn}. For this work we utilize APE smearing \cite{Albanese:1987ds, Falcioni:1984ei} and hypercubic blocking (HYP smearing) \cite{Hasenfratz:2001hp, DellaMorte:2005yc}.
In case of APE smearing with smearing parameter $\alpha=1$, one computes all staples for a given link, adds them and projects the result finally back onto $SU(3)$. HYP smearing consists of three steps of APE smearing restricted to the links within the hypercube surrounding a given link. For each step $i$ we have a choice for the smearing parameters $\alpha_i$. Following Della Morte et al.~\cite{DellaMorte:2005yc} we choose $(\alpha_1, \alpha_2, \alpha_3)=(1.0,\, 1.0,\, 0.5)$, which is commonly referred to as ``HYP2''. The required $SU(3)$ projection is not unique: in the case of APE smearing it is performed by the unit circle projection method \cite{Kamleh:2004xk}, while for HYP smearing we employ an iterative procedure which yields the $SU(3)$ projection of a matrix $V$ by seeking $U_\txt{max} \in SU(3)$ such that $\txt{Re}\, \txt{Tr}(U_\txt{max}V^\dagger)$ is maximal \cite{Bali:1992ab}. Both projections are equivalent in the weak coupling limit \cite{Kamleh:2004xk, ThePaper}.
\section{Lattice calculation} \label{sec:LatticeRenormalizationChiral}\vspace{-2mm}
The perturbative renormalization of the heavy-light axial current and $\Delta B = 2$ four fermion operator follow a two-step matching procedure. First we match the continuum QCD operators at a scale $\mu_b$ in the $\overline{MS}$-scheme using naive dimensional regularization onto operators in the continuum static effective theory at a scale $\mu$. Next, we match the operators in the continuum static effective theory to operators on the lattice \cite{Talk:Taku}. Combining the results of the two matching steps we obtain the perturbatively computed matching coefficients $c_A$, $Z_\Phi$, $Z_{VA}$, and $Z_{SP}$, which are listed in Table \ref{tab:Matching} \cite{PT_Oa}. We account for the truncation of the perturbative series in our estimate of the systematic errors.
\begin{table}[bt]
\begin{center}
\begin{tabular}{ccccc}
\toprule
smearing & $c_A$ &$Z_\Phi$ & $Z_{VA}$ & $Z_{SP}$ \\ \midrule
APE & 0.0653 & 0.9507 & 0.7485 & -0.1448 \\
HYP & 0.1204 & 0.9813 & 0.8108 & -0.1448 \\
\bottomrule
\end{tabular}
\caption{Perturbative matching coefficients for bilinear and four-quark operators evaluated for APE- and HYP-smeared static-quark gauge links.}
\label{tab:Matching}
\end{center}\vspace{-5mm}
\end{table}
For the bilinear operators we include the $O(\alpha_s pa)$ improvement term in our analysis
\begin{align}
\Phi^\text{ren}_B = Z_\Phi(1+c_A \sinh(m^*_B))\Phi_B^\text{lat},
\end{align}
but we have not yet implemented $O(pa)$ improvement for the four-quark operator. The renormalized decay amplitude $\Phi^\text{ren}_B$ is related to the decay constant $f_{B_q} = \Phi^\text{ren}_B/\sqrt{m_{B_q}}$ and is obtained by computing on the lattice the ratio of two-point functions of the heavy-light axial current using local (L) and wall (W) sources/sinks:
\begin{align}
\Phi_B^\text{lat} = \lim_{t\gg t_0}\sqrt{\frac{2}{L^3}}\frac{\abs{{\cal C}^{LW}(t,t_0)}}{\sqrt{{\cal C}^{WW}(t,t_0)\text{e}^{-m_{B_q}^*(t-t_0)}}},
\end{align}
where $m_{B_q}^*$ is the unphysical $B_q$-meson rest mass. For the matrix elements containing the four-fermion operator we compute additional three-point functions ${\cal C}_{\cal O}$ using box sources/sinks (B) \cite{Christ:2007cn} in case of APE smearing:
\begin{align}
M_{\cal O}^\text{lat} = \lim_{t_f\gg t \gg t_0}2\, \frac{{\cal C}_{\cal O}^B(t_f,t, t_0) \text{e}^{m_{B_q}^*(t_f-t_0)/2}}{\sqrt{{\cal C}^{BB}(t,t_f){\cal C}^{BB}(t,t_0)}},
\end{align}
and wall sources/sinks (W) for HYP smearing:
\begin{align}
M_{\cal O}^\text{lat} = \lim_{t_f\gg t \gg t_0} L^3 \frac{{\cal C}_{\cal O}^W(t_f,t, t_0)}{{\cal C}^{LW}(t,t_f){\cal C}^{LW}(t,t_0)} \cdot\left( \Phi_{B_q}^\text{ren}\right)^2,
\end{align}
Because the four-quark operators of different chiralities mix under renormalization, we obtain the renormalized expression for the matrix element by
\begin{align}
M^\text{ren}_{B_q} = Z_{VA} M^\text{lat}_{VV+AA} + Z_{SP}M^\text{lat}_{SS+PP}.
\end{align}
Finally, we extrapolate our lattice data to the physical quark masses and the continuum using next-to-leading order partially quenched $SU(3)$ heavy-light meson chiral perturbation theory. Schematically, the expressions for the $SU(3)$ breaking ratios are given by
\begin{align}
\frac{\Phi_{B_{s^\prime}}^\textrm{ren}}{\Phi_{B_l}^\textrm{ren}} & = 1 + \textrm{``chiral logs"} + \frac{2\mu}{(4\pi f)^2}\tilde{c}_\textrm{val} (m_{s^\prime} - m_l), \\
\sqrt{\frac{M^\textrm{ren}_{B_{s^\prime}}}{M^\textrm{ren}_{B_l}}} & = 1 + \textrm{``chiral logs"} + \frac{\mu}{(4\pi f)^2} \tilde{d}_\textrm{val} (m_{s^\prime} - m_l),
\end{align}
where the quark masses are expressed as dimensionless ratios and ``chiral logs'' denote non-analytic functions of the pseudo-Goldstone meson masses. Performing a linear, one-parameter fit of our data with respect to the expressions above we are able to extract the physical value for $\Phi^\txt{ren}_s/\Phi^\text{ren}_d$ and $\xi$. As input parameters we use: $\Lambda_\chi = 1$ GeV, $\mu = 2.35(16)$ \cite{Allton:2008pn}, $f_\pi = 130.4$ MeV \cite{Amsler:2008zzb}, $g_{B^*B\pi} = 0.516$ \cite{Ohki:2008py}, $a m_{ud}+ am_{res} = 0.001300(62)$, $a m_s+a m_{res} = 0.0375(17)$ \cite{Allton:2008pn}. A detailed description of our chiral extrapolation will be included in \cite{ThePaper}, where we also will discuss the alternative of using heavy-light $SU(2)$ chiral perturbation theory.
\section{Results and Conclusion} \label{sec:Results}\vspace{-2mm}
We present the preliminary results of our chiral extrapolation in Fig.~\ref{fig:Fits}. The plot on the left shows the data and the fit for $\Phi_{B_s}/\Phi_{B_d}$, the plot on the right the outcome for $\sqrt{m_{B_s}/m_{B_d}}\cdot \xi$. In both cases only statistical errors are shown, which are computed following to Ref.~\cite{Wolff:2003sm}. In addition we estimate systematic errors by varying the input parameters around their uncertainty, considering a constrained linear fit as alternative fit function, and using power-counting for the discretization errors, the errors due to the renormalization factors, the finite volume errors $1/m_b$ corrections. All of these errors are listed in Table \ref{tab:TotalError} and discussed in detail in \cite{ThePaper}.
\begin{figure}[tb]
\begin{picture}(150,54)
\put(0,0){\includegraphics[width=0.52\textwidth,clip]{images/FitZeta}}
\put(75,0){\includegraphics[width=0.52\textwidth,clip]{images/FitXi}}
\end{picture}
\caption{Chiral extrapolation of $\Phi_{B_s}/\Phi_{B_d}=\sqrt{m_{B_s}/m_{B_d}}\cdot f_{B_s}/f_{B_d}$ (left) and $\sqrt{m_{B_s}/m_{B_d}} \cdot \xi$ (right). The APE data points are shown as blue triangles, whereas the HYP data points are shown as red squares. The color of the shaded (hatched) error bands match those of the APE (HYP) data points. The physical average $u-d$ quark mass is indicated by the dashed line and the black dot denotes the physical strange quark mass. Only statistical errors are shown.}
\label{fig:Fits}
\end{figure}
\begin{table}
\begin{center}\vspace{-5mm}
\begin{tabular}{lcccccc} \\ \toprule
& &\multicolumn{2}{c}{$f_{B_s}/f_{B_d}$} & &\multicolumn{2}{c}{$\xi$} \\
uncertainty & \quad\quad &APE & HYP & \quad\quad & APE & HYP \\ \midrule
statistics && 7\% & 4\% && 5\%& 4\% \\ \hline
chiral extrapolation && 11\% & 11\% && 12\%& 11\% \\
uncertainty in $g_{B^*B\pi}$ && 4\% & 4\%&& 3\%& 3\% \\
discretization error && 3\% & 3\%&& 4\% & 4\%\\
renormalization factors && 0\% & 0\%&& 2\% & 2\% \\
scale and quark mass uncertainties && 1\%& 1\% && 1\% & 1\% \\
finite volume error && 1\% & 1\%&& 1\% & 1\%\\
$1/m_b$ corrections && 2\% & 2\%&& 2\% & 2\%\\
\hline
total systematics && 12\% & 12\%&& 13\%& 12\%\\
\bottomrule
\end{tabular}
\caption{Total error budget for the $SU(3)$-breaking ratios $f_{B_s}/f_{B_d}$ and $\xi$ rounded to the nearest percentage.}
\label{tab:TotalError}
\end{center}\vspace{-6mm}
\end{table}
Finally, we use the experimentally-measured ratio of the masses $m_{B_s^0}/m_{B_d^0} = 5366.6 / 5279.5 = 1.0165$~\cite{Amsler:2008zzb} to obtain the following values for the $SU(3)$-breaking ratios of $B$-meson decay constants and mixing matrix elements:
\begin{align}
\frac{f_{B_s}}{f_{B_d}} =
\left\{ \begin{aligned}
& 1.20(08)(14) \quad\textrm{APE}\\
& 1.19(05)(14) \quad\textrm{HYP}
\end{aligned}\right. \, ,
\qquad \text{and} \qquad
\xi =
\left\{ \begin{aligned}
&1.19 (06)(15) \quad\textrm{APE}\\
&1.19 (05)(14) \quad\textrm{HYP}
\end{aligned}\right. \, ,
\end{align}
where the first errors are statistical, the second are the sum of all systematic errors added in quadrature. Currently, we are updating our analysis and hence these values may change in our publication \cite{ThePaper}. We find that both smearings used agree very well indicating that the discretization errors are small in the ratios. When comparing these new results to the ones in the literature published by the HPQCD collaboration \cite{Gamiz:2009ku} and presented by the FNAL-MILC collaboration at Lattice 2008 \cite{ToddEvans:2008}, we also find good agreement (see Fig.~\ref{fig:LatResults}). However, our errors are large and we look forward to improve upon them in future works. \vspace{-2mm}
\begin{figure}[b]
\vspace{-4mm}
\includegraphics[width=\textwidth]{images/LatticeResults}
\caption{Comparison of lattice QCD results for $f_{B_s}/f_{B_d}$ (left) and $\xi$ (right). Our new preliminary data points are marked as blue triangle (APE) and red square (HYP). The gray circle marks the values published by the HPQCD collaboration \cite{Gamiz:2009ku}, the beige diamond the preliminary value presented by the FNAL-MILC collaboration at Lattice 2008\cite{ToddEvans:2008}.}
\label{fig:LatResults}
\end{figure}
\section*{Acknowledgments}\vspace{-2mm}
I am thankful to all the members of the RBC and UKQCD
collaborations.
Numerical computations for this work were performed on
the QCDOC computers of the RIKEN-BNL Research Center
and the USQCD Collaboration, in part funded by the
Office of Science of the U.S.~Department of Energy.
This manuscript has been authored by
an employee of Brookhaven Science Associates, LLC
under Contract No.~DE-AC02-98CH10886 with the
U.S.~Department of Energy.
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BkiUdfM4uBhi_K2XLntv
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\section{INTRODUCTION}
\label{sec:intro}
The system design of the Adaptive Optics Module (AOM) of the MCAO Assisted Visible Imager and Spectrograph\cite{2021Msngr.185....7R} (MAVIS) has its roots in a idea from Esposito \textit{et al.}\cite{2016SPIE.9909E..3UE} and it is heavily supported by numerical simulations.
The main parameters of the systems, like the number of Laser Guide Stars (LGS), the number of Deformable Mirrors (DM) and the pitch of the DMs, were selected during the first phase of the design by studies presented in Ref. \citeonline{2020SPIE11447E..1RR,2020SPIE11448E..3RA,2020SPIE11448E..0DV,2020SPIE11448E..6WG,2020SPIE11448E..2LC,2020SPIE11448E..2CZ}.
We proved in these studies that the system is able to fulfill very tight requirements in terms of V band SR -- 10\% on a FoV of 30arcsec diameter (15\% goal) in standard atmospheric conditions at Paranal with bright Natural Guide Stars ($\mathrm{m_J}$=8) -- and sky coverage -- larger than 50\% in the south galactic pole guaranteeing an ensquared energy of at least 15\% on 50mas side in V band (in the same standard atmospheric conditions).
During the second phase of the design the work of performance estimation focused on new simulations with a refined set of the parameters -- we updated some parameters, like the laser spot characteristics and the ones related to the control strategy -- and sensitivity analysis on the various aspect of the atmosphere and of the system.
We relied on a few numerical tools to obtain the results presented in this work: end-to-end simulation tools like PASSATA \cite{doi:10.1117/12.2233963}, COMPASS\cite{2014SPIE.9148E..6OG,2016SPIE.9909E..71G} and YAO\cite{2013aoel.confE..18R}, Fourier-based tools like TIPTOP\cite{2021SPIE11448E..2TN} and other analytical tools like the ones used to estimate sky coverage presented in Ref. \citeonline{2020SPIE11448E..3RA}.
We crosschecked their results to be more confident in their performance prediction: in particular we verified that PASSATA and COMPASS give comparable results in the same configuration within an error of $<<$1\% V band SR.
Then, we refined these tools during this phase to allow us to study new features as we will show in the next sections.
This article is structured in a couple of sections: Sec. \ref{sec:params} focuses on the set of parameters, the control aspects and the baseline performance in the standard atmospheric conditions and Sec. \ref{sec:sens} shows how performance changes as a function of the main parameters and of mis-alignments.
\section{Baseline configuration}\label{sec:params}
In this section we report the parameters used in the baseline configuration. They are summarized in Tab. \ref{tab:params} and they are an update of what we presented in Ref. \citeonline{2020SPIE11448E..3RA}.
In particular we changed the total delay, the NGS pixel scale (see Sec. \ref{sec:LOpixel}) and the parameters of the laser spot.
We have also evaluated the impact of the influence functions of the deformable secondary mirror of the VLT\cite{10.1117/12.2057591} and spider shadows, but we found comparable results with respect to the DM and pupil used in this work.
We decided to keep two options for the control of the baseline configuration: the first one is the pseudo open loop control with Infinite Impulse Response filters and split tomography as described in Ref. \citeonline{2020SPIE11448E..3RA} and \citeonline{Busoni2019} and the second one is the predictive learn and apply approach reported in Ref. \citeonline{2020SPIE11448E..2LC} and \citeonline{2020SPIE11448E..2CZ}.
One important new feature is the adoption of ad-hoc mis-alignment between SH-WFS sub-aperture grid and the pupil to gain geometrical super-resolution\cite{2022JATIS...8b1514F}: this allows for a reduction of the error of 25nm with respect to the classical approach.
Another new feature is the use of noise priors\cite{Michel-Tallon:2008aa,2010JOSAA..27A...1B,Oberti2019} for the computation of the Minimum Mean Square Error (MMSE) reconstruction matrix.
MAVIS is not affected by significant truncation error, nevertheless it benefits from a more accurate model of the noise covariance matrix: thanks to this approach we are able to reduce the error of 12nm.
We present in Table \ref{tab:budget.HO} the residual WFE breakdown from end-to-end simulations in the baseline configuration for both high and low orders. We quantified each error source individually except for tomographic, generalized fitting and aliasing error that are presented together. The sum of individual HO terms is 107nm, of individual LO terms is 30nm (good NGS asterism with bright stars) and the overall error considering all the error sources (the terms that are not considered in the numerical simulations are reported in Tab. \ref{tab:budget.HO} and in more details in Ref. \citeonline{valentina2022aom}) is 127nm that means a V band SR of 12.3\% ($>$10\% that is the requirement).
Please note that when the predictive learn $\&$ apply control is used the performance improves because the temporal error (38nm, see Tab. \ref{tab:budget.HO}, that means a ratio of 0.83 of V band SR)
(see Tab. \ref{tab:budget.HO}) is reduced as can be seen in Fig. \ref{fig:MAVIS_SRV}.
Please note that in this figure the V band SR with the full error budget (blue dashed line) is slightly higher than the one presented above in the text (about 1\%).
We think that the discrepancy is mainly due to the finite simulated time, 5s, and to the finite sampling of the pupil (0.0222m/pix).
In particular this second features produces a slightly smaller fitting error of 10nm.
Another 10nm are due to the approximation of the DM pitch: in the error budget we considered a pitch of 0.22m, while in the simulation the exact value is 0.216m.
This shows how much the V band SR is sensible to small errors and approximations that have typically no impact on systems with a larger error budget.
Then in terms of sky coverage we get an Ensquared Energy (EE) in 50 mas at 550 nm of more than 19.4\% for half of the pointings at the South Galactic Pole (Fig. \ref{fig:sky_cov_EE}), which is above the requirement of EE = 15\% for half of the pointings. This corresponds to a Full Width at Half Maximum (FWHM) less or equal to 40.7 mas (Fig. \ref{fig:sky_cov_FWHM}). This result takes also into account the extra error of 58 nm (see Tab. \ref{tab:budget.HO}) as a SR reduction on the NGSs, following the Mar{\'e}chal approximation. The method to compute sky coverage is described in Agapito \textit{et al.} 2020 \cite{2020SPIE11448E..3RA}. Note that the computation of low-order residuals for a given NGS asterism is the same as in the TIPTOP software \cite{2021SPIE11448E..2TN}.
\begin{figure}[h]
\centering
\includegraphics[width=0.45\columnwidth]{MAVIS_SRV.png}
\caption{V band SR as a function of off-axis angle for the two control options of MAVIS (see Tab. \ref{tab:params}) considering the output of the end-to-end simulations and the full error budget.}
\label{fig:MAVIS_SRV}
\end{figure}
\begin{figure}[h]
\centering
\subfigure[V band ensquared energy in 50mas diameter.\label{fig:sky_cov_EE}]
{\includegraphics[width=0.48\columnwidth]{sky_cov_mavis_H_n1e4_fov60_mag20_4dirs_1x1_1x1_1x1_v7_EE50.png}}
\subfigure[V band FWHM.\label{fig:sky_cov_FWHM}]
{\includegraphics[width=0.48\columnwidth]{sky_cov_mavis_H_n1e4_fov60_mag20_4dirs_1x1_1x1_1x1_v7_FWHM.png}}
\caption{Sky coverage at the South Galactic Pole.\label{fig:sky_cov}}
\end{figure}
\begin{table}[h]
\caption{Summary of the baseline MAVIS parameters (in simulation).}
\label{tab:params}
\begin{center}
\begin{small}
\begin{tabular}{|l|c|}
\hline
\textbf{Parameter} & \textbf{value}\\
\hline
Telescope diameter & 8m (sampled by 360pixels)\\
Central obstruction & 1.25m (DSM wind screen external diameter)\\
Pupil mask & round, no spiders\\
Zenith angle & 30deg\\
Science FoV (diameter) & 30arcsec\\
Technical FoV (diameter) & 120arcsec\\
\hline
Atmospheric turbulence & 1 profile with 10 layers (Paranal median)\\
$r_0$ & 0.126m\\
$L_0$ & 25m\\
\hline
NGS full throughput & 0.22\\
LGS full throughput (with laser splitting) & 0.18\\
\hline
Ground DM & conjugated at 0m with $\sim$0.22m pitch (circular act. grid)\\
Post focal DM no 1 & conjugated at 6000m with $\sim$0.25m pitch\\
Post focal DM no 2 & conjugated at 13500m with $\sim$0.32m pitch\\
\hline
NGS WFS number & 3\\
NGS WFS off-axis angle (good asterism) & 20arcsec\\
NGS WFS nSA & 1 or 2\\
NGS WFS pixel scale & 20 (1SA) or 40 (2SA) mas\\
NGS WFS detector RON & 0.5 $\mathrm{e^-/pixel/frame}$\\
NGS WFS detector dark current & 20 $\mathrm{e^-/pixel/s}$\\
NGS (H band) sky background & 2100 $\mathrm{e^-/m^2/arcsec^2/s}$ (no moon)\\
NGS flux (full aperture, good asterism) & 1000 ph/ms\\
\hline
LGS WFS number & 8\\
LGS WFS off-axis angle & 17.5 arcsec\\
LGS WFS nSA & 40$\times$40\\
LGS WFS FoV & 5.0 arcsec\\
LGS WFS pixel scale & 0.866 mas\\
LGS WFS detector RON & 0.2 $\mathrm{e^-/pixel/frame}$\\
LGS WFS detector charge diffusion FWHM & 0.5pixel \\
LGS launcher number & 4\\
LGS launcher off-axis distance & 5.5m\\
LGS flux per sub-aperture (0.04 m$^2$) & 75ph/ms\\
LGS rotations (for super res.)& [133.7, 118.7, 61.3, 46.3, 313.7, 298.7, 241.3, 226.3] deg \\
LGS shifts (for super res.)& RMS = 0.1SA and absolute values $\leq$ 0.2SA\\
\hline
Sodium profile & ``multi peak''\cite{2014A&A...565A.102P} with short axis of 1.35arcsec\cite{haguenauer2022}\\
Laser jitter & 150mas RMS \\
\hline
Control (option 1) & POLC with split tomography\cite{Busoni2019}, IIR filters\cite{2020SPIE11448E..3RA} and noise priors\cite{Michel-Tallon:2008aa,2010JOSAA..27A...1B,Oberti2019}\\
Control (option 2) & predictive learn \& apply\cite{2020SPIE11448E..2LC,2020SPIE11448E..2CZ}\\
Centroiding algorithm & CoG (LGS), windowed CoG (NGS)\\
Framerate & 1000Hz\\
Total delay & 2.6ms\\
\hline
\multicolumn{2}{c}{\scriptsize Note: DSM is Deformable Secondary Mirror, NGS is Natural Guide Star, LGS is Laser Guide Star, DM is Deformable Mirror,}\\
\multicolumn{2}{c}{\scriptsize WFS is Wavefront Sensor, SA is Sub-Aperture, FoV is Field of View, RON is Read-Out Noise, POLC is pseudo-open loop }\\\multicolumn{2}{c}{\scriptsize control, IIR is Infinite Impulse Response and CoG is Center of Gravity.}\\
\end{tabular}
\end{small}
\end{center}
\end{table}
\begin{table}[h]
\caption{Breakdown of MAVIS AO residual wavefront error.}
\label{tab:budget.HO}
\smallskip
\begin{minipage}{.5\linewidth}
\centering
\begin{tabular}{| l | c |}
\hline
\multicolumn{2}{| l |}{\textbf{High Orders}}\\\hline
Error term & Error [nm]\\\hline
High-frequency fitting\cite{Rigaut_1998} & 65.3\\ \hline
Tomogr. + gen. fitt. + alias. & 58.1\\ \hline
Measurement noise & 40.5\\ \hline
Temporal & 37.9\\ \hline
Sodium elongation/truncation & 26.4\\ \hline
LGS jitter & 5.4\\ \hline
\end{tabular}\smallskip
\end{minipage}
\begin{minipage}{.5\linewidth}
\centering
\begin{tabular}{| l | c |}
\hline
\multicolumn{2}{| l |}{\textbf{Low Orders (bright)}}\\\hline
Error term & Error [nm]\\\hline
Tomographic & 27.9\\ \hline
Measurement noise & $\sim$0\\ \hline
Temporal & 11.3\\ \hline
\end{tabular}
\begin{tabular}{| l | c |}
\hline
\multicolumn{2}{| l |}{\textbf{Other errors}}\\\hline
Error term & Error [nm]\\\hline
Extra & 58.0\\ \hline
Vibrations (all modes) & 21.0\\ \hline
\end{tabular}\smallskip
\end{minipage}
\end{table}
\section{SENSITIVITY ANALYSIS}\label{sec:sens}
\subsection{Sensitivity to atmospheric profile}\label{sec:srMap}
In this section we present how MAVIS performance changes with the atmospheric profile.
We based this analysis on the release 2019B of the Stereo-SCIDAR profiles\cite{2020SPIE11448E..1WB,10.1093/mnras/sty1070}: we compute the average V band SR in the science FoV for 19356 profiles with TIPTOP and we summarize the results in Fig. \ref{fig:srMapAll}.
Maximum SR values can be found in the top-left part of the plot, where seeing is small and $\theta_0$ is large, and minimum SR values can be found in the bottom-right part, where seeing, $\epsilon$, is large and $\theta_0$ is small.
Iso-performance curves are oriented approximately as lines $\theta_0-k \epsilon = 0$.
We did a further analysis considering these profiles: we selected 529 profiles with seeing close (±1\%) to the median value, 0.72arcsec, and we computed tomography and general fitting error in the science FoV, $\Phi$=30arcsec, for combination of post focal altitudes 4-8 and 12-18km.
We found that best conjugation altitudes are 5 and 14km, close to the current baseline values, and sensitivity is small with a large combination of altitudes close to an average error of 60nm.
\begin{figure}[h]
\centering
\subfigure[Average V band SR map.\label{fig:srMap1}]
{\includegraphics[width=0.48\columnwidth]{seeing_theta0_space_SR.png}}
\subfigure[Standard deviation of the V band SR map.\label{fig:srMap2}]
{\includegraphics[width=0.49\columnwidth]{seeing_theta0_space_SR_stdev.png}}
\caption{V band SR (average on science FoV) map as a function of seeing and $\theta_0$. Bins size is 0.05arcsec seeing $\times$ 0.15arcsec $\theta_0$. The values reported are the average SR for all the profiles that fall in the bin.\label{fig:srMapAll}}
\end{figure}
\subsection{Sensitivity to zenith angle and seeing}
Sensitivity to zenith angle and seeing was evaluated with end-to-end simulations. We found that MAVIS is strongly sensitive to these parameters:
\begin{itemize}
\item Going from 50 to 10deg of zenith we gain a factor 9 in V band SR (see Fig. \ref{fig:sens_seeing}). Performance changes because effective seeing is a function of the airmass, but also because distance from the pupil of the atmospheric layers is a function of the airmass. In fact, atmospheric layers conjugation altitude impacts generalized fitting and tomographic error. Note that error variance changes following approximately a power of 5/3 of the airmass.
\item From the worst seeing case to the baseline one there is a factor 20 on V band SR (see Fig. \ref{fig:sens_zenith}). Performance change is dominated by the fitting error and error variance changes following approximately a power of 5/3 of the seeing value.
\end{itemize}
We think that these results are a good occasion to emphasize once again the fact that V band SR is highly sensitive to the simulation parameters.
\begin{figure}[h]
\centering
\subfigure[Sensitivity to zenith angle. Error variance changes following approximately a power of 5/3 of the airmass.\label{fig:sens_zenith}]
{\includegraphics[width=0.43\columnwidth]{sens_zenith.png}}
\subfigure[Sensitivity to seeing value (each point is the average of 5 atmospheric and noise realizations). Error variance changes following approximately a power of 5/3 of the seeing value.\label{fig:sens_seeing}]
{\includegraphics[width=0.42\columnwidth]{sens_seeing.png}}
\caption{V band SR as a function of zenith angle and seeing.\label{fig:sens_zen_see}}
\end{figure}
\subsection{Sensitivity to $\theta_0$}
We derived sensitivity to $\theta_0$ from the analysis presented in Sec.\ref{sec:srMap}.
We chose those profiles having seeing in the interval $\pm$2\% of a given input seeing: 0.6, 0.8 and 1.0arcsec.
We binned the error as a function of isoplanatic angle at steps of 0.5arcsec and fit a power law to the data (see Fig.~\ref{fig:SR_vs_theta0_bins_05}): we found that the error variance changes following approximately a power of -4/5 of the $\theta_0$ value.
\begin{figure}[h]
\centering
\includegraphics[width=0.90\linewidth]{SR_vs_theta0_bins_05.png}
\caption{V band SR as a function of $\theta_0$ from the analysis presented in Sec.\ref{sec:srMap} (each point is the average of the atmospheric profiles with $\theta_0$ in bins of 0.5arcsec) for three different seeings: 0.6, 0.8 and 1.0arcsec. Error variance follows approximately a power of $-4/5$ of the $\theta_0$ value.}
\label{fig:SR_vs_theta0_bins_05}
\end{figure}
\subsection{Sensitivity to wind speed}
Sensitivity to this parameter is relatively low with respect to other atmospheric parameters, nevertheless without a predictive control the V band SR could decrease by a factor 2 going from the best conditions to the bad ones.
Error follows approximately a power of 2 of the average wind speed because no specific optimization is used here and most of the atmospheric turbulence is below the bandwidth of the temporal filters.
\begin{figure}[h]
\centering
\includegraphics[width=0.45\linewidth]{sens_wind.png}
\caption{V band SR as a function of wind speed (each point is the average of 5 atmospheric and noise realizations). Error variance follows approximately a power of 2 of the average wind speed.}
\label{fig:sens_wind}
\end{figure}
\subsection{LGS return flux}
We consider here the performance sensitivity to LGS flux. Sensitivity to this parameter is relatively low, in particular when the most probable values are considered: baseline value is 3500ph/ms/m$^2$, while 90 percentile value is approximately half of this value\cite{haguenauer2022}. We have a reasonable margin in case of the requirement of 10\% V band SR (see dashed line at 1750ph/ms/m$^2$ it is about 12\%), and with good return flux it is possible to fulfill the goal of 15\% V band SR.
Performance does not scale simply as a function of the square root of the flux because noise covariance matrix and temporal filter gains are optimized for each flux considered.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.45\linewidth]{Flux_performance.png}
\caption{V band SR as a function of laser return flux, dashed line is considering the full error budget. Please note: from ph/ms/m$^2$ to ph/s/m$^2$ at M1 entrance: multiply by 2777, given by reciprocal of full throughput (1/0.18), the splitting factor (0.5) and conversion in seconds (1000).}
\label{fig:Flux_performance}
\end{figure}
\subsection{LGS spot size and jitter}
We studied the sensitivity to the main parameters of the LGS spot: size of the major axis (sodium elongation), size of the minor axis (uplink propagation) and jitter (residual tip-tilt).
We already know\cite{2020SPIE11448E..3RA} that MAVIS has a relatively low sensitivity to sodium elongation when the most elongated spots fall on the diagonal of the sub-aperture, with a difference of a few nm of errors between ``multi-peak'' and ``very wide'' sodium profiles\cite{2014A&A...565A.102P}.
Then, in the last months, we introduced noise priors\cite{Michel-Tallon:2008aa,2010JOSAA..27A...1B,Oberti2019} that are able to further reduce this sensitivity and we focused on the value of the LGS WFS FoV: we verified that there is no improvement in increasing it from 5.0 to 5.5arcsec.
Then we focused on sodium laser spot minor axis and laser jitter.
Sensitivity to laser spot minor axis is reported in Fig.~\ref{fig:short_axis}. Here we expect values of about 1.35arcsec\cite{haguenauer2022}.
Then, we simulated uplink propagation and jitter compensation using the tip-tilt mirror coupled to each launcher, with bandwidths from 100 to 1000Hz and seeing values up to 1.5arcsec, and considering that a single mirror has to correct for both LGS generated by a single launcher. We found that best case residual jitter is of the order of 100mas and worst case is of the order of 300mas (minimum and maximum abscissa values of Fig.~\ref{fig:jitter}).
Note that the sensitivity to laser jitter is used to define requirements to the bandwidth of the jitter compensation loop and to the jitter mirrors.
\begin{figure}[h]
\centering
\subfigure[V band SR as a function of sodium laser spot minor axis.\label{fig:short_axis}]
{\includegraphics[width=0.44\columnwidth]{short_axis.png}}
\subfigure[V band SR as a function of sodium laser jitter and for different spot minor axis values ($\phi$).\label{fig:jitter}]
{\includegraphics[width=0.42\columnwidth]{jitter.png}}
\caption{Sensitivity to laser parameters.\label{fig:laser}}
\end{figure}
\subsection{LO WFS pixel scale}\label{sec:LOpixel}
We present here the study on the LO WFS pixel scale: we want to find the best pixel scale for both configurations of the LO WFS the 2$\times$2 and 1$\times$1 and we use an analytical approach to evaluate the measurement noise error for different flux levels and pixel scales.
We consider either a 1$\times$1 or 2$\times$2 configuration in two cases:
\begin{itemize}
\item NGS close to the axis (best): SR(H) = 0.8 and Frame rate = 500 Hz.
\item NGS at the edge of the technical FoV (worse): SR(H) = 0.2 and Frame rate = 100 Hz.
\end{itemize}
We report the noise on the centroid in Fig. \ref{fig:noise_LO}.
In both considered cases the pixel scale of 20 mas for the 1$\times$1 (40 mas for the 2$\times$2) provides the best result.
We also see that we can accept a slightly larger pixel scale without a significant loss of performance.
A tolerance of 10\% is considered.
\begin{figure}[h]
\centering
\subfigure[Curve considering the 2$\times$2 configuration, 500Hz framerate and SR(H)=0.8 on the NGS star.\label{fig:noise_LO_2x2}]
{\includegraphics[width=0.48\columnwidth]{noise_LO_openloop_2x2_500Hz_sr0.80_mavis_v3_jitter_unbias.png}}
\subfigure[Curve considering the 1$\times$1 configuration, 100Hz framerate and SR(H)=0.2 on the NGS star.\label{fig:noise_LO_1x1}]
{\includegraphics[width=0.48\columnwidth]{noise_LO_openloop_1x1_100Hz_sr0.20_mavis_v3_jitter_unbias.png}}
\caption{Measurement noise tilt jitter in mas as a function of the H band magnitude.\label{fig:noise_LO}}
\end{figure}
\subsection{Vibrations}
We estimated the level of vibrations coming form ESO VLT from a set of closed loop PSDs of VLT-UT4 NEAR data (full telemetry data collected on 2016-09 at UT4 Cassegrain by the NEAR WFS working at 1kHz).
Modal PSDs show strong peaks at $\sim$48Hz in particular on trefoils (see solid lines in Fig. \ref{fig:cum_PSD_vib}).
We used this data set to model the vibrations in the MAVIS simulations introducing more than 20 peaks.
Then we used the adaptive vibration cancellation (AVC) algorithm\cite{7320616} developed at ESO to compensate for them.
The simulation with vibrations showed an additional error of 50nm that is partially recovered by the AVC: this method is capable of reducing this error from 50 to 21nm (see Fig. \ref{fig:modal_plot_vib})).
\begin{figure}[htpb]
\centering
\subfigure[Cumulated PSDs of the first 10 modes for the three simulated cases.\label{fig:cum_PSD_vib}]
{\includegraphics[width=0.42\columnwidth]{cum_PSD_vib.png}}
\subfigure[On-axis RMS of turbulence and residual projected on a modal base.\label{fig:modal_plot_vib}]
{\includegraphics[width=0.42\columnwidth]{modal_plot_vib.png}}
\caption{Comparison between case without vibrations, with vibrations and with vibrations and the adaptive vibration cancellation algorithm. Modal base used here is made by 5 Zernikes and 4000 Karhunen-Lo{\'e}ve modes\cite{Wang:78}.\label{fig:vibrations}}
\end{figure}
\subsection{Mis-registrations}
Here we report sensitivity analysis to registration/alignement errors.
We consider that these mis-registrations are not calibrated so the reconstruction and projection matrices are the nominal ones.
We analysed 5 kinds of mis-registration errors that are shown in Fig. \ref{fig:misal_legend}.
The results are summarized in Fig. \ref{fig:shift}, \ref{fig:rotation}, \ref{fig:DM_shift}, \ref{fig:DM_rotation} and \ref{fig:DM_magnification}.
Please note that in the figures red line shows the sensitivity of the second half of DM modes and it is reported to focus on the effect of mis-registration on the higher spatial frequencies.
These results are used by the system engineering to set up requirements on the alignment (static and dynamic parts) pupil/DMs/WFSs.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\linewidth]{misal_legend.png}
\caption{Legend of possible mis-registrations.}
\label{fig:misal_legend}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.66\linewidth]{shift.png}
\caption{Sensitivity to shift pupil/lenslet array in sub-apertures (SA). Left, shift of a single WFS (note that for a value of 1.4SA the system is unstable), right, shift of all WFSs.}
\label{fig:shift}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.66\linewidth]{rotation.png}
\caption{Sensitivity to rotation pupil/lenslet array. Left, rotation of a single WFS (note that for a value of 4deg the system is unstable), right, rotation of all WFSs.}
\label{fig:rotation}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.99\linewidth]{DM_shift.png}
\caption{Sensitivity to shift DM/pupil for, left, DSM (note that this is not realistic because DSM is also the pupil), center, medium altitude post focal DM, right high altitude post focal DM.}
\label{fig:DM_shift}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.99\linewidth]{DM_rotation.png}
\caption{Sensitivity to rotation DM/pupil for, left, DSM, center, medium altitude post focal DM, right high altitude post focal DM.}
\label{fig:DM_rotation}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.99\linewidth]{DM_magnification.png}
\caption{Sensitivity to magnification DM/pupil (the values are the ratio between DM and meta-pupil size) for, left, DSM (note that this is not realistic because DSM is also the pupil), center, medium altitude post focal DM, right high altitude post focal DM.}
\label{fig:DM_magnification}
\end{figure}
\section{Conclusion}
MAVIS design is progressing and now we have a more accurate performance estimation than a year and a half ago: we refined our tools for system modelling and performance prediction, we reviewed and updated the control strategy and a few parameters, like total delay and laser spots geometry, and, finally, we explored the performance sensitivity to many parameters.
In particular, now we are more confident that the system can meet the requirements and we have an as complete as possible view on MAVIS performance over a large set of conditions.
In this phase performance estimation is not used only to verify top level requirements but also to derive sub-system requirements: this is the case of the study on the effect of mis-registrations.
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train/arxiv
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BkiUfm3xK6-gDz87SQyy
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\section*{Introduction}
The infinite dihedral group is both a free product and an extension
of the infinite cyclic group $\mathbb{Z}$ by the cyclic group $\mathbb{Z}_2$
of order 2
\[
D_{\infty}~=~\mathbb{Z}_2*\mathbb{Z}_2~=~\mathbb{Z}\rtimes \mathbb{Z}_2
\]
with $\mathbb{Z}_2$ acting on $\mathbb{Z}$ by $-1$.
A group $G$ is said to be {\it over $D_\infty$} if it is
equipped with an epimorphism $p:G \to D_{\infty}$.
We study the algebraic $K$-theory of $R[G]$, for any ring $R$ and
any group $G$ over $D_{\infty}$.
Such a group $G$ inherits from $D_{\infty}$ an injective amalgamated
free product structure $G=G_1*_FG_2$ with $F$ an index 2 subgroup of
$G_1$ and $G_2$. Furthermore, there is a canonical index 2
subgroup $\overline{G} \subset G$ with an injective $HNN$ structure
$\overline{G}=F\rtimes_{\alpha}\mathbb{Z}$ for an automorphism $\alpha:F\to F$.
The various groups fit into a commutative braid of short exact sequences:
$$\xymatrix@C-25pt{ &&\mathbb{Z}~\ar@{>->}[dr] && \\
& \overline{G}=F\rtimes_{\alpha}\mathbb{Z}\ar@{->>}[ur] \ar@{>->}[dr]^-{\displaystyle{\theta}}
&& D_{\infty}=\mathbb{Z}_2*\mathbb{Z}_2~~ \ar@{->>}[dr]^-{\displaystyle{\pi}}\\
F=G_1 \cap G_2 \ar@{>->}[ur] \ar@{>->}@/_2pc/[rr]&&
G=G_1*_FG_2\ar@{->>}[ur]^-{\displaystyle{p}}\ar@{->>}@/_2pc/[rr]^-{\displaystyle{\pi \circ p}} &&
~~\mathbb{Z}_2 }$$
\bigskip
The algebraic $K$-theory decomposition theorems of Waldhausen for
injective amalgamated free products and $HNN$ extensions give
\[
K_*(R[G])~=~K_*(R[F] \to R[G_1] \times R[G_2]) \oplus
\widetilde{\mathrm{Nil}}_{*-1}(R[F];R[G_1-F],R[G_2-F])\eqno{(*)}
\]
and
\[
K_*(R[\overline{G}])~=~K_*(1-\alpha:R[F] \to R[F]) \oplus
\widetilde{\mathrm{Nil}}_{*-1}(R[F],\alpha)\oplus\widetilde{\mathrm{Nil}}_{*-1}(R[F],\alpha^{-1})~.
\eqno{(**)}\]
We establish isomorphisms
\[
\widetilde{\mathrm{Nil}}_*(R[F];R[G_1-F],R[G_2-F]) ~\cong~
\widetilde{\mathrm{Nil}}_*(R[F],\alpha)~\cong~\widetilde{\mathrm{Nil}}_*(R[F],\alpha^{-1})
\]
which we use to prove that the Farrell--Jones isomorphism conjecture in
algebraic $K$-theory can be reduced to the family of finite-by-cyclic
groups, so that virtually cyclic groups of infinite dihedral type need
not be considered.
\subsection{Algebraic semi-splitting}
A homotopy equivalence $f:M \to X=X_1\cup_Y X_2$ of finite $CW$
complexes is {\it split along $Y \subset X$} if it is a cellular map
and the restriction
$$g~=~f\vert~:~N~=~f^{-1}(Y) \to Y$$
is also a homotopy equivalence. The $\widetilde{\mathrm{Nil}}_*$-groups arise
as the obstruction groups to splitting homotopy equivalences of finite
$CW$ complexes in the case of injective $\pi_1(Y) \to \pi_1(X)$
(Farrell--Hsiang, Waldhausen). In this paper we introduce the
considerably weaker notion of a homotopy equivalence being {\it semi-split},
as defined in \S \ref{semi}. The following is a special case of our
main algebraic result (\ref{Thm_HigherDKR}, \ref{Thm_LowerDKR}) which
shows that there is no obstruction to semi-splitting.
\begin{thm} \label{first}
Let $G$ be a group over $D_\infty$, with
$$F~=~G_1 \cap G_2~=~\overline{G}~=~F\rtimes_{\alpha}\mathbb{Z} \subset
G~=~G_1*_FG_2~.$$
{\rm (i)} For any ring $R$ and $n \in \mathbb{Z}$ the
corresponding reduced $\mathrm{Nil}$-groups are naturally isomorphic:
\[\widetilde{\mathrm{Nil}}_n(R[F];R[G_1- F],R[G_2- F])
~\cong~\widetilde{\mathrm{Nil}}_n(R[F],\alpha)~\cong~\widetilde{\mathrm{Nil}}_n(R[F],\alpha^{-1})
\]
{\rm (ii)} The inclusion $\theta:R[\overline{G}]\to R[G]$ determines induction
and transfer maps
\[
\theta_!~:~K_n(R[\overline{G}]) \to K_n(R[G])~,~\theta^!~:~K_n(R[G]) \to
K_n(R[\overline{G}]).
\]
For all $n \leqslant 1$, the maps $\theta_!$ and $\theta^!$ restrict to
isomorphisms on the $\widetilde{\mathrm{Nil}}$-$\widetilde{\mathrm{Nil}}$-components
in the decompositions $(*)$ and $(**)$.
\end{thm}
\begin{proof}
Part~(i) is a special case of Theorem~\ref{maink}.
Part~(ii) follows from Proposition~\ref{induction}(ii) (induction)
and Proposition~\ref{transfer} (transfer).
\end{proof}
The $n=0$ case will be discussed in more detail in \S\ref{exp} and
\S\ref{First}.
\begin{rem} We do not seriously doubt that a more assiduous application
of higher $K$-theory would extend Theorem \ref{maink} (ii) to all $n \in \mathbb{Z}$ (see also \cite{DQR}).
\end{rem}
As an application of Theorem \ref{first}, we shall prove the following theorem.
\begin{thm}
Let $\Gamma$ be any group, and let $R$ be any ring. Then the following map
of equivariant homology groups with coefficients in the algebraic
$K$-theory functor $\mathbf{K}_R$ is an isomorphism:
\[
H^\Gamma_*(E_{\matheurm{fbc}}\Gamma;\mathbf{K}_R) \longrightarrow H^\Gamma_*(E_{\matheurm{vc}}\Gamma;\mathbf{K}_R) ~.
\]
\end{thm}
In fact, this is a special case of our more general fibered
version (Theorem \ref{isom_conj}).
The original reduced $\mathrm{Nil}$-groups $\widetilde{\mathrm{Nil}}_*(R)
=\widetilde{\mathrm{Nil}}_*(R,1)$ feature in the decompositions of Bass
\cite{Bass} and Quillen \cite{Grayson}:
\begin{eqnarray*}
K_*(R[t]) &~=~& K_*(R) \oplus \widetilde{\mathrm{Nil}}_{*-1}(R)~,\\
K_*(R[\mathbb{Z}]) &~=~& K_*(R) \oplus K_{*-1}(R) \oplus \widetilde{\mathrm{Nil}}_{*-1}(R)\oplus \widetilde{\mathrm{Nil}}_{*-1}(R)~.
\end{eqnarray*}
In \S \ref{Applications} we shall compute several examples which require Theorem \ref{first}:
\begin{eqnarray*}
K_*(R[\mathbb{Z}_2 * \mathbb{Z}_2]) &~=~& \frac{K_*(R[\mathbb{Z}_2]) \oplus K_*(R[\mathbb{Z}_2])}{K_*(R)} \oplus \widetilde{\mathrm{Nil}}_{*-1}(R)\\
K_*(R[\mathbb{Z}_2 * \mathbb{Z}_3]) &~=~& \frac{K_*(R[\mathbb{Z}_2]) \oplus K_*(R[\mathbb{Z}_3])}{K_*(R)} \oplus \widetilde{\mathrm{Nil}}_{*-1}(R)^{\infty}\\
\mathrm{Wh}(G_0 \times \mathbb{Z}_2 *_{G_0} G_0 \times \mathbb{Z}_2) &~=~& \frac{\mathrm{Wh}(G_0 \times \mathbb{Z}_2) \oplus \mathrm{Wh}(G_0 \times \mathbb{Z}_2) }{\mathrm{Wh}(G_0) } \oplus \widetilde{\mathrm{Nil}}_0(\mathbb{Z}[G_0])
\end{eqnarray*}
where $G_0 = \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}$. The point here is that
$\widetilde{\mathrm{Nil}}_0(\mathbb{Z}[G_0])$ is an infinite torsion abelian group.
This provides the first example (Example \ref{Exm_NonzeroWaldNil})
of a non-zero $\mathrm{Nil}$ group in the amalgamated product case and hence
the first example of a non-zero obstruction to splitting a homotopy
equivalence in the two-sided case $(A)$.
\subsection{Topological semi-splitting}\label{semi}
Let $(X,Y)$ be a separating, codimension 1, finite $CW$ pair, with
$X=X_1\cup_YX_2$ a union of connected $CW$ complexes such that
$\pi_1(Y) \to \pi_1(X)$ is injective. Let $\bar{X}$ (resp.
$\bar{X}_1,\bar{X}_2)$ be the connected cover of $X$ (resp. $X_1$,
$X_2$) classified by $\pi_1(Y) \subset \pi_1(X)$ (resp. $\pi_1 (Y)
\subset \pi_1(X_1)$, $\pi_1 (Y)\subset \pi_1(X_2)$). Then
$\bar{X}=\bar{X}^-\cup_Y \bar{X}^+$ with
$\pi_1(\bar{X}^-)=\pi_1(\bar{X}^+)=\pi_1(Y)$. Note $\bar{X}_1\subset
\bar{X}^-$ and $\bar{X}_2\subset \bar{X}^+$ with $\bar{X}_1\cap
\bar{X}_2=Y \subset \bar{X}$.
A homotopy equivalence $f:M \to X$ from a finite $CW$ complex is
\emph{semi-split along $Y \subset X$} if $N=f^{-1}(Y) \subset M$ is
a subcomplex and the restriction $(f,g):(M,N) \to (X,Y)$ is a map of
pairs such that the relative homology kernel $\mathbb{Z}[\pi_1(Y)]$-modules
$K_*(\bar{M}_2,N)$ vanish. Equivalently, $f$ is semi-split along
$Y$ if the following induced $\mathbb{Z}[\pi_1(Y)]$-module morphisms are
isomorphisms:
\[
\rho_2~:~K_*(\bar{M}^+,N) \longrightarrow K_*(\bar{M}^+,\bar{M}_2)
= \mathbb{Z}[\pi_1(X_2)-\pi_1(Y)]\otimes_{\mathbb{Z}[\pi_1(Y)]} K_*(\bar{M}^-,N) ~.
\]
The notation of $CW$-splitting is explained more in \S
\ref{Top_exp}.
We refer to \S\ref{Defn_almostnormal} for the definition of an
almost-normal subgroup. In particular, finite-index subgroups and
normal subgroups are almost-normal.
\begin{thm}\label{Thm_TopSemisplit}
Let $(X,Y)$ be a separating, codimension 1, finite $CW$ pair, with
$X=X_1\cup_YX_2$ a union of connected $CW$ complexes such that
$\pi_1(Y) \to \pi_1(X)$ is injective. Suppose $\pi_1(Y)$ is an
almost-normal subgroup of $\pi_1(X_2)$. Then, for any finite $CW$
complex $M$, any homotopy equivalence $h: M \to X$ is simple
homotopic to a semi-split homotopy equivalence $h': M \to X$ along $Y$.
\end{thm}
\subsection{Algebraic exposition}\label{exp}
For any ring $R$, we establish isomorphisms between two types of
codimension 1 splitting obstruction nilpotent class groups. The
first type, for separated splitting, arises in the decompositions of
the algebraic $K$-theory of the $R$-coefficient group ring $R[G]$ of
a group $G$ over $D_\infty$, with an epimorphism $p:G\to D_{\infty}$ onto the
infinite dihedral group $D_{\infty}$. The second type, for
non-separated splitting, arises from the $\alpha$-twisted polynomial
ring $R[F]_{\alpha}[t]$, with $F={\rm ker}(p)$ and $\alpha:F \to F$
an automorphism such that
\[
\overline{G}~=~{\rm ker}(\pi\circ p:G \to \mathbb{Z}_2)~=~F\rtimes_{\alpha}\mathbb{Z}
\]
where $\pi:D_{\infty} \to \mathbb{Z}_2$ is the unique epimorphism with
infinite cyclic kernel. Note:
\begin{itemize}
\item[(A)] $D_{\infty}=\mathbb{Z}_2*\mathbb{Z}_2$ is the free product of two
cyclic groups of order 2, whose generators will be denoted
$t_1,t_2$.
\item[(B)] $D_{\infty} = \langle ~t_1, t_2 ~|~ t_1^2 = 1 = t_2^2~ \rangle$ contains the infinite cyclic group $\mathbb{Z}
=\langle t \rangle$ as a subgroup of index 2 with
$t=t_1t_2$. In fact there is a short exact sequence with a
split epimorphism
$$\xymatrix{\{1\} \ar[r] & \mathbb{Z} \ar[r] &
D_{\infty} \ar[r]^-{\displaystyle{\pi}} & \mathbb{Z}_2 \ar[r]& \{1\}~.}$$
\end{itemize}
More generally, if $G$ is a group over $D_\infty$, with an epimorphism $p:G \to
D_{\infty}$, then:
\begin{itemize}
\item[(A)] $G=G_1*_FG_2$ is a free product with amalgamation
of two groups
\[
G_1~=~{\rm ker}(p_1:G \to \mathbb{Z}_2)~,~G_2~=~{\rm ker}(p_2:G \to \mathbb{Z}_2) \subset G
\]
amalgamated over their common subgroup $F={\rm ker}(p)=G_1
\cap G_2$ of index 2 in both $G_1$ and $G_2$.
\item[(B)] $G$ has a subgroup $\overline{G} = \ker(\pi \circ p: G
\to \mathbb{Z}_2)$ of index 2 which is an $HNN$ extension $\overline{G}
=F\rtimes_{\alpha} \mathbb{Z}$ where $\alpha:F \to F$ is
conjugation by an element $t \in \overline{G}$ with $p(t)=t_1t_2 \in D_{\infty}$.
\end{itemize}
\subsubsection*{The $K$-theory of type (A)}
For any ring $R$ and $R$-bimodules $\mathscr{B}_1,\mathscr{B}_2$, the $\mathrm{Nil}$-groups
$\mathrm{Nil}_*(R;\mathscr{B}_1,\mathscr{B}_2)$ are defined to be the algebraic $K$-groups
$K_*(\mathrm{Nil}(R;\mathscr{B}_1,\mathscr{B}_2))$ of the exact category
$\mathrm{Nil}(R;\mathscr{B}_1,\mathscr{B}_2)$ with objects quadruples
$(P_1,P_2,\rho_1,\rho_2)$ with $P_1,P_2$ f.g.~projective $R$-modules
and
\[
\rho_1~:~P_1 \to \mathscr{B}_1\otimes_RP_2~,~\rho_2~:~P_2 \to
\mathscr{B}_2\otimes_RP_1
\]
$R$-module morphisms such that $\rho_2\rho_1:P_1
\to \mathscr{B}_1\otimes_R\mathscr{B}_2\otimes_RP_1$ is nilpotent (or equivalently
such that $\rho_1\rho_2:P_2 \to \mathscr{B}_2\otimes_R\mathscr{B}_1\otimes_RP_2$ is
nilpotent). The reduced $\mathrm{Nil}$-groups $\widetilde{\mathrm{Nil}}_*$ are such
that \[ \mathrm{Nil}_*(R;\mathscr{B}_1,\mathscr{B}_2)~=~K_*(R)\oplus K_*(R) \oplus
\widetilde{\mathrm{Nil}}_*(R;\mathscr{B}_1,\mathscr{B}_2)~.\]
As already noted above,
Waldhausen \cite{Waldhausen_Rings} decomposed the algebraic
$K$-theory of $R[G]$ for an injective amalgamated free product
$G=G_1*_FG_2$ as \[K_*(R[G])~=~K_*(R[F] \to R[G_1]\times R[G_2])
\oplus \widetilde{\mathrm{Nil}}_{*-1}(R[F];R[G_1- F],R[G_2- F])~.\]
In particular, there is defined a split monomorphism
$$\sigma_A~:~\widetilde{\mathrm{Nil}}_{*-1}(R[F];R[G_1- F],R[G_2- F])
\xymatrix{\ar@{>->}[r]&} K_*(R[G]) ~,$$
which for $*=1$ is given by
\begin{align*}
&\sigma_A~:~\widetilde{\mathrm{Nil}}_0(R[F];R[G_1- F],R[G_2- F])
\xymatrix{\ar@{>->}[r]&}K_1(R[G])~;\\
&\hphantom{\sigma_A~:~}
[P_1,P_2,\rho_1,\rho_2] \mapsto \left[ R[G]\otimes_{R[F]}(P_1 \oplus P_2), (\begin{pmatrix} 1 & \rho_2 \\ \rho_1 & 1\end{pmatrix} \right]~.
\end{align*}
\subsubsection*{The $K$-theory of type (B)}
Given a ring $R$ and an $R$-bimodule $\mathscr{B}$, let
\[ T_R(\mathscr{B})~=~R \;\oplus\; \mathscr{B} \;\oplus\; \mathscr{B}\otimes_R\mathscr{B} \;\oplus\; \cdots\]
be the tensor algebra of $\mathscr{B}$ over $R$. The $\mathrm{Nil}$-groups
$\mathrm{Nil}_*(R;\mathscr{B})$ are defined to be the algebraic $K$-groups
$K_*(\mathrm{Nil}(R;\mathscr{B}))$ of the exact category $\mathrm{Nil}(R;\mathscr{B})$ with objects
pairs $(P,\rho)$ with $P$ a f.g.~(finitely generated) projective
$R$-module and $\rho:P \to \mathscr{B}\otimes_RP$ a $R$-module morphism,
nilpotent in the sense that for some $k$, we have
\[
\rho^k = 0 : P \to \mathscr{B} \otimes_R P \to \cdots \to \mathscr{B} \otimes_R \cdots \otimes_R \mathscr{B} \otimes_R P.
\]
The reduced $\mathrm{Nil}$-groups $\widetilde{\mathrm{Nil}}_*$ are such that
\[\mathrm{Nil}_*(R;\mathscr{B})~=~K_*(R) \oplus \widetilde{\mathrm{Nil}}_*(R;\mathscr{B})~.\]
Waldhausen
\cite{Waldhausen_Rings} proved that if $\mathscr{B}$ is f.g.~projective as a
left $R$-module and free as a right $R$-module, then
\[K_*(T_R(\mathscr{B}))~=~K_*(R) \oplus \widetilde{\mathrm{Nil}}_{*-1}(R;\mathscr{B})\]
with a split monomorphism
$$\sigma_B~:~\widetilde{\mathrm{Nil}}_{*-1}(R;\mathscr{B}) \xymatrix{\ar@{>->}[r]&} K_*(T_R(\mathscr{B})) ~,$$
which for $*=1$ is given by
\[\sigma_B~:~\widetilde{\mathrm{Nil}}_0(R;\mathscr{B}) \xymatrix{\ar@{>->}[r]&} K_1(T_R(\mathscr{B}))~;~
[P,\rho] \mapsto \left[ T_R(\mathscr{B})\otimes_RP, 1-\rho \right]~.\]
In particular, for $\mathscr{B}=R$
\begin{align*}
&\mathrm{Nil}_*(R;R)~=~\mathrm{Nil}_*(R)~,~\widetilde{\mathrm{Nil}}_*(R;R)~=~\widetilde{\mathrm{Nil}}_*(R)~,\\
&T_R(\mathscr{B})~=~R[t]~,~K_*(R[t])~=~K_*(R) \oplus \widetilde{\mathrm{Nil}}_{*-1}(R)~.
\end{align*}
\subsubsection*{Relating the $K$-theory of types (A) and (B)}
Recall that a category $I$ is \emph{filtered} if:
\begin{itemize}
\item
for any pair of objects $\alpha, \alpha'$ in $I$, there exist an
object $\beta$ and morphisms $\alpha \to \beta$ and $\alpha' \to
\beta$ in $I$, and
\item
for any pair of morphisms $u, v: \alpha \to \alpha'$
in $I$, there exists an object $\beta$ and morphism $w: \alpha'
\to \beta$ such that $w \circ u = w \circ v$.
\end{itemize}
Note that any directed poset $I$ is a filtered category.
\begin{thm}[General Algebraic Semi-splitting] \label{maink}
Let $R$ be a ring. Let $\mathscr{B}_1,\mathscr{B}_2$ be $R$-bimodules. Suppose that
$I$ is a small, filtered category and $\mathscr{B}_2 = \colim_{\alpha \in I}
\mathscr{B}_2^\alpha$ is a direct limit of $R$-bimodules such that each
$\mathscr{B}_2^\alpha$ is a f.g.~projective left $R$-module.
Then, for all $n \in \mathbb{Z}$, the $\mathrm{Nil}$-groups of the triple
$(R;\mathscr{B}_1,\mathscr{B}_2)$ are related to the $\mathrm{Nil}$-groups of the pair
$(R;\mathscr{B}_1\otimes_R\mathscr{B}_2)$ by isomorphisms
\begin{eqnarray*}
\mathrm{Nil}_n(R;\mathscr{B}_1,\mathscr{B}_2) &\cong &\mathrm{Nil}_n(R;\mathscr{B}_1\otimes_R\mathscr{B}_2) \oplus K_n(R)~,\\
\widetilde{\mathrm{Nil}}_n(R;\mathscr{B}_1,\mathscr{B}_2) &\cong& \widetilde{\mathrm{Nil}}_n(R;\mathscr{B}_1\otimes_R\mathscr{B}_2)~.
\end{eqnarray*}
In particular, for $n=0$ there are defined inverse isomorphisms
\begin{align*}
&i_*~:~\mathrm{Nil}_0(R;\mathscr{B}_1\otimes_R\mathscr{B}_2)\oplus K_0(R)
\xymatrix{\ar[r]^-{\displaystyle{\cong}}&}\mathrm{Nil}_0(R;\mathscr{B}_1,\mathscr{B}_2)~;\\
&([P_1,\rho_{12}:P_1 \to \mathscr{B}_1 \otimes_R\mathscr{B}_2\otimes_RP_1],[P_2])
\mapsto [P_1, \mathscr{B}_2 \otimes_R P_1\oplus P_2,\begin{pmatrix}
\rho_{12}\\ 0\end{pmatrix},(1~0)]~,\\
&j_*~:~\mathrm{Nil}_0(R;\mathscr{B}_1,\mathscr{B}_2)\xymatrix{\ar[r]^-{\displaystyle{\cong}}&}
\mathrm{Nil}_0(R;\mathscr{B}_1\otimes_R \mathscr{B}_2)\oplus K_0(R)~;\\
&[P_1,P_2,\rho_1:P_1 \to \mathscr{B}_1\otimes_RP_2,\rho_2:
P_2 \to \mathscr{B}_2\otimes_RP_1]\mapsto ([P_1,\rho_2\circ \rho_1],
[P_2] - [\mathscr{B}_2\otimes_RP_1])~.
\end{align*}
The reduced versions are the inverse isomorphisms
\begin{align*}
&i_*~:~\widetilde{\mathrm{Nil}}_0(R;\mathscr{B}_1\otimes_R\mathscr{B}_2) \xymatrix{\ar[r]^-{\displaystyle{\cong}}&} \widetilde{\mathrm{Nil}}_0(R;\mathscr{B}_1,\mathscr{B}_2)~;~
[P_1,\rho_{12}] \mapsto [P_1, \mathscr{B}_2 \otimes_R P_1,\rho_{12},1]~,\\
&j_*~:~\widetilde{\mathrm{Nil}}_0(R;\mathscr{B}_1,\mathscr{B}_2)\xymatrix{\ar[r]^-{\displaystyle{\cong}}&}
\widetilde{\mathrm{Nil}}_0(R;\mathscr{B}_1\otimes_R \mathscr{B}_2)~;~
[P_1,P_2,\rho_1,\rho_2]\mapsto [P_1,\rho_2\circ \rho_1]~.
\end{align*}
\end{thm}
\begin{proof}
This follows immediately from Theorems \ref{Thm_HigherDKR} and
\ref{Thm_LowerDKR}.
\end{proof}
\begin{rem} Theorem \ref{maink} was already known to Pierre Vogel in
1990 - see \cite{Vogel}.
\end{rem}
\subsection{Topological exposition} \label{Top_exp}
The proof of Theorem \ref{maink} is motivated by the obstruction
theory of Waldhausen \cite{Waldhausen_1969} for splitting homotopy
equivalences of finite $CW$ complexes $X$ along codimension 1
subcomplexes $Y \subset X$ with $\pi_1(Y) \to \pi_1(X)$ injective,
and the subsequent algebraic $K$-theory decomposition theorems of
Waldhausen \cite{Waldhausen_Rings}.
A \emph{codimension 1 pair} $(X,Y \subset X)$ is a pair of spaces
such that the inclusion $Y=Y \times \{0\}\subset X$ extends to an
open embedding $Y \times \mathbb{R} \subset X$. A \emph{map of codimension 1
pairs} $(f,g):(M,N) \to (X,Y)$ is a cellular map $f: M \to X$ with
$g=f\vert:N=f^{-1}(Y) \to Y$.
Let $(X,Y)$ be a codimension 1 finite $CW$ pair. A homotopy
equivalence $f:M \to X$ from a finite $CW$ complex \emph{splits
along $Y \subset X$} if there is a map of codimension 1 pairs
$(f,g):(M,N) \to (X,Y)$ so that $g: N \to Y$ is also a homotopy
equivalence. A map $f : M \to X$ between finite $CW$ complexes is
\emph{simple homotopic} to a map $f': M' \to X$ if $M'$ is a finite
CW complex, $s : M' \to M$ is a simple homotopy equivalence and
$f\circ s$ is homotopic to $f'$. A homotopy equivalence $f:M \to X$
from a finite $CW$ complex is \emph{splittable along $Y \subset X$}
if $f$ is simple homotopic to a map which splits along $Y \subset
X$.
A codimension 1 pair $(X,Y)$ is \emph{injective} if $X,Y$ are
connected and $\pi_1(Y) \to \pi_1(X)$ is injective. Let
$\widetilde{X}$ be the universal cover of $X$. As in \S\ref{semi}
let $\bar{X}=\widetilde{X}/\pi_1(Y)$, so that $(\bar{X},Y)$ is a
codimension 1 pair with $\bar{X}=\bar{X}^-\cup_Y\bar{X}^+$ for
connected subspaces $\bar{X}^-,\bar{X}^+\subset \bar{X}$ with
$\pi_1(\bar{X})=\pi_1(\bar{X}^-)=\pi_1(\bar{X}^+)=\pi_1(Y)$. As
usual, there are two cases, according as to whether $Y$ separates
$X$ or not:
\begin{enumerate}
\item[(A)] $X- Y$ is disconnected, so
\[X~=~X_1\cup_YX_2\] with $X_1,X_2$ connected. By the Seifert-van
Kampen theorem
\[\pi_1(X)~=~\pi_1(X_1)*_{\pi_1(Y)}\pi_1(X_2)\]
is the amalgamated free product, with $\pi_1(Y) \to
\pi_1(X_1)$, $\pi_1(Y) \to \pi_1(X_2)$ injective. The labeling
is chosen such that
\[\bar{X}_1~=~\widetilde{X}_1/\pi_1(Y) \subset \bar{X}^-~,~
\bar{X}_2~=~\widetilde{X}_2/\pi_1(Y) \subset \bar{X}^+~,~
\bar{X}_1 \cap \bar{X}_2 = Y~.
\]
\item[(B)] $X- Y$ is connected, so
\[X~=~X_1/\{y\sim ty\vert y \in Y\}\] for a connected space $X_1$
(a deformation retract of $X- Y$) which contains two disjoint
copies $Y,tY \subset X_1$ of $Y$. We shall only consider the
case when $\pi_1(Y) \to \pi_1(X_1)$, $\pi_1(tY) \to \pi_1(X_1)$
are isomorphisms, so that \[\pi_1(X)~=~\pi_1(Y)\rtimes_{\alpha}
\mathbb{Z}\] for an automorphism $\alpha:\pi_1(Y) \to\pi_1(Y)$ and
$\bar{X}$ is an infinite cyclic cover of $X$ with a generating
covering translation $t:\bar{X} \to \bar{X}$. The labeling is
chosen such that
\[
\bar{X}_1~=~\widetilde{X}_1/\pi_1(Y) \subset \bar{X}^- ~,~ t\bar{X}_1
\subset \bar{X}^+ ~,~ \bar{X}_1 \cap t \bar{X}_1 = Y ~.
\]
\end{enumerate}
In both cases $(\bar{X},Y)$ is an injective codimension 1 pair of
type (A).
The \emph{kernel $\mathbb{Z}[\pi_1(X)]$-modules} of a map $f:M \to X$ are
the relative homology $\mathbb{Z}[\pi_1(X)]$-modules
\[K_r(M)~=~H_{r+1}(\widetilde{f}:\widetilde{M} \to \widetilde{X})\]
with $\widetilde{X}$ the universal cover of $X$,
$\widetilde{M}=f^*\widetilde{X}$ the pullback cover of $M$, and
$\widetilde{f}:\widetilde{M} \to \widetilde{X}$ a
$\pi_1(X)$-equivariant lift of $f$. For a map of injective
codimension 1 $CW$ pairs $(f,g):(M,N) \to (X,Y)$ the kernel
$\mathbb{Z}[\pi_1(Y)]$-modules fit into an exact sequence
\[\dots \to K_r(N) \to K_r(\bar{M}) \to K_r(\bar{M}^+,N)\oplus
K_r(\bar{M}^-,N) \to K_{r-1}(N) \to \dots~.\] If $f$ is a homotopy
equivalence and $g_*:\pi_1(N) \to \pi_1(Y)$ is an isomorphism, then
$g$ is a homotopy equivalence if and only if $K_*(N)=0$, which
occurs if and only if $K_*(\bar{M}^+,N)=K_*(\bar{M}^-,N)=0$. In
particular, if $f$ is of type (A) and split, then $f$ is semi-split.
\begin{thm}[Waldhausen \cite{Waldhausen_1969} \emph{for} type (A),
Farrell--Hsiang \cite{FarrellHsiang} \emph{for} type (B)]\label{W}
Let $(X,Y)$ be an injective, codimension 1, finite $CW$ pair.
Suppose $f':M' \to X$ is a homotopy equivalence from a finite $CW$ complex.\\
{\rm (i)} The homotopy equivalence $f'$ is simple homotopic to a map
of pairs $(f,g):(M,N) \to (X,Y)$ such that $g_*:\pi_1(N) \to
\pi_1(Y)$ is an isomorphism and for some $n \geqslant 2$ we have
\[K_r(N)~=~0~\hbox{\it for}~r \neq n~.\]
Moreover, the $\mathbb{Z}[\pi_1(Y)]$-modules $K_{n+1}(\bar{M}^\pm,N)$ are
f.g.~projective, and the direct sum
\[K_n(N)~=~K_{n+1}(\bar{M}^-,N)\oplus K_{n+1}(\bar{M}^+,N)\] is
stably f.g.~free. Hence the projective classes are complementary:
\[[K_{n+1}(\bar{M}^-,N)]~=~-[K_{n+1}(\bar{M}^+,N)] \in
\widetilde{K}_0(\mathbb{Z}[\pi_1(Y)])~.\]
{\rm (ii)} In the separating case {\rm (A)} there is defined an
exact sequence
\begin{multline*}
\cdots \longrightarrow \mathrm{Wh}(\pi_1(X_1)) \oplus \mathrm{Wh}(\pi_1(X_2)) \longrightarrow
\mathrm{Wh}(\pi_1(X))\\
\longrightarrow \widetilde{K}_0(\mathbb{Z}[\pi_1(Y)]) \oplus
\widetilde{\mathrm{Nil}}_0(\mathbb{Z}[\pi_1(Y)];\mathscr{B}_1,\mathscr{B}_2) \longrightarrow \cdots
\end{multline*}
where
\[
\mathscr{B}_1~=~\mathbb{Z}[\pi_1(X_1)- \pi_1(Y)]~,\enspace \mathscr{B}_2~=~\mathbb{Z}[\pi_1(X_2)- \pi_1(Y)]~.
\]
The Whitehead torsion $\tau(f) \in \mathrm{Wh}(\pi_1(X))$ has image
\[
[\tau(f)]~=~([K_{n+1}(\bar{M}^-,N)],[K_{n+1}(\bar{M}^-,N),
K_{n+1}(\bar{M}^+,N),\rho_1,\rho_2])
\]
where
\begin{gather*}
\rho_1~:~K_{n+1}(\bar{M}^-,N) \longrightarrow K_{n+1}(\bar{M}^-,\bar{M}_1)~=~
\mathscr{B}_1\otimes_{\mathbb{Z}[\pi_1(Y)]}K_{n+1}(\bar{M}^+,N)~,\\
\rho_2~:~K_{n+1}(\bar{M}^+,N) \longrightarrow K_{n+1}(\bar{M}^+,\bar{M}_2)~=~
\mathscr{B}_2\otimes_{\mathbb{Z}[\pi_1(Y)]}K_{n+1}(\bar{M}^-,N)~.
\end{gather*}
The homotopy equivalence $f$ is splittable along $Y$ if and only
if $[\tau(f)]=0$.\\
{\rm (iii)} In the non-separating case {\rm (B)} there is defined
an exact sequence
\begin{multline*}
\cdots \longrightarrow \mathrm{Wh}(\pi_1(Y)) \xymatrix{\ar[r]^-{1-\alpha}&}
\mathrm{Wh}(\pi_1(Y)) \longrightarrow \mathrm{Wh}(\pi_1(X))\\
\longrightarrow \widetilde{K}_0(\mathbb{Z}[\pi_1(Y)]) \oplus
\widetilde{\mathrm{Nil}}_0(\mathbb{Z}[\pi_1(Y)],\alpha)\oplus
\widetilde{\mathrm{Nil}}_0(\mathbb{Z}[\pi_1(Y)],\alpha^{-1}) \longrightarrow \cdots~.
\end{multline*}
The Whitehead torsion $\tau(f) \in \mathrm{Wh}(\pi_1(X))$ has image
\[
[\tau(f)]~=~([K_{n+1}(\bar{M}^+,N)],[K_{n+1}(\bar{M}^+,N),\rho_1],
[K_{n+1}(\bar{M}^-,N),\rho_2])
\]
where
\begin{gather*}
\rho_1~:~K_{n+1}(\bar{M}^-,N) \longrightarrow K_{n+1}(\bar{M}^-,\bar{M}_1)~=~
t^{-1} K_{n+1}(\bar{M}^-,N)~,\\
\rho_2~:~K_{n+1}(\bar{M}^+,N) \longrightarrow
K_{n+1}(\bar{M}^+,t\bar{M}_1)~=~ t K_{n+1}(\bar{M}^+,N)~.
\end{gather*}
The homotopy equivalence $f$ is splittable along $Y$ if and only
if $[\tau(f)]=0$.
\end{thm}
\begin{proof}[Proof of Theorem \ref{Thm_TopSemisplit} (outline)]
The proof of \cite[Theorem \ref{W}(i)]{Waldhausen_1969} was based on
a one-one correspondence between the elementary operations in the
algebraic $K$-theory of the nilpotent categories and the elementary
operations (`surgeries' or cell-exchanges) for maps of injective
codimension 1 pairs. The proof of our Theorem \ref{maink} shows
that there is no algebraic obstruction to making a homotopy
equivalence semi-split by elementary operations, and hence there is
no geometric obstruction.
\end{proof}
\section{Higher $\mathrm{Nil}$-groups}\label{nilsection}
In this section, we shall prove Theorem \ref{maink} for
non-negative degrees.
Quillen \cite{Quillen} defined the $K$-theory space $K\mathscr{E} := \Omega
BQ(\mathscr{E})$ of an exact category $\mathscr{E}$. The space $BQ(\mathscr{E})$ is the
geometric realization of the simplicial set $N_\bullet Q(\mathscr{E})$,
which is the nerve of a certain category $Q(\mathscr{E})$ associated to
$\mathscr{E}$. The algebraic $K$-groups of $\mathscr{E}$ are defined for $* \in \mathbb{Z}$
$$K_*(\mathscr{E})~:=~\pi_*(K\mathscr{E})$$
using a nonconnective delooping for $* \leqslant -1$.
In particular, the algebraic $K$-groups of a ring $R$ are the
algebraic $K$-groups
$$K_*(R)~:=~K_*(\mathrm{Proj}(R))$$
of the exact category $\mathrm{Proj}(R)$ of f.g. projective $R$-modules.
The $\mathrm{Nil}$-categories defined in the Introduction all
have the structure of exact categories.
\begin{thm}\label{Thm_HigherDKR}
Let $R$ be a ring. Let $\mathscr{B}_1, \mathscr{B}_2$ be $R$-bimodules. Suppose
that $I$ is a filtered category and $\mathscr{B}_2=\colim_{\alpha \in I}
\mathscr{B}_2^\alpha$ is a direct limit of $R$-bimodules $\mathscr{B}_2^\alpha$,
each of which is a f.~g.~projective left $R$-module.
Let $i$ be the exact functor of exact categories of projective nil-objects:
\[
i: \mathrm{Nil}(R;\mathscr{B}_1 \otimes_R \mathscr{B}_2) \longrightarrow \mathrm{Nil}(R;\mathscr{B}_1,\mathscr{B}_2) ~;~ (Q,\sigma: Q \to \mathscr{B}_1 \mathscr{B}_2^\alpha Q) \longmapsto (Q,\mathscr{B}_2^\alpha Q, \sigma, 1).
\]
Then the induced map of $K$-theory spaces is a homotopy equivalence:
\[
\ol{K}i: K\mathrm{Nil}(R;\mathscr{B}_1 \otimes_R \mathscr{B}_2) \longrightarrow K\mathrm{Nil}(R;\mathscr{B}_1, \mathscr{B}_2)/(0 \times K(R)).
\]
In particular, for all $n \in \mathbb{N}$, there is an induced isomorphism
of abelian groups:
\[
i_*: \mathrm{Nil}_n(R;\mathscr{B}_1 \otimes_R \mathscr{B}_2) \oplus K_n(R) \longrightarrow \mathrm{Nil}_n(R;\mathscr{B}_1,\mathscr{B}_2).
\]
The exact functor
\[
j: \mathrm{Nil}(R;\mathscr{B}_1,\mathscr{B}_2) \longrightarrow \mathrm{Nil}(R;\mathscr{B}_1 \otimes_R \mathscr{B}_2)~;~ (P_1,P_2,\rho_1,\rho_2) \longmapsto (P_1, \rho_2 \circ \rho_1)
\]
satisfies $j \circ i = 1$.
\end{thm}
\begin{proof}
It is straightforward to show that tensor product commutes with
colimits over a category. Moreover, for any object $x =
(P_1,P_2,\rho_1: P_1 \to \mathscr{B}_1 P_2, \rho_2: P_2 \to \mathscr{B}_2 P_1)$,
since $P_2$ is finitely generated, there exists $\alpha \in I$ such
that $\rho_2$ factors through a map $P_2 \to \mathscr{B}_2^\alpha P_1$, and
similarly for short exact sequences of nil-objects. We thus obtain induced
isomorphisms of exact categories:
\begin{eqnarray*}
\colim_{\alpha \in I} \mathrm{Nil}(R;\mathscr{B}_1 \otimes_R \mathscr{B}_2^\alpha) &\longrightarrow& \mathrm{Nil}(R;\mathscr{B}_1 \otimes_R \mathscr{B}_2)\\
\colim_{\alpha \in I} \mathrm{Nil}(R;\mathscr{B}_1,\mathscr{B}_2^\alpha) &\longrightarrow& \mathrm{Nil}(R;\mathscr{B}_1,\mathscr{B}_2).
\end{eqnarray*}
So, by Quillen's colimit observation \cite[\S 2, Equation (9), page
20]{Quillen}, we obtain induced weak homotopy equivalences of
$K$-theory spaces:
\begin{eqnarray*}
\colim_{\alpha \in I} K\mathrm{Nil}(R;\mathscr{B}_1 \otimes_R \mathscr{B}_2^\alpha) &\longrightarrow& K\mathrm{Nil}(R;\mathscr{B}_1 \otimes_R \mathscr{B}_2)\\
\colim_{\alpha \in I}
K\mathrm{Nil}(R;\mathscr{B}_1,\mathscr{B}_2^\alpha) &\longrightarrow& K\mathrm{Nil}(R;\mathscr{B}_1,\mathscr{B}_2).
\end{eqnarray*}
Therefore, for each $\alpha \in I$, it suffices to show that the
restriction $\ol{K}i^\alpha$ is a homotopy equivalence.
Our setting is the exact category $\mathrm{Nil}(R;\mathscr{B}_1,\mathscr{B}_2^\alpha)$. By
assumption, we may consider objects
\begin{eqnarray*}
x &:=& (P_1, P_2, \rho_1, \rho_2)\\
x' &:=& (P_1, \mathscr{B}_2^\alpha P_1 \oplus P_2, \begin{pmatrix}0\\ \rho_1\end{pmatrix},
\begin{pmatrix}1 & \rho_2\end{pmatrix})\\
x'' &:=& (P_1, \mathscr{B}_2^\alpha P_1, \rho_2 \circ \rho_1, 1)\\
a &:=& (0,P_2,0,0)\\
a' &:=& (0,\mathscr{B}_2^\alpha P_1,0,0).
\end{eqnarray*}
Define morphisms
\begin{eqnarray*}
f &:=& (1, \begin{pmatrix}0\\ 1\end{pmatrix}) :x \longrightarrow x'\\
f' &:=& (1, \begin{pmatrix}1 & \rho_2\end{pmatrix}) :x' \longrightarrow x''\\
g &:=& (0, \begin{pmatrix}-\rho_2 \\ 1\end{pmatrix}): a \longrightarrow x'\\
g' &:=& (0, \begin{pmatrix}1 & 0\end{pmatrix}) :x' \longrightarrow a'\\
h &:=& (0, \rho_2) :a \longrightarrow a'.
\end{eqnarray*}
There are exact sequences
\begin{gather*}
\begin{CD}
0 @>>> x \oplus a @>{\begin{pmatrix}f & g\\ 0 & 1\end{pmatrix}}>> x' \oplus a
@>{\begin{pmatrix}g' & h\end{pmatrix}}>> a' @>>> 0
\end{CD}\\
\begin{CD}
0 @>>> a @>{\displaystyle{g}}>> x' @>{\displaystyle{f'}}>> x'' @>>> 0.
\end{CD}
\end{gather*}
Define functors $F', F'', G, G' : \mathrm{Nil}(R;\mathscr{B}_1,\mathscr{B}_2^\alpha) \longrightarrow
\mathrm{Nil}(R;\mathscr{B}_1,\mathscr{B}_2^\alpha)$ by
\[
F'(x) = x', \quad F''(x) = x'', \quad G(x) = a, \quad G'(x) = a'.
\]
Thus we have two exact sequences of exact functors
\begin{gather*}
\begin{CD}
0 @>>> 1 \oplus G @>>> F' \oplus G @>>> G' @>>> 0
\end{CD}\\
\begin{CD}
0 @>>> G @>>> F' @>>> F'' @>>> 0.
\end{CD}
\end{gather*}
Recall $j \circ i = 1$, and note $i \circ j = F''$. By Quillen's
Additivity Theorem \cite[p. 98, Cor. 1]{Quillen}, we obtain
homotopies $KF' \simeq 1 + KG'$ and $KF' \simeq KG + KF''$. Then
$$Ki \circ Kj~=~KF'' \simeq 1 + (KG'-KG)~,$$
where the subtraction uses the loop space structure.
Observe $G,G': \mathrm{Nil}(R;\mathscr{B}_1,\mathscr{B}_2^\alpha)
\to 0 \times \mathrm{Proj}(R)$. Therefore the functor $i^\alpha$ induces a
homotopy equivalence
\[\ol{K}i^\alpha: K\mathrm{Nil}(R;\mathscr{B}_1 \otimes_R
\mathscr{B}_2^\alpha) \longrightarrow K\mathrm{Nil}(R;\mathscr{B}_1, \mathscr{B}_2^\alpha)/(0 \times K(R)).\]
\end{proof}
\begin{rem}
The proof of Theorem \ref{Thm_HigherDKR} is best understood in
terms of finite chain complexes $x=(P_1,P_2,\rho_1,\rho_2)$ in
the category $\mathrm{Nil}(R;\mathscr{B}_1,\mathscr{B}_2)$, assuming that $\mathscr{B}_2$ is
a f.~g. projective left $R$-module. Any such $x$ represents a
class
\[[x]~=~\sum\limits^{\infty}_{r=0}(-1)^r[(P_1)_r,(P_2)_r,\rho_1,\rho_2]
\in \mathrm{Nil}_0(R;\mathscr{B}_1,\mathscr{B}_2)~.\]
The key observation is that $x$ determines a finite chain complex
$x'=(P'_1,P'_2,\rho'_1,\rho'_2)$ in $\mathrm{Nil}(R;\mathscr{B}_1,\mathscr{B}_2)$ such
that $\rho'_2:P'_2\to \mathscr{B}_2\otimes_RP_1$ is a chain equivalence
and
\[[x]~=~[x'] \in \widetilde{\mathrm{Nil}}_0(R;\mathscr{B}_1,\mathscr{B}_2)~.\eqno{(*)}\]
Specifically, let $P'_1=P_1$, $P'_2=\mathscr{M}(\rho_2)$, the algebraic
mapping cylinder of the chain map $\rho_2:P_2 \to
\mathscr{B}_2\otimes_RP_1$, and let
\[\begin{array}{rcccccl}
\rho'_1 &=&
\begin{pmatrix} 0 \\ 0 \\ \rho_1 \end{pmatrix} &:&
P'_1=P_1 &\longrightarrow& \mathscr{B}_1\otimes_RP'_2=\mathscr{M}(1_{\mathscr{B}_1} \otimes \rho_2)~,\\[2ex]
\rho'_2 &=& \begin{pmatrix} 1 & 0 & \rho_2\end{pmatrix} &:& P'_2~=~\mathscr{M}(\rho_2) &\longrightarrow& \mathscr{B}_2\otimes_RP_1~,
\end{array}\]
so that $P'_2/P_2=\mathscr{C}(\rho_2)$ is the algebraic mapping cone of
$\rho_2$. Moreover, the proof of $(*)$ is sufficiently functorial
to establish not only that the following maps of the reduced
nilpotent class groups are inverse isomorphisms:
\[\begin{array}{l}
i~:~\widetilde{\mathrm{Nil}}_0(R;\mathscr{B}_1\otimes_R\mathscr{B}_2)\xymatrix{\ar[r]^-{\displaystyle{\cong}}&}
\widetilde{\mathrm{Nil}}_0(R;\mathscr{B}_1,\mathscr{B}_2) ~;~(P,\rho) \mapsto (P,\mathscr{B}_2\otimes_RP,\rho,1)~,\\[1ex]
j~:~\widetilde{\mathrm{Nil}}_0(R;\mathscr{B}_1,\mathscr{B}_2)
\xymatrix{\ar[r]^-{\displaystyle{\cong}}&}\widetilde{\mathrm{Nil}}_0(R;\mathscr{B}_1\otimes_R\mathscr{B}_2)~;~
[x] \mapsto [x']~,
\end{array}
\]
but also that there exist isomorphisms of $\widetilde{\mathrm{Nil}}_n$ for
all higher dimensions $n > 0$, as shown above. In order to prove
equation $(*)$, note that $x$ fits into the sequence
\[\xymatrix@C+10pt{0 \ar[r] & x \ar[r]^-{\displaystyle{(1,u)}}& x'\ar[r]^-{\displaystyle{(0,v)}}&y
\ar[r]& 0} \eqno{(**)}\]
with
\[\begin{array}{rcl}
y &=&(0,\mathscr{C}(\rho_2),0,0)~,\\[2ex]
u &=& \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}~:~P_2
\to P'_2~=~\mathscr{M}(\rho_2)~,\\[3ex]
v &=& \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}~:~
P'_2 = \mathscr{M}(\rho_2) \to \mathscr{C}(\rho_2)
\end{array}\]
and
\[[y]~=~\sum\limits^{\infty}_{r=0}(-)^r[0,(\mathscr{B}_2\otimes_RP_1)_{r-1}\oplus(P_2)_r,0,0]~=~0
\in \widetilde{\mathrm{Nil}}_0(R;\mathscr{B}_1,\mathscr{B}_2)~.\] The projection
$\mathscr{M}(\rho_2) \to \mathscr{B}_2\otimes_RP_1$ defines a chain equivalence \[x'
~\simeq~ (P_1,\mathscr{B}_2\otimes_RP_1,\rho_2\circ \rho_1,1)~=~ij(x)\] so
that
\[[x]~=~[x']- [y]~=~[P_1,\mathscr{B}_2\otimes_RP_1,\rho_2\circ \rho_1,1]~
=~ij[x] \in \widetilde{\mathrm{Nil}}_0(R;\mathscr{B}_1,\mathscr{B}_2)~.\]
Now suppose that $x$ is a 0-dimensional chain complex in
$\mathrm{Nil}(R;\mathscr{B}_1,\mathscr{B}_2)$, that is, an object exactly as in the proof
of Theorem \ref{Thm_HigherDKR}. Let $x',x'',a,a',f,f',g,g',h$ be
as defined there. The exact sequence of $(**)$ can be written as
the short exact sequence of chain complexes
\[\xymatrix{ & & a \ar[d]^-{\displaystyle{g}} \ar@{=}[r] & a \ar[d]^-{\displaystyle{-h}} & \\
0 \ar[r] & x \ar[r]^-{\displaystyle{f}} & x' \ar[r]^-{\displaystyle{g'}} & a' \ar[r] &
0~.}\]
The first admissible exact sequence of the proof of Theorem
\ref{Thm_HigherDKR} is now immediate:
\[\xymatrix@C+15pt{0 \ar[r] &x \oplus a \ar[r]^-{\begin{pmatrix} f & g \\
0 & 1 \end{pmatrix}} & x'\oplus a \ar[r]^-{\begin{pmatrix} g' & h
\end{pmatrix}} & a' \ar[r] &0~.}\]
The second admissible exact sequence is self-evident:
\[\xymatrix{0 \ar[r] &a \ar[r]^-{\displaystyle{g}} & x' \ar[r]^-{\displaystyle{f'}} & x'' \ar[r]
&0~.}\]
\end{rem}
\section{Lower $\mathrm{Nil}$-groups}\label{Sec_Lower}
\subsection{Cone and suspension rings}
Let us recall some additional structures on the tensor product of
modules.
Originating from ideas of Karoubi--Villamayor
\cite{KaroubiVillamayor}, the following concept was studied
independently by S.M. Gersten \cite{Gersten} and J.B. Wagoner
\cite{Wagoner} in the construction of the non-connective
$K$-theory spectrum of a ring.
\begin{defn}[Gersten, Wagoner]
Let the \textbf{cone ring} $\Lambda\mathbb{Z}$ be the ring of $(\mathbb{N} \times
\mathbb{N})$-matrices over $\mathbb{Z}$ such that each row and column has only a
finite number of non-zero entries. Let the \textbf{suspension
ring} $\Sigma\mathbb{Z}$ be the quotient ring of $\Lambda\mathbb{Z}$ by the
two-sided ideal of matrices with only a finite number of non-zero
entries. For each $n \in \mathbb{N}$, write $\Sigma^n\mathbb{Z} :=
\underbrace{\Sigma\mathbb{Z} \otimes_\mathbb{Z} \cdots \otimes_\mathbb{Z} \Sigma\mathbb{Z}}_{n \text{
copies}}$ with $\Sigma^0\mathbb{Z} = \mathbb{Z}$. For a ring $R$ and for $n \in \mathbb{N}$, define the ring
$\Sigma^n R := \Sigma^n\mathbb{Z} \otimes_\mathbb{Z} R$.
\end{defn}
Roughly speaking, the suspension should be regarded
as the ring of ``bounded modulo compact operators.'' Gersten and
Wagoner showed that $K_i(\Sigma^n R)$ is naturally isomorphic to
$K_{i-n}(R)$ for all $i, n \in \mathbb{Z}$, in the sense of Quillen when
the subscript is positive, in the sense of Grothendieck when the
subscript is zero, and in the sense of Bass when the subscript is
negative.
For an $R$-bimodule $\mathscr{B}$, define the $\Sigma^n
R$-bimodule $\Sigma^n \mathscr{B} := \Sigma^n\mathbb{Z} \otimes_\mathbb{Z} \mathscr{B}$.
\begin{lem}\label{Lem_Transpose}
Let $R$ be a ring. Let $\mathscr{B}_1, \mathscr{B}_2$ be $R$-bimodules. Then, for
each $n \in \mathbb{N}$, there is a natural isomorphism of $\Sigma^n
R$-bimodules:
\[
t_n: \Sigma^n(\mathscr{B}_1 \otimes_R \mathscr{B}_2) \longrightarrow \Sigma^n \mathscr{B}_1 \otimes_{\Sigma^n R} \Sigma^n \mathscr{B}_2~;~
s \otimes (b_1 \otimes b_2) \longmapsto (s \otimes b_1) \otimes (1_{\Sigma^n \mathbb{Z}} \otimes b_2).
\]
\end{lem}
\begin{proof}
By transposition of the middle two factors, note that
\[
\Sigma^n \mathscr{B}_1 \otimes_{\Sigma^n R} \Sigma^n \mathscr{B}_2 = (\Sigma^n \mathbb{Z} \otimes_\mathbb{Z} \mathscr{B}_1)
\otimes_{(\Sigma^n\mathbb{Z} \otimes_\mathbb{Z} R)} (\Sigma^n \mathbb{Z} \otimes_\mathbb{Z} \mathscr{B}_2)
\]
is isomorphic to
\[
(\Sigma^n\mathbb{Z} \otimes_{\Sigma^n\mathbb{Z}} \Sigma^n\mathbb{Z}) \otimes_{\mathbb{Z}}
(\mathscr{B}_1 \otimes_R \mathscr{B}_2) = \Sigma^n\mathbb{Z} \otimes_\mathbb{Z} (\mathscr{B}_1 \otimes_R \mathscr{B}_2) = \Sigma^n(\mathscr{B}_1 \otimes_R \mathscr{B}_2).
\]
\end{proof}
\subsection{Definition of lower $\mathrm{Nil}$-groups}
\begin{defn}\label{Defn_NilTensorAlgebra}
Let $R$ be a ring. Let $\mathscr{B}$ be an $R$-bimodule. For all $n \in
\mathbb{N}$, define
\begin{eqnarray*}
\mathrm{Nil}_{-n}(R;\mathscr{B}) &:=& \mathrm{Nil}_0(\Sigma^nR;\Sigma^n \mathscr{B})\\
\wt{\mathrm{Nil}}_{-n}(R;\mathscr{B}) &:=& \wt \mathrm{Nil}_0(\Sigma^nR;\Sigma^n \mathscr{B}).
\end{eqnarray*}
\end{defn}
\begin{defn}\label{Defn_NilAmalgam}
Let $R$ be a ring. Let $\mathscr{B}_1, \mathscr{B}_2$ be $R$-bimodules. For all
$n \in \mathbb{N}$, define
\begin{eqnarray*}
\mathrm{Nil}_{-n}(R;\mathscr{B}_1,\mathscr{B}_2) &:=& \mathrm{Nil}_0(\Sigma^nR;\Sigma^n\mathscr{B}_1,\Sigma^n\mathscr{B}_2)\\
\wt{\mathrm{Nil}}_{-n}(R;\mathscr{B}_1,\mathscr{B}_2) &:=& \wt \mathrm{Nil}_0(\Sigma^nR;\Sigma^n\mathscr{B}_1,\Sigma^n\mathscr{B}_2)).
\end{eqnarray*}
\end{defn}
The next two theorems follow from the definitions and
\cite[Theorems 1,3]{Waldhausen_Rings}.
\begin{thm}[Waldhausen]\label{Thm_WaldhausenTensorRing}
Let $R$ be a ring and $\mathscr{B}$ be an $R$-bimodule. Consider the tensor
ring
\[
T_R(\mathscr{B}) = R \oplus \mathscr{B} \oplus (\mathscr{B}\otimes_R \mathscr{B}) \oplus (\mathscr{B}\otimes_R \mathscr{B}\otimes_R \mathscr{B}) \oplus \cdots.
\]
Suppose $\mathscr{B}$ is finitely generated projective as a left $R$-module
and free as a right $R$-module. Then, for all $n \in \mathbb{N}$, there is a
split monomorphism
\[
\sigma_B~:~\wt{\mathrm{Nil}}_{-n}(R;\mathscr{B}) \xymatrix{\ar@{>->}[r]&} K_{1-n}( T_R(\mathscr{B}))
\]
given for $n=0$ by the map
\[\sigma_B~:~\mathrm{Nil}_0(R;\mathscr{B}) \xymatrix{\ar@{>->}[r]&} K_1( T_R(\mathscr{B}))~;~
[P,\rho] \longmapsto \left[~ T_R(\mathscr{B}) \otimes_RP, 1-\widehat{\rho} ~\right]~,
\]
where $\widehat{\rho}$ is defined using $\rho$ and
multiplication in $ T_R(\mathscr{B})$.
Furthermore, there is a natural decomposition
\[
K_{1-n}( T_R(\mathscr{B})) = K_{1-n}(R) \oplus \wt{\mathrm{Nil}}_{-n}(R;\mathscr{B}).
\]
\end{thm}
For example, the last assertion of the theorem follows from the
equations:
\begin{align*}
K_{1-n}(T_R(\mathscr{B})) & = K_1(\Sigma^n T_R(\mathscr{B})) \\
& = K_1(T_{\Sigma^nR}(\Sigma^n\mathscr{B})) \\
& = K_1(\Sigma^n R) \oplus \wt{\mathrm{Nil}}_0(\Sigma^nR; \Sigma^n \mathscr{B}) \\
& = K_{1-n}(R) \oplus \wt{\mathrm{Nil}}_{-n}(R; \mathscr{B}).
\end{align*}
\begin{thm}[Waldhausen]\label{Waldhausen}
Let $R, A_1, A_2$ be rings. Let $R \to A_i$ be ring monomorphisms
such that $A_i = R \oplus \mathscr{B}_i$ for some $R$-bimodule $\mathscr{B}_i$.
Consider the pushout of rings
\begin{multline*}
A = A_1 *_R A_2 = R \oplus (\mathscr{B}_1 \oplus \mathscr{B}_2) \oplus
(\mathscr{B}_1\otimes \mathscr{B}_2 \oplus \mathscr{B}_2 \otimes \mathscr{B}_1) \\ \oplus (\mathscr{B}_1
\otimes\mathscr{B}_2 \otimes\mathscr{B}_1 \oplus \mathscr{B}_2 \otimes\mathscr{B}_1\otimes \mathscr{B}_2)
\oplus \cdots.
\end{multline*}
Suppose each $\mathscr{B}_i$ is free as a right $R$-module. Then, for all
$n \in \mathbb{N}$, there is a split monomorphism
\[
\sigma_A~:~\wt{\mathrm{Nil}}_{-n}(R;\mathscr{B}_1,\mathscr{B}_2) \xymatrix{\ar@{>->}[r]&} K_{1-n}(A),
\]
given for $n=0$ by the map
$$\begin{array}{l}
{\mathrm{Nil}}_0(R;\mathscr{B}_1,\mathscr{B}_2) \xymatrix{\ar@{>->}[r]&} K_1(A)~;\\[1ex]
[P_1,P_2,\rho_1,\rho_2] \longmapsto \left[ (A \otimes_R P_1)\oplus (A \otimes_R P_2), \begin{pmatrix} 1 &
\widehat{\rho}_2\\ \widehat{\rho}_1 & 1\end{pmatrix} \right]~,
\end{array}$$
where $\widehat{\rho}_i$ is defined using $\rho_i$ and
multiplication in $A_i$ for $i = 1,2$.
Furthermore, there is a natural Mayer--Vietoris type exact
sequence
$$\begin{array}{l}
\xymatrix{\cdots \ar[r]^-{\displaystyle{\partial}} & K_{1-n}(R) \ar[r] &
K_{1-n}(A_1) \oplus K_{1-n}(A_2)\ar[r]& }\\
\hskip125pt
\xymatrix{
\dfrac{K_{1-n}(A)}{\wt{\mathrm{Nil}}_{-n}(R;\mathscr{B}_1,\mathscr{B}_2)}
\ar[r]^-{\displaystyle{\partial}} &K_{-n}(R) \ar[r] & \cdots}
\end{array}$$
\end{thm}
\subsection{The isomorphism for lower $\mathrm{Nil}$-groups}
\begin{thm}\label{Thm_LowerDKR}
Let $R$ be a ring. Let $\mathscr{B}_1, \mathscr{B}_2$ be $R$-bimodules. Suppose
that $I$ is a filtered category and $\mathscr{B}_2=\colim_{\alpha \in I}
\mathscr{B}_2^\alpha$ is a direct limit of $R$-bimodules $\mathscr{B}_2^\alpha$,
each of which is a f.g.~projective left $R$-module. Then, for all $n
\in \mathbb{N}$, there is an induced isomorphism:
\[
\mathrm{Nil}_{-n}(R;\mathscr{B}_1 \otimes_R \mathscr{B}_2) \oplus K_{-n}(R) \longrightarrow \mathrm{Nil}_{-n}(R;\mathscr{B}_1,\mathscr{B}_2).
\]
\end{thm}
\begin{proof}
Let $n \in \mathbb{N}$. By Lemma \ref{Lem_Transpose} and Theorem
\ref{Thm_HigherDKR}, there are induced isomorphisms:
\begin{multline*}
\mathrm{Nil}_{-n}(R;\mathscr{B}_1 \otimes_R \mathscr{B}_2) \oplus K_{-n}(R) =
\mathrm{Nil}_0(\Sigma^nR;\Sigma^n(\mathscr{B}_1 \otimes_R \mathscr{B}_2)) \oplus K_0\Sigma^n(R)\\
\longrightarrow
\mathrm{Nil}_0(\Sigma^n R; \Sigma^n \mathscr{B}_1 \otimes_{\Sigma^n R} \Sigma^n
\mathscr{B}_2) \oplus K_0\Sigma^n(R)\\
\longrightarrow
\mathrm{Nil}_0(\Sigma^nR;\Sigma^n\mathscr{B}_1,\Sigma^n\mathscr{B}_2) =
\mathrm{Nil}_{-n}(R;\mathscr{B}_1,\mathscr{B}_2).
\end{multline*}
\end{proof}
\section{Applications} \label{Applications}
We indicate some applications of our main theorem (Theorem
\ref{maink}). In \S\ref{First} we prove Theorem~\ref{first}(ii),
which describes the restrictions of the maps
$$\theta_!~:~K_*(R[\overline{G}]) \to
K_*(R[G])~,~\theta^!~:~K_*(R[G]) \to K_*(R[\overline{G}])$$ to the
$\widetilde{\mathrm{Nil}}$-terms, with $\theta:\overline{G} \to G$ the
inclusion of the canonical index 2 subgroup $\overline{G}$ for any
group $G$ over $D_\infty$. In \S\ref{Second} we give the first known
example of a non-zero $\mathrm{Nil}$ group occurring in the $K$-theory of an
integral group ring of an amalgamated free product. In
\S\ref{Third} we sharpen the Farrell--Jones Conjecture in
$K$-theory, replacing the family of virtually cyclic groups by the
smaller family of finite-by-cyclic groups. In \S\ref{Subsec_PSL2Z}
we compute the $K_*(R[\Gamma])$ for the modular group $\Gamma = PSL_2(\mathbb{Z})$.
\subsection{Algebraic $K$-theory over $D_\infty$}\label{First}
The overall goal of this section is to show that the abstract
isomorphisms $i_*$ and $j_*$ coincide with the restrictions of the
induction and transfer maps $\theta_!$ and $\theta^!$ in the group
ring setting.
\subsubsection{Twisting}
We start by recalling the algebraic $K$-theory of twisted polynomial
rings.
\begin{state}
Consider any (unital, associative) ring $R$ and any ring
automorphism $\alpha:R \to R$. Let $t$ be an indeterminate over $R$
such that
\[rt~=~t\alpha(r)~~(r \in R)~.\]
For any $R$-module $P$, let $tP := \{tx ~|~ x \in P\}$ be the set
with left $R$-module structure
\[tx + t y~=~t (x+y)~,~r(tx)~=~t(\alpha(r)x) \in tP~.\]
Further endow the left $R$-module $tR$ with the $R$-bimodule
structure
\[R \times tR \times R \to tR~;~(q,tr,s) \mapsto t
\alpha(q)rs~.\]
The $\mathrm{Nil}$-category of $R$ with respect to $\alpha$ is the exact
category defined by
\[\mathrm{Nil}(R,\alpha)~:=~\mathrm{Nil}(R;tR).\]
The objects $(P,\rho)$ consist of any f.g. projective $R$-module $P$
and any nilpotent morphism $\rho:P \to tP=tR\otimes_RP$. The
$\mathrm{Nil}$-groups are written
\[\mathrm{Nil}_*(R,\alpha)~:=~\mathrm{Nil}_*(R;tR)~,~\widetilde{\mathrm{Nil}}_*(R,\alpha)~:=~
\widetilde{\mathrm{Nil}}_*(R;tR)~,\] so that
\[\mathrm{Nil}_*(R,\alpha)~=~K_*(R) \oplus \widetilde{\mathrm{Nil}}_*(R,\alpha)~.\]
\end{state}
\begin{state}
The tensor algebra on $tR$ is the $\alpha$-twisted polynomial
extension of $R$
\[T_R(tR)~=~R_{\alpha}[t]~=~\sum\limits^{\infty}_{k=0}t^kR~.\]
Given an $R$-module $P$ there is induced an $R_{\alpha}[t]$-module
$$R_{\alpha}[t]\otimes_RP~=~P_{\alpha}[t]$$
whose elements are finite linear combinations
$\sum\limits^{\infty}_{j=0}t^jx_j$ ($x_j \in P$). Given $R$-modules
$P,Q$ and an $R$-module morphism $\rho:P \to tQ$, define its
extension as the $R_{\alpha}[t]$-module morphism
$$\widehat{\rho}~=~t\rho~:~P_{\alpha}[t] \to Q_{\alpha}[t]~;~
\sum\limits^{\infty}_{j=0}t^jx_j
\mapsto\sum\limits^{\infty}_{j=0}t^j\rho(x_j)~.$$
\end{state}
\begin{state}
By Bass \cite{Bass}, Farrell--Hsiang \cite{FarrellHsiang}, and
Quillen \cite{Grayson}, there are decompositions
\begin{align*}
&K_n(R_{\alpha}[t])~=~K_n(R) \oplus \widetilde{\mathrm{Nil}}_{n-1}(R,\alpha)~,\\
&K_n(R_{\alpha^{-1}}[t^{-1}])~=~K_n(R) \oplus \widetilde{\mathrm{Nil}}_{n-1}(R,\alpha^{-1})~,\\
&K_n(R_{\alpha}[t,t^{-1}])~=~K_n(1-\alpha:R \to R)\oplus
\widetilde{\mathrm{Nil}}_{n-1}(R,\alpha)\oplus \widetilde{\mathrm{Nil}}_{n-1}(R,\alpha^{-1})~.
\end{align*}
In particular for $n = 1$, by Theorem
\ref{Thm_WaldhausenTensorRing}, there are defined split
monomorphisms
\begin{align*}
&\sigma^+_B~:~\widetilde{\mathrm{Nil}}_0(R,\alpha) \xymatrix{~\ar@{>->}[r]&} K_1(R_{\alpha}[t])~;~
[P,\rho] \longmapsto \left[ P_{\alpha}[t], 1-t\rho \right]~,\\[1ex]
&\sigma^-_B~:~\widetilde{\mathrm{Nil}}_0(R,\alpha^{-1}) \xymatrix{~\ar@{>->}[r]&} K_1(R_{\alpha^{-1}}[t^{-1}])~;~
[P,\rho] \longmapsto \left[ P_{\alpha^{-1}}[t^{-1}], 1-t^{-1}\rho \right]~,\\[1ex]
&\sigma_B~=~\begin{pmatrix}\psi^+\sigma^+_B & \psi^-\sigma^-_B\end{pmatrix}~:~\widetilde{\mathrm{Nil}}_0(R,\alpha)\oplus
\widetilde{\mathrm{Nil}}_0(R,\alpha^{-1}) \xymatrix{~\ar@{>->}[r]&} K_1(R_{\alpha}[t,t^{-1}])~;\\
&([P_1,\rho_1],[P_2,\rho_2]) \longmapsto \left[ (P_1\oplus P_2)_{\alpha}[t,t^{-1}], \begin{pmatrix}1-t\rho_1 & 0\\ 0 & 1-t^{-1}\rho_2 \end{pmatrix} \right]~.
\end{align*}
These definitions extend to all integers $n \leqslant 1$ by the
suspension isomorphisms of Section \ref{Sec_Lower}.
\end{state}
\subsubsection{Scaling}
Next, consider the effect an inner automorphism on $\alpha$.
\begin{state}
Suppose $\alpha,\alpha':R \to R$ are automorphisms satisfying
$$\alpha'(r)~=~u\alpha(r)u^{-1}\in R~~(r \in R)$$
for some unit $u \in R$, and that $t'$ is an indeterminate over $R$
satisfying
$$rt'~=~t'\alpha'(r)~~(r \in R)~.$$
Denote the canonical inclusions
\[
\begin{array}{ll}
\psi^+~:~R_{\alpha}[t] \longrightarrow R_{\alpha}[t,t^{-1}] &
\psi^-~:~R_{\alpha^{-1}}[t^{-1}] \longrightarrow R_{\alpha}[t,t^{-1}]\\[1ex]
\psi'^+~:~R_{\alpha'}[t'] \longrightarrow R_{\alpha'}[t',t'^{-1}] &
\psi'^-~:~R_{\alpha'^{-1}}[t'^{-1}] \longrightarrow R_{\alpha'}[t',t'^{-1}]~.
\end{array}
\]
\end{state}
\begin{state}\label{scaling1}
The various polynomial rings are related by \emph{scaling
isomorphisms}
$$\begin{array}{l}
\beta^+_u~:~R_{\alpha}[t] \xymatrix{\ar[r]^-{\displaystyle{\cong}}&} R_{\alpha'}[t']~;~
t \longmapsto t'u~,\\[1ex]
\beta^-_u~:~R_{\alpha^{-1}}[t^{-1}] \xymatrix{\ar[r]^-{\displaystyle{\cong}}&} R_{\alpha'^{-1}}[t'^{-1}]~;~
t^{-1} \longmapsto u^{-1}t'^{-1}~,\\[1ex]
\beta_u~:~R_{\alpha}[t,t^{-1}] \xymatrix{\ar[r]^-{\displaystyle{\cong}}&} R_{\alpha'}[t',t'^{-1}]~;~
t \longmapsto t'u
\end{array}$$
satisfying the equations
\begin{eqnarray*}
\beta_u \circ \psi^+ &=& \psi'^+ \circ \beta_u^+ ~:~ R_\alpha[t] \longrightarrow R_{\alpha'}[t',t'^{-1}]\\[1ex]
\beta_u \circ \psi^- &=& \psi'^- \circ \beta_u^- ~:~ R_{\alpha^{-1}}[t^{-1}] \longrightarrow R_{\alpha'}[t',t'^{-1}] ~.
\end{eqnarray*}
\end{state}
\begin{state}\label{scaling2}
There are corresponding scaling isomorphisms of exact categories
$$\begin{array}{l}
\beta^+_u~:~\mathrm{Nil}(R,\alpha) \xymatrix{\ar[r]^-{\displaystyle{\cong}}&} \mathrm{Nil}(R,\alpha')~;~
(P,\rho) \longmapsto (P,t'u \rho t^{-1}:P\to t'P)\\[1ex]
\beta^-_u~:~\mathrm{Nil}(R,\alpha^{-1})\xymatrix{\ar[r]^-{\displaystyle{\cong}}&} \mathrm{Nil}(R,\alpha'^{-1}) ~;~
(P,\rho) \longmapsto (P,t'^{-1}u \rho t:P'\to t'^{-1}P')~,
\end{array}$$
where we mean
\[
(t'u \rho t^{-1})(tx) := t'ux,\quad
(t'^{-1}u\rho t)(t^{-1}x) := t'^{-1}ux~.
\]
\end{state}
\begin{state}\label{scaling3}
For all $n \leqslant 1$, the various scaling isomorphisms are related by
equations
\begin{eqnarray*}
(\beta_u^+)_* \circ \sigma_B^+ &=& \sigma_B'^+ \circ \beta_u^+ ~:~
\widetilde{\mathrm{Nil}}_{n-1}(R,\alpha) \longrightarrow K_n(R_{\alpha'}[t'])\\[1ex]
(\beta_u^-)_* \circ \sigma_B^- &=& \sigma_B'^- \circ \beta_u^- ~:~
\widetilde{\mathrm{Nil}}_{n-1}(R,\alpha^{-1}) \longrightarrow K_n(R_{\alpha'^{-1}}[t'^{-1}])\\[1ex]
(\beta_u)_* \circ \sigma_B &=& \sigma_B' \circ \left(\begin{smallmatrix}\beta_u^+ & 0\\ 0 & \beta_u^-\end{smallmatrix}\right) ~:~
\widetilde{\mathrm{Nil}}_{n-1}(R,\alpha)\oplus \widetilde{\mathrm{Nil}}_{n-1}(R,\alpha^{-1}) \longrightarrow K_n(R_{\alpha'}[t',t'^{-1}]) ~.
\end{eqnarray*}
\end{state}
\subsubsection{Group rings}
We now adapt these isomorphisms to the case of group rings $R[G]$ of
groups $G$ over the infinite dihedral group $D_\infty$. In order to
prove Propositions \ref{induction} and \ref{transfer}, the overall
idea is to transform information about the product $t_2 t_1$ arising
from the transposition $\mathscr{B}_2 \otimes \mathscr{B}_1$ into information about
the product $t_2^{-1} t_1^{-1}$ arising in the second
$\tilde{\mathrm{Nil}}$-summand of the twisted Bass decomposition. We
continue to discuss the ingredients in a sequence of statements.
\begin{state}
Let $F$ be a group, and let $\alpha:F \to F$ be an automorphism.
Recall that the injective $HNN$ extension $F\rtimes_{\alpha} \mathbb{Z}$ is
the set $F \times \mathbb{Z}$ with group multiplication
$$(x,n)(y,m)~:=~(\alpha^m(x)y,m+n) \in F\rtimes_{\alpha}\mathbb{Z}~.$$
Then, for any ring $R$, writing $t=(1_F,1)$ and $(x,n)=t^nx \in
F\rtimes_{\alpha}\mathbb{Z}$, we have
$$R[F\rtimes_{\alpha} \mathbb{Z}]~=~R[F]_{\alpha}[t,t^{-1}]~.$$
\end{state}
\begin{state}
Consider any group $G=G_1 *_F G_2$ over $D_\infty$, where
$$F~=~G_1 \cap G_2~\subset~\overline{G}
~=~F\rtimes_{\alpha}\mathbb{Z}~=~F\rtimes_{\alpha'}\mathbb{Z} ~\subset~
G~=~G_1*_FG_2~.$$ Fix elements $t_1 \in G_1-F$, $t_2\in G_2-F$, and
define elements
$$t~:=~t_1t_2 \in \overline{G},~t'~:=~t_2t_1 \in \overline{G}~,~u~:=~(t')^{-1} t^{-1} \in F~.$$
Define the automorphisms
\begin{align*}
&\alpha_1~:~F \longrightarrow F~;~x \mapsto (t_1)^{-1}x t_1~,\\
&\alpha_2~:~F \longrightarrow F~;~x \mapsto (t_2)^{-1}x t_2~,\\
&\alpha~:=~\alpha_2\circ\alpha_1~:~F \longrightarrow F~;~x \mapsto t^{-1}xt~,\\
&\alpha'~:=~\alpha_1\circ\alpha_2~:~F \longrightarrow F~;~x \mapsto t'^{-1}xt'
\end{align*}
such that
$$xt~=~t \alpha(x)~,~xt'=~t'\alpha'(x)~,~
\alpha'(x)~=~u \alpha^{-1}(x)u^{-1}~~(x \in F)~.$$ In particular,
note that $\alpha'$ and $\alpha^{-1}$ (not $\alpha$) are related by
left inner automorphism by $u$.
\end{state}
\begin{state}\label{Statement_RingMaps}
Denote the canonical inclusions
\[
\begin{array}{ll}
\psi^+~:~R_{\alpha}[t] \longrightarrow R_{\alpha}[t,t^{-1}] &
\psi^-~:~R_{\alpha^{-1}}[t^{-1}] \longrightarrow R_{\alpha}[t,t^{-1}]\\[1ex]
\psi'^+~:~R_{\alpha'}[t'] \longrightarrow R_{\alpha'}[t',t'^{-1}] &
\psi'^-~:~R_{\alpha'^{-1}}[t'^{-1}] \longrightarrow R_{\alpha'}[t',t'^{-1}]~.
\end{array}
\]
The inclusion $R[F] \to R[G]$ extends to ring monomorphisms
\[
\begin{array}{ll}
\theta~:~R[F]_{\alpha}[t,t^{-1}] \longrightarrow R[G] &
\theta'~:~R[F]_{\alpha'}[t',t'^{-1}] \longrightarrow R[G]
\end{array}
\]
such that
\[
{\rm im}(\theta)~=~{\rm im}(\theta')~=~ R[\overline{G}]
\subset R[G]~=~R[G_1]*_{R[F]}R[G_2]~.
\]
Furthermore, the inclusion $R[F] \to R[G]$ extends to ring
monomorphisms
\[
\begin{array}{ll}
\phi = \theta \circ \psi^+~:~R[F]_{\alpha}[t] \longrightarrow R[G] & \phi' = \theta' \circ \psi'^+~:~R[F]_{\alpha'}[t'] \longrightarrow R[G]~.
\end{array}
\]
\end{state}
\begin{state}\label{scalingG1}
By Statement \ref{scaling1}, there are defined scaling isomorphisms
of rings
\[
\begin{array}{l} \beta^+_u~:~R[F]_{\alpha^{-1}}[t^{-1}]
\xymatrix{\ar[r]^-{\displaystyle{\cong}}&}
R[F]_{\alpha'}[t'] ~;~t^{-1} \longmapsto t'u~,\\[1ex]
\beta^-_u~:~R[F]_{\alpha}[t] \xymatrix{\ar[r]^-{\displaystyle{\cong}}&}
R[F]_{\alpha'^{-1}}[t'^{-1}] ~;~
t \longmapsto u^{-1} t'^{-1}~,\\[1ex]
\beta_u~:~R[F]_{\alpha}[t,t^{-1}]
\xymatrix{\ar[r]^-{\displaystyle{\cong}}&} R[F]_{\alpha'}[t',t'^{-1}]~;~
t \longmapsto u^{-1} t'^{-1}
\end{array}
\]
which satisfy the equations
\begin{eqnarray*}
\beta_u \circ \psi^- &=& \psi'^+ \circ \beta_u^+ ~:~ R[F]_{\alpha^{-1}}[t^{-1}] \longrightarrow R[F]_{\alpha'}[t',t'^{-1}]\\[1ex]
\beta_u \circ \psi^+ &=& \psi'^- \circ \beta_u^- ~:~ R[F]_{\alpha}[t] \longrightarrow R[F]_{\alpha'}[t',t'^{-1}]\\[1ex]
\theta &=& \theta' \circ \beta_u ~:~ R[F]_{\alpha}[t,t^{-1}] \longrightarrow R[G]~.
\end{eqnarray*}
\end{state}
\begin{state}\label{scalingG2}
By Statement \ref{scaling2}, there are scaling isomorphisms of exact
categories
$$\begin{array}{l}
\beta^+_u~:~\mathrm{Nil}(R[F],\alpha^{-1}) \xymatrix{\ar[r]^-{\displaystyle{\cong}}&} \mathrm{Nil}(R[F],\alpha')~;~
(P,\rho) \longmapsto (P,t'u t\rho)~,\\[1ex]
\beta^-_u~:~\mathrm{Nil}(R[F],\alpha) \xymatrix{\ar[r]^-{\displaystyle{\cong}}&} \mathrm{Nil}(R[F],\alpha'^{-1})~;~
(P,\rho) \longmapsto (P,t'^{-1}u t^{-1}\rho)~.
\end{array}$$
\end{state}
\begin{state}\label{scalingG3}
By Statement \ref{scaling3}, for all $n \leqslant 1$, the various scaling
isomorphisms are related by the equations
\begin{eqnarray*}
(\beta_u^+)_* \circ \sigma_B^- &=& \sigma_B'^+ \circ \beta_u^+ ~:~ \widetilde{\mathrm{Nil}}_{*-1}(R[F],\alpha^{-1}) \longrightarrow K_*(R[F]_{\alpha'}[t'])\\[1ex]
(\beta_u^-)_* \circ \sigma_B^+ &=& \sigma_B'^- \circ \beta_u^- ~:~ \widetilde{\mathrm{Nil}}_{*-1}(R[F],\alpha) \longrightarrow K_*(R[F]_{\alpha'^{-1}}[t'^{-1}])\\[1ex]
(\beta_u)_* \circ \sigma_B &=& \sigma_B' \circ \left(\begin{smallmatrix} 0 & \beta^+_u \\
\beta^-_u & 0 \end{smallmatrix}\right) ~:~ \widetilde{\mathrm{Nil}}_{*-1}(R[F],\alpha)\oplus \widetilde{\mathrm{Nil}}_{*-1}(R[F],\alpha^{-1}) \longrightarrow K_*(R[F]_{\alpha'}[t',t'^{-1}]) ~.
\end{eqnarray*}
\end{state}
\subsubsection{Transposition}
Next, we study the effect of transposition of the bimodules $\mathscr{B}_1$
and $\mathscr{B}_2$ in order to relate $\alpha$ and $\alpha'$. In
particular, there is no mention of $\alpha^{-1}$ in this section.
\begin{state}
The $R[F]$-bimodules
$$\mathscr{B}_1~=~R[G_1-F]~=~t_1R[F]~,~\mathscr{B}_2~=~R[G_2-F]~=~t_2R[F]$$
are free left and right $R[F]$-modules of rank one. We shall use the
$R[F]$-bimodule isomorphisms
\begin{align*}
&\mathscr{B}_1\otimes_{R[F]}\mathscr{B}_2 \longrightarrow tR[F]~;~t_1x_1\otimes t_2x_2 \longmapsto
t \alpha_2(x_1)x_2\\
&\mathscr{B}_2\otimes_{R[F]}\mathscr{B}_1 \longrightarrow t'R[F]~;~t_2x_2\otimes t_1x_1 \longmapsto
t' \alpha_1(x_2)x_1
\end{align*}
to make the identifications
$$\begin{array}{ll}
\mathscr{B}_1\otimes_{R[F]}\mathscr{B}_2~=~tR[F]~,&\mathrm{Nil}(R[F];\mathscr{B}_1\otimes_{R[F]}\mathscr{B}_2)~=~\mathrm{Nil}(R[F],\alpha)~,\\[1ex]
\mathscr{B}_2\otimes_{R[F]}\mathscr{B}_1~=~t'R[F]~,&\mathrm{Nil}(R[F];\mathscr{B}_2\otimes_{R[F]}\mathscr{B}_1)~=~\mathrm{Nil}(R[F],\alpha')~.
\end{array}$$
\end{state}
\begin{state}
Theorem \ref{maink} gives inverse isomorphisms
\begin{align*}
&i_*~:~\widetilde{\mathrm{Nil}}_*(R[F],\alpha) \xymatrix{\ar[r]^-{\displaystyle{\cong}}&} \widetilde{\mathrm{Nil}}_*(R[F];\mathscr{B}_1,\mathscr{B}_2)~,\\
&j_*~:~\widetilde{\mathrm{Nil}}_*(R[F];\mathscr{B}_1,\mathscr{B}_2)\xymatrix{\ar[r]^-{\displaystyle{\cong}}&}
\widetilde{\mathrm{Nil}}_*(R[F],\alpha)
\end{align*}
which for $*=0$ are given by
\begin{align*}
&i_*~:~\widetilde{\mathrm{Nil}}_0(R[F],\alpha) \xymatrix{\ar[r]^-{\displaystyle{\cong}}&} \widetilde{\mathrm{Nil}}_0(R[F];\mathscr{B}_1,\mathscr{B}_2)~;~
[P,\rho] \longmapsto [P, t_2P,\rho,1]~,\\
&j_*~:~\widetilde{\mathrm{Nil}}_0(R[F];\mathscr{B}_1,\mathscr{B}_2)
\xymatrix{\ar[r]^-{\displaystyle{\cong}}&} \widetilde{\mathrm{Nil}}_0(R[F],\alpha)~;~
[P_1,P_2,\rho_1,\rho_2]\longmapsto [P_1,\rho_2\circ \rho_1]~.
\end{align*}
\end{state}
\begin{state}
Similarly, there are defined inverse isomorphisms
\begin{align*}
&i'_*~:~\widetilde{\mathrm{Nil}}_*(R[F],\alpha') \xymatrix{\ar[r]^-{\displaystyle{\cong}}&} \widetilde{\mathrm{Nil}}_*(R[F];\mathscr{B}_2,\mathscr{B}_1)~,\\
&j'_*~:~\widetilde{\mathrm{Nil}}_*(R[F];\mathscr{B}_2,\mathscr{B}_1)
\xymatrix{\ar[r]^-{\displaystyle{\cong}}&} \widetilde{\mathrm{Nil}}_*(R[F],\alpha')
\end{align*}
which for $*=0$ are given by
\begin{align*}
&i'_*~:~\widetilde{\mathrm{Nil}}_0(R[F],\alpha') \xymatrix{\ar[r]^-{\displaystyle{\cong}}&} \widetilde{\mathrm{Nil}}_0(R[F];\mathscr{B}_2,\mathscr{B}_1)~;~
[P',\rho'] \longmapsto [P',t_1P',\rho',1]~,\\
&j'_*~:~\widetilde{\mathrm{Nil}}_0(R[F];\mathscr{B}_2,\mathscr{B}_1)\xymatrix{\ar[r]^-{\displaystyle{\cong}}&}
\widetilde{\mathrm{Nil}}_0(R[F],\alpha')~;~
[P_2,P_1,\rho_2,\rho_1]\longmapsto [P_2,\rho_1\circ \rho_2]~.
\end{align*}
\end{state}
\begin{state}\label{Statement_Transposition}
The transposition isomorphism of exact categories
\[
\tau_A ~:~\mathrm{Nil}(R[F];\mathscr{B}_1,\mathscr{B}_2) \xymatrix{\ar[r]^-{\displaystyle{\cong}}&} \mathrm{Nil}(R[F];\mathscr{B}_2,\mathscr{B}_1)~;~
(P_1,P_2,\rho_1,\rho_2) \longmapsto (P_2,P_1,\rho_2,\rho_1)
\]
induces isomorphisms
$$\begin{array}{l}
\tau_A~:~\mathrm{Nil}_*(R[F];\mathscr{B}_1,\mathscr{B}_2)~\cong~\mathrm{Nil}_*(R[F];\mathscr{B}_2,\mathscr{B}_1)~,\\[1ex]
\tau_A~:~\widetilde{\mathrm{Nil}}_*(R[F];\mathscr{B}_1,\mathscr{B}_2)~\cong~\widetilde{\mathrm{Nil}}_*(R[F];\mathscr{B}_2,\mathscr{B}_1)~.
\end{array}$$
Note, by Theorem \ref{maink}, the composites
$$\begin{array}{l}
\tau_B~:=~j'_* \circ \tau_A \circ i_*~:~\widetilde{\mathrm{Nil}}_*(R[F],\alpha) \xymatrix{\ar[r]^-{\displaystyle{\cong}}&}
\widetilde{\mathrm{Nil}}_*(R[F],\alpha')~,\\[1ex]
\tau'_B~:=~j_* \circ \tau^{-1}_A \circ i'_*~:~\widetilde{\mathrm{Nil}}_*(R[F],\alpha') \xymatrix{\ar[r]^-{\displaystyle{\cong}}&}
\widetilde{\mathrm{Nil}}_*(R[F],\alpha)
\end{array}$$
are inverse isomorphisms, which for $*=0$ are given by
$$\begin{array}{l}
\tau_B ~:~\widetilde{\mathrm{Nil}}_0(R[F],\alpha) \xymatrix{\ar[r]^-{\displaystyle{\cong}}&}
\widetilde{\mathrm{Nil}}_0(R[F],\alpha')~;~[P,\rho] \longmapsto
[t_2P,t_2\rho]~,\\[1ex]
\tau'_B ~:~\widetilde{\mathrm{Nil}}_0(R[F],\alpha') \xymatrix{\ar[r]^-{\displaystyle{\cong}}&}
\widetilde{\mathrm{Nil}}_0(R[F],\alpha)~;~[P',\rho'] \longmapsto
[t_1P',t_1\rho']~.
\end{array}$$
Furthermore, note that the various transpositions are related by the
equation
\begin{eqnarray*}
\tau_A \circ i_* &=& i'_* \circ \tau_B ~:~ \widetilde{\mathrm{Nil}}_*(R[F],\alpha) \longrightarrow \widetilde{\mathrm{Nil}}_*(R[F];\mathscr{B}_2,\mathscr{B}_1)~.
\end{eqnarray*}
\end{state}
\begin{state}\label{Statement_SigmaA}
Recall from Theorem \ref{Waldhausen} that there is a split
monomorphism
$$\sigma_A~:~\widetilde{\mathrm{Nil}}_{n-1}(R[F];\mathscr{B}_1,\mathscr{B}_2) \longrightarrow K_n(R[G]) $$
such that the $n=1$ case is given by
\begin{multline*}
\sigma_A~:~\widetilde{\mathrm{Nil}}_0(R[F];\mathscr{B}_1,\mathscr{B}_2) \longrightarrow K_1(R[G])~;\\
[P_1,P_2,\rho_1,\rho_2] \longmapsto \left[ P_1[G] \oplus P_2[G],
\begin{pmatrix} 1 & t_2\rho_2 \\ t_1\rho_1 & 1\end{pmatrix}
\right]~.
\end{multline*}
Elementary row and column operations produce an equivalent
representative:
\[
\begin{pmatrix}
1 & - t_2 \rho_2\\
0 & 1
\end{pmatrix}
\begin{pmatrix}
1 & t_2 \rho_2\\
t_1 \rho_1 & 1
\end{pmatrix}
\begin{pmatrix}
1 & 0\\
-\rho_1 & 1
\end{pmatrix}
= \begin{pmatrix}
1 - t \rho_2 \rho_1 & 0\\
0 & 1
\end{pmatrix}~.
\]
Thus the $n=1$ case satisfies the equations (with a similar argument
for the second equality):
\[
\sigma_A[P_1,P_2,\rho_1,\rho_2] ~=~ \left[ P_1[G], 1-t\rho_2\rho_1 \right]
~=~ \left[ P_2[G], 1-t'\rho_1\rho_2 \right]~.
\]
Therefore for all $n \leqslant 1$, the split monomorphism $\sigma_A'$,
associated to the amalgamated free product $G=G_2*_FG_1$, satisfies
the equation
\[
\sigma_A~=~\sigma_A' \circ \tau_A~:~
\widetilde{\mathrm{Nil}}_{n-1}(R[F];\mathscr{B}_1,\mathscr{B}_2) \longrightarrow K_n(R[G]) ~.
\]
\end{state}
\subsubsection{Induction}
We consider the effect of induction maps on
$\widetilde{\mathrm{Nil}}$-summands (Prop. \ref{induction}).
\begin{state}\label{Statement_meat}
Recall from Theorem \ref{maink} the isomorphism
\begin{multline*}
i_*~:~\widetilde{\mathrm{Nil}}_{*-1}(R[F],\alpha)~=~
\widetilde{\mathrm{Nil}}_{*-1}(R[F];\mathscr{B}_1\otimes_{R[F]}\mathscr{B}_2)
\longrightarrow \widetilde{\mathrm{Nil}}_{*-1}(R[F];\mathscr{B}_1,\mathscr{B}_2)~;\\
[P,\rho] \longmapsto [P,t_2P,\rho,1]~.
\end{multline*}
Let $(P,\rho)$ be an object in the exact category
$\mathrm{Nil}(R[F],\alpha)$. By Statement \ref{Statement_SigmaA}, note
\[
\sigma_A i_* [P,\rho] = \sigma_A [P,t_2 P, \rho,1] = [P[G],1-t\rho] = \phi_!\sigma_B^+ [P,\rho]~.
\]
Thus, for all $n \leqslant 1$, we obtain the key equality
\[
\sigma_A \circ i_* ~=~ \phi_! \circ \sigma_B^+ ~:~ \widetilde{\mathrm{Nil}}_{n-1}(R[F],\alpha) \longrightarrow K_n(R[G]) ~.
\]
\end{state}
\begin{prop}\label{induction} Let $n \leqslant 1$ be an integer.
{\rm (i)} The split monomorphisms
$\sigma_A,\sigma'_A,\sigma_B^+,\sigma_B'^+$ are related by a
commutative diagram
$$\xymatrix@R-5pt{
\widetilde{\mathrm{Nil}}_{n-1}(R[F],\alpha)
~\ar@{>->}[rrr]^-{\displaystyle{\sigma^+_B}} \ar[dddddd]_-{\displaystyle{\tau_B}}^-{\displaystyle{\cong}}
\ar[ddr]^-{\displaystyle{i_*}}_-{\displaystyle{\cong}} &&&
K_n(R[F]_{\alpha}[t]) \ar[dddl]_-{\displaystyle{\phi_!}} \ar[dd]^-{\displaystyle{\psi_!^+}} \\
&&&\\
&\widetilde{\mathrm{Nil}}_{n-1}(R[F];\mathscr{B}_1,\mathscr{B}_2)
~\ar@{>->}[dr]^-{\displaystyle{\sigma_A}}
\ar[dd]^-{\displaystyle{\cong}}_-{\displaystyle{\tau_A}}
&& K_n(R[F]_{\alpha}[t,t^{-1}])\ar[dl]_-{\displaystyle{\theta_!}}
\ar[dd]_-{\displaystyle{\cong}}^-{\displaystyle{(\beta_u)_!}}\\
& & K_n(R[G]) & \\
&\widetilde{\mathrm{Nil}}_{n-1}(R[F];\mathscr{B}_2,\mathscr{B}_1)
~\ar@{>->}[ur]^-{\displaystyle{\sigma'_A}} &&K_n(R[F]_{\alpha'}[t',t'^{-1}])\ar[ul]_-{\displaystyle{\theta'_!}}
\\
&&&\\
\widetilde{\mathrm{Nil}}_{n-1}(R[F],\alpha')
~\ar@{>->}[rrr]^-{\displaystyle{\sigma_B'^+}} \ar[uur]^-{\displaystyle{i'_*}}_-{\displaystyle{\cong}} &&&
K_n(R[F]_{\alpha'}[t'])\ar[uuul]^-{\displaystyle{\phi_!'}} \ar[uu]_-{\displaystyle{\psi_!'^+}}}
$$
{\rm (ii)} The induced map $\theta_!$ is such that there is a
commutative diagram
$$\begin{array}{l}
\xymatrix@C+10pt@R+10pt{\widetilde{\mathrm{Nil}}_{n-1}(R[F],\alpha)\oplus
\widetilde{\mathrm{Nil}}_{n-1}(R[F],\alpha^{-1})
\ar[d]_-{\begin{pmatrix} i_* & \tau_A^{-1} i'_* \beta_u^+ \end{pmatrix}}
~~\ar@{>->}[r]^-{\displaystyle{\sigma_B}} & K_n(R[\overline{G}])
\ar[d]^-{\displaystyle{\theta_!}}\\
\widetilde{\mathrm{Nil}}_{n-1}(R[F];\mathscr{B}_1,\mathscr{B}_2)
~~\ar@{>->}[r]^-{\displaystyle{\sigma_A}} & K_n(R[G])}
\end{array}$$
\end{prop}
\begin{proof} Part~(i) follows from the following implications:
\begin{itemize}
\item
Statement~\ref{Statement_RingMaps} gives $\phi_! = \theta_!
\circ \psi_!^+$ and $\phi_!' = \theta_!' \circ \psi_!'^+$
\item
Statement~\ref{scalingG1} gives $\theta_! = \theta_!' \circ
(\beta_u)_!$
\item
Statement~\ref{Statement_Transposition} gives $\tau_A \circ
i_* = i_*' \circ \tau_B$
\item
Statement~\ref{Statement_SigmaA} gives $\sigma_A = \sigma_A'
\circ \tau_A$
\item
Statement~\ref{Statement_meat} gives $\sigma_A \circ i_* =
\phi_! \circ \sigma_B^+$ and $\sigma_A' \circ i_*' = \phi_!'
\circ \sigma_B'^+$.
\end{itemize}
Part~(ii) follows from Part~(i):
\begin{itemize}
\item
$\sigma_A \circ i_* = \phi_! \circ \sigma_B^+ = \theta_! \circ
\psi_!^+ \circ \sigma_B^+ = \theta_! \circ
\sigma_B|\widetilde{\mathrm{Nil}}_{n-1}(R[F],\alpha)$
\item
Statement \ref{scalingG3} and Statement \ref{scalingG1} give
$\sigma_A \circ \tau_A^{-1} \circ i_*' \circ \beta_u^+ =
\sigma_A' \circ i_*' \circ \beta_u^+ = \phi_!' \circ \sigma_B'^+
\circ \beta_u^+ = \theta_! \circ (\beta_u)_!^{-1} \circ
\psi_!'^+ \circ (\beta_u^+)_! \circ \sigma_B^- = \theta_! \circ
\psi_!^- \circ \sigma_B^- = \theta_! \circ \sigma_B |
\widetilde{\mathrm{Nil}}_{n-1}(R[F],\alpha^{-1})$.
\end{itemize}
\end{proof}
\subsubsection{Transfer}
We consider the effect of induction maps on
$\widetilde{\mathrm{Nil}}$-summands (Prop. \ref{transfer}).
\begin{state}
Given an $R[G]$-module $M$, let $M^!$ be the
$R[\overline{G}]$-module defined by $M$ with the $R[G]$-action
restricted to the subring $R[\overline{G}]$. The transfer functor
of exact categories
$$\theta^!~:~\mathrm{Proj}(R[G]) \longrightarrow \mathrm{Proj}(R[\overline{G}])~;~M \longmapsto M^!$$
induces the transfer maps in algebraic $K$-theory
$$\theta^!~:~K_*(R[G]) \longrightarrow K_*(R[\overline{G}])~.$$
The exact functors of Theorem \ref{maink} combine to an exact
functor
$$\begin{array}{l}
\begin{pmatrix}j\\ j'\end{pmatrix} ~:~\mathrm{Nil}(R[F];\mathscr{B}_1,\mathscr{B}_2) \longrightarrow \mathrm{Nil}(R[F],\alpha)
\times \mathrm{Nil}(R[F],\alpha')~;\\[1ex]
\hskip150pt
[P_1,P_2,\rho_1,\rho_2] \longmapsto \big([P_1,\rho_2\circ \rho_1],[P_2,\rho_1\circ \rho_2]\big)
\end{array}$$
inducing a map between reduced $\mathrm{Nil}$-groups
\[
\begin{pmatrix}j_* \\ j'_*\end{pmatrix}~:~\widetilde{\mathrm{Nil}}_*(R[F];\mathscr{B}_1,\mathscr{B}_2) \longrightarrow \widetilde{\mathrm{Nil}}_*(R[F],\alpha)
\oplus \widetilde{\mathrm{Nil}}_*(R[F],\alpha')~.
\]
\end{state}
\begin{prop}\label{transfer}
Let $n \leqslant 1$ be an integer. The transfer map $\theta^!$ restricts
to the isomorphism $j_*$ in a commutative diagram
\[
\begin{array}{l} \xymatrix@C+10pt@R+10pt{
\widetilde{\mathrm{Nil}}_{n-1}(R[F];\mathscr{B}_1,\mathscr{B}_2) \ar[d]_-{\begin{pmatrix}
j_* \\ (\beta_u^+)^{-1}j'_* \end{pmatrix}}
~~\ar@{>->}[r]^-{\displaystyle{\sigma_A}} & K_n(R[G]) \ar[d]^-{\displaystyle{\theta^!}} \\
\widetilde{\mathrm{Nil}}_{n-1}(R[F],\alpha)\oplus
\widetilde{\mathrm{Nil}}_{n-1}(R[F],\alpha^{-1})
~~\ar@{>->}[r]^-{\displaystyle{\begin{pmatrix}\psi^+\sigma_B^+ & \beta_u \psi^- \sigma_B^-\end{pmatrix}}} & K_n(R[\overline{G}])~.}
\end{array}
\]
\end{prop}
\begin{proof}
Using the suspension isomorphisms of Section \ref{Sec_Lower}, we may
assume $n=1$. Let $(P_1,P_2,\rho_1,\rho_2)$ be an object in
$\mathrm{Nil}(R[F];\mathscr{B}_1,\mathscr{B}_2)$. Define an $R[G]$-module automorphism
\[
f~:=~\begin{pmatrix} 1 & t_2 \rho_2 \\ t_1 \rho_1 & 1
\end{pmatrix}~: P_1[G] \oplus P_2[G] \longrightarrow P_1[G] \oplus P_2[G] ~.
\]
By Theorem \ref{Waldhausen}, we have $[f] =
\sigma_A[P_1,P_2,\rho_1,\rho_2] \in K_1(R[G])$. Note the transfer is
\[
\theta^!(f)~=~\begin{pmatrix} 1 & t_2 \rho_2 & 0 & 0 \\
t_1 \rho_1 & 1 & 0 & 0 \\
0 & 0 & 1 & t_1 \rho_1 \\
0 & 0 & t_2 \rho_2 & 1 \end{pmatrix}
\]
as an $R[\overline{G}]$-module automorphism of $P_1[\overline{G}]
\oplus t_1P_2[\overline{G}] \oplus P_2[\overline{G}] \oplus
t_1P_1[\overline{G}]$. Furthermore, elementary row and column
operations produce a diagonal representation:
\[
\begin{pmatrix} 1 & -t_2 \rho_2 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & -t_1 \rho_1 \\
0 & 0 & 0 & 1 \end{pmatrix} \theta^!(f) \begin{pmatrix} 1 & 0 & 0 & 0 \\
-t_1 \rho_1 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & -t_2 \rho_2 & 1 \end{pmatrix}~=~
\begin{pmatrix} 1-t'\rho_2\rho_1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1-t\rho_1\rho_2 & 0 \\
0 & 0 & 0 & 1 \end{pmatrix}~.
\]
So $\theta^![f] = [1-t'\rho_2\rho_1] + [1 - t\rho_1\rho_2]$. Thus we
obtain a commutative diagram
$$\begin{array}{l}
\xymatrix@C+10pt@R+10pt{
\widetilde{\mathrm{Nil}}_{0}(R[F];\mathscr{B}_1,\mathscr{B}_2)
\ar[d]_-{\begin{pmatrix} j_* \\ j'_* \end{pmatrix}}
~~\ar@{>->}[r]^-{\displaystyle{\sigma_A}} & K_1(R[G]) \ar[d]^-{\displaystyle{\theta^!}} \\
\widetilde{\mathrm{Nil}}_{0}(R[F],\alpha)\oplus
\widetilde{\mathrm{Nil}}_{0}(R[F],\alpha')
~~\ar@{>->}[r]^-{\displaystyle{\begin{pmatrix}\psi^+ \sigma^+_B & \psi'^+ \sigma_B'^+\end{pmatrix}}} & K_1(R[\overline{G}])}
\end{array}$$
Finally, by Statement \ref{scalingG3} and Statement \ref{scalingG1},
note
\[
\psi'^+ \circ \sigma_B'^+ \circ \beta_u^+ = \psi'^+ \circ \beta_u^+ \circ \sigma_B^-
= \beta_u \circ \psi^- \circ \sigma_B^- ~.
\]
\end{proof}
\subsection{Waldhausen $\mathrm{Nil}$}\label{Second}
Natural examples of bimodules originate from group rings of
amalgamated product of groups.
\begin{defn}\label{Defn_almostnormal}
A subgroup $H$ of a group $G$ is \textbf{almost-normal} if $[H : H
\cap x H x^{-1}] < \infty$ for every $x \in G$. Equivalently, $H$ is
an almost-normal subgroup of $G$ if every $(H,H)$-double coset $HxH$
is a (disjoint) union of finitely many left cosets $gH$ and a
(disjoint) union of finitely many right cosets $Hg$.
\end{defn}
\begin{rem} Almost-normal subgroups arise in the Shimura theory of
automorphic functions, with $(G,H)$ called a Hecke pair. Here are two
sufficient conditions for a subgroup $H \subset G$ to be almost-normal:
if $H$ is a finite-index subgroup of $G$, or if $H$ is a normal
subgroup of $G$. Interesting examples of almost-normal subgroups are
given in \cite[p.~9]{Krieg}.
\end{rem}
Here is our reduction for a certain class of group rings,
specializing the General Algebraic Semi-splitting of Theorem
\ref{maink}.
\begin{cor}\label{Cor_GroupNil}
Let $R$ be a ring. Let $G = G_1 *_F G_2$ be an injective amalgamated
product of groups over a common subgroup $F$ of $G_1$ and $G_2$.
Suppose $F$ is an almost-normal subgroup of $G_2$. Then, for all $n
\in \mathbb{Z}$, there is an isomorphism of abelian groups:
\[
j_*: \widetilde {\mathrm{Nil}}_n(R[F];R[G_1-F], R[G_2-F]) \longrightarrow \widetilde \mathrm{Nil}_{n}(R[F]; R[G_1-F]
\otimes_{R[F]} R[G_2-F]).
\]
\end{cor}
\begin{proof}
Consider the set $J := (F \backslash G_2 \slash F) - F$ of
non-trivial double cosets. Let $\mathcal{I}$ be the poset of all
finite subsets of $J$, partially ordered by inclusion. Note, as
$R[F]$-bimodules, that
\[
R[G_2-F] = \colim_{I \in \mathcal{I}} R[I] \quad\text{where}\quad
R[I] := \bigoplus_{FgF \in I} R[FgF].
\]
Since $F$ is an almost-normal subgroup of $G_2$, each
$R[F]$-bimodule $R[I]$ is a finitely generated, free (hence
projective) left $R[F]$-module. Observe that $\mathcal{I}$ is a filtered poset: if
$I, I' \in \mathcal{I}$ then $I \cup I' \in \mathcal{I}$. Therefore
we are done by Theorem \ref{maink}.
\end{proof}
The case of $G=D_\infty = \mathbb{Z}_2 * \mathbb{Z}_2$ has a particularly simple
form.
\begin{cor} \label{dcor}
Let $R$ be a ring and $n \in \mathbb{Z}$. There are natural isomorphisms:
\begin{enumerate}
\item $ \widetilde {\mathrm{Nil}}_n(R;R,R) \cong \widetilde
\mathrm{Nil}_{n}(R) $
\item $K_n(R[D_\infty]) \cong (K_n(R[\mathbb{Z}_2]) \oplus K_n(R[\mathbb{Z}_2]))/K_n(R) \oplus\widetilde{\mathrm{Nil}}_{n-1}(R)
$.
\end{enumerate}
\end{cor}
\begin{proof}
Part (i) follows from Corollary \ref{Cor_GroupNil} with $F=1$ and
$G_i=\mathbb{Z}_2$. Then Part (ii) follows from Waldhausen's exact sequence
(Thm. \ref{Waldhausen}), where the group retraction $\mathbb{Z}_2 \to 1$
induces a splitting of the map $K_n(R) \to K_n(R[C_2]) \times
K_n(R[C_2])$.
\end{proof}
\begin{exm}\label{Exm_NonzeroWaldNil}
Consider the group $G = G_0 \times D_{\infty}$ where $G_0 = \mathbb{Z}_2
\times \mathbb{Z}_2 \times \mathbb{Z}$. Since $G$ surjects onto the infinite
dihedral group, there is a corresponding amalgamated product
decomposition
\[
G = (G_0 \times \mathbb{Z}_2) *_{G_0} (G_0 \times \mathbb{Z}_2)
\]
with the corresponding index 2 subgroup
\[
\bar G = G_0 \times \mathbb{Z}.
\]
Corollary \ref{dcor}(1) gives an isomorphism
\[
\widetilde{\mathrm{Nil}}_{-1}(\mathbb{Z}[G_0]; \mathbb{Z}[G_0], \mathbb{Z}[G_0]) \cong \widetilde{\mathrm{Nil}}_{-1}(\mathbb{Z}[G_0]).
\]
On the other hand, Bass showed that the latter group is an
infinitely generated, abelian group of exponent a power of two
\cite[XII, 10.6]{Bass}. Hence, by Waldhausen's algebraic
$K$-theory decomposition result, $\mathrm{Wh}(G)$ is infinitely generated
due to $\mathrm{Nil}$ elements. Now construct a codimension one, finite
$CW$ pair $(X,Y)$ with $\pi_1 X = G$ realizing the above
amalgamated product decomposition -- for example, let $Y \to Z$ be
a map of connected $CW$ complexes inducing the first factor
inclusion $G_0 \to G_0 \times \mathbb{Z}_2$ on the fundamental group and
let $X$ be the double mapping cylinder of $Z \leftarrow Y
\rightarrow Z$. Next construct a homotopy equivalence $f : M \to
X$ of finite $CW$ complexes whose torsion $\tau(f) \in \mathrm{Wh}(G)$ is
a non-zero $\mathrm{Nil}$ element. Then $f$ is non-splittable along $Y$ by
Waldhausen \cite{Waldhausen_1969} (see Theorem \ref{W}).
\emph{This is the first explicit example of a non-zero Waldhausen
$\mathrm{Nil}$ group and a non-splittable homotopy equivalence in the
two-sided case.}
\end{exm}
\subsection{Farrell--Jones Conjecture}\label{Third}
The Farrell--Jones Conjecture says that the family of virtually
cyclic subgroups is a ``generating'' family for $K_n(R[G])$. In
this section we apply our main theorem to show that the Farrell-Jones Conjecture holds if
and only if the smaller family of finite-by-cyclic subgroups is a
generating family for $K_n(R[G])$.
Let $\mathrm{Or} G$ be the orbit category of a group $G$; objects are
$G$-sets $G/H$ where $H$ is a subgroup of $G$ and morphisms are
$G$-maps. Davis--L\"uck \cite{DL} defined a functor $\mathbf{K}_R : \mathrm{Or} G
\to \matheurm{Spectra}$ with the key property $\pi_n\mathbf{K}_R(G/H) = K_n(R[H])$.
The utility of such a functor is that it allows the definition of
an equivariant homology theory, indeed for a $G$-$CW$-complex $X$,
one defines
\[
H^G_n(X; \mathbf{K}_R) = \pi_n(\mathrm{map}_G(-,X)_+ \wedge_{\mathrm{Or} G} \mathbf{K}_R(-))
\]
(see \cite[section 4 and 7]{DL} for basic properties). Note that
the ``coefficients'' of the homology theory are given by
$H^G_n(G/H; \mathbf{K}_R) = K_n(R[H])$.
A \emph{family} $\mathcal{F}$ of subgroups of $G$ is a nonempty set of
subgroups closed under subgroups and conjugation. For such a
family, $E_{\mathcal{F}}G$ is the classifying space for $G$-actions with
isotropy in $\mathcal{F}$. It is characterized up to $G$-homotopy type as
a $G$-$CW$-complex so that $(E_{\mathcal{F}}G)^H$ is contractible for
subgroups $H \in \mathcal{F}$ and is empty for subgroups $H \not \in \mathcal{F}$.
Four relevant families are $\matheurm{fin} \subset \matheurm{fbc} \subset \matheurm{vc} \subset
\matheurm{all}$, the families of finite subgroups, finite-by-cyclic,
virtually cyclic subgroups and all subgroups respectively. Here
\begin{align*}
\matheurm{fbc} & =\matheurm{fin} \cup \{H < G : H \cong F \rtimes \mathbb{Z} \text{ with $F$ finite}\}\\
\matheurm{vc} & = \{H < G : \exists \text{ cyclic } C < H \text{ with finite index}\}.
\end{align*}
The Farrell--Jones conjecture in $K$-theory for the group $G$
\cite{FJiso,DL} states that
\[
H^G_n(E_{\matheurm{vc}}G;\mathbf{K}_R) \longrightarrow H^G_n(E_{\matheurm{all}}G;\mathbf{K}_R) = K_n(R[G])
\]
is an isomorphism.
We now state a more general version, the fibered Farrell--Jones
conjecture. If $\varphi: \Gamma \to G$ is a group homomorphism and if
$\mathcal{F}$ is a family of subgroups of $G$, define the family of
subgroups
\[
\varphi^*\mathcal{F} = \{ H < \Gamma : \varphi(H) \in \mathcal{F}\} .
\]
The fibered Farrell--Jones conjecture in $K$-theory for the group
$G$ states that for every group epimorphism $\varphi: \Gamma \to G$ and
for every ring $R$, that following induced map is an isomorphism:
\[
H^\Gamma_n(E_{\varphi^*\matheurm{vc}(G)}\Gamma;\mathbf{K}_R) \longrightarrow H^\Gamma_n(E_{\varphi^*\matheurm{all}(G)}\Gamma;\mathbf{K}_R) = K_n(R[\Gamma]).
\]
The following theorem was inspired by Frank Quinn. It is proven
below.
\begin{thm} \label{isom_conj}
Let $\varphi: \Gamma \to G$ be an epimorphism of groups. Let $R$ be any
ring. Then the following induced map is an isomorphism:
\[
H^\Gamma_n(E_{\varphi^*\matheurm{fbc}(G)}\Gamma;\mathbf{K}_R) \longrightarrow H^\Gamma_n(E_{\varphi^*\matheurm{vc}(G)}\Gamma;\mathbf{K}_R).
\]
\end{thm}
Hence we have the following conjecture.
\begin{conj}
Let $G$ be a discrete group, and let $R$ be a ring.
\begin{enumerate}
\item There is an isomorphism:
\[
H_n^G(E_{\matheurm{fbc}}G; \mathbf{K}_R) \longrightarrow H_n^G(E_{\matheurm{all}}G; \mathbf{K}_R) = K_n(R[G]).
\]
\item For any epimorphism $\varphi: \Gamma \to G$ of groups, there
is an isomorphism:
\[
H^\Gamma_n(E_{\varphi^*\matheurm{fbc}(G)}\Gamma;\mathbf{K}_R) \longrightarrow H^\Gamma_n(E_{\matheurm{all}}\Gamma;\mathbf{K}_R) = K_n(R[\Gamma]).
\]
\end{enumerate}
\end{conj}
We have shown that first statement is equivalent to the
Farrell--Jones conjecture in $K$-theory and the second statement is
equivalent to the fibered Farrell--Jones conjecture in $K$-theory.
The proof of Theorem \ref{isom_conj} will require three
auxiliary results, some of which we quote from other sources. The
first is Theorem A.10 of Farrell--Jones \cite{FJiso}.
\begin{tp}
Let $\mathcal{F} \subset \mathcal{G}$ be families of subgroups of a group $\Gamma$.
Let $\mathbf{E} : \mathrm{Or} \Gamma \to \matheurm{Spectra}$ be a functor. Then for every $H
\in \mathcal{G} - \mathcal{F}$, if the assembly map
\[
H_*^H(E_{\mathcal{F} \cap H}H; \mathbf{E}) \longrightarrow H_*^H(E_\matheurm{all} H; \mathbf{E})
\]
is an isomorphism, then the following map is an isomorphism:
\[
H_*^\Gamma(E_\mathcal{F} \Gamma;\mathbf{E}) \longrightarrow H_*^\Gamma(E_\mathcal{G} \Gamma;\mathbf{E}).
\]
\end{tp}
Of course, we shall apply this principle to the families $\matheurm{fbc}
\subset \matheurm{vc}$. The following lemma is well-known (see \cite[Theorem
5.12]{ScottWall}), but we offer an alternative proof.
\begin{lem}\label{lem_vctypes}
Let $G$ be a virtually cyclic group. Then either
\begin{enumerate}
\item $G$ is finite.
\item $G$ admits an epimorphism to $\mathbb{Z}$, hence $G = F
\rtimes_{\alpha} \mathbb{Z}$ with $F$ finite.
\item $G$ admits an epimorphism to $D_{\infty}$, hence $G =
G_1 *_F G_2$ with $| G_i : F| = 2$ and $F$ finite.
\end{enumerate}
\end{lem}
\begin{proof} Assume $G$ is an infinite virtually cyclic group.
The intersection of the conjugates of a finite index, infinite
cyclic subgroup is a normal, finite index, infinite cyclic subgroup
$C$. Let $Q$ be the finite quotient group. Embed $C$ as an index
$|Q|$ subgroup of an infinite cyclic subgroup $C'$. The image of
the obstruction cocycle under the map $H^2(Q;C) \to H^2(Q;C')$ is
trivial, so $G$ embeds as a finite index subgroup of a semidirect
product $G' = C' \rtimes Q$. Note $G'$ maps epimorphically to $\mathbb{Z}$
(if $Q$ acts trivially) or to $D_{\infty}$ (if $Q$ acts
non-trivially). In either case, $G$ maps epimorphically to a
subgroup of finite index in $D_{\infty}$, which must be either
infinite cyclic or infinite dihedral.
\end{proof}
In order to see how the Farrell--Hsiang and Waldhausen $\mathrm{Nil}$-groups relate
to equivariant homology (and hence to the Farrell--Jones
Conjecture), we need \cite[Lemma 3.1, Theorem 1.5]{DQR}, as follows.
\begin{lem}[Davis--Quinn--Reich] \label{DW}
Let $\overline{G}$ be a group of the form $F \rtimes_{\alpha} \mathbb{Z}$,
and let $\bar{\mathcal{F}}$ be the smallest family of subgroups of
$\overline{G}$ containing $F$. Let $G$ be a group of the form $G_1
*_F G_2$ with $|G_i : F | = 2$, and let $\mathcal{F}$ be the smallest family
of subgroups of $G$ containing $G_1$ and $G_2$. Note that $F$ need
not be finite.
\begin{enumerate}
\item The following exact sequences are split, and hence
short exact:
\begin{align*}
H^{\overline{G}}_n(E_{\bar{\mathcal{F}}}\overline{G};\mathbf{K}_R) \longrightarrow &H^{\overline{G}}_n(E_{\matheurm{all}}\overline{G};\mathbf{K}_R) \longrightarrow H^{\overline{G}}_n(E_{\matheurm{all}}\overline{G},E_{\bar{\mathcal{F}}}\overline{G};\mathbf{K}_R) \\
H^{G}_n(E_{\mathcal{F}}G;\mathbf{K}_R) \longrightarrow &H^{G}_n(E_{\matheurm{all}}G;\mathbf{K}_R) \longrightarrow H^{G}_n(E_{\matheurm{all}}G,E_{\mathcal{F}}G;\mathbf{K}_R).
\end{align*}
\item Moreover, the relative terms are:
\begin{align*}
H^{\overline{G}}_n(E_{\matheurm{all}}\overline{G},E_{\bar{\mathcal{F}}}\overline{G};\mathbf{K}_R) & \cong \widetilde {\mathrm{Nil}}_{n-1}(R[F],\alpha) \oplus \widetilde {\mathrm{Nil}}_{n-1}(R[F],\alpha^{-1}) \\
H^{G}_n(E_{\matheurm{all}}G,E_{\mathcal{F}}G;\mathbf{K}_R) & \cong \widetilde {\mathrm{Nil}}_{n-1}(R[F];R[G_1-F],R[G_2-F]).
\end{align*}
\item Let $p: G \to D_{\infty}$ be the epimorphism
induced the amalgamated product splitting above. Then
\[
H^G_n(E_{p^*\matheurm{fbc}}G, E_{p^*\matheurm{fin}}G; \mathbf{K}_R) \cong \widetilde{\mathrm{Nil}}_{n-1}(R[F], \alpha),
\]
where the ring automorphism $\alpha: R[F] \to R[F]$ is induced
by conjugation of an indivisible element of infinite order in
$D_{\infty}$.
\end{enumerate}
\end{lem}
Furthermore, it is not difficult to compute
$H^{\overline{G}}_n(E_{\bar{\mathcal{F}}}\overline{G};\mathbf{K}_R)$ and
$H^{G}_n(E_{\mathcal{F}}G;\mathbf{K}_R)$ in terms of a Wang sequence and a
Mayer--Vietoris sequence respectively. We shall focus on a
particular example in Subsection \ref{Subsec_PSL2Z}.
\begin{proof}[Proof of Theorem \ref{isom_conj}]
Let $\varphi : \Gamma \to G$ be an epimorphism of groups. We will use
the Transitivity Principle applied to the families $\varphi^*\matheurm{fbc}
\subset \varphi^*\matheurm{vc}$. By Lemma \ref{lem_vctypes}, if $H \in
\varphi^*\matheurm{vc} - \varphi^*\matheurm{fbc}$, then $\varphi(H)$ admits an
epimorphism $p: \varphi(H) \to D_{\infty}$ with finite kernel. Thus
$\varphi(H)$ is an injective amalgam $\varphi(H) = G_1' *_{F'} G_2'$
of finite groups, where $F'$ is an index two subgroup of each
$G_i'$. It is not difficult to show
\begin{align*}
p^*\matheurm{fin} & = \matheurm{fin}\\
p^*\matheurm{fbc} & = \matheurm{fbc} ~.
\end{align*}
For clarity we denote the restriction by $\varphi | : H \to
\varphi(H)$. The decomposition of $H$ induced by $p \circ \varphi|$
is an injective amalgam $H = G_1 *_F G_2$ where $F := \varphi
|^{-1}(F')$ is an index two subgroup of each $G_i := \varphi
|^{-1}(G_i')$. Let $\mathcal{F}$ be the smallest family of subgroups of
$H$ containing $G_1$ and $G_2$. Using the above two displayed
equalities and the fact that every finite subgroup of $G_1' *_{F'}
G_2'$ is conjugate to a subgroup of $G_1'$ or $G_2'$ (see
\cite[Corollary to Theorem 4.3.8]{Serre}), one can show
\begin{align*}
\mathcal{F} &= (p \circ \varphi|)^*\matheurm{fin} \\
(\varphi^*\matheurm{fbc}) \cap H & = (p \circ \varphi|)^* \matheurm{fbc} ~.
\end{align*}
By Lemma \ref{DW} parts (2) and (3), we have
\begin{eqnarray*}
H_n^H(E_{\matheurm{all}}H,E_{\mathcal{F}}H; \mathbf{K}_R) &=& \widetilde \mathrm{Nil}_{n-1}(R[F]; R[G_1 - F],R[G_2 - F])\\
H_n^H(E_{(p \circ \varphi|)^* \matheurm{fbc}}H,E_{(p \circ \varphi|)^* \matheurm{fin}}H; \mathbf{K}_R) &=& \widetilde \mathrm{Nil}_{n-1}(R[F]; \alpha).
\end{eqnarray*}
Then the map
\begin{multline*}
H_n^H(E_{(p \circ \varphi|)^* \matheurm{fbc}}H,E_{(p \circ \varphi|)^* \matheurm{fin}}H;
\mathbf{K}_R) = H_n^H(E_{(\varphi^*\matheurm{fbc}) \cap H}H,E_{\mathcal{F}}H; \mathbf{K}_R)\\
\longrightarrow H_n^H(E_{\matheurm{all}}H,E_{\mathcal{F}}H; \mathbf{K}_R)
\end{multline*}
induces the isomorphism of Theorem \ref{first}. Hence, by the exact
sequence of a triple, $H_n^H(E_{\matheurm{all}}H,E_{(\varphi^*\matheurm{fbc}) \cap H}H;
\mathbf{K}_R) = 0 $. Therefore, by the Transitivity Principle, we have
proven
$H_n^\Gamma(E_{\varphi^*\matheurm{vc}}\Gamma,E_{\varphi^*\matheurm{fbc}}\Gamma;\mathbf{K}_R) =
0$.
\end{proof}
\subsection{$K$-theory of the modular group}\label{Subsec_PSL2Z}
Let $\Gamma = \mathbb{Z}_2 * \mathbb{Z}_3 = PSL_2(\mathbb{Z})$. The following theorem follows
from applying our main theorem and the recent proof \cite{BLR} of
the Farrell--Jones conjecture in $K$-theory for word hyperbolic
groups.
The Cayley graph for $\mathbb{Z}_2 * \mathbb{Z}_3$ with respect to the generating
set given by the nonzero elements of $\mathbb{Z}_2$ and $\mathbb{Z}_3$ has the
quasi-isometry type of the usual Bass--Serre tree for the
amalgamated product (Figure \ref{Fig_Tree}).
\begin{figure}[!ht]\begin{center}$\xymatrix@C+10pt{
\bullet \ar@{-}[dr] & & & & & & \bullet \\
& \bullet \ar@{-}[dr] & & & & \bullet \ar@{-}[ur] & \\
& & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[ur] \ar@{-}[dr]& &\\
& \bullet \ar@{-}[ur] & & & & \bullet \ar@{-}[dr] &\\
\bullet \ar@{-}[ur] & & & & & & \bullet }$\end{center}
\caption{Bass--Serre tree for $PSL_2(\mathbb{Z})$}\label{Fig_Tree}
\end{figure}
This is an infinite tree with alternating vertices of valence two
and three. The group $\Gamma$ acts on the tree, with the generator of
order two acting by reflection through an valence two vertex and the
generator of order three acting by rotation through an adjoining
vertex of valence three.
Any geodesic triangle in the Bass--Serre tree has the property
that the union of two sides is the union of all three sides. It
follows that the Bass--Serre graph is $\delta$-hyperbolic for any
$\delta > 0$, the Cayley graph is $\delta$-hyperbolic for some
$\delta > 0$, and hence $\Gamma$ is a hyperbolic group.
\begin{thm} \label{Jim}
For any ring $R$ and integer $n$,
\begin{multline*}
K_n(R[\Gamma]) = ( K_n(R[\mathbb{Z}_2]) \oplus K_n(R[\mathbb{Z}_3]))/K_n(R) \\
\oplus \bigoplus_{\mathcal{M}_C} \widetilde {\mathrm{Nil}}_{n-1}(R) \oplus
\widetilde {\mathrm{Nil}}_{n-1}(R) \oplus \bigoplus_{\mathcal{M}_D} \widetilde
{\mathrm{Nil}}_{n-1}(R)
\end{multline*}
where $\mathcal{M}_C$ and $\mathcal{M}_D$ are the set of conjugacy classes of
maximal infinite cyclic subgroups and maximal infinite dihedral
subgroups, respectively. Moreover, all virtually cyclic subgroups
of $\Gamma$ are cyclic or infinite dihedral.
\end{thm}
\begin{proof}
By Lemma \ref{DW}, the homology exact sequence of the pair
$(E_\matheurm{all}\Gamma,E_\matheurm{fin}\Gamma)$ is short exact and split:
\[
H_n^{\Gamma}(E_\matheurm{fin}\Gamma; \mathbf{K}_R) \to H_n^{\Gamma}(E_\matheurm{all}\Gamma; \mathbf{K}_R) \to H_n^{\Gamma}(E_\matheurm{all}\Gamma, E_\matheurm{fin}\Gamma; \mathbf{K}_R).
\]
Then, by the Farrell--Jones Conjecture \cite{BLR} for word
hyperbolic groups, we obtain
\begin{align*}
K_n(R[\Gamma]) &= H_n^{\Gamma}(E_\matheurm{fin}\Gamma; \mathbf{K}_R) \oplus H_n^{\Gamma}(E_\matheurm{all}\Gamma, E_\matheurm{fin}\Gamma; \mathbf{K}_R)\\
\label{vc,fin} & = H_n^{\Gamma}(E_\matheurm{fin}\Gamma; \mathbf{K}_R) \oplus H_n^{\Gamma}(E_\matheurm{vc}\Gamma, E_\matheurm{fin}\Gamma; \mathbf{K}_R).
\end{align*}
Note that $E_\matheurm{fin}\Gamma$ is constructed as a pushout of $\Gamma$-spaces
\[
\begin{CD}
\Gamma \sqcup \Gamma @>>> \Gamma/\mathbb{Z}_2 \sqcup \Gamma/\mathbb{Z}_3\\
@VVV @VVV\\
\Gamma \times D^1 @>>> E_\matheurm{fin}\Gamma.
\end{CD}
\]
Then $E_\matheurm{fin}\Gamma$ is the Bass--Serre tree for $\Gamma = \mathbb{Z}_2 * \mathbb{Z}_3$.
Note that $H^{\Gamma}_*(\Gamma/H; \mathbf{K}_R) = K_*(R[H])$. The pushout gives,
after canceling a $K_n(R)$ term, a long exact sequence
\[
\cdots \to K_n(R) \to K_n(R[\mathbb{Z}_2]) \oplus K_n(R[\mathbb{Z}_3]) \to H_n^{\Gamma}(E_{\matheurm{fin}}\Gamma; \mathbf{K}_R) \to K_{n-1}(R) \to \cdots.
\]
Hence
\[
H_n^{\Gamma}(E_\matheurm{fin}\Gamma; \mathbf{K}_R) = ( K_n(R[\mathbb{Z}_2]) \oplus K_n(R[\mathbb{Z}_3]))/K_n(R).
\]
For a word hyperbolic group $G$,
\[
H^G_n(E_{\matheurm{vc}}G,E_{\matheurm{fin}}G; \mathbf{K}) \cong \bigoplus_{[V] \in \mathcal{M}(G)} H^V_n(E_{\matheurm{vc}}V,E_{\matheurm{fin}}V; \mathbf{K})
\]
where $\mathcal{M}(G)$ is the set of conjugacy classes of maximal
virtually cyclic subgroups of $G$ (see L\"uck \cite[Theorem
8.11]{Lueck} and \cite{Juan-PinedaLeary}). The geometric
interpretation of this result is that $E_{\matheurm{vc}}G$ is
obtained by coning off each geodesic in the tree $E_{\matheurm{fin}}G$; then
apply excision.
The Kurosh subgroup theorem implies that a subgroup of $\mathbb{Z}_2 *
\mathbb{Z}_3$ is a free product of $\mathbb{Z}_2$'s, $\mathbb{Z}_3$'s, and $\mathbb{Z}$'s. Note
that $\mathbb{Z}_2 * \mathbb{Z}_3 = \langle a,b ~|~ a^2 = 1 = b^3 \rangle$, $\mathbb{Z}_3 *
\mathbb{Z}_3 = \langle c,d ~|~ c^3 = 1 = d^3 \rangle$, and $\mathbb{Z}_2 * \mathbb{Z}_2 *
\mathbb{Z}_2 = \langle e,f,g ~|~ e^2 = f^2 = g^2= 1 \rangle$ have free
subgroups of rank 2, for example $\langle ab,ab^2\rangle$, $\langle
cd, cd^2\rangle$, and $\langle ef, fg \rangle$. On the other hand
the rank 2 free group $F(2)$ is not a virtually cyclic group since
its first Betti number $\beta_1(F(2)) = \text{rank}~H_1(F(2)) =
2$, while for a virtually cyclic group $V$, transferring to the
cyclic subgroup $C\subset V$ of finite index shows that
$\beta_1(V)$ is 0 or 1. Subgroups of virtually cyclic groups are
also virtually cyclic. Therefore all virtually cyclic subgroups of
$\Gamma$ are cyclic or infinite dihedral.
By the fundamental theorem of $K$-theory and Waldhausen's Theorem
(see also Lemma \ref{DW})
\begin{align*}
H_n^{\mathbb{Z}}(E_{\matheurm{vc}}\mathbb{Z}, E_{\matheurm{fin}}\mathbb{Z}; \mathbf{K}_R) & = \widetilde {\mathrm{Nil}}_{n-1}(R) \oplus \widetilde {\mathrm{Nil}}_{n-1}(R) \\
H_n^{D_{\infty}}(E_{\matheurm{vc}}D_{\infty}, E_{\matheurm{fin}}D_{\infty}; \mathbf{K}_R) &= \widetilde \mathrm{Nil}_{n-1}(R;R,R)
\end{align*}
Finally, by Corollary \ref{dcor}(1), we obtain exactly one type of
Nil-group:
\[
\widetilde \mathrm{Nil}_{n-1}(R;R,R) \cong \widetilde \mathrm{Nil}_{n-1}(R).
\]
\end{proof}
\begin{rem}
The sets $\mathcal{M}_C$ and $\mathcal{M}_D$ are countably infinite. This can be
shown by parameterizing these subsets either: combinatorially (using
that elements in $\Gamma$ are words in $a,b,b^2$), geometrically
(maximal virtually cyclic subgroups correspond to stabilizers of
geodesics in the Bass--Serre tree $E_{\matheurm{fin}}\Gamma$, where the geodesic
may or may not be invariant under an element of order 2), or number
theoretically (using solutions to Pell's equation and Gauss' theory
of binary quadratic forms \cite{S}).
\end{rem}
Let us give an overview and history of some related work. The
Farrell--Jones Conjecture and the classification of virtually cyclic
groups (see Lemma \ref{lem_vctypes}) focused attention on the
algebraic $K$-theory of groups mapping to the infinite dihedral
group. Several years ago James Davis and Bogdan Vajiac outlined a
unpublished proof of Theorem \ref{first} when $n \leqslant 0$ using
controlled topology and hyperbolic geometry. Lafont and Ortiz
\cite{LafontOrtiz} proved that $\widetilde{\mathrm{Nil}}_n(\mathbb{Z}[F];\mathbb{Z}[V_1-
F],\mathbb{Z}[V_2- F])=0$ if and only if
$\widetilde{\mathrm{Nil}}_n(\mathbb{Z}[F],\alpha)=0$ for any virtually cyclic group
$V$ with an epimorphism $V \to D_{\infty}$ and $n=0,1$. More
recently, Lafont--Ortiz \cite{LO2} have studied the more general
case of the $K$-theory $K_n(R[G_1 *_F G_2])$ of an injective
amalgam, where $F, G_1, G_2$ are finite groups. Finally, we
mentioned the paper \cite{DQR}, which was written in parallel with
this one. It an alternate proof of Theorem \ref{first} assuming a
controlled topology result of Frank Quinn. Also, \cite{DQR}
provides several auxiliary results used in Subsection \ref{Third} of
this paper.
\section*{Acknowledgements}\label{ackref}
We would like to thank the participants of the workshop \emph{Nil
Phenomena in Topology} (14--15 April 2007, Vanderbilt University),
where our interests intersected and motivated the development of
this paper. Moreover, Chuck Weibel helped us to formulate the
filtered colimit hypothesis in Theorem \ref{maink}, and Dan Ramras
communicated the concept of almost-normal subgroup in Definition
\ref{Defn_almostnormal} to the second author. Both of these concepts
facilitated the formulation of the class of injective amalgams of groups in Theorem
\ref{Thm_TopSemisplit} and Corollary \ref{Cor_GroupNil}.
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\section{Appendix: finite AIC\ groups are nilpotent}\label{app:ladisch}
We present a proof of Theorem~\ref{t:ladisch}, namely that a finite AIC\ group must be nilpotent. The argument is paraphrased from one shown to the second author by F.~Ladisch~\cite{MO_Ladisch}, and is included here with his kind permission.
We have tried to make the presentation accessible to non-specialists in character theory.
Let us recall some of the relevant definitions and basic facts.
When $\chi$ is an irreducible group character on a finite group~$G$, the centre of $\chi$, denoted by $\operatorname{\bf Z}(\chi)$ of $\chi$ is a normal subgroup of~$G$.
We say $\chi$ is absolutely idempotent\ if and only if $\operatorname{\bf Z}(\chi)=\operatorname{supp}(\chi)$.
The following lemma, which appears to be well known to specialists, uses no special properties of~$G$.
\begin{lem}\label{l:normal-meets-support}
Let $N$ be a normal subgroup of $G$ and let $\chi$ be an irreducible character of $G$. If $N$ contains a non-identity element, then so does $N\cap \operatorname{supp}(\chi)$.
\end{lem}
\begin{proof}
Consider the character $\chi\vert_N$. We have two cases to consider. If $\chi\vert_N$ is not orthogonal to the trivial character $\varepsilon_N$, then since $N$ is normal it follows from a theorem of Clifford that $\chi$ is proportional to $\varepsilon_N$ (see, e.g.~\cite[Corollary 6.7]{Isaacs_CTbook}). In particular, $N\cap\operatorname{supp}(\chi) = N$ contains a non-identity element.
On the other hand, if $\chi\vert_N$ is orthogonal to $\varepsilon_N$, then
$0 = \pair{\chi\vert_N}{\varepsilon_N} = \frac{1}{\abs{N}}\sum_{x\in N} \chi(x)$.
Since $\chi(e) >0$, $\chi$ must be non-zero on at least one non-identity element of~$N$.
\end{proof}
\begin{cor}\label{c:SOFA}
Let $G$ be a group with at least two elements, and let $S$ be a set of absolutely idempotent\ characters on $G$. Then $\bigcap_{\chi\in S} \operatorname{\bf Z}(\chi)$ contains a non-identity element.
\end{cor}
\begin{proof}
We induct on the size of $S$. If $S$ is empty there is nothing to prove. Otherwise, suppose $\chi_1,\dots, \chi_{n-1}$ are absolutely idempotent\ characters for which $N\defeq \operatorname{\bf Z}(\chi_1)\cap\dots\cap \operatorname{\bf Z}(\chi_{n-1})$ contains a non-identity element. $N$ is a normal subgroup of $G$, since $\operatorname{\bf Z}(\chi_i)$ is for each~$i$; and since $\chi_n$ is absolutely idempotent\ , $\operatorname{supp}(\chi)=\operatorname{\bf Z}(\chi_n)$. By Lemma~\ref{l:normal-meets-support}, $N\cap \operatorname{\bf Z}(\chi_n)$ therefore contains a non-identity element, completing the inductive step.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{t:ladisch}]
We argue by strong induction on the order of the group. Every group of order $\leq 5$ is abelian, hence in particular both AIC\ and nilpotent.
Let $n\geq 6$ and suppose inductively that all AIC\ groups of order $<n$ are nilpotent. Let $G$ be an AIC\ group of order $n$. Now $Z(G)=\bigcap_{\chi\in\operatorname{Irr}(G)} \operatorname{\bf Z}(\chi)$ -- this is true for \emph{any} finite group, see \cite[Corollary 2.28]{Isaacs_CTbook} -- and therefore by Corollary~\ref{c:SOFA}, $Z(G)$ contains a non-identity element.
Since quotients of finite AIC\ groups are themselves AIC, $G/Z(G)$ is AIC\ and has order $\leq n/2 < n$, and so it is nilpotent by the inductive hypothesis. But then $G$ is a central extension of a nilpotent group, and so is itself nilpotent.
\end{proof}
\subsection*{Acknowledgements}
Some of the results here are taken from the PhD thesis of the first author (MA), who was supported by a Dean's Scholarship from the University of Saskatchewan. The second author (YC) was partially supported by NSERC Discovery Grant 402153-2011, and the third author (ES) by NSERC Discovery Grant 366066-2009.
The second author thanks F. Ladisch for useful exchanges, and I. M. Isaacs for supplying the proof of Theorem~\ref{t:MO_Isaacs}. The authors also thank the referee for several suggestions which improved the paper, in particular the proof of Lemma~\ref{l:zl1-of-binary-product}, and for bringing the prior study of absolutely idempotent characters and AIC groups to the authors' attention.
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\section{Introduction}
Astronomical data archives and catalogues have become a new paradigm in the astrophysics research. Reduced (photometrically and astrometrically corrected images, spectra ready for immediate scientific exploitation,...) and high-level (catalogues, mosaics, stacked images,...) data products are of fundamental importance for archives as they enhance their use by the community. Moreover, these science-ready data products provide a higher visibility of the project results as clearly demonstrated by the large number of refereed papers based on archived data from projects like SDSS \citep{York00}, 2MASS \citep{2003yCat.2246....0C}, UKIDSS \citep{Lawrence07} and WISE \citep{Wright10}, to name a few. By using these resources, astronomers are able to conduct research projects that would otherwise be very time-consuming or completely unaffordable.
The Gran Telescopio de Canarias\footnote{\url{http://www.gtc.iac.es/}} ({\it GTC }), with its 36 individual hexagonal segments acting as a 10.4\,m diameter single mirror, is currently the largest optical-infrared telescope in the world. Operated by the Instituto de Astrof\'isica de Canarias, it is located at the Observatorio del Roque de Los Muchachos in La Palma (Spain), which provides excellent observing conditions.
{\it OSIRIS } (Optical System for Imaging and low-Intermediate-Resolution Integrated Spectroscopy) is an imager and spectrograph for the optical wavelength range, located in the Nasmyth-B focus of {\it GTC}. {\it OSIRIS } allows broadband imaging over a field of view of 7.8 $\times$ 7.8 arcmin unvignetted covering a spectral range from $\lambda$\,3\,650\,\AA \, to $\lambda$\,10\,000\,\AA, with a high transmission coefficient, in particular at longer wavelengths. The spectral range is covered by the Sloan system broadband filters: u' ($\lambda$\,3\,500\,\AA), g' ($\lambda$\,4\,750\,\AA), r' ($\lambda$\,6\,300\,\AA), i' ($\lambda$\,7\,800\,\AA), z' ($\lambda$\,9\,250\,\AA).
The filter tranmission and detector efficiency curves, taken from the Filter Profile Service maintained by the Spanish Virtual Observatory\footnote{\url{http://svo2.cab.inta-csic.es/theory/fps/}}, are shown in Figure~\ref{fig:curvesfilt}.
More information on the {\it OSIRIS } capabilities can be found at the {\it GTC } web page\footnote{\url{http://www.gtc.iac.es/instruments/osiris/}}.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figures/gtc_grizQEtot_sep19.png}
\caption{Transmission curves of the Sloan g' (blue), r' (yellow), i' (green) and z' (red) filters and the {\it OSIRIS } detection efficiency curve (dashed black).}
\label{fig:curvesfilt}
\end{figure}
The {\it GTC } archive\footnote{\url{http://gtc.sdc.cab.inta-csic.es/}} is in operation since 2011 and is hosted by the Spanish Virtual Observatory (SVO\footnote{\url{http://svo.cab.inta-csic.es/main/index.php}}), one of the 21 national and trans-national Virtual Observatory initiatives distributed worldwide and coordinated by the International Virtual Observatory Alliance\footnote{\url{http://ivoa.net}}.
This paper presents the first release of the {\it OSIRIS } Broad Band data. We begin in Sections~\ref{sec:data_curation} and \ref{sec:data_processing} with a description of the data curation, preparation and processing of the images, followed by the source extraction in Section~\ref{sec:sources_extraction}. We present astrometric and photometric calibrations in Sections~\ref{sec:astrometric_calibration} and ~\ref{sec:flux_calibration}. We then turn, in Section~\ref{sec:catalogue_sources}, to a description of the catalogue sources and a photometric validation exercise in which Pan-Starrs DR1 data are compared to {\it OSIRIS } observations. In Section~\ref{sec:science}, we introduce two science cases carried out with the catalogue as an attempt to show its potential for science exploitation. The paper ends with a description on how to access the data and a summary in Sections~\ref{sec:data_access} and \ref{sec:conclusions}, respectively.
\section{Data curation and preparation}\label{sec:data_curation}
The first step consisted in the data selection and curation. Large facilities such as {\it GTC } produce vast amount of data coming from different instruments and observing modes. Since the goal of the present project is to deliver science grade products of the broad-band images, we therefore rejected spectroscopic observations and tunable filter observations taken with {\it OSIRIS }.
We obtain 27\,470 raw broad-band images from the {\it GTC } Public Archive acquired between April 2009 and January 2014 in the $griz$ bands.
A total of 5\,638 among them were obtained using windowing (moving target images) and were discarded for the present release. The vast majority of them were targeting a few transiting targets requiring a high temporal frequency, and the loss in spatial coverage is minimum.
Thus, it is important to remark that the absence of a source in the catalogue does not imply the absence of {\it GTC } {\it OSIRIS } observations in that position of the sky. To check this point, the {\it GTC } Archive must be used.
A significant number (13\,230) of early images were missing the standard \verb|MJD-OBS| keyword giving the precise acquisition date, which is crucial for many studies focusing on the time-domain.
Although it is possible to recover the missing information through other non-standard FITS keywords (e.g. \verb|OPENTIME|), we will focus in this first data release on the the 8\,602 (27\,470 - 5\,638 - 13\,230) images with the standard \verb|MJD-OBS| keyword.
Among them, there are 858 calibration images that will not be included in the catalogue. Hence, we process 7\,744 {\it OSIRIS } images in the $griz$ filters.
\section{Data processing}\label{sec:data_processing}
The individual raw images were processed using an upgraded version of \emph{Alambic} \citep{2002SPIE.4847..123V}, a software suite developed and optimized for the processing of large multi-CCD imagers, which was adapted for {\it OSIRIS }.
\emph{Alambic} includes standard processing procedures such as overscan, bias and dark subtraction for each individual readout ports of each CCD, flat-field correction, bad pixel identification and masking, CCD-to-CCD gain harmonization, and fringing correction in the $z$-band. \emph{Alambic} also combines the flat-field and bad pixel mask into a weight map for each individual dataset, and estimates the sky background in each image using an iterative multi-resolution median filtering. The method is very efficient, fast and robust for most stellar or extragalactic fields, but will produce artefacts for input images including large extended sources, as well as around the halo of very bright stars. Images and catalogues including large extended sources (i.e. covering a significant fraction of the field-of-view) should therefore be considered with caution, and probably reprocessed from scratch by the user.
Flat-fields were computed using the best twilight flat-field frames obtained over time windows of 15 days, since the {\it OSIRIS } flat-fields are considered to be very stable over periods of several weeks.
\section{Source extraction}\label{sec:sources_extraction}
Sources brighter than the 3-$\sigma $ noise of the local background were detected and their photometry and position were measured using {\sc SExtractor} \citep{1996A&AS..117..393B} and {\sc PSFEx} \citep{2013ascl.soft01001B}. The weight-maps were used to properly modulate the detection threshold over the image. Three fixed apertures were used:
\begin{itemize}
\item 21, 31 and 41 pixels in 1$\times$1 binning mode, corresponding to 2.66, 3.94 and 5.21\,arcsec, respectively
\item 11, 15 and 21 pixels in 2$\times$2 binning mode, corresponding to 2.79, 3.81 and 5.33\,arcsec, respectively.
\end{itemize}
In addition to these three fixed apertures, the Kron or automatic aperture as by {\sc SExtractor} \verb|MAG_AUTO|, the PSF (\verb|MAG_PSF|) and the model (\verb|MAG_MODEL|) photometry were also measured. A PSF model was fitted to the data with {\sc PSFEX} using a second order polynomial to model the PSF variations across the field-of-view. The PSF model was then used to fit every source using:
\begin{itemize}
\item standard PSF-fitting
\item a two-dimensional S\'ersic model convolved with the PSF model
\end{itemize}
The {\sc SExtractor} model (i.e., PSF + S\'ersic) parameters offer the advantage of being suited for both point-like and extended sources. Most galaxies are indeed resolved under sub-arcsecond seeing, and the PSF will not give a good fit.
Saturation was set at 62\,000 counts, even though the detector is expected to behave linearly up to the 16 bits encoding limit.
This is an approximate value and, therefore, some saturated sources may still be present in the catalogue of extracted sources (see Section~\ref{sec:removal}). For a better identification of saturated pixels we will use {\sc MaxiMask} \citep{maximask} in future data releases.
A few morphometric parameters were also extracted. {\sc SExtractor} offers the possibility to measure the Full-width at half maximum (FWHM), the flux radius (defined as the circular aperture radius enclosing half the total flux), the elongation and the ellipticity. The model fitting also provides useful information about the morphometry and is particularly interesting for extended and saturated sources \citep{2013A&A...554A.101B}.
Figure~\ref{fig:seeing} shows the distribution of FWHM as measured by {\sc PSFEx}. The distribution peaks around 1\,arcsec and more than half of the images were obtained under sub-arcsecond seeing. Note that the distribution extends far towards large values thanks to the {\it GTC } filler program specifically designed to make use of poor ambient conditions.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figures/seeing.png}
\caption{Distribution of average FWHM measured in the images themselves by {\sc PSFEx}
\label{fig:seeing}}
\end{figure}
The {\sc SExtractor} and {\sc PSFEx} configuration files used to produce the catalogues, as well as the parameters obtained are given in Tables~\ref{tab.def_sex}, \ref{tab.def_psfex} and \ref{tab.def_params}.
\section{Astrometric calibration}\label{sec:astrometric_calibration}
A precise astrometric calibration was computed using {\sc Scamp} \citep{2006ASPC..351..112B}. The 2MASS catalogue was used as reference for most frames, except when too few 2MASS sources were available in the field-of-view to derive a proper astrometric solution. In those cases, the SDSS (DR10) \citep{2014ApJS..211...17A} or USNO-B1 \citep{Monet03} catalogues were used.
The absolute astrometric accuracy is therefore set by these catalogues and is expected to be better than 0.1\,arcsec. Non-linear geometric distortions were fitted using a third order polynomial, as illustrated in Fig.~\ref{fig:distortions}. The internal accuracy is estimated to be better than 30\,mas in all cases, and better than 15\,mas in most cases.
Celestial coordinates are given at the observing epoch and Equinox J2000.0.
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{Figures/distort_1.png}
\caption{Distortion map for the $i$-band filter as measured by {\sc Scamp}. \label{fig:distortions}}
\end{figure*}
The astrometric calibration allowed us to detect 530 problematic images, typically including technical images with no sources at all, images obtained during a tracking failure or having suffered a read-out failure. The total number of remaining images added to 7\,214, distributed over the entire northern sky accessible from La Palma, as illustrated in Fig.~\ref{fig:coverage}
This figure reflects the patchy nature of the {\it OSIRIS } observations, with a modest fraction (0.02\%) of the full sky being covered (8.05\,deg$^{2}$). This is a primary difference with typical survey projects covering a regular geometric pattern in the sky and with uniform properties, such as exposure time and filter set.
\begin{figure*}
\centering
\includegraphics[width=0.95\textwidth]{Figures/osiris_planck_8072.png}
\caption{Location of the 7\,214 {\it OSIRIS } pointings overplotted on an all-sky 857GHz {\it Planck} map.}
\label{fig:coverage}
\end{figure*}
Individual astrometrically calibrated images can be easily downloaded from the {\it GTC } archive and stacked to make a larger and/or deeper mosaic of a given field. Because the astrometric calibration was obtained using the Astr{\it O}matic software suite, we recommend to use {\sc SWarp} \citep{2002ASPC..281..228B} for the stacking. {\sc Swarp} will indeed properly interpret the astrometric solution included in the image headers and the weight map delivered with every individual image by simply using the "\verb|-WEIGHT_TYPE MAP_WEIGHT|" option. \\
\section{Flux calibration} \label{sec:flux_calibration}
The set of {\it OSIRIS } images used to build the catalogue constitute, by definition, a heterogeneous collection where the variety of observational parameters (e.g. exposure times) reflect the diversity of science cases carried out with them. While images with the shortest exposure times typically present bright sources and few or no faint objects, images with the largest exposure times, aimed at reaching fainter sources, usually include saturated objects. This wide range of exposure times reflects both the potential and the challenge of the associated catalogue.
We adopted the Pan-STARRS DR1 survey (PS1 hereafter) \citep{Kaiser10,Chambers16} as the reference to photometrically calibrate {\it GTC } {\it OSIRIS } images.
The PS1 catalogue includes PSF and Kron photometric measurements, which we will use to calibrate our \verb|MAG_PSF| and \verb|MAG_AUTO| photometry, respectively.
In order to obtain the calibration parameters, we searched for counterparts to each {\it OSIRIS } source in PS1 within 1.0\,arcsec. If more than one counterpart exists in the search radius we took the nearest one.
Each individual CCD subimage has been calibrated independently.
For a source to be considered as a calibration source, it must fulfill the following requirements:
\begin{itemize}
\item PS1 sources must be point-like sources with \verb|PSF|-\verb|Kron|<0.05\,mag \citep{Chambers16}, and {\it Qual} flag equal to 52.
\item PS1 sources must be within the saturation and detection limits of the survey as stated in \cite{Chambers16}.
\item Magnitude errors, both in PS1 and {\it OSIRIS }, must be smaller than 0.2\,mag.
\item In order to avoid spikes and extended sources, we select in our catalogue sources whose FWHM, elongation and ellipticity do not significantly deviate from the median value of all sources in the subimage.
\end{itemize}
With the selected sources, we carried out a sigma-clipping linear fit in each subimage, after which we performed the calibration only if there were more than six calibrating stars satisfying the criteria listed above. Besides, we provide calibrated magnitudes only if the Pearson correlation coefficient ($r$) of the linear fit is greater than 0.98.
Of the 14\,428 subimages available (7\,214$\times$2), we calibrated 13\,555 and 13\,403 in \verb|MAG_PSF| and \verb|MAG_AUTO| photometry, respectively. Of them, 12\,454 and 11\,614 have a correlation coefficient $r \geq 0.98$ (82\% and 87\%, respectively). A total number of 11\,045 subimages have both, PSF and automatic photometric calibrations with $r \geq 0.98$.
The differences in the number of calibrated subimages arise, on one hand, from the discrepancies in magnitude errors in PSF and AUTO photometry, which prevent the same detection to pass the magnitude cuts in both photometries. On the other hand, the different procedures of flux measurements by {\sc SExtractor} in few cases lead to a wrong measure of the \verb|FLUX_PSF| but not of \verb|FLUX_AUTO| for the same source and viceversa. We could therefore obtain a varying number of sources in the same subimage for PSF and AUTO calibrations.
The fact that the {\it OSIRIS } observations lie within the Pan-STARRS footprint ensures that, in all cases, the lack of photometric calibration is not due to a different spatial coverage of the two surveys but to the absence of sources fulfilling the criteria listed above or to the bad quality of the fit.
\begin{figure*}
\centering
\includegraphics[width=0.246\textwidth]{Figures/g_osiris-ps1_psf_20082019_wmean.png}
\includegraphics[width=0.246\textwidth]{Figures/r_osiris-ps1_psf_20082019_wmean.png}
\includegraphics[width=0.246\textwidth]{Figures/i_osiris-ps1_psf_20082019_wmean.png}
\includegraphics[width=0.246\textwidth]{Figures/z_osiris-ps1_psf_20082019_wmean.png}
\includegraphics[width=0.246\textwidth]{Figures/g_osiris-ps1_histpsf_18072019.png}
\includegraphics[width=0.246\textwidth]{Figures/r_osiris-ps1_histpsf_18072019.png}
\includegraphics[width=0.246\textwidth]{Figures/i_osiris-ps1_histpsf_18072019.png}
\includegraphics[width=0.246\textwidth]{Figures/z_osiris-ps1_histpsf_18072019.png}
\includegraphics[width=0.246\textwidth]{Figures/g_osiris-ps1_auto_20082019_wmean.png}
\includegraphics[width=0.246\textwidth]{Figures/r_osiris-ps1_auto_20082019_wmean.png}
\includegraphics[width=0.246\textwidth]{Figures/i_osiris-ps1_auto_20082019_wmean.png}
\includegraphics[width=0.246\textwidth]{Figures/z_osiris-ps1_auto_20082019_wmean.png}
\includegraphics[width=0.246\textwidth]{Figures/g_osiris-ps1_histauto_18072019.png}
\includegraphics[width=0.246\textwidth]{Figures/r_osiris-ps1_histauto_18072019.png}
\includegraphics[width=0.246\textwidth]{Figures/i_osiris-ps1_histauto_18072019.png}
\includegraphics[width=0.246\textwidth]{Figures/z_osiris-ps1_histauto_18072019.png}
\caption{Comparison between PS1 and {\it OSIRIS } magnitudes of the sources used for the PSF (top first and second panels) and AUTO (third and fourth panels) photometric calibrations with $r >= 0.98$. Black filled circles represent the average difference of magnitudes in bins of 0.2\,mag. Only bins with more than ten sources are represented.
\label{fig:cal_comparison}}
\end{figure*}
Figure~\ref{fig:cal_comparison} shows the differences between PS1 and {\it OSIRIS } calibrated magnitudes of the calibration sources as a function of PS1 magnitudes and the distributions of the absolute values of the magnitude differences taken from the 12\,454 and 11\,614 subimages with good linear fit in PSF (top first and second panels) and AUTO (third and fourth panels) photometry, respectively. The few number of outliers existent in the first and the third panels, typically associated to sources with larger magnitude errors that pass the criteria listed above and that remain after the linear fit within $3\sigma$, have a minor impact in the calibration as we are using a weighted linear fit where the weight is inversely proportional to the error in magnitude.
We show in the first and third panels the average magnitude differences in bins of 0.2\,mag. Only bins with more than ten points are shown.
In all bands, these values are larger for faint sources (over $\sim$21.5\,mag in $g$, $r$ and $i$ bands, and over $\sim$20.0\,mag in the $z$ band). We also observe an increasing trend towards the bright edge of the plots. This behaviour is observed in magnitude intervals with less than 100 sources, very few compared to the several thousands included at intermediate magnitude bins.
These averaged differences of magnitudes remain under 0.1\,mag in PSF and under 0.2\,mag in AUTO comparisons for all bands.
On average, AUTO photometry presents larger scatter compared with PSF.
Mean magnitude absolute differences (second and fourth panels) are marginal. They vary from 0.063 to 0.079\,mag in PSF photometry and from 0.096 to 0.098\,mag in AUTO photometry, depending on the filter.
The difference in the filter transmission curves of the PS1 and {\it OSIRIS } filters do also contribute to the discrepancy between magnitudes. Figure~\ref{fig:tcurves_comp} shows the comparison between the transmission curves for each filter. In both cases, the throughput of the instrument is considered. In the case of the PS1 curves, taken from the SVO Filter Profile Service, we removed the contribution of the atmosphere, for comparison with {\it OSIRIS }' curves. While the $g$-band filters match fairly well, {\it OSIRIS } includes redder wavelengths in the $r$, $i$ and $z$ bands.
To assess these differences in the calibrated photometry, we address a colour dependence test in Section~\ref{sec.colour}.
\begin{figure*}
\centering
\includegraphics[width=0.4\textwidth]{Figures/ps1_gtc_g_sep19.png}
\includegraphics[width=0.4\textwidth]{Figures/ps1_gtc_r_sep19.png}
\includegraphics[width=0.4\textwidth]{Figures/ps1_gtc_i_sep19.png}
\includegraphics[width=0.4\textwidth]{Figures/ps1_gtc_z_sep19.png}
\caption{Comparison between the transmission curves of {\it OSIRIS } and PS1 in each band.
\label{fig:tcurves_comp}}
\end{figure*}
The number of calibration sources per subimage ranges from 7 to 1\,510 for \verb|MAG_PSF| and from 7 to 1\,323 for \verb|MAG_AUTO|, with a mean number of 68 and 65 stars, and a standard deviation of 113 and 107, respectively.
The contribution to the total error budget of the calibration errors is, on average, between 0.05\,mag and 0.07\,mag in \verb|MAG_PSF| and between 0.09\,mag and 0.12\,mag for $z$ in \verb|MAG_AUTO|, depending on the filter.
Figure~\ref{fig:exptimes_cal} shows the exposure time distributions of the calibrated subimages in \verb|MAG_PSF| and \verb|MAG_AUTO| photometry. In the whole catalogue, exposure time ranges from 0.5 to 900\,s.
From the mean value of the distributions ($\sim$ 100\,s) we can estimate the mean limiting magnitudes of the catalogue at 24.4, 23.9, 23.3, 22.1\,mag at S/N=5, and 23.6, 23.1, 22.6, 21.4\,mag at S/N=10 for $g$, $r$, $i$ and $z$ respectively. However, for the longest exposures, these magnitudes can reach 25.8, 24.1, 23.6 and 22.4 magnitudes for $griz$ at S/N=5.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figures/hist_exptimes_psfnew098xytt_and_autonewmod.png}
\cprotect\caption{Histogram of the exposure times of \verb|MAG_PSF| (red) and \verb|MAG_AUTO| (blue) calibrated subimages with $r >= 0.98$.}
\label{fig:exptimes_cal}
\end{figure}
To assess the photometric quality of our calibrations, we selected all calibration sources with magnitudes in the range $(18.5,18.5,18.5,18.0) < ($g$,$r$,$i$,$z$) < (22.0,22.0,22.0,21.5)$ and with at least five detections with PSF and AUTO photometry, separately. These intervals of magnitudes were chosen in order to get a representative sample of the sources used for the calibration (over 90\% for each filter and photometry), and to avoid saturation and very faint sources. We obtained the standard deviations of the measured magnitudes for each of the repeated sources. The median of the standard deviations in each filter are $(\sigma_g,\sigma_r,\sigma_i,\sigma_z)=(0.034,0.026,0.038,0.034)$ magnitudes for PSF and $(\sigma_g,\sigma_r,\sigma_i,\sigma_z)=(0.043,0.038,0.053,0.056)$ magnitudes for AUTO photometry. These low values reflect the good performance of the calibration parameters over time despite the assorted character of the observations.
We summarize in Table~\ref{tab.calibration_prop} the detailed numbers of the associated contribution of the calibration to the photometric errors, the range of exposure time, the mean limiting magnitudes at SNR=5 and SNR=10, and the typical magnitude deviations for the repeated sources in the above defined sample, for each filter.
\begin{table*}
\centering
\caption {Summary of the photometric properties of the calibrated subimages.}
\label{tab.calibration_prop}
\begin{tabular}{l cccc cccc }
\hline \hline
\noalign{\smallskip}
& \multicolumn{4}{c}{PSF} & \multicolumn{4}{c}{AUTO} \\
& $g$ & $r$ & $i$ & $z$ & $g$ & $r$ & $i$ & $z$ \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\noalign{\smallskip}
Contribution to the & & & & & & & & \\
photometric error (mag) & 0.06 & 0.07 & 0.05 & 0.05 & 0.12 & 0.012 & 0.09 & 0.10 \\
\noalign{\smallskip}
Exposure times (s) & 1--900 & 1--360 & 0.5--360 & 0.5--300 & 1-900 & 1--542 & 0.5--360 & 0.5--30 \\
\noalign{\smallskip}
Mean limiting magnitudes & & & & & & & & \\
at SNR=5 (mag) & 24.4 & 23.9 & 23.3 & 22.1 & & & & \\
\noalign{\smallskip}
Mean limiting magnitudes & & & & & & & & \\
at SNR=10 (mag) & 23.6 & 23.1 & 22.6 & 21.4 & & & & \\
\noalign{\smallskip}
Limiting magnitudes for the & & & & & & & & \\
longest exposures and SNR=5 (mag) & 25.8 & 24.1 & 23.6 & 22.4 & & & & \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
& \multicolumn{8}{c}{Photometric quality} \\
\noalign{\smallskip}
Magnitude interval (mag) & 18.5--22.0 & 18.5--22.0 & 18.5--22.0 & 18.0--21.5 & 18.5--22.0 & 18.5--22.0 & 18.5--22.0 & 18.0--21.5 \\
Median of the std. dev. $\sigma$ (mag) & 0.034 & 0.026 & 0.038 & 0.034 & 0.043 & 0.038 & 0.053 & 0.056 \\
\noalign{\smallskip}
\hline
\end{tabular}
\end{table*}
\section{Catalogue sources}\label{sec:catalogue_sources}
\subsection{Construction}
\subsubsection{Removal of spurious and saturated detections}\label{sec:removal}
To build the cleanest possible catalogue, we first removed sources with PSF instrumental magnitude errors equal to zero, larger than 1\,mag, or equal to 99 (meaning that {\sc SExtractor} PSF fit did not converge). We also discarded sources with the {\sc SExtractor} keywords \verb|FLUX_MAX|, \verb|FLUX_RADIUS| (defined as the half-light radius in Section~\ref{sec:sources_extraction}) and \verb|FWHM_IMAGE| smaller than or equal to zero, \verb|FLAGS_WEIGHT| (weighted extraction flag related to the presence of close neighbours bright enough to significantly bias the photometry, bad pixels, blended objects, saturated pixels or other features) equal to two (note that we do not impose any condition on the extraction flags \verb|FLAGS| parameter), and \verb|SNR_WIN| (the window-based signal-to-noise ratio) smaller than five or equal to 1e30 (the latter, related to a bad extraction of the source).
The limit in \verb|SNR_WIN| at five is set to avoid sources with very poor photometry.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figures/fmax_vs_fpsf_0000117506_2rep.png}
\cprotect\caption{Maximum flux at the peak (\verb|FLUX_MAX|) versus \verb|FLUX_PSF| in logarithmic scale for detections in one subimage. Blue points represent spurious detections, red points represent saturated sources and green points stand for valid detections.}
\label{fig:sat}
\end{figure}
Although a saturation limit was set at 62\,000 counts when extracting the sources from the images, few saturated sources remain. Therefore, for each subimage in the catalogue, we removed saturated sources as well as bad pixels, cosmic rays and artifacts by accounting for the linear relation between the flux at the peak of the distribution (\verb|FLUX_MAX|) and the integrated flux (\verb|FLUX_PSF|), as shown in Figure~\ref{fig:sat}. The procedure to remove spurious sources on each subimage consisted of running an iterative process that discards first all detections which ratio \verb|FLUX_MAX|/\verb|FLUX_PSF| deviates by more than 2$\sigma$ from the mean value. With the mean and standard deviation values of the remaining detections, we define a critical value of \verb|FLUX_MAX|/\verb|FLUX_PSF| equal to the mean value plus 3$\sigma$. From the original catalogue of sources of the subimage, we removed all detections which flux ratio is greater than this critical value. The rejected detections are represented with blue points in Figure~\ref{fig:sat}.
Once the removal of spurious sources has been completed, we removed saturated sources by identifying the position in the relation \verb|FLUX_MAX| vs. \verb|FLUX_PSF| at which the detector breaks linearity.
To determine this position, we iteratively performed a linear fit starting with sources with the lowest \verb|FLUX_MAX| and increasing towards higher values of \verb|FLUX_MAX|. We consider that linearity starts to fail when the Pearson correlation coefficient $r$ is lower than 0.98 and keeps decreasing. When there were no data available, we decreased the correlation coefficient to 0.96. We then selected the upper half of the sample with highest \verb|FLUX_MAX| to avoid non-linear behaviour at low fluxes (i.e., faint sources) and followed two complementary approaches:
\begin{itemize}
\item On one hand, we took the maximum number of sources which linear fit provides a correlation coefficient greater than 0.98 (or 0.96 if it is the case). Since this condition itself does not ensure the removal of all saturated sources, we defined a subset with the detections above that maximum number (i.e., sources with higher flux) and established a cut in \verb|FLUX_MAX| defined as the minimum value of the \verb|FLUX_MAX| in the subset minus its standard deviation.
\item On the other hand, we looked for the position at which the slope of the fit starts to decrease and defined a subset with detections starting at that point and with increasing fluxes. We again established a cut in \verb|FLUX_MAX| defined as the minimum value of the \verb|FLUX_MAX| in the subset minus its standard deviation. When the standard deviation was smaller than 2\,000 counts, we subtracted twice the value of the standard deviation to ensure the removal of all controversial sources.
\end{itemize}
In both cases, we removed all detections with \verb|FLUX_MAX| above those limits.
Despite the completion of these steps, we still found a few saturated sources that happen to follow the linear relation between fluxes as shown in Figure~\ref{fig:sat9000} and, therefore, have not been automatically removed. We observed that saturated sources have fluxes (\verb|FLUX_MAX|) more than $\approx$9\,000\,counts higher than the corresponding flux to the brightest-non saturated source in the subimage. Hence, we removed detections which \verb|FLUX_MAX| is at least 9\,000\,counts greater than the brightest, non-saturated source in the subimage.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figures/fmax_fpsf_9000.png}
\cprotect\caption{Maximum flux at the peak (\verb|FLUX_MAX|) versus \verb|FLUX_PSF| in logarithmic scale for detections in one subimage after the removal of spurious and saturated detections following the first two steps. Green points represent valid non-saturated source detections and the red point stands for a saturated source that follows the linear behaviour between fluxes and that accounts with more than 9\,000\,counts in \verb|FLUX_MAX| with respect to the maximum flux of the brightest-non saturated source in the subimage.}
\label{fig:sat9000}
\end{figure}
The above described procedure does only relate to PSF photometry. Hence, in parallel to this, we built a second catalogue considering exclusively calibrated \verb|MAG_AUTO| magnitudes and following similar steps than before. The only difference comes from the identification of saturated sources, because the relation between \verb|FLUX_MAX| and \verb|FLUX_AUTO| displays a higher dispersion and, therefore, executing the procedure to remove saturated sources as described above prevented us from obtaining a complete catalogue in AUTO photometry, since sources in more than 50\% of the calibrated subimages did not fulfill the required criteria. Therefore, for each subimage, we took the cuts in \verb|FLUX_MAX| determined when removing saturated sources from PSF photometry and eliminated all sources in this second catalogue with \verb|FLUX_MAX| higher than those values. This approach limits the AUTO photometry to the 11\,045 subimages that have both, PSF and AUTO calibrated photometry.
Moreover, we included a photometric flag to each detection in the catalogue to account for magnitudes beyond the magnitude coverage of the PS1 sources used in the photometric calibration in PSF (\verb|Flag_psf|) and AUTO (\verb|Flag_auto|) photometry. "B" and "C" stand for magnitudes fainter and brighter than the magnitude coverage of PS1, respectively. Detections with magnitudes in between are flagged with "A".
\subsubsection{Morphologic classification}\label{subsec:morph}
In a second step, we evaluated in our catalogue the magnitude difference limit \verb|PSF| $-$ \verb|AUTO| at 0.05\,mag proposed in the PS1 catalogue \citep{Farrow2014} to separate point-like (\verb|PSF| $-$ \verb|AUTO| $<0.05$\,mag) and extended sources (\verb|PSF| $-$ \verb|AUTO| $>0.05$\,mag).
Of the {\it OSIRIS } detections in the $g$-band under 21\,mag (limit above which the high dispersion makes this classification unreliable) and over $\sim$17\,mag (magnitude limit imposed by PS1) with PS1 counterparts and that satisfy PS1 \verb|gmag| $-$ \verb|gKmag| $<0.05$\,mag (i.e., point-like sources), 95.5\% were also classified as point-like sources following the same criterion with {\it OSIRIS } photometry (i.e., had \verb|PSF| $-$ \verb|AUTO| $<0.05$\,mag).
Hence, we could consider this photometric criterion to be valid to select point-like sources. However and due to its magnitude restriction, we do not ascribe an extended or point-like source flag in the catalogue following this rule. We evaluated instead the separation between point-like and extended sources at any magnitude range by looking into the \verb|FWHM_IMAGE|, \verb|elongation| and \verb|ellipticity| parameters.
To do so, we selected sources in each subimage with PSF calibrated magnitudes between 17 and 21\,mag (regardless the filter) and with magnitude differences \verb|PSF| $-$ \verb|AUTO|<0.05\,mag, and compute their \verb|FWHM_IMAGE|, \verb|elongation| and \verb|ellipticity| mean values and standard deviations.
Sources in the subimages which parameter values differed by more than the mean value within 2$\sigma$ were tagged as extended sources and as point-like sources otherwise.
To assess the goodness of this criterion, we compared both, the {\it OSIRIS } photometric and morphometric classifications, in the interval between 17 and 21\,mag. In all bands, more than 92\% of the sources identified photometrically as point-like were also labeled as point-like using morphometric parameters. Moreover, extended sources were also in agreement in more than 75, 83, 74, and 72 per cent of the cases in the $g$, $r$, $i$, and $z$ bands, respectively. We therefore conclude that the morphometric criterion can be extended to all detections in the catalogue, regardless their magnitudes, to tabulate them as point-like or extended sources.
For 1\,378 subimages in the four bands, we could not apply this selection because of the lack of automatic (Kron) magnitudes due to the absence of photometric calibration, or the lack of sources with magnitude differences under 0.05\,mag and magnitudes between 17 and 21\,mag. Since the number of subimages without any classification was significantly large, we decided to use PS1 magnitudes to identify point-like sources and compute their \verb|FWHM_IMAGE|, \verb|elongation| and \verb|ellipticity| mean and standard deviations values to be used as a reference for the classification of all detections in the subimage. In these cases, the classification is noted in the catalogue as "P*" for point-like and "E*" for extended sources.
After this, there are still 77 subimages (39\,246 detections representing 0.6\% of the catalogue) for which we were not able to ascribe any classification.
The privileged location and weather conditions of the Observatory favour astronomical observations with a typical seeing of 1\,arsec and reaching often lower values. Detections with FWHM below 0.5\,arcsec were removed as they were just simply artifacts. Detections with FWHM over 7\,arcsec and/or with ellipticities over 0.7 were also removed. The mean ellipticity measured in point-like sources is 0.11$\pm$0.07 with just a 1.9\% showing an ellipticity worse than {\it e} >0.3. This confirms the good performance of the {\it GTC } tracking capability.
\subsubsection{Resulting catalogue}\label{subsec:resulting_cat}
The next step is to build a science-ready, user-friendly catalogue containing both astrometric and photometric information as well as flags to warm about quality issues.
In summary, the catalogue contains 6\,226\,520 detections corresponding to 633\,559 different sources in 12\,409 subimages. Detections were merged into sources by performing an internal match of the whole catalogue (this is, regardless the photometric band) within 0.5\,arcsec using STILTS \citep{Taylor06}. This value is a trade-off between completeness and reliability. Larger values may allow unrelated detections to be linked in the same source while smaller values would pose problems for faint sources with large centroiding errors. A detailed description of each column in our detection catalogue is given in Table~\ref{tab.catalogue_description}.
Table~\ref{tab.cat-info} lists, for each filter, the number of subimages and detections contained in the catalogue.
\begin{table}
\centering
\caption {Number of subimages and detections per filter in the catalogue.
\label{tab.cat-info}
\begin{tabular}{c c c}
\hline \hline
\noalign{\smallskip}
Filter & Number of & Number of \\
& subimages & detections \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\noalign{\smallskip}
Sloan g & 1\,328 & 532\,760 \\
Sloan r & 2\,866 & 1\,585\,249 \\
Sloan i & 3\,825 & 2\,041\,184 \\
Sloan z & 4\,390 & 2\,067\,327 \\
\noalign{\smallskip}
\hline
\end{tabular}
\end{table}
Table~\ref{tab.bands-sources} summarizes the number of sources that have been detected in one, two, three or the four bands, regardless which bands are they. Near 30\% of the sources in the catalogue has been detected in two or more bands and only 3.2\% has been detected in the four bands.
\begin{table}
\centering
\caption {Number of sources detected in one to four bands.}
\label{tab.bands-sources}
\begin{tabular}{c c }
\hline \hline
\noalign{\smallskip}
Number of & Number of \\
bands & sources \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\noalign{\smallskip}
4 & 19\,991 (3.2\%)\\
3 & 64\,350 (10.2\%)\\
2 & 106\,747 (16.8\%)\\
1 & 442\,471 (69.8\%)\\
\noalign{\smallskip}
\hline
\end{tabular}
\end{table}
Astrometric errors in the catalogue were computed from the quadratic sum of the windowing position errors (i.e., the errors in the ellipse parameters reported by {\sc SExtractor}) and the estimated absolute astrometric calibration uncertainty mentioned in Section~\ref{sec:astrometric_calibration}. The mean accuracy in the catalogue is 0.12\,arcsec.
In order to provide accurate photometry at a single epoch, we identified the best detection for each source, hereafter refereed to as {\it primary} detection. The criteria used to select the best detection were to have signal to noise ratio above the mean value for that source, and the least relative PSF or AUTO magnitude error.
A total of 1\,209\,058 primary detections satisfied the above criteria. Primary detections are flagged in the catalogue as {\it p} while the rest of detections are flagged with {\it s} under {\it Flag\_source}.
In addition, we composed a source catalogue with these {\it primary} detections of each source, for the user to easily access the best photometry. It contains selected data such as the source identifier, equatorial coordinates, PSF and AUTO calibrated magnitudes with their corresponding flags, epochs and url of the associated image in each band, and a source class parameter ({\it cl}) defined as the ratio between the number of detections classified as point-like ("P" or "P*") and the total number of detections. The source class parameter takes values between 0 and 1, being 0 when the source has always been identified as extended and 1 when it has always been identified as point-like.
This source catalogue is complementary to the previously defined one containing all detections, magnitudes and parameters. Table~\ref{tab.primarycatalogue_description} contains a detailed description of each column in the source catalogue.
Typical saturation and limiting magnitudes in the catalogue in each filter are 13.1--24.6\,mag in $g$, 13.6--24.5\,mag in $r$, 13.1--23.9\,mag in $i$, and 12.0--22.7\,mag in $z$, respectively. Saturation magnitudes correspond to the minimum value of the PSF magnitude in the catalogue and limiting magnitudes correspond to the 90th percentile, per each filter.
This means that the magnitude limits in the {\it OSIRIS } catalogue are between 1.4 and 2 magnitudes brighter and between 0.4 and 1.4 magnitudes fainter than PS1, depending on the filter. Figure~\ref{fig:campo_gtcps1} shows a {\it GTC } {\it OSIRIS } image together with all PS1 and {\it GTC } sources in the field of view. The increased depth of the {\it OSIRIS } catalogue (1\,378 objects) compared to PS1 (366 objects) is clearly seen. The catalogues have 208 sources in common. There are 158 sources that are in PS1 and not in our catalogue due to saturation or \verb|Flags_weight| parameter equal to two (see the first paragraph in Section~\ref{sec:removal}). Magnitudes of the 1\,170 sources in our catalogue that are not in PS1 range from 20.0 to 25.4\,mag, which depending on the filters, would be beyond the magnitude limit of PS1.
\begin{figure*}
\centering
\includegraphics[width=0.5\textwidth]{Figures/0000113133fits.png}\\
\includegraphics[width=0.48\textwidth]{Figures/0000113133fits_ps1.png}
\includegraphics[width=0.48\textwidth]{Figures/0000113133fits_gtc.png}
\caption{Example of an {\it OSIRIS } image centered at 17:06:59.64 +58:46:37.5 (top panel) overlayed with PS1 sources (bottom left panel, blue open circles) and {\it OSIRIS } sources (bottom right panel, red open circles).
\label{fig:campo_gtcps1}}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.246\textwidth]{Figures/meang_vs_g_psf_08072019.png}
\includegraphics[width=0.246\textwidth]{Figures/meanr_vs_r_psf_08072019.png}
\includegraphics[width=0.246\textwidth]{Figures/meani_vs_i_psf_08072019.png}
\includegraphics[width=0.246\textwidth]{Figures/meanz_vs_z_psf_08072019.png}
\includegraphics[width=0.246\textwidth]{Figures/meang_vs_g_auto_08072019.png}
\includegraphics[width=0.246\textwidth]{Figures/meanr_vs_r_auto_08072019.png}
\includegraphics[width=0.246\textwidth]{Figures/meani_vs_i_auto_08072019.png}
\includegraphics[width=0.246\textwidth]{Figures/meanz_vs_z_auto_08072019.png}
\cprotect\caption{Photometric errors versus magnitudes for \verb|MAG_PSF| (top) and \verb|MAG_AUTO| (bottom) in the four bands.
Bullets are the average values of the magnitude errors in bin sizes of 0.5\,mag and the error bars show the standard deviation.}
\label{fig:sensitivity}
\end{figure*}
Figure~\ref{fig:sensitivity} represents averaged PSF and AUTO magnitude errors versus magnitude in bins of 0.5\,mag, illustrating the sensitivity reached in the catalogue on each of the four filters. The trend is to increase with increasing magnitudes (i.e., towards fainter sources).
In all bands, mean magnitude errors are under 0.1\,mag up to 21.5 / 19\,mag and do not exceed 0.30 / 0.35\,mag, in PSF /AUTO photometry, respectively. In PSF photometry, the mean photometric accuracy of the catalogue is 0.09\,mag and 0.15\,mag in AUTO photometry.
\subsection {Catalogue quality assessment}
\subsubsection{Comparison with PS1}
Top panel in Figure~\ref{fig:ps1_vs_gtc} compares good quality PS1 sources ({\it Qual} equal to 52) versus {\it OSIRIS } PSF magnitudes. We observe good agreement up to PS1 $\sim$21.5\,mag in the $g$ and $r$ bands, $\sim$21.0\,mag in the $i$ band and $\sim$20.5\,mag in the $z$ band, magnitudes above which we observe a tendency towards fainter {\it OSIRIS } magnitudes. Most outliers in these plots do also missmatch when comparing their PS1 magnitudes with clean ({\it q\_mode} equal to '+') SDSS DR12 photometry (see bottom panel in this figure). Note that the number of sources represented in the plots of the bottom panel is lower than the number of sources represented in the plots of the top panel due to the requirement of having good PS1 and SDSS DR12 photometry. In the comparison of PS1 with SDSS DR12 we observe as well the tendency towards fainter magnitudes above 20--21.5\,mag, depending on the filter. Hence, we ascribe these magnitude differences to an intrinsic feature of PS1 photometry.
Figure~\ref{fig:diff_mags} shows, for each filter, the normalized cumulative distribution of the PSF magnitude absolute differences between {\it OSIRIS } and PS1 sources with good quality flags ({\it Qual} equal to 52).
For 90\% of the sample of sources in the magnitude interval of linear behaviour, the photometric scatter in absolute values is below 0.16, 0.19, 0.18, and 0.16 magnitudes in $g$, $r$, $i$ and $z$, respectively. This reflects the good photometric agreement between catalogues.
\begin{figure*}
\centering
\includegraphics[width=0.246\textwidth]{Figures/deltag_vs_ggtc__density_20ago2019.png}
\includegraphics[width=0.246\textwidth]{Figures/deltar_vs_rgtc__density_20ago2019.png}
\includegraphics[width=0.246\textwidth]{Figures/deltai_vs_igtc__density_20ago2019.png}
\includegraphics[width=0.246\textwidth]{Figures/deltaz_vs_zgtc__density_20ago2019.png}
\includegraphics[width=0.246\textwidth]{Figures/deltag_vs_ggtc__density_20ago2019_sdss.png}
\includegraphics[width=0.246\textwidth]{Figures/deltar_vs_rgtc__density_20ago2019_sdss.png}
\includegraphics[width=0.246\textwidth]{Figures/deltai_vs_igtc__density_20ago2019_sdss.png}
\includegraphics[width=0.246\textwidth]{Figures/deltaz_vs_zgtc__density_20ago2019_sdss.png}
\caption{Comparison, for each filter, of PSF PS1 magnitudes with PSF {\it OSIRIS } (top panel) and of PSF PS1 magnitudes with PSF SDSS DR12 (bottom panel).
\label{fig:ps1_vs_gtc}}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figures/cumulativehist_deltamag_fig13.png}
\caption{Normalized cumulative distribution of PSF absolute magnitude differences between {\it OSIRIS } and PS1 in logarithmic scale in the $g$ (red), $r$ (blue), $i$ (green) and $z$ (gray) bands.
\label{fig:diff_mags}}
\end{figure}
\subsubsection{Colour dependence}\label{sec.colour}
To investigate the effect of a colour term in the photometric calibration, we compared the colour differences between {\it OSIRIS } and PS1 as a function of the calibrated magnitudes in the top panel of Figure~\ref{fig:colour_dep}. The large dispersion observed towards fainter sources is likely associated to the already noticed magnitude differences over 20--21.5\,mag when comparing {\it OSIRIS } with PS1 (see Figure~\ref{fig:ps1_vs_gtc}). Moreover, the number of sources with significantly high colour differences is not statistically representative (less than 1\% of the plotted sample).
The bottom panel of the figure shows the difference of magnitudes in the z band with respect to the PS1 $r-z$ (left) and $i-z$ colours (right). In light of these plots, we do not observe any colour dependence.
\begin{figure*}
\centering
\includegraphics[width=0.32\textwidth]{Figures/delta_gr_gtcps1_11mar2019.png}
\includegraphics[width=0.32\textwidth]{Figures/delta_ri_gtcps1_11mar2019.png}
\includegraphics[width=0.32\textwidth]{Figures/delta_iz_gtcps1_11mar2019.png}
\includegraphics[width=0.32\textwidth]{Figures/colour_rz_fig14.png}
\includegraphics[width=0.32\textwidth]{Figures/colour_iz_fig14.png}
\caption{Top panel: Dependence of the difference of colours ({\it OSIRIS } -- PS1) with {\it OSIRIS } PSF magnitudes. Bottom panel: Dependence of the difference of magnitudes in the z band with PS1 $r-z$ (left) and $i-z$ (right) colours.
\label{fig:colour_dep}}
\end{figure*}
\subsubsection{Binning mode and pixel position dependence}
We verify whether the observed magnitude differences of sources with PSF magnitudes fainter than 20.5--21.5\,mag (depending on the filter) when comparing with PS1 photometry have a relation with the pixel position of the source in the CCD.
Figure~\ref{fig:xy_dep} shows, for each filter, the $xy$ pixel position of sources with PSF magnitudes fainter than 21.5\,mag in $g$ and $r$, 21.0\,mag in $i$, and 20.5\,mag in $z$ (i.e., sources with the largest magnitude deviations) and with good quality PS1 photometry. {\it OSIRIS } standard observing mode uses 2$\times$2 binned pixels, this is, images of $\sim$ 1\,024$\times$2\,048\,pixels size. This is the binning mode where the majority of the observations were performed. In general, binning the pixels aims to increase the signal to noise of measured images at a cost of losing spatial resolution. In light of the random arrangement of the sources in the CCD, we can conclude that there is no dependence of the magnitude differences either with the binning mode of observation or the pixel position.
\begin{figure*}
\centering
\includegraphics[width=0.246\textwidth]{Figures/g_xy_gt215mag.png}
\includegraphics[width=0.246\textwidth]{Figures/r_xy_gt215mag.png}
\includegraphics[width=0.246\textwidth]{Figures/i_xy_gt210mag.png}
\includegraphics[width=0.246\textwidth]{Figures/z_xy_gt205mag.png}
\caption{CCD $xy$ position of sources with PS1 photometry and fainter PSF magnitudes than 21.5\,mag in $g$ (red panel) and $r$ (blue panel), 21.0\,mag in $i$ (green panel), and 20.5\,mag in $z$ (gray panel).
\label{fig:xy_dep}}
\end{figure*}
\subsubsection{Internal photometric precision
In order to estimate the internal photometric precision of the catalogue, we compared for each filter the magnitudes of sources observed more than five times.
The relation between their standard deviations and magnitudes is shown in Figure~\ref{fig:stdmag_mag}.
We also show in this figure the averaged photometric errors of Figure~\ref{fig:sensitivity}.
The scatter in the standard deviation of magnitudes increases towards fainter magnitudes in both, PSF and AUTO calibrated magnitudes. Also AUTO photometry shows slightly higher dispersion and error bars than PSF photometry.
Since {\sc SExtractor} uses flexible elliptical apertures for extracting AUTO photometry, this effect can be attributed to the possibly different configurations applied to the same source in different images and epochs that lead to different flux integrations.
In general, the variations of magnitudes of repeated sources is of the order of or lower than the mean accuracy of the catalogue within the errorbars at each interval of magnitudes with the exception of the bright end of both, PSF and AUTO photometry, in the $g$ and $z$ bands. Also the faintest interval of magnitudes in the PSF panels in all bands and the AUTO photometry in the $i$ band show this behaviour, explained by the lower number of sources in this regime.
\begin{figure*}
\centering
\includegraphics[width=0.246\textwidth]{Figures/meanstd_vs_mag_repetitividad_gpsf_23jul2019.png}
\includegraphics[width=0.246\textwidth]{Figures/meanstd_vs_mag_repetitividad_rpsf_23jul2019.png}
\includegraphics[width=0.246\textwidth]{Figures/meanstd_vs_mag_repetitividad_ipsf_23jul2019.png}
\includegraphics[width=0.246\textwidth]{Figures/meanstd_vs_mag_repetitividad_zpsf_23jul2019.png}
\includegraphics[width=0.246\textwidth]{Figures/meanstd_vs_mag_repetitividad_gauto_23jul2019.png}
\includegraphics[width=0.246\textwidth]{Figures/meanstd_vs_mag_repetitividad_rauto_23jul2019.png}
\includegraphics[width=0.246\textwidth]{Figures/meanstd_vs_mag_repetitividad_iauto_23jul2019.png}
\includegraphics[width=0.246\textwidth]{Figures/meanstd_vs_mag_repetitividad_zauto_23jul2019.png}
\caption{Photometric repeatability as a function of magnitudes for sources observed more than five times for PSF (top) and AUTO (bottom) photometry.
Bullets are the average values of the standard deviations of magnitudes of repeated sources in bin sizes of 0.5\,mag and the error bars show their standard deviations.
Mean values of the average photometric errors shown in Figure~\ref{fig:sensitivity} are displayed as black crosses.
\label{fig:stdmag_mag}}
\end{figure*}
While this exercise is a good approach to assess the homogeneity and quality of the entire dataset, it can also be misleading since the tails of the distributions are generally caused by faint sources with low signal-to-noise ratio and a small number of artifacts that passed previous tests. Figure~\ref{fig:hist_rep} shows, for each filter, the normalized cumulative histogram of the signal-to-noise ratio of sources with PSF magnitude standard deviations larger than the mean value plus 1$\sigma$ (this is, typically larger than 0.2\,mag).
Per filter, between 75 and 85\% of these problematic sources have \verb|SNR_WIN| under 25.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figures/hist_repetitividad_snr_cum.png}
\caption{Normalized cumulative distributions of the signal-to-noise ratios of sources observed more than five times and with large standard deviation of PSF magnitudes. The $g$, $r$, $i$ and $z$ filters are represented in red, blue, green and gray, respectively.
\label{fig:hist_rep}}
\end{figure}
\subsubsection{Variability}
Real variable objects can also contribute to the tail of the distribution.
We therefore looked for PSF magnitude variations according to the following equation:
\begin{equation}
S= mag_{min}+3*e\_mag_{min}-(mag_{max}-3*e\_mag_{max})
\end{equation}
where $mag_{min}$, $mag_{max}$, $e\_mag_{min}$ and $e\_mag_{max}$ are the lower and higher magnitudes measured for the same source and their associated errors. If $S$ is lower than zero, then the magnitude deviation of the source is outside the reach of three times the errorbars and cannot be explained by the photometric errors.
Of the 731\,685 sources that have been detected more than once in the same filter we found that, between 10 and 13\%, fulfill this criterion, depending on the filter. Figure~\ref{fig:hist_var} shows the PSF magnitude distributions of these sources in the four bands. They peak well over the 20.5--21.5\,mag limits stated for precise photometry and therefore, these magnitude variations could be related in some cases to the lower quality photometry of faint sources rather than to real photometric variation. Nonetheless, 33\,310 out of the 731\,685 repeated sources (4.5\%) are brighter than those magnitude limits (21.5\,mag in $g$ and $r$, 21.0\,mag in $i$ and 20.5\,mag in $z$) and could actually be real photometric variable sources. Of them, only 16 have already been identified as variable in the literature. A proper analysis of the true nature of the variability detected in these sources is beyond the scope of this paper.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figures/hist_mag_vairiables.png}
\caption{PSF magnitude distributions of sources with $S$<0 in the $g$ (red), $r$ (blue), $i$ (green) and $z$ (gray) bands. Vertical black dashed lines indicate the 20.5--21.5\,mag limits of precise photometry.
\label{fig:hist_var}}
\end{figure}
We also represent in Figure~\ref{fig:snr_dist_var} the distributions of the \verb|SNR_WIN|, elongation and ellipticity of the sources that show $S$ values smaller and greater than zero for comparison and in order to evaluate any drift of these parameters favouring variability.
Although sources with $S>0$ register a mean signal-to-noise ratio slightly higher compared to the mean value of sources with $S<0$ (85 vs. 64), the peak in both signal-to-noise ratio distributions is similar. The behaviour of morphometric parameters is comparable in both samples as well. Hence, we do not see any dependence of the photometric variability with lower signal-to-noise ratio or flattening of the sources.
\begin{figure*}
\centering
\includegraphics[width=0.32\textwidth]{Figures/snr_dist_var.png}
\includegraphics[width=0.32\textwidth]{Figures/elongation_dist_var.png}
\includegraphics[width=0.32\textwidth]{Figures/ellipticity_dist_var.png}
\caption{Distribution of the SNR (left), elongation (middle) and ellipticity (right) of repeated sources in the catalogue with $S<0$ (red) and with $S>0$ (blue).
\label{fig:snr_dist_var}}
\end{figure*}
On the other hand, we also investigated if known variable sources do also show variability in our catalogue.
To do so, we cross-matched our catalogue with {\it Gaia} DR2 and the AAVSO International Variable Star Index VSX \citep{2006SASS...25...47W}. Only 73 different sources in our catalogue are tagged as variables in these catalogues. We looked in the SIMBAD astronomical database \citep{Wenger00} and identified another 319 sources classified as variables or suspected of variability.
We show in Figure~\ref{fig:known_var} the PSF standard deviation of magnitudes for each of these sources with respect to their PSF magnitudes in the corresponding filter. The majority of these sources do not present significant variations except perhaps the five sources labeled in the plot with standard deviations over 0.2\,mag and larger than their magnitude errors.
Sources \#1 and \#2 correspond to the same source (NSV 13246) observed in the $g$ and $r$ bands, and classified in {\it Gaia} DR2 and SIMBAD as Fundamental-mode RR Lyrae star; source \#3 (SN 2011by) is a SN Ia in the AAVSO International Variable Star Index VSX catalogue; source \#4 (CRTS J154326.0-212800) is classified in {\it Gaia} DR2 and SIMBAD as a fundamental-mode RR Lyrae star; and source \#5 (2MASS J20325377+4115134) is found in SIMBAD as an eclipsing binary candidate, variable and X ray emitting source. Sources \#1 and \#2 have been detected within 0.83 years four and two times in our catalogue, respectively. Sources \#3, \#4 and \#5 have been detected twice in 56 days, six times within 10 days and six times within 18 days, respectively. Due to the time span of the observations, these magnitude variations could probably be associated to real photometric variations in all cases.
Unfortunately, the cadence in the observations might not be suitable for variability detection in the catalogue.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figures/std_vs_mag_variableslabel_mar2019.png}
\caption{Standard deviation of PSF magnitudes of known variable sources. Red, blue, green and gray filled circles stand for $g$, $r$, $i$ and $z$ bands respectively.
\label{fig:known_var}}
\end{figure}
\section{Scientific exploitation}\label{sec:science}
To illustrate the science capabilities of the catalogue, we defined two cases to prove the goodness of the photometry (identification of cool stars) and the astrometry (identification of asteroids) provided in the catalogue.
\subsection{Looking for cool dwarfs}
Cool dwarfs of M spectral type constitute around the 40\% of the stellar mass in the Galaxy \citep{Gould96, Boc10}.
Their ubiquity and lifetimes that exceed the current age of the Universe \citep{Bar98, Hen06} makes them excellent targets to broadly study the formation and evolution processes at the bottom of the main-sequence.
To identify cool dwarfs in the {\it OSIRIS } catalogue, we used the $i$ and $z$ bands and selected all sources with $i-z> 0.38$\,mag, which would correspond to spectral types later than M0 according to \cite{West08}. We then cross-matched the selected sources with {\it Gaia} DR2 within 1.5\,arcsec and kept all sources with relative errors in parallax and proper motion below 20\%. Later we removed all sources with proper motions $\mu < 30$\,mas/yr as in \cite{Solano19} to avoid contamination of giants and subgiants in the sample. This way, we ended up with 52 cool dwarf candidates, for which we derived effective temperatures in order to confirm their cool nature. To do so, we used VOSA \citep{Bayo08} which allowed us to gather photometry from the DENIS \citep{DENIS1999,DENIS2000}, 2MASS \citep{2MASS2006}, UKIDSS \citep{UKIDSS2006,UKIDSS2007,UKIDSS2007_2,UKIDSS2009}, IPHAS \citep{IPHAS2008,IPHAS2014}, WISE \citep{WISE2010}, SDSS DR9 \citep{SDSS2012}, Pan-STARRS DR1 \citep{Chambers16, Flewelling2016,Magnier2016, Magnier2016_1, Magnier2016_2, Tony12, Waters2016} and Gaia DR2 catalogues. This information was used together the photometric points in the $griz$ bands from our {\it OSIRIS } catalogue making use of the SVO Filter Profile Service to build the corresponding Spectral Energy Distributions (SEDs). We applied the BT-Settl collection of theoretical models \citep{BTSettl} with solar metallicity, $\log{g}$ between 4.5 and 6.0 and $T_{eff}$ between 1\,000 and 5\,000\,K. We left extinction $A_V$ as a free parameter varying from 0 to 1\,mag (assuming that extinction could be up to 1\,mag at 1\,kpc) since it can strongly modify the shape of the SED and, therefore, the parameters determination. VOSA fittings were visually inspected to confirm that {\it OSIRIS } photometry does not deviate from the SED.
The lack of good photometry prevented us from performing a reliable fit for three out of the 52 candidates.
One of these three turned out to be a non-catalogued resolved physical binary with a late-type primary separated 1.24\,arcsec at 195\,pc (241.7\,AU). Near a hundred binaries with late M/L dwarf primaries have been identified up to date \citep{Bouy03, Close03, Dupuy17, Gagliuffi15}. 2MASS, {\it Gaia} DR2 and {\it GTC } {\it OSIRIS } are able to resolve the pair and we can therefore confirm common proper motion. We could estimate the spectral type of the components from the {\it Gaia} DR2 $G-RP$ colours and the updated version of Table~5 in \cite{PecautMamajek13}\footnote{\url{http://www.pas.rochester.edu/~emamajek/EEM_dwarf_UBVIJHK_colors_Teff.txt}}. The primary (GTC$\_$OSIRIS$\_$BBI$\_$DR1$\_$J203146.23+411437.0) would be an M9.5\,V with $G-RP$=1.627\,mag.
For the secondary (GTC$\_$OSIRIS$\_$BBI$\_$DR1$\_$J203146.19+411437.6), there is no RP photometry. From the $\Delta G = 1.0$\,mag of the system, we estimate the secondary to be an L1-L2 dwarf.
Another one of these three sources (GTC$\_$OSIRIS$\_$BBI$\_$DR1$\_$J181631.68+691152.9) has a close bright companion in our {\it OSIRIS } images at 2.3\,arcsec that saturates (and is therefore not in the catalogue) and that is not resolved by {\it Gaia} DR2, PS1, 2MASS or WISE. The combined photometry of the two sources in these catalogues prevent us from obtaining the SED of our cool dwarf candidate. We can not suggest nor discard physical binding due to the lack of available information.
In addition, we obtained effective temperatures for 49 sources ranging from 2\,400 to 3\,700 K with an uncertainty of 50\,K. Their positions shown in the colour-magnitude diagram of Figure~\ref{fig:hrd} built with {\it Gaia} DR2 sources also agree with being sources later than M0. Their distances range between 75.9 and 854.6\,pc. The coolest dwarf in the sample lies in the M/L transition region at 85.9\,pc. With an effective temperature of 2\,400\,K we estimate it's spectral type to be an M9.0-M9.5 dwarf. Again, using the {\it Gaia} $G-RP$ colour and the updated version of Table~5 in \cite{PecautMamajek13}, we infer a spectral type between M9.5 and L0.
Among the 49 cool sources, we found another non previously reported close binary separated 3.84\,arcsec (1\,940.4\,AU). We confirm common proper motion from the detection of both components by UKIDSS, Pan-STARRS DR1 and {\it Gaia} DR2. The pair is located at 505.3\,pc. In this work, we identified the secondary source of the system (GTC$\_$OSIRIS$\_$BBI$\_$DR1$\_$J203324.41+410751.7) and obtained an effective temperature of 3\,300\,K. For the primary (not included in the {\it OSIRIS } catalogue because of saturation in all images), we determined an effective temperature of 4\,200\,K using VOSA. From the {\it Gaia} DR2 $G-RP$ colours and the updated version of Table~5 in \cite{PecautMamajek13}, we estimated the spectral type of the components to be a K7 and an M4.0-M4.5. Spectral types are in agreement with the derived temperatures.
Only three out of the 49 sources were found in SIMBAD and none of them have been reported as cool type stars. As a curiosity, 36 out of the 49 sources cover less than one square degree of the sky in the region of the Cygnus OB association, although they are found at less than half its distance. This is a region of high scientific interest and may have therefore been often observed in several bands. This would explain the large fraction of cool objects found in this region.
Table~\ref{tab.cooldwarfs} lists the {\it OSIRIS } and {\it Gaia} DR2 identifiers and effective temperatures of the 49 dwarf candidates and the identifiers of the late-type binary components.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figures/HRD_cooldwarfs_sep2019.png}
\cprotect\caption{Colour-magnitude diagram using {\it Gaia} DR2 sources with parallaxes larger than 10\,mas (black dots). Light blue filled circles represent our 48 single dwarf candidates later than M0, yellow filled diamond represents the M type secondary belonging to the K+M close binary and the magenta filled diamond stands for the M+L close binary system. L and T dwarfs with {\it Gaia} counterparts identified in \cite{Smart17} are displayed with dark blue dots.}
\label{fig:hrd}
\end{figure}
\subsection{Identification of asteroids}
We applied the \texttt{ssos}\footnote{\url{https://pypi.org/project/ssos/}} pipeline \citep{Max19} to detect and identify Solar System Objects (SSOs) serendipitously observed in the {\it OSIRIS } images. The pipeline detects both known and unknown SSOs primarily based on their linear apparent motion in subsequent exposures. Source detection and association are performed by {\sc SExtractor} and {\sc SCAMP} respectively, while the separation of SSOs from other sources in the image catalogues is performed by a chain of user-configurable filter algorithms. A more detailed discussion of the pipeline and its application to the {\it OSIRIS } images can be found in \cite{Max19}.
To apply the pipeline, the images were grouped by observation night and overlapping field-of-views. Of the full sample of images in the DR1, we built 420 groups made up of 6\,982 images. The remaining images had to be discarded as there were fewer than 4 exposures in the respective visits, a requirement for a reliable detection of SSOs.
Applying the \texttt{ssos} pipeline to these 420 groups revealed 204 unique SSOs present in a total of 2\,828 images. 63 objects could be identified as known SSOs using the IMCCE's SkyBoT service, which computes the ephemerides of SSOs within a given field-of-view and observation epoch. The returned computed ephemerides were cross-matched within a radius of 40\,arcsec with the positions of the recovered SSOs. Table~\ref{table:skybot_gtc} lists the classes of the 63 identified objects. The majority are Main-Belt (MB) asteroids. Four comets were retrieved as well, however, they were the targets of the respective observations. Figure~\ref{fig:sso_example} depicts four detections of \textit{ (355891) 2008 WE46} to illustrate the serendipitous observations.
The remaining 141 SSOs are either unknown or had a discrepancy between predicted and observed position larger than 40\,arcsec, meaning that their observation will greatly improve the accuracy of their orbit. The astrometric and photometric properties of all 2\,828 SSO detections have been reported to the Minor Planet Centre\footnote{\url{https://www.minorplanetcenter.net/}} (MPC). Of the 1\,002 observations of known SSOs, 872 were ingested into the MPC database\footnote{\url{https://minorplanetcenter.net/iau/ECS/MPCArchive/2018/MPS_20181118.pdf}, observatory code Z18}. The remaining 130 observations have not been published due to unknown reasons.
The 1\,826 observations of unknown SSOs consist of single-night observations only. Therefore, they will not receive temporary designations, however, they are ingested into the MPC database and might eventually be associated to a newly discovered object.
Unfortunately, the acquired photometry did not allow for spectrophotometric classification of the SSOs as the observations typically spanned more than 1 hour, rendering the determination of colours unreliable. Moreover, the temporal baseline was not long enough to estimate rotation periods from the light curve analysis. The observations will still be useful in combination with other data, however, to determine e.g. the SSO phase function parameters.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figures/sso_example.png}
\caption{Four observations of \textit{ (355891) 2008 WE46} recovered from the DR1 images. The main-belt asteroid was serendipitously observed and recovered using the \textit{ssos} pipeline. Its position is marked by the dotted white ellipse in the center of each frame. The text indicates the time difference between the frames in seconds.}
\label{fig:sso_example}
\end{figure}
\begin{table}
\centering
\caption {Distribution of previously known SSOs detected in the DR1 images over SSO classes. The 4 comets were retrieved from targeted observations. MB stands for Main-Belt.}
\label{table:skybot_gtc}
\begin{tabular}{lllll}
\hline \hline
\noalign{\smallskip}
Comet & Inner MB &Middle MB &Outer MB & Trojan\\
\noalign{\smallskip}
\hline
4 & 21 & 17 & 20 & 1 \\
\noalign{\smallskip}
\hline
\end{tabular}
\end{table}
\section{Data Access}\label{sec:data_access}
The photometrically and astrometrically corrected {\it OSIRIS } broadband images as well as the associated catalogue are available to the community through the {\it GTC } Archive Portal\footnote{\url{http://gtc.sdc.cab.inta-csic.es}} or the associated Virtual Observatory services (SIAP for images and ConeSearch for the catalogue). The {\it GTC } archive is maintained by Centro de Astrobiologia (INTA-CSIC) in the framework of the Spanish Virtual Observatory\footnote{\url{http://svo.cab.inta-csic.es}}. The results provided by the portal or the VO services can be sent using the SAMP protocol to other VO tools for its further visualization and/or analysis.
In order to help the astronomical community on using the detection and source catalogues built from the {\it OSIRIS } broadband images, we have developed an archive system that can be accessed from a webpage\footnote{\url{http://svo2.cab.inta-csic.es/vocats/v2/gtc-osiris/}} or through a Virtual Observatory ConeSearch\footnote{e.g. \url{http://svo2.cab.inta-csic.es/vocats/v2/gtc-osiris-primary/cs.php?RA=0.107&DEC=44.636&SR=0.1&VERB=2}}.
The archive system implements a very simple search interface (see Fig~\ref{fig.svoquery}) that permits queries by position or PSF magnitude interval in both, detection and source catalogues, and also by colour range only in the source catalogue. The system implements aswell a link to the images in {\it GTC } Public Archive.
\begin{figure*}
\centering
\includegraphics[width=\hsize]{Figures/svoquery_detection_oct2019.png}
\includegraphics[width=\hsize]{Figures/svoquery_source_oct2019.png}
\caption{Screenshots of the archive search interfaces that permit simple queries to the detection (top) and source (bottom) catalogues.}
\label{fig.svoquery}
\end{figure*}
The result of the query is a HTML table with all the sources found in the archive fulfilling the search criteria up to a limit of 10\,000 lines. The result can also be downloaded as a VOTable or a CSV file. Detailed information on the output fields can be obtained placing the mouse over the question mark (``?") located close to the name of the column. The archive also implements the SAMP\footnote{\url{http://www.ivoa.net/documents/SAMP/}} (Simple Application Messaging) Virtual Observatory protocol. SAMP allows Virtual Observatory applications to communicate with each other in a seamless and transparent manner for the user. This way, the results of a query can be easily transferred to other VO applications, such as, for instance, TOPCAT.
\section{Conclusions and future work}\label{sec:conclusions}
We have presented the database of processed and scientific grade broadband images and the associated catalogue obtained with the {\it GTC } {\it OSIRIS } instrument from April 2009 to January 2014. In this first release, the database includes 6\,788 images in the Sloan $griz$ bands with exposure times ranging from 0.5 to 900\,s, and 6\,226\,520 entries in the catalogue corresponding to 633\,559 astronomical sources. The catalogue is astrometrically and photometrically calibrated in standard PSF and automatic (Kron equivalent) photometry using 2MASS, SDSS DR10 or USNO-B1 for the astrometry and Pan-STARRS DR1 for the photometry, allowing a broad range of scientific activities.
Relative astrometric residuals typically are within 30\,mas and typical positional uncertainty is of 0.12\,arcsec. Only well correlated sources between instrumental and Pan-STARRS DR1 magnitudes were used for photometric calibration, providing a precision of 0.034--0.056\,mag, depending on the filter.
The mean photometric accuracy of the whole catalogue is better than 0.09\,mag and 0.15\,mag in PSF and AUTO photometry, respectively. Besides, a comparison with PAN-STARRS DR1 good quality photometry presents magnitude differences under 0.18\,mag in all bands.
Additional tests did not reveal any colour term in the photometric calibration nor dependence with the binning mode of observations nor the pixel position in the CCD.
Saturation and typical magnitude limits in the catalogue in each filter are 13.1--24.6\,mag in $g$, 13.6--24.5\,mag in $r$, 13.1--23.9\,mag in $i$, and 12.0--22.7\,mag in $z$, respectively.
We present two science cases aiming to prove the scientific capabilities of the catalogue. In the first case, we looked for cool dwarfs in the catalogue using a photometric criterium in the optical and {\it Gaia} DR2 astrometry. We identified 49 dwarfs of spectral types later than M0 according to their positions in the colour-magnitude diagram and their effective temperatures, with distances between 75.9 and 854.6\,pc, which need spectroscopic confirmation. Among them, we identified a new resolved binary system separated 3.84\,arcsec (1\,940.4\,AU) composed by a late K primary and a mid type M secondary. Additionally, we identified another new resolved binary system separated 1.24\,arcsec (247.7\,AU) with estimated spectral types M9.5 + L1/L2 dwarfs. We propose them for follow up and spectroscopic confirmation.
In the second case, we looked for Solar System Objects from their apparent linear motion and identified 59 known asteroids and four known comets plus 141 unknown objects or objects with poor accuracy in their orbital elements. For them, we provide 2\,828 detections that have been reported to the Minor Planet Center Database.
The catalogue includes celestial coordinates, sources and detections IDs, calibrated PSF and Kron photometry in the $griz$ Sloan bands, various morphometric measurements useful to assess the nature of the sources (point-like or extended), although we provide a point-like/extended classification and a point-like coefficient.
In addition, it contains two quality flags associated to the photometric coverage of the calibration using PAN-STARRS DR1 catalogue.
Instrumental magnitudes of \verb|MODEL| and aperture photometry, and epoch of observation are also included in the catalogue.
The user can access the raw and processed images through an url given in the catalogue.
New releases of the catalogue including the new public broadband images will be delivered over the life of the instrument. Also, a number of improvements and enhancements are planned for these future releases. One of the most important ones will be the building of stacked images to develop a much deeper catalogue. We also plan to improve the absolute astrometry of the catalogue by linking it to the {\it Gaia} DR2 catalogue. New versions of the Pan-Starrs will also be used for the photometric calibration.
Like in any other photometric survey, the complete absence of errors and problems in the catalogue cannot be guaranteed, in particular for detections close to the limiting magnitude. In these cases, users are strongly encouraged to download and check the associated images to assess the reliability of a given catalogue measurement. The {\it News} section of the catalogue website will contain a list of Frequently Asked Questions as well as a description of caveats that may arise with the scientific exploitation of the catalogue.
\section*{Acknowledgments}
H. Bouy acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 682903, P.I. H. Bouy), and from the French State in the framework of the "Investments for the future" Program, IdEx Bordeaux, reference ANR-10-IDEX-03-02.
This research has been financed by ASTERICS, a project supported by the European Commission Framework Programme Horizon 2020 Research and Innovation action under grant agreement n. 653477.
This research has been partially funded by the Spanish State Research Agency (AEI) Project No. ESP2017-87676-C5-1-R and No. MDM-2017-0737 Unidad de Excelencia ``Mar\'ia de Maeztu''- Centro de Astrobiolog\'ia (CSIC-INTA).
This publication makes use of VOSA, developed under the Spanish Virtual Observatory project supported by the Spanish MINECO through grant AyA2017-84089.
This research has made use of the SVO Filter Profile Service (http://svo2.cab.inta-csic.es/theory/fps/) supported from the Spanish MINECO through grant AYA2017-84089.
This research made use of the cross-match service, the SIMBAD database \citep{Wenger00}, VizieR catalogue access tool \citep{Och00}, "Aladin sky atlas" \citep{Bonn00, BF14} provided by CDS, Strasbourg, France.
This research has also made use of the TOPCAT \citep{Taylor05} and STILTS \citep{Taylor06}.
MCC and FJE acknowledge financial support from the Tec2Space-CM project (P2018/NMT-4291).
MM is funded by the European Space Agency under the research contract C4000122918.
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\section{Synthesizing Non-members}
\label{synthesizenonmembers}
\begin{figure}[!h]
\centering
\includegraphics[width=0.33\textwidth]{figs/revise/location_explore/compare_with_generate_figure.pdf}\\
\caption{Inference accuracy of the NN attack as the confidence score distortion budget increases on the Location dataset when synthesizing non-members for training the defense classifier (MemGuard-S).}
\label{fig:synthesize-app}
\end{figure}
When training the defense classifier, we can use $D_1$ as members and synthesize non-members based on $D_1$. For instance, for each data sample in $D_1$ and each of its feature, we keep the feature value with a probability 0.9 and randomly sample a value from the corresponding data domain for the feature with a probability 0.1, which synthesizes a non-member data sample. Then, we train the defense classifier using $D_1$ as members and the synthesized data samples as non-members.
\autoref{fig:synthesize-app}
shows the comparison results on the Location dataset (binary features), where MemGuard-S is the scenario where we synthesize the non-members for training the defense classifier. We observe that MemGuard and MemGuard-S achieve similar performance. Our results show that MemGuard does not necessarily need to split the training dataset in order to train the defense classifier.
\section{Conclusion and Future Work}
In this work, we propose MemGuard to defend against black-box membership inference attacks. MemGuard is the first defense that has formal utility-loss guarantees on the confidence score vectors predicted by the target classifier. MemGuard works in two phases. In Phase I, MemGuard leverages a new algorithm to find a carefully crafted noise vector to turn a confidence score vector into an adversarial example. The new algorithm considers the unique utility-loss constraints on the noise vector. In Phase II, MemGuard adds the noise vector to the confidence score vector with a certain probability, for which we derive an analytical solution. Our empirical evaluation results show that MemGuard can effectively defend against black-box membership inference attacks and outperforms existing defenses.
An interesting future work is to extend MemGuard to defend against other types of machine learning based inference attacks such as white-box membership inference attacks, website fingerprinting attacks, and side-channel attacks.
\section{Discussion and Limitations}
\label{discussion}
On one hand, machine learning can be used by attackers to perform automated inference attacks. On the other hand, machine learning has various vulnerabilities, e.g., \emph{adversarial examples}~\cite{CW17,PMJFCS16,PMGJCS17,SZSBEGF13,PMSW18,PMG16,KGB16,GSS15}. Therefore, attackers who rely on machine learning also share its vulnerabilities and we can exploit such vulnerabilities to defend against them. For instance, we can leverage adversarial examples to mislead attackers who use machine learning classifiers to perform automated inference attacks~\cite{JG19}. One key challenge in this research direction is how to extend existing adversarial example methods to address the unique challenges of privacy protection. For instance, how to achieve formal utility-loss guarantees.
In this work, we focus on membership inference attacks under the black-box setting, in which an attacker uses a binary classifier to predict a data sample to be a member or non-member of a target classifier's training dataset. In particular, the attacker's classifier takes a data sample's confidence score vector predicted by the target classifier as an input and predicts member or non-member. Our defense adds carefully crafted noise to a confidence score vector to turn it into an adversarial example, such that the attacker's classifier is likely to predict member or non-member incorrectly. To address the challenges of achieving formal utility-loss guarantees, e.g., 0 label loss and bounded confidence score distortion, we design new methods to find adversarial examples.
Other than membership inference attacks, many other attacks rely on machine learning classifiers, e.g., \emph{attribute inference attacks}~\cite{CAK12,GL162,JWZG17}, \emph{website fingerprinting attacks}~\cite{CZJJ12,JAADG14,WCNJG14,PNZE11,HWF09}, \emph{side-channel attacks}~\cite{ZJRR12}, \emph{location attacks}~\cite{BHPZ17,OTP17,PTC18,ZHRLPB18}, and \emph{author identification attacks}~\cite{NPGBSSS12,CYDHRGN18}. For instance, online social network users are vulnerable to {attribute inference attacks}, in which an attacker leverages a machine learning classifier to infer users' private attributes (e.g., gender, political view, and sexual orientation) using their public data (e.g., page likes) on social networks. The Facebook data privacy scandal in 2018\footnote{\url{https://bit.ly/2IDchsx}} is a notable example of attribute inference attack. In particular, Cambridge Analytica leveraged a machine learning classifier to automatically infer a large amount of Facebook users' various private attributes using their public page likes. Jia and Gong proposed AttriGuard~\cite{JG18}, which leverages adversarial examples to defend against attribute inference attacks. In particular, AttriGuard extends an existing adversarial example method to incorporate the unique challenges of privacy protection. The key difference between MemGuard and AttriGuard is that finding adversarial examples for confidence score vectors is subject to unique constraints, e.g., an adversarial confidence score vector should still be a probability distribution and the predicted label should not change. Such unique constraints require substantially different methods to find adversarial confidence score vectors.
Other studies have leveraged adversarial examples to defend against traffic analysis~\cite{ZHRZ19} and author identification~\cite{QMR19,MMJ18}. However, these studies did not consider formal utility-loss guarantees.
We believe it is valuable future work to extend MemGuard to defend against other machine learning based inference attacks such as website fingerprinting attacks, side-channel attacks, and membership inference attacks in the white-box setting. Again, a key challenge is how to achieve formal utility-loss guarantees with respect to certain reasonable utility-loss metrics.
Our MemGuard has a parameter $\epsilon$, which controls a tradeoff between membership privacy and confidence score vector distortion. The setting of $\epsilon$ may be dataset-dependent. One way to set $\epsilon$ is to leverage an inference accuracy vs. $\epsilon$ curve as shown in Figure~\ref{fix_k_query_infer_acc}. Specifically, given a dataset, we draw the inference accuracy vs. $\epsilon$ curves for various attack classifiers. Suppose we desire the inference accuracy to be less than a threshold. Then, we select the smallest $\epsilon$ such that the inference accuracies of all the evaluated attack classifiers are no larger than the threshold.
\section{Evaluation}
\label{evaluation_section}
\subsection{Experimental Setup}
\subsubsection{Datasets} We use three datasets that represent different application scenarios.
\myparatight{Location} This dataset was preprocessed from the Foursquare dataset\footnote{https://sites.google.com/site/yangdingqi/home/foursquare-dataset} and we obtained it from~\cite{SSSS17}.
The dataset has 5,010 data samples with 446 binary features, each of which represents whether a user visited a particular region or location type.
The data samples are grouped into $30$ clusters. This dataset represents a 30-class classification problem, where each cluster is a class.
\myparatight{Texas100} This dataset is based on the Discharge Data public use files published by the Texas
Department of State Health Services.\footnote{https://www.dshs.texas.gov/THCIC/Hospitals/Download.shtm} We obtained the preprocessed dataset from~\cite{SSSS17}.
The dataset has $67,330$ data samples with $6,170$ binary features.
These features represent the external causes of injury (e.g., suicide, drug misuse), the diagnosis, the procedures the patient underwent, and some generic information (e.g., gender, age, and race).
Similar to~\cite{SSSS17}, we focus on the $100$ most frequent procedures and the classification task is to predict a procedure for a patient using the patient's data. This dataset represents a 100-class classification problem.
\myparatight{CH-MNIST} This dataset is used for classification of different tissue types on histology tile from patients with colorectal cancer. The dataset contains $5,000$ images from $8$ tissues. The classification task is to predict tissue for an image, i.e., the dataset is a 8-class classification problem. The size of each image is $64\times 64$.
We obtained a preprocessed version from Kaggle.~\footnote{\url{https://www.kaggle.com/kmader/colorectal-histology-mnist}}.
\myparatight{Dataset splits}
For each dataset, we will train a target classifier, an attack classifier, and a defense classifier. Therefore, we split each dataset into multiple folds. Specifically, for the Location (or CH-MNIST) dataset, we randomly sample 4 disjoint sets, each of which includes 1,000 data samples. We denote them as ${D_1}$, ${D_2}$, ${D_3}$, and ${D_4}$, respectively. For the Texas100 dataset, we also randomly sample such 4 disjoint sets, but each set includes 10,000 data samples as the Texas100 dataset is around one order of magnitude larger. Roughly speaking, for each dataset, we use ${D_1}$, ${D_2}$, and ${D_3}$ to learn the target classifier, the attack classifier, and the defense classifier, respectively; and we use $D_1\cup{D_4}$ to evaluate the accuracy of the attack classifier. We will describe more details on how the sets are used when we use them.
\begin{table}[t]\renewcommand{\arraystretch}{1}
\centering
\caption{Neural network architecture of the target classifier for CH-MNIST.}
\begin{tabular}{|c|c|} \hline
Layer Type & Layer Parameters \\ \hline
\multicolumn{2}{|c|}{Input $64\times 64$} \\ \hline
Convolution& $32\times 3 \times 3$, strides=$(1, 1)$, padding=same \\
Activation& ReLU \\ \hline
Convolution& $32\times 3 \times 3$, strides=$(1, 1)$ \\
Activation& ReLU \\
Pooling& MaxPooling$(2\times 2)$ \\ \hline
Convolution& $32\times 3 \times 3$, strides=$(1, 1)$, padding=same \\
Activation& ReLU \\ \hline
Convolution& $32\times 3 \times 3$, strides=$(1, 1)$ \\
Activation& ReLU \\ \hline
Pooling& MaxPooling$(2\times 2)$ \\ \hline
Flatten& \\ \hline
Fully Connected& 512 \\ \hline
Fully Connected& 8 \\ \hline
Activation& softmax \\ \hline
\multicolumn{2}{|c|}{Output} \\ \hline
\end{tabular}
\label{architecture_ch_mnist}
\end{table}
\subsubsection{Target Classifiers} For the Location and Texas100 datasets, we use a fully-connected neural network with $4$ hidden layers as the target classifier. The number of neurons for the four layers are 1024, 512, 256, and 128, respectively. We use the popular activation function ReLU for the neurons in the hidden layers. The activation function in the output layer is softmax. We adopt the cross-entropy loss function and use Stochastic Gradient Descent (SGD) to learn the model parameters. We train $200$ epochs with a learning rate $0.01$, and we decay the learning rate by $0.1$ in the $150$th epoch for better convergence. For the CH-MNIST dataset, the neural network architecture of the target classifier is shown in~\autoref{architecture_ch_mnist}. Similarly, we also adopt the cross-entropy loss function and use SGD to learn the model parameters. We train $400$ epochs with a learning rate $0.01$ and decay the learning rate by $0.1$ in the $350$th epoch.
For each dataset, we use $D_1$ to train the target classifier.~\autoref{accuracy_of_target_classifier} shows the training and testing accuracies of the target classifiers on the three datasets, where the testing accuracy is calculated by using the target classifier to make predictions for the data samples that are not in $D_1$.
\subsubsection{Membership Inference Attacks}
\label{membershipattack}
In a membership inference attack, an attacker trains an attack classifier, which predicts \emph{member} or \emph{non-member} for a query data sample. The effectiveness of an attack is measured by the \emph{inference accuracy} of the attack classifier, where the inference accuracy is the fraction of data samples in $D_1\cup{D_4}$ that the attack classifier can correctly predict as member or non-member. In particular, data samples in $D_1$ are members of the target classifier's training dataset, while data samples in $D_4$ are non-members. We call the dataset $D_1\cup{D_4}$ \emph{evaluation dataset}. We consider two categories of state-of-the-art black-box membership inference attacks, i.e., \emph{non-adaptive attacks} and \emph{adaptive attacks}. In non-adaptive attacks, the attacker does not adapt its attack classifier based on our defense, while the attacker adapts its attack classifier based on our defense in adaptive attacks.
\begin{figure*}[!t]
\centering
\subfloat[Location]{\includegraphics[width=0.330\textwidth]{figs/revise/evaluate_attack/location_median_figure.pdf}}
\subfloat[Texas100]{\includegraphics[width=0.330\textwidth]{figs/revise/evaluate_attack/texas100_median_figure.pdf}}
\subfloat[CH-MNIST]{\includegraphics[width=0.330\textwidth]{figs/revise/evaluate_attack/ch_mnist_median_figure.pdf}}
\caption{Inference accuracies of different attacks as the confidence score distortion budget (i.e., $\epsilon$) increases.}
\label{fix_k_query_infer_acc}
\end{figure*}
\myparatight{Non-adaptive attacks} We consider the \emph{random guessing} attack and state-of-the-art attacks as follows.
{\bf Random guessing (RG) attack}. For any query data sample, this attack predicts it to be a member of the target classifier's training dataset with probability 0.5. The inference accuracy of the RG attack is 0.5.
\begin{table}
\centering
\caption{Training and testing accuracies of the target classifier on the three datasets.}
\begin{tabular}{|c|c|c|c|}
\hline
& Location & Texas100 & CH-MNIST \\
\hline
Training Accuracy & 100.0\% & 99.98\% & 99.0\% \\
\hline
Testing Accuracy & 60.32\% & 51.59\% & 72.0\% \\
\hline
\end{tabular}
\label{accuracy_of_target_classifier}
\end{table}
{\bf Neural Network (NN) attack~\cite{SSSS17,SZHBFB19}}. This attack assumes that the attacker knows the distribution of the target classifier's training dataset and the architecture of the target classifier. We further split the dataset $D_2$ into two halves denoted as $D_2'$ and $D_2''$, respectively. The attacker uses $D_2'$ to train a shadow classifier that has the same neural network architecture as the target classifier. After training the shadow classifier, the attacker calculates the confidence score vectors for the data samples in $D_2'$ and $D_2''$, which are members and non-members of the shadow classifier. Then, the attacker ranks each confidence score vector and treats the ranked confidence score vectors of members and non-members as a ``training dataset'' to train an attack classifier. The attack classifier takes a data sample's ranked confidence score vector as an input and predicts member or non-member. For all three datasets, we consider the attack classifier is a fully-connected neural network with three hidden layers, which have 512, 256, and 128 neurons, respectively. The output layer just has one neuron. The neurons in the hidden layers use the ReLU activation function, while the neuron in the output layer uses the sigmoid activation function. The attack classifier predicts member if and only if the neuron in the output layer outputs a value that is larger than 0.5. We train the attack classifier for 400 epochs with a learning rate 0.01 using SGD and decay the learning rate by 0.1 at the 300th epoch.
{\bf Random Forest (RF) attack}. This attack is the same as the NN attack except that RF attack uses random forest as the attack classifier, while NN uses a neural network as the attack classifier. We use scikit-learn with the default setting to learn random forest classifiers. We consider this RF attack to demonstrate that our defense mechanism is still effective even if the attack classifier and the defense classifier (a neural network) use different types of algorithms, i.e., the noise vector that evades the defense classifier can also evade the attack classifier even if the two classifiers use different types of algorithms.
{\bf NSH attack~\cite{NSH18}}. Nasr, Shokri, and Houmansadr~\cite{NSH18} proposed this attack, which we abbreviate as NSH. This attack uses multiple neural networks. One network operates on the confidence score vector. Another one operates on the label which is one hot encoded. Both networks are fully-connected and have the same number of input dimension, i.e., the number of classes of the target classifier. Specifically, NSH assumes the attacker knows some members and non-members of the target classifier's training dataset. In our experiments, we assume the attacker knows $30\%$ of data samples in $D_1$ (i.e., members) and 30\% of data samples in $D_4$ (i.e., non-members). The attacker uses these data samples to train the attack classifier. We adopt the neural network architecture in~\cite{NSH18} as the attack classifier. The remaining 70\% of data samples in $D_1$ and $D_4$ are used to calculate the inference accuracy of the attack classifier. We train the attack classifier for $400$ epochs with an initial learning rate $0.01$ and decay the learning rate by $0.1$ after $300$ epochs.
\myparatight{Adaptive attacks} We consider two attacks that are customized to our defense.
{\bf Adversarial training (NN-AT)}. One adaptive attack is to train the attack classifier via adversarial training, which was considered to be the most empirically robust method against adversarial examples so far~\cite{ACW18}. We adapt the NN attack using adversarial training and denote the adapted attack as NN-AT. Specifically, for each data sample in $D_2'$ and $D_2''$, the attacker calculates its confidence score vector using the shadow classifier. Then, the attacker uses the Phase I of our defense to find the representative noise vector and adds it to the confidence score vector to obtain a noisy confidence score vector. Finally, the attacker trains the attack classifier via treating the true confidence score vectors and their corresponding noisy versions of data samples in $D_2'$ and $D_2''$ as a training dataset.
{\bf Rounding (NN-R)}. Since our defense adds carefully crafted small noise to the confidence score vector, an adaptive attack is to \emph{round} each confidence score before using the attack classifier to predict member/non-member. Specifically, we consider the attacker rounds each confidence score to be one decimal and uses the NN attack. Note that rounding is also applied when training the NN attack classifier. We denote this attack NN-R.
\autoref{accuracy_without_defense} shows the inference accuracies of different attacks when our defense is not used. All attacks except RG have inference accuracies that are larger or substantially larger than 0.5.
\begin{table}
\centering
\caption{Inference accuracies of different attacks on the three datasets when our defense is not used.}
\begin{tabular}{|c|c|c|c|}
\hline
& Location & Texas100 & CH-MNIST \\
\hline
RG & 50.0\% & 50.0\% & 50.0\% \\
\hline
NN & 73.0\% & 68.9\% & 62.9\% \\
\hline
RF & 73.7\% & 67.3\% & 58.7\% \\
\hline
NSH & 81.1\% & 74.0\% & 58.4\% \\
\hline
NN-AT & 64.6\% & 68.3\% & 63.3\% \\
\hline
NN-R & 72.9\% & 69.2\% & 63.0\% \\
\hline
\end{tabular}
\label{accuracy_without_defense}
\end{table}
\subsubsection{Defense Setting} \label{defensesetting}
In our defense, we need to specify a defense classifier and the parameters in Algorithm~\autoref{algorithml1}.
\myparatight{Defense classifier} The defender itself trains a classifier to perform membership inference. We consider the defense classifier is a neural network. However, since the defender does not know the attacker's attack classifier, we
assume the defense classifier and the attack classifier use different neural network architectures. Specifically, we consider three different defense classifiers in order to study the impact of defense classifier on MemGuard. The three defense classifiers are fully-connected neural networks with 2, 3, and 4 hidden layers, respectively. The hidden layers of the three defense classifiers have (256, 128), (256, 128, 64), and (512, 256, 128, 64) neurons, respectively.
The output layer has just one neuron. The activation function for the neurons in the hidden layers is $ReLU$, while the neuron in the output layer uses the sigmoid activation function.
Unless otherwise mentioned, we use the defense classifier with 3 hidden layers.
The defender calculates the confidence score vector for each data sample in $D_1$ and $D_3$ using the target classifier. The confidence score vectors for data samples in $D_1$ and $D_3$ have labels ``member'' and ``non-member'', respectively. The defender treats these confidence score vectors as a training dataset to learn a defense classifier, which takes a confidence score vector as an input and predicts member or non-member. We train a defense classifier for $400$ epochs with a learning rate $0.001$. We note that we can also synthesize data samples based on $D_1$ as non-members (Appendix~\ref{synthesizenonmembers} shows details).
\myparatight{Parameter setting} We set $max\_iter=300$ and $\beta=0.1$ in Algorithm~\autoref{algorithml1}. We found that once $max\_iter$ is larger than some threshold, MemGuard's effectiveness does not change. Since we aim to find representative noise vector that does not change the predicted label, we assign a relatively large value to $c_2$, which means that the objective function has a large value if the predicted label changes (i.e., the loss function $L_2$ is non-zero). In particular, we set $c_2=10$. Our Algorithm~\autoref{algorithml1} searches for a large $c_3$ and we set the initial value of $c_3$ to be 0.1. We also compare searching $c_2$ with searching $c_3$.
\subsection{Experimental Results}
\label{experimentalresults}
\begin{figure*}[!t]
\centering
\subfloat[Location, without defense]{\includegraphics[width=0.330\textwidth]{figs/revise/entropy/location_origin_distribution.pdf}}
\subfloat[Texas100, without defense]{\includegraphics[width=0.330\textwidth]{figs/revise/entropy/texas100_origin_distribution.pdf}}
\subfloat[CH-MNIST, without defense]{\includegraphics[width=0.330\textwidth]{figs/revise/entropy/ch_mnist_origin_distribution.pdf}}
\subfloat[Location, with defense]{\includegraphics[width=0.330\textwidth]{figs/revise/entropy/location_defense_distribution.pdf}}
\subfloat[Texas100, with defense]{\includegraphics[width=0.330\textwidth]{figs/revise/entropy/texas100_defense_distribution.pdf}}
\subfloat[CH-MNIST, with defense]{\includegraphics[width=0.330\textwidth]{figs/revise/entropy/ch_mnist_defense_distribution.pdf}}
\caption{Distribution of the normalized entropy of the confidence score vectors for members and non-members of the target classifier. Figures on the upper side are results without our defense, and figures on the lower side are results with our defense.}
\label{entropy_distribution}
\end{figure*}
\myparatight{MemGuard is effective} ~\autoref{fix_k_query_infer_acc} shows the inference accuracies of different attacks as the confidence score distortion budget increases on the three datasets. Since we adopt the expected $L_1$-norm of the noise vector to measure the confidence score distortion, the confidence score distortion is in the range [0, 2]. Note that our defense is guaranteed to achieve 0 label loss as our Algorithm~\ref{algorithml1} guarantees that the predicted label does not change when searching for the representative noise vector. We observe that our MemGuard can effectively defend against membership inference attacks, i.e., the inference accuracies of all the evaluated attacks decrease as our defense is allowed to add larger noise to the confidence score vectors. For instance, on Location, when our defense is allowed to add noise whose expected $L_1$-norm is around 0.8, our defense can reduce all the evaluated attacks to the random guessing (RG) attack; on CH-MNIST,
our defense can reduce the NSH attack (or the remaining attacks) to random guessing when allowed to add noise whose expected $L_1$-norm is around 0.3 (or 0.7).
\myparatight{Indistinguishability between the confidence score vectors of members and non-members} We follow previous work~\cite{NSH18} to study the distribution of confidence score vectors of members vs. non-members of the target classifier. Specifically, given a confidence score vector $\mathbf{s}$, we compute its \emph{normalized entropy} as follows:
\begin{align}
\text{\bf Normalized entropy: } -\frac{1}{\log k}\sum_{j}s_j\log(s_j),
\end{align}
where $k$ is the number of classes in the target classifier.
Figure~\ref{entropy_distribution} shows the distributions of the normalized entropy of the confidence score vectors for members (i.e., data samples in $D_1$) and non-members (i.e., data samples in $D_4$) of the target classifier, where we set the confidence score distortion budget $\epsilon$ to be 1 when our defense is used. The gap between the two curves in a graph corresponds to the information leakage of the target classifier's training dataset. Our defense substantially reduces such gaps. Specifically, the \emph{maximum gap} between the two curves (without defense vs. with defense) is (0.27 vs. 0.11), (0.41 vs. 0.05), and (0.30 vs. 0.06) on the Location, Texas100, and CH-MNIST datasets, respectively. Moreover, the \emph{average gap} between the two curves (without defense vs. with defense) is (0.062 vs. 0.011), (0.041 vs. 0.005), and (0.030 vs. 0.006) on the three datasets, respectively.
\begin{figure}[!t]
\centering
\subfloat[Searching $c_3$]{\includegraphics[width=0.24\textwidth]{figs/revise/location_explore/compare_with_c2_figure.pdf}\label{searchc3-main}}
\subfloat[Searching $c_2$]{\includegraphics[width=0.24\textwidth]{figs/revise/location_explore/compare_with_search_compare_figure.pdf}\label{searchc2-main}}
\caption{Inference accuracy of the NN attack as the confidence score distortion budget increases on the Location dataset when searching $c_3$ or $c_2$.}
\end{figure}
\myparatight{Searching $c_2$ vs. searching $c_3$} Figure~\ref{searchc3-main} shows the inference accuracy of the NN attack as the confidence score distortion budget increases when fixing $c_2$ to different values and searching $c_3$. Figure~\ref{searchc2-main} shows the results when fixing $c_3$ and searching $c_2$. We observe that MemGuard is insensitive to the setting of $c_2$ when searching $c_3$. Specifically, MemGuard has almost the same effectiveness when fixing $c_2$ to different values, i.e., the different curves overlap in Figure~\ref{searchc3-main}. This is because when our Phase I stops searching the noise vector, the predicted label is preserved, which means that the loss function $L_2$ is 0. However, MemGuard is sensitive to the setting of $c_3$ when searching $c_2$. Specifically, when fixing $c_3$ to be 0.1, searching $c_2$ achieves the same effectiveness as searching $c_3$. However, when fixing $c_3$ to be 1.0, searching $c_2$ is less effective. Therefore, we decided to search $c_3$ while fixing $c_2$.
\myparatight{Impact of defense classifiers} Figure~\ref{defenseclassifier} shows the inference accuracy of the NN attack as the confidence score distortion budget increases on the Location dataset when using different defense classifiers. We observe that MemGuard has similar effectiveness for different defense classifiers, which means that our carefully crafted noise vectors can transfer between classifiers.
\myparatight{MemGuard outperforms existing defenses} We compare with state-of-the-art defenses including $L_2$-Regularizer~\cite{SSSS17}, Min-Max Game~\cite{NSH18}, Dropout~\cite{SZHBFB19}, Model Stacking~\cite{SZHBFB19}, and DP-SGD~\cite{ACGMMTZ16}. Each compared defense (except Model Stacking) has a hyperparameter to control the privacy-utility tradeoff. For instance, the hyperparameter that balances between the loss function and the $L_2$ regularizer in $L_2$-Regularizer, the hyperparameter that balances between the loss function and the adversarial regularizer in Min-Max Game, the dropout rate in Dropout, the privacy budget in DP-SGD, and $\epsilon$ in MemGuard.
We also compare with MemGuard-Random in which we use the \emph{Random} method (refer to Section~\ref{sec:phaseI}) to generate the noise vector in Phase I.
\begin{figure}[!t]
\centering
{\includegraphics[width=0.33\textwidth]{figs/revise/location_explore/compare_with_defense_model_figure.pdf}}
\caption{Inference accuracy of the NN attack as the confidence score distortion budget increases on the Location dataset when using different defense classifiers.}
\label{defenseclassifier}
\end{figure}
Before deploying any defense, we use the undefended target classifier to compute the confidence score vector for each data sample in the evaluation dataset $D_1\cup D_4$. For each defense and a given hyperparameter, we apply the defense to the target classifier and use the defended target classifier to compute the confidence score vector for each data sample in $D_1\cup D_4$. Then, we compute the confidence score distortion for each data sample and obtain the \emph{average confidence score distortion} on the evaluation dataset $D_1\cup D_4$. Moreover, we compute the inference accuracy of the attack classifier (we consider NN in these experiments) on the evaluation dataset after the defense is used. Therefore, for each defense and a given hyperparameter, we can obtain a pair (inference accuracy, average confidence score distortion). Via exploring different hyperparameters, we can obtain a set of such pairs for each defense. Then, we plot these pairs on a graph, which is shown in~\autoref{fix_infer_acc}.
Specifically, we tried the hyperparameter of $L_2$-Regularizer in the range $[0,0.05]$ with a step size $0.005$, $0.001$, and $0.005$ for Location, Texas100, and CH\_MNIST datasets, respectively.
We tried the hyperparameter of Min-Max Game in the range $[0,3]$ with a step size $0.5$.
We tried the dropout rate of Dropout in the range $[0,0.9]$ with a step size $0.1$.
We use a publicly available implementation\footnote{https://github.com/tensorflow/privacy} of DP-SGD. We tried the parameter $noise\_multiplier$ that controls the privacy budget in the range $[0,0.2]$ with a step size $0.05$.
We tried $[0,0.1,0.3,0.5,0.7,1.0]$ as the $\epsilon$ in MemGuard and MemGuard-Random.
\begin{table}
\centering
\caption{Results of Model Stacking.}
\begin{tabular}{|c|c|c|c|}
\hline
& Location & Texas100 & CH-MNIST \\
\hline
Inference Acc. & 50.0\% & 50.8\% & 50.0\% \\
\hline
Average Distortion & 1.63 & 1.28 & 0.81 \\
\hline
Label Loss & 56.3\% & 37.9\% & 18.3\% \\
\hline
\end{tabular}
\label{accuracy_of_target_model}
\end{table}
\begin{figure*}[!t]
\vspace{-2mm}
\centering
\subfloat[Location]{\includegraphics[width=0.330\textwidth]{figs/revise/comparison_050819/distortion/location_epsilon_acc_figure_sgd.pdf}}
\subfloat[Texas100]{\includegraphics[width=0.330\textwidth]{figs/revise/comparison_050819/distortion/texas100_epsilon_acc_figure_sgd.pdf}}
\subfloat[CH-MNIST]{\includegraphics[width=0.330\textwidth]{figs/revise/comparison_050819/distortion/ch_mnist_epsilon_acc_figure_sgd.pdf}}
\caption{Inference accuracy vs. average confidence score distortion of the compared defenses. Our MemGuard achieves the best privacy-utility tradeoff.}
\label{fix_infer_acc}
\vspace{-1mm}
\end{figure*}
\begin{figure*}[!t]
\centering
\subfloat[Location]{\includegraphics[width=0.330\textwidth]{figs/revise/comparison_050819/label/location_label_loss_figure.pdf}}
\subfloat[Texas100]{\includegraphics[width=0.330\textwidth]{figs/revise/comparison_050819/label/texas100_label_loss_figure.pdf}}
\subfloat[CH-MNIST]{\includegraphics[width=0.330\textwidth]{figs/revise/comparison_050819/label/ch_mnist_label_loss_figure.pdf}}
\caption{Inference accuracy vs. label loss of the compared defenses. Both MemGuard-Random and MemGuard achieve 0 label loss, while the other defenses incur large label losses in order to substantially reduce the attacker's inference accuracy.}
\label{labellossinfer}
\vspace{-1mm}
\end{figure*}
Our results show that MemGuard achieves the best privacy-utility tradeoff. In particular, given the same average confidence score distortion, MemGuard achieves the smallest inference accuracy. According to the authors of Model Stacking, it does not have a hyperparameter to easily control the privacy-utility tradeoff. Therefore, we just obtain one pair of (inference accuracy, average confidence score distortion) and~\autoref{accuracy_of_target_model} shows the results. Model Stacking reduces the inference accuracy to be close to 0.5, but the utility loss is intolerable.
Similarly, we can obtain a set of pairs (inference accuracy, label loss) for the compared defenses and~\autoref{labellossinfer} shows inference accuracy vs. label loss on the three datasets. Label loss is the fraction of data samples in the evaluation dataset whose predicted labels are changed by a defense. MemGuard-Random and MemGuard achieve 0 label loss. However, other defenses incur large label losses in order to substantially reduce the attacker's inference accuracy.
\section{Introduction}
\label{section:introduction}
Machine learning (ML) is transforming many aspects of our society. We consider a model provider deploys an ML classifier (called \emph{target classifier}) as a black-box software or service, which returns a \emph{confidence score vector} for a query data sample from a user. The confidence score vector is a probability distribution over the possible labels and the label of the query data sample is predicted as the one that has the largest confidence score. Multiple studies have shown that such black-box ML classifier is vulnerable to \emph{membership inference attacks}~\cite{SSSS17,NSH19,SZHBFB19,SSM19}.
Specifically, an attacker trains a binary classifier, which takes a data sample's confidence score vector predicted by the target classifier as an input and predicts whether the data sample is a \emph{member} or \emph{non-member} of the target classifier's training dataset. Membership inference attacks pose severe privacy and security threats to ML. In particular, in application scenarios where the training dataset is sensitive (e.g., biomedical records and location traces), successful membership inference leads to severe privacy violations. For instance, if an attacker knows her victim's data is used to train a medical diagnosis classifier,
then the attacker can directly infer the victim's health status. Beyond privacy, membership inference also damages the model provider's intellectual property of the training dataset as collecting and labeling the training dataset may require lots of resources.
Therefore, defending against membership inference attacks is an urgent research problem and multiple defenses~\cite{SSSS17,NSH18,SZHBFB19} have been explored. A major reason why membership inference attacks succeed is that the target classifier is overfitted. As a result, the confidence score vectors predicted by the target classifier are distinguishable for members and non-members of the training dataset. Therefore, state-of-the-art defenses~\cite{SSSS17,NSH18,SZHBFB19} essentially regularize the training process of the target classifier to reduce overfitting and the gaps of the confidence score vectors between members and non-members of the training dataset. For instance, $L_2$ regularization~\cite{SSSS17}, min-max game based adversarial regularization~\cite{NSH18}, and dropout~\cite{SZHBFB19} have been explored to regularize the target classifier. Another line of defenses~\cite{CMS11,KST12,INSTTW19,SCS13,BST14,WYX17,ACGMMTZ16,YLPGT19} leverage differential privacy~\cite{DMNS06} when training the target classifier.
Since tampering the training process has no guarantees on the confidence score vectors, these defenses have no formal utility-loss guarantees on the confidence score vectors. Moreover, these defenses achieve suboptimal tradeoffs between the membership privacy of the training dataset and utility loss of the confidence score vectors. For instance, Jayaraman and Evans~\cite{JE14} found that existing differentially private machine learning methods rarely offer acceptable privacy-utility tradeoffs for complex models.
\myparatight{Our work} In this work, we propose \emph{MemGuard}, the first defense with formal utility-loss guarantees against membership inference attacks under the black-box setting. Instead of tampering the training process of the target classifier, MemGuard randomly adds noise to the confidence score vector predicted by the target classifier for any query data sample.
MemGuard can be applied to an existing target classifier without retraining it. Given a query data sample's confidence score vector, MemGuard aims to achieve two goals: 1) the attacker's classifier is inaccurate at inferring member or non-member for the query data sample after adding noise to the confidence score vector, and 2) the utility loss of the confidence score vector is bounded. Specifically, the noise should not change the predicted label of the query data sample, since even 1\% loss of the label accuracy may be intolerable in some critical applications such as finance and healthcare. Moreover, the confidence score distortion introduced by the noise should be bounded by a budget since a confidence score vector intends to tell a user more information beyond the predicted label. We formulate achieving the two goals as solving an optimization problem. However, it is computationally challenging to solve the optimization problem as the noise space is large. To address the challenge, we propose a two-phase framework to approximately solve the problem.
We observe that an attacker uses an ML classifier to predict member or non-member and classifier can be misled by \emph{adversarial examples}~\cite{CW17,PMJFCS16,PMGJCS17,SZSBEGF13,PMSW18,PMG16,KGB16,GSS15}. Therefore, in Phase I, MemGuard finds a carefully crafted noise vector that can turn the confidence score vector into an adversarial example. Specifically, MemGuard aims to find a noise vector such that the attacker's classifier is likely to make a random guessing at inferring member or non-member based on the noisy confidence score vector. Since the defender does not know the attacker's classifier as there are many choices, the defender itself trains a classifier for membership inference and crafts the noise vector based on its own classifier. Due to \emph{transferability}~\cite{SZSBEGF13,KGB16,LCLS16,PMG16} of adversarial examples, the noise vector that misleads the defender's classifier is likely to also mislead the attacker's classifier. The adversarial machine learning community has developed many algorithms (e.g.,~\cite{CW17,PMJFCS16,GSS15,MMSTV18,KGB16,MFF16,TKPGBM17,MFFF17}) to find adversarial noise/examples. However, these algorithms are insufficient for our problem because they did not consider the unique constraints on utility loss of the confidence score vector. Specifically, the noisy confidence score vector should not change the predicted label of the query data sample and should still be a probability distribution. To address this challenge, we design a new algorithm to find a small noise vector that satisfies the utility-loss constraints.
In Phase II, MemGuard adds the noise vector found in Phase I to the true confidence score vector with a certain probability. The probability is selected such that the expected confidence score distortion is bounded by the budget and the defender's classifier is most likely to make random guessing at inferring member or non-member. Formally, we formulate finding this probability as solving an optimization problem and derive an analytical solution for the optimization problem.
We evaluate MemGuard and compare it with state-of-the-art defenses~\cite{SSSS17,NSH18,SZHBFB19,ACGMMTZ16} on three real-world datasets. Our empirical results show that MemGuard can effectively defend against state-of-the-art black-box membership inference attacks~\cite{NSH19,SZHBFB19}. In particular, as MemGuard is allowed to add larger noise (we measure the magnitude of the noise using its $L_1$-norm), the inference accuracies of all evaluated membership inference attacks become smaller. Moreover, MemGuard achieves better privacy-utility tradeoffs than state-of-the-art defenses. Specifically, given the same average confidence score distortion, MemGuard reduces the attacker's inference accuracy at inferring member/non-members by the most.
In summary, our key contributions are as follows:
\begin{itemize}
\item We propose MemGuard, the first defense with formal utility-loss guarantees against membership inference attacks under the black-box setting.
\item We propose a new algorithm to find a noise vector that satisfies the unique utility-loss constraints in Phase I of MemGuard. Moreover, in Phase II, we derive an analytical solution of the probability with which MemGuard adds the noise vector to the confidence score vector.
\item We evaluate MemGuard on three real-world datasets. Our results show that MemGuard is effective and outperforms existing defenses.
\end{itemize}
\section{Our M\lowercase{em}G\lowercase{uard}}
\subsection{Overview}
Finding the randomized noise addition mechanism is to solve the optimization problem in~\autoref{originalproblem}. We consider two scenarios depending on whether $g(\mathbf{s})$ is 0.5 or not.
\myparatight{Scenario I} In this scenario, $g(\mathbf{s})=0.5$. For such scenario, it is easy to solve the optimization problem in~\autoref{originalproblem}. Specifically, the mechanism that adds the noise vector $\mathbf{0}$ with probability 1 is the optimal randomized noise addition mechanism, with which the objective function has a value of 0.
\myparatight{Scenario II}
In this scenario, $g(\mathbf{s})$ is not 0.5.
The major challenge to solve the optimization problem in this scenario is that the randomized noise addition mechanism is a probability distribution over the continuous noise space for a given true confidence score vector. The noise space consists of the noise vectors that satisfy the four constraints of the optimization problem. As a result, it is challenging to represent the probability distribution and solve the optimization problem. To address the challenge, we observe that the noise space can be divided into two groups depending on the output of the defense classifier's decision function $g$.
Specifically, for noise vectors in one group, if we add any of them to the true confidence score vector, then the decision function $g$ outputs 0.5 as the probability of being member. For noise vectors in the other group, if we add any of them to the true confidence score vector, then the decision function $g$ outputs a probability of being member that is not 0.5.
Based on this observation, we propose a \emph{two-phase framework} to approximately solve the optimization problem. Specifically, in Phase I, for each noise group, we find the noise vector with minimum confidence score distortion (i.e., $d(\mathbf{s},\mathbf{s}+\mathbf{n})$ is minimized) as a \emph{representative} noise vector for the noise group. We select the noise vector with minimum confidence score distortion in order to minimize the confidence score distortion. Since $g(\mathbf{s})\neq 0.5$, the selected representative noise vector for the second noise group is $\mathbf{0}$. We denote by $\mathbf{r}$ the selected representative noise vector for the first noise group. In Phase II, we assume the randomized noise addition mechanism is a probability distribution over the two representative noise vectors instead of the overall noise space. Specifically, the defender adds the representative noise vector $\mathbf{r}$ to the true confidence score vector with a certain probability and does not add any noise with the remaining probability.
Next, we introduce our Phase I and Phase II.
\subsection{Phase I: Finding $\mathbf{r}$}
\label{sec:phaseI}
\myparatight{Finding $\mathbf{r}$ as solving an optimization problem} Our goal essentially is to find a noise vector $\mathbf{r}$ such that 1) the utility loss of the confidence score vector is minimized and 2) the decision function $g$ outputs 0.5 as the probability of being member when taking the noisy confidence score vector as an input. Formally, we find such noise vector via solving the following optimization problem:
\begin{align}
\label{optimization_p_1}
\min_{\mathbf{r}}&\ d(\mathbf{s},\mathbf{s}+\mathbf{r}) \\
\label{prediction_label}
\text{subject to: } & \argmax_j \{s_j+r_j\} =\argmax_j \{s_j\} \\
\label{noise_goal_constraint}
& g(\mathbf{s}+\mathbf{r})=0.5 \\
\label{probability_constraint_1}
&s_j+r_j \geq 0,\forall j \\
\label{probability_constraint_2}
&\sum_j r_j=0,
\end{align}
where $\mathbf{s}$ is the true confidence score vector, the objective function means that the confidence score distortion is minimized, the first constraint means that the noise does not change the predicted label of the query data sample, the second constraint means that the defense classifier's decision function outputs 0.5 (i.e., the defense classifier's prediction is random guessing), and the last two constraints mean that the noisy confidence score vector is still a probability distribution.
Solving the optimization problem in~\autoref{optimization_p_1} can be viewed as finding an \emph{adversarial example} to evade the defense classifier. In particular, $\mathbf{s}$ is a normal example and $\mathbf{s+r}$ is an adversarial example. The adversarial machine learning community has developed many algorithms (e.g.,~\cite{CW17,PMJFCS16,GSS15,MMSTV18,KGB16,MFF16,TKPGBM17,MFFF17}) to find adversarial examples. However, these algorithms are insufficient to our problem because they did not consider the unique challenges of privacy protection. In particular, they did not consider the utility-loss constraints, i.e., the constraints in~\autoref{prediction_label},~\autoref{probability_constraint_1}, and~\autoref{probability_constraint_2}.
One naive method (we call it \emph{Random}) to address the challenges is to generate a {random} noise vector that satisfies the utility-loss constraints. In particular, we can generate a random vector $\mathbf{r}^{\prime}$ whose entries are non-negative and sum to 1. For instance, we first sample a number ${r'_1}$ from the interval [0,1] uniformly at random as the first entry. Then, we sample a number ${r'_2}$ from the interval [0, 1-${r'_1}$] uniformly at random as the second entry. We repeat this process until the last entry is 1 minus the sum of the previous entries. Then, we exchange the largest entry of $\mathbf{r}^{\prime}$ to the position $j$ to satisfy the constraint~\ref{prediction_label}. Finally, we treat $\mathbf{r}=\mathbf{r}^{\prime}-\mathbf{s}$ as the noise vector, which is a solution to the optimization problem in~\autoref{optimization_p_1}. However, as we will show in experiments, this Random method achieves suboptimal privacy-utility tradeoffs because the noise vector is not optimized and it is challenging to satisfy the constraint~\autoref{probability_constraint_1}.
We propose to solve the optimization problem via change of variables and adding the constraints to the objective function.
\myparatight{Eliminating the constraints on probability distribution via change of variables} Since we consider the target classifier to be a neural network, whose output layer is a softmax layer, the true confidence score vector $\mathbf{s}$ is a {softmax function} of some vector $\mathbf{z}$. The vector $\mathbf{z}$ is the output of the neurons in the second-to-last layer of the neural network and is often called \emph{logits} of the neural network. Formally, we have:
\begin{align}
\mathbf{s}&=softmax(\mathbf{z}).
\end{align}
Moreover, we model the noisy confidence score vector as follows:
\begin{align}
\mathbf{s}+\mathbf{r}&=softmax(\mathbf{z}+\mathbf{e}),
\end{align}
where $\mathbf{e}$ is a new vector variable. For any value of $\mathbf{e}$, the noisy confidence score vector $\mathbf{s}+\mathbf{r}$ is a probability distribution, i.e., the constraints in~\autoref{probability_constraint_1} and~\autoref{probability_constraint_2} are satisfied. Therefore, in the optimization problem in~\autoref{optimization_p_1}, we change the true confidence score vector $\mathbf{s}$ as $softmax(\mathbf{z})$ and change the variable $\mathbf{r}$ as $softmax(\mathbf{z}+\mathbf{e}) - softmax(\mathbf{z})$. Then, we obtain the following optimization problem:
\begin{align}
\label{l_1_loss_constraints}
\min_{\mathbf{e}}&\ d(softmax(\mathbf{z}),softmax(\mathbf{z}+\mathbf{e})) \\
\label{constraints_1}
\text{subject to: } & \argmax_j \{z_j+e_j\} =\argmax_j \{z_j\} \\
\label{constraints_2}
&g(softmax(\mathbf{z}+\mathbf{e}))= 0.5.
\end{align}
After solving $\mathbf{e}$ in the above optimization problem,
we can obtain the noise vector $\mathbf{r}$ as follows:
\begin{align}
\label{noise_equation}
\mathbf{r}=softmax(\mathbf{z}+\mathbf{e})-softmax(\mathbf{z}).
\end{align}
The optimization problem without the constraints on probability distribution is still challenging to solve because the remaining two constraints are highly nonlinear. To address the challenge, we turn the constraints into the objective function.
\myparatight{Turning the constraint in~\autoref{constraints_2} into the objective function} We consider the defender's binary defense classifier is a neural network whose output layer has a single neuron with sigmoid activation function. Therefore, we have:
\begin{align}
g(softmax(\mathbf{z}+\mathbf{e}))=\frac{1}{1+\exp(-h(softmax(\mathbf{z}+\mathbf{e})))},
\end{align}
where $h(softmax(\mathbf{z}+\mathbf{e}))$ is the output of the neuron in the second-to-last layer of the defense classifier when the defense classifier takes the noisy confidence score vector $softmax(\mathbf{z}+\mathbf{e})$ as an input. In other words, $h$ is the logit of the defense classifier.
$g(softmax(\mathbf{z}+\mathbf{e}))=0.5$ implies $h(softmax(\mathbf{z}+\mathbf{e}))=0$. Therefore, we transform the constraint in~\autoref{constraints_2} to the following loss function:
\begin{align}
L_1=|h(softmax(\mathbf{z}+\mathbf{e}))|,
\end{align}
where $L_1$ is small when $h(softmax(\mathbf{z}+\mathbf{e}))$ is close to 0.
\begin{algorithm}[t]
\caption{Phase I of MemGuard}
\begin{algorithmic}[1]
\REQUIRE $\mathbf{z}$, $max\_iter$, $c_2$, $c_3$, and $\beta$ (learning rate). \\
\ENSURE $\mathbf{e}$ \\
\STATE //Predicted label \;
\STATE $l=\argmax_{j}\{z_j\}$
\WHILE {$True$} \;
\STATE //A new iteration to search $c_3$ \;
\STATE $\mathbf{e}=\mathbf{0}$
\STATE $\mathbf{e}'=\mathbf{e}$
\STATE $i=1$
\WHILE {$i<max\_iter$ and ($\argmax_{j}\{z_j+e_j\}\neq l$ or $h(softmax(\mathbf{z}))\cdot h(softmax(\mathbf{z}+\mathbf{e})) > 0$)} \;
\label{conditioncheck}
\STATE //Gradient descent with normalized gradient \;
\STATE $\mathbf{u}=\frac{\partial L}{\partial \mathbf{e}}$
\STATE $\mathbf{u}=\mathbf{u}/||\mathbf{u}||_2$
\STATE $\mathbf{e}=\mathbf{e}-\beta \cdot \mathbf{u}$
\STATE $i=i+1$
\ENDWHILE \;
\STATE //Return the vector in the previous iteration if the predicted label changes or the sign of $h$ does not change in the current iteration
\IF{$\argmax_{j}\{z_j+e_j\}\neq l$ or $h(softmax(\mathbf{z}))\cdot h(softmax(\mathbf{z}+\mathbf{e})) > 0$} \;
\RETURN $\mathbf{e}'$
\ENDIF \;
\STATE $c_3=10\cdot c_3$
\ENDWHILE \;
\end{algorithmic}
\label{algorithml1}
\end{algorithm}
\myparatight{Turning the constraint in~\autoref{constraints_1} into the objective function}
We denote by $l$ the predicted label for the query data sample, i.e., $l=argmax_{j}\{s_j\}=\argmax_{j}\{z_j\}$.
The constraint in~\autoref{constraints_1} means that $z_l + e_l$ is the largest entry in the vector $\mathbf{z+e}$. Therefore, we enforce the inequality constraint
$z_l + e_l \geq max_{j|j\neq l}\{z_j+e_j\}$. Moreover, we further transform the inequality constraint to the following loss function:
\begin{equation}
L_2=\text{ReLU}(-z_l-e_l+max_{j|j\neq l}\{z_j+e_j\}),
\end{equation}
where the function ReLU is defined as ReLU$(v)$=$\max\{0, v\}$.
The loss function $L_2$ is 0 if the inequality $z_l + e_l \geq max_{j|j\neq l}\{z_j+e_j\}$ holds.
\myparatight{Unconstrained optimization problem} After transforming the constraints into the objective function, we have the following unconstrained optimization problem:
\begin{align}
\min_{\mathbf{e}}\ L=L_1+c_2\cdot L_2 + c_3\cdot L_3,
\end{align}
where $L_3=d(softmax(\mathbf{z}),softmax(\mathbf{z}+\mathbf{e}))$, while $c_2$ and $c_3$ balance between the three terms.
\myparatight{Solving the unconstrained optimization problem} We design an algorithm based on gradient descent to solve the unconstrained optimization problem. Algorithm~\autoref{algorithml1} shows our algorithm. Since we aim to find a noise vector that has a small confidence score distortion, we iteratively search a large $c_3$. For each given $c_3$, we use gradient descent to find $\mathbf{e}$ that satisfies the constraints in \autoref{constraints_1} and~\autoref{constraints_2}. The process of searching $c_3$ stops when we cannot find a vector $\mathbf{e}$ that satisfies the two constraints. Specifically, given $c_2$, $c_3$, and a learning rate $\beta$, we iteratively update the vector variable $\mathbf{e}$ (i.e., the inner while loop in Algorithm~\autoref{algorithml1}). Since we transform the constraints in \autoref{constraints_1} and~\autoref{constraints_2} into the objective function, there is no guarantee that they are satisfied during the iterative gradient descent process. Therefore, in each iteration of gradient descent, we check whether the two constraints are satisfied (i.e., Line~\autoref{conditioncheck} in Algorithm~\autoref{algorithml1}). Specifically, we continue the gradient descent process when the predicted label changes or the sign of the logit $h$ does not change. In other words, we stop the gradient descent process when both constraints are satisfied. We use $h(softmax(\mathbf{z}))\cdot h(softmax(\mathbf{z}+\mathbf{e})) \leq 0$ to approximate the constraint in~\autoref{constraints_2}. In particular, the constraint in~\autoref{constraints_2} is equivalent to $h(softmax(\mathbf{z}+\mathbf{e}))=0$. Once we find a vector $\mathbf{e}$ such that $h(softmax(\mathbf{z}))$ and $h(softmax(\mathbf{z}+\mathbf{e}))$ have different signs (e.g., $h(softmax(\mathbf{z})) >0$ and $h(softmax(\mathbf{z}+\mathbf{e}))<0$), $h(softmax(\mathbf{z}+\mathbf{e}))$ just crosses 0 and should be close to 0 since we use a small learning rate. Note that we could also iteratively search $c_2$, but it is computationally inefficient to search both $c_2$ and $c_3$.
\subsection{Phase II}
After Phase I, we have two representative noise vectors. One is $\mathbf{0}$ and the other is $\mathbf{r}$. In Phase II, we assume the randomized noise addition mechanism is a probability distribution over the two representative noise vectors instead of the entire noise space. Specifically, we assume that the defender picks the representative noise vectors $\mathbf{r}$ and $\mathbf{0}$ with probabilities $p$ and $1-p$, respectively; and the defender adds the picked representative noise vector to the true confidence score vector. With such simplification, we can simplify the optimization problem in~\autoref{originalproblem} to the following optimization problem:
\begin{align}
p=\argmin_{p}& |p\cdot g(\mathbf{s}+\mathbf{r}) + (1-p)\cdot g(\mathbf{s+0})-0.5| \\
\text{subject to: }&p\cdot d(\mathbf{s},\mathbf{s}+\mathbf{r}) + (1-p)\cdot d(\mathbf{s},\mathbf{s}+\mathbf{0}) \leq \epsilon,
\end{align}
where the constraint means that the expected confidence score distortion is bounded by the budget. Note that we omit the other three constraints in~\autoref{originalproblem-c1},~\autoref{originalproblem-c3}, and~\autoref{originalproblem-c4}. This is because both of our representative noise vectors already satisfy those constraints. Moreover, we can derive an analytical solution to the simplified optimization problem. The analytical solution is as follows:
\begin{align}
\label{computep}
p=\begin{cases}
0, &\text{ if }|g(\mathbf{s})-0.5|\leq |g(\mathbf{s}+\mathbf{r})-0.5|\\
\min(\frac{\epsilon}{d(\mathbf{s},\mathbf{s}+\mathbf{r})},1.0), &\text{ otherwise.}
\end{cases}
\end{align}
\myparatight{One-time randomness} If the defender randomly samples one of the two representative noise vectors every time for the same query data sample, then an attacker could infer the true confidence score vector via querying the same data sample multiple times. We consider the attacker knows our defense mechanism including the confidence score distortion metric $d$, the budget $\epsilon$, and that the noise vector is sampled from two representative noise vectors, one of which is $\mathbf{0}$.
Suppose the attacker queries the same data sample $n$ times from the target classifier. The attacker receives a confidence score vector $\mathbf{s}_1$ for $m$ times and a confidence score vector $\mathbf{s}_2$ for $n-m$ times. One confidence score vector is $\mathbf{s+r}$ and the other is the true confidence score vector $\mathbf{s}$. Since the attacker receives two different confidence score vectors, the attacker knows $0<p<1$. Moreover, given the two confidence score vectors, the attacker can compute $p$ according to~\autoref{computep} since the distance $d(\mathbf{s},\mathbf{s}+\mathbf{r})$ does not depend on the ordering of $\mathbf{s}$ and $\mathbf{s}+\mathbf{r}$, i.e., $d(\mathbf{s},\mathbf{s}+\mathbf{r})=d(\mathbf{s}_1,\mathbf{s}_2)$. The attacker can also estimate the probabilities that the defender returns the confidence score vectors $\mathbf{s}_1$ and $\mathbf{s}_2$ as $\frac{m}{n}$ and $\frac{n-m}{n}$, respectively. If $\frac{m}{n}$ is closer to $p$, then the attacker predicts that $\mathbf{s}_2$ is the true confidence score vector, otherwise the attacker predicts $\mathbf{s}_1$ to be the true confidence score vector.
To address this challenge, we propose to use one-time randomness when the defender samples the representative noise, with which the defender always returns the same confidence score vector for the same query data sample. Specifically,
for a query data sample, the defender quantizes each dimension of the query data sample and computes the hash value of the quantized data sample. Then, the defender generates a random number $p^{\prime}$ in the range $[0,1]$ via a pseudo random number generator with the hash value as the seed. If $p^{\prime}<p$, the defender adds the representative noise vector $\mathbf{r}$ to the true confidence score vector, otherwise the defender does not add noise. The random number $p^{\prime}$ is the same for the same query data sample, so the defender always returns the same confidence score vector for the same query data sample. We compute the hash value of the quantized query data sample as the seed such that the attacker cannot just slightly modify the query data sample to generate a different $p^{\prime}$.
The attacker can compute the random number $p^{\prime}$ as we assume the attacker knows the defense mechanism including the hash function and pseudo random number generator. However, the attacker does not know $p$ any more because the defender always returns the same confidence score vector for the same query data sample. Therefore, the attacker does not know whether the returned confidence score vector is the true one or not.
\section{Problem Formulation}
\label{section:problem}
In our problem formulation, we have three parties, i.e., \emph{model provider}, \emph{attacker}, and \emph{defender}. \autoref{table:notations} shows some important notations used in this paper.
\begin{table}[!t]
\centering
\caption{Notations}
\label{table:notations}
\addtolength{\tabcolsep}{-2pt}
\begin{tabular}{c|l}
\toprule
Notation & Description\\
\midrule
$\mathbf{x}$ & A data sample\\
$\mathbf{s}$ & A true confidence score vector\\
$\mathbf{s}^{\prime}$ & A noisy confidence score vector\\
$\mathbf{n}$ & A noise vector\\
$f$ & Decision function of the target classifier\\
$\mathbf{z}$ & Logits of the target classifier\\
$C$ & Attacker's attack classifier for membership inference\\
$g$ & Decision function of defender's defense classifier\\
$h$& Logits of the defender's defense classifier\\
$\mathcal{M}$ & Randomized noise addition mechanism\\
$\epsilon$ & Confidence score distortion budget\\
\bottomrule
\end{tabular}
\end{table}
\subsection{Model Provider}
We assume a model provider has a proprietary training dataset (e.g., healthcare dataset, location dataset). The model provider trains a machine learning classifier using the proprietary training dataset. Then, the model provider deploys the classifier as a cloud service or a client-side AI software product (e.g., a mobile or IoT app), so other users can leverage the classifier to make predictions for their own data samples. In particular, we consider the deployed classifier returns a {confidence score vector} for a query data sample.
Formally, we have:
\[
f: \mathbf{x} \mapsto \mathbf{s},
\]
where $f$, $\mathbf{x}$, and $\mathbf{s}$ represent the classifier's decision function, the query data sample, and the confidence score vector, respectively. The confidence score vector essentially is the predicted posterior probability distribution of the label of the query data sample, i.e., $s_j$ is the predicted posterior probability that the query data sample has label $j$. The label of the query data sample is predicted to be the one that has the largest confidence score, i.e., the label is predicted as $\argmax_j \{s_j\}$. For convenience, we call the model provider's classifier \emph{target classifier}. Moreover, we consider the target classifier is neural network in this work.
\subsection{Attacker}
An attacker aims to infer the proprietary training dataset of the model provider. Specifically, we consider the attacker only has \emph{black-box} access to the target classifier, i.e., the attacker can send query data samples to the target classifier and obtain their confidence score vectors predicted by the target classifier. The attacker leverages black-box \emph{membership inference attacks}~\cite{LBWBWTGC18,NSH18,SSSS17,SZHBFB19} to infer the members of the target classifier's training dataset. Roughly speaking, in membership inference attacks, the attacker trains a binary classifier, which takes a query data sample's confidence score vector as input and predicts whether the query data sample is in the target classifier's training dataset or not. Formally, we have:
\[
C: \mathbf{s} \mapsto \{0, 1\},
\]
where $C$ is the attacker's binary classifier, $\mathbf{s}$ is the confidence score vector predicted by the target classifier for the query data sample $\mathbf{x}$, 0 indicates that the query data sample $\mathbf{x}$ is not a member of the target classifier's training dataset, and 1 indicates that the query data sample $\mathbf{x}$ is a member of the target classifier's training dataset. For convenience, we call the attacker's binary classifier $C$ \emph{attack classifier}. We will discuss more details about how the attacker could train its attack classifier in Section~\ref{evaluation_section}. Note that, to consider strong attacks, we assume the attacker knows our defense mechanism, but the defender does not know the attack classifier since the attacker has many choices for the attack classifier.
\subsection{Defender}
The defender aims to defend against black-box membership inference attacks. The defender could be the model provider itself or a trusted third party. For any query data sample from any user, the target classifier predicts its confidence score vector and the defender adds a \emph{noise vector} to the confidence score vector before returning it to the user. Formally, we have:
\[
\mathbf{s}'=\mathbf{s}+\mathbf{n},
\]
where $\mathbf{s}$ is the true confidence score vector predicted by the target classifier for a query data sample, $\mathbf{n}$ is the noise vector added by the defender, and $\mathbf{s}'$ is the noisy confidence score vector that is returned to a user. Therefore, an attacker only has access to the noisy confidence score vectors.
The defender aims to add noise to achieve the following two goals:
\begin{itemize}
\item {\bf Goal I.} The attacker's attack classifier is inaccurate at inferring the members/non-members of the target classifier's training dataset, i.e., protecting the privacy of the training dataset.
\item {\bf Goal II.} The utility loss of the confidence score vector is bounded.
\end{itemize}
However, achieving these two goals faces several challenges which we discuss next.
\myparatight{Achieving Goal I} The first challenge to achieve Goal I is that the defender does not know the attacker's attack classifier.
To address the challenge, the defender itself trains a binary classifier to perform membership inference and adds noise vectors to the confidence score vectors such that its own classifier is inaccurate at inferring members/non-members. In particular, the defender's classifier takes a confidence score vector as input and predicts member or non-member for the corresponding data sample. We call the defender's binary classifier \emph{defense classifier} and denote its decision function as $g$. Moreover, we consider the decision function $g(\mathbf{s})$ represents the probability that the corresponding data sample, whose confidence score vector predicted by the target classifier is $\mathbf{s}$, is a member of the target classifier's training dataset. In particular, we consider the defender trains a neural network classifier, whose output layer has one neuron with sigmoid activation function. For such classifier, the decision function's output (i.e., the output of the neuron in the output layer) represents probability of being a member. Formally, we have:
\[
g: \mathbf{s} \mapsto [0, 1].
\]
The defense classifier predicts a data sample to be member of the target classifier's training dataset if and only if $g(\mathbf{s})>0.5$.
To make the defense classifier inaccurate, one method is to add a noise vector to a true confidence score vector such that the defense classifier makes an incorrect prediction. Specifically, if the defense classifier predicts member (or non-member) for the true confidence score vector, then the defender adds a noise vector such that the defense classifier predicts non-member (or member) for the noisy confidence score vector. However, when an attacker knows the defense mechanism, the attacker can easily adapt its attack to achieve a high accuracy. In particular, the attacker predicts member (or non-member) when its attack classifier predicts non-member (or member) for a data sample. Another method is to add noise vectors such that the defense classifier always predicts member (or non-member) for the noisy confidence score vectors. However, for some true confidence score vectors, such method may need noise that violates the utility-loss constraints of the confidence score vectors (we will discuss utility-loss constraints later in this section).
{\bf Randomized noise addition mechanism.} Therefore, we consider the defender adopts a \emph{randomized noise addition mechanism} denoted as $\mathcal{M}$. Specifically, given a true confidence score vector $\mathbf{s}$, the defender samples a noise vector $\mathbf{n}$ from the space of possible noise vectors with a probability $\mathcal{M}(\mathbf{n}|\mathbf{s})$ and adds it to the true confidence score vector.
Since random noise is added to a true confidence score vector, the decision function $g$ outputs a random probability of being member. We consider the defender's goal is to make the expectation of the probability of being member predicted by $g$ close to 0.5. In other words, the defender's goal is to add random noise such that the defense classifier randomly guesses member or non-member for a data sample on average. Formally, the defender aims to find a randomized noise addition mechanism $\mathcal{M}$ such that $|E_{\mathcal{M}}(g(\mathbf{s} + \mathbf{n}))-0.5|$ is minimized.
\myparatight{Achieving Goal II} The key challenge to achieve Goal II is how to quantify the utility loss of the confidence score vector. To address the challenge, we introduce two utility-loss metrics.
{\bf Label loss.}
Our first metric concentrates on the query data sample's label predicted by the target classifier. Recall that the label of a query data sample is predicted as the one that has the largest confidence score. If the true confidence score vector and the noisy confidence score vector predict the same label for a query data sample, then the \emph{label loss} is 0 for the query data sample, otherwise the label loss is 1 for the query data sample. The overall label loss of a defense mechanism is the label loss averaged over all query data samples. In some critical applications such as finance and healthcare, even 1\% of label loss may be intolerable. In this work, we aim to achieve 0 label loss, i.e., our noise does not change the predicted label for any query data sample. Formally, we aim to achieve $\argmax_j \{s_j\} = \argmax_j \{s_j + n_j\}$, where $\argmax_j \{s_j\}$ and $\argmax_j \{s_j + n_j\}$ are the labels predicted based on the true and noisy confidence score vectors, respectively.
{\bf Confidence score distortion.} The confidence score vector for a query data sample tells the user more information about the data sample's label beyond the predicted label. Therefore, the added noise should not substantially distort the confidence score vector. First, the noisy confidence score vector should still be a probability distribution. Formally, we have $s_j+n_j\geq 0$ for $\forall j$ and $\sum_j (s_j+n_j)=1$. Second, the distance $d(\mathbf{s}, \mathbf{s}+\mathbf{n})$ between the true confidence score vector and the noisy confidence score vector should be small. In particular, we consider the model provider specifies a confidence score distortion budget called $\epsilon$, which indicates the upper bound of the expected confidence score distortion that the model provider can tolerate. Formally, we aim to achieve $E_{\mathcal{M}}(d(\mathbf{s},\mathbf{s}+\mathbf{n}))\leq \epsilon$. While any distance metric can be used to measure the distortion, we consider $L_1$-norm of the noise vector as the distance metric, i.e., $d(\mathbf{s},\mathbf{s}+\mathbf{n})=||\mathbf{n}||_1$. We adopt $L_1$-norm of the noise vector because it is easy to interpret. Specifically, the $L_1$-norm of the noise vector is simply the sum of the absolute value of its entries.
\myparatight{Membership inference attack defense problem} After quantifying Goal I and Goal II, we can formally define our problem of defending against membership inference attacks.
\begin{definition}[Membership-Inference-Attack Defense Problem]
Given the decision function $g$ of the defense classifier, a confidence score distortion budget $\epsilon$, a true confidence score vector $\mathbf{s}$, the defender aims to find a randomized noise addition mechanism $\mathcal{M}^{\ast}$ via solving the following optimization problem:
\begin{align}
\label{originalproblem}
\mathcal{M}^{\ast}&=\argmin_{\mathcal{M}} |E_{\mathcal{M}}(g(\mathbf{s}+\mathbf{n}))-0.5| \\
\label{originalproblem-c1}
\text{subject to: } & \argmax_j \{s_j+{n}_j\} =\argmax_j \{s_j\} \\
\label{originalproblem-c2}
&E_{\mathcal{M}}(d(\mathbf{s},\mathbf{s}+\mathbf{n}))\leq \epsilon \\
\label{originalproblem-c3}
&s_j+{n}_j \geq 0,\forall j \\
\label{originalproblem-c4}
&\sum_j s_j + {n}_j=1,
\end{align}
where the objective function of the optimization problem is to achieve Goal I and the constraints are to achieve Goal II. Specifically, the first constraint means that the added noise does not change the predicted label of the query data sample; the second constraint means that the confidence score distortion is bounded by the budget $\epsilon$; and the last two constraints mean that the noisy confidence score vector is still a probability distribution. Note that the last constraint is equivalent to $\sum_j {n}_j=0$ since $\sum_j {s}_j=1$. Moreover, we adopt $L_1$-norm of the noise vector to measure the confidence score distortion, i.e., $d(\mathbf{s},\mathbf{s}+\mathbf{n})=||\mathbf{n}||_1$.
\label{optimization_original}
\end{definition}
\section{Related Work}
\label{related}
\subsection{Membership Inference}
\myparatight{Membership inference attacks}
The goal of membership inference is to determine whether a certain data sample is inside a dataset.
Homer et al.~\cite{HSRDTMPSNC08} proposed the first membership inference attack in the biomedical setting,
in particular on genomic data. Specifically, they showed that an attacker can compare a user's genomic data with the summary statistics
of the target database, such as mean and standard deviation,
to determine the presence of the user in the database.
The comparison can be done by using statistical testing methods such as log-likelihood ratio test.
Later, several works performed similar membership inference attacks against other types of biomedical data such as MicroRNA~\cite{BBHM16}
and DNA methylation~\cite{HZHBTWB19}.
Recently, Pyrgelis et al.~\cite{PTC18,PTC19} further showed that
membership inference can also be performed effectively against location databases. In particular, they showed that an attacker can infer whether a user's location dataset was used for computing a given aggregate location dataset.
\myparatight{Membership inference attacks against ML models}
Shokri et al.~\cite{SSSS17} introduced membership inference in the ML setting.
The goal here is to determine whether a data sample is in the training dataset of a target black-box ML classifier.
To achieve the goal, the attacker trains binary ML classifiers, which take a data sample's {confidence score vector} predicted by the target classifier as input and infer the data sample to be a member or non-member of the target classifier's training dataset. We call these classifiers \emph{attack classifiers} and they are trained using \emph{shadow classifiers}. Specifically, the attacker is assumed to have a dataset coming from the same distribution as the target classifier's training dataset and the attacker uses the dataset to train {shadow classifiers}, each of which aims to replicate the target classifier. Then, the attacker trains the attack classifiers by using the confidence score vectors predicted by the shadow classifiers for some members and non-members of the shadow classifiers' training datasets.
Salem et al.~\cite{SZHBFB19} recently proposed new membership inference attacks for black-box target classifiers, which relax the assumptions of the attacks proposed by Shokri et al.
from both model and data angles. For instance, they showed that the attacker can rank the entries in a confidence score vector before feeding it into an attack classifier, which improves the attack effectiveness. Moreover, they showed that it is sufficient for the attacker to train just one shadow classifier. These results
indicate that membership inference threat is even larger than previously thought.
More recently, Nasr et al.~\cite{NSH19} proposed membership inference attacks against white-box ML models.
For a data sample,
they calculate the corresponding gradients over the white-box target classifier's parameters
and use these gradients as the data sample's feature for membership inference. Moreover, both Nasr et al.~\cite{NSH19} and Melis et al.~\cite{MSCS19} proposed membership inference attacks against federated learning.
While most of the previous works concentrated on classification models~\cite{SSSS17,LBG17,LBWBWTGC18,NSH18,YGFJ18,SZHBFB19,NSH19},
Hayes et al.~\cite{HMDC19} studied membership inference against generative models,
in particular generative adversarial networks (GANs)~\cite{GPMXWOCB14}.
They designed attacks for both white- and black-box settings.
Their results showed that generative models are also vulnerable to membership inference.
\myparatight{Defense mechanisms against membership inference}
Multiple defense mechanisms have been proposed to mitigate the threat of membership inference in the ML setting.
We summarize them as the following.
{\bf $L_2$-Regularizer~\cite{SSSS17}.} Overfitting, i.e., ML classifiers are more confident when facing data samples they are trained on (members)
than others, is one major reason why membership inference is effective. Therefore, to defend against membership inference, people have explored to reduce overfitting using regularization.
For instance, Shokri et al.~\cite{SSSS17} explored using conventional $L_2$ regularizer when training the target classifier.
{\bf Min-Max Game~\cite{NSH18}.} Nasr et al.~\cite{NSH18} proposed a min-max game-theoretic method to train a target classifier. Specifically, the method formulates a min-max optimization problem that aims to minimize the target classifier's prediction loss while maximizing the membership privacy. This formulation is equivalent to adding a new regularization term called \emph{adversarial regularization} to the loss function of the target classifier.
{\bf Dropout~\cite{SZHBFB19}.} Dropout is a recently proposed technique to regularize neural networks~\cite{SHKSS14}. Salem et al.~\cite{SZHBFB19} explored using dropout to mitigate membership inference attacks. Roughly speaking, dropout drops a neuron with a certain probability in each iteration of training a neural network.
{\bf Model Stacking~\cite{SZHBFB19}.}
Model stacking is a classical ensemble method which combines multiple weak classifiers' results as a strong one. Salem et al.~\cite{SZHBFB19} explored using model stacking to mitigate membership inference attacks.
Specifically, the target classifier consists of three classifiers organized into a two-level tree structure.
The first two classifiers on the bottom of the tree take
the original data samples as input, while the third one's input is the outputs of the first two classifiers.
The three classifiers are trained using disjoint sets of data samples,
which reduces the chance for the target classifier to remember any specific data sample, thus preventing overfitting.
{\bf Differential privacy.} Differential privacy~\cite{DMNS06} is a classical method for privacy-preserving machine learning.
Most differential privacy based defenses add noise to the objective function that is used to learn a model~\cite{CMS11,KST12,INSTTW19}, or the gradient in each iteration of gradient descent or stochastic gradient descent that is used to minimize the objective function~\cite{SCS13,BST14,WYX17,ACGMMTZ16,YLPGT19}. Shokri and Shmatikov~\cite{SS15} designed a differential privacy method for collaborative learning of deep neural networks.
{\bf Limitations.} Existing defenses suffer from two key limitations: 1) they do not have formal utility loss guarantee of the confidence score vector; and 2) they achieve suboptimal privacy-utility tradeoffs. Our defense addresses these two limitations. For instance, as we will show in experiments, with the same utility loss of the confidence score vector (e.g., the same $L_1$-norm distortion of the confidence score vector), our defense reduces the attack classifier's
accuracy at inferring members/non-members to a larger extent than existing defenses.
\myparatight{Other privacy/confidentiality attacks against ML}
There exist multiple other types of privacy/confidentiality attacks against ML models~\cite{FLJLPR14,FJR15,AFMSVV13,GWYGB18,MSCS19,TZJRR16,WG18,OASF18,SBBFZ19}.
Fredrikson et al.~\cite{FLJLPR14,FJR15} proposed \emph{model inversion attacks}. For instance,
they can infer the missing values of an input feature vector
by leveraging a classifier's prediction on the input feature vector. Several works~\cite{AFMSVV13,GWYGB18,MSCS19} studied \emph{property inference attacks}, which aim to infer a certain property (e.g., the fraction of male and female users) of a target classifier's training dataset.
Tram{\`{e}}r et al.~\cite{TZJRR16}
proposed \emph{model stealing attacks}.
They designed different techniques tailored to different ML models
aiming at stealing the parameters of the target models.
Another line of works studied \emph{hyperparameter stealing attacks}~\cite{WG18,OASF18}, which aim to steal the hyperparameters such as the neural network architecture and the hyperparameter that balances between the loss function and the regularization term.
\subsection{Adversarial Examples}
\label{adversarialexample}
Given a classifier and an example, we can add carefully crafted noise to the example such that the classifier predicts its label as we desire. The example with carefully crafted noise is called an \emph{adversarial example}. Our MemGuard adds carefully crafted noise to a confidence score vector to turn it into an adversarial example, which is likely to mislead the attack classifier to make a random guessing at member or non-member. The adversarial machine learning community has developed many algorithms (e.g.,~\cite{CW17,PMJFCS16,GSS15,MMSTV18,KGB16,MFF16,TKPGBM17,MFFF17}) to find adversarial examples. However, these algorithms are insufficient to our problem because they did not consider the utility-loss constraints on the confidence score vectors. We address these challenges via designing a new algorithm to find adversarial examples.
Since our defense leverages adversarial examples to mislead the attacker's attack classifier, an adaptive attacker can leverage a classifier that is more robust against adversarial examples as the attack classifier. Although different methods (e.g., adversarial training~\cite{GSS15,TKPGBM17,MMSTV18}, defensive distillation~\cite{PMWJS16}, Region-based Classification~\cite{CG17}, MagNet~\cite{MC17}, and Feature Squeezing~\cite{XEQ18}) have been explored to make classifiers robust against adversarial examples, it is still considered an open challenge to design such robust classifiers. Nevertheless, in our experiments, we will consider the attacker uses adversarial training to train its attack classifier, as adversarial training was considered to be the most empirically robust method against adversarial examples so far~\cite{ACW18}.
\section{Preliminaries}
\label{section:model}
In this section, we first define machine learning classification.
Then, we present the threat model.
In the end, we discuss our system model.
\subsection{Machine Learning}
In this work, we focus on machine learning classification,
one of the most common ML applications~\cite{SSSS17,SZHBFB19,SBBFZ19}.
A machine learning classifier, denoted by $\mathcal{C}$, is essentially a function
which maps a data sample $\mathbf{x}$ to
a vector of confidence scores $\mathbf{s}$.
Formally, we have:
\[
\mathcal{C}: \mathbf{x} \mapsto \mathbf{s}
\]
A confidence score vector $\mathbf{s}$ is also referred to as posterior probability:
Each entry of $\mathbf{s}$ indicates the probability of the corresponding data sample
being classified into a certain class or affiliated with a certain label.
The class with the highest probability is the final predicted class $\hat{\mathbf{y}}$
for the sample.
Meanwhile, we denote the real class label of the sample by $\mathbf{y}$.
For presentation purposes, we summarize the notations introduced here and later in this paper
in~\autoref{table:notations}.
\begin{table}[!t]
\centering
\caption{Notations, \YZ{we need add more notations}}
\label{table:notations}
\begin{tabular}{r|l}
\toprule
Notation & Description\\
\midrule
$\mathbf{x}$ & A data sample\\
$\mathbf{s}$ & A confidence score vector\\
$\mathbf{y}$ & Class label of a sample\\
$\mathcal{C}$ & Target machine learning model\\
$\mathcal{A}$ & Membership inference attack classifier\\
$f$ & Defender's membership inference classifier\\
\bottomrule
\end{tabular}
\end{table}
\subsection{Threat Model}
Membership inference attack in the setting of machine learning
aims to determine whether a target data sample $\mathbf{x}$
is in the training set of the target ML model $\mathcal{C}$.
In this work, we concentrate on the black-box ML setting,
i.e., the adversary can only query the target model with the sample
and obtain the corresponding confidence score vector.
The adversary can only decide the membership of her target sample
based on the confidence score.
As pointed out by previous works~\cite{SSSS17,SBBFZ19},
this is the most difficult setting for the adversary.
A membership inference attack model, denoted by $\mathcal{A}$,
can be formalized as the following map.
\[
\mathcal{A}: \mathbf{s} \mapsto \{0, 1\}
\]
Here, $\mathbf{s}$ denotes the confidence score vector of the target data sample $\mathbf{x}$, i.e., $\mathbf{s}=\mathcal{C}(\mathbf{x})$,
$1$ indicates $\mathbf{x}$
is a member of the target model $\mathcal{C}$'s training set and vice versa.
The attack model $\mathcal{A}$ is also realized by an ML classifier.
An adversary can train $\mathcal{A}$ in multiple ways depending on her knowledge
which is summarized as the following.
\begin{itemize}
\item As previous works~\cite{SSSS17,SZHBFB19},
we can assume that the adversary has some data samples coming from the same distribution as the training data of the target model.
Then, she can establish a shadow model~\cite{SZHBFB19} or multiple shadow models~\cite{SSSS17}
to mimic the behavior of the target model.
As the adversary controls the whole training process of her shadow models,
she can derive the labeled data with respect to members and non-members,
and utilize the data to train her attack model.
One further assumption made by previous works is that the attacker knows the architecture of the target model.
This can be realized by either using the same MLaaS API as the target model to establish shadow models~\cite{SSSS17}
or performing model hyperparameter stealing attacks~\cite{WG18,OASF18}.
%
\item A different but stronger assumption in this direction is recently proposed by Nasr et al.~\cite{NSH18}.
Specifically, they assume the attacker knows part of member and non-member data samples of target model.
This indicates the attacker can directly train her attack model without building any shadow model.
This attack is referred to as the sub-network attack.
%
\item The adversary can also know the underlying defense mechanism enforced by the target model.
In this case, the adversary can establish her shadow model
with the defense in place, and derive the adversarial training data for her attack model.
We hypothesize that this attack, termed as the adaptive attack,
results in a more robust membership inference against defense mechanisms.
\end{itemize}
Following these assumptions, we focus on 7 different membership inference attacks in this paper
such that our proposed defense mechanism can be evaluated comprehensively.
\subsection{System Model}
There are two parties in our system including user and defender.
\medskip
\myparatight{User}
We assume an user $u$ has a training set and
denote it by $D_{\text{train}}^{u}=\{\mathbf{X}_{\text{train}},\mathbf{Y}_{\text{train}}\}$.
Here, $\mathbf{X}_{\text{train}}=\{\mathbf{x}_1,\cdots,\mathbf{x}_{|\mathbf{X}_{\text{train}}|}\}$
contains all the data samples and $\mathbf{Y}_{\text{train}}$ contains labels for each sample.
The user can use the training set to train a target model $\mathcal{C}_{u}$
and uses a testing set $D_{\text{test}}^{u}=\{\mathbf{X}_{\text{test}}, \mathbf{Y}_{\text{test}}\}$
to evaluate the performance of the model.
The goal of the user is to deploy $\mathcal{C}_{u}$ as a service
for others to query.
Specifically, the service will return a confidence score vector given a data sample.
The user may suffer from recently proposed membership inference attack~\cite{SSSS17,SZHBFB19}
by providing the original confidence score vector.
Instead, she relies on a defender to sanitize the confidence score vector.
\medskip
\myparatight{Defender}
The defender is on the user side,
it can be an extra component of the user's deployed MLaaS,
or it can be an independent service provided by a trusted party.
The defender defends membership inference attack by adding a noise vector $\mathbf{n}$ to the output of $\mathcal{C}_{u}$
given a data sample $\mathbf{x}$.
This indicates the output of $\mathcal{C}_{u}$ on $\mathbf{x}$ will be:
\[
\mathbf{s}'=\mathbf{s}+\mathbf{n}
\]
To calculate the noise,
the defender first builds a local membership inference classifier denoted by $f$
to assess the membership threat.
As the defender is on the user side,
we assume it can access the user's training and testing sets as well as the target model.
This allows the defender to build the strongest membership inference classifier,
as it is trained on the ground truth data.
In detail, the defender first uses
$\mathbf{X}_{\text{train}}$ and $\mathbf{X}_{\text{test}}$
to query $\mathcal{C}_{u}$,
and obtains the corresponding confidence scores
denoted by $\mathbf{S}_{\text{train}}$ and $\mathbf{S}_{\text{test}}$, respectively.
All confidence score vectors in $\mathbf{S}_{\text{train}}$ are labeled as $1$, i.e., member samples,
and $\mathbf{S}_{\text{test}}$ as $0$, non-member samples.
Then, $f$ is trained on this dataset.
|
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\begin{document}
\baselineskip16pt
\title{Constructions of Optical Queues With a Limited Number of
Recirculations--Part~II: Optimal Constructions}
\author{Xuan-Chao Huang and Jay Cheng \\
\thanks{This work was supported in part by the National Science Council, Taiwan, R.O.C.,
under Contract NSC 96-2221-E-007-076, Contract NSC 97-2221-E-007-105-MY3,
and the Program for Promoting Academic Excellence of Universities NSC 94-2752-E-007-002-PAE.}
\thanks{The authors are with the Department of Electrical Engineering and
the Institute of Communications Engineering,
National Tsing Hua University, Hsinchu 30013, Taiwan, R.O.C.
(e-mails: [email protected]; [email protected]).}
}
\maketitle
\thispagestyle{empty}
\begin{abstract}
\baselineskip12pt
One of the main problems in all-optical packet-switched networks
is the lack of optical buffers,
and one feasible technology for the constructions of optical buffers
is to use optical crossbar Switches and fiber Delay Lines (SDL).
In this two-part paper, we consider SDL constructions of optical queues
with a limited number of recirculations through
the optical switches and the fiber delay lines.
Such a problem arises from practical feasibility considerations.
In Part~I, we have proposed a class of greedy constructions
for certain types of optical queues, including linear compressors,
linear decompressors, and 2-to-1 FIFO multiplexers,
and have shown that every optimal construction among our previous constructions
of these types of optical queues under the constraint of
a limited number of recirculations must be a greedy construction.
Specifically, given $M\geq 2$ and $1\leq k\leq M-1$,
we have shown that to find an optimal construction,
it suffices to find an optimal sequence ${\dbf^*}_1^M\in \Gcal_{M,k}$ such that
$B({\dbf^*}_1^M;k)=\max_{\dbf_1^M\in \Gcal_{M,k}}B(\dbf_1^M;k)$,
where $B(\dbf_1^M;k)$ is the maximum representable integer
with respect to $\dbf_1^M$ and $k$ (defined in Part~I)
and $\Gcal_{M,k}$ is the set of sequences $\dbf_1^M=(d_1,d_2,\ldots,d_M)$
given by $d_{s_i+j}=B(\dbf_1^{s_i+j-1};i+1)+1$
for $i=0,1,\ldots,k-1$ and $j=1,2,\ldots,n_{i+1}$
for some sequence $\nbf_1^k=(n_1,n_2,\ldots,n_k)$
with $n_1\geq 2$, $n_2,n_3,\ldots,n_k\geq 1$, and $\sum_{i=1}^{k}n_i=M$,
in which $s_0=0$ and $s_i=\sum_{\ell=1}^{i}n_{\ell}$ for $i=1,2,\ldots,k$.
In Part~II, the present paper,
we further show that there are at most two optimal constructions
and give a simple algorithm to obtain the optimal construction(s).
The main idea in Part~II is to use \emph{pairwise comparison}
to remove a sequence $\dbf_1^M\in \Gcal_{M,k}$ such that
$B(\dbf_1^M;k)<B({\dbf'}_1^M;k)$ for some ${\dbf'}_1^M\in \Gcal_{M,k}$.
To our surprise, the simple algorithm for obtaining the optimal construction(s)
is related to the well-known \emph{Euclid's algorithm}
for finding the greatest common divisor (gcd) of two integers.
In particular, we show that if $\gcd(M,k)=1$, then there is only one optimal construction;
if $\gcd(M,k)=2$, then there are two optimal constructions;
and if $\gcd(M,k)\geq 3$, then there are at most two optimal constructions.
\end{abstract}
\begin{keywords}
Euclid's algorithm, FIFO multiplexers, integer representation,
linear compressors, linear decompressors, maximum representable integer,
optical buffers, optical queues, packet switching.
\end{keywords}
\pagestyle{empty}
\pagestyle{headings}
\pagenumbering{arabic}
\newpage
\bsection{Introduction}{introduction}
One of the bottlenecks toward all-optical packet-switched networks
is the O-E-O (optical-electrical-optical) conversion
due to the lack of optical buffers.
Currently, the only known way to ``store'' optical packets
without converting them into other media
is to use optical Switches and fiber Delay Lines (SDL)
to direct optical packets to the right place at the right time.
Although the optical buffers constructed by the SDL approach
can only be used as sequential buffers with fixed storage times
so that they do not have the random access capability,
research results in the SDL literature (see the references in Part~I \cite{CCCLL10} of this paper)
show that they can still be used to construct many types of optical queues
commonly encountered in practice:
including output-buffered switches, FIFO multiplexers, FIFO queues, LIFO queues, priority queues,
time slot interchanges, linear compressors, linear decompressors,
non-overtaking delay lines, and flexible delay lines.
\bpdffigure{LC-2-to-1-FIFO-MUX.pdf}{6.0in}
\epdffigure{LC-2-to-1-FIFO-MUX}
{(a) A construction of a linear compressor.
(b) A construction of a 2-to-1 FIFO multiplexer.}
In this two-part paper, we address an important practical feasibility issue
that is of great concern in the SDL constructions of optical queues:
the constructions of optical queues with a limited number of
recirculations through the optical switches and the fiber delay lines.
We recall that it was shown in \cite{CCCL09} (resp., \cite{CCLC06}) that the construction
in \rfigure{LC-2-to-1-FIFO-MUX}(a)/mirror image of \rfigure{LC-2-to-1-FIFO-MUX}(a)
(resp., \rfigure{LC-2-to-1-FIFO-MUX}(b))
can be operated as a linear compressor/decompressor
(resp., 2-to-1 FIFO multiplexer) under a simple packet routing scheme.
Suppose that there is a limitation on the number, say $k$,
of recirculations through the $M$ fibers in \rfigure{LC-2-to-1-FIFO-MUX}
due to practical feasibility considerations.
For the nontrivial case that $M\geq 2$ and $1\leq k\leq M-1$,
we have proposed in Part~I a class of greedy constructions
by specifying a class $\Gcal_{M,k}$ of sequences
of the delays of the $M$ fiber delay lines in \rfigure{LC-2-to-1-FIFO-MUX}
such that every sequence $\dbf_1^M=(d_1,d_2,\ldots,d_M)$ in $\Gcal_{M,k}$ is given by
\beqnarray{OQ-LR-delays-greedy-1}
d_{s_i+j}=B(\dbf_1^{s_i+j-1};i+1)+1,
\emph{ for } i=0,1,\ldots,k-1 \textrm{ and } j=1,2,\ldots,n_{i+1},
\eeqnarray
for some sequence $\nbf_1^k=(n_1,n_2,\ldots,n_k)$
with $n_1\geq 2$, $n_2,n_3,\ldots,n_k\geq 1$, and $\sum_{i=1}^{k}n_i=M$,
where $s_0=0$, $s_i=\sum_{\ell=1}^{i}n_{\ell}$ for $i=1,2,\ldots,k$,
and $B(\dbf_1^{s_i+j-1};i+1)$ is the maximum representable integer
with respect to $\dbf_1^{s_i+j-1}$ and $i+1$
(see \cite{CCCLL10} for a definition of the maximum representable integer).
Furthermore, we have shown that every optimal construction among our previous constructions
of linear compressors/decompressors in \cite{CCCL09}
and 2-to-1 FIFO multiplexers in \cite{CCLC06}
under the constraint of a limited number of recirculations
must be a greedy construction.
Specifically, let
\beqnarray{G-M-k}
\Gcal_{M,k}=
\left\{\dbf_1^M\in (\Zbf^+)^M: \dbf_1^M \textrm{ is given by } \reqnarray{OQ-LR-delays-greedy-1}
\textrm{ for some } \nbf_1^k\in \Ncal_{M,k}\right\},
\eeqnarray
where
\beqnarray{N-M-k}
\Ncal_{M,k}=
\left\{\nbf_1^k\in (\Zbf^+)^k: n_1\geq 2 \textrm{ and } \sum_{i=1}^{k}n_i=M\right\},
\eeqnarray
then to find an optimal construction,
it suffices to find an optimal sequence over $\Gcal_{M,k}$,
i.e., to find a sequence ${\dbf^*}_1^M\in \Gcal_{M,k}$ such that
$B({\dbf^*}_1^M;k)=\max_{\dbf_1^M\in \Gcal_{M,k}}B(\dbf_1^M;k)$.
We call a sequence ${\nbf^*}_1^M\in \Ncal_{M,k}$
an \emph{optimal} sequence over $\Ncal_{M,k}$
if the sequence ${\dbf^*}_1^M\in \Gcal_{M,k}$
obtained by using ${\nbf^*}_1^M$ in \reqnarray{OQ-LR-delays-greedy-1}
is an optimal sequence over $\Gcal_{M,k}$.
Therefore, to find an optimal construction,
it suffices to find an optimal sequence over $\Ncal_{M,k}$.
Our contribution in Part~II, the present paper,
is to show in \rsection{optimal constructions} and \rsection{proof of the main result}
that there are at most two optimal sequences over $\Ncal_{M,k}$
and give a simple algorithm to obtain the optimal sequence(s).
The main idea in \rsection{optimal constructions} and \rsection{proof of the main result}
is to use \emph{pairwise comparison} to remove a sequence $\nbf_1^M\in \Ncal_{M,k}$
such that $B(\dbf_1^M;k)<B({\dbf'}_1^M;k)$ for some ${\nbf'}_1^M\in \Ncal_{M,k}$,
where $\dbf_1^M$ and ${\dbf'}_1^M$ are obtained by using $\nbf_1^M$ and ${\nbf'}_1^M$,
respectively, in \reqnarray{OQ-LR-delays-greedy-1}.
To our surprise, the simple algorithm for obtaining the optimal sequence(s)
is related to the well-known \emph{Euclid's algorithm}
for finding the greatest common divisor (gcd) of two integers.
In particular, we show that if $\gcd(M,k)=1$, then there is only one optimal sequence;
if $\gcd(M,k)=2$, then there are two optimal sequences;
and if $\gcd(M,k)\geq 3$, then there are at most two optimal sequences.
We conclude this paper in \rsection{conclusion}.
\bsection{The Optimal Constructions}{optimal constructions}
To simplify the presentation in this paper,
in the following we first define left-imbedded sequences, left pre-sequences,
right-imbedded sequences, and right pre-sequences.
\bpdffigure{left-right-imbedded-sequences.pdf}{4.5in}
\epdffigure{left-right-imbedded-sequences}
{(a) An illustration of \reqnarray{left-imbedded sequences-2}
in the definition of left-imbedded sequences in \rdefinition{left-imbedded sequences}
and \reqnarray{left pre-sequences-2}
in the definition of left pre-sequences in \rdefinition{left pre-sequences}.
(b) An illustration of \reqnarray{right-imbedded sequences-2}
in the definition of right-imbedded sequences in \rdefinition{right-imbedded sequences}
and \reqnarray{right pre-sequences-2}
in the definition of right pre-sequences in \rdefinition{right pre-sequences}.}
\bdefinition{left-imbedded sequences}\textbf{(Left-imbedded sequences)}
Suppose that $M\geq 2$ and $1\leq k\leq M-1$.
Let $M=qk+r$, where $q$ and $r$ are the quotient and the remainder,
respectively, of $M$ divided by $k$.
Suppose that $r\neq 0$ and $\nbf_1^k=(n_1,n_2,\ldots,n_k)$ is a sequence of positive integers
such that
\beqnarray{left-imbedded sequences-1}
n_i=
\bselection
q+1, &\textrm{if } i=i_1,i_2,\ldots,i_r, \\
q, &\textrm{otherwise},
\eselection
\eeqnarray
for some $1=i_1<i_2<\cdots <i_r\leq k$ (note that $\sum_{i=1}^{k}n_i=qk+r=M$).
The \emph{left-imbedded sequence} $\mbf_1^r=(m_1,m_2,\ldots,m_r)$ of the sequence $\nbf_1^k$
with respect to $M$ and $k$, denoted $\mbf_1^r=L_{M,k}^I(\nbf_1^k)$,
is a sequence of positive integers given by
(see \rfigure{left-right-imbedded-sequences}(a) for an illustration)
\beqnarray{left-imbedded sequences-2}
m_j=
\bselection
|\{i_j,i_j+1,\ldots,i_{j+1}-1\}|=i_{j+1}-i_j, &\textrm{if } j=1,2,\ldots,r-1, \\
|\{i_r,i_r+1,\ldots,k\}|=k-i_r+1, &\textrm{if } j=r.
\eselection
\eeqnarray
Note that $\sum_{j=1}^{r}m_j=k-i_1+1=k$.
\edefinition
\bdefinition{left pre-sequences}\textbf{(Left pre-sequences)}
Suppose that $M\geq 2$ and $1\leq k\leq M-1$.
Let $M=qk+r$, where $q$ and $r$ are the quotient and the remainder,
respectively, of $M$ divided by $k$.
Suppose that $r\neq 0$ and $\mbf_1^r=(m_1,m_2,\ldots,m_r)$ is a sequence of positive integers
such that $\sum_{j=1}^{r}m_j=k$.
The \emph{left pre-sequence} $\nbf_1^k=(n_1,n_2,\ldots,n_k)$ of the sequence $\mbf_1^r$
with respect to $M$ and $k$, denoted $\nbf_1^k=L_{M,k}(\mbf_1^r)$,
is a sequence of positive integers given by
\beqnarray{left pre-sequences-1}
n_i=
\bselection
q+1, &\textrm{if } i=i_1,i_2,\ldots,i_r, \\
q, &\textrm{otherwise},
\eselection
\eeqnarray
where
\beqnarray{left pre-sequences-2}
i_j=\sum_{\ell=1}^{j-1}m_{\ell}+1, \textrm{ for } j=1,2,\ldots,r,
\eeqnarray
(see \rfigure{left-right-imbedded-sequences}(a) for an illustration).
Note that $i_1=\sum_{\ell=1}^{0}m_{\ell}+1=1$ and $\sum_{i=1}^{k}n_i=qk+r=M$.
\edefinition
\bexample{left-imbedded sequences and left pre-sequences}
Suppose that $M=13$ and $k=5$.
Then the quotient and the remainder of $M$ divided by $k$
are $q=2$ and $r=3$, respectively.
(i) Suppose that $\nbf_1^k=(3,3,2,3,2)$, then the left-imbedded sequence $\mbf_1^r$
of the sequence $\nbf_1^k$ with respect to $M$ and $k$ is given by
\beqnarray{}
\mbf_1^r=L_{M,k}^I(\nbf_1^k)=L_{13,5}^I((3,3,2,3,2))=(1,2,2). \nn
\eeqnarray
(ii) Suppose that $\mbf_1^r=(1,2,2)$, then the left pre-sequence $\nbf_1^k$
of the sequence $\mbf_1^r$ with respect to $M$ and $k$ is given by
\beqnarray{}
\nbf_1^k=L_{M,k}(\mbf_1^r)=L_{13,5}((1,2,2))=(3,3,2,3,2). \nn
\eeqnarray
\eexample
Clearly, left-imbedded sequences and left pre-sequences are closely related
as can be seen in the following lemma.
\blemma{left-imbedded sequences-left pre-sequences}
Suppose that $M\geq 2$ and $1\leq k\leq M-1$.
Let $M=qk+r$, where $q$ and $r$ are the quotient and the remainder,
respectively, of $M$ divided by $k$.
Suppose that $r\neq 0$.
(i) If $\mbf_1^r=L_{M,k}^I(\nbf_1^k)$,
where $\nbf_1^k$ is given by \reqnarray{left-imbedded sequences-1}
for some $1=i_1<i_2<\cdots <i_r\leq k$,
then we have $\nbf_1^k=L_{M,k}(\mbf_1^r)$.
(ii) Conversely, if $\nbf_1^k=L_{M,k}(\mbf_1^r)$,
where $\mbf_1^r$ is a sequence of positive integers
such that $\sum_{j=1}^{r}m_j=k$,
then we have $\mbf_1^r=L_{M,k}^I(\nbf_1^k)$.
\elemma
\bproof
(i) If $\mbf_1^r=L_{M,k}^I(\nbf_1^k)$,
where $\nbf_1^k$ is given by \reqnarray{left-imbedded sequences-1}
for some $1=i_1<i_2<\cdots <i_r\leq k$,
then we see from \reqnarray{left-imbedded sequences-2} that
\beqnarray{}
\alignspace
\sum_{\ell=1}^{j-1}m_{\ell}+1=
\bselection
\sum_{\ell=1}^{0}m_{\ell}+1=1=i_1=i_j, &\textrm{if } j=1, \\
\sum_{\ell=1}^{j-1}(i_{\ell+1}-i_{\ell})+1=i_j-i_1+1=i_j, &\textrm{if } j=2,3,\ldots,r,
\eselection
\label{eqn:proof-left-imbedded sequences-left pre-sequences-111} \\
\alignspace
\sum_{j=1}^{r}m_j=\sum_{\ell=1}^{r-1}(i_{\ell+1}-i_{\ell})+k-i_r+1=k-i_1+1=k.
\label{eqn:proof-left-imbedded sequences-left pre-sequences-222}
\eeqnarray
As we have $\sum_{j=1}^{r}m_j=k$ in
\reqnarray{proof-left-imbedded sequences-left pre-sequences-222},
the left pre-sequence of $\mbf_1^r$ with respect to $M$ and $k$ is well defined,
say ${\nbf'}_1^k=L_{M,k}(\mbf_1^r)$.
From the definition of left pre-sequences
in \rdefinition{left pre-sequences}
and \reqnarray{proof-left-imbedded sequences-left pre-sequences-111},
we have
\beqnarray{proof-left-imbedded sequences-left pre-sequences-333}
n_i'=
\bselection
q+1, &\textrm{if } i=i_1',i_2',\ldots,i_r', \\
q, &\textrm{otherwise},
\eselection
\eeqnarray
where
\beqnarray{proof-left-imbedded sequences-left pre-sequences-444}
i_j'=\sum_{\ell=1}^{j-1}m_{\ell}+1=i_j, \textrm{ for } j=1,2,\ldots,r,
\eeqnarray
As such, it follows from \reqnarray{left-imbedded sequences-1},
\reqnarray{proof-left-imbedded sequences-left pre-sequences-333},
\reqnarray{proof-left-imbedded sequences-left pre-sequences-444},
and ${\nbf'}_1^k=L_{M,k}(\mbf_1^r)$ that $\nbf_1^k={\nbf'}_1^k=L_{M,k}(\mbf_1^r)$.
(ii) Conversely, if $\nbf_1^k=L_{M,k}(\mbf_1^r)$,
where $\mbf_1^r$ is a sequence of positive integers
such that $\sum_{j=1}^{r}m_j=k$,
then we see from \reqnarray{left pre-sequences-2} that
\beqnarray{}
\alignspace
i_1=\sum_{\ell=1}^{0}m_{\ell}+1=1,
\label{eqn:proof-left-imbedded sequences-left pre-sequences-555} \\
\alignspace
i_{j+1}-i_j=\left(\sum_{\ell=1}^{j}m_{\ell}+1\right)
-\left(\sum_{\ell=1}^{j-1}m_{\ell}+1\right)=m_j,
\textrm{ for } j=1,2,\ldots,r-1,
\label{eqn:proof-left-imbedded sequences-left pre-sequences-666}
\eeqnarray
\beqnarray{}
\hspace*{-1.36in}
k-i_r+1=\sum_{j=1}^{r}m_j-\left(\sum_{\ell=1}^{r-1}m_{\ell}+1\right)+1=m_r.
\label{eqn:proof-left-imbedded sequences-left pre-sequences-777}
\eeqnarray
\iffalse
\beqnarray{}
\alignspace
i_1=\sum_{\ell=1}^{0}m_{\ell}+1=1,
\label{eqn:proof-left-imbedded sequences-left pre-sequences-555} \\
\alignspace
i_{j+1}-i_j=\left(\sum_{\ell=1}^{j}m_{\ell}+1\right)
-\left(\sum_{\ell=1}^{j-1}m_{\ell}+1\right)=m_j,
\textrm{ for } j=1,2,\ldots,r-1,
\label{eqn:proof-left-imbedded sequences-left pre-sequences-666} \\
\alignspace
k-i_r+1=\sum_{j=1}^{r}m_j-\left(\sum_{\ell=1}^{r-1}m_{\ell}+1\right)+1=m_r.
\label{eqn:proof-left-imbedded sequences-left pre-sequences-777}
\eeqnarray
\fi
From $i_1=1$ in \reqnarray{proof-left-imbedded sequences-left pre-sequences-555}
and \reqnarray{left pre-sequences-1}
we see that $\nbf_1^k$ satisfies \reqnarray{left-imbedded sequences-1}
and hence the left-imbedded sequence of $\nbf_1^k$
with respect to $M$ and $k$ is well defined,
say ${\mbf'}_1^r=L_{M,k}^I(\nbf_1^k)$.
From the definition of left-imbedded sequences
in \rdefinition{left-imbedded sequences},
we have
\beqnarray{proof-left-imbedded sequences-left pre-sequences-888}
m_j'=
\bselection
i_{j+1}-i_j, &\textrm{if } j=1,2,\ldots,r-1, \\
k-i_r+1, &\textrm{if } j=r.
\eselection
\eeqnarray
As such, it follows from \reqnarray{proof-left-imbedded sequences-left pre-sequences-666},
\reqnarray{proof-left-imbedded sequences-left pre-sequences-777},
\reqnarray{proof-left-imbedded sequences-left pre-sequences-888},
and ${\mbf'}_1^r=L_{M,k}^I(\nbf_1^k)$ that $\mbf_1^r={\mbf'}_1^r=L_{M,k}^I(\nbf_1^k)$.
\eproof
\bdefinition{right-imbedded sequences}\textbf{(Right-imbedded sequences)}
Suppose that $M\geq 2$ and $1\leq k\leq M-1$.
Let $M=qk+r$, where $q$ and $r$ are the quotient and the remainder,
respectively, of $M$ divided by $k$.
Suppose that $r\neq 0$ and $\nbf_1^k=(n_1,n_2,\ldots,n_k)$ is a sequence of positive integers
such that
\beqnarray{right-imbedded sequences-1}
n_i=
\bselection
q+1, &\textrm{if } i=i_1,i_2,\ldots,i_r, \\
q, &\textrm{otherwise},
\eselection
\eeqnarray
for some $1\leq i_1<i_2<\cdots <i_r=k$ (note that $\sum_{i=1}^{k}n_i=qk+r=M$).
The \emph{right-imbedded sequence} $\mbf_1^r=(m_1,m_2,\ldots,m_r)$ of the sequence $\nbf_1^k$
with respect to $M$ and $k$, denoted $\mbf_1^r=R_{M,k}^I(\nbf_1^k)$,
is a sequence of positive integers given by
(see \rfigure{left-right-imbedded-sequences}(b) for an illustration)
\beqnarray{right-imbedded sequences-2}
m_j=
\bselection
|\{1,2,\ldots,i_1\}|=i_1, &\textrm{if } j=1, \\
|\{i_{j-1}+1,i_{j-1}+2,\ldots,i_j\}|=i_j-i_{j-1}, &\textrm{if } j=2,3,\ldots,r.
\eselection
\eeqnarray
Note that $\sum_{j=1}^{r}m_j=i_r=k$.
\edefinition
\bdefinition{right pre-sequences}\textbf{(Right pre-sequences)}
Suppose that $M\geq 2$ and $1\leq k\leq M-1$.
Let $M=qk+r$, where $q$ and $r$ are the quotient and the remainder,
respectively, of $M$ divided by $k$.
Suppose that $r\neq 0$ and $\mbf_1^r=(m_1,m_2,\ldots,m_r)$ is a sequence of positive integers
such that $\sum_{j=1}^{r}m_j=k$.
The \emph{right pre-sequence} $\nbf_1^k=(n_1,n_2,\ldots,n_k)$ of the sequence $\mbf_1^r$
with respect to $M$ and $k$, denoted $\nbf_1^k=R_{M,k}(\mbf_1^r)$,
is a sequence of positive integers given by
\beqnarray{right pre-sequences-1}
n_i=
\bselection
q+1, &\textrm{if } i=i_1,i_2,\ldots,i_r, \\
q, &\textrm{otherwise},
\eselection
\eeqnarray
where
\beqnarray{right pre-sequences-2}
i_j=\sum_{\ell=1}^{j}m_{\ell}, &\textrm{for } j=1,2,\ldots,r,
\eeqnarray
(see \rfigure{left-right-imbedded-sequences}(b) for an illustration).
Note that $i_r=\sum_{\ell=1}^{r}m_{\ell}=k$ and $\sum_{i=1}^{k}n_i=qk+r=M$.
\edefinition
\bexample{right-imbedded sequences and right pre-sequences}
Suppose that $M=13$ and $k=5$.
Then the quotient and the remainder of $M$ divided by $k$
are $q=2$ and $r=3$, respectively.
(i) Suppose that $\nbf_1^k=(2,3,2,3,3)$, then the right-imbedded sequence $\mbf_1^r$
of the sequence $\nbf_1^k$ with respect to $M$ and $k$ is given by
\beqnarray{}
\mbf_1^r=R_{M,k}^I(\nbf_1^k)=R_{13,5}^I((2,3,2,3,3))=(2,2,1). \nn
\eeqnarray
(ii) Suppose that $\mbf_1^r=(2,2,1)$, then the right pre-sequence $\nbf_1^k$
of the sequence $\mbf_1^r$ with respect to $M$ and $k$ is given by
\beqnarray{}
\nbf_1^k=R_{M,k}(\mbf_1^r)=R_{13,5}((2,2,1))=(2,3,2,3,3). \nn
\eeqnarray
\eexample
Right-imbedded sequences and right pre-sequences are also closely related
as can be seen in the following lemma.
\blemma{right-imbedded sequences-right pre-sequences}
Suppose that $M\geq 2$ and $1\leq k\leq M-1$.
Let $M=qk+r$, where $q$ and $r$ are the quotient and the remainder,
respectively, of $M$ divided by $k$.
Suppose that $r\neq 0$.
(i) If $\mbf_1^r=R_{M,k}^I(\nbf_1^k)$,
where $\nbf_1^k$ is given by \reqnarray{right-imbedded sequences-1}
for some $1\leq i_1<i_2<\cdots <i_r=k$,
then we have $\nbf_1^k=R_{M,k}(\mbf_1^r)$.
(ii) Conversely, if $\nbf_1^k=R_{M,k}(\mbf_1^r)$,
where $\mbf_1^r$ is a sequence of positive integers
such that $\sum_{j=1}^{r}m_j=k$,
then we have $\mbf_1^r=R_{M,k}^I(\nbf_1^k)$.
\elemma
\bproof
(i) If $\mbf_1^r=R_{M,k}^I(\nbf_1^k)$,
where $\nbf_1^k$ is given by \reqnarray{right-imbedded sequences-1}
for some $1\leq i_1<i_2<\cdots <i_r=k$,
then we see from \reqnarray{right-imbedded sequences-2} that
\beqnarray{proof-right-imbedded sequences-right pre-sequences-111}
\alignspace
\sum_{\ell=1}^{j}m_{\ell}=i_j, \textrm{ for } j=1,2,\ldots,r.
\eeqnarray
From $i_r=k$ and \reqnarray{proof-right-imbedded sequences-right pre-sequences-111},
we can see that $\sum_{j=1}^{r}m_j=i_r=k$
and hence the right pre-sequence of $\mbf_1^r$ with respect to $M$ and $k$ is well defined,
say ${\nbf'}_1^k=R_{M,k}(\mbf_1^r)$.
From the definition of right pre-sequences
in \rdefinition{right pre-sequences}
and \reqnarray{proof-right-imbedded sequences-right pre-sequences-111},
we have
\beqnarray{proof-right-imbedded sequences-right pre-sequences-222}
n_i'=
\bselection
q+1, &\textrm{if } i=i_1',i_2',\ldots,i_r', \\
q, &\textrm{otherwise},
\eselection
\eeqnarray
where
\beqnarray{proof-right-imbedded sequences-right pre-sequences-333}
i_j'=\sum_{\ell=1}^{j}m_{\ell}=i_j, \textrm{ for } j=1,2,\ldots,r,
\eeqnarray
As such, it follows from \reqnarray{right-imbedded sequences-1},
\reqnarray{proof-right-imbedded sequences-right pre-sequences-222},
\reqnarray{proof-right-imbedded sequences-right pre-sequences-333},
and ${\nbf'}_1^k=R_{M,k}(\mbf_1^r)$ that $\nbf_1^k={\nbf'}_1^k=R_{M,k}(\mbf_1^r)$.
(ii) Conversely, if $\nbf_1^k=R_{M,k}(\mbf_1^r)$,
where $\mbf_1^r$ is a sequence of positive integers
such that $\sum_{j=1}^{r}m_j=k$,
then we see from \reqnarray{right pre-sequences-2} that
\beqnarray{}
\alignspace
i_1=m_1,
\label{eqn:proof-right-imbedded sequences-right pre-sequences-444} \\
\alignspace
i_j-i_{j-1}=\sum_{\ell=1}^{j}m_{\ell}-\sum_{\ell=1}^{j-1}m_{\ell}=m_j,
\textrm{ for } j=2,3,\ldots,r,
\label{eqn:proof-right-imbedded sequences-right pre-sequences-555} \\
\alignspace
i_r=\sum_{j=1}^{r}m_j=k.
\label{eqn:proof-right-imbedded sequences-right pre-sequences-666}
\eeqnarray
From \reqnarray{right pre-sequences-1}
and $i_r=k$ in \reqnarray{proof-right-imbedded sequences-right pre-sequences-666},
we see that $\nbf_1^k$ satisfies \reqnarray{right-imbedded sequences-1}
and hence the right-imbedded sequence of $\nbf_1^k$
with respect to $M$ and $k$ is well defined,
say ${\mbf'}_1^r=R_{M,k}^I(\nbf_1^k)$.
From the definition of right-imbedded sequences
in \rdefinition{right-imbedded sequences},
we have
\beqnarray{proof-right-imbedded sequences-right pre-sequences-777}
m_j'=
\bselection
i_1, &\textrm{if } j=1, \\
i_j-i_{j-1}, &\textrm{if } j=2,3,\ldots,r.
\eselection
\eeqnarray
As such, it follows from \reqnarray{proof-right-imbedded sequences-right pre-sequences-444},
\reqnarray{proof-right-imbedded sequences-right pre-sequences-555},
\reqnarray{proof-right-imbedded sequences-right pre-sequences-777},
and ${\mbf'}_1^r=R_{M,k}^I(\nbf_1^k)$ that $\mbf_1^r={\mbf'}_1^r=R_{M,k}^I(\nbf_1^k)$.
\eproof
In the following theorem,
we state the main result in this paper on optimal sequences over $\Ncal_{M,k}$.
The proof of the theorem will be given in \rsection{proof of the main result}.
\btheorem{main result}
Let $M\geq 2$ and $1\leq k\leq M-1$.
(i) Suppose that $\gcd(M,k)=1$.
Then there is only one optimal sequence over $\Ncal_{M,k}$,
and the optimal sequence is given by the sequence $\nbf_1^k(1)$
obtained in Step 2 or Step 3 of \ralgorithm{main result} below
(depending on which of the two steps is executed in \ralgorithm{main result}).
(ii) Suppose that $\gcd(M,k)=2$.
Then there are two optimal sequences over $\Ncal_{M,k}$,
and the two optimal sequences are given by the two sequences
$\nbf_1^k(1)$ and $\mbf_1^k(1)$
obtained in Step 2 or Step 3 of \ralgorithm{main result}
(depending on which of the two steps is executed in \ralgorithm{main result}).
(iii) Suppose that $\gcd(M,k)\geq 3$.
Then there are at most two optimal sequences over $\Ncal_{M,k}$,
and the two possible optimal sequences are given by the two sequences
$\nbf_1^k(1)$ and $\mbf_1^k(1)$
obtained in Step 2 or Step 3 of \ralgorithm{main result}
(depending on which of the two steps is executed in \ralgorithm{main result}).
\etheorem
\balgorithm{main result}
Given $M\geq 2$ and $1\leq k\leq M-1$.
Step 1. (Euclid's algorithm) Let $r_{-1}=M$, $r_0=k$.
If $r_{i-1}\neq 0$ for $i\geq 1$, then define $q_i\geq 1$ and $0\leq r_i<r_{i-1}$ recursively
as the quotient and the remainder, respectively, of $r_{i-2}$ divided by $r_{i-1}$ so that
\beqnarray{Euclid's algorithm}
r_{i-2}=q_i\cdot r_{i-1}+r_i.
\eeqnarray
It is well known that such a recursive process will stop after a finite number of recursions
when the remainder is zero, say $r_N=0$ for some $N\geq 1$,
at which point we obtain the greatest common divisor of $M$ and $k$ as $\gcd(M,k)=r_{N-1}$.
If $N$ is an odd integer, then go to Step 2;
otherwise, if $N$ is an even integer, then go to Step 3.
Step 2. (i) Let $\nbf_1^{r_{N-1}}(N)$ be given by $n_j(N)=q_N$ for $1\leq j\leq r_{N-1}$.
For $i=N-2,N-4,\ldots,1$ (in that order), recursively compute
\beqnarray{}
\nbf_1^{r_{i}}(i+1)\aligneq R_{r_{i-1},r_{i}}(\nbf_1^{r_{i+1}}(i+2)),\label{eqn:main result-111}\\
\nbf_1^{r_{i-1}}(i)\aligneq L_{r_{i-2},r_{i-1}}(\nbf_1^{r_{i}}(i+1)).\label{eqn:main result-222}
\eeqnarray
(ii) If $r_{N-1}\geq 2$, let $\mbf_1^{r_{N-1}}(N)$ be given by
$m_1(N)=q_N+1$, $m_{r_{N-1}}(N)=q_N-1$
(note that $q_N\geq 2$ as $r_{N-2}=q_N\cdot r_{N-1}+r_N=q_N\cdot r_{N-1}$ and $r_{N-2}>r_{N-1}$),
and $m_j(N)=q_N$ for $2\leq j\leq r_{N-1}-1$.
For $i=N-2,N-4,\ldots,1$ (in that order), recursively compute
\beqnarray{}
\mbf_1^{r_{i}}(i+1)\aligneq R_{r_{i-1},r_{i}}(\mbf_1^{r_{i+1}}(i+2)),\label{eqn:main result-333}\\
\mbf_1^{r_{i-1}}(i)\aligneq L_{r_{i-2},r_{i-1}}(\mbf_1^{r_{i}}(i+1)).\label{eqn:main result-444}
\eeqnarray
Step 3. (i) Let $\nbf_1^{r_{N-1}}(N)$ be given by $n_j(N)=q_N$ for $1\leq j\leq r_{N-1}$.
First compute
\beqnarray{main result-555}
\nbf_1^{r_{N-2}}(N-1)=L_{r_{N-3},r_{N-2}}(\nbf_1^{r_{N-1}}(N)).
\eeqnarray
Then for $i=N-3,N-5,\ldots,1$ (in that order), recursively compute
\beqnarray{}
\nbf_1^{r_{i}}(i+1)\aligneq R_{r_{i-1},r_{i}}(\nbf_1^{r_{i+1}}(i+2)),\label{eqn:main result-666}\\
\nbf_1^{r_{i-1}}(i)\aligneq L_{r_{i-2},r_{i-1}}(\nbf_1^{r_{i}}(i+1)).\label{eqn:main result-777}
\eeqnarray
(ii) If $r_{N-1}\geq 2$, let $\mbf_1^{r_{N-1}}(N)$ be given by
$m_1(N)=q_N-1$ (note that $q_N\geq 2$), $m_{r_{N-1}}(N)=q_N+1$,
and $m_j(N)=q_N$ for $2\leq j\leq r_{N-1}-1$.
First compute
\beqnarray{main result-888}
\mbf_1^{r_{N-2}}(N-1)=L_{r_{N-3},r_{N-2}}(\mbf_1^{r_{N-1}}(N)).
\eeqnarray
Then for $i=N-3,N-5,\ldots,1$ (in that order), recursively compute
\beqnarray{}
\mbf_1^{r_{i}}(i+1)\aligneq R_{r_{i-1},r_{i}}(\mbf_1^{r_{i+1}}(i+2)),\label{eqn:main result-999}\\
\mbf_1^{r_{i-1}}(i)\aligneq L_{r_{i-2},r_{i-1}}(\mbf_1^{r_{i}}(i+1)).\label{eqn:main result-aaa}
\eeqnarray
\ealgorithm
Note that in Step~2(i) (for the case that $N$ is an odd integer),
we begin with $\nbf_1^{r_{N-1}}(N)=(q_N,q_N,\ldots,q_N)$ and then compute
\beqnarray{}
\nbf_1^{r_{N-2}}(N-1) \aligneq R_{r_{N-3},r_{N-2}}(\nbf_1^{r_{N-1}}(N)), \nn\\
\nbf_1^{r_{N-3}}(N-2) \aligneq L_{r_{N-4},r_{N-3}}(\nbf_1^{r_{N-2}}(N-1)), \nn\\
\nbf_1^{r_{N-4}}(N-3) \aligneq R_{r_{N-5},r_{N-4}}(\nbf_1^{r_{N-3}}(N-2)), \nn\\
\nbf_1^{r_{N-5}}(N-4) \aligneq L_{r_{N-6},r_{N-5}}(\nbf_1^{r_{N-4}}(N-3)), \nn\\
&\vdots& \nn\\
\nbf_1^{r_1}(2) \aligneq R_{r_0,r_1}(\nbf_1^{r_2}(3)), \nn\\
\nbf_1^{r_0}(1) \aligneq L_{r_{-1},r_0}(\nbf_1^{r_1}(2)). \nn
\eeqnarray
As $r_N=0$, we have $\sum_{i=1}^{r_{N-1}}n_i(N)=q_N\cdot r_{N-1}=q_N\cdot r_{N-1}+r_N=r_{N-2}$,
and hence the right pre-sequence $\nbf_1^{r_{N-2}}(N-1)=R_{r_{N-3},r_{N-2}}(\nbf_1^{r_{N-1}}(N))$
of the sequence $\nbf_1^{r_{N-1}}(N)$ with respect to $r_{N-3}$ and $r_{N-2}$ is well defined.
From the definition of right pre-sequences in \rdefinition{right pre-sequences},
we can see that $\sum_{i=1}^{r_{N-2}}n_i(N-1)=r_{N-3}$,
and thus the left pre-sequence $\nbf_1^{r_{N-3}}(N-2)=L_{r_{N-4},r_{N-3}}(\nbf_1^{r_{N-2}}(N-1))$
of the sequence $\nbf_1^{r_{N-2}}(N-1)$ with respect to $r_{N-4}$ and $r_{N-3}$ is well defined.
We can repeat the above argument and see that
the right pre-sequence $\nbf_1^{r_{N-4}}(N-3)=R_{r_{N-5},r_{N-4}}(\nbf_1^{r_{N-3}}(N-2))$
of the sequence $\nbf_1^{r_{N-3}}(N-2)$ with respect to $r_{N-5}$ and $r_{N-4}$ is well defined,
the left pre-sequence $\nbf_1^{r_{N-5}}(N-4)=L_{r_{N-6},r_{N-5}}(\nbf_1^{r_{N-4}}(N-3))$
of the sequence $\nbf_1^{r_{N-4}}(N-3)$ with respect to $r_{N-6}$ and $r_{N-5}$ is well defined, $\ldots$,
the right pre-sequence $\nbf_1^{r_1}(2)=R_{r_0,r_1}(\nbf_1^{r_2}(3))$
of the sequence $\nbf_1^{r_2}(3)$ with respect to $r_0$ and $r_1$ is well defined,
and the left pre-sequence $\nbf_1^{r_0}(1)=L_{r_{-1},r_0}(\nbf_1^{r_1}(2))$
of the sequence $\nbf_1^{r_1}(2)$ with respect to $r_{-1}$ and $r_0$ is well defined.
Similarly, in Step~2(ii) (for the case that $N$ is an odd integer and $r_{N-1}\geq 2$),
we begin with $\mbf_1^{r_{N-1}}(N)=(q_N+1,q_N,\ldots,q_N,q_N-1)$
and we can see that the left pre-sequences and the right pre-sequences in Step~2(ii) are all well defined.
Also note that in Step~3(i) (for the case that $N$ is an even integer),
we begin with $\nbf_1^{r_{N-1}}(N)=(q_N,q_N,\ldots,q_N)$ and then compute
\beqnarray{}
\nbf_1^{r_{N-2}}(N-1) \aligneq L_{r_{N-3},r_{N-2}}(\nbf_1^{r_{N-1}}(N)), \nn\\
\nbf_1^{r_{N-3}}(N-2) \aligneq R_{r_{N-4},r_{N-3}}(\nbf_1^{r_{N-2}}(N-1)), \nn\\
\nbf_1^{r_{N-4}}(N-3) \aligneq L_{r_{N-5},r_{N-4}}(\nbf_1^{r_{N-3}}(N-2)), \nn\\
&\vdots& \nn\\
\nbf_1^{r_1}(2) \aligneq R_{r_0,r_1}(\nbf_1^{r_2}(3)), \nn\\
\nbf_1^{r_0}(1) \aligneq L_{r_{-1},r_0}(\nbf_1^{r_1}(2)). \nn
\eeqnarray
As in Step~2(i), we can argue that the left pre-sequences and the right pre-sequences in Step~3(i)
are all well defined.
Similarly, in Step~3(ii) (for the case that $N$ is an even integer and $r_{N-1}\geq 2$),
we begin with $\mbf_1^{r_{N-1}}(N)=(q_N-1,q_N,\ldots,q_N,q_N+1)$
and we can see that the left pre-sequences and the right pre-sequences in Step~3(ii) are all well defined.
In the following, we give a few examples to illustrate how
\rtheorem{main result} and \ralgorithm{main result} work.
\bexample{main result-1}
Suppose that $M=11$ and $k=3$.
In Step 1 of \ralgorithm{main result},
we obtain $r_{-1}=11$, $r_0=3$, $q_1=3$, $r_1=2$, $q_2=1$, $r_2=1$, $q_3=2$, and $r_3=0$.
As a result, we have $N=3$ and $\gcd(M,k)=r_{N-1}=r_2=1$.
Since $N=3$ is an odd integer and $r_{N-1}=1$,
we proceed to Step 2(i) of \ralgorithm{main result} and obtain
\beqnarray{}
\alignspace \nbf_1^{r_2}(3)=(q_3)=(2), \nn\\
\alignspace \nbf_1^{r_1}(2)=R_{r_0,r_1}(\nbf_1^{r_2}(3))=R_{3,2}((2))=(1,2), \nn\\
\alignspace \nbf_1^k(1)=\nbf_1^{r_0}(1)=L_{r_{-1},r_0}(\nbf_1^{r_1}(2))=L_{11,3}((1,2))=(4,4,3). \nn
\eeqnarray
It follows from \rtheorem{main result}(i) that
there is only one optimal sequence over $\Ncal_{M,k}$,
and the optimal sequence is given by $\nbf_1^k(1)=(4,4,3)$.
\eexample
\bexample{main result-2}
Suppose that $M=13$ and $k=5$.
In Step 1 of \ralgorithm{main result},
we obtain $r_{-1}=13$, $r_0=5$, $q_1=2$, $r_1=3$, $q_2=1$, $r_2=2$,
$q_3=1$, $r_3=1$, $q_4=2$, and $r_4=0$.
Consequently, we have $N=4$ and $\gcd(M,k)=r_{N-1}=r_3=1$.
Since $N=4$ is an even integer and $r_{N-1}=1$,
we proceed to Step 3(i) of \ralgorithm{main result} and obtain
\beqnarray{}
\alignspace \nbf_1^{r_3}(4)=(q_4)=(2), \nn\\
\alignspace \nbf_1^{r_2}(3)=L_{r_1,r_2}(\nbf_1^{r_3}(4))=L_{3,2}((2))=(2,1), \nn\\
\alignspace \nbf_1^{r_1}(2)=R_{r_0,r_1}(\nbf_1^{r_2}(3))=R_{5,3}((2,1))=(1,2,2), \nn\\
\alignspace \nbf_1^k(1)=\nbf_1^{r_0}(1)=L_{r_{-1},r_0}(\nbf_1^{r_1}(2))=L_{13,5}((1,2,2))=(3,3,2,3,2). \nn
\eeqnarray
It follows from \rtheorem{main result}(i) that
there is only one optimal sequence over $\Ncal_{M,k}$,
and the optimal sequence is given by $\nbf_1^k(1)=(3,3,2,3,2)$.
\eexample
\bexample{main result-3}
Suppose that $M=16$ and $k=6$.
In Step 1 of \ralgorithm{main result},
we obtain $r_{-1}=16$, $r_0=6$, $q_1=2$, $r_1=4$, $q_2=1$, $r_2=2$,
$q_3=2$, and $r_3=0$.
As a result, we have $N=3$ and $\gcd(M,k)=r_{N-1}=r_2=2$.
Since $N=3$ is an odd integer and $r_{N-1}=2$,
we proceed to Step 2(i) and Step 2(ii) of \ralgorithm{main result} and obtain
\beqnarray{}
\alignspace \nbf_1^{r_2}(3)=(q_3,q_3)=(2,2), \nn\\
\alignspace \nbf_1^{r_1}(2)=R_{r_0,r_1}(\nbf_1^{r_2}(3))=R_{6,4}((2,2))=(1,2,1,2), \nn\\
\alignspace \nbf_1^k(1)=\nbf_1^{r_0}(1)=L_{r_{-1},r_0}(\nbf_1^{r_1}(2))=L_{16,6}((1,2,1,2))=(3,3,2,3,3,2); \nn\\
\alignspace \mbf_1^{r_2}(3)=(q_3+1,q_3-1)=(3,1), \nn\\
\alignspace \mbf_1^{r_1}(2)=R_{r_0,r_1}(\mbf_1^{r_2}(3))=R_{6,4}((3,1))=(1,1,2,2), \nn\\
\alignspace \mbf_1^k(1)=\mbf_1^{r_0}(1)=L_{r_{-1},r_0}(\mbf_1^{r_1}(2))=L_{16,6}((1,1,2,2))=(3,3,3,2,3,2). \nn
\eeqnarray
It follows from \rtheorem{main result}(ii) that
there are two optimal sequences over $\Ncal_{M,k}$,
and the two optimal sequences are given by
$\nbf_1^k(1)=(3,3,2,3,3,2)$ and $\mbf_1^k(1)=(3,3,3,2,3,2)$.
\eexample
\bexample{main result-4}
Suppose that $M=26$ and $k=10$.
In Step 1 of \ralgorithm{main result},
we obtain $r_{-1}=26$, $r_0=10$, $q_1=2$, $r_1=6$, $q_2=1$, $r_2=4$,
$q_3=1$, $r_3=2$, $q_4=2$, and $r_4=0$.
Consequently, we have $N=4$ and $\gcd(M,k)=r_{N-1}=r_3=2$.
Since $N=4$ is an even integer and $r_{N-1}=2$,
we proceed to Step 3(i) and Step 3(ii) of \ralgorithm{main result} and obtain
\beqnarray{}
\alignspace \nbf_1^{r_3}(4)=(q_4,q_4)=(2,2), \nn\\
\alignspace \nbf_1^{r_2}(3)=L_{r_1,r_2}(\nbf_1^{r_3}(4))=L_{6,4}((2,2))=(2,1,2,1), \nn\\
\alignspace \nbf_1^{r_1}(2)=R_{r_0,r_1}(\nbf_1^{r_2}(3))=R_{10,6}((2,1,2,1))=(1,2,2,1,2,2), \nn\\
\alignspace \nbf_1^k(1)=\nbf_1^{r_0}(1)=L_{r_{-1},r_0}(\nbf_1^{r_1}(2))=L_{26,10}((1,2,2,1,2,2))=(3,3,2,3,2,3,3,2,3,2); \nn\\
\alignspace \mbf_1^{r_3}(4)=(q_4-1,q_4+1)=(1,3), \nn\\
\alignspace \mbf_1^{r_2}(3)=L_{r_1,r_2}(\mbf_1^{r_3}(4))=L_{6,4}((1,3))=(2,2,1,1), \nn\\
\alignspace \mbf_1^{r_1}(2)=R_{r_0,r_1}(\mbf_1^{r_2}(3))=R_{10,6}((2,2,1,1))=(1,2,1,2,2,2), \nn\\
\alignspace \mbf_1^k(1)=\mbf_1^{r_0}(1)=L_{r_{-1},r_0}(\mbf_1^{r_1}(2))=L_{26,10}((1,2,1,2,2,2))=(3,3,2,3,3,2,3,2,3,2). \nn
\eeqnarray
It follows from \rtheorem{main result}(ii) that
there are two optimal sequences over $\Ncal_{M,k}$,
and the two optimal sequences are given by
$\nbf_1^k(1)=(3,3,2,3,2,3,3,2,3,2)$ and $\mbf_1^k(1)=(3,3,2,3,3,2,3,2,3,2)$.
\eexample
\bexample{main result-5}
Suppose that $M=24$ and $k=9$.
In Step 1 of \ralgorithm{main result},
we obtain $r_{-1}=24$, $r_0=9$, $q_1=2$, $r_1=6$, $q_2=1$, $r_2=3$,
$q_3=2$, and $r_3=0$.
As a result, we have $N=3$ and $\gcd(M,k)=r_{N-1}=r_2=3$.
Since $N=3$ is an odd integer and $r_{N-1}=3$,
we proceed to Step 2(i) and Step 2(ii) of \ralgorithm{main result} and obtain
\beqnarray{}
\alignspace \nbf_1^{r_2}(3)=(q_3,q_3,q_3)=(2,2,2), \nn\\
\alignspace \nbf_1^{r_1}(2)=R_{r_0,r_1}(\nbf_1^{r_2}(3))=R_{9,6}((2,2,2))=(1,2,1,2,1,2), \nn\\
\alignspace \nbf_1^k(1)=\nbf_1^{r_0}(1)=L_{r_{-1},r_0}(\nbf_1^{r_1}(2))=L_{24,9}((1,2,1,2,1,2))=(3,3,2,3,3,2,3,3,2); \nn\\
\alignspace \mbf_1^{r_2}(3)=(q_3+1,q_3,q_3-1)=(3,2,1), \nn\\
\alignspace \mbf_1^{r_1}(2)=R_{r_0,r_1}(\mbf_1^{r_2}(3))=R_{9,6}((3,2,1))=(1,1,2,1,2,2), \nn\\
\alignspace \mbf_1^k(1)=\mbf_1^{r_0}(1)=L_{r_{-1},r_0}(\mbf_1^{r_1}(2))=L_{24,9}((1,1,2,1,2,2))=(3,3,3,2,3,3,2,3,2). \nn
\eeqnarray
It follows from \rtheorem{main result}(iii) that
there are at most two optimal sequences over $\Ncal_{M,k}$,
and the two possible optimal sequences are given by
$\nbf_1^k(1)=(3,3,2,3,3,2,3,3,2)$ and $\mbf_1^k(1)=(3,3,3,2,3,3,2,3,2)$.
\eexample
\bexample{main result-6}
Suppose that $M=39$ and $k=15$.
In Step 1 of \ralgorithm{main result},
we obtain $r_{-1}=39$, $r_0=15$, $q_1=2$, $r_1=9$, $q_2=1$, $r_2=6$,
$q_3=1$, $r_3=3$, $q_4=2$, and $r_4=0$.
Consequently, we have $N=4$ and $\gcd(M,k)=r_{N-1}=r_3=3$.
Since $N=4$ is an even integer and $r_{N-1}=3$,
we proceed to Step 3(i) and Step 3(ii) of \ralgorithm{main result} and obtain
\beqnarray{}
\alignspace \nbf_1^{r_3}(4)=(q_4,q_4,q_4)=(2,2,2), \nn\\
\alignspace \nbf_1^{r_2}(3)=L_{r_1,r_2}(\nbf_1^{r_3}(4))=L_{9,6}((2,2,2))=(2,1,2,1,2,1), \nn\\
\alignspace \nbf_1^{r_1}(2)=R_{r_0,r_1}(\nbf_1^{r_2}(3))=R_{15,9}((2,1,2,1,2,1))=(1,2,2,1,2,2,1,2,2), \nn\\
\alignspace \nbf_1^k(1)=\nbf_1^{r_0}(1)=L_{r_{-1},r_0}(\nbf_1^{r_1}(2))=L_{39,15}((1,2,2,1,2,2,1,2,2)) \nn\\
\alignspace \hspace*{0.43in} =(3,3,2,3,2,3,3,2,3,2,3,3,2,3,2); \nn\\
\alignspace \mbf_1^{r_3}(4)=(q_4-1,q_4,q_4+1)=(1,2,3), \nn\\
\alignspace \mbf_1^{r_2}(3)=L_{r_1,r_2}(\mbf_1^{r_3}(4))=L_{9,6}((1,2,3))=(2,2,1,2,1,1), \nn\\
\alignspace \mbf_1^{r_1}(2)=R_{r_0,r_1}(\mbf_1^{r_2}(3))=R_{15,9}((2,2,1,2,1,1))=(1,2,1,2,2,1,2,2,2), \nn\\
\alignspace \mbf_1^k(1)=\mbf_1^{r_0}(1)=L_{r_{-1},r_0}(\mbf_1^{r_1}(2))=L_{39,15}((1,2,1,2,2,1,2,2,2))\nn\\
\alignspace \hspace*{0.48in} =(3,3,2,3,3,2,3,2,3,3,2,3,2,3,2). \nn
\eeqnarray
It follows from \rtheorem{main result}(iii) that
there are at most two optimal sequences over $\Ncal_{M,k}$,
and the two possible optimal sequences are given by
$\nbf_1^k(1)=(3,3,2,3,2,3,3,2,3,2,3,3,2,3,2)$ and $\mbf_1^k(1)=(3,3,2,3,3,2,3,2,3,3,2,3,2,3,2)$.
\eexample
\iffalse
Before we give the proof of \rtheorem{main result} in \rsection{proof of the main result} below,
we note that when $\gcd(M,k)=2$,
we see from \rtheorem{main result}(ii) that
the two sequences $\nbf_1^k(1)$ and $\mbf_1^k(1)$
obtained in Step 2 or Step 3 of \ralgorithm{main result}
(depending on which of the two steps is executed in \ralgorithm{main result})
are both optimal sequences over $\Ncal_{M,k}$
and there are no other optimal sequences over $\Ncal_{M,k}$.
When $\gcd(M,k)\geq 3$,
we see from \rtheorem{main result}(iii) that
they are the only two possible optimal sequences over $\Ncal_{M,k}$.
However, our numerical results show that when $\gcd(M,k)\geq 3$,
they both are optimal sequences over $\Ncal_{M,k}$.
Therefore, we make the following conjecture.
\bconjecture{main result}
Suppose that $M\geq 2$ and $1\leq k\leq M-1$.
Suppose that $\gcd(M,k)\geq 2$.
Then there are two optimal sequences over $\Ncal_{M,k}$,
and the two optimal sequences are given by the two sequences
$\nbf_1^k(1)$ and $\mbf_1^k(1)$
obtained in Step 2 or Step 3 of \ralgorithm{main result}
(depending on which of the two steps is executed in \ralgorithm{main result}).
\econjecture
In \rappendix{N=1} and \rappendix{N=2}
(note that \rappendix{proof of adjacent distance larger than one with h=1}--\rappendix{proof of main lemma II}
will be devoted to the proofs of the lemmas that are needed for the proof of \rtheorem{main result}),
we show that \rconjecture{main result} holds for the two special cases $N=1$ and $N=2$, respectively,
where $N$ is obtained as in Step~1 of \ralgorithm{main result}.
\fi
\bsection{Proof of \rtheorem{main result}}{proof of the main result}
The main idea in our proof of \rtheorem{main result}
is to use \emph{pairwise comparison} to remove a sequence $\nbf_1^M\in \Ncal_{M,k}$
such that $B(\dbf_1^M;k)<B({\dbf'}_1^M;k)$ for some ${\nbf'}_1^M\in \Ncal_{M,k}$,
where $\dbf_1^M$ and ${\dbf'}_1^M$ are obtained by using $\nbf_1^M$ and ${\nbf'}_1^M$,
respectively, in \reqnarray{OQ-LR-delays-greedy-1}.
To simplify the presentation of the proof,
we first introduce a few notations that will be used in the proof of \rtheorem{main result}.
Suppose that $M\geq 2$ and $1\leq k\leq M-1$. Let $r_{-1}=M$, $r_0=k$, and let $q_i$ and $r_i$, $i=1,2,\ldots,N$,
be recursively obtained as in Step 1 of \ralgorithm{main result}.
For $1\leq h\leq N$, let $\Ncal_{M,k}(h)$ be the set of sequences of positive integers
$\nbf_1^{r_{h-1}}(h)=(n_1(h),n_2(h),\ldots,n_{r_{h-1}}(h))$
such that: (i) $\sum_{i=1}^{r_{h-1}}n_i(h)=r_{h-2}$, and (ii) $n_1(h)\geq 2$ in the case that $h=1$,
i.e.,
\beqnarray{N-M-k-h}
\Ncal_{M,k}(h)=
\bselection
\{\nbf_1^{r_{h-1}}(h)\in (\Zbf^+)^{r_{h-1}}:
\sum_{i=1}^{r_{h-1}}n_i(h)=r_{h-2} \textrm{ and } n_1(h)\geq 2\}, &\textrm{if } h=1, \\
\{\nbf_1^{r_{h-1}}(h)\in (\Zbf^+)^{r_{h-1}}:
\sum_{i=1}^{r_{h-1}}n_i(h)=r_{h-2}\}, &\textrm{if } 2\leq h\leq N.
\eselection
\eeqnarray
Note that as $r_{-1}=M$ and $r_0=k$,
it is clear from \reqnarray{N-M-k-h} and \reqnarray{N-M-k} that $\Ncal_{M,k}(1)=\Ncal_{M,k}$.
Let $\nbf_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$ and ${\nbf'}_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$,
where $1\leq h\leq N$.
If $2\leq h\leq N$ and $h$ is an odd integer,
then we recursively compute for $i=h-2,h-4,\ldots,1$ (in that order)
\beqnarray{}
\nbf_1^{r_i}(i+1)\aligneq R_{r_{i-1},r_i}(\nbf_1^{r_{i+1}}(i+2)), \label{eqn:order relation-111}\\
\nbf_1^{r_{i-1}}(i)\aligneq L_{r_{i-2},r_{i-1}}(\nbf_1^{r_i}(i+1)). \label{eqn:order relation-222}
\eeqnarray
If $2\leq h\leq N$ and $h$ is an even integer,
then we first compute
\beqnarray{order relation-333}
\nbf_1^{r_{h-2}}(h-1)=L_{r_{h-3},r_{h-2}}(\nbf_1^{r_{h-1}}(h)),
\eeqnarray
and then we recursively compute for $i=h-3,h-5,\ldots,1$ (in that order)
\beqnarray{}
\nbf_1^{r_i}(i+1)\aligneq R_{r_{i-1},r_i}(\nbf_1^{r_{i+1}}(i+2)), \label{eqn:order relation-444}\\
\nbf_1^{r_{i-1}}(i)\aligneq L_{r_{i-2},r_{i-1}}(\nbf_1^{r_i}(i+1)). \label{eqn:order relation-555}
\eeqnarray
Therefore, by using $\nbf_1^{r_{h-1}}(h)$,
we obtain sequences of positive integers $\nbf_1^{r_{i-1}}(i)$, $i=1,2,\ldots,h-1$,
such that: (i) $\sum_{j=1}^{r_{i-1}}n_j(i)=r_{i-2}$
(according to the definitions of left pre-sequences in \rdefinition{left pre-sequences}
and right pre-sequences in \rdefinition{right pre-sequences}),
and (ii) $n_1(i)=q_i+1\geq 2$ in the case that $i$ is an odd integer
(according to the definition of left pre-sequences in \rdefinition{left pre-sequences}).
As such, we see from \reqnarray{N-M-k-h} that
$\nbf_1^{r_{i-1}}(i)\in \Ncal_{M,k}(i)$ for $i=1,2,\ldots,h-1$.
Similarly, by using ${\nbf'}_1^{r_{h-1}}(h)$,
we can also obtain ${\nbf'}_1^{r_{i-1}}(i)\in \Ncal_{M,k}(i)$ for $i=1,2,\ldots,h-1$.
Let $\dbf_1^M$ and ${\dbf'}_1^M$ be obtained by using $\nbf_1^k(1)=\nbf_1^{r_0}(1)$
and ${\nbf'}_1^k(1)={\nbf'}_1^{r_0}(1)$, respectively, in \reqnarray{OQ-LR-delays-greedy-1}.
We define the binary relation $\prec$ (resp., $\equiv$, $\succ$, $\preceq$, $\succeq$)
on $\Ncal_{M,k}(h)$ as follows:
\beqnarray{order relation-666}
\nbf_1^{r_{h-1}}(h)\prec (\textrm{resp.}, \equiv, \succ, \preceq, \succeq)\
{\nbf'}_1^{r_{h-1}}(h)
\textrm{ if }
B(\dbf_1^M;k)< (\textrm{resp.}, =, >, \leq, \geq)\ B({\dbf'}_1^M;k).
\eeqnarray
We call a sequence of positive integers $\nbf_1^{r_{h-1}}(h)$
an \emph{optimal} sequence over $\Ncal_{M,k}(h)$
if $\nbf_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$
and $\nbf_1^{r_{h-1}}(h)\succeq{\nbf'}_1^{r_{h-1}}(h)$
for all ${\nbf'}_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$.
It is clear that if $\nbf_1^{r_{i-1}}(i)$ (resp., ${\nbf'}_1^{r_{i-1}}(i)$),
where $1\leq i\leq h-1$,
is given by \reqnarray{order relation-111}--\reqnarray{order relation-222}
or \reqnarray{order relation-333}--\reqnarray{order relation-555}
(depending on whether $h$ is an odd or an even integer),
then we have from the definition of the binary relation
$\prec$ (resp., $\equiv$, $\succ$, $\preceq$, $\succeq$)
in \reqnarray{order relation-666} that
\beqnarray{order relation-777}
\nbf_1^{r_{h-1}}(h)\prec
(\textrm{resp.}, \equiv, \succ, \preceq, \succeq)\ {\nbf'}_1^{r_{h-1}}(h)
\textrm{ iff } \nbf_1^{r_{i-1}}(i)\prec
(\textrm{resp.}, \equiv, \succ, \preceq, \succeq)\ {\nbf'}_1^{r_{i-1}}(i).
\eeqnarray
We need the following theorem from Part~I to prove \rtheorem{main result}.
\btheorem{OQ-LR-delays-greedy} \cite{CCCLL10}
Suppose that $M\geq 2$ and $1\leq k\leq M-1$.
Let $\dbf_1^M\in \Gcal_{M,k}$ so that there exists a sequence $\nbf_1^k\in \Ncal_{M,k}$
such that $d_{s_i+j}$ is given by \reqnarray{OQ-LR-delays-greedy-1},
i.e., $d_{s_i+j}=B(\dbf_1^{s_i+j-1};i+1)+1$,
for $i=0,1,\ldots,k-1$ and $j=1,2,\ldots,n_{i+1}$,
where $s_0=0$ and $s_i=\sum_{\ell=1}^{i}n_{\ell}$ for $i=1,2,\ldots,k$.
Then $d_1,d_2,\ldots,d_M$ can be recursively expressed as
\beqnarray{}
\alignspace
d_j=j, \textrm{ for } j=1,2,\ldots,s_1,
\label{eqn:OQ-LR-delays-greedy-2}\\
\alignspace
d_{s_i+j}=2d_{s_i}+(j-1)(d_{s_1}+d_{s_2}+\cdots+d_{s_i}+1), \nn\\
\alignspace \hspace*{0.2in}
\textrm{ for } i=1,2,\ldots,k-1 \textrm{ and } j=1,2,\ldots,n_{i+1},
\label{eqn:OQ-LR-delays-greedy-3}
\eeqnarray
and we have
\beqnarray{OQ-LR-delays-greedy-4}
\dbf_1^{s_i+j}\in \Bcal_{s_i+j},
\textrm{ for } i=0,1,\ldots,k-1 \textrm{ and } j=1,2,\ldots,n_{i+1}.
\eeqnarray
Furthermore, we have
\beqnarray{}
\alignspace
B(\dbf_1^j;1)=j, \textrm{ for } j=1,2,\ldots,s_1,
\label{eqn:OQ-LR-delays-greedy-5}\\
\alignspace
B(\dbf_1^{s_i+j};i+1)=d_{s_i+j}+d_{s_1}+d_{s_2}+\cdots+d_{s_i}, \nn\\
\alignspace \hspace*{0.1in}
\textrm{ for } i=1,2,\ldots,k-1 \textrm{ and } j=1,2,\ldots,n_{i+1}.
\label{eqn:OQ-LR-delays-greedy-6}
\eeqnarray
In particular, we have
\beqnarray{OQ-LR-delays-greedy-7}
B(\dbf_1^{s_i};i)=d_{s_1}+d_{s_2}+\cdots+d_{s_i}, \textrm{ for } i=1,2,\ldots,k.
\eeqnarray
\etheorem
We also need the following eight lemmas
(i.e., \rlemma{adjacent distance larger than one}, \rlemma{comparison rule A},
\rlemma{nonadjacent distance larger than one}, \rlemma{main lemma},
\rlemma{adjacent distance larger than one II}, \rlemma{comparison rule B},
\rlemma{nonadjacent distance larger than one II}, and \rlemma{main lemma II})
to prove \rtheorem{main result}.
The first four lemmas are for the case that $1\leq h\leq N$ is an odd integer
and the last four lemmas are for the case that $1\leq h\leq N$ is an even integer.
The proofs of these lemmas are given in
\rappendix{proof of adjacent distance larger than one with h=1}--\rappendix{proof of main lemma II}.
Note that if $r_{h-1}=1$,
then it is clear from \reqnarray{N-M-k-h} that $\Ncal_{M,k}(h)=\{(r_{h-2})\}$.
As $(r_{h-2})$ is the only sequence in $\Ncal_{M,k}(h)$,
it is the only optimal sequence over $\Ncal_{M,k}(h)$.
Therefore, we only consider the nontrivial case that $r_{h-1}\geq 2$
in the following eight lemmas.
In the following lemma, we show some pairwise comparison results
for a sequence $\nbf_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$,
where $1\leq h\leq N$ is an odd integer and $r_{h-1}\geq 2$,
such that the absolute value of the difference of two ``adjacent'' entries
of $\nbf_1^{r_{h-1}}(h)$ is greater than or equal to two.
\blemma{adjacent distance larger than one}
Suppose that $M\geq 2$ and $1\leq k\leq M-1$. Let $r_{-1}=M$, $r_0=k$, and let $q_i$ and $r_i$, $i=1,2,\ldots,N$, be recursively obtained
as in Step 1 of \ralgorithm{main result}.
Assume that $1\leq h\leq N$ is an odd integer and $r_{h-1}\geq 2$.
Let $\nbf_1^{r_{h-1}}(h)=(n_1(h),n_2(h),\ldots,n_{r_{h-1}}(h))\in \Ncal_{M,k}(h)$.
(i) Suppose that $n_a(h)-n_{a+1}(h)\leq -2$ for some $1\leq a\leq r_{h-1}-1$.
Let ${\nbf'}_1^{r_{h-1}}(h)=(n'_1(h),n'_2(h),\ldots,n'_{r_{h-1}}(h))$ be a sequence of positive integers such that
$n'_a(h)=n_a(h)+1$, $n'_{a+1}(h)=n_{a+1}(h)-1$, and $n'_i(h)=n_i(h)$ for $i\neq a$ and $a+1$.
Then we have ${\nbf'}_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$ and
\beqnarray{adjacent distance larger than one-1}
\nbf_1^{r_{h-1}}(h)\prec{\nbf'}_1^{r_{h-1}}(h).
\eeqnarray
(ii) Suppose that $n_a(h)-n_{a+1}(h)\geq 2$ for some $1\leq a\leq r_{h-1}-1$.
Let ${\nbf'}_1^{r_{h-1}}(h)=(n'_1(h),n'_2(h),\ldots,n'_{r_{h-1}}(h))$ be a sequence of positive integers such that
$n'_a(h)=n_a(h)-1$, $n'_{a+1}(h)=n_{a+1}(h)+1$, and $n'_i(h)=n_i(h)$ for $i\neq a$ and $a+1$.
Then we have ${\nbf'}_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$ and
\beqnarray{adjacent distance larger than one-2}
\nbf_1^{r_{h-1}}(h)\preceq{\nbf'}_1^{r_{h-1}}(h),
\eeqnarray
where $\nbf_1^{r_{h-1}}(h)\equiv{\nbf'}_1^{r_{h-1}}(h)$ if and only if $r_{h-1}=2$ and $n_1(h)=n_2(h)+2$.
\elemma
\bexample{adjacent distance larger than one}
Suppose that $M=16$ and $k=6$.
In Step 1 of \ralgorithm{main result},
we obtain $r_{-1}=16$, $r_0=6$, $q_1=2$, $r_1=4$, $q_2=1$, $r_2=2$,
$q_3=2$, and $r_3=0$.
(i) Assume that $h=1$ and hence $r_{h-1}=r_0=6\geq 2$.
Let
\beqnarray{}
{\nbf''}_1^{r_0}(1) \aligneq (3,3,2,1,5,2), \nn\\
{\nbf'}_1^{r_0}(1) \aligneq (3,3,2,2,4,2), \nn\\
\nbf_1^{r_0}(1) \aligneq (3,3,2,3,3,2), \nn\\
{\mbf'}_1^{r_0}(1) \aligneq (3,3,2,4,2,2), \nn\\
{\mbf''}_1^{r_0}(1) \aligneq (3,3,2,5,1,2). \nn
\eeqnarray
Then it follows from \reqnarray{adjacent distance larger than one-1}
in \rlemma{adjacent distance larger than one}(i) (with $h=1$ and $a=4$) that
\beqnarray{}
{\nbf''}_1^{r_0}(1)\prec{\nbf'}_1^{r_0}(1)\prec\nbf_1^{r_0}(1). \nn
\eeqnarray
We also have from \reqnarray{adjacent distance larger than one-2}
in \rlemma{adjacent distance larger than one}(ii) (with $h=1$ and $a=4$) that
\beqnarray{}
\nbf_1^{r_0}(1)\succ{\mbf'}_1^{r_0}(1)\succ{\mbf''}_1^{r_0}(1). \nn
\eeqnarray
These results can be verified by the numerical results in \rtable{adjacent distance larger than one},
where we compute the maximum representable integers $B(\dbf_1^M;k)$ with $\dbf_1^M$ obtained by using
${\nbf''}_1^{r_0}(1)$, ${\nbf'}_1^{r_0}(1)$, $\nbf_1^{r_0}(1)$, ${\mbf'}_1^{r_0}(1)$,
and ${\mbf''}_1^{r_0}(1)$, respectively, in \reqnarray{OQ-LR-delays-greedy-1}.
\btable{htbp}{|r|c|}
\hline & $B(d_1^M;k)$ \\
\hline ${\nbf''}_1^{r_0}(1)=(3,3,2,1,5,2)$ & $3543$ \\
\hline ${\nbf'}_1^{r_0}(1)=(3,3,2,2,4,2)$ & $4327$ \\
\hline $\nbf_1^{r_0}(1)=(3,3,2,3,3,2)$ & $4599$ \\
\hline ${\mbf'}_1^{r_0}(1)=(3,3,2,4,2,2)$ & $4359$ \\
\hline ${\mbf''}_1^{r_0}(1)=(3,3,2,5,1,2)$ & $3607$ \\
\hline
\etable{adjacent distance larger than one}
{The maximum representable integers $B(\dbf_1^M;k)$ with $\dbf_1^M$ obtained by using
${\nbf''}_1^{r_0}(1)$, ${\nbf'}_1^{r_0}(1)$, $\nbf_1^{r_0}(1)$, ${\mbf'}_1^{r_0}(1)$,
and ${\mbf''}_1^{r_0}(1)$, respectively, in \reqnarray{OQ-LR-delays-greedy-1},
where $M=16$ and $k=6$.}
(ii) Assume that $h=3$ and hence $r_{h-1}=r_2=2$.
Let $\nbf_1^{r_2}(3)=(3,1)$ and ${\nbf'}_1^{r_2}(3)=(2,2)$.
As $r_2=2$ and $n_1(3)=n_2(3)+2$,
it follows from \reqnarray{adjacent distance larger than one-2}
in \rlemma{adjacent distance larger than one}(ii) (with $h=3$ and $a=1$) that
\beqnarray{}
\nbf_1^{r_2}(3)\equiv{\nbf'}_1^{r_2}(3). \nn
\eeqnarray
This result can also be verified numerically.
To see this, note that from \reqnarray{order relation-111}
and \reqnarray{order relation-222} with $h=3$, we have
\beqnarray{}
\alignspace \nbf_1^{r_1}(2)=R_{r_0,r_1}(\nbf_1^{r_2}(3))=R_{6,4}((3,1))=(1,1,2,2), \nn\\
\alignspace \nbf_1^{r_0}(1)=L_{r_{-1},r_0}(\nbf_1^{r_1}(2))=L_{16,6}((1,1,2,2))=(3,3,3,2,3,2); \nn\\
\alignspace {\nbf'}_1^{r_1}(2)=R_{r_0,r_1}({\nbf'}_1^{r_2}(3))=R_{6,4}((2,2))=(1,2,1,2), \nn\\
\alignspace {\nbf'}_1^{r_0}(1)=L_{r_{-1},r_0}({\nbf'}_1^{r_1}(2))=L_{16,6}((1,2,1,2))=(3,3,2,3,3,2). \nn
\eeqnarray
Let $\dbf_1^M$ and ${\dbf'}_1^M$ be obtained by using $\nbf_1^{r_0}(1)=(3,3,3,2,3,2)$
and ${\nbf'}_1^{r_0}(1)=(3,3,2,3,3,2)$, respectively, in \reqnarray{OQ-LR-delays-greedy-1}.
We then compute that $B(\dbf_1^M;k)=B({\dbf'}_1^M;k)=4599$
and this verifies that $\nbf_1^{r_2}(3)\equiv{\nbf'}_1^{r_2}(3)$.
\eexample
We have the following corollary to \rlemma{adjacent distance larger than one}.
\bcorollary{adjacent distance larger than one}
Suppose that $M\geq 2$ and $1\leq k\leq M-1$. Let $r_{-1}=M$, $r_0=k$, and let $q_i$ and $r_i$, $i=1,2,\ldots,N$, be recursively obtained
as in Step 1 of \ralgorithm{main result}.
Assume that $1\leq h\leq N$ is an odd integer and $r_{h-1}\geq 2$.
(i) Suppose that $r_{h-1}\neq 2$ or $r_h\neq 0$.
Then an optimal sequence $\nbf_1^{r_{h-1}}(h)$ over $\Ncal_{M,k}(h)$
must satisfy the condition that the absolute value of the difference
of any two adjacent entries of $\nbf_1^{r_{h-1}}(h)$ is less than or equal to one, i.e.,
\beqnarray{adjacent distance larger than one-3}
|n_i(h)-n_{i+1}(h)|\leq 1, \textrm{ for } i=1,2,\ldots,r_{h-1}-1.
\eeqnarray
(ii) Suppose that $r_{h-1}=2$ and $r_h=0$.
Then there are two optimal sequences over $\Ncal_{M,k}(h)$,
and the two optimal sequences, say $\nbf_1^{r_{h-1}}(h)$ and $\mbf_1^{r_{h-1}}(h)$,
are given by
\beqnarray{adjacent distance larger than one-4}
\nbf_1^{r_{h-1}}(h)=(q_h,q_h) \textrm{ and } \mbf_1^{r_{h-1}}(h)=(q_h+1,q_h-1)
\eeqnarray
(note that $q_h\geq 2$ as $r_{h-2}=q_h\cdot r_{h-1}+r_h=q_h\cdot r_{h-1}$ and $r_{h-2}>r_{h-1}$).
\ecorollary
\bproof
(i) Let $\nbf_1^{r_{h-1}}(h)$ be an optimal sequence over $\Ncal_{M,k}(h)$.
We show that $|n_i(h)-n_{i+1}(h)|\leq 1$ for $i=1,2,\ldots,r_{h-1}-1$ by contradiction.
Assume on the contrary that $|n_a(h)-n_{a+1}(h)|\geq 2$ for some $1\leq a\leq r_{h-1}-1$.
Note that in \rcorollary{adjacent distance larger than one}(i),
we have $r_{h-1}\neq 2$ or $r_h\neq 0$.
As such, if $r_{h-1}=2$, then we have $r_h\neq 0$
and it must be the case that $n_1(h)\neq n_2(h)+2$.
Otherwise, if $n_1(h)=n_2(h)+2$,
then we see from $\nbf_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$ and \reqnarray{N-M-k-h} that
\beqnarray{}
r_{h-2}=\sum_{i=1}^{r_{h-1}}n_i(h)=n_1(h)+n_2(h)=2n_2(h)+2=(n_2(h)+1)\cdot r_{h-1}. \nn
\eeqnarray
Thus, the remainder $r_h$ of $r_{h-2}$ divided by $r_{h-1}$ is equal to zero,
contradicting to $r_h\neq 0$.
Since it cannot be the case that $r_{h-1}=2$ and $n_1(h)=n_2(h)+2$,
we see from \rlemma{adjacent distance larger than one}
that there exists ${\nbf'}_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$
such that $\nbf_1^{r_{h-1}}(h)\prec{\nbf'}_1^{r_{h-1}}(h)$,
contradicting to the optimality of $\nbf_1^{r_{h-1}}(h)$.
(ii) Let $\nbf_1^{r_{h-1}}(h)$ be an optimal sequence over $\Ncal_{M,k}(h)$.
As we have from $r_{h-1}=2$ and $r_h=0$
that $r_{h-2}=q_h\cdot r_{h-1}+r_h=2q_h$
and we have from $\nbf_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$, \reqnarray{N-M-k-h}, and $r_{h-1}=2$
that $r_{h-2}=\sum_{i=1}^{r_{h-1}}n_i(h)=n_1(h)+n_2(h)$,
we see that $n_1(h)+n_2(h)=2q_h$ is an even integer.
As such, it follows that $n_1(h)-n_2(h)$ is also an even integer.
If $n_1(h)-n_2(h)\leq -2$ or $n_1(h)-n_2(h)\geq 4$,
then we see from \rlemma{adjacent distance larger than one}
that there exists ${\nbf'}_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$
such that $\nbf_1^{r_{h-1}}(h)\prec{\nbf'}_1^{r_{h-1}}(h)$,
contradicting to the optimality of $\nbf_1^{r_{h-1}}(h)$.
Therefore, we must have $n_1(h)-n_2(h)=0$ or 2,
i.e., $\nbf_1^{r_{h-1}}(h)=(q_h,q_h)$ or $\nbf_1^{r_{h-1}}(h)=(q_h+1,q_h-1)$.
It is easy to see from \rlemma{adjacent distance larger than one}(ii)
that $(q_h,q_h)\equiv (q_h+1,q_h-1)$,
and hence both $(q_h,q_h)$ and $(q_h+1,q_h-1)$ are optimal sequences over $\Ncal_{M,k}(h)$.
\eproof
In the following lemma, we show some pairwise comparison results
for a sequence $\nbf_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$,
where $1\leq h\leq N$ is an odd integer and $r_{h-1}\geq 2$,
such that the absolute value of the difference of two ``adjacent'' entries
of $\nbf_1^{r_{h-1}}(h)$ is equal to one.
\blemma{comparison rule A} (\textbf{Comparison rule A})
Suppose that $M\geq 2$ and $1\leq k\leq M-1$. Let $r_{-1}=M$, $r_0=k$, and let $q_i$ and $r_i$, $i=1,2,\ldots,N$, be recursively obtained
as in Step 1 of \ralgorithm{main result}.
Assume that $1\leq h\leq N$ is an odd integer and $r_{h-1}\geq 2$.
Let $\nbf_1^{r_{h-1}}(h)=(n_1(h),n_2(h),\ldots,n_{r_{h-1}}(h))\in \Ncal_{M,k}(h)$,
$n_a(h)-n_{a+1}(h)=1$ for some $1\leq a\leq r_{h-1}-1$,
and $n_1(h)\geq 3$ in the case that $h=1$ and $a=1$.
Let ${\nbf'}_1^{r_{h-1}}(h)=(n'_1(h),n'_2(h),\ldots,n'_{r_{h-1}}(h))$
be a sequence of positive integers such that $n'_a(h)=n_a(h)-1$, $n'_{a+1}(h)=n_{a+1}(h)+1$,
and $n'_i(h)=n_i(h)$ for $i\neq a$ and $a+1$.
Then we have ${\nbf'}_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$.
Furthermore, we have the following pairwise comparison results.
(i) Suppose that $a=1$ or $a=r_{h-1}-1$.
Then we have
\beqnarray{comparison rule A-1}
\nbf_1^{r_{h-1}}(h)\succ {\nbf'}_1^{r_{h-1}}(h).
\eeqnarray
(ii) Suppose that $2\leq a\leq r_{h-1}-2$ and there exists a positive integer $j$ such that
$1\leq j\leq \min\{a-1,r_{h-1}-a-1\}$, $n_{a-j'}(h)=n_{a+1+j'}(h)$ for $j'=1,2,\ldots,j-1$,
and $n_{a-j}(h)\neq n_{a+1+j}(h)$.
If $n_{a-j}(h)<n_{a+1+j}(h)$, then we have
\beqnarray{comparison rule A-2}
\nbf_1^{r_{h-1}}(h)\succ{\nbf'}_1^{r_{h-1}}(h).
\eeqnarray
On the other hand, if $n_{a-j}(h)>n_{a+1+j}(h)$, then we have
\beqnarray{comparison rule A-3}
\nbf_1^{r_{h-1}}(h)\preceq{\nbf'}_1^{r_{h-1}}(h),
\eeqnarray
where $\nbf_1^{r_{h-1}}(h)\equiv {\nbf'}_1^{r_{h-1}}(h)$
if and only if $a-j=1$, $a+1+j=r_{h-1}$, and $n_1(h)=n_{r_{h-1}}(h)+1$.
(iii) Suppose that $2\leq a\leq r_{h-1}-2$ and $n_{a-j'}(h)=n_{a+1+j'}(h)$
for $j'=1,2,\ldots,\min\{a-1,r_{h-1}-a-1\}$.
Then we have
\beqnarray{comparison rule A-4}
\nbf_1^{r_{h-1}}(h)\succ{\nbf'}_1^{r_{h-1}}(h).
\eeqnarray
\elemma
\bexample{comparison rule A}
Suppose that $M=16$ and $k=6$.
In Step 1 of \ralgorithm{main result},
we obtain $r_{-1}=16$, $r_0=6$, $q_1=2$, $r_1=4$, $q_2=1$, $r_2=2$,
$q_3=2$, and $r_3=0$.
Assume that $h=1$ and hence $r_{h-1}=r_0=6\geq 2$.
Let
\beqnarray{}
{\nbf'''''}_1^{r_0}(1) \aligneq (2,3,2,3,3,3), \nn\\
{\nbf''''}_1^{r_0}(1) \aligneq (3,2,2,3,3,3), \nn\\
{\nbf'''}_1^{r_0}(1) \aligneq (3,2,3,2,3,3), \nn\\
{\nbf''}_1^{r_0}(1) \aligneq (3,2,3,3,2,3), \nn\\
{\nbf'}_1^{r_0}(1) \aligneq (3,2,3,3,3,2), \nn\\
\nbf_1^{r_0}(1) \aligneq (3,3,2,3,3,2), \nn\\
\mbf_1^{r_0}(1) \aligneq (3,3,3,2,3,2). \nn
\eeqnarray
Then we have
\beqnarray{}
{\nbf'''''}_1^{r_0}(1)\prec{\nbf''''}_1^{r_0}(1)\prec{\nbf'''}_1^{r_0}(1)\prec{\nbf''}_1^{r_0}(1)
\prec{\nbf'}_1^{r_0}(1)\prec\nbf_1^{r_0}(1)\equiv\mbf_1^{r_0}(1). \nn
\eeqnarray
This can be proved by using Comparison rule A in \rlemma{comparison rule A}.
First, it is easy to see from $n_1''''(1)-n_2''''(1)=1$ and \reqnarray{comparison rule A-1}
in \rlemma{comparison rule A}(i) (with $h=1$ and $a=1$) that
\beqnarray{}
{\nbf'''''}_1^{r_0}(1)\prec{\nbf''''}_1^{r_0}(1). \nn
\eeqnarray
From $n_3'''(1)-n_4'''(1)=1$, $n_2'''(1)<n_5'''(1)$,
and \reqnarray{comparison rule A-2} in \rlemma{comparison rule A}(ii) (with $h=1$, $a=3$, and $j=1$)
we see that
\beqnarray{}
{\nbf''''}_1^{r_0}(1)\prec{\nbf'''}_1^{r_0}(1). \nn
\eeqnarray
From $n_4''(1)-n_5''(1)=1$, $n_3''(1)=n_6''(1)$,
and \reqnarray{comparison rule A-4} in \rlemma{comparison rule A}(iii) (with $h=1$ and $a=4$),
we have
\beqnarray{}
{\nbf'''}_1^{r_0}(1)\prec{\nbf''}_1^{r_0}(1). \nn
\eeqnarray
From $n_5'(1)-n_6'(1)=1$ and \reqnarray{comparison rule A-1}
in \rlemma{comparison rule A}(i) (with $h=1$ and $a=r_0-1=5$),
we have
\beqnarray{}
{\nbf''}_1^{r_0}(1)\prec{\nbf'}_1^{r_0}(1). \nn
\eeqnarray
Since $n_2(1)-n_3(1)=1$ and $n_1(1)=n_4(1)$,
it follows from \reqnarray{comparison rule A-4}
in \rlemma{comparison rule A}(iii) (with $h=1$ and $a=2$)
that
\beqnarray{}
{\nbf'}_1^{r_0}(1)\prec\nbf_1^{r_0}(1). \nn
\eeqnarray
Finally, as $m_3(1)-m_4(1)=1$, $m_2(1)=m_5(1)$, and $m_1(1)=m_6(1)+1$,
it follows from \reqnarray{comparison rule A-3} in \rlemma{comparison rule A}(ii) (with $h=1$, $a=3$, and $j=2$)
that
\beqnarray{}
\nbf_1^{r_0}(1)\equiv\mbf_1^{r_0}(1). \nn
\eeqnarray
These results can also be verified by the numerical results in \rtable{comparison rule A},
where we compute the maximum representable integers $B(\dbf_1^M;k)$ with $\dbf_1^M$ obtained by using
${\nbf'''''}_1^{r_0}(1)$, ${\nbf''''}_1^{r_0}(1)$, ${\nbf'''}_1^{r_0}(1)$, ${\nbf''}_1^{r_0}(1)$,
${\nbf'}_1^{r_0}(1)$, $\nbf_1^{r_0}(1)$, and $\mbf_1^{r_0}(1)$, respectively, in \reqnarray{OQ-LR-delays-greedy-1}.
\btable{htbp}{|r|c|}
\hline & $B(d_1^M;k)$ \\
\hline ${\nbf'''''}_1^{r_0}(1)=(2,3,2,3,3,3)$ & $4231$ \\
\hline ${\nbf''''}_1^{r_0}(1)=(3,2,2,3,3,3)$ & $4395$ \\
\hline ${\nbf'''}_1^{r_0}(1)=(3,2,3,2,3,3)$ & $4439$ \\
\hline ${\nbf''}_1^{r_0}(1)=(3,2,3,3,2,3)$ & $4455$ \\
\hline ${\nbf'}_1^{r_0}(1)=(3,2,3,3,3,2)$ & $4579$ \\
\hline $\nbf_1^{r_0}(1)=(3,3,2,3,3,2)$ & $4599$ \\
\hline $\mbf_1^{r_0}(1)=(3,3,3,2,3,2)$ & $4599$ \\
\hline
\etable{comparison rule A}
{The maximum representable integers $B(\dbf_1^M;k)$ with $\dbf_1^M$ obtained by using
${\nbf'''''}_1^{r_0}(1)$, ${\nbf''''}_1^{r_0}(1)$, ${\nbf'''}_1^{r_0}(1)$, ${\nbf''}_1^{r_0}(1)$,
${\nbf'}_1^{r_0}(1)$, $\nbf_1^{r_0}(1)$, and $\mbf_1^{r_0}(1)$, respectively,
in \reqnarray{OQ-LR-delays-greedy-1}, where $M=16$ and $k=6$.}
\eexample
From \rcorollary{adjacent distance larger than one}(i),
we know that if $1\leq h\leq N$ is an odd integer and $r_{h-1}\geq 3$,
then an optimal sequence $\nbf_1^{r_{h-1}}(h)$ over $\Ncal_{M,k}(h)$
must satisfy the condition that the absolute value of the difference
of any two adjacent entries of $\nbf_1^{r_{h-1}}(h)$ is less than or equal to one.
In the following lemma, we show some pairwise comparison results
for a sequence $\nbf_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$,
where $1\leq h\leq N$ is an odd integer and $r_{h-1}\geq 3$,
such that the absolute value of the difference of any two adjacent entries
of $\nbf_1^{r_{h-1}}(h)$ is less than or equal to one
and the absolute value of the difference of two ``nonadjacent'' entries
of $\nbf_1^{r_{h-1}}(h)$ is greater than or equal to two.
\blemma{nonadjacent distance larger than one}
Suppose that $M\geq 2$ and $1\leq k\leq M-1$. Let $r_{-1}=M$, $r_0=k$, and let $q_i$ and $r_i$, $i=1,2,\ldots,N$, be recursively obtained
as in Step 1 of \ralgorithm{main result}.
Assume that $1\leq h\leq N$ is an odd integer and $r_{h-1}\geq 3$.
Let $\nbf_1^{r_{h-1}}(h)=(n_1(h),n_2(h),\ldots,n_{r_{h-1}}(h))\in \Ncal_{M,k}(h)$
and $|n_i(h)-n_{i+1}(h)|\leq 1$ for $i=1,2,\ldots,r_{h-1}-1$.
(i) Suppose that $n_a(h)-n_b(h)\geq 2$ for some $1\leq a<b\leq r_{h-1}$ and $b\geq a+2$.
If $n_1(h)\neq n_{r_{h-1}}(h)+2$ or $n_i(h)\neq n_{r_{h-1}}(h)+1$ for some $2\leq i\leq r_{h-1}-1$,
then there exists a sequence of positive integers
${\nbf'}_1^{r_{h-1}}(h)=(n'_1(h),n'_2(h),\ldots,n'_{r_{h-1}}(h))\in \Ncal_{M,k}(h)$
such that
\beqnarray{nonadjacent distance larger than one-1}
{\nbf'}_1^{r_{h-1}}(h)\succ\nbf_1^{r_{h-1}}(h).
\eeqnarray
(ii) Suppose that $n_a(h)-n_b(h)\leq -2$ for some $1\leq a<b\leq r_{h-1}$ and $b\geq a+2$.
Then there exists a sequence of positive integers
${\nbf'}_1^{r_{h-1}}(h)=(n'_1(h),n'_2(h),\ldots,n'_{r_{h-1}}(h))\in \Ncal_{M,k}(h)$
such that
\beqnarray{nonadjacent distance larger than one-2}
{\nbf'}_1^{r_{h-1}}(h)\succ\nbf_1^{r_{h-1}}(h).
\eeqnarray
\elemma
We have the following corollary to \rlemma{nonadjacent distance larger than one}.
\bcorollary{nonadjacent distance larger than one}
Suppose that $M\geq 2$ and $1\leq k\leq M-1$. Let $r_{-1}=M$, $r_0=k$, and let $q_i$ and $r_i$, $i=1,2,\ldots,N$, be recursively obtained
as in Step 1 of \ralgorithm{main result}.
Assume that $1\leq h\leq N$ is an odd integer.
(i) Suppose that $r_{h-1}\geq 3$ and $r_h\neq 0$.
Then an optimal sequence $\nbf_1^{r_{h-1}}(h)$ over $\Ncal_{M,k}(h)$
must satisfy the condition that the absolute value of the difference
of any two entries (adjacent or nonadjacent) of $\nbf_1^{r_{h-1}}(h)$
is less than or equal to one, i.e.,
\beqnarray{nonadjacent distance larger than one-3}
|n_a(h)-n_b(h)|\leq 1, \textrm{ for all } 1\leq a<b\leq r_{h-1}.
\eeqnarray
(ii) Suppose that $r_{h-1}\geq 3$ and $r_h=0$.
Then there are at most two optimal sequences over $\Ncal_{M,k}(h)$,
and the two possible optimal sequences, say $\nbf_1^{r_{h-1}}(h)$ and $\mbf_1^{r_{h-1}}(h)$,
are given by
\beqnarray{nonadjacent distance larger than one-4}
\nbf_1^{r_{h-1}}(h)=(q_h,q_h,\ldots,q_h)
\textrm{ and } \mbf_1^{r_{h-1}}(h)=(q_h+1,q_h,\ldots,q_h,q_h-1)
\eeqnarray
(note that $q_h\geq 2$ as $r_{h-2}=q_h\cdot r_{h-1}+r_h=q_h\cdot r_{h-1}$ and $r_{h-2}>r_{h-1}$).
\ecorollary
\bproof
(i) Let $\nbf_1^{r_{h-1}}(h)$ be an optimal sequence over $\Ncal_{M,k}(h)$.
As we have $r_{h-1}\geq 3$ and hence $r_{h-1}\neq 2$,
it follows from \rcorollary{adjacent distance larger than one}(i)
that $\nbf_1^{r_{h-1}}(h)$ must satisfy the condition that
\beqnarray{proof-nonadjacent distance larger than one-corollary-111}
|n_i(h)-n_{i+1}(h)|\leq 1, \textrm{ for all } i=1,2,\ldots,r_{h-1}-1.
\eeqnarray
Furthermore, as we have $r_h\neq 0$, it must be the case that $n_1(h)\neq n_{r_{h-1}}(h)+2$
or $n_i(h)\neq n_{r_{h-1}}(h)+1$ for some $2\leq i\leq r_{h-1}-1$.
Otherwise, if $n_1(h)=n_{r_{h-1}}(h)+2$ and $n_i(h)=n_{r_{h-1}}(h)+1$
for all $2\leq i\leq r_{h-1}-1$,
then we see from $\nbf_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$ and \reqnarray{N-M-k-h} that
\beqnarray{}
r_{h-2}=\sum_{i=1}^{r_{h-1}}n_i(h)=(n_{r_{h-1}}(h)+1)\cdot r_{h-1}. \nn
\eeqnarray
Thus, the remainder $r_h$ of $r_{h-2}$ divided by $r_{h-1}$ is equal to zero,
contradicting to $r_h\neq 0$.
To show \reqnarray{nonadjacent distance larger than one-3},
it is clear from \reqnarray{proof-nonadjacent distance larger than one-corollary-111}
that it suffices to show that $|n_a(h)-n_b(h)|\leq 1$
for all $1\leq a<b\leq r_{h-1}$ and $b\geq a+2$ by contradiction.
Assume on the contrary that $|n_a(h)-n_b(h)|\geq 2$
for some $1\leq a<b\leq r_{h-1}$ and $b\geq a+2$.
Since $n_1(h)\neq n_{r_{h-1}}(h)+2$
or $n_i(h)\neq n_{r_{h-1}}(h)+1$ for some $2\leq i\leq r_{h-1}-1$,
we see from \rlemma{nonadjacent distance larger than one}
that there exists ${\nbf'}_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$
such that ${\nbf'}_1^{r_{h-1}}(h)\succ \nbf_1^{r_{h-1}}(h)$,
contradicting to the optimality of $\nbf_1^{r_{h-1}}(h)$.
(ii) Let $\nbf_1^{r_{h-1}}(h)$ be an optimal sequence over $\Ncal_{M,k}(h)$.
As we have $r_{h-1}\geq 3$,
\reqnarray{proof-nonadjacent distance larger than one-corollary-111} still holds.
Furthermore, as we have $r_h=0$,
it is clear that $r_{h-2}=q_h\cdot r_{h-1}+r_h=q_h\cdot r_{h-1}$.
It then follows from $\nbf_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$ and \reqnarray{N-M-k-h} that
\beqnarray{proof-nonadjacent distance larger than one-corollary-222}
\sum_{i=1}^{r_{h-1}}n_i(h)=r_{h-2}=q_h\cdot r_{h-1}.
\eeqnarray
We need to consider the following two cases.
\emph{Case 1: $|n_a(h)-n_b(h)|\leq 1$ for all $1\leq a<b\leq r_{h-1}$.}
In this case, it is easy to see from
\reqnarray{proof-nonadjacent distance larger than one-corollary-222} that
\beqnarray{proof-nonadjacent distance larger than one-corollary-333}
n_i(h)=q_h, \textrm{ for } i=1,2,\ldots,r_{h-1}.
\eeqnarray
\emph{Case 2: $|n_a(h)-n_b(h)|\geq 2$ for some $1\leq a<b\leq r_{h-1}$.}
In this case, it is clear from
\reqnarray{proof-nonadjacent distance larger than one-corollary-111} that $b\geq a+2$.
If $n_a(h)-n_b(h)\geq 2$ and $n_1(h)\neq n_{r_{h-1}}(h)+2$
or $n_i(h)\neq n_{r_{h-1}}(h)+1$ for some $2\leq i\leq r_{h-1}-1$,
then it follows from \rlemma{nonadjacent distance larger than one}(i)
that there exists ${\nbf'}_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$
such that ${\nbf'}_1^{r_{h-1}}(h)\succ \nbf_1^{r_{h-1}}(h)$,
contradicting to the optimality of $\nbf_1^{r_{h-1}}(h)$.
Also, if $n_a(h)-n_b(h)\leq -2$,
then it follows from \rlemma{nonadjacent distance larger than one}(ii)
that there exists ${\nbf'}_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$
such that ${\nbf'}_1^{r_{h-1}}(h)\succ \nbf_1^{r_{h-1}}(h)$,
contradicting to the optimality of $\nbf_1^{r_{h-1}}(h)$.
As such, it must be the case that
\beqnarray{proof-nonadjacent distance larger than one-corollary-444}
n_a(h)-n_b(h)\geq 2,\ n_1(h)=n_{r_{h-1}}(h)+2,
\textrm{ and } n_i(h)=n_{r_{h-1}}(h)+1 \textrm{ for } 2\leq i\leq r_{h-1}-1.
\eeqnarray
It follows from \reqnarray{proof-nonadjacent distance larger than one-corollary-222}
and \reqnarray{proof-nonadjacent distance larger than one-corollary-444} that
\beqnarray{proof-nonadjacent distance larger than one-corollary-555}
n_1(h)=q_h+1,\ n_{r_{h-1}}(h)=q_h-1,
\textrm{ and } n_i(h)=q_h \textrm{ for } 2\leq i\leq r_{h-1}-1.
\eeqnarray
By combining \reqnarray{proof-nonadjacent distance larger than one-corollary-333}
and \reqnarray{proof-nonadjacent distance larger than one-corollary-555},
we see that $(q_h,q_h,\ldots,q_h)$ and $(q_h+1,q_h,\ldots,q_h,q_h-1)$
are the two possible optimal sequences over $\Ncal_{M,k}(h)$,
and the proof is completed.
\eproof
Suppose that $1\leq h\leq N$ is an odd integer, $r_{h-1}\geq 2$, $r_h\neq 0$,
and $\nbf_1^{r_{h-1}}(h)$ is an optimal sequence over $\Ncal_{M,k}(h)$.
If $r_{h-1}=2$, then it follows from \rcorollary{adjacent distance larger than one}(i)
that $|n_1(h)-n_2(h)|\leq 1$, i.e., \reqnarray{nonadjacent distance larger than one-3} holds.
On the other hand, if $r_{h-1}\geq 3$,
then it follows from \rcorollary{nonadjacent distance larger than one}(i)
that \reqnarray{nonadjacent distance larger than one-3} also holds.
As such, we see from \reqnarray{nonadjacent distance larger than one-3},
$\sum_{i=1}^{r_{h-1}}n_i(h)=r_{h-2}$ in \reqnarray{N-M-k-h},
and $r_{h-2}=q_h\cdot r_{h-1}+r_h$ that
\beqnarray{}
n_i(h)=
\bselection
q_h+1, &\textrm{if } i=i_1,i_2,\ldots,i_{r_h}, \\
q_h, &\textrm{otherwise},
\eselection \nn
\eeqnarray
for some $1\leq i_1<i_2<\cdots <i_{r_h}\leq r_{h-1}$.
In \rlemma{main lemma} below, we further show that $i_1$ must be equal to 1.
\blemma{main lemma}
Suppose that $M\geq 2$ and $1\leq k\leq M-1$. Let $r_{-1}=M$, $r_0=k$, and let $q_i$ and $r_i$, $i=1,2,\ldots,N$, be recursively obtained
as in Step 1 of \ralgorithm{main result}.
Assume that $1\leq h\leq N$ is an odd integer, $r_{h-1}\geq 2$, and $r_h\neq 0$.
Then an optimal sequence $\nbf_1^{r_{h-1}}(h)$ over $\Ncal_{M,k}(h)$
must satisfy the condition that
\beqnarray{main lemma-1}
n_i(h)=
\bselection
q_h+1, &\textrm{if } i=i_1,i_2,\ldots,i_{r_h}, \\
q_h, &\textrm{otherwise},
\eselection
\eeqnarray
for some $1=i_1<i_2<\cdots <i_{r_h}\leq r_{h-1}$.
\elemma
The following four lemmas (for the case that $1\leq h\leq N$ is an even integer)
are the counterparts of the above four lemmas (for the case that $1\leq h\leq N$ is an odd integer).
In the following lemma, we show some pairwise comparison results
for a sequence $\nbf_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$,
where $1\leq h\leq N$ is an even integer and $r_{h-1}\geq 2$,
such that the absolute value of the difference of two ``adjacent'' entries
of $\nbf_1^{r_{h-1}}(h)$ is greater than or equal to two.
\blemma{adjacent distance larger than one II}
Suppose that $M\geq 2$ and $1\leq k\leq M-1$. Let $r_{-1}=M$, $r_0=k$, and let $q_i$ and $r_i$, $i=1,2,\ldots,N$, be recursively obtained
as in Step 1 of \ralgorithm{main result}.
Assume that $1\leq h\leq N$ is an even integer and $r_{h-1}\geq 2$.
Let $\nbf_1^{r_{h-1}}(h)=(n_1(h),n_2(h),\ldots,n_{r_{h-1}}(h))\in \Ncal_{M,k}(h)$.
(i) Suppose that $n_a(h)-n_{a+1}(h)\geq 2$ for some $1\leq a\leq r_{h-1}-1$.
Let ${\nbf'}_1^{r_{h-1}}(h)=(n'_1(h),n'_2(h),\ldots,n'_{r_{h-1}}(h))$
be a sequence of positive integers such that
$n'_a(h)=n_a(h)-1$, $n'_{a+1}(h)=n_{a+1}(h)+1$, and $n'_i(h)=n_i(h)$ for $i\neq a$ and $a+1$.
Then we have ${\nbf'}_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$ and
\beqnarray{adjacent distance larger than one II-1}
\nbf_1^{r_{h-1}}(h)\prec{\nbf'}_1^{r_{h-1}}(h).
\eeqnarray
(ii) Suppose that $n_a(h)-n_{a+1}(h)\leq -2$ for some $1\leq a\leq r_{h-1}-1$.
Let ${\nbf'}_1^{r_{h-1}}(h)=(n'_1(h),n'_2(h),\ldots,n'_{r_{h-1}}(h))$
be a sequence of positive integers such that
$n'_a(h)=n_a(h)+1$, $n'_{a+1}(h)=n_{a+1}(h)-1$, and $n'_i(h)=n_i(h)$ for $i\neq a$ and $a+1$.
Then we have ${\nbf'}_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$ and
\beqnarray{adjacent distance larger than one II-2}
\nbf_1^{r_{h-1}}(h)\preceq{\nbf'}_1^{r_{h-1}}(h),
\eeqnarray
where $\nbf_1^{r_{h-1}}(h)\equiv {\nbf'}_1^{r_{h-1}}(h)$ if and only if $r_{h-1}=2$ and $n_1(h)=n_2(h)-2$.
\elemma
\bexample{adjacent distance larger than one II}
Suppose that $M=26$ and $k=10$.
In Step 1 of \ralgorithm{main result},
we obtain $r_{-1}=26$, $r_0=10$, $q_1=2$, $r_1=6$, $q_2=1$, $r_2=4$,
$q_3=1$, $r_3=2$, $q_4=2$, and $r_4=0$.
(i) Assume that $h=2$ and hence $r_{h-1}=r_1=6\geq 2$.
Let
\beqnarray{}
{\nbf''}_1^{r_1}(2) \aligneq (1,1,5,1,1,1), \nn\\
{\nbf'}_1^{r_1}(2) \aligneq (1,1,4,2,1,1), \nn\\
\nbf_1^{r_1}(2) \aligneq (1,1,3,3,1,1), \nn\\
{\mbf'}_1^{r_1}(2) \aligneq (1,1,2,4,1,1), \nn\\
{\mbf''}_1^{r_1}(2) \aligneq (1,1,1,5,1,1). \nn
\eeqnarray
Then it follows from \reqnarray{adjacent distance larger than one II-1}
in \rlemma{adjacent distance larger than one II}(i) (with $h=2$ and $a=3$) that
\beqnarray{}
{\nbf''}_1^{r_1}(2)\prec{\nbf'}_1^{r_1}(2)\prec\nbf_1^{r_1}(2). \nn
\eeqnarray
From \reqnarray{adjacent distance larger than one II-2}
in \rlemma{adjacent distance larger than one II}(ii) (with $h=2$ and $a=3$),
we also have
\beqnarray{}
\nbf_1^{r_1}(2)\succ{\mbf'}_1^{r_1}(2)\succ{\mbf''}_1^{r_1}(2). \nn
\eeqnarray
These results can be verified by the numerical results in \rtable{adjacent distance larger than one II},
where we compute the maximum representable integers $B(\dbf_1^M;k)$ with $\dbf_1^M$ obtained by using
\beqnarray{}
{\nbf''}_1^{r_0}(1)=L_{r_{-1},r_0}({\nbf''}_1^{r_1}(2))
=L_{26,10}((1,1,5,1,1,1)) \aligneq (3,3,3,2,2,2,2,3,3,3), \nn\\
{\nbf'}_1^{r_0}(1)=L_{r_{-1},r_0}({\nbf'}_1^{r_1}(2))
=L_{26,10}((1,1,4,2,1,1)) \aligneq (3,3,3,2,2,2,3,2,3,3), \nn\\
\nbf_1^{r_0}(1)=L_{r_{-1},r_0}(\nbf_1^{r_1}(2))
=L_{26,10}((1,1,3,3,1,1)) \aligneq (3,3,3,2,2,3,2,2,3,3), \nn\\
{\mbf'}_1^{r_0}(1)=L_{r_{-1},r_0}({\mbf'}_1^{r_1}(2))
=L_{26,10}((1,1,2,4,1,1)) \aligneq (3,3,3,2,3,2,2,2,3,3), \nn\\
{\mbf''}_1^{r_0}(1)=L_{r_{-1},r_0}({\mbf''}_1^{r_1}(2))
=L_{26,10}((1,1,1,5,1,1)) \aligneq (3,3,3,3,2,2,2,2,3,3), \nn
\eeqnarray
respectively, in \reqnarray{OQ-LR-delays-greedy-1}.
\btable{htbp}{|r|r|c|}
\hline & & $B(d_1^M;k)$ \\
\hline ${\nbf''}_1^{r_1}(2)=(1,1,5,1,1,1)$ & ${\nbf''}_1^{r_0}(1)=(3,3,3,2,2,2,2,3,3,3)$ & $1072727$ \\
\hline ${\nbf'}_1^{r_1}(2)=(1,1,4,2,1,1)$ & ${\nbf'}_1^{r_0}(1)=(3,3,3,2,2,2,3,2,3,3)$ & $1084591$ \\
\hline $\nbf_1^{r_1}(2)=(1,1,3,3,1,1)$ & $\nbf_1^{r_0}(1)=(3,3,3,2,2,3,2,2,3,3)$ & $1086295$ \\
\hline ${\mbf'}_1^{r_1}(2)=(1,1,2,4,1,1)$ & ${\mbf'}_1^{r_0}(1)=(3,3,3,2,3,2,2,2,3,3)$ & $1084655$ \\
\hline ${\mbf''}_1^{r_1}(2)=(1,1,1,5,1,1)$ & ${\mbf''}_1^{r_0}(1)=(3,3,3,3,2,2,2,2,3,3)$ & $1073111$ \\
\hline
\etable{adjacent distance larger than one II}
{The maximum representable integers $B(\dbf_1^M;k)$ with $\dbf_1^M$ obtained by using
${\nbf''}_1^{r_0}(1)=L_{r_{-1},r_0}({\nbf''}_1^{r_1}(2))$,
${\nbf'}_1^{r_0}(1)=L_{r_{-1},r_0}({\nbf'}_1^{r_1}(2))$,
$\nbf_1^{r_0}(1)=L_{r_{-1},r_0}(\nbf_1^{r_1}(2))$,
${\mbf'}_1^{r_0}(1)=L_{r_{-1},r_0}({\mbf'}_1^{r_1}(2))$,
and ${\mbf''}_1^{r_0}(1)=L_{r_{-1},r_0}({\mbf''}_1^{r_1}(2))$,
respectively, in \reqnarray{OQ-LR-delays-greedy-1},
where $M=26$ and $k=10$.}
(ii) Assume that $h=4$ and hence $r_{h-1}=r_3=2$.
Let $\nbf_1^{r_3}(4)=(1,3)$ and ${\nbf'}_1^{r_3}(4)=(2,2)$.
As $r_3=2$ and $n_1(4)=n_2(4)-2$,
it follows from \reqnarray{adjacent distance larger than one II-2}
in \rlemma{adjacent distance larger than one II}(ii) (with $h=4$ and $a=1$) that
\beqnarray{}
\nbf_1^{r_3}(4)\equiv{\nbf'}_1^{r_3}(4). \nn
\eeqnarray
This result can also be verified numerically.
To see this, note that from \reqnarray{order relation-333}--\reqnarray{order relation-555}
with $h=4$, we have
\beqnarray{}
\alignspace \nbf_1^{r_2}(3)=L_{r_1,r_2}(\nbf_1^{r_3}(4))=L_{6,4}((1,3))=(2,2,1,1), \nn\\
\alignspace \nbf_1^{r_1}(2)=R_{r_0,r_1}(\nbf_1^{r_2}(3))=R_{10,6}((2,2,1,1))=(1,2,1,2,2,2), \nn\\
\alignspace \nbf_1^{r_0}(1)=L_{r_{-1},r_0}(\nbf_1^{r_1}(2))=L_{26,10}((1,2,1,2,2,2))=(3,3,2,3,3,2,3,2,3,2); \nn\\
\alignspace {\nbf'}_1^{r_2}(3)=L_{r_1,r_2}({\nbf'}_1^{r_3}(4))=L_{6,4}((2,2))=(2,1,2,1), \nn\\
\alignspace {\nbf'}_1^{r_1}(2)=R_{r_0,r_1}({\nbf'}_1^{r_2}(3))=R_{10,6}((2,1,2,1))=(1,2,2,1,2,2), \nn\\
\alignspace {\nbf'}_1^{r_0}(1)=L_{r_{-1},r_0}({\nbf'}_1^{r_1}(2))=L_{26,10}((1,2,2,1,2,2))=(3,3,2,3,2,3,3,2,3,2). \nn
\eeqnarray
Let $\dbf_1^M$ and ${\dbf'}_1^M$ be obtained by using $\nbf_1^{r_0}(1)=(3,3,2,3,3,2,3,2,3,2)$
and ${\nbf'}_1^{r_0}(1)=(3,3,2,3$, $2,3,3,2,3,2)$, respectively, in \reqnarray{OQ-LR-delays-greedy-1}.
We then compute that $B(\dbf_1^M;k)=B({\dbf'}_1^M;k)=1141023$
and this verifies that $\nbf_1^{r_2}(3)\equiv{\nbf'}_1^{r_2}(3)$.
\eexample
We have the following corollary to \rlemma{adjacent distance larger than one II}.
\bcorollary{adjacent distance larger than one II}
Suppose that $M\geq 2$ and $1\leq k\leq M-1$. Let $r_{-1}=M$, $r_0=k$, and let $q_i$ and $r_i$, $i=1,2,\ldots,N$, be recursively obtained
as in Step 1 of \ralgorithm{main result}.
Assume that $1\leq h\leq N$ is an even integer and $r_{h-1}\geq 2$.
(i) Suppose that $r_{h-1}\neq 2$ or $r_h\neq 0$.
Then an optimal sequence $\nbf_1^{r_{h-1}}(h)$ over $\Ncal_{M,k}(h)$
must satisfy the condition that the absolute value of the difference
of any two adjacent entries of $\nbf_1^{r_{h-1}}(h)$ is less than or equal to one, i.e.,
\beqnarray{adjacent distance larger than one II-3}
|n_i(h)-n_{i+1}(h)|\leq 1, \textrm{ for } i=1,2,\ldots,r_{h-1}-1.
\eeqnarray
(ii) Suppose that $r_{h-1}=2$ and $r_h=0$.
Then there are two optimal sequences over $\Ncal_{M,k}(h)$,
and the two optimal sequences, say $\nbf_1^{r_{h-1}}(h)$ and $\mbf_1^{r_{h-1}}(h)$,
are given by
\beqnarray{adjacent distance larger than one II-4}
\nbf_1^{r_{h-1}}(h)=(q_h,q_h) \textrm{ and } \mbf_1^{r_{h-1}}(h)=(q_h-1,q_h+1)
\eeqnarray
(note that $q_h\geq 2$ as $r_{h-2}=q_h\cdot r_{h-1}+r_h=q_h\cdot r_{h-1}$ and $r_{h-2}>r_{h-1}$).
\ecorollary
\bproof
(i) Let $\nbf_1^{r_{h-1}}(h)$ be an optimal sequence over $\Ncal_{M,k}(h)$.
We show that $|n_i(h)-n_{i+1}(h)|\leq 1$ for $i=1,2,\ldots,r_{h-1}-1$ by contradiction.
Assume on the contrary that $|n_a(h)-n_{a+1}(h)|\geq 2$ for some $1\leq a\leq r_{h-1}-1$.
Note that in \rcorollary{adjacent distance larger than one II}(i),
we have $r_{h-1}\neq 2$ or $r_h\neq 0$.
As such, if $r_{h-1}=2$, then we have $r_h\neq 0$
and it must be the case that $n_1(h)\neq n_2(h)-2$.
Otherwise, if $n_1(h)=n_2(h)-2$,
then we see from $\nbf_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$ and \reqnarray{N-M-k-h} that
\beqnarray{}
r_{h-2}=\sum_{i=1}^{r_{h-1}}n_i(h)=n_1(h)+n_2(h)=2n_2(h)-2=(n_2(h)-1)\cdot r_{h-1}. \nn
\eeqnarray
Thus, the remainder $r_h$ of $r_{h-2}$ divided by $r_{h-1}$ is equal to zero,
contradicting to $r_h\neq 0$.
Since it cannot be the case that $r_{h-1}=2$ and $n_1(h)=n_2(h)-2$,
we see from \rlemma{adjacent distance larger than one II}
that there exists ${\nbf'}_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$
such that $\nbf_1^{r_{h-1}}(h)\prec{\nbf'}_1^{r_{h-1}}(h)$,
contradicting to the optimality of $\nbf_1^{r_{h-1}}(h)$.
(ii) Let $\nbf_1^{r_{h-1}}(h)$ be an optimal sequence over $\Ncal_{M,k}(h)$.
As we have from $r_{h-1}=2$ and $r_h=0$
that $r_{h-2}=q_h\cdot r_{h-1}+r_h=2q_h$
and we have from $\nbf_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$, \reqnarray{N-M-k-h}, and $r_{h-1}=2$
that $r_{h-2}=\sum_{i=1}^{r_{h-1}}n_i(h)=n_1(h)+n_2(h)$,
we see that $n_1(h)+n_2(h)=2q_h$ is an even integer.
As such, it follows that $n_1(h)-n_2(h)$ is also an even integer.
If $n_1(h)-n_2(h)\geq 2$ or $n_1(h)-n_2(h)\leq -4$,
then we see from \rlemma{adjacent distance larger than one II}
that there exists ${\nbf'}_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$
such that $\nbf_1^{r_{h-1}}(h)\prec{\nbf'}_1^{r_{h-1}}(h)$,
contradicting to the optimality of $\nbf_1^{r_{h-1}}(h)$.
Therefore, we must have $n_1(h)-n_2(h)=0$ or $-2$,
i.e., $\nbf_1^{r_{h-1}}(h)=(q_h,q_h)$ or $\nbf_1^{r_{h-1}}(h)=(q_h-1,q_h+1)$.
It is easy to see from \rlemma{adjacent distance larger than one II}(ii)
that $(q_h,q_h)\equiv (q_h-1,q_h+1)$,
and hence both $(q_h,q_h)$ and $(q_h-1,q_h+1)$ are optimal sequences over $\Ncal_{M,k}(h)$.
\eproof
In the following lemma, we show some pairwise comparison results
for a sequence $\nbf_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$,
where $1\leq h\leq N$ is an even integer and $r_{h-1}\geq 2$,
such that the absolute value of the difference of two ``adjacent'' entries
of $\nbf_1^{r_{h-1}}(h)$ is equal to one.
\blemma{comparison rule B} (\textbf{Comparison rule B}).
Suppose that $M\geq 2$ and $1\leq k\leq M-1$. Let $r_{-1}=M$, $r_0=k$, and let $q_i$ and $r_i$, $i=1,2,\ldots,N$, be recursively obtained
as in Step 1 of \ralgorithm{main result}.
Assume that $1\leq h\leq N$ is an even integer and $r_{h-1}\geq 2$.
Let $\nbf_1^{r_{h-1}}(h)=(n_1(h),n_2(h),\ldots,n_{r_{h-1}}(h))\in \Ncal_{M,k}(h)$
and $n_a(h)-n_{a+1}(h)=1$ for some $1\leq a\leq r_{h-1}-1$.
Let ${\nbf'}_1^{r_{h-1}}(h)=(n'_1(h),n'_2(h),$ $\ldots,n'_{r_{h-1}}(h))$
be a sequence of positive integers such that
$n'_a(h)=n_a(h)-1$, $n'_{a+1}(h)=n_{a+1}(h)+1$, and $n'_i(h)=n_i(h)$ for $i\neq a$ and $a+1$.
Then we have ${\nbf'}_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$.
Furthermore, we have the following pairwise comparison results.
(i) Suppose that $a=1$ or $a=r_{h-1}-1$.
Then we have
\beqnarray{comparison rule B-1}
\nbf_1^{r_{h-1}}(h)\prec{\nbf'}_1^{r_{h-1}}(h).
\eeqnarray
(ii) Suppose that $2\leq a\leq r_{h-1}-2$ and there exists a positive integer $j$ such that
$1\leq j\leq \min\{a-1,r_{h-1}-a-1\}$, $n_{a-j'}(h)=n_{a+1+j'}(h)$ for $j'=1,2,\ldots,j-1$,
and $n_{a-j}(h)\neq n_{a+1+j}(h)$.
If $n_{a-j}(h)>n_{a+1+j}(h)$, then we have
\beqnarray{comparison rule B-2}
\nbf_1^{r_{h-1}}(h)\prec{\nbf'}_1^{r_{h-1}}(h).
\eeqnarray
On the other hand, if $n_{a-j}(h)<n_{a+1+j}(h)$, then we have
\beqnarray{comparison rule B-3}
\nbf_1^{r_{h-1}}(h)\succeq{\nbf'}_1^{r_{h-1}}(h),
\eeqnarray
where $\nbf_1^{r_{h-1}}(h)\equiv {\nbf'}_1^{r_{h-1}}(h)$
if and only if $a-j=1$, $a+1+j=r_{h-1}$, and $n_1(h)=n_{r_{h-1}}(h)-1$.
(iii) Suppose that $2\leq a\leq r_{h-1}-2$ and $n_{a-j'}(h)=n_{a+1+j'}(h)$
for $j'=1,2,\ldots,\min\{a-1,r_{h-1}-a-1\}$.
Then we have
\beqnarray{comparison rule B-4}
\nbf_1^{r_{h-1}}(h)\prec{\nbf'}_1^{r_{h-1}}(h).
\eeqnarray
\elemma
\bexample{comparison rule B}
Suppose that $M=26$ and $k=10$.
In Step 1 of \ralgorithm{main result},
we obtain $r_{-1}=26$, $r_0=10$, $q_1=2$, $r_1=6$, $q_2=1$, $r_2=4$,
$q_3=1$, $r_3=2$, $q_4=2$, and $r_4=0$.
Assume that $h=2$ and hence $r_{h-1}=r_1=6\geq 2$.
Let
\beqnarray{}
{\nbf'''''}_1^{r_1}(2) \aligneq (2,2,2,1,2,1), \nn\\
{\nbf''''}_1^{r_1}(2) \aligneq (2,2,1,2,2,1), \nn\\
{\nbf'''}_1^{r_1}(2) \aligneq (2,2,1,2,1,2), \nn\\
{\nbf''}_1^{r_1}(2) \aligneq (2,1,2,2,1,2), \nn\\
{\nbf'}_1^{r_1}(2) \aligneq (1,2,2,2,1,2), \nn\\
\nbf_1^{r_1}(2) \aligneq (1,2,2,1,2,2), \nn\\
\mbf_1^{r_1}(2) \aligneq (1,2,1,2,2,2). \nn
\eeqnarray
Then we have
\beqnarray{}
{\nbf'''''}_1^{r_1}(2)\prec{\nbf''''}_1^{r_1}(2)\prec{\nbf'''}_1^{r_1}(2)
\prec{\nbf''}_1^{r_1}(2)\prec{\nbf'}_1^{r_1}(2)\prec\nbf_1^{r_1}(2)\equiv\mbf_1^{r_1}(2). \nn
\eeqnarray
This can be proved by using Comparison rule B in \rlemma{comparison rule B}.
First, it is easy to see from $n'''''_3(2)-n'''''_4(2)=1$, $n'''''_2(2)=n'''''_5(2)$,
$n'''''_1(2)>n'''''_6(2)$, and \reqnarray{comparison rule B-2}
in \rlemma{comparison rule B}(ii) (with $h=2$, $a=3$, and $j=2$) that
\beqnarray{}
{\nbf'''''}_1^{r_1}(2)\prec{\nbf''''}_1^{r_1}(2). \nn
\eeqnarray
From $n''''_5(2)-n''''_6(2)=1$ and \reqnarray{comparison rule B-1} in
\rlemma{comparison rule B}(i) (with $h=2$ and $a=r_1-1=5$), we have
\beqnarray{}
{\nbf''''}_1^{r_1}(2)\prec{\nbf'''}_1^{r_1}(2). \nn
\eeqnarray
From $n'''_2(2)-n'''_3(2)=1$, $n'''_1(2)=n'''_4(2)$,
and \reqnarray{comparison rule B-4} in \rlemma{comparison rule B}(iii)
(with $h=2$ and $a=2$), we have
\beqnarray{}
{\nbf'''}_1^{r_1}(2)\prec{\nbf''}_1^{r_1}(2). \nn
\eeqnarray
From $n''_1(2)-n''_2(2)=1$ and \reqnarray{comparison rule B-1} in
\rlemma{comparison rule B}(i) (with $h=2$ and $a=1$), we have
\beqnarray{}
{\nbf''}_1^{r_1}(2)\prec{\nbf'}_1^{r_1}(2). \nn
\eeqnarray
Since $n'_4(2)-n'_5(2)=1$ and $n'_3(2)=n'_6(2)$,
it follows from \reqnarray{comparison rule B-4}
in \rlemma{comparison rule B}(iii) (with $h=2$ and $a=4$) that
\beqnarray{}
{\nbf'}_1^{r_1}(2)\prec\nbf_1^{r_1}(2). \nn
\eeqnarray
Finally, as $n_3(2)-n_4(2)=1$, $n_2(2)=n_5(2)$, and $n_1(2)=n_6(2)-1$,
it follows from \reqnarray{comparison rule B-3}
in \rlemma{comparison rule B}(ii) (with $h=2$, $a=3$, and $j=2$) that
\beqnarray{}
\nbf_1^{r_1}(2)\equiv\mbf_1^{r_1}(2). \nn
\eeqnarray
These results can also be verified by the numerical results in \rtable{comparison rule B},
where we compute the maximum representable integers $B(\dbf_1^M;k)$ with $\dbf_1^M$ obtained by using
\beqnarray{}
{\nbf'''''}_1^{r_0}(1)=L_{r_{-1},r_0}({\nbf'''''}_1^{r_1}(2))
=L_{26,10}((2,2,2,1,2,1)) \aligneq (3,2,3,2,3,2,3,3,2,3), \nn\\
{\nbf''''}_1^{r_0}(1)=L_{r_{-1},r_0}({\nbf''''}_1^{r_1}(2))
=L_{26,10}((2,2,1,2,2,1)) \aligneq (3,2,3,2,3,3,2,3,2,3), \nn\\
{\nbf'''}_1^{r_0}(1)=L_{r_{-1},r_0}({\nbf'''}_1^{r_1}(2))
=L_{26,10}((2,2,1,2,1,2)) \aligneq (3,2,3,2,3,3,2,3,3,2), \nn\\
{\nbf''}_1^{r_0}(1)=L_{r_{-1},r_0}({\nbf''}_1^{r_1}(2))
=L_{26,10}((2,1,2,2,1,2)) \aligneq (3,2,3,3,2,3,2,3,3,2), \nn\\
{\nbf'}_1^{r_0}(1)=L_{r_{-1},r_0}({\nbf'}_1^{r_1}(2))
=L_{26,10}((1,2,2,2,1,2)) \aligneq (3,3,2,3,2,3,2,3,3,2), \nn\\
\nbf_1^{r_0}(1)=L_{r_{-1},r_0}(\nbf_1^{r_1}(2))
=L_{26,10}((1,2,2,1,2,2)) \aligneq (3,3,2,3,2,3,3,2,3,2), \nn\\
\mbf_1^{r_0}(1)=L_{r_{-1},r_0}(\mbf_1^{r_1}(2))
=L_{26,10}((1,2,1,2,2,2)) \aligneq (3,3,2,3,3,2,3,2,3,2), \nn
\eeqnarray
respectively, in \reqnarray{OQ-LR-delays-greedy-1}.
\btable{htbp}{|r|r|c|}
\hline & & $B(d_1^M;k)$ \\
\hline ${\nbf'''''}_1^{r_1}(2)=(2,2,2,1,2,1)$ & ${\nbf'''''}_1^{r_0}(1)=(3,2,3,2,3,2,3,3,2,3)$ & $1104735$ \\
\hline ${\nbf''''}_1^{r_1}(2)=(2,2,1,2,2,1)$ & ${\nbf''''}_1^{r_0}(1)=(3,2,3,2,3,3,2,3,2,3)$ & $1104799$ \\
\hline ${\nbf'''}_1^{r_1}(2)=(2,2,1,2,1,2)$ & ${\nbf'''}_1^{r_0}(1)=(3,2,3,2,3,3,2,3,3,2)$ & $1136415$ \\
\hline ${\nbf''}_1^{r_1}(2)=(2,1,2,2,1,2)$ & ${\nbf''}_1^{r_0}(1)=(3,2,3,3,2,3,2,3,3,2)$ & $1136495$ \\
\hline ${\nbf'}_1^{r_1}(2)=(1,2,2,2,1,2)$ & ${\nbf'}_1^{r_0}(1)=(3,3,2,3,2,3,2,3,3,2)$ & $1140511$ \\
\hline $\nbf_1^{r_1}(2)=(1,2,2,1,2,2)$ & $\nbf_1^{r_0}(1)=(3,3,2,3,2,3,3,2,3,2)$ & $1141023$ \\
\hline $\mbf_1^{r_1}(2)=(1,2,1,2,2,2)$ & $\mbf_1^{r_0}(1)=(3,3,2,3,3,2,3,2,3,2)$ & $1141023$ \\
\hline
\etable{comparison rule B}
{The maximum representable integers $B(\dbf_1^M;k)$ with $\dbf_1^M$ obtained by using
${\nbf'''''}_1^{r_0}(1)=L_{r_{-1},r_0}({\nbf'''''}_1^{r_1}(2))$,
${\nbf''''}_1^{r_0}(1)=L_{r_{-1},r_0}({\nbf''''}_1^{r_1}(2))$,
${\nbf'''}_1^{r_0}(1)=L_{r_{-1},r_0}({\nbf'''}_1^{r_1}(2))$,
${\nbf''}_1^{r_0}(1)=L_{r_{-1},r_0}({\nbf''}_1^{r_1}(2))$,
${\nbf'}_1^{r_0}(1)=L_{r_{-1},r_0}({\nbf'}_1^{r_1}(2))$,
$\nbf_1^{r_0}(1)=L_{r_{-1},r_0}(\nbf_1^{r_1}(2))$,
and $\mbf_1^{r_0}(1)=L_{r_{-1},r_0}(\mbf_1^{r_1}(2))$, respectively,
in \reqnarray{OQ-LR-delays-greedy-1}, where $M=26$ and $k=10$.}
\eexample
From \rcorollary{adjacent distance larger than one II}(i),
we know that if $1\leq h\leq N$ is an even integer and $r_{h-1}\geq 3$,
then an optimal sequence $\nbf_1^{r_{h-1}}(h)$ over $\Ncal_{M,k}(h)$
must satisfy the condition that the absolute value of the difference
of any two adjacent entries of $\nbf_1^{r_{h-1}}(h)$ is less than or equal to one.
In the following lemma, we show some pairwise comparison results
for a sequence $\nbf_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$,
where $1\leq h\leq N$ is an even integer and $r_{h-1}\geq 3$,
such that the absolute value of the difference of any two adjacent entries
of $\nbf_1^{r_{h-1}}(h)$ is less than or equal to one
and the absolute value of the difference of two ``nonadjacent'' entries
of $\nbf_1^{r_{h-1}}(h)$ is greater than or equal to two.
\blemma{nonadjacent distance larger than one II}
Suppose that $M\geq 2$ and $1\leq k\leq M-1$. Let $r_{-1}=M$, $r_0=k$, and let $q_i$ and $r_i$, $i=1,2,\ldots,N$, be recursively obtained
as in Step 1 of \ralgorithm{main result}.
Assume that $1\leq h\leq N$ is an even integer and $r_{h-1}\geq 3$.
Let $\nbf_1^{r_{h-1}}(h)=(n_1(h),n_2(h),\ldots,n_{r_{h-1}}(h))\in \Ncal_{M,k}(h)$
and $|n_i(h)-n_{i+1}(h)|\leq 1$ for $i=1,2,\ldots,r_{h-1}-1$.
(i) Suppose that $n_a(h)-n_b(h)\leq -2$ for some $1\leq a<b\leq r_{h-1}$ and $b\geq a+2$.
If $n_{r_{h-1}}(h)\neq n_1(h)+2$ or $n_i(h)\neq n_1(h)+1$ for some $2\leq i\leq r_{h-1}-1$,
then there exists a sequence of positive integers
${\nbf'}_1^{r_{h-1}}(h)=(n'_1(h),n'_2(h),\ldots,n'_{r_{h-1}}(h))\in \Ncal_{M,k}(h)$
such that
\beqnarray{nonadjacent distance larger than one II-1}
{\nbf'}_1^{r_{h-1}}(h)\succ\nbf_1^{r_{h-1}}(h).
\eeqnarray
(ii) Suppose that $n_a(h)-n_b(h)\geq 2$ for some $1\leq a<b\leq r_{h-1}$ and $b\geq a+2$.
Then there exists a sequence of positive integers
${\nbf'}_1^{r_{h-1}}(h)=(n'_1(h),n'_2(h),\ldots,n'_{r_{h-1}}(h))\in \Ncal_{M,k}(h)$
such that
\beqnarray{nonadjacent distance larger than one II-2}
{\nbf'}_1^{r_{h-1}}(h)\succ\nbf_1^{r_{h-1}}(h).
\eeqnarray
\elemma
We have the following corollary to \rlemma{nonadjacent distance larger than one}.
\bcorollary{nonadjacent distance larger than one II}
Suppose that $M\geq 2$ and $1\leq k\leq M-1$. Let $r_{-1}=M$, $r_0=k$, and let $q_i$ and $r_i$, $i=1,2,\ldots,N$, be recursively obtained
as in Step 1 of \ralgorithm{main result}.
Assume that $1\leq h\leq N$ is an even integer.
(i) Suppose that $r_{h-1}\geq 3$ and $r_h\neq 0$.
Then an optimal sequence $\nbf_1^{r_{h-1}}(h)$ over $\Ncal_{M,k}(h)$
must satisfy the condition that the absolute value of the difference
of any two entries (adjacent or nonadjacent) of $\nbf_1^{r_{h-1}}(h)$
is less than or equal to one, i.e.,
\beqnarray{nonadjacent distance larger than one II-3}
|n_a(h)-n_b(h)|\leq 1, \textrm{ for all } 1\leq a<b\leq r_{h-1}.
\eeqnarray
(ii) Suppose that $r_{h-1}\geq 3$ and $r_h=0$.
Then there are at most two optimal sequences over $\Ncal_{M,k}(h)$,
and the two possible optimal sequences, say $\nbf_1^{r_{h-1}}(h)$ and $\mbf_1^{r_{h-1}}(h)$,
are given by
\beqnarray{nonadjacent distance larger than one II-4}
\nbf_1^{r_{h-1}}(h)=(q_h,q_h,\ldots,q_h)
\textrm{ and } \mbf_1^{r_{h-1}}(h)=(q_h-1,q_h,\ldots,q_h,q_h+1)
\eeqnarray
(note that $q_h\geq 2$ as $r_{h-2}=q_h\cdot r_{h-1}+r_h=q_h\cdot r_{h-1}$ and $r_{h-2}>r_{h-1}$).
\ecorollary
\bproof
(i) Let $\nbf_1^{r_{h-1}}(h)$ be an optimal sequence over $\Ncal_{M,k}(h)$.
As we have $r_{h-1}\geq 3$ and hence $r_{h-1}\neq 2$,
it follows from \rcorollary{adjacent distance larger than one II}(i)
that $\nbf_1^{r_{h-1}}(h)$ must satisfy the condition that
\beqnarray{proof-nonadjacent distance larger than one II-corollary-111}
|n_i(h)-n_{i+1}(h)|\leq 1, \textrm{ for all } i=1,2,\ldots,r_{h-1}-1.
\eeqnarray
Furthermore, as we have $r_h\neq 0$, it must be the case that $n_{r_{h-1}}(h)\neq n_1(h)+2$
or $n_i(h)\neq n_1(h)+1$ for some $2\leq i\leq r_{h-1}-1$.
Otherwise, if $n_{r_{h-1}}(h)=n_1(h)+2$ and $n_i(h)=n_1(h)+1$
for all $2\leq i\leq r_{h-1}-1$,
then we see from $\nbf_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$ and \reqnarray{N-M-k-h} that
\beqnarray{}
r_{h-2}=\sum_{i=1}^{r_{h-1}}n_i(h)=(n_1(h)+1)\cdot r_{h-1}. \nn
\eeqnarray
Thus, the remainder $r_h$ of $r_{h-2}$ divided by $r_{h-1}$ is equal to zero,
contradicting to $r_h\neq 0$.
To show \reqnarray{nonadjacent distance larger than one II-3},
it is clear from \reqnarray{proof-nonadjacent distance larger than one II-corollary-111}
that it suffices to show that $|n_a(h)-n_b(h)|\leq 1$
for all $1\leq a<b\leq r_{h-1}$ and $b\geq a+2$ by contradiction.
Assume on the contrary that $|n_a(h)-n_b(h)|\geq 2$
for some $1\leq a<b\leq r_{h-1}$ and $b\geq a+2$.
Since $n_{r_{h-1}}(h)\neq n_1(h)+2$
or $n_i(h)\neq n_1(h)+1$ for some $2\leq i\leq r_{h-1}-1$,
we see from \rlemma{nonadjacent distance larger than one II}
that there exists ${\nbf'}_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$
such that ${\nbf'}_1^{r_{h-1}}(h)\succ \nbf_1^{r_{h-1}}(h)$,
contradicting to the optimality of $\nbf_1^{r_{h-1}}(h)$.
(ii) Let $\nbf_1^{r_{h-1}}(h)$ be an optimal sequence over $\Ncal_{M,k}(h)$.
As we have $r_{h-1}\geq 3$,
\reqnarray{proof-nonadjacent distance larger than one II-corollary-111} still holds.
Furthermore, as we have $r_h=0$,
it is clear that $r_{h-2}=q_h\cdot r_{h-1}+r_h=q_h\cdot r_{h-1}$.
It then follows from $\nbf_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$ and \reqnarray{N-M-k-h} that
\beqnarray{proof-nonadjacent distance larger than one II-corollary-222}
\sum_{i=1}^{r_{h-1}}n_i(h)=r_{h-2}=q_h\cdot r_{h-1}.
\eeqnarray
We need to consider the following two cases.
\emph{Case 1: $|n_a(h)-n_b(h)|\leq 1$ for all $1\leq a<b\leq r_{h-1}$.}
In this case, it is easy to see from
\reqnarray{proof-nonadjacent distance larger than one II-corollary-222} that
\beqnarray{proof-nonadjacent distance larger than one II-corollary-333}
n_i(h)=q_h, \textrm{ for } i=1,2,\ldots,r_{h-1}.
\eeqnarray
\emph{Case 2: $|n_a(h)-n_b(h)|\geq 2$ for some $1\leq a<b\leq r_{h-1}$.}
In this case, it is clear from
\reqnarray{proof-nonadjacent distance larger than one II-corollary-111} that $b\geq a+2$.
If $n_a(h)-n_b(h)\leq -2$ and $n_{r_{h-1}}(h)\neq n_1(h)+2$
or $n_i(h)\neq n_1(h)+1$ for some $2\leq i\leq r_{h-1}-1$,
then it follows from \rlemma{nonadjacent distance larger than one II}(i)
that there exists ${\nbf'}_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$
such that ${\nbf'}_1^{r_{h-1}}(h)\succ \nbf_1^{r_{h-1}}(h)$,
contradicting to the optimality of $\nbf_1^{r_{h-1}}(h)$.
Also, if $n_a(h)-n_b(h)\geq 2$,
then it follows from \rlemma{nonadjacent distance larger than one II}(ii)
that there exists ${\nbf'}_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$
such that ${\nbf'}_1^{r_{h-1}}(h)\succ \nbf_1^{r_{h-1}}(h)$,
contradicting to the optimality of $\nbf_1^{r_{h-1}}(h)$.
As such, it must be the case that
\beqnarray{proof-nonadjacent distance larger than one II-corollary-444}
n_a(h)-n_b(h)\leq -2,\ n_{r_{h-1}}(h)=n_1(h)+2,
\textrm{ and } n_i(h)=n_1(h)+1 \textrm{ for } 2\leq i\leq r_{h-1}-1.
\eeqnarray
It follows from \reqnarray{proof-nonadjacent distance larger than one II-corollary-222}
and \reqnarray{proof-nonadjacent distance larger than one II-corollary-444} that
\beqnarray{proof-nonadjacent distance larger than one II-corollary-555}
n_1(h)=q_h-1,\ n_{r_{h-1}}(h)=q_h+1,
\textrm{ and } n_i(h)=q_h \textrm{ for } 2\leq i\leq r_{h-1}-1.
\eeqnarray
By combining \reqnarray{proof-nonadjacent distance larger than one II-corollary-333}
and \reqnarray{proof-nonadjacent distance larger than one II-corollary-555},
we see that $(q_h,q_h,\ldots,q_h)$ and $(q_h-1,q_h,\ldots,q_h,q_h+1)$
are the two possible optimal sequences over $\Ncal_{M,k}(h)$,
and the proof is completed.
\eproof
Suppose that $1\leq h\leq N$ is an even integer, $r_{h-1}\geq 2$, $r_h\neq 0$,
and $\nbf_1^{r_{h-1}}(h)$ is an optimal sequence over $\Ncal_{M,k}(h)$.
If $r_{h-1}=2$, then it follows from \rcorollary{adjacent distance larger than one II}(i)
that $|n_1(h)-n_2(h)|\leq 1$, i.e., \reqnarray{nonadjacent distance larger than one II-3} holds.
On the other hand, if $r_{h-1}\geq 3$,
then it follows from \rcorollary{nonadjacent distance larger than one II}(i)
that \reqnarray{nonadjacent distance larger than one II-3} also holds.
As such, we see from \reqnarray{nonadjacent distance larger than one II-3},
$\sum_{i=1}^{r_{h-1}}n_i(h)=r_{h-2}$ in \reqnarray{N-M-k-h},
and $r_{h-2}=q_h\cdot r_{h-1}+r_h$ that
\beqnarray{}
n_i(h)=
\bselection
q_h+1, &\textrm{if } i=i_1,i_2,\ldots,i_{r_h}, \\
q_h, &\textrm{otherwise},
\eselection \nn
\eeqnarray
for some $1\leq i_1<i_2<\cdots <i_{r_h}\leq r_{h-1}$.
In \rlemma{main lemma II} below, we further show that $i_{r_h}$ must be equal to $r_{h-1}$.
\blemma{main lemma II}
Suppose that $M\geq 2$ and $1\leq k\leq M-1$. Let $r_{-1}=M$, $r_0=k$, and let $q_i$ and $r_i$, $i=1,2,\ldots,N$, be recursively obtained
as in Step 1 of \ralgorithm{main result}.
Assume that $1\leq h\leq N$ is an even integer, $r_{h-1}\geq 2$, and $r_h\neq 0$.
Then an optimal sequence $\nbf_1^{r_{h-1}}(h)$ over $\Ncal_{M,k}(h)$
must satisfy the condition that
\beqnarray{main lemma II-1}
n_i(h)=
\bselection
q_h+1, &\textrm{if } i=i_1,i_2,\ldots,i_{r_h}, \\
q_h, &\textrm{otherwise},
\eselection
\eeqnarray
for some $1\leq i_1<i_2<\cdots <i_{r_h}=r_{h-1}$.
\elemma
In the appendices,
we first show that \rlemma{adjacent distance larger than one}
and Comparison rule A in \rlemma{comparison rule A} hold for $h=1$
in \rappendix{proof of adjacent distance larger than one with h=1}
and \rappendix{proof of comparison rule A with h=1}, respectively.
We also show that if Comparison rule A in \rlemma{comparison rule A}
holds for some odd integer $h-1$, where $1\leq h-1\leq N-1$,
then \rlemma{adjacent distance larger than one II}
and Comparison rule B in \rlemma{comparison rule B} hold for the even integer $h$
in \rappendix{proof of adjacent distance larger than one II for an even integer h
by using comparison rule A for the odd integer h-1}
and \rappendix{proof of comparison rule B for an even integer h
by using comparison rule A for the odd integer h-1}, respectively.
Similarly, we show that if Comparison rule B in \rlemma{comparison rule B}
holds for some even integer $h-1$, where $2\leq h-1\leq N-1$,
then \rlemma{adjacent distance larger than one}
and Comparison rule A in \rlemma{comparison rule A} hold for the odd integer $h$
in \rappendix{proof of adjacent distance larger than one for an odd integer h
by using comparison rule B for the even integer h-1}
and \rappendix{proof of comparison rule A for an odd integer h
by using comparison rule B for the even integer h-1}, respectively.
Therefore, \rlemma{adjacent distance larger than one},
Comparison rule A in \rlemma{comparison rule A},
\rlemma{adjacent distance larger than one II},
and Comparison rule B in \rlemma{comparison rule B}
are proved by induction on $h$
(see \rfigure{proof-of-lemmas-by-induction} for an illustration).
\bpdffigure{proof-of-lemmas-by-induction.pdf}{5.5in}
\epdffigure{proof-of-lemmas-by-induction}
{Proof of \rlemma{adjacent distance larger than one},
Comparison rule A in \rlemma{comparison rule A},
\rlemma{adjacent distance larger than one II},
and Comparison rule B in \rlemma{comparison rule B} by induction on $h$:
(a) $N$ is an odd integer. (b) $N$ is an even integer.}
Then we use \rlemma{adjacent distance larger than one} and Comparison rule A in \rlemma{comparison rule A}
to prove \rlemma{nonadjacent distance larger than one} in \rappendix{proof of nonadjacent distance larger than one},
and use \rcorollary{adjacent distance larger than one}(i) (corollary to \rlemma{adjacent distance larger than one}),
\rcorollary{nonadjacent distance larger than one}(i) (corollary to \rlemma{nonadjacent distance larger than one}),
and Comparison rule A in \rlemma{comparison rule A}
to prove \rlemma{main lemma} in \rappendix{proof of main lemma}
(see \rfigure{proof-of-the-main-lemmas}(a) for an illustration).
Finally, we use \rlemma{adjacent distance larger than one II} and Comparison rule B in \rlemma{comparison rule B}
to prove \rlemma{nonadjacent distance larger than one II} in \rappendix{proof of nonadjacent distance larger than one II},
and use \rcorollary{adjacent distance larger than one II}(i) (corollary to \rlemma{adjacent distance larger than one II}),
\rcorollary{nonadjacent distance larger than one II}(i) (corollary to \rlemma{nonadjacent distance larger than one II}),
and Comparison rule B in \rlemma{comparison rule B}
to prove \rlemma{main lemma II} in \rappendix{proof of main lemma II}
(see \rfigure{proof-of-the-main-lemmas}(b) for an illustration).
\bpdffigure{proof-of-the-main-lemmas.pdf}{6.0in}
\epdffigure{proof-of-the-main-lemmas}
{(a) Proof of \rlemma{nonadjacent distance larger than one} and \rlemma{main lemma}.
(b) Proof of \rlemma{nonadjacent distance larger than one II} and \rlemma{main lemma II}.}
We now use \rcorollary{adjacent distance larger than one}(ii),
\rcorollary{nonadjacent distance larger than one}(ii), \rlemma{main lemma},
\rcorollary{adjacent distance larger than one II}(ii),
\rcorollary{nonadjacent distance larger than one II}(ii), and \rlemma{main lemma II}
to prove \rtheorem{main result}.
\bproof \textbf{(Proof of \rtheorem{main result})}
Let $r_{-1}=M$, $r_0=k$, and let $q_i$ and $r_i$, $i=1,2,\ldots,N$, be recursively obtained
as in Step 1 of \ralgorithm{main result}.
Then we have $\gcd(M,k)=r_{N-1}$.
Note that as $r_{-1}>r_0>r_1>\cdots>r_{N-1}>r_N$ and $r_N=0$,
it is easy to see that
\beqnarray{proof-main result-111}
r_{h-1}\geq 2 \textrm{ and } r_h\neq 0, \textrm{ for } h=0,1,\ldots,N-1.
\eeqnarray
(i) Note that in \rtheorem{main result}(i), we have $\gcd(M,k)=r_{N-1}=1$.
Thus, it is easy to see from \reqnarray{N-M-k-h} that $\Ncal_{M,k}(N)=\{(r_{N-2})\}$.
Since $r_{N-1}=1$ and $r_N=0$,
we have $r_{N-2}=q_N\cdot r_{N-1}+r_N=q_N$
and it follows that $\Ncal_{M,k}(N)=\{(q_N)\}$.
We need to consider the following three cases.
\emph{Case 1: $N=1$.}
In this case, we have $\Ncal_{M,k}(1)=\{(q_1)\}$.
Let $\nbf_1^{r_0}(1)=(q_1)$.
As $\nbf_1^{r_0}(1)=(q_1)$ is the only sequence in $\Ncal_{M,k}(1)$,
it follows that there is only one optimal sequence over $\Ncal_{M,k}(1)$
and the optimal sequence is given by $\nbf_1^{r_0}(1)=(q_1)$.
It is also clear that $\nbf_1^{r_0}(1)=(q_1)$ is the sequence obtained
in Step 2(i) of \ralgorithm{main result} (note that $N=1$ is an odd integer).
\emph{Case 2: $N\geq 2$ and $N$ is an odd integer.}
Let $\nbf_1^{r_0}(1)\in \Ncal_{M,k}(1)$ be an optimal sequence over $\Ncal_{M,k}(1)$.
From \reqnarray{proof-main result-111}, we have $r_0\geq 2$ and $r_1\neq 0$.
It then follows from \rlemma{main lemma} (for the odd integer $h=1$)
that $\nbf_1^{r_0}(1)$ must satisfy the condition that
\beqnarray{}
n_i(1)=
\bselection
q_1+1, &\textrm{if } i=i_1,i_2,\ldots,i_{r_1}, \\
q_1, &\textrm{otherwise},
\eselection \nn
\eeqnarray
for some $1=i_1<i_2<\cdots <i_{r_1}\leq r_0$.
Thus, the left-imbedded sequence of $\nbf_1^{r_0}(1)$
with respect to $r_{-1}$ and $r_0$ is well defined, say
\beqnarray{proof-main result-(i)-case-2-111}
\nbf_1^{r_1}(2)=L_{r_{-1},r_0}^I(\nbf_1^{r_0}(1)).
\eeqnarray
From the definition of left-imbedded sequences in \rdefinition{left-imbedded sequences},
we know that $\nbf_1^{r_1}(2)$ is a sequence of positive integers such that $\sum_{i=1}^{r_1}n_i(2)=r_0$,
i.e., $\nbf_1^{r_1}(2)\in \Ncal_{M,k}(2)$.
We prove by contradiction that $\nbf_1^{r_1}(2)$ is an optimal sequence over $\Ncal_{M,k}(2)$.
Assume that there exists a sequence ${\nbf'}_1^{r_1}(2)\in \Ncal_{M,k}(2)$
such that ${\nbf'}_1^{r_1}(2)\succ\nbf_1^{r_1}(2)$.
As ${\nbf'}_1^{r_1}(2)\in \Ncal_{M,k}(2)$, we have $\sum_{i=1}^{r_1}n_i'(2)=r_0$.
Thus, the left pre-sequence of ${\nbf'}_1^{r_1}(2)$
with respect to $r_{-1}$ and $r_0$ is well defined, say
\beqnarray{proof-main result-(i)-case-2-222}
{\nbf'}_1^{r_0}(1)=L_{r_{-1},r_0}({\nbf'}_1^{r_1}(2)).
\eeqnarray
From \reqnarray{proof-main result-(i)-case-2-111} and \rlemma{left-imbedded sequences-left pre-sequences}(i),
we have
\beqnarray{proof-main result-(i)-case-2-333}
\nbf_1^{r_0}(1)=L_{r_{-1},r_0}(\nbf_1^{r_1}(2)).
\eeqnarray
As such, we have from
${\nbf'}_1^{r_1}(2)\succ\nbf_1^{r_1}(2)$,
\reqnarray{proof-main result-(i)-case-2-222},
\reqnarray{proof-main result-(i)-case-2-333},
and \reqnarray{order relation-777} that ${\nbf'}_1^{r_0}(1)\succ\nbf_1^{r_0}(1)$,
and we have reached a contradiction to the optimality of $\nbf_1^{r_0}(1)$.
As $N\geq 2$ and $N$ is an odd integer, we have $N\geq 3$.
Thus, it is clear from \reqnarray{proof-main result-111} that $r_1\geq 2$ and $r_2\neq 0$.
As $\nbf_1^{r_1}(2)$ is an optimal sequence over $\Ncal_{M,k}(2)$,
it then follows from \rlemma{main lemma II} (for the even integer $h=2$)
that $\nbf_1^{r_1}(2)$ must satisfy the condition that
\beqnarray{}
n_i(2)=
\bselection
q_2+1, &\textrm{if } i=i_1,i_2,\ldots,i_{r_2}, \\
q_2, &\textrm{otherwise},
\eselection \nn
\eeqnarray
for some $1\leq i_1<i_2<\cdots <i_{r_2}=r_1$.
Thus, the right-imbedded sequence of $\nbf_1^{r_1}(2)$
with respect to $r_0$ and $r_1$ is well defined, say
\beqnarray{proof-main result-(i)-case-2-444}
\nbf_1^{r_2}(3)=R_{r_0,r_1}^I(\nbf_1^{r_1}(2)).
\eeqnarray
From the definition of right-imbedded sequences in \rdefinition{right-imbedded sequences},
we know that $\nbf_1^{r_2}(3)$ is a sequence of positive integers such that
$\sum_{i=1}^{r_2}n_i(3)=r_1$, i.e., $\nbf_1^{r_2}(3)\in \Ncal_{M,k}(3)$.
We prove by contradiction that $\nbf_1^{r_2}(3)$ is an optimal sequence over $\Ncal_{M,k}(3)$.
Assume that there exists a sequence ${\nbf'}_1^{r_2}(3)\in \Ncal_{M,k}(3)$
such that ${\nbf'}_1^{r_2}(3)\succ\nbf_1^{r_2}(3)$.
As ${\nbf'}_1^{r_2}(3)\in \Ncal_{M,k}(3)$, we have $\sum_{i=1}^{r_2}n_i'(3)=r_1$.
Thus, the right pre-sequence of ${\nbf'}_1^{r_2}(3)$
with respect to $r_0$ and $r_1$ is well defined, say
\beqnarray{proof-main result-(i)-case-2-555}
{\nbf'}_1^{r_1}(2)=R_{r_0,r_1}({\nbf'}_1^{r_2}(3)).
\eeqnarray
From \reqnarray{proof-main result-(i)-case-2-444} and \rlemma{right-imbedded sequences-right pre-sequences}(i),
we have
\beqnarray{proof-main result-(i)-case-2-666}
\nbf_1^{r_1}(2)=R_{r_0,r_1}(\nbf_1^{r_2}(3)).
\eeqnarray
As such, we have from
${\nbf'}_1^{r_2}(3)\succ\nbf_1^{r_2}(3)$,
\reqnarray{proof-main result-(i)-case-2-555},
\reqnarray{proof-main result-(i)-case-2-666},
and \reqnarray{order relation-777} that ${\nbf'}_1^{r_1}(2)\succ\nbf_1^{r_1}(2)$,
and we have reached a contradiction to the optimality of $\nbf_1^{r_1}(2)$.
Clearly, for the optimal sequence $\nbf_1^{r_0}(1)$ over $\Ncal_{M,k}(1)$,
we can repeat the above argument and obtain a corresponding optimal sequence
$\nbf_1^{r_{h-1}}(h)$ over $\Ncal_{M,k}(h)$ for $h=2,3,\ldots,N$, where
\beqnarray{}
\nbf_1^{r_1}(2) \aligneq L_{r_{-1},r_0}^I(\nbf_1^{r_0}(1)),
\label{eqn:proof-main result-(i)-case-2-777} \\
\nbf_1^{r_2}(3) \aligneq R_{r_0,r_1}^I(\nbf_1^{r_1}(2)),
\label{eqn:proof-main result-(i)-case-2-888} \\
\nbf_1^{r_3}(4) \aligneq L_{r_1,r_2}^I(\nbf_1^{r_2}(3)),
\label{eqn:proof-main result-(i)-case-2-999} \\
\nbf_1^{r_4}(5) \aligneq R_{r_2,r_3}^I(\nbf_1^{r_3}(4)),
\label{eqn:proof-main result-(i)-case-2-aaa} \\
&\vdots& \nn\\
\nbf_1^{r_{N-2}}(N-1) \aligneq L_{r_{N-4},r_{N-3}}^I(\nbf_1^{r_{N-3}}(N-2)),
\label{eqn:proof-main result-(i)-case-2-bbb} \\
\nbf_1^{r_{N-1}}(N) \aligneq R_{r_{N-3},r_{N-2}}^I(\nbf_1^{r_{N-2}}(N-1)).
\label{eqn:proof-main result-(i)-case-2-ccc}
\eeqnarray
As $\Ncal_{M,k}(N)=\{(q_N)\}$, we see that $\nbf_1^{r_{N-1}}(N)=(q_N)$.
It then follows from \reqnarray{proof-main result-(i)-case-2-777}--\reqnarray{proof-main result-(i)-case-2-ccc},
\rlemma{left-imbedded sequences-left pre-sequences}(i),
and \rlemma{right-imbedded sequences-right pre-sequences}(i)
that $\nbf_1^{r_0}(1)$ can be obtained from $\nbf_1^{r_{N-1}}(N)=(q_N)$ as follows:
\beqnarray{}
\nbf_1^{r_{N-2}}(N-1) \aligneq R_{r_{N-3},r_{N-2}}(\nbf_1^{r_{N-1}}(N)),
\label{eqn:proof-main result-(i)-case-2-ddd} \\
\nbf_1^{r_{N-3}}(N-2) \aligneq L_{r_{N-4},r_{N-3}}(\nbf_1^{r_{N-2}}(N-1)),
\label{eqn:proof-main result-(i)-case-2-eee} \\
&\vdots& \nn\\
\nbf_1^{r_3}(4) \aligneq R_{r_2,r_3}(\nbf_1^{r_4}(5)),
\label{eqn:proof-main result-(i)-case-2-fff} \\
\nbf_1^{r_2}(3) \aligneq L_{r_1,r_2}(\nbf_1^{r_3}(4)),
\label{eqn:proof-main result-(i)-case-2-ggg} \\
\nbf_1^{r_1}(2) \aligneq R_{r_0,r_1}(\nbf_1^{r_2}(3)),
\label{eqn:proof-main result-(i)-case-2-hhh} \\
\nbf_1^{r_0}(1) \aligneq L_{r_{-1},r_0}(\nbf_1^{r_1}(2)).
\label{eqn:proof-main result-(i)-case-2-iii}
\eeqnarray
Suppose that there exists an optimal sequence ${\nbf'}_1^{r_0}(1)$
over $\Ncal_{M,k}(1)$ such that ${\nbf'}_1^{r_0}(1)\neq \nbf_1^{r_0}(1)$.
For the optimal sequence ${\nbf'}_1^{r_0}(1)$,
we can obtain a corresponding optimal sequence ${\nbf'}_1^{r_{N-1}}(N)$ over $\Ncal_{M,k}(N)$
in a similar way that the optimal sequence $\nbf_1^{r_{N-1}}(N)$ is obtained from $\nbf_1^{r_0}(1)$
by using \reqnarray{proof-main result-(i)-case-2-777}--\reqnarray{proof-main result-(i)-case-2-ccc}.
As $\Ncal_{M,k}(N)=\{(q_N)\}$, we have ${\nbf'}_1^{r_{N-1}}(N)=(q_N)=\nbf_1^{r_{N-1}}(N)$.
Since ${\nbf'}_1^{r_0}(1)$ can be obtained from ${\nbf'}_1^{r_{N-1}}(N)$
in a similar way that $\nbf_1^{r_0}(1)$ is obtained from $\nbf_1^{r_{N-1}}(N)$
by using \reqnarray{proof-main result-(i)-case-2-ddd}--\reqnarray{proof-main result-(i)-case-2-iii},
we see from $\nbf_1^{r_{N-1}}(N)={\nbf'}_1^{r_{N-1}}(N)=(q_N)$ that
$\nbf_1^{r_0}(1)={\nbf'}_1^{r_0}(1)$, contradicting to $\nbf_1^{r_0}(1)\neq {\nbf'}_1^{r_0}(1)$.
Therefore, we conclude that $\nbf_1^{r_0}(1)$
is the only optimal sequence over $\Ncal_{M,k}(1)$.
As $\nbf_1^{r_{N-1}}(N)=(q_N)$ and \reqnarray{proof-main result-(i)-case-2-ddd}--\reqnarray{proof-main result-(i)-case-2-iii}
are the same as \reqnarray{main result-111}--\reqnarray{main result-222}
in Step 2(i) of \ralgorithm{main result} (note that $N$ is an odd integer),
it then follows that the optimal sequence $\nbf_1^{r_0}(1)$ obtained from $\nbf_1^{r_{N-1}}(N)=(q_N)$
by using \reqnarray{proof-main result-(i)-case-2-ddd}--\reqnarray{proof-main result-(i)-case-2-iii}
is the sequence obtained in Step 2(i) of \ralgorithm{main result}.
\emph{Case 3: $N\geq 2$ and $N$ is an even integer.}
Let $\nbf_1^{r_0}(1)\in \Ncal_{M,k}(1)$ be an optimal sequence over $\Ncal_{M,k}(1)$.
By a similar argument as in Case~2 above,
for the optimal sequence $\nbf_1^{r_0}(1)$ over $\Ncal_{M,k}(1)$,
we can obtain a corresponding optimal sequence
$\nbf_1^{r_{h-1}}(h)$ over $\Ncal_{M,k}(h)$ for $h=2,3,\ldots,N$, where
\beqnarray{}
\nbf_1^{r_1}(2) \aligneq L_{r_{-1},r_0}^I(\nbf_1^{r_0}(1)),
\label{eqn:proof-main result-(i)-case-3-111} \\
\nbf_1^{r_2}(3) \aligneq R_{r_0,r_1}^I(\nbf_1^{r_1}(2)),
\label{eqn:proof-main result-(i)-case-3-222} \\
\nbf_1^{r_3}(4) \aligneq L_{r_1,r_2}^I(\nbf_1^{r_2}(3)),
\label{eqn:proof-main result-(i)-case-3-333} \\
\nbf_1^{r_4}(5) \aligneq R_{r_2,r_3}^I(\nbf_1^{r_3}(4)),
\label{eqn:proof-main result-(i)-case-3-444} \\
&\vdots& \nn\\
\nbf_1^{r_{N-3}}(N-2) \aligneq L_{r_{N-5},r_{N-4}}^I(\nbf_1^{r_{N-4}}(N-3)),
\label{eqn:proof-main result-(i)-case-3-555} \\
\nbf_1^{r_{N-2}}(N-1) \aligneq R_{r_{N-4},r_{N-3}}^I(\nbf_1^{r_{N-3}}(N-2)),
\label{eqn:proof-main result-(i)-case-3-666} \\
\nbf_1^{r_{N-1}}(N) \aligneq L_{r_{N-3},r_{N-2}}^I(\nbf_1^{r_{N-2}}(N-1)).
\label{eqn:proof-main result-(i)-case-3-777}
\eeqnarray
As $\Ncal_{M,k}(N)=\{(q_N)\}$, we see that $\nbf_1^{r_{N-1}}(N)=(q_N)$.
It then follows from \reqnarray{proof-main result-(i)-case-3-111}--\reqnarray{proof-main result-(i)-case-3-777},
\rlemma{left-imbedded sequences-left pre-sequences}(i),
and \rlemma{right-imbedded sequences-right pre-sequences}(i)
that $\nbf_1^{r_0}(1)$ can be obtained from $\nbf_1^{r_{N-1}}(N)=(q_N)$ as follows:
\beqnarray{}
\nbf_1^{r_{N-2}}(N-1) \aligneq L_{r_{N-3},r_{N-2}}(\nbf_1^{r_{N-1}}(N)),
\label{eqn:proof-main result-(i)-case-3-888} \\
\nbf_1^{r_{N-3}}(N-2) \aligneq R_{r_{N-4},r_{N-3}}(\nbf_1^{r_{N-2}}(N-1)),
\label{eqn:proof-main result-(i)-case-3-999} \\
\nbf_1^{r_{N-4}}(N-3) \aligneq L_{r_{N-5},r_{N-4}}(\nbf_1^{r_{N-3}}(N-2)),
\label{eqn:proof-main result-(i)-case-3-aaa} \\
&\vdots& \nn\\
\nbf_1^{r_3}(4) \aligneq R_{r_2,r_3}(\nbf_1^{r_4}(5)),
\label{eqn:proof-main result-(i)-case-3-bbb} \\
\nbf_1^{r_2}(3) \aligneq L_{r_1,r_2}(\nbf_1^{r_3}(4)),
\label{eqn:proof-main result-(i)-case-3-ccc} \\
\nbf_1^{r_1}(2) \aligneq R_{r_0,r_1}(\nbf_1^{r_2}(3)),
\label{eqn:proof-main result-(i)-case-3-ddd} \\
\nbf_1^{r_0}(1) \aligneq L_{r_{-1},r_0}(\nbf_1^{r_1}(2)).
\label{eqn:proof-main result-(i)-case-3-eee}
\eeqnarray
Furthermore, we can argue as in Case~2 above that
$\nbf_1^{r_0}(1)$ is the only optimal sequence over $\Ncal_{M,k}(1)$.
As $\nbf_1^{r_{N-1}}(N)=(q_N)$ and \reqnarray{proof-main result-(i)-case-3-888}--\reqnarray{proof-main result-(i)-case-3-eee}
are the same as \reqnarray{main result-555}--\reqnarray{main result-777}
in Step 3(i) of \ralgorithm{main result} (note that $N$ is an even integer),
it then follows that the optimal sequence $\nbf_1^{r_0}(1)$ obtained from $\nbf_1^{r_{N-1}}(N)=(q_N)$
by using \reqnarray{proof-main result-(i)-case-3-888}--\reqnarray{proof-main result-(i)-case-3-eee}
is the sequence obtained in Step 3(i) of \ralgorithm{main result}.
(ii) Note that in \rtheorem{main result}(ii), we have $\gcd(M,k)=r_{N-1}=2$.
We consider the following three cases.
\emph{Case 1: $N=1$.}
As $r_{N-1}=2$ and $r_N=0$,
it follows from \rcorollary{adjacent distance larger than one}(ii) (for the odd integer $h=N=1$)
that there are two optimal sequences over $\Ncal_{M,k}(1)$,
and the two optimal sequences, say $\nbf_1^{r_0}(1)$ and $\mbf_1^{r_0}(1)$,
are given by $\nbf_1^{r_0}(1)=(q_1,q_1)$ and $\mbf_1^{r_0}(1)=(q_1+1,q_1-1)$.
Also, it is clear that $\nbf_1^{r_0}(1)=(q_1,q_1)$
and $\mbf_1^{r_0}=(q_1+1,q_1-1)$
are the sequences obtained in Step 2(i) and Step 2(ii), respectively,
of \ralgorithm{main result} (note that $N=1$ is an odd integer).
\emph{Case 2: $N\geq 2$ and $N$ is an odd integer.}
As $r_{N-1}=2$ and $r_N=0$,
it follows from \rcorollary{adjacent distance larger than one}(ii) (for the odd integer $h=N$)
that there are two optimal sequences over $\Ncal_{M,k}(N)$,
and the two optimal sequences, say $\nbf_1^{r_{N-1}}(N)$ and $\mbf_1^{r_{N-1}}(N)$,
are given by $\nbf_1^{r_{N-1}}(N)=(q_N,q_N)$
and $\mbf_1^{r_{N-1}}(N)=(q_N+1,q_N-1)$.
Let $\nbf_1^{r_0}(1)\in \Ncal_{M,k}(1)$ be an optimal sequence over $\Ncal_{M,k}(1)$.
As in Case~2 of (i) above,
for the optimal sequence $\nbf_1^{r_0}(1)$,
we can obtain a corresponding optimal sequence over $\Ncal_{M,k}(N)$,
say $\nbf_1^{r_{N-1}}(N)=(q_N,q_N)$,
and $\nbf_1^{r_0}(1)$ can be uniquely obtained from $\nbf_1^{r_{N-1}}(N)=(q_N,q_N)$
by using \reqnarray{proof-main result-(i)-case-2-ddd}--\reqnarray{proof-main result-(i)-case-2-iii}.
Furthermore, let
\beqnarray{}
\mbf_1^{r_{N-2}}(N-1) \aligneq R_{r_{N-3},r_{N-2}}(\mbf_1^{r_{N-1}}(N)),
\label{eqn:proof-main result-(ii)-case-2-111} \\
\mbf_1^{r_{N-3}}(N-2) \aligneq L_{r_{N-4},r_{N-3}}(\mbf_1^{r_{N-2}}(N-1)),
\label{eqn:proof-main result-(ii)-case-2-222} \\
&\vdots& \nn\\
\mbf_1^{r_3}(4) \aligneq R_{r_2,r_3}(\mbf_1^{r_4}(5)),
\label{eqn:proof-main result-(ii)-case-2-333} \\
\mbf_1^{r_2}(3) \aligneq L_{r_1,r_2}(\mbf_1^{r_3}(4)),
\label{eqn:proof-main result-(ii)-case-2-444} \\
\mbf_1^{r_1}(2) \aligneq R_{r_0,r_1}(\mbf_1^{r_2}(3)),
\label{eqn:proof-main result-(ii)-case-2-555} \\
\mbf_1^{r_0}(1) \aligneq L_{r_{-1},r_0}(\mbf_1^{r_1}(2)).
\label{eqn:proof-main result-(ii)-case-2-666}
\eeqnarray
As $\nbf_1^{r_{N-1}}(N)=(q_N,q_N)$ and $\mbf_1^{r_{N-1}}(N)=(q_N+1,q_N-1)$
are optimal sequences over $\Ncal_{M,k}(N)$,
we have $\nbf_1^{r_{N-1}}(N)\equiv \mbf_1^{r_{N-1}}(N)$.
It then follows from $\nbf_1^{r_{N-1}}(N)\equiv \mbf_1^{r_{N-1}}(N)$,
\reqnarray{proof-main result-(i)-case-2-ddd}--\reqnarray{proof-main result-(i)-case-2-iii},
\reqnarray{proof-main result-(ii)-case-2-111}--\reqnarray{proof-main result-(ii)-case-2-666},
and \reqnarray{order relation-777} that $\nbf_1^{r_0}(1)\equiv \mbf_1^{r_0}(1)$,
i.e., $\mbf_1^{r_0}(1)$ is also an optimal sequence over $\Ncal_{M,k}(1)$.
Since there are two optimal sequences over $\Ncal_{M,k}(N)$,
we see that there are two optimal sequences over $\Ncal_{M,k}(1)$,
and the two optimal sequences are $\nbf_1^{r_0}(1)$
obtained from $\nbf_1^{r_{N-1}}(N)=(q_N,q_N)$
by using \reqnarray{proof-main result-(i)-case-2-ddd}--\reqnarray{proof-main result-(i)-case-2-iii}
and $\mbf_1^{r_0}(1)$ obtained from $\mbf_1^{r_{N-1}}(N)=(q_N+1,q_N-1)$
by using \reqnarray{proof-main result-(ii)-case-2-111}--\reqnarray{proof-main result-(ii)-case-2-666}.
It is clear that the two optimal sequences $\nbf_1^{r_0}(1)$ and $\mbf_1^{r_0}(1)$
are the sequences obtained in Step 2(i) and Step 2(ii), respectively,
of \ralgorithm{main result} (note that $N$ is an odd integer).
\emph{Case 3: $N\geq 2$ and $N$ is an even integer.}
As $r_{N-1}=2$ and $r_N=0$,
it follows from \rcorollary{adjacent distance larger than one II}(ii) (for the even integer $h=N$)
that there are two optimal sequences over $\Ncal_{M,k}(N)$,
and the two optimal sequences, say $\nbf_1^{r_{N-1}}(N)$ and $\mbf_1^{r_{N-1}}(N)$,
are given by $\nbf_1^{r_{N-1}}(N)=(q_N,q_N)$
and $\mbf_1^{r_{N-1}}(N)=(q_N-1,q_N+1)$.
Let $\nbf_1^{r_0}(1)\in \Ncal_{M,k}(1)$ be an optimal sequence over $\Ncal_{M,k}(1)$.
As in Case~3 of (i) above,
for the optimal sequence $\nbf_1^{r_0}(1)$,
we can obtain a corresponding optimal sequence over $\Ncal_{M,k}(N)$,
say $\nbf_1^{r_{N-1}}(N)=(q_N,q_N)$,
and $\nbf_1^{r_0}(1)$ can be uniquely obtained from $\nbf_1^{r_{N-1}}(N)=(q_N,q_N)$
by using \reqnarray{proof-main result-(i)-case-3-888}--\reqnarray{proof-main result-(i)-case-3-eee}.
Furthermore, let
\beqnarray{}
\mbf_1^{r_{N-2}}(N-1) \aligneq L_{r_{N-3},r_{N-2}}(\mbf_1^{r_{N-1}}(N)),
\label{eqn:proof-main result-(ii)-case-3-111} \\
\mbf_1^{r_{N-3}}(N-2) \aligneq R_{r_{N-4},r_{N-3}}(\mbf_1^{r_{N-2}}(N-1)),
\label{eqn:proof-main result-(ii)-case-3-222} \\
\mbf_1^{r_{N-4}}(N-3) \aligneq L_{r_{N-5},r_{N-4}}(\mbf_1^{r_{N-3}}(N-2)),
\label{eqn:proof-main result-(ii)-case-3-333} \\
&\vdots& \nn\\
\mbf_1^{r_3}(4) \aligneq R_{r_2,r_3}(\mbf_1^{r_4}(5)),
\label{eqn:proof-main result-(ii)-case-3-444} \\
\mbf_1^{r_2}(3) \aligneq L_{r_1,r_2}(\mbf_1^{r_3}(4)),
\label{eqn:proof-main result-(ii)-case-3-555} \\
\mbf_1^{r_1}(2) \aligneq R_{r_0,r_1}(\mbf_1^{r_2}(3)),
\label{eqn:proof-main result-(ii)-case-3-666} \\
\mbf_1^{r_0}(1) \aligneq L_{r_{-1},r_0}(\mbf_1^{r_1}(2)).
\label{eqn:proof-main result-(ii)-case-3-777}
\eeqnarray
As $\nbf_1^{r_{N-1}}(N)=(q_N,q_N)$ and $\mbf_1^{r_{N-1}}(N)=(q_N-1,q_N+1)$
are optimal sequences over $\Ncal_{M,k}(N)$,
we have $\nbf_1^{r_{N-1}}(N)\equiv \mbf_1^{r_{N-1}}(N)$.
It then follows from $\nbf_1^{r_{N-1}}(N)\equiv \mbf_1^{r_{N-1}}(N)$,
\reqnarray{proof-main result-(i)-case-3-888}--\reqnarray{proof-main result-(i)-case-3-eee},
\reqnarray{proof-main result-(ii)-case-3-111}--\reqnarray{proof-main result-(ii)-case-3-777},
and \reqnarray{order relation-777} that $\nbf_1^{r_0}(1)\equiv \mbf_1^{r_0}(1)$,
i.e., $\mbf_1^{r_0}(1)$ is also an optimal sequence over $\Ncal_{M,k}(1)$.
Since there are two optimal sequences over $\Ncal_{M,k}(N)$,
we see that there are two optimal sequences over $\Ncal_{M,k}(1)$,
and the two optimal sequences are $\nbf_1^{r_0}(1)$
obtained from $\nbf_1^{r_{N-1}}(N)=(q_N,q_N)$
by using \reqnarray{proof-main result-(i)-case-3-888}--\reqnarray{proof-main result-(i)-case-3-eee}
and $\mbf_1^{r_0}(1)$ obtained from $\mbf_1^{r_{N-1}}(N)=(q_N-1,q_N+1)$
by using \reqnarray{proof-main result-(ii)-case-3-111}--\reqnarray{proof-main result-(ii)-case-3-777}.
It is clear that the two optimal sequences $\nbf_1^{r_0}(1)$ and $\mbf_1^{r_0}(1)$
are the sequences obtained in Step 3(i) and Step 3(ii), respectively,
of \ralgorithm{main result} (note that $N$ is an even integer).
(iii) Note that we assume that $\gcd(M,k)=r_{N-1}\geq 3$.
The proof is similar to that of (ii)
and we consider the following three cases.
\emph{Case 1: $N=1$.}
As $r_{N-1}\geq 3$ and $r_N=0$,
we have from \rcorollary{nonadjacent distance larger than one}(ii) (for the odd integer $h=N=1$)
that there are at most two optimal sequences over $\Ncal_{M,k}(1)$,
and the two possible optimal sequences, say $\nbf_1^{r_0}(1)$ and $\mbf_1^{r_0}(1)$,
are given by $\nbf_1^{r_0}(1)=(q_1,q_1,\ldots,q_1)$ and $\mbf_1^{r_0}(1)=(q_1+1,q_1,\ldots,q_1,q_1-1)$.
Clearly, $\nbf_1^{r_0}(1)=(q_1,q_1,\ldots,q_1)$ and $\mbf_1^{r_0}=(q_1+1,q_1,\ldots,q_1,q_1-1)$
are the sequences obtained in Step 2(i) and Step 2(ii), respectively,
of \ralgorithm{main result} (note that $N=1$ is an odd integer).
\emph{Case 2: $N\geq 2$ and $N$ is an odd integer.}
As $r_{N-1}\geq 3$ and $r_N=0$,
we have from \rcorollary{nonadjacent distance larger than one}(ii) (for the odd integer $h=N$)
that there are at most two optimal sequences over $\Ncal_{M,k}(N)$,
and the two possible optimal sequences, say $\nbf_1^{r_{N-1}}(N)$ and $\mbf_1^{r_{N-1}}(N)$,
are given by $\nbf_1^{r_{N-1}}(N)=(q_N,q_N,\ldots,q_N)$
and $\mbf_1^{r_{N-1}}(N)=(q_N+1,q_N,\ldots,q_N,q_N-1)$.
Therefore, there are at most two optimal sequences over $\Ncal_{M,k}(1)$,
and the two possible optimal sequences are $\nbf_1^{r_0}(1)$
obtained from $\nbf_1^{r_{N-1}}(N)=(q_N,q_N,\ldots,q_N)$
by using \reqnarray{proof-main result-(i)-case-2-ddd}--\reqnarray{proof-main result-(i)-case-2-iii}
and $\mbf_1^{r_0}(1)$ obtained from $\mbf_1^{r_{N-1}}(N)=(q_N+1,q_N,\ldots,q_N,q_N-1)$
by using \reqnarray{proof-main result-(ii)-case-2-111}--\reqnarray{proof-main result-(ii)-case-2-666}.
It is clear that the two optimal sequences $\nbf_1^{r_0}(1)$ and $\mbf_1^{r_0}(1)$
are the sequences obtained in Step 2(i) and Step 2(ii), respectively,
of \ralgorithm{main result} (note that $N$ is an odd integer).
\emph{Case 3: $N\geq 2$ and $N$ is an even integer.}
As $r_{N-1}\geq 3$ and $r_N=0$,
we have from \rcorollary{nonadjacent distance larger than one II}(ii) (for the even integer $h=N$)
that there are at most two optimal sequences over $\Ncal_{M,k}(N)$,
and the two possible optimal sequences, say $\nbf_1^{r_{N-1}}(N)$ and $\mbf_1^{r_{N-1}}(N)$,
are given by $\nbf_1^{r_{N-1}}(N)=(q_N,q_N,\ldots,q_N)$
and $\mbf_1^{r_{N-1}}(N)=(q_N-1,q_N,\ldots,q_N,q_N+1)$.
Therefore, there are at most two optimal sequences over $\Ncal_{M,k}(1)$,
and the two possible optimal sequences are $\nbf_1^{r_0}(1)$
obtained from $\nbf_1^{r_{N-1}}(N)=(q_N,q_N,\ldots,q_N)$
by using \reqnarray{proof-main result-(i)-case-3-888}--\reqnarray{proof-main result-(i)-case-3-eee}
and $\mbf_1^{r_0}(1)$ obtained from $\mbf_1^{r_{N-1}}(N)=(q_N-1,q_N,\ldots,q_N,q_N+1)$
by using \reqnarray{proof-main result-(ii)-case-3-111}--\reqnarray{proof-main result-(ii)-case-3-777}.
It is clear that the two optimal sequences $\nbf_1^{r_0}(1)$ and $\mbf_1^{r_0}(1)$
are the sequences obtained in Step 3(i) and Step 3(ii), respectively,
of \ralgorithm{main result} (note that $N$ is an even integer).
\eproof
\bsection{Conclusion}{conclusion}
In this two-part paper, we address an important practical feasibility issue
that is of great concern in the SDL constructions of optical queues:
the constructions of optical queues with a limited number of
recirculations through the optical switches and the fiber delay lines.
In Part~I, we have proposed a class of greedy constructions
for certain types of optical queues, including linear compressors,
linear decompressors, and 2-to-1 FIFO multiplexers,
and have shown that every optimal construction among our previous constructions
of these types of optical queues under the constraint of
a limited number of recirculations must be a greedy construction.
In Part~II, we have further shown that there are at most two optimal constructions
and give a simple algorithm to obtain the optimal construction(s).
\iffalse
\section*{Acknowledgment}
The authors wish to thank their colleague Professor Cheng-Shang Chang
for inspiring discussions during the preparation of the manuscript.
\fi
\appendices
\bappendix{Proof of \rlemma{adjacent distance larger than one} with $h=1$}
{proof of adjacent distance larger than one with h=1}
In this appendix, we prove \rlemma{adjacent distance larger than one} for the case that $h=1$.
We need the following lemma for the proof of \rlemma{adjacent distance larger than one} with $h=1$.
\blemma{d>B+1}
Suppose that $M\geq 2$ and $1\leq k\leq M-1$.
Let $\nbf_1^k\in \Ncal_{M,k}$,
and let $s_0=0$ and $s_i=\sum_{\ell=1}^{i}n_{\ell}$ for $i=1,2,\ldots,k$.
Let $\dbf_1^M$ be obtained by using $\nbf_1^k$ in \reqnarray{OQ-LR-delays-greedy-1}.
Then
\beqnarray{d>B+1}
d_{s_{i+1}}>B(\dbf_1^{s_i};i)+1,
\eeqnarray
for $i=0,1,\ldots,k-1$.
\elemma
\bproof
We prove this lemma by induction on $i$.
As $\nbf_1^k\in \Ncal_{M,k}$,
we have $n_1\geq 2$, $n_2,n_3,\ldots,n_k\geq 1$, and $\sum_{i=1}^{k}n_{i}=M$.
From \reqnarray{OQ-LR-delays-greedy-2}, $n_1\geq 2$, and $B(\dbf_1^{s_0};0)=0$,
we have
\beqnarray{proof-d>B+1-111}
d_{s_1}=s_1=n_1\geq 2>1=B(\dbf_1^{s_0};0)+1.
\eeqnarray
It follows from \reqnarray{proof-d>B+1-111} that \reqnarray{d>B+1} holds for $i=0$.
Suppose as the induction hypothesis that \reqnarray{d>B+1} holds for some $0\leq i\leq k-2$.
From \reqnarray{OQ-LR-delays-greedy-3} and \reqnarray{OQ-LR-delays-greedy-7},
we have
\beqnarray{proof-d>B+1-222}
\alignspace d_{s_{i+2}}-(B(\dbf_1^{s_{i+1}};i+1)+1) \nn \\
\alignspace =2d_{s_{i+1}}+(n_{i+2}-1)(B(\dbf_1^{s_{i+1}};i+1)+1)-(B(\dbf_1^{s_{i+1}};i+1)+1) \nn \\
\alignspace =2d_{s_{i+1}}+(n_{i+2}-2)(B(\dbf_1^{s_{i+1}};i+1)+1).
\eeqnarray
If $n_{i+2}\geq 2$, then it follows from \reqnarray{proof-d>B+1-222} that
\beqnarray{proof-d>B+1-333}
d_{s_{i+2}}-(B(\dbf_1^{s_{i+1}};i+1)+1) \geq 2d_{s_{i+1}}>0.
\eeqnarray
On the other hand, if $n_{i+2}=1$, then we have from
\reqnarray{proof-d>B+1-222}, \reqnarray{OQ-LR-delays-greedy-7}, and the induction hypothesis that
\beqnarray{proof-d>B+1-444}
d_{s_{i+2}}-(B(\dbf_1^{s_{i+1}};i+1)+1)
\aligneq 2d_{s_{i+1}}-(B(\dbf_1^{s_{i+1}};i+1)+1) \nn\\
\aligneq 2d_{s_{i+1}}-(B(\dbf_1^{s_{i}};i)+d_{s_{i+1}}+1)\nn\\
\aligneq d_{s_{i+1}}-(B(\dbf_1^{s_{i}};i)+1)>0.
\eeqnarray
The induction is completed by combining
\reqnarray{proof-d>B+1-333} and \reqnarray{proof-d>B+1-444}.
\eproof
Now we use \rlemma{d>B+1} to prove \rlemma{adjacent distance larger than one}
for the case that $h=1$.
Suppose that $h=1$ in \rlemma{adjacent distance larger than one}.
For simplicity, let $\nbf_1^k=\nbf_1^{r_0}(1)$ and ${\nbf'}_1^k={\nbf'}_1^{r_0}(1)$
(note that $r_0=k$), i.e., $n_i=n_i(1)$ and $n'_i=n'_i(1)$ for $i=1,2,\ldots,k$.
Let $\dbf_1^M$ and ${\dbf'}_1^M$ be obtained by using
$\nbf_1^k$ and ${\nbf'}_1^k$, respectively,
in \reqnarray{OQ-LR-delays-greedy-1}.
Let $s_0=0$ and $s_i=\sum_{\ell=1}^{i}n_{\ell}$ for $i=1,2,\ldots,k$,
and let $s'_0=0$ and $s'_i=\sum_{\ell=1}^{i}n'_{\ell}$ for $i=1,2,\ldots,k$.
Let
\beqnarray{}
\alpha_i \aligneq d_{s_i}-d'_{s'_i}, \textrm{ for } i=1,2,\ldots,k,
\label{eqn:proof-adjacent distance larger than one-111}\\
\beta_i \aligneq B(\dbf_1^{s_i};i)-B({\dbf'}_1^{s'_i};i), \textrm{ for } i=0,1,2,\ldots,k.
\label{eqn:proof-adjacent distance larger than one-222}
\eeqnarray
It follows from \reqnarray{proof-adjacent distance larger than one-222},
\reqnarray{OQ-LR-delays-greedy-7},
and \reqnarray{proof-adjacent distance larger than one-111} that
\beqnarray{proof-adjacent distance larger than one-333}
\beta_i
\aligneq B(\dbf_1^{s_i};i)-B({\dbf'}_1^{s'_i};i)\nn \\
\aligneq B(\dbf_1^{s_{i-1}};i-1)+d_{s_i}-B({\dbf'}_1^{s'_{i-1}};i-1)-d'_{s'_i}\nn \\
\aligneq \alpha_i+\beta_{i-1},
\eeqnarray
for $i=1,2,\ldots,k$.
Note that in both \rlemma{adjacent distance larger than one}(i)
and \rlemma{adjacent distance larger than one}(ii),
we have $n'_i=n_i$ for $i=a+2,a+3,\ldots,k$,
and hence it follows from \reqnarray{proof-adjacent distance larger than one-111},
\reqnarray{OQ-LR-delays-greedy-3},
and \reqnarray{proof-adjacent distance larger than one-222} that
\beqnarray{proof-adjacent distance larger than one-444}
\alpha_i
\aligneq d_{s_i}-d'_{s'_i} \nn \\
\aligneq 2d_{s_{i-1}}+(n_i-1)(B(\dbf_1^{s_{i-1}};i-1)+1)
-2d_{s'_{i-1}}-(n'_i-1)(B({\dbf'}_1^{s'_{i-1}};i-1)+1) \nn \\
\aligneq 2\alpha_{i-1}+(n_i-1)\beta_{i-1},
\eeqnarray
for $i=a+2,a+3,\ldots,k$.
(i) Note that in \rlemma{adjacent distance larger than one}(i),
we have $\nbf_1^k \in \Ncal_{M,k}(1)$,
$n_a-n_{a+1}\leq -2$ for some $1\leq a\leq k-1$,
$n'_a=n_a+1$, $n'_{a+1}=n_{a+1}-1$, and $n'_i=n_i$ for $i\neq a$ and $a+1$.
It is clear that
\beqnarray{proof-adjacent distance larger than one-(i)-111}
n'_a=n_a+1\geq 2,\ n'_{a+1}=n_{a+1}-1\geq n_a+1\geq 2,
\textrm{ and } n'_i=n_i\geq 1 \textrm{ for } i\neq a, a+1.
\eeqnarray
Also, we have from $n'_a=n_a+1$, $n'_{a+1}=n_{a+1}-1$, $n'_i=n_i$ for $i\neq a$ and $a+1$,
$\nbf_1^k \in \Ncal_{M,k}(1)$, and \reqnarray{N-M-k-h} that
\beqnarray{proof-adjacent distance larger than one-(i)-222}
\sum_{i=1}^{k}n'_i
\aligneq \sum_{i\neq a,a+1}n'_i+n'_a+n'_{a+1} \nn\\
\aligneq \sum_{i\neq a,a+1}n_i+(n_a+1)+(n_{a+1}-1) \nn\\
\aligneq \sum_{i=1}^{k}n_i=M=r_{-1}.
\eeqnarray
As such, it follows from \reqnarray{proof-adjacent distance larger than one-(i)-111},
\reqnarray{proof-adjacent distance larger than one-(i)-222},
and \reqnarray{N-M-k-h} that ${\nbf'}_1^k\in \Ncal_{M,k}(1)$.
To show \reqnarray{adjacent distance larger than one-1} with $h=1$,
i.e., $\nbf_1^k\prec {\nbf'}_1^k$,
we see from the definition of the binary relation
$\prec$ in \reqnarray{order relation-666}
that we need to show that $B(\dbf_1^M;k)<B({\dbf'}_1^M;k)$.
It follows from \reqnarray{proof-adjacent distance larger than one-222}
that we need to show that
\beqnarray{proof-adjacent distance larger than one-(i)-333}
\beta_k=B(\dbf_1^{s_k};k)-B({\dbf'}_1^{s_k};k)=B(\dbf_1^M;k)-B({\dbf'}_1^M;k)<0.
\eeqnarray
We discuss the two cases $a=1$ and $2\leq a \leq k-1$ separately.
\emph{Case 1: $a=1$.}
In this case, we have from \reqnarray{proof-adjacent distance larger than one-111},
\reqnarray{proof-adjacent distance larger than one-222},
\reqnarray{OQ-LR-delays-greedy-7}, \reqnarray{OQ-LR-delays-greedy-2}, and $n'_1=n_1+1$ that
\beqnarray{proof-adjacent distance larger than one-(i)-case-1-111}
\alpha_1=\beta_1=d_{s_1}-d'_{s'_1}=s_1-s'_1=n_1-n'_1=-1.
\eeqnarray
From \reqnarray{proof-adjacent distance larger than one-111},
\reqnarray{OQ-LR-delays-greedy-3}, $n'_2=n_2-1$,
\reqnarray{proof-adjacent distance larger than one-222},
\reqnarray{proof-adjacent distance larger than one-(i)-case-1-111},
\reqnarray{OQ-LR-delays-greedy-7}, \reqnarray{OQ-LR-delays-greedy-2},
and $n_1-n_2\leq -2$, we have
\beqnarray{proof-adjacent distance larger than one-(i)-case-1-222}
\alpha_2
\aligneq d_{s_2}-d'_{s'_2} \nn\\
\aligneq 2d_{s_1}+(n_2-1)(B(\dbf_1^{s_{1}};1)+1)-2d'_{s'_1}-(n'_2-1)(B({\dbf'}_1^{s'_{1}};1)+1) \nn\\
\aligneq 2\alpha_1+(n_2-2)\beta_1+(B(\dbf_1^{s_{1}};1)+1) \nn\\
\aligneq -2-(n_2-2)+(d_{s_1}+1) \nn\\
\aligneq n_1-n_2+1 \nn\\
\alignleq -1.
\eeqnarray
As a result of $\beta_1<0$ in \reqnarray{proof-adjacent distance larger than one-(i)-case-1-111},
$\alpha_2<0$ in \reqnarray{proof-adjacent distance larger than one-(i)-case-1-222},
and $n_i\geq 1$ for $i=3,4,\ldots,k$,
we can use \reqnarray{proof-adjacent distance larger than one-333}
and \reqnarray{proof-adjacent distance larger than one-444}
(note that \reqnarray{proof-adjacent distance larger than one-444}
holds for $i=3,4,\ldots,k$ as we have $a=1$ in this case)
repeatedly to show that
$\beta_2<0,\ \alpha_3<0,\ \beta_3<0,\ \alpha_4<0, \ldots,\ \beta_{k-1}<0,\ \alpha_k<0$,
and $\beta_k<0$.
\emph{Case 2: $2\leq a\leq k-1$.}
In this case, we have from $n_i=n'_i$ for $i=1,2,\ldots,a-1$
that $s_i=s'_i$ for $i=1,2,\ldots,a-1$.
Thus, it is easy to see from \reqnarray{OQ-LR-delays-greedy-2}
and \reqnarray{OQ-LR-delays-greedy-3} that
\beqnarray{proof-adjacent distance larger than one-(i)-case-2-111}
d_i=d'_i, \textrm{ for } i=1,2,\ldots,s_{a-1}.
\eeqnarray
It follows from \reqnarray{proof-adjacent distance larger than one-111},
\reqnarray{proof-adjacent distance larger than one-222},
and \reqnarray{proof-adjacent distance larger than one-(i)-case-2-111} that
\beqnarray{proof-adjacent distance larger than one-(i)-case-2-222}
\alpha_i=\beta_i=0, \textrm{ for } i=1,2,\ldots,a-1.
\eeqnarray
From \reqnarray{proof-adjacent distance larger than one-111},
\reqnarray{OQ-LR-delays-greedy-3}, $n'_a=n_a+1$,
\reqnarray{proof-adjacent distance larger than one-222},
and $\alpha_{a-1}=\beta_{a-1}=0$ in \reqnarray{proof-adjacent distance larger than one-(i)-case-2-222},
we have
\beqnarray{proof-adjacent distance larger than one-(i)-case-2-333}
\alpha_a
\aligneq d_{s_a}-d'_{s'_a} \nn \\
\aligneq 2d_{s_{a-1}}+(n_a-1)(B(\dbf_1^{s_{a-1}};a-1)+1)-2d'_{s'_{a-1}}-(n'_a-1)(B({\dbf'}_1^{s'_{a-1}};a-1)+1) \nn\\
\aligneq 2\alpha_{a-1}+n_a\beta_{a-1}-(B(\dbf_1^{s_{a-1}};a-1)+1) \nn\\
\aligneq -(B(\dbf_1^{s_{a-1}};a-1)+1).
\eeqnarray
It then follows from \reqnarray{proof-adjacent distance larger than one-333},
$\beta_{a-1}=0$ in \reqnarray{proof-adjacent distance larger than one-(i)-case-2-222},
and \reqnarray{proof-adjacent distance larger than one-(i)-case-2-333} that
\beqnarray{proof-adjacent distance larger than one-(i)-case-2-444}
\beta_a=\alpha_a+\beta_{a-1}=-(B(\dbf_1^{s_{a-1}};a-1)+1).
\eeqnarray
From \reqnarray{proof-adjacent distance larger than one-111},
\reqnarray{OQ-LR-delays-greedy-3}, $n'_{a+1}=n_{a+1}-1$,
\reqnarray{proof-adjacent distance larger than one-222},
\reqnarray{OQ-LR-delays-greedy-7},
\reqnarray{proof-adjacent distance larger than one-(i)-case-2-333},
\reqnarray{proof-adjacent distance larger than one-(i)-case-2-444},
\reqnarray{OQ-LR-delays-greedy-3}, \reqnarray{OQ-LR-delays-greedy-7},
and $n_a-n_{a+1}\leq -2$, we have
\beqnarray{}
\alpha_{a+1}
\aligneq d_{s_{a+1}}-d'_{s'_{a+1}} \nn \\
\aligneq 2d_{s_a}+(n_{a+1}-1)(B(\dbf_1^{s_{a}};a)+1)-2d'_{s'_a}-(n'_{a+1}-1)(B({\dbf'}_1^{s'_{a}};a)+1) \nn\\
\aligneq 2\alpha_a+(n_{a+1}-2)\beta_a+(B(\dbf_1^{s_{a}};a)+1) \nn\\
\aligneq 2\alpha_a+(n_{a+1}-2)\beta_a+(B(\dbf_1^{s_{a-1}};a-1)+d_{s_a}+1) \nn\\
\aligneq -(n_{a+1}-1)(B(\dbf_1^{s_{a-1}};a-1)+1)+d_{s_a} \nn \\
\aligneq -(n_{a+1}-1)(B(\dbf_1^{s_{a-1}};a-1)+1)+2d_{s_{a-1}}+(n_a-1)(B(\dbf_1^{s_{a-1}};a-1)+1) \nn\\
\aligneq (n_a-n_{a+1})(B(\dbf_1^{s_{a-1}};a-1)+1)+2d_{s_{a-1}} \nn
\eeqnarray
\beqnarray{proof-adjacent distance larger than one-(i)-case-2-555}
\phantom{\alpha_{a+1}}
\alignspace \phantom{-(n_{a+1}-1)(B(\dbf_1^{s_{a-1}};a-1)+1)+2d_{s_{a-1}}+(n_a-1)(B(\dbf_1^{s_{a-1}};a-1)+1)} \nn\\
\aligneq (n_a-n_{a+1})(B(\dbf_1^{s_{a-2}};a-2)+d_{s_{a-1}}+1)+2d_{s_{a-1}}\nn\\
\aligneq (n_a-n_{a+1})(B(\dbf_1^{s_{a-2}};a-2)+1)+(n_a-n_{a+1}+2)d_{s_{a-1}}\nn\\
\alignleq -2(B(\dbf_1^{s_{a-2}};a-2)+1).
\eeqnarray
\iffalse
\beqnarray{proof-adjacent distance larger than one-(i)-case-2-555}
\alpha_{a+1}
\aligneq d_{s_{a+1}}-d'_{s'_{a+1}} \nn \\
\aligneq 2d_{s_a}+(n_{a+1}-1)(B(\dbf_1^{s_{a}};a)+1)-2d'_{s'_a}-(n'_{a+1}-1)(B({\dbf'}_1^{s'_{a}};a)+1) \nn\\
\aligneq 2\alpha_a+(n_{a+1}-2)\beta_a+(B(\dbf_1^{s_{a}};a)+1) \nn\\
\aligneq 2\alpha_a+(n_{a+1}-2)\beta_a+(B(\dbf_1^{s_{a-1}};a-1)+d_{s_a}+1) \nn\\
\aligneq -(n_{a+1}-1)(B(\dbf_1^{s_{a-1}};a-1)+1)+d_{s_a} \nn \\
\aligneq -(n_{a+1}-1)(B(\dbf_1^{s_{a-1}};a-1)+1)+2d_{s_{a-1}}+(n_a-1)(B(\dbf_1^{s_{a-1}};a-1)+1) \nn\\
\aligneq (n_a-n_{a+1})(B(\dbf_1^{s_{a-1}};a-1)+1)+2d_{s_{a-1}}\nn\\
\aligneq (n_a-n_{a+1})(B(\dbf_1^{s_{a-2}};a-2)+d_{s_{a-1}}+1)+2d_{s_{a-1}}\nn\\
\aligneq (n_a-n_{a+1})(B(\dbf_1^{s_{a-2}};a-2)+1)+(n_a-n_{a+1}+2)d_{s_{a-1}}\nn\\
\alignleq -2(B(\dbf_1^{s_{a-2}};a-2)+1).
\eeqnarray
\fi
As a result of $\beta_a<0$ in \reqnarray{proof-adjacent distance larger than one-(i)-case-2-444},
$\alpha_{a+1}<0$ in \reqnarray{proof-adjacent distance larger than one-(i)-case-2-555},
and $n_i\geq 1$ for $i=a+2,a+3,\ldots,k$,
we can use \reqnarray{proof-adjacent distance larger than one-333}
and \reqnarray{proof-adjacent distance larger than one-444}
(note that \reqnarray{proof-adjacent distance larger than one-444} holds for $i=a+2,a+3,\ldots,k$)
repeatedly to show that
$\beta_{a+1}<0,\ \alpha_{a+2}<0,\ \beta_{a+2}<0,\ \alpha_{a+3}<0, \ldots,\ \beta_{k-1}<0,\ \alpha_k<0$,
and $\beta_k<0$.
(ii) Note that in \rlemma{adjacent distance larger than one}(ii),
we have $\nbf_1^k \in \Ncal_{M,k}(1)$,
$n_a-n_{a+1}\geq 2$ for some $1\leq a\leq k-1$,
$n'_a=n_a-1$, $n'_{a+1}=n_{a+1}+1$, and $n'_i=n_i$ for $i\neq a$ and $a+1$.
It is clear that
\beqnarray{proof-adjacent distance larger than one-(ii)-111}
n'_a=n_a-1\geq n_{a+1}+1\geq 2,\ n'_{a+1}=n_{a+1}+1\geq 2,
\textrm{ and } n'_i=n_i\geq 1 \textrm{ for } i\neq a,a+1.
\eeqnarray
Also, we have from $n'_a=n_a-1$, $n'_{a+1}=n_{a+1}+1$, $n'_i=n_i$ for $i\neq a$ and $a+1$,
$\nbf_1^k \in \Ncal_{M,k}(1)$, and \reqnarray{N-M-k-h} that
\beqnarray{proof-adjacent distance larger than one-(ii)-222}
\sum_{i=1}^{k}n'_i
\aligneq \sum_{i\neq a,a+1}n'_i+n'_a+n'_{a+1} \nn\\
\aligneq \sum_{i\neq a,a+1}n_i+(n_a-1)+(n_{a+1}+1) \nn\\
\aligneq \sum_{i=1}^{k}n_i=M=r_{-1}.
\eeqnarray
As such, it follows from \reqnarray{proof-adjacent distance larger than one-(ii)-111},
\reqnarray{proof-adjacent distance larger than one-(ii)-222},
and \reqnarray{N-M-k-h} that ${\nbf'}_1^k\in \Ncal_{M,k}(1)$.
To show \reqnarray{adjacent distance larger than one-2} with $h=1$,
i.e., $\nbf_1^k\preceq {\nbf'}_1^k$,
where $\nbf_1^k\equiv {\nbf'}_1^k$ if and only if $k=2$ and $n_1=n_2+2$,
we see from \reqnarray{order relation-666}
and \reqnarray{proof-adjacent distance larger than one-222}
that we need to show that
\beqnarray{proof-adjacent distance larger than one-(ii)-333}
\beta_k=B(\dbf_1^{s_k};k)-B({\dbf'}_1^{s_k};k)=B(\dbf_1^M;k)-B({\dbf'}_1^M;k) \leq 0,
\eeqnarray
where $\beta_k=0$ if and only if $k=2$ and $n_1=n_2+2$.
We discuss the two cases $a=1$ and $2\leq a \leq k-1$ separately.
\emph{Case 1: $a=1$.}
In this case, we have from \reqnarray{proof-adjacent distance larger than one-111},
\reqnarray{proof-adjacent distance larger than one-222},
\reqnarray{OQ-LR-delays-greedy-7}, \reqnarray{OQ-LR-delays-greedy-2}, and $n'_1=n_1-1$ that
\beqnarray{proof-adjacent distance larger than one-(ii)-case-1-111}
\alpha_1=\beta_1=d_{s_1}-d'_{s'_1}=s_1-s'_1=n_1-n'_1=1.
\eeqnarray
From \reqnarray{proof-adjacent distance larger than one-111},
\reqnarray{OQ-LR-delays-greedy-3}, $n'_2=n_2+1$,
\reqnarray{proof-adjacent distance larger than one-222},
\reqnarray{proof-adjacent distance larger than one-(ii)-case-1-111},
\reqnarray{OQ-LR-delays-greedy-7}, \reqnarray{OQ-LR-delays-greedy-2},
and $n_1-n_2\geq 2$, we have
\beqnarray{}
\alpha_2
\aligneq d_{s_2}-d'_{s'_2} \nn\\
\aligneq 2d_{s_1}+(n_2-1)(B(\dbf_1^{s_{1}};1)+1)-2d'_{s'_1}-(n'_2-1)(B({\dbf'}_1^{s'_{1}};1)+1) \nn\\
\aligneq 2\alpha_1+n_2\beta_1-(B(\dbf_1^{s_{1}};1)+1) \nn\\
\aligneq 2+n_2-(d_{s_1}+1)\nn\\
\aligneq n_2-n_1+1
\label{eqn:proof-adjacent distance larger than one-(ii)-case-1-222}\\
\alignleq -1,
\label{eqn:proof-adjacent distance larger than one-(ii)-case-1-333}
\eeqnarray
where the inequality holds with equality if and only if $n_1=n_2+2$.
It follows from \reqnarray{proof-adjacent distance larger than one-333},
\reqnarray{proof-adjacent distance larger than one-(ii)-case-1-333},
and \reqnarray{proof-adjacent distance larger than one-(ii)-case-1-111} that
\beqnarray{proof-adjacent distance larger than one-(ii)-case-1-444}
\beta_2=\alpha_2+\beta_1\leq -1+1=0,
\eeqnarray
where the inequality holds with equality if and only if $n_1=n_2+2$.
If $k=2$, then we have from \reqnarray{proof-adjacent distance larger than one-(ii)-case-1-444}
that $\beta_k=\beta_2\leq 0$,
where $\beta_k=\beta_2=0$ if and only if $n_1=n_2+2$.
On the other hand, if $k>2$,
then as a result of $\alpha_2<0$ in \reqnarray{proof-adjacent distance larger than one-(ii)-case-1-333},
$\beta_2\leq 0$ in \reqnarray{proof-adjacent distance larger than one-(ii)-case-1-444},
and $n_i\geq 1$ for $i=3,4,\ldots,k$,
we can use \reqnarray{proof-adjacent distance larger than one-444}
(note that \reqnarray{proof-adjacent distance larger than one-444}
holds for $i=3,4,\ldots,k$ as we have $a=1$ in this case)
and \reqnarray{proof-adjacent distance larger than one-333}
repeatedly to show that $\alpha_3<0,\ \beta_3<0,\ \alpha_4<0,\ \beta_4<0, \ldots,\ \alpha_k<0,\ \beta_k<0$.
\emph{Case 2: $2\leq a\leq k-1$.}
As in Case~2 of (i) above,
\reqnarray{proof-adjacent distance larger than one-(i)-case-2-222} also holds in this case.
From \reqnarray{proof-adjacent distance larger than one-111},
\reqnarray{OQ-LR-delays-greedy-3}, $n'_a=n_a-1$,
\reqnarray{proof-adjacent distance larger than one-222},
and $\alpha_{a-1}=\beta_{a-1}=0$ in \reqnarray{proof-adjacent distance larger than one-(i)-case-2-222},
we have
\beqnarray{proof-adjacent distance larger than one-(ii)-case-2-111}
\alpha_a
\aligneq d_{s_a}-d'_{s'_a} \nn \\
\aligneq 2d_{s_{a-1}}+(n_a-1)(B(\dbf_1^{s_{a-1}};a-1)+1)-2d'_{s'_{a-1}}-(n'_a-1)(B({\dbf'}_1^{s'_{a-1}};a-1)+1) \nn\\
\aligneq 2\alpha_{a-1}+(n_a-2)\beta_{a-1}+(B(\dbf_1^{s_{a-1}};a-1)+1) \nn\\
\aligneq B(\dbf_1^{s_{a-1}};a-1)+1.
\eeqnarray
It then follows from \reqnarray{proof-adjacent distance larger than one-333},
$\beta_{a-1}=0$ in \reqnarray{proof-adjacent distance larger than one-(i)-case-2-222},
and \reqnarray{proof-adjacent distance larger than one-(ii)-case-2-111} that
\beqnarray{proof-adjacent distance larger than one-(ii)-case-2-222}
\beta_a=\alpha_a+\beta_{a-1}=B(\dbf_1^{s_{a-1}};a-1)+1.
\eeqnarray
From \reqnarray{proof-adjacent distance larger than one-111},
\reqnarray{OQ-LR-delays-greedy-3}, $n'_{a+1}=n_{a+1}+1$,
\reqnarray{proof-adjacent distance larger than one-222},
\reqnarray{OQ-LR-delays-greedy-7},
\reqnarray{proof-adjacent distance larger than one-(ii)-case-2-111},
\reqnarray{proof-adjacent distance larger than one-(ii)-case-2-222},
\reqnarray{OQ-LR-delays-greedy-3}, \reqnarray{OQ-LR-delays-greedy-7},
and $n_a-n_{a+1}\geq 2$, we have
\beqnarray{}
\alpha_{a+1}
\aligneq d_{s_{a+1}}-d'_{s'_{a+1}} \nn \\
\aligneq 2d_{s_a}+(n_{a+1}-1)(B(\dbf_1^{s_{a}};a)+1)-2d'_{s'_a}-(n'_{a+1}-1)(B({\dbf'}_1^{s'_{a}};a)+1) \nn\\
\aligneq 2\alpha_a+n_{a+1}\beta_a-(B(\dbf_1^{s_{a}};a)+1) \nn\\
\aligneq 2\alpha_a+n_{a+1}\beta_a-(B(\dbf_1^{s_{a-1}};a-1)+d_{s_a}+1) \nn\\
\aligneq (n_{a+1}+1)(B(\dbf_1^{s_{a-1}};a-1)+1)-d_{s_a}\nn\\
\aligneq (n_{a+1}+1)(B(\dbf_1^{s_{a-1}};a-1)+1)-2d_{s_{a-1}}-(n_a-1)(B(\dbf_1^{s_{a-1}};a-1)+1) \nn\\
\aligneq (n_{a+1}-n_a+2)(B(\dbf_1^{s_{a-1}};a-1)+1)-2d_{s_{a-1}} \nn\\
\aligneq (n_{a+1}-n_a+2)(B(\dbf_1^{s_{a-2}};a-2)+d_{s_{a-1}}+1)-2d_{s_{a-1}}\nn\\
\aligneq (n_{a+1}-n_a+2)(B(\dbf_1^{s_{a-2}};a-2)+1)+(n_{a+1}-n_a)d_{s_{a-1}}
\label{eqn:proof-adjacent distance larger than one-(ii)-case-2-333}\\
\alignleq -2d_{s_{a-1}}.
\label{eqn:proof-adjacent distance larger than one-(ii)-case-2-444}
\eeqnarray
Also, from \reqnarray{proof-adjacent distance larger than one-333},
\reqnarray{proof-adjacent distance larger than one-(ii)-case-2-333},
\reqnarray{proof-adjacent distance larger than one-(ii)-case-2-222},
\reqnarray{OQ-LR-delays-greedy-7},
$n_a-n_{a+1}\geq 2$, and \rlemma{d>B+1}, we have
\beqnarray{}
\beta_{a+1}
\aligneq \alpha_{a+1}+\beta_a \nn\\
\aligneq (n_{a+1}-n_a+2)(B(\dbf_1^{s_{a-2}};a-2)+1)+(n_{a+1}-n_a)d_{s_{a-1}}+B(\dbf_1^{s_{a-1}};a-1)+1 \nn\\
\aligneq (n_{a+1}-n_a+2)(B(\dbf_1^{s_{a-2}};a-2)+1)+(n_{a+1}-n_a)d_{s_{a-1}}+B(\dbf_1^{s_{a-2}};a-2)+d_{s_{a-1}}+1 \nn
\eeqnarray
\beqnarray{}
\phantom{\beta_{a+1}}
\alignspace \phantom{(n_{a+1}-n_a+2)(B(\dbf_1^{s_{a-2}};a-2)+1)+(n_{a+1}-n_a)d_{s_{a-1}}+B(\dbf_1^{s_{a-2}};a-2)+d_{s_{a-1}}+1} \nn\\
\aligneq (n_{a+1}-n_a+3)(B(\dbf_1^{s_{a-2}};a-2)+1)+(n_{a+1}-n_a+1)d_{s_{a-1}}
\label{eqn:proof-adjacent distance larger than one-(ii)-case-2-555}\\
\alignleq B(\dbf_1^{s_{a-2}};a-2)+1-d_{s_{a-1}} \nn\\
\alignless 0.
\label{eqn:proof-adjacent distance larger than one-(ii)-case-2-666}
\eeqnarray
\iffalse
\beqnarray{}
\beta_{a+1}
\aligneq \alpha_{a+1}+\beta_a \nn\\
\aligneq (n_{a+1}-n_a+2)(B(\dbf_1^{s_{a-2}};a-2)+1)+(n_{a+1}-n_a)d_{s_{a-1}}+B(\dbf_1^{s_{a-1}};a-1)+1 \nn\\
\aligneq (n_{a+1}-n_a+2)(B(\dbf_1^{s_{a-2}};a-2)+1)+(n_{a+1}-n_a)d_{s_{a-1}}+B(\dbf_1^{s_{a-2}};a-2)+d_{s_{a-1}}+1 \nn\\
\aligneq (n_{a+1}-n_a+3)(B(\dbf_1^{s_{a-2}};a-2)+1)+(n_{a+1}-n_a+1)d_{s_{a-1}}
\label{eqn:proof-adjacent distance larger than one-(ii)-case-2-555}\\
\alignleq B(\dbf_1^{s_{a-2}};a-2)+1-d_{s_{a-1}} \nn\\
\alignless 0.
\label{eqn:proof-adjacent distance larger than one-(ii)-case-2-666}
\eeqnarray
\fi
As a result of $\alpha_{a+1}<0$ in \reqnarray{proof-adjacent distance larger than one-(ii)-case-2-444},
$\beta_{a+1}<0$ in \reqnarray{proof-adjacent distance larger than one-(ii)-case-2-666},
and $n_i\geq 1$ for $i=a+2,a+3,\ldots,k$,
we can use \reqnarray{proof-adjacent distance larger than one-444}
(note that \reqnarray{proof-adjacent distance larger than one-444} holds for $i=a+2,a+3,\ldots,k$)
and \reqnarray{proof-adjacent distance larger than one-333} repeatedly to show that
$\alpha_{a+2}<0,\ \beta_{a+2}<0,\ \alpha_{a+3}<0,\ \beta_{a+3}<0,\ldots,\ \alpha_k<0,\ \beta_k<0$.
\eproof
\bappendix{Proof of Comparison rule A in \rlemma{comparison rule A} with $h=1$}
{proof of comparison rule A with h=1}
In this appendix, we prove Comparison rule A in \rlemma{comparison rule A} for the case that $h=1$.
Suppose that $h=1$ in Comparison rule A in \rlemma{comparison rule A}.
For simplicity, let $\nbf_1^k=\nbf_1^{r_0}(1)$ and ${\nbf'}_1^k={\nbf'}_1^{r_0}(1)$
(note that $r_0=k$).
Let $\dbf_1^M$ and ${\dbf'}_1^M$ be obtained by using
$\nbf_1^k$ and ${\nbf'}_1^k$, respectively, in \reqnarray{OQ-LR-delays-greedy-1}.
Let $s_0=0$ and $s_i=\sum_{\ell=1}^{i}n_{\ell}$ for $i=1,2,\ldots,k$,
and let $s'_0=0$ and $s'_i=\sum_{\ell=1}^{i}n'_{\ell}$ for $i=1,2,\ldots,k$.
Also let $\alpha_i=d_{s_i}-d'_{s'_i}$ for $i=1,2,\ldots,k$
as in \reqnarray{proof-adjacent distance larger than one-111},
and $\beta_i=B(\dbf_1^{s_i};i)-B({\dbf'}_1^{s'_i};i)$ for $i=0,1,2,\ldots,k$
as in \reqnarray{proof-adjacent distance larger than one-222}.
Note that in \rlemma{comparison rule A},
we have $\nbf_1^k \in \Ncal_{M,k}(1)$,
$n_a-n_{a+1}=1$ for some $1\leq a\leq k-1$,
$n_1\geq 3$ in the case that $a=1$,
$n'_a=n_a-1$, $n'_{a+1}=n_{a+1}+1$, and $n'_i=n_i$ for $i\neq a$ and $a+1$.
As ${\nbf'}_1^k$ is obtained from $\nbf_1^k$ in exactly the same way as that
in \rlemma{adjacent distance larger than one}(ii),
it is clear that \reqnarray{proof-adjacent distance larger than one-333}--\reqnarray{proof-adjacent distance larger than one-444},
\reqnarray{proof-adjacent distance larger than one-(ii)-222}--\reqnarray{proof-adjacent distance larger than one-(ii)-case-1-222},
\reqnarray{proof-adjacent distance larger than one-(ii)-case-2-111}--\reqnarray{proof-adjacent distance larger than one-(ii)-case-2-333},
and \reqnarray{proof-adjacent distance larger than one-(ii)-case-2-555}
in the proof of \rlemma{adjacent distance larger than one}(ii) still hold.
It is also clear that
\beqnarray{proof-comparison rule A-111}
n'_a=n_a-1=n_{a+1}\geq 1,\ n'_{a+1}=n_{a+1}+1\geq 2,
\textrm{ and } n'_i=n_i\geq 1,\ i\neq a, a+1.
\eeqnarray
In the case that $a=1$, we have $n_1\geq 3$ and it follows that
\beqnarray{proof-comparison rule A-222}
n'_a=n_a-1=n_1-1\geq 3-1=2.
\eeqnarray
As such, it follows from \reqnarray{proof-comparison rule A-111},
\reqnarray{proof-comparison rule A-222},
\reqnarray{proof-adjacent distance larger than one-(ii)-222},
and \reqnarray{N-M-k-h} that ${\nbf'}_1^k\in \Ncal_{M,k}(1)$.
(i) Note that in \rlemma{comparison rule A}(i), we have $a=1$ or $a=k-1$.
To show \reqnarray{comparison rule A-1} with $h=1$,
i.e., $\nbf_1^k\succ {\nbf'}_1^k$,
we see from \reqnarray{order relation-666}
and \reqnarray{proof-adjacent distance larger than one-222}
that we need to show that $\beta_k>0$.
For the case that $a=1$,
we have from \reqnarray{proof-adjacent distance larger than one-(ii)-case-1-222} and $n_1-n_2=1$ that
\beqnarray{proof-comparison rule A-(i)-111}
\alpha_2=n_2-n_1+1=0.
\eeqnarray
It then follows from \reqnarray{proof-adjacent distance larger than one-333},
\reqnarray{proof-comparison rule A-(i)-111},
and \reqnarray{proof-adjacent distance larger than one-(ii)-case-1-111} that
\beqnarray{proof-comparison rule A-(i)-222}
\beta_2=\alpha_2+\beta_1=0+1=1.
\eeqnarray
As a result of $\alpha_2=0$ in \reqnarray{proof-comparison rule A-(i)-111},
$\beta_2>0$ in \reqnarray{proof-comparison rule A-(i)-222},
and $n_i\geq 1$ for $i=3,4,\ldots,k$,
we can use \reqnarray{proof-adjacent distance larger than one-444}
(note that \reqnarray{proof-adjacent distance larger than one-444}
holds for $i=3,4,\ldots,k$ as we have $a=1$ in this case)
and \reqnarray{proof-adjacent distance larger than one-333} repeatedly to show that
$\alpha_3\geq 0,\ \beta_3>0,\ \alpha_4\geq 0,\ \beta_4>0, \ldots,\ \alpha_k\geq 0,\ \beta_k>0$.
For the case that $a=k-1\geq 2$ (note that the case $a=1$ has just been discussed),
we have from \reqnarray{proof-adjacent distance larger than one-(ii)-case-2-555} and $n_{k-1}-n_k=1$ that
\beqnarray{}
\beta_k=2(B(\dbf_1^{s_{k-3}};k-3)+1)>0. \nn
\eeqnarray
(ii) Note that in \rlemma{comparison rule A}(ii),
we have $2\leq a\leq r_{h-1}-2$ and there exists a positive integer $j$ such that
$1\leq j\leq \min\{a-1,r_{h-1}-a-1\}$,
$n_{a-j'}=n_{a+1+j'}$ for $j'=1,2,\ldots,j-1$, and $n_{a-j}\neq n_{a+1+j}$.
First we use $n_{a-j'}=n_{a+1+j'}$ for $j'=1,2,\ldots,j-1$ to show that
\beqnarray{}
\alpha_{a+j'}\aligneq 2^{j'-1}(B(\dbf_1^{s_{a-j'-1}};a-j'-1)+1-d_{s_{a-j'}}),
\label{eqn:proof-comparison rule A-(ii)-111}\\
\beta_{a+j'}\aligneq 2^{j'}(B(\dbf_1^{s_{a-j'-1}};a-j'-1)+1),
\label{eqn:proof-comparison rule A-(ii)-222}
\eeqnarray
for $j'=1,2,\ldots,j$.
We prove \reqnarray{proof-comparison rule A-(ii)-111} and \reqnarray{proof-comparison rule A-(ii)-222}
by induction on $j'$.
From \reqnarray{proof-adjacent distance larger than one-(ii)-case-2-333},
\reqnarray{proof-adjacent distance larger than one-(ii)-case-2-555},
and $n_a-n_{a+1}=1$, we have
\beqnarray{}
\alpha_{a+1}\aligneq B(\dbf_1^{s_{a-2}};a-2)+1-d_{s_{a-1}},
\label{eqn:proof-comparison rule A-(ii)-333}\\
\beta_{a+1}\aligneq 2(B(\dbf_1^{s_{a-2}};a-2)+1).
\label{eqn:proof-comparison rule A-(ii)-444}
\eeqnarray
It follows from
\reqnarray{proof-comparison rule A-(ii)-333} and \reqnarray{proof-comparison rule A-(ii)-444}
that \reqnarray{proof-comparison rule A-(ii)-111} and \reqnarray{proof-comparison rule A-(ii)-222}
hold for $j'=1$.
Assume as the induction hypothesis that \reqnarray{proof-comparison rule A-(ii)-111}
and \reqnarray{proof-comparison rule A-(ii)-222} hold for some $1\leq j'\leq j-1$.
From \reqnarray{proof-adjacent distance larger than one-444},
the induction hypothesis, \reqnarray{OQ-LR-delays-greedy-3} (note that $a-j'-1\geq a-j\geq 1$),
$n_{a-j'}=n_{a+1+j'}$ (as $1\leq j'\leq j-1$), and \reqnarray{OQ-LR-delays-greedy-7}, we have
\beqnarray{proof-comparison rule A-(ii)-555}
\alignspace \hspace*{-0.3in} \alpha_{a+(j'+1)} \nn\\
\aligneq 2\alpha_{a+j'}+(n_{a+1+j'}-1)\beta_{a+j'} \nn \\
\aligneq 2\cdot 2^{j'-1}(B(\dbf_1^{s_{a-j'-1}};a-j'-1)+1-d_{s_{a-j'}}) \nn\\
\alignspace +(n_{a+1+j'}-1)\cdot 2^{j'}(B(\dbf_1^{s_{a-j'-1}};a-j'-1)+1) \nn \\
\aligneq 2^{j'}\left(B(\dbf_1^{s_{a-j'-1}};a-j'-1)+1-2d_{s_{a-j'-1}}
-(n_{a-j'}-1)(B(\dbf_1^{s_{a-j'-1}};a-j'-1)+1)\right)\nn\\
\alignspace +2^{j'}(n_{a+1+j'}-1)(B(\dbf_1^{s_{a-j'-1}};a-j'-1)+1) \nn \\
\aligneq 2^{j'}(B(\dbf_1^{s_{a-j'-1}};a-j'-1)+1-2d_{s_{a-j'-1}})\nn \\
\aligneq 2^{j'}(B(\dbf_1^{s_{a-j'-2}};a-j'-2)+d_{s_{a-j'-1}}+1-2d_{s_{a-j'-1}})\nn \\
\aligneq 2^{j'}(B(\dbf_1^{s_{a-j'-2}};a-j'-2)+1-d_{s_{a-j'-1}}).
\eeqnarray
From \reqnarray{proof-adjacent distance larger than one-333},
\reqnarray{proof-comparison rule A-(ii)-555}, the induction hypothesis,
and \reqnarray{OQ-LR-delays-greedy-7}, we have
\beqnarray{proof-comparison rule A-(ii)-666}
\alignspace \hspace*{-0.3in} \beta_{a+(j'+1)} \nn\\
\aligneq \alpha_{a+j'+1}+\beta_{a+j'} \nn\\
\aligneq 2^{j'}(B(\dbf_1^{s_{a-j'-2}};a-j'-2)+1-d_{s_{a-j'-1}})+2^{j'}(B(\dbf_1^{s_{a-j'-1}};a-j'-1)+1) \nn\\
\aligneq 2^{j'}(B(\dbf_1^{s_{a-j'-2}};a-j'-2)+1-d_{s_{a-j'-1}})+2^{j'}(B(\dbf_1^{s_{a-j'-2}};a-j'-2)+d_{s_{a-j'-1}}+1) \nn\\
\aligneq 2^{j'+1}(B(\dbf_1^{s_{a-j'-2}};a-j'-2)+1).
\eeqnarray
The induction is completed by combining \reqnarray{proof-comparison rule A-(ii)-555}
and \reqnarray{proof-comparison rule A-(ii)-666}.
Now we show that if $n_{a-j}<n_{a+1+j}$,
then \reqnarray{comparison rule A-2} holds with $h=1$,
i.e., $\beta_k>0$;
on the other hand, if $n_{a-j}>n_{a+1+j}$,
then \reqnarray{comparison rule A-3} holds with $h=1$,
i.e., $\beta_k\leq 0$,
where $\beta_k=0$ if and only if $a-j=1$, $a+1+j=k$, and $n_1=n_k+1$.
Note that as $j\leq \min\{a-1,k-a-1\}$, we have $j+1\leq a\leq k-j-1$.
We consider the two cases $a=j+1$ and $j+2\leq a\leq k-j-1$ separately.
\emph{Case 1: $a=j+1$.}
In this case, we have from \reqnarray{proof-comparison rule A-(ii)-222} (with $j'=j$),
$a=j+1$, and $B(\dbf_1^{s_0};0)=0$ that
\beqnarray{proof-comparison rule A-(ii)-case-1-111}
\beta_{a+j}=2^j(B(\dbf_1^{s_{a-j-1}};a-j-1)+1)=2^j(B(\dbf_1^{s_0};0)+1)=2^j.
\eeqnarray
We also have from \reqnarray{proof-adjacent distance larger than one-444},
\reqnarray{proof-comparison rule A-(ii)-111} (with $j'=j$),
\reqnarray{proof-comparison rule A-(ii)-case-1-111}, $a=j+1$,
$B(\dbf_1^{s_0};0)=0$, and \reqnarray{OQ-LR-delays-greedy-2} that
\beqnarray{proof-comparison rule A-(ii)-case-1-222}
\alpha_{a+j+1}
\aligneq 2\alpha_{a+j}+(n_{a+1+j}-1)\beta_{a+j} \nn \\
\aligneq 2\cdot 2^{j-1}(B(\dbf_1^{s_{a-j-1}};a-j-1)+1-d_{s_{a-j}})+(n_{a+1+j}-1)\cdot 2^j \nn \\
\aligneq 2^j(B(\dbf_1^{s_0};0)+1-s_1)+2^j(n_{a+1+j}-1) \nn\\
\aligneq 2^j(1-n_1)+2^j(n_{a+1+j}-1) \nn\\
\aligneq 2^j(n_{a+1+j}-n_{a-j}).
\eeqnarray
If $n_{a-j}<n_{a+1+j}$,
then it follows from \reqnarray{proof-comparison rule A-(ii)-case-1-222}
that $\alpha_{a+j+1}>0$.
As a result of $\beta_{a+j}>0$ in \reqnarray{proof-comparison rule A-(ii)-case-1-111},
$\alpha_{a+j+1}>0$, and $n_i\geq 1$ for $i=a+j+2,a+j+3,\ldots,k$,
we can use \reqnarray{proof-adjacent distance larger than one-333}
and \reqnarray{proof-adjacent distance larger than one-444}
(note that \reqnarray{proof-adjacent distance larger than one-444} holds for $i=a+2,a+3,\ldots,k$)
repeatedly to show that $\beta_{a+j+1}>0,\ \alpha_{a+j+2}>0,\ \beta_{a+j+2}>0,\ \alpha_{a+j+3}>0,
\ldots,\ \beta_{k-1}>0,\ \alpha_k>0$, and $\beta_k>0$.
On the other hand, if $n_{a-j}>n_{a+1+j}$, then it follows from \reqnarray{proof-comparison rule A-(ii)-case-1-222}
that $\alpha_{a+j+1}<0$.
From \reqnarray{proof-adjacent distance larger than one-333},
\reqnarray{proof-comparison rule A-(ii)-case-1-222}, \reqnarray{proof-comparison rule A-(ii)-case-1-111},
and $n_{a-j}>n_{a+1+j}$, we have
\beqnarray{}
\beta_{a+j+1}
\aligneq \alpha_{a+j+1}+\beta_{a+j} \nn\\
\aligneq 2^j(n_{a+1+j}-n_{a-j}+1)
\label{eqn:proof-comparison rule A-(ii)-case-1-333}\\
\alignleq 0,
\label{eqn:proof-comparison rule A-(ii)-case-1-444}
\eeqnarray
where the inequality holds with equality if and only if $n_{a-j}=n_{a+1+j}+1$.
For the case that $a=k-j-1$,
we see from \reqnarray{proof-comparison rule A-(ii)-case-1-444} that $\beta_k=\beta_{a+j+1}\leq 0$,
where $\beta_k=\beta_{a+j+1}=0$ if and only if $n_{a-j}=n_{a+1+j}+1$,
i.e., $n_1=n_k+1$ (as we have $a=j+1$ and $a=k-j-1$ in this case).
For the case that $a<k-j-1$,
we see that as a result of $\alpha_{a+j+1}<0$,
$\beta_{a+j+1}\leq 0$ in \reqnarray{proof-comparison rule A-(ii)-case-1-444},
and $n_i\geq 1$ for $i=a+j+2,a+j+3,\ldots,k$,
we can use \reqnarray{proof-adjacent distance larger than one-444}
(note that \reqnarray{proof-adjacent distance larger than one-444} holds for $i=a+2,a+3,\ldots,k$)
and \reqnarray{proof-adjacent distance larger than one-333} repeatedly
to show that $\alpha_{a+j+2}<0,\ \beta_{a+j+2}<0,\ \alpha_{a+j+3}<0,\ \beta_{a+j+3}<0,
\ldots,\ \alpha_k<0,\ \beta_k<0$.
\emph{Case 2: $j+2\leq a\leq k-j-1$.}
In this case, we have from \reqnarray{proof-comparison rule A-(ii)-222} (with $j'=j$) that
\beqnarray{proof-comparison rule A-(ii)-case-2-111}
\beta_{a+j}=2^j(B(\dbf_1^{s_{a-j-1}};a-j-1)+1).
\eeqnarray
We also have from \reqnarray{proof-adjacent distance larger than one-444},
\reqnarray{proof-comparison rule A-(ii)-111} (with $j'=j$),
\reqnarray{proof-comparison rule A-(ii)-case-2-111},
\reqnarray{OQ-LR-delays-greedy-3}, and \reqnarray{OQ-LR-delays-greedy-7} that
\beqnarray{proof-comparison rule A-(ii)-case-2-222}
\alignspace \hspace*{-0.3in} \alpha_{a+j+1} \nn \\
\aligneq 2\alpha_{a+j}+(n_{a+1+j}-1)\beta_{a+j} \nn \\
\aligneq 2\cdot 2^{j-1}(B(\dbf_1^{s_{a-j-1}};a-j-1)+1-d_{s_{a-j}})
+(n_{a+1+j}-1)\cdot 2^j(B(\dbf_1^{s_{a-j-1}};a-j-1)+1) \nn \\
\aligneq 2^j\Bigl(B(\dbf_1^{s_{a-j-1}};a-j-1)+1-2d_{s_{a-j-1}}-(n_{a-j}-1)(B(\dbf_1^{s_{a-j-1}};a-j-1)+1)\Bigr)\nn\\
\alignspace +2^j(n_{a+1+j}-1)(B(\dbf_1^{s_{a-j-1}};a-j-1)+1) \nn \\
\aligneq 2^j\Bigl((n_{a+1+j}-n_{a-j}+1)(B(\dbf_1^{s_{a-j-1}};a-j-1)+1)-2d_{s_{a-j-1}}\Bigr)\nn\\
\aligneq 2^j\Bigl((n_{a+1+j}-n_{a-j}+1)(B(\dbf_1^{s_{a-j-2}};a-j-2)+d_{s_{a-j-1}}+1)-2d_{s_{a-j-1}}\Bigr)\nn\\
\aligneq 2^j\Bigl((n_{a+1+j}-n_{a-j}+1)(B(\dbf_1^{s_{a-j-2}};a-j-2)+1)+(n_{a+1+j}-n_{a-j}-1)d_{s_{a-j-1}}\Bigr).
\eeqnarray
If $n_{a-j}<n_{a+1+j}$, then it follows from \reqnarray{proof-comparison rule A-(ii)-case-2-222} that
\beqnarray{proof-comparison rule A-(ii)-case-2-333}
\alpha_{a+j+1} \geq 2^j\cdot 2(B(\dbf_1^{s_{a-j-2}};a-j-2)+1)>0.
\eeqnarray
As we have $\beta_{a+j}>0$ in \reqnarray{proof-comparison rule A-(ii)-case-2-111},
$\alpha_{a+j+1}>0$ in \reqnarray{proof-comparison rule A-(ii)-case-2-333},
and $n_i\geq 1$ for $i=a+j+2,a+j+3,\ldots,k$,
we can use \reqnarray{proof-adjacent distance larger than one-333}
and \reqnarray{proof-adjacent distance larger than one-444}
(note that \reqnarray{proof-adjacent distance larger than one-444} holds for $i=a+2,a+3,\ldots,k$)
repeatedly to show that $\beta_{a+j+1}>0,\ \alpha_{a+j+2}>0,\ \beta_{a+j+2}>0,\ \alpha_{a+j+3}>0,
\ldots,\ \beta_{k-1}>0,\ \alpha_k>0$, and $\beta_k>0$.
On the other hand, if $n_{a-j}>n_{a+1+j}$, then it follows from \reqnarray{proof-comparison rule A-(ii)-case-2-222} that
\beqnarray{proof-comparison rule A-(ii)-case-2-444}
\alpha_{a+j+1} \leq 2^j(-2d_{s_{a-j-1}})< 0.
\eeqnarray
From \reqnarray{proof-adjacent distance larger than one-333},
\reqnarray{proof-comparison rule A-(ii)-case-2-222}, \reqnarray{proof-comparison rule A-(ii)-case-2-111},
\reqnarray{OQ-LR-delays-greedy-7}, $n_{a-j}>n_{a+1+j}$,
and \rlemma{d>B+1} (note that $0\leq a-j-2\leq k-2j-3\leq k-1$),
we have
\beqnarray{proof-comparison rule A-(ii)-case-2-555}
\alignspace \hspace*{-0.3in} \beta_{a+j+1} \nn\\
\aligneq \alpha_{a+j+1}+\beta_{a+j} \nn\\
\aligneq 2^j\Bigl((n_{a+1+j}-n_{a-j}+1)(B(\dbf_1^{s_{a-j-2}};a-j-2)+1)+(n_{a+1+j}-n_{a-j}-1)d_{s_{a-j-1}}\Bigr) \nn\\
\alignspace + 2^j(B(\dbf_1^{s_{a-j-1}};a-j-1)+1) \nn\\
\aligneq 2^j\Bigl((n_{a+1+j}-n_{a-j}+1)(B(\dbf_1^{s_{a-j-2}};a-j-2)+1)+(n_{a+1+j}-n_{a-j}-1)d_{s_{a-j-1}}\Bigr) \nn\\
\alignspace + 2^j(B(\dbf_1^{s_{a-j-2}};a-j-2)+d_{s_{a-j-1}}+1) \nn\\
\aligneq 2^j\Bigl((n_{a+1+j}-n_{a-j}+2)(B(\dbf_1^{s_{a-j-2}};a-j-2)+1)+(n_{a+1+j}-n_{a-j})d_{s_{a-j-1}}\Bigr) \nn\\
\alignleq 2^j(B(\dbf_1^{s_{a-j-2}};a-j-2)+1-d_{s_{a-j-1}}) \nn\\
\alignless 0.
\eeqnarray
As a result of $\alpha_{a+j+1}<0$ in \reqnarray{proof-comparison rule A-(ii)-case-2-444},
$\beta_{a+j+1}< 0$ in \reqnarray{proof-comparison rule A-(ii)-case-2-555},
and $n_i\geq 1$ for $i=a+j+2,a+j+3,\ldots,k$,
we can use \reqnarray{proof-adjacent distance larger than one-444}
(note that \reqnarray{proof-adjacent distance larger than one-444} holds for $i=a+2,a+3,\ldots,k$)
and \reqnarray{proof-adjacent distance larger than one-333} repeatedly
to show that $\alpha_{a+j+2}<0,\ \beta_{a+j+2}<0,\ \alpha_{a+j+3}<0,\ \beta_{a+j+3}<0,
\ldots,\ \alpha_k<0,\ \beta_k<0$.
(iii) Note that in \rlemma{comparison rule A}(iii),
we have $2\leq a\leq k-2$ and
$n_{a-j'}=n_{a+1+j'}$ for $j'=1,2,\ldots,\min\{a-1,k-a-1\}$.
To show \reqnarray{comparison rule A-4} with $h=1$,
i.e., $\nbf_1^k\succ {\nbf'}_1^k$,
we see from \reqnarray{order relation-666}
and \reqnarray{proof-adjacent distance larger than one-222}
that we need to show that $\beta_k>0$.
If $a-1\leq k-a-1$, then $\min\{a-1,k-a-1\}=a-1$
and we have $n_{a-j'}=n_{a+1+j'}$ for $j'=1,2,\ldots,a-1$.
From $n_{a-j'}=n_{a+1+j'}$, $j'=1,2,\ldots,a-1$,
we can show as in (ii) above that
\reqnarray{proof-comparison rule A-(ii)-111} and \reqnarray{proof-comparison rule A-(ii)-222}
hold for $j'=1,2,\ldots,a$,
and \reqnarray{proof-comparison rule A-(ii)-case-1-222} and \reqnarray{proof-comparison rule A-(ii)-case-1-333}
hold for $j=a-1$.
As $n_{a-j'}=n_{a+1+j'}$ for $j'=a-1$, we have $n_1=n_{2a}$.
It then follows from \reqnarray{proof-comparison rule A-(ii)-case-1-222} (with $j=a-1$),
\reqnarray{proof-comparison rule A-(ii)-case-1-333} (with $j=a-1$),
and $n_1=n_{2a}$ that
\beqnarray{proof-comparison rule A-(iii)-111}
\alpha_{2a}=0 \textrm{ and } \beta_{2a}=2^{a-1}>0.
\eeqnarray
As a result of \reqnarray{proof-comparison rule A-(iii)-111}
and $n_i\geq 1$ for $i=2a+1,2a+2,\ldots,k$,
we can use \reqnarray{proof-adjacent distance larger than one-444}
(note that \reqnarray{proof-adjacent distance larger than one-444}
holds for $i=a+2,a+3,\ldots,k$ and $2a+1\geq a+2$)
\reqnarray{proof-adjacent distance larger than one-333} repeatedly
to show that $\alpha_{2a+1}\geq 0,\ \beta_{2a+1}>0,\ \alpha_{2a+2}\geq 0,\ \beta_{2a+2}>0,
\ldots,\ \alpha_k\geq 0,\ \beta_k>0$.
On the other hand, if $a-1>k-a-1$,
then $\min\{a-1,k-a-1\}=k-a-1$
and we have $n_{a-j'}=n_{a+1+j'}$ for $j'=1,2,\ldots,k-a-1$.
We can show as in (ii) above that
\reqnarray{proof-comparison rule A-(ii)-111} and \reqnarray{proof-comparison rule A-(ii)-222}
hold for $j'=1,2,\ldots,k-a$.
It then follows from \reqnarray{proof-comparison rule A-(ii)-222} (with $j'=k-a$)
and $2a-k-1\geq 0$ (as $a-1>k-a-1$) that
\beqnarray{}
\beta_k=2^{k-a}(B(\dbf_1^{s_{2a-k-1}};2a-k-1)+1)>0. \nn
\eeqnarray
\bappendix{Proof of \rlemma{adjacent distance larger than one II} for an even integer $2\leq h\leq N$
by using Comparison rule A in \rlemma{comparison rule A} for the odd integer $h-1$}
{proof of adjacent distance larger than one II for an even integer h
by using comparison rule A for the odd integer h-1}
In this appendix, we assume that Comparison rule A in \rlemma{comparison rule A} holds
for some odd integer $h-1$, where $1\leq h-1\leq N-1$,
and show that \rlemma{adjacent distance larger than one II} holds for the even integer $h$.
Let
\beqnarray{}
\nbf_1^{r_{h-2}}(h-1)\aligneq L_{r_{h-3},r_{h-2}}(\nbf_1^{r_{h-1}}(h)),
\label{eqn:proof of adjacent distance larger than one II-111} \\
{\nbf'}_1^{r_{h-2}}(h-1)\aligneq L_{r_{h-3},r_{h-2}}({\nbf'}_1^{r_{h-1}}(h)).
\label{eqn:proof of adjacent distance larger than one II-222}
\eeqnarray
For simplicity, let $\mbf_1^{r_{h-1}}=\nbf_1^{r_{h-1}}(h)$,
${\mbf'}_1^{r_{h-1}}={\nbf'}_1^{r_{h-1}}(h)$,
$\nbf_1^{r_{h-2}}=\nbf_1^{r_{h-2}}(h-1)$,
and ${\nbf'}_1^{r_{h-2}}={\nbf'}_1^{r_{h-2}}(h-1)$.
Then we have from \reqnarray{proof of adjacent distance larger than one II-111},
\reqnarray{proof of adjacent distance larger than one II-222},
and \reqnarray{order relation-777} that
\beqnarray{proof of adjacent distance larger than one II-333}
\mbf_1^{r_{h-1}}\prec
(\textrm{resp.}, \equiv, \succ, \preceq, \succeq)\ {\mbf'}_1^{r_{h-1}}
\textrm{ iff } \nbf_1^{r_{h-2}}\prec
(\textrm{resp.}, \equiv, \succ, \preceq, \succeq)\ {\nbf'}_1^{r_{h-2}}.
\eeqnarray
Furthermore, from \reqnarray{proof of adjacent distance larger than one II-111},
\reqnarray{proof of adjacent distance larger than one II-222},
and the definition of left pre-sequences in \rdefinition{left pre-sequences},
we have
\beqnarray{proof of adjacent distance larger than one II-444}
n_i=
\bselection
q_{h-1}+1, &\textrm{if } i=i_1,i_2,\ldots,i_{r_{h-1}}, \\
q_{h-1}, &\textrm{otherwise},
\eselection
\eeqnarray
where
\beqnarray{proof of adjacent distance larger than one II-555}
i_j=\sum_{\ell=1}^{j-1}m_{\ell}+1, \textrm{ for } j=1,2,\ldots,r_{h-1},
\eeqnarray
and
\beqnarray{proof of adjacent distance larger than one II-666}
n'_i=
\bselection
q_{h-1}+1, &\textrm{if } i=i'_1,i'_2,\ldots,i'_{r_{h-1}}, \\
q_{h-1}, &\textrm{otherwise},
\eselection
\eeqnarray
where
\beqnarray{proof of adjacent distance larger than one II-777}
i'_j=\sum_{\ell=1}^{j-1}m'_{\ell}+1, \textrm{ for } j=1,2,\ldots,r_{h-1}.
\eeqnarray
Note that in \rlemma{adjacent distance larger than one II}, we have $r_{h-1}\geq 2$.
As such, it follows from $r_{h-2}>r_{h-1}$ that
\beqnarray{proof of adjacent distance larger than one II-888}
r_{h-2}\geq 2.
\eeqnarray
(i) Note that in \rlemma{adjacent distance larger than one II}(i),
we have $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
$m_a-m_{a+1}\geq 2$ for some $1\leq a\leq r_{h-1}-1$,
$m'_a=m_a-1$, $m'_{a+1}=m_{a+1}+1$, and $m'_i=m_i$ for $i\neq a$ and $a+1$.
It is easy to see that
\beqnarray{proof of adjacent distance larger than one II-(i)-111}
m'_a=m_a-1\geq m_{a+1}+1\geq 2,\ m'_{a+1}=m_{a+1}+1\geq 2,
\textrm{ and } m'_i=m_i \textrm{ for } i\neq a, a+1.
\eeqnarray
Also, we have from $m'_a=m_a-1$, $m'_{a+1}=m_{a+1}+1$, and $m'_i=m_i$ for $i\neq a$ and $a+1$,
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$, and \reqnarray{N-M-k-h} that
\beqnarray{proof of adjacent distance larger than one II-(i)-222}
\sum_{i=1}^{r_{h-1}}m'_i=\sum_{i=1}^{r_{h-1}}m_i=r_{h-2}.
\eeqnarray
As such, it follows from \reqnarray{proof of adjacent distance larger than one II-(i)-111},
\reqnarray{proof of adjacent distance larger than one II-(i)-222},
and \reqnarray{N-M-k-h} that ${\mbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
As $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$ and ${\mbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
we see from \reqnarray{proof of adjacent distance larger than one II-111},
\reqnarray{proof of adjacent distance larger than one II-222},
and the argument in the paragraph after \reqnarray{N-M-k-h} that
\beqnarray{proof of adjacent distance larger than one II-(i)-333}
\nbf_1^{r_{h-2}}\in \Ncal_{M,k}(h-1) \textrm{ and } {\nbf'}_1^{r_{h-2}}\in \Ncal_{M,k}(h-1).
\eeqnarray
To show \reqnarray{adjacent distance larger than one II-1},
i.e., $\mbf_1^{r_{h-1}}\prec{\mbf'}_1^{r_{h-1}}$,
we see from \reqnarray{proof of adjacent distance larger than one II-333}
that it suffices to show that
\beqnarray{proof of adjacent distance larger than one II-(i)-444}
\nbf_1^{r_{h-2}}\prec{\nbf'}_1^{r_{h-2}}.
\eeqnarray
\bpdffigure{appendix-C-i.pdf}{5.5in}
\epdffigure{appendix-C-(i)}
{An illustration of
\reqnarray{proof of adjacent distance larger than one II-(i)-666}--\reqnarray{proof of adjacent distance larger than one II-(i)-bbb}:
(a) $1\leq a\leq r_{h-1}-2$
(note that in this case we have $i_a<i'_{a+1}<i_{a+1}$
in \reqnarray{proof of adjacent distance larger than one II-(i)-ccc}
and $i'_{a+1}<i'_{a+1}+1<i'_{a+2}$
in \reqnarray{proof of adjacent distance larger than one II-(i)-ddd});
(b) $a=r_{h-1}-1$
(note that in this case we have $i_a<i'_{a+1}<i_{a+1}$
in \reqnarray{proof of adjacent distance larger than one II-(i)-ccc}
and $i'_{r_{h-1}}<i'_{a+1}+1\leq r_{h-2}$
in \reqnarray{proof of adjacent distance larger than one II-(i)-eee}
and \reqnarray{proof of adjacent distance larger than one II-(i)-fff}).}
Note that from $m'_a=m_a-1$, $m'_{a+1}=m_{a+1}+1$, $m'_i=m_i$ for $i\neq a$ and $a+1$,
\reqnarray{proof of adjacent distance larger than one II-555},
and \reqnarray{proof of adjacent distance larger than one II-777},
it is easy to see that
\beqnarray{proof of adjacent distance larger than one II-(i)-555}
i'_j=
\bselection
i_j-1, &\textrm{if } j=a+1, \\
i_j, &\textrm{otherwise}.
\eselection
\eeqnarray
In the following, we show that
\beqnarray{}
\alignspace n_{i'_{a+1}}=q_{h-1},
\label{eqn:proof of adjacent distance larger than one II-(i)-666}\\
\alignspace n'_{i'_{a+1}+1}=q_{h-1}.
\label{eqn:proof of adjacent distance larger than one II-(i)-777}
\eeqnarray
It then follows from \reqnarray{proof of adjacent distance larger than one II-444},
\reqnarray{proof of adjacent distance larger than one II-666},
\reqnarray{proof of adjacent distance larger than one II-(i)-555},
\reqnarray{proof of adjacent distance larger than one II-(i)-666},
and \reqnarray{proof of adjacent distance larger than one II-(i)-777} that
\beqnarray{}
\alignspace n'_{i'_{a+1}}-n'_{i'_{a+1}+1}=(q_{h-1}+1)-q_{h-1}=1,
\label{eqn:proof of adjacent distance larger than one II-(i)-888}\\
\alignspace n_{i'_{a+1}}=q_{h-1}=n'_{i'_{a+1}}-1,
\label{eqn:proof of adjacent distance larger than one II-(i)-999}\\
\alignspace n_{i'_{a+1}+1}=n_{i_{a+1}}=q_{h-1}+1=n'_{i'_{a+1}+1}+1,
\label{eqn:proof of adjacent distance larger than one II-(i)-aaa}\\
\alignspace n_i=n'_i, \textrm{ for } i\neq i'_{a+1} \textrm{ and } i'_{a+1}+1.
\label{eqn:proof of adjacent distance larger than one II-(i)-bbb}
\eeqnarray
An illustration of \reqnarray{proof of adjacent distance larger than one II-(i)-666}--\reqnarray{proof of adjacent distance larger than one II-(i)-bbb}
is given in \rfigure{appendix-C-(i)}.
To prove \reqnarray{proof of adjacent distance larger than one II-(i)-666},
note from \reqnarray{proof of adjacent distance larger than one II-(i)-555} that
\beqnarray{proof of adjacent distance larger than one II-(i)-ccc}
i'_{a+1}>i'_a=i_a \textrm{ and } i'_{a+1}=i_{a+1}-1<i_{a+1}.
\eeqnarray
Thus, \reqnarray{proof of adjacent distance larger than one II-(i)-666}
follows from \reqnarray{proof of adjacent distance larger than one II-444}
and $i_a<i'_{a+1}<i_{a+1}$ in \reqnarray{proof of adjacent distance larger than one II-(i)-ccc}.
To prove \reqnarray{proof of adjacent distance larger than one II-(i)-777},
note that if $1\leq a\leq r_{h-1}-2$,
then we have from \reqnarray{proof of adjacent distance larger than one II-(i)-555} that
\beqnarray{proof of adjacent distance larger than one II-(i)-ddd}
i'_{a+1}+1=i_{a+1}<i_{a+2}=i'_{a+2}.
\eeqnarray
Thus, \reqnarray{proof of adjacent distance larger than one II-(i)-777}
follows from \reqnarray{proof of adjacent distance larger than one II-666}
and $i'_{a+1}<i'_{a+1}+1<i'_{a+2}$ in \reqnarray{proof of adjacent distance larger than one II-(i)-ddd}.
On the other hand, if $a=r_{h-1}-1$, then we have
\beqnarray{proof of adjacent distance larger than one II-(i)-eee}
i'_{a+1}+1>i'_{a+1}=i'_{r_{h-1}},
\eeqnarray
and we have from \reqnarray{proof of adjacent distance larger than one II-(i)-555} that
\beqnarray{proof of adjacent distance larger than one II-(i)-fff}
i'_{a+1}+1=i_{a+1}=i_{r_{h-1}}\leq r_{h-2}.
\eeqnarray
Thus, \reqnarray{proof of adjacent distance larger than one II-(i)-777}
also follows from \reqnarray{proof of adjacent distance larger than one II-666}
and $i'_{r_{h-1}}<i'_{a+1}+1\leq r_{h-2}$
in \reqnarray{proof of adjacent distance larger than one II-(i)-eee}
and \reqnarray{proof of adjacent distance larger than one II-(i)-fff}.
Note that from \reqnarray{proof of adjacent distance larger than one II-777}, we have
\beqnarray{proof of adjacent distance larger than one II-(i)-ggg}
i'_{a+1}=\sum_{\ell=1}^{a}m'_{\ell}+1\geq m'_a+1 \geq 2,
\eeqnarray
and
\beqnarray{proof of adjacent distance larger than one II-(i)-hhh}
i'_{j+1}=\sum_{\ell=1}^{j}m'_{\ell}+1
=\left(\sum_{\ell=1}^{j-1}m'_{\ell}+1\right)+m'_j
=i'_j+m'_j, \textrm{ for } j=1,2,\ldots,r_{h-1}-1.
\eeqnarray
We then consider the two cases $1\leq a\leq r_{h-1}-2$ and $a=r_{h-1}-1$ separately.
\emph{Case 1: $1\leq a\leq r_{h-1}-2$.}
From \reqnarray{proof of adjacent distance larger than one II-777},
\reqnarray{proof of adjacent distance larger than one II-(i)-222},
$a+1\leq r_{h-1}-1$,
and $m'_{a+1}\geq 2$ in \reqnarray{proof of adjacent distance larger than one II-(i)-111},
we have
\beqnarray{proof of adjacent distance larger than one II-(i)-case-1-111}
i'_{a+1}
\aligneq \sum_{\ell=1}^{a}m'_{\ell}+1=\sum_{\ell=1}^{r_{h-1}}m'_{\ell}-\sum_{\ell=a+1}^{r_{h-1}}m'_{\ell}+1 \nn\\
\alignleq r_{h-2}-(m'_{a+1}+m'_{r_{h-1}})+1\nn\\
\alignleq r_{h-2}-(2+1)+1 \nn\\
\aligneq r_{h-2}-2.
\eeqnarray
As such, we see from \reqnarray{proof of adjacent distance larger than one II-(i)-ggg}
and \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-111} that
\beqnarray{proof of adjacent distance larger than one II-(i)-case-1-222}
2\leq i'_{a+1}\leq r_{h-2}-2.
\eeqnarray
\bpdffigure{appendix-C-i-case-1.pdf}{5.5in}
\epdffigure{appendix-C-(i)-case-1}
{(a) An illustration of \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-333}
in the case that $1\leq a\leq r_{h-1}-2$ and $m_{a+1}=1$
(note that in this case we have $i'_a<i'_{a+1}-1<i'_{a+1}$
in \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-444}
and $(i'_{a+1}+1)+1=i'_{a+2}$
in \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-666});
(b) An illustration of \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-999}
and \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-aaa}
in the case that $1\leq a\leq r_{h-1}-2$ and $m_{a+1}\geq 2$
(note that in this case we have $i'_a<i'_{a+1}-(m'_{a+1}-1)<i'_{a+1}$
in \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-ddd}
and $(i'_{a+1}+1)+(m'_{a+1}-1)=i'_{a+2}$
in \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-hhh}).}
If $m_{a+1}=1$, then we show that
\beqnarray{proof of adjacent distance larger than one II-(i)-case-1-333}
n'_{i'_{a+1}-1}=q_{h-1}<n'_{(i'_{a+1}+1)+1}=q_{h-1}+1.
\eeqnarray
An illustration of \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-333}
is given in \rfigure{appendix-C-(i)-case-1}(a).
Therefore, it follows from
\reqnarray{proof of adjacent distance larger than one II-888},
${\nbf'}_1^{r_{h-2}}\in \Ncal_{M,k}(h-1)$ in
\reqnarray{proof of adjacent distance larger than one II-(i)-333},
\reqnarray{proof of adjacent distance larger than one II-(i)-888}--\reqnarray{proof of adjacent distance larger than one II-(i)-bbb},
\reqnarray{proof of adjacent distance larger than one II-(i)-case-1-222},
\reqnarray{proof of adjacent distance larger than one II-(i)-case-1-333},
and \reqnarray{comparison rule A-2} in \rlemma{comparison rule A}(ii)
(for the odd integer $h-1$) that ${\nbf'}_1^{r_{h-2}}\succ\nbf_1^{r_{h-2}}$,
i.e., \reqnarray{proof of adjacent distance larger than one II-(i)-444} holds.
To prove \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-333},
note that from \reqnarray{proof of adjacent distance larger than one II-(i)-hhh}, $1\leq a\leq r_{h-1}-2$,
and $m'_a\geq 2$ in \reqnarray{proof of adjacent distance larger than one II-(i)-111}, we shave
\beqnarray{proof of adjacent distance larger than one II-(i)-case-1-444}
i'_{a+1}-1=(i'_a+m'_a)-1\geq i'_a+1.
\eeqnarray
It follows from \reqnarray{proof of adjacent distance larger than one II-666}
and $i'_a<i'_{a+1}-1<i'_{a+1}$ in
\reqnarray{proof of adjacent distance larger than one II-(i)-case-1-444} that
\beqnarray{proof of adjacent distance larger than one II-(i)-case-1-555}
n'_{i'_{a+1}-1}=q_{h-1}.
\eeqnarray
Also, from $m'_{a+1}=m_{a+1}+1=1+1=2$,
\reqnarray{proof of adjacent distance larger than one II-(i)-hhh},
and $1\leq a\leq r_{h-1}-2$, we have
\beqnarray{proof of adjacent distance larger than one II-(i)-case-1-666}
(i'_{a+1}+1)+1=i'_{a+1}+2=i'_{a+1}+m'_{a+1}=i'_{a+2}.
\eeqnarray
It follows from \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-666},
\reqnarray{proof of adjacent distance larger than one II-666},
and $1\leq a\leq r_{h-1}-2$ that
\beqnarray{proof of adjacent distance larger than one II-(i)-case-1-777}
n'_{(i'_{a+1}+1)+1}=n'_{i'_{a+2}}=q_{h-1}+1.
\eeqnarray
Thus, \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-333}
follows from \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-555}
and \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-777}.
On the other hand, if $m_{a+1}\geq 2$,
then we show that
\beqnarray{}
\alignspace 1\leq m'_{a+1}-1\leq \min\{i'_{a+1}-1,r_{h-2}-i'_{a+1}-1\},
\label{eqn:proof of adjacent distance larger than one II-(i)-case-1-888}\\
\alignspace n'_{i'_{a+1}-j'}=n'_{(i'_{a+1}+1)+j'}=q_{h-1}, \textrm{ for } j'=1,2,\ldots,m'_{a+1}-2,
\label{eqn:proof of adjacent distance larger than one II-(i)-case-1-999}\\
\alignspace n'_{i'_{a+1}-(m'_{a+1}-1)}=q_{h-1}<n'_{(i'_{a+1}+1)+(m'_{a+1}-1)}=q_{h-1}+1,
\label{eqn:proof of adjacent distance larger than one II-(i)-case-1-aaa}
\eeqnarray
where we note that $m'_{a+1}-2=(m_{a+1}+1)-2\geq 1$ (as $m_{a+1}\geq 2$).
An illustration of \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-999}
and \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-aaa}
is given in \rfigure{appendix-C-(i)-case-1}(b).
Therefore, it follows from
\reqnarray{proof of adjacent distance larger than one II-888},
${\nbf'}_1^{r_{h-2}}\in \Ncal_{M,k}(h-1)$ in
\reqnarray{proof of adjacent distance larger than one II-(i)-333},
\reqnarray{proof of adjacent distance larger than one II-(i)-888}--\reqnarray{proof of adjacent distance larger than one II-(i)-bbb},
\reqnarray{proof of adjacent distance larger than one II-(i)-case-1-222},
\reqnarray{proof of adjacent distance larger than one II-(i)-case-1-888}--\reqnarray{proof of adjacent distance larger than one II-(i)-case-1-aaa},
and \reqnarray{comparison rule A-2} in \rlemma{comparison rule A}(ii)
(for the odd integer $h-1$) that ${\nbf'}_1^{r_{h-2}}\succ\nbf_1^{r_{h-2}}$,
i.e., \reqnarray{proof of adjacent distance larger than one II-(i)-444} holds.
To prove \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-888}--\reqnarray{proof of adjacent distance larger than one II-(i)-case-1-aaa},
observe from $m'_{a+1}=m_{a+1}+1$ that
\beqnarray{proof of adjacent distance larger than one II-(i)-case-1-bbb}
m'_{a+1}-1=m_{a+1}\geq 1.
\eeqnarray
From \reqnarray{proof of adjacent distance larger than one II-(i)-hhh},
$1\leq a\leq r_{h-1}-2$, $m'_a=m_a-1$, $m'_{a+1}=m_{a+1}+1$,
and $m_a-m_{a+1}\geq 2$, we have
\beqnarray{}
i'_{a+1}-(m'_{a+1}-1)
\aligneq i'_a+m'_a-m'_{a+1}+1 \nn\\
\aligneq i'_a+(m_a-1)-(m_{a+1}+1)+1 \nn\\
\aligneq i'_a+m_a-m_{a+1}-1
\label{eqn:proof of adjacent distance larger than one II-(i)-case-1-ccc} \\
\aligngeq i'_a+1.
\label{eqn:proof of adjacent distance larger than one II-(i)-case-1-ddd}
\eeqnarray
We immediately see from \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-ddd} that
\beqnarray{}
\alignspace m'_{a+1}-1\leq i'_{a+1}-i'_a-1<i'_{a+1}-1,
\label{eqn:proof of adjacent distance larger than one II-(i)-case-1-eee}\\
\alignspace
i'_a<i'_{a+1}-j'<i'_{a+1}, \textrm{ for } j'=1,2,\ldots,m'_{a+1}-1.
\label{eqn:proof of adjacent distance larger than one II-(i)-case-1-fff}
\eeqnarray
It is clear from \reqnarray{proof of adjacent distance larger than one II-666}
and \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-fff} that
\beqnarray{proof of adjacent distance larger than one II-(i)-case-1-ggg}
n'_{i'_{a+1}-j'}=q_{h-1}, \textrm{ for } j'=1,2,\ldots,m'_{a+1}-1.
\eeqnarray
Furthermore, we have from \reqnarray{proof of adjacent distance larger than one II-(i)-hhh}
and $1\leq a\leq r_{h-1}-2$ that
\beqnarray{proof of adjacent distance larger than one II-(i)-case-1-hhh}
(i'_{a+1}+1)+(m'_{a+1}-1)=i'_{a+1}+m'_{a+1}=i'_{a+2}.
\eeqnarray
We immediately see from \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-hhh}
and $i'_{a+2}\leq r_{h-2}$ that
\beqnarray{proof of adjacent distance larger than one II-(i)-case-1-iii}
m'_{a+1}-1=i'_{a+2}-i'_{a+1}-1\leq r_{h-2}-i'_{a+1}-1.
\eeqnarray
Thus, \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-888}
follows from \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-bbb},
\reqnarray{proof of adjacent distance larger than one II-(i)-case-1-eee},
and \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-iii}.
Also, it is clear from \reqnarray{proof of adjacent distance larger than one II-666}
and \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-hhh} that
\beqnarray{}
\alignspace n'_{(i'_{a+1}+1)+j'}=q_{h-1}, \textrm{ for } j'=1,2,\ldots,m'_{a+1}-2,
\label{eqn:proof of adjacent distance larger than one II-(i)-case-1-jjj} \\
\alignspace n'_{(i'_{a+1}+1)+(m'_{a+1}-1)}=n'_{i'_{a+2}}=q_{h-1}+1.
\label{eqn:proof of adjacent distance larger than one II-(i)-case-1-kkk}
\eeqnarray
Thus, \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-999}
and \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-aaa}
follow from \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-ggg},
\reqnarray{proof of adjacent distance larger than one II-(i)-case-1-jjj},
and \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-kkk}.
\emph{Case 2: $a=r_{h-1}-1$.}
\bpdffigure{appendix-C-i-case-2.pdf}{5.5in}
\epdffigure{appendix-C-(i)-case-2}
{An illustration of \reqnarray{proof of adjacent distance larger than one II-(i)-case-2-444}
in the case that $a=r_{h-1}-1$ and $m_{a+1}\geq 2$
(note that in this case we have
$\min\{i'_{a+1}-1,r_{h-2}-i'_{a+1}-1\}=m'_{a+1}-2$
in \reqnarray{proof of adjacent distance larger than one II-(i)-case-2-777},
$i'_a<i'_{a+1}-(m'_{a+1}-2)<i'_{a+1}$
in \reqnarray{proof of adjacent distance larger than one II-(i)-case-2-999},
and $(i'_{a+1}+1)+(m'_{a+1}-2)=r_{h-2}$
in \reqnarray{proof of adjacent distance larger than one II-(i)-case-2-ccc}).}
If $m_{a+1}=1$, then we have from $a=r_{h-1}-1$,
\reqnarray{proof of adjacent distance larger than one II-777},
\reqnarray{proof of adjacent distance larger than one II-(i)-222},
and $m'_{a+1}=m_{a+1}+1=1+1=2$ that
\beqnarray{}
i'_{a+1}
\aligneq i'_{r_{h-1}}=\sum_{\ell=1}^{r_{h-1}-1}m'_{\ell}+1 \nn\\
\aligneq \sum_{\ell=1}^{r_{h-1}}m'_{\ell}-m'_{r_{h-1}}+1 \nn\\
\aligneq r_{h-2}-m'_{a+1}+1
\label{eqn:proof of adjacent distance larger than one II-(i)-case-2-111} \\
\aligneq r_{h-2}-1.
\label{eqn:proof of adjacent distance larger than one II-(i)-case-2-222}
\eeqnarray
Therefore, it follows from
\reqnarray{proof of adjacent distance larger than one II-888},
${\nbf'}_1^{r_{h-2}}\in \Ncal_{M,k}(h-1)$ in
\reqnarray{proof of adjacent distance larger than one II-(i)-333},
\reqnarray{proof of adjacent distance larger than one II-(i)-888}--\reqnarray{proof of adjacent distance larger than one II-(i)-bbb},
\reqnarray{proof of adjacent distance larger than one II-(i)-case-2-222},
and \reqnarray{comparison rule A-1} in \rlemma{comparison rule A}(i)
(for the odd integer $h-1$) that ${\nbf'}_1^{r_{h-2}}\succ\nbf_1^{r_{h-2}}$,
i.e., \reqnarray{proof of adjacent distance larger than one II-(i)-444} holds.
On the other hand, if $m_{a+1}\geq 2$,
then we show that
\beqnarray{}
\alignspace 2\leq i'_{a+1}\leq r_{h-2}-2,
\label{eqn:proof of adjacent distance larger than one II-(i)-case-2-333}\\
\alignspace n'_{i'_{a+1}-j'}=n'_{(i'_{a+1}+1)+j'}=q_{h-1},
\textrm{ for } j'=1,2,\ldots,\min\{i'_{a+1}-1,r_{h-2}-i'_{a+1}-1\}.
\label{eqn:proof of adjacent distance larger than one II-(i)-case-2-444}
\eeqnarray
An illustration of \reqnarray{proof of adjacent distance larger than one II-(i)-case-2-444}
is given in \rfigure{appendix-C-(i)-case-2}.
Therefore, it follows from
\reqnarray{proof of adjacent distance larger than one II-888},
${\nbf'}_1^{r_{h-2}}\in \Ncal_{M,k}(h-1)$ in
\reqnarray{proof of adjacent distance larger than one II-(i)-333},
\reqnarray{proof of adjacent distance larger than one II-(i)-888}--\reqnarray{proof of adjacent distance larger than one II-(i)-bbb},
\reqnarray{proof of adjacent distance larger than one II-(i)-case-2-333},
\reqnarray{proof of adjacent distance larger than one II-(i)-case-2-444},
and \reqnarray{comparison rule A-4} in \rlemma{comparison rule A}(iii)
(for the odd integer $h-1$) that ${\nbf'}_1^{r_{h-2}}\succ\nbf_1^{r_{h-2}}$,
i.e., \reqnarray{proof of adjacent distance larger than one II-(i)-444} holds.
To prove \reqnarray{proof of adjacent distance larger than one II-(i)-case-2-333},
note that from \reqnarray{proof of adjacent distance larger than one II-(i)-case-2-111}
and $m'_{a+1}=m_{a+1}+1\geq 2+1=3$, we have
\beqnarray{proof of adjacent distance larger than one II-(i)-case-2-555}
i'_{a+1}=r_{h-2}-m'_{a+1}+1\leq r_{h-2}-2.
\eeqnarray
Thus, \reqnarray{proof of adjacent distance larger than one II-(i)-case-2-333}
follows from \reqnarray{proof of adjacent distance larger than one II-(i)-ggg}
and \reqnarray{proof of adjacent distance larger than one II-(i)-case-2-555}.
To prove \reqnarray{proof of adjacent distance larger than one II-(i)-case-2-444},
note that from \reqnarray{proof of adjacent distance larger than one II-(i)-case-2-111},
\reqnarray{proof of adjacent distance larger than one II-777},
$m'_a=m_a-1$, $m'_{a+1}=m_{a+1}+1$, and $m_a-m_{a+1}\geq 2$, we have
\beqnarray{proof of adjacent distance larger than one II-(i)-case-2-666}
(i'_{a+1}-1)-(r_{h-2}-i'_{a+1}-1)
\aligneq (i'_{a+1}-1)-(m'_{a+1}-2)=\sum_{\ell=1}^{a}m'_{\ell}-m'_{a+1}+2\nn\\
\aligngeq m'_a-m'_{a+1}+2=(m_a-1)-(m_{a+1}+1)+2 \nn\\
\aligneq m_a-m_{a+1}>0.
\eeqnarray
It then follows from \reqnarray{proof of adjacent distance larger than one II-(i)-case-2-666}
and \reqnarray{proof of adjacent distance larger than one II-(i)-case-2-111} that
\beqnarray{proof of adjacent distance larger than one II-(i)-case-2-777}
\min\{i'_{a+1}-1,r_{h-2}-i'_{a+1}-1\}=r_{h-2}-i'_{a+1}-1=m'_{a+1}-2.
\eeqnarray
From \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-ccc}
and $m_a-m_{a+1}\geq 2$, we have
\beqnarray{proof of adjacent distance larger than one II-(i)-case-2-888}
i'_{a+1}-(m'_{a+1}-2)=i'_a+m_a-m_{a+1}>i'_a.
\eeqnarray
From \reqnarray{proof of adjacent distance larger than one II-(i)-case-2-888},
we immediately see that
\beqnarray{proof of adjacent distance larger than one II-(i)-case-2-999}
i'_a<i'_{a+1}-j'<i'_{a+1}, \textrm{ for } j'=1,2,\ldots,m'_{a+1}-2.
\eeqnarray
It then follows from \reqnarray{proof of adjacent distance larger than one II-666}
and \reqnarray{proof of adjacent distance larger than one II-(i)-case-2-999} that
\beqnarray{proof of adjacent distance larger than one II-(i)-case-2-aaa}
n'_{i'_{a+1}-j'}=q_{h-1}, \textrm{ for } j'=1,2,\ldots,m'_{a+1}-2.
\eeqnarray
Also, we have from $a=r_{h-1}-1$ that
\beqnarray{proof of adjacent distance larger than one II-(i)-case-2-bbb}
i'_{a+1}+1=i'_{r_{h-1}}+1>i'_{r_{h-1}},
\eeqnarray
and we have from \reqnarray{proof of adjacent distance larger than one II-(i)-case-2-111} that
\beqnarray{proof of adjacent distance larger than one II-(i)-case-2-ccc}
(i'_{a+1}+1)+(m'_{a+1}-2)=r_{h-2}.
\eeqnarray
It then follows from \reqnarray{proof of adjacent distance larger than one II-666},
\reqnarray{proof of adjacent distance larger than one II-(i)-case-2-bbb},
and \reqnarray{proof of adjacent distance larger than one II-(i)-case-2-ccc} that
\beqnarray{proof of adjacent distance larger than one II-(i)-case-2-ddd}
\alignspace n'_{(i'_{a+1}+1)+j'}=q_{h-1}, \textrm{ for } j=1,2,\ldots,m'_{a+1}-2.
\eeqnarray
Thus, \reqnarray{proof of adjacent distance larger than one II-(i)-case-2-444}
follows from \reqnarray{proof of adjacent distance larger than one II-(i)-case-2-777},
\reqnarray{proof of adjacent distance larger than one II-(i)-case-2-aaa},
and \reqnarray{proof of adjacent distance larger than one II-(i)-case-2-ddd}.
(ii) Note that in \rlemma{adjacent distance larger than one II}(ii),
we have $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
$m_a-m_{a+1}\leq -2$ for some $1\leq a\leq r_{h-1}-1$,
$m'_a=m_a+1$, $m'_{a+1}=m_{a+1}-1$, and $m'_i=m_i$ for $i\neq a$ and $a+1$.
It is easy to see that
\beqnarray{proof of adjacent distance larger than one II-(ii)-111}
m'_a=m_a+1\geq 2,\ m'_{a+1}=m_{a+1}-1\geq m_a+1\geq 2,
\textrm{ and } m'_i=m_i \textrm{ for } i\neq a, a+1.
\eeqnarray
Also, we have from $m'_a=m_a+1$, $m'_{a+1}=m_{a+1}-1$, and $m'_i=m_i$ for $i\neq a$ and $a+1$,
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$, and \reqnarray{N-M-k-h} that
\beqnarray{proof of adjacent distance larger than one II-(ii)-222}
\sum_{i=1}^{r_{h-1}}m'_i=\sum_{i=1}^{r_{h-1}}m_i=r_{h-2}.
\eeqnarray
As such, it follows from \reqnarray{proof of adjacent distance larger than one II-(ii)-111},
\reqnarray{proof of adjacent distance larger than one II-(ii)-222},
and \reqnarray{N-M-k-h} that ${\mbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
As $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$ and ${\mbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
we see from \reqnarray{proof of adjacent distance larger than one II-111},
\reqnarray{proof of adjacent distance larger than one II-222},
and the argument in the paragraph after \reqnarray{N-M-k-h} that
\beqnarray{proof of adjacent distance larger than one II-(ii)-333}
\nbf_1^{r_{h-2}}\in \Ncal_{M,k}(h-1) \textrm{ and } {\nbf'}_1^{r_{h-2}}\in \Ncal_{M,k}(h-1).
\eeqnarray
To show \reqnarray{adjacent distance larger than one II-2},
i.e., $\mbf_1^{r_{h-1}}\preceq {\mbf'}_1^{r_{h-1}}$,
where $\mbf_1^{r_{h-1}}\equiv {\mbf'}_1^{r_{h-1}}$ if and only if
$r_{h-1}=2$ and $m_1=m_2-2$,
we see from \reqnarray{proof of adjacent distance larger than one II-333}
that it suffices to show that
\beqnarray{proof of adjacent distance larger than one II-(ii)-444}
\nbf_1^{r_{h-2}}\preceq {\nbf'}_1^{r_{h-2}},
\eeqnarray
where $\nbf_1^{r_{h-2}}\equiv {\nbf'}_1^{r_{h-2}}$ if and only if
\beqnarray{proof of adjacent distance larger than one II-(ii)-555}
r_{h-1}=2 \textrm{ and } m_1=m_2-2.
\eeqnarray
Note that from $m_a=m'_a-1$, $m_{a+1}=m'_{a+1}+1$, $m_i=m'_i$ for $i\neq a$ and $a+1$,
\reqnarray{proof of adjacent distance larger than one II-444}--\reqnarray{proof of adjacent distance larger than one II-888},
we can show as in the proof of
\reqnarray{proof of adjacent distance larger than one II-(i)-555}--\reqnarray{proof of adjacent distance larger than one II-(i)-bbb}
in (i) above
(with the roles of $\mbf_1^{r_{h-1}}$ and ${\mbf'}_1^{r_{h-1}}$ interchanged
and the roles of $\nbf_1^{r_{h-2}}$ and ${\nbf'}_1^{r_{h-2}}$ interchanged) that
\beqnarray{}
\alignspace n_{i_{a+1}+1}=q_{h-1},
\label{eqn:proof of adjacent distance larger than one II-(ii)-666} \\
\alignspace n'_{i_{a+1}}=q_{h-1},
\label{eqn:proof of adjacent distance larger than one II-(ii)-777}\\
\alignspace n_{i_{a+1}}-n_{i_{a+1}+1}=(q_{h-1}+1)-q_{h-1}=1,
\label{eqn:proof of adjacent distance larger than one II-(ii)-888}\\
\alignspace n'_{i_{a+1}}=q_{h-1}=n_{i_{a+1}}-1,
\label{eqn:proof of adjacent distance larger than one II-(ii)-999}\\
\alignspace n'_{i_{a+1}+1}=n'_{i'_{a+1}}=q_{h-1}+1=n_{i_{a+1}+1}+1,
\label{eqn:proof of adjacent distance larger than one II-(ii)-aaa}\\
\alignspace n'_i=n_i, \textrm{ for } i\neq i_{a+1} \textrm{ and } i_{a+1}+1.
\label{eqn:proof of adjacent distance larger than one II-(ii)-bbb}
\eeqnarray
\bpdffigure{appendix-C-ii.pdf}{5.5in}
\epdffigure{appendix-C-(ii)}
{An illustration of \reqnarray{proof of adjacent distance larger than one II-(ii)-eee}
and \reqnarray{proof of adjacent distance larger than one II-(ii)-fff}:
(a) $1\leq a\leq r_{h-1}-2$
(note that in this case we have $i_{a+1}-m_a=i_a$
in \reqnarray{proof of adjacent distance larger than one II-(ii)-2222}
and $i_{a+1}<(i_{a+1}+1)+m_a<i_{a+2}$
in \reqnarray{proof of adjacent distance larger than one II-(ii)-8888});
(b) $a=r_{h-1}-1$
(note that in this case we have $i_{a+1}-m_a=i_a$
in \reqnarray{proof of adjacent distance larger than one II-(ii)-2222}
and $i_{r_{h-1}}=i_{a+1}<(i_{a+1}+1)+m_a\leq r_{h-2}$
in \reqnarray{proof of adjacent distance larger than one II-(ii)-cccc}).}
In the following, we show that
\beqnarray{}
\alignspace 2\leq i_{a+1}\leq r_{h-2}-2,
\label{eqn:proof of adjacent distance larger than one II-(ii)-ccc}\\
\alignspace 1\leq m_a\leq \min\{i_{a+1}-1,r_{h-2}-i_{a+1}-1\},
\label{eqn:proof of adjacent distance larger than one II-(ii)-ddd}\\
\alignspace n_{i_{a+1}-j'}=n_{(i_{a+1}+1)+j'}=q_{h-1}, \textrm{ for } j'=1,2,\ldots,m_a-1,
\label{eqn:proof of adjacent distance larger than one II-(ii)-eee}\\
\alignspace n_{i_{a+1}-m_a}=q_{h-1}+1>n_{(i_{a+1}+1)+m_a}=q_{h-1}.
\label{eqn:proof of adjacent distance larger than one II-(ii)-fff}
\eeqnarray
An illustration of \reqnarray{proof of adjacent distance larger than one II-(ii)-eee}
and \reqnarray{proof of adjacent distance larger than one II-(ii)-fff}
is given in \rfigure{appendix-C-(ii)}.
Therefore, it follows from
\reqnarray{proof of adjacent distance larger than one II-888},
$\nbf_1^{r_{h-2}}\in \Ncal_{M,k}(h-1)$ in
\reqnarray{proof of adjacent distance larger than one II-(ii)-333},
\reqnarray{proof of adjacent distance larger than one II-(ii)-888}--\reqnarray{proof of adjacent distance larger than one II-(ii)-fff},
and \reqnarray{comparison rule A-3} in \rlemma{comparison rule A}(ii)
(for the odd integer $h-1$) that $\nbf_1^{r_{h-2}}\preceq{\nbf'}_1^{r_{h-2}}$,
where $\nbf_1^{r_{h-2}}\equiv{\nbf'}_1^{r_{h-2}}$ if and only if
\beqnarray{proof of adjacent distance larger than one II-(ii)-ggg}
i_{a+1}-m_a=1,\ (i_{a+1}+1)+m_a=r_{h-2}, \textrm{ and } n_1=n_{r_{h-2}}+1.
\eeqnarray
To prove \reqnarray{proof of adjacent distance larger than one II-(ii)-ccc}--\reqnarray{proof of adjacent distance larger than one II-(ii)-fff},
note that from \reqnarray{proof of adjacent distance larger than one II-555}
and $1\leq a\leq r_{h-1}-1$, we have
\beqnarray{proof of adjacent distance larger than one II-(ii)-1111}
i_{a+1}=\sum_{\ell=1}^{a}m_{\ell}+1\geq m_a+1,
\eeqnarray
and
\beqnarray{proof of adjacent distance larger than one II-(ii)-2222}
i_{j+1}=\sum_{\ell=1}^{j}m_{\ell}+1
=\left(\sum_{\ell=1}^{j-1}m_{\ell}+1\right)+m_j
=i_j+m_j, \textrm{ for } j=1,2,\ldots,r_{h-1}-1.
\eeqnarray
From $i_{a+1}-m_a=i_a$ in \reqnarray{proof of adjacent distance larger than one II-(ii)-2222},
we immediately see that
\beqnarray{proof of adjacent distance larger than one II-(ii)-3333}
i_a<i_{a+1}-j'<i_{a+1}, \textrm{ for } j'=1,2,\ldots,m_a-1.
\eeqnarray
It then follows from \reqnarray{proof of adjacent distance larger than one II-444},
\reqnarray{proof of adjacent distance larger than one II-(ii)-3333},
and $i_{a+1}-m_a=i_a$ in \reqnarray{proof of adjacent distance larger than one II-(ii)-2222} that
\beqnarray{}
\alignspace n_{i_{a+1}-j'}=q_{h-1}, \textrm{ for } j'=1,2,\ldots,m_a-1,
\label{eqn:proof of adjacent distance larger than one II-(ii)-4444}\\
\alignspace n_{i_{a+1}-m_a}=n_{i_a}=q_{h-1}+1.
\label{eqn:proof of adjacent distance larger than one II-(ii)-5555}
\eeqnarray
If $1\leq a\leq r_{h-1}-2$, then we have from
\reqnarray{proof of adjacent distance larger than one II-(ii)-2222},
$i_{a+2}\leq r_{h-2}$, and $m_a-m_{a+1}\leq -2$ that
\beqnarray{proof of adjacent distance larger than one II-(ii)-6666}
i_{a+1}=i_{a+2}-m_{a+1}\leq r_{h-2}-m_a-2.
\eeqnarray
Thus, \reqnarray{proof of adjacent distance larger than one II-(ii)-ccc}
follows from $i_{a+1}\geq m_a+1\geq 2$
in \reqnarray{proof of adjacent distance larger than one II-(ii)-1111}
and $i_{a+1}\leq r_{h-2}-m_a-2\leq r_{h-2}-2$
in \reqnarray{proof of adjacent distance larger than one II-(ii)-6666},
and \reqnarray{proof of adjacent distance larger than one II-(ii)-ddd}
follows from $m_a\leq i_{a+1}-1$
in \reqnarray{proof of adjacent distance larger than one II-(ii)-1111}
and $m_a\leq r_{h-2}-i_{a+1}-2\leq r_{h-2}-i_{a+1}-1$
in \reqnarray{proof of adjacent distance larger than one II-(ii)-6666}.
Also, we have from $m_a-m_{a+1}\leq -2$,
\reqnarray{proof of adjacent distance larger than one II-(ii)-2222},
and $1\leq a\leq r_{h-1}-2$ that
\beqnarray{proof of adjacent distance larger than one II-(ii)-7777}
(i_{a+1}+1)+m_a\leq i_{a+1}+m_{a+1}-1=i_{a+2}-1.
\eeqnarray
From \reqnarray{proof of adjacent distance larger than one II-(ii)-7777},
we immediately see that
\beqnarray{proof of adjacent distance larger than one II-(ii)-8888}
i_{a+1}<(i_{a+1}+1)+j'<i_{a+2}, \textrm{ for } j'=1,2,\ldots,m_a.
\eeqnarray
It then follows from \reqnarray{proof of adjacent distance larger than one II-444}
and \reqnarray{proof of adjacent distance larger than one II-(ii)-8888} that
\beqnarray{proof of adjacent distance larger than one II-(ii)-9999}
n_{(i_{a+1}+1)+j'}=q_{h-1}, \textrm{ for } j'=1,2,\ldots,m_a.
\eeqnarray
Thus, \reqnarray{proof of adjacent distance larger than one II-(ii)-eee}
and \reqnarray{proof of adjacent distance larger than one II-(ii)-fff}
follow from \reqnarray{proof of adjacent distance larger than one II-(ii)-4444},
\reqnarray{proof of adjacent distance larger than one II-(ii)-5555},
and \reqnarray{proof of adjacent distance larger than one II-(ii)-9999}.
On the other hand, if $a=r_{h-1}-1$,
then we have from
\reqnarray{proof of adjacent distance larger than one II-555},
\reqnarray{proof of adjacent distance larger than one II-(ii)-222},
and $m_a-m_{a+1}\leq -2$ that
\beqnarray{proof of adjacent distance larger than one II-(ii)-aaaa}
i_{a+1}
\aligneq i_{r_{h-1}}=\sum_{\ell=1}^{r_{h-1}-1}m_{\ell}+1
=\sum_{\ell=1}^{r_{h-1}}m_{\ell}-m_{r_{h-1}}+1 \nn\\
\aligneq r_{h-2}-m_{a+1}+1 \leq r_{h-2}-m_a-1,
\eeqnarray
where the equality holds if and only if $m_a-m_{a+1}=-2$.
Thus, \reqnarray{proof of adjacent distance larger than one II-(ii)-ccc}
follows from $i_{a+1}\geq m_a+1\geq 2$
in \reqnarray{proof of adjacent distance larger than one II-(ii)-1111}
and $i_{a+1}\leq r_{h-2}-m_a-1\leq r_{h-2}-2$
in \reqnarray{proof of adjacent distance larger than one II-(ii)-aaaa},
and \reqnarray{proof of adjacent distance larger than one II-(ii)-ddd}
follows from $m_a\leq i_{a+1}-1$
in \reqnarray{proof of adjacent distance larger than one II-(ii)-1111}
and $m_a\leq r_{h-2}-i_{a+1}-1$
in \reqnarray{proof of adjacent distance larger than one II-(ii)-aaaa}.
Also, we have from \reqnarray{proof of adjacent distance larger than one II-(ii)-aaaa} that
\beqnarray{proof of adjacent distance larger than one II-(ii)-bbbb}
(i_{a+1}+1)+m_a\leq r_{h-2},
\eeqnarray
where the equality holds if and only if $m_a-m_{a+1}=-2$.
From $a=r_{h-1}-1$ and \reqnarray{proof of adjacent distance larger than one II-(ii)-bbbb},
we immediately see that
\beqnarray{proof of adjacent distance larger than one II-(ii)-cccc}
i_{r_{h-1}}=i_{a+1}<(i_{a+1}+1)+j'\leq r_{h-2}, \textrm{ for } j'=1,2,\ldots,m_a.
\eeqnarray
It then follows from \reqnarray{proof of adjacent distance larger than one II-444}
and \reqnarray{proof of adjacent distance larger than one II-(ii)-cccc} that
\beqnarray{proof of adjacent distance larger than one II-(ii)-dddd}
n_{(i_{a+1}+1)+j'}=q_{h-1}, \textrm{ for } j'=1,2,\ldots,m_a.
\eeqnarray
As such, \reqnarray{proof of adjacent distance larger than one II-(ii)-eee}
and \reqnarray{proof of adjacent distance larger than one II-(ii)-fff}
follow from \reqnarray{proof of adjacent distance larger than one II-(ii)-4444},
\reqnarray{proof of adjacent distance larger than one II-(ii)-5555},
and \reqnarray{proof of adjacent distance larger than one II-(ii)-dddd}.
To complete the proof,
we need to show that the condition in \reqnarray{proof of adjacent distance larger than one II-(ii)-ggg}
is equivalent to the condition in \reqnarray{proof of adjacent distance larger than one II-(ii)-555}.
Note that if $i_{a+1}-m_a=1$ and $(i_{a+1}+1)+m_a=r_{h-2}$,
then we have from $n_1=n_{i_{a+1}-m_a}=q_{h-1}+1$ and $n_{r_{h-2}}=n_{(i_{a+1}+1)+m_a}=q_{h-1}$
in \reqnarray{proof of adjacent distance larger than one II-(ii)-fff} that
\beqnarray{}
n_1=n_{r_{h-2}}+1.\nn
\eeqnarray
As such, we see that the condition in \reqnarray{proof of adjacent distance larger than one II-(ii)-ggg}
is equivalent to the following condition:
\beqnarray{proof of adjacent distance larger than one II-(ii)-eeee}
i_{a+1}-m_a=1 \textrm{ and } (i_{a+1}+1)+m_a=r_{h-2}.
\eeqnarray
As we have $i_{a+1}-m_a=i_a$ in \reqnarray{proof of adjacent distance larger than one II-(ii)-2222}
and it is clear from \reqnarray{proof of adjacent distance larger than one II-555}
that $i_a=1$ if and only if $a=1$,
it follows that
\beqnarray{proof of adjacent distance larger than one II-(ii)-ffff}
i_{a+1}-m_a=1 \textrm{ iff } a=1.
\eeqnarray
Furthermore, if $1\leq a\leq r_{h-1}-2$,
then we see from \reqnarray{proof of adjacent distance larger than one II-(ii)-7777}
and $i_{a+2}\leq r_{h-2}$ that
\beqnarray{}
(i_{a+1}+1)+m_a\leq i_{a+2}-1 \leq r_{h-2}-1. \nn
\eeqnarray
On the other hand, if $a=r_{h-1}-1$, then we see from
\reqnarray{proof of adjacent distance larger than one II-(ii)-bbbb} that
\beqnarray{}
(i_{a+1}+1)+m_a\leq r_{h-2}, \nn
\eeqnarray
where the equality holds if and only if $m_a=m_{a+1}-2$.
As such, it is easy to see that
\beqnarray{proof of adjacent distance larger than one II-(ii)-gggg}
(i_{a+1}+1)+m_a=r_{h-2} \textrm{ iff } a=r_{h-1}-1 \textrm{ and } m_a=m_{a+1}-2.
\eeqnarray
From \reqnarray{proof of adjacent distance larger than one II-(ii)-ffff}
and \reqnarray{proof of adjacent distance larger than one II-(ii)-gggg},
we deduce that the condition in \reqnarray{proof of adjacent distance larger than one II-(ii)-eeee}
is equivalent to the following condition:
\beqnarray{proof of adjacent distance larger than one II-(ii)-hhhh}
a=1,\ a=r_{h-1}-1, \textrm{ and } m_a=m_{a+1}-2.
\eeqnarray
It is clear that if $a=1$, $a=r_{h-1}-1$, and $m_a=m_{a+1}-2$,
then we have $r_{h-1}=2$ and $m_1=m_2-2$.
Conversely, if $r_{h-1}=2$ and $m_1=m_2-2$,
then it follows from $1\leq a\leq r_{h-1}-1$ that $a=1$
and hence we have $a=1=r_{h-1}-1$ and $m_a=m_{a+1}-2$.
Therefore, the condition in \reqnarray{proof of adjacent distance larger than one II-(ii)-hhhh}
is equivalent to the condition that $r_{h-1}=2$ and $m_1=m_2-2$
in \reqnarray{proof of adjacent distance larger than one II-(ii)-555},
and the proof is completed.
\bappendix{Proof of Comparison rule B in \rlemma{comparison rule B} for an even integer $2\leq h\leq N$
by using Comparison rule A in \rlemma{comparison rule A} for the odd integer $h-1$}
{proof of comparison rule B for an even integer h by using comparison rule A for the odd integer h-1}
In this appendix, we assume that Comparison rule A in \rlemma{comparison rule A} holds
for some odd integer $h-1$, where $1\leq h-1\leq N-1$,
and show that Comparison rule B in \rlemma{comparison rule B} holds for the even integer $h$.
Let
\beqnarray{}
\nbf_1^{r_{h-2}}(h-1)\aligneq L_{r_{h-3},r_{h-2}}(\nbf_1^{r_{h-1}}(h)),
\label{eqn:proof of comparison rule B-111} \\
{\nbf'}_1^{r_{h-2}}(h-1)\aligneq L_{r_{h-3},r_{h-2}}({\nbf'}_1^{r_{h-1}}(h)).
\label{eqn:proof of comparison rule B-222}
\eeqnarray
For simplicity, let $\mbf_1^{r_{h-1}}=\nbf_1^{r_{h-1}}(h)$,
${\mbf'}_1^{r_{h-1}}={\nbf'}_1^{r_{h-1}}(h)$,
$\nbf_1^{r_{h-2}}=\nbf_1^{r_{h-2}}(h-1)$,
and ${\nbf'}_1^{r_{h-2}}={\nbf'}_1^{r_{h-2}}(h-1)$.
Then \reqnarray{proof of adjacent distance larger than one II-333}--\reqnarray{proof of adjacent distance larger than one II-777}
in \rappendix{proof of adjacent distance larger than one II for an even integer h
by using comparison rule A for the odd integer h-1} still hold.
Note that in \rlemma{comparison rule B},
we have $r_{h-1}\geq 2$, $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
$m_a-m_{a+1}=1$ for some $1\leq a\leq r_{h-1}-1$,
$m'_a=m_a-1$, $m'_{a+1}=m_{a+1}+1$, and $m'_i=m_i$ for $i\neq a$ and $a+1$.
As $r_{h-1}\geq 2$,
we see that \reqnarray{proof of adjacent distance larger than one II-888}
in \rappendix{proof of adjacent distance larger than one II for an even integer h
by using comparison rule A for the odd integer h-1} also holds.
It is easy to see that
\beqnarray{proof of comparison rule B-333}
m'_a=m_a-1=m_{a+1}\geq 1,\ m'_{a+1}=m_{a+1}+1\geq 2,
\textrm{ and } m'_i=m_i \textrm{ for } i\neq a, a+1.
\eeqnarray
From $m'_a=m_a-1$, $m'_{a+1}=m_{a+1}+1$, and $m'_i=m_i$ for $i\neq a$ and $a+1$,
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$, and \reqnarray{N-M-k-h},
we can see that \reqnarray{proof of adjacent distance larger than one II-(i)-222}
in \rappendix{proof of adjacent distance larger than one II for an even integer h
by using comparison rule A for the odd integer h-1} also holds.
As such, it follows from \reqnarray{proof of comparison rule B-333},
\reqnarray{proof of adjacent distance larger than one II-(i)-222},
$2\leq h\leq N$, and \reqnarray{N-M-k-h} that ${\mbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
From $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$, ${\mbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of comparison rule B-111}, and \reqnarray{proof of comparison rule B-222},
we can see that \reqnarray{proof of adjacent distance larger than one II-(i)-333}
in \rappendix{proof of adjacent distance larger than one II for an even integer h
by using comparison rule A for the odd integer h-1} also holds.
Furthermore, since ${\mbf'}_1^{r_{h-1}}$ is obtained from $\mbf_1^{r_{h-1}}$
in exactly the same way as that in \rlemma{adjacent distance larger than one II}(i),
it is clear that \reqnarray{proof of adjacent distance larger than one II-(i)-555}--\reqnarray{proof of adjacent distance larger than one II-(i)-hhh}
in \rappendix{proof of adjacent distance larger than one II for an even integer h
by using comparison rule A for the odd integer h-1} also hold.
We also note that from $m'_a=m_a-1$, $m'_{a+1}=m_{a+1}+1$, and $m_a-m_{a+1}=1$, we have
\beqnarray{proof of comparison rule B-444}
m'_a-m'_{a+1}=(m_a-1)-(m_{a+1}+1)=m_a-m_{a+1}-2=-1.
\eeqnarray
(i) Note that in \rlemma{comparison rule B}(i), we have $a=1$ or $a=r_{h-1}-1$.
To show \reqnarray{comparison rule B-1},
i.e., $\mbf_1^{r_{h-1}}\prec {\mbf'}_1^{r_{h-1}}$,
we see from \reqnarray{proof of adjacent distance larger than one II-333}
that it suffices to show that
\beqnarray{proof of comparison rule B-(i)-111}
\nbf_1^{r_{h-2}}\prec{\nbf'}_1^{r_{h-2}}.
\eeqnarray
We consider the two cases $a=1\neq r_{h-1}-1$ and $a=r_{h-1}-1$ separately.
\emph{Case 1: $a=1\neq r_{h-1}-1$.}
In this case, we have $a=1$ and $a\leq r_{h-1}-2$.
It follows that
\reqnarray{proof of adjacent distance larger than one II-(i)-case-1-111}
and \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-222}
in \rappendix{proof of adjacent distance larger than one II for an even integer h
by using comparison rule A for the odd integer h-1} also hold.
\bpdffigure{appendix-D-i-case-1.pdf}{3.5in}
\epdffigure{appendix-D-(i)-case-1}
{An illustration of \reqnarray{proof of comparison rule B-(i)-case-1-111}
in the case that $a=1\neq r_{h-1}-1$
(note that in this case we have
$\min\{i'_{a+1}-1,r_{h-2}-i'_{a+1}-1\}=m'_a$
in \reqnarray{proof of comparison rule B-(i)-case-1-555},
$i'_{a+1}-m'_a=i'_a$ in \reqnarray{proof of comparison rule B-(i)-case-1-333},
and $(i'_{a+1}+1)+m'_a=i'_{a+2}$ in \reqnarray{proof of comparison rule B-(i)-case-1-444}).}
In the following, we will show that
\beqnarray{proof of comparison rule B-(i)-case-1-111}
n'_{i'_{a+1}-j'}=n'_{(i'_{a+1}+1)+j'}, \textrm{ for } j'=1,2,\ldots,\min\{i'_{a+1}-1,r_{h-2}-i'_{a+1}-1\}.
\eeqnarray
An illustration of \reqnarray{proof of comparison rule B-(i)-case-1-111}
is given in \rfigure{appendix-D-(i)-case-1}.
Therefore, it follows from
\reqnarray{proof of adjacent distance larger than one II-888},
${\nbf'}_1^{r_{h-2}}\in \Ncal_{M,k}(h-1)$ in
\reqnarray{proof of adjacent distance larger than one II-(i)-333},
\reqnarray{proof of adjacent distance larger than one II-(i)-888}--\reqnarray{proof of adjacent distance larger than one II-(i)-bbb},
\reqnarray{proof of adjacent distance larger than one II-(i)-case-1-222},
\reqnarray{proof of comparison rule B-(i)-case-1-111},
and \reqnarray{comparison rule A-4} in \rlemma{comparison rule A}(iii)
(for the odd integer $h-1$) that ${\nbf'}_1^{r_{h-2}}\succ\nbf_1^{r_{h-2}}$,
i.e., \reqnarray{proof of comparison rule B-(i)-111} holds.
To prove \reqnarray{proof of comparison rule B-(i)-case-1-111},
note that from \reqnarray{proof of adjacent distance larger than one II-(i)-hhh}
and $a\leq r_{h-1}-2$, we have
\beqnarray{proof of comparison rule B-(i)-case-1-333}
i'_{a+1}-m'_a=i'_a.
\eeqnarray
From \reqnarray{proof of comparison rule B-444},
\reqnarray{proof of adjacent distance larger than one II-(i)-hhh},
and $a\leq r_{h-1}-2$, we have
\beqnarray{proof of comparison rule B-(i)-case-1-444}
(i'_{a+1}+1)+m'_a=i'_{a+1}+m'_{a+1}=i'_{a+2}.
\eeqnarray
As we have from $i'_1=1$ in \reqnarray{proof of adjacent distance larger than one II-777},
$a=1$, \reqnarray{proof of comparison rule B-(i)-case-1-333}, \reqnarray{proof of comparison rule B-(i)-case-1-444},
and $i'_{a+2}\leq r_{h-2}$ that
\beqnarray{}
i'_{a+1}-1=i'_{a+1}-i'_1=i'_{a+1}-i'_a=m'_a=i'_{a+2}-(i'_{a+1}+1)\leq r_{h-2}-i'_{a+1}-1, \nn
\eeqnarray
it is clear that
\beqnarray{proof of comparison rule B-(i)-case-1-555}
\min\{i'_{a+1}-1,r_{h-2}-i'_{a+1}-1\}=i'_{a+1}-1=m'_a.
\eeqnarray
It is easy to see from \reqnarray{proof of adjacent distance larger than one II-666}
and \reqnarray{proof of comparison rule B-(i)-case-1-333} that
\beqnarray{}
\alignspace n'_{i'_{a+1}-j'}=q_{h-1}, \textrm{ for } j'=1,2,\ldots,m'_a-1,
\label{eqn:proof of comparison rule B-(i)-case-1-666} \\
\alignspace n'_{i'_{a+1}-m'_a}=n'_{i'_a}=q_{h-1}+1,
\label{eqn:proof of comparison rule B-(i)-case-1-777}
\eeqnarray
and it is also easy to see from \reqnarray{proof of adjacent distance larger than one II-666}
and \reqnarray{proof of comparison rule B-(i)-case-1-444} that
\beqnarray{}
\alignspace n'_{(i'_{a+1}+1)+j'}=q_{h-1}, \textrm{ for } j'=1,2,\ldots,m'_a-1,
\label{eqn:proof of comparison rule B-(i)-case-1-888} \\
\alignspace n'_{(i'_{a+1}+1)+m'_a}=n'_{i'_{a+2}}=q_{h-1}+1.
\label{eqn:proof of comparison rule B-(i)-case-1-999}
\eeqnarray
Thus, \reqnarray{proof of comparison rule B-(i)-case-1-111}
follows from \reqnarray{proof of comparison rule B-(i)-case-1-555}--\reqnarray{proof of comparison rule B-(i)-case-1-999}.
\emph{Case 2: $a=r_{h-1}-1$.}
If $m_{a+1}=1$, then \reqnarray{proof of adjacent distance larger than one II-(i)-case-2-222} in
\rappendix{proof of adjacent distance larger than one II for an even integer h
by using comparison rule A for the odd integer h-1} also holds,
and hence it follows from
\reqnarray{proof of adjacent distance larger than one II-888},
${\nbf'}_1^{r_{h-2}}\in \Ncal_{M,k}(h-1)$ in
\reqnarray{proof of adjacent distance larger than one II-(i)-333},
\reqnarray{proof of adjacent distance larger than one II-(i)-888}--\reqnarray{proof of adjacent distance larger than one II-(i)-bbb},
\reqnarray{proof of adjacent distance larger than one II-(i)-case-2-222},
and \reqnarray{comparison rule A-1} in \rlemma{comparison rule A}(i)
(for the odd integer $h-1$) that ${\nbf'}_1^{r_{h-2}}\succ\nbf_1^{r_{h-2}}$,
i.e., \reqnarray{proof of comparison rule B-(i)-111} holds.
On the other hand, if $m_{a+1}\geq 2$,
then it is easy to see that
\reqnarray{proof of adjacent distance larger than one II-(i)-case-2-111}--\reqnarray{proof of adjacent distance larger than one II-(i)-case-2-ddd}
still hold (as we only need $m_a-m_{a+1}>0$ to prove
\reqnarray{proof of adjacent distance larger than one II-(i)-case-2-666}
and \reqnarray{proof of adjacent distance larger than one II-(i)-case-2-888}),
and hence we can show that $\nbf_1^{r_{h-2}}\prec{\nbf'}_1^{r_{h-2}}$,
i.e., \reqnarray{proof of comparison rule B-(i)-111} holds,
as in the proof of Case~2 of \rlemma{adjacent distance larger than one II}(i)
in \rappendix{proof of adjacent distance larger than one II for an even integer h
by using comparison rule A for the odd integer h-1}.
(ii) Note that in \rlemma{comparison rule B}(ii),
we have $2\leq a\leq r_{h-1}-2$ and there exists a positive integer $j$ such that
$1\leq j\leq \min\{a-1,r_{h-1}-a-1\}$, $m_{a-\ell}=m_{a+1+\ell}$ for $\ell=1,2,\ldots,j-1$,
and $m_{a-j}\neq m_{a+1+j}$.
As $2\leq a\leq r_{h-1}-2$,
we see that \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-222}
in \rappendix{proof of adjacent distance larger than one II for an even integer h
by using comparison rule A for the odd integer h-1} also holds.
Furthermore, from $m'_i=m_i$ for $i\neq a$ and $a+1$
and $m_{a-\ell}=m_{a+1+\ell}$ for $\ell=1,2,\ldots,j-1$,
it is clear that
\beqnarray{proof of comparison rule B-(ii)-111}
m'_{a-\ell}=m'_{a+1+\ell}, \textrm{ for } \ell=1,2,\ldots,j-1.
\eeqnarray
\bpdffigure{appendix-D-ii.pdf}{6.0in}
\epdffigure{appendix-D-(ii)}
{An illustration of \reqnarray{proof of comparison rule B-(ii)-222}
(note that we have
$i'_{a+1}-(\sum_{\ell=1}^{j''}m'_{a+\ell}-1)=i'_{a-j''+1}$
for $1\leq j''\leq j$ in \reqnarray{proof of comparison rule B-(ii)-444},
$(i'_{a+1}+1)+(\sum_{\ell=1}^{j''}m'_{a+\ell}-1)=i'_{a+j''+1}$
for $1\leq j''\leq r_{h-1}-a-1$ in \reqnarray{proof of comparison rule B-(ii)-666},
and $j\leq \min\{a-1,r_{h-1}-a-1\}\leq r_{h-1}-a-1$).}
By using \reqnarray{proof of comparison rule B-(ii)-111},
we can show that
\beqnarray{proof of comparison rule B-(ii)-222}
n'_{i'_{a+1}-j'}=n'_{(i'_{a+1}+1)+j'},
\textrm{ for } j'=1,2,\ldots,\sum_{\ell=1}^{j}m'_{a+\ell}-1.
\eeqnarray
An illustration of \reqnarray{proof of comparison rule B-(ii)-222}
is given in \rfigure{appendix-D-(ii)}.
To prove \reqnarray{proof of comparison rule B-(ii)-222},
observe that for $1\leq j''\leq j$,
we have from \reqnarray{proof of comparison rule B-444}
and $m'_{a-\ell}=m'_{a+1+\ell}$ for $\ell=1,2,\ldots,j''-1$
in \reqnarray{proof of comparison rule B-(ii)-111} that
\beqnarray{proof of comparison rule B-(ii)-333}
\sum_{\ell=1}^{j''}m'_{a+\ell}-1
\aligneq m'_{a+1}+\sum_{\ell=2}^{j''}m'_{a+\ell}-1=m'_a+\sum_{\ell=1}^{j''-1}m'_{a+1+\ell} \nn\\
\aligneq m'_a+\sum_{\ell=1}^{j''-1}m'_{a-\ell}=\sum_{\ell=0}^{j''-1}m'_{a-\ell}
=\sum_{\ell=a-j''+1}^{a}m'_{\ell}.
\eeqnarray
As such, for $1\leq j''\leq j$,
we have from \reqnarray{proof of adjacent distance larger than one II-777}
and \reqnarray{proof of comparison rule B-(ii)-333} that
\beqnarray{proof of comparison rule B-(ii)-444}
i'_{a+1}-\left(\sum_{\ell=1}^{j''}m'_{a+\ell}-1\right)
=\left(\sum_{\ell=1}^{a}m'_{\ell}+1\right)-\sum_{\ell=a-j''+1}^{a}m'_{\ell}
=\sum_{\ell=1}^{a-j''}m'_{\ell}+1=i'_{a-j''+1}.
\eeqnarray
It follows from \reqnarray{proof of adjacent distance larger than one II-666}
and \reqnarray{proof of comparison rule B-(ii)-444} that
\beqnarray{proof of comparison rule B-(ii)-555}
n'_{i'_{a+1}-j'}=
\bselection
q_{h-1}+1, &\textrm{for } j'=m'_{a+1}-1, \sum_{\ell=1}^{2}m'_{a+\ell}-1,\ldots,\sum_{\ell=1}^{j}m'_{a+\ell}-1, \\
q_{h-1}, &\textrm{for } 1\leq j'\leq \sum_{\ell=1}^{j}m'_{a+\ell}-1 \\
&\textrm{and } j'\neq m'_{a+1}-1, \sum_{\ell=1}^{2}m'_{a+\ell}-1,\ldots,\sum_{\ell=1}^{j}m'_{a+\ell}-1.
\eselection
\eeqnarray
Furthermore, for $1\leq j''\leq r_{h-1}-a-1$,
we have from \reqnarray{proof of adjacent distance larger than one II-777} that
\beqnarray{proof of comparison rule B-(ii)-666}
(i'_{a+1}+1)+\left(\sum_{\ell=1}^{j''}m'_{a+\ell}-1\right)
=\left(\sum_{\ell=1}^{a}m'_{\ell}+1\right)+\sum_{\ell=a+1}^{a+j''}m'_{\ell}
=\sum_{\ell=1}^{a+j''}m'_{\ell}+1=i'_{a+j''+1}.
\eeqnarray
It then follows from \reqnarray{proof of adjacent distance larger than one II-666}
and \reqnarray{proof of comparison rule B-(ii)-666} that
\beqnarray{proof of comparison rule B-(ii)-777}
n'_{(i'_{a+1}+1)+j'}=
\bselection
q_{h-1}+1, &\textrm{for } j'=m'_{a+1}-1, \sum_{\ell=1}^{2}m'_{a+\ell}-1,\ldots,\sum_{\ell=1}^{r_{h-1}-a-1}m'_{a+\ell}-1, \\
q_{h-1}, &\textrm{for } 1\leq j'\leq \sum_{\ell=1}^{r_{h-1}-a-1}m'_{a+\ell}-1 \\
&\textrm{and } j'\neq m'_{a+1}-1, \sum_{\ell=1}^{2}m'_{a+\ell}-1,\ldots,\sum_{\ell=1}^{r_{h-1}-a-1}m'_{a+\ell}-1.
\eselection
\eeqnarray
As we have $j\leq \min\{a-1,r_{h-1}-a-1\}\leq r_{h-1}-a-1$,
it is clear that \reqnarray{proof of comparison rule B-(ii)-222}
follows from \reqnarray{proof of comparison rule B-(ii)-555}
and \reqnarray{proof of comparison rule B-(ii)-777}.
From \reqnarray{proof of adjacent distance larger than one II-333},
we see that:
(a) If $m_{a-j}>m_{a+1+j}$,
then to show \reqnarray{comparison rule B-2},
i.e., $\mbf_1^{r_{h-1}}\prec{\mbf'}_1^{r_{h-1}}$,
it suffices to show that $\nbf_1^{r_{h-2}}\prec{\nbf'}_1^{r_{h-2}}$;
(b) If $m_{a-j}<m_{a+1+j}$,
then to show \reqnarray{comparison rule B-3},
i.e., $\mbf_1^{r_{h-1}}\succeq {\mbf'}_1^{r_{h-1}}$,
where $\mbf_1^{r_{h-1}}\equiv {\mbf'}_1^{r_{h-1}}$
if and only if $a-j=1$, $a+1+j=r_{h-1}$, and $m_1=m_{r_{h-1}}-1$,
it suffices to show that $\nbf_1^{r_{h-2}}\succeq{\nbf'}_1^{r_{h-2}}$,
where $\nbf_1^{r_{h-2}}\equiv{\nbf'}_1^{r_{h-2}}$
if and only if $a-j=1$, $a+1+j=r_{h-1}$, and $m_1=m_{r_{h-1}}-1$.
(a) First we assume that $m_{a-j}>m_{a+1+j}$ and show that
\beqnarray{proof of comparison rule B-(ii)-(a)-111}
\nbf_1^{r_{h-2}}\prec{\nbf'}_1^{r_{h-2}}.
\eeqnarray
As $j\leq \min\{a-1,r_{h-1}-a-1\}$, we have $j+1\leq a\leq r_{h-1}-j-1$.
We consider the two cases $j+1\leq a\leq r_{h-1}-j-2$ and $a=r_{h-1}-j-1$ separately.
\bpdffigure{appendix-D-ii-a.pdf}{6.0in}
\epdffigure{appendix-D-(ii)-(a)}
{(a) An illustration of \reqnarray{proof of comparison rule B-(ii)-(a)-case-1-222}
and \reqnarray{proof of comparison rule B-(ii)-(a)-case-1-333}
in the case that $m_{a-j}>m_{a+1+j}$ and $j+1\leq a\leq r_{h-1}-j-2$
(note that in this case we have
$i'_{a+1}-(\sum_{\ell=1}^{j''}m'_{a+\ell}-1)=i'_{a-j''+1}$
for $1\leq j''\leq j$ in \reqnarray{proof of comparison rule B-(ii)-444},
$i'_{a+1}-(\sum_{\ell=1}^{j+1}m'_{a+\ell}-1)>i'_{a-j}$
in \reqnarray{proof of comparison rule B-(ii)-(a)-case-1-999},
$(i'_{a+1}+1)+(\sum_{\ell=1}^{j''}m'_{a+\ell}-1)=i'_{a+j''+1}$
for $1\leq j''\leq r_{h-1}-a-1$ in \reqnarray{proof of comparison rule B-(ii)-666},
and $j+1\leq r_{h-1}-a-1$).
(b) An illustration of \reqnarray{proof of comparison rule B-(ii)-(a)-case-2-111}
in the case that $m_{a-j}>m_{a+1+j}$ and $a=r_{h-1}-j-1$
(note that in this case we have
$i'_{a+1}-(\sum_{\ell=1}^{j''}m'_{a+\ell}-1)=i'_{a-j''+1}$
for $1\leq j''\leq j$ in \reqnarray{proof of comparison rule B-(ii)-444},
$i'_{a+1}-(\sum_{\ell=1}^{j+1}m'_{a+\ell}-2)>i'_{a-j}$
in \reqnarray{proof of comparison rule B-(ii)-(a)-case-1-999},
$(i'_{a+1}+1)+(\sum_{\ell=1}^{j''}m'_{a+\ell}-1)=i'_{a+j''+1}$
for $1\leq j''\leq r_{h-1}-a-1=j$ in \reqnarray{proof of comparison rule B-(ii)-666},
and $(i'_{a+1}+1)+(\sum_{\ell=1}^{j+1}m'_{a+\ell}-2)=r_{h-2}$
in \reqnarray{proof of comparison rule B-(ii)-(a)-case-2-666}).}
\emph{Case 1: $j+1\leq a\leq r_{h-1}-j-2$.}
In this case, we show that
\beqnarray{}
\alignspace
1\leq \sum_{\ell=1}^{j+1}m'_{a+\ell}-1\leq \min\{i'_{a+1}-1,r_{h-2}-i'_{a+1}-1\},
\label{eqn:proof of comparison rule B-(ii)-(a)-case-1-111}\\
\alignspace
n'_{i'_{a+1}-j'}=n'_{(i'_{a+1}+1)+j'},
\textrm{ for } j'=1,2,\ldots,\sum_{\ell=1}^{j+1}m'_{a+\ell}-2,
\label{eqn:proof of comparison rule B-(ii)-(a)-case-1-222}\\
\alignspace
n'_{i'_{a+1}-(\sum_{\ell=1}^{j+1}m'_{a+\ell}-1)}=q_{h-1}
<n'_{(i'_{a+1}+1)+(\sum_{\ell=1}^{j+1}m'_{a+\ell}-1)}=q_{h-1}+1.
\label{eqn:proof of comparison rule B-(ii)-(a)-case-1-333}
\eeqnarray
An illustration of \reqnarray{proof of comparison rule B-(ii)-(a)-case-1-222}
and \reqnarray{proof of comparison rule B-(ii)-(a)-case-1-333}
is given in \rfigure{appendix-D-(ii)-(a)}(a).
Therefore, it follows from \reqnarray{proof of adjacent distance larger than one II-888},
${\nbf'}_1^{r_{h-2}}\in \Ncal_{M,k}(h-1)$ in
\reqnarray{proof of adjacent distance larger than one II-(i)-333},
\reqnarray{proof of adjacent distance larger than one II-(i)-888}--\reqnarray{proof of adjacent distance larger than one II-(i)-bbb},
\reqnarray{proof of adjacent distance larger than one II-(i)-case-1-222},
\reqnarray{proof of comparison rule B-(ii)-(a)-case-1-111}--\reqnarray{proof of comparison rule B-(ii)-(a)-case-1-333},
and \reqnarray{comparison rule A-2} in \rlemma{comparison rule A}(ii)
(for the odd integer $h-1$) that ${\nbf'}_1^{r_{h-2}}\succ\nbf_1^{r_{h-2}}$,
i.e., \reqnarray{proof of comparison rule B-(ii)-(a)-111} holds.
To prove \reqnarray{proof of comparison rule B-(ii)-(a)-case-1-111}, note that
\beqnarray{proof of comparison rule B-(ii)-(a)-case-1-444}
\sum_{\ell=1}^{j+1}m'_{a+\ell}-1\geq m'_{a+1}+m'_{a+j+1}-1\geq 1.
\eeqnarray
From \reqnarray{proof of comparison rule B-(ii)-444} (with $j''=j$),
$m'_{a-j}=m_{a-j}>m_{a+1+j}=m'_{a+1+j}$,
\reqnarray{proof of adjacent distance larger than one II-(i)-hhh},
and $i'_{a-j}\geq 1$, we have
\beqnarray{}
\sum_{\ell=1}^{j+1}m'_{a+\ell}-1
\aligneq \left(\sum_{\ell=1}^{j}m'_{a+\ell}-1\right)+m'_{a+1+j}=(i'_{a+1}-i'_{a-j+1})+m'_{a+1+j} \nn\\
\alignless i'_{a+1}-i'_{a-j+1}+m'_{a-j}=i'_{a+1}-i'_{a-j}
\label{eqn:proof of comparison rule B-(ii)-(a)-case-1-555}\\
\alignleq i'_{a+1}-1.
\label{eqn:proof of comparison rule B-(ii)-(a)-case-1-666}
\eeqnarray
As in this case we have $j+1\leq r_{h-1}-a-1$,
it follows from \reqnarray{proof of comparison rule B-(ii)-666} (with $j''=j+1$)
and $i'_{a+j+2}\leq r_{h-2}$ that
\beqnarray{proof of comparison rule B-(ii)-(a)-case-1-777}
\sum_{\ell=1}^{j+1}m'_{a+\ell}-1
=i'_{a+j+2}-(i'_{a+1}+1)
\leq r_{h-2}-i'_{a+1}-1.
\eeqnarray
Thus, \reqnarray{proof of comparison rule B-(ii)-(a)-case-1-111} follows from
\reqnarray{proof of comparison rule B-(ii)-(a)-case-1-444},
\reqnarray{proof of comparison rule B-(ii)-(a)-case-1-666},
and \reqnarray{proof of comparison rule B-(ii)-(a)-case-1-777}.
To prove \reqnarray{proof of comparison rule B-(ii)-(a)-case-1-222}
and \reqnarray{proof of comparison rule B-(ii)-(a)-case-1-333},
note that from \reqnarray{proof of comparison rule B-(ii)-444} (with $j''=j$), we have
\beqnarray{proof of comparison rule B-(ii)-(a)-case-1-888}
i'_{a+1}-\sum_{\ell=1}^{j}m'_{a+\ell}=i'_{a-j+1}-1<i'_{a-j+1},
\eeqnarray
and from \reqnarray{proof of comparison rule B-(ii)-(a)-case-1-555}, we have
\beqnarray{proof of comparison rule B-(ii)-(a)-case-1-999}
i'_{a+1}-\left(\sum_{\ell=1}^{j+1}m'_{a+\ell}-1\right)>i'_{a-j}.
\eeqnarray
Thus, we see from \reqnarray{proof of adjacent distance larger than one II-666},
\reqnarray{proof of comparison rule B-(ii)-(a)-case-1-888},
and \reqnarray{proof of comparison rule B-(ii)-(a)-case-1-999} that
\beqnarray{proof of comparison rule B-(ii)-(a)-case-1-aaa}
n'_{i'_{a+1}-j'}=q_{h-1},
\textrm{ for } \sum_{\ell=1}^{j}m'_{a+\ell}\leq j'\leq \sum_{\ell=1}^{j+1}m'_{a+\ell}-1.
\eeqnarray
Since in this case we have $j+1\leq r_{h-1}-a-1$,
it follows from \reqnarray{proof of comparison rule B-(ii)-777} that
\beqnarray{proof of comparison rule B-(ii)-(a)-case-1-bbb}
n'_{(i'_{a+1}+1)+j'}=
\bselection
q_{h-1}, &\textrm{ for } \sum_{\ell=1}^{j}m'_{a+\ell}\leq j'\leq \sum_{\ell=1}^{j+1}m'_{a+\ell}-2,\\
q_{h-1}+1, &\textrm{ for } j'=\sum_{\ell=1}^{j+1}m'_{a+\ell}-1.
\eselection
\eeqnarray
By combining \reqnarray{proof of comparison rule B-(ii)-222},
\reqnarray{proof of comparison rule B-(ii)-(a)-case-1-aaa},
and \reqnarray{proof of comparison rule B-(ii)-(a)-case-1-bbb},
we obtain \reqnarray{proof of comparison rule B-(ii)-(a)-case-1-222}
and \reqnarray{proof of comparison rule B-(ii)-(a)-case-1-333}.
\emph{Case 2: $a=r_{h-1}-j-1$.}
In this case, we show that
\beqnarray{proof of comparison rule B-(ii)-(a)-case-2-111}
n'_{i'_{a+1}-j'}=n'_{(i'_{a+1}+1)+j'},
\textrm{ for } j'=1,2,\ldots,\min\{i'_{a+1}-1,r_{h-2}-i'_{a+1}-1\}.
\eeqnarray
An illustration of \reqnarray{proof of comparison rule B-(ii)-(a)-case-2-111}
is given in \rfigure{appendix-D-(ii)-(a)}(b).
Therefore, it follows from \reqnarray{proof of adjacent distance larger than one II-888},
${\nbf'}_1^{r_{h-2}}\in \Ncal_{M,k}(h-1)$ in
\reqnarray{proof of adjacent distance larger than one II-(i)-333},
\reqnarray{proof of adjacent distance larger than one II-(i)-888}--\reqnarray{proof of adjacent distance larger than one II-(i)-bbb},
\reqnarray{proof of adjacent distance larger than one II-(i)-case-1-222},
\reqnarray{proof of comparison rule B-(ii)-(a)-case-2-111},
and \reqnarray{comparison rule A-4} in \rlemma{comparison rule A}(iii)
(for the odd integer $h-1$) that ${\nbf'}_1^{r_{h-2}}\succ\nbf_1^{r_{h-2}}$,
i.e., \reqnarray{proof of comparison rule B-(ii)-(a)-111} holds.
To prove \reqnarray{proof of comparison rule B-(ii)-(a)-case-2-111}, observe that
\reqnarray{proof of comparison rule B-(ii)-(a)-case-1-555}--\reqnarray{proof of comparison rule B-(ii)-(a)-case-1-666}
and \reqnarray{proof of comparison rule B-(ii)-(a)-case-1-888}--\reqnarray{proof of comparison rule B-(ii)-(a)-case-1-aaa}
still hold in this case.
From \reqnarray{proof of adjacent distance larger than one II-(i)-222},
$a+1+j=r_{h-1}$, and \reqnarray{proof of adjacent distance larger than one II-777},
we obtain
\beqnarray{proof of comparison rule B-(ii)-(a)-case-2-222}
r_{h-2}\aligneq \sum_{\ell=1}^{r_{h-1}}m'_{\ell}=\sum_{\ell=1}^{a+1+j}m'_{\ell}
=\sum_{\ell=1}^{a}m'_{\ell}+\sum_{\ell=a+1}^{a+1+j}m'_{\ell}
= i'_{a+1}-1+\sum_{\ell=1}^{j+1}m'_{a+\ell}.
\eeqnarray
From \reqnarray{proof of comparison rule B-(ii)-(a)-case-2-222},
\reqnarray{proof of comparison rule B-(ii)-(a)-case-1-555},
and $i'_{a-j}\geq i'_1=1$, we have
\beqnarray{proof of comparison rule B-(ii)-(a)-case-2-333}
r_{h-2}-i'_{a+1}-1=\sum_{\ell=1}^{j+1}m'_{a+\ell}-2
<i'_{a+1}-i'_{a-j}\leq i'_{a+1}-1.
\eeqnarray
It then follows from \reqnarray{proof of comparison rule B-(ii)-(a)-case-2-333} that
\beqnarray{proof of comparison rule B-(ii)-(a)-case-2-444}
\min\{i'_{a+1}-1,r_{h-2}-i'_{a+1}-1\}=r_{h-2}-i'_{a+1}-1=\sum_{\ell=1}^{j+1}m'_{a+\ell}-2.
\eeqnarray
From \reqnarray{proof of comparison rule B-(ii)-666} (with $j''=j$)
and $a=r_{h-1}-j-1$, we have
\beqnarray{proof of comparison rule B-(ii)-(a)-case-2-555}
(i'_{a+1}+1)+\left(\sum_{\ell=1}^{j}m'_{a+\ell}-1\right)=i'_{a+j+1}=i'_{r_{h-1}},
\eeqnarray
and from \reqnarray{proof of comparison rule B-(ii)-(a)-case-2-222}, we have
\beqnarray{proof of comparison rule B-(ii)-(a)-case-2-666}
(i'_{a+1}+1)+\left(\sum_{\ell=1}^{j+1}m'_{a+\ell}-2\right)=r_{h-2}.
\eeqnarray
Thus, we see from \reqnarray{proof of adjacent distance larger than one II-666},
\reqnarray{proof of comparison rule B-(ii)-(a)-case-2-555},
and \reqnarray{proof of comparison rule B-(ii)-(a)-case-2-666} that
\beqnarray{proof of comparison rule B-(ii)-(a)-case-2-777}
n'_{(i'_{a+1}+1)+j'}=q_{h-1},
\textrm{ for } \sum_{\ell=1}^{j}m'_{a+\ell}\leq j'\leq \sum_{\ell=1}^{j+1}m'_{a+\ell}-2.
\eeqnarray
By combining \reqnarray{proof of comparison rule B-(ii)-222},
\reqnarray{proof of comparison rule B-(ii)-(a)-case-1-aaa},
\reqnarray{proof of comparison rule B-(ii)-(a)-case-2-777},
and \reqnarray{proof of comparison rule B-(ii)-(a)-case-2-444},
we obtain \reqnarray{proof of comparison rule B-(ii)-(a)-case-2-111}.
(b) Now we assume that $m_{a-j}<m_{a+1+j}$ and show that
\beqnarray{proof of comparison rule B-(ii)-(b)-111}
\nbf_1^{r_{h-2}}\succeq {\nbf'}_1^{r_{h-2}},
\eeqnarray
where $\nbf_1^{r_{h-2}}\equiv{\nbf'}_1^{r_{h-2}}$ if and only if
\beqnarray{proof of comparison rule B-(ii)-(b)-222}
a-j=1,\ a+1+j=r_{h-1}, \textrm{ and } m_1=m_{r_{h-1}}-1.
\eeqnarray
\bpdffigure{appendix-D-ii-b.pdf}{6.0in}
\epdffigure{appendix-D-(ii)-(b)}
{An illustration of \reqnarray{proof of comparison rule B-(ii)-(b)-333}
and \reqnarray{proof of comparison rule B-(ii)-(b)-444}
in the case that $m_{a-j}<m_{a+1+j}$
(note that in this case we have
$i'_{a+1}-(\sum_{\ell=1}^{j''}m'_{a+\ell}-1)=i'_{a-j''+1}$
for $1\leq j''\leq j$ in \reqnarray{proof of comparison rule B-(ii)-444},
$i'_{a+1}-(\sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-1)=i'_{a-j}$
in \reqnarray{proof of comparison rule B-(ii)-(b)-999},
and $(i'_{a+1}+1)+(\sum_{\ell=1}^{j''}m'_{a+\ell}-1)=i'_{a+j''+1}$
for $1\leq j''\leq r_{h-1}-a-1$ in \reqnarray{proof of comparison rule B-(ii)-666}):
(a) $j+1\leq a\leq r_{h-1}-j-2$
(note that in this case we have
$(i'_{a+1}+1)+(\sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-1)<i'_{a+j+2}$
in \reqnarray{proof of comparison rule B-(ii)-(b)-ccc});
(b) $a=r_{h-1}-j-1$
(note that in this case we have
$(i'_{a+1}+1)+(\sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-1)\leq r_{h-2}$
in \reqnarray{proof of comparison rule B-(ii)-(b)-fff}).}
In the following, we show that
\beqnarray{}
\alignspace
n'_{i'_{a+1}-j'}=n'_{(i'_{a+1}+1)+j'},
\textrm{ for } j'=1,2,\ldots,\sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-2,
\label{eqn:proof of comparison rule B-(ii)-(b)-333} \\
\alignspace
n'_{i'_{a+1}-(\sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-1)}=q_{h-1}+1
>n'_{(i'_{a+1}+1)+(\sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-1)}=q_{h-1}.
\label{eqn:proof of comparison rule B-(ii)-(b)-444}
\eeqnarray
An illustration of \reqnarray{proof of comparison rule B-(ii)-(b)-333}
and \reqnarray{proof of comparison rule B-(ii)-(b)-444}
is given in \rfigure{appendix-D-(ii)-(b)}.
Therefore, it follows from \reqnarray{proof of adjacent distance larger than one II-888},
${\nbf'}_1^{r_{h-2}}\in \Ncal_{M,k}(h-1)$ in
\reqnarray{proof of adjacent distance larger than one II-(i)-333},
\reqnarray{proof of adjacent distance larger than one II-(i)-888}--\reqnarray{proof of adjacent distance larger than one II-(i)-bbb},
\reqnarray{proof of adjacent distance larger than one II-(i)-case-1-222},
\reqnarray{proof of comparison rule B-(ii)-(b)-333}--\reqnarray{proof of comparison rule B-(ii)-(b)-444},
and \reqnarray{comparison rule A-3} in \rlemma{comparison rule A}(ii)
(for the odd integer $h-1$) that ${\nbf'}_1^{r_{h-2}}\preceq\nbf_1^{r_{h-2}}$,
where ${\nbf'}_1^{r_{h-2}}\equiv\nbf_1^{r_{h-2}}$ if and only if
\beqnarray{}
\alignspace i'_{a+1}-\left(\sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-1\right)=1,
\label{eqn:proof of comparison rule B-(ii)-(b)-555}\\
\alignspace (i'_{a+1}+1)+\left(\sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-1\right)=r_{h-2},
\label{eqn:proof of comparison rule B-(ii)-(b)-666}\\
\alignspace n'_1=n'_{r_{h-2}}+1.
\label{eqn:proof of comparison rule B-(ii)-(b)-777}
\eeqnarray
To prove \reqnarray{proof of comparison rule B-(ii)-(b)-333}
and \reqnarray{proof of comparison rule B-(ii)-(b)-444},
note that from \reqnarray{proof of comparison rule B-(ii)-444} (with $j''=j$)
and \reqnarray{proof of adjacent distance larger than one II-(i)-hhh}, we have
\beqnarray{}
\alignspace
i'_{a+1}-\sum_{\ell=1}^{j}m'_{a+\ell}=i'_{a-j+1}-1<i'_{a-j+1},
\label{eqn:proof of comparison rule B-(ii)-(b)-888}\\
\alignspace
i'_{a+1}-\left(\sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-1\right)=i'_{a-j+1}-m'_{a-j}=i'_{a-j}.
\label{eqn:proof of comparison rule B-(ii)-(b)-999}
\eeqnarray
Thus, we see from \reqnarray{proof of adjacent distance larger than one II-666},
\reqnarray{proof of comparison rule B-(ii)-(b)-888},
and \reqnarray{proof of comparison rule B-(ii)-(b)-999} that
\beqnarray{proof of comparison rule B-(ii)-(b)-aaa}
n'_{i'_{a+1}-j'}=
\bselection
q_{h-1}, &\textrm{ for } \sum_{\ell=1}^{j}m'_{a+\ell}\leq j'\leq \sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-2,\\
q_{h-1}+1, &\textrm{ for } j'=\sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-1.
\eselection
\eeqnarray
If $j+1\leq a\leq r_{h-1}-j-2$,
then we have from \reqnarray{proof of comparison rule B-(ii)-666}
(with $j''=j$), $m'_{a-j}=m_{a-j}<m_{a+1+j}=m'_{a+1+j}$,
and \reqnarray{proof of adjacent distance larger than one II-(i)-hhh} that
\beqnarray{}
\alignspace \hspace*{-0.2in}
(i'_{a+1}+1)+\sum_{\ell=1}^{j}m'_{a+\ell}=i'_{a+j+1}+1>i'_{a+j+1},
\label{eqn:proof of comparison rule B-(ii)-(b)-bbb}\\
\alignspace \hspace*{-0.2in}
(i'_{a+1}+1)+\left(\sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-1\right)
=i'_{a+j+1}+m'_{a-j}< i'_{a+j+1}+m'_{a+1+j}=i'_{a+j+2}.
\label{eqn:proof of comparison rule B-(ii)-(b)-ccc}
\eeqnarray
Thus, we see from \reqnarray{proof of adjacent distance larger than one II-666},
\reqnarray{proof of comparison rule B-(ii)-(b)-bbb},
and \reqnarray{proof of comparison rule B-(ii)-(b)-ccc} that
\beqnarray{proof of comparison rule B-(ii)-(b)-ddd}
n'_{(i'_{a+1}+1)+j'}=q_{h-1},
\textrm{ for } \sum_{\ell=1}^{j}m'_{a+\ell}\leq j'\leq \sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-1.
\eeqnarray
By combining \reqnarray{proof of comparison rule B-(ii)-222},
\reqnarray{proof of comparison rule B-(ii)-(b)-aaa},
and \reqnarray{proof of comparison rule B-(ii)-(b)-ddd},
we obtain \reqnarray{proof of comparison rule B-(ii)-(b)-333}
and \reqnarray{proof of comparison rule B-(ii)-(b)-444}.
On the other hand, if $a=r_{h-1}-j-1$,
then we have from \reqnarray{proof of comparison rule B-(ii)-666} (with $j''=j$) that
\beqnarray{proof of comparison rule B-(ii)-(b)-eee}
(i'_{a+1}+1)+\sum_{\ell=1}^{j}m'_{a+\ell}=i'_{a+j+1}+1=i'_{r_{h-1}}+1>i'_{r_{h-1}},
\eeqnarray
and we have from $m'_{a-j}=m_{a-j}<m_{a+1+j}=m'_{a+1+j}$
and \reqnarray{proof of comparison rule B-(ii)-(a)-case-2-222} that
\beqnarray{proof of comparison rule B-(ii)-(b)-fff}
(i'_{a+1}+1)+\left(\sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-1\right)
\alignleq i'_{a+1}+\sum_{\ell=1}^{j}m'_{a+\ell}+(m'_{a+1+j}-1) \nn\\
\aligneq i'_{a+1}+\sum_{\ell=1}^{j+1}m'_{a+\ell}-1=r_{h-2},
\eeqnarray
where the equality holds if and only if $m_{a-j}=m_{a+1+j}-1$.
Thus, we see from \reqnarray{proof of adjacent distance larger than one II-666},
\reqnarray{proof of comparison rule B-(ii)-(b)-eee},
and \reqnarray{proof of comparison rule B-(ii)-(b)-fff} that
\beqnarray{proof of comparison rule B-(ii)-(b)-ggg}
n'_{(i'_{a+1}+1)+j'}=q_{h-1},
\textrm{ for }\sum_{\ell=1}^{j}m'_{a+\ell}\leq j'\leq \sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-1.
\eeqnarray
By combining \reqnarray{proof of comparison rule B-(ii)-222},
\reqnarray{proof of comparison rule B-(ii)-(b)-aaa},
and \reqnarray{proof of comparison rule B-(ii)-(b)-ggg},
we obtain \reqnarray{proof of comparison rule B-(ii)-(b)-333}
and \reqnarray{proof of comparison rule B-(ii)-(b)-444}.
To complete the proof,
we need to show that the condition in
\reqnarray{proof of comparison rule B-(ii)-(b)-555}--\reqnarray{proof of comparison rule B-(ii)-(b)-777}
is equivalent to the condition in \reqnarray{proof of comparison rule B-(ii)-(b)-222}.
Note that if $i'_{a+1}-(\sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-1)=1$
and $(i'_{a+1}+1)+(\sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-1)=r_{h-2}$,
then we have from $n'_1=n'_{i'_{a+1}-(\sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-1)}=q_{h-1}+1$
and $n'_{r_{h-2}}=n'_{(i'_{a+1}+1)+(\sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-1)}=q_{h-1}$
in \reqnarray{proof of comparison rule B-(ii)-(b)-444} that
\beqnarray{}
n'_1=n'_{r_{h-2}}+1.\nn
\eeqnarray
As such, we see that the condition in
\reqnarray{proof of comparison rule B-(ii)-(b)-555}--\reqnarray{proof of comparison rule B-(ii)-(b)-777}
is equivalent to the following condition:
\beqnarray{proof of comparison rule B-(ii)-(b)-hhh}
i'_{a+1}-\left(\sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-1\right)=1
\textrm{ and } (i'_{a+1}+1)+\left(\sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-1\right)=r_{h-2}.
\eeqnarray
As we have $i'_{a+1}-\left(\sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-1\right)=i'_{a-j}$
in \reqnarray{proof of comparison rule B-(ii)-(b)-999}
and it is clear from \reqnarray{proof of adjacent distance larger than one II-777}
that $i'_{a-j}=1$ if and only if $a-j=1$,
it follows that
\beqnarray{proof of comparison rule B-(ii)-(b)-iii}
i'_{a+1}-\left(\sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-1\right)=1 \textrm{ iff } a-j=1.
\eeqnarray
Furthermore, if $j+1\leq a\leq r_{h-1}-j-2$,
then we see from \reqnarray{proof of comparison rule B-(ii)-(b)-ccc}
and $i'_{a+j+2}\leq r_{h-2}$ that
\beqnarray{}
(i'_{a+1}+1)+\left(\sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-1\right)<i'_{a+j+2}\leq r_{h-2}. \nn
\eeqnarray
On the other hand, if $a=r_{h-1}-j-1$, then we see from
\reqnarray{proof of comparison rule B-(ii)-(b)-fff} that
\beqnarray{}
(i'_{a+1}+1)+\left(\sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-1\right)\leq r_{h-2}, \nn
\eeqnarray
where the equality holds if and only if $m_{a-j}=m_{a+1+j}-1$.
As such, it is easy to see that
\beqnarray{proof of comparison rule B-(ii)-(b)-jjj}
(i'_{a+1}+1)+\left(\sum_{\ell=1}^{j}m'_{a+\ell}+m'_{a-j}-1\right)=r_{h-2}
\textrm{ iff } a=r_{h-1}-j-1 \textrm{ and } m_{a-j}=m_{a+1+j}-1.
\eeqnarray
Therefore, we deduce from \reqnarray{proof of comparison rule B-(ii)-(b)-iii}
and \reqnarray{proof of comparison rule B-(ii)-(b)-jjj}
that the condition in \reqnarray{proof of comparison rule B-(ii)-(b)-hhh}
is equivalent to the following condition:
\beqnarray{}
a-j=1,\ a=r_{h-1}-j-1, \textrm{ and } m_{a-j}=m_{a+1+j}-1, \nn
\eeqnarray
which is clearly equivalent to the condition
that $a-j=1$, $a+1+j=r_{h-1}$, and $m_1=m_{r_{h-1}}-1$
in \reqnarray{proof of comparison rule B-(ii)-(b)-222},
and the proof is completed.
(iii) Note that in \rlemma{comparison rule B}(iii),
we have $2\leq a\leq r_{h-1}-2$
and $m_{a-\ell}=m_{a+1+\ell}$ for $\ell=1,2,\ldots,\min\{a-1,r_{h-1}-a-1\}$.
To show \reqnarray{comparison rule B-4},
i.e., $\mbf_1^{r_{h-1}}\prec{\mbf'}_1^{r_{h-1}}$,
we see from \reqnarray{proof of adjacent distance larger than one II-333}
that it suffices to show that
\beqnarray{proof of comparison rule B-(iii)-111}
\nbf_1^{r_{h-2}}\prec{\nbf'}_1^{r_{h-2}}.
\eeqnarray
Note that as we have $a\leq r_{h-1}-2$,
it follows that \reqnarray{proof of adjacent distance larger than one II-(i)-case-1-222}
in \rappendix{proof of adjacent distance larger than one II for an even integer h
by using comparison rule A for the odd integer h-1} also holds.
\bpdffigure{appendix-D-iii.pdf}{6.0in}
\epdffigure{appendix-D-(iii)}
{An illustration of \reqnarray{proof of comparison rule B-(iii)-222}
(note that we have
$i'_{a+1}-(\sum_{\ell=1}^{j''}m'_{a+\ell}-1)=i'_{a-j''+1}$
for $1\leq j''\leq j^*+1$ in \reqnarray{proof of comparison rule B-(ii)-444}
and $(i'_{a+1}+1)+(\sum_{\ell=1}^{j''}m'_{a+\ell}-1)=i'_{a+j''+1}$
for $1\leq j''\leq r_{h-1}-a-1$ in \reqnarray{proof of comparison rule B-(ii)-666}):
(a) $a-1<r_{h-1}-a-1$
(note that in this case we have $\min\{i'_{a+1}-1,r_{h-2}-i'_{a+1}-1\}=\sum_{\ell=1}^{j^*+1}m'_{a+\ell}-1$
in \reqnarray{proof of comparison rule B-(iii)-case-1-666},
and $j^*=a-1$ and $j^*+1\leq r_{h-1}-a-1$ in \reqnarray{proof of comparison rule B-(iii)-case-1-111});
(b) $a-1\geq r_{h-1}-a-1$
(note that in this case we have
$\min\{i'_{a+1}-1,r_{h-2}-i'_{a+1}-1\}=\sum_{\ell=1}^{j^*+1}m'_{a+\ell}-2$
in \reqnarray{proof of comparison rule B-(iii)-case-2-333},
$j^*=r_{h-1}-a-1$ in \reqnarray{proof of comparison rule B-(iii)-case-2-111},
and $(i'_{a+1}+1)+(\sum_{\ell=1}^{j^*+1}m'_{a+\ell}-2)=r_{h-2}$
in \reqnarray{proof of comparison rule B-(ii)-(a)-case-2-666}).}
In the following, we show that
\beqnarray{proof of comparison rule B-(iii)-222}
n'_{i'_{a+1}-j'}=n'_{(i'_{a+1}+1)+j'},
\textrm{ for } j'=1,2,\ldots,\min\{i'_{a+1}-1,r_{h-2}-i'_{a+1}-1\}.
\eeqnarray
An illustration of \reqnarray{proof of comparison rule B-(iii)-222}
is given in \rfigure{appendix-D-(iii)}.
Therefore, it follows from
\reqnarray{proof of adjacent distance larger than one II-888},
${\nbf'}_1^{r_{h-2}}\in \Ncal_{M,k}(h-1)$ in
\reqnarray{proof of adjacent distance larger than one II-(i)-333},
\reqnarray{proof of adjacent distance larger than one II-(i)-888}--\reqnarray{proof of adjacent distance larger than one II-(i)-bbb},
\reqnarray{proof of adjacent distance larger than one II-(i)-case-1-222},
\reqnarray{proof of comparison rule B-(iii)-222},
and \reqnarray{comparison rule A-4} in \rlemma{comparison rule A}(iii)
(for the odd integer $h-1$) that ${\nbf'}_1^{r_{h-2}}\succ\nbf_1^{r_{h-2}}$,
i.e., \reqnarray{proof of comparison rule B-(iii)-111} holds.
To prove \reqnarray{proof of comparison rule B-(iii)-222},
let $j^*=\min\{a-1,r_{h-1}-a-1\}$.
Since $m'_{a-\ell}=m_{a-\ell}=m_{a+1+\ell}=m'_{a+1+\ell}$ for $\ell=1,2,\ldots,j^*$,
we see from the same argument as in (ii) above that
\reqnarray{proof of comparison rule B-(ii)-444} holds for $1\leq j''\leq j^*+1$
and \reqnarray{proof of comparison rule B-(ii)-555} holds for $j=j^*+1$.
It is clear that \reqnarray{proof of comparison rule B-(ii)-777} also holds.
We then consider the two cases
$a-1<r_{h-1}-a-1$ and $a-1\geq r_{h-1}-a-1$ separately.
\emph{Case 1: $a-1<r_{h-1}-a-1$.}
In this case, we have
\beqnarray{proof of comparison rule B-(iii)-case-1-111}
j^*=a-1 \textrm{ and } j^*<r_{h-1}-a-1.
\eeqnarray
From \reqnarray{proof of adjacent distance larger than one II-(i)-222},
$j^*\leq r_{h-1}-a-2$ in \reqnarray{proof of comparison rule B-(iii)-case-1-111},
and \reqnarray{proof of adjacent distance larger than one II-777},
we see that
\beqnarray{proof of comparison rule B-(iii)-case-1-222}
r_{h-2}=\sum_{\ell=1}^{r_{h-1}}m'_{\ell}
=\sum_{\ell=1}^{a}m'_{\ell}+\sum_{\ell=a+1}^{r_{h-1}}m'_{\ell}
=(i'_{a+1}-1)+\sum_{\ell=1}^{r_{h-1}-a}m'_{a+\ell}
\geq i'_{a+1}-1+\sum_{\ell=1}^{j^*+2}m'_{a+\ell}.
\eeqnarray
From \reqnarray{proof of comparison rule B-(ii)-444} (with $j''=j^*+1$),
$a-j^*=1$ in \reqnarray{proof of comparison rule B-(iii)-case-1-111},
and $i'_1=1$ in \reqnarray{proof of adjacent distance larger than one II-777}, we have
\beqnarray{}
i'_{a+1}
\aligneq \left(\sum_{\ell=1}^{j^*+1}m'_{a+\ell}-1\right)+i'_{a-j^*}
\label{eqn:proof of comparison rule B-(iii)-case-1-333}\\
\aligneq \sum_{\ell=1}^{j^*+1}m'_{a+\ell}-1+i'_1=\sum_{\ell=1}^{j^*+1}m'_{a+\ell}.
\label{eqn:proof of comparison rule B-(iii)-case-1-444}
\eeqnarray
It follows from \reqnarray{proof of comparison rule B-(iii)-case-1-222}
and \reqnarray{proof of comparison rule B-(iii)-case-1-444} that
\beqnarray{proof of comparison rule B-(iii)-case-1-555}
r_{h-2}-i'_{a+1}-1
\aligngeq \sum_{\ell=1}^{j^*+2}m'_{a+\ell}-2
=\sum_{\ell=1}^{j^*+1}m'_{a+\ell}+m'_{a+j^*+2}-2 \nn\\
\aligneq i'_{a+1}+m'_{a+j^*+2}-2 \nn\\
\aligngeq i'_{a+1}-1.
\eeqnarray
Thus, we see from \reqnarray{proof of comparison rule B-(iii)-case-1-555}
and \reqnarray{proof of comparison rule B-(iii)-case-1-444} that
\beqnarray{proof of comparison rule B-(iii)-case-1-666}
\min\{i'_{a+1}-1,r_{h-2}-i'_{a+1}-1\}=i'_{a+1}-1=\sum_{\ell=1}^{j^*+1}m'_{a+\ell}-1.
\eeqnarray
From \reqnarray{proof of comparison rule B-(ii)-555} (with $j=j^*+1$),
\reqnarray{proof of comparison rule B-(ii)-777},
and $j^*+1\leq r_{h-1}-a-1$ in \reqnarray{proof of comparison rule B-(iii)-case-1-111},
we see that \reqnarray{proof of comparison rule B-(ii)-222}
holds for $j=j^*+1$, i.e.,
\beqnarray{proof of comparison rule B-(iii)-case-1-777}
n'_{i'_{a+1}-j'}=n'_{(i'_{a+1}+1)+j'},
\textrm{ for } j'=1,2,\ldots,\sum_{\ell=1}^{j^*+1}m'_{a+\ell}-1.
\eeqnarray
As such, \reqnarray{proof of comparison rule B-(iii)-222}
follows from \reqnarray{proof of comparison rule B-(iii)-case-1-666}
and \reqnarray{proof of comparison rule B-(iii)-case-1-777}.
\emph{Case 2: $a-1\geq r_{h-1}-a-1$.}
In this case, we have
\beqnarray{proof of comparison rule B-(iii)-case-2-111}
j^*=r_{h-1}-a-1 \textrm{ and } j^*\leq a-1.
\eeqnarray
As \reqnarray{proof of comparison rule B-(ii)-666} holds for $j''=j^*$
and we have $a=r_{h-1}-j^*-1$ in \reqnarray{proof of comparison rule B-(iii)-case-2-111},
it is easy to see that \reqnarray{proof of comparison rule B-(ii)-(a)-case-2-222}
and \reqnarray{proof of comparison rule B-(ii)-(a)-case-2-555}--\reqnarray{proof of comparison rule B-(ii)-(a)-case-2-777}
hold with $j=j^*$.
From \reqnarray{proof of comparison rule B-(ii)-(a)-case-2-222} (with $j=j^*$)
and \reqnarray{proof of comparison rule B-(ii)-444} (with $j=j^*+1$), we have
\beqnarray{proof of comparison rule B-(iii)-case-2-222}
r_{h-2}-i'_{a+1}-1=\sum_{\ell=1}^{j^*+1}m'_{a+\ell}-2
=i'_{a+1}-i'_{a-j^*}-1<i'_{a+1}-1.
\eeqnarray
Thus, we see from \reqnarray{proof of comparison rule B-(iii)-case-2-222} that
\beqnarray{proof of comparison rule B-(iii)-case-2-333}
\min\{i'_{a+1}-1,r_{h-2}-i'_{a+1}-1\}=r_{h-2}-i'_{a+1}-1=\sum_{\ell=1}^{j^*+1}m'_{a+\ell}-2.
\eeqnarray
From \reqnarray{proof of comparison rule B-(ii)-555} (with $j=j^*+1$),
\reqnarray{proof of comparison rule B-(ii)-777} (note that $r_{h-1}-a-1=j^*$),
and \reqnarray{proof of comparison rule B-(ii)-(a)-case-2-777} (with $j=j^*$),
we can see that
\beqnarray{proof of comparison rule B-(iii)-case-2-444}
n'_{i'_{a+1}-j'}=n'_{(i'_{a+1}+1)+j'},
\textrm{ for } j'=1,2,\ldots,\sum_{\ell=1}^{j^*+1}m'_{a+\ell}-2.
\eeqnarray
As such, \reqnarray{proof of comparison rule B-(iii)-222}
follows from \reqnarray{proof of comparison rule B-(iii)-case-2-333}
and \reqnarray{proof of comparison rule B-(iii)-case-2-444}.
\bappendix{Proof of \rlemma{adjacent distance larger than one} for an odd integer $3\leq h\leq N$
by using Comparison rule B in \rlemma{comparison rule B} for the even integer $h-1$}
{proof of adjacent distance larger than one for an odd integer h by using comparison rule B for the even integer h-1}
In this appendix, we assume that Comparison rule B in \rlemma{comparison rule B} holds
for some even integer $h-1$, where $2\leq h-1\leq N-1$,
and show that \rlemma{adjacent distance larger than one} holds for the odd integer $h$.
Let
\beqnarray{}
\nbf_1^{r_{h-2}}(h-1)\aligneq R_{r_{h-3},r_{h-2}}(\nbf_1^{r_{h-1}}(h)),
\label{eqn:proof of adjacent distance larger than one-111} \\
{\nbf'}_1^{r_{h-2}}(h-1)\aligneq R_{r_{h-3},r_{h-2}}({\nbf'}_1^{r_{h-1}}(h)).
\label{eqn:proof of adjacent distance larger than one-222}
\eeqnarray
For simplicity, we let $\mbf_1^{r_{h-1}}=\nbf_1^{r_{h-1}}(h)$,
${\mbf'}_1^{r_{h-1}}={\nbf'}_1^{r_{h-1}}(h)$,
$\nbf_1^{r_{h-2}}=\nbf_1^{r_{h-2}}(h-1)$,
and ${\nbf'}_1^{r_{h-2}}={\nbf'}_1^{r_{h-2}}(h-1)$.
Then we have from \reqnarray{proof of adjacent distance larger than one-111},
\reqnarray{proof of adjacent distance larger than one-222}, and \reqnarray{order relation-777} that
\beqnarray{proof of adjacent distance larger than one-333}
\mbf_1^{r_{h-1}}\prec
(\textrm{resp.}, \equiv, \succ, \preceq, \succeq)\ {\mbf'}_1^{r_{h-1}}
\textrm{ iff } \nbf_1^{r_{h-2}}\prec
(\textrm{resp.}, \equiv, \succ, \preceq, \succeq)\ {\nbf'}_1^{r_{h-2}}.
\eeqnarray
Furthermore, from \reqnarray{proof of adjacent distance larger than one-111},
\reqnarray{proof of adjacent distance larger than one-222},
and the definition of right pre-sequences in \rdefinition{right pre-sequences},
we have
\beqnarray{proof of adjacent distance larger than one-444}
n_i=
\bselection
q_{h-1}+1, &\textrm{if } i=i_1,i_2,\ldots,i_{r_{h-1}}, \\
q_{h-1}, &\textrm{otherwise},
\eselection
\eeqnarray
where
\beqnarray{proof of adjacent distance larger than one-555}
i_j=\sum_{\ell=1}^{j}m_{\ell}, \textrm{ for } j=1,2,\ldots,r_{h-1},
\eeqnarray
and
\beqnarray{proof of adjacent distance larger than one-666}
n'_i=
\bselection
q_{h-1}+1, &\textrm{if } i=i'_1,i'_2,\ldots,i'_{r_{h-1}}, \\
q_{h-1}, &\textrm{otherwise},
\eselection
\eeqnarray
where
\beqnarray{proof of adjacent distance larger than one-777}
i'_j=\sum_{\ell=1}^{j}m'_{\ell}, \textrm{ for } j=1,2,\ldots,r_{h-1}.
\eeqnarray
Note that in \rlemma{adjacent distance larger than one}, we have $r_{h-1}\geq 2$.
As such, it follows from $r_{h-2}>r_{h-1}$ that
\beqnarray{proof of adjacent distance larger than one-888}
r_{h-2}\geq 2.
\eeqnarray
(i) Note that in \rlemma{adjacent distance larger than one}(i),
we have $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
$m_a-m_{a+1}\leq -2$ for some $1\leq a\leq r_{h-1}-1$,
$m'_a=m_a+1$, $m'_{a+1}=m_{a+1}-1$, and $m'_i=m_i$ for $i\neq a$ and $a+1$.
It is easy to see that
\beqnarray{proof of adjacent distance larger than one-(i)-111}
m'_a=m_a+1\geq 2,\ m'_{a+1}=m_{a+1}-1\geq m_{a}+1\geq 2,
\textrm{ and } m'_i=m_i \textrm{ for } i\neq a, a+1.
\eeqnarray
Also, we have from $m'_a=m_a+1$, $m'_{a+1}=m_{a+1}-1$, and $m'_i=m_i$ for $i\neq a$ and $a+1$,
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$, and \reqnarray{N-M-k-h} that
\beqnarray{proof of adjacent distance larger than one-(i)-222}
\sum_{i=1}^{r_{h-1}}m'_i=\sum_{i=1}^{r_{h-1}}m_i=r_{h-2}.
\eeqnarray
As such, it follows from \reqnarray{proof of adjacent distance larger than one-(i)-111},
\reqnarray{proof of adjacent distance larger than one-(i)-222},
and \reqnarray{N-M-k-h} that ${\mbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
As $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$ and ${\mbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
we see from \reqnarray{proof of adjacent distance larger than one-111},
\reqnarray{proof of adjacent distance larger than one-222},
and the argument in the paragraph after \reqnarray{N-M-k-h} that
\beqnarray{proof of adjacent distance larger than one-(i)-333}
\nbf_1^{r_{h-2}}\in \Ncal_{M,k}(h-1) \textrm{ and } {\nbf'}_1^{r_{h-2}}\in \Ncal_{M,k}(h-1).
\eeqnarray
To show \reqnarray{adjacent distance larger than one-1},
i.e., $\mbf_1^{r_{h-1}}\prec{\mbf'}_1^{r_{h-1}}$,
we see from \reqnarray{proof of adjacent distance larger than one-333}
that it suffices to show that
\beqnarray{proof of adjacent distance larger than one-(i)-444}
\nbf_1^{r_{h-2}}\prec{\nbf'}_1^{r_{h-2}}.
\eeqnarray
\bpdffigure{appendix-E-i.pdf}{5.5in}
\epdffigure{appendix-E-(i)}
{An illustration of
\reqnarray{proof of adjacent distance larger than one-(i)-666}--\reqnarray{proof of adjacent distance larger than one-(i)-bbb}:
(a) $a=1$
(note that in this case we have $1\leq i_a<i'_1$
in \reqnarray{proof of adjacent distance larger than one-(i)-ddd}
and $i_a<i_a+1<i_{a+1}$
in \reqnarray{proof of adjacent distance larger than one-(i)-ccc});
(b) $2\leq a\leq r_{h-1}-1$
(note that in this case we have $i'_{a-1}<i_a<i'_a$
in \reqnarray{proof of adjacent distance larger than one-(i)-eee}
and $i_a<i_a+1<i_{a+1}$
in \reqnarray{proof of adjacent distance larger than one-(i)-ccc}).}
Note that from $m_a=m'_a-1$, $m_{a+1}=m'_{a+1}+1$, $m_i=m'_i$ for $i\neq a$ and $a+1$,
\reqnarray{proof of adjacent distance larger than one-555},
and \reqnarray{proof of adjacent distance larger than one-777},
it is easy to see that
\beqnarray{proof of adjacent distance larger than one-(i)-555}
i_j=
\bselection
i'_j-1, &\textrm{if } j=a, \\
i'_j, &\textrm{otherwise}.
\eselection
\eeqnarray
In the following, we show that
\beqnarray{}
\alignspace n'_{i_a}=q_{h-1},
\label{eqn:proof of adjacent distance larger than one-(i)-666}\\
\alignspace n_{i_a+1}=q_{h-1}.
\label{eqn:proof of adjacent distance larger than one-(i)-777}
\eeqnarray
It then follows from \reqnarray{proof of adjacent distance larger than one-444},
\reqnarray{proof of adjacent distance larger than one-666},
and \reqnarray{proof of adjacent distance larger than one-(i)-555}--\reqnarray{proof of adjacent distance larger than one-(i)-777} that
\beqnarray{}
\alignspace n_{i_a}-n_{i_a+1}=(q_{h-1}+1)-q_{h-1}=1,
\label{eqn:proof of adjacent distance larger than one-(i)-888}\\
\alignspace n'_{i_a}=q_{h-1}=n_{i_a}-1,
\label{eqn:proof of adjacent distance larger than one-(i)-999}\\
\alignspace n'_{i_a+1}=n'_{i'_a}=q_{h-1}+1=n_{i_a+1}+1,
\label{eqn:proof of adjacent distance larger than one-(i)-aaa}\\
\alignspace n'_i=n_i, \textrm{ for } i\neq i_a \textrm{ and } i_a+1.
\label{eqn:proof of adjacent distance larger than one-(i)-bbb}
\eeqnarray
An illustration of \reqnarray{proof of adjacent distance larger than one-(i)-666}--\reqnarray{proof of adjacent distance larger than one-(i)-bbb}
is given in \rfigure{appendix-E-(i)}.
To prove \reqnarray{proof of adjacent distance larger than one-(i)-666},
note that if $a=1$, then we have from
\reqnarray{proof of adjacent distance larger than one-(i)-555} that
\beqnarray{proof of adjacent distance larger than one-(i)-ddd}
1\leq i_a=i'_a-1<i'_a=i'_1.
\eeqnarray
Thus, \reqnarray{proof of adjacent distance larger than one-(i)-666}
follows from \reqnarray{proof of adjacent distance larger than one-666}
and $1\leq i_a<i'_1$ in \reqnarray{proof of adjacent distance larger than one-(i)-ddd}.
On the other hand, if $2\leq a\leq r_{h-1}-1$,
then we have from \reqnarray{proof of adjacent distance larger than one-(i)-555} that
\beqnarray{proof of adjacent distance larger than one-(i)-eee}
i_a>i_{a-1}=i'_{a-1} \textrm{ and } i_a=i'_a-1<i'_a.
\eeqnarray
Thus, \reqnarray{proof of adjacent distance larger than one-(i)-666}
follows from \reqnarray{proof of adjacent distance larger than one-666} and $i'_{a-1}<i_a<i'_a$
in \reqnarray{proof of adjacent distance larger than one-(i)-eee}.
To prove \reqnarray{proof of adjacent distance larger than one-(i)-777},
note that from \reqnarray{proof of adjacent distance larger than one-(i)-555} we have
\beqnarray{proof of adjacent distance larger than one-(i)-ccc}
i_a<i_a+1=i'_a<i'_{a+1}=i_{a+1}.
\eeqnarray
Thus, \reqnarray{proof of adjacent distance larger than one-(i)-777}
follows from \reqnarray{proof of adjacent distance larger than one-444}
and $i_a<i_a+1<i_{a+1}$ in \reqnarray{proof of adjacent distance larger than one-(i)-ccc}.
Note that from \reqnarray{proof of adjacent distance larger than one-555},
$a\leq r_{h-1}-1$,
$\sum_{\ell=1}^{r_{h-1}}m_{\ell}=r_{h-2}$ in \reqnarray{proof of adjacent distance larger than one-(i)-222},
and $m_a-m_{a+1}\leq -2$, we have
\beqnarray{proof of adjacent distance larger than one-(i)-fff}
i_a=\sum_{\ell=1}^{a}m_{\ell}=\sum_{\ell=1}^{r_{h-1}}m_{\ell}-\sum_{\ell=a+1}^{r_{h-1}}m_{\ell}
\leq r_{h-2}-m_{a+1} \leq r_{h-2}-m_a-2\leq r_{h-2}-2,
\eeqnarray
and
\beqnarray{proof of adjacent distance larger than one-(i)-ggg}
i_j=\sum_{\ell=1}^{j}m_{\ell}=\sum_{\ell=1}^{j-1}m_{\ell}+m_j=i_{j-1}+m_j,
\textrm{ for } j=2,3,\ldots,r_{h-1}.
\eeqnarray
We then consider the two cases $a=1$ and $2\leq a\leq r_{h-1}-1$ separately.
\emph{Case 1: $a=1$}.
\bpdffigure{appendix-E-i-case-1.pdf}{4.0in}
\epdffigure{appendix-E-(i)-case-1}
{An illustration of \reqnarray{proof of adjacent distance larger than one-(i)-case-1-333}
for the case that $a=1$ and $m_a\geq 2$
(note that in this case we have
$\min\{i_a-1,r_{h-2}-i_a-1\}=m_a-1$
in \reqnarray{proof of adjacent distance larger than one-(i)-case-1-555},
$i_a-(m_a-1)=1$
in \reqnarray{proof of adjacent distance larger than one-(i)-case-1-666},
and $(i_a+1)-(m_a-1)<i_{a+1}$
in \reqnarray{proof of adjacent distance larger than one-(i)-case-1-999}).}
In this case, we have from
\reqnarray{proof of adjacent distance larger than one-555} and $a=1$ that
\beqnarray{proof of adjacent distance larger than one-(i)-case-1-111}
i_a=\sum_{\ell=1}^{a}m_{\ell}=m_1=m_a.
\eeqnarray
If $m_a=1$, then we have from \reqnarray{proof of adjacent distance larger than one-(i)-case-1-111} that $i_a=1$.
Therefore, it follows from
\reqnarray{proof of adjacent distance larger than one-888},
$\nbf_1^{r_{h-2}}\in \Ncal_{M,k}(h-1)$ in
\reqnarray{proof of adjacent distance larger than one-(i)-333},
\reqnarray{proof of adjacent distance larger than one-(i)-888}--\reqnarray{proof of adjacent distance larger than one-(i)-bbb},
$i_a=1$, and \reqnarray{comparison rule B-1} in \rlemma{comparison rule B}(i)
(for the even integer $h-1$) that $\nbf_1^{r_{h-2}}\prec{\nbf'}_1^{r_{h-2}}$,
i.e., \reqnarray{proof of adjacent distance larger than one-(i)-444} holds.
On the other hand, if $m_a\geq 2$,
then we show that
\beqnarray{}
\alignspace
2\leq i_a\leq r_{h-2}-2,
\label{eqn:proof of adjacent distance larger than one-(i)-case-1-222}\\
\alignspace
n_{i_a-j}=n_{(i_a+1)+j}=q_{h-1}, \textrm{ for } j=1,2,\ldots,\min\{i_a-1,r_{h-2}-i_a-1\}.
\label{eqn:proof of adjacent distance larger than one-(i)-case-1-333}
\eeqnarray
An illustration of \reqnarray{proof of adjacent distance larger than one-(i)-case-1-333}
is given in \rfigure{appendix-E-(i)-case-1}.
Therefore, it follows from
\reqnarray{proof of adjacent distance larger than one-888},
$\nbf_1^{r_{h-2}}\in \Ncal_{M,k}(h-1)$ in
\reqnarray{proof of adjacent distance larger than one-(i)-333},
\reqnarray{proof of adjacent distance larger than one-(i)-888}--\reqnarray{proof of adjacent distance larger than one-(i)-bbb},
\reqnarray{proof of adjacent distance larger than one-(i)-case-1-222},
\reqnarray{proof of adjacent distance larger than one-(i)-case-1-333},
and \reqnarray{comparison rule B-4} in \rlemma{comparison rule B}(iii)
(for the even integer $h-1$) that $\nbf_1^{r_{h-2}}\prec{\nbf'}_1^{r_{h-2}}$,
i.e., \reqnarray{proof of adjacent distance larger than one-(i)-444} holds.
To prove \reqnarray{proof of adjacent distance larger than one-(i)-case-1-222},
note that from \reqnarray{proof of adjacent distance larger than one-(i)-case-1-111} and $m_a\geq 2$,
we have $i_a\geq 2$.
Thus, \reqnarray{proof of adjacent distance larger than one-(i)-case-1-222}
follows from $i_a\geq 2$ and \reqnarray{proof of adjacent distance larger than one-(i)-fff}.
To prove \reqnarray{proof of adjacent distance larger than one-(i)-case-1-333},
note that from \reqnarray{proof of adjacent distance larger than one-(i)-case-1-111},
$\sum_{\ell=1}^{r_{h-1}}m_{\ell}=r_{h-2}$ in \reqnarray{proof of adjacent distance larger than one-(i)-222},
and $m_a-m_{a+1}\leq -2$, we have
\beqnarray{proof of adjacent distance larger than one-(i)-case-1-444}
(i_a-1)-(r_{h-2}-i_a-1)
\aligneq 2i_a-r_{h-2}=2m_a-\sum_{\ell=1}^{r_{h-1}}m_{\ell} \nn\\
\alignleq 2m_a-(m_a+m_{a+1})=m_a-m_{a+1}<0.
\eeqnarray
It then follows from \reqnarray{proof of adjacent distance larger than one-(i)-case-1-444}
and \reqnarray{proof of adjacent distance larger than one-(i)-case-1-111} that
\beqnarray{proof of adjacent distance larger than one-(i)-case-1-555}
\min\{i_a-1,r_{h-2}-i_a-1\}=i_a-1=m_a-1.
\eeqnarray
From \reqnarray{proof of adjacent distance larger than one-(i)-case-1-111},
we have
\beqnarray{proof of adjacent distance larger than one-(i)-case-1-666}
i_a-(m_a-1)=1.
\eeqnarray
From \reqnarray{proof of adjacent distance larger than one-(i)-case-1-666}
and $a=1$, we immediately see that
\beqnarray{proof of adjacent distance larger than one-(i)-case-1-777}
1\leq i_a-j<i_a=i_1, \textrm{ for } j=1,2,\ldots,m_a-1.
\eeqnarray
It then follows from \reqnarray{proof of adjacent distance larger than one-444}
and \reqnarray{proof of adjacent distance larger than one-(i)-case-1-777} that
\beqnarray{proof of adjacent distance larger than one-(i)-case-1-888}
n_{i_a-j}=q_{h-1}, \textrm{ for } j=1,2,\ldots,m_a-1.
\eeqnarray
Also, we have from $m_a-m_{a+1}\leq -2$ and
$i_a+m_{a+1}=i_{a+1}$ in \reqnarray{proof of adjacent distance larger than one-(i)-ggg} that
\beqnarray{proof of adjacent distance larger than one-(i)-case-1-999}
(i_a+1)+(m_a-1)\leq i_a+m_{a+1}-2=i_{a+1}-2<i_{a+1}.
\eeqnarray
It then follows from \reqnarray{proof of adjacent distance larger than one-444}
and $i_a<(i_a+1)+(m_a-1)<i_{a+1}$ in
\reqnarray{proof of adjacent distance larger than one-(i)-case-1-999} that
\beqnarray{proof of adjacent distance larger than one-(i)-case-1-aaa}
\alignspace n_{(i_a+1)+j}=q_{h-1}, \textrm{ for } j=1,2,\ldots,m_a-1.
\eeqnarray
Thus, \reqnarray{proof of adjacent distance larger than one-(i)-case-1-333}
follows from \reqnarray{proof of adjacent distance larger than one-(i)-case-1-555},
\reqnarray{proof of adjacent distance larger than one-(i)-case-1-888},
and \reqnarray{proof of adjacent distance larger than one-(i)-case-1-aaa}.
\textbf{Remark:} It is to be noted that if $m_a-m_{a+1}=-1$
(instead of $m_a-m_{a+1}\leq -2$ as in \rlemma{adjacent distance larger than one}(i)),
then \reqnarray{proof of adjacent distance larger than one-(i)-fff}--\reqnarray{proof of adjacent distance larger than one-(i)-case-1-aaa}
still hold (as we only need $m_a-m_{a+1}\leq -1$ to prove
\reqnarray{proof of adjacent distance larger than one-(i)-fff},
\reqnarray{proof of adjacent distance larger than one-(i)-case-1-444},
and \reqnarray{proof of adjacent distance larger than one-(i)-case-1-999}),
and hence $\nbf_1^{r_{h-2}}\prec{\nbf'}_1^{r_{h-2}}$ in
\reqnarray{proof of adjacent distance larger than one-(i)-444} also holds in such a case.
As such, we conclude that if $3\leq h\leq N$ is an odd integer,
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$, $m_a-m_{a+1}=-1$,
$m'_a=m_a+1$, $m'_{a+1}=m_{a+1}-1$, and $m'_i=m_i$ for $i\neq a$ and $a+1$,
where $a=1$, then we have $\mbf_1^{r_{h-1}}\prec{\mbf'}_1^{r_{h-1}}$.
This result will be used later in the proof of Case 1 of \rlemma{comparison rule A}(i) in
\rappendix{proof of comparison rule A for an odd integer h by using comparison rule B for the even integer h-1}.
\emph{Case 2: $2\leq a\leq r_{h-1}-1$}.
First note that from $a\geq 2$ and \reqnarray{proof of adjacent distance larger than one-555},
we have
\beqnarray{proof of adjacent distance larger than one-(i)-case-2-111}
i_a=\sum_{\ell=1}^{a}m_a\geq m_1+m_a\geq 2.
\eeqnarray
As such, we see from \reqnarray{proof of adjacent distance larger than one-(i)-fff}
and \reqnarray{proof of adjacent distance larger than one-(i)-case-2-111} that
\beqnarray{proof of adjacent distance larger than one-(i)-case-2-222}
2\leq i_a\leq r_{h-2}-2.
\eeqnarray
\bpdffigure{appendix-E-i-case-2.pdf}{5.5in}
\epdffigure{appendix-E-(i)-case-2}
{(a) An illustration of \reqnarray{proof of adjacent distance larger than one-(i)-case-2-333}
for the case that $2\leq a\leq r_{h-1}-1$ and $m_a=1$
(note that in this case we have $i_{a-1}=i_a-1$
in \reqnarray{proof of adjacent distance larger than one-(i)-case-2-444}
and $i_a<(i_a+1)+1<i_{a+1}$
in \reqnarray{proof of adjacent distance larger than one-(i)-case-2-666});
(b) An illustration of \reqnarray{proof of adjacent distance larger than one-(i)-case-2-888}
and \reqnarray{proof of adjacent distance larger than one-(i)-case-2-999}
for the case that $2\leq a\leq r_{h-1}-1$ and $m_a\geq 2$
(note that in this case we have $i_{a-1}=i_a-m_a$
in \reqnarray{proof of adjacent distance larger than one-(i)-case-2-aaa}
and $i_a<(i_a+1)+m_a<i_{a+1}$
in \reqnarray{proof of adjacent distance larger than one-(i)-case-2-ddd}).}
If $m_a=1$, then we show that
\beqnarray{proof of adjacent distance larger than one-(i)-case-2-333}
n_{i_a-1}=q_{h-1}+1>n_{(i_a+1)+1}=q_{h-1}.
\eeqnarray
An illustration of \reqnarray{proof of adjacent distance larger than one-(i)-case-2-333}
is given in \rfigure{appendix-E-(i)-case-2}(a).
Therefore, it follows from
\reqnarray{proof of adjacent distance larger than one-888},
$\nbf_1^{r_{h-2}}\in \Ncal_{M,k}(h-1)$ in
\reqnarray{proof of adjacent distance larger than one-(i)-333},
\reqnarray{proof of adjacent distance larger than one-(i)-888}--\reqnarray{proof of adjacent distance larger than one-(i)-bbb},
\reqnarray{proof of adjacent distance larger than one-(i)-case-2-222},
\reqnarray{proof of adjacent distance larger than one-(i)-case-2-333},
and \reqnarray{comparison rule B-2} in \rlemma{comparison rule B}(ii)
(for the even integer $h-1$) that $\nbf_1^{r_{h-2}}\prec{\nbf'}_1^{r_{h-2}}$,
i.e., \reqnarray{proof of adjacent distance larger than one-(i)-444} holds.
To prove \reqnarray{proof of adjacent distance larger than one-(i)-case-2-333},
note that from $m_a=1$, \reqnarray{proof of adjacent distance larger than one-(i)-ggg},
and $2\leq a\leq r_{h-1}-1$, we have
\beqnarray{proof of adjacent distance larger than one-(i)-case-2-444}
i_a-1=i_a-m_a=i_{a-1}.
\eeqnarray
It follows from \reqnarray{proof of adjacent distance larger than one-444}
and $i_a-1=i_{a-1}$ in
\reqnarray{proof of adjacent distance larger than one-(i)-case-2-444} that
\beqnarray{proof of adjacent distance larger than one-(i)-case-2-555}
n_{i_a-1}=n_{i_{a-1}}=q_{h-1}+1.
\eeqnarray
Also, from $m_{a+1}\geq 3$ (as $m_a-m_{a+1}\leq -2$),
\reqnarray{proof of adjacent distance larger than one-(i)-ggg},
and $2\leq a\leq r_{h-1}-1$,
we have
\beqnarray{proof of adjacent distance larger than one-(i)-case-2-666}
i_a<(i_a+1)+1<i_a+m_{a+1}=i_{a+1}.
\eeqnarray
It follows from \reqnarray{proof of adjacent distance larger than one-444}
and $i_a<(i_a+1)+1<i_{a+1}$ in \reqnarray{proof of adjacent distance larger than one-(i)-case-2-666} that
\beqnarray{proof of adjacent distance larger than one-(i)-case-2-777}
n_{(i_a+1)+1}=q_{h-1}.
\eeqnarray
Thus, \reqnarray{proof of adjacent distance larger than one-(i)-case-2-333}
follows from \reqnarray{proof of adjacent distance larger than one-(i)-case-2-555}
and \reqnarray{proof of adjacent distance larger than one-(i)-case-2-777}.
On the other hand, if $m_a\geq 2$,
then we show that
\beqnarray{}
\alignspace n_{i_a-j}=n_{(i_a+1)+j}=q_{h-1}, \textrm{ for } j=1,2,\ldots,m_a-1,
\label{eqn:proof of adjacent distance larger than one-(i)-case-2-888} \\
\alignspace n_{i_a-m_a}=q_{h-1}+1>n_{(i_a+1)+m_a}=q_{h-1},
\label{eqn:proof of adjacent distance larger than one-(i)-case-2-999}
\eeqnarray
An illustration of \reqnarray{proof of adjacent distance larger than one-(i)-case-2-888}
and \reqnarray{proof of adjacent distance larger than one-(i)-case-2-999}
is given in \rfigure{appendix-E-(i)-case-2}(b).
Therefore, it follows from
\reqnarray{proof of adjacent distance larger than one-888},
$\nbf_1^{r_{h-2}}\in \Ncal_{M,k}(h-1)$ in
\reqnarray{proof of adjacent distance larger than one-(i)-333},
\reqnarray{proof of adjacent distance larger than one-(i)-888}--\reqnarray{proof of adjacent distance larger than one-(i)-bbb},
\reqnarray{proof of adjacent distance larger than one-(i)-case-2-222},
\reqnarray{proof of adjacent distance larger than one-(i)-case-2-888},
\reqnarray{proof of adjacent distance larger than one-(i)-case-2-999},
and \reqnarray{comparison rule B-2} in \rlemma{comparison rule B}(ii)
(for the even integer $h-1$) that $\nbf_1^{r_{h-2}}\prec{\nbf'}_1^{r_{h-2}}$,
i.e., \reqnarray{proof of adjacent distance larger than one-(i)-444} holds.
To prove \reqnarray{proof of adjacent distance larger than one-(i)-case-2-888}
and \reqnarray{proof of adjacent distance larger than one-(i)-case-2-999},
observe from \reqnarray{proof of adjacent distance larger than one-(i)-ggg}
and $2\leq a\leq r_{h-1}-1$ that
\beqnarray{proof of adjacent distance larger than one-(i)-case-2-aaa}
i_a-m_a=i_{a-1}.
\eeqnarray
It then follows from \reqnarray{proof of adjacent distance larger than one-444}
and \reqnarray{proof of adjacent distance larger than one-(i)-case-2-aaa} that
\beqnarray{}
\alignspace n_{i_a-j}=q_{h-1}, \textrm{ for } j=1,2,\ldots,m_a-1,
\label{eqn:proof of adjacent distance larger than one-(i)-case-2-bbb} \\
\alignspace n_{i_a-m_a}=n_{i_{a-1}}=q_{h-1}+1.
\label{eqn:proof of adjacent distance larger than one-(i)-case-2-ccc}
\eeqnarray
Also, from $m_a-m_{a+1}\leq -2$,
\reqnarray{proof of adjacent distance larger than one-(i)-ggg},
and $2\leq a\leq r_{h-1}-1$, we see that
\beqnarray{proof of adjacent distance larger than one-(i)-case-2-ddd}
(i_a+1)+m_a\leq i_a+m_{a+1}-1=i_{a+1}-1<i_{a+1}.
\eeqnarray
From \reqnarray{proof of adjacent distance larger than one-(i)-case-2-ddd},
we immediately see that
\beqnarray{proof of adjacent distance larger than one-(i)-case-2-eee}
i_a<(i_a+1)+j<i_{a+1}, \textrm{ for } j=1,2,\ldots,m_a.
\eeqnarray
It then follows from \reqnarray{proof of adjacent distance larger than one-444}
and \reqnarray{proof of adjacent distance larger than one-(i)-case-2-eee} that
\beqnarray{proof of adjacent distance larger than one-(i)-case-2-fff}
n_{(i_a+1)+j}=q_{h-1}, \textrm{ for } j=1,2,\ldots,m_a.
\eeqnarray
Thus, \reqnarray{proof of adjacent distance larger than one-(i)-case-2-888}
and \reqnarray{proof of adjacent distance larger than one-(i)-case-2-999}
follow from \reqnarray{proof of adjacent distance larger than one-(i)-case-2-bbb},
\reqnarray{proof of adjacent distance larger than one-(i)-case-2-ccc},
and \reqnarray{proof of adjacent distance larger than one-(i)-case-2-fff}.
(ii) Note that in \rlemma{adjacent distance larger than one}(ii),
we have $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
$m_a-m_{a+1}\geq 2$ for some $1\leq a\leq r_{h-1}-1$,
$m'_a=m_a-1$, $m'_{a+1}=m_{a+1}+1$, and $m'_i=m_i$ for $i\neq a$ and $a+1$.
It is easy to see that
\beqnarray{proof of adjacent distance larger than one-(ii)-111}
m'_a=m_a-1\geq m_{a+1}+1\geq 2,\ m'_{a+1}=m_{a+1}+1\geq 2,
\textrm{ and } m'_i=m_i \textrm{ for } i\neq a, a+1.
\eeqnarray
Also, we have from $m'_a=m_a-1$, $m'_{a+1}=m_{a+1}+1$, and $m'_i=m_i$ for $i\neq a$ and $a+1$,
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$, and \reqnarray{N-M-k-h} that
\beqnarray{proof of adjacent distance larger than one-(ii)-222}
\sum_{i=1}^{r_{h-1}}m'_i=\sum_{i=1}^{r_{h-1}}m_i=r_{h-2}.
\eeqnarray
As such, it follows from \reqnarray{proof of adjacent distance larger than one-(ii)-111},
\reqnarray{proof of adjacent distance larger than one-(ii)-222},
and \reqnarray{N-M-k-h} that ${\mbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
As $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$ and ${\mbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
we see from \reqnarray{proof of adjacent distance larger than one-111},
\reqnarray{proof of adjacent distance larger than one-222},
and the argument in the paragraph after \reqnarray{N-M-k-h} that
\beqnarray{proof of adjacent distance larger than one-(ii)-333}
\nbf_1^{r_{h-2}}\in \Ncal_{M,k}(h-1) \textrm{ and } {\nbf'}_1^{r_{h-2}}\in \Ncal_{M,k}(h-1).
\eeqnarray
To show \reqnarray{adjacent distance larger than one-2},
i.e., $\mbf_1^{r_{h-1}}\preceq {\mbf'}_1^{r_{h-1}}$,
where $\mbf_1^{r_{h-1}}\equiv {\mbf'}_1^{r_{h-1}}$ if and only if
$r_{h-1}=2$ and $m_1=m_2+2$,
we see from \reqnarray{proof of adjacent distance larger than one-333}
that it suffices to show that
\beqnarray{proof of adjacent distance larger than one-(ii)-444}
\nbf_1^{r_{h-2}}\preceq {\nbf'}_1^{r_{h-2}},
\eeqnarray
where $\nbf_1^{r_{h-2}}\equiv {\nbf'}_1^{r_{h-2}}$ if and only if
\beqnarray{proof of adjacent distance larger than one-(ii)-555}
r_{h-1}=2 \textrm{ and } m_1=m_2+2.
\eeqnarray
Note that from $m_a=m'_a+1$, $m_{a+1}=m'_{a+1}-1$, $m_i=m'_i$ for $i\neq a$ and $a+1$,
\reqnarray{proof of adjacent distance larger than one-444}--\reqnarray{proof of adjacent distance larger than one-888},
we can show as in the proof of
\reqnarray{proof of adjacent distance larger than one-(i)-666}--\reqnarray{proof of adjacent distance larger than one-(i)-bbb}
in (i) above (with the roles of $\mbf_1^{r_{h-1}}$ and ${\mbf'}_1^{r_{h-1}}$ interchanged
and the roles of $\nbf_1^{r_{h-2}}$ and ${\nbf'}_1^{r_{h-2}}$ interchanged) that
\beqnarray{}
\alignspace n_{i'_a}=q_{h-1},
\label{eqn:proof of adjacent distance larger than one-(ii)-666} \\
\alignspace n'_{i'_a+1}=q_{h-1},
\label{eqn:proof of adjacent distance larger than one-(ii)-777}\\
\alignspace n'_{i'_a}-n'_{i'_a+1}=(q_{h-1}+1)-q_{h-1}=1,
\label{eqn:proof of adjacent distance larger than one-(ii)-888}\\
\alignspace n_{i'_a}=q_{h-1}=n'_{i'_a}-1,
\label{eqn:proof of adjacent distance larger than one-(ii)-999}\\
\alignspace n_{i'_a+1}=n_{i_a}=q_{h-1}+1=n'_{i'_a+1}+1,
\label{eqn:proof of adjacent distance larger than one-(ii)-aaa}\\
\alignspace n_i=n'_i, \textrm{ for } i\neq i'_a \textrm{ and } i'_a+1.
\label{eqn:proof of adjacent distance larger than one-(ii)-bbb}
\eeqnarray
\bpdffigure{appendix-E-ii.pdf}{5.5in}
\epdffigure{appendix-E-(ii)}
{An illustration of \reqnarray{proof of adjacent distance larger than one-(ii)-eee}
and \reqnarray{proof of adjacent distance larger than one-(ii)-fff}:
(a) $a=1$
(note that in this case we have $1\leq i'_a-(m'_{a+1}-1)<i'_1$
in \reqnarray{proof of adjacent distance larger than one-(ii)-8888}
and $(i'_a+1)+(m'_{a+1}-1)=i'_{a+1}$
in \reqnarray{proof of adjacent distance larger than one-(ii)-3333});
(b) $2\leq a\leq r_{h-1}-1$
(note that in this case we have $i'_{a-1}<i'_a-(m'_{a+1}-1)<i'_a$
in \reqnarray{proof of adjacent distance larger than one-(ii)-bbbb}
and $(i'_a+1)+(m'_{a+1}-1)=i'_{a+1}$
in \reqnarray{proof of adjacent distance larger than one-(ii)-3333}).}
In the following, we show that
\beqnarray{}
\alignspace 2\leq i'_a\leq r_{h-2}-2,
\label{eqn:proof of adjacent distance larger than one-(ii)-ccc}\\
\alignspace 1\leq m'_{a+1}-1\leq \min\{i'_a-1,r_{h-2}-i'_a-1\},
\label{eqn:proof of adjacent distance larger than one-(ii)-ddd}\\
\alignspace n'_{i'_a-j'}=n'_{(i'_a+1)+j'}=q_{h-1}, \textrm{ for } j'=1,2,\ldots,m'_{a+1}-2,
\label{eqn:proof of adjacent distance larger than one-(ii)-eee}\\
\alignspace n'_{i'_a-(m'_{a+1}-1)}=q_{h-1}<n'_{(i'_a+1)+(m'_{a+1}-1)}=q_{h-1}+1.
\label{eqn:proof of adjacent distance larger than one-(ii)-fff}
\eeqnarray
An illustration of \reqnarray{proof of adjacent distance larger than one-(ii)-eee}
and \reqnarray{proof of adjacent distance larger than one-(ii)-fff}
is given in \rfigure{appendix-E-(ii)}.
Therefore, it follows from
\reqnarray{proof of adjacent distance larger than one-888},
${\nbf'}_1^{r_{h-2}}\in \Ncal_{M,k}(h-1)$ in
\reqnarray{proof of adjacent distance larger than one-(ii)-333},
\reqnarray{proof of adjacent distance larger than one-(ii)-888}--\reqnarray{proof of adjacent distance larger than one-(ii)-fff},
and \reqnarray{comparison rule B-3} in \rlemma{comparison rule B}(ii)
(for the even integer $h-1$) that ${\nbf'}_1^{r_{h-2}}\succeq\nbf_1^{r_{h-2}}$,
where ${\nbf'}_1^{r_{h-2}}\equiv\nbf_1^{r_{h-2}}$ if and only if
\beqnarray{proof of adjacent distance larger than one-(ii)-ggg}
i'_a-(m'_{a+1}-1)=1,\ (i'_a+1)+(m'_{a+1}-1)=r_{h-2}, \textrm{ and } n'_1=n'_{r_{h-2}}-1.
\eeqnarray
To prove \reqnarray{proof of adjacent distance larger than one-(ii)-ccc}--\reqnarray{proof of adjacent distance larger than one-(ii)-fff},
note that from \reqnarray{proof of adjacent distance larger than one-777},
$m'_a=m_a-1$,
$m_a-m_{a+1}\geq 2$,
and $m'_{a+1}=m_{a+1}+1$,
we have
\beqnarray{proof of adjacent distance larger than one-(ii)-1111}
i'_a
=\sum_{\ell=1}^{a}m'_{\ell}\geq m'_a
=m_a-1 \geq m_{a+1}+1=m'_{a+1},
\eeqnarray
where the first inequality holds with equality if and only if $a=1$
and the second inequality holds with equality if and only if $m_a-m_{a+1}=2$.
Also, we have from \reqnarray{proof of adjacent distance larger than one-777},
$1\leq a\leq r_{h-1}-1$,
and \reqnarray{proof of adjacent distance larger than one-(ii)-222} that
\beqnarray{proof of adjacent distance larger than one-(ii)-2222}
i'_a
=\sum_{\ell=1}^{a}m'_{\ell}
=\sum_{\ell=1}^{r_{h-1}}m'_{\ell}-\sum_{\ell=a+1}^{r_{h-1}}m'_{\ell}
\leq r_{h-2}-m'_{a+1}.
\eeqnarray
Thus, \reqnarray{proof of adjacent distance larger than one-(ii)-ccc}
follows from $i'_a\geq m'_{a+1}$ in \reqnarray{proof of adjacent distance larger than one-(ii)-1111},
$i'_a\leq r_{h-2}-m'_{a+1}$ in \reqnarray{proof of adjacent distance larger than one-(ii)-2222},
and $m'_{a+1}\geq 2$ in \reqnarray{proof of adjacent distance larger than one-(ii)-111},
and \reqnarray{proof of adjacent distance larger than one-(ii)-ddd}
follows from $m'_{a+1}\leq i'_a$ in \reqnarray{proof of adjacent distance larger than one-(ii)-1111}
and $m'_{a+1}\leq r_{h-2}-i_a$ in \reqnarray{proof of adjacent distance larger than one-(ii)-2222}.
From \reqnarray{proof of adjacent distance larger than one-777} we see that
\beqnarray{proof of adjacent distance larger than one-(ii)-3333}
i'_j=\sum_{\ell=1}^{j}m'_{\ell}
=\sum_{\ell=1}^{j-1}m'_{\ell}+m'_j
=i'_{j-1}+m'_j, \textrm{ for } j=2,3,\ldots,r_{h-1}.
\eeqnarray
From $i'_a+m'_{a+1}=i'_{a+1}$ in \reqnarray{proof of adjacent distance larger than one-(ii)-3333},
we immediately see that
\beqnarray{proof of adjacent distance larger than one-(ii)-4444}
i'_a<(i'_a+1)+j'<i'_{a+1}, \textrm{ for } j'=1,2,\ldots,m'_{a+1}-2.
\eeqnarray
It then follows from \reqnarray{proof of adjacent distance larger than one-666},
\reqnarray{proof of adjacent distance larger than one-(ii)-4444},
and $i'_a+m'_{a+1}=i'_{a+1}$ in \reqnarray{proof of adjacent distance larger than one-(ii)-3333} that
\beqnarray{}
\alignspace n'_{(i'_a+1)+j'}=q_{h-1}, \textrm{ for } j'=1,2,\ldots,m'_{a+1}-2,
\label{eqn:proof of adjacent distance larger than one-(ii)-5555}\\
\alignspace n'_{(i'_a+1)+(m'_{a+1}-1)}=n'_{i'_{a+1}}=q_{h-1}+1.
\label{eqn:proof of adjacent distance larger than one-(ii)-6666}
\eeqnarray
If $a=1$,
then we see from \reqnarray{proof of adjacent distance larger than one-(ii)-1111} that
\beqnarray{proof of adjacent distance larger than one-(ii)-7777}
i'_a-(m'_{a+1}-1)\geq 1,
\eeqnarray
where the inequality holds if and only if $a=1$ and $m_a-m_{a+1}=2$.
It then follows from $a=1$ and \reqnarray{proof of adjacent distance larger than one-(ii)-7777} that
\beqnarray{proof of adjacent distance larger than one-(ii)-8888}
1\leq i'_a-j'<i'_a=i'_1, \textrm{ for } j'=1,2,\ldots,m'_{a+1}-1.
\eeqnarray
From \reqnarray{proof of adjacent distance larger than one-666} and \reqnarray{proof of adjacent distance larger than one-(ii)-8888}
we have
\beqnarray{proof of adjacent distance larger than one-(ii)-9999}
n'_{i'_a-j'}=q_{h-1}, \textrm{ for } j'=1,2,\ldots,m'_{a+1}-1.
\eeqnarray
Thus, \reqnarray{proof of adjacent distance larger than one-(ii)-eee}
and \reqnarray{proof of adjacent distance larger than one-(ii)-fff}
follow from \reqnarray{proof of adjacent distance larger than one-(ii)-5555},
\reqnarray{proof of adjacent distance larger than one-(ii)-6666},
and \reqnarray{proof of adjacent distance larger than one-(ii)-9999}.
On the other hand, if $2\leq a\leq r_{h-1}-1$,
then from $m'_a=m_a-1\geq (m_{a+1}+2)-1=m'_{a+1}$,
and \reqnarray{proof of adjacent distance larger than one-(ii)-3333} we have
\beqnarray{proof of adjacent distance larger than one-(ii)-aaaa}
i'_a-(m'_{a+1}-1)\geq i'_a-(m'_a-1)=i'_{a-1}+1>i'_{a-1}.
\eeqnarray
From \reqnarray{proof of adjacent distance larger than one-(ii)-aaaa},
we immediately see that
\beqnarray{proof of adjacent distance larger than one-(ii)-bbbb}
i'_{a-1}<i'_a-j'<i'_a, \textrm{ for } j'=1,2,\ldots,m'_{a+1}-1.
\eeqnarray
It then follows from \reqnarray{proof of adjacent distance larger than one-666}
and \reqnarray{proof of adjacent distance larger than one-(ii)-bbbb} that
\beqnarray{proof of adjacent distance larger than one-(ii)-cccc}
n'_{i'_a-j'}=q_{h-1}, \textrm{ for } j'=1,2,\ldots,m'_{a+1}-1.
\eeqnarray
As such, \reqnarray{proof of adjacent distance larger than one-(ii)-eee}
and \reqnarray{proof of adjacent distance larger than one-(ii)-fff}
follow from \reqnarray{proof of adjacent distance larger than one-(ii)-5555},
\reqnarray{proof of adjacent distance larger than one-(ii)-6666},
and \reqnarray{proof of adjacent distance larger than one-(ii)-cccc}.
To complete the proof,
we need to show that the condition in \reqnarray{proof of adjacent distance larger than one-(ii)-ggg}
is equivalent to the condition in \reqnarray{proof of adjacent distance larger than one-(ii)-555}.
Note that if $i'_a-(m'_{a+1}-1)=1$ and $(i'_a+1)+(m'_{a+1}-1)=r_{h-2}$,
then we have from $n'_1=n'_{i'_a-(m'_{a+1}-1)}=q_{h-1}$ and $n'_{r_{h-2}}=n'_{(i'_a+1)+(m'_{a+1}-1)}=q_{h-1}+1$
in \reqnarray{proof of adjacent distance larger than one-(ii)-fff} that
\beqnarray{}
n'_1=n'_{r_{h-2}}-1.\nn
\eeqnarray
As such, we see that the condition in \reqnarray{proof of adjacent distance larger than one-(ii)-ggg}
is equivalent to the following condition:
\beqnarray{proof of adjacent distance larger than one-(ii)-dddd}
i'_a-(m'_{a+1}-1)=1 \textrm{ and } (i'_a+1)+(m'_{a+1}-1)=r_{h-2}.
\eeqnarray
From \reqnarray{proof of adjacent distance larger than one-(ii)-7777}, we see that
\beqnarray{proof of adjacent distance larger than one-(ii)-eeee}
i'_a-(m'_{a+1}-1)=1 \textrm{ iff } a=1 \textrm{ and } m_a=m_{a+1}+2.
\eeqnarray
As we have $i'_a+m'_{a+1}=i'_{a+1}$ in \reqnarray{proof of adjacent distance larger than one-(ii)-3333}
and it is clear from \reqnarray{proof of adjacent distance larger than one-777} and \reqnarray{proof of adjacent distance larger than one-(ii)-222}
that $i'_{a+1}=r_{h-2}$ if and only if $a+1=r_{h-1}$,
it follows that
\beqnarray{proof of adjacent distance larger than one-(ii)-ffff}
(i'_a+1)+(m'_{a+1}-1)=r_{h-2} \textrm{ iff } a+1=r_{h-1}.
\eeqnarray
From \reqnarray{proof of adjacent distance larger than one-(ii)-eeee}
and \reqnarray{proof of adjacent distance larger than one-(ii)-ffff},
we deduce that the condition in \reqnarray{proof of adjacent distance larger than one-(ii)-dddd}
is equivalent to the following condition:
\beqnarray{proof of adjacent distance larger than one-(ii)-gggg}
a=1,\ a+1=r_{h-1}, \textrm{ and } m_a=m_{a+1}+2.
\eeqnarray
It is clear that if $a=1$, $a+1=r_{h-1}$, and $m_a=m_{a+1}+2$,
then we have $r_{h-1}=2$ and $m_1=m_2+2$.
Conversely, if $r_{h-1}=2$ and $m_1=m_2+2$,
then it follows from $1\leq a\leq r_{h-1}-1$ that $a=1$
and hence we have $a=1$, $a+1=r_{h-1}$, and $m_a=m_{a+1}+2$.
Therefore, the condition in \reqnarray{proof of adjacent distance larger than one-(ii)-gggg}
is equivalent to the condition that $r_{h-1}=2$ and $m_1=m_2+2$
in \reqnarray{proof of adjacent distance larger than one-(ii)-555},
and the proof is completed.
\bappendix{Proof of Comparison rule A in \rlemma{comparison rule A} for an odd integer $3\leq h\leq N$
by using Comparison rule B in \rlemma{comparison rule B} for the even integer $h-1$}
{proof of comparison rule A for an odd integer h by using comparison rule B for the even integer h-1}
In this appendix, we assume that Comparison rule B in \rlemma{comparison rule B} holds
for some even integer $h-1$, where $2\leq h-1\leq N-1$,
and show that Comparison rule A in \rlemma{comparison rule A} holds for the odd integer $h$.
Let
\beqnarray{}
\nbf_1^{r_{h-2}}(h-1)\aligneq R_{r_{h-3},r_{h-2}}(\nbf_1^{r_{h-1}}(h)),
\label{eqn:proof of comparison rule A-111} \\
{\nbf'}_1^{r_{h-2}}(h-1)\aligneq R_{r_{h-3},r_{h-2}}({\nbf'}_1^{r_{h-1}}(h)).
\label{eqn:proof of comparison rule A-222}
\eeqnarray
For simplicity, let $\mbf_1^{r_{h-1}}=\nbf_1^{r_{h-1}}(h)$,
${\mbf'}_1^{r_{h-1}}={\nbf'}_1^{r_{h-1}}(h)$,
$\nbf_1^{r_{h-2}}=\nbf_1^{r_{h-2}}(h-1)$,
and ${\nbf'}_1^{r_{h-2}}={\nbf'}_1^{r_{h-2}}(h-1)$.
Then \reqnarray{proof of adjacent distance larger than one-333}--\reqnarray{proof of adjacent distance larger than one-777}
in \rappendix{proof of adjacent distance larger than one for an odd integer h by using comparison rule B for the even integer h-1} still hold.
It follows from \reqnarray{proof of adjacent distance larger than one-777} that
\reqnarray{proof of adjacent distance larger than one-(ii)-3333}
in \rappendix{proof of adjacent distance larger than one for an odd integer h by using comparison rule B for the even integer h-1} also holds.
Note that in \rlemma{comparison rule A},
we have $r_{h-1}\geq 2$, $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
$m_a-m_{a+1}=1$ for some $1\leq a\leq r_{h-1}-1$,
$m'_a=m_a-1$, $m'_{a+1}=m_{a+1}+1$, and $m'_i=m_i$ for $i\neq a$ and $a+1$.
As $r_{h-1}\geq 2$,
we see that \reqnarray{proof of adjacent distance larger than one-888}
in \rappendix{proof of adjacent distance larger than one for an odd integer h by using comparison rule B for the even integer h-1} also holds.
It is easy to see that
\beqnarray{proof of comparison rule A-333}
m'_a=m_a-1=m_{a+1}\geq 1,\ m'_{a+1}=m_{a+1}+1\geq 2,
\textrm{ and } m'_i=m_i \textrm{ for } i\neq a, a+1.
\eeqnarray
From $m'_a=m_a-1$, $m'_{a+1}=m_{a+1}+1$, and $m'_i=m_i$ for $i\neq a$ and $a+1$,
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$, and \reqnarray{N-M-k-h},
we can see that \reqnarray{proof of adjacent distance larger than one-(ii)-222}
in \rappendix{proof of adjacent distance larger than one for an odd integer h by using comparison rule B for the even integer h-1} also holds.
As such, it follows from \reqnarray{proof of comparison rule A-333},
\reqnarray{proof of adjacent distance larger than one-(ii)-222},
$2\leq h\leq N$,
and \reqnarray{N-M-k-h} that ${\mbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
Note that from $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$, ${\mbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of comparison rule A-111}, and \reqnarray{proof of comparison rule A-222},
we can see that \reqnarray{proof of adjacent distance larger than one-(ii)-333}
in \rappendix{proof of adjacent distance larger than one for an odd integer h by using comparison rule B for the even integer h-1} also holds.
Furthermore, since ${\mbf'}_1^{r_{h-1}}$ is obtained from $\mbf_1^{r_{h-1}}$
in exactly the same way as that in \rlemma{adjacent distance larger than one}(ii),
it is clear that \reqnarray{proof of adjacent distance larger than one-(ii)-666}--\reqnarray{proof of adjacent distance larger than one-(ii)-bbb} also hold.
We also note that from $m'_a=m_a-1$, $m'_{a+1}=m_{a+1}+1$, and $m_a-m_{a+1}=1$, we have
\beqnarray{proof of comparison rule A-444}
m'_a-m'_{a+1}=(m_a-1)-(m_{a+1}+1)=m_a-m_{a+1}-2=-1.
\eeqnarray
(i) Note that in \rlemma{comparison rule A}(i), we have $a=1$ or $a=r_{h-1}-1$.
To show \reqnarray{comparison rule A-1},
i.e., $\mbf_1^{r_{h-1}}\succ {\mbf'}_1^{r_{h-1}}$,
we see from \reqnarray{proof of adjacent distance larger than one-333}
that it suffices to show that
\beqnarray{proof of comparison rule A-(i)-111}
\nbf_1^{r_{h-2}}\succ{\nbf'}_1^{r_{h-2}}.
\eeqnarray
We consider the two cases $a=1$ and $a=r_{h-1}-1\neq 1$ separately.
\emph{Case 1: $a=1$.}
As $3\leq h\leq N$ is an odd integer, ${\mbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
$m'_a-m'_{a+1}=-1$ in \reqnarray{proof of comparison rule A-444},
$m_a=m'_a+1$, $m_{a+1}=m'_{a+1}-1$, and $m_i=m'_i$ for $i\neq a$ and $a+1$,
where we have $a=1$ in this case,
it then follows from the remark after \reqnarray{proof of adjacent distance larger than one-(i)-case-1-aaa}
in \rappendix{proof of adjacent distance larger than one for an odd integer h by using comparison rule B for the even integer h-1}
(with the roles of $\mbf_1^{r_{h-1}}$ and ${\mbf'}_1^{r_{h-1}}$ interchanged)
that ${\mbf'}_1^{r_{h-1}}\prec\mbf_1^{r_{h-1}}$.
\emph{Case 2: $a=r_{h-1}-1\neq 1$.}
\bpdffigure{appendix-F-i-case-2.pdf}{3.5in}
\epdffigure{appendix-F-(i)-case-2}
{An illustration of \reqnarray{proof of comparison rule A-(i)-case-2-222}
in the case that $a=r_{h-1}-1\neq 1$
(note that in this case we have
$\min\{i'_{a+1}-1,r_{h-2}-i'_{a+1}-1\}=m'_a$
in \reqnarray{proof of comparison rule A-(i)-case-2-666},
$i'_a-m'_a=i'_{a-1}$ in \reqnarray{proof of adjacent distance larger than one-(ii)-3333},
and $(i'_a+1)+m'_a=i'_{a+1}$ in \reqnarray{proof of comparison rule A-(i)-case-2-999}).}
In this case, we have $a=r_{h-1}-1\geq 2$.
In the following, we will show that
\beqnarray{}
\alignspace 2\leq i'_a\leq r_{h-2}-2,
\label{eqn:proof of comparison rule A-(i)-case-2-111} \\
\alignspace n'_{i'_a-j'}=n'_{(i'_a+1)+j'}, \textrm{ for } j'=1,2,\ldots,\min\{i'_a-1,r_{h-2}-i'_a-1\}.
\label{eqn:proof of comparison rule A-(i)-case-2-222}
\eeqnarray
An illustration of \reqnarray{proof of comparison rule A-(i)-case-2-222}
is given in \rfigure{appendix-F-(i)-case-2}.
Therefore, it follows from
\reqnarray{proof of adjacent distance larger than one-888},
${\nbf'}_1^{r_{h-2}}\in \Ncal_{M,k}(h-1)$ in
\reqnarray{proof of adjacent distance larger than one-(ii)-333},
\reqnarray{proof of adjacent distance larger than one-(ii)-888}--\reqnarray{proof of adjacent distance larger than one-(ii)-bbb},
\reqnarray{proof of comparison rule A-(i)-case-2-111},
\reqnarray{proof of comparison rule A-(i)-case-2-222},
and \reqnarray{comparison rule B-4} in \rlemma{comparison rule B}(iii)
(for the even integer $h-1$) that ${\nbf'}_1^{r_{h-2}}\prec\nbf_1^{r_{h-2}}$,
i.e., \reqnarray{proof of comparison rule A-(i)-111} holds.
To prove \reqnarray{proof of comparison rule A-(i)-case-2-111},
note that from \reqnarray{proof of adjacent distance larger than one-777} and $a\geq 2$,
we have
\beqnarray{proof of comparison rule A-(i)-case-2-333}
i'_a=\sum_{\ell=1}^{a}m'_{\ell}\geq m'_1+m'_a.
\eeqnarray
Also, from $a=r_{h-1}-1$, \reqnarray{proof of adjacent distance larger than one-777},
$\sum_{\ell=1}^{r_{h-1}}m'_{\ell}=r_{h-2}$
in \reqnarray{proof of adjacent distance larger than one-(ii)-222},
and $m'_a-m'_{a+1}=-1$ in \reqnarray{proof of comparison rule A-444},
we have
\beqnarray{proof of comparison rule A-(i)-case-2-444}
i'_a=i'_{r_{h-1}-1}=\sum_{\ell=1}^{r_{h-1}-1}m'_{\ell}
=\sum_{\ell=1}^{r_{h-1}}m'_{\ell}-m'_{r_{h-1}}
=r_{h-2}-m'_{a+1}=r_{h-2}-m'_a-1.
\eeqnarray
As such, \reqnarray{proof of comparison rule A-(i)-case-2-111} follows from
$i'_a\geq m'_1+m'_a\geq 2$ in \reqnarray{proof of comparison rule A-(i)-case-2-333}
and $i'_a\leq r_{h-2}-m'_a-1\leq r_{h-2}-2$ in \reqnarray{proof of comparison rule A-(i)-case-2-444}.
To prove \reqnarray{proof of comparison rule A-(i)-case-2-222},
note that from \reqnarray{proof of comparison rule A-(i)-case-2-444}
and \reqnarray{proof of comparison rule A-(i)-case-2-333},
we have
\beqnarray{proof of comparison rule A-(i)-case-2-555}
r_{h-2}-i'_a-1=m'_a\leq i'_a-m'_1\leq i'_a-1.
\eeqnarray
It is clear from \reqnarray{proof of comparison rule A-(i)-case-2-555} that
\beqnarray{proof of comparison rule A-(i)-case-2-666}
\min\{i'_a-1,r_{h-2}-i'_a-1\}=r_{h-2}-i'_a-1=m'_a.
\eeqnarray
From \reqnarray{proof of adjacent distance larger than one-666}
and $i'_a-m'_a=i'_{a-1}$ in \reqnarray{proof of adjacent distance larger than one-(ii)-3333}
(note that $a\geq 2$ in this case), we immediately see that
\beqnarray{}
\alignspace n'_{i'_a-j'}=q_{h-1}, \textrm{ for } j'=1,2,\ldots,m'_a-1,
\label{eqn:proof of comparison rule A-(i)-case-2-777} \\
\alignspace n'_{i'_a-m'_a}=n'_{i'_{a-1}}=q_{h-1}+1.
\label{eqn:proof of comparison rule A-(i)-case-2-888}
\eeqnarray
Also, from $m'_a-m'_{a+1}=-1$ in \reqnarray{proof of comparison rule A-444}
and \reqnarray{proof of adjacent distance larger than one-(ii)-3333},
we have
\beqnarray{proof of comparison rule A-(i)-case-2-999}
(i'_a+1)+m'_a=i'_a+m'_{a+1}=i'_{a+1}.
\eeqnarray
It then follows from \reqnarray{proof of adjacent distance larger than one-666}
and \reqnarray{proof of comparison rule A-(i)-case-2-999} that
\beqnarray{}
\alignspace n'_{(i'_a+1)+j'}=q_{h-1}, \textrm{ for } j'=1,2,\ldots,m'_a-1,
\label{eqn:proof of comparison rule A-(i)-case-2-aaa} \\
\alignspace n'_{(i'_a+1)+m'_a}=n'_{i'_{a+1}}=q_{h-1}+1.
\label{eqn:proof of comparison rule A-(i)-case-2-bbb}
\eeqnarray
Thus, \reqnarray{proof of comparison rule A-(i)-case-2-222} follows from
\reqnarray{proof of comparison rule A-(i)-case-2-666}--\reqnarray{proof of comparison rule A-(i)-case-2-888} and
\reqnarray{proof of comparison rule A-(i)-case-2-aaa}--\reqnarray{proof of comparison rule A-(i)-case-2-bbb}.
(ii) Note that in \rlemma{comparison rule A}(ii),
we have $2\leq a\leq r_{h-1}-2$ and there exists a positive integer $j$ such that
$1\leq j\leq \min\{a-1,r_{h-1}-a-1\}$, $m_{a-\ell}=m_{a+1+\ell}$ for $\ell=1,2,\ldots,j-1$, and $m_{a-j}\neq m_{a+1+j}$.
As $a\geq 2$, it is clear that \reqnarray{proof of comparison rule A-(i)-case-2-333} holds.
As $a\leq r_{h-1}-2$, we see from \reqnarray{proof of adjacent distance larger than one-777}
and $\sum_{\ell=1}^{r_{h-1}}m'_{\ell}=r_{h-2}$
in \reqnarray{proof of adjacent distance larger than one-(ii)-222} that
\beqnarray{proof of comparison rule A-(ii)-111}
i'_a=\sum_{\ell=1}^{a}m'_{\ell}
=\sum_{\ell=1}^{r_{h-1}}m'_{\ell}-\sum_{\ell=a+1}^{r_{h-1}}m'_{\ell}
\leq r_{h-2}-m'_{a+1}-m'_{r_{h-1}}.
\eeqnarray
It follows from
$i'_a\geq m'_1+m'_a\geq 2$ in \reqnarray{proof of comparison rule A-(i)-case-2-333}
and $i'_a\leq r_{h-2}-m'_{a+1}-m'_{r_{h-1}}\leq r_{h-2}-2$ in \reqnarray{proof of comparison rule A-(i)-case-2-444} that
\beqnarray{proof of comparison rule A-(ii)-222}
2\leq i'_a\leq r_{h-2}-2.
\eeqnarray
Furthermore, from $m'_i=m_i$ for $i\neq a$ and $a+1$
and $m_{a-\ell}=m_{a+1+\ell}$ for $\ell=1,2,\ldots,j-1$,
it is clear that
\beqnarray{proof of comparison rule A-(ii)-333}
m'_{a-\ell}=m'_{a+1+\ell}, \textrm{ for } \ell=1,2,\ldots,j-1.
\eeqnarray
\bpdffigure{appendix-F-ii.pdf}{6.0in}
\epdffigure{appendix-F-(ii)}
{An illustration of \reqnarray{proof of comparison rule A-(ii)-444}
(note that we have
$i'_a-\sum_{\ell=0}^{j''-1}m'_{a-\ell}=i'_{a-j''}$
for $1\leq j''\leq a-1$ in \reqnarray{proof of comparison rule A-(ii)-555},
$(i'_a+1)+\sum_{\ell=0}^{j''-1}m'_{a-\ell}=i'_{a+j''}$
for $1\leq j''\leq j$ in \reqnarray{proof of comparison rule A-(ii)-888},
and $j\leq \min\{a-1,r_{h-1}-a-1\}\leq a-1$).}
By using \reqnarray{proof of comparison rule A-(ii)-333},
we can show that
\beqnarray{proof of comparison rule A-(ii)-444}
n'_{i'_a-j'}=n'_{(i'_a+1)+j'}, \textrm{ for } j'=1,2,\ldots,\sum_{\ell=0}^{j-1}m'_{a-\ell}.
\eeqnarray
An illustration of \reqnarray{proof of comparison rule A-(ii)-444}
is given in \rfigure{appendix-F-(ii)}.
To prove \reqnarray{proof of comparison rule A-(ii)-444},
observe that for $1\leq j''\leq a-1$,
we have from \reqnarray{proof of adjacent distance larger than one-777} that
\beqnarray{proof of comparison rule A-(ii)-555}
i'_a-\sum_{\ell=0}^{j''-1}m'_{a-\ell}
=\sum_{\ell=1}^{a}m'_{\ell}-\sum_{\ell=a-j''+1}^{a}m'_{\ell}
=\sum_{\ell=1}^{a-j''}m'_{\ell}=i'_{a-j''}.
\eeqnarray
It follows from \reqnarray{proof of adjacent distance larger than one-666}
and \reqnarray{proof of comparison rule A-(ii)-555} that
\beqnarray{proof of comparison rule A-(ii)-666}
n'_{i'_a-j'}=
\bselection
q_{h-1}+1, &\textrm{for } j'=m'_a, \sum_{\ell=0}^{1}m'_{a-\ell},\ldots,\sum_{\ell=0}^{a-2}m'_{a-\ell}, \\
q_{h-1}, &\textrm{for } 1\leq j'\leq \sum_{\ell=0}^{a-2}m'_{a-\ell} \\
&\textrm{and } j'\neq m'_a, \sum_{\ell=0}^{1}m'_{a-\ell},\ldots,\sum_{\ell=0}^{a-2}m'_{a-\ell}.
\eselection
\eeqnarray
Furthermore, for $1\leq j''\leq j$,
we have from $m'_a-m'_{a+1}=-1$ in \reqnarray{proof of comparison rule A-444}
and $m'_{a-\ell}=m'_{a+1+\ell}$ for $\ell=1,2,\ldots,j''-1$
in \reqnarray{proof of comparison rule A-(ii)-333} that
\beqnarray{proof of comparison rule A-(ii)-777}
\sum_{\ell=0}^{j''-1}m'_{a-\ell}
=m'_a+\sum_{\ell=1}^{j''-1}m'_{a-\ell}
=m'_{a+1}-1+\sum_{\ell=1}^{j''-1}m'_{a+1+\ell}
=\sum_{\ell=0}^{j''-1}m'_{a+1+\ell}-1.
\eeqnarray
As such, for $1\leq j''\leq j$,
we have from \reqnarray{proof of adjacent distance larger than one-777}
and \reqnarray{proof of comparison rule A-(ii)-777} that
\beqnarray{proof of comparison rule A-(ii)-888}
(i'_a+1)+\sum_{\ell=0}^{j''-1}m'_{a-\ell}
=\sum_{\ell=1}^{a}m'_{\ell}+\sum_{\ell=0}^{j''-1}m'_{a+1+\ell}
=\sum_{\ell=1}^{a+j''}m'_{\ell}
= i'_{a+j''}.
\eeqnarray
It then follows from \reqnarray{proof of adjacent distance larger than one-666}
and \reqnarray{proof of comparison rule A-(ii)-888} that
\beqnarray{proof of comparison rule A-(ii)-999}
n'_{(i'_a+1)+j'}=
\bselection
q_{h-1}+1, &\textrm{for } j'=m'_a, \sum_{\ell=0}^{1}m'_{a-\ell},\ldots,\sum_{\ell=0}^{j-1}m'_{a-\ell}, \\
q_{h-1}, &\textrm{for } 1\leq j'\leq \sum_{\ell=0}^{j-1}m'_{a-\ell} \\
&\textrm{and } j'\neq m'_a, \sum_{\ell=0}^{1}m'_{a-\ell},\ldots,\sum_{\ell=0}^{j-1}m'_{a-\ell}.
\eselection
\eeqnarray
As we have $j\leq \min\{a-1,r_{h-1}-a-1\}\leq a-1$,
it is clear that \reqnarray{proof of comparison rule A-(ii)-444}
follows from \reqnarray{proof of comparison rule A-(ii)-666}
and \reqnarray{proof of comparison rule A-(ii)-999}.
From \reqnarray{proof of adjacent distance larger than one-333},
we see that:
(a) If $m_{a-j}<m_{a+1+j}$,
then to show \reqnarray{comparison rule A-2},
i.e., $\mbf_1^{r_{h-1}}\succ{\mbf'}_1^{r_{h-1}}$,
it suffices to show that $\nbf_1^{r_{h-2}}\succ{\nbf'}_1^{r_{h-2}}$;
(b) If $m_{a-j}>m_{a+1+j}$,
then to show \reqnarray{comparison rule A-3},
i.e., $\mbf_1^{r_{h-1}}\preceq {\mbf'}_1^{r_{h-1}}$,
where $\mbf_1^{r_{h-1}}\equiv {\mbf'}_1^{r_{h-1}}$
if and only if $a-j=1$, $a+1+j=r_{h-1}$, and $m_1=m_{r_{h-1}}+1$,
it suffices to show that $\nbf_1^{r_{h-2}}\preceq{\nbf'}_1^{r_{h-2}}$,
where $\nbf_1^{r_{h-2}}\equiv{\nbf'}_1^{r_{h-2}}$
if and only if $a-j=1$, $a+1+j=r_{h-1}$, and $m_1=m_{r_{h-1}}+1$.
(a) First we assume that $m_{a-j}<m_{a+1+j}$ and show that
\beqnarray{proof of comparison rule A-(ii)-(a)-111}
\nbf_1^{r_{h-2}}\succ{\nbf'}_1^{r_{h-2}}.
\eeqnarray
As $j\leq \min\{a-1,r_{h-1}-a-1\}$, we have $j+1\leq a\leq r_{h-1}-j-1$.
We consider the two cases $a=j+1$ and $j+2\leq a\leq r_{h-1}-j-1$ separately.
\bpdffigure{appendix-F-ii-a.pdf}{6.0in}
\epdffigure{appendix-F-(ii)-(a)}
{(a) An illustration of \reqnarray{proof of comparison rule A-(ii)-(a)-case-1-111}
in the case that $m_{a-j}<m_{a+1+j}$ and $a=j+1$
(note that in this case we have
$i'_a-\sum_{\ell=0}^{j''-1}m'_{a-\ell}=i'_{a-j''}$
for $1\leq j''\leq a-1=j$ in \reqnarray{proof of comparison rule A-(ii)-555},
$i'_a-(\sum_{\ell=0}^{j}m'_{a-\ell}-1)=1$
in \reqnarray{proof of comparison rule A-(ii)-(a)-case-1-555},
$(i'_a+1)+\sum_{\ell=0}^{j''-1}m'_{a-\ell}=i'_{a+j''}$
for $1\leq j''\leq j$ in \reqnarray{proof of comparison rule A-(ii)-888},
and $(i'_a+1)+(\sum_{\ell=0}^{j}m'_{a-\ell}-1)<i'_{a+1+j}$
in \reqnarray{proof of comparison rule A-(ii)-(a)-case-1-888}.
(b) An illustration of \reqnarray{proof of comparison rule A-(ii)-(a)-case-2-222}
and \reqnarray{proof of comparison rule A-(ii)-(a)-case-2-333}
in the case that $m_{a-j}<m_{a+1+j}$ and $j+2\leq a\leq r_{h-1}-j-1$
(note that in this case we have $j+1\leq a-1$,
$i'_a-\sum_{\ell=0}^{j''-1}m'_{a-\ell}=i'_{a-j''}$
for $1\leq j''\leq a-1$ in \reqnarray{proof of comparison rule A-(ii)-555},
$(i'_a+1)+\sum_{\ell=0}^{j''-1}m'_{a-\ell}=i'_{a+j''}$
for $1\leq j''\leq j$ in \reqnarray{proof of comparison rule A-(ii)-888},
and $(i'_a+1)+\sum_{\ell=0}^{j}m'_{a-\ell}<i'_{a+1+j}$
in \reqnarray{proof of comparison rule A-(ii)-(a)-case-1-888}.}
\emph{Case 1: $a=j+1$.}
In this case, we show that
\beqnarray{proof of comparison rule A-(ii)-(a)-case-1-111}
n'_{i'_a-j'}=n'_{(i'_a+1)+j'}, \textrm{ for } j'=1,2,\ldots,\min\{i'_a-1,r_{h-2}-i'_a-1\}.
\eeqnarray
An illustration of \reqnarray{proof of comparison rule A-(ii)-(a)-case-1-111}
is given in \rfigure{appendix-F-(ii)-(a)}(a).
Therefore, it follows from \reqnarray{proof of adjacent distance larger than one-888},
${\nbf'}_1^{r_{h-2}}\in \Ncal_{M,k}(h-1)$ in
\reqnarray{proof of adjacent distance larger than one-(ii)-333},
\reqnarray{proof of adjacent distance larger than one-(ii)-888}--\reqnarray{proof of adjacent distance larger than one-(ii)-bbb},
\reqnarray{proof of comparison rule A-(ii)-222},
\reqnarray{proof of comparison rule A-(ii)-(a)-case-1-111},
and \reqnarray{comparison rule B-4} in \rlemma{comparison rule B}(iii)
(for the even integer $h-1$) that ${\nbf'}_1^{r_{h-2}}\prec\nbf_1^{r_{h-2}}$,
i.e., \reqnarray{proof of comparison rule A-(ii)-(a)-111} holds.
To prove \reqnarray{proof of comparison rule A-(ii)-(a)-case-1-111},
note that from \reqnarray{proof of adjacent distance larger than one-777},
$\sum_{\ell=1}^{r_{h-1}}m'_{\ell}=r_{h-2}$ in \reqnarray{proof of adjacent distance larger than one-(ii)-222},
$a\leq r_{h-1}-j-1$, $a=j+1$,
\reqnarray{proof of comparison rule A-444} ,
\reqnarray{proof of comparison rule A-(ii)-333},
and $m'_{a-j}=m_{a-j}<m_{a+1+j}=m'_{a+1+j}$, we have
\beqnarray{proof of comparison rule A-(ii)-(a)-case-1-222}
(i'_a-1)-(r_{h-2}-i'_a-1)
\aligneq 2i'_a-r_{h-2}=2\sum_{\ell=1}^{a}m'_{\ell}-\sum_{\ell=1}^{r_{h-1}}m'_{\ell}
=\sum_{\ell=1}^{a}m'_{\ell}-\sum_{\ell=a+1}^{r_{h-1}}m'_{\ell} \nn\\
\alignleq \sum_{\ell=1}^{a}m'_{\ell}-\sum_{\ell=a+1}^{a+1+j}m'_{\ell}
=\sum_{\ell=0}^{a-1}m'_{a-\ell}-\sum_{\ell=0}^{j}m'_{a+1+\ell} \nn\\
\aligneq m'_a+\sum_{\ell=1}^{j}m'_{a-\ell}-m'_{a+1}-\sum_{\ell=1}^{j}m'_{a+1+\ell}\nn\\
\aligneq -1+m'_{a-j}-m'_{a+1+j}<0.
\eeqnarray
It follows from \reqnarray{proof of comparison rule A-(ii)-(a)-case-1-222},
\reqnarray{proof of adjacent distance larger than one-777},
and $a=j+1$ that
\beqnarray{proof of comparison rule A-(ii)-(a)-case-1-333}
\min\{i'_a-1,r_{h-2}-i'_a-1\}
=i'_a-1=\sum_{\ell=1}^{a}m'_{\ell}-1
=\sum_{\ell=0}^{a-1}m'_{a-\ell}-1=\sum_{\ell=0}^{j}m'_{a-\ell}-1.
\eeqnarray
As $a=j+1$, we have from \reqnarray{proof of comparison rule A-(ii)-555} (with $j''=a-1=j$) that
\beqnarray{proof of comparison rule A-(ii)-(a)-case-1-444}
i'_a-\sum_{\ell=0}^{j-1}m'_{a-\ell}=i'_{a-j}=i'_1.
\eeqnarray
From \reqnarray{proof of comparison rule A-(ii)-(a)-case-1-444},
$a=j+1$, and \reqnarray{proof of adjacent distance larger than one-777},
we have
\beqnarray{proof of comparison rule A-(ii)-(a)-case-1-555}
i'_a-\left(\sum_{\ell=0}^{j}m'_{a-\ell}-1\right)=i'_{a-j}-m'_{a-j}+1=i'_1-m'_1+1=1.
\eeqnarray
Thus, we see from \reqnarray{proof of adjacent distance larger than one-666},
\reqnarray{proof of comparison rule A-(ii)-(a)-case-1-444},
and \reqnarray{proof of comparison rule A-(ii)-(a)-case-1-555} that
\beqnarray{proof of comparison rule A-(ii)-(a)-case-1-666}
n'_{i'_a-j'}=q_{h-1},
\textrm{ for } \sum_{\ell=0}^{j-1}m'_{a-\ell}+1\leq j'\leq \sum_{\ell=0}^{j}m'_{a-\ell}-1.
\eeqnarray
Furthermore, from \reqnarray{proof of comparison rule A-(ii)-888} (with $j''=j$),
we have
\beqnarray{proof of comparison rule A-(ii)-(a)-case-1-777}
(i'_a+1)+\sum_{\ell=0}^{j-1}m'_{a-\ell}=i'_{a+j}.
\eeqnarray
From \reqnarray{proof of comparison rule A-(ii)-(a)-case-1-777},
$m'_{a-j}=m_{a-j}<m_{a+1+j}=m'_{a+1+j}$,
and \reqnarray{proof of adjacent distance larger than one-(ii)-3333},
we have
\beqnarray{proof of comparison rule A-(ii)-(a)-case-1-888}
(i'_a+1)+\sum_{\ell=0}^{j}m'_{a-\ell}
= i'_{a+j}+m'_{a-j}
< i'_{a+j}+m'_{a+1+j}=i'_{a+1+j}.
\eeqnarray
Thus, we see from \reqnarray{proof of adjacent distance larger than one-666},
\reqnarray{proof of comparison rule A-(ii)-(a)-case-1-777},
and \reqnarray{proof of comparison rule A-(ii)-(a)-case-1-888} that
\beqnarray{proof of comparison rule A-(ii)-(a)-case-1-999}
n'_{(i'_a+1)+j'}=q_{h-1},
\textrm{ for } \sum_{\ell=0}^{j-1}m'_{a-\ell}+1\leq j'\leq \sum_{\ell=0}^{j}m'_{a-\ell}.
\eeqnarray
By combining \reqnarray{proof of comparison rule A-(ii)-444},
\reqnarray{proof of comparison rule A-(ii)-(a)-case-1-333},
\reqnarray{proof of comparison rule A-(ii)-(a)-case-1-666},
and \reqnarray{proof of comparison rule A-(ii)-(a)-case-1-999},
we obtain \reqnarray{proof of comparison rule A-(ii)-(a)-case-1-111}.
\emph{Case 2: $j+2\leq a\leq r_{h-1}-j-1$.}
In this case, we show that
\beqnarray{}
\alignspace
1\leq \sum_{\ell=0}^{j}m'_{a-\ell}\leq \min\{i'_a-1,r_{h-2}-i'_a-1\},
\label{eqn:proof of comparison rule A-(ii)-(a)-case-2-111}\\
\alignspace
n'_{i'_a-j'}=n'_{(i'_a+1)+j'},\
j'=1,2,\ldots,\sum_{\ell=0}^{j}m'_{a-\ell}-1,
\label{eqn:proof of comparison rule A-(ii)-(a)-case-2-222}\\
\alignspace
n'_{i'_a-\sum_{\ell=0}^{j}m'_{a-\ell}}=q_{h-1}+1>n'_{(i'_a+1)+\sum_{\ell=0}^{j}m'_{a-\ell}}=q_{h-1}.
\label{eqn:proof of comparison rule A-(ii)-(a)-case-2-333}
\eeqnarray
An illustration of \reqnarray{proof of comparison rule A-(ii)-(a)-case-2-222}
and \reqnarray{proof of comparison rule A-(ii)-(a)-case-2-333}
is given in \rfigure{appendix-F-(ii)-(a)}(b).
Therefore, it follows from \reqnarray{proof of adjacent distance larger than one-888},
${\nbf'}_1^{r_{h-2}}\in \Ncal_{M,k}(h-1)$ in
\reqnarray{proof of adjacent distance larger than one-(ii)-333},
\reqnarray{proof of adjacent distance larger than one-(ii)-888}--\reqnarray{proof of adjacent distance larger than one-(ii)-bbb},
\reqnarray{proof of comparison rule A-(ii)-222},
\reqnarray{proof of comparison rule A-(ii)-(a)-case-2-111}--\reqnarray{proof of comparison rule A-(ii)-(a)-case-2-333},
and \reqnarray{comparison rule B-2} in \rlemma{comparison rule B}(ii)
(for the even integer $h-1$) that ${\nbf'}_1^{r_{h-2}}\prec\nbf_1^{r_{h-2}}$,
i.e., \reqnarray{proof of comparison rule A-(ii)-(a)-111} holds.
To prove \reqnarray{proof of comparison rule A-(ii)-(a)-case-2-111},
note that as $j+2\leq a$,
we have from \reqnarray{proof of comparison rule A-(ii)-555} (with $j''=j+1\leq a-1$) that
\beqnarray{proof of comparison rule A-(ii)-(a)-case-2-444}
i'_a-\sum_{\ell=0}^{j}m'_{a-\ell}=i'_{a-j-1}.
\eeqnarray
Also, from \reqnarray{proof of comparison rule A-(ii)-(a)-case-1-888}, we have
\beqnarray{proof of comparison rule A-(ii)-(a)-case-2-555}
(i'_a+1)+\sum_{\ell=0}^{j}m'_{a-\ell}< i'_{a+1+j}\leq r_{h-2}.
\eeqnarray
Thus, \reqnarray{proof of comparison rule A-(ii)-(a)-case-2-111} follows from
$\sum_{\ell=0}^{j}m'_{a-\ell}=i'_a-i'_{a-j-1}\leq i'_a-1$
in \reqnarray{proof of comparison rule A-(ii)-(a)-case-2-444}
and $\sum_{\ell=0}^{j}m'_{a-\ell}\leq r_{h-2}-i'_a-1$
in \reqnarray{proof of comparison rule A-(ii)-(a)-case-2-555}.
To prove \reqnarray{proof of comparison rule A-(ii)-(a)-case-2-222}
and \reqnarray{proof of comparison rule A-(ii)-(a)-case-2-333},
note that from \reqnarray{proof of comparison rule A-(ii)-555} (with $j''=j\leq a-2$),
we have
\beqnarray{proof of comparison rule A-(ii)-(a)-case-2-666}
i'_a-\sum_{\ell=0}^{j-1}m'_{a-\ell}=i'_{a-j}.
\eeqnarray
Thus, we see from
\reqnarray{proof of adjacent distance larger than one-666},
\reqnarray{proof of comparison rule A-(ii)-(a)-case-2-666},
and \reqnarray{proof of comparison rule A-(ii)-(a)-case-2-444} that
\beqnarray{proof of comparison rule A-(ii)-(a)-case-2-777}
n'_{i'_a-j'}=
\bselection
q_{h-1}, &\textrm{for } \sum_{\ell=0}^{j-1}m'_{a-\ell}+1\leq j'\leq \sum_{\ell=0}^{j}m'_{a-\ell}-1. \\
q_{h-1}+1, &\textrm{for } j'=\sum_{\ell=0}^{j}m'_{a-\ell}.
\eselection
\eeqnarray
By combining \reqnarray{proof of comparison rule A-(ii)-444},
\reqnarray{proof of comparison rule A-(ii)-(a)-case-1-999},
and \reqnarray{proof of comparison rule A-(ii)-(a)-case-2-777},
we obtain \reqnarray{proof of comparison rule A-(ii)-(a)-case-2-222}
and \reqnarray{proof of comparison rule A-(ii)-(a)-case-2-333}.
(b) Now we assume that $m_{a-j}>m_{a+1+j}$ and show that
\beqnarray{proof of comparison rule A-(ii)-(b)-111}
\nbf_1^{r_{h-2}}\preceq {\nbf'}_1^{r_{h-2}},
\eeqnarray
where $\nbf_1^{r_{h-2}}\equiv{\nbf'}_1^{r_{h-2}}$ if and only if
\beqnarray{proof of comparison rule A-(ii)-(b)-222}
a-j=1,\ a+1+j=r_{h-1}, \textrm{ and } m_1=m_{r_{h-1}}+1.
\eeqnarray
\bpdffigure{appendix-F-ii-b.pdf}{6.0in}
\epdffigure{appendix-F-(ii)-(b)}
{An illustration of \reqnarray{proof of comparison rule A-(ii)-(b)-333}
and \reqnarray{proof of comparison rule A-(ii)-(b)-444}
in the case that $m_{a-j}>m_{a+1+j}$
(note that in this case we have
$(i'_a+1)+\sum_{\ell=0}^{j''-1}m'_{a-\ell}=i'_{a+j''}$
for $1\leq j''\leq j$ in \reqnarray{proof of comparison rule A-(ii)-888},
$(i'_a+1)+(\sum_{\ell=0}^{j-1}m'_{a-\ell}+m'_{a+1+j})=i'_{a+1+j}$
in \reqnarray{proof of comparison rule A-(ii)-(b)-999},
$i'_a-\sum_{\ell=0}^{j''-1}m'_{a-\ell}=i'_{a-j''}$
for $1\leq j''\leq a-1$ in \reqnarray{proof of comparison rule A-(ii)-555}):
(a) $a=j+1$
(note that in this case we have
$i'_a-(\sum_{\ell=0}^{j-1}m'_{a-\ell}+m'_{a+1+j})\geq 1$
in \reqnarray{proof of comparison rule A-(ii)-(b)-ccc});
(b) $j+2\leq a\leq r_{h-1}-j-1$
(note that in this case we have
$i'_a-(\sum_{\ell=0}^{j-1}m'_{a-\ell}+m'_{a+1+j})>i'_{a-j-1}$
in \reqnarray{proof of comparison rule A-(ii)-(b)-eee}).}
In the following, we show that
\beqnarray{}
\alignspace
n'_{i'_a-j'}=n'_{(i'_a+1)+j'},
\textrm{ for } j'=1,2,\ldots,\sum_{\ell=0}^{j-1}m'_{a-\ell}+m'_{a+1+j}-1,
\label{eqn:proof of comparison rule A-(ii)-(b)-333} \\
\alignspace
n'_{i'_a-(\sum_{\ell=0}^{j-1}m'_{a-\ell}+m'_{a+1+j})}=q_{h-1}
<n'_{(i'_a+1)+(\sum_{\ell=0}^{j-1}m'_{a-\ell}+m'_{a+1+j})}=q_{h-1}+1.
\label{eqn:proof of comparison rule A-(ii)-(b)-444}
\eeqnarray
An illustration of \reqnarray{proof of comparison rule A-(ii)-(b)-333}
and \reqnarray{proof of comparison rule A-(ii)-(b)-444}
is given in \rfigure{appendix-F-(ii)-(b)}.
Therefore, it follows from \reqnarray{proof of adjacent distance larger than one-888},
${\nbf'}_1^{r_{h-2}}\in \Ncal_{M,k}(h-1)$ in
\reqnarray{proof of adjacent distance larger than one-(ii)-333},
\reqnarray{proof of adjacent distance larger than one-(ii)-888}--\reqnarray{proof of adjacent distance larger than one-(ii)-bbb},
\reqnarray{proof of comparison rule A-(ii)-222},
\reqnarray{proof of comparison rule A-(ii)-(b)-333}--\reqnarray{proof of comparison rule A-(ii)-(b)-444},
and \reqnarray{comparison rule B-3} in \rlemma{comparison rule B}(ii)
(for the even integer $h-1$) that ${\nbf'}_1^{r_{h-2}}\succeq\nbf_1^{r_{h-2}}$,
where ${\nbf'}_1^{r_{h-2}}\equiv\nbf_1^{r_{h-2}}$ if and only if
\beqnarray{}
\alignspace i'_a-\left(\sum_{\ell=0}^{j-1}m'_{a-\ell}+m'_{a+1+j}\right)=1,
\label{eqn:proof of comparison rule A-(ii)-(b)-555}\\
\alignspace (i'_a+1)+\left(\sum_{\ell=0}^{j-1}m'_{a-\ell}+m'_{a+1+j}\right)=r_{h-2},
\label{eqn:proof of comparison rule A-(ii)-(b)-666}\\
\alignspace n'_1=n'_{r_{h-2}}-1.
\label{eqn:proof of comparison rule A-(ii)-(b)-777}
\eeqnarray
To prove \reqnarray{proof of comparison rule A-(ii)-(b)-333}
and \reqnarray{proof of comparison rule A-(ii)-(b)-444},
note that from \reqnarray{proof of comparison rule A-(ii)-888} (with $j''=j$)
and \reqnarray{proof of adjacent distance larger than one-(ii)-3333},
we have
\beqnarray{}
\alignspace
(i'_a+1)+\sum_{\ell=0}^{j-1}m'_{a-\ell}=i'_{a+j},
\label{eqn:proof of comparison rule A-(ii)-(b)-888}\\
\alignspace
(i'_a+1)+\left(\sum_{\ell=0}^{j-1}m'_{a-\ell}+m'_{a+1+j}\right)=i'_{a+j}+m'_{a+1+j}=i'_{a+1+j}.
\label{eqn:proof of comparison rule A-(ii)-(b)-999}
\eeqnarray
Thus, we see from \reqnarray{proof of adjacent distance larger than one-666},
\reqnarray{proof of comparison rule A-(ii)-(b)-888},
and \reqnarray{proof of comparison rule A-(ii)-(b)-999} that
\beqnarray{proof of comparison rule A-(ii)-(b)-aaa}
n'_{(i'_a+1)+j'}=
\bselection
q_{h-1}, &\textrm{ for } \sum_{\ell=0}^{j-1}m'_{a-\ell}+1\leq j'\leq \sum_{\ell=0}^{j-1}m'_{a-\ell}+m'_{a+1+j}-1,\\
q_{h-1}+1, &\textrm{ for } j'=\sum_{\ell=0}^{j-1}m'_{a-\ell}+m'_{a+1+j}.
\eselection
\eeqnarray
As $j\leq \min\{a-1,r_{h-1}-a-1\}\leq a-1$,
we have from \reqnarray{proof of comparison rule A-(ii)-555} (with $j''=j\leq a-1$)
and $m'_{a-j}=m_{a-j}>m_{a+1+j}=m'_{a+1+j}$ that
\beqnarray{proof of comparison rule A-(ii)-(b)-bbb}
i'_a-\left(\sum_{\ell=0}^{j-1}m'_{a-\ell}+m'_{a+1+j}\right)=i'_{a-j}-m'_{a+1+j}\geq i'_{a-j}-m'_{a-j}+1,
\eeqnarray
where the equality holds if and only if $m_{a-j}=m_{a+1+j}+1$.
If $a=j+1$, then we have from \reqnarray{proof of comparison rule A-(ii)-(b)-bbb}
and \reqnarray{proof of adjacent distance larger than one-777} that
\beqnarray{proof of comparison rule A-(ii)-(b)-ccc}
i'_a-\left(\sum_{\ell=0}^{j-1}m'_{a-\ell}+m'_{a+1+j}\right)\geq i'_1-m'_1+1=1,
\eeqnarray
where the equality holds if and only if $m_{a-j}=m_{a+1+j}+1$.
Thus, we see from \reqnarray{proof of adjacent distance larger than one-666},
\reqnarray{proof of comparison rule A-(ii)-(a)-case-1-444} (note that $a=j+1$),
and \reqnarray{proof of comparison rule A-(ii)-(b)-ccc} that
\beqnarray{proof of comparison rule A-(ii)-(b)-ddd}
n'_{i'_a-j'}=q_{h-1},
\textrm{ for } \sum_{\ell=0}^{j-1}m'_{a-\ell}+1\leq j'\leq \sum_{\ell=0}^{j-1}m'_{a-\ell}+m'_{a+1+j}.
\eeqnarray
By combining \reqnarray{proof of comparison rule A-(ii)-444},
\reqnarray{proof of comparison rule A-(ii)-(b)-aaa},
and \reqnarray{proof of comparison rule A-(ii)-(b)-ddd},
we obtain \reqnarray{proof of comparison rule A-(ii)-(b)-333}
and \reqnarray{proof of comparison rule A-(ii)-(b)-444}.
On the other hand, if $j+2\leq a\leq r_{h-1}-j-1$,
then we have from \reqnarray{proof of comparison rule A-(ii)-(b)-bbb} and \reqnarray{proof of adjacent distance larger than one-(ii)-3333} that
\beqnarray{proof of comparison rule A-(ii)-(b)-eee}
i'_a-\left(\sum_{\ell=0}^{j-1}m'_{a-\ell}+m'_{a+1+j}\right)\geq i'_{a-j}-m'_{a-j}+1=i'_{a-j-1}+1>i'_{a-j-1}.
\eeqnarray
Thus, we see from \reqnarray{proof of adjacent distance larger than one-666},
\reqnarray{proof of comparison rule A-(ii)-(a)-case-2-666}
(note that $j+2\leq a\leq r_{h-1}-j-1$),
and \reqnarray{proof of comparison rule A-(ii)-(b)-eee} that
\beqnarray{proof of comparison rule A-(ii)-(b)-fff}
n'_{i'_a-j'}=q_{h-1},
\textrm{ for } \sum_{\ell=0}^{j-1}m'_{a-\ell}+1\leq j'\leq \sum_{\ell=0}^{j-1}m'_{a-\ell}+m'_{a+1+j}.
\eeqnarray
By combining \reqnarray{proof of comparison rule A-(ii)-444},
\reqnarray{proof of comparison rule A-(ii)-(b)-aaa},
and \reqnarray{proof of comparison rule A-(ii)-(b)-fff},
we obtain \reqnarray{proof of comparison rule A-(ii)-(b)-333}
and \reqnarray{proof of comparison rule A-(ii)-(b)-444}.
To complete the proof,
we need to show that the condition in
\reqnarray{proof of comparison rule A-(ii)-(b)-555}--\reqnarray{proof of comparison rule A-(ii)-(b)-777}
is equivalent to the condition in \reqnarray{proof of comparison rule A-(ii)-(b)-222}.
Note that if $i'_a-(\sum_{\ell=0}^{j-1}m'_{a-\ell}+m'_{a+1+j})=1$
and $(i'_a+1)+(\sum_{\ell=0}^{j-1}m'_{a-\ell}+m'_{a+1+j})=r_{h-2}$,
then we have from $n'_1=n'_{i'_a-(\sum_{\ell=0}^{j-1}m'_{a-\ell}+m'_{a+1+j})}=q_{h-1}$
and $n'_{r_{h-2}}=n'_{(i'_a+1)+(\sum_{\ell=0}^{j-1}m'_{a-\ell}+m'_{a+1+j})}=q_{h-1}+1$
in \reqnarray{proof of comparison rule A-(ii)-(b)-444} that
\beqnarray{}
n'_1=n'_{r_{h-2}}-1.\nn
\eeqnarray
As such, we see that the condition in
\reqnarray{proof of comparison rule A-(ii)-(b)-555}--\reqnarray{proof of comparison rule A-(ii)-(b)-777}
is equivalent to the following condition:
\beqnarray{proof of comparison rule A-(ii)-(b)-ggg}
i'_a-\left(\sum_{\ell=0}^{j-1}m'_{a-\ell}+m'_{a+1+j}\right)=1
\textrm{ and } (i'_a+1)+\left(\sum_{\ell=0}^{j-1}m'_{a-\ell}+m'_{a+1+j}\right)=r_{h-2}.
\eeqnarray
Note that from \reqnarray{proof of comparison rule A-(ii)-(b)-ccc}
and \reqnarray{proof of comparison rule A-(ii)-(b)-eee},
it is easy to see that
\beqnarray{proof of comparison rule A-(ii)-(b)-hhh}
i'_a-\left(\sum_{\ell=0}^{j-1}m'_{a-\ell}+m'_{a+1+j}\right)=1
\textrm{ iff } a=j+1 \textrm{ and } m_{a-j}=m_{a+1+j}+1.
\eeqnarray
As we have $(i'_a+1)+\left(\sum_{\ell=0}^{j-1}m'_{a-\ell}+m'_{a+1+j}\right)=i'_{a+1+j}$
in \reqnarray{proof of comparison rule A-(ii)-(b)-999}
and it is clear from \reqnarray{proof of adjacent distance larger than one-777}
and \reqnarray{proof of adjacent distance larger than one-(ii)-222}
that $i'_{a+1+j}=r_{h-2}$ if and only if $a+1+j=r_{h-1}$,
it follows that
\beqnarray{proof of comparison rule A-(ii)-(b)-iii}
(i'_a+1)+\left(\sum_{\ell=0}^{j-1}m'_{a-\ell}+m'_{a+1+j}\right)=r_{h-2} \textrm{ iff } a+1+j=r_{h-1}.
\eeqnarray
Therefore, we deduce from \reqnarray{proof of comparison rule A-(ii)-(b)-hhh}
and \reqnarray{proof of comparison rule A-(ii)-(b)-iii}
that the condition in \reqnarray{proof of comparison rule A-(ii)-(b)-ggg}
is equivalent to the following condition:
\beqnarray{}
a=j+1,\ a+1+j=r_{h-1}, \textrm{ and } m_{a-j}=m_{a+1+j}+1, \nn
\eeqnarray
which is clearly equivalent to the condition
that $a-j=1$, $a+1+j=r_{h-1}$, and $m_1=m_{r_{h-1}}+1$
in \reqnarray{proof of comparison rule A-(ii)-(b)-222},
and the proof is completed.
(iii) Note that in \rlemma{comparison rule A}(iii),
we have $2\leq a\leq r_{h-1}-2$
and $m_{a-\ell}=m_{a+1+\ell}$ for $\ell=1,2,\ldots,\min\{a-1,r_{h-1}-a-1\}$.
To show \reqnarray{comparison rule A-4},
i.e., $\mbf_1^{r_{h-1}}\succ{\mbf'}_1^{r_{h-1}}$,
we see from \reqnarray{proof of adjacent distance larger than one-333}
that it suffices to show that
\beqnarray{proof of comparison rule A-(iii)-111}
\nbf_1^{r_{h-2}}\succ{\nbf'}_1^{r_{h-2}}.
\eeqnarray
Note that as we have $2\leq a\leq r_{h-1}-2$,
it follows that \reqnarray{proof of comparison rule A-(ii)-222} also holds.
\bpdffigure{appendix-F-iii.pdf}{6.0in}
\epdffigure{appendix-F-(iii)}
{An illustration of \reqnarray{proof of comparison rule A-(iii)-222}
(note that we have
$i'_a-\sum_{\ell=0}^{j''-1}m'_{a-\ell}=i'_{a-j''}$
for $1\leq j''\leq a-1$ in \reqnarray{proof of comparison rule A-(ii)-555}
and $(i'_a+1)+\sum_{\ell=0}^{j''-1}m'_{a-\ell}=i'_{a+j''}$
for $1\leq j''\leq j^*+1$ in \reqnarray{proof of comparison rule A-(ii)-888}):
(a) $a-1>r_{h-1}-a-1$
(note that in this case we have
$j^*=r_{h-1}-a-1$ and $j^*+1\leq a-1$ in \reqnarray{proof of comparison rule A-(iii)-case-1-111}
and $\min\{i'_a-1,r_{h-2}-i'_a-1\}=\sum_{\ell=0}^{j^*}m'_{a-\ell}$
in \reqnarray{proof of comparison rule A-(iii)-case-1-555});
(b) $a-1\leq r_{h-1}-a-1$
(note that in this case we have
$i'_a-(\sum_{\ell=0}^{j*}m'_{a-\ell}-1)=1$
in \reqnarray{proof of comparison rule A-(ii)-(a)-case-1-555},
$j^*=a-1$ in \reqnarray{proof of comparison rule A-(iii)-case-2-111},
and $\min\{i'_a-1,r_{h-2}-i'_a-1\}=\sum_{\ell=1}^{j^*}m'_{a-\ell}-1$
in \reqnarray{proof of comparison rule A-(iii)-case-2-222}).}
In the following, we show that
\beqnarray{proof of comparison rule A-(iii)-222}
n'_{i'_a-j'}=n'_{(i'_a+1)+j'},
\textrm{ for } j'=1,2,\ldots,\min\{i'_a-1,r_{h-2}-i'_a-1\}.
\eeqnarray
An illustration of \reqnarray{proof of comparison rule A-(iii)-222}
is given in \rfigure{appendix-F-(iii)}.
Therefore, it follows from \reqnarray{proof of adjacent distance larger than one-888},
${\nbf'}_1^{r_{h-2}}\in \Ncal_{M,k}(h-1)$ in
\reqnarray{proof of adjacent distance larger than one-(ii)-333},
\reqnarray{proof of adjacent distance larger than one-(ii)-888}--\reqnarray{proof of adjacent distance larger than one-(ii)-bbb},
\reqnarray{proof of comparison rule A-(ii)-222},
\reqnarray{proof of comparison rule A-(iii)-222},
and \reqnarray{comparison rule B-4} in \rlemma{comparison rule B}(iii)
(for the even integer $h-1$) that ${\nbf'}_1^{r_{h-2}}\prec\nbf_1^{r_{h-2}}$,
i.e., \reqnarray{proof of comparison rule A-(iii)-111} holds.
To prove \reqnarray{proof of comparison rule A-(iii)-222},
let $j^*=\min\{a-1,r_{h-1}-a-1\}$.
As $a\geq 2$, it is clear from the same argument as in (ii) above
that \reqnarray{proof of comparison rule A-(ii)-666} still holds.
Furthermore, from $m'_i=m_i$ for $i\neq a$ and $a+1$
and $m_{a-\ell}=m_{a+1+\ell}$ for $\ell=1,2,\ldots,j^*$,
it is clear that
\beqnarray{proof of comparison rule A-(iii)-333}
m'_{a-\ell}=m'_{a+1+\ell}, \textrm{ for } \ell=1,2,\ldots,j^*.
\eeqnarray
From \reqnarray{proof of comparison rule A-(iii)-333},
it is easy to see from the same argument as in (ii) above that
\reqnarray{proof of comparison rule A-(ii)-777} and \reqnarray{proof of comparison rule A-(ii)-888}
hold for $1\leq j''\leq j^*+1$,
and \reqnarray{proof of comparison rule A-(ii)-999} holds for $j=j^*+1$.
We then consider the two cases
$a-1>r_{h-1}-a-1$ and $a-1\leq r_{h-1}-a-1$ separately.
\emph{Case 1: $a-1>r_{h-1}-a-1$.}
In this case, we have
\beqnarray{proof of comparison rule A-(iii)-case-1-111}
j^*=r_{h-1}-a-1 \textrm{ and } j^*<a-1.
\eeqnarray
From $\sum_{\ell=1}^{r_{h-1}}m'_{\ell}=r_{h-2}$ in \reqnarray{proof of adjacent distance larger than one-(ii)-222},
\reqnarray{proof of adjacent distance larger than one-777},
$j^*=r_{h-1}-a-1$ in \reqnarray{proof of comparison rule A-(iii)-case-1-111},
$m'_a-m'_{a+1}=-1$ in \reqnarray{proof of comparison rule A-444},
and \reqnarray{proof of comparison rule A-(iii)-333},
we have
\beqnarray{proof of comparison rule A-(iii)-case-1-222}
r_{h-2}
\aligneq \sum_{\ell=1}^{r_{h-1}}m'_{\ell}=\sum_{\ell=1}^{a}m'_{\ell}+\sum_{\ell=a+1}^{r_{h-1}}m'_{\ell} \nn\\
\aligneq i'_a+\sum_{\ell=0}^{r_{h-1}-a-1}m'_{a+1+\ell}=i'_a+m'_{a+1}+\sum_{\ell=1}^{j^*}m'_{a+1+\ell} \nn\\
\aligneq i'_a+m'_{a}+1+\sum_{\ell=1}^{j^*}m'_{a-\ell}=i'_a+1+\sum_{\ell=0}^{j^*}m'_{a-\ell}.
\eeqnarray{}
As we have $j^*+1\leq a-1$ in \reqnarray{proof of comparison rule A-(iii)-case-1-111},
we see from \reqnarray{proof of comparison rule A-(ii)-555} (with $j''=j^*+1\leq a-1$) that
\beqnarray{proof of comparison rule A-(iii)-case-1-333}
i'_a-\sum_{\ell=0}^{j^*}m'_{a-\ell}=i'_{a-j^*-1}.
\eeqnarray
It follows from \reqnarray{proof of comparison rule A-(iii)-case-1-222}
and \reqnarray{proof of comparison rule A-(iii)-case-1-333} that
\beqnarray{proof of comparison rule A-(iii)-case-1-444}
r_{h-2}-i'_a-1=\sum_{\ell=0}^{j^*}m'_{a-\ell}=i'_a-i'_{a-j^*-1}\leq i'_a-1.
\eeqnarray
Thus, we see from \reqnarray{proof of comparison rule A-(iii)-case-1-444} that
\beqnarray{proof of comparison rule A-(iii)-case-1-555}
\min\{i'_a-1,r_{h-2}-i'_a-1\}=r_{h-2}-i'_a-1=\sum_{\ell=0}^{j^*}m'_{a-\ell}.
\eeqnarray
As such, it follows from \reqnarray{proof of comparison rule A-(ii)-666},
\reqnarray{proof of comparison rule A-(ii)-999} (with $j=j^*+1$),
and $j^*+1\leq a-1$ in \reqnarray{proof of comparison rule A-(iii)-case-1-111}
that \reqnarray{proof of comparison rule A-(ii)-444} holds for $j=j^*+1$, i.e.,
\beqnarray{proof of comparison rule A-(iii)-case-1-777}
n'_{i'_a-j'}=n'_{(i'_a+1)+j'},
\textrm{ for } j'=1,2,\ldots,\sum_{\ell=0}^{j^*}m'_{a-\ell}.
\eeqnarray
Therefore, \reqnarray{proof of comparison rule A-(iii)-222}
follows from \reqnarray{proof of comparison rule A-(iii)-case-1-555}
and \reqnarray{proof of comparison rule A-(iii)-case-1-777}.
\emph{Case 2: $a-1\leq r_{h-1}-a-1$.}
In this case, we have
\beqnarray{proof of comparison rule A-(iii)-case-2-111}
j^*=a-1 \textrm{ and } j^*\leq r_{h-1}-a-1.
\eeqnarray
As we have $a=j^*+1$ and $a\leq r_{h-1}-j^*-1$
in \reqnarray{proof of comparison rule A-(iii)-case-2-111}
and $m'_{a-\ell}=m'_{a+1+\ell}$ for $\ell=1,2,\ldots,j^*$
in \reqnarray{proof of comparison rule A-(iii)-333},
it is easy to see that $(i'_a-1)-(r_{h-2}-i'_a-1)<0$
in \reqnarray{proof of comparison rule A-(ii)-(a)-case-1-222} still holds
and hence \reqnarray{proof of comparison rule A-(ii)-(a)-case-1-333} holds with $j=j^*$,
i.e.,
\beqnarray{proof of comparison rule A-(iii)-case-2-222}
\min\{i'_a-1,r_{h-2}-i'_a-1\}=\sum_{\ell=0}^{j^*}m'_{a-\ell}-1.
\eeqnarray
From \reqnarray{proof of comparison rule A-(ii)-999} (with $j=j^*+1$), we have
\beqnarray{proof of comparison rule A-(iii)-case-2-333}
n'_{(i'_a+1)+j'}=
\bselection
q_{h-1}+1, &\textrm{ for } j'=m'_a, \sum_{\ell=0}^{1}m'_{a-\ell},\ldots,\sum_{\ell=0}^{j^*}m'_{a-\ell}, \\
q_{h-1}, &\textrm{ for } 1\leq j'\leq \sum_{\ell=0}^{j^*}m'_{a-\ell} \\
&\textrm{ and } j'\neq m'_a, \sum_{\ell=0}^{1}m'_{a-\ell},\ldots,\sum_{\ell=0}^{j^*}m'_{a-\ell}.
\eselection
\eeqnarray
As we have $j^*=a-1$ in \reqnarray{proof of comparison rule A-(iii)-case-2-111},
it follows from \reqnarray{proof of comparison rule A-(ii)-666} that
\beqnarray{proof of comparison rule A-(iii)-case-2-444}
n'_{i'_a-j'}=
\bselection
q_{h-1}+1, &\textrm{ for } j'=m'_a, \sum_{\ell=0}^{1}m'_{a-\ell},\ldots,\sum_{\ell=0}^{j^*-1}m'_{a-\ell}, \\
q_{h-1}, &\textrm{ for } 1\leq j'\leq \sum_{\ell=0}^{j^*-1}m'_{a-\ell} \\
&\textrm{ and } j'\neq m'_a, \sum_{\ell=0}^{1}m'_{a-\ell},\ldots,\sum_{\ell=0}^{j^*-1}m'_{a-\ell}.
\eselection
\eeqnarray
Also, it is clear from $j^*=a-1$ in \reqnarray{proof of comparison rule A-(iii)-case-2-111} that
\reqnarray{proof of comparison rule A-(ii)-(a)-case-1-444}--\reqnarray{proof of comparison rule A-(ii)-(a)-case-1-666}
hold for $j=j^*$,
and hence we see from \reqnarray{proof of comparison rule A-(ii)-(a)-case-1-666} (with $j=j^*$) that
\beqnarray{proof of comparison rule A-(iii)-case-2-555}
n'_{i'_a-j'}=q_{h-1},
\textrm{ for } \sum_{\ell=0}^{j^*-1}m'_{a-\ell}+1\leq j'\leq \sum_{\ell=0}^{j^*}m'_{a-\ell}-1.
\eeqnarray
As such, it follows from
\reqnarray{proof of comparison rule A-(iii)-case-2-333}--\reqnarray{proof of comparison rule A-(iii)-case-2-555} that
\beqnarray{proof of comparison rule A-(iii)-case-2-666}
n'_{i'_a-j'}=n'_{(i'_a+1)+j'},
\textrm{ for } j'=1,2,\ldots,\sum_{\ell=0}^{j^*}m'_{a-\ell}-1.
\eeqnarray
Therefore, \reqnarray{proof of comparison rule A-(iii)-222}
follows from \reqnarray{proof of comparison rule A-(iii)-case-2-222}
and \reqnarray{proof of comparison rule A-(iii)-case-2-666}.
\bappendix{Proof of \rlemma{nonadjacent distance larger than one}}
{proof of nonadjacent distance larger than one}
In this appendix, we use \rlemma{adjacent distance larger than one}
and Comparison rule A in \rlemma{comparison rule A}
to prove \rlemma{nonadjacent distance larger than one}.
For simplicity, let $\nbf_1^{r_{h-1}}=\nbf_1^{r_{h-1}}(h)$.
Note that in \rlemma{nonadjacent distance larger than one}, we have
\beqnarray{proof of nonadjacent distance larger than one-111}
r_{h-1}\geq 3,\ \nbf_1^{r_{h-1}}\in \Ncal_{M,k}(h),
\textrm{ and } |n_i-n_{i+1}|\leq 1 \textrm{ for } i=1,2,\ldots,r_{h-1}-1.
\eeqnarray
(i) Note that in \rlemma{nonadjacent distance larger than one}(i),
we have $n_a-n_b\geq 2$ for some $1\leq a<b\leq r_{h-1}$ and $b\geq a+2$.
For ease of presentation, let $n_b=p$.
Then we have from $n_a-n_b\geq 2$ that $n_a\geq p+2$.
Note that the condition $|n_{i+1}-n_i|\leq 1$ for $i=1,2,\ldots,r_{h-1}-1$
in \reqnarray{proof of nonadjacent distance larger than one-111}
says that the absolute value of the difference of any two adjacent entries
of $\nbf_1^{r_{h-1}}$ is at most equal to one.
As such, it is easy to see from $n_a\geq p+2$, $n_b=p$, $b\geq a+2>a$,
and the condition $|n_{i+1}-n_i|\leq 1$ for $i=1,2,\ldots,r_{h-1}-1$
in \reqnarray{proof of nonadjacent distance larger than one-111}
that there must exist a positive integer $c$ such that $a\leq c<b$ and $n_c=p+2$.
Let
\beqnarray{}
a'\aligneq \max\{i:n_i=p+2,\ c\leq i<b\},
\label{eqn:proof of nonadjacent distance larger than one-(i)-111}\\
b'\aligneq \min\{i:n_i=p,\ a'<i\leq b\}.
\label{eqn:proof of nonadjacent distance larger than one-(i)-222}
\eeqnarray
In other words, $a'$ is the largest positive integer $i$ such that $c\leq i<b$ and $n_i=p+2$,
and $b'$ is the smallest positive integer $i$ such that $a'<i\leq b$ and $n_i=p$.
Note that $a'$ and $b'$ are well defined as we have $n_c=p+2$ and $n_b=p$.
Since we have from \reqnarray{proof of nonadjacent distance larger than one-(i)-111}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-222}
that $n_{a'}=p+2$ and $n_{b'}=p$,
it is easy to see from the condition $|n_{i+1}-n_i|\leq 1$ for $i=1,2,\ldots,r_{h-1}-1$
in \reqnarray{proof of nonadjacent distance larger than one-111} that $b'\geq a'+2$.
In summary, we have
\beqnarray{}
\alignspace a\leq c\leq a'<b'\leq b \textrm{ and } b'\geq a'+2,
\label{eqn:proof of nonadjacent distance larger than one-(i)-333}\\
\alignspace n_a\geq p+2,\ n_c=n_{a'}=p+2, \textrm{ and } n_{b'}=n_b=p.
\label{eqn:proof of nonadjacent distance larger than one-(i)-444}
\eeqnarray
\bpdffigure{appendix-G-i.pdf}{4.5in}
\epdffigure{appendix-G-(i)}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one-(i)-333}--\reqnarray{proof of nonadjacent distance larger than one-(i)-555}.}
We claim that
\beqnarray{proof of nonadjacent distance larger than one-(i)-555}
n_i=p+1, \textrm{ for } a'<i<b'.
\eeqnarray
An illustration of
\reqnarray{proof of nonadjacent distance larger than one-(i)-333}--\reqnarray{proof of nonadjacent distance larger than one-(i)-555}
is given in \rfigure{appendix-G-(i)}.
We prove \reqnarray{proof of nonadjacent distance larger than one-(i)-555} by contradiction.
First assume that $n_i\geq p+2$ for some $a'<i<b'$.
From $c\leq a'<b'\leq b$ in \reqnarray{proof of nonadjacent distance larger than one-(i)-333} and $a'<i<b'$,
we have $c\leq a'<i<b'\leq b$ and hence it follows from the definition of $a'$
in \reqnarray{proof of nonadjacent distance larger than one-(i)-111} that $n_i\neq p+2$.
Since we assume that $n_i\geq p+2$, it is clear that we must have $n_i>p+2$.
As such, we see from $n_i>p+2$,
$n_{b'}=p$ in \reqnarray{proof of nonadjacent distance larger than one-(i)-444},
$i<b'$, and the condition $|n_{i+1}-n_i|\leq 1$ for $i=1,2,\ldots,r_{h-1}-1$
in \reqnarray{proof of nonadjacent distance larger than one-111}
that there must exist a positive integer $a''$ such that
\beqnarray{proof of nonadjacent distance larger than one-(i)-666}
i<a''<b' \textrm{ and } n_{a''}=p+2.
\eeqnarray
From $c\leq a'<b'\leq b$ in \reqnarray{proof of nonadjacent distance larger than one-(i)-333},
$a'<i<b'$, and $i<a''<b'$ in \reqnarray{proof of nonadjacent distance larger than one-(i)-666},
we have $c\leq a'<i<a''<b'\leq b$ and hence it follows from the definition of $a'$
in \reqnarray{proof of nonadjacent distance larger than one-(i)-111} that $n_{a''}\neq p+2$,
contradicting to $n_{a''}=p+2$ in \reqnarray{proof of nonadjacent distance larger than one-(i)-666}.
Now assume that $n_i\leq p$ for some $a'<i<b'$.
From $a'<b'\leq b$ in \reqnarray{proof of nonadjacent distance larger than one-(i)-333} and $a'<i<b'$,
we have $a'<i<b'\leq b$ and hence it follows from the definition of $b'$
in \reqnarray{proof of nonadjacent distance larger than one-(i)-222} that $n_i\neq p$.
Since we assume that $n_i\leq p$, it is clear that we must have $n_i<p$.
As such, we see from $n_{a'}=p+2$ in \reqnarray{proof of nonadjacent distance larger than one-(i)-444},
$n_i<p$, $a'<i$, and the condition $|n_{i+1}-n_i|\leq 1$ for $i=1,2,\ldots,r_{h-1}-1$
in \reqnarray{proof of nonadjacent distance larger than one-111}
that there must exist a positive integer $b''$ such that
\beqnarray{proof of nonadjacent distance larger than one-(i)-777}
a'<b''<i \textrm{ and } n_{b''}=p.
\eeqnarray
From $a'<b''<i$ in \reqnarray{proof of nonadjacent distance larger than one-(i)-777},
$a'<i<b'$, and $a'<b'\leq b$ in \reqnarray{proof of nonadjacent distance larger than one-(i)-333},
we have $a'<b''<i<b'\leq b$ and hence it follows from the definition of $b'$
in \reqnarray{proof of nonadjacent distance larger than one-(i)-222} that $n_{b''}\neq p$,
contradicting to $n_{b''}=p$ in \reqnarray{proof of nonadjacent distance larger than one-(i)-777}.
The proof of \reqnarray{proof of nonadjacent distance larger than one-(i)-555} is completed.
Note that as we have $b'\geq a'+2$ in \reqnarray{proof of nonadjacent distance larger than one-(i)-333},
it is clear that $a'<a'+1<b'$ and $a'<b'-1<b'$.
It then follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-555} that
\beqnarray{proof of nonadjacent distance larger than one-(i)-888}
n_{a'+1}=p+1 \textrm{ and } n_{b'-1}=p+1.
\eeqnarray
To prove \rlemma{nonadjacent distance larger than one}(i),
we need to show that if $n_1\neq n_{r_{h-1}}+2$
or $n_i\neq n_{r_{h-1}}+1$ for some $2\leq i\leq r_{h-1}-1$,
then there exists a sequence of positive integers
${\nbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$ such that
\beqnarray{proof of nonadjacent distance larger than one-(i)-999}
{\nbf'}_1^{r_{h-1}}\succ\nbf_1^{r_{h-1}}.
\eeqnarray
Note that if $n_1\neq n_{r_{h-1}}+2$
or $n_i\neq n_{r_{h-1}}+1$ for some $2\leq i\leq r_{h-1}-1$,
then we have $a'\geq 2$ or $b'\leq r_{h-1}-1$.
To see this, suppose on the contrary that $a'=1$ and $b'=r_{h-1}$.
Then it follows from $a'=1$, $b'=r_{h-1}$,
$n_{a'}=p+2$ and $n_{b'}=p$
in \reqnarray{proof of nonadjacent distance larger than one-(i)-444},
and $n_i=p+1$ for $a'+1\leq i\leq b'-1$
in \reqnarray{proof of nonadjacent distance larger than one-(i)-555} that
\beqnarray{proof of nonadjacent distance larger than one-(i)-aaa}
n_1=p+2,\ n_{r_{h-1}}=p,
\textrm{ and } n_i=p+1 \textrm{ for } 2\leq i\leq r_{h-1}-1.
\eeqnarray
It is clear from \reqnarray{proof of nonadjacent distance larger than one-(i)-aaa}
that $n_1=n_{r_{h-1}}+2$ and $n_i=n_{r_{h-1}}+1$ for all $2\leq i\leq r_{h-1}-1$,
and a contradiction is reached.
In the following, we show that if $a'\geq 2$ or $b'\leq r_{h-1}-1$,
then there exists a sequence of positive integers ${\nbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$
such that \reqnarray{proof of nonadjacent distance larger than one-(i)-999} holds,
and hence \rlemma{nonadjacent distance larger than one}(i) is proved.
(a) First, we assume that $a'\geq 2$
and show that there exists a sequence of positive integers
${\nbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$ such that
\reqnarray{proof of nonadjacent distance larger than one-(i)-999} holds.
Note that from $b'\geq a'+2$
in \reqnarray{proof of nonadjacent distance larger than one-(i)-333}
and $b'\leq r_{h-1}$, we have
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-111}
a'\leq b'-2\leq r_{h-1}-2.
\eeqnarray
As we assume that $a'\geq 2$,
it follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-111} that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-222}
2\leq a'\leq r_{h-1}-2.
\eeqnarray
Furthermore, we have from $n_{a'}=p+2$
in \reqnarray{proof of nonadjacent distance larger than one-(i)-444}
and $n_{a'+1}=p+1$
in \reqnarray{proof of nonadjacent distance larger than one-(i)-888} that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-333}
n_{a'}-n_{a'+1}=(p+2)-(p+1)=1.
\eeqnarray
We need to consider the following three possible cases.
\emph{Case 1: There exists a positive integer $j$ such that
$1\leq j\leq \min\{a'-1,b'-a'-1\}$,
$n_{a'-j'}=p+1$ for $j'=1,2,\ldots,j-1$, and $n_{a'-j}>p+1$.}
Let $\mbf_1^{r_{h-1}}$ be a sequence of positive integers such that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-111}
m_{a'}=n_{a'}-1,\ m_{a'+1}=n_{a'+1}+1, \textrm{ and } m_i=n_i \textrm{ for } i\neq a', a'+1.
\eeqnarray
As before, it is easy to show that $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
As we have $j\leq \min\{a'-1,b'-a'-1\}\leq b'-a'-1$,
we consider the two subcases $j<b'-a'-1$ and $j=b'-a'-1$ separately.
\emph{Subcase 1(a): $j<b'-a'-1$.}
\bpdffigure{appendix-G-i-a-case-1.pdf}{5.5in}
\epdffigure{appendix-G-(i)-(a)-case-1}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-333}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-444}.}
In this subcase, we show that
\beqnarray{}
\alignspace 1\leq j\leq \min\{a'-1,r_{h-1}-a'-1\}
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-1-222}\\
\alignspace n_{a'-j'}=n_{(a'+1)+j'}, \textrm{ for } j'=1,2,\ldots,j-1,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-1-333}\\
\alignspace n_{a'-j}>n_{(a'+1)+j}.
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-1-444}
\eeqnarray
An illustration of
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-333}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-444}
is given in \rfigure{appendix-G-(i)-(a)-case-1}.
Therefore, it follows from
$\nbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$
in \reqnarray{proof of nonadjacent distance larger than one-111},
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-222}--\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-444},
and \reqnarray{comparison rule A-3} in \rlemma{comparison rule A}(ii) that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-555}
\nbf_1^{r_{h-1}}\preceq \mbf_1^{r_{h-1}},
\eeqnarray
where $\nbf_1^{r_{h-1}}\equiv \mbf_1^{r_{h-1}}$
if and only if $a'-j=1$, $(a'+1)+j=r_{h-1}$,
and $n_1=n_{r_{h-1}}+1$ (i.e., $n_{a'-j}=n_{(a'+1)+j}+1$).
From $j<b'-a'-1$ in this subcase and $b'\leq r_{h-1}$,
we see that $(a'+1)+j<b'\leq r_{h-1}$.
This implies that $(a'+1)+j\neq r_{h-1}$ and hence it cannot be the case that
$\nbf_1^{r_{h-1}}\equiv \mbf_1^{r_{h-1}}$.
As such, we see from \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-555}
that $\nbf_1^{r_{h-1}}\prec \mbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one-(i)-999} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
From $1\leq j\leq \min\{a'-1,b'-a'-1\}$ and $b'\leq r_{h-1}$, we see that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-666}
1\leq j\leq \min\{a'-1,b'-a'-1\}\leq \min\{a'-1,r_{h-1}-a'-1\}.
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-222}
follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-666}.
To prove \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-333}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-444},
note that in this subcase we have $a'<(a'+1)+j<b'$,
and hence it follows from
\reqnarray{proof of nonadjacent distance larger than one-(i)-555} that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-777}
n_{(a'+1)+j'}=p+1, \textrm{ for } j'=1,2,\ldots,j.
\eeqnarray
By combining $n_{a'-j'}=p+1$ for $j'=1,2,\ldots,j-1$, $n_{a'-j}>p+1$,
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-777},
we obtain \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-333}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-444}.
\emph{Subcase 1(b): $j=b'-a'-1$.}
In this subcase, we have $a'<(a'+1)+j=b'$
and hence it follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-555}
and $n_{b'}=p$ in \reqnarray{proof of nonadjacent distance larger than one-(i)-444} that
\beqnarray{}
\alignspace n_{(a'+1)+j'}=p+1, \textrm{ for } j'=1,2,\ldots,j-1,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-1-888}\\
\alignspace n_{(a'+1)+j}=n_{b'}=p.
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-1-999}
\eeqnarray
By using \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-888}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-999},
we can argue as in Subcase~1(a) above that
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-222}--\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-555}
still hold.
Since it is clear from $n_{a'-j}>p+1$ and $n_{(a'+1)+j}=p$
in \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-999}
that $n_{a'-j}\neq n_{(a'+1)+j}+1$,
it cannot be the case that $\nbf_1^{r_{h-1}}\equiv \mbf_1^{r_{h-1}}$.
As such, we see from \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-555}
that $\nbf_1^{r_{h-1}}\prec \mbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one-(i)-999} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
\emph{Case 2: There exists a positive integer $j$ such that
$1\leq j\leq \min\{a'-1,b'-a'-1\}$,
$n_{a'-j'}=p+1$ for $j'=1,2,\ldots,j-1$, and $n_{a'-j}<p+1$.}
In this case, we can show that $j\geq 2$.
To see this, suppose on the contrary that $j=1$,
then we have $n_{a'-1}<p+1$ in this case.
As it is easy to see from $n_{a'}=p+2$
in \reqnarray{proof of nonadjacent distance larger than one-(i)-444}
and the condition $|n_{i+1}-n_i|\leq 1$ for $i=1,2,\ldots,r_{h-1}-1$
in \reqnarray{proof of nonadjacent distance larger than one-111}
that $n_{a'-1}$ must be equal to $p+1$, $p+2$, or $p+3$,
we have reached a contradiction.
Let $\mbf_1^{r_{h-1}}$ be a sequence of positive integers such that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-111}
m_{a'-1}=n_{a'-1}+1,\ m_{a'}=n_{a'}-1,
\textrm{ and } m_i=n_i \textrm{ for } i\neq a'-1, a'.
\eeqnarray
As before, it is easy to show that $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
As $j\geq 2$, we have $n_{a'-1}=p+1$ in this case.
It then follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-111},
$n_{a'-1}=p+1$, and $n_{a'}=p+2$
in \reqnarray{proof of nonadjacent distance larger than one-(i)-444} that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-222}
m_{a'-1}-m_{a'}=(n_{a'-1}+1)-(n_{a'}-1)=(p+1+1)-(p+2-1)=1.
\eeqnarray
\bpdffigure{appendix-G-i-a-case-2.pdf}{5.5in}
\epdffigure{appendix-G-(i)-(a)-case-2}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-555}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-666}.}
In the following, we show that
\beqnarray{}
\alignspace
2\leq a'-1\leq r_{h-1}-2,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-2-333}\\
\alignspace
1\leq j-1\leq \min\{(a'-1)-1,r_{h-1}-(a'-1)-1\},
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-2-444}\\
\alignspace m_{(a'-1)-j'}=m_{a'+j'}, \textrm{ for } j'=1,2,\ldots,j-2,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-2-555}\\
\alignspace m_{(a'-1)-(j-1)}<m_{a'+(j-1)}.
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-2-666}
\eeqnarray
An illustration of
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-555}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-666}
is given in \rfigure{appendix-G-(i)-(a)-case-2}.
Therefore, it follows from
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-111}--\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-666},
and \reqnarray{comparison rule A-2} in \rlemma{comparison rule A}(ii) that
$\mbf_1^{r_{h-1}}\succ \nbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one-(i)-999} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
From $2\leq j\leq \min\{a'-1,b'-a'-1\}$,
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-222},
and $b'\leq r_{h-1}$, we can see that
\beqnarray{}
\alignspace
2\leq \min\{a'-1,b'-a'-1\}\leq a'-1<a'\leq r_{h-1}-2,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-2-777}\\
\alignspace
1\leq j-1\leq \min\{a'-2,b'-a'-2\}\leq \min\{(a'-1)-1,r_{h-1}-(a'-1)-1\}.
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-2-888}
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-333}
follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-777},
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-444}
follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-888}.
To prove \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-555}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-666},
note that as we have $n_{a'-j'}=p+1$ for $j'=1,2,\ldots,j-1$ and $n_{a'-j}<p+1$ in this case,
it is clear from \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-111} that
\beqnarray{}
\alignspace m_{(a'-1)-j'}=n_{(a'-1)-j'}=p+1, \textrm{ for } j'=1,2,\ldots,j-2,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-2-999}\\
\alignspace m_{(a'-1)-(j-1)}=n_{(a'-1)-(j-1)}=n_{a'-j}<p+1.
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-2-aaa}
\eeqnarray
Furthermore, we have from $2\leq j\leq \min\{a'-1,b'-a'-1\}$ that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-bbb}
a'< a'+j-1\leq a'+(b'-a'-1)-1=b'-2<b'.
\eeqnarray
It then follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-111},
\reqnarray{proof of nonadjacent distance larger than one-(i)-555},
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-bbb} that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-ccc}
m_{a'+j'}=n_{a'+j'}=p+1, \textrm{ for } j'=1,2,\ldots,j-1.
\eeqnarray
By combining \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-999},
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-aaa},
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-ccc},
we obtain \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-555}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-666}.
\emph{Case 3: $n_{a'-j'}=p+1$ for $j'=1,2,\ldots,\min\{a'-1,b'-a'-1\}$.}
We consider the two subcases $a'-1<b'-a'-1$ and $a'-1\geq b'-a'-1$ separately.
\emph{Subcase 3(a): $a'-1<b'-a'-1$.}
Let $\mbf_1^{r_{h-1}}$ be a sequence of positive integers as given in
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-111}.
As in Case~2 above, we have $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
As it is clear from $a'\geq 2$ and $b'\geq a'+2$
in \reqnarray{proof of nonadjacent distance larger than one-(i)-333}
that $\min\{a'-1,b'-a'-1\}\geq 1$,
we have $n_{a'-1}=p+1$ in this case and hence it is easy to see that
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-222}
still holds in this subcase.
If $a'=2$, then we have $a'-1=1$ and
it follows from $r_{h-1}\geq 3$
in \reqnarray{proof of nonadjacent distance larger than one-111},
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-111},
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-222},
and \reqnarray{comparison rule A-1} in \rlemma{comparison rule A}(i)
that $\mbf_1^{r_{h-1}}\succ \nbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one-(i)-999} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
\bpdffigure{appendix-G-i-a-case-3-1.pdf}{4.5in}
\epdffigure{appendix-G-(i)-(a)-case-3-1}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-222}
(note that we have $\min\{(a'-1)-1, r_{h-1}-(a'-1)-1\}=a'-2$
in \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-555}).}
On the other hand, if $a'\geq 3$, then we show that
\beqnarray{}
\alignspace \hspace*{-0.2in}
2\leq a'-1\leq r_{h-1}-2,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-3-111}\\
\alignspace \hspace*{-0.2in}
m_{(a'-1)-j'}=m_{a'+j'}=p+1, \textrm{ for } j'=1,2,\ldots,\min\{(a'-1)-1, r_{h-1}-(a'-1)-1\}.
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-3-222}
\eeqnarray
An illustration of \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-222}
is given in \rfigure{appendix-G-(i)-(a)-case-3-1}.
Therefore, it follows from
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-111},
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-222},
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-111},
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-222},
and \reqnarray{comparison rule A-4} in \rlemma{comparison rule A}(iii) that
$\mbf_1^{r_{h-1}}\succ \nbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one-(i)-999} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
From $a'\geq 3$ and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-222},
we see that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-333}
2\leq a'-1<a'\leq r_{h-1}-2.
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-111}
follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-333}.
To prove \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-222},
note that from $a'-1<b'-a'-1$ in this subcase and $b'\leq r_{h-1}$, we have
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-444}
(a'-1)-1<(b'-a'-1)-1\leq r_{h-1}-a'-2<r_{h-1}-(a'-1)-1.
\eeqnarray
It follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-444} that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-555}
\min\{(a'-1)-1, r_{h-1}-(a'-1)-1\}=(a'-1)-1=a'-2.
\eeqnarray
As in this subcase we have
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-666}
n_{a'-j'}=p+1, \textrm{ for } j'=1,2,\ldots,\min\{a'-1,b'-a'-1\}=a'-1,
\eeqnarray
it is clear from \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-111}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-666} that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-777}
m_{(a'-1)-j'}=n_{(a'-1)-j'}=p+1, \textrm{ for } j'=1,2,\ldots,a'-2.
\eeqnarray
Furthermore, we have from $a'\geq 3$ and $a'-1<b'-a'-1$ that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-888}
a'<a'+(a'-2)<a'+(b'-a'-2)=b'-2<b'.
\eeqnarray
It then follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-2-111},
\reqnarray{proof of nonadjacent distance larger than one-(i)-555},
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-888} that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-999}
m_{a'+j'}=n_{a'+j'}=p+1, \textrm{ for } j'=1,2,\ldots,a'-2.
\eeqnarray
By combining
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-555},
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-777},
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-999},
we obtain \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-222}.
\emph{Subcase 3(b): $a'-1\geq b'-a'-1$.}
\bpdffigure{appendix-G-i-a-case-3-2.pdf}{4.5in}
\epdffigure{appendix-G-(i)-(a)-case-3-2}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-bbb}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-ccc}.}
Let $\mbf_1^{r_{h-1}}$ be a sequence of positive integers as given in
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-111}.
As in Case~1 above, we have $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
In this subcase, we show that
\beqnarray{}
\alignspace 1\leq b'-a'-1\leq \min\{a'-1,r_{h-1}-a'-1\}
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-3-aaa}\\
\alignspace n_{a'-j'}=n_{(a'+1)+j'}=p+1, \textrm{ for } j'=1,2,\ldots,b'-a'-2,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-3-bbb}\\
\alignspace n_{a'-(b'-a'-1)}=p+1>n_{(a'+1)+(b'-a'-1)}=p.
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-3-ccc}
\eeqnarray
An illustration of
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-bbb}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-ccc}
is given in \rfigure{appendix-G-(i)-(a)-case-3-2}.
Therefore, it follows from
$\nbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$
in \reqnarray{proof of nonadjacent distance larger than one-111},
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-222}--\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-111},
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-aaa}--\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-ccc},
and \reqnarray{comparison rule A-3} in \rlemma{comparison rule A}(ii)
(with $j=b'-a'-1$) that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-ddd}
\nbf_1^{r_{h-1}}\preceq \mbf_1^{r_{h-1}}.
\eeqnarray
From $b'\geq a'+2$ in \reqnarray{proof of nonadjacent distance larger than one-(i)-333},
$a'-1\geq b'-a'-1$, and $b'\leq r_{h-1}$, we can see that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-eee}
1\leq b'-a'-1\leq \min\{a'-1,b'-a'-1\}\leq \min\{a'-1,r_{h-1}-a'-1\}.
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-aaa}
follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-eee}.
To prove \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-bbb}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-ccc},
note that in this case we have
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-fff}
n_{a'-j'}=p+1, \textrm{ for } j'=1,2,\ldots,\min\{a'-1,b'-a'-1\}=b'-a'-1.
\eeqnarray
Also, it is clear from
\reqnarray{proof of nonadjacent distance larger than one-(i)-555},
$(a'+1)+(b'-a'-1)=b'$,
and $n_{b'}=p$ in \reqnarray{proof of nonadjacent distance larger than one-(i)-444} that
\beqnarray{}
\alignspace n_{(a'+1)+j'}=p+1, \textrm{ for } j'=1,2,\ldots,b'-a'-2,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-3-ggg}\\
\alignspace n_{(a'+1)+(b'-a'-1)}=n_{b'}=p.
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-3-hhh}
\eeqnarray
By combining
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-fff},
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-ggg},
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-hhh},
we obtain \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-bbb}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-ccc}.
Now let ${\mbf'}_1^{r_{h-1}}$ be a sequence of positive integers such that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-iii}
m'_{a'+1}=m_{a'+1}-1,\ m'_{a'+2}=m_{a'+2}+1, \textrm{ and } m'_i=m_i \textrm{ for } i\neq a'+1, a'+2.
\eeqnarray
Again, it is easy to show that ${\mbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
If $b'=a'+2$, then we see from
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-111},
$n_{a'+1}=p+1$ in \reqnarray{proof of nonadjacent distance larger than one-(i)-888},
and $n_{b'}=p$ in \reqnarray{proof of nonadjacent distance larger than one-(i)-444} that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-jjj}
\alignspace m_{a'+1}-m_{a'+2}=(n_{a'+1}+1)-n_{a'+2}=n_{a'+1}+1-n_{b'}=(p+1)+1-p=2.
\eeqnarray
Therefore, it follows from
$r_{h-1}\geq 3$ in \reqnarray{proof of nonadjacent distance larger than one-111},
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-iii},
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-jjj},
and \reqnarray{adjacent distance larger than one-2} in \rlemma{adjacent distance larger than one}(ii) that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-kkk}
\mbf_1^{r_{h-1}}\preceq {\mbf'}_1^{r_{h-1}},
\eeqnarray
where $\mbf_1^{r_{h-1}}\equiv {\mbf'}_1^{r_{h-1}}$
if and only if $r_{h-1}=2$ and $m_1=m_2+2$.
Since it is clear from $r_{h-1}\geq 3$ in
\reqnarray{proof of nonadjacent distance larger than one-111}
that $r_{h-1}\neq 2$, it cannot be the case that $\mbf_1^{r_{h-1}}\equiv {\mbf'}_1^{r_{h-1}}$.
As such, we see from \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-ddd}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-kkk}
that $\nbf_1^{r_{h-1}}\preceq \mbf_1^{r_{h-1}}\prec {\mbf'}_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one-(i)-999} holds
with ${\nbf'}_1^{r_{h-1}}={\mbf'}_1^{r_{h-1}}$.
On the other hand, if $b'\geq a'+3$, then we have $a'<a'+2<b'$
and it follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-555} that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-1111}
n_{a'+1}=n_{a'+2}=p+1.
\eeqnarray
From \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-111}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-1111},
we see that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-2222}
\alignspace m_{a'+1}-m_{a'+2}=(n_{a'+1}+1)-n_{a'+2}=1.
\eeqnarray
\bpdffigure{appendix-G-i-a-case-3-3.pdf}{4.5in}
\epdffigure{appendix-G-(i)-(a)-case-3-3}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-5555}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-6666}.}
We will show that
\beqnarray{}
\alignspace
2\leq a'+1\leq r_{h-1}-2,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-3-3333}\\
\alignspace
1\leq b'-a'-2\leq \min\{(a'+1)-1,r_{h-1}-(a'+1)-1\},
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-3-4444}\\
\alignspace
m_{(a'+1)-j'}=m_{(a'+2)+j'}=p+1, \textrm{ for } j'=1,2,\ldots,b'-a'-3,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-3-5555}\\
\alignspace
m_{(a'+1)-(b'-a'-2)}=p+1>m_{(a'+2)+(b'-a'-2)}=p.
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-3-6666}
\eeqnarray
An illustration of
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-5555}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-6666}
is given in \rfigure{appendix-G-(i)-(a)-case-3-3}.
Therefore, it follows from
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-iii},
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-2222}--\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-6666},
and \reqnarray{comparison rule A-3} in \rlemma{comparison rule A}(ii)
(with $j=b'-a'-2$) that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-7777}
\mbf_1^{r_{h-1}}\preceq {\mbf'}_1^{r_{h-1}},
\eeqnarray
where $\mbf_1^{r_{h-1}}\equiv {\mbf'}_1^{r_{h-1}}$
if and only if $(a'+1)-(b'-a'-2)=1$, $(a'+1)+1+(b'-a'-2)=r_{h-1}$, and $m_1=m_{r_{h-1}}+1$.
Since in this subcase we have $a'-1\geq b'-a'-1$,
it is clear that $(a'+1)-(b'-a'-2)\geq (b'-a'+1)-(b'-a'-2)=3$.
This implies that $(a'+1)-(b'-a'-2)\neq 1$
and hence it cannot be the case that $\mbf_1^{r_{h-1}}\equiv {\mbf'}_1^{r_{h-1}}$.
As such, we see from \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-ddd}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-7777}
that $\nbf_1^{r_{h-1}}\preceq \mbf_1^{r_{h-1}}\prec {\mbf'}_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one-(i)-999} holds
with ${\nbf'}_1^{r_{h-1}}={\mbf'}_1^{r_{h-1}}$.
To prove \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-3333}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-4444},
note from $a'\geq 2$, $b'\geq a'+3$, $b'\leq r_{h-1}$,
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-aaa} that
\beqnarray{}
\alignspace \hspace*{-0.3in}
2\leq a'<a'+1\leq b'-2\leq r_{h-1}-2,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-3-8888}\\
\alignspace \hspace*{-0.3in}
1\leq b'-a'-2\leq \min\{a'-2,r_{h-1}-a'-2\}\leq \min\{(a'+1)-1,r_{h-1}-(a'+1)-1\}.
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-3-9999}
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-3333}
follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-8888},
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-4444}
follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-9999}.
To prove \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-5555}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-6666},
note that from \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-1-111},
$n_{a'}=p+2$ and $n_{b'}=p$ in \reqnarray{proof of nonadjacent distance larger than one-(i)-444},
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-fff}--\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-hhh},
we have
\beqnarray{}
\alignspace m_{(a'+1)-1}=m_{a'}=n_{a'}-1=(p+2)-1=p+1,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-3-aaaa}\\
\alignspace m_{(a'+1)-j'}=n_{(a'+1)-j'}=p+1, \textrm{ for } j'=2,3,\ldots,b'-a',
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-3-bbbb}\\
\alignspace m_{(a'+2)+j'}=n_{(a'+2)+j'}=p+1, \textrm{ for } j'=1,2,\ldots,b'-a'-3,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-3-cccc}\\
\alignspace m_{(a'+2)+(b'-a'-2)}=n_{(a'+2)+(b'-a'-2)}=n_{b'}=p.
\label{eqn:proof of nonadjacent distance larger than one-(i)-(a)-case-3-dddd}
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-5555}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-6666}
follow from
\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-aaaa}--\reqnarray{proof of nonadjacent distance larger than one-(i)-(a)-case-3-dddd}.
(b) Now we assume that $b'\leq r_{h-1}-1$ and show that there exists a sequence of positive integers
${\nbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$ such that
\reqnarray{proof of nonadjacent distance larger than one-(i)-999} holds.
Note that from $b'\geq a'+2$
in \reqnarray{proof of nonadjacent distance larger than one-(i)-333}
and $a'\geq 1$, we have
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-111}
b'-1\geq a'+1\geq 2.
\eeqnarray
As we assume that $b'\leq r_{h-1}-1$,
it follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-111} that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-222}
2\leq b'-1\leq r_{h-1}-2.
\eeqnarray
Furthermore, we have from $n_{b'-1}=p+1$
in \reqnarray{proof of nonadjacent distance larger than one-(i)-888}
and $n_{b'}=p$
in \reqnarray{proof of nonadjacent distance larger than one-(i)-444} that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-333}
n_{b'-1}-n_{b'}=(p+1)-p=1.
\eeqnarray
We need to consider the following three possible cases.
\emph{Case 1: There exists a positive integer $j$ such that
$1\leq j\leq \min\{b'-a'-1,r_{h-1}-b'\}$,
$n_{b'+j'}=p+1$ for $j'=1,2,\ldots,j-1$, and $n_{b'+j}<p+1$.}
Let $\mbf_1^{r_{h-1}}$ be a sequence of positive integers such that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-111}
m_{b'-1}=n_{b'-1}-1,\ m_{b'}=n_{b'}+1, \textrm{ and } m_i=n_i \textrm{ for } i\neq b'-1, b'.
\eeqnarray
As before, it is easy to show that $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
As we have $j\leq \min\{b'-a'-1,r_{h-1}-b'\}\leq b'-a'-1$,
we consider the two subcases $j<b'-a'-1$ and $j=b'-a'-1$ separately.
\emph{Subcase 1(a): $j<b'-a'-1$.}
\bpdffigure{appendix-G-i-b-case-1.pdf}{5.5in}
\epdffigure{appendix-G-(i)-(b)-case-1}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-333}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-444}.}
In this subcase, we show that
\beqnarray{}
\alignspace 1\leq j\leq \min\{(b'-1)-1,r_{h-1}-(b'-1)-1\}
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-1-222}\\
\alignspace n_{(b'-1)-j'}=n_{b'+j'}, \textrm{ for } j'=1,2,\ldots,j-1,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-1-333}\\
\alignspace n_{(b'-1)-j}>n_{b'+j}.
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-1-444}
\eeqnarray
An illustration of
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-333}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-444}
is given in \rfigure{appendix-G-(i)-(b)-case-1}.
Therefore, it follows from
$\nbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$
in \reqnarray{proof of nonadjacent distance larger than one-111},
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-222}--\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-444},
and \reqnarray{comparison rule A-3} in \rlemma{comparison rule A}(ii) that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-555}
\nbf_1^{r_{h-1}}\preceq \mbf_1^{r_{h-1}},
\eeqnarray
where $\nbf_1^{r_{h-1}}\equiv \mbf_1^{r_{h-1}}$
if and only if $(b'-1)-j=1$, $b'+j=r_{h-1}$,
and $n_1=n_{r_{h-1}}+1$ (i.e., $n_{(b'-1)-j}=n_{b'+j}+1$).
As we have $j<b'-a'-1$ in this subcase and $a'\geq 1$,
we immediately see that $(b'-1)-j>a'\geq 1$.
This implies that $(b'-1)-j\neq 1$ and hence it cannot be the case that
$\nbf_1^{r_{h-1}}\equiv \mbf_1^{r_{h-1}}$.
As such, we see from \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-555}
that $\nbf_1^{r_{h-1}}\prec \mbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one-(i)-999} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
From $1\leq j\leq \min\{b'-a'-1,r_{h-1}-b'\}$ and $a'\geq 1$, we see that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-666}
1\leq j\leq \min\{b'-a'-1,r_{h-1}-b'\}\leq \min\{(b'-1)-1,r_{h-1}-(b'-1)-1\}.
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-222}
follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-666}.
To prove \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-333}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-444},
note that in this subcase we have $a'<(b'-1)-j<b'$,
and hence it follows from
\reqnarray{proof of nonadjacent distance larger than one-(i)-555} that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-777}
n_{(b'-1)-j'}=p+1, \textrm{ for } j'=1,2,\ldots,j.
\eeqnarray
By combining \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-777},
$n_{b'+j'}=p+1$ for $j'=1,2,\ldots,j-1$, and $n_{b'+j}<p+1$,
we obtain \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-333}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-444}.
\emph{Subcase 1(b): $j=b'-a'-1$.}
In this subcase, we have $(b'-1)-j=a'$
and it follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-555}
and $n_{a'}=p+2$ in \reqnarray{proof of nonadjacent distance larger than one-(i)-444} that
\beqnarray{}
\alignspace n_{(b'-1)-j'}=p+1, \textrm{ for } j'=1,2,\ldots,j-1,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-1-888}\\
\alignspace n_{(b'-1)-j}=n_{a'}=p+2.
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-1-999}
\eeqnarray
By using \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-888}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-999},
we can argue as in Subcase~1(a) above that
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-222}--\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-555}
still hold.
Since it is clear from $n_{(b'-1)-j}=p+2$
in \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-999}
and $n_{b'+j}<p+1$ that $n_{(b'-1)-j}\neq n_{b'+j}+1$,
it cannot be the case that $\nbf_1^{r_{h-1}}\equiv \mbf_1^{r_{h-1}}$.
As such, we see from \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-555}
that $\nbf_1^{r_{h-1}}\prec \mbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one-(i)-999} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
\emph{Case 2: There exists a positive integer $j$ such that
$1\leq j\leq \min\{b'-a'-1,r_{h-1}-b'\}$,
$n_{b'+j'}=p+1$ for $j'=1,2,\ldots,j-1$, and $n_{b'+j}>p+1$.}
In this case, we can show that $j\geq 2$.
To see this, suppose on the contrary that $j=1$,
then we have $n_{b'+1}>p+1$ in this case.
As it follows from $n_{b'}=p$ in \reqnarray{proof of nonadjacent distance larger than one-(i)-444}
and the condition $|n_{i+1}-n_i|\leq 1$ for $i=1,2,\ldots,r_{h-1}-1$
in \reqnarray{proof of nonadjacent distance larger than one-111}
that $n_{b'+1}$ must be equal to $p-1$ (provided that $p\geq 2$), $p$, or $p+1$,
we have reached a contradiction.
Let $\mbf_1^{r_{h-1}}$ be a sequence of positive integers such that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-111}
m_{b'}=n_{b'}+1,\ m_{b'+1}=n_{b'+1}-1,
\textrm{ and } m_i=n_i \textrm{ for } i\neq b', b'+1.
\eeqnarray
As before, it is easy to show that $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
As $j\geq 2$, we have $n_{b'+1}=p+1$ in this case.
It then follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-111},
$n_{b'+1}=p+1$, and $n_{b'}=p$
in \reqnarray{proof of nonadjacent distance larger than one-(i)-444} that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-222}
m_{b'}-m_{b'+1}=(n_{b'}+1)-(n_{b'+1}-1)=(p+1)-(p+1-1)=1.
\eeqnarray
\bpdffigure{appendix-G-i-b-case-2.pdf}{5.5in}
\epdffigure{appendix-G-(i)-(b)-case-2}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-555}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-666}.}
In the following, we show that
\beqnarray{}
\alignspace
2\leq b'\leq r_{h-1}-2,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-2-333}\\
\alignspace
1\leq j-1\leq \min\{b'-1,r_{h-1}-b'-1\},
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-2-444}\\
\alignspace m_{b'-j'}=m_{(b'+1)+j'}, \textrm{ for } j'=1,2,\ldots,j-2,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-2-555}\\
\alignspace m_{b'-(j-1)}<m_{(b'+1)+(j-1)}.
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-2-666}
\eeqnarray
An illustration of
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-555}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-666}
is given in \rfigure{appendix-G-(i)-(b)-case-2}.
Therefore, it follows from
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-111}--\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-666},
and \reqnarray{comparison rule A-2} in \rlemma{comparison rule A}(ii) that
$\mbf_1^{r_{h-1}}\succ \nbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one-(i)-999} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
From $2\leq j\leq \min\{b'-a'-1,r_{h-1}-b'\}$ and $a'\geq 1$,
we can see that
\beqnarray{}
\alignspace
2\leq \min\{b'-a'-1,r_{h-1}-b'\}\leq b'-a'-1\leq b',
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-2-777}\\
\alignspace
2\leq \min\{b'-a'-1,r_{h-1}-b'\}\leq r_{h-1}-b',
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-2-888}\\
\alignspace
1\leq j-1\leq \min\{b'-a'-2,r_{h-1}-b'-1\}\leq \min\{b'-1,r_{h-1}-b'-1\}.
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-2-999}
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-333}
follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-777}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-888},
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-444}
follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-999}.
To prove \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-555}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-666},
note that we have from $2\leq j\leq \min\{b'-a'-1,r_{h-1}-b'\}\leq b'-a'-1$ that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-aaa}
a'< a'+2\leq b'-(j-1)\leq b'-1<b'.
\eeqnarray
It then follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-111},
\reqnarray{proof of nonadjacent distance larger than one-(i)-555},
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-aaa} that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-bbb}
m_{b'-j'}=n_{b'-j'}=p+1, \textrm{ for } j'=1,2,\ldots,j-1.
\eeqnarray
Furthermore, as we have $n_{b'+j'}=p+1$ for $j'=1,2,\ldots,j-1$ and $n_{b'+j}>p+1$ in this case,
it is clear from \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-111} that
\beqnarray{}
\alignspace m_{(b'+1)+j'}=n_{(b'+1)+j'}=p+1, \textrm{ for } j'=1,2,\ldots,j-2,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-2-ccc}\\
\alignspace m_{(b'+1)+(j-1)}=n_{(b'+1)+(j-1)}=n_{b'+j}>p+1.
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-2-ddd}
\eeqnarray
By combining \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-bbb},
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-ccc},
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-ddd},
we obtain \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-555}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-666}.
\emph{Case 3: $n_{b'+j'}=p+1$ for $j'=1,2,\ldots,\min\{b'-a'-1,r_{h-1}-b'\}$.}
We consider the two subcases $b'-a'-1>r_{h-1}-b'$ and $b'-a'-1\leq r_{h-1}-b'$ separately.
\emph{Subcase 3(a): $b'-a'-1>r_{h-1}-b'$.}
Let $\mbf_1^{r_{h-1}}$ be a sequence of positive integers as given in
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-111}.
As in Case~2 above, we have $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
As it is clear from $b'\geq a'+2$
in \reqnarray{proof of nonadjacent distance larger than one-(i)-333}
and $b'\leq r_{h-1}-1$ that $\min\{b'-a'-1,r_{h-1}-b'\}\geq 1$,
we have $n_{b'+1}=p+1$ in this case and hence it is easy to see that
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-222}
still holds in this subcase.
If $b'=r_{h-1}-1$,
then it follows from $r_{h-1}\geq 3$
in \reqnarray{proof of nonadjacent distance larger than one-111},
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-111},
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-222},
and \reqnarray{comparison rule A-1} in \rlemma{comparison rule A}(i)
that $\mbf_1^{r_{h-1}}\succ \nbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one-(i)-999} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
\bpdffigure{appendix-G-i-b-case-3-1.pdf}{4.5in}
\epdffigure{appendix-G-(i)-(b)-case-3-1}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-222}
(note that we have $\min\{b'-1, r_{h-1}-b'-1\}=r_{h-1}-b'-1$
in \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-555}).}
On the other hand, if $b'\leq r_{h-1}-2$, then we show that
\beqnarray{}
\alignspace \hspace*{-0.2in}
2\leq b'\leq r_{h-1}-2,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-3-111}\\
\alignspace \hspace*{-0.2in}
m_{b'-j'}=m_{(b'+1)+j'}=p+1, \textrm{ for } j'=1,2,\ldots,\min\{b'-1, r_{h-1}-b'-1\}.
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-3-222}
\eeqnarray
An illustration of \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-222}
is given in \rfigure{appendix-G-(i)-(b)-case-3-1}.
Therefore, it follows from
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-111},
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-222},
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-111},
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-222},
and \reqnarray{comparison rule A-4} in \rlemma{comparison rule A}(iii) that
$\mbf_1^{r_{h-1}}\succ \nbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one-(i)-999} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
From $b'\geq a'+2>2$ and $b'\leq r_{h-1}-2$, we see that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-333}
2\leq b'\leq r_{h-1}-2.
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-111}
follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-333}.
To prove \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-222},
note that from $b'-a'-1>r_{h-1}-b'$ in this subcase and $a'\geq 1$, we have
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-444}
r_{h-1}-b'-1<(b'-a'-1)-1<b'-1.
\eeqnarray
It follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-444} that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-555}
\min\{b'-1, r_{h-1}-b'-1\}=r_{h-1}-b'-1.
\eeqnarray
From $b'-a'-1>r_{h-1}-b'$ and $b'\leq r_{h-1}-2$,
we can see that
\beqnarray{}
\alignspace
b'-(r_{h-1}-b'-1)>b'-(b'-a'-2)=a'+2>a',
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-3-666}\\
\alignspace
b'-(r_{h-1}-b'-1)\leq b'-1<b'.
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-3-777}
\eeqnarray
It follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-111},
\reqnarray{proof of nonadjacent distance larger than one-(i)-555},
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-666},
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-777} that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-888}
m_{b'-j'}=n_{b'-j'}=p+1, \textrm{ for } j'=1,2,\ldots,r_{h-1}-b'-1.
\eeqnarray
Furthermore, in this subcase we have
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-999}
n_{b'+j'}=p+1, \textrm{ for } j'=1,2,\ldots,\min\{b'-a'-1,r_{h-1}-b'\}=r_{h-1}-b'.
\eeqnarray
It then follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-2-111}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-999} that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-aaa}
m_{(b'+1)+j'}=n_{(b'+1)+j'}=p+1, \textrm{ for } j'=1,2,\ldots,r_{h-1}-b'-1.
\eeqnarray
By combining
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-555},
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-888},
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-aaa},
we obtain \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-222}.
\emph{Subcase 3(b): $b'-a'-1\leq r_{h-1}-b'$.}
\bpdffigure{appendix-G-i-b-case-3-2.pdf}{4.5in}
\epdffigure{appendix-G-(i)-(b)-case-3-2}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-ccc}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-ddd}.}
Let $\mbf_1^{r_{h-1}}$ be a sequence of positive integers as given in
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-111}.
As in Case~1 above, we have $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
In this subcase, we show that
\beqnarray{}
\alignspace 1\leq b'-a'-1\leq \min\{(b'-1)-1,r_{h-1}-(b'-1)-1\}
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-3-bbb}\\
\alignspace n_{(b'-1)-j'}=n_{b'+j'}=p+1, \textrm{ for } j'=1,2,\ldots,b'-a'-2,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-3-ccc}\\
\alignspace n_{(b'-1)-(b'-a'-1)}=p+2>n_{b'+(b'-a'-1)}=p+1.
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-3-ddd}
\eeqnarray
An illustration of
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-ccc}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-ddd}
is given in \rfigure{appendix-G-(i)-(b)-case-3-2}.
Therefore, it follows from
$\nbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$
in \reqnarray{proof of nonadjacent distance larger than one-111},
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-222}--\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-111},
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-bbb}--\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-ddd},
and \reqnarray{comparison rule A-3} in \rlemma{comparison rule A}(ii)
(with $j=b'-a'-1$) that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-eee}
\nbf_1^{r_{h-1}}\preceq \mbf_1^{r_{h-1}}.
\eeqnarray
From $b'\geq a'+2$ in \reqnarray{proof of nonadjacent distance larger than one-(i)-333},
$b'-a'-1\leq r_{h-1}-b'$, and $a'\geq 1$, we can see that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-fff}
1\leq b'-a'-1\leq \min\{b'-a'-1,r_{h-1}-b'\}\leq \min\{(b'-1)-1,r_{h-1}-(b'-1)-1\}.
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-bbb}
follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-fff}.
To prove \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-ccc}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-ddd},
note that it is clear from
\reqnarray{proof of nonadjacent distance larger than one-(i)-555},
$(b'-1)-(b'-a'-1)=a'$,
and $n_{a'}=p+2$ in \reqnarray{proof of nonadjacent distance larger than one-(i)-444} that
\beqnarray{}
\alignspace n_{(b'-1)-j'}=p+1, \textrm{ for } j'=1,2,\ldots,b'-a'-2,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-3-ggg}\\
\alignspace n_{(b'-1)-(b'-a'-1)}=n_{a'}=p+2.
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-3-hhh}
\eeqnarray
Also, in this case we have
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-iii}
n_{b'+j'}=p+1, \textrm{ for } j'=1,2,\ldots,\min\{b'-a'-1,r_{h-1}-b'\}=b'-a'-1.
\eeqnarray
By combining
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-ggg},
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-hhh},
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-iii},
we obtain \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-ccc}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-ddd}.
Now let ${\mbf'}_1^{r_{h-1}}$ be a sequence of positive integers such that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-jjj}
m'_{b'-2}=m_{b'-2}-1,\ m'_{b'-1}=m_{b'-1}+1, \textrm{ and } m'_i=m_i \textrm{ for } i\neq b'-2, b'-1.
\eeqnarray
Again, it is easy to show that ${\mbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
If $b'=a'+2$, then we see from
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-111},
$n_{a'}=p+2$ in \reqnarray{proof of nonadjacent distance larger than one-(i)-444},
and $n_{b'-1}=p+1$ in \reqnarray{proof of nonadjacent distance larger than one-(i)-888} that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-kkk}
\alignspace m_{b'-2}-m_{b'-1}=n_{b'-2}-(n_{b'-1}-1)=n_{a'}-n_{b'-1}+1=(p+2)-(p+1)+1=2.
\eeqnarray
Therefore, it follows from
$r_{h-1}\geq 3$ in \reqnarray{proof of nonadjacent distance larger than one-111},
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-jjj},
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-kkk},
and \reqnarray{adjacent distance larger than one-2} in \rlemma{adjacent distance larger than one}(ii) that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-1111}
\mbf_1^{r_{h-1}}\preceq {\mbf'}_1^{r_{h-1}},
\eeqnarray
where $\mbf_1^{r_{h-1}}\equiv {\mbf'}_1^{r_{h-1}}$
if and only if $r_{h-1}=2$ and $m_1=m_2+2$.
Since it is clear from $r_{h-1}\geq 3$
in \reqnarray{proof of nonadjacent distance larger than one-111}
that $r_{h-1}\neq 2$, it cannot be the case that $\mbf_1^{r_{h-1}}\equiv {\mbf'}_1^{r_{h-1}}$.
As such, we see from \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-eee}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-1111}
that $\nbf_1^{r_{h-1}}\preceq \mbf_1^{r_{h-1}}\prec {\mbf'}_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one-(i)-999} holds
with ${\nbf'}_1^{r_{h-1}}={\mbf'}_1^{r_{h-1}}$.
On the other hand, if $b'\geq a'+3$, then we have $a'<b'-2<b'$
and it follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-555} that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-2222}
n_{b'-2}=n_{b'-1}=p+1.
\eeqnarray
From \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-111}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-2222},
we see that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-3333}
\alignspace m_{b'-2}-m_{b'-1}=n_{b'-2}-(n_{b'-1}-1)=1.
\eeqnarray
\bpdffigure{appendix-G-i-b-case-3-3.pdf}{4.5in}
\epdffigure{appendix-G-(i)-(b)-case-3-3}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-6666}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-7777}.}
We will show that
\beqnarray{}
\alignspace 2\leq b'-2\leq r_{h-1}-2,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-3-4444}\\
\alignspace 1\leq b'-a'-2\leq \min\{(b'-2)-1,r_{h-1}-(b'-2)-1\}.
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-3-5555}\\
\alignspace m_{(b'-2)-j'}=m_{(b'-1)+j'}=p+1, \textrm{ for } j'=1,2,\ldots,b'-a'-3,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-3-6666}\\
\alignspace m_{(b'-2)-(b'-a'-2)}=p+2>m_{(b'-1)+(b'-a'-2)}=p+1.
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-3-7777}
\eeqnarray
An illustration of
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-6666}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-7777}
is given in \rfigure{appendix-G-(i)-(b)-case-3-3}.
Therefore, it follows from
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-jjj},
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-3333}--\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-7777},
and \reqnarray{comparison rule A-3} in \rlemma{comparison rule A}(ii)
(with $j=b'-a'-2$) that
\beqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-8888}
\mbf_1^{r_{h-1}}\preceq {\mbf'}_1^{r_{h-1}},
\eeqnarray
where $\mbf_1^{r_{h-1}}\equiv {\mbf'}_1^{r_{h-1}}$
if and only if $(b'-2)-(b'-a'-2)=1$, $(b'-1)+(b'-a'-2)=r_{h-1}$, and $m_1=m_{r_{h-1}}+1$.
Since in this subcase we have $b'-a'-1\leq r_{h-1}-b'$,
it is clear that $(b'-1)+(b'-a'-2)\leq (b'-1)+(r_{h-1}-b'-1)=r_{h-1}-2$.
This implies that $(b'-1)+(b'-a'-2)\neq r_{h-1}$
and hence it cannot be the case that $\mbf_1^{r_{h-1}}\equiv {\mbf'}_1^{r_{h-1}}$.
As such, we see from \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-eee}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-8888}
that $\nbf_1^{r_{h-1}}\preceq \mbf_1^{r_{h-1}}\prec {\mbf'}_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one-(i)-999} holds
with ${\nbf'}_1^{r_{h-1}}={\mbf'}_1^{r_{h-1}}$.
To prove \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-4444}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-5555},
note from $a'\geq 1$, $b'\geq a'+3$, $b'\leq r_{h-1}-1$,
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-bbb} that
\beqnarray{}
\alignspace \hspace*{-0.3in}
2\leq a'+1\leq b'-2\leq r_{h-1}-2,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-3-9999}\\
\alignspace \hspace*{-0.3in}
1\leq b'-a'-2\leq \min\{b'-3,r_{h-1}-b'-1\}\leq \min\{(b'-2)-1,r_{h-1}-(b'-2)-1\}.
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-3-aaaa}
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-4444}
follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-9999},
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-5555}
follows from \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-aaaa}.
To prove \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-6666}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-7777},
note that from \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-1-111},
$n_{a'}=p+2$ and $n_{b'}=p$ in \reqnarray{proof of nonadjacent distance larger than one-(i)-444},
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-ggg}--\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-iii},
we see that
\beqnarray{}
\alignspace m_{(b'-2)-j'}=n_{(b'-2)-j'}=p+1, \textrm{ for } j'=1,2,\ldots,b'-a'-3,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-3-bbbb}\\
\alignspace m_{(b'-2)-(b'-a'-2)}=n_{(b'-2)-(b'-a'-2)}=n_{a'}=p+2.
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-3-cccc}\\
\alignspace m_{(b'-1)+1}=m_{b'}=n_{b'}+1=p+1,
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-3-dddd}\\
\alignspace m_{(b'-1)+j'}=n_{(b'-1)+j'}=p+1, \textrm{ for } j'=2,3,\ldots,b'-a',
\label{eqn:proof of nonadjacent distance larger than one-(i)-(b)-case-3-eeee}
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-6666}
and \reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-7777}
follow from
\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-bbbb}--\reqnarray{proof of nonadjacent distance larger than one-(i)-(b)-case-3-eeee}.
(ii) Note that in \rlemma{nonadjacent distance larger than one}(ii),
we have $n_a-n_b\leq -2$ for some $1\leq a<b\leq r_{h-1}$ and $b\geq a+2$.
For ease of presentation, let $n_a=p$.
Then we have from $n_a-n_b\leq -2$ that $n_b\geq p+2$.
It is easy to see from $n_a=p$, $n_b\geq p+2$, $b\geq a+2>a$,
and the condition $|n_{i+1}-n_i|\leq 1$ for $i=1,2,\ldots,r_{h-1}-1$
in \reqnarray{proof of nonadjacent distance larger than one-111}
that there must exist a positive integer $c$ such that $a< c\leq b$ and $n_c=p+2$.
Let
\beqnarray{}
a'\aligneq \max\{i:n_i=p,\ a\leq i<c\},
\label{eqn:proof of nonadjacent distance larger than one-(ii)-111}\\
b'\aligneq \min\{i:n_i=p+2,\ a'<i\leq c\}.
\label{eqn:proof of nonadjacent distance larger than one-(ii)-222}
\eeqnarray
In other words, $a'$ is the largest positive integer $i$ such that $a\leq i<c$ and $n_i=p$,
and $b'$ is the smallest positive integer $i$ such that $a'<i\leq c$ and $n_i=p+2$.
Note that $a'$ and $b'$ are well defined as we have $n_a=p$ and $n_c=p+2$.
Since we have from \reqnarray{proof of nonadjacent distance larger than one-(ii)-111}
and \reqnarray{proof of nonadjacent distance larger than one-(ii)-222}
that $n_{a'}=p$ and $n_{b'}=p+2$, it is easy to see from the condition
$|n_{i+1}-n_i|\leq 1$ for $i=1,2,\ldots,r_{h-1}-1$
in \reqnarray{proof of nonadjacent distance larger than one-111} that $b'\geq a'+2$.
In summary, we have
\beqnarray{}
\alignspace a\leq a'<b'\leq c\leq b \textrm{ and } b'\geq a'+2,
\label{eqn:proof of nonadjacent distance larger than one-(ii)-333}\\
\alignspace n_a=n_{a'}=p,\ n_{b'}=n_c=p+2, \textrm{ and } n_b\geq p+2.
\label{eqn:proof of nonadjacent distance larger than one-(ii)-444}
\eeqnarray
\bpdffigure{appendix-G-ii.pdf}{4.5in}
\epdffigure{appendix-G-(ii)}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one-(ii)-333}--\reqnarray{proof of nonadjacent distance larger than one-(ii)-555}.}
We claim that
\beqnarray{proof of nonadjacent distance larger than one-(ii)-555}
n_i=p+1, \textrm{ for } a'<i<b'.
\eeqnarray
An illustration of
\reqnarray{proof of nonadjacent distance larger than one-(ii)-333}--\reqnarray{proof of nonadjacent distance larger than one-(ii)-555}
is given in \rfigure{appendix-G-(ii)}.
We prove \reqnarray{proof of nonadjacent distance larger than one-(ii)-555} by contradiction.
First assume that $n_i\leq p$ for some $a'<i<b'$.
From $a\leq a'<b'\leq c$ in \reqnarray{proof of nonadjacent distance larger than one-(ii)-333} and $a'<i<b'$,
we have $a\leq a'<i<b'\leq c$ and hence it follows from the definition of $a'$
in \reqnarray{proof of nonadjacent distance larger than one-(ii)-111} that $n_i\neq p$.
Since we assume that $n_i\leq p$, it is clear that we must have $n_i<p$.
As such, we see from $n_i<p$,
$n_{b'}=p+2$ in \reqnarray{proof of nonadjacent distance larger than one-(ii)-444},
$i<b'$, and the condition $|n_{i+1}-n_i|\leq 1$ for $i=1,2,\ldots,r_{h-1}-1$
in \reqnarray{proof of nonadjacent distance larger than one-111}
that there must exist a positive integer $a''$ such that
\beqnarray{proof of nonadjacent distance larger than one-(ii)-666}
i<a''<b' \textrm{ and } n_{a''}=p.
\eeqnarray
From $a\leq a'<b'\leq c$ in \reqnarray{proof of nonadjacent distance larger than one-(ii)-333},
$a'<i<b'$, and $i<a''<b'$ in \reqnarray{proof of nonadjacent distance larger than one-(ii)-666},
we have $a\leq a'<i<a''<b'\leq c$ and hence it follows from the definition of $a'$
in \reqnarray{proof of nonadjacent distance larger than one-(ii)-111} that $n_{a''}\neq p$,
contradicting to $n_{a''}=p$ in \reqnarray{proof of nonadjacent distance larger than one-(ii)-666}.
Now assume that $n_i\geq p+2$ for some $a'<i<b'$.
From $a'<b'\leq c$ in \reqnarray{proof of nonadjacent distance larger than one-(ii)-333} and $a'<i<b'$,
we have $a'<i<b'\leq c$ and hence it follows from the definition of $b'$
in \reqnarray{proof of nonadjacent distance larger than one-(ii)-222} that $n_i\neq p+2$.
Since we assume that $n_i\geq p+2$, it is clear that we must have $n_i>p+2$.
As such, we see from $n_{a'}=p$ in \reqnarray{proof of nonadjacent distance larger than one-(ii)-444},
$n_i>p+2$, $a'<i$, and the condition $|n_{i+1}-n_i|\leq 1$ for $i=1,2,\ldots,r_{h-1}-1$
in \reqnarray{proof of nonadjacent distance larger than one-111}
that there must exist a positive integer $b''$ such that
\beqnarray{proof of nonadjacent distance larger than one-(ii)-777}
a'<b''<i \textrm{ and } n_{b''}=p+2.
\eeqnarray
From $a'<b''<i$ in \reqnarray{proof of nonadjacent distance larger than one-(ii)-777},
$a'<i<b'$, and $a'<b'\leq c$ in \reqnarray{proof of nonadjacent distance larger than one-(ii)-333},
we have $a'<b''<i<b'\leq c$ and hence it follows from the definition of $b'$
in \reqnarray{proof of nonadjacent distance larger than one-(ii)-222} that $n_{b''}\neq p+2$,
contradicting to $n_{b''}=p+2$ in \reqnarray{proof of nonadjacent distance larger than one-(ii)-777}.
The proof of \reqnarray{proof of nonadjacent distance larger than one-(ii)-555} is completed.
To prove \rlemma{nonadjacent distance larger than one}(ii),
we need to show that there exists a sequence of positive integers
${\nbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$ such that
\beqnarray{proof of nonadjacent distance larger than one-(ii)-888}
{\nbf'}_1^{r_{h-1}}\succ\nbf_1^{r_{h-1}}.
\eeqnarray
We consider the following four possible cases.
Note that in Case~2--Case~4 below,
we have $b'\leq r_{h-1}-1$ and hence it follows from $a'\geq 1$
and $b'\geq a'+2$ in \reqnarray{proof of nonadjacent distance larger than one-(ii)-333} that
\beqnarray{proof of nonadjacent distance larger than one-(ii)-999}
2\leq b'-1\leq r_{h-1}-2.
\eeqnarray
\emph{Case 1: $b'=r_{h-1}$.}
Let $\mbf_1^{r_{h-1}}$ be a sequence of positive integers such that
\beqnarray{proof of nonadjacent distance larger than one-(ii)-case-1-111}
m_{b'-1}=n_{b'-1}+1,\ m_{b'}=n_{b'}-1, \textrm{ and } m_i=n_i \textrm{ for } i\neq b'-1, b'.
\eeqnarray
As before, it is easy to show that $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
From \reqnarray{proof of nonadjacent distance larger than one-(ii)-555}
and $b'\geq a'+2$ in \reqnarray{proof of nonadjacent distance larger than one-(ii)-333},
we see that $n_{b'-1}=p+1$.
It then follows from \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-1-111},
$n_{b'-1}=p+1$, and $n_{b'}=p+2$ in \reqnarray{proof of nonadjacent distance larger than one-(ii)-444} that
\beqnarray{proof of nonadjacent distance larger than one-(ii)-case-1-222}
m_{b'-1}-m_{b'}=(n_{b'-1}+1)-(n_{b'}-1)=(p+1+1)-(p+2-1)=1.
\eeqnarray
Therefore, it follows from
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of nonadjacent distance larger than one-(ii)-case-1-111},
\reqnarray{proof of nonadjacent distance larger than one-(ii)-case-1-222},
$b'-1=r_{h-1}-1$, and \reqnarray{comparison rule A-1} in \rlemma{comparison rule A}(i)
that $\mbf_1^{r_{h-1}}\succ\nbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one-(ii)-888} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
\emph{Case 2: $b'\leq r_{h-1}-1$ and there exists a positive integer $j$ such that
$1\leq j\leq \min\{b'-a'-1,r_{h-1}-b'\}$, $n_{b'+j'}=p+1$ for $j'=1,2,\ldots,j-1$, and $n_{b'+j}>p+1$.}
Let $\mbf_1^{r_{h-1}}$ be a sequence of positive integers as given in
\reqnarray{proof of nonadjacent distance larger than one-(ii)-case-1-111}.
As in Case~1 above, we have $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$
and \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-1-222}
also holds in this case.
\bpdffigure{appendix-G-ii-case-2.pdf}{5.5in}
\epdffigure{appendix-G-(ii)-case-2}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-2-333}
and \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-2-444}:
(a) $j<b'-a'-1$; (b) $j=b'-a'-1$.}
In the following, we show that
\beqnarray{}
\alignspace 1\leq j\leq \min\{(b'-1)-1,r_{h-1}-(b'-1)-1\},
\label{eqn:proof of nonadjacent distance larger than one-(ii)-case-2-222}\\
\alignspace m_{(b'-1)-j'}=m_{b'+j'}, \textrm{ for } j'=1,2,\ldots,j-1,
\label{eqn:proof of nonadjacent distance larger than one-(ii)-case-2-333}\\
\alignspace m_{(b'-1)-j}<m_{b'+j}.
\label{eqn:proof of nonadjacent distance larger than one-(ii)-case-2-444}
\eeqnarray
An illustration of
\reqnarray{proof of nonadjacent distance larger than one-(ii)-case-2-333}
and \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-2-444}
is given in \rfigure{appendix-G-(ii)-case-2}.
Therefore, it follows from
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of nonadjacent distance larger than one-(ii)-999}--\reqnarray{proof of nonadjacent distance larger than one-(ii)-case-2-444},
and \reqnarray{comparison rule A-2} in \rlemma{comparison rule A}(ii) that
$\mbf_1^{r_{h-1}}\succ \nbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one-(ii)-888} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
From $1\leq j\leq \min\{b'-a'-1,r_{h-1}-b'\}$ and $a'\geq 1$, we have
\beqnarray{proof of nonadjacent distance larger than one-(ii)-case-2-555}
1\leq j\leq \min\{b'-a'-1,r_{h-1}-b'\}\leq \min\{(b'-1)-1,r_{h-1}-(b'-1)-1\}.
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-2-222}
follows from \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-2-555}.
To prove \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-2-333}
and \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-2-444},
note that we have $j\leq \min\{b'-a'-1, r_{h-1}-b'\}\leq b'-a'-1$.
If $j<b'-a'-1$, then we have $a'<b'-1-j<b'$ and it follows from
\reqnarray{proof of nonadjacent distance larger than one-(ii)-case-1-111}
and \reqnarray{proof of nonadjacent distance larger than one-(ii)-555} that
\beqnarray{proof of nonadjacent distance larger than one-(ii)-case-2-666}
m_{(b'-1)-j'}=n_{(b'-1)-j'}=p+1, \textrm{ for } j'=1,2,\ldots,j.
\eeqnarray
On the other hand, if $j=b'-a'-1$, then we have $a'=b'-1-j<b'$
and it follows from \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-1-111},
\reqnarray{proof of nonadjacent distance larger than one-(ii)-555},
and $n_{a'}=p$ in \reqnarray{proof of nonadjacent distance larger than one-(ii)-444} that
\beqnarray{}
\alignspace
m_{(b'-1)-j'}=n_{(b'-1)-j'}=p+1, \textrm{ for } j'=1,2,\ldots,j-1,
\label{eqn:proof of nonadjacent distance larger than one-(ii)-case-2-777}\\
\alignspace
m_{(b'-1)-j}=n_{(b'-1)-j}=n_{a'}=p.
\label{eqn:proof of nonadjacent distance larger than one-(ii)-case-2-888}
\eeqnarray
As in this case we have $n_{b'+j'}=p+1$ for $j'=1,2,\ldots,j-1$, and $n_{b'+j}>p+1$,
we immediately see from \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-1-111} that
\beqnarray{}
\alignspace
m_{b'+j'}=n_{b'+j'}=p+1, \textrm{ for } j'=1,2,\ldots,j-1,
\label{eqn:proof of nonadjacent distance larger than one-(ii)-case-2-999}\\
\alignspace
m_{b'+j}=n_{b'+j}>p+1.
\label{eqn:proof of nonadjacent distance larger than one-(ii)-case-2-aaa}
\eeqnarray
By combining
\reqnarray{proof of nonadjacent distance larger than one-(ii)-case-2-666}--\reqnarray{proof of nonadjacent distance larger than one-(ii)-case-2-aaa},
we obtain \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-2-333}
and \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-2-444}.
\emph{Case 3: $b'\leq r_{h-1}-1$ and there exists a positive integer $j$ such that
$1\leq j\leq \min\{b'-a'-1,r_{h-1}-b'\}$, $n_{b'+j'}=p+1$ for $j'=1,2,\ldots,j-1$, and $n_{b'+j}<p+1$.}
In this case, we can show that $j\geq 2$.
To see this, suppose on the contrary that $j=1$,
then we have $n_{b'+1}<p+1$ in this case.
As it follows from $n_{b'}=p+2$ in \reqnarray{proof of nonadjacent distance larger than one-(ii)-444}
and the condition $|n_{i+1}-n_i|\leq 1$ for $i=1,2,\ldots,r_{h-1}-1$
in \reqnarray{proof of nonadjacent distance larger than one-111}
that $n_{b'+1}$ must be equal to $p+1$, $p+2$, or $p+3$,
we have reached a contradiction.
Since $j\geq 2$, we have $n_{b'+j-1}=p+1$ in this case.
It then follows from the condition $|n_{i+1}-n_i|\leq 1$ for $i=1,2,\ldots,r_{h-1}-1$
in \reqnarray{proof of nonadjacent distance larger than one-111}
that $n_{b'+j}$ must be equal to $p$, $p+1$, or $p+2$.
As we also have $n_{b'+j}<p+1$ in this case, we immediately see that $n_{b'+j}=p$.
From $n_{b'}=p+2$, $n_{b'+j'}=p+1$ for $j'=1,2,\ldots,j-1$, $n_{b'+j}=p$,
and $b'\geq 3$ in \reqnarray{proof of nonadjacent distance larger than one-(ii)-999},
we can argue in the same way as in the proof of (i) above
(with the roles of $a'$ and $b'$ in the proof of (i) replaced by $b'$ and $b'+j$, respectively)
that there exists a sequence of positive integers ${\nbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$
such that ${\nbf'}_1^{r_{h-1}}\succ\nbf_1^{r_{h-1}}$.
\emph{Case 4: $b'\leq r_{h-1}-1$ and $n_{b'+j'}=p+1$ for $j'=1,2,\ldots,\min\{b'-a'-1,r_{h-1}-b'\}$.}
Let $\mbf_1^{r_{h-1}}$ be a sequence of positive integers as given in
\reqnarray{proof of nonadjacent distance larger than one-(ii)-case-1-111}.
As in Case~1 above, we have $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$
and \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-1-222}
also holds in this case.
We then consider the two subcases $b'-a'-1>r_{h-1}-b'$ and $b'-a'-1\leq r_{h-1}-b'$ separately.
\emph{Subcase 4(a): $b'-a'-1>r_{h-1}-b'$.}
\bpdffigure{appendix-G-ii-case-4-1.pdf}{4.5in}
\epdffigure{appendix-G-(ii)-case-4-1}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-111}
(note that we have $\min\{(b'-1)-1, r_{h-1}-(b'-1)-1\}=r_{h-1}-b'$
in \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-444}).}
In this subcase, we show that
\beqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-111}
m_{(b'-1)-j'}=m_{b'+j'}=p+1, \textrm{ for } j'=1,2,\ldots,\min\{(b'-1)-1, r_{h-1}-(b'-1)-1\}.
\eeqnarray
An illustration of \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-111}
is given in \rfigure{appendix-G-(ii)-case-4-1}.
Therefore, it follows from
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of nonadjacent distance larger than one-(ii)-999}--\reqnarray{proof of nonadjacent distance larger than one-(ii)-case-1-222},
\reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-111},
and \reqnarray{comparison rule A-4} in \rlemma{comparison rule A}(iii) that
$\mbf_1^{r_{h-1}}\succ \nbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one-(ii)-888} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
To prove \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-111},
note that from $b'-a'-1>r_{h-1}-b'$, $a'\geq 1$, and $b'\leq r_{h-1}-1$, we have
\beqnarray{}
\alignspace
r_{h-1}-b'<b'-a'-1\leq b'-2,
\label{eqn:proof of nonadjacent distance larger than one-(ii)-case-4-222}\\
\alignspace
a'<(b'-1)-(r_{h-1}-b')\leq (b'-1)-1<b'.
\label{eqn:proof of nonadjacent distance larger than one-(ii)-case-4-333}
\eeqnarray
From \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-222},
we see that
\beqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-444}
\min\{(b'-1)-1, r_{h-1}-(b'-1)-1\}=\min\{b'-2, r_{h-1}-b'\}=r_{h-1}-b'.
\eeqnarray
From \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-1-111},
\reqnarray{proof of nonadjacent distance larger than one-(ii)-555},
and \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-333},
we have
\beqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-555}
m_{(b'-1)-j'}=n_{(b'-1)-j'}=p+1, \textrm{ for } j'=1,2,\ldots,r_{h-1}-b'.
\eeqnarray
Furthermore, in this subcase we have from
\reqnarray{proof of nonadjacent distance larger than one-(ii)-case-1-111}
and $b'-a'-1>r_{h-1}-b'$ that
\beqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-666}
m_{b'+j'}=n_{b'+j'}=p+1, \textrm{ for } j'=1,2,\ldots,\min\{b'-a'-1,r_{h-1}-b'\}=r_{h-1}-b'.
\eeqnarray
By combining \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-444},
\reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-555},
and \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-666},
we obtain \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-111}.
\emph{Subcase 4(b): $b'-a'-1\leq r_{h-1}-b'$.}
\bpdffigure{appendix-G-ii-case-4-2.pdf}{4.5in}
\epdffigure{appendix-G-(ii)-case-4-2}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-888}
and \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-999}.}
In this subcase, we show that
\beqnarray{}
\alignspace
1\leq b'-a'-1\leq \min\{(b'-1)-1, r_{h-1}-(b'-1)-1\},
\label{eqn:proof of nonadjacent distance larger than one-(ii)-case-4-777}\\
\alignspace
m_{(b'-1)-j'}=m_{b'+j'}=p+1, \textrm{ for } j'=1,2,\ldots,b'-a'-2,
\label{eqn:proof of nonadjacent distance larger than one-(ii)-case-4-888}\\
\alignspace
m_{(b'-1)-(b'-a'-1)}=p<m_{b'+(b'-a'-1)}=p+1.
\label{eqn:proof of nonadjacent distance larger than one-(ii)-case-4-999}
\eeqnarray
An illustration of
\reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-888}
and \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-999}
is given in \rfigure{appendix-G-(ii)-case-4-2}.
Therefore, it follows from
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of nonadjacent distance larger than one-(ii)-999}--\reqnarray{proof of nonadjacent distance larger than one-(ii)-case-1-222},
\reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-777}--\reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-999},
and \reqnarray{comparison rule A-2} in \rlemma{comparison rule A}(ii) that
$\mbf_1^{r_{h-1}}\succ \nbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one-(ii)-888} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
From $b'\geq a'+2$ in \reqnarray{proof of nonadjacent distance larger than one-(ii)-333}
and $a'\geq 1$, we see that
\beqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-aaa}
1\leq b'-a'-1\leq b'-2.
\eeqnarray
It then follows from \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-aaa}
and $b'-a'-1\leq r_{h-1}-b'$ that
\beqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-bbb}
1\leq b'-a'-1\leq \min\{b'-2, r_{h-1}-b'\}=\min\{(b'-1)-1, r_{h-1}-(b'-1)-1\}.
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-777}
follows from \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-bbb}.
To prove \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-888}
and \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-999},
note that from \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-1-111},
\reqnarray{proof of nonadjacent distance larger than one-(ii)-555},
$a'=(b'-1)-(b'-a'-1)<b'$,
and $n_{a'}=p$ in \reqnarray{proof of nonadjacent distance larger than one-(ii)-444},
we have
\beqnarray{}
\alignspace
m_{(b'-1)-j'}=n_{(b'-1)-j'}=p+1, \textrm{ for } j'=1,2,\ldots,b'-a'-2,
\label{eqn:proof of nonadjacent distance larger than one-(ii)-case-4-ccc}\\
\alignspace
m_{(b'-1)-(b'-a'-1)}=n_{(b'-1)-(b'-a'-1)}=n_{a'}=p.
\label{eqn:proof of nonadjacent distance larger than one-(ii)-case-4-ddd}
\eeqnarray
Furthermore, in this subcase we have from
\reqnarray{proof of nonadjacent distance larger than one-(ii)-case-1-111}
and $b'-a'-1\leq r_{h-1}-b'$ that
\beqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-eee}
m_{b'+j'}=n_{b'+j'}=p+1, \textrm{ for } j'=1,2,\ldots,\min\{b'-a'-1,r_{h-1}-b'\}=b'-a'-1.
\eeqnarray
By combining \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-ccc},
\reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-ddd},
and \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-eee},
we obtain \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-888}
and \reqnarray{proof of nonadjacent distance larger than one-(ii)-case-4-999}.
\bappendix{Proof of \rlemma{main lemma}}{proof of main lemma}
In this appendix,
we use \rcorollary{adjacent distance larger than one}(i) (corollary to \rlemma{adjacent distance larger than one}),
\rcorollary{nonadjacent distance larger than one}(i) (corollary to \rlemma{nonadjacent distance larger than one}),
and Comparison rule A in \rlemma{comparison rule A} to prove \rlemma{main lemma}.
Let $\nbf_1^{r_{h-1}}(h)$ be an optimal sequence over $\Ncal_{M,k}(h)$.
As commented before the statement of \rlemma{main lemma},
we can use \rcorollary{adjacent distance larger than one}(i)
and \rcorollary{nonadjacent distance larger than one}(i) to show that
\beqnarray{proof of main lemma-111}
n_i(h)=
\bselection
q_h+1, &\textrm{if } i=i_1,i_2,\ldots,i_{r_h}, \\
q_h, &\textrm{otherwise},
\eselection
\eeqnarray
for some $1\leq i_1<i_2<\cdots <i_{r_h}\leq r_{h-1}$.
In the following, we show that $i_1=1$ by contradiction.
Assume on the contrary that $i_1\geq 2$.
If $h=1$ and $q_h=1$,
then we see from $\nbf_1^{r_{h-1}}(1)=\nbf_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)=\Ncal_{M,k}(1)$
and the definition of $\Ncal_{M,k}(1)$ in \reqnarray{N-M-k-h} that $n_1(1)\geq 2$.
We also see from \reqnarray{proof of main lemma-111} and $i_1\geq 2$ that $n_1(1)=n_1(h)=q_h=1$,
and a contradiction is reached.
On the other hand, if $h\neq 1$ or $q_h\neq 1$,
then we will use Comparison rule A in \rlemma{comparison rule A} to show that
there exists a sequence ${\nbf'}_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$
such that ${\nbf'}_1^{r_{h-1}}(h)\succ\nbf_1^{r_{h-1}}(h)$,
contradicting to the optimality of $\nbf_1^{r_{h-1}}(h)$.
For simplicity, let $\nbf_1^{r_{h-1}}=\nbf_1^{r_{h-1}}(h)$.
Let ${\nbf'}_1^{r_{h-1}}$ be a sequence of positive integers such that
\beqnarray{proof of main lemma-222}
n'_{i_1-1}=n_{i_1-1}+1,\ n'_{i_1}=n_{i_1}-1,
\textrm{ and } n'_i=n_i \textrm{ for } i\neq i_1-1, i_1.
\eeqnarray
As $i_1\geq 2$, we have $1\leq i_1-1<i_1$ and it is easy to see from
\reqnarray{proof of main lemma-222}, \reqnarray{proof of main lemma-111},
$\nbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$, and \reqnarray{N-M-k-h} that
\beqnarray{}
\alignspace
n'_1=
\bselection
n_1+1=q_h+1, &\textrm{if } i_1-1=1, \\
n_1=q_h, &\textrm{otherwise},
\eselection
\label{eqn:proof of main lemma-333}\\
\alignspace
\sum_{i=1}^{r_{h-1}}n'_i=\sum_{i=1}^{r_{h-1}}n_i=r_{h-2}.
\label{eqn:proof of main lemma-444}
\eeqnarray
In the case that $h=1$, we must have $q_h\neq 1$, i.e., $q_h\geq 2$,
and hence it is clear from \reqnarray{proof of main lemma-333}
that $n'_1\geq q_h\geq 2$.
As such, it follows from $\nbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of main lemma-111}, \reqnarray{proof of main lemma-222},
and \reqnarray{proof of main lemma-444} that ${\nbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
Note that from
\reqnarray{proof of main lemma-222} and \reqnarray{proof of main lemma-111},
we have
\beqnarray{proof of main lemma-555}
n'_{i_1-1}-n'_{i_1}=(n_{i_1-1}+1)-(n_{i_1}-1)=(q_h+1)-(q_h+1-1)=1.
\eeqnarray
Furthermore, note that in the case that $h=1$ and $i_1-1=1$,
we have $q_h\geq 2$ and it follows from \reqnarray{proof of main lemma-333}
that $n'_1=q_h+1\geq 3$.
Now we have ${\nbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
$n'_{i_1-1}-n'_{i_1}=1$ in \reqnarray{proof of main lemma-555},
$n'_1\geq 3$ in the case that $h=1$ and $i_1-1=1$,
and $n_{i_1-1}=n'_{i_1-1}-1$, $n_{i_1}=n'_{i_1}+1$, and $n_i=n'_i$ for $i\neq i_1-1, i_1$
in \reqnarray{proof of main lemma-222}.
As such, we are in a position to use Comparison rule A in \rlemma{comparison rule A} (with $a=i_1-1$)
to show that ${\nbf'}_1^{r_{h-1}}\succ\nbf_1^{r_{h-1}}$.
We need to consider the two cases $r_h=1$ and $r_h\geq 2$ separately.
\emph{Case 1: $r_h=1$}.
In this case, we have from \reqnarray{proof of main lemma-222},
\reqnarray{proof of main lemma-111}, and $i_1\geq 2$ that
\beqnarray{proof of main lemma-case-1-111}
n'_i=
\bselection
q_h+1, &\textrm{if } i=i_1-1, \\
q_h, &\textrm{otherwise}.
\eselection
\eeqnarray
If $i_1=2$ or $i_1=r_{h-1}$, then we have $i_1-1=1$ or $i_1-1=r_{h-1}-1$,
and it follows from \reqnarray{comparison rule A-1} in \rlemma{comparison rule A}(i)
that ${\nbf'}_1^{r_{h-1}}\succ\nbf_1^{r_{h-1}}$.
On the other hand, if $3\leq i_1\leq r_{h-1}-1$,
then we have
\beqnarray{proof of main lemma-case-1-222}
2\leq i_1-1\leq r_{h-1}-2
\eeqnarray
From \reqnarray{proof of main lemma-case-1-111}, it is easy to see that
\beqnarray{proof of main lemma-case-1-333}
n'_{(i_1-1)-j'}=n'_{i_1+j'}=q_h, \textrm{ for } j'=1,2,\ldots,\min\{(i_1-1)-1,r_{h-1}-(i_1-1)-1\}.
\eeqnarray
Therefore, it follows from
\reqnarray{proof of main lemma-case-1-222}, \reqnarray{proof of main lemma-case-1-333},
and \reqnarray{comparison rule A-4} in \rlemma{comparison rule A}(iii)
that ${\nbf'}_1^{r_{h-1}}\succ\nbf_1^{r_{h-1}}$.
\emph{Case 2: $r_h\geq 2$}.
As $2\leq i_1<i_2<\cdots <i_{r_h}\leq r_{h-1}$ and $r_h\geq 2$,
we have $2\leq i_1\leq i_2-1\leq r_{h-1}-1$ in this case.
If $i_1=2$, then we have $i_1-1=1$, and it follows from \reqnarray{comparison rule A-1}
in \rlemma{comparison rule A}(i) that ${\nbf'}_1^{r_{h-1}}\succ\nbf_1^{r_{h-1}}$.
On the other hand, if $3\leq i_1\leq r_{h-1}-1$,
then \reqnarray{proof of main lemma-case-1-222} holds
and we also have from \reqnarray{proof of main lemma-222}
and \reqnarray{proof of main lemma-111} that
\beqnarray{proof of main lemma-case-2-111}
n'_i(h)=
\bselection
q_h+1, &\textrm{if } i=i_1-1,i_2,\ldots,i_{r_h}, \\
q_h, &\textrm{otherwise},
\eselection
\eeqnarray
We then consider the following two subcases.
\emph{Subcase 2(a): $i_1-2<i_2-i_1$.}
In this subcase, we have $i_1-2<i_2-i_1\leq r_{h-1}-i_1$,
and it follows that
\beqnarray{proof of main lemma-case-2-222}
\min\{(i_1-1)-1,r_{h-1}-(i_1-1)-1\}=\min\{i_1-2,r_{h-1}-i_1\}=i_1-2.
\eeqnarray
As $(i_1-1)-(i_1-2)=1$ and $i_1+(i_1-2)<i_1+(i_2-i_1)=i_2$,
it is easy to see from \reqnarray{proof of main lemma-case-2-111}
and \reqnarray{proof of main lemma-case-2-222} that
\beqnarray{proof of main lemma-case-2-333}
n'_{(i_1-1)-j'}=n'_{i_1+j'}=q_h,\ j'=1,2,\ldots,i_1-2=\min\{(i_1-1)-1,r_{h-1}-(i_1-1)-1\}.
\eeqnarray
Therefore, it follows from
\reqnarray{proof of main lemma-case-1-222}, \reqnarray{proof of main lemma-case-2-333},
and \reqnarray{comparison rule A-4} in \rlemma{comparison rule A}(iii)
that ${\nbf'}_1^{r_{h-1}}\succ\nbf_1^{r_{h-1}}$.
\emph{Subcase 2(b): $i_1-2\geq i_2-i_1$.}
In this subcase, we see from $i_1<i_2$, $i_1-2\geq i_2-i_1$, and $i_2\leq r_{h-1}$ that
\beqnarray{proof of main lemma-case-2-444}
1\leq i_2-i_1\leq \min\{i_1-2,r_{h-1}-i_1\}=\min\{(i_1-1)-1,r_{h-1}-(i_1-1)-1\}.
\eeqnarray
As it is clear that $1\leq (i_1-1)-(i_2-i_1)\leq i_1-2$ and $i_1+(i_2-i_1)=i_2$,
we see from \reqnarray{proof of main lemma-case-2-111} that
\beqnarray{}
\alignspace
n'_{(i_1-1)-j'}=n'_{i_1+j'}=q_h, \textrm{ for } j'=1,2,\ldots,i_2-i_1-1,
\label{eqn:proof of main lemma-case-2-555}\\
\alignspace
n'_{(i_1-1)-(i_2-i_1)}=q_h<n'_{i_1+(i_2-i_1)}=n'_{i_2}=q_h+1.
\label{eqn:proof of main lemma-case-2-666}
\eeqnarray
Therefore, it follows from
\reqnarray{proof of main lemma-case-1-222},
\reqnarray{proof of main lemma-case-2-444}--\reqnarray{proof of main lemma-case-2-666},
and \reqnarray{comparison rule A-2} in \rlemma{comparison rule A}(ii)
that ${\nbf'}_1^{r_{h-1}}\succ\nbf_1^{r_{h-1}}$.
\bappendix{Proof of \rlemma{nonadjacent distance larger than one II}}
{proof of nonadjacent distance larger than one II}
In this appendix, we use \rlemma{adjacent distance larger than one II}
and Comparison rule B in \rlemma{comparison rule B}
to prove \rlemma{nonadjacent distance larger than one II}.
For simplicity, let $\nbf_1^{r_{h-1}}=\nbf_1^{r_{h-1}}(h)$.
Note that in \rlemma{nonadjacent distance larger than one II}, we have
\beqnarray{proof of nonadjacent distance larger than one II-111}
r_{h-1}\geq 3,\ \nbf_1^{r_{h-1}}\in \Ncal_{M,k}(h),
\textrm{ and } |n_i-n_{i+1}|\leq 1 \textrm{ for } i=1,2,\ldots,r_{h-1}-1.
\eeqnarray
(i) Note that in \rlemma{nonadjacent distance larger than one II}(i),
we have $n_a-n_b\leq -2$ for some $1\leq a<b\leq r_{h-1}$ and $b\geq a+2$.
For ease of presentation, let $n_a=p$.
Then we have from $n_a-n_b\leq -2$ that $n_b\geq p+2$.
Note that the condition $|n_{i+1}-n_i|\leq 1$ for $i=1,2,\ldots,r_{h-1}-1$
in \reqnarray{proof of nonadjacent distance larger than one II-111}
says that the absolute value of the difference of any two adjacent entries
of $\nbf_1^{r_{h-1}}$ is at most equal to one.
As such, from $n_a=p$, $n_b\geq p+2$, $b\geq a+2>a$,
and the condition $|n_{i+1}-n_i|\leq 1$ for $i=1,2,\ldots,r_{h-1}-1$
in \reqnarray{proof of nonadjacent distance larger than one II-111},
we can argue as that for \reqnarray{proof of nonadjacent distance larger than one-(ii)-111}--\reqnarray{proof of nonadjacent distance larger than one-(ii)-555}
in the proof of \rlemma{nonadjacent distance larger than one}(ii)
in \rappendix{proof of nonadjacent distance larger than one} that there exist two positive integers $a'$ and $b'$ such that
\beqnarray{}
\alignspace a\leq a'<b'\leq b, \textrm{ and } b'\geq a'+2,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-111} \\
\alignspace n_{a'}=p,\ n_{b'}=p+2, \textrm{ and } n_i=p+1 \textrm{ for } a'<i<b'.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-222}
\eeqnarray
An illustration of
\reqnarray{proof of nonadjacent distance larger than one II-(i)-111}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-222}
is given in \rfigure{appendix-I-(i)}.
Note that from $b'\geq a'+2$ in \reqnarray{proof of nonadjacent distance larger than one II-(i)-111}
and $n_i=p+1$ for $a'<i<b'$ in \reqnarray{proof of nonadjacent distance larger than one II-(i)-222},
it is clear that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-333}
n_{a'+1}=p+1 \textrm{ and } n_{b'-1}=p+1.
\eeqnarray
\bpdffigure{appendix-I-i.pdf}{4.0in}
\epdffigure{appendix-I-(i)}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one II-(i)-111}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-222}.}
To prove \rlemma{nonadjacent distance larger than one II}(i),
we need to show that if $n_{r_{h-1}}\neq n_1+2$
or $n_i\neq n_1+1$ for some $2\leq i\leq r_{h-1}-1$,
then there exists a sequence of positive integers
${\nbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$ such that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-444}
{\nbf'}_1^{r_{h-1}}\succ\nbf_1^{r_{h-1}}.
\eeqnarray
Note that if $n_{r_{h-1}}\neq n_1+2$
or $n_i\neq n_1+1$ for some $2\leq i\leq r_{h-1}-1$,
then we have $a'\geq 2$ or $b'\leq r_{h-1}-1$.
To see this, suppose on the contrary that $a'=1$ and $b'=r_{h-1}$.
Then it follows from $a'=1$, $b'=r_{h-1}$, and \reqnarray{proof of nonadjacent distance larger than one II-(i)-222} that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-555}
n_1=p,\ n_{r_{h-1}}=p+2,
\textrm{ and } n_i=p+1 \textrm{ for } 2\leq i\leq r_{h-1}-1.
\eeqnarray
It is clear from \reqnarray{proof of nonadjacent distance larger than one II-(i)-555}
that $n_{r_{h-1}}=n_1+2$ and $n_i=n_1+1$ for all $2\leq i\leq r_{h-1}-1$,
and a contradiction is reached.
In the following, we show that if $a'\geq 2$ or $b'\leq r_{h-1}-1$,
then there exists a sequence of positive integers ${\nbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$
such that \reqnarray{proof of nonadjacent distance larger than one II-(i)-444} holds,
and hence \rlemma{nonadjacent distance larger than one II}(i) is proved.
(a) First, we assume that $a'\geq 2$
and show that there exists a sequence of positive integers
${\nbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$ such that
\reqnarray{proof of nonadjacent distance larger than one II-(i)-444} holds.
Note that from $b'\geq a'+2$
in \reqnarray{proof of nonadjacent distance larger than one II-(i)-111}
and $b'\leq r_{h-1}$, we have
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-111}
a'\leq b'-2\leq r_{h-1}-2.
\eeqnarray
As we assume that $a'\geq 2$,
it follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-111} that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-222}
2\leq a'\leq r_{h-1}-2.
\eeqnarray
We need to consider the following three possible cases.
\emph{Case 1: There exists a positive integer $j$ such that
$1\leq j\leq \min\{a'-1,b'-a'-1\}$,
$n_{a'-j'}=p+1$ for $j'=1,2,\ldots,j-1$, and $n_{a'-j}<p+1$.}
Let $\mbf_1^{r_{h-1}}$ be a sequence of positive integers such that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-111}
m_{a'}=n_{a'}+1,\ m_{a'+1}=n_{a'+1}-1, \textrm{ and } m_i=n_i \textrm{ for } i\neq a', a'+1.
\eeqnarray
As before, it is easy to show that $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
From \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-111},
$n_{a'}=p$ in \reqnarray{proof of nonadjacent distance larger than one II-(i)-222},
and $n_{a'+1}=p+1$ in \reqnarray{proof of nonadjacent distance larger than one II-(i)-333},
we have
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-222}
m_{a'}-m_{a'+1}=(n_{a'}+1)-(n_{a'+1}-1)=(p+1)-(p+1-1)=1.
\eeqnarray
As we have $j\leq \min\{a'-1,b'-a'-1\}\leq b'-a'-1$,
we consider the two subcases $j<b'-a'-1$ and $j=b'-a'-1$ separately.
\emph{Subcase 1(a): $j<b'-a'-1$.}
\bpdffigure{appendix-I-i-a-case-1.pdf}{5.5in}
\epdffigure{appendix-I-(i)-(a)-case-1}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-444}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-555}.}
In this subcase, we show that
\beqnarray{}
\alignspace 1\leq j\leq \min\{a'-1,r_{h-1}-a'-1\}
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-1-333}\\
\alignspace m_{a'-j'}=m_{(a'+1)+j'}, \textrm{ for } j'=1,2,\ldots,j-1,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-1-444}\\
\alignspace m_{a'-j}<m_{(a'+1)+j}.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-1-555}
\eeqnarray
An illustration of
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-444}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-555}
is given in \rfigure{appendix-I-(i)-(a)-case-1}.
Therefore, it follows from
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-222}--\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-555},
and \reqnarray{comparison rule B-3} in \rlemma{comparison rule B}(ii) that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-666}
\mbf_1^{r_{h-1}}\succeq \nbf_1^{r_{h-1}},
\eeqnarray
where $\mbf_1^{r_{h-1}}\equiv \nbf_1^{r_{h-1}}$
if and only if $a'-j=1$, $(a'+1)+j=r_{h-1}$,
and $m_1=m_{r_{h-1}}-1$ (i.e., $m_{a'-j}=m_{(a'+1)+j}-1$).
From $j<b'-a'-1$ in this subcase and $b'\leq r_{h-1}$,
we see that $(a'+1)+j<b'\leq r_{h-1}$.
This implies that $(a'+1)+j\neq r_{h-1}$ and hence it cannot be the case that
$\mbf_1^{r_{h-1}}\equiv \nbf_1^{r_{h-1}}$.
As such, we see from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-666}
that $\mbf_1^{r_{h-1}}\succ \nbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one II-(i)-444} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
From $1\leq j\leq \min\{a'-1,b'-a'-1\}$ and $b'\leq r_{h-1}$, we see that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-777}
1\leq j\leq \min\{a'-1,b'-a'-1\}\leq \min\{a'-1,r_{h-1}-a'-1\}.
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-333}
follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-777}.
To prove \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-444}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-555},
note that as we have $n_{a'-j'}=p+1$ for $j'=1,2,\ldots,j-1$, and $n_{a'-j}<p+1$ in this case,
it is clear from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-111} that
\beqnarray{}
\alignspace m_{a'-j'}=n_{a'-j'}=p+1, \textrm{ for } j'=1,2,\ldots,j-1,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-1-888}\\
\alignspace m_{a'-j}=n_{a'-j}<p+1.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-1-999}
\eeqnarray
Furthermore, as in this subcase we have $a'<(a'+1)+j<b'$,
it follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-111}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-222} that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-aaa}
m_{(a'+1)+j'}=n_{(a'+1)+j'}=p+1, \textrm{ for } j'=1,2,\ldots,j.
\eeqnarray
By combining
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-888}--\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-aaa},
we obtain \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-444}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-555}.
\emph{Subcase 1(b): $j=b'-a'-1$.}
In this subcase, we have $a'<(a'+1)+j=b'$
and hence it follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-111}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-222} that
\beqnarray{}
\alignspace m_{(a'+1)+j'}=n_{(a'+1)+j'}=p+1, \textrm{ for } j'=1,2,\ldots,j-1,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-1-bbb}\\
\alignspace m_{(a'+1)+j'}=n_{(a'+1)+j}=n_{b'}=p+2.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-1-ccc}
\eeqnarray
By using \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-bbb}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-ccc},
we can argue as in Subcase~1(a) above that
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-333}--\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-666}
still hold.
Since it is clear from $m_{a'-j}<p+1$ in \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-999}
and $m_{(a'+1)+j}=p+2$ in \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-ccc}
that $m_{a'-j}\neq m_{(a'+1)+j}-1$,
it cannot be the case that $\mbf_1^{r_{h-1}}\equiv \nbf_1^{r_{h-1}}$.
As such, we see from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-666}
that $\mbf_1^{r_{h-1}}\succ \nbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one II-(i)-444} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
\emph{Case 2: There exists a positive integer $j$ such that
$1\leq j\leq \min\{a'-1,b'-a'-1\}$,
$n_{a'-j'}=p+1$ for $j'=1,2,\ldots,j-1$, and $n_{a'-j}>p+1$.}
In this case, we can show that $j\geq 2$.
To see this, suppose on the contrary that $j=1$,
then we have $n_{a'-1}>p+1$ in this case.
As it is easy to see from $n_{a'}=p$
in \reqnarray{proof of nonadjacent distance larger than one II-(i)-222}
and the condition $|n_{i+1}-n_i|\leq 1$ for $i=1,2,\ldots,r_{h-1}-1$
in \reqnarray{proof of nonadjacent distance larger than one II-111}
that $n_{a'-1}$ must be equal to $p-1$ (provided that $p\geq 2$), $p$, or $p+1$,
we have reached a contradiction.
As $j\geq 2$, we have $n_{a'-1}=p+1$ in this case.
It then follows from
$n_{a'-1}=p+1$ and $n_{a'}=p$
in \reqnarray{proof of nonadjacent distance larger than one II-(i)-222} that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-111}
n_{a'-1}-n_{a'}=(p+1)-p=1.
\eeqnarray
Let $\mbf_1^{r_{h-1}}$ be a sequence of positive integers such that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-222}
m_{a'-1}=n_{a'-1}-1,\ m_{a'}=n_{a'}+1,
\textrm{ and } m_i=n_i \textrm{ for } i\neq a'-1, a'.
\eeqnarray
As before, it is easy to show that $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
\bpdffigure{appendix-I-i-a-case-2.pdf}{5.5in}
\epdffigure{appendix-I-(i)-(a)-case-2}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-555}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-666}.}
In the following, we show that
\beqnarray{}
\alignspace
2\leq a'-1\leq r_{h-1}-2,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-2-333}\\
\alignspace
1\leq j-1\leq \min\{(a'-1)-1,r_{h-1}-(a'-1)-1\},
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-2-444}\\
\alignspace n_{(a'-1)-j'}=n_{a'+j'}, \textrm{ for } j'=1,2,\ldots,j-2,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-2-555}\\
\alignspace n_{(a'-1)-(j-1)}>n_{a'+(j-1)}.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-2-666}
\eeqnarray
An illustration of
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-555}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-666}
is given in \rfigure{appendix-I-(i)-(a)-case-2}.
Therefore, it follows from
$\nbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$ in \reqnarray{proof of nonadjacent distance larger than one II-111},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-111}--\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-666},
and \reqnarray{comparison rule B-2} in \rlemma{comparison rule B}(ii) that
$\nbf_1^{r_{h-1}}\prec \mbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one II-(i)-444} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
From $2\leq j\leq \min\{a'-1,b'-a'-1\}$,
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-222},
and $b'\leq r_{h-1}$, we can see that
\beqnarray{}
\alignspace
2\leq \min\{a'-1,b'-a'-1\}\leq a'-1<a'\leq r_{h-1}-2,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-2-777}\\
\alignspace
1\leq j-1\leq \min\{a'-2,b'-a'-2\}\leq \min\{(a'-1)-1,r_{h-1}-(a'-1)-1\}.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-2-888}
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-333}
follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-777},
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-444}
follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-888}.
To prove \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-555}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-666},
note that as we have $n_{a'-j'}=p+1$ for $j'=1,2,\ldots,j-1$ and $n_{a'-j}>p+1$ in this case,
it is clear that
\beqnarray{}
\alignspace n_{(a'-1)-j'}=p+1, \textrm{ for } j'=1,2,\ldots,j-2,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-2-999}\\
\alignspace n_{(a'-1)-(j-1)}=n_{a'-j}>p+1.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-2-aaa}
\eeqnarray
Furthermore, we have from $2\leq j\leq \min\{a'-1,b'-a'-1\}$ that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-bbb}
a'< a'+j-1\leq a'+(b'-a'-1)-1=b'-2<b'.
\eeqnarray
It then follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-222}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-bbb} that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-ccc}
n_{a'+j'}=p+1, \textrm{ for } j'=1,2,\ldots,j-1.
\eeqnarray
By combining \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-999},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-aaa},
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-ccc},
we obtain \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-555}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-666}.
\emph{Case 3: $n_{a'-j'}=p+1$ for $j'=1,2,\ldots,\min\{a'-1,b'-a'-1\}$.}
We consider the two subcases $a'-1<b'-a'-1$ and $a'-1\geq b'-a'-1$ separately.
\emph{Subcase 3(a): $a'-1<b'-a'-1$.}
Let $\mbf_1^{r_{h-1}}$ be a sequence of positive integers as given in
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-222}.
As in Case~2 above, we have $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
As it is clear from $a'\geq 2$ and $b'\geq a'+2$
in \reqnarray{proof of nonadjacent distance larger than one II-(i)-111}
that $\min\{a'-1,b'-a'-1\}\geq 1$,
we have $n_{a'-1}=p+1$ in this case and hence it is easy to see that
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-111}
still holds in this subcase.
If $a'=2$, then we have $a'-1=1$ and
it follows from $r_{h-1}\geq 3$
in \reqnarray{proof of nonadjacent distance larger than one II-111},
$\nbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$ in \reqnarray{proof of nonadjacent distance larger than one II-111},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-111},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-222},
and \reqnarray{comparison rule B-1} in \rlemma{comparison rule B}(i)
that $\nbf_1^{r_{h-1}}\prec \mbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one II-(i)-444} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
\bpdffigure{appendix-I-i-a-case-3-1.pdf}{4.5in}
\epdffigure{appendix-I-(i)-(a)-case-3-1}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-222}
(note that we have $\min\{(a'-1)-1, r_{h-1}-(a'-1)-1\}=a'-2$
in \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-555}).}
On the other hand, if $a'\geq 3$, then we show that
\beqnarray{}
\alignspace \hspace*{-0.2in}
2\leq a'-1\leq r_{h-1}-2,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-3-111}\\
\alignspace \hspace*{-0.2in}
n_{(a'-1)-j'}=n_{a'+j'}=p+1, \textrm{ for } j'=1,2,\ldots,\min\{(a'-1)-1, r_{h-1}-(a'-1)-1\}.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-3-222}
\eeqnarray
An illustration of \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-222}
is given in \rfigure{appendix-I-(i)-(a)-case-3-1}.
Therefore, it follows from
$\nbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$ in \reqnarray{proof of nonadjacent distance larger than one II-111},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-111},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-2-222},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-111},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-222},
and \reqnarray{comparison rule B-4} in \rlemma{comparison rule B}(iii) that
$\nbf_1^{r_{h-1}}\prec \mbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one II-(i)-444} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
From $a'\geq 3$ and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-222},
we see that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-333}
2\leq a'-1<a'\leq r_{h-1}-2.
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-111}
follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-333}.
To prove \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-222},
note that from $a'-1<b'-a'-1$ in this subcase and $b'\leq r_{h-1}$, we have
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-444}
(a'-1)-1<(b'-a'-1)-1\leq r_{h-1}-a'-2<r_{h-1}-(a'-1)-1.
\eeqnarray
It follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-444} that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-555}
\min\{(a'-1)-1, r_{h-1}-(a'-1)-1\}=(a'-1)-1=a'-2.
\eeqnarray
Note that this subcase we have
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-666}
n_{a'-j'}=p+1, \textrm{ for } j'=1,2,\ldots,\min\{a'-1,b'-a'-1\}=a'-1.
\eeqnarray
From $a'\geq 3$ and $a'-1<b'-a'-1$, we have that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-777}
a'<a'+(a'-2)<a'+(b'-a'-2)=b'-2<b'.
\eeqnarray
It then follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-222}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-777} that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-888}
n_{a'+j'}=p+1, \textrm{ for } j'=1,2,\ldots,a'-2.
\eeqnarray
by combining
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-555},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-666},
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-888},
we obtain \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-222}.
\emph{Subcase 3(b): $a'-1\geq b'-a'-1$.}
Let $\mbf_1^{r_{h-1}}$ be a sequence of positive integers as given in
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-111}.
As in Case~1 above, we have $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
Also note that \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-222} still holds in this subcase.
\bpdffigure{appendix-I-i-a-case-3-2.pdf}{4.5in}
\epdffigure{appendix-I-(i)-(a)-case-3-2}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-aaa}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-bbb}.}
In the following, we show that
\beqnarray{}
\alignspace 1\leq b'-a'-1\leq \min\{a'-1,r_{h-1}-a'-1\}
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-3-999}\\
\alignspace m_{a'-j'}=m_{(a'+1)+j'}=p+1, \textrm{ for } j'=1,2,\ldots,b'-a'-2,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-3-aaa}\\
\alignspace m_{a'-(b'-a'-1)}=p+1<m_{(a'+1)+(b'-a'-1)}=p+2.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-3-bbb}
\eeqnarray
An illustration of
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-aaa}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-bbb}
is given in \rfigure{appendix-I-(i)-(a)-case-3-2}.
Therefore, it follows from
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-222}--\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-222},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-999}--\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-bbb},
and \reqnarray{comparison rule B-3} in \rlemma{comparison rule B}(ii)
(with $j=b'-a'-1$) that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-ccc}
\mbf_1^{r_{h-1}}\succeq \nbf_1^{r_{h-1}}.
\eeqnarray
From $b'\geq a'+2$ in \reqnarray{proof of nonadjacent distance larger than one II-(i)-111},
$a'-1\geq b'-a'-1$, and $b'\leq r_{h-1}$, we can see that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-ddd}
1\leq b'-a'-1\leq \min\{a'-1,b'-a'-1\}\leq \min\{a'-1,r_{h-1}-a'-1\}.
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-999}
follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-ddd}.
To prove \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-aaa}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-bbb},
note that in this case we have
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-eee}
n_{a'-j'}=p+1, \textrm{ for } j'=1,2,\ldots,\min\{a'-1,b'-a'-1\}=b'-a'-1.
\eeqnarray
It follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-111}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-eee} that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-fff}
m_{a'-j'}=n_{a'-j'}=p+1, \textrm{ for } j'=1,2,\ldots,b'-a'-1.
\eeqnarray
Also, it is clear from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-111},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-222},
$(a'+1)+(b'-a'-1)=b'$,
and $n_{b'}=p+2$ in \reqnarray{proof of nonadjacent distance larger than one II-(i)-222} that
\beqnarray{}
\alignspace m_{(a'+1)+j'}=n_{(a'+1)+j'}=p+1, \textrm{ for } j'=1,2,\ldots,b'-a'-2,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-3-ggg}\\
\alignspace m_{(a'+1)+(b'-a'-1)}=n_{(a'+1)+(b'-a'-1)}=n_{b'}=p+2.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-3-hhh}
\eeqnarray
By combining
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-fff},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-ggg},
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-hhh},
we obtain \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-aaa}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-bbb}.
Now let ${\mbf'}_1^{r_{h-1}}$ be a sequence of positive integers such that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-iii}
m'_{a'+1}=m_{a'+1}+1,\ m'_{a'+2}=m_{a'+2}-1, \textrm{ and } m'_i=m_i \textrm{ for } i\neq a'+1, a'+2.
\eeqnarray
Again, it is easy to show that ${\mbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
If $b'=a'+2$, then we see from
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-111},
$n_{a'+1}=p+1$ in \reqnarray{proof of nonadjacent distance larger than one II-(i)-333},
and $n_{b'}=p+2$ in \reqnarray{proof of nonadjacent distance larger than one II-(i)-222} that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-jjj}
m_{a'+1}-m_{a'+2}
\aligneq (n_{a'+1}-1)-n_{a'+2}=n_{a'+1}-1-n_{b'} \nn\\
\aligneq (p+1)-1-(p+2)=-2.
\eeqnarray
Therefore, it follows from
$r_{h-1}\geq 3$ in \reqnarray{proof of nonadjacent distance larger than one II-111},
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-iii},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-jjj},
and \reqnarray{adjacent distance larger than one II-2} in \rlemma{adjacent distance larger than one II}(ii) that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-kkk}
\mbf_1^{r_{h-1}}\preceq {\mbf'}_1^{r_{h-1}},
\eeqnarray
where $\mbf_1^{r_{h-1}}\equiv {\mbf'}_1^{r_{h-1}}$
if and only if $r_{h-1}=2$ and $m_1=m_2-2$.
Since it is clear from $r_{h-1}\geq 3$ in
\reqnarray{proof of nonadjacent distance larger than one II-111}
that $r_{h-1}\neq 2$, it cannot be the case that $\mbf_1^{r_{h-1}}\equiv {\mbf'}_1^{r_{h-1}}$.
As such, we see from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-ccc}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-kkk}
that $\nbf_1^{r_{h-1}}\preceq \mbf_1^{r_{h-1}}\prec {\mbf'}_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one II-(i)-444} holds
with ${\nbf'}_1^{r_{h-1}}={\mbf'}_1^{r_{h-1}}$.
On the other hand, if $b'\geq a'+3$, then we have $a'<a'+2<b'$
and it follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-222} that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-1111}
n_{a'+1}=n_{a'+2}=p+1.
\eeqnarray
From \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-iii},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-111},
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-1111},
we see that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-2222}
m'_{a'+1}-m'_{a'+2}
\aligneq (m_{a'+1}+1)-(m_{a'+2}-1)=m_{a'+1}-m_{a'+2}+2 \nn\\
\aligneq (n_{a'+1}-1)-n_{a'+2}+2=1.
\eeqnarray
\bpdffigure{appendix-I-i-a-case-3-3.pdf}{4.5in}
\epdffigure{appendix-I-(i)-(a)-case-3-3}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-5555}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-6666}.}
We will show that
\beqnarray{}
\alignspace
2\leq a'+1\leq r_{h-1}-2,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-3-3333}\\
\alignspace
1\leq b'-a'-2\leq \min\{(a'+1)-1,r_{h-1}-(a'+1)-1\},
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-3-4444}\\
\alignspace
m'_{(a'+1)-j'}=m'_{(a'+2)+j'}=p+1, \textrm{ for } j'=1,2,\ldots,b'-a'-3,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-3-5555}\\
\alignspace
m'_{(a'+1)-(b'-a'-2)}=p+1<m'_{(a'+2)+(b'-a'-2)}=p+2.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-3-6666}
\eeqnarray
An illustration of
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-5555}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-6666}
is given in \rfigure{appendix-I-(i)-(a)-case-3-3}.
Therefore, it follows from
${\mbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-iii},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-2222}--\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-6666},
and \reqnarray{comparison rule B-3} in \rlemma{comparison rule B}(ii)
(with $j=b'-a'-2$) that
\beqnarray{proof of nonadjacent distance larger than one- II(i)-(a)-case-3-7777}
{\mbf'}_1^{r_{h-1}}\succeq \mbf_1^{r_{h-1}},
\eeqnarray
where ${\mbf'}_1^{r_{h-1}}\equiv \mbf_1^{r_{h-1}}$
if and only if $(a'+1)-(b'-a'-2)=1$, $(a'+1)+1+(b'-a'-2)=r_{h-1}$, and $m'_1=m'_{r_{h-1}}-1$.
Since in this subcase we have $a'-1\geq b'-a'-1$,
it is clear that $(a'+1)-(b'-a'-2)\geq (b'-a'+1)-(b'-a'-2)=3$.
This implies that $(a'+1)-(b'-a'-2)\neq 1$
and hence it cannot be the case that ${\mbf'}_1^{r_{h-1}}\equiv \mbf_1^{r_{h-1}}$.
As such, we see from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-ccc}
and \reqnarray{proof of nonadjacent distance larger than one- II(i)-(a)-case-3-7777}
that $\nbf_1^{r_{h-1}}\preceq \mbf_1^{r_{h-1}}\prec {\mbf'}_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one II-(i)-444} holds
with ${\nbf'}_1^{r_{h-1}}={\mbf'}_1^{r_{h-1}}$.
To prove \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-3333}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-4444},
note from $a'\geq 2$, $b'\geq a'+3$, $b'\leq r_{h-1}$,
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-999} that
\beqnarray{}
\alignspace \hspace*{-0.3in}
2\leq a'<a'+1\leq b'-2\leq r_{h-1}-2,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-3-8888}\\
\alignspace \hspace*{-0.3in}
1\leq b'-a'-2\leq \min\{a'-2,r_{h-1}-a'-2\}\leq \min\{(a'+1)-1,r_{h-1}-(a'+1)-1\}.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-3-9999}
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-3333}
follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-8888},
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-4444}
follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-9999}.
To prove \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-5555}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-6666},
note that from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-iii},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-1-111},
$n_{a'}=p$ in \reqnarray{proof of nonadjacent distance larger than one II-(i)-222},
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-fff}--\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-hhh},
we have
\beqnarray{}
\alignspace m'_{(a'+1)-1}=m_{(a'+1)-1}=m_{a'}=n_{a'}+1=p+1,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-3-aaaa}\\
\alignspace m'_{(a'+1)-j'}=m_{(a'+1)-j'}=p+1, \textrm{ for } j'=2,3,\ldots,b'-a',
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-3-bbbb}\\
\alignspace m'_{(a'+2)+j'}=m_{(a'+2)+j'}=p+1, \textrm{ for } j'=1,2,\ldots,b'-a'-3,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-3-cccc}\\
\alignspace m'_{(a'+2)+(b'-a'-2)}=m_{(a'+2)+(b'-a'-2)}=p+2.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(a)-case-3-dddd}
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-5555}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-6666}
follow from
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-aaaa}--\reqnarray{proof of nonadjacent distance larger than one II-(i)-(a)-case-3-dddd}.
(b) Now we assume that $b'\leq r_{h-1}-1$ and show that there exists a sequence of positive integers
${\nbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$ such that
\reqnarray{proof of nonadjacent distance larger than one II-(i)-444} holds.
Note that from $b'\geq a'+2$
in \reqnarray{proof of nonadjacent distance larger than one II-(i)-111}
and $a'\geq 1$, we have
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-111}
b'-1\geq a'+1\geq 2.
\eeqnarray
As we assume that $b'\leq r_{h-1}-1$,
it follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-111} that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-222}
2\leq b'-1\leq r_{h-1}-2.
\eeqnarray
We need to consider the following three possible cases.
\emph{Case 1: There exists a positive integer $j$ such that
$1\leq j\leq \min\{b'-a'-1,r_{h-1}-b'\}$,
$n_{b'+j'}=p+1$ for $j'=1,2,\ldots,j-1$, and $n_{b'+j}>p+1$.}
Let $\mbf_1^{r_{h-1}}$ be a sequence of positive integers such that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-111}
m_{b'-1}=n_{b'-1}+1,\ m_{b'}=n_{b'}-1, \textrm{ and } m_i=n_i \textrm{ for } i\neq b'-1, b'.
\eeqnarray
As before, it is easy to show that $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
Also, noth that from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-111},
$n_{b'-1}=p+1$ in \reqnarray{proof of nonadjacent distance larger than one II-(i)-333},
and $n_{b'}=p+2$ in \reqnarray{proof of nonadjacent distance larger than one II-(i)-222},
we have
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-222}
m_{b'-1}-m_{b'}=(n_{b'-1}+1)-(n_{b'}-1)=(p+1+1)-(p+2-1)=1
\eeqnarray
As we have $j\leq \min\{b'-a'-1,r_{h-1}-b'\}\leq b'-a'-1$,
we consider the two subcases $j<b'-a'-1$ and $j=b'-a'-1$ separately.
\emph{Subcase 1(a): $j<b'-a'-1$.}
\bpdffigure{appendix-I-i-b-case-1.pdf}{5.5in}
\epdffigure{appendix-I-(i)-(b)-case-1}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-444}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-555}.}
In this subcase, we show that
\beqnarray{}
\alignspace 1\leq j\leq \min\{(b'-1)-1,r_{h-1}-(b'-1)-1\}
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-1-333}\\
\alignspace m_{(b'-1)-j'}=m_{b'+j'}, \textrm{ for } j'=1,2,\ldots,j-1,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-1-444}\\
\alignspace m_{(b'-1)-j}<m_{b'+j}.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-1-555}
\eeqnarray
An illustration of
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-444}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-555}
is given in \rfigure{appendix-I-(i)-(b)-case-1}.
Therefore, it follows from
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-222}--\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-555},
and \reqnarray{comparison rule B-3} in \rlemma{comparison rule B}(ii) that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-666}
\mbf_1^{r_{h-1}}\succeq \nbf_1^{r_{h-1}},
\eeqnarray
where $\mbf_1^{r_{h-1}}\equiv \nbf_1^{r_{h-1}}$
if and only if $(b'-1)-j=1$, $b'+j=r_{h-1}$,
and $n_1=n_{r_{h-1}}-1$ (i.e., $n_{(b'-1)-j}=n_{b'+j}-1$).
As we have $j<b'-a'-1$ in this subcase and $a'\geq 1$,
we immediately see that $(b'-1)-j>a'\geq 1$.
This implies that $(b'-1)-j\neq 1$ and hence it cannot be the case that
$\mbf_1^{r_{h-1}}\equiv \nbf_1^{r_{h-1}}$.
As such, we see from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-666}
that $\mbf_1^{r_{h-1}}\succ \nbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one II-(i)-444} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
From $1\leq j\leq \min\{b'-a'-1,r_{h-1}-b'\}$ and $a'\geq 1$, we see that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-777}
1\leq j\leq \min\{b'-a'-1,r_{h-1}-b'\}\leq \min\{(b'-1)-1,r_{h-1}-(b'-1)-1\}.
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-333}
follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-777}.
To prove \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-444}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-555},
note that in this subcase we have $a'<(b'-1)-j<b'$,
and hence it follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-111}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-222} that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-888}
m_{(b'-1)-j'}=n_{(b'-1)-j'}=p+1, \textrm{ for } j'=1,2,\ldots,j.
\eeqnarray
Also, as we have $n_{b'+j'}=p+1$ for $j'=1,2,\ldots,j-1$ and $n_{b'+j}>p+1$ in this case,
it follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-111} that
\beqnarray{}
\alignspace m_{b'+j'}=n_{b'+j'}=p+1, \textrm{ for } j'=1,2,\ldots,j-1,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-1-999}\\
\alignspace m_{b'+j}=n_{b'+j}>p+1.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-1-aaa}
\eeqnarray
By combining \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-888},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-999},
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-aaa},
we obtain \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-444}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-555}.
\emph{Subcase 1(b): $j=b'-a'-1$.}
In this subcase, we have $(b'-1)-j=a'$
and it follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-111}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-222} that
\beqnarray{}
\alignspace m_{(b'-1)-j'}=n_{(b'-1)-j'}=p+1, \textrm{ for } j'=1,2,\ldots,j-1,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-1-bbb}\\
\alignspace m_{(b'-1)-j'}=n_{(b'-1)-j}=n_{a'}=p.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-1-ccc}
\eeqnarray
By using \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-bbb}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-ccc},
we can argue as in Subcase~1(a) above that
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-333}--\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-666}
still hold.
Since it is clear from $m_{(b'-1)-j}=p$
in \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-ccc}
and $m_{b'+j}>p+1$ in \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-aaa}
that $m_{(b'-1)-j}\neq m_{b'+j}-1$,
it cannot be the case that $\mbf_1^{r_{h-1}}\equiv \nbf_1^{r_{h-1}}$.
As such, we see from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-666}
that $\mbf_1^{r_{h-1}}\succ \nbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one II-(i)-444} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
\emph{Case 2: There exists a positive integer $j$ such that
$1\leq j\leq \min\{b'-a'-1,r_{h-1}-b'\}$,
$n_{b'+j'}=p+1$ for $j'=1,2,\ldots,j-1$, and $n_{b'+j}<p+1$.}
In this case, we can show that $j\geq 2$.
To see this, suppose on the contrary that $j=1$,
then we have $n_{b'+1}<p+1$ in this case.
As it follows from $n_{b'}=p+2$ in \reqnarray{proof of nonadjacent distance larger than one II-(i)-222}
and the condition $|n_{i+1}-n_i|\leq 1$ for $i=1,2,\ldots,r_{h-1}-1$
in \reqnarray{proof of nonadjacent distance larger than one II-111}
that $n_{b'+1}$ must be equal to $p+1$, $p+2$, or $p+3$,
we have reached a contradiction.
Let $\mbf_1^{r_{h-1}}$ be a sequence of positive integers such that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-111}
m_{b'}=n_{b'}-1,\ m_{b'+1}=n_{b'+1}+1,
\textrm{ and } m_i=n_i \textrm{ for } i\neq b', b'+1.
\eeqnarray
As before, it is easy to show that $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
As $j\geq 2$, we have $n_{b'+1}=p+1$ in this case.
It then follows from $n_{b'+1}=p+1$ and $n_{b'}=p+2$ in \reqnarray{proof of nonadjacent distance larger than one II-(i)-222} that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-222}
n_{b'}-n_{b'+1}=(p+2)-(p+1)=1.
\eeqnarray
\bpdffigure{appendix-I-i-b-case-2.pdf}{5.5in}
\epdffigure{appendix-I-(i)-(b)-case-2}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-555}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-666}.}
In the following, we show that
\beqnarray{}
\alignspace
2\leq b'\leq r_{h-1}-2,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-2-333}\\
\alignspace
1\leq j-1\leq \min\{b'-1,r_{h-1}-b'-1\},
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-2-444}\\
\alignspace n_{b'-j'}=n_{(b'+1)+j'}, \textrm{ for } j'=1,2,\ldots,j-2,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-2-555}\\
\alignspace n_{b'-(j-1)}>n_{(b'+1)+(j-1)}.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-2-666}
\eeqnarray
An illustration of
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-555}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-666}
is given in \rfigure{appendix-I-(i)-(b)-case-2}.
Therefore, it follows from $\nbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$ in \reqnarray{proof of nonadjacent distance larger than one II-111},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-111}--\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-666},
and \reqnarray{comparison rule B-2} in \rlemma{comparison rule B}(ii) that
$\nbf_1^{r_{h-1}}\prec \mbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one II-(i)-444} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
From $2\leq j\leq \min\{b'-a'-1,r_{h-1}-b'\}$ and $a'\geq 1$,
we can see that
\beqnarray{}
\alignspace
2\leq \min\{b'-a'-1,r_{h-1}-b'\}\leq b'-a'-1\leq b',
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-2-777}\\
\alignspace
2\leq \min\{b'-a'-1,r_{h-1}-b'\}\leq r_{h-1}-b',
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-2-888}\\
\alignspace
1\leq j-1\leq \min\{b'-a'-2,r_{h-1}-b'-1\}\leq \min\{b'-1,r_{h-1}-b'-1\}.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-2-999}
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-333}
follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-777}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-888},
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-444}
follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-999}.
To prove \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-555}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-666},
note that we have from $2\leq j\leq \min\{b'-a'-1,r_{h-1}-b'\}\leq b'-a'-1$ that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-aaa}
a'< a'+2\leq b'-(j-1)\leq b'-1<b'.
\eeqnarray
It then follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-222}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-aaa} that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-bbb}
n_{b'-j'}=p+1, \textrm{ for } j'=1,2,\ldots,j-1.
\eeqnarray
By combining \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-bbb},
$n_{b'+j'}=p+1$ for $j'=1,2,\ldots,j-1$, and $n_{b'+j}<p+1$,
we obtain \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-555}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-666}.
\emph{Case 3: $n_{b'+j'}=p+1$ for $j'=1,2,\ldots,\min\{b'-a'-1,r_{h-1}-b'\}$.}
We consider the two subcases $b'-a'-1>r_{h-1}-b'$ and $b'-a'-1\leq r_{h-1}-b'$ separately.
\emph{Subcase 3(a): $b'-a'-1>r_{h-1}-b'$.}
Let $\mbf_1^{r_{h-1}}$ be a sequence of positive integers as given in
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-111}.
As in Case~2 above, we have $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
As it is clear from $b'\geq a'+2$
in \reqnarray{proof of nonadjacent distance larger than one II-(i)-111}
and $b'\leq r_{h-1}-1$ that $\min\{b'-a'-1,r_{h-1}-b'\}\geq 1$,
we have $n_{b'+1}=p+1$ in this case and hence it is easy to see that
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-222}
still holds in this subcase.
If $b'=r_{h-1}-1$,
then it follows from $r_{h-1}\geq 3$ and $\nbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$
in \reqnarray{proof of nonadjacent distance larger than one II-111},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-111},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-222},
and \reqnarray{comparison rule B-1} in \rlemma{comparison rule B}(i)
that $\nbf_1^{r_{h-1}}\prec \mbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one II-(i)-444} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
\bpdffigure{appendix-I-i-b-case-3-1.pdf}{4.5in}
\epdffigure{appendix-I-(i)-(b)-case-3-1}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-222}
(note that we have $\min\{b'-1, r_{h-1}-b'-1\}=r_{h-1}-b'-1$
in \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-555}).}
On the other hand, if $b'\leq r_{h-1}-2$, then we show that
\beqnarray{}
\alignspace \hspace*{-0.2in}
2\leq b'\leq r_{h-1}-2,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-3-111}\\
\alignspace \hspace*{-0.2in}
n_{b'-j'}=n_{(b'+1)+j'}=p+1, \textrm{ for } j'=1,2,\ldots,\min\{b'-1, r_{h-1}-b'-1\}.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-3-222}
\eeqnarray
An illustration of \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-222}
is given in \rfigure{appendix-I-(i)-(b)-case-3-1}.
Therefore, it follows from
$\nbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$ in \reqnarray{proof of nonadjacent distance larger than one II-111},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-111},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-2-222},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-111},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-222},
and \reqnarray{comparison rule B-4} in \rlemma{comparison rule B}(iii) that
$\nbf_1^{r_{h-1}}\prec \mbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one II-(i)-444} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
From \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-222}
and $b'\leq r_{h-1}-2$, we see that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-333}
2\leq b'-1<b'\leq r_{h-1}-2.
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-111}
follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-333}.
To prove \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-222},
note that from $b'-a'-1>r_{h-1}-b'$ in this subcase and $a'\geq 1$, we have
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-444}
r_{h-1}-b'-1<(b'-a'-1)-1<b'-1.
\eeqnarray
It follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-444} that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-555}
\min\{b'-1, r_{h-1}-b'-1\}=r_{h-1}-b'-1.
\eeqnarray
From $b'-a'-1>r_{h-1}-b'$ and $b'\leq r_{h-1}-2$,
we can see that
\beqnarray{}
\alignspace
b'-(r_{h-1}-b'-1)>b'-(b'-a'-2)=a'+2>a',
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-3-666}\\
\alignspace
b'-(r_{h-1}-b'-1)\leq b'-1<b'.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-3-777}
\eeqnarray
It follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-222},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-666},
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-777} that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-888}
n_{b'-j'}=p+1, \textrm{ for } j'=1,2,\ldots,r_{h-1}-b'-1.
\eeqnarray
Furthermore, in this subcase we have
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-999}
n_{b'+j'}=p+1, \textrm{ for } j'=1,2,\ldots,\min\{b'-a'-1,r_{h-1}-b'\}=r_{h-1}-b'.
\eeqnarray
It then follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-999} that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-aaa}
n_{(b'+1)+j'}=p+1, \textrm{ for } j'=1,2,\ldots,r_{h-1}-b'-1.
\eeqnarray
By combining
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-555},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-888},
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-aaa},
we obtain \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-222}.
\emph{Subcase 3(b): $b'-a'-1\leq r_{h-1}-b'$.}
Let $\mbf_1^{r_{h-1}}$ be a sequence of positive integers as given in
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-111}.
As in Case~1 above, we have $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
Also, note that \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-222} still holds in this subcase.
\bpdffigure{appendix-I-i-b-case-3-2.pdf}{4.5in}
\epdffigure{appendix-I-(i)-(b)-case-3-2}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-ccc}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-ddd}.}
In the following, we show that
\beqnarray{}
\alignspace 1\leq b'-a'-1\leq \min\{(b'-1)-1,r_{h-1}-(b'-1)-1\}
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-3-bbb}\\
\alignspace m_{(b'-1)-j'}=m_{b'+j'}=p+1, \textrm{ for } j'=1,2,\ldots,b'-a'-2,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-3-ccc}\\
\alignspace m_{(b'-1)-(b'-a'-1)}=p<m_{b'+(b'-a'-1)}=p+1.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-3-ddd}
\eeqnarray
An illustration of
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-ccc}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-ddd}
is given in \rfigure{appendix-I-(i)-(b)-case-3-2}.
Therefore, it follows from
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-222}--\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-222},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-bbb}--\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-ddd},
and \reqnarray{comparison rule B-3} in \rlemma{comparison rule B}(ii)
(with $j=b'-a'-1$) that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-eee}
\mbf_1^{r_{h-1}}\succeq \nbf_1^{r_{h-1}}.
\eeqnarray
From $b'\geq a'+2$ in \reqnarray{proof of nonadjacent distance larger than one II-(i)-111},
$b'-a'-1\leq r_{h-1}-b'$, and $a'\geq 1$, we can see that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-fff}
1\leq b'-a'-1\leq \min\{b'-a'-1,r_{h-1}-b'\}\leq \min\{(b'-1)-1,r_{h-1}-(b'-1)-1\}.
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-bbb}
follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-fff}.
To prove \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-ccc}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-ddd},
note that from
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-111},
$(b'-1)-(b'-a'-1)=a'$,
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-222},
we have
\beqnarray{}
\alignspace m_{(b'-1)-j'}=n_{(b'-1)-j'}=p+1, \textrm{ for } j'=1,2,\ldots,b'-a'-2,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-3-ggg}\\
\alignspace m_{(b'-1)-(b'-a'-1)}=n_{(b'-1)-(b'-a'-1)}=n_{a'}=p.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-3-hhh}
\eeqnarray
Also, in this case we have
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-iii}
n_{b'+j'}=p+1, \textrm{ for } j'=1,2,\ldots,\min\{b'-a'-1,r_{h-1}-b'\}=b'-a'-1.
\eeqnarray
It follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-111}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-iii} that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-jjj}
m_{b'+j'}=n_{b'+j'}=p+1, \textrm{ for } j'=1,2,\ldots,b'-a'-1.
\eeqnarray
By combining
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-ggg},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-hhh},
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-jjj},
we obtain \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-ccc}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-ddd}.
Now let ${\mbf'}_1^{r_{h-1}}$ be a sequence of positive integers such that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-kkk}
m'_{b'-2}=m_{b'-2}+1,\ m'_{b'-1}=m_{b'-1}-1, \textrm{ and } m'_i=m_i \textrm{ for } i\neq b'-2, b'-1.
\eeqnarray
Again, it is easy to show that ${\mbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
If $b'=a'+2$, then we see from
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-111},
$n_{a'}=p$ in \reqnarray{proof of nonadjacent distance larger than one II-(i)-222},
and $n_{b'-1}=p+1$ in \reqnarray{proof of nonadjacent distance larger than one II-(i)-333} that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-1111}
\alignspace m_{b'-2}-m_{b'-1}=n_{b'-2}-(n_{b'-1}+1)=n_{a'}-n_{b'-1}-1=p-(p+1)-1=-2.
\eeqnarray
Therefore, it follows from
$r_{h-1}\geq 3$ in \reqnarray{proof of nonadjacent distance larger than one II-111},
$\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-kkk},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-1111},
and \reqnarray{adjacent distance larger than one II-2} in \rlemma{adjacent distance larger than one II}(ii) that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-2222}
\mbf_1^{r_{h-1}}\preceq {\mbf'}_1^{r_{h-1}},
\eeqnarray
where $\mbf_1^{r_{h-1}}\equiv {\mbf'}_1^{r_{h-1}}$
if and only if $r_{h-1}=2$ and $m_1=m_2-2$.
Since it is clear from $r_{h-1}\geq 3$
in \reqnarray{proof of nonadjacent distance larger than one II-111}
that $r_{h-1}\neq 2$, it cannot be the case that $\mbf_1^{r_{h-1}}\equiv {\mbf'}_1^{r_{h-1}}$.
As such, we see from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-eee}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-2222}
that $\nbf_1^{r_{h-1}}\preceq \mbf_1^{r_{h-1}}\prec {\mbf'}_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one II-(i)-444} holds
with ${\nbf'}_1^{r_{h-1}}={\mbf'}_1^{r_{h-1}}$.
On the other hand, if $b'\geq a'+3$, then we have $a'<b'-2<b'$
and it follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-222} that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-3333}
n_{b'-2}=n_{b'-1}=p+1.
\eeqnarray
From \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-kkk},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-111},
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-3333},
we see that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-4444}
m'_{b'-2}-m'_{b'-1}
\aligneq (m_{b'-2}+1)-(m_{b'-1}-1)=m_{b'-2}-m_{b'-1}+2 \nn\\
\aligneq n_{b'-2}-(n_{b'-1}+1)+2=(p+1)-(p+1+1)+2=1.
\eeqnarray
\bpdffigure{appendix-I-i-b-case-3-3.pdf}{4.5in}
\epdffigure{appendix-I-(i)-(b)-case-3-3}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-7777}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-8888}.}
We will show that
\beqnarray{}
\alignspace 2\leq b'-2\leq r_{h-1}-2,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-3-5555}\\
\alignspace 1\leq b'-a'-2\leq \min\{(b'-2)-1,r_{h-1}-(b'-2)-1\}.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-3-6666}\\
\alignspace m'_{(b'-2)-j'}=m'_{(b'-1)+j'}=p+1, \textrm{ for } j'=1,2,\ldots,b'-a'-3,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-3-7777}\\
\alignspace m'_{(b'-2)-(b'-a'-2)}=p<m'_{(b'-1)+(b'-a'-2)}=p+1.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-3-8888}
\eeqnarray
An illustration of
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-7777}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-8888}
is given in \rfigure{appendix-I-(i)-(b)-case-3-3}.
Therefore, it follows from
${\mbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-kkk},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-4444}--\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-8888},
and \reqnarray{comparison rule B-3} in \rlemma{comparison rule B}(ii)
(with $j=b'-a'-2$) that
\beqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-9999}
{\mbf'}_1^{r_{h-1}}\succeq \mbf_1^{r_{h-1}},
\eeqnarray
where ${\mbf'}_1^{r_{h-1}}\equiv \mbf_1^{r_{h-1}}$
if and only if $(b'-2)-(b'-a'-2)=1$, $(b'-1)+(b'-a'-2)=r_{h-1}$, and $m'_1=m'_{r_{h-1}}-1$.
Since in this subcase we have $b'-a'-1\leq r_{h-1}-b'$,
it is clear that $(b'-1)+(b'-a'-2)\leq (b'-1)+(r_{h-1}-b'-1)=r_{h-1}-2$.
This implies that $(b'-1)+(b'-a'-2)\neq r_{h-1}$
and hence it cannot be the case that ${\mbf'}_1^{r_{h-1}}\equiv \mbf_1^{r_{h-1}}$.
As such, we see from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-eee}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-9999}
that $\nbf_1^{r_{h-1}}\preceq \mbf_1^{r_{h-1}}\prec {\mbf'}_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one II-(i)-444} holds
with ${\nbf'}_1^{r_{h-1}}={\mbf'}_1^{r_{h-1}}$.
To prove \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-5555}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-6666},
note from $a'\geq 1$, $b'\geq a'+3$, $b'\leq r_{h-1}-1$,
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-bbb} that
\beqnarray{}
\alignspace \hspace*{-0.3in}
2\leq a'+1\leq b'-2\leq r_{h-1}-2,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-3-aaaa}\\
\alignspace \hspace*{-0.3in}
1\leq b'-a'-2\leq \min\{b'-3,r_{h-1}-b'-1\}\leq \min\{(b'-2)-1,r_{h-1}-(b'-2)-1\}.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-3-bbbb}
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-5555}
follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-aaaa},
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-6666}
follows from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-bbbb}.
To prove \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-7777}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-8888},
note that from \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-kkk},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-ggg}--\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-hhh},
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-1-111},
$n_{b'}=p+2$ in \reqnarray{proof of nonadjacent distance larger than one II-(i)-222},
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-jjj},
we see that
\beqnarray{}
\alignspace m'_{(b'-2)-j'}=m_{(b'-2)-j'}=p+1, \textrm{ for } j'=1,2,\ldots,b'-a'-3,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-3-cccc}\\
\alignspace m'_{(b'-2)-(b'-a'-2)}=m_{(b'-2)-(b'-a'-2)}=p.
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-3-dddd}\\
\alignspace m'_{(b'-1)+1}=m'_{b'}=m_{b'}=n_{b'}-1=p+1,
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-3-eeee}\\
\alignspace m'_{(b'-1)+j'}=m_{(b'-1)+j'}=p+1, \textrm{ for } j'=2,3,\ldots,b'-a',
\label{eqn:proof of nonadjacent distance larger than one II-(i)-(b)-case-3-ffff}
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-7777}
and \reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-8888}
follow from
\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-cccc}--\reqnarray{proof of nonadjacent distance larger than one II-(i)-(b)-case-3-ffff}.
(ii) Note that in \rlemma{nonadjacent distance larger than one II}(ii),
we have $n_a-n_b\geq 2$ for some $1\leq a<b\leq r_{h-1}$ and $b\geq a+2$.
For ease of presentation, let $n_b=p$.
Then we have from $n_a-n_b\geq 2$ that $n_a\geq p+2$.
As we have $n_a\geq p+2$, $n_b=p$, $b\geq a+2>a$,
and the condition $|n_{i+1}-n_i|\leq 1$ for $i=1,2,\ldots,r_{h-1}-1$
in \reqnarray{proof of nonadjacent distance larger than one II-111},
we can argue as that for \reqnarray{proof of nonadjacent distance larger than one-(i)-111}--\reqnarray{proof of nonadjacent distance larger than one-(i)-555}
in the proof of \rlemma{nonadjacent distance larger than one}(i)
in \rappendix{proof of nonadjacent distance larger than one} that there exist two positive integers $a'$ and $b'$ such that
\beqnarray{}
\alignspace a\leq a'<b'\leq b \textrm{ and } b'\geq a'+2,
\label{eqn:proof of nonadjacent distance larger than one II-(ii)-111} \\
\alignspace n_{a'}=p+2,\ n_{b'}=p, \textrm{ and } n_i=p+1 \textrm{ for } a'<i<b'.
\label{eqn:proof of nonadjacent distance larger than one II-(ii)-222}
\eeqnarray
An illustration of
\reqnarray{proof of nonadjacent distance larger than one II-(ii)-111} and \reqnarray{proof of nonadjacent distance larger than one II-(ii)-222}
is given in \rfigure{appendix-I-(ii)}.
\bpdffigure{appendix-I-ii.pdf}{4.0in}
\epdffigure{appendix-I-(ii)}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one II-(ii)-111} and \reqnarray{proof of nonadjacent distance larger than one II-(ii)-222}.}
To prove \rlemma{nonadjacent distance larger than one II}(ii),
we need to show that there exists a sequence of positive integers
${\nbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$ such that
\beqnarray{proof of nonadjacent distance larger than one II-(ii)-333}
{\nbf'}_1^{r_{h-1}}\succ\nbf_1^{r_{h-1}}.
\eeqnarray
Note that from \reqnarray{proof of nonadjacent distance larger than one II-(ii)-222}
and $b'\geq a'+2$ in \reqnarray{proof of nonadjacent distance larger than one II-(ii)-111},
we see that $n_{b'-1}=p+1$.
It then follows from $n_{b'-1}=p+1$ and $n_{b'}=p$
in \reqnarray{proof of nonadjacent distance larger than one II-(ii)-222} that
\beqnarray{proof of nonadjacent distance larger than one II-(ii)-444}
n_{b'-1}-n_{b'}=(p+1)-p=1.
\eeqnarray
We consider the following four possible cases.
Note that in Case~2--Case~4 below,
we have $b'\leq r_{h-1}-1$ and hence it follows from $a'\geq 1$
and $b'\geq a'+2$ in \reqnarray{proof of nonadjacent distance larger than one II-(ii)-111} that
\beqnarray{proof of nonadjacent distance larger than one II-(ii)-555}
2\leq b'-1\leq r_{h-1}-2.
\eeqnarray
\emph{Case 1: $b'=r_{h-1}$.}
Let $\mbf_1^{r_{h-1}}$ be a sequence of positive integers such that
\beqnarray{proof of nonadjacent distance larger than one II-(ii)-case-1-111}
m_{b'-1}=n_{b'-1}-1,\ m_{b'}=n_{b'}+1, \textrm{ and } m_i=n_i \textrm{ for } i\neq b'-1, b'.
\eeqnarray
As before, it is easy to show that $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
It follows from
$\nbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$ in \reqnarray{proof of nonadjacent distance larger than one II-111},
\reqnarray{proof of nonadjacent distance larger than one II-(ii)-444},
\reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-1-111},
$b'-1=r_{h-1}-1$, and \reqnarray{comparison rule B-1} in \rlemma{comparison rule B}(i)
that $\nbf_1^{r_{h-1}}\prec\mbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one II-(ii)-333} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
\emph{Case 2: $b'\leq r_{h-1}-1$ and there exists a positive integer $j$ such that
$1\leq j\leq \min\{b'-a'-1,r_{h-1}-b'\}$, $n_{b'+j'}=p+1$ for $j'=1,2,\ldots,j-1$, and $n_{b'+j}<p+1$.}
Let $\mbf_1^{r_{h-1}}$ be a sequence of positive integers as given in
\reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-1-111}.
As in Case~1 above, we have $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
\bpdffigure{appendix-I-ii-case-2.pdf}{5.5in}
\epdffigure{appendix-I-(ii)-case-2}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-2-333}
and \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-2-444}:
(a) $j<b'-a'-1$; (b) $j=b'-a'-1$.}
In the following, we show that
\beqnarray{}
\alignspace 1\leq j\leq \min\{(b'-1)-1,r_{h-1}-(b'-1)-1\},
\label{eqn:proof of nonadjacent distance larger than one II-(ii)-case-2-222}\\
\alignspace n_{(b'-1)-j'}=n_{b'+j'}, \textrm{ for } j'=1,2,\ldots,j-1,
\label{eqn:proof of nonadjacent distance larger than one II-(ii)-case-2-333}\\
\alignspace n_{(b'-1)-j}>n_{b'+j}.
\label{eqn:proof of nonadjacent distance larger than one II-(ii)-case-2-444}
\eeqnarray
An illustration of
\reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-2-333}
and \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-2-444}
is given in \rfigure{appendix-I-(ii)-case-2}.
Therefore, it follows from
$\nbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$ in \reqnarray{proof of nonadjacent distance larger than one II-111},
\reqnarray{proof of nonadjacent distance larger than one II-(ii)-444}--\reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-2-444},
and \reqnarray{comparison rule B-2} in \rlemma{comparison rule B}(ii) that
$\nbf_1^{r_{h-1}}\prec \mbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one II-(ii)-333} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
From $1\leq j\leq \min\{b'-a'-1,r_{h-1}-b'\}$ and $a'\geq 1$, we have
\beqnarray{proof of nonadjacent distance larger than one II-(ii)-case-2-555}
1\leq j\leq \min\{b'-a'-1,r_{h-1}-b'\}\leq \min\{(b'-1)-1,r_{h-1}-(b'-1)-1\}.
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-2-222}
follows from \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-2-555}.
To prove \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-2-333}
and \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-2-444},
note that we have $j\leq \min\{b'-a'-1, r_{h-1}-b'\}\leq b'-a'-1$.
If $j<b'-a'-1$, then we have $a'<b'-1-j<b'$ and it follows from
\reqnarray{proof of nonadjacent distance larger than one II-(ii)-222} that
\beqnarray{proof of nonadjacent distance larger than one II-(ii)-case-2-666}
n_{(b'-1)-j'}=p+1, \textrm{ for } j'=1,2,\ldots,j.
\eeqnarray
On the other hand, if $j=b'-a'-1$, then we have $a'=b'-1-j<b'$
and it follows from
\reqnarray{proof of nonadjacent distance larger than one II-(ii)-222} that
\beqnarray{}
\alignspace
n_{(b'-1)-j'}=p+1, \textrm{ for } j'=1,2,\ldots,j-1,
\label{eqn:proof of nonadjacent distance larger than one II-(ii)-case-2-777}\\
\alignspace
n_{(b'-1)-j}=n_{a'}=p+2.
\label{eqn:proof of nonadjacent distance larger than one II-(ii)-case-2-888}
\eeqnarray
By combining \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-2-666}--\reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-2-888},
$n_{b'+j'}=p+1$ for $j'=1,2,\ldots,j-1$, and $n_{b'+j}<p+1$,
we obtain \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-2-333}
and \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-2-444}.
\emph{Case 3: $b'\leq r_{h-1}-1$ and there exists a positive integer $j$ such that
$1\leq j\leq \min\{b'-a'-1,r_{h-1}-b'\}$, $n_{b'+j'}=p+1$ for $j'=1,2,\ldots,j-1$, and $n_{b'+j}>p+1$.}
In this case, we can show that $j\geq 2$.
To see this, suppose on the contrary that $j=1$,
then we have $n_{b'+1}>p+1$ in this case.
As it follows from $n_{b'}=p$ in \reqnarray{proof of nonadjacent distance larger than one II-(ii)-222}
and the condition $|n_{i+1}-n_i|\leq 1$ for $i=1,2,\ldots,r_{h-1}-1$
in \reqnarray{proof of nonadjacent distance larger than one II-111}
that $n_{b'+1}$ must be equal to $p-1$ (provided that $p\geq 2$), $p$, or $p+1$,
we have reached a contradiction.
Since $j\geq 2$, we have $n_{b'+j-1}=p+1$ in this case.
It then follows from the condition $|n_{i+1}-n_i|\leq 1$ for $i=1,2,\ldots,r_{h-1}-1$
in \reqnarray{proof of nonadjacent distance larger than one II-111}
that $n_{b'+j}$ must be equal to $p$, $p+1$, or $p+2$.
As we also have $n_{b'+j}>p+1$ in this case, we immediately see that $n_{b'+j}=p+2$.
From $n_{b'}=p$, $n_{b'+j'}=p+1$ for $j'=1,2,\ldots,j-1$, $n_{b'+j}=p+2$,
and $b'\geq 3$ in \reqnarray{proof of nonadjacent distance larger than one II-(ii)-555},
we can argue in the same way as in the proof of (i) above
(with the roles of $a'$ and $b'$ in the proof of (i) replaced by $b'$ and $b'+j$, respectively)
that there exists a sequence of positive integers ${\nbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$
such that ${\nbf'}_1^{r_{h-1}}\succ\nbf_1^{r_{h-1}}$.
\emph{Case 4: $b'\leq r_{h-1}-1$ and $n_{b'+j'}=p+1$ for $j'=1,2,\ldots,\min\{b'-a'-1,r_{h-1}-b'\}$.}
Let $\mbf_1^{r_{h-1}}$ be a sequence of positive integers as given in
\reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-1-111}.
As in Case~1 above, we have $\mbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
We then consider the two subcases $b'-a'-1>r_{h-1}-b'$ and $b'-a'-1\leq r_{h-1}-b'$ separately.
\emph{Subcase 4(a): $b'-a'-1>r_{h-1}-b'$.}
\bpdffigure{appendix-I-ii-case-4-1.pdf}{4.5in}
\epdffigure{appendix-I-(ii)-case-4-1}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-111}
(note that we have $\min\{(b'-1)-1, r_{h-1}-(b'-1)-1\}=r_{h-1}-b'$
in \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-444}).}
In this subcase, we show that
\beqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-111}
n_{(b'-1)-j'}=n_{b'+j'}=p+1, \textrm{ for } j'=1,2,\ldots,\min\{(b'-1)-1, r_{h-1}-(b'-1)-1\}.
\eeqnarray
An illustration of \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-111}
is given in \rfigure{appendix-I-(ii)-case-4-1}.
Therefore, it follows from
$\nbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$ in \reqnarray{proof of nonadjacent distance larger than one II-111},
\reqnarray{proof of nonadjacent distance larger than one II-(ii)-444}--\reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-1-111},
\reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-111},
and \reqnarray{comparison rule B-4} in \rlemma{comparison rule B}(iii) that
$\nbf_1^{r_{h-1}}\prec \mbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one II-(ii)-333} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
To prove \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-111},
note that from $b'-a'-1>r_{h-1}-b'$, $a'\geq 1$, and $b'\leq r_{h-1}-1$, we have
\beqnarray{}
\alignspace
r_{h-1}-b'<b'-a'-1\leq b'-2,
\label{eqn:proof of nonadjacent distance larger than one II-(ii)-case-4-222}\\
\alignspace
a'<(b'-1)-(r_{h-1}-b')\leq (b'-1)-1<b'.
\label{eqn:proof of nonadjacent distance larger than one II-(ii)-case-4-333}
\eeqnarray
From \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-222},
we see that
\beqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-444}
\min\{(b'-1)-1, r_{h-1}-(b'-1)-1\}=\min\{b'-2, r_{h-1}-b'\}=r_{h-1}-b'.
\eeqnarray
From \reqnarray{proof of nonadjacent distance larger than one II-(ii)-222}
and \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-333},
we have
\beqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-555}
n_{(b'-1)-j'}=p+1, \textrm{ for } j'=1,2,\ldots,r_{h-1}-b'.
\eeqnarray
Furthermore, in this subcase we have from $b'-a'-1>r_{h-1}-b'$ that
\beqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-666}
n_{b'+j'}=p+1, \textrm{ for } j'=1,2,\ldots,\min\{b'-a'-1,r_{h-1}-b'\}=r_{h-1}-b'.
\eeqnarray
By combining \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-444},
\reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-555},
and \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-666},
we obtain \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-111}.
\emph{Subcase 4(b): $b'-a'-1\leq r_{h-1}-b'$.}
\bpdffigure{appendix-I-ii-case-4-2.pdf}{4.5in}
\epdffigure{appendix-I-(ii)-case-4-2}
{An illustration of \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-888}
and \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-999}.}
In this subcase, we show that
\beqnarray{}
\alignspace
1\leq b'-a'-1\leq \min\{(b'-1)-1, r_{h-1}-(b'-1)-1\},
\label{eqn:proof of nonadjacent distance larger than one II-(ii)-case-4-777}\\
\alignspace
n_{(b'-1)-j'}=n_{b'+j'}=p+1, \textrm{ for } j'=1,2,\ldots,b'-a'-2,
\label{eqn:proof of nonadjacent distance larger than one II-(ii)-case-4-888}\\
\alignspace
n_{(b'-1)-(b'-a'-1)}=p+2>n_{b'+(b'-a'-1)}=p+1.
\label{eqn:proof of nonadjacent distance larger than one II-(ii)-case-4-999}
\eeqnarray
An illustration of
\reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-888}
and \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-999}
is given in \rfigure{appendix-I-(ii)-case-4-2}.
Therefore, it follows from
$\nbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$ in \reqnarray{proof of nonadjacent distance larger than one II-111},
\reqnarray{proof of nonadjacent distance larger than one II-(ii)-444}--\reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-1-111},
\reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-777}--\reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-999},
and \reqnarray{comparison rule B-2} in \rlemma{comparison rule B}(ii) that
$\nbf_1^{r_{h-1}}\prec \mbf_1^{r_{h-1}}$,
i.e., \reqnarray{proof of nonadjacent distance larger than one II-(ii)-333} holds
with ${\nbf'}_1^{r_{h-1}}=\mbf_1^{r_{h-1}}$.
From $b'\geq a'+2$ in \reqnarray{proof of nonadjacent distance larger than one II-(ii)-111}
and $a'\geq 1$, we see that
\beqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-aaa}
1\leq b'-a'-1\leq b'-2.
\eeqnarray
It then follows from \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-aaa}
and $b'-a'-1\leq r_{h-1}-b'$ that
\beqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-bbb}
1\leq b'-a'-1\leq \min\{b'-2, r_{h-1}-b'\}=\min\{(b'-1)-1, r_{h-1}-(b'-1)-1\}.
\eeqnarray
Thus, \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-777}
follows from \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-bbb}.
To prove \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-888}
and \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-999},
note that from \reqnarray{proof of nonadjacent distance larger than one II-(ii)-222}
and $a'=(b'-1)-(b'-a'-1)<b'$ we have
\beqnarray{}
\alignspace
n_{(b'-1)-j'}=p+1, \textrm{ for } j'=1,2,\ldots,b'-a'-2,
\label{eqn:proof of nonadjacent distance larger than one II-(ii)-case-4-ccc}\\
\alignspace
n_{(b'-1)-(b'-a'-1)}=n_{a'}=p+2.
\label{eqn:proof of nonadjacent distance larger than one II-(ii)-case-4-ddd}
\eeqnarray
Furthermore, in this subcase we have from $b'-a'-1\leq r_{h-1}-b'$ that
\beqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-eee}
n_{b'+j'}=p+1, \textrm{ for } j'=1,2,\ldots,\min\{b'-a'-1,r_{h-1}-b'\}=b'-a'-1.
\eeqnarray
By combining \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-ccc},
\reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-ddd},
and \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-eee},
we obtain \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-888}
and \reqnarray{proof of nonadjacent distance larger than one II-(ii)-case-4-999}.
\bappendix{Proof of \rlemma{main lemma II}} {proof of main lemma II}
In this appendix,
we use \rcorollary{adjacent distance larger than one II}(i) (corollary to \rlemma{adjacent distance larger than one II}),
\rcorollary{nonadjacent distance larger than one II}(i) (corollary to \rlemma{nonadjacent distance larger than one II}),
and Comparison rule B in \rlemma{comparison rule B} to prove \rlemma{main lemma II}.
Let $\nbf_1^{r_{h-1}}(h)$ be an optimal sequence over $\Ncal_{M,k}(h)$.
As commented before the statement of \rlemma{main lemma II},
we can use \rcorollary{adjacent distance larger than one II}(i)
and \rcorollary{nonadjacent distance larger than one II}(i) to show that
\beqnarray{proof of main lemma II-111}
n_i(h)=
\bselection
q_h+1, &\textrm{if } i=i_1,i_2,\ldots,i_{r_h}, \\
q_h, &\textrm{otherwise},
\eselection
\eeqnarray
for some $1\leq i_1<i_2<\cdots <i_{r_h}\leq r_{h-1}$.
In the following, we show that $i_{r_h}=r_{h-1}$ by contradiction.
Assume on the contrary that $i_{r_h}\leq r_{h-1}-1$.
We will use Comparison rule B in \rlemma{comparison rule B} to show that
there exists a sequence ${\nbf'}_1^{r_{h-1}}(h)\in \Ncal_{M,k}(h)$
such that ${\nbf'}_1^{r_{h-1}}(h)\succ\nbf_1^{r_{h-1}}(h)$,
contradicting to the optimality of $\nbf_1^{r_{h-1}}(h)$.
For simplicity, let $\nbf_1^{r_{h-1}}=\nbf_1^{r_{h-1}}(h)$.
As $i_{r_h}\leq r_{h-1}-1$, we have $i_{r_h}<i_{r_h}+1\leq r_{h-1}$.
Let ${\nbf'}_1^{r_{h-1}}$ be a sequence of positive integers such that
\beqnarray{proof of main lemma II-222}
n'_{i_{r_h}}=n_{i_{r_h}}-1,\ n'_{i_{r_h}+1}=n_{i_{r_h}+1}+1,
\textrm{ and } n'_i=n_i \textrm{ for } i\neq i_{r_h}, i_{r_h}+1.
\eeqnarray
It is easy to see from \reqnarray{proof of main lemma II-222},
\reqnarray{proof of main lemma II-111},
$\nbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$, and \reqnarray{N-M-k-h} that
\beqnarray{proof of main lemma II-333}
\sum_{i=1}^{r_{h-1}}n'_i=\sum_{i=1}^{r_{h-1}}n_i=r_{h-2}.
\eeqnarray
As such, it follows from $\nbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
\reqnarray{proof of main lemma II-111}, \reqnarray{proof of main lemma II-222},
and \reqnarray{proof of main lemma II-333} that ${\nbf'}_1^{r_{h-1}}\in \Ncal_{M,k}(h)$.
Note that from \reqnarray{proof of main lemma II-111} we have
\beqnarray{proof of main lemma II-444}
n_{i_{r_h}}-n_{i_{r_h}+1}=(q_h+1)-q_h=1.
\eeqnarray
Now we have $\nbf_1^{r_{h-1}}\in \Ncal_{M,k}(h)$,
$n_{i_{r_h}}-n_{i_{r_h}+1}=1$ in \reqnarray{proof of main lemma II-444},
and $n'_{i_{r_h}}=n_{i_{r_h}}-1$, $n'_{i_{r_h}+1}=n_{i_{r_h}+1}+1$, and $n'_i=n_i$ for $i\neq i_{r_h}, i_{r_h}+1$
in \reqnarray{proof of main lemma II-222}.
As such, we are in a position to use Comparison rule B in \rlemma{comparison rule B} (with $a=i_{r_h}$)
to show that ${\nbf'}_1^{r_{h-1}}\succ\nbf_1^{r_{h-1}}$.
We need to consider the two cases $r_h=1$ and $r_h\geq 2$ separately.
\emph{Case 1: $r_h=1$}.
In this case, we have from \reqnarray{proof of main lemma II-111} and $i_{r_h}\leq r_{h-1}-1$ that
\beqnarray{proof of main lemma II-case-1-111}
n_i=
\bselection
q_h+1, &\textrm{if } i=i_{r_h}, \\
q_h, &\textrm{otherwise}.
\eselection
\eeqnarray
If $i_{r_h}=1$ or $i_{r_h}=r_{h-1}-1$,
then it follows from \reqnarray{comparison rule B-1} in \rlemma{comparison rule B}(i)
that $\nbf_1^{r_{h-1}}\prec{\nbf'}_1^{r_{h-1}}$.
On the other hand, if $2\leq i_{r_h}\leq r_{h-1}-2$,
then it is easy to see from \reqnarray{proof of main lemma II-case-1-111} that
\beqnarray{proof of main lemma II-case-1-222}
n_{i_{r_h}-j}=n_{(i_{r_h}+1)+j}=q_h, \textrm{ for } j=1,2,\ldots,\min\{i_{r_h}-1,r_{h-1}-i_{r_h}-1\}.
\eeqnarray
Therefore, it follows from $2\leq i_{r_h}\leq r_{h-1}-2$,
\reqnarray{proof of main lemma II-case-1-222},
and \reqnarray{comparison rule B-4} in \rlemma{comparison rule B}(iii)
that $\nbf_1^{r_{h-1}}\prec{\nbf'}_1^{r_{h-1}}$.
\emph{Case 2: $r_h\geq 2$}.
As $1\leq i_1<i_2<\cdots <i_{r_h}\leq r_{h-1}-1$ and $r_h\geq 2$,
we have $2\leq i_1+1\leq i_{r_h}\leq r_{h-1}-1$ in this case.
If $i_{r_h}=r_{h-1}-1$, then it follows from \reqnarray{comparison rule B-1}
in \rlemma{comparison rule B}(i) that $\nbf_1^{r_{h-1}}\prec{\nbf'}_1^{r_{h-1}}$.
On the other hand, if $2\leq i_{r_h}\leq r_{h-1}-2$,
then we consider the following two subcases.
\emph{Subcase 2(a): $i_{r_h}-i_{r_h-1}>r_{h-1}-i_{r_h}-1$.}
In this subcase, we have $i_{r_h}-1\geq i_{r_h}-i_{r_h-1}>r_{h-1}-i_{r_h}-1$,
and it follows that
\beqnarray{proof of main lemma II-case-2-111}
\min\{i_{r_h}-1,r_{h-1}-i_{r_h}-1\}=r_{h-1}-i_{r_h}-1.
\eeqnarray
From $i_{r_h}-i_{r_h-1}>r_{h-1}-i_{r_h}-1$ in this subcase, we have
\beqnarray{}
\alignspace
i_{r_h}-(r_{h-1}-i_{r_h}-1)>i_{r_h}-(i_{r_h}-i_{r_h-1})=i_{r_h-1},
\label{eqn:proof of main lemma II-case-2-222}\\
\alignspace
(i_{r_h}+1)+(r_{h-1}-i_{r_h}-1)=r_{h-1}.
\label{eqn:proof of main lemma II-case-2-333}
\eeqnarray
It is easy to see from \reqnarray{proof of main lemma II-111}
and \reqnarray{proof of main lemma II-case-2-111}--\reqnarray{proof of main lemma II-case-2-333} that
\beqnarray{proof of main lemma II-case-2-444}
n_{i_{r_h}-j}=n_{(i_{r_h}+1)+j}=q_h,\ j=1,2,\ldots,r_{h-1}-i_{r_h}-1=\min\{i_{r_h}-1,r_{h-1}-i_{r_h}-1\}.
\eeqnarray
Therefore, it follows from $2\leq i_{r_h}\leq r_{h-1}-2$,
\reqnarray{proof of main lemma II-case-2-444},
and \reqnarray{comparison rule B-4} in \rlemma{comparison rule B}(iii)
that $\nbf_1^{r_{h-1}}\prec{\nbf'}_1^{r_{h-1}}$.
\emph{Subcase 2(b): $i_{r_h}-i_{r_h-1}\leq r_{h-1}-i_{r_h}-1$.}
In this subcase, we see from $i_{r_h-1}<i_{r_h}$, $i_{r_h}-1\geq i_{r_h}-i_{r_h-1}$,
and $i_{r_h}-i_{r_h-1}\leq r_{h-1}-i_{r_h}-1$ that
\beqnarray{proof of main lemma II-case-2-555}
1\leq i_{r_h}-i_{r_h-1}\leq \min\{i_{r_h}-1,r_{h-1}-i_{r_h}-1\}.
\eeqnarray
As it is clear that
\beqnarray{}
\alignspace
i_{r_h}-(i_{r_h}-i_{r_h-1})=i_{r_h-1},
\label{eqn:proof of main lemma II-case-2-666}\\
\alignspace
i_{r_h}<(i_{r_h}+1)+(i_{r_h}-i_{r_h-1})\leq (i_{r_h}+1)+(r_{h-1}-i_{r_h}-1)=r_{h-1},
\label{eqn:proof of main lemma II-case-2-777}
\eeqnarray
we see from \reqnarray{proof of main lemma II-111},
\reqnarray{proof of main lemma II-case-2-666},
and \reqnarray{proof of main lemma II-case-2-777} that
\beqnarray{}
\alignspace
n_{i_{r_h}-j}=n_{(i_{r_h}+1)+j}=q_h, \textrm{ for } j=1,2,\ldots,i_{r_h}-i_{r_h-1}-1,
\label{eqn:proof of main lemma II-case-2-888}\\
\alignspace
n_{i_{r_h}-(i_{r_h}-i_{r_h-1})}=n_{i_{r_h-1}}=q_h+1>n_{(i_{r_h}+1)+(i_{r_h}-i_{r_h-1})}=q_h.
\label{eqn:proof of main lemma II-case-2-999}
\eeqnarray
Therefore, it follows from
$2\leq i_{r_h}\leq r_{h-1}-2$,
\reqnarray{proof of main lemma II-case-2-555},
\reqnarray{proof of main lemma II-case-2-888},
\reqnarray{proof of main lemma II-case-2-999},
and \reqnarray{comparison rule B-2} in \rlemma{comparison rule B}(ii)
that $\nbf_1^{r_{h-1}}\prec{\nbf'}_1^{r_{h-1}}$.
\iffalse
\bappendix{}{N=1}
In this appendix,
we prove that \rconjecture{main result} holds for the special case that $N=1$,
where $N$ is obtained as in Step~1 of \ralgorithm{main result}.
Let $r_{-1}=M$, $r_0=k$, and let $q_i$ and $r_i$, $i=1,2,\ldots,N$,
be recursively obtained as in Step~1 of \ralgorithm{main result}.
Note that in \rconjecture{main result} we have $\gcd(M,k)\geq 2$,
it then follows from \rtheorem{main result}(ii) and (iii) that there are at most two optimal sequences over $\Ncal_{M,k}$.
Since in this appendix we have an additional condition that $N=1$,
in Step~1 of \ralgorithm{main result} we obtain
\beqnarray{proof of N=1-111}
\gcd(M,k)=r_{N-1}=r_0=k,
\eeqnarray
and then we go to Step~2 of \ralgorithm{main result} and obtain the two possible optimal sequences
\beqnarray{proof of N=1-222}
\nbf_1^k(1)=\nbf_1^{r_0}(1)=(q_1,q_1,\ldots,q_1)\ \textrm{ and }\ \mbf_1^k(1)=\mbf_1^{r_0}(1)=(q_1+1,q_1,\ldots,q_1,q_1-1).
\eeqnarray
Note that $\mbf_1^k(1)=(q_1+1,q_1-1)$ for the case that $k=2$,
also note that $q_1\geq 2$ as explained in Step 2 (ii) of \ralgorithm{main result}.
It suffices to show that
\beqnarray{proof of N=1-333}
\nbf_1^k(1)\equiv\mbf_1^k(1).
\eeqnarray
We will prove \reqnarray{proof of N=1-333} by induction on $k\geq 2$.
First consider the case that $k=2$.
Note that in this case we have from \reqnarray{proof of N=1-111} that $\gcd(M,k)=k=2$.
It then follows from \rtheorem{main result}(ii) that the $\nbf_1^k(1)$ and $\mbf_1^k(1)$ are both optimal sequences over $\Ncal_{M,k}$,
i.e., \reqnarray{proof of N=1-333} holds.
Suppose as the induction hypothesis that \reqnarray{proof of N=1-333} holds up to some $k-1\geq 2$.
Now consider the case $k$.
For simplicity, let $\nbf_1^k=\nbf_1^k(1)$ and $\mbf_1^k=\mbf_1^k(1)$.
To prove \reqnarray{proof of N=1-333},
let $\dbf_1^M=(d_1,d_2,\ldots,d_M)$ and $\ebf_1^M=(e_1,e_2,\ldots,e_M)$ be given by using
$n_1,n_2,\ldots,n_k$ and $m_1,m_2,\ldots,$ $m_k$, respectively, in \reqnarray{OQ-LR-delays-greedy-1}.
Let $s_0=0$ and $s_i=\sum_{\ell=0}^{i}n_{\ell}$ for $i=1,2,\ldots,k$,
and let $t_0=0$ and $t_i=\sum_{\ell=0}^{i}m_{\ell}$ for $i=1,2,\ldots,k$.
Let
\beqnarray{}
\alignspace \alpha_i=e_{t_i}-d_{s_i}, \textrm{ for } i=1,2,\ldots,k,
\label{eqn:proof of N=1-444}\\
\alignspace \beta_i=B(\ebf_1^{t_i};i)-B(\dbf_1^{s_i};i), \textrm{ for } i=0,1,2,\ldots,k.
\label{eqn:proof of N=1-555}
\eeqnarray
It follows from \reqnarray{proof of N=1-555}, \reqnarray{OQ-LR-delays-greedy-7},
and \reqnarray{proof of N=1-444} that
\beqnarray{proof of N=1-666}
\beta_i
\aligneq B(\ebf_1^{t_i};i)-B(\dbf_1^{s_i};i)\nn \\
\aligneq B(\ebf_1^{t_{i-1}};i-1)+e_{t_i}-B(\dbf_1^{s_{i-1}};i-1)-d_{s_i}\nn \\
\aligneq \alpha_i+\beta_{i-1},
\eeqnarray
where $i=1,2,\ldots,k$.
We will prove \reqnarray{proof of N=1-333} by showing that $B(\dbf_1^M;k)=B(\ebf_1^M;k)$,
i.e., $\beta_k=0$.
To utilize the induction hypothesis,
for some $\tilde{M}\geq 2$ and $1\leq \tilde{k}\leq \tilde{M}-1$,
let $\tilde{r}_{-1}=\tilde{M}$, $\tilde{r}_0=\tilde{k}$, and let $\tilde{q}_i$ and $\tilde{r}_i$, $i=1,2,\ldots,\tilde{N}$,
be recursively obtained as in Step~1 of \ralgorithm{main result}.
We choose $\tilde{M}$ and $\tilde{k}$ such that
\beqnarray{proof of N=1-777}
2\leq \tilde{k}\leq k-1,\ \tilde{q}_1=q_1, \textrm{ and } \tilde{r}_1=r_1=0.
\eeqnarray
Note that $\tilde{r}_1=0$ implies that $\tilde{N}=1$.
As such,
it follows from \rtheorem{main result} and \ralgorithm{main result} the two optimal sequences over $\Ncal_{\tilde{M},\tilde{k}}$
are given by
\beqnarray{proof of N=1-888}
\tilde{\nbf}_1^{\tilde{k}}(1)=\tilde{\nbf}_1^{\tilde{r}_0}(1)=(\tilde{q}_1,\tilde{q}_1,\ldots,\tilde{q}_1)\
\textrm{ and }\ \tilde{\mbf}_1^{\tilde{k}}(1)=\tilde{\mbf}_1^{\tilde{r}_0}(1)=(\tilde{q}_1+1,\tilde{q}_1,\ldots,\tilde{q}_1,\tilde{q}_1-1).
\eeqnarray
As we have $2\leq \tilde{k}\leq k-1$ in \reqnarray{proof of N=1-777},
it follows from \reqnarray{proof of N=1-888} and the induction hypothesis for \reqnarray{proof of N=1-333} that
\beqnarray{proof of N=1-999}
\tilde{\nbf}_1^{\tilde{k}}(1)\equiv \tilde{\mbf}_1^{\tilde{k}}(1).
\eeqnarray
For simplicity, let $\tilde{\nbf}_1^{\tilde{k}}=\tilde{\nbf}_1^{\tilde{k}}(1)$ and $\tilde{\mbf}_1^{\tilde{k}}=\tilde{\mbf}_1^{\tilde{k}}(1)$.
Let $\tilde{\dbf}_1^{\tilde{M}}=(\tilde{d}_1,\tilde{d}_2,\ldots,\tilde{d}_{\tilde{M}})$
and $\tilde{\ebf}_1^{\tilde{M}}=(\tilde{e}_1,\tilde{e}_2,\ldots,\tilde{e}_{\tilde{M}})$ be given by using
$\tilde{n}_1,\tilde{n}_2,\ldots,\tilde{n}_{\tilde{k}}$ and $\tilde{m}_1,\tilde{m}_2,\ldots,\tilde{m}_{\tilde{k}}$, respectively, in \reqnarray{OQ-LR-delays-greedy-1}.
Let $\tilde{s}_0=0$ and $\tilde{s}_i=\sum_{\ell=0}^{i}\tilde{n}_{\ell}$ for $i=1,2,\ldots,\tilde{k}$,
and let $\tilde{t}_0=0$ and $\tilde{t}_i=\sum_{\ell=0}^{i}\tilde{m}_{\ell}$ for $i=1,2,\ldots,\tilde{k}$.
Let
\beqnarray{proof of N=1-aaa}
\tilde{\beta}_i=B(\tilde{\ebf}_1^{\tilde{t}_i};i)-B(\tilde{\dbf}_1^{\tilde{s}_i};i) \textrm{ for } i=0,1,2,\ldots,\tilde{k}.
\eeqnarray
Then \reqnarray{proof of N=1-999} implies that
\beqnarray{proof of N=1-bbb}
\tilde{\beta}_{\tilde{k}}=B(\tilde{\ebf}_1^{\tilde{M}};\tilde{k})-B(\tilde{\dbf}_1^{\tilde{M}};\tilde{k})=0.
\eeqnarray
We can use the information above to show that
\beqnarray{proof of N=1-ccc}
\beta_{\tilde{k}}=B(\ebf_1^{t_{\tilde{k}-1}};\tilde{k}-1)+1.
\eeqnarray
To see this, note that from \reqnarray{proof of N=1-222}, \reqnarray{proof of N=1-888},
and $\tilde{q}_1=q_1$ and $2\leq \tilde{k}\leq k-1$ in \reqnarray{proof of N=1-777},
we have
\beqnarray{proof of N=1-ddd}
n_i=\tilde{n}_i \textrm{ for } i=1,2,\ldots,\tilde{k},
\eeqnarray
it follows that $s_i=\tilde{s}_i$ for $i=1,2,\ldots,\tilde{k}$, and $d_i=\tilde{d}_i$ for $i=1,2,\ldots,s_{\tilde{k}}$,
and hence we obtain
\beqnarray{proof of N=1-eee}
B(\dbf_1^{s_{\tilde{k}}};\tilde{k})=B(\tilde{\dbf}_1^{\tilde{s}_{\tilde{k}}};\tilde{k}).
\eeqnarray
Similarly, from \reqnarray{proof of N=1-222}, \reqnarray{proof of N=1-888},
and $\tilde{q}_1=q_1$ and $2\leq \tilde{k}\leq k-1$ in \reqnarray{proof of N=1-777}, we have
\beqnarray{proof of N=1-fff}
m_i=\tilde{m}_i \textrm{ for } i=1,2,\ldots,\tilde{k}-1, \textrm{ and } m_{\tilde{k}}=\tilde{m}_{\tilde{k}}+1.
\eeqnarray
It then follows that $t_i=\tilde{t}_i$ for $i=1,2,\ldots,\tilde{k}-1$, $t_{\tilde{k}}=\tilde{t}_{\tilde{k}}+1$,
and $e_i=\tilde{e}_i$ for $i=1,2,\ldots,\tilde{t}_{\tilde{k}}$.
From \reqnarray{OQ-LR-delays-greedy-7}, \reqnarray{OQ-LR-delays-greedy-3},
$t_{\tilde{k}}=\tilde{t}_{\tilde{k}}+1$, $t_{\tilde{k}-1}=\tilde{t}_{\tilde{k}-1}$, $e_i=\tilde{e}_i$ for $i=1,2,\ldots,\tilde{t}_{\tilde{k}}$.
and \reqnarray{OQ-LR-delays-greedy-7}, we see that
\beqnarray{proof of N=1-ggg}
B(\ebf_1^{t_{\tilde{k}}};\tilde{k})
\aligneq e_{t_{\tilde{k}}}+B(\ebf_1^{t_{\tilde{k}-1}};\tilde{k}-1) \nn\\
\aligneq \left(e_{t_{\tilde{k}}-1}+B(\ebf_1^{t_{\tilde{k}-1}};\tilde{k}-1)+1\right)+B(\ebf_1^{t_{\tilde{k}-1}};\tilde{k}-1) \nn\\
\aligneq \left(e_{\tilde{t}_{\tilde{k}}}+B(\ebf_1^{\tilde{t}_{\tilde{k}-1}};\tilde{k}-1)\right)+B(\ebf_1^{t_{\tilde{k}-1}};\tilde{k}-1)+1 \nn\\
\aligneq \left(\tilde{e}_{\tilde{t}_{\tilde{k}}}+B(\tilde{\ebf}_1^{\tilde{t}_{\tilde{k}-1}};\tilde{k}-1)\right)+B(\ebf_1^{t_{\tilde{k}-1}};\tilde{k}-1)+1 \nn\\
\aligneq B(\tilde{\ebf}_1^{\tilde{t}_{\tilde{k}}};\tilde{k})+B(\ebf_1^{t_{\tilde{k}-1}};\tilde{k}-1)+1.
\eeqnarray
Now, by using \reqnarray{proof of N=1-555}, \reqnarray{proof of N=1-ggg}, \reqnarray{proof of N=1-eee}, \reqnarray{proof of N=1-aaa},
and \reqnarray{proof of N=1-bbb}, we obtain
\beqnarray{proof of N=1-hhh}
\beta_{\tilde{k}}
\aligneq B(\ebf_1^{t_{\tilde{k}}};\tilde{k})-B(\dbf_1^{s_{\tilde{k}}};\tilde{k}) \nn\\
\aligneq B(\tilde{\ebf}_1^{\tilde{t}_{\tilde{k}}};\tilde{k})+B(\ebf_1^{t_{\tilde{k}-1}};\tilde{k}-1)+1-B(\tilde{\dbf}_1^{\tilde{s}_{\tilde{k}}};\tilde{k}) \nn\\
\aligneq \left(B(\tilde{\ebf}_1^{\tilde{t}_{\tilde{k}}};\tilde{k})-B(\tilde{\dbf}_1^{\tilde{s}_{\tilde{k}}};\tilde{k})\right)+B(\ebf_1^{t_{\tilde{k}-1}};\tilde{k}-1)+1 \nn\\
\aligneq \tilde{\beta}_{\tilde{k}}+B(\ebf_1^{t_{\tilde{k}-1}};\tilde{k}-1)+1 \nn\\
\aligneq B(\ebf_1^{t_{\tilde{k}-1}};\tilde{k}-1)+1.
\eeqnarray
Thus, \reqnarray{proof of N=1-ccc} follows from \reqnarray{proof of N=1-hhh}.
Recall that in \reqnarray{proof of N=1-777} we choose some $2\leq \tilde{k}\leq k-1$ so that we obtain \reqnarray{proof of N=1-ccc}.
Since $\tilde{k}$ can be any of the positive integers $2,3,\ldots,k-1$,
it then follows from \reqnarray{proof of N=1-ccc} that
\beqnarray{proof of N=1-iii}
\beta_i\aligneq B(\ebf_1^{t_{i-1}};i-1)+1 \textrm{ for } i=1,2,\ldots,k-1.
\eeqnarray
Note that we have claimed that \reqnarray{proof of N=1-iii} also holds for $i=1$.
We prove this as follows.
From \reqnarray{proof of N=1-555},
\reqnarray{OQ-LR-delays-greedy-7}, \reqnarray{OQ-LR-delays-greedy-2},
and $m_1=q_1+1$ and $n_1=q_1$ in \reqnarray{proof of N=1-222}, we have
\beqnarray{}
\beta_{1}=B(\ebf_1^{t_1};1)-B(\dbf_1^{s_1};1)=e_{t_1}-d_{s_1}=t_1-s_1=m_1-n_1=(q_1+1)-q_1=1. \nn
\eeqnarray
This proves \reqnarray{proof of N=1-iii} for $i=1$ (note that $B(\ebf_1^{t_0};0)=0$).
We need to find $\alpha_{k-1}$ so that we can compute $\alpha_k$ and $\beta_k$.
To do this, we use \reqnarray{proof of N=1-666},
\reqnarray{proof of N=1-iii} (with $i=k-1$ and $i=k-2$),
and \reqnarray{OQ-LR-delays-greedy-7} to see that
\beqnarray{proof of N=1-jjj}
\alpha_{k-1}
\aligneq \beta_{k-1}-\beta_{k-2} \nn\\
\aligneq B(\ebf_1^{t_{k-2}};k-2)+1-(B(\ebf_1^{t_{k-3}};k-3)+1) \nn\\
\aligneq B(\ebf_1^{t_{k-2}};k-2)-B(\ebf_1^{t_{k-3}};k-3) \nn\\
\aligneq \left(e_{t_{k-2}}+B(\ebf_1^{t_{k-3}};k-3)\right)-B(\ebf_1^{t_{k-3}};k-3) \nn\\
\aligneq e_{t_{k-2}}.
\eeqnarray
Now we can compute $\alpha_k$ and $\beta_k$.
From \reqnarray{proof of N=1-444},
\reqnarray{OQ-LR-delays-greedy-3}, $m_k=q_1-1$ and $n_k=q_1$ in \reqnarray{proof of N=1-222},
\reqnarray{proof of N=1-555},
\reqnarray{proof of N=1-jjj},
\reqnarray{proof of N=1-iii} (with $i=k-1$),
and \reqnarray{OQ-LR-delays-greedy-7}, we have
\beqnarray{}
\alpha_k
\aligneq e_{t_k}-d_{s_k} \nn\\
\aligneq 2e_{t_{k-1}}+(m_k-1)(B(\ebf_1^{t_{k-1}};k-1)+1)-2d_{s_{k-1}}-(n_k-1)(B(\dbf_1^{s_{k-1}};k-1)+1) \nn\\
\aligneq 2\alpha_{k-1}+(q_1-2)(B(\ebf_1^{t_{k-1}};k-1)+1)-(q_1-1)(B(\dbf_1^{s_{k-1}};k-1)+1) \nn\\
\aligneq 2\alpha_{k-1}+(q_1-1)\beta_{k-1}-(B(\ebf_1^{t_{k-1}};k-1)+1) \nn\\
\aligneq 2e_{t_{k-2}}+(q_1-1)(B(\ebf_1^{t_{k-2}};k-2)+1)-(e_{t_{k-1}}+B(\ebf_1^{t_{k-2}};k-2)+1) \nn \\
\aligneq 2e_{t_{k-2}}+(q_1-2)(B(\ebf_1^{t_{k-2}};k-2)+1)-e_{t_{k-1}}. \nn
\eeqnarray
To continue, we use \reqnarray{OQ-LR-delays-greedy-3} and $m_{k-1}=q_1$ in \reqnarray{proof of N=1-222} to see that
\beqnarray{}
\alpha_k
\aligneq 2e_{t_{k-2}}+(q_1-2)(B(\ebf_1^{t_{k-2}};k-2)+1)-2e_{t_{k-2}}-(m_{k-1}-1)(B(\ebf_1^{t_{k-2}};k-2)+1) \nn\\
\aligneq -B(\ebf_1^{t_{k-2}};k-2)-1. \nn
\eeqnarray
Finally, by combining \reqnarray{proof of N=1-555},
\reqnarray{proof of N=1-iii} (with $i=k-1$), and the equation above,
we obtain the desired result that $\beta_k=\beta_{k-1}+\alpha_k=0$.
The proof is then completed.
\bappendix{}{N=2}
In this appendix,
we prove that \rconjecture{main result} holds for the special case that $N=2$,
where $N$ is obtain as in Step~1 of \ralgorithm{main result}.
Let $r_{-1}=M$, $r_0=k$, and let $q_i$ and $r_i$, $i=1,2,\ldots,N$,
be recursively obtained as in Step~1 of \ralgorithm{main result}.
Note that in \rconjecture{main result} we have $\gcd(M,k)\geq 2$,
it then follows from \rtheorem{main result}(ii) and (iii) that there are at most two optimal sequences over $\Ncal_{M,k}$.
Since in this appendix we have an additional condition that $N=2$,
in Step~1 of \ralgorithm{main result} we obtain
\beqnarray{proof of N=2-111}
\gcd(M,k)=r_{N-1}=r_1,
\eeqnarray
and then we go to Step~3 of \ralgorithm{main result} and obtain the two possible optimal sequences
\beqnarray{proof of N=2-222}
\nbf_1^k(1)=\nbf_1^{r_0}(1)=L_{r_{-1},r_0}(\nbf_1^{r_1}(2))\ \textrm{ and }\ \mbf_1^k(1)=\mbf_1^{r_0}(1)=L_{r_{-1},r_0}(\mbf_1^{r_1}(2)),
\eeqnarray
where
\beqnarray{proof of N=2-333}
\nbf_1^{r_1}(2)=(q_2,q_2,\ldots,q_2)\ \textrm{ and }\ \mbf_1^{r_1}(2)=(q_2-1,q_2,\ldots,q_2,q_2+1).
\eeqnarray
Note that $\mbf_1^{r_1}(2)=(q_2-1,q_2+1)$ for the case that $r_1=2$,
also note that $q_2\geq 2$ as noted in Step~3(ii) of \ralgorithm{main result}.
Furthermore, note that we have
\beqnarray{proof of N=2-444}
\sum_{\ell=1}^{r_1}n_{\ell}(2)=\sum_{\ell=1}^{r_1}m_{\ell}(2)=r_0=k.
\eeqnarray
As such, to prove \rconjecture{main result} holds for the special case that $N=2$,
we see that it suffices to show that
\beqnarray{proof of N=2-555}
\nbf_1^k(1)\equiv\mbf_1^k(1).
\eeqnarray
We will prove \reqnarray{proof of N=2-555} by induction on $r_1$.
First consider the case that $r_1=2$.
Note that in this case we have from \reqnarray{proof of N=2-111} that $\gcd(M,k)=r_1=2$.
It then follows from \rtheorem{main result}(ii) that the $\nbf_1^k(1)$ and $\mbf_1^k(1)$ are both optimal sequences over $\Ncal_{M,k}$,
i.e., \reqnarray{proof of N=2-555} holds.
Suppose as the induction hypothesis that \reqnarray{proof of N=2-555} holds up to some $r_1-1\geq 2$.
Now consider the case $r_1$.
To prove \reqnarray{proof of N=2-555},
let $\dbf_1^M=(d_1,d_2,\ldots,d_M)$ and $\ebf_1^M=(e_1,e_2,\ldots,e_M)$ be given by using
$n_1(1),n_2(1),\ldots,n_k(1)$ and $m_1(1),m_2(1),\ldots,$ $m_k(1)$, respectively, in \reqnarray{OQ-LR-delays-greedy-1}.
Let $s_0=0$ and $s_i=\sum_{\ell=0}^{i}n_{\ell}(1)$ for $i=1,2,\ldots,k$,
and let $t_0=0$ and $t_i=\sum_{\ell=0}^{i}m_{\ell}(1)$ for $i=1,2,\ldots,k$.
Also, let
\beqnarray{}
\alignspace \alpha_i=e_{t_i}-d_{s_i}, \textrm{ for } i=1,2,\ldots,k, \nn\\
\alignspace \beta_i=B(\ebf_1^{t_i};i)-B(\dbf_1^{s_i};i), \textrm{ for } i=0,1,2,\ldots,k, \nn
\eeqnarray
as in \reqnarray{proof of N=1-444} and \reqnarray{proof of N=1-555}, respectively.
It follows that \reqnarray{proof of N=1-666} still holds.
We will prove \reqnarray{proof of N=2-555} by showing that $B(\dbf_1^M;k)=B(\ebf_1^M;k)$, i.e., $\beta_k=0$.
Note that from \reqnarray{proof of N=2-222}, \reqnarray{proof of N=2-333},
and the definition of left pre-sequences in \rdefinition{left pre-sequences},
we have
\beqnarray{proof of N=2-666}
n_i(1)=
\bselection
q_1+1, &\textrm{if } i=i_1,i_2,\ldots,i_{r_1}, \\
q_1, &\textrm{otherwise},
\eselection
\eeqnarray
where
\beqnarray{proof of N=2-777}
i_j=\sum_{\ell=1}^{j-1}n_{\ell}(2)+1, \textrm{ for } j=1,2,\ldots,r_1,
\eeqnarray
and
\beqnarray{proof of N=2-888}
m_i(1)=
\bselection
q_1+1, &\textrm{if } i=i'_1,i'_2,\ldots,i'_{r_1}, \\
q_1, &\textrm{otherwise},
\eselection
\eeqnarray
where
\beqnarray{proof of N=2-999}
i'_j=\sum_{\ell=1}^{j-1}m_{\ell}(2)+1, \textrm{ for } j=1,2,\ldots,r_1,
\eeqnarray
Also, from \reqnarray{proof of N=2-444} and \reqnarray{proof of N=2-999} we have
\beqnarray{proof of N=2-aaa}
k=\sum_{\ell=1}^{r}m_{\ell}(2)=\sum_{\ell=1}^{r-1}m_{\ell}(2)+m_r(2)=(i'_r-1)+(q_2+1)=i'_r+q_2.
\eeqnarray
We then divided the proof for $\beta_k=0$ into two parts.
(i) In this part, we will use the induction hypothesis to show that
\beqnarray{proof of N=2-(i)-111}
\beta_{i'_j}=B(\ebf_1^{t_{i'_j-1}};i'_j-1)+1, \textrm{ for } j=2,3,\ldots,r_1.
\eeqnarray
To utilize the induction hypothesis,
for some $\tilde{M}\geq 2$ and $1\leq \tilde{k}\leq \tilde{M}-1$,
let $\tilde{r}_{-1}=\tilde{M}$, $\tilde{r}_0=\tilde{k}$, and let $\tilde{q}_i$ and $\tilde{r}_i$, $i=1,2,\ldots,\tilde{N}$,
be recursively obtained as in Step~1 of \ralgorithm{main result}.
We choose $\tilde{M}$ and $\tilde{k}$ such that
\beqnarray{proof of N=2-(i)-222}
\tilde{q}_1=q_1,\ 2\leq \tilde{r}_1\leq r_1-1,\ \tilde{q}_2=q_2, \textrm{ and } \tilde{r}_2=r_2=0.
\eeqnarray
Note that $\tilde{r}_2=0$ implies that $\tilde{N}=2$.
As such,
it follows from \rtheorem{main result} and \ralgorithm{main result} the two possible optimal sequences over $\Ncal_{\tilde{M},\tilde{k}}$
are given by
\beqnarray{proof of N=2-(i)-333}
\tilde{\nbf}_1^{\tilde{k}}(1)=\tilde{\nbf}_1^{\tilde{r}_0}(1)=L_{\tilde{r}_{-1},\tilde{r}_0}(\tilde{\nbf}_1^{\tilde{r}_1}(2))\
\textrm{ and }\ \tilde{\mbf}_1^{\tilde{k}}(1)=\tilde{\mbf}_1^{\tilde{r}_0}(1)=L_{\tilde{r}_{-1},\tilde{r}_0}(\tilde{\mbf}_1^{\tilde{r}_1}(2)),
\eeqnarray
where
\beqnarray{proof of N=2-(i)-444}
\tilde{\nbf}_1^{\tilde{r}_1}(2)=(\tilde{q}_2,\tilde{q}_2,\ldots,\tilde{q}_2)\ \textrm{ and }\ \tilde{\mbf}_1^{\tilde{r}_1}(2)=(\tilde{q}_2-1,\tilde{q}_2,\ldots,\tilde{q}_2,\tilde{q}_2+1).
\eeqnarray
Note that we have
\beqnarray{proof of N=2-(i)-555}
\sum_{\ell=1}^{\tilde{r}_1}\tilde{n}_{\ell}(2)=\sum_{\ell=1}^{\tilde{r}_1}\tilde{m}_{\ell}(2)=\tilde{r}_0=\tilde{k}.
\eeqnarray
As we have $2\leq \tilde{r}_1\leq r_1-1$ in \reqnarray{proof of N=2-(i)-222},
it follows from \reqnarray{proof of N=2-(i)-333}, \reqnarray{proof of N=2-(i)-444},
and the induction hypothesis for \reqnarray{proof of N=2-555} that
\beqnarray{proof of N=2-(i)-666}
\tilde{\nbf}_1^{\tilde{k}}(1)\equiv \tilde{\mbf}_1^{\tilde{k}}(1).
\eeqnarray
Let $\tilde{\dbf}_1^{\tilde{M}}=(\tilde{d}_1,\tilde{d}_2,\ldots,\tilde{d}_{\tilde{M}})$
and $\tilde{\ebf}_1^{\tilde{M}}=(\tilde{e}_1,\tilde{e}_2,\ldots,\tilde{e}_{\tilde{M}})$ be given by using
$\tilde{n}_1(1),\tilde{n}_2(1),\ldots,\tilde{n}_{\tilde{k}}(1)$ and $\tilde{m}_1(1),\tilde{m}_2(1),\ldots,\tilde{m}_{\tilde{k}}(1)$, respectively, in \reqnarray{OQ-LR-delays-greedy-1}.
Let $\tilde{s}_0=0$ and $\tilde{s}_i=\sum_{\ell=0}^{i}\tilde{n}_{\ell}(1)$ for $i=1,2,\ldots,\tilde{k}$,
and let $\tilde{t}_0=0$ and $\tilde{t}_i=\sum_{\ell=0}^{i}\tilde{m}_{\ell}(1)$ for $i=1,2,\ldots,\tilde{k}$.
Let
\beqnarray{proof of N=2-(i)-777}
\tilde{\beta}_i=B(\tilde{\ebf}_1^{\tilde{t}_i};i)-B(\tilde{\dbf}_1^{\tilde{s}_i};i) \textrm{ for } i=0,1,2,\ldots,\tilde{k}.
\eeqnarray
Then \reqnarray{proof of N=2-(i)-666} implies that
\beqnarray{proof of N=2-(i)-888}
\tilde{\beta}_{\tilde{k}}=B(\tilde{\ebf}_1^{\tilde{M}};\tilde{k})-B(\tilde{\dbf}_1^{\tilde{M}};\tilde{k})=0.
\eeqnarray
We can use the information above to show that
\beqnarray{proof of N=2-(i)-999}
\beta_{\tilde{k}}=B(\ebf_1^{t_{\tilde{k}-1}};\tilde{k}-1)+1.
\eeqnarray
To see this, first we show that
\beqnarray{}
\alignspace n_i(1)=\tilde{n}_i(1) \textrm{ for } i=1,2,\ldots,\tilde{k},
\label{eqn:proof of N=2-(i)-aaa}\\
\alignspace m_i(1)=\tilde{m}_i(1) \textrm{ for } i=1,2,\ldots,\tilde{k}-1, \textrm{ and } m_{\tilde{k}}(1)=\tilde{m}_{\tilde{k}}(1)+1.
\label{eqn:proof of N=2-(i)-bbb}
\eeqnarray
To prove \reqnarray{proof of N=2-(i)-aaa},
we see from \reqnarray{proof of N=2-(i)-333}, \reqnarray{proof of N=2-(i)-444},
and the definition of left pre-sequences in \rdefinition{left pre-sequences} that
\beqnarray{proof of N=2-(i)-ccc}
\tilde{n}_i(1)=
\bselection
\tilde{q}_1+1, &\textrm{if } i=\tilde{i}_1,\tilde{i}_2,\ldots,\tilde{i}_{\tilde{r}_1}, \\
\tilde{q}_1, &\textrm{otherwise},
\eselection
\eeqnarray
where
\beqnarray{proof of N=2-(i)-ddd}
\tilde{i}_j=\sum_{\ell=1}^{j-1}\tilde{n}_{\ell}(2)+1, \textrm{ for } j=1,2,\ldots,\tilde{r}_1.
\eeqnarray
As we have $q_2=\tilde{q}_2$ and $2\leq \tilde{r}_1\leq r_1-1$ in \reqnarray{proof of N=2-(i)-222},
it follows from \reqnarray{proof of N=2-333} and \reqnarray{proof of N=2-(i)-444} that
$n_i(2)=\tilde{n}_i(2)$ for $i=1,2,\ldots,\tilde{r}_1$,
and hence we see from \reqnarray{proof of N=2-777} and \reqnarray{proof of N=2-(i)-ddd} that
$i_j=\tilde{i}_j$ for $j=1,2,\ldots,\tilde{r}_1$.
Thus, \reqnarray{proof of N=2-(i)-aaa} follows from $q_1=\tilde{q}_1$ in \reqnarray{proof of N=2-(i)-222},
$i_j=\tilde{i}_j$ for $j=1,2,\ldots,\tilde{r}_1$, \reqnarray{proof of N=2-666} and \reqnarray{proof of N=2-(i)-ccc}.
To prove \reqnarray{proof of N=2-(i)-bbb},
we see again from \reqnarray{proof of N=2-(i)-333}, \reqnarray{proof of N=2-(i)-444},
and the definition of left pre-sequences in \rdefinition{left pre-sequences} that
\beqnarray{proof of N=2-(i)-eee}
\tilde{m}_i(1)=
\bselection
\tilde{q}_1+1, &\textrm{if } i=\tilde{i}'_1,\tilde{i}'_2,\ldots,\tilde{i}'_{\tilde{r}_1}, \\
\tilde{q}_1, &\textrm{otherwise},
\eselection
\eeqnarray
where
\beqnarray{proof of N=2-(i)-fff}
\tilde{i}'_j=\sum_{\ell=1}^{j-1}\tilde{m}_{\ell}(2)+1, \textrm{ for } j=1,2,\ldots,\tilde{r}_1.
\eeqnarray
As we have $q_2=\tilde{q}_2$ and $2\leq \tilde{r}_1\leq r_1-1$ in \reqnarray{proof of N=2-(i)-222},
it follows from \reqnarray{proof of N=2-333} and \reqnarray{proof of N=2-(i)-444} that
$m_i(2)=\tilde{m}_i(2)$ for $i=1,2,\ldots,\tilde{r}_1-1$, and $m_{\tilde{r}_1}(2)=\tilde{m}_{\tilde{r}_1}(2)-1$.
As such, we see from \reqnarray{proof of N=2-999},
$m_i(2)=\tilde{m}_i(2)$ for $i=1,2,\ldots,\tilde{r}_1-1$,
and \reqnarray{proof of N=2-(i)-fff} that
\beqnarray{proof of N=2-(i)-ggg}
i'_j=\sum_{\ell=1}^{j-1}m_{\ell}(2)+1=\sum_{\ell=1}^{j-1}\tilde{m}_{\ell}(2)+1=\tilde{i}'_j, \textrm{ for } j=1,2,\ldots,\tilde{r}_1.
\eeqnarray
Also, we see from \reqnarray{proof of N=2-999},
$m_i(2)=\tilde{m}_i(2)$ for $i=1,2,\ldots,\tilde{r}_1-1$,
$m_{\tilde{r}_1}(2)=\tilde{m}_{\tilde{r}_1}(2)-1$,
and \reqnarray{proof of N=2-(i)-555} that
\beqnarray{proof of N=2-(i)-hhh}
i'_{\tilde{r}_1+1}
=\sum_{\ell=1}^{\tilde{r}_1}m_{\ell}(2)+1
=\sum_{\ell=1}^{\tilde{r}_1-1}\tilde{m}_{\ell}(2)+(\tilde{m}_{\tilde{r}_1}(2)-1)+1
=\sum_{\ell=1}^{\tilde{r}_1}\tilde{m}_{\ell}(2)
=\tilde{k}.
\eeqnarray
From $q_1=\tilde{q}_1$ in \reqnarray{proof of N=2-(i)-222},
\reqnarray{proof of N=2-888}, \reqnarray{proof of N=2-(i)-eee},
and \reqnarray{proof of N=2-(i)-ggg}, we have
\beqnarray{proof of N=2-(i)-iii}
m_i(1)=\tilde{m}_i(1) \textrm{ for } i=1,2,\ldots,i'_{\tilde{r}_1}.
\eeqnarray
Furthermore, from \reqnarray{proof of N=2-888} we have
\beqnarray{proof of N=2-(i)-jjj}
m_i(1)=
\bselection
q_1, &\textrm{ for } i=i'_{\tilde{r}_1}+1,i'_{\tilde{r}_1}+2,\ldots,i'_{\tilde{r}_1+1}-1, \\
q_1+1, &\textrm{ for } i=i'_{\tilde{r}_1+1}.
\eselection
\eeqnarray
and from \reqnarray{proof of N=2-(i)-eee} we have
\beqnarray{proof of N=2-(i)-kkk}
\tilde{m}_i(1)=\tilde{q}_1, \textrm{ for } i=\tilde{i}'_{\tilde{r}_1}+1,\tilde{i}'_{\tilde{r}_1}+2,\ldots,\tilde{k}.
\eeqnarray
As we have $q_1=\tilde{q}_1$ in \reqnarray{proof of N=2-(i)-222},
$i'_{\tilde{r}_1}=\tilde{i}'_{\tilde{r}_1}$ in \reqnarray{proof of N=2-(i)-ggg},
and $i'_{\tilde{r}_1+1}=\tilde{k}$ in \reqnarray{proof of N=2-(i)-hhh},
it follows from \reqnarray{proof of N=2-(i)-jjj} and \reqnarray{proof of N=2-(i)-kkk} that
\beqnarray{proof of N=2-(i)-1111}
m_i(1)=\tilde{m}_i(1) \textrm{ for } i=i'_{\tilde{r}_1}+1,i'_{\tilde{r}_1}+2,\ldots,\tilde{k}-1,
\textrm{ and } m_{\tilde{k}}(1)=\tilde{m}_{\tilde{k}}(1)+1
\eeqnarray
Thus, \reqnarray{proof of N=2-(i)-bbb} follows from \reqnarray{proof of N=2-(i)-iii} and \reqnarray{proof of N=2-(i)-1111}.
Now we can prove \reqnarray{proof of N=2-(i)-999}.
To see this, note that the condition in \reqnarray{proof of N=2-(i)-aaa} is the same as that in \reqnarray{proof of N=1-ddd}
(with $n_i$ replaced by $n_i(1)$)
and the condition in \reqnarray{proof of N=2-(i)-bbb} is the same as that in \reqnarray{proof of N=1-fff}
(with $m_i$ replaced by $m_i(1)$),
it follows that \reqnarray{proof of N=1-eee} and \reqnarray{proof of N=1-ggg} still hold.
Also, since the definition of $\beta_i$ in this appendix is the same as the $\beta_i$ defined in \reqnarray{proof of N=1-555},
$\tilde{\beta}_i$ in \reqnarray{proof of N=2-(i)-777} is the same as the $\tilde{\beta}_i$ defined in \reqnarray{proof of N=1-aaa},
and $\tilde{\beta}_{\tilde{k}}=0$ in \reqnarray{proof of N=2-(i)-888} is the same as \reqnarray{proof of N=1-bbb},
we see that \reqnarray{proof of N=1-hhh} still holds,
and hence \reqnarray{proof of N=2-(i)-999} follows from \reqnarray{proof of N=1-hhh}.
Note that from \reqnarray{proof of N=2-(i)-hhh} we have $\tilde{k}=i'_{\tilde{r}_1+1}$.
As we have $2\leq \tilde{r}_1\leq r_1-1$ in \reqnarray{proof of N=2-(i)-222},
it follows that $\tilde{k}$ can by any of $i'_3,i'_4,\ldots,i'_{r_1}$,
and hence we obtain from \reqnarray{proof of N=2-(i)-999} that
\beqnarray{proof of N=2-(i)-2222}
\beta_{i'_j}=B(\ebf_1^{t_{i'_j-1}};i'_j-1)+1, \textrm{ for } j=3,4,\ldots,r_1.
\eeqnarray
Thus, \reqnarray{proof of N=2-(i)-2222} proves that \reqnarray{proof of N=2-(i)-111} holds for $j=3,4,\ldots,r_1$.
It remains to prove that \reqnarray{proof of N=2-(i)-111} also holds for $j=2$.
To see this,
note that from \reqnarray{proof of N=2-333}, \reqnarray{proof of N=2-777},
and \reqnarray{proof of N=2-999} we have $i_2=n_1(2)+1=q_2+1$ and $i'_2=m_1(2)+1=q_2$,
it follows that
\beqnarray{proof of N=2-(i)-3333}
i'_2=i_2-1.
\eeqnarray
From \reqnarray{proof of N=2-666} we have
\beqnarray{proof of N=2-(i)-4444}
n_1(1)=q_1+1 \textrm{ and } n_i(1)=q_1 \textrm{ for } i=2,3,\ldots,i_2-1.
\eeqnarray
Also, from \reqnarray{proof of N=2-888} we have
\beqnarray{proof of N=2-(i)-5555}
m_1(1)=q_1+1,\ m_i(1)=q_1 \textrm{ for } i=2,3,\ldots,i'_2-1, \textrm{ and } m_{i'_2}(1)=q_1+1.
\eeqnarray
By combining \reqnarray{proof of N=2-(i)-3333}--\reqnarray{proof of N=2-(i)-5555} we obtain
\beqnarray{proof of N=2-(i)-6666}
m_i(1)=n_i(1) \textrm{ for } i=1,2,\ldots,i'_2-1, \textrm{ and } m_{i'_2}(1)=n_{i'_2}(1)+1.
\eeqnarray
Note that the condition in \reqnarray{proof of N=2-(i)-6666} is again the same as that in \reqnarray{proof of N=1-fff}
(with the roles of $m_i$, $\tilde{m}_i$, and $\tilde{k}$ replaced by $m_i(1)$, $n_i(1)$, and $i'_2$, respectively).
As such, by the same argument as that for \reqnarray{proof of N=1-ggg}
(with $\tilde{t}_i$ and $\tilde{e}_i$ replaced by $s_i$ and $d_i$ in the derivation, respectively)
we obtain
\beqnarray{proof of N=2-(i)-7777}
B(\ebf_1^{t_{i'_2}};i'_2)=B(\dbf_1^{s_{i'_2}};i'_2)+B(\ebf_1^{t_{i'_2-1}};i'_2-1)+1.
\eeqnarray
It then follows from \reqnarray{proof of N=2-(i)-7777} that
\beqnarray{proof of N=2-(i)-8888}
\beta_{i'_2}
\aligneq B(\ebf_1^{t_{i'_2}};i'_2)-B(\dbf_1^{s_{i'_2}};i'_2) \nn\\
\aligneq \left(B(\dbf_1^{s_{i'_2}};i'_2)+B(\ebf_1^{t_{i'_2-1}};i'_2-1)+1\right)-B(\dbf_1^{s_{i'_2}};i'_2) \nn\\
\aligneq B(\ebf_1^{t_{i'_2-1}};i'_2-1)+1.
\eeqnarray
Thus, we see that \reqnarray{proof of N=2-(i)-111} also holds for $j=2$.
The proof for \reqnarray{proof of N=2-(i)-111} is completed by combining \reqnarray{proof of N=2-(i)-2222} and \reqnarray{proof of N=2-(i)-8888}.
(ii) In this part, we will show that $\beta_k=0$.
For simplicity, let $r=r_1$, $\nbf_1^k=\nbf_1^k(1)$, and $\mbf_1^k=\mbf_1^k(1)$.
We first show the following equations
\beqnarray{}
\alignspace m_{i'_r-j}=q_1, \textrm{ for } j=1,2,\ldots,q_2-1,
\label{eqn:proof of N=2-(ii)-111}\\
\alignspace n_{i'_r-j}=q_1, \textrm{ for } j=1,2,\ldots,q_2-2,
\label{eqn:proof of N=2-(ii)-222}\\
\alignspace m_{i'_r}=q_1+1 \textrm{ and } n_{i'_r}=q_1,
\label{eqn:proof of N=2-(ii)-333}\\
\alignspace m_{i'_r+1}=q_1 \textrm{ and } n_{i'_r+1}=q_1+1,
\label{eqn:proof of N=2-(ii)-444}\\
\alignspace m_{i'_r+j}=n_{i'_r+j}=q_1, \textrm{ for } j=2,3,\ldots,q_2.
\label{eqn:proof of N=2-(ii)-555}
\eeqnarray
To prove \reqnarray{proof of N=2-(ii)-111},
we note from \reqnarray{proof of N=2-999} and \reqnarray{proof of N=2-333} that
\beqnarray{proof of N=2-(ii)-666}
i'_j=\sum_{\ell=1}^{j-1}m_{\ell}(2)+1=(j-1)q_2, \textrm{ for } j=2,3,\ldots,r.
\eeqnarray
It follows from \reqnarray{proof of N=2-(ii)-666} that $i'_r=q_2+i'_{r-1}$,
and hence we have
\beqnarray{proof of N=2-(ii)-777}
i'_{r-1}<i'_r-j<i'_r, \textrm{ for } j=1,2,\ldots,q_2-1.
\eeqnarray
Thus, \reqnarray{proof of N=2-(ii)-111} follows from \reqnarray{proof of N=2-(ii)-777} and \reqnarray{proof of N=2-888}.
To prove \reqnarray{proof of N=2-(ii)-222}, similarly,
we have from \reqnarray{proof of N=2-777} and \reqnarray{proof of N=2-333} that
\beqnarray{proof of N=2-(ii)-888}
i_j=\sum_{\ell=1}^{j-1}n_{\ell}(2)+1=(j-1)q_2+1, \textrm{ for } j=2,3,\ldots,r.
\eeqnarray
As such, we see from \reqnarray{proof of N=2-(ii)-666} and \reqnarray{proof of N=2-(ii)-888} that
\beqnarray{proof of N=2-(ii)-999}
i'_r=i_r-1.
\eeqnarray
Also, from \reqnarray{proof of N=2-(ii)-888} we have $i_r=q_2+i_{r-1}$,
it then follows from \reqnarray{proof of N=2-(ii)-999} and $i_r=q_2+i_{r-1}$ that
\beqnarray{proof of N=2-(ii)-aaa}
i'_r-(q_2-2)=(i_r-1)-(q_2-2)=i_{r-1}+1>i_{r-1}.
\eeqnarray
Therefore, we see from \reqnarray{proof of N=2-(ii)-999} and \reqnarray{proof of N=2-(ii)-aaa} that
\beqnarray{proof of N=2-(ii)-bbb}
i_{r-1}<i'_r-j<i_r, \textrm{ for } j=1,2,\ldots,q_2-2.
\eeqnarray
Thus, \reqnarray{proof of N=2-(ii)-222} follows from \reqnarray{proof of N=2-(ii)-bbb} and \reqnarray{proof of N=2-666}.
To prove \reqnarray{proof of N=2-(ii)-333},
first note that $m_{i'_r}=q_1+1$ immediately follows from \reqnarray{proof of N=2-888}.
From \reqnarray{proof of N=2-(ii)-999}, $q_2\geq 2$, and $i_r-q_2=i_{r-1}$ in \reqnarray{proof of N=2-(ii)-888} we have
\beqnarray{proof of N=2-(ii)-ccc}
i'_r=i_r-1<i_r \textrm{ and } i'_r=i_r-1>i_r-q_2=i_{r-1}.
\eeqnarray
Thus, $n_{i'_r}=q_1$ in \reqnarray{proof of N=2-(ii)-333} follows from \reqnarray{proof of N=2-666} and $i_{r-1}<i'_r<i_r$ in \reqnarray{proof of N=2-(ii)-ccc}.
To prove \reqnarray{proof of N=2-(ii)-444} and \reqnarray{proof of N=2-(ii)-555},
first note that $n_{i'_r+1}=q_1+1$ in \reqnarray{proof of N=2-(ii)-444} immediately follows from \reqnarray{proof of N=2-666} and $i'_r+1=i_r$ in \reqnarray{proof of N=2-(ii)-999}.
From $i'_r+q_2=k$ in \reqnarray{proof of N=2-aaa} we have
\beqnarray{proof of N=2-(ii)-ddd}
i'_r<i'_r+j\leq k, \textrm{ for } j=1,2,\ldots,q_2.
\eeqnarray
It follows from \reqnarray{proof of N=2-888} and \reqnarray{proof of N=2-(ii)-ddd} that
\beqnarray{proof of N=2-(ii)-eee}
m_{i'_r+j}=q_1, \textrm{ for } j=1,2,\ldots,q_2.
\eeqnarray
Furthermore, from \reqnarray{proof of N=2-(ii)-999} we have $i'_r+2=(i_r-1)+2=i_r+1>i_r$,
and it then follows from $i'_r+2>i_r$, $q_2\geq 2$, and $i'_r+q_2=k$ in \reqnarray{proof of N=2-aaa} that
\beqnarray{proof of N=2-(ii)-fff}
i_r<i'_r+j\leq k, \textrm{ for } j=2,3,\ldots,q_2.
\eeqnarray
Therefore we have from \reqnarray{proof of N=2-666} and \reqnarray{proof of N=2-(ii)-fff} that
\beqnarray{proof of N=2-(ii)-ggg}
n_{i'_r+j}=q_1, \textrm{ for } j=2,3,\ldots,q_2.
\eeqnarray
Thus, $m_{i'_r+1}=q_1$ in \reqnarray{proof of N=2-(ii)-444} follows from \reqnarray{proof of N=2-(ii)-eee},
and \reqnarray{proof of N=2-(ii)-555} follows from \reqnarray{proof of N=2-(ii)-eee} and \reqnarray{proof of N=2-(ii)-ggg}.
We are now in position to show that $\beta_k=0$.
Note that $\beta_{i'_r}$ is given by \reqnarray{proof of N=2-(i)-111}.
We need to find $\alpha_{i'_r}$ so that we can recursively compute $\alpha_i$ and $\beta_i$ for $i=i'_r+1,i'_r+2,\ldots,k$.
To do this,
we note from \reqnarray{proof of N=1-666} and \reqnarray{proof of N=2-(i)-111} (with $j=r$) that
\beqnarray{proof of N=2-(ii)-iii}
\alpha_{i'_r}=\beta_{i'_r}-\beta_{i'_r-1}=B(\ebf_1^{t_{i'_r-1}};i'_r-1)+1-\beta_{i'_r-1}.
\eeqnarray
However, we find something useful if we compute $\alpha_{i'_r}$ and $\beta_{i'_r}$ in another way as follows.
By using \reqnarray{OQ-LR-delays-greedy-3} and $m_{i'_r}=q_1+1$ and $n_{i'_r}=q_1$ in \reqnarray{proof of N=2-(ii)-333},
we have
\beqnarray{proof of N=2-(ii)-jjj}
\alpha_{i'_r}
\aligneq e_{t_{i'_r}}-d_{s_{i'_r}} \nn\\
\aligneq 2e_{t_{{i'_r}-1}}+(m_{i'_r}-1)(B(\ebf_1^{t_{{i'_r}-1}};{i'_r}-1)+1)
-2d_{s_{{i'_r}-1}}-(n_{i'_r}-1)(B(\dbf_1^{s_{{i'_r}-1}};{i'_r}-1)+1) \nn\\
\aligneq 2\alpha_{{i'_r}-1}+(q_1-1)\beta_{{i'_r}-1}+(B(\ebf_1^{t_{{i'_r}-1}};{i'_r}-1)+1).
\eeqnarray
As such, we have from \reqnarray{proof of N=1-666} and \reqnarray{proof of N=2-(ii)-jjj} that
\beqnarray{proof of N=2-(ii)-kkk}
\beta_{i'_r}
=\alpha_{i'_r}+\beta_{i'_r-1}
=2\alpha_{{i'_r}-1}+q_1\beta_{{i'_r}-1}+(B(\ebf_1^{t_{{i'_r}-1}};{i'_r}-1)+1).
\eeqnarray
By comparing \reqnarray{proof of N=2-(ii)-kkk} with \reqnarray{proof of N=2-(i)-111} (with $j=r$),
we see that
\beqnarray{proof of N=2-(ii)-1111}
2\alpha_{{i'_r}-1}+q_1\beta_{{i'_r}-1}=0.
\eeqnarray
This equation will be used later.
In the following, we recursively compute $\alpha_i$ and $\beta_i$ for $i=i'_r+1,i'_r+2,\ldots,k$.
From \reqnarray{OQ-LR-delays-greedy-3} and $m_{i'_r+1}=q_1$ and $n_{i'_r+1}=q_1+1$ in \reqnarray{proof of N=2-(ii)-444},
we have
\beqnarray{}
\alpha_{i'_r+1}
\aligneq e_{t_{i'_r+1}}-d_{s_{i'_r+1}} \nn\\
\aligneq 2e_{t_{i'_r}}+(m_{i'_r+1}-1)(B(\ebf_1^{t_{i'_r}};{i'_r})+1)
-2d_{s_{i'_r}}-(n_{i'_r+1}-1)(B(\dbf_1^{s_{i'_r}};{i'_r})+1) \nn\\
\aligneq 2\alpha_{i'_r}+q_1\beta_{i'_r}-(B(\ebf_1^{t_{i'_r}};{i'_r})+1), \nn
\eeqnarray
and then we use \reqnarray{proof of N=2-(ii)-iii} and \reqnarray{OQ-LR-delays-greedy-7} to see that
\beqnarray{}
\alpha_{i'_r+1}
\aligneq 2\left(B(\ebf_1^{t_{i'_r-1}};i'_r-1)+1-\beta_{i'_r-1}\right)+q_1\beta_{i'_r}
-(B(\ebf_1^{t_{i'_r}};{i'_r})+1), \nn\\
\aligneq 2\left(B(\ebf_1^{t_{i'_r-1}};i'_r-1)+1-\beta_{i'_r-1}\right)+q_1\beta_{i'_r}
-(e_{t_{i'_r}}+B(\ebf_1^{t_{i'_r-1}};{i'_r-1})+1) \nn\\
\aligneq B(\ebf_1^{t_{i'_r-1}};i'_r-1)+1-2\beta_{i'_r-1}+q_1\beta_{i'_r}-e_{t_{i'_r}}. \nn
\eeqnarray
To continue, we use \reqnarray{OQ-LR-delays-greedy-3}, $m_{i'_r}=q_1+1$ in \reqnarray{proof of N=2-(ii)-333},
and \reqnarray{proof of N=2-(i)-111} (with $j=r$) to obtain
\beqnarray{proof of N=2-(ii)-2222}
\alpha_{i'_r+1}
\aligneq B(\ebf_1^{t_{i'_r-1}};i'_r-1)+1-2\beta_{i'_r-1}+q_1\beta_{i'_r}
-2e_{t_{i'_r-1}}-(m_{i'_r}-1)(B(\ebf_1^{t_{i'_r-1}};i'_r-1)+1) \nn\\
\aligneq B(\ebf_1^{t_{i'_r-1}};i'_r-1)+1-2\beta_{i'_r-1}-2e_{t_{i'_r-1}}
+q_1\left(\beta_{i'_r}-(B(\ebf_1^{t_{i'_r-1}};i'_r-1)+1)\right) \nn\\
\aligneq B(\ebf_1^{t_{i'_r-1}};i'_r-1)+1-2\beta_{i'_r-1}-2e_{t_{i'_r-1}}.
\eeqnarray
As such, we can compute from \reqnarray{proof of N=1-666},
\reqnarray{proof of N=2-(i)-111} (with $j=r$),
and \reqnarray{proof of N=2-(ii)-2222} that
\beqnarray{proof of N=2-(ii)-3333}
\beta_{i'_r+1}
=\alpha_{i'_r+1}+\beta_{i'_r}
=2\left(B(\ebf_1^{t_{i'_r-1}};i'_r-1)+1-\beta_{i'_r-1}-e_{t_{i'_r-1}}\right).
\eeqnarray
To compute $\alpha_i$ and $\beta_i$ for $i=i'_r+2,i'_r+3,\ldots,k$,
note that from \reqnarray{OQ-LR-delays-greedy-3} and $m_{i'_r+j}=n_{i'_r+j}=q_1$ for $j=2,3,\ldots,q_2$ in \reqnarray{proof of N=2-(ii)-555}
we have
\beqnarray{proof of N=2-(ii)-4444}
\alpha_{i'_r+j}
\aligneq e_{t_{i'_r+j}}-d_{s_{i'_r+j}} \nn \\
\aligneq 2e_{t_{i'_r+j-1}}+(m_{i'_r+j}-1)(B(\ebf_1^{t_{i'_r+j-1}};i'_r+j-1)+1) \nn\\
\alignspace -2d_{s_{i'_r+j-1}}-(n_{i'_r+j}-1)(B(\dbf_1^{s_{i'_r+j-1}};i'_r+j-1)+1) \nn \\
\aligneq 2\alpha_{i'_r+j-1}+(q_1-1)\beta_{i'_r+j-1},
\eeqnarray
for $j=2,3,\ldots,q_2$.
We claim that for $2\leq j\leq q_2$, we have
\beqnarray{}
\alignspace
\alpha_{i'_r+j}=2^{j-1}\left(B(\ebf_1^{t_{i'_r-j}};i'_r-j)+1-\beta_{i'_r-j}-2e_{t_{i'_r-j}}+\alpha_{i'_r-j+1}\right),
\label{eqn:proof of N=2-(ii)-5555} \\
\alignspace
\beta_{i'_r+j}=2^j\left(B(\ebf_1^{t_{i'_r-j}};i'_r-j)+1-\beta_{i'_r-j}-e_{t_{i'_r-j}}\right).
\label{eqn:proof of N=2-(ii)-6666}
\eeqnarray
Note that $i'_r+q_2=k$ as we have shown in \reqnarray{proof of N=2-aaa}.
We prove \reqnarray{proof of N=2-(ii)-5555} and \reqnarray{proof of N=2-(ii)-6666} by induction on $j$.
First consider the case that $j=2$.
From \reqnarray{proof of N=2-(ii)-4444} (with $j=2$),
\reqnarray{proof of N=2-(ii)-2222}, and \reqnarray{proof of N=2-(ii)-3333}, we have
\beqnarray{}
\alpha_{i'_r+2}
\aligneq 2\alpha_{i'_r+1}+(q_1-1)\beta_{i'_r+1} \nn\\
\aligneq 2\left(B(\ebf_1^{t_{i'_r-1}};i'_r-1)+1-2\beta_{i'_r-1}-2e_{t_{i'_r-1}}\right) \nn\\
\alignspace +2(q_1-1)\left(B(\ebf_1^{t_{i'_r-1}};i'_r-1)+1-\beta_{i'_r-1}-e_{t_{i'_r-1}}\right) \nn\\
\aligneq 2q_1(B(\ebf_1^{t_{i'_r-1}};i'_r-1)+1)-2(q_1+1)(\beta_{i'_r-1}+e_{t_{i'_r-1}}), \nn
\eeqnarray
and then we use \reqnarray{OQ-LR-delays-greedy-7} to obtain
\beqnarray{}
\alpha_{i'_r+2}
\aligneq 2q_1\left(e_{t_{i'_r-1}}+B(\ebf_1^{t_{i'_r-2}};i'_r-2)+1\right)-2(q_1+1)(\beta_{i'_r-1}+e_{t_{i'_r-1}}) \nn\\
\aligneq 2q_1(B(\ebf_1^{t_{i'_r-2}};i'_r-2)+1)-2(q_1+1)\beta_{i'_r-1}-2e_{t_{i'_r-1}}. \nn
\eeqnarray
To continue, from \reqnarray{OQ-LR-delays-greedy-3} and $m_{i'_r-1}=q_1$ in \reqnarray{proof of N=2-(ii)-111}
we have
\beqnarray{}
\alpha_{i'_r+2}
\aligneq 2q_1(B(\ebf_1^{t_{i'_r-2}};i'_r-2)+1)-2(q_1+1)\beta_{i'_r-1} \nn\\
\alignspace -2\left(2e_{t_{i'_r-2}}+(m_{i'_r-1}-1)(B(\ebf_1^{t_{i'_r-2}};i'_r-2)+1)\right) \nn\\
\aligneq 2(B(\ebf_1^{t_{i'_r-2}};i'_r-2)+1)-2(q_1+1)\beta_{i'_r-1}-4e_{t_{i'_r-1}}. \nn
\eeqnarray
Finally, by using \reqnarray{proof of N=2-(ii)-1111} and \reqnarray{proof of N=1-666},
we see that
\beqnarray{proof of N=2-(ii)-7777}
\alpha_{i'_r+2}
\aligneq 2(B(\ebf_1^{t_{i'_r-2}};i'_r-2)+1)-2\beta_{i'_r-1}+4\alpha_{i'_r-1}-4e_{t_{i'_r-1}} \nn\\
\aligneq 2(B(\ebf_1^{t_{i'_r-2}};i'_r-2)+1)-2(\beta_{i'_r-2}+\alpha_{i'_r-1})+4\alpha_{i'_r-1}-4e_{t_{i'_r-1}} \nn\\
\aligneq 2\left(B(\ebf_1^{t_{i'_r-2}};i'_r-2)+1-\beta_{i'_r-2}-2e_{t_{i'_r-2}}+\alpha_{i'_r-1}\right).
\eeqnarray
This proves \reqnarray{proof of N=2-(ii)-5555} for the case that $j=2$.
To prove \reqnarray{proof of N=2-(ii)-6666} for the case that $j=2$,
note that from \reqnarray{proof of N=2-(ii)-3333}, \reqnarray{OQ-LR-delays-greedy-7},
and \reqnarray{proof of N=1-666}, we have
\beqnarray{proof of N=2-(ii)-8888}
\beta_{i'_r+1}
\aligneq 2\left(B(\ebf_1^{t_{i'_r-1}};i'_r-1)+1-\beta_{i'_r-1}-e_{t_{i'_r-1}}\right). \nn\\
\aligneq 2\left(e_{t_{i'_r-1}}+B(\ebf_1^{t_{i'_r-2}};i'_r-2)+1-\beta_{i'_r-1}-e_{t_{i'_r-1}}\right) \nn\\
\aligneq 2\left(B(\ebf_1^{t_{i'_r-2}};i'_r-2)+1-\beta_{i'_r-1}\right) \nn\\
\aligneq 2\left(B(\ebf_1^{t_{i'_r-2}};i'_r-2)+1-\beta_{i'_r-2}-\alpha_{i'_r-1}\right).
\eeqnarray
Now we use \reqnarray{proof of N=1-666}, \reqnarray{proof of N=2-(ii)-7777},
and \reqnarray{proof of N=2-(ii)-8888} to see that
\beqnarray{}
\beta_{i'_r+2}
=\alpha_{i'_r+2}+\beta_{i'_r+1}
=2^2\left(B(\ebf_1^{t_{i'_r-2}};i'_r-2)+1-\beta_{i'_r-2}-e_{t_{i'_r-2}}\right). \nn
\eeqnarray
This proves \reqnarray{proof of N=2-(ii)-6666} for the case that $j=2$.
Suppose as the induction hypothesis that
\reqnarray{proof of N=2-(ii)-5555} and \reqnarray{proof of N=2-(ii)-6666} hold for some $2\leq j\leq q_2-1$.
Now consider the case $j+1$.
First note that we have from \reqnarray{proof of N=2-(ii)-111} and \reqnarray{proof of N=2-(ii)-222} that $m_{i'_r-j}=n_{i'_r-j}=q_1$ for $j=1,2,\ldots,q_1-2$.
As such, and by a similar derivation as that for \reqnarray{proof of N=2-(ii)-4444} (with $j$ replaced by $-j$) we have
\beqnarray{proof of N=2-(ii)-9999}
\alpha_{i'_r-j}=2\alpha_{i'_r-j-1}+(q_1-1)\beta_{i'_r-j-1}, \textrm{ for } j=1,2,\ldots,q_1-2.
\eeqnarray
This equation will be used later.
Now we compute from \reqnarray{proof of N=2-(ii)-4444},
the induction hypothesis, and \reqnarray{OQ-LR-delays-greedy-7} that
\beqnarray{}
\alpha_{i'_r+j+1}
\aligneq 2\alpha_{i'_r+j}+(q_1-1)\beta_{i'_r+j} \nn\\
\aligneq 2\cdot 2^{j-1}\left(B(\ebf_1^{t_{i'_r-j}};i'_r-j)+1-\beta_{i'_r-j}-2e_{t_{i'_r-j}}+\alpha_{i'_r-j+1}\right) \nn\\
\alignspace +2^j(q_1-1)\left(B(\ebf_1^{t_{i'_r-j}};i'_r-j)+1-\beta_{i'_r-j}-e_{t_{i'_r-j}}\right) \nn\\
\aligneq 2^j\left(q_1(B(\ebf_1^{t_{i'_r-j}};i'_r-j)+1)-q_1\beta_{i'_r-j}
-(q_1+1)e_{t_{i'_r-j}}+\alpha_{i'_r-j+1}\right) \nn\\
\aligneq 2^j\left(q_1(e_{t_{i'_r-j}}+B(\ebf_1^{t_{i'_r-j-1}};i'_r-j-1)+1)-q_1\beta_{i'_r-j}
-(q_1+1)e_{t_{i'_r-j}}+\alpha_{i'_r-j+1}\right) \nn\\
\aligneq 2^j\left(q_1(B(\ebf_1^{t_{i'_r-j-1}};i'_r-j-1)+1)-q_1\beta_{i'_r-j}-e_{t_{i'_r-j}}+\alpha_{i'_r-j+1}\right), \nn
\eeqnarray
and then we use \reqnarray{OQ-LR-delays-greedy-3}
and $m_{i'_r-j}=q_1$ in \reqnarray{proof of N=2-(ii)-111} (note that $2\leq j\leq q_2-1$) to see that
\beqnarray{}
\alpha_{i'_r+j+1}
\aligneq 2^j\Bigl(q_1(B(\ebf_1^{t_{i'_r-j-1}};i'_r-j-1)+1)-q_1\beta_{i'_r-j} \nn\\
\alignspace -2e_{t_{i'_r-j-1}}-(m_{i'_r-j}-1)(B(\ebf_1^{t_{i'_r-j-1}};i'_r-j-1)+1)+\alpha_{i'_r-j+1}\Bigr) \nn\\
\aligneq 2^j\left(B(\ebf_1^{t_{i'_r-j-1}};i'_r-j-1)+1-q_1\beta_{i'_r-j}-2e_{t_{i'_r-j-1}}+\alpha_{i'_r-j+1}\right). \nn
\eeqnarray
From \reqnarray{proof of N=2-(ii)-9999} (note that $2\leq j\leq q_2-1$ implies that $1\leq j-1\leq q_2-2$),
the above equation can be further arranged as
\beqnarray{}
\alpha_{i'_r+j+1}
\aligneq 2^j\left(B(\ebf_1^{t_{i'_r-j-1}};i'_r-j-1)+1-q_1\beta_{i'_r-j}-2e_{t_{i'_r-j-1}}
+2\alpha_{i'_r-j}+(q_1-1)\beta_{i'_r-j}\right) \nn\\
\aligneq 2^j\left(B(\ebf_1^{t_{i'_r-j-1}};i'_r-j-1)+1-\beta_{i'_r-j}-2e_{t_{i'_r-j-1}}+2\alpha_{i'_r-j}\right). \nn
\eeqnarray
Finally, by using \reqnarray{proof of N=1-666} we see that
\beqnarray{proof of N=2-(ii)-aaaa}
\alpha_{i'_r+j+1}
\aligneq 2^j\left(B(\ebf_1^{t_{i'_r-j-1}};i'_r-j-1)+1-(\beta_{i'_r-j-1}+\alpha_{i'_r-j})
-2e_{t_{i'_r-j-1}}+2\alpha_{i'_r-j}\right) \nn\\
\aligneq 2^j\left(B(\ebf_1^{t_{i'_r-j-1}};i'_r-j-1)+1-\beta_{i'_r-j-1}-2e_{t_{i'_r-j-1}}+\alpha_{i'_r-j}\right).
\eeqnarray
To compute $\beta_{i'_r+j+1}$,
note that from the induction hypothesis, \reqnarray{OQ-LR-delays-greedy-7},
and \reqnarray{proof of N=1-666} we have
\beqnarray{proof of N=2-(ii)-bbbb}
\beta_{i'_r+j}
\aligneq 2^j\left(B(\ebf_1^{t_{i'_r-j}};i'_r-j)+1-\beta_{i'_r-j}-e_{t_{i'_r-j}}\right). \nn\\
\aligneq 2^j\left(e_{t_{i'_r-j}}+B(\ebf_1^{t_{i'_r-j-1}};i'_r-j-1)+1-\beta_{i'_r-j}-e_{t_{i'_r-j}}\right) \nn\\
\aligneq 2^j\left(B(\ebf_1^{t_{i'_r-j-1}};i'_r-j-1)+1-\beta_{i'_r-j}\right) \nn\\
\aligneq 2^j\left(B(\ebf_1^{t_{i'_r-j-1}};i'_r-j-1)+1-\beta_{i'_r-j-1}-\alpha_{i'_r-j}\right).
\eeqnarray
Now we use \reqnarray{proof of N=1-666}, \reqnarray{proof of N=2-(ii)-aaaa},
and \reqnarray{proof of N=2-(ii)-bbbb} to see that
\beqnarray{proof of N=2-(ii)-cccc}
\beta_{i'_r+j+1}
\aligneq \alpha_{i'_r+j+1}+\beta_{i'_r+j} \nn\\
\aligneq 2^{j+1}\left(B(\ebf_1^{t_{i'_r-j-1}};i'_r-j-1)+1-\beta_{i'_r-j-1}-e_{t_{i'_r-j-1}}\right).
\eeqnarray
The induction for \reqnarray{proof of N=2-(ii)-5555} and \reqnarray{proof of N=2-(ii)-6666} is completed by combining \reqnarray{proof of N=2-(ii)-aaaa} and \reqnarray{proof of N=2-(ii)-cccc}.
Now we can conclude that $\beta_k=0$ as follows.
From \reqnarray{proof of N=2-(ii)-6666} (with $j=q_2$) we see that we have
\beqnarray{proof of N=2-dddd}
\beta_{i'_r+q_2}=2^{q_2}\left(B(\lbf_1^{t_{i'_r-q_2}};i'_r-q_2)+1-\beta_{i'_r-q_2}-l_{t_{i'_r-q_2}}\right).
\eeqnarray
Note that we have $i'_r+q_2=k$ in \reqnarray{proof of N=2-aaa}.
Also, we have from \reqnarray{proof of N=2-(ii)-666} that $i'_r-q_2=i'_{r-1}$.
As such, it follows from \reqnarray{proof of N=2-dddd} that
\beqnarray{}
\beta_k=2^{q_2}\left(B(\ebf_1^{t_{i'_{r-1}}};i'_{r-1})+1-\beta_{i'_{r-1}}-e_{t_{i'_{r-1}}}\right). \nn
\eeqnarray
From \reqnarray{OQ-LR-delays-greedy-7}, we can arrange the above equation as
\beqnarray{}
\beta_k
\aligneq 2^{q_2}\left(e_{t_{i'_{r-1}}}+B(\ebf_1^{t_{i'_{r-1}-1}};i'_{r-1}-1)+1-\beta_{i'_{r-1}}-e_{t_{i'_{r-1}}}\right) \nn\\
\aligneq 2^{q_2}\left(B(\ebf_1^{t_{i'_{r-1}-1}};i'_{r-1}-1)+1-\beta_{i'_{r-1}}\right). \nn
\eeqnarray
Finally, it follows from \reqnarray{proof of N=2-(i)-111} (with $j=r-1$) that
the term in bracket on right hand side of the above equation is equal to zero.
We thus obtain the desired result that $\beta_k=0$.
The proof is then completed.
\fi
|
train/arxiv
|
BkiUfoTxK1fBGsWmx20U
| 5
| 1
|
\section{Introduction}\label{sec:intro}
Let $\K$ be the real or complex field. We consider a class $\E$ of Banach spaces over $\K$
which is closed under isometric isomorphisms, i.\,e. if $X\in \E$ and $Y$ is isometrically
isomorphic to $X$, then $Y\in \E$.\par
For a given Banach space $X$, we denote by $X^*$ its dual space, by $B_X$ its closed unit ball
and by $S_X$ its unit sphere. Furthermore, $B_X^{\text{fin}}$ and $S_X^{\text{fin}}$ will denote
the sets of all finite sequences in $B_X$ and $S_X$. For fixed $n\in \N$, $B_X^n$ and $S_X^n$ will
stand for the sets of all sequences of length $n$ in $B_X$ and $S_X$. Given $\mathbf{x}=(x_1,\dots,x_n)\in B_X^n$,
we set $\norm{\mathbf{x}}_{\infty}:=\max_{i=1,\dots,n}\norm{x_i}$. Finallly, $\U(X)$ denotes the set of all closed, nontrivial subspaces of $X$.\par
The following is our main definition.
\begin{definition}\label{def:test}
Let $X$ be a Banach space. A family of real-valued functions
$F_{\eps,U}$ on $B_U^{\text{fin}}\times B_{U^*}^{\text{fin}}\times B_U^{\text{fin}}\times B_{U^*}^{\text{fin}}$
with $U\in \U(X)$ and $\eps>0$ is said to be a test family for $\E$ in $X$ if the following conditions are satisfied:
\begin{enumerate}[\upshape(i)]
\item For every $U\in \U(X)$ one has that $U\in \E$ if and only if for every $\eps>0$ and all $\mathbf{x}\in S_U^{\text{fin}}$
and $\mathbf{x}^*\in S_{U^*}^{\text{fin}}$ there exist $\mathbf{y}\in S_U^{\text{fin}}$ and $\mathbf{y}^*\in S_{U^*}^{\text{fin}}$
such that $F_{\eps,U}(\mathbf{x},\mathbf{x^*},\mathbf{y},\mathbf{y}^*)\leq \eps$.
\item If $0<\eps_1<\eps_2$ and $U\in \U(X)$, then $F_{\eps_1,U}\geq F_{\eps_2,U}$.
\item There exists $c>0$ such that for all $U\in \U(X)$, all $\eps>0$, every $\mathbf{x},\mathbf{y}\in B_U^{\text{fin}}$ and
every $\mathbf{x}^*, \mathbf{y}^*\in B_{X^*}^{\text{fin}}$ one has
\begin{equation*}
F_{\eps,X}(\mathbf{x},\mathbf{x}^*,\mathbf{y},\mathbf{y}^*)\leq cF_{\eps,U}(\mathbf{x},\mathbf{x}^*|_U,\mathbf{y},\mathbf{y}^*|_U),
\end{equation*}
where for $\mathbf{x}^*=(x_1^*,\dots,x_n^*)$ we define $\mathbf{x}^*|_U=(x_1^*|_U,\dots,x_n^*|_U)$ (and analogously for $\mathbf{y}^*$).
\item For every $\eps>0$, all $\tau>0$, each $\mathbf{x}^*\in B_{X^*}^{\text{fin}}$, all $n\in \N$ and all $\mathbf{x}\in B_X^n$ there exists
a $\delta>0$ such that
\begin{equation*}
|F_{\varepsilon,X}(\mathbf{x},\mathbf{x}^*,\mathbf{y},\mathbf{y}^*)-F_{\varepsilon,X}(\mathbf{z},\mathbf{x}^*,\mathbf{y},\mathbf{y}^*)|\leq \tau
\end {equation*}
holds for all $\mathbf{y}\in B_X^{\text{fin}}$, all $\mathbf{y}^*\in B_{X^*}^{\text{fin}}$ and every $\mathbf{z}\in B_X^n$ with
$\norm{\mathbf{x}-\mathbf{z}}_{\infty}\leq \delta$.
\item For every $\eps>0$, all $n,m\in \N$ and all $\eta>0$ there exists $\theta>0$ such that for every $U\in \U(X)$ one has
\begin{equation*}
|F_{\varepsilon,U}(\mathbf{x},\mathbf{x}^*,\mathbf{y},\mathbf{y}^*)-F_{\varepsilon,U}(\mathbf{x},\mathbf{z}^*,\mathbf{y},\mathbf{y}^*)|\leq \eta
\end{equation*}
for all $\mathbf{x}\in B_U^n$, all $\mathbf{y}\in B_U^{\text{fin}}$, all $\mathbf{y}^*\in B_{U^*}^{\text{fin}}$ and all $\mathbf{x}^*, \mathbf{z}^*\in B_{U^*}^m$
with $\norm{\mathbf{x}^*-\mathbf{z}^*}_{\infty}\leq \theta$.
\end{enumerate}
\end{definition}
Roughly speaking, we want to show that if a Banach space property can be characterised in terms of test families
and is stable under certain finite, absolute sums, then it is also stable under the formation of corresponding
K\"othe-Bochner function spaces.\par
Examples of Banach space properties which can be described by test families will be presented in the next section
(the constant $c$ in the above definition will be 1 for all these examples). Here we continue with the necessary
basics on absolute sums and K\"othe-Bochner spaces.\par
Let $I$ be a non-empty set, $E$ a subspace of $\R^I$ with $e_i\in E$ for all $i\in I$ and $\norm*{\,.\,}_E$ a
complete norm on $E$ (here $e_i$ denotes the characteristic function of $\set*{i}$).\par
\noindent The norm $\norm*{\,.\,}_E$ is called {\em absolute} if
\begin{align*}
&(a_i)_{i\in I}\in E, \ (b_i)_{i\in I}\in \R^I \ \mathrm{and} \ \abs*{a_i}=\abs*{b_i} \ \forall i\in I \\
&\Rightarrow \ (b_i)_{i\in I}\in E \ \mathrm{and} \ \norm*{(a_i)_{i\in I}}_E=\norm*{(b_i)_{i\in I}}_E.
\end{align*}
The norm is called {\em normalised} if $\norm*{e_i}_E=1$ for every $i\in I$.\par
Standard examples of subspaces of $\R^I$ with absolute normalised norm are of course the spaces $\ell^p(I)$
for $1\leq p\leq \infty$ and the space $c_0(I)$.\par
We note the following lemma on absolute norms (see e.\,g. \cite{lee}*{Remark 2.1}).
\begin{lemma}\label{lemma:abs norms}
Let $(E,\norm*{\,.\,}_E)$ be a subspace of $\R^I$ with an absolute normalised norm. Then the following is true.
\begin{align*}
&(a_i)_{i\in I}\in E, \ (b_i)_{i\in I}\in \R^I \ \mathrm{and} \ \abs*{b_i}\leq\abs*{a_i} \ \forall i\in I \\
&\Rightarrow \ (b_i)_{i\in I}\in E \ \mathrm{and} \ \norm*{(b_i)_{i\in I}}_E\leq\norm*{(a_i)_{i\in I}}_E.
\end{align*}
\end{lemma}
If $(X_i)_{i\in I}$ is a family of (real or complex) Banach spaces we put
\begin{equation*}
\Bigl[\bigoplus_{i\in I}X_i\Bigr]_E:=\set*{(x_i)_{i\in I}\in \prod_{i\in I}X_i: (\norm*{x_i})_{i\in I}\in E}.
\end{equation*}
This defines a subspace of the product space $\prod_{i\in I}X_i$ which becomes a Banach space when endowed with the norm
\begin{equation*}
\norm*{(x_i)_{i\in I}}_E:=\norm*{(\norm*{x_i})_{i\in I}}_E \ \forall (x_i)_{i\in I}\in \Bigl[\bigoplus_{i\in I}X_i\Bigr]_E.
\end{equation*}
We call this Banach space the absolute sum of the family $(X_i)_{i\in I}$ with respect to $E$. For $p\in [1,\infty]$ and
$E=\ell^p(I)$ one obtains the usual $p$-sums of Banach spaces.\par
The ``continuous counterpart'' to absolute sums are the K\"othe-Bochner function spaces, whose definition we will recall now.
Let $(S,\A,\mu)$ be a complete, $\sigma$-finite measure space. For $A\in \A$ we denote by $\chi_A$ the characteristic function
of $A$. A K\"othe function space over $(S,\A,\mu)$ is a Banach space $(E,\norm{\cdot}_E)$ of real-valued measurable
functions on $S$ (modulo equality $\mu$-almost everywhere) such that
\begin{enumerate}[(i)]
\item $\chi_A\in E$ for every $A\in \A$ with $\mu(A)<\infty$,
\item for every $f\in E$ and every set $A\in \A$ with $\mu(A)<\infty$ $f$ is $\mu$-integrable over $A$,
\item if $g$ is measurable and $f\in E$ such that $\abs*{g(t)}\leq\abs*{f(t)}$ $\mu$-a.\,e. then $g\in E$
and $\norm{g}_E\leq\norm{f}_E$.
\end{enumerate}
Standard examples are the spaces $L^p(\mu)$ for $1\leq p\leq\infty$.\par
Further recall that, given a Banach space $X$, a function $f:S \rightarrow X$ is called simple if there are finitely many disjoint
measurable sets $A_1,\dots ,A_n\in \A$ such that $\mu(A_i)<\infty$ for all $i=1,\dots,n$, $f$ is constant on each $A_i$ and $f(t)=0$
for every $t\in S\sm \bigcup_{i=1}^nA_i$. The function $f$ is said to be Bochner-measurable if there exists a sequence $(f_n)_{n\in \N}$
of simple functions such that $\lim_{n\to \infty}\norm{f_n(t)-f(t)}=0$ $\mu$-a.\,e.\par
For a K\"othe function space $E$ and a Banach space $X$, we denote by $E(X)$ the space of all Bochner-measurable functions
$f:S\rightarrow X$ (modulo equality a.\,e.) such that $\norm{f(\cdot)}\in E$. Endowed with the norm $\norm{f}_{E(X)}=
\norm*{\norm{f(\cdot)}}_E$ $E(X)$ becomes a Banach space, the so called K\"othe-Bochner space induced by $E$ and $X$.
For $E=L^p(\mu)$ we obtain the usual Lebesgue-Bochner spaces $L^p(\mu,X)$ for $1\leq p\leq\infty$. For more information on
K\"othe-Bochner spaces the reader is referred to the book \cite{lin}.
\section{Examples}\label{sec:examples}
We will now discuss a number of examples of Banach space properties which can be described via test families.
We start with the octahedral spaces and their relatives.
\subsection{Octahedrality}\label{sub:octa}
A real Banach space $X$ is called octahedral (OH) (see \cite{godefroy}) if the following holds: for every
finite-dimensional subspace $F$ of $X$ and every $\eps>0$ there is some $y\in S_X$ such that
\begin{equation*}
\norm{x+y}\geq (1-\eps)(\norm{x}+1) \ \ \forall x\in F.
\end{equation*}
$\ell^1$ is the model example of an OH space. It is known that a Banach space has an equivalent OH norm if and only if
it contains an isomorphic copy of $\ell^1$ (see \cite{deville}*{Theorem 2.5, p. 106}).\par
In the paper \cite{haller3}, two variants of octahedrality where introduced.\par
$X$ is called locally octahedral (LOH) if for every $x\in X$ and every $\eps>0$ there exists $y\in S_X$ such that
\begin{equation*}
\norm{sx+y}\geq (1-\eps)(|s|\norm{x}+1) \ \ \forall s\in \R.
\end{equation*}
$X$ is called weakly octahedral (WOH) if for every finite-dimensional subspace $F$ of $X$, every $x^*\in B_{X^*}$
and each $\eps>0$ there is some $y\in S_X$ such that
\begin{equation*}
\norm{x+y}\geq (1-\eps)(|x^*(x)|+1) \ \ \forall x\in F.
\end{equation*}
The motivation for this definition in \cite{haller3} was the study of so called diameter-two-properties.
Given $x^*\in S_{X^*}$ and $\alpha>0$, the slice of $B_X$ induced by $x^*$ and $\alpha$ is
$S(x^*,\alpha):=\set*{z\in B_X,x^*(z)>1-\alpha}$. According to \cite{abrahamsen}, the space $X$ is said
to have the local diameter-two-property (LD2P) if every slice of $B_X$ has diamter 2;
$X$ has the diameter-two-property (D2P) if every nonempty, relatively weakly open subset of $B_X$ has diameter 2;
$X$ has the strong diameter-two-property (SD2P) if every convex combination of slices of $B_X$ has diameter 2.\par
The following results were proved in \cite{haller3}:
\begin{enumerate}[(a)]
\item $X$ has LD2P $\iff$ $X^*$ is LOH.
\item $X$ has D2P $\iff$ $X^*$ is WOH.
\item $X$ has SD2P $\iff$ $X^*$ is OH.
\end{enumerate}
The equivalence (c) was also proved independently in \cite{becerra-guerrero2}.\par
It is known that the three diameter-two-properties are really different. For example, it follows from the
results on direct sums in \cite{haller3} that $c_0\oplus_2 c_0$ has the D2P but not the SD2P (we will
recall these results in Section \ref{sec:appl}).\par
Concerning the nonequivalence of the LD2P and the D2P, it has been shown in \cite{becerra-guerrero4} that there is
a Banach space with the LD2P whose unit ball contains relatively weakly open subsets of arbitrarily small diameter
(every Banach space containing an isomorphic copy of $c_0$ can be renormed to become such a space
\cite{becerra-guerrero4}*{Theorem 2.4}).\footnote{Note that the abbreviation SD2P in \cite{becerra-guerrero4} does
not stand for ``strong diameter-two-property'' but for ``slice diameter-two-property'', which coincides
with the LD2P of \cite{abrahamsen}.}\par
In \cite{kubiak} it was shown that Ces\`aro function spaces have the D2P.\par
It is possible to characterise all three octahedrality properties in terms of test families. To do so,
we make use of the following equivalent formulations proved in \cite{haller3} (other equivalent
characterisations in terms of coverings of the unit ball were proved in \cite{haller2}).\par
A Banach space $X$ is OH if and only if for every $n\in \N$, all
$x_1,\dots,x_n\in S_X$ and every $\eps>0$ there exists an element $y\in S_X$
such that $\norm{x_i+y}\geq 2-\eps$ for all $i=1,\dots,n$.\par
$X$ is LOH if and only if for every $x\in S_X$ and all $\eps>0$
there exists $y\in S_X$ such $\norm{x\pm y}\geq 2-\eps$.\par
Of course, the same characterisations also hold for all closed subspaces of $X$.
Thus if we put
\begin{equation*}
F_{\eps,U}(\mathbf{x},\mathbf{x}^*,\mathbf{y},\mathbf{y}^*):=
\max\set*{2-\norm{x_i+y_1}:i=1,\dots,n}
\end{equation*}
for $U\in \U(X)$, $\mathbf{x}=(x_1,\dots,x_n),\mathbf{y}=(y_1,\dots,y_m)\in B_U^{\text{fin}}$
and $\mathbf{x}^*,\mathbf{y}^*\in B_{U^*}^{\text{fin}}$,
we obtain a test family for the class of octahedral spaces in $X$.\par
If we put instead
\begin{equation*}
F_{\eps,U}(\mathbf{x},\mathbf{x}^*,\mathbf{y},\mathbf{y}^*):=
\max\set*{2-\norm{x_1+y_1},2-\norm{x_1-y_1}},
\end{equation*}
we obtain a test family for LOH in $X$.\par
(In both cases, condition (i) in Definition \ref{def:test} follows from the above characterisations,
while the conditions (ii)--(v) are easily verified.)\par
For weak octahedrality, the following was proved in \cite{haller3}: $X$ is WOH if and only if
for every $n\in \N$, all $x_1,\dots,x_n\in S_X$ , every $x^*\in S_{X^*}$ and every $\eps>0$ there
exists a $y\in S_X$ such that $\norm{x_i+ty}\geq (1-\eps)(|x^*(x_i)|+t)$ for all $i=1,\dots,n$ and
every $t\geq \eps$.\par
(The original formulation in \cite{haller3} reads ``for every $x^*\in B_{X^*}$'', but it clearly
suffices to take $x^*\in S_{X^*}$.)\par
Thus if we define
\begin{equation*}
F_{\eps,U}(\mathbf{x},\mathbf{x}^*,\mathbf{y},\mathbf{y}^*):=
\max_{i=1,\dots,n}\sup_{t\geq \eps}\paren*{1-\frac{\norm{x_i+ty_1}}{|x_1^*(x_i)|+t}}
\end{equation*}
for $U\in \U(X)$, $\mathbf{x}=(x_1,\dots,x_n),\mathbf{y}=(y_1,\dots,y_m)\in B_U^{\text{fin}}$
and $\mathbf{x}^*=(x_1^*,\dots,x_k^*)$, $\mathbf{y}^*\in B_{U^*}^{\text{fin}}$, then
condition (i) in the definition of a test family for the class of WOH spaces is satisfied and (ii)
and (iii) are clearly true as well. The conditions (iv) and (v) easily follow from the next auxiliary lemma.\par
\begin{lemma}\label{lemma:aux}
If $Y$ is a real Banach space and $\eps>0$, define the function $f:B_Y\times B_Y\times B_{Y^*} \rightarrow \R$ by
\begin{equation*}
f(x,y,x^*):=\sup_{t\geq \eps}\paren*{1-\frac{\norm{x+ty}}{|x^*(x)|+t}} \ \ \forall x,y\in B_Y, \forall x^*\in B_{Y^*}.
\end{equation*}
If $\delta>0$, $x,\tilde{x},y,\tilde{y}\in B_Y$ with $\norm{x-\tilde{x}}, \norm{y-\tilde{y}}\leq \delta$ and
$x^*,\tilde{x}^*\in B_{Y^*}$ with $\norm{x^*-\tilde{x}^*}\leq \delta$, then
\begin{equation*}
|f(x,y,x^*)-f(\tilde{x},\tilde{y},\tilde{x}^*)|\leq \delta (3/\eps+2/\eps^2+1).
\end{equation*}
\end{lemma}
\begin{Proof}
We have $||\tilde{x}^*(\tilde{x})|-|x^*(x)||\leq |\tilde{x}^*(\tilde{x})-\tilde{x}^*(x)|+|\tilde{x}^*(x)-x^*(x)|\leq 2\delta$.\par
Thus, for every $t\geq \eps$ we have
\begin{align*}
&1-\frac{\norm{\tilde{x}+t\tilde{y}}}{|\tilde{x}^*(\tilde{x})|+t}-\paren*{1-\frac{\norm{x+ty}}{|x^*(x)|+t}} \\
&=\frac{\norm{x+ty}(|\tilde{x}^*(\tilde{x})|+t)-\norm{\tilde{x}+t\tilde{y}}(|x^*(x)|+t)}{(|x^*(x)|+t)(|\tilde{x}^*(\tilde{x})|+t)} \\
&\leq \frac{\norm{x+ty}(|x^*(x)|+2\delta+t)-\norm{\tilde{x}+t\tilde{y}}(|x^*(x)|+t)}{(|x^*(x)|+t)(|\tilde{x}^*(\tilde{x})|+t)} \\
&\leq \frac{2\delta}{t^2}\norm{x+ty}+\frac{\norm{x+ty}-\norm{\tilde{x}+t\tilde{y}}}{|\tilde{x}^*(\tilde{x})|+t} \\
&\leq \frac{2\delta}{t^2}(1+t)+\frac{\norm{x-\tilde{x}}+t\norm{y-\tilde{y}}}{|\tilde{x}^*(\tilde{x})|+t} \\
&\leq \frac{2\delta}{t^2}(1+t)+\frac{(1+t)\delta}{t}=\delta(3/t+2/t^2+1) \\
&\leq \delta (3/\eps+2/\eps^2+1).
\end{align*}
By symmetry we also have
\begin{equation*}
1-\frac{\norm{x+ty}}{|x^*(x)|+t}-\paren*{1-\frac{\norm{\tilde{x}+t\tilde{y}}}{|\tilde{x}^*(\tilde{x})|+t}}\leq \delta (3/\eps+2/\eps^2+1)
\end{equation*}
for all $t\geq \eps$. This implies the desired inequality.
\end{Proof}
The above-mentioned dual characterisations from \cite{haller3} allow us to write the diameter-two-properties
in terms of test families as well.\par
Since a Banach space has the LD2P if and only if its dual is LOH, we obtain a test family for the LD2P in $X$ by setting
\begin{equation*}
F_{\eps,U}(\mathbf{x},\mathbf{x}^*,\mathbf{y},\mathbf{y}^*):=
\max\set*{2-\norm{x_1^*+y_1^*},2-\norm{x_1^*-y_1^*}}
\end{equation*}
for $U\in \U(X)$, $\mathbf{x},\mathbf{y}\in B_U^{\text{fin}}$ and
$\mathbf{x}^*=(x_1^*,\dots,x_n^*)$, $\mathbf{y}^*=(y_1^*,\dots,y_m^*)\in B_{U^*}^{\text{fin}}$.\par
Likewise, since a Banach has the SD2P if and only its dual is OH, a test family for the SD2P in $X$ is given by
\begin{equation*}
F_{\eps,U}(\mathbf{x},\mathbf{x}^*,\mathbf{y},\mathbf{y}^*):=
\max\set*{2-\norm{x_i^*+y_1^*}:i=1,\dots,n}
\end{equation*}
for $U\in \U(X)$, $\mathbf{x},\mathbf{y}\in B_U^{\text{fin}}$ and
$\mathbf{x}^*=(x_1^*,\dots,x_n^*)$, $\mathbf{y}^*=(y_1^*,\dots,y_m^*)\in B_{U^*}^{\text{fin}}$.
(In both cases the conditions (i)--(v) are easily checked.)\par
We also know that a Banach space has the D2P if and only if its dual is WOH. Then we can make use of
the following characterisation (see \cite{haller3}) for the property WOH in dual spaces, which does
not involve the bidual:\par
$X^*$ is WOH if and only if for every $n\in \N$, all $x_1^*,\dots,x_n^*\in S_{X^*}$, every $x\in S_X$
and every $\eps>0$ there exists $y^*\in S_{X^*}$ such that
\begin{equation*}
\norm{x_i^*+ty^*}\geq (1-\eps)(|x_i^*(x)|+t) \ \ \forall i\in \set*{1,\dots,n}\ \forall t\geq \eps.
\end{equation*}
Thus we can define a test family for the D2P in $X$ by
\begin{equation*}
F_{\eps,U}(\mathbf{x},\mathbf{x}^*,\mathbf{y},\mathbf{y}^*):=
\max_{i=1,\dots,n}\sup_{t\geq \eps}\paren*{1-\frac{\norm{x_i^*+ty_1^*}}{|x_i^*(x_1)|+t}}
\end{equation*}
for $U\in \U(X)$, $\mathbf{x}=(x_1,\dots,x_k),\mathbf{y}\in B_U^{\text{fin}}$
and $\mathbf{x}^*=(x_1^*,\dots,x_n^*)$, $\mathbf{y}^*=(y_1^*,\dots,y_m^*)\in B_{U^*}^{\text{fin}}$.\par
(The conditions (i)--(iii) are clear, and the conditions (iv) and (v) are proved by using an auxiliary lemma
similar to Lemma \ref{lemma:aux}).\par
We remark that it is also possible to describe the LD2P via a different test family, using directly the definition of the LD2P
(and not its dual characterisation). It is easily checked that a Banach space $X$ has the LD2P if and only if the following holds:
for every $x^*\in S_{X^*}$ and every $\eps>0$ there exist $y_1,y_2\in S_X$ such that $x^*(y_1),x^*(y_2)\geq 1-\eps$ and
$\norm{y_1-y_2}\geq 2-\eps$.\par
Thus we can define a test family for the LD2P in $X$ as follows:
\begin{equation*}
F_{\eps,U}(\mathbf{x},\mathbf{x}^*,\mathbf{y},\mathbf{y}^*):=
\max\set*{1-x_1^*(y_1),1-x_1^*(y_2),2-\norm{y_1-y_2}}
\end{equation*}
for $U\in \U(X)$, $\mathbf{x},\mathbf{y}=(y_1,\dots,y_m)\in B_U^{\text{fin}}$
and $\mathbf{x}^*=(x_1^*,\dots,x_n^*),\mathbf{y}^*\in B_{U^*}^{\text{fin}}$, where $y_2:=y_1$
if $m=1$ (once again, the conditions (i)--(v) in Definition \ref{def:test} are easily verified).\par
Finally, there is yet another weakening of the definition of octahedral spaces, which was introduced in \cite{haller4}:
$X$ is called alternatively octahedral (AOH) if for every $n\in \N$, all $x_1,\dots,x_n\in S_X$ and
every $\eps>0$ there is some $y\in S_X$ such that
\begin{equation*}
\max\set*{\norm{x_i+y},\norm{x_i-y}}\geq 2-\eps \ \ \forall i=1,\dots,n.
\end{equation*}
Every octahedral space is alternatively octahedral, while for example $c_0$ is alternatively
octahedral but not locally octahedral (see \cite{haller4}).\par
It is easily checked that
\begin{equation*}
F_{\eps,U}(\mathbf{x},\mathbf{x}^*,\mathbf{y},\mathbf{y}^*):=
\max\set*{2-\max\set*{\norm{x_i+y_1},\norm{x_i-y_1}}:i=1,\dots,n},
\end{equation*}
where $U\in \U(X)$, $\mathbf{x}=(x_1,\dots,x_n),\mathbf{y}=(y_1,\dots,y_m)\in B_U^{\text{fin}}$
and $\mathbf{x}^*,\mathbf{y}^*\in B_{U^*}^{\text{fin}}$, defines a test family for AOH in $X$.
\subsection{Almost square spaces}\label{sub:asq}
Next we turn to the classes of almost square and locally almost square Banach spaces. These notions
were introduced in \cite{abrahamsen2}.\par
A real Banach space $X$ is said to be almost square (ASQ) if the following holds: for all $n\in \N$
and all $x_1,\dots,x_n\in S_X$ there exists a sequence $(y_k)_{k\in \N}$ in $B_X$ such that
$\norm{y_k}\to 1$ and $\norm{x_i+y_k}\to 1$ for all $i=1,\dots,n$.\par
$X$ is called locally almost square (LASQ) if for every $x\in S_X$ there is a sequence $(y_k)_{k\in \N}$
in $B_X$ such that $\norm{y_k}\to 1$ and $\norm{x\pm y_k}\to 1$.\par
According to \cite{abrahamsen2} $X$ is ASQ if and only if for every $\eps>0$, every $n\in \N$ and all
$x_1,\dots,x_n\in S_X$ there exists a $y\in S_X$ such that $\norm{x_i-y}\leq 1+\eps$ for all $i=1,\dots,n$,
and $X$ is LASQ if and only if for every $\eps>0$ and every $x\in S_X$ there is some $y\in S_X$ such that
$\norm{x\pm y}\leq 1+\eps$.\par
$c_0$ is the model example of an ASQ space. It was further proved in \cite{abrahamsen2} that every ASQ space
contains an isomorphic copy of $c_0$ and that every separable Banach spaces containing an isomorphic copy of
$c_0$ has an equivalent ASQ norm. In \cite{becerra-guerrero3} it was proved that the same holds also for
nonseparable spaces.\par
In \cite{abrahamsen2} it was also proved that $X^*$ is OH (i.\,e. $X$ has the SD2P) whenever $X$ is ASQ.
By \cite{kubiak}*{Proposition 2.5} every LASQ space has the LD2P.\par
If we define
\begin{equation*}
F_{\eps,U}(\mathbf{x},\mathbf{x}^*,\mathbf{y},\mathbf{y}^*):=
\max\set*{\norm{x_i-y_1}-1:i=1,\dots,n}
\end{equation*}
for $U\in \U(X)$, $\mathbf{x}=(x_1,\dots,x_n),\mathbf{y}=(y_1,\dots,y_m)\in B_U^{\text{fin}}$ and
$\mathbf{x}^*,\mathbf{y}^*\in B_{U^*}^{\text{fin}}$, then we obtain a test family for ASQ in $X$,
as is easily checked.\par
Likewise, a test family for LASQ in $X$ is given by
\begin{equation*}
F_{\eps,U}(\mathbf{x},\mathbf{x}^*,\mathbf{y},\mathbf{y}^*):=
\max\set*{\norm{x_1+y_1}-1,\norm{x_1-y_1}-1}.
\end{equation*}
There is also an intermediate notion of weakly almost square (WASQ) spaces defined in \cite{abrahamsen2}
(by \cite{kubiak}*{Proposition 2.6} theses spaces have the D2P) but it is not clear whether this notion
can be phrased in terms of test families.
\subsection{The Daugavet property}\label{sub:daugavet}
We now consider spaces with the Daugavet and the alternative Daugavet property.\par
A real Banach space $X$ is said to have the Daugavet property (DP) if the equality
$\norm{\text{id}+T}=1+\norm{T}$ holds for every rank-one operator $T:X \rightarrow X$
(see for example \cites{kadets, werner}).\par
Examples of such spaces include $C(K)$ for compact Hausdorff spaces $K$ without isolated points, and
$L^1(\mu)$ for atomless measures $\mu$ (see the examples in \cite{werner}). In \cite{kadets} the following
remarkable result was proved: if $X$ has the DP, then $\norm{\text{id}+T}=1+\norm{T}$ actually
holds for all weakly compact operators on $X$.\par
According to \cite{kadets}*{Lemma 2}, $X$ has the DP if and only if for every $x\in S_X$,
every $x^*\in S_{X^*}$ and all $\eps>0$ there exists $y\in S_X$ such that $x^*(y)\geq 1-\eps$
and $\norm{x+y}\geq 2-\eps$.\par
Thus a test family for the Daugavet property in $X$ is given by
\begin{equation*}
F_{\eps,U}(\mathbf{x},\mathbf{x}^*,\mathbf{y},\mathbf{y}^*):=
\max\set*{1-x_1^*(y_1),2-\norm{x_1+y_1}}
\end{equation*}
for all $U\in \U(X)$, $\eps>0$, $\mathbf{x}=(x_1,\dots,x_n), \mathbf{y}=(y_1,\dots,y_m)\in B_U^{\text{fin}}$
and all $\mathbf{x}^*=(x_1^*,\dots,x_k^*), \mathbf{y}^*\in B_{U^*}^{\text{fin}}$ (again the conditions (i)--(v)
are easily verified).\par
The following weaker version of the DP was introduced in \cite{martin}: a real or complex Banach space $X$ is said to have
the alternative Daugavet property (ADP) if $\max_{\omega\in \T}\norm{\text{id}+\omega T}=1+\norm{T}$ holds for every rank-one
operator $T$ on $X$, where $\T:=\set*{\omega\in \K:|\omega|=1}$.\par
Again it was proved in \cite{martin} that the above equality holds for all weakly compact opertaors if it holds
for all rank-one operators. It was also proved in \cite{martin} that $X$ has the ADP if and only if for every $\eps>0$,
every $x\in S_X$ and every $x^*\in S_{X^*}$ there is some $y\in S_X$ such that $\operatorname{Re}x^*(y)\geq 1-\eps$ and
$\max_{\omega\in \T}\norm{y+\omega x}\geq 2-\eps$.\par
We can thus define a test family for the ADP in $X$ as follows:
\begin{equation*}
F_{\eps,U}(\mathbf{x},\mathbf{x}^*,\mathbf{y},\mathbf{y}^*):=
\max\set*{1-\operatorname{Re}x_1^*(y_1),2-\max_{\omega\in \T}\norm{y_1+\omega x_1}}.
\end{equation*}
\subsection{Lush spaces}\label{sub:lush}
Next we consider the class of lush Banach spaces which was introduced in
\cite{boyko1} (in connection with the study of the numerical index of Banach spaces).
A Banach space $X$ is called lush provided that for every $\eps>0$ and
all $x_1,x_2\in S_X$ there exists a functional $y^*\in S_{X^*}$ such that
$x_1\in S(y^*,\eps)$ and $\dist{x_2}{\aco{S(y^*,\eps)}}<\eps$, where $\aco$ denotes
the absolutely convex hull and $d$ is the usual inf-distance.\par
For example, if $K$ is a compact Hausdorff space, then $C(K)$, and more generally every so called
$C$-rich subspace of $C(K)$, is lush (see \cite{boyko1}).\par
We can define a test family for lushness in $X$ by
\begin{equation*}
F_{\eps,U}(\mathbf{x},\mathbf{x}^*,\mathbf{y},\mathbf{y}^*):=
\max\set*{1-y_1^*(x_1),\dist{x_2}{\aco{S(y_1^*,\eps)}}}
\end{equation*}
for $\eps>0$, $U\in \U(X)$, $\mathbf{x}=(x_1,\dots,x_n),\mathbf{y}\in B_U^{\text{fin}}$ and
$\mathbf{x}^*,\mathbf{y}^*=(y_1^*,\dots,y_m^*)\in B_{U^*}^{\text{fin}}$
(where we set $x_2:=x_1$ if $n=1$ and $\dist{x_2}{\aco{S(y_1^*,\eps)}}:=2$ if $\norm{y_1^*}<1$).
The conditions (i)--(v) in Definition \ref{def:test} are easily verified.\par
In \cite{tan} the following related notion was introduced: the space $X$
is called generalised lush (GL) if for every $x\in S_X$ and every $\eps>0$
there is some functional $y^*\in S_{X^*}$ such that $x\in S(y^*,\eps)$ and
$\dist{z}{S(y^*,\eps)}+\dist{z}{-S(y^*,\eps)}<2+\eps$ for every $z\in S_X$.\par
It was shown in \cite{tan} that every separable lush space is GL, and that $\R^2$ equipped
with the hexagonal norm $\norm{(a,b)}=\max\set*{\abs{b},\abs{a}+1/2\abs{b}}$ is GL but not lush.
It is not known whether every nonseparable lush space is GL.\par
The main result in \cite{tan} is that every GL-space $X$ has the Mazur-Ulam property (MUP), i.\,e. if
$Y$ is any Banach space and $T:S_X \rightarrow S_Y$ is a surjective isometry, then $T$ can be extended
to an isometric isomorphism between $X$ and $Y$.\par
It is not obvious whether the property GL can be described via test families. However, there is
the following (at least formally) weaker version of GL-spaces: $X$ is said to have the property $(**)$
if for all $x_1,x_2\in S_X$ and each $\eps>0$ one can find $y^*\in S_{X^*}$ such that $x_1\in S(y^*,\eps)$
and $\dist{x_2}{S(y^*,\eps)}+\dist{x_2}{-S(y^*,\eps)}<2+\eps$.\par
This notion was introduced in the author's paper \cite{hardtke} (with the help of an anonymous referee)
and the following observations were made:
\begin{enumerate}[(a)]
\item Every lush space has property $(**)$.
\item For separable spaces, $(**)$ is equivalent to GL.
\item Every space with property $(**)$ has the MUP.
\end{enumerate}
A test family for $(**)$ in $X$ can be defined by
\begin{equation*}
F_{\eps,U}(\mathbf{x},\mathbf{x}^*,\mathbf{y},\mathbf{y}^*):=
\max\set*{1-y_1^*(x_1),\dist{x_2}{S(y_1^*,\eps)}+\dist{x_2}{-S(y_1^*,\eps)}-2}
\end{equation*}
for $\eps>0$, $U\in \U(X)$, $\mathbf{x}=(x_1,\dots,x_n),\mathbf{y}\in B_U^{\text{fin}}$ and
$\mathbf{x}^*,\mathbf{y}^*=(y_1^*,\dots,y_m^*)\in B_{U^*}^{\text{fin}}$, where $x_2:=x_1$ if $n=1$
and $\dist{x_2}{S(y_1^*,\eps)}:=\dist{x_2}{-S(y_1^*,\eps)}:=2$ if $\norm{y_1^*}<1$.
\section{Main result}\label{sec:mainresult}
Given a complete, $\sigma$-finite measure space $(S,\A,\mu)$, a K\"othe function space $E$
over $(S,\A,\mu)$ and pairwise disjoint sets $A_1,\dots,A_N\in \A$ with
$0<\mu(A_i)<\infty$ for $i=1,\dots,N$, we define
\begin{equation*}
\norm{(a_1,\dots,a_N)}_{E(A_1,\dots,A_N)}:=\norm*{\sum_{i=1}^N\frac{a_i}{\norm{\chi_{A_i}}}_E\chi_{A_i}}_E \ \ \forall (a_1,\dots,a_N)\in \R^N.
\end{equation*}
Then $\norm{\cdot}_{E(A_1,\dots,A_N)}$ is an absolute, normalised norm on $\R^N$.
If $p\in [1,\infty]$ and $E=L^p(\mu)$, then this norm coincides with the usual $p$-norm on $\R^N$,
regardless of the choice of $A_1,\dots,A_N$.\par
For a Banach space $X$, we denote by $E(A_1,\dots,A_N,X)$ the $N$-fold absolute sum
of $X$ with respect to $\norm{\cdot}_{E(A_1,\dots,A_N)}$.\par
The following theorem is the main result of this paper.
\begin{theorem}\label{thm:reduction}
Let $(S,\A,\mu)$ be a complete, $\sigma$-finite measure space and $E$ a K\"othe function space over $(S,\A,\mu)$.
Suppose that $X$ is a Banach space such that the simple functions are dense in $E(X)$ and
$E(A_1,\dots,A_N,X)\in \E$ for every $N\in \N$ and all pairwise disjoint sets $A_1,\dots,A_N\in \A$
with $0<\mu(A_i)<\infty$ for each $i$. Suppose further that there exists a test family for $\E$ in $E(X)$.
Then $E(X)\in \E$.
\end{theorem}
\begin{Proof}
Let $(F_{\eps,U})_{\eps>0,U\in \U(E(X))}$ be a test family for $\E$ in $E(X)$.
Let $\mathbf{f}=(f_1,\dots,f_n)\in S_{E(X)}^n$, $\Phi=(\varphi_1,\dots,\varphi_m)\in S_{E(X)^*}^m$
and $\eps>0$. Choose $\delta>0$ such that:
\begin{enumerate}[(a)]
\item For all $\mathbf{y}\in B_{E(X)}^{\text{fin}}$, all $\mathbf{y}^*\in B_{E(X)^*}^{\text{fin}}$
and all $\mathbf{z}\in B_{E(X)}^n$ with $\norm{\mathbf{f}-\mathbf{z}}_{\infty}\leq\delta$ we have
\begin{equation*}
\abs{F_{\eps,E(X)}(\mathbf{f},\Phi,\mathbf{y},\mathbf{y}^*)-F_{\eps,E(X)}(\mathbf{z},
\Phi,\mathbf{y},\mathbf{y}^*)}\leq\frac{\eps}{2}.
\end{equation*}
\item For every $U\in \U(E(X))$, for all $\mathbf{x}\in B_U^n$, all
$\mathbf{y}\in B_U^{\text{fin}}$, every $\mathbf{y}^*\in B_{U^*}^{\text{fin}}$ and all
$\mathbf{x}^*,\mathbf{z}^*\in B_{U^*}^m$ with $\norm{\mathbf{x}^*-\mathbf{z}^*}_{\infty}\leq\delta$
we have
\begin{equation*}
\abs{F_{\eps,U}(\mathbf{x},\mathbf{x}^*,\mathbf{y},\mathbf{y}^*)-
F_{\eps,U}(\mathbf{x},\mathbf{z}^*,\mathbf{y},\mathbf{y}^*)}\leq \frac{\eps}{4c},
\end{equation*}
where $c$ is the constant from Definition \ref{def:test} (iii).
\end{enumerate}
(This is possible because of (iv) and (v) in Definition \ref{def:test}).\par
Put $\tilde{\eps}:=\min\set*{\eps,\eps/4c}$.\par
We can find simple functions $h_1,\dots,h_n\in E(X)$ such that $\norm{h_i}_{E(X)}=1$
and $\norm{f_i-h_i}_{E(X)}\leq\delta$ for all $i=1,\dots,n$.\par
Also, there are simple functions $g_1,\dots,g_m\in E(X)$ with $\norm{g_j}_{E(X)}=1$
and $\abs{\varphi_j(g_j)}\geq 1-\delta$ for all $j=1,\dots,m$.\par
Fix pairwise disjoint sets $A_1,\dots,A_N\in \A$ with $0<\mu(A_i)<\infty$ such that
each $h_i$ and each $g_j$ belongs to the subspace
\begin{equation*}
U:=\set*{\sum_{k=1}^Nx_k\chi_{A_k}:x_1,\dots,x_N\in X}\ssq E(X).
\end{equation*}
By considering the map $T:E(A_1,\dots,A_N,X) \rightarrow U$ defined by
\begin{equation*}
T(x_1,\dots,x_N):=\sum_{k=1}^N\frac{x_k}{\norm{\chi_{A_k}}_E}\chi_{A_k},
\end{equation*}
we see that $U$ is isometrically isomorphic to $E(A_1,\dots,A_N,X)$.\par
By assumption we have $E(A_1,\dots,A_N,X)\in \E$, thus $U\in \E$.\par
Since $g_j\in S_U$ we have $1\geq\norm{\varphi_j|_U}\geq 1-\delta$ for each $j$.
Hence $\psi_j:=\varphi_j|_U/\norm{\varphi_j|_U}\in S_{U^*}$ with
\begin{equation}\label{eq:psi}
\norm{\psi_j-\varphi_j|_U}=\abs{1-\norm{\varphi_j|_U}}\leq\delta \ \ \forall j=1,\dots,m.
\end{equation}
Put $\Psi:=(\psi_1,\dots,\psi_m)\in S_{U^*}^m$ and $\mathbf{h}=(h_1,\dots,h_n)\in S_U^n$. Since $U\in \E$
we can find $\mathbf{u}=(u_1,\dots,u_l)\in S_U^{\text{fin}}$ and $\mathbf{u}^*=(u_1^*,\dots,u_s^*)\in S_{U^*}^{\text{fin}}$ such that
$F_{\tilde{\eps},U}(\mathbf{h},\Psi,\mathbf{u},\mathbf{u}^*)\leq \tilde{\eps}$.\par
Because of $\tilde{\eps}\leq \eps$ and (ii) in Definition \ref{def:test}, it follows that
$F_{\eps,U}(\mathbf{h},\Psi,\mathbf{u},\mathbf{u}^*)\leq \tilde{\eps}$.\par
Then (b) and \eqref{eq:psi} imply
$F_{\eps,U}(\mathbf{h},\Phi|_U,\mathbf{u},\mathbf{u}^*)\leq \tilde{\eps}+\eps/4c\leq \eps/2c$.\par
By the Hahn-Banach theorem there are functionals $\omega_1,\dots,\omega_s\in S_{E(X)^*}$
such that $\omega_i|_U=u_i^*$ for $i=1,\dots,s$. Let $\Omega:=(\omega_1,\dots,\omega_s)$.\par
Now it follows from (iii) in Definition \ref{def:test} that
$F_{\eps,E(X)}(\mathbf{h},\Phi,\mathbf{u},\Omega)\leq\eps/2$.\par
Since $\norm{\mathbf{f}-\mathbf{h}}_{\infty}\leq\delta$, (a) imlpies
$F_{\eps,E(X)}(\mathbf{f},\Phi,\mathbf{u},\Omega)\leq\eps/2+\eps/2=\eps$
and the proof is finished.
\end{Proof}
Every K\"othe function space $E$ is a Banach lattice in its natural
ordering ($f\leq g$ if and only if $f(s)\leq g(s)$ for a.\,e. $s\in S$).
It is well-known that if $(E,\leq)$ is order continuous, then for every
Banach space $X$ the simple functions lie dense in $E(X)$. This includes
in particular the case of $L^p$-spaces for $1\leq p<\infty$. So from the
above theorem we obtain the following corollary ($\ell^p_N(X)$ denotes the
$N$-fold $p$-sum of $X$).
\begin{corollary}\label{cor:reductionp}
Let $(S,\A,\mu)$ be a complete, $\sigma$-finite measure space and $1\leq p<\infty$. If $X$ is
a Banach space such that $\ell^p_N(X)\in \E$ for every $N\in \N$ and there exists a
test family for $\E$ in $L^p(\mu,X)$, then $L^p(\mu,X)\in \E$.
\end{corollary}
In the case $p=\infty$, it is well-known that one still has the density of
$\set*{f\in L^{\infty}(\mu,X):\text{ran}(f)\ \text{is\ countable}}$ in $L^{\infty}(\mu,X)$,
where $\text{ran}(f)$ denotes the range of $f$. Thus one can prove the following Theorem
in an analogous way to the proof of Theorem \ref{thm:reduction} (we omit the details).
\begin{theorem}\label{thm:Linfty}
Let $(S,\A,\mu)$ be a complete, $\sigma$-finite measure space. If $X$ is a
Banach space such that $\ell_N^{\infty}(X)\in \E$ for every $N\in \N$ and
$\ell^{\infty}(X)\in\E$ and there exists a test family for $\E$ in $L^{\infty}(\mu,X)$,
then $L^{\infty}(\mu,X)\in \E$.
\end{theorem}
Here $\ell^{\infty}(X)$ stands for $\big[\bigoplus_{n\in \N}X\big]_{\ell^{\infty}}$.\par
We also have a reduction result for the case of infinite absolute sums to finite sums,
which reads as follows.
\begin{proposition}\label{prop:sums}
Let $I$ be an index set and $E$ a subspace of $\R^I$ endowed with an absolute,
normalised norm such that $\operatorname{span}\set*{e_i:i\in I}$ is dense in $E$.
Let $(X_i)_{i\in I}$ be a family of Banach spaces such that $\big[\bigoplus_{i\in J}X_i\big]_E\in \E$
for every nonempty, finite subset $J\ssq I$. If there is a test family for $\E$ in
$\big[\bigoplus_{i\in I}X_i\big]_E$, then $\big[\bigoplus_{i\in I}X_i\big]_E\in \E$.
\end{proposition}
The notation $\big[\bigoplus_{i\in J}X_i\big]_E$ means that all summands with index in $I\sm J$ are $\set*{0}$.
The proof is similar to the one of Theorem \ref{thm:reduction} and will therefore be omitted.\par
As an immediate consequence of Proposition \ref{prop:sums} we get the following results for $p$-sums
and $c_0$-sums.
\begin{corollary}\label{cor:sum-p}
If $I$ is any index set, $1\leq p<\infty$, $(X_i)_{i\in I}$ is a family of Banach spaces
such that $\big[\bigoplus_{i\in J}X_i\big]_p\in \E$ for every nonempty, finite subset $J\ssq I$,
and there exists a test family for $\E$ in $\big[\bigoplus_{i\in I}X_i\big]_p$, then
$\big[\bigoplus_{i\in I}X_i\big]_p\in \E$.
\end{corollary}
\begin{corollary}\label{cor:c0-sum}
If $I$ is any index set, $(X_i)_{i\in I}$ is a family of Banach spaces such that
$\big[\bigoplus_{i\in J}X_i\big]_{\infty}\in \E$ for every nonempty, finite subset
$J\ssq I$, and there exists a test family for $\E$ in $\big[\bigoplus_{i\in I}X_i\big]_{c_0}$,
then $\big[\bigoplus_{i\in I}X_i\big]_{c_0}\in \E$.
\end{corollary}
\section{Applications}\label{sec:appl}
In this section we will apply the abstract results to the examples discussed earlier. This
will yield some new results as well as some alternative proofs of already known results.\par
We first collect what is known about sums of octahedral spaces and their relatives. The following
results were proved in \cite{haller3}: if $X$ and $Y$ are real Banach spaces, then
\begin{enumerate}[(a)]
\item $X$ or $Y$ is LOH/WOH/OH $\Rightarrow$ $X\oplus_1 Y$ is LOH/WOH/OH,
\item $X$ and $Y$ are LOH/WOH $\Rightarrow$ $X\oplus_p Y$ is LOH/WOH for every $p\in (1,\infty]$,
\item $X$ and $Y$ are OH $\Rightarrow$ $X\oplus_{\infty} Y$ is OH,
\item For $p\in (1,\infty)$ $X\oplus_p Y$ is never OH.
\end{enumerate}
In \cite{abrahamsen2} the following generalisation was obtained: if $I$ is any index set and $E$ a
subspace of $\R^I$ with an absolute, normalised norm, and $(X_i)_{i\in I}$ is a family of LOH spaces,
then $\big[\bigoplus_{i\in I}X_i\big]_E$ is also LOH. If each $X_i$ is WOH and moreover
$\text{span}\set*{e_i:i\in I}$ is dense in $E$, then $\big[\bigoplus_{i\in I}X_i\big]_E$ is also WOH.\par
It is also easily checked that $\ell^{\infty}(X)$ is OH whenever $X$ is OH (the proof is analogous to the proof
of (c) above that was given in \cite{haller3}).\par
Combining all this with our Theorems \ref{thm:reduction} resp. \ref{thm:Linfty} and the fact that OH, WOH
and LOH can be described by test families (see Section \ref{sec:examples}), we obtain the following results.
\begin{theorem}\label{thm:LOH-WOH}
If $(S,\A,\mu)$ is a complete, $\sigma$-finite measure space, $E$ a K\"othe function space over $(S,\A,\mu)$
and $X$ an LOH/WOH space such that the simple functions are dense in $E(X)$ (for instance, if $E$ is order continuous),
then $E(X)$ is also LOH/WOH.\par
In particular, if $p\in [1,\infty)$ and $X$ is LOH/WOH, then so is $L^p(\mu,X)$.\par
Also, $L^{\infty}(\mu,X)$ is LOH if $X$ is LOH.
\end{theorem}
\begin{proposition}\label{prop:OH}
If $(S,\A,\mu)$ is a complete, $\sigma$-finite measure space and $X$ is an OH space, then
$L^1(\mu,X)$ and $L^{\infty}(\mu,X)$ are also OH.
\end{proposition}
This result is not optimal. In fact, it is not difficult to see that $L^1(\mu,X)$ is OH for
{\it any} Banach space $X$ (provided that $L^1(\mu)$ is infinite-dimensional), see the examples
at the end of \cite{langemets1}.\par
Now we turn to the diameter-two-properties. In \cite{haller3} the following results were derived
via duality from the corresponding results on octahedrality in direct sums.
\begin{enumerate}[(a)]
\item $X$ or $Y$ has the LD2P/D2P/SD2P $\Rightarrow$ $X\oplus_{\infty} Y$ has the LD2P/D2P/SD2P,
\item $X$ and $Y$ have the LD2P/D2P $\Rightarrow$ $X\oplus_p Y$ has the LD2P/D2P for every $p\in [1,\infty)$,
\item $X$ and $Y$ have the SD2P $\Rightarrow$ $X\oplus_1 Y$ has the SD2P,
\item For $p\in (1,\infty)$ $X\oplus_p Y$ never has the SD2P.
\end{enumerate}
All these results have been known before (they are scattered in \cites{abrahamsen, acosta, becerra-guerrero, haller, lopez-perez},
see \cite{haller3} for a detailed account), but the previous proofs were based on different methods. In \cite{acosta} it was shown
that the LD2P and the D2P are stable under sums with respect to an arbitrary absolute norm.\par
Since LD2P, D2P and SD2P can be described by test families (see Section \ref{sec:examples}), we obtain the following stability
result from Theorem \ref{thm:reduction}.
\begin{theorem}\label{thm:diam2}
Let $(S,\A,\mu)$ be a complete, $\sigma$-finite measure space, $E$ a K\"othe function space over $(S,\A,\mu)$ and $X$ a Banach
space such that the simple functions are dense in $E(X)$ (for instance, if $E$ is order continuous). If $X$ has the LD2P/D2P, then
$E(X)$ also has the LD2P/D2P.\par
In particular, if $p\in [1,\infty)$ and $X$ has the LD2P/D2P, then so does $L^p(\mu,X)$.\par
Further, if $X$ has the SD2P, then $L^1(\mu,X)$ also has the SD2P.
\end{theorem}
In \cite{acosta} it was already proved that $L^p(\mu,X)$ has the D2P whenever $1\leq p <\infty$, $\mu$ is a finite measure and
$X$ has the D2P (this proof also uses simple functions). Also, for the special case $p=1$, better results are already known, for
instance, it has been proved in \cite{becerra-guerrero}*{Theorem 2.13} that for a finite measure $\mu$ the space $L^1(\mu,X)$ has
the D2P if and only if $X$ has the D2P or $\mu$ has no atoms (and $L^{\infty}(\mu,X)$ has the D2P if and only if $L^{\infty}(\mu)$
is infinite-dimensional or $X$ has the D2P).\par
Even more, it is known that the Daugavet property implies the SD2P (see \cite{abrahamsen}*{Theorem 4.4}) and that
$L^1(\mu,X)$ and $L^{\infty}(\mu,X)$ have the Daugavet property for any atomless measure $\mu$ and any Banach space $X$
(\cite{werner}, see the discussion for the DP below).\par
Also, if $X$ or $Y$ has the LD2P, then so does $X\hat{\otimes}_{\pi}Y$ (see \cite{abrahamsen}*{Theorem 2.7}) and if $X$
and $Y$ have the SD2P, then so does $X\hat{\otimes}_{\pi}Y$ (see \cite{becerra-guerrero5}), where $\hat{\otimes}_{\pi}$ denotes the projective
tensor product, and it is well-known that $L^1(\mu,X)=L^1(\mu)\hat{\otimes}_{\pi}X$. For more information on octahedrality and related
properties in tensor products see also \cites{langemets1,langemets2}.\par
For AOH spaces, the following equivalent characterisation can be proved:
$X$ is AOH if and only if for every $n\in \N$, all $x_1,\dots,x_n\in S_X$ and each $\eps>0$ there is some $y\in S_X$
such that
\begin{equation*}
\max\set*{\norm{x_i+ty},\norm{x_i-ty}}\geq (1-\eps)(1+t) \ \ \forall t>0, \forall i\in \set*{1,\dots,n}.
\end{equation*}
The proof is analogous to the proof of the corresponding characterisation for octahedral spaces in \cite{haller3}
and will therefore be skipped.\par
Using this characterisation, one can show that $X\oplus_1 Y$ is AOH if $X$ or $Y$ is AOH and that $X\oplus_{\infty} Y$
is AOH if $X$ and $Y$ are AOH. The latter result also extends to $\ell^{\infty}(X)$. Again the proofs are analogous
to the ones for the corresponding results on OH spaces in \cite{haller3} and thus we will skip them.\par
Using our Theorems \ref{thm:reduction} and \ref{thm:Linfty} we can now obtain the following result.
\begin{proposition}\label{prop:AOH}
If $(S,\A,\mu)$ is a complete, $\sigma$-finite measure space and $X$ is an AOH space, then
$L^1(\mu,X)$ and $L^{\infty}(\mu,X)$ are also AOH.
\end{proposition}
Again, if $L^1(\mu)$ is infinite-dimensional, then $L^1(\mu,X)$ is even OH for {\it any} Banach space $X$ (\cite{langemets1}).\par
Concerning sums of ASQ and LASQ spaces, the following was proved in \cite{abrahamsen2}: if $I$ is any index
set and $E$ a subspace of $\R^I$ with an absolute, normalised norm, and $(X_i)_{i\in I}$ is a family of LASQ
spaces, then $\big[\bigoplus_{i\in I}X_i\big]_E$ is also LASQ. Further, $X\oplus_{\infty} Y$ is ASQ/LASQ if
and only if $X$ or $Y$ is ASQ/LASQ. Analogously to the proof of the ``if'' part in \cite{abrahamsen2} one can
show that $\ell^{\infty}(X)$ is ASQ/LASQ whenever $X$ is ASQ/LASQ.\footnote{It has also been proved in
\cite{abrahamsen2} that for $p\in [1,\infty)$ the sum $X\oplus_p Y$ is never ASQ.}\par
If we combine these facts with Theorem \ref{thm:reduction} resp. \ref{thm:Linfty} and the fact that ASQ and LASQ
can be expressed in terms of test families (Section \ref{sec:examples}), we obtain the following stability result.
\begin{theorem}\label{ASQ-LASQ}
If $(S,\A,\mu)$ is a complete, $\sigma$-finite measure space, $E$ a K\"othe function space over $(S,\A,\mu)$
and $X$ an LASQ space such that the simple functions are dense in $E(X)$ (for instance, if $E$ is order continuous),
then $E(X)$ is also LASQ.\par
In particular, if $p\in [1,\infty)$ and $X$ is LASQ, then so is $L^p(\mu,X)$.\par
Moreover, $L^{\infty}(\mu,X)$ is ASQ/LASQ whenever $X$ is ASQ/LASQ.
\end{theorem}
Now we consider spaces with the Daugavet property. It has been shown in \cite{khalil} that
$L^1([0,1],X)$ and $L^{\infty}([0,1],X)$ have the DP if $X$ has it.
More generally, $L^1(\mu,X)$ has the DP for every {\it atomless} measure $\mu$ and
{\it every} Banach space $X$, see \cite{werner}*{p.81}.\par
In \cite{wojtaszczyk} it was already proved that the $\ell^1$- and $\ell^{\infty}$-sum of any (finite or infinite)
sequence of Banach spaces with the Daugavet property has again the Daugavet property (the Daugavet property for
weakly compact operators was considered in \cite{wojtaszczyk}, but this is equivalent to considering just rank-one
operators by \cite{kadets}*{Theorem 2.3}). In \cite{kadets} a different proof for the stability of the DP by finite
or infinite $\ell^1$- and $c_0$-sums has been given.\footnote{The cases of infinite sums are reduced to the corresponding
finite sums by a density argument, similar to the general reduction results for sums that we have stated in Section \ref{sec:mainresult}.}\par
Putting everything together, the following characterisation was obtained in \cite{martin00}*{Remark 9}: $L^1(\mu,X)$ has
the DP if and only if $X$ has the DP or $\mu$ has no atoms. Likewise, $L^{\infty}(\mu,X)$ has the DP if and only if $X$ has
the DP or $\mu$ has no atoms (see \cite{martin0}).\par
Analogous results also hold for the alternative Daugavet property: the space $L^1(\mu,X)$ has the ADP if and only if $X$ has the ADP
or $\mu$ has no atoms if and only if $L^{\infty}(\mu,X)$ has the ADP (see \cite{martin}). Also, the ADP is stable under arbitrary
$\ell^1$-, $c_0$- and $\ell^{\infty}$-sums (see again \cite{martin}).\par
Using the stability results for sums and our Theorems \ref{thm:reduction} and \ref{thm:Linfty}, we obtain an alternative
proof of the following known result.\par
\begin{theorem}\label{thm:daugavet}
If $(S,\A,\mu)$ is a complete, $\sigma$-finite measure space and $X$ a Banach space with the DP/ADP,
then $L^1(\mu,X)$ and $L^{\infty}(\mu,X)$ also have the DP/ADP.
\end{theorem}
Concerning lush spaces, the following has been proved in \cite{boyko2}: if $\norm{\cdot}_E$ is an absolute norm on $\R^n$,
then the sum of every collection $X_1,\dots,X_n$ of lush spaces with respect to $\norm{\cdot}_E$ is again lush if and only if
$(\R^n,\norm{\cdot}_E)$ is lush.\par
It was also proved in \cite{boyko2} that the $\ell^1$-, $c_0$- and $\ell^{\infty}$-sums of any family $(X_i)_{i\in I}$ of
lush spaces are again lush.\footnote{Also here the cases of $\ell^1$- and $c_0$-sums are reduced to the corresponding
finite sums (cf. footnote 3).}\par
Very recently, the following stability result has been proved in \cite{kadets2}*{Corollaries 8.9 and 8.12}.
\begin{theorem}[\cite{kadets2}]\label{thm:lush}
Let $(S,\A,\mu)$ be a $\sigma$-finite measure space and $X$ a Banach space. Then $L^{\infty}(\mu,X)$ is lush if and only if $X$
is lush if and only if $L^1(\mu,X)$ is lush.
\end{theorem}
In fact, even more general results are proved in \cite{kadets2} for so called lush operators.\par
If we use instead the above-mentioned results on sums of lush spaces in combination with our Theorems \ref{thm:reduction}
and \ref{thm:Linfty}, we obtain an alternative proof for the fact that lushness of $X$ is sufficient for lushness of
$L^1(\mu,X)$ and $L^{\infty}(\mu,X)$ (the proofs in \cite{kadets2} did not use a reduction to sums, but they also used
the density of the simple functions (resp. functions with countable range) in $L^1(\mu,X)$ (resp. $L^{\infty}(\mu,X)$).\par
Let us now turn to generalised lushness. It was proved in \cite{tan} that the property GL is stable under arbitrary
$\ell^1$-, $c_0$- and $\ell^{\infty}$-sums. The same results also hold for the property $(**)$, with completely analogous proofs.\par
Now we can apply our Theorems \ref{thm:reduction} and \ref{thm:Linfty} to obtain the following result.\par
\begin{theorem}\label{thm:starstar}
If $(S,\A,\mu)$ is a complete, $\sigma$-finite measure space and $X$ a Banach space with property $(**)$, then
$L^1(\mu,X)$ and $L^{\infty}(\mu,X)$ also have the property $(**)$.
\end{theorem}
We recall (see Subsection \ref{sub:lush}) that $(**)$ implies the MUP and $(**)$ is equivalent to GL for separable spaces, but it is
not known whether this equivalence is true in general nor if it is in general possible to describe the property GL by test families.
Thus we cannot apply our general reduction theorems directly to GL-spaces. However, it is still possible to show that GL is stable
with respect to $L^1$-Bochner spaces by a similar proof technique. This is carried out in the next section.
\section{GL-spaces}\label{sec:GL}
Here we show directly that $L^1(\mu,X)$ is GL whenever $X$ is GL. The argument is similar to the proof for $\ell^1$-sums
given in \cite{tan}, in combination with an approximation by simple functions.
\begin{theorem}\label{thm:GL}
Let $(S,\A,\mu)$ be a complete, $\sigma$-finite measure space. If $X$ is a GL-space,
then so is $L^1(\mu,X)$.
\end{theorem}
\begin{Proof}
Let $f\in L^1(\mu,X)$ with $\norm{f}_1=1$ and let $\eps\in (0,1)$. Choose $\eta\in (0,1)$ such that
\begin{align*}
&(2+\eps/2)(1+\eta)+4\eta<2+\eps, \\
&(1-\eps/2)(1-\eta)-\eta>1-\eps, \\
&(1-\eps/2)\frac{1-\eta}{1+\eta}>1-\eps.
\end{align*}
We can find a simple function $g$ on $S$ such $\norm{f-g}_1\leq \eta$.
Write $g=\sum_{i=1}^Nx_i\chi_{A_i}$ with pairwise disjoint sets $A_1,\dots,A_N\in \A$
and $x_1,\dots,x_N\in X$.\par
Since $X$ is GL, we can find functionals $x_1^*,\dots,x_N^*\in S_{X^*}$ such that
$x_i^*(x_i)\geq (1-\eps/2)\norm{x_i}$ and
\begin{equation}\label{eq:GL}
\dist{y}{S(x_i^*,\eps/2)}+\dist{y}{-S(x_i^*,\eps/2)}<2+\frac{\eps}{2} \ \ \forall y\in S_X.
\end{equation}
Let $h=\sum_{i=1}^Nx_i^*\chi_{A_i}$ and
$\varphi(v)=\int_Sh(s)(v(s))\,\text{d}\mu(s)$ for $v\in L^1(\mu,X)$. Then
$\varphi\in L^1(\mu,X)^*$ with $\norm{\varphi}=1$.\par
We further have
\begin{equation*}
\varphi(g)=\sum_{i=1}^N\int_{A_i}x_i^*(x_i)\,\text{d}\mu(s)\geq
(1-\eps/2)\sum_{i=1}^N\int_{A_i}\norm{x_i}\,\text{d}\mu(s)=(1-\eps/2)\norm{g}_1.
\end{equation*}
Since $\norm{f-g}_1\leq \eta$ and $\norm{f}_1=1$, it follows that
$\varphi(f)\geq \varphi(g)-\eta\geq (1-\eps/2)(1-\eta)-\eta$. Thus the choice of $\eta$
implies $f\in S(\varphi,\eps)$.\par
Now take any function $w\in L^1(\mu,X)$ with $\norm{w}_1=1$. There exists a simple
function $\tilde{w}$ on $S$ such that $\norm{w-\tilde{w}}_1\leq \eta$. Write $\tilde{w}=\sum_{j=1}^My_j\chi_{B_j}$
with pairwise disjoint sets $B_1,\dots,B_M\in \A$ and $y_1,\dots,y_M\in X$.\par
We put $C_{ij}:=A_i\cap B_j$ for $(i,j)\in I:=\set*{1,\dots,N}\times \set*{1,\dots,M}$.
By \eqref{eq:GL} we can find, for each pair $(i,j)\in I$, vectors $u_{ij},v_{ij}\in B_X$ such that
$x_i^*(u_{ij})>1-\eps/2$, $-x_i^*(v_{ij})>1-\eps/2$ and
\begin{equation}\label{eq:GL2}
\norm{y_j-\norm{y_j}u_{ij}}+\norm{y_j-\norm{y_j}v_{ij}}\leq(2+\eps/2)\norm{y_j}.
\end{equation}
Let $u=\sum_{(i,j)\in I}\norm{y_j}u_{ij}\chi_{C_{ij}}$ and
$v=\sum_{(i,j)\in I}\norm{y_j}v_{ij}\chi_{C_{ij}}$.\par
Since $\norm{u_{ij}}\leq 1$ we have $\norm{u(s)}\leq \norm{y_j}=\norm{\tilde{w}(s)}$
for all $s\in C_{ij}$ and all $(i,j)\in I$. Hence $\norm{u}_1\leq \norm{\tilde{w}}_1
\leq \norm{w-\tilde{w}}_1+\norm{w}_1\leq 1+\eta$. Analogously, one can see that
$\norm{v}_1\leq 1+\eta$.\par
Thus we have $\tilde{u}:=u/(1+\eta)\in B_{L^1(\mu,X)}$ and
$\tilde{v}:=v/(1+\eta)\in B_{L^1(\mu,X)}$.\par
We further have
\begin{align*}
&\varphi(\tilde{u})=\frac{1}{1+\eta}\sum_{(i,j)\in I}\int_{C_{ij}}x_i^*(u_{ij})\norm{y_j}\,\text{d}\mu(s)\geq
\frac{1-\eps/2}{1+\eta}\sum_{(i,j)\in I}\int_{C_{ij}}\norm{y_j}\,\text{d}\mu(s) \\
&=\norm{\tilde{w}}_1\frac{1-\eps/2}{1+\eta}\geq (1-\eta)\frac{1-\eps/2}{1+\eta}
>1-\eps.
\end{align*}
Thus $\tilde{u}\in S(\varphi,\eps)$ and analogously one can show that
$\tilde{v}\in -S(\varphi,\eps)$.\par
It further follows from \eqref{eq:GL2} that
\begin{equation*}
\norm{\tilde{w}(s)-u(s)}+\norm{\tilde{w}(s)-v(s)}\leq (2+\eps/2)\norm{\tilde{w}(s)}
\ \ \forall s\in S.
\end{equation*}
Hence $\norm{\tilde{w}-u}_1+\norm{\tilde{w}-v}_1\leq (2+\eps/2)\norm{\tilde{w}}_1\leq (2+\eps/2)(1+\eta)$.\par
Since $\norm{w-\tilde{w}}_1\leq \eta$ we get
$\norm{w-u}_1+\norm{w-v}_1\leq (2+\eps/2)(1+\eta)+2\eta$.\par
We also have $\norm{u-\tilde{u}}_1\leq \eta$ and $\norm{v-\tilde{v}}_1\leq \eta$.
Thus $\norm{w-\tilde{u}}_1+\norm{w-\tilde{v}}_1\leq (2+\eps/2)(1+\eta)+4\eta
<2+\eps$ and we are done.
\end{Proof}
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\address
\email
\end{document}
|
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BkiUbHTxK1UJ-rRH-V2k
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\section{Introduction}
The prolonged and devastating COVID-19 pandemic has affected every aspect of people's lives as well as the global economy. In an attempt to curb the spread of highly contagious variants, governments around the world have contemplated or adopted vaccination mandates (VMs) and vaccination certificates (or passports) (VCs) in schools, hospitals, public transportation, and other social contexts~\cite{IsraeliWorkplace,CollegeMandate,CanadaVaxpass,EUGreenCert,Excelsior,HawaiiVC,BritainBarVCPassport}. COVID VMs and VCs challenge established societal norms and conventions. While vaccination mandates and certificates are not new (e.g., vaccination mandates for children attending schools, ``yellow cards'' for travel to or from a country with a high risk of diseases such as yellow fever~\cite{yellowcard}),
the sudden and unprecedented requirement to show proof of vaccination to gain access to public venues or engage in a range of daily activities has triggered a fierce global debate on the appropriateness of COVID-19 VMs and VCs in light of established societal norms and conventions, perceived privacy harms, and civil liberty expectations~\cite{SupremeNews,APAustrian,GermanMandate,Kofler2021TenReasons,WEForumVMDebate}.
Some proponents of VMs and VCs argue for overriding these values in the short term to accommodate urgent public health needs. Those opposing these measures warn against grave long-term repercussions that could result from bending traditional norms and civil liberty expectations. The arguments of both sides have merits. On the one hand, with over 6 million COVID-related deaths worldwide at the time of writing~\cite{WhoCOVIDDashboard}, enforcing VMs and requiring VCs is compelling, with data showing that vaccinations have a proven record of reducing the number of infections and hospitalizations~\cite{CDCVaccineEffective}. On the other hand, without proper policies and technology-backed measures, the collection and use of the information contained in VCs, such as an individual's ID number, full name, date of birth, gender, nationality, and vaccination records, may not be restricted to its intended context or purposes. Risks of the digitization and re-purposing of this information have prompted warnings against large-scale adoption of VCs~\cite{IPThreatPrivacy, ExpertPrivacy2021} that could result in privacy violations, widening inequalities, and discrimination~\cite{Cofone2021,Baylis2021,Liz2021}.
In our view, the privacy implications of VC-related and other technologies depend on the context in which they operate. For example, it might be appropriate for vaccination status and date of birth to be shared with a doctor in a health context for a medical evaluation, while not so, when the same information is shared with an event organizer to gain access. Against the background of the global pandemic and increasing adoptions of VMs and VCs, it is important to ensure that the information flow rules embodied in the new technologies adhere to prevailing societal norms for each given context.
The present study was conducted to develop a deeper understanding of what the respective societal norms are and whether the information flows that VC deployments give rise to are appropriate, with the help of the Contextual Integrity (CI) framework~\cite{nissenbaum2009privacy}.
We use an established CI-based vignette survey methodology~\cite{apthorpe2019evaluating,Shvartzshnaider2016} to analyze data from a US-based demographically-stratified sample ($N=890$) about how they perceive sharing VC information with various recipients in different contexts such as education, health, or public transportation, under different conditions and for various purposes.
Our analysis reveals that perceived appropriateness is contextual and varies depending on CI's five parameters for information flows (i.e., sender, attribute, subject, recipient, transmission principle). There is also a significant difference in acceptance of first-hand information sharing compared to later re-sharing and re-purposing of originally collected information. Overall, we find that information re-sharing with entities other than public health agencies is widely viewed as unacceptable. These findings are relevant both to policy and technology design because they may reveal that even though VCs alter information flows, not all of these alterations constitute violations of privacy; benefits may be enjoyed without---in these instances---a need to curtail societal values.
\section{Background and Related Work}
In this section, we describe how this study builds upon related work around the topic of certifying immunity, recent user research on COVID-19 pandemic mitigation technologies, and the privacy framework of Contextual Integrity.
\subsection{Social and Privacy Considerations for Certifying Immunity}
As COVID-19 vaccines are increasingly available, several nations and industries have begun to develop digital solutions to certify people's immunity towards COVID-19~\cite{IsraelPass,EUGreenCert,Excelsior}. These efforts draw on the existing paper-based proof of vaccination counterparts~\cite{WHO1969} for diseases like smallpox, typhus, and cholera~\cite{VaccinePassports85Years}.
Although these solutions vary in what they certify (e.g., the presence of an antibody, a negative virus test, or a vaccination record), they introduce some digital verification mechanisms that may restrict people's access to different social activities or venues, such as traveling, attending large public events, or entering restaurants or bars.
Deploying these solutions at scale has profound social implications: They could widen the existing inequities in access to healthcare resources and technologies; they could also lead to increased and systemic discrimination, especially for the already vulnerable~\cite{Baylis2021,baylis2020covid,Brown2021,Brown2021Reply,Phelan2020,Kofler2021TenReasons}. These risks have prompted the World Health Organization to discourage the use of VCs for international travel~\cite{WHOVaccineCertificates}.
More relevant to our work, mechanisms to certify one's vaccination status also potentially increase privacy risks. In addition to vaccination records, a typical VC includes an individual's national ID (e.g., passport number), full name, date of birth, gender, and nationality, most of which is personally identifiable information~\cite{yellowcard,IsraelPass,CanadaVaxpass}.
Although both paper and digital VCs ostensibly have similar features, digital VCs introduce new privacy challenges.
Digital information is inherently easier to collect and share on a much greater scale, possibly outside its intended context. For example, if people need to show their VCs to gain access to public venues and social events such as swimming pools, gyms, and concerts~\cite{IsraelPass}, data about their daily activities may be easily tracked and open to potential misuse. To gain a deeper and more comprehensive understanding of how deploying VCs and enforcing VMs could affect privacy and social life, our work explores the potential privacy violations of VC usage across a number of different contexts, which also enriches the discussion around the social implications of VMs and VCs.
\subsection{Surveying Public Opinions about Vaccination Certificates}
The use of technology in COVID-19 pandemic mitigation and containment, such as digital contact tracing apps, has spurred a number of studies into public opinion about these solutions, whose success relies upon broad user adoption~\cite{AustralianCOVIDTracking2021Garrett, williams2021public,o2021national, lohar2021irish, altmann2020acceptability,HowGoodCOVIDApps2020}.
Although these studies have revealed generally positive attitudes towards some of these solutions~\cite{lohar2021irish,williams2021public}, people often hesitate to install these apps out of concerns about cybersecurity and privacy~\cite{altmann2020acceptability, lohar2021irish}, as well as greater surveillance by governments and big technology companies after the pandemic~\cite{altmann2020acceptability, williams2021public}.
With increasing COVID-19 vaccination rates around the world, a number of non-academic surveys have attempted to gauge public attitudes towards and concerns around potential mechanisms, such as VCs, to verify people's vaccination status against COVID-19~\cite{QualtricsAmerican2020Haney, SwissImmunitySerology2020Nehme, BritonsImmunityCert2020Redfield, GermanyImmunitySnapshot2020, HillHarris}.
In Germany, 45\% of the population was reported to oppose the introduction of VCs~\cite{GermanyImmunitySnapshot2020}. In Australia, 75\% of Australians were reported to support the use of VCs with only ~10\% opposing it~\cite{AustralianCOVIDTracking2021Garrett}. High levels of approval were also reported in Switzerland and in the UK, 60\% of Swiss~\cite{SwissImmunitySerology2020Nehme} and 69\% of Brits indicated they supported VCs~\cite{BritonsImmunityCert2020Redfield}.
A survey of the US population in April 2021 has found a mixed-level of support for using VCs: 53\% of Americans expressed support for government-issued vaccine passports, while 47\% reported being against the use of vaccine certificates~\cite{HillHarris}.
Although the Brookings Institute's report~\cite{ZhangBB2021Brookings} outlining principles to build robust and ethical vaccination verification systems is an important contribution, nevertheless, comprehensive and rigorous academic research to supplement such contributions with insights on attitudes and conditions of acceptance behind potential support for the use of VCs is much needed.
In our work, we aimed to take a rigorous and nuanced approach to understand the privacy concerns around VCs in their different contexts of use. Guided by the theory of Contextual Integrity, we surveyed a representative sample of the US population and studied their normative perceptions of sharing VC information with different entities, for varying purposes, and in diverse conditions.
\subsection{Studying Privacy through Contextual Integrity}\label{sec:CI_def}
The theory of Contextual Integrity (CI)~\cite{nissenbaum2004privacy,nissenbaum2009privacy} provides a practical way to study privacy and assess the ethical implications of data handling practices. CI defines privacy in terms of the appropriate and legitimate flow of information. Appropriate flow, generally, is a function of conformance with established contextual norms, which are expressible in terms of five CI parameters: three actor parameters (\verb|sender|, \verb|recipient|, \verb|information subject|), an \verb|attribute| parameter, specifying the type of information, and the \verb|transmission principle| parameter, constraining the conditions under which information flows. Being able to specify the values for all 5 parameters is imperative to evaluating the privacy implication of any practice involving information flows. CI posits that a potential privacy violation occurs when one, or more of the information flow parameters, deviates from an established norm. For example, it might be considered appropriate for a store owner (recipient) to collect VC information (attribute) from a customer (sender) before letting them into the store (transmission principle). However, if the business owner were to collect this information for advertising purposes or get the VC information from a third party, the resulting flow---with a different transmission principle and sender---would deviate from the established expectation. According to CI, a deviation such as this may be experienced as a norm violation; in turn norm violations raise a red flag, signaling the possibility of a privacy violation. Although a complete analysis of the ethical implications of privacy norm violations requires a comparative philosophical assessment of norms versus novel flows, for the studies we report on, here, our focus is on people's judgments of appropriateness (or, people's privacy expectations.)
Previous efforts~\cite{Apthorper_2018,apthorpe2019evaluating,Shvartzshnaider2016,Martin_Nissenbaum_2015,zhang2021did,ZhangSOUPS2021} aimed at discovering privacy expectations have used the CI norm structure to inform vignette studies for investigating privacy implications of particular technologies in different contexts.
These studies have also shown that varying or omitting any of the five CI parameters has a significant effect on subjects' judgments of the appropriateness of particular information flows.
Among studies using the CI framework, some have applied it to investigate users' perception of applications and data handling practices concerning COVID-19. In 2019, \citet{gerdon2021individual}, for example, conducted a CI-based longitudinal study in Germany, before the pandemic, examining people's acceptance of using individual health data during a pandemic, for public health or for private purposes. In 2020, in the wake of the pandemic, they were able to perform another such (opportunistic) study. Through the lens of CI their findings revealed that the COVID-19 pandemic altered German individuals' perspective on sharing health data with a public agency, from least acceptable before the pandemic to acceptable in the wake of the COVID-19 pandemic. Open questions remain on whether the perception will swing back after the pandemic subsides. The COVID-19 pandemic has prompted the development of ``corona apps'' for contact tracing, symptom checks, quarantine enforcement, and health certificates to help stop the spread of the virus. Using a CI-based study,~\citet{utz2021apps} examined how these applications handle health information and people's willingness to adopt them in Germany, the US, and China. They found that participants from Germany and the US perceived sharing ``corona app'' data with law enforcement agencies as inappropriate. Nevertheless, a restrictive transmission principle (e.g., limited purpose or use) increases the overall appropriateness of information flows. Additionally, compared to Germans and Americans, Chinese respondents considered sharing unique IDs with government servers and digital health certificates overall as more acceptable, highlighting the cultural differences in social norms and privacy expectations.
Our study complements the increasing body of work to examine the perceived social norms and privacy implications around pandemic mitigation technologies by focusing on VCs.
There are increasing privacy concerns about pandemic mitigation technologies re-sharing people's personal information, such as controversies related to contact tracing data being shared with law enforcement~\cite{GermanyPolice2020, AustralianPolice2021}. Building on the insights from prior studies structured by CI~\cite{Apthorper_2018,apthorpe2019evaluating,Shvartzshnaider2016,Martin_Nissenbaum_2015}, our work focuses on assessments of appropriateness that explicitly distinguish between initial information flows (i.e., when the data subject is the sender) and the subsequent re-distribution practices (when sender is a different party from subject.) Our study draws on CI to uncover the factors that are likely to affect people's attitudes and acceptance of re-sharing of information associated with VCs. Accordingly, our study draws on CI to compare reactions both to the initial information flows as well as to the subsequent re-sharing of VC information. The outcome we seek is a comprehensive understanding of people's attitudes towards the complicated information sharing practice associated with VCs.
\section{Study Methodology}
Our study explores the privacy and societal implications of information flows resulting from the use of vaccination certificates (VCs) in enforcing vaccination mandates (VMs).
We survey a demographically-stratified US sample on Prolific~\cite{Prolific} to investigate how various VC information sharing practices affect people's perceptions of norms.
\subsection{CI-Based Vignette Survey}\label{sec:method:ci}
We use a CI-based vignette survey method~\cite{Shvartzshnaider2016, apthorpe2019evaluating} to gauge the effects of contextual factors on the perceived appropriateness of information sharing practices associated with common VC usage scenarios. We generated vignettes using the five CI parameters (see Table~\ref{tab:vignettes} and Figure~\ref{fig:second-hand}), based on a review of existing VC proposals~\cite{IsraelPass,Excelsior,EUGreenCert,CommonPass} and related news articles~\cite{AustralianPolice2021,Kofler2021TenReasons,GermanyPolice2020,FranceVaccinePassport,Excelsior,EUGreenCert,Martichoux2021, CAGOV2021, BritainBarVCPassport, Kiesnoski2021, Howard2021}.
Our study included vignettes describing two types of VC information sharing practices:
(1)``first-hand'' VC information sharing, where the sender shares their own VC information, and (2) VC information re-sharing, where the sender shares someone else's VC information. These hypothetical vignettes reflect a wide range of real-world scenarios regarding the use of VCs.
\subsubsection{First-hand information sharing vignettes}\label{sec:first-hand}
Using the following template with the CI parameters in Table~\ref{tab:vignettes}, we generated 21 vignettes describing ``first-hand'' information sharing when people present their VCs, as~\textit{de facto} passports, to gain access or use services potentially on a regular basis:
\begin{framed}
\verb+[Recipient]+ ask \verb+[Sender]+ to show their
\verb+(Subject)+ vaccination certificates \verb+(Attribute)+ to \verb+[Transmission Principle]+. Would such a practice be acceptable?
\end{framed}
To avoid potential respondent fatigue~\cite{RespondentFagitue2008Lavrakas,WebSurveyLengthQuality2009Galesic} and limit survey completion time, we presented each participant with three randomly selected vignettes out of the 21.
In addition, we curated another nine ``first-hand'' \emph{VC mandate vignettes} pertaining to in-person education, employment, international travel, and apartment rental. These nine vignettes, shown at the bottom half of Table~\ref{tab:vignettes}, are based on relevant and/or debated contexts where people comply with a VM by sharing their VCs~\cite{CAGOV2021, Kelleher2021, Faircloth2021, HalfEmployerVM, NYCEmploymentVM,FloridaLandlordVM,ApartmentVMCanada,MAAssistedLivingVM}. We showed these nine VC mandate vignettes to all participants in randomized order with an attention check.
\subsubsection{VC information re-sharing vignettes}\label{sec:second-hand}
To analyze the perceptions towards possible VC information re-sharing outside the context of the original collection, we used the following question template:
\begin{framed}
Would it be acceptable for \verb+[Sender]+ to share \verb+[Subject]+ \verb+[Attribute]+ with \verb+[Recipient]+ for \verb+[Transmission Principle]+?
\end{framed}
For the sender values in the above question template, we used the recipient values from the first-hand VC information sharing vignettes, listed in Table~\ref{tab:vignettes}, alongside additional CI parameter values in Figure~\ref{fig:second-hand}. Figure~\ref{fig:fis-sis} shows an example of the two types of vignette questions presented to participants.
\begin{center}
\begin{figure*}[h]
\centering
\scalebox{0.84}{
\begin{minipage}{\textwidth}
\centering
\begin{dependency}[edge style={red,densely dotted},text only label]
\begin{deptext}
1. \&[0.1cm] {\bf Large indoor event organizers} \&[0.1cm] ask \&[0.1cm] {\bf attendees} \&[0.1cm] to show \& {\bf their} \&[0.2cm] {\bf vaccination certificates} \&[0.1cm] to \&[0.1cm] {\bf gain indoor access}.\\
\end{deptext}
\deproot[inner sep=.5ex, edge height=5ex]{2}{Recipient}
\deproot[inner sep=.5ex, edge height=5ex]{4}{Sender}
\deproot[inner sep=.5ex, edge height=5ex]{6}{Subject}
\deproot[inner sep=.5ex, edge height=5ex]{7}{Attribute}
\deproot[inner sep=.5ex, edge height=5ex]{9}{Transmission Principle}
\wordgroup[inner sep=.3ex]{1}{2}{2}{recipient}
\wordgroup[inner sep=.3ex]{1}{4}{4}{sender}
\wordgroup[inner sep=.05ex]{1}{6}{6}{subject}
\wordgroup {1}{7}{7}{attribute}
\wordgroup[inner sep=.1ex]{1}{9}{9}{TP}
\end{dependency}
\newline
\textit{Would such a practice be acceptable?}
\medskip\\
\begin{dependency}[edge style={red,densely dotted}, text only label]
\begin{deptext}
2. \&[0.1cm] Would it be acceptable for \&[0.1cm] {\bf large indoor event organizers} \&[0.1cm] to share \&[0.1cm] {\bf attendees'} \&[0.1cm] {\bf vaccination certificate information} \\
\end{deptext}
\deproot[inner sep=.5ex, edge height=5ex]{3}{Sender}
\deproot[inner sep=.5ex, edge height=5ex]{5}{Subject}
\deproot[inner sep=.5ex, edge height=5ex]{6}{Attribute}
\wordgroup[inner sep=.05ex]{1}{3}{3}{sender}
\wordgroup[inner sep=.05ex]{1}{5}{5}{subject}
\wordgroup[inner sep=.05ex]{1}{6}{6}{attribute}
\end{dependency}
\newline
\newline
\begin{dependency}[edge style={red,densely dotted}, text only label]
\begin{deptext}
\& with \&[0.1cm] {\bf health insurers} \&[0.1cm] for \&[0.1cm] {\bf public health purposes such as contact tracing}?\\
\end{deptext}
\deproot[inner sep=.5ex, edge height=5ex]{3}{Recipient}
\deproot[inner sep=.5ex, edge height=5ex]{5}{Transmission Principle}
\wordgroup {1}{3}{3}{recipient}
\wordgroup[inner sep=.1ex]{1}{5}{5}{TP}
\end{dependency}
\end{minipage}
}%
\caption{Example of first-hand sharing (top) and re-sharing (bottom) of VC information vignette questions with marked CI parameters. Note that, as per CI theory, in the re-sharing template, the sender value does not match the subject, indicating that the sender is not sharing their own information. }\label{fig:fis-sis}
\Description{This image shows two vignettes marked with respective CI parameters. The top one reads: large indoor event organizers (recipient) ask attendees (sender) to show their (subject) vaccination certificates (attribute) to gain indoor access (transmission principle). Would such a practice be acceptable? The bottom one reads: Would it be acceptable for large indoor event organizers (sender) to share attendees' (subject) vaccination certificate information (attribute) with health insurers (recipient) for public health purposes such as contact tracing (transmission principle)?}
\end{figure*}
\end{center}
\input{table1}
\input{table2}
\subsubsection{Free-text Questions}
We asked participants additional questions about their attitudes related to COVID-19 and their vaccination status, given the divided public opinion on COVID-19 vaccines and VCs in the US~\cite{Pew2021}.
To contextualize participants' responses to the vignettes, we included optional free-text questions to allow participants to explain their choices.
\subsection{Survey Deployment}
We administered our survey using Qualtrics~\cite{Qualtrics} and ran two pilot surveys with 75 participants each in June 2021 on the Prolific platform~\cite{Prolific}. We chose Prolific because prior findings show that their participants provide high-quality data and are relatively diverse~\cite{BeyondTurk2017Peer}. We used the results from the pilots only to improve the survey questions.
For our study, we used Prolific's ``representative sample'' option to recruit a demographically-stratified sample of 1,000 participants based on the age, gender, and ethnicity of the 2015 US Census data~\cite{ProlificRepresentative}.
The data collection took approximately four days to complete in July 2021, and the median time spent on the survey was 13 minutes.
Out of the 1,006 respondents recruited, we rejected six low-quality submissions and compensated the remaining participants \$2.00 for completing the survey. To further ensure data quality, we excluded results from the 110 respondents who failed one of the attention questions. In total, we analyzed valid responses from 890 participants. Their reported demographics can be found in Table~\ref{table:demographics} in the Appendix. The survey study protocol was approved by the Institutional Review Board at Carnegie Mellon.
\subsubsection{Timing of the Survey}
We conducted our survey in July 2021. At the time of the study, vaccines were widely available to all adults aged over 16, and 48.3\% of the US population was fully vaccinated (55.9\% had received at least one dose)~\cite{COVIDVaccinationRate}.
By early July 2021, the relaxed COVID-19 measures and the Delta variant had led to a resurgence of positive cases and hospitalizations.
At the time of the survey,
states across the US had adopted or were about to adopt widely diverging policies regarding VCs. States such as California, New York, Louisiana, and Hawaii started to use digital vaccination records~\cite{Excelsior, HawaiiVC}, whereas states like Florida and Georgia had passed a state-wide ban on digital vaccination records~\cite{floridaBan, georgiaBan}. The debate over the use of VCs or similar vaccination verification systems remains a timely and controversial topic in public discourse~\cite{ZhangBB2021Brookings}. This study should be viewed within this particular context
\subsection{Data Analysis}
In our study, we measured people's acceptance levels towards CI-based VC usage scenarios using the 5-point Likert scale and performed a qualitative analysis of the free texts about respondents' attitudes related to COVID and VCs.
\subsubsection{Acceptance Levels for VC Information Sharing Practices}
We first compiled and graphed participants' acceptance levels towards various VC usage scenarios, which provided an overall picture of the survey responses.
Then, we ran Wilcoxon and Mann-Whitney U tests, which do not assume normal distributions, to compare ordinal distributions means of first-hand sharing and re-sharing vignettes.
\subsubsection{Regression Analysis on Acceptance Levels}\label{sec:method:clmm} We constructed a regression model of the five essential CI parameters (i.e., sender, attribute, subject, recipient, transmission principle) to measure their effects on the perceived acceptance of the respective information flow these parameters define.
For the re-sharing vignettes, we set up a cumulative link mixed model~\cite{RPackageOrdinal} (CLMM), treating perceived acceptance levels as ordinal dependent variables. The CI parameters are independent variables, and every participant is treated as a random effect. The model was fitted with the adaptive Gauss-Hermite quadrature approximation with five quadrature points. The resulting model is well defined with a condition number of the Hessian less than $10^4$~\cite{RPackageOrdinal}. The re-sharing vignettes are more suitable for a regression analysis than first-hard vignettes because the CI parameter values in the re-sharing vignettes are relatively independent of each other.
\subsubsection{Analysis of the Free-text Responses}
For the qualitative analysis of free-text responses, we conducted a streamlined thematic analysis~\cite{braun2006using} of 6,230 responses. The first author open coded all free-text responses and discussed the coded data with two other authors. We discuss the resulting themes in Section~\ref{sec:attitudes}.
\subsection{Limitations}
Our study has several limitations.
First, similar to previous efforts in CI-based surveys~\cite{Shvartzshnaider2016, apthorpe2019evaluating, zhang2021did}, our study is limited to the information flow space defined by the CI parameter values. As we discussed in Section~\ref{sec:method:ci}, we purposefully elicited the CI parameter values from relevant news on COVID and VC deployments. These values are not comprehensive and might change as the real-world situation evolves. Future work can examine these changes. Second, our results may not be generalizable to the US population, as crowd workers recruited from Prolific may differ from the general public.
We tried to mitigate this issue by recruiting a large demographically-stratified sample based on the US census data. Our sample has a vaccination rate of 75\% compared with the national rate of 56\% at the time of the survey~\cite{COVIDVaccinationRate}, which might induce bias in our results. Also, we only surveyed US participants, which means the results may not apply to other nations, as information norms may vary across cultures. Finally, as with all survey work, we rely on participants' self-reported data, which may be prone to biases such as social desirability bias.
\section{Results}\label{sec:results}
This section details our analysis of vignettes as discussed in Section~\ref{sec:first-hand} where individuals are asked to share their VC information with a range of entities for various purposes and under different constraints.
As discussed in Section~\ref{sec:second-hand}, we also examine vignettes that describe the possible re-sharing of one's VC information by the receiving entity beyond the context of the original data collection. By varying the different contextual parameters across vignettes (see Section~\ref{sec:method:ci}), we can better understand the privacy expectations and converging norms regarding VC information sharing around the following research questions:
\begin{itemize}
\item {\bf In what contexts are VC deployments and mandates perceived appropriate?} In Sections~\ref{sec:daily} and \ref{sec:occational}, we report and compare the levels of acceptance towards VC deployment and mandate under different contexts.
\item {\bf How does the practice of re-sharing VC information affect the perceived appropriateness?} In Section~\ref{sec:sis}, we compare the levels of acceptance of first-hand VC information sharing (when the sender is also the subject of the information) to the re-sharing of VC information (when the sender shares someone else's information).
\end{itemize}
\input{04_results_01_passport}
\input{04_results_02_mandate}
\input{clmm_table2}
\subsection{Examining Scenarios on Re-sharing VC Information}\label{sec:sis}
We examined respondents' perceived appropriateness regarding VC information re-sharing practices: a situation in which a VC shown in a given context is being shared by the original recipient with a different entity for a new purpose or under a new condition. For example, when businesses share their customers' VC information with the health protection agency for public health purposes such as contact tracing. For a full list of vignettes and CI parameters, see Figure~\ref{fig:second-hand}.
Figure~\ref{fig:table2_2row} shows a heat map of average acceptance levels of vignettes describing VC information re-sharing. Overall, the practice of re-sharing and re-purposing of VC information is perceived as less appropriate compared with the first-hand VC information exchange in the original context. We found a statistically significant difference between the two types of information flows using a Wilcoxon matched-pair signed rank test ($Z=4.80, p<10^{-7}$).
\subsubsection{Regression analysis of vignettes' CI parameters}
A closer examination of CI parameters in the re-sharing vignettes reveals varied levels of perceived appropriateness.
Table~\ref{tab:clmm_tbl2} shows the results of the CLMM regression analysis (see Section~\ref{sec:method:clmm}) of factors affecting participants' perceived acceptance levels of re-sharing VC information. We found that values of three CI parameters---sender, recipient, and transmission principle---have a statistically significant effect on participants' perceived appropriateness.
\paragraph{Sender}
We used ``customs and border control agencies'' as the baseline in our regression analysis of the sender parameter as such a sender is the most accepted among all senders.
Out of all 14 senders of the vignettes, rideshare drivers/companies and residential management were perceived as the most unacceptable sender, with respective odds ratios of 1.6 (=$e^{0.6041}$) and 1.4 (=$e^{0.5122}$) compared to the baseline value. In other words, VC information sharing by rideshare companies is 1.6 times less acceptable than customs and border controls, holding constant all other variables.
\paragraph{Recipient}
Our results show that sharing VC information with public health protection agencies (which we used as the baseline) is significantly more acceptable than sharing VC information with other receiving entities. The least unacceptable recipients included advertising and marketing partners, followed by technology companies, and business partners, as indicated by the decreasing coefficients.
\paragraph{Transmission Principle}
Our results indicate that sharing VC-related information for public health purposes (the baseline) is significantly more acceptable than for other purposes or conditions.
\paragraph{Attribute}
In addition to the information in the VC itself, we looked at the meta-data associated with VC information sharing, such as the location and the timestamp. This information, when shared with other entities, could be further used to surveil or track individuals.
Our analysis, however, shows no statistically significant difference in perceptions of re-sharing the VC information or the residual meta-data (location and time) associated with the VC check.
\begin{comment}
\begin{figure*}
\centering
\begin{subfigure}[b]{0.495\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/table2_sender.pdf}
\caption[]%
{{\small Sender}}
\label{fig:ci:sender}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.495\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/table2_recipient.pdf}
\caption[]%
{{\small Recipient}}
\label{fig:ci:recipient}
\end{subfigure}
\caption[ The average and standard deviation of critical parameters ]
{\small Box plots of perceived appropriateness organized by values of different CI parameters}
\label{fig:ciparam}
\end{figure*}
\end{comment}
\begin{figure*}[htbp]
\centering
\includegraphics[width=\textwidth]{figures/table2.pdf}\hfill
\caption{A heat map of the average of all participants' responses under a combination of four CI parameters (sender, recipient, attribute, and transmission principle). For instance, the color of the top left cell represents the acceptance level of the information flow---customs and border control agencies share their customers' vaccination certificate information with the local government for public health purposes. }\label{fig:table2_2row}
\Description{The x-axis of the heat map lists the different senders and transmission principles. The y-axis of the heat map lists the various recipients and attributes. The color red indicates unacceptable, whereas the color blue indicates acceptable. Sharing VC information with public health protection agencies for public health purposes seems to be the most acceptable. }
\end{figure*}
\paragraph*{\textbf{Summary}}
Our analysis shows that contextual factors captured by the CI framework affect the degree to which participants judged a VC practice acceptable. Some combinations of sender, subject, recipient of the VC information and the condition/constraint of the transfer (transmission principle) have a statistically significant effect on the perceived acceptance of the information flow that these parameters define. This aligns with prior work that leverages CI to evaluate privacy violations in other contexts~\cite{Shvartzshnaider2016, apthorpe2019evaluating}. Notably, the \verb|subject| parameter of the information flow is particularly important, as it distinguishes re-sharing practices. Our participants found VC-related information re-sharing practices less acceptable than their providing VC directly to recipients.
\input{04_results_free_response}
\subsection{VCs as \textit{de facto} passports}\label{sec:daily}
The 21 first-hand VC information sharing vignettes reflected the scenarios in which people show their VCs, as~\textit{de facto} passports, to gain access to a service, venue, or facility.
Figure~\ref{fig:place_norms} summarizes the acceptance levels of providing VC information to 21 different CI recipients in this particular context. A majority of respondents viewed ``VC as passport" scenarios as acceptable or somewhat acceptable. For scenarios involving gaining access to assisted living facilities, cruises, and airlines, respondents expressed on average high levels of acceptance, where over 75\% of participants considered those at least somewhat acceptable. The least acceptable scenarios involve asking visitors to show their VCs to enter apartment buildings or visit worship places. Fewer than 50\% of responses indicate requiring VCs in apartment buildings as acceptable (27\%) or somewhat acceptable (21\%).
Worship places elicit a similar reaction with only 34\% and 16\% of responses suggesting requiring VCs is ``acceptable'' and ``somewhat acceptable''. Furthermore, asking to show VCs in public transportation, government buildings, shops, worship places, and apartments elicited more diverse reactions.
\paragraph{Variances in perceptions}
We analyze variances in perceptions across 21 vignettes, which is an indicator of norm formation. A low variance is a sign of a relative agreement within the scenario. Figure~\ref{fig:place_norms} shows the variances of appropriateness scores among the scenarios. We observe low variances in perceptions for scenarios in assisted living facilities, cruises, airlines, and indoor events. This is in contrast to the high overall variances in perceptions associated with hospitals, workplaces, shops, worship places, and apartments.
\subsubsection{\textbf{Essential services and basic facilities}}\label{sec:fis_necessity}
Our results showed that asking for VCs in a non-essential facility is considered significantly more appropriate than asking for VCs in an essential facility.
For example, 68\% of responses indicate it is ``acceptable'' (47\%) or ``somewhat acceptable'' (21\%) to show VCs in eateries compared with the lower 57\% (37\% ``acceptable'' and 20\% ``somewhat acceptable'') for showing VCs in stores.
Several accompanying free-text comments potentially explained the discrepancy. P213 commented on restaurants requiring VCs: \textit{``The spaces are just too small, and the ambient air is not efficiently exchanged. This is the number one place for requiring people to be vaccinated. People are voluntarily choosing to go, so should have to show a pass.''} Yet, P156 noted: \textit{``Freedom to access a source of food such as a supermarket should be effortless. Having to show vaccination certificates to enter would cause mayhem.''}
Noticeably, asking visitors to show their VCs in hospitals is significantly less acceptable than that in assisted living facilities (Mann–Whitney U $= 7924.5$, $n_1 = 117$, $n_2 = 116$, $p < 0.01$, $Cohen'd = 0.293$, two-tailed), although both are places with COVID-19 vulnerable populations.
P72 provided a possible line of reasoning:~\textit{``Even though some people may not feel comfortable getting the vaccine, they should still be granted access to hospital resources indoors. If an individual lacks certification, they should be wearing a mask.''}
The results also reveal a similar contrast between public transportation and other forms of transportation such as airlines, trains, and taxis or ride-sharing services. Only about 60\% of respondents found it ``acceptable''(39\%) or ``somewhat acceptable''(22\%) to show their VCs to use public transportation. P209's open-ended comment provides some context to the reported contrast: \textit{ ``Safety is important here too, but unlike flying, public transportation is more of a necessity and shouldn't be hindered by this.''}
In summary, the results highlight the relationship between the nature of the context---whether it is deemed essential or non-essential---and the perceived appropriateness. This suggests a need for nuanced policy making with regard to using VCs as passports.
\subsubsection{GROUP B: \textbf{Acceptance Levels for Most FIS Occasional Vignettes Are High}}~\label{sec:occational}
\subsection{Examining VC mandate vignettes}~\label{sec:occational}
The nine VC mandate vignettes reflected publicly debated scenarios in the context for which governments around the world are seeking mandates to require VCs, such as for international travel, returning to in-person learning, applying for a job, and renting an apartment, as mentioned in Section~\ref{sec:first-hand}. Figure~\ref{fig:vignette_norms} summarizes the acceptance levels for each of the nine vignettes from all participants. Overall, 74\% of participants found the selected vignettes to be ``acceptable'' (58\%) or ``somewhat acceptable'' (16\%).
\paragraph{\textbf{A VC mandate for international travel is perceived appropriate to take a flight or use at the border.}}
Our results show that requesting VCs for international travel is largely perceived as appropriate: 82\% of all participants stated that it is acceptable (68\%) or somewhat acceptable (14\%) for passengers to show VCs to take an international flight. Similarly, 85\% of respondents perceived showing VCs to customs and border control agencies as acceptable (70\%) or somewhat acceptable (15\%), both for entering the US or a foreign country.
\begin{figure*}[htbp]
\centering
\includegraphics[width=\textwidth]{figures/vignettes.pdf}\hfill
\caption{Participants' acceptance levels for nine vignettes. The top row displays the averaged response across nine vignettes. The right graph shows a box plot of the ordinal data with the mean marked in orange.}
\label{fig:vignette_norms}
\end{figure*}
\paragraph{\textbf{A VC mandate for employment: Perceived appropriate to apply for a job at assisted living facilities or hospitals.}}
81\% of respondents expressed similar levels of acceptability for sharing vaccination certificate information with employers to be considered for a job in assisted living facilities and hospitals, with 14\% stating it was somewhat acceptable and 67\% viewing it as acceptable. In comparison, when it comes to applying for a general position, only 60\% participants considered showing vaccination certificates to potential employers for a job as acceptable (37\%) or somewhat acceptable (23\%).
A Wilcoxon Signed-Rank test shows that levels of acceptance for the general case were statistically significantly lower than the levels of acceptance for the cases involving hospitals and assisted living facilities ($Z=5.42, p < 10^{-26}$).
\paragraph{\textbf{A VC mandate for education: Perceived appropriate for teachers, less so for students}}
When asked whether it is appropriate to share VC information with schools for returning to in-person learning, the acceptance levels depended on whether the sender is the students or teachers. These two vignettes involved the same CI parameters except for the sender.
Using a Wilcoxon Signed-Rank test ($Z=9.89, p<0.000001$), we noted the perceived levels of acceptability for students were statistically significantly lower than those for teachers. This means that even though the majority of our survey respondents considered the VC mandate in schools acceptable, they regarded asking students to share their VC information with the school as less acceptable than asking teachers to do so.
\paragraph{\textbf{A VC mandate in residential settings: Perceived as inappropriate overall}}
Respondents viewed showing VCs to building management to rent an apartment as the least acceptable. With only 17\% stating that it was somewhat acceptable, a slightly higher percentage of the respondents (26\%) saw it as acceptable. Such low acceptance is also consistent Section~\ref{sec:fis_necessity} where respondents considered showing VCs to visit an apartment as the least acceptable.
\subsection{Different Views on VCs: Qualitative Analysis}\label{sec:attitudes}
The open-ended comments accompanying the vignettes provide insight into the motivating factors behind the stated perceptions of appropriateness. Our thematic analysis of the free texts reveals three main attitudes.
\subsubsection{In favor of VCs}
Over half of participants (57\%) noted that VCs would make them feel safer or curb the spread of the virus.
Some (12.5\%) also mentioned VCs can show proof and prevent counterfeit CDC cards, or carrying digital VCs are easy and safe from losing them. 7.6\% of participants referred to communitarian ethics in helping protect others and their community, while others (3.8\%) saw no difference from existing practices like showing IDs to buy alcohol. 2.5\% of participants commented that VCs would serve as an incentive to get more people vaccinated.
\subsubsection{Opposing VCs}
Some (11.7\%) participants indicated they regard VCs as an invasion of their privacy or, more generally, a restriction on their personal freedom. Others (7.3\%) believed that deciding on whether to receive the vaccine should be left to their individual discretion instead of being imposed by the government or some other organization. 6.6\% of participants considered vaccination status as private medical information and thus information that should not be shared with anyone other than their doctors. 4.3\% claimed that VCs are illegal and/or unconstitutional and/or violated their HIPAA rights.
Concerned about potential harms, some (6.2\%) perceived VCs as a form of government overreach, compared the practice to identity control measures such as the ones under the Nazi regime (``Papers, please'').
A few participants (4.7\%) referred to information privacy, indicating that they were not comfortable with some information included in VCs or would not want such data retained or shared. Some (4.0\%) noted that employing VCs would result in discrimination against the unvaccinated.
\subsubsection{Context-sensitive views}
37\% of participants expressed mixed reactions and considerations dependent on the contexts of VC information sharing.
For example, 11.2\% thought that private businesses are free to require VCs at their own discretion, and 7.6\% were against requiring VCs at places to which people need access, such as public transportation and stores. 5.7\% of participants believed that other methods such as mask wearing, negative COVID tests, and occupancy limits should also be accepted if some would not want to present their VCs. 4.6\% of participants mentioned the need to accommodate people who may not be able to receive vaccines when deploying VCs. 3.9\% of participants thought that VCs are particularly controversial and could elicit strong objections and potentially violent behaviors.
\section{Discussion}
As we write this paper, VMs and VCs remain a highly contentious and politically polarizing subject. Faced with the new and highly infectious omicron variant, many governments around the world have introduced vaccination mandates or the use of vaccination certificates across a number of different contexts~\cite{IsraeliWorkplace,CanadaVaxpass,APAustrian}.
The intensity and polarization of the debate is vividly reflected in the views expressed by the participants in our study. At the one extreme, a handful of participants left profanities in the free-text responses, aimed at the authors whom they mistakenly thought were conducting research to shore up support for VC mandates and deployments. At the other extreme, a few participants left equally strong responses about people's collective responsibility to protect one another, asserting that those who refuse vaccinations are selfishly neglecting their responsibility.
Aside from the extremes, at an aggregate level, the percentage of people who find appropriate many of the VC sharing scenarios presented to them, could be taken as potential support for a fairly broad VC mandate. A closer look, however, reveals a more nuanced picture in which contextual factors had significantly affected participants' attitudes. It mattered whether the VC information is shared with the school to facilitate in-person classroom, with a grocery store owner or with a gym operator as a condition of admittance, or with a customs agent to enter a country. The recipient with whom VC data is shared, the purpose(s) for sharing, as well as guarantees (or lack thereof) about the processing of VC information all have a significant effect on people's acceptance of VC deployments. It is worth noting, too, that our study found the \verb|subject| parameter of the information flow to be important, lending credence to our initial question about first-hand use versus re-sharing practices.
When the values for all the parameters are clearly stated, our results indicate a negative sentiment towards requiring VCs for access to essential services and activities, places of worship, and apartment buildings. Further, perhaps not surprisingly, the practice of re-sharing VC information is perceived as largely inappropriate. These empirical results illustrate the importance of organizing a survey like this one by systematically sampling different contextual values, especially when it comes to understanding people's acceptance of information flows associated with different possible VC deployments and their implications.
Finally, as posited by the CI theory~\cite{nissenbaum2014respect}, newly-formed information flows that challenge established norms can affect the ultimate realization of a range of societal values such as equality, equity, and civil liberties. The assessment of the appropriateness of new flows includes: 1) a cost and benefit analysis of the information flow related to all the affected parties: Who benefits? What risks are involved? 2) a review of moral and ethical values such as fairness, autonomy, and informational harm; 3) considerations around how the new information flow contributes to fulfilling the ``context-specific values, ends and purposes''~\cite{nissenbaum2014respect}.
The qualitative analysis of the open-ended responses in Section~\ref{sec:attitudes} reveals that the ethical and societal values indeed are part of the normative assessment of the perceived appropriateness of VCs. The open-ended comments included different aspects related to the appropriateness assessment. We observed the weighing of public health interests against the expectations of freedom and privacy in various contexts. Many participants reported viewing enhanced public health as a societal benefit, while some were concerned about potential harm brought by heightened government surveillance. Some participants also expressed concerns about their bodily autonomy, the violation of personal freedom, and the intrusion of privacy on their health information, while others also warned of potential discrimination against the unvaccinated and restrictions on their rights to access essential facilities such as stores and hospitals.
\section{Conclusion}
We presented a US-based online survey study aimed at gauging people's acceptance of VCs across a diverse collection of possible deployment scenarios. This work is unique in its recognition of the fundamental privacy questions entailed by the deployment of VCs and differs from other surveys in its use of Contextual Integrity as an organizational framework to systematically explore possible deployment scenarios and contextual parameters.
Our study illustrates how Context Integrity (CI) provides an effective framework for approaching controversial societal practices, such as VC deployment. It suggests that the multifactorial insights that CI yields can inform richer and more nuanced responses to challenges confronting society in today's fight against COVID-19, and potentially other similar challenges going forward. Our study shows that contextual parameters can significantly affect people's judements about what is and isn't appropriate in the deployment of VCs.
In the context of vaccination mandates and certificates, beyond the blunt approach one often hears---that privacy must be traded off against public health---our findings open the door to more informed and nuanced alternatives that allow the pursuit of public health even as we reinforce appropriate information-flow practices that conform with the wide attitudes of ordinary people.
\section{Appendix}
\subsection{Survey}\label{apx:survey}
\begin{comment}
\subsubsection{Consent Form}
\begin{itemize}
\item I am age 18 or order.
\item I have read and understand the information above.
\item I want to participate in this research and continue with the survey.
\end{itemize}
\end{comment}
\subsubsection{Introduction}
With ongoing COVID-19 vaccination efforts, governments and other organizations around the world have proposed the use of "vaccination certificates" as a way to verify that a person has been vaccinated against the coronavirus, received a negative test or has recovered from the virus. Some vaccination certificates are already in use today. Researchers at Carnegie Mellon University are conducting a study to understand people's opinions and perceptions of these vaccination certificate proposals. Please answer the survey honestly. There are no right or wrong answers to any of the questions.
\begin{comment}
Do you agree or disagree with the following statements?
\begin{itemize}
\item Vaccination certificates will be commonplace in the next 3 to 6 months.
\item Vaccination certificates are useful for a safe return to normal.
\item Would you prefer these vaccination certificates to be $\_\_\_\_$?
\end{itemize}
\end{comment}
\subsubsection{Fist-hand Information Sharing: Vaccination Passport Vignettes}
\begin{itemize}
\item Pre-COVID, how often did you visit \verb+[place]+?
\item Assume that you have a vaccination certificate similar to the one below.
\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth]{figures/vc.png}
\caption{An example vaccination certificate shown to survey participants.}
\Description{The image displays a COVID-19 vaccination certificate with a QR code, the holder's name listed as John Doe, his date of birth listed as July 8th, 1973, and expiration date Feb. 1st, 2021. At the bottom of the certificate, it says: Please have a photo ID available when presenting this pass for validation. There is a text box to the right of the pass that says: The entity scanning this QR code can view your full name, date of birth, and your Vaccination Certificate Status (valid, invalid, expired).}
\label{fig:my_label}
\end{figure}
\item Template: \verb+[Recipient]+ ask \verb+[Sender]+ to show \texttt{[Subject +Attribute]} to \verb+[Transmission Principle]+? Would such a practice be acceptable?
\item Example: [Gyms] ask [members] to show [their vaccination certificates] to [gain indoor access]. Would such a practice be acceptable? Please explain.
\item If such a certificate were to be required to [gain indoor access], how much more likely would you be to go to [gyms] over the next 6 months? Please explain.
\end{itemize}
\subsubsection{Fist-hand Information Sharing: Vaccination Mandate Vignettes}
\begin{itemize}
\item Passengers are asked to show their vaccination certificates to airline companies to take an international flight. Is this acceptable?
\item Foreign travelers are asked to show their vaccination certificates to customs and border controls to enter the United States. Is this acceptable?
\item Us nationals are asked to show their vaccination certificates to customs and border controls to enter a foreign country. Is this acceptable?
\item Teachers are asked to show their vaccination certificates to schools (K-12 and higher education) to return to in-person learning. Is this acceptable?
\item Students are asked to show their vaccination certificates to schools (K-12 and higher education) to return to in-person learning. Is this acceptable?
\item Job applicants are asked to their show vaccination certificates to employers to be considered for a job. Is this acceptable?
\item Job applicants are asked to show their vaccination certificates to employers to apply for jobs in hospitals. Is this acceptable?
\item Job applicants are asked to show their vaccination certificates to employers to apply for or retain jobs in assisted living facilities. Is this acceptable?
\item Potential renters asked to show their vaccination certificates to building management to rent an apartment. Is this acceptable?
\item Word count: Please select the answer choice with the largest number of words in the list below.
\end{itemize}
\subsubsection{VC Information Re-sharing Vignettes}
\begin{itemize}
\item Template: Would it be acceptable for \verb+[Sender]+ to share \verb|[Subject Attribute]| with the following entities for \\\texttt{[Transmission Principle]}? \\
\ Example: Would it be acceptable for [recreational services or facilities (e.g., bars, gyms, salons)] to share [information on a person's vaccination certificate] with the following entities [for public health purposes such as contact tracking]?
\end{itemize}
\subsubsection{Vaccination Certificate Questions}
\begin{itemize}
\item Do you agree or disagree with the following statements?\\
The government (federal or state)
\begin{itemize}
\item should promote vaccination against COVID-19.
\item has no right to impose vaccination certificates.
\item should issue vaccination certificates and require them to be used in different contexts.
\end{itemize}
\item Which entity do you consider trustworthy to develop a vaccination certificate? Please select all that apply. Please explain.
\item Would you prefer to have a single certificate issued by the federal government and recognized by everyone or different certificates issued by different organizations from which you can choose?
\end{itemize}
\subsubsection{COVID Related Questions}
\begin{itemize}
\item Have you been vaccinated against COVID-19?
\item Have you contracted COVID-19?
\item Do you personally know anyone who got seriously ill due to COVID-19?
\item If vaccination certificates were to be used, would you be more or less likely to get vaccinated?
\end{itemize}
\subsubsection{Demographics}
\begin{itemize}
\item What is your age?
\item What is your gender?
\item What is the highest level of education you have completed?
\item What was your total household income before taxes during the past 12 months?
\item What is your marital status?
\item Which of the following best describes your primary occupation?
\item Please specify your ethnicity.
\item In general, would you describe your political views as \_\_\_?
\item Have you ever held a job in assisted living facilities or hospitals?
\item Do you use a smartphone?
\item In which state do you currently reside [drop-down]?
\item Which category best describes where you live?
\item Have you used the following tools in the past year? Please select all that apply.
\item Not including this survey, approximately how many surveys related to privacy or security have you completed in the past year?
\item Anything else you'd like to say about the situation and/or your concerns?
\end{itemize}
\subsection{Sample Demographics}
\begin{table}[hb]
\resizebox{\columnwidth}{!}{%
\begin{tabular}{lr|lr|lr}
\multicolumn{2}{c|}{\cellcolor{gray!20}Gender} & \multicolumn{2}{c|}{\cellcolor{gray!20}Age} & \multicolumn{2}{c}{\cellcolor{gray!20}Ethnicity} \\ \bottomrule
Female & 51.0\% & 18--27 & 19.6\% & Asian & 6.6\% \\
Male & 47.4\% & 28--37 & 18.8\% & African American & 12.5\% \\
Other & 1.2\% & 38--47 & 16.6\% & Caucasian & 71.6\% \\
Decline to answer & 0.3\% & 48--57 & 16.3\% & Hispanic & 4.7\% \\
& & 58+ & 28.8\% & Other & 3.6\%\\
& & & & Decline to answer& 1.0\% \\
\bottomrule
\end{tabular}%
}
\caption{Demographics of our study participants $N=890$}
\label{table:demographics}
\end{table}
|
train/arxiv
|
BkiUdRI5qWTD6faZ2z3l
| 5
| 1
|
\section{Introduction}
In this paper we define and study quasi $n$-absorbing elements in
multiplicative lattices. A multiplicative lattice is a complete lattice $L$
with the least element $0$ and compact greatest element $1,$ on which there
is defined a commutative, associative, completely join distributive product
for which $1$ is a multiplicative identity. Notice that $L(R)$ the set of
all ideals of a commutative ring $R$ is a special example for multiplicative
lattices which is principally generated, compactly generated and modular.
However, there are several examples of non-modular multiplicative lattices
(see \cite{fd}) Weakly prime ideals \cite{ds} were generalized to
multiplicative lattices by introducing weakly prime elements in \cite{fct}.
While 2-absorbing, weakly 2-absorbing ideals in commutative rings, and $n
-absorbing ideals were introduced in \cite{b}, \cite{bd}, and \cite{Dand},
2-absorbing and weakly 2-absorbing elements in multiplicative lattices were
studied in \cite{JTY}.
We begin by recalling some background material which will be needed. An
element $a$ of $L$ is said to be compact if whenever $a\leq
\dbigvee\limits_{\alpha \in I}a_{\alpha }$ implies $a\leq
\dbigvee\limits_{\alpha \in I_{0}}a_{\alpha }$ for some finite subset $I_{0}$
of $I$. By a $C$-lattice we mean a (not necessarily modular) multiplicative
lattice which is generated under joins by a multiplicatively closed subset
C $ of compact elements of $L$. We note that in a $C$-lattice, a finite
product of compact elements is again compact. Throughout this paper $L$ and
L_{\ast }$ denotes a multiplicative lattice and the set of compact elements
of the lattice $L$, respectively. An element $a$ of $L$ is said to be proper
if $a<1.$ A proper element $p$ of $L$ is said to be prime (resp. weakly
prime) if $ab\leq p$ (resp. $0\neq ab\leq p)$ implies either $a\leq p$ or
b\leq p.$ If $0$ is prime, then $L$ is said to be a domain. A proper element
$m$ of $L$ is said to be maximal if $m<x\leq 1$ implies $x=1.$ The jacobson
radical of a lattice $L$ is defined as $J(L)=\wedge \{m$ $:$ $m$ is a
maximal element of $L\}.$ $L$ is said to be quasi-local if it contains a
unique maximal element. If $L=\left\{ 0,1\right\} $, then $L$ is called a
field. For $a\in L,$ we define radical of $a$ as $\sqrt{a}=\wedge \{p\in L:p$
is prime and $a\leq p\}.$ Note that in a $C$-lattice $L$, $\sqrt{a}=\wedge
\{p\in L:p$ is prime and $a\leq p\}=\vee \{x\in L_{\ast }\mid x^{n}\leq a$
for some $n\in Z^{+}\}$ by (Theorem 3.6 of \cite{nk}). The nilpotent
elements of $L$ is the set of$\ Nil(L)=\sqrt{0}$. $C$-lattices can be
localized. For any prime element $p$ of $L$, $L_{p}$ denotes the
localization at $F$ $=$ $\{x\in C\mid x\nleq p\}$. For details on $C
-lattices and their localization theory, the reader is referred to \cite{jw}
and \cite{js}. An element $e\in L$ is said to be principal \cite{dil}, if it
satisfies the dual identities (i) $a\wedge be=((a:e)\wedge b)e$ and (ii)
(ae\vee b):e=(b:e)\vee a$. Elements satisfying the identity (i) are called
meet principal and elements satisfying the identity (ii) are called join
principal. Note that by \cite[Lemma 3.3 and Lemma 3.4]{dil}, a finite
product of meet (join) principal elements of $L$ is again meet (join)
principal. If every element of $L$ is principal, then $L$ is called a
principal element lattice which is studied in \cite{dd2}.
Recall from \cite{JTY} that a proper element $q$ of $L$ is called a $2
-absorbing (resp. weakly $2$-absorbing) element of $L$ if whenever $a,b,c\in
L$ with $abc\leq q$ (resp. $0\neq abc\leq q),$ then either $ab\leq q$ or
ac\leq q$ or $bc\leq q$. We say that $\ (a,b,c)$ is a triple zero element of
$q$ if $abc=0$, $ab\not\leq q$, $ac\not\leq q$ and $bc\not\leq q$. Observe
that if $q$ is a\ weakly 2-absorbing element which is not a 2-absorbing,
then there exist a triple zero of $q.$ As a generalization of 2-absorbing
elements, a proper element $q$ of $L$ is called a $n$-absorbing (resp.
weakly $n$-absorbing) element of $L$ if whenever $a_{1}a_{2}...a_{n+1}\leq q$
$($ resp. $0\neq a_{1}a_{2}...a_{n+1}\leq q)$ for some $a_{1}a_{2}...a_{n+1
\in L_{\ast }$ implies that $a_{1}a_{2}...a_{k-1}a_{k+1}...a_{n+1}\leq q$
for some $k=1,...,n+1.$
\section{Quasi $n$-absorbing and weakly quasi $n$-absorbing elements}
Let $L$ be a multiplicative lattice and $n$ be a positive integer. In this
section, we introduce quasi $n$-absorbing and weakly quasi $n$-absorbing
elements of $L$ and investigate their basic properties.
\begin{definition}
\label{d1}Let $q$ be a proper element of $L.$
\end{definition}
\begin{enumerate}
\item $q$ is said to be a quasi $n$-absorbing element of $L$ if whenever
a^{n}b\leq q$ for some $a,b\in L_{\ast }$ implies that either $a^{n}\leq q$
or $a^{n-1}b\leq q.$
\item $q$ is said to be a weakly quasi $n$-absorbing element of $L$ if
whenever $0\neq a^{n}b\leq q$ for some $a,$ $b\in L_{\ast }$ implies that
either $a^{n}\leq q$ or $a^{n-1}b\leq q$.
\end{enumerate}
We can obtain some relations among the concepts of prime, 2-absorbing, $n
-absorbing, quasi $n$-absorbing elements and weakly quasi $n$-absorbing
elements by the following:
\begin{theorem}
\label{relations}Let $q$ be a proper element of $L$ and $n\geq 1.$ Then the
following statements hold:
\end{theorem}
\begin{enumerate}
\item $q$ is a prime element of $L$ if and only if $\ q$ is a quasi $1
-absorbing element of $L.$
\item $q$ is a weakly prime element of $L$ if and only if $\ q$ is a weakly
quasi $1$-absorbing element of $L.$
\item If $q$ is $n$-absorbing, then $q$ is a quasi $n$-absorbing element of
L$.
\item If $q$ is quasi $n$-absorbing, then $q$ is a weakly quasi $n
-absorbing element of $L$.
\item If $q$ is a quasi $n$-absorbing element of $L$, then $q$ is a quasi $m
-absorbing element of $L$ for all $m\geq n.$
\item If $q$ is weakly quasi $n$-absorbing, then $q$ is a weakly quasi $m
-absorbing element of $L$ for all $m\geq n.$
\end{enumerate}
\begin{proof}
(1) and (2) are clear from definition \ref{d1}. Since (4) and (6) can be
shown very similar to (3) and (5), respectively, it is sufficient to prove
(3) and (5) only.
(3) Suppose that $q$ is a $n$-absorbing element of $L$ and assume that
a^{n}b\leq q$ for some $a,b\in L_{\ast }.$ Then $\underset{n\text{ times}}
\underbrace{a...a}b}$ $\leq q$ implies that either $a^{n}\leq q$ or
a^{n-1}b\leq q$, we are done.
(5) Suppose that $q$ is a quasi $n$-absorbing element of $L,$ and let
a,b\in L_{\ast }$ with $a^{m}b\leq q$ for some $m\geq n$. Hence
a^{n}(a^{m-n}b)\leq q$. Since $q$ is quasi $n$-absorbing, we have either
a^{n}\leq q$ or $a^{n-1}(a^{m-n}b)\leq q$. It means either $a^{m}\leq q$ or
a^{m-1}b\leq q$. This shows that $q$ is a quasi $m$-absorbing element of $L.$
\end{proof}
We have the following corollary as a direct result of Theorem \ref{relations
:
\begin{corollary}
\label{c1}Let $q$ be a proper element of $L.$
\end{corollary}
\begin{enumerate}
\item If $q$ is a prime element of $L$, then $q$ is a quasi $n$-absorbing
element of $L$ for all $n\geq 1.$
\item If $q$ is weakly prime, then $q$ is a weakly quasi $n$-absorbing
element of $L$ for all $n\geq 1.$
\item If $q$ is a 2-absorbing element of $L,$ then $q$ is a quasi $n
-absorbing element of $L$ for all $n\geq 2.$
\item If $q$ is weakly 2-absorbing$,$ then $q$ is a weakly quasi $n
-absorbing element of $L$ for all $n\geq 2.$
\end{enumerate}
\begin{proof}
(1), (2) Since a (weakly) prime element of $L$ is exactly a (weakly) quasi
1-absorbing element of $L$, this result is clear by \ref{relations} (5), (6).
(3), (4) Since a (weakly) 2-absorbing element of $L$ is a (weakly) quasi
2-absorbing element of $L$, so we are done by again (5), (6).
\end{proof}
However the converses of these relations above are not true in general.
\begin{example}
Consider the lattice of ideals of the ring of integers $L=L
\mathbb{Z}
)$. Note that the element $3
\mathbb{Z}
$ of $L$ is a quasi $2$-absorbing element, and so quasi $n$-absorbing
element for all $n\geq 2$ by Corollary \ref{c1}, but it is not a 2-absorbing
element of $L$ by Theorem 2.6 in \cite{fct}.
\end{example}
\begin{proposition}
Let $q$ be a proper element of $L$. Then the following statements are
equivalent:
\end{proposition}
\begin{enumerate}
\item $q$ is a quasi $n$-absorbing element of $L$.
\item $(q:a^{n})=(q:a^{n-1})$ where $a\in L_{\ast },$ $a^{n}\nleq q$.
\end{enumerate}
In paticular $0$ is a quasi $n$-absorbing element of $L$ if and only if$\
for each $a\in L_{\ast }$, $a^{n}=0$ or $ann(a^{n})=ann(a^{n-1}).$
\begin{proof}
It is clear from Definition \ref{d1}.
\end{proof}
Notice that if $q$ is a weakly quasi $n$-absorbing element which is not
quasi $n$-absorbing, then there are some elements $a,$ $b\in L_{\ast }$ such
that $a^{n}b=0$, $a^{n}\nleq q$ and $a^{n-1}b\nleq q$. We call the elements
(a,b)$ with this property as a quasi $n$-zero element of $q$. Notice that a
zero divisor element of $L$ is a quasi $1$-zero element of $0_{L},$ and
(a,a,b)$ is a triple zero element of $q$ if and only if $(a,b)$ is a quasi
2-zero element of $q.$
\begin{theorem}
Let $q$ be a weakly quasi $n$-absorbing element of $L$. If $(a,b)$ is a
quasi $n$-zero element of $q$ for some $a,b\in L_{\ast },$ then $a^{n}\in
ann(q)$ and $b^{n}\in ann(q).$
\end{theorem}
\begin{proof}
Suppose that $a^{n}\notin ann(q).$ Hence $a^{n}q_{1}\neq 0$ for some$\
q_{1}\in L_{\ast }$ where $q_{1}\leq q.$ It follows $0\neq a^{n}(b\vee
q_{1})\leq q$. Since $a^{n}\nleq q$, and $q$ is weakly quasi $n$-absorbing,
we conclude that $a^{n-1}(b\vee q_{1})\leq q.$ So $a^{n-1}b\leq q,$ a
contradiction. Thus $a^{n}q=0$, and so $a^{n}\in ann(q)$. Similarly we
conclude that $b^{n}\in ann(q)$.
\end{proof}
\begin{theorem}
\label{pintersect}
\end{theorem}
\begin{enumerate}
\item Let $\{p_{\lambda }\}_{\lambda \in \Lambda }$ be a family of prime
elements of $L,$ then $\underset{\lambda \in \Lambda }{\overset{}{\tbigwedge
}}p_{\lambda }$ is a quasi $m$-absorbing element for all $m\geq 2.$
\item Let $\{p_{\lambda }\}_{\lambda \in \Lambda }$ be a family of weakly
prime elements of $L,$ then $\underset{\lambda \in \Lambda }{\overset{}
\tbigwedge }}p_{\lambda }$ is a weakly quasi $m$-absorbing element for all
m\geq 2.$
\end{enumerate}
\begin{proof}
(1) To prove this argument above, we need to verify that $\underset{\lambda
\in \Lambda }{\overset{}{\tbigwedge }}p_{\lambda }$ is a quasi 2-absorbing
element of $L$ by Corollary \ref{c1} (3). Let $a,b\in L_{\ast }$ with
a^{2}b\leq \underset{\lambda \in \Lambda }{\overset{}{\tbigwedge }
p_{\lambda }.$ Since $a^{2}b\leq p_{i}$ for all $p_{i}$ prime elements, we
have $a\leq p_{i}$ or $b\leq p_{i}.$ Thus $ab\leq p_{i}$ for all $i=1,...,n$
and so $ab\leq $ $\underset{\lambda \in \Lambda }{\overset{}{\tbigwedge }
p_{\lambda }$, which completes the proof.
(2) It can be easily obtained that $\underset{\lambda \in \Lambda }{\overset
}{\tbigwedge }}p_{\lambda }$ is a weakly quasi 2-absorbing element by the
similar argument in (1). Consequently it is a weakly quasi $m$-absorbing
element$\ $or all $m\geq 2$ by Corollary \ref{c1} (4).
\end{proof}
\begin{corollary}
Let $q$ be a proper element of $L.$ Then $\sqrt{q},$ $Nil(L)$ and $Jac(L)$
are quasi $n$-absorbing elements of $L$ for all $n\geq 2.$
\end{corollary}
\begin{proof}
It is clear from Theorem \ref{pintersect}.
\end{proof}
\begin{theorem}
Let $L$ be a totally ordered lattice and $m$ a positive integer.
\end{theorem}
\begin{enumerate}
\item If $\{q_{\lambda }\}_{\lambda \in \Lambda }$ is a family of quasi $m
-absorbing elements of $L,$ then $\underset{\lambda \in \Lambda }{\overset{}
\tbigwedge }}q_{\lambda }$ is a quasi $m$-absorbing element of $L$.
\item If $\{q_{\lambda }\}_{\lambda \in \Lambda }$ is a family of weakly
quasi $m$-absorbing elements of $L,$ then $\underset{\lambda \in \Lambda }
\overset{}{\tbigwedge }}q_{\lambda }$ is a weakly quasi $m$-absorbing
element of $L.$
\end{enumerate}
\begin{proof}
(1) Assume that $\{q_{\lambda }\}_{\lambda \in \Lambda }$ is an ascending
chain and $a^{m}\nleq \underset{\lambda \in \Lambda }{\overset{}{\tbigwedge
}q_{\lambda }$and $a^{m-1}b\nleq \underset{\lambda \in \Lambda }{\overset{}
\tbigwedge }}q_{\lambda }$. We show that $a^{m}b\nleq \underset{\lambda \in
\Lambda }{\overset{}{\tbigwedge }}q_{\lambda }$. Hence $a^{m}\nleq q_{j}$
and $a^{m-1}b\nleq q_{k}$ for some $j,k=1,...,n.$ Put $t=\min \{j,k\}$. Then
$a^{m}\nleq q_{t}$ and $a^{m-1}b\nleq q_{t}$. Since $q_{t}$ is a quasi $m
-absorbing element, it follows $a^{m}b\nleq q_{t}$. Thus $a^{m}b\nleq
\underset{\lambda \in \Lambda }{\overset{}{\tbigwedge }}q_{\lambda }$, we
are done.
(2) It can be easily shown similar to (1).
\end{proof}
\begin{enumerate}
\item If $q_{1},$...$,q_{n}$ are quasi $m_{i}$-absorbing elements of $L$ for
all $i=1,2,...,n$, then $\underset{i=1}{\overset{n}{\tbigwedge }}q_{i}$ is a
quasi $m$-absorbing element of $L$ where $m=\max \{m_{1},...,m_{n}\}+1.$
\item If $q_{1},$...$,q_{n}$ are weakly quasi $m_{i}$-absorbing elements of
L$ for all $i=1,2,...,n,$ then $\underset{i=1}{\overset{n}{\tbigwedge }
q_{i} $ is a weakly quasi $m$-absorbing element of $L$ where $m=\max
\{m_{1},...,m_{n}\}+1.$
\end{enumerate}
\begin{proof}
(1) Let $a,b\in L_{\ast }$ such that $a^{m}b\leq $ $\underset{i=1}{\overset{
}{\tbigwedge }}q_{i}.$ Hence $a^{m_{i}}\leq q_{i}$ or $a^{m_{i}-1}b\leq
q_{i} $ for all $i=1,..,n.$ Now assume that $a^{m}\not\leq \underset{i=1}
\overset{n}{\tbigwedge }}q_{i}.$ Without loss generality we can suppose that
$a^{m_{i}}\leq q_{i}$ for all $1\leq i\leq j,$ and $a^{m_{i}}\not\leq q_{i}$
for all $j+1\leq i\leq n.$ Hence we have $a^{m_{i}-1}b\leq q_{i}$ for all
j+1\leq i\leq n.$ Then we get clearly $a^{m-1}b\leq q_{i}$ for $m=\max
\{m_{1},...,m_{n}\}+1$ and for all $1\leq i\leq n.$ Thus $a^{m-1}b\leq
\underset{i=1}{\overset{n}{\tbigwedge }}q_{i}$, so we are done.
(2) The proof is obtained similar to (1).
\end{proof}
If $x\in L,$ the interval $[x,1]$ is denoted $L/x.$ The elemets of
\overline{a}$ and $L/x$ is again a multiplicative lattice with $\overline{a
\circ \overline{b}=ab\vee x$ for all $\overline{a},\overline{b}\in L/x.$
\begin{theorem}
Let $x$ and $q$ be proper elements of $L$ with $x\leq q.$
\end{theorem}
\begin{enumerate}
\item If $q$ is a quasi $n$-absorbing element of $L,$ then $\overline{q}$ is
a quasi $n$-absorbing element of $L/x.$
\item If $q$ is a weakly quasi $n$-absorbing element of $L,$ then $\overline
q}$ is a weakly quasi $n$-absorbing element of $L/x.$
\end{enumerate}
\begin{proof}
(1) Suppose that $\overline{a}=a\vee x,$ $\overline{b}=b\vee x\in L$ with
\overline{a}^{n}\overline{b}\leq \overline{q}.$ Then $a^{n}b\vee x\leq q$,
and so $a^{n}b\leq q.$ Since $q$ is quasi 2-absorbing, we get either
a^{n}\leq q$ or $a^{n-1}b\leq q$. Thus $\overline{a}^{n}=(a\vee x)^{n}\leq
\overline{q}$ or $\overline{a}^{n-1}\overline{b}=\left( a\vee x\right)
^{n-1}(b\vee x)\leq \overline{q}$, as needed.
(2) Let $\overline{a}=a\vee x,$ $\overline{b}=b\vee x\in L$ with $x\neq
\overline{a}^{n}\overline{b}\leq \overline{q}.$ Then $x\neq a^{n}b\vee x\leq
q$, and so $0\neq a^{n}b\leq q.$ Since $q$ is weakly quasi 2-absorbing, we
conclude that either $a^{n}\leq q$ or $a^{n-1}b\leq q$. Consequently,
\overline{a}^{n}=(a\vee x)^{n}\leq \overline{q}$ or $\overline{a}^{n-1
\overline{b}=\left( a\vee x\right) ^{n-1}(b\vee x)\leq \overline{q}$, we are
done.
\end{proof}
Recall that any $C$-lattice can be localized at a multiplicatively closed
set. Let $L$ be a $C$-lattice and $S$ a multiplicatively closed subset of
L_{\ast }$.Then for $a\in L,$ $a_{S}=\vee \{x\in L_{\ast }:$ $xs\leq a$ for
some $s\in S\}$ and $L_{S}=\{a_{S}:a\in L\}.$ $L_{S}$ is again a
multiplicative lattice under the same order as $L$ with the product
a_{S}\circ b_{s}=(a_{S}b_{S})_{S}$ where the right hand side is evaluated in
$L$. If $p\in L$ is prime and $S=\{x\in L_{\ast }:x\nleq p\}$, then $L_{S}$
is denoted by $L_{p}.$ \cite{jw}
\begin{theorem}
Let $m$ be a maximal element of $L$ and $q$ be a proper element of $L.$
\end{theorem}
\begin{enumerate}
\item If $q$ is a quasi $n$-absorbing element of $L,$ then $q_{m}$ is a
quasi $n$-absorbing element of $L_{m}.$
\item If $q$ is a weakly quasi $n$-absorbing element of $L,$ then $q_{m}$ is
a weakly quasi $n$-absorbing element of $L_{m}.$
\end{enumerate}
\begin{proof}
(1) Let $a,b\in L_{\ast }$ such that $a_{m}^{n}b_{m}\leq q_{m}$. Hence
ua^{n}b\leq q$ for some $u\nleq m.$ It implies that $a^{n}\leq q$ or
a^{n-1}(ub)\leq q.$ Since $u_{m}=1_{m},$ we get $a_{m}^{n}\leq q_{m}$ or
a_{m}^{n-1}b_{m}\leq q_{m}$, we are done.
(2) It can be verified by the similar argument in (1).
\end{proof}
\begin{theorem}
\label{t1}Let$\ L$ be a principal element lattice. Then the following
statements are equivalent:
\end{theorem}
\begin{enumerate}
\item Every proper element of $L$ is a quasi $n$-absorbing element of $L.$
\item For every $a,$ $b\in L_{\ast }$, $a^{n}=ca^{n}b$ or $a^{n-1}b=da^{n}b$
for some $c,$ $d\in L$.
\item For all $a_{1},a_{2},...,a_{n+1}\in L_{\ast },$ $(a_{1}\wedge
a_{2}\wedge ...\wedge a_{n})^{n}\leq ca_{1}a_{2}...a_{n+1}$ or $(a_{1}\wedge
a_{2}\wedge ...\wedge a_{n})^{n-1}a_{n+1}\leq da_{1}a_{2}...a_{n+1}$ for
some $c,$ $d\in L$.
\end{enumerate}
\begin{proof}
(1)$\Leftrightarrow $(2) Suppose that every proper element of $L$ is a quasi
$n$-absorbing element of $L.$ Hence $a^{n}b\leq (a^{n}b)$ implies that
a^{n}\leq (a^{n}b)$ or $a^{n-1}b\leq (a^{n}b)$. Since $L$ is a principal
element lattice, there is some element $c\in L$ with $a^{n}=ca^{n}b$ or
there is some element $d\in L$ with $a^{n-1}b=da^{n}b$. The converse is
clear.
(2)$\Rightarrow $(3) Put $a=a_{1}\wedge a_{2}\wedge ...\wedge a_{n}$ and
b=a_{n+1}.$ Hence the result follows from (2).
(3)$\Rightarrow $(2) For all $a,b\in L_{\ast },$ we can write $a^{n}
\underset{n\text{ times}}{(\underbrace{a\wedge a\wedge ...\wedge a})}$ $\leq
ca^{n}b$ or $a^{n-1}b=\underset{n-1\text{ times}}{(\underbrace{a\wedge
a\wedge ...\wedge a})b}$ $\leq da^{n}b.$
\end{proof}
\begin{theorem}
Let $L=L_{1}\times L_{2}$ where $L_{1}$ and $L_{2}$ are $C$-lattices. Then
the following statements hold:
\end{theorem}
\begin{enumerate}
\item $q_{1}$ is a quasi $n$-absorbing element of $L_{1}$ if and only if
(q_{1},1_{L_{2}})$ is a quasi $n$-absorbing element of $L.$
\item $q_{2}$ is a quasi $n$-absorbing element of $L_{2}$ if and only if
(1_{L_{1}},q_{2})$ is a quasi $n$-absorbing element of $L.$
\end{enumerate}
\begin{proof}
(1) Suppose that $q_{1}$ is a quasi $n$-absorbing element of $L_{1}.$ Let
(a_{1},a_{2})^{n}(b_{1},b_{2})\leq (q_{1},1_{L_{2}})$ for some $a_{1},$
b_{1}\in L_{1_{\ast }}$ and $a_{2},$ $b_{2}\in L_{2_{\ast }}$.$\ $Then
a_{1}^{n}b_{1}\leq q_{1}$ implies that either $a_{1}^{n}\leq q_{1}$ or
a_{1}^{n-1}b_{1}\leq q_{1}.$ It follows either $(a_{1},a_{2})^{n}\leq
(q_{1},1_{L_{2}})$ or $(a_{1},a_{2})^{n-1}(b_{1},b_{2})\leq
(q_{1},1_{L_{2}}).$ Thus $(q_{1},1_{L_{2}})$ is a quasi $n$-absorbing
element of $L.$ Conversely suppose that $(q_{1},1_{L_{2}})$ is a quasi $n
-absorbing element of $L$ and $a^{n}b\leq q_{1}$ for some $a,b\in L_{1_{\ast
}}$. Hence $(a,1_{L_{2}})^{n}(b,1_{L_{2}})\leq (q_{1},1_{L_{2}})$ which
implies that either $(a,1_{L_{2}})^{n}\leq (q_{1},1_{L_{2}})$ or
(a,1_{L_{2}})^{n-1}(b,1_{L_{2}})\leq (q_{1},1_{L_{2}}).$ So $a_{1}^{n}\leq
q_{1}$ or $a_{1}^{n-1}b_{1}\leq q_{1}$, as needed.
(2) It can be verified easily similar to (1).
\end{proof}
\begin{theorem}
\label{tcartes}Let $L=L_{1}\times ...\times L_{k}$ where $L_{i}^{{}}$ s are
C$-lattices for all $i=1,...,k$. If $q_{i}$ is a quasi $n_{i}$-absorbing
element of $L_{i}$ for all $i=1,...,k,$ then $(q_{1},...,q_{k})$ is a quasi
m$-absorbing element of $L$ where $m=\max \{n_{1},$...,$n_{k}\}+1.$
\end{theorem}
\begin{proof}
Suppose that $(a_{1},...,a_{k})^{m}(b_{1},...,b_{k})\leq (q_{1},...,q_{k})$
for $(a_{1},...,a_{k}),(b_{1},...,b_{k})\in L_{\ast }$ and $m=\max \{n_{1},
...,$n_{k}\}+1$. Hence $a_{i}^{m}b_{i}=a_{i}^{n_{i}}(a_{i}^{m-n_{i}}b_{i}
\leq q_{i}$ for all $i=1,..,k$. Since each $q_{i}$ is a quasi $n_{i}
-absorbing element, we have either $a_{i}^{n_{i}}\leq q_{i}$ or
a_{i}^{m-1}b_{i}=a_{i}^{n_{i}-1}(a_{i}^{m-n_{i}}b_{i})\leq q_{i}$ for all
i=1,..,k$. If $a_{i}^{n_{i}}\leq q_{i}$ for all $i=1,...,k,$ then
(a_{1},...,a_{k})^{m}\leq (q_{1},...,q_{k}).$ Without loss generality,
suppose that $a_{i}^{n_{i}}\leq q_{i}$ for all $1\leq i\leq j$ and
a_{i}^{m-1}b_{i}\leq q_{i}$ for all $j+1\leq i\leq k,$ for some $j=1,..k.$
Thus $(a_{1},...,a_{k})^{m-1}(b_{1},...,b_{k})\leq (q_{1},...,q_{k}),$ so we
are done.
\end{proof}
\begin{definition}
\label{d2}A proper element $q$ of $L$ is said to be a strongly quasi $n
-absorbing element of $L$ if whenever $a,b\in L$ (not necessarily compact)
with $a^{n}b\leq q$ implies that either $a^{n}\leq q$ or $a^{n-1}b\leq q$.
\end{definition}
It is clearly seen that every strongly quasi $n$-absorbing element of $L$ is
quasi $n$-absorbing.
\begin{theorem}
Let$\ L$ be a principal element lattice. The following statements are
equivalent:
\end{theorem}
\begin{enumerate}
\item Every proper element of $L$ is a strongly quasi $n$-absorbing element
of $L.$
\item For all $a,b\in L$, $a^{n}=a^{n}b$ or $a^{n-1}b=a^{n}b$,
\item $(a_{1}\wedge a_{2}\wedge ...\wedge a_{n})^{n}\leq
a_{1}a_{2}...a_{n+1} $ or $(a_{1}\wedge a_{2}\wedge ...\wedge
a_{n})^{n-1}a_{n+1}\leq a_{1}a_{2}...a_{n+1}$ for all $a_{1},$ $a_{2},$
...,a_{n+1}\in L.$
\end{enumerate}
\begin{proof}
This can be easily shown using the similar argument in Theorem \ref{t1}.
\end{proof}
\begin{theorem}
Let $q$ be a proper element of $L.$ Then the following statements hold:
\end{theorem}
\begin{enumerate}
\item If $a^{n}b\leq q\leq a\wedge b$ where $a,b\in L$ implies that
a^{n}\leq q$ or $a^{n-1}b\leq q,$ then $q$ is a strongly quasi $n$-absorbing
element of $L.$
\item If $a_{1}a_{2}...a_{n+1}\leq q\leq a_{1}\wedge a_{2}\wedge ...\wedge
a_{n+1}$ where $a_{1},a_{2},...,a_{n+1}\in L$ implies that
a_{1}...a_{i-1}a_{i+1}...a_{n+1}\leq q,$ for some $1\leq i\leq n+1$ then $q$
is a strongly quasi $n$-absorbing element of $L.$
\end{enumerate}
\begin{proof}
(1) Let $x,y\in L$ with $x^{n}y\leq q$. We show that $x^{n}\leq q$ or
x^{n-1}y\leq q.$ Now put $a=x\vee q$ and $b=y\vee q.$ Hence we conclude
a^{n}b\leq q\leq a\wedge b,$ and so $a^{n}\leq q$ or $a^{n-1}b\leq q$ by
(1). It follows $x^{n}\leq q$ or $x^{n-1}y\leq q.$
(2) It can be easily verified similar to (1).
\end{proof}
|
train/arxiv
|
BkiUfbE25V5jZT5RBB1T
| 5
| 1
|
\section{Introduction}\label{sec:intro}
Block encoding \cite{low2017optimal} is a widely used technique in quantum computing and a crucial component of many quantum algorithms with a potentially exponential advantage over classical algorithms, such as quantum phase estimation (QPE) \cites{kitaev2002classical, nielsen2001quantum}, the HHL algorithm \cite{harrow2009quantum}, quantum singular value transformation (QSVT) \cite{gilyen2019quantum} and various quantum linear system solvers \cites{lin2020optimal, an2022quantum, CostaPRXQ22}, to name a few. The idea of block encoding is to embed a linear operator $A$ into a unitary operator $U_A$ with larger dimensions after appropriate scaling. The unitary $U_A$ is then converted into a quantum circuit, allowing a quantum computer to access $U_A$ for actual computations.
The potential advantage of quantum algorithms depends critically on efficient and practical quantum circuits for block-encoding of the operators involved, and the construction of such circuits can be non-trivial in general. Researchers have constructed block encoding schemes leveraging different structures of the operators studied.
For example, a block encoding scheme is provided in \cites{camps2022explicit, gilyen2019quantum} for sparse matrices, and a recipe is presented for hierarchical matrices in \cite{nguyen2022quantum}.
In this paper, we consider the problem of block encoding a large family of dense operators: the pseudo-differential operators (PDOs). PDO is a rich family of linear operators that include many commonly used examples in scientific problems, which is typically given in the following form:
\begin{equation}\label{eq:pdo}
A f({\bm{x}}) =\int_{\mathbb{R}^d} e^{2 \pi i {\bm{x}} \cdot {\bm{\xi}}} a({\bm{x}}, {\bm{\xi}}) \widehat{f}({\bm{\xi}}) d {\bm{\xi}},
\end{equation}
where $a({\bm{x}}, {\bm{\xi}})\in C^\infty(\mathbb{R}^d\times\mathbb{R}^d)$ is called the symbol of $A$ and $\widehat{f}$ is the Fourier transform of $f$. A major motivation for studying operators with the form \eqref{eq:pdo} is that differential operators often enjoy a simple representation in the Fourier domain. For example, the elliptic operator:
\begin{equation}\label{eq:ellip}
A = I - \nabla\cdot(\omega({\bm{x}})\nabla)
\end{equation}
with $\omega({\bm{x}})>0$ can be represented in the form of \eqref{eq:pdo} with the symbol
\begin{equation}\label{eq:ellipsym}
a({\bm{x}}, {\bm{\xi}}) = 1-2\pi i\nabla\omega({\bm{x}})\cdot{\bm{\xi}}+4\pi^2\omega({\bm{x}})|{\bm{\xi}}|^2.
\end{equation}
More generally, an $m$-th order linear partial differential operator $P({\bm{x}}, D) = \sum_{|\alpha|\leq m} a_\alpha({\bm{x}})D^\alpha$ with $D = -\frac{i}{2\pi}\nabla_{\bm{x}}$ can be represented by
\[
Pf({\bm{x}}) = \int_{\mathbb{R}^d} e^{2 \pi i {\bm{x}} \cdot {\bm{\xi}}} \sum_{|\alpha|\leq m} a_\alpha({\bm{x}}) {\bm{\xi}}^\alpha \widehat{f}({\bm{\xi}}) d {\bm{\xi}} = \int_{\mathbb{R}^d} e^{2 \pi i {\bm{x}} \cdot {\bm{\xi}}} P({\bm{x}}, {\bm{\xi}}) \widehat{f}({\bm{\xi}}) d {\bm{\xi}},
\]
where $\alpha=(\alpha_1, \ldots, \alpha_d)$ is the $d$-dimensional multi-index and $|\alpha|=\sum_{j=1}^d\alpha_j$.
Another popular example is the translation-invariant operator. If an operator $A$ is translation-invariant, i.e., $A(e^{2\pi i{\bm{x}}\cdot{\bm{\xi}}})=a({\bm{\xi}})e^{2\pi i{\bm{x}}\cdot{\bm{\xi}}}$, then
\begin{equation}\label{eq:multiplier}
Af({\bm{x}}) = \int_{\mathbb{R}^d} e^{2 \pi i {\bm{x}} \cdot {\bm{\xi}}}a({\bm{\xi}})\widehat{f}({\bm{\xi}}) d {\bm{\xi}},
\end{equation}
in which case we say that the symbol $a({\bm{\xi}})$ is a multiplier. Apart from the examples mentioned above, the PDO family also contains other operators such as convolution operators, singular integral operators, etc. Moreover, a space of PDOs is often closed with respect to many elementary operations under certain conditions. For example, for the operator $A$ in \eqref{eq:multiplier} with symbol $a({\bm{\xi}})\not=0$, the inverse of $A$ can be simply represented by
\[
A^{-1}f({\bm{x}})= \int_{\mathbb{R}^d} e^{2 \pi i {\bm{x}} \cdot {\bm{\xi}}} \frac{1}{a({\bm{\xi}})}\widehat{f}({\bm{\xi}}) d {\bm{\xi}}.
\]
In general, the operator defined by $C^\infty$ function $a({\bm{x}},{\bm{\xi}})$ as in \eqref{eq:pdo} is called a pseudo-differential operator only if $a({\bm{x}},{\bm{\xi}})$ satisfies some additional requirements such as
\[
|\partial_{\bm{x}}^\alpha\partial_{\bm{\xi}}^\beta a({\bm{x}}, {\bm{\xi}})|\leq C_{\alpha\beta}\langle{\bm{\xi}}\rangle^{m-\alpha-\beta},
\]
where $\langle{\bm{\xi}}\rangle:=\sqrt{1+|{\bm{\xi}}|^2}$, and the space of the corresponding PDOs is denoted by $S^m$. There are multiple monographs on PDOs that interested readers can refer to such as \cites{stein1993harmonic, wong1991anitro}.
The PDOs considered in this paper are equipped with a periodic boundary condition on the space domain $\Omega = [0, 1]^d$.
The frequency variable ${\bm{\xi}}$ thus takes the value on the integer grid and the operator $A$ becomes
\begin{equation}\label{eq:pdoZZ}
A f({\bm{x}}) = \sum_{{\bm{\xi}} \in \mathbb{Z}^d} e^{2 \pi i {\bm{x}} \cdot {\bm{\xi}}} a({\bm{x}},{\bm{\xi}}) \hat{f}({\bm{\xi}}),
\end{equation}
where $\hat{f}$ is the coefficient of the Fourier series of $f$. In this paper, we derive block encoding schemes for the PDO \eqref{eq:pdoZZ} based on different additional structures of the symbol $a({\bm{x}}, {\bm{\xi}})$.
First, we present a block encoding scheme for generic symbols $a({\bm{x}}, {\bm{\xi}})$ without additional structures. We then point out that the success probability of the quantum circuit can be significantly improved if the symbol $a({\bm{x}}, {\bm{\xi}})$ can be expanded into series:
\begin{equation}
a({\bm{x}}, {\bm{\xi}}) = \sum_j \alpha_j({\bm{x}})\beta_j({\bm{\xi}}),
\end{equation}
with only $\mathcal{O}(1)$ terms. Furthermore, the circuit can be constructed in a much more explicit way with the help of quantum signal processing (QSP) and quantum eigenvalue transformation (QET) if the symbol is a sum of \emph{fully separable} terms, i.e., it can be expanded as
\begin{equation}\label{eq:fullysepex}
a({\bm{x}}, {\bm{\xi}}) = \sum_j \alpha_{j1}(x_1)\cdots\alpha_{jd}(x_d)\beta_{j1}(\xi_1)\cdots\beta_{jd}(\xi_d),
\end{equation}
with $\mathcal{O}(1)$ terms, where ${\bm{x}}=(x_1,\ldots,x_d)$ and ${\bm{\xi}}=(\xi_1,\ldots,\xi_d)$. See \Cref{def:fullysep} and \Cref{sec:fully sep} for details. Complexity analysis is included for all block encoding schemes and their applications are showcased with specific examples. The contributions of this paper can be summarized as follows:
\begin{itemize}
\item We provide practical block encoding schemes for pseudo-differential operators, including algorithms applicable to generic PDOs (see \Cref{fig:PDO1}), efficient block encoding for separable PDOs (see \Cref{fig:PDO2}) and explicit circuits for fully separable PDOs (see \Cref{fig:PDO3}). Novel ideas of circuit design such as the phase multiplication circuit (see \Cref{fig:lemma2}) and the prototype for diagonal multiplication (see \Cref{fig:D}) are included in the block encoding schemes.
\item We conduct comprehensive complexity analysis for the block encoding schemes proposed. The result for complexity analysis includes the success probability, the number of ancilla qubits needed and the number of gates used. In addition to theorems applicable to general cases, we also demonstrate possibilities of improving the complexity results by leveraging particular structures of the problem (see \Cref{subsec:varellip} for example).
\item We demonstrate the usage of our results with explicit examples. One can use our block-encoding scheme not only as an integrated part of established quantum algorithms but also as an option for conducting operations directly on certain operators. For the example shown in \Cref{subsec:appinverse}, we use the idea of symbol calculus to directly block-encode the inverse of an elliptic operator, and the dependence of the complexity on $P$ (the number of discretization points used for each dimension) is at least quadratically improved compared to previous results for block-encoding the inverse matrix (see \Cref{rem:inv}).
\end{itemize}
\subsection{Contents}
The paper is organized as follows. In \Cref{sec:prelim}, we specify the notation used in this paper and provide preliminary results needed in subsequent sections such as quantum Fourier transform (QFT), the linear combination of unitaries (LCU), quantum signal processing (QSP) and quantum eigenvalue transformation (QET). In \Cref{sec:arithmD}, an algorithm is given for block encoding of generic symbols. For a separable symbol $a({\bm{x}},{\bm{\xi}})= \alpha({\bm{x}})\beta({\bm{\xi}})$, a more efficient block encoding scheme is provided in \Cref{sec:separable}. Then a more explicit and practically feasible block encoding is developed in \Cref{sec:fully sep} for fully separable symbols of the form displayed in \eqref{eq:fullysepex}. Finally, in \Cref{sec:app} we present the application of the proposed block encoding method to two types of widely used PDOs including a variable coefficient second-order elliptic operator and the inverse of a constant coefficient elliptic operator. The paper is ended with a conclusion and discussion for future directions in \Cref{sec:con}.
\subsection{Related works}
\subsubsection{Block encoding}
Most of the previous work \cites{childs2017quantum, berry2015simulating, low2017optimal} assumes that we have access to a matrix by querying two oracles that encode the locations and values of the non-zero elements of the objective matrix. Among them, \cite{gilyen2019quantum}*{Lemma 48} provides a general framework to explicitly construct the block encoding of sparse matrices if we are given these two oracles. Following this routine, \cite{camps2022explicit} constructs the block encoding of banded circulant matrices, extended binary tree matrices, and quantum walk operators.
For general non-sparse matrices, it is clearly impossible to block-encode them in logarithmic time, and \cite{camps2022fable} proposes a near-optimal scheme for block encoding general unstructured matrices. Many methods are also proposed to implement the block encoding for full-rank dense matrices with certain structures, such as Toeplitz and Hankel systems \cite{mahasinghe2016efficient}, and linear group convolutions \cite{castelazo2022quantum} based on quantum Fourier transforms. The authors of \cite{nguyen2022quantum} introduce a new method for kernel matrices with a hierarchical structure, which can be applied to non-uniform grids the Fourier transform cannot be used.
\subsubsection{Quantum PDE solvers}
Along with the development of quantum linear system solvers \cites{harrow2009quantum, childs2017quantum, gilyen2019quantum, lin2020optimal, costa2022optimal}, many quantum PDE solvers are proposed to take advantage of exponential acceleration. Quantum counterparts of finite element method \cite{montanaro2016quantum} and finite difference method \cites{cao2013quantum, costa2019quantum} emerged for solving Poisson's equation and wave equation. In \cite{childs2021high}, adaptive FDM and spectral methods are proposed to improve the dependence of the complexity on the error $\epsilon$ from $\mathcal{O}(\poly(1 / \epsilon))$ to $\mathcal{O}(\polylog (1 / \epsilon))$. It is worth noting that the process of block encoding the discretized differential operator is in general not provided in these works, and constructing the block encoding for generic partial differential operators is highly non-trivial.
\subsubsection{Numerical algorithms for PDOs}
There are also various classical numerical algorithms that compute PDOs efficiently. For example, \cite{demanet2011discrete} exploits the following expansion of symbols:
\[
a({\bm{x}},{\bm{\xi}}) \approx \sum_{j} \alpha_j({\bm{x}}) \beta_j({\bm{\xi}}).
\]
The paper presents efficient numerical approximations of $\beta_j({\bm{\xi}})$ with Chebyshev polynomials and hierarchical splines and further reduces the number of terms in the expansion by SVD or QR decomposition. However, a naive extension to PDOs in high dimensions leads to exponential overhead, as is the case for most classical methods. This is also one of the reasons why a quantum implementation of PDOs can be potentially useful.
\section{Preliminaries and notations}\label{sec:prelim}
\subsection{Notations}\label{subsec:note}
We adopt the commonly used notation for binary numbers: for an integer power of two $P=2^p$ and any $y\in\{0,\ldots, P-1\}$, if $y = y_0 + 2y_1 + \cdots+2^{p-1}y_{p-1} = (y_{p-1}y_{p-2}\cdots y_0.)$ in the binary system, the corresponding quantum state is $\ket{y} \equiv \ket{y_{p-1}\ldots y_0}$.
This extends to an $m$-tuple ${\bm{x}} = (x_1, \ldots, x_m)$ with $x_j\in\{0,\ldots,P-1\}$. The corresponding quantum state is given by $\ket{x_m}\cdots\ket{x_1}$, where $\ket{x_j} = \ket{x_{j,p-1} \cdots x_{j,0}}$ for each $j$. For a multivariate function $g:\{0,\ldots, P-1\}^m\rightarrow\mathbb{R}$, we denote by $D_{g}$ the diagonal multiplication operator on the Hilbert space $\mathbb{C}^{mp}$:
\begin{equation}\label{eq:diagg}
\ket{x_{m}}\cdots\ket{x_{1}} \rightarrow g(x_{1}, \ldots, x_{m})\ket{x_{m}}\cdots\ket{x_{1}}.
\end{equation}
For a vector $\bm{v} = (v_1, v_2, \ldots, v_m)$, we denote by $\mathrm{diag}~(\bm{v})$ the diagonal matrix with diagonal elements $(v_1,v_2, \ldots, v_m)$.
The notation $|\bm{v}|$ for a $d$-dimensional vector $\bm{v}$ stands for the Euclidean norm $\sqrt{\sum_{j=1}^d|v_j|^2}$, where $v_j$ is the $j$-th coordinate of $\bm{v}$.
We also use the single qubit rotations $R_y(\theta)=e^{-i\frac{\theta}{2} Y} = \begin{bmatrix}\cos\frac{\theta}{2}&-\sin(\frac{\theta}{2})\\\sin\frac{\theta}{2}&\cos\frac{\theta}{2}\end{bmatrix}$ and $R_z(\theta)=e^{i\frac{\theta}{2}}e^{-i\frac{\theta}{2} Z} = \begin{bmatrix}1&\\&e^{i\theta}\end{bmatrix}$, where $Y$ and $Z$ are the Pauli matrices $Y=\begin{bmatrix}0&-i\\i&0\end{bmatrix}$ and $Z = \begin{bmatrix}1&\\&-1\end{bmatrix}$. The phase gate $S=\sqrt{Z}$ denotes the matrix $S = \begin{bmatrix}1&\\&i\end{bmatrix}$.
An $(m+n)$-qubit unitary operator $U$ is called a $(\gamma, m, \epsilon)$-block-encoding of an $n$-qubit operator $A$, if
$$
\left\|A-\gamma\left(\left\langle 0^m\right| \otimes I_n\right) U\left(\left|0^m\right\rangle \otimes I_n\right)\right\| \leq \epsilon,
$$
where $I_n$ denotes the $n$-qubit identity operator.
In the matrix form, a $(\gamma, m, \epsilon)$-block-encoding is a $2^{m+n}$ dimensional unitary matrix
$$
U=\left(\begin{array}{cc}
\widetilde{A} / \gamma & * \\
* & *
\end{array}\right)
$$
where $*$ can be any block matrices of the correct size and $\|\widetilde{A}-A\| \leq \epsilon$. In addition, when $A$ is a Hermitian matrix, it is possible to construct $U_A$ such that it is also Hermitian, in which case it is called a $(\gamma, m, \epsilon)$-Hermitian-block-encoding of $A$. The error $\epsilon$ is omitted in the notation of block encodings if $\epsilon=0$.
For an $n$-qubit system, the quantum Fourier transform (QFT) is a circuit $U_{\text{FT}}$ with $\mathcal{O}(n^2)$ elementary gates and no ancilla qubit that implements the transformation:
\begin{equation}\label{eq:QFT}
U_{\text{FT}}\ket{j} = \frac{1}{\sqrt{N}}\sum_{k=0}^{N-1} e^{2\pi i \frac{kj}{N}}\ket{k},
\end{equation}
where $N=2^n$. The elementary gates involved include $2$-qubit swap gates and $2$-qubit controlled rotation gates. We refer the readers to \cites{coppersmith2002approximate, nielsen2001quantum} for more details on QFT.
\subsection{Linear combination of unitaries (LCU)}\label{subsec:LCU}
Given a few block-encoded matrices, a block encoding of a certain linear combination of them is often needed in practice. To this end, the linear combination of unitaries (LCU) technique has been developed (\cites{berry2015simulating, childs2017quantum, gilyen2019quantum}). As an example, for two matrices $A$ and $B$, a block encoding of $A+B$ can be given by the circuit in \Cref{fig:LCU}, where $U_A$ and $U_B$ are block encodings of $A$ and $B$, respectively.
\begin{figure}[ht]
\centerline{
\Qcircuit @C=1em @R=1.5em {
\lstick{\ket{0}} & \gate{H} & \ctrlo{1} & \ctrl{1} & \gate{H}&\qw \\
\lstick{\ket{\psi}}& \qw & \gate{U_A} & \gate{U_B} & \qw& \qw \\
}
}
\caption{LCU for two unitaries} \label{fig:LCU}
\end{figure}
For general linear combinations, we adapt the following result from \cites{gilyen2019quantum}.
\begin{lemma}\label{lem:LCU}
For a vector $y\in\mathbb{C}^m$ with $\|y\|_1\leq\beta$, assume we have
\begin{enumerate}
\item a pair of unitaries $(U_L, U_R)$ with shape $2^b\times2^b$ such that $U_L\ket{0^b} = \sum_{j=0}^{2^b-1}c_j\ket{j}$, $U_R\ket{0^b} = \sum_{j=0}^{2^b-1}d_j\ket{j}$, $\sum_{j=0}^{m-1}|\beta c_j^*d_j - y_j|<\epsilon_1$ and $\sum_{j=m}^{2^b-1}|c_j^*d_j|=0$, and
\item a unitary $W=\sum_{j=0}^{m-1}\ket{j}\bra{j}\otimes U_j+\sum_{j=m}^{2^b-1}\ket{j}\bra{j}\otimes I_{a+s}$ where each $U_j$ is an $(\alpha, a, \epsilon_2)$-block-encoding of $A_j$ for $j=0, 1, \ldots, m-1$,
\end{enumerate}
then $(U_L^\dag\otimes I_{a+s})W(U_R\otimes I_{a+s})$ is an $(\alpha\beta, a+b, \alpha\epsilon_1+\beta\epsilon_2)$-block-encoding of $\sum_{j=0}^{m-1}y_jA_j$, where $I_{a+s}$ denotes the identity operator with size $2^{a+s}\times2^{a+s}$.
\end{lemma}
\subsection{Quantum eigenvalue transformation and quantum signal processing}\label{subsec:QET}
Given a Hermitian block encoding of Hermitian matrix $A$, one can construct a block encoding of $f(A)$ for a certain function $f$ using the quantum eigenvalue transformation (QET) technique \cites{low2017optimal,gilyen2019quantum}. Let $f^e({\bm{x}})=\frac{1}{2}(f({\bm{x}})+f(-{\bm{x}}))$ and $f^o({\bm{x}})=\frac{1}{2}(f({\bm{x}})-f(-{\bm{x}}))$ be the even and odd part of $f({\bm{x}})$, respectively. The standard procedure consists of the following steps, where we assume that $f({\bm{x}})$ is properly scaled such that $\|f^e\|<1$, $\|f^o\|<1$, and $\|\cdot\|$ denotes the $L^\infty$ norm on $[-1,1]$.
\begin{enumerate}
\item Approximate $f^e$ and $f^o$ with degree $\deg_{f^e}(\epsilon)$ even polynomial $\tilde{f}^e$ and degree $\deg_{f^o}(\epsilon)$ odd polynomial $\tilde{f}^o$, respectively, such that $\|\tilde{f}^e-f^e\|+\|\tilde{f}^o-f^o\|<\epsilon$ and $\|\tilde{f}^e\|\leq1$, $\|\tilde{f}^o\|\leq1$.
\item Find two sequences of phase factors $(\phi_0^e, \ldots, \phi_{\deg_{f^e}(\epsilon)}^e), (\phi_0^o, \ldots, \phi_{\deg_{f^o}(\epsilon)}^o)$ with each element in$[-\pi, \pi]$ using quantum signal processing (QSP) such that $\tilde{f}^e({\bm{x}})=\mathrm{Re}(p^e({\bm{x}}))$, $\tilde{f}^o({\bm{x}})=\mathrm{Re}(p^o({\bm{x}}))$, where $p^e$ and $p^o$ are complex polynomials with degree $\deg_{f^e}(\epsilon)$ and $\deg_{f^o}(\epsilon)$, respectively, given by
\begin{equation}\label{eq:QSP}
\begin{bmatrix}
p({\bm{x}})&r({\bm{x}})\\r^*({\bm{x}})&p^*({\bm{x}})
\end{bmatrix}=e^{i\phi_0Z}e^{i\arccos x X}e^{i\phi_1Z}e^{i\arccos x X}\cdots e^{i\phi_{\deg_f(\epsilon)-1}Z}e^{i\arccos x X} e^{i\phi_{\deg_f(\epsilon)}Z}.
\end{equation}
Here, the superscripts $e$ and $o$ are omitted for simplicity. The phase factors are then used in the QET circuit shown in \Cref{fig:QET} to construct block encodings $U_{f^e(A)}$ and $U_{f^o(A)}$, where the controlled rotation gate $\text{CR}_\phi$ is described in \Cref{fig:cr}
\item Combine the block encodings $U_{f^e(A)}$ and $U_{f^o(A)}$ with the LCU circuit in \Cref{fig:LCU} to form the block encoding $U_{f(A)}$
\end{enumerate}
\begin{figure}[ht]
\centering
\subfigure[$\text{CR}_{\phi}$\label{fig:cr}.]{
\Qcircuit @C=0.4em @R=1.5em {
& \targ & \gate{e^{-i \phi Z}} & \targ & \qw \\
& \ctrlo{-1} & \qw & \ctrlo{-1} & \qw\\
} }\hspace{3em}
\subfigure[Quantum eigenvalue transformation.\label{fig:QET}]{
\Qcircuit @C=0.35em @R=1.2em {
\lstick{\ket{0}} &\gate{H}& \multigate{1}{\text{CR}_{{\phi}_{\t}}} & \qw & \multigate{1}{\text{CR}_{{\phi}_{\t-1}}} & \qw & \qw &&&\cdots&&&&\qw& \multigate{1}{\text{CR}_{{\phi}_0}}&\gate{H}&\qw\\
\lstick{\ket{0^m}} &\qw&\ghost{\text{CR}_{{\phi}_d}}& \multigate{1}{U_A} & \ghost{\text{CR}_{{\phi}_{d-1}}} & \multigate{1}{U_A} &\qw&&&\cdots &&&&\multigate{1}{U_A}&\ghost{\text{CR}_{{\phi}_0}}&\qw&\qw\\
\lstick{\ket{\psi}}&\qw& \qw& \ghost{U_A}& \qw& \ghost{U_A}& \qw&&&\cdots&&&&\ghost{U_A}&\qw&\qw&\qw\\
}}
\caption{(a): Circuit for the controlled rotation gate $\text{CR}_\phi$. (b): Circuit for quantum eigenvalue transformation. The circuit in (b) gives a block encoding for $U_{\mathrm{Re}(p(A))}$ based on the block encoding $U_A$ and phase factors $(\phi_0, \ldots, \phi_{\t})$, where $\t=\deg_f(\epsilon)$ and the polynomial $p({\bm{x}})$ and the phase factors satisfy \eqref{eq:QSP}.}
\label{fig:QETcr}
\end{figure}
There are several methods to find the phase factors $(\phi_0^e, \ldots, \phi_{\deg_f(\epsilon)}^e)$ and $(\phi_0^o, \ldots, \phi_{\deg_f(\epsilon)}^o)$ in $[-\pi, \pi]^{\deg_f(\epsilon)+1}$ in a stable and efficient way. For example, we refer to \cites{haah2019product,chao2020finding,dong2021efficient,ying2022stable,dong2022infinite} for more details. We summarize the procedure given above in \Cref{lem:QET} below, where we assume that $f$ is either even or odd for simplicity.
\begin{lemma}\label{lem:QET}
For an even (resp. odd) function $f:\mathbb{R}\rightarrow \mathbb{R}$ and an $(\alpha,m)$-Hermitian-block-encoding of $A$ denoted as $U_A$, there is an $(\alpha C_f, m+1, \epsilon)$-block-encoding of $f(A)$ with gate complexity $\mathcal{O}(\deg_f(\epsilon) G_A)$ using the circuit shown in \Cref{fig:QET}, where $C_f\geq\max\{1, \|f\|\}$ is a scaling factor, $G_A$ is the gate complexity of $U_A$ and $\deg_f(\epsilon)$ is the smallest integer such that there exists an even (resp. odd) polynomial $\tilde{f}$ with a degree bounded by $\deg_f(\epsilon)$ satisfying $\|{f}-C_f\tilde{f}\|<\epsilon$ and $\|\tilde{f}\|\leq1$. The phase factors $(\phi_0, \ldots, \phi_{\deg_f(\epsilon)})$ in \Cref{fig:QET} are related with $\tilde{f}$ through \eqref{eq:QSP} and $\tilde{f}=\mathrm{Re}(p)$.
\end{lemma}
\subsection{Discretization of pseudo-differential operators}\label{subsec:disc}
As mentioned in \Cref{sec:intro}, the PDO considered in this paper is defined for periodic functions on $\Omega = [0, 1]^d$:
\[
A f({\bm{x}}) = \sum_{{\bm{\xi}} \in \mathbb{Z}^d} e^{2 \pi i {\bm{x}} \cdot {\bm{\xi}}} a({\bm{x}},{\bm{\xi}}) \hat{f}({\bm{\xi}}).
\]
In most numerical treatments, the function $f$ is given on a discrete grid $X=\{\frac{{\bm{x}}}{P}\equiv(\frac{x_{1}}{P}, \ldots, \frac{x_{d}}{P}): x_j \in\{0, 1, \ldots, P-1\}\}$, where $P = 2^p$ is the number of discrete points used for each dimension. Notice that here we slightly abuse the notation by reusing ${\bm{x}}$ for the integer index of the grid points. Since the space variable takes value on the Cartesian grid $X$, the frequency domain is discretized correspondingly on $\{-\frac{P}{2}, \ldots, \frac{P}{2}-1\}^d$, which leads to the discretized PDO:
\begin{equation}\label{eq:pdoZ}
A f({\bm{x}}) \equiv \sum_{{\bm{\xi}} \in \{-\frac{P}{2}, \ldots, \frac{P}{2}-1\}^d} e^{2 \pi i {\bm{x}}\cdot {\bm{\xi}}/P} a(\frac{{\bm{x}}}{P},{\bm{\xi}}) \hat{f}({\bm{\xi}}), \quad {\bm{x}}\in \Xi,
\end{equation}
where $\Xi=\{0,1, \ldots, P-1\}^d$ and we adopt an abuse of notation and denote the discretized PDO by $A$ as well.
Though the frequency variable ${\bm{\xi}}$ is discretized on $\{-\frac{P}{2}, \ldots, \frac{P}{2}-1\}^d$ in \eqref{eq:pdoZ}, by the convention of discrete Fourier transform (DFT) and fast Fourier transform (FFT), the frequency $(P/2, \ldots, P-1)$ is often identified with $(-P/2, \ldots, -1)$, respectively, since $P$ is a period for the frequency variable after DFT. In other words, the discretized PDO can be written as
\[
A f({\bm{x}}) = \sum_{{\bm{\xi}} \in \Xi} e^{2 \pi i {\bm{x}}\cdot {\bm{\xi}}/P} \tilde{a}(\frac{{\bm{x}}}{P},{\bm{\xi}}) \hat{f}({\bm{\xi}}), \quad {\bm{x}}\in \Xi=\{0,1, \ldots, P-1\}^d,
\]
where
\[
\tilde{a}({\bm{x}}, {\bm{\xi}}) \equiv a\left({\bm{x}}, {\bm{\xi}}-P\sum_{{\bm{\xi}}_j\geq P/2}\bm{e}_j\right),
\]
and $\bm{e}_j$ is the $j$-th standard basis vector in $\mathbb{C}^d$. As an example, when $d=1$ we have
\[
\tilde{a}({\bm{x}}, {\bm{\xi}}) =
\begin{cases}
a({\bm{x}}, {\bm{\xi}}), \quad 0\leq{\bm{\xi}}<P/2,\\
a({\bm{x}}, {\bm{\xi}}-P), \quad P/2\leq{\bm{\xi}}<P.
\end{cases}
\]
In order to simplify the notation and avoid repetitive use of $P$, we further define
\begin{equation}
\a({\bm{x}}, {\bm{\xi}}) \equiv \tilde{a}(\frac{{\bm{x}}}{P}, {\bm{\xi}}),
\end{equation}
and the discretized PDO becomes
\begin{equation}\label{eq:pdoZgen}
A f({\bm{x}}) = \sum_{{\bm{\xi}} \in \Xi} e^{2 \pi i {\bm{x}}\cdot {\bm{\xi}}/P} \a({{\bm{x}}},{\bm{\xi}}) \hat{f}({\bm{\xi}}), \quad {\bm{x}}\in \Xi.
\end{equation}
It is clear that $\sup|\a|=\sup|a|$. In what follows, we also refer to $\a$ as the symbol of the PDO to be computed.
\section{Block encoding for generic symbols}\label{sec:arithmD}
This section is concerned with the block encoding of the PDO \eqref{eq:pdoZ} (or rather \eqref{eq:pdoZgen}) with a generic symbol $a({\bm{x}},{\bm{\xi}})$, without assuming any additional structure. In order to compute the PDO in \eqref{eq:pdoZgen}, a simple strategy is to first lift the state to the phase space $\Xi\times\Xi$. Then the multiplication of $\a({\bm{x}}, {\bm{\xi}})$ in \eqref{eq:pdoZgen} can be performed by diagonal matrix block encodings. Combining the QFT circuit and the block encoding of diagonal matrices, one can construct the entire circuit as illustrated in \Cref{fig:PDO1}.
\begin{figure}[ht]
\centerline{
\Qcircuit @C=1.5em @R=1.5em {
\lstick{\ket{0^{pd}}}&\gate{H^{\otimes pd}}&\multigate{1}{U_{\text{ph}}^{\otimes d}}&\multigate{2}{U_\a}&\qw&&&\frac{1}{|Af|}\sum_{{\bm{x}},{\bm{\xi}}}\a({\bm{x}}, {\bm{\xi}})e^{\frac{2\pi i {\bm{x}}\cdot{\bm{\xi}}}{P}}\hat{f}({\bm{\xi}})\ket{{\bm{x}}}\\
\lstick{\frac{1}{|f|}\sum_{{\bm{x}}}f(\frac{\x}{P})\ket{{\bm{x}}}} & \gate{{U_{\text{FT}}^\dag}^{\otimes d}}&\ghost{U_{\text{ph}}^{\otimes d}}& \ghost{U_\a}&\gate{H^{\otimes pd}} &\qw&\meter \\
\lstick{\ket{0^{b}}} & \qw&\qw& \ghost{U_\a}&\qw &\qw&\meter \\
}}
\caption{Circuit that implements the PDO in \eqref{eq:pdoZgen} with a generic symbol. Here $b$ is the number of ancilla qubits needed for $U_\a$, $H$ is the Hadamard gate, $\frac{1}{|f|}\sum_{{\bm{x}}}f(\frac{\x}{P})\ket{{\bm{x}}}$ is the normalized input data , $U_{\text{FT}}$ is the QFT circuit, $U_{\text{ph}}$ and $U_\a$ are the circuits that perform the multiplication of $e^{2\pi i {\bm{x}} \cdot{\bm{\xi}}/P}$ and $\a({\bm{x}},{\bm{\xi}})$ in \eqref{eq:pdoZgen}, and the desired output is obtained with normalizing factor $\frac{1}{|Af|}$ when getting $\ket{0^{pd+b}}$ for the $pd+b$ qubits on the bottom.}
\label{fig:PDO1}
\end{figure}
Now we explain the circuit displayed in \Cref{fig:PDO1} in more detail. First, we assume that the information of the function $f$ is prepared by a normalized vector $\frac{1}{|f|}\sum_{{\bm{x}}}f(\frac{\x}{P})\ket{{\bm{x}}}$, where $|f|$ is the normalization factor
\[|f|=\sqrt{\sum_{{\bm{x}}\in\Xi}\left|f\left(\frac{\x}{P}\right)\right|^2},\]
and $\Xi=\{0,1, \ldots, P-1\}^d$. For functions $f$ with certain properties such as integrability, the state $\frac{1}{|f|}\sum_{{\bm{x}}}f(\frac{\x}{P})\ket{{\bm{x}}}$ can be constructed efficiently (see \cite{grover2002creating} for more details). For the rest of the paper, we assume the accessibility of the state $\frac{1}{|f|}\sum_{{\bm{x}}}f(\frac{\x}{P})\ket{{\bm{x}}}$ as an input.
\textbf{Step 1. Apply QFT and lift the input state to the phase space.}
We first obtain the representation of $f$ in the frequency domain by QFT. After applying the (inverse) QFT to the state $\frac{1}{|f|}\sum_{{\bm{x}}}f(\frac{\x}{P})\ket{{\bm{x}}}$ for $d$ times, we get
\begin{equation}\label{eq:getxi}
\frac{1}{|f|}\frac{1}{\sqrt{P^d}}\sum_{\bm{\xi}} \sum_{{\bm{x}}} f(\frac{\x}{P})e^{-2\pi i {\bm{\xi}} \cdot {\bm{x}}/P}\ket{{\bm{\xi}}} = \frac{\sqrt{P^d}}{|f|}\sum_{\bm{\xi}}\hat{f}({\bm{\xi}})\ket{{\bm{\xi}}}.
\end{equation}
Then the state $\frac{1}{|f|}\sum_{{\bm{x}},{\bm{\xi}}}\hat{f}({\bm{\xi}})\ket{{\bm{x}}}\ket{{\bm{\xi}}}$ is obtained by applying the Hadamard gates $H^{\otimes pd}$ to the ${\bm{x}}$-register and putting both registers together.
\textbf{Step 2. Multiply the phase $e^{2\pi i {\bm{x}}\cdot{\bm{\xi}}/P}$ with $U_{\text{ph}}$}.
A naive way of multiplying the phase $e^{2\pi i {\bm{x}}\cdot{\bm{\xi}}/P}$ is to use \Cref{prop:naivemult}, which involves many ancilla qubits and reduces the success probability. Here, we develop an efficient implementation for multiplication without involving any extra error or ancilla qubits in the following lemma.
\begin{lemma}\label{lem:dot}
The $2p$-qubit circuit $U_{\text{ph}}$ displayed in \Cref{fig:lemma2} implements the unitary operator:
\begin{equation}
\ket{x}\ket{\xi}\mapsto e^{2\pi i x\xi/P}\ket{x}\ket{\xi}, \quad 0\le x,\xi<P,
\end{equation}
with $\mathcal{O}(p^2)$ gate complexity precisely without ancilla qubits.
\begin{figure}[ht]
\centerline{
\Qcircuit @C=.9em @R=0em @!R{
\lstick{\ket{x_0}} & \gate{R_p} & \gate{R_{p-1}} & \qw & \cdots & & \gate{R_1}\ & \qw & \qw& \qw & \cdots & & \qw& \qw & \cdots & & \qw& \qw \\
\lstick{\ket{x_1}} & \qw & \qw & \qw & \cdots & & \qw\ & \gate{R_{p-1}} & \gate{R_{p-2}} & \qw & \cdots & & \gate{R_1} &\qw & \cdots & & \qw\ & \qw \\
\lstick{\vdots}&&&&\vdots &&&&&&\vdots &&&&\vdots \\
\lstick{\ket{x_{p-1}}} & \qw & \qw & \qw & \cdots & & \qw\ & \qw & \qw & \qw & \cdots & & \qw&\qw &\cdots & & \gate{R_1} & \qw \\
\lstick{\ket{\xi_0}} & \ctrl{-4} & \qw & \qw & \cdots & & \qw\ & \ctrl{-3} & \qw& \qw & \cdots & & \qw& \qw & \cdots & & \ctrl{-1} & \qw \\
\lstick{\ket{\xi_1}} & \qw & \ctrl{-5} & \qw & \cdots & & \qw\ & \qw & \ctrl{-4} & \qw & \cdots & & \qw &\qw & \cdots & & \qw\ & \qw \\
\lstick{\vdots}&&&&\vdots &&&&&&\vdots &&&&\vdots \\
\lstick{\ket{\xi_{p-2}}} & \qw & \qw & \qw & \cdots & & \qw\ & \qw & \qw & \qw & \cdots & & \ctrl{-6} &\qw &\cdots & & \qw & \qw \\
\lstick{\ket{\xi_{p-1}}} & \qw & \qw & \qw & \cdots & & \ctrl{-8}\ & \qw & \qw & \qw & \cdots & & \qw&\qw &\cdots & & \qw & \qw \\
}}
\caption{Circuit for $U_{\text{ph}}$, the phase multiplication $\ket{x}\ket{\xi}\mapsto e^{2\pi i x\xi/P}\ket{x}\ket{\xi}$. Here $R_{j} = \ket{0}\bra{0}+e^{2\pi i \cdot2^{-j}}\ket{1}\bra{1}$ is a rotation operator.}
\label{fig:lemma2}
\end{figure}
\end{lemma}
\begin{proof}
The idea of the construction is similar to the implementation of QFT, which is based on bit-wise controlled rotation. We first write the binary representation of integers
\begin{equation*}
x = (x_{p-1}\cdots x_0.),\quad \xi = (\xi_{p-1}\cdots\xi_0.),
\end{equation*}
and do the following calculation:
\begin{equation}\label{eq:Uph}
\begin{aligned}
&e^{2\pi i x\xi/P}\ket{x}\ket{\xi}
= \left(\prod_{j=0}^{p-1}\prod_{k=0}^{p-1}e^{2\pi i x_j\xi_k \cdot2^{j+k-p}}\right)\ket{x_{p-1}\cdots x_0}\ket{\xi_{p-1}\cdots \xi_0}\\
=& \left(\prod_{0\le j+k<p}e^{2\pi i x_j\xi_k \cdot2^{j+k-p}}\right)\ket{x_{p-1}\cdots x_0}\ket{\xi_{p-1}\cdots \xi_0}\\
=& \left(\left(\prod_{k=0}^{p-1}e^{2\pi i x_{0}\xi_{k} \cdot 2^{k-p}}\right)\ket{x_{0}}\right)\otimes\cdots\otimes\left(\left(\prod_{k=0}^{0}e^{2\pi i x_{p-1}\xi_{k} \cdot 2^{k-1}}\right)\ket{x_{p-1}}\right)\otimes\ket{\xi_{p-1}\cdots \xi_0},
\end{aligned}
\end{equation}
where the second equality is true because $e^{2\pi i x_j\xi_k \cdot2^{j+k-p}}=1$ when $j+k\ge p$. The circuit corresponding to the unitary in \eqref{eq:Uph} can be implemented by a series of controlled rotations
\begin{equation*}
\Qcircuit @C=1em @R=1em {
\lstick{\ket{x_j}} & \gate{R_{p-j-k}} & \rstick{e^{2\pi i x_j\xi_k \cdot2^{j+k-p}}\ket{x_j}} \qw \\
\lstick{\ket{\xi_k}} & \ctrl{-1} & \rstick{\ket{\xi_k}} \qw
}.
\end{equation*}
where $R_{j} =R_z(\pi/2^{j-1})= \ket{0}\bra{0}+e^{2\pi i \cdot2^{-j}}\ket{1}\bra{1}$. Finally, the circuit shown in \Cref{fig:lemma2} is obtained after arranging the controlled rotations in the corresponding places.
\end{proof}
Rewriting the state $\frac{1}{|f|}\sum_{{\bm{x}},{\bm{\xi}}}e^{2\pi i {\bm{x}}\cdot{\bm{\xi}}/P}\hat{f}({\bm{\xi}})\ket{{\bm{x}}}\ket{{\bm{\xi}}}$ as
\[
\frac{1}{|f|}\sum_{{\bm{x}},{\bm{\xi}}}e^{2\pi i x_{d}\xi_{d}/P}\cdots e^{2\pi i x_{1}\xi_{1}/P}\hat{f}({\bm{\xi}})\ket{{\bm{x}}}\ket{{\bm{\xi}}},
\]
then the map from $\frac{1}{|f|}\sum_{{\bm{x}},{\bm{\xi}}}\hat{f}({\bm{\xi}})\ket{{\bm{x}}}\ket{{\bm{\xi}}}$ to $\frac{1}{|f|}\sum_{{\bm{x}},{\bm{\xi}}}e^{2\pi i {\bm{x}}\cdot{\bm{\xi}}/P}\hat{f}({\bm{\xi}})\ket{{\bm{x}}}\ket{{\bm{\xi}}}$ can be performed by applying \Cref{lem:dot} for $d$ times to the register pairs $(x_{d}, \xi_{d}), \ldots, (x_{1}, \xi_{1})$. The corresponding circuit is denoted as $U_\text{ph}^{\otimes d}$ and involves $\mathcal{O}(p^2d)$ elementary gates with no ancilla qubits. After the multiplication of $U_{\text{ph}}^{\otimes d}$, one obtains the state $\frac{1}{|f|}\sum_{{\bm{x}},{\bm{\xi}}}e^{2\pi i {\bm{x}}\cdot{\bm{\xi}}/P}\hat{f}({\bm{\xi}})\ket{{\bm{x}}}\ket{{\bm{\xi}}}$.
\textbf{Step 3. Multiply the symbol $\a({\bm{x}},{\bm{\xi}})$. }
The next component in \Cref{fig:PDO1} is the diagonal multiplication $U_\a$, which is designed to approximate the map
\begin{equation}\label{eq:Ua}
\frac{1}{|f|}\sum_{{\bm{x}},{\bm{\xi}}}e^{2\pi i {\bm{x}}\cdot{\bm{\xi}}/P}\hat{f}({\bm{\xi}})\ket{{\bm{x}}}\ket{{\bm{\xi}}}\ket{0^b} \mapsto \frac{1}{C_a|f|}\sum_{{\bm{x}},{\bm{\xi}}}\a({\bm{x}}, {\bm{\xi}})e^{2\pi i {\bm{x}}\cdot{\bm{\xi}}/P}\hat{f}({\bm{\xi}})\ket{{\bm{x}}}\ket{{\bm{\xi}}}\ket{0^b}+\perp,
\end{equation}
where $C_a>0$ is a constant that depends on $\a({\bm{x}},{\bm{\xi}})$, $b$ is the number of ancilla qubits used for $U_\a$ and $\perp$ is an unnormalized state that is orthogonal to any state of the form $\ket{{\bm{x}}}\ket{{\bm{\xi}}}\ket{0^b}$.
As mentioned earlier, the idea is to treat $({\bm{x}}, {\bm{\xi}})$ as a $2d$-dimensional variable and utilize the result from arithmetic circuit construction. Leveraging the reversible computational model and the uncomputation technique, any classical arithmetic operation can be implemented by a quantum circuit efficiently. More specifically, one can construct a corresponding quantum circuit using $\mathcal{O}(\polylog(\frac{1}{\epsilon}))$ ancilla qubits and $\mathcal{O}(\polylog(\frac{1}{\epsilon}))$ gates, where $\epsilon$ is the precision one wants to achieve (Cf. \cites{nielsen2001quantum,rieffel2011quantum}). We state a general result for an efficient block encoding of diagonal matrices $D_g$, as defined in \eqref{eq:diagg}, which is summarized in \Cref{prop:diag1}. A similar idea has been used in \cite{grover2002creating} to create a given state, in \cite{harrow2009quantum} to construct the reciprocals of the eigenvalues and in \cite{tong2021fast} to implement the diagonal preconditioner.
\begin{prop}\label{prop:diag1}
Assume that $g:\mathbb{R}^m\rightarrow\mathbb{R}$ is a smooth arithmetic function with $\sup |g| <\infty$. Then there is an $(C, \mathcal{O}(\polylog(\frac{1}{\epsilon})+\poly(mp)), \epsilon)$-block-encoding of $D_g$ with $\mathcal{O}(\polylog(\frac{1}{\epsilon})+\poly(mp))$ gates, where $D_g$ is a diagonal operator on the Hilbert space $\mathbb{C}^{mp}$ defined in \eqref{eq:diagg}, and $C\geq\sup|g|$.
\end{prop}
\begin{proof}
The circuit $U_g$ is constructed as follows. Let $t = \lceil \log_2(\frac{C\pi}{\epsilon})\rceil$, and let $\theta(x_1, \ldots, x_m)$ be a map that gives $\ket{\theta_{\text{sgn}}\theta_{t-1}\theta_{t-2}\ldots\theta_{0}}$, where $(.\theta_{t-1}\theta_{t-2}\ldots\theta_{0})$ is the closest $t$-bit fixed-point representation of $\frac{1}{\pi}\arcsin(|g(x_1, \ldots, x_m)|/C)$, and $\theta_{\text{sgn}}$ is assigned the value $0$ if $g\geq0$ and the value $1$ otherwise.
For an arbitrary basis state
$\ket{x_{m}}\cdots\ket{x_{1}}$, we consider the system with $t+2$ ancilla qubits $\ket{0}\ket{x_{m}}\cdots\ket{x_{1}}\ket{0^{t+1}}$. Here $\ket{x_j} = \ket{x_{j,p-1} \cdots x_{j,0}}$ for each $j$. Using the reversible computational model and uncomputation (\cites{nielsen2001quantum,rieffel2011quantum}), the classical circuit:
\[
\ket{0}\ket{x_{m}}\cdots\ket{x_{1}}\ket{0^{t+1}} \rightarrow \ket{0}\ket{x_{m}}\cdots\ket{x_{1}}\ket{\theta_{\text{sgn}}\theta_{t-1}\theta_{t-2}\ldots\theta_{0}}
\]
can be constructed with $\mathcal{O}(\poly(t)+\poly(mp))$ gates and $\mathcal{O}(\poly(mp))$ ancilla qubits. Then we apply the circuit in Figure \ref{fig:CR} on the $t+2$ ancilla qubits. The state obtained is:
\[(-1)^{\theta_{\text{sgn}}}(\cos(\pi(.\theta_{t-1}\ldots\theta_0))\ket{1}+\sin(\pi(.\theta_{t-1}\ldots\theta_0))\ket{0})\ket{x_{m}}\cdots\ket{x_{1}}\ket{\theta_{\text{sgn}}\theta_{t-1}\theta_{t-2}\ldots\theta_{0}},\]
which can then be mapped to
\[(-1)^{\theta_{\text{sgn}}}(\cos(\pi(.\theta_{t-1}\ldots\theta_0))\ket{1}+\sin(\pi(.\theta_{t-1}\ldots\theta_0))\ket{0})\ket{x_{m}}\cdots\ket{x_{1}}\ket{0^{t+1}},\]
via uncomputation.
Since $|(.\theta_{t-1}\ldots\theta_0)-\frac{1}{\pi}\arcsin(|g(x_1, \ldots, x_m)|/C)|<\frac{\epsilon}{C\pi}$, we have
$|(-1)^{\theta_{\text{sgn}}}\sin(\pi(.\theta_{t-1}\ldots\theta_0))-\frac{1}{C}g(x_1, \ldots, x_m))|<\frac{\epsilon}{C}$, which means the desired state $\frac{1}{C}g(x_1, \ldots, x_m)\ket{x_{m}}\cdots\ket{x_{1}}$ is obtained with error at most $\frac{\epsilon}{C}$ upon measuring the ancilla qubits and getting $\ket{0^{t+2}}$.
\end{proof}
\begin{figure}[ht]
\centerline{
\Qcircuit @C=1.5em @R=1.5em {
\lstick{\ket{0}}&\gate{Z}& \gate{R_{y}(\pi)} & \gate{R_{y}(\pi/2)} &\qw&\cdots& & \gate{R_{y}(\pi/2^{t-1})}&\gate{X}&\qw\\
\lstick{\ket{\theta_{\text{sgn}}}}& \ctrl{-1}& \qw & \qw&\qw&\qw&\qw&\qw&\qw&\qw\\
\lstick{\ket{\theta_{t-1}}}& \qw& \ctrl{-2} & \qw&\qw&\qw&\qw&\qw&\qw&\qw\\
\lstick{\ket{\theta_{t-2}}}& \qw& \qw& \ctrl{-3} &\qw& \qw&\qw&\qw&\qw&\qw\\
\lstick{\cdots}&\\
\lstick{\ket{\theta_{0}}}& \qw& \qw & \qw&\qw&\qw &\qw& \ctrl{-5} & \qw&\qw\\
}}
\caption{The controlled rotation part of $U_g$.}
\label{fig:CR}
\end{figure}
As a result of applying the block encoding given in \Cref{prop:diag1} with $m=2d$, $g = \a$ and $\epsilon$ replaced by $\epsilon/\sqrt{P^d}$ to the state $\frac{1}{|f|}\sum_{{\bm{x}},{\bm{\xi}}}e^{2\pi i {\bm{x}}\cdot{\bm{\xi}}/P}\hat{f}({\bm{\xi}})\ket{{\bm{x}}}\ket{{\bm{\xi}}}$, one obtains the state
\[
\ket{\phi}\ket{0^b}+\perp,
\]
using $\mathcal{O}(\polylog(\frac{1}{\epsilon})+\poly(pd))$ ancilla qubits and $\mathcal{O}(\polylog(\frac{1}{\epsilon})+\poly(pd))$ gates where
\begin{equation}\label{eq:ineqphi}
\left\|\ket{\phi}-\frac{1}{C_a|f|}\sum_{{\bm{x}},{\bm{\xi}}}\a({\bm{x}}, {\bm{\xi}})e^{2\pi i {\bm{x}}\cdot{\bm{\xi}}/P}\hat{f}({\bm{\xi}})\ket{{\bm{x}}}\ket{{\bm{\xi}}}\right\|<\frac{\epsilon}{\sqrt{P^d}C_a},
\end{equation}
and $C_a\geq\sup|a|=\sup|\a|$ is a constant and $\perp$ is an unnormalized state orthogonal to all state with the form $\ket{{\bm{x}}}\ket{{\bm{\xi}}}\ket{0^b}$.
\textbf{Step 4. Sum over the frequency variable.}
Finally, after applying the Hadamard gate $H^{\otimes pd}$ to the ${\bm{\xi}}$ registers, we obtain the state
\begin{equation}
\left((I_{pd}\otimes\ket{0^{pd}}) (I_{pd}\otimes\bra{0^{pd}})(I_{pd}\otimes H^{\otimes pd})\ket{\phi}\right)\ket{0^b}+\tilde{\perp},
\end{equation}
where $\tilde{\perp}$ is an unnormalized state that is orthogonal to all states of the form $\ket{{\bm{x}}}\ket{0^{pd+b}}$ and
\begin{equation}\label{eq:gen_final}
\begin{aligned}
\bigg\|&(I_{pd}\otimes\ket{0^{pd}}) (I_{pd}\otimes\bra{0^{pd}})(I_{pd}\otimes H^{\otimes pd})\ket{\phi}\\
&-\frac{1}{\sqrt{P^d}|f|C_a}\sum_{{\bm{x}},{\bm{\xi}}}\a({\bm{x}},{\bm{\xi}})e^{2\pi i {\bm{x}}\cdot{\bm{\xi}}/P}\hat{f}({\bm{\xi}})\ket{{\bm{x}}}\ket{0^{pd}}\ket{0^b}\bigg\|<\frac{\epsilon}{\sqrt{P^d}C_a},
\end{aligned}
\end{equation}
which can be seen from \eqref{eq:ineqphi}.
Therefore, one obtains the desired state on the ${\bm{x}}$ registers upon measuring $\ket{0^{pd}}$ for the ${\bm{\xi}}$ registers and $\ket{0^{b}}$ for the ancilla qubits. Notice that there is an extra scaling factor $\frac{1}{\sqrt{P^d}}$ due to the application of Hadamard gates, so the success probability is $\mathcal{O}(2^{-pd})$. The complete circuit we use is exactly the one shown in \Cref{fig:PDO1}.
The block encoding scheme of the PDO \eqref{eq:pdoZgen} constructed in this section can be summarized in the following theorem:
\begin{theorem}\label{thm:generica}
For a generic symbol $a({\bm{x}},{\bm{\xi}})$, a block encoding of the corresponding discretized PDO defined by \eqref{eq:pdoZgen} can be $(2^{\frac{pd}{2}}C_a, \mathcal{O}(\poly(pd)+\polylog(1/\epsilon)), \epsilon)$-block-encoded using the circuit displayed in \Cref{fig:PDO1} with gate complexity $\mathcal{O}(\poly(pd)+\polylog(1/\epsilon))$, where $C_a\geq\sup|a|$ is a constant.
\end{theorem}
\textbf{Challenge.}
Despite being applicable to generic symbols, one can observe from \eqref{eq:gen_final} that the success probability of the circuit in \Cref{fig:PDO1} can be low when $pd$ is large. In the following sections, we show that this challenge can be overcome when the symbol $a({\bm{x}},{\bm{\xi}})$ has additional structures.
\section{Efficient block encoding for separable symbols}\label{sec:separable}
As explained in \Cref{sec:arithmD}, the circuit designed as in \Cref{fig:PDO1} suffers from exponentially small success probability, despite being simple and applicable to generic PDOs. In this section, we are concerned with symbols with particular structures, and an efficient block encoding of the corresponding PDOs with $\mathcal{O}(1)$ success probability is constructed.
\subsection{Separable symbols}
We first give the following definition for separable symbols.
\begin{definition}\label{def:sep}
A symbol $a({\bm{x}},{\bm{\xi}})$ is \emph{separable} if $a({\bm{x}}, {\bm{\xi}}) = \alpha({\bm{x}})\beta({\bm{\xi}})$.
\end{definition}
As explained in \Cref{subsec:disc}, we identify the frequency $(P/2, \ldots, P-1)$ with $(-P/2, \ldots, -1)$, respectively, since $P$ is a period for the frequency variable after DFT. We also define
\begin{equation}\label{eq:sepnote}
{\breve{\alpha}}({\bm{x}}) = \alpha(\frac{\x}{P}), ~ {\breve{\beta}}({\bm{\xi}}) = \beta\left({\bm{\xi}}-P\sum_{{\bm{\xi}}_j\geq P/2} \bm{e}_j\right),
\end{equation}
where $\bm{e}_j$ is the $j$-th standard basis vector in $\mathbb{C}^d$.
With help of the notations \eqref{eq:sepnote}, the PDO \eqref{eq:pdoZ} becomes
\begin{equation}\label{eq:pdoZsep}
A f({\bm{x}}) = \sum_{{\bm{\xi}} \in \Xi} e^{2 \pi i {\bm{x}}\cdot {\bm{\xi}}/P} {\breve{\alpha}}({\bm{x}}){\breve{\beta}}({\bm{\xi}}) \hat{f}({\bm{\xi}}), \quad {\bm{x}}\in \Xi=\{0,1, \ldots, P-1\}^d,
\end{equation}
It is clear from the definition that ${\breve{\alpha}}$ is $P$-periodic since $\alpha$ is $1$-periodic, and we also have $\sup|\alpha|=\sup|{\breve{\alpha}}|$, $\sup|\beta|=\sup|{\breve{\beta}}|$.
For the PDO \eqref{eq:pdoZsep}, we propose an efficient block encoding illustrated by the following circuit.
\begin{figure}[ht]
\centerline{
\Qcircuit @C=1.5em @R=1.5em {
\lstick{\ket{0^{b_2}}} & \qw&\qw& \qw&\multigate{1}{U_{\breve{\alpha}}}\qw &\qw&\meter \\
\lstick{\frac{1}{|f|}\sum_{{\bm{x}}}f(\frac{\x}{P})\ket{{\bm{x}}}} & \gate{{U_{\text{FT}}^\dag}^{\otimes d}}&\multigate{1}{U_{\breve{\beta}}}& \gate{{U_{\text{FT}}}^{\otimes d}} &\ghost{U_{\breve{\alpha}}}&\qw&&&&\frac{1}{|Af|}\sum_{{\bm{x}},{\bm{\xi}}}{\breve{\alpha}}({\bm{x}}){\breve{\beta}}({\bm{\xi}})e^{2\pi i {\bm{x}}\cdot{\bm{\xi}}/P}\hat{f}({\bm{\xi}})\ket{{\bm{x}}} \\
\lstick{\ket{0^{b_1}}} & \qw& \ghost{U_{\breve{\beta}}}&\qw &\qw\qw&\qw&\meter \\
}
}
\caption{Circuit for an efficient block encoding for separable PDOs \eqref{eq:pdoZsep}. Here $b_1$, $b_2$ are the number of ancilla qubits needed for $U_{\breve{\beta}}$ and $U_{\breve{\alpha}}$, respectively, $\frac{1}{|f|}\sum_{{\bm{x}}}f(\frac{\x}{P})\ket{{\bm{x}}}$ is the normalized input data, $U_{\text{FT}}$ is the QFT circuit, $U_{\breve{\beta}}$ and $U_{\breve{\alpha}}$ are the block encodings of $D_{\breve{\beta}}$ and $D_{\breve{\alpha}}$, respectively, and the desired output is obtained with normalizing factor $\frac{1}{|Af|}$ when getting $\ket{0^{b_1+b_2}}$ for the $b_1+b_2$ ancilla qubits. $D_{\breve{\beta}}$ and $D_{\breve{\alpha}}$ are diagonal matrices defined in \eqref{eq:diagg}.}\label{fig:PDO2}
\end{figure}
For the rest of this section, we explain the circuit in \Cref{fig:PDO2} with more details and show that it significantly improves the success probability compared with \Cref{fig:PDO1}. The circuit begins with a QFT step similar to that in \Cref{fig:PDO1}.
\textbf{Step 1. Apply QFT and multiply the factor ${\breve{\beta}}({\bm{\xi}})$.}
With the same argument as \eqref{eq:getxi}, one obtains the state:
\[\frac{\sqrt{P^d}}{|f|}\sum_{\bm{\xi}}\hat{f}({\bm{\xi}})\ket{{\bm{\xi}}}\ket{0^{b_1+b_2}},\]
after applying the QFT gates ${U_{\text{FT}}^\dag}^{\otimes d}$. Now that the input is represented on the frequency domain, one can naturally implement the multiplication of the factor ${\breve{\beta}}({\bm{\xi}})$ since it only depends on the frequency variable ${\bm{\xi}}$. The corresponding block $U_{\breve{\beta}}$ can be constructed using \Cref{prop:diag1} with $m=d$, $g={\breve{\beta}}$ and $\epsilon$ replaced by $\frac{\epsilon}{2C_\alpha}$, where the constant $C_\alpha\geq\sup|\alpha|$. Then one obtains the state
\begin{equation}\label{eq:state1}
(\ket{\phi_1}\ket{0^{b_1}}+\perp_1)\ket{0^{b_2}},
\end{equation}
where $\perp_1$ is an unnormalized state orthogonal to all states of the form $\ket{{\bm{\xi}}}\ket{0^{b_1}}$ and $\ket{\phi_1}$ satisfies
\begin{equation}\label{eq:b}
\left\|\ket{\phi_1}-\frac{\sqrt{P^d}}{C_\beta|f|}\sum_{\bm{\xi}}{\breve{\beta}}({\bm{\xi}})\hat{f}({\bm{\xi}})\ket{{\bm{\xi}}}\right\|<\frac{\epsilon}{2C_\alpha C_\beta},
\end{equation}
using $\mathcal{O}(\poly(pd)+\polylog(1/\epsilon))$ gates and $b_1=\mathcal{O}(\poly(pd)+\polylog(1/\epsilon))$ ancilla qubits. Here $C_\beta\geq\sup|\beta|$ is a constant.
\textbf{Step 2. Apply QFT and multiply the factor ${\breve{\alpha}}({\bm{x}})$. }
In order to multiply the factor ${\breve{\alpha}}({\bm{x}})$ in the symbol, we apply QFT and convert the state to the space domain. Since $\sqrt{P^d}{U_{\text{FT}}}^{\otimes d}\sum_{\bm{\xi}}{\breve{\beta}}({\bm{\xi}})\hat{f}({\bm{\xi}})\ket{{\bm{\xi}}} = \sum_{{\bm{x}},{\bm{\xi}}}{\breve{\beta}}({\bm{\xi}})e^{2\pi i {\bm{x}}\cdot{\bm{\xi}}/P}\hat{f}({\bm{\xi}})\ket{{\bm{x}}}$, we have
\begin{equation}\label{eq:phib}
\begin{aligned}
&\left\|{U_{\text{FT}}}^{\otimes d}\ket{\phi_1}-\frac{1}{C_\beta|f|}\sum_{{\bm{x}},{\bm{\xi}}}{\breve{\beta}}({\bm{\xi}})e^{2\pi i {\bm{x}}\cdot{\bm{\xi}}/P}\hat{f}({\bm{\xi}})\ket{{\bm{x}}}\right\|\\
&=\left\|{U_{\text{FT}}}^{\otimes d}\ket{\phi_1}-{U_{\text{FT}}}^{\otimes d}\frac{\sqrt{P^d}}{C_\beta|f|}\sum_{\bm{\xi}}{\breve{\beta}}({\bm{\xi}})\hat{f}({\bm{\xi}})\ket{{\bm{\xi}}}\right\|\\
&=\left\|\ket{\phi_1}-\frac{\sqrt{P^d}}{C_\beta|f|}\sum_{\bm{\xi}}{\breve{\beta}}({\bm{\xi}})\hat{f}({\bm{\xi}})\ket{{\bm{\xi}}}\right\|<\frac{\epsilon}{2C_\alpha C_\beta},
\end{aligned}
\end{equation}
where we have used \eqref{eq:b} in the last line.
By using \Cref{prop:diag1} again with $m=d$, $g={\breve{\alpha}}$ and $\epsilon$ replaced by $\frac{\epsilon}{2C_\beta}$, the state $\ket{\phi_1}\ket{0^{b_2}}$ is mapped to $\ket{\phi_2}\ket{0^{b_2}}+\perp_2$, where $\perp_2$ is an unnormalized state orthogonal to all state of the form $\ket{{\bm{x}}}\ket{0^{b_2}}$ and $\ket{\phi_2}$ satisfies $\|\ket{\phi_2}-\frac{1}{C_\alpha}{D}_{\breve{\alpha}}{U_{\text{FT}}}^{\otimes d}\ket{\phi_1}\|<\frac{\epsilon}{2C_\alpha C_\beta}$. In this step, $\mathcal{O}(\poly(pd)+\polylog(1/\epsilon))$ gates and $b_2=\mathcal{O}(\poly(pd)+\polylog(1/\epsilon))$ ancilla qubits are used. The image of $\perp_1$ is still orthogonal to all states of the form $\ket{{\bm{\xi}}}\ket{0^{b_1}}$ since the $b_1$ ancilla qubits used in the previous step are unchanged. Therefore, the final state is
\begin{equation}\label{eq:state2}
\ket{\phi_2}\ket{0^{b_1+b_2}}+\perp,
\end{equation}
where $\perp$ is an unnormalized state orthogonal to all states of the form $\ket{{\bm{x}}}\ket{0^{b_1+b_2}}$ and $\ket{\phi_2}$ satisfies
\begin{equation}\label{eq:a}
\begin{aligned}
&\left\|C_\alpha C_\beta\ket{\phi_2}-\frac{1}{|f|}\sum_{{\bm{x}},{\bm{\xi}}}{\breve{\alpha}}({\bm{x}}){\breve{\beta}}({\bm{\xi}})e^{\frac{2\pi i {\bm{x}}\cdot{\bm{\xi}}}{P}}\hat{f}({\bm{\xi}})\ket{{\bm{x}}}\right\|\\
\leq&C_\alpha C_\beta\left\|\ket{\phi_2}-\frac{1}{C_\alpha}{D}_{\breve{\alpha}}{U_{\text{FT}}}^{\otimes d}\ket{\phi_1}\right\|+C_\beta\left\|D_{\breve{\alpha}}{U_{\text{FT}}}^{\otimes d}\ket{\phi_1}-\frac{1}{C_\beta|f|}\sum_{{\bm{x}},{\bm{\xi}}}{\breve{\alpha}}({\bm{x}}){\breve{\beta}}({\bm{\xi}})e^{\frac{2\pi i {\bm{x}}\cdot{\bm{\xi}}}{P}}\hat{f}({\bm{\xi}})\ket{{\bm{x}}}\right\|\\
<& \frac{\epsilon}{2} + {C_\beta}\left\|D_{\breve{\alpha}}{U_{\text{FT}}}^{\otimes d}\ket{\phi_1}-D_{\breve{\alpha}}\frac{1}{C_\beta|f|}\sum_{{\bm{x}},{\bm{\xi}}}{\breve{\beta}}({\bm{\xi}})e^{2\pi i {\bm{x}}\cdot{\bm{\xi}}/P}\hat{f}({\bm{\xi}})\ket{{\bm{x}}}\right\|\\
\leq& \frac{\epsilon}{2} + C_\beta\sup |{\breve{\alpha}}|\left\|{U_{\text{FT}}}^{\otimes d}\ket{\phi_1}-\frac{1}{C_\beta|f|}\sum_{{\bm{x}},{\bm{\xi}}}{\breve{\beta}}({\bm{\xi}})e^{2\pi i {\bm{x}}\cdot{\bm{\xi}}/P}\hat{f}({\bm{\xi}})\ket{{\bm{x}}}\right\|<\frac{\epsilon}{2} + \frac{\epsilon}{2} =\epsilon,
\end{aligned}
\end{equation}
where we have used the inequality \eqref{eq:phib} and the fact that $C_\alpha \geq \sup|\alpha| = \sup|{\breve{\alpha}}|$ in the last line.
By checking the definition of block encoding and adding up the gates and ancilla qubits used, we obtain the following theorem.
\begin{theorem}\label{thm:sep}
If $a({\bm{x}},{\bm{\xi}})=\alpha({\bm{x}})\beta({\bm{\xi}})$ is a separable symbol as defined in \Cref{def:sep}, then the discretized PDO \eqref{eq:pdoZsep} can be $({C_\alpha C_\beta}, \mathcal{O}(\poly(pd)+\polylog(1/\epsilon)), \epsilon)$-block-encoded using the circuit displayed in \Cref{fig:PDO2} with gate complexity $\mathcal{O}(\poly(pd)+\polylog(1/\epsilon))$, where $C_\alpha, C_\beta > 0$ are constants such that $C_\alpha\geq\sup|\alpha|$ and $C_\beta\geq\sup|\beta|$.
\end{theorem}
\begin{remark}
In contrast to the result in \Cref{thm:generica}, one can observe that the exponential factor $2^{\frac{pd}{2}}$ is removed and thus the success probability for the circuit in \Cref{fig:PDO2} is improved exponentially compared to the one in \Cref{fig:PDO1}.
\end{remark}
\subsection{Linear combination of separable terms}
With the block encoding for PDOs with separable symbols ready, the PDO for a linear combination of separable terms, i.e.,
\[
a({\bm{x}},{\bm{\xi}}) = \sum_{j=0}^{m-1} y_ja_j({\bm{x}},{\bm{\xi}}) =\sum_{j=0}^{m-1} y_j\alpha_j({\bm{x}})\beta_j({\bm{\xi}}),
\]
can also be block-encoded, thanks to LCU (see \Cref{subsec:LCU}). More precisely, we have the following corollary.
\begin{cor}\label{cor:sep}
For a linear combination of separable symbols $a({\bm{x}},{\bm{\xi}}) = \sum_{j=0}^{m-1} y_ja_j({\bm{x}},{\bm{\xi}}) =\sum_{j=0}^{m-1} y_j\alpha_j({\bm{x}})\beta_j({\bm{\xi}})$, where $\sup|\alpha_j|\leq1$, $\sup|\beta_j|\leq1$ and $y\in\mathbb{C}^m$ with $\|y\|_1\leq\delta$, assume that $(U_L, U_R)$ is a pair of unitaries described in \Cref{lem:LCU} with $\epsilon_1=\epsilon$, and $W=\sum_{j=0}^{m-1}\ket{j}\bra{j}\otimes U_j+\sum_{j=m}^{2^b-1}\ket{j}\bra{j}\otimes I_{a+s}$, where each $U_j$ is a $(1,a, \epsilon)$-block-encoding of the discretized PDO $A_j$ associated with symbol $a_j({\bm{x}},{\bm{\xi}})$ (see \eqref{eq:pdoZsep}) constructed in \Cref{thm:sep} with $a=\mathcal{O}(\poly(pd)+\polylog(1/\epsilon))$. Then $(U_L^\dag\otimes I_{a+s})W(U_R\otimes I_{a+s})$ is a $(\delta, a+b, (1+\delta)\epsilon)$-block-encoding of $\sum_{j=0}^{m-1}y_jA_j$, where $I_{a+s}$ denotes the identity operator with size $2^{a+s}\times2^{a+s}$. The gate complexity of the corresponding circuit is $\mathcal{O}(m(\poly(pd)+\polylog(1/\epsilon)))$.
\end{cor}
\section{Efficient block encoding for fully separable symbols with explicit circuits}\label{sec:fully sep}
For separable symbols, the circuit presented in \Cref{fig:PDO2} significantly increases the success probability compared to the one in \Cref{fig:PDO1}. However, this relies on circuits for arithmetic functions (see \Cref{prop:diag1}), which can still be challenging to construct in practice. In this section, we develop a more explicit circuit with the help of QSP and QET (see \Cref{subsec:QET}).
\subsection{Dimension-wise fully separable symbols}
To begin with, we consider the fully separable symbols defined as follows.
\begin{definition}\label{def:fullysep}
A symbol $a({\bm{x}},{\bm{\xi}})$ is called \emph{fully separable} if $a({\bm{x}},{\bm{\xi}}) = \alpha({\bm{x}})\beta({\bm{\xi}})$ with $\alpha({\bm{x}}) = \alpha_1(x_1)\cdots\alpha_d(x_d)$ and $\beta({\bm{\xi}}) = \beta_1(\xi_1)\cdots\beta_d(\xi_d)$ where each function in the set $\{\alpha_k\}_{k=1}^d\cup\{\beta_k\}_{k=1}^d$ is a real even function, a real odd function or an exponential function of the form $f(y)=\exp(i\theta y)$ for some real parameter $\theta$.
\end{definition}
Similar with \eqref{eq:sepnote}, we introduce the following notations:
\begin{equation}\label{eq:fullynote}
{\breve{\alpha}}_k(x_k) = \alpha_k(\frac{x_k}{P}), ~
{\breve{\beta}}_k(\xi_k) = \begin{cases}
\beta_k(\xi_k), & 0\leq\xi_k<P/2\\
\beta_k(\xi_k-P), & P/2\leq\xi_k<P
\end{cases},\quad k= 1, 2, \ldots, d.
\end{equation}
Then the discretized PDO \eqref{eq:pdoZ} becomes
\begin{equation}\label{eq:pdoZfully}
A f({\bm{x}}) = \sum_{{\bm{\xi}} \in \Xi} e^{2 \pi i {\bm{x}}\cdot {\bm{\xi}}/P} \left(\prod_{k=1}^d{\breve{\alpha}}_k(x_k)\right)\left(\prod_{k=1}^d{\breve{\beta}}_k(\xi_k)\right) \hat{f}({\bm{\xi}}), \quad {\bm{x}}\in \Xi=\{0,1, \ldots, P-1\}^d.
\end{equation}
In order to block-encode the PDO \eqref{eq:pdoZfully}, we adopt the following circuit, as shown in \Cref{fig:PDO3}, which is similar with the one in \Cref{fig:PDO2}:
\begin{figure}[ht]
\centerline{
\Qcircuit @C=1.5em @R=1.5em {
\lstick{\ket{0^{2d}}} & \qw&\qw& \qw&\multigate{1}{\bigotimes_{k=1}^d U_{{\breve{\alpha}}_k}}\qw &\qw&\meter \\
\lstick{\frac{1}{|f|}\sum_{{\bm{x}}}f(\frac{\x}{P})\ket{{\bm{x}}}} & \gate{{U_{\text{FT}}^\dag}^{\otimes d}}&\multigate{1}{\bigotimes_{k=1}^d U_{{\breve{\beta}}_k}}& \gate{{U_{\text{FT}}}^{\otimes d}} &\ghost{\bigotimes_{k=1}^d U_{{\breve{\alpha}}_k}}&\qw&&\frac{1}{|Af|}\sum_{{\bm{x}}}Af(\frac{\x}{P})\ket{{\bm{x}}} \\
\lstick{\ket{0^{2d}}} & \qw& \ghost{\bigotimes_{k=1}^d U_{{\breve{\beta}}_k}}&\qw &\qw\qw&\qw&\meter \\
}
}
\caption{Circuit for an explicit and efficient block encoding for fully separable PDOs \eqref{eq:pdoZfully}.
Here $\frac{1}{|f|}\sum_{{\bm{x}}}f(\frac{\x}{P})\ket{{\bm{x}}}$ is the normalized input data, $U_{\text{FT}}$ is the QFT circuit, $U_{{\breve{\beta}}_k}$ and $U_{{\breve{\alpha}}_k}$ are the block encodings of $D_{{\breve{\beta}}_k}$ and $D_{{\breve{\alpha}}_k}$, respectively, and the desired output is obtained with normalizing factor $\frac{1}{|Af|}$ when getting $\ket{0^{b_1+b_2}}$ for the $b_1+b_2$ ancilla qubits. Here $D_{{\breve{\beta}}_k}$ and $D_{{\breve{\alpha}}_k}$ are diagonal matrices defined in \eqref{eq:diagg}.}\label{fig:PDO3}
\end{figure}
Exploiting the fully separable structure of the symbol, one can construct explicit circuits for the diagonal multiplications shown in \Cref{fig:PDO3} by leveraging QSP and QET (see \Cref{subsec:QET}). To this end, we first introduce a lemma that gives Hermitian block encodings for two diagonal multiplication prototypes. During the preparation of this paper, we notice that a similar result is built in \cite{mcardle2022quantum} independently.
\begin{lemma}\label{lem:diagv}
For a fixed $P=2^p$, let $\bm{v}_-$, $\bm{v}_+$ be the vectors:
$$\bm{v}_- = \left(0,1,\ldots,\frac{P}{2}-1,-\frac{P}{2},-\frac{P}{2}+1,\ldots,-1\right),$$
and
$$\bm{v}_+ = \left(0,1,\ldots,\frac{P}{2}-1,\frac{P}{2},\frac{P}{2}+1,\ldots,P-1\right),$$
respectively, and $D_-$, $D_+$ be the diagonal matrices $\mathrm{diag}~\!(\bm{v}_-)$, $\mathrm{diag}~\!(\bm{v}_+)$, respectively. Then there is an $(1, 1)$-Hermitian-block-encoding of $\sin(\theta D_\sigma)$ with gate complexity $2p+5$, where $\sigma\in\{-,+\}$ and $\theta>0$ is a parameter.
\end{lemma}
\begin{proof}
For an arbitrary parameter $\theta>0$, we first construct matrices $R_- = \exp( i\theta D_-)$ and $R_+ = \exp( i\theta D_+)$ with quantum circuits. Using the binary representation of $\xi$, we get
\[
R_-\ket{\xi_{p-1}\cdots\xi_0} = e^{i((-\xi_{p-1})\xi_{p-2}\cdots\xi_0.)\theta}\ket{\xi_{p-1}\cdots\xi_0},
\]
and
\[
R_+\ket{\xi_{p-1}\cdots\xi_0} = e^{i(\xi_{p-1}\cdots\xi_0.)\theta}\ket{\xi_{p-1}\cdots\xi_0}.
\]
Then
\[
\begin{aligned}
R_+\ket{\xi_{p-1}\cdots\xi_0} &= e^{i(\xi_{p-1}\cdots\xi_0.)\theta}\ket{\xi_{p-1}\cdots\xi_0}=e^{i\sum_j\xi_j2^j\theta}\ket{\xi_{p-1}\cdots\xi_0}\\
& = \prod_je^{i\xi_j2^j\theta}\bigotimes_j\ket{\xi_j} = \bigotimes_je^{i\xi_j2^j\theta}\ket{\xi_j} = \bigotimes_jR_z(2^j\theta)\ket{\xi_j},
\end{aligned}
\]
and
\[
R_-\ket{\xi_{p-1}\cdots\xi_0} = e^{-i\xi_{p-1}2^p\theta}R_+\ket{\xi_{p-1}\cdots\xi_0} = \bigotimes_jR_z((-1)^{\delta_{j, p-1}}2^j\theta)\ket{\xi_j},
\]
so $R_+ = \bigotimes_jR_z(2^j\theta)$ and $R_- = \bigotimes_jR_z((-1)^{\delta_{j, p-1}}2^j\theta)$, where $R_z$ is the single qubit rotation defined in \Cref{subsec:note}. Let $U_\sigma$ be the circuit displayed in \Cref{fig:D}, where $\sigma\in\{-,+\}$, then $U_\sigma$ is a $(1, 1)$-Hermitian-block-encoding of $\sin(\theta D_\sigma)$.
\begin{figure}[ht]
\centerline{
\Qcircuit @C=1em @R=1.5em {
\lstick{\ket{0}}&\gate{S} & \gate{H} & \ctrlo{1} & \ctrl{1} & \gate{H}& \gate{X}& \gate{S} &\meter \\
\lstick{\ket{\psi}}&\qw & \qw & \gate{R_\sigma} & \gate{R_\sigma^\dag} & \qw& \qw& \qw &\qw &&\sin(\theta D_\sigma)\ket{\psi} \\
}
}
\caption{Hermitian block encoding of $\sin(\theta D_\sigma)$.} \label{fig:D}
\end{figure}
In fact, the matrix corresponding to $U_\sigma$ is
\[
\begin{aligned}
U_\sigma &= \begin{bmatrix}I&\\&iI\end{bmatrix}\begin{bmatrix}&I\\ I&\end{bmatrix}\frac{1}{\sqrt{2}}\begin{bmatrix}I&I\\ I&-I\end{bmatrix}\begin{bmatrix}R_\sigma&\\&R_\sigma^\dag\end{bmatrix}\frac{1}{\sqrt{2}}\begin{bmatrix}I&I\\ I&-I\end{bmatrix}\begin{bmatrix}I&\\&iI\end{bmatrix}\\
&= \begin{bmatrix}I&\\&iI\end{bmatrix}\begin{bmatrix}&I\\ I&\end{bmatrix}\frac{1}{2}\begin{bmatrix}R_\sigma+R_\sigma^\dag&R_\sigma-R_\sigma^\dag\\mathbb{R}_\sigma-R_\sigma^\dag&R_\sigma+R_\sigma^\dag\end{bmatrix}\begin{bmatrix}I&\\&iI\end{bmatrix}\\
&= \frac{1}{2}\begin{bmatrix}R_\sigma-R_\sigma^\dag&i(R_\sigma+R_\sigma^\dag)\\i(R_\sigma+R_\sigma^\dag)&-(R_\sigma-R_\sigma^\dag)\end{bmatrix} = i\begin{bmatrix}\sin(\theta D_\sigma)&\cos(\theta D_\sigma)\\\cos(\theta D_\sigma)&-\sin(\theta D_\sigma)\end{bmatrix},\\
\end{aligned}
\]
and this is a Hermitian matrix with the first diagonal block being $\sin(\theta D_\sigma)$ after ignoring the global phase factor $i$, since $\begin{bmatrix}\sin(\theta D_\sigma)&\cos(\theta D_\sigma)\\\cos(\theta D_\sigma)&-\sin(\theta D_\sigma)\end{bmatrix}$ is Hermitian. It can be seen from \Cref{fig:D} that the number of elementary gates used is $2p+5$.
\end{proof}
Now, we aim to construct the block encoding of diagonal matrices $D_{{\breve{\alpha}}_j}={\breve{\alpha}}_j(D_+)$ and $D_{{\breve{\beta}}_j}={\breve{\beta}}_j(D_+)=\beta_j(D_-)$, namely $U_{{\breve{\alpha}}_k}$ and $U_{{\breve{\beta}}_k}$ in \Cref{fig:PDO3}. For the case where $\alpha_j(x_j)=\exp(i\theta x_j)$ or $\beta_j(\xi_j)=\exp(i\theta \xi_j)$, the matrices $R_- = \exp( i\theta D_-)$ and $R_+ = \exp( i\theta D_+)$ constructed in \Cref{lem:diagv} are exactly the diagonal matrices $D_{{\breve{\beta}}_j}=\beta_j(D_-)$ and $D_{{\breve{\alpha}}_j}={\breve{\alpha}}_j(D_+)$, respectively. Therefore, we devote the rest of this section to the case where $\beta_j$ and $\alpha_j$ are even or odd real functions. Thanks to the block encodings $U_+$ and $U_-$ introduced in \Cref{lem:diagv}, what remains to do is to find polynomial approximations of ${\breve{\alpha}}$ and $\beta$ so as to complete the QET procedure described in \Cref{subsec:QET}. Specifically, we restrict the parameter $\theta$ to be $0<\theta<\frac{\pi}{2P}$ in order to recover $\theta D_\sigma$ from $\sin(\theta D_\sigma)$ with the arcsin function. Going through the QET procedure, one obtains the following result.
\begin{prop}\label{prop:QET}
Let $U_-$ and $U_+$ be the $(1, 1)$-Hermitian-block-encodings of $\sin(\theta D_-)$ and $\sin(\theta D_+)$ constructed in \Cref{lem:diagv} for $0<\theta<\frac{\pi}{2P}$ with $P=2^p$. Assume that $g$ is an even (resp. odd) continuous real function on $[-P,P]$, and that $\deg_g(\epsilon)$ is the smallest positive integer such that there exists an even (resp. odd) polynomial $\tilde{g}$ with the degree bounded by $\deg_g(\epsilon)$ satisfying
\begin{equation}\label{eq:qspasp}
\sup_{-\sin(P\theta)\leq x\leq \sin(P\theta)}\left|C_g\tilde{g}({\bm{x}})-g\left(\frac{1}{\theta}\arcsin x\right)\right|<\epsilon, ~ \|\tilde{g}\|\leq 1,
\end{equation}
where $\|\cdot\|$ is the $L^\infty$ norm on $[-1,1]$ and $C_g$ is a constant such that $C_g\geq\sup|g|$.
Then there is a $(C_g, 2, \epsilon)$-block-encoding for both $g(D_-)$ and $g(D_+)$ with $\mathcal{O}(p\deg_g(\epsilon))$ gates, where $D_-$ and $D_+$ are defined in \Cref{lem:diagv}.
\end{prop}
\begin{proof}
The circuit in \Cref{fig:QET} with $U_A$ replaced by $U_\sigma$ gives a $(C_g, 2)$-block-encoding for $\tilde{g}(\sin(\theta D_\sigma))$, and since
\[
\sup_{-\sin(P\theta)\leq x\leq \sin(P\theta)}\left|C_g\tilde{g}({\bm{x}})-g\left(\frac{1}{\theta}\arcsin x\right)\right|<\epsilon,
\]
we have
\[
\left\|C_g\tilde{g} (\sin(\theta D_\sigma))-g\left(\frac{1}{\theta}\arcsin (\sin(\theta D_\sigma))\right)\right\|<\epsilon,
\]
where the operator $2$-norm is used.
Thus the circuit in \Cref{fig:QET} with $U_A$ replaced by $U_\sigma$ gives a $(C_g, 2, \epsilon)$-block-encoding for
\[
g\left(\frac{1}{\theta}\arcsin (\sin(\theta D_\sigma))\right) = g\left(\frac{1}{\theta}\cdot\theta D_\sigma\right) = g(D_\sigma),
\]
where $\sigma\in\{-,+\}$ and we have used the fact that $0<\theta<\frac{\pi}{2P}$ in the first equality. Since $\mathcal{O}(p)$ gates are used in $U_\sigma$, the gate complexity of the circuit described above is $\mathcal{O}(p\deg_g(\epsilon))$, which closes the proof.
\end{proof}
The gate complexity of the circuit built with QET in \Cref{prop:QET} depends on the smoothness of $g$. For instance, we have the following corollary.
\begin{cor}\label{cor:analytic}
Assume that $g$ is an even (resp. odd) differentiable real function on $[-\frac{\pi}{2\theta},\frac{\pi}{2\theta}]$. Let $\dg{g}{\epsilon}{\theta}$ be the smallest integer such that there exists a polynomial $u$ with degree $\dg{g}{\epsilon}{\theta}$ satisfying $\sup_{|y|<\frac{\pi}{2\theta}}|u(y)-g(y)|<\frac{\epsilon}{3}$, then the gate complexity of the circuit used in \Cref{prop:QET} is
$$\mathcal{O}\left(p\log\left(\frac{C_g'}{\theta\epsilon}\right)\dg{g}{\epsilon}{\theta}\right),$$
where $C_g' = \sup_{|y|<\frac{\pi}{2\theta}}|g'(y)|$. In particular, if $g$ is a polynomial, the gate complexity reduces to $\mathcal{O}\left(p\log\left(\frac{C_g'}{\theta\epsilon}\right)\deg g\right)$.
\end{cor}
\begin{proof}
Without loss of generality, assume $1>\frac{\epsilon}{2C_g}$, otherwise we can just let $\tilde{g} = 0$ \eqref{eq:qspasp}. Since $\frac{1}{\theta}\arcsin(x)$ is an analytic function whose power series centered at $x=0$ has convergence radius $1>\sin(P\theta)$, there is a truncation $v$ of the Taylor series of $\frac{1}{\theta}\arcsin(x)$ with degree $\mathcal{O}(\log\left(\frac{C_g'}{\theta\epsilon}\right))$ such that $\sup_{|x|< \sin(P\theta)}|v(x)-\frac{1}{\theta}\arcsin(x)|<\frac{\epsilon}{3C_g'}$. Now since the coefficients of the Taylor series of $\frac{1}{\theta}\arcsin(x)$ at $x=0$ are all non-negative, it holds that $v([-1,1])\subset [-\frac{\pi}{2\theta},\frac{\pi}{2\theta}]$. Let $\bar{g}(x)=\frac{1}{C_g}u(v(x))$ and $\tilde{g} = (1-\frac{\epsilon}{3C_g})\bar{g}$, then $\tilde{g}$ has degree $\mathcal{O}(\log\left(\frac{C_g'}{\theta\epsilon}\right)\dg{g}{\epsilon}{\theta})$, and
\[
\begin{aligned}
\abs{C_g\bar{g}(x)-g\left(\frac{1}{\theta}\arcsin(x)\right)}
&\leq \abs{u(v(x))-g(v(x))}+\abs{g(v(x))-g\left(\frac{1}{\theta}\arcsin(x)\right)}\\
&< \frac{\epsilon}{3} + \sup_{|y|<\frac{\pi}{2\theta}}|g'(y)|\abs{v(x)-\frac{1}{\theta}\arcsin(x)}
< \frac{\epsilon}{3} + C_g'\cdot\frac{\epsilon}{3C_g'}=\frac{2\epsilon}{3},\\
\end{aligned}
\]
for $x\in [-\sin(P\theta), \sin(P\theta)]$. In addition, since $v$ maps $[-1,1]$ into $[-\frac{\pi}{2\theta},\frac{\pi}{2\theta}]$, we have
\[
\begin{aligned}
|\bar{g}(x)|&\leq\sup_{|y|<\frac{\pi}{2\theta}}\abs{\frac{1}{C_g}u(y)}
\leq \sup_{|y|<\frac{\pi}{2\theta}}\left|\frac{1}{C_g}u(y)-\frac{1}{C_g}g(y)\right|+\sup_{|y|<\frac{\pi}{2\theta}}\left|\frac{1}{C_g}g(y)\right|
\le\frac{\epsilon}{3C_g}+1,
\end{aligned}
\]
and therefore $\abs{\tilde{g}(x)} = (1-\frac{\epsilon}{3C_g})\abs{\bar{g}(x)}<1$ for $x\in[-1,1]$. In addition, we have
\[
\begin{aligned}
\abs{C_g\tilde{g}(x)-g\left(\frac{1}{\theta}\arcsin(x)\right)}
&= \abs{(1-\frac{\epsilon}{3C_g})\left(C_g\bar{g}(x)-g\left(\frac{1}{\theta}\arcsin(x)\right)\right)+\frac{\epsilon}{3C_g}g\left(\frac{1}{\theta}\arcsin(x)\right)}\\
&\leq (1-\frac{\epsilon}{3C_g})\frac{2\epsilon}{3}+\frac{\epsilon}{3C_g}C_g
< \frac{2\epsilon}{3} + \frac{\epsilon}{3}=\epsilon,\\
\end{aligned}
\]
for $x\in [-\sin(P\theta), \sin(P\theta)]$. According to \Cref{prop:QET}, we have $\deg_g(\epsilon)=\mathcal{O}(\log\left(\frac{C_g'}{\theta\epsilon}\right)\dg{g}{\epsilon}{\theta})$ and the gate complexity of the circuit used in \Cref{prop:QET} is $\mathcal{O}\left(p\log\left(\frac{C_g'}{\theta\epsilon}\right)\dg{g}{\epsilon}{\theta}\right)$, where the factor $p$ comes from preparing $\sin(\theta D_{\sigma})$ as mentioned in \cref{lem:diagv}. In the case that $g$ is a polynomial, we can simply let $u=g$ and thus $\dg{g}{\epsilon}{\theta} \leq \deg g$.
\end{proof}
For the final step, we first introduce the notation
\begin{equation}\label{eq:de}
\de{\epsilon} \equiv \sum_{k=1}^d\left[\deg_{{\breve{\alpha}}_k}(\epsilon)+\deg_{\beta_k}(\epsilon)\right],
\end{equation}
where $\deg_{{\breve{\alpha}}_k}(\epsilon)$ and $\deg_{\beta_k}(\epsilon)$ are defined in \Cref{prop:QET}. Denote by $U_{{\breve{\alpha}}_k}$ and $U_{{\breve{\beta}}_k}$ the block encodings of ${\breve{\alpha}}_k(D_+)$ and ${\breve{\beta}}_k(D_+)=\beta_k(D_-)$ obtained in \Cref{prop:QET} with $\epsilon$ replaced by $\epsilon/2dC$, respectively, where
\begin{equation}\label{eq:const}
C = \widetilde{C}_\alpha \widetilde{C}_\beta, ~ \widetilde{C}_\alpha=\prod_{k=1}^d C_{{\breve{\alpha}}_k}, ~ \widetilde{C}_\beta=\prod_{k=1}^dC_{{\breve{\beta}}_k},
\end{equation}
and $C\geq\prod_{k=1}^d(\sup|{\breve{\alpha}}_k|\sup|{\breve{\beta}}_k|)$ since $C_{{\breve{\alpha}}_k}\geq\sup|{\breve{\alpha}}_k|$ and $C_{{\breve{\beta}}_k}\geq\sup|{\breve{\beta}}_k|$.
Now we are ready to prove the following theorem that relies on the block encodings $\bigotimes_{k=1}^d U_{{\breve{\alpha}}_k}$ and $\bigotimes_{k=1}^d U_{{\breve{\beta}}_k}$ in \Cref{fig:PDO3}.
\begin{theorem}\label{thm:fullysep}
If $a({\bm{x}}, {\bm{\xi}})=\alpha({\bm{x}})\beta({\bm{\xi}})= \alpha_1(x_1)\cdots\alpha_d(x_d)\beta_1(\xi_1)\cdots\beta_d(\xi_d)$ is a fully separable symbol as defined in \Cref{def:fullysep}, then the corresponding PDO defined by \eqref{eq:pdoZfully} can be $(C, \mathcal{O}(d), \epsilon)$-block-encoded with gate complexity $\mathcal{O}(dp\de{\frac{\epsilon}{2dC}}+dp^2)$ using the circuit displayed in \Cref{fig:PDO3}, where $C > 0$ is a constant defined in \eqref{eq:const}, and $\de{\frac{\epsilon}{2dC}}$ is defined in \eqref{eq:de}.
\end{theorem}
\begin{proof}
Since each $U_{{\breve{\alpha}}_k}$ is a $(C_{{\breve{\alpha}}_k}, 2, \epsilon/2dC)$-block-encoding for ${\breve{\alpha}}_k(D_+)$ according to \Cref{prop:QET}, $\bigotimes_{k=1}^d U_{{\breve{\alpha}}_k}$ is a $(\prod_{k=1}^d C_{{\breve{\alpha}}_k}, 2d, \epsilon/2\widetilde{C}_\beta)$-block-encoding for
\[\bigotimes_{k=1}^d{\breve{\alpha}}_k(D_+)=\bigotimes_{k=1}^d D_{{\breve{\alpha}}_k}=D_{\breve{\alpha}}.\]
Similarly, $\bigotimes_{k=1}^d U_{{\breve{\beta}}_k}$ is a $(\prod_{k=1}^d C_{{\breve{\beta}}_k}, 2d, \epsilon/2\widetilde{C}_\alpha)$-block-encoding for
\[\bigotimes_{k=1}^d{\breve{\beta}}_k(D_+)=\bigotimes_{k=1}^d D_{{\breve{\beta}}_k}=D_{\breve{\beta}}.\]
Hence by the same argument as the proof of \Cref{thm:sep} (especially \eqref{eq:state1}, \eqref{eq:b}, \eqref{eq:phib}, \eqref{eq:state2} and \eqref{eq:a}), the circuit in \Cref{fig:PDO3} gives a $(C, 4d, \epsilon)$ block encoding of the PDO \eqref{eq:pdoZfully} with gate complexity $\mathcal{O}(p\de{\frac{\epsilon}{2dC}}+p^2d)$, where the $\mathcal{O}(p^2d)$ term comes from the QFT part of the circuit.
\end{proof}
\subsection{Linear combination of fully separable terms}
Similar to \Cref{cor:sep}, we can block-encode the PDO for a linear combination of fully separable terms, i.e.,
\[
a({\bm{x}},{\bm{\xi}}) = \sum_{j=0}^{m-1} y_ja_j({\bm{x}},{\bm{\xi}}) =\sum_{j=0}^{m-1} y_j\alpha_{j1}(x_1)\cdots\alpha_{jd}(x_d)\beta_{j1}(\xi_1)\cdots\beta_{jd}(\xi_d),
\]
with LCU (see \Cref{subsec:LCU}) and \Cref{thm:fullysep}, which is stated in the following corollary.
\begin{cor}\label{cor:fullysep}
For a linear combination of fully separable symbols $a({\bm{x}},{\bm{\xi}}) = \sum_{j=0}^{m-1} y_ja_j({\bm{x}},{\bm{\xi}}) =\sum_{j=0}^{m-1} y_j\alpha_{j1}(x_1)\cdots\alpha_{jd}(x_d)\beta_{j1}(\xi_1)\cdots\beta_{jd}(\xi_d)$, where $\underset{[0,1]}{\sup}~|\alpha_{jk}|\leq1$, $\underset{[-P/2,P/2]}{\sup}|\beta_{jk}|\leq1$ and $y\in\mathbb{C}^m$ with $\|y\|_1\leq\delta$, assume that $(U_L, U_R)$ is a pair of unitaries described in \Cref{lem:LCU} with $\epsilon_1=\epsilon$, and $W=\sum_{j=0}^{m-1}\ket{j}\bra{j}\otimes U_j+\sum_{j=m}^{2^b-1}\ket{j}\bra{j}\otimes I_{a+s}$, where each $U_j$ is a $(1,a, \epsilon)$-block-encoding of the discretized PDO $A_j$ associated with symbol $a_j({\bm{x}},{\bm{\xi}})$ (see \eqref{eq:pdoZfully}) constructed in \Cref{thm:fullysep} with $a=\mathcal{O}(d)$. Then $(U_L^\dag\otimes I_{a+s})W(U_R\otimes I_{a+s})$ is an $(\delta, a+b, (1+\delta)\epsilon)$-block-encoding of $\sum_{j=0}^{m-1}y_jA_j$, where $I_{a+s}$ denotes the identity operator with size $2^{a+s}\times2^{a+s}$. The gate complexity of the corresponding circuit is $\mathcal{O}(dp\sum_{j=0}^{m-1}\de{\frac{\epsilon}{2d}}+dp^2m)$.
\end{cor}
\section{Applications}\label{sec:app}
In this section, we provide worked examples for particular symbols using the circuit shown in \Cref{fig:PDO3} and provide complexity analysis, beginning with a variable coefficient second-order elliptic operator.
\subsection{Second-order elliptic operator with variable coefficients}\label{subsec:varellip}
Recall that the elliptic operator introduced in \eqref{eq:ellip} is of the following form:
\begin{equation}
(Au)({\bm{x}}) = u({\bm{x}})-\nabla\cdot (\omega({\bm{x}})\nabla u({\bm{x}})).\label{eq:elliptic}
\end{equation}
In this section, we assume that $\omega({\bm{x}})>0$ has a low-rank Fourier expansion
\begin{equation}\label{eq:lowrank}
\omega({\bm{x}}) = \sum_{j=1}^r c_j \exp\left(2\pi i \q_j \cdot {\bm{x}} \right),\quad \q_j\in\mathbb{Z}^d.
\end{equation}
Many commonly seen functions have low-rank expansions or approximations. For instance, $\omega({\bm{x}}) = 2+\sin(2\pi\sum_{l=1}^d x_l) > 0$ can be written in the rank-3 form
\[\omega({\bm{x}}) = 2 + \frac{ i}{2}(\exp(-2\pi i(x_1+\cdots+x_d))-\exp(2\pi i(x_1+\cdots+x_d))).\]
Plugging the form \eqref{eq:lowrank} of $\omega$ into \eqref{eq:ellipsym}, one obtains the symbol associated with the PDO above
\[
a({\bm{x}},{\bm{\xi}}) = 1+\sum_{j=1}^r\sum_{l=1}^d(4\pi^2Pq_{jl}c_j)e^{2\pi i \q_j\cdot {\bm{x}}}\frac{\xi_l}{P}+\sum_{j=1}^r\sum_{l=1}^d(4\pi^2P^2c_j)e^{2\pi i \q_j\cdot {\bm{x}}}\frac{\xi_l^2}{P^2},
\]
where $P=2^p$ is the number of discrete points used for each dimension (see \Cref{subsec:disc}). Notice that the terms $e^{2\pi i \q_j\cdot {\bm{x}}}\frac{\xi_l}{P}$ and $e^{2\pi i \q_j\cdot {\bm{x}}}\frac{\xi_l^2}{P^2}$ above are fully separable by \Cref{def:fullysep}, thus by \Cref{cor:fullysep}, we know that the corresponding PDO can be block-encoded. As explained in \Cref{sec:fully sep}, the multiplication of $e^{2\pi i \q_j\cdot {\bm{x}}}=\prod_{l=1}^de^{2\pi i q_{jl} x_l}$ can be implemented directly using $R_-$ and $R_+$ constructed in \Cref{lem:diagv}.
Consequently, the number of gates needed for multiplying each $e^{2\pi iq_{jl} x_l}$ factor without error is $\mathcal{O}(p)$, and no ancilla qubits are used. Since $\frac{\xi_l}{P}$ and $\frac{\xi_l^2}{P^2}$ are polynomials, by \Cref{cor:analytic}, the multiplication of each $\frac{\xi_l}{P}$ and $\frac{\xi_l^2}{P^2}$ factor can be implemented with $\mathcal{O}\left(p\log\left(\frac{1}{\epsilon}\right)\right)$ gates to $\mathcal{O}(\epsilon)$ precision, and $\mathcal{O}(d)$ ancilla qubits are used. Going through the proof of \Cref{thm:fullysep}, one can see that the PDO associated with $e^{2\pi i \q_j\cdot {\bm{x}}}\frac{\xi_l}{P}$ and $e^{2\pi i \q_j\cdot {\bm{x}}}\frac{\xi_l^2}{P^2}$ can be $(1, \mathcal{O}(d),\epsilon)$-block-encoded with gate complexity $\mathcal{O}(p\log(\frac{1}{\epsilon})+p^2+dp)$, where the three terms account for implementing the polynomials of $\xi_l$, the QFT of the $l$-th component, and the multiplication of $e^{2\pi i \q_j\cdot {\bm{x}}}$, respectively. Finally, going through the LCU step as in \Cref{cor:fullysep} with $\mathcal{O}(dr)$ terms, one obtains a $(\gamma, \mathcal{O}(d+\log(dr)), (1+\gamma)\epsilon)$-block-encoding of the PDO \eqref{eq:ellip} with total gate complexity $\mathcal{O}(dr(p\log\left(\frac{1}{\epsilon}\right)+p^2+dp))=\mathcal{O}\left(dpr(\log\frac{1}{\epsilon}+d+p)\right)$, where $\gamma=1+4\pi^2(P\sum_{j=1}^r|c_j|\norm{\q_j}_1+P^2d\sum_{j=1}^r |c_j|)$. This result is summarized in the following theorem, where we have used $\mathcal{O}(d+\log(dr))=\mathcal{O}(d+\log(r))$.
\begin{theorem}\label{thm:elliptic}
For the elliptic operator \eqref{eq:ellip} with variable coefficient, where $\omega({\bm{x}})$ has a low-rank expansion \eqref{eq:lowrank}, there exists a $(\gamma, \mathcal{O}(d+\log(r)), (1+\gamma)\epsilon)$-block-encoding for the corresponding discretized PDO defined in \eqref{eq:pdoZfully} with gate complexity
\[\mathcal{O}\left(dr\log P\left(\log\frac{P}{\epsilon}+d\right)\right),\]
where $\gamma=1+4\pi^2(P\sum_{j=1}^r|c_j|\norm{q_j}_1+P^2d\sum_{j=1}^r |c_j|)$.
\end{theorem}
Similar to previous sections and by a slight abuse of notation, we denote the discretization of the operator defined in \eqref{eq:elliptic} also by $A$. Now we can use the QLSA in \cite{costa2022optimal} to get the following corollary:
\begin{cor}
Let $(A, b)$ be the discretization of the operator and the right-hand side of \eqref{eq:elliptic}, respectively, there is a quantum algorithm finding the normalized state $\ket{A^{-1}b}=\frac{A^{-1}b}{\|A^{-1}b\|}$ within error $\epsilon$ with gate complexity
\[
\mathcal{O}\left(\gamma dr\log\frac{1}{\epsilon}\log P\left(\log\frac{\gamma P}{\epsilon}+d\right)\right).
\]
\end{cor}
\begin{proof}
It is clear that $\norm{A} \le \gamma$ and $\norm{A^{-1}} \le 1$. According to \cref{thm:elliptic}, one can construct $U_A$, the $(\gamma, \mathcal{O}(d+\log(r)), \epsilon/\gamma)$-block-encoding of $A$, with complexity
\[\mathcal{O}\left(dr\log P\left(\log\frac{\gamma P}{\epsilon}+d\right)\right).\]
In other words, $U_A$ is a $(1, \mathcal{O}(d+\log(r)), 0)$-block-encoding of some matrix $\tilde{A}/\gamma$ such that $\|A-\tilde{A}\|<\epsilon/\gamma$. Therefore, we have $\norm{\tilde{A}} = \mathcal{O}(\gamma)$, $\norm{\tilde{A}^{-1}} = \mathcal{O}(1)$, and thus $\kappa(\tilde{A}) = \mathcal{O}(\gamma)$. The main theorem of \cite{costa2022optimal} gives a quantum algorithm that can output a state $\ket{y}$ that is $\mathcal{O}(\epsilon)$ close to $\ket{(\tilde{A}/\gamma)^{-1}b} = \ket{\tilde{A}^{-1}b} = \frac{\tilde{A}^{-1}b}{\|\tilde{A}^{-1}b\|}$, using
\[
\mathcal{O}\left(\kappa(\tilde{A})\log{\frac{1}{\epsilon}}\right) = \mathcal{O}\left(\gamma\log{\frac{1}{\epsilon}}\right)
\]
queries of $U_A$. Finally, we have the estimation
\begin{equation*}
\begin{aligned}
\norm{\ket{A^{-1}b}-\ket{\tilde{A}^{-1}b}} &= \left\|\frac{A^{-1}b}{\norm{A^{-1}b}}-\frac{\tilde{A}^{-1}b}{\norm{\tilde{A}^{-1}b}}\right\|\\
&\le \frac{\norm{(A^{-1}-\tilde{A}^{-1})b}}{\norm{A^{-1}b}} + \norm{\tilde{A}^{-1}b}\left|\frac{1}{\norm{A^{-1}b}}-\frac{1}{\norm{\tilde{A}^{-1}b}}\right|\\
&\le \frac{\norm{A^{-1}}\norm{\tilde{A}^{-1}}\norm{A-\tilde{A}}\norm{b}}{\norm{A}^{-1}\norm{b}} + \frac{\norm{A^{-1}}\norm{\tilde{A}^{-1}}\norm{A-\tilde{A}}\norm{b}}{\norm{A}^{-1}\norm{b}}\\
&= \mathcal{O}\left(\gamma\cdot\frac{\epsilon}{\gamma}\right) = \mathcal{O}(\epsilon).
\end{aligned}
\end{equation*}
So $\ket{y}$ is also an $\mathcal{O}(\epsilon)$ approximation of $\ket{A^{-1}b}$ and the overall gate complexity is
\[
\mathcal{O}\left(\gamma dr\log\frac{1}{\epsilon}\log P\left(\log\frac{\gamma P}{\epsilon}+d\right)\right).
\]
\end{proof}
\subsection{Application for constant coefficient elliptic operator}\label{subsec:appinverse}
In this part, we investigate the multiplier operators (see \eqref{eq:multiplier}), i.e., the PDOs with symbols of the form $a({\bm{x}},{\bm{\xi}}) = \beta({\bm{\xi}})$. In particular, we showcase how to directly block-encode the inverse of multiplier operators based on the results obtained in previous sections, so that one can solve a discretized system of a PDE without invoking QLSAs (quantum linear system algorithms).
In practice, it is quite often the case that $\beta({\bm{\xi}})$ is radially symmetric. For example, many operators related to Laplacian have radial symmetric symbols, since the symbol of $\Delta$ is $-4\pi^2|{\bm{\xi}}|^2$. For the rest of this section, we focus on the radially symmetric symbol $\beta({\bm{\xi}})=\varphi(|{\bm{\xi}}|)$. The main idea of dealing with radially symmetric $\beta$ is to consider an approximation in the following form
\begin{equation}\label{eq:fullysepidea}
\beta({\bm{\xi}})=\varphi(|{\bm{\xi}}|)\approx \sum_{m=1}^M w_m e^{-a_m |{\bm{\xi}}|^2}=\sum_{m=1}^M w_m\prod_{i=1}^d e^{-a_m \xi_i^2}.
\end{equation}
Notice that the right-hand side is in the fully separable form by \Cref{def:fullysep}, thus the results from \Cref{sec:fully sep} can be used. In fact, the authors of \cite{beylkin2005approximation} developed an efficient algorithm to find a low-rank approximation of a single-variable function $f(y)$ by exponential sums
\begin{equation}
f(y)\approx\sum_{m=1}^M w_m e^{-a_m y^2},\label{eq:exp approx}
\end{equation}
for a large range of even functions $f(y)$, especially those whose amplitude decrease as $y\to\infty$.
Before diving into concrete examples, we first introduce a near-optimal polynomial approximation to the exponential functions in the following lemma, which is a direct corollary of \cite{sachdeva2014faster}*{Theorem 4.1}.
\begin{lemma}\label{lem:approx}
For every $a, b>0$, and $0<\delta \leq 1$, there exists an even polynomial $r(y)$ satisfying
\[
\sup _{y \in[0, b]}\left|e^{-ay^2}-r(y)\right| \leq \delta,\quad \sup _{y \in[0, b]}\left|r(y)\right| \leq 1,
\]
and has degree $\mathcal{O}\left(\sqrt{\max \{ab^2, \log 1 / \delta\} \cdot \log 1 / \delta}\right)$.
\end{lemma}
With \Cref{lem:approx} and established results in \Cref{sec:fully sep}, we are ready to give the following block encoding result for the diagonal multiplication $D_\beta$ associated with $\beta({\bm{\xi}})$ (see \eqref{eq:diagg}).
\begin{theorem}\label{thm:ex2}
If $\beta({\bm{\xi}})$ can be approximated in the following sense
\[
\left|\beta({\bm{\xi}}) - \sum_{m=1}^M w_m e^{-a_m |{\bm{\xi}}|^2}\right| \leqslant \epsilon, \quad \text { for all } {\bm{\xi}}\in\left[-\frac{P}{2},\frac{P}{2}\right]^d,
\]
with $w_m, a_m\ge 0$ and $R := \max_m a_m$, $W := \sum_{m=1}^M|w_m|$, then we can implement a $(\gamma,q,\epsilon)$-block-encoding for the PDO associated with $\beta$ (defined in \eqref{eq:pdoZ}) with
\[
\gamma = \mathcal{O}(W),\quad q = \mathcal{O}\left(d+\log M\right),
\]
and gate complexity
\[
\mathcal{O}\left(dM\log P\log\left(\frac{dWRP}{\epsilon}\right)\sqrt{\max\left\{RP^2,\log\frac{Wd}{\epsilon}\right\}\log\frac{Wd}{\epsilon}}\right).
\]
\end{theorem}
\begin{proof}
For each exponential term $g_m(z) = e^{-a_mz^2}$, we know
\[
\dg{g_m'}{\epsilon}{\theta}= \mathcal{O}\left(\sqrt{\max\left\{\frac{R}{\theta^2},\log\frac{1}{\epsilon}\right\}\log\frac{1}{\epsilon}}\right)
\]
according to \Cref{lem:approx}, where $\dg{g_m'}{\epsilon}{\theta}$ is defined in \Cref{cor:analytic}. Moreover, a simple calculation shows that
\begin{equation}
C_{g_m'} \leq \sup |g_m'(z)| = \sup \abs{2a_m z e^{-a_mz^2}} < \sqrt{a_m}\le \sqrt{R}.
\end{equation}
Therefore, if we let $\theta = \frac{\pi}{3P}$, then the complexity of implementing a $(1,1,\frac{\epsilon}{2Wd})$-block-encoding of matrix $\exp(-a_m (D_-)^2)$ is
$$\mathcal{O}\left(\log P\log\left(\frac{dWRP}{\epsilon}\right)\sqrt{\max\left\{RP^2,\log\frac{Wd}{\epsilon}\right\}\log\frac{Wd}{\epsilon}}\right),$$
according to \cref{cor:analytic}.
After implementing $\exp(-a_m (D_-)^2)$ for each $\xi_k$ register ($k=1,\ldots,d$), we get a $(1,d,\frac{\epsilon}{2W})$-block-encoding of $e^{-a_m|{\bm{\xi}}|^2}$. Since there are $M$ such terms, after conducting the LCU step the total gate complexity becomes
\begin{equation}\label{eq:pdeconst}
\mathcal{O}\left(dM\log P\log\left(\frac{dWRP}{\epsilon}\right)\sqrt{\max\left\{RP^2,\log\frac{Wd}{\epsilon}\right\}\log\frac{Wd}{\epsilon}}\right).
\end{equation}
Since only one QFT and one iQFT are needed, which has complexity $\mathcal{O}(d\log^2 P)$, the dominating term in the above complexity formula remains unchanged. The total error of this block encoding is $\sum_{m=1}^M |w_m|\frac{\epsilon}{2W} < \epsilon$ as desired. Since $\mathcal{O}(d)$ ancillas are used when encoding $e^{-a_m|{\bm{\xi}}|^2}$ and $\mathcal{O}(\log M)$ ancillas are used for LCU, the total number of ancilla is $\mathcal{O}(d+\log M)$.
\end{proof}
With \Cref{thm:ex2}, we are ready to work on a concrete example. Consider the following $d$-dimensional elliptic equation with periodic boundary conditions:
\begin{equation}\label{eq:PDEex}
-\frac{1}{4\pi^2}\Delta u({\bm{x}}) + u({\bm{x}}) = b({\bm{x}}),\quad x\in[0,1]^d.
\end{equation}
As explained by \eqref{eq:fullysepidea} and \eqref{eq:exp approx}, the idea is to expand $\varphi(y)=\frac{1}{1+y^2}$ as the sum of a series of exponential functions.
To this end, we introduce a
result for exponential approximations in the form of \eqref{eq:fullysepidea} and \eqref{eq:exp approx}.
\begin{lemma}\label{lem:exp1}
For any $0<\delta \leqslant 1$ and $0<\epsilon \leqslant \frac{1}{2}$, there exist positive numbers $p_m$ and $v_m$ such that
\begin{equation}
\left|r^{-1}-\sum_{m=1}^M v_m e^{-p_m r}\right| \leqslant r^{-1} \epsilon, \quad \text { for all } \delta \leqslant r \leqslant 1, \label{eq:exp2}
\end{equation}
with
\begin{equation}
M=\mathcal{O}\left(\log \epsilon^{-1}\left(\log \epsilon^{-1}+\log \delta^{-1}\right)\right),\label{eq:exp_m}
\end{equation}
and
\begin{equation}
\max_m p_m = \mathcal{O}\left(\delta^{-1}\log \epsilon^{-1}\left(\log \epsilon^{-1}+\log \delta^{-1}\right)\right).\label{eq:exp_p}
\end{equation}
\end{lemma}
The proof of \eqref{eq:exp_m} can be found in \cite{beylkin2005approximation}*{Theorem A.1}, and \eqref{eq:exp_p} is also implied in the proof there. We summarize the construction given in \cite{beylkin2005approximation} in passing. Denote $f_r(t) = e^{-r e^t+t}$, then $r^{-1} = \int_{-\infty}^{\infty}f_r(t)\d t$ and $f_r(t)$ decays rapidly when $|t|\rightarrow\infty$. Therefore one can approximate this integral using the trapezoidal rule on a finite interval $[a,b]$ with step size $h$, which is
\begin{equation}
h\left(\sum_{k=1}^{K-1} f_r(a+k h)+\frac{f_r(a)+f_r(b)}{2}\right),\label{eq:trap}
\end{equation}
where $K = (b-a)/h$. Since each term of \eqref{eq:trap} is of the form $ve^{-pr}$, the approximate integral formula above actually provides an approximation of the form \eqref{eq:exp2}. Finally, by choosing $a = -\log\frac{4}{\epsilon}$, $b = \log \left(4\log\frac{4}{\epsilon} \delta^{-1} \log \left(2\log\frac{4}{\epsilon}(\delta \epsilon)^{-1}\right)\right)$ and $h\le \pi/(2\log\frac{4}{\epsilon}+1)$, one can show that the condition \eqref{eq:exp2} is satisfied, and from $M = K+1 = (b-a)/h+1$ one can check that \eqref{eq:exp_m} and \eqref{eq:exp_p} hold.
Now, with the substitutions $\delta = (\frac{dP^2}{4}+1)^{-1}$ and $r = (1+y^2)/(\frac{dP^2}{4}+1)$ in \Cref{lem:exp1} we get the following approximation:
\begin{equation}\label{eq:exp3}
\left|\frac{1}{1+y^2}-\sum_{m=1}^M w_m e^{-a_m y^2}\right| \leqslant \epsilon, \quad \text { for all } -\frac{\sqrt{d}P}{2} \leqslant y \leqslant \frac{\sqrt{d}P}{2},
\end{equation}
where $a_m = p_m/(\frac{dP^2}{4}+1)$, $w_m = e^{-a_m}v_m/(\frac{dP^2}{4}+1)$, and
\begin{equation}
\begin{aligned}
M &= \mathcal{O}\left(\log \epsilon^{-1}\left(\log \epsilon^{-1}+\log (dP)\right)\right),\\
R &= \max_m a_m = \mathcal{O}\left(\log \epsilon^{-1}\left(\log \epsilon^{-1}+\log (dP)\right)\right),\\
W &= \sum_{m=1}^M |w_m| = \mathcal{O}(1).
\end{aligned}
\end{equation}
Here the estimation of $W$ is deduced from plugging $y=0$ into \eqref{eq:exp3} and the positivity of $w_m$. Finally, after plugging the estimations of $M$, $R$ and $W$ into \Cref{thm:ex2}, we obtain the following result:
\begin{cor}\label{thm:invmult}
For the pseudo-differential operator associated with the symbol $\beta({\bm{\xi}})=1+|{\bm{\xi}}|^2$ (see \eqref{eq:pdeconst}), there is a $(\gamma,q,\epsilon)$-block-encoding for its inverse with $\gamma = \mathcal{O}(1)$, $q = \mathcal{O}\left(d+\log\log\frac{dP}{\epsilon}\right)$, and gate complexity $\mathcal{O}\left(dP\left(\log\frac{dP}{\epsilon}\right)^{2.5}\left(\log\frac{1}{\epsilon}\right)^{1.5}\left(\log\frac{d}{\epsilon}\right)^{0.5}\log P\right)$.
\end{cor}
\begin{remark}\label{rem:inv}
If one uses the uniform grid to discretize the operator $(-\frac{1}{4\pi^2}\Delta + 1)$ in equation \eqref{eq:PDEex}, then the condition number of the matrix obtained is at least $\kappa=\mathcal{O}(dP^2)$. This indicates that the complexity is at least $\mathcal{O}(dP^2)$ when block-encoding its inverse matrix using the previous method, such as LCU or QSVT. However, here we achieved $\tilde{\mathcal{O}}_{\epsilon}(dP)=\tilde{\mathcal{O}}_{\epsilon}(\sqrt{d\kappa})$ complexity for encoding the discretization of $(-\frac{1}{4\pi^2}\Delta + 1)^{-1}$, where we omit the logarithm terms in $\tilde{\mathcal{O}}$ and denote the dependence on $\epsilon$ by the subscript. This improvement also demonstrates the potential of directly working on the symbol level as we did in this section. When solving the corresponding problem \eqref{eq:pdeconst} with a particular right-hand side $b$, the worst case gate complexity becomes $\tilde{\mathcal{O}}_{\epsilon}(\kappa\sqrt{d\kappa})$ since the worst case success probability is $\mathcal{O}(1/\kappa)$, which is inferior comparing with the dependence on $\kappa$ obtained in \cite{costa2022optimal}. However, the block encoding scheme in this paper is independent of the right-hand side $b$, and thus and be used repeatedly for different $b$ without constructing the circuit again. Also, the circuit is simpler compared with the one in \cite{costa2022optimal}, making it more applicable in practice.
\end{remark}
\section{Conclusion and Discussion}\label{sec:con}
This paper systematically investigates block encodings for pseudo-differential operators (PDOs) under different structural assumptions. For PDOs with generic symbols, a block encoding scheme is developed in \Cref{sec:arithmD} and the quantum circuit is illustrated in \Cref{fig:PDO1}. For PDOs with linear combinations of separable symbols, we improve the success probability exponentially and present an efficient block encoding algorithm in \Cref{sec:separable}. Then a more explicit and practical block encoding scheme is derived in \Cref{sec:fully sep} with the help of QSP and QET, along with which the complexity analysis is provided. Plenty of worked examples are given in \Cref{sec:app}, including the block encoding of elliptic operators with a variable coefficient that is difficult to deal with for quantum solvers that use finite difference schemes, and the block encoding of the inverse of constant-coefficient elliptic operators without using quantum linear system algorithms. The block encoding schemes presented in this paper enrich the study of the block encoding of dense operators and shed new light on designing practical quantum circuits for scientific computing.
For future directions, one can apply the established results in this paper to other PDOs besides the ones presented in \Cref{sec:app}. One can also use the idea of symbol calculus to implement different operations on the PDO such as taking square root or exponential, which can be useful in solving certain PDEs in practice.
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\section{Introduction}\label{Intro}
The minimum distance of a Reed-Muller code is a parameter that have played an important role in their study, in particular, is related with the error correction conditions \cite{G, Geil_2012}. Mart\'inez-Bernal, Pitones and Villarreal \cite{MBPV} introduced an algebraic invariant analogue for a homogeneous ideal in a standard graded polynomial ring. In this manuscript, we study the generalized Hamming
distance and its algebraic version \cite{GSMBVV}. The generalized Hamming distance plays an important role in coding theory; however, determining this distance is a difficult problem. Wei \cite{Wei91} gives a series of results on generalized Hamming distance and indicates that it completely characterizes the performance of a linear code.
The $\ell$-th generalized Hamming distance of a $\mathbb{K}$-linear code $C$ is
$$
\delta_{\ell}(C)=\min_{D\subseteq C\; \&\; \dim_\mathbb{K} D=\ell}^{} \mid \chi(D)\mid,
$$
where $\chi(D)=\{i\mid \exists (v_1,\ldots, v_n)\in D, v_i\neq 0\}$.
With this motivation in mind, the generalized minimum distance function \cite{GSMBVV}
of a homogeneous ideal $I\subseteq S=K[x_1,\ldots,x_n]$ is defined by
$$
\delta_I(t,\ell)=\left\{\begin{array}{ll}\e(S/I)-\max\{\e(S/(I,F))\vert\,
F\in\mathcal{F}_{t,\ell}\}&\mbox{if }\mathcal{F}_{t,\ell}\neq\varnothing,\\
\e(S/I)&\mbox{if\ }\mathcal{F}_{t,\ell}=\varnothing,
\end{array}\right.
$$
where
$$\mathcal{F}_{t,\ell}=\{F=\{f_1,\ldots, f_\ell\}\subseteq S_t \mid f_1,\ldots, f_{\ell} \text{ are } \mathbb{K}\text{-linearly independent in } S/I, (I\colon (F))\neq I\}.$$
We note that this numerical invariant depends only of the structure of $R=S/I$ as a graded algebra, and not on its presentation (see Definition \ref{Def MD Algebras}). We chose the approach of finitely generated algebras to obtain results that involves the geometry of $\Proj(R)$ without considering the embedding into a projective space.
If $\ell=1$, several properties of the behavior of the minimum distance function have been obtained
\cite{MBPV,NBPV,MBPV-CI}. In particular, it is know that $\delta_R (t,1)$ eventually stabilizes for
radical ideals. Furthermore, there are bounds of its regularity index, which is the value where
$\delta_R (t,1)$ stabilizes \cite{NBPV}.
If there exists a linear form that is a nonzero divisor for $R$, it is know that $\delta_R(t,\ell)$ stabilizes
for $t\gg 0$ \cite{GSMBVV}.
\begin{theoremx}[{\autoref{ThmStabilization}}]\label{MainStab}
If $R$ is a reduced graded algebra, then $\delta_R (t,\ell)\geq \delta_R (t+1,\ell)$.
Consequently, $\delta_R (t,\ell)$ stabilizes for $t\gg 0$.
\end{theoremx}
As a consequence of \autoref{MainStab},
one can study the value where the generalized minimum distance stabilizes, $s_{R}(\ell)=\lim\limits_{t\to\infty}\delta_R (t,\ell)$, and
its regularity index $r_{R}(\ell)=\min\{t\;|\; \delta_{R}(t,\ell)=s_{R}(\ell)\}.$
In general, it is a difficult problem even if $\mathbb{K}$ is a finite field and $\ell= 1$, which are simplified for the stabilization numerical invariants.
In \autoref{ThmStabValue}, we compute the stabilization values of $\delta_{R}(t,\ell)$.
We also study the behavior of the regularity index with respect to the growth of $\ell$.
\begin{theoremx}[\autoref{ThmIneqRI}]\label{MainReg}
If $R$ is a reduced graded algebra with $\depth(R)\geq 2$, then $r_{R}(\ell+1)\leq r_{R}(\ell) +1$.
\end{theoremx}
As a consequence of \autoref{MainReg}, we obtain the following bounds for the regularity index in terms of algebraic invariants.
\begin{theoremx}\label{MainRegBounds}
Let $R$ be a standard graded $\mathbb{K}$-algebra.
We have the following bounds for the regularity index
\begin{enumerate}
\item If $R$ is a Stanley-Reisner ring corresponding to either a shellable or a Gorenstein simplicial complex,
then $
r_{R}(\ell)\leq \reg(R)+\ell-1$ {\rm (\autoref{CorBoundRegSR})}.
\item If $R$ is Gorenstein and $F$-pure,
then $
r_{R}(\ell)\leq \reg(R)+\ell-1$ {\rm (\autoref{CorBoundRegGor})}.
\item If $R$ is an Stanley-Reisner ring and $\Proj(R)$ is connected, then
$r_{R}(\ell)\leq \dim(R)+\ell-1$ {\rm (\autoref{ThmBoundSR})}.
\item If $R$ is $F$-pure, $\mathbb{K}$ is a separably closed field, and $\Proj(R)$ is connected, then
$r_{R}(\ell)\leq \dim(R)+\ell-1$ {\rm (\autoref{ThmBoundFpure})}.
\end{enumerate}
\end{theoremx}
\begin{convention}
Throughout this manuscript we assume that $\mathbb{K}$ is a field and $R$ is a standard graded $\mathbb{K}$-algebra of dimension $d$. For an $R$-module $M$, $\e(M)$ denotes the Hilbert-Samuel multiplicity of $M$ as an $R$-module. If we use the multiplicity with respect to another $\mathbb{K}$-algebra, $T$, we write $\e_T(M)$.
\end{convention}
\section{Preliminaries}\label{prelim-section}
In this section we recall some well known notion and results that are needed throughout this manuscript.
Let $R$ be a standard graded $\mathbb{K}$-algebra. Given a graded $R$-module, $M$, with $\dim(M)=\theta$,
its {\it Hilbert function}, denoted $h_M$, is given by
$$
h_M(t)=\dim_\mathbb{K}[M]_{t}.
$$
By a classical theorem of Hilbert there is a unique polynomial of degree $\theta-1$ that agrees with $h_M (t)$ for $t\gg 0$.
Then, there is a polynomial of degree $\theta$ that agrees with $\sum^t_{i=0} h_M(i)$.
The {\it Hilbert-Samuel multiplicity of\/} $M$ is the positive integer defined by,
$$
\e_R(M)= \lim\limits_{t\rightarrow\infty}\frac{\theta\sum^t_{i=0} h_M(i)}{t^{\theta}}.
$$
If the ring $R$ is clear from the context, we just write $\e(M)$.
Given $t,\ell\in \mathbb{N}_+$, we define the following set,
$$
\mathcal{F}_{t,\ell}=\{F=\{f_1,\ldots, f_\ell\}\subseteq R_t \mid f_1,\ldots, f_{\ell} \text{ are } \mathbb{K}\text{-linearly independent and } \Ann_{R}(F)\neq 0\},$$
where $\Ann_{R}(F)$ is the annihilator of $F$ as R-module.
\begin{definition}\label{Def MD Algebras}
The function $\delta:\mathbb{N}_{+}\times \mathbb{N}_+ \rightarrow \mathbb{Z}$ given by
$$
\delta_{R}(t,\ell)=\left\{\begin{array}{ll}\e(R)-\max\{\e(R/(F))\vert\,
F\in\mathcal{F}_{t,\ell}\}&\mbox{if }\mathcal{F}_{t,\ell}\neq\varnothing,\\
\e(R)&\mbox{if\ }\mathcal{F}_{t,\ell}=\varnothing,
\end{array}\right.
$$
is called the {\it generalized minimum distance function} of $R$, or simply the {\it GMD function} of $R$.
\end{definition}
Let $R$ be a commutative Noetherian ring with identity and let $I$ be a homogeneous ideal generated by the forms $f_1,\ldots,f_{\ell}\in R$. Consider the \v{C}ech complex, \v{C}$^{\star}(\bar{f};R)$:
$$
0\rightarrow R \rightarrow \bigoplus_{i}R_{f_i} \rightarrow \bigoplus_{i,j}R_{f_{i}f_{j}}
\rightarrow \cdots \rightarrow R_{f_1,\ldots,f_{\ell}}\rightarrow 0.
$$
where \v{C}$^{i}(\bar{f};R)=\bigoplus_{1\leq j_1\leq\ldots \leq j_i\leq \ell}R_{f_{j_1},\ldots,f_{j_i}}$ and the homomorphism in every summand is a localization map with appropriate sign.
\begin{definition}
Let $M$ be a graded $R$-modue. The {\it $i$-th local cohomology of $M$ with support in $I$} is defined as
\begin{center}
$H_{I}^{i}(M)= H^{i}($\v{C}$^{\star}(\bar{f};R)\otimes_{R} M)$.
\end{center}
\end{definition}
\begin{remark}\rm
Since $M$ is a graded $R$-module and $I$ is homogeneous the local cohomology module $H_{I}^{i}(M)$ is graded.
\end{remark}
\begin{definition}
Let $M$ be an $R$-module with dimension $\theta$. The {\it $a_i$-invariants}, $a_i(M)$, for $i=0,\ldots,\theta$ are defined as follows.
If $H_{\mathfrak{m}}^{i}(M)\neq 0$,
\begin{center}
$a_i(M)=\max\{\alpha \mid H_{\mathfrak{m}}^{i}(M)_{\alpha}\neq 0\}$,
\end{center}
for $0\leq i \leq \theta$, where $H_{\mathfrak{m}}^{i}(M)$ denotes the local cohomology module with support in the maximal ideal $\mathfrak{m}$.
If $H_{\mathfrak{m}}^{i}(M)= 0$, we set $a_i(M)=-\infty.$
If $\theta=\dim(M)$, then, $a_\theta(M)$, is often just called the {\it $a$-invariant} of $M$.
\end{definition}
The $a$-invariant, is a classical invariant \cite{GW1}, and is closely related to the Castelnuovo--Mumford regularity.
\begin{definition}
Let $M$ be a finitely generated $R$-module. The {\it Castelnuovo--Mumford regularity of $M$}, $\reg(M)$, is defined as
$$
\reg(M)=\max\{a_i(M)+i \mid 1\leq i\leq d\}.
$$
\end{definition}
\begin{remark}
The Castelnuovo--Mumford regularity can also be defined in terms of the Betti numbers of $M$, taken from a minimal graded free resolution of $M$ as an
$R$-module, that is, ${\rm reg}(M)=\max\{j-i\vert\,\beta_{ij}\neq 0\}.$
\end{remark}
\begin{definition}
Suppose that $\Char(\mathbb{K})=p\geq 0$.
Let $F:R\to R$ be the Frobenius map.
We say that $R$ is $F$-pure if for every $R$-module, $M$, we have that
$$
\xymatrix{
M\otimes_R R\ar[rr]^{1_M\otimes_R F}
&&
M\otimes_R R
}
$$
is injective.
We say that $R$ is $F$-finite if $R$ is finitely generated as $R^p$-module.
\end{definition}
\begin{definition}
Suppose that $R$ has prime characteristic $p$.
The Frobenius map $F:R \rightarrow R$ is defined by $r\mapsto r^p$.
\end{definition}
\begin{remark}
If $R$ is reduced, $R^{1/p^{e}}$ the ring of the $p^{e}$-th roots of $R$ is well defined, and $R\subseteq R^{1/p^{e}}$.
\end{remark}
A ring $R$ is called {\it unmixed} if all its associated primes have the same height, in other case $R$ is {\it mixed}.
About the asymptotic behavior of the GMD function of $R$, we have the following result.
\begin{theorem}[{\cite[Theorem~3.9]{CSTVPV}}]
Suppose that $R$ is unmixed. Let $t\geq 1$ and $\ell\geq 1$ be integers. The following hold:
\begin{enumerate}
\item[{\rm (i)}] $\delta_{R}(t,\ell)\leq \delta_{R}(t,\ell +1)$.
\item[{\rm (ii)}] If there is $h\in R_1$ regular on $R$, then $\delta_{R}(t,\ell)\geq \delta_{R}(t+1,\ell)\geq 1$.
\end{enumerate}
\end{theorem}
\section{Asymptotic behavior of the GMD function}\label{section-mindis}
In this section we prove that the generalized minimum distance function $\delta_{R}(t,\ell)$ is a non-increasing function. Then, the notion of stabilization value of $\delta_{R}(t,\ell)$, and its regularity index, are well defined. We start this section establishing notation.
\begin{notation}
Let $X\subseteq R$ any subset and $(X)$ its generated ideal, we set
\begin{align*}
\mathcal{A}(R)&=\{\mathfrak{p}\in\Min(R)\; | \; \dim(R)=\dim(R/\mathfrak{p})\},\\
\mathcal{V}(X)&=\{\mathfrak{p}\in\Spec(R)\; |\; (X)\subseteq \mathfrak{p}\}.
\end{align*}
In addition, $ \lambda_{R_\mathfrak{p}}(M_\mathfrak{p})$ denotes the length of $M_\mathfrak{p}$ as $R_\mathfrak{p}$-module.
\end{notation}
\begin{remark}[Additivity Formula]\label{RemAdditivity}
For any finitely generated $R$-module, $M$, we have
$$
\e(M)=\sum_{\mathfrak{p}\in\mathcal{A}(R)} \lambda_{R_\mathfrak{p}}(M_\mathfrak{p})\e(R/\mathfrak{p}).
$$
In particular, if $R$ is a reduced algebra, then
$$
\e(R)=\sum_{\mathfrak{p}\in\mathcal{A}(R)} \e(R/\mathfrak{p}).
$$
\end{remark}
The following lemmas are needed to prove that the GMD function stabilizes.
\begin{lemma}\label{LemmaAdditivity}
Let $F\in\mathcal{F}_{t,\ell}$ such that $\dim(R/(F))=\dim(R)$. Then,
\[
\e(R/(F))=\sum_{\mathfrak{p}\in \mathcal{A}(R)\cap\mathcal{V}(F)}\e(R/\mathfrak{p}).
\]
\end{lemma}
\begin{proof}
Using \autoref{RemAdditivity} we have
\[
\e(R/(F))=\sum_{\mathfrak{p}\in \mathcal{A}(R)}\lambda_{R_{\mathfrak{p}}}(R_{\mathfrak{p}}/(F)R_{\mathfrak{p}})\e(R/\mathfrak{p}).
\]
Fix $\mathfrak{p}\in\mathcal{A}(R)$. Since $R$ is reduced and $R\twoheadrightarrow R_{\mathfrak{p}}/(F)R_{\mathfrak{p}}$, we have that
\[
1=\lambda_{R_{\mathfrak{p}}}(R_{\mathfrak{p}})\geq \lambda_{R_{\mathfrak{p}}}(R_{\mathfrak{p}}/(F)R_{\mathfrak{p}})\geq 0.
\]
If $(F)\nsubseteq\mathfrak{p}$, then $(F)R_{\mathfrak{p}}=R_{\mathfrak{p}}$, and so, $\lambda_{R_{\mathfrak{p}}}(R_{\mathfrak{p}}/(F)R_{\mathfrak{p}})=0$. If $(F)\subseteq\mathfrak{p}$, then $\lambda_{R_{\mathfrak{p}}}(R_{\mathfrak{p}}/(F)R_{\mathfrak{p}})>0$, and so
$\lambda_{R_{\mathfrak{p}}}(R_{\mathfrak{p}}/(F)R_{\mathfrak{p}})=1$. Therefore,
\[
\e(R/(F))=\sum_{\mathfrak{p}\in \mathcal{A}(R)\cap\mathcal{V}(F)}\e(R/\mathfrak{p}).
\]
\end{proof}
\begin{lemma}\label{dis.ig}
Let $y$ be a variable over $R$. Set $\tilde{R}= R\otimes_{\mathbb{K}}\mathbb{K}(y)$. Then, $\delta_{R}(t,\ell)=\delta_{\tilde{R}}(t,\ell)$.
\end{lemma}
\begin{proof}
By construction of $\tilde{R}$ it follows that $\delta_{R}(t,\ell)\geq \delta_{\tilde{R}}(t,\ell)$.
Let $\mathfrak{p}_{1},\ldots,\mathfrak{p}_{a}$ be the minimal primes of $R$. Then, $\tilde{\mathfrak{p}}_{1},\ldots,\tilde{\mathfrak{p}}_{a}$ are the minimal primes of $\tilde{R}$.
Let $G\in\mathcal{F}_{t,\ell}$ such that
\[
\delta_{\tilde{R}}(t,\ell)=\e(\tilde{R})-\e(\tilde{R}/(G)).
\]
Then, there exists $\tilde{\mathfrak{p}}_{1},\ldots,\tilde{\mathfrak{p}}_{b}\in\Ass_{\tilde{R}}(\tilde{R})$ with $b<a$ such that
$G\subseteq\tilde{\mathfrak{p}}_{1}\cap\cdots\cap\tilde{\mathfrak{p}}_{b}$. Thus,
\[
\dim_{\mathbb{K}}\left[\mathfrak{p}_{1}\cap\cdots\cap\mathfrak{p}_{b} \right]_{t}
=\dim_{\mathbb{K}(y)}\left[\tilde{\mathfrak{p}}_{1}\cap\cdots\cap\tilde{\mathfrak{p}}_{b}\right]_{t}
\geq \ell.
\]
Therefore, $\delta_{R}(t,\ell)\leq \delta_{\tilde{R}}(t,\ell)$.
\end{proof}
We are ready to prove one of our main results.
\begin{theorem}\label{ThmStabilization}
If $R$ is a reduced graded algebra, then $\delta_R (t,\ell)\geq \delta_{R}(t+1,\ell)$.
As a consequence, $\delta_R (t,\ell)$ stabilizes for $t\gg 0$.
\end{theorem}
\begin{proof}
If $\mathcal{F}_{t,\ell}=\varnothing$, then $\delta_R (t,\ell) =\e(R)$.
Then, by definition,
$\delta_R (t+1,\ell)\leq \e(R)=\delta_{R} (t,\ell)$.
We now assume that $\mathcal{F}_{t,\ell}\neq \varnothing$ and take $F=\{f_1,\ldots,f_\ell\}\in \mathcal{F}_{t,\ell}$ such that
$$
\delta_{R}(t,\ell)=\e(R)-\e(R/(F)).
$$
Let $\mathfrak{p}_{1},\ldots,\mathfrak{p}_{a}$ be the minimal primes of $R$ with $\dim(R)=\dim(R/\mathfrak{p}_i)$ containing $F$. Then, by \autoref{LemmaAdditivity},
$$
\e(R/(F))=\sum^a_{i=1} \e(R/\mathfrak{p}_i).
$$
Since $F\in \mathfrak{p}_{1}\cap\ldots\cap\mathfrak{p}_{a}$ and the $f_1,\ldots,f_\ell$ are linearly independent in $R$, we have that
$$
\dim_{\mathbb{K}}\left[\mathfrak{p}_{1}\cap\cdots\cap\mathfrak{p}_{a}\right]_{t}\geq \ell.
$$
Let $\tilde{R}$ as in \autoref{dis.ig}. Since $\depth(\tilde{R})=\depth(R)\geq 1$,
by prime avoidance, there exists $g\in [\tilde{R}]_{1}$ which is a nonzero divisor in $\tilde{R}$.
Then, $\tilde{R}\xhookrightarrow{g} \tilde{R}$ is injective. In particular,
$gF=\{gf_1,\ldots, gf_\ell\}\in\mathcal{F}_{t+1,\ell}$.
Therefore,
\begin{align*}
\delta_{\tilde{R}}(t+1,\ell)&\leq e(\tilde{R})-e(\tilde{R}/gF))\\
&\leq e(\tilde{R})-e(\tilde{R}/(F)\tilde{R})\\
&=\delta_{\tilde{R}}(t,\ell).
\end{align*}
Using \autoref{dis.ig} and the previous inequality we conclude that $\delta_{R}(t+1,\ell)\leq \delta_{R}(t,\ell)$.
\end{proof}
\autoref{ThmStabilization} implies that $\delta_R (t,\ell)$ eventually stabilizes. This motivates the following definition.
\begin{definition}
The {\it stabilization value of $R$}, denoted by $s_{R}(\ell)$, is defined by
$$
s_R(\ell)=\lim_{t\to \infty}\delta_{R}(t,\ell).
$$
The {\it regularity index of $R$}, denoted by $r_{R}(\ell)$, is defined by
$$
r_{R}(\ell) = \min\{t\in \mathbb{N} \mid \delta_R (t,\ell) = s_R(\ell) \}.
$$
\end{definition}
The next lemma is useful to study the stabilization of the GMD function.
\begin{lemma}\label{H-increasing}
Let $M$ be an $R$-module. If $\depth(M)\geq 1$, then its Hilbert function is
non-decreasing. If $\depth(M)\geq 2$, then its Hilbert function
strictly increasing for starting at $\alpha=\min\{t\; \mid\; [M]_t\neq 0\}$.
\end{lemma}
\begin{proof}
Since the depth and the Hilbert function are non-affected by field extensions, we can consider
$\mathbb{K}$ as an infinite field.
Let $f\in [R]_{1}$ which is a nonzero divisor on $M$. Consider the following exact sequence
\[
\xymatrix{
0\ar[r] & M(-1)\ar[r]^-{f} & M\ar[r] & \overline{M}\ar[r] & 0,
}
\]
where $\overline{M}=M/fM$.
Since the Hilbert function is an additivity function it follows that
\[
h_{\overline{M}}(t)=h_{M}(t)-h_{M}(t-1).
\]
This implies that $\dim_\mathbb{K} [M]_t\leq \dim_\mathbb{K} [M]_{t+1},$ and prove our first claim.
We now assume that $\depth(M)\geq 2$.
We have that $\depth(\overline{M})\geq 1$. Hence, $h_{\overline{M}}(t)$ is non-decreasing.
Let $g\in [R]_{1}$ which is a nonzero divisor on $\overline{M}$. Consider the following injective sequence
\[
\xymatrix{
0\ar[r] & \overline{M}(-1)\ar[r]^-{g} & \overline{M},
}
\]
this implies that $h_{\overline{M}}(t)\geq h_{\overline{M}}(t-1)$. Set $\alpha=\{t\mid M_t\neq 0\}$.
Since $f$ has degree $1$, the injective homomorphism $\xymatrix{M\ar[r]^-{f} & M}$ is not surjective over $M_t$.
Hence, $\overline{M}_\alpha\neq 0$. This implies that, $\dim_{\mathbb{K}}[\overline{M}]_t\geq 1$ for all $t\geq \alpha$.
Thus,
\[
h_{\overline{M}}(t)=h_{M}(t)-h_{M}(t-1)\geq 1,
\]
for all $t\geq\alpha$. Therefore, $h_{M}(t)>h_{M}(t-1)$ for all $t\geq \alpha$.
\end{proof}
The following theorem is another main results of this work, for several cases we compute the stabilization value of the GMD function.
\begin{theorem}\label{ThmStabValue}
Suppose that $R$ is a reduced graded algebra with minimal primes
$$\mathfrak{q}_{1},\ldots, \mathfrak{q}_{a}, \mathfrak{p}_1,\ldots, \mathfrak{p}_b$$ such that $\dim(R/\mathfrak{q}_i)=\dim(R)$,
$\dim(R/\mathfrak{p}_i)<\dim(R)$, and $\e(R/\mathfrak{q}_{i})\leq \e(R/\mathfrak{q}_{j})$ for $i\leq j$.
\begin{enumerate}
\item\label{ThmStabValue-1} If $R$ is mixed and $\dim(R/\mathfrak{p}_{j})\geq 2$ for some $j$, then $s_{R}(t,\ell)=0$.
\item\label{ThmStabValue-2} If $R$ is a domain, then $s_{R}(t,\ell)=\e(R)$ .
\item\label{ThmStabValue-3} If $R$ is unmixed and $\dim(R)\geq 2$, then $s_{R}(t, \ell)=\e(R/\mathfrak{q}_{1})$.
\item\label{ThmStabValue-4} If $R$ is one-dimensional and $\ell\leq \e(R)-\e(R/\mathfrak{q}_1)$, then
$$
s_{R}(t,\ell)=\min\left\{\sum_{i\in \sigma} \e(R/\mathfrak{q}_i)\;\mid\; \sigma\subsetneq [a] \text{ and } \ell \leq \sum_{i\in \sigma} \e(R/\mathfrak{q}_i)\right\}.
$$
\item\label{ThmStabValue-5} If $R$ is one-dimensional and $\ell> \e(R)-\e(R/\mathfrak{q}_1)$, then $s_{R}(t,\ell)=\e(R)$.
\item\label{ThmStabValue-6} If $R$ is mixed, $\dim(R/\mathfrak{p}_{j})=1$ for all $j$, and $\ell\leq \e_{T}(T)$ with
$T= R/\bigcap^b_{i=1}\mathfrak{p}_i$, then $s_{R}(t,\ell)=0$.
\item\label{ThmStabValue-7} If $R$ is mixed, $\dim(R/\mathfrak{p}_{j})=1$ for all $j$,
and $\ell>\e_{T}(T)$ with $T= R/\bigcap^b_{i=1}\mathfrak{p}_i$, then $s_{R}(t,\ell)=\e(R/\mathfrak{q}_1)$.
\end{enumerate}
\end{theorem}
\begin{proof}
\hfill\medskip
\begin{itemize}
\item[(1)] Let $N=\mathfrak{q}_{1}\cap\cdots\cap \mathfrak{q}_{a}$. Then, $\Ass(N)=\{\mathfrak{p}_1,\ldots,\mathfrak{p}_b\}$. This implies that $\dim(N)\geq 2$.
Then for some degree $\alpha$, there exists
$$F=\{f_{1},\ldots,f_{\ell}\}\subseteq [N]_{\alpha}$$
such that they are linearly independent in $R$ and $\Ann_R (F)\neq 0$. Thus, $F\in\mathcal{F}_{t,\ell}$.
Furthermore, $\dim(R)=\dim(R/(F))$.
Hence $\e(R)=\e(R/(F))$. Therefore, $\delta_{R}(\alpha,\ell)=0$. Since $\delta_{R}(-,\ell)$ is non-increasing
it follows that $\delta_{R}(t,\ell)=0$ for all $t\geq \alpha$.
\item[(2)] Since $R$ is a domain it follows that $\mathcal{F}_{t,\ell}=\varnothing$ for all $\ell,t$. Therefore,
$\delta_{R}(t,\ell)=\e(R)$ for all $\ell,t$.
\item[(3)]
We suppose that $R$ is not a domain, because this case is covered in Part (\ref{ThmStabValue-2}). Since $R$ is unmixed, $\mathcal{A}(R)=\Ass(R)$.
For all $F\in \mathcal{F}_{t,\ell}$, there exists $\mathfrak{p}\in\Ass(R)$ such that
$F\nsubseteq \mathfrak{p}$.
We take $F\in \mathcal{F}_{t,\ell}$ that gives the generalized minimum distance function. Then,
$$
\delta_{R}(t,\ell)=\e(R)-\e(R/(F))\geq \e(R/\mathfrak{p})\geq \min\{\e(R/\mathfrak{p})\mid \mathfrak{p}\in\Ass(R)\}.
$$
Let $N=\mathfrak{q}_{2}\cap\cdots\cap\mathfrak{q}_{a}$. Since $\Ass(N)=\{\mathfrak{q}_{1}\}$ it follows that $\dim(N)\geq 2$.
Then for some degree $\alpha$, there exists $F=\{f_{1},\ldots,f_{\ell}\}\subseteq [N]_{\alpha}$
such that they are linearly independent.
Furthermore,
$\Ann_R (F)\neq 0$. Hence, $F\in\mathcal{F}_{\alpha,\ell}$. Thus,
\[
\delta_{R}(\alpha,\ell)\leq \e(R)-\e(R/(F))=\e(R/\mathfrak{q}_{1}).
\]
Since $\delta_{R}(-,\ell)$ is non-increasing it follows that $\delta_{R}(t,\ell)=e(R/\mathfrak{q}_{1})$ for all $t\geq \alpha$.
\item[(4)]
We suppose that $R$ is not a domain because this case is covered in Part (\ref{ThmStabValue-2}).
Observe that $R$ is unmixed.
Let $\sigma\subsetneq [a]$ non-empty. We set $J_{\sigma}=\bigcap_{i\notin\sigma}\mathfrak{q}_{i}$.
Then, $\Ass(J_{\sigma})=\{\mathfrak{q}_{i}\mid i\in\sigma\}$, as an $R$-module. Hence $\dim(J_{\sigma})=1$. From the short exact sequence
\begin{equation*}
\xymatrix{
0\ar[r] & J_{\sigma}\ar[r] & R\ar[r] & R/J_\sigma\ar[r] & 0,
}
\end{equation*}
we have that $\e(J_{\sigma})=\sum_{i\in\sigma} \e(R/\mathfrak{q}_i)$.
We note that for every $\sigma$, there exists $\alpha_\sigma\in\mathbb{N}$ such that $\dim_\mathbb{K}[J_\sigma]_t=\e(J_{\sigma})$ for $t\geq \alpha_\sigma$.
Let $\alpha=\max\{\alpha_{\sigma}\}$.
If $\ell\leq \e(J_{\sigma})$ and $t\geq \alpha$, there exists a set of $\mathbb{K}$-linearly
independent elements $F=\{f_1,\ldots,f_\ell\}\subseteq [J_\sigma]_t$. Then,
$$
\e(R)-\e(R/(F)) = \sum_{F\not\in \mathfrak{q}_i} \e(R/\mathfrak{q}_i)\geq \sum_{J_\sigma\not\subseteq \mathfrak{q}_i} \e(R/\mathfrak{q}_i)= \e(J_\sigma).
$$
Hence, $$\delta_R(t,\ell)\geq \min\left\{\e(J_\sigma)\; \mid \; \e(J_\sigma)\geq \ell\right\}.$$
We fix $t\geq \alpha$, and set $\gamma\subsetneq [a]$ such that
$$\e(J_\gamma)=\min\left\{\e(J_\sigma)\; \mid \; \e(J_\sigma)\geq \ell\right\}$$
and $G=\{g_1,\ldots,g_\ell\}\subseteq [J_\gamma]_t$ a set of $\mathbb{K}$-linearly
independent elements.
We note that $(G)\not\subseteq J_\gamma \cap \mathfrak{q}_i$ for any $i\in \gamma$; otherwise,
$$\dim_\mathbb{K} [(G)]_t\leq \dim_\mathbb{K} [J_\gamma \cap \mathfrak{q}_i]_t=\e(J_\gamma\cap\mathfrak{q}_i)=\e(J_\gamma)-\e(R/\mathfrak{q}_i)<\ell,$$ which is not possible by our choice of $\gamma$.
Hence,
$$
\delta_R(t,\ell)\leq \e(R)-\e(R/(F)) = \e(R)-\e(R/J_\gamma)= \min\left\{\e(J_\sigma)\;\mid\; \e(J_\sigma)\geq \ell\right\}
$$
by \autoref{LemmaAdditivity}.
We conclude that
$$
\delta_R(t,\ell)=\min\left\{\e(J_\sigma)\;\mid\; \e(J_\sigma)\geq \ell\right\}.
$$
\item[(5)]
Note that in this case $R$ is unmixed.
We show by contradiction that $\mathcal{F}_{t,\ell}=\varnothing$.
Suppose that there exists
$G=\{g_1,\ldots, g_\ell\}\subseteq R$ a set of $\mathbb{K}$-linearly independent homogeneous elements of degree $t\geq 1$
such that $\Ann_R (G)\neq 0$.
Then, there exists $i$ such that $G\subseteq \mathfrak{q}_{i}$ by prime avoidance.
Thus, $\Ass(\mathfrak{q}_i)=\{\mathfrak{q}_{j}\mid j\neq i\}$. Hence $\dim_R(\mathfrak{q}_i)=1$, where we consider $\mathfrak{q}_i$ as an $R$-module. From the short exact sequence
\begin{equation*}
\xymatrix{
0\ar[r] & \mathfrak{q}_i\ar[r] & R\ar[r] & R/\mathfrak{q}_i\ar[r] & 0,
}
\end{equation*}
we have that $\e(\mathfrak{q}_i)=\e(R)-\e(R/\mathfrak{q}_i)\leq \e(R)-\e(R/\mathfrak{q}_1)$.
There exists $\alpha\in\mathbb{N}$ such that $\dim_\mathbb{K}[\mathfrak{q}_i]_t=\e(R)-\e(R/\mathfrak{q}_i)\leq \e(R)-\e(R/\mathfrak{q}_1)$ for $t\geq \alpha$.
Since $\depth(\mathfrak{q}_i)>0$, we have that the Hilbert function of $\mathfrak{q}_i$
is non-decreasing. Then, $\dim_\mathbb{K} [\mathfrak{q}_i]_t\leq \e(R)-\e(R/\mathfrak{q}_1)$ for all $t$, this contradicts the fact that
$G\subseteq R$ is a $\mathbb{K}$-linearly independent set, as $\ell>\e(R)-\e(R/\mathfrak{q}_1)$.
\item[(6)] We set $B= \e_T(T)$. Let $N=\mathfrak{q}_{1}\cap\cdots\cap \mathfrak{q}_{a}$. Then, $\Ass(N)=\{\mathfrak{p}_{1},\ldots,\mathfrak{p}_{b}\}$.
This implies that $\dim(N)=1$. Hence, there exists $\alpha\in\mathbb{N}$ such that
$\dim_{\mathbb{K}}[N]_{t}=\e_T(T)=B$
for all $t\geq \alpha$. Thus, there exists a linearly independent set $F=\{f_{1},\ldots,f_{\ell}\}\subseteq [N]_{\alpha}$. Furthermore, since $\Ann_R (F)\neq 0$ we have that $F\in\mathcal{F}_{\alpha,\ell}$. Then,
\[
\delta_{R}(\alpha,\ell)\leq \e(R)-\e(R/(F))=0.
\]
Therefore, $\delta_{R}(t,\ell)=0$ for all $t\geq \alpha$.
\item[(7)]
We set $B= \e_T(T)$. Let $N=\mathfrak{q}_{1}\cap\cdots\cap \mathfrak{q}_{a}$.
As in the previous part,
$\dim(N)=1$ and $\e_T(N)=\e_T(T)$.
Then, it does not exist a set $F=\{ f_1,\ldots,f_\ell \}$ of $\mathbb{K}$-linearly independent homogeneous elements in $N$.
Thus,
$\e(R/(G))>0$
for every set $G=\{ g_1,\ldots,g_\ell\}$ of $\mathbb{K}$-linearly independent homogeneous elements of the same degree such that $\Ann_R(G)\neq 0$ by \autoref{LemmaAdditivity}.
Thus,
\[
s_{R}(\ell)\geq \e(S/\mathfrak{q}_{1}).
\]
We now proceed by cases. We first assume that $a>1$.
Let $V=\mathfrak{q}_{2}\cap\cdots\cap \mathfrak{q}_{a}$. Then, $\Ass(V)=\{\mathfrak{q}_{1},\mathfrak{p}_{1},\ldots,\mathfrak{p}_{b}\}$.
Thus, $\dim(V)\geq 2$. This implies that for some degree $\alpha_{1}$, there exists a linearly independent set
$F=\{f_{1},\ldots,f_{\ell}\}\subseteq [V]_{\alpha_{1}}$
such that $\Ann_R (F)\neq 0$. Hence $F\in\mathcal{F}_{\alpha_{1},\ell}$.
Furthermore, observe that $\dim(R)=\dim(R/(F))$. Hence,
\[
\delta_R (\alpha_{1},\ell)\leq \e(R)-\e(R/(F))=\e(R/\mathfrak{q}_{1}).
\]
Therefore, $s_{R}(\ell)=\e(R/\mathfrak{q}_{1})$.
We now assume that $a=1$.
Let $F=\{f_1,\ldots, f_\ell\}$ be
$\mathbb{K}$-linearly independent homogeneous elements of the same degree such that $\Ann_R(F)\neq 0$.
Let $\sigma\subsetneq [b]$ non-empty. We set $J_{\sigma}=\bigcap_{i\notin\sigma}\mathfrak{p}_{i}$.
Then, $\Ass(J_{\sigma})=\{\mathfrak{p}_{i}\mid i\in\sigma\}\cup \{\mathfrak{q}_1\}$, as an $R$-module.
If $F\subseteq J_{\sigma}$, then $\e(R/(F))=\e(\mathfrak{q}_1)$ by \autoref{LemmaAdditivity}.
We have that $J_{\sigma}\cap \mathfrak{q}_1=J_\sigma \bigcap N$, and $F\nsubseteq J_{\sigma}\cap \mathfrak{q}_1$
because $\dim_\mathbb{K} [J_{\sigma}\cap \mathfrak{q}_1]_t\leq \dim_\mathbb{K} [N]_t<\ell$. Hence, $s_R(\ell)=\e(\mathfrak{q}_1)$.
\end{itemize}
\end{proof}
\begin{example}
Consider the ring $S=\mathbb{F}_{2}[x,y,z]$ and $I=(x^3+y^2z,xy+z^{2})$. Set $R=S/I$. Using {\it Macaulay2} \cite{M2}, the minimal primes of $R$ are
\[
\mathfrak{p}_{1}=(x,z), \quad \mathfrak{p}_{2}=(y+z,x+z), \text{ and }\; \mathfrak{p}_{3}=(xy+z^{2},x^{2}+y^{2}+xz+yz+z^{2}).
\]
Thus, we have that $\dim(R/p_{i})=1$ for all $i$, $\e(R)=6$, $\e(R/\mathfrak{p}_{1})=\e(R/\mathfrak{p}_{2})=1$
and $\e(R/\mathfrak{p}_{3})=4$. Let $\mu\geq0$. For $\ell=1$, we can take $f=y^{\mu}(y^{3}+x^{2}z)\in \mathcal{F}_{\mu+3,1}$.
Since $f\in\mathfrak{p}_{2}\cap\mathfrak{p}_{3}$ by \autoref{ThmStabValue} (\ref{ThmStabValue-4}), it follows that
\[
s_{R}(1)=1.
\]
For $\ell=5$, we have that
\[
s_{R}(5)=5,
\]
which is obtained via $F=\{x^{2+\mu}z,x^{\mu+1}z^{2},x^{\mu}y^{2}z,x^{\mu}yz^{2},x^{\mu}z^{3}\}\in\mathcal{F}_{\mu+3,5}\subseteq\mathfrak{p}_{1}$.
Since the Hilbert series of $R$ is
\[
1+3t+5t^{2}+\sum_{i=3}^{\infty}6t^{i}.
\]
For $\ell>7$, we have that $\mathcal{F}_{t,r}=\varnothing$ for all $t$. For any linearly independent set of homogeneous elements
of degree $t$, $F=\{f_{1},\ldots,f_{6}\}$, we have that there exist
$a_{i}\in\mathbb{F}_{2}$ such that $y^{3+(t-3)}=\sum_{i} a_{i}f_{i}$. Since $y^{3+(t-3)}\notin\mathfrak{p}_{i}$ for all $i$,
it follows that $\mathcal{F}_{t,6}=\varnothing$. Therefore, by \autoref{ThmStabValue} (\ref{ThmStabValue-5}), we have that
\[
s_{R}(\ell)=\e(R)=6 \text{ for }\ell>5.
\]
\end{example}
\begin{example}
Let $S=\mathbb{F}_{3}[x,y,z]$ and $I=(y^{2}-yz,x^{2}y-yz^{2})$.
Set $R=S/I$. Using {\it Macaulay2} \cite{M2}, the minimal primes of $R$ are
\[
\mathfrak{q}_{1}=(y), \quad \mathfrak{p}_{2}=(y-z,x-z), \text{ and }\; \mathfrak{p}_{2}=(y-z,x+z),
\]
and we have that $\dim(R/\mathfrak{q}_{1})=2$, $\dim(R/\mathfrak{p}_{1})=\dim(R/\mathfrak{p}_{2})=1$, $\e(R)=\e(R/\mathfrak{q}_{1})=1$ and $\e(R/\mathfrak{p}_{1})=\e(R/\mathfrak{p}_{2})=0$. For $\ell\in\{1,2\}$ there exists $F\in\mathcal{F}_{t,\ell}$ such that $F\subseteq\mathfrak{q}_{1}$. Therefore, by \autoref{ThmStabValue} (\ref{ThmStabValue-6}),
\[
s_{R}(\ell)=0.
\]
For $\ell>2$, we can find a set $F=\{f_{1},\ldots,f_{\ell}\}\subseteq [R]_{t}$ of linearly independent elements for $t\gg 0$.
However, using the Hilbert series of $\mathfrak{q}_{1}$ we have that $F\nsubseteq\mathfrak{q}_{1}$. This implies that $\e(R/(F))=0$. Therefore, by \autoref{ThmStabValue} (\ref{ThmStabValue-7}),
\[
s_{R}(\ell)=e(R)=1.
\]
\end{example}
The next proposition shows that the GMD function is non-decreasing when $\ell$ grows.
\begin{proposition}
Suppose that $R$ is a reduced graded algebra. Then, $\delta_{R}(t,\ell)\leq \delta_{R}(t,\ell+1)$.
\end{proposition}
\begin{proof}
By definition we have that $\delta_{R}(t,\ell)\leq \e(R)$. Suppose that $\mathcal{F}_{t,\ell+1}=\varnothing$. Then, $\delta_{R}(t,\ell+1)=\e(R)$, and so,
$\delta_{R}(t,\ell)\leq \delta_{R}(t,\ell+1)$.
Now, suppose $\mathcal{F}_{t,\ell +1}\neq\varnothing$. We take $F=\{f_1,\ldots,f_{\ell+1}\}$ such that $\delta_{R}(t,\ell+1)=\e(R)-\e(R/(F))$.
Let $F'=\{f_1,\ldots,f_\ell\}$ be linearly independent in $R$.
We have $0\neq \Ann_R (F)\subseteq \Ann_R (F')$. This implies that $F'\in \mathcal{F}_{t,\ell}$. Since $\Ann_R(F')\subseteq \Ann_R (F)$, we have that $\e(R/(F))\leq \e(R/(F'))$.
Therefore,
$$
\delta_{R}(t,\ell+1)=\e(R)-\e(R/(F))\geq \e(R)-\e(R/(F'))\geq \delta_{R}(t,\ell).
$$
\end{proof}
The next lemma show the grow of the regularity index of $\delta_R (t,\ell)$ is at most linear with respect to $\ell$.
\begin{lemma}\label{LemmaRImixed}
Suppose that $R$ is a reduced graded mixed algebra with minimal primes
$$\mathfrak{q}_{1},\ldots, \mathfrak{q}_{a}, \mathfrak{p}_1,\ldots, \mathfrak{p}_b$$ such that $\dim(R/\mathfrak{q}_i)=\dim(R)$ and
$\dim(R/\mathfrak{p}_i)<\dim(R)$.
Let $N_{1}=(\cap_{i=1}^{a}\mathfrak{q}_{i})$ and $N_{2}=(\cap_{i=1}^{b}\mathfrak{p}_{i})$.
If $\dim(N_1)\geq 2$, then
$r_{R}(\ell)=\min\{t\; | \;\dim_{\mathbb{K}}[N_{1}]_{t}\geq \ell\}$.
\end{lemma}
\begin{proof}
Set $\alpha=\min\{t\; | \;\dim_{\mathbb{K}}[N_{1}]_{t}\geq \ell\}$.
By \autoref{ThmStabValue} (1),
we have $\delta_{R}(t,\ell)=0$ for all $t\geq\alpha$. This implies that $r_{R}(\ell)\leq\alpha$.
Suppose there exists $F\in\mathcal{F}_{t,\ell}$ with $t<\alpha$. Then $F\nsubseteq N_{1}$. This implies
that there exists a prime ideal $\mathfrak{q}_i$ such that $F\nsubseteq\mathfrak{q}_i$. Hence, $\delta_{R}(t,\ell)\geq e(R/\mathfrak{q}_i)>0$.
Thus, $r_{R}(\ell)\geq\alpha$. However, if such $F$ does not exist, then by \autoref{H-increasing} we have that
$\dim_{\mathbb{K}}[N_1]_{t}<\ell$ for all $t<\alpha$. Hence, $\mathcal{F}_{t,\ell}=\varnothing$ for all $t<\alpha$,
and so, $\delta_{R}(t,\ell)=\e(R)>0$. Since the function is non-increasing, it follows that
$\delta_{R}(t,\ell)\leq e(R)$ for all $t\geq \alpha$. Thus, $r_{R}(\ell)\geq\alpha$.
Therefore, $r_{R}(\ell)=\alpha$.
\end{proof}
\begin{lemma}\label{LemmaRIunmixed}
Suppose that $R$ is an unmixed standard graded algebra such that $d\geq 2$.
Let $\mathfrak{p}_1,\ldots,\mathfrak{p}_a$ be the minimal primes of $R$. Let $\mathcal{A}=\{i\; \mid\; \e(R/\mathfrak{p}_i)\leq \e(R/\mathfrak{p}_j) \hbox{ for every }j\}.$
Then,
$$
r_{R}(\ell)=\min\{t\; | \;\dim_{\mathbb{K}}[\cap_{j\neq i} \mathfrak{p}_j]_{t}\geq \ell \hbox{ for some }i\in \mathcal{A}\}.
$$
\end{lemma}
\begin{proof}
We set $\theta_{i} =\min\{t\; | \;\dim_{\mathbb{K}}[\bigcap_{j\neq i} \mathfrak{p}_j]_{t}\geq \ell \hbox{ for some }i\in \mathcal{A}\}.$
We fix $s\in\mathcal{A}$ such that $\theta_{s}=\min\{\theta_{i}\}$.
Then, there exists a set $F=\{f_1,\ldots,f_\ell\}\subseteq [\cap_{j\neq i} \mathfrak{p}_j]_{\theta_s}$ of linearly independent elements.
Then,
$$
\delta_{R}(\theta_s,\ell)\leq \e(R)-\e(R/(F))=\e(R/\mathfrak{p}_s).
$$
By \autoref{ThmStabValue} (3), $\delta_{R}(r_{R}(\ell),\ell)=\e(R/\mathfrak{p}_s)$. Since $\delta_{R}(t,\ell)$ is a non-decreasing function by \autoref{ThmStabilization}, we have that $r_R(\ell)\leq \theta_s$.
If $t< \theta_s$, we have that do not exists a linearly independent set $G=\{g_1,\ldots,g_\ell\}\subseteq [R]_t$ such that $G\not\subseteq [\bigcap_{j\neq i} \mathfrak{p}_j]_{t}$ for any $i\in\mathcal{A}$.
Hence,
$$
\e(R)-\e(R/(G))> \e(R/\mathfrak{p}_s).
$$
Thus, $\delta_{R}(t,\ell)> \e(R/\mathfrak{p}_s)$, and so, $t<r_{R}(\ell)$. We conclude that $r_{R}(\ell)\geq \theta_s$.
\end{proof}
\begin{theorem}\label{ThmIneqRI}
Suppose that $\depth(R)\geq 2$. Then, $ r_{R}(\ell+1)\leq r_{R}(\ell)+1$.
\end{theorem}
\begin{proof}
Let $\mathfrak{p}_1,\ldots,\mathfrak{p}_a$ be the minimal primes of $R$.
Let $\sigma\subsetneq [a]$ and $N_\sigma=\bigcap_{i\not\in \sigma} \mathfrak{p}_i$.
From the short exact sequence
$$
0\to N_\sigma \to R\to R/N_\sigma\to 0,
$$
we obtain the long exact sequence
$$
0\to H^0_\mathfrak{m}(N_\sigma)\to H^0_\mathfrak{m}(R)\to H^0_\mathfrak{m}(R/N_\sigma)\to
H^1_\mathfrak{m}(N_\sigma)\to H^1_\mathfrak{m}(R)\to H^1_\mathfrak{m}(R/N_\sigma)\to \cdots .
$$
Since $\depth(R)\geq 2$, we have that $ H^0_\mathfrak{m}(R)= H^1_\mathfrak{m} (R)=0$.
Since $N_\sigma$ is a radical ideal, we have that $H^0_\mathfrak{m} (R/N_\sigma)=0$.
From the long exact sequence, we have that
$H^0_\mathfrak{m}(N_\sigma)=H^1_\mathfrak{m}(N_\sigma)=0$, whence, $\depth(N_\sigma)\geq 2$.
Then, the Hilbert function of $N$ is strictly increasing once it is positive by \autoref{H-increasing}.
We set $N$ a intersection of prime ideals such that $r_{R}(\ell)=\min\{t\; \mid \; [N]_t\neq 0\}$
in \autoref{LemmaRImixed} if $R$ is mixed or in \autoref{LemmaRIunmixed} if $R$ is unmixed.
Then, $\dim_\mathbb{K} [N]_t\geq \ell$. By \autoref{H-increasing}, we have that
$\dim_\mathbb{K} [N]_{t+1}\geq \ell+1$. Hence, $r_{R}(\ell+1)\leq r_{R}(\ell)+1.$
\end{proof}
From previous bounds of the regularity index when $\ell=1$, we obtain bounds for any $\ell\geq 1$.
\begin{corollary}\label{CorBoundDim}
Suppose that $R$ is either Stanley-Reisner ring or an $F$-pure ring.
If $\depth(R)\geq 2$, then
$$
r_{R}(\ell)\leq \dim(R)+\ell-1.
$$
\end{corollary}
\begin{proof}
We have that
$r_{R}(\ell+1)\leq r_{R}(\ell)+1$, by \autoref{ThmIneqRI}.
Since $r_{R}(1)\leq \dim(R)$ \cite[Theorem 5.5 \& 5.7]{Ovidius},
we conclude that $r_{R}(\ell)\leq \dim(R)+\ell-1$.
\end{proof}
\begin{corollary}\label{CorBoundRegSR}
Suppose that $R$ is a Stanley-Reisner ring corresponding to either a shellable or a Gorenstein simplicial complex.
Then,
$$
r_{R}(\ell)\leq \reg(R)+\ell-1.
$$
\end{corollary}
\begin{proof}
We have that
$r_{R}(\ell+1)\leq r_{R}(\ell)+1$, by \autoref{ThmIneqRI}.
Since $r_{R}(1)\leq \reg(R)$ \cite[Theorem 5.5 \& 5.7]{Ovidius},
we conclude that $r_{R}(\ell)\leq \reg(R)+\ell-1$.
\end{proof}
\begin{corollary}\label{CorBoundRegGor}
Suppose that $R$ is a Gorenstein $F$-pure ring.
Then,
$$
r_{R}(\ell)\leq \reg(R)+\ell-1.
$$
\end{corollary}
\begin{proof}
We have that
$r_{R}(\ell+1)\leq r_{R}(\ell)+1$, by \autoref{ThmIneqRI}.
Since $r_{R}(1)\leq \reg(R$ \cite[Theorem 5.7]{Ovidius},
we conclude that $r_{R}(\ell)\leq \reg(R)+\ell-1$.
\end{proof}
We now seek to rephrase the condition on depth in \autoref{CorBoundDim}
to the topology of the projective space associated to $R,$ in certain cases.
\begin{lemma}\label{LemmaSRConnected}
Let $S=\mathbb{K}[x_{1},\ldots,x_{n}]$, $I\subseteq S$ a Stanley-Reisner ideal and $R=S/I$.
Then, $\depth(R)\geq 2$ if and only if $\Proj(R)$ is connected.
\end{lemma}
\begin{proof}
We first assume that $\mathbb{K}$ has prime characteristic.
We note that neither $\depth(R)$ nor the connectedness of $\Proj(R)$ change after field extensions.
We may assume that $\mathbb{K}$ is a separably closed field.
Let $\widehat{S}$ and $\widehat{R}$ denote the completion of $S$ and $R$ at the maximal homogeneous ideal.
We have that $\depth(R)=\depth(\widehat{R})$. In addition, $\Proj(R)$ is connected if and only if the punctured spectrum of $\widehat{R}$, $\Spec^\circ(\widehat{R})$, is connected.
We have that $\depth(\widehat{R})\geq 2$ if and only if $H^{n-1}_{I}(\widehat{S})=0$ \cite[Theorem 4.3]{LyuVan}, as $F\text{-}\depth(R)=\depth(R)$ for Stanley-Reisner rings.
In addition, $H^{n-1}_{I}(\widehat{S})=0$ if and only if $\Spec^\circ(\widehat{R})$ is connected \cite[III, Corollairie 5.5]{P-S} (see also \cite{HartCD,H-L}).
We now assume that $\mathbb{K}$ has characteristic zero.
We note that neither $\depth(R)$ nor the connectedness of $\Proj(R)$ change after field extensions.
We may assume that $\mathbb{K}=\mathbb{Q}$.
Then, the result follows from reduction to prime characteristic \cite[Theorem 2.3.5]{HHCharZero}.
\end{proof}
\begin{lemma}\label{LemmaFpureConnected}
Let $S=\mathbb{K}[x_{1},\ldots,x_{n}]$, $I\subseteq S$ a homogeneous ideal and $R=S/I$.
Suppose that $\mathbb{K}$ is a separably closed field and $R$ is a $F$-pure ring.
Then, $\depth(R)\geq 0$ if and only if $\Proj(R)$ is connected.
\end{lemma}
\begin{proof}
Let $\widehat{S}$ and $\widehat{R}$ denote the completion of $S$ and $R$ at the maximal homogeneous ideal.
We have that $\depth(R)=\depth(\widehat{R})$. In addition, $\Proj(R)$ is connected if and only if the punctured spectrum of $\widehat{R}$, $\Spec^\circ(\widehat{R})$, is connected.
We have that $\depth(\widehat{R})\geq 2$ if and only if $H^{n-1}_{I}(\widehat{S})=0$ \cite[Theorem 4.3]{LyuVan}, as $F\text{-}\depth(R)=\depth(R)$ for $F$-pure rings.
In addition, $H^{n-1}_{I}(\widehat{S})=0$ if and only if $\Spec^\circ(\widehat{R})$ is connected \cite[III, Corollairie 5.5]{P-S} (see also \cite{HartCD,H-L}).
\end{proof}
We now rephrase \autoref{CorBoundDim}
in terms of the connectedness of $\Proj(R)$.
\begin{theorem}\label{ThmBoundSR}
Suppose that $R$ is an Stanley-Reisner ring.
If $\Proj(R)$ is connected, then
$$
r_{R}(\ell)\leq \dim(R)+\ell-1.
$$
\end{theorem}
\begin{proof}
This result follows from \autoref{LemmaSRConnected}, \autoref{LemmaFpureConnected}, and \autoref{CorBoundDim}.
\end{proof}
\begin{theorem}\label{ThmBoundFpure}
Suppose that $R$ is an $F$-pure graded algebra over a separably closed field.
If $\Proj(R)$ is connected, then
$$
r_{R}(\ell)\leq \dim(R)+\ell-1.
$$
\end{theorem}
\begin{proof}
This result follows from \autoref{LemmaSRConnected}, \autoref{LemmaFpureConnected}, and \autoref{CorBoundDim}.
\end{proof}
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\section{Introduction}
\label{sec:introduction}
Coinvestment allows several stakeholders to share expenses and revenues when deploying of certain projects, otherwise non-beneficial for them.
We model coinvestment via coalitional game theory. Our model can be applied in scenarios when (i)~costly resources must be deployed in some nodes, (ii)~only one entity, hereafter named Network Owner (NO), has access to these nodes and (iii)~such resources are beneficial to third party Service Providers (SPs).
One such scenario is Edge Computing (EC), which is the main application of this paper.
In EC, physical nodes at the edge are possessed by an NO, which can be a network operator, like AT\&T, or a tower company~\cite{Guillemin2021}; in such nodes, the NO deploys computational resources,\footnote{
Cloud providers, e.g. Amazon, distribute ``edge'' resources for rent, but their edge locations go as far as ``data-centers at the edge of the 5G network''(\url{aws.amazon.com/fr/edge}). We instead consider edge nodes much closer to users, e.g. base stations or road side units, which are owned by the NO.
}
which are used by third party SPs, e.g. video streaming services, as YouTube, Netflix, or car manufacturers offering in the future automated driving services, as Tesla or Renault~\cite{AraldoSAC2020,Araldo2016}. The SPs use the resources at the Edge, closer to their end-users, to distribute the load or to satisfy their low latency applications.
The deployment costs however can be very high and cannot be supported solely by the NO. This explains today's impediment for the widespread of EC.
The deployment costs however can be very high and cannot be supported solely by the NO.
This explains today's impediment for the widespread of EC. On the other hand, SPs offering new services, can make large benefits by offering good quality services to their end-users. Hence, it is reasonable to assume that SPs may be willing to contribute to the cost of the deployment of EC resources together with the NO~\cite{Sabella2019}.
On the other hand, SPs offering new services, can make large benefits by offering good quality services to their end-users. Hence, it is reasonable to assume that SPs may be willing to contribute to the cost of the deployment of EC resources together with the NO~\cite{Sabella2019}.
The goal of this paper is to understand how coinvestment can occur, i.e. how cost of deployment and benefits should be shared among the NO and SPs.
Our contributions are:
\begin{compactitem}
\item We propose a model based on coalitional game theory, and describe the discrepancy between the two categories of stakeholders in terms of revenues (brought by SPs) and infrastructure deployment (by the NO).
\item We assess the existence of the core, i.e. the fact that the grand coalition composed of all the stakeholders exists and is stable, by showing the convexity of our game.
\item We propose sharing of the payoff between the stakeholders based on the Shapley value solution concept, which lies in the core in this case.
\item We evaluate the performance of our proposal, through numerical examples for different scenarios and configurations, considering the case of EC.
\end{compactitem}
The remainder of the paper is as follows. \S\ref{sec:relatedwork} introduces the related work. In \S\ref{sec:model}, we formulate our model. \S\ref{sec: analysis} provides an analysis for the coinvestment plan between the stakeholders.
In \S\ref{sec: numerical results}, we show numerical results for several scenarios and configurations. \S\ref{sec: conclusion} concludes the paper.
\section{Related Work}
\label{sec:relatedwork}
The advantage of sharing resources between several tenants has been shown in~\cite{Afraz2018}, for the case of passive optical networks. However, they assume resources are already deployed and allocation to tenants is determined via auctions. We instead consider the case where no resources are deployed yet (as for EC) and thus devise a mechanism for all the players to coinvest to buy and deploy resources, jointly taking into account the resource allocation among players.
An example of coinvestment is given from~\cite{Kiedanski2020}. Investors are producers/consumers of energy, and a battery exists that must be sized to meet everyone's needs. Each of them can decide whether to charge or discharge the battery to use stored energy later, when energy is more expensive (daily hours). The result of coinvestment is the price that each one has to pay to buy the battery. The problem is solved using coalitional game theory~\cite{Tamer2009}, as we shall do in this work. However, we cannot directly use these results, because~\cite{Kiedanski2020} is a linear production problem, valid to model the cost of electricity, while in our case the benefit of using resources at the Edge can be non-linear, due to the diminishing return in resource deployment at the Edge~\cite{Salchow2008}. Another peculiarity of our study is that we have a veto player, which corresponds to the NO. In~\cite{Sereno2012} a coinvestment between Network Providers (NPs) is realized to improve energy efficiency in cellular access networks, i.e. in some parts of the day, except the peak hours, the resources of a single NP can be oversized, so, just a subset of NPs can serve all the load allowing the others to save energy. The resultant benefit of this sharing is later divided among the players. This paper is very close to the problem in this work, but does not match exactly because: (i) the players are at the same level (all NPs), while in our case players have different roles (NO and SPs), (ii) in our problem we do not care about exchanging load between players to save energy. Coalitional Game Theory has been applied to EC~\cite{Moura2019}, e.g. for pricing, spectrum or content sharing in D2D communication, task offloading~\cite{Pham2019}. To the best of our knowledge we are the first to apply it for co-investment in EC, and thus there are no other works with which we can compare our proposal.
\section{Coinvestment model}
\label{sec:model}
Our coinvestment problem is modeled as a coalitional game with transferable payoff, i.e. players can share a common amount of utility/cost~\cite{Osborne1994}. The game is defined by the tuple $(\mathcal{N}, v)$, where $\mathcal{N}$ is the set of players and the~\emph{coalitional value} $v(\mathcal{S})$ is a function that associates a value to any subset $\mathcal{S}\subseteq\mathcal{N}$, called \emph{coalition}.
The players are one physical Network Owner (NO) and $N$ Service providers (SPs). The NO is the entity that owns the network nodes. It could be a network operator, who owns the location of an antenna or a central office. It could be a separated tower company~\cite{Guillemin2021}. The set of players is $\mathcal{N}=\{1,\dots,N, NO\}$.
The first question to be asked in coalitional games is whether the grand coalition $\mathcal{N}$, formed by all players, is stable, i.e. all the players have an incentive to be part of it, in terms of individual payoff.
The payoff of any player $i\in\mathcal{N}$ is
$x^i=r^i-p^i$, where $r^i$ and $p^i$ are the revenues and the payment, i.e. the capital cost, respectively.
We assume that the NO is willing to host physical resources on its nodes. To fix ideas, such nodes are edge nodes and resources are CPU in our case. They might also refer to GPU, etc.
After the deployment, the NO virtualizes and allocates the resources to SPs. SPs do not own physical resources, only the NO does, in base stations for instance. Whenever players form a coalition, they invest together in deploying an amount of computational capacity $C$, measured in \emph{millicores}.
Denoting by $h^i$ the resources allocated to player $i$, $\sum_{i \in S}h^i = C$. For $d$ denoting the price expressed in \emph{dollars} per resource unit, the sum of all players' payments should be such that $\sum_{i \in S}p^i=d\cdot C$. As the NO is only player that hosts the capacity, if it does not join the coalition, the coinvestment cannot take place, and so, no computational capacity can be deployed and no EC can be realized. So, for any coalition $\mathcal{S}$, capacity is subject to:
\begin{equation}
\label{eq:C}
C=\left\{\begin{aligned}
\frac{1}{d } \cdot \sum_{i \in S}p^i\ \text{if}\ \text{NO} \in \mathcal{S}\\
0 \ \text{otherwise}
\end{aligned}\right .
\end{equation}
At every timeslot $t$, each player $i$ has an expected load, $l_t^i$, i.e. the average number of requests coming from users of SP $i$ at time $t$, and which is exogenous to the problem. We assume that the average load profile is the same every day. Each player $i$ has an instantaneous \textbf{utility} $u^i_t$, in monetary units, e.g. \emph{dollars}, which represents the revenues coming from the end users of the services. In other words, the utility is what users pay to a SP to consume its services.
We assume the load and the shape of the utility function are known and truthful inputs.
Similarly to~\cite{Misra2015}, the utility is function of the expected load $l^i_t$ (the more users consume the service, the more they pay) and of the allocated resources $h^i$ (the more resources, the better the service, the more users are willing to pay):
\begin{align}
\label{eq:null_resources_null_utility}
u^i_t=u^i(l^i_t,h^i);\ \ \ \ \ u^i(l_t^i,0)=0
\end{align}
We thus assume (Eqn.\ref{eq:null_resources_null_utility} on the right) that the utility is null if no resources are allocated.
Observe that the case where NO uses some Edge resources for itself can be easily modeled in our framework by introducing a fictitious SP representing the NO using resources of EC.
The NO does not offer any Edge service to the end users, so, its load is null $(l_t^\text{NO}=0)$ and thus it does not need Edge resources for itself ($h^\text{NO}=0$), which implies that $u_t^\text{NO}=0$ and $ \sum_{i \in \mathcal{S}\setminus\{\text{NO}\}}h^i = C$. However, the NO gets a fraction of the value of the grand coalition, if it exists.
The revenues of a coalition is the sum of the utilities of the SPs over the investment period: $\sum_{i \in \mathcal{S}}r^i= D \cdot \sum_{i \in \mathcal{S}}\sum_{t \in [T]} u^i_t$, where $D = 365 \cdot Y$, $Y$ is the duration in years of the investment, $T=96$ is the number of timeslots in one day, considering each timeslot has duration $15 \ minutes$.
We now define the value $v(\mathcal{S})$ of any coalition $\mathcal{S}\subseteq\mathcal{N}$. When forming coalition $\mathcal{S}$, the players involved choose allocation vector $\vec h$ and capacity $C$ so as to maximize the coalition value $v(\mathcal{S})$:
\begin{align}
\label{eq:max1}
v(\mathcal{S})
&= \max_{\vec h, C} v^{\Vec{h}, C,\mathcal{S}}
\defeq
\max_{\vec h, C} D \sum_{i \in \mathcal{S}}\sum_{t=1}^T u^i(l_t^i, h^i)
- d \cdot C
\\
\text{s.t.} &
\label{eq: capacity_constr}
\sum_{i \in \mathcal{S}\setminus\{\text{NO}\}}h^i=C; \,\,\,\, h^\text{NO}=0.
\\
\label{eq:variables_constr}
& C, h^i\geq 0, \ \forall t \in [T], \forall i \in \mathcal{S}.
\end{align}
Observe that NO is a \textbf{veto player}. Indeed, withouth NO, the investment does not take place~\cite[\S13.2]{Osborne1994} (see~\eqref{eq:C}). Therefore, applying~\eqref{eq: capacity_constr}, \eqref{eq:null_resources_null_utility} and~\eqref{eq:max1}, if $\text{NO}\notin\mathcal{S}$, we get $v(\mathcal{S})=0$. We will see also that the SPs form altogether a veto player too.
\section{Analysis}
\label{sec: analysis}
We now assess the cooperative structure of our game and its stability. We show the existence of the core, and hence the formation of the grand coalition.
\subsection{Core and convexity of the game}
\label{sec: Core}
Let us define a payoff vector $(x^i)_{i \in \mathcal{N}}$.
The core is a set of payoff vectors, such that a payoff vector is in the core if the payoffs of each player are such that no subgroup can gain by quitting the grand coalition and forming a different coalition~\cite{Osborne1994}. A well known result in coalitional game theory affirms that the core is non empty if the game is convex~\cite{Shapley1971} and that a particular payoff vector having some ``fairness'' properties, i.e. the Shapley value~\cite{Tamer2009,Shapley1971} lies in the core.
\begin{theo}
\label{theo: the game is convex}
(The game is convex).
Our game $(\mathcal{N}, v)$, whose value function $v$ is described by the optimization problem~\eqref{eq:max1}-\eqref{eq:variables_constr} is convex.
\end{theo}
\begin{proof}
We can rewrite $v(\mathcal{S})$, via~\eqref{eq:max1},\eqref{eq: capacity_constr}, as
\begin{align}
\label{eq: separated_value_function}
v(\mathcal{S}) & =\sum_{i \in \mathcal{S}}\max_{h^i}v^{h^i}
&
\text{ where }
&
v^{h^i} & \triangleq D \cdot \sum_{t=1}^T u^i(l_t^i, h^i)- d \cdot h^i.
\end{align}
The maximum over $h^i$ of Eqn.~\eqref{eq: separated_value_function} gives the contribution of a single player $i$ to the coalitional value, considering its part of the revenues due to its utility function, and the cost of the resources $h^i$ it uses to produce this utility.
Observe that such a contribution is independent from the coalition $\mathcal{S}$ in which $i$ participate.
Eqns.~\eqref{eq: separated_value_function} show that the contribution is separable, i.e. it is the summation of the value functions of the individual players.
We now prove that the game is supermodular, which implies its convexity, thanks to~\cite{Driessen1988}. A game is supermodular if
\begin{align}
\label{eq: check_supermodularity}
\Delta_i(\mathcal{T})
&\leq \Delta_i(\mathcal{S}),
\forall \mathcal{T}\subseteq \mathcal{S}\subseteq \mathcal{N}\setminus\{i\}, \ \forall i \in \mathcal{N}
\\
\text{where }
\Delta_i(\mathcal{S})
&=
v(\mathcal{S} \cup \{ i \}) - v(\mathcal{S})
\underset{\eqref{eq: separated_value_function}}{=}
\max_{h^i} v^{h^i}
\end{align}
is the marginal contribution of player $i$ to coalition $\mathcal{S}$.
Let us fix any $i\in\mathcal{N}$. Given two coalitions, $\mathcal{S}$ and $\mathcal{T}$, such that $\mathcal{T}\subseteq\mathcal{S}\subseteq\mathcal{N}\setminus\{i\}$, we calculate the marginal contribution of player $i$ to both coalitions. Consider the case in which $\text{NO} \in \mathcal{T}$ and $i=\text{SP}^i$, for coalition $\mathcal{T}$ we have (see~\eqref{eq: separated_value_function}): $ \Delta_i(\mathcal{T})=
\max_{h^i} v^{h^i}$.
For coalition $\mathcal{S}$ we have: $\Delta_i(\mathcal{S})=\max_{h^i}v^{h^i}$
so, the marginal contributions are $\Delta_i(\mathcal{S})=\Delta_i(\mathcal{T})$.
Now, we consider the case $i=\text{NO}$. In this case the proof is trivial, in fact $\text{NO} \notin \mathcal{T}\cup \mathcal{S}$, so, for the fact that NO is a veto player
\begin{align}
\label{eq: step_00}
v(\mathcal{T} \cup \{i\})
&\geq 0, v(\mathcal{T})=0, \ \forall \mathcal{T}\setminus\{\text{NO}\}
\\
\label{eq: step_01}
\text{and } v(\mathcal{S} \cup \{i\})
& \geq 0, v(\mathcal{S})=0, \ \forall \mathcal{S}\setminus\{\text{NO}\}.
\end{align}
Therefore, Eqn.~\eqref{eq: check_supermodularity} is verified if and only if
\begin{equation}
v(\mathcal{S} \cup \{i\}) - v(\mathcal{T} \cup \{i\}) \geq 0
\end{equation}
which is equivalent to $\sum_{j \in \mathcal{S}\setminus \mathcal{T}}\max_{h^j}v^{h^j} \geq 0$
where the last inequality is obviously true, since we have a sum of non-negative terms. Another case is the following: $\text{NO} \notin \mathcal{T}\cup \mathcal{S}$ and $i=\text{SP}^i$. This case is easy to prove because we get
$
v(\mathcal{T} \cup \{i\}) - v(\mathcal{T}) = v(\mathcal{S} \cup \{i\}) - v(\mathcal{S})=0
$,
which satisfies the definition of supermodularity. The last case to prove is: $i=\text{SP}^i$, $\text{NO} \notin \mathcal{T}$. In this case the marginal contribution of $i$ to coalition $\mathcal{T}$ is null and Eqn.~\eqref{eq: check_supermodularity} becomes
$
\Delta_i(\mathcal{S}) \geq 0
$. To verify this, we observe that, thanks to~\eqref{eq: separated_value_function}:
\[
\Delta_i(\mathcal{S})=v(\mathcal{S}\cup\{i\})-v(\mathcal{S})=
\begin{cases}\max_{h^i} v^{h_i} & \text{if NO}\in\mathcal{S} \\ 0 & \text{otherwise}\end{cases}\ge 0.
\]
This completes the supermodularity proof and thus convexity. So, the grand coalition can be always formed.
\end{proof}
\subsection{Shapley value}
Finding a mechanism to share the payoff among players is not trivial. One idea would be to divide the payoff equally among players. However, this would not be accepted, since some players contribute to the coalition more than others. First, the NO is a veto player, and its contribution is of primary importance. Second, the SPs do not contribute equally to the coalition: some SPs have more users than others. A second idea would be to share payoffs proportionally to the request load of each SP. However, this would be still unfair, as the benefits collected by SPs do not only depend on the quantity of requests, but also on their type (see \S\ref{sec:utility-function-and-price}).
Fortunately, the proof of convexity in the Th.~\ref{theo: the game is convex} gives us the certainty that
there is a somehow fair way to share the payoff: the Shapley value, which considers the marginal contribution of each player to all the possible coalitions and is computed as~\cite{Osborne1994}: $x^i=\phi^i=\frac{1}{|\mathcal{N}|!}\sum_{\mathcal{S} \subseteq \mathcal{N}\setminus\{i\}}|\mathcal{S}|! \cdot (|\mathcal{N}|-|\mathcal{S}|-1)!\cdot\Delta_i(\mathcal{S})$.
\subsection{Initial investment of players}
\label{sec:initialinvest}
Now that we derived the payoff $x^i$ for each player, we need to calculate how much each player must pay at the beginning of the investment, i.e. $p^i$. This is obtained by solving the following equations:
\begin{flalign}
\label{eq:payments1}
r^i-p^i &=x^i, \forall i \in \mathcal{N}
\\
\text{s.t.} \ \ \sum_{i \in \mathcal{N}} r^i &= D \cdot \sum_{i \in \mathcal{N}}\sum_{t=1}^T u^i(l_t^i, h^{*i})
\\
\label{eq:payments3}
\text{where:} \ \ \vec h^*,C^* &= \argmax_{\vec h, C} v^{\vec h, C,\mathcal{N}} \ \text{s.t.\eqref{eq: capacity_constr}-\eqref{eq:variables_constr}}.
\end{flalign}
\subsection{Relevant properties of our game}
\label{sec:properties of the model}
If player $i$ does not produce revenues and makes not payments, then it is a null player, i.e. $v(\mathcal{S} \cup \{i\})=v(\mathcal{S})$~\cite{Brink2007}. Note that there can be players that do not pay or are even paid ($p^i\le 0$), which still positively contribute to the coalition. For instance, any SP $i$ can positively contribute to the coalition collecting large revenues $r^i$. The NO is never null player, because it is veto player and contributes always to any coalition.
\begin{theo}(Payoff sharing)
\label{theo: payoff_proportions}
The Shapley outcome of the game $(\mathcal{N}, v)$, where $v$ is described by the problem~\eqref{eq:max1}-\eqref{eq:variables_constr}, is divided equally between the NO and the set of all SPs.
\end{theo}
\begin{proof}
Consider the Shapley value of the game $(\mathcal{N}, v)$, i.e. the payoff vector $(\phi_i)_{ i \in \mathcal{N}}$.
We want to prove that the Shapley value of the NO is equal to the sum of the Shapley values of all SPs. To calculate the Shapley value, we need the value of the marginal contribution of any player $i$ to the coalition, i.e. $\Delta_i(\mathcal{S})= \max_{h^i} v^{h^i}$. The NO is a veto player, and so, the $v$ function is null for coalitions without it.
\begin{multline}
\label{eq:deltaNO}
\Delta_\text{NO}(\mathcal{S})= v(\mathcal{S} \cup \{\text{NO}\}) =
\\\sum_{i \in \mathcal{S}} \max_{h^i} v^{h^i}=\sum_{i \in \mathcal{S}}\Delta_j(\mathcal{S}),
\forall \mathcal{S}\subseteq\mathcal{N}\setminus\{\text{NO}\}.
\end{multline}
Now, we can show that the Shapley value of the NO is equal to the sum of the SPs Shapley values. We know that the Shapley value is in the core, which is subject to the efficiency property, $\sum_{i \in \mathcal{N}}\phi_i = v(\mathcal{N})$. Hence, $v(\mathcal{N})= \phi_\text{NO} + \sum_{ j \in \mathcal{N}\setminus\{\text{NO}\}}\phi_j$. The Shapley value of the NO is
\begin{multline}
\phi_\text{NO}=\frac{1}{|\mathcal{N}|!}\sum_{\mathcal{S} \subseteq \mathcal{N}\setminus\{NO\}}|\mathcal{S}|!\cdot (|\mathcal{N}|-|\mathcal{S}|-1)!\cdot\Delta_{\text{NO}}(\mathcal{S}) \underset{\eqref{eq:deltaNO}}{=} \\ \frac{1}{|\mathcal{N}|!}\sum_{\mathcal{S} \subseteq \mathcal{N}\setminus\{NO\}}|\mathcal{S}|!\cdot (|\mathcal{N}|-|\mathcal{S}|-1)!\cdot\sum_{j \in \mathcal{S}}\Delta_j(\mathcal{S}).
\end{multline}
This implies the coalitional value is divided equally between the NO and the set of SPs, $\phi_\text{NO} = \sum_{ j \in \mathcal{N}\setminus\{\text{NO}\}}\phi_j =\frac{v(\mathcal{N})}{2}.
$ This completes the proof.
\end{proof}
The intuition behind this equal sharing of the Shapley value between the NO and the SPs is based on~\cite{Beal2014}:{\it ``each game is decomposed into a weighted sum of unanimity games in which the Shapley value assigns an equal share of a unit to each veto player''.} In our case if the set of SPs is considered as one super-player, it is actually a veto player as well, because the value function is zero if no SP is in the coalition, since it would not be possible to collect revenues from users utilization.
\section{Application to Edge Computing}
\label{sec: numerical results}
\subsection{Parameters}
\subsubsection{Load}
\label{sec:load}
We define the load as an exogenous variable (\S\ref{sec:model}). To reproduce a realistic trend, we consider the daily traffic profile of a SP serving residential users, as modeled in~\cite{Vela2016}, i.e.
$l_t^i=a_0+\sum_{k=1}^K a_k\sin{(2k\pi\frac{t-t_k}{T})}$,
where $t$ is the timeslot and $T$ is the number of timeslots in one day; $a_k$ and $t_k$ are hyperparameters determining the amplitude and the offset of each of the $K$ sinusoidal components.
We take their values from~\cite[Fig.2]{Vela2016}.
\subsubsection{Utility function and price}
\label{sec:utility-function-and-price}
As often observed in reality, we assume the utility~\eqref{eq:null_resources_null_utility} of any~$\text{SP}^i$ is characterized by a diminishing return effect~\cite{Salchow2008}: the marginal utility increment becomes smaller by increasing the $h^i$. For this reason, we model the utility with the following increasing and concave function, similar to (1) of~\cite{Misra2015}:
\begin{equation}
\label{eq: utility_func}
u^i(l_t^i, h^i)=\beta^i \cdot l^i_t \cdot (1- \emph{e}^{- \xi \cdot h^i}).
\end{equation}
The term $\beta^i$ is the \emph{benefit factor} of player $i$ which represents the benefit that a SP gets from serving one unit of load at the Edge. It is a multiplicative constant, null for the NO, $\beta^\text{NO}= 0$. The term $\xi$ models the shape of the diminishing return, i.e. how fast it saturates to its upper bound $\beta^i \cdot l^i_t$. Note that this utility function follows property~\eqref{eq:null_resources_null_utility}.
\subsection{Scenario with 2 SPs of the same type}
\label{sec:secnatio2SPssame}
In this case there are two SPs of the same type: $\beta^{\text{SP}^1}=\beta^{\text{SP}^2}=\hat p$ where $\hat p \triangleq\frac{d}{D \cdot T}$ is the price, $d=0.05$~\emph{dollars/millicores}, amortized over each of $T$ time slots over the investment duration, $\text{D}$. $\text{SP}^1$ and $\text{SP}^2$ have the same temporal trends, but $l_t^1=4l_t^2 \ \forall t$.
In Fig.~\ref{fig: twoSPs capacity and coalitional value}, we show the capacity of purchased CPU and the value of the grand coalition, as a function of the daily total load, $l^\text{tot}=\sum_{t=1}^T l_t^\text{tot}$. We observe that, the more the load the more the capacity installed to serve it. However, recall that the utility functions follow a diminishing return with respect to the resources, so, the trend of the capacity $C$ is sublinear. We observe a linear trend for coalitional value because the value function is linearly dependent on the load (see Eqn.~\eqref{eq: utility_func}).
\begin{figure}[t]
\centering
{\includegraphics[width=0.44\textwidth, height=90pt]{images/Results/coal_val_capacity.png}}
\caption{Capacity and coalitional value as a function of the overall daily load}
\label{fig: twoSPs capacity and coalitional value}
\end{figure}
We observe in Fig.~\ref{fig: twoSPs capacity split and revenues contribution} capacity sharing between the SPs, $\text{SP}^1$ receives a larger capacity: it has to serve a larger part of the requests. Note that, even if the load of $\text{SP}^1$ is 4 times the load of $\text{SP}^2$, the difference between the resource allocated to them is not that big: a consequence of the diminishing return.
The contribution of $\text{SP}^i$ to the coalitional revenues is defined as $\hat r^i=D \cdot \sum_{t=1}^T u^i(l_t^i,h^i)$.
We denote the grand coalitional revenues as $r^\mathcal{N}=\sum_{i\in\mathcal{N}} r^i$, where $r^i$ is the result obtained in~\eqref{eq:payments1}-\eqref{eq:payments3}, i.e. the payoff of each player without considering the component of the payment. The term $\hat r^i$ is the amount of revenues produced by SP $i$, due to the served load during the overall duration of the coinvestment.
Fig.~\ref{fig: twoSPs capacity split and revenues contribution} shows that most contribution comes from $\text{SP}^1$, since its load is four times higher than that of $\text{SP}^2$, and the utility of any SP (and thus its contribution to the grand coalitional revenues) is proportional to the served load; indeed, we observe that $\hat r^{\text{SP}^1}= \frac{1}{4}\hat r^{\text{SP}^2}$.
Note that $h^\text{NO}$ and $r^\text{NO}$ are not in the figure, as the NO does not use resources, because its load is null; this implies that its utility is null so it does not produce revenues to the grand coalition by serving a load.
\begin{figure}[t]
\centering
{\includegraphics[width=0.44\textwidth]{images/Results/twoSPs/stacked_shapley_2SPs.jpg}}
\caption{Split of capacity and contributions to coalitional value}
\label{fig: twoSPs capacity split and revenues contribution}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}[b]{0.44\textwidth}
\centering
{\includegraphics[width=\textwidth]{images/Results/shapley_2SPs_NO_separate.png}}
\caption{Payoff, revenues and payment by NO}
\label{fig: two Sps shapley NO}
\end{subfigure}
\begin{subfigure}[b]{0.44\textwidth}
\centering
{\includegraphics[width=\textwidth]{images/Results/shapley_2SPs_SP1_separate.png}}
\caption{Payoff, revenues and payment by $\text{SP}^1$}
\label{fig: two Sps shapley SP1}
\end{subfigure}
\begin{subfigure}[b]{0.44\textwidth}
\centering
{\includegraphics[width=\textwidth]{images/Results/shapley_2SPs_SP2_separate.png}}
\caption{Payoff, revenues and payment by $\text{SP}^2$}
\label{fig: two Sps shapley SP2}
\end{subfigure}
\caption{Shapley value: payoffs, revenues and payments.}
\label{fig: twoSPs payoffs Shapley}
\end{figure}
Fig.~\ref{fig: twoSPs payoffs Shapley} shows the outcome of the game, i.e. the payment $p^i$, the revenue $r^i$ and the payoff $x^i$ of each player $i\in\mathcal{N}$ given by the Shapley value. We first observe that the payoff of anyone increases with the total load, which is obvious as the revenues of the grand coalition is the sum of the utilities of each player, which in turn increase with the number of served requests. It is interesting to notice that only $\text{SP}^2$ actually pays to deploy the resources at the Edge, while the $\text{NO}$ and $\text{SP}^1$ have negative payments, so, they are not paying, but are being paid. In the case of $\text{SP}^1$, this means that it enjoys an additional gain from the coinvestment in the Edge resources, which sums to the revenue $r^{\text{SP}^1}$ directly coming from its customers. The ``privilege'' of the NO and $\text{SP}^1$ can be explained by the fact that they are the most important for the coalition: NO is the veto player; $\text{SP}^1$ brings to the coalition most of the revenues collected from users (Fig.~\ref{fig: twoSPs capacity split and revenues contribution}).
\subsection{Scenario with 2 SPs of different types}
In this case we have two SPs offering different types of services. For instance, $\text{SP}^2$ may offer an extremely low-latency service, e.g. augmented reality, while $\text{SP}^1$ may offer a less stringent service, e.g. online gaming. In this case, the utility coming from serving a request at the Edge is much higher for $\text{SP}^2$ than for $\text{SP}^1$, since the latter could serve some of the requests (for instance requests not related with interactions with the player) from the Cloud without degrading too much the perceived user experience. Therefore, we consider now that $\beta^{\text{SP}^2}\ge \beta^{\text{SP}^1}$. As in the previous scenario, $\beta^\text{tot}=\beta^{\text{SP}^1}+\beta^{\text{SP}^2}=2\hat p$.
We assume further that $\beta^{\text{SP}^1}=(1-\omega)\cdot \beta^\text{tot}$ and $\beta^{\text{SP}^2}=\omega\cdot \beta^\text{tot}$. We make $\omega$ vary in $[0.5,1]$. The case $\omega=0.5$ corresponds to the previous scenario, where the SPs were of the same type. Increasing $\omega$, the two SPs become more heterogeneous, and in particular $\text{SP}^2$ has higher benefits per unit of load than $\text{SP}^1$. We keep the load as before.
We now show how resource allocation and payoff change with $\omega$. We observe in Fig.~\ref{fig: two Sps shapley compact changing omega avg capacity} that increasing $\omega$, the percentage of CPU given to the $\text{SP}^1$ decreases and is null for $\omega=1$, at which point it is given entirely for $\text{SP}^2$. This is due to the fact that, despite it attracts most of the user load, $\text{SP}^1$ is less useful to assign resources to, as its benefit factor becomes smaller with $\omega$. This tells us, as expected, that resource allocation at the Edge must be taken not only based on load, but also on the nature of the services, and in particular on time-sensitivity, which is reflected in a different benefit per unit of load satisfied at the Edge. In Fig.~\ref{fig: two Sps shapley omega} the payoff sharing reflects what we mentioned above. The marginal contribution brought by $\text{SP}^2$ increases since it produces most of the coalition revenues. In all the cases, the NO has 1/2 the coalitional value, (Th.~\ref{theo: payoff_proportions}).
\begin{figure}
\centering
\begin{subfigure}[b]{0.4\textwidth}
\centering
{\includegraphics[width=0.9\textwidth]{images/Results/piechart_resources_omega.png}}
\caption{Capacity subdivision among the players}
\label{fig: two Sps shapley compact changing omega avg capacity}
\end{subfigure}
\begin{subfigure}[b]{0.4\textwidth}
\centering
{\includegraphics[width=0.9\textwidth]{images/Results/piechart_omega.png}}
\caption{Payoff sharing}
\label{fig: two Sps shapley omega}
\end{subfigure}
\caption{Coalitional value and capacity function of $\omega$}
\label{fig: two Sps shapley compact changing omega}
\end{figure}
\subsection{Price sensitivity analysis}
Here, we assess our model to show its behavior varying the number of SPs.
The consequence of increasing the price is that (Fig.~\ref{fig:manySpsSensitivity}) (i) the purchased capacity is reduced but in a sub-linear way, because of~\eqref{eq: utility_func} and the coalitional value decreases linearly. These trends remain consistent when changing the number $N$ of SPs. Fig.~\ref{fig:manySpsSensitivity} also confirms that adding a player in the game, brings a higher benefit in terms of the value of the coalition $v$. This is in line with the supermodularity of $v$ and hence the convexity of the game (Th.~\ref{theo: the game is convex}) and its stability.
\begin{figure}
\centering
\begin{subfigure}[b]{0.4\textwidth}
\centering
{\includegraphics[width=0.85\textwidth]{images/Results/price_severalSPs/simulations_results2SPs_price.jpg}}
\caption{NO and $N=2$ SPs}
\label{fig:2SpsSensitivity}
\end{subfigure}
\begin{subfigure}[b]{0.4\textwidth}
\centering
{\includegraphics[width=0.85\textwidth]{images/Results/price_severalSPs/simulations_results4SPs_price.jpg}}
\caption{NO and $N=4$ SPs}
\label{fig:4SpsSensitivity}
\end{subfigure}
\begin{subfigure}[b]{0.4\textwidth}
\centering
{\includegraphics[width=0.85\textwidth]{images/Results/price_severalSPs/simulations_results7SPs_price.jpg}}
\caption{NO and $N=7$ SPs}
\label{fig:7SpsSensitivity}
\end{subfigure}
\caption{Coalitional value and capacity function of the price $d$ for different numbers of players}
\label{fig:manySpsSensitivity}
\end{figure}
\section{Conclusion}
\label{sec: conclusion}
We proposed a coalitional game theory solution to enable coinvestment between heterogeneous players (NO and SPs)
and applied it to the of deployment of Edge Computing. By showing the convexity of the game, we proved that the core is non-empty and that the Shapley value, which provides a fair way to divide income among players, lies in the core. So, it is always possible to form the grand coalition, made of all players. We
studied numerically the solution under different scenarios. For future work, we will consider adding a strategy-proof enforcement feature to ensure that players are truthful.
\vspace{-0.9cm}
\section{Introduction}
\label{sec:introduction}
Coinvestment allows several stakeholders to share expenses and revenues when deploying of certain projects, otherwise non-beneficial for them.
We model coinvestment via coalitional game theory. Our model can be applied in scenarios when (i)~costly resources must be deployed in some nodes, (ii)~only one entity, hereafter named Network Owner (NO), has access to these nodes and (iii)~such resources are beneficial to third party Service Providers (SPs).
One such scenario is Edge Computing (EC), which is the main application of this paper.
In EC, physical nodes at the edge are possessed by an NO, which can be a network operator, like AT\&T, or a tower company~\cite{Guillemin2021}; in such nodes, the NO deploys computational resources,\footnote{
Cloud providers, e.g. Amazon, distribute ``edge'' resources for rent, but their edge locations go as far as ``data-centers at the edge of the 5G network''(\url{aws.amazon.com/fr/edge}). We instead consider edge nodes much closer to users, e.g. base stations or road side units, which are owned by the NO.
}
which are used by third party SPs, e.g. video streaming services, as YouTube, Netflix, or car manufacturers offering in the future automated driving services, as Tesla or Renault~\cite{AraldoSAC2020,Araldo2016}. The SPs use the resources at the Edge, closer to their end-users, to distribute the load or to satisfy their low latency applications.
The deployment costs however can be very high and cannot be supported solely by the NO. This explains today's impediment for the widespread of EC.
The deployment costs however can be very high and cannot be supported solely by the NO.
This explains today's impediment for the widespread of EC. On the other hand, SPs offering new services, can make large benefits by offering good quality services to their end-users. Hence, it is reasonable to assume that SPs may be willing to contribute to the cost of the deployment of EC resources together with the NO~\cite{Sabella2019}.
On the other hand, SPs offering new services, can make large benefits by offering good quality services to their end-users. Hence, it is reasonable to assume that SPs may be willing to contribute to the cost of the deployment of EC resources together with the NO~\cite{Sabella2019}.
The goal of this paper is to understand how coinvestment can occur, i.e. how cost of deployment and benefits should be shared among the NO and SPs.
Our contributions are:
\begin{compactitem}
\item We propose a model based on coalitional game theory, and describe the discrepancy between the two categories of stakeholders in terms of revenues (brought by SPs) and infrastructure deployment (by the NO).
\item We assess the existence of the core, i.e. the fact that the grand coalition composed of all the stakeholders exists and is stable, by showing the convexity of our game.
\item We propose sharing of the payoff between the stakeholders based on the Shapley value solution concept, which lies in the core in this case.
\item We evaluate the performance of our proposal, through numerical examples for different scenarios and configurations, considering the case of EC.
\end{compactitem}
The remainder of the paper is as follows. \S\ref{sec:relatedwork} introduces the related work. In \S\ref{sec:model}, we formulate our model. \S\ref{sec: analysis} provides an analysis for the coinvestment plan between the stakeholders.
In \S\ref{sec: numerical results}, we show numerical results for several scenarios and configurations. \S\ref{sec: conclusion} concludes the paper.
\section{Related Work}
\label{sec:relatedwork}
The advantage of sharing resources between several tenants has been shown in~\cite{Afraz2018}, for the case of passive optical networks. However, they assume resources are already deployed and allocation to tenants is determined via auctions. We instead consider the case where no resources are deployed yet (as for EC) and thus devise a mechanism for all the players to coinvest to buy and deploy resources, jointly taking into account the resource allocation among players.
An example of coinvestment is given from~\cite{Kiedanski2020}. Investors are producers/consumers of energy, and a battery exists that must be sized to meet everyone's needs. Each of them can decide whether to charge or discharge the battery to use stored energy later, when energy is more expensive (daily hours). The result of coinvestment is the price that each one has to pay to buy the battery. The problem is solved using coalitional game theory~\cite{Tamer2009}, as we shall do in this work. However, we cannot directly use these results, because~\cite{Kiedanski2020} is a linear production problem, valid to model the cost of electricity, while in our case the benefit of using resources at the Edge can be non-linear, due to the diminishing return in resource deployment at the Edge~\cite{Salchow2008}. Another peculiarity of our study is that we have a veto player, which corresponds to the NO. In~\cite{Sereno2012} a coinvestment between Network Providers (NPs) is realized to improve energy efficiency in cellular access networks, i.e. in some parts of the day, except the peak hours, the resources of a single NP can be oversized, so, just a subset of NPs can serve all the load allowing the others to save energy. The resultant benefit of this sharing is later divided among the players. This paper is very close to the problem in this work, but does not match exactly because: (i) the players are at the same level (all NPs), while in our case players have different roles (NO and SPs), (ii) in our problem we do not care about exchanging load between players to save energy. Coalitional Game Theory has been applied to EC~\cite{Moura2019}, e.g. for pricing, spectrum or content sharing in D2D communication, task offloading~\cite{Pham2019}. To the best of our knowledge we are the first to apply it for co-investment in EC, and thus there are no other works with which we can compare our proposal.
\section{Coinvestment model}
\label{sec:model}
Our coinvestment problem is modeled as a coalitional game with transferable payoff, i.e. players can share a common amount of utility/cost~\cite{Osborne1994}. The game is defined by the tuple $(\mathcal{N}, v)$, where $\mathcal{N}$ is the set of players and the~\emph{coalitional value} $v(\mathcal{S})$ is a function that associates a value to any subset $\mathcal{S}\subseteq\mathcal{N}$, called \emph{coalition}.
The players are one physical Network Owner (NO) and $N$ Service providers (SPs). The NO is the entity that owns the network nodes. It could be a network operator, who owns the location of an antenna or a central office. It could be a separated tower company~\cite{Guillemin2021}. The set of players is $\mathcal{N}=\{1,\dots,N, NO\}$.
The first question to be asked in coalitional games is whether the grand coalition $\mathcal{N}$, formed by all players, is stable, i.e. all the players have an incentive to be part of it, in terms of individual payoff.
The payoff of any player $i\in\mathcal{N}$ is
$x^i=r^i-p^i$, where $r^i$ and $p^i$ are the revenues and the payment, i.e. the capital cost, respectively.
We assume that the NO is willing to host physical resources on its nodes. To fix ideas, such nodes are edge nodes and resources are CPU in our case. They might also refer to GPU, etc.
After the deployment, the NO virtualizes and allocates the resources to SPs. SPs do not own physical resources, only the NO does, in base stations for instance. Whenever players form a coalition, they invest together in deploying an amount of computational capacity $C$, measured in \emph{millicores}.
Denoting by $h^i$ the resources allocated to player $i$, $\sum_{i \in S}h^i = C$. For $d$ denoting the price expressed in \emph{dollars} per resource unit, the sum of all players' payments should be such that $\sum_{i \in S}p^i=d\cdot C$. As the NO is only player that hosts the capacity, if it does not join the coalition, the coinvestment cannot take place, and so, no computational capacity can be deployed and no EC can be realized. So, for any coalition $\mathcal{S}$, capacity is subject to:
\begin{equation}
\label{eq:C}
C=\left\{\begin{aligned}
\frac{1}{d } \cdot \sum_{i \in S}p^i\ \text{if}\ \text{NO} \in \mathcal{S}\\
0 \ \text{otherwise}
\end{aligned}\right .
\end{equation}
At every timeslot $t$, each player $i$ has an expected load, $l_t^i$, i.e. the average number of requests coming from users of SP $i$ at time $t$, and which is exogenous to the problem. We assume that the average load profile is the same every day. Each player $i$ has an instantaneous \textbf{utility} $u^i_t$, in monetary units, e.g. \emph{dollars}, which represents the revenues coming from the end users of the services. In other words, the utility is what users pay to a SP to consume its services.
We assume the load and the shape of the utility function are known and truthful inputs.
Similarly to~\cite{Misra2015}, the utility is function of the expected load $l^i_t$ (the more users consume the service, the more they pay) and of the allocated resources $h^i$ (the more resources, the better the service, the more users are willing to pay):
\begin{align}
\label{eq:null_resources_null_utility}
u^i_t=u^i(l^i_t,h^i);\ \ \ \ \ u^i(l_t^i,0)=0
\end{align}
We thus assume (Eqn.\ref{eq:null_resources_null_utility} on the right) that the utility is null if no resources are allocated.
Observe that the case where NO uses some Edge resources for itself can be easily modeled in our framework by introducing a fictitious SP representing the NO using resources of EC.
The NO does not offer any Edge service to the end users, so, its load is null $(l_t^\text{NO}=0)$ and thus it does not need Edge resources for itself ($h^\text{NO}=0$), which implies that $u_t^\text{NO}=0$ and $ \sum_{i \in \mathcal{S}\setminus\{\text{NO}\}}h^i = C$. However, the NO gets a fraction of the value of the grand coalition, if it exists.
The revenues of a coalition is the sum of the utilities of the SPs over the investment period: $\sum_{i \in \mathcal{S}}r^i= D \cdot \sum_{i \in \mathcal{S}}\sum_{t \in [T]} u^i_t$, where $D = 365 \cdot Y$, $Y$ is the duration in years of the investment, $T=96$ is the number of timeslots in one day, considering each timeslot has duration $15 \ minutes$.
We now define the value $v(\mathcal{S})$ of any coalition $\mathcal{S}\subseteq\mathcal{N}$. When forming coalition $\mathcal{S}$, the players involved choose allocation vector $\vec h$ and capacity $C$ so as to maximize the coalition value $v(\mathcal{S})$:
\begin{align}
\label{eq:max1}
v(\mathcal{S})
&= \max_{\vec h, C} v^{\Vec{h}, C,\mathcal{S}}
\defeq
\max_{\vec h, C} D \sum_{i \in \mathcal{S}}\sum_{t=1}^T u^i(l_t^i, h^i)
- d \cdot C
\\
\text{s.t.} &
\label{eq: capacity_constr}
\sum_{i \in \mathcal{S}\setminus\{\text{NO}\}}h^i=C; \,\,\,\, h^\text{NO}=0.
\\
\label{eq:variables_constr}
& C, h^i\geq 0, \ \forall t \in [T], \forall i \in \mathcal{S}.
\end{align}
Observe that NO is a \textbf{veto player}. Indeed, withouth NO, the investment does not take place~\cite[\S13.2]{Osborne1994} (see~\eqref{eq:C}). Therefore, applying~\eqref{eq: capacity_constr}, \eqref{eq:null_resources_null_utility} and~\eqref{eq:max1}, if $\text{NO}\notin\mathcal{S}$, we get $v(\mathcal{S})=0$. We will see also that the SPs form altogether a veto player too.
\section{Analysis}
\label{sec: analysis}
We now assess the cooperative structure of our game and its stability. We show the existence of the core, and hence the formation of the grand coalition.
\subsection{Core and convexity of the game}
\label{sec: Core}
Let us define a payoff vector $(x^i)_{i \in \mathcal{N}}$.
The core is a set of payoff vectors, such that a payoff vector is in the core if the payoffs of each player are such that no subgroup can gain by quitting the grand coalition and forming a different coalition~\cite{Osborne1994}. A well known result in coalitional game theory affirms that the core is non empty if the game is convex~\cite{Shapley1971} and that a particular payoff vector having some ``fairness'' properties, i.e. the Shapley value~\cite{Tamer2009,Shapley1971} lies in the core.
\begin{theo}
\label{theo: the game is convex}
(The game is convex).
Our game $(\mathcal{N}, v)$, whose value function $v$ is described by the optimization problem~\eqref{eq:max1}-\eqref{eq:variables_constr} is convex.
\end{theo}
\begin{proof}
We can rewrite $v(\mathcal{S})$, via~\eqref{eq:max1},\eqref{eq: capacity_constr}, as
\begin{align}
\label{eq: separated_value_function}
v(\mathcal{S}) & =\sum_{i \in \mathcal{S}}\max_{h^i}v^{h^i}
&
\text{ where }
&
v^{h^i} & \triangleq D \cdot \sum_{t=1}^T u^i(l_t^i, h^i)- d \cdot h^i.
\end{align}
The maximum over $h^i$ of Eqn.~\eqref{eq: separated_value_function} gives the contribution of a single player $i$ to the coalitional value, considering its part of the revenues due to its utility function, and the cost of the resources $h^i$ it uses to produce this utility.
Observe that such a contribution is independent from the coalition $\mathcal{S}$ in which $i$ participate.
Eqns.~\eqref{eq: separated_value_function} show that the contribution is separable, i.e. it is the summation of the value functions of the individual players.
We now prove that the game is supermodular, which implies its convexity, thanks to~\cite{Driessen1988}. A game is supermodular if
\begin{align}
\label{eq: check_supermodularity}
\Delta_i(\mathcal{T})
&\leq \Delta_i(\mathcal{S}),
\forall \mathcal{T}\subseteq \mathcal{S}\subseteq \mathcal{N}\setminus\{i\}, \ \forall i \in \mathcal{N}
\\
\text{where }
\Delta_i(\mathcal{S})
&=
v(\mathcal{S} \cup \{ i \}) - v(\mathcal{S})
\underset{\eqref{eq: separated_value_function}}{=}
\max_{h^i} v^{h^i}
\end{align}
is the marginal contribution of player $i$ to coalition $\mathcal{S}$.
Let us fix any $i\in\mathcal{N}$. Given two coalitions, $\mathcal{S}$ and $\mathcal{T}$, such that $\mathcal{T}\subseteq\mathcal{S}\subseteq\mathcal{N}\setminus\{i\}$, we calculate the marginal contribution of player $i$ to both coalitions. Consider the case in which $\text{NO} \in \mathcal{T}$ and $i=\text{SP}^i$, for coalition $\mathcal{T}$ we have (see~\eqref{eq: separated_value_function}): $ \Delta_i(\mathcal{T})=
\max_{h^i} v^{h^i}$.
For coalition $\mathcal{S}$ we have: $\Delta_i(\mathcal{S})=\max_{h^i}v^{h^i}$
so, the marginal contributions are $\Delta_i(\mathcal{S})=\Delta_i(\mathcal{T})$.
Now, we consider the case $i=\text{NO}$. In this case the proof is trivial, in fact $\text{NO} \notin \mathcal{T}\cup \mathcal{S}$, so, for the fact that NO is a veto player
\begin{align}
\label{eq: step_00}
v(\mathcal{T} \cup \{i\})
&\geq 0, v(\mathcal{T})=0, \ \forall \mathcal{T}\setminus\{\text{NO}\}
\\
\label{eq: step_01}
\text{and } v(\mathcal{S} \cup \{i\})
& \geq 0, v(\mathcal{S})=0, \ \forall \mathcal{S}\setminus\{\text{NO}\}.
\end{align}
Therefore, Eqn.~\eqref{eq: check_supermodularity} is verified if and only if
\begin{equation}
v(\mathcal{S} \cup \{i\}) - v(\mathcal{T} \cup \{i\}) \geq 0
\end{equation}
which is equivalent to $\sum_{j \in \mathcal{S}\setminus \mathcal{T}}\max_{h^j}v^{h^j} \geq 0$
where the last inequality is obviously true, since we have a sum of non-negative terms. Another case is the following: $\text{NO} \notin \mathcal{T}\cup \mathcal{S}$ and $i=\text{SP}^i$. This case is easy to prove because we get
$
v(\mathcal{T} \cup \{i\}) - v(\mathcal{T}) = v(\mathcal{S} \cup \{i\}) - v(\mathcal{S})=0
$,
which satisfies the definition of supermodularity. The last case to prove is: $i=\text{SP}^i$, $\text{NO} \notin \mathcal{T}$. In this case the marginal contribution of $i$ to coalition $\mathcal{T}$ is null and Eqn.~\eqref{eq: check_supermodularity} becomes
$
\Delta_i(\mathcal{S}) \geq 0
$. To verify this, we observe that, thanks to~\eqref{eq: separated_value_function}:
\[
\Delta_i(\mathcal{S})=v(\mathcal{S}\cup\{i\})-v(\mathcal{S})=
\begin{cases}\max_{h^i} v^{h_i} & \text{if NO}\in\mathcal{S} \\ 0 & \text{otherwise}\end{cases}\ge 0.
\]
This completes the supermodularity proof and thus convexity. So, the grand coalition can be always formed.
\end{proof}
\subsection{Shapley value}
Finding a mechanism to share the payoff among players is not trivial. One idea would be to divide the payoff equally among players. However, this would not be accepted, since some players contribute to the coalition more than others. First, the NO is a veto player, and its contribution is of primary importance. Second, the SPs do not contribute equally to the coalition: some SPs have more users than others. A second idea would be to share payoffs proportionally to the request load of each SP. However, this would be still unfair, as the benefits collected by SPs do not only depend on the quantity of requests, but also on their type (see \S\ref{sec:utility-function-and-price}).
Fortunately, the proof of convexity in the Th.~\ref{theo: the game is convex} gives us the certainty that
there is a somehow fair way to share the payoff: the Shapley value, which considers the marginal contribution of each player to all the possible coalitions and is computed as~\cite{Osborne1994}: $x^i=\phi^i=\frac{1}{|\mathcal{N}|!}\sum_{\mathcal{S} \subseteq \mathcal{N}\setminus\{i\}}|\mathcal{S}|! \cdot (|\mathcal{N}|-|\mathcal{S}|-1)!\cdot\Delta_i(\mathcal{S})$.
\subsection{Initial investment of players}
\label{sec:initialinvest}
Now that we derived the payoff $x^i$ for each player, we need to calculate how much each player must pay at the beginning of the investment, i.e. $p^i$. This is obtained by solving the following equations:
\begin{flalign}
\label{eq:payments1}
r^i-p^i &=x^i, \forall i \in \mathcal{N}
\\
\text{s.t.} \ \ \sum_{i \in \mathcal{N}} r^i &= D \cdot \sum_{i \in \mathcal{N}}\sum_{t=1}^T u^i(l_t^i, h^{*i})
\\
\label{eq:payments3}
\text{where:} \ \ \vec h^*,C^* &= \argmax_{\vec h, C} v^{\vec h, C,\mathcal{N}} \ \text{s.t.\eqref{eq: capacity_constr}-\eqref{eq:variables_constr}}.
\end{flalign}
\subsection{Relevant properties of our game}
\label{sec:properties of the model}
If player $i$ does not produce revenues and makes not payments, then it is a null player, i.e. $v(\mathcal{S} \cup \{i\})=v(\mathcal{S})$~\cite{Brink2007}. Note that there can be players that do not pay or are even paid ($p^i\le 0$), which still positively contribute to the coalition. For instance, any SP $i$ can positively contribute to the coalition collecting large revenues $r^i$. The NO is never null player, because it is veto player and contributes always to any coalition.
\begin{theo}(Payoff sharing)
\label{theo: payoff_proportions}
The Shapley outcome of the game $(\mathcal{N}, v)$, where $v$ is described by the problem~\eqref{eq:max1}-\eqref{eq:variables_constr}, is divided equally between the NO and the set of all SPs.
\end{theo}
\begin{proof}
Consider the Shapley value of the game $(\mathcal{N}, v)$, i.e. the payoff vector $(\phi_i)_{ i \in \mathcal{N}}$.
We want to prove that the Shapley value of the NO is equal to the sum of the Shapley values of all SPs. To calculate the Shapley value, we need the value of the marginal contribution of any player $i$ to the coalition, i.e. $\Delta_i(\mathcal{S})= \max_{h^i} v^{h^i}$. The NO is a veto player, and so, the $v$ function is null for coalitions without it.
\begin{multline}
\label{eq:deltaNO}
\Delta_\text{NO}(\mathcal{S})= v(\mathcal{S} \cup \{\text{NO}\}) =
\\\sum_{i \in \mathcal{S}} \max_{h^i} v^{h^i}=\sum_{i \in \mathcal{S}}\Delta_j(\mathcal{S}),
\forall \mathcal{S}\subseteq\mathcal{N}\setminus\{\text{NO}\}.
\end{multline}
Now, we can show that the Shapley value of the NO is equal to the sum of the SPs Shapley values. We know that the Shapley value is in the core, which is subject to the efficiency property, $\sum_{i \in \mathcal{N}}\phi_i = v(\mathcal{N})$. Hence, $v(\mathcal{N})= \phi_\text{NO} + \sum_{ j \in \mathcal{N}\setminus\{\text{NO}\}}\phi_j$. The Shapley value of the NO is
\begin{multline}
\phi_\text{NO}=\frac{1}{|\mathcal{N}|!}\sum_{\mathcal{S} \subseteq \mathcal{N}\setminus\{NO\}}|\mathcal{S}|!\cdot (|\mathcal{N}|-|\mathcal{S}|-1)!\cdot\Delta_{\text{NO}}(\mathcal{S}) \underset{\eqref{eq:deltaNO}}{=} \\ \frac{1}{|\mathcal{N}|!}\sum_{\mathcal{S} \subseteq \mathcal{N}\setminus\{NO\}}|\mathcal{S}|!\cdot (|\mathcal{N}|-|\mathcal{S}|-1)!\cdot\sum_{j \in \mathcal{S}}\Delta_j(\mathcal{S}).
\end{multline}
This implies the coalitional value is divided equally between the NO and the set of SPs, $\phi_\text{NO} = \sum_{ j \in \mathcal{N}\setminus\{\text{NO}\}}\phi_j =\frac{v(\mathcal{N})}{2}.
$ This completes the proof.
\end{proof}
The intuition behind this equal sharing of the Shapley value between the NO and the SPs is based on~\cite{Beal2014}:{\it ``each game is decomposed into a weighted sum of unanimity games in which the Shapley value assigns an equal share of a unit to each veto player''.} In our case if the set of SPs is considered as one super-player, it is actually a veto player as well, because the value function is zero if no SP is in the coalition, since it would not be possible to collect revenues from users utilization.
\section{Application to Edge Computing}
\label{sec: numerical results}
\subsection{Parameters}
\subsubsection{Load}
\label{sec:load}
We define the load as an exogenous variable (\S\ref{sec:model}). To reproduce a realistic trend, we consider the daily traffic profile of a SP serving residential users, as modeled in~\cite{Vela2016}, i.e.
$l_t^i=a_0+\sum_{k=1}^K a_k\sin{(2k\pi\frac{t-t_k}{T})}$,
where $t$ is the timeslot and $T$ is the number of timeslots in one day; $a_k$ and $t_k$ are hyperparameters determining the amplitude and the offset of each of the $K$ sinusoidal components.
We take their values from~\cite[Fig.2]{Vela2016}.
\subsubsection{Utility function and price}
\label{sec:utility-function-and-price}
As often observed in reality, we assume the utility~\eqref{eq:null_resources_null_utility} of any~$\text{SP}^i$ is characterized by a diminishing return effect~\cite{Salchow2008}: the marginal utility increment becomes smaller by increasing the $h^i$. For this reason, we model the utility with the following increasing and concave function, similar to (1) of~\cite{Misra2015}:
\begin{equation}
\label{eq: utility_func}
u^i(l_t^i, h^i)=\beta^i \cdot l^i_t \cdot (1- \emph{e}^{- \xi \cdot h^i}).
\end{equation}
The term $\beta^i$ is the \emph{benefit factor} of player $i$ which represents the benefit that a SP gets from serving one unit of load at the Edge. It is a multiplicative constant, null for the NO, $\beta^\text{NO}= 0$. The term $\xi$ models the shape of the diminishing return, i.e. how fast it saturates to its upper bound $\beta^i \cdot l^i_t$. Note that this utility function follows property~\eqref{eq:null_resources_null_utility}.
\subsection{Scenario with 2 SPs of the same type}
\label{sec:secnatio2SPssame}
In this case there are two SPs of the same type: $\beta^{\text{SP}^1}=\beta^{\text{SP}^2}=\hat p$ where $\hat p \triangleq\frac{d}{D \cdot T}$ is the price, $d=0.05$~\emph{dollars/millicores}, amortized over each of $T$ time slots over the investment duration, $\text{D}$. $\text{SP}^1$ and $\text{SP}^2$ have the same temporal trends, but $l_t^1=4l_t^2 \ \forall t$.
In Fig.~\ref{fig: twoSPs capacity and coalitional value}, we show the capacity of purchased CPU and the value of the grand coalition, as a function of the daily total load, $l^\text{tot}=\sum_{t=1}^T l_t^\text{tot}$. We observe that, the more the load the more the capacity installed to serve it. However, recall that the utility functions follow a diminishing return with respect to the resources, so, the trend of the capacity $C$ is sublinear. We observe a linear trend for coalitional value because the value function is linearly dependent on the load (see Eqn.~\eqref{eq: utility_func}).
\begin{figure}[t]
\centering
{\includegraphics[width=0.44\textwidth, height=90pt]{images/Results/coal_val_capacity.png}}
\caption{Capacity and coalitional value as a function of the overall daily load}
\label{fig: twoSPs capacity and coalitional value}
\end{figure}
We observe in Fig.~\ref{fig: twoSPs capacity split and revenues contribution} capacity sharing between the SPs, $\text{SP}^1$ receives a larger capacity: it has to serve a larger part of the requests. Note that, even if the load of $\text{SP}^1$ is 4 times the load of $\text{SP}^2$, the difference between the resource allocated to them is not that big: a consequence of the diminishing return.
The contribution of $\text{SP}^i$ to the coalitional revenues is defined as $\hat r^i=D \cdot \sum_{t=1}^T u^i(l_t^i,h^i)$.
We denote the grand coalitional revenues as $r^\mathcal{N}=\sum_{i\in\mathcal{N}} r^i$, where $r^i$ is the result obtained in~\eqref{eq:payments1}-\eqref{eq:payments3}, i.e. the payoff of each player without considering the component of the payment. The term $\hat r^i$ is the amount of revenues produced by SP $i$, due to the served load during the overall duration of the coinvestment.
Fig.~\ref{fig: twoSPs capacity split and revenues contribution} shows that most contribution comes from $\text{SP}^1$, since its load is four times higher than that of $\text{SP}^2$, and the utility of any SP (and thus its contribution to the grand coalitional revenues) is proportional to the served load; indeed, we observe that $\hat r^{\text{SP}^1}= \frac{1}{4}\hat r^{\text{SP}^2}$.
Note that $h^\text{NO}$ and $r^\text{NO}$ are not in the figure, as the NO does not use resources, because its load is null; this implies that its utility is null so it does not produce revenues to the grand coalition by serving a load.
\begin{figure}[t]
\centering
{\includegraphics[width=0.44\textwidth]{images/Results/twoSPs/stacked_shapley_2SPs.jpg}}
\caption{Split of capacity and contributions to coalitional value}
\label{fig: twoSPs capacity split and revenues contribution}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}[b]{0.44\textwidth}
\centering
{\includegraphics[width=\textwidth]{images/Results/shapley_2SPs_NO_separate.png}}
\caption{Payoff, revenues and payment by NO}
\label{fig: two Sps shapley NO}
\end{subfigure}
\begin{subfigure}[b]{0.44\textwidth}
\centering
{\includegraphics[width=\textwidth]{images/Results/shapley_2SPs_SP1_separate.png}}
\caption{Payoff, revenues and payment by $\text{SP}^1$}
\label{fig: two Sps shapley SP1}
\end{subfigure}
\begin{subfigure}[b]{0.44\textwidth}
\centering
{\includegraphics[width=\textwidth]{images/Results/shapley_2SPs_SP2_separate.png}}
\caption{Payoff, revenues and payment by $\text{SP}^2$}
\label{fig: two Sps shapley SP2}
\end{subfigure}
\caption{Shapley value: payoffs, revenues and payments.}
\label{fig: twoSPs payoffs Shapley}
\end{figure}
Fig.~\ref{fig: twoSPs payoffs Shapley} shows the outcome of the game, i.e. the payment $p^i$, the revenue $r^i$ and the payoff $x^i$ of each player $i\in\mathcal{N}$ given by the Shapley value. We first observe that the payoff of anyone increases with the total load, which is obvious as the revenues of the grand coalition is the sum of the utilities of each player, which in turn increase with the number of served requests. It is interesting to notice that only $\text{SP}^2$ actually pays to deploy the resources at the Edge, while the $\text{NO}$ and $\text{SP}^1$ have negative payments, so, they are not paying, but are being paid. In the case of $\text{SP}^1$, this means that it enjoys an additional gain from the coinvestment in the Edge resources, which sums to the revenue $r^{\text{SP}^1}$ directly coming from its customers. The ``privilege'' of the NO and $\text{SP}^1$ can be explained by the fact that they are the most important for the coalition: NO is the veto player; $\text{SP}^1$ brings to the coalition most of the revenues collected from users (Fig.~\ref{fig: twoSPs capacity split and revenues contribution}).
\subsection{Scenario with 2 SPs of different types}
In this case we have two SPs offering different types of services. For instance, $\text{SP}^2$ may offer an extremely low-latency service, e.g. augmented reality, while $\text{SP}^1$ may offer a less stringent service, e.g. online gaming. In this case, the utility coming from serving a request at the Edge is much higher for $\text{SP}^2$ than for $\text{SP}^1$, since the latter could serve some of the requests (for instance requests not related with interactions with the player) from the Cloud without degrading too much the perceived user experience. Therefore, we consider now that $\beta^{\text{SP}^2}\ge \beta^{\text{SP}^1}$. As in the previous scenario, $\beta^\text{tot}=\beta^{\text{SP}^1}+\beta^{\text{SP}^2}=2\hat p$.
We assume further that $\beta^{\text{SP}^1}=(1-\omega)\cdot \beta^\text{tot}$ and $\beta^{\text{SP}^2}=\omega\cdot \beta^\text{tot}$. We make $\omega$ vary in $[0.5,1]$. The case $\omega=0.5$ corresponds to the previous scenario, where the SPs were of the same type. Increasing $\omega$, the two SPs become more heterogeneous, and in particular $\text{SP}^2$ has higher benefits per unit of load than $\text{SP}^1$. We keep the load as before.
We now show how resource allocation and payoff change with $\omega$. We observe in Fig.~\ref{fig: two Sps shapley compact changing omega avg capacity} that increasing $\omega$, the percentage of CPU given to the $\text{SP}^1$ decreases and is null for $\omega=1$, at which point it is given entirely for $\text{SP}^2$. This is due to the fact that, despite it attracts most of the user load, $\text{SP}^1$ is less useful to assign resources to, as its benefit factor becomes smaller with $\omega$. This tells us, as expected, that resource allocation at the Edge must be taken not only based on load, but also on the nature of the services, and in particular on time-sensitivity, which is reflected in a different benefit per unit of load satisfied at the Edge. In Fig.~\ref{fig: two Sps shapley omega} the payoff sharing reflects what we mentioned above. The marginal contribution brought by $\text{SP}^2$ increases since it produces most of the coalition revenues. In all the cases, the NO has 1/2 the coalitional value, (Th.~\ref{theo: payoff_proportions}).
\begin{figure}
\centering
\begin{subfigure}[b]{0.4\textwidth}
\centering
{\includegraphics[width=0.9\textwidth]{images/Results/piechart_resources_omega.png}}
\caption{Capacity subdivision among the players}
\label{fig: two Sps shapley compact changing omega avg capacity}
\end{subfigure}
\begin{subfigure}[b]{0.4\textwidth}
\centering
{\includegraphics[width=0.9\textwidth]{images/Results/piechart_omega.png}}
\caption{Payoff sharing}
\label{fig: two Sps shapley omega}
\end{subfigure}
\caption{Coalitional value and capacity function of $\omega$}
\label{fig: two Sps shapley compact changing omega}
\end{figure}
\subsection{Price sensitivity analysis}
Here, we assess our model to show its behavior varying the number of SPs.
The consequence of increasing the price is that (Fig.~\ref{fig:manySpsSensitivity}) (i) the purchased capacity is reduced but in a sub-linear way, because of~\eqref{eq: utility_func} and the coalitional value decreases linearly. These trends remain consistent when changing the number $N$ of SPs. Fig.~\ref{fig:manySpsSensitivity} also confirms that adding a player in the game, brings a higher benefit in terms of the value of the coalition $v$. This is in line with the supermodularity of $v$ and hence the convexity of the game (Th.~\ref{theo: the game is convex}) and its stability.
\begin{figure}
\centering
\begin{subfigure}[b]{0.4\textwidth}
\centering
{\includegraphics[width=0.85\textwidth]{images/Results/price_severalSPs/simulations_results2SPs_price.jpg}}
\caption{NO and $N=2$ SPs}
\label{fig:2SpsSensitivity}
\end{subfigure}
\begin{subfigure}[b]{0.4\textwidth}
\centering
{\includegraphics[width=0.85\textwidth]{images/Results/price_severalSPs/simulations_results4SPs_price.jpg}}
\caption{NO and $N=4$ SPs}
\label{fig:4SpsSensitivity}
\end{subfigure}
\begin{subfigure}[b]{0.4\textwidth}
\centering
{\includegraphics[width=0.85\textwidth]{images/Results/price_severalSPs/simulations_results7SPs_price.jpg}}
\caption{NO and $N=7$ SPs}
\label{fig:7SpsSensitivity}
\end{subfigure}
\caption{Coalitional value and capacity function of the price $d$ for different numbers of players}
\label{fig:manySpsSensitivity}
\end{figure}
\section{Conclusion}
\label{sec: conclusion}
We proposed a coalitional game theory solution to enable coinvestment between heterogeneous players (NO and SPs)
and applied it to the of deployment of Edge Computing. By showing the convexity of the game, we proved that the core is non-empty and that the Shapley value, which provides a fair way to divide income among players, lies in the core. So, it is always possible to form the grand coalition, made of all players. We
studied numerically the solution under different scenarios. For future work, we will consider adding a strategy-proof enforcement feature to ensure that players are truthful.
\vspace{-0.9cm}
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| 5
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\section{Introduction}
Data with missing values are ubiquitous in many applications, as in health or business: some observations come with missing features. There is a rich statistical literature on imputation as well as inference with missing values \citep{rubin1976inference,little2002statistical}. Most of the theory and practices build upon the \emph{Missing At Random} (MAR) assumption that allows to maximize the likelihood of observed data while ignoring the missing-values mechanism, for instance using expectation maximization \citep{dempster1977maximum}. On the opposite, Missing Not At Random settings, where missingness depends on the unobserved values, may not be identifiable and require dedicated methods with a model of the missing-values mechanism.
Learning predictive models with missing values poses distinct challenges compared to inference tasks \citep{josse2019consistency}. Indeed, when the input is an arbitrary subset of variables in dimension $d$, there are $2^d$ potential missing data patterns and as many sub-models to learn. Consequently, even simple data-generating mechanisms lead to complex decision rules \citep{LeMorvan2020Linear}. To date, there are few supervised-learning models natively suited for partially-observed data. A notable exception is found with tree-based models \citep{Twala2008_mia,chen2016xgboost}, widely used in data-science practice.
The most common practice however remains by far to use off-the-shelf methods first for imputation of missing values and second for supervised-learning on the resulting completed data. Such a procedure may benefit from progress in missing-value imputation with machine learning [\citealt{VanBuuren2018Flexible}, \mbox{\citealt{yoon2018gain}}].
However, there is a lack of learning theory to support such Impute-then-Regress procedures: Under what conditions are they Bayes consistent? Which aspects of the imputation are important?
There is empirical realization that the choice of imputation matters for predictive performance. The NADIA R package \citep{nadia_R} can select an imputation method to minimize a prediction error on a test set.
Auto-ML is used to optimize full pipelines, including imputation \cite[eg][]{jarrett2021clairvoyance}.
\citet{ipsen2020deal} optimize a constant imputation for supervised learning. However, the imputation is only weakly guided by the target in these approaches, it is set either from a family of black-box methods using gradient-free model selection, or from trivial imputation functions. In addition, there is a lack of insight on what drives a good imputation for prediction.
We contribute a systematic analysis of Impute-the-Regress procedures in a general setting: non-linear response function and any missingness mechanism (no MAR assumptions).
We show that:
\begin{itemize}[leftmargin=1.5em, itemsep=.5ex, topsep=-.5ex, parsep=.5ex, partopsep=-.5ex]
\item Impute-then-Regress procedures are Bayes optimal for \emph{all missing data mechanisms} and for \emph{almost all imputation functions}, whatever the number of variables that may be missing. This very general result gives theoretical grounding to such widespread procedures.
\item We study ``natural'' choices of imputation and regression functions: the oracle imputation by the conditional expectation and oracle regression function on the complete data. We show that chaining these oracles is not Bayes optimal in general and quantify its excess risk. We show that in both cases, choosing an oracle for one step, imputation or regression, imposes discontinuities on the other step, thus making it harder to learn.
\item As these results suggest that imputation and regression should be adapted to one another, we contribute a method that jointly optimizes imputation and regression, using NeuMiss networks \citep{LeMorvan2020NeuMiss} as a differentiable imputation procedure.
\item We compare empirically a number of Impute-then-Regress procedures on simulated non-linear regression tasks. Joint optimization of both steps provides the best performance.
\end{itemize}
\section{Problem setting}
\paragraph{Notations} We consider a dataset of i.i.d.\ realizations of the random variable $(X, M, Y) \in \mathbb{R}^d \times \cbr{0, 1}^d \times \mathbb{R}$ where $X$ are the complete covariates, $M$ a missingness indicator, and $Y$ a response of interest. For each realization $(x, m, y)$, $m_j=1$ indicates that $x_j$ is missing, and $m_j=0$ that it is observed. We denote by $mis(m) \subset \bbr{1, d}$ the indices corresponding to the missing covariates (and similarly $obs(m)$ the observed indices), so that $x_{obs(m)}$ corresponds to the entries actually observed. We define the incomplete covariate vector $\widetilde{X} \in \br{\mathbb{R} \cup \br{\mathtt{NA}}}^d$ as $\widetilde{X}_j = X_j$ if $M_j = 0$ and $\widetilde{X}_j = \mathtt{NA}$ otherwise, where $\mathtt{NA}$ represents a missing value.
\paragraph{Assumptions} We assume that $X$ admits a density on $\mathbb{R}^d$ and that, for all $j \in \bbr{1, d},$ each component $X_j$ has finite expectation and variance, that is $ \mathbb{E}\sqb{X_j^2}<\infty$. Moreover, we assume that the response $Y$ is generated according to:
\begin{equation}
\label{eq:generation}
Y = f^\star(X) + \epsilon, \qquad \textrm{with}~~\mathbb{E}\sqb{\epsilon|X_{obs(M)}, M} = 0 \quad \textrm{and}~~\mathbb{E}\sqb{Y^2} < \infty.
\end{equation}%
where $f^\star: \mathbb{R}^d \to \mathbb{R}$ is a function of the complete input data $X$, $\epsilon \in \mathbb{R}$ is a random noise variable.
\subsection{Supervised learning with missing values}
\paragraph{Optimization problem} In practice, in the presence of missing values, we do not have access to the complete data $(X, M, Y)$ but only to the subset of it that is observed, i.e, $(X_{obs(M)}, M, Y)$. Thus instead of learning a mapping from $\mathbb{R}^d$ to $\mathbb{R}$, we need to learn a mapping from $\br{\mathbb{R} \cup \br{\mathtt{NA}}}^d$ to $\mathbb{R}$, where the set of observed covariates can be any subset of $\bbr{1, d}$. It is this unusual input space, partly discrete, that makes supervised learning with missing values challenging and different from classical supervised learning problems. Formally, the optimization problem we wish to solve is:
\begin{equation}
\label{eq:minimization_pb}
\min_{f: \br{\mathbb{R} \cup \br{\mathtt{NA}}}^d \mapsto \mathbb{R}} \mathcal{R}(f):= \mathbb{E}\sqb{\br{Y - f(\widetilde X)}^2}
\end{equation}
\paragraph{Bayes predictor} The function which minimizes \eqref{eq:minimization_pb}, called the \emph{Bayes predictor}, is given by:
\begin{equation}
\tilde{f}^{\star}(\widetilde X) = \mathbb{E} \sqb{Y | X_{obs(M)}, M}
= \mathbb{E} \sqb{f^\star(X) | X_{obs(M)}, M}. \label{eq:BP}
\end{equation}
As $\widetilde X$ is a function of $X_{obs}$ and $M$, we will sometimes slightly abuse notations and write $\tilde f^\star(\widetilde X) = \tilde f^\star(X_{obs}, M)$. The risk of the Bayes predictor is called the \emph{Bayes risk}, which we denote as $\mathcal{R}^{\star}$. It is the lowest achievable risk for a given supervised learning problem.
\begin{definition}[Bayes optimality]
A \emph{Bayes optimal} function $f$ achieves the Bayes rate, i.e, $\mathcal{R}(f) = \mathcal{R}^{\star}$.
\end{definition}
As can be seen from \eqref{eq:BP}, the Bayes predictor is a function of $M$, a discrete random variable that can take one of $2^d$ values since $M \in \cbr{0, 1}^d$. The function $\tilde{f}^{\star}$ can thus be viewed as $2^d$ different functions, one for each possible subset of variables. This view raises questions that are central to this paper: How should we parametrize functions on such input domains? And which function families should we consider to approximate $\tilde{f}^{\star}$? These questions have been studied in the case where $f^\star$ is assumed to be a linear function, and $X$ follows a Gaussian distribution. Indeed, under these assumptions, \citet{LeMorvan2020Linear,LeMorvan2020NeuMiss} have derived analytical expressions for the Bayes predictor and deduced appropriate parametric estimators. However, aside from specific cases, it is impossible to derive an analytical expression for the Bayes predictor, and novel arguments are needed to understand which classes of functions should be considered in general.
\section{Asymptotic analysis of Impute-then-regress procedures}
\subsection{Impute-then-regress procedures}
Let $|mis(m)|$ (resp. $|obs(m)|$) be the number of missing entries (resp. observed) for any missing data pattern $m$. For each $m \in \{0,1\}^d$, we define an \emph{imputation function} $\phi^{(m)}: \mathbb{R}^{|obs(m)|} \to \mathbb{R}^{|mis(m)|}$ which outputs values for the missing entries based on the observed ones. We denote by $\phi_j^{(m)}: \mathbb{R}^{|obs(m)|} \to \mathbb{R}$ the component function of $\phi^{(m)}$ that imputes the $j$-th component in $X$ if it is missing. Classical choices of imputation functions include constant functions or linear functions. Finally, we introduce the family of functions $\mathcal{F}^I$ that transform an incomplete vector into a complete one, precisely:
\begin{equation}
\label{eq:imputation}
\mathcal{F}^I = \cbr{\Phi: \br{\mathbb{R} \cup \cbr{\mathtt{NA}}}^d \to \mathbb{R}^d: \forall j \in \bbr{1, d},\: \Phi_j(\widetilde X) =
\begin{cases}
X_j \: &\text{if} \: M_j = 0\\
\phi^{(M)}_j(X_{obs(M)}) \: &\text{if} \: M_j = 1
\end{cases}
}.
\end{equation}
Let us define $\mathcal{F}^I_{\infty}$ in the exact same way but for imputation functions $\phi^{(m)} \in \mathcal{C}^{\infty}$, for all $m \in \{0,1\}^d$. Here we study \emph{Impute-then-regress procedures}, which we define as two-step procedures where the data is first imputed using a function $\Phi \in \mathcal{F}^I$, and then a regression is performed on the imputed data. Such a procedure is quite natural to deal with arbitrary subsets of inputs variables. It embeds the data into $\mathbb{R}^d$ to reduce the problem to a classical one. In practice, impute-then-regress procedures are widely used. However, the choice of function class is so far mostly ad-hoc and raises multiple questions: How close to the Bayes rate can functions obtained via such procedures be? Should we prefer some choices of imputation functions over others? What happens when the missing data mechanism is missing not at random? In this section, we will give answers to these questions.
Below, we write $obs$ (resp.~$mis$) instead of $obs(M)$ (resp.~$mis(M)$) to lighten notations.
\subsection{Impute-then-regress procedures are Bayes optimal}
\begin{definition}[Universal consistency]
An estimator $f_n$ is \emph{Bayes consistent} if $\lim_{n \to \infty}\mathcal{R}(f_n) = \mathcal{R}^{\star}$. It is said to be \emph{universally consistent} if the previous statement holds for all distributions of $(X,Y)$.
\end{definition}
The following theorem shows that Impute-then-regress procedures are Bayes optimal for almost all imputation functions. In other words, it means that a universal learner trained on imputed data provides optimal performances asymptotically for almost all imputation functions. Let us nos define, for all imputation function $\Phi \in \mathcal{F}^I$, the function $g^\star_{\Phi} \in \underset{g: \mathbb{R}^d \mapsto \mathbb{R}}{\text{argmin}} \quad \mathbb{E}\sqb{\br{Y-g \circ \Phi(\widetilde X)}^2}$.
\begin{restatable}[Bayes consistency of Impute-then-regress procedures]{theorem}{fbp}
\label{th:fbp}
Assume the data is generated according to \eqref{eq:generation}.
Then, for almost all imputation function $\Phi \in \mathcal{F}^I_{\infty}$, the function $g^\star_{\Phi} \circ \Phi$ is Bayes optimal.
In other words, for almost all imputation functions $\Phi\in \mathcal{F}^I_{\infty}$, a universally consistent algorithm trained on the imputed data $\Phi(\widetilde X)$ is Bayes consistent.
\end{restatable}
Appendix~\ref{ss:bayes_optimality} gives the proof. Theorem~\ref{th:fbp} states a very general result: Impute-then-regress procedures are Bayes consistent for all missing data mechanisms, almost all imputation functions, regardless of the distribution of $(X,Y)$ and the number of missing covariates. Since Theorem~\ref{th:fbp} holds for almost all imputation functions, it implies that good imputations are not required to obtain good predictive performances, at least asymptotically. Note that here, the notion of \emph{almost all} is to be understood in its topological sense, and not in its measure theory sense. Moreover, this theorem does not make any assumption on the missing data mechanism, and is therefore valid for Missing Not At Random (MNAR) data. This contrasts with most methods for inference and imputation with missing values, valid only for MAR data. Finally, the theorem remains valid for any configuration of variables that may contain missing values, including the case in which all variables may contain missing values.
Bayes consistency of Impute-the-Regress procedures has already been
studied, but in much more restricted settings.
\citet{josse2019consistency} proved that such procedures are Bayes consistent under the MAR assumption, for constant imputations functions and for only one potentially missing variable.
\citet{Bertsimas2021Prediction} refined this result to almost surely continuous imputation functions. While these two prior works build on very similar proofs, we use here very different arguments summarized in the next paragraph.
The first key idea of the proof is that, after imputation, all data points with a given missing data pattern $m$ are mapped to a manifold $\mathcal{M}^{(m)}$ of dimension $|obs(m)|$. For example in 2D, data points are mapped to $\mathbb{R}^2$ when completely observed, to 1D manifolds when they have one value missing, and to one point when all values are missing (see fig.\,\ref{fig:example_manifolds}). Thus, Impute-then-Regress procedures first map data points to various manifolds depending on their missing data patterns and then apply a prediction function defined on the whole space including manifolds. The second key idea of the proof is to ensure that the missing data patterns of imputed points can almost surely be identified. For this, the proof requires that all manifolds of the same dimension are pairwise transverse. This assumption is sufficient, though not necessary, to ensure that the intersection of two manifolds of dimension $|obs(m)|$ cannot itself be of dimension $|obs(m)|$. Transversality is a weak assumption. In fact, Thom's transversality theorem, (which we rely on in our proof) says that it is a generic property: it holds for ``typical examples'', i.e \emph{almost all} imputation functions will lead to transverse manifolds.
The proof is constructive and exhibits a function $g^\star_{\Phi}$ which achieves the Bayes rate for a given set of imputation functions. For each manifold $\mathcal{M}^{(m)}$, ordered from smallest dimension to largest, we require that $g^\star_{\Phi}$ on $\mathcal{M}^{(m)}$ equals the Bayes predictor for missing data pattern $m$ except on points for which $g^\star_{\Phi}$ has already been defined, i.e, the points where $\mathcal{M}^{(m)}$ intersects with the manifolds ranked before it. Thus, we obtain a function $g^\star_{\Phi}$ that does not depend on $m$, and which for each manifold, equals the Bayes predictor except on subsets of measure zero under the assumption that manifolds of the same dimension are pairwise transverse. Refer to appendix~\ref{ss:bayes_optimality} for more details.
\begin{figure}
\begin{minipage}{.7\linewidth}
\caption{\textbf{Example - Imputation manifolds in two dimensions} --- Manifolds represented for linear imputation functions. $\mathcal{M}^{(0, 0)}$ is the whole plane. Note that $\mathcal{M}^{(1, 1)}$ need not be at the intersection of the two lines, it depends on the imputation function chosen. Here, $\mathcal{M}^{(0, 1)}$ and $\mathcal{M}^{(1, 0)}$ are transverse if and only if the two lines are not coincident.}
\label{fig:example_manifolds}
\end{minipage}%
\hfill
\begin{minipage}{.28\linewidth}
\scalebox{.21}{\subfile{tikz_manifolds}}
\end{minipage}
\end{figure}
While this theorem is a very general result, it does not say what the optimal function associated to a given imputation looks like. In fact, depending on the imputation function it may be non-continuous, vary widely, and require a very large number of samples to be approximated.
\section{Imputation versus regression: choosing one may break the other}
Theorem~\ref{th:fbp} gives a theoretical grounding to Impute-then-regress procedures. As it holds for almost any imputation function, one could very well choose simple and cheap imputations such as imputing by a constant. However, the difficulty of the ensuing learning problem will depend on the choice of imputation function. Indeed, the function $g^\star_{\Phi}$ that achieves the Bayes rate depends on the imputation function $\Phi$. In general, it may not be continuous or smooth. Thus $g^\star_{\Phi}$ can be more or less difficult to approximate by machine learning algorithms depending on the chosen imputation function.
\citet{LeMorvan2020Linear} showed that even if $Y$ is a linear function of $X$, imputing by a constant leads to a complicated Bayes predictor: piecewise affine but with $2^d$ regions. This result highlights how imputations neglecting the structure of covariates can result in additional complexity for the regression function $g^\star_{\Phi}$. Rather, another common practice is to impute by the conditional expectation: it minimizes the mean squared error between the imputed matrix and the complete one and is the target of most imputation methods. One hope may be that if $f^\star$ has desirable properties, such as smoothness, conditional imputation will lead to a function $g^\star_{\Phi}$ which inherits from these properties.
In this section we first show that replacing missing values by their conditional expectation in the oracle regression function $f^\star$ gives a small but non-zero risk. Characterizing the optimal function on the conditionally-imputed data, we find that it suffers from discontinuities and thus forms a difficult estimation problem. Rather, we study whether the imputation can be corrected for $f^\star$ to form the Bayes predictor on partially-observed data.
\subsection{Applying $f^\star$ on conditional imputations: chaining oracles isn't without risks.}
The conditional imputation function $\Phi^{CI}: \br{\mathbb{R} \cup \cbr{\mathtt{NA}}}^d \to \mathbb{R}^d$ is defined as follows:
\begin{equation*}
\forall j \in \bbr{1, d},\: \Phi^{CI}_j(\widetilde X) =
\begin{cases}
X_j \: &\text{if} \: M_j = 0\\
\mathbb{E} \sqb{X_j | X_{obs}, M} \: &\text{if} \: M_j = 1
\end{cases}
\end{equation*}
Note that $\Phi^{CI} \in \mathcal{F}^I$. To lighten notations, we will write $X^{CI}:= \Phi^{CI}(\widetilde X)$ to denote the conditionally imputed data.
\begin{restatable}[First order approximation]{lemma}{fimpth}
\label{le:fimp}
Assume that the data is generated according to \eqref{eq:generation}. Moreover assume that (i) $f^\star \in \mathcal{C}^2(\mathcal{S}, \mathbb{R})$ where $\mathcal{S} \subset \mathbb{R}^d$ is the support of the data, and that (ii) there exists positive semidefnite matrices $\bar{H}^+ \in P_d^+$ and $\bar{H}^- \in P_d^+$ such that for all $X$ in $\mathcal{S}$, $\bar{H}^- \preccurlyeq H(X) \preccurlyeq \bar{H}^+$ with $H(X)$ the Hessian of $f^\star$ at $X$. Then for all $X$ in $\mathcal{S}$ and for all missing data patterns:
\begin{equation}
\frac{1}{2}\text{tr}\left(\bar{H}^-_{mis, mis} \Sigma_{mis|obs, M}\right) \leq \tilde f^\star( \widetilde X) - f^\star(X^{CI}) \leq \frac{1}{2}\text{tr}\left(\bar{H}^+_{mis, mis} \Sigma_{mis|obs, M}\right)
\end{equation}
where $\Sigma_{mis|obs, M}$ is the covariance matrix of the distribution of $X_{mis}$ given $X_{obs}$ and $M$.
\end{restatable}
Appendix \ref{ss:proof_fimp} gives the proof. The assumption that $\bar{H}^- \preccurlyeq H(X) \preccurlyeq \bar{H}^+$ for any $X$ means that the minimum and maximum curvatures of $f^\star$ in any direction are uniformly bounded over the entire space. Lemma~\ref{le:fimp} shows that applying $f^\star$ to the conditionally imputed (CI) data is a good approximation of the Bayes predictor when there is no direction in which both the curvature of $f^\star$ and the conditional variance of the missing data given the observed one are high. Intuitively, if a low quality imputation is compensated by a flat function, or conversely, if a fast varying function is compensated by a high quality imputation, then $f^\star$ applied to the CI data approximates well the Bayes predictor.
\begin{restatable}[(Non-)Consistency of chaining oracles]{proposition}{propchain}
\label{prop:chain}
The function $f^\star \circ \Phi^{CI}$ is Bayes optimal if and only if the function $f^{\star}$ and the imputed data $X^{CI}$ satisfy:
\begin{equation}
\label{eq:bo_cond}
\forall M \; \text{s.t.} \; P(M)>0, \quad \mathbb{E}\sqb{f^\star(X)|X_{obs}, M} = f^\star(X^{CI}) \quad \text{almost everywhere}.
\end{equation}
Besides,
under the assumptions of Lemma~\ref{le:fimp}, the excess risk of chaining oracles compared to the Bayes risk $\mathcal{R}^\star$ is upper-bounded by:
\begin{equation}
\mathcal{R}(f^\star \circ \Phi^{CI}) - \mathcal{R}^\star \leq \frac{1}{4} \mathbb{E}_M \sqb{ \max \br{tr \br{\bar{H}^-_{mis, mis} \Sigma_{mis|obs, M}}^2, tr\br{\bar{H}^+_{mis, mis} \Sigma_{mis|obs, M}}^2}}
\label{eq:excess_risk}
\end{equation}
\end{restatable}
Appendix \ref{ss:proof_prop_chain} gives the proof.
Condition~\eqref{eq:bo_cond} for Bayes optimality is clearly stringent.
Indeed, if one variable is missing, condition~\eqref{eq:bo_cond} says that the expectation of the regression function should be equal to the regression function applied at the expected entry. Although such functions are difficult to characterize precisely, it is clear that condition~\eqref{eq:bo_cond} is difficult to fulfill for generic regression functions (linear functions are among the few examples that do satisfy it). Therefore, for most functions $f^\star$, $f^\star \circ \Phi^{CI}$ is not Bayes optimal. Proposition~\ref{prop:chain} also gives an upper bound for the excess risk of the predictor $f^\star(X^{CI})$ compared to the Bayes rate, showing here again that if there is no direction in which both the curvature and the variance of the missing data given the observed one are high, the excess risk is small.
\emph{The special case of linear regression:} When $f^\star$ is a linear function, the curvature is 0, hence eq.\,(\ref{eq:excess_risk}) implies no excess risk. This is also visible from the expression of the Bayes predictor \eqref{eq:BP}, where the expectation on unobserved data can be pushed inside $f^\star$ as it is linear. The Bayes predictor can thus be exactly written as $f^\star$ applied to conditionally-imputed data.
\subsection{Regressing on conditional imputations, a good idea?}
\begin{restatable}[Regression function discontinuities]{proposition}{propdisc}
\label{prop:disc}
Suppose that $f^\star \circ \Phi^{CI}$ is not Bayes optimal, and that the probability of observing all variables is strictly positive, i.e., for all $x$, $P(M=(0, \dots , 0), X=x)>0$. Then there is no \underline{continuous} function $g$ such that $g \circ \Phi^{CI}$ is Bayes optimal.
\end{restatable}
In other words, when conditional imputation is used, the optimal regression function experiences discontinuities unless it is $f^\star$. The proof is given in appendix \ref{ss:proof_prop_disc}. From a finite-sample learning standpoint, discontinuous functions are in general harder to learn: in the general case, non-parametric regression requires more samples to achieve a given error on functions without specific regularities as opposed to functions with a form of smoothness
\citep[see e.g.,][chap 3]{gyorfi2006distribution}. Hence, while regression on conditional imputation may be consistent (\autoref{th:fbp}), it can require an inordinate number of samples.
\subsection{Fasten your seat belt: corrected imputations may experience discontinuities.}
\begin{figure}[t]
\includegraphics[width=.4\linewidth, clip, trim={1.9cm 1.9cm .2cm 2.6cm}]{corrected_imputation_bowl.png}%
\llap{\raisebox{.3\linewidth}{\rlap{\sffamily Bowl}\hspace*{.4\linewidth}}}
\hfill%
\includegraphics[width=.4\linewidth, clip, trim={1.9cm 1.9cm .2cm 2.6cm}]{corrected_imputation.png}%
\llap{\raisebox{.3\linewidth}{\rlap{\sffamily Wave}\hspace*{.4\linewidth}}}%
\caption{\textbf{Left: corrected imputation} The regression function is $f^\star(x_1, x_2) \mapsto x_1^2 + x_2^2$. When $x_2$ is missing, chaining perfect conditional imputation with the regression function ($f^\star \circ \Phi^{CI}$) gives a biased predictor, shown in red, as the unexplained variance in $x_2$ is turned into bias. However, using as an imputation $\Phi(x_1) = \sqrt{\rho^2 x_1^2 + (1-\rho^2)}$ corrects this bias, with $\rho$ the correlation between $x_1$ and $x_2$.
\textbf{Right: no continuous corrected imputation exists}. The
function is defined as $f^\star(x_1, x_2) \mapsto x_2^2 -
3\,x_2$. No continuous corrected imputation is possible because the Bayes
predictor on the partially-observed data $\mathbb{E}[Y|X_1]$ is
monotonous, while the regression function $f^\star$ is not.}
\label{fig:2d_examples}
\end{figure}
Another possible route is to find \emph{corrected imputations} which we
define as imputation functions $\Phi$ such that, if $f^\star$ is used as
regression function, the impute-then-regress procedure $f^\star \circ
\Phi$ is Bayes optimal. Intuitively, given a
fixed regression function $f^\star$, the question is: can we "correct" an
imputation function and thus the manifold that it describes so that
$f^\star$ restricted to this manifold is equal to the Bayes
predictor?
Assuming $f^\star$ is continuous, the intermediate value theorem gives a first answer to this question by ensuring the existence of imputations functions satisfying
\begin{align*}
f^\star \circ
\Phi (X_{obs(M)}, M) = \mathbb{E} \sqb{f^\star(X) | X_{obs(M)}, M}.
\end{align*}
For the same reasons as above, determining that such imputations not only exist but are \emph{continuous} is important from a practical perspective. Indeed, assuming $f^\star$ is continuous, the Bayes predictor with missing values could then be tackled as the composition of two continuous functions, with an Impute-then-Regress strategy. Intuitively in 2D, the existence of a continuous corrected imputation can be seen as the existence of a continuous path in the 2D plane whose value by $f^\star$ equals the Bayes predictor. Figure~\ref{fig:2d_examples} (left) gives a simple example in 2D for which a continuous corrected imputation exists. However, as illustrated in Figure~\ref{fig:2d_examples} (right), \emph{continuous} corrected imputations do not always exist. Indeed, on this example the Bayes predictor is non-decreasing but there is no continuous path in the 2D plane on which $f^\star$ is non-decreasing and maps at some point to both the 'purple' and 'yellow values' (proof in Appendix~\ref{ss:no_cc}). It is thus of interest to clarify when continuous corrected imputations exist. Proposition~\ref{prop:cont_imp} establishes such conditions.
\begin{restatable}[Existence of continuous corrected imputations]{proposition}{prop:cont_imp}
\label{prop:cont_imp}
Assume that $f^{\star}$ is uniformly continuous, twice continuously differentiable and that, for all missing patterns $m$ and all $x_{obs}$, the support of $X_{mis} | X_{obs}=x_{obs}, M=m$ is connected.
Additionally, assume that for all missing patterns $m$, and all $(x_{obs}, x_{mis})$, the gradient of $f^{\star}$ with respect to the missing coordinates is nonzero:
\begin{equation}
\nabla_{x_{mis}} f^{\star} (x_{obs}, x_{mis}) \neq 0. \label{eq:regularity_assumption}
\end{equation}
Then, for all $m$, theres exist continuous imputation functions $\phi^{(m)} : \mathbb{R}^{|obs(m)|} \to \mathbb{R}^{|mis(m)|}$ such that $f^{\star} \circ \Phi$ is Bayes optimal.
\end{restatable}
Appendix~\ref{app:continuous_imputations} gives a proof based on a global implicit function theorem. Assumption~\ref{eq:regularity_assumption} is restrictive: it is for instance not met for our example in Figure~\ref{fig:2d_examples} (left), which still admits continuous corrected imputations. This highlights the fact that continuous corrected imputations also exist under weaker conditions, but it is difficult to conclude on ``how often'' it is the case.
\section{Jointly optimizing an impute-n-regress procedure: NeuMiss+MLP}
The above suggests that it is beneficial to adapt the regression function to the imputation procedure and vice versa. Hence, we introduce a method for the joint optimization of these two steps by chaining a NeuMiss network, which learns an imputation, with an MLP (multi-layer perceptron).
NeuMiss \citep{LeMorvan2020NeuMiss} is a neural-network architecture originally designed to approximate the Bayes predictor for a linear model with missing values.
It contains a theoretically-grounded Neumann block that can efficiently approximate the conditional expectation of the missing values given the observed ones. Here, we reuse the Neumann block as it outputs a learned imputation: each observed coordinate is mapped to its observed value and each missing coordinate is mapped to a function of the observed ones. The whole architecture can be seen as an Impute-then-Regress architecture, but that can be jointly optimized.
We performed a few minor improvements on the NeuMiss architecture. First, though
the theory behind NeuMiss points to using shared weights in the Neumann block as well as residual connections going from the input to each hidden layer of the Neumann block, \citet{LeMorvan2020NeuMiss} used neither. We found empirically that shared weights as well as residual connections improved performance. Moreover, while \citet{LeMorvan2020NeuMiss} initialized the weights of the Neumann block randomly, we chose to initialize them with sample estimates of quantities that should be targeted to perform well according to \citet{LeMorvan2020NeuMiss}. For clarity, the (non-linear) NeuMiss architecture and its initialization strategy are described in detail in Appendix~\ref{ss:NeuMiss_archi}.
\section{Empirical study of impute-n-regress procedures}
\subsection{Experimental setup}\label{sec:experimental_setup}
\paragraph{Data generation}
The data $X \in \mathbb{R}^{n \times d}$ are generated according to a multivariate Gaussian distribution $\mathcal{N}(\mu, \Sigma)$ where the mean is drawn from a standard Gaussian $\mathcal{N}(0, Id)$ and the covariance is generated as $\Sigma = B B^\top + D$. $B \in \mathbb{R}^{d \times q}$ is a matrix with entries drawn from a standard normal Gaussian distribution, and $D$ is a diagonal matrix with small entries that ensures that the covariance matrix is full rank. We study two correlation settings called \emph{high} and \emph{low} corresponding respectively to $q=\texttt{int}(0.3*d)$ and $q=\texttt{int}(0.7*d)$. The experiments are run with $d=50$.
\paragraph{Choice of $f^\star$} The response $Y$ is generated according to $Y = f^\star(X) + \epsilon$ with two choices of $f^\star$ named \emph{bowl} and \emph{wave}, depicted in Figure~\ref{fig:f_star} (exact expression in appendix \ref{ss:f_star}). $\beta$ is a vector of ones normalized such that the quantity $z = \beta^\top X + \beta_0$ follows a Gaussian distribution centered on 1 with variance 1. Note that $f^\star_{bowl}$ and $f^\star_{wave}$ were designed so that the desired variations occur over the support of the data. The noise $\epsilon$ is chosen so as to have a signal-to-noise ratio of 10.
\begin{figure}[h]
\begin{minipage}{.38\linewidth}
\caption{\textbf{Bowl and wave functions} used for $f^\star$ in the empirical study.}
\label{fig:f_star}
\end{minipage}%
\hfill%
\begin{minipage}{.2\linewidth}
\includegraphics[width=\linewidth]{f_star_bowl.pdf}
\end{minipage}%
\hfill%
\begin{minipage}{.2\linewidth}
\includegraphics[width=\linewidth]{f_star_wave.pdf}
\end{minipage}%
\end{figure}
\paragraph{Missing values} 50\% of the entries of $X$ were deleted according to one of two missing data mechanisms: Missing Completely At Random (MCAR) or Gaussian self-masking \citep[GSM, see ][]{LeMorvan2020NeuMiss}. Gaussian self-masking is a Missing Not At Random (MNAR) mechanism, where the probability that a variable $j$ is missing depends on $X_j$ via a Gaussian function.
\paragraph{Baseline methods benchmarked}
For each level of correlation (\emph{low} or \emph{high}), for each function $f^\star$ (\emph{bowl} or \emph{wave}), and each missing data mechanism (MCAR or GSM), we compare a number of methods. First, for reference, we compute various oracle predictors:
\begin{itemize}[itemsep=0ex, topsep=0ex, partopsep=0.5ex, parsep=0.5ex, leftmargin=2.5ex]
\item \textbf{Bayes predictor}: This is the function that achieves the lowest achievable risk. In general cases, its expression cannot be derived analytically. However, we show that it can be derived for ridge functions, i.e. functions of the form $x \mapsto g(\beta^\top x)$, for some choices of $g$ including polynomials and the Gaussian cdf. We thus built $f^\star_{bowl}$ and $f^\star_{wave}$ as combination of these base functions which allows us to compute their corresponding Bayes predictors. Appendix~\ref{ss:f_star} gives their expressions.
\item \textbf{Chained oracles}: $f^\star \circ \Phi^{CI}$ consists in imputing by the conditional expectation and then applying $f^\star$. The analytical expression of $\Phi^{CI}$ can be derived analytically for both MCAR and GSM, and we thus use this analytical expression to impute.
\item \textbf{Oracle + MLP}: The data is imputed using the analytical expression of the conditional expectation, and then a MLP is fitted to the completed data.
\end{itemize}
These three predictors all use ground truth information (parameters $\mu$, $\Sigma$ of the data distribution, expression of $f^\star$ or of the missing data mechanism) which are unavailable in practice. They are mainly useful as reference points. We then compare the NeurMiss+MLP architecture and a number of classic Impute-then-Regress methods as well as gradient boosted regression trees:
\begin{itemize}[itemsep=0ex, topsep=0ex, partopsep=0.5ex, parsep=0.5ex, leftmargin=2.5ex]
\item \textbf{Mean + MLP} The data is imputed by the mean, and a multilayer perceptron (MLP) is fitted on the completed data.
\item \textbf{MICE + MLP} The data is imputed using Scikit-learn's \citep[BSD licensed]{Pedregosa2011Scikit} conditional imputer \texttt{IterativeImputer} that adapts the popular Multivariate Imputation by Chained Equations \citep[MICE,][]{VanBuuren2018Flexible} to be able to impute a test set. A multilayer perceptron (MLP) is then fitted on the completed data.
\item \textbf{GBRT}: Gradient boosted regression trees (Scikit-learn's \texttt{HistGradientBoostingRegressor} with default parameters). This predictor readily supports missing values: during training, missing values on the decision variable for a given split are sent to the left or right child depending on which provides the largest gain, the Missing Incorporated Attribute strategy \citep{Twala2008_mia}.
\end{itemize}
Finally, we also run \textbf{Mean + mask + MLP} as well as \textbf{MICE + mask + MLP} in which the mask is concatenated to the imputed data before fitting a MLP. Concatenating the mask is a widespread pratice to account for MNAR data.
All the MLPs used have the same hyperparameters: 0, 1 or 2 hidden layers (chosen on a validation set), ReLU activation functions, and a width of 100 hidden neurons. Adam is used with an adaptive learning rate: the learning rate is divided by 5 each time 2 consecutive epochs fail to decrease the training loss by at least 1e-4. Early stopping is triggered when the validation score does not improve by at least 1e-4 for 10 consecutive epochs. Finally for NeuMiss the depth of the Neumann block is 5 or 15 (chosen on a validation set). The experiments use training sets of $n=100\,000$ and $n=500\,000$ and validation and test sets of size $n=10\,000$. For $n=100,000$, running all methods in one setting of correlation, missing data mechanisms, and choice of $f^\star$, takes 3 hours on one core.
\subsection{Experimental results}
The results are presented in Figure~\ref{fig:boxplots}.
\begin{figure}[t!]
\includegraphics[width=\linewidth]{boxplots2}
\caption{Performances (R2 score on a test set) compared to that of the Bayes predictor across 10 repeated experiments. The dark plots are for $n=100\,000$ and the light ones for $n=500\,000$. }
\label{fig:boxplots}
\end{figure}
\paragraph{Chaining oracles fail when both curvature is high and correlation is low.} The chained oracle has a performance close to that of the Bayes predictor in all cases except when the wave function is applied to low correlation data. This observation illustrates well Proposition~\ref{prop:chain}. Intuitively, the Bayes predictor for each missing data pattern is a smoothed version of $f^\star$, and it is all the more smoothed that there is uncertainty around the likely values of the missing data. In the low correlation setting, the uncertainty is such that $f^\star$ is not a good proxy anymore for the Bayes predictor.
\paragraph{Regressing on oracle conditional imputation provide excellent performances.} Contrary to the chained oracles, \emph{Oracle + MLP} is close to the Bayes rate in all cases. This result should be put in perspective to Proposition~\ref{prop:disc}, which states that there is no \emph{continuous} regression function $g$ such that $g \circ \Phi^{CI}$ is Bayes optimal if $f^\star$ is not. Indeed, as the MLP can only learn continuous functions, it shows that there are continuous functions $g$ such that $g \circ \Phi$, albeit non-consistent, performs very well.
\paragraph{Adding the mask is critical in MNAR settings with \emph{mean} and \emph{MICE} imputations} In MNAR settings, missingness carries information that can be useful for prediction. However, both the mean and iterative conditional imputation discard this information and output an imputed dataset in which the missingness information is lost or more difficult to retrieve. For this reason, it is common practice to concatenate the mask with the imputed data to expose the missingness information to the predictor. Our experiments show that under self-masking (MNAR), adding the mask to the mean or iteratively imputed data markedly improve performances. Note that NeuMiss does not require adding the mask as an input since the missingness information is already incorporated via the non-linearities.
\paragraph{NeuMiss+MLP performs best among Impute-then-Regress predictors.} In \emph{all}
settings, NeuMiss performs best. GBRT perform poorly here possibly because they are not well adapted to approximate smooth functions. Finally, note that when the difficulty of the problem increases, for example with a lower correlation, then (i) the performance of the Bayes predictor decreases and (ii) the differences between the performances of methods is reduced, as in the lower right panel.
\section{Conclusion}
Impute-then-regress procedures assemble standard statistical routines to build predictors suited for data with missing values. However, we have shown that seeking the best prediction of the outcome leads to different tradeoffs compared to inferential purposes. Given a powerful learner, \emph{almost all imputations} lead asymptotically to the optimal prediction, \emph{whatever the missingness mechanism}. A good choice of imputation can however reduce the complexity of the function to learn.
Though conditional expectation can lead to discontinuous optimal regression functions, our experiments show that it still leads to easier learning problems compared to simpler imputations.
In order to adapt the imputation to the regression function, we proposed to jointly learn these two steps by chaining a trainable imputation via the NeuMiss networks and a classical MLP. An empirical study of non-linear regression shows that it outperforms impute-then-regress procedures built on standard imputation methods as well as gradient-boosted trees with incorporated handling of missing values. In further work, it would be useful to theoretically characterize the learning behaviors of Impute-then-Regress methods in finite sample regimes.
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\section{INTRODUCTION}
\nobreak
Type Ia supernovae (SNe~Ia) have proved to be an extremely valuable
tool for measuring the cosmological parameters, as they are the best
high-luminosity standard candles currently known to astronomy.
Studies of the peak $B$-band luminosities of high redshift SNe~Ia
led to the surprising discovery by two independent groups (the Supernova
Cosmology Project (SCP; Perlmutter et~al.\ 1998, Perlmutter et~al.\ 1999
(hereafter P99)) and the High-z Supernova Search Team (HZSST;
Garnavich et~al.\ 1998; Schmidt et~al.\ 1998; Riess et~al.\ 1998),
that the expansion of the Universe is accelerating. This acceleration is
consistent with some form of `dark energy', possibly Einstein's cosmological
constant $\Lambda$. The implications of this result for the future fate of
the Universe and our understanding of fundamental physics are profound;
therefore, it is extremely important that it be verified by
independent methods.
The best approach is to make use of alternative measurements that depend
on other physical processes. There are now several additional lines
of evidence that support the accelerating Universe, but most are based on
combining several different measurements. For example, the combination
of the angular size of
fluctuations on the surface of last scattering of the cosmic microwave
background (CMB) with measurements of the clustering of mass on large
scales \citep{Spergel:03,Tegmark:04,Eisenstein:05} provides strong
evidence for a dark energy component. There is also a direct detection
of dark energy using the integrated Sachs-Wolfe effect \citep{Padmanabhan:05}.
It is encouraging that these different lines of evidence,
which depend on very disparate physical processes and probe very
different cosmic epochs, are consistent with a $\ensuremath{\Omega_{m}} \sim 0.3$, $\ensuremath{\Omega_{\Lambda}} \sim 0.7$
Universe.
Still, SNe~Ia provide the best direct evidence for dark energy, and any
improvement in our understanding of their properties is very welcome.
There are several possible alternative explanations for the SN result.
Since dark energy
manifests itself in this context as high-redshift SNe~Ia being slightly dimmer
than expected, the most obvious alternative explanation is that this dimming
is caused by extragalactic dust, either in intergalactic space or in the host
galaxies of the SNe. Another possibility, and a
significantly more difficult one to quantify, is that high redshift SNe
are somehow dissimilar from low redshift SNe in a way that we have not
yet detected. This paper presents results based on an analysis of SNe~Ia
with a new method (CMAGIC, for Color-MAGnitude Intercept Calibration)
introduced in \citet{Wang:03} (hereafter W03) that partially addresses both
issues.
There is no unique choice for the magnitude to associate with an SN~Ia
because their luminosity varies in time.
For convenience, virtually all previous studies have used the $B$ magnitude
at maximum brightness, \ensuremath{m_{B}} , as the standardized candle,
but there is no {\it a priori} reason why this choice is optimal.
\ensuremath{m_{B}}\ is generally determined by fitting an empirical curve to the $B$-band
brightness as a function of time and reading off the peak value.
When available, observations in other passbands are frequently
incorporated into the fitting procedure. There is a well-established
empirical relation between absolute magnitude and the width of the light curve
as parameterized by stretch (Perlmutter et~al.\ 1997; P99;
Goldhaber et~al.\ 2001), \ensuremath{\Delta m_{15}\left( B \right)}\ \citep{Phillips:93,Phillips:99} or
the MLCS parameter $\Delta$ \citep{Riess:96} in the sense that SNe with wider,
more slowly declining light curves (high stretches) are intrinsically brighter.
Here the stretch parameterization is used.
Since ordinary interstellar dust both extinguishes and reddens light, P99
compared the distributions of \ensuremath{B-V}\ colors at maximum luminosity of the
low and high redshift SN samples, finding no significant evidence that the
high redshift sample is more reddened. It should be emphasized that the
SN measurement of \ensuremath{\Omega_{m}}\ and \ensuremath{\Omega_{\Lambda}}\ is relative -- as long as the low-
and high-redshift samples suffer the same amount of extinction (or any other
bias), there is no effect on the final result. \citet{Sullivan:02}
decomposed the SN sample into subsets based on the Hubble
type of their host galaxies, a powerful approach because early-type galaxies
are expected to have little or no dust, and found that \ensuremath{\Omega_{\Lambda}}\ was detected
in each subsample. A difficulty with
this analysis is that the resulting error bars on \ensuremath{\Omega_{m}}, \ensuremath{\Omega_{\Lambda}}\ are necessarily
much larger because the morphological subsets have considerably
fewer SNe than the full sample.
One may attempt to measure the reddening for each SN by measuring
its color and correcting for host galaxy extinction by assuming a dust
extinction law. The error in the extinction correction usually dominates
the statistical errors of each SN. In early work the HZSST
team made use of an asymmetric prior on the intrinsic extinction
distribution to limit the propagated uncertainties resulting from the
extinction correction \citep{Riess:98} while performing light-curve
fits, which can bias the results under some circumstances (P99).
More recent papers have made improvements in the form of the prior and
its application and corrected this problem \citep{Barris:04, Riess:04},
although at the potential cost of enhanced sensitivity to any evolution in
the extinction distribution. \citet{Knop:03} (hereafter K03) made use
of high quality color measurements made possible by the
{\it Hubble Space Telescope (HST)} to estimate the extinction values
of individual SN without making use of such a prior.
The evolution issue is extremely difficult to address. To first
order evolution should not be a concern because the diversity of the
environments in which local SNe~Ia occur is much larger than the mean
difference in environment between the high and low redshift samples.
While there are some properties of SNe~Ia that are known to correlate with host
environment, these correlations disappear once the width-luminosity
relation is taken into account \citep{Hamuy:00}. The analysis of
\citet{Sullivan:02} also has relevance for this question because it compares
SNe~Ia from similar host environments at high and low redshift. One can
also compare individual SNe in more detail spectroscopically
\citep{Hook:05}, although such measurements are taxing even for modern
\mbox{8-10m} class telescopes. In a spectroscopic study of 12 high
redshift SNe, \citet{Garavini:05} found no evidence for evolution.
CMAGIC offers some benefits with respect to dust and evolutionary models,
as described further in \S \ref{subsec:dust} and \S \ref{subsec:evolution}.
It is possible to define a standard candle magnitude with CMAGIC,
and because of the nature of the CMAGIC relationships, this magnitude is
affected by the same amount of dust by roughly half as much as \ensuremath{m_{B}} . On
the evolutionary front the situation is more complicated. There are some
potential evolutionary effects for which CMAGIC offers advantages, but
it is uncertain how important this is because the effects of these theories
have not been clearly delineated. Because CMAGIC depends on light-curve
data in a very different fashion than maximum magnitude fits, and in
particular because it is much more sensitive to later epochs relative
to maximum light, for some potential evolutionary effects we can expect
the CMAGIC magnitude to be affected differently. However, this is difficult
to quantify given the current lack of detailed predictions from theories
of SN evolution. Combining these two considerations,
CMAGIC can provide a powerful cross check of previous SNe~Ia cosmology
results. Because we are attempting to verify previous results, it is
important to prevent the analysis from being unintentionally biased towards
the expected outcome. To this end a blindness technique has been
developed and used during the cosmological analysis in this paper.
Perhaps for some of the above reasons, low redshift SNe~Ia analyzed
with CMAGIC have a smaller intrinsic variation than the maximum
magnitudes of the same SNe without extinction correction ($\ensuremath{\sigma_{int}}= 0.12$
mag, compared with approximately $0.17$ mag for \ensuremath{m_{B}} ). For many current
data sets, the intrinsic variation dominates over observational errors, so
CMAGIC may allow us to obtain tighter constraints on the cosmological
parameters for a similar observational expense in future surveys.
The goals of this paper are twofold: (1) to show that the CMAGIC
relations hold at high redshift for well measured SNe, and (2) to
measure the cosmological parameters from already existing data sets and
use this to cross-check previous results. We first describe CMAGIC in
more detail (\S \ref{sec:cmagic}). We then describe the data sample
(\S \ref{sec:data}) and the CMAGIC fitting procedures (\S \ref{sec:cmagfits}),
and then we use these to demonstrate that CMAGIC works for high redshift
SNe (\S \ref{sec:highzdemo}). Once this is established, we proceed to the
primary analysis of this paper, the cosmological fits. First we describe
the cosmological fitting techniques (\S \ref{sec:cosfits}), including a
discussion of the blindness technique (\S \ref{subsec:blindness}).
Finally, the cosmological results are presented (\S \ref{sec:cosresults}),
systematic effects are discussed (\S \ref{sec:systematics}), and
the results are analyzed (\S \ref{sec:analysis}).
\section{CMAGIC}
\label{sec:cmagic}
\nobreak
CMAGIC is described in considerably more depth in W03. Here we
provide a brief review of the relations, define the magnitude (\ensuremath{B_{BV0.6}} )
used in this study, and discuss the benefits of CMAGIC with respect to
extinction and evolution.
\subsection{CMAGIC Relations}
\label{subsec:cmagrelations}
\nobreak
CMAGIC is based on the behavior of SNe~Ia in color-magnitude diagrams.
Starting approximately 1 week after $B$ maximum and lasting approximately
3 weeks, the relation between the $B$ magnitude and \ensuremath{B-V}\ color is
strikingly linear. This holds true for other colors as well (at least $B-R$,
$B-I$). Some typical low redshift examples are shown in figure~\ref{fig:cmag}.
The temporal extent of this linear region is a function of stretch, with
slower, higher stretch light-curves starting and ending their linear behavior
later. The slope, $\beta$, of the linear region has a narrow distribution.
Currently very few rest-frame $R$ and $I$ observations are available for high
redshift SNe~Ia, so here we consider only $B$ versus \ensuremath{B-V} . The simplicity of
this behavior is so far not completely explained by theory, which gives it a
status similar to the empirical width-luminosity relation. Prior to the linear
region, the majority of SNe~Ia are less luminous than the linear extrapolation.
However, a minority (typically those with high stretch) display excess
luminosity, which is referred to as a `bump'. Standard light-curve template
fitting techniques (stretch, MLCS) do not adequately reproduce the CMAGIC
relations. Both issues are discussed in more detail in W03.
The distribution of slopes in this linear region is fairly narrow, with
$\left< \ensuremath{\beta_{BV}} \right> = 1.98$ and a RMS of 0.16, as shown in
figure~\ref{fig:slopes} for low-redshift SNe~Ia. To first order, \ensuremath{\beta_{BV}}\
is affected by \ensuremath{K}-corrections\ but not by extinction. W03 explored fixing the
slope at the mean value for all fits. The effects of this assumption are
quite minor, but it is possible to improve on this procedure by including
information about the distribution of slopes in the fitting procedure
(\S\ref{sec:cmagfits}).
The CMAGIC relation for $B$ versus \ensuremath{B-V}\ can be written conveniently in the form
\begin{equation}
B = \ensuremath{B_{BV0.6}} + \ensuremath{\beta_{BV}} \left( B - V - 0.6 \right),
\end{equation}
which defines \ensuremath{B_{BV0.6}}\ as the $B$ magnitude when $\ensuremath{B-V} = 0.6$; this is the
magnitude used as a standard candle in this paper. The particular \ensuremath{B-V}\
color is chosen to minimize the covariance between the standard candle
magnitude and the slope \ensuremath{\beta_{BV}} , as it is approximately the mean \ensuremath{B-V}\
color in the linear region of an unextinguished SN~Ia. Because the color
roughly measures the ejecta temperature, by evaluating the
magnitude at a fixed color we essentially ensure that all SNe are evaluated
at a point where their physical properties are similar.
The behavior of an SN Ia on a CMAGIC diagram can also be viewed temporally.
Proceeding in a clockwise fashion around the curves in figure~\ref{fig:cmag},
a typical, unextinguished SN Ia usually has a color of approximately $\ensuremath{B-V} = 0$
at maximum, and evolves rapidly to the red for about a month. After this it
enters the so-called nebular phase and evolves bluewards, again in a linear
fashion. This second linear region has some interesting properties, but
since data at such late epochs are very rarely available for high-redshift
SNe, we do not discuss it further here. With good time coverage it is possible
to determine the extent of the linear region by examination, but this is
generally not possible with current high redshift data. Fortunately, the
beginning and ending dates of the linear region relative to the date of
$B$ maximum form a well-defined sequence in terms of stretch and the presence
or absence of the bump feature. Using well-observed low-redshift SNe to
determine the earliest and latest points in the linear region as a function
of stretch, we find that the beginning date of the linear region is well
described by $t_b = 5 + 3 \left(s - 1\right)$ and the ending date by
$t_e = 29 + 40 \left(s - 1\right)$, where both are measured in
rest-frame days relative to $B$ maximum and $s$ is the stretch. SNe~Ia with
bumps (e.g., the lower panel of figure~\ref{fig:cmag}) do not fit smoothly into
this scheme and are well represented by $t_b = 13.5$ and $t_e = 30$. This
suggests a possible source of bias in the analysis of the high redshift sample,
since the presence or absence of a bump may be difficult to detect given
the typical quality of high redshift photometry. Fortunately, for this data
sample this issue proves to be unimportant (Appendix~\ref{apndx:bumps}).
Detailed studies (Appendix~\ref{apndx:correlations}) show that the fitting
procedure induces weak negative correlations between \ensuremath{B_{BV0.6}}\ and \ensuremath{m_{B}} ,
at least for current light-curve templates. Clearly, these templates have
missed some aspect of SNe~Ia behavior (or the correlations would be much
stronger), and \ensuremath{B_{BV0.6}}\ provides some additional information that can be
used to constrain the cosmological parameters. Peculiar velocities, stretch
correction, and extinction induce additional correlations between these
magnitudes.
\subsection{Host Galaxy Dust}
\label{subsec:dust}
\nobreak
Interstellar dust is a major component of our and other galaxies.
A good review can be found in \citet{Draine:03}. Ordinary dust both
extinguishes and reddens starlight because it absorbs blue light more
strongly than red light. The relative amount of absorption between wavelengths
is characterized by an absorption law such as that of \citet{Cardelli:89}.
For an object with a stellar spectrum, the extinction in the $B$-band $A_B$
(in magnitudes) is related to the amount of reddening \ensuremath{E(B\,-\,V)}\ by
$A_B = \ensuremath{{\cal R}_B} \ensuremath{E(B\,-\,V)}$. For SNe, which do not have stellar-like spectra,
and whose spectral features change with time, this is not strictly appropriate,
but \ensuremath{{\cal R}_B}\ is still useful as a parameterization of the extinction law. A
typical value in our Galaxy is $\ensuremath{{\cal R}_B} = 4.1$, although it varies considerably
along different lines of sight \citep{Fitzpatrick:99}. The characteristic
scatter of \ensuremath{{\cal R}_B}\ is not well constrained.
So far it has not been feasible to measure the extinction law directly
for the host galaxies of high redshift SNe, so the general approach has been
to assume that the \ensuremath{{\cal R}_B}\ values for the high and low redshift SNe samples
are identical. This assumption takes several forms. In the primary fit
of P99 (fit C) no extinction correction is performed, but it is argued that
the similarity of the observed \ensuremath{E(B\,-\,V)}\ distributions of the two samples
implies that host galaxy dust extinction is not contaminating the
cosmological results. Because \ensuremath{{\cal R}_B}\ is necessary to transform \ensuremath{E(B\,-\,V)}\ into
the amount of extinction, this is tantamount to assuming that \ensuremath{{\cal R}_B}\ is the
same for the two samples. There is a theoretical and empirical
expectation that the SN sample suffers from relatively little
extinction \citep{Hatano:98}. K03 perform an extinction correction by
comparing the measured \ensuremath{B-V}\ at maximum to an empirical model, then
converting this to $A_{B}$ by assuming a value for \ensuremath{{\cal R}_B} .
\citet{Riess:98,Riess:04,Tonry:03,Barris:04} use a similar procedure.
Previous analyses have generally performed a color cut on their SN
samples on the theory that large color excesses may represent SNe in dustier
environments where the value of \ensuremath{{\cal R}_B}\ is likely to depart from the fiducial
value. It is interesting to note that we may now have evidence
for higher mean extinction at high redshift. The recent SN sample
of \citet{Riess:04}, which represents the deepest, highest redshift SN survey
yet published, has much higher host galaxy extinction values than any other
available SN sample, although survey selection effects may explain this
result.
Because of the nature of the linear CMAGIC relations, the effective
${\cal R}$-value for \ensuremath{B_{BV0.6}}\ is approximately half of the value that it takes
for \ensuremath{m_{B}}\ (assuming a standard dust law), as shown schematically in
figure~\ref{fig:cmagdust}. The critical point is that the magnitude
is always evaluated at the same fixed color, and therefore the extinction
and reddening effects partially cancel. Since SNe~Ia redden as they
evolve along the linear relation, ${\cal R}_{\ensuremath{B_{BV0.6}}} = \ensuremath{{\cal R}_B} - \ensuremath{\beta_{BV}}$.
For normal dust, \ensuremath{B_{BV0.6}}\ is less affected than \ensuremath{m_{B}} , which results in
smaller uncertainties arising from the extinction correction, if a fixed
\ensuremath{{\cal R}_B}\ is assumed. Because the boundaries of the linear region are determined
by date relative to maximum and not color, \ensuremath{B_{BV0.6}}\ remains less affected
even if the amount of extinction is large enough that $\ensuremath{B-V} = 0.6$ does
not lie within the linear region. The precise epoch of maximum light is
not nearly as important for \ensuremath{B_{BV0.6}}\ as it is for \ensuremath{m_{B}}\ because the `roll-off'
at the edges of the linear region is much less severe than it is near peak
luminosity. Note that CMAGIC offers no benefits with respect to an evolving
\ensuremath{{\cal R}_B}\ -- the derivatives of \ensuremath{m_{B}}\ and \ensuremath{B_{BV0.6}}\ with respect to \ensuremath{{\cal R}_B}\ are
identical. Nor does it offer any advantages for the so-called 'gray dust'
($\ensuremath{{\cal R}_B} = \infty$) suggested by \citet{Aguirre:99}. Constraints on gray dust
have been explored by \citet{Riess:00, Riess:04}, but also see
\citet{Nobili:03, Nobili:05}.
Since \ensuremath{B_{BV0.6}}\ and \ensuremath{m_{B}}\ are affected by extinction differently, it is
possible to estimate the amount of extinction by comparing the
two magnitudes using the quantity \ensuremath{\mathcal{E}} , which is an estimator
of \mbox{ \ensuremath{E(B\,-\,V)} } :
\begin{equation}
\ensuremath{\mathcal{E}} = \frac{ \ensuremath{m_{B}} - \ensuremath{B_{BV0.6}} }{ \ensuremath{\beta_{BV}} } + \mbox{const}.
\end{equation}
Using this correction substantially increases the correlations
between \ensuremath{m_{B}}\ and \ensuremath{B_{BV0.6}} . Assuming a standard extinction law (\ensuremath{{\cal R}_B} = 4.1),
the correlation coefficient between these two magnitudes climbs to
$\rho > 0.7$ from $\left< \rho \right> = 0.15$
(Appendix~\ref{apndx:correlations}), even in the absence of significant
extinction. For this reason, this approach is not followed here. However,
for smaller values of \ensuremath{{\cal R}_B} , such as those found by \citet{Tripp:99} and
\citet{Guy:05}, this correlation is significantly reduced.
\subsection{Evolution of SNe~Ia}
\label{subsec:evolution}
\nobreak
The possibility that the average properties of SNe~Ia have evolved
between the current epoch and a redshift of 1 is of considerable
concern for SN cosmologists. So far it has been impossible
to demonstrate conclusively that evolution is not the cause of the claimed
cosmological results. The best that can be done is to continue to
quantitatively add ``to the list of ways in which they are similar
while failing to discern any way in which they are different''
\citep{Riess:99b}. One method to approach this problem is to compare
high and low redshift SNe in similar environments, as in \citet{Sullivan:02},
where we found no evidence for evolutionary biases. Since all
measured dependencies of SN Ia properties on local environment disappear
after stretch correction, and because of the diversity of environments in which
local SNe~Ia occur, concerns about evolution can be usefully restricted
to mechanisms that affect the width-luminosity relationship.
There are several theoretical models that predict possible avenues for
evolution. \citet{Dominguez:01} and \citet{Hoflich:00} have investigated the
effects of decreasing metallicity and changing progenitor mass on SN~Ia
properties by constructing models of the progenitor star and then following
them through detonation. If $\Delta$ is the change in \ensuremath{B-V}\ they
find that decreasing metallicity causes an SN to become slightly bluer
($\Delta = -0.05$ for an extreme case) without affecting the maximum
$B$ magnitude. Most extinction corrections compare observed
colors to empirically derived color relations to calculate the
amount of extinction. If the intrinsic colors change, then the extinction
correction will be incorrect. If no extinction correction is applied,
then \ensuremath{m_{B}}\ is unaffected, while \ensuremath{B_{BV0.6}}\ is overestimated by
$\ensuremath{\beta_{BV}} \Delta \sim 2 \Delta$. If an extinction correction is applied,
then for positive values of $\Delta$, the extinction correction for \ensuremath{m_{B}}\ is
overestimated and the SN is assigned an extinction-corrected magnitude that
is too bright by ${\cal R}_{B} \Delta \sim 4 \Delta$. \ensuremath{\mathcal{E}} , by contrast,
is underestimated, so once this correction is applied, \ensuremath{B_{BV0.6}}\ is too
dim by $\ensuremath{\beta_{BV}} \Delta - \left( \ensuremath{{\cal R}_B} - \beta \right)^2 \Delta / \ensuremath{\beta_{BV}}$.
For typical values of \ensuremath{\beta_{BV}}\ and \ensuremath{{\cal R}_B} , this cancels, and the extinction
corrected value of \ensuremath{B_{BV0.6}}\ is unaffected by this evolutionary effect.
In other words, either with or without extinction correction this particular
evolutionary model will have different effects on \ensuremath{m_{B}}\ and \ensuremath{B_{BV0.6}} , so
by comparing the two magnitudes this model can be evaluated against data.
We note that the range of metallicities considered in this study is
far greater than the expected change out to $z \sim 1$.
\section{DATA}
\label{sec:data}
\nobreak
Currently available SN data sets have not been observed in a
manner optimized for CMAGIC, particularly at high redshift. Out
of the roughly 100 SNe at $z > 0.1$ with light curves available
in the literature, only approximately 20 are useful for
CMAGIC purposes. High redshift SNe are frequently not observed
in the rest-frame $V$. Even when such observations do exist, they are
usually only intended to establish the color at maximum for the purposes of
applying an extinction correction, and therefore are usually concentrated too
close to the peak to lie within the CMAGIC linear region. Future high redshift
data sets (SNLS \citep{Astier:05}, ESSENCE \citep{Matheson:05},
SDSS Supernova Search \citep{Sako:05}, SNAP \citep{Aldering:04},
LSST \citep{Pinto:04}) will not suffer from this limitation, as they are
designed to obtain multi-color photometry for almost all observed epochs.
The current situation is considerably better
for low redshift data sets, as many of these SNe have excellent multi-color
coverage. There is an observational
cost associated with CMAGIC because the linear region is $\sim 1.2$ mag
dimmer than at peak, so the photometric error bars are larger for the same
observational effort. Whether or not this extra cost is outweighed by
the benefits with respect to dust and/or evolution depends on the specifics
of the survey design.
We have attempted to construct a data sample including all SNe~Ia with
published light curves.
In order to eliminate SNe that cannot be useful for the purposes of
this paper, we enforce the following requirements. First, an object
must be at least plausibly an SN~Ia based on either light-curve shape,
spectroscopic ID, or host galaxy morphology. Second, it must have at
least one rest-frame \ensuremath{B-V}\ observation. For this purpose we require that
the central wavelength of the redshifted $B$- or $V$-band lie within one
HWHM of the central wavelength of the observed filter, which improves the
reliability of the \ensuremath{K}-corrections\ by limiting the amount of extrapolation.
We also do not include
observations taken with extremely wide filters, such as F110W and
F160W NICMOS filters on {\it HST}. These filters are wide enough that
for many of the redshift ranges of interest they overlap considerably with
both $B$ and $V$ (and sometimes $R$), making it difficult to measure
\ensuremath{B-V}\ in a fashion that is not heavily influenced by the model used to
calculate the \ensuremath{K}-corrections . Clearly it must be possible to
use these data in some fashion for CMAGIC, but it will require extreme care.
Observations in $B$ and $V$ are only
combined to form \ensuremath{B-V}\ if they are within 0.5 rest frame days of each other;
the analysis is quite insensitive to this value.
This results in a sample of 131 SNe, of which one third are at redshifts greater
than 0.3. Note that we have not yet required that the \ensuremath{B-V}\ point lie in the
CMAGIC linear region, since this depends on the measured value of the stretch
and date of maximum, or that the SN lie in the Hubble flow.
The high-redshift portion of the sample comes from a fairly
diverse set of sources. There are 14 from P99, six from K03,
two from \citet{Garnavich:98}, one from \citet{Schmidt:98}, five from
\citet{Riess:98}, four from \citet{Tonry:03}, 13 from \citet{Barris:04},
and one from \citet{Riess:04}. The low-redshift sample is even more diverse,
but primarily comes from three sources: \citet{Hamuy:96},
\citet{Riess:99a} and \citet{Jha:05}. Source information is provided in
tables~\ref{tbl:primarysamplowz} and \ref{tbl:primarysamphighz}.
Once a reasonable series of cuts are applied to this sample
(\S \ref{subsec:cuts}), approximately half of the SNe remain and are
used in the cosmological analysis.
\section{CMAGIC FITTING PROCEDURES}
\label{sec:cmagfits}
\nobreak
In order to determine if an individual data point lies within the linear
CMAGIC region for a particular SN it is necessary to know
the stretch and the date of $B$ maximum, although not to a high degree
of accuracy. These are determined by performing a template fit to the $B$
and $V$ light curves in a manner similar to P99 and K03. Briefly,
light-curve fits are performed using a \ensuremath{\chi^{2}}\ minimization procedure based
on MINUIT \citep{James:75} with both \ensuremath{K}-corrections\ and corrections for
Milky Way dust extinction taken into account. The light-curve template
is that of K03 (which uses the $B$ template of \citet{Goldhaber:01}
but a different $V$ template). For the photometry from P99 and K03,
the photometric correlation matrices were used in the light-curve fits.
These reflect
the correlations between different observations of the same SN
induced by the subtraction of the final reference image(s).
For the literature objects, where this information was not available,
the observations are assumed to be uncorrelated.
In order to prevent systematic errors arising from
differences in fitting procedures, we have only
included SNe that we can treat consistently, i.e.\ with our own light-curve
fitting procedure and \ensuremath{K}-corrections .
The correlation of the bump feature with different $B$ and $V$ stretch values
complicates matters. As explained in Appendix~\ref{apndx:bumps},
SNe~Ia with bumps can be fitted by the standard stretch templates if
the ratio between $B$ and $V$ stretch values is allowed to vary.
In order to handle this situation, three light-curve fits
were performed for each SN -- joint $B$ and $V$, $B$ only, and $V$ only.
In joint fits the dates of maximum and stretch values of the two filters
are fixed relative to each other by the light-curve template. Except
when a bump is visible in the CMAGIC diagram, the joint fit is used.
The reduced detectability of the bump feature at high redshift
due to reduced data quality is a concern that is further discussed in
Appendix~\ref{apndx:bumps}.
\ensuremath{K}-corrections\ play a critical role in this procedure. At high redshift
cross-filter corrections are necessary \citep{Kim:96}, but even at
low redshift same-filter \ensuremath{K}-corrections\ are not insignificant. Erroneous
\ensuremath{K}-corrections\ alter the slope of the CMAGIC linear region,
unlike extinction. Those used in this paper are based on the prescription of
\citet{Nugent:02} but with the time series of spectral templates and
empirical stretch-color relation of K03.
Milky Way extinction is included in this calculation using
the dust map of \citet{Schlegel:98}. Our approach naturally takes into
account the non-stellar nature of SN spectra and their variation
with epoch. Errors associated with the \ensuremath{K}-corrections\ are discussed in
\S\ref{sec:systematics}, where we also discuss the effects of several
other modifications to the fitting procedure described here.
Since the \ensuremath{K}-correction\ is a function of stretch and epoch, the light-curve
fits must be performed in an iterative manner. On the first iteration the
stretch is set to 1 and the date of maximum is set to the date of the
brightest point. The combined Milky Way and \ensuremath{K}-corrections\ are calculated
and the light curve is fitted, and the new stretch and date of maximum
are used to calculate new corrections. This process is iterated until
convergence. The majority of SNe converge
within three iterations, but the maximum number allowed is 16. Those
SNe that do not converge within 16 iterations invariably have extremely
poor light-curve coverage and are excluded from the sample.
Because high-redshift SNe very rarely have data beyond day 30, in order
to prevent a bias between high and low redshift SNe in the fitting procedure
data between 30 and 200 rest-frame days of maximum are not included,
a similar procedure to that followed in P99 and K03.
Observations more than 200 days after maximum light are included because
they provide final reference information useful for setting the amount of
host galaxy light underlying the SN.
This data set contains observations in 14 filters.
$BVRI$ filter curves were obtained from \citet{Bessell:90}. We reiterate the
warning of \citet{Suntzeff:99} that these filter functions include a
linear function of $\lambda$, which we have removed. The same is true
of the redshifted $B$ and $V$ filters used for some observations by the HZSST
($B35,V35,B45,V45$), with filter curves given by \citet{Schmidt:98}.
Filter curves for the {\it HST} filters on WFPC2 and ACS were generated using
{\it synphot} \citep{Simon:96synphot}. There are two sets of ground-based
$z$-band observations: those from \citet{Tonry:03}, and the
$z^{\prime}$ observations taken with SuprimeCam on the Subaru telescope
presented in \citet{Barris:04}. The \citet{Tonry:03} $Z$-band response
curve is as presented in that paper, and the SuprimeCam $z^{\prime}$ system
response was provided by H.\ Furusawa (2004, private communication).
Once the date of maximum and stretch are measured, the points in the CMAGIC
linear region can be determined and the linear relation fitted. Note that the
CMAGIC fit is performed on the observed data points, not on the template
fit used to determine the stretch and date of maximum. Again a
\ensuremath{\chi^{2}}\ minimization routine is used based on MINUIT that allows for errors
in both $B$ and \ensuremath{B-V} . The narrowness of the CMAGIC slope distribution, as
shown in figure~\ref{fig:slopes}, led W03 to suggest fitting all
CMAGIC relations with a fixed slope set at the mean of this distribution.
This is particularly important when working with high-redshift SNe because
the observational error bars are sufficiently large that accurate slope
measurements are difficult. We can make better use
of the available data by assuming that low- and high-redshift SNe have
similar \ensuremath{\beta_{BV}}\ distributions, as determined by examining
low-redshift SNe. This is similar to the approach followed by
previous analyses based on maximum magnitudes, where light-curve templates
developed from low-redshift SNe are used to fit high redshift data.
This leaves only one parameter in the fit, \ensuremath{B_{BV0.6}} . However, it is possible
to test the assumption that the slope distributions are consistent
with the handful of high-redshift SNe with sufficiently small observational
errors (\S\ref{sec:highzdemo}).
We improve on the fixed slope assumption by numerically propagating the
additional error due to the observed distribution of slopes using
a Monte-Carlo style approach. The slope distribution is determined from
the low-redshift SN sample, which for this purpose includes SNe~Ia
that are not in the Hubble flow. We take care to apply the same cuts,
described in \S \ref{subsec:cuts}, on this sample as we
do on the sample used to directly determine the cosmological parameters,
except for the redshift cut. This approach slightly overestimates the
errors because the measured slope distribution includes observational
errors, but in any case the net effect is quite small, inflating the
errors on \ensuremath{B_{BV0.6}}\ by around 0.01-0.03 mag in quadrature without
affecting the central values. In other words, the assumption of a fixed
slope used in W03 works extremely well for current data sets,
although we do include the additional error term in this analysis.
\section{CMAGIC RELATIONS AT HIGH REDSHIFT}
\label{sec:highzdemo}
\nobreak
The first task in applying CMAGIC at high redshift is to determine if
SNe~Ia at high redshift follow the linear relations derived at low redshift.
A brief examination of the CMAGIC diagrams shows that high-redshift
SNe do obey linear relations between magnitude and color. However,
in order to put this statement on a more quantitative footing,
we investigate the consistency of the \ensuremath{\beta_{BV}}\ distributions.
Most high-redshift observations have sufficiently large
error bars that they do not provide useful slope constraints. However, there
are a handful of relatively well observed SNe~Ia that can be used to
investigate this question: SNe 1997ce, 1997cj, 1998aw, 1998ax,
and 1998ba. The requirement for membership in this set is that there
be at least three points in the CMAGIC linear region and that
$\sigma_{\ensuremath{\beta_{BV}}} < 0.5$. SN 1997ce is particularly interesting because it
clearly displays a bump feature. Whatever physical mechanism causes the
bump feature is still active at high redshift.
The best fit slopes for these SNe are tabulated in
table~\ref{tbl:highzslopes} and the CMAGIC diagrams are plotted
in figure~\ref{fig:hizcmagex}. The \ensuremath{\chi^{2}}\ values for these fits
are improbably low, suggesting that the photometric errors
have been overestimated, which is also true of the low redshift sample.
The slopes are histogrammed in figure~\ref{fig:slopehisto}.
The mean slope for the low redshift sample is
$\left< \ensuremath{\beta_{BV}} \right> = 1.98 \pm 0.03$ and for the high redshift sample
it is $\left< \ensuremath{\beta_{BV}} \right> = 1.96 \pm 0.11$, so there is no evidence
for disagreement. A stronger statement requires more high quality multicolor
observations of high redshift SNe~Ia.
\section{COSMOLOGY FITTING PROCEDURES}
\label{sec:cosfits}
\nobreak
We now proceed to the primary purpose of this paper, the cosmological
analysis. Here we describe our methodology for
performing these fits. The results presented here differ from previous
papers in several respects. First, we have attempted to formalize the
procedure whereby individual SNe are rejected or accepted into the data
sample to a greater extent than has been true previously. Second, we make
use of a blind analysis procedure in order to prevent experimenter bias
from affecting the results. To this end, the results of the
cosmological analysis have been hidden from the authors until the cuts and
fitting procedure were finalized.
\subsection{Determining the Cosmological Parameters}
\nobreak
The luminosity distance equation can be written (in magnitudes) as
\begin{equation}
m = 5 \log_{10} \left( {\cal D}_{L} \left( z, \ensuremath{\Omega_{m}}, \ensuremath{\Omega_{\Lambda}} \right ) \right)
+ \ensuremath{\mathcal{M}} - \alpha \left( s - 1 \right) \label{eqn:lumdist}
\end{equation}
where $m$ is the observed magnitude, $s$ is the stretch, \ensuremath{\mathcal{M}}\ is a
combination of the Hubble constant $H_{0}$ and the absolute magnitude of
an SN Ia, and ${\cal D}_{L}$ is the $H_{0}$ free luminosity distance
given in \citet{Perlmutter:97}.
Because of the somewhat complicated nature of this parameter space, the
most conservative approach to fitting this relation is to perform a grid
search over the four
fitting parameters \mbox{(\ensuremath{\Omega_{m}} , \ensuremath{\Omega_{\Lambda}}, \ensuremath{\alpha} , \ensuremath{\mathcal{M}})} and then marginalize
over the two nuisance parameters \mbox{(\ensuremath{\mathcal{M}} , \ensuremath{\alpha})}. This is the
procedure used in P99 and K03. Because of the highly nonlinear nature
of the problem and the large errors on the cosmological parameters,
looking for the point where the \ensuremath{\chi^{2}}\ has increased by 2.3 over its
minimum leads to an underestimate of the errors.
A \ensuremath{\chi^{2}}\ is calculated at each
point on the grid, making use of equation~\ref{eqn:lumdist},
and converted into a relative probability
$P \propto \exp \left( - \ensuremath{\chi^{2}} / 2 \right )$. The probabilities are then
normalized over the grid, and the nuisance dimensions are summed over.
The parameter ranges explored are $\ensuremath{\Omega_{m}} = [0,3]$, $\ensuremath{\Omega_{\Lambda}}=[-1,4]$,
$\ensuremath{\mathcal{M}} = [24.7, 25.5]$,\footnote{The definition of \ensuremath{\mathcal{M}}\ used here
differs slightly from that of P99 and K03 in that all of the constants have
been absorbed, including c.} and $\ensuremath{\alpha} = [-0.5, 2.0]$.
These ranges include more than 99.99\% of the probability.\footnote{This
could be verified prior to unblinding for \ensuremath{\mathcal{M}}\ and \ensuremath{\alpha} , but
the confirmation of this statement for \ensuremath{\Omega_{m}}\ and \ensuremath{\Omega_{\Lambda}}\ was only available
after unblinding. If the final cosmology had disagreed very strongly
with previous results, this would have led to problems with the
blindness procedure. Fortunately, this turned out not to be the case.}
We have also constructed fits to the equation of state parameter $w$.
In order to reduce the computational complexity of this problem, these
fits are restricted to the flat universe case. Here the four parameters
are \ensuremath{\Omega_{m}} , $w$, \ensuremath{\mathcal{M}} , and \ensuremath{\alpha} . ${\cal D}_L$ must be modified
appropriately, but in all other respects the fit procedure is identical.
The range of $w$ considered is $[0,-3.5]$.
The errors on each \ensuremath{B_{BV0.6}}\ include the following terms:
\begin{itemize}
\item The uncertainty from the CMAGIC fits, including a contribution from the
distribution of \ensuremath{\beta_{BV}} .
\item The uncertainty of the stretch from the lightcurve fits multiplied by
\ensuremath{\alpha} .
\item A term due to the uncertainty in redshift. This includes an
assumed peculiar velocity dispersion of 300 km $s^{-1}$ and
redshift measurement errors .
\item \ensuremath{\sigma_{int}}\ magnitudes of intrinsic variation determined by
fits to the low-redshift Hubble diagram.
\end{itemize}
At high redshift the redshift measurement errors are taken to be
0.001 when the redshift was measured from host galaxy lines and 0.01
when measured from SN features, as in P99 and K03.
The intrinsic variation is assumed to be distributed as a Gaussian, and
is determined by performing Hubble fits with low redshift SNe and finding
the value that results in a \ensuremath{\chi^{2}}\ per degree of freedom of 1.
A Monte-Carlo simulation was used to calculate the errors associated with
this estimate by generating 100,000 realizations of a nearby SN
sample with identical properties to the actual one (redshift distribution
and photometry errors). For \ensuremath{B_{BV0.6}}\ with stretch correction,
$\ensuremath{\sigma_{int}} = 0.12^{+0.03}_{-0.04}$ mag. Two additional estimators for
\ensuremath{\sigma_{int}}\ were considered: the RMS corrected for photometry errors and
peculiar velocities, and the maximum-likelihood (ML) estimator for this
problem. All three agree, although the ML and \ensuremath{\chi^{2}}\ estimators
are considerably more efficient than the corrected RMS. We note that this
value for \ensuremath{\sigma_{int}}\ is slightly higher than that given in W03; the values
there were based on samples with tighter color cuts.
\subsection{Cuts on the Supernova Sample}
\label{subsec:cuts}
\nobreak
The procedure used to estimate the systematic errors in this paper
is an extension of that used by P99 and K03 and differs only in
that we have endeavored to be even more methodical in our exploration of
changes to the fits. For this paper we specify a {\it primary}
fit defined by a set of cuts, which are designed to be fairly loose while still
removing SNe with obviously bad data or that provide no useful constraint
on the cosmological parameters. We then explore the effects of changing these
cuts in great detail and use the information thus gleaned to estimate the
systematic errors. As we discuss below, altering most of these cuts has
little effect on the final result, but this systematic exploration raises the
specter of an unconscious fine-tuning to obtain the
expected result. To circumvent this possibility we have
performed a blind analysis, as detailed in \S\ref{subsec:blindness}.
The cuts can roughly be split into two categories: data quality and
analysis cuts. Not all are used in every fit considered.
Their values for the primary fit are summarized in table~\ref{tbl:primarycuts}.
More complete descriptions are provided below. The same cuts
are applied when determining the sample of SNe that are
used to measure the intrinsic distribution of \ensuremath{\beta_{BV}} .
There are four data quality cuts:
\begin{itemize}
\item A cut on the minimum number of points in the linear cmagic region.
As long as the date of maximum is well known, it is not
necessary to have more than one point.\footnote{Technically a floor of 2
points is used when the slope distribution sample is determined, but
this has no effect because all of the low redshift SNe have 2 or more
points in the linear region.}
\item A cut on the maximum allowable error on \ensuremath{B_{BV0.6}} .
Objects with very poorly determined magnitudes add little statistical
weight to the cosmology fit but make the Hubble diagram more difficult to
read and in general obfuscate the result.
\item A cut on the maximum allowable error in the date of maximum.
This is used because the date of maximum is used to specify
the points that are in the linear CMAGIC region. Points that fail this cut
usually fail the next cut as well.
\item A cut on the maximum allowable gap (in rest frame days)
between the nearest point in either $B$ or $V$ and the date of $B$ maximum.
If this gap is too large, the date of maximum, stretch, and maximum
magnitude can easily be incorrect. This arises because the error in the
light-curve template itself is currently not fully taken into account.
\end{itemize}
There are four analysis cuts:
\begin{itemize}
\item A minimum redshift cut for the cosmology fit.
It is ignored when the sample of SNe used to determine the intrinsic
\ensuremath{\beta_{BV}}\ distribution is determined.
\item A maximum redshift cutoff for the cosmology fit, which is not
used in the primary fit.
\item A maximum allowable color excess at $B$ maximum when compared with
the color model of K03. This can be interpreted as an extinction cut.
\item A minimum allowable stretch value. SNe with best fit values below
this are removed from the sample for the reason discussed below.
\end{itemize}
We find that our estimates for the cosmological parameters from \ensuremath{B_{BV0.6}}\
are relatively insensitive to changes in the cut on the color excess, but the
same cannot be said of the \ensuremath{m_{B}}\ fits. Because we seek to compare
the CMAGIC results directly with the \ensuremath{m_{B}}\ results, it is useful to
choose a value of the color cut that can be used for both fits.
Therefore, we have chosen to use the same cut as \citet{Knop:03}
($< 0.25$) in the primary fit.
A minimum stretch cut of 0.7 is applied to our primary fit sample because
our \ensuremath{K}-corrections\ may not be reliable for extremely low stretch SNe, as their
spectra display strong Ti features that are not well represented by our
spectral template \citep{Nugent:02}. We require
spectroscopic identification for our sample. There is only one SN
that passes the other cuts but does not have a firm spectroscopic ID:
SN 2001fo from \citet{Barris:04}. As was the case in K03 and P99,
SN 1997O has been manually excluded from our sample. When
included it is a 7 \ensuremath{\sigma}\ outlier from the best fit cosmology.
Two of the low-redshift SN in
our sample (SN 1997br and SN 1997bp) appear to have internal
inconsistencies in their photometry, displaying a far higher degree
of scatter both in light-curve and CMAGIC fits than can be explained
by their quoted photometric errors.\footnote{The \ensuremath{\chi^{2}}\ per degree
of freedom for the CMAGIC fits to SN1997bp and SN1997br are around 4,
which is particularly striking because for the majority of SNe~Ia
the \ensuremath{\chi^{2}}\ per degree of
freedom is considerably less than one.} We have taken the conservative
approach of removing them from the sample. When included, they have
no impact on the cosmological parameters. In addition to these cuts,
the maximum redshift of SNe that are used to measure the
\ensuremath{\beta_{BV}}\ distribution is specified by another cut.
There are 119 SNe at redshifts greater than 0.01 of the 131 SNe in
our baseline sample. Lower redshift
SNe can also be included in our fits, but add essentially no statistical
weight because of the dominance of their peculiar velocity errors. They
are still useful for measuring the intrinsic slope distribution. The data
quality cuts
at the levels of the primary fit eliminate 62 of the SNe from the primary
sample, and the analysis cuts remove five more. 53 are at $z > 0.1$, of which
28 are eliminated by the quality cuts and four by the analysis cuts.
We have explored the effects of both relaxing and tightening the cuts in
a systematic fashion. Many of the SNe fail multiple cuts, and the cuts are
not applied in any order, so it would be misleading to specify the number of
SNe removed by each cut. However, a list of which SNe are removed by
each cut is potentially interesting, and is provided in
appendix~\ref{apndx:cutremoved}.
\subsection{Blindness}
\label{subsec:blindness}
\nobreak
``Experimenter bias'' occurs when an analysis is affected by the
expectations of the experimentalist. Such bias is frequently unconscious,
and can take quite subtle forms. For example, a result that disagrees
strongly with a previous result is frequently subject to more
scrutiny than one that appears to be in agreement. This may bias an
experimenter into being more likely to find errors that cause their
result to disagree with expectations while making it less likely
that they will discover errors that have the opposite effect. Since the
research process has a natural termination point (publication), if the
decision to stop analyzing a result is at all influenced by the value
of the result, a bias will be introduced. A nice
summary of these issues can be found in \citet{Heinrich:03}. It has
long been recognized that a useful technique for mitigating experimenter
bias is to hide the final results of the experiment
from the experimenter for as long as possible. This is known as blind analysis.
Such an approach is particularly useful in an analysis with a substantial
number of cuts, such as that presented here.
In the medical fields double blind procedures (which hide some details
of the experiment from both the test subject and the experimenters)
are used almost as a matter of course. Naturally,
hiding the details of the experiment from the subject is not of great
concern in astronomical research.
A critical point is that these techniques do not seek to completely
hide all information during the analysis. In fact, the goal is to hide
as little information as possible while still acting against experimenter
bias. Human judgment and scientific experience continue to play a
critical role in a blind analysis. One does not mechanically carry out
the steps of the analysis and then publish the results. All that a
blind analysis does is prevent unconscious misuse of particular types
of information during the analysis process. The kind of data that are
excluded from consideration (namely, the final answer derived from each
option under consideration) is invariably that which no reasonable
scientist would allow to consciously influence his or her decision making
process. However, subconscious effects are still present, and this
is what this approach helps prevent.
Specifically, it is important to design the blindness technique such that
subsidiary diagnostics are available even while hiding the
final answer. Errors are initially present in any analysis, and it is
important that even while the result remains blinded mechanisms are available
to catch these problems. Specifically, our goal is to hide the values of \ensuremath{\Omega_{m}}\
and \ensuremath{\Omega_{\Lambda}}\ until the cuts and fitting procedures have been finalized, while
preserving as much ancillary information as possible. In particular, our
method preserves the residuals of individual SNe with respect to the
Hubble line, which is extremely useful while diagnosing the fits. For example,
an error in the \ensuremath{K}-corrections\ might result in all SNe in a given redshift range
departing significantly from the Hubble line. This problem would still be
detectable in our blinded fits. In addition, the method preserves the
shifts in \ensuremath{\Omega_{m}} , \ensuremath{\Omega_{\Lambda}}\ between fits to different subsamples -- if excluding a
particular SN causes
the unblinded result to shift by $\Delta \ensuremath{\Omega_{m}} = 0.1,\ \Delta \ensuremath{\Omega_{\Lambda}} = 0.2$, the
blinded result shifts by the same amount, which is important when investigating
systematic errors.
The technique used here is based on altering the true fit estimates. Hidden,
but fixed, offsets are added to \ensuremath{\Omega_{m}}\ and \ensuremath{\Omega_{\Lambda}} , and this change is propagated
through to the \ensuremath{B_{BV0.6}}\ values. In essence the cosmological parameters
are fitted twice, with the magnitudes modified between fits, but the
results of the first fit are never output. Because it would be possible
to circumvent the blindness if the real \ensuremath{B_{BV0.6}}\ values were known,
these values must be kept hidden. All of the programs used to
plot CMAGIC diagrams add random offsets to the $B$ magnitudes for display
purposes. Furthermore, the CMAGIC fitter and cosmology fitter are integrated
so that the true \ensuremath{B_{BV0.6}}\ values are not output.
The expression for the luminosity distance cannot
be evaluated in terms of simple functions except in limited cases,
so the magnitude modification is calculated numerically. The results of the
first, unmodified, fit are marginalized to determine the secret true measured
values $\Omega_{mT}$ and $\Omega_{\Lambda T}$. The
hidden offsets are then applied to these values, and the difference in
magnitudes between the two cosmologies is calculated and applied.
If $\Delta \ensuremath{\Omega_{m}}$ and $\Delta \ensuremath{\Omega_{\Lambda}}$ are the hidden offsets,
then the following function is added to \ensuremath{B_{BV0.6}}\ for each SN:
\begin{equation}
\Delta \ensuremath{B_{BV0.6}} \left( z \right) = 5 \log_{10}
{\cal D}_{L} \left( z, \Omega_{mT} + \Delta \ensuremath{\Omega_{m}} ,
\Omega_{\Lambda T} + \Delta \ensuremath{\Omega_{\Lambda}} \right) -
5 \log_{10} {\cal D}_{L} \left( z, \Omega_{mT}, \Omega_{\Lambda T} \right),
\label{eqn:magshift}
\end{equation}
where ${\cal D}_{L}$ is as in equation~\ref{eqn:lumdist}.
The cosmological fit is then redone with the new magnitudes and this result
is output. It is safe to output the modified magnitudes, which can be used
to construct a Hubble diagram and to perform various tests on the fit.
The simplest method to choose the hidden offsets is to generate them randomly.
This performs poorly in this case because there are several non-physical
regions in the \ensuremath{\Omega_{m}} , \ensuremath{\Omega_{\Lambda}}\ parameter space. Negative values of \ensuremath{\Omega_{m}}\ result
in a non-convergent luminosity distance integral. For high values of \ensuremath{\Omega_{\Lambda}}\
the universe did not experience a Big Bang, but is instead rebounding from a
previous bout of contraction \citep{Carroll:92}. In such a universe
there is a maximum observable redshift, and if any of the SNe are at higher
redshifts the luminosity
distance expression cannot be evaluated. A randomly generated
offset could easily push the cosmological parameters into one of these
regions. Instead we have chosen to generate the hidden offsets by specifying
the desired values of \ensuremath{\Omega_{m}}\ and \ensuremath{\Omega_{\Lambda}}\ for a particular SN sample
(the primary fit). A special version of the cosmological fitter determines
the offsets between a fit to the primary sample and the chosen
value\footnote{These values were chosen to be sufficiently different from the
results of previous analyses to force internal reviewers to psychologically
confront the blindness scheme while remaining close enough to the expected
values that the resulting error contours were not overly distorted.} \ensuremath{\Omega_{m}} = 1,
\ensuremath{\Omega_{\Lambda}} = 1.1. These offsets are then used for all other fits.
As long as the resulting fit values for \ensuremath{\Omega_{m}}\ and \ensuremath{\Omega_{\Lambda}}\ are roughly
equal to ($\Omega_{mT} + \Delta \ensuremath{\Omega_{m}}$, $\Omega_{\Lambda T} + \Delta \ensuremath{\Omega_{\Lambda}}$)
this preserves the residuals with respect to the fit by construction.
Because the same hidden offsets are used
for all fits, this approximately preserves relative shifts between different
fits. The caveat is that, for a particular value of \ensuremath{\Omega_{m}}\ and \ensuremath{\Omega_{\Lambda}} , the
shape of the luminosity distance equation effectively weights SNe
depending on their redshift, and therefore altering the values of these
parameters may cause the relative shifts in the blinded fits to be slightly
different than for the true values. Therefore, the offsets are determined
iteratively. However, as long as the hidden offset
is relatively small, this effect is negligible. Tests on both previous
data sets (specifically, the low-extinction primary subset of K03) and
artificially generated data show that this procedure works in that
the resulting cosmological parameter estimates are equal to the unblinded
result plus the specified offset. The offset between the blind target
values and the actual estimates for this analysis was somewhat larger
than anticipated, so the specified offset does not quite match the
actual shift. However, the relative shifts are preserved accurately over
small distances, which allowed us to compare different fits to the same
data prior to unblinding.
A similar procedure is followed in the $w$ fits, although a different set of
offsets are used. Because problems related to non-physical regions
of the parameter space are not as severe in this case, the offsets to
\ensuremath{\Omega_{m}}\ and $w$ were randomly generated from the ranges $[-0.2,0.2]$
and $[-0.4,0.4]$.
Should a mistake in the analysis be found after the result is unblinded,
it should still be corrected. In this situation, one should
publish both the corrected and uncorrected results and note the effects
of the discovered error on the result. An example of this can be found in
\citet{Akerib:04}. We also note that it is important to determine the
systematic errors prior to unblinding, or it would be possible to explain
away any unexpected results by inflating them. This technique certainly
does not prevent all types of bias, but it does provide an
opportunity to improve the situation, and thus is worth pursuing.
\subsection{Complete Fitting Procedure (Blind)}
\nobreak
Our cosmological fits proceed in the following order:
\begin{itemize}
\item The SNe used to measure the intrinsic \ensuremath{\beta_{BV}}\ distribution
are determined by applying the specified cuts. The distribution of \ensuremath{\beta_{BV}}\
is then calculated from these SNe.
\item A one-parameter (\ensuremath{B_{BV0.6}} ) CMAGIC fit is performed for all SNe in the data
sample using a Monte-Carlo fitting technique that takes into account the
distribution of \ensuremath{\beta_{BV}}\ from the distribution calculated in the previous
step. The fitted \ensuremath{B_{BV0.6}}\ values are not output.
\item The cuts are applied again to determine the SNe used to
measure \ensuremath{\Omega_{m}}\ and \ensuremath{\Omega_{\Lambda}} . The same cuts are used,
except for the redshift ranges in \S \ref{subsec:cuts}.
\item A cosmological fit is performed. Estimates for \ensuremath{\Omega_{m}}\ and
\ensuremath{\Omega_{\Lambda}}\ are calculated but not output.
\item The hidden offsets are read in and added to \ensuremath{\Omega_{m}}\ and \ensuremath{\Omega_{\Lambda}} . A
magnitude offset is applied to each SN based on equation~\ref{eqn:magshift}.
\item The cosmology is refitted with the new magnitudes. These results
are output.
\item The altered magnitudes are used to construct a Hubble diagram.
\end{itemize}
Once the blindness was removed, the fits were redone without the secret
offset step. We have also performed fits using the maximum $B$ magnitude,
\ensuremath{m_{B}} . Since these fits are not a principal result of this paper they can be
performed in an unblinded fashion, allowing us to test our procedures.
\section{COSMOLOGICAL RESULTS}
\label{sec:cosresults}
\nobreak
Figure~\ref{fig:baselinecontour} shows the \ensuremath{\Omega_{m}} , \ensuremath{\Omega_{\Lambda}}\ confidence regions of
our primary fit, based on 31 nearby and 21 distant SNe~Ia. An additional nine
very nearby SNe ($z < 0.01$) are used while determining the \ensuremath{\beta_{BV}}\
distribution (for a total of 40). The resulting estimates for
the cosmological parameters are
$\ensuremath{\Omega_{m}} = 1.26^{+0.38}_{-0.51}$ and $\ensuremath{\Omega_{\Lambda}} = 2.20^{+0.41}_{-0.67}$. If we
require a flat universe, consistent with recent CMB results, then
$\ensuremath{\Omega_{m}} = 0.19^{+0.06}_{-0.06}$. These confidence regions are comparable to
those from P99 (but not as good as those from K03), despite the fact that
fewer SNe are involved, due to the smaller value of \ensuremath{\sigma_{int}}\ for CMAGIC.
The fit residuals are shown in figure~\ref{fig:baselinehubble}.
\ensuremath{\Omega_{m}}\ and \ensuremath{\Omega_{\Lambda}}\ are not the natural variables for this measurement, as they
are not independent for this data set. The result of our analysis is
better expressed in the principal axes frame of the error ellipse
$\Omega_1 \equiv 0.790 \ensuremath{\Omega_{m}} - 0.613 \ensuremath{\Omega_{\Lambda}}$ (the short axis) and
$\Omega_2 \equiv 0.613 \ensuremath{\Omega_{m}} + 0.790 \ensuremath{\Omega_{\Lambda}}$ (the long axis).
Roughly, $\Omega_1$ can be thought of as measuring acceleration and
$\Omega_2$ as measuring geometry. Analyzing the results in this frame
has considerable benefits while calculating systematic errors and when
comparing the CMAGIC results to those derived from maximum magnitudes.
In this frame the results of the primary fit are
$\Omega_1 = -0.349^{+0.117}_{-0.131}$ and
$\Omega_2 = 2.502^{+0.530}_{-0.838}$.
The values of the nuisance parameters are
$\ensuremath{\alpha} = 0.516^{+0.193}_{-0.206}$ and $\ensuremath{\mathcal{M}} = 25.166^{+0.049}_{-0.045}$,
and they are almost completely statistically independent.
Magnitudes and redshifts are provided in table~\ref{tbl:primarysamplowz}
for the low-redshift sample, and in table~\ref{tbl:primarysamphighz} for the
high redshift sample. The \ensuremath{\chi^{2}}\ of this fit is 49.5 for
52 degrees of freedom. In the next section we discuss variations of the
cuts, which produce different sets of SNe. The stretch-luminosity
relation is shown in figure~\ref{fig:stretchlum}. When compared
with the \ensuremath{m_{B}}\ relation (Fig.\ 13 of \citet{Knop:03}, for example), the
evidence for the utility of a stretch correction is much weaker for \ensuremath{B_{BV0.6}} .
Our estimates for $w$ in a flat universe are shown in
figure~\ref{fig:wcontour}. These are combined with the measurement of
the angular size of the baryon acoustic peak (BAP) in SDSS galaxy clustering
statistics at $z=0.35$ \citep{Eisenstein:05}, which are quite complementary
to the SN measurements. The resulting constraint is
$w = -1.21^{+0.15}_{-0.12}$ and $\ensuremath{\Omega_{m}} = 0.25^{+0.02}_{-0.02}$ (statistical
errors only).
This is the first analysis that treats
the combined data from the different SN groups in a fully consistent manner.
Unlike \citet{Leibundgut:01} or \citet{Riess:04}, we find no significant
evidence for anomalously blue colors in the high-redshift SN, even though
this sample contains many of the same objects as those studies.
Figure~\ref{fig:maxcolor} shows the \ensuremath{B-V}\ color at $B$ maximum for the
low- and high-redshift primary fit sample. The highly negative color
point from the high-redshift sample is due to (by far) the most poorly
measured SN, SN 1997af, which has $\ensuremath{E(B\,-\,V)_{Bmax}} = -0.24 \pm 0.24$. Excluding
this point, the mean color of the low redshift sample is
$\ensuremath{(B-V)_{Bmax}} = 0.045 \pm 0.027$ and that of the high redshift sample is
$\ensuremath{(B-V)_{Bmax}} = 0.027 \pm 0.019$, where the standard errors are quoted.
\section{SYSTEMATICS}
\label{sec:systematics}
\nobreak
We explore various systematic errors by performing alternate fits and
comparing the results with our primary fit. Because of the way in which our
blindness scheme is constructed, this comparison was possible before the
final answer was known. As was the case in \cite{Knop:03}, we find that
the effects of most of the systematics act along the long axis of our error
ellipse. They therefore do not significantly affect the value of the SN
measurements for determining if the Universe is accelerating, but do
substantially limit our ability to measure geometry. Fortunately, this is
the dimension in which CMB measurements are extremely powerful.
There are two types of systematic error possible in this analysis.
First, there are the systematics arising from alterations in the fitting
procedures, \ensuremath{K}-corrections , etc. Second, there are those arising from the
cuts applied to the sample. Ideally this second set would be handled
by a complete Monte-Carlo simulation of the SN sample.
Unfortunately, there are far too many pieces of information missing to
make the results of such a study at all useful. In order to construct
a believable Monte-Carlo, it would be necessary to have a reasonable
understanding of the intrinsic luminosity and extinction distributions,
which have not been convincingly measured. To make matters substantially
worse, it would also be necessary to have a good understanding of the search
and follow up strategy used to construct the SN sample. Because the
sample used in this paper is primarily constituted of literature SN,
a clear definition of the search techniques and procedures is simply not
available. Providing the results of such a procedure would provide a
misleading sense of accuracy. We therefore proceed by
calculating the effects of changing the cuts applied to our sample over
what we consider to be a reasonable range and combining the resulting
shifts as an estimate of the systematic error. Clearly this procedure
is somewhat subjective, but any credible improvement requires the availability
of large, well defined SN samples such as those that should be
provided by the SNfactory, SNLS, SDSS Supernova Survey, and ESSENCE.
The effects of these shifts can most precisely be stated in terms of
the principal axes of the primary fit error ellipse, $\Omega_1$
and $\Omega_2$, which is the primary justification for their
use. Recall that for the primary fit $\Omega_1 = -0.349^{+0.117}_{-0.131}$
and $\Omega_2 = 2.502^{+0.530}_{-0.838}$ (statistical errors only).
We follow the standard practice of adding the negative and positive
shifts in quadrature when handling asymmetric errors (however,
see \cite{Barlow:03} for criticism of this procedure). The resulting
systematic errors are $^{+0.060}_{-0.062}$ on $\Omega_1$ (the short axis),
$^{+0.476}_{-0.545}$ on $\Omega_2$ (the long axis), and
$^{+0.029}_{-0.049}$ on the value of \ensuremath{\Omega_{m}}\ in a flat universe. The shifts
are summarized in table~\ref{tbl:identifiedsystematics}, and detailed
individually in the following sections. Some representative examples
can be seen in figure~\ref{fig:baseline_comp}. An essentially identical
procedure has been carried out for the fit to $w$, \ensuremath{\Omega_{m}}\ in a
flat Universe, including the BAP constraint, resulting in systematics
error estimates of $^{+0.07}_{-0.12}$ on $w$ and $^{+0.01}_{-0.01}$ on \ensuremath{\Omega_{m}} .
Note that this only includes the systematics from the SN measurement.
Unlike the \ensuremath{\Omega_{m}} , \ensuremath{\Omega_{\Lambda}}\ fits, here the statistical errors are dominant,
reflecting the more challenging nature of the $w$ measurement.
\subsection{Variation of Fitting Procedures}
\label{subsec:systematicsfitting}
\nobreak
There are many reasonable ways to alter the CMAGIC fitting procedures
that result in slightly different values of the cosmological parameters.
We have attempted to explore some of these variations.
P99 found that using a floating value of \ensuremath{\alpha}\ when propagating
the stretch error into the fit magnitude artificially inflates \ensuremath{\alpha} ,
as this decreases the \ensuremath{\chi^{2}}\ by increasing the magnitude errors.
Therefore, \ensuremath{\alpha}\ was fixed for the purposes of error propagation.
As in K03, we find no evidence for this effect. Fixing \ensuremath{\alpha}\ at the estimate
from the primary fit ($\ensuremath{\alpha} = 0.5$) has essentially no effect on the
\ensuremath{\Omega_{m}} , \ensuremath{\Omega_{\Lambda}}\ values except to shrink the error bars slightly, as expected.
Not performing a stretch correction ($\ensuremath{\alpha} = 0$) shifts the error ellipse
primarily along $\Omega_2$ by 0.06. This is not included in the final
value for the systematic error.
It is possible to include estimates about the error in the stretch
and date of maximum in the CMAGIC fitting procedure, since they influence
which points are included in the CMAGIC fit. A modified version of
the fitting code has been used to investigate this possibility.
This approach is considerably more expensive computationally, and for
this data sample it turns out to make no difference. In our fits we
have effectively assumed that $B$ and \ensuremath{B-V}\ are independent variables.
An alternative formulation of the linear relations that treats $B$ and $V$
as independent variables is possible. This also has no
effect on the fit values (less than 0.005 mag for any SN).
The light-curve fitting procedure used in P99 differs slightly from that
used here (and by K03) in that the fits to the $V$ band were performed fixing
the stretch and date of maximum to the values derived from a $B$ only
fit. This procedure arose from concerns that the rest frame $V$
light curves for the high-redshift sample are more poorly sampled
than the rest frame $B$ light curves, which is not the case for the
low-redshift sample. Thus, a light-curve fitting procedure that treats both
bands on an equal footing might effectively introduce a bias in the
fits. This is of considerably less concern for this data sample, since
by its nature CMAGIC demands good $V$-band coverage, but to guard
against this problem we re-calculated all of the lightcurve fits following
this prescription, which affects the CMAGIC fits because it changes the
values of the stretch and date of maximum. The resulting effect on the
error contours was minor, and primarily towards larger values
of $\Omega_2$ by 0.144.
Variations in the \ensuremath{K}-corrections\ are investigated by considering alternative
versions of the spectral template. In particular, we follow K03 by making use
of a $U$-enhanced version of the template with $U - B = -0.5$ instead
of $-0.4$ as in our primary fit. This shifts the error ellipse primarily
along the short axis, with
$\Delta \Omega_1 = -0.052$ (towards smaller values of \ensuremath{\Omega_{m}} ).
The \ensuremath{\chi^{2}}\ worsens slightly to 50.9. This is, by far, the
most significant source of uncertainty related to alterations in the
fitting procedures. Simply treating this error as a statistical contribution
to each SN is a completely inadequate representation of its effect
on the cosmological results. Clearly, future projects would benefit
substantially from additional constraints on the $U$-band behavior of
SNe~Ia.
\subsection{Variation of Cuts}
\nobreak
We considered both increasing and decreasing the cut values for
all of the cuts described in \S \ref{subsec:cuts}. Here we only
present those that had a measurable effect on the error ellipse
or are interesting for some other reason.
Requiring SNe to have observations within 5 rest-frame days of maximum
eliminates two low redshift SNe (SN 1998ab and SN 2000fa)
and one at high redshift (SN 1996E), and induces a shift along the
long axis by $\Delta \Omega_2 = +0.139$. Loosening the requirement
to 10 days adds one high-redshift SN (SN 2001jp), and results in a shift
along the $\Omega_1$ axis of +0.024 towards higher values of \ensuremath{\Omega_{m}} .
Changing the minimum allowable redshift to 0.015 from 0.01 has an
extremely small effect on the fit results while eliminating six
low redshift SNe. Halving (to 0.25) or tripling (to 1.5) the cut on the
maximum allowable magnitude error alternately removes five high
redshift SNe or adds one, but does not affect the results substantially,
as one would expect given the low weight given SNe with such
large errors.
Placing a substantially tighter cut on the color at maximum
[$\ensuremath{E(B\,-\,V)_{Bmax}} \le 0.1$, similar to that used for the low-extinction
subset of K03] shifts the error contours by
a substantial amount along the long axis (towards a flat universe)
by $\Delta \Omega_2 = -0.467$, eliminating three high
and eight low redshift SNe. Using a color cut of 0.125 (half of the primary
fit value) is not substantially different than using 0.1.
Relaxing the color cut to 0.5 adds two high-redshift
(SN 1998aw and SN 2002ad) and four low redshift SNe, and
moves the contours principally along
the short axis by $\Delta \Omega_1 = -0.048$.
While less affected by extinction than \ensuremath{m_{B}} , CMAGIC is not
completely unaffected. The analysis presented in this paper suggests
that assumptions about the extinction law are not a significant systematic
bias, and therefore future studies, including those that use CMAGIC,
may benefit by applying an extinction correction. This must
be weighed against the decrease in independence of the two magnitudes
after correction.
Requiring that the date of maximum be known
to better than 0.5 days removes a large number of high redshift
SNe from the sample (nine), but has little effect except to inflate
the error contours along the long axis.
Relaxing the requirement to 2 days adds eight
poorly measured high-redshift SNe and shifts the ellipse outwards
along the long axis by $\Delta \Omega_2 = +0.115$.
Requiring that there be at least two observations in the CMAGIC
linear region, and hence providing some level of confidence that
the CMAGIC relations are being obeyed, does have a non-negligible
effect on the cosmological parameters. Three high-redshift
SNe are eliminated (SN 1998as, SN 2002ab, and
SN 2002kd), and the error ellipse shifts primarily outward
along the long axis by $\Delta \Omega_2 = +0.23$. Even when two points are
required in the linear region, the quality of the high redshift data
is such that the CMAGIC slope \ensuremath{\beta_{BV}}\ cannot be usefully fitted to
each SN.
As can be seen from the above discussion, the primary systematic
effect related to the cuts on the SN sample is associated
with the extinction cut. A better understanding of the extinction
distribution would help reduce this systematic considerably.
Note that we do not apply an extinction correction,
so we are more sensitive to the extinction cut than some other analyses --
although they trade this off with sensitivity to extinction and the
intrinsic peak color of SNe~Ia. Fortunately, the systematics arising from the
cut selection are primarily along the long axis of the error ellipse,
and hence have little effect on our detection of acceleration.
\subsection{Other Systematics}
\label{subsec:othersys}
\nobreak
We have also considered limiting our low-redshift SN sample to only those
from large, systematic SN studies in order to limit any systematic
errors arising from differences in calibration. There are three major
low-redshift samples: \citet{Hamuy:96}, \citet{Riess:99a} and
\citet{Jha:05}. Excluding all nearby SNe that are not from one of the
above three sources has a very minor effect.
To test the sensitivity of our results to individual SNe, we have performed
a jack-knife test by removing each of the 21 high-redshift SNe individually
and recalculating the cosmological fit. Our values for \ensuremath{\Omega_{m}}\ and \ensuremath{\Omega_{\Lambda}}\ are
sensitive to the removal of SN 2001ix and SN 2002kd, both at the very high
redshift end of the sample. Removing either of these SNe shifts the
contours primarily along the long axis, although in opposite senses.
Removing SN 2001ix results in a shift inward of $\Delta \Omega_2 = -0.28$,
and removing SN 2002kd shifts the contour outward by $\Delta \Omega_2 = 0.31$.
Interestingly, their effects on the cosmological parameters nearly cancel.
This analysis would benefit from additional SNe in
this redshift range, but overall the results are reasonably robust.
Properly speaking, \ensuremath{\sigma_{int}}\ should be another quantity that is marginalized
over while performing the cosmological fits. To determine if this is
necessary, we performed fits in which \ensuremath{\sigma_{int}}\ was varied by 1 \ensuremath{\sigma}\ in each
direction, and found that the effects on the cosmological parameters
were negligible (less than 0.1 \ensuremath{\sigma}\ in $\Omega_1$ and $\Omega_2$).
Since all of the high-redshift supernovae (and many of those at
low redshift) come from flux-limited samples, they suffer
from Malmquist bias \citep{Malmquist:36}. We note that only
a difference in the amount of Malmquist bias between the low- and
high-redshift SN samples can affect the cosmological results.
This effect is discussed extensively in P99 and K03, and we adopt
the estimates contained therein for these samples: 0.01 mag for P99 and
0.03 mag for K03. P99 also estimated the Malmquist bias for the
\citet{Hamuy:96} sample as 0.04 mag. The \citet{Riess:99a} and
\citet{Jha:05} samples were primarily discovered using a
galaxy catalog search, so they may suffer from little or no Malmquist
bias \citep{Li:01}. We therefore adopt a Malmquist bias of 0 mag
for these samples. It is difficult to estimate the Malmquist bias
for the remaining SNe in the low redshift sample, since they were
discovered in a rather inhomogeneous fashion. However, since they constitute
only a small fraction of the sample, the effects of any Malmquist bias
on the cosmological parameters from this sample are expected to be negligible,
and so we adopt a value of 0 mag. For the remaining portion of
the high-redshift sample (approximately half) we provisionally use
the same value as for the P99 SNe, 0.01 mag. To test the effects of
this bias on our estimate, we apply the offsets to each sample and
recalculate the fit. The resulting shift in the cosmological parameters is
quite small, less than $0.1\ \sigma$ in both dimensions.
Appendix~\ref{apndx:bumps} contains a discussion of the effects
of the `bump' in the CMAGIC diagram exhibited by some SNe. The effects
of this systematic are negligible along both axes (less than 0.05 \ensuremath{\sigma} ).
\section{ANALYSIS OF RESULTS}
\label{sec:analysis}
\nobreak
There are two channels available for analyzing the results of
this paper. First, the estimates of the cosmological parameters
can be considered in isolation. Second, the CMAGIC results can
be compared with a maximum magnitude fit to the same SN.
Several of the systematics should affect both samples equally
(e.g., Malmquist bias); therefore, this comparison should
be more precise. However, this requires that the covariance
between \ensuremath{m_{B}}\ and \ensuremath{B_{BV0.6}}\ be determined.
\subsection{Constraints on the Cosmological Parameters}
\nobreak
The results of a CMAGIC fit to currently published SN data
strongly favor an accelerating Universe --- in fact, more strongly
than previous results based on \ensuremath{m_{B}} . Perhaps more interesting is that
the fit contours depart mildly from a flat universe.
In the principal axis frame, a flat universe corresponds
to $\Omega_2 = 0.756 \pm 0.010$ for $\ensuremath{\Omega_{m}} = 0.191$. Once systematics
are taken into account, the disagreement is 1.75\ensuremath{\sigma} , which
is expected to occur approximately 8\% of the time
due to random chance. A similar result was seen in the SN sample of
\cite{Tonry:03}, although at a somewhat lower level of significance.
Both results are interesting, but not yet strong enough to be of
serious concern. One of the lessons of blind analyses is that
1.5+\ensuremath{\sigma}\ disagreements occur in science more frequently
than our intuition, developed from exposure to non-blind experiments,
often expects.\footnote{See \citet{Heinrich:03} \S4 for further discussion.}
The departure from flatness is driven by SNe at moderate redshifts
$0.3 < z < 0.5$. The three with the highest pull are
SNe 1998as, 1996k, and 1997ce. It is difficult to find any common
thread between them. They come from three different papers, were
observed with different telescopes (although SN 1998as and SN 1997ce
were both partially observed with {\it HST}), and their photometry
was reduced by
different authors using different techniques. Since they constitute
the low-redshift end of their respective surveys, there may be a suspicion
that they suffer from unusually high extinction.
While SN 1998as does suffer from considerable host galaxy extinction
($A_V = 0.49$; K03), the other two suffer from negligible extinction
($A_V=0.02$ and 0.08 for SN 1996K and SN 1997ce, respectively;
Riess et~al.\ 2004).
Note that removing each of these SNe individually has little effect on our
results, as explained in \S \ref{subsec:othersys}.
\subsection{Comparison of \ensuremath{B_{BV0.6}}\ and \ensuremath{m_{B}}\ Results}
\label{subsec:compare}
\nobreak
The results of an \ensuremath{m_{B}}\ fit to the same SN as the primary
are compared with the \ensuremath{B_{BV0.6}}\ fit in figure~\ref{fig:contcompare}.
The \ensuremath{\chi^{2}}\ of this fit is 44.32 for 52 degrees of freedom,
and the resulting estimates are $\ensuremath{\Omega_{m}} = 1.08^{+0.49}_{-0.69}$ and
$\ensuremath{\Omega_{\Lambda}} = 1.65^{+0.65}_{-0.91}$, with a flat universe value of
$\ensuremath{\Omega_{m}} = 0.32^{+0.07}_{-0.07}$.
The principal axes of this fit are almost identical to those of
the CMAGIC fit, so it is useful to express them in this frame.
Here they correspond to $\Omega_1 = -0.167^{+0.146}_{-0.133}$ and
$\Omega_2 = 1.969^{+0.787}_{-1.146}$ (statistical errors only).
Note that the \ensuremath{m_{B}}\ fits agree somewhat better with a flat universe
than the \ensuremath{B_{BV0.6}}\ fits.
If \ensuremath{m_{B}}\ and \ensuremath{B_{BV0.6}}\ were equivalent (given current templates) we
would expect \ensuremath{\alpha}\ to be identical for the two methods. When comparing
these numbers the marginalized, one-dimensional errors are appropriate
instead of the outer extent of the 1 \ensuremath{\sigma}\ error contours quoted
previously. For \ensuremath{B_{BV0.6}}\ $\ensuremath{\alpha} = 0.516^{+0.193}_{-0.206}$,
and for \ensuremath{m_{B}}\ it is $\ensuremath{\alpha} = 0.995^{+0.253}_{-0.226}$, a difference
of 1.6 \ensuremath{\sigma} . They are marginally inconsistent, but not at a
significant level.
Directly comparing the \ensuremath{m_{B}}\ and \ensuremath{B_{BV0.6}}\ cosmological results
requires that the correlation between the two methods be measured,
and then propagated into the cosmological parameter space.
The details of this process are presented in
Appendix~\ref{apndx:correlations}. The result is that the
correlation coefficients between the
two fits are 0.34 along the $\Omega_1$ axis and 0.15
along $\Omega_2$.
While many of the systematic errors should affect \ensuremath{m_{B}}\ and \ensuremath{B_{BV0.6}}\
equally, not all apply to both fits. For example,
the number of points in the CMAGIC linear region is meaningless
in an \ensuremath{m_{B}}\ context. Furthermore, individual SNe may have quite different
weights in the two fits, which partially removes the insensitivity
to systematics. Both issues must be addressed before the results
can be compared. The number of points in the linear region
and the detectability of CMAGIC bumps at high redshift have
already been discussed, and are summarized in
table~\ref{tbl:identifiedsystematics}. In addition, we expect that
the effects of the $U-B$ color of the spectral templates
will not be the same for both methods, since \ensuremath{m_{B}}\ and \ensuremath{B_{BV0.6}}\ depend
on color information in a very different fashion. Comparing the
results of \ensuremath{m_{B}}\ and \ensuremath{B_{BV0.6}}\ fits using the $U$-enhanced spectral templates
as discussed in \S\ref{subsec:systematicsfitting}, we find that
the residual difference due to this systematic is
$\Delta \Omega_1 = 0.010$, $\Delta \Omega_2 = 0.151$. The
effects of the differing weights can be addressed by performing a fit
to \ensuremath{m_{B}}\ where each SN is given the weight it has in the \ensuremath{B_{BV0.6}}\ fit,
and vice-versa. It is not fair to include both values as systematics
errors, since they are essentially measuring the same effect.
Fortunately, they turn out to have almost identical effects.
The short axis is brought into better agreement by a
shift of $\Delta \Omega_1 = 0.054$ and the long axis by
$\Delta \Omega_2 = 0.31$.
Putting these contributions together, and using the correlations
given above, we find that the difference between the \ensuremath{m_{B}}\ and
\ensuremath{B_{BV0.6}}\ fits is
\begin{eqnarray*}
\Delta \Omega_1 & = & -0.182 \pm 0.097 \mbox{(stat)}
\pm 0.058 \mbox{(sys)} \\
\Delta \Omega_2 & = & 0.530 \pm 0.661 \mbox{(stat)}
\pm 0.414 \mbox{(sys)}.
\end{eqnarray*}
The difference along the $\Omega_1$ axis amounts to 1.6 \ensuremath{\sigma} ,
and along the $\Omega_2$ axis to 0.7 \ensuremath{\sigma} . The major
disagreement is along the short axis, as is obvious from
figure~\ref{fig:contcompare}, and a disagreement of this size or larger
is expected to occur in 11\% of measurements. Since $\Omega_1$ is
essentially sensitive to acceleration, this amounts to the statement
that the \ensuremath{B_{BV0.6}}\ results favor more acceleration at the 1.6 \ensuremath{\sigma}\ level.
The differences along both axes can be combined into one measure by
projecting them along the difference vector, defined by
$\Omega_3 \equiv -0.325 \Omega_1 + 0.946 \Omega_2$. Then the difference
between the two fits is $\Delta \Omega_3 = 0.560 \pm 0.657 \mbox{(stat)} \pm
0.410 \mbox{(syst)}$, a difference of 0.7 \ensuremath{\sigma} .
A similar comparison is possible with the \ensuremath{\Omega_{m}} , $w$ fits.
The result is shown in figure~\ref{fig:wcompare}. The
same sort of detailed comparison is not carried out here
for several reasons. First, the difference is certainly not
independent from the difference observed in \ensuremath{\Omega_{m}} , \ensuremath{\Omega_{\Lambda}}\ space,
so little additional information would be gained from this
procedure. Second, because the current constraints on
\ensuremath{\Omega_{m}} , $w$ from SN data alone are not well behaved
(not closing off until very negative values of $w$), it is not
useful to compare the two fits without the addition of additional
constraints, here the BAP measurement, which
is the same between the two fits.
\section{CONCLUSIONS}
\nobreak
CMAGIC provides some additional information that is not
captured by the standard light-curve template fitting techniques
used to estimate \ensuremath{m_{B}} . This allows us to provide some additional
constraints on the cosmological parameters. Furthermore, \ensuremath{B_{BV0.6}}\
should be affected differently by several potential evolutionary
effects.
We have carried out the first blind analysis of the cosmological
parameters using SN data, developing a technique to prevent
experimenter bias by hiding the final result until the data cuts
and analysis procedures are finalized. We
find that the results of a CMAGIC fit broadly confirm our picture
of an accelerating Universe. In fact, they favor a higher amount of
acceleration than the \ensuremath{m_{B}}\ results by approximately 1.6 \ensuremath{\sigma}\ (including
systematics and the correlations between the two measurements).
The \ensuremath{B_{BV0.6}}\ error contours differ from a flat Universe
by 1.7 \ensuremath{\sigma}\ (including systematics), which would be interesting if it
were more statistically significant.
The constraints on the cosmological parameters from
a CMAGIC fit to 31 nearby and 21 distant SNe~Ia are
$\ensuremath{\Omega_{m}} = 1.26^{+0.38}_{-0.51}$, $\ensuremath{\Omega_{\Lambda}} = 2.20^{+0.41}_{-0.67}$
(statistical errors only). However, this is a poor frame for
expressing the results. It is significantly more useful to
instead quote the results as
\begin{displaymath}
\Omega_1 = 0.790 \ensuremath{\Omega_{m}} - 0.613 \ensuremath{\Omega_{\Lambda}} = -0.349^{+0.117}_{-0.131}
\left( \mbox{stat} \right) ^{+0.060}_{-0.062}
\left( \mbox{syst} \right)
\end{displaymath}
\begin{displaymath}
\Omega_2 = 0.613 \ensuremath{\Omega_{m}} + 0.790 \ensuremath{\Omega_{\Lambda}} = 2.502^{+0.530}_{-0.838}
\left( \mbox{stat} \right) ^{+0.476}_{-0.545}
\left( \mbox{syst} \right)
\end{displaymath}
with
\begin{displaymath}
\Omega_{m} = 0.19^{+0.06}_{-0.06} \left( \mbox{stat} \right)
^{+0.03}_{-0.05} \left( \mbox{syst}\right)
\end{displaymath}
for a flat Universe, where the dark energy has been assumed to
have a constant equation of state with $w = -1$, as is the case for
a cosmological constant. The systematic errors have been estimated by
considering a wide range of alternatives to the primary fit of this
paper. The largest systematic error is the extinction cut,
indicating that while CMAGIC has some benefits with respect
to extinction by interstellar dust, we still have a great deal to
learn about this issue. A direct comparison is also possible
with an \ensuremath{m_{B}}\ fit to the same SN, which requires that the correlations
between the two methods be estimated. After including the systematics
and correlations, the difference between the two fits is almost exclusively
along the short axis, with the CMAGIC fits favoring more acceleration by
1.6\ensuremath{\sigma} . Fitting for a constant value of $w$ in a flat Universe,
the combination of the CMAGIC results with the angular scale of
the BAP measured in \citet{Eisenstein:05} yields
$w = -1.21^{+0.15}_{-0.12} \left( \mbox{stat} \right)
^{+0.07}_{-0.12} \left( \mbox{supernova syst} \right)$,
$\ensuremath{\Omega_{m}} = 0.25^{+0.02}_{-0.02} \left( \mbox{stat} \right)
^{+0.01}_{-0.01} \left( \mbox{supernova syst} \right)$, consistent
with a cosmological constant at the $1.2 \sigma$ level.
The currently available high redshift SN
sample was not observed in an optimal fashion for CMAGIC.
Out of the approximately 100 published high-redshift SNe light curves,
only about 20 are useful for \ensuremath{B_{BV0.6}} . As a result, the current data set
does not place strong constraints on dust or evolutionary effects.
This situation will change
in this decade; within the next 5 years it should be possible to
measure both \ensuremath{B_{BV0.6}}\ and \ensuremath{m_{B}}\ for 1000 high-redshift SNe, at which point the
comparison between \ensuremath{m_{B}}\ and \ensuremath{B_{BV0.6}}\ will be extremely interesting.
\acknowledgments
The authors would like to thank Brian Schmidt for providing non-$K$-corrected
light curves for SN 1997ce and SN 1997cj. This research has
made use of the NASA/IPAC Extragalactic Database (NED) which is operated
by the Jet Propulsion Laboratory, California Institute of Technology,
under contract with the National Aeronautics and Space Administration.
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\section{INTRODUCTION}
The life of a massive star ends in a supernova (SN). The detection of neutrinos from SN 1987A verified the idea that some SNe are set off by the gravitational collapse of the iron core of their progenitor star \citep{Krauss1987}. However, theory predicts that very massive stars with helium cores between $\sim$64 and 133 $M_{\odot}$ could find another way to blow up, through the thermonuclear explosion of oxygen via the pair-production instability \citep{Rakavy1967, Barkat1967, Heger2002}. The production of electron/positron pairs in the core softens the equation of state, leading to collapse and the ignition of explosive oxygen burning. The subsequent thermonuclear runaway reverses the collapse and ejects the entire star, leaving no remnant behind. Unlike iron core-collapse supernovae (CCSNe), which involves poorly constrained physical processes such as turbulence, pulsations, perhaps rotation and magnetic fields, the physics involved in pair-instability supernovae (PISNe) is fairly well understood and can be modeled with fewer uncertainties \citep{Langer2009}.
Due to the extremely large stellar mass required, the progenitors of
PISNe are expected to be rare, and may only form under unusual
conditions. One such condition existed in the early universe, when
metal-free Population III stars were born \citep{Loeb2010}. In star
formation, it is the accretion process that ultimately sets the final
mass of a star. From dimensional arguments, the mass growth rate is
simply given by the Jeans mass $M_{J} \sim c_{s}^3 G^{-\frac{3}{2}}
\rho^{-\frac{1}{2}}$ over the free-fall time $t_{ff} \sim
1/\sqrt{G\rho}$, implying $dM/dt \propto c_{s}^3/G \propto
T^{\frac{3}{2}}$, where the sound speed $c_{s} \sim \sqrt{kT/m_{p}}$.
In present day star-forming regions, heavy elements radiatively cool
the gas to a temperature as low as $T\sim 10$K. However, in
primordial clouds, the primary coolant at low temperatures is
molecular hydrogen, which can only cool the gas to $T\sim 200-300$K,
implying an accretion rate higher than present day by two orders of
magnitude. Hence, theoretical studies suggest that the initial mass
function (IMF) of Pop III stars might have been biased toward masses
much higher than today, e.g. several hundred $M_{\odot}$
\citep{Bromm2004}. The nucleosynthesis imprints of this top heavy IMF
have been seen in globular clusters and damped Lyman alpha systems
\citep{Cooke2011, Puzia2006}. Moreover, massive stars have strong
winds driven by radiation pressure through spectral lines, with a mass
loss rate scaling with stellar metallicity $\dot{M}\propto Z^{0.5\sim
0.7}$ \citep{Vink2001, Kudritzki2002}. Most PISNe should therefore be
from Pop III stars, which have weak radiation-driven winds due to their
extremely low metallicities, and retain enough of their initial masses
at the end of their lives to undergo a pair-instability explosion.
Naturally, studies of the rates and detectability of PISNe
focused on high redshifts before reionization. \citet{Mackey2003}
found the PISNe rate to be $\sim 50$ deg$^{-2}$ yr$^{-1}$ at $z > 15$,
while \citet{Weinmann2005}, using more conservative assumptions for
the number of PISNe produced per unit Pop III stellar mass formed,
found the PISNe rate to be $\sim 4$ deg$^{-2}$ yr$^{-1}$ at similar
redshifts. Assuming that only one supermassive Pop III star forms in
unenriched minihalos, and that none form in protogalaxies,
\citet{Wise2005} found the PISNe rate be $\sim 0.34$ deg$^{-2}$
yr$^{-1}$ at $z\sim 20$. After our paper was submitted,
\citet{Hummel2011} presented a complimentary analysis of the source
density of PISNe from pristine minihalos, and determined the
observability of such events with the \emph{James Webb Space
Telescope} (JWST), finding approximately $\sim 0.4$ PISNe visible per
JWST field of view at any given time. PISNe after the epoch of
reionization were also considered; \citet{Scannapieco2005a} calculated
a suite of PISNe model light curves with blackbody spectra, and
analyzed the detectability and rates of PISNe from Pop III stars
formed from leftover pristine gas at $z \lesssim 6$. As for CCSNe
during reionization, \citet{Mesinger2006} presented detailed
predictions for the number of core collapse SNe that JWST could
observe as a function of different survey parameters.
In this paper, we present light curves and spectral time series for
PISNe from our multi-wavelength radiation-hydrodynamics simulations.
As the stellar population responsible for reionization is currently
unknown, instead of predicting a fixed SNe rate, we normalize the star
formation rate by requiring that enough ionizing photons must be
produced by either Pop III or Pop II stars in protogalaxies to
complete reionization by $z \sim 6$, and calculate the rates of
pair-instability, core-collapse, and Type Ia SNe and their
detectability with JWST for these two scenarios; the actual SNe rates
will be in between these limiting cases. We show that using the
observed rates of these SNe, it is possible to distinguish the
contribution of Pop III and Pop II stars toward reionization by
characterizing the IMF at that time.
\section{LIGHT CURVES AND SPECTRA}
The stellar evolution and explosion of PISN models, and the resulting broadband light curves and spectral time-series are described in detail in \citet{Kasen2011}; here we summarize the results. Models R150, R175, R200, R225, R250 represent explosions of 150-250 $M_{\odot}$ red supergiant stars, respectively, each with their hydrogen envelope intact. In principle, blue supergiants are also possible progenitors of PISNe, but convective mixing of metals into the hydrogen envelope makes it more likely that the progenitor dies as a red supergiant. Models He80, He100, He130 were explosions of 80, 100, 130 $M_{\odot}$ bare helium cores. Such models may represent stars that lost they hydrogen envelope due to a prior pulsational phase or through binary interactions. Here we use an approximate empirical relation between the helium core mass and the progenitor main-sequence mass \citep{Heger2002}:
\begin{equation}
M_{He} \approx \frac{13}{24}(M_{ZAMS}-20M_{\odot}).
\end{equation}
Properties of all presupernova stars and their explosions are given in Table \ref{ProgenitorModelTable}.
\begin{table}
\caption{Parameters of supernova explosion models. The R-prefix models refer to red supergiant progenitor PISNe, and the He-prefix models refer to the exposed helium core PISNe. The proxy core-collapse SN model (CC), a model Type~IIP supernova, is shown for comparison. $R_0$ is the presupernova radius. $M_i$, $M_f$ are the initial and final masses of the progenitor, respectively, while $M_{He}$ is the helium core mass and $M_{Ni}$ is the amount of $^{56}$Ni synthesized in the explosion. All mass units are in $M_{\odot}$.}
\begin{tabular}{ | l | l | l | l | l | l |}
\hline
Name & $M_i$ & $M_f$ & $M_{He}$ & $M_{Ni}$ & $R_0$ ($10^{12}$cm)\\ \hline
R150 & 150 & 142.9 & 72.0 & 0.07 & 162 \\
R175 & 175 & 163.8 & 84.4 & 0.70 & 174 \\
R200 & 200 & 181.1 & 96.7 & 5.09 & 184 \\
R225 & 225 & 200.3 & 103.5 & 16.5 & 333 \\
R250 & 250 & 236.3 & 124.0 & 37.86 & 225 \\ \hline
He80 & 80 & 80 & 80 & 0.19 & - \\
He100 & 100 & 100 & 100 & 5.00 & - \\
He130 & 130 & 130 & 130 & 40.32 & - \\ \hline
CC & 15 & 13.3 & - & 0.28 & 44 \\ \hline
\end{tabular}
\label{ProgenitorModelTable}
\end{table}
A few days after the explosion, hydrodynamical processes subside and the ejected material reaches a phase of nearly free expansion. The energy powering the subsequent light curve may derive from three possible sources: (\emph{i}) Lingering thermal energy from the explosion itself; (\emph{ii}) The radioactive decay of synthesized $^{56}$Ni; (\emph{iii}) The interaction of the ejecta with a dense circumstellar medium. Thermal energy suffers adiabatic losses on the expansion timescale $t_{ex} = R_{0}/v$, and so source (\emph{i}) is only significant for stars with large initial radii $R_{0}$. Circumstellar interaction has not been included in the models discussed here.
We have computed light curves and spectral time series of the explosion models using the time dependent radiative transfer code SEDONA \citep{Kasen2006}. All models shown here assume spherical symmetry, and calculations of atomic level populations assume local thermodynamic equilibrium. Using Monte Carlo methods, we solve the full multi-wavelength radiative transfer problem using realistic opacities as a function of wavelength, composition and temperature, over millions of line transitions. Unlike previous blackbody models \citep{Wise2005, Scannapieco2005a}, our results allow us to calculate more accurate light curves for any given color bands and to study the time evolution of the supernova colors and spectral features.
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{Rmodels_light_curve_rband.eps}
\caption{Rest frame R-band light curves for the red supergiant progenitor models. In some models, a brief spike in luminosity occurs at the end plateau when radiation is released by hydrogen recombination. The sharpness of the spike may be exaggerated by the lack of numerical convergence of the ionization front recession. }
\label{restframeRlightcurve}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{Hemodels_light_curve_rband.eps}
\caption{ Rest frame R-band light curves for the helium core progenitor models.}
\label{restframeHelightcurve}
\end{figure}
The shape and duration of PISN light curves depend on the mass and radius of their progenitors. Model R250 shows a weak and then strong peak in its light curve (Figure \ref{restframeRlightcurve}), the initial peak powered by thermal energy and the second by the radioactive decay of $^{56}$Ni. The heating from radioactive decay delays the inward-propagating recombination wave from ejecta cooling, regulating the electron scattering opacity (and thus the release of thermal energy), and causing the second peak to rise at 200-300 days, which reaches a spectacular brightness of $\sim -21.5$ mag. However, model R150 produces very little $^{56}$Ni, and therefore lacks a prominent second peak; the light curve is essentially thermally powered and reaches a brightness less than that of a Type Ia SN.
The helium core models are more compact and hence lack an initial thermal peak (Figure \ref{restframeHelightcurve}). Model He130 reaches an exceptional peak brightness of $2\times 10^{44}$ ergs s$^{-1}$, whereas Model He80 demonstrates that despite being massive and energetic, not all PISNe are bright. This steep mass-luminosity relation for PISNe suggests that to increase the sheer number of SNe detected, it is better to conduct a wide rather than deep survey of the sky \citep{Weinmann2005}.
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{R250_SpectraVsTime_50Ares.eps}
\caption{ Time evolution of the rest frame spectra for the R250 red supergiant model.}
\label{restframeR250}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{He130_SpectraVsTime_50Ares.eps}
\caption{ Time evolution of the rest frame spectra for the He130 helium core model.}
\label{restframeHe130}
\end{figure}
The spectra of a PISN resemble that of average SNe, with P-Cygni line profiles on top of a blackbody, see Figures \ref{restframeR250}, \ref{restframeHe130}. For the RSG models, at early times, the spectrum is rather featureless with only weak Balmer and calcium lines, reflecting the low abundance of metals in unburned ejecta. The spectral energy distributions of the models are blue at earlier times ($\la 50$ days) but become redder over time as the expanding ejecta cools. In addition, line blanketing of the bluer wavelengths becomes more prominent over time, as the photosphere recedes into the deepest layers which are abundant in freshly synthesized iron group elements. For PISNe at the redshifts of reionization, JWST will mostly be observing in the rest frame UV, so it is important to use more accurate spectral models, rather than the blackbody models of \citet{Scannapieco2005a}.
Spectroscopic or rest frame UV observations of PISNe may be able to constrain the metallicity of the progenitor star. However, the hydrogen envelope may be polluted by newly synthesized metals mixed out during the explosion. \citet{Chen2011} simulated multi-dimensional models of PISNe to predict the degree of mixing. They found relatively small fluid instabilities generated from burning at the boundaries of the oxygen shell, and concluded that PISNe keep their onion-shell structure in the explosion, until the reverse shock passes which generates Rayleigh-Taylor instabilities. This is in contrast with CCSNe, in which a shock runs through the inner metal-rich core, inducing the growth of instabilities and mixing. Also, ordinary Pop II/I CCSNe have non-zero metallicity in their hydrogen envelopes to begin with. Hence, metal lines in early-time spectroscopy might be able to distinguish PISNe from CCSNe, before the photosphere has receded deep into the ejecta. With a little mixing, N and possibly some C and O might appear in the early spectra of PISNe, but PISNe will not have any Si, Ni, Fe lines \citep{Joggerst2011}. This best applies to the red supergiant models, as the Helium core models undergo significant burning and have spectra that show many metal lines at maximum light.
\section{Supernovae During Reionization}
Observations of quasar absorption spectra \citep{Fan2006} indicate
that reionization was completed by $z=6$. It is believed that most of
the ionizing photons came from stars \citep{Loeb2010, Bouwens2011a}.
Although the very first stars could have ignited as early as $z\sim
30-40$, due to the exponential nature of structure formation, most of
ionizing photons originated from stars born in the later stages of
reionization at $z\sim 10$. Although it is not known which population
of stars dominated at this epoch, an unusual stellar mass function
10-20 times more efficient than the standard Salpeter IMF in producing
ionizing photons is required at $z\sim 6$ \citep{Cen2010}. This
favors the existence of a top-heavy Pop III stars at these redshifts,
which may be observable via their extraordinary deaths as PISNe if the
IMF included mostly stars between 140 and 260 $M_{\odot}$. Moreover,
the observed rates of PISNe, CCSNe, and Type Ia SNe may be used to
infer the IMF responsible for reionization at $z\gtrsim 6$.
\subsection{The Initial Mass Function}
The ionizing photon yield per baryon incorporated into present day stars with a Salpeter IMF is $\overline{\eta_{\gamma}}\sim 4000$. However, if the IMF is dominated by massive metal free stars ($M > 100 M_{\odot}$), then $\overline{\eta}$ can be up to a factor of 20 higher \citep{Bromm2001,Raiter2010}. The transition from Pop III to Pop II/I star formation is thought to occur at a critical metallicity of $Z_{crit}\sim 5 \times 10^{-4}Z_{\odot}$, above which cooling and fragmentation become efficient, which stops the preferential formation of massive stars \citep{Bromm2003}.
This transition can be associated with the assembly of atomic H cooling halos with virial temperatures $> 10^4$K \citep{Haiman2009}. Molecular hydrogen is fragile to photodissociation, and the molecular coolant in halos are likely depleted after a single episode of metal free star formation. Therefore, molecular hydrogen halos are unlikely to allow continued formation of stars above $Z_{crit}$. Subsequent star formation only occurs when the deeper gravitational potential wells of atomic H cooling halos are assembled, corresponding to a virial temperature of $T_{vir}\approx 10^4$K and a minimum halo mass of $M_{halo}\approx 10^8M_{\odot}$ The gas in these halos will thus have already gone through a burst of primordial star formation, and contain traces of metals, leading to Pop II star formation. Most of the photons responsible for reionization will come from the profusion of these Pop II stars in this scenario, although without contribution from Pop III stars, this may require an unrealistic star formation efficiency, see Figure \ref{SFRchart}.
There is another possibility. Most molecular $H_{2}$ cooling halos may not have formed stars at all, due to global $H_{2}$ photodissociation by an early cosmic background of 11.2-13.6 eV photons (the Lyman-Werner band), to which the universe is otherwise transparent. In this scenario, the majority of primordial star formation will appear in atomic H cooling halos with $M_{halo}\approx 10^8M_{\odot}$. During blowouts from repeated SN explosions, these halos allow most of their self-generated metals to be accelerated into the IGM as SN ejecta, but, in contrast to smaller molecular $H_{2}$ cooling halos, these halos hold on to most of their interstellar gas \citep{MacLow1999}, and can have significant Pop III star formation. Coupled with the high ionizing efficiency of massive metal free stars, in this scenario Pop III stars will make a significant contribution to reionization.
Hence, we consider two mutually exclusive scenarios for reionization, where either Pop III or Pop II stars reionize the universe; the actual star formation history of reionization will be in between these limiting cases. As for the IMF in each scenario, for massive, metal free Pop III stars, we use either a Salpeter IMF slope $dN/d\log M\propto M^{-1.35}$, or a flat IMF slope $dN/d\log M\propto M^0$ hinted by recent simulations \citep{Clark2011, Greif2011}, with $M_{upper} = 500M_{\odot}$ and $M_{lower}=1M_{\odot}$. Note that the resulting PISN rates are not sensitive to the upper and lower mass bounds of reasonable Pop III IMFs. As long as $M_{upper} > 260 M_{\odot}$, due to the steepness of $dN/dM$, there are not enough stars at the most massive end to affect the overall normalization. Moreover, for our SFR model described in the next section, any reasonable $M_{lower}$ ranging from $0.1$ - $10 M_{\odot}$ makes negligible difference to the PISN rates of Pop III stars. This is because we normalize star formation by requiring the generation of enough stellar UV photons necessary to reionize the universe, and massive stars $M \gtrsim 10 M_{\odot}$ are drastically more efficient at producing ionizing photons. In essence, we fix the number of massive stars produced in any IMF, but are free to vary the number of low mass stars, as the latter do not contribute to reionization anyway. $M_{lower} = 1 M_{\odot}$ was chosen to match the smallest Pop III stars seen in recent simulations by \citet{Clark2011}.
For Population II stars forming with traces of metals, we use a Salpeter IMF with $M_{upper} = 125M_{\odot}$ and $M_{lower} = 0.1M_{\odot}$, where we include a factor 0.7 in the mass integral to account for the reduced number of low mass stars in a realistic IMF \citep{Fukugita1998}, compared to the original Salpeter IMF. The different IMF models are tabulated in Table \ref{IMFTable}.
\begin{table}
\caption{Model parameters of the different IMFs. Here $\alpha$ is the slope of the stellar mass function, i.e. $dN/d\log M\propto M^{\alpha}$, and the slope of the Pop II IMF flattens at $M<0.5M_{\odot}$. All mass units are in $M_{\odot}$.}
\begin{tabular}{ | l | l | l | l | l | l |}
\hline
IMF model $\phi(M)$ & $M_{lower}$ & $M_{upper}$ & $\alpha$ & $\overline{\eta}_{\gamma}$ \\ \hline
Pop III Salpeter & 1 & 500 & -1.35 & 28683 \\
Pop III Flat & 1 & 500 & 0 & 77087 \\
Pop II & 0.1 & 125 & -1.35 & 5761 \\ \hline
\end{tabular}
\label{IMFTable}
\end{table}
\subsection{The Star Formation Rate}
We calibrate the SFR by requiring enough UV photons are produced by stars so as to ionize the intergalactic medium (IGM) by the end of reionization. This requires $C \sim 10$ ionizing photons per baryon in the IGM, accounting for recombinations \citep{Trac2007}. Using the time-averaged ionizing flux and stellar lifetime for individual stars from \cite{Schaerer2002}, we find the number of ionizing photons per baryon incorporated into stars $\eta_{\gamma}(M)$ as a function of stellar mass, for Pop III stars and early Pop II stars ($Z = 1/50 Z_{\odot}$). For a given stellar track, the average ionizing photon per baryon in star is thus:
\begin{equation}
\overline{\eta}_{\gamma} = \frac{\int \eta_{\gamma}(M)\phi(M) M dM}{\int \phi(M) M dM},
\end{equation}
where $\phi(M)$ denotes the IMF. Then, the fraction of total baryons in the universe that are in stars $F_s(z)$ must satisfy:
\begin{equation}
\frac{F_s(z_{end}) \overline{\eta}_{\gamma} f_{esc}}{C}=1,
\label{FractionOfTotalBaryonsEquation}
\end{equation}
where $z_{end}$ is the redshift at the end of Reionization, chosen to be $z_{end}=6$ in our model, and where $f_{esc}$ is the escape fraction of ionizing photons from the host galaxy into the IGM. In the calibration of the SFR as a function of redshift, we assume the stars instantaneously produce all the ionizing photons at birth that they would normally produce during their lifetimes. For a fixed redshift of reionization, this will underestimate the SFR. Nevertheless, since most of the ionizing radiation was dominated by the massive stars ($M\gg 10 M_{\odot}$), with lifetimes $<10$ Myr, this is an adequate approximation.
The mass in stars per comoving volume as a function of redshift, $\rho_{*}(z)$, can be related to the fraction of gas in halos which converts to stars, i.e. the star formation efficiency $f_{*}$, using the Sheth-Tormen mass function $\frac{dn}{dM}$ \citep{Sheth1999} of halos:
\begin{equation}
\rho_{*}(z) = F_s(z)\rho_{b} = f_{*} \frac{\Omega_b}{\Omega_m} \int_{M_{min}}^\infty M \frac{dn(z)}{dM} dM.
\label{MassFormedIntoStarPerVolumeEq}
\end{equation}
Here $\rho_{b}$ is the cosmological baryon density, and $M_{min} \sim 10^{8} M_{\odot}$ for both Pop II and Pop III scenarios, corresponding to halos with atomic H cooling. For cosmological parameters used in generating the Sheth-Tormen mass function, we adopt $h=0.71$, $\Omega_m=0.27$, $\Omega_{\Lambda}=0.73$, and $\Omega_b=0.045$, where $h$ is the Hubble constant in units of 100 km s$^{-1}$ and $\Omega_m$, $\Omega_{\Lambda}$, and $\Omega_b$ are the total matter, vacuum, and baryonic densities in units of the critical density \citep{Komatsu2011}. Since $F_s(z_{end})$ is known via equation (\ref{FractionOfTotalBaryonsEquation}), by plugging $z=z_{end}$ into equation (\ref{MassFormedIntoStarPerVolumeEq}), we can calibrate the value of $f_{*}$, and evaluate $\rho_{*}(z)$ at any redshift. Although $f_{*}$ will generally vary with redshift, here we take $f_{*}$ as a constant for simplicity of calibration. The star formation rate is then simply:
\begin{equation}
SFR(z) = \frac{d\rho_{*}(z)}{dz}.
\end{equation}
Figure \ref{SFRchart} shows the resulting star formation rates, using $C=10$ and $f_{esc}=0.1$. Observations of Lyman-break galaxies around $z\sim 3$ suggests that $f_{esc}$ could be larger at higher redshifts \citep{Steidel2001}. However, theoretically the high gas densities at the redshifts of the first galaxies could decrease the escape fraction down to $f_{esc}\lesssim 0.01$ \citep{Wood2000}, in which case using $f_{esc}=0.1$ is a conservative choice that may underestimate the SFR and the corresponding SN rates. To consider different choices of these parameters, note that the SFR in our model linearly scales with $C$ and $f_{esc}^{-1}$.
\begin{figure}
\begin{center}
\includegraphics[width=0.8\columnwidth]{SFRPlot.eps}
\caption{ The SFR for Pop II or Pop III stars required to reionize the universe by $z_{end}=6$. The calibrated star formation rate is $f_{*}=$ 0.3\%, 0.8\%, and the unrealistically high 24.1\% for the Pop III Flat, Pop III Salpeter, and Pop II IMF models, respectively, suggesting that Pop II stars could not have driven reionization by themselves. The SFR inferred by \citet{Bouwens2011b} from integrating the observed galaxy UV luminosity densities to $M_{AB}\approx 18$ is plotted for comparison; their substantially lower SFR is not surprising, as the contribution from the very steep faint-end slope of lower-luminosity galaxies was omitted \citep{Bouwens2011}. The SFR for our models and the resulting SN rates all linearly scale with $C$ and $f_{esc}^{-1}$. $C=10$ and $f_{esc} = 0.1$ were used throughout this paper.}
\label{SFRchart}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=1.0\columnwidth]{AllSNeRatePlot.eps}
\caption{ The rate $R(z)$ of PISNe, CCSNe, and Type Ia SNe for our different IMF models for the stellar population responsible for reionization, per year per Mpc$^3$.}
\label{SNRates}
\end{center}
\end{figure}
\subsection{Snapshot Rate with JWST}
The \emph{James Webb Space Telescope}\footnote{http://www.jwst.nasa.gov/} (JWST) will include a Near Infrared Camera (NIRCam), with a spectral coverage from 0.6-5 $\mu$m with $\sim$10 nJy sensitivities in 10$^4$ s of integration time (10$\sigma$); a Near Infrared Spectrograph (NIRSpec) which operates at approximately the same wavelength range. The Mid InfraRed Instrument (MIRI) covers 5-27 $\mu$m, but is an order of magnitude less sensitive than NIRCam. Since isolated Pop III stars are likely beyond the reach of JWST, to test the prediction that metal-free stars had a top-heavy IMF (which has been recently debated, see \citet{Hosokawa2011}), we can either observe the cumulative properties of the first stars by imaging Pop III galaxies \citep{Zackrisson2011}, or detect their deaths as extraordinary bright supernova.
The number of new events at a given redshift that can be observed per unit solid angle is \citep{Woods1998}:
\begin{equation}
N(z) = R(z) \: (1+z)^{-1} \: \frac{dV_{c}}{dz}, \mbox{ for } z<z_{max(F;\nu)},
\label{RatePerRedshiftEquation}
\end{equation}
where $z_{max(F;\nu)}$ is the maximum redshift at which a source will appear brighter than limiting flux $F$ at an observed frequency $\nu$, $R(z)$ is the event rate per unit comoving volume, and $dV_{c}$ is the cosmology-dependent comoving volume element corresponding to a redshift interval $dz$. The above expression includes the (1+z) reduction in apparent rate owing to cosmic time dilation.
The `snapshot rate', i.e. the total number of events (not per unit time) observed at limiting flux $F$ is:
\begin{equation}
N(F;\nu) = \int_{0}^\infty dz \: R(z) \: t(z; F; \nu) \: \frac{dV_{c}}{dz},
\label{SnapshotRateEquation}
\end{equation}
where $t(z; F; \nu)$ is the rest-frame duration over which an event will be brighter than the limiting flux $F$ at redshift $z$ for an observed frequency $\nu$. We find this duration from our spectral time series calculated with SEDONA. There is an implicit $(1+z)$ factor in equation (\ref{SnapshotRateEquation}) due to the time dilation of the light curve, but that cancels with the $(1+z)^{-1}$ reduction in apparent rate. Although $t(z; F; \nu)$ of PISNe will generally be longer for more massive progenitors, the snapshot rate is not necessarily dominated by the highest mass stars, as they are less numerous, see Figure \ref{snapshot_rate_with_z:globfig}. For CCSNe, it is not clear how the brightness of a Type IIP SN should depend on the mass of the progenitor star; here we use light curves of a $15M_{\odot}$ red giant progenitor generated with SEDONA \citep{Kasen2009}, whose broadband light curves and spectra agree very well with observed Type IIP SNe, which are observed to be the most common, at least in the nearby universe. However, this single CCSN model means we do not capture the variation in CCSN peak flux from different progenitors, which we do so for PISNe.
\begin{figure*}
\centering
\subfloat[Subfigure 1 list of figures text][Pop III Salpeter IMF model]{
\includegraphics[width=0.45\textwidth]{plot_snapshot_rate_with_z_PopIIISalpeter_v2.eps}
\label{snapshot_rate_with_z:PopIIISalpeter}}
\qquad
\subfloat[Subfigure 2 list of figures text][Pop III Flat IMF model]{
\includegraphics[width=0.45\textwidth]{plot_snapshot_rate_with_z_PopIIIFlat_v2.eps}
\label{snapshot_rate_with_z:PopIIIFlat}}
\caption{ Differential snapshot rate $\frac{dN(F;\nu)}{dz}$ in the 10 arcsec$^2$ field-of-view of NIRCam on JWST, calculated using the sensitivities of the F444W filter (44400 \AA) with $t=3\times 10^4$s integration time at 5$\sigma$ (a flux threshold of 2 nJy). At this sensitivity, each type of SN appears in rough proportion to their actual event rates (Figure \ref{SNRates}) up to $z\sim 8$, past which the ratio of detected PISNe versus CCSNe turns over, with only the brighter PISNe staying in view. As we have not accounted for the intrinsic scatter in the luminosity of CCSNe, the actual turnover will be less sharp. A similar turnover exists for the less massive PISN models R175, R150 starting at $z\sim 11$, past which the more massive progenitors are more likely to be seen in the field of view despite being less numerous in number given the IMF. The snapshot rate of CCSNe in Figure \ref{snapshot_rate_with_z:PopIIISalpeter} is less than that implied in \citet{Mesinger2006}, as their assumed IMF is closer to our Pop II IMF model.}
\label{snapshot_rate_with_z:globfig}
\end{figure*}
\begin{table}
\caption{Snapshot rate in NIRCam's field of view (10 arcsec$^2$) on JWST, using the same survey parameters as Figure \ref{snapshot_rate_with_z:globfig}. This is the total number of PISNe and CCSNe in each NIRCam snapshot, integrated across $z \geq 6$, for the different IMF models. One can multiply the values below by 360 to get the snapshot rate per deg$^2$. It is not clear what fraction of PISNe explode from red supergiants (RSG) versus exposed helium cores; the rates shown in the RSG and He Core columns assume all PISN explode as that type. Here the snapshot rate of PISNe from red supergiants is higher than the helium core model due to the longer duration of the former. The snapshot rate for CCSNe in our Pop II IMF scenario is in good agreement with the high end estimate of 24 SNe per field found by \citet{Mesinger2006} under similar survey parameters.}
\begin{tabular}{ | l | l | l | l |}
\hline
IMF model & He Core & RSG & CC \\ \hline
Pop III Salpeter & 0.28 & 0.42 & 10.43 \\
Pop III Flat & 0.74 & 1.03 & 0.64 \\
Pop II & 0 & 0 & 31.83 \\ \hline
\end{tabular}
\label{SnapshotRateNIRCam}
\end{table}
\begin{table}
\caption{Snapshot rate in MIRI's field of view (2.35 arcsec$^2$) on JWST, using the F770W filter (77000 \AA) with $3\times 10^4$s exposure (5$\sigma$). The resulting snapshot rate is an order of magnitude worse than NIRCam; however, these results suggest that MIRI can be used as a follow-up instrument to distinguish bright PISN events from core-collapse events. Since MIRI is much less sensitive, the brighter He core models are more readily observable, while CCSNe cannot be seen at all in this integration time.}
\begin{tabular}{ | l | l | l | l |}
\hline
IMF model & He Core & RSG & CC \\ \hline
Pop III Salpeter & 0.05 & 0.04 & 0 \\
Pop III Flat & 0.15 & 0.12 & 0 \\
Pop II & 0 & 0 & 0 \\ \hline
\end{tabular}
\label{SnapshotRateMIRI}
\end{table}
PISNe and CCSNe occur for stars with main-sequence masses between $\sim$140-260$M_{\odot}$ \citep{Heger2002} and 8-25$M_{\odot}$ \citep{Smartt2009}, respectively. The fate of stars between 25-140$M_{\odot}$ is uncertain; due to fallback, progenitors more massive than $\sim 40M_{\odot}$ may form black holes directly with no SN explosion \citep{Fryer1999}. Notably, stars in the mass range 95-130$M_{\odot}$ may reach the pair production instability in the core, but the thermonuclear explosion is insufficient to unbind the star \citep{Woosley2007}, and the star undergoes pulsations of matter ejecta which may produce a very bright light curve when the shells of ejected matter collide with each other, before the star dies as a normal CCSN. The resulting pulsation pair-instability supernova can be ultra-luminous and are presumably detectable by JWST. However, we do not consider such events here.
Using a progenitor mass range of 8-25$M_{\odot}$ to calculate the CCSN rates is likely an underestimate; for detailed predictions on the number of CCSNe detectable by JWST at the redshifts of reionization, see \citet{Mesinger2006}, who also take into account the variation in peak magnitude of observed CCSNe and the effects of dust extinction. For a fixed progenitor mass range, we calculate the SN rate per comoving volume $R(z)$ and find the snapshot rate shown in Tables \ref{SnapshotRateNIRCam}, \ref{SnapshotRateMIRI} and Figures \ref{SNRates}, \ref{snapshot_rate_with_z:globfig}. In Figure \ref{snapshot_rate_with_z:globfig}, we use each red supergiant model as a proxy for the light curves of all progenitors similar in mass, e.g. R250 represents all progenitors in mass range 226-260$M_{\odot}$.
If both Pop III and Pop II stars contributed to reionization, the actual IMF will be a mixture of the Pop III and Pop II IMF used above. By counting the number of each type of SN found in JWST snapshots, the IMF of these early stellar populations can be constrained, and the relative contribution of Pop III and Pop II stars toward reionizing the universe can be inferred. To reduce selection effects due to the different intrinsic luminosity of the SN, a sufficiently deep exposure is needed, to enable observations of both types of SNe at the peak of their light curves should they exist at the target redshift.
For the integration time assumed in Figure \ref{snapshot_rate_with_z:globfig}, one can directly characterize the ratio of PISNe to CCSNe before $z\sim 8$, and set existence limits on top heavy Pop III stars up to $z\sim 10$ with a survey of $\sim 10$ JWST fields.
\subsection{Probing Intermediate Mass Stars with Type Ia SNe}
To probe the intermediate mass range ($\sim 1-8 M_{\odot}$) of the IMF during reionization, one may use Type Ia SN rates. Type Ia supernovae (SNe Ia) are thought to occur when a white dwarf nears the Chandrasekhar mass, resulting in a thermonuclear explosion. This requires the white dwarf to accrete mass from a binary companion. Although the physics behind SNe Ia have been widely studied using both observations and theoretical simulations, there is still no consensus on the mechanisms that proceed the supernova. The single degenerate model proposes the companion to be a main sequence or giant star, which donates mass via Roche lobe overflow, whereas the double degenerate scenario considers the merger of two white dwarf stars; the latter may be necessary for at least some observed Type Ia SNe \citep{Bloom2012, Schaefer2012}. Either way, after stellar birth it takes the main sequence lifetime of the progenitor star plus an additional delay time for the Type Ia SN to proceed.
Hence, the rate of SNe Ia is empirically parametrized to follow the star formation rate (SFR), but shifted toward lower redshift after taking the delay time into account. The SN rate at a redshift $z$ or cosmic time $t$, $R(z)=R(t)$, is given by a convolution of the SFR over delay times,
\begin{equation}
R(t) = \int_{0}^{t} SFR(t-\tau) {\rm DTD}(\tau) d\tau,
\end{equation}
where ${\rm DTD}(\tau)$ is the delay time distribution (SNe per unit time per unit stellar mass formed), in which $\tau$ is the time elapsed between the formation of the progenitor star and the explosion of the SN Ia. Note that since the ${\rm DTD}(\tau)$ is normalized to the total stellar mass formed, it only indirectly reflects the physical efficiency of SNe Ia from their actual progenitors of $3-8 M_{\odot}$ stars \citep{Nomoto1994}.
In previous reionization literature \citep{Haiman2009}, Type Ia SNe were expected to be extremely rare at high redshifts ($z> 6$), as the delay between the formation of the progenitor and the SN event was thought to be longer than the age of the Universe at these redshifts. However, this view should be reconsidered in light of recent converging evidence for a prompt population of SNe Ia, see recent work by \citet{Maoz2010b}, \citet{Graur2011}, and references within.
\citet{Scannapieco2005} calibrated the prompt rate via the `B' parameter, a constant of proportionality between the SFR and the prompt SN Ia rate, equivalent to the number of prompt SNe per unit stellar mass formed. The delayed component is characterized via the parameter `A' which is the constant of proportionality between galaxy mass and the delayed SN Ia rate. We ignore the A component as this delay exceeds the age of the universe during reionization. The value of B is calibrated at low redshifts, for example $B=2.7-11\times 10^{-3} M_{\odot}^{-1}$ in \citet{Maoz2010}, for prompt delay times $T \in (35,330)$ Myr; here we adopt $B=3\times 10^{-3} M_{\odot}^{-1}$. From B, we set a uniform ${\rm DTD}(\tau)=B/\Delta T$. Using the Pop II SFR and this ${\rm DTD}(\tau)$, we calculate the event rate $R(t)$ of Type Ia SNe shown in Figures \ref{SNRates} and \ref{TypeIa_rate}. Since the validity of these estimates depend on ${\rm DTD}(\tau)$, we assume that the astrophysics involved in shaping the forming efficiency and delay time of Type Ia SNe is not very sensitive to the cosmological epoch.
\begin{figure}
\begin{center}
\includegraphics[width=1\columnwidth]{IaRatePlot.eps}
\caption{Type Ia SNe rate in NIRCam's field of view, per year of observation per $dz$. For the Pop II IMF model for reionization, $\sim$1 new SNe Ia will occur every year per unit redshift at $z\sim 6-7$ in NIRCam's field of view. The 20 days around the SN Ia peak flux at that redshift is equivalent to half a year in observer frame. Every NIRCam snapshot of the sky, with sufficient integration time (e.g. $3\times 10^4$ s to get absolute magnitude above -18 at $z=8$ with signal-to-noise ratio of 5), will have a $\sim 50$\% probability of finding a Type Ia SN near peak flux.}
\label{TypeIa_rate}
\end{center}
\end{figure}
As seen in Figure \ref{TypeIa_rate}, in the scenario where Pop II stars dominated reionization, we expect $\sim$1 new SNe Ia every year per unit redshift at $z\sim 6-7$ in NIRCam's field of view of $\approx 10$ square arcseconds. As the AB magnitude of SNe Ia during peak \citep{Hillebrandt2000} is $M_{B} \sim M_{V} \sim -19.30$ with a dispersion of 0.3, several hours integration time on JWST will be sufficient to catch a Type Ia SN near peak flux at these redshifts.
Regardless of which population of stars dominated reionization, Type Ia SNe offers a way to probe the intermediate mass range of the reionization IMF. In addition, the $z\gtrsim 6$ sky offers an unambiguous way of isolating a prompt population of SNe Ia, as the universe was not sufficiently old in the epoch of reionization for the delayed component of Type Ia SNe to contribute any events. Therefore, Type Ia SNe at $z>6$ could be used to test whether the delay times are indeed connected to SN Ia formation mechanisms and properties. Finally, the existence of standard candles during the epoch of reionization could be useful for cosmological measurements. In the Appendix, we discuss that probing reionization history with Type Ia SNe by constraining the global ionization fraction using Thomson optical depth measurements requires an unrealistic survey time for JWST.
\subsection{Typing the Supernovae}
At lower redshifts, SNe are usually typed by spectral lines. Furthermore, the smoking gun evidence for a PISN is the measurement of a large core ($>50M_{\odot}$) composed of helium or other heavier elements. In the case of SN 2007bi, \citet{Gal-Yam2009} analyzed the nebular spectrum 16 months after peak light to infer $\sim 4M_{\odot}$ of $^{56}$Ni, implying a large core mass $>50M_{\odot}$ as in a pair-instability explosion (but see \citet{Moriya2010} for a CCSN model for SN 2007bi that ejects $6.1 M_{\odot}$ of $^{56}$Ni). However, spectroscopic typing of high redshift SNe seen by JWST may be unrealistic; for example, at around 3 $\mu$m on NIRSpec, achieving a signal-to-noise ratio of 5 at redshift $\sim 8$ would require one full day of integration time. Alternatively, in anticipation of Pan-STARRS\footnote{http://pan-starrs.ifa.hawaii.edu/public} and LSST\footnote{http://www.lsst.org/lsst/} increasing the number of photometrically detected SNe to a few hundred thousand in the next two decades, much work has been done in the photometric identification and classification of SNe \citep{Kessler2010}. As photometric classification of SNe matures, it could complement or replace the spectroscopic typing of high-z SNe, reducing the required JWST time.
The light curves of the more massive PISNe are very luminous ($10^{43}$-$10^{44}$ erg s$^{-1}$) and long-lasting ($\sim 300$ days), characteristics that do not exist for most other types of SNe. As long as the SN redshift is known, multi-epoch observations can determine its rest-frame luminosity and duration, and identify the more massive PISNe explosions. Aside from the most energetic events, typing PISNe using their magnitude and color will be difficult. Despite their enormous kinetic energies of $\sim 10^{53}$ ergs, the peak optical luminosities of PISNe are similar to those of other SNe, even falling below the Ia and II curves for smaller mass progenitors. The majority of PISNe will actually be these dimmer events. Also, since PISNe spend most of their lives in the same temperature range as other SNe, their colors are also similar.
An extended light curve, rather than an extreme luminosity or unusual color, may therefore be the most important signature of PISNe. In particular, the distinguishing feature of PISNe is its exceptionally long rise time, $\gg 100$ days in the rest frame. Also, the detection of a slow decline rate that follows the decay rate ($\sim$0.01 mag/day) of $^{56}$Co, the product of $^{56}$Ni decay, would provide strong evidence that the SN synthesized significant amounts of $^{56}$Ni. At $z=8$, even with time dilation, this results in $\sim$0.4 mag variation per year, which should be within the sensitivity of a multi-year JWST survey; the decline in the bluer bands are 2-3 times larger, due to the onset of iron group line blanketing. The detection of a secondary maximum in the light curve also supports the synthesis of $^{56}$Ni. However, the lack of a secondary peak does not rule out a large presence of $^{56}$Ni, as strong radial mixing could smear out the two bumps \citep{Kasen2006a}.
\subsection{Survey Strategies}
The long duration of high redshift PISN light curves, prolonged by cosmological time dilation, poses a great challenge for detecting them as transients. At z$\sim$7 the light curve of a PISN can last for over 1000 days in the observer's frame. Without spectroscopic measurements, the telltale sign of a massive progenitor PISN is an incredibly long plateau in its light curve. Therefore, instead of a threshold experiment, we suggest a search strategy that involves taking a series of `snapshots' of a field, each snapshot separated by $\sim 1$ year, and searching for variations in the flux of objects in successive images. Since Pop III star formation occurs in the smallest galaxies, blank-field surveys should be the sufficient for searching for PISNe.
The total number of SNe detected in a survey of total integration time $t_{surv}$ is
\begin{equation}
N_{surv} = \frac{1}{2}\frac{t_{surv}}{t_{exp}} \frac{\Delta \Omega_{FOV}}{4\pi} N_{exp},
\end{equation}
where $\Delta \Omega_{FOV}$ is the instrument's field of view, $t_{surv}/t_{exp}$ is the number of fields which
can be tiled within the survey time $t_{surv}$, and $N_{exp}$ is the snapshot rate from equation (\ref{SnapshotRateEquation}), i.e. the number of SNe bright enough to be detected in an exposure of duration $t_{exp}$ \citep{Haiman2009}. The factor of $\frac{1}{2}$ is included to account for observations in $4$ color bands (2 pairs of filters, as NIRCam observes in two bands simultaneously using a dichroic) for determining photometric redshift and typing of the SNe. To detect SNe by their variability, each field requires repeated observations, and therefore any survey should piggyback on fields that have already been observed.
In the case where several fields are already available from other JWST surveys, the snapshot rates given by Table \ref{SnapshotRateNIRCam} suggest that a dedicated, long program may not be required to detect dozens of high redshift SNe. Due to the order-of-magnitude difference in the snapshot rate of PISNe vs CCSNe for the different IMF models, more than $10$ fields with followup repeated imaging should already help constrain the stellar population responsible for reionization. Cosmic variance will affect the total number of SNe for small number of fields, but the ratio of PISNe to CCSNe would still be indicative of the IMF. For example, if Pop III (Flat IMF model) and Pop II stars had equal contribution to reionization (which means Pop II stars dominate Pop III stars by roughly 20-to-1 in total mass), one could use 20 images conducted for other programs as references and only revisit the same image twice for a total of 3 snapshots per field of view. Observing in 4 bands, for a total of 28 days integration time over 2 years, such a survey expects to see $\sim$10 red supergiant PISNe and $\sim$300 CCSNe.
To the extent that PISN spectra can be represented as a distribution of blackbodies at different temperatures, since the temperature and redshift would be degenerate, it will be impossible to acquire photometric redshifts without further information about the SN epoch. However, our simulated spectra show significant deviations from a blackbody in the UV ($l < 3500$\AA) due to metal-line blanketing in the SN photosphere, providing spectral and photometric signatures that could be used as redshift indicators, depending on their strength.
\subsection{Luminosity Function}
Although the UV flux of PISNe is relatively short lived, the more massive PISNe stay bright in its rest frame visible band for over a year. Given this brightness and long intrinsic duration, coupled with the $(1+z)$ time dilation at high redshifts, it is conceivable that PISNe could contribute to the luminosity function of all objects at high redshifts when galaxies were dim. Figure \ref{luminosity_function} illustrates the luminosity function of PISNe at $\sim$4000\AA, calculated using the helium core progenitor models for PISN luminosity, and the Pop III Flat or Pop III Salpeter models for the star formation rate. Shown for comparison are the projected galaxy luminosity functions at high redshifts, using the \citet{Bouwens2011b} best fit Schechter parameterization for the UV luminosity function, and shifting to the visible band using $U-V \approx 0.4$, $0.3$ for $z=7,$ $8$ respectively, measured using the Spitzer Infrared Array Camera \citep{Labb'e2010, Labb'e2010a}. Applying this U-V shift is a crude approximation, as luminous and faint galaxies have different rest frame UV-to-optical color; however, we are most interested in the bright end of the luminosity function, where this current U-V measurement is applicable.
The luminosity function for PISNe implied by our Pop III IMF models overlaps with the galaxy luminosity function at the brightest magnitudes. If a top-heavy Pop III IMF was solely responsible for reionization, PISNe will contaminate the brightest end of the galaxy luminosity function, unless great care is taken to remove these supernovae. Since the volumetric count of the brightest galaxies and PISNe is very low, it will take a wide infrared survey to observe this effect.
\begin{figure}
\centering
\includegraphics[width=1\columnwidth]{LuminosityFunction.eps} \\
\caption{Luminosity functions of PISNe at 4000\AA for $z=7$ and $8$, for the Pop III Flat IMF or Salpeter IMF star formation rate models, compared with the galaxy luminosity function. For the Pop III Flat model, the volumetric count of PISNe exceeds galaxies past $M_{AB} \sim -21.5$; the dominance of PISNe should become greater at higher redshifts, as galaxies decrease in luminosity while PISNe stay the same. The PISN luminosity functions at $z=7,$ $8$ overlaps coincidentally because the increase in time dilation compensates for the decrease in PISN event rate.}
\label{luminosity_function}
\end{figure}
\section{Discussion}
In our discussion we ignored complicating factors such as metallicity and rotation, and calculated the PISN and CCSN event rate using only the SN progenitor mass range along with the star formation rate. However, at low redshifts $z<1$, the measured CCSN rate is a factor of $\sim 2$ smaller than that predicted by the analogous calculation using the measured cosmic star formation rate. The discrepancy is likely due to many intrinsically low-luminosity or obscured SNe being missed in surveys \citep{Horiuchi2011}. As this discrepancy is lower than the uncertainty in our SFR model parameters, and we already account for lower intrinsic luminosities for the lower progenitor mass PISNe, we do not take obscuration into account for our predictions of the SN rate as seen by JWST.
The IMF of early stellar populations responsible for reionization should also leave an imprint on the metal enrichment pattern via their SN products. So far, the abundance patterns observed to date in extremely metal-deficient stars in the Galactic halo \citep{Beers2005} are more consistent with an IMF that produced much more CCSNe instead of PISNe \citep{Joggerst2010}. However, in previous surveys, subtle selection effects might have disfavored finding PISN-enriched stars; the metal yields of PISNe are so high that the metal abundances of stars formed out of PISN ejecta \citep{Greif2008} are already higher than the metallicity range targeted by metal-deficient star surveys \citep{Karlsson2008}.
Large carbon enhancements observed in metal-poor stars, when interpreted as the outcome of pollution by winds from binary companions that have gone through the AGB phase, suggest the existence of a large number of intermediate-mass stars ($\sim 1-8 M_{\odot}$) at high redshifts \citep{Tumlinson2007,Tumlinson2007a}. Alternatively, nucleosynthesis in faint CCSNe from higher mass stars could also explain the observed carbon enhancement in metal-poor stars \citep{Iwamoto2005}. Observing the Type Ia SN rate during the epoch of reionization will be an complementary way to test these models, and constrain the number of intermediate-mass stars at high redshifts.
The predicted initial mass range of $\sim$140 to 260 $M_{\odot}$ for PISN progenitors assumed the stars to be non-rotating \citep{Heger2002}. However, observations find that at very low metallicities, stars rotate faster \citep{Martayan2007}. The fast rotation of the first stars is supported by the latest hydrodynamic simulations of their formation \citep{Stacy2011}, and also by observations of anomalously high abundances of Ba and La with respect to Fe in ancient low-mass stars \citep{Chiappini2011}, which could originate in metal-poor fast-rotating massive stars. Generally, rotation should increase the required PISN progenitor mass by increasing mass loss. \citet{Meynet2006} found that, contrary to the usual $\dot{M}\propto Z^{0.5}$ scaling relation, rotating stars at very low metallicity $Z \sim 10^{-5}$ to $10^{-8} Z_{\odot}$ show a large mass loss, up to $\sim 50\%$, mainly resulting from efficient mixing of stellar nucleosynthesis products into the stellar surface. However, \citet{Ekstrom2008} found that for strictly $Z=0$ stars, the mass loss is very low, even for models that reach critical velocity in the main sequence. These results imply that, for rapidly rotating Pop III stars to die as a PISN, the required progenitor mass is extremely sensitive to whether the star is truly metal-free or not.
At much lower redshifts, PISNe have likely already been observed, most persuasively in the case of the very luminous and long duration event SN 2007bi \citep{Gal-Yam2009}. Other more recent candidates include PTF 10nmn (Gal-Yam in preparation; Yaron et al. in preparation) and PS1-11ap (Rubina Kotak et al. in preparation). As pristine gas was recently observed at redshifts after reionization \citep{Fumagalli2011}, it is possible that some low redshift PISNe have Pop III progenitors born out of surviving pockets of metal-free gas; the rates of PISNe in this scenario was considered by \citet{Scannapieco2005a}. However, the metallicities of the host galaxies of SN 2007bi and PTF 10nmn are well above the metallicity threshold required to form Pop III stars \citep{Young2010}. Therefore, it is plausible that PISNe can have very massive Pop II/I progenitors as well, perhaps born via the merger of stars in collision runaways in young, dense star clusters \citep{Pan2011}.
\section{Conclusions}
We analyzed simulated light curves and spectra of pair-instability supernovae for a variety of progenitor masses and envelope types, and found that the supernovae from the more massive progenitors are super-luminous and have extended light curves, traits that would help photometrically distinguish pair-instability supernovae from other types of supernovae using repeated snapshots. We calculated the rates and detectability of pair-instability, core collapse, and Type Ia supernovae during the redshifts of reionization, and showed that it is possible to constrain the initial mass function of stars at that time, and identify the stellar population responsible for reionization. If Pop III stars made the dominant contribution of ionizing photons during reionization, the bright end of the galaxy luminosity function will be contaminated by pair-instability supernovae.
\bigskip
\section*{Acknowledgments.}
We thank Bob Kirshner, Kaisey Mandel, and Jonathan Pritchard for helpful discussions. TP was supported by the Hertz Foundation. This work was supported in part by NSF grant AST-0907890 and NASA grants NNX08AL43G and NNA09DB30A. This work is supported by the Director, Office of Energy
Research, Office of High Energy and Nuclear Physics, Divisions of Nuclear Physics, of the U.S. Department of Energy under Contract No.DE-AC02-05CH11231. This research has been supported by the DOE SciDAC Program (DE-FC02-06ER41438). We are grateful for computer time provided by ORNL through an INCITE award and by NERSC.
\bibliographystyle{mn2e}
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\section*{Conflict of interest}
\end{document}
\section{Equilibration algorithm for higher order}
\renewcommand{\thesection}{\Alph{section}}
\label{app:equilibration}
\FloatBarrier
\newcommand{a}{a}
\newcommand{{q}}{{q}}
\newcommand{\tilde\sigma_z^\Delta}{\tilde\sigma_z^\Delta}
\newcommand{\sigma_z^\Delta}{\sigma_z^\Delta}
\newcommand{\sigma_z}{\sigma_z}
\newcommand{RT^{0}_k}{RT^{0}_k}
\newcommand{RT_{\kp}^{{\rm pw}, 0}}{RT_{{q}}^{{\rm pw}, 0}}
\newcommand{RT_{\kp}^{{\rm pw}}}{RT_{{q}}^{{\rm pw}}}
\renewcommand{\Pi_{\RTzk}}{\Pi_{RT}}
\newcommand{\Lambda_T}{\Lambda_T}
\newcommand{\mathrm{dx}}{\mathrm{dx}}
\newcommand{\mathrm{ds}}{\mathrm{ds}}
\newcommand{\lambda_{T, \alpha}^{\ell, m}}{\lambda_{T, \alpha}^{\ell, m}}
\newcommand{\lambda_{T, \ddiv}^{\ell, m}}{\lambda_{T, \ddiv}^{\ell, m}}
\newcommand{\lambda_{T, E}^j}{\lambda_{T, E}^j}
\newcommand{\lambda_T^{r}}{\lambda_T^{r}}
\newcommand{\widetilde{\dofBasis}}{\widetilde{\Lambda_T}}
\newcommand{{\mathcal{B}}_{RT}}{{\mathcal{B}}_{RT}}
\newcommand{{\mathcal{B}}_{RT,T}}{{\mathcal{B}}_{RT,T}}
\newcommand{c_{T, E}^j}{c_{T, E}^j}
\newcommand{c_{T, \ddiv}^{\ell, m}}{c_{T, \ddiv}^{\ell, m}}
\newcommand{c_{T_\inda, \ddiv}^{\ell, m}}{c_{T_a, \ddiv}^{\ell, m}}
\newcommand{c_{T}^r}{c_{T}^r}
\newcommand{\varphi_{T, E}^j}{\varphi_{T, E}^j}
\newcommand{\varphi_{T, \ddiv}^{\ell, m}}{\varphi_{T, \ddiv}^{\ell, m}}
\newcommand{\varphi_{T}^r}{\varphi_{T}^r}
\newcommand{\mathcal S(z)}{\mathcal S(z)}
\newcommand{V(z)}{V(z)}
\newcommand{\sigma_z^\Delta}{\sigma_z^\Delta}
The post-processed quantity $Q_p\in RT_{k+p}(\mathcal{T})$ from Subsection \ref{sub:Computation_Q} enters the equilibrated error estimator $\eta_{{\rm eq}, p}(\mathcal{T})$ in Theorem \ref{thm:GUB-HHO} and could be computed by a minimization problem on the vertex patches.
The solution property \eqref{eqn:HHO_solution_property} gives rise to the two cases $r = 0$ if $k = 0$ and $r = k+p$ if $k \geq 1$ for the polynomial degree $r$ in the equilibrium $\ddiv Q_p + \Pi_{r} f = 0$ in $\Omega$ from \eqref{eqn:div_Q_f}.
This appendix follows \cite{verfurth_note_2009,cai_robust_2012,bertrand_weakly_2019,braess_finite_2007} to compute the quantity of interest
$Q_p - \nabla_\mathrlap{\pd}{w} R u_h$ directly in an efficient two-step procedure in 2D.
Throughout this appendix, fix $k,p\in \mathbb N_0$ and abbreviate ${q}\coloneqq k+p$ and $G_h \coloneqq \nabla_\mathrlap{\pd}{w} R u_h\in P_{k}(\mathcal{T};\mathbb R^2)$.
Let the data $f\in L^2(\Omega)$ be given and assume, for the sake of brevity, that $f \in P_0(\mathcal{T})$ if $k = 0$.
\subsection{Overview}
Recall the definition \eqref{eq:discrete_minimisation} of the summand $\QQ_{z,h}$ in $Q_p \coloneqq\sum_{z\in \mathcal V}\QQ_{z,h}$ from Subsection \ref{sub:Computation_Q}
with the piecewise Raviart-Thomas interpolation $\mathcal I_{\rm RT}:H^1(\mathcal{T};\mathbb R^2)\to RT_{\kp}^{{\rm pw}}(\mathcal{T})$ \cite[Section III.3.1]{boffi_mixed_2013}.
The focus is on one vertex $z\in\mathcal V$ with vertex-patch $\omega(z)$ and its triangulation $\mathcal{T}(z)=\{T\in\mathcal{T}\ |\ z\in T\}$.
Consider set of edges $\mathcal{F}$ and the facet-spider $\mathcal{F}(z) =\{E\in\mathcal{F}\ |\ z\in E\}$ as in Figure \ref{fig:NodePatch}. The nodal basis function $\varphi_z\in S^1(\mathcal{T}(z))$ gives rise to the discrete spaces
\begin{align*}
&RT_{\kp}^{{\rm pw}, 0}(\mathcal{T}(z))\coloneqq\{\sigma_z\in RT_{{q}}^{\text{pw}}(\mathcal{T}(z))\ :\ \sigma_{z}\cdot \nu_E=0 \text{ for all }E\in\mathcal{F}\setminus\mathcal{F}(z)\},\\
&\mathcal S(z)\coloneqq\left\{\sigma_z \in RT_{\kp}^{{\rm pw}, 0}(z)\ :\ \begin{array}{llr}
\ddiv\sigma_z &=-\Pi_{T,{{q}}}(\varphi_z(f+\ddiv G_h))&\text{ for all } T\in\mathcal{T}(z)\\
{[}\sigma_z\cdot \nu_E{]_E} &= -\Pi_{E, {{q}}}(\varphi_z[G_h \cdot \nu_E]_E)&\text{ for all } E\in\mathcal{F}(\omega(z))
\end{array}\right\}.
\end{align*}
\begin{proposition}[alternative minimization]\label{prop:RzhD_def}
It holds
\begin{align}\label{eqn:RzhD_def}
\Rzh^\Delta\coloneqq\QQ_{z,h} - \mathcal I_{\rm RT}(\varphi_z G_h) = \argmin\limits_{%
\sigma_z \in\mathcal S(z)
}
\| \sigma_z\|_{\Lz}.
\end{align}
\end{proposition}
\begin{proof}
Recall $f_z = \Pi_{\tilde p}(\varphi_z f - G_h \cdot \nabla \varphi_z)$ from \eqref{def:f-z}. (Notice that this formula coincide with the definition \eqref{def:f-z} for $k = 0$ because $f \in P_0(\mathcal{T})$.) Given any $\sigma_z \in \mathcal{S}(z)$, the commuting diagram property ${\ddiv}_{\rm pw}\circ \mathcal I_{\rm RT} = \Pi_{{q}}\circ{\ddiv}_{\rm pw}$ \cite[Proposition 2.5.2]{boffi_mixed_2013} shows
\begin{align}\label{eq:divergence-S(z)-IRT}
\mathrm{div}_\mathrlap{\pd}{w} (\sigma_z + \mathcal{I}_{RT} (\varphi_z G_h)) = \mathrm{div}_\mathrlap{\pd}{w} \sigma_z + \Pi_{\tilde p}\,\mathrm{div}_\mathrlap{\pd}{w} (\varphi_z G_h) = - f_z.
\end{align}
By design of the interpolation $\mathcal{I}_{RT}$, $(\mathcal{I}_{RT} (\varphi_z G_h)_{|T}\cdot\nu_E)_{|E} = \Pi_{E,{q}}(\varphi_z G_h)_{|T}\cdot\nu_E$ holds and so
$[\mathcal I_{\rm RT} (\varphi_z G_h)\cdot \nu_E]_E = \Pi_{E, {{q}}}(\varphi_z[G_h \cdot \nu_E]_E)$ follows for any $E \in \mathcal{F}(T)$ and $T\in\mathcal{T}$.
Therefore, the jump $[\sigma_z + \mathcal I_{\rm RT}(\varphi_z G_h)]_E\cdot\nu_E \equiv 0$ vanishes on $E \in \mathcal{F}(\omega(z))$, whence $\sigma_z + \mathcal I_{\rm RT}(\varphi_z G_h) \in RT_{\tilde{p}}(\mathcal{T}(z))$.
Since $RT_{\tilde{p}}^{\mathrlap{\pd}{w},0}(\mathcal{T}(z)) \cap H(\mathrm{div},\omega(z)) = RT_{\tilde{p}}^0(\mathcal{T}(z))$, this and \eqref{eq:divergence-S(z)-IRT} imply $\sigma_z + \mathcal I_{\rm RT}(\varphi_z G_h) \in \mathcal{Q}_h(z)$ for any $\sigma_z \in \mathcal{S}(z)$.
In particular, $\mathcal S(z) + \mathcal I_{\rm RT}(\varphi_z G_h) \subseteq \mathcal{Q}_h(z)$.
On the other hand, similar arguments verify the reverse inclusion $\mathcal{Q}_h(z) \subseteq \mathcal S(z) + \mathcal I_{\rm RT}(\varphi_z G_h)$. %
The substitution $\mathcal S(z) + \mathcal I_{\rm RT}(\varphi_z G_h) = \mathcal{Q}_h(z)$ in \eqref{eq:discrete_minimisation} concludes the proof.\qed
\end{proof}
This establishes that the norm $\|Q^\Delta_p\|$ of $Q^\Delta_p\coloneqq\sum_{z\in \mathcal V}\Rzh^\Delta = Q_p - G_h$ contributes to the equilibrated error estimator and the remaining parts of this
appendix compute the minimizer $\Rzh^\Delta$ of \eqref{eqn:RzhD_def} in a two-step procedure.
\begin{figure}
\centering
\scalebox{.6}{\includegraphics{./Figures/NodePatch_out.pdf}}
\scalebox{.6}{\includegraphics{./Figures/NodePatchBoundary_out.pdf}}
\caption{Triangulation $\mathcal{T}(z)$ and enumeration of the edges $\mathcal{F}(z)$ of the vertex-patch $\omega(z)$ for an internal vertex $z\in\mathcal V(\Omega)$ with $N=|\mathcal{T}(z)|=5$ (left) or boundary vertex $z\in\mathcal V(\partial \Omega)$ with $N=|\mathcal{T}(z)|=4$ (right).}
\label{fig:NodePatch}
\end{figure}
First, Algorithm \ref{alg:eq} generates the coefficients of a particular solution $\tilde\sigma_z^\Delta\in \mathcal S(z)$ in terms of the finite element basis ${\mathcal{B}}_{RT}$ of $RT_{\kp}^{{\rm pw}}(\mathcal{T}(z))$ from Subsection \ref{sub:basisRT}.
The second step computes the correction \begin{align}\label{eqn:linear_min}
\sigma_z^\Delta\coloneqq\Rzh^\Delta - \tilde\sigma_z^\Delta= \argmin_{\sigma_z\in V(z)}\|\sigma_z +\tilde\sigma_z^\Delta \|_{L^2(\omega(z))}
\end{align}
in terms of the low-dimensional unconstrained
minimization problem
over the linear space $V(z)\coloneqq\mathcal S(z)-\tilde\sigma_z^\Delta=RT_{\kp}^{{\rm pw}}(\mathcal{T}(z))\cap H(\ddiv=0, \omega(z))$ characterized in Lemma \ref{lem:appendixB}.
Because \cite[Lemma 3.1]{cai_robust_2012} is wrong (take, e.g., $\tau_K=\mathrm{curl}\,b_K\ne 0$ for the element bubble function $b_K$ in their notation to see that uniqueness for general polynomial degrees ${q}$ cannot hold) and \cite{cai_robust_2012} omits algorithmic details, this appendix focuses on the explicit characterization of the degrees of freedom for the minimization problem \eqref{eqn:linear_min} over $V(z)$.
\subsection{Degrees of freedom for $RT_{\kp}^{{\rm pw}}(T)$}
\label{sub:basisRT}
This subsection introduces a basis for the Raviart-Thomas finite element on $T\in\mathcal{T}(z)$ and starts with the definition of some linear functionals on $H(\ddiv, \mathcal{T})$. For any $\sigma\in H(\ddiv, \mathcal{T})$, set
\begin{align*}
\lambda_{T, \alpha}^{\ell, m}(\sigma) \coloneqq& \int_T \sigma\cdot e_\alpha\ x_1^\ell x_2^m \mathrm{dx}&& 0\leq \ell+m\leq {q}-1,\alpha=1, 2,\\
\lambda_{T, \ddiv}^{\ell, m}(\sigma) \coloneqq& \int_T \ddiv \sigma \ x_1^\ell x_2^m \mathrm{dx}&& 0\leq \ell+m\leq {q},\\
\lambda_{T, E}^j(\sigma) \coloneqq& \int_E \sigma_{|T} \cdot \nu_T\ s^j \mathrm{ds}&& 0\leq j\leq {q}, E\in\mathcal{F}(T).
\end{align*}
Here and throughout, $e_1=(1,0)$ and $e_2=(0,1)$ denote the canonical unit vectors in $\mathbb R^2$.
Note that the (classical) degrees of freedom for the Raviart-Thomas finite element $RT_{q}(T)$ of degree ${q}\in\mathbb N_0$ from \cite{brezzi_mixed_1991} read
$$\widetilde{\dofBasis}\coloneqq\{\lambda_{T, E}^j, \lambda_{T, \alpha}^{\ell, m}\ :\ \text{for }0\leq j\leq {q}, 0\leq \ell+m\leq {q}-1, \alpha=1,2, E\in\mathcal{F}(T)\}.$$
This appendix requires, for the construction of $\tilde\sigma_z^\Delta\in\mathcal S(z)$, a different set of (unisolvent) degrees of freedom $\Lambda_T$ for $RT_{q}(T)$ that includes the edge and divergence moments
\begin{align}\label{eqn:dof_basis}
\Lambda_T^0\coloneqq \{\lambda_{T, E}^j, \lambda_{T, \ddiv}^{\ell, m}\ :\ \text{for }0\leq j\leq {q}, 1\leq \ell+m\leq {q}, E\in\mathcal{F}(T)\}\subseteq \Lambda_T.
\end{align}
(The set $\Lambda_T^0$ itself is linear independent \cite[Lemma 3.1]{verfurth_note_2009}.)
Given any $\Lambda_T$ with \eqref{eqn:dof_basis}, denote the remaining $N_{q}={q}({q}-1)/2$ degrees of freedom $\Lambda_T\setminus\Lambda_T^0$ by $\lambda_T^{r}$ for $r=1,\dots,N_{q}$.
Let ${\mathcal{B}}_{RT,T}=\{\varphi_{T, E}^j, \varphi_{T, \ddiv}^{\ell, m}, \varphi_{T}^r\}$ be the unique basis of $RT_{q}(T)$ dual to $\Lambda_T$ with inferred indices from $\Lambda_T$.
Then, the collection ${\mathcal{B}}_{RT}\coloneqq \bigcup_{T\in\mathcal{T}(z)} {\mathcal{B}}_{RT,T}$ is a basis of $RT_{\kp}^{{\rm pw}}(\mathcal{T}(z))$ and any function $\sigma_z\in RT_{\kp}^{{\rm pw}}(\mathcal{T}(z))$ has the representation
\begin{align}\label{eqn:basis_RTkpwz}
\sigma_z \coloneqq \sum_{T\in\mathcal{T}(z)}\left(\sum_{E\in\mathcal{F}(T)}\sum_{j=0}^{q} c_{T, E}^j\varphi_{T, E}^j + \sum_{1\leq \ell+m\leq {q}}c_{T, \ddiv}^{\ell, m}\varphi_{T, \ddiv}^{\ell, m} + \sum_{r=1}^{N_{q}}c_{T}^r\varphi_{T}^r\right)
\end{align}
with coefficients $c_{T, E}^j = \lambda_{T, E}^j(\sigma_z), c_{T, \ddiv}^{\ell, m} = \lambda_{T, \ddiv}^{\ell, m}(\sigma_z)$, and $c_{T}^r = \lambda_T^{r}(\sigma_z)$ for all $T\in\mathcal{T}(z), E\in\mathcal{F}(T)$, and $0\leq j\leq {q}, 1\leq \ell+m\leq {q}, 1\leq r\leq N_{q}$.
By duality, the coefficients $c_{T, E}^j$ with $0\leq j\leq {q}$ uniquely determine the normal trace $(\sigma_{z|T})_{|E}\cdot \nu_E\in P_{q}(E)$ on the edge $E\in\mathcal{F}(T)$ of $T\in\mathcal{T}(z)$.
Any set of degrees of freedom $\Lambda_T$ with \eqref{eqn:dof_basis} works with the equilibration algorithm in \ref{sub:app:alg}.
\newline
\begin{example}[Construction of $\Lambda_T$]
This example presents a generic procedure to obtain such a set from $\widetilde{\dofBasis}$.
The integration by parts formula shows that the lowest-order divergence moment $\lambda_{T, \div}^{0,0} = \sum_{E\in\mathcal{F}(T)} \lambda_{T, E}^0$ depends linearly on the lowest-order edge moments and, similarly, the sums
\begin{align}
\label{eqn:ibp}
\lambda_{T, \ddiv}^{\ell, m} + \ell\lambda_{T, 1}^{\ell-1, m} + m\lambda_{T, 2}^{\ell, m-1}\in (P_{\ell + m}(\mathcal{F}(T)))^*
\end{align}
are functionals on $P_{\ell + m}(\mathcal{F}(T))$ (summands with negative indices are understood as zero).
This relation allows for the substitution of volume moments in $\widetilde{\dofBasis}$ for divergence moments $\lambda_{T, \ddiv}^{\ell, m}$, $1\leq \ell+m\leq {q}$, and leads to $\Lambda_T$ with \eqref{eqn:dof_basis}.
The remaining degrees of freedom $\Lambda_T\setminus\Lambda_T^0$ are volume moments of the form $\lambda_T^{r}=\lambda_{T, \alpha}^{\ell, m}$ for a fixed $\alpha \in \{1,2\}$, e.g.,
$$\Lambda_T \setminus \Lambda_T^0=\{\lambda_{T, 2}^{\ell, m}\ :\ 1\leq \ell\leq {q}-1, 0\leq m\leq {q}-1-\ell\}.$$
\end{example}
\subsection{Equilibration algorithm}\label{sub:app:alg}
\FloatBarrier
\begin{algorithm}
\caption{Particular solution in $\mathcal S(z)$}
\label{alg:eq}
\textbf{Input:} Data $f\in L^2(\omega(z))$ and $G_h \in H^1(\mathcal{T}(z))^2$ for vertex $z\in\mathcal V$.
\begin{algorithmic}[1]
\State Initialize all coefficients $c_{T, E}^j,c_{T, \ddiv}^{\ell, m}, c_{T}^r$ in \eqref{eqn:basis_RTkpwz} with zero.
\For{$a\coloneqq 1:N$}
\State $c_{T_a, E_{a-1}}^{0}\coloneqq \begin{cases}
0%
&\text{if } a=1,\\
([G_h]_E\cdot \nu_E, \varphi_z)_{L^2(E_{a-1})} - c_{T_{a-1}, E_{a-1}}^{0} &\text{else}
\end{cases}$
\State $c_{T_a, E_{a\phantom {-1}}}^{0}\coloneqq (f+\ddiv G_h, \varphi_z)_{L^2(T_a)} - c_{T_a, E_{a-1}}^{0}$
\For{$1\leq \ell+m\leq {q}$}
\State $c_{T_a, \div}^{\ell,m}\coloneqq (f+\ddiv G_h, \varphi_zx_1^\ell x_2^m)_{L^2(T_a)}$
\EndFor
\For{$1\leq j\leq {q}$}
\State $c_{T_a, E_{a-1}}^{j}\coloneqq 0 %
$
\State $c_{T_a, E_{a\phantom {-1}}}^{j}\coloneqq ([G_h]_E\cdot \nu_E, \varphi_zs^j)_{L^2(E_a)}%
$
\EndFor
\EndFor
\end{algorithmic}
\textbf{Output:} $\tilde\sigma_z^\Delta\inRT_{\kp}^{{\rm pw}}(\mathcal{T}(z))$ defined %
by \eqref{eqn:basis_RTkpwz} with coefficients $c_{T, E}^j,c_{T, \ddiv}^{\ell, m}, c_{T}^r$.
\end{algorithm}
This subsection presents the equilibration procedure, starting with Algorithm \ref{alg:eq}, that computes an admissible function $\tilde\sigma_z^\Delta\in\mathcal S(z)$ in terms of the representation \eqref{eqn:basis_RTkpwz}.
Enumerate the $N\coloneqq|\mathcal{T}(z)|$ triangles $T\in\mathcal{T}(z)$ from $1$ to $N$ as in Figure \ref{fig:NodePatch}.
Any two neighbouring triangles $T_a, T_{a+1}$ share an edge $E_a\coloneqq T_a\cap T_{a+1}$ for $a=1,...,N-1$.
If $z\in\mathcal V(\Omega)$ is an interior vertex, $T_1$ and $T_{N}$ share an additional edge $E_0\coloneqq E_{N}\coloneqq T_1\cap T_{N}$.
For a boundary vertex $z\in\mathcal V(\partial\Omega)$, $T_1, T_{N}$ have the distinct boundary edges $E_0, E_{N}\in \mathcal{F}(z)\cap \mathcal{F}(\partial\Omega)$.
The following lemma shows correctness of Algorithm \ref{alg:eq} under the compatibility condition \eqref{eq:solution_property} and represents step one of the equilibration algorithm.
The final step is the local minimization problem in Lemma \ref{lem:appendixB} that provides $\Rzh^\Delta$ from \eqref{eqn:RzhD_def}.
Both proofs are provided in \ref{sub:app:proofs}.
\begin{lemma}\label{lem:appendix}
Given $z\in\mathcal V$, let $\{\varphi_{T, E}^j, \varphi_{T, \ddiv}^{\ell, m}, \varphi_{T}^r\}$ be the basis of $RT_{q}(T)$ dual to $\Lambda_T$ with \eqref{eqn:dof_basis} for all $T\in\mathcal{T}(z)$.
Suppose $f \in L^2(\omega(z))$ and $G_h\in P_{q}(\mathcal{T}(z);\mathbb{R}^2)$ satisfy %
\begin{equation}\label{eqn:crutial_identity}
(G_h, \nabla \varphi_z)_{L^2(\omega(z))}=(f,\varphi_z)_{L^2(\omega(z))}\quad\text{if }z\in\mathcal V(\Omega).
\end{equation}
Then the output of Algorithm \ref{alg:eq} with input $f$ and $G_h$ defines a function $\tilde\sigma_z^\Delta\in\mathcal S(z)$.
\end{lemma}
Note that \eqref{eqn:crutial_identity} is a local version of \eqref{eq:solution_property} and therefore holds for the HHO method with the choice $G_h \coloneqq \nabla_\mathrlap{\pd}{w} R u_h$ as proven in \eqref{eqn:HHO_solution_property}.
This allows for the computation of $Q^\Delta_p \coloneqq\sum_{z\in \mathcal V}\Rzh^\Delta = Q_p - G_h$ in terms of local and unconstrained minimization problems on the vertex-patches $\omega(z)$.
\begin{lemma}\label{lem:appendixB}
Given $z\in\mathcal V$, let $\{\varphi_{T, E}^j, \varphi_{T, \ddiv}^{\ell, m}, \varphi_{T}^r\}$ be as in Lemma \ref{lem:appendix} for all $T\in\mathcal{T}(z)$ and let $\tilde\sigma_z^\Delta\in \mathcal S(z)$ be arbitrary.
Then $V(z)\coloneqq \mathcal S(z) - \tilde\sigma_z^\Delta$ is a linear vector space and consists of all functions of the form
\begin{align} \label{eqn:sigma}
\sum_{a=1}^{N}\left(d_0 (\varphi_{T_a, E_{a-1}}^0-\varphi_{T_{a}, E_{a}}^0)
+ \sum_{\ell=1}^{q}(d_{E_{a-1}}^\ell\varphi_{T_a, E_{a-1}}^\ell-d_{E_{a}}^\ell\varphi_{T_a, F_a}^\ell)
+\sum_{r=1}^{N_{q}}d_{T_a}^r\varphi_{T_a}^r\right)%
\end{align}
for arbitrary $d_0, d_{E_a}^\ell, d_{T_a}^r\in\mathbb R$ with $\ell=1,...,{q}, r=1,...,N_{q}, a=1,...,N$ (and $d_{E_0}^\ell=d_{E_{N}}^\ell$ for $z\in\mathcal V(\Omega)$) and the enumeration of $\mathcal{T}(z)$ as in Figure \ref{fig:NodePatch}.
Furthermore, $\Rzh^\Delta=\tilde\sigma_z^\Delta + \sigma_z^\Delta$ holds for the solution
$\sigma_z^\Delta\in V(z)$
to the $1+{q}|\mathcal{F}(z)|+{q}({q}-1)/2N$-dimensional minimization problem \eqref{eqn:linear_min}.
\end{lemma}
\subsection{Proofs}\label{sub:app:proofs}
The remaining parts of this appendix are devoted to the verification of Lemmas \ref{lem:appendix}--\ref{lem:appendixB}.
\begin{proof}[of Lemma \ref{lem:appendix}]
Enumerate $\mathcal{T}(z)$ as in \ref{sub:app:alg} and
recall the definition of the jump $[G_h]_E = G_h|_{T_+} - G_h|_{T_-}$ on the interior edge $E=T_+\cap T_-$ shared by $T_+, T_-\in\mathcal{T}$, and $[G_h]_E = G_h|_{T_+}$ for the unique triangle $T_+\in\mathcal{T}$ with $E\subset T_+$ for the boundary edge $E\in\mathcal{F}(\partial \Omega)$.
First, observe that $\sigma_z\inRT_{\kp}^{{\rm pw}}(\mathcal{T}(z))$ lies in $RT_{\kp}^{{\rm pw}, 0}(\mathcal{T}(z))$ if and only if the coefficients $c_{T, E}^j=0$ in the representation \eqref{eqn:basis_RTkpwz} are zero for $0\leq j\leq {q}$ at the edge $E\in\mathcal{F}(T)\setminus\mathcal{F}(z)$ in $T\in\mathcal{T}$ opposing $z$.
By definition, $\sigma_z\inRT_{\kp}^{{\rm pw}, 0}(\mathcal{T}(z))$ belongs to $\mathcal S(z)$ if and only if
\begin{align}
\lambda_{T, \ddiv}^{\ell, m}(\sigma_z) &= (f+\ddiv G_h, \varphi_zx_1^\ell x_2^m)_{L^2(T)}&&\text{for all }0\leq \ell+m\leq {q},T\in\mathcal{T}(z),\label{eqn:var_div}\\
(\lambda_{T_+, E}^{j} + \lambda_{T_-, E}^{j})(\sigma_z) &= ([G_h]_E\cdot \nu_E, \varphi_zs^j)_{L^2(E)}&&\text{for all }0\leq j\leq {q}, E\in \mathcal{F}(\omega(z)).\label{eqn:var_jump_cond}
\end{align}
This translates into equivalent conditions on the coefficients of $\sigma_z$ in the representation \eqref{eqn:basis_RTkpwz}, namely, for all $a=1, ..., N$,
\begin{align}
c_{T_a, E_{a}}^{0} &= (f+\ddiv G_h, \varphi_z)_{L^2(T_a)} - c_{T_a, E_{a-1}}^{0},&&\label{eqn:cond0}\\
c_{T_\inda, \ddiv}^{\ell, m} &= (f+\ddiv G_h, \varphi_zx_1^\ell x_2^m)_{L^2(T_a)}&&\text{for all }1\leq \ell+m\leq {q},\label{eqn:c_div}\\
c_{T_{a}, E_{a-1}}^j &= d_{E_{a-1}}^j&&\text{for all }0\leq j\leq {q},\label{eqn:jump1}\\
c_{T_a, E_a}^j &= ([G_h]_E\cdot \nu_E, \varphi_zs^j)_{L^2(E_a)} - d_{E_a}^j&&\text{for all }0\leq j\leq {q},\label{eqn:jump2}
\end{align}
where $d_{E_a}^\ell\in\mathbb R$.
Since $\sigma_z\inRT_{\kp}^{{\rm pw}, 0}(\mathcal{T}(z))$ vanishes at the other edges $E\in\mathcal{F}\setminus \mathcal{F}(z)$, $\lambda_{T_a, \ddiv}^{0,0}(\sigma_z) = c_{T_a, E_{a-1}}^{0} + c_{T_a, E_{a}}^{0}$ and \eqref{eqn:cond0}--\eqref{eqn:c_div} are equivalent to \eqref{eqn:var_div}.
The identification $d_{E_0}^\ell=d_{E_{N}}^\ell$ for an interior vertex $z\in\mathcal V(\Omega)$ with $E_0=E_N\in\mathcal{F}(z)$ shows that \eqref{eqn:jump1}--\eqref{eqn:jump2} are equivalent to \eqref{eqn:var_jump_cond}.
This identification is well defined.
Note that, whereas there is no condition on $d_{E_a}^\ell$ for $1\leq\ell\leq {q}$, the combination of \eqref{eqn:cond0} and \eqref{eqn:jump1} with \eqref{eqn:jump2} shows the implicit extra condition
$$d_{E_a}^0 = d_{E_0}^0 + \sum_{\alpha=1}^a \left(([G_h]_E\cdot \nu_E, \varphi_z)_{L^2(E_\alpha)} - (f+\ddiv G_h, \varphi_z)_{L^2(T_\alpha)}\right)\quad\text{for }a=0,...,N.$$
For an interior vertex $z\in\mathcal V(\Omega)$, an integration by parts and \eqref{eqn:crutial_identity} show that the sum on the right-hand side above vanishes for $a=N$, whence $d_{E_N}^0 = d_{E_0}^0$ is indeed well defined.
Furthermore, there is no condition on the coefficients $c_{T_a}^{r}$ for all $T_a\in\mathcal{T}(z)$ and $r=1,...,N_{q}$ and $c_{T_a}^{r}=d_{T_a}^r$ is a further degree of freedom.
Algorithm \ref{alg:eq} finds coefficients that satisfy \eqref{eqn:cond0}--\eqref{eqn:jump2} in a loop over $a=1,...,N$ and therefore defines $\tilde\sigma_z^\Delta\in\mathcal S(z)$ by \eqref{eqn:basis_RTkpwz}.\qed
\end{proof}
\begin{proof}[of Lemma \ref{lem:appendixB}]
This follows immediately after revisiting the proof of Lemma \ref{lem:appendix} for an arbitrary function $\sigma_z\in RT_{\kp}^{{\rm pw}, 0}(\mathcal{T}(z))$.
Since $\sigma_z\in \mathcal S(z)$ is equivalent to \eqref{eqn:cond0}--\eqref{eqn:jump2} for the representation \eqref{eqn:basis_RTkpwz} of $\sigma_z$ in the given basis, all functions $\sigma_z\in \mathcal S(z) -\tilde\sigma_z^\Delta$ are of the form \eqref{eqn:sigma}
for arbitrary $d_0, d_{E_a}^\ell, d_{T_a}^r\in\mathbb R$ with $\ell=1,...,{q}$, $r=1,...,N_{q}$, and $a=1,...,N$ (and $d_{E_0}^\ell=d_{E_{N}}^\ell$ for $z\in\mathcal V(\Omega)$).
Hence, the dimension of the linear space $V(z)=\mathcal S(z) -\tilde\sigma_z^\Delta$ is $1+{q}|\mathcal{F}(z)|+{q}({q}-1)/2N$.
The claim follows from Proposition \ref{prop:RzhD_def} by observing
\begin{align*}
\Rzh^\Delta
\coloneqq
&
\argmin\limits_{%
\sigma_z\in \mathcal S(z)
}
\| \sigma_z \|_{\Lz} = \tilde\sigma_z^\Delta + \argmin\limits_{%
\sigma_z\in \mathcal S(z) - \tilde\sigma_z^\Delta
}
\| \tilde\sigma_z^\Delta + \sigma_z \|_{\Lz}.\quad\qed
\end{align*}
\end{proof}
\section{Equilibrium-based a posteriori HHO error analysis}
\label{sec:eq}
\newcommand{\nabla_{\rm pw} Ru_h}{\nabla_{\rm pw} Ru_h}
\newcommand{q^\Delta}{q^\Delta}
\newcommand{\tilde f}{\tilde f}
\newcommand{\supp}{\text{supp}}
\newcommand{\mz}{|\partial \omega(z) \cap \partial \Omega|}
\newcommand{\RTz}{
RT_{k+1}^0(\mathcal T (z)) %
}
\newcommand{\RTzk}{
RT_{k}^0(\mathcal T (z)) %
}
\newcommand{\RTzkp}{
RT_{k+1}^0(\mathcal T (z)) %
}
\newcommand{\Pi_{\RTzk}}{\Pi_{\RTzk}}
\renewcommand{\Pi_{\RTzk}}{\Pi_{RT}^k}
\newcommand{\Pi_{\RTzk}}{\Pi_{\RTzk}}
\newcommand{\mathcal T (z)}{\mathcal T (z)}
\newcommand{L^2(\omega(z))}{L^2(\omega(z))}
\newcommand{_{\Lz}}{_{L^2(\omega(z))}}
\newcommand{_{L^2(T)}}{_{L^2(T)}}
\newcommand{\nLz}[1]{||#1||_{\Lz}}
\newcommand{RT^{k+1}_{{\rm pw},0}(\mathcal T (z))}{RT^{k+1}_{{\rm pw},0}(\mathcal T (z))}
The residual-based guaranteed upper bound (GUB) of the error $\trb{u - R u_h}_\mathrlap{\pd}{w}$ from Subsection \ref{sub:HHO_eta} employs explicit constants that may lead to overestimation in higher dimensions and for different triangular shapes.
This section utilizes equilibrated flux reconstructions \cite{Ainsworth2005,Ain:07,ErnVohralik2015,bertrand_weakly_2019,bertrand_opt} to establish a constant-free guaranteed upper bounds for a tight error control.
\subsection{Guaranteed error control}
\label{sub:Motivation_eq}
The guaranteed upper bounds of this section involves \emph{two post-processings} of the potential reconstruction $R u_h \in P_{k+1}(\mathcal{T})$ of the discrete solution $u_h$ to \eqref{eqn:HHO}.
First, the patch-wise design of a flux reconstruction $Q_p \in RT_{k+p}(\mathcal{T})$ with $p \in \mathbb{N}_0$ from \cite{AinOde:93,BraPilSch:09,ErnVohralik2020} provides an $H(\mathrm{div},\Omega)$-conforming approximation to $\nabla_{\pw} R u_h$ with the equilibrium $\Pi_{r} f + \div Q_p = 0$ in $\Omega$
and $r$ from \eqref{def:r} below.
Second, the nodal average $\mathcal{A} R u_h \in S^{k+1}_0(\mathcal{T})\subset V$ results in an $V$-conforming approximation of $R u_h$ by averaging all values of the discontinuous function $R u_h$ at each Lagrange point of $S^{k+1}_0(\mathcal{T})$. %
This, the split \eqref{eqn:decomp}, and the solution property \eqref{eqn:HHO_solution_property} give rise to the guaranteed upper bound (GUB)
\begin{align}
\label{eq:definitionetaq}
\eta_{\mathrm{eq},p}^2(\mathcal{T}) := \left(C_P\osc{r}(f,\mathcal{T}) + \| Q_p- \nabla_{\pw} R u_h\|\right)^2 + \trb{(1 - \mathcal{A})R u_h}_\mathrlap{\pd}{w}^2
\end{align}
with $r \in \mathbb{N}_0$ defined by
\begin{align}\label{def:r}
r \coloneqq 0 \text{ if } k = 0 \quad\text{and}\quad r \coloneqq k + p \text{ if } k \geq 1.
\end{align}
The main result of this section states the reliability and efficiency (up to data oscillations) of $\eta_{\mathrm{eq},p}$ for all parameters $p \in \mathbb{N}_0$.
\begin{theorem}[equilibrium-based GUB for HHO]\label{thm:GUB-HHO}
Let $u \in V$ resp.\ $u_h \in V_h$ solve \eqref{eq:poisson} resp.\ \eqref{eqn:HHO}.
Given a parameter $p \in \mathbb{N}_0$,
there exists $Q_p\in RT_{k+p}(\mathcal{T})$ such that
the error estimator $\eta_{\mathrm{eq},p}(\mathcal{T})$ from \eqref{eq:definitionetaq}
is an efficient GUB
%
\begin{align}\label{ineq:equilibrium-equivalence}
\trb{u-R u_h}_\mathrlap{\pd}{w} \le\eta_{\mathrm{eq},p}(\mathcal{T}) \leq \cnst{cnst:equilibrium-efficiency}\big(\trb{u-R u_h}_\mathrlap{\pd}{w} + \osc{q}(f, \mathcal{T})\big).
\end{align}
for any $q\in \mathbb N_0$ and $\osc{k-1}(f, \mathcal{T}) \leq\cnst{cnst:equilibrium-reliability}\eta_{\mathrm{eq},p}(\mathcal{T})$.
%
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%
%
%
%
%
The constants $\newcnst\label{cnst:equilibrium-efficiency}$ and $\newcnst\label{cnst:equilibrium-reliability}$ exclusively depend on the polynomial degree $k \in \mathbb{N}_0$, the parameter $q \in \mathbb{N}_0$, and the shape-regularity of $\mathcal{T}$.
%
%
%
\end{theorem}
At least two technical contributions for the proof of Theorem~\ref{thm:GUB-HHO} are of broader interest.
A first contribution to the HHO literature
is the local equivalence
of the original HHO stabilization $s_h$ from \eqref{eqn:s_h0} and the alternative stabilization $\tilde{s}_h(v_h,v_h) \coloneqq \sum_{T \in \mathcal{T}} \tilde{s}_T(v_h,v_h)$ from \cite{DiPietroDroniou2020} defined, for $v_h=(v_\mathcal{T}, v_\mathcal{F})\in V_h$, by
\begin{align}
\label{def:s-tilde}
\tilde{s}_T(v_h,v_h)\coloneqq h_T^{-2} \|\Pi_{T,k} (v_\mathcal{T} &- R v_h)\|_{L^2(T)}^2 \\&+ \sum_{F \in \mathcal{F}(T)} h_F^{-1} \|\Pi_{F,k} (v_\mathcal{F} - R v_{h}|_{T})\|_{L^2(F)}^2.\nonumber
\end{align}
A second result of separate interest in the HHO literature (cf.~\cite{DiPietroDroniou2020} where the efficiency in \eqref{eqn:efficient_stabilization} is left open) is the efficiency of the stabilizations from Theorems \ref{thm:equivalence-stabilization}--\ref{thm:best-approximation} below,
\begin{align}\label{eqn:efficient_stabilization}
\tilde{s}_h(v_h,v_h)^{1/2}\approx s_h(u_h,u_h)^{1/2} \lesssim \trb{u - R u_h}_\mathrlap{\pd}{w} + \osc{k+p}(f,\mathcal T).
\end{align}
The subsequent subsection continues with some explanations on the flux reconstruction $Q_p\in RT_{k+p}(\mathcal{T})$ that is defined by local minimization problems on each vertex patch.
Appendix A complements the discussion with an algorithmic two-step procedure for the computation of $Q_{p}-\nabla_{\pw} R u_h$ in 2D.
The efficiency of the averaging $\trb{(1 - \mathcal{A})R u_h}_\mathrlap{\pd}{w}$ up to data oscillations follows in Subsections \ref{sub:local-equivalence-of-stabilizations}--\ref{sub:efficiency_stabilization}.
Subsection \ref{sub:proof-efficiency-GUB} concludes with the proof of Theorem \ref{thm:GUB-HHO}.
\subsection{Construction of equilibrated flux}
\renewcommand{\Pi_{\RTzk}}{\Pi_{RT_k^{\rm pw}}}
\label{sub:Computation_Q}
This subsection defines the post-processed $H(\mathrm{div},\Omega)$-conforming equilibrated flux $Q_p\in RT_{k+p}(\mathcal{T})$ that enters the GUB $\eta_{\mathrm{eq},p}$ from \eqref{eq:definitionetaq} based on local patch-wise minimization problems in the spirit of \cite{BraPilSch:09,ErnVohralik2015,ErnVohralik2020}.
Consider the shape-regular vertex-patch $\omega(z)\coloneqq \mathrm{int}(\bigcup\{T\in\mathcal{T}(z)\})$ covered by the neighbouring simplices $\mathcal{T}(z)\coloneqq\{T\in\mathcal{T} : z\in T\}$ sharing a given vertex $z \in \mathcal{V}$ with the facet spider $\mathcal{F}(z)\coloneqq\{F\in\mathcal{F}\ :\ z\in F\}$.
Recall the space of piecewise Raviart-Thomas functions $RT_{k}^{\mathrlap{\pd}{w}}(\mathcal{T})$ from Subsection \ref{sub:notation} and define
\begin{align*}
L^2_0(\omega(z)) &\coloneqq \{f\in L^2(\omega(z)) : (f,1)_{\Lz} = 0\},\\
L^2_*(\omega(z)) &\coloneqq \begin{cases}
L^2_0(\omega(z))& \text{ if } z \in \mathcal{V}(\Omega),\\
L^2(\omega(z)) &\text{ else},
\end{cases}\\
H^1_*(\omega(z)) &\coloneqq
\begin{cases}
H^1(\omega(z)) \cap L^2_0(\omega(z))\quad\text{ if } z \in \mathcal{V}(\Omega),\\
\{v \in H^1(\omega(z)) : v = 0 \text{ on } \partial \Omega \cap \bigcup\mathcal{F}(z)\}\quad\text{ else},
\end{cases}\\
H_0(\ddiv,\omega(z)) &\coloneqq
\begin{cases}
\{r \in H(\mathrm{div},\omega(z)) : r \cdot \nu = 0 \text{ on } \partial \omega(z)\} \quad\text{ if } z \in \mathcal{V}(\Omega),\\
\{r \in H(\mathrm{div},\omega(z)) : r \cdot \nu = 0 \text{ on } \partial \omega(z) \setminus \bigcup\mathcal{F}(z)\}\quad\text{ else},
\end{cases}\\
RT_k^0(\mathcal{T}(z))&\coloneqq\;RT_{k}^{\mathrlap{\pd}{w}}(\mathcal{T}(z)) \cap H_0(\ddiv,\omega(z)).
\end{align*}
Throughout the remaining parts of this section, abbreviate $G_h \coloneqq \nabla_\mathrlap{\pd}{w} R u_h \in P_k(\mathcal{T};\mathbb R^n)$.
Given a vertex $z\in\mathcal V$ with the $P_1$-conforming nodal basis function $\varphi_z\in S^1(\mathcal{T})$,
the property \eqref{eqn:HHO_solution_property} provides compatible data
\begin{align}\label{def:f-z}
f_z \coloneqq\,
\begin{cases}
\Pi_{p}(\varphi_z \Pi_0 f- G_h \cdot \nabla \varphi_z) &\mbox{if } k = 0,\\
\Pi_{k+p}(\varphi_z f- G_h \cdot \nabla \varphi_z) &\mbox{if } k \geq 1
\end{cases}\quad\in L^2_*(\omega(z))
\end{align}
such that the discrete affine space \begin{align}\label{eqn:Qhz_def}
\mathcal{Q}_h(z) \coloneqq \{\tau_z \in RT^0_{k+p}(\mathcal{T}(z))\ :\ \mathrm{div}\, \tau_z + f_z = 0 \text{ in } \Omega\} \neq \emptyset
\end{align}
is not empty. Consequently,
\begin{align}
\label{eq:discrete_minimisation}
\QQ_{z,h} \coloneqq \argmin\limits_{\tau_z \in \mathcal{Q}_h(z)}
\| \tau_z - \mathcal{I}_{RT} (\varphi_z G_h)\|_{\Lz} = \Pi_{\mathcal{Q}_h(z)} \mathcal{I}_{RT} (\varphi_z G_h)
\end{align}
is well defined as the $L^2$ projection $\Pi_{\mathcal{Q}_h(z)} \mathcal{I}_{RT} (\varphi_z G_h)$ of $\mathcal{I}_{RT} (\varphi_z G_h)$ onto $\mathcal{Q}_h(z)$
with the piecewise Raviart-Thomas interpolation $\mathcal{I}_{RT} : H^1(\mathcal{T};\mathbb R^n)\to RT^\mathrlap{\pd}{w}_{k+p}(\mathcal{T})$ \cite[Section III.3.1]{boffi_mixed_2013}.
In the case $p \geq 1$, $\varphi_z G_h \in P_{k+1}(\mathcal{T}(z)) \subset RT_{k+p}^\mathrlap{\pd}{w}(\mathcal{T}(z))$
is a piecewise Raviart-Thomas function of degree $k+p$.
Hence $\mathcal{I}_{RT}(\varphi_z G_h) = \varphi_z G_h$ and
the interpolation $\mathcal{I}_{RT}$
could be omitted in the formula \eqref{eq:discrete_minimisation}.
The partition of unity $\sum_{z\in\mathcal V}\varphi_z\equiv 1$ shows that the sum $Q_p = \sum_{z \in \mathcal V} \QQ_{z,h}\in H(\ddiv, \Omega)$ of the patch-wise contributions
satisfies
\begin{align}
&\ddiv Q_p =
\begin{cases}
\sum\limits_{z \in \mathcal V} \Pi_{p} (G_h \cdot\nabla \varphi_z - \varphi_z \Pi_0 f) = -\Pi_{0} f &\mbox{if } k = 0,\\
\sum\limits_{z \in \mathcal V} \Pi_{k+p} (G_h \cdot\nabla \varphi_z - \varphi_z f) = -\Pi_{k+p} f &\mbox{if } k \geq 1,
\end{cases}\label{eqn:div_Q_f}\\
&\|Q_{p}-G_h \|_{L^2(\Omega)}^2 = \sum_{z\in\mathcal V}
\|\QQ_{z,h} - %
\mathcal{I}_{RT}(\varphi_z G_h)
%
\|_{L^2(\omega(z))}^2.\label{eqn:Q_q_small}
\end{align}
This establishes the flux reconstruction $Q_p$.
The efficiency of the flux reconstruction will be based on the following general equivalence.
\newcommand{C_{\rm s}}{C_{\rm s}}
\begin{lemma}[control of $H(\ddiv)$ minimization by residual {\cite{BraPilSch:09,ErnVohralik2020}}]\label{lem:uniform_stability_mixed}
\newcommand{\sigma_z}{\sigma_z}
Given any vertex $z\in\mathcal V$, a piecewise Raviart-Thomas function $\sigma_z \in RT_{q}^{{\rm pw}}(\mathcal{T}(z))$ and a piecewise polynomial $r_z \in {P}_{q}\left(\mathcal{T}(z)\right)$ of degree $q \in \mathbb{N}_0$, define the residual
\begin{align}\label{eq:compatibility}
\mathrm{Res}_z(v)\coloneqq \sum_{T \in \mathcal{T}(z)}\Big(
\left(r_{z}, v\right)_{L^2(T)}+\left\langle\sigma_z\cdot \nu_T, v\right\rangle_{L^2(\partial T)}\Big)
\end{align}
for all $v\in H^1(\Omega)$. If $z\in \mathcal V(\Omega)$ is an interior vertex, then suppose additionally that
$\mathrm{Res}_z(1) = 0$.
Then
\begin{align}
\label{lem:uniform_stability_mixedineq}
\min\limits_{\substack{\tau_z \in RT^0_q(\mathcal{T}(z))\\\mathrm{div} \tau_z = r_z + \mathrm{div}_\mathrlap{\pd}{w} \sigma_z}}
\left\|\tau_z-\sigma_z\right\|_{L^2(\omega(z))}
\leq C_{\rm s}
\max\limits_{%
\substack{%
v \in H^{1}_*(\omega(z))
\\
\|\nabla v\|_{L^2(\omega(z))}=1
}
} \mathrm{Res}_z(v)
\end{align}
holds for a constant $C_{\rm s}$ that exclusively depends on the shape-regularity (and is in particular independent of the polynomial degree $q$).
\end{lemma}
\begin{proof}
The assertion follows from \cite[Theorem 7]{BraPilSch:09} in $n=2$ dimensions and \cite[Corollaries 3.3, 3.6, and 3.8]{ErnVohralik2020} in $n=3$ dimensions.\qed
\end{proof}
\begin{remark}\label{rem:local_osc_eff}
The patch-wise construction of $Q_p$ in Subsection \ref{sub:Computation_Q} typically
generates local data oscillation $\osc{k+p}(\varphi_z f,\mathcal{T}(z))$ in the error analysis as in the proof of Theorem \ref{thm:GUB-HHO} in Subsection \ref{sub:proof-efficiency-GUB} below or, e.g., \cite[Theorem 3.17]{ErnVohralik2015}.
A straightforward computation $\osc{k+p}(\varphi_z f,\mathcal{T}(z)) \leq \osc{k+p-1}(f,\mathcal{T}(z))$ apparently leads to a loss of one degree
in the data oscillation but Lemma \ref{lem:local-data-oscillation} verifies
\begin{align}\label{eqn:local_osc_eff}
\osc{k+p}^2(\varphi_z f, \mathcal T(z))\lesssim \min_{v_{h}\in P_{k+1}(\mathcal{T}(z))} \|\nabla_{\pw}(u - v_{h})\|_{L^2(\omega(z)}^2 + \osc{q}^2(f,\mathcal T(z))
\end{align}
for any $p, q\in{\mathbb N}_0$.
This allows for efficiency of the data oscillations
on the right-hand side of the efficiency estimate \cite[Formula (3.42)]{ErnVohralik2015} and leads to a corresponding refinement in
\cite[Theorem 3.17]{ErnVohralik2015}.
\end{remark}
\subsection{Local equivalence of stabilizations}
\label{sub:local-equivalence-of-stabilizations}
The first improvement to the current HHO literature is the local equivalence of the two stabilizations $\tilde{s}_h$ from \eqref{def:s-tilde} and $s_h$ from \eqref{eqn:s_h0}.
The authors of this paper could not find any motivation for the alternative stabilization $\tilde{s}_h$ in the error analysis of \cite[Section 4]{DiPietroDroniou2020} and suggest to apply Theorem \ref{thm:equivalence-stabilization} below to \cite[Theorem 4.7]{DiPietroDroniou2020} to recover the results therein for the original HHO stabilization $s_h$.
Recall the local stabilization $\tilde{s}_T$ in $\tilde{s}_h(v_h,v_h) \coloneqq \sum_{T \in \mathcal{T}} \tilde{s}_T(v_h,v_h)$ from \eqref{def:s-tilde}
and $S_{TF} v_h = \Pi_{F,k}(R v_{h|T} - v_\mathcal{F}|_F - \varphi_k)$ in the definition of $s_h$ from \eqref{eqn:s_h0}.
\begin{theorem}[local equivalence of stabilizations]\label{thm:equivalence-stabilization}
Any $v_h = (v_{\mathcal T},v_\mathcal{F}) \in V_h$ and $T \in \mathcal T$ satisfy
\begin{align}
\cnst{cnst:equivalence-norms-LHS}^{-1}\tilde{s}_T(v_h,v_h) \leq \sum_{F \in \mathcal{F}(T)} h_F^{-1} \|S_{TF} v_h\|_{L^2(F)}^2 \leq \cnst{cnst:equivalence-norms-RHS}\tilde{s}_T(v_h,v_h).
\label{ineq:efficiency-trace}
\end{align}
The constants $\newcnst\label{cnst:equivalence-norms-LHS}$ and $\newcnst\label{cnst:equivalence-norms-RHS}$ exclusively depend on the polynomial degree $k$ and the shape regularity of $\mathcal{T}$.
\end{theorem}
\begin{proof}
The second inequality in \eqref{ineq:efficiency-trace} follows directly from a triangle inequality and an inverse estimate. Therefore, the proof focuses on the first inequality in \eqref{ineq:efficiency-trace}.
Given $v_h = (v_\mathcal T,v_\mathcal{F}) \in V_h$ and $T \in \mathcal T$, let $\varphi_k \coloneqq (\Pi_k R v_h - v_\mathcal{T})|_{T} \in P_k(T)$.
Since $S_{TF} v_h = \Pi_{F,k}(R v_{h|T} - v_\mathcal{F}|_F - \varphi_k)$, the triangle inequality $\|\Pi_{F,k}(R v_{h|T} - v_{\mathcal{F}})\|_{L^2(F)} \leq \|S_{TF} v_h\|_{L^2(F)} + \|\varphi_k\|_{L^2(F)}$, the discrete trace inequality $\|\varphi_k\|_{L^2(F)} \lesssim h_F^{-1/2}\|\varphi_k\|_{L^2(T)}$,
%
and the shape-regularity $h_F \approx h_T$ for all $F \in \mathcal{F}(T)$ reveal
\begin{align}
&\sum_{F \in \mathcal{F}(T)} h_F^{-1}\|\Pi_{F,k}(R v_{h|T} - v_{\mathcal{F}})\|_{L^2(F)}^2\nonumber\\
&\qquad\qquad\lesssim \sum_{F \in \mathcal{F}(T)} h_F^{-1} \|S_{TF} v_h\|_{L^2(F)}^2 + h_T^{-2}\|\varphi_k\|_{L^2(T)}^2.
\label{ineq:proof-efficiency-trace-triangle-inequality}
\end{align}
Since $\Pi_0 \varphi_k = 0$ (from the design of $Rv_h$), a Poincar\'e inequality shows
\begin{align}
h_T^{-2}\|\varphi_k\|_{L^2(T)}^2 \leq C_P^2\|\nabla \varphi_k\|_{L^2(T)}^2.
\label{ineq:proof-efficiency-trace-Poincare}
\end{align}
On the one hand, an integration by parts provides
\begin{align}
\|\nabla \varphi_k\|_{L^2(T)}^2 = - (\Pi_k R v_h - v_\mathcal{T}, \Delta \varphi_k)_{L^2(T)} + \langle\varphi_k, \nabla \varphi_k \cdot \nu_T\rangle_{L^2(\partial T)}.
\label{eq:proof-upper-bound-trace-ibp-1}
\end{align}
On the other hand, an integration by parts and the definition of $R$ imply
\begin{align}
&- (R v_h, \Delta \varphi_k)_{L^2(T)} = (\nabla R v_h, \nabla \varphi_k)_{L^2(T)} - \langle R v_{h|T}, \nabla \varphi_k \cdot \nu_T\rangle_{L^2(\partial T)}\nonumber\\
&\qquad= - (v_{\mathcal{T}}, \Delta \varphi_k)_{L^2(T)} + \sum_{F \in \mathcal{F}(T)} \langle v_{\mathcal{F}} - R v_{h|T}, \nabla \varphi_k \cdot \nu_T\rangle_{L^2(F)}.
\label{eq:proof-upper-bound-trace-ibp-2}
\end{align}
Since $\Delta \varphi_k \in P_k(T)$, the $L^2$ projection $\Pi_k$ on the right-hand side of \eqref{eq:proof-upper-bound-trace-ibp-1} is redundant. Hence, the combination of \eqref{eq:proof-upper-bound-trace-ibp-1}--\eqref{eq:proof-upper-bound-trace-ibp-2} with $\nabla \varphi_k \cdot \nu_{T|F} \in P_k(F)$ for all $F \in \mathcal{F}(T)$
results in
\begin{align}
\|\nabla \varphi_k\|_{L^2(T)}^2 = \sum_{F \in \mathcal{F}(T)} \langle \Pi_{F,k}(v_{\mathcal{F}} - R v_{h|T} + \varphi_k), \nabla \varphi_k \cdot \nu_T \rangle_{L^2(F)}.
\label{eq:proof-upper-bound-trace-ibp-3}
\end{align}
A Cauchy inequality on the right-hand side of \eqref{eq:proof-upper-bound-trace-ibp-3}, a discrete trace inequality, and $S_{TF} v_h = \Pi_{F,k}(R v_{h|T} - v_{\mathcal{F}|F} - \varphi_k)$ for all $F \in \mathcal{F}(T)$ lead to
\begin{align}
\|\nabla \varphi_k\|_{L^2(T)}^2 \lesssim \sum_{F \in \mathcal{F}(T)} h_F^{-1} \|S_{TF} v_h\|_{L^2(F)}^2.
\label{ineq:bound-gradient-phi-k}
\end{align}
Since $\tilde{s}_T(v_h,v_h) = \sum_{F \in \mathcal{F}(T)} h_F^{-1}\|\Pi_{F,k}(R v_{h|T} - v_{\mathcal{F}})\|_{L^2(F)}^2 + h_T^{-2}\|\varphi_k\|_{L^2(T)}^2$,
the combination of \eqref{ineq:proof-efficiency-trace-triangle-inequality}--\eqref{ineq:proof-efficiency-trace-Poincare} with \eqref{ineq:bound-gradient-phi-k} concludes the proof of \eqref{ineq:efficiency-trace}.\qed
\end{proof}
\subsection{Efficiency of the stabilization}\label{sub:efficiency_stabilization}
The second improvement to the HHO literature
is a quasi-best approximation estimate along the lines of the seminal paper \cite[Theorem 4.10]{ErnZanotti2020}.
In combination with Theorem \ref{thm:equivalence-stabilization}, this, in particular, provides the efficiency \eqref{eqn:efficient_stabilization} of the stabilization up to data oscillation.
\begin{theorem}[quasi-best approximation up to data oscillation]\label{thm:best-approximation}
For any $p\in\mathbb N_0$, the solution $u$ to \eqref{eq:poisson} and the discrete solution $u_h$ to \eqref{eqn:HHO} satisfy
\begin{align*}
\trb{u - R u_h}_\mathrlap{\pd}{w} &+ s_h(u_h,u_h)^{1/2}\\
&\leq \cnst{cnst:best-approximation}\Big(\min_{v_{k+1} \in P_{k+1}(\mathcal T)} \trb{u - v_{k+1}}_\mathrlap{\pd}{w} + \osc{k+p}(f,\mathcal T)\Big).
%
\end{align*}
The constant $\newcnst\label{cnst:best-approximation}$ exclusively depends on $k$, $p$, and the shape regularity of $\mathcal{T}$.
\end{theorem}
\begin{proof}
Given $k,p \in \mathbb{N}_0$, let $\widetilde{u} \in V$ solve the Poisson model problem $-\Delta \widetilde{u} = \Pi_{k+p} f$ with the right-hand side $\Pi_{k+p} f$.
Subsection 4.3 in \cite{ErnZanotti2020} constructs a stable enriching operator $\mathcal{J} : V_h \to V$ with local bubble-functions from \cite{Ver:13}.
A modification, where the polynomial degree $k-1$ in \cite[Eq.~(4.16)]{ErnZanotti2020} is replaced by $k+p$, leads to a right-inverse $\mathcal{J} : V_h \to V$ of the interpolation $\mathrm{I}: V_h \to V$ with the stability $\trb{\mathcal{J} v_h}^2 \lesssim a_h(v_h,v_h)$ and the additional $L^2$ orthogonality $\mathcal{J} v_h - v_\mathcal{T} \perp P_{k+p}(\mathcal{T})$ for all $v_h = (v_\mathcal{T},v_\mathcal{F}) \in V_h$.
The extra orthogonality allows for
$$
(\Pi_{k+p} f, \mathcal{J} v_h)_{L^2(\Omega)} = (\Pi_{k+p} f, v_\mathcal{T})_{L^2(\Omega)} = (f,v_\mathcal{T})_{L^2(\Omega)}\quad\text{for all } (v_\mathcal{T}, v_\mathcal{F})\in V_h.
$$
Consequently, the smoother $\mathcal{J}$ leads to a discrete solution $u_h=(u_\mathcal{T},u_\mathcal{F})\in V_h$ in the modified HHO discretization of \cite{ErnZanotti2020} as a quasi-best approximation of the above solution $\tilde u$. The point is that $u_h\in V_h$ coincides with the original HHO solution $u_h$ from \eqref{eqn:HHO}.
The arguments from the proof of \cite[Theorem 4.10]{ErnZanotti2020} reveal the quasi-best approximation
$$\trb{\tilde{u} - R u_h}_\mathrlap{\pd}{w} + s_h(u_h,u_h)^{1/2} \lesssim \min_{v_{k+1} \in P_{k+1}(\mathcal{T})} \trb{\tilde{u} - v_{k+1}}_\mathrlap{\pd}{w}$$ also for the above modified smoother $\mathcal{J}$.
This, the triangle inequalities $\trb{u - R u_h}_\mathrlap{\pd}{w} \leq \trb{u - \tilde{u}} + \trb{\tilde{u} - R u_h}_\mathrlap{\pd}{w}$ and
$\trb{\tilde{u} - v_{k+1}}_\mathrlap{\pd}{w} \leq \trb{u - \tilde{u}} + \trb{u - v_{k+1}}_\mathrlap{\pd}{w}$ for any $v_{k+1} \in P_{k+1}(\mathcal{T})$, and the standard perturbation bound $\trb{u - \tilde{u}} \leq C_P\osc{k+p}(f,\mathcal{T})$ conclude the proof. \qed
\end{proof}
\subsection{Stabilization-free efficiency of averaging}\label{sub:averaging-efficiency}
The main result of this subsection establishes the stabilization-free efficiency of the nodal averaging technique.
\begin{theorem}[averaging is efficient]\label{thm:averaging-efficiency}
Let $u \in V$ resp.\ $u_h \in V_h$ solve \eqref{eq:poisson} resp.\ \eqref{eqn:HHO}. Then $R u_h$ and $\mathcal{A} R u_h$ satisfy, for any $p \in \mathbb N_0$, that
\begin{align}
\cnst{cnst:efficiency-averaging}^{-1}\trb{(1 - \mathcal{A}) R u_h}_\mathrlap{\pd}{w}
%
%
&\leq \trb{u - R u_h}_\mathrlap{\pd}{w} + \osc{k+p}(f,\mathcal T).
\label{ineq:averaging-efficiency-oscillation}
\end{align}
The constant $\newcnst\label{cnst:efficiency-averaging}$ exclusively depends on $k$, $p$, and the shape regularity of the triangulation $\mathcal{T}$.
\end{theorem}
The proof can follow the proof of \cite[Theorem 4.7]{DiPietroDroniou2020} but additionally utilizes the two significant improvements from Subsections
\ref{sub:local-equivalence-of-stabilizations}--\ref{sub:efficiency_stabilization} that allow a stabilization-free efficiency in \eqref{ineq:averaging-efficiency-oscillation}.
\begin{proof}
Theorem 4.7 in \cite{DiPietroDroniou2020} shows $\trb{(1 - \mathcal{A})R u_h}_\mathrlap{\pd}{w}^2 \lesssim \trb{u - R u_h}_\mathrlap{\pd}{w}^2 + \tilde{s}_h(u_h,u_h)$. This, the equivalence $\tilde{s}_h(u_h,u_h) \approx s_h(u_h,u_h)$ of stabilizations (from Theorem \ref{thm:equivalence-stabilization}), and
the efficiency $s_h(u_h,u_h) \lesssim \trb{u - R u_h}_\mathrlap{\pd}{w}^2 + \osc{k+p}^2(f,\mathcal{T})$ (from Theorem \ref{thm:best-approximation}) imply the assertion. \qed
\end{proof}
\begin{remark}[$p$-robustness]
The $H^1(\Omega)$-conforming post-processing of $R u_h$ from \cite{ErnVohralik2020} provides an efficiency constant independent of the polynomial degree $k$.
The right-hand side of \cite[Corollary 4.2]{ErnVohralik2020} involves the stabilization-related term $\sum_{F \in \mathcal{F}}h_F^{-1}\|\Pi_{F,0}[R u_h]_F\|_{L^2(F)}^2$.
It remains an open question whether this term can be controlled $p$-robustly by $\trb{u - R u_h}_\mathrlap{\pd}{w} + \osc{k}(f,\mathcal{T})$ (with a multiplicative
constant independent of the polynomial degree $k$).
\end{remark}
\subsection{Proof of Theorem \ref{thm:GUB-HHO}}\label{sub:proof-efficiency-GUB}
Let $p \in \mathbb{N}_0$ and $r = 0$ if $k = 0$ and $r = k+p$ if $k \geq 1$ as in \eqref{def:r} be given. Recall the abbreviation $G_h = \nabla_\mathrlap{\pd}{w} R u_h \in P_k(\mathcal{T})$ with the discrete solution $u_h \in V_h$ to \eqref{eqn:HHO} and let $Q_p = \sum_{z \in \mathcal V} \QQ_{z,h}\in H(\ddiv, \Omega)$ denote the flux reconstruction from Subsection \ref{sub:Computation_Q}. The proof establishes \eqref{ineq:equilibrium-equivalence} in five steps.\medskip
\emph{Step 1} provides the GUB $\trb{u - R u_h}_\mathrlap{\pd}{w} \leq \eta_{\mathrm{eq},p}(\mathcal{T})$.
This can follow from the paradigm of \cite{Ainsworth2005,Ain:07,ErnVohralik2015} as outlined below.
The choice ${G} \coloneqq G_h$ and $w \coloneqq \mathcal{A} R u_h$ in \eqref{eqn:decomposition_full} and a triangle inequality lead to
\begin{align*}
\trb{u - R u_h}_\mathrlap{\pd}{w}^2
\leq (\trb{f + \ddiv Q_p}_* + \trb{\mathrm{div}(Q_p - G_h)}_*)^2 + \trb{(1 - \mathcal{A}) R u_h}_\mathrlap{\pd}{w}^2.%
\end{align*}
Since $\ddiv Q_p + \Pi_{r} f = 0$ vanishes in $\Omega$ by \eqref{eqn:div_Q_f}, a piecewise Poincar\'e inequality shows $\trb{f + \ddiv Q_p}_* \leq C_P\osc{r}(f,\mathcal{T})$. This, the bound $\trb{\mathrm{div}(Q_p - G_h)}_* \leq \|Q_p - G_h\|$ from the definition of $\trb{\bullet}_*$, and the previously displayed formula result in the reliability $\trb{u - R u_h}_\mathrlap{\pd}{w} \leq \eta_{\mathrm{eq,p}}(\mathcal{T})$.\medskip
\emph{Step 2} establishes $\osc{k-1}(f,\mathcal{T}) \lesssim \eta_{\mathrm{eq},p}(\mathcal{T})$.
Lemma \ref{lem:local-data-oscillation} provides
\begin{align}
\osc{k-1}(f,\mathcal{T})\lesssim \trb{u - R u_h}_\mathrlap{\pd}{w} +\osc r(f, \mathcal{T}).
\end{align}
This, Step 1 and $\osc r(f, \mathcal{T})\lesssim\eta_{{\rm eq}, p}(\mathcal{T})$ from \eqref{eq:definitionetaq} conclude the proof of Step 2.\medskip
\newcommand{\mathcal I_{\rm RT}}{\mathcal I_{\rm RT}}
\emph{Step 3} reveals, for any $q\in{\mathbb N}_0$, the efficiency of the flux reconstruction
\begin{align}\label{ineq:proof-GUB-efficiency-flux}
\|Q_p - G_h\| \lesssim \trb{u-R u_h}_\mathrlap{\pd}{w} + \osc{q}(\mathcal T, f)
\end{align}
for any polynomial degree $k \geq 1$.
The case $k=0$ follows in Step 4 below.
Recall $f_z = \Pi_{k+p}(\varphi_z f - G_h\cdot \nabla \varphi_z)$ from \eqref{def:f-z} for any vertex $z\in\mathcal V$ in the construction of $Q_p$ from Subsection \ref{sub:Computation_Q} and set
$\sigma_z \coloneqq \mathcal I_{\rm RT}(\varphi_z G_h)\in RT_{k+p}^{{\rm pw}}(\mathcal{T}(z))$. %
The piecewise Raviart-Thomas interpolation $\mathcal I_{\rm RT}:H^1(\mathcal{T};\mathbb R^n)\to RT_{k+p}^{\rm pw}(\mathcal{T})$ satisfies the well-known commuting diagram properties \cite[Section 2.5.1]{boffi_mixed_2013}
\begin{align}\label{eq:commuting-diagram-I-RT}
{\ddiv}_{\rm pw}\circ\mathcal I_{\rm RT} = \Pi_{k+p}\circ{\ddiv}_{\rm pw}&&\text{and}&&\gamma_T\circ\mathcal I_{\rm RT}|_F=\Pi_{F,k+p}\circ \gamma_T
\end{align}
for any facet $F\in\mathcal{F}(T)$ of a simplex $T\in\mathcal{T}$ and the normal trace $\gamma_T:H^1(T;\mathbb R^n)\to L^2(\partial T)$ with $\gamma_T \sigma \coloneqq \sigma \cdot \nu_T$ for $\sigma\in H^1(T;\mathbb R^n)$.
This and elementary algebra with the product rule ${\ddiv}_{\rm pw} (\varphi_z G_h)=\varphi_z{\ddiv}_{\rm pw} G_h + \nabla \varphi_z\cdot G_h \in P_k(\mathcal{T}(z))$ imply
\begin{align}\label{ineq:proof-GUB-efficiency-p=0-divergence}
r_z \coloneqq -\mathrm{div}_\mathrlap{\pd}{w} \sigma_z - f_z = - \varphi_z \mathrm{div}_\mathrlap{\pd}{w} G_h- \Pi_{k+p}(\varphi_zf)\in P_{k+p}(\mathcal{T}(z)).
\end{align}
Recall the residual $\mathrm{Res}_z(v)$ with $v\in H^1(\Omega)$ for the vertex $z\in\mathcal V$ from \eqref{eq:compatibility}. The commuting diagram property \eqref{eq:commuting-diagram-I-RT}--\eqref{ineq:proof-GUB-efficiency-p=0-divergence} establish the identity
\begin{align*}
\mathrm{Res}_z(1) = -(\mathrm{div}_\mathrlap{\pd}{w} G_h + f, \varphi_z)_{L^2(\omega(z))} + \sum_{T \in \mathcal{T}(z)} \langle G_h \cdot \nu_T, \varphi_z\rangle_{L^2(\partial T)}.
\end{align*}
This, a piecewise integration by parts, and the property \eqref{eqn:HHO_solution_property} verify $\mathrm{Res}(1) = (G_h,\nabla \varphi_z)_{L^2(\omega(z))} - (f,\varphi_z)_{L^2(\omega(z))}=0$ for any interior vertex $z \in \mathcal{V}(\Omega)$.
Hence, Lemma \ref{lem:uniform_stability_mixed} applies for any vertex $z\in\mathcal V$ and provides
\begin{align}\label{eq:efficiencyHdiv}
\|\QQ_{z,h} - \mathcal{I}_{RT}(\varphi_z G_h) \|_{L^2(\omega(z))} \leq C_{\rm s}
\sup\limits_{
\substack{v \in H^{1}_*(\omega(z))\\\|\nabla v\|_{L^2(\omega(z))}=1}}
\mathrm{Res}_z(v)
\end{align}
for the local contributions $Q_{z,h}$ of $Q_p = \sum_{z \in \mathcal{V}} Q_{z,h}$ from \eqref{eq:discrete_minimisation}.
The identity $\mathcal I_{\rm RT}(\varphi_z G_h) = \varphi_z G_h$ for $p\geq 1$ from $\varphi_zG_h\in P_{k+1}(\mathcal{T}(z))\subset RT_{k+p}^{{\rm pw}}(\mathcal{T}(z))$ allows for a $k$- and $p$-robust efficiency control of the flux reconstruction error
\begin{align}\label{eqn:alt_etaeq_eff}
\cnst{cnst:efficiency-flux-reconstruction}^{-1} \|Q_p - G_h\|^2 \leq \|\nabla u-G_h\|^2 + \sum_{z \in \mathcal{V}} \osc{r}^2(\varphi_z f,\mathcal T(z))
\end{align}
with a constant $\newcnst\label{cnst:efficiency-flux-reconstruction}$ that solely depends on the shape regularity of $\mathcal{T}$.
This is deemed noteworthy and motivates two different approaches for the bound of the residual $\mathrm{Res}_z(v)$ on the right-hand side of \eqref{eq:efficiencyHdiv} for $p=0$ and $p \geq 1$ below.\medskip
\emph{Step 3.1} provides \eqref{ineq:proof-GUB-efficiency-flux} for $p=0$.
Given any normalized $v \in H^1_*(\omega(z))$ with $\|\nabla v\|_{L^2(\omega(z))}=1$,
the product $\mathcal{I}_{RT}(\varphi_z G_h) \cdot \nu_F\,v$ vanishes on any boundary facet $F \in \mathcal{F}(\partial \omega(z))$ of the patch $\omega(z)$.
This, the commuting diagram property \eqref{eq:commuting-diagram-I-RT}, and \eqref{ineq:proof-GUB-efficiency-p=0-divergence} with $\varphi_z{\ddiv}_{\rm pw} G_h \in P_k(\mathcal{T}(z))$ in the definition of the residual $\mathrm{Res}_z(v)$ from \eqref{eq:compatibility} verify
\begin{align}\label{ineq:proof-efficiency-flux-p=0-Res}
\mathrm{Res}_z(v) &= -(\varphi_z f + \varphi_z{\ddiv}_{\rm pw} G_h,\Pi_k v)_{L^2(\omega(z))}\nonumber\\
&\qquad\qquad+ \sum_{F\in\mathcal{F}(z) \cap \mathcal{F}(\Omega)} \langle \varphi_z [G_h]_F\cdot\nu_F, \Pi_{F,k} v\rangle_{L^2(F)}.
\end{align}
The shape regularity of $\mathcal{T}$, a Poincar\'e inequality for interior vertices $z \in \mathcal{V}(\Omega)$, and a Friechrichs inequality for boundary vertices $z \in \mathcal{V}(\partial \Omega)$ provide
\begin{align}\label{ineq:proof-efficiency-GUB-p=0-scaling}
\|h_\mathcal{T}^{-1} v\|_{L^2(\omega(z))} \approx \mathrm{diam}(\omega(z))^{-1} \|v\|_{L^2(\omega(z))} \lesssim \|\nabla v\|_{L^2(\omega(z))} = 1.
\end{align}
Given any facet $F \in \mathcal{F}(z) \cap \mathcal{F}(\Omega)$ in the facet spider $F\in\mathcal{F}(z)$ with facet patch $\omega(F)$,
a trace inequality thus shows \begin{align}
\label{eqn:v_F_bound1}h_F^{-1/2}\|v\|_{L^2(F)} \lesssim \|h_\mathcal{T}^{-1} v\|_{L^2(\omega(F))}+\|\nabla v\|_{L^2(\omega(F))}\lesssim 1.
\end{align}
Cauchy inequalities on the right-hand side of \eqref{ineq:proof-efficiency-flux-p=0-Res}, the stability of the $L^2$ projection,
$\|\varphi_z\|_{L^\infty(\omega_z)} = 1$, and \eqref{ineq:proof-efficiency-GUB-p=0-scaling}--\eqref{eqn:v_F_bound1} prove that $\mathrm{Res}_z(v)$ is controlled by
\begin{align*}
\|h_\mathcal{T}(f + {\ddiv}_{\rm pw} G_h)\|_{L^2(\omega(z))}+
\Big(\sum_{F \in \mathcal{F}(z) \cap \mathcal{F}(\Omega)} h_F\|[G_h]_F \cdot \nu_F\|_{L^2(F)}^2\Big)^{1/2}.
\end{align*}
The efficiency of those residual terms follows from the proof of Theorem \ref{thm:HHO_equivalence} in Section \ref{sec:residual}. In combination with \eqref{eqn:Q_q_small} and \eqref{eq:efficiencyHdiv}, this results in the global efficiency \eqref{ineq:proof-GUB-efficiency-flux} for $p = 0$ and concludes the proof of Step 3.1.\medskip
\emph{Step 3.2} provides \eqref{ineq:proof-GUB-efficiency-flux} for $p\geq 1$.
Given any normalized $v \in H^1_*(\omega(z))$ with $\|\nabla v\|_{L^2(\omega(z))}=1$, \eqref{ineq:proof-GUB-efficiency-p=0-divergence} and the identity $\mathcal I_{\rm RT}(\varphi_z G_h) = \varphi_z G_h$ in the definition of the residual $\mathrm{Res}_z(v)$ from \eqref{eq:compatibility} verify that $\mathrm{Res}_z(v)$ is equal to
\begin{align*}
- (\varphi_z \mathrm{div}_\mathrlap{\pd}{w} G_h + \Pi_{k+p}(\varphi_z f) , v)_{L^2(\omega(z))} + \sum_{T \in \mathcal{T}(z)} \langle G_h \cdot \nu_T, \varphi_z v \rangle_{L^2(\partial T)}.
\end{align*}
This, a piecewise integration by parts, and the product rule $\nabla(\varphi_z v)=\varphi_z\nabla v+ v\nabla\varphi_z$ reveal
\begin{align}
\mathrm{Res}_z(v) %
%
&= - (\Pi_{k+p}(\varphi_z f) , v)_{\Lz}
+ ( G_h, \nabla (\varphi_z v))_{\Lz}.
\label{eq:Res_z(v)}
\end{align}
The weak formulation \eqref{eq:poisson} with the test function $\varphi_z v \in H^1_0(\omega(z))\subset V$ shows $(\nabla u, \nabla(\varphi_z v))_{L^2(\omega(z))} = (f, \varphi_z v)_{L^2(\omega(z))}$. Consequently, \eqref{eq:Res_z(v)} implies
\begin{align*}
\mathrm{Res}_z(v) =&\,
((1 - \Pi_{k+p})(\varphi_z f), v)_{\Lz}
- (\nabla u - G_h, \nabla (\varphi_z v))_{\Lz}.
\end{align*}
This, a Cauchy-Schwarz, and a piecewise Poincar\'e inequality with the normalization $\|\nabla v\|_{L^2(\omega(z))} = 1$ provide
\begin{align}
\label{ineq:upper-bound-Res_z(v)}
\mathrm{Res}_z(v)
& \leq
\|\nabla u - G_h\|_{\Lz}
\|\nabla(\varphi_zv)\|_{L^2(\omega(z))}
+ C_P\osc{k+p}(\varphi_z f,\mathcal{T}(z)).
\end{align}
Since $\|\nabla \varphi_z\|_{L^\infty(\omega(z))}\approx h^{-1}$, the Leibniz rule, a triangle inequality, and \eqref{ineq:proof-efficiency-GUB-p=0-scaling} show that
\begin{align*}
\|\nabla(\varphi_zv)\|_{L^2(\omega(z))}&\leq \|\varphi_z\|_{L^\infty(\omega(z))}\|\nabla v\|_{L^2(\omega(z))}+\|\nabla \varphi_z\|_{L^\infty(\omega(z))}\|v\|_{L^2(\omega(z))}\\ &\lesssim \|\nabla v\|_{L^2(\omega(z))} = 1
\end{align*}
can be bounded by a constant independent of $h$.
Thus, the combination of \eqref{eqn:Q_q_small} with \eqref{eq:efficiencyHdiv} and \eqref{ineq:upper-bound-Res_z(v)} results in the efficiency of $\|Q_p - G_h\|$ in \eqref{eqn:alt_etaeq_eff}
with a $k$- and $p$-robust constant \cnst{cnst:efficiency-flux-reconstruction}.
Since $\varphi_z \Pi_{k-1} f \in P_k(T) \subset P_{k+p}(T)$, the Pythagoras theorem and $\|\varphi_z\|_{L^\infty(\omega(z))} = 1$ show
$\|(1 - \Pi_{k+p})(\varphi_z f)\|_{L^2(T)} \leq \|\varphi_z(1 - \Pi_{k-1}) f\|_{L^2(T)} \leq \|(1-\Pi_{k-1})f\|_{L^2(T)}$ for all $T \in \mathcal{T}(z)$,
whence
$$\osc{k+p}(\varphi_z f,\mathcal{T}(z)) \leq \osc{k-1}(f,\mathcal{T}(z)).$$
This and Lemma \ref{lem:local-data-oscillation} implies the efficiency
$$\osc{k+p}(\varphi_z f, \mathcal T(z))
\lesssim \|\nabla_{\pw}(u - R u_h)\|_{L^2(\omega(z))} + \osc{q}(f,\mathcal T(z))$$
of the local oscillations from \eqref{ineq:upper-bound-Res_z(v)} for any $q\in\mathbb N_0$ as in Remark \ref{rem:local_osc_eff}.
This and \eqref{eqn:alt_etaeq_eff} prove \eqref{ineq:proof-GUB-efficiency-flux} for $p \geq 1$ and conclude the proof of Step 3.2.\medskip
\emph{Step 4} affirms the efficiency of the flux reconstruction
\begin{align}\label{ineq:proof-efficiency-GUB-flux-k=0}
\cnst{cnst:efficiency-flux-k=0}^{-1}\|Q_p - G_h\| \leq \trb{u - R u_h}_\mathrlap{\pd}{w} + \osc{q}(f,\mathcal{T})
\end{align}
for the polynomial degree $k = 0$ and any $q\in\mathbb N_0$.
Let $\widetilde{u} \in V$ solve the Poisson model problem $-\Delta \widetilde{u} = \Pi_0 f$ with piecewise constant right-hand side $\Pi_0 f \in P_0(\mathcal{T})$.
A careful inspection reveals that all arguments from Step 3 apply to the case $k = 0$ for $f$ replaced by $\Pi_0 f$.
This leads to $\|Q_p - G_h\| \leq \cnst{cnst:efficiency-flux-k=0}\trb{\widetilde{u} - R u_h}_\mathrlap{\pd}{w}$
with a constant $\newcnst\label{cnst:efficiency-flux-k=0}$ that solely depends on the shape of $\mathcal{T}$ (because the data oscillations on the right-hand side of \eqref{eqn:alt_etaeq_eff} vanish).
Therefore, the triangle inequality $\trb{\widetilde{u} - R u_h}_\mathrlap{\pd}{w} \leq \trb{u - \widetilde{u}} + \trb{u - R u_h}_\mathrlap{\pd}{w}$, the standard bound $\trb{u - \widetilde{u}} \leq C_P\osc{0}(f,\mathcal{T})$, and $C_P = 1/\pi < 1$ on simplicial domains lead to \eqref{ineq:proof-efficiency-GUB-flux-k=0}.
This and the efficiency of the data oscillations from Lemma \ref{lem:local-data-oscillation} conclude the proof of Step 4.
Notice that the constant $\cnst{cnst:efficiency-flux-k=0}$ is independent of the parameter $p$.
\medskip
\emph{Step 5} finishes the proof.
On the one hand,
the reliability $\trb{u - R u_h}_\mathrlap{\pd}{w} + \osc{k-1}(f,\mathcal{T}) \lesssim \eta_{\mathrm{eq},p}$ is established in Step 1--2.
On the other hand, the efficiency of the averaging operator from Theorem \ref{thm:averaging-efficiency}, Lemma \ref{lem:local-data-oscillation}, and the efficiency of the flux reconstruction in \eqref{ineq:proof-GUB-efficiency-flux} for $k \geq 1$ and in \eqref{ineq:proof-efficiency-GUB-flux-k=0} for $k = 0$ imply the efficiency
$\eta_{\mathrm{eq,p}} \lesssim \trb{u - R u_h}_\mathrlap{\pd}{w} + \osc{q}(f,\mathcal{T})$ for any $q \in \mathbb{N}_0$.
This concludes the proof of Theorem \ref{thm:GUB-HHO}.\qed
\section{Introduction}
\label{intro}
Hybrid high-order methods (HHO) were introduced in \cite{Di-Pietro.Ern:15,Di-Pietro.Ern.ea:14} and are examined in the textbooks \cite{DiPietroDroniou2020,ern_finite_2021-2}
as a promising class of flexible nonconforming discretization methods for partial differential equations
that involve a parameter-free stabilization term for the link between the volume and skeletal variables.
\subsection{Known a posteriori error estimator}
The a priori error analysis of HHO involves the stability terms in extended norms as part of the methodology and motivated a first explicit residual-based
a~posteriori error estimator in \cite{DiPietroDroniou2020} with a reformulation of the stabilization in the upper bound.
Let $s_h(u_h,u_h)$ denote the stabilization at the discrete solution $u_h\in V_h$ and let the (elliptic) reconstruction $Ru_h$ of $u_h$
denote a piecewise polynomial of degree at most $k+1$ that approximates $u\in H^1(\Omega)$, cf.\ \eqref{eqn:s_h0} and Section \ref{sec:residual} below for further details.
Then a possible error term reads
\begin{align}\label{eqn:tot_error}
\textup{total error}^2:=\| \nabla_\textup{pw}(u-Ru_h)\|_{L^2(\Omega)}^2 +s_h(u_h,u_h).
\end{align}
It is disputable if $s_h(u_h,u_h)\geq0$ is an error contribution, but
if the total error includes $s_h(u_h,u_h) $ (or an equivalent form), then the error estimator
may also include this term (or a computable equivalent) for a reliable and efficient a~posteriori error control.
Amongst the many skeletal schemes like (nonconforming) virtual elements, hybridized (weak) discontinuous Galerkin schemes et al., the HHO methodology has a clear and efficacious stabilization
\begin{align}\label{eqn:s_h0}
s_h(v_h, w_h) \coloneqq \sum_{T\in\mathcal{T}}\sum_{F\in\mathcal{F}(T)} h_F^{-1}\langle S_{TF} v_h, S_{TF} w_h\rangle_{L^2(F)}
\end{align}
with the abbreviation
$
S_{TF} v_h
\coloneqq \Pi_{F,k}
\left(
v_{\mathcal{T}} +
(1-\Pi_{T,k})R v_{h}
\right)|_{T}-v_{\mathcal{F}}|_{F}
$
for $v_h=(v_\mathcal{T}, v_\mathcal{F})\in V_h$ in terms of the $L^2$ projections $\Pi_{K,k}$ onto polynomials of degree at most $k$ on a facet or simplex $K\in\mathcal{F}\cup\mathcal{T}$ of diameter $h_K = \mathrm{diam}(K)$; cf. Subsection \ref{sub:notation} for further details.
The original residual-based estimator $\eta_{\rm HHO}$ from the textbook \cite{DiPietroDroniou2020} for the Poisson model problem $-\Delta u = f$ includes \eqref{eqn:s_h0} and an interpolation $\mathcal{A} Ru_h\in V$ of $Ru_h$ by nodal averaging in
\begin{align*}
\eta_{\rm HHO}^2 =\ & \| h_{\mathcal{T}}(1 - \Pi_0)(f+{\Delta}_{\rm pw} Ru_h)\|_{L^2(\Omega)}^2
+ \|\nabla_{\pw} (1-\mathcal{A})Ru_h\|_{L^2(\Omega)}^2\\
&+ s_h(u_h, u_h).
\end{align*}
(Multiplicative constants are undisplayed in this introduction for simplicity.)
The results from Theorem 4.3 and 4.7 in \cite{DiPietroDroniou2020} show reliability and efficiency for the total error \eqref{eqn:tot_error} and piecewise polynomial source terms $f\in P_{k+1}(\mathcal{T})$,
$$\text{total error}^2 \approx \eta_{\rm HHO}^2.$$
\subsection{Stabilization-free a posteriori error control}
There are objections against
the double role of $s_h(u_h,u_h)$ on both sides of the efficiency and reliability estimate. First, the term $s_h(u,u_h) $ may dominate both sides of the error estimate.
In other words, the total error might be equivalent to $s_h(u_h,u_h)$,
but the quantity of interest may exclusively be
\[
\textup{error}^2:=\| \nabla_\textup{pw}(u-Ru_h)\|_{L^2(\Omega)}^2.
\]
Second, since the stabilization \eqref{eqn:s_h0} incorporates a negative power of the mesh-size, a reduction property for local refinements remains unclear but is inevitable in the proofs of optimal convergence of an adaptive algorithm \cite{bertrand_opt,carstensen_axioms_2014}.
This paper, therefore, asks a different question about the control of the $\textup{error}$
without the stabilization term \eqref{eqn:s_h0} in the upper bound and introduces two stabilization-free error estimators (multiplicative constants are undisplayed)
\begin{align*}
\eta_{\text{res}}^2=\ & \|h_\mathcal{T}( f+{\Delta}_{\rm pw} Ru_h)\|_{L^2(\Omega)}^2
+\sum_{F\in {\mathcal F}} h_F\| [ \nabla_{\pw} Ru_h ]_F \|_{L^2(F)}^2,\\
\eta_{\text{eq},p}^2=\ & \osc{k+p}^2(f, \mathcal{T}) + \| Q_p - \nabla_\mathrlap{\pd}{w} R u_h \|_{L^2(\Omega)}^2 + \|\nabla_{\pw} (1-\mathcal{A})Ru_h\|_{L^2(\Omega)}^2
\end{align*}
for some parameter $p \in \mathbb{N}_0$.
The explicit residual-based a posteriori error estimator $\eta_{\rm res}$ follows from the a posteriori methodology in the spirit of \cite{b9ffd40a,c6537ecf,5be62542,normOfdGrad4Hdiv2015ccdpas} with a piecewise volume residual $f+\Delta_\mathrlap{\pd}{w} Ru_h$ and the jumps $[\nabla_{\pw} Ru_h]_F$ across a facet $F$ (on the boundary this is only the tangential component of $\nabla_{\pw} Ru_h$).
The equilibrated error estimator $\eta_{\mathrm{eq},p}$ includes the post-processed quantity $Q_p \in RT_{k+p}(\mathcal{T})$ in the space $ RT_{k+p}(\mathcal{T})$ of Raviart-Thomas functions of degree $k+p$ for $p \in \mathbb{N}_0$ and the nodal average $\mathcal{A} R u_h \in S^{k+1}_0(\mathcal{T})$ of $R u_h$.
The main results establish reliability and efficiency
$$\text{error}^2 + \osc{k-1}^2(f, \mathcal{T}) \lesssim \eta_{\rm res}^2 \approx \eta_{\mathrm{eq},p}^2\lesssim \text{error}^2 + \osc{q}^2(f, \mathcal{T})$$
for any $p, q\in \mathbb{N}_0$
up to data-oscillations $\osc{q}^2(f, \mathcal{T})\coloneqq \|h_\mathcal{T} (1-\Pi_{q})f\|^2_{L^2(\Omega)}$ of the source term $f\in L^2(\Omega)$ and {\em without any stabilization terms}.
Computational benchmarks with adaptive mesh-refinement driven by any of these estimators provide numerical evidence for optimal convergence rates.
\subsection{Further contributions and outline}
The higher-order Crouzeix-Raviart finite element schemes are complicated at least in 3D \cite{Ciarlet2018} and then the HHO methodology is an attractive
alternative even for simplicial triangulations with partly unexpected advantages like the computation of higher-order guaranteed eigenvalue bounds
\cite{CEP21}. Higher convergence rates rely on an appropriate adaptive mesh-refining algorithm and hence stabilization-free
a posteriori error estimators are of particular interest. The recent paper \cite{daveiga2021adaptive} establishes the latter for virtual elements with an over-penalization strategy
as an extension of \cite{bonito_quasi-optimal_2010} for the discontinuous Galerkin schemes. A disadvantage is the quantification of the restriction on the stabilization parameter in
practise and poor condition for larger parameters. The stabilization-free a posteriori error control in this paper is based on two observations for the HHO schemes on simplicial triangulations.
First, the $P_1$-conforming finite element functions let the stabilization vanish and, second, the divergence-free lowest-order Raviart-Thomas functions are $L^2$ perpendicular to the piecewise gradients $\nabla_{\pw} R u_h$. In fact, those two fairly general properties lead in Section 2 to a reliable explicit residual-based a posteriori error estimator.
In contrast to the simplified introduction above, the paper also focuses on multiplicative constants that lead to the GUB
$$\textup{error}\leq \eta_{\rm res} \quad\text{and}\quad \textup{error}\leq \eta_{\mathrm{eq},p};$$
cf. Table \ref{tab:expl_const} for explicit quantities and Theorem \ref{thm:reliability} and Theorem \ref{thm:GUB-HHO} for further details.
\begin{table}
\centering
\begin{tabular}{c|ccc}
$\omega_{\rm max}$ & $\pi$ & $3\pi/2$ & $2\pi$\\\hline
$M_{\rm bd}$ & 4 & 6 & 8\\
$c_{\rm apx}$ & 2.9568 & 6.4642 & 11.3771\\\hline
$C_{\rm st}$ & 26.0893 & 55.8498 & 97.5374\\
$\newcnst\label{cnst:C1}$ & 2.9718 & 6.4710 & 11.3810\\
$\newcnst\label{cnst:C2}$ & 7.0495 & 15.2341 & 26.7317
\end{tabular}
\caption{Explicit constants $\cnst{cnst:C1}, C_{\rm st}, \cnst{cnst:C2}$ for right-isosceles triangles with respect to the maximum interior angle $\omega_{\rm max}$ of the polygonal domain $\Omega$.}
\label{tab:expl_const}
\end{table}
Numerical comparisons of $\eta_{\rm HHO}$ with $\eta_{\rm res}$ and $\eta_{\mathrm{eq},p}$ favour the latter.
Section \ref{sec:blocks} identifies general building blocks of the a posteriori error analysis for discontinuous schemes with emphasis on explicit constants.
An application to HHO leads to the new stabilization-free residual-based estimator $\eta_{\rm res}$
in Section \ref{sec:residual}.
The alternative stabilization-free error estimator $\eta_{\mathrm{eq},p}$ follows from an equilibration strategy plus post-processing in Section \ref{sec:eq}.
This paper also contributes to the HHO literature a local equivalence of two stabilizations and the efficiency of the stabilization terms up to data-oscillations in extension of \cite{ErnZanotti2020}.
Numerical comparisons of the different error estimators and an error estimator competition for guaranteed error control of the piecewise energy norm in 2D conclude this paper in Section \ref{sec:Numerical results}.
Three computational benchmarks provide striking numerical evidences for the optimality of the associated adaptive algorithms.
The appendix provides algorithmic details on the computation of the post-processed contribution $\|Q_p-\nabla_{\pw} Ru_h\|_{L^2(\Omega)}$ in $\eta_{{\rm eq}, p}$.
\subsection{Overall notation}\label{sub:notation}
Standard notation for Sobolev and Lebesgue spaces and norms apply with $\|\bullet\|\coloneqq\|\bullet\|_{L^2(\Omega)}$ and $\trb{\bullet}\coloneqq\|\nabla\bullet\|_{L^2(\Omega)}$.
In particular,
$H(\ddiv, \Omega)$ is the space of Sobolev functions with weak divergence in $L^2(\Omega)$ and $H(\ddiv=0, \Omega)$ contains only divergence-free functions in $H(\ddiv, \Omega)$.
Throughout this paper, $\mathcal{T}$ denotes a shape-regular triangulation of the polyhedral bounded Lipschitz domain $\Omega\subset \mathbb R^n$ into $n$-simplices with facets $\mathcal{F}$ (edges for $n=2$ and faces for $n=3$) and vertices $\mathcal V$.
Let $\mathcal{F}(\Omega)$ (resp.~$\mathcal{V}(\Omega)$) denote the set of interior facets (resp.~vertices).
Given $v \in \Omega \to \mathbb{R}^n$ and $w : \Omega \to \mathbb{R}^{2n-3}$, let $\text{curl } v \coloneqq \partial_1 v_2 - \partial_2 v_1$ and $\Curl w \coloneqq (\partial_2 w, -\partial_1 w)^t$ if $n = 2$ and $\text{curl } v \coloneqq (\partial_2 v_3 - \partial_3 v_2, \partial_3 v_1 - \partial_1 v_3, \partial_1 v_2 - \partial_2 v_1)^t$ and $\Curl w \coloneqq \text{curl } w$ if $n = 3$.
For $s\in\mathbb R$, let $H^s(\mathcal{T})$, $H(\ddiv,\mathcal{T})$, and $H(\text{curl },\mathcal{T})$ denote the space of piecewise Sobolev functions with restriction to $T \in \mathcal{T}$ in $H^s(T)$, $H(\ddiv,T)$, and $H(\text{curl },T)$.
To simplify notation, $H^s(K)$ abbreviates $H^s(\mathrm{int}(K))$ for the open interior $\mathrm{int}(K)$ of a compact set $K$.
The $L^2$-scalar product reads \((\bullet, \bullet)_{L^{2}(\omega)}\) for volumes $\omega\subseteq\Omega$ and \(\langle\bullet, \bullet\rangle_{L^2(\gamma)}\) for surfaces $\gamma\subset \overline{\Omega}$ of co-dimension one; the same symbol applies to scalars and to vectors.
For $V \coloneqq H^1_0(\Omega)$ and $V^* \coloneqq H^{-1}(\Omega)$, let $\langle\bullet, \bullet\rangle$ denote the duality-brackets in $V^* \times V$ for the dual space $V^*$ of $V$ equipped with the operator norm
$\trb{F}_{*}\coloneqq \sup_{v\in V\setminus\{0\}}|F v|/\trb{v}$ for $F\in V^*$.
Define the energy scalar product $a(v,w) \coloneqq (\nabla v, \nabla w)_{L^2(\Omega)}$ for $v, w\in H^1(\Omega)$ and its piecewise version $a_{\pw}(v, w)=(\nabla_{\pw} v,\nabla_{\pw} w)_{L^2(\Omega)}$ for $v, w\in H^1(\mathcal{T})$.
The latter induces the piecewise energy norm $\trb{\bullet}_\mathrlap{\pd}{w} \coloneqq a_\mathrlap{\pd}{w}(\bullet,\bullet)^{1/2}$ in $H^1(\mathcal{T})$.
Here and throughout the paper, $\nabla_{\pw}$, ${\ddiv}_{\rm pw}$, $\text{curl }_\mathrlap{\pd}{w}$, ${\Delta}_{\rm pw}$, denote the piecewise evaluation of the differential operators $\nabla$, $\ddiv$, $\text{curl }_\mathrlap{\pd}{w}$, $\Delta$ without explicit reference to the underlying shape-regular triangulation $\mathcal{T}$.
The vector space $P_k(K)$ of polynomials of degrees at most $k\in{\mathbb N}_0$ over a facet or simplex $K\in\mathcal{F}\cup\mathcal{T}$ defines the piecewise polynomial spaces
\begin{align*}
P_k(\mathcal{T}) &\coloneqq\{ p \in L^2(\Omega) \ :\ p_{|T} \in \dPT{k} \text{ for all } T \in \mathcal{T} \}, \\
P_k(\mathcal{F}) &\coloneqq
\{ p \in L^2(\mathcal F) \ :\ p_{|F} \in P_k(F) \text{ for all } F \in \mathcal{F}
\} \
\end{align*}
and the space of piecewise Raviart-Thomas functions
\begin{align*}
RT_{k}^\mathrlap{\pd}{w}(\mathcal{T}) \coloneqq P_k(\mathcal{T};\mathbb R^n) + xP_k(\mathcal{T}).
\end{align*}
The associated $L^2$ projections read
$\Pi_{K,k}:L^2(\Omega)\to P_k(K), \Pi_k : L^2(\Omega) \rightarrow P_k(\mathcal{T}),$ and $\Pi_{\mathcal F, k} : L^2(\Omega) \rightarrow P_k(\mathcal{F})$ with the convention $\Pi_{-1}\coloneqq0$.
Abbreviate $S^{k+1}_0(\mathcal{T}) \coloneqq P_{k+1}(\mathcal{T}) \cap V$ and $RT_k(\mathcal{T}) \coloneqq RT^\mathrlap{\pd}{w}_k(\mathcal{T}) \cap H(\ddiv,\Omega)$ for all $k \in \mathbb{N}_0$.
The piecewise constant mesh-size function $h_\mathcal{T}\in P_0(\mathcal{T})$ satisfies $h_{\mathcal{T}|T}\coloneqq h_T$ for $T\in\mathcal{T}$ with the diameter $h_K \coloneqq \mathrm{diam}(K)\in P_0(K)$ of $K\in \mathcal{F}\cup\mathcal{T}$.
If not explicitly stated otherwise, constants are independent of the mesh-size in the triangulation but may depend on the shape-regularity and on the polynomial degree $k$.
The abbreviation $A\lesssim B$ hides a generic constant $C$ (independent of the mesh-size) in $A\leq C\; B$; $A\approx B$ abbreviates $A\lesssim B\lesssim A$.
\section{Foundations of the a~posteriori error analysis}
\label{sec:blocks}
This section investigates general building blocks of the a posteriori error analysis and revisits arguments from \cite{b9ffd40a,c6537ecf,5be62542,normOfdGrad4Hdiv2015ccdpas} with emphasis on multiply connected domains $\Omega\subset \mathbb R^n$ for $n=2,3$.
The general setting of this section results in reliability for an error estimator that is applicable beyond the HHO methodology.
Consider the weak solution $u\in V = H^1_0(\Omega)$ to the Poisson model problem $-\Delta u=f$ a.e.~in $\Omega$ for a given source $f\in L^2(\Omega)$; i.e., $u\in V$ satisfies
\begin{align}
\label{eq:poisson}
a(u,v)= (f,v)_{L^2(\Omega)} \quad \text{for all } v\in V .
\end{align}
An approximation ${G} \in L^2(\Omega;\mathbb R^n)$ of the gradient $\nabla u\in H(\ddiv, \Omega)$ gives rise to the residual $f+\ddiv {G}\in V^* = H^{-1}(\Omega)$ seen as a linear functional on $V$, i.e.,
\[
\langle f+\ddiv {G},\varphi\rangle := (f,\varphi)_{L^2(\Omega)} - ({G}, \nabla \varphi)_{L^2(\Omega)}
\quad\text{for all }\varphi\in V.
\]
Let $\nu_T$ denote the unit outer normal along the boundary $\partial T$ of each simplex $T\in\mathcal{T}$ and fix the orientation of a unit normal $\nu_F=\pm \nu_T$ for each facet $F\in\mathcal{F}(T)$ of $T$ such
that it matches the outer unit normal $\nu$ of $\partial \Omega$ at the boundary.
The jump $[{G}]_F$ of a piecewise function in $m\in{\mathbb N}$ components ${G}\in H^1(\mathcal{T};\mathbb R^m)$ reads $[{G}]_F\coloneqq {G}_{|T_+}-{G}_{|T_-}$ on interior facets $F=T_+\cap T_-\in\mathcal{F}(\Omega)$ (with $T_\pm$ labelled such that $\nu_{T_+|F} = \nu_F = -\nu_{T_-|F})$ and $[{G}]_F\coloneqq {G}$ on the boundary $F\in\mathcal{F}(\partial \Omega)$.
The main result of this section establishes the residual-based error estimator
\begin{align}
\eta^2(\mathcal{T}, {G}):= & \left(\cnst{cnst:C1}\|h_\mathcal{T}( f+{\ddiv}_{\rm pw} {G})\|
+\cnst{cnst:C2}\sqrt{\sum_{F\in {\mathcal F}(\Omega)} \ell(F)\| [ {G} ]_F\cdot \nu_F \|_{L^2(F)}^2}\right)^2\label{eq:eta}\\
&+C_{\rm H}^2\left(\cnst{cnst:C1}\| h_\mathcal{T}{\text{curl }}_{\rm pw} {G} \|
+\cnst{cnst:C2}\sqrt{\sum_{F\in {\mathcal F}} \ell(F)\| [ {G} ]_F\times \nu_F \|_{L^2(F)}^2}\right)^2\notag
\end{align}
as a GUB $\|\nabla u - {G}\| \le \eta(\mathcal{T}, {G})$ under minimal assumptions on the approximation ${G}\in H^1(\mathcal{T};\mathbb R^n)\subset L^2(\Omega; \mathbb R^n)$. The constants $C_1$, $C_2$, and $C_H$ (or upper bounds thereof) are computable; cf. Table \ref{tab:expl_const} for an example in 2D with details in Example \ref{ex:constants} at the end of Section \ref{sec:blocks}.
The first assumption is a weakened discrete solution property \begin{equation}\label{eq:solution_property}
({G}, \nabla w_C)_{L^2(\Omega)}=(f,w_C)_{L^2(\Omega)}\quad\text{for all }w_C\in S_0^1(\mathcal T).
\end{equation}
The second assumption is the orthogonality to the lowest-order divergence-free Raviart-Thomas functions
\begin{align}\label{eq:div_0_property}
({G}, r)_{L^2(\Omega)}=0\quad\text{for all } r\in RT_0(\mathcal{T})\cap H(\ddiv=0, \Omega).
\end{align}
\begin{theorem}[residual-based GUB]
\label{thm:reliability}
Suppose that ${G}\in H^1(\mathcal T;\mathbb R^n)$ and $f\in L^2(\Omega)$ satisfy \eqref{eq:solution_property}--\eqref{eq:div_0_property}.
Then the error estimator
$\eta(\mathcal{T}, {G})$ from \eqref{eq:eta}
is a GUB
$$
\|\nabla u-{G}\| \le \eta(\mathcal{T}, {G})
$$
of the error $\|\nabla u-{G}\|$ for the solution $u\in V$ to \eqref{eq:poisson}.
The constants $C_{\rm 1}, C_{2}, C_{\rm H}$ exclusively
depend on $\Omega$ and the shape-regularity of $\mathcal T$.
\end{theorem}
The remaining parts of this section are devoted to the proof of Theorem~\ref{thm:reliability} and the computation of (upper bounds of) the constants $C_{\rm 1}, C_{\rm 2}$, and $C_{\rm H}$ in \eqref{eq:eta}. %
The point of departure is the subsequent decomposition that appears necessary in the
nonconforming and mixed finite element a~posteriori error analysis.
It leads to a split of the error $\|\nabla u-{G}\|$ into some divergence part and some consistency part.
\begin{lemma}[decomposition]\label{lemmadecomposition}
Any $v\in V$ and ${G}\in L^2(\Omega; \mathbb R^n)$ satisfy the decomposition
\begin{align}\label{eqn:decomp}
\|\nabla v-{G} \|^2 = \trb{v-w}^2+\| {G}- \nabla w\|^2
\end{align}
with the (unique) minimizer $w\in V$ of the distance
\[
\delta:= \min_{\varphi \in V}\|{G}- \nabla \varphi\|
\]
of ${G}$ to the gradients $\nabla V$ of Sobolev functions.
The solution $u\in V$ to \eqref{eq:poisson} satisfies
\begin{align}
\mu\coloneqq&\;\trb{ f+\ddiv {G}}_{*}=\trb{u-w}&&\text{and} \notag\\\label{eqn:decomposition_full}
\|\nabla u-{G} \|
^2 =&\;\trb{ f+\ddiv {G}}_{*}^2+\| {G}- \nabla w\|
^2=\mu^2+\delta^2.
\end{align}
\end{lemma}
\begin{proof}
The minimizer $w \in V$ of $\|{G} - \nabla \varphi\|$ among $\varphi \in V$ satisfies the variational formulation $a(w, \varphi)_{L^2(\Omega)} = ({G},\nabla \varphi)_{L^2(\Omega)}$ for all $\varphi \in V$.
(Notice that $w$ is the unique weak solution to the Poisson model problem $-\Delta w = -\ddiv {G} \in V^*$.)
In particular, ${G}- \nabla w$ is $L^2$ orthogonal onto $ \nabla V$ and the Pythagoras
theorem proves \eqref{eqn:decomp}.
Given $\varphi\in V$ with $ \trb{\varphi} =1$, the orthogonality of ${G}-\nabla w$ to $\nabla\varphi$ and \eqref{eq:poisson} show
\begin{align}\label{eqn:split}
a(u-w,\varphi)_{L^2(\Omega)}= (\nabla u- {G},\nabla\varphi)_{L^2(\Omega)}=\langle f+\ddiv {G},\varphi\rangle
\end{align}
with the duality brackets $\langle\bullet, \bullet\rangle$ in $V^* \times V$.
Since the supremum of \eqref{eqn:split} over all $\varphi\in V$ with $ \trb{\varphi} = 1$ is equal to $\trb{u-w}= \trb{ f+\ddiv {G}}_{*} $,
this and \eqref{eqn:decomp} conclude the proof of \eqref{eqn:decomposition_full}.
\qed
\end{proof}
The split \eqref{eqn:decomp} of the error $\|\nabla u-{G} \|$ allows for and enforces a separate estimation of the equilibrium and consistency contribution in residual-based a posteriori error estimators.
In order to derive explicit constants, two lemmas are recalled.
The first has a long tradition in the a~posteriori error control in form of a Helmholtz
decomposition on simply connected domains \cite{c43f5cd9,alonso1996error} and introduces the constant $C_{\rm H}$ from Theorem \ref{thm:reliability}.
The following version includes the general case of multiply
connected domains as in \cite{GirRav:86} for $n=2$ or $n=3$ dimensions and weak assumptions
on a divergence-free function
$\varrho\in H(\ddiv=0,\Omega)$.
\begin{lemma}[Helmholtz-decomposition]\label{lem:lemmacc2a}
Suppose the divergence-free function $\varrho\in H(\ddiv=0,\Omega)$ is $L^2$ orthogonal onto $RT_0(\mathcal T)\cap H(\ddiv=0, \Omega)$.
Then there exists $\beta\in H^1(\Omega;\mathbb R^N)$, $N=2n-3$, such that any $\beta_C\in S^1(\mathcal T)^N$
satisfies
\begin{align}\label{eqn:lemmacc2a}
\| \varrho\|^2=\int_\Omega \varrho\cdot\mathrm{Curl}(\beta-\beta_C)\ \mathrm{d} x
\quad\text{and}\quad
\trb{\beta}\le C_{\rm H}\, \| \varrho\|.
\end{align}
The constant $C_{\rm H}>0$ exclusively depends on $\Omega$.
\end{lemma}
\begin{proof}
The compact polyhedral boundary $\partial\Omega$ of the bounded Lipschitz domain $\Omega$
has $J+1$ connectivity components $\Gamma_0,\dots,\Gamma_J$ for some finite
$J\in\mathbb N_0$. Those connectivity components have a positive surface measure $|\Gamma_j|$ and
a positive distance of each other. So the
integral mean
\[
\gamma_j:= \int_{\Gamma_j} \varrho\cdot\nu\, \ \mathrm{d} s /|\Gamma_j|
\]
is well defined and depends continuously on $\varrho\in H(\ddiv=0,\Omega)$ in the sense that
$|\gamma_j|\le c_1\| \varrho\|$ (recall $\ddiv\varrho=0$) for each $j=0,\dots, J$ and $c_1>0$. This constant $c_1$ and the constants $c_2,c_3,c_4$ below exclusively depend on the domain $\Omega$.
The finite real numbers
$\gamma_0,\dots,\gamma_J$ define the Neumann data for the harmonic
function $z\in H^1(\Omega)/\mathbb R$ with
\[
\Delta z=0 \text{ in }\Omega \quad\text{and}\quad
\partial z/\partial\nu = \gamma_j\text{ on }\Gamma_j \text{ for all }j=0,\dots, J.
\]
The elliptic regularity theory for polyhedral domains lead to $z\in H^{1+\alpha}(\Omega)$
for some $\alpha>1/2$ and $c_2>0$ with $\| z \|_{H^{1+\alpha}(\Omega)}\le c_2\, (|\gamma_0|+\dots +|\gamma_J|)$.
The Fortin interpolation operator defines a bounded linear operator on
$H(\ddiv,\Omega)\cap L^p(\Omega;\mathbb R^n)$ for $p>2$.
It is generally accepted that, for $\alpha>0$ and
$ \nabla z\in H(\ddiv=0,\Omega)\cap H^\alpha(\Omega;\mathbb R^n)$, the Fortin interpolation
$I_{\rm F} \nabla z \in RT_0(\mathcal T)\cap H(\ddiv=0,\Omega)$ is well defined
and $\| I_{\rm F} \nabla z\|\le c_3 \| \nabla z\|_{H^\alpha(\Omega)}$ follows for some $c_3>0$.
The additional property
$ \nabla z\in L^p(\Omega;\mathbb R^n)$ for some $p>2$ allows the definition of
$\int_F \nabla z\cdot \nu_F \ \mathrm{d} s$ as a Lebesgue integral over a facet $F\in\mathcal{F}$.
One consequence for the boundary facets is the vanishing integral
\[
\int_{\Gamma_j} (\varrho-I_{\rm F} \nabla z)\cdot\nu\, \mathrm{d} s =0\quad\text{for all }j=0,\dots, J.
\]
Since $\varrho-I_{\rm F} \nabla z\in H(\ddiv=0,\Omega)$ is divergence-free, Theorems 3.1 and 3.4 in \cite{GirRav:86} prove the existence of $c_4>0$ and $\beta\in H^1(\Omega;\mathbb R^N)$
with
\[
\varrho=I_{\rm F} \nabla z+ \Curl\beta
\quad\text{and}\quad
\trb{\beta}\le c_4 \| \varrho-I_{\rm F} \nabla z\|.
\]
Recall that $\varrho\perp I_{\rm F}\nabla z$ and
$\varrho\perp\text{curl }\beta_C\in RT_0(\mathcal{T})\cap H(\ddiv=0, \Omega)$. This concludes the proof of
\eqref{eqn:lemmacc2a}
with $C_{\rm H}\coloneqq\sqrt{1 + (c_1c_2c_3(1+J))^2}c_4$.\qed
\end{proof}
The subsequent version of the trace inequality on the facets $\mathcal{F}$ leads to the piecewise constant $\ell\in P_0(\mathcal{F})$ defined by
$$\ell(F)\coloneqq \begin{cases}{}
(n+1)h_{T}^{2}|F|/|T|&\text{for }F\in\mathcal{F}(\partial\Omega)\cap \mathcal{F}(T),\\
(n+1) |F|/(h_{T_+}^{-2}|T_+| + h_{T_-}^{-2}|T_-|)&\text{for }F=\partial T_+\cap \partial T_-\in\mathcal{F}(\Omega).
\end{cases}$$
\begin{lemma}[trace inequality]\label{lem:Skeletal trace inequality}
Any $f\in H^1(\Omega)$ satisfies
$$\sum^{}_{F\in\mathcal{F}} \ell(F)^{-1}\left\Vert f\right\Vert_{L^2(F)}^{2} \leq \left\Vert
h_\mathcal{T}^{-1}f\right\Vert
%
^{2} + \frac{2C_{\rm tr}}{n}\left\Vert
h_\mathcal{T}^{-1}f\right\Vert
%
\ENorm{f}$$
with the constant $C_{\rm tr}\coloneqq \max_{T\in\mathcal{T}}\max_{x\in T}|x-\mathrm{mid}(T)|/h_T<n/(n+1)$.
\end{lemma}
\begin{proof}
The center of inertia $\mathrm{mid}(T)=\sum_{j=0}^n P_j/(n+1)$ of the $n$-simplex $T=\mathrm{conv}\{P_0, ..., P_n\}\in\mathcal{T}$ and the $n+1$ faces $F_j=\mathrm{conv}\{P_0, ..., P_{j-1},P_{j+1}, ..., P_n\}\in \mathcal F(T)$ for $j=0, ..., n$ give rise to the decomposition of $T$ into $n+1$ sub-simplices $T_j' = \mathrm{conv}(F_j, \mathrm{mid}(T))$ with volume
%
$|T_j'|=|T|/(n+1)$.
%
%
%
%
%
Standard arguments like the trace identity on $T_j'\subset T$ \cite[Lemma
2.1]{carstensen_explicit_2012} for $|f|^2\in
W^{1,1}(T')$ and a Cauchy inequality show
$$\frac1{|F_j|}\left\Vert f\right\Vert_{L^2(F_j)}^{2}\leq \frac{1}{|T_j'|}\left\Vert
f\right\Vert_{L^2(T_j')}^{2}+ \frac{2}{n|T_j'|}\max_{x\in T_j'}|x-\mathrm{mid}(T)|\left\Vert
f\right\Vert_{L^2({T_j'})}^{}\|\nabla f\|_{L^2(T')}.$$
The distance $\max_{x\in T_j'}|x-\mathrm{mid}(T)|=|P_k - \mathrm{mid}(T)|\leq C_{\rm tr} h_T$ is attained at a vertex $P_k$ for $k\in \{0, ..., j-1, j+1, ..., n\}$.
Since the centroid $\mathrm{mid}(T)$ divides each median of $T$ in the ratio $n$ to $1$ and the length of each median
is strictly bounded by $h_T$,
%
the bound $C_{\rm tr}<n/(n+1)$ follows and cannot be improved in the absence of further assumptions on the shape of the simplex $T$.
Since $|T_j'| = |T|/(n+1)$, %
the previously displayed estimate leads to
\begin{align*}
\frac{|T|}{(n+1)h_{T}^2|F_j|}\left\Vert f\right\Vert_{L^2(F_j)}^{2}&\leq\left\Vert
h_{T}^{-1}f\right\Vert_{L^2(T_j')}^{2}+ \frac{2C_{\rm tr}}{n}\left\Vert
h_T^{-1}f\right\Vert_{L^2({T_j'})}^{}\|\nabla f\|_{L^2(T_j')}.%
\end{align*}
Let $\mathcal{T}'$ be the refinement of $\mathcal{T}$, obtained by replacing $T\in\mathcal{T}$ with $T_0', ..., T_d'$ from above.
The triangulation $\mathcal{T}'$ allows for the facet based decomposition $\{\omega'(F)\}_{F\in\mathcal{F}}$ of $\Omega$, where $\omega'(F)$
is either the patch $\omega'(F)=\mathrm{int}(T'_+\cup T'_-)$ for an interior facet
$F=T'_+\cap T'_-$ or $\omega'(F)=\mathrm{int}(T')$ for
$F\in\mathcal{F}(\partial\Omega)\cap\mathcal{F}(T')$.
This establishes, for any $F\in\mathcal{F}$, the estimate
\begin{align*}
\ell(F)^{-1}\left\Vert f\right\Vert_{L^2(F)}^{2}
&\leq \left\Vert
h_{\mathcal{T}}^{-1}f\right\Vert_{L^2(\omega'(F))}^{2} + \frac{2C_{\rm tr}}{n}\left\Vert
h_{\mathcal{T}}^{-1}f\right\Vert_{L^2(\omega'(F))}\|\nabla f\|_{L^2(\omega'(F))}.
\end{align*}
Since the family $\{\omega'(F) : F\in\mathcal{F}\}$ has no overlap, the sum of the last displayed inequality over all $F \in \mathcal{F}$ and a Cauchy inequality conclude the proof of Lemma \ref{lem:Skeletal trace inequality}.\qed
\end{proof}
The next lemma utilizes a quasi-interpolation operator $J: H^1(\Omega) \to S^1(\mathcal{T})$ with the restriction $J(V) \subset S^1_0(\mathcal{T})$, e.g., $J=J_1\circ I_{\rm NC}$ with explicit constants for $n=2$ from \cite{carstensen_constants_2018}, and the approximation and stability properties
\begin{align}\label{eqn:approx_stab}
\left\Vert h_\mathcal{T}^{-1}(\varphi-J\varphi)\right\Vert
%
&\leq \cnst{cnst:C1} \ENorm{\varphi}\quad\text{and}\quad
\ENorm{\varphi-J\varphi}\leq C_{\rm st} \ENorm{\varphi}
\end{align}
for constants $\cnst{cnst:C1}$ and $C_{\rm st}$ exclusively depending on the shape-regularity of $\mathcal{T}$.
Recall the constant $C_{\rm tr}$ from Lemma \ref{lem:Skeletal trace inequality} and set $\cnst{cnst:C2}\coloneqq (\cnst{cnst:C1}(\cnst{cnst:C1}+2\CtrC_{\rm st}\,/n))^{1/2}$.
\begin{lemma}[equilibrium]\label{lem:eq1}
Suppose that ${G} \in H(\ddiv,\mathcal T )$ and $f\in L^2(\Omega)$ satisfy \eqref{eq:solution_property} and suppose
$({G}|_T)|_F\cdot \nu_F\in L^2(F)$ for all $F\in{\mathcal F}(T)$ and $T\in \mathcal T$.
Then
\[
\trb{f+\ddiv {G}}_* \le \cnst{cnst:C1}\| h_\mathcal T( f+{\ddiv}_{\rm pw} {G}) \|
+ \cnst{cnst:C2}\sqrt{\sum_{F\in{\mathcal F}(\Omega)} \ell(F) \| [ {G} ]_F\cdot \nu_F \|_{L^2(F)}^2} .
\]
\end{lemma}
\begin{proof}
Given $\varphi\in V$ with $ \trb{\varphi} =1$, set $\psi:=\varphi-\varphi_C$ for some quasi-interpolation $\varphi_C\coloneqq J\varphi\in S^1_0(\mathcal T)$ with \eqref{eqn:approx_stab}.
Since \eqref{eq:solution_property} implies $\langle f+\ddiv {G},\varphi\rangle =\langle f+\ddiv {G},\psi\rangle$, a piecewise integration by parts and the collection of jump contributions show
\begin{align}
\langle f+\ddiv {G},\varphi\rangle =
%
%
(f + {\ddiv}_{\rm pw} {G}, \psi)_{L^2(\Omega)}
-\sum_{F\in{\mathcal F}(\Omega)} \langle[{G}]_F\cdot \nu_F, \psi\rangle_{L^2(F)}.
\label{eq:proof-equilibrium-ibp}
\end{align}
The first bound follows from a Cauchy inequality and \eqref{eqn:approx_stab},
\begin{align}
(f + {\ddiv}_{\rm pw} {G}, \psi)_{L^2(\Omega)} &\leq \|h_\mathcal{T}(f+{\ddiv}_{\rm pw} {G})\|\, \|h_\mathcal{T}^{-1}\psi\|
\label{ineq:proof-equilibrium-1}\\
&\leq \cnst{cnst:C1} \|h_\mathcal{T}(f+{\ddiv}_{\rm pw} {G})\|\,
\trb{\varphi}.\nonumber
\end{align}
The second bound additionally exploits the trace inequality of
Lemma \ref{lem:Skeletal trace inequality},
\begin{align}
\sum_{F\in{\mathcal F}(\Omega)} &\langle[{G}]_F\cdot \nu_F, \psi\rangle_{L^2(F)}
\label{ineq:proof-equilibrium-2}\\
&\leq \sqrt{\sum_{F\in{\mathcal F}(\Omega)} \ell(F)\|[{G}]_F\cdot \nu_F\|_{L^2(F)}^2} \sqrt{\sum_{F\in {\mathcal F}(\Omega)}\ell(F)^{-1}\|\psi\|_{L^2(F)}^2}\nonumber\\
&\leq \sqrt{\cnst{cnst:C1}^2 +
\frac{2C_{\rm tr}}{n}C_{\rm st}\cnst{cnst:C1}}\sqrt{\sum_{F\in{\mathcal F}(\Omega)}
\ell(F)\|[{G}]_F\cdot \nu_F\|_{L^2(F)}^2}\nonumber
\end{align}
with $\ENorm{\varphi}=1$ in the last step.
Since \eqref{ineq:proof-equilibrium-1}--\eqref{ineq:proof-equilibrium-2} hold for all $\varphi\in V$ with $\trb{\varphi}=1$, the supremum in \eqref{eq:proof-equilibrium-ibp} over all such $\varphi$ concludes the proof.\qed
\end{proof}
The final ingredient for the proof of Theorem \ref{thm:reliability} controls the second term $\delta$ in the decomposition of Lemma \ref{lemmadecomposition} for $\varrho\coloneqq {G}-\nabla w$. Recall $C_{\rm H}$ from Lemma \ref{lem:lemmacc2a} and $\cnst{cnst:C1}, \cnst{cnst:C1}$ from Lemma \ref{lem:eq1}.
\begin{lemma}[conformity]\label{lem:eq2}
Suppose the divergence-free function $\varrho\in H(\ddiv=0,\Omega)\cap H(\text{curl },\mathcal T)$ is $L^2$ orthogonal onto $RT_0(\mathcal T) \cap H(\ddiv=0,\Omega)$ and satisfies
$(\varrho|_T)|_F\times \nu_F\in L^2(F)$ for all $F\in{\mathcal F}(T)$ and $T\in T$.
Then
\[
C_{\rm H}^{-1}\|\varrho\| \le \cnst{cnst:C1}\| h_\mathcal T{\text{curl }}_{\rm pw} \varrho \|
+ \cnst{cnst:C2}\sqrt{\sum_{F\in{\mathcal F}} \ell(F) \| [ \varrho]_F \times \nu_F \|_{L^2(F)}^2}.\]
\end{lemma}
\begin{proof}
Lemma \ref{lem:lemmacc2a} provides $\beta\in H^1(\Omega; \mathbb R^N)$ with
\eqref{eqn:lemmacc2a}
for a
(component-wise) quasi-interpolation $\beta_C\in S^1(\mathcal{T})^N$ with \eqref{eqn:approx_stab} as in the proof of Lemma \ref{lem:eq1}; set $\psi:=\beta-\beta_C$.
A piecewise integration by parts and the collection of jump contributions shows
\[
\| \varrho\| ^2 = \int_\Omega \varrho\cdot\Curl(\beta-\beta_C)dx =
\int_\Omega \psi \cdot {\text{curl }}_{\rm pw} \varrho \, dx
+\sum_{F\in{\mathcal F}} \int_F \psi [ \varrho]_F\times\nu_F \, ds.
\]
Stability and approximation properties of the
quasi-interpolation \eqref{eqn:approx_stab} and the trace inequality of Lemma \ref{lem:Skeletal trace inequality} eventually lead to
\begin{align*}
\left\Vert \varrho\right\Vert^{2}& \leq \cnst{cnst:C1}\left\Vert h_\mathcal{T}{\text{curl }}_{\rm pw} \varrho\right\Vert_{}^{}\trb{\beta} + \cnst{cnst:C2} \sqrt{\sum_{F\in{\mathcal F}}
\ell(F)\|[\varrho]_F\times \nu_F\|_{L^2(F)}^2}\trb{\beta}.
\end{align*}
In fact, the routine estimation with element and jump terms is completely analogous to the proof of Lemma \ref{lem:eq1} and leads to the same constants $\cnst{cnst:C1}, \cnst{cnst:C2}$.
This and $\trb{\beta}\leq C_{\rm H}\|\varrho\|$ conclude the proof.\qed
\end{proof}
\begin{proof}[Theorem \ref{thm:reliability}]
The trace of ${G}|_T\in H^1(T; \mathbb R^n)$ is well defined on any facet $F\in\mathcal{F}(T)$ of the simplex $T\in\mathcal{T}$.
Lemma \ref{lemmadecomposition} provides $w\in V=H^1_0(\Omega)$ with $\| \nabla u-{G} \|^2 = \trb{ f + \ddiv {G}}_{*}^2+\| {G}- \nabla w\|^2$.
Lemma \ref{lem:eq1} establishes
$$\trb{ f + \ddiv {G}}_{*} \le \Ca\| h_\mathcal T( f+{\ddiv}_{\rm pw} {G}) \|
+ C_{2}\sqrt{\sum_{F\in{\mathcal F}(\Omega)} \ell(F) \| [ {G} ]_F\cdot \nu_F \|_{L^2(F)}^2}.$$
Since $\text{curl }\nabla w = 0$ and $\nabla w\times \nu = 0$ vanish, Lemma \ref{lem:eq2}
applies to $\varrho\coloneqq {G}-\nabla w\in H(\ddiv=0, \Omega)\cap H(\text{curl }, \mathcal{T}; \mathbb R^n)$ and reveals
$$C_{\rm H}^{-1}\| {G}- \nabla w\|\leq\Ca\| h_\mathcal T{\text{curl }}_{\rm pw} {G} \|
+ C_{2}\sqrt{\sum_{F\in{\mathcal F}} \ell(F) \| [{G}]_F \times \nu_F \|_{L^2(F)}^2}.
$$
The above estimates together with the decomposition of Lemma \ref{lemmadecomposition} establish $\eta(\mathcal{T}, {G})$ as a GUB for the error $\|\nabla u - {G}\|$. \qed
\end{proof}
\begin{example}[constants for right-isosceles triangles]\label{ex:constants}
In two space dimensions, $\|\Curl \bullet\| = \trb{\bullet}$ and so $C_{\rm H}\leq 1$ for a simply connected domain $\Omega$ in Lemma \ref{lem:lemmacc2a}.
The choice $J\coloneqq J_1\circ I_{\rm NC}$ from \cite[Section 5]{carstensen_constants_2018} of the quasi-interpolation operator $J$ in the proof of Lemma \ref{lem:eq1} allows for the explicit estimates $$\cnst{cnst:C1}\leq\sqrt{48^{-1} + j_{1,1}^{-2} + c_{\rm apx}^2} \text{ and } C_{\rm st}\leq 1+\sqrt{72}c_{\rm apx}.$$
For
triangulations into right-isosceles triangles, the constant $c_{\rm apx}\leq\sqrt{3}/(2 - 2\cos(\pi/\max\{4,M_{\rm bd}\}))$ %
from \cite[Lemma 4.8]{carstensen_constants_2018} depends on the domain by the maximal number $M_{\rm bd}\leq 4\max\{\pi,\omega_{\rm max}\}/\pi\leq 8$ of triangles sharing a boundary vertex. %
Given the maximal interior angle $\omega_{\rm max}$ of $\Omega$, Table \ref{tab:expl_const} displays those constants for the maximal possible value $M_{\rm bd}= 4\max\{\pi,\omega_{\rm max}\}/\pi$.
The geometric quantity $\max_{x\in T}|x-\mathrm{mid}(T)|$ equals two-thirds of the maximum median of $T$.
Thus, $C_{\rm tr}=\sqrt{5}/(3\sqrt{2})\leq 0.5271$ and $\ell(F) = 6h_F$ for interior edges $F\in\mathcal{F}(\Omega)$ and $\ell(F)=12h_F$ for boundary edges $F\in\mathcal{F}(\partial\Omega)$ of triangulations into right-isosceles triangles.
Consequently,
\begin{align}\label{eqn:C_T}
\cnst{cnst:C1}&\leq \sqrt{48^{-1} + j_{1,1}^{-2} + c_{\rm apx}^2}=:C_\mathcal{T},\\
\cnst{cnst:C2}&\leq \sqrt{C_\mathcal{T}(C_\mathcal{T}+0.5271(1+\sqrt{72}c_{\rm apx}))}=:C_\mathcal{F}\label{eqn:C_F}.
\end{align}
\end{example}
\section{Explicit residual-based a posteriori HHO error estimator}
\label{sec:residual}
The arguments from Section \ref{sec:blocks} apply to the HHO method and result in a stabilization-free reliable a posteriori error control.
In combination with the efficiency estimate from this section, this leads to a new explicit residual-based a posteriori error estimator for the HHO method that is equivalent to the error up to data oscillations.
\subsection{Hybrid high-order methodology}\label{sec:HHO}
The HHO ansatz space reads $ V_h \coloneqq P_k(\mathcal{T}) \times P_k(\mathcal{F}(\Omega))$
for $k\in{\mathbb N}_0$ with the subspace $P_k(\mathcal{F}(\Omega)) \subset P_k(\mathcal{F})$ of piecewise polynomials $p\in P_k(\mathcal{F})$ under the convention $p_{|\partial\Omega}=0$.
The interpolation $\mathrm{I}: V \to V_h$ maps $v \in V$ onto $\mathrm{I} v \coloneqq (\Pi_k v, \Pi_{\mathcal{F},k} v) \in V_h$.
Given any $v_h = (\vT,\vF) \in V_h$, the reconstruction operator $R: V_h \rightarrow P_{k+1}(\mathcal{T})$ defines the unique piecewise polynomial $Rv_h \in P_{k+1}(\mathcal{T})$ with $\Pi_{0}(Rv_h - v_{\mathcal T}) = 0$ such that, for all $w_{k+1} \in P_{k+1}(\mathcal{T})$,
\begin{align}
\label{eqn:R}
&a_{\pw}(Rv_h,w_{k+1})\nonumber\\
&\qquad= a_{\pw}(v_{\mathcal T}, w_{k+1})
-\sum_{T\in\mathcal{T}}\langle v_{\mathcal{T}|T}-v_{\mathcal F}, \nabla w_{k+1|T} \cdot \nu_T \rangle_{L^2(\partial T)}.
\end{align}
Let $u_h\in V_h$ solve the HHO discrete formulation of \eqref{eq:poisson} with
\begin{align}\label{eqn:HHO}
a_h(u_h,v_h)= (f,v_{\mathcal T})_{L^2(\Omega)} \qquad \text{for all } v_h=(\vT,\vF) \in V_h
\end{align}
for the HHO bilinear form
\begin{align}%
a_h(u_h,v_h) &\coloneqqa_{\pw}( Ru_h, Rv_h)+
s_h(u_h, v_h)%
\end{align}
and the stabilization term $s_h(u_h, v_h)$ from \eqref{eqn:s_h0}.
Given any $w_C \in S^1_0(\mathcal{T})= P_1(\mathcal{T})\cap H^1_0(\Omega)$,
the definition of the reconstruction operator $R$ in \eqref{eqn:R} verifies $R \mathrm{I} w_C=w_C$ with the interpolation $\mathrm{I}$ onto $V_h$.
Hence, $S_{TF} \mathrm{I} w_C=0$ vanishes for all $F \in \mathcal{F}(T)$ and $T \in \mathcal{T}$.
This and \eqref{eqn:HHO} show, for all $w_C\in S^1_0(\mathcal{T})$, that
\begin{align}\label{eqn:HHO_solution_property}
a_{\pw}(Ru_h, w_C) = (f, \Pi_k w_C)_{L^2(\Omega)} = (\Pi_k f, w_C)_{L^2(\Omega)}.
\end{align}
\subsection{Explicit a posteriori error estimator}\label{sub:HHO_eta}
As a result of \eqref{eqn:HHO_solution_property}, $\nabla_{\pw} Ru_h$ satisfies the solution property \eqref{eq:solution_property} if $k\geq 1$ and, in the lowest order case $k=0$, \eqref{eq:solution_property} holds with $f$ replaced by $\Pi_0 f$.
This allows the application of the theory from Section \ref{sec:blocks} to the HHO method with minor modifications for the case $k=0$.
Define the error estimator contributions
\begin{equation}\label{eqn:etares_parts}
\begin{split}
\eta_{\text{res}, 1}(\mathcal{T})^2&\coloneqq \begin{cases}\|h_\mathcal{T}( f+\Delta_{\rm pw} Ru_h)\|^2&\text{for } k\geq1,\\
\|h_\mathcal{T}\Pi_0 f\|^2&\text{for } k=0,
\end{cases}\\
\eta_{\text{res}, 2}(\mathcal{T})^2&\coloneqq \begin{cases}0&\text{for } k\geq1,\\
\osc0^2(f, \mathcal{T})&\text{for } k=0,
\end{cases}\\
\eta_{\text{res}, 3}(\mathcal{T})^2&\coloneqq\sum_{F\in {\mathcal F}(\Omega)} \ell(F)\| [ \nabla_{\pw} Ru_h ]_F\cdot \nu_F \|_{L^2(F)}^2,\\
\eta_{\text{res}, 4}(\mathcal{T})^2&\coloneqq\sum_{F\in {\mathcal F}} \ell(F)\| [ \nabla_{\pw} Ru_h ]_F\times \nu_F \|_{L^2(F)}^2.
\end{split}
\end{equation}
Since $\nabla_{\pw} Ru_h$ is a piecewise gradient, its piecewise $\text{curl }$ vanishes.
This leads to the explicit residual-based a posteriori error estimator
\begin{equation}
\label{eq:residualdef}
\eta_{\text{res}}(\mathcal{T})^2\coloneqq \left(\cnst{cnst:C1}\eta_{\text{res}, 1}(\mathcal{T}) + C_P\eta_{\text{res}, 2}(\mathcal{T})
+\cnst{cnst:C2}\eta_{\text{res}, 3}(\mathcal{T})\right)^2+ C_{\rm H}^2\cnst{cnst:C2}^2\eta_{\text{res}, 4}^2(\mathcal{T}).
\end{equation}
(Recall $\cnst{cnst:C1}, \cnst{cnst:C2} $ from Lemma \ref{lem:eq1} and $C_{\rm H}$ from Lemma \ref{lem:lemmacc2a} as well as the Poincar\'e constant $C_P\leq \pi^{-1}$.)
The main result of this section verifies the assumptions in Theorem \ref{thm:reliability} and proves reliability and efficiency of $\eta_{\rm res}(\mathcal{T})$.
\begin{theorem}[residual-based GUB for HHO]\label{thm:HHO_equivalence}
Let $u\in V$ solve the
Poisson equation \eqref{eq:poisson} and let
$u_h \in V_h$ solve the discrete
formulation \eqref{eqn:HHO}.
Then
\begin{align*}
\npw{u- R u_h} \leq \eta_{\rm res}(\mathcal{T})\leq \cnst{cnst:res_eff}\big( \npw{u- R u_h}+\osc q(f, \mathcal T)\big)
\end{align*}
and $\osc {k-1}(f, \mathcal T)\leq \cnst{cnst:oscres_rel} \eta_{\rm res}(\mathcal{T})$ hold for any $q\in{\mathbb N}_0$.
The constants $\newcnst\label{cnst:res_eff}$ and $\newcnst\label{cnst:oscres_rel}$ exclusively depend on $k, q$ and on the shape-regularity of the triangulation $\mathcal T$.
\end{theorem}
\subsection{Proof of Theorem \ref{thm:HHO_equivalence}}
The orthogonality of $\nabla_\mathrlap{\pd}{w} R u_h$ to the divergence-free Raviart-Thomas space {of} lowest degree is an assumption in Theorem \ref{thm:reliability} and verified below.
\begin{lemma}[orthogonality]
\label{lem:GuhorthRTd0}
The piecewise gradients $\nabla_{\pw} R V_h$
are $L^2$ orthogonal to the space {~$RT_0(\mathcal{T})\cap H(\mathrm{div} = 0,\Omega)$}, i.e., any $v_h\in V_h$ and $q_{RT} \in RT_0(\mathcal{T})\cap H(\mathrm{div} = 0,\Omega)$ satisfy
\begin{align}
(\nabla_{\pw} Rv_h, q_{RT})_{L^2(\Omega)} = 0.
\end{align}
\end{lemma}
\begin{proof}
Given any $q_{RT} \in RT_0(\mathcal{T})\cap H(\mathrm{div} = 0,\Omega)$, $\ddiv q_{RT}=0$ shows
$q_{RT} \in P_0(\mathcal{T};\mathbb R^n)$ \cite[Lemma 14.9]{ErnGuermond2021}.
Since $P_0(\mathcal{T};\mathbb R^n) = \nabla_{\pw} P_1(\mathcal{T})$, there exists a piecewise affine function $\phi_1 \in P_1(\mathcal{T})$ with $q_{RT} = \nabla_\mathrlap{\pd}{w} \phi_1$ a.e.~in $\Omega$.
This and the definition of $ R v_h$ from \eqref{eqn:R} imply, for any $v_h = (\vT,\vF)\in V_h$, that
\begin{align*}
(\nabla_{\pw} Rv_h,q_{RT})_{L^2(\Omega)} &= a_\mathrlap{\pd}{w}(Rv_h, \phi_1)\\
&= a_\mathrlap{\pd}{w}(v_\mathcal{T},\phi_1) - \sum\limits_{T\in \mathcal T} \langle v_{\mathcal T}|_T-v_{\mathcal F}, \nabla_\mathrlap{\pd}{w} \phi_1 \cdot \bn_T \rangle_{L^2(\partial T)}.
\end{align*}
This, a piecewise integration by parts, and $\Delta_\mathrlap{\pd}{w} \phi_1 \equiv 0$ lead to
\begin{align}
(\nabla_{\pw} R v_h,q_{RT})_{L^2(\Omega)} &= \sum\limits_{T\in \mathcal T}\langle v_{\mathcal F},\nabla_\mathrlap{\pd}{w} \phi_1 \cdot \bn_T \rangle_{L^2(\partial T)}\nonumber\\
&= \sum\limits_{F\in \mathcal F}\langle v_{\mathcal F}, [q_{RT}\cdot\bn_F]_F \rangle_{L^2(F)}.\label{eq:proof-orthogonality-RT-div=0-ibp}
\end{align}
Since $v_{\mathcal F}$ is single-valued on $F$ and $q_{RT}\in RT_0(\mathcal{T})$ has continuous normal components, the jump term $\langle v_{\mathcal F}, [q_{RT}]_F\cdot \bn_F\rangle_{L^2(F)} =0$ vanishes for all $F\in \mathcal F(\Omega)$. This, $v_{\mathcal F}\equiv 0$ on $\partial\Omega$, and \eqref{eq:proof-orthogonality-RT-div=0-ibp} conclude
$(\nabla_{\pw} R v_h,q_{RT})_{L^2(\Omega)} = 0$. \qed
\end{proof}
The following lemma concerns the efficiency of the jump contributions. %
Each facet $F\in\mathcal{F}$ has at most two adjacent simplices that define a triangulation $\mathcal{T}(F)\coloneqq\{T\in\mathcal{T} : F\in\mathcal{F}(T)\}$ of the facet-patch $\omega(F)\coloneqq \mathrm{int}(\bigcup_{T\in\mathcal{T}(F)}T)$.
\begin{lemma}[efficiency of jumps]
\label{lem:efficiency}
The solution $u \in V$ to \eqref{eq:poisson} and the discrete solution $u_h \in V_h$ to \eqref{eqn:HHO} satisfy (a) for all $F \in \mathcal{F}$ and (b) for all $F \in \mathcal{F}(\Omega)$.
\begin{enumerate}
\item[(a)] $h_F^{1/2}\|[\nabla_{\pw} R u_h]_F\times \nu_F \|_{L^2(F)} \lesssim \min_{v \in V} \|\nabla v - \nabla_{\pw} R u_h\|_{L^2(\omega(F))}$,\\
\item[(b)] $h_F^{1/2}\|[\nabla_{\pw} R u_h]_F\cdot \nu_F \|_{L^2(F)} \lesssim \|\nabla u - \nabla_{\pw} R u_h\|_{L^2(\omega(F))} + \mathrm{osc}_k(f, \mathcal{T}(F))$.
\end{enumerate}
\end{lemma}
\begin{proof}
\newcommand{\mathrm{conv}}{\mathrm{conv}}
The proof is based on the following extension argument.
Given a polynomial $p\in P_k(F)$ of degree at most $k$ along the side $F\in\mathcal{F}$, the coefficients determine a polynomial (also denoted by $p$)
along the hyperplane $H$ that enlarges $F$. The intersection $\widehat{F}:= H\cap \mathrm{conv}({ \omega(F)}) $ of the hyperplane
$H$ with the convex hull of the facet-patch $\omega(F)$ may be strictly larger than $F$.
The shape-regularity of $\mathcal{T}$ bounds the size of $\widehat{F}$ in terms of $F$ and an inverse estimate leads to a bound $\| p \|_{L^\infty(\widehat{F})} \le C(k)\| p \|_{L^\infty(F)}$
with a constant $C(k)$ that depends on the shape-regularity of $\mathcal{T}$ and on $k$. The extension of $p$ from $H$ to $\mathbb R^n$ by constant values along the side normal $\nu_F$ leads to a polynomial $\widehat{p}\in P_k(\mathbb R^
n)$ with
\begin{align}\label{eqn:L_infty_p}\| \widehat{p} \|_{L^\infty(\omega(F))}
\le
\| p \|_{L^\infty(\widehat{F})} \le C(k)\| p \|_{L^\infty(F)}.
\end{align}
\emph{Proof of (a).}
The %
tangential jump $\varrho_F:= [\nabla_{\pw} R v_h]_F\times \nu_F \in P_{k}(F;\mathbb{R}^N)$
is a polynomial in $N=2n-3$ components on $F\in\mathcal{F}$ for $n=2,3$.
Let $p=\varrho_F(j)$ be one of the components of $\varrho_F\in P_k(F)^N$, for $j=1,...,N$, and extend it as explained above to $\hat p\in P_k(\mathbb R^n)$ and call this $\widehat\varrho(j)$ in the vector
$\widehat{\varrho_F}\in P_k(\mathbb R^n;\mathbb{R}^N)$.
The proof involves the piecewise polynomial facet-bubble function $b_F\coloneqq n^n \Pi_{j=1}^n \varphi_j $ for the $n$
nodal basis function $\varphi_1, \dots, \varphi_n \in S^1(\mathcal T)$ associated with the vertices of $F$.
An inverse estimate \cite[Proposition 3.37]{Ver:13} shows
\begin{align}\label{eqn:lemma_7_a}
\| \varrho_F \|_{L^2(F)}^2 \lesssim \| b_F^{1/2}\varrho_F \|_{L^2(F)}^2=\langle b_F\varrho_F, \varrho_F\rangle_{L^2(F)}.
\end{align}
Since $\varrho:= b_F\widehat{\varrho_F} \in S^{k+n}_0(\mathcal{T}(F);\mathbb{R}^N)$ vanishes on $\partial \omega(F)\setminus\mathrm{int}(F)$, \eqref{eqn:lemma_7_a} and a piecewise integration by parts show
\begin{align}
\|\varrho_F \|_{L^2(F)}^2 \lesssim \langle
\varrho,
[\nabla_{\pw} R v_h]_F\times \nu_F
\rangle_{L^2(F)} = (\text{curl } \varrho, \nabla_{\pw} R v_h )_{L^2(\omega(F))}.
\label{ineq:bound-tangential-jump-ibp}
\end{align}
This and $(\text{curl }\varrho, \nabla v)_{L^2(\Omega)} = 0$ for any $v\in V$ imply
\begin{align}
\label{eq:lemma2_s2ls}
\|\varrho_F \|_{L^2(F)}^2 &\lesssim (\text{curl }\varrho,\nabla_{\pw}(R v_h-v))_{L^2(\omega(F))}\notag\\
&\leq \| \text{curl }\varrho \|_{L^2(\omega(F))} \| \nabla_{\pw}(R v_h-v) \|_{L^2(\omega(F))}.
\end{align}
An inverse estimate, $\|b_F\|_{L^\infty(\omega(F))} = 1$, and \eqref{eqn:L_infty_p} imply
\begin{align}
\| \text{curl }\varrho \|_{L^2(\omega(F))}
&\lesssim \|\nabla\varrho\|_{L^2(\omega(F))}
\lesssim h_F^{-1+n/2} \| \varrho \|_{L^\infty(\omega(F))}\nonumber\\
&\lesssim h_F^{-1+n/2} h_F^{-(n-1)/2}\| \varrho_F \|_{L^2(F)} = h_F^{-1/2}\| \varrho_F \|_{L^2(F)}\label{ineq:bound-curl-tangential-jump}.
\end{align}
%
In combination with \eqref{eq:lemma2_s2ls}, this concludes the proof of (a).
\medskip
\noindent\emph{Proof of (b).}
The efficiency of normal jumps (b) follows from the arguments for conforming FEMs, cf.~\cite[Section 1.4.5]{Ver:13}; further details are omitted. \qed
\end{proof}
The following lemma reveals that the order $k \geq \mathbb{N}_0$ of the oscillations $\osc k(f, T)$ in Lemma \ref{lem:efficiency} (b) can be any natural number.
It is certainly known to the experts but hard to find in the literature.
Recall the convention $\Pi_{-1}\coloneqq0$.
\begin{lemma}[efficiency of lower-order oscillations]
\label{lem:local-data-oscillation}
Given any simplex $T \in \mathcal{T}$ and parameters $k, q \in \mathbb{N}_0$,
the solution $u \in V$ to \eqref{eq:poisson} satisfies
\begin{align}
&\cnst{cnst:local-data-oscillation}^{-1}\osc{k-1}^2( f,T)\label{ineq:osc-best-order}
\leq \min_{v_{k+1} \in P_{k+1}(T)}\|\nabla_\mathrlap{\pd}{w}(u - v_{k+1})\|_{L^2(T)}^2+\osc{q}^2(f,T).
\end{align}
The constant $\newcnst\label{cnst:local-data-oscillation}$ exclusively depends on $q$ and the shape of $T$.
\end{lemma}
\begin{proof}
The assertion \eqref{ineq:osc-best-order} is trivial for $q \leq k-1$,
so suppose $k \leq q$.
Any $v_{k+1}\in P_{k+1}(T)$ and $\varrho_T\coloneqq \Pi_q f + \Delta v_{k+1}\in P_q(T)$ satisfy
\begin{align}\label{eqn:osc_split}
\osc{k-1}^2(f, T) \leq h_T^2\|f + \Delta v_{k+1}\|_{L^2(T)}^2 = \osc{q}^2(f, T) + h_T^2\|\varrho_T\|_{L^2(T)}^2.
\end{align}
Let $b_T\in S^{n+1}_0(T)$ with $0\leq b_T\leq 1=\max b_T$ denote the volume bubble-function on $T\in\mathcal{T}$.
The equivalence of norms in the finite-dimensional space $P_q(T)$ provides
\begin{align}\label{eqn:bT_equivalence}
\|b_T^{1/2}\varrho_T\|_{L^2(T)} \leq \|\varrho_T\|_{L^2(T)}\leq \cnst{cnst:bT_efficiency} \|b_T^{1/2}\varrho_T\|_{L^2(T)}.
\end{align}
A more detailed analysis of the mass matrices reveals that the constant $\newcnst\label{cnst:bT_efficiency}$ exclusively depends on the polynomial degree $q$. An integration by parts with $b_T\varrho_T\in S^{q+n+1}_0(T)\subset V$ and the weak formulation \eqref{eq:poisson} result in
\begin{align*}
\|b_T^{1/2}\varrho_T\|_{L^2(T)}^2 &= (\Pi_q f + \Delta v_{k+1}, b_T\varrho_T)_{L^2(T)}\\
&= (\nabla(u-v_{k+1}), \nabla (b_T\varrho_T))_{L^2(T)} - (f - \Pi_q f, b_T\varrho_T)_{L^2(T)}.
\end{align*}
A Cauchy inequality, the inverse estimate $h_T\|\nabla (b_T\varrho_T)\|_{L^2(T)}\leq \newcnst\label{cnst:bT_inverse}\|\varrho_T\|_{L^2(T)}$
with a constant $\cnst{cnst:bT_inverse}$ that exclusively depends on $q+n+1$ and the shape of $T$, and \eqref{eqn:bT_equivalence} lead to
\begin{align}\label{eqn:osc_efficiency_proof}
\cnst{cnst:bT_efficiency}^{-2}h_T\|\varrho_T\|_{L^2(T)}\leq \cnst{cnst:bT_inverse}\|\nabla(u-v_{k+1})\|_{L^2(T)} + \osc q(f, T)\big.
\end{align}
The combination of \eqref{eqn:osc_split} with \eqref{eqn:osc_efficiency_proof} and a Cauchy inequality conclude the proof of \eqref{ineq:osc-best-order}, e.g., with $\cnst{cnst:local-data-oscillation}= 1 + \cnst{cnst:bT_efficiency}^4(1 + \cnst{cnst:bT_inverse}^2).$\qed
\end{proof}
\begin{proof}[of Theorem \ref{thm:HHO_equivalence}]
Recall the definition of $\eta_{\rm res}(\mathcal{T})$ for $k\geq 1$ and $k=0$ in \eqref{eqn:etares_parts}.
Since
$
\osc{k-1}^2(f, \mathcal{T})\leq \eta_{\text{res},1}^2(\mathcal{T}) + \eta_{\text{res},2}^2(\mathcal{T})\lesssim \eta_{\rm res}^2(\mathcal{T})$,
the remaining parts of this proof discuss the reliability and efficiency of $\eta_{\rm res}(\mathcal{T})$.
Lemma \ref{lem:GuhorthRTd0} provides
the orthogonality of $\nabla_{\pw} Ru_h\in H^1(\mathcal{T}; \mathbb R^n)$ to the divergence-free Raviart-Thomas function $RT_0(\mathcal{T})\cap H(\mathrm{div} = 0,\Omega)$.
This and \eqref{eqn:HHO_solution_property} show that the assumptions
in Theorem \ref{thm:reliability} hold for ${G} \coloneqq \nabla_{\pw} Ru_h$ and $k\geq 1$, whence the reliability of $\eta_{\rm res}(\mathcal{T})$ follows with a reliability constant 1.
Minor modifications to the proof of Theorem \ref{thm:reliability} lead to reliability in the case $k=0$.
In fact, the only modifications required concern the upper bound of $\trb{f + \mathrm{div}\,\nabla_{\pw} R u_h}_* \leq \trb{(1 - \Pi_0) f}_* + \trb{\Pi_0 f + \mathrm{div}\,\nabla_{\pw} R u_h}_*$.
A piecewise Poincar\'e inequality shows $\trb{(1 - \Pi_0) f}_* \leq C_P\osc{0}(\mathcal{T},f)$ with the Poincar\'e constant $C_P$($\leq 1/\pi$ for simplices). Lemma \ref{lem:eq1} proves $\trb{\Pi_0 f + \mathrm{div}\,\nabla_{\pw} R u_h}_* \leq C_{\rm 1}\eta_{\text{res},1}(\mathcal{T}) + C_{\rm 2} \eta_{\text{res},3}(\mathcal{T})$. Hence, the decomposition of Lemma \ref{lemmadecomposition} and Lemma \ref{lem:eq2} result in $\trb{u-Ru_h}_\mathrlap{\pd}{w}\leq\eta_{\rm res}(\mathcal{T})$.
This provides the reliability and it remains to verify the efficiency $\eta_{\rm res}(\mathcal{T})\lesssim \trb{u-Ru_h}_\mathrlap{\pd}{w} + \osc q(f,\mathcal{T})$ for any $q\in\mathbb N_0$.
The Pythagoras theorem and \eqref{eqn:osc_efficiency_proof} with $\varrho_T\coloneqq \Pi_kf + \Delta R u_h\in P_k(T)$ and $v_{k+1}\coloneqq Ru_h\in P_{k+1}(T)$ lead to the local efficiency of the volume contributions
\begin{align*}
\|h_T(f+\Delta_\mathrlap{\pd}{w} Ru_h)\|_{L^2(T)}^2&=\mathrm{osc}_k^2(f, T) + \|h_T\varrho_T\|_{L^2(T)}^2\\
&\lesssim \|\nabla(u- Ru_h)\|_{L^2(T)}^2 + \mathrm{osc}_k^2(f, T).
\end{align*}
Lemma \ref{lem:efficiency} considers the remaining terms in the error estimator and establishes their efficiency namely,
\begin{align*}
\sum_{F\in\mathcal{F}}h_F\| [ \nabla_\mathrlap{\pd}{w} R u_h ]_F\|_{L^2(F)}\lesssim \trb{u-Ru_h}_\mathrlap{\pd}{w} + \mathrm{osc}_k(f,\mathcal{T})
\end{align*}
with the modified jump $[\nabla_\mathrlap{\pd}{w} R u_h]_F = \nabla_\mathrlap{\pd}{w} R u_h \times \nu_F$ on boundary facets $F \in \mathcal{F}(\partial \Omega)$.
This and Lemma \ref{lem:local-data-oscillation} establish the existence of some mesh-independent constant $\cnst{cnst:res_eff}>0$ with $\cnst{cnst:res_eff}^{-1}\eta_{\rm res}(\mathcal{T})\leq \trb{u-Ru_h}_\mathrlap{\pd}{w} + \osc q(f,\mathcal{T})$ for arbitrary $q\in\mathbb N_0$. This concludes the proof.\qed
\end{proof}
\section{Numerical experiments}%
\label{sec:Numerical results}
This section provides numerical evidence for optimal convergence and
a comparison of the stabilization-free GUBs $\eta_{\rm res}$ and $\eta_{\rm eq,p}$ from the Sections \ref{sec:residual} and \ref{sec:eq} with the original error estimator $\eta_{\rm HHO}$ from \cite[Theorem 4.3]{DiPietroDroniou2020} for the HHO method
in three 2D benchmarks. %
\subsection{A posteriori error estimation with explicit constants}
All triangulations in this section consist of right-isosceles triangles with the Poincar\'e constant $C_P = (\sqrt{2}\pi)^{-1}$ \cite{kikuchi_estimation_2007}.
With the estimates $C_{\rm 1}\leq C_\mathcal{T}$, $C_{\rm H}\leq 1$, and $C_{\rm 2}\leq C_\mathcal{F}$ %
from Example \ref{ex:constants}, the residual-based error estimator from \eqref{eq:residualdef} reads
\begin{align*}
\eta_{\text{res}}(\mathcal{T})^2\coloneqq& \left(C_\mathcal{T}\eta_{\text{res}, 1}(\mathcal{T}) + C_P\eta_{\text{res}, 2}(\mathcal{T})
+C_\mathcal{F}\eta_{\text{res}, 3}(\mathcal{T})\right)^2+ C_\mathcal{F}^2\eta_{\text{res}, 4}^2(\mathcal{T}).
\end{align*}
The equilibrated GUB from Theorem \ref{thm:GUB-HHO},
\begin{align*}
\eta_{\mathrm{eq},p}^2(\mathcal{T})\coloneqq \left(C_P\osc{k+p}(f, \mathcal{T}) + \| Q_p^\Delta \|_{L^2(\Omega)}\right)^2 + \trb{(1-\mathcal{A}) Ru_h}_\mathrlap{\pd}{w}^2,
\end{align*}
depends on the post-processed quantity $Q_p^\Delta\coloneqq Q_p-\nabla_{\pw} Ru_h$ with the Raviart-Thomas function $Q_p\in RT_{k+p}(\mathcal{T})$ of degree $k+p$ for $p\in\mathbb N_0$ from Subsection \ref{sub:Computation_Q}.
An algorithmic description of the computation of $Q_p^\Delta$ for arbitrary $p$ follows in Appendix A.
Theorem \ref{thm:GUB-HHO} shows that $\eta_{{\rm eq}, p}$ is efficient and reliable for all $p\in\mathbb N_0$.
The residual-based error estimator $\eta_{\rm HHO}$ from \cite[Theorem 4.3]{DiPietroDroniou2020} reads
\begin{align*}
\eta_{\rm HHO}^2(\mathcal{T}) \coloneqq& \sum_{T\in\mathcal{T}} \bigg( C_Ph_{T}\| (I - \Pi_{T,0})(f+{\Delta}_{\rm pw} Ru_h)\|_{L^2(T)}\\
&\quad+ \sqrt{\sum_{F\in\mathcal{F}(T)} C_{\partial T} h_T \|R_{T, F}^k u_h\|_{L^2(F)}^2}\bigg)^2
+ \trb{(I-\mathcal{A}) Ru_h}_\mathrlap{\pd}{w}^2
\end{align*}
with the operator $R_{T, F}^k$ from \cite[Eq.~(2.59)]{DiPietroDroniou2020} for the original HHO stabilization \cite[Eq.~(2.22)]{DiPietroDroniou2020} that induces the global stabilization $s_h$.
The constant %
$C_{\partial T}=12C_P(C_P+C_{\rm tr})\leq 2.0315$ for right-isosceles triangles bounds the trace of $f-\Pi_0 f$ for $f\in H^1(T)$ and improves on the estimate $C_{\partial T}\leq (C_P(h_T|\partial T|/|T|)(1+C_P))^2=2.524$ from \cite[Subs.~4.1.1]{DiPietroDroniou2020}.
\begin{lemma}[Poincar\'e-type inequality on trace]
Given a simplex $T \subset \mathbb{R}^n$, any $f\in H^1(T)$ and $C_{\partial T}\coloneqq (n+1)h_T^2|T|^{-1} C_P(C_P + 2C_{\rm tr}/n)$ satisfy
\begin{align}\label{eqn:C_pT}
\|f-\Pi_{0} f\|_{L^2(\partial T)}^2 \leq C_{\partial T}h_T\|\nabla f\|_{L^2(T)}^2.
\end{align}
\end{lemma}
\begin{proof}
Abbreviate $\tilde \ell(F)\coloneqq (n+1)h_Fh_T^2|T|^{-1}$ for the facet $F\in\mathcal{F}(T)$ of $T$ and apply Lemma \ref{lem:Skeletal trace inequality} to the singleton triangulation $\{T\}$ and $f-\Pi_0f\in H^1(T)$.
This and the Poincar\'e inequality $\|f-\Pi_{0, T}f\|_{L^2(T)}\leq C_P h_T\|\nabla f\|_{L^2(T)}$ show
\begin{align*}
\sum_{F\in\mathcal{F}(T)} \tilde \ell(F)^{-1}\|f-\Pi_{0, T} f\|_{L^2(F)}^2
&\leq(C_P+2C_{\rm tr}/n)C_P \|\nabla f\|_{L^2(T)}^2.
\end{align*}
The assertion \eqref{eqn:C_pT} follows from the observation that $\tilde \ell(F)$ is maximized on the facet $F\in\mathcal{F}(T)$ with $h_F=h_T$.\qed
\end{proof}
\subsection{Implementation and adaptive algorithm} %
\label{sub:Implementation}
Our implementation of the HHO method in MATLAB uses nodal bases for the spaces $P_k(\mathcal{F}), P_k(\mathcal{T})$, and $P_{k+1}(\mathcal{T})$ and the direct solver \textit{mldivide} (behind the \textbackslash-operator) for the discrete system of equations representing \eqref{eqn:HHO}.
For implementation details on the HHO method itself we refer to \cite[Appendix B]{DiPietroDroniou2020}.
For the numerical integration over the reference triangle $T_{\rm ref}\coloneqq\mathrm{conv}\{(0,0),(1,0),(0,1)\}$, the product quadrature rule
\begin{align}\label{eqn:Qf}
Q(g)\coloneqq \frac18\sum_{j=1}^mw_j^0\sum_{k=1}^mw_k^1 \;g\left(F(q_j^0,q_k^1)\right)&&\text{for }g\in C(T_{\rm ref})
\end{align}
with the Duffy transform $F:(-1,1)^2\to T_{\rm ref}$ from the square to $T_{\rm ref}$, defined by $F(X,Y)\coloneqq((1+X)(1-Y)/4, (1+Y)/2)$, is applied.
Here, $q_j^\alpha\in(-1,1)$ (resp.\ $w_j^\alpha\in\mathbb R$) for $j=1,..., m$ denote the $m\in{\mathbb N}$ points (resp.\ weights) of the Gauß-Jacobi quadrature over the interval $(-1,1)$ with respect to the weight function $w(x)=(1-x)^\alpha, \alpha=0,1$.
It is well-known that the Gauß-Jacobi quadrature is exact up to degree $2m-1$ and an integration by substitution verifies
\renewcommand{\d}{\,\mathrm{d}}
\begin{align*}
\int_{T_{\rm ref}}g\d x = \frac18\int_{-1}^1\int_{-1}^1(1-Y) g(F(X,Y))\d Y\d X.
\end{align*}
This shows that the quadrature rule \eqref{eqn:Qf} for the integration of $g$ over $T_{\rm ref}$
is exact for polynomials $g\in P_{2m-1}(T_{\rm ref})$ of total degree at most $2m-1$ and an integration by substitution allows the exact numerical integration of polynomials over $T\in\mathcal{T}$.
The errors in approximating non-polynomial expressions, such as exact solutions $u$ and source terms $f$, by polynomials of sufficiently high degree are expected to be very small and are neglected for simplicity.
Algorithm \ref{alg:afem} displays the standard adaptive algorithm (AFEM) \cite{carstensen_axioms_2014,carstensen_axioms_2017} %
driven by the refinement indicators, for any triangle $T\in \mathcal{T}$,
\begin{align}\label{eqn:etares_T}
\eta_{\rm res}^2(T)&\coloneqq |T|\|f+{\Delta}_{\rm pw} Ru_h\|^2_{L^2(T)}
+|T|^{1/2}\sum_{F\in {\mathcal F}(T)}\|[ {\nabla}_{\rm pw} Ru_h ]_F \|_{L^2(F)}^2 %
\end{align}
with the modified jump %
$[\bullet]_F\coloneqq \bullet\times n_F$ along a boundary side $F\in\mathcal{F}(\partial \Omega)$.
\newcommand{\mathcal M}{\mathcal M}
\begin{algorithm}
\caption{AFEM algorithm}
\label{alg:afem}
\textbf{Input:} Initial regular triangulation $\mathcal{T}_0$ and polynomial degree $k\in\mathbb N_0$ of the HHO method
\begin{algorithmic}
\For{levels $\ell\coloneqq 0, 1, 2...$}
\State \textbf{Solve} \eqref{eqn:HHO} for discrete solution $u_\ell\in V_\ell$ exactly on $\mathcal{T}_\ell$ and compute $Ru_\ell$
\State \textbf{Compute} (refinement indicators) $\eta_{\rm res}^2(T)$ for all $T\in\mathcal{T}_\ell$ %
\State \textbf{Mark} minimal subset $\mathcal M_\ell\subset\mathcal{T}_\ell$ with $\frac12\sum_{T\in\mathcal{T}_\ell}\eta_{\rm res}(T)^2 \leq \sum_{T\in\mathcal M_\ell}\eta_{\rm res}(T)^2$
\State \textbf{Refine} $\mathcal{T}_\ell$ to smallest NVB refinement $\mathcal{T}_{\ell+1}$ with $\mathcal M_\ell\subseteq\mathcal{T}_\ell\setminus\mathcal{T}_{\ell+1}$
\EndFor
\end{algorithmic}
\textbf{Output:} sequences of triangulations $\mathcal{T}_\ell$ and $Ru_\ell$
\end{algorithm}
\subsection{High oscillations on the unit square}%
\label{sub:Smooth solution}
This benchmark on the unit square $\Omega\coloneqq(0,1)^2$ considers the Laplace equation $-\Delta u=f$ with source term $f$ matching the smooth exact solution
$$u(x, y) = x(x-1)y(y-1) e^{-100((x-1/2)^2 + (y-117/1000)^2)}\in C^\infty(\Omega).$$
Figure \ref{fig:SquareConvergence} displays the energy norm $\trb{e}_\mathrlap{\pd}{w}$ of the error $e\coloneqq u-Ru_h$ for uniform and adaptive refinement by Algorithm \ref{alg:afem} on the left.
The smooth solution allows for optimal convergence rates $(k+1)/2$ in the number {\rm ndof}\ of degrees of freedom, while the adaptive mesh sequence leads to a lower energy error with respect to {\rm ndof}.
The GUBs $\eta_{\rm res}, \etaeqnewa p$, and $\eta_{\rm HHO}$ are efficient and therefore equivalent to the energy error $\trb{e}_\mathrlap{\pd}{w}$, see Figure \ref{fig:SquareConvergence} on the right for $k=0, 2$.
Figure \ref{fig:SquareGUB} shows the efficiency indices $EF(\eta)\coloneqq \eta/\trb{e}_\mathrlap{\pd}{w}$ for the residual-based GUBs $\eta=\eta_{\rm res}, \eta_{\rm HHO}$ and the equilibration-based GUB $\eta=\etaeqnewa p$ for $p=0,1$. Higher values of $p$ for a more expensive postprocessing in $\eta_{{\rm eq}, p}$ do not significantly improve on $\etaeqnewa1$.
\begin{figure}
\centering
\includegraphics{./Figures/Legend_k_short_out.pdf}\\
\hspace*{-2em}\hbox{\includegraphics{./Figures/SquareConvergenceUniform_out.pdf}
\includegraphics{./Figures/SquareConvergenceGUB_out.pdf}}
\caption{Convergence history of the energy error $\trb{e}_{\rm pw}$ (left) and the GUBs $\eta_{\rm res}, \eta_{\rm HHO}, \etaeqnewa{0}$ (right) on the square domain}
\label{fig:SquareConvergence}
\end{figure}
\begin{figure}
\centering
\includegraphics{./Figures/Legend_k_short_out.pdf}\\
\includegraphics{./Figures/Legend_ErrorV2_out.pdf}\\
%
\hbox{
\includegraphics{./Figures/SquareEfficiencyV2_out.pdf}
\includegraphics{./Figures/SquareEfficiency_EQ_out.pdf}}
\caption{History of the overestimation factor
$EF(\eta)=\eta/\trb{e}_\mathrlap{\pd}{w}$ for the residual-based error estimators $\eta_{\rm res}, \eta_{\rm HHO}$ (left) and $\etaeqnewa0, \etaeqnewa1$ (right) on the unit square}
\label{fig:SquareGUB}
\end{figure}
\subsection{Analytical solution for the slit domain}%
\label{sub:Slit singularity on the slit domain}
The source term $f\in L^2(\Omega)$ in the second benchmark on the slit domain $\Omega\coloneqq(0,1)^2\setminus([0,1)\times\{0\})$
matches the singular solution (in polar coordinates)
$$u(r, \varphi) = r^{1/2}(r^2\sin(\varphi)^2 - 1)(r^2\cos(\varphi)^2 - 1)\sin(\varphi/2).$$
The singularity of $u$ at the origin $(0,0)$ leads to reduced convergence rates $1/4$ under uniform refinement, regardless of the polynomial degree $k$.
Figure \ref{fig:SlitConvergence} shows that the adaptive algorithm recovers optimal rates and verifies the equivalence of the GUBs $\eta_{\rm res}, \etaeqnewa p$, and $\eta_{\rm HHO}$ to the energy error $\trb{e}_\mathrlap{\pd}{w}$.
The efficiency indices in Figure \ref{fig:SlitGUB} show a strong overestimaton by $\eta_{\rm res}$ in the preasymptotic regime (undisplayed) with values $EF(\eta_{\rm res})>60$.
However, asymptotically the quotients $EF(\eta)=\eta/\trb{e}_\mathrlap{\pd}{w}$ for the two residual-based GUBs $\eta_{\rm res}, \eta_{\rm HHO}$ differ only by a factor $10$, while the equilibrated GUBs $\eta_{{\rm eq}, p}$ provide the closest values to $1$.
\begin{figure}
\centering
\includegraphics{./Figures/Legend_k_short_out.pdf}\\
\hspace*{-2em}\hbox{\includegraphics{./Figures/SlitConvergenceAdaptive_out.pdf}
\includegraphics{./Figures/SlitConvergenceGUB_out.pdf}}
\caption{Convergence history of the energy error $\trb{e}_{\rm pw}$ (left) and the GUBs $\eta_{\rm res}, \eta_{\rm HHO}, \etaeqnewa{0}$ (right) on the slit domain}
\label{fig:SlitConvergence}
\end{figure}
\begin{figure}
\centering
\includegraphics{./Figures/Legend_k_short_out.pdf}\\
\includegraphics{./Figures/Legend_ErrorV2_out.pdf}\\
%
\hbox{
\includegraphics{./Figures/SlitEfficiencyV2adaptive_out.pdf}
\includegraphics{./Figures/SlitEfficiency_EQ_out.pdf}}
\caption{History of the overestimation factor
$EF(\eta)=\eta/\trb{e}_\mathrlap{\pd}{w}$ for the residual-based error estimators $\eta_{\rm res}, \eta_{\rm HHO}$ (left) and $\etaeqnewa0, \etaeqnewa1$ (right) on the slit domain}
\label{fig:SlitGUB}
\end{figure}
\subsection{Corner singularity in the L-shaped domain}%
\label{sub:Singular solution}
The third benchmark problem is set in the
L-shaped domain $\Omega = (-1, 1)^2\setminus [0, 1)^2$ with constant right-hand side $f\equiv1$.
Figure \ref{fig:LshapeConvergence}
displays the convergence history of the error $e\coloneqq u-Ru_h$ and compares the adaptive scheme, Algorithm \ref{alg:afem}, driven by the refinement indicators $\eta_{\rm res}(T)$ from \eqref{eqn:etares_T} and
\begin{align*}
\eta_{\rm HHO}^2(T)&\coloneqq |T|\| (I - \Pi_0)(f+{\Delta}_{\rm pw} Ru_h)\|_{L^2(T)}^2 + \|\nabla(1-\mathcal{A}) Ru_h\|_{L^2(T)}^2\notag \\
&\quad+
|T|^{1/2}\sum_{F\in\mathcal{F}(T)} \| R_{T, F}^k u_h\|_{L^2(F)}^2,\\%\label{eqn:etaHHO_ref}
\etaeqnewa0^2(T)&\coloneqq\osc{k}^2(f, T) + \| Q_0^\Delta \|_{L^2(T)}^2 + \|\nabla(1-\mathcal{A}) Ru_h\|_{L^2(T)}^2
\end{align*}
for $T\in\mathcal{T}$ that are induced from the GUB $\eta_{\rm res}, \eta_{\rm HHO}$, and $\etaeqnewa0$.
Here, the norm $\trb{e}_\mathrlap{\pd}{w}$ of the distance $e$ from the discrete solution $Ru_h\in P_{k+1}(\mathcal{T})$ over $\mathcal{T}$ to the unknown solution $u\in H^1(\Omega)$ is approximated by $\trb{\widehat u-Ru_h}_\mathrlap{\pd}{w}$, where $\widehat u\in P_{k+1}(\widehat \mathcal{T})$ is the HHO approximation of $u$ on an adaptive refinement $\widehat \mathcal{T}$ of $\mathcal{T}$ with at least $2|\mathcal{T}|\leq|\widehat\mathcal{T}|$ elements.
The three adaptive schemes (Algorithm \ref{alg:afem}, driven by $\eta_{\rm res}(T), \eta_{\rm HHO}(T)$, or $\etaeqnewa0(T)$) recover optimal rates of convergence and lead to similar local refinement of the adaptive mesh sequences as in Figure~\ref{fig:LshapeMesh}.
\begin{figure}
\centering
\includegraphics{./Figures/Legend_k_short_out.pdf}\\
\hspace*{-2em}\hbox{\includegraphics{./Figures/LshapeConvergence_out.pdf}
\includegraphics{./Figures/LshapeConvergence2_out.pdf}}
\caption{Convergence history plot of the energy error $\trb{e}_\mathrlap{\pd}{w}$ on the L-shaped domain with uniform and adaptive refinement with AFEM, driven by $\eta_{\rm res}(T)$, (left) and for AFEM, driven by $\eta_{\rm res}(T), \eta_{\rm HHO}(T)$, and $\etaeqnewa0(T)$, (right).}
\label{fig:LshapeConvergence}
\end{figure}
\begin{figure}
\centering
\hbox{%
\includegraphics{./Figures/E3_Mesh_res_out.pdf}
\includegraphics{./Figures/E3_Mesh_dp_out.pdf}
\includegraphics{./Figures/E3_Mesh_eq_out.pdf}}
\caption{Adaptive triangulations on the L-shaped domain for $k=3$ from AFEM, driven by $\eta_{\rm res}(T)$ (left, $|\mathcal{T}|=882$), driven by $\eta_{\rm HHO}(T)$ (middle, $|\mathcal{T}|=907$), and driven by $\etaeqnewa0(T)$ (right, $|\mathcal{T}|=919$).}
\label{fig:LshapeMesh}
\end{figure}
\subsection{Conclusion}
The adaptive mesh-refining algorithm recovers optimal convergence rates in all three benchmarks.
This holds for
AFEM driven by any of the three refinement indicators derived from the GUB $\eta_{\rm res}, \eta_{\rm HHO}$, and $\eta_{{\rm eq}, p}$.
The generated mesh sequences from the adaptive schemes, driven by the different estimators, display a very similar concentration of the local mesh-refinement as in Figure \ref{fig:LshapeMesh}.
All three benchmarks verify that the considered error estimators are GUB with reliability constant $1$ while the post-processing in the equilibrated GUB $\eta_{{\rm eq}, p}$ produces minimal overestimation.
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train/arxiv
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BkiUdgc5qYVBbRfeuyi8
| 5
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\section{Introduction}
The brain is a multi-rhythmic system with spatiotemporal activity spanning several orders of magnitude which range from fast localized spiking at the single unit level to macroscopic oscillations that involve many units firing in synchronous patterns. Neural oscillations are believed to underlie a wide spectrum of brain functions (see Buszaki et al. for a review \cite{buszakiscience}). The coordination of motor patterns utilized in animal locomotion and maintenance processes such as breathing are controlled in part by central pattern generators \cite{golubitsky,delcomyn}. There is a growing view that rhythmic neural activity plays an active role in shaping neural processing and behavior by transiently binding cells into synchronized assemblies \cite{singernature}. Different rhythms may shape the effective connectivities of local circuits and alter the way information is processed and routed. Recently, experiments in rodents have shown that theta (8 $Hz$ to 9 $Hz$) oscillations in the rats hippocampus encode the animals position in an arena \cite{buzsakilfp}. The emerging view is that neuronal oscillators may provide a basis for a variety of neural functions including cognition \cite{buszakilibro}.
Nonlinear resonances in periodically-driven neural circuits have previously been studied both theoretically and experimentally in several contexts. Thalamo-cortical interactions were studied using models of weakly connected oscillators in which communication between cortical columns is enabled by resonances \cite{izifm}. Additional examples in the neuroscience literature include the amplification of gamma rhythms in inhibition-stabilized networks \cite{sejnowski}. Recently, studies of periodic stimulation in models of spiking neurons have shown that intrinsic network oscillations can be shaped by changing the amplitude and frequency of the stimulation, suggesting possible implications for trans-cranial stimulation methods \cite{lefebvrejon}. Periodic stimulation of nonlinear physical and biological systems constitutes a traditional approach to characterizing their intrinsic dynamical properties. This approach was used to investigate the intrinsic dynamics of a small network of electrically coupled neurons in the pyloric central pattern generator of the lobster \cite{abarbanelpyloric}. More recently, it was shown that periodic stimulation of telencephalic nuclei in songbirds creates subharmonic entrainments of the respiratory network \cite{mendez}.
The notion that chaotic dynamics might underlie neural dynamics has been explored by many authors. Two aspects of chaotic dynamics that are argued to be relevant for brain function are sparse exploration of the phase space and built-in multipurpose flexibility. Even though the dimension of a chaotic attractor is in general smaller than the allowed state space, chaotic motions explore a broad region of the behaviors available to the system, providing a mean to seize opportunities upon environmental changes \cite{rabinovichroleofchaos}. Neural systems are required for different purposes at different times and under different conditions. A chaotic system can accommodate this type of multipurpose flexibility by switching the temporal programming of a small parametric perturbation in order to stabilize the different orbits embedded within the attractor \cite{ottcontrol}. More recently, Mindlin et al. explored the possibility that the diversity of respiratory motor gestures in birdsong can be partly explained by nonlinear entrainments and bifurcations of a driven hierarchy of neural nuclei \cite{prlsubarmonicos,granada,pranama,alonso09,goldinmindlin,gogui}.
This article introduces a numerical procedure designed to tune dynamical models of neural activity toward special regimes in which many qualitatively different behaviors are available. The procedure is applied to the case of simple circuits composed of interacting subpopulations of neurons which receive oscillatory input from elsewhere. The response of a circuit to a given stimuli will vary depending on the parameters which define the architecture. Additionally, the response of a given circuit may be quite different as the frequency or the amplitude of the incoming stimuli change. Here we are interested in a hypothetical scenario in which the response of the circuit exhibits maximal diversity when receiving external stimuli within a given range. This scenario can be attained by asking that the circuits can be entrained to many different nonlinear resonances. The procedure specifically consists of computing a low resolution approximation of the Arnold tongues diagram of the circuit over a fixed range of stimuli parameters. Each point in the range is assigned an integer corresponding to the period of the response (if it is periodic) in units of the stimuli period. The resulting diagrams are scored by counting how many subharmonic solutions occur in the inspected range and asking that there is an equal number of solutions for each subharmonic type. This approach yields simple circuits that exhibit patterns of periodic activity with multiple timescales and diverse waveforms, as well as complex non-periodic behavior. The procedure can be used to further explore links between chaos theory and neural oscillations.
This work is organized as follows. A numerical procedure to generate circuits which respond sub-harmonically to a family of periodic stimuli is presented in Sections 2a and 2b. The procedure is applied in a simple setting of two interacting populations of neurons described in Section 2c. Several examples of such circuits are presented and discussed in Section 3. Section 4 contains final remarks and future directions.
\section{Methods}
\subsection{Definition of locking period. Description of the algorithm.}
This section introduces a numerical procedure designed to tune the parameters of periodically-driven dynamical systems towards regimes in which several subharmonic entrainment regimes are possible. Let $F(x,p) : \mathbb{R}^{m \times q} \rightarrow \mathbb{R}^m$ be a smooth vector field with state variables $x \in \mathbb{R}^m $ and parameters $p \in \mathbb{R}^q$. It is further assumed that the system is driven by a family of periodic signals of period $\tau$, $\gamma_{\alpha}(t)$ parameterized by $\alpha$,
\begin{eqnarray}
\dot{x} &=& F(x,p + \gamma_{\alpha}(t)) \nonumber \\
x(0) &=& x_0.
\label{ode}
\end{eqnarray}
The purpose of this procedure is to find parameters $p$ such that the system will exhibit subharmonic entrainments with the driving signal $\gamma$, ie: that there is an integer number of periods $p \geq 1$ such that
\begin{eqnarray}
x(t + k (p \tau)) = x(t) \qquad \forall k \in \mathbb{Z} \quad \forall t \in \mathbb{R}.
\end{eqnarray}
In order to test if a given solution is entrained to the driving signal, a Poincar\'e section of the flow is computed by annotating the state of the system $x$ stroboscopically using the frequency of the driving signal \cite{gucken,wiggins}. This is done by taking regularly spaced timestamps every period of the forcing $\tau$ for a maximum number of periods $M$. This yields a sequence of states $\{x_0, x_1, . . ., x_M\}$ which are used to compute the mismatch between the first state $x_0$ and successive states $x_n$,
\begin{eqnarray}
E_n = \parallel x_n - x_0 \parallel^2 .
\end{eqnarray}
Finally, the locking period $l_p$ is defined as the lower $n$ such that $E_n < \epsilon$,
\begin{eqnarray}
l_p = \min_n \{n \in [1,M]: E_n < \epsilon\},
\end{eqnarray}
where $\epsilon$ is a parameter of the procedure corresponding to the numerical accuracy utilized to determine the mismatch between the state of the system in the control sections. If there is no $n$ that satisfies $E_n < \epsilon$, we assign the value $M+1$ indicating that the system does not entrain up to period $M$.
In order to compute the Poincar\'e sections, the system is integrated using a Runge Kutta $O(4)$ method using $dt = \frac{\tau}{100}$ \cite{numericalrecipes}. Before the Poincar\'e section is computed, the system is allowed to relax to the attractor for a number of transient periods $M_t$. This provides a simple numerical way to check if the system is entrained, but it may provide the wrong answer if the transient decays to the attractors are too slow or if the numerical resolution parameter $\epsilon$ is set too high. An additional caveat is that a fixed point solution will be identified as a period $1$ locking. This ambiguity was found to be unimportant for the purposes of this work. This definition provides a map between the parameters of the driving signal $\alpha$ and an integer number corresponding the period of the resulting solution. It is employed here to roughly quantify the diversity of dynamical responses available to a given set of inputs. In the case of weakly interacting oscillators these diagrams have specific shapes in the space of frequencies and amplitudes which resemble a V. They are known as Arnold tongues after V.I. Arnold and they are the objects which inspired this procedure \cite{arnold}.
\subsection{Objective function}
Next we introduce an objective function to drive the parameters of system (\ref{ode}) towards nontrivial entrainment regimes. In general, nonlinear systems will exhibit complex resonances when driven with periodic input. Small variations in the features of the input may result in different locking regimes which can be partially characterized by an integer. This maps the space of stimuli features into a set of integers which define the different entrainment regions. The set of external stimuli is specified by defining a domain for the stimuli parameters $\alpha$, and the locking period is computed in a grid of $N$ regularly spaced points in this domain. A discrete distribution of locking periods $\{L_j\}$ is built by counting how many solutions of each period were found over the chosen domain. The goal of this procedure is to obtain circuits that are close to many nonlinear resonances. One way to achieve this is to ask that the distribution of locking periods is flat, ie: that there is an equal number of solutions for each locking period (up to period $M$) within range. This is attained by minimizing an objective function,
\begin{eqnarray}
\label{costfun}
C(x_0,p) = \sum^M_{j=1} (\frac{L_j}{N} - \frac{1}{M})^2.
\end{eqnarray}
Evaluation of the objective function requires determining the locking period of a number $N$ of solutions. This is the computationally intensive part of the procedure. To summarize, the computational effort required to evaluate the objective function (\ref{costfun}) scales linearly with the number of solutions to evaluate $N$, and linearly with the maximum number of periods $M$ and with the transient periods $M_t$. As discussed in the next section, there are choices of these values for which evaluation of this function is fast enough so that the problem can be tackled by several heuristic optimization approaches.
\subsection{The model}
As a case study, this article considers dynamical models of neural activity which can be cast in the form of an ordinary differential equation or vector field. The procedure is applied to study the dynamics of neural populations using the celebrated Wilson-Cowan model \cite{wilsoncowan72}. This model and its extensions have been widely used to model neural populations and it is a common approach to address several problems in computational neuroscience \cite{destexhewilsoncowan}. Here we consider the simplest case of two populations of interconnected neurons of excitatory and inhibitory subtypes. The state variables are a measure of the activity of each population. In the absence of stimulus, this system can present limit cycles and fixed-point behavior. However, when driven by periodic input, the dynamics can be extremely complex depending on the precise weights of the connections and other parameters which define the architecture. Here we study a family of circuits given by
\begin{eqnarray}
\label{model}
\frac{1}{\tau_1} \dot{x_1} &=& -x_1 + S(C_{11} x_1 + C_{12} x_{2} + \rho_1 + \gamma(t) ) \\ \nonumber
\frac{1}{\tau_2} \dot{x_2} &=& -x_2 + S(C_{21} x_1 + C_{22} x_{2} + \rho_2 ), \\ \nonumber
\end{eqnarray}
where $S$ is the sigmoid function,
\begin{equation}
S(x) = \frac{1}{1 + e^{-x}}.
\end{equation}.
The timescale of each population is controlled by parameters $\tau_i$. Each population receives input from the rest via the connectivities $C_{ij}$ and a constant input $\rho_i$. We assume that an external input $\gamma(t)$ is injected into population $1$ and we are interested in the possibility that the circuits will respond in qualitatively different ways when driven by similar stimuli. Alternatively, the incoming signal can be thought of as produced by another neural oscillator and therefore the full circuit would present multiple stable patterns of periodic activity which can be switched by changing the weights of the connectivities or the offsets activities. We assume a particular family of periodic signals $\gamma$ that models an external neural oscillation defined by
\begin{eqnarray}
\label{forcing}
\gamma(t) = \rho + A S(\eta(\cos(\omega t)-\mu))).
\end{eqnarray}
Here $A$ is the amplitude, $\omega$ is the frequency and $\rho$ is a constant offset. Parameter $\rho$ is included in equation (\ref{forcing}) in order to study modulations both in the amplitude and frequency $(\omega, A)$ as well as modulations in the amplitude and offset $(\rho, A)$. Parameters $\eta =0.75$ and $\mu=-1$ are used to control the waveform of the oscillation and are kept fixed in this work. These parameters were chosen so that the forcing is close to a purely sinusoidal function. Because the focus of this article is to obtain circuits that respond differently to similar stimuli, these parameters were kept fixed for clarity but they can be included in the optimization procedure in the same way as the rest of the parameters. These parameters were kept fixed so that the forcing signal is the same for all the circuits discussed in this work. Therefore, differences in the dynamical properties of each circuit arise from different architectures and not from changes in the spectral content of the forcing signal $\gamma$.
\section{Results}
In this section we discuss circuits defined by equations (\ref{model}) which were obtained by optimization of the objective function (\ref{costfun}). For visualization purposes, two scenarios are considered independently. In the first case, the driving signal $\gamma$ can be modulated in amplitude and frequency, $\alpha=(\omega,A)$. In the second case, the amplitude and constant offsets are allowed to vary, $\alpha=(\rho,A)$. The connectivities are allowed to take a wide range of values $C_{ij} \in [-20,20]$, $\rho_j \in [-20,20]$ and the timescales $\tau_i = 1$ are kept fixed for simplicity. In order to evaluate the cost function we specified a domain for the stimulus parameters $\alpha={ A \in [0,10] , \omega \in [1-0.2 , 1+0.2], \rho \in [-5,5]}$. For each scenario, we took a regular grid of $10 \times 10$ values in the corresponding domains. We set the maximum number of locking periods $M=10$ and the numerical resolution parameter $\epsilon =0.001$. The solutions were allowed to decay to the attractors for $M_t = 10$ periods. Evaluation of the cost function thus entails determining the locking period for $N = 100$ regularly spaced points in the corresponding $\alpha$ domain up to period $M = 10$. For these parameters and in our current implementation we achieve about $15$ evaluations per second in a commercially available linux server. This makes the problem tractable by many heuristic approaches which do not require knowledge of the target function. The objective function (\ref{costfun}) is optimized by a genetic algorithm starting from $50$ random seeds in the search domain which were evolved for $200$ generations \cite{holland}. Each of the circuits presented here were found in about $10$ minutes of computer work and their parameters are summarized in table I.
Figure 1 shows the locking diagrams corresponding to several evaluations of the objective function (\ref{costfun}). The locking period is computed in a $10 \times 10$ grid and indicated in colors (color bar in Fig. 2). Values close to $1$ correspond to circuits which do not respond to the periodic stimuli or wherein the response is trivial, while lower values correspond to more colorful diagrams for which several different entrainments occur. The key realization is that the underlying structure of the locking regions is such that it can be detected by sampling only a few points. The system can accommodate locking regions which consist of well-defined, connected large \enquote{blobs} in the space of stimulus features $\alpha$. The problem of optimizing the objective function is computationally feasible because the structure of the locking diagrams can be partially characterized by sampling the space of stimuli features with low resolutions. The main result of this article is that optimization of this function is possible and yields a procedural mechanism to generate neural circuits with rich dynamical properties.
\begin{figure*}
\includegraphics[width=140mm]{figure.0.pdf}
\caption{\textbf{Evaluations of the objective function.} The circuits are obtained by minimizing the objective function. A single evaluation of the objective function consists of computing the locking period between the circuits response and the stimulus in a $10 \times 10$ fixed grid of stimulus parameters $(A,\omega)$ within a specified domain. This yields a map between stimulus parameters and integers, and a distribution of locking periods over the computed domain. The objective function measures how far this distribution is from being flat (the red line corresponds to a flat distribution). \textbf{(A)} Distribution of locking periods over the computed domain (range shown in figure). \textbf{(B)} Map between stimulus parameters and locking period. Locking periods are color coded ranging from period 1 (blue) to period higher than 10 or no locking (red). The color bar is shown in Fig. 2. \textbf{(C)} Increasing the resolution of (B) reveals the intricacies of the response diagrams.}
\end{figure*}
Figure 2 shows the response of a circuit as the amplitude and the frequency of the periodic forcing are allowed to change. We are interested in the case in which multiple timescales emerge out of the interaction of the populations. Figure 2A shows the response of the circuit for different values of the amplitude $A$. The system can be entrained to a multitude of oscillatory patterns with diverse temporal structure while the frequency of the stimuli remains fixed. Figure 2B corresponds to the locking period diagram over the inspected domain. Note that while the structure of the diagram is complex, there are large connected regions for each locking period. These regions are in turn separated by smaller nested structures of bands that exhibit beautiful patterns. In many cases, the waveform of the oscillations is different for each region. The dashed line in Figure 2B indicates the frequency of the stimulus used in 2A. This value was chosen so that several different regions became available by changing only the amplitude.
The dynamics of the system in areas of the diagrams in-between the larger regions is often associated with strange attractors. Transitions from periodic to chaotic dynamics are known to exhibit universal scaling rules and there are specific routes by which chaotic attractors can develop in low dimensional systems \cite{feigenbaum,libchaber}. Figure 2C shows the dynamics of the circuit as the amplitude values are changed in small decrements towards a transition between regions ($A \in [2.705,2.740]$). The solutions are shown in full phase space $(x_1,x_2,\gamma)$ in order to visualize changes. As the amplitude is increased, the initial curve seems to split in two copies of itself which fail to intersect, in what appears to be a period doubling bifurcation. Further decrements of the amplitude result in successive folds of this mechanism until the rightmost attractor which is non-periodic and possibly chaotic. Period doubling bifurcations and possible chaotic behavior in neural models of the sort studied here, were previously reported by Ermentrout \cite{ermentroutchaos}. The underlying dynamical mechanisms by which these different oscillations emerge is beyond the scope this work as it depends on the specific details of each circuit. While in many cases the transitions from one region to another seem to correspond to local bifurcations, such as period doubling, there are other mechanisms by which complicated dynamics may emerge. Despite the complexity of the solutions there are a finite number of ways by which this can occur due to the low dimensionality of the system. The mechanisms underlying the generation of complex dynamics in low dimensional systems can be partly characterized and classified by topological analysis \cite{alicestretchland,classification}.
\begin{figure*}
\includegraphics[width=140mm]{figure.1.pdf}
\caption{\textbf{Emergence of multiple stable rhythms with complex organization.} When neural circuits are close to nonlinear resonances many dynamical behaviors become possible. The figure shows the response of a neural circuit consisting of two interacting populations of neurons $(x_1,x_2)$ receiving periodic input $\gamma(t)$, for increasing values of the amplitude of the input. \textbf{(A)} Responses of the circuit (red) as the amplitude of the stimulus (green) is increased. Both for low and high values of $A$ the system entrains 1 to 1 and the amplitude of the response is low. For intermediate values of $A$, the system displays a multiplicity of stable periodic rhythms with complex waveforms. \textbf{(B)} Locking period diagram for the considered domain of stimulus parameters $A$ and $\omega$. The locking period is color coded ranging from 1 (blue) to 11 (red, no locking or higher than 10). The dashed line indicates the frequency of the stimulus for all shown solutions. The symbols indicate solutions shown in (A). There are large connected regions for any considered locking period separated by nested structures of bands. This structures are in turn associated with complex dynamics shown in (C). \textbf{(C)} Simulations for decreasing values of $A$ near a transition between large regions plotted in phase space ($A \approx 2.705$). As the amplitude is decreased the solutions undergo period doubling bifurcations and possibly low-dimensional chaos.}
\end{figure*}
This procedure can also be applied to the potentially more interesting scenario in which the stimulus frequency is fixed and only the connectivities are changed. This results in circuits which can be slowly modulated to produce patterns on multiple timescales while being driven at a unique fixed frequency. We obtained circuits with this property by allowing $A$ and $\rho$ to change and optimizing the objective function (\ref{costfun}). Figure 3 shows the locking diagrams of one such circuit. The leftmost panel corresponds to the domain used in the optimization, while the successive panels zoom into the spiral. As before, the diagrams exhibit beautiful nested patterns which look self-similar. There are wide connected regions which are associated with a particular locking period, and typically, these regions are separated by thin bands that correspond to different periods. We can think of an extended system by including $\dot{A}=0 ,\dot{\omega}=0,\dot{\rho}=0$ and then the diagrams can be interpreted as different basins of attraction in the space of initial conditions $(A_0, \omega_0, \rho_0)$. From this point of view, the observed nested band structure is reminiscent of the Wada property for the basins of attraction of multi-stable nonlinear systems \cite{yorke}.
\begin{figure*}
\includegraphics[width=170mm]{figure.2.pdf}
\caption{\textbf{Locking period diagrams for modulations in amplitude and offset.} Circuits can also be tuned close to nonlinear resonances by considering modulations in other parameters. The diagrams show the locking period as the amplitude $A$ and the constant offset $\rho$ of the stimuli are allowed to change. There are large regions corresponding to the same locking period which are separated by complex structures of nested bands. This is highlighted in the zoomed in diagrams (b,c,d) to the right. The rightmost panel is zooming to the point $(A, \rho) = (8.010, 1.4965) \pm 0.001$.}
\end{figure*}
Finally, in order to allow for more biological realism in the resulting circuits, the optimization was performed in an extended domain including the possibility that the time scales of the populations are different. This additional degree of freedom results in lower values of the cost function and into the stunningly diverse locking diagrams shown in Figure 4. It is noteworthy that these type of models are able to accommodate such remarkable properties despite being relatively simple. This scenario might not be achievable in other systems for meaningful ranges of their parameters. Neural models of the sort studied here are remarkably flexible not only because they can be connected in many different ways, but also because the intrinsic properties of the nodes can be different.
In the case considered here, the dynamics of the non-driven autonomous systems differ radically from the driven case. Because the circuits are composed of two populations, the dimension of the autonomous systems is $2$, implying that their solutions are either fixed points or limit cycles, for all values the parameters that define the architectures. As discussed in this article, this is not the case when the system receives periodic input. Since the dimension of the driven system is $3$, the system can in principle accommodate complex dynamical attractors, as those shown in Figure 2. The results presented here show that optimization of function (\ref{costfun}) constitutes an efficient procedure to achieve rich dynamical properties in circuits composed of two neural populations, and suggest that the procedure may be applicable to other systems of interacting nonlinear oscillators.
\begin{figure*}
\includegraphics[width=160mm]{figure.3.pdf}
\caption{\textbf{Stunning complexity of locking diagrams as more diversity is allowed in the circuits.} The diagrams correspond to circuits which were obtained by optimization of the objective function in a more general domain which includes the possibility of having different timescales for each population. The resulting circuits yield better scores in the optimization function and they exhibit a remarkable diversity of behaviors which are available by small modulations of the input.}
\end{figure*}
\begin{table*}
\label{parameters}
\begin{tabular}{ l c c c c c c c c }
Location & $\tau_1$ & $C_{11}$ & $C_{12}$ & $\rho_1$ & $\tau_2$ & $C_{21}$ & $C_{22}$ & $\rho_2$ \\
\hline \hline
\textbf{Fig 1.1} & 1.00 & 5.84 & 12.84 & -16.00 & 1.00 & -12.48 & 5.44 & 6.60 \\
Fig 1.2 & 1.00 & 15.64 &-14.32 &-0.32 & 1.00 & 19.12 & 10.64 & -8.36 \\
Fig 1.3 & 1.00 & 1.08 & -10.72 &4.32 & 1.00 & 16.28 & 8.72 & -18.96 \\
Fig 1.4 & 1.00 & 15.56 &-9.32 &-4.80 & 1.00 & 13.08 & 14.00 & -15.32 \\
Fig 1.5 & 1.00 & 8.04 & 6.80 & -10.00 & 1.00 & -19.92 & 11.52 & -3.28 \\
\hline
\textbf{Fig 2} & 1.00 & 4.92 & -6.76 & -3.00 & 1.00 & 14.96 & 18.76 & -14.96 \\
\hline
\textbf{Fig 3} & 1.00 & 2.32 & -17.32 &8.52 & 1.00 & 15.16 & 16.44 &-18.88 \\
\hline
\textbf{Fig 4.1.1} & 1.838 & 11.44 & -8.76 & -3.64 & 1.751 & 19.40 &10.28 & -7.12 \\
Fig 4.1.2 & 1.8395 & 10.96 & -12.00 & -3.68 & 1.088 & 8.4 & 10.00 & -6.84\\
Fig 4.1.3 & 1.7705 & 14.28 & -9.72 & -3.64 & 0.701 & 16.96 & 7.60 & -4.92 \\
Fig 4.2.1 & 1.9145 & 14.68 & -9.72 & -3.84 & 1.9085 & 11.32 & 10.56 & -7.80 \\
Fig 4.2.2 & 1.802 & 10.36 & -9.44 & 0.56 & 0.545 & 16.60 & 7.24 & 5.00 \\
Fig 4.2.3 & 1.9355 & 11.20 & -8.24 & -3.56 & 0.7745 & 12.88 & 19.28 & -16.76 \\
\end{tabular}
\caption{The parameters for all the circuits shown in this article are listed here. Circuits in Figure 1 are numbered 1 through 5 from left to right. The circuits in Figure 4 are numbered according to their row and column numbers.
}
\end{table*}
\section{Conclusions}
This article describes a numerical procedure designed to tune the parameters of periodically-driven dynamical systems towards regimes in which several subharmonic solutions are possible. The procedure was employed to tune the parameters of simple, low-dimensional models of neural circuits to a state in which an incoming periodic stimuli can result in many different dynamical behaviors. The procedure makes it possible to find simple architectures consisting of two populations of neurons that can accommodate such a scenario by nonlinearly entraining to the incoming signals. These circuits can be found as the result of optimizing an objective function that measures the diversity of entrainment types by computing a low resolution approximation of the locking regions of the circuits. While it is likely that a similar approach would be applicable to other systems, it is unclear whether a given system may be able to accommodate many nonlinear resonances in ranges of the parameters that are physically and biologically plausible. Neural circuits of the sort discussed in this article possess a remarkable flexibility to support this type of dynamical behavior, even in the simple case of only two interacting populations.
Neural circuits that support an array of functions might be desirable in several contexts \cite{hoppenstead}. The procedure presented here is useful to design multipurpose pattern generators \cite{kopellcpg}. This in turn can be used as an encoder: a long complex sequence of oscillatory commands can be encoded as different responses to simpler signals consisting of parametric modulations to the circuit \cite{paramest}. The procedure presented here facilitates exploration of the hypothesis that nonlinear resonances in neural circuits may play a role in neural function by providing a tool to find the special connectivities that give rise to these dynamical properties. This procedure also enables the investigation of the dynamics of large scale models of weakly interacting neural circuits when the parameters of each circuit are tuned close to nonlinear resonances. From a dynamical systems perspective it would also be interesting to see if circuits that exhibit these properties also share other dynamical properties such as the bifurcations diagrams of the non-driven system. Finally, the procedure makes it easy to identify transitions between regimes in which universal scaling rules are expected to arise. Therefore, the topological mechanisms by which complex dynamics emerge in simple neural circuits can be systematically investigated.
\section*{Acknowledgments}
Leandro M. Alonso's research was supported by funds from a Leon Levy Fellowship at The Rockefeller University.
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\section{Introduction}
\label{sec:intro}
The existence of phase transitions in nuclear matter was proposed more than 20
years ago in conjunction with the description of the structure of neutron
stars~\cite{Baym-75,Palmer-74}.
This led to the question whether phase separation~\cite{Jaqaman-83}, dynamical
instabilities~\cite{Bertsch-83} or critical phenomena~\cite{Bauer-85} play a
role in the disintegration of highly excited nuclei produced in proton or heavy
ion induced collisions.
The observation of a power law for the fragment size
distribution~\cite{Finn-82,Minich-82} was the first experimental hint for a
second order phase transition, but the determination of critical
exponents~\cite{Campi-86} or other parameters~\cite{Siemens-83} turned out to be
difficult with the mostly inclusive data available at that
time~\cite{Panagiotou-84,Porile-89,Campi-88}.
In the past 3 years, however, data from a new generation of experiments with an
almost complete coverage for the decay products of an excited system allowed for
rapid progress.
The observation of potential signals of a first order phase
transition~\cite{Pochodzalla-95,Moretto-96} and the attempt of a quantitative
determination of critical exponents~\cite{Gilkes-94} sparked an intense
discussion on what reliable signatures for a first or second order phase
transition in a small and dynamically evolving system are.
In the following we will take the recent claim~\cite{Gilkes-94,CRIS-Tincknell},
that a continuous phase transition with critical exponents consistent with those
of a liquid-gas system has been observed in the fragmentation of nuclei, at
face value and revisit the theoretical arguments used in this analysis.
\section{Critical Phenomena in a Nutshell}
\label{sec:nutshell}
Near a critical point, the order parameter $\phi$, associated susceptibility
$\chi$, specific heat $c$, and the correlation length $\xi$ exhibit a power law
dependence on two reduced control parameters $\epsilon$ and $h$:
\[
\begin{array}{r @{\;\;} c @{\;\;} l}
\phi_{(\epsilon,h=0)} &=& B \epsilon^{\beta} \\[0.5ex]
\phi_{(\epsilon=0,h)} &\sim& h^{1/\delta} \\[0.5ex]
\chi_{(\epsilon,h=0)} &=& \Gamma \epsilon^{-\gamma} \\[0.5ex]
c_{(\epsilon,h=0)} &=& A \epsilon^{-\alpha} \\[0.5ex]
\xi_{(\epsilon,h=0)} &=& \xi_0 \epsilon^{-\nu}
\end{array}
\mbox{\hspace{2em} with \hspace{2em}}
\begin{array}{c @{\hspace{1em}} c @{\hspace{1em}} c @{\hspace{1em}} c}
& \mbox{Liquid-Gas} & \mbox{Magnet} & \mbox{Percolation} \\[0.5ex]
\phi & \rho-\rho_c & M & P_\infty \\[0.5ex]
\chi & {\textstyle \frac{1}{\rho^2} \frac{\partial \rho}{\partial \mu}} &
{\textstyle \frac{\partial M}{\partial H}} &
{\textstyle \sum s^2 n_s} \\[0.5ex]
\epsilon & {\textstyle \frac{T-T_c}{T_c}} &
{\textstyle \frac{T-T_c}{T_c}} &
p_c - p \\[0.5ex]
h & \mu - \mu_{0(T)} & H & p_g
\end{array}
\]
For liquid-gas, magnetic and percolation systems the order parameter $\phi$ is
the difference between liquid and critical density, magnetization per spin $M$
or the fraction of sites in the largest cluster $P_\infty$, respectively.
The susceptibility $\chi$ corresponds to the isothermal compressibility
$\kappa_T = -\frac{1}{V} \frac{\partial V}{\partial p}$ and the second moment of
the cluster size distribution for the liquid-gas and percolation case.
The `thermal' control parameter $\epsilon$ is the difference of the temperature
$T$ or bond probability $p$ to the critical value while the `field' control
parameter $h$ is given by the chemical potential $\mu$,
the external field $H$ or the ghost bond~\cite{Reynolds-77,Reynolds-80}
probability $p_g$.
\section{From Thermodynamics to Percolation and Back}
\label{sec:td2perc}
The exponents $\beta$ and $\gamma$ can be determined from a measurement
of $\phi_{(\epsilon,h=0)}$ and $\chi_{(\epsilon,h=0)}$.
For liquids it is indeed possible to determine $\Delta\rho = \rho - \rho_c$ and
$\frac{\partial \rho}{\partial \mu}$ directly, e.g. by measuring the density
profile in the gravitational field~\cite{Wilcox-68,Moldover-79,Pestak-84}.
In the nuclear physics case, the only directly measured quantities are the
size and momenta of clusters produced in an interaction.
The standard argument to deduce $\Delta\rho$ and $\kappa_T$ from the cluster
size distribution uses the Fisher droplet model~\cite{Fisher-67}. In this
Ansatz the grandcanonical partition function ${\cal Z}$ is expressed as a sum
over cluster yields $Y_{(A)}$ which depend on a surface energy~$s$ and
chemical potential difference between liquid and gas phase $\mu$:
\[
\ln {\cal Z} = \sum Y_{(A)}
\mbox{\hspace{2em} with \hspace{2em}}
Y_{(A)} = q_0 A^{-\tau} \exp \left ( {\textstyle - \frac{1}{T}}
\left ( s A^\sigma + \mu A \right ) \right )
\]
From this follows immediately that $\kappa_T$ is proportional to the
second moment of the cluster size distribution
\begin{equation}
\kappa_T = \frac{T}{V \rho^2} \frac{\partial^2}{\partial \mu^2} \ln {\cal Z}
= \frac{1}{T V \rho^2} \sum A^2 Y_{(A)} .
\label{kappa_m2_fisher}
\end{equation}
It can also be shown~\cite{Fisher-67} that $\Delta\rho$ is proportional to the
fraction of constituents belonging to the largest cluster $P_\infty$.
This strong similarity to percolation, where $P_\infty$ and the second moment
also play the role of order parameter and susceptibility, has led to a
widespread use of percolation in the modeling and interpretation of nuclear
physics experiments.
In particular, percolation has served as reference model for the development
of methods to extract critical exponents~\cite{Elliott-94,Woerner-95}.
Even though there is a mathematical connection between percolation and the
thermodynamics of interacting systems (the $q \rightarrow 1$ limit of the
Potts model corresponds to bond percolation~\cite{Kasteleyn-69}) there are
also some significant differences:
\begin{licompact}
\item Percolation is usually discussed in terms of only one control
parameter, the bond or site probability, leaving the impression that
the `field' control parameter, corresponding to chemical potential
or average density in interacting systems, is of minor importance.
\item Finite size effects depend strongly on the ensemble in
interacting systems~\cite{Promberger-95,CRIS-Hueller,CRIS-Gross}, but
there is no direct equivalent to the concept of an ensemble in
percolation.
\item Percolation theory simply starts with the definition of what a
cluster is.
Thermal systems, on the other hand, are usually defined in terms of the
interaction between their constituents and the appropriate definition of
a cluster is, as will be shown, nontrivial.
\end{licompact}
The almost exclusive reliance on percolation as a reference model bears
therefore the risk that important parts of the physics are missed.
In the following we will consequently turn back to the thermodynamic basics
and use the Ising model as guidance.
\section{From Constituents to Clusters}
\label{sec:con2clu}
As a first step it is interesting to establish the connection between
$\kappa_T$ and the cluster size distribution with minimal assumptions and
without using a specific model.
We follow an idea given by Alexandrowicz~\cite{Alexandrowicz-89} for the
Ising model but generalize it to an arbitrary system.
In the grand canonical ensemble, $\kappa_T$ is related to the fluctuation of
the particle number $\sigma^2_N$ by
\[
\sigma^2_N = \left . kT \frac{\partial N}{\partial \mu} \right | _{T,V}
= kT \frac{N^2}{V} \kappa_T
\]
In the following, we consider a system with $N_S$~constituents coupled to a
reservoir with $N_R$~constituents:
\\[1ex]
\begin{center}
\begin{picture}(300,50)(0,0)
\put(100,10){$N_R$}
\put(130,25){\oval(100,50)}
\put(140,15){\framebox(30,20){$N_S$}}
\put(0,35){$N_S$: System}
\put(0,25){$N_R$: Reservoir}
\put(0,15){$N = N_R + N_S$}
\put(200,40){Define for constituent $i$:}
\put(200,15){$ q_i = \left \{ \begin{array}{ll}
1 & \mbox{if in $S$} \\
0 & \mbox{if in $R$}
\end{array} \right .
$}
\end{picture}
\end{center}
\vspace*{1ex}
To express $\sigma^2_{N_S}$ in terms of clusters we assume that the
constituents are grouped into clusters with the properties:
\\[0.5ex]
\hspace*{2em}{\bf P1} \hspace{1em}
A cluster is either completely in $S$ or in $R$.
\\
\hspace*{2em}{\bf P2} \hspace{1em}
Constituents in different clusters are uncorrelated.
\\[0.5ex]
$\sigma^2_{N_S}$ can now be rewritten as a $q$ correlation. The sum over
constituent pairs can be split into two parts, one where the pair is in the
same cluster and one where it is in different clusters:
\begin{eqnarray*}
\sigma^2_{N_S}
& = &
\left < N_S^2 \right > - \left < N_S \right >^2
\\
& = &
\left < \sum_{ij} (q_i-\overline{q})(q_j-\overline{q}) \right >
\mbox{\hspace{3em} with
$\overline{q} = \frac{\left < N_S \right >}{N}$
}
\\
& = &
\left < \sum_{\stackrel{ij}{\mbox{\scriptsize same}}}
(q_i-\overline{q})(q_j-\overline{q}) \right > +
\underbrace{
\left < \sum_{\stackrel{ij}{\mbox{\scriptsize diff}}}
(q_i-\overline{q})(q_j-\overline{q}) \right >
}_{\stackrel{\mbox{\tiny (P2)}}{=} \; 0}
\\
& \stackrel{\mbox{\tiny (P1)}}{=} &
(1-\overline{q})^2
\left < \sum_{\stackrel{c_i}{\mbox{\scriptsize S}}} |c_i|^2 \right > +
\overline{q}^2
\left < \sum_{\stackrel{c_i}{\mbox{\scriptsize R}}} |c_i|^2 \right >
\end{eqnarray*}
In the last line, the sum over constituents was rewritten as a sum over
clusters ${\displaystyle
\sum_{\stackrel{ij}{\mbox{\scriptsize same}}} \rightarrow
\sum_{c_i} |c_i|^2 }$
and split into two parts, running over $S$ and $R$, respectively.
In the thermodynamic limit, $N_R \rightarrow \infty$ , $\overline{q}
\rightarrow 0$, the second term goes to zero if the largest cluster in $R$
grows slower than $N_R$ (in other words: No condensation in $R$)
and one finally gets
\begin{equation}
\kappa_T = \frac{V}{k T}\frac{\sigma^2_{N_S}}{\left < N_S \right >^2}
= \frac{1}{k T \rho} \sum_s s^2 n_s
\label{kappa_cluster}
\end{equation}
where $n_s$ is the concentration of clusters of size $s$.
This is equivalent to Eqn.~(\ref{kappa_m2_fisher}) but was derived with only
one essential assumption, that constituents in different clusters are
uncorrelated.
\section{The Ising Model}
\label{sec:ising}
One of the simplest thermodynamic models with a phase transition is the Ising
model, given by the Hamiltonian:
\begin{equation}
{\cal H} = -J \sum_{<i,j>} \sigma_i \sigma_j - H \sum_i \sigma_i
\label{ising_hamiltonian}
\end{equation}
It can be interpreted as a model for a magnet, with $\sigma = \pm 1$
representing up and down spins, or as a lattice model of a gas~\cite{Yang-52}
where $\sigma = \pm 1$ now indicates whether a site is occupied or empty.
The number of sites $N$, magnetization $M$ and external field $H$ correspond
in the lattice gas interpretation to the volume $V$, density $\rho$ and
chemical potential $\mu$, respectively.
Order parameter $\left<|M|\right>$ and susceptibilities $\chi$ and $\chi^\prime$
for $T \ge T_c$ and $T < T_c$, respectively, are defined in terms of
constituents as:
\begin{equation}
\begin{array}{rcl}
\left<|M|\right> & = & {\textstyle \left < \left |
\frac{1}{N} \sum_i \sigma_i \right | \right > }
\\[1.0ex]
\chi & = & { \textstyle \frac{N}{kT} \left < M^2 \right > =
\frac{N}{kT} \left (\frac{1}{N^2}
\left < \sum_{ij} \sigma_i \sigma_j \right > \right ) }
\\[1.0ex]
\chi^\prime & = & { \textstyle \frac{N}{kT} \left (
\left < M^2 \right > - \left < \left | M \right | \right > ^2 \right ) }
\end{array}
\label{mchi_true}
\end{equation}
$M$ and $\chi$ can be expressed in terms of clusters using arguments
which are for $\chi$ analogous to the ones in the previous section.
A detailed derivation along those lines was given by de Meo~\cite{DeMeo-90}
while other authors arrived at the same result from different
viewpoints~\cite{Alexandrowicz-89,Wang-89,Hu-84,Roussenq-82}.
One obtains for the cluster observables $\widetilde{M}$, $\widetilde{\chi}$
and $\widetilde{\chi^\prime}$:
\begin{equation}
\begin{array}{cclcc}
\widetilde{M} & := &
\left < P_\infty \right > & \leq &
\left < \left | M \right | \right >
\\[0.7ex]
\widetilde{\chi} & := &
\frac{1}{kT} \: \chi_p & = &
\chi
\\[0.7ex]
\widetilde{\chi^\prime} & := &
\frac{1}{kT} \left ( \chi_p^\prime +
N (\Delta P)^2 \right ) & \geq &
\chi^\prime
\end{array}
\label{mchi_cluster}
\end{equation}
with the fluctuation of largest cluster $(\Delta P)^2 = \left < P^2_\infty
\right > - \left < P_\infty \right >^2$ and the percolation susceptibility
$\chi_p = {\textstyle \sum_m m^2 <n_m>}$ where $n_m$ is the number of clusters
per site with magnetization $m$.
While the order parameter is indeed given by the relative size of the
largest cluster, we observe two significant differences between
$\widetilde{\chi}$ and $\chi_p$:
\begin{licompact}
\item The trivial $\frac{1}{kT}$ factor (see also
Eqn.~(\ref{kappa_m2_fisher}) and (\ref{kappa_cluster})) makes sure that
the paramagnetic or ideal gas limit $\chi \propto \frac{1}{kT}$ is
approached for $T \rightarrow \infty$ where $\chi_p \rightarrow 1$.
Even though it does not affect the asymptotic power law behaviour
for $T \rightarrow T_c$ it will change the effective exponents if a wide
temperature range is considered, like in the nuclear exponent
analysis~\cite{Gilkes-94}.
\item The susceptibitity $\widetilde{\chi^\prime}$ for $T < T_c$ is given
by the sum of $\chi_p^\prime$, the second moment taken without the
largest cluster, and the fluctuation of the largest cluster $(\Delta P)^2$.
\end{licompact}
\begin{figure}[tb]
\centerline{
\epsfig{file=plot_g_p08_th_ws_pro.eps,height=9.5cm}
\epsfig{file=plot_chirat.eps,height=9.5cm}
}
\caption[]{
Left two columns: Comparison of cluster observables (Eqn.~(\ref{mchi_cluster}),
full dots) and true values (Eqn.~(\ref{mchi_true}), full lines) for an
Ising model with $8^3$ sites. The dotted lines indicate the bulk critical
temperature $T_c = 4.51152$~\cite{Bloete-95}.
\\
Right column: Susceptibility ratio
$\widetilde{\chi}/\widetilde{\chi^\prime}$ at $T_c$ as a function
of the linear system size $L$ with (full dots) and without (open dots)
the $(\Delta P)^2$ contribution to $\widetilde{\chi^\prime}$.
}
\label{fig_ising}
\end{figure}
\noindent
Numerical results for the 3d Ising model with periodic boundary conditions
are shown in Fig.~\ref{fig_ising}.
The calculations were done with the Swendsen-Wang~\cite{Swendsen-87} and
Wolff~\cite{Wolff-89} algorithms in implementations similar to
Ref.~\cite{Wang-90} using a random number generator proposed by
Ziff~\cite{Ziff-92,RNG-remark}.
In the left two columns we compare the cluster observables to the true values
given by Eqn.~(\ref{mchi_true}).
Consistent with Eqn.~(\ref{mchi_cluster}) we find $\widetilde{M}$ slightly
smaller than $\left < \left | M \right | \right >$
and $\widetilde{\chi}$ equal to $\chi$ while $\widetilde{\chi^\prime}$ is
only close to $\chi^\prime$ for $T < T_c$.
It is interesting to note, that the main contribution to
$\widetilde{\chi^\prime}$ is the fluctuation term $(\Delta P)^2$ while
the second moment term $\frac{1}{kT}\chi_p^\prime$ (indicated by open
dots in the top middle frame of Fig.~\ref{fig_ising}) carries only a small
part of the signal.
This can be seen more quantitatively in the right panel of Fig.~\ref{fig_ising}
where the susceptibility ratio $\widetilde{\chi}/\widetilde{\chi^\prime}$ at
$T_c$ is shown as a function of the linear system size $L$ with (full dots)
and without (open dots) the $(\Delta P)^2$ contribution to
$\widetilde{\chi^\prime}$.
Depending on system size, between 60 and 75\% of the susceptibility are
carried by the fluctuation term.
\section{Critical Clusters}
\label{sec:criclu}
Clusters can be defined in interacting systems in many different ways.
We call the result of a definition `critical clusters' if order parameter
and susceptibility correspond to size of largest cluster and second moment,
or formally, if Eqn.~(\ref{mchi_cluster}) holds.
We saw in the previous two sections that a necessary condition is the
noncorrelation of constituents in different clusters but have not given
so far an explicit definition.
The most obvious way to define a cluster is the `geometrical cluster' which
corresponds to a domain in the Ising model, or in the lattice gas picture
to a connected region of occupied sites surrounded by empty sites.
However, it is easy to see that this violates the noncorrelation
requirement~\cite{Alexandrowicz-89} and it has indeed been shown that
in 3d `geometrical clusters' lead to a percolation transition at temperatures
below the thermal transition~\cite{mueller-krum-74}, thus do not fulfill
Eqn.~(\ref{mchi_cluster}).
The proper prescription for `critical clusters' for the Ising model was given
by Coniglio and Klein~\cite{Coniglio-80} and requires that the geometrical
clusters are broken into smaller pieces with a bond percolation with
$p_b = 1 - \exp(-\frac{2J}{kT})$.
The definition of a `physical cluster', based on pairwise binding, proposed
by Hill~\cite{Hill-55} and recently used in a lattice gas model for
nuclear fragmentation~\cite{Pan-95}, results in bond probabilities quite
similar to the Coniglio-Klein values.
Even though both definitions give numerically similar results for small
system sizes they are clearly not equivalent in the thermodynamic limit.
\section{Summary}
\label{sec:summary}
Percolation describes remarkably well the cluster distributions and
correlations in nuclear fragmentation~\cite{Desbois-87,Kreutz-93}.
This, together with its simplicity and ease of use, made it the ideal
reference model for the study of signatures of critical behaviour.
But given the hints, that we observe a liquid-gas rather than a percolation
phase transition, we have to face the limitations of percolation.
Even scratching at the surface of a proper thermodynamic description raises
many issues:
\begin{licompact}
\item {\bf Critical clusters} have rather remarkable properties, they are
neither well separated nor compact \footnote{they are in general fractals
with a dimension $y_h = \frac{1}{\sigma\nu}$ or about 2.5 in the 3d Ising
class}, are interacting but have nevertheless no correlations between
constituents of different clusters.
So one might wonder whether critical clusters are mere mathematical
constructs, like the clusters in Mayer's cluster expansion, or real
physical entities.
All attempts to infer signatures of critical behaviour in nuclear physics
not only imply that critical clusters are observable objects, they also
assume that the distribution of clusters formed in the decay of a system
is representative of the equilibrium distribution at some freeze-out
condition.
Even though there are attempts to support this connection with model
calculations~\cite{Finocchiaro-96} it remains to be seen whether there is
a more rigorous way to justify this.
\item {\bf Ensembles:}
Although the experimental situation is certainly better represented
by a microcanonical treatment we used for simplicity for all arguments
in sections~\ref{sec:con2clu} and~\ref{sec:ising} the grand canonical
ensemble.
While the expectation values of extensive quantities do not depend on
the ensemble in the thermodynamic limit it is easy to show that this is
not the case for fluctuations~\cite{Moukarzel-89}.
The cluster observables (Eqn.~(\ref{mchi_cluster})) are therefore likely
to be ensemble dependent.
\item {\bf Role of control parameters:}
The exponent analysis rests on the precondition that the temperature of
the system is varied while the second control parameter, the density,
stays constant and close to its critical value.
Assuming that we observe conditions along a freeze-out line and taking
Papp's schematic model~\cite{Papp-95,CRIS-Papp} as a guide one sees that
the reverse might be true, that the temperature is almost constant but
that the density varies.
In this case one would determine different exponents, e.g.
$1/\delta$ rather than $\beta$ (see section~\ref{sec:nutshell}).
One should also keep in mind that the relative size of the largest
cluster correlates with the density of the liquid phase (thus the
order parameter) only on the critical isochore but is in general more
readily interpreted as a measure of the mass fraction of the liquid phase.
\item {\bf Corrections to scaling:}
The power laws listed in section~\ref{sec:nutshell} hold only
asymptotically for $\epsilon \rightarrow 0$ while a description in a
wider range is possible with correction to scaling terms~\cite{Wegner-72}
or a crossover approach~\cite{Anisimov-92}.
The size of the critical region depends strongly on the exact form of
the interaction and is rather small in real liquids~\cite{Anisimov-92}
but substantially larger in models with only next-neighbor or short
range interactions.
The leading correction term for the order parameter in the Ising
model~\cite{Talapov-96} is not only substantially smaller in magnitude
but has even a different sign as compared to typical
liquids~\cite{Pestak-84}.
It is therefore uncertain whether simple schematic models, like
percolation or Ising, are adequate for the modeling of finite size
and finite control parameter effects in the nuclear case.
\item {\bf Field gradients:}
The hallmark of critical phenomena is scale invariance which at least
requires homogeneous conditions throughout the system under study.
A Coulomb or a radial flow~\cite{CRIS-Lacey} field will prevent the
growth of fluctuations across the whole system much like the chemical
potential gradient caused by the gravitational field is limiting the
usable sample size in earthbound experiments on
liquids~\cite{Moldover-79}.
\end{licompact}
All those points will have to be addressed before the experimental results can
be connected to the parameters of bulk nuclear matter in a quantitative way.
\section*{Acknowledgments}
J.P. and M.B. acknowledge the financial support of the Deutsche
Forschungsgemeinschaft under Contract Nos. Po256/2-1 and
Be1634/1-1, respectively.
This work was supported in part by the European Community under Contracts
ERBCHGE-CT92-0003 and ERBCIPD-CT94-0091.
We like to thank D. Stauffer for bringing Ref.~\cite{Moukarzel-89} to our
attention.
\section*{References}
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\section{Introduction}
\IEEEPARstart{M}{olecular} communication is a biologically inspired form of communication, where chemical signals are used to transfer information \cite{far16ST,mole_book,weimag}. It is possible to modulate information on the particles using different techniques such as concentration \cite{kur12}, type, ratio \cite{kim13}, number \cite{far12NanoBio}, time of release \cite{eck09}, or a combination of these techniques \cite{far16ST}. Moreover, information particles can be transported from the transmitter to the receiver using diffusion \cite{mah10}, active transport \cite{far11Bionet}, bacteria \cite{lio12}, \revised{and/or flow} (or advection) \cite{bic13}. Between all these techniques, diffusion and flow-based propagation are the easiest to implement, and a few experimental platforms have been built to demonstrate molecular communication based on these transport mechanisms~\cite{far13,far14INFOCOM,koo16}.
In this work we focus on modulation techniques for molecular communication and their corresponding system models. Most prior work on modulation techniques rely on the concentration or the type of the released particles. For example in \cite{ata12}, the order of release of consecutive distinguishable particles is proposed for encoding information. In this work, we consider molecular timing (MT) channels where timing-based modulation is employed. Only a few works have considered this type of modulation: In \cite{eck09} the time of release of the particles is used for encoding information, while in \cite{kri13} the information is encoded in the time interval between two pulse releases of information particles \revised{in mirofluidic channels}.
The work \cite{sri12} showed that in the case of timing-based modulation, where information is encoded on the release timing of particles, and the transport mechanism is diffusion assisted by constant laminar flow, the channel can be represented as an additive noise channel. In this case the noise term follows the inverse Gaussian (IG) distribution.
Capacity bounds for the additive IG noise channel, in bits per channel use, under an average delay constraint, were derived in \cite{sri12, li14}.
In \cite{far15GLOBCOM}, we have shown that in the case of timing-based modulation, where information is encoded on the release timing of particles, and pure diffusive transport (i.e., diffusion without any flow) is employed, the channel can be represented as an additive noise channel where the noise follows the L\'evy distribution.
The capacity of this channel was studied in \cite{far16ISIT, rose16Part1, rose16Part2, far17TIT}, and it was shown that this capacity can increase poly-logarithmically with respect to the number of simultaneously released particles. A sequence detector for this modulation scheme was presented in \cite{mur17NanoComNet}.
\revised{In this work, we propose two asynchronous timing-based modulation techniques and compare them with the synchronized timing modulation considered in prior work \cite{mur17NanoComNet,mur16Globecom,mur17}.}
These systems can be represented by an additive noise channel, and for diffusion-based MT (DBMT) systems \revised{in a 1-dimensional environment} the noise falls in the stable distribution family~\cite{nol15}. \revised{The models considered in this paper can be used to represent molecular channels for communication on bio-chips. In bio-chips, components within a chip, such as a storage unit, a molecular processing unit, sorting unit, etc. are connected by narrow microfluidic links. Since these links are very narrow, they are well approximated as a 1-dimensional environment. Moreover, the transmitter and the receiver in these applications can be designed with great precision \cite{lee09,hor15}.
The three systems considered in this paper are as follows. First, we consider a synchronized MT system, where information is encoded in the release timing of information particles (system A); second, an asynchronous MT system is proposed where information is encoded in the time between two consecutive releases of {\em indistinguishable} information particles (system B); and finally, another asynchronous MT system is considered where information is encoded in the time between two consecutive releases of {\em distinguishable} information particles (system C).} Fig.~\ref{fig:chanModels} depicts all three systems.
One of the main motivations for proposing these new modulations is the challenge of synchronization. In particular, for some applications involving micro and nano-scale devices, it may be difficult to synchronize the transmitter and the receiver due to their small size and limited power. In this case, the modulation scheme in system A, which has been used in previous works, may be too difficult to implement in practice. The newly presented modulation schemes in systems B and C, however, do not require synchronization between the transmitter and the receiver. These modulations are analogous to differential phase-shift keying (PSK) in that the asynchronous MT modulations do not require an absolute time reference, while the differential PSK does not require an absolute phase reference.
It must be noted that stable distributed noise arises in system models for a number of different applications. Therefore, the results of this paper could also be applicable in those areas. Specifically, in \cite{he14}, alpha-stable distributed noise was used to model room acoustics. In radio communications, symmetric alpha-stable distributions were used to model impulsive non-Gaussian noise such as those that exists in ultra-wide bandwidth systems~\cite{nir09,fan12}. Capacity bounds for a special class of alpha-stable additive noise channels were provided in~\cite{wan11,fah12}. \revised{Although in this work we focus on additive stable distributed noise channels in the context of molecular timing channels, the analysis and the results are applicable to the general detection problem in additive stable distributed noise channels.}
There are only three classes of stable distributions with closed-form probability density functions (PDF) in terms of elementary functions: Gaussian, Cauchy, and L\'evy. In this work, we derive closed-form expressions for the PDFs of the noise terms in systems B and C in terms of the complex error function and Voigt functions \cite{voigt92, abr11}, which are used in other fields of science such as physics. Thus we develop new closed-form PDF results for a subclass of stable noise distributions in terms of the Voigt functions, which can be efficiently calculated numerically \cite{zag11, abr15}, and can be approximated using elementary functions in some special cases \cite{sch16}.
To compare the performance of the three proposed modulation schemes, we consider a binary communication system and derive the optimal detection rule for each modulation technique. Since the system noise in all three cases is heavy-tailed with infinite variance, the standard definition of signal power, used in electromagnetic communication, is not suitable. Instead, we derive the expressions for the geometric power \cite{gon06} of a large class of stable distributions, and use it to represent the noise power. Furthermore, instead of using the well known signal-to-noise ratio (SNR) metric, we use the geometric SNR (G-SNR)~\cite{gon06} metric, which is given by the geometric power of the signal divided by the geometric power of the noise with some normalization constants. Based on numerical evaluations we observe that for the modulations considered, the bit error rate (BER) is constant for a given G-SNR regardless of the geometric signal power and the geometric noise power.
Based on the above derivations, we next use numerical evaluations to compare the BER of all three systems. We show that system B with indistinguishable particles exhibits the highest BER, while system A achieves the lowest BER. This indicates that time-synchronized transmission over MT channels, i.e. Modulation A, works better than the other two modulations considered.
We further show that by adjusting the diffusion coefficients of the information particles in system C, which is an asynchronous transmission, the BER can approach the BER of system A, where full synchronization is assumed.
However, this comes at the cost of added system complexity where both the transmitter and receiver must be capable of transmitting and detecting two distinguishable particles.
The rest of this paper is organized as follows. In Section \ref{sec:model} we present the three timing-based modulation techniques, and derive an additive noise system model for each of them. In Section \ref{sec:noiseModel} we focus on the diffusion-based propagation and derive the PDF for the additive noise term for each system. In Section \ref{sec:PBE}, binary communication is studied, and the optimal detectors are derived. The geometric power of the noise and the G-SNR of each system are derived in Section \ref{sec:SNR}.
Numerical BER evaluations of the proposed modulation techniques are presented in Section \ref{sec:numEval}, and concluding remarks are provided in Section \ref{sec:conc}.
\textbf{\textit{Notation:}} We denote the set of real numbers by $\mathcal{R}$, and the set of positive real numbers by $\mathcal{R}^{+}$. Other than these sets, we denote sets with calligraphic letters, e.g., $\mathcal{T}$. We denote random variables (RV)s with upper case letters, e.g., $X$ and $Y$, and their realizations with the corresponding lower case letters, e.g., $x$ and $y$. We use $f_{Y}(y)$ to denote the PDF of a continuous RV $Y$ on $\mathcal{R}$, $f_{Y|X}(y|x)$ to denote the conditional PDF of $Y$ given $X$, and $F_{Y}(y)$ and $F_{Y|X}(y|x)$ to denote the corresponding cumulative distribution functions (CDF). We use $\varphi_{X} (\revised{\omega})$ to denote the characteristic function of the RV $X$ and we use the notation $X \overset{d}{=} Y$ to denote the equality in distribution, i.e., $X$ has the same PDF as $Y$.
We use $|\cdot|$ to denote the absolute value, $j \triangleq \sqrt{-1}$ to denote the imaginary number, and $\Re\{ z \}$ to denote the real part of the complex number $z$.
Finally, $\erfc\left( \cdot \right)$ is used to denote the complementary error function given by $\erfc(x) = \frac{2}{\sqrt{\pi}} \int_{x}^{\infty}{e^{-u^2} du}$.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{MolCom.pdf}
\end{center}
\vspace{-0.3cm}
\caption{\label{fig:chanModels} Summary of the System models corresponding to each modulation scheme. System A: a synchronized MT, system B: an asynchronous MT with indistinguishable information particles, and system C: an asynchronous MT with distinguishable information particles }
\vspace{-0.3cm}
\end{figure}
\section{System Models}
\label{sec:model}
In this section we present three different timing-based modulation techniques, which results in three different MT system models. Note that there are no unified channel models for all possible modulation schemes in molecular communication, and typically each modulation yields a different channel model. To develop our model, we make the following assumptions about the system:
\begin{enumerate}[label = {\bf A{\arabic*}})]
\item \label{assmp:perfectTxRx}
The transmitter perfectly controls the release time of each information particle, and the receiver perfectly measures the arrival times of the information particles.
Furthermore, the transmitter and the receiver are perfectly synchronized in time, when synchronization is required by the modulation scheme.
\item \label{assmp:Arrival}
Any information particle that arrives at the receiver is absorbed and hence is removed from the propagation medium.
\item \label{assmp:indep}
All information particles propagate independently of each other, and their trajectories are random according to an independent and identically distributed (i.i.d.) random process. This is a fair assumption for many different propagation schemes in molecular communication such as diffusion in dilute solutions, i.e., when the number of particles released is much smaller than the number of molecules of the solutions.
\item \label{assmp:noISI}
There is no inter-symbol interference (ISI) between consecutive channel uses.\footnote{In \cite{mur17NanoComNet} we study communication over DBMT channels in the presence of ISI.} In practice this assumption can be satisfied if the time between consecutive channel uses is large enough or if chemical reactions are used to dissipate the particles \cite{noe14}.
\item
\revised{Initially, there are no information particles in the environment and the particles that arrive at the receiver are the ones released by the transmitter.}
\end{enumerate}
\noindent \revised{Note that these assumptions are typical in the study of molecular communication systems \cite{ein2011,ein11ITW,ata13,nak12, cha12,eck12,pie13, li14, rose14}.
Moreover, as our system uses molecular timing channels through microfluidic links on bio-chips, the assumptions can be satisfied as follows. In bio-chips, it is possible to release and detect information particles with great precision \cite{lee09,hor15} (satisfies A1). Many detectors will remove the information particles as part of the detection process, e.g., by binding to a receptor, diffusion into a sensing layer, or chemical reactions (satisfies A2). The number of information particles released as part of the modulation techniques considered in this work are one or two (satisfies A3). The applications considered in this work are biological studies, where only a very few bits of information need to be transmitted followed by large periods of silence (satisfies A4). Finally, the microfluidic links can be carefully monitored and designed to exclude interfering molecules (satisfies A5).}
The first MT system that we consider is the one proposed in \cite{eck09,sri12}, where the information is encoded in the release timing of a single information particle. Let $T_{x} \in \mathcal{T}\subseteq\mathcal{R}^+$ be the release time of the information particle at the transmitter. In this scheme, the information is modulated onto the release time itself. The released particle is then transported from the transmitter to the receiver, where the transport process is random. Let $T_{y}$ be the time of arrival at the receiver. Then we have
\begin{align}
\label{eq:timingChA}
T_{y} = T_{x} + T_n,
\end{align}
where $T_n$ is the random propagation delay of the information particle, which is the {\em noise term} in this channel. One of the main challenges of this modulation scheme is the need for synchronization between the transmitter and the receiver. In this work, whenever system A is used we assume that the transmitter and the receiver are perfectly synchronized.
To overcome this synchronization challenge, we propose two new modulation schemes in which information is modulated on the time duration {\em between} two consecutive releases of information particles. The receiver decodes the information from the time between the arrivals of two molecules. Note that in this case synchronization between the transmitter and the receiver is not required.
Two cases are possible: either the two released information particles are {\em indistinguishable} at the receiver, or the two released information particles are {\em distinguishable} at the receiver.
We first consider the case where both information particles are {\em indistinguishable}. Without loss of generality, let $T_{x_1}$ be the release timing of the first information particle and let $T_{x_2}$ be the release timing for the second information particle, with $T_{x_2}>T_{x_1}$. Thus, the information is encoded in $L_x = T_{x_2}-T_{x_1}$. Using \eqref{eq:timingChA}, the system model for this modulation scheme is given by:
\begin{align}
\label{eq:timingChB}
|T_{y_2} - T_{y_1}|&= |T_{x_2}-T_{x_1} + T_{n_2} - T_{n_1}|, \nonumber \\
L_y &= | L_x+L_n|,
\end{align}
where $L_n = T_{n_2} - T_{n_1}$ is the random noise term in this system, and $T_{n_2}$ and $T_{n_1}$ are the random propagation delays for the first and the second particles as in (\ref{eq:timingChA}). Note that the absolute value in the system formulation is due to the fact that both information particles are indistinguishable, and therefore the receiver can observe only the absolute difference of arrival times.
The last modulation scheme uses the time between releases of two {\em distinguishable} information particles (i.e., two different particle types) to encode information. Let $T_{x}^a$ be the release timing of the type-$a$ information particle and let $T_{x}^b$ be the release timing of the type-$b$ information particle. We assume that the information is encoded in $Z_x = T_{x}^b-T_{x}^a$. Unlike (\ref{eq:timingChB}) where $L_x$ is always positive, $Z_x$ can be positive or negative depending on the order that the type-$a$ and type-$b$ information particles are released. Using (\ref{eq:timingChA}), the system model for this scheme is given by:
\begin{align}
\label{eq:timingChC}
T_{y}^b - T_{y}^a &= T_{x}^b-T_{x}^a + T_{n}^b - T_{n}^a, \nonumber \\
Z_y &= Z_x+ Z_n,
\end{align}
where $Z_n = T_{n}^b - T_{n}^a$ is the random additive noise term in this system, and $T_{n}^b$ and $T_{n}^a$ are the random propagation delays for the type-$a$ and type-$b$ particles as in (\ref{eq:timingChA}). Again, no synchronization is required between the transmitter and receiver. Fig.~\ref{fig:chanModels} summarizes all three modulation techniques.
Note that the proposed modulation schemes and their corresponding system models could be applied to any type of propagation model through the medium as long as Assumptions \ref{assmp:perfectTxRx}-\ref{assmp:noISI} are not violated. In the next section, we derive the distribution of the noise terms for the proposed MT systems, when diffusion propagation is used for particle transport.
\section{Noise Models for DBMT Systems}
\label{sec:noiseModel}
In the rest of this work we focus on DBMT systems where diffusion is used for particle transport. In particular, in this section we derive the PDF of the noise terms $T_n$, $L_n$, and $Z_n$ for DBMT systems in \eqref{eq:timingChA}-\eqref{eq:timingChC}, and discuss some of the properties of these RVs.
\subsection{System A}
To specify the random additive noise term $T_n$ in system A, we define a L\'evy-distributed RV as follows.
\begin{definition}[L\'evy Distribution] \label{def:levyRV}
Let the RV $X$ be L\'evy-distributed with location parameter $\mu$ and scale parameter $c$ \cite{nol15}. Then its PDF is given by:
\begin{align}
\label{eqn:LevyPDF_0}
f_X(x;\mu,c)=
\begin{cases}
\sqrt{\frac{c}{2 \pi (x-\mu)^3}}\exp \left( -\frac{c}{2(x-\mu)} \right), & x>\mu \\
0, & x\leq \mu
\end{cases},
\end{align}
its characteristic function is given by:
\begin{align}
\varphi (\revised{\omega};\mu,c) = \exp \left( j \mu \revised{\omega} -\sqrt{-2 j c \revised{\omega} } \right),
\end{align}
and its CDF is given by:
\begin{align}
\label{eqn:LevyCDF}
F_X(x;\mu,c) = \begin{cases} \erfc\left(\sqrt{\frac{c}{2(x-\mu)}}\right), & x>\mu \\ 0, & x\leq\mu \end{cases}.
\end{align}
\end{definition}
Throughout the paper, we use the notation $X \sim {\mathscr{L}}(\mu,c)$ to indicate a L\'evy RV with parameters $\mu$ and $c$.
Assumption \ref{assmp:Arrival} implies that the distribution of $T_n$ is the distribution of the first hitting time (first arrival at the receiver) of a particle transported via diffusion without flow.
In previous works, it was shown that the first hitting time for a diffusion channel with constant drift (i.e., flow) in 1-dimensional space follows the inverse Gaussian distribution \cite{sri12}. In this work, we consider the pure diffusion channel with no flow. Let $d$ denote the distance between the transmitter and the receiver, and $D$ denote the diffusion coefficient of the information particles in the propagation medium. Following along the lines of the derivations in \cite[Sec. II]{sri12}, and using \cite[Sec. 2.6.A]{karatzas-shreve}, it can be shown that for 1-dimensional pure diffusion, the propagation time of each of the information particles follows a L\'evy distribution, and therefore the noise in system A is distributed as $T_n \sim {\mathscr{L}}(0,c_{\text{A}})$ with $c_{\text{A}} = \frac{d^2}{2D}$. Similarly, the conditional PDF $P(T_{y}|T_{x}) \sim {\mathscr{L}}(T_{x},c_{\text{A}})$.
The PDF and CDF of the standardized (i.e., with $\mu=0$ and $c_{\text{A}} =1$) L\'evy noise are depicted in Figs. \ref{fig:stablePDF} and \ref{fig:stableCDF}, respectively.
\begin{rem}
In \cite{yilmaz20143dChannelCF} it is shown that for an infinite, three-dimensional homogeneous medium without flow, and a spherically absorbing receiver, the first arrival time follows a scaled L\'evy distribution. Therefore, the results presented in this paper can be extended to 3-D space by simply introducing a scalar multiple in the noise distribution.
\end{rem}
\subsection{System B}
To find the noise distribution of the system in (\ref{eq:timingChB}), we first discuss the class of probability distributions known as {\em stable distributions} \cite{zol86-book,nol15}. Note that the L\'evy distribution belongs to this class.
\begin{definition}[Stable Distributions]
\revised{A} RV $X$ has a stable distribution if for two independent copies $X_1$ and $X_2$, and positive constants $a_1, a_2, a_3 \in \mathcal{R}^{+}$ and $a_4 \in \mathcal{R}$, the following holds:
\begin{align*}
a_1X_1 + a_2X_2 \overset{d}{=} a_3X+a_4.
\end{align*}
\end{definition}
Stable distributions can also be defined via their characteristic function.
\begin{definition}[Characteristic Function of a Stable Distribution]
Let $-\infty < \mu < \infty, c\ge 0, 0 < \alpha \le 2$, and $-1 \le \beta \le 1$. Further define:
\begin{align*}
\Phi(\revised{\omega},\alpha) \triangleq \begin{cases}
\tan \left( \frac{\pi \alpha}{2}\right), & \alpha \ne 1 \\
-\frac{2}{\pi} \log (|\revised{\omega}|), & \alpha = 1
\end{cases}.
\end{align*}
\noindent Then, the characteristic function of a stable RV $X$, with location parameter $\mu$, scale parameter $c$, characteristic exponent $\alpha$, and skewness parameter $\beta$, is given by:
\begin{align}
\varphi (\revised{\omega};\mu,c,\alpha,\beta) = \exp \left[ j \mu \revised{\omega} -| c \revised{\omega} |^\alpha (1-j\beta \sgn(\revised{\omega}) \Phi(\revised{\omega}, \alpha) ) \right].
\label{eq:stableCharFunc}
\end{align}
\end{definition}
In the following, we use the notation $\mathscr{S}(\mu, c, \alpha, \beta)$ to represent a stable distribution with the parameters $\mu, c, \alpha$, and $\beta$.
Only the PDFs of three classes of stable distributions are known to have closed-form expressions in terms of elementary functions: the Gaussian distribution with $\alpha=2$ (the value of $\beta$ does not matter in this case and can be assumed to be zero), the L\'evy distribution with $\alpha = \tfrac{1}{2}$ and $\beta=1$, and the Cauchy distribution with $\alpha=1$ and $\beta=0$. Generally, the parameters $\alpha$ and $\beta$ define a subclass within the stable distribution family. Next, we introduce some important properties of stable distributions \cite{nol15}.
\begin{proper}
\label{prop:LinearStable}
Let $X \sim {\mathscr{S}}(\mu,c,\alpha,\beta)$, and define $Y = \frac{X-\mu}{c}$. Then $f(x)dx =f(y)dy$, and $Y$ is called the standard form of $X$.
\end{proper}
\begin{proper}
\label{prop:standStable}
Let $\tilde{X} \sim {\mathscr{S}}(0,1,\alpha,\beta)$ be the standard form of a stable RV with parameters $\alpha$ and $\beta$. Then the PDF and the CDF of any RV $X \sim {\mathscr{S}}(\mu,c,\alpha,\beta)$ can be calculated as
\begin{align}
f_X(x) &= \frac{f_{\tilde{X}}\big( \tfrac{x-\mu}{c} \big)}{c}, \label{eq:standPDFConv}\\
F_X(x) &= F_{\tilde{X}}( \tfrac{x-\mu}{c} ). \label{eq:standCDFConv}
\end{align}
\end{proper}
Using this property, the standard PDF and CDF of a stable RV can be used to calculate probabilities involving non-standard stable RVs just like the way the standard Gaussian PDF and CDF are used to calculate probabilities involving non-standard Gaussian RVs.
\begin{proper}
\label{prop:sym}
The PDFs of stable RVs with $\beta=0$ are symmetric around $\mu$.
\end{proper}
\begin{proper}
\label{prop:tailProbability}
If $X$ is a standardized (i.e., with $\mu=0$ and $c =1$) stable RV with parameters $0<\alpha<2$ and $\beta$, then as $x\rightarrow\infty$,
\begin{align}
P(X>x;\alpha,\beta) \approx \frac{1+\beta}{\pi x^\alpha} \Gamma(\alpha)\sin\bigg(\frac{\alpha \pi}{2}\bigg).
\end{align}
\end{proper}
\begin{rem}
Using this property it can be shown that for a stable distributed RV $X$ with parameter $\alpha$, the moments of order greater than $\alpha$ (i.e., $\mathbb{E}[|X|^\alpha]$) are infinite. Therefore, all stable distributions with $\alpha<2$ have infinite variances, and all stable distributions with $\alpha <1$ have infinite mean values.
\end{rem}
With these definitions we now model the noise term $L_n$ in (\ref{eq:timingChB}).
\begin{theorem}
Let $c_{\text{B}} = \frac{2d^2}{D} $, where $d$ is the distance between the transmitter and the receiver and $D$ is the diffusion coefficient of the information particles. Then, the characteristic function of the noise term $L_n$ is given by:
\begin{align*}
\varphi \left( \revised{\omega};c_{\text{B}} \right) = \exp \left[ -\sqrt{c_{\text{B}}| \revised{\omega} |} \right],
\end{align*}
which implies that $L_n \sim {\mathscr{S}}(0,c_{\text{B}},\frac{1}{2},0)$.
\end{theorem}
\begin{IEEEproof}
We know that $L_n = T_{n_2}+( - T_{n_1})$ with $T_{n_2}, T_{n_1} \sim {\mathscr{S}}(0,c_{\text{A}},\frac{1}{2},1)$, where $c_{\text{A}} = \frac{d^2}{2D}$. Since $T_{n_1}$ and $T_{n_2}$ are independent, the characteristic function for $L_n$ is given by
\begin{align}
\varphi_{L_n} (\revised{\omega}) &= \varphi_{T_{n_2}} (\revised{\omega}) \varphi_{T_{n_1}} (-\revised{\omega}) \\
&= \exp \left[ -\sqrt{| c_{\text{A}} \revised{\omega} |} (1- j \sgn(\revised{\omega})) \right] \times \nonumber \\
& ~~~~~~~~~~~~~~~~~~~~\exp \left[ -\sqrt{| c_{\text{A}} \revised{\omega} |} (1+ j \sgn(\revised{\omega})) \right] \\
&= \exp \left[ -\sqrt{|4c_{\text{A}} \revised{\omega} |} \right].
\end{align}
\noindent Thus, using the expression in \eqref{eq:stableCharFunc} we conclude that $L_n \sim {\mathscr{S}}(0,c_{\text{B}},\frac{1}{2},0)$.
\end{IEEEproof}
\begin{rem}
If the same type of particle is used in system A and system B, and the distance between the transmitter and the receiver is the same, then the scale parameter $c$ in the noise term for system B is four times greater than the scale parameter for system A, i.e., $c_{\text{B}} = 4c_{\text{A}}$.
\end{rem}
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{StablePDFs.pdf}
\end{center}
\caption{\label{fig:stablePDF} The probability density function of different standardized noise terms.}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{StableCDFs.pdf}
\end{center}
\caption{\label{fig:stableCDF} The cumulative distribution function of different standardized noise terms.}
\end{figure}
To find an expression for the PDF of the noise term $L_n$ in (\ref{eq:timingChB}), we first define the following functions.
Let $K(a,b)$ and $L(a,b), a \in \mathcal{R}, b \in \mathcal{R}^{+}$, be the complex and imaginary Voigt functions \cite{arm67}, given by :
\begin{align}
\label{eq:ReVoigt}
K(a,b) \triangleq \frac{1}{\sqrt{\pi}} \int_0^\infty \exp(-t^2/4)\exp(-bt)\cos(at) dt,
\end{align}
and
\begin{align}
\label{eq:ImVoigt}
L(a,b) \triangleq \frac{1}{\sqrt{\pi}} \int_0^\infty \exp(-t^2/4)\exp(-bt)\sin(at) dt.
\end{align}
The Voigt functions, which are widely used in the fields of physics, astronomy, and chemistry, can be computed efficiently and quickly numerically \cite{zag11, abr15}. Moreover, for some special cases (e.g., $b\gg0$), analytical approximations of these functions exist in terms of elementary functions \cite{sch16}. We further define:
\begin{align}
\label{eq:theGfunc}
G(u) &\triangleq \frac{1}{\sqrt{8\pi |u|^3}} \left[ K \left(-\tfrac{1}{\sqrt{8|u|}}, \tfrac{1}{\sqrt{8|u|}} \right) \right. \nonumber \\
&\qquad \qquad \qquad \left. + L\left(-\tfrac{1}{\sqrt{8|u|}}, \tfrac{1}{\sqrt{8|u|}} \right) \right].
\end{align}
The PDF of $L_n$ is stated in the following theorem:
\begin{theorem}
\label{thm:PDFNoiseChanB}
Let $L_n \sim {\mathscr{S}}(0,c_{\text{B}},\frac{1}{2},0)$. Then the PDF of $L_n$ is given by:
\begin{align}
\label{eq:noisePDFchanB}
f_{L_n}(\ell_n) = \begin{cases}
\frac{1}{c_{\text{B}}} G\left( \tfrac{\ell_n}{c_{\text{B}}} \right), & \ell_n \neq 0 \\
\frac{2}{c_{\text{B}}\pi}, & \ell_n = 0
\end{cases}.
\end{align}
\end{theorem}
\begin{IEEEproof}
The proof is provided in Appendix \ref{annex:ProofOfPDFchanB}.
\end{IEEEproof}
In this work, we do not provide an expression for the CDF of the noise terms $F_{L_n}(\ell_n)$, and it would be difficult to integrate \eqref{thm:PDFNoiseChanB} to obtain the CDF. However, the CDF can be calculated numerically using the methods described in \cite[Sec. 3]{nol97}. Moreover, tables of the standardized CDF could be used to calculate probabilities involving the noise term. Figs. \ref{fig:stablePDF} and \ref{fig:stableCDF} depict the PDF and CDF for the standardized noise term $L_n$ with $c_{\text{B}} =1$.
\subsection{System C}
We first note that the noise $Z_n$ given in (\ref{eq:timingChC}) is fundamentally different from the noise $L_n$ in system B since the two different types of information particles may have different diffusion coefficients. Let $D_a$ be the diffusion coefficient of information particle $a$, and $D_b$ be the diffusion coefficient for the information particle $b$. We define $c_{\text{C}} \triangleq \frac{d^2(\sqrt{D_a}+\sqrt{D_b})^2}{2D_aD_b}$, $\beta_{\text{C}} \triangleq \frac{\sqrt{D_a}-\sqrt{D_b}}{\sqrt{D_a}+\sqrt{D_b}}$. Furthemore, without loss of generality, we assume that particle $a$ is released before particle $b$. We now model the noise term $Z_n$ in (\ref{eq:timingChC}).
\begin{theorem}
The characteristic function for the noise term $Z_n$ is given by:
\small
\begin{align*}
&\varphi \left( \revised{\omega};c_{\text{C}}, \beta_{\text{C}}\right) = \exp \left[ -\sqrt{c_{\text{C}}| \revised{\omega} |} \left( 1- j\beta_{\text{C}}\sgn(\revised{\omega}) \right) \right],
\end{align*}
which implies that $Z_n \sim {\mathscr{S}} \left( 0,c_{\text{C}},\frac{1}{2},\beta_{\text{C}} \right)$.
\end{theorem}
\begin{IEEEproof}
First, note that $Z_n = T_{n_b}+( - T_{n_a})$ with $T_{n_a}, T_{n_b} \sim {\mathscr{S}}(0,c_i,\frac{1}{2},1)$, where $c_i = \frac{d^2}{2D_i}$ for $i \in\{ a,b \}$. Since $T_{n_a}$ and $T_{n_b}$ are independent, the characteristic function of $Z_n$ is given by:
\begin{align}
& \varphi_{Z_n} (\revised{\omega}) \nonumber \\
& \quad = \varphi_{T_{n_b}} (\revised{\omega}) \varphi_{T_{n_a}} (-\revised{\omega}) \\
& \quad = \exp \left[ -\sqrt{c_b}\sqrt{| \revised{\omega} |} (1- j \sgn(\revised{\omega})) \right] \times \nonumber \\
& \mspace{100mu} \exp \left[ -\sqrt{c_a}\sqrt{| \revised{\omega} |} (1+ j \sgn(\revised{\omega})) \right] \\
& \quad = \exp \left[ - \mspace{-3mu} \sqrt{|\revised{\omega}|}(\sqrt{c_b} \mspace{-3mu} + \mspace{-3mu} \sqrt{c_a} \mspace{-3mu} - \mspace{-3mu} j\sgn(\revised{\omega})(\sqrt{c_a} \mspace{-3mu} - \mspace{-3mu} \sqrt{c_b})) \right] \\
& \quad = \exp \left[ -(\sqrt{c_b}+\sqrt{c_a})\sqrt{|\revised{\omega}|} \times \right. \nonumber \\
& ~~~~~~~~~~~~~~~\left. \left( 1-j\sgn(\revised{\omega})\frac{\sqrt{c_a}-\sqrt{c_b}}{\sqrt{c_b}+\sqrt{c_a}}\right) \right] \\
& \quad = \exp \left[ -\frac{d(\sqrt{D_a}+\sqrt{D_b})}{\sqrt{2D_aD_b}}\sqrt{| \revised{\omega} |} \times \right. \nonumber \\
&~~~~~~~~~~~~~~~~~~~~\left. \left( 1- j \frac{\sqrt{D_a}-\sqrt{D_b}}{\sqrt{D_a}+\sqrt{D_b}}\sgn(\revised{\omega}) \right)\right].
\end{align}
\noindent Thus, using the expression in \eqref{eq:stableCharFunc} we conclude that $Z_n \sim {\mathscr{S}} \left( 0,c_{\text{C}},\frac{1}{2},\beta_{\text{C}} \right)$.
\end{IEEEproof}
\begin{rem}
When the diffusion coefficients of the two particles are approximately the same, i.e., $D_a \approx D_b$, the distribution of $Z_n$ approaches the distribution $L_n$ (i.e. $\beta_{\text{C}} \approx 0$). On the other hand, when $D_a \ll D_b$ or $D_a \gg D_b$, then $\beta_{\text{C}} \approx \pm1$ which implies that $Z_n$ is L\'evy distributed.
Therefore, when one information particle has a much higher diffusion coefficient than the other, system C can be reduced to system A with the added benefit that no synchronization is required between the transmitter and the receiver.
However, this comes at a cost of: 1) Using two particles instead of one; and 2) The resulting system A has a scaling parameter that corresponds to the smaller diffusion coefficient.
\end{rem}
To derive the PDF of $Z_n$ we first define the following two functions:
\begin{align}
G_+(u,\beta) &\triangleq \frac{1}{\sqrt{8\pi |u|^3}} \left[ (1+\beta)K \left(-\tfrac{1+\beta}{\sqrt{8|u|}}, \tfrac{1-\beta}{\sqrt{8|u|}} \right) \right. \nonumber \\
&\qquad \left. + (1-\beta)L\left(-\tfrac{1+\beta}{\sqrt{8|u|}}, \tfrac{1-\beta}{\sqrt{8|u|}} \right) \right], \label{eq:theGpfunc}\\
G_-(u,\beta) &\triangleq \frac{1}{\sqrt{8\pi |u|^3}} \left[ (1-\beta)K \left(\tfrac{1-\beta}{\sqrt{8|u|}},\tfrac{1+\beta}{\sqrt{8|u|}} \right) \right. \nonumber \\
&\qquad \left. + (1+\beta)L\left(\tfrac{1-\beta}{\sqrt{8|u|}}, \tfrac{1+\beta}{\sqrt{8|u|}} \right) \right], \label{eq:theGnfunc}
\end{align}
where $K(a,b)$ and $L(a,b)$ are the real and imaginary Voigt functions given in \eqref{eq:ReVoigt} and \eqref{eq:ImVoigt}, respectively. The PDF of $Z_n$ is now stated in the following theorem:
\begin{theorem}
\label{thm:PDFNoiseChanC}
Let $Z_n \sim {\mathscr{S}}(0,c_{\text{C}},\tfrac{1}{2},\beta_{\text{C}})$. Then the PDF of $Z_n$ is given by:
\begin{align}
\label{eq:noisePDFchanC}
f_{Z_n}(z_n) = \begin{cases}
\frac{1}{c_{\text{C}}} G_+\left( \tfrac{z_n}{c_{\text{C}}}, \beta_{\text{C}} \right), & z_n > 0 \\
\frac{2(1-\beta^2)}{c_{\text{C}}\pi(1+\beta^2)^2}, & z_n = 0 \\
\frac{1}{c_{\text{C}}} G_-\left( \tfrac{z_n}{c_{\text{C}}}, \beta_{\text{C}} \right), & z_n < 0
\end{cases}.
\end{align}
\end{theorem}
\begin{IEEEproof}
The proof is provided in Appendix \ref{annex:ProofOfPDFchanC}.
\end{IEEEproof}
Again, we do not provide an expression for the CDF of the noise terms $F_{Z_n}(z_n)$, instead we note that it can be numerically calculated using the methods of \cite[Sec. 3]{nol97}.
Figs. \ref{fig:stablePDF} and \ref{fig:stableCDF} depict the PDF and CDF for the standardized noise term $Z_n$ with $c_{\text{C}} =1$ and four different values for $\beta_{\text{C}}$. Note that for $\beta_{\text{C}} = 0$ the distribution and the density functions of the noise $Z_n$ in System C are the same as the noise $L_n$ in system B. This is due to the diffusion coefficient of type-$a$ and type-$b$ particles being equal, which means that both particle types have the same random propagation delay characteristics.
\revised{
\begin{rem}
Moving from a 1-D space to a 3-D space, there is a probability that the particle will never arrive at the receiver (see \cite{yilmaz20143dChannelCF}). For systems B and C, both particles must arrive for error free communication. Conditioned on the event that the two particles arrive, the noise distribution will be the same as the 1-D case. Therefore, in 3-D, the noise distribution will be scaled by the probability of the event that both particles arrive.
\end{rem}
}
\section{Optimal Detection in Binary DBMT Systems}
\label{sec:PBE}
In this section, we consider equiprobable binary transmission over the three different DBMT systems. Using the noise models developed in the previous section, we characterize the optimal detection rule for each modulation.
\subsection{System A}
For system A, \revised{we} assume that the transmission symbols are $T_x \in \{0,\Delta\}$, where $\Delta>0$.
Using Property \ref{prop:standStable}, we write the distribution of the output probability, conditioned on the input, in terms of the standard L\'evy distribution $\tilde{T}_n \sim {\mathscr{L}}(0,1)$ as follows:
\begin{align}
f_{T_y|T_x}(t_y|T_x=0) &= f_{T_n} (t_y) = \frac{f_{\tilde{T}_n}(t_y/c_{\text{A}})}{c_{\text{A}}} , \\
f_{T_y|T_x}(t_y|T_x=\Delta) &= f_{T_n} (t_y-\Delta) = \frac{f_{\tilde{T}_n}\big((t_y-\Delta)/c_{\text{A}}\big)}{c_{\text{A}}}.
\end{align}
\begin{figure*}
\begin{center}
\includegraphics[width=0.7\textwidth]{OptimalThreshold.pdf}
\end{center}
\caption{\label{fig:optiThreshold} The ML optimal decision threshold for the three systems.}
\end{figure*}
As the two transmitted symbols are equiprobable, the detector that minimizes the probability of error is the maximum likelihood (ML) detector.
In this work we assume both the 0-bit and the 1-bit are equiprobable and apply the ML detector. In this case, the likelihood ratio is given by:
\begin{align}
\Lambda_A (t_y) = \frac{f_{T_y|T_x}(t_y|T_x=0)}{f_{T_y|T_x}(t_y|T_x=\Delta)}, \label{eq:LR_A}
\end{align}
\noindent and optimal detection can be done by a comparison of the log likelihood ratio (LLR) to zero, i.e.,
\begin{align}
\log(\Lambda_A (t_y) ) \begin{matrix} T_x = 0 \\ \gtrless \\ T_x = \Delta \end{matrix} 0.
\end{align}
Note that the proof of the existence of the optimal threshold value is straightforward using the fact that stable distributions are unimodal \cite[Theorem 2.7.6]{zol86-book}, and that for the noise term $T_n$ the mode is at $c/3$. Therefore, there exists a threshold $\Delta<\mathsf{th}_{\text{A}}\leq c/3+\Delta$, such that $\Lambda_A (t)>1$ for $t<\mathsf{th}_{\text{A}}$ and $\Lambda_A (t)\leq1$ for $t\geq\mathsf{th}_{\text{A}}$ \cite{mur16Globecom,mur17}.
The top plot in Figure \ref{fig:optiThreshold} shows the optimal threshold for the case when $\Delta=1$, the distance is $d=1$ and the diffusion coefficient is $D=0.5$.
The probability of error for system A is now given by:
\begin{align}
P_e^A &= P(T_x=0) {\mathrm{Pr}}(t_y>\mathsf{th}_{\text{A}} | T_x = 0) \nonumber \\
&\qquad +P(T_x=\Delta) {\mathrm{Pr}}(t_y\leq\mathsf{th}_{\text{A}} | T_x= \Delta), \\
&= 0.5 {\mathrm{Pr}}(t_n>\mathsf{th}_{\text{A}})+0.5 {\mathrm{Pr}}(t_n\leq\mathsf{th}_{\text{A}}-\Delta) \\
&= 0.5[1-F_{\tilde{T}_n}(\tfrac{\mathsf{th}_{\text{A}}}{c_{\text{A}}})+F_{\tilde{T}_n}(\tfrac{\mathsf{th}_{\text{A}}-\Delta}{c_{\text{A}}})],
\end{align}
where $F_{\tilde{T}_n}(t)$ is the CDF of a standard L\'evy RV.
\subsection{System B}
For system B we assume that the input is $L_x\in\{0,\Delta\}$, where $L_x=0$ represents two particles released simultaneously, while $L_x=\Delta$ represents two particles released $\Delta$ seconds apart. Let $\tilde{L}_n \sim {\mathscr{S}}(0,1,\tfrac{1}{2},0)$ be the standard form of the noise term in \eqref{eq:timingChB}.
The PDF of the output $L_y$, given the input $L_x$, is provided in the following proposition:
\begin{proposition}
The system output $L_y$, given the system input $L_x$, has the PDF:
\begin{align}
f_{L_y|L_x}(\ell_y|L_x=0) &=
\begin{cases}
\frac{2f_{\tilde{L}_n}\big(\tfrac{\ell_y}{c_{\text{B}}}\big)}{c_{\text{B}}} & \ell_y>0 \\
\frac{2}{(c_{\text{B}}\pi)} & \ell_y=0 \\
0 & \ell_y<0
\end{cases}, \label{eq:LyGLx0}\\
f_{L_y|L_x}(\ell_y|L_x=\Delta) &=
\begin{cases}
\frac{f_{\tilde{L}_n}\big( \tfrac{\ell_y-\Delta}{c_{\text{B}}}\big)+f_{\tilde{L}_n}\big( \tfrac{-\ell_y-\Delta}{c_{\text{B}}}\big)}{c_{\text{B}}} & \ell_y>0 \\
\frac{f_{\tilde{L}_n}\big( \tfrac{\Delta}{c_{\text{B}}}\big)}{c_{\text{B}}} & \ell_y=0 \\
0 & \ell_y<0
\end{cases}. \label{eq:LyGLxD}
\end{align}
\end{proposition}
\begin{IEEEproof}
It is clear from the system definition that when $L_y<0$ the PDF is 0 (i.e. the time between two arrival times is not negative). When $L_y=0$, we have $f_{L_y|L_x}(0|L_x=0)=f_{L_n}(0)$, and $f_{L_y|L_x}(0|L_x=\Delta)=f_{L_n}(\Delta)$. To derive the PDF value for $L_y>0$, we use the fact that the CDF of $L_y$ given $L_x=x\geq0$ can be obtained from the CDF of $\tilde{L}_n$ as
%
\begin{align*}
F_{L_y|L_x}(\ell_y|L_x=x) &= {\mathrm{Pr}}(L_y \leq \ell_y|L_x=x) \\
&= {\mathrm{Pr}}(|x+L_n| \leq \ell_y) \\
&= {\mathrm{Pr}}(-\ell_y \leq x+L_n \leq \ell_y) \\
&= {\mathrm{Pr}}(\tfrac{-\ell_y-x}{c_{\text{B}}} \leq \tilde{L}_n \leq \tfrac{\ell_y-x}{c_{\text{B}}} ) \\
&= F_{\tilde{L}_n}\big(\tfrac{\ell_y-x}{c_{\text{B}}}\big)-F_{\tilde{L}_n}\big(\tfrac{-\ell_y-x}{c_{\text{B}}}\big).
\end{align*}
By differentiating with respect to $\ell_y$, and setting $x=0$ and $x=\Delta$, we obtain \eqref{eq:LyGLx0} and \eqref{eq:LyGLxD}, respectively.
\end{IEEEproof}
Similarly to \eqref{eq:LR_A}, the likelihood ratio for the ML detector for system B is given by:
\begin{align}
\Lambda_B (\ell_y) = \frac{f_{L_y|L_x}(\ell_y|L_x=0)}{f_{L_y|L_x}(\ell_y|L_x=\Delta)}.
\end{align}
\noindent The following theorem states that, just like in the case of system A, the ML detector can be implemented by comparing $\log (\Lambda_B (\ell_y))$ to zero:
\begin{theorem}
\label{thrm:ChanBthrExistance}
There exists a fixed threshold $\mathsf{th}_{\text{B}} >\tfrac{\Delta}{2}$ such that the ML detector in the case of system B is given by:
\begin{align}
\log(\Lambda_B (\ell_y) ) \begin{matrix} T_x = 0 \\ \gtrless \\ T_x = \Delta \end{matrix} 0.
\end{align}
\end{theorem}
\begin{IEEEproof}
The proof is provided in Appendix \ref{annex:ProofOfExistanceOfThreshold}.
\end{IEEEproof}
The middle plot in Fig. \ref{fig:optiThreshold} depicts the optimal threshold for the case when $\Delta=1$, the distance is $d=1$ and the diffusion coefficient is $D=0.5$. Since the closed-form expression for the CDF of the noise term is unknown, this threshold is calculated numerically.
Finally, the probability of error for binary communication over system B is given by:
\begin{align}
P_e^B &= P(L_x=0) {\mathrm{Pr}}(L_y>\mathsf{th}_{\text{B}} | L_x = 0) \nonumber \\
&\qquad +P(L_x=\Delta) {\mathrm{Pr}}(L_y\leq\mathsf{th}_{\text{B}} | L_x= \Delta), \nonumber \\
&= 0.5({\mathrm{Pr}}(L_n>\mathsf{th}_{\text{B}})+{\mathrm{Pr}}(L_n\leq -\mathsf{th}_{\text{B}}) )\nonumber \\
&\qquad +0.5{\mathrm{Pr}}(-\mathsf{th}_{\text{B}}-\Delta \leq L_n\leq \mathsf{th}_{\text{B}}-\Delta), \nonumber \\
&= 0.5({\mathrm{Pr}}(\tilde{L}_n>\tfrac{\mathsf{th}_{\text{B}}}{c_{\text{B}}})+{\mathrm{Pr}}(\tilde{L}_n\leq -\tfrac{\mathsf{th}_{\text{B}}}{c_{\text{B}}}) )\nonumber \\
&\qquad +0.5{\mathrm{Pr}}(\tfrac{-\mathsf{th}_{\text{B}}-\Delta}{c_{\text{B}}} \leq \tilde{L}_n\leq \tfrac{\mathsf{th}_{\text{B}}-\Delta}{c_{\text{B}}}), \nonumber \\
&= F_{\tilde{L}_n}\big(\tfrac{\mathsf{th}_{\text{B}}}{c_{\text{B}}}\big) \mspace{-2mu} + \mspace{-2mu} 0.5\big(F_{\tilde{L}_n}\big(\tfrac{\mathsf{th}_{\text{B}}-\Delta}{c_{\text{B}}}\big) \mspace{-2mu} - \mspace{-2mu} F_{\tilde{L}_n}\big(\tfrac{\mathsf{th}_{\text{B}}+\Delta}{c_{\text{B}}}\big)\big).
\end{align}
Thus, similarly to the case of system A, the probability of error can be calculated using the standard form of the noise term.
\subsection{System C}
Recall that for system C the two particles are {\em distinguishable}, and $Z_x=T_{x_b}-T_{x_a}$ is the time interval between the releases of particles $b$ and $a$. Here, we assume information is encoded in the {\em order} of release. The input $Z_x\in\{-\Delta,\Delta\}$ is now given by:
\begin{align}
Z_x = \begin{cases} \Delta, & T_{x_a} = 0, T_{x_b} = \Delta \\
-\Delta, & T_{x_b} = 0, T_{x_a} = \Delta \end{cases}.
\end{align}
\noindent Note that similarly to systems A and B, the information is encoded over the time period $\Delta$.
Let $\tilde{Z}_n \sim (0,1,\tfrac{1}{2},\beta_{\text{C}})$ be the standard form of the noise term in \eqref{eq:timingChC}. Then the PDF of the output given the input is given by
\begin{align}
f_{Z_y|Z_x}(z_y|Z_x=-\Delta) &= \frac{f_{\tilde{Z}_n}\big(\tfrac{z_y+\Delta}{c_{\text{C}}}\big)}{c_{\text{C}}}\\
f_{Z_y|Z_x}(z_y|Z_x=\Delta) &= \frac{f_{\tilde{Z}_n}\big(\tfrac{z_y-\Delta}{c_{\text{C}}}\big)}{c_{\text{C}}}.
\end{align}
Again, to minimize the probability of error at the receiver, the ML detector is used.
Let $\mathsf{th}_{\text{C}}$ be the optimal ML detection threshold for this system. It is easy to see that this threshold exists for system C since stable distributions are unimodal and the two PDFs are shifted versions of each other. The bottom plot in Figure \ref{fig:optiThreshold} shows the optimal threshold for the case when $\Delta=1$, the distance is $d=1$ and the diffusion coefficients are $D_a=1$ and $D_b=0.5$.
The probability of error is now given by:
\begin{align}
\label{eq:PeC}
P_e^C &= P(Z_x=-\Delta) {\mathrm{Pr}}(z_y>\mathsf{th}_{\text{C}} | Z_x = -\Delta) \nonumber \\
&\qquad +P(Z_x=\Delta) {\mathrm{Pr}}(z_y\leq\mathsf{th}_{\text{C}} | Z_x= \Delta), \nonumber \\
&= 0.5 {\mathrm{Pr}}(z_n>\mathsf{th}_{\text{C}}+\Delta)+0.5 {\mathrm{Pr}}(z_n\leq\mathsf{th}_{\text{C}}-\Delta) \nonumber \\
&= 0.5[1-F_{\tilde{Z}_n}(\mathsf{th}_{\text{C}}+\Delta)+F_{\tilde{Z}_n}(\mathsf{th}_{\text{C}}-\Delta)],
\end{align}
which can be calculated using the CDF of the standard form of the noise term.
\section{Geometric Power and G-SNR} \label{sec:SNR}
We first note that all stable distributions, apart from the case $\alpha = 2$, have infinite variance, and all stable distributions with $\alpha \le 1$ also have infinite mean. In fact, this statement can be generalized to moments of order $p \le \alpha$, see \cite{gon06}. Therefore, the conventional notion of power, which is based on the variance of a signal, is not informative in the case of stable RVs with $\alpha < 2$ as, regardless of the specific distribution, the conventional power is infinity.
In this section we use a more generalized definition of power, the {\em geometric power}, as proposed in \cite[Section III]{gon06}. This definition uses zero-order statistics, i.e., it is based on logarithmic ``moments" of the form $\mathbb{E}[\log|N|]$.
\begin{definition}[Geometric Power]
The geometric power of the RV $N$ is given by:
\begin{align}
S_0(N) \triangleq e^{\mathbb{E} [\log|N|]}.
\end{align}
\end{definition}
In the following we use the terms {\em noise power} and the geometric power of the noise interchangeably.\footnote{Note that the definition of geometric power/SNR we introduce here is different from the one widely used in RF communications.}
In \cite[Prop. 1]{gon06}, an expression for the geometric power of a symmetric stable distribution is presented. Property \ref{prop:sym} implies that symmetric stable distributions are in fact ${\mathscr{S}} (0,c,\alpha,0)$.
This expression can therefore be used to calculate the geometric power of the noise term $L_N$ in system B. Yet, this expression is not applicable for the noise terms of systems A and C in which $\beta \neq 0$. The following theorem characterizes the geometric power of almost all stable distributions:
\begin{theorem}
\label{thrm:GpowerStable}
Let $N \sim {\mathscr{S}} (0,c,\alpha, \beta)$, where $\alpha \neq 1$, or $\alpha = 1$ and $\beta=0$. Then, the geometric power of $N$ is given by:
\begin{align}
S_0(N) = c G_\gamma^{(1/\alpha-1)} \big(1+\beta^2\tan^2(\tfrac{\pi \alpha}{2})\big)^{1/(2\alpha)},
\end{align}
where $G_\gamma = e^\gamma$, and $\gamma \approx 0.5772$ is the Euler's constant \cite[Ch. 5.2]{nist10}.
\end{theorem}
\begin{IEEEproof}
The proof is provided in Appendix \ref{annex:ProofOfGeomPower}.
\end{IEEEproof}
\begin{rem}
For the systems considered in this paper, since $\alpha = \frac{1}{2}$, the noise power simplifies to:
\begin{align}
S_0(N) = cG_\gamma\big(1+\beta^2\big).
\end{align}
Note that in this case, the noise power increases with respect to $\beta$ (the degree of skewness) and $c$ (the scale parameter).
\end{rem}
We now define the geometric SNR (G-SNR) as in \cite[Section III]{gon06}:
\begin{definition}[Geometric Signal-to-Noise Ratio]
\label{def:GSNR}
Let $X$ be the input signal in an additive-noise channel with a random noise $N$. Then the G-SNR is defined as:
\begin{align}
\label{eq:GSNRdef}
\gsnr \triangleq \frac{1}{2 G_\gamma }\bigg(\frac{X_{\max} - X_{\min}}{S_0(N)}\bigg)^2,
\end{align}
where $X_{\max}$ and $X_{\min}$ are the maximum and minimum admissible values for the channel input $X$. The normalizing term $\frac{1}{2 G_\gamma}$ is used to ensure that the G-SNR corresponds to the standard SNR in the case of an additive Gaussian noise channel.
\end{definition}
Using this definition and Theorem \ref{thrm:GpowerStable}, the G-SNR for systems A and C is defined as follows:
\begin{align}
\gsnr_A &= \frac{1}{2 G_\gamma }\bigg(\frac{\Delta}{2 c_{\text{A}} G_\gamma }\bigg)^2, \label{eq:GSNRa}\\
\gsnr_C &= \frac{1}{2 G_\gamma }\bigg(\frac{2\Delta}{ c_{\text{C}} G_\gamma(1+\beta_{\text{C}}^2) }\bigg)^2. \label{eq:GSNRc}
\end{align}
\begin{rem} \label{rem:absValImpact}
Note that system B involves an absolute value operation, thus, the G-SNR of system B cannot be obtained based on the techniques used to derive the G-SNR for systems A and C. Since the absolute value operation can only degrade the detection performance, calculating the G-SNR of the system $L_y = L_x + L_n$ can serve as an upper bound on the G-SNR of system B. This upper bound is given by:
\begin{align}
\gsnr_B \le \gsnr_B^{\text{ub}} = \frac{1}{2 G_\gamma }\bigg(\frac{\Delta}{ c_{\text{B}} G_\gamma }\bigg)^2. \label{eq:GSNRb}
\end{align}
\noindent This implies that the BER of the ML detector for system B is higher than the BER of the ML detector for the system $L_y = L_x + L_n$, as indicated in Section \ref{sec:numEval}.
\end{rem}
\begin{rem}
When the diffusion coefficient and the distance between the transmitter and the receiver are the same, the G-SNR of system A is four times larger than the G-SNR of system B since $c_{\text{B}}=4c_{\text{A}}$.
This implies that on top of the fact that two information particles are released in system B while only a single particle is released in system A, the gain from synchronization is a factor of $\frac{1}{4}$ in the noise geometric power.
\end{rem}
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{ProofOfSNR1.pdf}
\end{center}
\caption{\label{fig:proofSNR1} This plot shows that for a constant G-SNR, the BER is constant. For each point, the parameter $c$ of the noise distribution is calculated using the corresponding value for $\Delta$ such that the G-SNR$=1$. }
\end{figure}
%
\begin{rem}
\label{rem:effectBeta}
For system C let $r=D_a/D_b$ be the ratio of the diffusion coefficient of the two information particles. Then the noise parameters can be written as $c_{\text{C}}= \frac{d^2(\sqrt{r}+1)^2}{2rD_b}$ and $\beta_{\text{C}} = \frac{\sqrt{r}-1}{\sqrt{r}+1}$. Next, assume that the diffusion coefficient $D_b$ is fixed, and the diffusion coefficient of $D_a$ can be changed. In this case the noise geometric power is proportional to $\frac{1}{r}$, which decreases as $r$ increases. This also implies that the G-SNR increases with $r$. From the expression for $\beta_{\text{C}}$ and $c_{\text{C}}$ we observe that $\beta_c \rightarrow 1$ and $c_{\text{C}} \rightarrow \tfrac{d^2}{2D_b}$, when $r \rightarrow \infty$. Thus, in this case, system C reduces to system A, while no synchronization is required between the transmitter and the receiver.
Yet, this comes at a cost of using two different information particles. Note that this cost is captured in the G-SNR expression since the geometric power of the transmitted signal in system C is four times that of systems A and B, which can result in as much as 4 times improvement in G-SNR.
\end{rem}
\section{Numerical Evaluation}
\label{sec:numEval}
We start this section by evaluating the affects of the G-SNR, provided in \eqref{eq:GSNRdef}, on the BER performance of the three modulation schemes. In the additive white Gaussian noise channel the BER of the ML detector is only a function of SNR, namely, for a fixed SNR, the individual values of the signal power and the noise power do not affect the BER.
To evaluate if this property also holds for the three MT systems, we consider system C which can be specialized to both systems A and B using different values of the parameter $\beta_{\text{C}}$, see \eqref{eq:GSNRa}--\eqref{eq:GSNRc}. Thus, we evaluate if a constant BER is observed for a fixed value of G-SNR.
Figs. \ref{fig:proofSNR1} and \ref{fig:proofSNR10} depict BER versus $\Delta$ for two values of G-SNR: 1 and 10. In these plots, the x-axis corresponds to the values of $\Delta$.
For each point in the plot, the value of the noise parameter $c_{\text{C}}$ is calculated such that G-SNR is either 1 (Fig. \ref{fig:proofSNR1}) or 10 (Fig. \ref{fig:proofSNR10}). The BER is then numerically calculated using these values based on \eqref{eq:PeC}. It can clearly be observed that the BER is constant for a given G-SNR {\em regardless} of the value of $\Delta$ and $c_{\text{C}}$. It can further be observed that the BER decreases as $\beta_{\text{C}} \rightarrow 1$, which is in agreement with Remark \ref{rem:effectBeta}.
%
%
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{ProofOfSNR10.pdf}
\end{center}
\caption{\label{fig:proofSNR10} This plot shows that for a constant G-SNR, the BER is constant. For each point, the parameter $c$ of the noise distribution is calculated using the corresponding value for $\Delta$ such that the G-SNR$=10$.}
\end{figure}
%
Fig. \ref{fig:PBEvsGSNR} depicts the BER versus G-SNR for the different modulation techniques. For system C, five different values of $\beta_{\text{C}} =0,0.25,0.5,0.75,0.95$ are considered. The asynchronous scheme in system B with indistinguishable particles achieves the highest BER, while system A, which assumes perfect synchronization, achieves the lowest BER. The gap between these can be thought of as the cost of having no synchronization. Note that in system A, a single particle is released, while in system B two particles are released.
For system C it can be observed that by using two distinguishable particles, the BER improves compared to system B. Note that when $\beta_{\text{C}} = 0$ the noise distribution is the same as that in system B. In this case, when the dispersion parameter $c$ is the same for both systems, the G-SNR of system C is four times larger than $\gsnr_B^{\text{ub}}$ in \eqref{eq:GSNRb}. Yet, Fig. \ref{fig:PBEvsGSNR} indicates that even for $\beta_{\text{C}} = 0$ the BER of system C is lower than the BER of system B. This demonstrates the destructive effect of the absolute value operation as indicated in Remark \ref{rem:absValImpact}.
Finally, we observe that as $\beta_{\text{C}}$ increases the BER of system C decreases, while when $\beta_{\text{C}} \rightarrow 1$ the BER of system C approaches the BER of system A. In this case, asynchronous communication is possible with the same BER performance as synchronized communication at the cost of using two distinguishable particles.
%
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{ThreeChans_VS_GSNR.pdf}
\end{center}
\caption{\label{fig:PBEvsGSNR} BER versus G-SNR in dB for each modulation scheme.}
\end{figure}
%
We conclude the numerical evaluations with a case study. We consider a DBMT system where the distance between the transmitter and the receiver is $d = 20$ $\mu$m. Assume that the receiver is capable of detecting insulin molecules, which has a diffusion coefficient of $D_I = 150$ $\mu$m$^2$/s \cite{far16ST}. From these values the noise parameters $c_{\text{A}}$ and $c_{\text{B}}$ can be calculated for the modulation techniques represented by systems A and B. For system C, we consider six different particles as candidates for the second distinguishable particle. These particles are assumed to have diffusion coefficients ranging from 30 $\mu$m$^2$/s (e.g., diffusion coefficient of DNA) to 930 $\mu$m$^2$/s (e.g., diffusion coefficient of glycerol).
Fig.~\ref{fig:PBEvsDelta} depicts the results. \revised{In order to further evaluate the correctness of the analytical results, we also perform Monte Carlo simulations. It can be observed that the theoretical results (line plots) match perfectly with the results of the Monte Carlo simulations (point plots).} The asynchronous modulation scheme in system B with indistinguishable particles has the highest BER. Note that even the modulation scheme in system C where the diffusion coefficient of the second particle is one fifth of the diffusion coefficient of the particles used in system B (i.e. 30 $\mu$m$^2$/s) has lower BER. For the modulation technique in system C, as the diffusion coefficient of the second particle increases, the BER decreases. The modulation in system A achieves the best BER performance, and this shows that transmitter-receiver synchronization could have a considerable effect on BER. Table \ref{tab:PBE} quantifies the BER for each case.
\begin{figure}
\begin{center}
\includegraphics[width=0.9\columnwidth]{CaseStudySecondSub.pdf}
\end{center}
\caption{\label{fig:PBEvsDelta} BER versus $\Delta$ under different modulation schemes. \revised{The line plots are based on the theoretical results derived in the paper, and the point plots are the corresponding Monte Carlo simulation results.}}
\end{figure}
\small
\begin{table}[h]
\caption{\revised{The BER of different modulations in the case study for different values of $\Delta$. For the modulation scheme in system C the term in parenthesis is the diffusion coefficient of the second particle.} \label{tab:PBE}}
\vspace{-0.5cm}
\begin{center}
\begin{tabular}[t]{|c|c|c|c|c|c|}
\hline
$\Delta$ & 1 & 25 & 50 & 75 & 100 \\
\hline
\hline
System A & 0.3590 & 0.0912 & 0.0648 & 0.0530 & 0.0460 \\
\hline
System B & 0.4778 & 0.2346 & 0.1799 & 0.1523 & 0.1348 \\
\hline
System C ($D=30$) & 0.4592 & 0.2202 & 0.1687 & 0.1428 & 0.1263 \\
\hline
System C ($D=150$) & 0.4073 & 0.1535 & 0.1145 & 0.0957 & 0.0841 \\
\hline
System C ($D=930$) & 0.3533 & 0.1109 & 0.0812 & 0.0674 & 0.0589 \\
\hline
\end{tabular}
\end{center}
\end{table}
\normalsize
\revised{Finally, recall that the above results are derived under the assumption of an ISI-free channel (see Assumption \ref{assmp:noISI}). This can be achieved by properly spacing the transmissions leading to relatively large symbol durations. We now discuss how large this duration should be for the different \nariman{considered} systems.
Let $T_{\mathrm{symbol}}$ denote the spacing between consecutive transmissions, and $0 < p_{\mathrm{clean}} < 1$. Further, let $T_{\mathrm{last}}$ denote the last particle arrival time {\em calculated over the particles released in the current channel use}.
We propose to choose $T_{\mathrm{symbol}}$ such that:
\begin{align*}
{\mathrm{Pr}}\{ T_{\mathrm{last}} \nariman{\le}
T_{\mathrm{symbol}} \} \nariman{=}
p_{\mathrm{clean}}.
\end{align*}
\noindent Hence, if $p_{\mathrm{clean}}$ is a (fixed) value close to 1, then with high probability all the released particles arrive \nariman{at}
the receiver \nariman{before}
$T_{\mathrm{symbol}}$, implying that after this idle duration the channel can be used for another transmission. Moreover, the symbol duration $T_{\mathrm{symbol}}$ can be found from the CDF of $T_{\mathrm{last}}$ as a function of $p_{\mathrm{clean}}$. Note that the CDF of $T_{\mathrm{last}}$ is different for \nariman{each}
system \nariman{considered}. Yet, Assumption \ref{assmp:indep} implies that for all three systems the CDF of $T_{\mathrm{last}}$ is the product of the CDFs of the propagation time of the individual particles (in the case of System A it is simply the CDF of the L\'evy distribution).
Fig. \ref{fig:SymbolDurationVsDelta} depicts the calculated $T_{\mathrm{symbol}}$ for the different systems with different diffusion coefficients. $p_{\mathrm{clean}}$ was set to $0.99$. It can be observed that due to the heavy tails of the propagation density, the symbol duration is almost constant as a function of $\Delta$, and is much larger than $\Delta$. It can further be observed that the order of the curves in Fig. \ref{fig:SymbolDurationVsDelta} is almost identical to the order of the curves in Fig. \ref{fig:PBEvsDelta}, where the exception is the curve corresponding to System B which requires almost the same $T_{\mathrm{symbol}}$ as System C with the same diffusion coefficients.
Note that the modulation in system A requires the smallest $T_{\mathrm{symbol}}$, thus, together with the results of Fig. \ref{fig:PBEvsDelta}, \nariman{we conclude}
that the performance gains obtained from transmitter-receiver synchronization are significant.
\begin{figure}
\begin{center}
\includegraphics[width=0.975\columnwidth]{SymbolDuration.pdf}
\end{center}
\caption{\label{fig:SymbolDurationVsDelta} $T_{\mathrm{symbol}}$ versus $\Delta$ under different modulation schemes, for $p_{\mathrm{clean}}=0.99$.}
\end{figure}
}
\section{Conclusions and Future Work}
\label{sec:conc}
In this work, we considered two new asynchronous timing-based modulation techniques based on the time between release of two similar information particles, and the time between release of two different information particles. For evaluation, we compared the performance of these systems to the synchronized modulation based on the time of release of information particles. We showed that the three modulation techniques can be modeled as systems with an additive noise term, where for diffusion-based propagation, the noise terms are stable distributed. For the asynchronous systems, we derived the PDF of the additive noise in terms of the Voigt functions, which can be calculated efficiently and in some special cases be approximated using elementary functions.
Using these PDFs we then characterized the ML detectors for each system.
Since stable distributions, with the exception of the Gaussian distribution, have infinite variance, we used geometric power as a measure of strength of the noise. Using this approach, we derived the G-SNR for each modulation scheme for comparison.
Numerical evaluations show that for a constant G-SNR the BER is constant. Therefore, the G-SNR in DBMT channels plays a similar role as the SNR in the additive Gaussian noise channels.
Finally, we showed that, as expected, synchronization has a considerable effect on BER, where the first modulation scheme achieves the lowest BER. Moreover, we showed that it is possible to achieve a similar BER asynchronously if two distinguishable particles are used per bit.
As part of future work, we will explore extending the results to the case where multiple information particles are released simultaneously instead of one. Note that some of our current ongoing work has shown that simultaneously releasing multiple particles can improve the performance of the first system significantly \cite{far17TIT,mur17}. We would like to extend these results to the asynchronous systems presented in this paper using order statistics.
\appendices
\section{Proof of Theorem \ref{thm:PDFNoiseChanB}} \label{annex:ProofOfPDFchanB}
We use Property \ref{prop:standStable}, and find an expression for the standardized distribution with $c_{\text{B}}=1$. Then the PDF for any value of $c_{\text{B}}$ can be calculated using \eqref{eq:standPDFConv}.
Let $X \sim {\mathscr{S}} (0,1,\tfrac{1}{2},0)$ be a standardized stable RV with parameters $\alpha = \tfrac{1}{2}$ and $\beta=0$. Then the PDF of $X$ is given by \cite[Eq. (7.1)]{hol73}:
\begin{align}
\label{eq:alphaHalf}
f(x;1/2,\beta) = \Re \left\{ \frac{z}{\pi x} [ \sqrt{\pi} e^{-z^2} - 2 j D(z)] \right\},
\end{align}
where
\begin{align}
D(z) = e^{-z^2} \int_0^z e^{t^2} dt
\end{align}
is the Dawson's Integral \cite[Eq. (7.2.5)]{nist10}, and
\begin{align}
\label{eq:z}
z = \frac{1+\beta - j(1-\beta)}{2\sqrt{2x}}.
\end{align}
\noindent It is possible to rewrite (\ref{eq:alphaHalf}) in terms of the complex error function, also known as Faddeeva function or the Kramp function \cite[Eq. (7.2.3)]{nist10}:
\begin{align}
w(z) = e^{-z^2} \left( 1+ \frac{2 j}{\sqrt{\pi}} \int_0^z e^{t^2} dt \right) = e^{-z^2} \erfc(-jz).
\end{align}
Using \cite[Eq. (7.5.1)]{nist10}:
\begin{align}
\label{eq:dawsonFadev}
D(z) =0.5 j \sqrt{\pi}( e^{-z^2} -w(z)),
\end{align}
and the property $w(-z) = 2 e^{-z^2} - w(z)$ , we rewrite (\ref{eq:alphaHalf}) as:
\begin{align}
\label{eq:alphaHalf2}
f(x;1/2,\beta) = \Re \left\{ \frac{z}{\sqrt{\pi} x} w(-z) \right\}.
\end{align}
One of the benefits of writing the PDF in terms of the complex error function is that there are a large body of works that considered calculating it numerically. Moreover, if $z = a+j b$, for $b>0$ the complex error function can be represented by its real and imaginary parts as \cite[Sec. 1]{abr11}:
\begin{align}
\label{eq:CompErToVoigt}
w(a+ j b) = K(a,b)+j L(a,b),\quad b>0,
\end{align}
where $K(a,b)$ and $L(a,b)$ are the real and imaginary Voigt functions given in \eqref{eq:ReVoigt} and \eqref{eq:ImVoigt}, respectively.
Using Property \ref{prop:sym}, the PDF of $X$ is symmetric. Hence, the density for $X\geq0$ is sufficient for characterizing the whole PDF.
Since $\beta = 0$, when $X>0$, we can write $z = p_{x} - j p_{x}$ where $p_{x}= 1/\sqrt{8x}$. Substituting \eqref{eq:CompErToVoigt} in \eqref{eq:alphaHalf2}, the density of $X$, when $X\geq0$, can be written as:
\begin{align}
\label{eq:fLn}
f(x) =
\begin{cases}
\frac{1}{\sqrt{8 \pi x^3}}\left[ K(-p_{x},p_{x})+L(-p_{x},p_{x})\right] & x>0 \\
\frac{2}{\pi} & x=0
\end{cases},
\end{align}
where the value for $x=0$ follows from \cite[Eq. (2.2.11)]{zol86-book}. Finally, the density for $X<0$ is obtained using symmetry. The proof is completed by applying \eqref{eq:standPDFConv}.
\section{Proof of Theorem \ref{thm:PDFNoiseChanC}} \label{annex:ProofOfPDFchanC}
We use Property \ref{prop:standStable}, and find an expression for the standardized distribution with $c_{\text{C}}=1$. Thus, the PDF for any value of $c_{\text{C}}$ can be calculated using \eqref{eq:standPDFConv}.
Let $X \sim {\mathscr{S}} (0,1,\tfrac{1}{2},\beta_{\text{C}})$ be the standardized stable RV with parameters $\alpha = \tfrac{1}{2}$ and $\beta_{\text{C}}$.
Using \eqref{eq:alphaHalf2}, and recalling that $\beta_{\text{C}} = (\sqrt{D_a}-\sqrt{D_b})/(\sqrt{D_a}+\sqrt{D_b})$, we write (\ref{eq:z}) as $z=p_{x} - jq_{x}$ when $x>0$, where $p_{x}=(1+\beta_{\text{C}})/(\sqrt{8|x|})$ and $q_{x}=(1-\beta_{\text{C}})/(\sqrt{8|x|})$. Similarly, we write (\ref{eq:z}) as $z=-q_{x} - jp_{x}$ when $x<0$. Using (\ref{eq:alphaHalf2}) and the Voigt functions decomposition of the Faddeeva function \eqref{eq:CompErToVoigt}, the PDF of the standardized distribution is given by:
\vspace{-0.15cm}
\begin{align*}
\label{eq:fZn}
f(x; \beta_{\text{C}}) =
\begin{cases}
\frac{1}{\sqrt{8 \pi x^3}} \bigg[ (1+\beta_{\text{C}}) K(-p_{x},q_{x}) & \\
\quad \quad \quad ~~+(1-\beta_{\text{C}})L(-p_{x},q_{x})\bigg], & x>0\\
\frac{2(1-\beta^2)}{\pi(1+\beta^2)^2}, & x=0 \\
\frac{1}{\sqrt{8 \pi |x|^3}}\bigg[(1-\beta_{\text{C}})K(q_{x},p_{x}) & \\
\quad \quad \quad ~~-(1+\beta_{\text{C}}) L(q_{x},p_{x}) \bigg], & x<0
\end{cases},
\end{align*}
\vspace{-0.1cm}
\noindent where, again, the value for $x=0$ follows from \cite[Eq. (2.2.11)]{zol86-book}. The proof is completed by applying \eqref{eq:standPDFConv}.
\vspace{-0.2cm}
\section{Proof of Theorem \ref{thrm:ChanBthrExistance}} \label{annex:ProofOfExistanceOfThreshold}
\vspace{-0.1cm}
We first observe that for $\ell_y = 0$, $f_{L_y|L_x}(0 | 0)>f_{L_y|L_x}( 0| \Delta)$. This follows from the fact that stable distributions are unimodal, and the mode of the noise term $L_n$ is at $\ell =0$. Therefore, the threshold is located at $\mathsf{th}_{\text{B}}>0$, and we focus of the case where $\ell_y>0$. Note that in this case, due to the continuity and unimodality of stable distributions\cite[Theorem 2.7.6]{zol86-book}, both $f_{L_y|L_x}(\ell_y | L_x = 0)$ and $f_{L_y|L_x}( \ell_y| L_x=\Delta)$ are continuous functions and unimodal.
We now have the following lemma:
\begin{lemma}
If $0 < \ell_y \leq \tfrac{\Delta}{2}$, then $f_{L_y|L_x}(\ell_y | L_x = 0)>f_{L_y|L_x}( \ell_y| L_x=\Delta)$.
\end{lemma}
\begin{IEEEproof}
We first consider the system in \eqref{eq:timingChB} without the absolute value: $\tilde{L}_y = L_x + L_n$. Here, $f_{\tilde{L}_y|L_x}(\tilde{l}_y | l_x) = f_{L_n}(\tilde{l}_y - l_x)$. Since stable distributions are unimodal, we have $f_{\tilde{L}_y|L_x}(\tilde{l}_y | L_x = 0) > f_{\tilde{L}_y|L_x}(\tilde{l}_y | L_x = \Delta), \forall \tilde{l}_y < \frac{\Delta}{2}$. Using the expression for the PDF of system \eqref{eq:timingChB} in \eqref{eq:LyGLx0}--\eqref{eq:LyGLxD} we obtain the desired result.
%
%
%
%
\end{IEEEproof}
\begin{lemma}
\label{lemma:existanceThreshold}
If $ \ell_y > \tfrac{\Delta}{2}$, then there exists a point $\mathsf{th}_{\text{B}}$ such that for all $\tfrac{\Delta}{2}< \ell_y <\mathsf{th}_{\text{B}}$, $f_{L_y|L_x}(\ell_y | L_x = 0)>f_{L_y|L_x}( \ell_y| L_x=\Delta)$ and for all $\ell_y >\mathsf{th}_{\text{B}}$, $f_{L_y|L_x}(\ell_y | L_x = 0) \leq f_{L_y|L_x}( \ell_y| L_x=\Delta)$.
\end{lemma}
\begin{IEEEproof}
Note that for $ \ell_y > \tfrac{\Delta}{2}$, $f_{L_n}(\ell_y) < f_{L_n}(\ell_y-\Delta)$. Moreover, note that $f_{L_n}(\ell)$ is a smooth function and it is decreasing for $\ell>0$. Then clearly there exists a $\mathsf{th}_{\text{B}} >\tfrac{\Delta}{2}$ such that:
\vspace{-0.15cm}
\begin{align}
\mspace{-5mu} \begin{cases}
f_{L_n}(\ell_y) \mspace{-3mu} - \mspace{-3mu} f_{L_n}(\ell_y-\Delta) \mspace{-3mu} > \mspace{-3mu} f_{L_n}(\ell_y+\Delta) \mspace{-3mu} - \mspace{-3mu} f_{L_n}(\ell_y) & \ell_y \mspace{-3mu} < \mspace{-3mu} \mathsf{th}_{\text{B}} \\
f_{L_n}(\ell_y) \mspace{-3mu} - \mspace{-3mu} f_{L_n}(\ell_y-\Delta) \mspace{-3mu} \leq \mspace{-3mu} f_{L_n}(\ell_y+\Delta) \mspace{-3mu} - \mspace{-3mu} f_{L_n}(\ell_y) & \ell_y \mspace{-3mu} \geq \mspace{-3mu} \mathsf{th}_{\text{B}},
\end{cases}
\end{align}
\vspace{-0.1cm}
\noindent which follows since the slope of $f_{L_n}(\ell)$ for $\ell>0$ decreases, reaches a minimum, and then increases. Combining both Lemmas, the theorem is proved.
\end{IEEEproof}
\vspace{-0.15cm}
\section{Proof of Theorem \ref{thrm:GpowerStable}} \label{annex:ProofOfGeomPower}
To prove this theorem, we first derive $\mathbb{E}[|N|^s]$. We write this expectation in integral form as:
\vspace{-0.15cm}
\begin{align*}
\mathbb{E}[|N|^s] &= \int_{-\infty}^{\infty} |n|^s f(n;0,c,\alpha, \beta) dn\\
& \stackrel{(a)}{=} \int_{0}^{\infty} n^s f(n;0,c,\alpha, \beta) dn \nonumber \\
&\qquad+ \int_{0}^{\infty} n^s f(n;0,c,\alpha, -\beta) dn,
\end{align*}
\noindent where (a) follows since $f(-x;0,c,\alpha, \beta)=f(x;0,c,\alpha, -\beta)$ \cite[Proposition 1.11]{nol15}. Taking the derivative with respect to $s$ we obtain:
\begin{align*}
\frac{d}{ds}\mathbb{E}[|N|^s] &= \int_{0}^{\infty} n^s \log n f(n;0,c,\alpha, \beta) dn \nonumber \\
&\qquad+ \int_{0}^{\infty} n^s \log n f(n;0,c,\alpha, -\beta) dn.
\end{align*}
\noindent Further setting $s=0$ results in:
\begin{align*}
\frac{d}{ds}\mathbb{E}[|N|^s] \bigg\rvert_ {s=0} &= \mathbb{E}[\log(|N|)].
\end{align*}
\noindent We now define $\lambda \triangleq c^\alpha \sqrt{1+\frac{\beta^2}{\cot^2(\tfrac{\pi \alpha}{2})}}$,
and let
\begin{align*}
\theta \triangleq 2 \arctan\bigg(\frac{\beta}{\cot(\tfrac{\pi\alpha}{2})} \bigg)/(\pi\alpha).
\end{align*}
\noindent Using \cite[Fact 3, pg. 117]{zol86-book}, and \cite[Theorem 2.6.4]{zol86-book} we have that for $N \sim {\mathscr{S}} (0,c,\alpha, \beta)$, $\alpha \neq 1$,
\begin{align}
\label{eq:ExpAbsNtoS}
\mathbb{E}[|N|^s] = \lambda^{s/\alpha} \frac{\cos(\tfrac{\pi}{2} \theta s) \Gamma (1-s/\alpha)}{\cos(\tfrac{\pi}{2} s) \Gamma (1-s)}.
\end{align}
By taking the derivative of \eqref{eq:ExpAbsNtoS} with respect to $s$ and evaluating the result at $s=0$ we obtain:
\begin{align}
\label{eq:ExpAbsLogN}
\mathbb{E}[\log(|N|)] &= \log(c) + \frac{1}{2\alpha}\log\bigg(1+\frac{\beta^2}{\cot^2(\tfrac{\pi \alpha}{2})}\bigg) \nonumber \\
& \mspace{120mu} +(1/\alpha-1)\gamma,
\end{align}
\noindent where $\gamma$ is the Euler's constant \cite[Ch. 5.2]{nist10}. Finally, recalling that $S_0(N) = e^{ \mathbb{E}[\log(|N|)] } $ we conclude the proof.
\section*{Acknowledgment}
The authors would like to thank Professor John P. Nolan at American University for providing valuable correspondence on stable distributions.
\bibliographystyle{IEEEtran}
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\section{Introduction}
Developing an enhanced understanding of mechanical behavior in materials relies upon sufficiently characterizing microstructure details at the relevant length scales that contribute to this behavior. Moreover, to truly enhance the predictive capability of processing-structure-property models that aim to improve material performance requires a quantitative stereological description of the relevant microstructure features and, thereby, the material itself. Predictive models that effectively capture the linkage between processing and properties (through microstructure) can be utilized within an integrated computational materials engineering (ICME) approach to design materials and accelerate their insertion into application.
The focus of the present work is on single crystal nickel-based superalloys, which are used in turbine blades within the high temperature section of the modern turbine engine \citep{Ree2006, Pol2006}. In single crystal nickel-based superalloys, there are a number of length scales of microstructure that contribute to mechanical behavior, ranging from the $\gamma^\prime$ precipitates to pores and eutectic particles to the dendrites themselves. At the largest microstructure length scale in directionally-solidified single crystal microstructures, the features of interest are the dendrites; many features at lower length scales (e.g., eutectic particles, precipitates, etc.) or at similar scales (e.g., porosity, freckle defects, etc.) are strongly associated with the dendrite arm spacing and morphology \citep{Whi2001,Ell2004,Mel2005,Lam2007,Bru2012}. Historically, the primary dendrite arm spacing (PDAS) has been found to correlate with processing (e.g., solidification rate) \citep{McC1981,Hui2002,Wan2003, Mil2012,Bru2011,Bru2012} as well as with properties (e.g., creep strength, fatigue properties)\citep{Wil2008,Lam2007}. For instance, Lamm and Singer \citep{Lam2007} produced single crystal nickel-based microstructures (PWA 1483) with a varied range of different dendrite arm spacings (250 $\mu$m to 600 $\mu$m) and found that decreasing the mean dendrite arm spacing was associated with an increased high-cycle fatigue life. The fatigue cracks were found to originate at shrinkage porosity and the largest pores correlated with a large PDAS.
The traditional approach for measuring primary dendrite arm spacing in single crystal metals, whereby the number of dendrite cores in a specified area is related to the dendrite arm spacing \citep{Fle1974,Jac1976,McC1981} is given by:
\begin{equation}
\lambda = c \sqrt{\frac{A}{n}}
\label{lambda}
\end{equation}
\noindent where $\lambda$ is primary dendrite arm spacing, $A$ is the area analyzed, $n$ is the number of dendrites, and $c$ is a coefficient that depends on the microstructure. McCartney and Hunt \cite{McC1981} showed that $c=0.5$ for a random array of points, $c=1$ for a square array of points, and $c=1.075$ for a hexagonal array of points; they had to apply a correction for the bulk dendrite arm spacing $\lambda$ as processing conditions caused a change in the local environment of the dendrites. However, this approach is insufficient for capturing local arm spacings or the dendrite arm spacing distribution, and may provide problems with complex geometries such as turbine blades. In fact, part of the motivation for quantifying the local PDAS is that a narrow distribution (i.e., low standard deviation) of local PDAS values may result in a more homogeneous distribution of interdendritic microstructure features and, more importantly, a narrow distribution of mechanical properties.
The research objective herein is to evaluate the capability of some recent approaches, as well as some modified versions of these approaches, for characterizing the local dendrite arm spacing within experimental dendritic microstructures. In this work, an experimental dendritic microstructure is used for this analysis along with three different techniques that are based on the nearest neighbor spacing and/or a Voronoi tessellation of the dendrite cores. Comparison of existing and new metrics with traditional primary dendrite arm spacing metrics is discussed for both local and global measures. The current methods investigated supply statistical information of local spacing and coordination number while introducing a technique for addressing edge effects and examining the parameter sensitivity of these different methods. In comparison to previous work \cite{Tsc2013}, this work introduces and compares the statistical distributions of local dendrite arm spacings for the four methods, for a more quantitative analysis. It was found that augmenting existing techniques with Voronoi tesselations to define the subset of first nearest neighbors or refining existing Voronoi-based techniques resulted in a more physical description of the local dendrite arm spacing. Moreover, for certain cases, the mean local dendrite statistics can adequately approximate the PDAS found with the traditional bulk characterization technique (Eq.~\ref{lambda}).
\section{Methodology}
The approach utilized herein to measure the local dendrite arm spacing is based on a Voronoi tessellation of the spatial array of dendrite cores. The following analysis techniques were implemented in MATLAB R2012a (The MathWorks, Inc.). To illustrate how the present method works and differs from some other published methods, we generated a synthetic 5x5 cubic pattern of points with a small degree of noise (100\% noise fraction, 0.20$a_0$ noise fraction [2]), as shown in Figure~\ref{5x5} \cite{Tsc2013}. For the purposes of describing several different methods shown in Figure~\ref{various_methods}, this synthetic pattern of points can be considered as the cores of primary dendrites.
\begin{figure}[hbt!]
\centering
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_1a}
\caption{}
\label{5x5}
\end{subfigure}%
\quad
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_1b}
\caption{}
\label{inner_circle}
\end{subfigure}
\quad
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_1c}
\caption{}
\label{voronoi_tesselation}
\end{subfigure}
\\
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_1d}
\caption{}
\label{mod_warnken_reed}
\end{subfigure}
\quad
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_1e}
\caption{}
\label{nearest_neighbor}
\end{subfigure}
\quad
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_1f}
\caption{}
\label{mod_tesselation}
\end{subfigure}
\caption{The difference between various methods for defining the nearest neighbors (red dots) and their spacing for a single point (large black dot). (a) Initial 5 x 5 cubic pattern with noise fraction of 100\% and noise level of 0.20$a_0$. (b) The Warnken--Reed method with $\alpha$ = 1.5 and $k_{initial} = 3$. The inner circle represents the average spacing, $d_{avg}$, of these neighbors and the outer circle represents the cutoff for adding the next neighbor, $d_{avg} + \alpha d_{std}$. (c) Voronoi tesselation diagram for the points. The potential first nearest neighbors are identified through shared vertices with each point. (d) The modified Warnken--Reed method with $\alpha = 1.5$ and $k_{initial} = 3$, whereby the neighbors are restricted to only those identified using the Voronoi tesselation. (e) Using only shared vertices (and connecting lines forming a polygon) of the Voronoi tesselation to identify the nearest neighbors ($d_{crit}=0.0$). (f) Modified tesselation-based technique whereby the nearest neighbors are identified as those with line lengths above a critical threshold fraction of the total perimeter line length $d_{crit}=0.10$ of the tesselated polygon for the point (Reprinted from \cite{Tsc2013}).}
\label{various_methods}
\end{figure}
One such method for measuring the local dendrite arm spacing is the Warnken--Reed method \citep{War2011,War2011a}. The Warnken--Reed method calculates the dendrite arm spacing for a single point (black dot) by starting with an initial number of nearest neighbors (3 closest neighbors) and iteratively adding potential nearest neighbors that are within a cutoff distance defined by the already-added nearest neighbors. For instance, the inner circle in Figure~\ref{inner_circle} represents the average spacing, $d_{avg}$, of these neighbors and the outer circle represents the cutoff for adding the next neighbor, $d_{avg} + \alpha d_{std}$, where $d_{std}$ is the standard deviation of the nearest neighbor spacings and $\alpha$ is a parameter that is typically between 1 and 2. Neighbors continue to be added until the cutoff does not include any new neighbors. The local coordination number and dendrite arm spacing is calculated from the neighbors added (shown as red dots). However, if the standard deviation of the distances of the nearest neighbors $d_{std}$ or the parameter $\alpha$ is large, this technique can continue to add nearest neighbors beyond the first nearest neighbors; our implementation stopped after 20 nearest neighbors. Clearly, a method for restricting the number of nearest neighbors using such a technique is necessary.
A simple way of identifying the potential first nearest neighbors is to perform a Voronoi tessellation of the space surrounding the points, as shown in Figure~\ref{voronoi_tesselation}. The polygon edges are equidistant between the points contained in the two adjacent polygons and the triple points (merging of three lines) are equidistant between the points contained in the three adjacent polygons. Therefore, the first nearest neighbors (FNNs, shown as open circles in Fig.~\ref{voronoi_tesselation}) correspond to the edges of the central polygon (that contains the black dot). This subset of points is the maximum number of nearest neighbors that the central point can have.
In this manner, several techniques have been identified to quantify a local dendrite arm spacing based on the Voronoi-identified FNNs \cite{Tsc2013}. For instance, the Voronoi Warnken--Reed method (Figure~\ref{mod_warnken_reed}) only includes the Voronoi FNNs as potential nearest neighbors and cannot expand beyond these, alleviating a potential problem of selecting second nearest neighbors or greater. Another method using the Voronoi FNNs is to consider all of these potential nearest neighbors as nearest neighbors (Figure~\ref{nearest_neighbor}), as in Brundidge et al.~\citep{Bru2011}. Unfortunately, this approach is sensitive to small perturbations in the spatial positions of the neighbors. For instance, if the lower right hand neighbor in Figure~\ref{nearest_neighbor} moves away from the central point, it no longer shares an edge with the polygon containing the black dot; in this scenario, the two adjacent polygons on either side effectively ``pinch'' off this neighbor. This scenario, however, has a physical basis as these two dendrite cores mainly compete with the central core, and the lower right core has a much less prominent effect on the central core. The last method, which is examined in the present paper, utilizes a criterion based on the edge lengths of the Voronoi polygon. In Figure~\ref{mod_tesselation}, those neighbors with edge lengths less than a critical fraction, $d_{crit}$, of the total polygon perimeter are excluded as nearest neighbors (e.g., 10\% in Figure~\ref{mod_tesselation}). In the present study, the local dendrite arm spacing statistics are evaluated using these four techniques: Warnken--Reed, Voronoi Warnken--Reed, and the Voronoi technique with ($d_{crit}>0$) and without ($d_{crit}=0$) a line length threshold.
As an example of a more disordered structure, Figure~\ref{various_methods2} plots the four different methods for a different configuration of surrounding points (dendrite cores). In Figure~\ref{inner_circle2}, the iterative Warnken--Reed method continues to non-physically add neighbors beyond the first nearest neighbors due to a large initial $d_{std}$ value from the initial three distances. The Voronoi-modified version in Figure~\ref{mod_warnken_reed2} stops at four nearest neighbors despite the fact that several points lie within the outer boundary computed by this method. The Voronoi method with $d_{crit}=0.0$ clearly overestimates the number of nearest neighbors, while the four nearest neighbors identified through $d_{crit}=0.10$ (Figure~\ref{mod_tesselation2}) perhaps offers a better approximation of the number of nearest neighbors. Interestingly, comparing the methods in Figure~\ref{mod_warnken_reed2} and \ref{mod_tesselation2}, the coordination number is the same, but the nearest neighbors identified is different. This is due to the Warnken--Reed method being a distance-based method, and identifying the four closest neighbors, while the modified Voronoi technique is based on the edge lengths of the Voronoi polygon, and hence utilizes this to identify nearest neighbors (which may not be the closest neighbors).
\begin{figure}[hbt!]
\centering
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_2a}
\caption{}
\label{5x52}
\end{subfigure}%
\quad
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_2b}
\caption{}
\label{inner_circle2}
\end{subfigure}
\quad
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_2c}
\caption{}
\label{voronoi_tesselation2}
\end{subfigure}
\\
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_2d}
\caption{}
\label{mod_warnken_reed2}
\end{subfigure}
\quad
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_2e}
\caption{}
\label{nearest_neighbor2}
\end{subfigure}
\quad
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_2f}
\caption{}
\label{mod_tesselation2}
\end{subfigure}
\caption{The difference between various methods for defining the nearest neighbors (red dots) and their spacing for a single point (large black dot) with a distorted local environment. Parts (a)-(f) are as Figure 1: (a) initial pattern, (b) the Warnken--Reed method with $\alpha$ = 1.5 and $k_{initial} = 3$, (c) the Voronoi tesselation diagram for the points, (d) the Voronoi-modified Warnken--Reed method with $\alpha = 1.5$ and $k_{initial} = 3$, (e) the Voronoi method ($d_{crit}=0.0$), and (f) the modified tesselation-based technique with $d_{crit}=0.10$.}
\label{various_methods2}
\end{figure}
The traditional PDAS metric does not consider the order or disorder of the dendrites within the microstructure. Figure~\ref{various_methods2} illustrates why a local metric for PDAS may be needed. For the field of view given in Figs.~\ref{5x5} and \ref{5x52}, the bulk PDAS metric would be the same since the number of dendrites $n$ and the area $A$ are equal (see Eq.~\ref{lambda}). However, the disorder of the dendritic structure in the case of Fig.~\ref{various_methods2} may yield (i) a more uneven distribution of solute elements, (ii) the formation of second phase particles, (iii) the formation of gas or shrinkage porosity, or (iv) the lateral growth of secondary dendrite arms. Hence, in addition to the bulk PDAS values, understanding how processing conditions may impact the disorder of the dendritic structure may be important for understanding the properties of directionally-solidified alloys.
Other techniques exist for quantifying the homogeneity or heterogeneity of primary dendrite arm spacing in directionally-solidified dendritic microstructures. For instance, the minimal spanning tree (MST) method \citep{Dus1986} provides a statistical analysis of the disorder in a system of points by connecting all points with the shortest line segments possible. In this manner, the mean distance of all line segments ($m$) and the standard deviation ($\sigma$) characterize the disorder of the system and casting these values into a $m$-$\sigma$ design space allows for comparison between different point systems \citep{Dus1986}. This has been effectively applied to characterize the mean dendrite arm spacing, PDAS distribution, and the disorder in first Pb-Tl alloys \citep{Bil1991} and subsequently in other alloy systems \citep[e.g.,][]{Tew2002,Hui2002,Pen2013}. As an example of this technique, Figure \ref{MST} plots the dendrite cores and connecting line segments for the single crystal nickel-based superalloy micrograph used in this study (Figure \ref{sx_nickel}). Moreover, other methods such as radial distribution functions, fast Fourier transforms, and/or correlation functions can also be used to characterize the dendrite arm spacing distribution. However, it should be noted that these approaches are not intended for local characterization of the dendrite arm spacing and are not as effective for correlating the local spacing with local microstructure features as shown herein. Moreover, these techniques do not quantify the number of nearest neighbors and are often coupled with Voronoi polygons to compute the nearest neighbor distributions. Rather, these analysis methods are more effective at characterizing and comparing the homogeneity/heterogeneity of the dendritic structure between different processing conditions. Hence, there will be limited discussion of these techniques in the present work.
\begin{figure}[bht!]
\centering
\includegraphics[width = 0.7\textwidth]{Figure_3}
\caption[]{Minimal spanning tree method \citep{Dus1986} for defining the spacing and homogeneity of a microstructure (set of points), whereby the lines represent the minimal distance of connecting line segments. The set of points selected for this example were selected from the dendrite cores shown in Figure \ref{sx_nickel}, where the white dots indicate `edge' dendrite cores. }
\label{MST}
\end{figure}
\section{Results}
\subsection{Application to dendritic microstructure}
A micrograph of a directionally-solidified single crystal nickel-based superalloy microstructure that is polished and imaged perpendicular to the solidification direction is shown in Figure~\ref{sx_nickel}. This microstructure was produced using the liquid metal cooling technique, as described in Miller \citep{Mil2011} and Elliott et al.~\citep{Ell2004}. First, the dendrite cores were identified manually (white and black dots). Automated methods to identify dendrite cores can be invaluable for future large scale analysis \citep{Tsc2010a,Tsc2010b}. Moreover, the white particles in this image are eutectic particles. A total of 393 dendrite cores are contained in this image over an area of 24.25 mm$^2$, giving a PDAS of 248.4 $\mu$m using $c=1$ (Equation~\ref{lambda}). The remainder of the analysis uses this micrograph as a template for characterizing the local dendrite arm spacing.
\begin{figure}[bht!]
\centering
\includegraphics[width = \textwidth]{Figure_4}
\caption[]{Dendritic structure normal to the withdrawal direction in a directionally-solidified single crystal nickel-based superalloy cast using the liquid metal cooling technique \citep{Mil2011}. The dots denote the dendrite cores, where the white dots indicate `edge' dendrite cores, as discussed in Figure~\ref{sx_nickel2} and the associated text. }
\label{sx_nickel}
\end{figure}
\subsection{Accounting for image/part edge effects}
The ability to handle edge effects when computing local dendrite arm spacings with dendrite cores is vital for quantifying statistics in thin sections, such as the wall of an airfoil blade that may only contain 1--3 dendrite cores across the section \citep[e.g.,][]{Tsc2010a,Tsc2010b}. As a first example of one such a technique, we have used a convex hull of the dendrite cores in Figure~\ref{sx_nickel} to identify ``edge'' dendrite cores and quantify the dendrite arm spacing. The dendrite core locations are first extracted from the experimental image, as shown in Figure~\ref{sx_nickel2}. Then, a convex hull is generated around the points; this is the minimum ``convex'' area that contains all the points. Next, the edge points (white dots in Fig.~\ref{sx_nickel}) are identified by finding those points with Voronoi vertices that lie outside of the convex hull (dotted blue line in Figure~\ref{hulla}). Then, to utilize Voronoi-based techniques for these points, a new polygon is generated by the intersection of the initial polygon from the Voronoi tessellation and the convex hull; the new polygon of the edge dendrite cores is colored red in \ref{hulla} to distinguish from the bulk dendrite cores. The polygons belonging to the interior and edge dendrites are shown in Figures \ref{hullb} and \ref{hullc}, with a random coloring scheme used to delineate the different polygons. Last, the neighbors can now be calculated using either a new criterion or the same criterion used for interior points. For the present analysis, the same criterion (polygon with edge length threshold) was used for all points; although herein the interior dendrite cores are used to compare statistics with other techniques and bulk PDAS values. More complicated techniques are needed to deal with complex geometries that include concave character and internal passages in order to eventually apply these techniques to complex structures such as turbine blades. Multiple instantiations of microstructures with edge effects can shed light on the appropriate method for determining the local PDAS at edges, which may be different from that used in the interior.
\begin{figure}[bht!]
\centering
\begin{subfigure}[b]{0.75\textwidth}
\includegraphics[width=\textwidth]{Figure_5a}
\caption{Convex hull}
\label{hulla}
\end{subfigure}%
\begin{subfigure}[b]{0.475\textwidth}
\includegraphics[width=\textwidth]{Figure_5b}
\caption{Interior dendrite cores}
\label{hullb}
\end{subfigure}%
\begin{subfigure}[b]{0.475\textwidth}
\includegraphics[width=\textwidth]{Figure_5c}
\caption{Edge dendrite cores}
\label{hullc}
\end{subfigure}
\caption[]{(a) Voronoi tessellation of dendritic structure from Figure \ref{sx_nickel}. The dotted blue line (surrounding the points) denotes the convex hull of the dendrite cores and the red polygons delineate the cores that intersect the convex hull. The interior and edge dendrites are shown in (b) and (c), respectively, with each polygon colored differently as a guide to the eye.}
\label{sx_nickel2}
\end{figure}
\subsection{Spatial distribution of local primary dendrite arm spacings}
\begin{figure}[bht!]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[trim=0 225 0 0, clip,width=\textwidth]{Figure_6a}
\phantomsubcaption
\end{subfigure}%
\quad
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[trim=0 225 0 0, clip,width=\textwidth]{Figure_6b}
\phantomsubcaption
\end{subfigure}
\setcounter{subfigure}{0}
\\
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_6c}
\caption{Primary dendrite arm spacing ($\mu$m)}
\end{subfigure}
\quad
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_6d}
\caption{Coordination number}
\end{subfigure}
\caption{(a) Local dendrite arm spacing ($\mu$m) and (b) coordination number based on the Voronoi tessellation with edge length threshold of $d_{crit}$=0.12 or 12\%.}
\label{dendrite}
\end{figure}
The spatial distribution of local dendrite arm spacing and coordination number can provide insight into the order/disorder of primary dendrites and can identify regions that could potentially contain more/less interdendritic features and/or contain different properties. For instance, the primary dendrite arm spacing and coordination number for the directionally-solidified superalloy micrograph (Figure~\ref{sx_nickel}) is shown in Figure~\ref{dendrite}. In this example, we used the Voronoi tessellation-based technique with an edge length threshold of $d_{crit} = 0.12$. Dendrite cores with local PDAS similar to the mean PDAS of the bulk (248.4 $\mu$m) are colored white and those with PDAS above (below) the mean PDAS are red (blue); the lower and upper bounds of the colorbar are -25\% and +25\% of the mean PDAS value, respectively. In general, the exterior dendrite cores have similar PDAS as the interior dendrite cores using this technique. A similar colorbar is used for the coordination number as well. As would be expected, the exterior dendrite cores tend to have a lower coordination number than the interior dendrite cores, with a few that only have 2 nearest neighbors. However, the dendrite cores with a low coordination number on the edges are not consistently over/under the mean PDAS (i.e., they do not significantly bias the statistics from the edge dendrite cores). Future work will examine what techniques may be most applicable for characterizing local dendrite arm spacings and coordination numbers for dendrite cores on free surfaces. It is envisioned that sectioning large numbers of instantiations of synthetically-generated microstructures of known bulk dendrite arm spacings can be used to understand the bias introduced by edge effects and to understand what are the best techniques for quantifying the local spacing.
\begin{figure}[bht!]
\centering
\begin{subfigure}[b]{0.75\textwidth}
\centering
\includegraphics[trim=0 225 0 0, clip,width=\textwidth]{Figure_7a}
\phantomsubcaption
\end{subfigure}%
\setcounter{subfigure}{0}
\\
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_7b}
\caption{Voronoi Tesselation ($d_{crit}=0.0$)}
\end{subfigure}
\quad
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_7c}
\caption{Warnken-Reed ($\alpha=2.0$)}
\label{wr}
\end{subfigure}
\quad
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_7d}
\caption{Voronoi Warnken-Reed ($\alpha=2.0$)}
\label{mwr}
\end{subfigure}
\caption{Local dendrite arm spacing ($\mu$m) for the three techniques not shown in Figure~\ref{dendrite}: (a) Voronoi tessellation with edge length threshold of $d_{crit}$=0.0, (b) Warnken--Reed technique with $\alpha=2.0$, and (c) Voronoi Warnken-Reed with $\alpha=2.0$.}
\label{dendrite2}
\end{figure}
The local dendrite arm spacing for the remaining three techniques is shown in Figure~\ref{dendrite2}. The same color bar for local PDAS as in Figure~\ref{dendrite} is used here. First, notice that the Voronoi tessellation-based technique with an edge length threshold of $d_{crit} = 0.0$ has a much larger fraction of dendrite cores with PDAS greater than the bulk PDAS than below the bulk PDAS (83.5\% above 248.4 $\mu$m). Clearly, the local primary dendrite arm spacing is overpredicted in this case. The Warnken--Reed and Voronoi Warnken--Reed methods are shown in Figures~\ref{wr} and \ref{mwr}. At first glance, a majority of the local PDAS values are very similar between the two methods ($\sim$79\%). However, $\sim$21\% of the cores resulted in a difference between the two techniques, which is caused by the original Warnken--Reed method using neighbors outside of those FNNs identified from the Voronoi polygons. In every case, the Warnken--Reed method resulted in higher local PDAS values than the Voronoi Warnken--Reed method, as would be expected since this is purely a distance-based method and subsequent additions can only increase the local PDAS.
\begin{figure}[bht!]
\centering
\begin{subfigure}[b]{0.75\textwidth}
\centering
\includegraphics[trim=0 225 0 0, clip,width=\textwidth]{Figure_8a}
\phantomsubcaption
\end{subfigure}%
\setcounter{subfigure}{0}
\\
\begin{subfigure}[b]{0.75\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_8b}
\caption{Primary dendrite arm spacing ($\mu$m) difference}
\end{subfigure}
\caption{Difference in the local dendrite arm spacing ($\mu$m) between the Warnken--Reed and Voronoi Warnken--Reed techniques with $\alpha=2.0$. The Warnken--Reed technique had a greater PDAS value in every case ($\sim$21\% of dendrite cores are different).}
\label{dendrite3}
\end{figure}
Figure \ref{dendrite3} shows the difference in local PDAS values between the two techniques. In several cases, the difference is greater than 250 $\mu$m and/or 100\% of the PDAS value quantified by the Voronoi Warnken-Reed method. The differing dendrite cores is approximately an equal percentage for edge dendrites as well as interior dendrite cores. For some cases, it is apparent that one of the closest three dendrite cores is significantly closer or further away than the other two, thereby resulting in a larger standard deviation $d_{std}$ and a greater chance to add multiple neighbors; this case is similar to that shown in Figure \ref{various_methods2}.
\subsection{Local primary dendrite arm spacing statistics}
The local dendrite arm spacing statistics are also calculated for the interior dendrite cores to compare with the traditional PDAS measurement. The cumulative distribution function plot for the local dendrite arm spacing is shown in Figure~\ref{probability} for the three different techniques over a range of parameter values, which are given in the legend. The bulk PDAS measurement is shown as a vertical black line and the hexagonal star shows the 50$^\textrm{th}$ percentile intersection point. The local dendrite arm spacing distributions are characterized in terms of mean, standard deviation, skewness, and kurtosis (Table \ref{table1}), while the coordination number distributions are characterized in terms of their mean and the percentages of 3, 4, 5, 6, and 7+ nearest neighbors (Table \ref{table2}). The skewness and kurtosis measure the asymmetry of the distribution and the sharpness of the peak/thickness of the tail, respectively. The skewness and kurtosis are 0 and 3, respectively, for a normal distribution.
\begin{figure}[bht!]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_9a}
\caption{}
\end{subfigure}%
\quad
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_9b}
\caption{}
\end{subfigure}
\caption[]{Probability distribution functions for the various local characterization methods compared within for the internal dendrites within the dendritic microstructure shown in Figure~\ref{sx_nickel}. The four different techniques are compared with the bulk PDAS for a range of parameter values. The upper bound of the parameter range for each technique is shown as a dotted line. To facilitate the comparison, the Warnken--Reed and the Voronoi technique ($d_{crit}=0.0$) are shown in (a), and the remaining Voronoi-modified techniques are shown in (b).}
\label{probability}
\end{figure}
There are distinct differences between the local dendrite arm spacing distributions calculated by the four techniques (Figure~\ref{probability}, Tables \ref{table1}). The Warnken--Reed and Voronoi Warnken--Reed are compared initially. In the case of the Warnken--Reed method, the PDAS distribution is shifted towards large PDAS values at high $\alpha$ values (a positive skewness value gives a long tail) and has a sharper peak and a longer, fatter tail (high kurtosis values), more so than the other methods. This skewness is caused by an overestimation of the number of nearest neighbors in some cases, due to large values of either $d_{std}$ or the parameter $\alpha$. Hence, while the calculated mean PDAS can approach the bulk-measured value of 248.4 $\mu$m (within 0.1\% for $\alpha$=1.8), this mean PDAS is highly sensitive to these large PDAS values. This overprediction of nearest neighbors, and their result on the local PDAS distribution, is also apparent by comparing this with the Voronoi Warnken--Reed method, whereby the potential nearest neighbors are restricted to only those FNNs defined by the Voronoi polygon. In this case, there is a lack of a long tail and the skewness/kurtosis of the distribution tends more towards normality. However, the calculated mean PDAS with this method tends to underestimate the bulk-measured PDAS. While the maximum number of nearest neighbors (8 for $\alpha \ge 1.2$) is more realistic, a large percentage of dendrite cores are predicted to have only 3 nearest neighbors, even in the case of a large $\alpha$ parameter (48.6\% for $\alpha =2.0$). It is also interesting that increasing the $\alpha$ parameter for the case of the Voronoi Warnken--Reed method tends to shift the slope of the probability distribution function without affecting either the minimum or maximum local dendrite arm spacings.
For comparison, the minimal spanning tree method (Fig.~\ref{MST}) was also included in Table \ref{table1}. Not surprisingly, the mean distance of the connecting line segments is much shorter than the bulk calculated PDAS using Eq.~\ref{lambda} with $c=1$. Remember that the MST method is composed of the shortest line segments to connect all dendrites. Both the MST standard deviation and kurtosis values are larger than the Voronoi tesselation method (for all $d_{crit}$) and the Voronoi Warnken-Reed method (for all $\alpha$), indicating a wider distribution and a larger deviation of the distribution from normality (kurtosis = 3). Moreover, the distribution is skewed towards a larger tail at lower distances (negative skewness) unlike the other techniques, which again is associated with the selection of the shortest line segments to characterize the spacing.
\begin{table}[ht]
\centering
\footnotesize
\caption{Local primary dendrite arm spacing statistics}
\begin{tabular}{c c c . c cc}
\addlinespace
\toprule
\multicolumn{1}{c}{} &
\multicolumn{1}{c}{} &
\multicolumn{5}{c}{Primary Dendrite Arm Spacing} \\
\multicolumn{1}{c}{Method} &
\multicolumn{1}{c}{Parameter} &
\multicolumn{1}{c}{mean, $\mu$m} &
\multicolumn{1}{c}{diff, \%} &
\multicolumn{1}{c}{std, $\mu$m} &
\multicolumn{1}{c}{skewness} &
\multicolumn{1}{c}{kurtosis} \\
\midrule
Bulk (Eq.~\ref{lambda}, $c=1$) & - & 248.4 & 0.0 & - & - & - \\
Voronoi Tesselation & $d_{crit}$ = 0.00 & 272.9 & 9.9 & 28.0 & 0.1 & 3.3 \\
Voronoi Tesselation & $d_{crit}$ = 0.02 & 270.0 & 8.7 & 27.2 & 0.1 & 3.2 \\
Voronoi Tesselation & $d_{crit}$ = 0.04 & 266.4 & 7.2 & 26.8 & 0.1 & 3.1 \\
Voronoi Tesselation & $d_{crit}$ = 0.06 & 263.0 & 5.9 & 25.7 & 0.1 & 3.1 \\
Voronoi Tesselation & $d_{crit}$ = 0.08 & 258.4 & 4.0 & 26.1 & 0.2 & 3.1 \\
Voronoi Tesselation & $d_{crit}$ = 0.10 & 253.3 & 2.0 & 25.3 & 0.2 & 3.0 \\
Voronoi Tesselation & $d_{crit}$ = 0.12 & 247.6 & -0.3 & 26.0 & 0.2 & 2.9 \\
Voronoi Warnken--Reed & \ensuremath{\alpha} = 1.0 & 230.0 & -7.4 & 25.0 & 0.4 & 3.5 \\
Voronoi Warnken--Reed & \ensuremath{\alpha} = 1.2 & 230.9 & -7.0 & 26.0 & 0.4 & 3.4 \\
Voronoi Warnken--Reed & \ensuremath{\alpha} = 1.4 & 232.7 & -6.3 & 27.4 & 0.4 & 3.2 \\
Voronoi Warnken--Reed & \ensuremath{\alpha} = 1.6 & 236.1 & -5.0 & 29.5 & 0.4 & 3.1 \\
Voronoi Warnken--Reed & \ensuremath{\alpha} = 1.8 & 239.0 & -3.8 & 30.5 & 0.4 & 3.0 \\
Voronoi Warnken--Reed & \ensuremath{\alpha} = 2.0 & 242.2 & -2.5 & 31.8 & 0.3 & 3.0 \\
Warnken--Reed & \ensuremath{\alpha} = 1.0 & 230.1 & -7.4 & 25.1 & 0.3 & 3.4 \\
Warnken--Reed & \ensuremath{\alpha} = 1.2 & 231.8 & -6.7 & 26.9 & 0.5 & 3.5 \\
Warnken--Reed & \ensuremath{\alpha} = 1.4 & 234.5 & -5.6 & 29.4 & 0.6 & 3.8 \\
Warnken--Reed & \ensuremath{\alpha} = 1.6 & 239.2 & -3.7 & 33.0 & 0.8 & 4.0 \\
Warnken--Reed & \ensuremath{\alpha} = 1.8 & 248.1 & -0.1 & 48.1 & 1.9 & 8.3 \\
Warnken--Reed & \ensuremath{\alpha} = 2.0 & 259.2 & 4.3 & 64.2 & 1.9 & 6.5 \\
Minimal spanning tree & N/A & 215.2 & -13.4 & 34.1 & -0.5 & 4.8 \\
\bottomrule
\end{tabular}%
\label{table1}%
\end{table}%
The Voronoi tessellation techniques are also compared. First, quantifying the coordination number and the local PDAS values using all FNNs identified by the Voronoi tessellation polygons ($d_{crit}=0$) clearly overestimates both measures; mean PDAS is $\sim$10\% off from the bulk-measured PDAS value and $\sim$20\% of dendrite cores have more than 6 nearest neighbors. As the edge length threshold parameter increases, less nearest neighbors are identified and the calculated mean PDAS approaches the bulk-measured PDAS value (within 0.3\% for $d_{crit}=0.12$). For $d_{crit}=0.12$, the majority of dendrite cores have 4 nearest neighbors ($>$50\%), followed by 5 nearest neighbors (26.9\%) and 3 nearest neighbors (17.3\%). Moreover, the local PDAS distribution has a low skewness value (0.2) and a kurtosis of 2.9, indicating an approximately normal distribution. In general, the Voronoi tessellation-based technique with an edge length threshold criterion of $d_{crit}=0.12$ tends to give the best agreement in terms of both bulk-measured PDAS and coordination number. Furthermore, this technique allows for calculating the local PDAS value and the local PDAS distribution, which may be important for assessing the homogeneity of dendrite growth and/or for identifying local regions where the local growth conditions/properties are different from the norm.
\begin{table}[ht]
\centering
\footnotesize
\caption{Local coordination number statistics}
\begin{tabular}{c c . . . . . c}
\toprule
\multicolumn{1}{c}{} &
\multicolumn{1}{c}{} &
\multicolumn{5}{c}{Coordination Number (\%)} &
\multicolumn{1}{c}{} \\
\multicolumn{1}{c}{Method} &
\multicolumn{1}{c}{Parameter} &
\multicolumn{1}{c}{3} &
\multicolumn{1}{c}{4} &
\multicolumn{1}{c}{5} &
\multicolumn{1}{c}{6} &
\multicolumn{1}{c}{\ensuremath{\ge} 7} &
\multicolumn{1}{c}{Mean} \\
\midrule
Voronoi Tesselation & $d_{crit}$ = 0.00 & 0.0 & 2.5 & 20.4 & 57.3 & 19.8 & 5.98 \\
Voronoi Tesselation & $d_{crit}$ = 0.02 & 0.0 & 4.0 & 26.0 & 57.0 & 13.0 & 5.80 \\
Voronoi Tesselation & $d_{crit}$ = 0.04 & 0.0 & 7.4 & 34.7 & 51.4 & 6.5 & 5.57 \\
Voronoi Tesselation & $d_{crit}$ = 0.06 & 0.0 & 11.8 & 45.2 & 39.9 & 3.1 & 5.35 \\
Voronoi Tesselation & $d_{crit}$ = 0.08 & 1.5 & 24.1 & 49.2 & 24.8 & 0.3 & 4.98 \\
Voronoi Tesselation & $d_{crit}$ = 0.10 & 5.3 & 43.0 & 41.5 & 10.2 & 0.0 & 4.57 \\
Voronoi Tesselation & $d_{crit}$ = 0.12 & 17.3 & 52.3 & 26.9 & 3.4 & 0.0 & 4.16 \\
Voronoi Warnken--Reed & \ensuremath{\alpha} = 1.0 & 94.1 & 4.6 & 0.9 & 0.0 & 0.3 & 3.08 \\
Voronoi Warnken--Reed & \ensuremath{\alpha} = 1.2 & 87.6 & 8.7 & 2.8 & 0.3 & 0.6 & 3.18 \\
Voronoi Warnken--Reed & \ensuremath{\alpha} = 1.4 & 79.3 & 11.8 & 5.3 & 2.5 & 1.2 & 3.35 \\
Voronoi Warnken--Reed & \ensuremath{\alpha} = 1.6 & 65.9 & 18.0 & 8.0 & 5.3 & 2.8 & 3.62 \\
Voronoi Warnken--Reed & \ensuremath{\alpha} = 1.8 & 56.3 & 19.8 & 12.1 & 8.4 & 3.4 & 3.84 \\
Voronoi Warnken--Reed & \ensuremath{\alpha} = 2.0 & 48.6 & 19.8 & 14.9 & 11.5 & 5.3 & 4.07 \\
Warnken--Reed & \ensuremath{\alpha} = 1.0 & 91.6 & 7.1 & 0.9 & 0.0 & 0.3 & 3.10 \\
Warnken--Reed & \ensuremath{\alpha} = 1.2 & 83.9 & 9.6 & 4.0 & 1.2 & 1.2 & 3.27 \\
Warnken--Reed & \ensuremath{\alpha} = 1.4 & 72.8 & 14.6 & 7.1 & 2.8 & 2.8 & 3.52 \\
Warnken--Reed & \ensuremath{\alpha} = 1.6 & 58.5 & 21.7 & 9.0 & 4.0 & 6.8 & 3.89 \\
Warnken--Reed & \ensuremath{\alpha} = 1.8 & 52.0 & 19.2 & 10.8 & 6.2 & 11.8 & 4.63 \\
Warnken--Reed & \ensuremath{\alpha} = 2.0 & 44.0 & 19.2 & 12.1 & 7.1 & 17.6 & 5.55 \\
\bottomrule
\end{tabular}%
\label{table2}%
\end{table}%
\subsection{Correlation with interdendritic features}
The relationship between the occurrence of interdendritic features (e.g., pores or eutectic particles) and the local dendrite arm spacing (or distance from cores, etc.) can provide insight into the importance of quantifying the local spacings. We have examined how these metrics may relate to the formation of eutectic particles in this work by first segmenting the interdendritic particles and then computing probability distribution functions.
The eutectic particles in Figure \ref{sx_nickel} were segmented using the following process. The particles were segmented by first leveling the intensity of the micrograph using a cubic polynomial with interaction terms. This step ensured that there wasn't a shift in contrast from one side of the micrograph to the other (due to uneven etching, etc.). The threshold intensity was then selected by maximizing the difference in the mean intensity between the two distributions (eutectic particle and matrix). Then, a size threshold was enforced by discarding eutectic particles with less than 5 pixels (1 pixel $\sim{1.7}$ $\mu$m, i.e., 5 pixels = $15.2$ $\mu$m$^2$). As an example of the segmentation, Figure \ref{subimages_a} shows a 1 mm x 1 mm region from Figure \ref{sx_nickel} and Figure \ref{subimages_b} shows the corresponding binary image of the segmented eutectic particles (in white).
The Euclidean distance to the nearest dendrite core and the nearest Voronoi vertex was then calculated for each pixel within the micrograph. The Euclidean distance is the distance from each pixel to the nearest feature, which in this case is either the centroids of the dendrite cores or the Voronoi vertices, and this metric is repeated over all pixels within the image to create a map. As an example, Figure \ref{subimages_c} shows the Euclidean distance map for the same 1 mm x 1 mm area utilizing the dendrite core centroids identified in Figure \ref{sx_nickel}. The darker intensity indicates closer Euclidean distances to the dendrite core and the lightest pixels between the dendrite cores actually correspond to the boundaries of the Voronoi tesselation.
\begin{figure}[bht!]
\centering
\begin{subfigure}[b]{0.31\textwidth}
\includegraphics[width=\textwidth]{Figure_10a}
\caption{Original image}
\label{subimages_a}
\end{subfigure}%
\quad
\begin{subfigure}[b]{0.31\textwidth}
\includegraphics[width=\textwidth]{Figure_10b}
\caption{Segmented image}
\label{subimages_b}
\end{subfigure}
\quad
\begin{subfigure}[b]{0.31\textwidth}
\includegraphics[width=\textwidth]{Figure_10c}
\caption{Euclidean map}
\label{subimages_c}
\end{subfigure}
\caption[]{(a) A 1 mm x 1 mm subregion from Figure \ref{sx_nickel} is shown along with two corresponding images of the same area: (b) a binary image with segmented eutectic particles (white) and (c) a Euclidean distance map from the dendrite core centroids, where lighter intensity refers to further distances from the dendrite cores.}
\label{subimages}
\end{figure}
The probability of encountering (or forming) a eutectic particle can then be calculated as a function of this Euclidean distance from the nearest dendrite core or the Voronoi vertex, as shown in Figure \ref{pdf}. Based on the image segmentation, the area fraction of eutectic particles in Figure \ref{sx_nickel} is 3.6\% and is shown as a red line in Figure \ref{pdf}. The pixels lying within 100 $\mu$m of the image boundaries were excluded to eliminate the possibility that dendrite cores just outside of the field of view could affect the statistics. As can be seen from Fig.~\ref{pdf_a}, the left (right) blue line indicates the distance whereby all distances below (above) have a probability of having a eutectic particle that is lower (higher) than the global area fraction (red line), i.e., it is less (more) favorable for a eutectic particle to form. The transition distance of eutectic particle favorability is between 86-93 $\mu$m, i.e., approximately ${1/3}$ of the primary dendrite arm spacing (248.4 $\mu$m). This plot shows that it is not favorable for eutectic particles to form close to the primary dendrite core.
Figure \ref{pdf_b} is a similar plot as a function of distance from the vertices of the Voronoi tessellation (see schematic). This plot was generated in a similar manner to Figure \ref{pdf_a}; a Euclidean distance map was first formed from the Voronoi vertices, then the boundary pixels within 100 $\mu$m of the image boundaries were excluded, etc. There is an increased occurrence of eutectic particles at vertices, regardless of their distance from the dendrite core. This observation (along with the fact that the probability of occurence is higher than in Figure \ref{pdf_a} by almost 2\%) suggests that solute is forced near the Voronoi vertices, thereby increasing the probability of eutectic particle occurrence. The transition distance in this plot is between 67-90 $\mu$m, i.e., at approximately ${1/3}$ of the primary dendrite arm spacing. While this analysis shows the preferential formation of eutectic particles based on the local distances, correlation with the size of particles is also important.
\begin{figure}[bht!]
\centering
\begin{subfigure}[b]{0.475\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_11a}\llap{\raisebox{5cm}{\includegraphics[height=2cm]{Figure_11b}}}
\caption{}
\label{pdf_a}
\end{subfigure}%
\quad
\begin{subfigure}[b]{0.475\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_11c}\llap{\raisebox{4.8cm}{\includegraphics[height=2.25cm]{Figure_11d}}}
\caption{}
\label{pdf_b}
\end{subfigure}
\caption[]{The probability of a eutectic particle as a function of the distance to (a) the nearest dendrite core or (b) the nearest Voronoi vertex. The inset schematic shows the refence point(s) for the Euclidean distance in each plot.}
\label{pdf}
\end{figure}
The eutectic particle size may be correlated with the distance from the dendrite cores or Voronoi vertices as well. Figure \ref{a50} shows the eutectic particle size as a function of the distance from the nearest dendrite core and the nearest Voronoi vertice. The solid line shows the 50$^{th}$-percentile area, $A_{50}=410$ $\mu{m}^2$, which refers to the eutectic particle size where 50\% of the eutectic particle area lies above/below this size. There is a noticeable tendency for the larger particles ($A>A_{50}$) to form further away from the dendrite core and closer to the Voronoi vertices, while the smaller eutectic particles ($A<A_{50}$) can form at all distances. However, it is difficult to quantitatively tell from the following plot what the preference is for smaller or larger particles as a function of distance. Therefore, to further quantify this relationship with respect to the size of the particles, the probability associated with a eutectic particle pixel belonging to either a small or large particle is calculated in Figure \ref{size}. Interestingly, in Figure \ref{size_a}, at distances closer to the dendrite cores, there is a clear preference for smaller particles ($A<A_{50}$) to form over larger particles ($A>A_{50}$). At a distance of 84.5 $\mu{m}$ ($\sim{1/3}$ PDAS), as denoted by the solid line, there is a crossover in the probability function and larger particles are statistically favored to form over smaller particles. In the case of distances from the Voronoi vertices, there is a similar behavior except that \textit{larger} particles are favored at smaller distances (closer to Voronoi vertices). The crossover in the probability functions occurs at 79.3 $\mu{m}$ ($\sim{1/3}$ PDAS again). At distances greater than this, there is not as definitive of a trend as with the dendrite cores, i.e., in some cases, there is a greater probability for smaller particles to form and, in some cases, for larger particles to form. This lack of a well-defined trend at larger distances may be caused by the fact that these larger distances could lie close or far away from the dendrite core, further obscuring the trend. Clearly, the distance from the dendrite cores and, hence, the local primary dendrite arm spacing affect the probabilities of interdendritic particles to form, though. In a similar manner, it is anticipated that a similar relationship may be associated with shrinkage porosity, gas porosity, and other interdendritic defects.
\begin{figure}[bht!]
\centering
\begin{subfigure}[b]{0.475\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_12a}
\caption{}
\label{a50_a}
\end{subfigure}%
\quad
\begin{subfigure}[b]{0.475\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_12b}
\caption{}
\label{a50_b}
\end{subfigure}
\caption[]{The eutectic particle size as a function of the distance to (a) the nearest dendrite core or (b) the nearest Voronoi vertex. The distance for each particle is the distance for the particle centroid. The 50$^{th}$-percentile area, $A_{50}=410$ $\mu{m}^2$, refers to the particle size where 50\% of the eutectic particle area lies above/below this size.}
\label{a50}
\end{figure}
\begin{figure}[bht!]
\centering
\begin{subfigure}[b]{0.475\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_13a}
\caption{}
\label{size_a}
\end{subfigure}%
\quad
\begin{subfigure}[b]{0.475\textwidth}
\centering
\includegraphics[width=\textwidth]{Figure_13b}
\caption{}
\label{size_b}
\end{subfigure}
\caption[]{The probability of a eutectic particle of a certain size occurring as a function of the distance to (a) the nearest dendrite core or (b) the nearest Voronoi vertex. Two particle sizes are considered: particle sizes below and above the 50$^{th}$-percentile area $A_{50}$. The solid line denotes the distance at which the probability functions first intercept, indicating a transition fromthe favorability of small particles to large particles (in \ref{size_a}) or vice versa (in \ref{size_b}).}
\label{size}
\end{figure}
\section{Conclusions}
In summary, characterizing the primary dendrite arm spacing in directionally-solidified microstructures is an important step for developing process-structure-property relationships by enabling the quantification of (i) the influence of processing on microstructure and (ii) the influence of microstructure on properties. Thin-walled directionally-solidified structures (e.g., a turbine blade) require new approaches for characterizing the dendrite arm spacing and the microstructure. In this work, we utilized a new Voronoi-based approach for spatial point pattern analysis that was applied to an experimental dendritic microstructure. This technique utilizes a Voronoi tessellation of space surrounding the dendrite cores to determine nearest neighbors and the local primary dendrite arm spacing. In addition, we compared this technique to a recent distance-based technique, the Warnken--Reed method, and a modification to this using Voronoi tesselations, along with the minimal spanning tree method. Moreover, a convex hull-based technique was used to include edge effects for such techniques, which can be important for thin specimens. These methods were used to quantify the distribution of local primary dendrite arm spacings as well as their spatial distribution for an experimental directionally-solidified superalloy micrograph. Last, eutectic particles were segmented to correlate distances from dendrite cores and Voronoi vertices to the occurence and size of these interdendritic features. Interestingly, with respect to the distance from the dendrite core, it was found that there is a greater probability of occurence of large eutectic particles ($>410$ $\mu$m) over small particles at distances greater than approximately ${1}/{3}$ of the bulk-measured primary dendrite arm spacing. In conclusion, this systematic study of the different techniques for quantifying local primary dendrite arm spacings, and their effect on microstructure, can be an important step for correlating with both processing and properties in single crystal nickel-based superalloys.
\section*{Acknowledgments}
MAT would like to acknowledge AFOSR for support for this research through contract FA9550-12-1-0135 (PM: Dr. David Stargel, AFOSR/RSA). MAT would like to acknowledge support from the U.S. Army Research Laboratory (ARL) administered by the Oak Ridge Institute for Science and Education through an interagency agreement between the U.S. Department of Energy and ARL.
\bibliographystyle{unsrt}
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\section{Introduction}
The components of the proton spin is one of key questions in the nuclear physics.
It has been proposed to use W boson production to decompose anti quark
contributions \cite{Wprogram}. In leading order process,
positive (negative) W boson is connected with u and $\bar{\rm d}$ (d and $\bar{\rm u}$) quarks.
Because of the parity violating process, it provides a unique way to access the
flavor dependence of polarized quarks in the proton.
It is a complementary approach to semi inclusive deep inelastic interaction experiments,
where the quark flavor is identified only via the fragmentation process.
In 2009, RHIC provided the first polarized proton proton collisions at $\sqrt{s}=500$GeV.
The proton polarization is achieved to be 39\% for both beams.
In this article, we report the observation of W boson production
and the measurement of
single longitudinal spin asymmetry of
$W^+$ production at RHIC-PHENIX.
\section{RHIC-PHENIX}
The data were collected by the PHENIX detector \cite{PHENIXNIM}.
The central arm detector covers a range of $|\eta|<0.35$ in pseudo-rapidity and 2 times 90$^\circ$ in azimuth.
The primary detector for this measurement is an electromagnetic calorimeter (EMCal). Each calorimeter tower covers $\Delta \eta \times \Delta \phi \sim 0.01 \times 0.01$. A tracking system consist of a drift chamber
and a pad chamber is used to identify the charge sign.
Events were collected with the EMCal trigger, which is fully efficient
at 12 GeV of the transverse energy. The calibration of EMCal energy
scale was done with two photons' invariant mass for $\pi^0$ and $\eta$
particles. For the tracking system, the detector position was
calibrated with zero magnetic field data. The angular resolution
of the tracking system was also checked with zero magnetic field data.
Compared with the bending angle from the magnetic filed, there is
2.1 $\sigma$ charge separation capability for 40 GeV/$c$ particles.
A coincidence of beam-beam counters (BBC) positioned at pseudo-rapidities
$3.1<|\eta|<3.9$ was used as the luminosity monitor.
The conversion factor from the count rate to the luminosity
was obtained via the van der Meer scan technique
\cite{vernierscan}, which measures the transverse profile of the beam
overlap.
A correction was necessary for the event overlap in a single crossing.
An integrated luminosity of 8.6 $\rm{pb}^{-1}$ after a vertex cut of
$\pm 30$ cm is used in this analysis.
\section{W boson signal}
In this analysis, W boson is tagged by its decay electron.
Electron candidates were selected from clusters in the EMCal within
$\pm 10$ mrad of the transverse position projected by a
charged track found in the central drift chamber.
It applied loose cuts on the time measured in the EMCal and track momentum not to
be much lower than the deposit energy in the EMCal.
Figure \ref{fig:spectra_pm} shows the transverse momentum spectra.
The transverse momentum ($p_T$) is calculated from the energy deposit
in the EMCal.
The histogram is overlaid with curves of background (QCD events)
and signal (weak boson decays).
Because decay electrons from Z bosons have
similar spectra to the ones from W bosons, they can not be separated
in the current PHENIX detector acceptance.
The main contributions from the QCD background are charged hadron
clusters, photons from hadrons' decay
converted to electrons before the tracking system, and some are from
track mis-association in the same jet event.
These backgrounds are estimated from all EMCal cluster distribution
for the photon contribution and the NLO pQCD distribution
folded by the EMCal response for the charged hadron contribution.
First the EMCal distribution was multiplied by the probability
of track association. Second the charged hadron distribution was scaled,
so that the range from 10 to 20 GeV/$c$ is explained by the sum of them.
The signal shape
is the decay electrons from W and Z bosons taken from PYTHIA MC
smeared by the EMCal resolution.
The dominant systematic uncertainty in the background estimation is
from the conversion probability. It has to
estimate not only for a single photon conversion, but also for a probability of two photons,
because two high energy clusters from $\pi^0$ decay merge in the EMCal.
A significant excess corresponding to the W boson Jacobian peak is observed in the spectra.
\begin{figure}[h]
\includegraphics[width=18pc]{spectra_pm.eps}\hspace{2pc}%
\begin{minipage}[b]{18pc}\caption{\label{label} Transverse momentum spectra of EMCal clusters associated with a track in the PHENIX central arm detector. It is overlaid with curves of background and signal. The region of 10-20GeV is used for the normalization.}
\label{fig:spectra_pm}
\end{minipage}
\end{figure}
\section{Single longitudinal spin asymmetry}
For the spin asymmetry measurement, an isolation cut was applied
to increase the signal to noise ratio. The cut is to require
less than 2 GeV of energy deposit around the electron candidate.
This cut should not depend on the spin state.
Figure \ref{fig:wiso_pos} and Fig. \ref{fig:wiso_neg} show the
spectra of inclusive and the one with
the isolation cut for positive and negative particles.
It is seen the background component from the QCD events
($p_T<25{\rm GeV}/c$) is suppressed by a factor of about 4,
and the signal from W boson ($p_T>30{\rm GeV}/c$) is mostly remained.
This is another evidence for the W boson signal is in the sample.
The single longitudinal spin asymmetry is defined as
$A_L^W=\frac{1}{P}\cdot \frac{N^+(W)-N^-(W)}{N^+(W)+N^-(W)}$,
where $P$ is the beam polarization, and $N^\pm$ is the number of
signals normalized by the integrated luminosity
in positive and negative helicity beam.
For the luminosity measurement,
number of BBC coincidence was used.
Since there are two beams polarized at RHIC,
the same sample can be used twice. Therefore the statistical uncertainty
follows
$\delta A_L=\frac{1}{P}\cdot\frac{1}{\sqrt{2(N^++N^-)}}$.
In the actual calculation, the sample was divided into 4 spin states
(2 beams $\times$ 2 spin states), then a
simultaneous fit was applied to get raw asymmetries ($\equiv P\cdot A_L$).
Table \ref{table:rawasym} shows the raw asymmetry of positive particles. It also shows the background
region for a sanity check expecting the asymmetry to be 0.
For the physics asymmetry of W bosons, it has to be corrected for
the contribution of Z boson
and QCD background in the sample. Those work as a dilution factor.
The QCD background contribution is estimated to
be $1\pm1$ event from the extrapolation of the lower part of
the spectra.
The ratio of Z boson to W boson is taken from PYTHIA MC ((W+Z)/W=1.08).
Figure \ref{fig:asym_plus} shows the physics asymmetry of $W^+$ production after the correction.
The systematic uncertainty is from the absolute polarization measurement ($\delta P/P$=9.2\%) and
the estimation of dilution effect ($\sim$4\%).
The asymmetry is consistent with predictions of various polarized
PDF parametrization within
the uncertainty and it is 2.7$\sigma$ away from 0.
\begin{figure}[h]
\begin{minipage}{18pc}
\includegraphics[width=18pc]{wiso_pos.eps}
\caption{\label{label} $p_T$ spectra of positive particles with and without the isolation cut.}
\label{fig:wiso_pos}
\end{minipage}\hspace{2pc}%
\begin{minipage}{18pc}
\includegraphics[width=18pc]{wiso_neg.eps}
\caption{\label{label} $p_T$ spectra of negative particles with and without the isolation cut.}
\label{fig:wiso_neg}
\end{minipage}
\end{figure}
\begin{table}
\centering
\caption{\label{book}Single longitudinal spin asymmetry for positive particles}
\begin{tabular}{@{}l*{15}{l}}
\br
$p_T$ Range [GeV/$c$]&Raw asymmetry \\
\mr
12-20 (Background) & $0.035\pm0.047$ \\
30-50 (Signal) & $-0.29\pm0.11$ \\
\br
\end{tabular}
\label{table:rawasym}
\end{table}
\begin{figure}[h]
\includegraphics[width=18pc]{asym_plus.eps}\hspace{2pc}%
\begin{minipage}[b]{18pc}\caption{\label{label} Longitudinal single spin asymmetry of $W^+ \rightarrow e^+$ in the mid rapidity region. It is compared with calculations from various polarized parton distribution functions.
(GRSV standard, GRSV valence \cite{GRSV}, DSSV \cite{DSSV}, and DNS \cite{DNS} using a maximal and minimal sea polarization scenario.)}
\label{fig:asym_plus}
\end{minipage}
\end{figure}
\section{Summary}
The W boson production is observed through its decay electrons at
RHIC-PHENIX in the data
collected in 2009 ($\int{L}=8.6$/pb, P=39\%). The single longitudinal
spin asymmetry of $W^+$ boson is measured for the first time and
it is consistent with various predictions within the uncertainty.
\section{Outlook}
The RHIC spin program is planning to accumulate $\sqrt{s}=500$ GeV
collision data for the next few years. There is also a plan to
measure W boson to muon channel in the forward muon detectors.
It has better
sensitivity to the polarized $\bar{\rm u}$ distribution than the central
arm measurement because of its decay kinematics.
A detector upgrade to improve the trigger capability
is the major challenge in the next data taking.
The measurement of beam polarization will be an
important systematics for non-zero helicity asymmetries.
The systematic uncertainty has been already achieved at the level of
$\sim5\%$ in the absolute polarization measurement in the past
200GeV data periods. It is expected to be the same level in the next
500GeV data period.
\section*{References}
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\section{Introduction}
The Dirac phase $\d$, which represents CP violation in the lepton sector, has been measured recently. Although the phase $\d$ is not determined experimentally,
a generalized CP symmetry (GCP) \cite{Ecker:1981wv, Ecker:1983hz, Gronau:1985sp, Ecker:1987qp,Neufeld:1987wa,Ferreira:2009wh,Feruglio:2012cw,Holthausen:2012dk,Ding:2013bpa,Girardi:2013sza,Nishi:2013jqa,Ding:2013hpa,Feruglio:2013hia,Chen:2014wxa,Ding:2014ora,Ding:2014hva,Chen:2014tpa,Li:2015jxa,Turner:2015uta, Rodejohann:2017lre, Penedo:2017vtf, Nath:2018fvw} can fix this CP violating phase.
One of a notable example is the $\m-\t$ reflection symmetry
\cite{Harrison:2002et,Grimus:2003yn, Grimus:2005jk, Farzan:2006vj, Joshipura:2007sf, Adhikary:2009kz, Joshipura:2009tg, Xing:2010ez, Ge:2010js, He:2011kn, Gupta:2011ct, Grimus:2012hu, He:2012yt, Joshipura:2015dsa, Xing:2015fdg, He:2015afa, Chen:2015siy, He:2015xha, Samanta:2017kce, Xing:2017cwb, Nishi:2018vlz, Nath:2018hjx, Sinha:2018xof, Huang:2018fog, Xing:2019edp, Pan:2019qcc, Chakraborty:2019rjc, Liao:2019qbb, Yang:2020qsa, Duarah:2020zjo, Zhao:2021dwc, Bao:2022kon}
that predicts the maximal Dirac phase $\d = \pm \pi/2$.
On the other hand, diagonal reflection symmetries \cite{ Yang:2020goc, Yang:2021smh, Yang:2021xob, Yang:2022gcs}
are GCPs that can predict relatively small $\d$ in a way that can unify quarks and leptons.
However, general properties of the symmetries are not yet well understood.
Thus, in this letter, we perform a systematic analysis of the DRS.
\section{Diagonal reflection symmetries}
In this section, we define the diagonal reflection symmetries (DRS).
First, a representation of the CKM matrix proposed by Fritzsch and Xing is \cite{Fritzsch:1997fw},
\begin{align}
V_{\rm CKM} = U_{u}^{\dagger} \def\del{\partial} \def\nn{\nonumber} U_{d} =
\begin{pmatrix}
c_{u} & s_{u} & 0 \\
- s_{u} & c_{u} & 0 \\
0 & 0 & 1 \\
\end{pmatrix}
\begin{pmatrix}
e^{- i \phi} & 0 & 0 \\
0 & c_{q} & s_{q} \\
0 & - s_{q} & c_{q} \\
\end{pmatrix}
\begin{pmatrix}
c_{d} & - s_{d} & 0 \\
s_{d} & c_{d} & 0 \\
0 & 0 & 1 \\
\end{pmatrix} ,
\label{VCKM1}
\end{align}
where $s_{\th} \equiv \sin \th , c_{\th} \equiv \cos \th$.
By determining the four parameters $s_{q,u,d}$ and $\phi$ from the observables,
$\phi$ is almost equal to $\pi/2$.
Thus, if we interpret unitary matrices $U_{u,d}$ diagonalizing the mass matrix of quarks $m_{u,d}$ as
\begin{align}
U_{u} & =
\begin{pmatrix}
+i & 0 & 0 \\
0 & c_{t} & s_{t} \\
0 & - s_{t} & c_{t} \\
\end{pmatrix}
\begin{pmatrix}
c_{u} & -s_{u} & 0 \\
s_{u} & c_{u} & 0 \\
0 & 0 & 1 \\
\end{pmatrix}
, ~~~
U_{d} =
\begin{pmatrix}
1 & 0 & 0 \\
0 & c_{b} & s_{b} \\
0 & - s_{b} & c_{b} \\
\end{pmatrix}
\begin{pmatrix}
c_{d} & - s_{d} & 0 \\
s_{d} & c_{d} & 0 \\
0 & 0 & 1 \\
\end{pmatrix} , \label{Ud}
\end{align}
the mass matrices $m_{u,d} = U_{u,d} m^{\rm diag}_{u,d} U_{u,d}^{\dagger} \def\del{\partial} \def\nn{\nonumber}$ reconstructed from $U_{u,d}$
have diagonal reflection symmetries defined as \cite{Yang:2020goc}
\begin{align}
R \, m_{u,\n}^{*} \, R = m_{u,\n} \, , ~~~
m_{d,e}^{*} = m_{d,e} \, ~~~
R = {\rm diag} (-1 \, , 1 \, , 1) \, .
\end{align}
Under these symmetries, the mass matrix of charged leptons $m_{e}$ is real and the that of neutrinos $m_{\n}$ has the following form;
\begin{align}
m_{\n} =
\begin{pmatrix}
m_{11} & i m_{12} & i m_{13} \\
i m_{12} & m_{22} & m_{23} \\
i m_{13} & m_{23} & m_{33} \\
\end{pmatrix} \, ,
\end{align}
with $m_{ij} \in \mathbb R$.
These remnant symmetries are almost renormalization-invariant and are easily realized
by scalar fields with vacuum expectation values $\vev{\th_{u}} = i v_{u} \, , \vev{\th_{d}} = v_{d}$
that couple to only the first generation \cite{Yang:2021smh}.
Since the singular value decomposition of a real matrix is done
by a real orthogonal matrix $O_{f}$, the MNS matrix $U$ is
\begin{align}
U &=
O_{e}^{T}
\Diag{i}{1}{1}
O_{\n} P, \label{UVn0} \\
~~~
O_{\n} P &=
\begin{pmatrix}
1 & 0 & 0 \\
0 & c_{\n} & s_{\n} \\
0 & - s_{\n} & c_{\n} \\
\end{pmatrix}
\begin{pmatrix}
c_{13} & 0 & s_{13} \\
0 & 1 & 0 \\
- s_{13} & 0 & c_{13} \\
\end{pmatrix}
\begin{pmatrix}
c_{12} & s_{12} & 0 \\
- s_{12} & c_{12} & 0 \\
0 & 0 & 1 \\
\end{pmatrix}
\Diag{e^{i \phi_{1}}}{e^{i \phi_{2}}}{e^{i \phi_{3}}} .
\label{UVn}
\end{align}
Here, phases $\phi_{i} = 0$ or $\pi /2$ in the phase matrix $P$ originate from positive or negative singular mass values after a real diagonalization by $O_{\n}$.
Under an approximation that the 13 mixing of $m_{e}$ is negligible,
a combination of the 23 mixings of $O_{\n}$ and $O_{e}$ yields
a representation of the MNS matrix as
\begin{align}
U =
\begin{pmatrix}
c_{e} & s_{e}& 0 \\
- s_{e} & c_{e} & 0 \\
0 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
i & 0 & 0 \\
0 & c_{23} & s_{23} \\
0 & - s_{23} & c_{23} \\
\end{pmatrix}
\begin{pmatrix}
c_{13} & 0 & s_{13} \\
0 & 1 & 0 \\
- s_{13} & 0 & c_{13} \\
\end{pmatrix}
\begin{pmatrix}
c_{12} & s_{12} & 0 \\
- s_{12} & c_{12} & 0 \\
0 & 0 & 1 \\
\end{pmatrix} \Diag{e^{i \phi_{1}}}{e^{i \phi_{2}}}{e^{i \phi_{3}}}
\, .
\label{MNSmatrix}
\end{align}
For a hierarchical $m_{e}$, the error from this approximation is at most $\sqrt{m_{\m} m_{e}} / m_{\t} \simeq 0.004$ and is safely neglected.
Let us consider sign degrees of freedom for these parameters $c_{ij}$ and $s_{ij}$.
First, the sign of the phase $i$ can be fixed to positive because it is absorbed to $s_{e}$.
Furthermore, the five signs can be made positive by the redefinition of phases.
For example, the sign of $c_{12}$ can be changed by multiplying diag $(-1,-1, 1)$ from the right.
For the later convenience, we choose the signs of $c_{12}, c_{13}, c_{23}$ and $s_{13}, s_{23}$ to be positive.
The other two signs can also be determined from the signs of $\cos \d$ and $\sin \d$ for the Dirac phase $\d$.
From the following calculation, we can choose those of $s_{12}$ and $s_{e} c_{e}$.
However, it is found that the signs of $c_{e}$ and $s_{e}$ cannot be determined independently.
\section{Systematic analysis of $U_{\rm MNS}$}
In this section, we analyze the MNS matrix with DRS.
By neglecting the phase matrix $P$ and performing the product of the matrices, $U$ is represented as
\begin{align}
U =
\begin{pmatrix}
-s_{e} c_{23} s_{12}+c_{12} (-s_{e} s_{13} s_{23}+i c_{e} c_{13}) & s_{e} c_{12} c_{23} +s_{12} (-s_{e} s_{13} s_{23}+i c_{e} c_{13}) & s_{e} c_{13} s_{23} + i c_{e} s_{13} \\
- c_{e} c_{23} s_{12} + c_{12} (- c_{e} s_{13} s_{23} - i s_{e} c_{13}) & c_{e} c_{12} c_{23}+s_{12} (-c_{e} s_{13} s_{23}-i s_{e} c_{13} ) & c_{e} c_{13} s_{23} - i s_{e} s_{13} \\
s_{12} s_{23}-c_{12} c_{23} s_{13} & -c_{12} s_{23}-c_{23} s_{12} s_{13} & c_{13} c_{23}
\end{pmatrix} \, .
\label{U}
\end{align}
On the other hand, the standard PDG parameterization is
\cite{ParticleDataGroup:2018ovx}
\begin{align}
U_{}^{\rm PDG} &=
\begin{pmatrix}
c_{12} c_{13} & s_{12} c_{13} & s_{13} e^{-i\d} \\
-s_{12} c_{23} - c_{12} s_{23} s_{13} e^{i \d} & c_{12} c_{23} - s_{12} s_{23} s_{13} e^{i \d} & s_{23} c_{13} \\
s_{12} s_{23} - c_{12} c_{23} s_{13} e^{i \d} & -c_{12} s_{23} - s_{12} c_{23} s_{13} e^{ i \d} & c_{23} c_{13}
\end{pmatrix} \nn \\
& \times {\rm diag} (1 , e^{ i \alpha} \def\b{\beta} \def\g{\gamma} \def\G{\Gamma} \def\d{\delta} \def\D{\Delta} \def\e{\epsilon} \def\z{\zeta} \def\h{\eta} \def\th{\theta} \def\Th{\Theta} \def\k{\kappa} \def\l{\lambda} \def\L{\Lambda} \def\m{\mu} \def\n{\nu} \def\x{\xi} \def\X{\Xi} \def\p{\pi} \def\P{\Pi} \def\r{\rho} \def\s{\sigma} \def\S{\Sigma} \def\t{\tau} \def\ph{\phi} \def\Ph{\Phi} \def\ps{\psi} \def\Ps{\Psi} \def\c{\chi} \def\o{\omega} \def\O{\Omega_{2} / 2} , e^{ i \alpha} \def\b{\beta} \def\g{\gamma} \def\G{\Gamma} \def\d{\delta} \def\D{\Delta} \def\e{\epsilon} \def\z{\zeta} \def\h{\eta} \def\th{\theta} \def\Th{\Theta} \def\k{\kappa} \def\l{\lambda} \def\L{\Lambda} \def\m{\mu} \def\n{\nu} \def\x{\xi} \def\X{\Xi} \def\p{\pi} \def\P{\Pi} \def\r{\rho} \def\s{\sigma} \def\S{\Sigma} \def\t{\tau} \def\ph{\phi} \def\Ph{\Phi} \def\ps{\psi} \def\Ps{\Psi} \def\c{\chi} \def\o{\omega} \def\O{\Omega_{3} / 2}) \, .
\label{PDG}
\end{align}
In this definition, $P$ and the phases $\phi_{i}$ have a potential to set the Majorana phases $\alpha} \def\b{\beta} \def\g{\gamma} \def\G{\Gamma} \def\d{\delta} \def\D{\Delta} \def\e{\epsilon} \def\z{\zeta} \def\h{\eta} \def\th{\theta} \def\Th{\Theta} \def\k{\kappa} \def\l{\lambda} \def\L{\Lambda} \def\m{\mu} \def\n{\nu} \def\x{\xi} \def\X{\Xi} \def\p{\pi} \def\P{\Pi} \def\r{\rho} \def\s{\sigma} \def\S{\Sigma} \def\t{\tau} \def\ph{\phi} \def\Ph{\Phi} \def\ps{\psi} \def\Ps{\Psi} \def\c{\chi} \def\o{\omega} \def\O{\Omega_{i}$ to $\alpha} \def\b{\beta} \def\g{\gamma} \def\G{\Gamma} \def\d{\delta} \def\D{\Delta} \def\e{\epsilon} \def\z{\zeta} \def\h{\eta} \def\th{\theta} \def\Th{\Theta} \def\k{\kappa} \def\l{\lambda} \def\L{\Lambda} \def\m{\mu} \def\n{\nu} \def\x{\xi} \def\X{\Xi} \def\p{\pi} \def\P{\Pi} \def\r{\rho} \def\s{\sigma} \def\S{\Sigma} \def\t{\tau} \def\ph{\phi} \def\Ph{\Phi} \def\ps{\psi} \def\Ps{\Psi} \def\c{\chi} \def\o{\omega} \def\O{\Omega_{i} + \pi$ and have no effect on any other physical quantity.
By comparing the absolute values of the third column of Eq.~(\ref{U})
with the PDG parameterization, $s_{13}$ and $s_{23}$ should satisfy
\begin{align}
| s_{e} c_{13} s_{23} + i c_{e} s_{13} |^{2} & = (s_{13}^{\rm PDG})^{2} \, , \\
| c_{e} c_{13} s_{23} - i s_{e} s_{13}|^{2} & = (s_{23}^{\rm PDG} c_{13}^{\rm PDG})^{2} \, ,
\end{align}
and we obtain the following solutions;
\begin{align}
s_{23} = \frac{\sqrt{(s_{e} s_{13}^{\text{PDG}})^{2} - (c_{e} c_{13}^{\text{PDG}} s_{23}^{\text{PDG}} )^{2} } }{\sqrt{-c_{e}^4 + (c_{e} s_{13}^{\text{PDG}})^{2} - (s_{e} c_{13}^{\text{PDG}} s_{23}^{\text{PDG}})^{2} + s_{e}^4}} \, ,
~~~
s_{13} = \frac{\sqrt{ (c_{e} s_{13}^{\text{PDG}})^{2} - (s_{e} c_{13}^{\text{PDG}} s_{23}^{\text{PDG}})^{2} } }{\sqrt{c_{e}^4-s_{e}^4}} \, .
\label{s2313}
\end{align}
Similarly, from $|U_{12}|^{2} = |U_{12}^{\rm PDG}|^{2}$,
the condition satisfied by $s_{12}$ is
\begin{align}
|s_{e} (c_{12} c_{23} - s_{12} s_{13} s_{23})+i s_{12} c_{e} c_{13}|^{2} = (s_{12}^{\rm PDG} c_{13}^{\rm PDG})^{2} \, . \label{13}
\end{align}
Solving $s_{12}$ from Eqs.~(\ref{s2313}) and (\ref{13}), we obtain four solutions.
However, only two solutions are physically inequivalent, and they differ in the sign of $\cos \d$.
Experiments favor the solution of $s_{12} < 0$,
because the sign of $s_{12}$ and $\cos \d$ are almost same as a result of drawing the plots (Figure 1).
%
Since the condition (\ref{13}) depends only on $s_{e}^{2}$ and $c_{e}^{2}$,
the solutions of $s_{12}$ is independent of signs of $s_{e}$ and $c_{e}$.
As input values, we use the latest global fit
without Super-Kamiokande (SK) in the Normal Hierarchy (NH) \cite{Gonzalez-Garcia:2021dve};
\begin{align}
\sin^{2} \th_{12}^{\rm PDG} = 0.304 \, , ~~~ \sin^{2} \th_{23}^{\rm PDG} = 0.573 \, , ~~~ \sin^{2} \th_{13}^{\rm PDG} = 0.0222 \, .
\end{align}
This is due to the reason that the values of Inverted hierarchy (IH) with or without SK are close to these values.
Although the inclusion of the SK data makes $s_{23}^{\rm PDG}$ about $0.1$ smaller for NH,
the qualitative behavior in the following discussion remains the same.
From this, $\sin \d$ and $\cos \d$ can be expressed as functions of $s_{e}$ and some sign degrees of freedom.
The parameter $\cos \d$ is given by
\begin{align}
\cos \d & = { |U_{22}^{\rm PDG}|^{2} - (s_{12}^{\rm PDG} s_{13}^{\rm PDG} s_{23}^{\rm PDG})^{2} - (c_{12}^{\rm PDG} c_{23}^{\rm PDG})^{2} \over - 2 s_{12}^{\rm PDG} s_{13}^{\rm PDG} s_{23}^{\rm PDG} c_{12}^{\rm PDG} c_{23}^{\rm PDG} } \\
&= {|U_{22}|^{2} (1 - |U_{13}|^{2})^{2} - |U_{13}|^{2} |U_{12}|^{2} |U_{23}|^{2} - |U_{11}|^{2} |U_{33}|^{2} \over - 2 |U_{13}| |U_{12}| |U_{23}| |U_{11}| |U_{33}|} \, .
\end{align}
Also, $\sin \d$ can be evaluated from the Jarlskog invariant;
\begin{align}
J = - {\rm Im} \, [U_{\m 3} U_{\t 2} U_{\m 2}^{*} U_{\t 3}^{*}]
&= \sin \d \, s_{12}^{\rm PDG} c_{12}^{\rm PDG} s_{13}^{\rm PDG} (c_{13}^{\rm PDG})^{2} s_{23}^{\rm PDG} c_{23}^{\rm PDG} \\
& = c_{13} c_{23} c_{e} s_{e} (c_{12} s_{23} + c_{23} s_{12} s_{13}) (s_{12} s_{23} - c_{12} c_{23} s_{13}) \, .
\label{exp}
\end{align}
It predicts a proportional relationship between $\sin \d$ and $s_{e}$. When $s_{13}$ is small, the invariant is roughly
\begin{align}
J \simeq c_{12} c_{23} c_{e} s_{12} s_{23}^2 s_{e} \, , ~~~
\sin \d \simeq - { c_{e} s_{e} s_{23} \over s_{13}^{\rm PDG} (c_{13}^{\rm PDG})^{2} }
\simeq \pm \, 5 \, s_{e} \, .
\end{align}
Then the sign of $s_{e} c_{e}$ and $\sin \d$ are opposite.
The minus sign comes from the choice $s_{12} \simeq - s_{12}^{\rm PDG}$.
Figure 1 shows plots of $\cos \d$ and $\sin \d$ expressed as functions of $s_{e}$.
In the plot of $\cos \d$, the red and green lines correspond to $s_{12} > 0$ and $s_{12} < 0$.
Since the signs are approximately equal (${\rm sign} (\cos \d) \simeq {\rm sign} (s_{12})$) in the parameter regions, experiments favor $s_{12} < 0$.
For $\sin \d$, the color of the lines depend on the sign of $c_{e}$.
\begin{figure}[t]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=7cm]{fig1.eps} ~&~
\includegraphics[width=7cm]{fig2.eps}
\end{tabular}
\caption{ Plots of $\cos \d$ and $\sin \d$ expressed as functions of $s_{e}$.
The bright green and orange region represent the $1 \s$ regions of NH and IH (without SK).
}
\end{center}
\end{figure}
The parameter $|\sin \d|$ becomes maximal around $|s_{e}| \simeq 0.2$.
In regions where $s_{e}$ is larger than this, $\sin \d$ and $s_{13}$ become complex numbers and have no real solution. This is due to the following reasons.
In the range $|s_{e}| \lesssim 0.2$, $s_{e}$ can be regarded as a perturbation.
From Eq.~(\ref{s2313}), $s_{12}$ and $s_{23}$ are approximately equal to those of PDG and
$s_{13}$ is constrained as
\begin{align}
s_{23} \simeq s_{23}^{\rm PDG} \, , ~~~
s_{12} \simeq - s_{12}^{\rm PDG} \, , ~~~
s_{13} \simeq \sqrt{ (s_{13}^{\text{PDG}})^{2} - (s_{e} s_{23}^{\text{PDG}})^{2} } \, .
\end{align}
Since the maximum value of $s_{e}$ in this range is realized by $s_{13} = 0$,
\begin{align}
s_{e}^{\rm max} \simeq {s_{13}^{\rm PDG} \over s_{23}^{\rm PDG}} \simeq 0.196 \, .
\end{align}
There exists other solutions with $s_{e} \simeq \pm 1$.
However, since these solutions imply that the eigenstates of the charged leptons $e$ and $\m$ are interchanged by diagonalization, it is excluded from a point of view of the natural mass matrix \cite{Peccei:1995fg}.
\subsection{Majorana phases}
A similar analysis is performed for the Majorana phases.
These phases can be evaluated from the following quantities \cite{Nieves:1987pp};
\begin{align}
I_{1} & = {\rm Im} \, [U_{e2}^{2} U_{e1}^{*2} / |U_{e2} U_{e1}|^{2}] =
\sin \alpha} \def\b{\beta} \def\g{\gamma} \def\G{\Gamma} \def\d{\delta} \def\D{\Delta} \def\e{\epsilon} \def\z{\zeta} \def\h{\eta} \def\th{\theta} \def\Th{\Theta} \def\k{\kappa} \def\l{\lambda} \def\L{\Lambda} \def\m{\mu} \def\n{\nu} \def\x{\xi} \def\X{\Xi} \def\p{\pi} \def\P{\Pi} \def\r{\rho} \def\s{\sigma} \def\S{\Sigma} \def\t{\tau} \def\ph{\phi} \def\Ph{\Phi} \def\ps{\psi} \def\Ps{\Psi} \def\c{\chi} \def\o{\omega} \def\O{\Omega_{2},
\label{I1} \\
I_{2} & = {\rm Im} \, [ U_{e3}^{2} U_{e1}^{*2} / |U_{e3} U_{e1}|^{2}] =
\sin (\alpha} \def\b{\beta} \def\g{\gamma} \def\G{\Gamma} \def\d{\delta} \def\D{\Delta} \def\e{\epsilon} \def\z{\zeta} \def\h{\eta} \def\th{\theta} \def\Th{\Theta} \def\k{\kappa} \def\l{\lambda} \def\L{\Lambda} \def\m{\mu} \def\n{\nu} \def\x{\xi} \def\X{\Xi} \def\p{\pi} \def\P{\Pi} \def\r{\rho} \def\s{\sigma} \def\S{\Sigma} \def\t{\tau} \def\ph{\phi} \def\Ph{\Phi} \def\ps{\psi} \def\Ps{\Psi} \def\c{\chi} \def\o{\omega} \def\O{\Omega_{3} - 2 \d) .
\label{I2}
\end{align}
Expansions of $\sin \alpha} \def\b{\beta} \def\g{\gamma} \def\G{\Gamma} \def\d{\delta} \def\D{\Delta} \def\e{\epsilon} \def\z{\zeta} \def\h{\eta} \def\th{\theta} \def\Th{\Theta} \def\k{\kappa} \def\l{\lambda} \def\L{\Lambda} \def\m{\mu} \def\n{\nu} \def\x{\xi} \def\X{\Xi} \def\p{\pi} \def\P{\Pi} \def\r{\rho} \def\s{\sigma} \def\S{\Sigma} \def\t{\tau} \def\ph{\phi} \def\Ph{\Phi} \def\ps{\psi} \def\Ps{\Psi} \def\c{\chi} \def\o{\omega} \def\O{\Omega_{2,3}$ for small $s_{e}$ is respectively
\begin{align}
\sin \alpha} \def\b{\beta} \def\g{\gamma} \def\G{\Gamma} \def\d{\delta} \def\D{\Delta} \def\e{\epsilon} \def\z{\zeta} \def\h{\eta} \def\th{\theta} \def\Th{\Theta} \def\k{\kappa} \def\l{\lambda} \def\L{\Lambda} \def\m{\mu} \def\n{\nu} \def\x{\xi} \def\X{\Xi} \def\p{\pi} \def\P{\Pi} \def\r{\rho} \def\s{\sigma} \def\S{\Sigma} \def\t{\tau} \def\ph{\phi} \def\Ph{\Phi} \def\ps{\psi} \def\Ps{\Psi} \def\c{\chi} \def\o{\omega} \def\O{\Omega_{2}^{0} \simeq - \frac{2 s_{e} c_{23} }{ c_{e} c_{12} s_{12} } \simeq + 3 {s_{e} \over c_{e}} \, , ~~~
\sin \alpha} \def\b{\beta} \def\g{\gamma} \def\G{\Gamma} \def\d{\delta} \def\D{\Delta} \def\e{\epsilon} \def\z{\zeta} \def\h{\eta} \def\th{\theta} \def\Th{\Theta} \def\k{\kappa} \def\l{\lambda} \def\L{\Lambda} \def\m{\mu} \def\n{\nu} \def\x{\xi} \def\X{\Xi} \def\p{\pi} \def\P{\Pi} \def\r{\rho} \def\s{\sigma} \def\S{\Sigma} \def\t{\tau} \def\ph{\phi} \def\Ph{\Phi} \def\ps{\psi} \def\Ps{\Psi} \def\c{\chi} \def\o{\omega} \def\O{\Omega_{3}^{0} \simeq - \frac{2 s_{e} c_{e} c_{12} c_{23} }{ c_{13} s_{12} } \simeq + 2 {s_{e} c_{e}} \, .
\end{align}
Since these signs depend on ${\rm sign} (c_{e} s_{e})$, we conclude that the signs of $c_{e}$ and $s_{e}$ cannot be determined independently.
Furthermore, $\sin \alpha} \def\b{\beta} \def\g{\gamma} \def\G{\Gamma} \def\d{\delta} \def\D{\Delta} \def\e{\epsilon} \def\z{\zeta} \def\h{\eta} \def\th{\theta} \def\Th{\Theta} \def\k{\kappa} \def\l{\lambda} \def\L{\Lambda} \def\m{\mu} \def\n{\nu} \def\x{\xi} \def\X{\Xi} \def\p{\pi} \def\P{\Pi} \def\r{\rho} \def\s{\sigma} \def\S{\Sigma} \def\t{\tau} \def\ph{\phi} \def\Ph{\Phi} \def\ps{\psi} \def\Ps{\Psi} \def\c{\chi} \def\o{\omega} \def\O{\Omega_{i}$ has sign degrees of freedom due to $\phi_{i} = 0 $ or $\pi / 2$ in Eq.~(\ref{UVn});
\begin{align}
\alpha} \def\b{\beta} \def\g{\gamma} \def\G{\Gamma} \def\d{\delta} \def\D{\Delta} \def\e{\epsilon} \def\z{\zeta} \def\h{\eta} \def\th{\theta} \def\Th{\Theta} \def\k{\kappa} \def\l{\lambda} \def\L{\Lambda} \def\m{\mu} \def\n{\nu} \def\x{\xi} \def\X{\Xi} \def\p{\pi} \def\P{\Pi} \def\r{\rho} \def\s{\sigma} \def\S{\Sigma} \def\t{\tau} \def\ph{\phi} \def\Ph{\Phi} \def\ps{\psi} \def\Ps{\Psi} \def\c{\chi} \def\o{\omega} \def\O{\Omega_{2} = \alpha} \def\b{\beta} \def\g{\gamma} \def\G{\Gamma} \def\d{\delta} \def\D{\Delta} \def\e{\epsilon} \def\z{\zeta} \def\h{\eta} \def\th{\theta} \def\Th{\Theta} \def\k{\kappa} \def\l{\lambda} \def\L{\Lambda} \def\m{\mu} \def\n{\nu} \def\x{\xi} \def\X{\Xi} \def\p{\pi} \def\P{\Pi} \def\r{\rho} \def\s{\sigma} \def\S{\Sigma} \def\t{\tau} \def\ph{\phi} \def\Ph{\Phi} \def\ps{\psi} \def\Ps{\Psi} \def\c{\chi} \def\o{\omega} \def\O{\Omega_{2}^{0} + 2(\phi_{2} - \phi_{1}) \, , ~~~
\alpha} \def\b{\beta} \def\g{\gamma} \def\G{\Gamma} \def\d{\delta} \def\D{\Delta} \def\e{\epsilon} \def\z{\zeta} \def\h{\eta} \def\th{\theta} \def\Th{\Theta} \def\k{\kappa} \def\l{\lambda} \def\L{\Lambda} \def\m{\mu} \def\n{\nu} \def\x{\xi} \def\X{\Xi} \def\p{\pi} \def\P{\Pi} \def\r{\rho} \def\s{\sigma} \def\S{\Sigma} \def\t{\tau} \def\ph{\phi} \def\Ph{\Phi} \def\ps{\psi} \def\Ps{\Psi} \def\c{\chi} \def\o{\omega} \def\O{\Omega_{3} = \alpha} \def\b{\beta} \def\g{\gamma} \def\G{\Gamma} \def\d{\delta} \def\D{\Delta} \def\e{\epsilon} \def\z{\zeta} \def\h{\eta} \def\th{\theta} \def\Th{\Theta} \def\k{\kappa} \def\l{\lambda} \def\L{\Lambda} \def\m{\mu} \def\n{\nu} \def\x{\xi} \def\X{\Xi} \def\p{\pi} \def\P{\Pi} \def\r{\rho} \def\s{\sigma} \def\S{\Sigma} \def\t{\tau} \def\ph{\phi} \def\Ph{\Phi} \def\ps{\psi} \def\Ps{\Psi} \def\c{\chi} \def\o{\omega} \def\O{\Omega_{3}^{0} + 2(\phi_{3} - \phi_{1}) \, .
\end{align}
Therefore, it is difficult to derive general results on the signs of $\sin \alpha} \def\b{\beta} \def\g{\gamma} \def\G{\Gamma} \def\d{\delta} \def\D{\Delta} \def\e{\epsilon} \def\z{\zeta} \def\h{\eta} \def\th{\theta} \def\Th{\Theta} \def\k{\kappa} \def\l{\lambda} \def\L{\Lambda} \def\m{\mu} \def\n{\nu} \def\x{\xi} \def\X{\Xi} \def\p{\pi} \def\P{\Pi} \def\r{\rho} \def\s{\sigma} \def\S{\Sigma} \def\t{\tau} \def\ph{\phi} \def\Ph{\Phi} \def\ps{\psi} \def\Ps{\Psi} \def\c{\chi} \def\o{\omega} \def\O{\Omega_{i}$ and $\cos \alpha} \def\b{\beta} \def\g{\gamma} \def\G{\Gamma} \def\d{\delta} \def\D{\Delta} \def\e{\epsilon} \def\z{\zeta} \def\h{\eta} \def\th{\theta} \def\Th{\Theta} \def\k{\kappa} \def\l{\lambda} \def\L{\Lambda} \def\m{\mu} \def\n{\nu} \def\x{\xi} \def\X{\Xi} \def\p{\pi} \def\P{\Pi} \def\r{\rho} \def\s{\sigma} \def\S{\Sigma} \def\t{\tau} \def\ph{\phi} \def\Ph{\Phi} \def\ps{\psi} \def\Ps{\Psi} \def\c{\chi} \def\o{\omega} \def\O{\Omega_{i}$ from the mixing matrix only.
Plots of $\sin \alpha} \def\b{\beta} \def\g{\gamma} \def\G{\Gamma} \def\d{\delta} \def\D{\Delta} \def\e{\epsilon} \def\z{\zeta} \def\h{\eta} \def\th{\theta} \def\Th{\Theta} \def\k{\kappa} \def\l{\lambda} \def\L{\Lambda} \def\m{\mu} \def\n{\nu} \def\x{\xi} \def\X{\Xi} \def\p{\pi} \def\P{\Pi} \def\r{\rho} \def\s{\sigma} \def\S{\Sigma} \def\t{\tau} \def\ph{\phi} \def\Ph{\Phi} \def\ps{\psi} \def\Ps{\Psi} \def\c{\chi} \def\o{\omega} \def\O{\Omega_{i}$ for $s_{e}$ are shown in Figure 2.
\begin{figure}[h]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=7cm]{fig3.eps} ~&~
\includegraphics[width=7cm]{fig4.eps}
\end{tabular}
\caption{Plots of $\sin_{\alpha} \def\b{\beta} \def\g{\gamma} \def\G{\Gamma} \def\d{\delta} \def\D{\Delta} \def\e{\epsilon} \def\z{\zeta} \def\h{\eta} \def\th{\theta} \def\Th{\Theta} \def\k{\kappa} \def\l{\lambda} \def\L{\Lambda} \def\m{\mu} \def\n{\nu} \def\x{\xi} \def\X{\Xi} \def\p{\pi} \def\P{\Pi} \def\r{\rho} \def\s{\sigma} \def\S{\Sigma} \def\t{\tau} \def\ph{\phi} \def\Ph{\Phi} \def\ps{\psi} \def\Ps{\Psi} \def\c{\chi} \def\o{\omega} \def\O{\Omega_{2,3}}$ for $s_{e}$.
Sign degrees of freedom come from $e^{i \phi_{i}} = 0$ or $\pi /2$.}
\end{center}
\end{figure}
As a result, the CP-violating observables have the following correlations.
\begin{align}
\abs {\sin \alpha} \def\b{\beta} \def\g{\gamma} \def\G{\Gamma} \def\d{\delta} \def\D{\Delta} \def\e{\epsilon} \def\z{\zeta} \def\h{\eta} \def\th{\theta} \def\Th{\Theta} \def\k{\kappa} \def\l{\lambda} \def\L{\Lambda} \def\m{\mu} \def\n{\nu} \def\x{\xi} \def\X{\Xi} \def\p{\pi} \def\P{\Pi} \def\r{\rho} \def\s{\sigma} \def\S{\Sigma} \def\t{\tau} \def\ph{\phi} \def\Ph{\Phi} \def\ps{\psi} \def\Ps{\Psi} \def\c{\chi} \def\o{\omega} \def\O{\Omega_{2} \over \sin \d} & \simeq
\abs{2 s_{13} c_{13}^{2} c_{23} \over c_{e}^{2} c_{12} s_{12} s_{23}} \simeq
{3 \sqrt 2 s_{13} } \simeq 0.6 \, , \\
\abs {\sin \alpha} \def\b{\beta} \def\g{\gamma} \def\G{\Gamma} \def\d{\delta} \def\D{\Delta} \def\e{\epsilon} \def\z{\zeta} \def\h{\eta} \def\th{\theta} \def\Th{\Theta} \def\k{\kappa} \def\l{\lambda} \def\L{\Lambda} \def\m{\mu} \def\n{\nu} \def\x{\xi} \def\X{\Xi} \def\p{\pi} \def\P{\Pi} \def\r{\rho} \def\s{\sigma} \def\S{\Sigma} \def\t{\tau} \def\ph{\phi} \def\Ph{\Phi} \def\ps{\psi} \def\Ps{\Psi} \def\c{\chi} \def\o{\omega} \def\O{\Omega_{3} \over \sin \d} & \simeq
\abs{2 c_{12} c_{23} c_{13} s_{13} \over s_{12} s_{23} } \simeq 2 \sqrt{2} s_{13} \simeq 0.4 \, .
\end{align}
Since $U$~(\ref{U}) is CP-symmetric in the limit where $s_{e}$ becomes zero,
it is a natural consequence that these CP phases have such correlations.
\section{Summary}
In this letter, we systematically analyzed the MNS matrix with diagonal reflection symmetries.
If the mass matrix of charged leptons $m_{e}$ is hierarchical, by neglecting the 13 mixing of $m_{e}$, the MNS matrix is represented by four parameters and several sign degrees of freedom.
By substituting the three observed mixing angles $\th_{ij}$ as input parameters,
the Dirac phase $\d$ and the Majorana phases $\alpha} \def\b{\beta} \def\g{\gamma} \def\G{\Gamma} \def\d{\delta} \def\D{\Delta} \def\e{\epsilon} \def\z{\zeta} \def\h{\eta} \def\th{\theta} \def\Th{\Theta} \def\k{\kappa} \def\l{\lambda} \def\L{\Lambda} \def\m{\mu} \def\n{\nu} \def\x{\xi} \def\X{\Xi} \def\p{\pi} \def\P{\Pi} \def\r{\rho} \def\s{\sigma} \def\S{\Sigma} \def\t{\tau} \def\ph{\phi} \def\Ph{\Phi} \def\ps{\psi} \def\Ps{\Psi} \def\c{\chi} \def\o{\omega} \def\O{\Omega_{2,3}$ become functions of the 12 mixing of charged leptons $s_{e}$.
As a result, we obtain a clear correlations between CP-violating phases
$|\sin \d| \simeq 1.6 |\sin \alpha} \def\b{\beta} \def\g{\gamma} \def\G{\Gamma} \def\d{\delta} \def\D{\Delta} \def\e{\epsilon} \def\z{\zeta} \def\h{\eta} \def\th{\theta} \def\Th{\Theta} \def\k{\kappa} \def\l{\lambda} \def\L{\Lambda} \def\m{\mu} \def\n{\nu} \def\x{\xi} \def\X{\Xi} \def\p{\pi} \def\P{\Pi} \def\r{\rho} \def\s{\sigma} \def\S{\Sigma} \def\t{\tau} \def\ph{\phi} \def\Ph{\Phi} \def\ps{\psi} \def\Ps{\Psi} \def\c{\chi} \def\o{\omega} \def\O{\Omega_{2}| \simeq 2.5 |\sin \alpha} \def\b{\beta} \def\g{\gamma} \def\G{\Gamma} \def\d{\delta} \def\D{\Delta} \def\e{\epsilon} \def\z{\zeta} \def\h{\eta} \def\th{\theta} \def\Th{\Theta} \def\k{\kappa} \def\l{\lambda} \def\L{\Lambda} \def\m{\mu} \def\n{\nu} \def\x{\xi} \def\X{\Xi} \def\p{\pi} \def\P{\Pi} \def\r{\rho} \def\s{\sigma} \def\S{\Sigma} \def\t{\tau} \def\ph{\phi} \def\Ph{\Phi} \def\ps{\psi} \def\Ps{\Psi} \def\c{\chi} \def\o{\omega} \def\O{\Omega_{3}|$.
\section*{Acknowledgment}
This study is financially supported
by JSPS Grants-in-Aid for Scientific Research
No.~JP18H01210
and MEXT KAKENHI Grant No.~JP18H05543.
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\section{Introduction}
Quantum metrology with light estimates unknown parameters of quantum processes and reveals the limits of the existing measurements from treating the measuring probes as physical systems in specific optimized quantum states \cite{giovannetti2004quantum,giovannetti2006quantum,giovannetti2011advances, pirandola2018advances}.
Optical interferometry is one the main examples of improvement brought in by using quantum states \cite{demkowicz2015quantum}. There, quantum noise entering the interferometer through the beam splitter's idle port can be reduced by using squeezed states of light \cite{caves1981quantum, olivares2009bayesian, berni2015ab, oh2019optimal}. The effect is robust enough to find a place in practical applications where high precision is required, such as detecting gravitational waves \cite{aasi2013enhanced}. A specific case of optical interferometry relies on homodyne detection \cite{schleich2011quantum}, where the reference arm of the interferometer is represented by a classical local oscillator beam \cite{olivares2009bayesian, berni2015ab, oh2019optimal}, which leaves only the probe to be prepared in a quantum state. Another approach takes advantage of controlling the full interferometer and preparing joint quantum NOON states in both of its arms \cite{lee2002quantum, dowling2008quantum, mccormick2019quantum}. The probes in quantum metrology are not limited to light. Atomic interferometry using collective states of atoms \cite{bongs2019taking, PhysRevLett.125.100402} and quantum optomechanics employing state of motion of a massive mirror \cite{qvarfort2018gravimetry, schneiter2020optimal} have both been studied for gravimetry. Similarly, quantum states of trapped ions \cite{dalvit2006quantum, wolf2019motional} can be applied towards the detection of weak electric fields.
One thing all of these methods have in common is the use of quantum states that are nonclassical \cite{tan2019nonclassical}, which means that they cannot be described by classical physics alone. The nonclassical states in optical systems can be further divided into Gaussian and non-Gaussian. Gaussian states, such as squeezed states \cite{caves1981quantum, olivares2009bayesian, berni2015ab, oh2019optimal} can be described by Gaussian functions in phase space, while the non-Gaussian cannot. One drawback shared by both kinds of nonclassical states is their vulnerability to imperfections. While the nonclassical quantum states of probes have been shown to significantly outperform the classical methods in ideal cases~\cite{olivares2009bayesian, mccormick2019quantum}, this improvement can vanish under realistic experimental conditions. In Gaussian scenarios \cite{genoni2013optimal, pinel2013quantum, vsafranek2015quantum, vsafranek2016optimal, nichols2018multiparameter, vsafranek2018estimation, oh2019optimal, oh2019optimal2}, such as employing squeezed states to boost interferometry, the losses of the quantum state simply reduce its effectiveness \cite{berni2015ab, oh2019optimal}. In non-Gaussian cases, such as employing so-called NOON states used to estimate phase, losses can completely remove the quantum advantage \cite{dowling2008quantum, escher2011general}. Even though recent progress tries to alleviate this effect \cite{slussarenko2017unconditional}, one question that needs to be addressed for all quantum metrology proposals is whether its benefits survive contact with practical reality.
Fluctuation of phase is one of the major sources of imperfection in optical interferometry \cite{genoni2011optical, genoni2012optical, escher2012quantum, vidrighin2014joint, szczykulska2017reaching, aguilar2020robust} and quantum communications \cite{fanizza2020classical, zhuang2020entanglement}. It can take the shape of random phase fluctuation in the sample, but also of the inability to lock the process in the sample to the phase of the probe. There are also situations in which there are no detectors to capitalize on the phase. This can happen in optics when broadband photon counting detectors are used instead of homodyne detection \cite{eckstein2011highly}.
However, control of the phase is not necessary for estimating the strength of the phase-sensitive operations, such as coherent displacement and nonclassical squeezing process.
It was shown in a recent trapped-ion experiment \cite{wolf2019motional}, where the size of displacement, a generally phase-sensitive operation, was estimated without any kind of phase reference, just relying on Fock state preparation and measurement.
In this paper, we extend this idea to optical experiments and show that optical estimation of the strength of unitary Gaussian operations, displacement and squeezing, can be indeed realized without any phase reference with Fock states and measurements on the photon number basis. For displacement, this approach surpasses even optimal Gaussian methods, which are based on homodyne detection and squeezed states with equivalent energy and which use the phase reference. In squeezing estimation, Fock states are generally comparable or even slightly inferior to Gaussian methods at low energies and overcoming them for larger energies of the probe states. Our probe states do not need to be pure and can have an advantage over the Gaussian probe states even for realistic losses of the order of 20\%. Finally, we show that this approach can be useful for the simultaneous estimation of both quantities \cite{chiribella2006joint}.
\section{Model of the process}
In our quantum sensing protocol for Gaussian processes without a stable phase reference, represented by a mixture of quantum evolution with all phases, we start with an ensemble of $M$ probes prepared in a well-defined quantum state that is fully under our control.
The probes then sequentially interact with a sample, undergoing weak phase-randomized Gaussian evolution in the process, and are measured by a specific measurement.
The measured data are then evaluated using maximum likelihood estimator (MLE) to extract the unknown parameters of the Gaussian operation.
In our analysis we will focus on photon number resolving detectors (PNRD).
Action of a general single-mode Gaussian channel \cite{serafini2017quantum} can be represented by a linear transformation of quadrature operators of the field $\hat{x}$ and $\hat{p}$, with $[\hat{x},\hat{p}] = i$. If we arrange the quadratures into a vector $\hat{\xi} = (\hat{x},\hat{p})^\text{T}$, the general Gaussian operation transforms them into a new vector
\begin{equation}\label{}
\hat{\xi}' = V \hat{\xi} + \alpha + (\hat{x}_{E},\hat{p}_E)^\text{T},
\end{equation}
where $V$ is a real matrix with $|V|<1$, $\alpha = (\alpha_x,\alpha_p)^\text{T}$ is a vector of real values, and $\hat{x}_E$ and $\hat{p}_E$ are zero mean value Hermitian operators of the environment that satisfy $[\hat{x}_E,\hat{p}_E] \ge i(1-|V|)$ and their statistics is Gaussian. Among these terms, vector $\alpha$ models displacement, matrix $V$ includes phase shift, squeezing, and losses, and the pair of environment operators contributes added noise. In the absence of imperfections, the operation is unitary and is fully described by the vector $\alpha$ and matrix $V$ with $|V|=1$.
Such operation can be decomposed into three separate processes. Phase shift, represented by operator $\hat{R}(\phi) = e^{-i\phi\hat{n}}$, displacement represented by $\hat{D}(\alpha) = \exp(\alpha \hat{a}^\dag - \alpha^*\hat{a})$ and squeezing represented by operator $\hat{S}(\xi) = \exp[(\xi \hat{a}^{\dag 2} - \xi^*\hat{a}^2)/2]$. Here, $\hat{a} = (\hat{x} + i\hat{p})/\sqrt{2}$ denotes the annihilation operator of the field and $\hat{n} = \hat{a}^{\dag}\hat{a}$ is the photon number operator. Although these operations do not commute in general, the order of these operations can be arbitrary - what matters is their final product and different orderings can lead to the same overall operation if the parameters are suitably changed.
Let us now consider a scenario without a well-defined phase.
The Gaussian effect on the mechanical probe can be decomposed by a sequence of displacement, phase shift, and squeezing, where the strengths of the displacement and squeezing are of main interest. In optical testing, it removes the need to lock the pump of the nonlinear process appearing in the sample, such as weakly nonlinear waveguide, to the phase of the signal.
Both displacement and squeezing are phase-sensitive operations.
If their phase cannot be locked to the signal or detectors are phase-insensitive, it is considered random.
In the extreme case the phase is assumed to be uniformly distributed on the interval $[0,2\pi)$ by the principle of maximum entropy \cite{jaynes1968prior}. In any case, the operation is no longer unitary and needs to be expressed in terms of map which transforms the probe state $\hat{\rho}_{in}$ as
$\mathcal{M}$:
\begin{align}\label{evolution_map}
&\hat{\rho}_f(N_c,N_s) = \mathcal{M}(\hat{\rho}_{in}) = \int_0^{2\pi} \frac{d\phi_1}{2\pi} \int_0^{2\pi} \frac{d\phi_2}{2\pi} \hat{D}(\sqrt{N_c}e^{i\phi_1})\hat{S}(\sqrt{N_s} e^{i\phi_2})\hat{\rho}_{in}\hat{S}^{\dag}(\sqrt{N_s} e^{i\phi_2}) \hat{D}^{\dag}(\sqrt{N_c}e^{i\phi_1}).
\end{align}
In contrast to a general pure Gaussian unitary operation with five parameters, the phase indeterminate operation has only two free parameters: $N_c$ related to the average linearized energy added by the displacement, and $N_s$ related to the energy added by the squeezing. On vacuum state $\hat{\rho}_{in}=|0\rangle\langle0|$, the added energies of the two operations are given as $N_c$ for displacement and $\sinh^2 (\sqrt{N_s})$ for squeezing. In the limit of weak strengths $N_s$ and $N_c$, which is the regime we are interested in, phase-insensitive displacement and squeezing operations commute and the average photon number is increased by $N_s+N_c$.
Two main effects that are not part of this model and yet play a significant role in practical situations are the fluctuations of parameters $N_S$ and $N_c$, and linear optical losses.
Fluctuation of the parameters arises due to instability of the estimation process. In this case, what is usually estimated is the mean value of the parameters with their known fluctuation. The other effect, optical losses, naturally appear when part of information contained in the probe is lost. In the case of fast short-time passage through the sample, the majority of losses will appear either during the in-coupling and out-coupling of the probe. As a consequence, they can often be estimated separately and we can treat them as a known parameter in the estimation of the displacement and squeezing.
We will discuss both of these imperfections more later in the text.
Before we proceed to the analysis of the protocol, let us establish some theoretical framework by briefly recalling the quantum Cram\'{e}r-Rao (QCR) inequality and the quantum Fisher information (FI).
QCR inequality states that for a given probe and channel that encodes an unknown parameter of interest $\theta$, the estimation error of any unbiased estimator is bounded by the inverse of quantum FI \cite{helstrom1976quantum, braunstein1994statistical},
\begin{align}\label{crb}
\Delta^2\theta \geq \frac{1}{M H(\theta)},
\end{align}
where the variance $\Delta^2\theta=\langle (\theta^\text{est}-\theta)^2\rangle$ is the estimation error, and $M$ is the number of trials in an experiment, and $H(\theta)=\text{Tr}(\hat{\rho}_\theta \hat{L}_\theta^2)$ is the FI of the probe state after the encoding.
It is known that the lower bound is asymptotically saturable by using MLE \cite{fisher1925theory, braunstein1992large, braunstein1992quantum}.
In experiments, the optimal precision suggested by equality in (\ref{crb}) is obtained by an optimal positive operator valued measurement (POVM),
which can be found by the eigenbasis of the symmetric logarithmic derivative operator $\hat{L}_\theta$ satisfying an equation,
\begin{align}
\frac{\partial \hat{\rho}_\theta}{\partial \theta}=\frac{\hat{\rho}_\theta \hat{L}_\theta+\hat{L}_\theta\hat{\rho}_\theta }{2}.
\end{align}
If the density matrix of the output state is diagonalized as $\hat{\rho}_\theta=\sum_n\rho_n |\psi_n\rangle\langle\psi_n|$, the symmetric logarithmic derivative operator is given $\hat{L}_\theta=2\sum_{n,m}\frac{\langle\psi_n|\partial_\theta\hat{\rho}_\theta|\psi_m\rangle}{\rho_n+\rho_m} |\psi_n\rangle\langle\psi_m|$~\cite{paris2009quantum}.
When the optimal measurement is chosen, the classical FI
\begin{align}
F(\theta)=\sum_n \frac{1}{p(n|\theta)}\left(\frac{\partial p(n|\theta)}{\partial \theta}\right)^2
\end{align}
becomes the quantum FI where $p(n|\theta)=\text{Tr}(\hat{\rho}_\theta \hat{\Pi}_n)$ with $\{\hat{\Pi}_n\}$ being an optimal POVM, i.e. the projectors of eigenstates of $\hat{L}_\theta$.
According to QCR inequality \eqref{crb}, FI lower-bounds the estimation error obtainable during the actual measurement by $\Delta^2 N_c=1/M F$ where $M$ is the number of trials.
In a general scenario, however, there is no guarantee that this bound can be achieved with a practical number of $M$ although it is achievable using the MLE in an asymptotic regime of $M\rightarrow \infty$.
In the following sections, we present the estimation error of our scenario obtained by the MLE for a finite number of $M$.
Map (\ref{evolution_map}) is phase-insensitive; it commutes with any phase shift applied to the state of the probe. Consequently, if the probe state is phase-insensitive it remains so.
This suggests that a well-defined phase of the probe may not be required for optimal estimation. This can be illustrated on an example of probe prepared in Fock state $\hat{\rho}_{in}=|m\rangle\langle m|$. Such probe is pure but completely phase-insensitive. In the limit of weak strengths $N_c, N_s \ll 1$, map (\ref{evolution_map}) transforms the initial pure state of the probe into a mixture of Fock states with weights as:
\begin{align} \label{eq:binary1}
p(m-2|N_c,N_s) &\simeq N_s m(m-1)/4 \\
p(m-1|N_c,N_s) &\simeq N_c m \\
p(m|N_c,N_s) &\simeq 1 - N_c(2m+1) - N_s(m^2+m+1)/2 \\
p(m+1|N_c,N_s) &\simeq N_c(m+1) \\
p(m+2|N_c,N_s) &\simeq N_s(m+2)(m+1)/4. \label{eq:binary2}
\end{align}
Since the state is diagonal in the Fock basis these weights can be perfectly measured by PNRD. In this important limit we can see that the displacement and squeezing unitaries act in a complementary way - displacement changes the photon number by one, squeezing changes it by two. This indicates that in this limit the two operations can be discerned independently.
The measured data, with the help of Eqs.~\eqref{eq:binary1}-\eqref{eq:binary2}, can be used to construct MLEs for each parameter. Let us denote the number of outcomes corresponding to detecting particular state $|k\rangle$ by $n_k$ with $k\in\{m-2,m-1,m,m+1,m+2\}$; thus, $\sum_{k=m-2}^{m+2} n_k=M$, where $M$ is again the total number of trials.
We simply ignore the outcome out of the above range since the probability is negligible for very small strength of the signal.
By maximizing the log-likelihood function $\log L(\mathbb{D}=\{n_k\}|N_c,N_s)=\sum_k n_k\log p(k|N_c,N_s)$ for each parameter $N_c$ and $N_s$, one can find that the MLE is written as
\begin{align}\label{weakML}
N_c^\text{est}=\frac{n_{m-1}+n_{m+1}}{M(2m+1)}, ~~~\text{and}~~~ N_s^\text{est}=\frac{2(n_{m-2}+n_{m+2})}{M(m^2+m+1)},
\end{align}
and that they are unbiased for any $M>0$, i.e., $\langle N_c^\text{est}\rangle=N_c$ and $\langle N_s^\text{est}\rangle=N_s$, where the bracket represents the average over all possible outcomes.
The form of the MLEs, in particular the exclusive use of the respective count numbers $n_k$, implies that each parameter can be estimated simultaneously without knowing the other parameter.
In addition, from the same equations we can derive classical FI to evaluate the estimation of $N_c$ and $N_s$ by using a PNRD, which we can then compared to the upper limit given by quantum FI for the optimal detector. For the case of phase-insensitive states, PNRD gives us full available information and the classical FI is equal to quantum FI. Based on Eqs.~\eqref{eq:binary1}-\eqref{eq:binary2}, the classical FI of Fock states can be approximately found to be:
\begin{equation}\label{disp_qfi}
F(N_c)\approx\frac{2m+1}{N_c},
\end{equation}
for displacement and
\begin{equation}\label{sq_qfi}
F(N_s)\approx \frac{m^2+m+1}{2N_s},
\end{equation}
for squeezing in the limit of $N_c, N_s \ll 1$.
In this limit, one can derive the average estimation error of ML estimators \eqref{weakML}
\begin{align}\label{weakVar}
\Delta^2 N_c\approx \frac{N_c}{M(2m+1)}~~~\text{and}~~~ \Delta^2 N_s\approx \frac{2N_s}{M(m^2+m+1)},
\end{align}
which is consistent with the FI.
For both of them the performance improves with the increased Fock number of the probe, linearly in the case of displacement while quadratically in the case of the squeezing.
The different scalings of Fisher information is ascribed by the fact that weak displacement and squeezing operations are single-photon and two-photon processes, respectively, as shown in Eqs.~\eqref{eq:binary1}-\eqref{eq:binary2}.
Both quantities diverge as the signal decreases, but the relative estimation errors, which are given as inverse of the Fisher information relative to the signal, $R = \frac{1}{F(N_i) N_i}$ with $i = c,s$, attain constant value.
In the next sections, we will analyze how these values can be obtained with a realistic number of probes and how is the procedure affected by realistic processing and imperfections in comparison to results obtainable with Gaussian resources.
\section{Displacement estimation with Fock state probe}
Let us first analyze situations in which the displacement operation is the only relevant effect.
In this case, the operation is represented by map (\ref{evolution_map}) with the squeezing parameter $N_s = 0$.
It transforms the initial Fock state $|m\rangle\langle m|$ into a mixed state
\begin{align}\label{displacement_channel}
\hat{\rho}_f(N_c)=\int_0^{2\pi}\frac{d\phi}{2\pi}\hat{D}(\sqrt{N_c}e^{i\phi})|m\rangle\langle m|\hat{D}^\dagger(\sqrt{N_c}e^{i\phi})=\sum_{n=0}^{\infty}p(n|N_c)|n\rangle\langle n|,
\end{align}
where
\begin{align}
p(n|N_c)=|\langle n|\hat{D}(\sqrt{N_c})|m\rangle|^2=\frac{m!}{n!}e^{-N_c}N_c^{n-m} L_m^{(n-m)}(N_c)^2,
\end{align}
is the conditional probability to detect $n$ photons for a given $N_c$ and $m$ with $L_m^{n}(x)$ being associated Laguerre polynomials \cite{de1990properties}.
Since the final state is diagonal in Fock basis, it has equal quantum and classical FI for PNRD:
\begin{equation}\label{nc_fisher}
F(N_c)=\sum_{n=0}^\infty \frac{1}{p(n|N_c)}\left(\frac{\partial p(n|N_c)}{\partial N_c} \right)^2
=\frac{2m+1}{N_c},
\end{equation}
which is exactly the value approached by the approximate relation (\ref{disp_qfi}). It implies that in the limit of weak strength $N_c\ll1$, the protocol of the MLE of Eq.~\eqref{weakML} is the optimal procedure.
The derivation of the FI is supplied in Appendix A.
It should be noted that the monotonous increase of FI with the energy of the state $m$ requires the measurement in the Fock basis.
If the measurement was replaced by measurement of the mean energy $\langle \hat{n} \rangle$, the size of displacement could still be inferred, but the error of the measurement increases with $m$ (see Appendix B for detail).
\begin{figure}[t!]
\includegraphics[width=500px]{dis_nc.eps}
\caption{Simulation for displacement estimation error $\Delta^2 N_c$.
(a) Against the number of copies $M$ for various strenths $N_c=0.1, 1.0, 2.0$ by a Fock state probe $|m=3\rangle$.
Lines and dots represent the inverse of FI (CR bound) and the estimation errors of simulations averaged over $3000$ trials using MLE.
(b) Against the signal energy $N_c$ for various input probe Fock number $m=0,1,2,3,4$ with the number of copies $M=500$.
Again the lines and dots are inverse of FI and errors obtained by simulation averaged over $3000$ trials using MLE.
Throughout the paper, error bars represent twice the standard deviation of the obtained estimation error divided by the square root of the number of simulation runs.
}
\label{fig:dis1}
\end{figure}
To see whether practical errors can reach the bounds given by classical FI, we have performed a numerical simulation of the full protocol of estimating displacement with Fock state probes and PNRD.
For each scenario given by a different combination of $m$ and $N_c$, we have generated $3000$ sets of simulated data $\mathbb{D}$ with probability distributions $p(n|N_c)$ and evaluated them with a MLE, $N_c^{\text{est}}$, obtained by numerical maximization over a finite range, corresponding to a prior knowledge, of the log-likelihood function:
\begin{align}\label{likeli}
\log L(\mathbb{D}\equiv\{n_k\}|N_c)=\sum_{k=0}^\infty n_k \log p(k|N_c),
\end{align}
where $n_k$ is the number of outcomes for $k$ photons.
The estimated value $N_c^\text{est}$ was then compared to the true value $N_c$ to obtain the estimation error $\Delta^2 N_c = \langle(N_c^\text{est} - N_c)^2\rangle$ and compared to the QCR bound.
The results of the simulations can be seen in Fig.~\ref{fig:dis1}, where the simulated runs, marked by points, are compared to the bounds derived from quantum FI, represented by lines. In Fig.~\ref{fig:dis1}a we can see that, for probe in state $|3\rangle$, the realistic estimation error shows the same scaling as the bounds given by FI, saturates this bound already for $M = 500$
and that this scaling does not depend on the estimated value.
Both the dependence on $M$ and $N_c$ show that the the QCR bound is practically achievable with a finite $M$. Fig.~\ref{fig:dis1}b then confirms that this behavior holds even for probes prepared in different Fock states. It is worth noting that when the signal stength is small $N_c\ll1$, the process becomes a binary outcome estimation problem of $n_{m-1}+n_{m+1}$ and $n_m$ in Eq.~\eqref{weakML} so that the CR inequality is saturated by the ML estimator for any number of copies $M$ as shown in \eqref{weakVar}.
We emphasize that phase is not the only parameter that has a fluctuation in realistic experiment.
Fluctuations of the strength of the displacement, which may arise from the fluctuation of an auxiliary pump, can also be considered.
In this case, we can assume that the strength of the signal is a random variable following a normal distribution with a mean $N_c$ and a given variance depending on the amount of fluctuation and that the aim is to estimation the mean value $N_c$. We have numerically checked that in this case, the estimation error is additively increased by the amount of the fluctuation, which is shown in Appendix C. Keeping this in mind, from now on, we assume the fluctuation is small enough to neglect.
\begin{figure}[b]
\includegraphics[width=250px]{dis_limit.eps}
\caption{Comparison between the lower bounds on estimation error given by the inverse of classical FI of Fock states with PNRD (dots), quantum FI of squeezed states with mean photon number equal to $m$ (solid lines), and quantum FI of coherent states (dashed lines) with mean photon number equal to $m$. Quantum FI for squeezed states was calculated numerically. Different colors correspond to different values of the estimated parameter, $N_c=0.1, 0.3, 0.5, 1.0, 2.0$, and are specified by the color bar.
The squeezing required for energies equivalent to $m = 2,4,6,8,10$ amounts to 10.0,12.5, 14.1, 15.3, 16.2 dB.
}
\label{fig:dis_limit}
\end{figure}
Since quantum FI for Fock state probes is achievable by estimation with PNRD, we can use it for further analysis. For comparison we can consider practical Gaussian estimation methods employing Gaussian probes and stable phase between the input and the measured state.
Notice that even if we assume stable phase for Gaussian probes and measurements, the phase of quantum operation of interest can be still random and the quantum operation is characterized by Eq.~\eqref{evolution_map}.
Even though the operation is random in phase, the stable phase is actually the most optimistic scenario for the Gaussian tools, because it enables noise reduction coming from squeezed vacuum fluctuations \cite{schafermeier2018deterministic}. If we lifted this assumptions and considered phase randomized Gaussian states or, equivalently, Gaussian states with phase randomized detection, their performance would be necessarily worse than that of Fock states.
For our comparison we shall therefore consider quantum FI of phase-sensitive Gaussian states which might be higher than what can be achievable by Gaussian measurements.
In this way we are comparing realistic estimation based on Fock states with the upper bound for Gaussian states.
Let us denote the output state from a channel with a parameter $\theta$ as $\hat{\rho}_\theta$.
In our case, the unknown parameter $\theta$ corresponds to the amount of energy $N_c$ pumped by the displacement.
The quantum FI of these density matrix can be found \cite{braunstein1994statistical}:
\begin{align}\label{eq:fisherfid}
H(\theta)=\frac{4[1-\mathcal{F}(\hat{\rho}_\theta,\hat{\rho}_{\theta+d\theta})]}{d\theta^2},
\end{align}
where $\mathcal{F}(\hat{\rho}_0,\hat{\rho}_1)=\left(\text{Tr}\sqrt{\hat{\rho}_0^{1/2}\hat{\rho}_1\hat{\rho}_0^{1/2}}\right)^2$ is the quantum fidelity between two quantum states $\hat{\rho}_0$ and $\hat{\rho}_1$.
Any Gaussian probe in a pure state can be expressed as a displaced squeezed vacuum state $\hat{D}(\beta)\hat{S}(\zeta)|0\rangle$.
Finding the optimal Gaussian probe requires maximization of the FI over the two parameters $\beta$ and $\zeta$ under the chosen constraints such as the total mean photon number in the input state.
One can easily check that the value of $\beta$ does not change the precision; thus, the optimal Gaussian probe is a squeezed state without any displacement possessing the lowest energy.
In Fig.~\ref{fig:dis_limit} we show the comparison of classical FI for Fock probes with PNRD, marked by dots, quantum FI of optimized Gaussian probes with equal energy, marked by solid lines, and quantum FI for vacuum state, marked by dashed lines, for the estimation of the unknown phase-insensitive displacement operations with various $N_c$.
We can see that the Fock state probes are superior to optimal Gaussian probes with the same mean photon number for the entire range of displacement strengths even though the former requires no phase stability and the latter may use arbitrary coherent detection schemes. This improvement is most prominent for large values $N_c$.
Note that since quantum FI is used to assess the achievable estimation precision of Gaussian states, and the final state from a squeezed state probe is generally phase-sensitive, we are implicitly assuming that a stable reference beam outside of the sensor is prepared and may be properly used for phase-sensitive measurement.
Without this reference the state needs to be treated as phase randomized squeezed state, which always performs worse than Fock state with equivalent energy, and is even definitely inferior to the vacuum state for low energies.
Let us now discuss the effects of optical imperfections, such as losses, to ensure the validity of the results in practical scenarios.
The photon-loss process, which is the main imperfection for light, can be described by quantum master equation in the interaction picture as \cite{walls2007quantum}
\begin{align}\label{mastereq}
\frac{d\hat{\rho}}{dt}=\frac{\gamma}{2}\left(2\hat{a}\hat{\rho}\hat{a}^{\dag}-\hat{\rho}\hat{a}^\dagger\hat{a}-\hat{a}^\dagger\hat{a}\hat{\rho}\right),
\end{align}
where $\gamma$ is the loss parameter.
The loss rate is defined as $1-\eta=1-e^{-\gamma t}$ with $t\ge 0$ describing the monotonous decay of the coherence terms.
This dynamics can be equivalently described with a virtual beam splitter interaction coupling the probe with a zero temperature bath.
The losses can manifest either before the sample and thus represent the degradation of the probe, or after the sample and be related to imperfection of the measurement.
In the case of displacement estimation, the two losses act in almost the exactly the same way, only the one after the channel also reduces the measured quantity. To be more exact, if we represent the losses by completely positive trace-preserving map \cite{escher2011general}
\begin{equation}\label{decoherencemap}
\mathcal{L}_{\eta}(\hat{\rho}) = \sum_{k = 0}^{\infty} \hat{A}_k \hat{\rho} \hat{A}_k^{\dag},\quad \hat{A}_k = \frac{\sqrt{1-\eta}^k}{\sqrt{k!}}\sqrt{\eta}^{\hat{a}^{\dag}\hat{a}}\hat{a}^k,
\end{equation}
then the losses after the displacing sample and before the displacing sample can be related as $\mathcal{L}_{\eta}[ \hat{D}(\sqrt{N_c})\hat{\rho}_{in} \hat{D}^{\dag}(\sqrt{N_c})] = \hat{D}(\sqrt{\eta N_c})\mathcal{L}_{\eta}(\hat{\rho}_{in} )\hat{D}^{\dag}(\sqrt{ \eta N_c})$. We can see that the losses affect the probe in exactly the same way and the only difference is in scaling of the estimated parameter. Displacement adding energy $N_c$ before the loss is equivalent to displacement adding energy $\eta N_c$ after the loss. In addition, this behavior remains also for the Gaussian states. For the sake of simplicity we can therefore consider only the losses before the sample. It should be noted that we assume $\eta$ is known through prior measurements and not a subject of the estimation.
\begin{figure}[t]
\includegraphics[width=480px]{dis_loss.eps}
\caption{Simulation for displacement estimation in the presence of photon-loss before the sample.
Dots and solid lines represent estimation errors and inverse quantum FI.
(a) Against the number of copies $M$ for different loss rates, $\eta = 1, 0.9, 0.7$ for $N_c = 1.0$ and $|m = 3\rangle$.
(b) Against the loss rate $1-\eta$ for different values of the measured displacement $N_c=0.1, 1.0, 2.0$ with input probe $|m=3\rangle$ and $M=500$.
Classical limits (dashed lines) given by coherent states, and optimal Gaussian limits (dotted lines) by squeezed state with the same energy are evaluated from the inverse quantum FI without the losses.
}
\label{fig:dis_loss_limit}
\end{figure}
In numerical simulation shown in Fig.~\ref{fig:dis_loss_limit}, the theoretical Fock state distribution $p(n|N_c,\eta)$ was used to generate 1000 sets of data, which were then used, through MLE algorithm, to obtain the unknown value $N_c$ and estimate the error $\Delta^2 N_c$. The theoretical distribution $p(n|N_c,\eta)$ was obtained by using virtual beam splitter model represented by Eq.~\eqref{mastereq}. In Fig.~\ref{fig:dis_loss_limit}(a) are the numerically obtained errors for probe state $|3\rangle$ for three different levels of loss, represented by points, compared to quantum FI of the same probe states represented by solid lines. We can see that the errors saturate the CR bound even in the presence of loss and keep the same scaling as the ideal scenario for various $M$. In Fig.~\ref{fig:dis_loss_limit}(b) are the same errors plotted with respect to range of losses and compared to optimal Gaussian states with the same mean photon number and without losses, represented by dashed lines, and to vacuum states, represented by dotted lines. We can see that even in the presence of losses, the estimation based on Fock states surpasses even the optimal methods using Gaussian states and optimal coherent measurements. We can also see different trends that appear for the comparison of Fock states to Gaussian and classical limits as the $N_c$ changes. As $N_c$ decreases, higher losses can be tolerated before the Fock state estimation falls behind the classical limit, or the shot-noise limit \cite{giovannetti2004quantum,giovannetti2006quantum,giovannetti2011advances, pirandola2018advances}, but at the same time lower losses are enough to fall behind methods using optimal Gaussian states. We have observed similar behavior for larger Fock states up to $m=7$ from a numerical calculation of Fisher information.
\section{Squeezing estimation with Fock state probe}
Let us now turn to the scenario in which we are interested only in the strength of an unknown squeezing operation that can be represented by map $\mathcal{M}$ in (\ref{evolution_map}) with $N_c = 0$.
We can analyze this scenario in the same way as the previous one. After this phase-insensitive squeezing operation, a Fock state $|m\rangle$ transforms to a mixed state
\begin{align}
\hat{\rho}_f(N_s)=\int_0^{2\pi}\frac{d\phi}{2\pi}\hat{S}(\sqrt{N_s}e^{i\phi})|m\rangle\langle m|\hat{S}^\dagger(\sqrt{N_s}e^{i\phi})=\sum_{n=0}^{\infty}p(n|N_s)|n\rangle\langle n|,
\end{align}
where \cite{kim1989properties}
\begin{align}
p(n|N_s)=|\langle n|\hat{S}(\sqrt{N_s})|m\rangle|^2&=\frac{n!m!}{2^{n-m}}\frac{\tanh^{n-m}\sqrt{N_s}}{\cosh^{2m+1}\sqrt{N_s}} S(\sqrt{N_s},m,n) ~~\text{ when } |m-n| \text{ is even} \\
&=0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{ when } |m-n| \text{ is odd},
\end{align}
with
\begin{align}
S(r,m,n)=\left|\sum_k \frac{(-1)^k \sinh^{2k}r}{2^{2k}k!(m-2k)![k+(n-m)/2]!}\right|.
\end{align}
Here, the sum is taken for integers $k$ for which the argument of the factorials is positive.
For input Fock states we can explicitly calculate the quantum FI, which is again equal to classical FI for measurement in Fock state basis,
\begin{align}
F(N_s)=\frac{m^2+m+1}{2N_s}.
\end{align}
The derivation is similar to that of displacement estimation and is supplied in Appendix A.
We can thus analyze the estimation errors again in realistic scenario by numerical simulation.
Similar to the displacement estimation case, as the strength of the signal $N_s$ decreases, the estimation error by ML estimator of \eqref{weakML} saturates the QCR bound for any number of $M$ as shown in \eqref{weakVar}.
\begin{figure}[t]
\includegraphics[width=500px]{sq2_r.eps}
\caption{Simulation for squeezing parameter estimation.
The solid lines and circle dots represent the estimation error based on CR bound and simulation averaged over 3000 trials using MLE with the full number statistics $p(n|N_s)$.
(a) Against the number of copies $M$ for $N_s=0.1, 0.3,0.5$.
(b) Against the energy added by squeezing $N_s$ for $m=0,1,2,3,4$.
The number of copies for simulation is $M=500$.}
\label{fig:sq}
\end{figure}
In Fig. \ref{fig:sq}(a) and (b), these numerically obtained errors, marked by dots, are shown relative to the number of copies $M$ and value of the measured squeezing $N_s$, respectively. They are again compared to quantum FI represented by solid lines.
In both cases we can see that the estimation with Fock states and PNRD again approaches the precision predicted by the CR bound for a broad range of parameters.
Increasing the photon number $m$ of the Fock state probe leads to better performance for the estimation of the phase-insensitive squeezing operation.
Again, we note that the fluctuation of $N_s$ can also be considered and the same behavior as displacement estimation is presented in Appendix C.
Again we can compare the performance of Fock states to the Gaussian methods. When the Gaussian methods cannot take advantage of stable phase, the Gaussian probes can be expressed as mixtures of Fock states and therefore exhibit inferior performance. Numerical tests have confirmed that, in contrast to the displacement estimation, phase randomized squeezed states with arbitrary energy always perform worse than the vacuum state.
To see the limits of the Fock-state-based estimation, we compare them to the optimal Gaussian estimation that takes advantage of phase reference. Here, in contrast to the displacement, a pre-displacement of the probe improves the estimation contrast while increasing mean photon number of the state of the probe. However, our numerical simulations revealed that the improvement gained by this displacement is, with regards to the number of photons added, smaller than what would be gained by additional squeezing with the same number of added photons.
We therefore compared the estimation error obtained by Fock states with the FI of Gaussian probes of coherent states or squeezed vacuum states without loss to obtain a strict threshold.
Fig.~\ref{fig:sq_comp} shows the mean photon number of (a) squeezed or (b) coherent states that have the same estimation error as the inverse of the classical FI exhibited by various Fock states and PNRD.
The comparison shows that the Fock states are more energy-efficient than the Gaussian states in most cases. This is not the case for estimation of squeezing for small values of $N_s$ and small $m$. This is a rather interesting realization; metrology with Gaussian states can be applicable even in scenarios seemingly favouring the symmetry of Fock states.
It should be noted that in optical sensing, Gaussian states are generally easier to prepare than Fock states, but the difficulty varies wildly. Coherent states can be prepared routinely and are significantly more feasible than impure squeezed states. Preparing completely pure squeezed states, on the other hand, has difficulties comparable to preparation of Fock states. Comparison of equal mean photon numbers in Fig.~\ref{fig:sq_comp} shows that to attain the same precision with Gaussian states, significantly higher energy is required, which might be an issue for some applications \cite{taylor2016quantum}.
It is worthwhile to emphasize that our numerical calculation of quantum FI of Gaussian states showed by fitting with respect to the mean photon number that the scaling of quantum FI of coherent state and squeezed state is linear and quadratic with the mean photon number of the probe.
\begin{figure}[b]
\includegraphics[width=500px]{sq_comp}
\caption{Average photons required to attain the same quantum FI as classical FI of Fock states with PNRD (a) using squeezed states, (b) using coherent states. The left (right) plot shows that in overall, squeezed (coherent) states require more photon numbers than Fock states.}
\label{fig:sq_comp}
\end{figure}
\begin{figure}[t]
\includegraphics[width=480px]{sq_loss}~~~~~~~~~~
\caption{Simulation for squeezing estimation with a Fock state $|m=3\rangle$ as an input probe in the presence of photon-loss.
The dots represent the estimation error based on simulation using MLE with the full number statistics $p(n|N_s)$, which are obtained by averaging over 1000 trials with $N_s=0.25$.
(a) Against the number of copies $M$.
Similarly to displacement estimation, the CR bound is saturated by MLE for a wide range of $M$ in the presence of photon-loss.
(b) Against loss rate $1-\eta$.
The dotted and dashed lines represent the estimation error limit of squeezed states and of coherent states without photon-loss, respectively. These classical estimation limits were evaluated as inverse of quantum FI for the states. The figure clearly showcases several scenarios in which Fock states subjected to loss provide better precision than pure Gaussian states.
}
\label{fig:sq_loss}
\end{figure}
The bounds for coherent and squeezed states can now help us in evaluating the performance of the estimation with Fock states under losses.
The first important observation is that there is no simple relation in the estimation of squeezing between the effect of losses before and after the sample.
This is because squeezing can lead to entanglement between the probe and the after-sample-bath, which then alters the properties of the probes.
However, in the limit of low values of estimated parameter $N_s \ll 1$, the effect of squeezing is linearized and this difference can be neglected.
In this regime, losses after the channel would alter the estimated value, but the qualitative behavior of the error rates would remain the same. Since this is the regime we are most often interested in, we can again, for the sake of simplicity of analysis, consider only the case with losses before the channel.
We again performed numerical simulation for the estimation errors of squeezing under loss, which are shown in Fig.~\ref{fig:sq_loss} (a) relative to number of copies $M$ and (b) relative to the loss rate $1-\eta$.
Both figures show that, similarly to the scenario of displacement estimation, the estimation errors under loss approach the CR bound and that the scaling with M remains consistent.
While for small values of $N_s$ and $m$ the Fock states and squeezed states with optimal coherent detection are comparable such as $N_s = 0.1$ and $m=3$, the Fock states enable attaining a better scaling of precision for large $N_s$ and $m$ as suggested by Fig.~\ref{fig:sq_comp} (b) and Fig.~\ref{fig:sq_loss} (b) for small $1-\eta$.
\section{Simultaneous estimation of displacement and squeezing}
Finally, let us consider a general scenario in which both quantities, $N_c$ and $N_s$, appear at the same time and are estimated simultaneously.
This can be part of characterization of a general Gaussian process.
Another way this scenario can arise is when squeezing, the nonlinear process we want to characterize, is accompanied by noise with Poissonian distribution that can not be separated from the process. In this scenario we need to attempt simultaneous estimation of both quantities even though we are only ultimately interested in one.
After the general channel (\ref{evolution_map}), the Fock state probe will be transformed to
\begin{align}\label{simul_map}
\hat{\rho}_f(N_c,N_s) = \mathcal{M}(\hat{\rho}_{in})
=\sum_{n,k=0}^\infty w(n|k)q(k|m)|n\rangle\langle n|=\sum_{n=0}^\infty p(n|m) |n\rangle\langle n|,
\end{align}
where $q(k|m)=|\langle k| \hat{S}(r)|m\rangle|^2$, $w(n|k)=|\langle n|\hat{D}(\alpha)|k\rangle|^2$, and $p(n|m)=\sum_{k=0}^\infty w(n|k)q(k|m)$.
The lower bound of error on simultaneous estimation of $N_c$ and $N_s$, or multiparameter CR bound, is given by the quantum FI matrix
$C\geq H^{-1}$ \cite{helstrom1976quantum, szczykulska2016multi, gessner2018sensitivity, liu2019quantum} with covariance matrix $C_{N_c,N_c}=\langle (N_c^\text{est}-N_c)^2\rangle, C_{N_s,N_s}=\langle (N_s^\text{est}-N_s)^2\rangle, C_{N_c,N_s}=C_{N_s,N_c}=\langle (N_c^\text{est}-N_c)(N_s^\text{est}-N_s)\rangle$.
Here the matrix inequality $A\geq B$ means that $A-B$ is a positive semi-definite matrix.
Since the final state is always diagonal in the Fock basis, the quantum FI matrix, which is the same as the classical FI matrix based on the PNRD, can be written as
\begin{align}
H(N_c,N_s)=
\begin{pmatrix}
H_{N_c,N_c} & H_{N_c,N_s} \\
H_{N_s,N_c} & H_{N_s,N_s}
\end{pmatrix},
\end{align}
with
\begin{align}
H_{x,y}&=\sum_{n=0}^\infty \frac{1}{p(n|m)}\left(\frac{\partial p(n|m)}{\partial x}\right)\left(\frac{\partial p(n|m)}{\partial y}\right), \\
\end{align}
where $x,y$ are in $\{N_c, N_s\}$.
The classical multiparameter CR bound can also be asymptotically saturated by ML estimator.
From the multiparameter CR bound, we can extract the estimation errors of each parameter,
\begin{align}
\Delta^2 N_c&\geq \frac{H_{N_s,N_s}}{H_{N_c,N_c}H_{N_s,N_s}-H_{N_c,N_s}^2}=H_{N_c,N_c}^{-1}\left(1-H_{N_c,N_s}^2/H_{N_c,N_c}H_{N_s,N_s}\right)^{-1},~~~ \label{mcrb1}\\
\Delta^2 N_s&\geq \frac{H_{N_c,N_c}}{H_{N_c,N_c}H_{N_s,N_s}-H_{N_c,N_s}^2}=H_{N_s,N_s}^{-1}\left(1-H_{N_c,N_s}^2/H_{N_c,N_c}H_{N_s,N_s}\right)^{-1}.\label{mcrb2}
\end{align}
When more than one parameter in the process are involved, two main difficulties arise that may degrade the estimation error \cite{helstrom1976quantum, proctor2018multiparameter}.
First of all, as shown in inequalities ~\eqref{mcrb1} and \eqref{mcrb2}, the off-diagonal elements of FI matrix decrease the estimation error for fixed diagonal elements.
Non-vanishing off-diagonal elements of FI matrix imply that the parameters interplay each other in the process, so that one needs to know the other parameters in order to estimate a parameter of interest precisely.
On the other hand, when the off-diagonal element of the FI matrix vanishes, the estimation errors reduce to
\begin{align}
\Delta^2 N_c&\geq H_{N_c,N_c}^{-1},~~~ \label{mcrb_approx1}\\
\Delta^2 N_s&\geq H_{N_s,N_s}^{-1}.\label{mcrb_approx2}
\end{align}
Thus, when the off-diagonal element of the FI matrix is much smaller than the diagonal elements, the estimation error of each parameter is bounded by the inverse of each diagonal element of the FI matrix.
In this case, we can interpret the inequalities as that of a single-parameter estimation where any information about the other parameters is not required to estimate the parameter of interest.
The second difficulty is that even when one is estimating a single parameter, since the other parameters are involved, the diagonal elements of quantum (classical) FI may decrease.
For instance, when we estimate the squeezing parameter $N_s$, the displacement process in Map \eqref{simul_map}, written as $w(n|k)$, plays a role of a noisy process in the measurement setup.
Similarly when we estimate the displacement parameter $N_c$, the squeezing process in Map \eqref{simul_map}, written as $q(k|m)$, plays a role as a preparation error.
Thus, generally when more than one parameter is involved, the estimation error may decrease.
We investigate our case by numerical simulation focusing on these difficulties.
\begin{figure}[t]
\includegraphics[width=480px]{sim}~~~~~~~~~~~~~
\caption{Simultaneous estimation errors for $N_c$ (blue lines) and $N_s$ (red lines).
More specifically, solid lines with circles represent the lower bound of the esimation error in Eq.~\eqref{mcrb1} and Eq.~\eqref{mcrb2}, respectively.
Dashed lines with triangles represent the quantum FI of Fock state probe for estimating $N_c$($N_s$) when $N_s$($N_c$) is $0$ and known.
(a) shows values for estimating $N_c = N_s = 0.01$, (b) shows values for estimating $N_c = N_s =0.05$.
}
\label{sim1}
\end{figure}
In Fig.~\ref{sim1}(a) and (b), we show the numerically calculated the lower bounds of the estimation errors for two different sets of measured values.
We numerically confirmed that the off-diagonal elements are very small compared to the diagonal elements ($H_{N_c,N_s}^2/H_{N_c,N_c}H_{N_s,N_s}<10^{-3}$ in Fig.~ \ref{sim1}(a) and $H_{N_c,N_s}^2/H_{N_c,N_c}H_{N_s,N_s}<1.2\times10^{-2}$ in Fig.~ \ref{sim1}(b)), which means that we do not suffer from the first difficulty in this regime and that the inverse of the diagonal components of FI matrix approximately give the lower bounds of the estimation errors of each quantity as written in Eqs.~\eqref{mcrb_approx1} and \eqref{mcrb_approx2}.
This is best seen for small values of estimated parameters in Fig.~\ref{sim1}(a) where the off-diagonal elements are truly negligible and errors are practically identical, whether both quantities are measured or just single ones.
The reason why simultaneous estimation for small values of estimated paramters works well is that when the estimated parameters are small, displacement and squeezing operations do not interfere each other because they are single-photon and two-photon processes, respectively, as emphasized in Eqs.~\eqref{eq:binary1}-\eqref{eq:binary2}.
In Fig.~\ref{sim1}(b), where the estimated parameters are not as small, the quantities start disturbing each other and the deviation from the individual estimations increase.
When the estimated parameters are not small enough, the role of the other parameter as a noisy process becomes so dominant that highly nonclassical states such as Fock states of large photon numbers may cease to give a small estimation error as shown in Fig.~\ref{sim1}(b).
\section{Conclusion}
In this work, we have investigated the possibility of using Fock states and photon number resolving detectors for parameter estimation of single mode Gaussian unitaries in the absence of stable phase reference.
This scenario is relevant in optical sensing when stable phase reference is unavailable through the sample, for example because of random nature of the examined operation and the light emitted by the sample is weak.
To accommodate both points of view, we have evaluated the performance of Fock states under realistic environment and compared them to the optimal performance of phase-sensitive Gaussian states with no loss and optimal quantum measurement. We found out that for estimation of both weak displacement and squeezing, the Fock states together with Fock basis detection can, already for ensemble of 500 trials saturate the Cram\'{e}r-Rao bound and provide error rates surpassing optimal Gaussian states with equivalent mean photon number.
Loss incurred in the sample or during preparation of the probes limits the quality of the estimation. The influence generally depends on the strength of the measured interaction. When $N_c,N_s \approx 0.5$, Fock states outperform the Gaussian bounds even when affected by 20\% losses. When $N_c,N_s\approx 0.1$, loss of 20\% can be tolerated when compared to coherent states, but less then 5\% loss brings the Fock states above the level of pure squeezed states. Interestingly enough, when estimating squeezing with low energy probes, Gaussian squeezed states surpass the Fock states even though the symmetry of the operation favors them.
Simultaneous estimation of both squeezing and displacement is also possible without being disturbed from each other and it works best in the limit of small parameters, $N_c,N_s <0.1$, when the operations are effectively independent. Together, these features can allow either multi-parameter estimation of an optical Gaussian process in various systems, including atomic physics and solid-state physics, or estimation of new squeezing processes under inherent Poissoninan noise. The method can also be extended for estimation of higher order processes which encompass joint $n$-photon effects.
Experimental application of the procedure relies on Fock states and measurements in Fock basis. The measurement requires photon number resolving detectors. Transition edge sensors (TES) \cite{lita2008counting, calkins2013high, marsili2013detecting, harder2016single, burenkov2017full} is well known to be promising in this area as they are already capable of resolving up to 12 photons with estimated 0.98 detection efficiency \cite{sperling2017detector}. Alternatively, detector with photon number resolving capability can be constructed from an array of on-off detectors \cite{achilles2006direct, avenhaus2008photon, usuga2010noise, yukawa2013generating, harder2016local, yukawa2013generating, cooper2013experimental, bohmann2018incomplete}, or it can be, for purposes of proof-of-concept tests, replaced altogether by homodyne tomography. The photon number resolving detectors can be also used for preparation of the Fock states for the probes. Detecting a specific Fock state in one mode of a two-mode squeezed state generated by Optical Parametric Oscillator (OPO) projects the other mode into the same Fock state and is a technique often employed in quantum optics. It is also possible to generate the necessary Fock states by merging single photon states \cite{motes2016efficient}, which can be generated by quantum dots \cite{bulgarini2014nanowire, ding2016demand, senellart2017high, dusanowski2019near, ollivier2020reproducibility}. A proof-of-principle experimental test of the estimation method could be immediately realized with Fock states $|1\rangle$ or $|2\rangle$, conditionally obtained from an OPO, measured by TES and homodyne tomography for the verification purposes.
Coherent displacement can appear by a weak crosstalk to a different mode occupied by a coherent state in an optical system \cite{jeong2014generation}, using the optomechanical coupling in an optomechanical system \cite{sekatski2014macroscopic}. Finally, the detection method can be also considered outside the area of quantum optics. For example,
Fock states were already employed for estimation of displacement and can be considered for estimation of squeezing on the same platform \cite{ge2019trapped, drechsler2020state}.
\section*{Acknowledgement}
We acknowledge project 19-19722J of the Grant Agency of Czech Republic (GA\v{C}R).
C. O. acknowledges support from NSF (OMA-1936118).
K. P. acknowledges Danish National Research Foundation through the Center of Excellence for Macroscopic Quantum States (bigQ, DNRF142).
R. F. also acknowledges the MEYS of the Czech Republic (grant agreements $02.1.01/0.0/0.0/16\_026/0008460$ and 8C20002) and the funding from European Union's Horizon 2020 (2014-2020) research and innovation framework programme under grant agreement No 731473 (ShoQC). Project ShoQC has received funding from the QuantERA ERA-NET Cofund in Quantum Technologies implemented within the European Union's Horizon 2020 Programme.
H. J. acknowledges the National Research Foundation of Korea through grants funded by the Korea Government (NRF-2018K2A9A1A06069933, NRF-2019M3E4A1080074 and NRF-2020R1A2C1008609).
\section*{Appendix A: quantum Fisher information}
\setcounter{equation}{0}
\renewcommand{\theequation}{A\arabic{equation}}
Let us consider a unitary operation $\hat{U}=e^{-i \hat{H} \theta}$ where the Hamiltonian $\hat{H}$ generates $\theta$.
In the case of displacement, $\hat{H}=\hat{a}+\hat{a}^\dagger$, and for squeezing, $\hat{H}=i(\hat{a}^2-\hat{a}^{\dagger2}$).
After the phase-randomized displacement or squeezing operation, the output state can be written as
\begin{align}
\hat{\rho}_f=\sum_{n=0}^\infty p(n|m)|n\rangle\langle n|=\sum_{n=0}^\infty |\langle n|\hat{U}|m\rangle|^2|n\rangle\langle n|.
\end{align}
Let us first consider displacement estimation, $\hat{U}=e^{\alpha \hat{a}^\dagger-\alpha^* \hat{a}}$.
Since the final state is diagonal in Fock basis, the derivation of classical Fisher information of $p(n|m)$ is sufficient.
The classical Fisher information for $|\alpha|$ is written as
\begin{align}
F(|\alpha|)=\sum_{n=0}^\infty \frac{1}{p(n|m)}\left(\frac{\partial p(n|m)}{\partial |\alpha|} \right)^2.
\end{align}
Assuming $\alpha$ to be real without loss of generality, the differential term is simplified as
\begin{align}
\left(\frac{\partial p(n|m)}{\partial |\alpha|} \right)^2&=\left(-\langle n|\hat{D}(\alpha)|m\rangle\langle m|\hat{D}^\dagger(\alpha)(\hat{a}^\dagger-\hat{a})|n\rangle +\langle n|\hat{D}(\alpha)(\hat{a}^\dagger-\hat{a})|m\rangle\langle m|\hat{D}^\dagger(\alpha)|n\rangle \right)^2 \\
&=\left(\langle n|\hat{D}(\alpha)|m\rangle\langle m|\hat{D}^\dagger(\alpha)(\hat{a}^\dagger-\hat{a})|n\rangle\right)^2+\left(\langle n|\hat{D}(\alpha)(\hat{a}^\dagger-\hat{a})|m\rangle\langle m|\hat{D}^\dagger(\alpha)|n\rangle\right)^2 \nonumber \\
&-2\langle n|\hat{D}(\alpha)|m\rangle\langle m|\hat{D}^\dagger(\alpha)(\hat{a}^\dagger-\hat{a})|n\rangle\langle n|\hat{D}(\alpha)(\hat{a}^\dagger-\hat{a})|m\rangle\langle m|\hat{D}^\dagger(\alpha)|n\rangle \\
&=4p(n|m)\langle m|\hat{D}^\dagger(\alpha)(\hat{a}^\dagger-\hat{a})|n\rangle\langle n|\hat{D}(\alpha)(\hat{a}-\hat{a}^\dagger)|m\rangle.
\end{align}
Finally, we obtain the Fisher information
\begin{align}
F(|\alpha|)=4\langle m|\hat{D}^\dagger(\alpha)(\hat{a}^\dagger-\hat{a})\hat{D}(\alpha)(\hat{a}-\hat{a}^\dagger)|m\rangle=4\langle m|(\hat{a}^\dagger-\hat{a})(\hat{a}-\hat{a}^\dagger)|m\rangle=8m+4.
\end{align}
Simiarly, one can check that $F(r)=2(m^2+m+1)$.
Since we are interested in the Fisher information about $N_c=|\alpha|^2$, one can use the chain-rule for the classical Fisher information such as,
\begin{align}
F(N_c)=\sum_{n=0}^\infty \frac{1}{p(n|m)}\left(\frac{\partial p(n|m)}{\partial N_c} \right)^2=\frac{1}{4N_c}\sum_{n=0}^\infty \frac{1}{p(n|m)}\left(\frac{\partial p(n|m)}{\partial |\alpha|} \right)^2=\frac{F(|\alpha|)}{4N_c}=\frac{2m+1}{N_c}.
\end{align}
Similarly, the Fisher information about $N_s=r^2$ is obtained by
\begin{align}
F(N_s)=\sum_{n=0}^\infty \frac{1}{p(n|m)}\left(\frac{\partial p(n|m)}{\partial N_s} \right)^2=\frac{1}{4N_s}\sum_{n=0}^\infty \frac{1}{p(n|m)}\left(\frac{\partial p(n|m)}{\partial r} \right)^2=\frac{F(r)}{4N_s}=\frac{m^2+m+1}{2N_s}.
\end{align}
\section*{Appendix B: Estimating displacement by mean intensity measurement}
\setcounter{equation}{0}
\renewcommand{\theequation}{B\arabic{equation}}
The advantage of the Fock state probes requires the Fock state measurement to be fully utilized. This can be shown by considering a different, more limited, measurement that measures only the average moments instead of the full Fock state distribution. This is the case of coarse grained intensity detectors which average over the individual results and allow acces only to moments
\begin{align}
\langle\hat{n}\rangle&=m+N_c \\
\langle\hat{n}^2\rangle&=m^2+2N_c(2m+1)+N_c^2 \\
\Delta^2 \hat{n}&=\langle\hat{n}^2\rangle-\langle\hat{n}\rangle^2=2N_c(m+1).
\end{align}
The signal-to-noise ratio (SNR) can be defined where the signal is the change in the average photon number of the probe and the noise is the uncertainty in the photon number as
\begin{align}
\text{SNR}=\frac{\langle \hat{n}\rangle-m}{\sqrt{\Delta^2 \hat{n}}}=\sqrt{\frac{N_c}{2(m+1)}}.
\end{align}
Here, we notice that SNR has completely inverse tendencies from the Fisher information against $m$ and $N_c$, i.e. a larger precision for for a small $m$ and a large $N_c$.
The linearized sensitivity $(\Delta^2 N_c)_{\hat{O}}=\Delta^2\hat{O}/|\partial\langle\hat{O}\rangle/\partial N_c|^2$ based on $\langle\hat{n}\rangle$ gives an information about the uncertainty of $N_c$ by measuring the observable $\hat{O}$ which is given as
\begin{align}
(\Delta^2 N_c)_{\hat{n}}=\frac{\Delta^2 \hat{n}}{\left(\partial \langle \hat{n}\rangle /\partial N_c \right)^2}=2N_c (m+1).
\end{align}
Similarly, we can test the linearized sensitivity based on the second moment,
\begin{align}
\langle\hat{n}^4\rangle-\langle\hat{n}^2\rangle^2&=2(4 m+1) N_c^3+\left(18 m^2+2 m+3\right) N_c^2+\left(8 m^3+2 m^2+6 m+1\right) N_c\simeq 8m^3 N_c, \\
\frac{\langle\hat{n}^4\rangle-\langle\hat{n}^2\rangle^2}{(\partial\langle\hat{n}^2\rangle/\partial N_c)^2}&\simeq \frac{1}{2}mN_c,
\end{align}
where the approximation is under the condition of a small $N_c$ and a large $m$.
It shows that the error based on the second moment also increases with $m$.
Note that a high linearized sensitivity indicates a less precise measurement, or a large error.
The error increases with $m$ and the best probe is the vacuum probe, and therefore the Fock state probes do not give any advantage.
Therefore, we know that the gain in Fisher information of Eq.~\eqref{nc_fisher} cannot be obtained by simply using the information about average photon number $\langle \hat{n}\rangle$.
These results imply that the information about mean values does not provide the scalings of sensitivity available by estimation strategies with PNRD.
\section*{Appendix C: Fluctuations of signal}
\setcounter{equation}{0}
\renewcommand{\theequation}{C\arabic{equation}}
In realistic experiments, fluctuations of pumps lead to a fluctuations of signals. Or, the signal itself may have a fluctuation. In this case, a natural approach is to estimate the mean value of the signal.
In this section, we analyze the effect of the fluctuation on the estimation error.
We have simulated the estimation of the strength of the displacement and squeezing, $N_c=1$ and $N_s=0.1$, with a Fock state $|m=3\rangle$.
In the Monte-Carlo simulation, each individual probe passes through a channel with different parameters, displacement and squeezing, and these fluctuate according a normal distribution with the mean $N_c$, $N_s$, respectively.
Fig. \ref{fluc} shows the additional estimation error due to fluctuations of the signals, characterized by the variance of the normal distribution.
Here, the additional estimation error is calculated by the difference between the average estimation error obtained by the Monte-Carlo simulation,
\begin{align}
\Delta^2 N_c\equiv \langle (N_c^\text{est}-N_c)^2\rangle,~~~\Delta^2 N_s\equiv \langle (N_s^\text{est}-N_s)^2\rangle
\end{align}
and the estimation error indicated by Fisher information.
Explicitly, the simulation is performed using the maximum likelihood estimation, and the estimation error from Fisher information is written as $1/MF(N_c)$ and $1/MF(N_s)$.
We can see from the figure that the amount of the fluctuation of the signals adds the estimation error.
It can be understood by noting that when fluctuations are small, Fisher information can be approximated by that at the mean so that the only the fluctuation changes the total estimation error.
Thus, when fluctuation is large as shown in Fig. \ref{fluc} (a), one observes the deviation because the Fisher information cannot be simply approximated by that at the mean.
In conclusion, assuming that the fluctuation of the signal strength is small enough, Fisher information at the mean is an essential quantity that determines the relevant estimation error.
\begin{figure}[t]
\includegraphics[width=400px]{fluctuation}~~~~~~~~~~~~~
\caption{Additional estimation error that arises from fluctuations of the signals of (a) displacement and (b) squeezing. The additional estimation errors are the same as the fluctuations of the signals (solid lines) when fluctuations are small. We have averaged over 1000 trials and used $M=500$ copies.
}
\label{fluc}
\end{figure}
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\section{Introduction}
Statistical mechanics, studying thermal properties of a many-body system from microscopic perspective, have gained huge success in the past century. However, the basic principles of statistical mechanics have not been fully understood; the establishment of micro-cannonical ensemble
has to rely on hypotheses\cite{HuangBook}.
Since microscopic particles --- elements of a macroscopic system --- are
governed by the Schr\"odinger equation, one feels obliged to address the problem
with quantum mechanics. Von Neumann was among the first physicists trying to use
quantum mechanics to understand the basic principles of statistical mechanics. In a 1929 paper\cite{Von1929}, von Neumann proposed a method to construct commutable
macroscopic momentum and position operators and, therefore, quantum phase space.
Within this framework, he introduced an entropy for quantum pure state
and proved two theorems, which he called quantum ergodic theorem and
quantum H-theorem, respectively.
These results are remarkable advances in the establishment of the micro-canonical ensemble,
the foundation of statistical mechanics, without hypothesis. However, von Neumann's
beautiful results have been largely forgotten likely due to
misunderstanding\cite{vonNeumann2010commentary}.
Probably due to the developments in ultra-cold atomic gas
experiments\cite{Kinoshita2006,Schmiedmayer2013njp,Yukalov2011lpl},
we have recently seen tremendous efforts to study the foundation of statistical
mechanics. Many new and beautiful results are obtained \cite{GemmerBook,Srednicki1994PRE,Typicality2006prl,Popescu2006Nphys,Sun2007pra,Rigol2008Nature,goldstein2010normaltypicality,linden2009thermalequilibrium,Reimann2007prl,reimann2008experimental,Kastner2012njp,Cho2010PRL,Ueda2011pre,Ji2011prls,Yukalov2011PLA,Sugiura2012prl,Snoke2012,Wang2012pre,Srednicki2012prl,Eisert2012prl,Emerson2013prl,zhuang2014,Goldstein2014short}.
These efforts have also led to renewed interest in von Neumann's forgotten work;
the English version of his paper is now available~\cite{vonNeumann2010qhtheorem}.
Von Neumann's quantum ergodic theorem has been re-exmained recently\cite{goldstein2010thermalequilibrium}.
In particular, a different version of quantum ergodic theorem was proved by
Reimann~\cite{reimann2008experimental,Short2011njp}. Reimann's ergodic theorem
does not involve any coarse-graining and can be subjected to
numerical study\cite{Zhuang2013pre}. In contrast, much less progress has been
made on the quantum H-theorem and the associated key concepts, such as
macroscopic momentum and position operators, and
entropy for quantum pure states, which were introduced in 1929.
In this work we define a different entropy for quantum pure states
and study its long-time dynamical fluctuation in attempt to improve on
von Neumann's quantum H-theorem\cite{Von1929}. Von Neumann proved his theorem
with the following steps: ({\it i}) construct commutable macroscopic
position and momentum operators; ({\it ii}) define an entropy for pure quantum states
with coarse-graining; ({\it iii}) investigate the long-time behavior of the entropy.
We follow von Neumann's steps with new theoretical tools and perspectives.
For step ({\it i}), we use Kohn's method~\cite{wkohn1973wannierfunctions} to
construct a complete set of Wannier functions that are
localized both in position and momentum space. Such a construction can be implemented numerically with great efficiency. With these Wannier functions, we are able to construct commutable macroscopic position and momentum operators and, therefore, a
quantum phase space, which is divided into cells of size of the Planck constant
and each of these Planck cells is assigned a Wannier function.
The success of step ({\it i}) allows us to map unitarily a pure quantum state
onto the phase space.
We accomplish step ({\it ii}) by defining
an entropy for a quantum pure state based on its probability distribution on the phase space.
Here we do not use coarse-graining used by von Neumann in the context of macroscopic
observables. For our entropy, the total system always has a larger entropy than its
subsystems even if the total system is described by a quantum pure state. This is not the case for
the conventional von Neumann's entropy for mixed states.
For step ({\it iii}), we introduce an ensemble entropy for a pure state and
prove an inequality regarding the dynamical fluctuation of our entropy, which is
similar to von Neumann's quantum H-theorem. This inequality includes a constant
$C$ that characterizes the correlation of probability fluctuations between different Planck
cells. When the correlation is small, the inequality dictates that
our entropy relax dynamically to the ensemble entropy
and stay at this value most of time with small fluctuations for macroscopic systems.
Our analysis shows that $C$ is small as long as the energy shell of microcanonical ensemble
is not too narrow and not sporadically populated. As a result,
a better understanding of the microscopic origin of the second law of
thermodynamics is achieved. The long-time dynamical evolution of our entropy
is illustrated numerically with an example.
\section{Quantum phase space}
To establish quantum phase space, von Neumann proposed to construct a macroscopic position operator $\bm{Q}$ and a macroscopic momentum operator $\bm{P}$ that satisfy\cite{vonNeumann2010qhtheorem}
\begin{eqnarray}
[\bm{Q}, \bm{P}] = 0 \label{eq:hyp1}\,,\\
\bm{Q} \sim \bm{q}, \quad \bm{P} \sim \bm{p}\,, \label{eq:hyp2}
\end{eqnarray}
where $\bm{q}$ and $\bm{p}$ are usual microscopic position and momentum operators, respectively, that have the commutator $[\bm{q}, \bm{p}] = i\hbar$. Eq.\;(\ref{eq:hyp2}) indicates that the macroscopic position and momentum operators are not identical but close to their microscopic counterparts.
Mathematically it is equivalent to finding a complete set of normalized orthogonal wave functions
$\{\mathit{w}_j\}$ localized in both position and momentum spaces.
The macroscopic position and momentum operators can then be expressed as
\begin{eqnarray}
\bm{P} = \sum_j |\mathit{w}_j\rangle\langle \mathit{w}_j|\bm{p}|\mathit{w}_j\rangle\langle \mathit{w}_j|\,, \\
\bm{Q} = \sum_j |\mathit{w}_j\rangle\langle \mathit{w}_j|\bm{q}|\mathit{w}_j\rangle\langle \mathit{w}_j|\,.
\end{eqnarray}
Eq.\;(\ref{eq:hyp2}) implies that the $i^\textrm{th}$ order central moments
\begin{eqnarray} \Delta^{(i)} p_j \equiv \langle \mathit{w}_j|(\bm{p} - \langle\bm{p}\rangle_j)^i|\mathit{w}_j\rangle^{1/i} \label{spreadp}\\
\Delta^{(i)} q_j \equiv \langle \mathit{w}_j|(\bm{q} - \langle\bm{q}\rangle_j)^i|\mathit{w}_j\rangle^{1/i} \label{spreadq}\end{eqnarray}
should be relatively small for all $i\ge 2$. $\langle f\rangle_j$ denotes $\langle w_j | f | w_j \rangle$. For convenience, we often denote $\Delta^{(2)}$ simply by $\Delta$.
For one-dimensional system in which $\bm{q} \equiv \bm{x}$,
$\bm{p} \equiv \hbar \bm{k} = - i \hbar \partial_x$, von Neumann
proposed to find $\{\mathit{w}_j\}$ by Schmidt orthogonalizing a set of Gaussian
wave packets of width $\zeta$\cite{Von1929}
\begin{equation}
\label{gaussian}
g_{j_x,j_k}\equiv\exp\big[-\frac{(x - j_x x_0)^2}{4 \zeta^2} + i j_k k_0 x\big]\,,
\end{equation}
where $j_x,j_k$ are integers. When $x_0 \,k_0=2\pi$, this set of Gaussian packets are
complete. We are at liberty to choose $x_0$, $k_0$, and $\zeta$ as long as
$x_0 \,k_0=2\pi$ is satisfied.
Unless otherwise specified, parameters are chosen as $x_0 = 1$, $k_0 = 2 \pi$ and $\zeta = (2 \pi)^{-1}$.
This method, which is called ``cumbersome" by von Neumann
himself~\cite{vonNeumann2010qhtheorem}, suffers from two major drawbacks.
First, it is not feasible numerically due to its high computational cost and
sensitivity to the order of the orthogonalization procedure.
Secondly, von Neumann argued \cite{Von1929} that the existence
of $\bm{P}$ and $\bm{Q}$ corresponds to the fact that
the position and momentum can be measured simultaneously in macroscopic measurements.
As there is no difference among measuring positions at different spatial points,
we expect that the constructed $\{\mathit{w}_j\}$ have spatial translational symmetry.
However, the wave packets constructed with von Neumann's method have no such symmetry.
\subsection{Wannier Basis}
Kohn suggested a method to construct Wannier functions out of Gaussian wave
packets~\cite{wkohn1973wannierfunctions}. We adapt Kohn's approach to
orthogonalize the Gaussian packets in Eq.(\ref{gaussian}) and construct a
complete set of Wannier functions $\{\mathit{w}_j\}$ whose translational symmetry
is guaranteed. The detailed procedure of construction is elaborated as follows.
\begin{enumerate}
\item Choose an initial set of localized wave packets such as
the Gaussian wave packets $g_{j_k}(x)\equiv g_{0,j_k}(x)$ in Eq.\;(\ref{gaussian}).
Find their Fourier transform $\tilde{g}_{j_k}(k) \equiv \mathcal{F}\{g_{j_k}(x)\} \equiv \frac{1}{\sqrt{2\pi}} \int g_{j_k}(x)
\mathrm{e}^{-i k x} \;\mathrm{d} x$.
\item At a fixed $k$, for every $j_k$, $(\tilde{g}_{j_k}(k + 2 n \pi))_{n \in \mathbb{Z}}$ is a
normalizable vector; we denote it by $u_{k, j_k}(n)$. Apply Schmidt orthogonalization procedure
$v_0 = u_0$ (the subscript $k$ is omitted), normalize $v_0$, $v_1 = u_1 - (u_1, v_0) v_0$, normalize $v_1$, and repeat for $u_2$, $u_3,\cdots$. We eventually
get an orthonormal basis $\{v_{k, j_k} \in l^2(\mathbb{Z})\}_{j_k \in \mathbb{Z}}$.
Define $\tilde{\mathit{w}}_{j_k}(k + 2 n \pi) \equiv v_{k, j_k}(n) / \sqrt{2 \pi}$.
\item For every $k$ (discrete in numerical calculations) on $[0, 2 \pi)$, repeat step 2. According to Proposition \ref{thm1} in Appendix A, $\mathit{w}_{j_x,j_k}(x)\equiv\mathit{w}_{j_k}(x - j_x)$
($\mathit{w}_{j_k}$ is the Fourier transform of $\tilde{\mathit{w}}_{j_k}$) are orthonormal.
$\{w_j\}$ is the desired orthonormal basis ($j=(j_x, j_k)$).
\end{enumerate}
We have thus established a quantum phase space which is different from
the classical phase space: (1) It is divided into phase cells of size
Planck constant $h$ (for one dimensional system) as illustrated in
Fig. \ref{phspace} (a); we call such a cell Planck cell for brevity.
(2) Each Planck cell is assigned a Wannier function $w_j$, which is
localized near site ($x=j_x$, $k=2j_k\pi$).
We are now able to map a pure wave function unitarily onto phase space. There has been tremendous efforts to
formulate quantum mechanics in phase space based on Wigner's quasi-distribution function and
Weyl's correspondence~\cite{qphasespace}. However, Wigner's
quasi-distribution is not positive-definite and cannot be interpreted
as probability in phase space. According to our construction, for a wave function $\psi$,
$|\braket{\psi|\mathit{w}_j}|^2$ is its probability
at Planck cell $j$ as $\{\mathit{w}_j\}$ is a set of complete orthonormal basis.
The generalization to higher dimensions is straightforward. With
the one-dimensional $\{\mathit{w}_j(x)\}$ that we have constructed, we simply define
\begin{equation}
\mathit{w}_{j_1 j_2 \ldots j_n}(x_1, x_2, \ldots, x_n) \equiv \mathit{w}_{j_1}(x_1) \mathit{w}_{j_2}(x_2) \ldots \mathit{w}_{j_n}(x_n)\,. \label{highd}
\end{equation}
Then $\{\mathit{w}_{j_1 j_2 \ldots j_n}\}$ is the localized orthonormal basis for an $n$-dimensional system.
Numerical results of one-dimensional Wannier functions are provided in Fig.\;\ref{phspace}.
A Wannier function localized near $(x=3,k=20\pi)$ is
plotted in the $k$ and $x$ spaces, respectively, in Fig.\;\ref{phspace}(c) and (d).
This Wannier function is obtained with the above procedure
using the Gaussian wave packets $g_{j_x,j_k}$ as initial functions. And
the order of Schmidt orthogonalization in our procedure is chosen to be
$j_k = 0, 1, -1, 2, -2, \ldots$. The result does not sensitively depend on the order.
Our numerical computation finds that the Wannier function spreads out slowly with
increasing momentum $k$. From Fig.\;\ref{phspace} (b) we can see that
both $\Delta^{(i)} k_j$ and $\Delta^{(i)} x_j$, which characterize the spreads of
the Wannier function, diverge as $j_k$ increases; $\Delta^{(i)} x_j$ appears to grow
more slowly.
Actually, it can be proved that the product of $\Delta x_j\cdot \Delta k_j$ diverges
as $j_k$ increases no matter what initial wave packets are chosen (see Appendix B).
This divergent behavior of $\Delta x \cdot \Delta k$ is called strong uncertainty
relation\cite{bourgain1988remark}.
However, the divergence is not very severe. As shown in Fig.\;\ref{phspace} (b)
where both axes are in logarithmic scales, all the growth slopes are much less than one.
Therefore,
all orders of the relative spreads $\Delta^{(i)} x_j / 2 \pi j_k$ and $\Delta^{(i)} k_j/ 2 \pi j_k$
fall to zero quickly as $j_k$ increases.
This suggests that for the one-dimensional system, the requirement (\ref{spreadp}) and (\ref{spreadq}) are satisfied in the sense
\begin{equation} \lim_{\langle p \rangle_j / p_0 \to \infty} \frac{\Delta^{(i)} p_j}{\langle p \rangle_j} = \lim_{\langle p \rangle_j / p_0 \to \infty} \frac{p_0 \cdot \Delta^{(i)} q_j }{\langle p \rangle_j q_0} = 0\,,
\label{localization}
\end{equation}
where we have used $p=\hbar k$, $q=x$, and
$\langle p\rangle_j \approx j_k p_0$.
\begin{figure*}
\includegraphics[width=0.35\textwidth]{phasegrid}
\includegraphics[width=0.45\textwidth]{sp}\\
\includegraphics[width=0.41\textwidth]{w10k}
\includegraphics[width=0.43\textwidth]{w10x}
\caption{Illustration of quantum phase space. (a) Schematic plot of quantum phase space (b)
Spreads of Wannier functions as a function of $j_k$. Both axes are in logarithmic
scales. Solid lines are for $i = 8, 6, 4, 2$, from top to bottom, respectively.
The dashed line of slope 1 is drawn to show that all solid lines have slope less than 1.
(c) Wannier function $\mathit{w}_{3,10}$ is shown localized near $k = 10\cdot 2 \pi$. (d) Wannier function $\mathit{w}_{3,10}$ is shown localized near $x = 3$.}
\label{phspace}
\end{figure*}
\subsection{Quantum Energy Shell}
In classical phase space, there is an important concept of energy
surface, where the dynamics of an isolated system is confined.
Energy surface, which is of no width, is no longer valid in
the quantum phase space which consists of cells of finite size.
However, a similar concept, energy shell of finite width, can be introduced.
For this purpose, we need to first show that each of our Planck cells is localized in energy
for most of the macroscopic systems of physical interest.
For an isolated system of fixed number of particles $N \gg 1$ with Hamiltonian $H(\bm{p}, \bm{r})$ where $\bm{p}$ and $\bm{r}$ are $3N$-dimensional vectors, define $p_C$ as the typical magnitude of momentum of any particle and $r_C$ as the typical length scale on which $H$ changes relatively significantly. For example, $r_C$ can be the mean free path of a particle or
the characteristic scale of the external potential. We define the index
\begin{equation} I \equiv p_C r_C / h \end{equation}
In this work we focus on the cases where $I$ is considerably large.
We expect that the quantum phase space is reduced to the classical phase space in
the limit $I \to \infty$ in the sense that the relative size of a Planck cell and
the relative spreads of the Wannier functions tend to zero. This is indeed the case.
We construct Planck cells defined by $p_0 = p_C / \sqrt{I}$ and $r_0 = r_C / \sqrt{I}$.
We immediately have $p_0 / p_C = r_0 / r_C = I^{-1/2} \to 0$ in the limit $I \to \infty$.
Suppose that $j_C$ is the momentum index such that $\langle p\rangle_{j_{C}} \approx p_C$.
For a typical Planck cell $j$ whose $|j_k| \lesssim j_C$, we have
according to Eq.\;(\ref{localization})
\begin{equation}
\frac{\Delta^{(i)} p_j} {p_C} \lesssim \frac{\Delta^{(i)} p_{j_C}}{p_C} \to 0 \,,
\end{equation}
and similarly,
\begin{equation}
\frac{\Delta^{(i)} r_j }{r_C} = \frac{p_0}{r_0} \frac{\Delta^{(i)} r_j }{p_C} \lesssim \frac{p_0}{r_0} \frac{\Delta^{(i)} r_{j_C} }{p_C} \to 0\,,
\end{equation} for $i = 2, 3, \dots$ in the limit $I \to \infty$.
We obtain the desirable picture, the quantum phase space becoming the classical phase space
as $I \to \infty$. We thus call $I \to \infty$ classical limit.
We will continue to use this choice of $p_0$ and $r_0$ in the following discussion.
Now we are ready to show that indeed our Wannier functions are localized in energy. To avoid cumbersome partial derivatives and summations, we illustrate the point with single-particle one-dimensional potential $V(x)$; the case of kinetic energy and multi-particle systems should be essentially the same. For a typical Planck cell $j$,
we expand $V$ at $\langle x\rangle_j$ where $|w_j\rangle$
is localized and compute its relative spread
\begin{equation}
\frac{\Delta V^2}{V_0^2}=\left\langle\frac{(V - V_0)^2}{V_0^2}\right\rangle_j = \sum_{i = 2}^\infty a_{j,i}
\left(\frac{\Delta^{(i)} x_j}{r_C}\right)^i
\end{equation}
where $V_0 \equiv V(\langle x\rangle_j)$. As $V$ varies on the scale $r_C$ ,
it is easy to see that $a_{j, i} = O(1)$. Therefore, the relative spread $\Delta V^2/V_0^2$
tends to zero in the classical limit $I \to \infty$.
As our Wannier functions are localized in energy, when we map an energy
eigenstate $\ket{\phi_\alpha}$ with eigen-energy $E_\alpha$
to the quantum phase space, only the Planck cells
with their energies $E_j=\braket{w_j|H|w_j}\sim E_\alpha$ are significantly occupied.
We say that energy eigenstate $\ket{\phi_\alpha}$ crosses Planck cell $j$ when $\langle w_j | \phi_\alpha \rangle$ is significantly non-zero.
As a result, we can define
an energy shell $a$ of energy interval $[E_a, E_a+\Delta E_a]$ as a set of phase
cells $\mathit{w}_{j}$'s such that $\langle\phi_\alpha|\bm{\Delta}_a|\phi_\alpha\rangle\sim 1$
when $E_\alpha\in [E_a, E_a+\Delta E_a]$. The projection operator
$\bm{\Delta}_a \equiv \sum_{j=1}^{N_a} |\mathit{w}_{j}\rangle\langle\mathit{w}_{j}|$,
where $N_a = \,\mathrm{tr}\, \bm{\Delta}_a$ is the number of Planck cells in energy shell $a$.
Energy shell $a$ is said to be significantly occupied by a quantum state $\psi(t)$ when $\overline{\langle\psi(t)|\bm{\Delta}_a|\psi(t)\rangle}$ is considerably larger than zero.
We draw the quantum phase space schematically in Fig.\;\ref{phspace}(a),
where squares are for Planck cells and circles represent eigen-energies.
Two energy shells are illustrated: one with blue Planck cells and the other with orange Planck cells.
Each energy eigenstate may cross many Planck cells; at the same time, one Planck cell
can be crossed by many energy eigenstates. The purple Planck cell is in the orange energy shell while the gray one is in neither shell colored.
\section{Hierarchy of Energy Scales}
In this section we examine the energy scales involved and establish a hierarchy
among them. It will become clear later that these energy scales and
their hierarchy play crucial roles in regulating the long time dynamics of the system.
One energy scale is $\Delta_{\textrm E} E$, the typical difference between adjacent eigen-energies.
The typical energy uncertainty $\Delta_{\textrm Q} E$ in a Planck cell is another energy scale.
For a typical Planck cell $j$, we have
\begin{equation}
\Delta_{\textrm Q} E = \big[\sum_{\alpha} (E_\alpha - E_j)^2 |\langle w_j | \phi_\alpha \rangle|^2\big]^{\frac{1}{2}}\,.\label{DeltaE}
\end{equation}
For a quantum system with large number of particles $N \gg 1$, it should be expected
that though $w_j$'s are localized in energy, eigenstates that cross every Planck cell
are numerous. To see this, we note that the density of state $\rho(E)$ grows
exponentially while $\Delta_{\textrm Q} E$ increases polynomially as $N \to \infty$.
Therefore, for a typical many-particle system, we have $\Delta_{\textrm E} E \ll \Delta_{\textrm Q} E$.
Consider a general quantum state
$
\ket{\psi(t)}=\sum_\alpha c_\alpha (t) \ket{\phi_\alpha}\,,
$
and denote $\wp_\alpha\equiv|c_\alpha (t)|^2$. For this quantum state,
there exists an energy scale $\Delta_{\textrm C} E$ defined as
\begin{equation}
\Delta_{\textrm C} E = \big[\sum_{j, \alpha} (E_\alpha - E_j)^2 \wp_\alpha |\langle w_j | \phi_\alpha \rangle|^2\big]^{\frac{1}{2}}\,,\label{RE}
\end{equation}
where $E_j\equiv\braket{w_j|H|w_j}$ is the average energy of Planck cell $j$. We call
$\Delta_{\textrm C} E$ the correlation energy scale. As we will show later, only the Planck cells
which are separated by energy less than $\Delta_{\textrm C} E$ are correlated. A comparison between
Eq.\;(\ref{DeltaE}) and Eq.\;(\ref{RE}) indicates that we have $\Delta_{\textrm Q} E\sim\Delta_{\textrm C} E$ for
a typical quantum state.
Many properties, in particular macroscopic properties of a system, are not sensitive
to the details of a quantum state. Since $\Delta_{\textrm E} E \ll \Delta_{\textrm Q} E$, we define a
smoothed function over energy scale $\Delta_{\textrm Q} E$ as follows
\begin{equation}
\langle f_\alpha \rangle_s(E) \equiv \sum_{|E_\alpha - E| < \Delta_{\textrm Q} E} f_\alpha \big/ \sum_{|E_\alpha - E| < \Delta_{\textrm Q} E} 1 \,,
\end{equation}
For example, $\langle \wp_\alpha\rangle_s(E)$ is the smoothed probabilities of the quantum state
$\psi$ at $E$. We can now introduce another energy scale $\Delta_{\textrm {mc}} E$ on which
$\langle \wp_\alpha\rangle_s$ can be regarded as constant. This energy scale $\Delta_{\textrm {mc}} E$
indicates the width of the energy shell which is significantly occupied by $\psi$.
In this work we focus on the quantum state such that the following hierarchy
of magnitudes is satisfied,
\begin{equation}
\Delta_{\textrm E} E \ll \Delta_{\textrm Q} E \sim \Delta_{\textrm C} E \ll \Delta_{\textrm {mc}} E \ll E \label{hierarchy}\,,
\end{equation}
where $E\equiv\braket{\psi|H|\psi}$. For a quantum state prepared in real experiments for
a many-body system, both $\Delta_{\textrm {mc}} E $ and $E$ are of macroscopic size while
$\Delta_{\textrm Q} E$ and $\Delta_{\textrm C} E$ are microscopic. Therefore, the hierarchy in Eq.\;(\ref{hierarchy})
are readily satisfied in real experiments.
In textbooks on quantum statistical mechanics\cite{HuangBook}, the micro-canonical ensemble
is established on an energy shell of width $\Delta_{\textrm {mc}} E \ll E$. Usually no lower bound is
given for $\Delta_{\textrm {mc}} E$. Here we see that it should have a quantum lower bound of $\Delta_{\textrm Q} E$,
which will be shown later to play a key role to guarantee the equilibration of the system.
Finally, we assume that the eigenstates are not highly concentrated in the highly occupied
energy shell $[E, E+\Delta_{\textrm {mc}} E]$. Mathematically, this means that
the density of states $\rho(E)$ satisfies
\begin{equation}
\int_{E_j - \Delta_{\textrm Q} E}^{E_j + \Delta_{\textrm Q} E} \mathrm{d} E \;\rho(E) \ll
\int_{E_j - \Delta_{\textrm {mc}} E}^{E_j + \Delta_{\textrm {mc}} E} \mathrm{d} E \;\rho(E)\,. \label{inevitable}
\end{equation}
Despite a few exceptions(flat band etc.), this assumption is not strong and should be satisfied by most of the macroscopic systems in high energy states.
\section{Entropy for Pure Quantum State and an inequality for its Fluctuations}
As we can now map a wave function unitarily to the quantum phase space, we can use
its probability distribution in the phase space to define an entropy.
For a pure quantum state $\psi(\bm{r})$, we define its entropy as
\begin{equation}\label{eq:basisentropy}
S_{\mathit{w}}(\psi) \equiv -\sum_{j} \langle\psi|\bm{W}_j|\psi\rangle \ln \langle\psi|\bm{W}_j|\psi\rangle
\end{equation}
where $\bm{W}_j \equiv |\mathit{w}_j\rangle\langle \mathit{w}_j|$ is the projection to Planck cell $j$ characterized by
Wannier function $\mathit{w}_j(\bm{r})$.
Consider an isolated quantum system described by $\psi$. As this state
evolves with time according to the Schr\"odinger equation, its entropy $S_{\mathit{w}}(\psi)$ will evolve in time.
Will the entropy increase and eventually approach a maximum in accordance with the second law of thermodynamics? The answer is yes for a large class of quantum systems in the sense established by
von Neumann in 1929\cite{Von1929}. In the 1929 paper, von Neumann introduced an entropy for pure quantum
states; he then proved an inequality concerning the long time dynamical behavior of this entropy.
According to this inequality, if the system starts with a low entropy state, the system will evolve
into high entropy states and stay there almost all the time with small fluctuations. Von Neumann
called this inequality quantum H-theorem. We will prove a similar inequality in this section.
As the system evolves,
the probability in each Planck cell $j$ will change with time ($\hbar = 1$)
\begin{eqnarray}
&&\wp_j(t)\equiv\braket{\psi(t)|\bm{W}_j|\psi(t)}\nonumber\\
&&=\sum_{\alpha,\beta}\braket{\psi(0)|\phi_\alpha}\braket{\phi_\alpha|\bm{W}_j|\phi_\beta}\braket{\phi_\beta|\psi(0)} e^{i(E_\alpha-E_\beta)t}\,.\nonumber\\
\end{eqnarray}
We define $\overline{\wp}_j$ as the long time averaging of $\wp_j(t)$ and introduce a corresponding entropy
\begin{equation}
S_E(\psi)\equiv-\sum_j \overline{\wp}_j\ln \overline{\wp}_j\,. \label{def:se}
\end{equation}
We
call it ensemble entropy for pure state $\psi$.
The ensemble entropy $S_E$ does not change with time. We find that under some reasonable conditions, the entropy $S_w(\psi)$ will approach
$S_E(\psi)$ and stay close to it almost all the time with small fluctuations. First we present a rather universal inequality concerning the long time behavior of our entropy, which will imply the equilibration of our entropy under reasonable conditions.
We leave details of the proof to Appendix C; the inequality is as follows.
\newtheorem*{hthm}{Theorem }
\begin{hthm}
For a quantum system governed by a Hamiltonian whose eigenvalues satisfy the following conditions 1, 2 and 3, and for every $j$, $0 \leq \overline{\wp}_j \leq 1 / \mathrm{e}$, we have
\begin{equation}
\frac{\overline{(S_w(\psi(t)) - S_E)^2}}
{S_E^2} \leq C + \frac{8} {S_E} + \frac{4}{S_E^2}\,,
\label{htheorem}
\end{equation}
where \begin{equation} C \equiv \sum_{j, j'} C_{j j'}\left(\overline{\wp}_j \ln \overline{\wp}_j\right)\left(\overline{\wp}_{j'} \ln \overline{\wp}_{j'}\right) \big/ \big(\sum_j \overline{\wp}_j \ln \overline{\wp}_j\big)^2 \label{average}\end{equation} and $C_{j j'} \equiv \overline{(\wp_j(t) - \overline{\wp}_j) (\wp_{j'}(t) - \overline{\wp}_{j'})} \big/ \overline{\wp}_j \overline{\wp}_{j'}$.
\end{hthm}
The three conditions are
\begin{itemize}
\item {\it Condition 1:}\; $E_\alpha = E_\beta \Rightarrow \alpha = \beta$;
\item {\it Condition 2:}\; $E_{\alpha} - E_{\beta} = E_{\alpha'} - E_{\beta'},
\alpha \neq \beta \Rightarrow \alpha = \alpha', \beta = \beta'$;
\item {\it Condition 3:}\; $E_\alpha + E_\chi - E_\beta - E_\gamma = E_{\alpha'} + E_{\chi'} - E_{\beta'} - E_{\gamma'}$, $\{\alpha, \chi\} \cap \{\beta, \gamma\} = \emptyset$
$\Rightarrow \{\alpha, \chi\} = \{\alpha', \chi'\}$ and $\{\beta, \gamma\} = \{\beta', \gamma'\}$.
\end{itemize}
Condition 1 and 2 are commonly used~\cite{Von1929,reimann2008experimental,Short2012njp}, representing no degeneracies of energies and energy gaps, respectively.
Condition 3 implies differences between energy gaps are also distinct. From the random matrix theory\cite{StockmannBook}, we believe condition 3 should be satisfied by most non-integrable systems; as a result, the inequality should hold for majority of quantum systems. These three conditions have a close connection with moments of $\wp_j(t)$ statistically, i.e. $\overline{\wp_j(t)}$, $\overline{\wp_j(t)^2}$ and $\overline{\wp_j(t)^4}$. For example, with condition 1, we have
\begin{equation}
\overline{\wp_j}=\sum_\alpha c_{\alpha j}^*c_{\alpha j}\,.
\end{equation}
where $c_{\alpha j}=\braket{\psi(0)|\phi_\alpha}\braket{\phi_\alpha|w_j}$. For the rest of
details, please see Appendix C.
We now discuss the physical interpretation of $C$, $C_{j j'}$ and the inequality. Clearly, $0 \leq |C_{j j'}| \leq 1$ signifies the fluctuation correlation between Planck cells $j$ and $j'$; $C$ can be regarded as some kind of averaging over $C_{j j'}$ with weight $-\overline{\wp} \ln \overline{\wp}$. Hence $C$ characterizes the averaged fluctuation correlation between cells.
With such understanding, the inequality can be understood intuitively: when $S_E$ is large, that is the probability distribution spreads over many Planck cells, the correlation of $\wp$ between the majority of Planck cells are small; the total entropy $S_w $ undergoes small fluctuations
most of time. In these situations, the inequality (\ref{htheorem}) implies a
quantum H-theorem similar to von Neumann's.
Indeed we can demonstrate that $S_E$ is large and $C$ is small under the following two conditions:
\begin{itemize}
\item The hierarchy (\ref{hierarchy}) and the assumption (\ref{inevitable}) hold.
\item For significantly occupied energy shells, the occupancy rate
\begin{equation}
R \equiv \langle \wp_\alpha \rangle^2_s / \langle \wp_\alpha^2 \rangle_s
\end{equation}
is high.
\end{itemize}
$R$ signifies the fluctuation of $\wp_\alpha$: if all eigenstates are equally occupied, $R = 1$;
if only one of $N_c$ consecutive eigenstates is occupied, $R = 1 / N_c$.
\paragraph{Estimate of $S_E$} We can show (see Appendix D)
\begin{equation}
S_E^{\max} - S_E \lesssim -\ln R \,, \label{SEmax}
\end{equation}
where
\begin{equation}
S_E^{\max} = -\int_{-\infty}^\infty \mathrm{d} E\; \rho(E) \langle \wp_\alpha \rangle_s(E) \ln \langle \wp_\alpha \rangle_s(E)\,.\end{equation}
By Jensen's inequality $ S_E^{\max} \geq \ln d_{\textrm{eff}} $
where $d_{\textrm{eff}}^{-1} \equiv \sum_\alpha \wp_\alpha^2$ is the effective number of eigenstates occupied~\cite{Short2011njp}. $d_{\textrm{eff}}$ can certainly also
be regarded as the microscopic states occupied in a macroscopic quantum state.
For a quantum state prepared in real experiments,
$d_{\textrm{eff}}$ is a very large number~\cite{reimann2008experimental,Short2011njp}.
When $R$ is reasonably high, $R\sim 1$,
we have $S_E\approx S_E^{\max}\geq \ln d_{\textrm{eff}}$.
Therefore, $S_E$ is indeed very large.
\paragraph{Estimate of C} As our Wannier functions are localized in energy as discussed in Section II, Planck cells $i$ and $j$ far apart are not likely to share energy eigenfunctions (that is, for energy eigenstate $\phi_\alpha$, $\langle w_i | \phi_\alpha \rangle$ and $\langle w_j | \phi_\alpha \rangle$ are not significant simultaneously); thus with condition 1 and 2, $\wp_i$ should not be considerably correlated with $\wp_j$. When hierarchy (\ref{hierarchy}) holds, $\Delta_{\textrm Q} E \ll \Delta_{\textrm {mc}} E$, pairs of Planck cells not significantly correlated should be the majority in Eq.\;(\ref{average}). As a result, $C$ should be small. In fact we estimate (see Appendix D)
\begin{equation} C \lesssim \Delta_{\textrm Q} E / \Delta_{\textrm {mc}} E \label{estimateC} \,.\end{equation}
When $S_E$ is maximized with $C\approx 0$, the inequality (\ref{htheorem}) shows that
the relative fluctuation of $S_w$ away from $S_E$ is small when it is averaged over a long time.
This means that when the system starts with a low entropy state, it will relax dynamically to
states whose entropies are very close to $S_E$. Otherwise the inequality would be violated. Note that
it is possible that the system can evolve into a state whose entropy is far away from $S_E$.
When this happens, the system will relax dynamically back in a short time to states whose entropies
are high and close to $S_E$. This reminds us the Poincar\'e recurrence in classical dynamic systems.
So, the morale is the same for both quantum and classical dynamics:
due to the time reversal symmetry inherently possessed by both quantum and classical systems,
it is impossible to rule out that the system evolves dynamically to a lower entropy state. However,
with conditions above we can assert that the large deviation from
the maximized entropy is possible only rarely in quantum dynamics.
As the quantum system equilibrates, not only its entropy reaches its maximum, other
observables such as momentum or density distribution also settle.
In our definition of entropy, it is clear when the entropy reaches its maximum, $\langle w_j |\psi\rangle$ can acquire distinct phase factors while not affecting the total entropy. When $\langle w_i | \bm{p} | w_j \rangle$ is small (relatively) for $i \neq j$, $\langle \psi | \bm{p} | \psi \rangle$ does not significantly depend on those phase factors, either. In macroscopic systems, if Planck cell $i$ and $j$ are close to each other (with $|p_i - p_j|$ much less than $p \equiv (p_i + p_j) / 2$), $\bm{p}$ can be regarded as a constant on the cells thus $\langle w_i | \bm{p} | w_j \rangle \approx p \langle w_i | w_j \rangle = 0$ for $i\neq j$; if Planck cell $i$ and $j$ are far apart, their overlapping is small and as a result, $\langle w_i | \bm{p} | w_j \rangle$ is relatively small. Similar argument applies to other observables (such as $\bm{x}$) as long as the observable varies on a scale much larger than $\Delta p$ and $\Delta x$ of Wannier functions. The inequality proved by Riemann for the fluctuations of observables
~\cite{reimann2008experimental,Short2011njp} is also
an indication that observables should equilibrate when the entropy approaches its maximum value.
\begin{figure}[h!]
\includegraphics[width=0.45\textwidth]{s1}
\includegraphics[width=0.45\textwidth]{s2}
\includegraphics[width=0.45\textwidth]{s3}
\caption{Time evolution of $S_w(\psi(t))$ for three different ripple billiards. The initial state is
a moving Gaussian wave packet. The dashed red lines indicate the theoretical recurrence time for a square infinite potential well of size $2 b \times 2 b$. The green lines are the ensemble entropy $S_E$.
The three ripple billiards shown in the insets are characterized by
$\epsilon =a/b= 0.25\%, 2.5\%, 25\%$, respectively. }
\label{ripple}
\end{figure}
As we are able to compute the Wannier functions ${\mathit{w}_j}$ numerically, the entropy for quantum
pure states and the relaxation of our entropy towards a maximum can now be illustrated with a concrete example.
We are trying to answer whether a macroscopic many-body quantum system
can equilibrate dynamically. However, as we have seen in this work and in
many others' work\cite{Von1929,reimann2008experimental},
the conclusion relies on only the structure of eigen-energies of the system (degeneracy,
energy gaps, etc.), which are shared by both single-particle and many-body systems according to
the random matrix theory\cite{StockmannBook}. This means that in many situations it is sufficient to use
single-particle systems to illustrate entropy for pure states and the quantum H-theorem.
We choose to use ripple billiard with which we are very familiar. The ripple billiard
is an infinite potential well with $V = 0$ in the area enclosed by $y = \pm b$, $x = \pm b \pm a \cos (\pi y / b)$ and $V = \infty$ otherwise\cite{Li2002PRE,Xiong2011lpl}.
In our numerical computation, the initial state is a moving Gaussian
wave packet and the simulation is carried out on a $64 \times 64$ grid.
The results for the entropy $S_w(\psi(t))$ are plotted in Fig. \ref{ripple} for ripple billiards
with three different values of $\epsilon = a / b$. When $\epsilon$ is small, the system is nearly integrable and $S_w$ is almost periodic but with a decaying oscillating amplitude (see Fig.\;\ref{ripple}(a)).
As $\epsilon$ becomes larger and the system gets far away from the integrable regime, the entropy
$S_w$ rises quickly to a maximum value and stays there with small fluctuations as discussed. The ensemble entropy $S_E$ is also plotted and it deviates visibly
from the long-time averaged value of $S_w$. The reason is that since this is a single-particle system, $S_w$ and $S_E$ are not large. As a result, the right-hand side of the inequality (\ref{htheorem})
is not very small.
A few remarks are warranted before we conclude this section. There seems to be
a hidden assumption in von Neumann's proof of his quantum H-theorm besides
two explicitly-stated conditions (identical to conditions 1 and 2 here).
This assumption is equivalent to eigenstate thermalization
hypothesis~\cite{Srednicki1994PRE,Rigol2008Nature} as
pointed out in Ref.\cite{Srednicki2012prl} and by an anonymous referee.
In our opinion, this assumption is linked directly to Eq.(27) in von Neumann's proof~\cite{vonNeumann2010qhtheorem}, which is highly questionable.
In contrast, we do not have any other assumption in our proof of the inequality
Eq. (\ref{htheorem}) besides the three conditions. The conditions for $C$ to be small,
such as the hierarchy of energy scales, have also been explicitly expressed.
Our effort here is to follow the line of von Neumann and Reimann to understand
the microscopic origin of the second law of thermodynamics without any hypothesis.
It is true that our inequality with $C\approx 0$ does not exclude the happening
of large deviation from the maximized entropy. However, this kind of large deviation
occurs rarely according to our analysis. More efforts are needed to find out exactly
how rare these events are. The usual fluctuation theorem seems not applicable here as
it depends on many concepts, such as temperature, heat bath, and entropy, whose
quantum origins are not clear themselves.
\section{Generalization to mixed states and comparison with von Neumann's entropy}
In quantum mechanics, we are all familiar with the von Neumann entropy
that is defined as
\begin{equation}
S_v(\rho) \equiv -\mathrm{tr}\; \rho \ln \rho\,,
\end{equation} where $\rho$ is the density matrix for mixed states. This entropy $S_v$
is zero for any pure state. This fact leads to a well-known dilemma: a large system
in a pure state has zero entropy while any of its subsystems that interacts or entangles with the rest
of the system has non-zero entropy $S_v$.
To compare our entropies to $S_v$,
we need to generalize our entropy for mixed states.
There is a straightforward way to accomplish the goal: for $N$-particle mixed states $\rho^N$,
we define
\begin{equation}
S_w(\rho^N) \equiv \sum_{j_1 j_2 \dots j_N} s_w(\,\mathrm{tr}\, \rho^N \bm{W}_{j_1} \otimes \bm{W}_{j_2} \otimes \cdots \otimes \bm{W}_{j_N})
\end{equation}
where $s_w(\wp) = - \wp \ln \wp$.
Several basic properties that $S_w(\rho^N)$ shares with $S_v$ \cite{Wehrl1978} are listed below :
\begin{description}
\item[Invariance] $S_w(\rho^N)$ depends on $\{w_j\}$ but not on the choice of basis in the Hilbert space. This is a result of the invariance of the trace.
\item[Positivity] $S_w \geq 0$ since $s_w(x) \geq 0$ for $0 \leq x \leq 1$.
\item[Concavity] For $\lambda_1, \lambda_2 > 0$, $\lambda_1 + \lambda_2 = 1$,
\begin{equation}
S_w(\lambda_1 \rho_1^N + \lambda_2 \rho_2^N) \geq \lambda_1 S_w(\rho_1^N) + \lambda_2 S_w(\rho_2^N). \label{eq:concavity}
\end{equation}
This originates from the concavity of $s_w(x)$.
\item[Additivity]
\begin{equation}
S_w(\rho_1^M \otimes \rho_2^N) = S_w(\rho_1^M) + S_w(\rho_2^N) \label{eq:additivity}
\end{equation}
The equality indicates that for two independent systems, the total entropy is the sum of the two. The property is also inherited from $s_w(x)$.
\end{description}
Proof of these properties is essentially the same as that in \cite{Wehrl1978} and hence omitted here.
Despite these similarities, there is one crucial difference between our entropy and $S_v$.
As we have mentioned, for $S_v$ there is a well-known dilemma: for a large system on a pure state, $S_v = 0$
while its subsystem has non-zero entropy. In stark contrast, as we shall show, for our entropy $S_w$,
a large system always has bigger entropy than its subsystem. To demonstrate this, we only need to prove
that the entropy $S_w$ decreases when one particle is traced out of an $N$-particle system.
Without loss of generality, we tend to trace out the $N^{\textrm{th}}$ particle and write
\begin{equation}
\rho^N = \sum_{i, i_N, i', i'_N} c_{i, i_N, i', i'_N} |\psi_i^{N-1},\psi_{i_N}\rangle
\langle\psi_{i'}^{N-1},\psi_{i'_N}|\,.
\end{equation}
Here $\psi$'s are general orthonormal basis, not energy eigenstates.
With the use of the inequality $s_w(\wp_1 + \wp_2) \leq s_w(\wp_1) + s_w(\wp_2)$ for $\wp_1, \wp_2 \geq 0$,
the proof is straightforward.
\begin{eqnarray}
&&S_w(\rho^N) \nonumber\\
&&= \sum_{j, j_N} s_w(\sum c \langle\psi_{i'}^{N-1}|\bm{W}_j^{N-1}|\psi_i^{N-1}\rangle\langle\psi_{i'_N}|\bm{W}_{j_N}|\psi_{i_N}\rangle)\nonumber\\
&&\geq \sum_{j} s_w(\sum_{j_N} \sum c \langle\psi_{i'}^{N-1}|\bm{W}_j^{N-1}|\psi_i^{N-1}\rangle\langle\psi_{i'_N}|\bm{W}_{j_N}|\psi_{i_N}\rangle)\nonumber\\
&&= \sum_{j} s_w( \sum c \langle\psi_{i'}^{N-1}|\bm{W}_j^{N-1}|\psi_i^{N-1}\rangle\langle\psi_{i'_N}|\sum_{j_N}\bm{W}_{j_N}|\psi_{i_N}\rangle)\nonumber\\
&&=\sum_{j} s_w( \sum c \langle\psi_{i'}^{N-1}|\bm{W}_j^{N-1}|\psi_i^{N-1}\rangle\langle\psi_{i'_N}|\psi_{i_N}\rangle)\nonumber\\
&&= S_w(\rho^{N-1})
\end{eqnarray}
where the subscripts $i, i_N, i', i'_N$ are omitted for brevity without causing confusion.
We introduce a density matrix
\begin{equation}
\rho_{\rm mc}=\sum_\alpha |\braket{\psi(0)|\phi_\alpha}|^2\ket{\phi_\alpha}\bra{\phi_\alpha}\,.
\label{mc}
\end{equation}
This density matrix can be regarded as a micro-canonical ensemble for two reasons:
(1) It is easy to check that $S_E=S_w(\rho_{\rm mc})$.
This means that the system's entropy is essentially given by $\rho_{\rm mc}$ at equilibrium.
(2) Reimann~\cite{reimann2008experimental} has also shown that the expectation of
all observables can be also be computed with $\rho_{\rm mc}$ at equilibrium.
This ensemble is clearly different from the conventional micro-canonical ensemble in
textbooks~\cite{HuangBook} as it depends on the initial condition.
The ensemble in Eq.\;(\ref{mc}) is also different from von Neumann's\cite{Von1929} that involves certain coarse-graining of energy. However,
both our ensemble and von Neumann's depend more or less on the choice of initial conditions.
This can lead to very interesting new physics: we are at liberty to choose an initial condition that
composes of energy-eigenstates from two very different energy shells, which can lead to
an equilibrium state with two distinct temperatures\cite{zhuang2014}.
\section{Conclusion}
In summary, we have used Kohn's method to construct a complete set of Wannier
functions which are localized at both given positions and momenta.
We then established a quantum phase space, where each Planck cell is represented by
one of these Wannier functions. By mapping unitarily a quantum pure state to
this quantum phase space, we have defined an entropy for pure states.
A hierarchy of energy scales is proposed and the properties of this entropy have been examined. In particular, we have shown that
for our entropy, a system always has larger entropy than its subsystems.
The long-time dynamical behavior of our entropy has been examined and found to obey
an inequality, which like the quantum H-theorem proved by von
Neumann\cite{Von1929}, along with reasonable hypotheses, indicates that majority of
isolated quantum systems equilibrate dynamically: starting with reasonable initial states,
the quantum system will evolve into a state with maximized entropy and stay there almost
all the time with small fluctuations. Due to the time reversal symmetry, the system does
sometimes undertake large fluctuations. However, the quantum H-theorem demands that
these large fluctuations happen rarely and are short-lived, which provides a quantum perspective of the second law of thermodynamics.
As already pointed out in the introduction, there have been renewed interests in
the foundation of quantum statistical mechanics. These new efforts have not only led to
better theoretical understanding of the issue but also to new physical predications and
challenges that await for answers from experimentalists.
For example, a quantum state which is at equilibrium but with multiple temperatures
was predicted based on the micro-cannonical ensemble established by von Neumann\cite{zhuang2014}.
And it was shown recently\cite{Goldstein2014short,Monnai2014,Short2014Rapid} that quantum
systems can relax much faster than what has been observed in reality.
Can this multiple temperature state be realized in experiments? Does it really exist
a quantum state that can relax as fast as what the theorists have predicted? The answers may
ultimately lie in understanding the borderline between the microscopic and the macroscopic
world.
\section{Acknowledgments}
We thank Hongwei Xiong and Michael Kastner for helpful discussion.
This work is supported by the NBRP of China (2013CB921903,2012CB921300) and
the NSF of China (11274024,11334001).
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\section{Introduction}\label{sec:intro}
Over the last few years, deep object detection has had an impressive evolution. Architectures such as \textsc{Faster-RCNN}~\citep{ren2015faster} and \textsc{DETR}~\citep{carion2020end} showed remarkable performances on challenging datasets. However, these architectures usually lack in assessing their own uncertainty associated with their predictions~\citep{feng2021review}. Often, the quality of probabilistic predictions is poor and should not be trusted out of the box. For example, consider a detection task in an autonomous driving environment. Here, over- or underconfident predictions leading to wrong decisions may result in accidents with cars or even more vulnerable road users such as cyclists~\citep{bieshaar_cooperative_2018}.
\begin{figure*}
\centering
\begin{subfigure}{0.33\linewidth}
\centering
\includegraphics[width=\linewidth]{graphical_abstract_a.pdf}
\caption{Top-Label Calibration Plot}
\label{fig:graphical_abstract:a}
\end{subfigure}%
\begin{subfigure}{.33\linewidth}
\centering
\includegraphics[width=\linewidth]{graphical_abstract_b.pdf}
\caption{Confident Prediction}
\label{fig:graphical_abstract:b}
\end{subfigure}%
\begin{subfigure}{.33\linewidth}
\centering
\includegraphics[width=\linewidth]{graphical_abstract_c.pdf}
\caption{Uncertain Prediction}
\label{fig:graphical_abstract:c}
\end{subfigure}%
\caption{Calibration plot and detector predictions from \textsc{DETR} trained on a subset of COCO.}
\label{fig:graphical_abstract}
\end{figure*}
Ideally, we wish to have a detector capable of providing well-calibrated probabilistic predictions while also being able to recognize object-like entities in images that do not originate from the training data distribution. In the case of well-calibrated probabilistic predictions such as class probabilities, we would like a probability to represent the actual occurrence frequency. For instance, if our detector predicts ``pedestrian'' with a probability of $0.8$, then we expect the ground truth to be a pedestrian $80$ percent of the time. The concept of how well a model can reflect this is referred to as \emph{uncertainty calibration}~\citep{guo2017calibration}. The latter property enables to identify distributional changes, which is crucial when deploying detectors in the real world, especially if the detector is faced data samples that are not well covered by the training data distribution~\citep{ovadia2019can}. Thereby, we can determine to which degree we can trust a prediction. Consider a scenario from autonomous driving in which a flock of sheep crosses a road and our detector was only trained on cars and pedestrians. In such a case, the network should correctly identify this problematic situation due to a distributional change of the samples and leave further decisions to the driver.
{
Figure~\ref{fig:graphical_abstract} illustrates the attempt of assessing the calibration quality for the popular pretrained object detector \textsc{DETR} finetuned on a subset of the COCO dataset~\citep{lin2014microsoft} with two object classes (giraffe and elephant).
Figure~\ref{fig:graphical_abstract}~(a) represents the \textit{Top-Label Calibration Plot} (TCP) that only considers the highest predicted probabilities of a detector as done in the literature (e.g., expected calibration error in~\citep{neumann2018relaxed}). As these should reflect the actual occurrence, this diagram shows accuracy as a function of predicted probability. A perfectly calibrated detector would yield a model calibration (red) lying on the diagonal (black), with the predicted probability equal to the number of correct predictions. Hence, in this example, we seem to observe a well-calibrated detector. Note, the red curve starts at about 0.5 because there are no lower maximum probabilities.
In Fig.~\ref{fig:graphical_abstract}~(b) and Fig.~\ref{fig:graphical_abstract}~(c), we see detector predictions on two images. The actual predictions are marked red, while the ground truth boxes are marked green. Figure~\ref{fig:graphical_abstract}~(b) demonstrates a very confident but reasonable prediction for the class giraffe. Accordingly, this prediction will positively influence the assessment in the calibration plot on the left. In contrast, the right figure shows an uncertain false prediction for the background class presumably because of the fence.
Here, the probability for the class background is only slightly higher than the one for the correct class giraffe. This outlines the importance of considering all available information for evaluation (i.e., all probabilities, not only the highest ones). Thus, the TCP should not be used for evaluation. In the remainder of this article, we will also see that we should assess the calibration of different classes individually.
}
\section{Contributions}
In this work, we investigate multi-class calibration properties of different pretrained state-of-the-art object detection architectures without and with distributional changes, i.e., shifted and out-of-distribution data. We provide detailed analyses of two different detector paradigms (i.e., two-stage and set-based) with two architectures (i.e., \textsc{Faster-RCNN} and \textsc{DETR}), examine their ability to identify distributional changes, and their calibration quality from two perspectives.
In the first perspective, we consider the inference of an object detection architecture as suggested by the literature with the aim to achieve a high generalization performance in the form of mean averge precsision (mAP)~\citep{lin2014microsoft}. More precisely, this means that the raw predictions of a detector are often \textit{post-processed} (e.g., removing duplicates) to obtain a more appropriate prediction set for an image. We often encounter such post-processing steps (e.g., non-maximum suppression, NMS) when working with popular object detection architectures such as \textsc{Faster-RCNN}. In the second perspective, we focus on the \textit{raw outputs of the DNN} with the goal to examine the quality of the probabilistic predictions before applying any post-processing steps. Consequently, we focus on assessing the DNN's calibration and not the calibration of the whole detection pipeline. This way, we intend to give deeper insights, hopefully guiding future research and making it easier to develop techniques for uncertainty modeling in object detection.
Furthermore, in contrast to related work evaluating only the highest class probability~\citep{schwaiger2021black}, we focus on a \textit{multi-class setting} in which all class probabilities are considered when evaluating the calibration of the detectors. As will be shown in this article, while detectors often appear to be well-calibrated regarding the highest predicted class probability, their probabilistic outputs of the remaining classes are not. This can lead to severe problems in many cases. For instance, suppose a detector's highest predicted probability is $0.6$ for the class pedestrian and its second-highest probability is $0.4$ for the class e-scooter driver. In such an example, it is evident that this probability must also be well-calibrated to avoid wrong decisions.
Moreover, since changes in the \textit{data distribution} are widespread in real applications, it is essential that detectors are able to recognize and handle them~\citep{ovadia2019can}. Accordingly, we evaluate the uncertainty of detectors under distributional shifts (i.e., different sample but approx.~same class distributions) and also their ability to identify out-of-distribution samples (i.e., both sample and class distributions differ).
More specifically, we construct shifted and out-of-distribution dataset versions to assess the properties of the detectors.
In summary, we investigate the following research questions:
\begin{itemize}
\item How can we build a modular calibration evaluation framework that is suitable for various object detection architectures and ensures an unbiased and repeatable evaluation?
\item Which metrics should be used to evaluate multi-class object detection architectures regarding their calibration and how should these metrics be applied?
\item How do post-processing steps in a detection pipeline influence the calibration and do architectures that avoid them deliver better calibrated predictions?
\item How well are the class probabilities of pretrained object detectors calibrated when the detectors are applied on samples from shifted versions of the training distribution?
\item How well can a detector identify new objects from out-of-distribution data by means of its probabilistic outputs?
\end{itemize}
Based on the experimental results and findings, we conduct a case study in which we re-calibrate an object detector by simply finetuning its last layers through changing the importance of the background class. Our implementation is publicly available at \url{https://github.com/ies-research/uncertainty-object-detection}.
The remainder of this article is organized as follows:
In Sect.~\ref{sec:problem_setting}, we give a formal definition of our problem setting and introduce the considered object detection architectures which will be exemplarily used as they are the most prominent representatives of their paradigms.
Section~\ref{sec:related_work} analyzes related research regarding uncertainty calibration in classification and object detection.
Afterward, in Sect.~\ref{sec:eval_method}, we propose our evaluation framework and the employed metrics, which allow us to assess the calibration and uncertainty of object detection architectures, and in Sect.~\ref{sec:experiments}, we address the aforementioned research questions by quantitative and qualitative analyses. Based on these insights, in Sect.~\ref{sec:recal}, we recalibrate a detector by finetuning its last layers. Finally, in Sect.~\ref{sec:conclusion}, we conclude our work and highlight potential future research directions.
\section{Problem Setting}\label{sec:problem_setting}
This section introduces the notation regarding object detection and calibration used throughout this article and the architectures we evaluate.
\subsection{Notation}
In our setting, we consider object detection problems for computer vision. We represent a color image (i.e., input sample to an object detector) by a tensor $\x \in \set{X}$, where $\set{X} = \mathbb{R}^{W \times H \times C}$ describes the space of all possible images with $H, W, C \in \mathbb{N}$ as height, width, and number of color channels.. An image $\x$ can contain an unknown number of objects, of which each is represented by a box $\b \in \set{B}$ describing its position and a label $\y \in \set{Y}$ explaining its class. The sets $\set{B} = [0, 1]^4$ and $\set{Y} = \{1, \dots, K\}$ define the space of all possible boxes and $K \in \mathbb{N}_{>1}$ class labels, respectively. We define the target set for a single image as $\set{T} \in \set{P}(\set{Y}\times\set{B})$ where $\set{P}(\cdot)$ denotes the power set. For example, an image $\x_n$ with two objects would have the target set $\set{T}_n = \{(y_1, \vec{b}_1), (y_2, \vec{b}_2)\}$. Finally, we describe a dataset consisting of images and targets as $\set{D} = \{(\vec{x}_n, \set{T}_n)\}_{n=1}^N$ where we have a total number of $N \in \mathbb{N}_{\gt 0}$ images. The images $\vec{x}_n$ are distributed according to the distribution $p(\vec{x})$ and boxes $\vec{b}_{nt}$ and labels $y_{nt}$ in the target set $\set{T}_n$ are assumed to be distributed according to $p(\vec{b} \vert \vec{x}_n)$ and $ p(y \vert \vec{b}_{nt}, \vec{x}_n)$, respectively.
Formally, an object detector is a function $f^{\detectorParameters}: \set{X} \to \mathcal{P}(\simplex{K} \times \mathcal{B})$, where $\vec{\omega}$ is the set of trainable parameters usually optimized with techniques such as gradient descent~\citep{bishop2006pattern} and $\simplex{K}$ is the $K$-simplex within the $K+1$ dimensional unit hypercube spanned by the $K$ classes and the additional background class $\otimes$~\citep{ren2015faster,carion2020end}. Most object detectors employ a background class to distinguish whether an object is present in a proposed region. Hence, an object detector is a function that takes an image $\x_n$ as input and outputs a set of predictions $\hat{\set{T}}_n = \{(\hat{\p}_{nt}, \hat{\b}_{nt})\}_{t=1}^{\vert\hat{\set{T}}_n\vert}$ where $\hat{\p}_n \in \simplex{K}$ are class probabilities for the corresponding box $\hat{\b}_n~\in~\set{B}$. The number of predictions per image $\vert\hat{\set{T}}_n\vert \in \mathbb{N}$ might vary depending on the architecture.
Calibration expresses the quality of the predicted probabilities of a model. Formally, for all probability vectors $\hat{\p}$ of a trained detector on the simplex $\simplex{K}$, we want to satisfy
\begin{align}
P(Y=y \vert \hat{\p}) = \hat{p}_y \text{ for all } y \in \set{Y}\cup\{\otimes\},\label{eq:calibration}
\end{align}
where $Y$ is the random variable for the true class.
Intuitively, for an object detector, this means that all predicted probabilities $\hat{p}_y$ for a prediction $(\hat{\p}, \hat{\b})$ should match their true (but unknown) probability. For example, collecting all predictions where the object detector returned a probability of $0.2$ for class pedestrian, we want $20$\% of them to actually be a pedestrian. Furthermore, this should hold for all classes $y\in\set{Y}$ and probabilities $p_y \in [0, 1]$. In contrast to similar works which assess the calibration based solely on the predicted class probability $\max \hat{\p}$~\citep{kuppers2020multivariate,neumann2018relaxed}, we focus on calibration always considering all class probabilities as suggested in~\cite{kumar2019verified}.
\subsection{Detection Architectures}
This work examines two essential kinds of object detection paradigms, namely \textit{two-stage} and \textit{set-based}. Specifically, we focus on the architectures \textsc{Faster-RCNN} and \textsc{DETR}, as they represent the paradigms' most prominent representatives. Since architectures of the \textit{one-stage} paradigm do not directly output multi-class probability vectors as defined in our problem setting, we leave its investigation for future work.
In the following, we briefly describe how the architectures and their detection pipeline work, including the respective post-processing steps.
\textsc{Faster-RCNN}~\citep{ren2015faster} is one of the most prominent object detection architectures and its forward-propagation is performed in two stages. First, it uses a so-called region proposal network (RPN) to predict multiple regions that may contain potential objects. The second stage utilizes these regions and deploys another DNN to solve a simple classification and regression problem. The output of this network is called prediction and is denoted by $f^{\detectorParameters}(\vec{x})$. In this article, we only focus on the calibration of the second stage and leave the classification problem in the RPN (first stage) for future work.
For specifying the post-processing steps, we consider the implementation of detectron2~\citep{wu_detectron2_2019}. Out of $1000$ predictions, we discard those where the maximum probability within $\hat{\vec{p}}$ is below a certain threshold. Additionally, to avoid duplicate predictions for the same ground truth object, we filter them out using NMS. That means, that we have a varying number of predictions per image $\vert\hat{\set{T}}_n\vert$.
\textsc{DETR}~\citep{carion2020end} is a set-based object detection architecture based on transformers~\cite{vaswani2017attention}. The term set-based arises as \textsc{DETR} views object detection as a direct set prediction problem in which we can detect objects directly without the need for anchors or region proposals. To achieve this, \textsc{DETR} learns attention weights in a transformer describing the pixel and object relationships in an image. In contrast to other object detection architectures, \textsc{DETR} learns to avoid duplicates through its set-based loss function and therefore does not require any post-processing steps.
This means, that it predicts a fixed number of $\vert\hat{\set{T}}_n\vert = 100$ objects per image, as suggested by the authors.
\section{Related Work}\label{sec:related_work}
Commonly, calibration properties are modeled and researched in the context of image classification problems~\citep{ovadia2019can}, and in recent years, considerable work has been done in that field.
For image classification, \cite{guo2017calibration} showed that although modern architectures such as ResNet~\citep{he2016deep} achieve outstanding generalization performance, they provide poorly calibrated and overconfident outputs. They introduced temperature scaling, a multi-class extension to Platt scaling, which improves a DNN's calibration by scaling its predictions with a parameter learned from a separate dataset. \cite{kull2019beyond} noticed that, while temperature scaling improves the calibration of the highest predicted probability of a vector containing all class probabilities, the remaining probabilities are still poorly calibrated. Accordingly, they proposed Dirichlet Calibration for the multi-class setting to improve the quality of all predicted probabilities. Furthermore, in a large-scale evaluation, \cite{ovadia2019can} assessed the behavior of various models under distributional changes and showed a deteriorated quality of their uncertainty estimates.
For classification problems, we only need to consider that the input of a model is assigned to a single class. In object detection, however, we need to consider an unknown number of objects per image, jointly solve regression and classification problems, and often use heuristics such as NMS to obtain our final predictions. These complexities make the transfer of existing uncertainty modeling and calibration evaluation concepts to the object detection task challenging. To the best of our knowledge, one of the first methods regarding uncertainty calibration in deep object detection was proposed by \cite{neumann2018relaxed}. Their article demonstrates the poor calibration of pedestrian detection models and proposes an extension of temperature scaling, avoiding the need for a separate calibration dataset. More recently, \cite{kuppers2020multivariate} discovered that the calibration of a model depends on an object's position in an image and proposed box-sensitive recalibration methods for improvement. As most works focus on the evaluation of the entire detection pipeline, \cite{schwaiger2021black} assessed the influence of NMS on the calibration. They examined the highest predicted probability of DNNs used in different detectors and showed the negative impact of NMS on the calibration quality.
In summary, the related work regarding calibration in object detection is sparse, often focuses only on the highest probability leading to a loss in information in multi-class settings, and frequently employs inappropriate metrics for evaluation.
In this article, we focus on these aspects regarding the evaluation of pretrained detectors. An in-depth study on recalibration techniques of detectors is subject to our future work.
\section{Evaluation Framework}\label{sec:eval_method}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{eval_framework.pdf}
\caption{Our modular evaluation framework for each architecture consists of three steps.}
\label{fig:eval_framework}
\end{figure*}
The literature suggests many techniques for evaluating the calibration and uncertainty of object detection architectures. However, most of these do not clearly define the steps toward an evaluation metric. In particular, it is not clearly described how to incorporate the predictions of object classes (i.e., true and false positives, TP and FP), the predictions of the background class (i.e., true negatives, TN) and missing predictions (i.e., false negatives, FN). \textit{As an attempt to answer our first research question and to make the evaluation understandable and repeatable, we propose a modular evaluation pipeline consisting of three steps} (cf.~Fig.~\ref{fig:eval_framework}). Starting point for the evaluation is the raw output $f^{\detectorParameters}(\vec{x})$ of an object detectors's forward propagation. To obtain an evaluation metric, we look at the subsequent three steps:
\begin{enumerate}
\item Post-Processing: Involves filtering out redundant predictions as done by NMS.
\item Creation of an Evaluation Set: Involves matching predictions with ground truth objects.
\item Metric Evaluation: Involves defining calibration metrics based on the evaluation set.
\end{enumerate}
Each of these steps can be adapted according to the requirements of a specific architecture. The upcoming subsections highlight essential design choices in each step that are necessary for a fair evaluation of all considered architectures.
\subsection{Post-Processing}
Since there exist a wide range of object detection paradigms and architectures, there are also a lot of different detector-specific ways to obtain a final prediction set for an image. Many detection pipelines employ NMS and filter out several predictions based on different criteria to achieve satisfying generalization performances (measured, e.g., with mAP)~\citep{ren2015faster}. However, these post-processing steps remove predictions from the predictions set $\hat{\set{T}}_n$, which might carry valuable information regarding calibration and uncertainty assessment. Thus, it is essential to be aware of specific post-processing steps that a detection pipeline uses as these might bias the evaluation, especially when evaluating uncertainties~\citep{schwaiger2021black}.
In our evaluation, we specify a set $\hat{\set{T}}_n^{\text{Eval}}$ to define the (final) prediction set for sample $\x_n$ that are considered during evaluation. For example, when considering NMS as a post-processing method, this set will be a subset of the original predictions, i.e.,~$\hat{\set{T}}_n^{\text{Eval}} \subseteq \hat{\set{T}}_n$.
On one hand, we consider this set from a perspective which treats a detector as a black box. In particular, we use post-processing steps as it would be done when deploying the detection pipeline in practice.
For example, for the detection architecture \textsc{Faster-RCNN}, the predictions $\hat{\set{T}}_n^{\text{Eval}}$ for image $\x_n$ after post-processing are defined by i)~removing all predictions $(\hat{\vec{p}}_{nt}, \hat{\vec{b}}_{nt})$ where the maximum of $\hat{\vec{p}}_{nt}$ is below a certain threshold, ii)~removing all duplicate predictions with NMS, and iii)~keeping a certain number of predictions with the highest maximum probability. At this point, it becomes apparent that with this perspective we will neglect many predictions in the subsequent course of the evaluation. These might have better calibrated class probabilities or we might lose potentially helpful predictions which we could exploit to identify distributional changes.
On the other hand, we also intend to evaluate a detector in a more detailed perspective without post-processing. Specifically, we determine the calibration quality of $f^{\detectorParameters}$ and not of the entire detection pipeline by considering the raw predictions that were made by the DNN (i.e., $f^{\detectorParameters}$) without filtering any predictions. Thus, our set for evaluation $\hat{\set{T}}_n^{\text{Eval}}$ is given by all the predictions $\hat{\set{T}}_n$ of the DNN.
\subsection{Evaluation Set}
In order to be able to evaluate the predictions, it is necessary to have a suitable assignment of predictions from the prediction set $\hat{\set{T}}^{\text{Eval}}_n$ to ground-truth objects in the target set $\set{T}_n$. We can realize such an assignment in multiple ways. However, it is important to note that the previously mentioned perspectives leading to our prediction set $\hat{\set{T}}^{\text{Eval}}_n$ play a critical role in that selection.
Generally, if the detector aims at avoiding duplicates during inference (e.g.,~\textsc{Faster-RCNN} with post-processing), then we have to force an assignment of a single prediction to a single ground truth object. Otherwise, we would not penalize duplicate predictions (FP) in this case.
Conversely, if our detector predicts duplicates (e.g.,~\textsc{Faster-RCNN} without post-processing), we must ensure that multiple predictions can be assigned to a single ground truth object. Otherwise, we would penalize duplicate predictions, in this case, even if the detector is supposed to make multiple per object.
As an example, consider evaluating the raw predictions of \textsc{Faster-RCNN}. Here, we must ensure that we assign multiple predictions to a single ground truth object. \textsc{Faster-RCNN} without NMS may correctly predict multiple boxes for a single ground truth object. On the other hand, it is essential to use a one-to-one assignment for \textsc{DETR}, since we expect that a \textsc{DETR} based detector learned to avoid duplicate predictions during inference.
To build our evaluation set $\set{E}_n$ for a sample, we therefore focus on a simple matching of predictions to ground truth objects based on the \textit{Intersection over Union} (IOU) between predicted and ground truth box. In particular, anything above an IOU threshold of $0.5$ is considered as matched. We match the pair with the highest IOU when we need a one-to-one assignment. Note that we prefer this simple matching over the matching used in the respective architecture (e.g.,~\textsc{DETR}'s Hungarian matcher) to compare different architectures against each other in the same evaluation setting.
We define the index sets of matches based on the predictions $\hat{\set{T}}^{\text{Eval}}_n$ and targets $\set{T}_n$ as
\newcommand{\operatorname{BS}}{\operatorname{BS}}
\newcommand{\operatorname{NLL}}{\operatorname{NLL}}
\newcommand{\operatorname{TCE}}{\operatorname{TCE}}
\newcommand{\operatorname{MCE}}{\operatorname{MCE}}
\newcommand{\operatorname{dTCE}}{\operatorname{dTCE}}
\newcommand{\operatorname{dMCE}}{\operatorname{dMCE}}
\newcommand{\set{E}}{\set{E}}
\newcommand{\set{M}_n}{\set{M}_n}
\newcommand{\overline{\set{M}}^{\set{T}}_n}{\overline{\set{M}}^{\set{T}}_n}
\newcommand{\overline{\set{M}}^{\hat{\set{T}}}_n}{\overline{\set{M}}^{\hat{\set{T}}}_n}
\newcommand{y^\mathrm{\otimes}}{y^\mathrm{\otimes}}
\newcommand{\b^\mathrm{\otimes}}{\b^\mathrm{\otimes}}
\newcommand{\hat{\p}^{\mathrm{\otimes}}}{\hat{\p}^{\mathrm{\otimes}}}
\begin{align}
\set{M}_n &= \{ (i,j) \mid (y_{ni}, \vec{b}_{ni}) \text{ matches } (\hat{\vec{p}}_{nj}, \hat{\vec{b}}_{nj}) \}, \\
\overline{\set{M}}^{\set{T}}_n &= \{ j \mid (\hat{\vec{p}}_{nj}, \hat{\vec{b}}_{nj}) \text{ is not matched}\}, \\
\overline{\set{M}}^{\hat{\set{T}}}_n &= \{ i \mid (y_{ni}, \vec{b}_{ni}) \text{ is not matched} \}.
\end{align}
Consequently, for our evaluation, we define the evaluation set as
\newcommand{\textunderbrace}[2]{{%
\underbrace{#1}_{\text{#2}}
}}
\begin{align}
\begin{gathered}
\set{E}_n = \underbrace{\{(y_{ni}, \b_{ni}, \hat{\vec{p}}_{nj}, \hat{\vec{b}}_{nj})\}_{(i, j) \in \set{M}_n}}_{\text{\makebox[0pt]{matched predictions (TP and FP)}}} \\
\cup \underbrace{\{(y^\mathrm{\otimes}, \b^\mathrm{\otimes}, \hat{\vec{p}}_{nj}, \hat{\vec{b}}_{nj})\}_{j \in \overline{\set{M}}^{\set{T}}_n}}_{\text{\makebox[0pt]{unmatched predictions (FP and TN)}}}\\
\cup \underbrace{\{(y_{ni}, \b_{ni}, \hat{\p}^{\mathrm{\otimes}}, \b^\mathrm{\otimes})\}_{i \in \overline{\set{M}}^{\hat{\set{T}}}_n}}_{\text{\makebox[0pt]{missing predictions (FN)}}},\label{eq:eval_set}
\end{gathered}
\end{align}
where $y^\mathrm{\otimes} = K+1$ denotes the label for the background class, $\b^\mathrm{\otimes} \in \set{B}$ is an arbitrary box, and $\hat{\p}^{\mathrm{\otimes}} \in \simplex{K}$ is a probability vector assigning all its probability mass to the background class $K+1$. With this set, our evaluation considers not only all predictions (TP, FP, and TN) but also missing ones (FN). Hence, we obtain an unbiased evaluation as we do not ignore potentially valuable predictions.
\subsection{Metric Definition}
\renewcommand{\b}{\vec{b}}
\newcommand{\hat{\b}}{\hat{\b}}
\renewcommand{\p}{\vec{p}}
\newcommand{\hat{\p}}{\hat{\p}}
\renewcommand{\det}{(\hat{\p}_{nt}, \hat{\b}_{nt})}
\newcommand{\operatorname{acc}}{\operatorname{acc}}
\newcommand{\operatorname{prob}}{\operatorname{prob}}
\renewcommand{\B}{\set{B}}
\newcommand{\vec{y}}{\vec{y}}
\newcommand{\indicator}[1]{\mathbb{I}(#1)}
The final step toward our framework consists of defining the required metrics based on the evaluation sets $\set{E}_n$. More specifically, we focus on proper scoring rules~\citep{feng2021review}, calibration plots and errors~\citep{kumar2019verified} for the evaluation of the calibration, and on the entropy~\citep{ovadia2019can} to assess if a detector can identify distributional changes. Note that we can easily incorporate additional metrics for evaluation if necessary.
To further simplify the notation, we define the evaluation set containing all predictions as $\set{E} = \bigcup_{n=1}^N\set{E}_n$, an element of that set as $d \in \set{E}$ (i.e., matched, unmatched, or missing predictions), and a one-hot encoded version for $y_{nt}$ as $\vec{y}_{nt}$.
\textbf{Proper scoring rules} are prevalent metrics for evaluating predictive probability distributions. Scoring rules are functions that map a probability distribution $\hat{\p}_{nt}$ and a ground truth label $y_{nt}$ to a score. This score expresses how well distribution $\hat{\p}_{nt}$ matches the ground truth distribution of $y_{nt}$. Furthermore, we call it proper if only the ground truth distribution leads to a minimum score value.
For our evaluation, we use the \textit{Negative Log-Likelihood} (NLL) defined as
\begin{align}
\operatorname{NLL}(\set{E}) = -\frac{1}{\vert\set{E}\vert}\sum_{d \in \set{E}} \vec{y}_{nt}^\mathrm{T} \ln \hat{\p}_{nt},
\end{align}
where $\ln$ is applied element-wise to the entries of $\hat{\p}_{nt}$.
Furthermore, we also use the \textit{Brier Score} (BS) defined as
\begin{align}
\operatorname{BS}(\set{E}) = \frac{1}{\vert\set{E}\vert} \sum_{d \in \set{E}} \|\vec{y}_{nt} - \hat{\p}_{nt} \|^2,
\end{align}
where $\|\cdot\|^2$ is the Euclidean distance.
Both the NLL and BS are proper scoring rules and we refer the reader to~\cite{feng2021review} for a more detailed explanation.
\textbf{Calibration plots} or reliability diagrams are often employed to assess a model's calibration visually~\citep{guo2017calibration}. Their idea is to visualize Eq.~\ref{eq:calibration} by plotting approximations of the true (but unknown) probability against the detector's predicted probability. Hence, we estimate both quantities empirically by splitting the probability space $[0, 1]$ into $M$ adjacent bins (i.e., intervals) of equal sizes, assigning all predictions based on their probability, and computing accuracy and mean probability as defined below. On the one hand, we employ the \textit{Top-Label Calibration Plot} (TCP), in which we consider a bin $m$ containing all predictions $\B_m \subseteq \set{E}$ where the maximum probability falls into the respective bin interval.
The mean probability and accuracy are therefore defined by
\begin{align}
\operatorname{prob}(\B_m) &= \frac{1}{\vert\B_m\vert} \sum_{d \in \B_m} \max\hat{\p}_{nt},\label{eq:tce_conf} \\
\operatorname{acc}(\B_m) &= \frac{1}{\vert\B_m\vert} \sum_{d \in \B_m} \indicator{\arg\max\hat{\p}_{nt} = y_{nt}}\label{eq:tce_acc},
\end{align}
where $\indicator{\cdot}$ is the indicator function.
Since the TCP only considers the calibration of the most probable class, it neglects to assess all other predicted probabilities. For this reason, we also employ the \textit{Marginal Calibration Plot} (MCP), which depicts the mismatch of all probabilities~\citep{kumar2019verified}.
In such a case, we define $\B_m$ for each class~$k$ individually and only consider the probabilities of the respective class when assigning predictions to a bin. Thus, the average mean probability and accuracy in bin $m$ for class $k$ is defined as
\begin{align}
\operatorname{prob}_k(\B_m) &= \frac{1}{\vert\B_m\vert} \sum_{d \in \B_m} \hat{p}_{ntk},\label{eq:mce_conf}\\
\operatorname{acc}_k(\B_m) &= \frac{1}{\vert\B_m\vert} \sum_{d \in \B_m} \indicator{k = y_{nt}}\label{eq:mce_acc}.
\end{align}
By plotting the accuracy (y-axis) against mean probability (x-axis) in a two-dimensional Cartesian coordinate system, we can observe the calibration qualities. A diagonal corresponds to a perfect calibration. In contrast, our detector is overconfident if the line is below the diagonal and underconfident if it is above. Considering the TCP, we only examine the highest predicted probabilities. This results in a single calibration line. In the case of multiple classes, we will have one line per class. As we will see in Sect.~\ref{sec:experiments}, it is crucial not to average the respective metrics per bin since this might bias the curves towards the diagonal.
\textbf{Calibration errors} are aggregated values derived from calibration plots, allowing us to simplify the comparison of different models against each other. Correspondingly, we employ Eq.~\ref{eq:tce_conf} and Eq.~\ref{eq:tce_acc} to define the \textit{Top-Label Calibration Error} (TCE) as
\begin{equation}
\begin{gathered}
\operatorname{TCE}(\set{E}) = \\\left(\sum_{m=1}^{M}\frac{\vert\B_m\vert}{\vert\set{E}\vert} (\operatorname{acc}(\B_m) - \operatorname{prob}(\B_m))^2\right)^{1/2}.
\end{gathered}
\end{equation}
The TCE is also referred to as the \emph{Expected Calibration Error} in literature~\citep{guo2017calibration}.
Similarly, we use Eq.~\ref{eq:mce_conf} and Eq.~\ref{eq:mce_acc} to define the \textit{Marginal Calibration Error} (MCE) as
\begin{equation}
\begin{gathered}
\operatorname{MCE}(\set{E}) = \\\left(\sum_{k=1}^{K+1}\sum_{m=1}^{M}\frac{\vert\B_m\vert}{\vert\set{E}\vert} (\operatorname{acc}_k(\B_m) - \operatorname{prob}_k(\B_m))^2\right)^{1/2}.
\end{gathered}
\end{equation}
In addition, we also report the recently proposed detection variants of TCP/TCE and MCP/MCE, which we refer to as dTCP/dTCE and dTCP/dMCE~\citep{kuppers2020multivariate}. Instead of the accuracy, they use the precision as an estimate for the true probability. We utilize an adjusted version of those metrics ignoring box positions to be able to compare them to the other plots and errors.
\renewcommand{\H}{\operatorname{H}}
\textbf{Entropy} is a measure which can be interpreted as the uncertainty of a distribution and is defined by
\begin{align}
\H(\hat{\p}) = - \sum_{k=1}^{K+1} \hat{p}_k \ln \hat{p}_k.
\end{align}
Similar to~\cite{ovadia2019can}, we compute it for all predictions in $\set{E}$ and plot a histogram of entropy values to visualize the uncertainty of a model on a whole dataset. This way, we expect to observe differences resulting from data distributional changes. Since we typically have many background predictions with high probability per image, we consider only matched and missing predictions as defined by Eq.~\ref{eq:eval_set}. This is because we intend to assess the uncertainty of predictions that were made for potential objects of unknown classes. We expect that these uncertainties are higher than the ones obtained from the in-distribution objects.
Note, the entropy is the only metric we can apply to out-of-distribution data, since all other metrics require the same label space $\set{Y}$.
\section{Experiments}\label{sec:experiments}
In this section, we introduce our experimental setup and discuss the results of the experiments that answer our research questions.
\subsection{Setup}
We focus on object detectors pretrained on the COCO dataset~\citep{lin2014microsoft} and will evaluate their capability to provide well-calibrated probabilistic predictions as well as their ability to identify distributional changes.
Besides using the 80 class test split of COCO, we want to be able to represent the calibration plots in a clear and understandable way and to investigate the dependence of calibration quality on the number of classes.
Therefore, we additionally construct two subsets of COCO and fine-tune the last layers of detectors by using their respective standard hyperparameter settings for training. In particular, we use a simple subset called \textsc{animals} consisting of images with the two classes giraffe and elephant, and a slightly more complex subset \textsc{traffic} consisting of the ten classes person, bicycle, car, motorcycle, bus, train, truck, traffic light, fire hydrant, and stop sign. We refer to \textsc{all} when talking about the original 80 class COCO test split. Note, as we consider the additional background class in each detector $f^{\detectorParameters}$, we have at least a three-class classification problem that needs to be solved.
Generally, a distributional shift occurs when the test sample distribution no longer matches the training sample distribution. To simulate distributional shifts within our evaluation, we follow the works of~\cite{ovadia2019can} and \cite{harakeh2021estimating} and adapt the distribution $p(\x)$ while keeping the distribution $p(y\vert\x, \b)$ approximately unchanged. More precisely, we construct a new dataset with approximately the same class distribution as our respective training dataset but with a different sample distribution. To realize this, we create two subsets of the Open-Images object detection dataset~\citep{kuznetsova2020open} with the same set of classes as in \textsc{animals} and \textsc{traffic} and evaluate them in an identical setting. Like~\cite{harakeh2021estimating}, we assume that these constructed datasets can be interpreted as a shifted version of the training dataset due to a different data collection process (i.e., image quality, sources, and difficulty).
The evaluation on out-of-distribution (OOD) data is commonly done in classification~\citep{huseljic2021separation} as it gives insights about the quality of uncertainty estimates of a model (e.g., probabilistic outputs or derived measures). The idea behind an OOD evaluation in an object detection setting, however, is fairly uncommon~\citep{du2022vos}. We argue that a detector should be able to identify object-like entities in OOD images while returning high uncertainty for them. For example, consider a flock of sheep crossing a road and a detector in an autonomous vehicle trained on traffic classes such as cars or pedestrians. Without returning highly uncertain predictions for the unknown objects (i.e., sheep), we would not be able to identify this situation and only predict the background class with a high probability. Optimally, in such a scenario, our detector should return high uncertainty for all predicted boxes. We evaluate the detectors on out-of-distribution data similar to~\cite{harakeh2021estimating}. Thus, we require a dataset that has not only a different sample distribution $p(\x)$ but also an unknown class distribution $p(y\vert\x,\b)$. Accordingly, we take a subset from Open Images containing images in which none of the 80 classes from COCO appear.
More details to the corresponding datasets, experimental setup, experiments, additional results as well as the implementation of our framework can be found in our implementation.
\subsection{Results}\label{sec:results}
\renewcommand{\rq}[1]{\textbf{#1}}
\newcommand{\ra}[1]{\textit{#1}}
In the following, we present the results from numerous conducted experiments to sequentially address our research questions. We report values of all numerical metrics (i.e., generalization and calibration) for \textsc{animals}, \textsc{traffic}, and \textsc{all}, and the corresponding shifted versions in Table~\ref{tab:results}. Additionally, we also show calibration plots (Fig.~\ref{fig:calibration_plots}) for the \textsc{animals} subset and entropy histograms (Fig.~\ref{fig:entropy_hists}) for the OOD subset. We will utilize these to underline the answers to research questions with intuitive explanations. For additional calibration plots, we refer to our implementation.
\begin{table*}[ht!]
\small
\caption{Results from \textsc{DETR} and \textsc{Faster-RCNN} on in-distribution datasets and their respective shifted versions. Arrows next to metrics indicate the direction of the optimal value. We abbreviate \textsc{Faster-RCNN} as \textsc{F-RCNN} and ``without post-processing'' as \textsc{w.o. po.}}
\label{tab:results}
\centering
\begin{tabular}{c|c|c|c|c|c|c|c}
\toprule
\multirow{2}{*}{Dataset} & \multirow{2}{*}{Architecture} & \multirow{2}{*}{mAP($\uparrow$)} & \multirow{2}{*}{NLL($\downarrow$)} & \multirow{2}{*}{BS($\downarrow$)} & \multirow{2}{*}{TCE($\downarrow$)} & \multirow{2}{*}{MCE($\downarrow$)} & \multirow{2}{*}{dMCE($\downarrow$)} \\
{} & {} & {} & {} & {} & {} & {} & {} \\
\midrule
\multirow{3}{*}{\shortstack{\textsc{animals} \\(In-Distribution)}}
& \textsc{DETR} & \pmb{0.701} & \pmb{0.120} & \pmb{0.036} & \pmb{0.019} & \pmb{0.053} & \pmb{0.046} \\
& \textsc{F-RCNN} & 0.639 & 0.549 & 0.268 & 0.061 & 0.140 & 0.122 \\
& \textsc{F-RCNN w.o.~po.} & 0.639 & 0.403 & 0.158 & 0.080 & 0.077 & 0.063 \\
\midrule
\multirow{3}{*}{\shortstack{\textsc{animals} \\(Shifted)}}
& \textsc{DETR} & \pmb{0.718} & \pmb{0.105} & \pmb{0.027} & \pmb{0.019} & \pmb{0.046} & \pmb{0.040} \\
& \textsc{F-RCNN} & 0.648 & 0.534 & 0.255 & 0.058 & 0.112 & 0.095 \\
& \textsc{F-RCNN w.o.~po.} & 0.648 & 0.327 & 0.121 & 0.067 & 0.070 & 0.058 \\
\midrule\midrule
\multirow{3}{*}{\shortstack{\textsc{traffic} \\(In-Distribution)}}
& \textsc{DETR} & \pmb{0.488} & \pmb{0.209} & \pmb{0.095} & \pmb{0.036} & 0.059 & 0.044 \\
& \textsc{F-RCNN} & 0.473 & 0.637 & 0.312 & 0.057 & 0.085 & 0.061 \\
& \textsc{F-RCNN w.o.~po.} & 0.473 & 0.360 & 0.155 & 0.065 & \pmb{0.030} & \pmb{0.021} \\
\midrule
\multirow{3}{*}{\shortstack{\textsc{traffic} \\(Shifted)}}
& \textsc{DETR} & 0.381 & 0.371 & 0.163 & 0.067 & 0.099 & 0.073 \\
& \textsc{F-RCNN} & 0.387 & 0.878 & 0.434 & 0.135 & 0.150 & 0.108 \\
& \textsc{F-RCNN w.o.~po.} & \pmb{0.387} & \pmb{0.335} & \pmb{0.156} & \pmb{0.056} & \pmb{0.046} & \pmb{0.035} \\
\midrule\midrule
\multirow{3}{*}{\shortstack{\textsc{all} \\(In-Distribution)}}
& \textsc{DETR} & \pmb{0.420} & \pmb{0.380} & \pmb{0.172} & \pmb{0.046} & 0.030 & 0.021 \\
& \textsc{F-RCNN} & 0.392 & 0.792 & 0.364 & 0.087 & 0.036 & 0.023 \\
& \textsc{F-RCNN w.o.~po.} & 0.392 & 0.510 & 0.213 & 0.077 & \pmb{0.011} & \pmb{0.006} \\
\bottomrule
\end{tabular}
\end{table*}
\newcommand\factor{.80}
\begin{figure*}
\centering
\begin{subfigure}{\factor\linewidth}
\centering
\includegraphics[width=\linewidth]{CP_DETR_animals.pdf}
\caption{\textsc{DETR} on \textsc{animals} subset.}
\label{fig:a_detr_animals}
\end{subfigure}%
\begin{subfigure}{\factor\linewidth}
\centering
\includegraphics[width=\linewidth]{CP_FRCNN_application_animals.pdf}
\caption{\textsc{Faster-RCNN} on \textsc{animals} subset with post-processing.}
\label{fig:b_frcnn_animals}
\end{subfigure}%
\begin{subfigure}{\factor\linewidth}
\centering
\includegraphics[width=\linewidth]{CP_FRCNN_modeling_animals.pdf}
\caption{\textsc{Faster-RCNN} on \textsc{animals} subset without post-processing.}
\label{fig:c_frcnn_animals}
\end{subfigure}%
\caption{Calibration plots for the object detection architectures. The red shaded area shows the bin quartiles calculated across classes.}
\label{fig:calibration_plots}
\end{figure*}
\rq{Which metrics should be used to evaluate multi-class object detection architectures regarding their calibration and how should these metrics be applied?}
\ra{When evaluating object detection architectures in a multi-class environment, it is essential to assess all predicted probabilities (instead of the maximum one) of our detector by using the MCP and MCE.}
An example of this is given in the calibration plots shown in Fig.~\ref{fig:calibration_plots} on the \textsc{animals} subset of COCO. The comparison between TCP and MCP demonstrates for all detectors that although their highest predicted probability seems to be fairly well-calibrated, the rest of the predicted probabilities is not. For example, the predictions for the two object classes, giraffe and elephant, are overconfident for \textsc{DETR} and \textsc{Faster-RCNN} with post-processing as their predicted probability lies below the diagonal. Vice versa, when looking at the background class, we can observe underconfidence. Such an interdependency between the object classes and the background class seems reasonable. In object detection, the neural network needs to learn an extremely imbalanced classification problem between background and objects. Therefore, it makes sense that this potentially biases the quality of our probabilistic predictions either towards under- or overconfidence. Furthermore, as shown by the red curves in Fig.~\ref{fig:calibration_plots}, it is important to not average the bin metrics across classes as this might lead to biased results improving the calibration performance. To further highlight the importance of a suitable metric, we also report the dMCP, which we can see on the right in Fig.~\ref{fig:calibration_plots}. Using precision instead of accuracy, we cannot capture the interdependency between background and objects. This is because the background predictions are ignored by only considering TPs and FPs for evaluation. Accordingly, we observe in Table~\ref{tab:results} that the MCE reports the highest error compared to TCE and dMCE. Note the importance of the number of predictions in a bin when examining the MCP. Comparing only the upper plots of the MCP, \textsc{Faster-RCNN} with post-processing appears to be better calibrated than \textsc{DETR}. However, it also has many predictions in bins between 0 and 1, leading to a higher error, as seen in Table~\ref{tab:results}. In summary, it is essential to consider the MCP and MCE to evaluate the multi-class calibration qualities of detectors as they consider TP, FP, TN, and FN.
\begin{figure*}
\centering
\includegraphics[width=.8\linewidth]{entropy_hists.pdf}
\caption{Entropy histograms for in-distribution, shifted, and OOD datasets on the \textsc{traffic} subset.}
\label{fig:entropy_hists}
\end{figure*}
\rq{How do post-processing steps in a detection pipeline influence the calibration and do architectures that avoid them deliver better calibrated predictions?}
\ra{The post-processing steps in a detection pipeline significantly influence a model's calibration quality.} We can see an example of this by comparing the calibration plots of \textsc{Faster-RCNN} with (Fig.~\ref{fig:calibration_plots}~(b)) and without (Fig.~\ref{fig:calibration_plots}~(c)) post-processing. When using post-processing steps, the probabilities for object classes (i.e., giraffe and elephant) are overconfident, and the background class probabilities are underconfident. Without post-processing, however, this property no longer holds. There, the probabilities for object classes become underconfident, whereas the probabilities for the background class become overconfident. This means that post-processing steps of \textsc{Faster-RCNN} modify the neural network's probabilistic predictions such that underconfidence of object classes changes to overconfidence.
Table~\ref{tab:results} also shows that we get much better calibration properties on all datasets from our neural network when avoiding post-processing. Again, we see that the averages across classes (red curves) would not be sufficient to identify this property.
\ra{Furthermore, we see that architectures avoiding post-processing steps, such as \textsc{DETR}, provide better calibrated probabilistic predictions.} Table~\ref{tab:results} demonstrates for all datasets that the proper scoring rules (NLL and BS) and calibration errors (TCE, MCE, and dMCE) of \textsc{DETR} are below the ones of \textsc{Faster-RCNN} with post-processing. When looking at the calibration plots, this looks surprising as \textsc{Faster-RCNN}'s mean probabilities seem to be closer to the diagonal. However, its predictions are more spread across bins, weighing calibration errors in the middle higher. \textsc{DETR}'s predictions, on the other hand, are concentrated near the edges, which results in overall lower errors. These properties are often referred to in the literature as sharpness and reliability~\citep{ovadia2019can}.
\rq{How well are the class probabilities of pretrained object detectors calibrated when the detectors are applied on samples from shifted versions of the training distribution?}
\ra{Based on our experiments, the calibration qualities of the pretrained detectors on a dataset based on a shifted version of original the training sample distribution seem to depend on the difficulty of the object detection problem.} Generally, we would expect that the calibration quality worsens as we shift. However, it is also possible that it improves the detector calibration. For example, consider the three-class scenario of the \textsc{animals} subset (COCO vs.~Open Images) in Table~\ref{tab:results}. Here, we observe that both \textsc{DETR} and \textsc{Faster-RCNN} achieve a slightly better mAP, i.e., their performance is better on the shifted test dataset. We believe this quite surprising result is due to the fact that the shifted version of the \textsc{animals} subset contains images on which objects are easier to detect (approx.~50\% bigger objects and fewer objects per image). Similarly, we also see that the calibration errors and proper scoring rules report better results for all detectors. Hence, we conclude that a shift of the sample distribution, which simplifies the problem, can lead to an improved calibration quality of our detections. Looking at the more complex problem \textsc{traffic}, we see that the mAP and the calibration metrics worsen when evaluating them under the shifted version. This is expected and matches the results from literature~\citep{ovadia2019can,harakeh2021estimating}.
The entropy histograms in Fig.~\ref{fig:entropy_hists} show no noticeable difference between the in-distribution dataset of \textsc{traffic} and its shifted version for all architectures. Although the calibration qualities deteriorate, the detectors cannot identify this shift. Optimally, the worse calibration should be reflected by the dissimilarity of these histograms.
\rq{How well is a detector able to identify new objects in the case of out-of-distribution data by means of its probabilistic outputs?}
\ra{Based on our experiments, pretrained object detectors cannot identify new objects through their probabilistic outputs as they only predict the background class with high probability.} To evaluate this, we visualize log entropy histograms for in-distribution, shifted, and out-of-distribution datasets in Fig.~\ref{fig:entropy_hists}. Instead of the entropy, we use its logarithm to better demonstrate the differences in distributions better. We can see for \textsc{DETR} and \textsc{Faster-RCNN} without post-processing that there are some distributional differences. However, these are due to the fact that the pretrained detectors predict actual object classes (e.g., giraffe or elephant) for the in-distribution and shifted dataset. In contrast, for the OOD dataset, the detector solely predicts the background class with high probability. As a result, we notice that the predictions on the OOD dataset are even more overconfident when compared to the other histograms. Unfortunately, we cannot leverage this distributional difference to simply detect out-of-distribution samples.
For instance, consider an autonomous vehicle in front of a flock of sheep. Since our detectors only predict the background class, we can not distinguish this situation from an empty road. Thus, we cannot identify out-of-distribution samples as detectors will just predict the background class.
\section{Simple Recalibration}\label{sec:recal}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{results_recalibration_study.pdf}
\caption{Generalization and calibration metrics with respect to the background weight. Note the different scaling range of the respective metrics. The default background weight (0.1) is the minimum value on the x-axis.}
\label{fig:results_recal_study}
\end{figure*}
As mentioned in Sect.~\ref{sec:experiments}, there seems to exist some interdependency between object classes and the background class in a trained detector. While the predictions for the background class are underconfident, the ones for the object classes are overconfident. To investigate this and potentially improve a model's calibration, we conduct a case study on \textsc{DETR} and the \textsc{animals} subset in which we adjust its hyperparameter for the importance of the background class by changing its classification weight during fine-tuning. Note that we can do this for any architecture by simply increasing the weight for the background class in the cross-entropy loss part of a objective function. Specifically, as the background predictions were underconfident, we finetuned 100 object detectors with higher background weights and plot the results in Fig.~\ref{fig:results_recal_study}. We can see that the model's calibration errors and proper scoring rules are improving by increasing the importance of the background class. Furthermore, we also note a slightly increasing generalization performance (mAP). Thus, it seems to be promising for calibration and generalization to raise the background class importance during fine-tuning of a specific detector. In contrast to the other metrics, the TCE does not capture this improvement, highlighting its inappropriateness for the calibration evaluation of multi-class problems again. Figure~\ref{fig:cp_recal_study} shows the MCP from the detector that achieved the best MCE in this study. Both object and background predictions are closer to the diagonal compared to Fig.~\ref{fig:calibration_plots}~(a) while their interdependencies are no longer recognizable.
We conclude that there seems to be a correlation between calibration quality and the imbalanced problem of object detection (i.e., background vs. objects) and leave further analyses regarding this for future work.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{CP_recalibration_study.pdf}
\caption{The MCP for the detector that achieved the best MCE (value of $8.6$ for the background weight) on the \textsc{animals} subset.}
\label{fig:cp_recal_study}
\end{figure}
\section{Conclusion and Outlook}\label{sec:conclusion}
This work assessed different pretrained object detectors regarding their capability of modeling uncertainty considering various factors. First, we proposed a modular framework for evaluating calibration properties of object detection architectures in a multi-class setting while considering all actual and missing predictions of a detector. We analyzed the importance of the choice of metrics and concluded that, besides proper scoring rules, MCP and MCE are the most appropriate metrics for evaluation. Subsequently, we investigated the influence of post-processing steps (e.g., NMS), which worsened the detector's calibration and revealed an interdependency between the confidences of the object class and background class predictions.
We also evaluated the detectors' calibration on datasets with changed distributions. When considering a shift in the sample distribution, we noticed that the calibration does not necessarily worsen as the detection problem may get easier. Furthermore, we saw that detectors could not identify OOD objects when considering an OOD dataset as they only predicted the background class with high probability.
Finally, based on the interdependency insights, we conducted a case study in which we demonstrated that \textsc{DETR}'s calibration can be improved by simply increasing the importance weight of the background class in the objective function during training.
For future work, we need to \textbf{extend the presented framework} such that it includes additional metrics (box-dependent calibration errors~\citep{kuppers2020multivariate}) or we can evaluate additional object detection paradigms. Currently, we assume that the neural network predictions in the classification task describe the parameters of a categorical distribution. This assumption, however, does not always hold. For example, the architecture Retinanet, which is a one-stage paradigm, assumes that every prediction describes a two-class problem between object and background. Hence, the network uses the binary cross-entropy loss function during training. To evaluate such an architecture, we need to extend our framework.
As seen in our experiments, post-processing steps such as NMS deteriorate the calibration quality of detectors. Therefore, it is vital to \textbf{research the dependency between post-processing and uncertainty modeling} if we want to use these architectures in tasks such as active learning. For example, it is hard to tell whether it makes sense to enhance an architecture with uncertainty modeling techniques such as Monte-Carlo Dropout~\cite{gal2016dropout} if the outputs are discarded anyway. Therefore, we need to examine whether we can use the direct predictions of a neural network instead of the ones from the detection pipeline or develop more suitable post-processing steps. Additionally, it seems reasonable to just avoid post-processing architectures and use ones such as \textsc{DETR}. They seem promising as we can intuitively recalibrate the predictions of neural networks instead of a detection pipeline that might ignore half of the predictions.
Additionally, further research is needed that focuses on the \textbf{class-imbalance problem in object detection} (i.e., interdependency between calibration of object classes and background class) and its \textbf{influence on the uncertainty modeling} of a detector. In our experiments, we solely employed pre-trained detectors and finetuned them on specific subsets. However, it would be interesting to investigate proper scoring rules and calibration errors when training a detector from scratch with different hyperparameter settings.
We assume that training the entire detector with a very high background weight would probably not lead to good generalization.
At last, we want to address the \textbf{missing ability of detectors to identify OOD objects}. We believe that it is vital for many tasks to distinguish between potential unknown objects and a natural background. With that addition, it would be possible, for instance, to improve the responses to different autonomous driving scenarios or perform better exploration during active learning~\citep{herde2021survey}. The training of such an object detection architecture might be realized by using out-of-distribution data~\citep{huseljic2021separation}.
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BkiUe4vxK4sA-5fmyxs6
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\section*{Introduction}
The notion of protomodular category was introduced by Bourn in \cite{Bourn}. The categories of groups, rings, Lie algebras, crossed modules, rings, the dual of the category of sets are examples of protomodular categories.
In the pointed case, a category $\mathcal{C}$ is protomodular if and only if the Split Short Five Lemma holds. This means that for any diagram of the form
\begin{equation}\label{SSFL}
\begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base),scale=0.8]
\node (A) at (0,0) {$A_1$};
\node (B) at (2.5,0) {$B_1$};
\node (C) at (-2.5,0) {$X_1$};
\node (A') at (0,-2) {$A_2$};
\node (B') at (2.5,-2) {$B_2$};
\node (C') at (-2.5,-2) {$X_2$};
\node (O1) at (-4.5,0) {$0$};
\node (O1') at (-4.5,-2) {$0$};
\node (O2) at (4.5,0) {$0$};
\node (O2') at (4.5,-2) {$0$};
\path[->,font=\scriptsize]
(O1) edge node[above] {$ $} (C)
(B) edge node[above] {$ $} (O2)
(B') edge node[above] {$ $} (O2')
(O1') edge node[above] {$ $} (C')
(B.south) edge node[right] {$ g$} (B'.north)
(C.south) edge node[left] {$ v $} (C'.north)
(A.south) edge node[left] {$ p$} (A'.north)
(C'.east) edge node[above] {$\kappa_2$} (A'.west)
([yshift=-4pt]A'.east) edge node[below] {$\alpha_2$} ([yshift=-4pt]B'.west)
([yshift=2pt]B'.west) edge node[above] {$e_2$} ([yshift=2pt]A'.east)
(C.east) edge node[above] {$\kappa_1$} (A.west)
([yshift=-4pt]A.east) edge node[below] {$\alpha_1$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$e_1$} ([yshift=2pt]A.east);
\end{tikzpicture}
\end{equation}
where $\kappa_i$ is the kernel of $\alpha_i$ and $\alpha_i \cdot e_i = 1_{B_i}$ for any $i \in \{ 1,2\}$, then $p$ is an isomorphism whenever $v$ and $g$ are, this property is often referred to as the ``Split Short Five Lemma'' holds in $\mathcal{C}$.
It is well-known that the category of internal groups in a finitely complete category is protomodular \cite{Bourn}. In particular, this result implies that the category of cocommutative Hopf algebras in a symmetric monoidal category, that has equalizers, is protomodular, seen as internal groups in the category of cocommutative coalgebras. The category of cocommutative Hopf algebras over a field is even semi-abelian \cite{GSV}.
In this paper, we are interested in cocommutative bialgebras in any symmetric monoidal category that has equalizers. As a matter of fact, it was proven in \cite{GVdL} that the category of cocommutative $K$-bialgebras is not a protomodular category.
Similarly, the category of monoids is not a protomodular category. However, a class of split epimorphisms of monoids, called \emph{Schreier split epimorphisms}, turned out to have some very interesting properties, similar to the ones of split epimorphisms of groups. For example, \emph{Schreier split epimorphisms} are equivalent to the actions of monoids, the Split Short Five Lemma holds, etc. \cite{BMS,BMMS}. The authors in \cite{BMS}, introduced the notion of $S$-protomodularity, that is the protomodularity with respect to a class $S$ of split epimorphisms. It implies that we have the Split Short Five Lemma as in \eqref{SSFL} whenever $(\alpha_1,e_1)$ and $(\alpha_2,e_2)$ are in the class $S$.
The main example is the category of monoids with the class of \emph{Schreier split epimorphisms}.
In \cite{Stercksplit}, we defined a notion of split extensions for (non-associative) bialgebras (and non-associative Hopf algebras) such that they have ``group-like'' properties. More precisely, we showed that this definition of split extension of (non-associative) bialgebras is equivalent to the notion of action of (non-associative) bialgebras. Moreover, we proved the validity of the Split Short Five
Lemma for these kinds of split extensions.
In particular, these results restrict to the case of cocommutative bialgebras.
Hence, it is natural to hope that the category of cocommutative bialgebras in any symmetric monoidal category is $S$-protomodular with respect to the class introduced in \cite{Stercksplit}.
In a finitely complete category, we can define two types of centrality: the centrality of equivalence relations in the sense of Smith and the centrality of normal monomorphisms in the sense of Huq. Note that these notions of centrality are not independent. If two relations of equivalence centralize each other (in the sense of Smith) then the corresponding normal monomorphisms necessarily centralize in the sense of Huq \cite{BG}.
The converse is not true in general. If $\mathcal{C}$ is a category such that any two equivalence relations always centralize each other as soon as their normalizations centralize in the sense of Huq, one says that $\mathcal{C}$ satisfies the so-called \textit{Smith is Huq} property \cite{BG,MFVdl}.
In \cite{MFM}, the authors introduced the notion of \emph{Smith is Huq} for pointed $S$-protomodular categories. In that context, the \emph{Smith is Huq} property can be expressed as follows: two $S$-equivalence relations centralize each other if and only if their associated normal subobjects commute (in the sense of Huq).
In this paper, we prove that the category of cocommutative bialgebras in any symmetric monoidal category is $S$-protomodular with respect to the class of split extensions introduced in \cite{Stercksplit}, we give a description of the centrality in the sense of Huq for two subbialgebras of a cocommutative bialgebra and we examine the \emph{Smith is Huq} condition for cocommutative bialgebras.
The results obtained in this paper generalize results in \cite{BMMS} and \cite{MFM} on the category of monoids.
The layout of this article is as follows: the first section contains some preliminaries on cocommutative bialgebras in a symmetric monoidal category.
In the second section, we prove that the category of cocommutative bialgebras in any symmetric monoidal category (that has equalizers) is $S_{coc}$-protomodular with respect to the class $S_{coc}$ of split extensions of cocommutative bialgebras defined in \cite{Stercksplit}.
In the third section, we describe the Huq commutator of two subbialgebras of a cocommutative bialgebra.
In the last section, we investigate the \emph{Smith is Huq} condition for cocommutative bialgebras by using the results of \cite{MFM}.
\section{Preliminaries}
\subsection{Bialgebras in a symmetric monoidal category}
We recall that a \textit{monoidal category} is given by a triple $(\mathcal{C}, \otimes, I)$ where $\mathcal{C}$ is a category, $\otimes \colon \mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C}$ a bifunctor and $I$ is the identity element (we omit to explicit the three natural isomorphisms, the associator, the right unit and the left unit).
A \textit{braided monoidal category} is a 4-tuple $(\mathcal{C}, \otimes, I, \sigma)$ where $(\mathcal{C}, \otimes, I)$ is a monoidal category and $\sigma$ is a \textit{braiding}. A braiding consists of a family of natural isomorphisms $\sigma_{X,Y} \colon X \otimes Y \rightarrow Y \otimes X$ satisfying \[ \sigma_{X \otimes Y, Z} = (\sigma_{X,Z} \otimes 1_Y)\cdot (1_X \otimes \sigma_{Y,Z} )\]
\[ \sigma_{X , Y \otimes Z} = (1_Y \otimes \sigma_{X,Z} )\cdot (\sigma_{X,Y} \otimes 1_Z).\]
A braided monoidal category is called \textit{symmetric} when \begin{equation}\label{symm}
\sigma_{Y,X}^{-1} = \sigma_{X,Y}.
\end{equation}
In this paper, we omit the indexes of the braiding when this does not bring any confusion.
An \textit{algebra} in a symmetric monoidal category $(\mathcal{C}, \otimes, I, \sigma)$ is given by an object $A \in \mathcal{C}$ endowed with a morphism $m \colon A \otimes A \rightarrow A$, called the multiplication. An algebra is associative and unital when there is a morphism $u_A \colon I \rightarrow A$ called the unit, such that the following equalities are satisfied
\begin{equation}\label{unital multiplication}
m \cdot (u_A \otimes 1_A) = 1_A = m \cdot (1_A \otimes u_A)
\end{equation}
\begin{equation}\label{ass}
m \cdot (m \otimes 1_A) = m \cdot (1_A \otimes m)
\end{equation}
\begin{center}
\begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base)]
\node (A) at (0,0) {$ A \otimes A$};
\node (B) at (2,0) {$A$};
\node (C) at (-2,0) {$A $};
\node (D) at (0,-1.5) {$ A.$};
\draw[commutative diagrams/.cd, ,font=\scriptsize]
(B.south west) edge[commutative diagrams/equal] (D.north east)
(C.south east) edge[commutative diagrams/equal] (D.north west);
\path[->,font=\scriptsize]
(A.south) edge node[descr] {$m$} (D.north)
(C.east) edge node[above] {$ u_A \otimes 1_A$} (A.west)
(B.west) edge node[above] {$1_A \otimes u_A$} (A.east)
;
\end{tikzpicture}
\begin{tikzpicture}[descr/.style={fill=white},baseline=(D.base)]
\node (A) at (0,0) {$ A \otimes A $};
\node (B) at (3,0) {$A.$};
\node (C) at (3,1.5) {$A \otimes A $};
\node (D) at (0,1.5) {$ A \otimes A \otimes A$};
\path[->,font=\scriptsize]
(C.south) edge node[right] {$m $} (B.north)
(D.south) edge node[descr] {$1_A \otimes m$} (A.north)
(A.east) edge node[above] {$ m $} (B.west)
(D.east) edge node[above] {$ m \otimes 1_A$} (C.west)
;
\end{tikzpicture}
\end{center}
All the algebras that we will consider in this paper are associative and unital.
A\textit{ morphism of algebras} $f \colon A \rightarrow B$ is a morphism in $\mathcal{C}$ such that the following diagrams commute
\begin{center}
\begin{tikzpicture}[descr/.style={fill=white},baseline=(D.base)]
\node (A) at (0,0) {$ A $};
\node (B) at (3,0) {$B$};
\node (C) at (3,1.5) {$B \otimes B $};
\node (D) at (0,1.5) {$ A \otimes A$};
\path[->,font=\scriptsize]
(C.south) edge node[descr] {$m$} (B.north)
(D.south) edge node[descr] {$m$} (A.north)
(A.east) edge node[above] {$ f$} (B.west)
(D.east) edge node[above] {$ f \otimes f$} (C.west)
;
\end{tikzpicture}
\begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base)]
\node (A) at (0,0) {$ A $};
\node (C) at (-2,0) {$I $};
\node (D) at (0,-1.5) {$ B.$};
\path[->,font=\scriptsize]
(A.south) edge node[descr] {$f $} (D.north)
(C.south east) edge node[descr] {$u_B $} (D.north west)
(C.east) edge node[above] {$ u_A $} (A.west)
;
\end{tikzpicture}
\end{center}
A coalgebra is the dual notion of the notion of an algebra. In other words, a coalgebra over $(\mathcal{C}, \otimes, I, \sigma)$ is an object $C \in \mathcal{C}$ with a comultiplication $\Delta \colon C \rightarrow C \otimes C$. From now on, the coalgebras will always be coassociative, i.e. the following equality holds
\begin{equation}\label{coass comultiplication}
(\Delta \otimes 1_C) \cdot \Delta = (1_C \otimes \Delta) \cdot \Delta
\end{equation}
\begin{center}
\begin{tikzpicture}[descr/.style={fill=white},baseline=(D.base)]
\node (A) at (0,0) {$ C \otimes C $};
\node (B) at (3,0) {$C \otimes C \otimes C.$};
\node (C) at (3,1.5) {$C \otimes C $};
\node (D) at (0,1.5) {$ C$};
\path[->,font=\scriptsize]
(C.south) edge node[descr] {$1_C \otimes \Delta$} (B.north)
(D.south) edge node[descr] {$\Delta$} (A.north)
(A.east) edge node[above] {$ \Delta \otimes 1_C$} (B.west)
(D.east) edge node[above] {$ \Delta$} (C.west)
;
\end{tikzpicture}
\end{center}
We will also assume that the coalgebras are counital, meaning that there exists a morphism $\epsilon_C \colon C \rightarrow I$, called counit, satisfying the condition:
\begin{equation}\label{counital comultiplication}
( \epsilon_C \otimes 1_C) \cdot \Delta = 1_C = (1_C \otimes \epsilon_C) \cdot \Delta,
\end{equation}
as expressed by the commutativity of the following diagram
\begin{center}
\begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base)]
\node (A) at (0,0) {$ C \otimes C$};
\node (B) at (2,0) {$C$};
\node (C) at (-2,0) {$C $};
\node (D) at (0,-1.5) {$ C$};
\draw[commutative diagrams/.cd, ,font=\scriptsize]
(B.south west) edge[commutative diagrams/equal] (D.north east)
(C.south east) edge[commutative diagrams/equal] (D.north west);
\path[->,font=\scriptsize]
(D.north) edge node[descr] {$\Delta$} (A.south)
(A.west) edge node[above] {$ \epsilon_C \otimes 1_C$} (C.east)
(A.east) edge node[above] {$1_C \otimes \epsilon_C$} (B.west)
;
\end{tikzpicture}
\end{center}
Similarly, a \textit{morphism of coalgebras} $g \colon C \rightarrow D$ is a morphism in $\mathcal{C}$ such that the following two diagrams commute
\begin{center}
\begin{tikzpicture}[descr/.style={fill=white},baseline=(D.base)]
\node (A) at (0,0) {$ D $};
\node (B) at (3,0) {$C$};
\node (C) at (3,1.5) {$C \otimes C $};
\node (D) at (0,1.5) {$ D \otimes D$};
\path[->,font=\scriptsize]
(B.north) edge node[descr] {$\Delta$} (C.south)
(A.north) edge node[descr] {$\Delta$} (D.south)
(B.west) edge node[above] {$ g$} (A.east)
(C.west) edge node[above] {$ g\otimes g$} (D.east)
;
\end{tikzpicture}
\begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base)]
\node (A) at (0,0) {$ D $};
\node (C) at (-2,0) {$I $};
\node (D) at (0,-1.5) {$ C.$};
\path[->,font=\scriptsize]
(D.north)edge node[descr] {$g $} (A.south)
(D.north west) edge node[descr] {$\epsilon_C $} (C.south east)
(A.west) edge node[above] {$ \epsilon_D $} (C.east)
;
\end{tikzpicture}
\end{center}
We also recall that a \textit{bialgebra} is a 5-tuple $(B,m,u_B,\Delta,\epsilon_B)$ where $(B,m,u_B)$ is an algebra, $(B,\Delta, \epsilon_B)$ is a coalgebra and $\Delta, \epsilon_B$ are algebra morphisms (which is equivalent of asking that $m$, $u_B$ are coalgebra morphisms) i.e.
\begin{equation}\label{delta et m}
\Delta \cdot m = (m \otimes m) \cdot (1_B \otimes \sigma \otimes 1_B) \cdot (\Delta \otimes \Delta)
\end{equation}
\begin{equation}\label{delta et u}
\Delta \cdot u_B = u_B \otimes u_B
\end{equation}
\begin{equation}\label{epsilon et m}
\epsilon_B \cdot m = \epsilon_B \otimes \epsilon_B
\end{equation}
\begin{equation}\label{epsilon et u}
\epsilon_B \cdot u_B = 1_I
\end{equation}
Moreover, a morphism in $\mathcal{C}$ is a \textit{morphism of bialgebras} if it is a morphism of algebras and coalgebras.
A bialgebra is called cocommutative when its underlying coalgebra structure is cocommutative, it means that \begin{equation}\label{coco}
\sigma \cdot \Delta = \Delta.
\end{equation}
The category of cocommutative bialgebras in $\mathcal{C}$, a symmetric monoidal category, is denoted by $\sf Bial_{\mathcal{C},coc}$.
\begin{examples}
(1) In the symmetric monoidal category $(\mathsf{Set}, \times , \{\star\})$ of sets where $\sigma$ is the twist morphism ( where $\sigma(x,y) = (y,x)$ for any element $x$ of a set $X$ and any element $y$ of a set $Y$), every object has a coalgebra structure with $\Delta$ being the diagonal and $\epsilon$ the morphism sending every element to the singleton. Hence, a (cocommutative) bialgebra (or algebra) is a monoid.
(2) In the symmetric monoidal category $(\mathsf{Vect_K},\otimes,K)$ of vector spaces over a field $K$ where $\sigma$ is the twist morphism (defined by $\sigma(x \otimes y) = y \otimes x$ for any $x \otimes y \in X \otimes Y$ ), we recover the notion of $K$-algebra, $K$-coalgebra and $K$-bialgebra.
(3) In \cite{CG}, a symmetric monoidal category was introduced such that Hom-algebras, Hom-coalgebras and Hom-bialgebras (see \cite{MS}) coincide with the algebras, coalgebras and bialgebras in this symmetric monoidal category.
\end{examples}
\subsection{Adjunction cocommutative bialgebras and cocommutative coalgebras}
It is well-known that we have the following adjunction between cocommutative bialgebras and cocommutative coalgebras.
\begin{equation}\label{adj}
\begin{tikzpicture}[descr/.style={fill=white},scale=1.2,baseline=(A.base)]
\node at (1.5,0) {$\perp$};
\node (A) at (0,0) {$\mathsf{BiAlg_{\mathcal{C},coc}}$};
\node (B) at (3,0) {$\mathsf{CoAlg}_{\mathcal{C},coc}$};
\path[->,font=\scriptsize]
([yshift=5pt]B.west) edge node[above] {$F$}([yshift=5pt]A.east)
([yshift=-5pt]A.east) edge node[below] {$U$} ([yshift=-5pt]B.west);
\end{tikzpicture}
\end{equation}
where $U$ is the forgetful functor and $F$ is the free algebra functor.
In particular, this adjunction implies that $U$ preserves the limits and that a monomorphism of bialgebras is in particular also a monomorphism of coalgebras. This observation will be useful several times in this paper.
\subsection{Limits in the category of cocommutative bialgebras}
We recall the constructions of products, equalizers and pullbacks in the category of cocommutative bialgebras in a symmetric monoidal category that has equalizers.
It is interesting to recall that the categorical product of cocommutative bialgebras is given by the monoidal product.
\begin{proposition}\label{tensor product}
In the category $\sf BiAlg_{\mathcal{C},coc}$, the categorical product of two cocommutative bialgebras $A$ and $B$ is given by the monoidal product
$(A \otimes B, \pi_A,\pi_B)$ where the projections $\pi_A \colon A \otimes B \rightarrow A$ and $\pi_B \colon A \otimes B \rightarrow B$ are defined by $\pi_A \coloneqq 1_A \otimes \epsilon_B $ and $\pi_B \coloneqq \epsilon_A \otimes 1_B$.
\begin{equation}\label{triangleproduit}
\begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base),scale=1.3]
\node (A) at (0,0) {$ A \otimes B$};
\node (B) at (2,0) {$B$};
\node (C) at (-2,0) {$A $};
\node (D) at (0,-1.5) {$ C.$};
\path[->,font=\scriptsize]
(D.north east)edge node[right,xshift=5pt] {$g$}(B.south west)
(D.north west)edge node[left,xshift=-5pt] {$f$}(C.south east)
(A.west)edge node[above] {$\pi_A$}(C.east)
(A.east)edge node[above] {$\pi_B$} (B.west)
;
\path[->,font=\scriptsize,dashed]
(D.north) edge node[descr] {$(f \otimes g) \cdot \Delta$}(A.south)
;
\end{tikzpicture}
\end{equation}
\end{proposition}
From now on, we are considering a symmetric monoidal category $\mathcal{C}$ that has equalizers.
The equalizers are defined as in \cite{Agore}.
Let $f,g \colon A \to B$ be two morphisms of bialgebras, then the following construction in $\mathcal{C}$ is the equalizer of $f$ and $g$.
\begin{equation}\label{equalizer}
\begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base)]
\node (A) at (-1,0) {$A$};
\node (B) at (2.5,0) {$B \otimes A $};
\node (C) at (-2.5,0) {$E$};
\path[->,font=\scriptsize]
([yshift=-4pt]A.east) edge node[below] {$(f \otimes 1_A) \cdot \Delta$} ([yshift=-4pt]B.west)
([yshift=0pt]C.east) edge node[above] {$\varepsilon $} ([yshift=0pt]A.west)
([yshift=4pt]A.east) edge node[above] {$( g \otimes 1_A) \cdot \Delta$} ([yshift=4pt]B.west)
;
\end{tikzpicture}.
\end{equation}
where $\varepsilon$ is the equalizer of $(f \otimes 1_A) \cdot \Delta$ and $( g \otimes 1_A) \cdot \Delta$ in $\mathcal{C}$. We can easily check that this construction is the equalizer of $f$ and $g$ in $\sf BiAlg_{\mathcal{C},coc}$.
Via the definitions of products and equalizers in $\sf BiAlg_{\mathcal{C},coc}$, we define the pullback in $\sf BiAlg_{\mathcal{C},coc}$ of two morphisms of bialgebras $f \colon A \to B$ and $g \colon C \to B$ as follows
\[
\begin{tikzpicture}[descr/.style={fill=white},baseline=(current bounding box.center),xscale=1.3]
\node (A) at (0,0) {$A \otimes_B C$};
\node (B) at (2.5,0) {$C$};
\node (A') at (0,-2) {$A$};
\node (B') at (2.5,-2) {$B$};
\path[->,font=\scriptsize]
(B) edge node[right] {$ g$} (B')
(A.south) edge node[left] {$ p_A$} (A'.north)
(A'.east) edge node[below] {$f$} (B'.west)
(A.east) edge node[above] {$p_C$} (B.west)
;
\end{tikzpicture} \]
where $A \otimes_B C$ is the object in the following equalizer
\[
\begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base),xscale=1.5]
\node (A) at (-1,0) {$A \otimes C$};
\node (B) at (2.5,0) {$A \otimes B \otimes C $};
\node (C) at (-2.5,0) {$A \otimes_B C$};
\path[->,font=\scriptsize]
([yshift=-4pt]A.east) edge node[below] {$(1_A \otimes f \otimes 1_C) \cdot ( \Delta \otimes 1_C) $} ([yshift=-4pt]B.west)
([yshift=0pt]C.east) edge node[above] {$\varepsilon$} ([yshift=0pt]A.west)
([yshift=4pt]A.east) edge node[above] {$(1_A \otimes g \otimes 1_C) \cdot (1_A \otimes \Delta) $} ([yshift=4pt]B.west)
;
\end{tikzpicture}.
\]
and the two projections are defined as $p_A \coloneqq (1_A \otimes \epsilon_C)\cdot \varepsilon $ and $p_C \coloneqq (\epsilon_A \otimes 1_C) \cdot \varepsilon$.
\subsection{Split extensions of cocommutative bialgebras}
In \cite{Stercksplit}, we introduced a notion of split extensions of (non-associative) bialgebras and we showed several properties. Here, we recall this notion and some results in the case of cocommutative bialgebras that will be useful later on.
\begin{definition}\label{definition split extension}
A \textit{split extension of cocommutative bialgebras} is given by a diagram \begin{equation}\label{split extension}
\begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base)]
\node (A) at (0,0) {$A$};
\node (B) at (2.5,0) {$B$};
\node (C) at (-2.5,0) {$X$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A.west) edge node[above] {$\lambda$} ([yshift=2pt]C.east);
\path[->,font=\scriptsize]
([yshift=-4pt]C.east) edge node[below] {$\kappa$} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$\alpha$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$e$} ([yshift=2pt]A.east);
\end{tikzpicture} ,
\end{equation}
where $X$, $A$, $B$ are cocommutative bialgebras, $\kappa$, $\alpha$, $e$ are morphisms of bialgebras, such that
\begin{itemize}
\item[(1)] $\lambda \cdot \kappa = 1_X$, $\alpha \cdot e =1_B$ ,
\item[(2)] $\lambda \cdot e = u_X\cdot \epsilon_B$, $\alpha \cdot \kappa = u_B \cdot\epsilon_X$,
\item[(3)] $m \cdot ((\kappa\cdot \lambda) \otimes (e \cdot \alpha)) \cdot \Delta = 1_A$,
\item[(4)]$ \lambda \cdot m \cdot (\kappa \otimes e) = 1_X \otimes \epsilon_B$,
\item[(5)] $\lambda$ is a morphism of coalgebras preserving the unit.
\end{itemize}
\end{definition}
We recall that the conditions $\lambda \cdot \kappa = 1_X$, $\lambda \cdot e = u_X \cdot \epsilon_B$ and the preservation of the unit by $\lambda$ are consequences of the axiom $(4)$.
\begin{definition}\label{morph split ext}
A \textit{morphism of split extensions} from the split extension
\begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base),xscale=0.7]
\node (A) at (0,0) {$A$};
\node (B) at (2.5,0) {$B$};
\node (C) at (-2.5,0) {$X$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A.west) edge node[above] {$\lambda$} ([yshift=2pt]C.east);
\path[->,font=\scriptsize]
([yshift=-4pt]C.east) edge node[below] {$\kappa$} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$\alpha$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$e$} ([yshift=2pt]A.east);
\end{tikzpicture}
to the split extension
\begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base),xscale=0.7]
\node (A) at (0,0) {$A'$};
\node (B) at (2.5,0) {$B'$};
\node (C) at (-2.5,0) {$X'$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A.west) edge node[above] {$\lambda'$} ([yshift=2pt]C.east);
\path[->,font=\scriptsize]
([yshift=-4pt]C.east) edge node[below] {$\kappa'$} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$\alpha'$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$e'$} ([yshift=2pt]A.east);
\end{tikzpicture}
is given by 3 morphisms of bialgebras $g\colon B \rightarrow B'$, $v \colon X \rightarrow X'$ and $p \colon A \rightarrow A'$ such that the following diagram commutes \begin{center}
\begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base),xscale=0.7]
\node (A) at (0,0) {$A$};
\node (B) at (2.5,0) {$B$};
\node (C) at (-2.5,0) {$X$};
\node (A') at (0,-2) {$A'$};
\node (B') at (2.5,-2) {$B'.$};
\node (C') at (-2.5,-2) {$X'$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A.west) edge node[above] {$\lambda$} ([yshift=2pt]C.east)
([yshift=2pt]A'.west) edge node[above] {$\lambda'$} ([yshift=2pt]C'.east);
\path[->,font=\scriptsize]
(B.south) edge node[right] {$ g$} (B'.north)
(C.south) edge node[left] {$ v $} (C'.north)
(A.south) edge node[left] {$ p$} (A'.north)
([yshift=-4pt]C'.east) edge node[below] {$\kappa'$} ([yshift=-4pt]A'.west)
([yshift=-4pt]A'.east) edge node[below] {$\alpha'$} ([yshift=-4pt]B'.west)
([yshift=2pt]B'.west) edge node[above] {$e'$} ([yshift=2pt]A'.east)
([yshift=-4pt]C.east) edge node[below] {$\kappa$} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$\alpha$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$e$} ([yshift=2pt]A.east);
\end{tikzpicture}
\end{center}
\end{definition}
The two definitions above give rise to the category $\mathsf{SplitExt(BiAlg_{\mathcal{C},coc})}$ of split extensions of bialgebras. In this paper, we will denote this class of split extensions of cocommutative bialgebras by $S_{coc}$.
We recall a convenient equality.
\begin{lemma}
Let
\begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base)]
\node (A) at (0,0) {$A$};
\node (B) at (2.5,0) {$B$};
\node (C) at (-2.5,0) {$X$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A.west) edge node[above] {$\lambda$} ([yshift=2pt]C.east);
\path[->,font=\scriptsize]
([yshift=-4pt]C.east) edge node[below] {$\kappa$} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$\alpha$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$e$} ([yshift=2pt]A.east);
\end{tikzpicture}
be a split extension of bialgebras, then the following identities hold,
\begin{equation}\label{lambda morph Prop 2.5}
\lambda \cdot m = m \cdot (\lambda \otimes \lambda) \cdot (1_A \otimes m) \cdot (1_A \otimes (e \cdot \alpha) \otimes (\kappa \otimes \lambda)) \cdot (\Delta \otimes 1_A),
\end{equation}
where $\triangleright = \lambda \cdot m \cdot (e \otimes \kappa)$.
\end{lemma}
Even if the category of cocommutative bialgebras is not protomodular (see \cite{GVdL}), we have an interesting result for the split extensions defined above: a relative form of the Split Short Five Lemma holds in $\sf BiAlg_{\mathcal{C},coc}$.
\begin{theorem}\label{Sec SSFL}
Let $(g,v,p)$ be a morphism of split extensions of bialgebras in a symmetric monoidal category $\mathcal{C}$
\begin{center}
\begin{tikzpicture}[descr/.style={fill=white}]
\node (A) at (0,0) {$A$};
\node (B) at (2.5,0) {$B$};
\node (C) at (-2.5,0) {$X$};
\node (A') at (0,-2) {$A'$};
\node (B') at (2.5,-2) {$B'$};
\node (C') at (-2.5,-2) {$X'$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A'.west) edge node[above] {$\lambda'$} ([yshift=2pt]C'.east)
([yshift=2pt]A.west) edge node[above] {$\lambda$} ([yshift=2pt]C.east)
;
\path[->,font=\scriptsize]
(B.south) edge node[right] {$ g$} (B'.north)
(C.south) edge node[left] {$ v $} (C'.north)
(A.south) edge node[left] {$ p$} (A'.north)
([yshift=-4pt]C'.east) edge node[below] {$\kappa'$} ([yshift=-4pt]A'.west)
([yshift=-4pt]A'.east) edge node[below] {$\alpha'$} ([yshift=-4pt]B'.west)
([yshift=2pt]B'.west) edge node[above] {$e'$} ([yshift=2pt]A'.east)
([yshift=-4pt]C.east) edge node[below] {$\kappa$} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$\alpha$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$e$} ([yshift=2pt]A.east);
\end{tikzpicture}
\end{center}
then $p$ is an isomorphism whenever $v$ and $g$ are.
\end{theorem}
In \cite{Stercksplit}, it was proved that there is an equivalence between the categories of actions and the one of split extensions of (cocommutative) bialgebras in a symmetric monoidal category. We recall this equivalence and the definition of the objects and morphisms of the category of actions of bialgebras.
\begin{definition}\label{def action}
Let $X$ and $B$ be cocommutative bialgebras in a symmetric monoidal category $(\mathcal{C}, \otimes, I, \sigma)$. An \textit{action of bialgebras} is a morphism in $\mathcal{C}$, $\triangleright \colon B \otimes X \rightarrow X $, such that
\begin{align*}
&\triangleright \cdot (u_B \otimes 1_X) = 1_X,\\
&\triangleright \cdot (1_B \otimes \triangleright) = \triangleright \cdot (m \otimes 1_X) ,\\
& \triangleright \cdot (1_B \otimes u_X) = u_X \cdot\epsilon_B,\\
&\triangleright \cdot (1_B \otimes m) = m \cdot (\triangleright \otimes \triangleright) \cdot (1_B \otimes \sigma \otimes 1_X) \cdot (\Delta \otimes 1_X \otimes 1_X),\\
&\epsilon_X \cdot \triangleright = \epsilon_B \otimes \epsilon_X,\\
&\Delta \cdot \triangleright = (\triangleright \otimes \triangleright) \cdot (1_B \otimes
\sigma \otimes 1_X) \cdot (\Delta \otimes \Delta).
\end{align*}
\end{definition}
In other words, an action of bialgebras of $B$ on $X$ is a cocommutative bialgebra in the symmetric monoidal category of $B$-modules, where $B$ is a cocommutative bialgebra.
\begin{definition}\label{morph action}
Let $\triangleright \colon B \otimes X \rightarrow X$ and $\triangleright' \colon B' \otimes X' \rightarrow X'$ be two actions of bialgebras. A \textit{morphism of actions of bialgebras} is defined as a pair of morphisms of bialgebras $g \colon B \rightarrow B'$ and $v \colon X \rightarrow X'$ such that \[v \cdot \triangleright = \triangleright' \cdot (g \otimes v).\]
\end{definition}
There is then the category of actions of cocommutative bialgebras that will be denoted by $\sf Act(BiAlg_{\mathcal{C},coc})$.
We recall the theorem obtained in \cite{Stercksplit} and the construction of the functor in the equivalence of categories between $\sf Act(BiAlg_{\mathcal{C},coc})$ and $\sf SplitExt(BiAlg_{\mathcal{C},coc})$.
\begin{theorem}\label{equi bialg}
There is an equivalence between the category $\mathsf{SplitExt(BiAlg_{\mathcal{C},coc})}$ of split extensions of cocommutative bialgebras and the category $\mathsf{Act(BiAlg_{\mathcal{C},coc})}$ of actions of cocommutative bialgebras.
\end{theorem}
\begin{proof}
The functor $ F \colon \mathsf{SplitExt(BiAlg_{\mathcal{C},coc})} \rightarrow \mathsf{Act(BiAlg_{\mathcal{C},coc})}$ is defined as
\[ F \left(
\begin{tikzpicture}[descr/.style={fill=white},baseline=(O.base)]
\node (O) at (1,-1) {$ $};
\node (A) at (0,0) {$A$};
\node (B) at (2.5,0) {$B$};
\node (C) at (-2.5,0) {$X$};
\node (A') at (0,-2) {$A'$};
\node (B') at (2.5,-2) {$B'$};
\node (C') at (-2.5,-2) {$X'$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A'.west) edge node[above] {$\lambda'$} ([yshift=2pt]C'.east)
([yshift=2pt]A.west) edge node[above] {$\lambda$} ([yshift=2pt]C.east)
;
\path[->,font=\scriptsize]
(C.south) edge node[left] {$ v$} (C'.north)
(B.south) edge node[left] {$ g$} (B'.north)
(A.south) edge node[left] {$ p$} (A'.north)
([yshift=-4pt]C'.east) edge node[below] {$\kappa'$} ([yshift=-4pt]A'.west)
([yshift=-4pt]A'.east) edge node[below] {$\alpha'$} ([yshift=-4pt]B'.west)
([yshift=2pt]B'.west) edge node[above] {$e'$} ([yshift=2pt]A'.east)
([yshift=-4pt]C.east) edge node[below] {$\kappa$} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$\alpha$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$e$} ([yshift=2pt]A.east);
\end{tikzpicture}
\right) =
\begin{tikzpicture}[descr/.style={fill=white},baseline=(O.base)]
\node (E) at (4,0) {$B \otimes X$};
\node (E') at (4,-2) {$B' \otimes X'$};
\node (F) at (6,0) {$X$};
\node (F') at (6,-2) {$X',$};
\path[->,font=\scriptsize]
(E.east) edge node[above] {$\triangleright$} (F.west)
(E'.east) edge node[above] {$\triangleright'$} (F'.west)
(E.south) edge node[descr] {$ g \otimes v$} (E'.north)
(F.south) edge node[left] {$ v$} (F'.north)
;
\end{tikzpicture}
\]
where $\triangleright \colon = \lambda \cdot m \cdot (e \otimes \kappa)$.
The functor $G \colon \mathsf{Act(BiAlg_{\mathcal{C},coc})} \rightarrow \mathsf{SplitExt(BiAlg_{\mathcal{C},coc})}$ is defined as
\[
G \left(
\begin{tikzpicture}[descr/.style={fill=white},baseline=(O.base)]
\node (O) at (-5,-1) {$ $};
\node (E) at (-6.5,0) {$B \otimes X$};
\node (E') at (-6.5,-2) {$B' \otimes X'$};
\node (F) at (-4.5,0) {$X$};
\node (F') at (-4.5,-2) {$X'$};
\path[->,font=\scriptsize]
(E.east) edge node[above] {$\triangleright$} (F.west)
(E'.east) edge node[above] {$\triangleright'$} (F'.west)
(E.south) edge node[descr] {$ g \otimes v$} (E'.north)
(F.south) edge node[left] {$ v$} (F'.north);
\end{tikzpicture}
\right) =
\begin{tikzpicture}[descr/.style={fill=white},baseline=(O.base)]
\node (O) at (-1,-1) {$ $};
\node (A) at (0,0) {$X \rtimes B$};
\node (B) at (2.5,0) {$B$};
\node (C) at (-2.5,0) {$X$};
\node (A') at (0,-2) {$X' \rtimes B'$};
\node (B') at (2.5,-2) {$B',$};
\node (C') at (-2.5,-2) {$X'$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A'.west) edge node[above] {$\pi_1'$} ([yshift=2pt]C'.east)
([yshift=2pt]A.west) edge node[above] {$\pi_1$} ([yshift=2pt]C.east);
\path[->,font=\scriptsize]
(C.south) edge node[left] {$ v$} (C'.north)
(B.south) edge node[left] {$ g$} (B'.north)
(A.south) edge node[left] {$ v \otimes g$} (A'.north)
([yshift=-4pt]C'.east) edge node[below] {$i_1'$} ([yshift=-4pt]A'.west)
([yshift=-4pt]A'.east) edge node[below] {$\pi_2'$} ([yshift=-4pt]B'.west)
([yshift=2pt]B'.west) edge node[above] {$i_2'$} ([yshift=2pt]A'.east)
([yshift=-4pt]C.east) edge node[below] {$i_1$} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$\pi_2$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$i_2$} ([yshift=2pt]A.east);
\end{tikzpicture}
\]
where $i_1 = 1_X \otimes u_B$, $i_2 = u_X \otimes 1_B $, $\pi_1 = 1_X \otimes \epsilon_B$, $\pi_2= \epsilon_X \otimes 1_B$ and $X \rtimes B$ is the object $X \otimes B$ where the bialgebra structure is given by the following morphisms of $\mathcal{C}$
\begin{align*}
m_{X \rtimes B} &= (m \otimes m) \cdot (1_X \otimes \triangleright \otimes 1_B \otimes 1_B) \cdot (1_X \otimes 1_B \otimes \sigma \otimes 1_B) \cdot (1_X \otimes \Delta \otimes 1_X \otimes 1_B)\\
u_{X \rtimes B} &=u_X \otimes u_B,\\
\Delta_{X \rtimes B} &= (1_X \otimes \sigma \otimes 1_B) \cdot (\Delta \otimes \Delta),\\
\epsilon_{X \rtimes B}&= \epsilon_X \otimes \epsilon_B.
\end{align*}
\end{proof}
Let $B$ and $X$ be two cocommutative bialgebras, since the trivial action $\epsilon_B \otimes 1_X \colon B \otimes X \to X$ is an action of bialgebras as defined in Definition \ref{def action}, we obtain the following corollary.
\begin{corollary}\label{product projection}
Le $B$ and $X$ be two cocommutative bialgebras, the projection $\pi_2$ of the product
\[ \begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base)]
\node (A) at (0,0) {$X \otimes B$};
\node (B) at (2.5,0) {$B$};
\node (C) at (-2.5,0) {$X$};
\path[->,font=\scriptsize]
([yshift=2pt]A.west) edge node[above] {$\pi_1$} ([yshift=2pt]C.east);
\path[->,font=\scriptsize]
([yshift=-4pt]C.east) edge node[below] {$i_1$} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$\pi_2$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$i_2$} ([yshift=2pt]A.east);
\end{tikzpicture} \]
where $i_1 = 1_X \otimes u_B$, $i_2 = u_X \otimes 1_B $, $\pi_1 = 1_X \otimes \epsilon_B$ and $\pi_2= \epsilon_X \otimes 1_B$, belongs to $S_{coc}$.
\end{corollary}
\section{$S_{coc}$-protomodularity}
It is known that the category of cocommutative bialgebras is not protomodular. However, we have interesting results with respect to the class $S_{coc}$ of split extensions of cocommutative bialgebras, as the Split Short Five Lemma (Theorem \ref{Sec SSFL}) and the equivalence with the actions (Theorem \ref{equi bialg}).
These results are the motivation to prove that the category is $S_{coc}$-protomodular, which means protomodular with respect to $S_{coc}$.
Let $\mathcal{C}$ be a pointed category, and $S$ be a class of split epimorphisms with their kernels (also called points), to recall the definition of an $S$-protomodular category we give the definition of a strong point.
\begin{definition}
A split epimophism wiht its kernel \begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base),xscale=0.7]
\node (A) at (0,0) {$A$};
\node (B) at (2.5,0) {$B$};
\node (C) at (-2.5,0) {$X$};
\path[->,font=\scriptsize]
(C.east) edge node[above] {$\kappa$} (A.west)
([yshift=-4pt]A.east) edge node[below] {$\alpha$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$e$} ([yshift=2pt]A.east);
\end{tikzpicture} is called a \emph{strong point} whenever the kernel $\kappa$ and the splitting $e$ are \emph{jointly strongly epimorphic}.
\end{definition}
\begin{definition}\label{stronglyepic}
The morphisms $v \colon X \to A$ and $w \colon B \to A$ are \emph{jointly strongly epimorphic} if for any morphism $\mu : M \to A$ that factors through $v$ and $w$, $\mu$ is an isomorphism.
\begin{center} \begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base)]
\node (A) at (0,0) {$A$};
\node (B) at (3,0) {$B$};
\node (C) at (-3,0) {$X$};
\node (D) at (-0,-1.5) {$M$};
\path[->,font=\scriptsize]
(C.east) edge node[above] {$ v $} (A.west)
(C.south east) edge node[left,xshift=-10pt] {$ \delta $} (D.north west)
(B.south west) edge node[right,xshift=10pt] {$\gamma$} (D.north east);
\path[->,font=\scriptsize]
(B.west) edge node[above] {$w$} (A.east);
\path[>->,font=\scriptsize]
(D.north) edge node[right] {$\mu$} (A.south);
\end{tikzpicture}\end{center}
\end{definition}
\begin{definition}\cite{BMS}
Let $\mathcal{C}$ be a pointed finitely complete category and $S$ a class of split epimorphisms stable under pullbacks. $\mathcal{C}$ is said to be \emph{$S$-protomodular} when the class $S$ is closed under finite limits and any point in $S$ is a strong point.
\end{definition}
\subsection{Pullback stable}
We prove that the class $S_{coc}$ of split extensions of cocommutative bialgebras is pullback stable.
We consider the pullback of \begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base),xscale=0.7]
\node (A) at (0,0) {$A$};
\node (B) at (2.5,0) {$B$};
\node (C) at (-2.5,0) {$X$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A.west) edge node[above] {$\lambda$} ([yshift=2pt]C.east);
\path[->,font=\scriptsize]
([yshift=-4pt]C.east) edge node[below] {$\kappa$} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$\alpha$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$e$} ([yshift=2pt]A.east);
\end{tikzpicture} along the morphism of bialgebras $g \colon C \to B$.
\begin{equation}\label{pullback}
\begin{tikzpicture}[descr/.style={fill=white},baseline=(current bounding box.center),xscale=1.3]
\node (A) at (0,0) {$A \otimes_B C$};
\node (B) at (2.5,0) {$C$};
\node (C) at (-2.5,0) {$X$};
\node (A') at (0,-2) {$A$};
\node (B') at (2.5,-2) {$B$};
\node (C') at (-2.5,-2) {$X$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A'.west) edge node[above] {$\lambda$} ([yshift=2pt]C'.east)
([yshift=2pt]A.west) edge node[above] {$\lambda \cdot p_A$} ([yshift=2pt]C.east)
;
\path[->,font=\scriptsize]
(B) edge node[right] {$ g$} (B')
(C.south) edge node[left] {$ 1_X$} (C'.north)
(A.south) edge node[left] {$ p_A$} (A'.north)
([yshift=-4pt]C'.east) edge node[below] {$\kappa$} ([yshift=-4pt]A'.west)
([yshift=-4pt]A'.east) edge node[below] {$\alpha$} ([yshift=-4pt]B'.west)
([yshift=2pt]B'.west) edge node[above] {$e$} ([yshift=2pt]A'.east)
([yshift=-4pt]C.east) edge node[below] {$(\kappa , u_C \cdot \epsilon_X) $} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$p_C$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$(e \cdot g , 1_C)$} ([yshift=2pt]A.east);
\end{tikzpicture}
\end{equation}
where $(\kappa, u_C \cdot \epsilon_X)$ and $(e \cdot g,1_C)$ are the morphisms induced by the universal property of the pullback.
\begin{proposition}\label{pullbackstable}
Let \eqref{pullback} be the pullback of \begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base),xscale=0.7]
\node (A) at (0,0) {$A$};
\node (B) at (2.5,0) {$B$};
\node (C) at (-2.5,0) {$X$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A.west) edge node[above] {$\lambda$} ([yshift=2pt]C.east);
\path[->,font=\scriptsize]
([yshift=-4pt]C.east) edge node[below] {$\kappa$} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$\alpha$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$e$} ([yshift=2pt]A.east);
\end{tikzpicture} along a morphism of bialgebras $g \colon C \to B$, then the upper row of \eqref{pullback}:
\begin{equation*}
\begin{tikzpicture}[descr/.style={fill=white},baseline=(current bounding box.center),scale=1.3]
\node (A) at (0,0) {$A \otimes_B C$};
\node (B) at (2.5,0) {$C$};
\node (C) at (-2.5,0) {$X$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A.west) edge node[above] {$\lambda \cdot p_A$} ([yshift=2pt]C.east)
;
\path[->,font=\scriptsize]
([yshift=-4pt]C.east) edge node[below] {$(\kappa , u_C \cdot \epsilon_X) $} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$p_C$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$(e \cdot g , 1_C)$} ([yshift=2pt]A.east);
\end{tikzpicture},
\end{equation*}
belongs to $S_{coc}$.
\end{proposition}
\begin{proof}
The conditions $(1)$, $(2)$, $(4)$ and $(5)$ of Definition \ref{split extension} are easily checked. We give the details of the condition $(3)$. To show that \[m_{A \otimes_B C} \cdot ((\kappa , u_C \cdot \epsilon_X) \cdot \lambda \cdot p_A \otimes(e \cdot g , 1_C) \cdot p_C ) \cdot \Delta_{A \otimes_B C} = 1_{A \otimes_B C},\] we compose the two sides of the equality with $p_A$ and $p_C$.
\ctikzfig{piA}
Since $p_A$ and $p_C$ are jointly monic in $\sf BiAlg_{\mathcal{C},coc}$ (and then also jointly monic in $\sf CoAlg_{\mathcal{C},coc}$ thanks to the adjunction \eqref{adj}) and the cocommutativity implies that $m_{A \otimes_B C} \cdot ((\kappa , u_C \cdot \epsilon_X) \cdot \lambda \cdot p_A \otimes(e \cdot g , 1_C) \cdot p_C ) \cdot \Delta_{A \otimes_B C}$ is a morphism of coalgebras, we can conclude that the condition $(3)$ holds.
Hence, the class $S_{coc}$ of split extensions of cocommutative bialgebras is stable under pullbacks.
\end{proof}
\subsection{Closure under finite limits}
To prove that the split extensions in $S_{coc}$ are closed under finite limits, we prove that they are closed under products and equalizers.
\begin{proposition}\label{closedlimit}
The class $S_{coc}$ of split extensions of bialgebras is closed under finite limits.
\end{proposition}
\begin{proof}
Let \begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base)]
\node (A) at (0,0) {$A$};
\node (B) at (2.5,0) {$B$};
\node (C) at (-2.5,0) {$X$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A.west) edge node[above] {$\lambda$} ([yshift=2pt]C.east);
\path[->,font=\scriptsize]
([yshift=-4pt]C.east) edge node[below] {$\kappa$} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$\alpha$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$e$} ([yshift=2pt]A.east);
\end{tikzpicture} and \begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base)]
\node (A) at (0,0) {$A'$};
\node (B) at (2.5,0) {$B'$};
\node (C) at (-2.5,0) {$X'$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A.west) edge node[above] {$\lambda'$} ([yshift=2pt]C.east);
\path[->,font=\scriptsize]
([yshift=-4pt]C.east) edge node[below] {$\kappa'$} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$\alpha'$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$e'$} ([yshift=2pt]A.east);
\end{tikzpicture} two split extensions of bialgebras. We can prove that
\begin{equation}\label{productdiag}
\begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base),xscale=1.4]
\node (A) at (0,0) {$A \otimes A'$};
\node (B) at (2.5,0) {$B \otimes B'$};
\node (C) at (-2.5,0) {$X \otimes X'$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A.west) edge node[above] {$\lambda \otimes \lambda'$} ([yshift=2pt]C.east);
\path[->,font=\scriptsize]
([yshift=-4pt]C.east) edge node[below] {$\kappa \otimes \kappa'$} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$\alpha \otimes \alpha'$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$e\otimes e'$} ([yshift=2pt]A.east);
\end{tikzpicture}
\end{equation}
belongs to $S_{coc}$. Since the construction is made component wise it is clear that \eqref{productdiag} belongs to $S_{coc}$ as we can explicitly see in the proof of condition (3) via the commutativity of the following diagram
\ctikzfig{product}
The other conditions hold via similar computations
The class $S_{coc}$ is also stable under equalizers. We construct the equalizer of two morphisms of split extensions of bialgebras $(g,v,p)$ and $(g',v',p')$.
\begin{equation*}
\begin{tikzpicture}[descr/.style={fill=white},baseline=(current bounding box.center),scale=1.1]
\node (A) at (0,0) {$A'$};
\node (B) at (2.5,0) {$B'$};
\node (C) at (-2.5,0) {$X'$};
\node (A') at (0,-2) {$A$};
\node (B') at (2.5,-2) {$B$};
\node (C') at (-2.5,-2) {$X$};
\node (A'') at (0,2) {$E$};
\node (B'') at (2.5,2) {$E'$};
\node (C'') at (-2.5,2) {$\hat{E}$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A'.west) edge node[above] {$\lambda$} ([yshift=2pt]C'.east)
([yshift=2pt]A.west) edge node[above] {$\lambda'$} ([yshift=2pt]C.east)
([yshift=2pt]A''.west) edge node[above] {$\tilde{\lambda}$} ([yshift=2pt]C''.east)
;
\path[->,font=\scriptsize]
(C'') edge node[left] {$ \hat{\varepsilon}$} (C)
(A'') edge node[left] {$ \varepsilon$} (A)
(B'') edge node[left] {$ \varepsilon'$} (B)
([xshift=-4pt]A.south) edge node[left] {$ g$} ([xshift=-4pt]A'.north)
([xshift=4pt]A.south) edge node[right] {$ g'$} ([xshift=4pt]A'.north)
([xshift=-4pt]C.south) edge node[left] {$ v$} ([xshift=-4pt]C'.north)
([xshift=4pt]C.south) edge node[right] {$ v'$} ([xshift=4pt]C'.north)
([xshift=-4pt]B.south) edge node[left] {$ p$} ([xshift=-4pt]B'.north)
([xshift=4pt]B.south) edge node[right] {$ p'$} ([xshift=4pt]B'.north)
([yshift=-4pt]C''.east) edge node[below] {$\tilde{\kappa}$} ([yshift=-4pt]A''.west)
([yshift=-4pt]A''.east) edge node[below] {$\tilde{\alpha}$} ([yshift=-4pt]B''.west)
([yshift=2pt]B''.west) edge node[above] {$\tilde{e}$} ([yshift=2pt]A''.east)
([yshift=-4pt]C'.east) edge node[below] {$\kappa$} ([yshift=-4pt]A'.west)
([yshift=-4pt]A'.east) edge node[below] {$\alpha$} ([yshift=-4pt]B'.west)
([yshift=2pt]B'.west) edge node[above] {$e$} ([yshift=2pt]A'.east)
([yshift=-4pt]C.east) edge node[below] {$\kappa'$} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$\alpha'$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$e'$} ([yshift=2pt]A.east);
\end{tikzpicture}
\end{equation*}
where $\tilde{\alpha}$, $\tilde{e}$, $\tilde{\kappa}$ and $\tilde{\lambda}$ are induced by the universal properties of the equalizers. Note that $\tilde{\lambda}$ is induced by the universal property of $\hat{\varepsilon}$ seen as a coequalizer in $\sf CoAlg_{\mathcal{C},coc}$ via \eqref{adj}.
By using the fact that $\varepsilon, \varepsilon'$ and $\hat{\varepsilon}$ are monomorphisms of bialgebras (and hence also of coalgebras) we can conclude that
\begin{equation}\label{equalizer}
\begin{tikzpicture}[descr/.style={fill=white},baseline=(current bounding box.center),scale=1.3]
\node (A'') at (0,2) {$E$};
\node (B'') at (2.5,2) {$E'$};
\node (C'') at (-2.5,2) {$\hat{E}$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A''.west) edge node[above] {$\tilde{\lambda}$} ([yshift=2pt]C''.east)
;
\path[->,font=\scriptsize]
([yshift=2pt]B''.west) edge node[above] {$\tilde{e}$} ([yshift=2pt]A''.east)
([yshift=-4pt]C''.east) edge node[below] {$\tilde{\kappa}$} ([yshift=-4pt]A''.west)
([yshift=-4pt]A''.east) edge node[below] {$\tilde{\alpha}$} ([yshift=-4pt]B''.west);
\end{tikzpicture}
\end{equation}
belongs to $S_{coc}$. We give an explicit proof of the condition (4) via the commutativity of this diagram:
\ctikzfig{equalizer}
Since $\hat{\varepsilon}$ is a monomorphism of coalgebras and $\tilde{\lambda}\cdot m \cdot (\tilde{\kappa} \otimes \tilde{e})$ is a coalgebra morphism, we can conclude that \eqref{equalizer} satisfies condition (4).
\end{proof}
\subsection{Strong points}
It was proven in \cite{Stercksplit} that for a split extension of (cocommutative) bialgebras
\[ \begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base)]
\node (A) at (0,0) {$A$};
\node (B) at (2.5,0) {$B$};
\node (C) at (-2.5,0) {$X$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A.west) edge node[above] {$\lambda$} ([yshift=2pt]C.east);
\path[->,font=\scriptsize]
([yshift=-4pt]C.east) edge node[below] {$\kappa$} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$\alpha$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$e$} ([yshift=2pt]A.east);
\end{tikzpicture} \]
$\kappa$ and $e$ are jointly epimorphic. Now we prove that for cocommutative biagebras, $\kappa$ and $e$ are jointly strongly epimorphic and hence any split extension of cocommutative bialgebras is a strong point.
\begin{proposition}\label{strong}
Any split extension of cocommutative bialgebras is a strong point.
\end{proposition}
\begin{proof}
Let \begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base)]
\node (A) at (0,0) {$A$};
\node (B) at (2.5,0) {$B$};
\node (C) at (-2.5,0) {$X$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A.west) edge node[above] {$\lambda$} ([yshift=2pt]C.east);
\path[->,font=\scriptsize]
([yshift=-4pt]C.east) edge node[below] {$\kappa$} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$\alpha$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$e$} ([yshift=2pt]A.east);
\end{tikzpicture} be a split extension of cocommutative bialgebras and $\mu \colon M \to A$ be a monomorphism of bialgebras. Let $\kappa$ and $e$ factor through $\mu$:
\begin{center} \begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base)]
\node (A) at (0,0) {$A$};
\node (B) at (3,0) {$B$};
\node (C) at (-3,0) {$X$};
\node (D) at (-0,-1.5) {$M$};
\path[->,font=\scriptsize]
(C.east) edge node[above] {$ \kappa $} (A.west)
(C.south east) edge node[left,xshift=-10pt] {$ \delta $} (D.north west)
(B.south west) edge node[right,xshift=10pt] {$\gamma$} (D.north east);
\path[->,font=\scriptsize]
(B.west) edge node[above] {$e$} (A.east);
\path[>->,font=\scriptsize]
(D.north) edge node[right] {$\mu$} (A.south);
\end{tikzpicture}\end{center}
We can form the following commutative diagram
\begin{equation*}
\begin{tikzpicture}[descr/.style={fill=white},baseline=(current bounding box.center),scale=1.1]
\node (A) at (0,0) {$M$};
\node (B) at (2.5,0) {$B$};
\node (C) at (-2.5,0) {$X$};
\node (A') at (0,-2) {$A$};
\node (B') at (2.5,-2) {$B$};
\node (C') at (-2.5,-2) {$X$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A'.west) edge node[above] {\normalsize $\lambda$} ([yshift=2pt]C'.east)
([yshift=2pt]A.west) edge node[above] {$\lambda \cdot \mu$} ([yshift=2pt]C.east)
;
\path[->,font=\scriptsize]
(B) edge node[right] {$ 1_B$} (B')
(C.south) edge node[left] {$ 1_X$} (C'.north)
(A.south) edge node[left] {$ \mu$} (A'.north)
([yshift=-4pt]C'.east) edge node[below] {$\kappa$} ([yshift=-4pt]A'.west)
([yshift=-4pt]A'.east) edge node[below] {$\alpha$} ([yshift=-4pt]B'.west)
([yshift=2pt]B'.west) edge node[above] {$e$} ([yshift=2pt]A'.east)
([yshift=-4pt]C.east) edge node[below] {$\delta$} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$\alpha \cdot \mu$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$\gamma$} ([yshift=2pt]A.east);
\end{tikzpicture}
\end{equation*}
We prove that the upper row is in the class $S_{coc}$ by checking all the conditions of Definition \ref{split extension}. In particular, via the commutativity of the following diagram
\ctikzfig{mu}
we conclude that \begin{tikzpicture}[descr/.style={fill=white},baseline=(current bounding box.center),scale=1.1]
\node (A) at (0,0) {$M$};
\node (B) at (2.5,0) {$B$};
\node (C) at (-2.5,0) {$X$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A.west) edge node[above] {$\lambda \cdot \mu$} ([yshift=2pt]C.east)
;
\path[->,font=\scriptsize]
([yshift=-4pt]C.east) edge node[below] {$\delta$} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$\alpha \cdot \mu$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$\gamma$} ([yshift=2pt]A.east);
\end{tikzpicture} satisfies condition $(3)$ since $\mu$ is a monomorphism of bialgebras (and then also a monomorphism of coalgebras \eqref{adj}) and $m \cdot ( \delta \cdot \lambda \cdot \mu \otimes \gamma \cdot \alpha \cdot \mu ) \cdot \Delta$ is a coalgebra morphism thanks to the cocommutativity.
\end{proof}
Thanks to the previous results we obtain the following theorem:
\begin{theorem}\label{thmproto}
The category of cocommutative bialgebras in any symmetric monoidal category is an $S_{coc}$-protomodular category with respect to the class $S_{coc}$ of split extensions of cocommutative bialgebras defined in Definition \ref{split extension}.
\end{theorem}
\begin{proof}
This theorem holds thanks to Proposition \ref{pullbackstable}, Proposition \ref{closedlimit} and Proposition \ref{strong}.
\end{proof}
\begin{remark}
This theorem generalizes the result of \cite{BMS} saying that the category of monoids is $S$-protomodular with respect to the class of Schreier split epimorphisms. Indeed, if $\mathcal{C}$ is the symmetric monoidal category $\sf (Set, \times, \{ \star \})$, then $S_{coc}$ becomes the class of the Schreier split epimorphisms and $\sf BiAlg_{Set,coc}$ the category of monoids.
\end{remark}
\section{Huq commutator}
In this section, we recall the notion of centrality in the sense of Huq in a pointed category with binary products. Moreover, we give an explicit description of this centrality for two subbialgebras of a cocommutative bialgebra.
Let $\mathcal{C}$ be a pointed category with binary products, we say that two subobjects $x \colon X \to A$ and $y \colon Y \to A$ commute (or centralize) in the sense of Huq \cite{Huq} if there exists a morphism $p \colon X \times Y \to A$ such that the following diagram commutes
\begin{equation*}
\begin{tikzpicture}[descr/.style={fill=white},yscale=0.9,xscale=1.1]
\node (A) at (0,0) {$A$};
\node (C) at (2,2) {$Y$};
\node (D) at (0,2) {$X \times Y$};
\node (B) at (-2,2) {$X$};
\path[-stealth]
(B.south) edge node[left] {${x\,} $} (A.north west)
(C.south) edge node[right] {${\, y} $} (A.north east)
(B.east) edge node[above] {$(1,0)$} (D.west)
(C.west) edge node[above] {$(0,1)$} (D.east);
\path[-stealth,dashed]
(D.south) edge node[right] {$p$} (A.north);
\end{tikzpicture}
\end{equation*}
Then we denote by $[X,Y] = 0$ the fact that the subobjects $X$ and $Y$ commute.
Note that in $\sf BiAlg_{\mathcal{C},coc}$, if it exists, $p$ is unique thanks to Corollary \ref{product projection} and Proposition \ref{strong}.
\begin{proposition}
In $\sf BiAlg_{\mathcal{C},coc}$, the following are equivalent for $x \colon X \to A$ and $y \colon Y \to A$ two subobjects of a cocommutative bialgebra $A$:
\begin{itemize}
\item[(i)] $m \cdot \sigma \cdot (x \otimes y) = m \cdot (x \otimes y) $
\item[(ii)] $[X,Y]$ = 0, i.e\ there exists a (unique) morphism of bialgebras $p \colon X \otimes Y \to A$ such that the following diagram commutes
\begin{equation}\label{connector}
\begin{tikzpicture}[descr/.style={fill=white},yscale=0.9,xscale=1]
\node (A) at (0,0) {$A$};
\node (C) at (2.5,2) {$Y$};
\node (D) at (0,2) {$X \otimes Y$};
\node (B) at (-2.5,2) {$X$};
\path[-stealth]
(B.south) edge node[left] {${x\,} $} (A.north west)
(C.south) edge node[right] {${\, y} $} (A.north east)
(B.east) edge node[above] {$1_X \otimes u_Y$} (D.west)
(C.west) edge node[above] {$u_X \otimes 1_Y$} (D.east);
\path[-stealth,dashed]
(D.south) edge node[right] {$p$} (A.north);
\end{tikzpicture}
\end{equation}
\end{itemize}
\end{proposition}
\begin{proof}
$(i) \Rightarrow (ii)$, we define $p : X \otimes Y \to A$ by $p \coloneqq m \cdot (x \otimes y)$. It is easy to see that this map makes the diagram \eqref{connector} commutes. Moreover, thanks to $(i)$ we can prove that $p$ is a morphism of bialgebras (this proof is straghforward and is left to the reader).
On the other way around, since $p$ is a morphism of bialgebras which makes \eqref{connector} commute, we can make the following diagram commute
\ctikzfig{huq}
and conclude that $(i)$ holds.
\end{proof}
Note that we obtained a similar result in the case of cocommutative Hopf algebras in the paper \cite{GSV}.
\section{Smith is Huq}
Now we would like to compare the notion of centrality in the sense of Huq and in the sense of Smith.
We first recall the centrality of two equivalence relations in the sense of Smith in any category with pullbacks \cite{BG}.
\begin{definition}\label{connectorSmith}
Let $(R,r_0,r_1)$ and $(S,s_0,s_1)$ be two equivalence relations over the same object $X$. We denote the pullback of $s_0$ along $r_1$ by
\begin{center}
\begin{tikzpicture}[descr/.style={fill=white},yscale=1.2]
\node (A) at (0,0) {$R$};
\node (B) at (0,1.5) {$R \times_X S$};
\node (C) at (3,1.5) {$S$};
\node (D) at (3,0) {$X.$};
\path[->,font=\scriptsize]
(C.south) edge node[right] {$s_0$} (D.north)
(B.south) edge node[right] {$p_1$} (A.north)
(B) edge node[above] {$p_2$} (C)
(A) edge node[above] {$r_1$} (D)
;
\end{tikzpicture}
\end{center}
A connector between $R$ and $S$ is an arrow $\hat{p} \colon R \times_X S \to X$ such that
\begin{itemize}
\item $xS\hat{p}(x,y,z)$ and $zR\hat{p}(x,y,z)$,
\item $\hat{p}(x,x,y) = y$ and $\hat{p}(x,y,y)=x$,
\item $\hat{p}(x,y,\hat{p}(y,u,v)) = \hat{p}(x,u,v) $ and $\hat{p}(\hat{p}(x,y,u),u,v) = \hat{p}(x,y,v)$,
\end{itemize}
where the above expressions are defined.
When such an arrow exists, we say that the relations $R$ and $S$ centralize (in the sense of Smith).
\end{definition}
Note that this notion of centrality of equivalence relations is not independent of the centrality in the sense of Huq. If two equivalence relations centralize each other (in the sense of Smith) then it is true that the normal subobjects associated with them, called normalizations \cite{Bourn}, centralize in the sense of Huq \cite{BG}.
The converse is not true in general. If $\mathcal{C}$ is a category such that any two equivalence relations always centralize each other as soon as their normalizations centralize in the sense of Huq, will say that $\mathcal{C}$ satisfies the \textit{Smith is Huq} property \cite{BG}.
For example, the categories of groups and cocommutative Hopf algebras over a field satisfy this condition \cite{GSV}.
In the paper \cite{MFM}, the authors studied this condition in the context of $S$-protomodular categories.
They considered pointed $S$-protomodular categories with respect to a class of points $S$ that are stable under composition and such that any product projection, i.e. any such diagram
\[\begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base)]
\node (A) at (0,0) {$X \times B$};
\node (B) at (2.5,0) {$B,$};
\node (C) at (-2.5,0) {$X$};
\path[->,font=\scriptsize]
([yshift=2pt]A.west) edge node[above] {$\pi_1$} ([yshift=2pt]C.east);
\path[->,font=\scriptsize]
([yshift=-4pt]C.east) edge node[below] {$i_1$} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$\pi_2$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$i_2$} ([yshift=2pt]A.east);
\end{tikzpicture} \]
where $i_1 \coloneqq (1_X,0)$ and $i_2 \coloneqq (0,1_B)$, belongs to the class $S$.
In that context, the condition \emph{Smith is Huq} means that two $S$-equivalence relations centralize each other (in the sense of Smith) if and only if their normalization commute in the sense of Huq. The category of monoids (but also the category of monoids with operations \cite{MFMS}) satisfies this property, with respect to the class of Schreier split epimorphisms.
First, we prove that the class $S_{coc}$ is stable under composition.
\begin{proposition}
The class $S_{coc}$ of split extensions of cocommutative bialgebras is closed under composition.
\end{proposition}
\begin{proof}
Let
\begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base)]
\node (A) at (0,0) {$A$};
\node (B) at (2.5,0) {$B$};
\node (C) at (-2.5,0) {$X$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A.west) edge node[above] {$\lambda$} ([yshift=2pt]C.east);
\path[->,font=\scriptsize]
([yshift=-4pt]C.east) edge node[below] {$\kappa$} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$\alpha$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$e$} ([yshift=2pt]A.east);
\end{tikzpicture}
and
\begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base)]
\node (A) at (0,0) {$B$};
\node (B) at (2.5,0) {$C$};
\node (C) at (-2.5,0) {$Y$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A.west) edge node[above] {$\lambda'$} ([yshift=2pt]C.east);
\path[->,font=\scriptsize]
([yshift=-4pt]C.east) edge node[below] {$\kappa'$} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$\alpha'$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$e'$} ([yshift=2pt]A.east);
\end{tikzpicture} be two composable split extensions. We can build the following diagram
\[
\begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base)]
\node (A) at (0,0) {$A$};
\node (B) at (2.5,0) {$C$};
\node (C) at (-2.5,0) {$Z$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A.west) edge node[above] {$\hat{\lambda}$} ([yshift=2pt]C.east);
\path[->,font=\scriptsize]
([yshift=-4pt]C.east) edge node[below] {$\hat{\kappa}$} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$\alpha' \cdot \alpha$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$e \cdot e'$} ([yshift=2pt]A.east);
\end{tikzpicture}
\]
where $\hat{\kappa} \colon Z \to A$ is the kernel of $\alpha' \cdot \alpha$ and $\hat{\lambda}$ is the factorization through $\hat{\kappa}$ of the morphism of coalgebras $ m \cdot (\kappa\cdot \lambda \otimes e \cdot \kappa' \cdot \lambda' \cdot \alpha) \cdot \Delta $.
In particular, we have the following equality
\begin{equation}\label{kernelaa'}
\hat{\kappa} \cdot \hat{\lambda} = m \cdot (\kappa\cdot \lambda \otimes e \cdot \kappa' \cdot \lambda' \cdot \alpha) \cdot \Delta .
\end{equation}
We prove that this construction belongs to $S_{coc}$.
The condition $(3)$ is proven via the commutativity of the following diagram.
\tikzfig{composition3}
The condition $(4)$ holds thanks to the commutativity of the two diagrams in Figure \ref{(4)composition},
where $\overline{\alpha \cdot \hat{\kappa}}$ is the factorization of $\alpha \cdot \hat{\kappa}$ through the kernel $\kappa'$ of $\alpha'$:
\begin{equation}\label{factorise kernel}
\kappa' \cdot \overline{\alpha \cdot \hat{\kappa}} = \alpha \cdot \hat{\kappa},
\end{equation}
and the fact that $\hat{\lambda} \cdot m \cdot (\hat{\kappa} \otimes e \cdot e') $ and $( 1_Z \otimes \epsilon_C) $ are coalgebras morphisms and $\hat{\kappa}$ is a monomorphism of coalgebras.
\end{proof}
Thanks to this proposition and Corollary \ref{product projection}, $\sf BiAlg_{\mathcal{C},coc}$ is a category in which we can apply the results obtained in \cite{MFM}. To apply the results of \cite{MFM}, we recall the notions of $S$-equivalence relation and reflexive-multiplicative graph.
\begin{definition}
An equivalence relation \[
\begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base)]
\node (A) at (0,0) {$A$};
\node (B) at (2.5,0) {$B$};
\path[->,font=\scriptsize]
([yshift=0pt]B.west) edge node[descr] {$e$} ([yshift=0pt]A.east)
([yshift=-6pt]A.east) edge node[below] {$\beta$} ([yshift=-6pt]B.west)
([yshift=6pt]A.east) edge node[above] {$\alpha$} ([yshift=6pt]B.west);
\end{tikzpicture}\] is an $S$-equivalence relation if the point $(\alpha, e)$ is in $S$.
\end{definition}
\begin{definition}\cite{CPP}
In a category $\mathcal{C}$ with pullbacks, a reflexive graph, denoted by
\begin{tikzpicture}[descr/.style={fill=white},xscale=0.8,yscale=1,baseline=(A.base)]
\node (A) at (0,0) {$A_1$};
\node (D) at (3,0) {$A_0$};
\path[->,font=\scriptsize]
([yshift=7pt]A.east) edge node[above] {$\delta$} ([yshift=7pt]D.west)
(D.west) edge node[descr] {$\iota$} (A.east)
([yshift=-7pt]A.east) edge node[below] {$\gamma$} ([yshift=-7pt]D.west);
\end{tikzpicture},
together with a morphism $c \colon A_1 \times_{A_0} A_1 \rightarrow A_1$ is called a \emph{reflexive-multiplicative graph}
\begin{equation}\label{mgraph}
\begin{tikzpicture}[descr/.style={fill=white},yscale=1.2,baseline=(A.base)]
\node (A) at (0,0) {$A_1$};
\node (D) at (3,0) {$A_0$};
\node (X) at (-3,0) {$A_1\times_{A_0}A_1$};
\path[->,font=\scriptsize]
(X.east) edge node[above] {$c$} (A.west)
([yshift=7pt]A.east) edge node[above] {$\delta$} ([yshift=7pt]D.west)
(D.west) edge node[descr] {$\iota$} (A.east)
([yshift=-7pt]A.east) edge node[below] {$\gamma$} ([yshift=-7pt]D.west);
\end{tikzpicture} ,
\end{equation}
where $A_1\times_{A_0}A_1$ is the (object part of the) following pullback
\[
\xymatrix{A_1\times_{A_0} A_1 \ar[d]_{p_1} \ar[r]^-{p_2} & A_1 \ar[d]^{ \gamma} \\
A_1 \ar[r]_-{\delta}& A_0
}
\]
and $c$ is a multiplication that is required to satisfy the identities
\begin{equation}\label{RGM}
c \cdot (1_{A_1} , \iota \cdot \delta ) = 1_{A_1} = c \cdot (\iota \cdot \gamma, 1_{A_1} ),
\end{equation}
where $(1_{A_1} , \iota \cdot \delta ) \colon A_1 \rightarrow A_1\times_{A_0}A_1 $ and $(\iota \cdot \gamma, 1_{A_1} ) \colon A_1 \rightarrow A_1\times_{A_0}A_1 $ are induced by the universal property of the pullback $A_1\times_{A_0}A_1$.
\end{definition}
To apply Theorem 4.2 in \cite{MFM} we need the following result.
\begin{proposition}
Every reflexive graph \[
\begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base)]
\node (A) at (0,0) {$A$};
\node (B) at (2.5,0) {$B$};
\node (C) at (-2.5,0) {$X$};
\node (C') at (0,2) {$X'$};
\path[->,dashed,font=\scriptsize]
([yshift=2pt]A.west) edge node[above] {$\lambda$} ([yshift=2pt]C.east)
([xshift=2pt]A.north) edge node[right] {$\lambda'$} ([xshift=2pt]C'.south);
\path[->,font=\scriptsize]
([yshift=0pt]B.west) edge node[descr] {$e$} ([yshift=0pt]A.east)
([yshift=-6pt]A.east) edge node[below] {$\beta$} ([yshift=-6pt]B.west)
([xshift=-4pt]C'.south) edge node[left] {$\kappa'$} ([xshift=-4pt]A.north)
([yshift=-4pt]C.east) edge node[below] {$\kappa$} ([yshift=-4pt]A.west)
([yshift=6pt]A.east) edge node[above] {$\alpha$} ([yshift=6pt]B.west);
\end{tikzpicture}\]
such that \begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base)]
\node (A) at (0,0) {$A$};
\node (B) at (2.5,0) {$B$};
\node (C) at (-2.5,0) {$X$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A.west) edge node[above] {$\lambda$} ([yshift=2pt]C.east);
\path[->,font=\scriptsize]
([yshift=-4pt]C.east) edge node[below] {$\kappa$} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$\alpha$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$e$} ([yshift=2pt]A.east);
\end{tikzpicture}
and
\begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base)]
\node (A) at (0,0) {$A$};
\node (B) at (2.5,0) {$B$};
\node (C) at (-2.5,0) {$X'$};
\path[dashed,->,font=\scriptsize]
([yshift=2pt]A.west) edge node[above] {$\lambda'$} ([yshift=2pt]C.east);
\path[->,font=\scriptsize]
([yshift=-4pt]C.east) edge node[below] {$\kappa'$} ([yshift=-4pt]A.west)
([yshift=-4pt]A.east) edge node[below] {$\beta$} ([yshift=-4pt]B.west)
([yshift=2pt]B.west) edge node[above] {$e$} ([yshift=2pt]A.east);
\end{tikzpicture} belong to $S_{coc}$ and $[ X , X'] = 0 $, is a reflexive-multiplicative graph.
\end{proposition}
\begin{proof}
Let consider the following pullback
\[
\begin{tikzpicture}[descr/.style={fill=white},baseline=(current bounding box.center),yscale=0.9,xscale=1.1]
\node (A) at (0,0) {$A \otimes_B A$};
\node (B) at (2.5,0) {$A$};
\node (A') at (0,-2) {$A$};
\node (B') at (2.5,-2) {$B$};
\path[->,font=\scriptsize]
(B) edge node[right] {$\beta$} (B')
(A.south) edge node[left] {$ p_1$} (A'.north)
(A'.east) edge node[below] {$\alpha$} (B'.west)
(A.east) edge node[below] {$p_2$} (B.west)
;
\end{tikzpicture} \]
where $A \otimes_B A$ is the object in the following equalizer
\begin{equation}\label{eq compo}
\begin{tikzpicture}[descr/.style={fill=white},baseline=(A.base),xscale=1.5]
\node (A) at (-1,0) {$A \otimes A$};
\node (B) at (2.5,0) {$A \otimes B \otimes A $};
\node (C) at (-2.5,0) {$A \otimes_B A$};
\path[->,font=\scriptsize]
([yshift=-4pt]A.east) edge node[below] {$(1_A \otimes \alpha \otimes 1_A) \cdot ( \Delta \otimes 1_A) $} ([yshift=-4pt]B.west)
([yshift=0pt]C.east) edge node[above] {$\varepsilon$} ([yshift=0pt]A.west)
([yshift=4pt]A.east) edge node[above] {$(1_A \otimes \beta \otimes 1_C) \cdot (1_A \otimes \Delta) $} ([yshift=4pt]B.west)
;
\end{tikzpicture}.
\end{equation}
By defining $c \colon A \otimes_B A \to A$ by
\[c = m \cdot (\kappa \cdot \lambda \otimes 1_A) \cdot (\varepsilon \otimes \varepsilon)\]
we can verify that this reflexive graph is multiplicative thanks to Figure \ref{cmorph1}
and Figure \ref{cmorph2}.
\end{proof}
This result and Theorem 4.2 in \cite{MFM}, implies the following result
\begin{theorem}\label{thmhuq}
In $\sf BiAlg_{\mathcal{C},coc}$, let $S_{coc}$ be the class of split extensions of cocommutative bialgebras as defined in Definition \ref{split extension}, two $S_{coc}$-equivalence relations centralize each other (in the sense of Smith) if and only if their normalization commute in the sense of Huq.
\end{theorem}
By restricting the above theorem to the symmetric monoidal category $\sf (Set, \times , \{ \star \})$, we obtain the result on monoids of \cite{MFM}. Note that, in \cite{MFM}, they also prove this result for every category of monoids with operations.
\section{Conclusion}
In \cite{Stercksplit}, we introduced a class of split extensions of (cocommutative) bialgebras, called $S_{coc}$, such that their category is equivalent to the category of actions of (cocommutative) bialgebras and we proved the Split Short Five Lemma when we restrict it to the split extensions of (cocommutative) bialgebras. In this paper, we use this class $S_{coc}$, to prove that $\sf BiAlg_{\mathcal{C},coc}$ is $S_{coc}$-protomodular. It implies that we can now apply the theory and results of \cite{BM3x3,Bournpartial} to obtain new results for cocommutative bialgebras. For example, it follows that $\sf BiAlg_{\mathcal{C},coc}$ is a $S_{coc}$-Mal'tsev category \cite{BM3x3,Bournpartial}, hence any $S_{coc}$-reflexive relation is transitive.
Another result of this paper is the description of the notion of centrality in the sense of Huq for cocommutative bialgebras. Moreover, we prove that $\sf BiAlg_{\mathcal{C},coc}$ satisfies the ``partial'' \emph{Smith is Huq} condition, meaning that two $S_{coc}$-equivalence relations centralize each other as soon as their normalization commute in the sense of Huq. Note that Theorem \ref{thmhuq} and Theorem \ref{thmproto} generalize results in \cite{BMS} and \cite{MFM} on the category of monoids.
\begin{figure}[b]
\tikzfig{composition4}
\caption{Condition (4) of the composition of two split extensions}
\label{(4)composition}
\end{figure}
\begin{landscape}
\begin{figure}
\tikzfig{cmorphisme}
\caption{The morphism $c$ is a morphism of bialgebras (part 1)}
\label{cmorph1}
\end{figure}
\end{landscape}
\begin{landscape}
\begin{figure}
\tikzfig{cmorphisme2}
\caption{The morphism $c$ is a morphism of bialgebras (part 2)}
\label{cmorph2}
\end{figure}
\end{landscape}
\bibliographystyle{mybibstyleJL}
|
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| 5
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\section{Introduction}
The concept of non-compactness measure was introduced by Kuratowski in \cite{Ku30} to characterize relative compact sets as those whose non-compactness measure equals zero. This measure was used by Sadowskii in \cite{Sa67} to define condensing function in a Banach space, whereby he generalized the well-known Schauder fixed point theorem \cite{Sc30}. Later, by extending the definition of non-compactness measure to locally convex spaces, in \cite{HiEtAl69} Himmelberg \emph{et al.} proved a fixed point theorem for condensing functions in locally convex spaces, generalizing even the corresponding result by Tychonov in \cite{Ty35}. Nowdays, condensing functions and multi-functions have been defined on metric and topological vector spaces and significant results have been obtained.
On the other hand, since the Banach contraction principle \cite{Ba22} and Brouwer work \cite{Br12} were published, fixed point theory has been developed for operators defined on complete metric and topological vector spaces. Some remarkable works on this subject, among others, are the articles of Darbo \cite{Da55}, Kakutani \cite{Ka41}, Schauder \cite{Sc30} and Tychonoff \cite{Ty35}. Even though completeness is defined for both metric and topological linear spaces, some metric spaces are not linear and some linear spaces are not metrizable. Due to the convenience of establishing results as the aforementioned in a unified way, the main aim of this paper is addressing attention in developing concepts such as non-compactness measure and condensing multi-function into a structure containing both of these class of spaces, namely the uniform structure. Uniform spaces have been studied by various authors in different contexts. The importance of these structures lies in the fact that a wide variety of classes of topological spaces are included in this category. For example, metric spaces, Hausdorff topological vector spaces and Menger (probabilistic) metric spaces are particular cases of uniform spaces. In \cite{Fa96} Fang proved that these classes are $F$-type topological spaces, which, according to Hamel \cite{HL03} (see also \cite{Ha05}), coincides with the category of uniform spaces.
We are interested in studying topological properties of multi-functions taking values in a uniform space. To this end, the Hausdorff topology on the closed and bounded subsets of the space is considered. It is well-known that this topology is generated by a family of pseudo metrics (Hausdorff pseudometrics), consequently it is uniform as well. We use these concepts to define weak lower and upper semi-continuity for multi-functions, moreover their relation with the continuity with respect to the Hausdorff topology is studied. By extending the definition of non-compactness measure to uniform spaces, we are able to characterize the relative compactness of subsets as those sets having non-compactness measure equal to zero and condensing functions and multi functions can be defined. Such extensions allow us to locate, in the context of uniform spaces, some classical results existing in nonlinear analysis. Caristi and Bishop and Phelps results have been extended, by some authors, for operators taking values in a uniform space (see for example, \cite{Br74,Ha05,HL03,Mi90,MT89}). We revisit some of these works in order to expand the range of situations where new versions of these results can be applied.
Including this introduction, the paper is divided in six sections. In
Section 2, some preliminary definitions, notations and facts are stated in order to prove results in subsequence
sections. Section 3 is devoted to study relations among the different type of semi-continuities and the continuity of multi-functions according to the Hausdorff topology. The non-compactness and condensing concepts for uniform spaces are introducing in Section 4, along with some of the main results of this paper. In Section 5, extended versions of Caristi's theorem are presented and finally, we devote Section 6 for an application of some our results to multi-functions defined on locally convex spaces.
\section{Preliminaries}
In all of this work, $(X,\mathcal{U})$ stands for a uniform space. For each $x\in X$, $A\subseteq X$ and $U\in \mathcal{U}$, we denote $U[x]=\{y\in X:(x,y)\in U\}$ and $U[A]=\bigcup_{x\in A}U[x]$. We consider $X$ endowed with the topology induced by $\mathcal{U}$.
A subset $B$ of $X$ is said to be bounded, if there exist $x\in X$ and $U\in \mathcal{U}$ such that $B\subseteq U[x]$. We denote by $\mathcal{B}(X)$ the family of all bounded subsets of $X$ and by $\mathcal{C}(X)$ the family of all closed and nonempty subsets of $X$. Let $\mathcal{CB}(X)=\mathcal{C}(X)\cap \mathcal{B}(X)$. On $\mathcal{CB}(X)$ a family $\mathcal{H}=\{H_U\}_{U\in\mathcal{U}}$ is defined as follows:
$$
H_U=\{(A,B)\in\mathcal{CB}(X)\times\mathcal{CB}(X):A\subseteq U[B]\mbox{ and }B\subseteq U[A]\}.
$$
It is well known, see \cite{CV77} for instance, that $\mathcal{H}$ is a fundamental system of entourages for a uniformity on $\mathcal{CB}(X)$. The topology on $\mathcal{CB}(X)$ induced by $\mathcal{H}$ is referred to as the $\mathcal{H}$-topology and continuity with respect to this topology is referred to as $\mathcal{H}$-continuity. Let $\mathcal{K}(X)$ denote the family of all nonempty compact subsets of $X$. Since $\mathcal{K}(X)\subseteq \mathcal{CB}(X)$, we consider $\mathcal{K}(X)$ endowed with the induced $\mathcal{H}$-topology. In \cite{SR69/70} it is proved that $\mathcal{K}(X)$ is $\mathcal{H}$-complete, if and only if, $(X,\mathcal{U})$ so is.
It is well-known that the uniformity $\mathcal{U}$ is generated by a family of pseudo metrics $\{d_\lambda\}_{\lambda\in\Lambda}$ on $X$. As usual, for each $\lambda\in\Lambda$, $a\in A$, $\epsilon>0$ and $A\in\mathcal{CB}(X)$, we denote $\mathrm{B}_\lambda(a,\epsilon)=\{x\in X:d_\lambda(a,x)<\epsilon\}$ and $\mathrm{B}_\lambda(A,\epsilon)=\bigcup_{x\in A}\mathrm{B}_\lambda(x,\epsilon)$.
It is assumed that the family $\{d_\lambda\}_{\lambda\in\Lambda}$ is saturated, which means that the following condition holds:
\begin{description}
\item[(S)] For each $\lambda,\mu\in\Lambda$, there exists $\nu\in\Lambda$ such that $\max\{d_\lambda(x,y),d_\mu(x,y)\}\leq d_\nu(x,y)$, for all $x,y\in X$.
\end{description}
This condition implies that a set $G\subseteq X$ is open, whenever for each $a\in G$ there exist $\lambda\in\Lambda$ and $\epsilon>0$ such that $\mathrm{B}_\lambda(a,\epsilon)\subseteq G$. As showed in Section \S1.2, Chapter IX in \cite{Bo66b}, condition (S) is not a restriction on the topology of $X$. That is, when the family of pseudo metrics does not satisfy condition (S), we can find such a family satisfying this condition and generating the same topology.
It is easy to see (see Theorem II-12 in \cite{CV77}, for instance) that the $\mathcal{H}$-topology on $\mathcal{CB}(X)$ is generated by the family of pseudo metrics $\{H^\lambda\}_{\lambda\in\Lambda}$ defined by
$$
H^\lambda(A,B)=\max\left\{\sup_{x\in A}d_\lambda(x,B),\sup_{y\in B}d_\lambda(y,A)\right\},
$$
where, as usual for $A\in\mathcal{CB}(X)$, $a\in A$ and $\lambda\in\Lambda$, $d_\lambda(a,A)=\inf_{x\in A}d_\lambda(a,x)$. In the sequel, these notations are maintained.
\begin{remark}
Observe that, for all $A,B\in\mathcal{CB}(X)$ and $\lambda\in\Lambda$,
$$
H^\lambda(A,B)=\inf\{\epsilon>0:A\subseteq \mathrm{B}_\lambda(B,\epsilon),B\subseteq \mathrm{B}_\lambda(A,\epsilon)\}.
$$
\end{remark}
Let $E$ be a set and $T:E\to \mathcal{C}(X)$ be a multi-function. For any $B\subset X$, the inverse image of $B$ under $T$ is defined by $T^{-1}(B)=\{x\in E:Tx\cap B\neq\emptyset\}$ and for each $A\subseteq E$ the image of $A$ by $T$ is defined as $T(A)=\bigcup_{x\in A}Tx$. Next, suppose $E$ is a
topological space. The multi-function $T$ is said to be lower (respectively, upper) semi-continuous, if for any open subset $G$ (respectively, closed subset
$F$) of $X$, $T^{-1}(G)$ is an open (respectively, $T^{-1}(F)$ is a closed) subset of $E$.
\section{Semi-continuity and $\mathcal{H}$-continuity}
\begin{proposition}\label{p1} Let $E$ be a topological space and $T:E\to\mathcal{CB}(X)$ an $\mathcal{H}$-continuous multi-function.
Then, $T$ is lower semi-continuous.
\end{proposition}
\begin{proof}
Let $a\in E$ and $G$ be an open subset of $X$ such that $Ta\cap G\neq\emptyset$. Hence,
there exist $x_0\in Ta$ and $U\in\mathcal{U}$ such that $Ta\cap U^{-1}[x_0]\neq\emptyset$ and $U^{-1}[x_0]\subseteq
G$. Since $T$ is continuous at $a$, with respect to the $\mathcal{H}$-topology, there exists $V_a$ neighborhood of $a$ such that $(Tu,Ta)\in H_U$, if $u\in V_a$. Hence for each $u\in V_a$, we have $x_0\in Ta\subseteq U[Tu]$ and consequently $\emptyset\neq Tu\cap U^{-1}[x_0]\subseteq Tu\cap G$. This concludes the proof.
\end{proof}
Let $\mathcal{K}(X)$ denote the family of all compact nonempty subsets of $X$. Since $\mathcal{K}(X)\subseteq \mathcal{CB}(X)$, we consider $\mathcal{K}(X)$ endowed with the
induced $\mathcal{H}$-topology.
\begin{theorem}\label{t2}
Let $E$ be a topological space and $T:E\to\mathcal{K}(X)$ a multi-function. Then, $T$ is lower and upper semi-continuous, if and only if, $T$
is continuous with respect to the induced $\mathcal{H}$-topology on $\mathcal{K}(X)$.
\end{theorem}
\begin{proof}
Let us assume $T$ is a lower and upper semi-continuous
multi-function. Let $a\in E$, $U\in\mathcal{U}$ and $V\in\mathcal{U}$ such that $V\circ V^{-1}\subseteq U$. We have
$Ta\subseteq V[Ta]$ and due to $T$ is upper semi-continuous, there exists $V_a$
neighborhood of $a$ such that, for each $u\in V_a$, $Tu\subseteq V[Ta]\subseteq (V\circ V^{-1})[Ta]$. On the other hand, $Ta$ is compact and hence there exist $x_1,\dots,x_r\in Ta$ such that $Ta\subseteq G$, where $G=V[x_1]\cup\dots\cup V[x_r]$. Since for all $i\in\{1,\dots,r\}$, $Ta\cap V[x_i]\neq\emptyset$, the lower semi-continuity of $T$ implies there exists
$W_a$ neighborhood of $a$ such that, for each $u\in W_a$, $Tu\cap V[x_i]\neq\emptyset$, for all $i\in\{1,\dots,r\}$. That is, for
each $u\in W_a$, there exist $y_1,\dots,y_r\in Tu$ such that for all $i\in\{1,\dots,r\}$, $y_i\in V[x_i]$. Hence
$$
Ta\subseteq\bigcup_{i=1}^rV[x_i]\subseteq\bigcup_{i=1}^r(V\circ V^{-1})[y_i]\subseteq\bigcup_{y\in Tu}U[y].
$$
We have proved that for each $u\in V_a\cap W_a$, $Tu\subseteq U[Ta]$ and $Ta\subseteq U[Tu]$. Thus, $T$ is continuous at $a$ according with the $\mathcal{H}$-topology.
Next, we assume $T$ is continuous according to the $\mathcal{H}$-topology on $\mathcal{K}(X)$. From Proposition \ref{p1}, $T$ is lower semi-continuous. Let us prove that
$T$ is upper semi-continuous. Let $a\in X$ and $G$ be an open subset of $X$ such that $Ta\subseteq G$. Since $X$ is a uniform space and $Ta$ is
compact (see Proposition 4, Section \S4.3, Chapter II in \cite{Bo66}, for instance), there exists $U\in\mathcal{U}$ such that $U[Ta]\subseteq G$. By assumption, there exists $V_a$ neighborhood of $a$ such that $Tu\subseteq U[Ta]$ (and $Ta\subseteq U[Tu]$) if $u\in
V_a$. Hence, for all $u\in V_a$, $ T(u)\subseteq G$, i.e., $T$ is upper semi-continuous and therefore the proof is
complete.
\end{proof}
Let $D$ be a subset of $X$ and $T: D\to\mathcal{C}(X)$ a multi-function. We say $T$ is weakly lower (respectively, upper) semi-continuous, if for each $\lambda\in\Lambda$ and $\alpha\geq0$, the set $\{x\in D:d_\lambda(x,T(x))<\alpha\}$ (respectively, $\{x\in D:d_\lambda(x,T(x))>\alpha\}$) is open, and we say $T$ is weakly continuous, if $T$ is both weakly lower and weakly upper semi-continuous.
\begin{theorem}\label{t3}
Let $D$ be a subset of $X$ and $T: D\to\mathcal{C}(X)$ a multi-function.
\begin{description}
\item[(\ref{t3}.1)]If $T$ is
lower semi-continuous, then $T$ is weakly lower semi-continuous.
\item[(\ref{t3}.2)] If $T$ is upper semi-continuous, then $T$ is weakly upper
semi-continuous.
\end{description}
\end{theorem}
\begin{proof}
Let $\lambda\in \Lambda$ and $\alpha \geq0$. Suppose $T$ is lower semi-continuous and
define $A=\{x\in D:d_\lambda(x,Tx)<\alpha \}$. In order to prove that $A$ is an open set, suppose $A\neq \emptyset $ and choose $a\in A$. Let
$\epsilon =\alpha -d_\lambda(a,Ta)$ and $y\in Ta$ such that $d_\lambda(a,y)<\epsilon /3+d_\lambda(a,Ta)$. Since $T$ is lower semi-continuous,
there exists a neighborhood $V'_a$ of $a$ such that $Tu\cap B_\lambda(y,\epsilon /3)\neq \emptyset $, for all $u\in V'_a$. Let
$V_a=V'_a\cap B_\lambda(a,\epsilon/3)$, $u\in V_a$ and $ b_{u}\in Tu\cap B_\lambda(y,\epsilon /3)$. We have $d_\lambda(u,b_{u})\leq
d_\lambda(u,a)+d_\lambda(a,y)+d_\lambda(y,b_{u})<\alpha $ and consequently, $d_\lambda(u,Tu)<\alpha$. This proves that $A$ is an open set and
therefore, (\ref{t3}.1) holds.
Next we define $A=\{x\in D:d_\lambda(x,Tx)>\alpha\}$. Let $a\in A$ and
choose $\beta, \gamma\in\mathbb{R}$ such that $\gamma>\beta>\alpha$ and $d_\lambda(a,Ta)>\gamma$. Let $G=\{x\in
D:d_\lambda(x,Ta)<(\gamma-\beta)/2\}$. Since $Ta\subseteq G$, $G$ is open and $T$ is upper semi-continuous, there exists $V'_a$ neighborhood of
$a$ such that for each $x\in V'_a$, $Tx\subseteq G$. This implies that for each $x\in V'_a$ and each $y\in Tx$,
$d_\lambda(y,Ta)<(\gamma-\beta)/2$. Hence, $$ \gamma<d_\lambda(a,Ta)\leq d_\lambda(a,y)+d_\lambda(y,Ta)<d_\lambda(a,y)+(\gamma-\beta)/2. $$
Thus, $d_\lambda(a,y)>(\gamma+\beta)/2$ and consequently, $d_\lambda(a,Tx)\geq(\gamma+\beta)/2$. Let $V_a=V'_a\cap B_\lambda(a,\beta-\alpha)$ and
note that for each $x\in V_a$, $$ \beta<d_\lambda(a,Tx)\leq d_\lambda(a,x)+d_\lambda(x,Tx)<\beta-\alpha+d_\lambda(x,Tx). $$ This proves that
$V_a\subseteq A$ and therefore, $A$ is an open set, which concludes the proof.
\end{proof}
\begin{remark}
As shown in \cite{FMO11}, even though $X$ is a metric space, there exist weakly continuous multi-functions, which are not
lower or upper semi-continuous multi-functions.
\end{remark}
\begin{theorem}\label{t4} Let $E$ be a subset of $X$ and $T:E\to\mathcal{CB}(X)$ be a $\mathcal{H}$-continuous multi-function. Then, $T$ is
weakly continuous.
\end{theorem}
\begin{proof}
Let $\lambda\in\Lambda$, $u,v\in E$, $\epsilon>0$ and $y_\epsilon\in T(v)$ such that
$d_\lambda(v,y_\epsilon)<d_\lambda(v,T(v))+\epsilon$. Hence,
$$
\begin{array}{ccl} d_\lambda(u,T(u)) & \leq & d_\lambda(u,v)+d_\lambda(v,y_\epsilon)+
d_\lambda(y_\epsilon,T(u))\\ &< & d_\lambda(u,v)+d_\lambda(v,T(v))+\epsilon+H_\lambda(T(v),T(u)).\\
\end{array}
$$
Consequently,
$$
|d_\lambda(u,T(u))-d_\lambda(v,T(v))|\leq d_\lambda(u,v)+H_\lambda(T(v),T(u))
$$
and therefore, the weak continuity of $T$ is directly obtained.
\end{proof}
\section{Non-compactness measure}
In accordance with the uniformity generating the topology of $X$, a subset $D$ of $X$ is precompact, if for each $\lambda\in\Lambda$ and $\epsilon>0$, there exist $x_1,\dots,x_r\in X$ such that $D\subseteq \mathrm{B}_\lambda(x_1,\epsilon)\cup\dots \cup\mathrm{B}_\lambda(x_r,\epsilon)$. Moreover, a filter $\mathcal{F}$ on $X$ is a Cauchy filter (c.f. \cite{Bo66}), if for each $\lambda\in\Lambda$ and $\epsilon>0$, there exist $A\in\mathcal{F}$ such that $A\times A\subseteq \mathrm{U}(\lambda,\epsilon)$, where $\mathrm{U}(\lambda,\epsilon)=\{(x,y)\in X\times X:d_\lambda(x,y)<\epsilon\}$.
A non-compactness measure $\alpha:\mathcal{B}(X)\to[0,\infty[$ is defined as
$\alpha(A)=\inf\{\epsilon>0:\mathcal{C}(\epsilon)\ni A\}$, where for $\epsilon>0$, $\mathcal{C}(\epsilon)$ is the family of all $A\in\mathcal{B}(X)$ such that for
each $\lambda\in\Lambda$, there exist $x_1,\dots, x_r\in X$ such that $A\subseteq \mathrm{B}_\lambda(x_i,\epsilon/2)\cup\cdots\cup \mathrm{B}_\lambda(x_r,\epsilon/2)$. From Theorem 3, Section \S 4.2, Chapter II in \cite{Bo66}, $A\in \mathcal{B}(X)$ is precompact, if and only if, $\alpha(A)=0$.
It is easy to see the following three properties hold:
\begin{description}
\item[a)]$\alpha(A)\leq\alpha(B)$ whenever $A\subseteq B$,\quad($B\in
\mathcal{B}(X)$).
\item[b)] $\alpha(A\cup B)=\max\{\alpha(A),\alpha(B)\}$,\quad ($A,B\in \mathcal{B}(X)$).
\item[c)] $\alpha(\overline{A})=\alpha(A)$,\quad ($A\in \mathcal{B}(X)$).
\item[d)] $\alpha\left(\bigcap_{\lambda\in\Lambda}\mathrm{\mathrm{B}}_\lambda(A,\epsilon)\right)\leq\alpha(A)+\epsilon$,\quad ($A\in \mathcal{B}(X),\epsilon>0$).
\end{description}
Theorem \ref{t5} below extends a classical result (Theorem 1') by Kuratowski in \cite{Ku30}.
\begin{theorem}\label{t5}
Suppose $X$ is complete. Let $\mathcal{B}$ be a filter base on $X$ such that $\mathcal{B}\subseteq \mathcal{CB}(X)$, $E=\bigcap_{B\in\mathcal{B}}B$,
and suppose $\inf\{\alpha(B):B\in\mathcal{B}\}=0$. Then, $E$ is compact and nonempty, and $\mathcal{B}$ converges to $E$ in the $\mathcal{H}$-topology, i.e.\ for each $\lambda\in\Lambda$ and $\epsilon>0$, there exists $B\in\mathcal{B}$ such that $H^\lambda(B,E)<\epsilon$.
\end{theorem}
\begin{proof}
Since $E$ is closed, $X$ is complete and $\alpha(E)=0$, we have $E$ is compact. Let $\mathcal{F}$ be the filter generated by
$\mathcal{B}$, i.e., $\mathcal{F}=\{A\subseteq X:\exists B\in\mathcal{B}, B\subseteq A\}$. In order to prove $E$ is nonempty, let us denote by $\mathcal{F}^*$ a
ultrafilter such that $\mathcal{F}\subseteq\mathcal{F}^*$. By assumption, for each $\epsilon>0$ there exists $B\in\mathcal{B}$ such that for each $\lambda\in\Lambda$, there exist $x_1,\dots,x_{r}\in X$ satisfying
$B\subseteq\mathrm{B}_\lambda(x_1,\epsilon/2)\cup\cdots\cup\mathrm{B}_\lambda(x_r,\epsilon/2)$.
Since $B\in\mathcal{F}^*$, one of these balls $\mathrm{B}_\lambda(x_1,\epsilon/2),\dots,\mathrm{B}_\lambda(x_r,\epsilon/2)$ belongs to $\mathcal{F}^*$ (see Corollary in Section \S 6.4, Chapter I in
\cite{Bo66}, for instance). Thus, $\mathcal{F}^*$ is a Cauchy filter and consequently it converges to some point $x^*\in X$. But,
$\{x^*\}=\bigcap_{F\in\mathcal{F}^*}F\subseteq \bigcap_{F\in\mathcal{F}}F= \bigcap_{B\in\mathcal{B}}B$ and therefore $E$ is non empty.
Next, we prove the convergence of $\mathcal{B}$ to $E$ in the $\mathcal{H}$-topology. Suppose there exist $\lambda\in\Lambda$ and $\epsilon>0$ such that for any $B\in\mathcal{B}$, $H^\lambda(B,E)>\epsilon$. Let $L(\epsilon,\lambda)=\{x\in X:d_\lambda(x,E)\geq\epsilon\}$. Since $H^\lambda(B,E)=\sup_{x\in
B}d_\lambda(x,E)$, there exists $x\in B$ such that $d_\lambda(x,E)>\epsilon$. Thus, for any $B\in\mathcal{B}$, $B\cap
L(\epsilon,\lambda)\ne\emptyset$. This fact implies, $\widetilde{\mathcal{B}}=\{B\cap L(\epsilon,\lambda):B\in\mathcal{B}\}$ is a filter base on the complete uniform space
$L(\epsilon,\lambda)$. Moreover $\inf\{\alpha(B):B\in\widetilde{\mathcal{B}}\}=0$. Hence, by applying to $\widetilde{\mathcal{B}}$ what we have shown at the first paragraph to $\mathcal{B}$, we have
$\emptyset\ne\bigcap_{B\in\widetilde{\mathcal{B}}}B=\bigcap_{B\in\mathcal{B}}B\cap L(\epsilon,\lambda)=E\cap L(\epsilon,\lambda)$, which is a contradiction. Therefore, the proof is complete.
\end{proof}
\begin{remark} Note that whether $\mathcal{B}$ is countable, in Theorem \ref{t5}, in order to
$\bigcap_{B\in\mathcal{B}}B$ is compact and nonempty, it suffices that $X$ is sequentially complete.
\end{remark}
Let $T:D\subseteq X\to 2^X$ be a multi-function. A point $x\in D$ is said to be a fixed point of $T$, whenever $x\in Tx$. We say that $T$ is
condensing, if $T$ is continuous and for each $A\subseteq D$ such that $A\in \mathcal{B}(X)$ and $\alpha(A)>0$, we have $T(A)\in \mathcal{B}(X)$ and $\alpha(T(A))<\alpha(A)$.
\begin{proposition}\label{p6}
Suppose $X$ is complete. Let $D$ be a closed subset of $X$, $T:D\subseteq X\to \mathcal{CB}(X)$ be a weakly upper semi-continuous
multi-function and for each $\lambda\in\Lambda$ and $\eta>0$, $B_{\lambda,\eta}=\{x\in D:d_\lambda(x,Tx)\leq\eta\}$. Suppose for each $\lambda\in\Lambda$ and $\eta>0$, $B_{\lambda,\eta}\neq\emptyset$ and $\inf\{\alpha(B_{\lambda,\eta}):\lambda\in\Lambda,\eta>0\}=0$. Then, $T$ has a fixed point.
\end{proposition}
\begin{proof}
Since $\{d_\lambda\}_{\lambda\in\Lambda}$ satisfies condition (S) and $T$ is a weakly upper semi-continuous multi-function,
$\mathcal{B}=\{B_{\lambda,\eta}:\lambda\in\Lambda,\eta>0\}$ is a filter base of closed and bounded subsets of the complete space $D$. Hence, Theorem \ref{t5} implies $\bigcap_{\lambda\in\Lambda,\eta>0}B_{\lambda,\eta}\neq\emptyset$ and therefore, due to $T$ has closed images, $T$ has a fixed point.
\end{proof}
Given a filter base $\mathcal{A}$ on a subset $D$ of $X$, we denote by $\mathcal{A}'$ the set of all its accumulation points, i.e.\
$x\in\mathcal{A}'$, if for any neighborhood $V$ of $x$ and $A\in\mathcal{A}$, $V\cap A\ne\emptyset$. An extension of an old result due to Furi Vignoli
\cite{FV69} can be stated as follows.
\begin{proposition}\label{p7}
Let $D$ be a complete subset of $X$, $T:D\subseteq X\to \mathcal{CB}(X)$ a condensing weakly upper semi-continuous multi-function and $\mathcal{A}\subseteq\mathcal{B}(X)$ a filter base on $D$. Suppose $\inf\{\sup_{x\in A,\lambda\in\Lambda}d_\lambda(x,Tx):A\in
\mathcal{A}\}=0$. Then, $\mathcal{A}'\neq\emptyset$ and each point in $\mathcal{A}'$ is a fixed point of $T$.
\end{proposition}
\begin{proof}
Let $\epsilon>0$. From assumption, there exists $A\in\mathcal{A}$ such that $\sup_{x\in A,\lambda\in\Lambda}d_\lambda(x,Tx)<\epsilon$. This fact
implies $A\subseteq \bigcap_{\lambda\in\Lambda}B_\lambda(T(A),\epsilon)$ and hence, property d) implies that $ \alpha(A)\leq\alpha(T(A))+\epsilon. $ Thus,
$\alpha(A)\leq\alpha(T(A))$ and since $T$ is condensing, we have $\alpha(A)=0$. The completeness of $D$ implies that $A$ is compact and consequently $\mathcal{A}'=\bigcap_{A\in\mathcal{A}}\overline{A}\neq\emptyset$. Note that from the assumption, for each $\eta>0$, there exists
$A_{\eta}\in\mathcal{A}$ such that $A_{\eta}\subseteq \bigcap_{\lambda}B_{\lambda,\eta}$, where $B_{\lambda,\eta}$ is defined as in Proposition \ref{p6}. Since $T$ is weakly upper semi-continuous, $\overline{A}_{\eta}\subseteq \bigcap_{\lambda}B_{\lambda,\eta}$ and therefore, $\mathcal{A}'\subseteq \bigcap_{\lambda\in\Lambda,\eta>0}B_{\lambda,\eta}$, concluding the proof due to $T$ has closed images.
\end{proof}
We state below an extension of a known result in metric spaces.
\begin{theorem}\label{t8}
Let $D\in \mathcal{CB}(X)$ and $T:D\to 2^D$ be a condensing multi-function. Then, there exists a compact subset $C$ of $D$ such that $T(C)\subseteq C$.
\end{theorem}
\begin{proof}
Let $x_0\in D$ and $\Sigma=\{K\in\mathcal{CB}(X):x_0\in K\subseteq D\mbox{ and } T(K)\subseteq K\}$. Due to $D\in\Sigma$, we have $\Sigma\neq\emptyset$. Let $B=\bigcap_{K\in\Sigma}K$ and $C=\overline{\{x_0\}\cup T(B)}$. We have $T(B)\subseteq\bigcap_{K\in\Sigma}T(K)\subseteq B$ and $x_0\in B$. Moreover, since $B$ is closed, $C\subseteq B$. Thus $T(C)\subseteq T(B)\subseteq C$, $C\in\Sigma$ and $B=C$. From properties b) and c), we obtain $\alpha(C)=\alpha(T(B))=\alpha(T(C))$ and due to $T$ is condensing, we have $\alpha(C)=0$ and the proof is complete.
\end{proof}
Let $\mathcal{F}=\{k_\lambda\}_{\lambda\in\Lambda}$ be a family of constants such that for each $\lambda\in\Lambda$, $0\leq
k_\lambda<1$, $D$ a subset of $X$ and $T:D\to \mathcal{CB}(X)$. We say $T$ is an $\mathcal{F}$-contractive multi-function, if for any $x,y\in D$ and $\lambda\in\Lambda$, $H^\lambda(Tx,Ty)\leq k_\lambda d_\lambda(x,y)$. Let $k$ be a constant such that $0<k<1$. We say $T$ is a $k$-set contraction, if $T$ is continuous and for each $A\subseteq D$ such that $A\in \mathcal{B}(X)$, we have $T(A)\in \mathcal{B}(X)$ and $\alpha(T(A))\leq k\alpha(A)$. Of course, every $k$-set contraction is condensing.
\begin{theorem}\label{t9}
Let $\mathcal{F}=\{k_\lambda\}_{\lambda\in\Lambda}$ be a family of nonnegative constants such that $k=\sup_{\lambda\in\Lambda}k_\lambda<1$ and $T:X\to \mathcal{K}(X)$ be an $\mathcal{F}$-contractive multi-function. Then, $T$ is a $k$-set contraction.
\end{theorem}
\begin{proof}
Let $A\in\mathcal{CB}(X)$ and $\epsilon=\alpha(A)$. For each $\lambda\in\Lambda$ and $\eta>0$, there exist $a_1,\dots,a_r\in X$ such that $A\subseteq A^\lambda_1\cup\cdots\cup A^\lambda_r$, where $A^\lambda_i=\mathrm{B}_\lambda(a_i,\eta+\epsilon/2)$, for $i\in\{1,\dots,r\}$. Hence, for each $x\in A^\lambda_i$, we have
$H^\lambda(Tx,Ta_i)\leq k_\lambda d_\lambda(x,a_i)<k_\lambda(\eta+\epsilon/2)< \eta+k\epsilon/2$. Thus, $Tx\subseteq \mathrm{B}_\lambda(Ta_i,\eta+k\epsilon/2)$ and $T(A^\lambda_i)\subseteq\mathrm{B}_\lambda(Ta_i,\eta+k\epsilon/2)$. Since $Ta_i$ is compact, there exist $b_{i1},\dots,b_{is}\in Ta_i$ such that $Ta_i\subseteq \mathrm{B}_\lambda(b_{i1},\eta)\cup\cdots\cup\mathrm{B}_\lambda(b_{is},\eta)$. Hence, $T(A^\lambda_i)\subseteq\mathrm{B}_\lambda(b_{i1},2\eta+k\epsilon/2)\cup\cdots\cup\mathrm{B}_\lambda(b_{is},2\eta+k\epsilon/2)$ and consequently, $\alpha(T(A^\lambda_i))\leq 2\eta +k\epsilon$. But $T(A)\subseteq T(A^\lambda_1)\cup\cdots\cup T(A^\lambda_r)$ along with property b) imply $\alpha(T(A))\leq 2\eta+k\epsilon$, for all $\eta>0$. Accordingly, $\alpha(T(A))\leq k\alpha(A)$ and the proof is complete.
\end{proof}
\begin{remark}
Let $B\in \mathcal{CB}(X)$ be a non compact set and $T:X\to\mathcal{CB}(X)$ be defined as $Tx=B$, for all $x\in X$. We have, for any family $\mathcal{F}$ of nonnegative constants, $T$ is an $\mathcal{F}$-contractive multi-function, which is not condensing. This example shows that it is not a strong condition, in Theorem \ref{t9}, to assume that $T$ has compact images.
\end{remark}
A well-known result by Nadler \cite{Na69} is generalized as follows.
\begin{theorem}\label{t15}
Suppose $X$ is sequentially complete and let $T:X\to \mathcal{CB}(X)$ be an $\mathcal{F}$-contractive multi-function with $\mathcal{F}=\{k_\lambda;\lambda\in\Lambda\}$ a family of nonnegative constants such that
$\sup_{\lambda\in\Lambda}k_\lambda<1$. Then, $T$ has a fixed point.
\end{theorem}
\begin{proof} Let
$\{x_n\}_{n\in\mathbb{N}}$ be a Cauchy sequence with respect to the metric $\rho$ defined, for $x,y\in X$, as
$\rho(x,y)=\sup_{\lambda\in\Lambda}d_\lambda(x,y)\wedge1$. Hence, $\{x_n\}_{n\in\mathbb{N}}$ is a Cauchy sequence with respect to the uniformity $\mathcal{U}= \{\mathrm{U}(\lambda,\epsilon);\lambda\in\Lambda, \epsilon>0\}$ and since $X$ is
sequentially complete, there exists $x\in X$ such that $\{x_n\}_{n\in\mathbb{N}}$ converges to $x$. Let
$\epsilon>0$ and $N\in\mathbb{N}$ such that $\rho(x_m,x_n)<\epsilon$ whether
$m,n\geq N$. From the lower semi-continuity of $\rho(x_m,\cdot)$, we
have $\rho(x_m,x)\leq\epsilon$, for each $m\geq N$. This proves that $\{x_n\}_{n\in\mathbb{N}}$ converges to $x$ according to $\rho$ and
consequently $(X,\rho)$ is a complete metric space.
Let $H^\rho$ be the Hausdorff metric on $\mathcal{CB}(X)$ defined from $\rho$. It is easy to see that for each $A,B\in\mathcal{CB}(X)$,
$H^\rho(A,B)=\sup_{\lambda\in\Lambda}H_\lambda(A,B)\wedge1$ and consequently, according to $H^\rho$, $T$ is a contraction. It follows from Theorem 5
by Nadler \cite{Na69} that there exists $x^*\in X$ such that $x^*\in T(x^*)$, which completes the proof.
\end{proof}
\section{Extended versions of Caristi's theorem}
In this section, the space $X$ is assumed complete with respect to $\mathcal{U}$, i.e.\ any Cauchy filter base in $X$ converges.
\begin{lemma}\label{le10}
Let $\{\varphi_\lambda\}_{\lambda\in\Lambda}$ be a family of lower semi-continuous and bounded below functions from $X$ to
$\mathbb{R}$ and $\preceq$ be a relation on $X$ defined as follows: $u\preceq v$, if and only if, for all $\lambda\in\Lambda$, $d_\lambda(u,v)\leq
\varphi_\lambda(u)-\varphi_\lambda(v)$. Then, $\preceq$ is a partial order relation on $X$ and for each $x_0\in X$, there exists a maximal element
$x^*\in X$ such that $x_0\preceq x^*$.
\end{lemma}
\begin{proof} It is easy to see that $\preceq$ is a partial order relation on $X$. For each $x\in
X$, let $I(x)=\{y\in X:x\preceq y\}$. Since for each $\lambda\in\Lambda$, $\varphi_\lambda$ is lower semi-continuous and
$I(x)=\bigcap_{\lambda\in\Lambda}\{y\in X: \varphi_\lambda(y)+d_\lambda(x,y)\leq \varphi_\lambda(x)\}$, we have $I(x)$ is closed. Fix $x_0\in X$, let $C$ be a
totally ordered subset of $I(x_0)$ and $\mathcal{B}=\{I(x)\cap C:x\in C\}$. Hence, $\mathcal{B}$ is a filter base in $I(x_0)$. Let $\epsilon>0$.
Because of for each $\lambda\in\Lambda$, $\varphi_\lambda$ is bounded below, there exists $L_\lambda=\inf\{\varphi_\lambda(x):x\in C\}$. Let us choose
$x_\lambda\in C$ such that $\varphi_\lambda(x_\lambda)<L_\lambda+\epsilon$ and notice $\varphi_\lambda$ is decreasing. Consequently, for
$x_{\lambda}\preceq u\preceq v$, we have $\varphi_\lambda(u)-\varphi_\lambda(v)<\epsilon$ and hence $d_\lambda(u,v)<\epsilon$. This fact proves
$\mathcal{B}$ is a Cauchy filter base in $I(x_0)$ and thus it converges to some $v\in I(x_0)$. Since for each $x\in C$, $I(x)$ is closed, we have
$\{v\}=\bigcap_{x\in C}\overline{I(x)\cap C}\subseteq \bigcap_{x\in C}I(x)$ and hence $v$ is an upper bound of $C$. Therefore, by Zorn's Lemma there
exists a maximal element $x^*\in I(x_0)$, concluding de proof.
\end{proof}
\begin{remark} Lemma of Mizoguchi in \cite{Mi90} could have been used to prove Lemma \ref{le10}. However, for the sake of completeness, we preferred to give an independent proof. On the other hand, when for each $\lambda\in\Lambda$, $\varphi_\lambda=\varphi$ does not depend on $\lambda$, the proof of the above lemma could be carried out by defining a suitable sequence instead of a filter base. In this case, this lemma requires only sequentially completeness instead of completeness.
\end{remark}
Theorem \ref{th11} below gives two extended versions of Caristi's theorem. In particular, Theorem 1 by Mizogushi in \cite{Mi90}, which is an improved version of Caristi's Theorem \cite{Ca76}, is generalized by means of (\ref{th11}.1) below.
\begin{theorem}\label{th11} Let $T:X\to \mathcal{C}(X)$ be a multivalued function and suppose for each $\lambda\in\Lambda$, $\varphi_\lambda:X\to\mathbb{R}$ is a lower
semi-continuous and bounded below function. Then, the following two propositions hold:
\begin{description}
\item[(\ref{th11}.1)] If for each $x\in X$, there exists $y\in Tx$ such that for each $\lambda\in\Lambda$, $d_\lambda(x,y)\leq \varphi_\lambda(x)-\varphi_\lambda(y)$, then, $T$ has a fixed point.
\item[(\ref{th11}.2)] If for each $x\in X$, each $y\in Tx$ and each $\lambda\in\Lambda$, $d_\lambda(x,y)\leq
\varphi_\lambda(x)-\varphi_\lambda(y)$, then, there exists $x^*\in X$ such that $\{x^*\}= Tx^*$.
\end{description}
\end{theorem}
\begin{proof}
From Lemma \ref{le10}, there exists a maximal element $x^*\in X$ and from assumption in (\ref{th11}.1), there exists $y\in Tx^*$ such that $x^*\preceq y$. Since $x^*$ is maximal, we have $x^*=y$ and hence $x^*\in Tx^*$. Therefore, (\ref{th11}.1) holds.
Let us prove (\ref{th11}.2). Let $x^*\in X$ the maximal element of $X$. Since $Tx^*$ is nonempty, assumption in (\ref{th11}.2) implies that there exists $y\in Tx^*$ such that $x^*\preceq y$. Thus $x^*=y$ and $\{x^*\}\subseteq Tx^*$. By applying assumption in (\ref{th11}.2) again and the maximality of $x^*$, we have $T(x^*)\subseteq \{x^*\}$ and the proof is complete.
\end{proof}
When the functions $\varphi_\lambda$, in Theorem \ref{th11}, does not depend on $\lambda\in\Lambda$, this theorem coincides with Theorem 2 by Hamel in \cite{Ha05}. Even, in this case, it is only required $X$ to be sequentially complete. However, as we think in the proof of Corollary \ref{c13} below, it is a great limitation to assume all the functions $\varphi_\lambda$ are equal.
The following corollary follows from Theorem \ref{th11} and is an extension of the well-known theorem by Caristi \cite{Ca67} for single valued functions.
\begin{corollary}\label{c12} Let $f:X\to X$ be an arbitrary function. Suppose for each $\lambda\in\Lambda$, $\varphi_\lambda:X\to\mathbb{R}$ is a lower
semi-continuous and bounded below function such that for each $x\in X$, $d_\lambda(x,f(x))\leq \varphi_\lambda(x)-\varphi_\lambda(f(x))$. Then, there
exists $x^*\in X$ such that $f(x^*)=x^*$.
\end{corollary}
Corollaries \ref{c13} and \ref{c14} below are extensions of the classical Banach contraction principle.
\begin{corollary}\label{c13}
Let $\mathcal{F}=\{k_\lambda\}_{\lambda\in\Lambda}$ be a family of constants such that for each $\lambda\in\Lambda$, $0\leq k_\lambda<1$ and $T:X\to \mathcal{K}(X)$ be an $\mathcal{F}$-contractive multi-function. Then, $T$ has a fixed point.
\end{corollary}
\begin{proof}
For each $x\in X$, $\lambda\in\Lambda$ and $\eta>0$, let $C_{\lambda,\eta}(x)= \{y\in Tx:d_\lambda(x,y)\leq d_\lambda(x,Tx)+\eta\}$. Since $\{d_\lambda\}_{\lambda\in\Lambda}$ satisfies condition (S), for each $x\in C$, $\lambda,\mu\in\Lambda$ and $\eta_1,\eta_2>0$, there exists $\nu\in\Lambda$ such that $C_{\nu,\eta}(x)\subseteq C_{\lambda,\eta_1}(x)\cap C_{\mu,\eta_2}(x)$, where $\eta=\min\{d_\lambda(x,Tx)+\eta_1,d_\mu(x,Tx)+\eta_2\}$. Thus, $\{C_{\lambda,\eta}(x)\}_{\lambda\in\Lambda}$ is a filter base of compact subsets of $Tx$ and hence $\bigcap_{\lambda\in\Lambda,\eta>0}C_{\lambda,\eta}(x)$ is nonempty. Consequently, for each $x\in X$, there exists $f(x)\in Tx$ such that $d_\lambda(x,Tx)=d_\lambda(x,f(x))$, for all $\lambda\in\Lambda$. We have
$$
d_\lambda(f(x),Tf(x))\leq H^\lambda(Tx,Tf(x))\leq k_\lambda d_\lambda(x,f(x))
$$
and hence, by defining $\varphi_\lambda:X\to\mathbb{R}$ as $\varphi_\lambda(x)=d_\lambda(x,Tx)/(1-k_\lambda)$, we obtain
$$
d_\lambda(x,f(x))\leq \varphi_\lambda(x)-\varphi_\lambda(f(x)).
$$
Moreover $T$ is $\mathcal{H}$-continuous. Hence, Theorems \ref{t2} and \ref{t3} imply that $\varphi_\lambda$ is lower semi-continuous and from Corollary \ref{c12}, there exists $x^*\in X$ such that $f(x^*)=x^*$. Therefore, $x^*\in Tx^*$ and the proof is complete.
\end{proof}
\begin{remark}
Note that the functions $\varphi_\lambda$ in Corollary \ref{c13} depend on $\lambda$ not only through $\nu_\lambda=1-k_\lambda$, but also due to $ d_\lambda $. Hence, this corollary cannot be obtained from Theorem 1 by Mizoguchi in \cite{Mi90}.
\end{remark}
Corollary \ref{c13} enables us to obtain the following old result by Tarafdar in \cite{Ta74}.
\begin{corollary}\label{c14} Let $\mathcal{F}=\{k_\lambda\}_{\lambda\in\Lambda}$ be a family of constants such that for each $\lambda\in\Lambda$, $0\leq
k_\lambda<1$ and $f:X\to X$ be an $\mathcal{F}$-contractive function, i.e.\ for any $x,y\in X$, $d_\lambda(f(x),f(y))\leq k_\lambda d_\lambda(x,y)$. Then, there
exists a unique $x^*\in X$ such that $f(x^*)=x^*$.
\end{corollary}
\begin{proof} From Corollary \ref{c13}, there exists $x^*\in X$ such that $f(x^*)=x^*$.
Suppose $x^{**}$ is another fixed point of $f$. Hence
$
d_\lambda(x^*,x^{**})=d_\lambda(f(x^*),f(x^{**}))\leq k_\lambda
d_\lambda(x^*,x^{**})
$
and hence uniqueness follows due to for all $\lambda\in\Lambda$, $d_\lambda(x^*,x^{**})=0$. Therefore, the proof is complete.
\end{proof}
Next we state an extended version of the Bishop-Phelps theorem as follows.
\begin{theorem}\label{t16} For each $\lambda\in\Lambda$, let $\varphi_\lambda:X\to\mathbb{R}$ be a lower semi-continuous and bounded below function. Then, for
each $x_0\in X$, there exists $x^*\in X$ such that the following two conditions hold:
\begin{description}
\item[(\ref{t16}.1)] for any $\lambda\in\Lambda$, $\varphi_\lambda(x^*)+d_\lambda(x_0,x^*)\leq\varphi_\lambda(x_0)$; and
\item[(\ref{t16}.2)] for each $x\in X$ with $x\neq x^*$, there exists $\mu\in \Lambda$ satisfying $\varphi_\mu(x^*)<\varphi_\mu(x)+d_\mu(x,x^*)$.
\end{description}
\end{theorem}
\begin{proof}
Let $x_0\in X$. From Lemma \ref{le10}, there exists a maximal element $x^*\in X$ such that (\ref{t16}.1) is satisfied. The maximality of $x^*$ implies condition (\ref{t16}.2), therefore the proof is complete.
\end{proof}
Also, a generalization of Ekeland's variational principle is given below. A function $\varphi:X\to\mathbb{R}\cup\{\infty\}$ is said to be proper, whenever
there exists $x_0\in X$ such that $\varphi(x_0)<\infty$.
\begin{theorem}\label{t17} For each $\lambda\in\Lambda$, let $\varphi_\lambda:X\to\mathbb{R}\cup\{\infty\}$ be a proper, lower semi-continuous and bounded
below function. Then, for each $x_0\in X$ and every family $\{\delta_\lambda\}_{\lambda\in\Lambda}$ of positive real numbers such that
$\varphi_\lambda(x_0)\leq\inf\varphi_\lambda(X)+\delta_\lambda$, there exists $x^*\in X$ satisfying the following three conditions:
\begin{description}
\item[(\ref{t17}.1)] for any $\lambda\in\Lambda$, $\varphi_\lambda(x^*)\leq\varphi_\lambda(x_0)$;
\item[(\ref{t17}.2)] for any $\lambda\in\Lambda$, $d_\lambda(x_0,x^*)\leq\delta_\lambda$; and
\item[(\ref{t17}.3)] for each $x\in X$ with $x\neq x^*$, there exists $\mu\in \Lambda$ satisfying $\varphi_\mu(x^*)<\varphi_\mu(x)+d_\mu(x,x^*)$.
\end{description}
\end{theorem}
\begin{proof}
Let $x_0\in X$ and $\{\delta_\lambda\}_{\lambda\in\Lambda}$ be a family of positive real numbers such that
$\varphi_\lambda(x_0)\leq\inf\varphi_\lambda(X)+\delta_\lambda$. Since each $\varphi_\lambda$ is proper, for each $\lambda\in\Lambda$,
$\inf\varphi_\lambda(X)<\infty$ and thus, $\varphi_\lambda(x_0)<\infty$. Let $Y=\bigcap_{\lambda\in\Lambda}\{x\in
X:\varphi_\lambda(x)\leq\varphi_\lambda(x_0)\}$ and $\widetilde{\varphi}_\lambda:Y\to\mathbb{R}$ the restriction of $\varphi_\lambda$ to $Y$. Notice $Y$ is
closed in $X$ and consequently $Y$ is complete. Hence, we can apply Theorem \ref{t16} to the family
$\{\widetilde{\varphi}_\lambda\}_{\lambda\in\Lambda}$ and, since $x_0\in Y$, there exists $x^*\in Y$ such that for any $\lambda\in\Lambda$,
\begin{equation}\label{e1}
\widetilde{\varphi}_\lambda(x^*)+d_\lambda(x_0,x^*)\leq\widetilde{\varphi}_\lambda(x_0).
\end{equation}
Moreover, for each $x\in Y$ with $x\neq x^*$, there exists $\mu\in\Lambda$ such that
\begin{equation}\label{e2}
\widetilde{\varphi}_\mu(x^*)<\widetilde{\varphi}_\mu(x)+d_\mu(x,x^*).
\end{equation}
Since $x^*\in Y$, condition (\ref{t17}.1) holds. From (\ref{e1}) and the
assumption, for each $\lambda\in\Lambda$, we have $$ \widetilde{\varphi}_\lambda(x^*)+d_\lambda(x_0,x^*)\leq\widetilde{\varphi}_\lambda(x_0)\leq
\inf\varphi_\lambda(X)+\delta_\lambda \leq \varphi_\lambda(x^*)+\delta_\lambda, $$ which implies condition (\ref{t17}.2). Let us prove condition (\ref{t17}.3). From
(\ref{e2}) this condition holds for all $x\in Y$. Let $x\in X\setminus Y$. Hence, there exists $\mu\in\Lambda$ such that
$\varphi_\mu(x_0)<\varphi_\mu(x)\leq \varphi_\mu(x)+d_\mu(x,x^*)$ and since $\varphi_\mu(x^*)\leq\varphi_\mu(x_0)$, we have $
\varphi_\mu(x^*)<\varphi_\mu(x)+d_\mu(x,x^*)$. Therefore, for each $x\in X$, $\varphi_\mu(x^*)<\varphi_\mu(x)+d_\mu(x,x^*)$, which concludes the
proof.
\end{proof}
The following corollary is useful to obtain some fixed point results under some inwardness conditions.
\begin{corollary}\label{c18} Let $K$ be a closed subset of $X$, $f:K\to K$ an arbitrary function and $T:K\to\mathcal{C}(X)$ a multi-function. Suppose the following two conditions hold:
\begin{description}
\item[(\ref{c18}.1)] $T$ is weakly upper semi-continuous, and
\item[(\ref{c18}.2)] for each $\lambda\in\Lambda$ and $x\in X$, $d_\lambda(f(x),Tf(x))\leq d_\lambda(x,Tx) +r_\lambda d_\lambda(x,f(x))$, whenever $r_\lambda<0$.
\end{description}
Then, there exists $x^*\in X$ such that $f(x^*)=x^*$.
\end{corollary}
\begin{proof}
Since $K$ is a closed subset of $X$ and $X$ is complete, $K$ is
complete. For each $\lambda\in\Lambda$, let $\varphi_\lambda:K\to\mathbb{R}$ such that $\varphi_\lambda(x)=-d_\lambda(x,Tx)/r_\lambda$. From
(\ref{c18}.1), $\varphi_\lambda$ is lower semi-continuous and bounded below, and from (\ref{c18}.2) we have, $ d_\lambda(x,f(x))\leq
\varphi_\lambda(x)-\varphi_\lambda(f(x))$. Therefore, from Corollary \ref{c12}, there exists $x^*\in X$ such that $f(x^*)=x^*$ and the proof is
complete.
\end{proof}
\section{Applications to locally convex spaces}
In this section, $X$ denotes a real or complex locally convex space endowed with a family $\{\n{\cdot}_\lambda\}_{\lambda\in\Lambda}$ of semi norms defining the topology of $X$ and for each $Y\subseteq X$, $\mathcal{Q}(Y)$ stands for the family of all nonempty, compact, convex subsets of $X$. Here, for each $\lambda\in\Lambda$, $d_\lambda$ is defined as $d_\lambda(x,y)=\n{x-y}_\lambda$, for $(x,y)\in X\times X$.
\begin{lemma}\label{l20}
Let $A$ and $B$ be two subsets of $X$ and $\beta$ be a scalar. Then
\begin{itemize}
\item [(i)] $\alpha(A+B)\leq\alpha(A)+\alpha(B)$.
\item [(ii)] $\alpha(\beta A)=|\beta|\alpha(A)$.
\item [(iii)] $\alpha(\mathrm{co}(A))=\alpha(A)$.
\end{itemize}
\end{lemma}
\begin{proof}
Conditions (i) and (ii) are easy to verify and we only prove condition (iii). We need to prove that $\alpha(co(A))\leq\alpha(A)$. Let $\epsilon=\alpha(A)$, $\lambda\in\Lambda$, $\eta>0$ and $x_1,\dots,x_r\in X$ such that $A\subseteq \mathrm{B}_\lambda(x_1,\epsilon+\eta)\cup\cdots\cup\mathrm{B}_\lambda(x_r,\epsilon+\eta)$. We have
$$
\mathrm{co}(\mathrm{B}_\lambda(x_1,\epsilon+\eta)\cup\mathrm{B}_\lambda(x_2,\epsilon+\eta))=\{tx_1+(1-t)x_2:0\leq t\leq1\}+\mathrm{B}_\lambda(0,\epsilon+\eta)
$$
and there exists $0=t_0<t_1\cdots<t_s=1$ a partition of $[0,1]$ such that $(t_i-t_{i-1})\n{x_1-x_2}_\lambda<\eta$. Hence
\begin{equation}\label{e3}
\mathrm{co}(\mathrm{B}_\lambda(x_1,\epsilon+\eta)\cup\mathrm{B}_\lambda(x_2,\epsilon+\eta))\subseteq \bigcup_{i=1}^s\{t_ix_1+(1-t_i)x_2:0\leq t\leq1\}+\mathrm{B}_\lambda(0,\epsilon+2\eta)
\end{equation}
and since
$$
\mathrm{co}(\mathrm{B}_\lambda(x_1,\epsilon+\eta)\cup\cdots\cup\mathrm{B}_\lambda(x_r,\epsilon+\eta))\subseteq
\mathrm{co}\{\mathrm{B}_\lambda(x_1,\epsilon+\eta)\cup\mathrm{co}(\mathrm{B}_\lambda(x_2,\epsilon+\eta)\cup\cdots\cup\mathrm{B}_\lambda(x_r,\epsilon+\eta))\},
$$
it follows from induction and \eqref{e3} that there exist a finite subset $F$ of $X$ such that
$$
\mathrm{co}(\mathrm{B}_\lambda(x_1,\epsilon+\eta)\cup\cdots\cup\mathrm{B}_\lambda(x_r,\epsilon+\eta))\subseteq F+\mathrm{B}_\lambda(0,\epsilon+2\eta).
$$
Consequently, $\alpha(\mathrm{co}(A))\leq\alpha(F+\mathrm{B}_\lambda(0,\epsilon+2\eta))\leq \epsilon+2\eta$, which completes the proof due to $\eta>0$ is arbitrary.
\end{proof}
\begin{lemma}\label{l19}
Let $C\in\mathcal{Q}(X)$ and $T:C\to\mathcal{Q}(C)$ be a continuous multi-function. Then, $T$ has a fixed point.
\end{lemma}
\begin{proof}
It directly follows from Theorem 2 by Fan in \cite{Fa61}.
\end{proof}
The following result extends, to locally convex spaces and for continuous multi-functions, a known theorem by Darbo in \cite{Da55}.
\begin{theorem}
Let $D$ be a bounded, closed and convex subset of $X$ and $T:D\to \mathcal{Q}(D)$ be a condensing multi-function. Then, $T$ has a fixed point.
\end{theorem}
\begin{proof}
Let $x_0\in X$ and $\Sigma=\{K\in\mathcal{CB}(X):x_0\in K\subset D, K\mbox{ convex and } T(K)\subseteq K\}$. Due to $D\in \Sigma$, we have $\Sigma\neq\emptyset$. Let $B=\bigcap_{K\in\Sigma}K$ and $C=\overline{\mathrm{co}}(\{x_0\}\cup T(B))$. We have $T(B)\subseteq\bigcap_{K\in\Sigma}T(K)\subseteq B$ and $x_0\in B$. Moreover, since $B$ is closed and convex, $C\subseteq B$. Thus $T(C)\subseteq T(B)\subseteq C$, $C\in\Sigma$ and $B=C$. From property c) and (iii) in Lemma \ref{l20}, we obtain $\alpha(C)=\alpha(T(B))=\alpha(T(C))$ and since $T$ is condensing, we have $\alpha(C)=0$. Since $C\in\mathcal{Q}(X)$ and $T(C)\in\mathcal{Q}(C)$, it follows from Lemma \ref{l19} that $T$ has a fixed point, which completes the proof.
\end{proof}
For each $K$ subset of $X$ and $x\in K$, the inner set of $K$ at $x$ is defined as
$$
I_K(x)=x+\{c(y-x):y\in K,c\geq1\}.
$$
Also we define the envelope of $I_K(x)$ as
$$
\widetilde{I}_K(x)=\bigcup_{z\in I_K(x)}\bigcap_{\lambda\in\Lambda,\eta>0}\{t\in X:\n{t-z}_\lambda\leq\eta\n{t-x}_\lambda\}.
$$
Let $T:K\to \mathcal{C}(X)$ be a function. We say $T$ is inward (respectively, weakly inward), if for each $x\in K$, $T(x)\in I_K(x)$ (respectively, $T(x)\in \widetilde{I}_K(x)$).
Theorem \ref{tt} below is an extension to locally convex spaces of an result by Mart\'inez-Y\'a\~{n}ez in \cite{Ma91}. See also \cite{Re72}.
\begin{theorem}\label{tt}
Let $\mathcal{F}=\{k_\lambda;\lambda\in\Lambda\}$ be a family of nonnegative constants and $T:K\subseteq X\to X$ be a weakly inward $\mathcal{F}$-contractive function. Then, $T$ has a fixed point.
\end{theorem}
\begin{proof}
For each $\lambda\in\Lambda$, choose $\epsilon_\lambda>0$ such that $k_\lambda<(1-\epsilon_\lambda)/(1+\epsilon_\lambda)$. Since $T$ is weakly inward, there exists $f:K\to K$ and $c:K\to\mathbb{R}$ such that, for any $\lambda\in\Lambda$ and $x\in X$,
\begin{equation}\label{e4}
\n{T(x)-x-c(x)(f(x)-x)}_\lambda\leq\epsilon_\lambda \n{T(x)-x}_\lambda
\end{equation}
with $c(x)\geq1$. Let $h(x)=1/c(x)$, $w(x)=(1-h(x))x+h(x)T(x)$, fix $\lambda\in\Lambda$ and observe that
$$
\n{w(x)-x}_\lambda=h(x)\n{T(x)-x}_\lambda\quad\mbox{and}\quad\n{w(x)-f(x)}_\lambda\leq\epsilon_\lambda h(x)\n{T(x)-x}_\lambda.
$$
Hence
$
\n{x-f(x)}_\lambda\leq(1+\epsilon_\lambda)\n{w(x)-x}_\lambda
$
and accordingly
$$
\begin{array}{ccl}
\n{f(x)-T(f(x))}_\lambda & \leq & \n{f(x)-w(x)}_\lambda+\n{w(x)-T(x)}_\lambda+\n{Tx-T(f(x))}_\lambda \\
& \leq & \epsilon_\lambda h(x)\n{T(x)-x}_\lambda+(1-h(x))\n{T(x)-x}_\lambda +k_\lambda\n{x-f(x)} \\
& = & \n{T(x)-x}_\lambda +k_\lambda\n{x-f(x)}-(1-\epsilon_\lambda)\n{w(x)-x}_\lambda.\\
\end{array}
$$
Consequently,
\begin{equation}\label{e5}
\n{f(x)-T(f(x))}_\lambda \leq \n{T(x)-x}_\lambda+\left(k_\lambda -\frac{1-\epsilon_\lambda}{1+\epsilon_\lambda}\right)\n{x-f(x)}_\lambda,
\end{equation}
and thus, Corollary \ref{c18} implies that there exists $x^*\in X$ such that $f(x^*)=x^*$. Since $\epsilon_\lambda\leq1$, from \eqref{e4} we have, for any $\lambda\in\Lambda$, $\n{T(x^*)-x^*}_\lambda=0$. Therefore, $x^*$ is a fixed point of $T$ and the proof is complete.
\end{proof}
\paragraph{Acknowledgments} This work was partially supported by FONDECYT grant 1120879 from the Chilean government.
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\section{Introduction}
We have reported results of $B_K$ calculated using improved staggered
fermions with $N_f=2+1$ flavors in Ref.~\cite{wlee-10-3}.
In Ref.~\cite{wlee-10-3}, we use the diagonal approximation
(uncorrelated fitting) instead of the full covariance fitting.
This is due to the fact that the $\chi^2$ value was out of range,
which indicates that the full covariance fitting fails manifestly.
One of the most frequently asked questions on Ref.~\cite{wlee-10-3} is
why we do the uncorrelated fitting instead of the full covariance fitting.
Here, we provide an elaborate answer to why we use the diagonal
approximation.
In addition, we propose a new method, named the eigenmode shift (ES)
method, which fine-tunes the fitting function while keeping the
covariance matrix untouched.
More details on this issue will be reported in
Ref.~\cite{wlee-2011-1}.
\section{Covariance fitting}
First, we review the covariance fitting. Then, we would like to address
the possible failure of the covariance fitting, which originates from
the truncation error of the fitting function in the series expansion
of the staggered chiral perturbation theory (SChPT).
Let us consider $N$ samples of unbiased estimates of quantity
$y_i$ with $i=1,2,3,\ldots,D$.
Here, the data set is $\{y_i(n) | n=1,2,3,\ldots,N \}$.
Let us assume that the samples $y_i(n)$ are statistically independent
in $n$ for fixed $i$ but are substantially correlated in $i$.
An introduction to this subject is given in
Ref.~\cite{milc-1988-1,toussaint-1990-1,anderson-2003,johnson-2007}.
We are interested in the probability distribution of the average
$\bar{y}_i$ of the data $y_i(n)$, defined by
$\bar{y}_i = \frac{1}{N} \sum_{n=1}^{N} y_i(n)$.
We assume that the measured values of $\bar{y}_i$ have a normal
distribution $P(\bar{y})$ by the central limit theorem for the
multivariate statistical analysis as follows:
\begin{equation}
\label{eq:pdf_normal}
P(\bar{y}) = \frac{1}{Z} \exp\left[ -\frac{1}{2} \sum_{i,j=1}^{D}
(\bar{y}_i - \mu_i) (N \ \Gamma^{-1}_{ij}) (\bar{y}_j - \mu_j)
\right] \,,
\end{equation}
where $\mu_i$ represents the true mean value of $y_i$, which is, in
general, unknown and can be obtained as $N \rightarrow \infty$,
and $Z$ is the normalization constant.
Here, $\Gamma_{ij}$ is the true covariance matrix, which is, in
general, unknown in our problems.
The maximum likelihood estimator of $\dfrac{1}{N}\Gamma_{ij}$ turns out
to be the sample covariance matrix of mean, $C_{ij}$, defined as
follows,
\begin{eqnarray}
C_{ij} &=& \frac{1}{N(N-1)} \sum_{n=1}^{N} [y_i(n) - \bar{y}_i]
[y_j(n) - \bar{y}_j]
\,.
\label{eq:cov_mat}
\end{eqnarray}
Let us consider a fitting function, $f_\text{th}(X_i; c_a)$. Here,
$X_i$ are the input variables which define data points and $c_a$ are
fitting parameters. What we want to do is to determine the fitting
parameters to give the best fit and to test whether the fitting
function describes the data reliably from the standpoint of
statistics.
Here, the best fit is defined by minimizing the $T^2$, where $T^2$ is
\begin{eqnarray}
\label{eq:def_Tsq}
T^2 = \sum_{i,j=1}^{D}
[\bar{y}_i - f_\text{th}(X_i)] [C^{-1}_{ij}] [\bar{y}_j - f_\text{th} (X_j)]
\,.
\end{eqnarray}
In ideal case, the best fit gives the true mean of the
data, $\mu_i$, in Eq.~\eqref{eq:pdf_normal}.
We notice that $\sqrt{N} [\bar{y}_i - f_\text{th} (X_i)]$ is distributed
according to the multivariate normal distribution,
$\mathcal{N}(\rho, \Gamma)$, where
$\rho_i = \sqrt{N} [\mu_i - f_\text{th} (X_i)]$.
In this case, $[T^2/(N-1)] [(N-d)/d]$ is distributed as the noncentral
$F$ distribution of $F_{d, N-d}$, which is defined in
Ref.~\cite{anderson-2003}, with noncentrality parameter $\kappa$, defined by
$\kappa = \sum_{i,j} \rho_i \Gamma_{ij}^{-1} \rho_{j} $.
Here, $d$ is the degrees of freedom of the fitting.
In Ref.~\cite{anderson-2003}, it is proved that the limiting
distribution of $T^2$ as $N \rightarrow \infty$ is the
$\chi^2$-distribution with $d$ degrees of freedom if
$f_\text{th} (X_i) = \mu_i$.
The multivariate statistical theory predicts the following
\cite{schervish-1995}:
\begin{eqnarray}
{\cal E} (T^2) &=& (d + \kappa) \left[ 1 + \frac{d+1}{N} +
{\cal O}(\frac{1}{N^2}) \right]
\label{eq:exp_tsq}\\
{\cal V} (T^2) &=& 2 (d + 2\kappa)
\biggl[ 1 + \frac{1}{N}
\Big(2 d + 4 + \frac{(d+\kappa)^2}{d + 2 \kappa} \Big)
+ {\cal O}(\frac{1}{N^2} ) \biggl]
\label{eq:var_tsq}
\,,
\end{eqnarray}
where ${\cal E} (T^2)$ and ${\cal V} (T^2)$ represent the expectation
value and variance of the $T^2$, defined in Eq.~\eqref{eq:def_Tsq}.
Here, $d$ is the degrees of freedom and $\kappa$ is the noncentrality
parameter.
If the fitting function is exact (which means $f_\text{th}(X_i) = \mu_i$),
the noncentrality parameter is zero. In that case, if we have large
enough number of data samples to ignore the $\mathcal{O}(1/N)$ terms,
we expect that the $T^2$ has a value around the degrees of freedom,
$T^2 = d \pm \sqrt{2d}$.
\subsection{Inexact fitting function}
One caveat is that the covariance fitting works only if the fitting
function is precise enough.
In practice, we determine the fitting function based on the SChPT
and it is given as a series of ${\cal O}(p^{2n})$.
Since we can include only the finite number of terms in the series,
we usually truncate the series at a certain higher order.
As a consequence, the fitting function has a truncation error which
makes it inexact in some high precision.
This usually does not cause much trouble.
However, if the covariance matrix has a very small eigenvalue,
$\lambda_l$, the truncation error can be amplified by a factor of
$\dfrac{1}{\lambda_l}$, and then, sometimes, causes failure of the
covariance fitting.
To see this, let us rewrite the Eq.~\eqref{eq:def_Tsq} using the
eigenmode decomposition:
\begin{equation}
\label{eq:eig_mod_decomp}
[C_{ij}^{-1}] = \sum_{k=1}^D \frac{1}{\lambda_k}
| v_k \rangle \langle v_k |,
\qquad
T^2 = \sum_{k=1}^D \frac{1}{\lambda_k}
\langle \bar{y} - f_\text{th} | v_k \rangle^2
\,,
\end{equation}
where $\lambda_k$ and $| v_k \rangle$ are eigenvalues and eigenvectors
of the covariance matrix $C_{ij}$, respectively.
Here, the average data points and
the fitting function values are also written in bra-ket vector notation,
$| \bar{y} \rangle_i \equiv \bar{y}_i$ and
$| f_\text{th} \rangle_i \equiv f_\text{th}(X_i)$.
If an eigenvalue $\lambda_l$ is very small, the $T^2$ is dominated by
the corresponding eigenmode.
The fitting procedure works very hard to minimize the difference
between the average data points and the fitting function value,
$(\bar{y} - f_\text{th})$, in $|v_l\rangle$ direction.
If the fitting function has error in $|v_l\rangle$ direction, the fitting
procedure endeavor to fit in wrong direction, losing precisions
in other directions.
Even if the error of fitting function is small, the lost precisions
in other directions can yield significant error of fitting result.
Section \ref{subsec:bk_cov_fit} exemplifies this situation.
If we have large number of samples, Eq.~\eqref{eq:exp_tsq} and
Eq.~\eqref{eq:var_tsq} can be approximated by
\begin{equation}
T^2 = d+\kappa \pm \sqrt{2(d+2\kappa)}
\,,
\end{equation}
where $d$ is the degrees of freedom of the fitting and $\kappa$ is
the noncentrality parameter.
Using the eigenmode decomposition, the $\kappa$ can be written as
\begin{equation}
\kappa = \sum_{k=1}^D \frac{1}{\lambda_k}
\langle \mu - f_\text{th} | v_k \rangle^2
\,,
\end{equation}
where $\mu_i$ are the true mean of $\bar{y}_i$.\footnote{Here, we
assume that we have large enough number of data samples so that
the $\lambda_k$ and $|v_k\rangle$ of sample covariance matrix
$C_{ij}$ are fairly representing those of the true covariance matrix.}
Therefore, the error of fitting function, $(\mu - f_\text{th})$,
increases the minimized value of $T^2$. Even if the error is small,
tiny eigenvalues amplify the $\kappa$.
\subsection{Trouble with covariance fitting for $B_K$}
\label{subsec:bk_cov_fit}
To demonstrate the problem, we choose the $B_K$ data on the C3
(coarse) ensemble of Ref.~\cite{wlee-10-3}.
This ensemble is particularly a good sample, because it has relatively
large statistics.
It contains 671 configurations and we measured 9 times for each
configuration.
Details are given in Ref.~\cite{wlee-10-3}.
The fitting functional form suggested by the SU(2) staggered chiral
perturbation theory (SChPT) is linear as follows:
\begin{equation}
f_\text{th} (X) = \sum_{a=1}^{P} c_a F_a(X)
\,,
\label{eq:fit-func-1}
\end{equation}
where $c_a$ are the low energy constants (LECs) and $F_a$ are
functions of $X$, which represents collectively $X_P$ (pion squared
mass of light valence (anti-)quarks), $Y_P$ (pion squared mass of
strange valence (anti-)quarks), and so on.
The details on $F_a$ and $X$ are given in Ref.~\cite{wlee-10-3}.
Here, we focus on the X-fit of 4X3Y-NNLO fitting of the SU(2) SChPT,
which is explained in great detail in Ref.~\cite{wlee-10-3}.
Since we have only 4 data points, we truncated higher order terms
in the fitting function and we have three LECs so $P=3$.
The neglected highest order term in the $f_\text{th} (X)$ is
$X^2 (\ln(X))^2 \approx 0.006$,
where $X = X_P/\Lambda^2 \approx 0.02$.
Hence, the fitting function has an error in that order.
In the X-fit, we fix $am_y = 0.05$ and select 4 data points
of $am_x =$ 0.005, 0.010, 0.015, 0.020 to fit to the functional
form suggested by the SU(2) SChPT as in Ref.~\cite{wlee-10-3}.
Hence, the covariance matrix $C_{ij}$ is a $4 \times 4$ matrix.
Its eigenvalues are
\begin{equation}
\lambda_i
= \{\ 1.95\times 10^{-5},\ 1.92\times 10^{-6},
\ 7.58\times 10^{-8},\ 1.11\times 10^{-9}
\}
\,.
\end{equation}
Due to the high correlation of data, the smallest eigenvalue is
smaller than the largest eigenvalue by four orders of magnitude.
Let us look into the eigenvectors,
\begin{equation}
|v_1\rangle = \left[ \begin{array}{r}
0.837 \\
0.429 \\
0.276 \\
0.200
\end{array} \right] \quad
|v_2\rangle = \left[ \begin{array}{r}
-0.508 \\
0.387 \\
0.542 \\
0.546
\end{array} \right] \quad
|v_3\rangle = \left[ \begin{array}{r}
0.202 \\
-0.739 \\
0.0725 \\
0.639
\end{array} \right] \quad
|v_4\rangle = \left[ \begin{array}{r}
-0.0378 \\
0.347 \\
-0.790 \\
0.503
\end{array} \right]
\,.
\end{equation}
The eigenvector $|v_4\rangle$ corresponds to the smallest eigenvalue
and it dominates the fitting completely.
\begin{figure}[t!]
\centering
\subfigure[Full covariance fit]{\includegraphics[width=0.45\textwidth]
{bk_sxpt_su2_4X3Y_NNLO_Xfit_cov_MILC_2064f21b676m010m050}
\label{subfig:full_cov_fit}}
\subfigure[Diagonal approximation]{\includegraphics[width=0.45\textwidth]
{bk_sxpt_su2_4X3Y_NNLO_Xfit_diag_MILC_2064f21b676m010m050}
\label{subfig:diag_fit}}
\caption{
$B_K(1/a)$ vs. $X_P$ on the C3 ensemble. The fit type is
4X3Y-NNLO in the SU(2) analysis. The red
line of left figure represents the results of fitting with the
full covariance matrix. The red line of right figure represents
the results of fitting with the uncorrelated fitting using
diagonal approximation. The red diamond corresponds to the $B_K$ value
obtained by extrapolating $m_x$ to the physical light valence
quark mass after setting all the pion multiplet splittings to
zero.}
\label{fig:full_and_diag}
\end{figure}
In Fig.~\ref{subfig:full_cov_fit}, we show the fitting results with the
full covariance matrix.
As one can see, the fitting curve does not pass through the data
points.
The $T^2$ value is $7.2$ with degrees of freedom $1$, which indicates
that the fitting fails manifestly.
Let us perform the eigenmode decomposition on $|\bar{y}\rangle$ and
$|f_\text{th}\rangle$ as follows:
\begin{equation}
|\bar{y}\rangle = \sum_{i=1}^4 a_i |v_i\rangle
\,, \qquad
|f_\text{th}\rangle = \sum_{i=1}^4 b_i |v_i\rangle
\end{equation}
where $a_i$ and $b_i$ are the eigenmode projection coefficients.
As we can see in Table \ref{tab:y-f-eigen-cov}, the difference is
$1.75 \sigma$ for $|v_1\rangle$, and $1.7\sigma$ for $|v_2\rangle$,
whereas it is only $0.33\sigma$ for $|v_4\rangle$.
Hence, the procedure of the covariance fitting works hard for the
coefficient of $|v_4\rangle$ but works less precisely for the
coefficients of $|v_1\rangle$ and $|v_2\rangle$, mainly because the
eigenvalue $\lambda_4$ is significantly smaller than $\lambda_1$ and
$\lambda_2$.
The irony is that the average data points, $|\bar{y}\rangle$, has only
$0.015\%$ overlap with $|v_4\rangle$ while more than $99\%$ of them
are dominated by $|v_1\rangle$ and $|v_2\rangle$.
As a result, the fitting function misses the average data points.
In this sense, the failure of the full covariance fitting is obviously
due to the fact that the covariance fitting tries to determine the
coefficient of $|v_4\rangle$ very precisely, while losing precisions
in $|v_1\rangle$ and $|v_2\rangle$ direction.
If the fitting function is exact, this procedure should yield a fitting
result reasonably describing the data.
However, if the fitting function has error in $|v_4\rangle$ direction,
this failing situation can happen.
\begin{table}[htbp]
\caption{ Eigenmode decomposition of $|\bar{y}\rangle$ and
$|f_\text{th}\rangle$ for the full covariance fitting.}
\label{tab:y-f-eigen-cov}
\begin{center}
\begin{tabular}{c | c c c c }
\hline
\hline
$i$ & 1 & 2 & 3 & 4 \\
\hline
$a_i$ & 1.021(4) & 0.5655(14) & 0.1061(3) & 0.01442(3) \\
$b_i$ & 1.014(4) & 0.5679(11) & 0.1058(3) & 0.01443(3) \\
\hline
\hline
\end{tabular}
\end{center}
\end{table}
\section{Prescriptions for the trouble}
If the covariance matrix has small eigenvalues, even a small error of
fitting function may yield large error in fitting result.
To circumvent this problem, we need some approximation methods, such as
diagonal approximation or cutoff method.
In subsection \ref{subsec:es-method}, we propose a new method which we
call the eigenmode shift (ES) method.
One simple solution to the problem is to use the diagonal
approximation (uncorrelated fitting).
In this method, we neglect the off-diagonal covariance as follows:
$C_{ij} = 0 \text{ if } i \ne j \,.$
In this way, the small eigenvalue problem disappears.
The fitting results are shown in Fig.~\ref{subfig:diag_fit}.
Another possible solution is to exclude the eigenmodes corresponding to
the small eigenvalues from the inverse covariance matrix, $C^{-1}_{ij}$.
In our example, $|v_4\rangle$ is removed by setting $\dfrac{1}{\lambda_4} = 0$
in Eq.~\eqref{eq:eig_mod_decomp}.
We call this the cutoff method.
A number of lattice QCD groups \cite{fnal-2010-1,lanl-1999-1}
use this method in the popular name
of the SVD (singular value decomposition) method.
In Fig.~\ref{subfig:cutoff_fit}, we show the results of the covariance
fitting using the cutoff method.
\begin{figure}[t!]
\centering
\subfigure[Cutoff method]{\includegraphics[width=0.45\textwidth]
{bk_sxpt_su2_4X3Y_NNLO_Xfit_cutoff_MILC_2064f21b676m010m050}
\label{subfig:cutoff_fit}}
\subfigure[Eigenmode shift method]{\includegraphics[width=0.45\textwidth]
{bk_sxpt_su2_4X3Y_NNLO_Xfit_es_MILC_2064f21b676m010m050}
\label{subfig:es_fit}}
\caption{
$B_K(1/a)$ vs. $X_P$ on the C3 ensemble.
The left figure shows the result of cutoff method and
the right figure shows the result of eigenmode shift method.}
\label{fig:cutoff_and_es}
\end{figure}
\subsection{Eigenmode shift method}
\label{subsec:es-method}
We know that the whole trouble comes from the error of fitting function
in $|v_4\rangle$ direction.
Hence, we can think of a new fitting function $f_\text{th}'$
defined as follows:
\begin{equation}
f_\text{th}'(X) = f_\text{th} (X) + \eta |v_4\rangle
\,.
\end{equation}
Here, $\eta$ is a tiny parameter that can be determined
by the Bayesian method.
Hence, we modify the $\chi^2$ as follows,
\begin{equation}
\chi^2_\text{aug} = \chi^2 + \frac{(\eta-a_\eta)^2}{\sigma_\eta^2}
\,, \qquad
\chi^2 = \sum_{i,j} [\bar{y}_i - f_\text{th}'(X_i)] C^{-1}_{ij}
[\bar{y}_j - f_\text{th}'(X_j)]
\,.
\end{equation}
We know that $\eta$ is very tiny so we choose $a_\eta = 0$.
As mentioned in section \ref{subsec:bk_cov_fit}, the order of the
neglected highest order term in the $f_\text{th} (X)$ is $0.006$.
Hence, we set $\sigma_\eta = 0.006$.
Then we can do the full covariance fitting with an extra fitting
parameter, $\eta$.
When we do the extrapolation to the physical pion mass,
we use only the $f_\textrm{th}(X)$ function, dropping out the
$\eta$ terms.
We call this the eigenmode shift (ES) method.
This is the same procedure as following:
First, find a shifting vector, $\eta |v_4\rangle$, which minimizes the
$\chi^2_\text{aug}$.
Then fit with the tuned(shifted) fitting function.
To consider the statistical error of $\eta$, do this procedure over
jackknife or bootstrap samples.
In our example, the fitted $\eta = -0.00082(31)$, which is much
smaller by an order of magnitude than truncated highest order terms in
$f_\text{th}$.
In Fig.~\ref{subfig:es_fit}, we show the fitting results obtained
using the ES method.
This method tunes the fitting function by a tiny amount so that
minimizes the small eigenvalue contribution.
In this sense, it looks similar to the cutoff method.
However, unlike the cutoff method, the ES method determines the
shifting parameter, $\eta$, using the Bayesian method and
the full covariance matrix remains untouched.
\section{Conclusion}
Here, we address an issue of covariance fitting on the highly
correlated $B_K$ data.
It turns out that the small error of fitting function can make the
fitting fail if the covariance matrix has small eigenvalues.
In order to get around the trouble, we have used approximations: the
diagonal approximation and the cutoff method.
Here, we propose a new method, the eigenmode shift method, which
fine-tunes the fitting function, while keeping the covariance matrix
untouched.
\section{Acknowledgments}
C.~Jung is supported by the US DOE under contract DE-AC02-98CH10886.
The research of W.~Lee is supported by the Creative Research
Initiatives Program (3348-20090015) of the NRF grant funded by the
Korean government (MEST).
W.~Lee would like to acknowledge the support from KISTI supercomputing
center through the strategic support program for the supercomputing
application research [No. KSC-2011-C3-03].
Computations were carried out in part on QCDOC computing facilities of
the USQCD Collaboration at Brookhaven National Lab, and on the DAVID
GPU clusters at Seoul National University. The USQCD Collaboration are
funded by the Office of Science of the U.S. Department of Energy.
|
train/arxiv
|
BkiUbTQ5qX_AYzDLsDNE
| 5
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\section{Introduction}\label{S0}
Let $p$ be a fixed prime number and let $F$ be a field such that
$F$ contains a primitive $p$-th root of unity $\zeta_p$. Let $F_s$
be the separable closure of $F$ and $\Gal(F_s/F)$ be the absolute
Galois group of $F$. Let $H^*(F) = H^*(\Gal(F_s/F),
\mathbb{Z}/p\mathbb{Z})$ be the Galois cohomology of $F$ in
coefficients $\mathbb{Z}/p\mathbb{Z}$. (From now on we shall omit
the coefficients of our cohomology groups as they will always be
$\mathbb{Z}/p\mathbb{Z}$.) In \cite{mi}, Milnor defined the group
$K_nF$ by generators and relations for any $n\in \nn \cup \{0\}$.
The generators are $n$-tuples $\{a_1,\dots ,a_n\}$ of elements of
$F^*$, and the defining relations are the multiplicativity in each
component and the Steinberg relation $\{a_1,\dots ,a_n\} = 0$ if
$a_i + a_j = 1$ for some $i\not= j$. Set $k_nF = K_nF/pK_nF$ and
$k_*F = \bigoplus_{n\geq 0} k_nF$. Thus $k_*F$ is the Milnor
K-theory of $F$ modulo $p$. In the same paper Milnor defined a
graded homomorphism $h_*:k_*F\to H^*(F)$ and implicitly
conjectured that it is an isomorphism when $p=2$. Because of the
later important work of Bloch and Kato for any prime $p$, the
general case is now known as the Bloch-Kato conjecture.
Let $F(p)$ be the maximal $p$-extension of $F$. This means that
$F(p)$ is a union of Galois extensions $K/F$, such that
$\Gal(K/F)$ is a $p$-group, in a fixed $F_s$ of $F$. Let $G$ be
the Galois group of $F(p)/F$. Observe that $\inf:H^1(G)\to
H^1(F)$ is an isomorphism and the Steinberg relations hold in
$H^2(G)$. Then $h_*:k_*F\to H^*(F)$ factors through $h_*:k_*F\to
H^*(G)$. A simple application of the Hochschild-Serre spectral
sequence shows that if $h_n:k_nF\to H^n(F)$ is isomorphic for all
fields $F$ then $h_n:k_nF\to H^n(G)$ is an isomorphism for all
fields $F$. Also see \cite[page 97, for the case $p=2$]{g-m}. In
the paper \cite{m-s}, Merkurjev and Suslin proved that $h_2$ is an
isomorphism. Recently Rost and Voevodsky, with Weibel's patch,
proved that $h_n$ is an isomorphism for all $n\in \nn$. (See
\cite{ha-w}, \cite{ro1}, \cite{ro2}, \cite{voe1}, \cite{voe2},
\cite{voe3}, and \cite{wei1}, \cite{wei2}.) The cohomology of
Galois groups of maximal pro-$p$-extensions of fields reflects
some properties that the class of these groups share. The
relationship between the structure of groups and their cohomology
groups is in general nontrivial and it is quite mysterious. We are
interested in the group theoretic meaning of the Bloch-Kato
conjecture.
We investigate some strong, and at the same time simple conditions
on the relations of $\Gal(F(p)/F)$ for any field $F$ which is
closely related to the Bloch-Kato conjecture. These conditions say
that all relations of $G=\Gal(F(p)/F)$ are generated by relations
of small weight. This will be made more precise in section 4. For
related work on the structure of $\Gal(F(p)/F)$ and its relations
with cohomology, see \cite{a-k-m}, \cite{j-w}, \cite{ko1},
\cite{ko2}, \cite{mi-sp1} and \cite{n-s-w}.
Let $\{\sigma_i\}_{i\in I}$ be a minimal set of generators of $G$. (See
\cite[Chapter~4]{ko1}.) We assume that $I$ is well ordered. Let $S$ be a free
pro-$p$-group with a minimal set of generators $\{s_i\}_{i\in I}$. By
sending $s_i$ to $\sigma_i$ we obtain a continuous homomorphism $\pi :S\to
G$. We set $R = Ker(\pi)$.
For a pro-$p$-group $G$, we define $G^{(n+1)} =
(G^{(n)})^p[G^{(n)},G]$ and $G^{(1)} = G$. Then $G^{(n)}$ is a
closed normal subgroup of $G$. We denote the quotient by
$G^{[n]}$. We set $R^{(1,S)} = R$ and $R^{(n+1,S)} =
(R^{(n,S)})^p[R^{(n,S)},S]$ for $n\geq 1$. Since $S$ and $G$ have
the same cardinality of the minimal set of generators, then
$R\subset S^{(2)}$. In general, we see by induction on $n$ that
$R^{(n,S)}\subset R\cap S^{(n+1)}$. Lemma~\ref{L2.2} says that
if $h_2$ is surjective, then $R^{(2,S)} = R\cap S^{(3)}$.
{\bf Example A.} The equality $R^{(2,S)}=R \cap S^{(3)}$ implies
that every $\sigma \in G=\Gal(F(p)/F)$ of finite order $\ne1$ has
order $p$ as follows. Suppose that $\sigma$ has order $\ge p^2$.
The subgroup of $G$ generated by $\sigma$ is a closed subgroup of
$G$ and if $L$ is its fixed field, we have $L(p) = F(p)$ and
$\Gal(L(p)/L) = \langle\sigma\rangle$. Hence we may assume that $F
= L$ and $G = \langle\sigma\rangle$. Moreover by taking a suitable
power of $\sigma$ we may assume that $\sigma$ has an order $p^2$.
Then $\sigma^{p^2}\in R \cap S^{(3)}$ but $\sigma^{p^2} \notin
R^{(2,S)}$ as $\sigma^{p^2}$ generates $R$ as a normal subgroup of
$S$. Hence the order of $\sigma$ is $p$. This fact also follows
from the work of Becker~\cite{be} where he also shows that $p=2$
and that two elements of $G$ order $2$ cannot commute. In fact
the only non-trivial finite subgroup of $G$ is the cyclic group
of order $2$.
{\bf Example B.} (See \cite[Section~9]{ch-e-m} for more examples.)
One can also deduce from the equality $R^{(2,S)}=R \cap S^{(3)}$ that a minimal se of
relations among the $\sigma_i$ cannot contain a relation of the
form $[[\sigma_1, \sigma_2],\sigma_3]$ with $\sigma_1,
\sigma_2,\sigma_3$ distinct. Indeed, such a relator would be in
$R\cap S^{(3)}$ but not in $R^p[R,S]=R^{(2,S)}$.
Our first result is
\begin{theorem}\label{T1}
Assume that $G$ is $G_F(p)$ for some field $F$ containing a
primitive $p$th-root of $1$. Then $R^{(3,S)} = R\cap S^{(4)}$.
\end{theorem}
Theorem \ref{T1} is proved in section \ref{S2}. It is natural
to ask the following question.
\begin{question}\label{Q2}
{\it{Let $G$ be isomorphic to a Galois group of a maximal p-extension of
a field. Under which condition is $R^{(n,S)} = R\cap S^{(n+1)}$ for all
$n\geq 1$?}}
\end{question}
More generally we ask:
\medskip
\begin{question}\label{Q3}
{\it{Let $G$ be isomorphic to the Galois group of the maximal
$p$-extension of a field. Describe the quotients $R\cap
S^{(n+1)}/R^{(n,S)}$ for all $n\geq 1$. }}
\end{question}
In the first two sections we show that $R^{(n,S)} = R\cap
S^{(n+1)}$ for $n\leq 3$. In section \ref{S3}, we give an
equivalent description of Question \ref{Q2} in the language of Lie
algebras. We prove that, for quadratically defined pro-$2$-groups
as well as for $G=\Gal(F(2)/F)$ when $F$ is a totally imaginary
number field, the relation $R^{(n,S)}=R\cap S^{(n+1)}$ is valid
for each $n\geq 1$. For all odd primes $p$ we show the same holds
for $G=\Gal(F(p)/F)$ where $F$ is any local or global field.
\section{Preliminaries}\label{S1}
We use the following usual notation: $[a]$ means
both an element of $F^*/F^{*p}$ and its corresponding element
$(a)$ in $H^1(F)$ or more generally in $H^1(\Gal(T/F))$ where
$T/F$ is any Galois extension of $F$ which contains $F^{(2)}$: =
compositum of all cyclic extensions of degree $p$ of $F$, and is a subfield
of $F(p)$. Observe that by Kummer
theory we have the canonical isomorphism $F^*/F^{*p}\to
H^1(\Gal(T/F))$, which justifies our identification mentioned
above.
We will work in the category of pro-$p$-groups and make the usual
conventions that by ``subgroup'' we mean ``closed subgroup'', by
``generated'' we mean ``topologically generated'' and by ``morphism''
we mean ``continuous morphism''. We shall always work with $\mod p$
cohomology except when we explicitly mention other coefficients.
For the standard facts for Galois cohomology we refer to \cite{se2}
and \cite{n-s-w}.
For an extension $1\to A\to B\to C\to 1$ of profinite groups and a
discrete $B$-module $M$ we have the corresponding Lyndon-Hochschild-Serre
(LHS, for short) spectral sequences $\{E^{p,q}_r,d^{p,q}_r\}$ where
$E^{p,q}_2 = H^p(C,H^q(A,M))$ (\cite{h-s} and \cite{n-s-w}).
Using LHS we also have the five-term exact sequence
\begin{align}
& 0\to H^1(C,M^A) \overset{\inf}{\longrightarrow} H^1(B,M)\overset
{\res}{\longrightarrow} H^1(A,M)^C \overset{d^{0,1}_2}{\longrightarrow}
H^2(C,M^A) \notag \\
& \overset{\inf}{\longrightarrow} H^2(B,M) \notag
\end{align}
where $\inf$ is the inflation map and $\res$ is the restriction
map.
Observe that $k_nF$ is isomorphic to the factor group of
\[ F^*/F^{*p}\otimes\cdots\otimes F^*/F^{*p} \]
by the subgroup generated by
$[a_1]\otimes\cdots\otimes [a_n]$ where $a_1$ is a norm in the
extension $F(\root{p}\of{a_j})/F$ of degree $p$, for
some $j = 2,\dots,n$. (If $F(\root{p}\of{a_j})/F$ is not an extension
of degree $p$, then $[a_j]=[1]$ and $[a_1]\otimes \dots \otimes
[a_n]=0$.)
Indeed let $l_nF$ be this factor group. Because all relations in
$l_nF$ are valid in $k_nF$, we see that $k_nF$ is a naturally
homomorphic image of $l_nF$. (See \cite[page~303, Exercise~6]{f-v}.)
In order to show that this homomorphism
is actually an isomorphism, it is enough to show that the Steinberg
relations which generate the relations in $k_nF$ are valid in $l_nF$
also. Let $a_1,\dots,a_n\in F^*$. We denote as $\langle a_1,\dots,a_n
\rangle$ the image of $[a_1]\otimes\dots\otimes [a_n], [a_i]\in
F^*/F^{*p}$ in $l_nF$.
We want to show that if $1=a_i + a_j, 1\le i<j\le n$, then
\[
\langle a_1,\dots,a_n\rangle = 0 \mbox{ in }l_nF.
\]
Because this is trivially true if $a_j\in F^{*p}$ we shall
assume that $a_j\notin F^{*p}$. If $i=1$ then this is true
by our definition of the relations in $l_nF$ as
\begin{align*}
1-a_j &= \prod^{p-1}_{i=0}(1-\zeta_p^i \root{p}\of{a_j})\\
&= N_{F(\root{p}\of{a_j})/F}(1-\root{p}\of{a_j}).
\end{align*}
Hence it is enough to show that if
\[
1 < i < j \le n
\]
then
\[
\langle a_1,\dots,a_i,\dots,a_j,\dots,a_n \rangle =
-\langle a_i,\dots,a_1,\dots,a_j,\dots,a_n \rangle.
\]
However using the equation
\[
-a = (1-a)/(1-a^{-1}) \mbox{ for }a \ne 1,
\]
we see that
\[
\langle a,\dots,-a,\dots \rangle = 0 \mbox{ in }l_nF
\mbox{ for all }a\in F^*.
\]
Hence
\begin{align*}
0 &= \langle a_1 a_i,\dots,-a_1 a_i,\dots a_n \rangle \\
&= \langle a_1,\dots,-a_1,\dots \rangle + \langle
a_1,\dots,a_i,\dots \rangle + \\ & \langle a_i,\dots,
a_1,\dots \rangle + \langle a_i,\dots,-a_i,\dots \rangle \\
&= \langle a_1,\dots,a_i,\dots \rangle + \langle a_i,\dots,
a_1,\dots \rangle,
\end{align*}
as required.
Hence $l_nF = k_nF$.
\section{The proof of Theorem~\ref{T1}}\label{S2}
We divide our proof into several lemmas. Lemma~\ref{L2.3}
shows that it is enough to prove that $\inf:H^2(S^{[3]})
\to H^2(S/R^{(2,S)})$ is surjective. This is proved by
considering certain spectral sequences below.
\begin{lemma}\label{L2.1}
$R^{(n,S)}\subset R\cap S^{(n+1)}$.
\end{lemma}
\begin{proof}
Our inclusion is true if $n = 1$. Assume that it is true for $k\leq n$.
Then $R^{(n+1,S)} = (R^{(n,S)})^p[R^{(n,S)},S]\subset
(S^{(n+1)})^p[S^{(n+1)},S] = S^{(n+2)}$. So $R^{(n+1,S)}\subset R\cap
S^{(n+2)}$.
\end{proof}
Lemma \ref{L2.2} below was observed in \cite[page 102]{wur} under
an additional hypothesis, and in
\cite[page 57]{mi-sp2} for the case $p=2$. This lemma was also
generalized in \cite[page 207]{g-m}
in the case $p=2$ and in \cite{ch-e-m} for all $p$.
\begin{lemma}\label{L2.2}
If $h_2$ is surjective then $R^p[R,S] = R^{(2,S)} = R\cap
S^{(3)}$.
\end{lemma}
We consider the following pair of extensions
\begeq\label{CD2.3}\begin{CD}
1 @>>> R^{(2,S)} @>>> S @>>> S/R^{(2,S)} @>>> 1 \\
@. @VVV @VVV @VVV \\
1 @>>> S^{(3)} @>>> S @>>> S^{[3]} @>>> 1\; .
\end{CD}
\end{equation}
By applying the five-term exact sequence to \eqref{CD2.3}, we obtain a
commutative diagram
\begeq\label{CD2.4}\begin{CD}
H^1(R^{(2,S)})^{S/R^{(2,S)}} @>{\cong}>> H^2(S/R^{(2,S)}) \\
@AA\res A @AA\inf A \\
H^1(S^{(3)})^{S^{[3]}} @>{\cong}>> H^2(S^{[3]})\; .
\end{CD}
\end{equation}
The surjectivities of both isomorphisms follow from the fact that
$H^2(S) = 0$, because $S$ is a free pro-$p$-group. The injectivities
follow from the fact that $H^1(S/R^{(2,S)})\cong H^1(S)$ and
$H^1(S^{[3]})\cong H^1(S)$, since both $R^{(2,S)}$ and $S^{(3)}$ are
normal subgroups of $S^{(2)}$.
Observe that $H^1(S^{(3)}/S^{(4)})\cong H^1(S^{(3)})^{S^{[3]}}$
and $H^1(R^{(2,S)}/R^{(3,S)})\cong H^1(R^{(2,S)})^{S/R^{(2,S)}}$.
The restriction $H^1(S^{(3)}/S^{(4)})\to H^1(R^{(2,S)}/R^{(3,S)})$
is given by the composite
\begin{align}
&H^1(S^{(3)}/S^{(4)})\overset{\res}{\longrightarrow}H^1(R^{(2,S)}S^{(4)}/S^{(4)})\overset{\cong}{\longrightarrow}H^1(R^{(2,S)}/R^{(2,S)}\cap S^{(4)}) \notag \\
&\overset{\inf}{\longrightarrow}H^1(R^{(2,S)}/R^{(3,S)})\; .
\notag
\end{align}
The restriction $H^1(S^{(3)}/S^{(4)})\to
H^1(R^{(2,S)}S^{(4)}/S^{(4)})$ is surjective. If $R^{(2,S)} =
R\cap S^{(3)}$, then $R^{(2,S)}\cap S^{(4)} = (R\cap S^{(4)}) =
R\cap S^{(4)}$. Then $R^{(3,S)} = R\cap S^{(4)}$ iff $R^{(3,S)} =
R^{(2,S)}\cap S^{(4)}$. Also the last inflation map is surjective
iff $R^{(2,S)} \cap S^{(4)} = R^{(3,S)}$. Thus the last map $\inf:
H^1(R^{(2,S)}/R^{(2,S)}\cap S^{(4)} \to H^1(R^{(2,S)}/R^{(3,S)})$
is surjective iff $R^{(3,S)} = R \cap S^{(4)}$.
Therefore we obtain
\begin{lemma}\label{L2.3
Assume that $R^{(2,S)} = R\cap S^{(3)}$. The following are equivalent
\begin{enumerate}
\item $R^{(3,S)} = R\cap S^{(4)}$.
\item $R^{(3,S)} = R^{(2,S)}\cap S^{(4)}$.
\item $\inf:H^1(R^{(2,S)}/R^{(2,S)}\cap S^{(4)})\to H^1(R^{(2,S)}/R^{(3,S)})$ is surjective.
\item $\res:H^1(S^{(3)}/S^{(4)})\to H^1(R^{(2,S)}/R^{(3,S)})$ is surjective.
\item $\inf:H^2(S^{[3]})\to H^2(S/R^{(2,S)})$ is surjective.
\end{enumerate}
\end{lemma}
We have a pair of extensions
\begeq\label{CD2.6}\begin{CD}
1 @>>> R/R^{(2,S)} @>>> S/R^{(2,S)} @>>> S/R = G @>>> 1 \\
@. @VVV @VVV @VVV \\
1 @>>> S^{(2)}/S^{(3)} @>>> S^{[3]} @>>> S^{[2]} @>>> 1\; .
\end{CD}
\end{equation}
To prove (5) in Lemma~\ref{L2.3} we can compare the LHS spectral
sequences corresponding to \eqref{CD2.6}. We set
$E^{p,q}_r(S^{[3]})$ (or $E^{p,q}_r(S/R^{(2,S)})$) the $E^{p,q}_r
-$term corresponding to the bottom extension (or the top
extension).
Observe that $H^1(R/R^{(2,S)})\cong H^1(R)^G$ and $H^1(R)^G\cong
H^2(G)$, the last isomorphism follows from the five-term exact
sequence corresponding to the extension $1\to R\to S\to G\to 1$.
Then $H^1(R/R^{(2,S)})^G = H^1(R/R^{(2,S)})\cong H^2(G)$.
Similarly we have $H^1(S^{(2)}/S^{(3)})\cong H^2(S^{[2]})$.
Therefore the transgression maps $d_2^{0,1}$ in both spectral
sequences are surjective. Thus we obtain
\begin{lemma}\label{L2.7
$E^{2,0}_\infty(S^{[3]}) = E^{2,0}_\infty(S/R^{(2,S)}) = 0$.
\end{lemma}
To prove the surjectivity of $\inf:H^2(S^{[3]})\to
H^2(S/R^{(2,S)})$, it is enough to show the surjectivities of the
homomorphism corresponding to $E^{1,1}_\infty$-terms and the
homomorphism corresponding to $E^{0,2}_\infty$-terms.
Assume that $R^{(2,S)} = R\cap S^{(3)}$. Observe that
\[ H^1(R/R^{(2,S)}) = H^1(R/R\cap S^{(3)})\cong H^1(RS^{(3)}/S^{(3)})\;
. \]
Thus we obtain
\begin{lemma}\label{L2.8
$\res\colon H^1(S^{(2)}/S^{(3)})\to H^1(R/R^{(2,S)})$ is surjective.
\end{lemma}
Since both extensions in \ref{CD2.6} are central extensions, then
$E^{0,2}_2(S^{[3]})\cong H^2(S^{(2)}/S^{(3)})$ and
$E^{0,2}_2(S/R^{(2,S)})\cong H^2(R/R^{(2,S)})$. Let $\{z_c\}_{c\in C}$
be a basis for $H^1(R/R^{(2,S)})$ and $\{w_d\}_{d\in D}$ be a basis for
$H^1(S^{(2)}/S^{(3)})$. Let $\beta w_d$ and $\beta z_c$ be Bocksteins
of $w_d$ and $z_c$ in $H^2(R/R^{(2,S)})=E_3^{0,2}(S/R^{(2,S)})$ and
$H^2(S^{(2)}/S^{(3)})=E_3^{0,2}(S^{[3]})$ respectively. Then
$\{\beta w_d\}_{d\in D}$ and $\{\beta z_c\}_{c\in C}$ generate
$E_3^{0,2}(S/R^{(2,S)})$ and $E_3^{0,2}(S^{[3]})$ respectively.
This follows from a standard argument exploiting the fact that
$d_2$ is a derivation
with respect to the multiplicative structure of $E_2^{p,q}$ and
the fact that $d_2^{0,1}$ is injective. However using the
fact that $d_2^{1,1}$ is surjective, we
see that $d_3^{0,2}(S^{[3]})=0$. Hence $E_3^{0,2}(S^{[3]})=
E_\infty^{0,2}(S^{[3]})$. In Lemma~3.9 we denote the natural map
$E^{0,2}_3(S^{[3]})\to E^{0,2}_3(S/R^{(2,S)})$ as $\res$ because
it is induced by the restriction map
$S^{(2)}/S^{(3)}\lra R/R^{(2,S)}$.
By Lemma \ref{L2.8}, and our discussion above, we obtain
\begin{lemma}\label{L2.9}
$\res\colon E^{0,2}_3(S^{[3]})\to E^{0,2}_3(S/R^{(2,S)})$ is surjective
and $E_3^{0,2}(S^{[3]})=E_\infty^{0,2}(S^{[3]})$.
\end{lemma}
We have a commutative diagram
\[ \begin{CD}
E^{0,2}_\infty (S/R^{(2,S)}) @>>> E^{0,2}_3(S/R^{(2,S)}) \\
@AAA @AAA \\
E^{0,2}_\infty (S^{[3]}) @= E^{0,2}_3(S^{[3]})\; .
\end{CD} \]
Using Lemma~\ref{L2.9} we deduce the following corollaries.
\begin{corollary}\label{C2.11}
$\res:E^{0,2}_\infty(S^{[3]})\to E^{0,2}_\infty(S/R^{(2,S)})$ is
surjective.
\end{corollary}
\begin{corollary}\label{C2.12}
$E^{0,2}_\infty(S/R^{(2,S)}) = E^{0,2}_3(S/R^{(2,S)})$.
\end{corollary}
Since both extensions in the above are central, then
\[
d^{1,1}_2:H^1(G,H^1(R/R^{(2,S)}))\to H^3(G)
\]
is the composite $H^1(G,H^1(R/R^{(2,S)}))
\overset{(1,d^{0,1}_2)}{\underset{\cong}{\longrightarrow}}
H^1(G)\otimes H^2(G)\overset{\cup}{\longrightarrow} H^3(G)$.
Also $d^{1,1}_2:H^1(S^{[2]}, H^1(S^{(2)}/S^{(3)}))\to H^3(S^{[3]})$
is the composite
\[ H^1(S^{[2]}, H^1(S^{(2)}/S^{(3)}))
\overset{(1,d^{0,1}_2)}{\underset{\cong}{\longrightarrow}}
H^1(S^{[2]})\otimes H^2(S^{[2]})\overset{\cup}{\longrightarrow}
H^3(S^{[2]})\; . \] We may abuse notation by setting
$$
E^{1,1}_\infty (S/R^{(2,S)}) = \Ker(\cup : H^1(G)\otimes
H^2(G)\to H^3(G))
$$
and $E^{1,1}_\infty (S^{[3]}) = \Ker(\cup :H^2(S^{[2]})\otimes
H^2(S^{[2]})\to H^3(S^{[2]}))$. We have a commutative diagram
\begeq\label{CD2.13}\begin{CD}
0 @>>> E^{1,1}_\infty (S/R^{(2,S)}) @>>> H^1(G)\otimes H^2(G) @>\cup >> H^3(G) @>>> 0 \\
@. @AA\text{\it u}A @AA\text{\it v}A @AA\text{\it inf}A \\
0 @>>> E^{1,1}_\infty (S^{[3]}) @>>> H^1(S^{[2]})\otimes H^2(S^{[2]}) @>\cup >>
H^3(S^{[2]}) @>>> 0.
\end{CD}
\end{equation}
Here $u$ and $v$ are natural maps induced by inflation maps.
Recall that $S^{[2]}\cong G^{[2]}$ under our projection map
$S \to G$. In the next lemma we use both the injectivity of
$h_3$ and the surjectivity of $h_2$.
\begin{lemma}\label{L2.14}
The map $u\colon E^{1,1}_{\infty}(S^{[3]})\lra E^{1,1}_{\infty}(S/R^{(2,S)})$ is surjective.
\end{lemma}
\begin{proof} We consider $E^{1,1}_{\infty}(S/R^{(2,S)})$ as
a subgroup of $H^1(G)\otimes H^2(G)$. Using the surjectivity
of $h_2$, we see that each element in $H^1(G)\otimes H^2(G)$
can be written as a sum of elements of the
form $\alpha\otimes(\beta\cup\gamma)$ where $\alpha,\beta,\gamma
\in H^1(G)$. As we remarked at the end of Section 2, the injectivity
of $h_3$ implies that $E^{1,1}_{\infty}(S/R^{(2,S)})$ is generated
by the elements $z_1\otimes (z_2\cup z_3)$ such that $z_1\cup z_2=0$
in $H^2(G)$. (Since $h_2$ is the isomorphism, we can work in
$H^1(G)$ rather than in $k_2F$.) Because the map $\inf\colon H^1
(S^{[2]})\lra H^1(G)$ is an isomorphism, we see that for the element
$z_1\otimes(z_2\cup z_3)$ as above we can find $y_i\in H^1(S^{[2]}),
i=1,2,3$ such that $\inf(y_i)=z_i$. Using our assumption that
$z_1\cup z_2=0$ in $H^2(G)$ we see that
$$u((y_1\otimes(y_2\cup y_3)-y_3\otimes(y_1\cup y_2))=z_1\otimes(z_2\cup z_3).$$
Therefore we see that our map $u$ is surjective.
\end{proof}
\section{Graded Lie Algebras}\label{S3}
Here we give an equivalent description of Question~1.2 in Lie
algebra language. A convenient reference is Lazard's paper
\cite{laz1}. As usual, we consider a minimal presentation of $G$.
\begeq\label{CD3.2}\begin{CD}
1\longrightarrow R\longrightarrow S\longrightarrow
G\overset{\pi}\longrightarrow 1\; .
\end{CD}
\end{equation}
Let $S$ and $G$ admit the usual filtrations
\[ S^{(1)} = S,\dots ,S^{(n+1)} = (S^{(n)})^p[S^{(n)},S],\dots \]
\[ G^{(1)} = G,\dots ,G^{(n+1)} = (G^{(n)})^p[G^{(n)},G],\dots\; . \]
The formulae $G^{(n+1)}\subset G^{(n)},[G^{(n)},G^{(m)}]\subset
G^{(n+m)}$ imply that $\gr_n(G) = G^{(n)}/G^{(n+1)}$ (denoted
additively) is an vector space over $\F_p$ and that the graded
algebra $\gr(G) = \sum \gr_n(G)$ is an algebra over $\F_p$ where
multiplication of homogenous elements of $\gr(G)$ is induced by
the commutator operation. This operation satisfies the Jacobi
identity and hence is a Lie bracket. The $p$-th power map in $G$
induces an operator $P$ on $\gr(G)$ making it a Lie algebra over
$\F_p[\pi]$ if $p\ne2$ and a mixed Lie algebra if $p=2$ (cf.
\cite{laz1}). Similarly, the induced filtration $\{R\cap
S^{(n)}\}$ yields the graded Lie algebra $\gr(R)_{ind}$. The
extension \ref{CD3.2} induces an exact sequence of (mixed) Lie
algebras
\[ 0\to \gr(R)_{ind}\to \gr(S)\overset{\phi}{\to} \gr(G)\lra 0. \]
Since the filtration of $G$ is discrete the map $\phi$ is
surjective. (See pages 428-430 in \cite{laz1} for details.)
On the other hand, the filtration $\{R^{(n,S)}\}$ yields another
Lie algebra $\gr(R,S)$. The inclusion $R^{(n,S)}\subset R\cap
S^{(n+1)}$ (see Lemma \ref{L2.1} induces the homomorphism
$\iota_n:\gr_n(R,S)\to \gr_{n+1}(R)_{ind}$ and therefore a Lie
algebra homomorphism $\iota:\gr(R,S)\to \gr(R)_{ind}$. Let $U$ be
the enveloping algebra of $\gr(S)$. Then $U$ is the free
associative $\F_p[\pi]$-algebra on the free generators of $S$.
There is a canonical embedding of $\gr(S)$ into $U$ with $P(x)=\pi
x$ if $p\ne2$. If $p=2$ we have $P(x)=x^2+\pi x$ if $x$ is of
degree $1$ and $P(x)=\pi x$ if $x$ is of degree $>1$. Note that
$\gr(R,S)$ and $\gr(R)_{ind}$ are $U$-modules via the adjoint
representation and that $\iota$ is a homomorphism of $U$-modules.
\begin{theorem}\label{equiv}
The following are equivalent
\begin{enumerate}[{\rm(A)}]
\item We have $R^{(n,S)} = R\cap S^{(n+1)}$ for $n\ge1$.
\item The homomorphism $\iota$ is injective.
\item The homomorphism $\iota$ is surjective.
\end{enumerate}
\end{theorem}
\begin{proof}
Note that (A) holds for $n=1$ since $R\subset S^{(2)}$.
Assume that $\iota$ is injective and that (A) holds for some
$n\ge1$. Let $x\in R\cap S^{(n+2)}$. Then $x\in R\cap
S^{(n+1)}=R^{(n,S)}$. If $\xi$ is the image of $x$ in $\gr_n(R,S)$
we have $\iota_n(\xi)=0$ which implies $x\in R^{(n+1,S)}$. Hence
(A) holds for $n+1$ and by induction for all $n$.
Assume that $\iota$ is surjective and let $x\in R\cap S^{(n+1)}$.
Then there exists $y_0\in R^{(n,S)}$ such that $x_1=y_0^{-1}x\in
R\cap S^{(n+2)}$. In the same way we define inductively $y_i$ such
that $y_i\in R^{(n+i,S)}$ and $x_{i+1}=y_i^{-1}x_i\in R\cap
S^{(n+2+i)}$ for $i\ge0$ with $x_0=x$. Then $x=\prod y_i\in
R^{(n,S)}$.
\end{proof}
\begin{corollary}
Let $G=S/R$ be a minimal presentation of $G$ of finite type. Then
$i$ is surjective if $\iota(\gr_1(R,S))$ generates $\gr(R)_{ind}$
as an ideal of $\gr(S)$.
\end{corollary}
The elements of $i(\gr_1(R,S))\subset \gr_2(S)$ are images of
elements $r$ of $R$ under the canonical mapping of $S^{(2)}$ onto
$\gr_2(S)$. These images are called initial forms of the elements
$r$. If $\iota$ is bijective and $R\ne 1$ then $R$ has a minimal
generating set whose initial forms are of degree $2$. In this case
the presentation $G=S/R$ is called {\bf quadratic}. If $G=F/R$ is
of finite type we say it is {\bf quadratically defined} if it is
quadratic and the set of initial forms of a minimal generating set
for $R$ generate $\gr(R)_{ind}$ as an ideal of $\gr(S)$. The
latter is true if the set of initial forms is strongly free, cf.
\cite{lm}, \cite{lab2}. The group $G$ is said to be quadratically
defined if it has a minimal presentation which is quadratically
defined.
\begin{theorem}
Let $G=S/R$ be a minimal presentation of $G$ of finite type. Then
$i$ is bijective iff $R=1$ or $G=S/R$ is quadratically defined.
\end{theorem}
\begin{proof}
If $G=S/R$ is quadratically defined then $i(\gr_1(R,S))$ is a
generating set for $\gr(R)_{ind}$ as an ideal of $\gr(S)$. Then
$i$ is surjective since it is a $U$-module homomorphism and hence
bijective by Theorem~\ref{equiv}.
Conversely, suppose that $i$ is bijective and identify $\gr(R,S)$
with its image in $\gr(S)$. To prove that $G=S/R$ is quadratically
defined it suffices to prove that $\gr_1(R,S)$ generates
$\gr(R,S)$ as a $U$-module. Let $M$ be the $U$-submodule of
$\gr(R,S)$ generated by $\gr_1(R,S)$. We have $M_1=\gr_1(R,S)$.
Suppose that $M_n=\gr_n(R,S)$ and let $\xi\in\gr_{n+1}(R,S)$,
$\xi\ne0$. If $x\in R^{(n+1,S)}$ is a representative of $\xi$ then
$x$ is a product of elements of the form $u^p$, $[u,v]$ with $u\in
R^{(n,S)}$, $v\in S$. Since $i$ is injective these elements lie in
$\gr_{n+1}(S)$ unless the degree of $u$ is $n$ and the degree of
$v$ is $1$. It follows that $\xi$ is a linear combination of
elements of the form $\pi\eta$, $[\eta,\zeta]$ with $\eta\in M_n$,
$\zeta\in U_1$ and hence that $\xi\in M_{n+1}.$
\end{proof}
\begin{question}
Let $G$ be isomorphic to the Galois group of a maximal
$p$-extension of a field, and let $i$ be a natural graded Lie
algebra homomorphism $i:\gr(R,S)\to \gr(R)_{ind}$. When is $i$ an
isomorphism?
\end{question}
If $F$ is a global field of characteristic $\ne p$ which is
totally imaginary if $F$ is a number field and $p=2$, then by
the results of \cite{lm} and \cite{sch}, $G=\Gal(F(p)/F)$ is a
projective limit of quadratically defined presentations. More
precisely, the group $G$ has a presentation $S/R$ where $S=\cup
S_i$ {$i\ge 1$} with $S_i\subset S_{i+1}$ finitely generated and,
if $R_i$ is the image of $R$ under the canonical projection of $S$
onto $S_i$, we have $S_i/R_i$ quadratically defined for all $i$.
By Theorem~\ref {equiv} this means that $R_i^{(n,S_i)}=R_i\cap
S_i^{(n+1)}$ for all $i,n$ which implies $R^{(n,S)}=R\cap
S^{(n+1)}$ for all $n$ since $S$ is the projective limit of the
$S_i$. Hence, by Theorem~\ref{equiv}, the map $\iota$ is an
isomorphism. The same is true if $F$ is a local field with
$\zeta_p$ in $F$ since then $G=\Gal(F(p)/F)$ is a Demushkin group
which is quadratically defined by \cite{lm}, \cite{lab2}. We thus
obtain the following result.
\begin{theorem} Let $F$ be a field containing a primitive $p$-th root of unity. If
$F$ is a global field, which is totally imaginary if $F$ is a
number field and $p=2$, or a local field containing a primitive
$p$-th root of unity then $F$ is quadratically defined.
\end{theorem}
\begin{question}
Is $\Gal(\Q(2)/\Q)$ quadratically defined?
\end{question}
\section{Acknowledgements}\label{S4}
We are very grateful to Alejandro Adem and Wenfeng Gao, who
made important contributions towards this work in its early
stages. Alejandro Adem's continuous interest, and discussions,
have been a considerable encouragement for our work.
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\section{Introduction}
It has been known that nuclei in the $Z=82$ region are rich in shape
coexistence. In particular, the important deformation driving
orbitals has been assigned as the $h_{9/2}$ and $ i_{13/2}$ proton
shells \cite{Heyde83,Wood92}. In odd-mass Tl isotopes (with $Z=81$),
one-particle--two-hole (1p-2h) intruder states and shape coexistence
have been discovered through the observation of low-lying $9/2^{-}$
isomeric states \cite{DS63}. The structure of these isomeric states
was confirmed to be decided by the odd proton occupying the
$h_{9/2}$ intruder orbital \cite{Newton70,BBJ85}. Later, the
rotational bands associated with both oblate ($\pi $h$_{9/2}$, $\pi
$i$_{13/2})$ and prolate ($\pi $h$_{9/2}$ , $\pi $i$_{13/2}$, $\pi
$f$_{7/2})$ structures have been observed in lighter isotopes
$^{185,187}$Tl \cite{Lane95}. The band-head of the 1p-2h oblate $\pi
$h$_{9/2}$ intruder band has been observed to lie lowest in energy
near $N$=108. In contrast, the band-head of the prolate intruder
band based on the $i_{13/2}$ structure has been predicted to
decrease continuously in excitation energy as the neutron number
decreases beyond the neutron mid-shell. This prolate structure is
presumably formed by coupling the odd $i_{13/2}$ proton to the
prolate Hg core with 4p-6h structure \cite{Lane95}. Recently, a
rotational-like yrast cascade was established in $^{183}$Tl and
assigned to associate with the prolate $i_{13/2}$
structure \cite{Reviol00}. Furthermore the band-head energy of its
yrast band was later determined \cite{Muiku01}.
Besides the coexistence of prolate and oblate shapes mentioned
above, the signature splitting observed in the $[505]9/2^{-}$ band
in $^{187}$Tl which is significantly larger than that observed in
its heavier odd-mass isotopes with $A \ge 191$ suggests that there
may exist triaxial deformation \cite{Heyde83,Lane95} and the
discrepancy between the calculated equilibrium energy and the
experimental data of the band-head energy of the
$[606]{\frac{13}{2}}^{+}$ band in $^{187}$Tl hints that there may
also involve triaxial deformation \cite{Lane95}. However, no concrete
investigations on the triaxiality in $^{185,187}$Tl have been
reported up to now. Furthermore, there does not exist, at present,
a systematic theoretical investigation on the structure of $^{183}$Tl.
In addition, whether the $[532]{\frac{3}{2}}^{-}$ ($h_{9/2}$) state
in $^{185,187}$Tl can be distinguished from the
$[530]{\frac{1}{2}}^{-}$ ($f_{7/2}$) state (the band originated from
such a configuration has not yet been observed in $^{185}$Tl) has
not yet been determined definitely \cite{Lane95}.
On the theoretical side, it has been well established that the total
energy surface calculation is quite successful in studying the
equilibrium shape of a nucleus and shape coexistence (see for
example Refs. \cite{Bengtsson,Xu,Frauend}). In addition, the projected
shell model \cite{HaraSun} and particle triaxial-rotor
model \cite{Meyer745,Larsson78,Ring80} are also suitable to study
triaxial deformation and configuration mixing \cite{Sheikh,Hamamoto}.
However, the triaxial deformation in the light odd-{\it A} Tl-isotopes has
not yet been studied. Because of its simplicity, we take the
particle triaxial-rotor model with variable moment of inertia of
the core to analyze the structure and deformation of the energy bands in
$^{183,185,187}$Tl systemically and to identify their microscopic
configuration.
The paper is organized as follows. After this introduction, we
describe briefly the formalism of the particle triaxial-rotor model
in Section II. In Section III, we describe our calculation and obtained
results. In Section IV, we give a summary and brief remark.
\section{Particle Triaxial-Rotor Model }
In the particle rotor model, the Hamiltonian of an odd-{\it A} nucleus
is usually written as \cite{Meyer745,Larsson78,Ring80}
\begin{equation}
\label{eq1} \hat {H} = \hat {H}_{core} + \hat {H}_{s.p.} + \hat
{H}{ }_{pair} \quad .
\end{equation}
In the case of triaxial deformation, the Hamiltonian of the
even-even core is given as
\begin{equation}
\label{eq2}
\hat {H}_{core} = \sum\limits_{i = 1}^3 {\frac{\hbar ^2R_i ^2}{2\Im _i }} =
\sum\limits_{i = 1}^3 {\frac{\hbar ^2(I_i - j_i )^2}{2\Im _i }} ,
\end{equation}
\noindent
where $R$, $I$ and $j$ are the angular momentum of the core, the nucleus and the
single particle, respectively. The three rotational moments of inertia are
assumed to be connected by a relation of hydrodynamical type
\begin{equation}
\label{eq3} \Im _\kappa = \frac{4}{3}\Im _0 (I) \sin ^2(\gamma +
\frac{2\pi }{3}\kappa ) \quad ,
\end{equation}
\noindent
with
\begin{equation}
\label{eq4}
\Im _0 (I) = \Im _0 \sqrt {1 + bI(I + 1)}
\end{equation}
\noindent being the variable moment of inertia \cite{MSB69} of the
core to replace the original constant $\Im _0$ to improve the
calculation. In present calculation, we take $b = 0.013$ as the
same as that in Refs. \cite{ZS02,SLZ04,CSL05}.
$\hat {H}_{s.p.} $ describes the Hamiltonian of the unpaired
single particle. In the triaxial deformed field of the even-even
core , $\hat {H}_{s.p.} $ is given by
\begin{eqnarray} \label{eq5}
\hat {H}_{s.p.}& = &- \frac{\hbar ^2}{2m}\nabla ^2 +
\frac{1}{2}m\omega _0^2 \{1 - 2\beta [Y_{20} \cos \gamma +
\frac{1}{\sqrt 2 }(Y_{22} + Y_{2 - 2} )\sin \gamma ]\} \nonumber \\
& & - \kappa \hbar \omega _0 \{2 l \cdot s + \mu (l^2 - < l_N >
^2)\} \, ,
\end{eqnarray}
\noindent where $\kappa $ and $\mu $ are Nilsson parameters,
$Y_{2q} $ is the rank-2 spherical harmonic function.
$\hat {H}_{pair}$ is the Hamiltonian to represent the pairing
correlation which can be treated in the Bardeen-Cooper-Schrieffer
(BCS) formalism.
The single-particle wavefunction can be expressed as
\begin{equation}
\label{eq6}
\left| \nu \right\rangle = \sum\limits_{Nlj\Omega } {C_{Nlj\Omega }^{(\nu )}
\left| {Nlj\Omega } \right\rangle } \quad ,
\end{equation}
\noindent where $\nu $ is the sequence number of the single-particle
orbitals, $\vert N l j \Omega \rangle $ represents the corresponding
Nilsson state, $C_{Nlj\Omega }^{(\nu )} $ is the coefficient to
identify the configuration mixing. Diagonalizing the single-particle
Hamiltonian in the basis $\left| {Nlj\Omega } \right\rangle $, we
can obtain the $C^{(\nu )}_{Nlj\Omega} $ and the single-particle
eigenvalue $\varepsilon _\nu $. The corresponding quasi-particle
energy can then be determined by $E_{\nu} = \sqrt {(\varepsilon _\nu
- \lambda )^2 + \Delta ^2} $, with $\lambda $ and $\Delta $ being
the Fermi energy and the energy gap, respectively.
The total Hamiltonian in Eq.(\ref{eq1}) can be diagonalized in the
symmetrically strong coupling basis
\begin{equation}
\label{eq7} \left| {IKM\nu } \right\rangle = \sqrt {\frac{2I +
1}{16\pi ^2}} \left[ D_{MK}^{I} \alpha _{\nu} ^{\dag} \left|
\tilde {0} \right\rangle + (-1)^{I - K}D_{M - K}^{I} \alpha
_{\tilde {\nu }}^{\dag} \left| \tilde {0} \right\rangle \right]
\, ,
\end{equation}
\noindent where $\alpha _\nu ^ + $ is creation operator of the
single nucleon (in present case, proton) in the orbital $\vert \nu
\rangle $, $D_{MK}^I $ is the rotational matrix.
\section{Calculation and Results}
In the present calculation to investigate the property of
$^{183,185,187}$Tl, we take the $\kappa $ and $\mu $ in standard
values \cite{BR85}, i.e., 0.054, 0.690, respectively, and the pairing
gap parameter as $\Delta = 12 / \sqrt A $. To improve the agreement
between calculated results and experimental data, we introduce a
Coriolis attenuation factor {$\xi $} and take value as that giving
the best agreement between the calculated and experimental energy
spectra. We found that, when {$\xi =0.95$}, the calculated results
agree best with the experimental data of $^{183,185,187}$Tl. In
general principle, in order to describe the nuclear property more
accurately and to make better agreement between calculated and
experimental data, it is necessary to involve sufficient
single-particle orbitals near the Fermi surface in the calculation.
Then we take 13 orbitals near the Fermi surface to couple with the
core for $^{183}$Tl, $^{185}$Tl, $^{187}$Tl, respectively. Practical
calculation shows that the Fermi levels of the bands 6 (we denote
the band labels here as the same as those for nucleus $^{187}$Tl in
Ref.~\cite{Lane95}, so that the similar bands can be compared) of
the nuclei lie between the 20th and the 21st single particle
orbitals, and the others lie between the 19th and the 20th. For the
deformation parameters {$\beta $} and $\gamma$ of $^{185,187}$Tl, we
take those given in Ref.\cite{Lane95} as the trial initial values to
fit. For the deformation parameters of $^{183}$Tl, since there does
not exist any report to discuss them, we take the values of its
neighbor nucleus $^{185}$Tl \cite{Lane95} as the trial initial ones.
Then we accomplished a series diagonalization of the total
Hamiltonian with various values of $\beta $ and $\gamma$ to make the
calculation error $\chi ^2 = \frac{1}{N} \sum _{j} (E^{cal}_{j} -
E^{exp}_{j} )^2 $ of the spectrum of a band (where $N$ is the number
of levels in the band) smaller (in such a process, the band-head
energy is fixed artificially with the definite angular momentum
assigned in experiment. The best fit is, in fact, focused on the
energy separations). Meanwhile, it should be noted that the
parameter sector we used in the present work is the same as that
taken in the book by Nilsson and Ragnarsson \cite{NR95}, where the
value of $\beta$ can be positive or negative, the value of $\gamma$
varies from 0 to 30 degrees. The best fitted values of the $\beta$
and $\gamma$ are listed in Table 1, 2, 3 for nucleus $^{183}$Tl,
$^{185}$Tl, $^{187}$Tl, respectively. At the same time, we obtain the
total wavefunctions in terms of the single-particle orbitals which,
as mentioned above, is fixed by diagonalizing the single particle
Hamiltonian at each set of deformation parameters ($\beta$,
$\gamma$). The calculated main components of the single-particle
orbitals in terms of the Nilsson levels (in the case of best fitted
deformation parameters) of nucleus $^{183}$Tl, $^{185}$Tl,
$^{187}$Tl are also listed in Table 1, 2, 3, respectively. The
resulting energy spectra for the nuclei $^{183,185,187}$Tl,
as obtained from the best fit, are illustrated in Fig.1.
From inspecting the results shown in Fig.~1, we observe a good agreement
with the experimental data.
\begin{table}[htbp]
\begin{footnotesize}
\caption{The deformation parameters and the main components of the
single-particle levels $\vert \nu \rangle $ near the Fermi surface
in terms of the Nilsson levels of the bands in $^{183}$Tl (the
initial values of the deformation parameters are taken as those of
$^{185}$Tl in Ref.~\cite{Lane95}.) } \vspace*{-2mm}
\begin{center}
\begin{tabular}{|c|c|c|c|c|l|}
\hline {} & \multicolumn{2}{|c|}{$\beta$} &
\multicolumn{2}{|c|}{$\gamma$} & {} \\ \cline{2-5} band & initial
& fitted & initial & fitted & $\nu \rangle \;\; $ wave function
in terms of $\vert N l j \Omega \rangle $ \\[-2mm]
{} & value & value & value & value & {} \\ \hline
{} & {} & {} & {} & {} & $\vert 19 \rangle \; \; 0.856 \vert 5h_{11/2}
\frac{1}{2} \rangle + 0.425 \vert 5 f_{5/2} \frac{1}{2} \rangle
-0.182 \vert 5 f_{7/2} \frac{1}{2} \rangle $ \\
{} & {} & {} & {} & {} & $\vert 20 \rangle \; \; 0.986 \vert 5h_{9/2}
\frac{9}{2} \rangle + 0.103 \vert 5 h_{9/2} \frac{5}{2} \rangle $ \\
band 3 & $-0.162$ & $-0.168$ & 0 & $15^{\circ}$ & $\vert 21 \rangle \; \;
0.764 \vert 5h_{9/2} \frac{7}{2} \rangle + 0.601 \vert 5 f_{7/2}
\frac{7}{2} \rangle -0.169 \vert 5 h_{9/2} \frac{3}{2} \rangle $ \\
($[505]\frac{9}{2}^{-}$) & {} & {} & {} & {} & $\vert 22 \rangle
\;\; 0.819 \vert 5h_{9/2}\frac{5}{2} \rangle + 0.385 \vert 5 h_{9/2}
\frac{1}{2} \rangle -0.210 \vert 5 f_{5/2} \frac{5}{2} \rangle $ \\
{} & {} & {} & {} & {} & $\vert 23 \rangle \; \; 0.733 \vert 5h_{9/2}
\frac{3}{2} \rangle + 0.392 \vert 5 h_{9/2} \frac{1}{2} \rangle
+0.378 \vert 5 h_{9/2} \frac{5}{2} \rangle $ \\ \hline
{} & {} & {} & {} & {} & $\vert 19 \rangle \; \; 0.995 \vert 4g_{9/2}
\frac{7}{2} \rangle $ \\
{} & {} & {} & {} & {} & $\vert 20 \rangle \; \; 0.967 \vert 4d_{5/2}
\frac{5}{2} \rangle + 0.217 \vert 4 g_{7/2} \frac{5}{2} \rangle
+ 0.130 \vert 4 g_{9/2}\frac{5}{2} \rangle $ \\
band 6 & 0.267 & 0.270 & 0 & 0 & $\vert 21 \rangle \; \;
0.951 \vert 6i_{13/2} \frac{1}{2} \rangle + 0.545 \vert 6 g_{9/2}
\frac{1}{2} \rangle $ \\
($[660]\frac{1}{2}^{+}$) & {} & {} & {} & {} & $\vert 22 \rangle
\;\; 0.945 \vert 6i_{13/2}\frac{3}{2} \rangle + 0.511 \vert 6 g_{9/2}
\frac{3}{2} \rangle $ \\
{} & {} & {} & {} & {} & $\vert 23 \rangle \; \; 0.902 \vert 4d_{3/2}
\frac{3}{2} \rangle + 0.251 \vert 4 s_{1/2} \frac{1}{2} \rangle
+0.216 \vert 5 d_{5/2} \frac{3}{2} \rangle $ \\ \hline
\end{tabular}
\label{tab1}
\end{center}
\end{footnotesize}
\end{table}
\begin{table}[htbp]
\begin{footnotesize}
\caption{The deformation parameters and the main components of the
single-particle levels $\vert \nu \rangle $ near the Fermi surface
in terms of the Nilsson levels of the bands in $^{185}$Tl (the
initial values of the deformation parameters are taken from
Ref.~\cite{Lane95}.) } \vspace*{-2mm}
\begin{center}
\begin{tabular}{|c|c|c|c|c|l|}
\hline {} & \multicolumn{2}{|c|}{$\beta$} &
\multicolumn{2}{|c|}{$\gamma$} & {} \\ \cline{2-5} band & initial
& fitted & initial & fitted & $\nu \rangle \;\; $ wave function
in terms of $\vert N l j \Omega \rangle $ \\[-2mm]
{} & value & value & value & value & {} \\ \hline
{} & {} & {} & {} & {} & $\vert 18 \rangle \; \; 0.993 \vert 5h_{11/2}
\frac{9}{2} \rangle $ \\
{} & {} & {} & {} & {} & $\vert 19 \rangle \; \; 0.872 \vert 5h_{9/2}
\frac{1}{2} \rangle - 0.420 \vert 5 f_{5/2} \frac{1}{2} \rangle
+ 0.183 \vert 5 f_{7/2} \frac{1}{2} \rangle $ \\
band 2 & 0.245 & 0.247 & 0 & $0$ & $\vert 20 \rangle \; \;
0.910 \vert 5h_{9/2} \frac{3}{2} \rangle + 0.335 \vert 5 f_{5/2}
\frac{3}{2} \rangle +0.195 \vert 5 f_{7/2} \frac{3}{2} \rangle $ \\
($[532]\frac{3}{2}^{-}$) & {} & {} & {} & {} & $\vert 21 \rangle
\;\; 0.997 \vert 5h_{11/2}\frac{11}{2} \rangle $ \\
{} & {} & {} & {} & {} & $\vert 22 \rangle \; \; 0.764 \vert 5f_{7/2}
\frac{1}{2} \rangle + 0.475 \vert 5 p_{3/2} \frac{1}{2} \rangle
+0.321 \vert 5 h_{9/2} \frac{1}{2} \rangle $ \\ \hline
{} & {} & {} & {} & {} & $\vert 19 \rangle \; \; 0.681 \vert 5h_{11/2}
\frac{1}{2} \rangle + 0.449 \vert 5 f_{5/2} \frac{1}{2} \rangle
-0.352 \vert 5 f_{7/2} \frac{1}{2} \rangle $ \\
{} & {} & {} & {} & {} & $\vert 20 \rangle \; \; 0.976 \vert 5h_{9/2}
\frac{9}{2} \rangle + 0.121 \vert 5 h_{9/2} \frac{5}{2} \rangle $ \\
band 3 & $-0.162$ & $-0.164$ & 0 & $15^{\circ}$ & $\vert 21 \rangle \; \;
0.770 \vert 5h_{9/2} \frac{9}{2} \rangle + 0.332 \vert 5 f_{7/2}
\frac{7}{2} \rangle -0.182 \vert 5 h_{9/2} \frac{3}{2} \rangle $ \\
($[505]\frac{9}{2}^{-}$) & {} & {} & {} & {} & $\vert 22 \rangle
\;\; 0.815 \vert 5h_{9/2}\frac{5}{2} \rangle + 0.394 \vert 5 h_{9/2}
\frac{1}{2} \rangle -0.221 \vert 5 f_{5/2} \frac{5}{2} \rangle $ \\
{} & {} & {} & {} & {} & $\vert 23 \rangle \; \; 0.752 \vert 5h_{9/2}
\frac{3}{2} \rangle - 0.392 \vert 5 h_{9/2} \frac{1}{2} \rangle
-0.372 \vert 5 h_{9/2} \frac{5}{2} \rangle $ \\ \hline
{} & {} & {} & {} & {} & $\vert 19 \rangle \; \; 0.998 \vert 4g_{9/2}
\frac{7}{2} \rangle $ \\
{} & {} & {} & {} & {} & $\vert 20 \rangle \; \; 0.935 \vert 4d_{5/2}
\frac{5}{2} \rangle + 0.235 \vert 4 g_{7/2} \frac{5}{2} \rangle
+ 0.151 \vert 4 g_{9/2}\frac{5}{2} \rangle $ \\
band 6 & 0.267 & 0.268 & 0 & 0 & $\vert 21 \rangle \; \;
0.941 \vert 6i_{13/2} \frac{1}{2} \rangle + 0.337 \vert 6 g_{9/2}
\frac{1}{2} \rangle $ \\
($[660]\frac{1}{2}^{+}$) & {} & {} & {} & {} & $\vert 22 \rangle
\;\; 0.952 \vert 6i_{13/2}\frac{3}{2} \rangle + 0.298 \vert 6 g_{9/2}
\frac{3}{2} \rangle $ \\
{} & {} & {} & {} & {} & $\vert 23 \rangle \; \; 0.921 \vert 4d_{3/2}
\frac{3}{2} \rangle + 0.238 \vert 4 s_{1/2} \frac{1}{2} \rangle
+0.208 \vert 5 d_{5/2} \frac{3}{2} \rangle $ \\ \hline
\end{tabular}
\label{tab2}
\end{center}
\end{footnotesize}
\end{table}
\begin{table}[htbp]
\begin{footnotesize}
\caption{The deformation parameters and the main components of the
single-particle levels $\vert \nu \rangle $ near the Fermi surface
in terms of the Nilsson levels of the bands in $^{187}$Tl (the
initial values of the deformation parameters are taken from
Ref.~\cite{Lane95}.) } \vspace*{-2mm}
\begin{center}
\tabcolsep=4.5pt
\begin{tabular}{|c|c|c|c|c|l|}
\hline {} & \multicolumn{2}{|c|}{$\beta$} &
\multicolumn{2}{|c|}{$\gamma$} & {} \\ \cline{2-5} band & initial
& fitted & initial & fitted & $\nu \rangle \;\; $ wave function
in terms of $\vert N l j \Omega \rangle $ \\[-2mm]
{} & value & value & value & value & {} \\ \hline
{} & {} & {} & {} & {} & $\vert 18 \rangle \; \; 0.997 \vert 5h_{11/2}
\frac{9}{2} \rangle $ \\
{} & {} & {} & {} & {} & $\vert 19 \rangle \; \; 0.849 \vert 5h_{9/2}
\frac{1}{2} \rangle + 0.429 \vert 5 f_{5/2} \frac{1}{2} \rangle
- 0.169 \vert 5 f_{7/2} \frac{1}{2} \rangle $ \\
band 1 & 0.250 & 0.253 & 0 & $0$ & $\vert 20 \rangle \; \;
0.909 \vert 5h_{9/2} \frac{3}{2} \rangle + 0.325 \vert 5 f_{5/2}
\frac{1}{2} \rangle +0.191 \vert 5 f_{7/2} \frac{3}{2} \rangle $ \\
($[530]\frac{1}{2}^{-}$) & {} & {} & {} & {} & $\vert 21 \rangle
\;\; 1.000 \vert 5h_{11/2}\frac{11}{2} \rangle $ \\
{} & {} & {} & {} & {} & $\vert 22 \rangle \; \; 0.741 \vert 5f_{7/2}
\frac{1}{2} \rangle + 0.480 \vert 5 p_{3/2} \frac{1}{2} \rangle
+0.327 \vert 5 h_{9/2} \frac{1}{2} \rangle $ \\ \hline
{} & {} & {} & {} & {} & $\vert 18 \rangle \; \; 0.997 \vert 5h_{11/2}
\frac{9}{2} \rangle $ \\
{} & {} & {} & {} & {} & $\vert 19 \rangle \; \; 0.868 \vert 5h_{9/2}
\frac{1}{2} \rangle + 0.410 \vert 5 f_{5/2} \frac{1}{2} \rangle
- 0.162 \vert 5 f_{7/2} \frac{1}{2} \rangle $ \\
band 2 & 0.234 & 0.237 & 0 & $0$ & $\vert 20 \rangle \; \;
0.916 \vert 5h_{9/2} \frac{3}{2} \rangle + 0.312 \vert 5 f_{5/2}
\frac{3}{2} \rangle +0.189 \vert 5 f_{7/2} \frac{3}{2} \rangle $ \\
($[532]\frac{3}{2}^{-}$) & {} & {} & {} & {} & $\vert 21 \rangle
\;\; 1.000 \vert 5h_{11/2}\frac{11}{2} \rangle $ \\
{} & {} & {} & {} & {} & $\vert 22 \rangle \; \; 0.766 \vert 5f_{7/2}
\frac{1}{2} \rangle + 0.469 \vert 5 p_{3/2} \frac{1}{2} \rangle
+0.300 \vert 5 h_{9/2} \frac{1}{2} \rangle $ \\ \hline
{} & {} & {} & {} & {} & $\vert 19 \rangle \; \; 0.695 \vert 5h_{11/2}
\frac{1}{2} \rangle + 0.447 \vert 5 f_{5/2} \frac{1}{2} \rangle
-0.360 \vert 5 f_{7/2} \frac{1}{2} \rangle $ \\
{} & {} & {} & {} & {} & $\vert 20 \rangle \; \; 0.986 \vert 5h_{9/2}
\frac{9}{2} \rangle - 0.101 \vert 5 h_{9/2} \frac{5}{2} \rangle $ \\
band 3 & $-0.162$ & $-0.162$ & 0 & $15^{\circ}$ & $\vert 21 \rangle \; \;
0.780 \vert 5h_{9/2} \frac{9}{2} \rangle + 0.348 \vert 5 f_{7/2}
\frac{7}{2} \rangle -0.169 \vert 5 h_{9/2} \frac{3}{2} \rangle $ \\
($[505]\frac{9}{2}^{-}$) & {} & {} & {} & {} & $\vert 22 \rangle
\;\; 0.825 \vert 5h_{9/2}\frac{5}{2} \rangle + 0.382 \vert 5 h_{9/2}
\frac{1}{2} \rangle -0.201 \vert 5 f_{5/2} \frac{5}{2} \rangle $ \\
{} & {} & {} & {} & {} & $\vert 23 \rangle \; \; 0.741 \vert 5h_{9/2}
\frac{3}{2} \rangle - 0.386 \vert 5 h_{9/2} \frac{1}{2} \rangle
-0.384 \vert 5 h_{9/2} \frac{5}{2} \rangle $ \\ \hline
{} & {} & {} & {} & {} & $\vert 23 \rangle \; \; 0.997 \vert 6i_{13/2}
\frac{13}{2} \rangle $ \\
{} & {} & {} & {} & {} & $\vert 24 \rangle \; \; 0.991 \vert 6i_{13/2}
\frac{11}{2} \rangle $ \\
band 5 & $-0.189$ & $-0.192$ & 0 & $11.3^{\circ}$ & $\vert 25 \rangle \; \;
0.974 \vert 6i_{13/2} \frac{9}{2} \rangle $ \\
($[606]\frac{13}{2}^{+}$) & {} & {} & {} & {} & $\vert 26 \rangle
\;\; 0.931 \vert 6i_{13/2}\frac{7}{2} \rangle + 0.298 \vert 6 i_{13/2}
\frac{3}{2} \rangle $ \\
{} & {} & {} & {} & {} & $\vert 27 \rangle \; \; 0.805 \vert 6i_{13/2}
\frac{5}{2} \rangle + 0.471 \vert 6 i_{13/2} \frac{1}{2} \rangle
+0.219 \vert 5 i_{13/2} \frac{3}{2} \rangle $ \\ \hline
{} & {} & {} & {} & {} & $\vert 19 \rangle \; \; 0.995 \vert 4g_{7/2}
\frac{7}{2} \rangle $ \\
{} & {} & {} & {} & {} & $\vert 20 \rangle \; \; 0.967 \vert 4d_{5/2}
\frac{5}{2} \rangle + 0.217 \vert 4 g_{7/2} \frac{5}{2} \rangle
+ 0.131 \vert 4 g_{9/2}\frac{5}{2} \rangle $ \\
band 6 & 0.267 & 0.265 & 0 & 0 & $\vert 21 \rangle \; \;
0.935 \vert 6i_{13/2} \frac{1}{2} \rangle + 0.334 \vert 6 g_{9/2}
\frac{1}{2} \rangle $ \\
($[660]\frac{1}{2}^{+}$) & {} & {} & {} & {} & $\vert 22 \rangle
\;\; 0.946 \vert 6i_{13/2}\frac{3}{2} \rangle + 0.303 \vert 6 g_{9/2}
\frac{3}{2} \rangle $ \\
{} & {} & {} & {} & {} & $\vert 23 \rangle \; \; 0.906 \vert 4d_{3/2}
\frac{3}{2} \rangle + 0.245 \vert 4 s_{1/2} \frac{1}{2} \rangle
+0.214 \vert 5 d_{5/2} \frac{3}{2} \rangle $ \\ \hline
\end{tabular}
\label{tab3}
\end{center}
\end{footnotesize}
\end{table}
\begin{figure}[ht]
\begin{center}
\includegraphics[scale=1.5,angle=0]{Spectrum-OddTl.eps}
\caption{Comparison of calculated energy levels of the rotational
bands in $^{183,185,187}$Tl with the experimental data (taken from
Refs.~\cite{Lane95,Muiku01}). }
\end{center}
\end{figure}
Because the experimentally observed rotational bands in $^{187}$Tl
are richer and more characteristic than those in $^{183}$Tl and
$^{185}$Tl, as a typical example, we analyze the band structure
and configuration in $^{187}$Tl in detail. To this end, we list
the total wavefunctions in terms of the single-particle orbitals
of the bands in $^{187}$Tl in Table 4.
\begin{table}[htpb]
\caption{The theoretically predicted main components of the
wavefunctions of the bands 1, 2, 3, 5 and 6 in $^{187}$Tl in terms
of the single-particle levels }
\begin{center}
\begin{tabular}{|c|l|}
\hline band & $I^{\pi}$ wavefunction $\vert \nu K \rangle $ \\
\hline
{} & $\frac{15}{2}^{-} \;\; - 0.981\vert 22 \, \frac{1}{2} \rangle
+ 0.189 \vert 19 \, \frac{1}{2} \rangle $ \\
{ } & $\frac{19}{2}^{-} \;\; - 0.981\vert 22 \, \frac{1}{2}\rangle
+ 0.190 \vert 19 \, \frac{1}{2} \rangle $ \\
band 1 & $\frac{23}{2}^{-}\;\; - 0.980\vert 22
\, \frac{1}{2} \rangle + 0.192 \vert 19 \, \frac{1}{2} \rangle $ \\
($[530]\frac{1}{2}^{-}$) & $\frac{27}{2}^{-} \;\; - 0.980\vert 22
\, \frac{1}{2}\rangle + 0.194 \vert 19 \, \frac{1}{2} \rangle $ \\
{ } & $\frac{31}{2}^{-} \;\; 0.979\vert 22 \, \frac{1}{2} \rangle
- 0.196\vert 19 \, \frac{1}{2} \rangle $ \\
{ } & $\frac{35}{2}^{-} \;\; 0.979\vert 22 \, \frac{1}{2} \rangle
- 0.197\vert 19 \, \frac{1}{2} \rangle $ \\
\hline
{ } & $\frac{17}{2}^{- }\;\; - 0.964\vert 20 \, \frac{3}{2} \rangle
+ 0.263 \vert 19 \, \frac{1}{2} \rangle $ \\
{ } & $\frac{21}{2}^{-}\;\; 0.955\vert 20 \, \frac{3}{2} \rangle
- 0.294\vert 19 \, \frac{1}{2} \rangle $ \\
band 2 & $\frac{25}{2}^{-} \;\; 0.947\vert 20 \,
\frac{3}{2} \rangle - 0.321\vert 19 \, \frac{1}{2} \rangle $ \\
($[532]\frac{3}{2}^{-}$) & $\frac{29}{2}^{-} \;\; - 0.939\vert 20
\, \frac{3}{2} \rangle + 0.343\vert 19 \, \frac{1}{2} \rangle $ \\
{ } & $\frac{33}{2}^{-}\;\; - 0.932\vert 20 \, \frac{3}{2} \rangle
+ 0.362\vert 19 \, \frac{1}{2} \rangle $ \\
{ } & $\frac{37}{2}^{-}\;\; - 0.925\vert 20 \, \frac{3}{2} \rangle
+ 0.378\vert 19 \, \frac{1}{2} > $ \\
\hline
{ } & $\frac{9}{2}^{-} \;\;\; 0.815\vert 20 \, \frac{9}{2} \rangle
+ 0.444\vert 21 \, \frac{7}{2} \rangle
+ 0.231\vert 23 \, \frac{5}{2} \rangle $ \\
band 3 & $\frac{11}{2}^{-} \;\; 0.728\vert 20 \,
\frac{9}{2} \rangle + 0.518\vert 21 \, \frac{7}{2} \rangle
- 0.299\vert 23 \, \frac{5}{2} \rangle $ \\
($[505]\frac{9}{2}^{-}$) & $\frac{13}{2}^{-}\;\; 0.500\vert 20 \,
\frac{9}{2} \rangle + 0.479\vert 21 \, \frac{7}{2} \rangle
+ 0.399\vert 23 \, \frac{5}{2} \rangle $ \\
{ } & $\frac{15}{2}^{-}\;\; -0.584\vert 20 \, \frac{9}{2} \rangle
+ 0.544\vert 21 \, \frac{7}{2} \rangle
- 0.388\vert 23 \, \frac{5}{2} \rangle $ \\
\hline
band 5 & $\frac{13}{2}^{+} \;\; 0.720\vert 23 \, \frac{13}{2}
\rangle + 0.520 \vert 24 \, \frac{11}{2} \rangle
+ 0.350 \vert 25 \, \frac{9}{2} \rangle $ \\
($[606]\frac{13}{2}^{+}$) & $\frac{15}{2}^{+} \;\; 0.580 \vert 23
\, \frac{13}{2} \rangle - 0.560\vert 24 \, \frac{11}{2} \rangle
+ 0.450\vert 25 \, \frac{9}{2} \rangle $ \\
{ } & $\frac{17}{2}^{+} \;\; - 0.620\vert 23 \, \frac{13}{2} \rangle
- 0.390\vert 27 \, \frac{5}{2} \rangle
+ 0.390 \vert 26 \, \frac{7}{2} \rangle $ \\
\hline
{ } & $\frac{17}{2}^{+} \;\; - 0.952\vert 21 \, \frac{1}{2} \rangle
- 0.298\vert 22 \, \frac{3}{2} \rangle $ \\
{ } & $\frac{21}{2}^{+} \;\; - 0.943\vert 21 \, \frac{1}{2} \rangle
- 0.326\vert 22 \, \frac{3}{2} \rangle $ \\
band 6 & $\frac{25}{2}^{+} \;\; 0.936\vert 21 \,
\frac{1}{2} \rangle + 0.347\vert 22 \, \frac{3}{2} \rangle $ \\
($[660]\frac{1}{2}^{+}$) & $\frac{29}{2}^{+}\;\; 0.930\vert 21 \,
\frac{1}{2} \rangle + 0.363\vert 22 \, \frac{3}{2} \rangle $ \\
{ } & $\frac{33}{2}^{+} \;\; - 0.924\vert 21 \, \frac{1}{2} \rangle
- 0.377\vert 22 \, \frac{3}{2} \rangle $ \\
{ } & $\frac{37}{2}^{+} \;\; 0.919\vert 21 \, \frac{1}{2} \rangle
+ 0.388\vert 22 \, \frac{3}{2} \rangle $ \\
{ } & $\frac{41}{2}^{+} \;\; 0.915\vert 21 \, \frac{1}{2} \rangle
+ 0.398\vert 22 \, \frac{3}{2} \rangle $ \\
\hline
\end{tabular}
\label{tab4}
\end{center}
\end{table}
From Table 4, we can recognize that the band 1 originates near
purely from the 22nd single particle orbital coupling with the
prolate even-even $^{186}$Hg core. Seen from Table 3, the 22nd
orbital contains mixing of 54.9{\%} $\vert 5 f_{7/2} \frac{1}{2}
\rangle $, 23.0{\%} $\vert 5 p_{3/2} \frac{1}{2} \rangle $ and
10.7{\%} $\vert 5 h_{9/2} \frac{1}{2} \rangle $ configurations.
Since the largest component is $\vert 5 f_{7/2} \frac{1}{2}
\rangle$, the band 1 can be assigned as the one arising mainly from
the configuration $[530]{\frac{1}{2}}^{-}$ ($\pi \, f_{7/2})$.
Meanwhile, from Table 4, we can see that the band 2 consists of
mixing of about 93{\%} 20th and 7{\%} 19th orbitals. Seen from Table
3, the 20th orbital contains 84.3{\%} of $\vert 5 h_{9/2}
\frac{3}{2}\rangle$ configuration. Thus, we can infer that the band
2 originates from the $[532]{\frac{3}{2}}^{-}$ configuration.
Similarly, combining Table 4 with Table 3, we can recognize that the
band 5, 6 originates mainly from the 23rd, 21st single particle
orbital, respectively, the 21st orbital contains 87.4{\%} of $\vert
6i_{13/2} \frac{1}{2} \rangle$ configuration, and the 23rd consists
of almost purely the $\vert 6 i_{13/2} {\frac{13}{2}} \rangle $
configuration. Therefore, the bands 5, 6 are based on the
configuration $\pi[606]{\frac{13}{2}}^{+}$, $\pi [660]
{\frac{1}{2}}^{+}$, respectively. The above calculated results of
the bands 1, 2, 5 and 6 are consistent with the prediction of
Ref.~\cite{Lane95}. And the configuration assignment of the bands 1
and 2 is a corroboration of that in Ref.~\cite{Lane95}. Furthermore,
the $[530]{\frac{1}{2}}^{-}$ ($f_{7/2}$) band involves about $10\%$
$h_{9/2}$ configuration and the $[532]{\frac{3}{2}}^{-}$ ($h_{9/2}$)
band involves only about $4\%$ $f_{7/2}$ configuration. On the other
hand, similar results are obtained for the corresponding bands in
$^{183,185}$Tl.
\begin{figure}[ht]
\begin{center}
\includegraphics[scale=1.0,angle=0]{Band3Tl187.eps}
\caption{Upper panel: calculation error $\chi^{2}$ against the
deformation parameter $\gamma$ at several axial deformation
parameter $\beta$'s of the band 3 in $^{187}$Tl. Lower panel:
variation of the calculated energy spectrum of the band 3
in $^{187}$Tl with fixed $\beta = -0.162$ against the value of
$\gamma$ and comparison with experimental data.
The experimental data are taken from Ref.\cite{Lane95}. }
\end{center}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[scale=1.0,angle=0]{Band3Tl183.eps}
\caption{Upper panel: calculation error $\chi^{2}$ against the
deformation parameter $\gamma$ at several axial deformation
parameter $\beta$'s of the band 3 in $^{183}$Tl. Lower panel:
variation of the calculated energy spectrum of the band 3
in $^{183}$Tl with fixed $\beta = -0.168$ against the value of
$\gamma$ and comparison with experimental data.
The experimental data are taken from Ref.\cite{Muiku01}. }
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=1.0,angle=0]{Band3Tl185.eps}
\caption{Upper panel: calculation error $\chi^{2}$ against the
deformation parameter $\gamma$ at several axial deformation
parameter $\beta$'s of the band 3 in $^{185}$Tl. Lower panel:
variation of the calculated energy spectrum of the band 3 in
$^{185}$Tl with fixed $\beta = -0.164$ against the value of
$\gamma$ and comparison with experimental data. The experimental
data are taken from Ref.~\cite{Lane95}. }
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=1.0,angle=0]{Band5Tl187.eps}
\caption{Upper panel: calculation error $\chi^{2}$ against the
deformation parameter $\gamma$ at several axial deformation
parameter $\beta$'s of the band 5 in $^{187}$Tl. Lower panel:
variation of the calculated energy spectrum of the band 5 in
$^{187}$Tl with fixed $\beta = -0.192$ against the value of
$\gamma$ and comparison with experimental data. The experimental
data are taken from Ref.~\cite{Lane95}. }
\end{center}
\end{figure}
In order to investigate the deformation nature of band 3 in
$^{183,185,187}$Tl, we illustrate at first the calculation error
$\chi^2 = \frac{1}{N} \sum_{j} (E^{cal.}_{j} - E^{exp.}_{j} )^2 $
of the energy spectrum of band 3 in $^{187}$Tl (where $N$ is the
number of levels in the band) with respect to the value of
$\gamma$ at several $\beta$'s in the upper panel of Fig.~2. We
also display the variation of the calculated energy spectrum
against the value of $\gamma$ at the best fitted $\beta \,
(-0.162)$ and the comparison with experimental data of the band in
the lower panel of Fig.~2. The upper panel of Fig.~2 shows that
the variation of the axial deformation parameter $\beta$ (except
for that with angular deformation parameter $\gamma$ in special
region) does not affect the calculation error $\chi ^{2}$ so
drastically as that of the $\gamma$ does. Combining the upper
panel and the lower panel of Fig.~2, one can notice clearly that,
for zero $\gamma$, the calculation error $\chi ^{2}$ is quite
large (about 60) and the calculated level sequence is not
consistent with experiments. As the $\gamma$ increases to 3-5
degrees, the calculated level sequence becomes consistent with the
experimental one and the $\chi ^{2}$ decreases to about $10^{-1}$.
For the value of $\gamma$ in the region 3 to 14 degrees, the
calculation error $\chi ^{2}$ maintains around $10^{-1}$. When
$\gamma = 15 ^{\circ}$, the $\chi^{2}$ with $\beta = - 0.162$
becomes suddenly the minimum ($\sim 10^{-4}$) of the $\chi^{2}
(\beta , \gamma)$ and the calculated energy spectrum agrees with
experimental data very well. As $\gamma$ increases from 15 degrees
further, the $\chi^{2}$ increases to around $10 ^{-3}$, even to
$10^{-1}$. Moreover, in the case of $\beta = -0.162$, even though
the calculated energies of the states with lower angular momentum
do not deviate from experimental data obviously, the ones with
higher angular momentum do drastically. It is then evident that,
when the deformation parameters $(\beta,\gamma) = (-0.162,
15^{\circ})$, the calculated energy spectrum agrees best with
experimental data. It indicates that the band 3 of $^{187}$Tl is
in triaxial oblate deformation. In addition, from Table 4, we
notice that the band 3 in $^{187}$Tl originates mainly form the
20th single particle orbital. As can be seen from inspecting Table
3, the 20th orbital contains 97.2{\%} of $\vert 5 h_{9/2}
\frac{9}{2} \rangle$ configuration. Therefore, the band 3 can be
identified as the one arising from the proton configuration
$[505]{\frac{9}{2}}^{-}$ ({$\pi \,h_{9/2})$ coupled to a triaxial
oblate deformed core. It provides then a corroboration of the
conjecture in Ref.~\cite{Lane95}. Similar results for the bands 3
in $^{183,185}$Tl are obtained, too (the calculation errors
$\chi^{2}$ of the energy separations against the value of $\gamma$
at several $\beta$'s and the comparison of the calculated energy
spectrum with $\gamma \in (0^{\circ}, 29^{\circ})$ and $\beta =
-0.168$ ($-0.164$) with experimental data are illustrated in
Fig.~3 (4) for $^{183}$Tl ($^{185}$Tl) ). The deformation
parameters can then be fixed as $(-0.168, 15^{\circ})$, $(-0.164,
15^{\circ})$ for the band~3 of $^{183}$Tl, $^{185}$Tl,
respectively. These results confirm the assumption that the band
originated from orbital $[505]{\frac{9}{2}}^{-}$ ($\pi \,
h_{9/2})$ may be in triaxial oblate deformation\cite{Lane95}.
Besides, from the calculation error $\chi ^{2}$ and the comparison
between the obtained energy spectrum with $\beta = -0.192$ and the
experimental data of the band $[606]\frac{13}{2}^{+}$ shown in
Fig.~5, one can recognize that the band~5 ($[606]\frac{13}{2}^{+}$
band) is also a triaxial oblate deformation band (with $(\beta ,
\gamma ) = (-0.192, 11.3^{\circ})$ ).
From the calculations one may recognize that the deformation parameter
$\gamma$ influences the results more drastically than the axial
deformation parameter $\beta$. Analyzing the obtained single
particle configuration we know that the states in the bands
$[505]\frac{9}{2}^{-}$ and $[606]\frac{13}{2}^{+}$ involve much more
complicated single particle Nilsson configurations than the other
bands. Such a result is consistent with that once given for nucleus
$^{127}$I in Ref.~\cite{SLZ04}. We infer then that the reason for
the $\gamma$-degree of freedom to play more important role than
$\beta$ may be that it induces more obvious mixing among the single
particle Nilsson configurations.
\section{Conclusion and Remarks}
In summary, we have systemically calculated the energy spectra, the
deformations and wavefunctions of the rotational bands in nuclei
$^{183,185,187}$Tl in the particle triaxial-rotor model with
variable moment of inertia. The calculated energy spectra of the
bands agree quite well with the experimental data. The configuration
of the bands in $^{187}$Tl is analyzed in detail as an example.
Meanwhile we have also calculated the variation of the configuration
of single-particle levels against the deformation parameters $\beta$
and $\gamma $. Considering both the parameters fitted and the
agreement between calculated results and experimental data, we
conclude that the rotation-aligned band structures observed in
$^{183,185,187}$Tl are due to one of the $[530]{\frac{1}{2}}^{-}$,
$[532]{\frac{3}{2}}^{-}$, $[660]{\frac{1}{2}}^{+}$ proton
configurations coupled to a prolate deformed core. Meanwhile, the
negative parity bands built upon the ${\frac{9}{2}}^{-}$ isomeric
states in $^{183,185,187}$Tl are formed by a proton with the
$[505]{\frac{9}{2}}^{-}$ configuration coupled to a core with
triaxial oblate deformation $(\beta , \gamma) = (-0.168,
15^{\circ}), \, (-0.164, 15^{\circ}), \, (-0.162, 15^{\circ})$,
respectively, and the positive parity band on the
${\frac{13}{2}}^{+}$ isomeric state in $^{187}$Tl is generated by a
proton with configuration $[606]{\frac{13}{2}}^{+}$ coupled to a
triaxial oblate core with deformation parameters $(\beta , \gamma) =
(-0.192 , 11.3^{\circ})$. In short, the nuclei $^{183,185,187}$Tl
involve quite rich shape coexistence.
Meanwhile our present calculation provides a clue that the triaxial
deformation may arise from the mixing of single particle Nilsson
configurations. To understand it much better, more investigations
are required.
\bigskip
This work was supported partially by the Natural Science Foundation
of Guangdong Province with contract No. 04011642, partially by the
Natural Science Research Foundation of the Education Department of
Guangdong Province with contract No. Z02069, and partially by the
National Natural Science Foundation of China with contract Nos.
10425521, 10135030 and 10075002. One of the authors (YXL) thanks
also the support by the Major State Basic Research Development
Program under Grant No. G2000077400, the Key Grant Project of
Chinese Ministry of Education (CMOE) under contract No.~305001, the
Foundation for University Key Teacher by the CMOE and the Research
Fund for the Doctoral Programme of Higher Education of China with
grant No. 20040001010.
|
train/arxiv
|
BkiUbfQ4dbjiU5gJucHg
| 5
| 1
|
\section{Introduction}
A real number is normal to an integer base if, in its infinite expansion in
that base, all blocks of digits of the same length have the same limiting
frequency. Émile Borel~\cite{Borel09} defined normality more than one
hundred years ago to formalize the most basic form of randomness for real
numbers. Many of his questions are still open, such as whether any of
$\pi, e$ or~$\sqrt{2}$ is normal in some base, as well as his conjecture
that the irrational algebraic numbers are normal to each
base~\cite{Borel50}. This motivates the search for new characterizations
of the concept of normality.
One characterization is based on finite state machines. A sequence of
digits is normal if and only if it cannot be compressed by lossless finite
transducers (also known as finite-state compressors). These are
deterministic finite automata augmented with an output tape with injective
input-output behavior. The compression ratio of an infinite run of a
transducer is defined as the $\liminf$, over all its finite prefixes, of
the ratio between the number of symbols written and the number of symbols
read so far. A given sequence is said to be compressed by a given
transducer if the compression ratio it achieves is less than~$1$.
A direct proof of the incompressibility characterization of normal
sequences can be found in~\cite{BecherHeiber13}. However, the result was
already known, although by indirect and more involved arguments. For
instance, combining results of Schnorr and Stimm~\cite{Schnorr71} and Dai,
Lathrop, Lutz and Mayordomo~\cite{Dai04} yields an earlier proof.
The notion of incompressibility by finite state machines is quite robust:
adding some feature to one-to-one transducers does not allow them to
compress normal sequences. It is proved in \cite{BecherCartonHeiber15}
that non-deterministic non-real-time transducers, with no extra memory or
just a single counter, cannot compress any normal sequence. Non-real-time
means here that the value of the counter can be incremented and decremented
without consuming any input symbol. It is also shown in
\cite{CartonHeiber15} that two-way transducers cannot compress normal
sequences. Adding too much memory yields compressibility results:
non-real-time transducers with at least two counters are Turing complete
and thus can compress some normal sequence. Combining non-determinism with
a single stack yields also compressibility of some normal sequence.
Results given in~\cite{BecherCartonHeiber15} are summarized in
Table~\ref{tbl:compress}. The only question left open was whether a
deterministic pushdown transducer can compress a normal sequence, that is
the question mark in Table~\ref{tbl:compress}. In this paper, we answer
this question positively.
\begin{table} \label{tbl:compress}
\begin{center}
\begin{tabular}{|l|ccc|} \hline
Finite-state transducer & det. & non-det. & non-real-time \\ \hline
No extra memory & N & N & N \\ \hline
One counter & N & N & N \\ \hline
More than one counter & N & N & Y \\ \hline
One stack & \textbf{?} & Y & Y \\ \hline
One stack and one counter & Y & Y & Y \\ \hline
\end{tabular}
\end{center}
\caption{Compressibility by different kinds of transducers.}
\end{table}
\begin{theorem} \label{thm:informal}
There is a deterministic pushdown transducer that can compress some
normal sequence.
\end{theorem}
A more precise statement is given in Proposition~\ref{pro:formal} where
the pushdown transducer and the normal sequence compressed by it are made
explicit.
\section{Precise statement}
Before giving a more precise statement, we recall a few definitions. Let
$A$ be a finite alphabet. Let $A^*$ and $A^ℕ$ be respectively the set of
finite words and the set of (infinite) sequences over~$A$. The positions of
words and sequences are numbered starting at~$1$. To denote the symbol at
position~$i$ of a word (respectively sequence) $w$ we write $w[i]$ and to
denote the substring of $w$ from position~$i$ to~$j$ we write $w[i{:}j]$.
The length of a finite word~$w$ is denoted by~$|w|$. The empty word is
denoted by~$\emptyword$. For a word $w = a_1 ⋯ a_n$, let $\tilde{w}$ be
the \emph{reverse} of~$w$ defined by $\tilde{w} = a_n ⋯ a_1$. We write
$\#E$ for the cardinality of a finite set~$E$. For $w$ and $u$ two words,
let us denote by $|w|_u$ the number of \emph{occurrences} of~$u$ in~$w$. A
sequence~$x ∈ A^ℕ$ over alphabet~$A$ is \emph{normal} if
\begin{displaymath}
\lim_{n→∞}\frac{|x[1{:}n]|_w}{n} = \frac{1}{(\#A)^{|w|}}
\end{displaymath}
holds for each word $w ∈ A^*$.
A \emph{pushdown transducer} is made of input and output alphabets $A$
and~$B$, a stack alphabet~$Z$ containing the starting symbol~$z_0$, a
finite state set~$Q$ containing the initial state~$q_0$ and a finite set of
transitions of the form $p,z \trans{a|v} q,h$ where $p,q∈ Q$, $a ∈ A$, $v ∈
B^*$, $z ∈ Z$ and $h ∈ Z^*$. The states $p$ and~$q$ are the starting and
ending states of the transition. The symbol~$a$ and the word~$v$ are its
input and output labels. The stack symbol~$z$ and the word~$h$ are
respectively the symbol popped from the stack and the word pushed to the
stack. Note that the transition $p,z \trans{a|v} q,h$ replaces the top
symbol~$z$ by the word~$h$. If $h$ is empty, it just pops the symbol~$z$.
A configuration~$C$ of the transducer is a pair $\tuple{q,h}$ where $q ∈ Q$
is its state and $h ∈ Z^*$ is its stack content. Note that the stack
content is written bottom up: the top symbol is the last symbol of~$h$.
The starting configuration is the pair $\tuple{w_0,z_0}$ where $q_0$ is the
initial state and $z_0$ the starting symbol.
A \emph{run step} is a pair of configuration $\tuple{C,C'}$ denoted
$C \trans{a|v} C'$ such that $C = \tuple{p, wz}$, $C' = \tuple{q,wh}$ for
some word $w ∈ Z^*$ and $p,z \trans{a|v} q,h$ is a transition of the
transducer. A finite (respectively infinite) \emph{run} is a finite
(respectively infinite) sequence of consecutive run steps
\begin{displaymath}
C_0 \trans{a_1|v_1} C_1 \trans{a_2|v_2} \cdots \trans{a_n|v_n} C_n.
\end{displaymath}
The input and output labels of the run are respectively $a_1 ⋯ a_n$ and
$v_1 ⋯ v_n$. Note that a transition $p,z \trans{a|v} q,h$ can be seen as a
run step whose starting stack content is reduced to a single symbol~$z$.
Conversely, each run step is obtained from a transition
$p,z \trans{a|v} q,h$ by adding a stack content~$w$ below the top
symbol~$z$.
Let $A$ be the alphabet $\{0, …, k-1\}$ for some positive integer~$k$ and
let $B$ be the alphabet $A ⊎ \{\extraone, \extratwo\}$ where $\extraone$
and $\extratwo$ are two new symbols not in~$A$. Now we give the
deterministic pushdown transducer~$𝒯_k$ with input alphabet~$A$ and output
alphabet~$B$. We first describe it informally and second we give a more
formal description of its transitions. The transducer $𝒯_k$ proceeds as
follows whenever it reads a symbol~$a ∈ A$ from the input tape. If the
symbol~$a$ is different from the top symbol of the stack, the symbol~$a$ is
pushed onto the stack and it is also written to the output tape. If the
symbol~$a$ is equal to the top symbol of the stack, this top symbol is
popped. Every two symbols consecutively popped from the stack, a
symbol~$\extratwo$ is written to the output tape. An additional
symbol~$\extraone$ is also written to the output tape if the run of popped
symbols is of odd length. In other words, after a run of $n$ consecutive
pops, is written to the output tape either the word $\extratwo^{n/2}$ if
$n$ is even or the word $\extratwo^{(n-1)/2}\extraone$ if $n$ is odd. This
coding of the length~$n$ is far from being optimal but it is sufficient to
get compression. More formally the state set of~$𝒯_k$ is $Q = \{0, 1\}$.
Its stack alphabet is $A ⊎ \{⊥\}$ and the start symbol~$z_0$ is the new
symbol~$⊥$. As the symbol~$⊥$ is different from any input symbol, it is
never popped from the stack. Therefore, the symbol~$⊥$ always remains at
the bottom of the stack and it is used to mark it. The transitions set~$E$
of~$𝒯_k$ is defined as follows.
\begin{alignat*}{2}
E = & \{ 0,z \trans{a|a} 0,za : z ≠ a \}
& \qquad & \text{Pushing $a$ and outputting $a$} \\
& \{ 0,z \trans{a|\emptyword} 1,\emptyword : z = a\}
& \qquad & \text{Popping $z = a$ and outputting $\emptyword$} \\
& \{ 1,z \trans{a|\extratwo} 0,\emptyword : z = a\}
& \qquad & \text{Popping $z = a$ and outputting $\extratwo$} \\
& \{ 1,z \trans{a|\extraone a} 0,za: z ≠ a \}
& \qquad & \text{Pushing $a$ and outputting $\extraone a$}
\end{alignat*}
The function realized by this transducer is obviously one-to-one and the
inverse function can even be computed by another deterministic pushdown
transducer.
Let $𝒯$ be a pushdown transducer with input alphabet~$A$ and output
alphabet~$B$. The \emph{compression ratio}~$ρ$ of an infinite run
\begin{displaymath}
C_0 \trans{a_1|v_1} C_1 \trans{a_2|v_2} C_2 \trans{a_3|v_3} \cdots
\end{displaymath}
is
\begin{displaymath}
ρ = \liminf_{n→∞}\frac{|v_1 ⋯ v_n|\log\#B}{n\log\#A}
\end{displaymath}
The factors $\log\#A$ and $\log\#B$ take into account the alphabet sizes.
Without them, it would be to easy to compress by taking a larger
alphabet~$B$. The transducer~$𝒯$ is said to \emph{compress} a sequence~$x$
if it realizes a one-to-one function and if there is an infinite run in~$𝒯$
with input label~$x$ and whose compression ratio~$ρ$ satisfies $ρ < 1$.
The following proposition is a more precise reformulation of
Theorem~\ref{thm:informal}.
\begin{proposition} \label{pro:formal}
Let $A$ be the alphabet $\{0, …, k-1\}$ for some great enough integer~$k$.
Let $w_n$ be, for each integer $n ⩾ 1$, the concatenation in some order
of all words of length~$n$ over~$A$. The deterministic pushdown
transducer~$𝒯_k$ given above compresses the normal sequence
$x = w_1\tilde{w}_1w_2\tilde{w}_2w_3\tilde{w}_3⋯$.
\end{proposition}
Before proving the proposition, we make some comments. The proof that the
sequence~$x$ is normal is an easy adaptation that the Champernowne sequence
is normal \cite[Thm~7.7.1]{BecherCarton18}.
The proposition states the result for $k$ great enough. The proof below
shows that the condition $\log(k+2)/\log k < 20/19$ is sufficient but
numerical experiments show that $k ⩾ 5$ is actually sufficient.
Some other normal sequences are compressible by the same transducer. For
each integer $n ⩾ 1$, let $u_1,…,u_{ℓ_n}$ be an enumeration in some order
of all words of length~$n$ over~$A$. This means that $ℓ_n = (\#A)^n$. Let
$w_n$ be the word $u_1\tilde{u}_1u_2\tilde{u}_2⋯ u_{ℓ_n}\tilde{u}_{ℓ_n}$
for each integer $n ⩾ 1$. The sequence $x = w_1w_2w_3⋯$ is also
compressible by the same transducer~$𝒯_k$. It seems that this result can be
proved using the same techniques. However, our numerical experiments
suggest that the compression ratio of this latter sequence is worse than
the one given in the proposition.
Our numerical experiments show that the compression ratio converges to~$3/4$
when the alphabet size~$k$ goes to infinity. It seems that the same ideas used
in the proof of the proposition can achieve this result, but we preferred
simplicity in our presentation.
\section{Proof}
Now we introduce a congruence~$∼$ on~$A^*$ which is used to characterize
stack contents of the pushdown transducer~$𝒯_k$. Let $→$ be the relation
defined on~$A^*$ as follows. Two words $w$ and~$w'$ satisfy $w → w'$ if
there are two words $u$ and~$v$ and a symbol $a ∈ A$ such that $w = uaav$
and $w' = uv$. A word~$w$ is \emph{irreducible} for~$→$ if it contains no
consecutive occurrences of the same symbol. Let $\trans{*}$ be the
transitive and reflexive closure of the relation~$→$. It is clear that
this relation is confluent and terminating. These two properties imply
that for each word~$w$, there is a unique irreducible word~$\bar{w}$ such
that $w \trans{*} \bar{w}$. Let $∼$ be the congruence generated by the set
of pairs $\{ (\emptyword, aa) : a ∈ A \}$. Two words satisfy $w ∼ w'$ if
and only if they both reduce to the same irreducible word, that is
$\bar{w} = \bar{w}'$. Note that each palindrome of even length, that is
each word of the form~$w\tilde{w}$, satisfies $w\tilde{w} ∼ \emptyword$.
The following lemma is easily proved by induction on the length of~$w$.
\begin{lemma} \label{lem:stack}
After reading a word~$w$, the stack content of~$𝒯_k$ is $⊥\bar{w}$ where
$\bar{w}$ is the unique irreducible word such that $w \trans{*} \bar{w}$.
\end{lemma}
Let us recall that the input sequence is
$w_1\tilde{w}_1w_2\tilde{w}_2w_3\tilde{w}_3⋯$. The lemma just stated above
implies that the stack only contains the bottom symbol~$⊥$ after reading
the prefix $w_1\tilde{w}_1⋯ w_n\tilde{w}_n$ because $w_i\tilde{w}_i ∼
\emptyword$ for each integer $i ⩾ 1$.
Let $P = \{1,…,|w_n\tilde{w}_n|\}$ be the set of positions of symbols in
$w_n\tilde{w}_n$. Each symbol of $w_n\tilde{w}_n$ is consumed by either a
pushing transition or a popping transition. In the former case, the
consumed symbol is pushed to the stack. In the latter case, the same
symbol as the one consumed is popped from the stack. This dichotomy
induces the partition $P = P_0 ⊎ P_1$ where $P_0$ is the set of positions
of symbols being pushed and $P_1$ is the set of positions of symbols
popping. Since the stack only contains the bottom symbol~$⊥$ before and
after reading $w_n\tilde{w}_n$, each pushed symbol is popped later. Then,
the run of~$𝒯_k$ also induces a function $f$ from~$P_0$ to~$P_1$ which maps
each position of a pushed symbol to the position of the symbol that pops
it. This function~$f$ is of course, one-to-one and onto because each
pushed symbol is popped by exactly one symbol. By definition, the
function~$f$ satisfies that $i < f(i)$ for each $i$ in~$P_0$ and that the
symbols at positions $i$ and $f(i)$ are the same. The stack policy implies
that if two positions $i$ and~$j$ in~$P_0$ satisfy $i < j$, then $f(i) >
f(j)$. Let us call an \emph{edge} a pair $(i,f(i))$. An edge is
respectively called a \emph{short} edge (respectively \emph{long}) edge if
$f(i) - i = 1$ (respectively $f(i) - i > 1$).
\begin{figure}[htp]
\begin{center}
\begin{tikzpicture}[yscale=1.4]
\foreach \x/\xtext in {0/0,0.25/0,0.5/0,0.75/1,1/2,1.25/2,1.5/1,1.75/1}
\node[anchor=base] at (\x,0) {$\xtext$};
\foreach \x/\xtext in {0/1,0.25/2,0.5/3,0.75/4,1/5,1.25/6,1.5/7,1.75/8}
\node[anchor=base] at (\x,-0.2) {$\scriptscriptstyle\xtext$};
\foreach \x/\xtext in {2.0/1,2.25/1,2.5/2,2.75/2,3/1,3.25/0,3.5/0,3.75/0}
\node[anchor=base] at (\x,0) {$\xtext$};
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\node[anchor=base] at (\x,-0.2) {$\scriptscriptstyle{\xtext}$};
\path[->] (0.00,-0.25) edge[bend left=-50] (0.25,-0.25);
\path[->] (0.50,-0.25) edge[bend left=-30] (3.25,-0.25);
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\end{tikzpicture}
\end{center}
\caption{Example of a function $f$: $f(3) = 14$.}
\end{figure}
Let us call a \emph{block} a maximal set $\{i, i+1, …, j\}$ of consecutive
positions with the same symbol at each position. Maximal means here that
the set cannot be expanded to the left because either $i = 1$ or symbols at
positions $i-1$ and~$i$ are different and that it cannot be expanded to the
right because either $j = |w_n\tilde{w}_n|$ or symbols at positions $j$
and~$j+1$ are different. The following lemma states a link between the
number of long edges and the length of the output of~$𝒯_k$.
\begin{lemma} \label{lem:output}
While reading $w_n\tilde{w}_n$, the transducer~$𝒯_k$ writes at most
$|w_n\tilde{w}_n| - h/4$ symbols where $h$ is the number of blocks of
length~$1$ in $w_n\tilde{w}_n$.
\end{lemma}
\begin{proof}
Let $d$ be the difference between the length of $w_n\tilde{w}_n$ and the
number of symbols written by~$𝒯_k$ while reading $w_n\tilde{w}_n$. We have
to prove that $d ⩾ h/4$. Each symbol pushed to the stack by~$𝒯_k$ is also
written to the output tape. For $n$ consecutive pops, $𝒯_k$ writes
$\extratwo^{n/2}$ if $n$ is even and $\extratwo^{(n-1)/2}\extraone$ if
$n$ is odd. This shows that a pair of consecutive pops contributes $1$
to~$d$. Furthermore $k$ disjoint pairs of consecutive pops contribute $k$
to~$d$.
For each block of length~$1$, there is a long edge $(i,f(i))$ such that
either $i$ or $f(i)$ belongs to the block. This shows that the number of
long edges is at least $h/2$. Due to the nesting of edges, the position
to the left is also the arrival of another edge which can be either short
or long. Note that these pairs of consecutive pops might be not
disjoint. At most half of the pairs have to be removed to make the
remaining pairs disjoint. This completes the proof that $d ⩾ h/4$.
\end{proof}
\begin{lemma} \label{lem:oddblocks}
The number of blocks of length~$1$ in $w_n\tilde{w}_n$ is greater than
$|w_n\tilde{w}_n|/5$ for $n$ great enough.
\end{lemma}
\begin{proof}
The number of words $abc ∈ A^3$ such that $a ≠ b$ and $b ≠ c$ is
$k(k-1)^2$. Therefore, the symbol~$b$ is a block of length~$1$ with
probability $(k-1)^2/k^2$ which is greater or equal to~$1/4$ for each~$k
⩾ 2$. The average number of blocks of length~$1$ in a random word of
length~$n$ is thus greater than $n/4$.
Now in $w_n\tilde{w}_n$, which is twice the concatenation of all $k^n$
words of length $n$: the total number of blocks of length~$1$ is greater
than $nk^n/2 - 4k^n$ where the second term $4k^n$ takes into account the
blocks that can be lost while concatenating two words of length~$n$. For
$n$ great enough, this is larger than $|w_n\tilde{w}_n|/5$.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{pro:formal}]
Combining Lemmas \ref{lem:output} and~\ref{lem:oddblocks} yields that,
for $n$ great enough, the number of symbols written by~$𝒯_k$ while
reading $w_n\tilde{w}_n$ is at most $19|w_n\tilde{w}_n|/20$. For $k$
great enough, the value of $\log(k+2)/\log k$ is less that $20/19$ and
therefore the transducer~$𝒯_k$ given above compresses the normal sequence
$x = w_1\tilde{w}_1w_2\tilde{w}_2w_3\tilde{w}_3⋯$.
\end{proof}
\bibliographystyle{plain}
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\subsubsection*{Other proofs}
\enlargethispage{\baselineskip}
\textbf{Preparata and Shamos} provide another proof that it takes $\Omega(n \log n)$ steps to decide whether a set of points is in convex position (Theorem 3.3 in Section 3.2 in their book~\cite{PS}). Their proof is based directly on the lower bounds for membership tests for a set $W$ in high-dimensional space with $n!$ connected components---the same lower bounds that also underly the lower bounds for $\eps$-closeness that we used above. The core of their proof is therefore an analysis of the number of connected components of $W$, the subset of possible inputs that corresponds to sets of points in convex position (\cite{PS}, p102--103). The key argument considers pairs of inputs in convex position that must lie in different components, because they cannot be continuously transformed into each other without passing through a configuration in which three points are collinear\footnote{The reader who wants to verify the details of the proof should beware of minor typing mistakes that make it look as if permutations on $N$ integers are applied to integers from 0 to $2N-1$ or from 0 to $N^2-1$.}. Note that a triple of collinear points immediately rules out convex position, but it is does not imply that one of the points lies in the interior of the convex hull (they could all lie on the edge). Thus, the proof by Preparata and Shamos is a bit more specific than ours: they prove our Theorem~\ref{thm:convexposition}, but not our Theorem~\ref{thm:boundary}.
\textbf{Kirkpatrick and Seidel}~\cite{KS} prove the following (Theorem 5.3): given a set $S$ of $n$ distinct points and a natural number $h \leq n$, any fixed-order algebraic decision tree algorithm requires, in the worst case, $\Omega(n \log h)$ steps to verify that the convex hull of $S$ has $h$ vertices. When $h$ is polynomial in $n$, this bound is equivalent to $\Omega(n \log n)$. Clearly, if we could compute in $o(n \log n)$ time which points of $S$ are vertices of the convex hull, then we could also count them in $o(n \log n)$ time. Thus, the theorem by Kirkpatrick and Seidel implies an $\Omega(n \log n)$ lower bound for the problem of computing the vertices of the convex hull.
Like the proof of our Lemma~\ref{lem:catching-identical-points}, the proof by Kirkpatrick and Seidel uses a reduction from a problem on a multiset $A$ of real numbers, namely the problem of verifying the number of distinct numbers in the set (in the proof of Lemma~\ref{lem:catching-identical-points}, the problem is to verify that \emph{all} numbers of the set are distinct). To this end, each number $a_i$ in $A$ is mapped to a point on a parabola and perturbed by moving it over a distance that grows with $i$. Thus, whenever there are $k > 1$ points that represent the same number, these points all become distinct; moreover, the perturbations are carefully chosen such that $k-1$ of these points are no longer a vertex of the convex hull. The perturbations could be implemented by adapting all nodes that evaluate the coordinates of the input points in the decision tree, so that they evaluate the perturbed coordinates instead of the original coordinates. The perturbations must be small enough, so that points that were distinct already have no effect on each other---in our proof of Lemma~\ref{lem:catching-identical-points}, we ensure this by limiting the perturbation in the horizontal direction to $\eps/2$.
Of course, smaller perturbations would also work and lead to the same end result. This is where Kirkpatrick and Seidel use a clever trick: rather than using perturbations of a fixed size, they describe how to adapt the decision tree (making it only slightly higher) so that it effectively takes the same decisions that it would take with \emph{any} small enough perturbations. Thus, Kirkpatrick and Seidel can realize a reduction from the multiset size verification problem, where no prespecified difference threshold $\eps$ can be used to determine the size of the perturbations.
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\section{Introduction}
The assumption that the turbulent stress is proportional to
pressure in accretion disks (the $\alpha$-model) is a
fundamental ingredient of classic
accretion disk theory (Shakura \& Sunyaev
1973). It has been partially justified by
successful application to quasi-steady systems, such as the thermal
spectrum of dwarf novae and X-ray binaries
(e.g.\ Warner 1995, Gierli\'{n}ski \& Done 2004);
but when
applied to more delicate time-dependent dynamics,
such as instabilities, the model has encountered difficulties.
For instance, the modelling of dwarf novae outbursts requires
different alphas for the high and low states (Smak 1984).
Another example involves
radiation-pressure dominated accretion flows which the alpha model predicts
are subject to
thermal and viscous instability
(Shakura \&
Sunyaev 1976, Lightman \& Eardley 1974). X-ray observations, however, fail to find
variability on the timescales expected
(Gierli\'{n}ski \& Done 2004), with only the exceptional luminous source
GRS 1915+105 and the intermediate black hole HLX-1 exhibiting
anything like cyclic behaviour driven by thermal instability (Belloni
et al.~1997, Done
et al.~2004, Sun et al.~2016, Wu et al.~2016). While it is possible
the disks are stabilised by an additional but unknown
cooling mechanism, it may be that the heating
depends on temperature in a weaker way than the alpha model
assumes. For example, the turbulent stresses may be proportional to gas
pressure rather than total pressure, or not on pressure at all
(Gierlinski \& Done 2004).
Of course, one can bypass the alpha model (and its assumptions) and
simulate the MHD turbulence in these disks self-consistently, the
turbulence then supplied directly by the magnetorotational instability (MRI, Balbus
\& Hawley 1991).
And in fact early work indicated that radiation-pressure dominated flows were
thermally stable (Hirose et al.~2009), in agreement with most
observations.
However, recent local and global simulations \emph{do} exhibit thermal runaways
(Jiang et al.~2013, Sadowski 2016, Mishra et al.~2016),
though these expose additional complications that may
suppress instability, such as
numerical effects (especially box size) and the impact of a mean
magnetic flux.
Obviously, both observations and simulations indicate that the onset and
development of thermal instability is far less straightforward than
predicted by the classical laminar theory.
One very clear complication is the fact that the
stress-pressure relationship can change,
for both numerical and physical reasons, yielding
instability or stability depending on conditions in the disk and
in the simulation.
A quite separate issue is the assumption that thermal instability can
be appropriately defined at all, at least when dealing with quasi-steady
turbulent states. The alpha-theory treats the turbulence as a static
eddy viscosity, and hence the equilibrium state as laminar. However, if the state hosts
vigorous fluctuations it may not be well-defined, or even make sense,
to add a small linear
perturbation on top the stochastic background field
and subsequently
calculate
a growth rate.
One
envisages that, at the very least, non-model, non-exponential behaviour
ensues. Indeed, Jiang et al.~(2013) report delayed runaway and algebraic
growth rather than exponential growth in their MHD simulations, while
Janiuk \& Mishra (2012) show via a stochastic 1D model
that fluctuations induce random luminosity variations rather than the
regular outbursts expected.
It is to this aspect of the problem that this paper is devoted,
focussing
on the constructive and destructive interference of turbulence
on thermal instability.
In order to isolate the essence of the problem
we employ an idealised model of MRI turbulence and of
thermal instability. Unstratified shearing box simulations are performed
using the code RAMSES on a state that is MRI turbulent and thermally
unstable (at least according to the laminar alpha theory). Note that radiation pressure
is omitted and the gas cools due to a simple cooling function.
We find that the turbulent fluctuations induce
thermal behaviour substantially different to that expected from
laminar theory.
In
particular, the evolution of the temperature resembles more a biased random
walk than an exponential runaway, with
a wide range of trajectories possible: the temperature in some simulations
departs from the laminar equilibrium relatively rapidly,
whereas in others it can `hang around' for several thermal times.
This motivates a probabilistic interpretation of instability, and we
develop a simple statistical framework based on
the model of geometric Brownian
motion.
A key idea is that of the `escape time' $t_{\text{esc}}$ (which replaces the e-folding
time). It describes how long it takes for the system to deviate
significantly from the equilibrium. Reduced models involving both
white noise and the power spectrum of the MRI show that the
probability distribution of $t_{\text{esc}}$ possesses a long tail.
Thus there is a reasonable chance in any given
simulation that thermal runaway is delayed.
It should be stressed that ultimately realistic models of
disks still undergo
thermal runaways: turbulent fluctuations can impede instability but it
cannot destroy it. The stabilisation witnessed in Janiuk \& Misra
(2012) we attribute to the peculiarities of their stochastic model and
a very large noise amplitude.
Another feature of our MHD simulations is thermal fragmentation
when the cooling rate is too small,
and hence the laminar thermal instability timescale too short. The
disk can then break up into hot and cold clouds. This occurs when the
thermal mixing (by turbulence or radiative diffusion) is inefficient
compared to thermal instability. On a sufficiently long lengthscale
this is always presumably the case, but how this relates to the onset
of instability in hot
accretion flows is unclear. Estimates of both radiative diffusion and turbulent mixing suitable
for the inner regions of X-ray binaries indicate that fragmentation is
a marginal possibility.
The paper is organised as follows. In Section 2 we discuss a number of
issues pertinent to thermal instability, stochastic
fluctuations, and the limits of the alpha theory (not all of which we
take up in the paper). The third and fourth sections contain our numerical MHD
model and the corresponding results, respectively. We then explore
stochastic models in Section 5 and construct a statistical theory to
help explain the MHD
simulations. Our conclusions are then presented in Section 6.
\section{Theoretical issues}
\subsection{Stress-pressure relationship}
The exact dependence of the turbulent stress, $\Pi_{xy}$, on pressure
is key to the onset of thermal stability in radiation pressure dominated flows.
The instability occurs when $\Pi_{xy}\propto
(P_{\text{gas}}+P_{\text{rad}})$, but does not occur
when $\Pi_{xy}\propto P_{\text{gas}}$ (Piran
1978). Here $P_{\text{gas}}$ and $P_{\text{rad}}$ denote gas and
radiation pressure, respectively. The failure to observe
signatures of thermal instability in most X-ray observations have been
attributed to the stress depending on gas pressure alone, or possibly
the geometric mean of gas and radiation pressure (Gierlinski \& Done
2004).
It has
also been speculated that only exceptionally luminous flows could lead to a
situation where $\Pi_{xy}\propto P_{\text{rad}}$, explaining the
outbursts of GRS 1915+105 (though what exactly causes this shift in the
stress's behaviour is unclear) (Gierlinski \& Done 2004, but see also
King and
Ritter 1998).
Numerical simulations of the MRI have since
complicated this picture, as they indeed exhibit instability
(Jiang et al.~2013),
and we are left with the task of numerically tracing out
the non-straightforward behaviour of $\Pi_{xy}$ in different conditions.
While earlier unstratified shearing box simulations without radiation
pressure found only a weak dependence of stress on pressure (Sano et
al. 2004), recent work has shown that $\Pi_{xy}\propto P_{\text{gas}}$
when the following conditions are satisfied: the computational
domain is sufficiently large, explicit dissipation is included, and
there is no net magnetic field (Ross et al.~2016, hereafter RLG16).
Small boxes restrict the size of the turbulent eddies and
prevent them from fully responding to an increase in
pressure (be it gas or radiation). This restriction no doubt played a role
in the failure of early radiation-pressure dominated simulations to
show thermal instability (Hirose et al.~2009): the radial
domain was too small, the eddies unnaturally
confined, and as a result the stress unable to respond to changes in
the total pressure.
When larger radial boxes are used, as in recent work, the instability
does in fact materialise (Jiang et al.~2013).
Another numerical effect uncovered by RLG16 was a sensitivity to the
grid in simulations with no net magnetic field and
no explicit diffusion. As in isothermal runs
(Fromang \& Papaloizou 2007), the stress is proportional to grid
size and this leads to a significantly weaker stress-pressure
relationship.
Local boxes are not the only domains that exhibit the effect;
recent vertically stratified simulations also show that
the stress depends on the grid length (Ryan et al.~2017).
The weaker dependence, both from numerical dissipation and from the
box size, leads to artificially more stable systems (as can be shown
from dimensional analysis). It
is therefore necessary for both of these numerical complications
to be considered when simulating thermal instability involving gas pressure.
It is likely that global disk simulations of the
MRI also suffer from strong numerical effects, though these have
yet to be fully explored.
Another intriguing result from RLG16 is that the stress-pressure
relationship depends on the existence and strength of any imposed magnetic flux.
The stronger a mean toroidal flux, the weaker the relationship. If this
behaviour generalises to other field configurations and to
the radiation-dominated regime, then one
might speculate that highly magnetised flows are less prone to thermal
instability. Indeed global simulations of the MRI
suggest that strong magnetic fields impede thermal instability (Sadowski 2016),
possibly because they weaken the connection between the stress and
pressure. This raises the interesting prospect that to assess
susceptibility to thermal instability we must also account for the
build up and evacuation of large-scale magnetic flux, in addition to
the turbulent dynamics (e.g.\ Guilet \& Ogilvie 2012a, 2012b).
\subsection{Time lags}
The alpha theory is a turbulence closure model,
supplying a simple eddy viscosity in place of the complicated
and chaotic time-dependent dynamics of the flow. On timescales and
lengthscales much longer than the characteristic scales of the turbulence,
this approximation provides an adequate description, but its
performance worsens the shorter the scales of interest. Since thermal
instability can possess growth rates of tens of orbits (or less), then the
detailed turbulent dynamics could potentially interfere with its onset.
Simulations, as expected, show that on shorter timescales the stress-pressure
relationship is more complicated than a mere proportionality.
One interesting feature is a time lag of a few orbits
between the stress and
pressure. Moreover, it is the pressure that follows the stress on
shorter times,
rather than the other way around. And so the dependency is opposite
to that assumed by the alpha theory:
bursts in stress are followed by jumps in pressure (Hirose et
al.~2009).
What is happening here is that
the bursts in stress drive fluctuations in the heating rate
(once their associated energy has reached the dissipation scale)
and hence cause bursts in pressure a short time later.
The effect of this time delay on a stable thermal
equilibrium was first explored by Hirose et al.~(2009), who argued
that if the direction of causation was from the stress to the pressure
then thermal instability may not work. The argument fails, however,
to acknowledge that the stress-pressure dynamics exhibit
different timescales, with longer timescales ($>10$ orbits)
characterised by
a pressure-dependent stress (as in the classical theory), while the
short timescales show the lags described above (Latter \&
Papaloizou 2012, RLG16).
Follow up work by Lin et al.~(2011) and Ciesielki et al.~(2012)
present linear instability analyses of an alpha disk model with a
time-delayed alpha. These show that the simulated delay of 1-10 orbits is
insufficiently long to stabilise a thermally unstable state. They also
reveal potential inconsistencies in such simple models: for example,
infinite growth rates are possible for certain time-lags,
a physical impossibility. Of course,
these models are also incomplete, as they include only the
either the short term or long term dependencies, but not both concurrently.
\subsection{Temporal fluctuations}
MRI turbulence has strong variation over a range of timescales, from
tens of orbits
to a few
shear times (Sano
et al.~2004, Lesur \& Ogilvie 2008). In what sense does a thermal equilibrium exist in such a
system? Time varying perturbations are constantly emerging which
lead to a shifting balance of heating and cooling
that the system continually responds
to. On the other hand, if we assume that there is well-defined mean
equilibrium,
then it is
awash in finite amplitude fluctuations. How can one then undertake a linear
instability analysis? Is it meaningful to add a tiny perturbation
ontop of a sea of
finite amplitude perturbations and check if it grows or not?
On long length and time scales this might work, but
certainly not on shorter scales.
Putting aside the difficulty of interpreting linear stability
analyses, a stochastic system exhibits a range of complicated and
sometimes
unexpected behaviour. A classic example is the destabilisation of
fixed points deemed stable by laminar theory. Originally studied in
biological population dynamics (e.g.\ Levins 1969, May 1973), this
feature of noisy systems appears in numerous
applications, such as atmospheric modelling (e.g.\ De Swart \& Grasman
1987), where the unresolved short time and lengths dynamics are
represented by stochastic terms (see Majda et al.\ 1999, 2003).
On the other hand,
the influence of stochasticity on an otherwise \emph{unstable} fixed point
has been studied in financial mathematics, where geometric Brownian motion can
be used to model volatile stock prices in a rising market. Despite the
mean trend of increasing prices, stochasticity can depress the price
of some stock dramatically, if not stabilising the fixed point then
delaying a runaway in price for some period of time.
The very last example is perhaps the most relevant for our study of
thermal instability, as it possesses the key ingredients of (a) an
unstable fixed point (according to a deterministic or `laminar'
theory) and (b) stochastic fluctuations. The competition between them
gives rise to behaviour one might
liken to a biased random walk. In between the kicks delivered by the
turbulence, the system drifts according to the deterministic unstable
dynamics. One can then imagine certain limits: when the characteristic
frequency of the turbulence is much greater than the thermal
instability growth rate then we may expect an unbiased random walk,
and the system will only weakly sense the underlying thermal physics.
In the opposite limit, when the growth rate is much greater than the
turbulent frequency, the deterministic laminar
dynamics should be reproduced. It is in the intermediate regime, explored in this paper,
that interesting nontrivial behaviour manifests. There are
also other key ratios, such as the size of the kicks relative to the
magnitude of the fixed point or the initial condition.
If these are too small, then we
return to the laminar case. But for intermediate values, as exhibited
by our MRI simulations, system trajectories can deviate markedly from
both the laminar behaviour and a simple unbiased random walk.
\subsection{Spatial fluctuations}
In the previous subsection we considered only temporal fluctuations on
the system variables, implicitly regarding them as `box averaged' or
mean quantities. Indeed, the $\alpha$ model assumes a homogenised
temperature over $\sim H$, the disk scale height.
Turbulent heating, however, is spatially inhomogeneous with strong
dissipation occurring in current and vorticity sheets and minimal
dissipation in the surrounding regions.
These spatial fluctuations
also complicate the picture of thermal instability, especially when
the instability growth rate is large.
Some form of thermal mixing is
necessary to homogenise the temperature of the fluid. This can take
multiple forms, such as turbulent advection, radiative diffusion, or
thermal conduction. When thermal instability is present, the assumption of a
uniform temperature is reasonable as long as the instability time scale is
longer than the mixing timescale, $t_{\text{inst}}>t_{\text{mix}}$. For less
efficient mixing or stronger instability, there exists a maximal thermal coupling
length scale, $l\sim v_{\text{mix}}t_{\text{inst}}$. Regions separated by more than
this only weakly interact thermally during an efolding time. This implies
that regions of a disk separated by more than $l$ can undergo thermal runaways
independently, and the disk fragments into cold and hot clouds.
Regions of strong kinetic and magnetic dissipation are
likely to heat catastrophically, while those with weak dissipation will
cool catastrophically. How relevant this
scenario is in realistic disks is unclear, though perhaps marginally
possible in X-ray binaries. It is certainly possible in
numerical simulations as we show later.
\begin{figure}
\includegraphics[width=8cm]{images/HandC1-eps-converted-to.pdf}
\caption{For illustrative purposes, the red curve shows the fitted
heating rate calculated from a zero net-flux simulation with
resolution $\frac{1}{64}H_{0}=\Delta$ from RLG16 (shown in green).
The blue line is an example cooling function $\propto P^{0.25}$.
The fixed points are shown by black dotes.}
\label{Fig:HandC}
\end{figure}
\section{Numerical tools and setup}
\subsection{Formulation}
We wish to explore the essential features of thermal stability and its
onset in MRI driven turbulence and so we choose an idealised set-up to
isolate it. We adopt the local shearing box model (Goldreich \&
Lynden-Bell 1965). To prevent complications such as buoyancy, mass
loss and disk expansion we consider the unstratified case. With this
model, MRI driven turbulence can be obtained when a Keplerian flow
profile is assumed (Hawley et al.~1995). As is conventional, $x,y,z$
are the radial, azimuthal and vertical spatial variables and
$\boldsymbol{\hat{e}}_{x}$, $\boldsymbol{\hat{e}}_{y}$,
$\boldsymbol{\hat{e}}_{z}$ are the corresponding unit vectors. This
frame of reference co-rotates with the disk at some radius
with angular frequency
$\mathbf{\Omega}=\Omega\boldsymbol{\hat{e}}_z$.
The ideal compressible MHD equations are hence
\begin{align}
\label{eq::1}
&\frac{\partial \rho}{ \partial t} + \nabla \cdot ( \rho \boldsymbol{v}
) = 0, \\
&\rho \frac{\partial \boldsymbol{v}}{\partial t} +\rho (\boldsymbol{v}
\cdot \nabla)\boldsymbol{v}
= - 2 \rho \mathbf{\Omega} \times \boldsymbol{v} +
3x\rho \Omega^{2}\boldsymbol{\hat{e}}_{x}-\nabla P \notag\\
& \hskip4cm +( \nabla \times \boldsymbol{B} ) \times \boldsymbol{B} \\
&\frac{\partial \boldsymbol{B}}{\partial t} = \nabla \times
(\boldsymbol{v} \times \boldsymbol{B}), \\
&\frac{\partial \varepsilon}{\partial t} + \boldsymbol{v}\cdot\nabla
\varepsilon= -P\nabla\cdot\boldsymbol{v} +Q -\Lambda.
\label{eq::4}
\end{align}
where $\rho$ is the mass density, $\boldsymbol{v}$ is the velocity,
$P$ is the gas pressure, $\boldsymbol{B}$ is the magnetic field and the internal energy is
denoted by $\varepsilon$. Heating is represented by $Q$ and cooling by
$\Lambda$.
This set of equations is then closed by relating the internal energy to the pressure by assuming an ideal gas so that
\begin{equation}
\varepsilon= P/(\gamma-1),
\end{equation}
where $\gamma$ is the adiabatic index, taken to be $7/5$. The sound speed is then given by $c_{s}=(\gamma P/\rho)^{1/2}$ and the
pressure scale height by $H=(2/\gamma)^{1/2}c_{s}/\Omega$.
Ideally we would include viscosity and Ohmic diffusion and so $Q$ would
be given by the sum of the physical dissipative processes;
however, to resolve
the diffusion length scales requires a higher resolution than is
practical for this study. Instead, we rely on numerical dissipation
for heating. By solving Equations \eqref{eq::1} - \eqref{eq::4} in
conservative form, the kinetic and magnetic energy dissipated by the
grid is converted to internal energy. Energy that is extracted from
the background shear is converted to internal energy and ultimately
removed via the cooling function $\Lambda$.
Finally, for our cooling function, we take a power law of pressure,
\begin{equation}
\Lambda = \theta\,P^{m},
\end{equation}
where $\theta$ and $m$ are both constants. Though this choice is
mainly for convenience, it might crudely approximate an optically thin medium.
\subsection{Numerical methods}
All of the simulations that we perform are carried out using RAMSES, a finite-volume Godunov code based on the MUSCL-Hancock algorithm
(Teyssier 2002; Fromang et al.~2006). The HLLD Riemann solver
(Miyoshi \& Kusano 2005),
and the multidimensional slope limiter described in Suresh (2000) are used in all the simulations presented in this paper.
Rather than solving for the total $y$-momentum, we evolve the equivalent
conservation law for the angular momentum fluctuation $\rho
v_y^{\prime} =\rho( v_y - v_K)$,
with $v_K$ the Keplerian velocity. An upwind solver is used for solving the azimuthal advection arising from $v_{K}$.
The tidal and Coriolis forces are treated as source terms and
implemented following the Crank-Nicholson algorithm described in
Stone \& Gardiner (2010).
The algorithm solves for the total fluctuation energy $E^{\prime} \equiv \varepsilon + \rho
v^{\prime 2}/2 + B^2/2 $ and its conservation law is written as
\begin{align} \label{eq::7}
\frac{\partial E^\prime}{\partial t} + \nabla\cdot &\left(E^\prime{
\boldsymbol{v}^\prime} + {\boldsymbol{v}^\prime} \cdot \mathsf{P} \right) = -v_K\frac{\partial E^\prime}{\partial y} \notag\\
&+ \left(B_xB_y - \rho v_x v_y^{\prime} \right)\frac{\partial v_K}{\partial x} -\Lambda,
\end{align}
where $\mathsf{P}$ is the total pressure tensor
\begin{align}
\mathsf{P} = (P + B^2/2)\sf{I} - \boldsymbol{B}\boldsymbol{B}.
\end{align}
The left hand side of Equation \eqref{eq::7} comprises the usual energy conservation law,
which we solve using the MUSCL-Hancock algorithm.
The treatment of the two terms on the right hand
have been modified: the azimuthal
advection of energy is solved with an upwind solver, and the second
term involving the Maxwell and
Reynolds stresses is added as a source term.
For the set of simulations shown in this paper, we used a box size of
$(L_{x},L_{y},L_{z})=(4,5,4)H_{0}$ with a resolution of
$(N_{x},N_{y},N_{z})=(128,100,128)$. $H_{0}$ is a reference scale
height which is close to but not exactly the same as the scale height at the start of
a simulation. It will be defined in detail later.
The grid scale is defined to be $\Delta=L_{x}/N_{x}$. We
set $\Omega=10^{-3}$ and $c_{s0}=10^{-3}$ in code units, where
$c_{s0}$ is the initial sound speed. The resolution is low in
comparison to other MHD shearing box simulations, however it is
sufficient to capture the basic properties of the MRI, in particular
the turbulent fluctuations. Importantly, it is computationally
inexpensive
allowing for the simulations to run for $\gtrsim 500$ orbits.
\subsection{Thermal equilibrium}\label{Sec:TE}
In our shearing box model, the energy is injected into the
computational domain by the second term on the right side of Equation \eqref{eq::7} which
represents the liberation of shear energy by the total stress:
\begin{equation}
\Pi_{xy}=B_xB_y - \rho v_x v_y^{\prime}.
\end{equation}
Simulations have found that the box-averaged stress $\langle \Pi_{xy}
\rangle$ is roughly proportional to the
box-averaged gas pressure to a given power (Sano et al.~2004, RLG16),
though be aware of the caveats given in Section 2.
Therefore, in numerical simulations, the averaged
heating rate may be approximated by
\begin{equation}
\langle Q\rangle\propto \langle\Pi_{xy}\rangle =\tilde{\alpha}\langle P\rangle^{q},
\label{eq::9}
\end{equation}
where $\tilde{\alpha}$ and $q$ may be calculated from the simulation.
Note that the former is
\textit{not} the same as the $\alpha$ parameter.
The exponent $q$ depends on the field geometry as well as the
numerical parameters (RLG16).
For our set-up, $q=0.5$, as long as the pressure is sufficiently
small (see later).
Combining this approximation with our cooling prescription,
the evolution of the volume averaged pressure, $\langle P\rangle$, is determined by
\begin{align}
\label{eqn::10}
\frac{d\langle P\rangle}{dt}&\approx\left(\gamma-1\right)\frac{3}{2}\Omega\langle\Pi_{r\phi}\rangle-(\gamma-1)\theta \langle P\rangle^{m} \\
&\cong a_1\langle P\rangle^{q}-a_2 \langle P\rangle^{m}.
\label{eqn::11}
\end{align}
where
\begin{align}
a_1=\frac{3}{2}\Omega(\gamma-1)\tilde{\alpha}, \qquad a_2=(\gamma-1)\theta.
\end{align}
In the formulation of Equation \eqref{eqn::10} we have made the
approximation that $\langle P^{m}\rangle\cong\langle P\rangle^{m}$,
which is reasonable as long as the variance of $P$ within the box at
fixed time is not too large. This system has two fixed points
\begin{align}
P_{1}&=0, \\
P_{2}&=\left(\frac{a_{1}}{a_{2}}\right)^{1/(m-q)}=\left(\frac{3\Omega\tilde{\alpha}}{3\theta}\right)^{1/(m-q)}.
\end{align}
In shearing box simulations the thermodynamics have an additional
complication: stress is independent of pressure once the scale height
is larger than the radial and vertical box sizes (RLG16). The exponent
$q$ may then be viewed as a function of the disk temperature, equal to
$0.5$ for cool disks, while asymptoting to 0 as the gas heats and the
scale height $H$ equals the box size $L_z$.
This behaviour introduces the possibility of
an additional stable fixed point if $m<\max\{q\}$. In Fig.
\ref{Fig:HandC} we plot (a) the heating rate as a function of pressure from a $L=4H_{0},
\Delta=1/64$ simulation (appearing in RLG16), (b) a smooth fit to
this curve, and (c)
overlay a $m=0.25$ power law cooling. Where the latter two curves
intersect
give three thermal equilibria. These are the $P_1$ and $P_2$ fixed
points, described above, in addition to a
third equilibrium, $P_3$, which arises from the finite size of the
box.
Note that a similar fixed point
will also appear in vertically stratified simulations.
The above volume averaged analysis is deterministic, but, turbulent
flows are not. The stress fluctuates around its mean value as a result
of the formation and break up of coherent structures within the
flow. This means that $Q$ can no longer be expressed in as simple a
form as Equation \eqref{eq::9}. Instead, the stress is determined by the
sum of a deterministic term and a fluctuating term. By using Equation
\eqref{eqn::10} we can obtain an estimate of the equilibrium pressure
that the system feels at any given time
\begin{equation}
P_{\text{exp}}=\left(\frac{3\Omega\langle \Pi_{xy}\rangle}{2\theta}\right)^{1/m}.
\label{Eqn::pexp}
\end{equation}
\subsection{Thermal instability}\label{Sec::Instab}
According to a linear analysis of Equation \eqref{eq::4},
the thermal instability criterion is
\begin{equation}
\frac{dQ}{dP}>\frac{d\Lambda}{dP}
\end{equation}
which, given our power law expressions for $Q$ and $\Lambda$,
leads to the simple condition $$q>m$$ for $P_{2}$, and the opposite
stability for $P_{1}$ and $P_3$.
We calculate the growth rate as
\begin{equation}
s_g=(q-m)(\gamma-1)\left(\frac{3\tilde{\alpha}\Omega}{2}\right)^{(1-m)/(q-m)}\theta^{(q-1)/(q-m)},
\label{Eqn::therm_lam}
\end{equation}
which can be used for comparison with the MHD simulations. The
associated instability timescale we define to be $t_{\text{inst}} =1/s_g$.
If the system was truly laminar the instability criterion would be
fully determined by $p$ and $q$ and the instability
timescale from \eqref{Eqn::therm_lam}. Fluctuating systems, however,
exhibit non-exponential and indeed non-monotonic behaviour which is
poorly approximated by the laminar model. In particular, $t_{\text{inst}}$ may
be an unsuitable measure for the instability timescale. In its place we introduce
the `escape time' $t_{\text{esc}}$ of a simulation,
which is defined to be the last instance that the system lies within a
pressure interval containing the fixed point.
Mathematically it may be defined via
\begin{equation}
t_{\text{esc}}=\max\left\lbrace t>0 : \vert P(t)-P_{\text{eqm}}\vert=\delta\right\rbrace,
\label{Eqn::Tesc}
\end{equation}
where $P_{\text{eqm}}$ is one of the three equilibria introduced
earlier, and $\delta$ is the interval size, outside of which we
consider the system to have unequivocally departed from the fixed
point. We are free to specify the size of $\delta$, and it might
reflect the particular problem of interest. For disk transitions
between states differing by many orders of magnitude in temperature
we might be generous with $\delta$, permitting it to be up to 10 times
the fixed point pressure. For smaller transitions, then $\delta$ must
be smaller.
Note that the system may stochastically dip in and out of this interval, but
$t_{\text{esc}}$ will capture the time when the system finally leaves it forever.
If the disk is laminar, then
\begin{align}
t_{\text{esc}}=t_{\text{inst}}\ln\left(\frac{\delta}{|P_{\text{eqm}}-P_{\text{init}}|}\right)
\end{align}
where $P_{\text{init}}$ is the
initial pressure of the system.
\subsection{Initial conditions}
If we want a thermal equilibrium to be achievable within our box and
for the stress to depend appreciably on pressure, the initial
conditions must be chosen carefully. An initial state that is too hot
means the box size will unduly influence the stress and weaken
$Q$'s dependence on $P$.
As a consequence, thermal instability will fail to occur.
Stress is only observed to be
a strong function of pressure ($\Pi_{xy}\propto P^{1/2}$)
when $\Delta\ll H<L$ (Sano et al.~2004,
RLG16). Therefore, $P_{2}$ must be sufficiently low so that the stress is
increasing with pressure, but sufficiently high so that
the characteristic lengthscale of the turbulence is not on (or below)
the grid.
That way we can be assured that the thermal instability is able to
appear in our numerical set-up. We choose
\begin{equation}
\frac{\sqrt{P_{2}}}{\sqrt{P_{\text{box}}}}=\frac{H_{2}}{L}\approx 0.5
\label{eqn::P0Peqa}
\end{equation}
where $P_{\text{box}}$ is the pressure at which the scale height equals the
box size and $H_{2}$ is the scale height when $\langle P\rangle
=P_{2}$. From this we can obtain the required value of $\theta$ for a given choice of $m$:
\begin{equation}
\theta=\frac{(3/2)\tilde{\alpha}}{(0.25P_{box})^{(m-q)}}
\end{equation}
To achieve a turbulent state we first initialize a zero net-flux
simulation with an initial field of $\boldsymbol{B}=B_{0}\sin(2\pi
x)\mathbf{\hat{e}}_z$ and no cooling. In code units
$B_{0}=\sqrt{2/\beta}$ which we set with $\beta=10^{3}$. Once a
turbulent state is reached we switch on cooling with $\theta=10$ and
$m=2$. This choice of $m$ is to obtain a $P_{2}$ which we expect to be
stable, as discussed in Section \ref{Sec::Instab}. These parameters
lead to $P_{2}\approx 3.2P_{0}$ where $P_{0}$ is the initial pressure,
having used that $P_{\text{box}}\approx 16P_{0}$. During this steady state we
calculate $\tilde{\alpha}\approx2.3\times10^{-5}$.
This fully turbulent state in thermal equilibrium is used as our
initial condition. The parameters $m$ and $\theta$ are then changed as
appropriate.
\begin{figure}
\includegraphics[width=10cm]{images/Run1-eps-converted-to.pdf}
\caption{The evolution of the box-averaged pressure and stress from simulation R1.}
\label{Fig::r1_p}
\end{figure}
\begin{figure}
\includegraphics[width=10cm]{images/Run1_zoom-eps-converted-to.pdf}
\caption{A zoom-in of the pressure and stress evolution in
Fig.~\ref{Fig::r1_p}.
The blue dashed vertical line indicates the time at which the stress
achieves its maximum in this interval.}
\label{Fig::R1_P_Z}
\end{figure}
\section{Results}
\subsection{Stable system}
\begin{figure}
\includegraphics[width=8cm]{images/Stable_2-eps-converted-to.pdf}
\caption{The evolution of the logarithm of the box averaged pressure (red) along with the logarithm of the instantaneous expected equilibrium pressure calculated using an average stress over the previous $9t_{\text{orb}}$ (blue) for the thermally stable simulation R1. The black dashed line indicates the mean equilibrium pressure. }
\label{fig::r1_pexp}
\end{figure}
To fix ideas we first consider the stable thermal equilibrium
associated with the initial condition, $\theta=10$, $m=2$ (simulation R1).
We run the simulation for over $500$ orbits and the resulting stresses
and pressures are shown in Fig. \ref{Fig::r1_p}. As predicted from
the laminar
linear analysis, the equilibrium is `stable' in the sense that both
the stress and pressure fluctuate within some interval enclosing
$P_{2}$.
Though the stress shows substantial variation during the simulation (with a
maximum value of $\approx3$ times its mean), there is no runaway or
mean drift during the $500$ orbits. The variability in the pressure is
less extreme, with a maximum value of $\approx 1.5$ times its
mean value.
The turbulence administers
random `kicks' to the system, but the deterministic physics always
draws it back to the vicinity of $P_2$.
The stochastic fluctuations in stress are a result of
dynamo cycles and the formation and breakup of coherent
structures. The energy in the flow then cascades down to the
dissipation scale where the magnetic and kinetic energies are
converted to thermal energy leading to changes in pressure.
A short time delay of a few orbits ($t_{\text{orb}}$) between the stress and pressure
is clearly visible in Fig. \ref{Fig::R1_P_Z}, a result of the finite
time taken for kinetic and magnetic energies to reach the dissipation
scale from the injection scales (Hirose et al.~2009).
The variations in pressure are smoother and longer
than those in the stress. However, this short time dependency of pressure on
stress is distinct to
the longer time dependence of stress on pressure that drives
instability/stability (Latter \& Papaloizou 2012).
Though the pressure seems to be drawn back to the mean of $P$,
in actual fact the equilibrium balance that the system feels at
any given instance is changing with time. A crude approximation to
this is the instantaneous fixed point $P_{\text{exp}}$,
calculated using Equation \eqref{Eqn::pexp}. But because the thermal
timescale is longer than the turbulent timescale, a better
approximation is the average of $P_{\text{exp}}$ over the last thermal time.
Using Equation
\eqref{Eqn::therm_lam} we estimate this time to be $\approx
9$ orbits. The resulting smoothed and time-dependent equlibrium is
plotted in Fig. \ref{fig::r1_pexp} in blue, alongside the actual
pressure of the system $\langle
P\rangle$ in red.
It is clear that $\langle
P\rangle$ follows the equilibrium with a time lag of
$\lesssim 10$ orbits. Despite the fluctuations the
system senses the stable fixed point and is attracted towards it,
though because of those same fluctuations it can never come to rest
upon it.
\subsection{Unstable systems}
\begin{figure*}
\includegraphics[width=15cm]{images/US_P-eps-converted-to.pdf}
\caption{Pressure verses time for the unstable simulations R2a-R2h shown by solid curves. The dashed red line is linear evolution. The filled area indicates the fixed point region $0.5P_{\text{eqm}}<P<1.5P_{\text{eqm}}$.}
\label{fig::UN_P_S}
\end{figure*}
Having explored the stable case we now proceed to an
unstable thermal equilibrium. For this set of simulations we choose
$m=0.25$ which we expect to be unstable
based on the argument in Section \ref{Sec::Instab}.
We present 8 simulations (R2a-R2h) that have the same fully turbulent
initial state. In order to vary the initial condition
between runs, these simulations each have slightly different
$\theta\approx1.31\times10^{-9}$ and so have slightly different
$P_{2}$. Each of these is within a few percent of the initial
pressure. In practice this means that though each simulation starts
from the same initial condition, each corresponds to a slightly
different perturbation from equilibrium.
In Fig. \ref{fig::UN_P_S} we show the evolution of the box-averaged
pressures alongside the trajectory derived from the laminar
linear theory. Unsurprisingly the
turbulent fluctuations lead to a diversity of outcomes but do not
indefinitely prevent thermal runaway. For example, a large kick can
cause the system to escape from the fixed point on a shorter timescale
than the laminar timescale, or the system may remain close to the
fixed point for extended periods of time, longer than
$t_{\text{inst}}$.
Our most `stable' simulation remains close to equilibrium for
$400t_{\text{orb}}$.
None
of the simulations can be well modelled by the laminar theory. For
$P\gtrsim5P_{0}$, the behaviour is closer to algebraic growth than
exponential runaway. In fact, the behaviour of the system is strongly
influenced by the fluctuations that occur on timescales comparable to
the instability timescale, $\sim 70 t_{\text{orb}}$ for these
parameters.
A more apt description of
the system could be a biased random walk, when strong fluctuations in
stress repeatedly perturb the system, while in between kicks the
system drifts according to the deterministic physics.
If we consider some characteristic interval around the fixed point
then we can find the very last time,
$t_{\text{esc}}$, the system was within this band, Equation
\eqref{Eqn::Tesc}. If we define the interval to be rather narrow,
$0.5P_{eq}<P<1.5P_{eq}$, then this `escape time' can be compared to
$t_{\text{inst}}$. We find a wide range of escape times in our MHD
simulations, from $0.25t_{\text{inst}}$ to $5t_{\text{inst}}$. Ideally, we would
calculate a probability distribution function for the escape time but
this would require substantially more simulations, which, at this time
is impractical. The choice of band is arbitrary, but it should be chosen to suit the system.
Once $P>10P_{0}$ the simulations become cooler than
the laminar model prediction. This can be attributed to the box size
beginning to influence the evolution. At this point, the dependence of
stress on pressure decreases (see Fig. \ref{Fig:HandC}) and hence
the laminar model is an over estimate. This numerical effect
introduces a third equilibrium point, $P_{3}>P_{2}$, as discussed in
Section \ref{Sec:TE}. We do not observe a plateau in pressure
associated with the system being attracted to $P_{3}$, but, we expect
that if run for a sufficient duration then a plateau would appear.
That a runaway heating is limited by
the box size is an important problem, both for these simulations and
stratified radiation MHD simulations. To explore this further, we
rescale the simulations by choosing a larger value of $\theta$, and initialize a
simulation with the same initial turbulent state as in
R2a-R2h. However, now the initial condition is much further away from
the putative equilibrium, by a factor of some 3, and yet box effects
remain negligible, because
Equation \eqref{eqn::P0Peqa} is still satisfied. In these runs
the thermal runaway was somewhat faster than witnessed in Fig 5 for
the hotter systems. These few simulations illustrate the numerical
limitations inherent in any simulation of catastrophic heating
undertaken in a finite domain.
\begin{figure}
\includegraphics[width=8cm]{images/UN_pdf1-eps-converted-to.pdf}
\caption{The pressure probability distribution function $PDF(P/P_{0})$ from a heating (red) and a cooling run (blue) at $t=340t_{\text{orb}}$. The green curve shows the initial distribution.}
\label{fig::UN_pdf_P}
\end{figure}
\subsection{Thermal fragmentation}
In a turbulent system, heat is not deposited uniformly throughout the
box, but
rather is localised in coherent structures such as current and
vorticity sheets. If the instability timescale, $t_{\text{inst}}$, is large
compared to the mixing timescale, $t_{\text{mix}}$, then the
heating inhomogeneity will have little effect as temperature
fluctuations will be smoothed out. Conversely, if $t_{\text{mix}}>t_{\text{inst}}$
then regions of fluid can evolve independently of each resulting
in localised runaway. In Fig. \ref{fig::UN_pdf_P} we plot the
pressure probability distribution function, $PDF(P)$, for two R2
simulations at $t=340t_{\text{orb}}$. Inefficient turbulent mixing would
result in the spreading of $PDF(P)$ during runaway. The figure exhibits
minimal evidence of spreading in the heating run, and the pressure ensemble
evolves with a well defined and relatively narrow shape. The pressure in the
cooling simulation is attracted to $P_{1}=0$ and in fact
we see a further
narrowing of the distribution as the equilibrium is approached.
To show localised runaway and fragmentation, we consider a case
which we expect to be very unstable, choosing $m=0.1$ and
$\theta\approx1.84\times10^{-10}$ (simulation R3). In Fig.
\ref{fig::UN_pdf_P_f} we plot $PDF(P)$ at $3$ instances of time.
The width of the distribution quickly increases,
indicating localised thermal runaway: initially Var($P/\langle
P\rangle$)$=0.0045$, but after $4$ and $14$ orbits this grows to
$0.019$ and $0.057$ respectively. During this time $\langle
P\rangle$ itself varies little, which
emphasises that the box-averaged properties no longer give a
satisfactory description of the state of the system. Soon after the
final snapshot very small pressures occur resulting in the termination
of the simulation, preventing further exploration. In Fig.
\ref{fig::UN_snaP} we show $x$-$z$ slices in pressure before and during
thermal fragmentation. Prior to thermal fragmentation there are strong
acoustic waves propagating in the
radial direction. These appear to break up into
patches that undergo rapid thermal runaway independent of each other.
Because this behaviour appears to be significantly nonlinear and
disordered, we do not attribute it to the action of a linear
instability mode with non-zero $k_x$ and $k_z$.
Though it is straightforward enough to achieve fragmentation in
an unstratified local box, how likely is this in the
inner regions of an X-ray binary?
In this context the two major contributors to
mixing are radiative diffusion and turbulent advection. We first consider
radiative diffusion. In the hot dense gas, the opacity is largely
dominated by Thomson scattering with
opacity $\kappa_{T}=0.33\text{cm}^{2}\text{g}^{-1}$. The radiative diffusion
time across a length of $l$ is then $t_{\text{rad}}\sim
l^{2}/c\lambda$ where $c$ is the speed of light and $\lambda$ is the
mean free path. The instability timescale we estimate to be of order,
but bounded below, by the thermal timescale, so that
$ t_{\text{inst}} \gtrsim t_{\text{th}} \sim (\alpha\Omega)^{-1}$.
If $t_{\text{rad}}\approx t_{th}$, we have the following condition on $l$
\begin{align}
\frac{l}{H}&\sim \left(\alpha\Sigma \kappa
\frac{c_s}{c}\right)^{-1/2}, \\
&\sim
0.1\,\left(\frac{\alpha}{0.1}\right)^{-1/2}\left(\frac{\Sigma}{10^5\,
\text{g cm}^{-2}}\right)^{-1/2}\left(\frac{T}{10^7 \,\text{K}}\right)^{-1/4},
\end{align}
where $\Sigma$ is surface density.
Regions separated by more than $l$ will be unable to mix
sufficiently well on the instability timescale. Typical values for an X-ray binary indicate that regions
$0.1 H$ apart my in fact thermally fragment.
What about turbulent mixing?
We assume that the turbulent transport of heat by the MRI is similar
in efficiency to its transport of angular momentum (though this is a
point that has not been studied in detail). If we are permitted this
assumption then the turbulent diffusion time scale is
$t_{\text{turb}}\sim l^{2}/(\alpha c_{s}H)$. If we next assume that the relevant eddies are of size $H$
(plausible if some form of MHD convection is operative)
then $t_{\text{turb}}\sim t_{\text{inst}}$ on these outer scales. These very rough
scalings indicate that turbulent heat transport is somewhat more
efficient than radiative transport, and moreover that it may be
sufficient to preclude fragmentation --- though a simple order of
magnitude treatment is unable to determine precisely when this might
occur. Only realistic simulations themselves can decide on this issue,
and in fact Jiang et al.~(2013) do not find fragmentation. Our
simulations
are marginally susceptible, in agreement with the above argument, but
they omit important physical effects such as buoyancy, which
may be crucial here, and enhanced compressibility effects in
radiation-dominated flow.
\begin{figure}
\includegraphics[width=8cm]{images/pdf-eps-converted-to.pdf}
\caption{The pressure probability distribution function from a
fragmenting run at $t=40t_{\text{orb}}$. The green, red and green curves show the distribution at $t=0,4,14t_{\text{orb}}$.}
\label{fig::UN_pdf_P_f}
\end{figure}
\begin{figure*}
\includegraphics[width=8cm]{images/xz_initial-eps-converted-to.pdf}
\includegraphics[width=8cm]{images/xz_fragment-eps-converted-to.pdf}
\includegraphics[width=8cm]{images/xz_heat-eps-converted-to.pdf}
\caption{Heat maps of $P/P_{0}$ in common $x-z$ slices.
The top left panel is from the initial turbulent state, the top right
is from the
the fragmenting simulation R3 at $t=4t_{\text{orb}}$. In the latter,
the difference in pressure between the coolest and hottest blobs is a factor 6.
For comparison we show a R2 simulation undergoing a slow heating
runaway at $t=340t_{\text{orb}}$ in the bottom panel. Here the
pressure difference is merely a factor 1.5. }
\label{fig::UN_snaP}
\end{figure*}
\section{Reduced stochastic models}
Because it is impossible to run a sufficient number of simulations to build
reliable statistics, especially regarding the distribution of $t_{\text{esc}}$,
we turn to simpler approximate models that illustrate
more fully the effects of the fluctuations on stability and which also
permit analytical results.
We work primarily with the averaged energy equation \eqref{eqn::11}
but model the
fluctuating turbulent stress via a random function $\zeta$. Our model
is related to the logistic equation, and may be written as
\begin{equation}
\frac{dx}{d\tau}=\left[1+\zeta(\tau)\right]x^{q}-\Theta x^{m}.
\label{eqn::M2}
\end{equation}
This can be derived from \eqref{eqn::11} by an appropriate rescaling,
with $x$ and $\tau$ representing pressure/temperature and time,
respectively. Constant parameters are $q$, $m$, and $\Theta$.
To simplify the analysis while not losing much generality, we set
$q=\Theta=1$ in much of what follows.
Equation \eqref{eqn::M2} admits the trivial steady state $x=0$ and the
more interesting equilibrium $x_{eq}=1$. In the
`laminar' case of $\zeta=0$ this equilibrium is unstable when $m<1$
with modes possessing the growth rate $1-m$.
For $\zeta(\tau)$ a random but \emph{continuous} function,
\eqref{eqn::M2} is the Bernoulli equation with analytic solution
\begin{dmath}\label{eq::sto}
\frac{x}{x_{eq}}=\phi(\tau)\left[\left(\frac{x_{0}}{x_{eq}}\right)^{1-m}-(1-m)\int^{\tau}_{0}\phi(s)^{m-1}ds\right]^{1/(1-m)}
\end{dmath}
where
\begin{equation}
\phi(\tau)=\exp\left\{\tau+\int^{\tau}_{0}\zeta(s) ds\right\},
\end{equation}
and $x_0$ is the initial value of $x$. As it stands, the analytic
solution is too unwieldy to be useful, even for basic prescriptions
for $\zeta$, but it does illustrate clearly the competition between
instability and stochasticity. These manifest as the two terms in
the exponent of $\phi(\tau)$. The first describes the deterministic
exponential runaway, while the second stochastic term
potentially impedes this
tendency. In fact, if $\int^{t}_{0}\zeta(s)
ds$ behaves like a random walk,
then its standard deviation will be proportional to $\sqrt{\tau}$,
and so on short to intermediate times the second stochastic term can
outcompete the first instability term. On longer times, however,
$\tau$ will always defeat $\sqrt{\tau}$ and the system will
approach the unstable laminar solution, given by
\begin{equation}\label{eq::det}
\frac{x}{x_{eq}}=\Bigg\{\left[\left(\frac{x_{0}}{x_{eq}}\right)^{1-m}-1\right]e^{(1-m)\tau}+1\Bigg\}^{1/(1-m)}.
\end{equation}
\subsection{Geometric Brownian motion (GBM)}
Before presenting an analysis of the full equation \eqref{eqn::M2},
it is worthwhile examining the simpler case of $\Theta=0$ and
$q=1$. The resulting system isolates cleanly all the main characteristics of
more realistic systems --- an unstable fixed point ($x=0$ now) and
stochastic noise --- while being analytically tractable.
Equation \eqref{eqn::M2} becomes
\begin{equation}
\frac{dx}{d\tau}=\left[1+\zeta(\tau)\right]x
\label{Eqn::M1}
\end{equation}
with initial condition $x_{0}>0$. For smooth random $\zeta$, the solution is
\begin{equation}
x(\tau)=x_{0}\phi(\tau).
\label{Eqn::GBM_gen}
\end{equation}
When studying stochastic dynamical systems, white noise is a convenient
choice for modelling the variability. For white noise to be a good
approximation, the fluctuation timescales of the system should be much
less than the characteristic time of interest. For MRI-driven
turbulence, this choice is not ideal given that the spectrum of
$\zeta$ has preferential frequencies. However, we use this as our
starting point as it makes a number of results especially clear. With this choice of
$\zeta$, Equation \eqref{Eqn::M1} must be written in differential form
\begin{equation}
dx=xd\tau+\sigma x dW
\label{Eqn::M1a}
\end{equation}
where $\sigma$ is the volatility coefficient (or noise amplitude)
and $dW$ is white
noise. Here, for simplicity, we have interpreted the calculus in the Ito
sense.
Equation \eqref{Eqn::M1a} actually describes geometric Brownian motion
and is frequently used in financial modelling. Its solution is
\begin{equation}
x(\tau)=x_{0}\exp\left\{\left[1-\frac{\sigma^{2}}{2}\right]\tau+\sigma W(\tau)\right\}.
\label{Eqn::M1sol}
\end{equation}
Given that $x_{0}>0$, the solution remains strictly positive as a
result of the multiplicative form of the noise. We plot sample
trajectories in Fig. \ref{Fig::GBM_Traj} along with the $10$th and
$90$th-percentiles. When
calculating these trajectories we use the Euler-Maruyama method
(Kloeden \& Planten 1992).
A feature of this collection of sample paths is
the wide variation between them, an attribute that is shared with the
simulations shown in Fig.
\ref{fig::UN_P_S}. They are also non-monotonic; fluctuations
`kick' the system towards or away from the fixed point.
The light blue curve
is particularly striking, exhibiting a trajectory that remain close
to equilibrium,
$x(\tau)<0.25$, up to time $\tau\approx 3$. Note that at $\tau=3$ a
purely deterministic model would have predicted $x$ to be $\approx 2$,
an order of magnitude greater.
The stochastic term in Equation \eqref{Eqn::M1sol} has
`balanced out' the deterministic drift, at least on these shorter
times.
The reader may note that when $\sigma>\sqrt{2}$, the stability of
$x=0$ switches. It becomes an attractor, and the
system is stabilised. We stress, however, this effect is an artefact of
multiplicative white
noise in combination with the Ito calculus, and is not to be expected
in real turbulent systems. For instance, the stabilisation vanishes
in the Stratonovich calculus and/or with noise models with memory and
which are not multiplicative. We
certainly do not expect MHD turbulence to exhibit the combination of special
features that leads to this stabilisation.
Indeed, it makes little physical sense that `shaking' an unstable
system more vigorously ultimately leads to zero
fluctuations. Moreover,
the required amplitude of the fluctuations
must be extremely large, in our case this would require negative $\alpha$ which is impossible.
It is worth pointing out that the auto-regressive stochastic model
employed by Janiuk \& Misra (2012) shares the same stabilising property as white
noise in the Ito calculus; consequently, we view their
stabilisation of thermal instability as an artefact of their
model, and
not representative of a real fluctuating disk system.
The probability distribution of the solution trajectories may be obtained by
solving the associated Fokker-Planck equation. If $f(x,\tau)dx$ is the
probability of finding a path lying between $x$ and $x+dx$ at time
$\tau$, then
\begin{align}
f(\tau,x)&=\frac{1}{\sigma x\sqrt{2\pi \tau}}\exp\left\{-\frac{(\log(x/x_{0})-\tau)^{2}}{2\sigma^{2}\tau}\right\}.
\end{align}
The expectation, $\mathbb{E}$, and variance, Var, can also be calculated
\begin{align}
\mathbb{E}\left(x\right)&=x_{0}e^{\tau}, \\
\textrm{Var}\left(x\right)&=x_{0}^{2}e^{2\tau}\left(e^{\sigma^{2}\tau}-1\right).
\end{align}
The mean is independent of the volatility and
hence agrees with the laminar model. However, the variance contains a
$(e^{\sigma^{2}\tau}-1)$ factor which grows exponentially! This
means that as time progresses the expectation value grows less and less
meaningful because the distribution becomes increasingly wide and flat.
Overall, trajectories move away from the unstable equilibrium point,
but
individual trajectories can deviate from the laminar model dramatically.
Next we turn to the statistics of the escape time (or `last hitting
time'), which
can be defined as follows:
\begin{equation}
t_{\text{esc}}=\max\left\lbrace \tau\geq0:x(\tau)= a\right\rbrace,
\end{equation}
where $a>x_{eq}=0$. This gives us the last time that a sample path is within the
interval $[0,a]$. Thus
$t_{\text{esc}}$ provides a measure for the effective instability timescale,
more accurate than the inverse of the laminar growth
rate. If the system was laminar ($\zeta=0$), however, the escape time
would be simply $t_{\text{esc}}^{\text{lam}}= \log(a/x_0)$.
For geometric Brownian motion, it is possible to
derive the probability distribution of $t_{\text{esc}}$ analytically (Kennedy 2010, Profeta 2010).
Because this is not a standard calculation we go through its details
in the Appendix. Denoted by $g(\xi)$ where $\xi=t_{\text{esc}}/t_{\text{esc}}^{\text{lam}}$, it is
a modified form of the Rayleigh distribution:
\begin{align}\label{escGBM}
g(\xi)&=\frac{1}{\sqrt{2\pi d^2 \xi}}
\exp\left\{ -\frac{1}{2d^2}\left(\xi^{1/2}-\xi^{-1/2}\right)^2 \right\},
\end{align}
where the parameters $\sigma$, $x_0$, and $a$ have conveniently combined
into $d=\sigma\log(a/x_0)^{-1/2}$. In Fig. \ref{Fig::GBM_ET}
we plot $g(\xi)$ for a range of noise amplitudes.
The mean and variance of $g$ can be calculated analytically
\begin{align}
\mathbb{E}(\xi)&=1+d^2 \\
\textrm{Var}(\xi)&= d^2(1+ 2d^2).
\end{align}
It is clear that the mean of the distribution does not depart greatly from
the laminar escape time. Though the variance does increase with the
volatality of the noise, the spread is not especially
dramatic. Perhaps a more illuminating quantity is the kurtosis:
\begin{equation}
\text{Kurt}(\xi) = 3\,\frac{20d^2+9d^2+1}{(2d^2+1)^2},
\end{equation}
which varies from 3, when $d=0$, to 9, when $d=1$. A Gaussian
possesses a kurtosis of 3, so our escape time distribution can in fact
be exceptionally `fat-tailed', or `leptokurtic', meaning it generates
a significant number of outliers. Fig \ref{Fig::GBM_ET}
illustrates this point, with larger $\sigma$ giving rise to very
skewed and broad distributions. Inserted in the upper right
of the figure we
also plot the cumulative distribution function for various values of
$d$. For $d=1$, in particular, we see that only 30\% of systems
possess an escape time equal to or less than the laminar escape time ($\xi=1$),
while over 10\% of systems possess an escape time of four times or
more the laminar escape time. In fact, the probability that a
trajectory possesses a $t_{\text{esc}}/ t_{\text{esc}}^{\text{lam}}$
greater than some value
$\xi_0 $ may be approximated by the expression
$$ \mathbb{P}(\xi>\xi_0)\approx \frac{d}{\sqrt{2\pi\xi_0}}
\text{exp}\left(\frac{2}{d^2}-\frac{\xi_0}{2d^2}\right),$$
for large $\xi_0$. When $d=1$ and $\xi_0=10$, the
probability is only about 1\%. However, this rises to 10\% when $d=2$,
indicating the strongly nonlinear dependence of the statistics on
noise amplitude.
In summary, through the statistics of the
escape time, one can observe stochasticity
impeding instability over some initial period of the evolutio.
\begin{figure}
\includegraphics[width=9cm]{images/GBM_traj-eps-converted-to.pdf}
\caption{ The solid curves are sample trajectories of geometric Brownian motion with $x_{0}=0.1$ and $\sigma=0.5$. The dashed black line is the laminar path and the blue dashed curves are the $10$th and $90$th percentiles.}
\label{Fig::GBM_Traj}
\end{figure}
\begin{figure}
\includegraphics[width=9cm]{images/GBM_ET_CDF-eps-converted-to.pdf}
\caption{Probability distribution function of the escape times for
Geometric Brownian Motion with $x_{0}=0.1$ and $a=1.5$.}
\label{Fig::GBM_ET}
\end{figure}
\subsection{Random logistic equation}
We return to the more general problem by reintroducing the cooling
term with $\Theta=1$
\begin{equation}
\frac{dx}{d\tau}=\left[1+\zeta(\tau)\right]x^q-x^{m}
\label{eqn::M2dfg}
\end{equation}
Again assuming white noise for $\zeta$ and setting $q=1$,
we rewrite the ODE in differential form
\begin{equation}
dx=\left(x- x^{m}\right)d\tau+\sigma x dW.
\end{equation}
The equation was solved numerically, and sample trajectories are shown in
Fig. \ref{Fig::TI_traj}. With a fixed initial condition, the system can either
undergo runaway heating or cooling. Note that the initial perturbation does
not give a good indication of which direction the sample paths
eventually go. The trajectories in
Fig. \ref{Fig::TI_traj} qualitatively resemble those in Fig.
\ref{fig::UN_P_S} from the MHD simulations. Both show trajectories
that remain close to their equilibrium values for multiple thermal
timescales along with trajectories that escape from the equilibrium
faster than the laminar model. One substantial difference exists
however: Fig. \ref{Fig::TI_traj} shows no indication that the
trajectories will be in general slower than the laminar model as
is the case in Fig. \ref{fig::UN_P_S}.
Unlike geometric Brownian motion, it is difficult to analytically
calculate
the probability
density function, and instead this is accomplished
by solving the Fokker-Planck equation numerically
\begin{equation}
\frac{\partial f(x,\tau)}{\partial \tau}=-\frac{\partial}{\partial
x}\left[\left(x-x^{m}\right) f(x,\tau)\right]+\frac{\sigma^{2}}{2}
\frac{\partial^2}{\partial x^{2}}\left[x^{2}f(x,\tau)\right].
\end{equation}
In order to do this, an approximation for the initial distribution
must be made. Rather than using a Dirac-$\delta$ function as the initial
condition it is necessary to use a somewhat smoother function,
a Gaussian distribution with variance $1000$.
An example solution for the probability distribution $f$ is plotted
in Fig. \ref{Fig::TI_FP}. As earlier, the distribution becomes
increasingly wide and flat as time progresses, indicating the broad
range in possible evolutionary paths.
With the introduction of cooling comes the chance of a sample path
going to zero, hence, our escape time definition must be modified to include a lower boundary
\begin{equation}
t_{\text{esc}}=\sup\left\lbrace \tau\geq0:(x= a) \text{ or } (x=b)\right\rbrace
\end{equation}
where $a>x_{eq}>b$.
For a given sample path, the escape time is the last instance
when the solution is within the interval $[a,b]$, containing
$x_{eq}$. In Fig. \ref{Fig::TI_ET} we show estimated probability
density functions $g$ of $t_{\text{esc}}$ for a range of $\sigma$. These are
calculated from $100,000$ sample paths for each $\sigma$. As expected,
for small $\sigma$ we approach the laminar escape time, while as
$\sigma$ increases, $g$ undergoes asymmetric broadening,
shifting its maximum to
lower $t_{\text{esc}}$ and developing a tail at long $t_{\text{esc}}$. Qualitatively
this behaviour mirrors that shown for
geometric Brownian motion, Fig \ref{Fig::GBM_ET}. And so our
conclusions for GBM carry over to the more realistic logistic case.
\subsubsection{Variable $q$ and $m$}
In Equation \eqref{eqn::M2}, $q$ and $m$ are free
parameters which can be chosen to fit the system of interest. This
flexibility
allows it to approximately describe a range of different scenarios.
For
instance, the classical Shakura \& Sunyaev disk (Shakura \& Sunyaev
1973) can be represented with $q=2$ and $m=1$. Alternatively, $q$ and
$m$ could be chosen so as to model the vertically stratified, radiation-pressure
dominated simulations of Jiang et al. (2013, 2016). In this case a
large surface density disk yields $q=1.6$ and $m=0.98$, while a less dense
disk gives $q=1.9$ and $m=0.9$. Finally, when the temperature of the
gas ensures the
opacity is influenced by the `iron bump', the scaling of cooling with central
pressure is found to greatly exceed that of the classical Shukura \&
Sunyaev model and $m=1.89$ (Jiang et al.~2016). In
Fig. \ref{Fig::qm_ET} we plot the probability distribution function
of the escape time $g$ for various choices of $q$ and $m$. As $m$
approaches $q$ from below, the escape time distribution becomes
increasingly elongated to large
$t_{\text{esc}}/t^{\text{lam}}_{\text{esc}}$.
The deterministic component
weakens and the stochastic drift becomes more important, dominating the
results.
To better quantify this effect we compute the probability
that a given trajectory possesses a $t_{\text{esc}}$ greater than
various multiples of $t^{\text{lam}}_{\text{esc}}$ and plot these
probabilities as a function of $m$ for fixed $q=2$. These results
appear in Fig.~\ref{Fig::remain}. The curves show that as $m$
approaches $q$ the escape time can be significantly enhanced. For
$m=1.9$ there is approximately $45\%$ chance that it is five times
the laminar prediction, and $5\%$ chance that is ten times greater.
It should be noted that as $m$ approaches $q$ the laminar instability
timescale itself can be significantly longer than the thermal
time $(\alpha\Omega)^{-1}$, which further separates the expected
turbulent $t_{\text{esc}}$ from the thermal time. The stability
uncovered in Jiang et al.~(2016), on the timescales of their
simulations, can be easily explained by the enhanced delay witnessed
in such
marginally unstable systems.
\begin{figure}
\includegraphics[width=9cm]{images/TI_traj-eps-converted-to.pdf}
\caption{Trajectories for the logistic model, $\sigma=0.2$ with $x_{0}=1.1$}
\label{Fig::TI_traj}
\end{figure}
\begin{figure}
\includegraphics[width=9cm]{images/TI_FP1-eps-converted-to.pdf}
\caption{Time evolution of the logarithm of the probability density
function $f(\tau,x)$ for the logistic equation with $x_{0}=1.1$, $\sigma=0.1$.}
\label{Fig::TI_FP}
\end{figure}
\begin{figure}
\includegraphics[width=9cm]{images/TI_ET-eps-converted-to.pdf}
\caption{Probability distribution of the escape time $g$
for the logistic model, with $x_{0}=1.4$ with $a=4$ and $b=0.25$.}
\label{Fig::TI_ET}
\end{figure}
\begin{figure}
\includegraphics[width=8cm]{images/qm-eps-converted-to.pdf}
\caption{Probability distribution of the escape time $g$
for the logistic model with various $q$ and $m$ but keeping fixed:
$x_{0}=1.4$, $\sigma=0.2$, $a=4$, and $b=0.25$.}
\label{Fig::qm_ET}
\end{figure}
\begin{figure}
\includegraphics[width=9cm]{images/Remain-eps-converted-to.pdf}
\caption{The probability that $t_{\text{esc}}$ is longer than a
multiple of $t_{\text{esc}}^{\text{lam}}$ as a function of $m$ for
fixed $q=2$.}
\label{Fig::remain}
\end{figure}
\begin{figure}
\includegraphics[width=8cm]{images/PSD-eps-converted-to.pdf}
\caption{Time averaged power spectral density normalised by $P_{0}$ from simulation R1. Slope $\propto f^{-1.2}$}
\label{Fig::PSD}
\end{figure}
\begin{figure}
\includegraphics[width=8cm]{images/PSD_ET_5-eps-converted-to.pdf}
\caption{Escape time distribution from our power spectral density model with $q=0.5$ and $m=0.25$}
\label{Fig::PSD_ET}
\end{figure}
\subsection{MHD power spectral density}
The next step is to replace the white noise with the power spectral
density (PSD) of $\Pi_{xy}$ calculated from the MHD simulations of
Section 4.
In Fig. \ref{Fig::PSD} we show the PSD of $\Pi_{xy}$ from
simulation R1. We then represent
the fluctuating term in Equation \eqref{eqn::M2} by
\begin{align}
\zeta(\tau)&=\frac{\sigma}{N}\Sigma_{i=1}^{200} \alpha_{i}(f_{i}) \cos\left( 2\pi f_{i}+\phi_{i}\right),\label{eqn::PDS} \\
N&=\left(\Sigma_{i=1}^{200}\alpha_{i}(f_{i})^{2}\right)^{1/2}
\end{align}
where $f_{i}$ are the frequencies, logarithmically spaced between $\left[0.05,
750\right]$, and $\phi_{i}$ are phase shifts, chosen randomly
from a
uniform distribution on the interval $\left[0, 2\pi\right]$.
Finally, the constant amplitudes $\alpha_{i}(f_{i})$ are
calculated from a two slope power law that fits our
MHD simulations (constant for $2\pi f/\Omega<0.1$ and $\propto
f^{-1.2}$ otherwise), which is then multiplied by a random number
generated from a uniform distribution on the interval $[0,1]$.
To improve the comparison with the simulations R2a-R2h, we set $q=0.5$
and $m=0.25$
and thus our model equation is
\begin{equation}
\frac{dx}{d\tau}=\left[1+\zeta(\tau)\right]x^{0.5}-x^{0.25},
\label{eqn::M2l}
\end{equation}
where $\zeta(\tau)$ is defined in Equation $\eqref{eqn::PDS}$. This
equation is evolved forward in time $50,000$ times for each choice of $\sigma$
in order to derive adequate statistics, especially for the escape time.
We show the resulting escape time distribution in
Fig. \ref{Fig::PSD_ET}. We find similar behaviour to both
the GBM and the random logistic equation. As $\sigma$ increases, the
distribution broadens and its tail at large $t_{\text{esc}}$
increases. We estimate that $\sigma\sim0.015$ gives approximately the
correct fluctuation amplitude when compared to the stress in
Fig. \ref{Fig::r1_p}. For this choice, Fig. \ref{Fig::PSD_ET}
shows a clear tail reaching $t_{\text{esc}}/t_{\text{esc}}^{\text{lam}}\sim2.5-3$.
We
conclude that fat-tailed distributions are a generic feature of the
escape time in turbulent but thermally unstable systems. Such systems
produce a broad range of outcomes, and instability can be delayed for
several instability timescales. Being fat-tailed they also exhibit
significant outliers --- systems that `hang around' for surprisingly
long times before wandering away.
\section{Discussion and Conclusions} \label{Sec::TIDC}
We have performed a set of idealised shearing box simulations of the MRI
in order to explore the effects of
turbulent variability on thermal instability. Our main aim was to
check if turbulence interferes with the thermal runaway
predicted by laminar theory. In our
simulations heating comes from numerical dissipation while
a power law cooling imitates optically thin radiative
cooling. Relatively large computational domains are used in
order to ensure a strong dependence of stress on pressure.
Simulations with
an expected stable thermal equilibrium are found to fluctuate around
their fixed points. When the cooling power law exponent is decreased,
and the laminar analysis predicts instability, our simulations
indeed show thermal runaways. However, the
system trajectories deviate from the laminar
theory. The turbulent fluctuations can stall the onset of instability
in a large number of cases, for multiple thermal times. To better
account for the instability timescale, we introduce the concept of the
escape time which we define to be the last time the system leaves
an interval encapsulating the equilibrium in phase space.
Our simulations show a large range of escape
times ranging from $\sim 1/4$ to $\sim 5$ times the laminar thermal
timescale, for relatively narrow intervals.
Further reducing the cooling power law exponent results in disk
fragmentation. This is a due to localised imbalances between
heating and cooling; the instability timescale is shorter than the
mixing timescale and hence distant pockets of fluid evolve
independently.
Very rough estimates indicate that thermal fragmentation on length scales larger
than $H$ are at best marginally possible in the inner regions
of X-ray binaries.
To better understand our results we construct a probabilistic theory centred
on
simple stochastic equations that approximate the box-averaged thermal
equation. These present us with a much larger sample of possible
system outcomes to analyse than the MHD simulations can afford.
First we analyse geometric Brownian motion (GBM),
which possesses the main ingredients of interest (an
unstable fixed point and stochastic variations), while remaining
analytically tractable. The distribution function of $t_{\text{esc}}$ exhibits
a variance proportional to the square of the noise amplitude and
considerable kurtosis. In general the distribution is `fat-tailed',
permitting many instances of delayed thermal instability, and outliers
for whom the escape time can be $\sim 10$ thermal times.
Our second model introduces a logistic
term to incorporate a power law cooling, and a third model
replaces the white noise with the power spectral density of
the stress from a thermally stable shearing box simulation. In both
cases we obtain qualitatively similar behaviour to before,
which instils confidence that its behaviour is generic to
systems sustaining noise and instability. GBM
may be thought of as the
model equation for such systems.
In our GBM model, very large
amplitude noise can stabilize the thermal instability, but we
believe this is physically implausible. In fact, this behaviour
arises from the special combination of Ito calculus and multiplicative white
noise. It is not generic.
While GBM is a convenient model for
`unresolved' turbulence, this dynamical peculiarity must be discarded when
applying results to real systems. The stochastic process
employed by Janiuk \& Misra (2012) shares the same properties and thus
suffers
the same dynamical artefact. The stabilisation witnessed in their
simulations we hence view as unphysical.
The bulk of X-ray binary observations show no indication of limit
cycles that could correspond to radiation-pressure induced
thermal instability
(Gierli\'{n}ski \& Done 2004). Notable exception are GRS 1915+105
(Done et al.~2004) and HLX-1 (Sun et al.~2016). We show that it is
unlikely that
turbulent fluctuations alone stabilise these disks.
However, turbulence can delay and weaken instability.
This weakening might be
significant
in combination with additional
stabilising mechanisms, for example: energy lost in the disk corona or by outflows,
magnetic buoyancy effects, alteration of the pressure-stress
relationship by strong magnetic fields, or opacity shifts near the
iron bump (Svensson \& Zdziarski 1994, Sadowski 2016, Jiang and Stone
2016). Finally, separate and important physical effects may arise from the
global nature of the flow, especially from accretion. The viscous timescale
is probably longer than a typical $t_{\text{esc}}$, as measured in this
paper. But if further delayed by additional physics, $t_{\text{esc}}$
could in some circumstances
approach the
accretion time. If this occurs then the classic limit cycle behaviour expected
from thermal instability could well be impeded, and/or pushed to smaller
radii.
\section*{Acknowledgements}
The authors thank the anonymous reviewer for
a set of helpful comments. They also thank Yanfei Jiang
and Jim Stone for discussions that helped sharpen the paper.
This work was partially funded by STFC
grants ST/L000636/1 and ST/K501906/1.
Some of the simulations were run on the
DiRAC Complexity system, operated by the University of Leicester
IT Services, which forms part of the STFC DiRAC HPC Facility
(www.dirac.ac.uk). This equipment is funded by BIS National E-
Infrastructure capital grant ST/K000373/1 and STFC DiRAC Op-
erations grant ST/K0003259/1. DiRAC is part of the UK National
E-Infrastructure.
\bibliographystyle{apalike}
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\section{Introduction}
\label{sec:introduction}
Partitionability (see \Cref{sec:preliminaries} for further details) is a combinatorial property of simplicial complexes that arose in the 1970s \cite{Ball:1977} \cite[App. 4]{Provan:1977} and appears early in the study of shellable and Cohen-Macaulay complexes \cite{Bjorner/Garsia/Stanley:1982,Stanley:1979}.
It is known from its occurrence in the Stanley Partitioning Conjecture, which asks whether every Cohen-Macaulay complex is partitionable \cite[p. 149]{Stanley:1979}.
Garsia asks for a similar question for order complexes in \cite[Remark 5.2]{Garsia:1980}.
If this conjecture were true, it would have explained combinatorially why the $h$-vector has non-negative entries.
The fact that shellable simplicial complexes are both Cohen-Macaulay and partitionable seems to reinforce the idea.
However, the Cohen-Macaulay yet non-partitionable counterexample constructed by Duval, Goeckner, Klivans and Martin in \cite{Duval/Goeckner/Klivans/Martin:2016} (see also \cite{Duval/Klivans/Martin:2017}) disproves the conjecture.
Although \cite{Duval/Goeckner/Klivans/Martin:2016} settles the Stanley Partitionability Conjecture, it raises some other questions.
For instance, we still do not know if the Stanley Conjecture holds when restricted to $2$-dimensional complexes.
This question motivates the current paper, where we show that all triangulations of certain well known $2$-dimensional, non-shellable, Cohen-Macaulay spaces are partitionable.
Our techniques are complementary to that of \cite{Duval/Goeckner/Klivans/Martin:2016}, where the authors built their counterexamples by gluing together several shellable $3$-balls along a common shellable $2$-ball.
Their gluing process carries over the Cohen-Macaulay property from the original complexes to the new one, but it breaks partitionability.
Their proof uses the framework of relative simplicial complexes to put restrictions on a partitioning scheme for the counterexample complex, eventually arriving at a contradiction.
In contrast, we will develop tools in \Cref{stt:gluing_lemma,stt:assymetrical_gluing_lemma,stt:folding_lemma} that give sufficient conditions for producing partitioning schemes on and from constituent relative complexes.
The first two lemmas carry over partitioning schemes of two relative complexes to a larger one, while the last lemma gives conditions for a quotient map to preserve a partitioning scheme.
These lemmas will allow us to use a ``splitting-then-gluing-back'' strategy that improves the naïve facet-by-facet approach.
Indeed, these techniques can be seen as a broad generalization of the partitioning aspects of shellability.
We use the splitting and gluing approach on triangulations of the real projective plane $\Pplane$ and of the dunce hat $\DunceH$.
Although particular triangulations of these spaces have long been known to be partitionable (see \cite{Hachimori:web}), we prove a considerably more general result.
\begin{theorem}
If $\Delta$ is any triangulation of the real projective plane $\Pplane$ or of the dunce hat $\DunceH$, then $\Delta$ is partitionable.
\label{stt:main_result}
\end{theorem}
As an intermediate result, and using the same machinery, we prove partitionability of any triangulation of the open Möbius strip.
Here the word \emph{open} means that we work relative to the boundary of the space.
This paper is structured as follows.
In \Cref{sec:preliminaries} we give a basic background on partitionability, relative complexes and related notions.
The main tools we use are introduced and developed in \Cref{sec:partitioning_preserving_tools}.
In \Cref{sec:partitionable_surfaces} we show that the $2$-disk and the Möbius strip are partitionable relative to certain portions of their respective boundaries.
Finally, in \Cref{sec:main_theorem} we prove \Cref{stt:main_result}.
\numberwithin{theorem}{section}
\section{Preliminaries}
\label{sec:preliminaries}
In this section we succinctly present the main notions we require throughout the paper.
We refer to \cite{Bjorner:1995, Stanley:1996} for additional background and notational conventions on simplicial combinatorics.
An \emph{(abstract) simplicial complex} $\Delta$ is a collection of subsets closed under inclusion whose elements are taken from a finite ground set.
For its standard geometric realization we use the notation $|\Delta|$.
The elements of $\Delta$ are called \emph{faces} and the maximal faces are called \emph{facets}.
By abuse of notation, when no confusion is induced, we denote the face $\{v_{1},v_{2},\dotsc,v_{k}\}$ as $v_{1}v_{2} \dotso v_{k}$, for $k \geq 1$.
To make this more visible in our concrete examples, we will use monospaced typeface numerals.
As usual, faces with one, two and three elements are called \emph{vertices}, \emph{edges} and \emph{triangles}, respectively.
The collection \emph{generated} by a family of sets $\Lambda$ consists of all subsets of the sets in $\Lambda$, and it is denoted by $\gSet{\Lambda}$.
This collection is a simplicial complex, sometimes called the \emph{combinatorial closure} of $\Lambda$.
The \emph{dimension} $\dim \sigma$ of a face $\sigma \in \Delta$ is one less than the cardinality of $\sigma$, and the \emph{dimension} of $\Delta$, denoted as $\dim \Delta$, is the maximum dimension among the faces of $\Delta$.
A complex is called \emph{pure} if all its facets have the same dimension.
We are mostly interested in pure complexes, although some of the results in this paper will also hold for non-pure complexes.
The \emph{link} of a face $\sigma \in \Delta$ is the complex defined by
$\lk_\Delta(\sigma)
\colonequals
\{\tau \in \Delta \, : \, \tau \cap \sigma = \varnothing,\, \tau \cup \sigma \in \Delta \}$,
and the \emph{deletion} of $\sigma$ is the complex
$\del_\Delta(\sigma)
\colonequals
\{\tau \in \Delta \, : \, \tau \cap \sigma = \varnothing \}$.
The \emph{cone} of a complex with \emph{apex} $u$ is defined by
$u*\Delta
\colonequals
\{\{u\} \cup \sigma \, : \, \sigma \in \Delta\} \, \cup \, \Delta$, $\{u\} \notin \Delta$.
\subsection{Partitionability and shellability}
\label{sec:partitionability_and_shellability}
A simplicial complex admits a partial order by inclusion in a natural way, and we regard a simplicial complex as equivalent to its \emph{face poset}.
We are interested in the problem of partitioning $\Delta$ (i.e.\ its face poset) into intervals of the form
$\left[R(\sigma),\sigma\right]
\colonequals
\{\rho \in \Delta \, : \, R(\sigma) \subseteq \rho \subseteq \sigma,\; \text{where } \sigma \text{ is a facet in } \Delta \}$.
Notice that an interval in the face poset of a simplicial complex is isomorphic to the Boolean lattice $B_{n}$, where $n$, the rank of $B_{n}$, is the cardinality of $\sigma \setminus R(\sigma)$.
If $\Delta$ can be partitioned in such a way, we say that $\Delta$ is \emph{partitionable}, and the set of intervals is a \emph{partitioning scheme} (or a \emph{partitioning}, for short) of $\Delta$.
\Cref{fig:graph_example} shows a partitionable simplicial complex with a partitioning scheme.
The \emph{$f$-vector} of a $(d-1)$-dimensional complex $\Delta$ is
$f(\Delta) \colonequals \left(f_{-1}, f_{0}, \dotsc, f_{d-1} \right)$,
where $f_i$ denotes the number of $i$-dimensional faces of $\Delta$.
The entry $f_{-1}$ is $0$ if $\Delta$ is empty and $1$ otherwise.
The \emph{(reduced) Euler characteristic} of $\Delta$ is defined as
$\tilde{\chi}(\Delta) \colonequals \sum\nolimits_{i=0}^{d} {(-1)^{i-1} f_{i-1}}$.
This value is a topological invariant of the space that $\Delta$ triangulates, so the choice of triangulation does not modify the Euler characteristic.
The $h$-vector of $\Delta$ is
$h(\Delta) \colonequals \left(h_{0}, h_{1}, \dotsc, h_{d}\right)$
where its entries are obtained from the relation
\begin{equation}
\sum_{i=0}^{d} {f_{i-1} (t-1)^{d-i}} = \sum_{i=0}^{d} {h_{i} t^{d-i}}.
\label{equ:h_pol_and_f_pol}
\end{equation}
In particular, $h_d = (-1)^{d-1}\tilde{\chi} (\Delta)$ and $f_{d-1} = \sum\nolimits_{i=0}^{d} h_i$.
The $h$-vector of a pure partitionable complex has a combinatorial meaning: the $i$th entry of $h(\Delta)$ counts how many intervals in a partitioning of $\Delta$ have a minimal face $R(\cdot)$ with dimension $i-1$.
This is true regardless of the chosen partitioning scheme.
\begin{example}
The graph ($1$-dimensional complex) appearing in \Cref{fig:graph_example} (a) has $h$-vector $\left(1, 3, 0\right)$.
Thus, if the graph is partitionable, any partitioning must match one edge to the empty set, three edges to one incident vertex each, and no edge matches to itself.
In this case, a partitioning exists and it is shown in \Cref{fig:graph_example} (b) and (c).
\begin{figure}[!ht]
\begin{center}
\centering
\begin{tabular}{ >{\centering\arraybackslash}m{4.5cm}
>{\centering\arraybackslash}m{5.0cm}
>{\centering\arraybackslash}m{3.0cm}
}
\vspace{ 0.20em} \includegraphics[scale=0.40]{./images/example_partitionable_non-shellable_01.pdf} &
\vspace{ 0.00em} \includegraphics[scale=0.46]{./images/example_partitionable_non-shellable_02.pdf} &
{$\!\begin{aligned}
\left[ \mathtt{\varnothing, 12} \right] \ & \sqcup \\
\left[ \mathtt{3, 34} \right] \ & \sqcup \\
\left[ \mathtt{4, 45} \right] \ & \sqcup \\
\left[ \mathtt{5, 35} \right] \ &
\end{aligned}$}
\vspace{ 0.20em}
\\
(a) & (b) & (c)
\end{tabular}
\caption{
A partitionable $1$-dimensional complex (a).
Its partitioning scheme is depicted in (b) and expressed as the union of intervals in (c).}
\label{fig:graph_example}
\end{center}
\end{figure}
In general, it is known that a graph is partitionable if and only if at most one of its connected components is a tree \cite[Problem 35]{Kaibel/Pfetsch:2003}.
See also \Cref{stt:part_graphs_resp_empty_face}.
\label{stt:graphs_are_partitionable_ex}
\end{example}
\begin{example}
The \emph{bow-tie} complex $\Delta = \gSet{\mathtt{125, 345}}$ is non-partitionable since $h(\Delta) = (1,2,-1,0)$ has negative entries.
\label{stt:bow-tie_partitionable_ex}
\end{example}
A notion related to partitionability is that of shellability.
A simplicial complex $\Delta$ is \emph{shellable} if its facets can be ordered linearly in such a way that the complex generated by the facet $\sigma_{j}$, for each $j > 1$, intersects in a pure $(\dim\sigma_j-1)$-dimensional complex with the complex generated by the previous $j-1$ facets in the ordering.
The $j$th step in the shelling process adds a unique minimal new face $R(\sigma_{j})$.
Thus, every time a facet $\sigma_{j}$ is attached to the shelling it introduces the interval $[R(\sigma_{j}), \sigma_{j}]$ with just new faces.
Hence, the shelling process inductively produces a partitioning scheme of $\Delta$.
In a recent work, Hachimori defines several properties satisfied by partitioning schemes induced from shellings \cite{Hachimori:2021}.
Although every shellable simplicial complex is partitionable, the converse is not true.
\Cref{stt:graphs_are_partitionable_ex} shows a partitionable but non-shellable simplicial complex.
In general, it is known that a graph is shellable if and only if it is connected \cite[Problem 34]{Kaibel/Pfetsch:2003}.
See a related result in \Cref{stt:shell_graphs_resp_empty_face}.
It is also useful to remember that the cone operation preserves shellability and partitionability.
Furthermore, a complex is partitionable (shellable) if and only if its cone is.
\subsection{Relative complexes}
\label{sec:relative_complexes}
A \emph{relative simplicial complex} $\Phi = \left(\Delta,\Gamma\right)$ is a pair of simplicial complexes, where $\Gamma$ is a subcomplex of $\Delta$.
If $\Gamma \neq \varnothing$, the relative complex $\Phi$ is said to be \emph{proper}.
Lacking a better way to refer to $\Gamma$, we say that $\Gamma$ is the \emph{relative part} of $\Phi$.
We regard the usual (non-relative) simplicial complex as the relative simplicial complex with $\Gamma = \varnothing$.
The \emph{faces} of $\left(\Delta,\Gamma\right)$ are the faces of $\Delta$ that are not in $\Gamma$, and the maximal faces are the \emph{facets} of $\left(\Delta,\Gamma\right)$.
We say that $\left(\Delta,\Gamma\right)$ is \emph{pure} if its facets have the same dimension.
We use the frequently overloaded symbol $\Delta \setminus \Gamma$ to denote the set of faces of $\left(\Delta,\Gamma\right)$, namely by taking $\Delta$ and $\Gamma$ as set systems.
We will use this notation even if $\Gamma$ is not a subcomplex of $\Delta$.
We define the \emph{minimal representation} of $\left(\Delta,\Gamma\right)$ as the complex $\left(\gSet{\Lambda},\gSet{\Lambda} \setminus \Lambda \right)$
where $\Lambda \colonequals \Delta \setminus \Gamma$.
The relative complex $\left( \Delta,\Gamma \right)$ can be seen as a combinatorial model of the topological pair of spaces $\left(|\Delta| , |\Gamma| \right)$
(see, e.g., \cite[\S 2.1--2.2]{Hatcher:2002} and \cite[\S 1.9]{Munkres:1984}).
This relative notion is used in algebraic topology to calculate the homology of the quotient space $|\Delta| \, / \, |\Gamma|$.
Furthermore, relative complexes offer a natural setting to build up new complexes with a given property, as in \cite{Duval:1999,Duval:2000,Duval/Goeckner/Klivans/Martin:2016,Juhnke-Kubitzke/Venturello:2019,Santamaria/Woodroofe:2021}.
The combinatorics of non-relative simplicial complexes carries over straightforwardly to their relative counterparts, including notions such as face poset and partitionability.
The \emph{face poset} of the relative complex $\Phi = \left( \Delta, \Gamma \right)$ is the set system $\Delta \setminus \Gamma$ ordered by inclusion.
This partial order of the elements of $\Phi$ with respect to the inclusion is order-convex, i.e.\ if $\sigma, \tau \in \Phi$, and $\sigma \subseteq \rho \subseteq \tau$, then $\rho \in \Phi$.
A relative complex $\Phi = \left( \Delta, \Gamma \right)$ is said to be \emph{partitionable} if its face poset can be written as the disjoint union of intervals in such a way that the top element of each is a facet of $\Phi$.
We also say that $\Delta$ \emph{is partitionable relative to} (or \emph{with respect to}) $\Gamma$.
Such a disjoint union of intervals is a \emph{partitioning scheme} of the relative complex (or a \emph{partitioning}, for short).
\Cref{fig:isomorphic_posets} shows a partitionable relative simplicial complex (d) with a partitioning scheme drawn over its face poset (e) and written as a disjoint union of intervals (f).
Similar to the non-relative setting, the \emph{$f$-vector} of a relative complex $f(\Delta,\Gamma)$ stores the number of faces of a given dimension of $\left(\Delta,\Gamma\right)$, and the \emph{$h$-vector} $h(\Delta,\Gamma)$ is determined by the $f$-vector through \labelcref{equ:h_pol_and_f_pol}.
When $(\Delta,\Gamma)$ is pure, the $h$-vector has the same combinatorial meaning as its non-relative pure counterpart, namely the $i$th entry counts the number of intervals in a partitioning scheme of $\left(\Delta,\Gamma\right)$ having minimal face $R(\cdot)$ with dimension $i - 1$.
Shellability can also be defined for relative complexes.
If $\Gamma = \varnothing$, we have the usual non-relative notion.
Let $\Gamma \neq \varnothing$.
An ordering $\sigma_{1}, \dotsc, \sigma_{m}$ of the facets of the complex $\left(\Delta,\Gamma\right)$ is a \emph{shelling} if, for each $j \geq 1$, the facet $\sigma_{j}$ intersects in a pure $(\dim\sigma_j-1)$-dimensional complex with the complex $\Lambda_{j-1}$
generated by $\sigma_{1}, \dotsc, \sigma_{j-1}$ together with $\Gamma$.
Set $\Lambda_{0} \colonequals \Gamma$.
If a shelling exists, we say that $\left(\Delta,\Gamma\right)$ is \emph{shellable}, or that $\Delta$ \emph{is shellable relative to} (or \emph{with respect to}) $\Gamma$.
From the definition, it is clear that the $j$th step of the shelling attaches all the faces from
$\langle \sigma_{j} \rangle \setminus \Lambda_{j-1}$ to $\Lambda_{j-1}$, and a unique minimal face $R(\sigma_{j}) \in \sigma_{j}$ is included into the shelling (just as in its non-relative counterpart).
Thus $\langle \sigma_{j} \rangle \setminus \Lambda_{j-1} = [R(\sigma_{j}), \sigma_{j}]$.
As a consequence, a shellable relative simplicial complex is also partitionable.
\begin{remark}
We digress here with an algorithmic description for the construction of partitioning schemes of relative simplicial complexes that fits with the approach of this paper.
Consider a relative simplicial complex $\left(\Delta, \Gamma \right)$.
We could think of a partitioning scheme as the outcome of a procedure that constantly takes facets from $\Delta$ to place them into $\Gamma$:
Start a partitioning of $\left(\Delta, \Gamma \right)$ by choosing a suitable facet $\sigma$.
Since continuing the partitioning is equivalent to considering the problem anew for the complex $\Phi = \left( \Delta, \Gamma \cup \gSet{\sigma} \right)$, we can naturally continue the process by recursively placing a new suitable facet into the relative part of $\Phi$.
The process ends when the first part and the second part of the pair coincide.
If any failure appears, backtrack and select the next suitable facet.
See the proofs of \Cref{stt:rel_disk_is_partitionable} and \Cref{stt:links_of_balls_as_balls} for a related approach.
Alternatively, we may consider the minimal representation of the complex every time a new facet is included.
In this case, the procedure ends when we reach the empty complex.
This alternative is also possible given the poset isomorphism between a relative simplicial complex and its minimal representation.
See \Cref{stt:isoposets_partitionable} and \Cref{stt:conseq_isoposets_part_ex} (\labelcref{stt:part_minimal_representation}).
We consider worth mentioning in this side note that an integer programming routine to decide partitionability has been implemented for SageMath \cite{sagemath}.
\label{stt:algorithm_feeding_rel_part}
\end{remark}
\subsection{Triangulations on surfaces}
\label{sec:triangulations_on_surfaces}
In this work we are concerned with partitionability of finite triangulations of concrete topological spaces.
We say that a \emph{(simplicial) triangulation} of a space $\SpaceX$ is a simplicial complex
$\TriangX$ whose underlying space is $\SpaceX$, i.e.\ $|\TriangX| \cong X$.
We relax the language to say that a topological space is \emph{partitionable} (or \emph{shellable}) when all of its finite triangulations are.
Again, we extend these notions to their relative counterparts when required.
In \Cref{sec:partitioning_pplane} we are going to need familiar results on surfaces with and without boundary.
Given that every surface can be triangulated (see e.g.\ \cite{Thomassen:1992}), the homeomorphism between surfaces can be reduced to the combinatorial comparison of certain topological invariants provided by the classification theorem for compact surfaces.
We refer the reader to \cite{Gallier/Xu:2013} for further details.
\section{Tools that preserve partitioning schemes}
\label{sec:partitioning_preserving_tools}
We start this section with two lemmas that give conditions under which we can glue complexes while preserving partitioning schemes.
They build a (relative) partitionable simplicial complex out of two partitionable relative simplicial complexes.
\begin{lemma}[Gluing Lemma]
Let $\Phi_{a} = \left(\Delta_{a}, \Gamma_{a} \right)$ and
$\Phi_{b} = \left(\Delta_{b}, \Gamma_{b} \right)$
be partitionable complexes such that no facet of one is properly contained in a facet of the other, and let
$\Sigma \colonequals
\left( \Gamma_{a} \setminus \Delta_{b} \right) \, \cup \,
\left( \Gamma_{b} \setminus \Delta_{a} \right) \, \cup \,
\left( \Gamma_{a} \cap \Gamma_{b} \right)$.
The complex
$\Phi = \left( \Delta_{a} \cup \Delta_{b}, \Sigma \right)$
is partitionable if the following conditions are met:
\begin{enumerate}[label=\tt(G\arabic*)]
\item \label{con:intersection_embedded}
$\Gamma_{a} \cup \Gamma_{b} \supseteq \Delta_{a} \cap \Delta_{b}$.
\item \label{con:partition}
$\Sigma$ is a subcomplex of $\Delta_{a} \cup \Delta_{b}$.
In particular, the set system $\Sigma$ is a simplicial complex.
\end{enumerate}
\label{stt:gluing_lemma}
\end{lemma}
\begin{proof}
It is sufficient to show that the union of the partitioning schemes of $\Phi_{a}$ and $\Phi_{b}$ forms a partitioning scheme of $\Phi$.
First, we want to prove that the faces of $\Phi_{a}$ are disjoint from those of $\Phi_{b}$.
By \labelcref{con:intersection_embedded} and properties of set difference
$\left(\Delta_{a} \setminus \Gamma_{a}\right) \, \cap \, \left(\Delta_{b} \setminus \Gamma_{b} \right) =
\left( \Delta_{a} \cap \Delta_{b} \right) \, \setminus \,
\left( \Gamma_{a} \cup \Gamma_{b} \right) = \varnothing$.
Thus, it is clear that no interval in the partitioning of $\Phi_{a}$ shares a face with an interval from $\Phi_{b}$.
Moreover, since the facets of $\Phi_{a}$ and $\Phi_{b}$ are facets in $\Phi$, the partitioning schemes of each can be directly carried over into $\Phi$.
Now we need to check that $\Phi$ is the union of the $\Phi_{a}$ and $\Phi_{b}$.
Notice that any face in
$\Lambda \colonequals \left(\Delta_{a} \setminus \Gamma_{a}\right) \, \cup \, \left(\Delta_{b} \setminus \Gamma_{b} \right)$
is a face of $\Delta_{a} \cup \Delta_{b}$.
Then, by elementary set theory, the faces of $\Delta_{a} \cup \Delta_{b}$ we are missing in $\Lambda$ are precisely those in $\Sigma$.
Hence, as a set system, the faces of $\Phi$ are those in $\left(\Delta_{a} \cup \Delta_{b}\right) \setminus \Sigma$.
The condition on $\Sigma$ in \labelcref{con:partition} makes $\Phi$ an actual relative simplicial complex.
\end{proof}
\begin{remark}
Unless $\Delta_{a}$ or $\Delta_{b}$ is the empty complex, the intersection $\Delta_{a} \cap \Delta_{b}$ is never empty since $\varnothing$ is always a face of a nonempty simplicial complex.
Hence, \Cref{stt:gluing_lemma} \labelcref{con:intersection_embedded} forces at least one of the complexes $\Phi_{a}$ or $\Phi_{b}$ to be a proper relative simplicial complex.
\label{stt:intersection_deltas_non_empty}
\end{remark}
Since the partitioning schemes of the relative simplicial complexes of $\Phi_{a}$ and $\Phi_{b}$ do not mingle with each other, it is immediate to see the following:
\begin{corollary}
If $\Phi_{a}, \Phi_{b}$ and $\Phi$ as in \Cref{stt:gluing_lemma} are pure and of the same dimension, then
$h(\Phi) = h(\Phi_{a}) + h(\Phi_{b})$.
\label{stt:adding_up_h-vectors}
\end{corollary}
We will often use a particular case of \Cref{stt:gluing_lemma}.
For the shelling portion of the statement, see the previous work of the author with Woodroofe in \cite[Lemma 2.3]{Santamaria/Woodroofe:2021}.
\begin{lemma}[Shelling-like Gluing Lemma]
Let
$\Phi_{a} = \left( \Delta_{a}, \Gamma_{a} \right)$ and
$\Phi_{b} = \left( \Delta_{b}, \left( \Delta_{a} \cap \Delta_{b} \right) \cup \Gamma_{b} \right)$
be partitionable complexes such that no facet of one is properly contained in a facet of the other, where $\Gamma_{b}$ is a subcomplex of $\Delta_{b}$.
If
$\Delta_{a} \cap \Gamma_{b} \subseteq \Gamma_{a}$, the complex
$\Phi = \left( \Delta_{a} \cup \Delta_{b}, \Gamma_{a} \cup \Gamma_{b} \right)$
is partitionable.
Furthermore, if $\Phi_{a}$ and $\Phi_{b}$ are shellable, then $\Phi$ is also shellable.
\label{stt:assymetrical_gluing_lemma}
\end{lemma}
\begin{proof}
The conditions in \Cref{stt:gluing_lemma} are met by $\Phi_{a}$ and $\Phi_{b}$ as follows:
the relative part of $\Phi_{b}$ includes $\Delta_{a} \cap \Delta_{b}$ by definition, so the condition \labelcref{con:intersection_embedded} holds.
It remains to prove that the complex $\Sigma$ in \labelcref{con:partition} (i.e.\ the relative part of $\Phi$) is actually $\Gamma_{a} \cup \Gamma_{b}$.
To do this, we will treat the complexes as set systems as in \Cref{stt:gluing_lemma}.
Let $\Gamma'_{b}$ be the relative part of $\Phi_{b}$, so $\Gamma'_{b} \colonequals \left( \Delta_{a} \cap \Delta_{b} \right) \cup \Gamma_{b}$, then
\begin{equation}
\Sigma
\, = \,
\left( \Gamma_{a} \setminus \Delta_{b} \right)
\, \cup \,
\left( \Gamma'_{b} \setminus \Delta_{a} \right)
\, \cup \,
\left( \Gamma_{a} \cap \Gamma'_{b} \right).
\label{equ:union_shelling_like_lemma}
\end{equation}
After some set-based computations, we have that
$\Gamma_{a} \cap \Gamma'_{b}
\, = \,
\left( \Delta_{b} \cap \Gamma_{a}\right)
\, \cup \,
\left( \Delta_{a} \cap \Gamma_{b} \right)$.
Then, since
$\Delta_{a} \cap \Gamma_{b} \subseteq
\Delta_{b} \cap \Gamma_{a} \subseteq
\Gamma_{a}$,
it follows that
$\Gamma_{a} \cap \Gamma'_{b}
\, = \,
\Delta_{b} \cap \Gamma_{a}$.
Thus, the union of the first and the last sets on the right-hand side of \labelcref{equ:union_shelling_like_lemma} yields $\Gamma_{a}$.
On the other hand, it is easy to see that the set $\Gamma'_{b} \setminus \Delta_{a}$, appearing as the middle set in the union \labelcref{equ:union_shelling_like_lemma}, is the same as $\Gamma_{b} \setminus \Delta_{a}$.
Moreover, since
$\Gamma_{b} \cap \Delta_{a} \subseteq
\Gamma_{a} \subseteq
\Sigma$,
it holds that
$\left( \Gamma_{b} \setminus \Delta_{a} \right)
\, \cup \,
\left( \Gamma_{b} \cap \Delta_{a} \right)
\, = \,
\Gamma_{b}
$.
Henceforth, $\Gamma_{a} \cup \Gamma_{b} = \Sigma$, as desired.
Finally, the case when $\Phi_{a}$ and $\Phi_{b}$ are shellable, was proved in \cite[Lemma 2.3]{Santamaria/Woodroofe:2021} (see also \Cref{stt:erratum}).
\end{proof}
\begin{remark
To ensure shellability, \cite[Lemma 2.3]{Santamaria/Woodroofe:2021} requires the complex $\Gamma'_{b} = \left( \Delta_{a} \cap \Delta_{b} \right) \cup \Gamma_{b}$ to be nonempty.
The condition is redundant, since $\Gamma'_{b}$ is never empty (see \Cref{stt:intersection_deltas_non_empty}).
Another condition that was silently assumed by \cite[Lemma 2.3]{Santamaria/Woodroofe:2021} was that the facets in the complexes $\Phi_{a}$ and $\Phi_{b}$ must be facets of $\Phi$.
The condition appears now explicitly stated in \Cref{stt:gluing_lemma,stt:assymetrical_gluing_lemma}.
\label{stt:erratum}
\end{remark}
\begin{remark}
The partitionings constructed by \Cref{stt:assymetrical_gluing_lemma} can be seen as generalizing those yielded by shellings, as we now explain.
Suppose that $\Delta$ has a shelling order $\sigma_{1}, \dotsc, \sigma_{m}$, and proceed inductively on $m$.
The base case with $m=1$ is trivial.
Now suppose
$\Delta_{t-1} \colonequals \langle \sigma_{1}, \dotsc, \sigma_{t-1} \rangle$ partitionable, and take
$\Phi_{a} = \left(\Delta_{t-1}, \varnothing \right)$
and
$\Phi_{b} = \left(\langle \sigma_{t} \rangle, \langle \sigma_{t} \rangle \cap \Delta_{t-1} \right)$.
Notice that the elements of $\Phi_{b}$, are precisely the same as $[R(\sigma_t), \sigma_t]$.
Since the conditions of \Cref{stt:assymetrical_gluing_lemma} are met, $\Delta_t$ is partitionable.
More generally, \Cref{stt:assymetrical_gluing_lemma} may be useful for finding partitioning schemes of nice enough constructible complexes.
\label{stt:alt_shellable_are_partitionable}
\end{remark}
We believe that \Cref{stt:gluing_lemma,stt:assymetrical_gluing_lemma} may have broader use.
Our main use in this paper will be to prove partitionability of certain $2$-dimensional spaces.
The strategy is as follows: we split each space into more digestible ``chunks'' where partitionability is easier to analyze, then we glue them back with the help of these lemmas.
In order to illustrate this strategy, we revisit the fact that the non-shellable Rudin's $3$-ball is partitionable \cite[\S III.2, Proposition 2.8]{Stanley:1996}.
\begin{example}[Rudin's non-shellable $3$-ball]
In 1958, Rudin constructed a non-shellable triangulation of the $3$-ball with a convex geometric realization \cite{Rudin:1958}.
It is also known to be non-evasive \cite{Benedetti/Lutz:2013}.
Rudin's ball has $f$-vector $(1,14,66,94,41)$ and $h$-vector $(1,10,30,0,0)$.
We will prove that it is partitionable by breaking it up into smaller partitionable relative complexes
$\left(\Delta_{a}, \varnothing \right)$
and
$\left(\Delta_{b}, \Delta_{a} \cap \Delta_{b} \right)$.
We use the list of facets given in \cite{Hachimori:web}.
Take $\{ \mathtt{1, \dotsc, 9, A, \dotsc, E} \}$ as ground set and let $\Delta_{a}$ be the shellable ball formed by the following facets (listed in a shelling order):
\begin{center}
\texttt{
\begin{tabular}{ l l l l l l l l l }
BCDE, &ABCE, &9BDE, &8CDE, &7BCD, &58CD, &3ABE, &36AE, &36AB, \\
6ABC, &26AC, &2ACE, &24AE, &28CE, &248E, &268C, &48DE, &4ADE, \\
458D, &458C, &68BC, &6ADE, &69DE, &48BC, &47BC, &17BD, &19BD.
\end{tabular}}
\end{center}
The remaining facets in Rudin's ball generate the complex $\Delta_{b}$:
\begin{center}
\texttt{
\begin{tabular}{ l l l l l l l l l}
137D, &139D, &39CD, &59CD, &37CD, &347C, &569D, &347B, &37BE,\\
157B, &159B, &59BE, &569E, &57BE. & & & &
\end{tabular}}
\end{center}
The intersection of the two complexes is shellable, with shelling order listed below:
\begin{align*}
\Delta_{a} \cap \Delta_{b}
= \left\langle
\mathtt{17B, 17D, 19B, 9BE, 7CD, 47B, 47C, 19D, 69D, 5CD, 69E, 3BE}
\right\rangle.
\end{align*}
Notice that the facets of $\Delta_{a} \cap \Delta_{b}$ belong to the boundary of $\Delta_{a}$ and $\Delta_{b}$, and that $\Delta_{a} \cap \Delta_{b}$ is homeomorphic to a $2$-disk.
The complex $\left(\Delta_{b}, \Delta_{a} \cap \Delta_{b} \right)$ has $h$-vector $(0,0,14,0,0)$.
This means that if the complex is partitionable, then any partitioning scheme for $\left(\Delta_{b}, \Delta_{a} \cap \Delta_{b} \right)$ matches the fourteen edges with the fourteen facets.
Such a partitioning is as follows:
\begin{align*}
\left(\Delta_{b}, \Delta_{a} \cap \Delta_{b} \right) =
&
\left[ \mathtt{13, 137D} \right] \, \sqcup \,
\left[ \mathtt{15, 159B} \right] \, \sqcup \,
\left[ \mathtt{34, 347B} \right] \, \sqcup \,
\left[ \mathtt{37, 37BE} \right] \, \sqcup \,
\left[ \mathtt{39, 139D} \right] \, \sqcup \,\\
&
\left[ \mathtt{3C, 347C} \right] \, \sqcup \,
\left[ \mathtt{3D, 37CD} \right] \, \sqcup \,
\left[ \mathtt{56, 569D} \right] \, \sqcup \,
\left[ \mathtt{57, 157B} \right] \, \sqcup \,
\left[ \mathtt{59, 59CD} \right] \, \sqcup \,\\
&
\left[ \mathtt{5B, 59BE} \right] \, \sqcup \,
\left[ \mathtt{5E, 569E} \right] \, \sqcup \,
\left[ \mathtt{7E, 57BE} \right] \, \sqcup \,
\left[ \mathtt{9C, 39CD} \right].
\end{align*}
We used computer assistance to find this partitioning.
The problem here is computationally tractable since it is reduced to that of finding a matching in a small bipartite graph.
The conditions of \Cref{stt:assymetrical_gluing_lemma} are met for
$\left(\Delta_{a}, \varnothing \right)$
and
$\left(\Delta_{b}, \Delta_{a} \cap \Delta_{b} \right)$ with $\Gamma_{a} = \Gamma_{b} = \varnothing$,
hence the complex $\Delta_{a} \cup \Delta_{b}$ is partitionable, as desired.
Finally, we notice that $\left(\Delta_{b}, \Delta_{a} \cap \Delta_{b} \right)$ is non-shellable, as otherwise the gluing would produce a shelling of Rudin's ball.
\label{stt:rudin_ball_ex_ex}
\end{example}
\Cref{stt:gluing_lemma,stt:assymetrical_gluing_lemma} build up one partitionable complex out of two.
In other situations we will instead want to modify a single partitionable complex in a manner that preserves its partitioning scheme.
The following lemma is completely immediate.
\begin{lemma}
Let $\Phi$ and $\Phi'$ be relative simplicial complexes with isomorphic face posets.
The complex $\Phi$ is partitionable if and only if $\Phi'$ is partitionable.
\label{stt:isoposets_partitionable}
\end{lemma}
\begin{example}
The following are direct consequences of \Cref{stt:isoposets_partitionable}:
\begin{enumerate}
\item
\label{stt:part_minimal_representation}
A first example of two complexes with identical posets appears when the complexes $\Phi$ and $\Phi'$ have the same minimal representation.
As a concrete example, the leftmost and middle complexes in \Cref{fig:minimal_representations} have the same minimal representation, namely the rightmost complex.
Observe that partitioning schemes (and also shellings) are preserved.
This is the reason why we may consider the minimal representation of a relative complex when we are deciding partitionability (shellability) (see \Cref{stt:algorithm_feeding_rel_part}).
\begin{figure}[!ht]
\begin{center}
\centering
\includegraphics[scale=0.40]{./images/relative_bowties.pdf}
\caption{
Three relative complexes with the same minimal representation.
They share the same combinatorial closure, namely the bow-tie simplicial complex seen in \Cref{stt:bow-tie_partitionable_ex}.
The bold light gray-colored vertices and edges, and the triangles bounded by them (left and middle), represent the relative part of each complex.}
\label{fig:minimal_representations}
\end{center}
\end{figure}
\item
\label{stt:part_isomorphic_posets}
More generally, \Cref{stt:isoposets_partitionable} does not depend on the dimension of the complexes involved, but strictly on their face poset structure.
As a concrete example, \Cref{fig:isomorphic_posets} shows two partitionable simplicial complexes.
Although the complexes do not have the same dimension, they do have isomorphic face posets.
\begin{figure}[!ht]
\begin{center}
\centering
\begin{tabular}{ >{\centering\arraybackslash}m{4.5cm}
>{\centering\arraybackslash}m{7.8cm}
>{\centering\arraybackslash}m{2.0cm}
}
\vspace{ 0.00em} \includegraphics[scale=0.40]{./images/isomorphic_face_posets_01.pdf} &
\vspace{ 0.00em} \includegraphics[scale=0.46]{./images/isomorphic_face_posets_03.pdf} &
\hfill
{$\!\begin{aligned}
\left[ \mathtt{\varnothing, 12} \right] \ & \sqcup \\
\left[ \mathtt{3, 23} \right] \ & \sqcup \\
\left[ \mathtt{4, 34} \right] \ & \sqcup \\
\left[ \mathtt{14, 14} \right] \ &
\end{aligned}$}
\vspace{ 0.20em}
\\
(a) & (b) & (c) \\[1em]
\vspace{ 0.00em} \includegraphics[scale=0.40]{./images/isomorphic_face_posets_02.pdf} &
\vspace{ 0.00em} \includegraphics[scale=0.46]{./images/isomorphic_face_posets_04.pdf} &
\hfill
{$\!\begin{aligned}
\left[ \mathtt{5, 125} \right] \ & \sqcup \\
\left[ \mathtt{35, 235} \right] \ & \sqcup \\
\left[ \mathtt{45, 345} \right] \ & \sqcup \\
\left[\mathtt{145, 145} \right] \ &
\end{aligned}$}
\vspace{ 0.20em}
\\
(d) & (e) & (f)
\end{tabular}
\caption{
The complexes in (a) and (d) have isomorphic face posets, as shown in (b) and (e).
The light gray-colored bold vertices and edges in the boundary of (d) form the relative part of the complex.
Partitioning schemes are highlighted in both posets by preserving the color code of the corresponding intervals, and they are explicitly written in (c) and (f), respectively.}
\label{fig:isomorphic_posets}
\end{center}
\end{figure}
\end{enumerate}
\label{stt:conseq_isoposets_part_ex}
\end{example}
We can improve \Cref{stt:isoposets_partitionable} by allowing relations to the face poset in a way that partitioning schemes are preserved.
\begin{definition}
A surjective dimension-preserving simplicial map $\varphi : \Delta \to \Delta'$ is a \emph{folding map} over the relative simplicial complex $\left( \Delta, \Gamma \right)$ if the following conditions hold for the map $\varphi$:
\begin{enumerate}[label=\tt(F\arabic*)]
\item \label{con:bijectivity}
It induces a bijection $\left( \Delta, \Gamma \right) \leftrightarrow \left( \Delta', \Gamma' \right)$, for some $\Gamma' \subseteq \Delta'$.
\item \label{con:facets_to_facets}
It sends facets of $\left( \Delta, \Gamma \right)$ to facets of $\left( \Delta', \Gamma' \right)$.
\end{enumerate}
%
We say that we \emph{fold} $\left( \Delta, \Gamma \right)$ to $\left( \Delta', \Gamma' \right)$.
%
\label{def:folding_map}
\end{definition}
We remark that the complex $\Gamma'$ in \Cref{def:folding_map} is completely determined by $\varphi$.
Indeed, we have $ \Gamma' \colonequals \Delta' \, \setminus \, \varphi(\Delta, \Gamma)$.
As a consequence, $\varnothing$ is a face of $\Gamma'$, except in trivial situations.
Additionally, if $\left( \Delta, \Gamma \right)$ is pure, the conditions of \Cref{def:folding_map} lead to the equality $h(\Delta, \Gamma) = h(\Delta', \Gamma')$.
The idea suggested by a folding map should be of a simplicial quotient map that twists and bends $\left( \Delta, \Gamma \right)$ onto itself.
As a consequence, the image of a folding map on $\left( \Delta, \Gamma \right)$ does not necessarily preserve the topology, homotopy type, nor homology of $\Delta$ or of $\left( \Delta, \Gamma \right)$.
See \Cref{stt:relative_cycle_ex}.
However, this feature is desirable for our purposes, since folding maps preserve partitioning schemes, as we now show.
\begin{lemma}[Folding Lemma]
Let $\left( \Delta, \Gamma \right)$ be a relative simplicial complex, and let $\varphi$ be a folding map on $\left( \Delta, \Gamma \right)$.
If the complex $\left( \Delta, \Gamma \right)$ is partitionable, then its image under $\varphi$ is also partitionable.
\label{stt:folding_lemma}
\end{lemma}
Thus, if we have a relative space where partitionability is well understood, then \Cref{stt:folding_lemma} allows us to transfer partitionings to new spaces popping up from precise manipulations of the original space.
\begin{proof}[Proof (of \Cref{stt:folding_lemma})]
Let $\left( \Delta', \Gamma' \right)$ be the image of $\left( \Delta, \Gamma \right)$ under $\varphi$.
We know that $\left( \Delta', \Gamma' \right)$ is a relative simplicial complex by \labelcref{con:bijectivity}.
Now, consider a partitioning of $\left( \Delta, \Gamma \right)$.
We need to check that $\varphi$ carries over that partitioning from $\left( \Delta, \Gamma \right)$ to $\left( \Delta', \Gamma' \right)$.
First, notice that $\varphi$ maps intervals into intervals.
For every facet $\sigma \in \left( \Delta, \Gamma \right)$ we have that
$\varphi([R(\sigma), \sigma]) = [R(\varphi(\sigma)), \varphi(\sigma)]$
is an interval in $\Delta'$.
This is provided by the simpliciality and dimension-preserving properties of $\varphi$.
Also, notice that the face $\varphi(\sigma)$ is a facet of $\left( \Delta', \Gamma' \right)$ by \labelcref{con:facets_to_facets}.
Finally, \labelcref{con:bijectivity} ensures that $[R(\varphi(\sigma)), \varphi(\sigma)]$ is an interval of $\left( \Delta', \Gamma' \right)$ and that every face in $\left( \Delta', \Gamma' \right)$ belongs to one and only one interval.
\end{proof}
As a cautionary remark, it is worthwhile saying that our folding maps are not related to the maps of topological pairs, where $\varphi (|\Gamma|) \subseteq |\Gamma'|$, viewed as a subspace under the subspace topology (as in \cite{Sato:1999}).
By contrast, in a typical application of a folding map, the preimage of $\Gamma'$ is a proper subset of $\Gamma$.
Even more, the preimage of any face in $\left( \Delta', \Gamma' \right)$ consists of one face in $\left( \Delta, \Gamma \right)$ and zero or more faces in $\Gamma$.
The next example hints at the way we usually apply \Cref{stt:folding_lemma}.
\begin{example}[Partitionability of $\left(\Triang{\Cycle}, \{\varnothing\} \right)$]
Let $P_{n}$ be a path (in the graph-theoretic sense) with $n > 3$ vertices, and consider the complex $\left( P_{n}, \gSet{x} \right)$, where $x$ is a vertex of degree one.
Let us label the vertices of $P_{n}$ like $v_1, v_2, \dotsc, v_n = x$ as we traverse from one vertex of degree one to the other.
With exactly $n-1$ vertices and $n-1$ edges, we get a partitioning scheme for $\left( P_{n}, \gSet{x} \right)$ by creating the intervals $[v_{i}, v_{i} v_{i+1}]$, $i = 1, \dotsc, n-1$.
By \Cref{stt:folding_lemma} we fold $\left( P_{n}, \gSet{x} \right)$ in such a way that $x$ is glued to any other vertex of choice but $v_{n-1}$ or $v_{n-2}$ (to preserve simpliciality).
So, the obtained complex keeps the original partitioning scheme (up to some relabeling).
In particular, when both vertices of degree one are glued to each other, the obtained complex is a cycle graph.
Thus, any simplicial triangulation of the $1$-sphere $\Cycle$ is partitionable relative to the complex $\{\varnothing\}$.
We observe that the face posets of $\left( P_{n}, \gSet{x} \right)$ and $\left(\Triang{\Cycle}, \{\varnothing\} \right)$ are not isomorphic despite the bijection between their facets. Hence, \Cref{stt:isoposets_partitionable} does not suffice, and we need the power of \Cref{stt:folding_lemma}.
\label{stt:relative_cycle_ex}
\end{example}
\subsection{Cutting before folding}
\label{sec:cutting_before_folding}
As we will use the same technique later on in the paper, we want to comment on the strategy used in \Cref{stt:relative_cycle_ex}.
We are seeking for partitionable preimages of the complex under consideration.
In \Cref{stt:relative_cycle_ex}, we began with a cycle and selected a particular vertex $x$.
We ``cut'' the complex at $x$ by replacing $x$ with two distinct copies of it.
We place one copy of $x$ into the relative part, while keeping the other as a face in the relative complex.
Each facet (edge) originally having $x$ as a vertex, now includes a distinct copy of it.
The resulting relative complex is shellable, hence partitionable, and \Cref{stt:folding_lemma} gives the desired partitionability result.
More generally, when we want to separate $k$ facets, we cut through the non-empty faces they have in common by making $k$ copies of each.
One copy is kept in as a face in the relative complex, while the remaining $k-1$ copies are placed in the relative part.
We seek to perform such cuts in a manner that yields a known shellable (or at least partitionable) relative complex.
Once \Cref{stt:folding_lemma} is applied, it turns the complex back to its original state by identifying the replicated faces along the cuts.
\subsection{A digression on relative partitionings of graphs}
\label{sec:partitionable_graphs}
The results we state here are easy to obtain and possibly well known.
However, we present proofs both for completeness and as an example of our techniques.
We want to generalize \Cref{stt:relative_cycle_ex}.
First, a negative result:
\begin{proposition}
Let $\Delta$ be a pure $k$-dimensional simplicial complex, with $k \geq 1$.
The complex $\left(\Delta, \{\varnothing\}\right)$ is not shellable.
\label{stt:shell_graphs_resp_empty_face}
\end{proposition}
\begin{proof}
Since the intersection of the complex induced by any facet of $\Delta$ and $\{\varnothing\}$ is not codimension one, there is no facet in $\Delta$ to start the shelling process.
\end{proof}
Although $1$-dimensional complexes (graphs) are never shellable relative to $\{\varnothing\}$, many are partitionable.
\begin{proposition}
Let $\Delta$ be a pure $1$-dimensional simplicial complex.
The complex $\left(\Delta, \{\varnothing\}\right)$ is partitionable if and only if no connected component of $\Delta$ is a tree.
\label{stt:part_graphs_resp_empty_face}
\end{proposition}
\begin{proof}
First, observe that a partitioning scheme of $\left(\Delta, \{\varnothing\}\right)$ must match each vertex to a unique incident edge, and the remaining edges, if any, to themselves.
In any case, the edges have to outnumber the vertices.
\noindent ($\Leftarrow$)
Suppose that no connected component of $\Delta$ is a tree.
To prove that $\left(\Delta, \{\varnothing\}\right)$ is partitionable, it is enough to see that there is a partitioning scheme for each connected component.
Then, without loss of any generality, we may assume that $\Delta$ is a connected graph.
Get a spanning tree $T$ of $\Delta$, and let $H$ be the subgraph generated by all the edges in $\Delta$ not included in $T$.
See $T$ and $H$ as simplicial complexes.
Denote as $V(H)$ the set of vertices of $H$ (seen as a $0$-dimensional complex).
Now, select a vertex $v \in V(H)$, and consider the complexes
$\left(T, \gSet{v} \right)$ and
$\left(H, V(H) \setminus \{v\} \right)$.
These two complexes meet the conditions of \Cref{stt:gluing_lemma} with $\Sigma = \{ \varnothing \}$, yielding the partitionable complex $\left(\Delta, \{\varnothing\}\right)$.
We need to check that $\left(T, \gSet{v} \right)$ and $\left(H, V(H) \setminus \{v\} \right)$ are partitionable.
The complex $\left(T, \gSet{v} \right)$ consists of the same number of vertices and edges, and we want to match each vertex with one of its incident edges to form intervals.
To do that, root $T$ at $v$ and match each vertex to the edge it lies in following a bottom-up orientation (starting from the leaves, upwards).
As for $\left(H, V(H) \setminus \{v\} \right)$, observe that it consists solely of edges and the vertex $v$.
Match $v$ with one of the edges having it as an end-point, and match the remaining edges to themselves.
\noindent ($\Rightarrow$)
If $\Delta$ has a tree as a connected component, there is no way to match vertices to edges since the former set outnumbers the latter.
\qedhere
\end{proof}
\section{Partitioning some relative surfaces}
\label{sec:partitionable_surfaces}
\subsection{Shellings on relative simplicial disks}
\label{sec:shellable_disks}
We are interested in partitioning $\Pplane$ and $\DunceH$.
A main building block will be partitioning schemes of the (relative) disk.
It is widely known that the disk (i.e.\ the $2$-ball) is shellable.
Furthermore, every partial shelling (a disk itself) can be extended to a complete shelling of the entire disk (see, e.g.\ \cite[\S 3.5 p. 35]{Danaraj/Klee:1978a}, \cite[\S III.2 p. 84]{Stanley:1996} and \cite[\S 3.4 p. 107]{Bing:1964}).
The following result is an easy consequence of these extendable shellings on the disk.
\begin{theorem}
Let $\TriangD$ be a triangulation of the $2$-dimensional disk $\Disk$ and let $\Upsilon$ be a pure connected $1$-dimensional subcomplex of its boundary.
The complex $\left( \TriangD, \Upsilon \right)$ is shellable.
\label{stt:rel_disk_is_partitionable}
\end{theorem}
This result is well known by experts in the field.
For completeness, and to avoid the side trip into extendable shellability, we give a proof of \Cref{stt:rel_disk_is_partitionable}.
Our proof is essentially that of Bing in \cite[\S 3.4 Theorem 3]{Bing:1964}.
\begin{proof}[Proof (of \Cref{stt:rel_disk_is_partitionable})]
Proceed inductively on $m$, the number of facets of $\left( \TriangD, \Upsilon \right)$.
The base case, when $m = 1$, is trivial.
For the inductive step, suppose the statement holds for relative simplicial disks with less than $m$ facets, and consider a facet $\sigma$ intersecting $\Upsilon$ in a pure $1$-dimensional subcomplex.
Such a face must exist.
The remaining complex, say $\Phi$, can be seen as the minimal representation of $\left( \TriangD, \Upsilon \cup \gSet{\sigma} \right)$ (see \Cref{stt:conseq_isoposets_part_ex} (\labelcref{stt:part_minimal_representation})).
Two cases may appear:
$\bullet$ \emph{Case 1}.
If $\sigma$ has two edges lying in the interior, and the vertex they share belongs to the boundary, $\Phi$ is split by $\sigma$ into two disks.
Call them $\TriangDi{1}$ and $\TriangDi{2}$, and let $v$ be the vertex they hinge at.
Let $uv$ be the edge of $\sigma$ shared with the former disk and $vw$ the one shared with the latter.
Hence, the complexes
$\Phi_{1} = \left(
\TriangDi{1},
\gSet{uv}
\cup
\left(\Upsilon \cap \TriangDi{1} \right)
\right)$
and
$\Phi_{2} = \left(
\TriangDi{2},
\gSet{vw}
\cup
\left(\Upsilon \cap \TriangDi{2} \right)
\right)$
meet the inductive hypothesis and they are shellable.
Now, build up the shelling of $\left( \TriangD, \Upsilon \right)$ by means of \Cref{stt:assymetrical_gluing_lemma}: first glue
$\left( \gSet{\sigma}, \gSet{\sigma} \cap \Upsilon \right) = \left( \gSet{\sigma}, \gSet{uw} \right)$
with $\Phi_{1}$, and then glue $\Phi_{2}$ to it.
$\bullet$ \emph{Case 2}.
If $\sigma$ does not split the complex, then $\Phi$ ends up as a smaller relative simplicial disk meeting the condition at the boundary.
Start the shelling with $\sigma$ and extend it through the inductive process.
\end{proof}
We comment that similar results to \Cref{stt:rel_disk_is_partitionable} may be given for relative cellular complexes in the sense of \cite[\S VI.6]{Bing:1983}.
\subsection{Partitionable relative triangulations of the Möbius strip}
\label{sec:partitioning_mobius}
It is known that no triangulation of the Möbius strip $\Mobius$ is shellable, since it is a $2$-dimensional space homotopy equivalent to the $1$-sphere $\Cycle$.
Nonetheless, $\Mobius$ is partitionable relative to certain subcomplexes.
This will be a useful ingredient in the proof of our main result.
\begin{figure}[!ht]
\begin{center}
\centering
\begin{tabular}{ >{\centering\arraybackslash}b{3.6cm}
>{\centering\arraybackslash}b{3.6cm}
>{\centering\arraybackslash}b{3.6cm} |
>{\centering\arraybackslash}b{3.6cm}
}
\includegraphics[scale=0.40]{./images/mobius_chunks_06.pdf} &
\includegraphics[scale=0.40]{./images/mobius_chunks_07.pdf} &
\includegraphics[scale=0.40]{./images/mobius_chunks_08.pdf} &
\includegraphics[scale=0.40]{./images/mobius_chunks_05.pdf} \\
(a) & (b) & (c) & (d) \\[1em]
\includegraphics[scale=0.40]{./images/mobius_chunks_02.pdf} &
\includegraphics[scale=0.40]{./images/mobius_chunks_03.pdf} &
\includegraphics[scale=0.40]{./images/mobius_chunks_04.pdf} &
\includegraphics[scale=0.40]{./images/mobius_chunks_01.pdf} \\
(f) & (g) & (h) & (i)
\end{tabular}
\caption{
The diagrams from (a) to (c) represent triangulations of the Möbius strip $\Mobius$ relative to $\dMobius$, whereas those from (f) to (h) represent triangulations of $\Mobius$ relative to the complex $\{\varnothing\}$.
We take out one triangle with at least one edge lying in the boundary and depict the three possible outcomes.
In the case of (a) and (f) we cut through $ux$ to get disks relative to a portion of their respective boundaries, as it is shown in (d) and (i).
}
\label{fig:folding_mobius_strip}
\end{center}
\end{figure}
\begin{theorem}
Let $\TriangMo$ be a triangulation of the Möbius strip $\Mobius$.
Then $\TriangMo$ is partitionable relative to its entire boundary.
\label{stt:partitionable_mobius_boundary}
\end{theorem}
\begin{proof}
We apply the following inductive argument.
Pick a triangle $\sigma = uvx$ with an edge lying on the boundary of $\TriangMo$.
Let $\dTriangMo$ be the boundary of $\TriangMo$.
We have three cases, all of them are drawn in \Cref{fig:folding_mobius_strip} (a) to (c).
If a case like \Cref{fig:folding_mobius_strip} (a) appears, namely, two edges in the interior and all vertices in the boundary, we can split the strip to turn it into a disk:
take the edge $ux$ in the interior, and cut through it as explained in \Cref{sec:cutting_before_folding} to get a disk relative to the path $x_1 u_2 x_2 u_1$ (see \Cref{fig:folding_mobius_strip} (d)).
Now, get a partitioning with \Cref{stt:rel_disk_is_partitionable} and fold back with \Cref{stt:folding_lemma}.
Use this as the base case for the induction.
In the case that such a triangle does not exist, consider the remaining cases (\Cref{fig:folding_mobius_strip} (b) and (c)).
Taking the triangle $\sigma$ out of $\TriangMo$ does not split the strip and preserves its topology.
Thus, we get a new triangulation of $\Mobius$ relative to its boundary, call it $\left(\TriangMo', \dTriangMo' \right)$.
More precisely, such a triangulation is better described as the minimal representation of $\left( \TriangMo, \dTriangMo \cup \gSet{\sigma} \right)$.
By the inductive hypothesis, $\left(\TriangMo', \dTriangMo' \right)$ is partitionable.
Use \Cref{stt:assymetrical_gluing_lemma} to glue back the relative triangle we took out from the original complex, i.e.\
$\left( \gSet{\sigma}, \gSet{\sigma} \cap \dTriangMo \right)$.
This yields a partitioning scheme of $\left(\TriangMo, \dTriangMo \right)$.
Observe that \Cref{stt:assymetrical_gluing_lemma} was invoked here with
$\Phi_{a} = \left( \gSet{\sigma}, \gSet{\sigma} \cap \dTriangMo \right)$,
and $\Gamma_{b} = \gSet{\dTriangMo \setminus \gSet{\sigma}}$.
\end{proof}
For the sake of completeness, we present an analogous result to that of \Cref{stt:partitionable_mobius_boundary}.
Use \Cref{fig:folding_mobius_strip} (f) to (i) as a pictorial reference.
\begin{proposition}
Let $\TriangMo$ be a triangulation of the Möbius strip $\Mobius$.
Then $\TriangMo$ is partitionable relative to the complex $\{\varnothing\}$.
\label{stt:partitionable_mobius_empty_face}
\end{proposition}
\begin{proof}
The proof is completely analogous to that of \Cref{stt:partitionable_mobius_boundary}.
However, for the base case, the complex we get is a disk relative to a single edge (see \Cref{fig:folding_mobius_strip} (i)).
Apply \Cref{stt:rel_disk_is_partitionable} and \Cref{stt:folding_lemma}.
As for the inductive step, with $uvx$ as before, we must consider the complexes
$\left(\TriangMo \setminus \left[ uv, uvx \right], \{ \varnothing \} \right)$
and
$\left( \gSet{uvx}, \gSet{ux, vx} \right)$
for the case illustrated in \Cref{fig:folding_mobius_strip} (g),
and the complexes
$\left(\TriangMo \setminus \left[ x, uvx \right], \{ \varnothing \} \right)$
and
$\left( \gSet{uvx}, \gSet{uv} \right)$
for the case in \Cref{fig:folding_mobius_strip} (h).
Glue with \Cref{stt:assymetrical_gluing_lemma}.
\end{proof}
By using the same method, we can obtain analogous results for the annulus $\Annulus$.
\begin{proposition}
Let $\TriangAn$ be a triangulation of the annulus $\Annulus$.
Then the following relative complexes are partitionable.
\begin{enumerate}
\item $\TriangAn$ relative to $\{\varnothing\}$.
\item $\TriangAn$ relative to its entire boundary.
\item $\TriangAn$ relative to one of the cycles in its boundary.
\end{enumerate}
\label{stt:partitionable_annulus}
\end{proposition}
\begin{proof}
Start with a simplicial triangulation $\TriangAn$ of $\Annulus$.
Then get the closest path between the two contours of $\TriangAn$ on its $1$-skeleton and cut through it.
Now, for each of the three introduced cases, get a partitioning scheme with \Cref{stt:rel_disk_is_partitionable} and apply the folding argument of \Cref{stt:folding_lemma}, as in \Cref{stt:partitionable_mobius_boundary} and \Cref{stt:partitionable_mobius_empty_face}.
\end{proof}
We are sure that the reader can apply our cut-and-fold techniques to obtain results on partitionability of other surfaces, like the sphere, the torus, or the Klein bottle, in relative terms.
\section{Proof of \Cref{stt:main_result}}
\label{sec:main_theorem}
\subsection{Partitionable triangulations of the projective plane}
\label{sec:partitioning_pplane}
The real projective plane $\Pplane$ is a non-orientable surface with Euler characteristic $0$.
No triangulation of $\Pplane$ is shellable.
In this section we prove that $\Pplane$ is partitionable.
\begin{figure}[!ht]
\begin{center}
\centering
\begin{tabular}{ >{\centering\arraybackslash}m{5.3cm
>{\centering\arraybackslash}m{5.3cm}
\vspace{ 0.0em} \includegraphics[scale=0.40]{./images/projective_plane_01.pdf} &
\vspace{ 0.0em} \includegraphics[scale=0.40]{./images/projective_plane_02.pdf} \\
(a) & (b) \\[1em]
\vspace{ 0.0em} \includegraphics[scale=0.40]{./images/projective_plane_03.pdf} &
\vspace{ 0.0em} \includegraphics[scale=0.40]{./images/projective_plane_04.pdf} \\
(c) & (d)
\end{tabular}
\caption{
The real projective plane $\Pplane$ as a CW-complex (a).
Sides labeled as $e$ must be identified.
A decomposition of $\Pplane$ into a Möbius strip (yellow) and a disk (dark blue) (b).
This decomposition is shown in a particular triangulation of $\Pplane$ (c).
We redraw in (d) the yellow-colored space in (b) to make clear it is a Möbius strip.}
\label{fig:projective_plane}
\end{center}
\end{figure}
\subsubsection{Decomposition of $\Pplane$}
\label{sec:decomposition_pplane}
We want to split $\Pplane$ into suitable partitionable subcomplexes, with the aim to glue them back by means of \Cref{stt:assymetrical_gluing_lemma}.
\begin{lemma}
Let $\TriangPp$ be a triangulation of the projective plane $\Pplane$, and $\sigma$ a facet of $\TriangPp$.
Then the complex $\TriangPp \setminus \sigma$ is a triangulation of the Möbius strip $\Mobius$.
\label{stt:decomposition_pplane}
\end{lemma}
\begin{proof}
Take an arbitrary facet $\sigma$ out of $\TriangPp$.
The complex $\gSet{\sigma}$ is obviously a simplicial disk, and its boundary is homeomorphic to a cycle $\Cycle$.
Notice that the remaining complex $\Delta \colonequals \TriangPp \setminus \sigma$ shares its entire boundary with $\gSet{\sigma}$.
Now, we want to recognize the space that $\Delta$ triangulates.
Since the geometric realization of $\gSet{\sigma}$ (homeomorphic to a disk) and the space $\Pplane$ wherein it lies are surfaces, then \cite[\S 3, Corollary 3.14$_{n+1}$]{Rourke/Sanderson:1972} ensures that $|\Delta| \cong \cl(\Pplane \setminus |\gSet{\sigma}|)$ is a surface.
On the other hand, the deletion of $\sigma$ reduces by one the Euler characteristic of $\Pplane$, so $\tilde{\chi} (|\Delta|) = -1$.
Then, by the classification theorem of compact surfaces (see \cite[\S 6.3, Theorem 6.2]{Gallier/Xu:2013}), the complex $\Delta$ is a triangulation of either the Möbius strip $\Mobius$ or the annulus $\Annulus$.
As the boundary of the annulus is disconnected, the only possible option is that $|\Delta| \cong \Mobius$.
\end{proof}
\begin{remark}
This decomposition of $\Pplane$ into a simplicial disk and a triangulation of a Möbius strip came as no surprise.
As a consequence of the classification of surfaces, any compact non-orientable surface is obtained from the sphere by deleting a disk and identifying the resulting boundary cycle with the boundary of a Möbius strip.
Also, all compact surfaces with boundary are obtained by deleting several disks from a closed surface (see \cite{Gallier/Xu:2013} for further reference).
However, there is a subtlety we had to take care of in our decomposition: deleting an arbitrary disk is not the same as deleting well-chosen one.
We were surprised not to find a result like \Cref{stt:decomposition_pplane} explicitly stated in the literature, although we believe it is well known by the experts in the field.
\end{remark}
\medskip
\phantomsection
\label{sec:main_result_pplane}
We are now ready to prove the first part of \Cref{stt:main_result}.
\begin{theorem}[\Cref{stt:main_result} for $\Pplane$]
Any triangulation of $\Pplane$ is partitionable.
\label{stt:pplane_partitionable}
\end{theorem}
\begin{proof}
\Cref{stt:decomposition_pplane} allows us to decompose a triangulation $\TriangPp$ of the projective plane into an arbitrary facet $\sigma$ and a triangulation $\TriangMo$ of the Möbius strip $\Mobius$.
Denote the boundary of $\TriangMo$ as $\dTriangMo$.
Following \Cref{stt:assymetrical_gluing_lemma} (and notation) we need partitionable complexes
$\Phi_{a} = \gSet{\sigma} = \left( \gSet{\sigma}, \varnothing \right)$ and
$\Phi_{b} = \left( \TriangMo, \dTriangMo \right)$.
The former complex is obviously shellable, and the latter is partitionable by \Cref{stt:partitionable_mobius_boundary}.
To check the set-theoretic conditions of \Cref{stt:assymetrical_gluing_lemma} observe that $\TriangMo \cap \gSet{\sigma} = \dTriangMo = \gSet{\sigma} \setminus \sigma$ and $\TriangMo \cup \gSet{\sigma} = \TriangPp$.
\end{proof}
By using the same techniques we can easily obtain a similar relative result.
\begin{theorem}
Any triangulation of $\Pplane$ is partitionable relative to $\{\varnothing\}$.
\label{stt:pplane_rel_empy_face_partitionable}
\end{theorem}
\begin{proof}
Entirely analogously to the previous proof, we consider the complexes
$\Phi_{a} = \left( \TriangMo, \{\varnothing\} \right)$ and
$\Phi_{b} = \left( \gSet{\sigma}, \gSet{\sigma} \setminus \sigma \right)$.
The latter is obviously shellable, and the former is partitionable by \Cref{stt:partitionable_mobius_empty_face}.
\end{proof}
\begin{remark}
It might be interesting to consider partitionability of $\Pspace{k}, k > 2$, with the use of our techniques.
However, we do not pursue this problem any further in the current work.
\label{stt:future_work_RPk}
\end{remark}
\subsection{Partitionable triangulations of the dunce hat}
\label{sec:partitioning_dunce_hat}
\begin{figure}[!ht]
\begin{center}
\centering
\begin{tabular}{ >{\centering\arraybackslash}m{5.3cm
>{\centering\arraybackslash}m{5.3cm}
\vspace{ 0.20em} \includegraphics[scale=0.40]{./images/dunce_hat_01.pdf} &
\vspace{ 0.00em} \includegraphics[scale=0.40]{./images/dunce_hat_02.pdf} \\
(a) & (b) \\%[1em]
\vspace{ 1.13em} \includegraphics[scale=0.40]{./images/dunce_hat_partitions_01.pdf} &
\vspace{ 1.00em} \includegraphics[scale=0.40]{./images/dunce_hat_partitions_02.pdf} \\
(c) & (d) \\%[1em]
\vspace{ 1.00em} \includegraphics[scale=0.40]{./images/dunce_hat_subcomplex_Y_01.pdf} &
\vspace{ 1.00em} \includegraphics[scale=0.40]{./images/dunce_hat_subcomplex_Y_02.pdf} \\
(e) & (f) \\
\vspace{ 1.13em} \includegraphics[scale=0.40]{./images/dunce_hat_pants_01.pdf} &
\vspace{ 0.90em} \includegraphics[scale=0.40]{./images/dunce_hat_pants_02.pdf} \\
(g) & (h)
\end{tabular}
\caption{
The dunce hat $\DunceH$ as a CW-complex (a) and a triangulation $\TriangDH$ of $\DunceH$ (b).
Sides labeled as $e$ must be identified preserving the orientation.
In (c) and (d) the space is decomposed into the cone of $v$ (dark blue) and the deletion of $v$ (yellow).
The boundary shared by these two spaces is shown in (e) and (f).
To obtain a partitioning for the deletion of $v$, we first get a disk cutting along $e$, as shown in (g) and (h).
Then we make the deletion of each copy of $v$.}
\label{fig:dunce_hat}
\end{center}
\end{figure}
The \emph{dunce hat} $\DunceH$ is the quotient space of a $2$-dimensional triangle, where the sides are identified in a non-cyclic manner.
It was introduced by Zeeman in \cite{Zeeman:1964} (hence $\DunceH$ for the notation).
The space $\DunceH$ is known to be contractible but non-collapsible; it is Cohen-Macaulay over any field, but it is not shellable \cite[\S III.2 p. 84]{Stanley:1996} nor even constructible \cite{Hachimori:2008}.
In \Cref{fig:dunce_hat} we depict the space $\DunceH$ (a) and one of its triangulations (b).
The sides labeled as $e$ are to be identified, as well as the $0$-cells labeled as $v$.
There are three kinds of points in the space $\DunceH$, and they are fully characterized by their own neighborhoods.
If a point lies in the interior of the triangle, its neighborhood is certainly homeomorphic to $\mathbb{R}^2$.
However, this is not going to happen to those points lying in the side $e$ nor to the special $0$-cell ``corner'' $v$.
The boundary of a sufficiently small ball on $\DunceH$ centered at $v$ looks precisely like \Cref{fig:dunce_hat} (e) (the simplicial case is detailed in \Cref{stt:description_link_of_corner}).
So, $v$ must be a vertex in any triangulation of $\DunceH$.
Furthermore, any triangulation of $\DunceH$ subdivides $e$ into edges.
For the forthcoming discussions, we consider \Cref{fig:dunce_hat} as a pictorial and notational reference.
\subsubsection{Decomposition of $\DunceH$}
\label{sec:decomposition_dunce_hat}
To prove the last part of \Cref{stt:main_result}, we will follow the strategy of the previous sections, namely, we decompose $\DunceH$ into two partitionable spaces, then glue them back using our toolkit.
Our decomposition will be as follows.
Let $\TriangDH$ be a triangulation of $\DunceH$, with $v$ corresponding to the corner vertex.
We use $v$ to split the facets of $\TriangDH$ into two subcomplexes:
on the one hand, we have all the facets that include $v$, and on the other, those that do not.
The corresponding subcomplexes are respectively the cone complex $v*\lk_{\TriangDH}(v)$ and the deletion complex $\del_{\TriangDH}(v)$ (see \Cref{fig:dunce_hat} (c)).
Notice that the boundary shared by these two complexes is precisely $\lk_{\TriangDH}(v)$.
We want to see what the link of $v$ looks like.
The following well know result can be proved by an easy shelling argument.
\begin{lemma}
The link of a boundary vertex in any triangulation of the $2$-disk $\Disk$ (i.e.\ the ball $\Ball^2$) is a triangulation of $\Ball^1$ (i.e.\ a path).
\label{stt:links_of_balls_as_balls}
\end{lemma}
\begin{remark}
\Cref{stt:links_of_balls_as_balls} cannot be propagated to higher dimensions.
For example, it is known that the double suspension of the Poincaré homology sphere is a $5$-sphere, but the link of a suspending vertex is not even a manifold (although it is a pseudomanifold with the same homologies of a sphere).
\label{stt:on_poincare_homology_sphere}
\end{remark}
Recall that $e$ must be simplicially subdivided by edges.
Hence, the corner vertex $v$ has to have distinct vertices $x$ and $y$ in its neighborhood lying on $e$ to ensure simpliciality.
Thus, we can lift the triangulation $\TriangDH$ to a triangulation $\Delta$ of the triangle whose quotient space is $\DunceH$.
To do that, we cut along $e$ as depicted in \Cref{fig:dunce_hat} (g).
Observe that this action lifts copies of $v$, say $v_1, v_2$ and $v_3$, and also copies of $x$ and $y$.
Denote as $\alpha, \beta$ and $\gamma$ to the respective quotients of $\lk_{\Delta}(v_1), \lk_{\Delta}(v_2)$ and $\lk_{\Delta}(v_3)$ once $\Delta$ is folded.
\begin{lemma}
Using the notation of the preceding paragraph, the subcomplexes $\alpha$ and $\beta$ triangulate $\Cycle$, while $\gamma$ triangulates $\Ball^1$.
The vertex $x$ is common to $\alpha$ and $\gamma$, and the vertex $y$ to $\beta$ and $\gamma$.
\label{stt:description_link_of_corner}
\end{lemma}
\begin{proof}
\Cref{stt:links_of_balls_as_balls} ensures that $\lk_{\Delta}(v_i)$ is a path over the $1$-skeleton of $\Delta$.
Given that the $x_i$'s and $y_i$'s are to be identified, respectively as $x$ and $y$, we get that $\alpha$ and $\beta$ become simplicial cycles and $\gamma$ the path joining them from $x$ to $y$.
We observe that there is no common vertex between $\alpha$ and $\beta$, otherwise we would break simpliciality of $\TriangDH$.
\end{proof}
\begin{remark}
Compare the link of $v$ described in \Cref{stt:description_link_of_corner} with the complex $L$ obtained in the first part of the proof of \cite[Theorem 4]{Zeeman:1964}; see also Figure 5 of the same paper.
\end{remark}
\begin{lemma}
Let $\TriangDH$ be a triangulation of the dunce hat $\DunceH$, and let $v$ be its distinguished corner vertex.
Then any triangulation of the complex $\left( \del_{\TriangDH}(v), \lk_{\TriangDH}(v) \right)$ is partitionable.
\label{stt:twisted_underwear_partitionable}
\end{lemma}
\begin{proof}
Since every edge in $\Delta$ belongs to at most two triangles, it follows that the complex $v_1 * \lk_{\Delta}(v_1)$ is a $2$-ball, and the link is a path (see \Cref{stt:links_of_balls_as_balls}).
It follows by the Jordan-Schoenflies Theorem that the deletion of $v_1$ in $\Delta$ is a $2$-ball, since it is bounded by $\Cycle$.
Use again the same argument for $v_2$ in the complex $\del_{\Delta}(v_1)$, and then for $v_3$ in the resulting complex.
This procedure yields a triangulation of a disk relative to a connected part of its boundary.
More precisely, in the way we labeled \Cref{fig:dunce_hat} (g), we end up with a triangulation of a disk relative to the path $\alpha' x_3 y_3 \gamma' x_2 y_2 \beta'$, where $\alpha', \gamma'$ and $\beta'$ are the link paths appearing after the iterative deletion of each $v_i$.
A partitioning scheme of this disk is given by \Cref{stt:rel_disk_is_partitionable}.
Then fold the complex with \Cref{stt:folding_lemma} to glue back the three copies of the path $xy$ lying on $e$.
This yields a partitioning scheme of
$\left( \del_{\TriangDH}(v), \lk_{\TriangDH}(v) \right)$.
\end{proof}
\begin{remark}
Cutting $\TriangDH$ along $e$, as in the proof of \Cref{stt:twisted_underwear_partitionable}, proves that $\left( \TriangDH, \{v\} \right)$ is partitionable.
Again, use the now-recurrent tandem of \Cref{stt:rel_disk_is_partitionable} and \Cref{stt:folding_lemma}.
\label{stt:fun_facts_twisted_underwear_space}
\end{remark}
\medskip
\phantomsection
\label{sec:main_resutl_dunce_hat}
Now we have the ingredients to finish the proof of \Cref{stt:main_result}.
\begin{theorem}[\Cref{stt:main_result} for $\DunceH$]
Any triangulation of $\DunceH$ is partitionable.
%
\label{stt:dunce_hat_partitionable}
\end{theorem}
\begin{proof}
We decompose a triangulation $\TriangDH$ of $\DunceH$ into $v*\lk_{\TriangDH}(v)$ and $\del_{\TriangDH}(v)$ as discussed earlier in this section.
We know that
$\left( \del_{\TriangDH}(v), \lk_{\TriangDH}(v) \right)$
is partitionable by \Cref{stt:twisted_underwear_partitionable}.
On the other hand, the cone $v*\lk_{\TriangDH}(v)$ is shellable hence partitionable:
\Cref{stt:description_link_of_corner} tells us that $\lk_{\TriangDH}(v)$ is connected, hence shellable; therefore, the cone of $\lk_{\TriangDH}(v)$ is also shellable.
Since
$\lk_{\TriangDH}(v) = (v*\lk_{\TriangDH}(v)) \, \cap \, \del_{\TriangDH}(v)$, \Cref{stt:assymetrical_gluing_lemma} gives us the desired partitionability of
$\left(\left(v*\lk_{\TriangDH}(v)\right)
\, \cup \,
\del_{\TriangDH}(v),\
\varnothing
\right) = \TriangDH.$
\qedhere
\end{proof}
\section*{Acknowledgments}
\label{sec:acknowledgments}
I am grateful to Russ Woodroofe for his guidance, ideas and careful reading of each draft of this paper.
I also thank Bennet Goeckner, Masahiro Hachimori, Caroline Klivans, Jeremy Martin, Bruno Benedetti and Lorenzo Venturello for their helpful feedback and comments.
I also thank the anonymous referees for their thoughtful comments.
The free open-source mathematics software system SageMath \cite{sagemath} and Hachimori's online library of simplicial complexes \cite{Hachimori:web} were valuable resources.
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\section{Introduction}\label{sec1}
Gamma-ray Bursts (GRBs) are the most luminescent phenomena in the universe. They result from the deaths of massive stars \citep{1993ApJ...405..273W,1998ApJ...494L..45P, 2006ARA&A..44..507W,Cano2017} or the merger of two compact objects, such as neutron stars \citep[NSs;][]{1992ApJ...392L...9D, 1992Natur.357..472U, 1994MNRAS.270..480T, 2011MNRAS.413.2031M} or a NS with a black hole \citep[BH,][]{1992ApJ...395L..83N}. GRBs are evaluated based on the phenomenology seen during their early and late phases and are often characterized by the fireball model \citep{1998ApJ...497L..17S} to distinguish their various sources. The principal and earliest emission, known as the ``prompt emission", is detected from hard X-rays to $\gamma$-rays. This phase can be explained by the interactions of internal shells of material launched forcefully from the central engine at various speeds \citep{1994ApJ...430L..93R, 1994ApJ...427..708P}, photospheric emission from the fireball \citep{2007ApJ...666.1012T,2013ApJ...765..103L,2011ApJ...732...26M} or discharges from a Poynting-flux dominated ejecta \citep{2008A&A...480..305G,2016MNRAS.459.3635B, 2015MNRAS.453.1820K, 2011ApJ...726...90Z}. Later emission, known as ``afterglow", \citep[e.g.,][]{1997Natur.387..783C, 1998ApJ...497L..17S,2002ApJ...568..820G, 1997Natur.386..686V,1998A&A...331L..41P,Gehrels2009ARA&A,Wang2015} is a long-lasting multi-wavelength emission detectable in gamma-rays, X-rays, optical, and radio. It is modeled using synchrotron radiation produced when the external environment decelerates the relativistic outflow, and a significant portion of its energy is transferred. Long GRBs (lGRBs) and short GRBs (sGRBs) are categorized based on their duration:\footnote{For a debate of controversial situations, see \cite{kann2011}.} $T_{90}\leq 2\mathrm{\,s}$ or $T_{90} \ge 2\mathrm{\,s}$,\footnote{$T_{90}$ is the time over which a GRB releases from $5\%$ to $95\%$ of the total measured counts.} respectively \citep{mazets1981catalog, kouveliotou1993identification}.
Synchrotron radiation is the fundamental emission mechanism in GRB afterglows in a forward-shock (FS) scenario \citep{Kumar,1997ApJ...476..232M}. Nevertheless, synchrotron is contingent on the existence of magnetic fields. The origin and arrangement of these magnetic fields behind the shock remain debatable.
They can originate from the compression of an existing magnetic field within the interstellar medium \citep[ISM;][]{1980MNRAS.193..439L,2021MNRAS.507.5340T} and shock-generated two-stream instabilities \citep{PhysRevLett.2.83, Medvedev}. The magnetic field generated by these plasma instabilities is random in orientation but mostly confined to the plane of the shock \citep{2020MNRAS.491.5815G}. Modeling the source and arrangement of those fields and other physical properties of GRBs presents a challenging task. This has necessitated the development of other methods for investigating these complicated systems. Among these techniques is linear polarization.
Linear polarization has been measured, up to a few percent, from the afterglow of several GRBs. Some examples include GRB 191221B \citep[$\Pi=1.2\%$;][]{Buckley} at the late afterglow, GRB 190114C \citep[$\Pi=0.8\pm0.13\%$;][]{Laskar_2019} on the radio band, and the upper limits determinations of GRB 991216 \citep[yielding $\Pi < 7\%$;][]{2005ApJ...625..263G} and GRB 170817A \citep[yielding $\Pi < 12\%$, on the 2.8 GHz radio band][]{2018ApJ...861L..10C}.
Since the degree of polarization relies on the configuration of the magnetic field, analyzing the degree of polarization permits us to investigate these configurations and, therefore, their origins. Previous works, including \cite{2003ApJ...594L..83G, Gill-1, 2004MNRAS.350.1288R, Lyutikov, Nakar, 2021MNRAS.507.5340T, 2020ApJ...892..131S}, have already investigated the practicality of utilizing polarization models to acquire source-related information.
Due to the unfortunate short number of orbital polarimeters and the normal difficulty of seeing these extreme events, collecting polarization data has been one of the most significant impediments. Despite this, progress has been made in the field as a result of initiatives like the POLAR project \citep{POLAR}, and it is anticipated that we will have abundant data to test various models in the coming years.
This study expands the analytical synchrotron afterglow scenario of the off-axis homogeneous jet in a stratified environment, which was required to characterize the multi-wavelength data of GRB 170817A \citep{2020ApJ...896...25F} and a sample of GRBs exhibiting off-axis emission.\footnote{We use the values of the cosmological constants $H_0=69.6\,{\rm km\,s^{-1}\,Mpc^{-1}}$, $\Omega_{\rm M}=0.286$ and $\Omega_\Lambda=0.714$ \citep{2016A&A...594A..13P}, which correspond to a spatially flat universe $\Lambda$CDM model.} The phenomenological model is extended from adiabatic to radiative regime, including the self-absorption synchrotron phase and the dimensionless factor, which provides information on the equal arrival time surface (EATS). We show the temporal development of polarization from the synchrotron afterglow stratification model and compute the expected polarization for GRB 080503 \citep{2009ApJ...696.1871P,2015ApJ...807..163G}, GRB 140903A \citep{2016ApJ...827..102T, 2017ApJ...835...73Z}, GRB 150101B \citep{2018NatCo...9.4089T}, GRB 160821B \citep{2019MNRAS.489.2104T} and GRB 170817A \citep{2017Sci...358.1559K, 2017MNRAS.472.4953L, mooley, 2018ApJ...867...95H, 2019ApJ...884...71F}.
For GRB 170817A in particular, we employ the available polarimetric upper limits from \cite{2018ApJ...861L..10C}.
Furthermore, taking into account the multi-wavelength upper limits of the closest Swift-detected bursts and the Gravitational Wave (GW) events that potentially produce electromagnetic emission,\footnote{These events were associated to at least one NS by Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO) and Advanced VIRGO detectors \citep{2021PhRvX..11b1053A, 2021arXiv211103606T}.} we create a polarization curve in order to constrain some of the parameters of our off-axis jet model.
Keeping this in mind, the following is the structure of the paper: In Section \ref{sec2}, we briefly describe the off-axis jet synchrotron model derived in \cite{2020ApJ...896...25F} with the extension. In Section \ref{sec3}, we introduce the polarization model used in this paper. In Section \ref{sec4}, we compute the assumed polarization and give the outcomes for a sample of off-axis afterglow-emitting bursts. In Section \ref{sec5} and Section \ref{sec6}, we give analogous analyses for the closest Swift-detected bursts and the GW events that could have emitted an electromagnetic signature, respectively. Finally, in Section \ref{sec7}, we present the conclusion and provide closing thoughts.
\section{Synchrotron Forward-shock model from a radially stratified off-axis jet}\label{sec2}
The multi-wavelength afterglow observations of GRB 170817A are consistent with the synchrotron FS scenario in the fully adiabatic regime from a radially stratified off-axis outflow decelerated in a homogeneous medium \citep{2019ApJ...871..123F}. \cite{2020ApJ...896...25F} extended the synchrotron FS approach to a stratified environment based on the immediate vicinity of a binary NS merger proposed to explain the gamma-ray flux in GRB 150101B. Additionally, \cite{2020ApJ...896...25F} successfully explained the multi-wavelength afterglow observations in GRB 080503, GRB 140903A and GRB 160821B using the synchrotron off-axis model.
In order to present a polarization model and perform a fully time-evolving analysis, we extend the synchrotron scenario described in \cite{2019ApJ...871..123F, 2020ApJ...896...25F} from adiabatic to radiative regime including the self-absorption phase and the dimensionless factor $\xi$ which provides information on the EATS \citep{1998ApJ...493L..31P, 2000ApJ...536..195C}.
\subsection{Synchrotron scenario}\label{subsec21}
Relativistic electrons are accelerated in the FS and cooled down mainly via synchrotron emission in the presence of a comoving magnetic field $B'= \sqrt{8\pi \varepsilon_Be}$, where $e$ is the energy density and $\varepsilon_B$ the fraction of magnetic energy given in the FS. Hereafter, we use the prime and unprimed quantities to refer them in the comoving and observer frames, respectively. The acceleration process leads to that electrons with Lorentz factors ($\gamma_e$) come by a distribution of the form $N(\gamma_e)\,d\gamma_e \propto \gamma_e^{-p}\,d\gamma_e$ with $p$ the electron power index. We consider a radially off-axis jet with an equivalent kinetic energy given by:
\bary\label{eq:ek}
E= \tilde{E}\,\Gamma^{-\alpha_s} \frac{1}{(1+ \Delta \theta^2\Gamma^2)^3}\,,
\eary
where $\tilde{E}$ is the characteristic energy, $\Delta \theta=\theta_{\rm obs} - \theta_{\rm j}$ corresponds to the viewing angle ($\theta_{\rm obs}$) and the half-opening angle of the jet ($\theta_{\rm j}$) and $\Gamma$ is the bulk Lorentz factor. We consider that the circumburst medium can be constant ($n$) or stratified (with a profile given by the stellar-wind $\propto A_{\rm W}r^{-2}$ with $A_{\rm W}$ the density parameter).
\subsubsection{Constant-density medium}
We assume an evolution of the FS with an isotropic equivalent-kinetic energy $E=\frac{4\pi}{3} m_pc^2 n r^{3}\Gamma_0^{\epsilon} \Gamma^{2-\epsilon}$ \citep[Blandford-McKee solution;][]{1976PhFl...19.1130B}, where $\epsilon=0$ corresponds to the adiabatic regime and $\epsilon=1$ to the fully radiative one, and a radial distance $r=c \xi \Gamma^2 t/(1+z)$ with $c$ the speed of light, $m_p$ is the proton mass and $z$ the redshift. Therefore, the evolution of the bulk Lorentz factor is given by:
\bary\label{Gamma}
\Gamma= 9.8\,\left(\frac{1+z}{1.022}\right)^{\frac{3}{\delta+8-\epsilon}}\,\xi^{-\frac{6}{\delta+8-\epsilon}}\, n^{-\frac{1}{\delta+8-\epsilon}}_{-4} \, \Delta \theta^{-\frac{6}{\delta+8-\epsilon}}_{15^\circ} \,\Gamma_0^{-\frac{\epsilon}{\delta+8-\epsilon}}\, \tilde{E}^{\frac{1}{\delta+8-\epsilon}}_{52} \,t^{-\frac{3}{\delta+8-\epsilon}}_{5}\,,
\eary
with $\delta=\alpha_s+6$. Using the bulk Lorentz factor (eq.~\ref{Gamma}) and the synchrotron afterglow theory introduced in \cite{1998ApJ...497L..17S} for the fully adiabatic regime, we derive, in this formalism, the relevant quantities of synchrotron emission originated from the FS. The minimum and cooling electron Lorentz factors can be written as:
{\small
\bary\label{eLor_syn_ism}
\gamma_m&=& 32.6\,\left(\frac{1+z}{1.022}\right)^{\frac{3}{\delta+8-\epsilon}}\, \xi^{-\frac{6}{\delta+8-\epsilon}}\,g(p)\, \varepsilon_{e,-2} \, n_{-4}^{-\frac{1}{\delta+8-\epsilon}}\,\Delta\theta^{-\frac{6}{\delta+8-\epsilon}}_{15^\circ}\,\Gamma_0^{-\frac{\epsilon}{\delta+8-\epsilon}}\, \tilde{E}^{\frac{1}{\delta+8-\epsilon}}_{52}\,t^{-\frac{3}{\delta+8-\epsilon}}_{5},\cr
\gamma_c&=& 4.0\times 10^8 \left(\frac{1+z}{1.022}\right)^{\frac{\delta-1-\epsilon}{\delta+8-\epsilon}}\xi^{\frac{2(1-\delta+\epsilon)}{\delta+8-\epsilon}} (1+Y)^{-1}\, \varepsilon_{B,-4}^{-1}\,n_{-4}^{-\frac{\delta+5-\epsilon}{\delta+8-\epsilon}}\,\Delta\theta^{\frac{18}{\delta+8-\epsilon}}_{15^\circ} \,\Gamma_0^{\frac{3\epsilon}{\delta+8-\epsilon}}\,\tilde{E}^{-\frac{3}{\delta+8-\epsilon}}_{52}\,t^{\frac{1-\delta+\epsilon}{\delta+8-\epsilon}}_{5}\,,
\eary
}
respectively. Here, $Y$ is the Compton parameter, $g(p)=(p-2)/(p-1)$ whereas $\epsilon_e$ is the fraction of energy given to accelerate the electron population. Using the electron Lorentz factors (eq. \ref{eLor_syn_ism}), the characteristic and cooling spectral breaks for synchrotron radiation are
{\small
\bary\label{En_br_syn_ism}
\nu_{\rm m}&\simeq& 2.0\times 10^{-3}\,{\rm GHz}\,\, \left(\frac{1+z}{1.022}\right)^{\frac{4-\delta+\epsilon}{\delta+8-\epsilon}}\,\xi^{-\frac{24}{\delta+8-\epsilon}}\, \varepsilon^2_{e,-2}\,\varepsilon_{B,-4}^{\frac12}\,n_{-4}^{\frac{\delta-\epsilon}{2(\delta+8-\epsilon)}}\Delta\theta_{15^\circ}^{-\frac{24}{\delta+8-\epsilon}}\,\Gamma_0^{-\frac{4\epsilon}{\delta+8-\epsilon}}\,E_{52}^{\frac{4}{\delta+8-\epsilon}}\, t_{5}^{-\frac{12}{\delta+8-\epsilon}},\cr
\nu_{\rm c}&\simeq& 7.6\times 10^{4}\,{\rm keV}\,\, \left(\frac{1+z}{1.022}\right)^{\frac{\delta-4-\epsilon}{\delta+8-\epsilon}}\,\xi^{-\frac{4(2+\delta-\epsilon)}{\delta+8-\epsilon}}\, (1+Y)^{-2} \,\varepsilon_{B,-4}^{-\frac32}\,n_{-4}^{-\frac{16+3\delta-3\epsilon}{2(\delta+8-\epsilon)}}\, \Delta\theta_{15^\circ}^{\frac{24}{\delta+8-\epsilon}}\,\Gamma_0^{\frac{4\epsilon}{\delta+8-\epsilon}}\, E_{52}^{-\frac{4}{\delta+8-\epsilon}}\, t_{5}^{-\frac{2(2+\delta-\epsilon)}{\delta+8-\epsilon}}\,,
\eary
}
respectively. Considering the maximum emissivity, the total number of radiating electrons and the luminosity distance $D_{\rm z}$, the maximum flux emitted by synchrotron radiation is given by
{\small
\bary\label{flux_syn}
F_{\rm max} &\simeq& 0.2\,{\rm mJy}\,\, \left(\frac{1+z}{1.022}\right)^{\frac{16-\delta+\epsilon}{\delta+8-\epsilon}}\,\xi^{\frac{6(\delta-\epsilon)}{\delta+8-\epsilon}}\,\varepsilon_{B,-4}^{\frac12}\,n_{-4}^{\frac{8+3\delta-3\epsilon}{2(\delta+8-\epsilon)}}\,\Delta\theta_{15^\circ}^{-\frac{48}{\delta+8-\epsilon}}\, D^{-2}_{\rm z, 26.3}\,\Gamma_0^{-\frac{8\epsilon}{\delta+8-\epsilon}}\,E_{52}^{\frac{8}{\delta+8-\epsilon}}\, t_{5}^{\frac{3(\delta-\epsilon)}{\delta+8-\epsilon}}\,.
\eary
}
The synchrotron spectral breaks in the self-absorption regime are derived from $\nu_{\rm a,1}=\nu_{\rm c}\tau^{\frac35}_{m}$, $\nu_{\rm a,2}=\nu_{\rm m}\tau^{\frac{2}{p+4}}_{m}$ and $\nu_{\rm a,3}=\nu_{\rm m}\tau^{\frac35}_{c}$ with the optical depths given by $\tau_{m}\simeq\frac{5}{3}\frac{q_e n r}{B'\gamma^5_{\rm m}}$ and $\tau_{c}\simeq\frac{5}{3}\frac{q_e n r}{B'\gamma^5_{\rm c}}$.
The light curves in the fast cooling regime are:
{\small
\begin{eqnarray}
\label{k0_fast}
F^{\rm syn}_{\nu}\propto \begin{cases}
t\,\nu^{\frac13},\hspace{1.5cm}
\hspace{1.2cm} {\rm for} \hspace{0.2cm} \nu < \nu_{\rm a,3}, \cr
t^{\frac{4+11(\delta-\epsilon)}{3(\delta+8-\epsilon)}}\nu^{-\frac12},
\hspace{1.3cm} {\rm for} \hspace{0.2cm} \nu_{\rm a,3} <\nu<\nu^{\rm syn}_{\rm c} ,\hspace{.1cm} \cr
t^{\frac{2(\delta-1-\epsilon)}{\delta+8-\epsilon}}\nu^{-\frac{p-1}{2}}, \hspace{1.1cm} {\rm for} \hspace{0.2cm} \nu^{\rm syn}_{\rm c}<\nu< \nu^{\rm syn}_{\rm m},\hspace{.1cm}\cr
t^{\frac{2(2-3p+\delta-\epsilon)}{\delta+8-\epsilon}}\,\nu^{-\frac{p}{2}},\hspace{0.95cm} {\rm for} \hspace{0.2cm} \nu^{\rm syn}_{\rm m}<\nu\,, \cr
\end{cases}
\end{eqnarray}
}
and in the slow cooling regime are:
{\small
\begin{eqnarray}
\label{k0_slow1}
F^{\rm syn}_{\nu}\propto \begin{cases}
t^{\frac{2(2+\delta-\epsilon)}{\delta+8-\epsilon}}\nu^{2},\hspace{1.5cm}
\hspace{0.22cm} {\rm for} \hspace{0.2cm} \nu < \nu_{\rm a,1}, \cr
t^{\frac{4+3(\delta-\epsilon)}{\delta+8-\epsilon}}\nu^{\frac13},
\hspace{1.65cm} {\rm for} \hspace{0.2cm} \nu_{\rm a,1} <\nu<\nu^{\rm syn}_{\rm m} ,\hspace{.1cm} \cr
t^{\frac{3(2-2p+\delta-\epsilon)}{\delta+8-\epsilon}}\nu^{-\frac{p-1}{2}}, \hspace{0.7cm} {\rm for} \hspace{0.2cm} \nu^{\rm syn}_{\rm m}<\nu< \nu^{\rm syn}_{\rm c},\hspace{.1cm}\cr
t^{\frac{2(2-3p+\delta-\epsilon)}{\delta+8-\epsilon}}\,\nu^{-\frac{p}{2}},\hspace{0.95cm} {\rm for} \hspace{0.2cm} \nu^{\rm syn}_{\rm c}<\nu\,. \cr
\end{cases}
\end{eqnarray}
}
{\small
\begin{eqnarray}
\label{k0_slow2}
F^{\rm syn}_{\nu}\propto \begin{cases}
t^{\frac{2(2+\delta-\epsilon)}{\delta+8-\epsilon}}\nu^{2},\hspace{1.5cm}
\hspace{0.23cm} {\rm for} \hspace{0.2cm} \nu < \nu^{\rm syn}_{\rm m}, \cr
t^{\frac{2(5+\delta-\epsilon)}{\delta+8-\epsilon}}\nu^{\frac52},
\hspace{1.65cm} {\rm for} \hspace{0.2cm} \nu^{\rm syn}_{\rm m} <\nu<\nu_{\rm a,2} ,\hspace{.1cm} \cr
t^{\frac{3(2-2p+\delta-\epsilon)}{\delta+8-\epsilon}}\nu^{-\frac{p-1}{2}}, \hspace{0.7cm} {\rm for} \hspace{0.2cm} \nu_{\rm a,2}<\nu< \nu^{\rm syn}_{\rm c},\hspace{.1cm}\cr
t^{\frac{2(2-3p+\delta-\epsilon)}{\delta+8-\epsilon}}\,\nu^{-\frac{p}{2}},\hspace{0.95cm} {\rm for} \hspace{0.2cm} \nu^{\rm syn}_{\rm c}<\nu\,. \cr
\end{cases}
\end{eqnarray}
}
\subsubsection{Stellar-wind medium}
In the case of a stratified stellar-wind like medium, the number density is given by $n(r)=\frac{\rho(r)}{m_p}=\frac{A}{m_p}\,r^{-2}$ where $A=\frac{\dot{M}}{4\pi\, v}=\,5\times 10^{11}\,A_{\rm W} \,{\rm g\,cm^{-1}}$, with $\dot{M}$ the mass-loss rate and $v$ the velocity of the outflow \citep[e.g., see][]{2016ApJ...818..190F}. Taking into account the Blandford-McKee solution for a stratified stellar-wind like medium, the bulk Lorentz factor derived through the adiabatic evolution \citep{1976PhFl...19.1130B, 1997ApJ...489L..37S} is given by
{\small
\bary\label{Gamma_wind}
\Gamma= 16.2 \left(\frac{1+z}{1.022}\right)^{\frac{1}{\delta+4-\epsilon}}\, \xi^{-\frac{2}{\delta+4-\epsilon}}\, A_{\rm W,-1}^{-\frac{1}{\delta+4-\epsilon}}\, \Delta \theta^{-\frac{6}{\delta+4-\epsilon}}_{15^\circ} \,\Gamma_0^{-\frac{\epsilon}{\delta+4-\epsilon}}\, \tilde{E}^{\frac{1}{\delta+4-\epsilon}}_{52} \,t^{-\frac{1}{\delta+4-\epsilon}}_{5}\,,
\eary
}
with the characteristic energy given by {\small $\tilde{E}= \frac{16\pi}{3}\, (1+z)^{-1}\,\xi^{2}\, A_{\rm W} \, \Delta \theta^6\,\Gamma_{0}^{\epsilon}\, \Gamma^{\delta+4-\epsilon}\,t\,$}.
Using the bulk Lorentz factor (eq.~\ref{Gamma_wind}) and the synchrotron afterglow theory for a wind-like medium \citep{2000ApJ...536..195C, 2000ApJ...543...66P}, we derive the relevant quantities of synchrotron emission for our model in the fully adiabatic regime. The minimum and cooling electron Lorentz factors are given by:
{\small
\bary\label{eLor_syn_wind}
\gamma_m&=& 41.5\left(\frac{1+z}{1.022}\right)^{\frac{1}{\delta+4-\epsilon}}\, \xi^{-\frac{2}{\delta+4-\epsilon}}\,g(p)\, \varepsilon_{\rm e,-2}\,\Delta\theta^{\frac{-6}{\delta+4-\epsilon}}_{15^\circ} \, A_{\rm W,-1}^{-\frac{1}{\delta+4-\epsilon}}\,\Gamma_0^{-\frac{\epsilon}{\delta+4-\epsilon}}\, \tilde{E}^{\frac{1}{\delta+4-\epsilon}}_{52}\,t^{-\frac{1}{\delta+4-\epsilon}}_{5}, \cr
\gamma_c&=& 52.1 \left(\frac{1+z}{1.022}\right)^{-\frac{\delta+3-\epsilon}{\delta+4-\epsilon}} (1+Y)^{-1}\,\xi^{\frac{2(\delta+3-\epsilon)}{\delta+4-\epsilon}}\, \varepsilon_{B,-4}^{-1}\,A_{\rm W,-1}^{-\frac{\delta+5-\epsilon}{\delta+4-\epsilon}}\,\Delta\theta^{-\frac{6}{\delta+4-\epsilon}}_{15^\circ} \,\Gamma_0^{-\frac{\epsilon}{\delta+4-\epsilon}}\,\tilde{E}^{\frac{1}{\delta+4-\epsilon}}_{52}\,t^{\frac{\delta+3-\epsilon}{\delta+4-\epsilon}}_{5}\,.
\eary
}
The characteristic and cooling spectral breaks for synchrotron emission are:
{\small
\bary\label{En_br_syn_wind}
\nu_{\rm m}&\simeq& 1.0\times 10^{14}{\rm Hz}\, \left(\frac{1+z}{1.022}\right)^{\frac{2}{\delta+4-\epsilon}}\, \xi^{-\frac{2(\delta+6-\epsilon)}{\delta+4-\epsilon}} \,\varepsilon^2_{\rm e,-2}\,\varepsilon_{B,-4}^{\frac12}\,A_{W,-1}^{\frac{\delta-\epsilon}{2(\delta+4-\epsilon)}}\, \,\Delta\theta_{15^\circ}^{-\frac{12}{\delta+4-\epsilon}}\,\Gamma_0^{-\frac{2\epsilon}{\delta+4-\epsilon}}\, E_{52}^{\frac{2}{\delta+4-\epsilon}}\, t_{5}^{-\frac{\delta+6-\epsilon}{\delta+4-\epsilon}},\cr
\nu_{\rm c}&\simeq& 1.1\times 10^{14}{\rm Hz}\,\, \left(\frac{1+z}{1.022}\right)^{-\frac{2(\delta +3-\epsilon)}{\delta+4-\epsilon}}\,\xi^{\frac{2(\delta+2-\epsilon)}{\delta+4-\epsilon}}\,(1+Y)^{-2}\, \,\varepsilon_{B,-4}^{-\frac32} A_{\rm W,-1}^{-\frac{3\delta+16-3\epsilon}{2(\delta+4-\epsilon)}}\,\Delta\theta_{15^\circ}^{-\frac{12}{\delta+4-\epsilon}}\,\Gamma_0^{-\frac{2\epsilon}{\delta+4-\epsilon}}\,E_{52}^{\frac{2}{\delta+4-\epsilon}}\, t_{5}^{\frac{\delta+2-\epsilon}{\delta+4-\epsilon}}\,,
\eary
}
respectively. Given the maximum emissivity in a stratified stellar-wind like medium, the maximum flux radiated by synchrotron emission is given by:
{\small
\bary\label{Flux_syn_wind}
F_{\rm max} &\simeq& 1.9\times 10^{3}\,{\rm mJy}\,\, \left(\frac{1+z}{1.022}\right)^{\frac{2(\delta+5-\epsilon)}{\delta+4-\epsilon}}\, \xi^{-\frac{4}{\delta+4-\epsilon}} \,\varepsilon_{B,-4}^{\frac12}\,A_{\rm W,-1}^{\frac{3\delta+8-3\epsilon}{2(\delta+4-\epsilon)}}\, D^2_{\rm z, 26.3}\,\Delta\theta_{15^\circ}^{-\frac{12}{\delta+4-\epsilon}}\,\Gamma_0^{-\frac{2\epsilon}{\delta+4-\epsilon}}\,E_{52}^{\frac{2}{\delta+4-\epsilon}}\, t_{5}^{-\frac{2}{\delta+4-\epsilon}}\,.
\eary
}
The synchrotron spectral breaks in the self-absorption regime are derived from $\nu_{\rm a,1}=\nu_{\rm c}\tau^{\frac35}_{m}$, $\nu_{\rm a,2}=\nu_{\rm m}\tau^{\frac{2}{p+4}}_{m}$ and $\nu_{\rm a,3}=\nu_{\rm m}\tau^{\frac35}_{c}$ with the optical depths given by $\tau_{m}\propto \frac{q_e A_{\rm W} r^{-1}}{B'\gamma^5_{\rm m}}$ and $\tau_{c} \propto \frac{q_e A_{\rm W} r^{-1}}{B'\gamma^5_{\rm c}}$.
The light curves in the fast cooling regime are:
{\small
\begin{eqnarray}
\label{k2_fast}
F^{\rm syn}_{\nu}\propto \begin{cases}
t^{\frac{8+3(\delta-\epsilon)}{\delta+4-\epsilon}}\nu^{\frac13},\hspace{1.5cm}
\hspace{0.35cm} {\rm for} \hspace{0.2cm} \nu < \nu_{\rm a,3}, \cr
t^{\frac{\epsilon-\delta-8}{3(\delta+4-\epsilon)}}\nu^{-\frac12},
\hspace{1.64cm} {\rm for} \hspace{0.2cm} \nu_{\rm a,3} <\nu<\nu^{\rm syn}_{\rm c} ,\hspace{.1cm} \cr
t^{\frac{\delta-2-\epsilon}{2(\delta+4-\epsilon)}}\nu^{-\frac{p-1}{2}}, \hspace{1.3cm} {\rm for} \hspace{0.2cm} \nu^{\rm syn}_{\rm c}<\nu< \nu^{\rm syn}_{\rm m},\hspace{.1cm}\cr
t^{\frac{2(2-3p)+(\epsilon-\delta)(p-2)}{2(\delta+4-\epsilon)}}\,\nu^{-\frac{p}{2}},\hspace{0.3cm} {\rm for} \hspace{0.2cm} \nu^{\rm syn}_{\rm m}<\nu\,, \cr
\end{cases}
\end{eqnarray}
}
whereas in the slow cooling regime are:
{\small
\begin{eqnarray}
\label{k2_slow1}
F^{\rm syn}_{\nu}\propto \begin{cases}
t^{\frac{2(2+\delta-\epsilon)}{\delta+4-\epsilon}}\nu^{2},\hspace{1.5cm}
\hspace{0.72cm} {\rm for} \hspace{0.2cm} \nu < \nu_{\rm a,1}, \cr
t^{\frac{\delta-\epsilon}{3(\delta+4-\epsilon)}}\nu^{\frac13},
\hspace{2.15cm} {\rm for} \hspace{0.2cm} \nu_{\rm a,1} <\nu<\nu^{\rm syn}_{\rm m} ,\hspace{.1cm} \cr
t^{\frac{2+\delta-\epsilon-p(6+\delta-\epsilon)}{2(\delta+4-\epsilon)}}\nu^{-\frac{p-1}{2}}, \hspace{0.7cm} {\rm for} \hspace{0.2cm} \nu^{\rm syn}_{\rm m}<\nu< \nu^{\rm syn}_{\rm c},\hspace{.1cm}\cr
t^{\frac{2(2-3p)+(\epsilon-\delta)(p-2)}{2(\delta+4-\epsilon)}}\,\nu^{-\frac{p}{2}},\hspace{0.63cm} {\rm for} \hspace{0.2cm} \nu^{\rm syn}_{\rm c}<\nu\,. \cr
\end{cases}
\end{eqnarray}
}
{\small
\begin{eqnarray}
\label{k2_slow2}
F^{\rm syn}_{\nu}\propto \begin{cases}
t^{\frac{2(2+\delta-\epsilon)}{\delta+4-\epsilon}}\nu^{2},\hspace{1.5cm}
\hspace{0.43cm} {\rm for} \hspace{0.2cm} \nu < \nu^{\rm syn}_{\rm m}, \cr
t^{\frac{14+5(\delta-\epsilon)}{2(\delta+4-\epsilon)}}\nu^{\frac52},
\hspace{1.75cm} {\rm for} \hspace{0.2cm} \nu^{\rm syn}_{\rm m} <\nu<\nu_{\rm a,2} ,\hspace{.1cm} \cr
t^{\frac{2+\delta-\epsilon-p(6+\delta-\epsilon)}{2(\delta+4-\epsilon)}}\nu^{-\frac{p-1}{2}}, \hspace{0.43cm} {\rm for} \hspace{0.2cm} \nu_{\rm a,2}<\nu< \nu^{\rm syn}_{\rm c},\hspace{.1cm}\cr
t^{\frac{2(2-3p)+(\epsilon-\delta)(p-2)}{2(\delta+4-\epsilon)}}\,\nu^{-\frac{p}{2}},\hspace{0.35cm} {\rm for} \hspace{0.2cm} \nu^{\rm syn}_{\rm c}<\nu\,. \cr
\end{cases}
\end{eqnarray}
}
\section{Polarization model}\label{sec3}
Since 1999, the phenomena of polarization, the confinement of wave vibrations to a certain geometrical direction, has been detected in GRBs \citep{1999A&A...348L...1C}. Further studies indicate that the polarization degree ($\Pi$) can have high variability, but the polarization angle (P.A.; $\theta_p$) remains roughly the same, for an observer outside the jet \citep{2021MNRAS.507.5340T}.
Polarization is commonly attributed to synchrotron radiation behind shock waves. This makes it dependent on the magnetic field configuration and the geometry of the shock, as they will define the the P.D. on each point and its integration over the whole image \citep{Gill-1}.
The Stokes parameters (I, Q, U, and V) control the approach to polarization calculation, and normally only linear polarization is considered.
From this point on, we refer to the observer and comoving frames as unprimed and primed, respectively.
The stokes parameters are expressed as
\begin{eqnarray}
&V = 0, \hspace{2cm} &\theta_p = \frac{1}{2}\arctan{\frac{U}{Q}},\,\cr
&\frac{U}{I} = \Pi'\sin{2\theta_p}, \hspace{1cm}
&\frac{Q}{I} = \Pi'\cos{2\theta_p}.
\end{eqnarray}
And the measured stokes parameters are the sum over the flux \citep{Granot-P2}, so
\begin{eqnarray}
&\frac{U}{I} = \frac{\int \mathrm{d}F_\nu\Pi'\sin{2\theta_p}}{\int \mathrm{d}F_\nu},\hspace{1cm}
\frac{Q}{I} = \frac{\int \mathrm{d}F_\nu\Pi'\cos{2\theta_p}}{\int \mathrm{d}F_\nu}, \\
&\Pi = \frac{\sqrt{Q^2+U^2}}{I}.
\end{eqnarray}
The relationship $\mathrm{d}F_\nu \propto \delta_D^3L'_{\nu'}\mathrm{d}\Omega$ -- where $L'_{\nu'}$ is the spectral luminosity and $\mathrm{d}\Omega$ is the element of solid angle of the fluid element in relation to the source -- allows the introduction of the factors regarding the geometry of the magnetic field and outflow by using \citep{ribicky}
\begin{eqnarray}
L'_{\nu'} \propto (\nu')^{-\alpha} (\sin{\chi'})^\epsilon r^m \propto (\nu')^{-\alpha} (1-\hat{n}' \cdot \hat{B}')^{\epsilon/2} r^m.
\end{eqnarray}
The parameter $\chi$ is the angle between the local magnetic field and the particle's direction of motion, and due to the highly beamed nature of synchrotron emission, this angle is also the pitch angle. The geometrical considerations of polarization can then be taken by averaging this factor over the local probability distribution of the magnetic field \citep[see Eq. 15 of][]{Gill-1},
\begin{eqnarray}
\Lambda = \expval{(1-\hat{n}' \cdot \hat{B}')^{\epsilon/2}}.
\end{eqnarray}
It is possible to do a Lorentz transformation on the unit vectors, like $\hat{n}$, or a certain configuration of $\hat{B}$ to express $\Lambda$ in terms of different magnetic field configurations \citep{Gill-1, Lyutikov, Granot-P2}:
\begin{eqnarray}
\Lambda_{\rm ord} &\approx& \left[\left(\frac{1-\Tilde{\xi}}{1+\Tilde{\xi}}\right)\cos^2{\varphi_B} + \sin^2{\varphi_B}\right]^{\epsilon/2}\label{eq:pd_ord},\\
\Lambda_{\perp} &\approx& \expval{\Lambda_{ord}(\Tilde{\xi}, \varphi_B)}_{\varphi_B}\label{eq:pd_perp},\\
\Lambda_{\parallel} &\approx& \left[\frac{\sqrt{4\Tilde{\xi}}}{1+\Tilde{\xi}}\right]^\epsilon \label{eq:pd_par},
\end{eqnarray}
where $\varphi_B$ as the azimuthal angle of the magnetic field measured from a reference point. $\tilde{\xi} \equiv (\Gamma\tilde{\theta})^2$, taking in consideration the approximations of $\Tilde{\mu} = \cos{\tilde{\theta}} \approx 1 - \tilde{\theta}^2/2$ and $\beta \approx 1 - 1/2\Gamma^2 $, which leads to $ \delta_D \approx \frac{2\Gamma}{1+\Tilde{\xi}}$ where $\tilde{\theta}$ the polar angle measured from the Line of Sight (LOS).
One of the still-unsolved mysteries of GRBs is the configuration of the magnetic field present at different regions of emission. As such, various possible configurations must be explored in a topic where magnetic field geometry is of paramount relevance, like polarization.
The considerations regarding the magnetic field geometry are varied based on the GRB epoch of relevance. For a scenario where the afterglow is described by a FS, two of the most suitable configurations are: a random perpendicular configuration -- where the anisotropy factor $b \equiv \frac{2\expval{B_\parallel^2}}{\expval{B_\perp^2}}= 0$ -- confined to the shock plane; and an ordered configuration parallel to the velocity vector, where $b\rightarrow\infty$. More complex configurations with multi-component, where the anisotropy is $b>0$, magnetic fields have been done \citep{2020MNRAS.491.5815G, 2021MNRAS.507.5340T, 2020ApJ...892..131S, 2018ApJ...861L..10C}, as it is warranted and needed, however, for the purposes of this paper we limit ourselves to the two following cases.
\paragraph*{Random magnetic field ($B_\perp,\,b=0$)}
In this scenario, the symmetry of the random magnetic field configuration, perpendicular to the shock plane, causes the polarization over the image to disappear when if the beaming cone is wholly contained within the jet aperture or if it is seen along the axis ($\theta_{\rm obs} = 0$). To break the symmetry, the jet must be viewed close to its edge ($q\equiv \frac{\theta_{\rm obs}}{\theta_j} \gtrsim 1+\xi_j^{-1/2}$), where missing emission (from $\theta > \theta_j$) results only in partial cancellation \citep{Waxman}.
The equation necessary to calculate this polarization is explicitly laid out as Eq. 5 in \citep{Granot-P2}.
\paragraph*{Ordered magnetic field ($B_\parallel, \,b\rightarrow\infty$)}
For the ordered magnetic field, a configuration parallel to the velocity vector, the same symmetry observations hold true and the calculation follows \citep{Granot-P2, Gill-1}, with $\Lambda(\tilxi) = \Lambda_\parallel$ from \Cref{eq:pd_par}.
By substituting the following integration limits
\begin{eqnarray}\label{conections}
\cos{\psi(\tilxi)} = \frac{(1-q)^2 \xi_j - \tilxi}{2q\sqrt{\xi_j\tilxi }}, \qquad \xi_j = (\Gamma\theta_j)^2, \qquad \xi_\pm = (1\pm q)^2\xi_j,
\end{eqnarray}
with an appropriate prescription of the bulk Lorentz factor $\Gamma(t)$, the evolution of the opening angle of the jet $\theta_j(t)$, and the parameters required to describe these expressions as described in Section \ref{sec2} and \cite{2020ApJ...896...25F}, we can obtain the temporal evolution of polarization.
\subsection{Polarization evolution for a Forward-Shock }
\Cref{fig:general_k0,fig:general_k2} show the temporal evolution of polarization degree for our chosen magnetic field configurations in two distinct scenarios regarding the density of the circumburst medium -- here considered a constant density and a wind-like medium. Each column of these figures represents a chosen combination of the $\epsilon$ and $\xi$ parameters. Table \ref{tab:table_general} shows the values required to generate \Cref{fig:general_k0,fig:general_k2}. We highlight that the generic values were determined based on the typical range reported for each parameter in the GRB synchrotron literature \citep[for reviews, see][]{Kumar, 2014ARA&A..52...43B}. The values of observation angle are varied over a range between $8$ and $15\deg$.\footnote{Over the course of this manuscript we will be using $\deg$ as the abbreviation of degree.} This range of values is shown in these figures with different colored lines, each one standing for a value of $q_0=\frac{\theta_{\rm obs}}{\theta_{\rm j,0}}$, the ratio between the observation angle and initial opening angle of the jet.
The synchrotron model chosen is a homogeneous off-axis jet, in which the equivalent kinetic energy is parameterized with a power-law velocity distribution (see \Cref{eq:ek}), that suffers sideways expansion (SE) with the comoving speed of sound given as Eq. 10 in \cite{2000ApJ...543...90H}. The homogeneous jet case has been studied by a few works now, such as \cite{2002ApJ...570L..61G, 2004MNRAS.354...86R}, however, these works only have explored random magnetic field on a fully adiabatic regime in a constant medium. Nonetheless, comparing results with the leftmost column of \Cref{fig:general_k0}, we see the typical double peak behavior for a homogeneous SE jet, reported by \cite{2004MNRAS.354...86R} for $q_0<5$, is presented for us as well. Some discrepancies are shown, with our polarization being initially higher at early times (increasingly so as $q_0 \rightarrow 1$) and overall in the magnitude of the peaks. The highest likelihood culprit for these differences is the choice of synchrotron model and parameter values. The center column of \Cref{fig:general_k0} presents the case for a partially radiative scenario, and it behaves quite similarly to the adiabatic case, with only a change in magnitude of the peaks being observable. The deceleration of the relativistic outflow by the circumburst medium is faster when it lies in the radiative regime rather than adiabatic one, and the temporal evolution of polarization is modified \citep{2000ApJ...532..281B, 2005ApJ...619..968W}. For our model, this has resulted in an enhancement of the increase in polarization as $q_0$ grows, but smaller second peaks. The rightmost side of \Cref{fig:general_k0} displays the case for an adiabatic regime with $\xi = 0.56$ \citep{2000ApJ...536..195C}. The variation on $\xi$ causes the emission to arrive earlier or later, and this produces a difference in the magnitudes of the peaks, as observed in \Cref{fig:general_k0} \citep{1997ApJ...491L..19W, 2000ApJ...536..195C, 1998ApJ...493L..31P}. The polarization behaviour flips in comparison with $\xi=1$, with the peak increasing as $q_0 \rightarrow 1$; comparatively, the second peak remains mostly the same. The parallel case presents similar behavior for all three considered cases. A small change is observed at the sharpness of decline of polarization at jet-break (where the synchrotron model bulk Lorentz factor changes regime to follow the on-axis calculations presented by \cite{2020ApJ...896...25F}) and post-break, with a stronger discontinuity happening with a decreasing value of $\xi$.
\Cref{fig:general_k2} shows the polarization evolution for the wind-like medium. \cite{2004A&A...422..121L} expected that the polarization evolved slower for a wind-like medium, as the relationship between afterglow timescale and density was $t\propto (E/n)^{\frac{1}{(3-k)}}$ \citep[][and with $k=2$ for a wind-like medium]{Kumar, 2022arXiv220502459F}, and this is observed here too. For a convenience of observation, the limits of the timescale have been expanded. Other significant differences between the constant-density medium and this scenario are the higher initial polarization peak and lower magnitude of the second peak, in all likelihood due to the lower value of bulk Lorentz factor at later times. Between the chosen values of $\xi$ and $\epsilon$, we see that a lower value of $\xi$ increases the magnitude of the first peak while decreasing the magnitude of the second one. This is similar to the constant-density medium case, with the addendum that the second peak is reduced further when compare to $\xi=1$. For the partially radiative case, the first polarization peak is similar to the adiabatic case with higher magnitude, but the second polarization peak is further reduced.
\section{Polarization from GRB Off-axis Afterglows}\label{sec4}
In this section, we describe the polarization for a group of GRBs that show similar characteristics on their afterglow: GRB 080503, GRB 140903A, GRB 150101B, GRB 160821B, and GRB 170817A. In \cite{2020ApJ...896...25F}, the authors have explored the similarities between those bursts. We use the parameter values obtained by \cite{2020ApJ...896...25F} via Markov Chain Monte Carlo (MCMC) simulations to calculate polarization. For this section, we will adopt the notation $f(q_0=x^{\pm y}_{\pm z}) = a^{\pm b}_{\pm c}$ when the chosen values of $q_0$ result in significant differentiation on polarization or peak time.
\paragraph{GRB 080503}
The first column in \Cref{fig:joined_GRB} shows the expected polarization evolution calculated for GRB 080503 for our two configurations. The parameters for calculating this polarization are presented in the first row of \Cref{tab:pol_grbs}. The granular increment of $q_0$ shows little effect in the polarization curves in either configuration, with the only major difference being the magnitude of the minimum located between peaks. The initial polarization for the time-frame we have chosen is $\abs{\Pi/\Pi_{\rm max}}\approx 5\%$ and $\Pi/\Pi_{\rm max}\approx 92\%$ for $B_\perp$ and $B_\parallel$, respectively. For $B_\perp$, the polarization evolves towards a peak of $\abs{\Pi/\Pi_{\rm max}}\approx 46\%$, with a second peak of $\abs{\Pi/\Pi_{\rm max}}\approx 33\%$ at $\sim 0.3$ and $\sim 0.8$ days, respectively. For the parallel configuration, the initial polarization decreases softly during the off-axis period by roughly $10\%$. After the jet break, the polarization drops sharply, and zero polarization is reached at $\sim 17 $ days.
\paragraph{GRB 140903A}
The second column in \Cref{fig:joined_GRB} shows the expected polarization evolution estimated for GRB 140903A. The parameters for calculating this polarization are presented in the second row of \Cref{tab:pol_grbs}. For this burst, a slightly higher degree of influence of $q_0$ is observed. The initial polarization values are $\abs{\Pi/\Pi_{\rm max}}(q_0=2.19^{+0.11}_{-0.11})\approx 7.1^{-0.6}_{+0.7}\%$ and $\Pi/\Pi_{\rm max}(q_0=2.19^{+0.11}_{-0.11})\approx 91^{+0.7}_{-1.0}\%$ for $B_\perp$ and $B_\parallel$, respectively. For the perpendicular field, the peak of $\abs{\Pi/\Pi_{\rm max}}\approx 42^{+0.4}_{-0.2}\%$ is seen at $t_{\rm peak}(q_0=2.19^{+0.11}_{-0.11}) \approx 4.4^{+0.6}_{-1.0}\times 10^{-2}$ days with a second peak of $\abs{\Pi/\Pi_{\rm max}}\approx 33^{+2}_{-1}\%$ at $t_{\rm peak} \approx 1.9^{+0.2}_{-0.5}\times 10^{-1}$ days, respectively. For the parallel configuration, the polarization at the break is $\Pi/\Pi_{\rm max}\approx 79^{+1.0}_{-0.8}\%$, and zero is achieved roughly at the same time of $\sim 6$ days.
\paragraph{GRB 150101B}
The third column in \Cref{fig:joined_GRB} shows the expected polarization calculated for GRB 150101B. The parameters for calculating this polarization are presented in the third row of \Cref{tab:pol_grbs}. The higher value of $q_0$ makes so the minute variation of the chosen values has little influence on the polarization. The initial values of polarization are $\abs{\Pi/\Pi_{\rm max}}\approx 1.5\%$ and $\Pi/\Pi_{\rm max}\approx 97\%$, for $B_\perp$ and $B_\parallel$, respectively. For $B_\perp$, the first polarization peak is $\abs{\Pi/\Pi_{\rm max}}(q_0=4.21^{+0.05}_{-0.05})\approx 40^{+0.5}_{-2.0}\%$, and the second $\abs{\Pi/\Pi_{\rm max}}\approx 42\%$ at $\sim2$ and $\sim 7$ days, respectively. For the parallel configuration, the polarization decreases by $\sim 13\%$ until the break is achieved and decreases to zero rapidly, reaching it at $\sim 40$ days.
\paragraph{GRB 160821B}
The fourth column in \Cref{fig:joined_GRB} shows the expected evolution of polarization calculated for GRB 160821B. The parameters for calculating this polarization are presented in the fourth row of \Cref{tab:pol_grbs}. The initial polarization values are $\abs{\Pi/\Pi_{\rm max}}(q_0=1.69^{+0.03}_{-0.03})\approx 17^{-0.6}_{+0.8}\%$ and $\Pi/\Pi_{\rm max}(q_0=1.69^{+0.03}_{-0.03})\approx 85.5^{+0.5}_{-0.5}\%$ for $B_\perp$ and $B_\parallel$, respectively. For the perpendicular case, the peak of $\abs{\Pi/\Pi_{\rm max}}\approx 50\%$ is seen at $t_{\rm peak}(q_0=1.69^{+0.03}_{-0.03}) \approx 2.0^{+0.2}_{-0.3}\times 10^{-2}$ days with a second peak of $\abs{\Pi/\Pi_{\rm max}}\approx 26\%$ at $t_{\rm peak} \approx 7.5^{+1.1}_{-0.9}\times 10^{-2}$ days. For the parallel configuration, the polarization at the break is $\Pi/\Pi_{\rm max}\approx 73^{+1.0}_{-0.8}\%$, and zero is achieved roughly at the same time of $\sim 6$ days.
\paragraph{GRB 170817A}
\Cref{fig:joined_170817A} shows the expected polarization, calculated with our model, for the different configurations of magnetic fields. GRB 170817A has been modelled by a variety of different synchrotron scenarios, while the more traditional top-hat off-axis jet has been disfavored, other models such as radially stratified ejecta \citep{2018Natur.554..207M, 2018ApJ...867...95H, 2019ApJ...871..123F}, and structured jets \citep{2017Sci...358.1559K, 2017MNRAS.472.4953L, 2018PhRvL.120x1103L} can properly describe the multiwavelength afterglow observations. One thing to note is that for the period starting two weeks after the burst, the flux can be described by a relativistic collimated jet \citep[see references above and][]{2019ApJ...884...71F}. As such, the angular structure of the jet is less relevant regarding whether (or not) a homogeneous jet can successfully describe the late afterglow.
We use the phenomenological model presented in this paper for a constant-density medium with $\xi=1$ and $\epsilon=0$ to obtain the polarization. These conditions reduce our model to the one used in \cite{2019ApJ...871..123F}, where the authors have fitted the synchrotron light curves. We have used the values reported in Table 3 in \cite{2019ApJ...871..123F} to generate the polarization curves. Based on these conditions, the polarization presents a similar behavior as the left side of \Cref{fig:general_k0}.
For the perpendicular configuration of magnetic field, we observe an initial $\sim 1.8\%$ polarization for all values of $q_0$. Then, the polarization begins its evolution towards a maximum of $\abs{\Pi/\Pi_{\rm max}}(q_0=4.05^{+0.15}_{-0.15})\approx 55^{-1}_{+1}\%$ at $t\approx 27$ days, with a second peak of $\abs{\Pi/\Pi_{\rm max}}\approx 57.5^{+0.5}_{-0.5}\%$ at $t\approx 100$ days.
The parallel configuration has an initially high degree of polarization across the board and low influence of $q_0$, with $\Pi/\Pi_{\rm max} \approx 97\%$, and $\Pi/\Pi_{\rm max} \approx 84\%$ at the break. The blue inverted triangles \Cref{fig:joined_170817A} show the upper limits, of $\abs{\Pi} \approx 12\%$ at $t\approx 243 $ days \citep[derived by][]{2018ApJ...861L..10C}, normalized by our arbitrarily chosen value of $\Pi_{\rm max} =70\%$. Upper limits are broken by the polarization curves, with $\abs{\Pi/\Pi_{\rm max}} (B_\perp) \approx 25\%$ and $\Pi/\Pi_{\rm max} (B_\parallel) \approx 33\%$. This indicates that the chosen configurations cannot successfully describe the polarization observed for GRB 170817A.
Several attempts at constraining the magnetic field configuration of GRB 170817A have been performed \citep[e.g., see][]{2018MNRAS.478.4128G, 2020ApJ...892..131S, 2020MNRAS.491.5815G, 2021MNRAS.507.5340T}, using the available polarimetric upper limits and multiple types of outflows. These works agree that a configuration with $b=0$ ($b\rightarrow\infty$) is ruled out. An exception is the case of a wide-angled quasi-spherical outflow with energy injection, calculated by \cite{2018MNRAS.478.4128G}, which does not break the upper limits. However, this particular model is disfavoured to describe the afterglow flux of the burst. \cite{2021MNRAS.507.5340T} and \cite{2020MNRAS.491.5815G} have constrained the anisotropy of the magnetic fields to a dominant perpendicular component with a sub-dominant parallel component ($0.85\lesssim b \lesssim1.16$ and $0.66 \lesssim b \lesssim 1.49$, for each paper, respectively).
More observations on a shorter post-burst period would be needed to constrain the magnetic field configuration further, and proper modeling of the afterglow light curve is necessary for breaking the degeneracy between models. Unfortunately, there were no polarization observations at any other frequency and time \citep{2018ApJ...861L..10C}.
\section{The closest sGRBs detected by Swift satellite}\label{sec5}
\cite{2020MNRAS.492.5011D} presented a systematic search for nearby sGRBs with similar features to GRB 170817A in the Swift database, covering 14 years of operations. A subset of four potential candidates: GRB 050906, GRB 070810B, GRB 080121, and GRB 100216A, were found between 100 and 200 ${\rm Mpc}$. These candidates were used to constrain the range of properties for X-ray counterparts of a merger of two NSs, and derived optical upper limits on the onset of a ``blue" KN, implying a low amount of lanthanide-poor ejecta (see Section 3.2 from \cite{2020MNRAS.492.5011D} and references therein).\\
\subsection{Light curves and Polarization}
Figure \ref{GRBs_Swift} presents three rows, where the first one corresponds to a set of Swift-identified bursts between 100 and 200 Mpc. Each panel in the row shows the optical upper limits with the synchrotron light curves expected from an off-axis jet decelerating in a constant-density medium for two different viewing angles; $\theta_{\rm obs}= 4\,$ (solid lines) and $15\,{\rm deg}$ (dashed lines). The synchrotron light curves are presented at the R-band (red) and the u-band (green). The parameter values are reported in Table \ref{tab:Best-Fit-Parameters} with $\Gamma=100$, $\varepsilon_{\rm e}=0.3$, $p=2.5$, $\zeta_e=1.0$ and $\varepsilon_{\rm B}=10^{-4}$. For the chosen afterglow values, higher viewing angles (more than $15\, {\rm deg}$) are favored by our model.
\Cref{fig:joined_swift} shows the expected polarization curves that could be present on account of the parameters used to obtain Figure \ref{GRBs_Swift}. We have the perpendicular and parallel configurations presented from left to right. Two values of $q_0$ were used for these calculations, and two curves were calculated based on the different angles constrained by fitting the upper flux limits. We can notice that the set of parameters for an observation angle of $\theta_{\rm obs}=4\deg$ violates the optical upper limits. As such, we will call this set of parameters ``disallowed", and the set for which the flux is below the upper limits, with $\theta_{\rm obs}=15\deg$ as ``allowed".
Looking at the perpendicular configuration, the disallowed set presents an initially high polarization ($\abs{\Pi/\Pi_{\rm max}} =22\%$) compared to the allowed set ($\abs{\Pi/\Pi_{\rm max}} =2\%$). The peak polarization also happens earlier for the disallowed set and reaches zero earlier. Considering both evolutions, rough limits can be set for these orphan afterglows of similar characteristics. The intersection between curves would set so that the polarization must be $\abs{\Pi/\Pi_{\rm max}} < 31\%$ for $t\approx1.5\times10^{-2}$. However, since the disallowed set decreases past this point (while the allowed set increases), the requirement is that $\abs{\Pi/\Pi_{\rm max}} > 31\%$ for $t > 1.5\times10^{-2}\,{\rm s}$.
For the parallel configuration, we consider the behavior that the $\Pi/\Pi_{\rm max}$ is higher than $q_0$ increases, with a slower descent until the jet break time, where the polarization plummets. The disallowed set faster decrease would indicate that the polarization at break times must remain high, if we consider the best fit option for the set of Swift-identified bursts is a sufficiently off-axis emission. As such, a polarization $\Pi/\Pi_{\rm max}>80\%$ would be required, at the time of the break, by our model.
\section{Promising GW events in the third observing run (O3) that could generate electromagnetic emission}\label{sec6}
\subsection{Multi-band observations}
During the O3 observing run (from 2019 April 01 to 2020 March 27), the Advanced LIGO and Advanced Virgo GW detectors reported 56 GW events. The run was homogeneously split into two periods called ``O3a'' (from 2019 April 01 to September 30) and ``O3b'' (from 2019 November 01 to 2020 March 27). The candidate GW events in the O3a and O3b runs are reported in Gravitational Wave Transient (GWTC-2) Catalog 2 and (GWTC-3) Catalog 3, respectively. The potential candidates reported that are consistent with a source with $m_2<3M_\odot$ -- where $m_2$ is the mass of the secondary component of the binary merger -- and that could generate electromagnetic emission are GW190425, GW190426\_152155, GW190814 in GWTC-2 \citep{2021PhRvX..11b1053A} and GW191219\_163120, GW200105\_162426, GW200115\_042309, GW200210\_092254 in GWTC-3 \citep{2021arXiv211103606T}.
\subsection{Light Curves and Polarization}
Figure \ref{GRBs_Swift} second and third rows presents the five promising GW events in the third observing run (O3) which are more likely to generate an electromagnetic counterpart, that is, in whose binary system there is at least one neutron star. Each panel shows the multi-band upper limits and the synchrotron light curves from the off-axis jet decelerating in a constant-density medium with different viewing angles $\theta_{\rm obs}= 6\,$ (solid) and $17\,{\rm deg}$ (dashed). The synchrotron light curves are presented at 1 keV (green), UVOT (orange), R-band (yellow) and 3 GHz (brown). Optical data were retrieved for the follow-up campaign carried out by the DDOTI collaboration \citep{2021MNRAS.507.1401B}. For the chosen values, the values of viewing angle less than $7\, {\rm deg}$ are ruled out in our model for the S190425z (GW190425), S190426C (GW190426\_152155) and S190814bv (GW190814) events which are consistent with the ones reported in \cite{2019ApJ...887L..13D,2020arXiv200201950A, 2019ApJ...884L..55G} using different off-axis models. More observations on duration ranging seconds from the burst trigger to months and years after the merging period are needed to infer tighter constraints.\\
\Cref{fig:joined_gw} shows the expected polarization curves that could be present because of the parameters used to satisfy the upper limits of the GW events. Similar to the Swift-identified bursts, we will be referring to the two sets of parameters as ``disallowed" (for $\theta_{\rm obs}=6\deg$) and ``allowed" (for $\theta_{\rm obs}=17\deg$).
Similar considerations can be taken as with the Swift-identified bursts; with the intersection happening at $t\approx 3.8\times 10^{-2}\,{\rm s}$ and $\abs{\Pi/\Pi_{\rm max}} \approx 47\%$, we can set the rough upper limit of $< 47\%$ for $t < 3.8\times 10^{-2}\,{\rm s}$, and the requirement of $\abs{\Pi/\Pi_{\rm max}}> 47\%$ for later times. Furthermore, the narrow $\theta_j$ and large $\theta_{\rm obs}$ constrain the allowed set at $q_0>5$. \cite{2004MNRAS.354...86R} have shown that for a homogeneous sideways expanding jet model, the value of $q_0>5$ threshold leads to a merging of the dual peaks present for $q_0<5$, which is consistent with \Cref{fig:joined_gw} and an extra condition imposed on the polarization for this burst.
Following the same procedure applied for the Swift-identified bursts, a rough limit for the parallel field would be $\Pi/\Pi_{\rm max}>83\%$ at the jet break time.
\section{Conclusions}\label{sec7}
We have introduced a polarization phenomenological model as an extension of the analytical synchrotron afterglow off-axis scenario presented in \cite{2019ApJ...871..200F, 2020ApJ...896...25F}. This synchrotron model can describe the multi-wavelength afterglow observations for both a constant-density and wind-like medium. We have shown the expected temporal evolution of polarization with a dependency on the physical parameters associated with afterglow GRB emission for two configurations of a magnetic field. Regarding our fiducial model, the calculated polarization took into consideration a broad set of parameters constrained within the typical values observed for off-axis GRBs.
We were able to see the differences in possible polarization caused by the two different ambient media and the chosen synchrotron model. We showed that our fiducial model generally agrees with previously found results for a homogeneous sideways expanding jet for the conditions of constant-density medium and adiabatic case with $\xi=1$ \citep{2002ApJ...570L..61G, 2004MNRAS.354...86R}. We have expanded the scenarios for a partially radiative regime and a case where $\xi < 1$. We expect that variation of these parameters present modifications on the temporal evolution of polarization; A partially radiative regime hastens the deceleration of the relativist outflow by the circumburst medium \citep{2000ApJ...532..281B, 2005ApJ...619..968W}, and this has exacerbated the baseline ($\xi=1,\, \epsilon = 0$) profile of polarization -- with peak $\Pi/\Pi_{\rm max}$ increasing further as $q_0$ grows, but second bump decreasing slightly. On the other hand, changing $\xi$ alters the arrival time of the emission \citep{1997ApJ...491L..19W, 2000ApJ...536..195C, 1998ApJ...493L..31P} and our chosen value of $\xi=0.56$ \citep{2000ApJ...536..195C} has caused the polarization behavior regarding $q_0$ to flip, with the magnitude of the peaks now decreasing as $q_0$ increases. Furthermore, we have calculated the same polarization for a wind-like medium to verify the possible differences. For the change in circumburst medium we have found that the polarization evolves slower in time and changes in the magnitude of polarization compared to the constant-density medium, in agreement with \cite{2004A&A...422..121L}.
We have obtained the expected polarization curves for a sample of bursts showing similar off-axis afterglow emissions -- GRB 080503, GRB 140903A, GRB 150101B, GRB 160821B, and GRB 170817A. In particular, we have used the available polarimetric upper limits of GRB 170817A; $\Pi < 12\%$ at 2.8 GHz and $t\approx244$ days \citep{2018ApJ...861L..10C} to rule out our chosen magnetic field configurations of anisotropy factors $b=0$ and $b\rightarrow\infty$.
Although the remaining bursts have neither detected polarization nor constrained upper limits, from our calculations, we can observe a few patterns that reinforce the similarity between these bursts. For the perpendicular field configuration, GRB 080503 and GRB 140903A showed similar magnitudes of polarization, but somewhat dephased in time. Regarding GRB 150101B, the second peak also has a similar polarization degree to the first one of the previously mentioned bursts; however, the peaks happen much later. GRB 160821B is the most distinct out of these bursts, as the polarization happens considerably faster, with a higher first peak (but not too dissimilar to the previous bursts) and a much lower second peak. This is likely due to the angular properties of the burst, as $q_0$ is closer to unity. GRB 170817A is immersed in a lower external density, with a somewhat more energetic jet seen at wider angles, which in combination causes the peaks to be higher than the other bursts by roughly $10\%$ and happens at later times. The peaks of polarization also roughly coincide with the afterglow flux peak in time \citep[see][for the flux fitting]{2019ApJ...871..200F, 2020ApJ...896...25F}, which is a result that agrees with the literature \citep{1999MNRAS.309L...7G, 2003ApJ...594L..83G, 2004MNRAS.354...86R, 2021MNRAS.507.5340T}. Overall, we could observe the similarities between the bursts' polarization. However, the particularities of each are sufficient to cause observable differences between them.
With the model presented in Section \ref{sec2} and \cite{2020ApJ...896...25F}, we have constrained the possible values of the physical parameters of our system. We take into consideration the upper limits of the four closest sGRBs detected by Swift - GRB 050906, GRB 070810B, GRB 080121 and GRB 100216A - and a set of five GW events that could produce an electromagnetic counterpart - S190425z, S190436c, S190814bv, S200105ae, and S200115j - under the condition they must be narrowly collimated jets and seen sufficiently off-axis. We have obtained two sets of parameters, one allowed by the upper limits and one disallowed, and the projected polarization for these values. We used these two sets to obtain what could be considered as a rough constrain on polarization degree, dependent on the geometry of the magnetic field chosen.
More observations, from seconds after the trigger to months and years, are needed to infer tighter constraints on polarization and adequate fitting of the light curves is necessary to obtain adequate parameter values and break degeneracy between synchrotron models.
\section*{Acknowledgement}
We thank Walas Oliveira, Rodolfo Barniol Duran, Tanmoy Laskar, Paz Beniamini and Bing Zhang for useful discussions. AP acknowledges financial support from CONACyT's doctorate fellowships, NF acknowledges financial support from UNAM-DGAPA-PAPIIT through grant IN106521. RLB acknowledges support from CONACyT postdoctoral fellowships and the support from the DGAPA/UNAM IG100820 and IN105921.
\section*{Data Availability}
The data used for this study was obtained from the respective credited references: upper limits of linear polarization for GRB 170817A \citep[obtained by][]{2018ApJ...861L..10C}; upper limits of GRB 050906, GRB 070810B, GRB 080121, and GRB 100216A \citep[taken from][]{2020MNRAS.492.5011D}; and upper limits for GW190425, GW190426\_152155, GW190814 in GWTC-2 \citep{2021PhRvX..11b1053A} and GW191219\_163120, GW200105\_162426, GW200115\_042309, GW200210\_092254 in GWTC-3 \citep{2021arXiv211103606T}. Optical upper limits were obtained by \cite{2021MNRAS.507.1401B}.
Other than cited sources, there is no new data generated or analysed in support of this research.
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\section{Introduction}
\label{sec:intro}
Hydrogen burning, the conversion of four protons into an alpha particle in the interior of stars, is the most important energy source in the universe and it is also responsible for the existence of several chemical elements. Besides the pp-chains powering e.g. our Sun, catalytic reactions cycles, like the various CNO cycles play the major role in hydrogen burning \cite{ade11,wie10}. Depending on the temperature and chemical composition of the stellar plasma, different CNO cycles can take place involving various isotopes of carbon, nitrogen, oxygen and fluorine.
The $^{17}$O(p,$\gamma$)$^{18}$F reaction, which competes with the $\alpha$-emission in $^{17}$O(p,$\alpha$)$^{14}$N \cite{str16} is the starting point of the third CNO cycle. This cycle is activated in various stellar conditions such as red giant and asymptotic giant stars and classical novae. The abundances of fluorine and the heavy oxygen isotopes are strongly related to the operation of this cycle and therefore the rates of the participating reactions must be known.
Below 1.5\,MeV the $^{17}$O(p,$\gamma$)$^{18}$F reaction is characterized by many broad and narrow resonances. Therefore, the temperature dependence of the $^{17}$O(p,$\gamma$)$^{18}$F thermonuclear reaction rate shows a complicated picture. The rate depends on the direct capture component as well as on the narrow low energy resonances and the tails of the higher energy broad resonances (see e.g. Fig.\,10 in ref.\,\cite{dil14} for the contribution of the different components to the reaction rate). An R-matrix fit to the experimental data is therefore inevitable to provide reaction rates at various temperatures for stellar models.
The first cross section measurement of $^{17}$O(p,$\gamma$)$^{18}$F was carried out several decades ago by C. Rolfs in a wide energy range between 300\,keV and 1.9\,MeV \cite{rol73}. After the turn of the century, several experimental studies were carried out mostly concentrating on the low energy region below about 500\,keV \cite{fox04,fox05,cha05,cha07,new10,kon12,hag12,sco12,dil14,buc15}. (The only exception was the work of A. Kontos \textit{et al.} \cite{kon12} which extended up to 1.6\,MeV.) The comparison of the new precise data with the results of \cite{rol73} revealed some discrepancy both in the absolute scale and the energy dependence of the cross section at the lowest energies studied by \cite{rol73}.
In most of the previous experiments the cross section of $^{17}$O(p,$\gamma$)$^{18}$F was measured with in-beam $\gamma$-spectroscopy: the prompt $\gamma$-radiation from the formed $^{18}$F nucleus was detected. The complicated level scheme of $^{18}$F (see e.g. Fig.\,1. in ref.\,\cite{buc15}) implies that the detection of many primary and secondary transitions is necessary for the cross section determination. This represents a source of uncertainty in the experiments. In order to provide the astrophysically relevant total cross section, all the transitions must be measured and care must be taken to measure even the weakest $\gamma$-lines. The angular distributions of all the $\gamma$-emissions must also be known. Moreover, in order to measure low cross sections, close target-detector geometries are typically used leading to strong true coincidence summing effects.
All these experimental difficulties can be avoided by the application of the activation method, which was used by only two experiments before at energies below 400\,keV \cite{cha05,cha07,sco12,dil14}. The reaction product of $^{17}$O(p,$\gamma$)$^{18}$F is radioactive, decays by positron emission \footnote{A weak electron capture decay branching also exist with 3\,\% probability} with a half-life of 109.77\,$\pm$\,0.05\,minutes \cite{til95}. The decay is entirely to the ground state of $^{18}$O, no $\gamma$-radiation follows thus the decay. The emission of the 511\,keV $\gamma$-radiation following the positron annihilation, on the other hand, allows the measurement of the decay by $\gamma$-detection. By measuring the $^{18}$F activity the number of reaction product and therefore the total reaction cross section can be determined directly. The activation measurement of $^{17}$O(p,$\gamma$)$^{18}$F provides therefore an independent means of cross section determination which can be used to check earlier experimental data and provide a constraint for R-matrix calculations regarding the total reaction cross section.
The aim of the present work is therefore to measure the $^{17}$O(p,$\gamma$)$^{18}$F cross section with the activation method in a wide energy range. The next section provides detailed information about the experimental technique, the results are presented in Sec.\,\ref{sec:results} while Sec.\,\ref{sec:summary} provides the summary and conclusions.
\section{Experimental procedure}
\label{sec:experiment}
\subsection{Target preparation and characterization}
\label{subsec:target}
Solid state oxygen targets were produced by anodic oxidation of tantalum disks in water enriched in $^{17}$O. With this technique Ta$_2$O$_5$ layers can be produced with well defined Ta:O ratio and the targets have high stability under beam bombardment. The anodization setup was the same as used recently by the LUNA collaboration for the low energy $^{17}$O(p,$\gamma$)$^{18}$F cross section measurements \cite{sco12,dil14}. Full details of the anodization device and the preparation procedure have been published by the LUNA collaboration \cite{cac12}, here only the most important features and the differences are summarized.
Two water samples were used for the target preparations. The isotopic abundances of the $^{16}$O, $^{17}$O and $^{18}$O isotopes, respectively, were the following: (15.5\,$\pm$\,0.6)\%, (77.8\,$\pm$\,0.6)\% and (6.7\,$\pm$\,0.2)\% (sample 1.) and (39.5\,$\pm$\,0.6)\%, (27.4\,$\pm$\,0.6)\% and (33.1\,$\pm$\,0.6)\% (sample 2.). These values are quoted by the supplier.
Applying two different anodization voltages (24V and 50V), targets with two different thicknesses were produced. Altogether seven targets were prepared from the two water samples and with the two thicknesses. Intercomparison of the different targets were done by carrying out activation at the same proton energy on targets with different isotopic composition and/or thickness.
As the determination of the number of target atoms is crucial for the precise cross section measurements, different experimental techniques were used to determine this quantity. First, the Ta:O stoichiometry ratio and the thickness of the oxide layer were measured with Rutherford Backscattering Spectrometry (RBS). The first set of RBS measurements were carried out before the start of the activation experiments at the microbeam setup installed at the 5\,MV Van de Graaff accelerator of Atomki \cite{hus16}. A 1.6\,MeV $\alpha$ beam bombarded the Ta$_2$O$_5$ targets and the scattered particles were detected by two ion implanted Si detectors positioned at 135 and 165 degrees with respect to the beam direction. Exploiting the high lateral resolution of the microbeam setup, spectra were recorded at several different positions on the target surfaces. This test proved that the thickness and stoichiometry of the targets are uniform along the whole surface of the targets. The spectra were analysed using the SIMNRA code \cite{SIMNRA} which provided the areal density of the O atoms as well as the Ta:O ratio.
A second set of RBS measurement was carried out after the activation experiments using a completely independent setup, namely the activation chamber itself (i.e. similar beam size and position to the proton beam used for the activations, see below). At the Tandetron accelerator a 10\,MeV $^{16}$O$^{4+}$ beam bombarded the Ta$_2$O$_5$ targets and a Si detector built into the activation chamber detected the backscattered ions.
Figure \ref{fig:RBS} shows typical spectra of the two RBS measurements. The measured data as well as the fits using the SIMNRA code are shown. The results of the $^{16}$O RBS measurements were in good agreement with the ones obtained with $\alpha$-RBS (see below).
The ratio of the Ta:O atoms was found to be 0.411\,$\pm$\,0.015 in agreement with the stoichiometric value of 0.4.
\begin{figure}
\includegraphics[angle=270,width=\columnwidth]{fig1a_alpha_RBS.eps}\\
\includegraphics[angle=270,width=\columnwidth]{fig1b_16O_RBS.eps
\caption{\label{fig:RBS} Relevant parts of the RBS spectra of a Ta$_2$O$_5$ target measured with $\alpha$-beam at the microprobe facility (upper panel) and with $^{16}$O$^{4+}$ beam in the activation chamber at the Tandetron accelerator.}
\end{figure}
If the Ta:O ratio is known, a totally independent target thickness value can be obtained by the measurement of the resonance profile on a suitable nuclear resonance. We have investigated the target thicknesses also by this method using both $^{17}$O and $^{18}$O isotope content of the targets. The E$_p$\,=\,1098\,keV and E$_p$\,=\,1925\,keV resonances in the $^{17}$O(p,$\gamma$)$^{18}$F and $^{18}$O(p,$\gamma$)$^{19}$F reactions, respectively, were used to measure the target profiles. A 100\,\% relative efficiency HPGe detector were placed next to the activation chamber at zero degree with its front face about 1\,cm distance from the target. The yield of the strongest transition was used for the measurement of the profiles which was the 937\,keV transition of the first excited state to the ground state in the case of $^{18}$F ($^{17}$O(p,$\gamma$)$^{18}$F reaction) and the 197 keV transition of the second excited state to the ground state in the case of $^{19}$F ($^{18}$O(p,$\gamma$)$^{19}$F reaction). The number of target atoms was obtained from the width of the target profiles using the Ta:O ratio given by the RBS measurements.
Figure\,\ref{fig:resonance} shows a typical resonance profile measured with the $^{17}$O(p,$\gamma$)$^{18}$F reaction. The target thickness obtained from the resonance profile measurement using the two reactions gave consistent results. The comparison with the RBS results, however, revealed a roughly 9\,\% systematic difference. The RBS measurements resulted in systematically higher thickness values. Table\,\ref{tab:target} summarizes the thickness results of a given target (prepared with 50\,V anodization voltage) obtained with the four measurements.
\begin{figure}
\includegraphics[angle=270,width=\columnwidth]{fig2_resonance.eps
\caption{\label{fig:resonance} Measured profile on the E$_p$\,=\,1098\,keV resonance in $^{17}$O(p,$\gamma$)$^{18}$F. The target thickness was obtained from the width of the fitted resonance profile.}
\end{figure}
The uncertainties quoted in the table are statistical only stemming form the fit of the RBS spectra and the resonance profiles. Taking into account only these errors, the two methods are in contradiction. If, however, one includes the uncertainty of the stopping power, the results can be considered to be consistent. It is difficult to quantify the uncertainty of stopping power in our experiment as the stopping of protons, $\alpha$-particles and $^{16}$O isotopes should be considered in O and Ta, and the related information \cite{SRIMerror} in the widely used SRIM code indicates uncertainties from about 2\,\% up to 6\,\%. Most likely the deviation of the thickness values has its origin in the uncertainty of the stopping power. Therefore, we have adopted the average of the two methods and assigned a conservative 6\,\% uncertainty to the number of target atoms.
\begin{table}
\caption{\label{tab:target} Results of the various thickness measurements on one of the targets. See text for details.
}
\begin{ruledtabular}
\begin{tabular}{ll}
Method & No. of O atoms \\
& [10$^{17}$ atoms/cm$^2$] \\
\hline
$\alpha$-RBS & 5.10\,$\pm$\,0.13 \\
$^{16}$O-RBS & 5.00\,$\pm$\,0.20 \\
$^{17}$O(p,$\gamma$)$^{18}$F resonance & 4.67\,$\pm$\,0.15 \\
$^{18}$O(p,$\gamma$)$^{19}$F resonance & 4.63\,$\pm$\,0.18 \\
\hline
RBS average & 5.07\,$\pm$\,0.11 \\
resonance average & 4.65\,$\pm$\,0.12 \\
\hline
adopted & 4.87\,$\pm$\,0.29 \\
\end{tabular}
\end{ruledtabular}
\end{table}
\subsection{Activations}
\label{subsec:activations}
The activations were carried out at the new Tandetron laboratory of Atomki where a 2\,MV Tandetron accelerator manufactured by High Voltage Engineering Europa B.V. was installed in 2015. The energy calibration of the accelerator was carried out by measuring resonances in the $^{27}$Al(p,$\gamma$)$^{28}$Si reaction and the neutron thresholds in $^{7}$Li(p,n)$^{7}$Be and $^{13}$C(p,n)$^{13}$O reactions \cite{raj16}.
The Tandetron provided proton beams in the energy range between 500\,keV and 1.8\,MeV and the beam current was limited to about 5\,$\mu$A in order to avoid target degradation. The lifetime of the targets was also increased by using an off-axis target chamber where the beam spot was shifted from the target center by 6\,mm. By rotating the target between the consecutive activations, fresh or not heavily bombarded target spots could be selected. The target chamber was insulated from the rest of the beam line and served as a Faraday cup in order to determine the number of projectiles by charge integration. A secondary electron suppression voltage of -300\,V was applied behind the 4\,mm diameter entrance aperture of the chamber.
Depending on the cross section, the length of the irradiations varied between 15\,minutes and 5 hours.
Although the beam intensity during the irradiations were typically very stable, in order to follow the possible fluctuations, the beam current was recorded in multichannel scaling mode with one minute time basis. The recorded time dependence of the beam current was then used in the analysis.
\subsection{Measurement of the $^{18}$F decay}
\label{subsec:decay}
After the irradiation the target was removed from the chamber and transported to the counting laboratory where a 100\,\% relative efficiency HPGe detector equipped with full 4\,$\pi$ lead shielding was used to measure the annihilation $\gamma$-radiation of the targets. The $\gamma$-countings started typically 15 minutes after the end of the irradiation and the spectra were recorded in every 10 minutes in order to follow the $^{18}$F decay.
Since the 511\,keV annihilation radiation is present also in the laboratory background and can come from many possible sources, it is crucial to determine the background. The length of the countings was therefore typically 16 hours. Towards the end of this counting period, the activity of $^{18}$F decayed to a negligible level and therefore the 511\,keV background level could be estimated. This was always found to be consistent with the laboratory background measured without target, indicating that no long-lived positron emitter was created in the targets.
In some cases excess in the 511\,keV activity was observed at the beginning of the counting period indicating the production of some short-lived positron emitter. From its decay rate it was identified as $^{13}$N produced by the $^{12}$C(p,$\gamma$)$^{13}$N reaction on carbon impurity of the target. This identification was also supported by the fact that such a deviation from the pure $^{18}$F decay was observed mostly around 500\,keV proton energy where the $^{12}$C(p,$\gamma$)$^{13}$N reaction has high cross section due to a broad resonance at about 420\,keV \cite{bur08}.
In such cases roughly the first one hour of the counting was omitted from the analysis.
The decay of the 511\,keV activity could always be fitted well using the literature half-life of $^{18}$F. As an example, Fig.\,\ref{fig:decay} shows the decay curve measured after the irradiation at 520\,keV. The figure indicates the above discussed $^{13}$N contribution, the $^{18}$F decay fitted with the literature half-life and the laboratory background level of the 511\,keV line.
\begin{figure}
\includegraphics[angle=270,width=\columnwidth]{fig3_decay.eps
\caption{\label{fig:decay} Decay of $^{18}$F in a target irradiated with an 520\,keV proton beam. Taking into account the initial short lived positron emitter and the laboratory background, the decay can be well fitted with the 109.77\,min half-life of $^{18}$F.}
\end{figure}
In order to maximize the detection efficiency, the targets were placed in close geometry onto the detector, directly on top of the detector end cap. Since the two annihilation photons are emitted from the source at opposite directions and no other X-ray or $\gamma$-radiation follows the $^{18}$F decay, true coincidence summing effect was not present in this measurement in spite of the close source-to-detector geometry \footnote{The true coincidence of the two annihilation gammas through a Compton scattering process was observed to cause less than 0.5\,\% loss of counts from the 511\,keV peak and was therefore neglected.}. The summing effect, on the other hand, is significant in the case of any multiline calibration source which could be used for the measurement of the absolute detection efficiency. The absolute efficiency was therefore measured in the counting geometry only with single line calibration sources. Calibrated $^{7}$Be, $^{65}$Zn, $^{85}$Sr and $^{137}$Cs sources were used to obtain the efficiency curve of the detector. The $^{85}$Sr source was especially useful for the efficiency determination as it has a single $\gamma$-line at 514\,keV, very close to the relevant 511\,keV. The absolute efficiency was measured with 3\,\% uncertainty which includes also the beam spot size and target positioning effect.
\section{Experimental results}
\label{sec:results}
The cross section of $^{17}$O(p,$\gamma$)$^{18}$F was measured between proton energies of 500\,keV and 1.8\,MeV. The selection of the actual proton energies was based on the structure of the $^{17}$O(p,$\gamma$)$^{18}$F excitation function. The two broad resonances at 590 and 717\,keV proton energies were measured with fine energy steps. The low energy tails of these resonances have significant contributions to the astrophysical reaction rate of $^{17}$O(p,$\gamma$)$^{18}$F especially at higher temperatures.
The energy regions where there are no resonances were covered with fewer data points. The aim of these measurements were to fix the absolute value of the direct capture part of the cross section which again has an important contribution to the reaction rate.
In the studied energy range there are several narrow resonances at proton energies of 517, 673, 741, 826, 926, 1098, 1240, 1270 and 1345 keV. Activation runs at these energies were also carried out with the aim of confirming their existence in the (p,$\gamma$) channel and check their resonance energies. The widths of these resonances, on the other hand, are often comparable with the target thicknesses used in the present work, the determination of the strengths of these resonances was therefore not aimed.
The obtained cross section results are listed in Table\,\ref{tab:results}. The first column shows the energy of the proton beam provided by the Tandetron accelerator. Based on the accelerator calibration, this value is known with a total uncertainty of less than 0.5\,keV. The energy loss of the beam in the target layer is given in the second column. Especially near the resonances the cross section changes significantly in the energy range covered by the target thickness. An effective proton energy was therefore calculated. For this calculation it was supposed that the cross section has a linear energy dependence in the energy range of the target. The slope of the cross section was estimated based on the adjacent experimental data points and on the shape of the excitation function as measured by previous works. The effective energy was then given by the median of the yield curve within the target thickness \cite{lem08}. The uncertainty of the effective energy as listed in the table was estimated based on the slope of the cross section function. Higher energy uncertainties were assigned to the data points near the narrow resonances where the cross section changes strongly within the target thickness.
\begingroup
\begin{table*}
\caption{\label{tab:results} Measured cross section of the $^{17}$O(p,$\gamma$)$^{18}$F reaction in the present work. The quoted cross section uncertainties are statistical only. For the total uncertainty, 7.6\,\% systematic uncertainty must be added quadratically to the relative statistical uncertainties.}
\begin{ruledtabular}
\begin{tabular}{llr@{\hspace{-2mm}}c@{\hspace{-2mm}}lr@{\hspace{-2mm}}c@{\hspace{-2mm}}l||llr@{\hspace{-2mm}}c@{\hspace{-2mm}}lr@{\hspace{-2mm}}c@{\hspace{-2mm}}l}
E$_p$ & Energy loss & \multicolumn{3}{c}{E$_{p\rm ,eff.}$} & \multicolumn{3}{c||}{Cross section} & E$_p$ & Energy loss & \multicolumn{3}{c}{E$_{p\rm ,eff.}$} & \multicolumn{3}{c}{Cross section}\\
& in target\footnote{See Section \ref{subsec:target} for information about the target thicknesses} & & & & & & & & in target$^a$ \\
keV & keV & \multicolumn{3}{c}{keV} & \multicolumn{3}{c||}{$\mu$barn} & keV & keV & \multicolumn{3}{c}{keV} & \multicolumn{3}{c}{$\mu$barn}\\
\hline
500.0 & 4.38 & 497.8 & $\pm$ & 1.4 & 0.592 & $\pm$ & 0.021 & 789.7 & 7.23 & 786.0 & $\pm$ & 2.1 & 3.30 & $\pm$ & 0.1 \\
509.8 & 9.05 & 505.4 & $\pm$ & 2.5 & 0.587 & $\pm$ & 0.118 & 819.8 & 3.41 & 818.2 & $\pm$ & 1.3 & 2.36 & $\pm$ & 1.0 \\
514.7 & 4.32 & 512.6 & $\pm$ & 1.4 & 0.738 & $\pm$ & 0.120 & 824.8 & 3.40 & 823.2 & $\pm$ & 1.3 & 9.51 & $\pm$ & 0.3 \\
519.8 & 4.30 & 517.7 & $\pm$ & 1.4 & 16.3 & $\pm$ & 1.8 & 829.7 & 3.38 & 828.0 & $\pm$ & 1.3 & 12.2 & $\pm$ & 0.3 \\
524.7 & 4.28 & 522.0 & $\pm$ & 2.0 & 6.13 & $\pm$ & 0.37 & 834.7 & 3.37 & 832.9 & $\pm$ & 1.3 & 2.5 & $\pm$ & 0.20 \\
529.7 & 4.26 & 527.1 & $\pm$ & 1.8 & 1.37 & $\pm$ & 0.38 & 880.0 & 3.28 & 878.4 & $\pm$ & 1.2 & 2.06 & $\pm$ & 0.13 \\
539.7 & 4.22 & 537.7 & $\pm$ & 1.4 & 1.96 & $\pm$ & 0.23 & 919.8 & 6.69 & 916.4 & $\pm$ & 1.8 & 2.18 & $\pm$ & 0.12 \\
549.7 & 8.72 & 545.9 & $\pm$ & 2.8 & 2.93 & $\pm$ & 0.14 & 924.8 & 3.20 & 924.0 & $\pm$ & 1.6 & 1.97 & $\pm$ & 0.09 \\
559.8 & 8.64 & 556.1 & $\pm$ & 2.9 & 4.95 & $\pm$ & 0.59 & 929.7 & 6.66 & 926.4 & $\pm$ & 1.9 & 12.1 & $\pm$ & 0.2 \\
569.8 & 4.11 & 568.0 & $\pm$ & 1.6 & 10.3 & $\pm$ & 1.0 & 934.7 & 3.19 & 933.1 & $\pm$ & 1.2 & 2.65 & $\pm$ & 0.09 \\
579.7 & 8.49 & 576.7 & $\pm$ & 3.4 & 30.5 & $\pm$ & 0.8 & 999.7 & 6.43 & 996.5 & $\pm$ & 1.8 & 2.67 & $\pm$ & 0.08 \\
584.7 & 4.06 & 582.9 & $\pm$ & 1.5 & 67.8 & $\pm$ & 1.2 & 1089.7 & 6.17 & 1086.8 & $\pm$ & 1.9 & 3.55 & $\pm$ & 0.08 \\
589.7 & 4.04 & 587.7 & $\pm$ & 1.3 & 107 & $\pm$ & 0.8 & 1096.8 & 2.95 & 1095.3 & $\pm$ & 1.1 & 11.7 & $\pm$ & 2.8 \\
599.7 & 8.35 & 595.0 & $\pm$ & 2.7 & 70.7 & $\pm$ & 1.2 & 1101.8 & 6.14 & 1098.8 & $\pm$ & 1.8 & 176 & $\pm$ & 0.8 \\
609.8 & 8.28 & 604.5 & $\pm$ & 3.3 & 25.0 & $\pm$ & 0.4 & 1106.7 & 6.13 & 1103.7 & $\pm$ & 1.7 & 35.5 & $\pm$ & 1.6 \\
619.8 & 8.21 & 614.9 & $\pm$ & 2.9 & 10.5 & $\pm$ & 0.2 & 1111.7 & 6.12 & 1108.1 & $\pm$ & 2.3 & 6.03 & $\pm$ & 0.4 \\
629.7 & 3.91 & 627.7 & $\pm$ & 1.4 & 4.14 & $\pm$ & 0.34 & 1150.0 & 2.89 & 1148.6 & $\pm$ & 1.1 & 4.24 & $\pm$ & 0.1 \\
639.7 & 3.88 & 637.8 & $\pm$ & 1.3 & 4.03 & $\pm$ & 0.21 & 1224.8 & 5.86 & 1221.9 & $\pm$ & 1.7 & 5.71 & $\pm$ & 0.1 \\
649.7 & 3.84 & 647.8 & $\pm$ & 1.3 & 3.39 & $\pm$ & 0.21 & 1239.7 & 5.82 & 1237.8 & $\pm$ & 2.6 & 12.2 & $\pm$ & 1.3 \\
659.8 & 3.81 & 657.9 & $\pm$ & 1.3 & 3.78 & $\pm$ & 0.15 & 1244.7 & 5.81 & 1241.9 & $\pm$ & 1.7 & 55.7 & $\pm$ & 0.7 \\
669.8 & 7.88 & 665.8 & $\pm$ & 2.1 & 5.00 & $\pm$ & 0.51 & 1249.7 & 2.79 & 1248.1 & $\pm$ & 1.3 & 19.2 & $\pm$ & 0.9 \\
672.5 & 3.78 & 671.0 & $\pm$ & 1.7 & 138 & $\pm$ & 1.7 & 1254.7 & 5.79 & 1251.4 & $\pm$ & 2.1 & 12.7 & $\pm$ & 0.4 \\
674.8 & 3.77 & 672.8 & $\pm$ & 1.3 & 218 & $\pm$ & 3.6 & 1259.7 & 5.78 & 1256.6 & $\pm$ & 1.8 & 9.34 & $\pm$ & 0.22 \\
677.3 & 3.76 & 674.6 & $\pm$ & 1.9 & 100 & $\pm$ & 2.3 & 1264.8 & 5.77 & 1262.2 & $\pm$ & 2.0 & 8.83 & $\pm$ & 0.36 \\
679.7 & 3.76 & 676.4 & $\pm$ & 1.9 & 16.9 & $\pm$ & 0.6 & 1274.8 & 5.75 & 1271.9 & $\pm$ & 1.7 & 19.5 & $\pm$ & 0.2 \\
689.7 & 7.76 & 685.8 & $\pm$ & 2.1 & 8.72 & $\pm$ & 0.20 & 1279.8 & 2.76 & 1278.3 & $\pm$ & 1.1 & 12.0 & $\pm$ & 0.3 \\
699.7 & 3.70 & 698.0 & $\pm$ & 1.4 & 14.9 & $\pm$ & 0.93 & 1284.7 & 2.75 & 1283.3 & $\pm$ & 1.1 & 7.75 & $\pm$ & 0.10 \\
704.7 & 7.68 & 702.1 & $\pm$ & 3.3 & 26.3 & $\pm$ & 0.83 & 1299.8 & 5.70 & 1296.8 & $\pm$ & 1.7 & 5.19 & $\pm$ & 0.22 \\
709.7 & 7.65 & 707.5 & $\pm$ & 3.7 & 47.0 & $\pm$ & 1.9 & 1339.7 & 2.70 & 1338.5 & $\pm$ & 1.1 & 4.22 & $\pm$ & 0.14 \\
714.8 & 3.66 & 713.1 & $\pm$ & 1.4 & 135 & $\pm$ & 4.2 & 1345.7 & 5.62 & 1343.0 & $\pm$ & 1.6 & 23.1 & $\pm$ & 0.2 \\
717.8 & 3.65 & 716.1 & $\pm$ & 1.4 & 170 & $\pm$ & 0.8 & 1349.7 & 2.69 & 1348.1 & $\pm$ & 1.3 & 5.42 & $\pm$ & 0.17 \\
719.8 & 7.59 & 715.6 & $\pm$ & 2.4 & 178 & $\pm$ & 1.0 & 1354.7 & 2.69 & 1353.3 & $\pm$ & 1.1 & 3.98 & $\pm$ & 0.16 \\
724.8 & 7.57 & 720.3 & $\pm$ & 2.7 & 121 & $\pm$ & 1.2 & 1359.7 & 2.68 & 1358.4 & $\pm$ & 1.0 & 4.33 & $\pm$ & 0.21 \\
729.7 & 3.62 & 727.6 & $\pm$ & 1.5 & 38.6 & $\pm$ & 1.7 & 1400.0 & 5.52 & 1397.2 & $\pm$ & 1.6 & 5.25 & $\pm$ & 0.12 \\
739.7 & 7.48 & 735.3 & $\pm$ & 2.7 & 18.6 & $\pm$ & 1.0 & 1500.0 & 2.57 & 1498.7 & $\pm$ & 1.0 & 5.76 & $\pm$ & 0.08 \\
744.7 & 3.58 & 743.3 & $\pm$ & 1.6 & 12.2 & $\pm$ & 0.8 & 1600.0 & 5.18 & 1597.4 & $\pm$ & 1.5 & 7.94 & $\pm$ & 0.31 \\
749.7 & 3.57 & 747.9 & $\pm$ & 1.2 & 49.6 & $\pm$ & 0.6 & 1700.0 & 5.01 & 1697.5 & $\pm$ & 1.5 & 10.7 & $\pm$ & 0.3 \\
754.7 & 3.56 & 752.9 & $\pm$ & 1.2 & 7.66 & $\pm$ & 0.2 & 1800.0 & 2.32 & 1798.8 & $\pm$ & 1.0 & 11.8 & $\pm$ & 0.2 \\
759.7 & 7.38 & 755.9 & $\pm$ & 2.1 & 7.34 & $\pm$ & 0.3 \\
\end{tabular}
\end{ruledtabular}
\end{table*}
\endgroup
In the table only the statistical uncertainty of the cross section values is quoted. This is obtained simply from the peak integration of the 511\,keV $\gamma$-peak and the background subtraction. Typically the statistical uncertainties are between 0.5\,\% and 5\,\%. Higher statistical uncertainties can be found in the case of the lowest cross sections and for those points where based on the literature data higher cross sections were expected at a resonance but the actual resonance was found at slightly shifted energy (see below).
In order to obtain the total uncertainty of the cross section values, 7.6\,\% systematic uncertainty must be added quadratically to the relative statistical uncertainties. This systematic uncertainty is the quadratic sum of the following components: number of oxygen atoms in the target (6\,\%), $\gamma$-detection efficiency (3\,\%), number of protons hitting the target (3\,\% from charge integration), $^{17}$O enrichment (2\,\%). Uncertainties well below 1\,\% - like the uncertainty of $^{18}$F decay parameters or the measurement of irradiation and counting times - were neglected.
In order to increase the reliability of our experiments, repeated activations were carried out at a few different proton energies using different targets. The results were always in agreement within the statistical uncertainties of the measurements. In table\,\ref{tab:results} either the weighted average of these points are shown or - if targets with different widths were used - the more precise value was adopted.
\section{Discussion}
\label{sec:discussion}
As most of the previous experiments yielded partial cross sections for the various transitions in $^{18}$F measured at a given angle, it is rather difficult to compare the results of the present work with previous experiments. Above 500\,keV proton energy the only total cross section (in the form of an astrophysical S-factor figure) is provided by C. Rolfs \cite{rol73}. Although C. Rolfs studied the reaction in a wide energy range, total S-factor is only provided in the energy regions far from the resonances, i.e. below 450\,keV (outside the energy range of the present work) and above 900\,keV.
Figure\,\ref{fig:results_highE} shows the total cross section determined in the present work and that of C. Rolfs in this high energy range. The points of C. Rolfs are taken from the EXFOR \cite{EXFOR} database where they were obtained by scanning Fig. 17 of \cite{rol73}. As one can see, the data of C. Rolfs are on average a factor of 1.5 higher than the present data although the agreement becomes somewhat better at the highest energies.
In addition to the total cross section data, the partial cross section involving the first excited state to ground state transition in $^{18}$F measured by C. Rolfs is also included in the figure taken from EXFOR. Obviously, this partial cross section is lower than the total one, but it is included in the figure in order to compare the observed resonances in the high energy region. The existence of the narrow resonances observed by C. Rolfs is confirmed by the present work. There is an apparent energy shift between the two datasets, the resonances in the present work are observed at slightly higher energies than in \cite{rol73} as given in EXFOR. However, the numerical values of the resonance energies, as given in \cite{rol73b} are in reasonably good agreement with the present work. The apparent discrepancy as can be seen in the figure can therefore most likely be attributed to the digitization uncertainty of the low resolution Fig. 11 of \cite{rol73}.
\begin{figure*}
\includegraphics[angle=270,width=0.8\textwidth]{fig4_results_highE.eps
\caption{\label{fig:results_highE} Experimental cross section of the $^{17}$O(p,$\gamma$)$^{18}$F reaction in the upper part of the studied energy range. Besides the present work, the total and partial cross sections of C. Rolfs \cite{rol73} are also shown as discussed in the text. The lines through the points are only to guide the eye.}
\end{figure*}
Between 500\,keV and 900\,keV proton energies no total cross section values are available in literature making the comparison of our data with the existing database even more difficult. Figure\,\ref{fig:results_lowE} shows the measured cross section in this energy range. In order to compare at least the energy dependence of the cross section, besides the present data, partial cross sections measured by C. Rolfs \cite{rol73} and A. Kontos \textit{et al.} \cite{kon12} are also included in the figure. As in Fig.\,\ref{fig:results_highE}, the data of C. Rolfs are taken from its Fig.\,11 as compiled in EXFOR. In the case of A. Kontos \textit{et al.}, capture to the first excited state of $^{18}$F measured at 135 degrees is arbitrarily chosen. The energy dependence of the cross section is very similar in the three datasets. The partial cross section of C. Rolfs exceeds the total cross section measured in the present work. This is similar to the observation on the direct capture cross section at higher energies.
A further comparison with literature data can be made at the lowest studied energy of the present work at E$_p$\,=\,500\,keV. Several recent low energy datasets extend up to this energy and some of them quote total cross section (or S-factor) which can be compared with the present work. Table\,\ref{tab:500keV} lists the experimental (or quasi-experimental, see below) cross section values at 500\,keV proton energy. The following literature data were considered: U. Hager \textit{et al.} \cite{hag12} measured the total cross section with the DRAGON recoil separator at E$_{c.m.}$\,=\,470\,keV corresponding to 497.9\,keV proton energy which matches exactly our lowest energy. The value is taken from table\,VI of \cite{hag12}. In-beam $\gamma$-spectroscopy measurement of J.R. Newton \textit{et al.} \cite{new10} provided total cross section at 500\,keV proton energy which again coincides with our data point taking into account the energy uncertainties. The value is taken from Table\,I of \cite{new10}. A. Kontos \textit{et al.} \cite{kon12} do not provide total cross section data directly at E$_p$\,=\,500\,keV. Measured partial cross section around this energy region, however, is available and based on these data the authors provide total S-factor values in tabular form in their Table\,V. Interpolated value for E$_{c.m.}$\,=\,470\,keV (corresponding to E$_p$\,=\,500\,keV) is put into Table\,\ref{tab:500keV} keeping the 12\,\% relative experimental uncertainty.
\begin{figure*}
\includegraphics[angle=270,width=0.8\textwidth]{fig5_results_lowE.eps
\caption{\label{fig:results_lowE} Experimental cross section of the $^{17}$O(p,$\gamma$)$^{18}$F reaction in the lower part of the studied energy range. Besides the present work, the partial cross sections of C. Rolfs \cite{rol73} and A. Kontos \textit{et al.} \cite{kon12} are also shown as discussed in the text. The lines through the points are only to guide the eye.}
\end{figure*}
\begin{table}
\caption{\label{tab:500keV} Experimental cross section of $^{17}$O(p,$\gamma$)$^{18}$F at E$_p$\,=\,500\,keV from the present and previous works.
}
\begin{ruledtabular}
\begin{tabular}{ll}
Reference & Cross section at\\
& E$_p$\,=\,500\,keV [nbarn] \\
\hline
U. Hager \textit{et al.} \cite{hag12} & 585\,$\pm$\,8$_{\rm stat.}$\,$\pm$\,75$_{\rm syst.}$ \\
J.R. Newton \textit{et al.} \cite{new10} & 488\,$\pm$\,49 \\
A. Kontos \textit{et al.} \cite{kon12} & 588\,$\pm$\,71\footnote{Not purely experimental value. See text.} \\
\hline
present work & 592\,$\pm$\,21$_{\rm stat.}$\,$\pm$\,45$_{\rm syst.}$ \\
\end{tabular}
\end{ruledtabular}
\end{table}
The result of the present work at E$_p$\,=\,500\,keV is in good agreement with U. Hager \textit{et al.} \cite{hag12} and A. Kontos \textit{et al.} \cite{kon12}. The cross section of J.R. Newton \textit{et al.} \cite{new10}, on the other hand, is almost 20\,\% lower than - and therefore barely consistent with - the other three values.
\section{R-matrix analysis}
\label{sec:rmat}
AZURE2, a multichannel and multilevel R-matrix code \cite{azuma10}, was used to simultaneously fit the total cross section, measured by J. R. Newton \textit{et al.} \cite{new10}, U. Hager \textit{et al.} \cite{hag12}, A. Di Leva \textit{et al.} \cite{dil14}, M. Q. Buckner \textit{et al.} \cite{buc15} and by the present work, as well as the primary transitions, measured by A. Kontos \textit{et al.} \cite{kon12}, of the $^{17}$O(p,$\gamma$)$^{18}$F reaction. The fit using these data will be referred to as ``our'' fit in the following.
\begin{figure}
\includegraphics[width=\columnwidth]{fig6_Rmat_KontosGyurkyTotalS.eps}
\caption{\label{fig:Rmat_KontosGyurkyTotalS} Total $S$ factor obtained from an R-matrix fit made by A. Kontos \textit{et al.} \cite{kon12} is compared with our experimental data of the $^{17}$O(p,$\gamma$)$^{18}$F reaction.}
\end{figure}
First, we have compared the total $S$ factor obtained from an R-matrix fit made by A. Kontos \textit{et al.} \cite{kon12} with our experimental data. One can see in Fig.\,\ref{fig:Rmat_KontosGyurkyTotalS} that there is a good agreement between our data and the calculated one of Kontos, although some narrow resonances are omitted from their plot. Values of Kontos are obtained from Fig.\,9 of \cite{kon12} by figure digitization using the software PlotDigitizer 2.6.8 \cite{PlotDig}.
In our R-matrix fit for the determination of direct capture, because of the nice agreement with the results of Kontos, we used same {\it asymptotic normalization coefficient} (ANC) values and high energy background poles as they used. Table\,\ref{tab:rmatANC} lists these fixed ANC values. In addition, 15 MeV as the excitation energy of the background poles was selected. There are no proton scattering data to provide restrictions for the proton partial widths of the poles, so they were fixed at $\Gamma_p = 6$ MeV, close to the Wigner limit. The R-matrix radius was taken as $r_c = r_0 \times ( A^{1/3}_t+A^{1/3}_p ) = 4.46$ fm, with $r_0 = 1.25$ fm. More details about the selected values are in \cite{kon12}.
\begin{table}
\caption{\label{tab:rmatANC} Fixed ANCs based on \cite{kon12}.}
\begin{ruledtabular}
\begin{tabular}{ c c c }
Energy (keV) & $\ell$ & ANC (fm$^{-1/2}$)\\[0.05cm]
\hline\\[-0.2cm]
~937 & 0 & 6.1 \\
~937 & 2 & 1.2 \\
1121 & 2 & 2.7 \\
2523 & 0 & 1.4 \\
3062 & 0 & 4.5 \\
3062 & 2 & 1.0 \\
3839 & 0 & 4.6 \\
3839 & 2 & 0.6 \\
4115 & 0 & 2.5 \\
4115 & 2 & 1.0 \\
4652 & 2 & 1.3 \\
4964 & 0 & 3.2 \\
4964 & 2 & 0.7 \\
\end{tabular}
\end{ruledtabular}
\end{table}
Our R-matrix analysis used the data set of table\,\ref{tab:results}.
As the quoted effective energies were used, no target effect was taken into account. No normalization of datasets was applied and for physical parameters the Brune parameterization \cite{bru02} was used. The full parameter list of our R-matrix fit is provided as Supplemental Material \cite{supmat}. It contains the used datasets and the AZURE2 input file with all parameters.
The estimated dependence of the R-matrix extrapolation on the choice of the channel radius, the position of the background poles and ANC values are $\sim$\,4\,\%, $\sim$\,7\,\% and $\sim$\,15\,\%, respectively. These values are estimated from the manual variation of the above parameters around their fixed values. The uncertainty of the extrapolation of the total $S$ factor to zero energy is $\sim$\,20\,\%.
Table\,\ref{tab:rmatS0} lists the calculated contributions of all the measured transitions to the total $S$ factor at zero energy. The second column is calculated by Kontos \textit{et al.} \cite{kon12}, the third one by Di Leva \textit{et al.} \cite{dil14} and the last one comes from our fit. The error of our data is $\sim$\,15\,\% because of the uncertainty of the choice of ANC. The uncertainties are statistical only. The total $S$(0) value of Kontos and Di Leva are $5.4 \pm \text{(th.)}1.0 \pm \text{(exp.)}0.6$ keV\,b and $5.0 \pm 0.3$ keV\,b, respectively. Our total $S$ factor value at zero energy is $4.7 \pm 1.0$ keV\,b where the error is statistical only.
\begin{table*}
\caption{\label{tab:rmatS0} Calculated $S$(0) values for each $\gamma$-ray transitions measured by Kontos \textit{et al.} \cite{kon12}.}
\begin{ruledtabular}
\begin{tabular}{ c c c c}
Transition (keV) & $S$(0)$^{\text{Kontos \cite{kon12}}}$ (keV\,b) & $S$(0)$^{\text{Di Leva \cite{dil14}}}$ (keV\,b) & $S$(0)$^{\text{Present}}$ (keV\,b)\footnote{The $S$(0) of the present work was obtained by using for the fit simultaneously the partial cross sections from the literature (see text) and the total cross section presented in this paper.}\\[0.05cm]
\hline\\[-0.2cm]
R/DC $\rightarrow 937~\,$ & $1.7\pm 0.3$ & $1.48\pm 0.08$ & $1.73 \pm 0.26$\\
R/DC $\rightarrow 1121$ & $0.66\pm 0.13$ & $0.47\pm 0.05$ & $0.65 \pm 0.10$ \\
R/DC $\rightarrow 1700$ & $0.013\pm 0.002$ & & $0.013 \pm 0.002$ \\
R/DC $\rightarrow 2523$ & $0.17\pm 0.03$ & $0.12\pm 0.03$ & $0.15 \pm 0.02$ \\
R/DC $\rightarrow 3062$ & $0.66\pm 0.1~\,$ & $0.59\pm 0.03$ & $0.45 \pm 0.07$ \\
R/DC $\rightarrow 3791$ & $0.032\pm 0.005$ & $0.20\pm 0.05$ & $0.030 \pm 0.005$ \\
R/DC $\rightarrow 3839$ & $0.93\pm 0.14$ & $0.92\pm 0.04$ & $0.66 \pm 0.10$ \\
R/DC $\rightarrow 4115$ & $0.55\pm 0.08$ & $0.50\pm 0.03$ & $0.51 \pm 0.08$ \\
R/DC $\rightarrow 4652$ & $0.21\pm 0.03$ & $0.10\pm 0.03$ & $0.19 \pm 0.03$ \\
R/DC $\rightarrow 4964$ & $0.49\pm 0.07$ & $0.43\pm 0.03$ & $0.35 \pm 0.05$ \\
\end{tabular}
\end{ruledtabular}
\end{table*}
Fig.\,\ref{fig:Rmat_FitGyurkyTotalS} shows the total $S$ factor obtained from our R-matrix fit (continuous line) as well as experimental datasets of J. R. Newton \textit{et al.} \cite{new10}, U. Hager \textit{et al.} \cite{hag12}, A. Di Leva \textit{et al.} \cite{dil14}, M. Q. Buckner \textit{et al.} \cite{buc15} and present work. Narrow resonances are also included. The $\chi^2$ value of our dataset is $7.3$ without any normalization of datasets. The contribution of the direct capture to the total $S$ factor at zero energy in our fit is $S_{\text{DC}}=4.3\pm1.0$ keV\,b, where the uncertainty is statistical only. Fig.\,\ref{fig:Rmat_FitS0} shows the low energy total $S$ factor obtained from our R-matrix fit with the above experimental datasets.
\begin{figure}
\includegraphics[width=\columnwidth]{fig7_Rmat_FitGyurkyTotalS.eps}
\caption{\label{fig:Rmat_FitGyurkyTotalS} Total $S$ factor obtained from our R-matrix fit (continuous line) is compared with experimental data of the $^{17}$O(p,$\gamma$)$^{18}$F reaction. The dashed line is the contribution of the direct capture to the total $S$ factor (background poles included).}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{fig8_Rmat_FitS0.eps}
\caption{\label{fig:Rmat_FitS0} Low energy total $S$ factor obtained from our R-matrix fit is compared with experimental data of the $^{17}$O(p,$\gamma$)$^{18}$F reaction. (A.: activation; P.: primary transitions; S.: secondary transitions)}
\end{figure}
\section{Summary and conclusions}
\label{sec:summary}
In the present work the total cross section of the $^{17}$O(p,$\gamma$)$^{18}$F reaction was measured with the activation method in a wide energy range for the first time with a total uncertainty of about 10\,\%. Since this method provides cross sections which are in several aspects independent from the ones obtained with in-beam $\gamma$-spectroscopy and some systematic errors are not present, our results can be used to check the validity of previous data.
In general, our data is in good qualitative agreement with the structure of the excitation function of previous works. The possibility of the comparison of total cross sections is limited owing to the scarcity of total cross section data in the literature in the studied energy range. At energies above 900\,keV our results are on average a factor of 1.5 lower than that of C. Rolfs \cite{rol73}. Similar deviation is found at lower energies where the partial cross sections of C. Rolfs exceed substantially our total cross section. Too high values of C. Rolfs were also pointed out earlier by A. Kontos \textit{et al.} \cite{kon12} in the case of individual transitions. This observation is confirmed by the present work.
A direct comparison of our cross section data with the literature was carried out also at a single proton energy of 500\,keV. It is found that our value agrees well with that of U. Hager \textit{et al.} \cite{hag12} and A. Kontos \textit{et al.} \cite{kon12}, while the result of J.R. Newton \textit{et al.} \cite{new10} is about 20\,\% (two standard deviations) lower.
An R-matrix analysis with the AZURE2 code was performed to check the conformity of our measured total cross section dataset and to extrapolate the astrophysical $S$ factor to lower energies. In this analysis all primary transitions observed from Ref. \cite{kon12} were simultaneously fitted with some total cross section datasets, included the present one. The resulting total $S$ factor is in good agreement with previous measurements and calculations within the experimental uncertainties.
Our total cross section data can be used to constrain any future theoretical description of the $^{17}$O(p,$\gamma$)$^{18}$F reaction.
\begin{acknowledgments}
We thank A. Formicola and the LUNA collaboration for giving us access to the Ta$_2$O$_5$ target preparation device at LNGS, Italy. We also thank R. J. deBoer for all of his helpful advices regarding the use of the R-matrix code, AZURE2. This work was supported by the SROP-4.2.2.B-15/1/KONV-2015-0001 project, by the European Union, co-financed by the European Social Fund and by OTKA grants No. K108459, K120666 and K112962
\end{acknowledgments}
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\section{Introduction}
Given $M>0$ and $a\in(-M,M)$, for $(v,r,\omega)\in\mathbb{R}\times(0,\infty)\times\mathbb{S}^2$ and $(\theta,\phi)$ spherical coordinates on $\mathbb{S}^2$, the Kerr metric in Eddington-Finkelstein coordinates takes the form
\begin{subequations}
\begin{align}
\mathring{g}
={}& \left(1-\frac{2Mr}{\Sigma}\right) \mathrm{d} v^2
+\frac{4Mra}{\Sigma}\sin^2\theta \mathrm{d} v\mathrm{d}\phi
-\frac{(r^2+a^2)^2-a^2\Delta\sin^2\theta}{\Sigma}\sin^2\theta\mathrm{d}\phi^2
\nonumber\\
&-\Sigma\mathrm{d}\theta^2
-2\mathrm{d} v \mathrm{d} r
+2a\sin^2\theta \mathrm{d} r\mathrm{d}\phi ,
\label{eq:KerrMetric} \\
\Sigma ={}& r^2+a^2\cos^2\theta,\quad
\Delta ={} r^2-2Mr +a^2 .
\end{align}
\end{subequations}
For $M>0$ and $a\in(-M,M)$, this metric describes a subextremal black hole geometry. As explained in many textbooks (e.g. \cite{ONeill:Kerr}), the metric \eqref{eq:KerrMetric} extends smoothly to the set $\mathcal{K}^*=\mathbb{R}\times(0,\infty)\times\mathbb{S}^2$, in particular to the north and south poles, and there is a further analytic extension, extending beyond $v=\pm\infty$ and (for $a\not=0$) to $r<0$. The Kerr space-time is of type D (or $\{2,2\}$), which means there are two, repeated principal null directions;
a future-directed ingoing (respectively outgoing) principal null vector is a positive multiple of $\hat{\vecN}$ (respectively $\hat{\vecL}$), where
\begin{subequations}
\label{eq:PNVs}
\begin{align}
\hat{\vecN} ={}& -\partial_r ,
\label{eq:ingoingPNV}\\
\hat{\vecL} ={}& \frac{\Delta}{2}\partial_r +\left((r^2+a^2)\partial_v+a\partial_\phi\right).
\label{eq:outgoingPNV}
\end{align}
\end{subequations}
Central to this paper is the following gauge condition:
\begin{definition}
\label{def:NLORG}
Let $M>0$ and $a\in(-M,M)$.
Let $\mathring{g}$ be the Kerr metric on $\mathcal{K}^*$, and let $n$ be a future-directed, ingoing principal null vector.
Let $U$ be an open subset of $\mathcal{K}^*$.
A symmetric tensor $g$ on $U$ is defined to satisfy the \defn{radiation gauge condition{}} iff
\begin{align}
\label{eq:NLnh}
n^ag_{ab} ={}& n^a\mathring{g}_{ab} .
\end{align}
\end{definition}
This paper uses the following technical choice of definition of a diffeomorphism gauge, which is synonymous with a local diffeomorphism. For convenience, we define a reference Riemannian metric on $\mathcal{K}^*$ from which we further define, for any $k\in\mathbb{N}$, the $C^k$ norm with respect to the reference metric on any subset of $\mathcal{K}^*$. It is well known that when dealing with diffeomorphism gauges, it is unfortunately common to lose regularity and to need to restrict to somewhat smaller sets. The relevant sets for the following definition are illustrated in figure \ref{fig:SetsForImposingGauge}.
\begin{definition}
\label{def:diffeomorphismGauge}
Within this paper,%
\footnote{Clearly, outside the context of this paper, a diffeomorphism gauge could be defined between open subsets of any manifold, not merely $\mathcal{K}^*$.}
given a nonnegative integer $k$ and an open set $V\subset\mathcal{K}^*$, a $C^{k}$ \defn{diffeomorphism gauge} is a map $\Phi:V\rightarrow \mathcal{K}^*$ such that $\Phi$ is a $C^{k}$ diffeomorphism of $V$ to its image.
Let $(X,Y,I,J,h,U,V)$ such that: $X$ is a bounded, open subset of $\mathbb{R}\times\mathbb{S}^2$; $Y$ is a open set such that its closure is a subset of $X$; $0<J<I<\infty$; $h:X\rightarrow(M/2,\infty)$ is smooth; and $U$ and $V$ are the spacetime slabs $U=\{(v,r,\omega):(v,\omega)\in X,h(v,\omega)-I<r<h(v,\omega)+I\}$, $V=\{(v,r,\omega):(v,\omega)\in Y, h(v,\omega)-J<r<h(v,\omega)+J\}$. A diffeomorphism $\Phi$ is defined to be \defn{compatible with $(X,Y,I,J,h,U,V)$} if $\Phi(V)\subset U$.
\end{definition}
Our first result is that for initial data that is close to data from the Kerr spacetime, it is possible to construct a diffeomorphism gauge so as to impose the radiation gauge condition{}.
\begin{theorem}[Enforceability of the radiation gauge condition]
\label{thm:NLnhEnforceability}
Let $M>0$ and $a\in(-M,M)$.
Let $\mathring{g}$ be the Kerr metric on $\mathcal{K}^*$, and let $n$ be a future-directed, ingoing principal null vector.
Let $(X,Y,I,J,h,U,V)$ be as in definition \ref{def:diffeomorphismGauge}, and let $k'$ be a sufficiently large integer.
There exist $\varepsilon_0>0$, $k>k'$, and $K>0$ such that,
if $g_{ab}$ is a symmetric $(0,2)$ tensor satisfying $|g-\mathring{g}|_{C^{k}(U)}<\varepsilon_0$,
then
there is a $C^{k'}$ diffeomorphism gauge $\Phi$ such that $\Phi^{-1}_*g$ satisfies the radiation gauge condition{} on $V$. Furthermore, there is the bound
$|\Phi^{-1}_*g-\mathring{g}|_{C^{k'}(h(Y))}$ $\leq K|g-\mathring{g}|_{C^{k}(h(X))}$.
\end{theorem}
\begin{figure}
\label{fig:SetsForImposingGauge}
\scalebox{0.75}{
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\newcommand{\innercylinderline}[1]{ ({2+sin(#1)/sqrt(2)}, {2.5+cos(#1)}, {2.2+1/4 + sin(#1)^2/8}) -- ({2+sin(#1)/sqrt(2)}, {2.5+cos(#1)}, {3.8+1/4 + sin(#1)^2/8})}
\newcommand{\bottomcylinderline}[1]{ ({2+sin(#1)}, {2.5+sqrt(2)*cos(#1)}, 0) -- ({2+sin(#1)}, {2.5+sqrt(2)*cos(#1)}, {2 + sin(#1)^2/4})}
\newcommand{\innerbottomcylinderline}[1]{ ({2+sin(#1)/sqrt(2)}, {2.5+cos(#1)}, {0}) -- ({2+sin(#1)/sqrt(2)}, {2.5+cos(#1)}, {2.2+1/4 + sin(#1)^2/8})}
\pgfmathsetmacro{\LeftAngle}{165}
\pgfmathsetmacro{\RightAngle}{-12}
\tdplotsetmaincoords{66}{110}
\begin{tikzpicture}[tdplot_main_coords,scale=2]
\draw[thick,->] (0,0,0) coordinate (O) -- (4,0,0) coordinate(X) node[pos=1.1]{};
\draw[thick,->] (O) -- (0,4,0) node[pos=1.1]{$v,\omega$};
\draw[thick,->] (O) -- (0,0,4) node[pos=1.1]{$r$};
\path[opacity=0.6,left color=blue!50,right color=blue!80,middle color=blue!20,shading angle=75]
plot[variable=\t,domain=0:360,smooth,samples=90] ({2+sin(\t)/sqrt(2)}, {2.5+cos(\t)}, {3.8+1/4 + sin(\t)^2/8}) -- cycle;
\path[opacity=0.4,left color=blue!50,right color=blue!80,middle color=blue!20,shading angle=75]
plot[variable=\t,domain=0:360,smooth,samples=90] ({2+sin(\t)}, {2.5+sqrt(2)*cos(\t)}, {4 + sin(\t)^2/4}) -- cycle;
\draw[dotted] plot[variable=\t,domain=-1:1,smooth,samples=90] ({2}, {2.5+sqrt(2)*\t}, {3+3/2 - (\t)^2/2});
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plot[variable=\t,domain=\RightAngle:\LeftAngle,smooth,samples=90] ({2+sin(\t)/sqrt(2)}, {2.5+cos(\t)}, {2.2+1/4 + sin(\t)^2/8}) --
plot[variable=\t,domain=\LeftAngle:\RightAngle,smooth,samples=90] ({2+sin(\t)/sqrt(2)}, {2.5+cos(\t)}, {3.8+1/4 + sin(\t)^2/8}) -- cycle;
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plot[variable=\t,domain=\RightAngle:\LeftAngle,smooth,samples=90] ({2+sin(\t)}, {2.5+sqrt(2)*cos(\t)}, {2 + sin(\t)^2/4}) --
plot[variable=\t,domain=\LeftAngle:\RightAngle,smooth,samples=90] ({2+sin(\t)}, {2.5+sqrt(2)*cos(\t)}, {4 + sin(\t)^2/4}) -- cycle;
\draw[thick] plot[variable=\t,domain=0:360,smooth,samples=90] ({2+sin(\t)}, {2.5+sqrt(2)*cos(\t)}, {4 + sin(\t)^2/4}) -- cycle;
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\draw[dashed] \innerbottomcylinderline{\LeftAngle};
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\draw plot[variable=\t,domain=0:360,smooth,samples=90] ({2+sin(\t)/sqrt(2)}, {2.5+cos(\t)}, {3.8+1/4 + sin(\t)^2/8}) ;
\draw plot[variable=\t,domain=0:360,smooth,samples=90] ({2+sin(\t)/sqrt(2)}, {2.5+cos(\t)}, {3+1/4 + sin(\t)^2/8}) ;
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\draw[thick, fill=gray, opacity=0.6] plot[variable=\t,domain=0:360,smooth,samples=90] ({2+sin(\t)/sqrt(2)}, {2.5+cos(\t)}, 0) ;
\node at (2,2.5,0) {$Y$};
\node at (2,2.5,3.25) {$h(Y)$};
\coordinate[label=right:$X$] (Xlabel) at (2,4,0);
\coordinate[label=right:$h(X)$] (hXlabel) at (2,4,3.1);
\node at (2,2.5,4.15) {$V$};
\node at (2,4,4.45) {$U$};
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\draw[thick, decorate,decoration={brace,amplitude=3pt} ] ({2+sin(\LeftAngle)}, {2.45+sqrt(2)*cos(\LeftAngle)}, {3 + sin(\LeftAngle)^2/4}) -- node[anchor=east] {$I\;$} ({2+sin(\LeftAngle)}, {2.45+sqrt(2)*cos(\LeftAngle)}, {4 + sin(\LeftAngle)^2/4}) ;
\end{tikzpicture}
}
\caption{The sets arising in definition \ref{def:diffeomorphismGauge}.}
\end{figure}
Our other main result is that the radiation gauge condition{} can be viewed as making the vacuum Einstein equation well-posed. We prove this result by constructing a first-order symmetric hyperbolic system. This involves using the GHP formalism \cite{GHP,PenroseRindler} to construct components of $g-\mathring{g}$, the difference between the connection coefficients of $g$ and of $\mathring{g}$, the difference of the corresponding curvatures, and some additional variables describing differential Lorentz transformations.
In applying the GHP formalism, it is necessary to make a choice of an equivalence class of frames, which we refer to as a choice of frame gauge. This is explained in section \ref{s:FOSH}. All calculations were done using the \emph{xAct} suite for Mathematica \cite{xAct}, and in particular the \emph{SpinFrames} package \cite{SpinframesPackage}.
\begin{theorem}[Well-posedness]
\label{thm:NLnhIsLWP}
Let $M>0$ and $a\in(-M,M)$.
Let $\mathring{g}$ be the Kerr metric on $\mathcal{K}^*$, and let $n$ be a future-directed, ingoing principal null vector.
\begin{enumerate}
\item The vacuum Einstein equation and the radiation gauge condition{} together imply a first-order symmetric hyperbolic system for the geometric variables in definition \ref{def:geometricVariables}.
\label{pt:FOSH}
\item The geometric variables in definition \ref{def:geometricVariables} uniquely determine a metric $g$.
\label{pt:systemDeterminesMetric}
\item If the initial data for first-order symmetric hyperbolic system in \ref{pt:FOSH} arise from initial data for the vacuum Einstein equation, then the metric determined by \ref{pt:FOSH}-\ref{pt:systemDeterminesMetric} satisfies the vacuum Einstein equation.
\end{enumerate}
\end{theorem}
It is well established that first-order symmetric-hyperbolic systems are well posed in suitable function spaces \cite{MR748308}. Note that the geometric nature of our variables ensures that the first-order symmetric hyperbolic system is well-defined for all $\omega\in\mathbb{S}^2$ and not just in a particular coordinate patch on the sphere.
In the final two sections of this paper, we go further in relating the radiation gauge condition{} for the Einstein equation to previously existing results for the linearized Einstein equation. In section \ref{s:traceCondition}, we apply a residual gauge transformation to further impose a condition on the trace $g^{ab}\mathring{g}_{ab}$ analogous to that imposed in the linear case by \cite{Chrzanowski, Price:2006ke}. In the final section of this paper, we linearize the Einstein equation in the radiation gauge and show that resulting linearized metric coefficients coincide with those constructed in our previous work on the linear stability of the Kerr metric \cite{Andersson:2019dwi}. However, in the previous and current works, we have made different choices in decomposing the linearized connection coefficients, which leads to a similar but different PDE system.
\subsection{Motivation and relation to existing literature}
In this paper, we introduce a new gauge choice to study the stability of Kerr black holes under the evolution generated by the vacuum Einstein equation. This gauge is inspired by what is called the ``outgoing radiation gauge (ORG)'' in \cite{Chrzanowski, Price:2006ke}, a so-called linearized gauge for the linearized Einstein equation.
The Kerr stability problem remains a central problem in the study of the Einstein equation as a hyperbolic differential equation. In brief, the problem is to show that, for any initial data that generates a solution containing a Kerr exterior, any sufficiently small perturbation of such initial data will generate a solution which contains a region that, in the future, converges to some Kerr exterior. So far, most work has focused on the linearized Einstein equation and models for it, such as the wave and Maxwell equations
\cite{bluesoffer03mora,BlueSterbenz,dafrod09red,blue:soffer:integral,blue2008decay,tataru2011localkerr,AnderssonBlue:KerrWave,larsblue15Maxwellkerr,Dafermos:2014cua,pasqualotto2019spin,Ma2017Maxwell} and the linearized gravity \cite{DHR:SchwarzschildStability,Hung:2017qop,Ma:2017bxq,Dafermos:2017yrz,Andersson:2019dwi,Hafner:2019kov}. Quite recently, a few works \cite{klainermanszeftel2020global,dafermos2021non,Klainerman:2021qzy} have made important progress on the full nonlinear stability of Kerr spacetimes.
We are particularly interested in the following approach to proving decay of solutions to the linearization of the Einstein equation on a Kerr background: The Kerr solutions admit a pair of principal null vectors. At least locally, one can construct a basis consisting of these principal null vectors, and an oriented orthonormal basis for the plane orthogonal to them. The GHP formalism uses spinors to construct the analogue of the Cartan formalism for such bases \cite{GHP,PenroseRindler}. Of central importance, in this set up, the two extreme components of the linearized curvature each satisfy a decoupled equation known as the Teukolsky master equation (TME) \cite{Teukolsky}. Chrzanowski \cite{Chrzanowski} introduced a linearized gauge transformation,
and showed that, in this linearized gauge, all linearized metric coefficients can be reconstructed from the linearized curvature. In the very slowly rotating case, uniform energy bounds and integrated local energy decay has been shown for the Teukolsky equation \cite{Ma:2017bxq,Dafermos:2017yrz}. Recently similar results have been obtained using physical-space methods \cite{Giorgi:2021skz}. In the full subextremal range, decay is proved for bounded frequencies in \cite{Shlapentokh-Rothman:2020vpj}.
From such results, we have shown that it follows that there are pointwise decay estimates for the linearized metric coefficients in the linear ORG \cite{Andersson:2019dwi}. In spherical symmetry, this linearized gauge choice uses the same choice of null tetrad as in the linearized gauge choice arising from double null coordinates, which has been used previously to show decay of linearized perturbations about Schwarzschild black holes \cite{DHR:SchwarzschildStability}. A significantly different approach to the linear stability problem was taken in \cite{Hafner:2019kov}.
As a geometric equation for curvature, the Einstein equation is invariant under changes of coordinate or, equivalently, diffeomorphisms. The freedom to apply diffeomorphisms is called diffeomorphism gauge freedom. As a consequence of this gauge freedom, for any solution of the Einstein equation $\mathring{g}$, any vector field $X$, and any solution $h$ of the linearization of the Einstein equation $\mathring{g}$, one finds that $h +\mathcal{L}_{X}\mathring{g}$ is also a solution of the linearization of the Einstein equation about $\mathring{g}$. The freedom to add any $\mathcal{L}_{X}\mathring{g}$ is called linearized gauge freedom.
For the linearized Einstein equation, the radiation gauge can be defined in the following way.
\begin{definition}
\label{def:linearORG}
Let $M>0$ and $a\in(-M,M)$.
Let $U$ be a subset of the maximal extension of the Kerr black hole with mass and angular momentum per unit mass $M,a$, and let $\mathring{g}$ be the metric on $U$.
Let $n$ denote the ingoing principal null vector on $U$.\footnote{Because equation \eqref{eq:linearnh} is homogeneous, the normalisation of $n$ does not need to be specified.}
Let $h$ be a symmetric $(0,2)$ tensor field on $U$.
$h$ is defined to satisfy the \defn{linear radiation gauge condition}
\footnote{Note that \cite{Price:2006ke} calls this the \defn{$n\cdoth$ gauge}.} if
\begin{subequations}
\begin{align}
n^ah_{ab} ={}& 0 ,
\label{eq:linearnh}
\end{align}
and to satisfy the \defn{linear trace condition} iff
\begin{align}
\mathring{g}^{ab} h_{ab} ={}& 0 .
\label{eq:linearTraceCondition}
\end{align}
\end{subequations}
$h$ is defined to satisfy the \defn{full radiation gauge of Chrzanowski} (\defn{ORG}) if it satisfies both the radiation gauge and the linear trace conditions.
\end{definition}
Essentially, this was first introduced in \cite{Chrzanowski} and then clarified in \cite{Price:2006ke}.
\cite{Price:2006ke} has shown that if $h$ satisfies the linear radiation gauge condition{}, then there is a linearized gauge transformation so that $h +\mathcal{L}_{X}\mathring{g}$ satisfies the full radiation gauge of Chrzanowski. From the perspective of naive function counting, it is surprising that all five of the conditions can be imposed, not merely the four of the linear null condition. A careful reading of \cite{Price:2006ke} shows that for any linearized metric (i.e. symmetric $(0,2)$ tensor), one can construct a linear gauge transformation so that the linear radiation gauge condition{} is satisfied on open sets. Furthermore, one can apply further residual gauge transformations that maintain the linear radiation gauge condition{}. From the perspective of naive function counting, it is convenient to consider residual gauge transformations as diffeomorphisms of the initial data set that can be applied in addition to the four gauge conditions that are applied within the spacetime and that generate a well-posed dynamics when combined with the Einstein equation.
While it is clear that if one has a smooth family of gauge transformations $\Phi_t$ then the linearization of this family determines a linear gauge transformation $\frac{\mathrm{d}}{\mathrm{d} t}\Phi_t^* h$, it is not clear that any so-called linear gauge transformation genuinely arises from the linearization of a family of gauge transformations, nor that even, if they did, the family of gauge transformations would have desirable properties. The main results of this paper, theorems \ref{thm:NLnhEnforceability}-\ref{thm:NLnhIsLWP}, show that the linear radiation gauge condition{} does arise from the linearization of a gauge for the full Einstein equation, namely the radiation gauge condition, and that this gauge gives a locally well-posed Cauchy problem for the Einstein equation. Furthermore, in section \ref{s:traceCondition}, we show that for the full Einstein equation, there is a diffeomorphism gauge that both satisfies the radiation gauge condition{} and such that $g^{ab}\mathring{g}_{ab}-4$ vanishes quadratically. Thus, the linearization of this can be seen as satisfying full radiation gauge of Chrzanowski.
The formalism we use to treat the radiation gauge condition{} has important similarities with and differences from the formalism based on principal geodesic structures in \cite{Klainerman:2021qzy}. Both formalisms specify one null vector field that is tangent to null geodesics. They are both frame formalisms based on a choice of a pair of null vector fields such that the orthogonal plane fails to be integrable in the sense of Frobenius.
By exclusively using properly weighted quantities, we can use the GHP formalism without specifying a choice of basis for the orthogonal plane and, hence, avoid the ``artificial gauge singularities'' noted in \cite[p27]{Klainerman:2021qzy}.
Perhaps in most striking contrast to the previous literature, both formalisms use not one but two classes of frame. In obtaining the first-order symmetric-hyperbolic form of the Einstein equations under the radiation gauge condition{}, we use the background principal null vectorfields $\mathring{\vecL},\mathring{\vecN}$ of the background Kerr geometry $\mathring{g}$ and a foreground pair of vectorfields $l,n=\mathring{\vecN}$ that are null with respect to the new, foreground geometry $g$. To each pair of null vectors, we associate the plane that is orthogonal in the relevant geometry. In contrast, the two frames used in the principal geodesic structures of \cite{Klainerman:2021qzy} share the same null legs, but one frame is completed by adjoining a basis for the (non-integrable) orthogonal plane while the other frame is completed by adjoining a basis for the (integrable) tangent space of the spheres that are $r,v$ level sets. Our two classes of frames coincide when the metric is exactly the Kerr metric, which suggests the possibility that the formalism based on the radiation gauge condition{} will provide significant simplifications, in addition to connecting with the previously existing physics literature.
\subsection{Structure of the proofs and of the paper}
Section \ref{s:ProofOfEnforecabilityOfTheNLORG} proves theorem \ref{thm:NLnhEnforceability} about the existence of a gauge transformation to impose the radiation gauge condition. Section \ref{s:FOSH} proves theorem \ref{thm:NLnhIsLWP} on the existence of a first-order symmetric hyperbolic system for the metric components and other geometric quantities; this section includes the definition of the frame gauge and the relevant geometric variables in terms of the GHP formalism. Section \ref{s:traceCondition} proves that perturbations of the trace, $g^{ab}\mathring{g}_{ab}-4$, can be made to vanish quadratically, in a quantifiable sense introduced in that section; this section is heavily inspired by \cite{Price:2006ke}. Section \ref{s:linearization} treats the linearization of the Einstein equation under our gauge choices.
\section{Imposing the radiation gauge condition{}}
\label{s:ProofOfEnforecabilityOfTheNLORG}
This section begins with some definitions to simplify discussion of the geometry in the directions orthogonal to the principal null vectors. There is then a lemma about metrics satisfying the radiation gauge condition, in particular that the flow along $n=-\partial_r$ generates affinely parameterized null geodesics, as is the case in the Kerr spacetime. Finally, there is a proof of the enforceability of the radiation gauge condition, which is based on appropriately constructing null geodesics. This completes the proof of theorem \ref{thm:NLnhEnforceability}.
Recall the notions of real null tetrad and complex null tetrads. These are given in appendix \ref {sec:worry}. Unless otherwise specified, a null tetrad is understood to mean an oriented complex null tetrad.
\begin{definition}
Let $M>0$ and $a\in(-M,M)$. Let $U$ be an open subset of $\mathcal{K}^*$ parameterized by $(v,r,\omega)$. In the domain of the standard spherical coordinates, define
\begin{subequations}
\begin{align}
\vecH ={}&\partial_\theta ,\\
\vecP ={}&\partial_\phi-a\sin^2\theta\partial_v .
\end{align}
\end{subequations}
\end{definition}
\begin{lemma}[Necessary results of the radiation gauge condition]
\label{lem:NLnhConsequences}
Let $M>0$ and $a\in(-M,M)$. Let $U$ be an open subset of $\mathcal{K}^*$ parameterized by $(v,r,\omega)$.
If $g$ is a Lorentzian metric on $U$ that satisfies the radiation gauge condition, then
\begin{enumerate}
\item $\partial_r$ is null.
\item In the portion of $U$ covered by spherical coordinates,
$\vecP$ and $\vecP$ are orthogonal to $\partial_r$.
\item At each point in the domain of the spherical coordinates, if $n=-\partial_r$ and $m$ is a complex linear combination of $\vecH$ and $\vecP$ such that $m$ and its complex conjugate $\bar{m}$ are a complex basis for the space spanned by $\vecH$ and $\vecP$ such that $g(m,m)=0$ and $g(m,\bar{m})=-1$, then there is a unique, future-directed null vector $l$ that is orthogonal to $m$ and $\bar{m}$ and that satisfies $g(l,n)=1$. Furthermore, if $g(m,m)=g(\bar{m},\bar{m})=0$ and $g(m,\bar{m})=-1$, then $(l,n,m,\bar{m})$ form a null tetrad.
\label{pt:NLnhHelper:ExistenceOfTetrad}
\item For all $(v_0,\omega_0)\in\mathbb{R}\times\mathbb{S}^2$, the curve $\gamma(s)=(v_0,s,\omega_0)$ is a (not necessarily affinely parameterized) geodesic.
\item If $\Sigma$ is $3$-submanifold of $U$ parameterized by $(v,\omega)$, and if $(\hat{v},\hat{\omega})$ are the restrictions of $(v,\omega)$ to $\Sigma$, $(\hat{\theta},\hat{\phi})$ denote the values of the standard spherical coordinate corresponding to $\hat{\omega}$, and $\hat{r}$ is the restriction of $r$ to $\Sigma$, then, in the domain of the standard spherical coordinates, $\partial_{\hat{v}}, \partial_{\hat{\theta}}, \partial_{\hat{\phi}} \in T\Sigma\subset TU$ satisfy
\begin{subequations}
\begin{align}
g(\partial_r,\partial_{\hat{v}}) ={}& -1 ,\\
g(\partial_r,\partial_{\hat{\theta}}) ={}& 0, \\
g(\partial_r,\partial_{\hat{\phi}}) ={}& -a\sin^2\theta .
\end{align}
\label{eq:surfaceInnerProducts}
\end{subequations}
\end{enumerate}
\end{lemma}
\begin{proof}
Unless otherwise specified, in this proof, we work in the domain of the spherical coordinates and then extend by continuity.
Since $g_{ra}\mathrm{d} x^a=(\mathrm{d} v+a\sin^2\theta\mathrm{d}\phi)$, it follows that $g(\partial_r,\partial_r)$ $=g_{ra}\mathrm{d} x^a(\partial_r)$ $=(\mathrm{d} v+a\sin^2\theta\mathrm{d}\phi)(\partial_r)$ $=0$, that $g(\partial_r,\vecP)$ $=(\mathrm{d} v+a\sin^2\theta\mathrm{d}\phi)(\partial_\theta)$ $=0$, and that $g(\partial_r,\vecP)$ $=(\mathrm{d} v+a\sin^2\theta\mathrm{d}\phi)(\partial_\phi-a\sin^2\theta\partial_v)$ $=0$, which establishes the first two claims in the domain of the spherical coordinates. By continuity, $\partial_r$ remains null at the poles of the spherical coordinates. The plane orthogonal to $m$ and $\bar{m}$ is a $1+1$-dimensional Lorentzian vector space with a time orientation, and, since $n$ is null but not zero, the existence of a unique $l$ as in the statement of point \ref{pt:NLnhHelper:ExistenceOfTetrad} holds.
To show that the curves $(v_0,s,\omega_0)$ are (not necessarily affinely parameterized) geodesics it is sufficient to show that $\ddot\gamma^b=n^a\nabla_\ian^b$ is parallel to $n$. This is equivalent to $\ddot\gamma^\ibn_b$ $=\ddot\gamma^\ibm_b$ $=\ddot\gamma^b\bar{m}_b=0$. Trivially,
\begin{align}
n_\ibn^a\nabla_\ian^b
={}&\frac12n^a\nabla_a(n_\ibn^b)
=0,
\end{align}
since $n_\ibn^b=0$. Before continuing, first observe that the commutator $[n,m]$ satisfies
\begin{subequations}
\begin{align}
[n,m]
={}&-[\partial_r,m^\Theta\vecH +m^\Phi\vecP] \nonumber\\
={}&-[\partial_r,m^\Theta\partial_\theta +m^\Phi(\partial_\phi-a\sin^2\theta\partial_v)] \nonumber\\
={}&-(\partial_rm^\Theta)\vecH -(\partial_rm^\Phi)\vecP ,\\
g(n,[n,m])
={}& 0.
\end{align}
\end{subequations}
Now, observe, from the orthogonality conditions and from properties of the commutator, that
\begin{subequations}
\begin{align}
m_\ibn^a\nabla_\ian^b
={}& n^a\nabla_a(m_\ibn^b) -n_\ibn^a\nabla_\iam^b
=-n_\ibn^a\nabla_\iam^b,\\
n_\ibn^a\nabla_\iam^b
={}& 0 +n_\ibm^a\nabla_\ian^b +n_b[n,m]^b\nonumber\\
={}&\frac12m^a\nabla_a(n_\ibn^b) +0
=0
\label{eq:llnablam}.
\end{align}
\end{subequations}
Observe that $\cos\phi\partial_\theta+\frac{\sin\phi}{\sin\theta}(\partial_\phi-a\sin^2\theta\partial_v)$ and $\sin\phi\partial_\theta-\frac{\cos\phi}{\sin\theta}(\partial_\phi-a\sin^2\theta\partial_v)$ form a basis for the planes they span, and that this combination extends smoothly to $\theta=0$ and to $\theta=\pi$. Thus, the results extend from the domain of the spherical coordinates to all of $U$.
From the chain rule, one finds $\partial_{\hat{v}}= \partial_v +\frac{\partial \hat{r}}{\partial \hat{v}}\partial_r$. From this and the fact that $\partial_r$ is null, it follows that $g(\partial_r,\partial_{\hat{v}}) = g(\partial_r,\partial_{v})$, which is equal to $-1$ by the radiation gauge condition. This proves the first equation of \eqref{eq:surfaceInnerProducts}. Replacing $\hat{v}$ by $\hat{\theta}$ and $\hat{\phi}$, one obtains the remaining two equations.
\end{proof}
\begin{proof}[Proof of the enforceability of the radiation gauge condition, theorem \ref{thm:NLnhEnforceability}]
To begin we construct the gauge transformation. In this paragraph $(v,r,\omega)$ denotes the original parameterization in $V$. On $h(X)$, define $(\hat{v},\hat{\omega})$ and $\hat{r}$ to be the restrictions of $(v,\omega)$ and $r$ respectively. By the closeness (in $C^0$) of $g$ to $\mathring{g}$, at each point $p\in h(X)$ in the domain of the spherical coordinates, there is a unique vector $n$ in $T_p W$ such that $n$ is null and satisfies the analogue of \eqref{eq:surfaceInnerProducts}, i.e.
\begin{subequations}
\label{eq:surfaceInnerProducts:imposed}
\begin{align}
g(n,\partial_{\hat{v}}) ={}& 1
\label{eq:surfaceInnerProducts:imposed:v},\\
g(n,\partial_{\hat{\theta}}) ={}& 0, \\
g(n,\partial_{\hat{\phi}}) ={}& a\sin^2\theta .
\end{align}
\end{subequations}
Furthermore, since $a\sin^2\theta\mathrm{d}\phi$ extends smoothly to $0$ in $T\mathbb{S}^2$, the vector field $n$ has a unique continuous extension from the portion of $h(X)$ covered by spherical coordinates to all of $h(X)$. Overloading notation, let $n$ denote this extension.
By the $C^2$ closeness of $g$ and $\mathring{g}$, $n$ is $C^2$ on $h(X)$, and there is an $\varepsilon_0>0$ and an open neighbourhood $W$ of $h(X)$ such that the geodesic flow defines a diffeomorphism $(-\varepsilon_0,\varepsilon_0)\times h(Y)\rightarrow U$. At $q\in W$, define $(v^{\text{new}},\omega^{\text{new}})$ to be the value of $(\hat{v},\hat{\omega})$ at the unique point $p\in h(X)$ such that $q$ is on the geodesic launched by $n$ at $p$. (The diffeomorphism guarantees the existence of such a point.) Let $\tilde{\gamma}_{(v^{\text{new}},\omega^{\text{new}})}(s)$ denote the geodesic corresponding to the values $(v^{\text{new}},\omega^{\text{new}})$ with, on $h(X)$, the initial conditions $s=r$ and $\frac{\mathrm{d}}{\mathrm{d} s}\tilde{\gamma} =-n$. Set $r^{\text{new}}=s$. Thus, $(v^{\text{new}},r^{\text{new}},\omega^{\text{new}})$ is a gauge choice.
In this parameterization, $-\partial_{r^{\text{new}}}$ is null, since it is the tangent to a geodesic launched from a null vector.
It remains to show the radiation gauge condition{} holds in this diffeomorphism gauge. For the remainder of this proof $(v,r,\omega)$ denotes the parameters in the new parameterization. In the domain of the spherical coordinates, the form $\lambda =i_\vecNg =-g_{ab}(\partial_r^b)\mathrm{d} x^a$ can be expanded, in $\mathrm{d} v$, $\mathrm{d} r$, $\mathrm{d}\theta$, and $\mathrm{d}\phi$. It is sufficient to show that $g(\partial_r,\partial_r)$ $=0$ $=g(\partial_r,\vecH)$ $=g(\partial_r,\vecP)$ and $g(-\partial_r,\partial_v)=1$.
Since $\partial_r$ is null, clearly $g(\partial_r,\partial_r)=0$.
From \eqref{eq:surfaceInnerProducts:imposed}, $g(\partial_r,\vecH)$ has the desired value on $h(X)$. Let $n$ denote $-\partial_r$. Observe that since $\partial_r$ is tangent to an affinely parameterized geodesic, $\nabla_{\partial_r}\partial_r=0$. Observe further that $[n,\vecH]$ $=[-\partial_r,\partial_\theta]$ $=0$. Thus,
\begin{align}
0
={}&g(\nabla_{n}n,\vecH) \nonumber\\
={}& \nabla_{n}\left(g(n,\vecH)\right)
-g(n,\nabla_{n}\vecH) \nonumber\\
={}& \nabla_{n}\left(g(n,\vecH)\right)
-g(n,\nabla_{\vecH}n) \nonumber\\
={}& \nabla_{n}\left(g(n,\vecH)\right)
-\frac12\nabla_{\vecH}\left(g(n,n)\right) .
\end{align}
The final term vanishes since $n$ is always a null vector. Thus, $g(n,\vecH)$ is constant, and, in particular, since it is initially zero, it remains zero along the entire geodesic. Since $[n,\vecP]$ $=[-\partial_r,\partial_\phi-a\sin^2\theta\partial_v]$ $=0$, the same argument applies with $\vecP$. Since $g(\partial_r,\partial_\theta)=0$, the $\mathrm{d}\theta$ component of $\lambda$ vanishes. Since $g(\partial_r,\vecP)=0$, the $\mathrm{d}\phi$ component of $\lambda$ is $a\sin^2\theta$ times the coefficient of $\mathrm{d} v$. Since $[n,\partial_v]=0$, a similar calculation shows that $g(n,\partial_v)$ is constantly $-1$. Since the parameterization is constructed smoothly, the construction extends from the domain of the spherical coordinates to the full sphere. Since the Kerr metric is itself a solution, from the continuity of solutions of ODE, it follows that for any $V\subset U$, if the initial data is sufficiently close (in a sufficiently high regularity class), the gauge transformation maps $V$ to a subset of $U$. Observe that the new metric on the initial hypersurface $h(X)$ depends only on the old metric on $h(X)$, which gives the desired norm property. This completes the proof.
\end{proof}
\section{Field equations}
\label{s:FOSH}
Within this section, we introduce geometric variables and a frame gauge condition, which are used to construct a first-order symmetric-hyperbolic system.
\subsection{GHP Notation}
In this subsection, we review the GHP notation \cite{GHP} for connection and curvature components, which we will use throughout this paper. Appendix \ref{sec:worry} explains the nature of GHP scalars and recalls the definition of tetrads.
\begin{definition}
\label{def:spincoeff}
Given any null tetrad $(l^a, n^a, m^a, \bar{m}^a)$ and the Levi-Civita connection $\nabla_a$ with respect to the corresponding metric, the spin coefficients are
\begin{subequations}
\begin{align}
\kappa ={}&l^{a} m^{b} \nabla_{a}l_{b},&
\kappa '={}&\bar{m}^{a} n^{b} \nabla_{b}n_{a},\\
\rho ={}&m^{a} \bar{m}^{b} \nabla_{b}l_{a},&
\rho '={}&m^{a} \bar{m}^{b} \nabla_{a}n_{b},\\
\sigma ={}&m^{a} m^{b} \nabla_{a}l_{b},&
\sigma '={}&\bar{m}^{a} \bar{m}^{b} \nabla_{b}n_{a},\\
\tau ={}&m^{a} n^{b} \nabla_{b}l_{a},&
\tau '={}&l^{a} \bar{m}^{b} \nabla_{a}n_{b},
\end{align}
\end{subequations}
and
\begin{subequations}
\begin{align}
\beta ={}&- \tfrac{1}{2} m^{a} \bar{m}^{b} \nabla_{a}m_{b}
- \tfrac{1}{2} l^{a} m^{b} \nabla_{b}n_{a},&
\beta '={}&\tfrac{1}{2} \bar{m}^{a} \bar{m}^{b} \nabla_{b}m_{a}
+ \tfrac{1}{2} l^{a} \bar{m}^{b} \nabla_{b}n_{a},\\
\epsilon ={}&- \tfrac{1}{2} l^{a} \bar{m}^{b} \nabla_{a}m_{b}
- \tfrac{1}{2} l^{a} l^{b} \nabla_{b}n_{a},&
\epsilon '={}&\tfrac{1}{2} \bar{m}^{a} n^{b} \nabla_{b}m_{a}
+ \tfrac{1}{2} l^{a} n^{b} \nabla_{b}n_{a}.
\end{align}
\end{subequations}
\end{definition}
\begin{definition}
Given any null tetrad $(l^a, n^a, m^a, \bar{m}^a)$ and the Weyl tensor $C_{abcd}$ with respect to the corresponding metric, we define the Weyl scalars
\begin{subequations}
\begin{align}
\Psi_{0}{}={}&l^{a} l^{c} m^{b} m^{d} C_{abcd},&
\Psi_{1}{}={}&l^{a} l^{c} m^{b} n^{d} C_{abcd},&
\Psi_{2}{}={}&l^{a} m^{b} \bar{m}^{c} n^{d} C_{abcd},\\
\Psi_{3}{}={}&l^{a} \bar{m}^{c} n^{b} n^{d} C_{abcd},&
\Psi_{4}{}={}&\bar{m}^{a} \bar{m}^{c} n^{b} n^{d} C_{abcd}.
\end{align}
\end{subequations}
\end{definition}
One of the central results of the GHP framework is that $\kappa,\tau,\rho,\sigma,\kappa',\tau',\rho',\sigma'$ and all the $\Psi_i$ are properly weighted, but $\beta,\epsilon,\beta',\epsilon'$ are not.
\subsection{Background and foreground metrics}
To begin our analysis of perturbations of the Kerr metric, we introduce the following hypotheses, which we typically use throughout the rest of this section.
\begin{definition}[The vacuum, radiation-gauge hypotheses]
\label{def:radiationGaugeHypotheses}
The \defn{background hypotheses} are defined as follows:
``Let $M>0$ and $a\in(-M,M)$.
Let $\mathring{g}_{ab}$ be the background Kerr metric as in equation \eqref{eq:KerrMetric} with parameters $(M, a)$.
Let $U$ be an open subset of $\mathcal{K}^*$.
Let $(\mathring{\vecL}^{a},\mathring{\vecN}^{a},\mathring{\vecM}^{a},\bar{\mathring{\vecM}}^{a})$ denote an arbitrary element of the set of local complex null tetrads such that $\mathring{\vecL}$ and $\mathring{\vecN}$ are outgoing and ingoing, future-directed principal null vectors.
Let $(\mathring{\vecL}_{a},\mathring{\vecN}_{a},\mathring{\vecM}_{a},\bar{\mathring{\vecM}}_{a})$ be the corresponding co-frame.
The spin coefficients and Weyl scalars with respect to this tetrad are indicated with the accent $\mathring{}$.''
The \defn{vacuum, radiation-gauge hypotheses} are defined to be the background hypotheses{}
together with the assumption that $g_{ab}$ is a Lorentzian metric satisfying the vacuum Einstein equation and the radiation gauge condition{}
\begin{align}
\mathring{\vecN}^{b} (g_{ab} - \mathring{g}_{ab})={}&0.
\end{align}
The \defn{background and foreground metrics} are defined to be $\mathring{g}_{ab}$ and $g_{ab}$ respectively with inverses $\mathring{g}{}^{ab}$ and $g^{\#}{}^{ab}$.
\end{definition}
Because $(\mathring{\vecL},\mathring{\vecN},\mathring{\vecM},\bar{\mathring{\vecM}})$ is used to denote an arbitrary element of the set of local tetrads in Kerr aligned with $(\hat{\vecL},\hat{\vecN})$, there is a freedom to apply spin and boost transformations. As long as our variables and operators are made so that they transform properly under such transformation, this allows us to introduce properly weighted quantities, which are globally defined. In the language of principal-$G$ bundles, as long as our variables transform equivariantly, we may use local tetrads to construct a globally defined section of an associated complex line bundle. In the language of gauge theory, we have a gauge freedom corresponding to choice of boost and spin transformation, and, as long as our variables transform correctly under such gauge transformations, they are globally defined gauge fields. This allows us to avoid problems at the poles in spherical coordinates that might arise from, for example, taking $\mathring{\vecM}=2^{-1/2}(r-ia\cos\theta)^{-1}(\partial_\theta+i(\sin\theta)^{-1}(\partial_\phi+a\sin^2\theta\partial_v))$ or any other explicit combination of $\vecH$ and $\vecP$.
\begin{definition}[Foreground metric coefficients in the background frame]
Assume the \radiationGaugeHypotheses{}.
Define the \defn{foreground metric coefficients in the background frame} to be
\begin{subequations}
\begin{align}
G_{2}={}&\bar{\mathring{\vecM}}^{a} \bar{\mathring{\vecM}}^{b}(g_{ab} - \mathring{g}_{ab})
= \bar{\mathring{\vecM}}^{a} \bar{\mathring{\vecM}}^{b}g_{ab},\\
G_{1}={}& \mathring{l}^{a} \bar{\mathring{\vecM}}^{b}(g_{ab} - \mathring{g}_{ab})
= \mathring{l}^{a} \bar{\mathring{\vecM}}^{b}g_{ab},\\
G_{0}={}& \mathring{l}^{a} \mathring{l}^{b}(g_{ab} - \mathring{g}_{ab})
= \mathring{l}^{a} \mathring{l}^{b}g_{ab},\\
\slashed{G}_{}={}&\mathring{g}^{ab} (g_{ab} - \mathring{g}_{ab})
=\mathring{g}^{ab} g_{ab} - 4,\\
G^{\#}_{2}={}& \bar{\mathring{\vecM}}_{a} \bar{\mathring{\vecM}}_{b}(g^{\#}{}^{ab} - \mathring{g}^{ab})
= \bar{\mathring{\vecM}}_{a} \bar{\mathring{\vecM}}_{b} g^{\#}{}^{ab},\\
G^{\#}_{1}={}&\mathring{l}_{a} \bar{\mathring{\vecM}}_{b}(g^{\#}{}^{ab} - \mathring{g}^{ab})
= \mathring{l}_{a} \bar{\mathring{\vecM}}_{b} g^{\#}{}^{ab},\\
G^{\#}_{0}={}&\mathring{l}_{a} \mathring{l}_{b}(g^{\#}{}^{ab} - \mathring{g}^{ab})
= \mathring{l}_{a} \mathring{l}_{b} g^{\#}{}^{ab},\\
\slashed{G}^{\#}={}&\mathring{g}_{ab}(g^{\#}{}^{ab} - \mathring{g}^{ab})
=\mathring{g}_{ab} g^{\#}{}^{ab} - 4.
\end{align}
\end{subequations}
\end{definition}
Observe that they vanish if the perturbation vanishes. They are all properly weighted with respect to background boost and spin transformations. The remaining metric coefficients vanish by the radiation gauge condition. The set $(G^{\#}_{2}, G^{\#}_{1}, G^{\#}_{0}, \slashed{G}^{\#})$ can be algebraically computed from the set $(G_{2}, G_{1}, G_{0}, \slashed{G}_{})$ and vice versa via
\begin{subequations}
\label{eq:GToInvG}
\begin{align}
G^{\#}_{2}={}&- \frac{G_{2}}{(1 + \tfrac{1}{2} \slashed{G}_{})^2 - |G_{2}|^2},\\
G^{\#}_{1}={}&- \frac{(1 + \tfrac{1}{2} \slashed{G}_{}) G_{1} + \overline{G_{1}} G_{2}}{(1 + \tfrac{1}{2} \slashed{G}_{})^2 - |G_{2}|^2},\\
G^{\#}_{0}={}&- G_{0}
- \frac{2 (1 + \tfrac{1}{2} \slashed{G}_{}) \overline{G_{1}} G_{1} + \overline{G_{2}} G_{1}{}^2 + \overline{G_{1}}{}^2 G_{2}}{(1 + \tfrac{1}{2} \slashed{G}_{})^2 - |G_{2}|^2},\\
\slashed{G}^{\#}={}&
\frac{1}{1 - |G_{2}| + \tfrac{1}{2} \slashed{G}_{}}
+ \frac{1}{1 + |G_{2}| + \tfrac{1}{2} \slashed{G}_{}}
- 2,
\end{align}
\end{subequations}
\begin{subequations}
\label{eq:InvGToG}
\begin{align}
G_{2}={}&- \frac{G^{\#}_{2}}{(1 + \tfrac{1}{2} \slashed{G}^{\#})^2- |G^{\#}_{2}|^2},\\
G_{1}={}&- \frac{(1 + \tfrac{1}{2} \slashed{G}^{\#}) G^{\#}_{1} + \overline{G^{\#}_{1}} G^{\#}_{2}}{(1 + \tfrac{1}{2} \slashed{G}^{\#})^2 - |G^{\#}_{2}|^2},\\
G_{0}={}&- G^{\#}_{0}
- \frac{2 (1 + \tfrac{1}{2} \slashed{G}^{\#}) \overline{G^{\#}_{1}} G^{\#}_{1} + \overline{G^{\#}_{2}} G^{\#}_{1}{}^2 + \overline{G^{\#}_{1}}{}^2 G^{\#}_{2}}{(1 + \tfrac{1}{2} \slashed{G}^{\#})^2- |G^{\#}_{2}|^2},\\
\slashed{G}_{}={}&
\frac{1}{1 - |G^{\#}_{2}| + \tfrac{1}{2} \slashed{G}^{\#}}
+ \frac{1}{1 + |G^{\#}_{2}| + \tfrac{1}{2} \slashed{G}^{\#}}
-2.
\end{align}
\end{subequations}
\subsection{Frame choice}
Given the set $(G^{\#}_{2}, G^{\#}_{1}, G^{\#}_{0}, \slashed{G}^{\#})$ of background frame components of the inverse foreground metric $g^{\#}{}^{ab}$, we can construct a null tetrad for the foreground metric. However, due to Lorentz gauge freedom this frame is not unique. Due to the fact that the radiation gauge condition{} singles out $\mathring{n}^{a}$, we choose to keep it also for the foreground tetrad, i.e. $n^a=\mathring{n}^{a}$.
With this leg fixed, the remaining group of Lorentz transformations are described by one real differential spin rotation parameter $\nu$ with $(p,q)$-weight $(0,0)$ and a complex parameter $\eta$ with $(p,q)$-weight $(2,0)$.
\begin{remark}
In principle one could instead demand that $n^a$ is merely proportional to $\mathring{n}^{a}$. Doing this would introduce a real differential boost parameter $\mu$ to the group of Lorentz transformations, so that $n^{a}=\mu^{-1}\mathring{n}^{a}$. However, as we later would like to set the $\tilde{\epsilon}'=\epsilon ' - \mu^{-1}\mathring{\epsilon}'$ to zero, and we find that $\tilde{\epsilon}' + \overline{\tilde{\epsilon}'}=- \mu^{-1}\thop \mu$, we conclude that $\mu=1$, i.e. $n^a=\mathring{n}^{a}$ is sensible.
\end{remark}
\begin{definition}[Foreground frame]
\label{def:ForegroundFrame}
Assume the \radiationGaugeHypotheses.
A choice of \defn{differential Lorentz transformation variables} is a choice of
$(\nu,\eta)$ with $(p,q)$-weights $(0,0)$ and $(2,0)$ respectively.
Assuming a choice of differential Lorentz transformation variables,
define the \defn{foreground frame} to be
\begin{subequations}
\begin{align}
l^{a}={}&\mathring{l}^{a}
+ (\eta \bar{\eta} + \tfrac{1}{2} G^{\#}_{0}) \mathring{n}^{a}
- \Bigl( G^{\#}_{1} + \frac{\eta G^{\#}_{2}}{2 e^{i \nu} \varsigma^{\#}{}} - e^{i \nu} \bar{\eta} \varsigma^{\#}{}\Bigr) \mathring{\vecM}^{a}
- \Bigl( \overline{G^{\#}_{1}} + \frac{e^{i \nu} \bar{\eta} \overline{G^{\#}_{2}}}{2 \varsigma^{\#}{}} - \frac{\eta \varsigma^{\#}{}}{e^{i \nu}}\Bigr) \bar{\mathring{\vecM}}^{a} ,\\
n^{a}={}&\mathring{n}^{a},\\
m^{a}={}&\eta \mathring{n}^{a}
+e^{i \nu} \varsigma^{\#}{} \mathring{\vecM}^{a}
- \frac{e^{i \nu} \overline{G^{\#}_{2}} \bar{\mathring{\vecM}}^{a}}{2 \varsigma^{\#}{}},
\end{align}
\end{subequations}
along with the auxiliary variables
\begin{subequations}
\label{eq:defVarsigma}
\begin{align}
\varsigma ={}&\tfrac{1}{2} \sqrt{1 - |G_{2}| + \tfrac{1}{2} \slashed{G}_{}}
+ \tfrac{1}{2} \sqrt{1 + |G_{2}| + \tfrac{1}{2} \slashed{G}_{}},\\
\varsigma^{\#}{}={}&\tfrac{1}{2} \sqrt{1 - |G^{\#}_{2}| + \tfrac{1}{2}\slashed{G}^{\#}}
+ \tfrac{1}{2} \sqrt{1 + |G^{\#}_{2}| + \tfrac{1}{2}\slashed{G}^{\#}}.
\end{align}
\end{subequations}
\end{definition}
\begin{lemma}
Assume the \radiationGaugeHypotheses{} and a choice of differential Lorentz transformation variables.
The foreground frame is a null tetrad for the foreground metric $g_{ab}$, i.e.
\begin{align}
g^{\#}{}^{ab}={}&2 l^{(a}n^{b)} - 2 m^{(a}\bar{m}^{b)}.
\end{align}
The corresponding co-frame is
\begin{subequations}
\begin{align}
l_{a}={}&\mathring{l}_{a}
+ \Bigl(\eta \bar{\eta} - \tfrac{1}{2} G^{\#}_{0} - e^{i \nu} \bar{\eta} \overline{G^{\#}_{1}} \varsigma - \frac{\eta G^{\#}_{1} \varsigma}{e^{i \nu}} - \frac{e^{i \nu} \bar{\eta} \overline{G^{\#}_{2}} G^{\#}_{1} \varsigma}{2 \varsigma^{\#}{}^2} - \frac{\eta \overline{G^{\#}_{1}} G^{\#}_{2} \varsigma}{2 e^{i \nu} \varsigma^{\#}{}^2}\Bigr)\mathring{n}_{a}\nonumber\\
& + \Bigl(e^{i \nu} \bar{\eta} \varsigma + \frac{\eta G^{\#}_{2} \varsigma}{2 e^{i \nu} \varsigma^{\#}{}^2}\Bigr)\mathring{\vecM}_{a}
+ \Bigl( \frac{\eta \varsigma}{e^{i \nu}} + \frac{e^{i \nu} \bar{\eta} \overline{G^{\#}_{2}} \varsigma}{2 \varsigma^{\#}{}^2}\Bigr)\bar{\mathring{\vecM}}_{a},\\
n_{a}={}&\mathring{n}_{a},\\
m_{a}={}& \Bigl( \eta - e^{i \nu} \overline{G^{\#}_{1}} \varsigma - \frac{e^{i \nu} \overline{G^{\#}_{2}} G^{\#}_{1} \varsigma}{2 \varsigma^{\#}{}^2}\Bigr)\mathring{n}_{a}
+ e^{i \nu} \varsigma \mathring{\vecM}_{a}
+ \frac{e^{i \nu} \overline{G^{\#}_{2}} \varsigma}{2 \varsigma^{\#}{}^2}\bar{\mathring{\vecM}}_{a} .
\end{align}
\end{subequations}
\end{lemma}
\begin{definition}
Assume the \radiationGaugeHypotheses{} and a choice of differential Lorentz transformation variables.
Define the \defn{foreground metric coefficients} to be
\begin{subequations}
\begin{align}
\tilde{\slashed{G}}={}&\mathring{g}_{ab} g^{\#}{}^{ab} - 4,&
\tilde{G}_{2}{}={}&\mathring{g}_{ab} \bar{m}^{a} \bar{m}^{b},&
\tilde{G}_{1}{}={}&\mathring{g}_{ab} l^{a} \bar{m}^{b},&
\tilde{G}_{0}{}={}&\mathring{g}_{ab} l^{a} l^{b},\\
\tilde{\slashed{G}}{}^{\#}_{\phantom{0}}{}={}&\mathring{g}^{ab} g_{ab} - 4,&
\tilde{G}^{\#}_2{}={}&\mathring{g}^{ab} \bar{m}_{a} \bar{m}_{b},&
\tilde{G}^{\#}_1{}={}&\mathring{g}^{ab} l_{a} \bar{m}_{b},&
\tilde{G}^{\#}_0{}={}&\mathring{g}^{ab} l_{a} l_{b}.
\end{align}
\end{subequations}
Unless otherwise specified, define \defn{metric coefficients} to be the foreground metric coefficients.
\end{definition}
Note that the background metric coefficients are the components of the foreground metric with respect to the background tetrad, and, conversely, the (foreground) metric coefficients are the components of the background metric with respect to the foreground frame.
We have the following useful relations
\begin{subequations}
\label{eq:GslashRelations}
\begin{align}
\slashed{G}{}^{\#}={}&\tilde{\slashed{G}}
=-2 + 4 \varsigma^{\#}{}^2 - \frac{2 \varsigma^{\#}{}}{\varsigma},&
| G^{\#}_{2}|^2={}&|\tilde{G}_{2}{}|^2
=4 \varsigma^{\#}{}^4 - \frac{4 \varsigma^{\#}{}^3}{\varsigma},\\
\slashed{G}_{}={}&\tilde{\slashed{G}}{}^{\#}_{\phantom{0}}{}
=-2 + 4 \varsigma^2 - \frac{2 \varsigma}{\varsigma^{\#}{}},&
|G_{2}|^2={}&|\tilde{G}^{\#}_2{}|^2
=4 \varsigma^4 - \frac{4 \varsigma^3}{\varsigma^{\#}{}}.
\end{align}
\end{subequations}
The relations between $(\tilde{\slashed{G}}, \tilde{G}_{2}{}, \tilde{G}_{1}{}, \tilde{G}_{0}{})$, $(\tilde{\slashed{G}}{}^{\#}_{\phantom{0}}{}\negthinspace, \tilde{G}^{\#}_2{}, \tilde{G}^{\#}_1{}, \tilde{G}^{\#}_0{})$ follows the pattern \eqref{eq:GToInvG}.
Given $\nu$ and $\eta$ we can express the sets $(\tilde{\slashed{G}}, \tilde{G}_{2}{}, \tilde{G}_{1}{}, \tilde{G}_{0}{})$, $(\tilde{\slashed{G}}{}^{\#}_{\phantom{0}}{}\negthinspace, \tilde{G}^{\#}_2{}, \tilde{G}^{\#}_1{}, \tilde{G}^{\#}_0{})$, $(G^{\#}_{2}, G^{\#}_{1}, G^{\#}_{0}, \slashed{G}^{\#}\hspace{-1pt})$ and $(G_{2}, G_{1}, G_{0}, \slashed{G}_{})$ in terms of each other. For instance
\begin{subequations}
\label{eq:InvGTildeInvGRelations}
\begin{align}
\tilde{G}^{\#}_2{}={}&\frac{G_{2}}{e^{2i \nu}} ,\\
\tilde{G}^{\#}_1{}={}&
\frac{G_{1} \varsigma^{\#}{}}{e^{i \nu}}
- \tfrac{1}{2} \bar{\eta} \slashed{G}_{}
+ \frac{\eta G_{2}}{e^{2i \nu}}
+ \frac{\overline{G_{1}} G_{2} \varsigma^{\#}{}}{2 e^{i \nu} \varsigma^2} ,\\
\tilde{G}^{\#}_0{}={}&
G_{0}
- \eta \bar{\eta} \slashed{G}_{}
+ e^{2i \nu} \bar{\eta}^2 \overline{G_{2}}
+ \frac{\eta^2 G_{2}}{e^{2i \nu}}
+ e^{i \nu} \bar{\eta} \varsigma^{\#}{} \Bigl(2 \overline{G_{1}} + \frac{\overline{G_{2}} G_{1}}{\varsigma^2}\Bigr)
+ \frac{\eta \varsigma^{\#}{}}{e^{i \nu}} \Bigl(2 G_{1} + \frac{\overline{G_{1}} G_{2}}{\varsigma^2}\Bigr)\nonumber\\
& + \frac{\varsigma^{\#}{}^2}{\varsigma^2} \bigl((2 + \slashed{G}_{}) \overline{G_{1}} G_{1} + \overline{G_{2}} G_{1}{}^2 + \overline{G_{1}}{}^2 G_{2}\bigr),\\
G^{\#}_{2}={}&- \frac{e^{2i \nu} \tilde{G}^{\#}_2{} \varsigma^{\#}{}^2}{\varsigma^2},\\
G^{\#}_{1}={}&- e^{i \nu} \tilde{G}^{\#}_1{} \varsigma^{\#}{}
+ e^{i \nu} \bar{\eta} (\varsigma^{\#}{} - \varsigma)
- \frac{e^{i \nu} \overline{\tilde{G}}{}^{\#}_1{} \tilde{G}^{\#}_2{} \varsigma^{\#}{}}{2 \varsigma^2}
+ \frac{e^{i \nu} \eta \tilde{G}^{\#}_2{} (\varsigma^{\#}{} + \varsigma)}{2 \varsigma^2},\\
G^{\#}_{0}={}&- \tilde{G}^{\#}_0{}
+ 2 \eta \tilde{G}^{\#}_1{}
+ 2 \bar{\eta} \overline{\tilde{G}}{}^{\#}_1{}
- \eta^2 \tilde{G}^{\#}_2{}
- \bar{\eta}^2 \overline{\tilde{G}}{}^{\#}_2{}
+ \eta \bar{\eta} \tilde{\slashed{G}}{}^{\#}_{\phantom{0}}{}.
\end{align}
\end{subequations}
\subsection{Geometric variables and operators}
In this section, we define differential spin coefficients and differential curvature components. The foreground spin coefficients carry all the information about the connection. However, several are not small for a small metric perturbation, because several of the background components are non-vanishing. Furthermore, not all of them are properly weighted with respect to spin and boost transformations of the background frame. Our choice of differential spin coefficients compensate for both of these issues. While the foreground curvature components are properly weighted, the middle curvature component is not small, since the middle curvature component in the background is non-vanishing. Our choice of differential curvature components compensates for this problem.
\begin{definition}
\label{def:DiffSpinCoeff}
\label{def:geometricVariables}
Assume the \radiationGaugeHypotheses{} and a choice of differential Lorentz transformation variables.
Define the \defn{differential spin coefficients}
\begin{subequations}
\begin{align}
\tilde{\beta}={}&\beta
- e^{i \nu} \varsigma^{\#}{} \mathring{\beta}
- \frac{e^{i \nu} \overline{G^{\#}_{2}} \mathring{\beta}'}{2 \varsigma^{\#}{}}
+ \eta \mathring{\epsilon}',\\
\tilde{\beta}'={}&\beta '
- \frac{G^{\#}_{2} \mathring{\beta}}{2 e^{i \nu} \varsigma^{\#}{}}
- \frac{\varsigma^{\#}{} \mathring{\beta}'}{e^{i \nu}}
- \bar{\eta} \mathring{\epsilon}',\\
\tilde{\epsilon}={}&\epsilon - \mathring{\epsilon}
+ \Bigl(G^{\#}_{1} + \frac{\eta G^{\#}_{2}}{2 e^{i \nu} \varsigma^{\#}{}} - e^{i \nu} \bar{\eta} \varsigma^{\#}{}\Bigr) \mathring{\beta}
- \Bigl( \overline{G^{\#}_{1}} + \frac{e^{i \nu} \bar{\eta} \overline{G^{\#}_{2}}}{2 \varsigma^{\#}{}} - \frac{\eta \varsigma^{\#}{}}{e^{i \nu}}\Bigr) \mathring{\beta}'
+ (\eta \bar{\eta} + \tfrac{1}{2} G^{\#}_{0}) \mathring{\epsilon}',\\
\tilde{\epsilon}'={}&\epsilon ' - \mathring{\epsilon}',\\
\tilde{\kappa}={}&\kappa ,\\
\tilde{\kappa}'={}&\kappa ',\\
\tilde{\rho}={}&\rho - \mathring{\rho},\\
\tilde{\rho}'={}&\rho ' - \mathring{\rho}',\\
\tilde{\sigma}={}&\sigma ,\\
\tilde{\sigma}'={}&\sigma ',\\
\tilde{\tau}={}&\tau - \mathring{\tau},\\
\tilde{\tau}'={}&\tau ' - \mathring{\tau}'.
\end{align}
\end{subequations}
Define the \defn{differential curvature coefficients} as
\begin{align}
\tilde{\Psi}_{0}{}={}&\Psi_{0}{},&
\tilde{\Psi}_{1}{}={}&\Psi_{1}{},&
\tilde{\Psi}_{2}{}={}&\Psi_{2}{} - \mathring{\Psi}_{2}{},&
\tilde{\Psi}_{3}{}={}&\Psi_{3}{},&
\tilde{\Psi}_{4}{}={}&\Psi_{4}{}.
\end{align}
The \defn{geometric variables} are defined to be
\begin{align*}
\mathfrak{u}=&{}(
\eta,\nu,
\tilde{G}^{\#}_2{}, \tilde{\slashed{G}}{}^{\#}_{\phantom{0}}{}, \tilde{G}^{\#}_1{}, \tilde{G}^{\#}_0{},
\tilde{\sigma}', \tilde{\rho}', \tilde{\tau}', \tilde{\beta},\tilde{\beta}', \tilde{\epsilon}, \tilde{\rho}, \tilde{\sigma}, \tilde{\kappa},
\tilde{\Psi}_{0},\tilde{\Psi}_{1},\tilde{\Psi}_{2},\tilde{\Psi}_{3},\tilde{\Psi}_{4}
)^T.
\end{align*}
\end{definition}
The differential variables are chosen so that they are properly weighted with respect to the background tetrad. This may initially seem surprising, since $\mathring{\beta},\mathring{\epsilon},\mathring{\beta}',\mathring{\epsilon}'$ are not. It may be helpful to recall this is similar to the fact that the Christoffel symbols for a connection do not transform as a tensor, but the difference between the Christoffel symbols for two different connections does transform as a tensor.
This choice of variables is not unique, and not all of them are properly weighted under differential Lorentz transformations. However, they are the simplest choices of variables that are properly weighted under spin and boost transformations of the background tetrad. We are going to use the differential Lorentz transformations to eliminate some of the differential spin coefficients. This would have been impossible if they were properly weighted under the differential Lorentz transformations.
\begin{definition}
Assume the \radiationGaugeHypotheses{} and a choice of differential Lorentz transformation variables.
Define the \defn{foreground GHP operators} acting on a $(p,q)$-weighted scalar $\varphi$ to be
\begin{subequations}
\begin{align}
\tho \varphi ={}&\thoBG \varphi
+ (\eta \bar{\eta}
+ \tfrac{1}{2} G^{\#}_{0}) \thopBG \varphi
- \Bigl(G^{\#}_{1}
+ \frac{\eta G^{\#}_{2}}{2 e^{i \nu} \varsigma^{\#}{}}
- e^{i \nu} \bar{\eta} \varsigma^{\#}{}\Bigr) \edtBG \varphi
- \Bigl(\overline{G^{\#}_{1}}
+ \frac{e^{i \nu} \bar{\eta} \overline{G^{\#}_{2}}}{2 \varsigma^{\#}{}}
- \frac{\eta \varsigma^{\#}{}}{e^{i \nu}}\Bigr) \edtpBG \varphi\nonumber\\
& - p \tilde{\epsilon} \varphi
- q \overline{\tilde{\epsilon}} \varphi ,\\
\thop \varphi ={}&\thopBG \varphi
+ p \tilde{\epsilon}' \varphi
+ q \overline{\tilde{\epsilon}'} \varphi ,\\
\edt \varphi ={}&\eta \thopBG \varphi
+ e^{i \nu} \varsigma^{\#}{} \edtBG \varphi
- \frac{e^{i \nu} \overline{G^{\#}_{2}} \edtpBG \varphi}{2 \varsigma^{\#}{}}
- p \tilde{\beta} \varphi
+ q \overline{\tilde{\beta}'} \varphi ,\\
\edtp \varphi ={}&\bar{\eta} \thopBG \varphi
- \frac{G^{\#}_{2} \edtBG \varphi}{2 e^{i \nu} \varsigma^{\#}{}}
+ \frac{\varsigma^{\#}{} \edtpBG \varphi}{e^{i \nu}}
- q \overline{\tilde{\beta}} \varphi
+ p \tilde{\beta}' \varphi,
\end{align}
\end{subequations}
where $\thoBG$, $\thopBG$, $\edtBG$ and $\edtpBG$ are the classical GHP operators with respect to the background tetrad.
\end{definition}
\begin{remark}
Observe that we define weight to be with respect to the background tetrad. Any background spin and boost transformation will induce the same spin and boost transformation on the foreground tetrad. Hence, any quantity which is properly weighted with respect to the foreground tetrad will become properly weighted with the same weights with respect to the background tetrad, when we have tied the frames together as in definition~\ref{def:ForegroundFrame}.
For any quantity which is properly weighted with respect to the foreground tetrad, our definition corresponds to the classical definition of GHP operators. Our definition can therefore be seen as an extension to quantities which are weighted only in terms of the background tetrad.
\end{remark}
These GHP operators satisfy the commutator relations in appendix \ref{sec:commutators}.
\subsection{Structure equations}
We now choose $\eta$ and $\nu$ so that two differential spin coefficients are eliminated (in addition to $\kappa'$, which vanishes as a result of $n$ being tangent to null geodesics in the radiation gauge condition{}) and so that the remaining connection coefficients satisfy transport equations.
\begin{definition}[The frame-gauge hypotheses]
\label{def:frameGaugeHypotheses}
Assume the \radiationGaugeHypotheses.
The \defn{frame-gauge hypotheses} are defined to hold if there is a choice of differential Lorentz transformation variables satisfying
\begin{subequations}
\label{eq:nuetaevoleq}
\begin{align}
\thop \nu ={}&- \frac{i}{2 \varsigma} (\varsigma^{\#}{}
- \varsigma) (\mathring{\rho}'
- \bar{\mathring{\rho}}')
+ \frac{i \overline{\tilde{G}}{}^{\#}_2{} \tilde{\sigma}'}{4 \varsigma^2}
- \frac{i \tilde{G}^{\#}_2{} \overline{\tilde{\sigma}'}}{4 \varsigma^2}
\label{eq:nuevoleq} ,\\
\thop \eta ={}&\tilde{\beta}
- \overline{\tilde{\beta}'}
+ \eta (\mathring{\rho}'
+ \tilde{\rho}')
+ \bar{\eta} \overline{\tilde{\sigma}'}
- \mathring{\tau}
+ e^{i \nu} \varsigma^{\#}{} \mathring{\tau}
+ \frac{\overline{\tilde{G}}{}^{\#}_2{} \varsigma^{\#}{} \bar{\mathring{\tau}}}{2 e^{i \nu} \varsigma^2}.
\label{eq:etaevoleq}
\end{align}
\end{subequations}
\end{definition}
\begin{lemma}[Structure equations]
\label{lem:structureEquations}
Assume the \radiationGaugeHypotheses{} and \frameGaugeHypotheses.
The structure equations take the form of a transport system for the metric coefficients
\begin{subequations}
\label{eq:thopmetric}
\begin{align}
(\thop{} + \mathring{\rho}' + 2 \tilde{\rho}' - \bar{\mathring{\rho}}')\tilde{G}^{\#}_2{}={}&(2 + \tilde{\slashed{G}}{}^{\#}_{\phantom{0}}{}) \tilde{\sigma}',
\label{eq:ThopInvGTilde2}\\
(\thop{} + \tilde{\rho}' + \overline{\tilde{\rho}'})\tilde{\slashed{G}}{}^{\#}_{\phantom{0}}{}={}&-2 \tilde{\rho}'
- 2 \overline{\tilde{\rho}'}
+ 2 \overline{\tilde{G}}{}^{\#}_2{} \tilde{\sigma}'
+ 2 \tilde{G}^{\#}_2{} \overline{\tilde{\sigma}'},
\label{eq:ThopInvGTrTilde}\\
(\thop{} + \tilde{\rho}' + \bar{\mathring{\rho}}')\tilde{G}^{\#}_1{}={}&
2 \tilde{\tau}'
-\Bigl(\overline{\tilde{G}}{}^{\#}_1{} \tilde{G}^{\#}_2{} \frac{\varsigma^{\#}{}}{\varsigma} + 2 \tilde{G}^{\#}_1{} \varsigma^{\#}{} \varsigma + 2 \bar{\eta} \varsigma^2\Bigr) (\mathring{\rho}' - \bar{\mathring{\rho}}')
- \bar{\eta} (2 + \tilde{\slashed{G}}{}^{\#}_{\phantom{0}}{}) \bar{\mathring{\rho}}'
- \overline{\tilde{G}}{}^{\#}_1{} \tilde{\sigma}'\nonumber\\
& + \eta \tilde{G}^{\#}_2{} (\mathring{\rho}' + \bar{\mathring{\rho}}')
+ \tfrac{1}{2} \tilde{\slashed{G}}{}^{\#}_{\phantom{0}}{} (\bar{\mathring{\tau}} + \mathring{\tau}' + \tilde{\tau}')
- \tilde{G}^{\#}_2{} ( \mathring{\tau} + \overline{\tilde{\tau}'} + \bar{\mathring{\tau}}')
+\frac{e^{i \nu} \tilde{G}^{\#}_2{}\bar{\mathring{\tau}}'}{\varsigma}\nonumber\\
& + 2 (1 - \varsigma e^{-i \nu}) \mathring{\tau}',
\label{eq:ThopInvGTilde1}\\
\thop \tilde{G}^{\#}_0{}={}&-2 \tilde{\epsilon} - 2 \overline{\tilde{\epsilon}}
+ \bigl(\eta^2 \tilde{G}^{\#}_2{} + \bar{\eta}^2 \overline{\tilde{G}}{}^{\#}_2{} - \eta \bar{\eta} (2 + \tilde{\slashed{G}}{}^{\#}_{\phantom{0}}{})\bigr) (\mathring{\rho}' + \bar{\mathring{\rho}}') \nonumber\\
& - \bigl(2 \eta \overline{\tilde{G}}{}^{\#}_1{} \tilde{G}^{\#}_2{} - 2 \bar{\eta} \tilde{G}^{\#}_1{} \overline{\tilde{G}}{}^{\#}_2{} + (\eta \tilde{G}^{\#}_1{} - \bar{\eta} \overline{\tilde{G}}{}^{\#}_1{}) (2 + \tilde{\slashed{G}}{}^{\#}_{\phantom{0}}{})\bigr) \varsigma^{\#}{} \varsigma^{-1} (\mathring{\rho}' - \bar{\mathring{\rho}}')\nonumber\\
& + e^{-i \nu} (2 \overline{\tilde{G}}{}^{\#}_1{} \varsigma^{\#}{} + \tilde{G}^{\#}_1{} \overline{\tilde{G}}{}^{\#}_2{} \varsigma^{\#}{} \varsigma^{-2} + \bar{\eta} \overline{\tilde{G}}{}^{\#}_2{} \varsigma^{-1} - 2 \eta \varsigma) \mathring{\tau}'
- 2 \overline{\tilde{G}}{}^{\#}_1{} (\bar{\mathring{\tau}} + \mathring{\tau}' + \tilde{\tau}')\nonumber\\
& + e^{i \nu} (2 \tilde{G}^{\#}_1{} \varsigma^{\#}{} + \overline{\tilde{G}}{}^{\#}_1{} \tilde{G}^{\#}_2{} \varsigma^{\#}{} \varsigma^{-2} + \eta \tilde{G}^{\#}_2{} \varsigma^{-1} - 2 \bar{\eta} \varsigma) \bar{\mathring{\tau}}'
- 2 \tilde{G}^{\#}_1{} (\mathring{\tau} + \overline{\tilde{\tau}'} + \bar{\mathring{\tau}}'),
\label{eq:ThopInvGTilde0}
\end{align}
\end{subequations}
algebraic relations for the spin coefficients
\begin{subequations}
\label{eq:AlgebraicSpincoeff1}
\begin{align}
\tilde{\kappa}'={}&0,\qquad
\tilde{\epsilon}'={}0,\qquad
\tilde{\tau}={}0,\\
\tilde{\rho}' - \overline{\tilde{\rho}'}={}&\frac{(\varsigma^{\#}{} - \varsigma) (\mathring{\rho}' - \bar{\mathring{\rho}}')}{\varsigma},
\label{eq:algebraicrho1}\\
\tilde{\beta}-\overline{\tilde{\beta}'}={}&
- \frac{(\overline{\tilde{G}}{}^{\#}_1{} \varsigma^{\#}{} - \bar{\eta} \overline{\tilde{G}}{}^{\#}_2{} \varsigma^{\#}{} - \eta \varsigma + 2 \eta \varsigma^{\#}{} \varsigma^2) (\mathring{\rho}' - \bar{\mathring{\rho}}')}{\varsigma}
+ \frac{\overline{\tilde{G}}{}^{\#}_2{} \varsigma^{\#}{} \mathring{\tau}'}{2 e^{i \nu} \varsigma^2}
- \overline{\tilde{\tau}'}
+ (e^{i \nu} \varsigma^{\#}{} - 1) \bar{\mathring{\tau}}',
\label{eq:algebraicbeta1}
\end{align}
\end{subequations}
and a supplementary set of equations displayed in \eqref{eq:StructureSpincoeff1}.
\end{lemma}
\begin{proof}
The foreground Levi-Civita connection $\nabla$ and background $\mathring{\nabla}$ connections are related via
\begin{align}
\widetilde\Gamma^{a}{}_{bc}={}&\tfrac{1}{2} g^{\#}{}^{ad} (\mathring{\nabla}_{b}g_{cd} + \mathring{\nabla}_{c}g_{bd} - \mathring{\nabla}_{d}g_{bc}).
\end{align}
Definition~\ref{def:spincoeff} lets us express the foreground spin coefficients in terms of the foreground $\nabla$ acting on the foreground tetrad. We can re-express this in terms of the background $\mathring{\nabla}$ as
\begin{subequations}
\begin{align}
\beta ={}&- \tfrac{1}{2} m^{a} \bar{m}^{b} (- \widetilde\Gamma^{c}{}_{ab} m_{c} + \mathring{\nabla}_{a}m_{b})
- \tfrac{1}{2} l^{a} m^{b} (- \widetilde\Gamma^{c}{}_{ba} n_{c} + \mathring{\nabla}_{b}n_{a}),\\
\beta '={}&\tfrac{1}{2} \bar{m}^{a} \bar{m}^{b} (- \widetilde\Gamma^{c}{}_{ba} m_{c} + \mathring{\nabla}_{b}m_{a})
+ \tfrac{1}{2} l^{a} \bar{m}^{b} (- \widetilde\Gamma^{c}{}_{ba} n_{c} + \mathring{\nabla}_{b}n_{a}),\\
\epsilon ={}&- \tfrac{1}{2} l^{a} \bar{m}^{b} (- \widetilde\Gamma^{c}{}_{ab} m_{c} + \mathring{\nabla}_{a}m_{b})
- \tfrac{1}{2} l^{a} l^{b} (- \widetilde\Gamma^{c}{}_{ba} n_{c} + \mathring{\nabla}_{b}n_{a}),\\
\epsilon '={}&\tfrac{1}{2} \bar{m}^{a} n^{b} (- \widetilde\Gamma^{c}{}_{ba} m_{c} + \mathring{\nabla}_{b}m_{a})
+ \tfrac{1}{2} l^{a} n^{b} (- \widetilde\Gamma^{c}{}_{ba} n_{c} + \mathring{\nabla}_{b}n_{a}),\\
\kappa ={}&l^{a} m^{b} (- \widetilde\Gamma^{c}{}_{ab} l_{c} + \mathring{\nabla}_{a}l_{b}),\\
\kappa '={}&\bar{m}^{a} n^{b} (- \widetilde\Gamma^{c}{}_{ba} n_{c} + \mathring{\nabla}_{b}n_{a}),\\
\rho ={}&m^{a} \bar{m}^{b} (- \widetilde\Gamma^{c}{}_{ba} l_{c} + \mathring{\nabla}_{b}l_{a}),\\
\rho '={}&m^{a} \bar{m}^{b} (- \widetilde\Gamma^{c}{}_{ab} n_{c} + \mathring{\nabla}_{a}n_{b}),\\
\sigma ={}&m^{a} m^{b} (- \widetilde\Gamma^{c}{}_{ab} l_{c} + \mathring{\nabla}_{a}l_{b}),\\
\sigma '={}&\bar{m}^{a} \bar{m}^{b} (- \widetilde\Gamma^{c}{}_{ba} n_{c} + \mathring{\nabla}_{b}n_{a}),\\
\tau ={}&m^{a} n^{b} (- \widetilde\Gamma^{c}{}_{ba} l_{c} + \mathring{\nabla}_{b}l_{a}),\\
\tau '={}&l^{a} \bar{m}^{b} (- \widetilde\Gamma^{c}{}_{ab} n_{c} + \mathring{\nabla}_{a}n_{b}).
\end{align}
\end{subequations}
Using the relation between the background and foreground tetrads, and expressing all background derivatives of background frame components in terms of background spin coefficients, we get expressions of all background tetrad components of $\widetilde\Gamma^{a}{}_{bc}$ in terms of the metric components $(G_{2}, G_{1}, G_{0}, \slashed{G}_{})$ or $(\tilde{\slashed{G}}{}^{\#}_{\phantom{0}}{}, \tilde{G}^{\#}_2{}, \tilde{G}^{\#}_1{}, \tilde{G}^{\#}_0{})$. See \eqref{eq:GammaComponents} below for explicit expressions. Putting it all together, we can express all differential spin coefficients in terms of background spin coefficients and GHP derivatives of the above mentioned metric components.
For instance, we get
\begin{subequations}
\label{eq:rhoTildePrimeSigmaTildePrime}
\begin{align}
\tilde{\rho}'={}&\frac{1}{2 \varsigma} (\varsigma^{\#}{} - \varsigma) (\mathring{\rho}' - \bar{\mathring{\rho}}')
- \frac{\varsigma^{\#}{} \thopBG \slashed{G}_{}}{4 \varsigma} (2 \varsigma^{\#}{} \varsigma - 1)
+ \frac{G_{2} \varsigma^{\#}{}^2 \thopBG \overline{G_{2}}}{4 \varsigma^2}
+ \frac{\overline{G_{2}} \varsigma^{\#}{}^2 \thopBG G_{2}}{4 \varsigma^2} ,\\
\tilde{\sigma}'={}&\frac{G_{2} \varsigma^{\#}{}}{2 e^{2i \nu} \varsigma} (\mathring{\rho}' - \bar{\mathring{\rho}}')
- \frac{G_{2} \varsigma^{\#}{}^2 \thopBG \slashed{G}_{}}{4 e^{2i \nu} \varsigma^2}
+ \frac{G_{2}{}^2 \varsigma^{\#}{}^2 \thopBG \overline{G_{2}}}{8 e^{2i \nu} \varsigma^4}
+ \frac{\varsigma^{\#}{}^2 \thopBG G_{2}}{2 e^{2i \nu}} .
\end{align}
\end{subequations}
This is equivalent to
\begin{subequations}
\label{eq:ThopBGG2GTr}
\begin{align}
\thopBG \slashed{G}_{}={}&- (2 + \slashed{G}_{}) (\tilde{\rho}' + \overline{\tilde{\rho}'})
+ 2 e^{2i \nu} \overline{G_{2}} \tilde{\sigma}'
+ \frac{2 G_{2} \overline{\tilde{\sigma}'}}{e^{2i \nu}} ,\\
\thopBG G_{2}={}&G_{2} (\bar{\mathring{\rho}}' - \mathring{\rho}' - \tilde{\rho}' - \overline{\tilde{\rho}'})
+ 2 e^{2i \nu} \varsigma^2 \tilde{\sigma}'
+ \frac{G_{2}{}^2 \overline{\tilde{\sigma}'}}{2 e^{2i \nu} \varsigma^2} ,\\
\overline{\tilde{\rho}'}={}&\tilde{\rho}'
- \frac{1}{\varsigma} (\varsigma^{\#}{} - \varsigma) (\mathring{\rho}' - \bar{\mathring{\rho}}').
\end{align}
\end{subequations}
Similarly, we get
\begin{subequations}
\begin{align}
\tilde{\epsilon}'={}&- \frac{\slashed{G}_{} \varsigma^{\#}{} \mathring{\rho}'}{8 \varsigma}
+ \frac{\slashed{G}_{} \varsigma^{\#}{} \bar{\mathring{\rho}}'}{8 \varsigma}
- \tfrac{1}{2} i \thopBG \nu
+ \frac{G_{2} \varsigma^{\#}{} \thopBG \overline{G_{2}}}{16 \varsigma^3}
- \frac{\overline{G_{2}} \varsigma^{\#}{} \thopBG G_{2}}{16 \varsigma^3} \\
={}&\frac{1}{4 \varsigma} (\varsigma^{\#}{} - \varsigma) (\mathring{\rho}' - \bar{\mathring{\rho}}')
- \frac{e^{2i \nu} \overline{G_{2}} \tilde{\sigma}'}{8 \varsigma^2}
+ \frac{G_{2} \overline{\tilde{\sigma}'}}{8 e^{2i \nu} \varsigma^2}
- \tfrac{1}{2} i \thopBG \nu .
\end{align}
\end{subequations}
As we have not yet fixed the differential spin rotation parameter $\nu$, we can use it to set $\tilde{\epsilon}'=0$. Translated into the foreground operators and $\tilde{G}^{\#}_2$, this condition is equivalent to the evolution equation \eqref{eq:nuevoleq} in the frame-gauge hypotheses.
Using this relation one can express the system \eqref{eq:ThopBGG2GTr} in terms of $\tilde{G}^{\#}_2$ and $\tilde{\slashed{G}}{}^{\#}_{\phantom{0}}{}$ to get \eqref{eq:ThopInvGTilde2}, \eqref{eq:ThopInvGTrTilde} and \eqref{eq:algebraicrho1}.
A similar calculation for $\tilde{\tau}'$ yields the following after reduction with \eqref{eq:ThopBGG2GTr}
\begin{align}
\tilde{\tau}'={}&\frac{\mathring{\rho}'}{4 e^{i \nu} \varsigma^2} (\overline{G_{1}} G_{2} \varsigma^{\#}{} + 4 e^{i \nu} \bar{\eta} \varsigma^2 + 4 G_{1} \varsigma^{\#}{} \varsigma^2)
+ \frac{\tilde{\rho}'}{2 e^{i \nu} \varsigma^2} (\overline{G_{1}} G_{2} \varsigma^{\#}{} + 2 e^{i \nu} \bar{\eta} \varsigma^2 + 2 G_{1} \varsigma^{\#}{} \varsigma^2)
- \frac{G_{1} \varsigma^{\#}{} \bar{\mathring{\rho}}'}{2 e^{i \nu}}\nonumber\\
& + \frac{\tilde{\sigma}'}{2 \varsigma^2} (e^{i \nu} \overline{G_{2}} G_{1} \varsigma^{\#}{} + 2 \eta \varsigma^2 + 2 e^{i \nu} \overline{G_{1}} \varsigma^{\#}{} \varsigma^2)
+ \frac{G_{2} \mathring{\tau}}{4 e^{i \nu} \varsigma^2} (\varsigma^{\#}{} + \varsigma)
+ \frac{\bar{\mathring{\tau}}}{2 e^{i \nu}} (\varsigma^{\#}{} - \varsigma)\nonumber\\
& - \frac{\mathring{\tau}'}{2 e^{i \nu}} (2 e^{i \nu} - \varsigma^{\#}{} - \varsigma)
+ \frac{G_{2} \bar{\mathring{\tau}}'}{4 e^{i \nu} \varsigma^2} (\varsigma^{\#}{} - \varsigma)
+ \frac{G_{2} \varsigma^{\#}{} \thopBG \overline{G_{1}}}{4 e^{i \nu} \varsigma^2}
+ \frac{\varsigma^{\#}{} \thopBG G_{1}}{2 e^{i \nu}} .
\end{align}
This equation can be used to solve for $\thopBG G_{1}$.
Similarly $\tilde{\tau}$ can be expressed as follows after substitution of the expressions for $\thopBG \slashed{G}_{}$, $\thopBG G_{2}$, $\thopBG G_{1}$ and $\thopBG \nu$ above
\begin{align}
\tilde{\tau}={}&- \frac{e^{i \nu} \varsigma^{\#}{}^2}{2 \varsigma^3} (\overline{G_{2}} G_{1} + 2 \overline{G_{1}} \varsigma^2) (\mathring{\rho}' - \bar{\mathring{\rho}}')
- \eta (2 \tilde{\epsilon}' - \overline{\tilde{\rho}'} - \bar{\mathring{\rho}}')
+ \bar{\eta} \overline{\tilde{\sigma}'}
+ \frac{e^{i \nu} \overline{G_{2}} \varsigma^{\#}{}}{2 \varsigma^2} (\bar{\mathring{\tau}} + \mathring{\tau}')
- \overline{\tilde{\tau}'}\nonumber\\
& + (e^{i \nu} \varsigma^{\#}{} - 1) (\mathring{\tau} + \bar{\mathring{\tau}}')
- \thopBG \eta .
\label{eq:tauexpr1}
\end{align}
In the frame-gauge hypotheses of definition \ref{def:frameGaugeHypotheses}, equation \eqref{eq:etaevoleq} was chosen so that $\tilde{\tau}=0$. This gives an expression for $\thopBG \eta$. Using this in the expression for $\tilde{G}{}^{\#}_1$, we can derive the evolution equation for $\tilde{G}{}^{\#}_1$, i.e. \eqref{eq:ThopInvGTilde1}.
Using all the previous relations, one can express $\tilde{\beta}-\overline{\tilde{\beta}'}$ as
\begin{align}
\tilde{\beta} - \overline{\tilde{\beta}'}={}&- \frac{\varsigma^{\#}{}}{2 \varsigma^3} (e^{i \nu} \overline{G_{2}} G_{1} \varsigma^{\#}{} + 2 \eta \varsigma^2 + 2 e^{i \nu} \overline{G_{1}} \varsigma^{\#}{} \varsigma^2) (\mathring{\rho}' - \bar{\mathring{\rho}}')
+ \frac{e^{i \nu} \overline{G_{2}} \varsigma^{\#}{} \mathring{\tau}'}{2 \varsigma^2}
- \overline{\tilde{\tau}'}
- \bar{\mathring{\tau}}'
+ e^{i \nu} \varsigma^{\#}{} \bar{\mathring{\tau}}'.
\end{align}
Using this relation, we can eliminate $G_{1}$ from \eqref{eq:tauexpr1} to obtain the evolution \eqref{eq:etaevoleq}. Translating to the $\tilde{G}{}^{\#}_i$ variables, we also get \eqref{eq:algebraicbeta1}.
Similarly, using the previous relations, we get a long expression
\begin{align}
\tilde{\epsilon} + \overline{\tilde{\epsilon}}={}&- \tfrac{1}{2} \thopBG G_{0} + \dots
\end{align}
where the dots indicates an expression depending on $G_{1}, G_{2}, \slashed{G}_{}, \tilde{\sigma }', \tilde{\tau }', \tilde{\rho }'$ and the background spin coefficients. Translating this to the $\tilde{G}{}^{\#}_i$ variables, we get \eqref{eq:ThopInvGTilde0}.
Similarly, one can express $\tilde{\beta} + \overline{\tilde{\beta}'}$ and $\tilde{\epsilon} - \overline{\tilde{\epsilon}}$ to obtain \eqref{eq:ethnuTobeta} and \eqref{eq:thoetaToepsilon}. Here however, the $\tilde{G}{}_i$ variables turned out to give shorter expressions, so we used them instead.
The remaining equations in \eqref{eq:StructureSpincoeff1} were derived in the same way using the expressions for $\tilde{\rho}$, $\tilde{\sigma}$ and $\tilde{\kappa}$. As the direct expressions for these spin coefficients became long and complicated, we found that solving for the left hand sides of \eqref{eq:StructureSpincoeff1} gave us shorter expressions. The expressions can be inverted though, so all spin coefficients are expressible in terms of derivatives of $\nu$, $\eta$ and the metric components.
\end{proof}
\subsection{Ricci relations}
\begin{lemma}[Ricci relations]
Assume the \radiationGaugeHypotheses{} and \frameGaugeHypotheses.
The Ricci relations take the form
\begin{subequations}
\label{eq:thopRicci}
\begin{align}
(\thop{} - \mathring{\rho}' - \tilde{\rho}' - \overline{\tilde{\rho}'} - \bar{\mathring{\rho}}')\tilde{\sigma}'={}&\tilde{\Psi}_{4}{},\\
(\thop{} - 2 \mathring{\rho}' - \tilde{\rho}')\tilde{\rho}'={}&\tilde{\sigma}' \overline{\tilde{\sigma}'},
\label{eq:transport:rhoTildePrime}\\
(\thop{} - \mathring{\rho}' - \tilde{\rho}')\tilde{\tau}'={}&\tilde{\Psi}_{3}{}
+ (\overline{\tilde{\tau}'} - \mathring{\tau} + \bar{\mathring{\tau}}') \tilde{\sigma}'
+ \tilde{\rho}' (- \bar{\mathring{\tau}} + \mathring{\tau}'),\\
(\thop{} - \mathring{\rho}' - \tilde{\rho}')\tilde{\beta}={}&- \tilde{\beta}' \overline{\tilde{\sigma}'}
+ (1 - e^{i \nu} \varsigma^{\#}{}) \mathring{\rho}' \mathring{\tau}
+ \tilde{\rho}' \mathring{\tau},
\label{eq:thopbeta}\\
(\thop{} - \overline{\tilde{\rho}'} - \bar{\mathring{\rho}}')\tilde{\beta}'={}&\tilde{\Psi}_{3}{}
- \tilde{\beta} \tilde{\sigma}'
- \frac{G^{\#}_{2} \mathring{\rho}' \mathring{\tau}}{2 e^{i \nu} \varsigma^{\#}{}}
- \tilde{\sigma}' \mathring{\tau},
\label{eq:thopbetap}\\
\thop \tilde{\epsilon}={}&- \tilde{\Psi}_{2}{}
+ (\tilde{\tau}' - \bar{\mathring{\tau}} + \mathring{\tau}') \tilde{\beta}
+ \Bigl(G^{\#}_{1} + \frac{\eta G^{\#}_{2}}{2 e^{i \nu} \varsigma^{\#}{}} - e^{i \nu} \bar{\eta} \varsigma^{\#}{}\Bigr) \mathring{\rho}' \mathring{\tau}
+ \mathring{\tau} \tilde{\tau}'\nonumber\\
& + \tilde{\beta}' (\mathring{\tau} - \overline{\tilde{\tau}'} - \bar{\mathring{\tau}}'),\\
(\thop{} - \overline{\tilde{\rho}'} - \bar{\mathring{\rho}}')\tilde{\rho}={}&- \tilde{\Psi}_{2}{}
+ \mathring{\rho} \overline{\tilde{\rho}'}
+ \tilde{\sigma} \tilde{\sigma}'
+ (\overline{\tilde{\beta}} + \tilde{\beta}') \mathring{\tau}
+ 2 \bar{\eta} \mathring{\rho}' \mathring{\tau}
- \frac{G^{\#}_{2} \mathring{\tau}^2}{2 e^{i \nu} \varsigma^{\#}{}}
- \Bigl(1 - \frac{\varsigma^{\#}{}}{e^{i \nu}}\Bigr) \edtpBG \mathring{\tau},\\
(\thop{} - \mathring{\rho}' - \tilde{\rho}')\tilde{\sigma}={}&(\mathring{\rho} + \tilde{\rho}) \overline{\tilde{\sigma}'}
- (\tilde{\beta} + \overline{\tilde{\beta}'}) \mathring{\tau}
+ 2 \eta \mathring{\rho}' \mathring{\tau}
- (1 - e^{i \nu} \varsigma^{\#}{}) \mathring{\tau}^2
- \frac{e^{i \nu} \overline{G^{\#}_{2}} \edtpBG \mathring{\tau}}{2 \varsigma^{\#}{}} ,\\
\thop \tilde{\kappa}={}&- \tilde{\Psi}_{1}{}
+ (\overline{\tilde{\tau}'} - \mathring{\tau} + \bar{\mathring{\tau}}') \tilde{\rho}
+ (\tilde{\tau}' - \bar{\mathring{\tau}} + \mathring{\tau}') \tilde{\sigma}
- (\tilde{\epsilon} - \overline{\tilde{\epsilon}}) \mathring{\tau}
+ (2 \eta \bar{\eta} + G^{\#}_{0}) \mathring{\rho}' \mathring{\tau}\nonumber\\
& - \Bigl(G^{\#}_{1} + \frac{\eta G^{\#}_{2}}{2 e^{i \nu} \varsigma^{\#}{}} - e^{i \nu} \bar{\eta} \varsigma^{\#}{}\Bigr) \mathring{\tau}^2
+ \mathring{\rho} \overline{\tilde{\tau}'}
- \Bigl(\overline{G^{\#}_{1}} + \frac{e^{i \nu} \bar{\eta} \overline{G^{\#}_{2}}}{2 \varsigma^{\#}{}} - \frac{\eta \varsigma^{\#}{}}{e^{i \nu}}\Bigr) \edtpBG \mathring{\tau},
\end{align}
\end{subequations}
together with the supplementary relations \eqref{eq:ExtraRicci}. Here $G^{\#}_{i}$ can be interpreted in terms of $\tilde{G}{}^{\#}_{i}$ via \eqref{eq:InvGTildeInvGRelations} and there is the background formula
\begin{align}
\label{eq:edtpBGtauBG}
\edtpBG \mathring{\tau}={}&\tfrac{1}{2} \mathring{\Psi}_{2}{}
- \frac{\bar{\mathring{\Psi}}_{2}{} \bar{\kappa}_{1'}{}}{2 \kappa_{1}{}}
+ \mathring{\rho} \mathring{\rho}'
- \mathring{\rho} \bar{\mathring{\rho}}'
+ \mathring{\tau} \mathring{\tau}',
\end{align}
where
\begin{align}
\label{eq:defKappa1}
\kappa_{1}={}&-\frac13(r-ia\cos\theta) .
\end{align}
\end{lemma}
\begin{proof}
To prove these relations, we begin with the Newman-Penrose (NP) version of the Ricci relations equations (4.11.12) in \cite{PenroseRindler} for both the foreground spin coefficients and operators. The foreground spin coefficients can then be written in terms of the differential spin coefficients from definition \ref{def:DiffSpinCoeff}.
When the foreground NP operators acts on the background spin coefficients, we express the operators in terms of the background operators, via the relations in definition~\ref{def:ForegroundFrame}. The resulting background NP operators acting on background spin coefficients can then be eliminated using the background Ricci relations. After this procedure, all non-properly weighted quantities have been eliminated, and the operators can be translated into the foreground GHP operators yielding \eqref{eq:thopRicci} and \eqref{eq:ExtraRicci} after reduction with the structure equations \eqref{eq:nuetaevoleq}, \eqref{eq:thopmetric}, \eqref{eq:AlgebraicSpincoeff1}, \eqref{eq:StructureSpincoeff1}. Here some background derivatives of background spin coefficients have been simplified due to the vacuum Bianchi type D property of the Kerr spacetime.
As an example we derive \eqref{eq:thopbetap} starting with the foreground NP-Ricci relation
\begin{align}
\Delta \beta '={}&\Psi_{3}{}
+ \bar{\beta} \epsilon '
- \beta ' \overline{\epsilon '}
+ \beta ' \overline{\rho '}
- \beta \sigma '
- \sigma ' \tau
- \epsilon ' \bar{\tau}
+ \bar\delta \epsilon '.
\end{align}
Translating to the differential spin coefficients and expressing the foreground NP derivatives in terms of the background NP derivatives when acting on background spin coefficients, we get
\begin{align}
\Delta \tilde{\beta}'={}&\tilde{\Psi}_{3}{}
+ \overline{\tilde{\beta}} \tilde{\epsilon}'
+ e^{-i \nu} \varsigma^{\#}{} \bar{\mathring{\beta}} (\mathring{\epsilon}'
+ \tilde{\epsilon}')
+ \tfrac{1}{2} e^{-i \nu} G^{\#}_{2} \varsigma^{\#}{}^{-1} \bar{\mathring{\beta}}' (\mathring{\epsilon}'
+ \tilde{\epsilon}')
- \tilde{\beta}' \overline{\tilde{\epsilon}'}
- ( \tilde{\beta}'
+ \bar{\eta} \tilde{\epsilon}') \bar{\mathring{\epsilon}}'
+ \tilde{\beta}' \overline{\tilde{\rho}'}
+ \bar\delta \tilde{\epsilon}'\nonumber\\
& + \tilde{\beta}' \bar{\mathring{\rho}}'
- \tilde{\beta} \tilde{\sigma}'
- \tilde{\sigma}' \mathring{\tau}
- \tilde{\sigma}' \tilde{\tau}
- \tilde{\epsilon}' \overline{\tilde{\tau}}
- \tilde{\epsilon}' \bar{\mathring{\tau}}
- \tfrac{1}{2} e^{-i \nu} G^{\#}_{2} \varsigma^{\#}{}^{-1} (\mathring{\Delta} \mathring{\beta}
+ \mathring{\delta} \mathring{\epsilon}')
- e^{-i \nu} \varsigma^{\#}{} ( \mathring{\Delta} \mathring{\beta}'
- \mathring{\bar\delta} \mathring{\epsilon}')\nonumber\\
&- \mathring{\beta} \bigl( e^{i \nu} \varsigma^{\#}{} \tilde{\sigma}'
+ \tfrac{1}{2} e^{-i \nu} \varsigma^{\#}{}^{-1} \Delta G^{\#}_{2}
- \tfrac{1}{2} e^{-i \nu} G^{\#}_{2} \varsigma^{\#}{}^{-1} (\varsigma^{\#}{}^{-1} \Delta \varsigma^{\#}{} + i \Delta \nu - \overline{\tilde{\epsilon}'} - \bar{\mathring{\epsilon}}' + \overline{\tilde{\rho}'} + \bar{\mathring{\rho}}')\bigr)
\nonumber\\
&+ \mathring{\beta}' \bigl(
e^{-i \nu} \varsigma^{\#}{} (i \Delta \nu - \overline{\tilde{\epsilon}'} - \bar{\mathring{\epsilon}}' + \overline{\tilde{\rho}'} + \bar{\mathring{\rho}}' )
- \tfrac{1}{2} e^{i \nu} \overline{G^{\#}_{2}} \varsigma^{\#}{}^{-1} \tilde{\sigma}'
- e^{-i \nu} \Delta \varsigma^{\#}{}\bigr)
\nonumber\\
& + \mathring{\epsilon}' (\overline{\tilde{\beta}}
- \bar{\eta} \overline{\tilde{\epsilon}'}
- 2 \bar{\eta} \bar{\mathring{\epsilon}}'
+ \bar{\eta} \overline{\tilde{\rho}'}
+ \bar{\eta} \bar{\mathring{\rho}}'
+ \eta \tilde{\sigma}'
- \overline{\tilde{\tau}}
- \bar{\mathring{\tau}}
- \Delta \bar{\eta}) .
\end{align}
Using the background Ricci relations, transforming the foreground NP derivatives into foreground GHP operators, and translating the metric coefficients to the $\tilde{G}{}^{\#}_i$ variables yield
\begin{align}
\thop \tilde{\beta}'={}&\tilde{\Psi}_{3}{}
+ \tilde{\beta}' \overline{\tilde{\rho}'}
+ \tilde{\beta}' \bar{\mathring{\rho}}'
- \tilde{\beta} \tilde{\sigma}'
+ \tfrac{1}{2} e^{i \nu} \tilde{G}^{\#}_2{} \varsigma^{\#}{} \varsigma^{-2} \mathring{\rho}' \mathring{\tau}
- \tilde{\sigma}' \mathring{\tau}
- \tilde{\sigma}' \tilde{\tau}
- \tilde{\epsilon}' \overline{\tilde{\tau}}
- \tilde{\epsilon}' \bar{\mathring{\tau}}
+ \edtp \tilde{\epsilon}'
\nonumber\\
&+ \mathring{\epsilon}' (\overline{\tilde{\beta}}
- \tilde{\beta}'
+ \bar{\eta} \tilde{\epsilon}'
+ \bar{\eta} \overline{\tilde{\epsilon}'}
+ \bar{\eta} \overline{\tilde{\rho}'}
+ \bar{\eta} \bar{\mathring{\rho}}'
+ \eta \tilde{\sigma}'
+ \tfrac{1}{2} e^{i \nu} \tilde{G}^{\#}_2{} \varsigma^{\#}{} \varsigma^{-2} \mathring{\tau}
- \overline{\tilde{\tau}}
- \bar{\mathring{\tau}}
+ e^{-i \nu} \varsigma^{\#}{} \bar{\mathring{\tau}}
- \thop \bar{\eta})
\nonumber\\
& + \mathring{\beta}' (e^{-i \nu} \varsigma^{\#}{} \tilde{\epsilon}'
- e^{-i \nu} \varsigma^{\#}{} \overline{\tilde{\epsilon}'}
+ e^{-i \nu} \varsigma^{\#}{} \overline{\tilde{\rho}'}
+ \tfrac{1}{2} e^{-i \nu} \overline{\tilde{G}}^{\#}_2{} \varsigma^{\#}{} \varsigma^{-2} \tilde{\sigma}'
+ i e^{-i \nu} \varsigma^{\#}{} \thop \nu
- e^{-i \nu} \thop \varsigma^{\#}{})
\nonumber\\
& + \mathring{\beta} \bigl(- e^{i \nu} \varsigma^{\#}{} \tilde{\sigma}'
+ \tfrac{1}{2} e^{i \nu} \varsigma^{\#}{} \varsigma^{-2} \thop \tilde{G}^{\#}_2{}
+ \tfrac{1}{2} e^{i \nu} \tilde{G}^{\#}_2{} \varsigma^{-2} \thop \varsigma^{\#}{}
+ \tfrac{1}{2} e^{i \nu} \tilde{G}^{\#}_2{} \varsigma^{\#}{} \varsigma^{-2} (\tilde{\epsilon}'
- \overline{\tilde{\epsilon}'}
+ \mathring{\rho}'
- \bar{\mathring{\rho}}'
\nonumber\\
&\qquad
- \overline{\tilde{\rho}'}
+ i \thop \nu
- 2 \varsigma^{-1} \thop \varsigma)\bigr).
\end{align}
The evolution equations \eqref{eq:nuetaevoleq} together with the structure equations \eqref{eq:thopmetric} and \eqref{eq:AlgebraicSpincoeff1} will reduce this to \eqref{eq:thopbetap}. Observe that the equation \eqref{eq:thopbetap} is properly weighted even though we started with non-properly weighted equation.
The other equations can be derived in the same way.
\end{proof}
\subsection{Bianchi system}
\begin{lemma}[Bianchi identities]
Assume the \radiationGaugeHypotheses{} and \frameGaugeHypotheses.
The Bianchi identities take the form
\begin{subequations}
\label{eq:Bianchi}
\begin{align}
0={}&(\tho{} - 4 \mathring{\rho} - 4 \tilde{\rho})\tilde{\Psi}_{1}{}
- (\edtp{} - \mathring{\tau}' - \tilde{\tau}')\tilde{\Psi}_{0}{}
+ 3 (\mathring{\Psi}_{2}{} + \tilde{\Psi}_{2}{}) \tilde{\kappa},\\
0={}&(\tho{} - 3 \mathring{\rho} - 3 \tilde{\rho})\tilde{\Psi}_{2}{}
- (\edtp{} - 2 \mathring{\tau}' - 2 \tilde{\tau}')\tilde{\Psi}_{1}{}
+ 2 \tilde{\Psi}_{3}{} \tilde{\kappa}
- 3 \mathring{\Psi}_{2}{} \tilde{\rho}
+ \tfrac{3}{2} (2 \eta \bar{\eta} + G^{\#}_{0}) \mathring{\Psi}_{2}{} \mathring{\rho}'
- \tilde{\Psi}_{0}{} \tilde{\sigma}'\nonumber\\
& - 3 \Bigl(G^{\#}_{1} + \frac{\eta G^{\#}_{2}}{2 e^{i \nu} \varsigma^{\#}{}} - e^{i \nu} \bar{\eta} \varsigma^{\#}{}\Bigr) \mathring{\Psi}_{2}{} \mathring{\tau}
- 3 \Bigl(\overline{G^{\#}_{1}} + \frac{e^{i \nu} \bar{\eta} \overline{G^{\#}_{2}}}{2 \varsigma^{\#}{}} - \frac{\eta \varsigma^{\#}{}}{e^{i \nu}}\Bigr) \mathring{\Psi}_{2}{} \mathring{\tau}',\\
0={}&(\tho{} - 2 \mathring{\rho} - 2 \tilde{\rho})\tilde{\Psi}_{3}{}
- (\edtp{} - 3 \mathring{\tau}' - 3 \tilde{\tau}')\tilde{\Psi}_{2}{}
+ \tilde{\Psi}_{4}{} \tilde{\kappa}
- 3 \bar{\eta} \mathring{\Psi}_{2}{} \mathring{\rho}'
- 2 \tilde{\Psi}_{1}{} \tilde{\sigma}'
+ \frac{3 G^{\#}_{2} \mathring{\Psi}_{2}{} \mathring{\tau}}{2 e^{i \nu} \varsigma^{\#}{}}\nonumber\\
& + 3 \Bigl(1 - \frac{\varsigma^{\#}{}}{e^{i \nu}}\Bigr) \mathring{\Psi}_{2}{} \mathring{\tau}'
+ 3 \mathring{\Psi}_{2}{} \tilde{\tau}',\\
0={}&(\tho{} - \mathring{\rho} - \tilde{\rho})\tilde{\Psi}_{4}{}
- (\edtp{} - 4 \mathring{\tau}' - 4 \tilde{\tau}')\tilde{\Psi}_{3}{}
- 3 (\mathring{\Psi}_{2}{} + \tilde{\Psi}_{2}{}) \tilde{\sigma}',\\
0={}&(\thop{} - \mathring{\rho}' - \tilde{\rho}')\tilde{\Psi}_{0}{}
- (\edt{} - 4 \mathring{\tau})\tilde{\Psi}_{1}{}
- 3 (\mathring{\Psi}_{2}{} + \tilde{\Psi}_{2}{}) \tilde{\sigma},\\
0={}&(\thop{} - 2 \mathring{\rho}' - 2 \tilde{\rho}')\tilde{\Psi}_{1}{}
- (\edt{} - 3 \mathring{\tau})\tilde{\Psi}_{2}{}
- 3 \eta \mathring{\Psi}_{2}{} \mathring{\rho}'
- 2 \tilde{\Psi}_{3}{} \tilde{\sigma}
+ 3 (1 - e^{i \nu} \varsigma^{\#}{}) \mathring{\Psi}_{2}{} \mathring{\tau}
+ \frac{3 e^{i \nu} \overline{G^{\#}_{2}} \mathring{\Psi}_{2}{} \mathring{\tau}'}{2 \varsigma^{\#}{}} ,\\
0={}&(\thop{} - 3 \mathring{\rho}' - 3 \tilde{\rho}')\tilde{\Psi}_{2}{}
- (\edt{} - 2 \mathring{\tau})\tilde{\Psi}_{3}{}
- 3 \mathring{\Psi}_{2}{} \tilde{\rho}'
- \tilde{\Psi}_{4}{} \tilde{\sigma},\\
0={}&(\thop{} - 4 \mathring{\rho}' - 4 \tilde{\rho}')\tilde{\Psi}_{3}{}
- (\edt{} - \mathring{\tau})\tilde{\Psi}_{4}{}.
\end{align}
\end{subequations}
Here $G^{\#}_{i}$ can be interpreted in terms of $\tilde{G}{}^{\#}_{i}$ via \eqref{eq:InvGTildeInvGRelations}.
\end{lemma}
\begin{proof}
A direct translation of the standard GHP Bianchi identities in \cite{GHP} to our differential variables gives the relations \eqref{eq:Bianchi}. Here we have also used the background type D Bianchi identities to handle the derivatives of $\mathring{\Psi}_{2}$.
\end{proof}
\begin{remark}
It is important to note that the full set of equations, i.e. the evolution equations for the differential Lorentz transformation variables \eqref{eq:nuetaevoleq}, the structure equations \eqref{eq:thopmetric}\eqref{eq:AlgebraicSpincoeff1}\eqref{eq:StructureSpincoeff1}, the Ricci relations \eqref{eq:thopRicci}\eqref{eq:ExtraRicci} and the Bianchi identities \eqref{eq:Bianchi}, are all properly weighted equations. This means that they make sense for the entire family of background principal null tetrads.
\end{remark}
\begin{remark}
From the Bianchi and Ricci equations, one can derive non-linear versions of the Teukolsky master equations (TME) \cite{Teukolsky}
\begin{subequations}
\begin{align}
\hspace{6ex}&\hspace{-6ex}\bigl((\thop{} - 4 \mathring{\rho}' - 4 \tilde{\rho}' - \overline{\tilde{\rho}'} - \bar{\mathring{\rho}}')(\tho{} - \mathring{\rho} - \tilde{\rho})
- (\edtp{} - \bar{\mathring{\tau}} - 4 \mathring{\tau}' - 4 \tilde{\tau}')(\edt{} - \mathring{\tau})
- 3 (\mathring{\Psi}_{2}{} + \tilde{\Psi}_{2}{} + \tilde{\sigma} \tilde{\sigma}')\bigr)\tilde{\Psi}_{4}{}\nonumber\\
={}&4 \tilde{\Psi}_{3}{} (\edt{} - \overline{\tilde{\tau}'} - \bar{\mathring{\tau}}')\tilde{\sigma}'
+ 4 \tilde{\sigma}' \edt \tilde{\Psi}_{3}{}
- 10 \tilde{\Psi}_{3}{}^2,\\
\hspace{6ex}&\hspace{-6ex}\bigl((\tho{} - 4 \mathring{\rho} - 4 \tilde{\rho} - \overline{\tilde{\rho}} - \bar{\mathring{\rho}})(\thop{} - \mathring{\rho}' - \tilde{\rho}')
- (\edt{} - 4 \mathring{\tau} - \overline{\tilde{\tau}'} - \bar{\mathring{\tau}}')(\edtp{} - \mathring{\tau}' - \tilde{\tau}')
- 3 (\mathring{\Psi}_{2}{} + \tilde{\Psi}_{2}{} + \tilde{\sigma} \tilde{\sigma}')\bigr)\tilde{\Psi}_{0}{}\nonumber\\
={}&-4 \tilde{\Psi}_{1}{} (\thop{} - \overline{\tilde{\rho}'} - \bar{\mathring{\rho}}')\tilde{\kappa}
+ 4 \tilde{\Psi}_{1}{} (\edtp{} - \bar{\mathring{\tau}})\tilde{\sigma}
- 4 \tilde{\kappa} \thop \tilde{\Psi}_{1}{}
+ 4 \tilde{\sigma} \edtp \tilde{\Psi}_{1}{}
- 10 \tilde{\Psi}_{1}{}^2.
\end{align}
\end{subequations}
\end{remark}
From the Bianchi equations, it follows that the differential curvatures satisfy the evolution system given in the following corollary.
\begin{corollary}[Evolution system for the differential curvature components]
\label{cor:evolsys}
Assume the \radiationGaugeHypotheses{} and \frameGaugeHypotheses.
Let $(t, x, y, z)$ be a real coordinate system such that constant $t$ hypersurfaces are spacelike.
The differential curvature components satisfy
\begin{subequations}
\label{eq:HyperbolicSystemForCurvature}
\begin{align}
B^t
\partial_t
\begin{pmatrix}
\tilde{\Psi}_{0}\\\tilde{\Psi}_{1}\\\tilde{\Psi}_{2}\\\tilde{\Psi}_{3}\\\tilde{\Psi}_{4}
\end{pmatrix}
={}&
-\sum_{i\in\{x,y,z\}}B^i\partial_i\begin{pmatrix}
\tilde{\Psi}_{0}\\\tilde{\Psi}_{1}\\\tilde{\Psi}_{2}\\\tilde{\Psi}_{3}\\\tilde{\Psi}_{4}
\end{pmatrix}
+
F,
\end{align}
where $F=F(\mathfrak{u})$ is a function of the geometric variables and
\begin{align}
B^i={}&
\begin{pmatrix}
n^t{}^3 n^i & - n^t{}^3 m^i & 0 & 0 & 0\\
- n^t{}^3\bar{m}^i & n^t{}^3l^i + l^t n^t{}^2 n^i & - l^t n^t{}^2m^i & 0 & 0\\
0 & - l^t n^t{}^2\bar{m}^i & l^t n^t{}^2l^i + l^t{}^2 n^t n^i & - l^t{}^2 n^t m^i & 0\\
0 & 0 & - l^t{}^2 n^t \bar{m}^i & l^t{}^2 n^t l^i + l^t{}^3 n^i & - l^t{}^3 m^i\\
0 & 0 & 0 & - l^t{}^3 \bar{m}^i & l^t{}^3 l^i
\end{pmatrix} \;\forall\; i\in \{t,x,y,z\} .
\end{align}
\end{subequations}
\end{corollary}
\begin{proof}
Consider the components of the foreground frame in terms of the coordinate co-frame, i.e. $l^t=l^a(dt)_a$ etc. The spacelike nature of the hypersurfaces means that the co-normal $(dt)_a$ is time-like, i.e. $0<g^{\#}{}^{ab}(dt)_a(dt)_b=2l^tn^t-2m^t\bar{m}^t$. In particular, we get $l^tn^t > m^t\bar{m}^t\geq 0$. Furthermore as we assume that $l^a$ and $n^a$ are future pointing, we get that $l^t>0$ and $n^t>0$.
We can write \eqref{eq:Bianchi} in the form
\begin{align*}
&\begin{pmatrix}
- \bar{m}^t & l^t & 0 & 0 & 0\\
0 & - \bar{m}^t & l^t & 0 & 0\\
0 & 0 & - \bar{m}^t & l^t & 0\\
0 & 0 & 0 & - \bar{m}^t & l^t\\
n^t & - m^t & 0 & 0 & 0\\
0 & n^t & - m^t & 0 & 0\\
0 & 0 & n^t & - m^t & 0\\
0 & 0 & 0 & n^t & - m^t
\end{pmatrix}\partial_t
\begin{pmatrix}
\tilde{\Psi}_{0}\\\tilde{\Psi}_{1}\\\tilde{\Psi}_{2}\\\tilde{\Psi}_{3}\\\tilde{\Psi}_{4}
\end{pmatrix}
\\&={}\sum_{i\in\{x,y,z\}}\begin{pmatrix}
\bar{m}^i & - l^i & 0 & 0 & 0\\
0 & \bar{m}^i & - l^i & 0 & 0\\
0 & 0 & \bar{m}^i & - l^i & 0\\
0 & 0 & 0 & \bar{m}^i & - l^i\\
- n^i & m^i & 0 & 0 & 0\\
0 & - n^i & m^i & 0 & 0\\
0 & 0 & - n^i & m^i & 0\\
0 & 0 & 0 & - n^i & m^i
\end{pmatrix}\partial_i\begin{pmatrix}
\tilde{\Psi}_{0}\\\tilde{\Psi}_{1}\\\tilde{\Psi}_{2}\\\tilde{\Psi}_{3}\\\tilde{\Psi}_{4}
\end{pmatrix}
+
l.o.
\end{align*}
where $l.o.$ denotes a function of the geometric variables $\mathfrak{u}$ but not their derivatives. The corollary follows from multiplying this by
\begin{align*}
\begin{pmatrix}
0 & 0 & 0 & 0 & n^t{}^3 & 0 & 0 & 0\\
n^t{}^3 & 0 & 0 & 0 & 0 & l^t n^t{}^2 & 0 & 0\\
0 & l^t n^t{}^2 & 0 & 0 & 0 & 0 & l^t{}^2 n^t & 0\\
0 & 0 & l^t{}^2 n^t & 0 & 0 & 0 & 0 & l^t{}^3\\
0 & 0 & 0 & l^t{}^3 & 0 & 0 & 0 & 0
\end{pmatrix} .
\end{align*}
\end{proof}
\subsection{First-order symmetric-hyperbolicity}
\begin{theorem}[First-order symmetric-hyperbolic system]
\label{thm:FOSH}
Assume the \radiationGaugeHypotheses{} and \frameGaugeHypotheses.
Assume $g$ is a solution of the vacuum Einstein equation.
Let $(t, x, y, z)$ be a real coordinate system such that constant $t$ hypersurfaces are spacelike.
The system \eqref{eq:nuetaevoleq},
\eqref{eq:thopmetric},
\eqref{eq:thopRicci}, and
\eqref{eq:HyperbolicSystemForCurvature}
forms a first-order symmetric-hyperbolic system for the geometric variables $\mathfrak{u}$,
and where $\tilde{G}_i$ and $\tilde{\slashed{G}}$ are given in terms of the geometric variables by equations \eqref{eq:InvGTildeInvGRelations} and \eqref{eq:GslashRelations} and where $\varsigma$ and $\varsigma^\#$ are given by equation \eqref{eq:defVarsigma}.
\end{theorem}
\begin{proof}
The goal is to show that, using the algebraic relations for $\tilde{G}_i$ and $\tilde{\slashed{G}}$ in \eqref{eq:InvGTildeInvGRelations} and for $\varsigma$ and $\varsigma^{\#}$ in \eqref{eq:defVarsigma},
the system \eqref{eq:nuetaevoleq},
\eqref{eq:thopmetric},
\eqref{eq:thopRicci}, and
\eqref{eq:HyperbolicSystemForCurvature} can be written in the form
\begin{align}
A(\mathfrak{u})^t \partial_t \mathfrak{u} ={}& \sum_{i\in\{x,y,z\}} A(\mathfrak{u})^i \partial_i \mathfrak{u} + F(\mathfrak{u}),
\label{eq:FullFOSH}
\end{align}
where $A^t$ and $A^i$ are Hermitian matrices, where $A^t$ is positive definite, and where $A^t$, each $A^i$, and $F$ are functions of $\mathfrak{u}$ and $\overline\mathfrak{u}$. Note also that in this equation $F(0)=0$, so $\mathfrak{u}=0$ is a solution of this system.
Since $n$ is future-directed, for any $\varphi\in\mathfrak{u}$, any transport equation of the form $\thop\varphi=f_1(\mathfrak{u})$ can be written in coordinates as $n^t\partial_t\varphi=-\sum_{i\in\{x,y,z\}}n^i\partial_i\varphi +f_2(\mathfrak{u})$, where $f_2$ is constructed from $f_1$ and from products of the connection coefficients appearing in $\thop$ and of $\varphi$. The equations \eqref{eq:nuetaevoleq},
\eqref{eq:thopmetric}, and \eqref{eq:thopRicci} are all of the form $\thop\varphi=f(\mathfrak{u})$. Therefore, the right hand side of the entire transport system has a diagonal principal part. Since $n$ is real, these diagonal parts are trivially Hermitian. The spacelike nature of the slice implies $n^t>0$, so the left hand side matrix is diagonal and positive definite. This gives equations for differential Lorentzian transformations, the metric components, and the spin components.
It remains to obtain equations for the curvature components. In equation \eqref{eq:HyperbolicSystemForCurvature}, the $B^i$ are clearly symmetric. It remains to show $B^t$ is positive definite. The determinant and sub-determinants of $B^t$ are $4 l^t{}^8 n^t{}^8 (l^t n^t - m^t \bar{m}^t) (2l^t n^t - m^t \bar{m}^t)>0$, $l^t{}^4 n^t{}^8 \bigl(l^t{}^2 n^t{}^2 + 6 l^t n^t (l^t n^t - m^t \bar{m}^t) + (l^t n^t - m^t \bar{m}^t)^2\bigr)>0$, $l^t{}^2 n^t{}^8 \bigl(l^t n^t + 3 (l^t n^t - m^t \bar{m}^t)\bigr)>0$, $n^t{}^6 (2 l^t n^t - m^t \bar{m}^t)>0$, $n^t{}^4>0$. Hence, $B^t$ is positive definite. This gives equations for the curvature components and hence all components of $\mathfrak{u}$. Thus, equations \eqref{eq:nuetaevoleq},
\eqref{eq:thopmetric},
\eqref{eq:thopRicci}, and
\eqref{eq:HyperbolicSystemForCurvature} form a first-order symmetric-hyperbolic system for $\mathfrak{u}$.
\end{proof}
\subsection{Completing the proof of theorem \ref{thm:NLnhIsLWP}}
\begin{proof}[Proof of theorem \ref{thm:NLnhIsLWP}]
The symmetric hyperbolicity in point \ref{pt:FOSH} is proved in theorem \ref{thm:FOSH}.
The geometric variables $\mathfrak{u}$ include the foreground metric coefficients $\tilde{G}^{\#}_i$ and $\tilde{\slashed{G}}^{\#}$ and the differential Lorentz transformations $\nu$ and $\eta$. From these, the components with respect to the background metric $G^{\#}$ and $\slashed{G}$ can be calculated using equations \eqref{eq:GslashRelations} and \eqref{eq:InvGTildeInvGRelations}. From these and the background tetrad $(\mathring{\vecL},\mathring{\vecN},\mathring{\vecM},\bar{\mathring{\vecM}})$, the original metric $g_{ab}$ can be calculated. This completes the proof of point \ref{pt:systemDeterminesMetric}.
Finally, suppose one has a set of initial data for the vacuum Einstein equation. Choose also a set of initial data for $\nu$ and $\eta$. On the one hand, the initial data for the Einstein equation launches a unique solution of the vacuum Einstein equation. From the results in section \ref{s:ProofOfEnforecabilityOfTheNLORG}, coordinates can be chosen so that this metric satisfies the radiation gauge condition. Let $\nu$ and $\eta$ satisfy the evolution equations \eqref{eq:nuetaevoleq} from the frame-gauge hypotheses. The lemmas from this section give that the geometric variables constructed from differential Lorentz transforms, the foreground metric, its connection coefficients, and its curvature (and from the background quantities) satisfy the system \eqref{eq:nuetaevoleq},
\eqref{eq:thopmetric},
\eqref{eq:thopRicci}, and
\eqref{eq:HyperbolicSystemForCurvature}. Call this solution $\mathfrak{u}_1$. On the other hand, the initial data for the vacuum Einstein equation, together with the choice of initial data for $\nu$ and $\eta$, launch a unique solution of the system \eqref{eq:nuetaevoleq},
\eqref{eq:thopmetric},
\eqref{eq:thopRicci}, and
\eqref{eq:HyperbolicSystemForCurvature}. Call this solution $\mathfrak{u}_2$. Since $\mathfrak{u}_1$ and $\mathfrak{u}_2$ have the same initial data, since they satisfy the same system, and since there is uniqueness of solutions to first-order symmetric-hyperbolic systems, it follows that $\mathfrak{u}_1$ and $\mathfrak{u}_2$ are the same. This means that the metric components coincide. In particular, the metric constructed from the solution of the first-order symmetric-hyperbolic system $\mathfrak{u}_2$ coincides with the solution of the vacuum Einstein equation launched from the corresponding initial data. In particular, the solution of the first-order symmetric-hyperbolic system determines a metric which satisfies the vacuum Einstein equation. This completes the proof of the final point in the theorem.
\end{proof}
\subsection{Initial data and residual gauge}
Before concluding this section, we make a few remarks about the initial data and the residual gauge.
\begin{remark}
With a coordinate system as in corollary~\ref{cor:evolsys}, one can interpret the equations \eqref{eq:AlgebraicSpincoeff1}, \eqref{eq:StructureSpincoeff1}, \eqref{eq:ExtraRicci} and the remaining Bianchi identities as a set of constraint equations, by expressing the derivatives in terms of coordinate derivatives and eliminating the time derivatives with \eqref{eq:FullFOSH}. By applying a $\thop$ derivative to this set of equations, commute the $\thop$ inside, use the evolution equations, and again the constraints, one finds that the constraints propagate.
\end{remark}
\begin{remark}
If one is given initial data only for the metric coefficients, $\nu$ and $\eta$, one can construct initial data for the differential spin coefficients via the full set of structure equations. Initial data for the curvature can be constructed from a subset of the Ricci relations. Note that the values for $\nu$ and $\eta$ on the initial slice are not constrained if we interpret \eqref{eq:StructureSpincoeff1} as equations giving initial data for differential spin coefficients. The initial data for the metric coefficients are constrained due to the fact that we are only considering vacuum perturbations.
\end{remark}
\begin{remark}
In this section, we use the radiation gauge condition{} in the open set on which we construct solutions. From the perspective of naive function counting, these specify the four free functions that can be specified by a gauge choice in an open set. This gives a unique solution for each choice of initial data. However, there remains a residual gauge freedom that can be treated as a diffeomorphism of the initial data.
In the next section we will see that the diffeomorphism part of the initial data gauge freedom can be partially fixed by making $\tilde{\slashed{G}}{}^{\#}_{\phantom{0}}{}$ small in appropriate sense. The initial data part of the differential frame gauge can be fixed by choosing the initial data for $\nu$ and $\eta$.
As discussed above, this can be done in an arbitrary way, but it is convenient to choose the initial data for $\nu$ to be $0$. As we will see below, $\nu$ will then stay quadratically small.
To also set the initial data for $\eta$ to zero is also possible, but it will not stay quadratically small during the evolution.
An alternative is to set the initial data for $\eta$ so that the initial data for $\tilde{\beta}$ is quadratically small. This has the advantage that $\tilde{\beta}$ will stay quadratically small. For details see section~\ref{sec:SmallnessBeta}.
\end{remark}
\section{Imposing the trace condition}
\label{s:traceCondition}
This section can be summarized as follows: Price-Shankar-Whiting \cite{Price:2006ke} have shown that, for the linearized Einstein equation, a linearized gauge transformation that satisfies the linear radiation gauge condition{} can be further transformed to satisfy the linear trace condition{} and hence the full radiation gauge of Chrzanowski; we show that the same result can be shown to quadratic order for the full Einstein equation.
The main result of this section is the following refinement of theorem \ref{thm:NLnhEnforceability}.
\begin{theorem}[Enforceability of the trace condition to quadratic order]
\label{thm:NLORGEnforceability}
Assume the \radiationGaugeHypotheses. Let $k'$ be a sufficiently large integer and let $(X,Y,I,J,h,U,V)$ be as in definition \ref{def:diffeomorphismGauge} for a diffeomorphism.
There exist $\varepsilon_0>0$, $k>k'$, and $K>0$ such that if $g_{ab}$ is a symmetric $(0,2)$ tensor $g_{ab}$ satisfying the vacuum Einstein equation and $|g-\mathring{g}|_{C^{k}(U)}<\varepsilon_0$, then:
\begin{enumerate}
\item
\label{pt:NLnhEnforceability}
There is a $C^{k'}$ diffeomorphism gauge transform $(U,V,\Phi)$ such that
$|\Phi^{-1}_*g-\mathring{g}|_{C^{k'}(h(Y))}$ $\leq K|g-\mathring{g}|_{C^{k}(h(X))}$, and $\Phi^{-1}_*g$ satisfies the radiation gauge condition{} on $V$.
\item
\label{pt:NLORGEnforceability}
Furthermore, $\Phi$ can be chosen such that
\begin{align}
\label{eq:NLTraceConditionToQuadraticOrder}
|\mathring{g}^{ab}(\Phi^{-1}_*g)_{ab}-\mathring{g}^{ab}\mathring{g}_{ab}|_{C^{k'}(V)}
\leq K |g-\mathring{g}|_{C^{k}(U)}^2 .
\end{align}
\end{enumerate}
\end{theorem}
\subsection{Review of the linear radiation gauge condition{} from Price-Shankar-Whiting \texorpdfstring{\cite{Price:2006ke}}{}}
In this subsection, we review the results of \cite{Price:2006ke} on the linear radiation gauge and linear trace conditions as well as the ORG, which appear in definition \ref{def:linearORG}. The radiation gauge in \cite{Price:2006ke} is based on the vector field $l$, while ours is based on $n$. Therefore, many of the formulas interchange primed and unprimed. We state the following result for a Kerr background, although \cite{Price:2006ke} show these results hold in the wider class of metrics.
To explain the linear theory, following \cite{Price:2006ke}, we introduce the Held integration technique first described in \cite{Held1974}.
We have re-derived all equations and made slight modifications to make sure that all expressions are properly weighted.
A spinor $\alpha$ is defined to be a Held spinor if $\thop\alpha=0$. For a spinor $\alpha$, the notation $\Held{\alpha}$ indicates that $\alpha$ is a Held spinor. For a vector field $X$ and a point $p$, define $\Flow{X}{s}(p)$ to be the flow along $X$, i.e. such that for any $p$, the function $\Flow{X}{s}(p)$ is the solution of $\frac{d}{ds}\Flow{X}{s}(p)=X$ and $\Flow{X}{0}(p)=p$; for sets $S$ and $P$ of $\mathbb{R}$ and the manifold respectively, define $\Flow{X}{S}(P)=\cup_{s\in S,p\in P}\Flow{X}{s}(p)$. For a spinor $\alpha$ defined on a hypersurface $\Sigma$ which is given as the graph of $r$ as a function of $(v,\omega)$, there is a unique extension of $\alpha$ as a Held spinor on $\Flow{n}{\mathbb{R}}(\Sigma)$, which we will denote by $\Held{\alpha}$. For Held spinors defined on an open set, the operators $\Heldtho$, $\Heldeth$, and $\Heldethp$ are defined to be
\begin{subequations}
\begin{align}
\Heldtho \varphi ={}&- \frac{p \mathring{\Psi}_{2}{} \varphi}{2 \mathring{\rho}'}
- \frac{q \bar{\mathring{\Psi}}_{2}{} \varphi}{2 \bar{\mathring{\rho}}'}
+ \thoBG \varphi
- \frac{\mathring{\tau}' \edtBG \varphi}{\mathring{\rho}'}
- \frac{\bar{\mathring{\tau}}' \edtpBG \varphi}{\bar{\mathring{\rho}}'} ,\\
\Heldeth \varphi ={}&- \frac{p \bar{\mathring{\tau}}' \varphi}{\bar{\mathring{\rho}}'}
+ \frac{\edtBG \varphi}{\mathring{\rho}'} ,\\
\Heldethp \varphi ={}&- \frac{q \mathring{\tau}' \varphi}{\mathring{\rho}'}
+ \frac{\edtpBG \varphi}{\bar{\mathring{\rho}}'} .
\end{align}
\end{subequations}
For a spinor $\alpha$ defined on $\Sigma$, the operator $\Heldtho$ denotes the operator defined by extending $\alpha$ to $\Held{\alpha}$, applying $\Heldtho$, and then restricting to $\Sigma$ again. For a spinor $\alpha$ defined on $\Sigma$, the operators $\Heldeth$ and $\Heldethp$ are defined analogously. Note that, when acting on Held spinors, the operator $\thop$ commutes with $\Heldtho$, $\Heldeth$, and $\Heldethp$.
The following lemma encapsulates the key results of \cite{Price:2006ke} regarding the linear radiation gauge condition. Equations \eqref{eq:NullGaugevcondGHP} and \eqref{eq:PSW23} correspond to equations (15) and (23) of \cite{Price:2006ke}.
\begin{lemma}[The linear radiation gauge condition{} \cite{Price:2006ke}]
Let $0<r_1<r_2<\infty$ and $v_1<v_2$. Let the background hypotheses of definition \ref{def:radiationGaugeHypotheses}{} hold with $U=(r_1,r_2)\times(v_1,v_2)\times\mathbb{S}^2$.
Let $h_{ab}$ be a symmetric $(0,2)$ tensor that satisfies the linear radiation gauge condition{} of definition \ref{def:linearORG}. Let $\xi$ be a vector field.
\begin{enumerate}
\item The tensor field $h_{ab}+\mathcal{L}_\xi\mathring{g}_{ab}$ satisfies the linear radiation gauge condition{} if
\begin{subequations}
\label{eq:NullGaugevcondGHP}
\begin{align}
\thopBG \xi_{\mathring{l}}={}&- \xi_{\mathring{\vecM}} (\bar{\mathring{\tau}} + \mathring{\tau}')
- \xi_{\bar{\mathring{\vecM}}} (\mathring{\tau} + \bar{\mathring{\tau}}')
- \thoBG \xi_{\mathring{n}},\\
\thopBG \xi_{\mathring{n}}={}&0,\\
\thopBG \xi_{\mathring{\vecM}}={}&- \xi_{\mathring{\vecM}} \mathring{\rho}'
- \xi_{\mathring{n}} \mathring{\tau}
- \edtBG \xi_{\mathring{n}},\\
\thopBG \xi_{\bar{\mathring{\vecM}}}={}&- \xi_{\bar{\mathring{\vecM}}} \bar{\mathring{\rho}}'
- \xi_{\mathring{n}} \bar{\mathring{\tau}}
- \edtpBG \xi_{\mathring{n}}.
\end{align}
\end{subequations}
\item The general solution of \eqref{eq:NullGaugevcondGHP} is given in terms of arbitrary Held spinors $\Held{\xi}_{\mathring{\vecL}}$, $\Held{\xi}_{\mathring{\vecN}}$, $\Held{\xi}_{\mathring{\vecM}}$, $\Held{\xi}_{\bar{\mathring{\vecM}}}$ by
\begin{subequations}
\label{eq:PSW23}
\begin{align}
\xi_{\mathring{l}}={}&\Held{\xi}_{\mathring{l}}
+ \frac{\Held{\xi}_{\mathring{\vecM}} \mathring{\tau}'}{\mathring{\rho}'^2}
+ \frac{\Held{\xi}_{\bar{\mathring{\vecM}}} \bar{\mathring{\tau}}'}{\bar{\mathring{\rho}}'^2}
+ \Held{\xi}_{\mathring{n}} \Bigl(\frac{\mathring{\Psi}_{2}{}}{2 \mathring{\rho}'^2} + \frac{\bar{\mathring{\Psi}}_{2}{}}{2 \bar{\mathring{\rho}}'^2} + \frac{\mathring{\tau}' \bar{\mathring{\tau}}'}{\mathring{\rho}' \bar{\mathring{\rho}}'}\Bigr)
+ \frac{1}{2}\Bigl(\frac{1}{\mathring{\rho}'} + \frac{1}{\bar{\mathring{\rho}}'}\Bigr) \Heldtho \Held{\xi}_{\mathring{n}}
- \frac{\mathring{\tau}' \Heldeth \Held{\xi}_{\mathring{n}}}{\mathring{\rho}'}
- \frac{\bar{\mathring{\tau}}' \Heldethp \Held{\xi}_{\mathring{n}}}{\bar{\mathring{\rho}}'} ,\\
\xi_{\mathring{n}}={}&\Held{\xi}_{\mathring{n}},\\
\xi_{\mathring{\vecM}}={}&\frac{\Held{\xi}_{\mathring{\vecM}}}{\mathring{\rho}'}
+ \frac{\Held{\xi}_{\mathring{n}} \bar{\mathring{\tau}}'}{\bar{\mathring{\rho}}'}
- \Heldeth \Held{\xi}_{\mathring{n}},\\
\xi_{\bar{\mathring{\vecM}}}={}&\frac{\Held{\xi}_{\bar{\mathring{\vecM}}}}{\bar{\mathring{\rho}}'}
+ \frac{\Held{\xi}_{\mathring{n}} \mathring{\tau}'}{\mathring{\rho}'}
- \Heldethp \Held{\xi}_{\mathring{n}}.
\end{align}
\end{subequations}
\end{enumerate}
\end{lemma}
The approach of \cite{Price:2006ke} to the full radiation gauge of Chrzanowski{} proceeds as follows. Since the linear radiation gauge condition{} has already been treated, it remains to treat the linear trace condition. In the linearization of the Einstein equation, the trace is the linearization of $\tilde{\slashed{G}}{}^{\#}_{\phantom{0}}{}$, so it satisfies a linearized version of \eqref{eq:ThopInvGTrTilde}, the linearization of which is a transport equation driven by the linearization of $\tilde{\rho}'$. In turn, $\tilde{\rho}'$ satisfies \eqref{eq:transport:rhoTildePrime}, the linearization of which is a homogeneous transport equation. Thus, the trace satisfies a second-order ordinary differential equation, which has a general solution involving two free parameters, denoted $\Held{a}$ and $\Held{b}$. If a linearized gauge transformation satisfies the linearized radiation gauge, then $\Held{a}$ and $\Held{b}$ can be expressed in terms of $\Held{\xi}_{\mathring{\vecL}}$, $\Held{\xi}_{\mathring{\vecM}}$, $\Held{\xi}_{\bar{\mathring{\vecM}}}$, $\Held{\xi}_{\mathring{\vecN}}$. Furthermore, $\Held{\xi}_{\mathring{\vecL}}$, $\Held{\xi}_{\mathring{\vecM}}$, $\Held{\xi}_{\bar{\mathring{\vecM}}}$, $\Held{\xi}_{\mathring{\vecN}}$ can be chosen so that the linear trace condition holds.
It is convenient for us to take a slightly different perspective on imposing the linear trace condition. This is based on considering the initial value problem for the second-order ODE satisfied by the linearized trace, rather than analyzing the general solution in terms of $\Held{a}$ and $\Held{b}$. A linearized gauge transformation takes $h_{ab}$ to $h_{ab}+\mathcal{L}_\xi\mathring{g}_{ab}$. Thus, to impose the trace condition, it is sufficient to be able to specify $\mathring{g}^{ab} \mathring{\nabla}_{(a}\xi_{b)}$. As noted in \cite{Price:2006ke}, this trace is given by
\begin{subequations}
\begin{align}
\mathring{g}^{ab} \mathring{\nabla}_{(a}\xi_{b)}={}&- \xi_{\mathring{n}} (\mathring{\rho} + \bar{\mathring{\rho}})
- \xi_{\mathring{l}} (\mathring{\rho}' + \bar{\mathring{\rho}}')
- \edtBG \xi_{\bar{\mathring{\vecM}}}
- \edtpBG \xi_{\mathring{\vecM}}.
\end{align}
Assuming that $\xi$ satisfies \eqref{eq:NullGaugevcondGHP}, the derivative along $n$ can be calculated as
\begin{align}
\thopBG (\mathring{g}^{ab} \mathring{\nabla}_{(a}\xi_{b)})={}& \xi_{\mathring{n}} (\mathring{\Psi}_{2}{} + \bar{\mathring{\Psi}}_{2}{} - 2 \mathring{\rho} \bar{\mathring{\rho}}')
- \xi_{\mathring{l}} (\mathring{\rho}'^2 + \bar{\mathring{\rho}}'^2)
+ 2 \xi_{\mathring{\vecM}} \mathring{\rho}' \mathring{\tau}'
+ 2 \xi_{\bar{\mathring{\vecM}}} \bar{\mathring{\rho}}' \bar{\mathring{\tau}}'\nonumber\\
& + (\mathring{\rho}' + \bar{\mathring{\rho}}') \thoBG \xi_{\mathring{n}}
- (\mathring{\rho}' - \bar{\mathring{\rho}}') \edtBG \xi_{\bar{\mathring{\vecM}}}
+ \edtBG \edtpBG \xi_{\mathring{n}}
+ (\mathring{\rho}' - \bar{\mathring{\rho}}') \edtpBG \xi_{\mathring{\vecM}}
+ \edtpBG \edtBG \xi_{\mathring{n}}.
\end{align}
\end{subequations}
Applying the general solution of the linear radiation gauge condition{} and further calculation leads to the pair of equations
\begin{subequations}
\label{eq:PSWHeldab}
\begin{align}
\hspace{6ex}&\hspace{-6ex}- \frac{\mathring{g}^{ab} \mathring{\nabla}_{(a}\xi_{b)}}{4 \kappa_{1}{} \bar{\kappa}_{1'}{}} (\kappa_{1}{}^2 + \bar{\kappa}_{1'}{}^2)
+ \frac{\thopBG (\mathring{g}^{ab} \mathring{\nabla}_{(a}\xi_{b)})}{4 \kappa_{1}{} \mathring{\rho}'} (\kappa_{1}{} + \bar{\kappa}_{1'}{})\nonumber\\
={}&\Heldtho \Held{\xi}_{\mathring{n}}
+ \tfrac{1}{2} \Heldeth \Held{\xi}_{\bar{\mathring{\vecM}}}
+ \tfrac{1}{2} \Heldethp \Held{\xi}_{\mathring{\vecM}},
\label{eq:PSWHeldab:ForN}\\
\hspace{6ex}&\hspace{-6ex}\frac{ (\kappa_{1}{} - \bar{\kappa}_{1'}{})^2\mathring{g}^{ab} \mathring{\nabla}_{(a}\xi_{b)}}{4 \kappa_{1}{}^2 \bar{\kappa}_{1'}{} \mathring{\rho}'} (\kappa_{1}{} + \bar{\kappa}_{1'}{})
- \frac{\thopBG (\mathring{g}^{ab} \mathring{\nabla}_{(a}\xi_{b)})}{4 \kappa_{1}{}^2 \mathring{\rho}'^2} (\kappa_{1}{}^2 + \bar{\kappa}_{1'}{}^2)\nonumber\\
={}&\Held{\xi}_{\mathring{l}}
- \tfrac{1}{2} \Heldeth \Heldethp \Held{\xi}_{\mathring{n}}
- \tfrac{1}{2} \Heldethp \Heldeth \Held{\xi}_{\mathring{n}}
+ \frac{(\kappa_{1}{} - \bar{\kappa}_{1'}{})}{2 \kappa_{1}{} \mathring{\rho}'}( \Heldethp \Held{\xi}_{\mathring{\vecM}}-\Heldeth \Held{\xi}_{\bar{\mathring{\vecM}}})
\nonumber\\
& + \frac{\Held{\xi}_{\mathring{n}}}{2 \mathring{\rho}'^2} \Bigl( \frac{\kappa_{1}{}}{\bar{\kappa}_{1'}{}}\mathring{\Psi}_{2}{} + \mathring{\Psi}_{2}{} + 2 \mathring{\rho} \mathring{\rho}' - 2 \mathring{\tau} \mathring{\tau}'\Bigr),
\label{eq:PSWHeldab:ForL}
\end{align}
\end{subequations}
where the Killing spinor coefficient $\kappa_{1}$ is given in equation \eqref{eq:defKappa1}.
The right-hand sides of these two equations loosely correspond to the quantities $\Held{a}$ and $\Held{b}$ from \cite{Price:2006ke}.
Set $\Held{\xi}_{\mathring{\vecM}}=0$ and $\Held{\xi}_{\bar{\mathring{\vecM}}}=0$. Set $\Held{\xi}_{\mathring{\vecN}}$ to satisfy the analogue of \eqref{eq:PSWHeldab:ForN} where $g^{ab}\mathring{\nabla}_{(a}\xi_{b)}$ and its $\thopBG$ derivative have been replaced by $\frac{1}{2}\mathring{g}^{ab}h_{ab}$ and its $\thopBG$ derivative on an initial hypersurface $\Sigma$. In a similar way, set $\Held{\xi}_{\mathring{\vecL}}$ to satisfy the analogue of \eqref{eq:PSWHeldab:ForL}. From this choice of $\Held{\xi}$, set $\xi$ to be the corresponding general solution of the linear radiation gauge condition. This has been chosen so that the trace of $h_{ab}+\mathcal{L}_\xi\mathring{g}_{ab}$ and the $\thopBG$ derivative of this trace both vanish on the initial hypersurface $\Sigma$. From the second-order, linear ODE that it satisfies, the trace remains zero. This imposes the linear trace condition, and hence the full radiation gauge of Chrzanowski.
\subsection{Proof of theorem \ref{thm:NLORGEnforceability}}
\begin{proof}[Proof of theorem \ref{thm:NLORGEnforceability}]
\begin{steps}
\step{Preliminaries.}
Note that the first point of the theorem is simply a restatement of theorem \ref{thm:NLnhEnforceability}. Thus, we may assume that a diffeomorphism gauge has already been chosen to impose that result. Within the proof, we will impose a pair of further diffeomorphism gauges. The first will impose the trace condition to quadratic order while potentially violating the radiation gauge condition, and the second will reimpose the radiation gauge condition{} while preserving the quadratic smallness of the trace term.
We assume the hypotheses of the theorem and initially consider what can be uniformly controlled. By taking $k$ sufficiently large with respect to $k'$, there is a constant $K$ such that $|\mathop{\mathrm{Riem}}{}[g]-\mathop{\mathrm{Riem}}{}[\mathring{g}]|_{C^{k'}(U)}$ $\leq K|g-\mathring{g}|_{C^{k}(U)}$. In this case, we can take $k=k'+2$, but this illustrates that to control any quantity to desired regularity $k'$, we can choose $k$ sufficiently large. We will use the notation $\varepsilon=|g-\mathring{g}|_{C^{k}(U)}$ and, for an exponent $p$, the notation $\alpha=\beta+O(\varepsilon^p)$ to indicate that there is a constant $K$, possibly depending on the open sets and regularity constants $k$ and $k'$, such that $|\alpha-\beta|_{C^{k'}(V)}\leq K|g-\mathring{g}|_{C^{k}(U)}^p$. We use $\alpha$ is $O(\varepsilon^p)$ to mean $\alpha=0+O(\varepsilon^p)$.
Within this proof, we shall use the ``noncurvature quantities'' to refer to the differential Lorentz transforms, metric, and spin coefficient components. The geometric variables as given before the diffeomorphism gauge is applied are called the geometric variables in the original gauge; the geometric variables after the diffeomorphism gauge has been applied are called the the regauged geometric variables.
There are three subtleties to address in this proof, all of which are resolved through the use of the smallness of the norms. The first subtlety is that, when constructing the diffeomorphisms, it is necessary that the image of $V$ remains in $U$.
The first diffeomorphism is generated by the flow along a vector field, and the image property is ensured by the $\varepsilon$ smallness of this vector field. The second diffeomorphism is generated using the argument from the geodesic flow from section \ref{s:ProofOfEnforecabilityOfTheNLORG}, and the image property is ensured by $\varepsilon$ smallness of the perturbation of the initial data in the geodesic flow.
The second subtlety is that the domain $V$ depends on the norm of the geometric variables, but the $C^{k'}(V)$ norm of the geometric variables depends on the choice of $V$. The regauged noncurvature quantities satisfy transport equations that are driven by both the regauged noncurvature quantities and the regauged $\tilde{\Psi}_i$; the regauged noncurvature quantities are determined by this evolution, while the $\tilde{\Psi}_i$ can be viewed as being calculated from the curvatures $\mathop{\mathrm{Riem}}{}[g]$ and $\mathop{\mathrm{Riem}}{}[\mathring{g}]$ in the original diffeomorphism gauge and from the regauged foreground tetrad, which is determined by the regauged differential Lorentz transformation variables. Since the curvatures in the original gauge are already given on $U$, the regauged noncurvature quantities can be determined from the transport equations from their initial data and from the curvature in the original gauge. Since the regauged noncurvature quantities are $\varepsilon$ small on the initial hypersurface, $h(X)$, it is possible to pass to a subset $h(Y)$ so that both the image under the transport equations remains in $V$ and the regauged noncurvature quantities remain $\varepsilon$ small on $V$, provided that the regauged $\tilde{\Psi}_i$ remain $\varepsilon$ small.
The third subtlety is that, a priori, the $C^{k}(U)$ norm of $\mathop{\mathrm{Riem}}{}[g]-\mathop{\mathrm{Riem}}{}[\mathring{g}]$ need not control the $C^{k}(U)$ of the $\tilde{\Psi}_i$ because there is not an a priori bound on the lengths of the foreground tetrad with respect to the reference Riemannian metric used to define the $C^{k}$ norms. This third subtlety is resolved by observing that as long as the Lorentz transformation variables remain bounded, the norms the regauged $\tilde{\Psi}_i$ are controlled by the corresponding norms of $\mathop{\mathrm{Riem}}{}[g]-\mathop{\mathrm{Riem}}{}[\mathring{g}]$ and the norms of the Lorentz transformation variables. Since the proof shows that the Lorentz transformation variables remain $\varepsilon$ small, we trivially recover the bootstrap assumption that they are bounded for the third subtlety, which then provides the necessary conditions for the second and first subtlety to be resolved.
Within this proof, we shall define a Held spinor to be a Held spinor with respect to $\mathring{g}$ and again use the notation $\Held{\alpha}$ to denote that $\alpha$ is a Held spinor.
\step{Define $\xi$.}
Set $\Held{\xi}_{m}=0$ and $\Held{\xi}_{\bar{m}}=0$. Set $\Held{\xi}_{n}$ to satisfy the analogue of \eqref{eq:PSWHeldab:ForN} where, on the initial hypersurface $h(X)$, the quantities $g^{ab}\mathring{\nabla}_{(a}v_{b)}$ and its $\thop$ derivative have been replaced by $\frac{1}{2}\mathring{g}^{ab}g_{ab}-2$ and its $\thop$ derivative respectively. In a similar way, set $\Held{\xi}_{l}$ to satisfy the analogue of \eqref{eq:PSWHeldab:ForL}. From this choice of $\Held{\xi}$, set $\xi$ to be the corresponding general solution of the linear radiation gauge condition{} given in equation \eqref{eq:PSW23}. This has been chosen so that the trace of $g_{ab}+\mathcal{L}_\xi\mathring{g}_{ab}$ and the $\thop$ derivative of this trace both vanish on the initial hypersurface $h(X)$. Note that from the smallness of $g$, the components of $\xi$ are $O(\varepsilon)$.
\step{Construct an initial gauge transformation from the flow along $\xi$.}
Recall $\Flow{\xi}{s}(p)$ denotes the flow along $\xi$, and that this defines a local diffeomorphism. For simplicity, denote by $\Phi_1$ the diffeomorphism such that $\Phi_1(p)=\Flow{\xi}{1}(p)$ for all $p$ for which this is defined. In particular, if $\varepsilon$ is sufficiently small on a scale dictated by $U$ and $V$, then $\Phi_1$ will define a bijection from $V$ to a subset of $U$. Since $\xi$ and $g-\mathring{g}$ are $O(\varepsilon)$, it follows that $\Phi_1^{*}g$ is also $O(\varepsilon)$.
Since $\xi$ and $g-\mathring{g}$ are both $O(\varepsilon)$, it follows that $\mathcal{L}_\vecPSWg_{ab}-\mathcal{L}_\xi\mathring{g}_{ab}$ is $O(\varepsilon^2)$. From the Price-Shankar-Whiting lemma on the linear theory, it follows that $\mathcal{L}_\xi\mathring{g}_{ab}$ satisfies the linear radiation gauge condition, so $n^a\mathcal{L}_\vecPSWg_{ab}$ is $O(\varepsilon^2)$. Similarly, on the initial hypersurface $h(X)$, the vector field $\xi$ was chosen so that $\mathring{g}^{ab}\mathcal{L}_{\xi}(g-\mathring{g})_{ab}$ and its $\thop$ derivative vanish. Thus, on the image of $h(X)$, they are $O(\varepsilon^2)$. From the transport equations \eqref{eq:ThopInvGTrTilde} and \eqref{eq:transport:rhoTildePrime} satisfied by $\tilde{\slashed{G}}{}^{\#}_{\phantom{0}}{}$ and $\tilde{\rho}'$, it follows that the perturbed trace $\tilde{\slashed{G}}{}^{\#}_{\phantom{0}}{}$ satisfies a second-order ODE in which all the terms that appear are either linear in $\tilde{\slashed{G}}{}^{\#}_{\phantom{0}}{}$ or of size $O(\varepsilon^2)$. Since the initial data is $O(\varepsilon^2)$ on the image of $h(X)$, this means that $\tilde{\slashed{G}}^{\#}$ remains $O(\varepsilon^2)$. Thus, $\mathring{g}^{ab}\Phi_{1}^{*}g_{ab}-4$ is also $O(\varepsilon^2)$.
\step{Reimpose the radiation gauge.}
From the enforceability of the radiation gauge condition{} in theorem \ref{thm:NLnhEnforceability}, it follows that there is a local diffeomorphism $\Phi_2$ such that $\Phi_2^{*}(\Phi_{1}^{*}g)$ satisfies the radiation gauge condition. From the previous step, we know that $\Phi_{1}^{*}g$ is already very close to satisfying the radiation gauge condition. In particular, following the proof of the enforceability of the radiation gauge condition{} in section \ref{s:ProofOfEnforecabilityOfTheNLORG}, one observes that the size of $\Phi_2^{*}(\Phi_{1}^{*}g)-\Phi_{1}^{*}g$ depends not on the size of all components of $\Phi_{1}^{*}g-\mathring{g}$, but only upon the size of the components of $n^a(\Phi_{1}^{*}g)_{ab}$. From this, it follows that $\Phi_2^{*}(\Phi_{1}^{*}g) -\Phi_{1}^{*}g$ is $O(\varepsilon^2)$. In particular, $\mathring{g}^{ab}(\Phi_2^{*}\Phi_{1}^{*}g)_{ab}-4$ is $O(\varepsilon^2)$. Defining $\Phi^{*}=\Phi_2^{*}\circ\Phi_{1}^{*}$, one obtains a $C^{k'}$ diffeomorphism of $V$ to a subset of $U$.
This completes the proof.
\end{steps}
\end{proof}
\section{Linearization}
\label{s:linearization}
\subsection{Definitions of linearizations}
To avoid ambiguity, we introduce two notions of linearization that are relevant for this paper. The second is a specialization arising in the case that a quantity vanishes quadratically under a gauge transformation, as occurs in section \ref{s:traceCondition}.
\begin{definition}[Direct linearization]
Let $A$ be a system of order-$m$ partial differential equations\footnote{This includes the case, for $m=0$, of algebraic systems.} for a set of variables $\mathfrak{u}$.
Assume that $\mathfrak{u}=0$ is a solution of the system $A$. Assume that all the coefficients in $A$ are functions in $\mathfrak{u}$ and smooth around $\mathfrak{u}=0$.
The \defn{direct linearization} of $A$ is defined to be the system constructed from treating all terms in $A$ as functions on the $m$-jet and then replacing each term by its first-order Taylor expansion in the $m$-jet variables.
\end{definition}
The variables in the $m$-jet in the linearization will typically denoted by the original variables with a dot above, e.g.{} $\dot\mathfrak{u}$. In some cases, we will consider the direct linearization as constrained to satisfy further linear equations.
\newcommand{\Phi}{\Phi}
\begin{definition}[Linearization under a gauge choice]
Let $(\mathcal{M},g)$ be a semi-Riemannian manifold. Let $U$ be a precompact open subset of $\mathcal{M}$ and $V$ a precompact open subset of $U$. Let $A$ be a system of first-order partial differential equations on $U$ for $\mathfrak{u}$. Let $\Phi$ be a family of diffeomorphisms depending pointwise on $\mathfrak{u}$.
A quantity $\varphi\in\mathfrak{u}$ is defined to \defn{vanish in the linearization under the gauge choice $\Phi$} if there exist $\varepsilon_0>0$, $k>k'$, and $K>0$ such that for all $\varepsilon<\varepsilon_0$ and $\Phi$ such that $\|\Phi\|_{C^k(U)}\leq \varepsilon$, the diffeomorphism $\Phi$ maps from $V$ to a subset of $U$ such that $\|\Phi^*\varphi\|_{C^{k'}(\Phi(V))}\leq K\varepsilon^2$.
The \defn{linearization under the gauge choice $\Phi$} of $A$ is defined to be the direct linearization of $A$ with $0$ substituted for all the quantities that vanish in the linearization under the gauge choice.
\end{definition}
\subsection{Scalar linearization}
The linearization of the systems considered in section \ref{s:FOSH} can now be computed. Recall that section \ref{s:FOSH} constructed a first-order symmetric hyperbolic system for the NP scalars defined in that section. Hence, we refer to this linearization as the scalar linearization.
\begin{theorem}[Scalar linearization]
\label{thm:scalarLinearization}
Assume the background hypotheses of definition \ref{def:radiationGaugeHypotheses}.
The direction linearization of the relations between the different versions of the metric components yield
\begin{subequations}
\label{eq:TildeGToInvTildeG:linearized}
\begin{align}
\dot{\tilde{G}}^{\#}_2{}={}&\dot{G}_2{}
=- \dot{\tilde{G}}_2{}
=- \dot{G}^{\#}_2{},\\
\dot{\tilde{G}}^{\#}_1{}={}&\dot{G}_1{}
=- \dot{\tilde{G}}_1{}
=- \dot{G}^{\#}_1{},\\
\dot{\tilde{G}}^{\#}_0{}={}&\dot{G}_0{}
=- \dot{\tilde{G}}_0{}
=- \dot{G}^{\#}_0{},\\
\dot{\tilde{\slashed{G}}}{}^{\#}{}={}&\dot{\slashed{G}}{}
=- \dot{\tilde{\slashed{G}}}{}
=- \dot{\slashed{G}}{}^{\#}{}.
\end{align}
\end{subequations}
Furthermore, the direct linearization of the system \eqref{eq:nuetaevoleq}, \eqref{eq:thopmetric}, \eqref{eq:thopRicci}, and \eqref{eq:Bianchi} consists of
\begin{subequations}
\begin{align}
\thopBG \dot{\nu}={}&\tfrac{1}{4} i \dot{\slashed{G}}{} (\mathring{\rho}' - \bar{\mathring{\rho}}') \label{eq:ThopFGDiffSpinLinEq1},\\
\thopBG \dot{\eta}={}&\dot{\tilde{\beta}}
- \overline{\dot{\tilde{\beta}}'}
+ \dot{\eta} \mathring{\rho}'
+ i \dot{\nu} \mathring{\tau}
- \tfrac{1}{4} \dot{\slashed{G}}{} \mathring{\tau}
+ \tfrac{1}{2} \overline{\dot{G}}_2{} \bar{\mathring{\tau}} \label{eq:ThopFGDiffLLinEq1},
\end{align}
\end{subequations}
\begin{subequations}
\begin{align}
\thopBG \dot{G}_2{}={}&- \dot{G}_2{} \mathring{\rho}'
+ \dot{G}_2{} \bar{\mathring{\rho}}'
+ 2 \dot{\tilde{\sigma}}' \label{eq:ThopFGInvGTilde2LinEq1},\\
\thopBG \dot{\slashed{G}}{}={}&-2 (\dot{\tilde{\rho}}' + \overline{\dot{\tilde{\rho}}'}) \label{eq:ThopFGInvGTrTildeLinEq1},\\
\thopBG \dot{G}_1{}={}&-2 \overline{\dot{\eta}} \mathring{\rho}'
- 2 \dot{G}_1{} \mathring{\rho}'
+ \dot{G}_1{} \bar{\mathring{\rho}}'
- \dot{G}_2{} \mathring{\tau}
+ \tfrac{1}{2} \dot{\slashed{G}}{} \bar{\mathring{\tau}}
+ 2i \dot{\nu} \mathring{\tau}'
+ 2 \dot{\tilde{\tau}}' \label{eq:ThopFGInvGTilde1LinEq1},\\
\thopBG \dot{G}_0{}={}&-2 \dot{\tilde{\epsilon}}
- 2 \overline{\dot{\tilde{\epsilon}}}
- 2 \dot{G}_1{} \mathring{\tau}
- 2 \overline{\dot{G}}_1{} \bar{\mathring{\tau}}
- 2 \dot{\eta} \mathring{\tau}'
- 2 \overline{\dot{\eta}} \bar{\mathring{\tau}}' \label{eq:ThopFGInvGTilde0LinEq1},
\end{align}
\end{subequations}
\begin{subequations}
\begin{align}
\thopBG \dot{\tilde{\sigma}}'={}&\dot{\Psi}_{4}{}
+ \mathring{\rho}' \dot{\tilde{\sigma}}'
+ \bar{\mathring{\rho}}' \dot{\tilde{\sigma}}' \label{eq:ThopFGSigmapLinEq1},\\
\thopBG \dot{\tilde{\rho}}'={}&2 \mathring{\rho}' \dot{\tilde{\rho}}' \label{eq:ThopFGRhopLinEq1},\\
\thopBG \dot{\tilde{\tau}}'={}&\dot{\Psi}_{3}{}
- \dot{\tilde{\sigma}}' \mathring{\tau}
- \dot{\tilde{\rho}}' \bar{\mathring{\tau}}
+ \dot{\tilde{\rho}}' \mathring{\tau}'
+ \mathring{\rho}' \dot{\tilde{\tau}}'
+ \dot{\tilde{\sigma}}' \bar{\mathring{\tau}}' \label{eq:ThopFGTaupLinEq1},\\
\thopBG \dot{\tilde{\beta}}={}&\dot{\tilde{\beta}} \mathring{\rho}'
- i \dot{\nu} \mathring{\rho}' \mathring{\tau}
+ \tfrac{1}{4} \dot{\slashed{G}}{} \mathring{\rho}' \mathring{\tau}
+ \dot{\tilde{\rho}}' \mathring{\tau} \label{eq:ThopFGBetaLinEq1},\\
\thopBG \dot{\tilde{\beta}}'={}&\dot{\Psi}_{3}{}
+ \dot{\tilde{\beta}}' \bar{\mathring{\rho}}'
+ \tfrac{1}{2} \dot{G}_2{} \mathring{\rho}' \mathring{\tau}
- \dot{\tilde{\sigma}}' \mathring{\tau} \label{eq:ThopFGBetapLinEq1},\\
\thopBG \dot{\tilde{\epsilon}}={}&- \dot{\Psi}_{2}{}
- \overline{\dot{\eta}} \mathring{\rho}' \mathring{\tau}
- \dot{G}_1{} \mathring{\rho}' \mathring{\tau}
+ \dot{\tilde{\beta}} (- \bar{\mathring{\tau}} + \mathring{\tau}')
+ \mathring{\tau} \dot{\tilde{\tau}}'
+ \dot{\tilde{\beta}}' (\mathring{\tau} - \bar{\mathring{\tau}}') \label{eq:ThopFGEpsilonLinEq1},\\
\thopBG \dot{\tilde{\rho}}={}&- \dot{\Psi}_{2}{}
+ \mathring{\rho} \overline{\dot{\tilde{\rho}}'}
+ \dot{\tilde{\rho}} \bar{\mathring{\rho}}'
+ \overline{\dot{\tilde{\beta}}} \mathring{\tau}
+ \dot{\tilde{\beta}}' \mathring{\tau}
+ 2 \overline{\dot{\eta}} \mathring{\rho}' \mathring{\tau}
+ \tfrac{1}{2} \dot{G}_2{} \mathring{\tau}^2\nonumber\\
& - \frac{i}{8 \kappa_{1}{}} (4 \dot{\nu} - i \dot{\slashed{G}}{}) \bigl(\mathring{\Psi}_{2}{} \kappa_{1}{} - \bar{\mathring{\Psi}}_{2}{} \bar{\kappa}_{1'}{} + 2 \kappa_{1}{} (\mathring{\rho} \mathring{\rho}' - \mathring{\rho} \bar{\mathring{\rho}}' + \mathring{\tau} \mathring{\tau}')\bigr) \label{eq:ThopFGRhoLinEq1},\\
\thopBG \dot{\tilde{\sigma}}={}&\mathring{\rho}' \dot{\tilde{\sigma}}
+ \mathring{\rho} \overline{\dot{\tilde{\sigma}}'}
- \dot{\tilde{\beta}} \mathring{\tau}
- \overline{\dot{\tilde{\beta}}'} \mathring{\tau}
+ 2 \dot{\eta} \mathring{\rho}' \mathring{\tau}
+ i \dot{\nu} \mathring{\tau}^2
- \tfrac{1}{4} \dot{\slashed{G}}{} \mathring{\tau}^2\nonumber\\
& + \frac{\overline{\dot{G}}_2{}}{4 \kappa_{1}{}} \bigl(\mathring{\Psi}_{2}{} \kappa_{1}{} - \bar{\mathring{\Psi}}_{2}{} \bar{\kappa}_{1'}{} + 2 \kappa_{1}{} (\mathring{\rho} \mathring{\rho}' - \mathring{\rho} \bar{\mathring{\rho}}' + \mathring{\tau} \mathring{\tau}')\bigr) \label{eq:ThopFGSigmaLinEq1},\\
\thopBG \dot{\tilde{\kappa}}={}&- \dot{\Psi}_{1}{}
- \dot{\tilde{\epsilon}} \mathring{\tau}
+ \overline{\dot{\tilde{\epsilon}}} \mathring{\tau}
- \dot{\tilde{\rho}} \mathring{\tau}
- \dot{G}_0{} \mathring{\rho}' \mathring{\tau}
+ \overline{\dot{\eta}} \mathring{\tau}^2
+ \dot{G}_1{} \mathring{\tau}^2
+ \dot{\tilde{\sigma}} (- \bar{\mathring{\tau}} + \mathring{\tau}')\nonumber\\
& + \frac{1}{2 \kappa_{1}{}} (\dot{\eta} + \overline{\dot{G}}_1{}) \bigl(\mathring{\Psi}_{2}{} \kappa_{1}{} - \bar{\mathring{\Psi}}_{2}{} \bar{\kappa}_{1'}{} + 2 \kappa_{1}{} (\mathring{\rho} \mathring{\rho}' - \mathring{\rho} \bar{\mathring{\rho}}' + \mathring{\tau} \mathring{\tau}')\bigr)
+ \mathring{\rho} \overline{\dot{\tilde{\tau}}'}
+ \dot{\tilde{\rho}} \bar{\mathring{\tau}}' \label{eq:ThopFGKappaLinEq1},
\end{align}
\end{subequations}
\begin{subequations}
\begin{align}
\thoBG \dot{\Psi}_{1}{} - \edtpBG \dot{\Psi}_{0}{}={}&-3 \mathring{\Psi}_{2}{} \dot{\tilde{\kappa}}
+ 4 \dot{\Psi}_{1}{} \mathring{\rho}
- \dot{\Psi}_{0}{} \mathring{\tau}',\\
\thoBG \dot{\Psi}_{2}{} - \edtpBG \dot{\Psi}_{1}{}={}&3 \dot{\Psi}_{2}{} \mathring{\rho}
+ 3 \mathring{\Psi}_{2}{} \dot{\tilde{\rho}}
+ \tfrac{3}{2} \dot{G}_0{} \mathring{\Psi}_{2}{} \mathring{\rho}'
- 3 \overline{\dot{\eta}} \mathring{\Psi}_{2}{} \mathring{\tau}
- 3 \dot{G}_1{} \mathring{\Psi}_{2}{} \mathring{\tau}
- 3 \dot{\eta} \mathring{\Psi}_{2}{} \mathring{\tau}'
- 3 \overline{\dot{G}}_1{} \mathring{\Psi}_{2}{} \mathring{\tau}'
- 2 \dot{\Psi}_{1}{} \mathring{\tau}',\\
\thoBG \dot{\Psi}_{3}{} - \edtpBG \dot{\Psi}_{2}{}={}&2 \dot{\Psi}_{3}{} \mathring{\rho}
+ 3 \overline{\dot{\eta}} \mathring{\Psi}_{2}{} \mathring{\rho}'
+ \tfrac{3}{2} \dot{G}_2{} \mathring{\Psi}_{2}{} \mathring{\tau}
- 3i \dot{\nu} \mathring{\Psi}_{2}{} \mathring{\tau}'
- \tfrac{3}{4} \dot{\slashed{G}}{} \mathring{\Psi}_{2}{} \mathring{\tau}'
- 3 \dot{\Psi}_{2}{} \mathring{\tau}'
- 3 \mathring{\Psi}_{2}{} \dot{\tilde{\tau}}',\\
\thoBG \dot{\Psi}_{4}{} - \edtpBG \dot{\Psi}_{3}{}={}&\dot{\Psi}_{4}{} \mathring{\rho}
+ 3 \mathring{\Psi}_{2}{} \dot{\tilde{\sigma}}'
- 4 \dot{\Psi}_{3}{} \mathring{\tau}',\\
\thopBG \dot{\Psi}_{0}{} - \edtBG \dot{\Psi}_{1}{}={}&\dot{\Psi}_{0}{} \mathring{\rho}'
+ 3 \mathring{\Psi}_{2}{} \dot{\tilde{\sigma}}
- 4 \dot{\Psi}_{1}{} \mathring{\tau},\\
\thopBG \dot{\Psi}_{1}{} - \edtBG \dot{\Psi}_{2}{}={}&3 \dot{\eta} \mathring{\Psi}_{2}{} \mathring{\rho}'
+ 2 \dot{\Psi}_{1}{} \mathring{\rho}'
+ 3i \dot{\nu} \mathring{\Psi}_{2}{} \mathring{\tau}
- \tfrac{3}{4} \dot{\slashed{G}}{} \mathring{\Psi}_{2}{} \mathring{\tau}
- 3 \dot{\Psi}_{2}{} \mathring{\tau}
+ \tfrac{3}{2} \overline{\dot{G}}_2{} \mathring{\Psi}_{2}{} \mathring{\tau}',\\
\thopBG \dot{\Psi}_{2}{} - \edtBG \dot{\Psi}_{3}{}={}&3 \dot{\Psi}_{2}{} \mathring{\rho}'
+ 3 \mathring{\Psi}_{2}{} \dot{\tilde{\rho}}'
- 2 \dot{\Psi}_{3}{} \mathring{\tau},\\
\thopBG \dot{\Psi}_{3}{} - \edtBG \dot{\Psi}_{4}{}={}&4 \dot{\Psi}_{3}{} \mathring{\rho}'
- \dot{\Psi}_{4}{} \mathring{\tau}.
\end{align}
\end{subequations}
Under the gauge transformation in theorem \ref{thm:NLORGEnforceability}, the linearization is such that $\dot{\slashed{G}}{}=0$.
\end{theorem}
\subsection{Tensor linearization}
There is another approach to linearizing the Einstein equation with the radiation gauge condition. This involves linearizing the equations for the metric $g$ and other tensorial quantities. We refer to this as tensor linearization.
One of the central motivations for this paper is to construct a gauge condition for the Einstein equation so that its linearization is the linear radiation gauge condition{} and, furthermore, to make a more restrictive gauge choice so that its linearization is the full radiation gauge of Chrzanowski, which has long been studied by, for example, \cite{Chrzanowski, Price:2006ke}. We have previously considered some of the consequences of the full radiation gauge of Chrzanowski{} in \cite{Andersson:2019dwi}. We encapsulate these results in the statement of the following theorem:
\begin{theorem}[Tensor linearization]
\label{thm:tensorLinearization}
Assume the background hypotheses of definition \ref{def:radiationGaugeHypotheses}.
The direct linearization of the radiation gauge condition{} \eqref{eq:NLnh} for a metric $g$ is the linear radiation gauge condition{} \eqref{eq:linearnh}. In the linearization under the gauge transformation in theorem \ref{thm:NLORGEnforceability}, the full radiation gauge of Chrzanowski{} of \ref{def:linearORG} holds.
Consider the system consisting of the definition of the connection, the definition of the curvature, the Bianchi relations, and the Einstein equations for the difference of metrics $g-\mathring{g}$, the tensor $\Gamma$ being the difference\footnote{Recall that although a connection is not tensorial, for the difference between two connections is a tensor.} between the Levi-Civita connections for $g$ and $\mathring{g}$, and for the difference of the curvature of $g$ and $\mathring{g}$. The direct linearization of this system constrained by the full radiation gauge of Chrzanowski{} together imply the system constructed in \cite{Andersson:2019dwi}.
\end{theorem}
The quantities in the scalar and tensor linearizations can be related. On the one hand, one can take the variables in the tensor linearization in our previous work \cite{Andersson:2019dwi}. On the other, one can take the variables in the scalar linearization under the gauge transformation in theorem \ref{thm:NLORGEnforceability} as give in theorem \ref{thm:scalarLinearization}. Most notably, in both the linearizations, the perturbation of the trace vanishes. Because the paper \cite{Andersson:2019dwi} is a tensorial linearization, it works solely with the background frame. The quantities appearing in our previous work \cite{Andersson:2019dwi} can be related to those in the current paper via the following table assuming that the trace condition holds.
\begin{center}
\begin{tabular}{|l|l|}
\hline
\cite{Andersson:2019dwi} & This paper\\
\hline
$G_{00'}$ & $\dot{G}_0{}$\\
$G_{10'}$ & $\dot{G}_1{}$\\
$G_{20'}$ &$\dot{G}_2{}$\\
$\tilde{\beta}$&$\dot{\tilde{\beta}}
+ \dot{\eta} \mathring{\rho}'
+ \tfrac{1}{2} \overline{\dot{G}}_1{} \mathring{\rho}'$\\
$\tilde{\beta}'$&$\dot{\tilde{\beta}}'$\\
$\tilde{\epsilon}$&$\dot{\tilde{\epsilon}}
+ \dot{\eta} \mathring{\tau}'
+ \tfrac{1}{2} \overline{\dot{G}}_1{} \mathring{\tau}'$\\
$\tilde{\kappa}$&$\dot{\tilde{\kappa}}
- \dot{\eta} \mathring{\rho}
- \overline{\dot{G}}_1{} \mathring{\rho}
+ \tfrac{1}{2} \dot{G}_0{} \mathring{\tau}
+ \thoBG \dot{\eta}
+ \tfrac{1}{2} \thoBG \overline{\dot{G}}_1{}$\\
$\tilde{\rho}$&$\dot{\tilde{\rho}}
- \overline{\dot{\eta}} \mathring{\tau}
+ \edtpBG \dot{\eta}
+ \tfrac{1}{2} \edtpBG \overline{\dot{G}}_1{}$\\
$\tilde{\sigma}$&$
\dot{\tilde{\sigma}}
- \tfrac{1}{2} \overline{\dot{G}}_2{} \mathring{\rho}
- \dot{\eta} \mathring{\tau}
+ \edtBG \dot{\eta}
+ \tfrac{1}{2} \edtBG \overline{\dot{G}}_1{}$\\
$\tilde{\sigma}'$&$
\dot{\tilde{\sigma}}'
- \tfrac{1}{2} \dot{G}_2{} \mathring{\rho}'$\\
$\tilde{\tau}'$&$
\dot{\tilde{\tau}}'
- \overline{\dot{\eta}} \mathring{\rho}'
- \dot{G}_1{} \mathring{\rho}'$\\
$\vartheta \Psi_i$ & $\dot{\Psi}_i$\\
\hline
\end{tabular}
\end{center}
Since, the main result of \cite{Andersson:2019dwi} is a stability statement, it is worth briefly discussing the stability of some of the quantities that appearing in the scalar linearization in this paper that do not appear in \cite{Andersson:2019dwi}.
One major difference between the scalar linearization in this paper and the linearization in \cite{Andersson:2019dwi} is the presence, in the scalar linearization of $\dot{\slashed{G}}{}$ and $\dot{\tilde{\rho}}'$. In the direct linearization, these quantities satisfies \eqref{eq:ThopFGInvGTrTildeLinEq1} and \eqref{eq:ThopFGRhopLinEq1} i.e.
\begin{subequations}
\begin{align}
\thopBG \dot{\slashed{G}}{}={}&-2 (\dot{\tilde{\rho}}' + \overline{\dot{\tilde{\rho}}'}),\\
\thopBG \dot{\tilde{\rho}}'={}&2 \mathring{\rho}' \dot{\tilde{\rho}}' .
\end{align}
\end{subequations}
There are two approaches to considering the stability of $\dot{\slashed{G}}{}$ and $\dot{\tilde{\rho}}'$. First, one could note that under the ORG, one simply has
$\dot{\slashed{G}}{}=0$ and hence $\dot{\tilde{\rho}}'=0$. Note that the above only determines the real part of $\dot{\tilde{\rho}}'$, but imaginary part must also vanish under the direct linearization of equation \eqref{eq:algebraicrho1}.
Alternatively, one could treat these ODEs as generating a dynamics and apply the method for proving decay of solutions of transport equations that was introduced in \cite{Andersson:2019dwi}. Qualitatively, the argument proceeds as follows. One works in Boyer-Lindquist coordinates and wishes to prove decay of a variable in $t$ for fixed $r$, assuming that the solution decays rapidly on the initial surface $\{t=0\}$ as $r\rightarrow\infty$. For a transport equation of the form $\thop\varphi=0$, one has that the value of $\varphi$ at $(t_1,r_1,\omega_1)$ is equal to the value of $\varphi$ at the intersection of the initial hypersurface $\{t=0\}$ with null geodesic tangent to $n$ going through $(t_1,r_1,\omega_1)$, which occurs at $t_0=0$ and $r_0-r_1$ is bounded above and below by positive multiples of $t_1$ for $t_1>1$. Thus, $|\varphi|$ decays in $t$ because the initial data decays in $r$. Similarly, for an equation of the form $\thop\varphi=c_1\mathring{\rho}'\varphi$, one can introduce an integrating factor $\mathring{\rho}'{}^{c_2}$, and the growth or decay arising from the change in value of this integrating factor can be more than compensated for if the decay of the initial data is sufficiently fast. Furthermore, for an inhomogeneous equation of the form $\thop\varphi=c_1\mathring{\rho}'\varphi+\vartheta$, applying the integrating factor and integrating, the contribution from integrating $\vartheta$ is like $t$ (the length of the integration along the geodesic) times the maximum of $|\vartheta|$ (the maximum on the geodesic), but if this also decays in $t+r$, then the additional factor of $t$ from the integration can be dominated by the decay in $t+r$, although the decay of $\varphi$ will be one power worse than that of $\vartheta$. Applying this method schematically, one sees that if $\dot{\tilde{\rho}}'$ decays rapidly as $r\rightarrow\infty$ on the initial hypersurface $\{t=0\}$, then it will also decay rapidly as $t\rightarrow\infty$ at fixed $r$. Integrating the transport equation for $\tilde{\slashed{G}}{}^{\#}_{\phantom{0}}{}$, one obtains that $\tilde{\slashed{G}}{}^{\#}_{\phantom{0}}{}$ also decays rapidly (although not quite as rapidly). In future work, we will investigate the quantitative behaviour. In doing so, we note that we will have at our disposal the diffeomorphism that allows us to set $\tilde{\slashed{G}}{}^{\#}_{\phantom{0}}{}$ to vanish quadratically, which one might plausibly expect to allow one to show that such a diffeomorphism could be chosen so that $\tilde{\slashed{G}}{}^{\#}_{\phantom{0}}{}$ and $\tilde{\rho}'$ vanish much more rapidly than the other metric and connection coefficients. As shown in \cite{Andersson:2019dwi}, these methods for proving decay apply not only in Boyer-Lindquist coordinates at fixed $r$ as $t\rightarrow\infty$, but also in hyperboloidal coordinates that allow for precise estimates near null infinity; such estimates are likely to be crucial for controlling nonlinear terms in the Einstein equation.
Another major difference arises from the presence of the differential Lorentz transformations $(\eta,\nu)$. Recall that \cite{Andersson:2019dwi} only uses the background frame $(\mathring{\vecL},\mathring{\vecN},\mathring{\vecM},\bar{\mathring{\vecM}})$, whereas most of the results in this paper use the foreground frame $(l,n,m,\bar{m})$. The difference between these two frames is given by the differential Lorentz transform variables $(\eta,\nu)$.
Before doing so, we note some other relations for other variables.
One finds that $\dot\varsigma=\dot{\slashed{G}}{}/4$ and $\dot\varsigma^\#=-\dot{\slashed{G}}{}/4$, which vanish under the full radiation gauge of Chrzanowski. Note that the background values of $\varsigma$ and $\varsigma^\#$ are both $1$.
The direct linearization of equation \eqref{eq:algebraicbeta1} is
\begin{align}
\dot{\tilde{\beta}} - \overline{\dot{\tilde{\beta}}'}={}&- \dot{\eta} \mathring{\rho}'
- \overline{\dot{G}}_1{} \mathring{\rho}'
+ \dot{\eta} \bar{\mathring{\rho}}'
+ \overline{\dot{G}}_1{} \bar{\mathring{\rho}}'
+ \tfrac{1}{2} \overline{\dot{G}}_2{} \mathring{\tau}'
- \overline{\dot{\tilde{\tau}}'}
+ i \dot{\nu} \bar{\mathring{\tau}}'
- \tfrac{1}{4} \dot{\slashed{G}}{} \bar{\mathring{\tau}}'.
\end{align}
Substituting this formula into equation \eqref{eq:ThopFGDiffLLinEq1}, one finds that the equations \eqref{eq:ThopFGDiffSpinLinEq1} and \eqref{eq:ThopFGDiffLLinEq1} become
\begin{subequations}
\begin{align}
\thopBG \dot{\nu}={}&\tfrac{1}{4} i \dot{\slashed{G}}{} (\mathring{\rho}' - \bar{\mathring{\rho}}'),\\
\thopBG \dot{\eta}={}&
\dot{\eta} \bar{\mathring{\rho}}'
- \overline{\dot{G}}_1{} (\mathring{\rho}' - \bar{\mathring{\rho}}')
+ \tfrac{1}{2} \overline{\dot{G}}_2{} (\bar{\mathring{\tau}} + \mathring{\tau}')
- \overline{\dot{\tilde{\tau}}'}
+ i \dot{\nu} (\mathring{\tau} + \bar{\mathring{\tau}}')
- \tfrac{1}{4} \dot{\slashed{G}}{} (\mathring{\tau} + \bar{\mathring{\tau}}').
\end{align}
\end{subequations}
The behaviour of $\dot{\nu}$ can be treated via either of the methods used to discuss $\dot{\slashed{G}}{}$ and $\dot{\tilde{\rho}}'$. On the one hand, one can set $\dot{\slashed{G}}{}=0$, so that $\dot{\nu}$ is constant and can be chosen to have zero initial data, so that it is globally zero. Alternatively, from integrating the transport equation for $\dot{\slashed{G}}{}$, one obtains that if $\dot{\slashed{G}}{}$ decays rapidly, then $\dot{\nu}$ also decays rapidly although not quite as fast. For $\dot{\eta}$, it is not clear how one could apply the first argument. However, assuming all the linearized quantities on the right of the transport equation vanish rapidly, then one obtains that $\dot{\eta}$ also vanishes rapidly although not quite as fast. Again, we intend to explore this quantitatively in future work.
\subsection{Smallness of \texorpdfstring{$\tilde\beta$}{beta tilde}}
\label{sec:SmallnessBeta}
In this section, we show that it is possible to choose initial data for $\eta$ (or its linearization) so that $\tilde{\beta}$ vanishes to linear order in both the linear and nonlinear settings. While it may seem natural to choose initial data with $\dot{\eta}=0$, this does not propagate, even when $\dot{\slashed{G}}{}=0$.
First, consider the linearized setting. The previous subsection argues that we can choose $\dot{\slashed{G}}{}=0$, $\dot{\tilde{\rho}}'=0$ and $\dot{\nu}=0$. With these choices \eqref{eq:ThopFGBetaLinEq1} gives a homogeneous evolution equation for $\dot{\tilde{\beta}}$. Thus, if $\dot{\tilde{\beta}}$ can be chosen to be initially zero, it remains so. Assuming that the linearized versions of equations \eqref{eq:ThopInvGTilde1}, \eqref{eq:algebraicbeta1}, and \eqref{eq:ethnuTobeta} hold, it follows that the vanishing of $\dot{\tilde{\beta}}$ is equivalent to each of the following
\begin{subequations}
\label{eq:smallBEtaLinearEquations}
\begin{align}
\dot{\eta}={}&- \tfrac{1}{4} \mathring{\rho}'^{-1} (\thopBG{} + 3 \mathring{\rho}' - 2 \bar{\mathring{\rho}}')\overline{\dot{G}}_1{}
+ \tfrac{1}{4} \mathring{\rho}'^{-1} (\edtpBG{} - \bar{\mathring{\tau}} + \mathring{\tau}')\overline{\dot{G}}_2{},\\
\dot{\tilde{\tau}}'={}&\dot{G}_1{} \mathring{\rho}'
+ (\tfrac{1}{2} - \tfrac{1}{4} \kappa_{1}{}^{-1} \bar{\kappa}_{1'}{}) (\thopBG{} - 2 \mathring{\rho}' - \bar{\mathring{\rho}}')\dot{G}_1{}
+ \tfrac{1}{4} \kappa_{1}{}^{-1} \bar{\kappa}_{1'}{} (\edtBG{} - \mathring{\tau} - \bar{\mathring{\tau}}')\dot{G}_2{},\\
\dot{\tilde{\beta}}'={}&\tfrac{1}{4} (\thopBG{} + 2 \mathring{\rho}' - \bar{\mathring{\rho}}')\dot{G}_1{}
+ \tfrac{1}{4} (\edtBG{} + \mathring{\tau} - \bar{\mathring{\tau}}')\dot{G}_2{}.
\end{align}
\end{subequations}
From the first of these, we see that $\dot{\eta}$ can be chosen initially so that $\dot{\tilde{\beta}}$ vanishes initially. Recall that the initial data for the metric, spin coefficients, and curvature components must satisfy constraint equations, so they cannot be all freely specified. Although equation \eqref{eq:ThopInvGTilde1} is an evolution equation, equations \eqref{eq:algebraicbeta1} and \eqref{eq:ethnuTobeta} can be viewed as constraints on the initial data, and these impose constraints in the linearization. Due to the fact that the evolution equation for $\dot{\tilde{\beta}}$ is homogeneous, it follows that $\dot{\tilde{\beta}}$ will remain zero, and hence that these three equations \eqref{eq:smallBEtaLinearEquations} will remain valid.
Now consider the nonlinear case. In this case, given initial data for the metric and its derivatives, one is free to choose initial data for $\nu$ and $\eta$ in the frame gauge. Once initial data for $\nu$ has been chosen, it is possible to compute $\tilde{\sigma}'$ and $\tilde{\rho}'$ via equation \eqref{eq:rhoTildePrimeSigmaTildePrime} purely in terms of quantities defined with respect to the background tetrad and $\nu$, without having specified $\eta$. Thus, for example, one may choose the initial value for $\eta$ so that
\begin{align}
\eta ={}&- \frac{3 \overline{G_{2}} G_{1} \varsigma^{\#}{}^3}{4 \varsigma^2 \mathring{\rho}'} (\mathring{\rho}' - \bar{\mathring{\rho}}')
+ \frac{\overline{G_{1}} \varsigma^{\#}{}^2}{2 \varsigma \mathring{\rho}'} (2 - 3 \varsigma^{\#}{} \varsigma) (\mathring{\rho}' - \bar{\mathring{\rho}}')
+ \frac{\overline{G_{2}} \varsigma^{\#}{}^2 \tilde{\sigma}'}{8 \varsigma^3 \mathring{\rho}'} (\overline{G_{2}} G_{1} + 2 \overline{G_{1}} \varsigma^2)\nonumber\\
& - \frac{\varsigma^{\#}{}^2 \overline{\tilde{\sigma}'}}{4 \varsigma \mathring{\rho}'} (\overline{G_{1}} G_{2} + 2 G_{1} \varsigma^2)
- \frac{\overline{G_{2}} \varsigma^{\#}{}}{4 \mathring{\rho}'} (\bar{\mathring{\tau}} - \mathring{\tau}')
+ \frac{1}{4 \mathring{\rho}'} (2 \varsigma^{\#}{} \varsigma^2 - \varsigma^{\#}{} - \varsigma ) (\mathring{\tau} - \bar{\mathring{\tau}}')\nonumber\\
& - \frac{\varsigma^{\#}{} (\thopBG{} + \mathring{\rho}' + 2 \tilde{\rho}')\overline{G_{1}}}{4 \mathring{\rho}'}
- \frac{\varsigma^{\#}{}^2 \edtBG \slashed{G}_{}}{8 \varsigma \mathring{\rho}'} (1 - 2 \varsigma^{\#}{} \varsigma)
+ \frac{G_{2} \varsigma^{\#}{}^2 \edtBG \overline{G_{2}}}{16 \varsigma^3 \mathring{\rho}'} (1 - 2 \varsigma^{\#}{} \varsigma)\nonumber\\
& - \frac{\overline{G_{2}} \varsigma^{\#}{}^2 \edtBG G_{2}}{16 \varsigma^3 \mathring{\rho}'} (1 + 2 \varsigma^{\#}{} \varsigma)
- \frac{\overline{G_{2}} \varsigma^{\#}{}^3 \edtpBG \slashed{G}_{}}{8 \varsigma^2 \mathring{\rho}'}
+ \frac{\varsigma^{\#}{}^3 \edtpBG \overline{G_{2}}}{4 \mathring{\rho}'}
+ \frac{\overline{G_{2}}{}^2 \varsigma^{\#}{}^3 \edtpBG G_{2}}{16 \varsigma^4 \mathring{\rho}'} .
\end{align}
With this choice, and trivial initial data for $\nu$, it follows from equations \eqref{eq:ThopInvGTilde1}, \eqref{eq:algebraicbeta1}, and \eqref{eq:ethnuTobeta}, that the initial data for $\tilde{\beta}$, $\tilde{\tau}'$ and $\tilde{\beta}'$ takes the form
\begin{subequations}
\label{eq:smallBEtaNonLinearEquations}
\begin{align}
\tilde{\beta}={}&- \eta \tilde{\rho}'
+ \frac{\bar{\eta} \overline{G_{2}} \bar{\mathring{\rho}}'}{2 \varsigma^2} ,\\
\tilde{\tau}'={}&\bar{\eta} (\mathring{\rho}' + \tilde{\rho}')
+ \frac{\varsigma^{\#}{}}{4 \varsigma^2} (\overline{G_{1}} G_{2} + 2 G_{1} \varsigma^2) (\mathring{\rho}' + 2 \tilde{\rho}')
+ \tfrac{1}{2} G_{1} \varsigma^{\#}{} (\mathring{\rho}' - \bar{\mathring{\rho}}')
+ \eta \tilde{\sigma}'
+ \frac{\varsigma^{\#}{} \tilde{\sigma}'}{2 \varsigma^2} (\overline{G_{2}} G_{1} + 2 \overline{G_{1}} \varsigma^2)\nonumber\\
& + \frac{G_{2} \mathring{\tau}}{4 \varsigma^2} (\varsigma^{\#}{} + \varsigma)
+ \tfrac{1}{2} (\varsigma^{\#}{} - \varsigma) \bar{\mathring{\tau}}
- \tfrac{1}{2} (2 - \varsigma^{\#}{} - \varsigma) \mathring{\tau}'
+ \frac{G_{2} \bar{\mathring{\tau}}'}{4 \varsigma^2} (\varsigma^{\#}{} - \varsigma)
+ \frac{G_{2} \varsigma^{\#}{} \thopBG \overline{G_{1}}}{4 \varsigma^2}
+ \tfrac{1}{2} \varsigma^{\#}{} \thopBG G_{1},\\
\tilde{\beta}'={}&\frac{\eta G_{2} \mathring{\rho}'}{2 \varsigma^2}
- \frac{\varsigma^{\#}{}^2}{2 \varsigma^3} (\overline{G_{1}} G_{2} + 2 G_{1} \varsigma^2) (\mathring{\rho}' - \bar{\mathring{\rho}}')
- \bar{\eta} (\mathring{\rho}' + \tilde{\rho}' - \bar{\mathring{\rho}}')
+ (1 - \varsigma^{\#}{}) \mathring{\tau}'
+ \tilde{\tau}'
- \frac{G_{2} \varsigma^{\#}{} \bar{\mathring{\tau}}'}{2 \varsigma^2} .
\end{align}
\end{subequations}
In particular, $\tilde{\beta}$ vanishes quadratically. The choice of $\eta$ is not unique, in that there are other choices of $\eta$ for which $\tilde{\beta}$ also vanishes quadratically. These equations will not propagate under the evolution, although $\tilde{\beta}$ will remain quadratically small for evolution under equation \eqref{eq:thopbeta}.
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| 5
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\section{Introduction}
Total masses of galaxy clusters are of central importance in multiple cosmological measurements, notably those involving the cluster baryon fraction and mass function (for a review, see {\it e.g.\ } \citealt{aem11}).
Historically, much of this work has used mass estimates from X-ray data, ultimately relying on the assumption of hydrostatic equilibrium.
In general, for a randomly selected cluster and arbitrary measurement radius, simulations predict that departures from equilibrium will induce bias at the tens of per cent level between such estimates and the true cluster masses \citep{nagai07, lau09}.
Restricting the analysis to intermediate measurement radii and the most massive, dynamically relaxed clusters reduces the expected bias and scatter considerably ($\ {\raise-.75ex\hbox{$\buildrel<\over\sim$}}\ 10$ per cent).
The small intrinsic scatter seen in gas mass fractions, $f_{\mathrm{gas}}$\ , for such clusters (e.g., Paper~II of this series, \citealt{mantz_fgas}) verifies that hydrostatic X-ray mass estimates trace the true mass well at these radii for these clusters.
However, some overall average bias may remain, whether due to residual non-thermal support or instrument calibration \mbox{\citep{ndg10, schellenberger14, nelson14}}.
Determining this bias requires independent mass measurements for the same clusters using an unbiased method.
Weak gravitational lensing (WL) provides a direct probe of the gravitational potential of a cluster, independent of baryonic physics.
While WL mass estimates are inherently noisy due to cluster triaxiality and line-of-sight structure, they can in principle provide unbiased mass estimates on average \citep{bas01, saasfee}.
In practice, systematic effects must be carefully controlled, as demonstrated by the Weighing the Giants project \citep[WtG;][]{paper1,paper2,paper3}.
But, with this in hand, WL can provide the necessary overall calibration of X-ray hydrostatic mass estimates.
A growing body of work has used WL to either directly calibrate cosmological tests or to understand the biases in published studies \citep{anja_planck, israel14, paper4, hoekstra2015}.
In addition to enhancing cluster cosmology directly, measurements of the lensing to X-ray mass ratio constrain the net effect of astrophysical and instrumental biases on the X-ray measurements\footnote{Unless otherwise explicitly stated, the term ``X-ray masses'' refer to hydrostatic mass estimates.}.
Several groups have previously compared lensing masses to X-ray masses measured in various ways \citep[{\it e.g.\ }][]{vbe09, zhang10, planck_i3, mahdavi13, israel14, donahue14}.
However, those efforts have used different X-ray telescopes, different calibration versions for those telescopes, varying sample selection, and varying lensing and X-ray analysis methods.
As a result, it is difficult to draw conclusions any broader than the relative calibration of each individual set of WL and X-ray mass estimates.
This only underscores the need for cluster mass estimation with strict tolerances on systematic uncertainties, which WL can, in principle, provide.
This is the fourth in a series of papers dedicated to studying the cosmology and astrophysics enabled by X-ray and robust lensing observations of dynamically relaxed clusters.
Paper~I \citep{mantz_morph} details the morphological selection of dynamically relaxed clusters from a sizable subset of cluster observations in the \emph{Chandra}{} data archive\footnote{\url{http://cxc.harvard.edu/cda}} and the X-ray data processing of the sample.
Paper~II \citep{mantz_fgas} derives cosmological constraints from the sample, using measurements of gas mass fractions, while Paper~III \citep{mantz_profiles} studies the thermodynamics and scaling relations of these clusters.
The current paper uses the 12 clusters in common between our relaxed sample and WtG to constrain the ratio of lensing to X-ray mass estimates, a calibration that was used for the cosmological measurements in Paper~II, and that impacts directly on the constraints on the cosmic mean matter density, $\Omega_\mathrm{m}$.
\label{sec:contents}
The paper is organized as follows.
In Section~\ref{sec:data}, we introduce our sample and mass measurements.
We calibrate our X-ray hydrostatic masses with lensing in Section~\ref{sec:xlratio} and investigate the robustness of the measurement in Section~\ref{sec:systematics}.
We measure the average concentration for the relaxed cluster sample in Section~\ref{sec:ave_concen}.
Finally, in Section~\ref{sec:discussion}, we provide physical explanations for our results and place them in context with other literature efforts.
Additionally in Section~\ref{sec:discussion}, we discuss our blinding strategy and examine the cosmology dependence of the lensing to X-ray mass ratio.
Concluding remarks are presented in Section~\ref{sec:summary}.
Unless otherwise noted, all mass measurements assume a flat
$\Lambda$CDM reference cosmology with $\Omega_{\rm m} = 0.3$,
$\Omega_{\Lambda} = 0.7$ and $H_0 = 100 \,h\, \mbox{km/s/Mpc}$, where
$h=0.7$.
\section{Observations, Mass Measurements, \& Statistical Methods}
\label{sec:data}
In this section, we describe how the sample was selected and review how the X-ray and lensing masses were measured.
\subsection{Sample Selection}
Our sample selection is motivated by the goal of measuring the lensing to X-ray mass ratio in as many massive, dynamically relaxed clusters as possible, while maintaining robust control of systematic uncertainties.
As described in Paper~I, we conducted a systematic search for relaxed clusters from 361 cluster observations in the \emph{Chandra}{} and ROSAT archives.
The dynamical state of each cluster was evaluated using an automated morphological classification.
Three discriminators were used to determine the morphological state of a cluster: sharpness of the surface brightness peak, offsets in the centers of neighboring isophotes, and offsets of individual isophotes from a global center.
Surface brightness levels used for the isophote measurements were selected to probe equivalent physical scales, assuming clusters evolve following the self-similar model of Kaiser \citetext{\citealp{kaiser86}; see also \citealp{santos08}}.
This procedure was designed to fairly compare clusters at varying redshifts and signal-to-noise levels, while also avoiding strong assumptions about cosmology (for example, the expansion history).
The employed criteria correlate with traditional measures of morphological disturbance such as surface brightness ``concentration'' and centroid variance (see Paper I for details).
Paper II imposed further requirements for the inclusion of targets in the cosmological analysis, namely a cut on the average gas temperature ($kT \ge 5 \mathrm{keV}$), as well as stricter data quality cuts.
For the lensing comparison, we use the Weighing the Giants (WtG) sample \citep{paper1}, which comprises 51 X-ray selected clusters with Subaru-SuprimeCam follow-up data \citep{mks02}.
The clusters used in this analysis are listed in Table~\ref{tab:targets}.
We refer to the 12 clusters in common between WtG and the dynamically relaxed cluster sample above as the ``relaxed WtG'' sample in this paper.
For our tests of morphology dependence of the lensing to X-ray ratio, we supplemented this sample with five additional clusters common to \citet{allen08} and WtG that fail at least one morphological criteria of Paper~I, which we call the ``marginal'' sample.
By using the WtG analysis, we benefit from uniformly derived, accurate WL masses for all clusters in the comparison.
\begin{table*}
\centering
\caption[]{%
The sample of massive, dynamically relaxed clusters used in this work. Column [1] name; [2] adopted redshift;
[3], [4] J2000 coordinates of adopted cluster center;
[6] \emph{Chandra}{} clean exposure time (ks); [7] Optical filter coverage; [8] Lensing Band (exposure in s, seeing). See Paper I for details of the X-ray data-set and processing; see \citet{paper1} for details of the lensing data-set, filter definitions, and processing.
%
}
\begin{tabular}{llcc c c c }
\hline
\textbf{Cluster} & \textbf{z} & \textbf{RA} & \textbf{Dec} & \textbf{X-ray Exposure} & \textbf{Optical Filters} & \textbf{Lensing Band} \\
\hline
\multicolumn{7}{c}{\textbf{Relaxed WtG Sample}}\\
Abell 2204 & 0.152 & 16:32:47.1 & 05:34:31.4 & 89.4 & {\it B}$_{\rm J}${\it V}$_{\rm J}${\it R}$_{\rm C}${\it g}$^{\star}${\it r}$^{\star}$ & {\it V}$_{\rm J}$ (1038, 0.58) \\
RXJ2129.6+0005 & 0.235 & 21:29:39.9 & 00:05:18.3 & 36.8 & {\it B}$_{\rm J}${\it V}$_{\rm J}${\it R}$_{\rm C}${\it i}$^{+}$ & {\it V}$_{\rm J}$ (1863, 0.58) \\
Abell 1835 & 0.252 & 14:01:01.9 & 02:52:39.0 & 205.3 & {\it V}$_{\rm J}${\it I}$_{\rm C}${\it i}$^{+}${\it g}$^{\star}${\it r}$^{\star}$ & {\it i}$^{+}$ (1944, 0.91) \\
MS2137.3-2353 & 0.313 & 21:40:15.2 & -23:39:40.0 & 63.2 & {\it B}$_{\rm J}${\it V}$_{\rm J}${\it R}$_{\rm C}${\it I}$_{\rm C}${\it z}$^{+}$ & {\it R}$_{\rm C}$ (5562, 0.57) \\
MACSJ1115.8+0129 & 0.355 & 11:15:51.9 & 01:29:54.3 & 45.3 & {\it V}$_{\rm J}${\it R}$_{\rm C}${\it I}$_{\rm C}$ & {\it R}$_{\rm C}$ (1944, 0.65) \\
RXJ1532.9+3021 & 0.363 & 15:32:53.8 & 30:20:58.9 & 102.2 & {\it B}$_{\rm J}${\it V}$_{\rm J}${\it R}$_{\rm C}${\it I}$_{\rm C}${\it z}$^{+}${\it u}$^{\star}$ & {\it R}$_{\rm C}$ (2106, 0.55) \\
MACSJ1720.3+3536 & 0.391 & 17:20:16.8 & 35:36:27.0 & 53.2 & {\it B}$_{\rm J}${\it V}$_{\rm J}${\it R}$_{\rm C}${\it I}$_{\rm C}${\it z}$^{+}$ & {\it V}$_{\rm J}$ (1944, 0.69) \\
MACSJ0429.6-0253 & 0.399 & 04:29:36.1 & -02:53:07.5 & 19.3 & {\it V}$_{\rm J}${\it R}$_{\rm C}${\it I}$_{\rm C}$ & {\it R}$_{\rm C}$ (2592, 0.73) \\
RXJ1347.5-1145 & 0.451 & 13:47:30.6 & -11:45:10.0 & 67.3 & {\it B}$_{\rm J}${\it V}$_{\rm J}${\it R}$_{\rm C}${\it I}$_{\rm C}${\it z}$^{+}${\it u}$^{\star}${\it g}$^{\star}${\it r}$^{\star}${\it i}$^{\star}${\it z}$^{\star}$ & {\it R}$_{\rm C}$ (2592, 0.69) \\
MACSJ1621.6+3810 & 0.461 & 16:21:24.8 & 38:10:09.0 & 134.0 & {\it B}$_{\rm J}${\it V}$_{\rm J}${\it R}$_{\rm C}${\it I}$_{\rm C}${\it z}$^{+}${\it u}$^{\star}$ & {\it I}$_{\rm C}$ (1568, 0.52) \\
MACSJ1427.3+4408 & 0.487 & 14:27:16.2 & 44:07:31.0 & 51.0 & {\it V}$_{\rm J}${\it R}$_{\rm C}${\it z}$^{+}$ & {\it R}$_{\rm C}$ (2544, 0.59) \\
MACSJ1423.8+2404 & 0.539 & 14:23:47.9 & 24:04:42.3 & 123.0 & {\it B}$_{\rm J}${\it V}$_{\rm J}${\it R}$_{\rm C}${\it I}$_{\rm C}${\it z}$^{+}${\it u}$^{\star}$ & {\it I}$_{\rm C}$ (1944, 0.73) \\
\hline
\multicolumn{7}{c}{\textbf{Marginal Sample}}\\
A963 & 0.206 & 10:17:03.562 & 39:02:51.51 & 38.3 &{\it V}$_{\rm J}${\it R}$_{\rm C}${\it I}$_{\rm C}$ & {\it I}$_{\rm C}$ (2700, 0.61) \\
A2390 & 0.233 & 21:53:37.070 & 17:41:45.39 & 79.4 &{\it B}$_{\rm J}${\it V}$_{\rm J}${\it R}$_{\rm C}${\it I}$_{\rm C}${\it i}$^{+}${\it z}$^{+}${\it u}$^{\star}$ & {\it R}$_{\rm C}$ (3420, 0.56)\\
A611 & 0.288 & 08:00:56.818 & 36:03:25.52 & 30 &{\it B}$_{\rm J}${\it V}$_{\rm J}${\it R}$_{\rm C}${\it I}$_{\rm C}${\it g}$^{\star}${\it r}$^{\star}$ & {\it I}$_{\rm C}$ (1896, 0.62) \\
MACSJ0329.6-0211 & 0.450 & 03:29:41.459 & -02:11:45.52 & 22.2 & {\it B}$_{\rm J}${\it V}$_{\rm J}${\it R}$_{\rm C}${\it I}$_{\rm C}${\it z}$^{+}${\it u}$^{\star}$ & {\it V}$_{\rm J}$ (1944, 0.55) \\
MACSJ0744.8+3927 & 0.698 & 07:44:52.310 & 39:27:26.80 & 73.2 &{\it B}$_{\rm J}${\it V}$_{\rm J}${\it R}$_{\rm C}${\it I}$_{\rm C}${\it i}$^{+}${\it z}$^{+}${\it u}$^{\star}$ & {\it R}$_{\rm C}$ (4869, 0.56) \\
\hline
\end{tabular}
\label{tab:targets}
\end{table*}
\subsection{X-ray Hydrostatic Masses}
Our X-ray mass estimates are derived from \emph{Chandra}{} data, using the analysis procedure described in Papers I and II.
However, we have used a newer version of the \emph{Chandra}{} calibration, resulting in updated hydrostatic masses.
For this paper, the data were processed with CIAO\footnote{\url{http://cxc.harvard.edu/ciao}} version 4.6.1 and CALDB\footnote{\url{http://cxc.harvard.edu/caldb}} version 4.6.2 following standard procedures.\footnote{\url{http://cxc.harvard.edu/ciao/guides/acis_data.html}}
The intracluster medium was modeled as a spherically symmetric, piece-wise isothermal atmosphere in hydrostatic equilibrium with a gravitational potential given by the Navarro-Frenk-White \citep[NFW; ][]{nfw97} model.
The center for this deprojection was chosen to maximize the symmetry of the emission at radii $\sim r_{2500}$, where the mass profiles are best constrained (and where we perform the comparison to lensing masses).
The radius $r_{2500}$ is where the mean enclosed density is 2500 times the critical density of the universe at the cluster's redshift, as determined by the X-ray observations.
Appropriate foreground and background components were included in the spectral analysis, along with a model for the cluster emission.
The model was fitted to the photon counts in the 0.6-7.0 keV band using the modified C-statistic in XSPEC \citep{xspec} in energy bins, defined to have at least one count per bin.
See Paper~II for more details.
\subsection{Weak Lensing Masses}
\label{sec:lensing_masses}
The lensing analysis for this work is similar to that presented in the Weighing the Giants papers \citep{paper1, paper2, paper3}.
Here, we summarize the essential elements of the mass measurement process that are referenced later in the text, and highlight all small changes in the modeling procedure.
WL masses were derived from multi-filter observations with SuprimeCam at Subaru and Megacam at CFHT.
We generated shear catalogs for each cluster field using the KSB$+$ \citep{ksb95, hfk98} moments-based code \textsc{analyseldac}, described in \citet{erben01}.
Absolute shear calibration was determined from the STEP2 simulations, which were designed to replicate SuprimeCam observations \citep{step2}.
The cluster shear profiles were modeled as NFW halos over a cluster-centric radial range 750\,kpc--3\,Mpc.
We adopted the X-ray center when constructing the 1D average shear profiles.
Note that the restriction of shear measurements to relatively large cluster radii makes our mass estimates insensitive to the exact choice of center \citep{paper1}.
Two different methods were used to measure the redshift distribution of galaxies selected as lensed sources: the ``color-cut'' method and the $P\left ( z\right )$\ method.
Both methods are described in \citet{paper3}, where we demonstrated that the $P\left ( z\right )$\ method is unbiased and that our implementation of the ``color-cut'' method yields consistent results.
We briefly describe each method below.
For the analysis of Section~\ref{sec:xlratio}, we adopted the color-cut masses cross-calibrated with $P\left ( z\right )$\ masses as our baseline, thus maximizing the sample size (12 clusters versus 6 with the $P\left ( z\right )$\ method) while maintaining uniform systematic uncertainties.
For the ``color-cut'' method, we matched the galaxy population of each cluster field to the COSMOS deep field \citep{ilbert09}, for which high quality 30+ filter photometric redshifts are available.
To combat signal dilution by contaminating cluster members, we removed the red sequence for each cluster.
We then performed a ``contamination correction'', where the measured signal was boosted using the average excess galaxy number counts observed in the WtG sample.
For the $P\left ( z\right )$\ method, we calculated photometric redshift probability functions for individual galaxies from five filter {\it B}$_{\rm J}${\it V}$_{\rm J}${\it R}$_{\rm C}${\it I}$_{\rm C}${\it z}$^{+}$ photometry \citep{paper2}.
We measured the NFW halo properties of the cluster with an unbinned Bayesian model fitted to the measured shear at each galaxy position.
During the fit, we marginalized over the redshift probability function for each galaxy.
$P\left ( z\right )$\ masses are used in this work to test the robustness of the lensing to X-ray mass ratio measurement and to measure the average concentration of the sample.
We adopted a prior on the NFW concentration from \citet{neto07}, using results for relaxed clusters from their largest mass bin, $\log_{10} M_{200}h^{-1}\mbox{$M_{\odot}$} = 14.875$--$15.125$.
The prior probability $P(\log_{10} c)$ was modelled as a Gaussian distribution with mean 0.664 and standard deviation 0.061.
Note that this differs slightly from \citet{paper3}, for which the analysis was not restricted to dynamically relaxed clusters, and where a Gaussian prior of $log_{10} c = 0.6 \pm 0.116$ representing clusters in all dynamic states, was used.
Previous work has shown that modelling a sample of cluster lensing observations as NFW halos following a fixed mass-concentration relation can lead to small biases in the average mass of the sample \citep{becker11, bahe12}.
This mass bias can be traced to the use of an incorrect mass-concentration relation and to deviations of the cluster density profile from NFW at radii beyond the cluster virial radius.
To correct both of these effects, we replicate our mass modeling procedure on mock lensing observations of the Millennium-XXL simulation \citep[S. Hilbert et al.~in prep; see also][]{angulo2012}.
Millennium-XXL features 303 billion particles in a 4.1 Gpc box, providing over one hundred halos in a mass range matching the clusters in this study.
We replicated our analysis on two simulation snapshots where mock lensing observations were available, $z=0.25$ and $z=1.0$.
For this study, we find that a correction factor constant in mass is sufficient to characterize the bias in $M_{2500}$.
At $z=0.25$, we find a correction factor of $1.03 \pm 0.01$ (stat only), increasing our masses by $\approx 3\%$.
At $z=1.0$, we find a correction factor of $1.10 \pm 0.02$.
We linearly interpolate the correction to each cluster redshift in our sample and marginalize over the statistical uncertainty when fitting for the lensing to X-ray mass ratio.
We discuss additional systematics associated with this simulation-based correction in Section~\ref{sec:concentration}.
A complete discussion of how we derive this correction and the related systematic error estimates will be presented in Applegate et al., in prep.
Note that this simulation correction is new since the publication of Paper II in the series.
In \citet{paper3}, we discussed the relevant systematic uncertainties in measuring the mean lensing mass of the full 51 cluster sample.
We attributed a 4\% systematic uncertainty to our shear measurements.
The determination of the photometric-redshift distribution contributed an additional 3\% systematic uncertainty.
We estimated a 4\% uncertainty in the calibration of the color-cut lensing masses by $P\left ( z\right )$\ masses.
The total systematic from these three sources is $\approx 6\%$.
We had previously included a 3\% systematic uncertainty attributed to modeling the mass distribution in the clusters.
We update the mass model uncertainty estimate, in light of the new simulation-based correction, in Section~\ref{sec:concentration}.
Triaxiality of cluster halos and line-of-sight structure contribute additional noise to lensing measurements, estimated to be $\approx 20\%$ \citep{hoe03, becker11}.
The Millennium-XXL mock observations allow us to estimate the intrinsic scatter from the matter distribution within 100 h$^{-1}$Mpc (physical) of the cluster center.
We model the intrinsic scatter in the simulations as a log-normal distribution, finding $\sigma_{\mathrm{int}} = 0.23 \pm 0.01$ at $z=0.25$ and $\sigma_{\mathrm{int}} = 0.28 \pm 0.01$ at $z=1.0$.
However, a 100 h$^{-1}$Mpc integrated length does not account for all scatter induced by line of sight structure.
While we could follow the method of \citet{hoe03} to estimate the scatter component induced by large scale structure, we elect to simply fit for the total magnitude of intrinsic scatter when modeling the lensing to X-ray mass ratio.
\subsection{Statistical Methods}
\label{sec:statmethods}
The X-ray to lensing mass ratio is poorly determined by a simple averaging procedure.
First, lensing masses generically have large uncertainties that are non-Gaussian (nor log-normal) in shape, even for the most massive clusters with the best data.
Second, the uncertainties in the lensing masses are correlated with the X-ray masses when measured within an aperture determined from the X-ray data.
Neglecting this correlation will artificially boost the apparent precision of the measurement.
Third, we expect an approximately log-normal intrinsic scatter between X-ray and lensing masses, since the lensing masses are sensitive to cluster triaxiality and line of sight structure \citep{becker11, bahe12}.
This scatter is of a comparable magnitude to the WL measurement uncertainties, so neither can be safely neglected.
While none of these issues are new, their combination limits the applicability of standard approaches.
Fitting packages such as \textsc{linmix\_err} from \citet{kelly07} or \textsc{BCES} from \citet{bces} assume that the forms of the measurement error and the intrinsic scatter are the same (Gaussian).
Both codes also restrict users to a simple two-parameter linear model.
To avoid these pitfalls, we can instead write down the posterior probability distribution for the lensing to X-ray mass ratio without approximation.
Specifically, we can incorporate the exact probability distributions for NFW halo parameters that describe the lensing and X-ray data sets.
This correctly incorporates both detections and non-detections into the ratio, while also allowing us to directly account for the correlation between lensing and X-ray masses.
We model the intrinsic scatter from triaxiality and line of sight structure as a log-normal distribution, independent of the form of the measurement errors.
Appendix~\ref{sec:likelihood} explicitly presents the posterior probability function for the lensing to X-ray mass ratio.
This basic ratio model can be easily extended to generalize the relationship between lensing and X-ray masses or to linearly regress the ratio against other variables.
When the independent variables in these fits have non-negligible uncertainty, for example the cluster morphology discriminators we consider in Section~\ref{sec:morphology}, we must also include a prior on the true, unobserved distribution of that variable.
For this task, we adopt the hierarchical mixture of Gaussians prior employed in \citet{kelly07}, with uniform priors on Gaussian scale parameters ({\it i.e.\ } $\sigma$) following \citet{gelman2006}.
By default, we report results for mixtures of two Gaussians, but have verified that results are not sensitive to the exact number employed.
Throughout the paper, we adopt uninformative priors when possible.
For example, we adopt a uniform prior on the log of lensing to X-ray mass ratio, which equally prefers ratios greater or less than one.
We also adopt uniform priors on intrinsic scatter $\sigma_{\mathrm{int}}$.
We quote maximum a posteriori parameter point estimates, transforming probability distributions and point estimators appropriately when log transforms are applied to variables.
For the majority of this paper, we consider the lensing to X-ray mass ratio at a fixed cosmology.
However, both the lensing and X-ray masses inferred from the data implicitly depend on the cosmological model.
We comment on the resulting dependence of the lensing to X-ray mass ratio on cosmological parameters in Section~\ref{sec:cosmo_dependence}.
When using the lensing data to calibrate cluster masses in a cosmological test, such as the gas-mass fraction ($f_{\mathrm{gas}}$\ ) constraints of Paper~II, it is best to directly incorporate the full model described above; for each trial cosmology, model parameters are inferred from the measured shear profiles (as a function of angle) and galaxy redshift distributions.
In the case of Paper~II (see also \citealt{allen08}), the model for the X-ray data includes a parameterized redshift-dependent scaling, $K(z)=K_0(1+K_1 \,z)$, encoding the average ratio of true cluster masses to X-ray mass estimates at a given cluster redshift.
Given the expectation that our lensing masses are unbiased within a 7\% tolerance, we can interpret this function as the lensing to X-ray mass ratio, and use the likelihood associated with the lensing data (Appendix~\ref{sec:likelihood}) to constrain the model for $K(z)$ (in practice, the current data can constrain $K_0$, but $K_1$ must be constrained by a prior).
For simplicity, the implementation in Paper~II neglects the cosmology dependence of the mass--concentration relation as well as the sub-dominant uncertainties from the contamination correction and the lensed galaxy redshift distribution (characterized by $\langle\beta\rangle$ and $\langle\beta^2\rangle$); due to the small size of the comparison sample, these effects are much smaller than the statistical uncertainties.
The same approach was also used in \citet{paper4}.
\section{Weak Lensing to Chandra X-ray Ratio}
\label{sec:xlratio}
In this section we present our measurements of the lensing to \emph{Chandra}{} X-ray mass ratio for our sample of 12 relaxed WtG clusters.
We first present results for the ratio measured within $r_{2500}$, where this radius is determined from the X-ray data.
This is the appropriate ratio to use in the $f_{\mathrm{gas}}$\ model described in Paper~II.
We then present results for the mass ratio $M_{2500}^L/M_{2500}^X$, where each data set independently determines its own value of $r_{2500}$.
Finally, we also report the WL to X-ray mass ratio measured within the X-ray defined $r_{500}$.
\begin{figure}
\includegraphics[width=\columnwidth]{figures/ratio_v_redshift_2014simcor}
\caption{Measured weak lensing to \emph{Chandra}{} X-ray mass ratio plotted against cluster redshift for the relaxed WtG sample of clusters. Lensing masses are measured within the X-ray defined $r_{2500}${} and use the color-cut analysis. Data points are for individual clusters, where error bars represent 68\% confidence intervals. The dashed line and orange bands indicate the ensemble mass ratio, 68\% and 95\% ranges. The ratio and statistical 68\% range is $0.967_{-0.092}^{+0.063}$, or $0.956 \pm 0.082$ when approximated as a Gaussian distribution. Additional systematic uncertainties of 9\% apply, for a total uncertainty of $\approx 12\%$. The y-axis and associated probability distributions are log transformed.}
\label{fig:ratio_v_redshift}
\end{figure}
We first consider the mass ratio within the X-ray defined $r_{2500}$, as plotted in Figure~\ref{fig:ratio_v_redshift}.
We measure a mass ratio of $M_L/M_X = 0.967_{-0.092}^{+0.063}$, where the quoted uncertainties are the statistical 68\% confidence interval after marginalizing over the intrinsic scatter.
The best Gaussian approximation of the posterior PDF is $M_L/M_X = 0.956 \pm 0.082$.
From the statistical uncertainties alone, the ratio is fully consistent with unity.
The measurement is also subject to systematic uncertainties from the WtG WL analysis (Section~\ref{sec:lensing_masses}).
Whereas the WtG analysis of \citet{paper3} measured masses within 1.5\,Mpc, the current measurement is at $r_{2500}$, typically a factor of $\sim3$ smaller.
This requires us to extrapolate our shear measurements inwards, thereby increasing our sensitivity to priors on the mass--concentration relation.
At this radius, the mass--concentration relation contributes an additional 6\% systematic uncertainty, which we discuss in Section~\ref{sec:concentration}.
Therefore, the total WL systematic uncertainty is 9\%, resulting in a combined statistical and systematic uncertainty for the mass ratio measurement of 12\%.
We measure an intrinsic scatter of $\sigma_{\mathrm{int}} = 0.146_{-0.068}^{+0.097}$ present in the lensing to X-ray ratio.
This amount of intrinsic scatter is consistent at $1\sigma$ with the intrinsic scatter predicted by the MXXL simulations and with the intrinsic scatter detected for the full WtG cluster sample \citep{paper4}.
Next, we measure the ratio within $r_{2500}$ apertures determined independently from the lensing and X-ray data.
This alternative ratio is relevant for calibrating mass proxies in some circumstances.\footnote{We reiterate, however, that the result applies only for our reference cosmology, and that in applications where cosmological parameters are allowed to vary, the only robust approach is to fit the mass ratio model simultaneously with the cosmology.}
These results should be noisier, as WL cannot easily measure $r_{2500}$.
The median mass ratio and 68\% statistical uncertainty is $0.90^{+0.12}_{-0.12}$, with a measurement of intrinsic scatter of $\sigma_{\mathrm{int}} = 0.25_{-0.11}^{+0.15}$
The same 9\% systematic uncertainty applies, resulting in a total measurement precision of 16\%.
Finally, we measure the ratio within the X-ray defined $r_{500}$.
Note that this is in some cases an extrapolation of the X-ray data to larger radii, but is well measured by the WL observations.
We measure a WL to X-ray ratio of $1.059_{-0.096}^{+0.092}$ when using a WL correction factor derived from the Millennium-XXL simulations that is appropriate for $r_{500}$.
Again, a total WL systematic uncertainty of 9\% applies.
\section{Discussion of Systematics}
\label{sec:systematics}
We investigate the possible influence of a number of systematics that in principle could alter the measured mass ratio.
We start with the adopted \emph{Chandra}{} temperature calibration.
We then verify the robustness of the measurement to the definition of the cluster sample and the details of the lensing analysis.
Finally, we search for trends in the X-ray to lensing ratio that correlate with cluster mass or redshift.
\subsection{Chandra Calibration}
\label{sec:calib_dependence}
X-ray hydrostatic masses, and therefore the lensing to X-ray mass ratio, are dependent on the effective area calibration adopted for \emph{Chandra}.
While \emph{Chandra}{} calibration updates usually refine the most recent observations, some retroactively affect temperatures measured from older data.
One such update happened after the publication of Paper II of this series.
To see how large an effect these calibration changes have on our measurements, we compared our measurements made above (using CIAO version 4.6.1 and CALDB version 4.6.2) with the calibration from May 2012 (CALDB version 4.4.10 and CIAO version 4.4) used in Paper~II.
When X-ray mass estimates were compared between the two calibration versions for the 40 clusters in the full relaxed sample from Paper~I, we saw that the newer calibration (the default used in this paper) lowers the average X-ray mass, $M_{2500}$, by $5\%$, with a scatter of $5\%$.
Restricting to only the 12 relaxed WtG clusters in the calibration sample examined in this paper, we see a downward shift of only $\approx3\%$.
The small shift in X-ray masses is reflected in an equally small shift in the lensing to X-ray mass ratio.
With the May 2012 calibration, the lensing to X-ray mass ratio at the X-ray measured $r_{2500}$ is $0.940_{-0.081}^{+0.065}$, in comparison to our baseline results of $0.967_{-0.092}^{+0.063}$.
This shows that our results are not overly sensitive to recent changes in the \emph{Chandra}{} effective area calibration.
More importantly, the use of lensing data to calibrate the X-ray masses in cosmological tests like that of Paper~II corrects for any overall mass bias potentially introduced by the \emph{Chandra}{} calibration.
\subsection{Sample Selection}
\label{sec:morphology}
In Papers I and II, clusters were included in the cosmological analysis with the $f_{\mathrm{gas}}$\ {} test based on X-ray morphology indicators.
Any detected trend with morphology would indicate that the lensing to X-ray mass ratio, and therefore the $f_{\mathrm{gas}}$\ test, is sensitive to the selection criteria.
In this section, we verify that no trend exists with any of these indicators.
Since by construction all clusters in the relaxed WtG sample qualify as relaxed, we extended our sample with five additional clusters that failed one or more morphology criteria in Paper~I (the ``marginal'' sample).
Each of the five clusters have previously been included in subjective selections of relaxed clusters \citep{allen08, donahue14}.
As a first check, we compared the measured lensing to X-ray mass ratio for the relaxed WtG sample against the marginal sample.
Using the five clusters in the marginal sample alone, the measured ratio within the X-ray measured $r_{2500}${} is $M_L/M_X = 1.05^{+0.24}_{-0.23}$.
In addition, we detect an intrinsic scatter of $\sigma_{\mathrm{int}} = 0.44^{+0.25}_{-0.21}$.
The ratio is consistent with the twelve clusters from the relaxed WtG sample, albeit with low precision.
When all seventeen clusters are considered, the ratio is $M_L/M_X = 0.979_{-0.076}^{+0.078}$ with an intrinsic scatter of $\sigma_{\mathrm{int}} = 0.22_{-0.075}^{+0.080}$, again consistent with the results of the relaxed-only sample.
\begin{figure*}
\centering
\begin{minipage}{0.32\linewidth}
\centering
\includegraphics[width=\columnwidth]{figures/ratio_v_sym_2014}
\end{minipage}
\begin{minipage}{0.32\linewidth}
\centering
\includegraphics[width=\columnwidth]{figures/ratio_v_peak_2014}
\end{minipage}
\begin{minipage}{0.32\linewidth}
\centering
\includegraphics[width=\columnwidth]{figures/ratio_v_alig_2014}
\end{minipage}
\caption{The sample ratio (black dotted), 68\% and 95\% confidence intervals (orange, shaded), and ratios for individual clusters, plotted against three X-ray morphology measurements: (left) symmetry, (center) peakiness, and (right) alignment. The vertical dashed red lines show the thresholds that define the relaxed region. Black data points are the relaxed WtG sample, while blue data points are the marginal sample.}
\label{fig:morphology}
\end{figure*}
\begin{table*}
\centering
\caption[]{Measured slopes and offsets for the best fitting linear relationship between the lensing to X-ray mass ratio versus three morphology measures, for the relaxed WtG and marginal samples. Quoted are the 1D 68\% marginalized uncertainties for each quantity.}
\begin{tabular}{l | c c c | c c c}
\hline
& \multicolumn{3}{ c |}{\textbf{Relaxed WtG Sample}} & \multicolumn{3}{ c }{\textbf{Relaxed WtG $+$ Marginal Sample}} \\
\textbf{Morphology Measure} & Slope & Offset & Pivot & Slope & Offset & Pivot \\
\hline
Symmetry & $0.42_{-0.24}^{+0.37}$ & $0.94_{-0.06}^{+0.07}$ & 1.18 & $0.07_{-0.45}^{+0.25}$ & $0.97_{-0.08}^{+0.08}$ & 1.10 \\
Peakiness & $-0.74_{-0.53}^{+1.02}$ & $0.95_{-0.08}^{+0.08}$ & -0.602 & $0.04_{-0.53}^{+0.61}$ & $0.99_{-0.09}^{+0.07}$ & -0.66\\
Alignment & $1.30_{-0.80}^{+0.96}$ & $0.95_{-0.10}^{+0.06}$ & 1.41 & $-0.40_{-0.57}^{+1.01}$ & $0.97_{-0.07}^{+0.08}$ & 1.35 \\
\hline
\end{tabular}
\label{table:morphology_fits}
\end{table*}
We also investigated linear trends in the lensing to X-ray mass ratio with each of the morphology measures used to define the sample (see Paper I).
The mass ratio is plotted against symmetry, peakiness, and alignment in figure~\ref{fig:morphology}.
To investigate trends in these parameters, we adopt a linear model between each parameter and the log mass ratio, $\ln M_{2500}^L/M_{2500}^X = \alpha + \beta X$ (see Section~\ref{sec:statmethods}).
We list the 1D marginalized uncertainties for the slope and intrinsic scatter for each morphology measure and both samples in table~\ref{table:morphology_fits}.
When considering the 12 clusters in our relaxed WtG sample, both Symmetry and Alignment exhibit an $\approx 1\sigma$ preference for a non-zero slope.
However, no trend is evident when the combined ``relaxed WtG'' and ``marginal'' samples are considered.
We note that the intrinsic scatter appears to increase for clusters closer to the multidimensional selection boundary, as seen in Figure~\ref{fig:morphology}.
From these tests, we conclude that our results on the WL to X-ray mass ratio are not overly sensitive to the exact morphological selection criteria used to select the sample.
\subsection{Lensing Analysis}
Weighing the Giants lensing masses measured within an aperture of at $1.5$ Mpc should be unbiased within a 7\% systematic uncertainty \citep{paper3}.
However, since the measurements here are at the smaller radius of $\approx r_{2500}$ and employ a new simulation-based correction factor, the systematic uncertainties of two components of the lensing analysis are worth reviewing.
First, we explore how the lensing to X-ray mass ratio varies when we change how we calculate the background redshift distribution of lensed sources and correct for cluster galaxy contamination.
Then, we quantify the systematic uncertainty from the mass model and the simulation correction.
\subsubsection{Lensed Source Redshift Distribution}
The Weighing the Giants project employed two methods to estimate the redshift distribution of lensed sources and address signal dilution by cluster galaxies, the color-cut method and the $P\left ( z\right )$\ method.
The color-cut method statistically matches the galaxy population in the cluster field with a reference deep field and corrects the shear profile for the diluting effects of cluster galaxies.
In contrast, the $P\left ( z\right )$\ method estimates a redshift probability function for each galaxy in the cluster field and only includes galaxies that are believed to be in the background of the galaxy cluster.
We used the color-cut method in this study to maximize our sample size while maintaining uniform systematic uncertainties.
We previously showed that our mass measurements using the $P\left ( z\right )$\ method were consistent with our implementation of the color-cut method used in this analysis \citep{paper3}.
Here we check if using the alternative $P\left ( z\right )$\ method produces consistent results for the six clusters in the relaxed WtG sample where a comparison is possible.
Figure~\ref{fig:cluster6} shows the constraints on the lensing to X-ray mass ratio within the X-ray measured $r_{2500}$ for the two sets of clusters in the analysis, those with and without $P\left ( z\right )$\ mass measurements.
For the clusters with $P\left ( z\right )$\ mass measurements, we calculated the lensing to X-ray mass ratio twice, once with the $P\left ( z\right )$\ method and once with the color-cut method.
Both analyses are statistically consistent with the independent set of color-cut only clusters.
For the six clusters where both the color-cut and $P\left ( z\right )$\ methods can be applied, the measured ratios differ by 11\%.
Scatter between the methods is expected: each method selects a somewhat different population of lensed galaxies for analysis, resulting in $\approx20\%$ scatter between these measurements at $z \approx 0.2$, growing to $\approx40\%$ at $z\approx 0.6$ \citep{paper3}.
To evaluate the statistical significance of the observed offset between methods for the six clusters in our sample, we created random samples of six clusters from the set of 27 clusters with $P\left ( z\right )$\ measurements in \citet{paper3}.
The distribution of offsets between $P\left ( z\right )$\ method masses and color-cut method masses for these subsamples is consistent with zero and has a population standard deviation of $\sigma=16\%$.
The offset detected between analyses for these six clusters is therefore within expectations.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{figures/cluster6}
\caption{Results for three subsets of six clusters, using both the color-cut analysis and the $P\left ( z\right )$\ analysis. Data points with error bars show 68\% confidence intervals for each cluster subset, while the dark orange and light yellow shaded regions are the 68\% and 95\% confidence intervals for the full sample. Groups one and two are identical clusters, but analyzed using the $P\left ( z\right )$\ and color-cut methods, respectively. Group 3 comprises six independent clusters where only the color-cut analysis can be applied. Groups 1 and 2 are statistically consistent with group 3 at the $1\sigma$ level. Groups 1 and 2 are also statistically consistent when the covariance between the two measurement methods is fully accounted for. Note that the correction for biases in the mass model has not been applied here. }
\label{fig:cluster6}
\end{figure}
\subsubsection{Lensing Mass Model}
\label{sec:concentration}
For this study, we have used the NFW density profile to model the observed shear signal of galaxy clusters.
We have also assumed a mass-concentration relation, both to reduce the nonlinearity of the model and to compensate for the inability of the lensing data to constrain the density profile shape.
However, studies of mock lensing observations of N-body simulations have shown that modeling a population of realistic clusters as isolated NFW halos can lead to small biases in the average mass of the sample \citep{becker11, bahe12}.
Lensing masses may also be sensitive to the choice of mass-concentration relation, either when modeling shear information at small cluster-centric radii or, as in this work, when extrapolating mass measurements to small radii.
To correct for these effects, we replicated our lensing measurement procedure on mock observations of the Millennium-XXL (MXXL) N-body simulation and introduced a correction factor (see Section~\ref{sec:lensing_masses}).
While the statistical uncertainty of the correction factor is small (less than 2\%), additional systematic uncertainties arise from mismatches between the simulation and reality.
The mass-concentration relation introduces the largest systematic uncertainty.
The simulation correction factor effectively fixes our choice of mass-concentration relation, on average altering the masses to what we would have measured using the mass-concentration relation realized in the MXXL.
However, a residual bias may remain due to the cosmology dependence of the mass-concentration relation \citep{ludlow2013, diemer15}.
We quantify our sensitivity to the mass-concentration relation by varying the peak of our prior on concentration by $\pm 20\%$.
This range is motivated by \citet{bhattacharya13}, who measured differences in the mass-concentration relation amplitude of $\approx 15\%$ between their results and \citet{duffy08}, as well as between results based on the Millennium Simulation \citep{neto07, gao08, hayashi08}.
The measured WL to X-ray mass ratio at $r_{2500}$ varied by $\pm 6\%$.
The simulation correction derived from MXXL is based on a mass-limited selection of halos.
In contrast, the morphology selection from Paper I should select clusters that have less substructure than the general population, yet has not significantly altered the ellipticity distribution of the clusters \citet{mantz_morph}.
This should on average decrease the magnitude of observed intrinsic scatter but not alter the correction factor \citep{bahe12}.
Based on the measured ellipticity distribution of the relaxed sample from Paper I, we also consider it unlikely that a conspiracy exists between a preferential selection of clusters projected along the line of sight, leading to overestimated lensing masses, and a biased Chandra temperature calibration, leading to overestimated X-ray masses.
Future work is required to replicate the Paper I selection function in cosmological simulations.
In principle, lensing mass measurements can be sensitive to the choice of center used to measure the shear profile.
However, we have previously shown that our shear measurements are minimally affected by the choice of cluster center \citep{paper1}.
We have also studied the effects of shear profile miscentering by introducing random offsets into the MXXL mock lensing observations.
When we perturb the assumed center following a probability distribution of BCG-X-ray offsets observed in Chandra archive observations of clusters (Paper I), we see a sensitivity in the correction factor of only $\approx 3\%$.
In addition, the relaxed WtG sample shows considerably reduced BCG-X-ray offsets than the general population, a further indication that these clusters are indeed dynamically relaxed (Paper I).
We therefore expect systematics from miscentering to be negligible for this sample.
We have used two simulated redshifts from the MXXL to measure our correction factor, at $z=0.25$ and $z=1.0$, and have linearly interpolated in redshift between the measurements.
However, the bulk of clusters in the sample are at low redshift.
To test the sensitivity of our measurements to the details of the interpolation scheme, we remeasured the lensing to X-ray ratio using only the correction factor from the $z=0.25$ simulation.
In this case, we measure the ratio to be $0.940_{-0.074}^{+0.081}$, where the best Gaussian approximation to the posterior PDF is $\mu = 0.948 \pm 0.082$, a shift of less than $\approx2\%$.
\subsection{Mass or Redshift Trend}
\label{sec:extensions}
In this section, we search for trends in the X-ray to lensing ratio with mass or redshift.
Such trends would suggest that a more complex calibration model is needed for the $f_{\mathrm{gas}}$\ experiment.
Previous work has suggested that a trend with mass may exist for hydrostatic masses from \emph{Chandra}, albeit at low statistical significance and for samples including all dynamic states \citep{israel14}.
In our case, we emphasize that the baseline ratio model is an adequate description of the data, and that the small size of our sample makes precise measurements of extended models difficult.
We first examine if the lensing to X-ray mass ratio exhibits a mass dependence.
To do this, we extended the ratio model from Section~\ref{sec:xlratio} to include a power-law scaling between lensing and X-ray masses (again including intrinsic scatter), and model the intrinsic distribution of X-ray masses as a sum of Gaussians (see Section~\ref{sec:statmethods}).
We used the mean X-ray mass as the pivot point for the power-law.
A power law index of $\beta=1$ indicates no mass dependence and reduces to the original ratio model.
Figure~\ref{fig:powerlaw} shows the inferred power-law.
The statistical 68\% confidence interval for the power law index, with all other variables marginalized over, is $\beta = 0.78_{-0.14}^{+0.17}$.
The power law index $\beta$ is less than unity at less than $2\sigma$ significance.
We therefore find no statistical evidence for a trend with mass.
\footnote{It is an interesting to note that when we consider the ``relaxed WtG'' plus the marginal cluster samples, the power law index $\beta$ is measured to be $\beta = 0.69 \pm 0.13$, which is less than unity at more than $2\sigma$ significance.
However, it is hard to interpret this measurement without a more robustly defined sample.}
\begin{figure}
\includegraphics[width=\columnwidth]{figures/lensing_v_xray_2014.pdf}\\
\caption{Lensing mass versus X-ray mass. The dark and light blue bands are the 68\% and 95\% confidence intervals allowed by the power-law model fit to the data. The dashed black line is the one-to-one line. The magnitude of the intrinsic scatter has been marginalized over.}
\label{fig:powerlaw}
\end{figure}
We next check for an explicit redshift dependence in the lensing to X-ray ratio.
We extended the ratio model from Section~\ref{sec:xlratio} to include a term linear in cluster redshift, such that $\ln M_{2500}^L/M_{2500}^X = \alpha + \beta z$, using the mean redshift as the pivot point in the fit.
We measured a best fit slope of $-0.80_{-0.69}^{+0.76}$, {\it i.e.\ } a decreasing mass ratio towards higher redshifts.
However, the results are statistically consistent with a constant value at approximately $1\sigma$.
\section{Average Concentration of Massive, Relaxed Clusters}
\label{sec:ave_concen}
The mass--concentration relation for massive halos influences the measured lensing to X-ray mass ratio (as discussed in Section~\ref{sec:concentration}).
It also reflects the formation history of halos and carries some weak information on cosmology \citep{wechsler2002, ludlow2013}.
By design, the WtG mass measurements are minimally sensitive to assumptions about the cluster concentration (Section~\ref{sec:concentration})
However, some partial sensitivity is still present, which we use here to measure the average concentration of the massive, relaxed clusters in our sample.
As opposed to the rest of this study, for this exercise we jointly fit the concentration of six clusters with $P\left ( z\right )$\ measurements.
We limit ourselves to the $P\left ( z\right )$\ clusters to avoid systematic uncertainties from the ``contamination correction'' required in the color cut analysis \citep[see][]{paper3}.
The contamination correction alters the slope of the shear profile, and is therefore degenerate with the measured concentration.
We also remove the prior on concentration from Section~\ref{sec:lensing_masses}.
Since we designed our measurements to be insensitive to concentration, we do not expect to measure the concentration of individual clusters with any useful precision.
In addition, we expect additional noise from triaxiality and large-scale structure \citep{bahe12}.
We therefore simultaneously model the population of clusters to measure an average concentration.
We assume that the concentrations of clusters will have a log-normal scatter around the average concentration.
Specifically, our model consists of a mass ($M(<1.5\mathrm{Mpc})$) and concentration for each cluster, the mean of the log-normal distribution of concentrations, $\mu_c$, and the scatter in concentration measurements, $\sigma_c$.
We assume flat priors on $\mu_c$ and $\sigma_c$, and restrict $\sigma_c$ to the range [0.286, 0.318], based on the largest mass bin reported in the simulation results of \citet{bahe12}.
We measure the average concentration to be $\mu_{c} = 3.0_{-1.8}^{+4.4}$.
While hardly discriminatory, this fit is consistent with our adopted prior on the concentration ($\mu_{c} = 4.6$) for measuring individual masses of clusters in previous sections.
Other groups have shown that WL observations can in principle achieve up to 10\% precision on average concentration measurements for an ensemble of clusters \citep{okabe13}.
However, this precision requires fitting NFW halo models to small radii (typically $\sim 150$kpc).
At these radii, the measured shear from these massive clusters exceeds $g\sim0.1$, and often exceeds $g\sim0.3$ \citep[see example shear profiles in][]{paper1}, while shear measurement codes have only been rigorously calibrated to shears of $g\sim0.05$ \citep{step2, great08, great10}.
Extending the shear calibration to this regime is an ongoing effort \citep{desc_whitepaper}.
\section{Discussion}
\label{sec:discussion}
\subsection{Interpreting the weak lensing to Chandra X-ray mass ratio}
We have measured a WL to \emph{Chandra}{} X-ray mass ratio of $0.956\pm 0.082$, with an additional 9\% systematic uncertainty from the lensing analysis.
Combining the statistical and systematic uncertainties in quadrature gives $\approx12\%$ precision, making the measured mass ratio consistent with unity at $< 1\sigma$.
Hydrodynamic simulations have examined X-ray hydrostatic masses with different methodologies and baryonic physics prescriptions.
Simulations agree that, even for the most relaxed clusters, hydrostatic mass estimates should slightly under-report the true mass within $r_{2500}$, with $\lesssim 10\%$ bias \citep{nagai07, lau09, rasia12, battaglia13}.
Therefore, these simulations would predict a lensing to X-ray mass ratio in the range of 1.0 - 1.1 at these radii.
Our results provide a 95\% confidence upper limit on the lensing to \emph{Chandra}{} X-ray mass ratio of $\approx 1.15$.
This disfavors models with stronger violations of hydrostatic equilibrium in massive, relaxed clusters than predicted by current simulations.
However, we can only establish a truly robust limit on departures from HSE if we also have a firm understanding of the systematic uncertainties in \emph{Chandra}{} cluster mass measurements.
In particular, the temperature calibration for in-orbit X-ray detectors is an area of active research by the community (see \url{http://web.mit.edu/iachec/} and the documents linked therein).
A pessimistic bound on \emph{Chandra}{} systematic errors would add an additional $\approx15\%$ systematic uncertainty to our results, permitting somewhat larger departures from HSE to be present in our clusters.
That would still disfavor a hydrostatic bias of $\approx20\%$ or more, which is one interpretation of the results from \citet{planck_clusters}.
Our results also allow the possibility that the current \emph{Chandra}{} calibration may bias temperatures measured from continuum emission high, though we cannot place a strong constraint on the size of such a bias.
It is well documented that a discrepancy exists between the XMM-Newton and \emph{Chandra}{} temperature measurements, with \emph{Chandra}{} temperatures currently being higher than XMM \citep{ndg10, mahdavi13, schellenberger14}.
One examination of broad-band versus line-emission measurements of cluster temperatures suggests that the true temperatures may lie between those inferred from fitting the continuum emission measured by the two telescopes \citep{schellenberger12}.
If \emph{Chandra}{} temperatures are indeed biased high, then so would be \emph{Chandra}{} hydrostatic masses.
If the calibration bias exceeds the bias due to non-thermal support, this would result in a lensing to X-ray mass ratio of less than 1, consistent with our results.
\subsection{Comparison to Literature}
Other groups have previously compared lensing masses to X-ray hydrostatic masses measured with \emph{Chandra}.
In particular, \citet{donahue14} have eleven clusters in common with our extended sample, and most of the raw \emph{Chandra}{} and lensing data are the same.
\citet{donahue14} independently processed the \emph{Chandra}{} data and adopted lensing masses from \citet{umetsu_clash}.
They use a weighted mean to measure a lensing to X-ray ratio of $r=1.14\pm 0.09$ at 0.5Mpc, approximately $r_{2500}$ for their clusters.
It is unclear why the results from \citet{donahue14} differ significantly from our measurements.
In part, the clusters analyzed in \citet{donahue14} are less relaxed on average than the clusters included in our sample, according to the measurements of Paper I.
However, even our measurements of the ``relaxed WtG plus marginal sample'' are still in some tension with \citet{donahue14}, given the nearly identical input data.
The differences are not obviously caused by discrepant lensing analyses.
\citet{umetsu_clash} compared their lensing masses with Weighing the Giants lensing masses at 1.5Mpc and found general agreement between the two works, with a median ratio (WtG/CLASH) of 1.02 and a geometric mean of 1.10.
Note that taking the geometric mean as the actual WtG/Clash offset only exacerbates the discrepancy between \citet{donahue14} and this work.
\citet{umetsu_clash} measure an average concentration consistent with our assumed mass--concentration model, making a radius-dependent offset between WtG and CLASH unlikely.
Turning to the X-ray measurements, we find some discrepancies between the \citet{donahue14} analysis and our analysis.
For the eleven clusters in common, X-ray masses at $r_{2500}$ in that work are $\approx20\%$ lower than our measurements, with $\approx 15\%$ scatter.
We do not expect the \emph{Chandra}{} calibration to be an issue, as \citet{donahue14} calibrate their data with CALDB 4.5.9, which should be comparable to the CALDB version 4.6.1 used in our work.
One clear methodological difference is that \citet{donahue14} bin their spectra and use a $\chi^2$ fitting statistic in their analysis, which is known to bias measurements of temperature when applied to relatively noisy data \citep{cash1979}.
In contrast, our procedure minimally bins the spectra, and instead uses the C-statistic to correctly describe the Poisson nature of the observed counts.
We estimate that perhaps as much as half of the discrepancy between our X-ray masses and those of \citet{donahue14} could originate in this difference, based on tests fitting mock spectra with realistic backgrounds and signal-to-noise.
However, a full resolution of the discrepancy would require more detailed comparisons.
The works of \citet{vbe09} and \citet{israel14} also feature WL calibrations of \emph{Chandra}{} derived X-ray masses.
However, no clusters are in common between this work and \citet{israel14}, while only 2 clusters are in common with the \citet{vbe09} calibration sample (Benson, priv comm).
Furthermore, both works use a custom \emph{Chandra}{} temperature calibration from \citet{vikhlinin_calib} that is not publicly available, and neither work restricts itself to relaxed clusters.
Both \citet{mahdavi13} and \citet{zhang10} compare WL masses to XMM-Newton derived hydrostatic masses for samples of relaxed clusters.
However, few clusters are in common between those samples and this work.
Also, both of those works use lensing data, from \citet{hoekstra12} and \citet{okabe_masses}, respectively, that are known to sharply disagree with Weighing the Giants masses.
Both analyses have been revised in the interim \citep[see ][]{hoekstra2015, okabe13}.
A comparison of the mass ratio results would therefore be of limited value.
Insufficient information is currently available to generalize measurements of the lensing to X-ray mass ratio from one X-ray telescope or X-ray calibration version to others.
Ideally, the community needs a large sample of relaxed clusters that are consistently analyzed with each calibration version and both \emph{Chandra}{} and XMM.
This would facilitate comparisons among X-ray based results measured at different times.
\subsection{Cosmology Dependence}
\label{sec:cosmo_dependence}
Both the X-ray and lensing masses in this analysis depend on cosmology.
The X-ray hydrostatic mass depends on the physical volume filled by gas of a given temperature, resulting in a dependence on the angular diameter distance $D$ to the cluster.
Lensing masses, on the other hand, depend on both the angular diameter distance to the cluster and on the angular diameter distances from the observer and cluster to the background lensed galaxy population.
For example, the 3D lensing mass within a fixed angular aperture will scale approximately as
\begin{equation}
M_L(<R) \propto D_{\mathrm{lens}}^2\Sigma_{\mathrm{crit}} = \frac{D_{\mathrm{lens}}D_{\mathrm{source}}}{D_{\mathrm{lens-source}}}.
\label{eq:cosmo_scaling}
\end{equation}
Equation~\ref{eq:cosmo_scaling} is exact for a Single Isothermal Sphere (SIS) model with all sources at one redshift.
The actual scaling depends on the type of mass calculated (SIS vs.\ NFW, aperture statistic vs.\ profile fit), the angular size of the measurement aperture, the functional form used to model the mass distribution, and the redshift distribution of lensed galaxies.
For flat models, this scaling is roughly independent of $\Omega_m$, but can vary by up to 10\% with $w$, as seen in figure~\ref{fig:cosmo_dependence}.
Since lensing masses depend on cosmology differently from X-ray hydrostatic masses, the lensing to X-ray mass ratio will also have a cosmology dependence.
To check this toy model, we re-evaluated the lensing to X-ray mass ratio from section~\ref{sec:xlratio} at four cosmologies, assuming flatness, that encompass the $2\sigma$ allowed region of the $f_{\mathrm{gas}}$\ experiment from Paper II, $\Omega_m = [0.21, 0.40]$ and $w = [-2.0, -0.35]$.
The mass ratio shifted as expected for clusters at $z \lesssim 0.5$, changing by less than $\approx 5\%$.
X-ray hydrostatic masses are only one example where a calibration with lensing is cosmology dependent.
X-ray gas mass and $Y_x$ mass proxies measured within a fixed aperture scale as $D^{5/2}$.
Therefore the lensing calibration of these mass proxies will be even more sensitive to cosmology than hydrostatic masses.
While this sensitivity to cosmological parameters is perhaps negligible for today's experiments, next generation experiments and measurements of high redshift clusters will need to account for it.
Our solution is to explicitly include the lensing data in the cosmological likelihood, by effectively fitting for lensing masses simultaneously with cosmology (see Section~\ref{sec:statmethods}).
In this way, we account for the exact cosmology dependence of the lensing masses and the relevant X-ray mass proxy.
\begin{figure}
\includegraphics[width=\columnwidth]{figures/cosmo_relativescaling_om}\\
\includegraphics[width=\columnwidth]{figures/cosmo_relativescaling_w}
\caption{Plotted curves show how lensing masses in fixed apertures (yellow) and the WL to X-ray mass ratio (hydrostatic masses in blue and gas mass proxies in green) scale with $\Omega_m$ (top) and $w$ (bottom) as a function of X-ray mass type (colors) and by cluster redshift (solid: z=0.25, dashed: z=1.0), as given by equation~\ref{eq:cosmo_scaling}. All curves are normalized to $\Omega_m = 0.3$, $w=-1$. Whereas the lensing to X-ray hydrostatic mass ratio does not vary with $\Omega_m$ at a level significant for this study, future work calibrating X-ray mass measurements for high redshift clusters will need to account for this dependence.}
\label{fig:cosmo_dependence}
\end{figure}
As an example of why one should explicitly include the lensing data in a cosmological likelihood, we investigate how the cosmology dependence of the lensing to X-ray ratio manifests itself in the $f_{\mathrm{gas}}$\ test.
In the context of the $f_{\mathrm{gas}}$\ test, the mass ratio scales as $K(z) \propto \Omega_\mathrm{m}D^{3/2}$ (see equation 2 from Paper II, where $K(z)$ is the mass ratio).
Figure~\ref{fig:cosmo2} shows the degeneracy between the mass ratio and $\Omega_\mathrm{m}$ that appears in practice when fitting the full model of Paper~II (for a flat constant-$w$ model), reflecting the $\Omega_\mathrm{m}D^{3/2}$ dependence.
Neglecting the cosmology dependence of the lensing calibration would introduce artificial priors on cosmological constraints from the $f_{\mathrm{gas}}$\ test (or any other cosmological test that uses lensing as a calibrator), potentially biasing results and underestimating uncertainties.
Conversely, the cosmological model space explored has an effect on the inferred mass ratio, albeit negligible with the current statistical uncertainties.
We find that measured lensing to X-ray mass ratio shifts by $\Delta r = 0.01$ between a flat constant-$w$ model and a $\Lambda$CDM model with curvature.
\begin{figure}
\includegraphics[width=\columnwidth]{figures/cosmodegen}
\caption{
Joint $1\sigma$ and $2\sigma$ confidence regions for $\Omega_\mathrm{m}$ and the lensing to X-ray mass ratio for the $f_{\mathrm{gas}}$\ {} measurement, marginalized over the full model used in Paper~II (specifically for a flat, constant-$w$ cosmology). In the $f_{\mathrm{gas}}$\ model, the mass ratio is proportional to $\Omega_\mathrm{m}D^{3/2}$, whereas the extent of the allowed region is primarily set by the precision of the mass ratio measurement. As these results come directly from the analysis of Paper~II, they use the May 2012 \emph{Chandra}{} calibration (CALDB 4.4.10) and do not include the correction for the lensing mass model. This has a small effect on the mass ratio; see Sections~\ref{sec:lensing_masses} and \ref{sec:calib_dependence}.}
\label{fig:cosmo2}
\end{figure}
\subsection{Blind Analysis}
\label{sec:blindanalysis}
Our aim in this work is to calibrate X-ray hydrostatic mass measurements for relaxed clusters to $\sim10\%$ accuracy, a level where ``observer's bias'' must be guarded against.
There is a clear expectation in the literature that X-ray hydrostatic masses should be biased low compared to the true mass \citep{nagai07}.
One group previously re-calibrated \emph{Chandra}{} X-ray masses downwards by 15\% to match these expectations \citep{mahdavi13}.
To combat observer's bias, our lensing and X-ray analyses were completed independently and by different members of the team.
For the X-ray analysis, gas and total mass profiles for individual clusters were blinded with random offsets until after the relaxed cluster sample was identified and their profiles analyzed (Paper~II).
For the lensing analysis, fit procedures and systematic error estimates were fixed before comparisons to literature lensing values occurred \citep{paper3}.
Comparison between lensing and X-ray hydrostatic masses occurred only after the sample of relaxed clusters was finalized.
For full disclosure, we briefly discuss changes that occurred in the analysis during and after the unblinding process.
Following the selection of relaxed clusters, but before $f_{\mathrm{gas}}$\ values were unblinded, we computed the lensing to X-ray mass ratio for 6 clusters using the $P\left ( z\right )$\ likelihood from \citet{paper3}, measuring a ratio of $r=0.95\pm 0.14$.
Following the unblinding of $f_{\mathrm{gas}}$\ values, we received the referee report for \citet{paper3}; however the report did not result in changes to the analysis.
Upgrades to the color-cut analysis described in Section~\ref{sec:data}, most notably marginalizing over a prior on concentration appropriate for relaxed clusters, were then finalized before lensing and X-ray masses for the full sample of 12 clusters were compared, resulting in a ratio of $r = 0.959^{+0.06}_{-0.06}$.
Subsequently, we concluded that the cosmology dependence of the lensing to X-ray mass ratio must be included in the $f_{\mathrm{gas}}$\ analysis, resulting in the lensing likelihood for color-cut masses being incorporated into the $f_{\mathrm{gas}}$\ analysis (see Section~\ref{sec:cosmo_dependence}).
The \emph{Chandra}{} effective area calibration was also updated, to CIAO version 4.6.1 and CALDB version 4.6.2 (Section~\ref{sec:calib_dependence}).
During that process, we fixed a bug in the importance sampling algorithm used to evaluate the ratio at fixed cosmology (Section~\ref{sec:xlratio}).
Immediately before submission, new simulations results became available indicating that the lensing masses were underreported by a few percent (Section~\ref{sec:lensing_masses}), after which a correction factor was introduced.
The cumulative effect of these changes shifted the ratio for our reference cosmology to $0.956\pm 0.082$.
\section{Summary}
\label{sec:summary}
In this work, we have calibrated \emph{Chandra}{} hydrostatic masses for relaxed clusters with accurate weak lensing measurements from the Weighing the Giants project \citep{paper1, paper2, paper3}.
Specifically, we measured the combined astrophysical and instrumental bias present for the \citet{mantz_fgas} sample.
Our approach fully captures lensing and X-ray measurement uncertainties, correlations between the measurements when using a common analysis aperture, the intrinsic scatter between X-ray and lensing masses, and the cosmology dependence of both.
For a fixed cosmology and a sample of 12 clusters, we measure a lensing to X-ray mass ratio of $r=0.96$ within the \emph{Chandra}-determined $r_{2500}$ aperature with a statistical precision of 9\% and additional systematic uncertainty from lensing of 9\%, for a combined precision of $\approx12\%$.
These results are for the specific \emph{Chandra}{} calibrations tested, and cannot be easily extrapolated to other versions of the \emph{Chandra}{} calibration.
We find that our results are robust to perturbations in the sample selection and lensing analysis.
We interpret the measured X-ray to lensing ratio as ruling out large departures from hydrostatic equilibrium, on the order of tens of percent, at the measurement radii in our relaxed target clusters.
Since the effects of biases in the X-ray effective area calibration are degenerate with biases due to non-thermal pressure support, more robust constraints on the level of non-thermal support from lensing and X-ray data will not be possible without an independent constraint on the X-ray calibration, {\it e.g.\ } from precisely measured emission line ratios.
By measuring the lensing to X-ray mass ratio within the cosmological framework of Paper II, we are able to render the results of that work independent of overall shifts in the \emph{Chandra}{} temperature calibration.
We are also able to naturally capture the cosmology dependence of the lensing to X-ray mass ratio by simultaneously modeling the lensing signal for the 12 clusters in this sample with the rest of the $f_{\mathrm{gas}}$\ data set.
The improvements to cosmological constraints that the lensing enables are significant, as discussed in detail in Paper II.
There are clear opportunities for improvement of the lensing-based calibration for the $f_{\mathrm{gas}}$\ experiment (as well as cluster counts; see discussion in \citealt{paper4}) in the next few years.
Efforts are under way to improve key lensing systematics, most notably with measuring shear \citep{great10} and with mass modeling of clusters \citep{desc_whitepaper}.
With these improvements coming into place, the remaining limitation to an improved measurement of the lensing to X-ray mass ratio is the number of clusters with sufficient optical filter coverage for $P\left ( z\right )$\ measurements, which eliminate a key systematic in the lensing masses used in this work \citep[see][]{paper3}.
Expanding the sample of $P\left ( z\right )$\ cluster masses from the current 6 clusters to $\sim24$ clusters would already improve the mass ratio precision, and thus constraints on $\Omega_m$, by $\approx 30\%$.
\section*{Acknowledgments}
We thank our other collaborators in the Weighing the Giants series, including Pat Burchat, Mark Allen, and David Donovan.
DEA would like to thank Lorenzo Lovisari and Tim Schrabback for helpful discussions.
SH thanks MPA and RZG for hospitality and for providing access to their computing facilities and the MXXL simulation data.
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\section{Introduction}
Science and engineering have often drawn inspiration from nature in addressing challenging problems. The adage ``the whole is greater than the sum of parts" has had a great impact on fields such as complex systems engineering, active matter physics and biomimetics towards understanding complex phenomena. Optimization has also witnessed a similar impact with a class of \textit{nature-inspired} computing techniques \cite{patnaik2017nature}. These techniques model the behaviour of agents in nature such as ants and birds that work collectively towards achieving a global objective with local (agent-specific) rules and extend them towards solving optimization problems. Such an approach has given rise to techniques such as evolutionary programming, genetic algorithms, simulated annealing and differential evolution. Swarm Intelligence \cite{fister2013brief} is one such class of nature-inspired optimization techniques that makes use of interacting agents towards optimizing a collective goal. Algorithms based on swarm intelligence include, among others, particle swarm optimization \cite{kennedy1995particle}, ant colony optimization \cite{dorigo2006ant}, firefly algorithm \cite{yang2008firefly}, cuckoo algorithm \cite{yang2009cuckoo}, bat algorithm \cite{yang2010new}, and squirrel-search algorithm \cite{jain2019novel}, which have been used in several applications \cite{mavrovouniotis2017survey,sundar2012swarm,ertenlice2018survey,nebti2017swarm}. These algorithms make use of recursive update rules drawn from natural agents and have proven application across a number of areas.
Particle Swarm Optimization (PSO) is inspired from the movement of a flock of birds and primarily works by combining an individual's cognizance and collective swarm intelligence \cite{kennedy1995particle}. PSO is a gradient-independent algorithm that has been identified as lying between genetic algorithm and evolutionary programming. The movement of its agents, based on individual and collective best strategy, is similar to crossover operations in genetic algorithms \cite{geneticalgointro1992}, and is dependent on stochastic processes similar to evolutionary programming \cite{evoluprogram1997}. PSO exhibits simple update rules and minimal set of tunable parameters along with a feedback control mechanism that make it a highly desirable choice for a diverse range of problems -- parameter estimation \cite{Schwaab2008}, dynamic optimization \cite{Zhou2014,Ourique2002}, forecasting properties of interest \cite{Wang2019}, clustering \cite{alam2014research}, and training feed-forward neural networks \cite{Zhang2007}. PSO has also been used in several applications including robotics \cite{camci2018aerial}, placement of distributed generators in smart grid \cite{el2011optimal}, astronomy \cite{jin2008analysis}, manufacturing \cite{navalertporn2011optimization} and several more \cite{Pluhacek2018}.
The majority of applications of PSO are characterized by a lack of complete knowledge, and at times an analytical formulation of the objective function. There have been several theoretical advances \cite{bonyadi2019theoretical} and algorithmic modifications to the standard PSO algorithm since its introduction. These include the importance of choosing adaptive inertia weights \cite{Shi1999} and the impact of maximum velocity on the algorithm's performance \cite{Shi1998}. Furthermore, PSO has been modified and generalized to incorporate constraints that arise in several optimization problems in practice \cite{ang2020constrained}. Extensions of the algorithm to multi-objective optimization \cite{bin2018multi} and hybridization approaches \cite{fatemeh2019shuffled} have also been proposed in the literature.
The impact of the communication network between agents in the swarm has also been studied in the literature. However, in certain applications, the agents face a hostile environment that can result in the loss of some of the agents during the search, as in search-and-rescue missions behind enemy lines, in drone swarms in modern warfare, and in targeted drug delivery using nano-technologies. For example, in targeted drug delivery using smart nano-particles that could communicate with each other to locate the diseased site, the carriers face a hostile environment wherein the body's natural defense mechanism (i.e., the immune system) perceives them as potential threats and attempts to kill them in the bloodstream. As a result, the number of agents as well as the information available to the remaining agents dynamically change during the search. This can have a detrimental impact on the performance of the swarm. In applications such as supply chain optimization \cite{shukla2011optimizing, shukla2007supply, shuklafunctional} and logistics \cite{meepetchdee2007logistical}, these problems become crucial. Graph-theoretic measures have been used to quantify efficiency and robustness of these network topologies \cite{venkatasubramanian2004spontaneous, ellens2013graph}. However, a generic framework unifying the performance and graph properties of optimal network topologies under hostile conditions has not been studied in the context of PSO, and is the primary focus of our work.
Here, we present such a study, where we consider an environment in which agents are killed at random during the search; we study the performance of different swarm topologies under such conditions and discover generalized properties of desirable network topologies. The rest of the paper is organized as follows -- In Section 2, we present the framework of PSO and the role of network topologies in driving its performance. In Section 3, we describe the agent-based simulation setup including the objective functions, parameters chosen for the PSO algorithm, the set of systematically generate network topologies primarily used in this work, and the probabilistic framework used for simulating hostile environments for the agents. In Section 4, we discuss graph-theoretic properties for efficiency and robustness, and introduce a set of measures used for quantifying performance of various network topologies in the PSO framework. The results corresponding to various levels of hostility on different objective functions are shown in Section 5. A discussion on the major findings of this study is presented in Section 6. Finally, in Section 7 we summarize the useful contributions of this work, alongside a few concluding remarks.
\section{Problem Formulation and Objectives}\label{sec:probformulation&objs}
Let us consider a function $f(\mathbf{x}):\mathbb{R}^n\to\mathbb{R}$ that we wish to optimize. For the sake of simplicity, we consider the range of $f(\cdot)$ to be $\mathbb{R}$ and the function to exhibit only one global optimum with several local optima. The objective of PSO is to optimize the objective function, i.e., to find the value $\mathbf{x}^*$, referred to as the optimal solution that results in the best value (global optimum) of $f(\cdot)$. We intend to study the impact of network topology on the performance of a swarm intelligence-based algorithm such as PSO in a hostile environment. In the following, we briefly introduce particle swarm optimization and highlight the importance of network configuration on the performance of the algorithm.
Particle Swarm Optimization is a nature-inspired algorithm that begins with a set of \textit{agents}, typically initialized at random locations in the search space. The positions of the agents are updated at each iteration according to a set of heuristic-based rules that account for the best known position of the agent and that of the swarm. The update rule in the standard PSO results in the agents moving in a direction that is the resultant of the direction of the best position of the swarm ($\vec{g}$) and the best position of the $i^th$ agent ($\vec{p}_i$) weighted by acceleration coefficients, and a stochastic component in each direction. The update rule for the position of the $i^{th}$ agent in a swarm at iteration $t$ can be expressed as follows:
\begin{eqnarray}
\vec{v}_i(t)&=&\chi[\vec{v}_i(t-1)+\phi_1.\mathrm{rand}(0,1)(\vec{p}_i(t-1)-\vec{x}_i(t-1)) + \phi_2.\mathrm{rand}(0,1) (\vec{n}_i(t-1)-\vec{x}_i(t-1))] \label{eq:velocity}
\\ \vec{x}_i(t)&=&\vec{x}_i(t-1) + \vec{v}_i(t) \label{eq:position}
\end{eqnarray}
In the above equations, $\vec{x}_i(t)$ and $\vec{v}_i(t)$ represent the position and velocity of the $i^{th}$ agent at iteration $t$, respectively; $\vec{p}_i(t-1)$ and $\vec{n}_i(t-1)$ represent the best known position of the $i^{th}$ agent and its neighborhood at iteration $(t-1)$, respectively; $\phi_1$ and $\phi_2$ are the acceleration coefficients, and $\chi$ represents the constriction coefficient. In a swarm where all agents communicate with each other (in standard PSO), $\vec{n}_i(t-1)$ represents the best position of the entire swarm ($\vec{g}$) and is the same for all agents, while for other configurations, $\vec{n}_i(t-1)$ represents the best position of the neighbors of the $i^{th}$ agent and can be different for different agents. These update rules result in exploration of the search space as well as exploitation of knowledge of the landscape over several iterations. Table \ref{table:standardPSO} presents the step-by-step procedure for the standard PSO algorithm.
\begin{table}[H]
\centering
\caption{Pseudo code for the standard PSO algorithm}
\label{table:standardPSO}
\scalebox{0.95}{
\begin{tabular}{|l|}
\hline
Algorithm $1$: Standard PSO \\ [0.5ex]
\hline
\textbf{Inputs:} \\
\texttt{nAgents}: the number of agents in the swarm\\
$f$: the function to be optimized\\
\texttt{maxIters}: the maximum number of iterations\\ \hline \\ [-2ex]
Initialize the agent positions and velocities randomly \\
while $t<$\texttt{numIters}:\\
\quad for each agent $i$ in \texttt{nAgents}:\\
\quad \quad Compute $\vec{p}_{i}$, position of best solution agent $i$ has found so far\\
\quad \quad Compute $\vec{g}$, position of best solution found by all the agents in the swarm so far\\
\quad \quad Update $\vec{v}_{i}$, velocity of agent $i$ updated using equation \ref{eq:velocity}\\
\quad \quad Update $\vec{x}_{i}$, position of agent $i$ updated using equation \ref{eq:position}\\ \hline
\textbf{Outputs:}\\
\quad $\vec{x}^{*}$, the position of the global optimum \\
\quad $f(\vec{x}^{*})$, the function values at the global optimum\\ [1ex]
\hline
\end{tabular}
}
\end{table}
It can clearly be observed from Equations \eqref{eq:velocity} and \eqref{eq:position} that the updated positions of the agents depend on the manner in which the best-known position of the neighbours, i.e., $\vec{n}_i(t-1)$ is obtained at each iteration. This is in turn determined by the topology of the communication network of the swarm. The network configuration thus influences the information received by each agent, determining the updated positions for the agents, playing a crucial role in convergence to the global optimum. The impact of network topology on the performance of PSO has been studied in the literature \cite{1004493, 1202252, 785509} and it has been established that network topologies play a significant role in determining the convergence of the algorithm. However, topological factors that result in superior performance have not been identified. Furthermore, the performance of the algorithm is also seen to have a strong dependence on the type of function being optimized. These studies are also limited to the case where agents are not lost to the environment, as a result of which the trade-off between efficiency of convergence and robustness to the environment is not studied. We conduct several computational experiments that simulate PSO in a hostile environment, which are discussed in detail in the next section.
\section{Agent-based Simulation Setup}
The computational experiments considered in this work have four crucial components, the choice of which determines the performance of the algorithms, and hence the generalizability of results. These factors include:
\begin{enumerate}[(i)]
\item the objective function
\item the hyperparameters of the PSO algorithm
\item the communication network configuration of agents in PSO
\item the severity and nature of the hostile environment
\end{enumerate}
The following sections present short discussions on the choice of the above factors and the rationale behind them.
\subsection{Objective Function}
We consider five standard benchmark objective functions -- Shekel, Ackley, Griewank, Schwefel and Rastrigin \cite{simulationlib, molga2005test} -- to evaluate and compare the performance of PSO for different network topologies. We perform extensive computational experiments with the Shekel function to identify common patterns in performance of PSO. We then choose a subset of topologies that exhibit a desirable trade-off between efficiency of convergence and robustness to hostile environment, and compare the performance of PSO for the remaining four functions. The first objective function considered in this work is the Shekel function, which can be expressed as:
\begin{equation}
f(\mathbf x) = \sum_{i=1}^m \bigg (c_i + \sum_{j=1}^n (x_j - a_{ji})^2 \bigg)^{-1} \label{eq:shekelfun}
\end{equation}
where, $m$ is the number of local maxima, $n$ is the dimension of the input space ($\mathbf{x}\in\mathbb{R}^n$), $c_i$ determines the magnitude of the $i^{th}$ local maximum and $a_{ji}$ is the $j^{th}$ coordinate of location of $i^{th}$ maximum. Figure \ref{fig:benchmarkfuns}(a) depicts the landscape of a two-dimensional Shekel function with $9$ local maxima and one global maximum. The remaining four objective functions are shown in Figure \ref{fig:benchmarkfuns}(b)-(e) for the two dimensional case. The functional form, number of dimensions, number of local optima and location of global optimum for the five functions are listed in Table \ref{tab:benchmarkfuns}.
\begin{figure}[H]
\centering
\subfigure[Shekel Function]{
\includegraphics[width=0.6\linewidth]{PSO/figs/shekel2.png}
}\\
\subfigure[Ackley Function]{\includegraphics[width=.48\textwidth]{PSO/figs/ackley2.png}} \quad
\subfigure[Griewank Function]{\includegraphics[width=.48\textwidth]{PSO/figs/griewank2.png}}\\
\subfigure[Schwefel Function]{\includegraphics[width=.48\textwidth]{PSO/figs/schwefel2.png}}\quad
\subfigure[Rastrigin Function]{\includegraphics[width=.48\textwidth]{PSO/figs/rastrigin2.png}}
\caption{Benchmark Objective Functions for $n=2$}
\label{fig:benchmarkfuns}
\end{figure}
\begin{table}[h]
\caption{Description of Objective Functions}
\label{tab:benchmarkfuns}
\centering
\scalebox{0.8}{
\begin{tabular}{cccccc}
\hline
\multirow{2}{*}{Function} & \multirow{2}{*}{Expression} & \multirow{2}{*}{$n$} & \multirow{2}{*}{Range} & Local & Location of \\
& & & & Optima & Global Optimum \\
\hline
Shekel & $f(\mathbf x) = \sum_{i=1}^m \bigg (c_i + \sum_{j=1}^n (x_j - a_{ji})^2 \bigg)^{-1}$ & 4 & $[0, 10]$ & 10 & $[4,4, 4, 4]^T$ \\
Ackley & $f(\mathbf x) = -a\exp\left(-b\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_i^2}\right) - \exp{\left(\frac{1}{n}\sum_{i=1}^{n}\cos(c x_i)\right)} + a + \exp(1)$ & 2 & $[-15, 30]$ & $>10$ & $[0,0]^T$ \\
Griewank & $f(\mathbf x) = \frac{1}{4000}\sum_{i=1}^n x_i^2 - \prod_{i=1}^{n} \cos \left(\frac{x_i}{\sqrt{i}}\right) + 1$ & 2 & $[-600, 600]$ & $>10$ & $[0,0]^T$ \\
Schwefel & $f(\mathbf x) = 418.9829 n - \sum_{i=1}^n \left[-x_i \sin \left(\sqrt{|x_i|}\right)\right]$ & 2 & $[-500, 500]$ & $>10$ & $[420.9687, 420.9687]^T$ \\
Rastrigin & $f(\mathbf x) = 10n + \sum_{i=1}^n \left[x_i^2 - 10\cos(2\pi x_i)\right]$ & 2 & $[-5.12, 5.12]$ & $>10$ & $[0,0]^T$ \\
\hline
\end{tabular}
}
\end{table}
\subsection{Hyperparameters for Particle Swarm Optimization}
The second component includes the set of hyperparameters employed by the PSO algorithm. In this work, we have chosen the following values of the hyperparameters:
\begin{enumerate}
\item $\chi=0.7298438$, a constriction coefficient that prevents the velocities from exploding
\item $\phi_1=0, ~ \phi_2=2.05$, acceleration coefficients for the agents' cognizance and neighborhood cognizance
\item $V_{min}=-10, V_{max}=10$, lower and upper bounds on the agents' velocities
\item $N=100$, number of agents in the swarm
\item $T=1000$, maximum number of iterations to allow convergence of all network configurations
\end{enumerate}
\textit{Remark 1:} In this study, we set $\phi_1=0$ so that the best known position of an agent is not incorporated in updating its position at any iteration. As a result, agents move in a direction that is solely driven by the information made available to them by their neighborhood.
\textit{Remark 2:} It is important to note here that while the hyperparameters of the algorithm could be tuned to achieve desired efficiency and robustness for each scenario, such an exercise can be practically infeasible and thus of little significance in several applications. Furthermore, owing to the uncertainty in the form of random loss of agents in our study and absence of complete information about the objective function in practical scenarios, a perfect tuning of hyperparameters can be extremely time-consuming, difficult, and computationally expensive. Therefore, we fix the values of the hyperparameters for all the experiments and focus the study towards discovering generalized features of the algorithm that are of interest in practical applications. With the objective function and tuning of the algorithm fixed, we next describe the different network configurations and the simulation of the hostile environment used in our study.
\subsection{Network Topology of Communicating Swarm}
Network topologies can broadly be classified as deterministic and random topologies. Since the objective of this study is to identify network configurations that result in a desired trade-off between efficiency and robustness of the algorithm, we focus our attention on deterministic graphs and present only limited results on random graphs. We consider three classes of graphs -- complete (all agents connected to each other), star (hub-and-spoke configuration) and ring (all agents connected to two neighbours, forming a closed ring) -- that represent different degrees and manners of connectivity of the graph. We create a spectrum of intermediate graphs by performing deterministic operations on the above graph classes as follows:
\begin{enumerate}[I.]
\item Complete--to--Star:
\begin{enumerate}[(a)]
\item The initial graph is a complete graph where all the agents constitute the fully connected core with each node connected to every other node
\item The core size of the graph is shrunk in steps of $1$ by removing an agent from the core and attaching it to one of the core nodes
\item In reducing the core size, it is ensured that the resultant core at each step is full connected, as shown in Figure \ref{fig:networks}(a).
\end{enumerate}
\item Star--to--Ring:
\begin{enumerate}[(a)]
\item The initial graph is a star graph which consists of only one core/hub node and $N-1$ non-core nodes.
\item The hub is expanded in steps of $1$ to connect one non-core node to the core at each step.
\item In expanding the core, it is ensured that the core is not fully connected but connected in a ring topology as shown in Figure \ref{fig:networks}(a).
\end{enumerate}
\item Ring--to--Complete:
\begin{enumerate}[(a)]
\item The initial graph is a ring-graph where each node is connected to its two immediate neighbors in a circular topology
\item In each step, an additional ring structure is introduced to the graph by symmetrically connecting agents to the non-immediate neighbors.
\item This results in a multi-ring graph with increasing connectivity as one moves from ring to complete graph as shown in Figure \ref{fig:networks}(a).
\end{enumerate}
\end{enumerate}
The above rules are used to generate $80$ configurations between each pair of basic graphs, resulting in a total of $240$ configurations. The colors shown in Figure \ref{fig:networks}(a) are used to demarcate different segments of the spectrum and are used for clarity in presentation of the results. In addition, we also use the von Neumann grid, scale-free graph, random graph and small-world graph as additional cases of network configurations as shown in Figure \ref{fig:networks}(b)--(e). These networks are generated using the NetworkX $2.4$ package in Python. Specifically, random graphs are generated in a manner that the probability of an edge between any two nodes is $0.1$, while small-world graphs are generated with a degree of $10$ for each node and a rewiring probability of $0.1$ for each edge.
\begin{figure}
\centering
\subfigure[]{\includegraphics[width = 0.8\linewidth]{PSO/figs/triangular_spectrum_less.png}}\\
\subfigure[]{\includegraphics[width = 0.12\linewidth]{PSO/figs/vonNeumann.PNG}} \hspace{5em}
\subfigure[]{\includegraphics[width = 0.15\linewidth]{PSO/figs/Scale-free.PNG}} \hspace{5em}
\subfigure[]{\includegraphics[width = 0.15\linewidth]{PSO/figs/Random.PNG}} \hspace{5em}
\subfigure[]{\includegraphics[width = 0.15\linewidth]{PSO/figs/Small-world.png}}
\caption{Network topologies used for swarm neighborhood connectivity: (a) $240$ network topologies are generated by traversal along the spectrum from complete-to-star (red), star-to-ring (blue), and ring-to-complete (green) (b) a von Neumann grid: each node is connected to 4 neighboring nodes, (c) Scale-free graph: the probability of finding a node with degree $k$ is proportional to $k^{-\gamma}$, (d) Random graph: edges between nodes are assigned at random with a predetermined probability $p_e$. (e) Watts-Strogatz (Small-world) graph: edges of a multi-ring graph are rewired with a rewiring probability $p_w$}
\label{fig:networks}
\end{figure}
\subsection{Loss of Agents in the Hostile Environment}
We consider a hostile environment where each agent has a probability of getting killed/deactivated at a given iteration. Let us consider a scenario where all agents have the same probability $p$ of being lost to the environment at any iteration of the algorithm. Then, the expected number $N_a$ of the alive/active agents at the end of $t$ iterations can be obtained as:
\begin{align}
N_a = N(1-p)^t
\end{align}
The expected fraction of active and deactivated agents, $F_a$ and $F_d$, respectively are given by,
\begin{align}
F_a &= (1-p)^t \\
F_d &= 1 - (1-p)^t
\end{align}
We study the performance of PSO in an environment that causes loss (death fractions) of $15\%$ and $30\%$ of the total number of agents at the end of $500$ iterations. The corresponding probabilities of deactivation for any agent for these cases is $p=0.00033$ and $p=0.0007$, respectively. The pseudo-code for this randomized death of agents is presented in Table \ref{table:randomDeath}.
\begin{table}[H]
\centering
\caption{Pseudo code for randomized death of agents}
\label{table:randomDeath}
\scalebox{0.95}{
\begin{tabular}{|l|}
\hline
Algorithm $2$: \texttt{Randomized Death} of agents \\ [0.5ex]
\hline
\textbf{Inputs:} \\
$p:$ the deactivation probability for each agent \\
\texttt{active agents}: the list of agents that are active \\
\hline \\ [-2ex]
for each \texttt{agent} in \texttt{active agents}:\\
\quad Generate a random number $r$ in the range $[0, 1]$\\
\quad if $r<p$: \\
\quad \quad deactivate \texttt{agent}\\
\quad Remove \texttt{agent} from the list \texttt{active agents}
\\\hline
\textbf{Outputs:}\\
\texttt{active~agents}: list of active agents after randomized death \\ [1ex]
\hline
\end{tabular}
}
\end{table}
The pseudo-code for the resultant algorithm that combines network topology with loss of agents for the PSO algorithm is presented in Table \ref{table:modifiedPSO}.
\begin{table}[H]
\centering
\caption{Pseudo code for PSO algorithm with network topology and dying agents}
\label{table:modifiedPSO}
\scalebox{0.95}{
\begin{tabular}{|l|}
\hline
Algorithm $3$: PSO with Network Topology and Dying Agents \\ [0.5ex]
\hline
\textbf{Inputs:} \\
\texttt{nAgents}: the number of agents in the swarm\\
$f$: the function to be optimized\\
\texttt{maxIters}: the maximum number of iterations\\
$p:$ the death probability for each agent \\
\texttt{adjMatrix}: the \texttt{nAgents} $\times$ \texttt{nAgents} matrix describing the network connectivity for each agent \\
\hline \\ [-2ex]
Initialize the agent positions and velocities randomly \\
while $t<$\texttt{numIters} and \texttt{numActiveAgents} $\neq 0$:\\
\quad for each agent $i$ in \texttt{active agents}:\\
\quad \quad Compute $\vec{p}_{i}$, position of best solution agent $i$ has found so far\\
\quad \quad Compute $\vec{n}_{i}$, position of best solution found by agent $i$'s neighborhood so far defined by the \texttt{adjMatrix}\\
\quad \quad Update $\vec{v}_{i}$, velocity of agent $i$ updated using equation \ref{eq:velocity}\\
\quad \quad Update $\vec{x}_{i}$, position of agent $i$ updated using equation \ref{eq:position}\\ [2ex]
\quad \quad Invoke the \texttt{Randomized Death} algorithm and update \texttt{active agents}, \texttt{adjMatrix}\\ [1ex]
\hline
\end{tabular}
}
\end{table}
In the next section, we present different graph-theoretic metrics considered in this study to quantify the robustness and efficiency of different network topologies followed by various measures that capture performance in the PSO framework.
\section{Graph Theoretic Properties and PSO Performance Metrics} \label{sec:robust-eff-tradeoff}
The convergence of PSO with different network topologies in a hostile environment depends on -- ($1$) the ability of the network to \textit{efficiently} transmit information between the nodes (agents), and ($2$) the ability of the network to sustain loss of agents with minimal deterioration in performance of PSO. We first present graph-theoretic metrics that quantify the efficiency and robustness of a network topology in itself (independent of PSO) followed by a set of four metrics that we use to quantify the performance of any topology on the PSO algorithm.
We wish to highlight here that the networks that maximize the graph-theoretic measures of efficiency and robustness do not necessarily result in the most efficient and robust performance on the PSO algorithm as we discuss in detail later in Section \ref{sec:results}.
\subsection{Graph Theoretic Properties} \label{sec:graphmetrics}
The efficiency of a graph is its ability to quickly communicate information between different nodes in the graph. The average geodesic distance (also called average path length) is a very commonly used metric to quantify efficiency of a graph. The average geodesic distance $L$ of a graph $G$ with $N$ nodes can be expressed as:
\begin{align}
L = \frac{1}{N(N-1)}\sum d_{ij}
\label{eq:apsp}
\end{align}
where $d_{ij}$ is the length of the shortest path between $i^{th}$ and $j^{th}$ nodes of $G$. The above quantity is equal to $1$ for a complete graph, signifying highly efficient transfer of information in the graph. On the other hand, the star graph exhibits an average geodesic distance of approximately $2$ while the ring graph has an average geodesic distance $\sim N/4$.
The robustness of a graph can be quantified using the natural connectivity which has been shown to be sufficient, superior, and is known to be a physically meaningful measure for quantifying the robustness of complex graphs \cite{jun2010natural}. Physically, it is proportional to the number of closed loops for each node of the graph. It is derived from the eigenvalues of the adjacency matrix of the graph as:
\begin{equation}\label{eq:naturalconnect}
\bar \lambda = \ln \bigg ( \frac{1}{N} \sum_{i=1}^N e^{\lambda_i} \bigg)
\end{equation}
where $\lambda_i$ is the $i^{th}$ eigenvalue of the adjacency matrix of the graph. The adjacency matrices of a complete, star and ring graphs (A, B, and C, respectively) with 5 nodes can be represented as:
\begin{equation*}
A = \begin{bmatrix}
0 & 1 & 1 & 1 & 1\\
1 & 0 & 1 & 1 & 1\\
1 & 1 & 0 & 1 & 1\\
1 & 1 & 1 & 0 & 1\\
1 & 1 & 1 & 1 & 0\\
\end{bmatrix}
B = \begin{bmatrix}
0 & 1 & 1 & 1 & 1 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0\\
1 & 0 & 0 & 0 & 0\\
1 & 0 & 0 & 0 & 0\\
\end{bmatrix}
C = \begin{bmatrix}
0 & 1 & 0 & 0 & 1 \\
1 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 1 \\
1 & 0 & 0 & 1 & 0 \\
\end{bmatrix}
\end{equation*}
The set of eigenvalues for the adjacency matrices (graph spectrum) for the three graphs above is given by,
\begin{equation*}
A_\lambda = \{4, -1, -1, -1, -1\} \quad B_\lambda = \{2, 0, 0, 0, -2\} \quad C_\lambda = \{2, 0.62, 0.62, -1.62, -1.62\}
\end{equation*}
Therefore, using Equation \ref{eq:naturalconnect}, the corresponding natural connectivities obtained for the given graphs are,
\begin{align*}
\bar \lambda_{complete} = 2.42 \quad \bar \lambda_{star}= 0.74 \quad \bar \lambda_{ring} = 0.83
\end{align*}
It can be clearly seen that the natural connectivity values for the three graphs are in agreement with the expected behavior. For example, the complete graph is the most robust network owing to the presence of multiple loops for a given node, while star and ring graphs -- with zero and one loops respectively -- are the least robust since the network can be fragmented with removal of only a few nodes. Figure \ref{fig:robustness_and_apsp} shows the computed measures of efficiency and robustness for the triangular spectrum of topologies described in Figure \ref{fig:networks}(a).
\begin{figure}[H]
\centering
\subfigure[]{{\includegraphics[width=0.42\linewidth]{PSO/figs/Average_Path_Length_vs_Topology_Index_grid.png} }}
\quad
\subfigure[]{{\includegraphics[width=0.42\linewidth]{PSO/figs/Natural_Connectivity_vs_Topology_Index_grid.png}}}%
\caption{Network topology metrics (a) Average path length and (b) Natural connectivity, as one goes along the spectrum of proposed networks, as a function of the topology index. Average path length is a measure of efficiency of the network, while natural connectivity is a measure of robustness. The color codes are corresponding to traversal from complete-to-star (red), star-to-ring (blue) and ring-to-complete (green)}%
\label{fig:robustness_and_apsp}%
\end{figure}
It can be seen that as we traverse the spectrum, the following patterns in the efficiency and robustness are observed:
\begin{itemize}
\item Complete--to--Star: The average path length increases and natural connectivity decreases. This is equivalent to decreasing efficiency and robustness, respectively. This is a result of removal of edges from the graph in this section of the spectrum.
\item Star--to--Ring: The average path length increases while natural connectivity remains same, suggesting decreasing efficiency and near-constant robustness. This is a result of increase in the core size for approximately the same number of edges along this section of the spectrum.
\item Ring--to--Complete: The average path length decreases and natural connectivity increases, resulting in an increase in network efficiency and robustness. This is a result of increased number of edges due to addition of multi-rings.
\end{itemize}
\subsection{PSO Performance Metrics} \label{sec:psometrics}
In order to compare the performance of PSO for different communication network configurations and different severity of the hostile environment, we note the following metrics:
\begin{enumerate}
\item Global Success Ratio (GSR): The fraction of times the \textit{entire swarm} reaches the global optimum
\item Global Success Time to Convergence (GS Time): Average number of iterations required for the \textit{entire swarm} to reach the global optimum. This metric is calculated only for the cases when \textit{global convergence} is achieved.
\item Number of Winners: Average number of agents that reach the global optimum
\item Trade-off metric: In addition to the performance measures defined above, we define the following performance metric that captures the trade-off between the number of winners and the time for convergence:
\begin{equation}\label{eq:performance-metric}
\text{Trade-off metric} = \alpha~\frac{{\mathrm{Winners}}}{{\mathrm{Winners}_{\mathrm{max}}}} - (1-\alpha)~\frac{{\mathrm {GS~Time}}}{{{\mathrm{GS~Time}}_{\mathrm{max}}}}
\end{equation}
where $\alpha$ determines the relative importance of the two measures towards the metric. In our simulations, we choose $\alpha=0.7$ to assign a relatively higher importance to success ratio, emphasizing on convergence of all agents to the global optimum. This implies that we desire all the agents to reach the global optimum while allowing for a larger time to convergence. These choices can be motivated for applications such as drug delivery where one desires all of the drug to be delivered to the subject at the desired location, even if it takes a (manageably) longer for the delivery.
\end{enumerate}
In the next section, we present the results of the study based on the above performance metrics for the benchmark objective functions, for different death fractions with respect to the graph-theoretic metrics.
\section{Results}\label{sec:results}
In order to balance out the effect of initial state of agents on the convergence of the algorithm and improve generalizability of results, $50$ different initializations of agents for each network topology and deactivation were considered. In each scenario, the performance metrics listed in the previous section were noted and the average metric for each configuration was computed. We first present the performance of PSO for different communication network topologies and death fractions, followed by an analysis of the performance in relation to graph-theoretic metrics of efficiency and robustness. As discussed earlier, we first perform exhaustive experiments with the 4D-Shekel function to identify common trends and then present results on additional benchmark functions with a subset of important topologies.
\subsection{Performance of PSO for Different Network Topologies}
The performance of PSO in terms of the four metrics listed above as a function of the network topology is presented in Figure \ref{fig:PerformancevsNetwork}. For the sake of clarity, all metrics corresponding to networks in complete-to-star, star-to-ring and ring-to-complete regimes of the spectrum are shown in red, blue and green colors, respectively.
It can be observed from Figure \ref{fig:PerformancevsNetwork} that in the case of no deactivation of agents, the global success ratio (GSR) increases as we move from complete graph to star graph, becomes nearly flat at the maximum value of $1$ while going from star graph to ring graph, and then decreases again while moving from the ring graph to complete graph. However, with deactivation of agents, GSR decreases in all the three regimes, with a significant increase in GSR after the initial few topologies in the ring-to-complete regime.
The global success time to convergence (GS Time) is lowest for the complete graph with only a marginal increase as we move towards the star topology. However, it rapidly increases while going from the star graph to the ring graph, before falling off again in the ring-to-complete regime. These trends are observed irrespective of the degree of hostility of the environment.
The number of winners for all the three regimes exhibits a similar trend as GSR. However, it is interesting to note that in the case of deactivation of agents, the number of winners in the complete-to-star regime follows an inverse trend with GSR suggesting a higher number of winners for the star-like topologies than completely connected topologies.
Finally, the trade-off metric that captures the weighted combination of number of winners and convergence time is maximized for an intermediate region in the star-to-ring and ring-to-complete regimes across all death fractions. It is also fascinating to note that the networks that maximize this metric under non-hostile conditions also result in the best performance under hostile conditions. This suggests an invariance in the performance of a network topology to hostility of the environment that can be exploited in designing robust network topologies. This is explored later in the article.
\begin{figure}[h]
\centering
\makebox[\textwidth][c]{\includegraphics[scale=0.52]{PSO/figs/perf_vs_topindex.png}}
\caption{Performance indicators: Global Success Rate (GSR), Global Success Time to Convergence (GS Time), Number of Winners, and Trade-off metric for different Death Rates for different network topologies}
\label{fig:PerformancevsNetwork}
\end{figure}
This section highlighted the dependence of PSO performance on various network architectures. While the complete graph is known to be robust and efficient graph in a purely graph-theoretic sense, such a topology exhibits consistently poor performance (GSR of $\sim50\%$) in terms of optimizing the objective function. In the following sections, we adopt a systematic approach to explore underlying patterns relating the properties of a graph with its performance in the PSO framework.
\subsection{Performance of PSO and Graph Efficiency}
We now study the variation of network performance as a function of its graph-theoretic efficiency. The four performance metrics of PSO defined in Section \ref{sec:psometrics} as a function of the average geodesic distance ($L$) of the network are shown in Figure \ref{fig:PerformancevsAPSP}. In the case of no deactivation of agents, the GSR initially increases with increase in $L$ after which it saturates close to the maximum value of $1$ for larger values of $L$. When agents are deactivated, the absolute values of GSR across all values of $L$ reduce with the magnitude of deterioration being higher for large values of $L$. Irrespective of the rate of deactivation, GSR reaches a maximum for $L\sim 3$, indicating toward the performance invariance of such graphs performance towards hostile environments.
The GS Time increases linearly with increase in average geodesic distance of the graph. This is expected since larger values of $L$ imply lower information transfer through the network resulting in longer times to convergence. The number of winners exhibits a trend that is similar to that observed for GSR but the sensitivity to loss of agents is relatively lower. The trade-off measure capturing the number of winners and the GS Time is also maximized for an average path length $\sim 3$ irrespective of the network topology, or the death fraction. This is suggestive of the existence of a class of networks that are inherently suited for maximizing performance under scenarios involving loss of agents.
It can be inferred from the above that there is a trade-off between the overall performance of a network in PSO and its average path length. Specifically, while highly efficient graphs take less time to converge, the GSR is much lower because of premature convergence to local optima. On the other hand, while graphs with low efficiency promise near-certain convergence, they take longer to reach the global optimum. The trade-off metric captures this effect well, and reveals that the performance peaks at an average path length of $\sim 3$.
\begin{figure}[h]
\centering
\makebox[\textwidth][c]{\includegraphics[scale=0.52]{PSO/figs/perf_vs_apsp.png}}
\caption{Performance indicators: Global Success Rate (GSR), Global Success Time to Convergence (GS Time), Number of Winners, and Trade-off metric for different Death Rates for different values of average path length}
\label{fig:PerformancevsAPSP}
\end{figure}
\subsection{Performance of PSO and Graph Robustness}
After studying the performance of PSO with respect to graph efficiency quantified by the average geodesic distance, we study the same with respect to the graph-theoretic robustness of networks. The performance of PSO as a function of the natural connectivity (robustness) of the network topology is shown in Figure \ref{fig:PerformancevsNatconnect}.
It is observed that, in the absence of a hostile environment, the GSR decreases with an increase in robustness of the graph. This is potentially due to the presence of larger number of closed-loops for highly robust graphs that result in a sub-optimal exploration of the search space and consequently results either in premature convergence, or no convergence at all. An interesting observation is that for the same value of natural connectivity, there exist multiple architectures that result in significantly different performances. For instance, networks with distributed architectures (such as multi-ring) with nearly the same connectivity for all nodes perform better than networks with hub-like structures (such as a star-graph) where a few hubs (central nodes) are highly connected to a large fraction of other nodes. This is because in hub-like networks, removal of the hubs would significantly limit the information transfer across the network.
Therefore, the foregoing sections have established that in order to maximize performance of a network topology in the PSO framework, the network should be sufficiently connected with an average geodesic distance of $\sim 3$ to ensure optimal information transfer and should have distributed architectures in order to ensure robustness towards hostile conditions where agents are deactivated at random.
\begin{figure}[h]
\centering
\makebox[\textwidth][c]{\includegraphics[scale=0.52]{PSO/figs/perf_vs_natconnect.png}}
\caption{Performance indicators: Global Success Rate (GSR), Global Success Time to Convergence (GS Time), Number of Winners, and Trade-off metric for different Death Rates for different values of Natural Connectivity (robustness)}
\label{fig:PerformancevsNatconnect}
\end{figure}
\subsection{Performance of PSO with Standard Topologies}
In addition to the 240 topologies from the triangular spectrum described in Figure \ref{fig:networks}(a), we study the performance of PSO with a few standard network topologies that do not fall under any regime in the above spectrum. Specifically, we consider the von Neumann grid, scale-free graphs, random graphs, and small-world graphs for our study. The results are presented in Table \ref{table:comparisontable}.
It is observed that the von Neumann grid has the highest number of winners and the largest GSR but the convergence time of this graph is the largest among all the four topologies considered. This behaviour is suggestive of lower information transfer (and hence, lower efficiency), which is also reflected in its largest average geodesic distance across all other networks. It also exhibits a highly sensitive GSR with respect to loss of agents, which is explained by the lowest value of natural connectivity ($\bar{\lambda}$) for this network.
The scale-free network exhibits a low average geodesic distance and hence higher efficiency, which results in lower convergence times, but a slightly lower GSR. However, the GSR of this network is extremely sensitive to loss of agents even with a relatively high natural connectivity because of the hub-like architecture similar to the networks in the star-to-ring regime. This results in a performance that is highly sensitive to the loss of agents in the network.
Random graphs exhibit the highest efficiency and hence the lowest convergence times. However, they also result in a lower GSR, potentially because of a relatively higher information overload in the network. Furthermore, due to lack of hubs in random graphs, the performance is seen to be robust to loss of agents.
Finally, small-world graphs are seen to result in a relatively higher GSR and lower convergence times as a result of higher efficiency. These graphs can also be observed to be relatively robust to loss of agents. The performance of small-world graphs is marginally better than random graphs in terms of the GSR and the number of winners in the PSO framework. The convergence times are, however, comparable with the random-graphs converging marginally faster than the small-world graphs.
In the final section, we present the results of PSO on standard benchmark optimization functions for selected networks, followed by a detailed discussion of the different inferences on desired network characteristics for maximizing PSO performance.
\begin{table}[H]
\centering
\caption{Comparison of mean performance metrics for standard topologies on the Shekel function}
\label{table:comparisontable}
\begin{tabular}{ccccccc}
\toprule
Network & Death Fraction (\%) & Number of Winners & GSR & GS Time & $L$ & $\bar{\lambda}$ \\\midrule
\multirow{3}{*}{von Neumann} & 0 & 100 & 1.00 & 189.1 & \multirow{3}{*}{6.67} & \multirow{3}{*}{1.51} \\
& 15 & 95 & 0.70 & 168.5 & & \\
& 30 & 87 & 0.68 & 174.1 & & \\
\hline
\multirow{3}{*}{Scale-free} & 0 & 96 & 0.96 & 138.7 & \multirow{3}{*}{2.75} & \multirow{3}{*}{20.01} \\
& 15 & 94 & 0.56 & 135.8 & & \\
& 30 & 75 & 0.34 & 139.4 & & \\
\hline
\multirow{3}{*}{Random} & 0 & 94 & 0.94 & 106.5 & \multirow{3}{*}{1.70} & \multirow{3}{*}{25.69} \\
& 15 & 91 & 0.78 & 105.0 & & \\
& 30 & 85 & 0.72 & 101.9 & & \\
\hline
\multirow{3}{*}{Small-world} & 0 & 98 & 0.98 & 124.7 & \multirow{3}{*}{2.72} & \multirow{3}{*}{6.31} \\
& 15 & 94 & 0.78 & 117.5 & & \\
& 30 & 88 & 0.78 & 121.5 & & \\ \bottomrule
\end{tabular}
\end{table}
\subsection{Effect of Objective Function}
In order to study the impact of objective function on network performance in PSO, we identified key network configurations that exhibit a desirable trade-off between efficiency, robustness, and performance to analyze their performance on four standard objective functions.
For this study, we choose the complete graph, star graph, and the ring graph topologies (corners of the triangular spectrum in Figure \ref{fig:networks}(a)) along with the standard topologies discussed in the previous section. In addition, we consider two graphs that result in the best performance among the 240 topologies in the triangular spectrum -- one in the star-to-ring regime with 8 hubs and the other in the ring-to-complete regime with 9 multi-rings as shown in Figure \ref{fig:bestTopologiesTriangle}.
\begin{figure}[h]
\centering
\subfigure[][Star-to-ring regime with $8$ hubs]{\includegraphics[width=.4\textwidth]{PSO/figs/s2rbest.png}} \quad
\subfigure[][Ring-to-complete regime with $9$ multi-rings]{\includegraphics[width=.4\textwidth]{PSO/figs/r2cbest.png}}
\caption{Best $2$ topologies from the triangular spectrum comprising 240 topologies}
\label{fig:bestTopologiesTriangle}
\end{figure}
We test the performance of these graphs on the Ackley, Griewank, Schwefel, and Rastrigin functions with function parameters and range as described in Table \ref{tab:benchmarkfuns}. The results for the respective functions are presented in Tables \ref{tab:ackley}, \ref{tab:griewank}, \ref{tab:schwefel}, and \ref{tab:rastrigin}.
It can be observed that, for each objective function, there exists a network topology that results in the best performance in terms of high GSR and number of winners, and low convergence times. For instance, across all death fractions, the Von Neumann topology results in the best performance for Ackley function, small-world topology for Griewank function, small-world for Schwefel function, and complete graph for Rastrigin function. These topologies maximize the number of winners and GSR while simultaneously minimizing the time to convergence (GS Time).
However, small-world graphs are observed to be consistently outperforming almost all the other topologies irrespective of the choice of objective function, or the death fraction. Even for the Ackley and the Rastrigin functions where the von Neumann graph and complete graph perform the best, small-world graphs result in a performance very close to the best performance achieved by both of these networks. We attribute such a superior performance to two factors -- first, the average geodesic distance of $2.72$ for small-world graphs as reported in Table \ref{table:comparisontable} is close to our observation of optimal information transfer occurring at $L\sim3$; second, the small-world networks exhibit a distributed architecture with almost equal connectivity for all the nodes in the network resulting in a robust performance in the presence of higher deactivation of agents as discussed in the foregoing sections.
Similar to small-world networks, the 9 multi-rings architecture of Figure \ref{fig:bestTopologiesTriangle}(b) also exhibits consistently better performance across all the objective functions and is robust to various levels of death fraction. The 9 multi-rings network architecture is characterized by an average geodesic distance of $3.27$ and has a distributed architecture -- validating the underlying network structures identified in the previous sections that are responsible for better and robust PSO performance. This is further reinforced by analyzing the performance of the 8-hub network architecture in Figure \ref{fig:bestTopologiesTriangle}(a) with a average geodesic distance of $3.86$ and with 8 hubs architecture. Although the multi-hub architecture results in faster convergence and better performance under non-hostile conditions, the performance degrades significantly when agents are deactivated at random due to the fragility of such an architecture towards random deactivation of agents.
\begin{table}[H]
\centering
\caption{Comparison of mean performance metrics on Ackley function}
\label{tab:ackley}
\begin{tabular}{ccccc}
\toprule
Network & Death Fraction (\%) & Number of Winners & GSR & GS Time \\\midrule
\multirow{3}{*}{Complete} & 0 & 100 & 1.00 & 146.2 \\
& 15 & 96 & 0.72 & 137.4 \\
& 30 & 91 & 0.72 & 136.4 \\
\hline
\multirow{3}{*}{Star} & 0 & 100 & 1.00 & 156.0 \\
& 15 & 93 & 0.64 & 145.9 \\
& 30 & 84 & 0.68 & 144.4 \\
\hline
\multirow{3}{*}{Ring} & 0 & 100 & 1.00 & 156.8 \\
& 15 & 95 & 0.76 & 149.9 \\
& 30 & 90 & 0.80 & 147.5 \\
\hline
\multirow{3}{*}{8-hub Graph} & 0 & 100 & 1.00 & 153.6 \\
& 15 & 92 & 0.60 & 151.5 \\
& 30 & 87 & 0.54 & 148.7 \\
\hline
\multirow{3}{*}{9-ring Graph} & 0 & 100 & 1.00 & 148.1 \\
& 15 & 95 & 0.78 & 141.3 \\
& 30 & 91 & 0.72 & 134.4 \\
\hline
\multirow{3}{*}{von Neumann} & 0 & 100 & 1.00 & 153.3 \\
& 15 & 95 & 0.80 & 144.6 \\
& 30 & 90 & 0.78 & 138.3 \\
\hline
\multirow{3}{*}{Scale-free} & 0 & 100 & 1.00 & 154.5 \\
& 15 & 94 & 0.60 & 148.6 \\
& 30 & 87 & 0.50 & 148.3 \\
\hline
\multirow{3}{*}{Random} & 0 & 100 & 1.00 & 147.0 \\
& 15 & 96 & 0.70 & 141.2 \\
& 30 & 91 & 0.64 & 137.8 \\
\hline
\multirow{3}{*}{Small-world} & 0 & 100 & 1.00 & 150.4 \\
& 15 & 96 & 0.84 & 142.4 \\
& 30 & 91 & 0.74 & 138.2 \\ \bottomrule
\end{tabular}
\end{table}
\begin{table}[H]
\centering
\caption{Comparison of mean performance metrics on Griewank function}
\label{tab:griewank}
\begin{tabular}{ccccc}
\toprule
Network & Death Fraction (\%) & Number of Winners & GSR & GS Time \\\midrule
\multirow{3}{*}{Complete} & 0 & 58 & 0.58 & 154.6 \\
& 15 & 53 & 0.52 & 147.7 \\
& 30 & 36 & 0.32 & 144.2 \\
\hline
\multirow{3}{*}{Star} & 0 & 66 & 0.66 & 183.3 \\
& 15 & 68 & 0.62 & 172.5 \\
& 30 & 52 & 0.42 & 174.5 \\
\hline
\multirow{3}{*}{Ring} & 0 & 75 & 0.38 & 829.7 \\
& 15 & 34 & 0 & -- \\
& 30 & 17 & 0 & -- \\
\hline
\multirow{3}{*}{8-hub Graph} & 0 & 82 & 0.82 & 257.3 \\
& 15 & 70 & 0.42 & 247.7 \\
& 30 & 60 & 0.32 & 260.1 \\
\hline
\multirow{3}{*}{9-ring Graph} & 0 & 70 & 0.70 & 223.2 \\
& 15 & 66 & 0.54 & 197.2 \\
& 30 & 59 & 0.52 & 227.0 \\
\hline
\multirow{3}{*}{von Neumann} & 0 & 90 & 0.90 & 354.7 \\
& 15 & 82 & 0.66 & 333.2 \\
& 30 & 72 & 0.58 & 347.3 \\
\hline
\multirow{3}{*}{Scale-free} & 0 & 80 & 0.80 & 218.5 \\
& 15 & 70 & 0.44 & 203.5 \\
& 30 & 64 & 0.28 & 196.6 \\
\hline
\multirow{3}{*}{Random} & 0 & 58 & 0.58 & 161.9 \\
& 15 & 43 & 0.38 & 160.8 \\
& 30 & 43 & 0.32 & 155.3 \\
\hline
\multirow{3}{*}{Small-world} & 0 & 86 & 0.86 & 201.6 \\
& 15 & 77 & 0.56 & 193.5 \\
& 30 & 73 & 0.52 & 187.8 \\ \bottomrule
\end{tabular}
\end{table}
\begin{table}[H]
\centering
\caption{Comparison of mean performance metrics on Schwefel function}
\label{tab:schwefel}
\begin{tabular}{ccccc}
\toprule
Network & Death Fraction (\%) & Number of Winners & GSR & GS Time \\\midrule
\multirow{3}{*}{Complete} & 0 & 56 & 0.56 & 102.4 \\
& 15 & 64 & 0.54 & 100.1 \\
& 30 & 58 & 0.46 & 93.8 \\
\hline
\multirow{3}{*}{Star} & 0 & 86 & 0.86 & 117.9 \\
& 15 & 75 & 0.66 & 111.0 \\
& 30 & 83 & 0.66 & 108.5 \\
\hline
\multirow{3}{*}{Ring} & 0 & 100 & 1.00 & 289.5 \\
& 15 & 80 & 0.42 & 234.4 \\
& 30 & 64 & 0.20 & 422.0 \\
\hline
\multirow{3}{*}{8-hub Graph} & 0 & 92 & 0.92 & 129.8 \\
& 15 & 88 & 0.64 & 123.7 \\
& 30 & 81 & 0.46 & 120.6 \\
\hline
\multirow{3}{*}{9-ring Graph} & 0 & 86 & 0.86 & 115.7 \\
& 15 & 83 & 0.74 & 113.6 \\
& 30 & 78 & 0.6 & 111.4 \\
\hline
\multirow{3}{*}{von Neumann} & 0 & 98 & 0.98 & 148.9 \\
& 15 & 94 & 0.80 & 137.3 \\
& 30 & 86 & 0.74 & 134.2 \\
\hline
\multirow{3}{*}{Scale-free} & 0 & 92 & 0.92 & 126.7 \\
& 15 & 90 & 0.58 & 117.4 \\
& 30 & 81 & 0.50 & 116.0 \\
\hline
\multirow{3}{*}{Random} & 0 & 76 & 0.76 & 107.8 \\
& 15 & 76 & 0.66 & 103.4 \\
& 30 & 74 & 0.58 & 100.5 \\
\hline
\multirow{3}{*}{Small-world} & 0 & 84 & 0.84 & 118.0 \\
& 15 & 92 & 0.72 & 113.4 \\
& 30 & 87 & 0.74 & 119.9 \\ \bottomrule
\end{tabular}
\end{table}
\begin{table}[H]
\centering
\caption{Comparison of mean performance metrics on Rastrigin function}
\label{tab:rastrigin}
\begin{tabular}{ccccc}
\toprule
Network & Death Fraction (\%) & Number of Winners & GSR & GS Time \\\midrule
\multirow{3}{*}{Complete} & 0 & 100 & 1.00 & 110.8 \\
& 15 & 97 & 0.86 & 107.5 \\
& 30 & 93 & 0.80 & 101.4 \\
\hline
\multirow{3}{*}{Star} & 0 & 100 & 1.00 & 128.5 \\
& 15 & 94 & 0.76 & 122.1 \\
& 30 & 90 & 0.56 & 119.9 \\
\hline
\multirow{3}{*}{Ring} & 0 & 100 & 1.00 & 249.1 \\
& 15 & 88 & 0.44 & 240.8 \\
& 30 & 78 & 0.30 & 220.3 \\
\hline
\multirow{3}{*}{8-hub Graph} & 0 & 100 & 1.00 & 144.3 \\
& 15 & 94 & 0.70 & 135.5 \\
& 30 & 89 & 0.42 & 132.0 \\
\hline
\multirow{3}{*}{9-ring Graph} & 0 & 100 & 1.00 & 114.6 \\
& 15 & 96 & 0.84 & 114.7 \\
& 30 & 93 & 0.78 & 112.9 \\
\hline
\multirow{3}{*}{von Neumann} & 0 & 100 & 1.00 & 149.2 \\
& 15 & 96 & 0.80 & 140.6 \\
& 30 & 92 & 0.66 & 140.6 \\
\hline
\multirow{3}{*}{Scale-free} & 0 & 100 & 1.00 & 139.8 \\
& 15 & 94 & 0.66 & 129.3 \\
& 30 & 88 & 0.44 & 124.7 \\
\hline
\multirow{3}{*}{Random} & 0 & 100 & 1.00 & 115.1 \\
& 15 & 97 & 0.78 & 110.9 \\
& 30 & 93 & 0.72 & 105.6 \\
\hline
\multirow{3}{*}{Small-world} & 0 & 100 & 1.00 & 123.5 \\
& 15 & 96 & 0.86 & 120.5 \\
& 30 & 92 & 0.72 & 116.1 \\ \bottomrule
\end{tabular}
\end{table}
\section{Discussion} \label{sec:discussion}
In the previous section, we presented the results of PSO-based optimization of different objective functions using various network topologies. A key insight derived from these results is that \textit{graph-theoretic metrics of efficiency and robustness do not extend directly towards quantifying the efficiency and robustness of a network topology in the PSO framework}. For instance, agents communicating in the PSO framework with highly-connected and hence efficient topologies exhibit poor global convergence rates. This can be attributed to the fact that with every node connected to every other node, each agent in the swarm is presented with an excessive amount of information which results in the swarm converging prematurely to one of the local optima. In general, such disagreements can be attributed to the differences in the definitions of graph-theoretic properties (efficiency and robustness) and the performance of PSO. The graph-theoretic efficiency and robustness are typically defined with respect to the speed of information transfer and the size of the largest connected component in a graph. On the other hand, the efficiency and robustness of a topology in the PSO framework also depends on additional crucial factors such as rate of convergence of the entire swarm to the global optimum, performance tolerance to various levels of hostility, and the characteristics of the objective function -- number of local and global optima, and function landscape around the global optimum.
In terms of performance in the PSO framework, it is required that the networks exhibit sufficiently high levels of connectivity between the nodes to facilitate faster information transfer among agents. However, as seen above, very high levels of connectivity also results in poor performance. It is evident from Figure \ref{fig:PerformancevsAPSP} that network topologies with an average path length $L \sim 3$ maximize the performance trade-off between the number of winners and the convergence time defined in Section \ref{sec:psometrics}. This trade-off measure is very low for lower values of average path length (very high information transfer) resulting in agents getting stuck in local optima. On the other hand, the trade-off measure decreases gradually as the average path length increases beyond 3 due to reduced information transfer that results in higher convergence times. Hence, \textit{there appears to be a negative utility of increasing the connectivity beyond a certain threshold and networks that exhibit a high enough connectivity are desirable from a PSO performance standpoint}.
In terms of robustness in the PSO framework, it is required that the networks retain their information transfer rates to exhibit robust performance under hostile environments. Networks in the star-to-ring regime are very likely to fragment under hostile conditions resulting in disruption of information transfer. The ring graph is the least robust in this regime since removal of even a few nodes fragments the entire topology. On the other hand, hub-like structures are relatively more robust with the robustness decreasing with increasing number of hubs (central nodes). This is because the loss of even a single hub fragments the graph and higher the number of hubs in the graph, higher is the probability of graph-fragmentation. Mathematically, the probability of a hub-like structure with $k$ hubs getting fragmented at the $i^{th}$ iteration can be expressed as $1-(1-p)^{ki}$ with $p$ being the probability of loss of an agent. However, the most robust graphs are characterized by high connectivity in addition to having a distributed architecture as observed in the ring-to-complete and complete-to-star regimes. For example, complete graphs exhibit the maximum robustness among all the graphs across the spectrum. Therefore, \textit{networks with distributed architectures and high connectivity are desirable from a robustness standpoint.}
In addition to the topologies across the triangular spectrum in Figure \ref{fig:networks}(a), our study on several standard topologies involving multiple objective functions revealed that small-world graphs achieve a high global success ratio while retaining the performance with loss of agents. Specifically, while the best performing graph for different objective functions varies, the success ratio of small-world graphs is consistently among the best two topologies. Moreover, the convergence time for small-world graphs is consistently close to the lowest convergence time achieved for all objective functions. Small-world network architectures appear to be optimizing information transfer and retention of performance under hostile conditions. In order to investigate this effect, we discuss a commonly used metric of \textit{small-world-ness} of networks \cite{telesford2011ubiquity}:
\begin{align}\label{eq:smallworldness}
\omega = \frac{L_{random}}{L} - \frac{C}{C_{lattice}}
\end{align}
Here $L$ is the average geodesic length (Equation \eqref{eq:apsp}) and $C$ is the average clustering coefficient of the network given as:
\begin{align*}
C = \frac{1}{N}\sum_{i=1}^N\frac{2e_i}{k_i(k_i-1)}
\end{align*}
with $e_i$ being the number of edges between the neighbors of a given node $i$ and $k_i (k_i-1)/2$ is the total number of edges possible between the neighbors. The quantities $L_{random}$ and $C_{lattice}$ represent the average path length of an equivalent random graph and clustering coefficient of an equivalent lattice graph, respectively. The \textit{small-world-ness} is close to zero for small-world networks. The above quantity suggests that for a network to exhibit small-world behavior, it must have an average path length close to that of an equivalent random graph and clustering coefficient close to that of an equivalent lattice graph. Therefore, it can be said that small-world graphs \textit{simultaneously} maximize the efficiency (low average path length) and robustness (high clustering coefficient) of a network. \textit{Therefore, small-world graphs exhibit a high enough connectivity with distributed architectures, and hence simultaneously maximize efficiency and robustness in the PSO framework}.
\section{Conclusions}\label{sec:conclusions}
In this article, we report a study of the performance of particle swarm optimization with various network topologies characterizing communication between agents under hostile environments. Such problems bear significance in applications such as targeted drug delivery and high value target localization. The major contribution of this work lies in studying the impact of hostile environments with respect to network topologies in PSO framework. Based on our study, we first conclude that maximizing the graph-theoretic measures of efficiency and robustness of a network do not necessarily result in efficient and robust performance in the PSO framework. This is attributed to factors such as local optima and nature of the function landscape around global optimum that are not incorporated in graph theoretic metrics. Second, we observe that in order to maximize the performance of PSO, a network topology has to be sufficiently connected to ensure optimal information transfer -- a higher information transfer results in over exploitation of the information limiting exploration, whereas a lower information transfer results in over exploration of the search space that results in higher convergence times. An average path length of $L \sim 3$ was observed to give the best performance in terms of higher global success ratio with lower convergence times -- consistent with our \textit{sufficient connectivity} hypothesis. Third, networks with distributed architectures with high connectivity between nodes were observed to be highly robust to deactivation of agents at random and retain their information transfer rates for various levels of severity of the hostile environment. We showed that scale-free and multi-hub networks are therefore less robust compared to multi-ring and von Neumann graphs.
Our study on the performance of the best topologies in the triangular spectrum indicated that two networks with 8-hub structure and 9 multi-rings exhibit the best performance among all the other structures. Further analysis of the performance of these topologies, the four standard topologies including small-world graphs on different objective functions revealed that small-world graphs consistently perform well in terms of a higher global success ratio and lower convergence time. Small-world graphs are in agreement with our observation of distributed architectures with high connectivity and an average path length of $\sim 3$, performing well in PSO. An analysis of a metric quantifying small-world-ness revealed that such networks inherently achieve a trade-off between efficient flow of information between agents and tolerance to hostile conditions. We therefore conclude that small-world like networks are well-suited for optimizing objective functions when the function landscape is unknown and the environment could be hostile towards the agents.
Although the findings of this work are based on performance of the PSO algorithm, the results are fairly generalizable, and should extend to algorithms that rely on efficient communication between agents while searching for an optimal solution. Future work in this area involves extending the current ideas to incorporate adaptive death rates as a function of time and space, and swarm reorganization to minimize the impact of agent deactivation on network topologies.
\section*{Acknowledgements} \label{sec:acknowledgements}
{The authors gratefully acknowledge the contributions of the following former students in our group for their earlier work on this topic - Arun V. Giridhar, Balachandra B. Krishnamurthy, Chunhua Zhao, Priyan R. Patkar, and Santhoji Katare. We also wish to thank the following research interns at the Complex Resilient Intelligent System (CRIS) laboratory at Columbia University - Xijao Li, Liyi Zhang, and Jia Wan.}
\bibliographystyle{model1-num-names}
\section{Introduction}
Science and engineering have often drawn inspiration from nature in addressing challenging problems. The adage ``the whole is greater than the sum of parts" has had a great impact on fields such as complex systems engineering, active matter physics and biomimetics towards understanding complex phenomena. Optimization has also witnessed a similar impact with a class of \textit{nature-inspired} computing techniques \cite{patnaik2017nature}. These techniques model the behaviour of agents in nature such as ants and birds that work collectively towards achieving a global objective with local (agent-specific) rules and extend them towards solving optimization problems. Such an approach has given rise to techniques such as evolutionary programming, genetic algorithms, simulated annealing and differential evolution. Swarm Intelligence \cite{fister2013brief} is one such class of nature-inspired optimization techniques that makes use of interacting agents towards optimizing a collective goal. Algorithms based on swarm intelligence include, among others, particle swarm optimization \cite{kennedy1995particle}, ant colony optimization \cite{dorigo2006ant}, firefly algorithm \cite{yang2008firefly}, cuckoo algorithm \cite{yang2009cuckoo}, bat algorithm \cite{yang2010new}, and squirrel-search algorithm \cite{jain2019novel}, which have been used in several applications \cite{mavrovouniotis2017survey,sundar2012swarm,ertenlice2018survey,nebti2017swarm}. These algorithms make use of recursive update rules drawn from natural agents and have proven application across a number of areas.
Particle Swarm Optimization (PSO) is inspired from the movement of a flock of birds and primarily works by combining an individual's cognizance and collective swarm intelligence \cite{kennedy1995particle}. PSO is a gradient-independent algorithm that has been identified as lying between genetic algorithm and evolutionary programming. The movement of its agents, based on individual and collective best strategy, is similar to crossover operations in genetic algorithms \cite{geneticalgointro1992}, and is dependent on stochastic processes similar to evolutionary programming \cite{evoluprogram1997}. PSO exhibits simple update rules and minimal set of tunable parameters along with a feedback control mechanism that make it a highly desirable choice for a diverse range of problems -- parameter estimation \cite{Schwaab2008}, dynamic optimization \cite{Zhou2014,Ourique2002}, forecasting properties of interest \cite{Wang2019}, clustering \cite{alam2014research}, and training feed-forward neural networks \cite{Zhang2007}. PSO has also been used in several applications including robotics \cite{camci2018aerial}, placement of distributed generators in smart grid \cite{el2011optimal}, astronomy \cite{jin2008analysis}, manufacturing \cite{navalertporn2011optimization} and several more \cite{Pluhacek2018}.
The majority of applications of PSO are characterized by a lack of complete knowledge, and at times an analytical formulation of the objective function. There have been several theoretical advances \cite{bonyadi2019theoretical} and algorithmic modifications to the standard PSO algorithm since its introduction. These include the importance of choosing adaptive inertia weights \cite{Shi1999} and the impact of maximum velocity on the algorithm's performance \cite{Shi1998}. Furthermore, PSO has been modified and generalized to incorporate constraints that arise in several optimization problems in practice \cite{ang2020constrained}. Extensions of the algorithm to multi-objective optimization \cite{bin2018multi} and hybridization approaches \cite{fatemeh2019shuffled} have also been proposed in the literature.
The impact of the communication network between agents in the swarm has also been studied in the literature. However, in certain applications, the agents face a hostile environment that can result in the loss of some of the agents during the search, as in search-and-rescue missions behind enemy lines, in drone swarms in modern warfare, and in targeted drug delivery using nano-technologies. For example, in targeted drug delivery using smart nano-particles that could communicate with each other to locate the diseased site, the carriers face a hostile environment wherein the body's natural defense mechanism (i.e., the immune system) perceives them as potential threats and attempts to kill them in the bloodstream. As a result, the number of agents as well as the information available to the remaining agents dynamically change during the search. This can have a detrimental impact on the performance of the swarm. In applications such as supply chain optimization \cite{shukla2011optimizing, shukla2007supply, shuklafunctional} and logistics \cite{meepetchdee2007logistical}, these problems become crucial. Graph-theoretic measures have been used to quantify efficiency and robustness of these network topologies \cite{venkatasubramanian2004spontaneous, ellens2013graph}. However, a generic framework unifying the performance and graph properties of optimal network topologies under hostile conditions has not been studied in the context of PSO, and is the primary focus of our work.
Here, we present such a study, where we consider an environment in which agents are killed at random during the search; we study the performance of different swarm topologies under such conditions and discover generalized properties of desirable network topologies. The rest of the paper is organized as follows -- In Section 2, we present the framework of PSO and the role of network topologies in driving its performance. In Section 3, we describe the agent-based simulation setup including the objective functions, parameters chosen for the PSO algorithm, the set of systematically generate network topologies primarily used in this work, and the probabilistic framework used for simulating hostile environments for the agents. In Section 4, we discuss graph-theoretic properties for efficiency and robustness, and introduce a set of measures used for quantifying performance of various network topologies in the PSO framework. The results corresponding to various levels of hostility on different objective functions are shown in Section 5. A discussion on the major findings of this study is presented in Section 6. Finally, in Section 7 we summarize the useful contributions of this work, alongside a few concluding remarks.
\section{Problem Formulation and Objectives}\label{sec:probformulation&objs}
Let us consider a function $f(\mathbf{x}):\mathbb{R}^n\to\mathbb{R}$ that we wish to optimize. For the sake of simplicity, we consider the range of $f(\cdot)$ to be $\mathbb{R}$ and the function to exhibit only one global optimum with several local optima. The objective of PSO is to optimize the objective function, i.e., to find the value $\mathbf{x}^*$, referred to as the optimal solution that results in the best value (global optimum) of $f(\cdot)$. We intend to study the impact of network topology on the performance of a swarm intelligence-based algorithm such as PSO in a hostile environment. In the following, we briefly introduce particle swarm optimization and highlight the importance of network configuration on the performance of the algorithm.
Particle Swarm Optimization is a nature-inspired algorithm that begins with a set of \textit{agents}, typically initialized at random locations in the search space. The positions of the agents are updated at each iteration according to a set of heuristic-based rules that account for the best known position of the agent and that of the swarm. The update rule in the standard PSO results in the agents moving in a direction that is the resultant of the direction of the best position of the swarm ($\vec{g}$) and the best position of the $i^th$ agent ($\vec{p}_i$) weighted by acceleration coefficients, and a stochastic component in each direction. The update rule for the position of the $i^{th}$ agent in a swarm at iteration $t$ can be expressed as follows:
\begin{eqnarray}
\vec{v}_i(t)&=&\chi[\vec{v}_i(t-1)+\phi_1.\mathrm{rand}(0,1)(\vec{p}_i(t-1)-\vec{x}_i(t-1)) + \phi_2.\mathrm{rand}(0,1) (\vec{n}_i(t-1)-\vec{x}_i(t-1))] \label{eq:velocity}
\\ \vec{x}_i(t)&=&\vec{x}_i(t-1) + \vec{v}_i(t) \label{eq:position}
\end{eqnarray}
In the above equations, $\vec{x}_i(t)$ and $\vec{v}_i(t)$ represent the position and velocity of the $i^{th}$ agent at iteration $t$, respectively; $\vec{p}_i(t-1)$ and $\vec{n}_i(t-1)$ represent the best known position of the $i^{th}$ agent and its neighborhood at iteration $(t-1)$, respectively; $\phi_1$ and $\phi_2$ are the acceleration coefficients, and $\chi$ represents the constriction coefficient. In a swarm where all agents communicate with each other (in standard PSO), $\vec{n}_i(t-1)$ represents the best position of the entire swarm ($\vec{g}$) and is the same for all agents, while for other configurations, $\vec{n}_i(t-1)$ represents the best position of the neighbors of the $i^{th}$ agent and can be different for different agents. These update rules result in exploration of the search space as well as exploitation of knowledge of the landscape over several iterations. Table \ref{table:standardPSO} presents the step-by-step procedure for the standard PSO algorithm.
\begin{table}[H]
\centering
\caption{Pseudo code for the standard PSO algorithm}
\label{table:standardPSO}
\scalebox{0.95}{
\begin{tabular}{|l|}
\hline
Algorithm $1$: Standard PSO \\ [0.5ex]
\hline
\textbf{Inputs:} \\
\texttt{nAgents}: the number of agents in the swarm\\
$f$: the function to be optimized\\
\texttt{maxIters}: the maximum number of iterations\\ \hline \\ [-2ex]
Initialize the agent positions and velocities randomly \\
while $t<$\texttt{numIters}:\\
\quad for each agent $i$ in \texttt{nAgents}:\\
\quad \quad Compute $\vec{p}_{i}$, position of best solution agent $i$ has found so far\\
\quad \quad Compute $\vec{g}$, position of best solution found by all the agents in the swarm so far\\
\quad \quad Update $\vec{v}_{i}$, velocity of agent $i$ updated using equation \ref{eq:velocity}\\
\quad \quad Update $\vec{x}_{i}$, position of agent $i$ updated using equation \ref{eq:position}\\ \hline
\textbf{Outputs:}\\
\quad $\vec{x}^{*}$, the position of the global optimum \\
\quad $f(\vec{x}^{*})$, the function values at the global optimum\\ [1ex]
\hline
\end{tabular}
}
\end{table}
It can clearly be observed from Equations \eqref{eq:velocity} and \eqref{eq:position} that the updated positions of the agents depend on the manner in which the best-known position of the neighbours, i.e., $\vec{n}_i(t-1)$ is obtained at each iteration. This is in turn determined by the topology of the communication network of the swarm. The network configuration thus influences the information received by each agent, determining the updated positions for the agents, playing a crucial role in convergence to the global optimum. The impact of network topology on the performance of PSO has been studied in the literature \cite{1004493, 1202252, 785509} and it has been established that network topologies play a significant role in determining the convergence of the algorithm. However, topological factors that result in superior performance have not been identified. Furthermore, the performance of the algorithm is also seen to have a strong dependence on the type of function being optimized. These studies are also limited to the case where agents are not lost to the environment, as a result of which the trade-off between efficiency of convergence and robustness to the environment is not studied. We conduct several computational experiments that simulate PSO in a hostile environment, which are discussed in detail in the next section.
\section{Agent-based Simulation Setup}
The computational experiments considered in this work have four crucial components, the choice of which determines the performance of the algorithms, and hence the generalizability of results. These factors include:
\begin{enumerate}[(i)]
\item the objective function
\item the hyperparameters of the PSO algorithm
\item the communication network configuration of agents in PSO
\item the severity and nature of the hostile environment
\end{enumerate}
The following sections present short discussions on the choice of the above factors and the rationale behind them.
\subsection{Objective Function}
We consider five standard benchmark objective functions -- Shekel, Ackley, Griewank, Schwefel and Rastrigin \cite{simulationlib, molga2005test} -- to evaluate and compare the performance of PSO for different network topologies. We perform extensive computational experiments with the Shekel function to identify common patterns in performance of PSO. We then choose a subset of topologies that exhibit a desirable trade-off between efficiency of convergence and robustness to hostile environment, and compare the performance of PSO for the remaining four functions. The first objective function considered in this work is the Shekel function, which can be expressed as:
\begin{equation}
f(\mathbf x) = \sum_{i=1}^m \bigg (c_i + \sum_{j=1}^n (x_j - a_{ji})^2 \bigg)^{-1} \label{eq:shekelfun}
\end{equation}
where, $m$ is the number of local maxima, $n$ is the dimension of the input space ($\mathbf{x}\in\mathbb{R}^n$), $c_i$ determines the magnitude of the $i^{th}$ local maximum and $a_{ji}$ is the $j^{th}$ coordinate of location of $i^{th}$ maximum. Figure \ref{fig:benchmarkfuns}(a) depicts the landscape of a two-dimensional Shekel function with $9$ local maxima and one global maximum. The remaining four objective functions are shown in Figure \ref{fig:benchmarkfuns}(b)-(e) for the two dimensional case. The functional form, number of dimensions, number of local optima and location of global optimum for the five functions are listed in Table \ref{tab:benchmarkfuns}.
\begin{figure}[H]
\centering
\subfigure[Shekel Function]{
\includegraphics[width=0.6\linewidth]{PSO/figs/shekel2.png}
}\\
\subfigure[Ackley Function]{\includegraphics[width=.48\textwidth]{PSO/figs/ackley2.png}} \quad
\subfigure[Griewank Function]{\includegraphics[width=.48\textwidth]{PSO/figs/griewank2.png}}\\
\subfigure[Schwefel Function]{\includegraphics[width=.48\textwidth]{PSO/figs/schwefel2.png}}\quad
\subfigure[Rastrigin Function]{\includegraphics[width=.48\textwidth]{PSO/figs/rastrigin2.png}}
\caption{Benchmark Objective Functions for $n=2$}
\label{fig:benchmarkfuns}
\end{figure}
\begin{table}[h]
\caption{Description of Objective Functions}
\label{tab:benchmarkfuns}
\centering
\scalebox{0.8}{
\begin{tabular}{cccccc}
\hline
\multirow{2}{*}{Function} & \multirow{2}{*}{Expression} & \multirow{2}{*}{$n$} & \multirow{2}{*}{Range} & Local & Location of \\
& & & & Optima & Global Optimum \\
\hline
Shekel & $f(\mathbf x) = \sum_{i=1}^m \bigg (c_i + \sum_{j=1}^n (x_j - a_{ji})^2 \bigg)^{-1}$ & 4 & $[0, 10]$ & 10 & $[4,4, 4, 4]^T$ \\
Ackley & $f(\mathbf x) = -a\exp\left(-b\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_i^2}\right) - \exp{\left(\frac{1}{n}\sum_{i=1}^{n}\cos(c x_i)\right)} + a + \exp(1)$ & 2 & $[-15, 30]$ & $>10$ & $[0,0]^T$ \\
Griewank & $f(\mathbf x) = \frac{1}{4000}\sum_{i=1}^n x_i^2 - \prod_{i=1}^{n} \cos \left(\frac{x_i}{\sqrt{i}}\right) + 1$ & 2 & $[-600, 600]$ & $>10$ & $[0,0]^T$ \\
Schwefel & $f(\mathbf x) = 418.9829 n - \sum_{i=1}^n \left[-x_i \sin \left(\sqrt{|x_i|}\right)\right]$ & 2 & $[-500, 500]$ & $>10$ & $[420.9687, 420.9687]^T$ \\
Rastrigin & $f(\mathbf x) = 10n + \sum_{i=1}^n \left[x_i^2 - 10\cos(2\pi x_i)\right]$ & 2 & $[-5.12, 5.12]$ & $>10$ & $[0,0]^T$ \\
\hline
\end{tabular}
}
\end{table}
\subsection{Hyperparameters for Particle Swarm Optimization}
The second component includes the set of hyperparameters employed by the PSO algorithm. In this work, we have chosen the following values of the hyperparameters:
\begin{enumerate}
\item $\chi=0.7298438$, a constriction coefficient that prevents the velocities from exploding
\item $\phi_1=0, ~ \phi_2=2.05$, acceleration coefficients for the agents' cognizance and neighborhood cognizance
\item $V_{min}=-10, V_{max}=10$, lower and upper bounds on the agents' velocities
\item $N=100$, number of agents in the swarm
\item $T=1000$, maximum number of iterations to allow convergence of all network configurations
\end{enumerate}
\textit{Remark 1:} In this study, we set $\phi_1=0$ so that the best known position of an agent is not incorporated in updating its position at any iteration. As a result, agents move in a direction that is solely driven by the information made available to them by their neighborhood.
\textit{Remark 2:} It is important to note here that while the hyperparameters of the algorithm could be tuned to achieve desired efficiency and robustness for each scenario, such an exercise can be practically infeasible and thus of little significance in several applications. Furthermore, owing to the uncertainty in the form of random loss of agents in our study and absence of complete information about the objective function in practical scenarios, a perfect tuning of hyperparameters can be extremely time-consuming, difficult, and computationally expensive. Therefore, we fix the values of the hyperparameters for all the experiments and focus the study towards discovering generalized features of the algorithm that are of interest in practical applications. With the objective function and tuning of the algorithm fixed, we next describe the different network configurations and the simulation of the hostile environment used in our study.
\subsection{Network Topology of Communicating Swarm}
Network topologies can broadly be classified as deterministic and random topologies. Since the objective of this study is to identify network configurations that result in a desired trade-off between efficiency and robustness of the algorithm, we focus our attention on deterministic graphs and present only limited results on random graphs. We consider three classes of graphs -- complete (all agents connected to each other), star (hub-and-spoke configuration) and ring (all agents connected to two neighbours, forming a closed ring) -- that represent different degrees and manners of connectivity of the graph. We create a spectrum of intermediate graphs by performing deterministic operations on the above graph classes as follows:
\begin{enumerate}[I.]
\item Complete--to--Star:
\begin{enumerate}[(a)]
\item The initial graph is a complete graph where all the agents constitute the fully connected core with each node connected to every other node
\item The core size of the graph is shrunk in steps of $1$ by removing an agent from the core and attaching it to one of the core nodes
\item In reducing the core size, it is ensured that the resultant core at each step is full connected, as shown in Figure \ref{fig:networks}(a).
\end{enumerate}
\item Star--to--Ring:
\begin{enumerate}[(a)]
\item The initial graph is a star graph which consists of only one core/hub node and $N-1$ non-core nodes.
\item The hub is expanded in steps of $1$ to connect one non-core node to the core at each step.
\item In expanding the core, it is ensured that the core is not fully connected but connected in a ring topology as shown in Figure \ref{fig:networks}(a).
\end{enumerate}
\item Ring--to--Complete:
\begin{enumerate}[(a)]
\item The initial graph is a ring-graph where each node is connected to its two immediate neighbors in a circular topology
\item In each step, an additional ring structure is introduced to the graph by symmetrically connecting agents to the non-immediate neighbors.
\item This results in a multi-ring graph with increasing connectivity as one moves from ring to complete graph as shown in Figure \ref{fig:networks}(a).
\end{enumerate}
\end{enumerate}
The above rules are used to generate $80$ configurations between each pair of basic graphs, resulting in a total of $240$ configurations. The colors shown in Figure \ref{fig:networks}(a) are used to demarcate different segments of the spectrum and are used for clarity in presentation of the results. In addition, we also use the von Neumann grid, scale-free graph, random graph and small-world graph as additional cases of network configurations as shown in Figure \ref{fig:networks}(b)--(e). These networks are generated using the NetworkX $2.4$ package in Python. Specifically, random graphs are generated in a manner that the probability of an edge between any two nodes is $0.1$, while small-world graphs are generated with a degree of $10$ for each node and a rewiring probability of $0.1$ for each edge.
\begin{figure}
\centering
\subfigure[]{\includegraphics[width = 0.8\linewidth]{PSO/figs/triangular_spectrum_less.png}}\\
\subfigure[]{\includegraphics[width = 0.12\linewidth]{PSO/figs/vonNeumann.PNG}} \hspace{5em}
\subfigure[]{\includegraphics[width = 0.15\linewidth]{PSO/figs/Scale-free.PNG}} \hspace{5em}
\subfigure[]{\includegraphics[width = 0.15\linewidth]{PSO/figs/Random.PNG}} \hspace{5em}
\subfigure[]{\includegraphics[width = 0.15\linewidth]{PSO/figs/Small-world.png}}
\caption{Network topologies used for swarm neighborhood connectivity: (a) $240$ network topologies are generated by traversal along the spectrum from complete-to-star (red), star-to-ring (blue), and ring-to-complete (green) (b) a von Neumann grid: each node is connected to 4 neighboring nodes, (c) Scale-free graph: the probability of finding a node with degree $k$ is proportional to $k^{-\gamma}$, (d) Random graph: edges between nodes are assigned at random with a predetermined probability $p_e$. (e) Watts-Strogatz (Small-world) graph: edges of a multi-ring graph are rewired with a rewiring probability $p_w$}
\label{fig:networks}
\end{figure}
\subsection{Loss of Agents in the Hostile Environment}
We consider a hostile environment where each agent has a probability of getting killed/deactivated at a given iteration. Let us consider a scenario where all agents have the same probability $p$ of being lost to the environment at any iteration of the algorithm. Then, the expected number $N_a$ of the alive/active agents at the end of $t$ iterations can be obtained as:
\begin{align}
N_a = N(1-p)^t
\end{align}
The expected fraction of active and deactivated agents, $F_a$ and $F_d$, respectively are given by,
\begin{align}
F_a &= (1-p)^t \\
F_d &= 1 - (1-p)^t
\end{align}
We study the performance of PSO in an environment that causes loss (death fractions) of $15\%$ and $30\%$ of the total number of agents at the end of $500$ iterations. The corresponding probabilities of deactivation for any agent for these cases is $p=0.00033$ and $p=0.0007$, respectively. The pseudo-code for this randomized death of agents is presented in Table \ref{table:randomDeath}.
\begin{table}[H]
\centering
\caption{Pseudo code for randomized death of agents}
\label{table:randomDeath}
\scalebox{0.95}{
\begin{tabular}{|l|}
\hline
Algorithm $2$: \texttt{Randomized Death} of agents \\ [0.5ex]
\hline
\textbf{Inputs:} \\
$p:$ the deactivation probability for each agent \\
\texttt{active agents}: the list of agents that are active \\
\hline \\ [-2ex]
for each \texttt{agent} in \texttt{active agents}:\\
\quad Generate a random number $r$ in the range $[0, 1]$\\
\quad if $r<p$: \\
\quad \quad deactivate \texttt{agent}\\
\quad Remove \texttt{agent} from the list \texttt{active agents}
\\\hline
\textbf{Outputs:}\\
\texttt{active~agents}: list of active agents after randomized death \\ [1ex]
\hline
\end{tabular}
}
\end{table}
The pseudo-code for the resultant algorithm that combines network topology with loss of agents for the PSO algorithm is presented in Table \ref{table:modifiedPSO}.
\begin{table}[H]
\centering
\caption{Pseudo code for PSO algorithm with network topology and dying agents}
\label{table:modifiedPSO}
\scalebox{0.95}{
\begin{tabular}{|l|}
\hline
Algorithm $3$: PSO with Network Topology and Dying Agents \\ [0.5ex]
\hline
\textbf{Inputs:} \\
\texttt{nAgents}: the number of agents in the swarm\\
$f$: the function to be optimized\\
\texttt{maxIters}: the maximum number of iterations\\
$p:$ the death probability for each agent \\
\texttt{adjMatrix}: the \texttt{nAgents} $\times$ \texttt{nAgents} matrix describing the network connectivity for each agent \\
\hline \\ [-2ex]
Initialize the agent positions and velocities randomly \\
while $t<$\texttt{numIters} and \texttt{numActiveAgents} $\neq 0$:\\
\quad for each agent $i$ in \texttt{active agents}:\\
\quad \quad Compute $\vec{p}_{i}$, position of best solution agent $i$ has found so far\\
\quad \quad Compute $\vec{n}_{i}$, position of best solution found by agent $i$'s neighborhood so far defined by the \texttt{adjMatrix}\\
\quad \quad Update $\vec{v}_{i}$, velocity of agent $i$ updated using equation \ref{eq:velocity}\\
\quad \quad Update $\vec{x}_{i}$, position of agent $i$ updated using equation \ref{eq:position}\\ [2ex]
\quad \quad Invoke the \texttt{Randomized Death} algorithm and update \texttt{active agents}, \texttt{adjMatrix}\\ [1ex]
\hline
\end{tabular}
}
\end{table}
In the next section, we present different graph-theoretic metrics considered in this study to quantify the robustness and efficiency of different network topologies followed by various measures that capture performance in the PSO framework.
\section{Graph Theoretic Properties and PSO Performance Metrics} \label{sec:robust-eff-tradeoff}
The convergence of PSO with different network topologies in a hostile environment depends on -- ($1$) the ability of the network to \textit{efficiently} transmit information between the nodes (agents), and ($2$) the ability of the network to sustain loss of agents with minimal deterioration in performance of PSO. We first present graph-theoretic metrics that quantify the efficiency and robustness of a network topology in itself (independent of PSO) followed by a set of four metrics that we use to quantify the performance of any topology on the PSO algorithm.
We wish to highlight here that the networks that maximize the graph-theoretic measures of efficiency and robustness do not necessarily result in the most efficient and robust performance on the PSO algorithm as we discuss in detail later in Section \ref{sec:results}.
\subsection{Graph Theoretic Properties} \label{sec:graphmetrics}
The efficiency of a graph is its ability to quickly communicate information between different nodes in the graph. The average geodesic distance (also called average path length) is a very commonly used metric to quantify efficiency of a graph. The average geodesic distance $L$ of a graph $G$ with $N$ nodes can be expressed as:
\begin{align}
L = \frac{1}{N(N-1)}\sum d_{ij}
\label{eq:apsp}
\end{align}
where $d_{ij}$ is the length of the shortest path between $i^{th}$ and $j^{th}$ nodes of $G$. The above quantity is equal to $1$ for a complete graph, signifying highly efficient transfer of information in the graph. On the other hand, the star graph exhibits an average geodesic distance of approximately $2$ while the ring graph has an average geodesic distance $\sim N/4$.
The robustness of a graph can be quantified using the natural connectivity which has been shown to be sufficient, superior, and is known to be a physically meaningful measure for quantifying the robustness of complex graphs \cite{jun2010natural}. Physically, it is proportional to the number of closed loops for each node of the graph. It is derived from the eigenvalues of the adjacency matrix of the graph as:
\begin{equation}\label{eq:naturalconnect}
\bar \lambda = \ln \bigg ( \frac{1}{N} \sum_{i=1}^N e^{\lambda_i} \bigg)
\end{equation}
where $\lambda_i$ is the $i^{th}$ eigenvalue of the adjacency matrix of the graph. The adjacency matrices of a complete, star and ring graphs (A, B, and C, respectively) with 5 nodes can be represented as:
\begin{equation*}
A = \begin{bmatrix}
0 & 1 & 1 & 1 & 1\\
1 & 0 & 1 & 1 & 1\\
1 & 1 & 0 & 1 & 1\\
1 & 1 & 1 & 0 & 1\\
1 & 1 & 1 & 1 & 0\\
\end{bmatrix}
B = \begin{bmatrix}
0 & 1 & 1 & 1 & 1 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0\\
1 & 0 & 0 & 0 & 0\\
1 & 0 & 0 & 0 & 0\\
\end{bmatrix}
C = \begin{bmatrix}
0 & 1 & 0 & 0 & 1 \\
1 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 1 \\
1 & 0 & 0 & 1 & 0 \\
\end{bmatrix}
\end{equation*}
The set of eigenvalues for the adjacency matrices (graph spectrum) for the three graphs above is given by,
\begin{equation*}
A_\lambda = \{4, -1, -1, -1, -1\} \quad B_\lambda = \{2, 0, 0, 0, -2\} \quad C_\lambda = \{2, 0.62, 0.62, -1.62, -1.62\}
\end{equation*}
Therefore, using Equation \ref{eq:naturalconnect}, the corresponding natural connectivities obtained for the given graphs are,
\begin{align*}
\bar \lambda_{complete} = 2.42 \quad \bar \lambda_{star}= 0.74 \quad \bar \lambda_{ring} = 0.83
\end{align*}
It can be clearly seen that the natural connectivity values for the three graphs are in agreement with the expected behavior. For example, the complete graph is the most robust network owing to the presence of multiple loops for a given node, while star and ring graphs -- with zero and one loops respectively -- are the least robust since the network can be fragmented with removal of only a few nodes. Figure \ref{fig:robustness_and_apsp} shows the computed measures of efficiency and robustness for the triangular spectrum of topologies described in Figure \ref{fig:networks}(a).
\begin{figure}[H]
\centering
\subfigure[]{{\includegraphics[width=0.42\linewidth]{PSO/figs/Average_Path_Length_vs_Topology_Index_grid.png} }}
\quad
\subfigure[]{{\includegraphics[width=0.42\linewidth]{PSO/figs/Natural_Connectivity_vs_Topology_Index_grid.png}}}%
\caption{Network topology metrics (a) Average path length and (b) Natural connectivity, as one goes along the spectrum of proposed networks, as a function of the topology index. Average path length is a measure of efficiency of the network, while natural connectivity is a measure of robustness. The color codes are corresponding to traversal from complete-to-star (red), star-to-ring (blue) and ring-to-complete (green)}%
\label{fig:robustness_and_apsp}%
\end{figure}
It can be seen that as we traverse the spectrum, the following patterns in the efficiency and robustness are observed:
\begin{itemize}
\item Complete--to--Star: The average path length increases and natural connectivity decreases. This is equivalent to decreasing efficiency and robustness, respectively. This is a result of removal of edges from the graph in this section of the spectrum.
\item Star--to--Ring: The average path length increases while natural connectivity remains same, suggesting decreasing efficiency and near-constant robustness. This is a result of increase in the core size for approximately the same number of edges along this section of the spectrum.
\item Ring--to--Complete: The average path length decreases and natural connectivity increases, resulting in an increase in network efficiency and robustness. This is a result of increased number of edges due to addition of multi-rings.
\end{itemize}
\subsection{PSO Performance Metrics} \label{sec:psometrics}
In order to compare the performance of PSO for different communication network configurations and different severity of the hostile environment, we note the following metrics:
\begin{enumerate}
\item Global Success Ratio (GSR): The fraction of times the \textit{entire swarm} reaches the global optimum
\item Global Success Time to Convergence (GS Time): Average number of iterations required for the \textit{entire swarm} to reach the global optimum. This metric is calculated only for the cases when \textit{global convergence} is achieved.
\item Number of Winners: Average number of agents that reach the global optimum
\item Trade-off metric: In addition to the performance measures defined above, we define the following performance metric that captures the trade-off between the number of winners and the time for convergence:
\begin{equation}\label{eq:performance-metric}
\text{Trade-off metric} = \alpha~\frac{{\mathrm{Winners}}}{{\mathrm{Winners}_{\mathrm{max}}}} - (1-\alpha)~\frac{{\mathrm {GS~Time}}}{{{\mathrm{GS~Time}}_{\mathrm{max}}}}
\end{equation}
where $\alpha$ determines the relative importance of the two measures towards the metric. In our simulations, we choose $\alpha=0.7$ to assign a relatively higher importance to success ratio, emphasizing on convergence of all agents to the global optimum. This implies that we desire all the agents to reach the global optimum while allowing for a larger time to convergence. These choices can be motivated for applications such as drug delivery where one desires all of the drug to be delivered to the subject at the desired location, even if it takes a (manageably) longer for the delivery.
\end{enumerate}
In the next section, we present the results of the study based on the above performance metrics for the benchmark objective functions, for different death fractions with respect to the graph-theoretic metrics.
\section{Results}\label{sec:results}
In order to balance out the effect of initial state of agents on the convergence of the algorithm and improve generalizability of results, $50$ different initializations of agents for each network topology and deactivation were considered. In each scenario, the performance metrics listed in the previous section were noted and the average metric for each configuration was computed. We first present the performance of PSO for different communication network topologies and death fractions, followed by an analysis of the performance in relation to graph-theoretic metrics of efficiency and robustness. As discussed earlier, we first perform exhaustive experiments with the 4D-Shekel function to identify common trends and then present results on additional benchmark functions with a subset of important topologies.
\subsection{Performance of PSO for Different Network Topologies}
The performance of PSO in terms of the four metrics listed above as a function of the network topology is presented in Figure \ref{fig:PerformancevsNetwork}. For the sake of clarity, all metrics corresponding to networks in complete-to-star, star-to-ring and ring-to-complete regimes of the spectrum are shown in red, blue and green colors, respectively.
It can be observed from Figure \ref{fig:PerformancevsNetwork} that in the case of no deactivation of agents, the global success ratio (GSR) increases as we move from complete graph to star graph, becomes nearly flat at the maximum value of $1$ while going from star graph to ring graph, and then decreases again while moving from the ring graph to complete graph. However, with deactivation of agents, GSR decreases in all the three regimes, with a significant increase in GSR after the initial few topologies in the ring-to-complete regime.
The global success time to convergence (GS Time) is lowest for the complete graph with only a marginal increase as we move towards the star topology. However, it rapidly increases while going from the star graph to the ring graph, before falling off again in the ring-to-complete regime. These trends are observed irrespective of the degree of hostility of the environment.
The number of winners for all the three regimes exhibits a similar trend as GSR. However, it is interesting to note that in the case of deactivation of agents, the number of winners in the complete-to-star regime follows an inverse trend with GSR suggesting a higher number of winners for the star-like topologies than completely connected topologies.
Finally, the trade-off metric that captures the weighted combination of number of winners and convergence time is maximized for an intermediate region in the star-to-ring and ring-to-complete regimes across all death fractions. It is also fascinating to note that the networks that maximize this metric under non-hostile conditions also result in the best performance under hostile conditions. This suggests an invariance in the performance of a network topology to hostility of the environment that can be exploited in designing robust network topologies. This is explored later in the article.
\begin{figure}[h]
\centering
\makebox[\textwidth][c]{\includegraphics[scale=0.52]{PSO/figs/perf_vs_topindex.png}}
\caption{Performance indicators: Global Success Rate (GSR), Global Success Time to Convergence (GS Time), Number of Winners, and Trade-off metric for different Death Rates for different network topologies}
\label{fig:PerformancevsNetwork}
\end{figure}
This section highlighted the dependence of PSO performance on various network architectures. While the complete graph is known to be robust and efficient graph in a purely graph-theoretic sense, such a topology exhibits consistently poor performance (GSR of $\sim50\%$) in terms of optimizing the objective function. In the following sections, we adopt a systematic approach to explore underlying patterns relating the properties of a graph with its performance in the PSO framework.
\subsection{Performance of PSO and Graph Efficiency}
We now study the variation of network performance as a function of its graph-theoretic efficiency. The four performance metrics of PSO defined in Section \ref{sec:psometrics} as a function of the average geodesic distance ($L$) of the network are shown in Figure \ref{fig:PerformancevsAPSP}. In the case of no deactivation of agents, the GSR initially increases with increase in $L$ after which it saturates close to the maximum value of $1$ for larger values of $L$. When agents are deactivated, the absolute values of GSR across all values of $L$ reduce with the magnitude of deterioration being higher for large values of $L$. Irrespective of the rate of deactivation, GSR reaches a maximum for $L\sim 3$, indicating toward the performance invariance of such graphs performance towards hostile environments.
The GS Time increases linearly with increase in average geodesic distance of the graph. This is expected since larger values of $L$ imply lower information transfer through the network resulting in longer times to convergence. The number of winners exhibits a trend that is similar to that observed for GSR but the sensitivity to loss of agents is relatively lower. The trade-off measure capturing the number of winners and the GS Time is also maximized for an average path length $\sim 3$ irrespective of the network topology, or the death fraction. This is suggestive of the existence of a class of networks that are inherently suited for maximizing performance under scenarios involving loss of agents.
It can be inferred from the above that there is a trade-off between the overall performance of a network in PSO and its average path length. Specifically, while highly efficient graphs take less time to converge, the GSR is much lower because of premature convergence to local optima. On the other hand, while graphs with low efficiency promise near-certain convergence, they take longer to reach the global optimum. The trade-off metric captures this effect well, and reveals that the performance peaks at an average path length of $\sim 3$.
\begin{figure}[h]
\centering
\makebox[\textwidth][c]{\includegraphics[scale=0.52]{PSO/figs/perf_vs_apsp.png}}
\caption{Performance indicators: Global Success Rate (GSR), Global Success Time to Convergence (GS Time), Number of Winners, and Trade-off metric for different Death Rates for different values of average path length}
\label{fig:PerformancevsAPSP}
\end{figure}
\subsection{Performance of PSO and Graph Robustness}
After studying the performance of PSO with respect to graph efficiency quantified by the average geodesic distance, we study the same with respect to the graph-theoretic robustness of networks. The performance of PSO as a function of the natural connectivity (robustness) of the network topology is shown in Figure \ref{fig:PerformancevsNatconnect}.
It is observed that, in the absence of a hostile environment, the GSR decreases with an increase in robustness of the graph. This is potentially due to the presence of larger number of closed-loops for highly robust graphs that result in a sub-optimal exploration of the search space and consequently results either in premature convergence, or no convergence at all. An interesting observation is that for the same value of natural connectivity, there exist multiple architectures that result in significantly different performances. For instance, networks with distributed architectures (such as multi-ring) with nearly the same connectivity for all nodes perform better than networks with hub-like structures (such as a star-graph) where a few hubs (central nodes) are highly connected to a large fraction of other nodes. This is because in hub-like networks, removal of the hubs would significantly limit the information transfer across the network.
Therefore, the foregoing sections have established that in order to maximize performance of a network topology in the PSO framework, the network should be sufficiently connected with an average geodesic distance of $\sim 3$ to ensure optimal information transfer and should have distributed architectures in order to ensure robustness towards hostile conditions where agents are deactivated at random.
\begin{figure}[h]
\centering
\makebox[\textwidth][c]{\includegraphics[scale=0.52]{PSO/figs/perf_vs_natconnect.png}}
\caption{Performance indicators: Global Success Rate (GSR), Global Success Time to Convergence (GS Time), Number of Winners, and Trade-off metric for different Death Rates for different values of Natural Connectivity (robustness)}
\label{fig:PerformancevsNatconnect}
\end{figure}
\subsection{Performance of PSO with Standard Topologies}
In addition to the 240 topologies from the triangular spectrum described in Figure \ref{fig:networks}(a), we study the performance of PSO with a few standard network topologies that do not fall under any regime in the above spectrum. Specifically, we consider the von Neumann grid, scale-free graphs, random graphs, and small-world graphs for our study. The results are presented in Table \ref{table:comparisontable}.
It is observed that the von Neumann grid has the highest number of winners and the largest GSR but the convergence time of this graph is the largest among all the four topologies considered. This behaviour is suggestive of lower information transfer (and hence, lower efficiency), which is also reflected in its largest average geodesic distance across all other networks. It also exhibits a highly sensitive GSR with respect to loss of agents, which is explained by the lowest value of natural connectivity ($\bar{\lambda}$) for this network.
The scale-free network exhibits a low average geodesic distance and hence higher efficiency, which results in lower convergence times, but a slightly lower GSR. However, the GSR of this network is extremely sensitive to loss of agents even with a relatively high natural connectivity because of the hub-like architecture similar to the networks in the star-to-ring regime. This results in a performance that is highly sensitive to the loss of agents in the network.
Random graphs exhibit the highest efficiency and hence the lowest convergence times. However, they also result in a lower GSR, potentially because of a relatively higher information overload in the network. Furthermore, due to lack of hubs in random graphs, the performance is seen to be robust to loss of agents.
Finally, small-world graphs are seen to result in a relatively higher GSR and lower convergence times as a result of higher efficiency. These graphs can also be observed to be relatively robust to loss of agents. The performance of small-world graphs is marginally better than random graphs in terms of the GSR and the number of winners in the PSO framework. The convergence times are, however, comparable with the random-graphs converging marginally faster than the small-world graphs.
In the final section, we present the results of PSO on standard benchmark optimization functions for selected networks, followed by a detailed discussion of the different inferences on desired network characteristics for maximizing PSO performance.
\begin{table}[H]
\centering
\caption{Comparison of mean performance metrics for standard topologies on the Shekel function}
\label{table:comparisontable}
\begin{tabular}{ccccccc}
\toprule
Network & Death Fraction (\%) & Number of Winners & GSR & GS Time & $L$ & $\bar{\lambda}$ \\\midrule
\multirow{3}{*}{von Neumann} & 0 & 100 & 1.00 & 189.1 & \multirow{3}{*}{6.67} & \multirow{3}{*}{1.51} \\
& 15 & 95 & 0.70 & 168.5 & & \\
& 30 & 87 & 0.68 & 174.1 & & \\
\hline
\multirow{3}{*}{Scale-free} & 0 & 96 & 0.96 & 138.7 & \multirow{3}{*}{2.75} & \multirow{3}{*}{20.01} \\
& 15 & 94 & 0.56 & 135.8 & & \\
& 30 & 75 & 0.34 & 139.4 & & \\
\hline
\multirow{3}{*}{Random} & 0 & 94 & 0.94 & 106.5 & \multirow{3}{*}{1.70} & \multirow{3}{*}{25.69} \\
& 15 & 91 & 0.78 & 105.0 & & \\
& 30 & 85 & 0.72 & 101.9 & & \\
\hline
\multirow{3}{*}{Small-world} & 0 & 98 & 0.98 & 124.7 & \multirow{3}{*}{2.72} & \multirow{3}{*}{6.31} \\
& 15 & 94 & 0.78 & 117.5 & & \\
& 30 & 88 & 0.78 & 121.5 & & \\ \bottomrule
\end{tabular}
\end{table}
\subsection{Effect of Objective Function}
In order to study the impact of objective function on network performance in PSO, we identified key network configurations that exhibit a desirable trade-off between efficiency, robustness, and performance to analyze their performance on four standard objective functions.
For this study, we choose the complete graph, star graph, and the ring graph topologies (corners of the triangular spectrum in Figure \ref{fig:networks}(a)) along with the standard topologies discussed in the previous section. In addition, we consider two graphs that result in the best performance among the 240 topologies in the triangular spectrum -- one in the star-to-ring regime with 8 hubs and the other in the ring-to-complete regime with 9 multi-rings as shown in Figure \ref{fig:bestTopologiesTriangle}.
\begin{figure}[h]
\centering
\subfigure[][Star-to-ring regime with $8$ hubs]{\includegraphics[width=.4\textwidth]{PSO/figs/s2rbest.png}} \quad
\subfigure[][Ring-to-complete regime with $9$ multi-rings]{\includegraphics[width=.4\textwidth]{PSO/figs/r2cbest.png}}
\caption{Best $2$ topologies from the triangular spectrum comprising 240 topologies}
\label{fig:bestTopologiesTriangle}
\end{figure}
We test the performance of these graphs on the Ackley, Griewank, Schwefel, and Rastrigin functions with function parameters and range as described in Table \ref{tab:benchmarkfuns}. The results for the respective functions are presented in Tables \ref{tab:ackley}, \ref{tab:griewank}, \ref{tab:schwefel}, and \ref{tab:rastrigin}.
It can be observed that, for each objective function, there exists a network topology that results in the best performance in terms of high GSR and number of winners, and low convergence times. For instance, across all death fractions, the Von Neumann topology results in the best performance for Ackley function, small-world topology for Griewank function, small-world for Schwefel function, and complete graph for Rastrigin function. These topologies maximize the number of winners and GSR while simultaneously minimizing the time to convergence (GS Time).
However, small-world graphs are observed to be consistently outperforming almost all the other topologies irrespective of the choice of objective function, or the death fraction. Even for the Ackley and the Rastrigin functions where the von Neumann graph and complete graph perform the best, small-world graphs result in a performance very close to the best performance achieved by both of these networks. We attribute such a superior performance to two factors -- first, the average geodesic distance of $2.72$ for small-world graphs as reported in Table \ref{table:comparisontable} is close to our observation of optimal information transfer occurring at $L\sim3$; second, the small-world networks exhibit a distributed architecture with almost equal connectivity for all the nodes in the network resulting in a robust performance in the presence of higher deactivation of agents as discussed in the foregoing sections.
Similar to small-world networks, the 9 multi-rings architecture of Figure \ref{fig:bestTopologiesTriangle}(b) also exhibits consistently better performance across all the objective functions and is robust to various levels of death fraction. The 9 multi-rings network architecture is characterized by an average geodesic distance of $3.27$ and has a distributed architecture -- validating the underlying network structures identified in the previous sections that are responsible for better and robust PSO performance. This is further reinforced by analyzing the performance of the 8-hub network architecture in Figure \ref{fig:bestTopologiesTriangle}(a) with a average geodesic distance of $3.86$ and with 8 hubs architecture. Although the multi-hub architecture results in faster convergence and better performance under non-hostile conditions, the performance degrades significantly when agents are deactivated at random due to the fragility of such an architecture towards random deactivation of agents.
\begin{table}[H]
\centering
\caption{Comparison of mean performance metrics on Ackley function}
\label{tab:ackley}
\begin{tabular}{ccccc}
\toprule
Network & Death Fraction (\%) & Number of Winners & GSR & GS Time \\\midrule
\multirow{3}{*}{Complete} & 0 & 100 & 1.00 & 146.2 \\
& 15 & 96 & 0.72 & 137.4 \\
& 30 & 91 & 0.72 & 136.4 \\
\hline
\multirow{3}{*}{Star} & 0 & 100 & 1.00 & 156.0 \\
& 15 & 93 & 0.64 & 145.9 \\
& 30 & 84 & 0.68 & 144.4 \\
\hline
\multirow{3}{*}{Ring} & 0 & 100 & 1.00 & 156.8 \\
& 15 & 95 & 0.76 & 149.9 \\
& 30 & 90 & 0.80 & 147.5 \\
\hline
\multirow{3}{*}{8-hub Graph} & 0 & 100 & 1.00 & 153.6 \\
& 15 & 92 & 0.60 & 151.5 \\
& 30 & 87 & 0.54 & 148.7 \\
\hline
\multirow{3}{*}{9-ring Graph} & 0 & 100 & 1.00 & 148.1 \\
& 15 & 95 & 0.78 & 141.3 \\
& 30 & 91 & 0.72 & 134.4 \\
\hline
\multirow{3}{*}{von Neumann} & 0 & 100 & 1.00 & 153.3 \\
& 15 & 95 & 0.80 & 144.6 \\
& 30 & 90 & 0.78 & 138.3 \\
\hline
\multirow{3}{*}{Scale-free} & 0 & 100 & 1.00 & 154.5 \\
& 15 & 94 & 0.60 & 148.6 \\
& 30 & 87 & 0.50 & 148.3 \\
\hline
\multirow{3}{*}{Random} & 0 & 100 & 1.00 & 147.0 \\
& 15 & 96 & 0.70 & 141.2 \\
& 30 & 91 & 0.64 & 137.8 \\
\hline
\multirow{3}{*}{Small-world} & 0 & 100 & 1.00 & 150.4 \\
& 15 & 96 & 0.84 & 142.4 \\
& 30 & 91 & 0.74 & 138.2 \\ \bottomrule
\end{tabular}
\end{table}
\begin{table}[H]
\centering
\caption{Comparison of mean performance metrics on Griewank function}
\label{tab:griewank}
\begin{tabular}{ccccc}
\toprule
Network & Death Fraction (\%) & Number of Winners & GSR & GS Time \\\midrule
\multirow{3}{*}{Complete} & 0 & 58 & 0.58 & 154.6 \\
& 15 & 53 & 0.52 & 147.7 \\
& 30 & 36 & 0.32 & 144.2 \\
\hline
\multirow{3}{*}{Star} & 0 & 66 & 0.66 & 183.3 \\
& 15 & 68 & 0.62 & 172.5 \\
& 30 & 52 & 0.42 & 174.5 \\
\hline
\multirow{3}{*}{Ring} & 0 & 75 & 0.38 & 829.7 \\
& 15 & 34 & 0 & -- \\
& 30 & 17 & 0 & -- \\
\hline
\multirow{3}{*}{8-hub Graph} & 0 & 82 & 0.82 & 257.3 \\
& 15 & 70 & 0.42 & 247.7 \\
& 30 & 60 & 0.32 & 260.1 \\
\hline
\multirow{3}{*}{9-ring Graph} & 0 & 70 & 0.70 & 223.2 \\
& 15 & 66 & 0.54 & 197.2 \\
& 30 & 59 & 0.52 & 227.0 \\
\hline
\multirow{3}{*}{von Neumann} & 0 & 90 & 0.90 & 354.7 \\
& 15 & 82 & 0.66 & 333.2 \\
& 30 & 72 & 0.58 & 347.3 \\
\hline
\multirow{3}{*}{Scale-free} & 0 & 80 & 0.80 & 218.5 \\
& 15 & 70 & 0.44 & 203.5 \\
& 30 & 64 & 0.28 & 196.6 \\
\hline
\multirow{3}{*}{Random} & 0 & 58 & 0.58 & 161.9 \\
& 15 & 43 & 0.38 & 160.8 \\
& 30 & 43 & 0.32 & 155.3 \\
\hline
\multirow{3}{*}{Small-world} & 0 & 86 & 0.86 & 201.6 \\
& 15 & 77 & 0.56 & 193.5 \\
& 30 & 73 & 0.52 & 187.8 \\ \bottomrule
\end{tabular}
\end{table}
\begin{table}[H]
\centering
\caption{Comparison of mean performance metrics on Schwefel function}
\label{tab:schwefel}
\begin{tabular}{ccccc}
\toprule
Network & Death Fraction (\%) & Number of Winners & GSR & GS Time \\\midrule
\multirow{3}{*}{Complete} & 0 & 56 & 0.56 & 102.4 \\
& 15 & 64 & 0.54 & 100.1 \\
& 30 & 58 & 0.46 & 93.8 \\
\hline
\multirow{3}{*}{Star} & 0 & 86 & 0.86 & 117.9 \\
& 15 & 75 & 0.66 & 111.0 \\
& 30 & 83 & 0.66 & 108.5 \\
\hline
\multirow{3}{*}{Ring} & 0 & 100 & 1.00 & 289.5 \\
& 15 & 80 & 0.42 & 234.4 \\
& 30 & 64 & 0.20 & 422.0 \\
\hline
\multirow{3}{*}{8-hub Graph} & 0 & 92 & 0.92 & 129.8 \\
& 15 & 88 & 0.64 & 123.7 \\
& 30 & 81 & 0.46 & 120.6 \\
\hline
\multirow{3}{*}{9-ring Graph} & 0 & 86 & 0.86 & 115.7 \\
& 15 & 83 & 0.74 & 113.6 \\
& 30 & 78 & 0.6 & 111.4 \\
\hline
\multirow{3}{*}{von Neumann} & 0 & 98 & 0.98 & 148.9 \\
& 15 & 94 & 0.80 & 137.3 \\
& 30 & 86 & 0.74 & 134.2 \\
\hline
\multirow{3}{*}{Scale-free} & 0 & 92 & 0.92 & 126.7 \\
& 15 & 90 & 0.58 & 117.4 \\
& 30 & 81 & 0.50 & 116.0 \\
\hline
\multirow{3}{*}{Random} & 0 & 76 & 0.76 & 107.8 \\
& 15 & 76 & 0.66 & 103.4 \\
& 30 & 74 & 0.58 & 100.5 \\
\hline
\multirow{3}{*}{Small-world} & 0 & 84 & 0.84 & 118.0 \\
& 15 & 92 & 0.72 & 113.4 \\
& 30 & 87 & 0.74 & 119.9 \\ \bottomrule
\end{tabular}
\end{table}
\begin{table}[H]
\centering
\caption{Comparison of mean performance metrics on Rastrigin function}
\label{tab:rastrigin}
\begin{tabular}{ccccc}
\toprule
Network & Death Fraction (\%) & Number of Winners & GSR & GS Time \\\midrule
\multirow{3}{*}{Complete} & 0 & 100 & 1.00 & 110.8 \\
& 15 & 97 & 0.86 & 107.5 \\
& 30 & 93 & 0.80 & 101.4 \\
\hline
\multirow{3}{*}{Star} & 0 & 100 & 1.00 & 128.5 \\
& 15 & 94 & 0.76 & 122.1 \\
& 30 & 90 & 0.56 & 119.9 \\
\hline
\multirow{3}{*}{Ring} & 0 & 100 & 1.00 & 249.1 \\
& 15 & 88 & 0.44 & 240.8 \\
& 30 & 78 & 0.30 & 220.3 \\
\hline
\multirow{3}{*}{8-hub Graph} & 0 & 100 & 1.00 & 144.3 \\
& 15 & 94 & 0.70 & 135.5 \\
& 30 & 89 & 0.42 & 132.0 \\
\hline
\multirow{3}{*}{9-ring Graph} & 0 & 100 & 1.00 & 114.6 \\
& 15 & 96 & 0.84 & 114.7 \\
& 30 & 93 & 0.78 & 112.9 \\
\hline
\multirow{3}{*}{von Neumann} & 0 & 100 & 1.00 & 149.2 \\
& 15 & 96 & 0.80 & 140.6 \\
& 30 & 92 & 0.66 & 140.6 \\
\hline
\multirow{3}{*}{Scale-free} & 0 & 100 & 1.00 & 139.8 \\
& 15 & 94 & 0.66 & 129.3 \\
& 30 & 88 & 0.44 & 124.7 \\
\hline
\multirow{3}{*}{Random} & 0 & 100 & 1.00 & 115.1 \\
& 15 & 97 & 0.78 & 110.9 \\
& 30 & 93 & 0.72 & 105.6 \\
\hline
\multirow{3}{*}{Small-world} & 0 & 100 & 1.00 & 123.5 \\
& 15 & 96 & 0.86 & 120.5 \\
& 30 & 92 & 0.72 & 116.1 \\ \bottomrule
\end{tabular}
\end{table}
\section{Discussion} \label{sec:discussion}
In the previous section, we presented the results of PSO-based optimization of different objective functions using various network topologies. A key insight derived from these results is that \textit{graph-theoretic metrics of efficiency and robustness do not extend directly towards quantifying the efficiency and robustness of a network topology in the PSO framework}. For instance, agents communicating in the PSO framework with highly-connected and hence efficient topologies exhibit poor global convergence rates. This can be attributed to the fact that with every node connected to every other node, each agent in the swarm is presented with an excessive amount of information which results in the swarm converging prematurely to one of the local optima. In general, such disagreements can be attributed to the differences in the definitions of graph-theoretic properties (efficiency and robustness) and the performance of PSO. The graph-theoretic efficiency and robustness are typically defined with respect to the speed of information transfer and the size of the largest connected component in a graph. On the other hand, the efficiency and robustness of a topology in the PSO framework also depends on additional crucial factors such as rate of convergence of the entire swarm to the global optimum, performance tolerance to various levels of hostility, and the characteristics of the objective function -- number of local and global optima, and function landscape around the global optimum.
In terms of performance in the PSO framework, it is required that the networks exhibit sufficiently high levels of connectivity between the nodes to facilitate faster information transfer among agents. However, as seen above, very high levels of connectivity also results in poor performance. It is evident from Figure \ref{fig:PerformancevsAPSP} that network topologies with an average path length $L \sim 3$ maximize the performance trade-off between the number of winners and the convergence time defined in Section \ref{sec:psometrics}. This trade-off measure is very low for lower values of average path length (very high information transfer) resulting in agents getting stuck in local optima. On the other hand, the trade-off measure decreases gradually as the average path length increases beyond 3 due to reduced information transfer that results in higher convergence times. Hence, \textit{there appears to be a negative utility of increasing the connectivity beyond a certain threshold and networks that exhibit a high enough connectivity are desirable from a PSO performance standpoint}.
In terms of robustness in the PSO framework, it is required that the networks retain their information transfer rates to exhibit robust performance under hostile environments. Networks in the star-to-ring regime are very likely to fragment under hostile conditions resulting in disruption of information transfer. The ring graph is the least robust in this regime since removal of even a few nodes fragments the entire topology. On the other hand, hub-like structures are relatively more robust with the robustness decreasing with increasing number of hubs (central nodes). This is because the loss of even a single hub fragments the graph and higher the number of hubs in the graph, higher is the probability of graph-fragmentation. Mathematically, the probability of a hub-like structure with $k$ hubs getting fragmented at the $i^{th}$ iteration can be expressed as $1-(1-p)^{ki}$ with $p$ being the probability of loss of an agent. However, the most robust graphs are characterized by high connectivity in addition to having a distributed architecture as observed in the ring-to-complete and complete-to-star regimes. For example, complete graphs exhibit the maximum robustness among all the graphs across the spectrum. Therefore, \textit{networks with distributed architectures and high connectivity are desirable from a robustness standpoint.}
In addition to the topologies across the triangular spectrum in Figure \ref{fig:networks}(a), our study on several standard topologies involving multiple objective functions revealed that small-world graphs achieve a high global success ratio while retaining the performance with loss of agents. Specifically, while the best performing graph for different objective functions varies, the success ratio of small-world graphs is consistently among the best two topologies. Moreover, the convergence time for small-world graphs is consistently close to the lowest convergence time achieved for all objective functions. Small-world network architectures appear to be optimizing information transfer and retention of performance under hostile conditions. In order to investigate this effect, we discuss a commonly used metric of \textit{small-world-ness} of networks \cite{telesford2011ubiquity}:
\begin{align}\label{eq:smallworldness}
\omega = \frac{L_{random}}{L} - \frac{C}{C_{lattice}}
\end{align}
Here $L$ is the average geodesic length (Equation \eqref{eq:apsp}) and $C$ is the average clustering coefficient of the network given as:
\begin{align*}
C = \frac{1}{N}\sum_{i=1}^N\frac{2e_i}{k_i(k_i-1)}
\end{align*}
with $e_i$ being the number of edges between the neighbors of a given node $i$ and $k_i (k_i-1)/2$ is the total number of edges possible between the neighbors. The quantities $L_{random}$ and $C_{lattice}$ represent the average path length of an equivalent random graph and clustering coefficient of an equivalent lattice graph, respectively. The \textit{small-world-ness} is close to zero for small-world networks. The above quantity suggests that for a network to exhibit small-world behavior, it must have an average path length close to that of an equivalent random graph and clustering coefficient close to that of an equivalent lattice graph. Therefore, it can be said that small-world graphs \textit{simultaneously} maximize the efficiency (low average path length) and robustness (high clustering coefficient) of a network. \textit{Therefore, small-world graphs exhibit a high enough connectivity with distributed architectures, and hence simultaneously maximize efficiency and robustness in the PSO framework}.
\section{Conclusions}\label{sec:conclusions}
In this article, we report a study of the performance of particle swarm optimization with various network topologies characterizing communication between agents under hostile environments. Such problems bear significance in applications such as targeted drug delivery and high value target localization. The major contribution of this work lies in studying the impact of hostile environments with respect to network topologies in PSO framework. Based on our study, we first conclude that maximizing the graph-theoretic measures of efficiency and robustness of a network do not necessarily result in efficient and robust performance in the PSO framework. This is attributed to factors such as local optima and nature of the function landscape around global optimum that are not incorporated in graph theoretic metrics. Second, we observe that in order to maximize the performance of PSO, a network topology has to be sufficiently connected to ensure optimal information transfer -- a higher information transfer results in over exploitation of the information limiting exploration, whereas a lower information transfer results in over exploration of the search space that results in higher convergence times. An average path length of $L \sim 3$ was observed to give the best performance in terms of higher global success ratio with lower convergence times -- consistent with our \textit{sufficient connectivity} hypothesis. Third, networks with distributed architectures with high connectivity between nodes were observed to be highly robust to deactivation of agents at random and retain their information transfer rates for various levels of severity of the hostile environment. We showed that scale-free and multi-hub networks are therefore less robust compared to multi-ring and von Neumann graphs.
Our study on the performance of the best topologies in the triangular spectrum indicated that two networks with 8-hub structure and 9 multi-rings exhibit the best performance among all the other structures. Further analysis of the performance of these topologies, the four standard topologies including small-world graphs on different objective functions revealed that small-world graphs consistently perform well in terms of a higher global success ratio and lower convergence time. Small-world graphs are in agreement with our observation of distributed architectures with high connectivity and an average path length of $\sim 3$, performing well in PSO. An analysis of a metric quantifying small-world-ness revealed that such networks inherently achieve a trade-off between efficient flow of information between agents and tolerance to hostile conditions. We therefore conclude that small-world like networks are well-suited for optimizing objective functions when the function landscape is unknown and the environment could be hostile towards the agents.
Although the findings of this work are based on performance of the PSO algorithm, the results are fairly generalizable, and should extend to algorithms that rely on efficient communication between agents while searching for an optimal solution. Future work in this area involves extending the current ideas to incorporate adaptive death rates as a function of time and space, and swarm reorganization to minimize the impact of agent deactivation on network topologies.
\section*{Acknowledgements} \label{sec:acknowledgements}
{The authors gratefully acknowledge the contributions of the following former students in our group for their earlier work on this topic - Arun V. Giridhar, Balachandra B. Krishnamurthy, Chunhua Zhao, Priyan R. Patkar, and Santhoji Katare. We also wish to thank the following research interns at the Complex Resilient Intelligent System (CRIS) laboratory at Columbia University - Xijao Li, Liyi Zhang, and Jia Wan.}
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\section{\label{Intro}Introduction}
Despite the relevance of non-equilibrium physics in many condensed matter systems\cite{RevModPhys.81.163,Le_Hur_2016} a robust and practical theoretical framework for studying these non-equilibrium systems is still lagging. One popular approach to studying non-equilibrium systems is to use two-time non-equilibrium Green's functions(NEGFs)\cite{stefanucci2013nonequilibrium, Kadanoff}. For a given system and set of initial conditions, the time evolution of the one particle NEGF can be computed using the Kadanoff-Baym equations(KBEs)\cite{Stan_2009}. This time evolution is exact, given the exact self-energy is known, but in practice, the self-energy is an approximate quantity.
Unfortunately, the KBEs are known to suffer from two major issues. Firstly, the KBEs have been reported to reach artificial steady states that are not present in the exact time evolution\cite{von_Friesen_2009,Friesen_2010,PhysRevB.90.125111}. The second major problem with the KBE approach is in the computational cost, specifically in the number of time propagation steps ($N_t$). When solving the KBEs, the two-time Green's function needs to be propagated at all points on a two-time grid, which leads to asymptotic computational scaling $O(N_t^3)$\cite{bonitz2015quantum}. This makes the use of the KBEs impractical outside of small systems and beyond short propagation times.
Because of this second issue, an approximate partial solution to the KBE, known as the Hartree-Fock generalized Kadanoff-Baym ansatz(HF-GKBA), is more commonly used for simulations of realistic problems\cite{PhysRevB.34.6933,Hermanns_2012,Hermanns_2012,PhysRevLett.128.016801}. With the HF-GKBA, only the KBE for the two-time Green's function at equal times is explicitly propagated. The time off-diagonal elements are then reconstructed from the information on the time-diagonal. The HF-GKBA has become widely used due to the speedup it offers over the full KBE, especially for long time propagation's. The HF-GKBA has even been argued to improve over the full KBE by removing spurious steady states and artificial damping\cite{von_Friesen_2009} and has even been claimed to outperform KBE in reconstructing particle densities in a simple Hubbard chain\cite{PhysRevB.90.125111}. However, the HF-GKBA only makes further approximations upon those already made in the KBE, and so any improvements are, at least in part, fortuitous.
Although faster than full KBE, in its original formulation the HF-GKBA still retains the $O(N_t^3)$ scaling (except when used with the Second-Born self-energy). Recently, a reformulation of the HF-GKBA, known as the G1-G2 scheme, has offered a method for propagating NEGFs within the HF-GKBA with scaling $O(N_t)$\cite{PhysRevB.101.245101}. While promising, the scheme still suffers from several drawbacks. Firstly, the method expresses the single particle self-energy in terms of the two-particle Green's function, thus upfolding the problem onto a larger space. This upfolding leads to $O(N_s^6)$ numerical scaling in the system size, making it difficult to use the G1-G2 scheme in large systems or systems with multiple bands. A second issue the G1-G2 scheme faces is that the time propagation is known to be unstable for long times and/or strong couplings since certain consistency relations for the two-particle Green's function break down\cite{PhysRevB.105.165155}. In the G1-G2 scheme removing this instability requires the repeated diagonalization of the two-particle Green's function making it impractical for large systems and long time evolutions\cite{PhysRevB.105.165155}. The former issue is well suited to the family of stochastic methods \cite{PhysRevLett.113.076402,doi:10.1021/acs.jctc.7b00770,PhysRevB.106.165129} which provides a seemingly straightforward strategy to reduce the computational cost in the system size. This, however, comes at a price: the time evolution trajectory is fundamentally not stable over extended propagation time.
All of this means in large scale or strongly interacting systems even HF-GKBA alone may be a computationally intractable tool to study NEGF over long-time trajectories.
In this paper, we test the ability of using dynamical mode decomposition(DMD) to reconstruct NEGF trajectories from partial samples of the full trajectory. For the case of large systems, this would be extremely advantageous since DMD is computationally much cheaper than HF-GKBA. Additionally, in the cases where we can successfully fit DMD on data before HF-GKBA becomes unstable, the need to diagonalize the two-particle Green's function at every step is removed. In both scenarios, DMD is a promising method of significantly speeding up the propagation of NEGF under HF-GKBA, especially for long propagation times in large scale realistic systems. For smaller systems where exact diagonalization is possible we compare these results to the HF-GKBA result. Further, we numerically demonstrate that (seemingly counterintuitively) the approximate methodology agrees better for the more realistic long-range Hamiltonian forms. This is rationalized as the approximate self-energy applied is well suited for realistic weakly and moderately correlated systems in which long-range interactions present significant contributions. This further motivates the use of DMD for creating computationally cheap long-time trajectories that match well with exact time propagation.
\section{Theory}\label{Theory}
\subsection{Model Systems}\label{model_systems}
To evaluate the ability of DMD to numerically extrapolate the Green's function trajectories, we use the following strategy illustrated in Fig \ref{fig:sys_setup}. The system is first prepared in a correlated stationary state and then driven from equilibrium via a quenching of specific sites of the Hamiltonian. More details on the initial state preparation are given in \ref{sec:Methods}. The discussion of system quenching is continued later in this section. After the non-equilibrium dynamics are initiated, DMD is used in an initial window of the trajectory, after which the DMD result is propagated and compared with the remainder of the unfitted trajectory. For a detailed discussion on the DMD procedure see section \ref{sub_sect:DMD}.
\FloatBarrier
\begin{figure}[h!]
\centering
\includegraphics[width=.4\textwidth]{Info_figure.png}
\caption{A) Model system with exponentially decaying interactions and an alternating local potential described by equations \eqref{MB_ham}, \eqref{two-body} and \eqref{single_particle}. B) Outline of non-equilibrium preparation and DMD fitting procedure}
\label{fig:sys_setup}
\end{figure}
\FloatBarrier
A generic many-body Hamiltonian can be written in the following form
\begin{equation}\label{MB_ham}
\mathcal{H} = \sum_{ij}h^{(0)}_{ij}(t)c^\dagger_ic_j + \frac{1}{2}\sum_{ijkl} w_{ijkl}(t) c^\dagger_ic_j^\dagger c_k c_l.
\end{equation}
Here $w_{ijkl}$ is the two-body interaction term and $h^{(0)}(t)$ is the single-particle Hamiltonian. For the model we study, the corresponding two-body interactions is given explicitly by
\begin{equation}\label{two-body}
\begin{split}
w_{ijkl}^{\sigma_i\sigma_j\sigma_k\sigma_l}(t) &= U(t)\delta_{ij}\delta_{ik}\delta_{il}\delta_{\sigma_i\sigma_k}\delta_{\sigma_j\sigma_l}(1-\delta_{\sigma_i\sigma_k}) \\&\hspace{5mm}+ U(t)\sum_{n=1}^{N_s}\mathrm{e}^{-\gamma|i-j|}\delta_{ij}^{(n)}\delta_{ik}\delta_{jl}\delta_{\sigma_i\sigma_k}\delta_{\sigma_j\sigma_l},
\end{split}
\end{equation}
where the $\sigma \in \{\uparrow,\downarrow\}$ are spin indices, $N_s$ is the number of sites in the chain and $\gamma$ determines the rate of decay of the long-range interactions. Here we define $\delta^{(n)}_{ij}$ to only be non-zero if $|i-j|=n$.
Later when using the HF-GKBA, since we prepare correlated initial states with adiabatic switching, we include an explicit time-dependence in the interaction terms above. The initial state preparation will be discussed in more detail in section \ref{sec:Methods}, but for now we note that for a model with the two-body term in equation \eqref{two-body} and nearest neighbour hopping only the gap between the grounds state and the first excited state tends to $0$ as the system size is increased. This makes the adiabatic switching procedure increasingly numerically unstable. To open a gap and allow for a numerically stable initial state preparation we add an alternating step potential to our model which is illustrated in figure \ref{fig:sys_setup}.
The single particle Hamiltonian for our model is given by
\begin{equation}\label{single_particle}
\begin{split}
h^{(0)}_{ij}(t) &= -J\delta^{(1)}_{ij} +\delta_{ij}(-1)^i V + h^{\textrm{quench}}_{ij}(t).
\end{split}
\end{equation}
$h^{\textrm{quench}}_{ij}(t)$ is the quench Hamiltonian that drives the system from equilibrium. We study two types of quenches here, written explicitly below as
\begin{equation}\label{quench}
\begin{split}
h^{\textrm{quench}}_{ij}(t) &= q\delta_{ij}f_\tau(t-t_0),\\
h^{\textrm{quench}}_{ij}(t) &= q\delta_{ij}f_\tau(t-t_0)(1-f_\tau(t-t_1)),
\end{split}
\end{equation}
where $f_\tau(t-t_0)$ is the Fermi-Dirac function,
\begin{equation}\label{fermi_dirac}
f_\tau(t-t_0) = \frac{1}{1+e^{-\frac{t-t_0}{\tau}}}.
\end{equation}
It should also be understood that the chosen quench acts only on a specific subset of the chain ($N_q)$.
\subsection{The Hartree-Fock Generalized\\Kadanoff-Baym Ansatz}
The HF-GKBA is an approximate partial solution to the full propagation of NEGF through the KBE. At zero temperature the full Kadanoff-Baym equations are a set of five integro-differential equations\cite{Stan_2009}
\begin{equation}\label{KBE}
\begin{split}
i\partial_t G^{</>}(t,t') &= h^{\textrm{HF}}(t)G^{</>}(t,t') + I_1^{</>}(t,t')\\
-i\partial_{t'} G^{</>}(t,t') &= G^{</>}(t,t')h^{\textrm{HF}}(t') + I_2^{</>}(t,t')\\
i\partial_t G^{<}(t,t) &= [h^{\textrm{HF}}(t),G^{<}(t,t)] + I_1^<(t,t) - I_2^{<}(t,t)
\end{split}
\end{equation}
with
\begin{equation}\label{coll_int}
\begin{split}
I_{1}^{</>}(t,t') &= \int_{0}^t \mathrm{d}\bar{t} \Sigma^R(t,\Bar{t})G^{</>}(\Bar{t},t') \\ &\hspace{20mm}+\int_{0}^{t'} \mathrm{d}\bar{t} \Sigma^{</>}(t,\Bar{t})G^{A}(\Bar{t},t')\\
I_{2}^{</>}(t,t') &= \int_{0}^t \mathrm{d}\bar{t} G^R(t,\Bar{t})\Sigma^{</>}(\Bar{t},t') \\&\hspace{20mm}+ \int_{0}^{t'} \mathrm{d}\bar{t} G^{</>}(t,\Bar{t})\Sigma^{A}(\Bar{t},t').
\end{split}
\end{equation}
Here $G^<(t,t')(G^>(t,t'))$ is the two time particle(hole) propagator. The collision integrals, $I^{</>}_{1,2}$, take into account many-body correlation effects as well as system memory. The two-time nature of the KBE combined with these integral terms leads to the cubic scaling of KBE mentioned in the introduction. The HF-GKBA is derived directly from the KBE and can be summarized in the following equations\cite{Hermanns_2012},
\begin{equation}\label{HF-GKBA}
\begin{split}
G^{</>}(t,t') &= G^R(t,t')G^{</>}(t,t) - G^{</>}(t,t)G^R(t,t'),\\
G^{R,A}&=\pm i \Theta(t_1 \pm t_2)T\{\mathrm{e}^{-i\int h^{\textrm{HF}}(t) dt}\}.
\end{split}
\end{equation}
In other words, at each time step only the final equation in equation \eqref{KBE} is explicitly evaluated. Equation \eqref{HF-GKBA} is then used to reconstruct the time off-diagonal components.
Apart from those approximations made to the self-energy, which HF-GKBA and KBE share, two additional approximations are made in the derivation of HF-GKBA. The first involves neglecting certain integrals, similar to those in equation \eqref{coll_int}, over products of different components of the Green's function and self-energy. These terms appear in the expression for reconstructing $G^{</>}(t,t')$ and are dropped, leading to the generalized Kadanoff-Baym ansatz(GKBA)\cite{PhysRevB.34.6933}. The HF-GKBA involves a further approximation where the full $G^{R/A}(t,t')$ are replaced by the retarded and advanced Hartree-Fock propagator. The HF-GKBA still leave important quantities such as energy and particle number conserved as well as retaining causal time evolution.
Recently a linear time scaling($\sim O(N_t))$ implementation of the HF-GKBA has been achieved, opening the door for long-time evolution's of NEGFs\cite{PhysRevB.101.245101}. The method removes the explicit appearance of integrals in equation \eqref{coll_int} from the differential equation for $G^<(t)$ by explicitly expressing them in terms of the correlated part of the equal time two-particle Green's function $\mathcal{G}(t)$. Within this formulation $\mathcal{G}(t)$ is propagated simultaneously with $G^<(t)$ using an equation analogous to the last line of equation \eqref{KBE}. Throughout this paper, we use this propagation scheme to generate HF-GKBA results for the models discussed in section \ref{model_systems}.
The exact equation of motion for $\mathcal{G}(t)$ depends on the self-energy approximation used. Throughout this paper, we use the $GW$ self-energy, due to its wide usage and its success in equilibrium condensed matter systems\cite{10.3389/fchem.2019.00377}. For the $GW$ self-energy the equations of motion for $G^<(t)$ and $\mathcal{G}(t)$ in the orbital basis are given below.
\begin{equation}\label{G1-G2}
\begin{split}
i \partial_t G^<_{ij}(t) &= [h^{\textrm{HF}}(t), G^<(t)]_{ij} + [I+I^\dagger]_{ij}(t)\\ i\partial_t \mathcal{G}_{ijkl}(t) &= [h^{(2),\textrm{HF}}(t),\mathcal{G}(t)]_{ijkl}\\&\hspace{13mm}+\Psi_{ijkl}(t) + \Pi_{ijkl}(t) - \Pi_{lkji}^*(t).
\end{split}
\end{equation}
Above, the following definitions are made,
\begin{equation}
\begin{split}
h_{ij}^{\textrm{HF}}(t) &= h^{(0)}_{ij}(t) - i\sum_{kl} (w_{ikjl} - w_{iklj})(t)G_{kl}^<(t),\\
I_{ij}(t)&=-i\sum_{klp} w_{iklp}(t)\mathcal{G}_{lpjk}(t),\\
h^{(2),\textrm{HF}}_{ijkl}(t) &= \delta_{jl}h^{\textrm{HF}}_{ik}(t) + \delta_{ik}h^{\textrm{HF}}_{jl}(t),\\
\Psi_{ijkl} &=\sum_{pqrs}w_{pqrs} \bigg[G^>_{ip}(t) G^<_{rk}(t) G^>_{jq}(t)G^<_{sl}(t)\\&\hspace{24mm}- G^<_{ip}(t)G^>_{rk}(t)G^<_{jq}(t)G^>_{sl}(t)\bigg],\\
\end{split}
\end{equation} \begin{equation*}
\begin{split} \Pi_{ijkl}&=\sum_{pqrs}w_{sqrp}(t)\bigg{[}G^>_{js}(t)G^<_{rl}(t)\\&\hspace{30mm}- G^<_{js}(t)G^>_{rl}(t)\bigg{]}\mathcal{G}_{ipkq}(t).
\end{split}
\end{equation*}
Here $\Pi_{ijkl}$ accounts for polarization in the system and $\Psi_{ijkl}$ accounts for pair correlations built up due to two-particle scattering events\cite{PhysRevB.101.245101}.
\subsection{DMD}\label{sub_sect:DMD}
DMD
is a data-driven dimension reduction technique used to predict observables of a nonlinear dynamical system with a large number of degrees of freedom by constructing a low dimensional linear
dynamical model\cite{DMD0,schmid2011applications,kutz2016dynamic,TuRowley}. The linear model
can be characterized by a number of spatial and temporal modes that can be obtained from the eigenvalues and eigenvectors of a linear operator known as a projected Koopman operator\cite{DMDtoKoop}.
To introduce the basic ideas of DMD, let us view~\eqref{G1-G2} as a general dynamical system of the form
\begin{equation}\label{eq:model}
\frac{d\mathbf{x}(t)}{dt} = \mathbf{f}(\mathbf{x}(t), t), \quad t\geq 0,
\end{equation}
where
\begin{equation}
\mathbf{x}(t) = [\mathbf{g}_1(t), \mathbf{g}_2(t), ..., \mathbf{g}_n(t)]^T,
\end{equation}
and
\begin{equation}
\mathbf{g}_i(t) = [G^<_{i1}(t), G^<_{i2}(t), ..., G^<_{in}(t)]. \;
\end{equation}
Here, $n$ is the number of sites, and
we simply consider the right-hand-side of\eqref{G1-G2} as a nonlinear function $\mathbf{f}: \mathbb{C}^n\otimes \mathbb{R}^+ \rightarrow \mathbb{C}^n$ of $\mathbf{x}$ and $t$.
The DMD method allows us to approximate \eqref{eq:model} by a linear model
\begin{equation}
\frac{d\mathbf{x}(t)}{dt} = \mathbf{A}\mathbf{x}(t),
\end{equation}
with a carefully constructed operator $\mathbf{A}$.
For problems that have
an explicit analytical expression of $\mathbf{f}(\mathbf{x}(t), t)$, it may be possible to linearize $\mathbf{f}(\mathbf{x}(t), t)$ and derive $\mathbf{A}$ explicitly. This linearization process essentially amounts to a linear response analysis. However, when the analytical form of $\mathbf{f}(\mathbf{x}(t), t)$ is unknown, performing such an analysis is difficult, if not impossible.
The linearization produced by DMD is based on the Koopman operator theory\cite{DMDtoKoop,Koopman1,Koopman2}, which is developed to characterize the evolution of a scalar observable function of $\mathbf{x}(t)$, denoted by $g(\mathbf{x}(t))$, to $g(\mathbf{x}(t+\Delta t))$ with $\Delta t>0$, i.e.
\[
g(\mathbf{x}(t+\Delta t)) = \mathcal{K}_{\Delta t} g(\mathbf{x}(t)).
\]
In the limit of $\Delta t \rightarrow 0$, the Koopman operator defines a linear dynamical system
\begin{equation}\label{eq:Koopman}
\frac{dg(\mathbf{x}(t))}{dt} = \mathcal{K} g(\mathbf{x}(t)).
\end{equation}
Because the Koopman operator $\mathcal{K}$ is a linear operator that maps from a function space to
another function space, it has an infinite number of eigenvalues $\lambda_j$ and eigenfunctions
$\varphi_j(\mathbf{x})$, $j = 1, 2, ..., \infty$.
If the observable functions of interest form an invariant subspace of
$\mathcal{K}$ spanned by a finite subset of eigenvalues and eigenvectors, then it is possible to
construct a finite-dimensional operator (matrix) approximation to $\mathcal{K}$.
To be specific, if $g_1(\mathbf{x})$, $g_2(\mathbf{x})$,...,$g_n(\mathbf{x})$
are $n$ observable functions so that
\begin{equation}\label{eq:KoopmanA}
\begin{bmatrix}
g_1(\mathbf{x}) \\
g_2(\mathbf{x}) \\
\vdots \\
g_n(\mathbf{x})
\end{bmatrix}
=
\mathbf{V}_1
\begin{bmatrix}
\varphi_1(\mathbf{x}) \\
\varphi_2(\mathbf{x}) \\
\vdots \\
\varphi_k(\mathbf{x})
\end{bmatrix}
=
\mathbf{V}_1\mathbf{V}_2
\begin{bmatrix}
g_1(\mathbf{x}) \\
g_2(\mathbf{x}) \\
\vdots \\
g_k(\mathbf{x})
\end{bmatrix}
\end{equation}
for some $k\in\mathbb{N}$ and matrices $\mathbf{V}_1 \in \mathbb{C}^{n\times k}$, $\mathbf{V}_2 \in\mathbb{C}^{k\times n}$,
then $\mathcal{K}$ can be approximated by a $n \times n$ matrix $\mathbf{A} = \mathbf{V}_1\mathbf{V}_2$ on these observable functions.
But in practice, we cannot assume that equation \eqref{eq:KoopmanA} holds for our observable functions, so we can only get a finite-dimension approximation of $\mathcal{K}$ represented by matrix $\mathbf{A}$. To construct such an approximation, observable functions are chosen to be the components of
$\mathbf{x}(t)$ defined in equation \eqref{eq:model}, we take snapshots of $\mathbf{x}(t)$ at $t_j = (j-1)\Delta t$, i.e., $\mathbf{x}_j = \mathbf{x}(t_j)$, for $j=1,...,m$, and use them to build two matrices
$\mathbf{X}_1$ and $\mathbf{X}_2$ of the form
\begin{equation}
\mathbf{X}_1 =\left( \mathbf{x}_1 \: \mathbf{x}_2 \: \cdots \: \mathbf{x}_{m-1} \right) \ \ \mbox{and} \ \
\mathbf{X}_2 =\left( \mathbf{x}_2 \: \mathbf{x}_3 \: \cdots \: \mathbf{x}_{m} \right).
\label{eq:mats}
\end{equation}
The finite-dimensional approximation to the Koopman operator can then be obtained by solving the following linear least squares problem
\begin{equation}
\min_{\mathbf{A}} \| \mathbf{A} \mathbf{X}_1 - \mathbf{X}_2 \|_F^2.
\label{eq:lsq}
\end{equation}
The solution to \eqref{eq:lsq} is
\begin{equation}\label{eq:A}
\mathbf{A} = \mathbf{X}_2\mathbf{X}_1^\dagger,
\end{equation}
where $\mathbf{X}_1^\dagger$ is the Moore-Penrose pseudoinverse of $\mathbf{X}_1$ that can be computed from the singular value decomposition (SVD)\cite{SVD} of $\mathbf{X}_1$.
If the nonzero singular values of $\mathbf{X}_1$, $\sigma_j$, $j = 1,2,...,m$, decrease rapidly with respect to $j$, which indicates that the numerical rank, denoted by $r$, of $\mathbf{X}_1$ is much smaller than $m$ and $n$, we can use a truncated SVD of $\mathbf{X}_1$ in the form of $\mathbf{X}_1 = \widetilde{\mathbf{U}}\widetilde{\mathbf{\Sigma}}\widetilde{\mathbf{V}}^T$, where the $r\times r$ diagonal matrix $\widetilde{\mathbf{\Sigma}}$ contains the leading $r$ dominant singular values of $\mathbf{X}_1$, and $\widetilde{\mathbf{U}}$ and $\widetilde{\mathbf{V}}$
contain the corresponding right and left singular vectors, to obtain an approximation of $\mathbf{A}$ as
\begin{equation}
\mathbf{A} \approx \mathbf{X}_2\widetilde{\mathbf{V}}\widetilde{\mathbf{\Sigma}}^{-1}\widetilde{\mathbf{U}}^*.
\label{eq:projA}
\end{equation}
We can now fully characterize the approximated reduced order linear dynamical system model by
diagonalizing the projected Koopman operator
$\widetilde{\mathbf{A}} = \widetilde{\mathbf{U}}^\ast \mathbf{A} \widetilde{\mathbf{U}}
= \widetilde{\mathbf{U}} \mathbf{X}_2 \widetilde{\mathbf{V}} \widetilde{\mathbf{\Sigma}}^{-1}\in\mathbb{C}^{r\times r}$.
Let
\begin{equation}
\widetilde{\mathbf{A}}\mathbf{W} = \mathbf{W}\mathbf{\Lambda}
\label{eq:dmdev}
\end{equation}
be the eigendecomposition of $\widetilde{\mathbf{A}}$,
where $\mathbf{\Lambda} = {\rm{diag}}(\lambda_1, ..., \lambda_r)$ is composed of the eigenvalues of $\widetilde{\mathbf{A}}$, and the columns of $\mathbf{W}$ are the corresponding eigenvectors.
The matrix
\begin{equation}
\mathbf{\Phi} = \mathbf{X}_2\widetilde{\mathbf{V}}\widetilde{\mathbf{\Sigma}}^{-1}\mathbf{W}
\label{eq:dmdmodes}
\end{equation}
contains the so-called DMD modes. If $\phi_{\ell}$ is the $\ell$th column of $\mathbf{\Phi}$, the DMD approximation to $\mathbf{x}$ can be represented by
\begin{equation}\label{eq:evol_DMD}
\mathbf{x}(t)\approx \sum_{\ell=1}^r\mathbf{\phi}_\ell\exp(i\omega_\ell^{\text{DMD}} t)b_\ell = \mathbf{\Phi}\exp(\mathbf{\Omega} t)\mathbf{b}.
\end{equation}
where $\omega_\ell^{\text{DMD}} = -i{\ln{\lambda_\ell}}/{\Delta t}$, $\ell = 1, ..., r$, $\mathbf{\Omega} = {\ln{\mathbf{\Lambda}}}/{\Delta t} = {\rm{diag}}(i\omega_1^{\rm{DMD}}, ..., i\omega_r^{\rm{DMD}})$, and the amplitude vector $\mathbf{b}:=[b_1, ..., b_r]^T$ is taken either as the projection of the initial value $\mathbf{x}_1$ onto the DMD modes, i.e.,
\begin{equation}\label{eq:b1}
\mathbf{b} = \mathbf{\Phi}^\dagger \mathbf{x}_1,
\end{equation}
or as the least squares fit of equation \eqref{eq:evol_DMD} on the sampled trajectories, i.e.,
\begin{equation}\label{eq:b2}
\mathbf{b} = \arg\min_{\tilde{\mathbf{b}}\in\mathbb{C}^n}\sum_{j=1}^m\|\mathbf{\Phi}\exp(\mathbf{\Omega} t_j)\tilde{\mathbf{b}}-\mathbf{x}_j\|^2,
\end{equation}
where $\|\cdot\|$ denotes the standard Euclidean norm of a vector. For more details on the numerical procedure, we refer readers to references\cite{DMD0,kutz2016dynamic,TuRowley,DMDdiag,DMDtwotime}.
The major computational cost of DMD computation is in the SVD of $\mathbf{X}_1$, which is $O(\min(m^2n, mn^2))$. The memory cost is $O(mn)$.
\section{Methods}\label{sec:Methods}
For systems with up to 8 sites at half-filling we prepare trajectories for the model and quenches described in section \ref{Theory} using both exact diagonalization and HF-GKBA. A publicly available version of the code used in these simulations is available online\footnote{https://github.com/VlcekGroup/G12KBA.git}. We also use HF-GKBA to create a trajectory for 16 sites, however exact time evolution was not possible for this system size. Equation \eqref{G1-G2} was propagated using fourth-order Runge-Kutte with a time step of $0.07 J^{-1}$, and we use the same time step for the exact diagonalization propagation. We performed calculations for two values of the decay parameter $\gamma$, firstly we take $\gamma=\infty$ which leads to a model with onsite interactions only. The second case we study has $\gamma = 0.7$ so that the nearest neighbour is subject to approximately half the interaction of the onsite interaction strength. For each model and quench, calculations were run for $U = 0.1J$, $0.3J$, $0.5J$ and $1.0J$. Additionally, we explored the effect of quenching different portions of the system, testing both half and quarter system quenches.
For both HF-GKBA and exact diagonalization we prepare the system in the respective correlated initial state before initiating the quench. In the case of exact diagonalization, we can trivially prepare the system in the exact ground state, by diagonalizing and finding the eigenstate with the lowest energy. For the HF-GKBA we start by preparing the system in the non-interacting ground state and then time evolves with equation \eqref{G1-G2} while slowly turning on the $U$ parameter. We choose the Fermi-Dirac function introduced in equation \eqref{fermi_dirac} as our switching function.
We found the values $t_0 = 25 $ and $\tau = 3$ gave a sufficiently slow rate of switching to converge the models and parameters presented here.
For the alternating step potential in equation \eqref{single_particle} we chose a value of $V=2$. We found for $N_s\leq16$, this value of $V$ opened the gap sufficiently to perform the adiabatic switching procedure successfully. However, we note that for $V = 2$ the adiabatic switching procedure became unstable as we went to larger systems($N_s=32)$.
\section{Results}
In total around 100 different system setups were tested, see section \ref{sec:Methods}, in this section we will present a small representative selection of these results. In section \ref{onsite_vs_lr} we look at the performance HF-GKBA for the long-range and onsite models. We compare trajectories for different values of $U$ and for the two quenches described in section \ref{model_systems}. In section \ref{DMD_results} we show results demonstrating the ability of the DMD to fit the Green's function in the long-range model with 16 sites.
\subsection{HF-GKBA Dynamics for Onsite and Long Range Models}\label{onsite_vs_lr}
First, we analyze the early time dynamics of both models and compare the trajectories to exact diagonalization. In Figure \ref{EHM_vs_HM} we show the dipole generated by HF-GKBA and the exact diagonalization for 8 sites for each of the Hamiltonians described by equations \eqref{two-body} and \eqref{single_particle}. The Green's function contains $N_s^2$ elements and so to compress this large quantity of time trajectories we choose the center of mass dipole as our figure of comparison, which we calculate as
\begin{equation}
p(t) = \frac{i}{N_s}\sum_{j=1}^{N_s}\left(\frac{N_s-1}{2} - j\right)[G_{jj}(t) - G_{N_s-j+1N_s-j+1}(t)],
\end{equation}
where $G_{jj}(t)$ is the density on site $j$ at time $t$.
We choose the dipole for two reasons, besides compressing the Green's function to a single number. Firstly, we believe it is most relevant to test the ability of the HF-GKBA to reproduce experimental observables. Secondly, the dipole is an integrated quantity and so provides a site-independent accumulated result of the density, which also removes some of the bias due to edge effects or the quench location. We found the dipole to be a representative quantity for the results in the following sections.
\FloatBarrier
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{Early_time_dynamics_8_sites.png}
\caption{Comparison of the early time dynamics of the dipole given by HF-GKBA(orange-dashed) to exact results(blue-solid) for $\gamma = \infty$(left) and $\gamma = 0.7$(right) with $U = 0.5J$ and $U = 1.0J$ and for $N_s=8$. Figures 1 (2) of each panel A,B,C and D show the dipole after the first half the system is pulse quenched (fully quenched). All figures are plotted on the same scale.}
\label{EHM_vs_HM}
\end{figure*}
\FloatBarrier
The results in Figure \ref{EHM_vs_HM} show the first $50$ time units after the quench at $t=50$, and we direct the reader to the SI for the full dipole trajectory over 1000 time units. After preparing the system in the correlated initial state we perform a full or pulsed quench of magnitude 1 to the first 4 sites. Quenches were also tested on the first 2 sites, however, little qualitative difference was observed between the two cases. For the full quench, the parameters chosen were $t_0 = 50$ and $\tau = 0.2$. In the case of the pulse quench, we fix $t_0$ and $\tau$ to be the same as for the full quench and take $t_1 = 55$. We note that for $U=1.0$ the HF-GKBA trajectory for the onsite model becomes unstable and diverges after between 500 and 800 time units depending on the quench type, whereas the extended model remains stable for the entire trajectory.
From the portion of the trajectory shown in Figure \ref{EHM_vs_HM}, we see that for $U = 0.5J$ the HF-GKBA captures the dynamics of the onsite model and long-range model quite well. However, already at around 40 time units after the pulse we see the amplitude of the HF-GKBA result begin to decrease relative to the exact. Meanwhile the long-range model continues to match the dynamics remarkably well for the entire trajectory shown. As we move to $U = 1.0J$ difference between the HF-GKBA in the onsite and long-ranged model becomes even more extreme. At around $20$ time units after the pulse the HF-GKBA fails completely to capture the true dipole dynamics, whereas in the long-range model it is matched almost exactly by the HF-GKBA.
To compare the HF-GKBA and exact results over the full dipole trajectory we look at the frequency spectrum for each of the trajectories shown in figure \ref{EHM_vs_HM}.
\FloatBarrier
\begin{figure*}
\centering
\includegraphics[width = \textwidth]{Frequency_spectrum_8_sites.png}
\caption{Comparison of frequency spectrum of the dipole for HF-GKBA(orange-dotted) and the exact result(blue-solid) for each of the cases in Figure \ref{EHM_vs_HM}. Figures 1 (2) of each panel A,B,C and D show the dipole frequency spectrum after the first half of the system is pulse quenched (fully quenched). All figures are plotted on the same scale. We also note that for sub-panel C, due to numerical instability only 500 time units of the exact and HF-GKBA trajectory were used to create the spectrum.}
\label{fig:freq_spectrum}
\end{figure*}
\FloatBarrier
\noindent
Due to the failure of HF-GKBA in the onsite model for $U=1.0$ we only use the first 500 time units to generate the frequency spectrum.
However, for the remaining results, we used the entire trajectory from the time after the quench. We found the frequency spectrum to be a more reliable measure of quality than the residual between HF-GKBA and the exact result. In particular, in the long-range model, the primary error that arose was a phase mismatch between the HF-GKBA and exact. In this case, the residual provides a misleading measure of the performance of HF-GKBA. In figure \ref{fig:freq_spectrum} we only show the frequency range $0.40J$ to $0.85 J$ as this range held the major spectrum peaks. The full spectrum is included in figure 2 of the SI.
For the onsite model, we see quite good agreement in the peak positions between the exact and HF-GKBA results. Clearly, the HF-GKBA consistently underestimates the magnitude of the spectrum peaks, and for the full quench the HF-GKBA incorrectly identifies the maximum frequency peak. As we go from $U = 0.5J$ to $U = 1.0J$ we see a broadening of the peaks both in exact and HF-GKBA results, which is partially related to the shorter trajectory used to create the spectrum. The HF-GKBA does however overestimate the broadening and even leads to the formation of additional peaks in the spectrum.
The HF-GKBA continues to capture the peak positions well in the long-range model. Furthermore, we now observe that the amplitudes of each frequency peak match the exact result far better than in the onsite model. Going to $U = 1.0J$ the amplitudes and peak positions continue to be matched very well. We point out there is a slight shift in the HF-GKBA peaks $U=1.0J$, this causes the dipole trajectories to move slowly in and out of phase with one another over the time evolution.
From these results, in the scenarios we have studied so far, we see a clear improvement of the HF-GKBA upon the inclusion of exponentially decaying interactions in the model. The improvement holds over relatively long times leading to a spectrum that matches the exact spectrum almost perfectly. We also note a similar improvement in the case of a $\frac{1}{r}$ decay, which is shown in figure 6 of the supplementary for a 4-site model.
In the following section, we will look at results obtained from applying DMD to fit the HF-GKBA Green's function.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{DMD_data.png}
\caption{DMD Trajectories(dashed-lines) for long-range models with 8 and 16 sites for different sizes of the fitting window with a pulse quench on half the sites compared to HF-GKBA(solid-orange). Sub-panels A and B show the dipole between 700-750 time units for HF-GKBA and the DMD reconstruction. Sub-panels C and D show the residual between DMD and HF-GKBA dipole for each size fitting window.}
\label{fig:DMD_fit}
\end{figure*}
\subsection{DMD Extrapolation of HF-GKBA Trajectories}\label{DMD_results}
In the previous section, we observed excellent agreement between the HF-GKBA and the exact time evolution for the long-range model. This motivates the use of DMD in conjunction with HF-GKBA to extrapolate long time trajectories from partial trajectories of HF-GKBA Green's functions. In this section, we present results investigating the effectiveness of DMD in predicting the dynamics of the HF-GKBA. We use the DMD procedure outlined in section \ref{sub_sect:DMD} and apply it to various portions of the total Green's function trajectory.
Relative to the long-range model we found for the onsite model a much larger portion of the trajectory was needed to produce reasonable results, and for most cases, DMD did not produce a successful reconstruction of the Green's function trajectory. Because of the poor performance of DMD as well as the poor performance of the HF-GKBA for the onsite model we omit these results and instead discuss possible reasons for the failure in section \ref{sec:discussion}.
In Figure \ref{fig:DMD_fit} we show results for the DMD extrapolation of $G^<(t)$ for the long-range model with 8 and 16 sites for $U = 1.0J$ and a pulse quench on half of the sites. For the smaller models we tested similar behaviour was found, so we omit these results. Panels A and B of Figure \ref{fig:DMD_fit} show the dipole for 4 different sized fitting windows. For each window, DMD is used to construct a reduced-order model for the total Green's function. The reduced order model is then used to extrapolate beyond the fitting window. We again choose to report the dipole of the system for the same reasons listed in section \ref{onsite_vs_lr}.
The DMD dipole is generated by using snapshots of the HF-GKBA Green's function to construct the DMD extrapolation model. This extrapolated result is then used to calculate the dipole. In the bottom two panels of figure \ref{DMD_results} the residuals between the DMD extrapolated dipole and the HF-GKBA dipole over the full trajectory for each size of the fitting window are shown.
We found the ability of DMD to reconstruct the Green's function was captured well by comparing the DMD dipole to the HF-GKBA dipole. Similar plots to that shown in figure \ref{fig:DMD_fit} are shown in figures 3,4 and 5 of the SI for a selection of components of the Green's function. As expected, we see the residual between the DMD extrapolated dipole and the HF-GKBA dipole decrease as the fitting window is increased. These figures suggest that for the 8-site long-range model, somewhere between 4\% and 8\% of the total trajectory is needed to have a good fit of $G^{<}(t)$, and this fraction increases to between 8\% and 16\% when we go to the 16 site model. This is likely due to more low-frequency modes being present in the 16-site model, which requires a longer fitting window for DMD. In section \ref{sec:discussion} the implications and prospects of DMD will be discussed.
\section{Discussion and Conclusions}\label{sec:discussion}
Returning to the results comparing the exact and HF-GKBA propagation, we first discuss the significant improvement of HF-GKBA upon going to the long-range model. We suggest two possible contributions to this observation. Firstly, we note the magnitude of oscillations is smaller for the extended model. We explain this by noting the higher degree of repulsive couplings between sites increases the localization of individual particles. This may in turn lead to dynamics that are easier to capture with HF-GKBA. A second possibility comes from the self-energy approximation used in these calculations. In systems where screening is important the $GW$ self-energy becomes the dominant contribution to the full self-energy and describes well the many-body interactions between particles. In the onsite model due to the completely local interactions the amount of screening will be quite small and so the $GW$ self-energy will not capture the physics well. However, for a more realistic setup with long-range interactions (encountered in most materials to which GW is meaningfully applied), we see that the HF-GKBA behavior is significantly closer to the ED results. We also see a similar behaviour for the case of a long-range interaction with $\frac{1}{r}$ decay, see figure 6 in the SI. We will investigate further the limitations of the various self-energy formulations, as recent works proposed a route to construct reliable higher order (i.e., beyond GW) schemes that help with the description of excited states in equilibrium\cite{PhysRevB.106.165129}.
We believe the poor performance of DMD in onsite model can be explained at least partially if we look at the full frequency spectrum for the dipole shown in figure 2 of the SI. Clearly, the onsite model has a much larger low-frequency component than the long-range models. Lower frequencies are more difficult to capture using DMD since a longer portion of the trajectory needs to be sampled to observe these long time modes. We believe this to be part of the reason why DMD tends to fail in the onsite model. We also point this out as one of the limitations of DMD, since for systems with very low-frequency modes DMD will have to be performed from a very large portion of the trajectory. Similarly, in the case of the 8 and 16 site models a downward shift in the low-frequency part of the spectrum going from $N_s=8$ to $N_s = 16$, which at least partially explains why a larger amount of the trajectory was required by DMD
We found that for the long-time trajectories prepared in this work, several exhibited numerical instabilities after several hundred time units. These errors typically arose for $U \geq 1.0$ and became more prominent as we moved to larger systems. We comment on two possible sources for these errors. The first possible way these errors arise is described in detail in \cite{PhysRevB.105.165155} and is related to certain consistency relations for the two-particle Green's function breaking down at long times and strong couplings. We point out that the procedure for enforcing the consistency relations function is extremely expensive as it requires the diagonalization of the two-particle Green's function throughout the time-stepping procedure, which scales as $O(N^6)$. If not corrected, these inconsistencies can lead to divergences of the HF-GKBA solution. The second possible source of error may arise from the adiabatic switching procedure. It is possible that for some of the parameters and models we tested small residual errors from the adiabatic switching preparation built up and contributed to the failures of the HF-GKBA time propagation.
These issues offer another opportunity for the use of DMD to assist in the propagation of NEGFs. If the trajectory fails after a sufficiently long time and one can clearly identify the point of failure, then DMD can be used to fit $G^{<}(t)$ on the portion of the trajectory before the solution breaks down. The DMD fitted result can be propagated in place of explicit propagation of equations \eqref{G1-G2}.
In this paper, we have presented results demonstrating a vast improvement of the HF-GKBA when long-range interactions are included. We also observed that DMD is a suitable tool for the reconstruction of long-time trajectories of the HF-GKBA Despite this we still believe DMD can be a powerful tool to be used alongside HF-GKBA and the G1-G2 scheme, especially in the long time propagation of Green's functions in large systems. In future work, we will continue to explore DMD as a way of preparing trajectories for large-scale systems. We will also investigate generalising existing stochastic techniques used in equilibrium systems\cite{PhysRevB.98.075107,doi:10.1021/acs.jctc.7b00770,PhysRevLett.113.076402} to non-equilibrium. Combining stochastic approaches in conjunction with DMD we hope push HF-GKBA to explore the physics of large multi-band systems.
\section*{Acknowledgements}
This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research and Office of Basic Energy Sciences, Scientific Discovery through Advanced Computing (SciDAC) program under Award Number DE-SC0022198
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\section{\label{}}
\section{Introduction}
Since the discovery of high-transition temperature ($T_c$) superconductivity in LaFeAsO$_{1-x}$F$_x$ at $T_c$ = 26 K\cite{Kamihara}, iron pnictides have become one of the most fascinating research areas in condensed-matter physics. The electron-doping suppresses structural and magnetic phase transitions in undoped ReFeAsO and superconductivity appears near the border of magnetism \cite{Kamihara} as seen in the high-$T_c$ cuprates. After the discovery of superconductivity in ReFeAsO, the high-$T_c$ superconductivity has also been found in a ThCr$_2$Si$_2$-type structure, BaFe$_2$As$_2$ by replacing Ba by K as hole doping\cite{Rotter}. One of the most remarkable features in these iron-pnictide superconductors is their superconducting gap structure. Previous nuclear-magnetic resonance (NMR) and nuclear-quadrupole resonance (NQR) measurements on ReFeAsOF and Ba$_{0.7}$K$_{0.3}$Fe$_2$As$_2$ consistently found multiple gap superconductivity\cite{MatanoPr,KawasakiLa,MatanoBa122}. These observations were also confirmed by angle-resolved photoemission spectroscopy. \cite{Ding} The multiple gap feature is believed to be relevant to their multiple electronic band structure.\cite{Singh}
On the other hand, the mechanism of the Cooper pair formation in iron pnictide is still unclear. Since the superconductivity in iron-pnictides is induced by chemical doping, systematic investigation of the relationship between electron correlations and superconductivity has been difficult.
Recent discoveries of pressure-induced superconductivity in RFe$_2$As$_2$ (R = Ca, Sr, and Ba) provide a new route to investigate superconductivity in iron pnictide\cite{Lonzarich,Canfield,Thompson,Kotegawa}. In RFe$_2$As$_2$, the parent compounds also show structural transition from a tetragonal (tetra.) to orthorhombic (orth.) structure with antiferromagnetic order\cite{KitagawaBa,KitagawaSr,CurroCa}. In BaFe$_2$As$_2$ and SrFe$_2$As$_2$, the structural phase transition and antiferromagnetic orders are both suppressed by pressure, and superconductivity was found around the critical pressure, $P_c$ = 40 - 60 kbar, with $T_c$ $\sim$ 30 K.\cite{Lonzarich,Kotegawa} On the other hand, pressure-induced superconductivity in CaFe$_2$As$_2$ has been observed with lower $P_c$ $\sim$ 5 kbar and lower $T_c(P)$ $\sim$ 10K.\cite{Canfield,Thompson} The most strikingly different feature in CaFe$_2$As$_2$ is the occurrence of another structural transition under pressure. Above $P$ $\sim$ 5 kbar, normal state tetra. phase changes to a collapsed tetragonal (c-tetra.) phase with drastic reduction in both the unit cell volume (5\%) and the $c/a$ ratio (11\%).\cite{Kreyssig,Goldman} Notably, when tetra. phase collapses, superconductivity disappears.\cite{Yu} Since these structural phase transitions are sensitive to external pressure, the detailed information about pressure-induced superconductivity in CaFe$_2$As$_2$ is still unknown.
In this paper, we report results of zero-field (ZF) NMR and NQR study in CaFe$_2$As$_2$. At $P$ = 4.7 and 10.8 kbar, pressure-induced superconductivity in the tetra. phase is confirmed by ac-susceptibility and nuclear-spin lattice relaxation time ($T_1$) measurements. The temperature dependences of 1/$T_1$ show no coherence peak just below $T_c(P)$. Below $T_c(P)$, the temperature dependences of $1/T_1$ indicate the unconventional nature of pressure-induced superconductivity in CaFe$_2$As$_2$. The systematic measurements indicate electron correlations play a vital role in inducing unconventional superconductivity in this compound.
\section{Experimental procedures}
The single crystals of CaFe$_2$As$_2$ are grown by a self-flux method and crushed into coarse powder for $^{75}$As ($I$ = 3/2, $\gamma$ = 7.292 MHz/T) ZF-NMR/NQR measurements under pressure.
The pressure was applied by utilizing a NiCrAl/BeCu piston-cylinder type cell filled with Daphne 7373 as the pressure-transmitting medium\cite{Murata}. The pressure at low temperatures was determined from the pressure dependence of the $T_{c}$ values of Sn metal measured by a conventional four-terminal method. The temperature dependence of ac-susceptibility is measured using an $in$-$situ$ NMR/NQR coil. The ZF-NMR/NQR spectra were taken by changing rf frequency and recording the spin echo intensity step by step. The value of $T_1$ was extracted by fitting the nuclear magnetization obtained by recording the spin echo intensity after the saturation pulse.
\begin{figure}[h]
\begin{center}
\includegraphics[width=6cm]{fig1.eps
\caption{\label{fig:t1t} Schematic phase diagram for structural phase transitions of CaFe$_2$As$_2$ under pressure (see text). }
\end{center}
\end{figure}
Figure 1 shows the schematic phase diagram of CaFe$_2$As$_2$ under pressure taken from the literature\cite{Kreyssig,Goldman,Yu,Baek}. Dashed lines indicate the first-order structural phase transitions, respectively. Arrows indicate the pressure at which the present experiments have been performed.
\section{Pressure dependence of zero-field NMR and NQR spectra}
\begin{figure}[!]
\begin{center}
\includegraphics[width=8.5cm]{fig2.eps
\caption{\label{fig:t1t} (Color online) (a) Pressure dependence of $^{75}$As-NMR/NQR spectra for CaFe$_2$As$_2$ measured at $T$ = 5 K and $H$ = 0. Solid arrows indicate the $^{75}$As ZF-NMR spectrum which comes from the orth. phase below $T_N$. Dotted and dashed arrows indicate NQR spectra at the tetra. and c-tetra. phases, respectively. Typical data sets of nuclear recovery curves measured at (b) orth. phase (ZF-NMR), (c) tetra. phase (NQR), and (d) c-tetra. phase (NQR), respectively. Solid curves are theoretical fittings to obtain $T_1$. (see text) }
\end{center}
\end{figure}
Figures 2 (a) shows the pressure dependence of ZF-NMR and NQR spectrum measured at $T$ = 5 K and $P$ = 0, 4.7, and 10.8 kbar, respectively. At $P$ = 0, three $^{75}$As-NMR lines are observed due to the internal magnetic field ($H_{int}$) induced by the Fe ordered moment below $T_N$, which comes from the orth. phase. Actually, as seen in Figs. 2 (b), the nuclear magnetization recovery curve measured at 19 MHz is well fitted by the theoretical curve for NMR ($-1/2\leftrightarrow +1/2$ transition) which is given by 1-$M(t)$/$M_0$ = 0.1$\exp$(-$t/T_1$)+0.9$\exp$(-6$t/T_1$), where $M_0$ and $M(t)$ are the nuclear magnetization at the thermal equilibrium and at a time $t$ after saturating pulse, respectively. Assuming that both $H_{int}$ at the As site and $\nu_Q$ for the As nuclei are along the $c$-axis direction, $H_{int}$ = 2.6 T and $\nu_Q$ = 12 MHz are obtained. Here, nuclear spin Hamiltonian is given as $\mathcal{H}_{AFM}=-\gamma\hbar\vec{I}\cdot\vec{H}_{int} + (h \nu_{Q}/6)[3{I_z}^2-I(I+1)]$. These parameters are in good agreement with previous As-NMR experiments on single crystalline CaFe$_2$As$_2$\cite{CurroCa}.
While the ground state of CaFe$_2$As$_2$ at ambient pressure is in a single orth. phase, a phase separation is observed under pressure due to the first-order transition\cite{CurroCa,Goldman} and pressure distribution\cite{Yu}. As seen in Figs.2(a), at $P$ = 4.7 kbar, a phase separation between orth. and tetra. phases is observed as the ground state. The peak around 18 MHz is due to the central transition ($-1/2\leftrightarrow +1/2$ transition) for ZF-NMR of the orth. phase as observed at $P$ = 0. However, the satellite peaks which are clearly observed at ambient pressure, due to nuclear quadrupole interaction is not observed, indicating an increase of $\nu_Q$ for the orth. phase under pressure. On the other hand, another peak appears around 24 MHz. Since the nuclear magnetization recovery curve measured at 24 MHz is well fitted by the nuclear magnetization recovery curve for $^{75}$As-NQR ($\pm 1/2\leftrightarrow \pm 3/2$ transition) given by the single exponential 1-$M(t)$/$M_0$ = $\exp$(-3$t/T_1$), as seen in Figs. 2 (c), we assigned this peak as coming from the tetra. phase which survives due to a pressure distribution. We have also confirmed this assignment by measuring As-NMR spectrum at $P$ = 5.0 kbar (not shown). Notably, it has been reported that the structural transition from tetra. to orth. under pressure is accompanied by a phase separation in a certain temperature range until a single otrh. phase is established as the magnetic ground state\cite{Goldman,Baek}. Since the present experiment is performed using coarse powdered single crystals, the local pressure distribution may cause the tetra. phase to coexist with the orth. phase even at the ground state.
As pressure reaches $P$ = 10.8 kbar another structural transition from tetra. to c-tetra. occurs\cite{Kreyssig,Goldman}, the NMR signal around 18 MHz and the NQR signal from tetra. around 25 MHz are still observed. In addition, a new peak appears around 30.4 MHz. As seen in Figs.2(d), since the nuclear magnetization recovery curve at 30.4 MHz indicates this new peak is also from NQR, we assigned that this peak as coming from the c-tetra. phase. Due to a pressure distribution, phase separation among orth., tetra., and c-tetra. is realized at $P$ = 10.8 kbar. Notably, an As-NQR frequency $\nu_Q$ probes the electric-field gradient (EFG) generated by the charge distribution surrounding the As site. The larger $\nu_Q$ for c-tetra. than tetra. phase is reasonable since the unit cell volume collapses in c-tetra. However, these pressure and structural dependences of $\nu_Q$ do not scale with the known unit cell volume for each phases\cite{Kreyssig}. This may be because the local charge distribution around the As site also contributes to EFG in addition to the lattice contribution. Since the NMR/NQR peaks are clearly separated even at $P$ = 10.8 kbar and the recovery curves measured at each phases are of a single $T_1$ component, we suggest that these phase separations under pressure are induced by a local pressure distribution and not by the sample inhomogeneity. Such a pressure distribution allowed us to investigate the pressure dependence of the electronic properties in each phase of CaFe$_2$As$_2$.
From the NMR/NQR spectra, the volume fraction of orth.:tetra.=54\%:46\% and orth.:tetra.:c-tetra.=45\%:18\%:37\% are estimated for $P$ = 4.7 and 10.8 kbar, respectively. It is clear that the effect of pressure distribution on the evolution of the ground states in CaFe$_2$As$_2$ is larger than a previous NMR study under pressure using a large single crystal\cite{Baek}.
\section{Pressure-induced superconductivity in CaFe$_2$As$_2$}
Figure 3 shows temperature dependence of ac-susceptibility measured using the $in$-$situ$ NMR/NQR coil. The pressure-induced superconducting transitions at $T_c(P)$ = 3.9 and 4.1 K at $P$ = 4.7 and 10.8 kbar are clearly observed. Although the $T_c(P)$s are relatively lower, the superconducting transitions are much sharper than previous reports\cite{Baek,Hanoh}.
\begin{figure}[h]
\begin{center}
\includegraphics[width=7cm]{fig3.eps
\caption{\label{fig:t1t} (Color online) Temperature dependence of ac-susceptibility measured using $in$-$situ$ NMR/NQR coil at $P$ = 4.7 and 10.8 kbar. Arrows indicate $T_c(P)$. }
\end{center}
\end{figure}
\section{Evolution of electronic property in CaFe$_2$As$_2$ under pressure}
To investigate the evolution of the electronic properties in CaFe$_2$As$_2$ under pressure, we measured the $^{75}$As nuclear spin lattice relaxation time ($T_1$) at each phase. Figs. 4 shows the temperature dependence of $1/T_1$ divided by temperature ($1/T_1T$) well below the structural transitions. In the orth. phases all of the data show a $1/T_1T$ = constant behavior, which is characteristic of a Fermi-liquid state. These results are consistent with previous NMR results on (Ba, Sr)Fe$_2$As$_2$\cite{KitagawaBa,KitagawaSr}, indicating that a small Fermi surface remains below antiferromagnetic order. Due to the large $H_{int}$ $\sim$ 2.5T in the orth. phase, the coexistence of antiferromagnetism and superconductivity, which has frequently been observed in heavy-fermion compounds\cite{Kitaoka}, could not be confirmed.
As discussed later, pressure-induced superconductivity is clearly observed as a reduction of $1/T_1T$ for the tetra. phases at which the onset of diamagnetism is observed at $P$ = 4.7 and 10.8 kbar, respectively.
In the normal state in the tetra. phases, $1/T_1T$ increases with decreasing temperature, which indicates that the antiferromagnetic correlation develops down to $T_c(P)$. To analyze the temperature dependence of $1/T_1T$ above $T_c(P)$, we employed the model for a weakly antiferromagnetically correlated metal, 1/$T_1T$ = const. + $C/(T+\theta)$ \cite{Moriya}. Here, the first term describes the contribution from the density of states (DOS) at the Fermi level, and the second term describes the contribution from the antiferromagnetic wave vector $Q$. As shown by the solid curves in Fig. 4, the temperature dependences of 1/$T_1T$ for tetra. phases are well fitted by this model; 1/$T_1T$ = 0.48 + $4.4/(T+\theta)$ with $\theta$ = 5.4$\pm$2.3 K for $P$ = 4.7 kbar and 1/$T_1T$ = 0.57 + $5.2/(T+\theta)$ with $\theta$ = 6.0$\pm$1.1 K for $P$ = 10.8 kbar, respectively. Surprisingly, the values of $\theta$, which is a measure of the distance to an antiferromagnetic quantum critical point (QCP), are not only one order of magnitude smaller than $\theta$ = 39 K observed in LaFeAsO$_{0.92}$F$_{0.08}$ ($T_c$ = 23 K)\cite{KawasakiLa}, but also comparable to that observed in unconventional superconductors in strongly correlated electron systems\cite{ZhengIr,KawasakiCeRhIrIn5,Kusano}. This indicates that superconductivity in CaFe$_2$As$_2$ is induced near an antiferromagnetic QCP. Since the value of $\theta$ is insensitive to pressure, the present results indicate that the quantum criticality in the tetra. phase is robust against pressure. This may be the reason why a robust superconducting dome was observed under pressure\cite{Canfield,Thompson}. Such a situation is somewhat different from heavy fermion superconductivity around QCP, at which both $T_c$ and electron correlations are enhanced.\cite{KawasakiCeRhIrIn5}
On the other hand, both the DOS at the Fermi level and the value of C in the antiferromagnetic correlation slightly increase with increasing pressure. The small increase of $T_c$, from 3.9 K at 4.7 kbar to 4.1 K at 10.8 kbar, may be due to this small increase of both the DOS at the Fermi level and the antiferromagnetic correlations. To describe the detailed relationship between QCP and superconductivity in CaFe$_2$As$_2$, further systematic measurements under pressure are in progress.
\begin{figure}[h]
\begin{center}
\includegraphics[width=7.5cm]{fig4.eps
\caption{\label{fig:t1t} (Color online) Temperature dependence of $1/T_1T$ below $T$ = 40 K in each phase at (a) $P$ = 0 and 4.7 kbar and (b) 10.8 kbar. Solid arrows indicate $T_c(P)$. Dotted lines indicates the relation of $1/T_1T$ = constant. The solid curves indicate relation, 1/$T_1T$ = const. + $C/(T+\theta)$. (see text) }
\end{center}
\end{figure}
The most important result is the difference of $1/T_1T$ between the tetra. and c-tetra. phases at $P$ = 10.8 kbar. As seen in Figs. 4(b), $1/T_1T$ = constant behavior is established even below $T_c$, indicating that the electron correlation and also superconductivity disappear in the c-tetra. phase. Importantly, recent electronic band structure calculations for CaFe$_2$As$_2$ have shown that the tetra. phase has the multiple band structure seen in other iron-pnictides\cite{Ding}, whereas the multiband nature along the $\Gamma$-M direction vanishes when it collapses.\cite{LonzarichBandcalc,Yildirim} It is thus suggested that the candidate for the antiferromagnetic wave vector $Q$ observed in the tetra. phases and the driving force of the Cooper pair formation in CaFe$_2$As$_2$ is the interband correlations, which has been suggested as the origin for the spin-density-wave order in the LaFeAsOF superconductor.\cite{Ma}
\section{Novel superconductivity in CaFe$_2$As$_2$}
\begin{figure}[h]
\begin{center}
\includegraphics[width=7.5cm]{fig5.eps
\caption{\label{fig:t1t} (Color online) Plot of $T_1(T)$$^{-1}$/$T_1(T_c)$$^{-1}$ versus $T/T_c$$(P)$ for $P$ = 4.7 and 10.8 kbar. The solid curve is a two-gap fit assuming a $d$-wave symmetry with parameters, $\Delta_1(0) = 3.9 k_{B}T_{c}$, $\Delta_2(0) = 1.7 k_{B}T_{c}$, and $\alpha = 0.65$ (see text). The dotted and dashed lines indicate the relations of $1/T_1$ $\propto$ $T$ and $1/T_1$ $\propto$ $T^3$, respectively. Inset shows plot of ($T_1T(T)$)$^{-1}$/($T_1T(T_c)$)$^{-1}$ versus $T/T_c$$(P)$ for 10.8 kbar. Solid line is eye-guide.}
\end{center}
\end{figure}
To focus on the superconducting gap structure for CaFe$_2$As$_2$, the plots of $T_1(T)$$^{-1}$/$T_1(T_c)$$^{-1}$ versus $T/T_c$$(P)$ are shown in Fig. 5. Here, the relaxation rate below $T_{c}$ ($1/T_{1s}$) can be expressed as, $\frac{T_{1N}}{T_{1s}}= \frac{2}{k_BT} \int \int N_s(E)N_s(E')f(E) \left[1-f(E') \right] \delta(E-E')dEdE'$. Where $Ns=\frac{E}{\sqrt{E^2- \Delta^2}}$ is the DOS in the superconducting state, and $f(E)$ is the Fermi distribution function. The coherence peak just below $T_c(P)$ is absent at both pressures. At $P$ = 4.7 kbar, $1/T_1$ decreases moderately and is saturated approaching $T$ = 0. This means that there is a residual density of states in the superconducting gap. Since the present experiments are performed in zero magnetic field, it is clear evidence for the occurrence of gapless superconductivity.
On the other hand, at $P$ = 10.8 kbar, $1/T_1$ continues to decrease steeply below $T_c$, as observed in other iron pnictide superconductors\cite{MatanoPr,KawasakiLa,MatanoBa122,Nakai,Mukuda,Grafe,KotegawaFeSe,Fukazawa,Kobayashi}. Notably, as clearly seen in the inset to Fig.5, $1/T_1T$ below $T_c$ has a hump structure around $T$ $\sim$ 0.5 $T_c$, which is a signature for multiple gap superconductivity, as observed in other pnictide superconductors\cite{MatanoPr,KawasakiLa,MatanoBa122}. By assuming two gaps of $d$-wave symmetry $\Delta(\phi) = \Delta_0 \cos(2\phi)$ with a mean-field temperature dependence with $\Delta(\phi)=\alpha \Delta_1+ (1-\alpha) \Delta_2$ and $\alpha = \frac{N_{s,1}}{N_{s,1}+N_{s,2}}$, we find that the $\Delta_1(0) = 3.9 k_{B}T_{c}$, $\Delta_2(0) = 1.7 k_{B}T_{c}$ and $\alpha = 0.65$ can fit the data reasonably well, as shown by the solid curve in Fig.5. These values of superconducting gaps and $\alpha$ are comparable to other iron pnictide superconductors\cite{MatanoPr,KawasakiLa,MatanoBa122}.
How can we understand this difference of the superconducting gap structure between $P$ = 4.7 and 10.8 kbar?
One possible scenario is the mechanism predicted in heavy fermion superconductivity around the antiferromagnetic QCP at which the gapless superconductivity has been observed\cite{YuKawasaki,Kawasaki115,Yamaguchi}. When the system locates at the vicinity of the antiferromagnetic QCP, odd-frequency $p$-wave spin singlet superconductivity (pSS) prevails over the $d$-wave singlet superconductivity (dSS)\cite{Fuseya}. Notably, for the pSS state, it is suggested that there is no gap in the quasiparticle spectrum anywhere on the Fermi surface due to its odd frequency, thus, gapless superconductivity is realized\cite{Fuseya}. In the present case, the values of $\theta$, which is the measure of closeness to the QCP are very small and comparable to the value of heavy fermion compounds around QCP.\cite{ZhengIr,KawasakiCeRhIrIn5} In this model, gapless pSS and dSS compete near a QCP.\cite{Fuseya} In addition, it would be difficult to realize gapless pSS when it competes against full-gap superconductivity, such as $\pm$$s$-wave pairing\cite{Fuseya2}. Thus, $d$-wave pairing is favored as the competing order against gapless pSS near a QCP\cite{Fuseya2}. In fact, it has been predicted that $d$-wave superconductivity can also be the candidate for iron-pnictide superconductivity, although $\pm$$s$-wave pairing has been suggested in other iron pnictide superconductors\cite{Kuroki}. The present results suggest that the pressure-induced superconductivity near a QCP in CaFe$_2$As$_2$ is a good candidate to investigate a variety of superconductivities in iron pnictides.
\section{Concluding remarks}
In conclusion, we report zero-field NMR/NQR experiments on the iron-pnictide pressure-induced superconductor CaFe$_2$As$_2$. Systematic measurements have revealed the evolution of the ground states under pressure and the electron correlations play a vital role in the formation of Cooper pairs in this compound. It is suggested that the electron correlation is induced by an interband correlation originating from its multiple bands structure. We found that gapless superconductivity is realized in the close vicinity of the antiferromagnetic quantum critical point. We believe that it is due to its closeness to a quantum critical point. The present results suggest a close relationship between antiferromagnetism and superconductivity in iron pnictides.\\
\section*{Acknowledgement}
S. K. thanks Yuki Fuseya for useful discussion and comments. This work was supported by a Grant-in-Aid for Scientific Research on Innovative Areas "Heavy Electrons" (No. 21102514) of The Ministry of Education, Culture, Sports, Science, and Technology, Japan.
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\section{Introduction}
\label{sec:Introduction}
Quantum Chromodynamics (QCD) is the fundamental theory of
the strong interaction among quarks.
Despite its essential roles in the elementary particle physics,
the lack of analytical method to treat its large coupling at
low energy (equivalently at long distance) prevents us from solving
QCD in general.
The lattice QCD, that is formulated on a 4-dimensional Euclidean
lattice, enables us to tackle such problems by numerical simulations
\cite{textbook}.
Quanti ing with the path integral formalism, the theory resembles
a statistical mechanical system to which the Monte Carlo methods
apply.
Typically the most time consuming part of the lattice QCD
simulations is solving a linear equation for a large sparse fermion
matrix that represents the interaction among quarks.
As the lattice size becomes large necessarily to precision
calculation, the numerical cost grows rapidly.
The lattice QCD simulations have been a typical problem
in high performance computing and conducted development of
supercomputers such as QCDPAX \cite{QCDPAX} and QCDOC \cite{QCDOC}.
There are two distinct trends in high performance machines.
One is accelerators, represented by GPUs, which is composed of
many cores of $O(1000)$.
The Intel Xeon Phi architecture is a variant of the other type,
massively parallel clusters, while it has potential to work
as an accelerator.
Knights Landing (KNL), the second generation of Xeon Phi series,
reinforced usability as a massive parallel machine.
Its performance is assured by the SIMD architecture.
Elaborated assignments of vector registers and exploiting
SIMD instructions are essential to achieve desired performance.
In this paper, we port a lattice QCD simulation code to
Intel Xeon Phi Knights Landing.
The aim of this work is not a state-of-the-art tuning on KNL,
but to establish prescriptions that are applicable to wide
part of the application with practical performance.
While a hot spot of the QCD simulation tends to concentrate in
small part of programs, there are plenty amount of code
for measuring various physical quantities that have been accumulated
in legacy codes.
One frequently needs to accelerate such a code on a new architecture.
Thus it is important to establish simple procedures to achieve
acceptable performance as well as implication to future development of code.
For this reason, we restrict ourselves in rather general methods:
change of data layout, application of Intel AVX-512 intrinsics
and prefetching.
As a testbed of our analysis, we choose two types of fermion
matrices that are widely used together with iterative linear
equation solver algorithms.
This paper is organized as follows.
The next section briefly introduce the linear equation system
in lattice QCD simulations with fermion matrices employed
in this work.
Features of KNL architecture are summarized in
Section~\ref{sec:KNL}.
Section~\ref{sec:implementation} describes the details of
our implementation.
In the following sections, we measure the performance of
individual fermion matrices and the whole linear equation solvers.
The last section discusses implication of our results.
\section{Lattice QCD simulation}
\label{sec:lattice_QCD}
\subsection{Structure of lattice QCD simulation}
For the formulation of lattice QCD and the principle of the numerical
simulation, there are many textbooks and reviews \cite{textbook}.
Thus we concentrate on the linear equation system for the fermion
matrix, to which high computational cost is required.
The lattice QCD theory consists of fermion (quark) fields
and a gauge (gluon) field.
The latter mediates interaction among quarks and are represented
by `link variables', $U_\mu(x)\in SU(3)$, where
$x=(x_1, x_2, x_3, x_4)$ stands for a lattice site and
$\mu$=1--4 is the spacetime direction.
In numerical simulations the lattice size is finite,
$x_\mu=1,2,\dots, L_\mu$.
The fermion field is represented as a complex vector on
lattice sites,
which carries 3 components of `color' and 4 components of `spinor',
thus in total 12,
degrees of freedom on each site.
The dynamics of fermion is governed by a functional
$S_F=\sum_{x,y} \psi^\dag(x) D[U]^{-1}(x,y) \psi(y)$,
where $D[U]$ is a fermion matrix.
A Monte Carlo algorithm is applied to generate an ensemble
of the gauge field $\{U_\mu(x) \}$, that requires to solve
a linear equation $x = D^{-1} \psi$ many times.
\subsection{Fermion matrix}
There is a variety of the fermion operator $D[U]$, since its
requirement is to coincide with that of QCD in the continuum limit,
the lattice spacing $a\rightarrow 0$.
Each formulation has advantages and disadvantages.
As a common feature, the matrix is sparse because of the locality
of the interaction.
In this paper, we examine the following two types of fermion matrices.
\subsubsection{Wilson fermion matrix}
\begin{figure}[!t]
\centering
\includegraphics[width=4.1cm]{Figs/Wilson_matrix.eps}
\vspace{-1mm}
\caption{
The schematic feature of the Wilson fermion matrix.}
\label{fig:Wilson_fermion_matrix}
\end{figure}
The first one called the Wilson fermion matrix has the form
\vspace{-7mm}
{\small
\begin{eqnarray}
D_W(x,y)\!\! &=& \!\!(m_0 + 4) \delta_{x,y} - \frac{1}{2} \sum_{\mu=1}^4
\big[ (1-\gamma_\mu) U_\mu(x) \delta_{x+\hat{\mu},y}
\nonumber \\
& &
\hspace{16mm}
+ (1+\gamma_\mu) U_\mu^\dag(x-\hat{\mu}) \delta_{x-\hat{\mu},y}
\big] ,
\label{eq:Wilson_fermion_operator}
\vspace{-1mm}
\end{eqnarray}
\vspace{-5mm}
}
\noindent
where $x$, $y$ are lattice sites,
$\hat{\mu}$ the unit vector along $\mu$-th axis,
and $m_0$ the quark mass.
Fig.~\ref{fig:Wilson_fermion_matrix} indicates how the interaction
to the neighboring sites are involved in the matrix.
As mentioned above, the link variable $U_\mu(x)$ is
a $3\times 3$ complex matrix acting on color
and $\gamma_\mu$ is a $4\times 4$
matrix acting on the spinor degrees of freedom.
Thus
$D_W$ is a complex matrix of the rank $4 N_c L_x L_y L_z L_t$.
It is standard to impose the periodic or antiperiodic boundary
conditions.
\subsubsection{Domain-wall fermion matrix}
The second fermion matrix we treat is called the domain-wall
fermion.
This matrix $D_{DW}$ is defined by extending the spacetime to
a 5-dimensional lattice.
The structure of $D_{DW}$ in the 5th direction reads
\vspace{-2mm}
{\small
\begin{equation}
D_{DW} = \left(
\begin{array}{ccccc}
D_W+1 & -P_- & 0 &\cdots & m P_+ \\
- P_+ & D_W+1 & - P_- & & 0 \\
0 & -P_+ & D_W+1 &\ddots & \vdots \\
\vdots & & \ddots &\ddots & -P_- \\
m P_- & 0 & \cdots & - P_+ & D_W+1 \\
\end{array}
\right) ,
\label{eq:domainwall_general}
\end{equation}
}
\vspace{-3mm}
\noindent
where $D_W(x,y)$ is the Wilson fermion matrix above
(with $m_0$ set to certain value), and $m$ instead represents
the quark mass.
$P_\pm$ are $4\times 4$ matrices
acting on the spinor components.
The size of the 5th direction, $N_s$, is also a parameter
of $D_{DW}$.
Eq.~(\ref{eq:domainwall_general}) means that $D_{DW}$ is a block
tridiagonal matrix including the boundary components
in the fifth direction.
Note that $U_\mu(x)$ is common in the 5th direction.
The domain-wall fermion is widely used because
of its good theoretical properties despite larger numerical
cost than the Wilson matrix.
There are several possible ways to implement $D_{DW}$.
The simplest way is to repeatedly use the implementation of $D_W$
to a set of 4-dimensional vectors.
Another way is to treat the structure in the 5th direction
as additional internal degrees of freedom, together with the
color and spinor ones.
The latter has an advantage in cache reuse.
We compare the both implementation and found that the latter
achieves better performance, and thus concentrate on it in the following.
\subsubsection{Features of the fermion matrices}
\begin{table}[!t]
\renewcommand{\arraystretch}{1.3}
\caption{
Features of fermion matrices:
The number of floating point operation and the data transfer
between memory and processor per site.
For the domain-wall matrix, the 5th directional size is set as $N_s=8$.}
\label{comparison_of_fermion_matrix}
\centering
\begin{tabular}{ccccc}
\hline
Fermion type & $N_{flop}$/site & data/site [B] (float) & Byte/Flop \\
\hline
Wilson & 1368 & 1536 B & 1.12 \\
Domain-wall & 11520 & 8256 B & 0.72 \\
\hline
\end{tabular}
\end{table}
Although these fermion matrices share the property of locality and
sparseness,
they have different features
in data size transferred between
the memory and the processor cores, number of arithmetic
operations, and data size of communication to other MPI process.
Table~\ref{comparison_of_fermion_matrix} summarizes the former
two values per site for single precision data.
For the domain-wall fermion matrix, these numbers
depend on the size of 5th direction, $N_s$.
Hereafter we adopt $N_s=8$ as a typical example.
As quoted in Table~\ref{comparison_of_fermion_matrix},
the domain-wall matrix tends to have smaller byte-per-flop
value, due to the independence of the link variable $U_\mu(x)$
in the 5th direction.
\subsection{Linear equation solver}
Since the fermion matrices are large and sparse,
iterative solvers based on the Krylov subspace method are
widely used.
For the Wilson fermion matrix, we employ the BiCGStab algorithm
for a nonhermitian matrix.
As for the domain-wall fermion, BiCGStab algorithm does not
work because its eigenvalues scatter also in the left side of
the imaginary axis.
We thus apply the conjugate gradient (CG) algorithm
to a hermitian positive matrix $D_{DW}^\dag D_{DW}$.
In practice a mixed precision solver is applicable.
In this case the single precision solver applied as the inner
solver determines the performance.
Therefore we measure the performance with the single precision.
While there are variety of improvement techniques for
a large-scale linear systems, such as a multi-grid or
domain-decomposition methods,
they are beyond the scope of this paper.
\section{Knights Landing architecture}
\label{sec:KNL}
The Knights Landing is the second generation of Intel Xeon Phi
architecture, whose details are found in \cite{KNLtextbook}.
Its maximal peak performance is 3 and 6 TFlops for double and single
precision, respectively.
It is composed of maximally 72 cores, in units of a
tile containing two cores.
Each tile has distributed L2 cache that is shared with 2 cores.
In addition to DDR4 with about 90 GB/s, MCDRAM of maximally 16 GB
accessible with 400 GB/s is available with one of three modes:
cache, flat, and hybrid.
Each core supports 4-way hyperthreading.
The SIMD instruction AVX-512 is available.
32 vector registers of 512-bit length are assigned to
8 double or 16 single precision numbers.
Our target machine is the Oakforest-PACS system hosted by
Joint Center for Advances High Performance Computing
(JCAHPC, University of Tokyo and University of Tsukuba)
\cite{Oakforest-PACS_website}.
The system is composed of 8208 nodes of Intel Xeon Phi
7250 (68 cores, 1.4 GHz) connected by full-bisection fat tree
network of the Intel Omni-Path interconnect.
It has 25 PFlops of peak performance in total, and started
public service in April 2017.
\section{Implementation}
\label{sec:implementation}
\subsection{Simulation code}
As the base code, we choose the Bridge++ code set
\cite{bridge_website,Ueda:2014zsa}, which is described
in C++ with the object-oriented design.
This code set allows us to replace fermion matrices
and solver algorithms independently.
Porting of Bridge++ to accelerators are performed in
\cite{Motoki:iccs2014}.
In the original Bridge++, hereafter called the
Bridge++ core library, the data is in double precision and
in fixed data layout.
Following the strategy employed in
\cite{Motoki:iccs2014},
we extend the vectors and matrices so as to enable any data
layout and changing the data type.
From the core library, the linear equation solver is offloaded
to the newly developed code.
This enables us to tune the hot spot on the specific architecture
with keeping the remaining part of Bridge++ available.
Following the implementation of Bridge++, we parallelized the
code with MPI and employ OpenMP for multi-threading.
\subsection{Related works}
Since the lattice QCD simulation, in particular the fermion
matrix multiplication, is considered a typical problem of
high performance computing, it is examined in a textbook
of KNL \cite{KNLtextbook}.
Its chapter 26 is devoted to performance tuning of
the Wilson-Dslash operator, corresponding to the Wilson matrix
above, using the QPhiX library \cite{QPhiX}.
On a single KNL with 68 cores at 1.4 GHz, the maximal performance
of the Wilson matrix multiplication achieved 272 and 587
GFlops for double and single precision, respectively.
As for the first generation of Xeon Phi, Knights Corner,
there are several works
\cite{Joo:2013,Li:2014kxa,Heybrock:2014iga,Arts:2015jia,
Boyle:2016lbp,Kobayashi:2016gog,Boku:2016dmw}.
Joo {\it et al.} \cite{Joo:2013} is the direct base of
the QPhiX library on KNL.
Ref.~\cite{Boyle:2016lbp} developed a library named `Grid'
for the SIMD architecture, and has largely affected our work.
These works would be extended to the KNL as well.
\subsection{Implementation}
To fully exploit the SIMD architecture of KNL, rearrangement
of data is inevitably important.
For double and single precision data types,
512-bit register corresponds to 8 and 16 floating point
numbers, respectively, so we rearrange the date in these units.
We implement the code in C++ template classes and instantiate
them for double and float data types individually.
Since the vector data in lattice QCD are complex, there are
several possibilities for the ordering of real and imaginary parts.
Considering the number of SIMD registers and the number of
the degree of freedom on each site, we decide to place
the real and imaginary parts as consecutive data on the memory.
The color and spinor components are distributed to separate
registers.
For float type, data on eight sites
are packed into a single register and processed simultaneously.
There is flexibility how to fold lattice sites into the data
array.
We have tested several reasonable types of site ordering for
the Wilson fermion matrix and found similar performance.
Throughout this paper, we adopt one of them in which the
implementation has been made most progress.
This site ordering is introduced in ~\cite{Boyle:2016lbp}.
Fig.~\ref{fig:GRID_site_ordering} displays how the index
of site is folded into SIMD-vector data.
This indexing does not require shuffling inside the
vector variables during the stencil calculation except for the boundary
sites in a local process, which makes the implementation easy.
It also allows us a flexible choice of local lattice volume.
In our implementation, local lattice sizes in $y$-, $z$-, and
$t$-directions must be even to pack 8 complex numbers into a single
SIMD-vector.
A possible disadvantage is a load imbalance between bulk and
boundary sites.
In most cases, however, this is hidden in the larger
imbalance due
to packing and unpacking of the data for MPI communication.
\begin{figure}[!t]
\centering
\includegraphics[width=8.0cm]{Figs/simd_lattice2.eps}
\caption{
The site index ordering. For float type complex variables, we use 3 dimensional
analogues of this figure.}
\label{fig:GRID_site_ordering}
\end{figure}
\subsection{Tuning procedure}
\subsubsection{Data alignment}
For a better performance with SIMD-vector data,
the data must be aligned by 64 bytes ({\it i.e.} 512 bits)
in AVX512 architecture.
To allocate the data, we use \texttt{std::vector} in the standard
C++ template library with providing an aligned allocator.
This allocation is done when only the master thread is
working.
The use of \texttt{std::vector} ensures that the data are
allocated on contiguous area of the memory.
For the KNL architecture, this may cause imbalance of
affinity between cores and memory and decrease of performance
in particular with large number of cores per process.
We nonetheless adopt this setup because of the least modification
of previously developed codes.
\subsubsection{Using Intrinsics}
The arithmetics for the SIMD-vector variables are implemented
with intrinsics.
For example, the following code is an implementation of
complex multiplication $a*b$ using AVX-512 intrinsics.
{ \small
\begin{lstlisting}
// a, b are vector variables
__m512 a_r=_mm512_moveldup_ps(a);
__m512 a_i=_mm512_movehdup_ps(a);
__m512 bt =_mm512_permute_ps(b,0xB1);
a_i=_mm512_mul_ps(a_i, bt);
__m512 c=_mm512_fmaddsub_ps(a_r,b,a_i);
\end{lstlisting}
}
\noindent
One can incorporate such codes with intrinsics by making use of
inline functions, C++ templates, or preprocessor macros.
We instead make use of \texttt{simd} directory in the `Grid'
library \cite{Boyle:2016lbp}, where a wrapper to the vector variable
(\texttt{\_\_m512}) and complex arithmetics are defined by using the intrinsics.
Using a template class defined there, operation $a*b$ can be written
such as \texttt{vComplexF c=a*b}.
\subsubsection{Prefetching}
We compare the manual prefetch and the automated prefetch by
compiler.
The most outer loop of the matrix is in the site index.
At each site, we accumulate 8 stencil contributions, from $+x$,
$-x$,...,$-t$ directions in order.
The prefetch to L1 and L2 cache is inserted
1 and 3 steps before the computation, respectively.
That is, before accumulating a $(+x)$-contribution, data
needed for $(-x)$-contribution is prefetched to the L1
cache and $(-y)$-contribution is to the L2 cache.
We use \texttt{\_mm\_prefetch} with
\texttt{\_MM\_HINT\_T0} and \texttt{\_MM\_HINT\_T1}
to generate the prefetch order.
The following is a pseudo code to show the prefetch insertions:
{\small
\begin{lstlisting}
for(s=0; s<num_of_sites; s++){
#pragma noprefetch
// +x
prefetch_to_L1(-x);
prefetch_to_L2(-y);
accumlate_from(+x);
// -x
prefetch_to_L1(+y);
prefetch_to_L2(+z);
accumlate_from(-x);
...
}
\end{lstlisting}
}
\noindent
It is not straightforward to insert prefetch commands
appropriately.
One needs to tune the variables and the place to insert
referring to a profiler, {\it e.g.} Intel Vtune amplifier.
The performance may sensitive to the problem size,
choice of parameters such as the number of threads,
and so on.
\subsubsection{Thread task assignment}
Since the lattice extends over the machine nodes,
the matrix and the reduction of vectors require communication
among nodes.
The function of matrix processes the following steps in order:
(1) Packing of the boundary data for communication,
(2-a) Doing communication,
(2-b) Operations of the bulk part, and
(3) Unpacking the boundary data.
(2-a) and (2-b) can be overlapped, and how efficient is this
is important for the total performance.
We restrict ourselves in the case that only the master thread
performs the communication, {\it i.e.} corresponding to
{\tt MPI\_THREAD\_FUNNELED}.
For the implementation of the steps (2-a) and (2-b) above,
there are two possibilities:
(i) arithmetic operational tasks are equally assigned to all
the available threads, and
(ii) the master thread concentrates the communication and
other threads bear the arithmetic tasks.
In this work, we adopt the case (ii).
\section{Performance of Wilson fermion matrix}
\subsection{Machine environment}
The performance is measured on the Oakforest-PACS
system.
We use the Intel compiler of version 17.0.4 with
options {\tt -O3 -ipo -no-prec-div -xMIC-AVX512}.
On execution, we use job classes with the cache mode of MCDRAM.
According to the tuning-guide provided by JCAHPC,
we adopt the following setup.
To avoid OS jitter, the 0th and 1st cores on each KNL card
are not assigned for execution.
{KMP\_AFFINITY=compact} is set if more than 1 thread is assigned
to a core (unset for 1 thread/core).
\subsection{Wilson fermion matrix}
\subsubsection{Task assignment to threads}
We start with the Wilson fermion matrix.
We first compare the performance for combinations
of a number of cores per MPI process and a number of threads
per core.
For the former,
(1) one core per process,
(2) two cores (one tile) per process, and
(3) 64 cores (whole KNL card) per process.
For the latter,
making use of the hyperthreading, the following three cases
are examined:
(a) one thread per core,
(b) two threads per core, and
(c) four threads per core.
The actual number of threads per MPI process is multiple of
numbers of cores and threads per core.
We compare one, two, and four threads per core cases.
Fig.~\ref{fig:number_of_thread_per_core} shows the results with
1-node and 16-node, generated with the strong scaling on
$32^3\times 64$ lattice.
For the case~(3), copy of the boundaries is enforced so as to
compare with the other cases on the same footing.
For the case~(1) and (2), dependence on the number of threads
per core is not strong.
Including the cases of other numbers of nodes,
there is a tendency
of achieving
the best performance with 2 threads per MPI process for
case~(1) and (2), and 1 thread per core for case~(3).
\subsubsection{Prefetching}
Effect of manual prefetch against the automated prefetch by
compiler is displayed in
Fig.~\ref{fig:Wilson_mult_effect_of_prefetch}.
In the single node case, where no inter-node communication is needed,
the manual prefetch improves the performance by more than 20\%.
Contrary to a statement in \cite{KNLtextbook}, manual prefetching
is effective to our case.
Increasing the number of nodes, however, the effect is gradually
washed and becomes a few percent at 16 nodes as shown in
Fig.~\ref{fig:Wilson_mult_effect_of_prefetch}.
Since our target lattice sizes assumes more than $O(10)$ KNL nodes,
the advantage of manual prefetch is not manifest compared to
involved tuning effort.
For this reason, we do not apply it to the
linear algebraic functions and the domain-wall fermion matrix,
while the following measurement of the Wilson matrix is done
with the tuned code.
Here we summarize the output of Intel Vtune Amplifier for the
Wilson matrix multiplication with a single process of 64 threads.
Applying the manual prefetch, L2 cache hit rate increases
from 76.7\% to 99.1\%.
With the manual prefetch, the UTLB overhead and the page walk
are 0.0\% and 0.2\% of clockticks, respectively.
The metric SIMD Compute-to-L2 Access Ratio reaches as large as 2,023.
For the whole BiCGStab solver examined below, however,
the L2 hit rate decreases to 84.4\%, as explained by the larger
byte-per-flop rate of the solver algorithm.
\begin{figure}[!t]
\centering
\includegraphics[width=7.7cm]{Figs/thread_dep_wilson_nomral_mult.eps}
\caption{
Comparison of numbers of threads per core.}
\label{fig:number_of_thread_per_core}
\end{figure}
\subsubsection{Comparison to other codes}
Now we compare the performance of the Wilson matrix
multiplication to other codes under the condition
of a single process on a single KNL.
As quoted already, QPhiX library achieves 587 GFlops for
single precision \cite{KNLtextbook} on a $32^3\times 96$ lattice.
The GRID library \cite{Boyle:2016lbp} provides a benchmark
of the Wilson matrix that we run on the same environment as
this work.
On $32^3\times 64$ lattice, based on v0.7.0,
it gives the best performance with one thread/core and amounts
to 348.6 GFlops that is comparable to our result.
While our result is not as fast as QPhiX, it shows that
large fraction of performance can be achieved with widely
applicable techniques.
These values are reasonable considering the memory bandwidth
of MCDRAM and the byte-per-flop in
Table~\ref{comparison_of_fermion_matrix}, while far below
the peak performance.
For reference, we also measure the performance of the
original Bridge++, that is implemented in not SIMD-oriented
manner and only in the double precision.
The best performance is obtained with 4 threads/core and
results in 60.0 GFlops, which roughly corresponds to 120 GFlops
in single precision.
This indicates the impact of the SIMD-oriented tuning.
\subsubsection{Scaling property of matrix multiplication}
Now we observe the scaling properties of the Wilson matrix
multiplication.
In the following, we adopt 1, 2, and 2 threads/core for
64, 2, and 1 cores/process, respectively.
The top panel of Fig.~\ref{fig:Wilson_mult_scaling}
shows the weak scaling plot for the $16^3 \times 32$
lattice in each node.
In the measurements, we do not enforce the copy of the boundary data
unless it is really needed.
For reference, if it is enforced on a single node with 64 cores/process,
the performance becomes 100.4 and 176.0 GFlops on $16^3 \times 32$
and $32^3 \times 64$ lattices, respectively.
For the 64 cores/process on multiple nodes, the performance is
about the half of the other two cases.
This may be explained by that all the 64 cores in a node share
the whole memory of the node so that imbalance of accessibility
to the memory among the cores exists.
The cases of 1 or 2 cores/process achieve more than 170 GFlops/node
up to 256 nodes.
The bottom panel of
Fig.~\ref{fig:Wilson_mult_scaling}
shows the strong scaling on a $32^3 \times 64$
lattice.
In this case, as the number of nodes increases, the local lattice
volume inside each node decreases so that the communication
overhead becomes more and more significant.
Again the 64 cores/process case exhibits less performance than
other two cases.
For the strong scaling, the performance depends on how the
lattice is divided into sublattices.
We plot several cases at each number of nodes.
For the case of 1 and 2 cores/process, the performance at 16 nodes
is about 2/3 of that of single node.
\begin{figure}[!t]
\centering
\includegraphics[width=7.7cm]{Figs/thread_dep_wilson_prefetch_mult.eps}
\caption{
Effect of manual prefetching.
The performance of the Wilson matrix multiplication is measured on
a $32^3 \times 64$ lattice.}
\label{fig:Wilson_mult_effect_of_prefetch}
\end{figure}
\subsection{Performance of BiCGStab solver}
For the Wilson matrix, the BiCGStab solver works efficiently.
We compose the BiCGStab algorithm with BLAS-like library.
For the linear algebraic functions, we apply neither manual
prefetch nor additional compiler option for prefetch.
While the Wilson matrix part is improved by manual prefetch,
this effect is small because the performance is determined
by the linear algebraic functions.
The top panel of Fig.~\ref{fig:Wilson_solver_scaling}
shows the weak scaling for the BiCGStab solver on
a $16^3 \times 32$ lattice in each node.
Because of larger byte-per-flop values of the linear algebraic
functions, the performance reduces to about 1/4 of the matrix
multiplication at 256 nodes.
The difference of 64 cores/process and other two cases also
decrease because of the linear algebraic functions.
The worse scaling as the number of node is caused by the
reduction operations.
The bottom panel of Fig.~\ref{fig:Wilson_solver_scaling}
shows the strong scaling plot of the solver on $32^3 \times 64$
lattice.
While the difference of 64 cores/process and other two cases
shrinks, smaller numbers of cores/process achieve better
performance for large number of nodes.
In total, these results indicate that small number of cores
per MPI process has advantage,
if the memory size and the local lattice volume allow.
\begin{figure}[!t]
\centering
\includegraphics[width=8.0cm]{Figs/weak_scaling_16x16x16x32_wilson_prefetch_mult.eps}
\includegraphics[width=8.0cm]{Figs/strong_scaling_32x32x32x64_wilson_prefetch_mult.eps}
\caption{
Scaling plots for the Wilson matrix multiplication.
Top: weak scaling with a $16^3 \times 32$ lattice in each node.
Bottom: strong scaling with a $32^3 \times 64$ lattice.
}
\label{fig:Wilson_mult_scaling}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=8.0cm]{Figs/weak_scaling_16x16x16x32_wilson_prefetch_solver.eps}
\includegraphics[width=8.0cm]{Figs/strong_scaling_32x32x32x64_wilson_prefetch_solver.eps}
\caption{
Scaling property of the BiCGStab solver with Wilson matrix.
Top: weak scaling plot for a $16^3 \times 32$ lattice in each node.
Bottom: strong scaling on a $32^3 \times 64$ lattice.
}
\label{fig:Wilson_solver_scaling}
\end{figure}
\section{Performance of domain-wall fermion matrix}
\subsection{Performance of matrix multiplication}
For the domain-wall fermion matrix, the condition of
performance measurement is the same as the Wilson matrix
except for no manual prefetch is applied.
The domain-wall multiplication achieves better performance
than the Wilson matrix, as expected from smaller byte-per-flop
value due to the reuse of link variable $U_\mu(x)$,
as well as the larger number of arithmetic operations per
lattice site.
The top panel of Fig.~\ref{fig:DW_mult_scaling} displays
a weak scaling plot of the domain-wall matrix
multiplication for a $16^3 \times 32$ lattice in each node.
On a single node with 64 cores/process,
if the boundary data copy is enforced, the performance becomes
155.2 GFlops on a $16^3 \times 32$ lattice and 211.8 GFlops on
a $32^3 \times 64$ lattice for the weak and strong scaling,
respectively.
While the 1 and 2 cores/process cases exhibit good scaling
behavior, multi-node result of 64 cores/process is quite bad.
What is different from the Wilson matrix is larger size of
boundary data transferred at the communication.
How such reduction of performance occurs and how can be
avoided is now under investigation.
The strong scaling on a $32^3 \times 64$ lattice in the
bottom panel of Fig.~\ref{fig:DW_mult_scaling} shows
similar tendency as the Wilson matrix, except for the reduction
of the 64 cores/process data at 16 nodes.
For the 64 cores/process data, we observe large fluctuations
up to almost factor 10 in the elapsed time.
We have not understood why such fluctuations occur
and are investigating the cause.
For this reason, we do not include the data for 64 cores/process
in the measurement of the CG solver below.
\begin{figure}[!t]
\centering
\includegraphics[width=8.0cm]{Figs/weak_scaling_16x16x16x32_domainwall_normal_mult.eps}
\includegraphics[width=8.0cm]{Figs/strong_scaling_32x32x32x64_domainwall_normal_mult.eps}
\caption{
The weak and strong scaling plots of the domain-wall matrix multiplication.
For the weak scaling, $16^3 \times 32$ lattice per node.
For the strong scaling, $32^3 \times 64$ lattice.}
\label{fig:DW_mult_scaling}
\end{figure}
\subsection{Performance of CG solver}
For the domain-wall fermion, we apply the CG algorithm to
the matrix $D^\dag D$.
Fig.~\ref{fig:DW_mult_scaling} displays the weak (top) and
strong (bottom) scaling plots for the CG solver.
Better scaling behaviors than those of the Wilson matrix
are explained by larger weight of
the matrix multiplication in the algorithm and larger size
of vectors.
Even for the strong scaling plot, the performance
of 1 and 2 cores/process cases almost unchanged
as increasing the number of nodes from 1 to 16.
\begin{figure}[!t]
\centering
\includegraphics[width=8.0cm]{Figs/weak_scaling_16x16x16x32_domainwall_normal_solver.eps}
\includegraphics[width=8.0cm]{Figs/strong_scaling_32x32x32x64_domainwall_normal_solver.eps}
\caption{
The weak and strong scaling plot for the CG solver with the
domain-wall fermion matrix.
For the weak scaling, $16^3 \times 32$ lattice per node.
For the strong scaling, $32^3 \times 64$ lattice.}
\label{fig:DW_solver_scaling}
\end{figure}
\section{Conclusion}
In this paper, we apply rather simple prescription to
make use of the SIMD architecture of the Intel Xeon
Phi KNL processor to a typical problem in lattice QCD
simulation.
Aiming at widely applying to existing codes,
we examine the rearrangement of data layout and AVX-512
intrinsics to arithmetic operations, and examined
the effect of manual prefetching.
The former two are inevitable to achieve acceptable
performance, compared to the original Bridge++ code.
The effect of manual prefetching is more restrictive.
It amounts to dedicated efforts only on single node
or small number of nodes.
Comparing the choices of numbers of cores per MPI
process, small numbers of cores per MPI process
have advantages as increasing number of nodes.
Our results indicate two efficient ways of using KNL.
On single KNL, multi-thread application without MPI
parallelization may works efficiently.
The manual prefetch is worth to try.
For a multi-node case, adopting small numbers of
cores per MPI process, like a massively parallel machine,
one can optimize the number of nodes against a given
problem size.
As for the code of application, it is essential to
employ design that enables flexible rearrangement of data
layout and incorporation of intrinsics.
\section*{Acknowledgment}
The authors would like to thank Peter Boyle, Guido Cossu,
Ken-Ichi Ishikawa, Daniel Richtmann, Tilo Wettig,
and the members of Bridge++ project for valuable discussion.
Numerical simulations were performed on Oakforest-PACS system
hosted by JCAHPC, with support of Interdisciplinary Computational
Science Program in CCS, University of Tsukuba.
This work is supported by JSPS KAKENHI (Grant Numbers
JP25400284, JP16H03988), and by
Priority Issue 9 to be tackled by Using Post K Computer,
and Joint Institute for Computational Fundamental Science (JICFuS).
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\title{Exact Amplitude Distributions of Sums of Stochastic Sinusoidals and
their Application in Bit Error Rate Analysis}
\author{Y. Maghsoodi \\
Scinance Analytics, www.ScinanceAnalytics.com, \\
[email protected]\linebreak
\and A. Al-Dweik, \textit{%
Senior Member, IEEE} \\
Khalifa University, P.O.Box 127788, Abu Dhabi, UAE\\
[email protected]\\
This version 2019\\
Copyright Y. Maghsoodi and A. Al-Dweik 2002}
\date{}
\maketitle
\vskip -15mm
\begin{abstract}
We consider a model applicable in many communication systems where the sum of
$n$ stochastic sinusoidal signals of the same frequency, but with random amplitudes as well as
phase angles is present. The exact probability distribution of the resulting
signal's amplitude is of particular interest in many important applications,
however the general problem has remained open. We derive new general formulae
for the resultant amplitude's distribution, in terms of any given general joint
density of the random variables involved. New exact probability densities of cases
of interest follow from the general formulae which are applied to the problem of the
exact evaluation of the bit error rate performance in narrowband multipath
fading channels that consist of a small number of jointly dependent
resolvable multipath components. The numerical results are compared
with those of the currently popular method of Gaussian approximation.
These are consistent with simulations, and show significant increase in accuracy
particularly for small number of components.\end{abstract}
\begin{keywords}
\vskip -2mm
\noin Stochastic sinusoidals, sums of sinusoidals, amplitude distribution, exact distribution, Rayleigh fading.
\end{keywords}
\evensidemargin 0in\oddsidemargin\evensidemargin
\section{Introduction}
In many communication systems the received signal can be expressed as \cite{Proakis-1}-\cite{Kah-1}
\begin{equation}
r(t)=\sum_{i=1}^{n}A_{i}\cos(\omega t+\phi_{i})+Z(t)\ . \label{1.1}
\end{equation}
The first term in (\ref{1.1}) represents the desired signal component with
amplitude $A_{1}$, frequency $\omega$, and phase $\phi_{1}$, mixed with
interference that is produced either unintentionally by a multiple access
process, reflections of the desired signal, or intentionally by a certain
jammer. The second term, $Z(t)$, is an additive white Gaussian
noise (AWGN) with two-sided power spectral density of $N_{0}/2$.
In general the amplitudes $(A_{1},A_{2},...,A_{n})$, and the phases $(\phi_{1},\phi_{2},...,\phi_{n})$%
, may all be random variables, with a given joint probability distribution
function, which is the framework treated in this paper. In specific
applications however, the statistical properties of these variables, and the
value of $n$, depend on the particular channel model and the application considered.
For example, in the case of narrowband
local area UHF and microwave propagation, the amplitudes are considered to
be constants and the phases are independent and uniformly distributed over $%
[-\pi ,\pi ]$ \cite{Proakis-1}-\cite{Rappaport-book}. The number of
multipath components $n$ is determined based on the terrain of the
surroundings at the receiver side. Similar assumptions are usually made for
synchronous frequency hopping spread spectrum systems (FH-SSS) with multiple
access interference (MAI) or multitone jamming (MTJ) with AWGN \cite{Kah-1}-%
\cite{Arafat-1}. The value of $n$ in multiple access systems corresponds to
the number of simultaneous users and it is modelled as a random variable
with binomial distribution.
The case where the amplitudes are random has also a wide range of
applications. For example, Beaulieu et al \cite{Beaulieu-1} have modelled the
amplitudes as a Nakagami-$m$ random variables to analyze the performance of
wireless networks with cochannel interference. The
Rayeligh-distributed amplitudes model was adapted for FH-SSS with orthogonal
frequency division multiplexing (OFDM) in \cite{Arafat-1}, \cite{Kim 1}.
In evaluation of the performance of communication systems with received
signals such as $r(t)$ in (\ref{1.1}), the knowledge of the pdf
(probability density function) of the resultant amplitude or envelope is of
crucial importance. Rice \cite{Rice-1} stated that the cdf (cumulative
distribution function) of the envelope of some special cases of (\ref{1.1})
may be represented by a Fourier-Bessel Transform. Kluyver \cite{Kluyver}
considered the case of $n$ randomly phased sine waves in the absence of
noise and gave a corresponding equation for the special case in \cite%
{Rice-1}. Simon \cite{Simon-1} considered a special recursive case of (\ref%
{1.1}) without noise, where all the amplitudes are fixed, and each new phase
angle is assumed to be equal to the resultant of all the previous ones, plus an
independent uniformly distributed angle. Thus, he deduced a recursive
formula for the pdf of the squared envelope in this special case (see also Section III below).
Helstrom \cite{Helstrom-1} derived the amplitude pdf of the sum of two sinusoidals of constant
amplitude, affected by a Gaussian noise, and later developed a numerical
scheme to approximate the cdf of the envelope when $n>10$, in the presence
of a narrow band Gaussian noise \cite{Helstrom-2}.
Beckman \cite{Beckmann-2}, considered and cited several other special cases
of the problem posed in model (\ref{1.1}), including constant amplitudes with a non-uniform
pdf for the $\phi_i$s, and $X$ and $Y$ having a non-Gaussian joint pdf, where
$X\den \sum_{i=1}^{n}A_{i}\cos(\phi_{i})$, and $Y\den \sum_{i=1}^{n}A_{i}\sin(\phi_{i})$.
The latter special case was also considered by Zabin and Wright \cite{Zabin}.
Abdi et al. \cite{Abdi} assumed independent and identically uniformly distributed phases,
with arbitrarily dependent amplitudes
and derived a multiple integral formula for the envelope pdf, for which they also gave
an infinite Laguerre expansion. They applied their results
to study the statistical behaviour of the scattering cross
section when the number of scatters is small and deterministic and all
amplitudes are equal. As also reported in \cite{Beckmann-2} and \cite{Abdi}, in most practical cases
the uniform pdf assumption for the phases is usually satisfied.
Other authors too, have made special assumptions of various forms,
about the distributions and independence of the phases and the amplitudes. A brief survey of these works
can also be found in Abdi et al. \cite{Abdi}. Maghsoodi \cite{Maghsoodi-1} derived exact formulae
for the envelope pdf for the general case, where the amplitudes and phases were not assumed to be
independent, nor were they assumed to have any particular probability distribution.
In this paper, we expand the work in \cite{Maghsoodi-1} and discard the restrictive assumptions other previous works,
and present general formulae for the envelope pdf in the most general case. We then apply the results
to bit error rate analysis. The extension of our formulae to include random number of signals is immediate under
independence, and still follows naturally under no independence, though slightly less immediately.
We first consider model (\ref{1.1}) in the absence of noise,
and while allowing all the amplitudes as well as phases to be random,
having a given general joint pdf, we derive two exact general formulae for the
probability distribution of the envelope. These are proved in
Theorems 1 and 2 \cite{Maghsoodi-1}. The presented formulae will be solely
in terms of integrals of any given general joint probability density of all
the amplitudes and phases in the sum. Examples of the implementations of the
general formulae, yield interesting new exact densities of some important
cases of interest. These are applied to evaluate the exact error probability of binary phase
shift keying (BPSK) systems over narrowband fading channels that consist of
a small number of multipath components. It turns out that significant gain in
accuracy is made, particularly over the currently popular method of Gaussian approximation, where
the application of the Central Limit Theorem is much less accurate for smaller number of components. Examples for using the proposed approach for evaluating the error probability can be found in \cite{Arafat-1, Arafat-2, Arafat-3}.
The paper is organized as follows. In section II Sinusoidals Addition
Theorem (SAT), cites the formulae for the resultant amplitude and phase
angle of sums of sinusoidals. Allowing for random amplitudes and phase angles, Theorem 1
presents EDDHAPT (Envelope Distribution from Density of
Half Angle Phase Tangents) formula, for the exact cdf of the envelope, in
terms of the joint pdf of the amplitudes and the tangents of the half phase angles. Two
examples of the implementation of this formula will follow. In section III,
Envelope Separation Theorem (EST) gives a recursive formula for the envelopes, followed by Theorem 2
which presents the EGED (Exact General Envelope Density) formula \cite{Maghsoodi-1}, which allows the calculation of the pdf of the
envelope, in terms of the given joint pdf of the amplitudes and the actual phase
angles themselves. In section IV we apply
the results of the preceding sections to evaluate the exact error
probability of BPSK systems in narrowband multipath fading channels.
Finally, the conclusions are presented in Section V and the mathematical
proofs are given in the appendix.
\section{New formulae for the envelope and its cdf via half-phase tangents}
In this section we assume that the joint pdf of all the amplitudes and the
tangents of all the half phase angles of model (\ref{1.1}) is given. The use of this joint pdf is
more convenient here due to the range $%
(-\infty,\infty)$ of the tangent. There is no loss of generality in this assumption,
since this joint pdf can always be obtained from that of the amplitudes and
the phase angles themselves by suitable transformations. In what follows
we shall denote by $|\mathbf{A}|$ the Euclidean norm of the vector $(A_{1},...,A_{n})$.
\vskip 4mm
The SAT states that, given the identity \begin{equation}
\sum_{i=1}^{n}A_{i}\cos (\omega t+\phi _{i})=B_{n}\cos (\omega t+\theta _{n})%
\text{\shs }t\geq 0\ , \label{E-2.1}
\end{equation}
\noin where $A_{i}\in (-\infty ,\infty )${\ and }$\phi _{i}\in
\lbrack -\pi ,\pi ]${, there exists a unique solution of \textrm{(\ref {E-2.1})} for }$\{B_{n},\theta _{n}\}${\ which is independent of }$%
\omega ${\ and }$t${\ and is given by }%
\begin{equation}B_{n}^{2}=|\mathbf{A}|^{2}+2\hskip -3mm\sum_{n\geq j>k\geq 1}\hskip -3mm A_{j}A_{k}\cos
\left( \phi _{j}-\phi _{k}\right) \label{2.2}
\end{equation}%
\begin{equation}
\theta _{n}=\tan ^{-1}{\frac{\sum_{i=1}^{n}A_{i}\sin \left( \phi _{i}\right)
}{\sum_{i=1}^{n}A_{i}\cos \left( \phi _{i}\right) }}\ , \label{E-2.3}
\end{equation}%
{where }$\theta _{n}${\ can be uniquely chosen such that }$%
B_{n}${\ is always positive.} The proof is by writing (\ref{E-2.1}) with $\sin$ as well, and expanding and solving
the resulting double identities for $(B_{n},\theta _{n})$.
It follows from formula (\ref{2.2}) ( see also (\ref{3.1}) below and \cite%
{Maghsoodi-1}) that if $\phi_{n}$ can take all possible values in $%
[-\pi,\pi] $, then
\begin{equation}
\left\vert B_{n-1}-|A_{n}|\right\vert \leq B_{n}\leq B_{n-1}+|A_{n}| \label{E-2.5}.
\end{equation}
Hence, we can have a recursion for the minimum and maximum possible values
of $B_{n}$, in terms of those of $B_{n-1}$, which we denote by $m_{n}$ and $M_{n}$
respectively. In each specific application the values of $m_{n}$
and $M_{n}$ would strictly depend on the specific range of the values of $A_{i}$
and $\phi_{i}$ for $i=1,2,...,n$, however the above inequalities would
determine $m_{i}$ and $M_{i}$ for $i=1,2,..,n$ recursively in each case. For
example when $A_{1}$ and $A_{2}$ are constants, $B_{1}=|A_{1}|$, $%
M_{2}=|A_{1}|+|A_{2}|$ and $m_{2}=\left\vert |A_{1}|-|A_{2}|\right\vert $,
and so on.
In what follows unless otherwise specified, upper case
letters such as $A_{1}$ and $T_{1}$ will denote random variables and lower
case letters such as $a_{1}$ and $t_{1}$ will represent their corresponding
possible values. In addition, bold-face letters such as $\mathbf{A}$ will
denote vectors with elements in regular fonts such as $\mathbf{A}%
=(A_{1},...,A_{n})$, and $d\mathbf{a}$ will denote $da_{1}da_{2}...da_{n}$
and so on. Further, $\bar{\mathbf{T}}$ and $\bar{\mathbf{t}}$ will denote
the truncated vectors $(T_{1},...,T_{n-1})$ and $(t_{1},...,t_{n-1})$
respectively.
\vskip 3mm\textbf{Theorem 1} ( The EDDHAPT Formula) \textit{Consider the sum of }$n$\textit{\ stochastic sinusoidals on the
LHS of \textrm{(\ref{E-2.1})}, with }$A_{i}\in(-\infty,\infty )$, \textit{\
and }$\phi_{i}\in\lbrack-\pi,\pi]$, \textit{\ random variables with a given
general joint pdf }$f_{_{\mathbf{A},\mathbf{T}}}(\mathbf{a},\mathbf{t}%
)\triangleq f_{A_{1}...A_{n},T_{1}...T_{n}}(a_{1},...,a_{n},t_{1},...,t_{n}),$%
\textit{\ where }$T_{i}\triangleq\tan(\phi_{i}/2), i=1,2,\cdots,n$\textit{. Then the cdf of
the resultant amplitude }$B_{n}$\textit{\ is given by}%
\vskip -6mm
\begin{equation}
P(B_n \leq b_n) =\int_{-\infty}^\infty d{\bf a} \int_{-\infty}^{\infty} d\bar {\bf t} {\bf 1}_{\{ a<0\}} f_{{\bf A},{\bf T}} ( {\bf a, \bar t}) +
\int_{-\infty}^{\infty} d{\bf a} \int_{-\infty}^{\infty} d\bar {\bf t} {\bf 1}_{\{\Delta_Q >0 \}} \int_{\bar a_n}^{\bar b_n}dt_n f_{{\bf A},{\bf T}} ( {\bf a, t}) \label{2.6}
\end{equation}
\noin {\it where}\ \ $\Delta_{Q}\triangleq4a_{n}^{2}\bar{S}_{n}^{2}-a(I_{n}+2a_{n}\bar{C}%
_{n})$, \ \ \ \ $\bar{S}_{n}\triangleq\sum_{i=1}^{n-1}\frac{2a_{i}t_{i}}{%
1+t_{i}^{2}}$, \ \ \ \ $\bar{C}_{n}\triangleq\sum_{i=1}^{n-1}\frac {%
a_{i}(1-t_{i}^{2}{)}}{1+t_{i}^{2}}$, \ \ \ \ $a\triangleq I_{n}-2a_{n}\bar {C%
}_{n}$
\vskip 3mm}\def\xvs{\vskip 2mm
$I_{n}=\sum_{i=1}^{n}a_{i}^{2}+\bar{k}_{n}-b_{n}^{2}, \ \ \bar{k}_{n}= 2 \sum_{(n-1)\geq j>k\geq1}\hskip -3mm\ \ {{ a_{j}a_{k}( (1+t_{j}t_{k})^{2}}-(t_j-t_k)^2)\over \left ( 1+t_{j}^{2}\right) \left( 1+t_{k}^{2}\right) },
\ \ \ \bar{a}_{n},\bar{b}_{n}\triangleq\lbrack{-2a_{n}\bar{S}_{n}\mp\sqrt{{\Delta}_{Q}}}]/{a},$
\vskip 2mm
\noin\textit{ and ${\bf 1}_{\{\cdot\}}$ denotes the indicator function of the set $\{\cdot\}$}.
\mn\textit
{Example 1}\ \ \ As an example of the implementation of the EDDHAPT formula (\ref {2.6})
assume that $n=2$, $A_{1}=A_{2}=A$, a constant, and $\phi_{1}$ and $%
\phi_{2}$ are independently uniformly distributed in $(-\pi,\pi)$. It can
then be easily deduced that the cdf $F_{T_{i}}(t_{i})$ and the pdf $%
f_{T_{i}}(t_{i})$ of $T_{i}$ are respectively given by \cite{Maghsoodi-1}
\vskip 3mm}\def\xvs{\vskip 2mm\shs $F_{T_{i}}(t_{i})={\frac{1}{\pi}}[\tan^{-1}(t_{i})+{\frac{\pi}{2}}]$\dhs and \dhs $%
f_{T_{i}}(t_{i})=\frac{1}{\pi(1+t_{i}^2)}$\dhs $\ \ t_{i}\in(-\infty
,\infty),\ \ i=1,2$.
\svs\noindent }\def\mn{\mvs\noin}\def\bn{\bvs\noin Application of the EDDHAPT formula (\ref {2.6}) immediately gives $\Delta_Q=b_2^2(4A^2-b_2^2)$
which is always positive since $b_{2}\in (0,2\vert A\vert)$ and $a=[(4A^{2}-b_{2}^{2})t_{1}^{2}-b_{2}^{2}]/(1+t_{1}^{2})$. Hence
\vskip -5mm
\begin {align} P(B_2 \leq b_2)
&=\int_{-\infty}^\infty {\bf 1}_{\{a<0\}}\ dt_1f_{T_1}(t_1) + \int_{-\infty}^\infty dt_1 \int_{\bar a_2}^{\bar b_2} dt_2 f_{T_1}(t_1)f_{T_2}(t_2)\cr
&={1\over \pi} \int_{-\infty}^\infty {\bf 1}_{\{\vert }\def\Ve{\Vert t_1\vert }\def\Ve{\Vert >\alpha\}} {dt_1\over 1+t_1^2} + {1\over \pi^2} \int_{-\infty}^\infty dt_1 {{\cal T}(\bar a_2,\bar b_2)\over 1+t_1^2} \label{2.7}
\end{align}
\vskip -3mm\noin where $\bar{a}_{2},\bar{b}_{2}=\frac{-4A^{2}t_{1}/(1+t_{1}^{2})\mp\sqrt {%
\Delta_{Q}}}{a}$, ${\cal T}(\bar a_2,\bar b_2)\den \tan^{-1}(\bar b_2)-\tan^{-1}(\bar a_2)$,
and $\alpha=\frac{b_{2}}{\sqrt{4A^{2}-b_{2}^{2}}}$ is the positive root of $a=0$.
Integrating the second integrand by parts, whilst noting the singular discontinuities of $\bar{a}_{2}$ and $\bar{b}_{2}$%
, at $t_{1}=\pm\alpha$, and the fact that
\vskip -3mm
$$\bar a_2'(1+\bar a_2^2)^{-1}=\bar b_2'(1+\bar b_2^2)^{-1}=(1+t_1^2)^{-1}\ ,$$
\vskip -4mm
\noin we obtain
\vskip -6mm
\begin{equation}
P(B_{2}\leq b_{2})={\frac{2}{\pi}}\tan^{-1}{\frac{b_{2}}{\sqrt{%
4A^{2}-b_{2}^{2}}}}\ , \label{2.8}
\end{equation}
which after differentiation gives the pdf of the amplitude%
\begin{equation}
f_{B_{2}}(b_{2})=\left\{
\begin{array}{cc}
\frac{{2}}{{\pi\sqrt{4A^{2}-b_{2}^{2}}}} & \text{ \ \ \ }0\ \leq{b}_{2}{\
\leq2|A|} \\
0\text{ \ \ \ \ \ \ \ } & \text{otherwise}\ .%
\end{array}
\right. \label{2.9}
\end{equation}
(See Figure 1 for a graph of this pdf).
\mn\textit
{Example 2}\ \ \ Assume that in example 1 above, $\phi_1$ and $\phi_2$ are dependently distributed with joint pdf
\vsn\begin{equation}
f(\phi_1,\phi_2)={1\over \pi^3}(\phi_1+\phi_2)\shs 0\le \phi_i\le \pi . \label{2.15}
\end{equation}
\noin It can easily be verified that the joint pdf of the corresponding $T_i$ random variables is \cite{Maghsoodi-1}
\begin{equation}
f_{T_1,T_2}(t_1,t_2)={8(\tan^{-1} t_1+ \tan^{-1} t_2)\over \pi^3 (1+t_1^2)(1+t_2^2)} \shs 0\le t_i< \infty\ ..\label{2.16}
\end{equation}
\noin Then, in this case formula (\ref{2.6}) above reduces to
\begin {equation}
P(B_2 \leq b_2)=\int_{-\infty}^\infty {\bf 1}_{\{\vert }\def\Ve{\Vert t_1\vert }\def\Ve{\Vert >\alpha }\def\Al{\Alpha }\def\be{\beta }\def\ep{\epsilon \}}\ dt_1 \int_{-\infty}^\infty dt_2 f_{T_1,T_2}(t_1,t_2)
+ \int_{-\infty}^\infty dt _1 \int_{\bar a_2}^{\bar b_2} dt_2 f_{T_1,T_2}(t_1,t_2). \label{2.17}
\end{equation}
\noin where the parameter values are as in example 1, except the joint pdf which is given in (\ref {2.16}). Calculating the integrals in (\ref {2.17}), by methods very similar to example
1 above we obtain
\begin{equation}
F_{B}(b)={4\over \pi^2} (\tan^{-1}\alpha)^2 \shs 0\le b\le 2\vert A\vert\ .\label{2.18}
\end{equation}
\noin Hence, the amplitude pdf for this example is
\begin{equation}
f_{B}(b)={8\over \pi^2} {\tan^{-1}\alpha\over\sqrt{(4A^2-b^2)}} \shs 0\le b\le 2\vert A\vert\ . \label{2.19}
\end{equation}
\noin From the pdf (\ref{2.19}) it is evident that it has a singularity at $b_2=2A$ which can also be seen in Fig. 1, where the pdf is illustrated for $A=1$.
Example 2 also illustrates that formula (\ref{2.6}) can find the envelope pdf in the more general, and practically more realistic situations where the phase angles may have any arbitrary joint law, and may not be limited to be independent or be uniform. This example can have applications where the phase angles are correlated, and their probability of occurrence would linearly increase with the angle, e.g. in signal propagation over correlated multipath fading channels, particularly the 2-ray models used for mobile radio channels.
\section{Envelope separation and its pdf in terms of the joint pdf}
In this section we represent the pdf of the envelope, directly in terms of any given joint pdf of the amplitudes and the phase
angles in the stochastic sinusoidals sum. First the Envelope Separation
Theorem (EST) is presented, which gives a recursive formula for the $n^{th}$ envelope in
terms of the $(n-1)^{st}$ envelope and resultant phase angle. This can also be viewed
as a type of second cosine theorem. Theorem 2 then derives a new Exact General
Envelope Distribution (EGED) formula for the envelope pdf. The advantage of this formula over (\ref{2.6}) is that the
regions of integration are explicitly determined, however the disadvantage
in applications is the complications of dealing with bounded regions of
integrations compared to $(-\infty,\infty)$ of $t_{i}$ in (\ref{2.6}).
\svs\noindent }\def\mn{\mvs\noin}\def\bn{\bvs\noin The EST expresses the $n^{th}$ envelope $B_{n}^{2}$ as
\begin{equation}
B_{n}^{2}=B_{n-1}^{2}+A_{n}^{2}+2A_{n}B_{n-1}\cos(\phi_{n}-\theta_{n-1}).
\label{3.1}
\end{equation}
\noin The proof is by separating all the terms involving $A_n$ in formula (\ref{2.2}), and writing the remaining
terms as $B_{n-1}^2$ \cite{Maghsoodi-1}.
It can immediately be seen from (\ref{3.1}) that, if we assume $\phi_{i}$s
to have a special form such that, $\{\xi_{i}\triangleq\phi_{i}-%
\theta_{i-1},\ i=2...n,\ \ \theta_{1}=\phi_{1}\}$ is an i.i.d. (independent and identically distributed)
sequence of random variables, then clearly the sequence $%
\{B_{i}^{2},\ i=1,2,...\}$ becomes Markovian, and a simple recursive formula
can be written for its distribution. This assumption was made by Simon \cite%
{Simon-1}, where the $\xi_{i}$s were further assumed to be independently
uniformly distributed. Though Simon's work is interesting, and his formula
can be derived as special cases of
the results of this paper, however the assumptions are very restrictive on
the general model considered here as well as on the scope of practical
applications, in particular these assumptions imply that, each received sinusoidal adapts
its phase to the resultant phase of all the previous sinusoidals in a
special way, which also leads to a particular joint law for the $\phi_{i}$s. Since
by our formula (\ref{E-2.3}), Simon's assumption is equivalent to saying that, $\{\phi_i,\ i=1\cdots,n\}$ are such that,
they satisfy
$$\xi_{i}\den \phi_i-\tan ^{-1}{\frac{\sum_{j=1}^{i-1}A_{j}\sin \left( \phi _{j}\right)
}{\sum_{j=1}^{i-1}A_{j}\cos \left( \phi _{j}\right) }}\sim U(-\pi,\pi)\mhs i=2,\cdots,n$$
\noin and the $\xi_{i}$\ s are i.i.d., which are complicated special modelling assumptions about the $\{\phi_i,\ i=1\cdots,n\}$.
Another immediate consequence of Simon's assumption is that, the phases can never be independent, since,
for example under this assumption we would have
\begin {align} P(\phi_2\le y/\phi_1=x)=
&P(\phi_2-\phi_1\le y-x/\phi_1=x)\cr=
&{{y-x+\pi\over 2\pi}}\cr\neq
&P(\phi_2\le y)\ . \nonumber
\end {align}
But, alternatively, in the absence of any additional (physical) information to the contrary, it would be more natural and physically meaningful, to assume that each signal's phase is independent of the others, and is uniformly distributed. Our general formulae allow us to model and solve these cases as well, as special cases. Moreover, at the expense of little extra mathematical complexity, we can still derive recursive formulae in the general independent cases without Simon's assumptions.
\vskip 3mm}\def\xvs{\vskip 2mm\textbf{Theorem 2} (Exact General Envelope Density (EGED) via the joint pdf )
\textit{\ \ Under the assumptions of Theorem 1, if the amplitudes and phases random
variables $\mathbf{A}$ and $\mathbf{\Phi}$, have a given general joint pdf }
$f_{_{\mathbf{A},\mathbf{\Phi}}}(\mathbf{a},\mathit{\Phi})\triangleq
f_{A_{1}...A_{n},\Phi_{1}...\Phi_{n}}(a_{1},...,a_{n},\phi_{1},...,\phi_{n})$%
\textit{, then the pdf of the envelope }$B_{n}$\textit{\ is given by }
\begin{equation}
f_{B_{n}}(b_{n})=2b_{n}E\left\{ \left\vert \sqrt{%
4A_{n}^{2}B_{n-1}^{2}-(b_{n}^{2}-B_{n-1}^{2}-A_{n}^{2})^{2}}\right\vert
^{-1}1_{\{\left\vert \Psi_{n}\right\vert \le 1,\ \Phi_{n}\in U_{n}\}}\right\}
\text{ \ \ }m_{n}\le b_{n}\le M_{n}\ , \label{3.3}
\end{equation}
\noin\textit{\ where the expectation is with respect to the given joint law }$%
f_{_{\mathbf{A},\mathbf{\Phi}}}$\textit{, and }$\mathbf{1}_{\{.\}}$\textit{\
denotes the indicator of the set }$\{.\}$\textit{\ and}%
\begin{equation}
\mathbf{\Psi}_{n}\triangleq\left( \frac{b_{n}^{2}-B_{n-1}^{2}-A_{n}^{2}}{%
2A_{n}B_{n-1}}\right) ,\text{ }U_{n}\triangleq\{\alpha_{i},\ |\alpha
_{i}|<\pi,i=1,...,4\},\text{ }\alpha_{i}\in\{\theta_{n-1}\pm|\cos^{-1}\Psi
_{n}|\mp2k\pi,\ k=0,1\}\ , \label{3.3b}
\end{equation}
\noin {\it and the expectation integration is in the $(\bf a,{\it\F})$ region where, $\phi_n$
can only take the four values listed in the set $U_n$, in terms of the remaining
integration variables.}
\vskip 3mm}\def\xvs{\vskip 2mm\textit{Example 3} \ \ As an example of the implementation of the EGED formula (\ref%
{3.3}), consider the case $n=3$, and suppose $\phi _{i},\
i=1,2,3$ are independently uniformly distributed in $[-\pi,\pi]$, and the $A_{i}$%
s are constants. Then, application of formula (\ref{3.3}), a change of variable of integration
and simplification, shows that if we le
\begin{equation}
f_{\mathbf{A},\mathbf{\Phi}}(\mathbf{a},\mathit{\Phi})=(8\pi^{3})^{-1},\ \ \
\bar{b}_{2}^{2}\triangleq|\mathbf{A}|^{2}+2A_{1}A_{2}\cos (\phi_{1}),\ \
\text{and }\ \ \bar{\psi}_{3}\triangleq\left( \frac{b_{3}^{2}-\bar{b}%
_{2}^{2}-A_{3}^{2}}{2A_{3}\bar{b}_{2}}\right)\ , \label{3.7}
\end{equation}
then we obtain \cite{Maghsoodi-1}
\begin{equation}
f_{B_{3}}(b_{3})=\left\{
\begin{array}{cc}
\hskip -3mm\nhs\tfrac{2b_{3}}{\pi^{2}}\int_{0}^{\pi}1_{\{\left\vert \bar{\psi}%
_{3}\right\vert \le 1\}}\frac{d\phi_{1}}{\sqrt{4A_{3}^{2}\bar{b}%
_{2}^{2}-(b_{3}^{2}-\bar{b}_{2}^{2}-A_{3}^{2})^{2}}} & m_{3}\leq b_{3}\leq
M_{3} \\
0\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ } & \text{otherwise}.%
\end{array}
\right. \label{3.8}
\end{equation}
Graphs of numerical examples of this pdf are also illustrated in Figure 2 for various amplitude values.
Moreover, when $A_{i}=1,\ i=1,2,3$, the pdf (\ref{3.8}) further simplifies to \cite{Maghsoodi-1}
\begin{equation}
f_{B_{3}}(b_{3})=\left\{
\begin{array}{cc}
\hskip -3mm\nhs\frac{2b_{3}}{\pi^{2}}\int_{\tilde{m}_{2}}^{\tilde{M}_{2}}\frac{d\phi_{1}}{%
\sqrt{4\bar{b}_{2}^{2}-(b_{3}^{2}-\bar{b}_{2}^{2}-1)^{2}}} & 0\leq b_{3}\leq3
\\
0\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } &
\text{otherwise}\ ,%
\end{array}
\right. \label{3.9}
\end{equation}
where in this case
\noin $\bar{b}_{2}^{2}\triangleq2+2\cos(\phi_{1})$,
$\tilde {M}_{2}\triangleq\left\vert \cos^{-1}\left( (b_{3}-1)^{2}/2-1\right))\right\vert $, and $\tilde{m}_{2}\triangleq\left\vert \cos^{-1}(\min
(b_{3}+1,2)^{2}/2-1)\right\vert $.
\noin It follows from formula (\ref{3.9}) that in this case the pdf has a singularity at $b_3=1$ which can also be observed in the graph of Fig. 2.
\noin Formula (\ref{3.9}) can also be written in terms of the EllipticF and EllipticK functions whose numerical values are
well tabulated and coded
\begin{equation}
f_{B_{3}}(b_{3})=\frac{{4}Ib_{3}}{\pi^{2}}U\left\{
\begin{array}{cc}
\hskip-8mm\text{EllipticK}(P)-\text{EllipticF}(1/P,P) & 0\leq b_{3}\leq1 \\
\text{EllipticK}(P)\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ } & 1<b_{3}\leq 3\ ,%
\end{array}
\right. \ \label{3.10}
\end{equation}
where
\begin{equation}
I\triangleq\sqrt{-1}\text{, }\ \ \ \ U\triangleq\frac{1}{\left\vert
1-b_{3}\right\vert \sqrt{(3+b_{3})(1-b_{3})}}\text{, }\ \ \ \ V\triangleq
(1+b_{3})^{-{3/2}}(3-b_{3})^{-\frac{1}{2}}\text{,}\ \ \ \ P\triangleq\frac {U%
}{V}\ , \label{3.11}
\end{equation}
and%
\begin{equation}
\text{EllipticF}(Q,P)=\int_{0}^{Q} [(1-P^{2}x^{2})(1-x^{2})]^{-\frac{1}{2}%
}dx,\ \ \ \ \text{EllipticK}(P)=\text{EllipticF}(1,P)\ ,
\end{equation}
\noin and formulae (\ref{3.9})-(\ref{3.11}) all give the required new
amplitude pdf. The pdf given in (\ref{3.9}) is illustrated in Fig. 2, which
matches the pdf obtained from simulation, also illustrated in Fig. 2.
The pdf for the $n=2$ case of example 3, with different amplitudes $A_1$ and $A_2$, and independent uniform phases $\phi_1$ and $\phi_2$, can also be directly obtained from
(\ref{3.7}) and (\ref{3.8}) above, by simply setting $A_1=0$, and replacing $b_3$ with $b_2$, $A_3$ with $A_2$, and $A_2$ with $A_1$, and noting that the integrand in (\ref{3.8}) becomes
independent of $\phi_1$, hence giving
\begin{equation}
f_{B_{2}}(b_{2})=\left\{
\begin{array}{cc}\hskip -2mm \frac{{2b_{2}}}
{{\pi \sqrt{4A_{1}^{2}A_{2}^{2}-(b_{2}^{2}-A_{1}^{2}-A_{2}^{2})^{2}}}} \hskip 20mm m_2 \le b_{2} \le M_2, \\
\hskip -4mm 0\text{\hskip 5.5cm otherwise.}%
\end{array} %
\right. \label{3.11b}
\end{equation}
\noin Examples of this pdf are also plotted in Figure 1, where the match with simulation is also observed.
\vskip 4mm\textit{Example 4} \ \ As another example of application of the EGED formula (\ref%
{3.3}), consider the $n=4$ case of example 1. Then, application of formula (\ref{3.3}), and again
a change of variable of integration and simplification, shows that if we let
\svs\noindent }\def\mn{\mvs\noin}\def\bn{\bvs\noin $f_{ {\bf A},{\bf \F} } ({\bf a},{\it \F})=(16\pi^4)^{-1},\ \ \bar b_3^2\den \vert }\def\Ve{\Vert {\bf A}\vert }\def\Ve{\Vert ^2+2\bigl}\def\br{\bigr}\def\Bl{\Bigl}\def\Br{\Bigr[A_1A_2\cos(\phi _1)+A_2A_3\cos(\f2)+A_1A_3\cos(\phi _2-\phi _1)\br]$,
\svs\noindent }\def\mn{\mvs\noin}\def\bn{\bvs\noin and \ \ $\bar\psi_4 \den {b_4^2-\bar b_3^2-A_4^2\over 2A_4 \bar b_3}$,\ \ then we obtain \cite{Maghsoodi-1}
\begin{equation}
f_{B_{4}}(b_{4})=\left\{
\begin{array}{cc}
\tfrac{b_{4}}{2\pi^{3}}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}1_{\{\left\vert
\bar{\psi}_{4}\right\vert \le 1\}}\frac{d\phi_{1}d\phi_{2}}{\sqrt{4A_{4}^{2}%
\bar{b}_{3}^{2}-(b_{4}^{2}-\bar{b}_{3}^{2}-A_{4}^{2})^{2}}} & m_{4}\leq
b_{4}\leq M_{4} \\
0\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ } & \text{otherwise}\ ,
\end{array}
\right. \label{3.13}
\end{equation}
The pdf (\ref{3.13}) is also another hitherto undiscovered pdf, applicable in practical situations such as multiple
access processes having one reference user and three interfering users, (see e.g. Fig. 4), multipath channels
with four taps, communication systems with cochannel interference, and so on. It is illustrated in Fig. 3, for $A_i=1,\ i=1,\cdots,4$. The graph
matches that of the pdf obtained from simulation, also illustrated in Fig. 3. In similar fashions we can apply the EGED formula to derive the exact amplitude pdfs for
all other values of $n$, in terms of multiple integrals of order $n-2$ of functions of the given parameters.
\vskip 4mm\textit{Example 5}\ \ (Random Amplitudes) If the amplitudes are random with the joint density $f_{\bf A}(\bf a)$, and are independent of the phases, then it
follows from formula (\ref{3.3}) that the envelope pdf, ${\tilde f}_{\tilde B}(\tilde b)$, of this case, is the integral of the envelope pdf, $f_B (b)$ of the corresponding constant amplitude case,
w.r.t. the joint density of the amplitudes, i.e.
\begin{equation}
{\tilde f}_{\tilde B}(\tilde b)=\int_{-\infty}^\infty\hskip -3mm\cdots \int_{-\infty}^\infty da_1\cdots da_n\ f_{\bf A}({\bf a}) \ f_{B}(b) \label{3.14}
\end{equation}
For example the extension of the pdf obtained in (\ref{3.8}) of example 3 above, to the random amplitude case is
\begin{equation}
{\tilde f}_{{\tilde B}_{3}}({\tilde b}_{3})=\tfrac{2b_{3}}{\pi^{2}} \int_{-\infty}^\infty \int_{-\infty}^\infty\int_{-\infty}^\infty\hskip -3mm da_1da_2da_3 f_{\bf A}({\bf a})
\int_{0}^{\pi} \frac{ 1_{\{\left\vert { \tilde \psi}_{3}\right\vert \le 1\}} d\phi_{1}}{\sqrt {4a_{3}^{2} \tilde {b}_{2}^{2}-(b_{3}^{2}-\tilde {b}_{2}^{2}-a_{3}^{2})^{2}}}\ \ \ m_{3}\leq b_{3}\leq M_{3}
\label{3.15}
\end{equation}
\noin where $\tilde b_2$ and $\tilde \psi_3$, respectively denote $\bar b_2$ and $\bar \psi_3$ of (\ref{3.7}), with $A_i$ replaced with $a_i$ for $i=1,2,3$.
\noin For example if the amplitudes ${\bf A}=(A_1,A_2,A_3)$ are jointly Gaussian, with mean vector $\boldsymbol {\mu}$ and covariance matrix $\bf C$, and the phases are distributed as in example 3, and are independent of the amplitudes, then the envelope density in this case is be given by (\ref{3.15}), with $f_{\bf A}(\bf a)$ substituted by
\begin{equation}
f_{\bf A}({\bf a}) ={1 \over {2\pi \sqrt {\vert }\def\Ve{\Vert {\bf C}\vert }\def\Ve{\Vert 2\pi} } } \exp \bigl}\def\br{\bigr}\def\Bl{\Bigl}\def\Br{\Bigr[ -{1\over 2 }}\def\inta{\int_{\tau_k}^{\tau_{k+1}} ({\bf a} -{\boldsymbol{\mu}})^T {\bf C}^{-1} ({\bf a} -{\boldsymbol{\mu}}) \br] \label{3.16}
\end{equation}
\noin For example in example 1 of section II, if the amplitudes are the same normally distributed random variable, with zero mean and variance $\sig^2$, then the pdf of the envelope is given by
\begin{align}
f_{B}(b)
&={2\over \sig\pi\sqrt {2\pi} } \int_{-\infty}^\infty da {\bf 1}_{\{\vert }\def\Ve{\Vert a\vert }\def\Ve{\Vert <b/2\}} { \exp -{1\over 2\sig^2}a^2 \over {\sqrt {4a^2-b^2} } } \cr
&={4\over \sig\pi\sqrt {2\pi} } \int_{b/2}^\infty da {\exp -{1\over 2\sig^2}a^2 \over {\sqrt {4a^2-b^2} }} \cr
&={\sqrt 2\over \sig\pi\sqrt {2\pi} } \int_0^2 dx {\exp -{b^2\over 4\sig^2 x} \over {x\sqrt {2-x} } } \cr
&=-{1\over \sig\pi\sqrt {2\pi} } e^{-{1\over 4}c^2} K_0 \bigl}\def\br{\bigr}\def\Bl{\Bigl}\def\Br{\Bigr({1\over 4}c^2\br) \label{3.17}
\end{align}
\noin where the third line in (\ref{3.17}) follows from the change of the variable of integration $a^2={b^2\over 2x}$, and in the last line, $c\den {b\over 2\sig}$, and $K_0$ denotes the modified Bessel function of the second kind of order zero.
\mvs\textit{Example 6} (Mixture of discrete and continuous distributions) The EGED formula
(\ref {3.3}) can also be applied to the cases where the amplitudes and phases have an arbitrary mixture of
discrete and continuous densities. To illustrate this power of the EGED formula, consider a two dimensional
example where the amplitudes have an arbitrary joint continuous pdf $f(a_1,a_2)$, and are independent of the phases, and assume
that the phases are mutually independent, taking only each of the discrete values $0$ or $\pi$, with probability ${1\over 2 }}\def\inta{\int_{\tau_k}^{\tau_{k+1}}$.
Thus in this case the EGED formula becomes
\begin{equation}
f_{B}(b)=2bE\left\{ \left\vert 2A_2A_1\sqrt{1-\Psi_{2}}\right\vert^{-1}
1_{\{\left\vert \Psi_{2}\right\vert \le 1,\ \Phi_{2}\in U_{2}\}}\right\}
\text{ \ \ }m_{2}\le b\le M_{2}\ , \label{3.50}
\end{equation}
and $\phi_1$ replaces $\theta _{n-1}$ in the definition (\ref {3.3b}) of the set $U_2$. Using the Dirac delta functions
we can represent the discrete phase densities and write
\begin{equation}
f_{B}(b)=2b\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} { da_1da_2f(a_1,a_2)\over \left\vert 2a_2a_1\sqrt{1-\psi_{2}}\right\vert}
{1\over 4}\prod_{i=1}^2\int_{-\pi}^\pi d\phi_i\bigl}\def\br{\bigr}\def\Bl{\Bigl}\def\Br{\Bigr[ \del(\phi_i)+\del(\phi_i-\pi)\br]{\bf 1}_{\{\left\vert \psi_{2}\right\vert \le 1,\ \phi_{2}\in U_{2}\}} \label{3.51}
\end{equation}
Let $C_2\den \vert }\def\Ve{\Vert \cos^{-1}\psi_2\vert }\def\Ve{\Vert \in [0,\pi] $, and implement the delta functions $[ \del(\phi_1)+\del(\phi_1-\pi)\br]$ into the set $U_2$, followed by
implementing the delta functions $[ \del(\phi_2)+\del(\phi_2-\pi)\br]$ into the result. The product term in the latter part of (\ref{3.51}), which we denote
by $\tilde P$, will then become
\begin{align}
\tilde P
&\den \prod_{i=1}^2 \int_{-\pi}^\pi d\phi_i \bigl}\def\br{\bigr}\def\Bl{\Bigl}\def\Br{\Bigr[ \del(\phi_i)+\del(\phi_i-\pi)\br] {\bf 1}_{\{\vert }\def\Ve{\Vert \psi_{2} \vert }\def\Ve{\Vert \le 1,\ \phi_{2}\in U_{2}\}}\cr
&= \int_{-\pi}^\pi d\phi_2 \bigl}\def\br{\bigr}\def\Bl{\Bigl}\def\Br{\Bigr[ \del(\phi_2)+\del(\phi_2-\pi)\br] {\bf 1}_{\{\vert }\def\Ve{\Vert \psi_{2} \vert }\def\Ve{\Vert \le 1\}} \Bl[ {\bf 1}_{ \{\phi_2\in \{\pm C_2,\ \pm(C_2-2\pi)\} \} } +
{\bf 1}_{ \{ \phi_2\in \{\pi\pm C_2,\ \pi\pm(C_2-2\pi)\} \} }\Br ]\cr
&={\bf 1}_{\{\left\vert \psi_{2}\right\vert \le 1\} \}} \Bl [ {\bf 1}_{\{C_2=0\}}+ {\bf 1}_{\{C_2=\pi\}} + {\bf 1}_{\{C_2=\pi\}} + {\bf 1}_{\{C_2=0\}}\Br] \cr
&={\bf 1}_{\{\left\vert \psi_{2}\right\vert \le 1\} \}} \Bl [ {\bf 1}_{\{\psi_2=1\}}+ {\bf 1}_{\{\psi_2=-1\}} + {\bf 1}_{\{\psi_2=-1\}} + {\bf 1}_{\{\psi_2=1\}}\Br] \label {3.52}
\end {align}
Substituting the last expression in (\ref{3.52}) back into (\ref{3.51}) whilst noting the denominator singularity at $\psi_2=\pm 1$ we have
\begin{equation}
f_{B}(b)=2b\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} da_1da_2{ f(a_1,a_2)\over \left\vert 2a_2a_1\right\vert}
{1\over 4}{\bf 1}_{\{\left\vert \psi_{2}\right\vert \le 1\} \}} \Bl [ 2\del(\psi_2-1) + 2\del(\psi_2+1)\Br] \label {3.53}
\end{equation}
Now using the formula
\begin{equation}
\del (f(x))=\sum_{i=1}^n {\del(x-x_i) \over \vert }\def\Ve{\Vert f'(x_i) \vert }\def\Ve{\Vert } \notag
\end{equation}
where the $x_i$ are the real roots of $f(x)$, the delta functions of $\psi_2$ can be written in terms of those of $a_2$, using the roots of $\psi_2=\pm 1$, which are $\{x_1=b-a_1,\ a_1\le b\}$,
$\{x_2=a_1-b,\ a_1\ge b\}$, and $x_3=a_1+b$ respectively. We thus obtain the required envelope pdf from (\ref{3.53})
\begin{align}
f_{B}(b)
&={1\over 2 }}\def\inta{\int_{\tau_k}^{\tau_{k+1}} b\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} da_1 da_2 { f(a_1,a_2)\over \left\vert a_2a_1\right\vert}
\sum_{i=1}^3{1\over b}\Bl [\vert }\def\Ve{\Vert a_1 x_i\vert }\def\Ve{\Vert \del(a_2-x_i)\Br ] \cr
&={1\over 2 }}\def\inta{\int_{\tau_k}^{\tau_{k+1}} \Bl [\int_{-\infty}^b f(a_1,b-a_1)+ \int_b^{\infty} f(a_1,a_1-b) + \int_{-\infty}^{\infty} f(a_1,a_1+b)\Br] da_1 \label{3.54}
\end{align}
A simple special case of this example is when the amplitudes are independently negative exponentially distributed, with parameter $\lam$, thus allowing higher probabilities
for lower amplitudes, we obtain an interesting envelope pdf
$$f_{B}(b)={1\over 2 }}\def\inta{\int_{\tau_k}^{\tau_{k+1}} \lam e^{-\lam b} ( \lam b + 1)$$
\vskip 6mm
\section{Application to BER performance}
As an application of the derived formulae, we consider the bit error rate
(BER) evaluation of BPSK systems over narrowband multipath fading channels.
In such channels, the delay spread $T_{m}$ of a channel is small relative to
the inverse signal bandwidth $B^{-1}$ of the transmitted signal, i.e. $%
T_{m}\ll B^{-1}$. This implies that the delay $\tau _{i}$ associated with
the $i$th multipath component $\tau _{i}\leq T_{m}$ $\forall $ $i$, so the
baseband signal $s(t)\approx s(t-\tau _{i})$ $\forall $ $i$. The received
signal representation with narrowband multipath fading and Gaussian noise in
the presence of $n$ multipath components is given by \cite{Goldsmith}\cite{Rappaport-book},
\begin{equation}
r(t)=\sum\limits_{i=1}^{n}\alpha _{i}\cos (2\pi f_{c}t+\phi _{i})+z(t),\text{%
\ \ \ \ \ \ \ \ }0\leq t<T_{s} \label{E-r-Fading}
\end{equation}%
where $\alpha _{i}$ is the amplitude and $\phi _{i}$ is the phase of the $i$th
multipath component, respectively, the carrier frequency is denoted as $%
f_{c}$ and the symbol duration as $T_{s}$. In practice, the amplitudes $%
\alpha _{i}s$ of the individual multipath components do not fluctuate widely
over a local area because the channel characteristics change slowly with
respect to $T_{s}$. However, the phases $\phi _{i}s$ vary greatly even for
very small values of time delays because the distances traversed by the
propagating waves are orders-of-magnitude larger than the wavelength of the
carrier frequency. Therefore, the phases are usually modelled as independent
random variables uniformly distributed over $[-\pi ,\pi ]$ \cite{Proakis-1}-\cite{Rappaport-book}. In such channels, each individual term $\alpha
_{i}\cos (2\pi f_{c}t+\phi _{i})$ in (\ref{E-r-Fading}) is referred to as
a specular component.
Using the SAT formula, we can express (\ref{E-r-Fading}) as $r(t)=B_{n}\cos (\omega
t+\theta _{n})+z(t)$ where the amplitude $B_{n}$ is a random variable with a
pdf that depends on the value of $n$. A considerable simplification for the
channel model is achieved by using the common assumption that $n$ is large
and all amplitudes and phases are mutually independent, thus the Central
Limit Theorem can be invoked to approximate the received signal as a
Gaussian random process, hence the pdf of $B_{n}$ becomes the well-known
Rayleigh distribution. However, applying the Gaussian approximation (GA)
to the modelling of fading channels that have small number of specular
components, will not be accurate enough to describe the effects of the fading
process. Thus, the formulae derived in this work can be of great benefit to
solve the exact model of the narrowband fading channels, and evaluate the BER
of such channels significantly more accurately.
The average bit error probability $P_{B}$ for BPSK in narrowband multipath
fading channels can be obtained by integrating the corresponding error
probability in AWGN over the fading distribution \cite{Proakis-1}\cite{Goldsmith},
\begin{equation}
P_{B}=\int_{0}^{\infty }Q\left( \sqrt{2b_{n}^{2}\frac{E_{b}}{N_{0}}}\right)
\,f_{B_{n}}(b_{n})\text{ }db_{n} \label{E-BER-Fading}
\end{equation}%
where $Q\left( \sqrt{2E_{b}/N_{0}}\right) $ is the probability of bit error
in AWGN channels. For large $n$, the pdf of $b_{n}$ is Rayleigh distributed
and (\ref{E-BER-Fading}) is reduced to a simple closed-form formula \cite%
{Proakis-1}. The computation of $P_{B}$ is usually performed versus the
average signal-to-noise ratio per bit $\bar{E}_{b}/N_{0}=E\left(
b_{n}^{2}\right) E_{b}/N_{0}$, where $E(b_{n}^{2})$ denotes the expected
value of $b_{n}^{2}$. For small $n$ values, the pdf of $b_{n}$ is given by (%
\ref{3.3}) which can be substituted in (\ref{E-BER-Fading}) to compute $P_{B}
$ for any $n$ value. The analytical and simulation results for $P_{B}$ as a
function of\ $\bar{E}_{b}/N_{0}$ are presented in Fig. $4$ for $n=1,2,3,4,$
the $n=2$ case with dependent phases is also included. All multipath
components are assumed to have equal average power. As demonstrated by Fig. $%
4$, the GA has a large discrepancy which is around $3$ dB for $n=3$, and it
is around $1.5$ dB for $n=4$. In the case of $n=2,$ which is known as the
Two-Ray model \cite{Goldsmith}, a large difference in $P_{B}$ is observed
between the dependent and the independent cases. Such behavior can be
understood with the aid of Fig. $1$ which shows that $P(b_{2}<1)$ with
independent phases is much larger than that with dependent phases, i.e., the
probability that the interference is destructive is much larger when the
phases are independent.
\mvs\section{Conclusion}
In this paper we considered the open problem of derivation of exact
distributions of the envelopes of general stochastic sinusoidal sums, with random
amplitudes and phase angles, and its application in an important communication problem.
We have seen that, in the most general case, it is possible to derive exact general formulae for the distribution
of the resultant envelope in terms of just the given joint distribution of the amplitudes
and the phase angles of the signals present in the sum. We derived two such general formulae,
EDDHAPT and EGED, depending on the particular applications at hand. Examples
showed that implementation of these formulae also lead to new explicit distributions
which we applied to compute the exact BER performance of BPSK systems in narrowband fading
channels. The extension of these envelope pdf formulae
to allow random number of signals is immediate under independence. Under no independence,
the extension still follows naturally, but less immediately.
The presented formulae were applied to compute the exact BER performance of
BPSK systems in narrowband multipath fading channels with small number of
resolvable multipath components. All the simulation results were consistent with the
exact theoretical findings, which also showed significant gain in accuracy over the currently
popular Gaussian approximation method, particularly for small number of components.
In all cases, the formulae presented in this paper render themselves to
accurate and efficient numerical implementations, since at worst they merely
involve numerical computation of multiple integrals of known functions, the various
algorithms for which are widely available and coded. In some cases the pdfs can be
written in terms of known integrals such as Elliptic and Bessel
functions. Further numerical implementations and applications may also form
part of future work.
\section{Appendix}
\vskip 3mm}\def\xvs{\vskip 2mm In this appendix we present the proofs of Theorem 1 and Theorem 2.
These were first reported in \cite{Maghsoodi-1}, where details of the proofs and
other related results may be found.
\mvs\subsection{Proof of Theorem {\rm 1 ( The EDDHAPT formula )}}
Formula (\ref{2.2}) can be written as%
\begin{equation}
B_{n}^{2}=\sum_{i=1}^{n}A_{i}^{2}+2A_{n}\sum_{k=1}^{n-1}A_{k}\cos(\phi
_{n}-\phi_{k})+2\hskip -3mm\sum_{(n-1)\geq j>k\geq1}\hskip -3mm A_{j}A_{k}\cos\left( \phi
_{j}-\phi_{k}\right)\ . \label{E_2.9}
\end{equation}
Expanding the $\cos$ terms and writing in terms of the half-angle tangents
we find that
\begin{equation}
B_{n}^{2}=\sum_{i=1}^{n}A_{i}^{2}+2A_{n}\sum_{k=1}^{n-1} {{(1+T_nT_k)^2-(T_n-T_k)^2}\over (1+T_n^2)(1+T_k^2) }A_{k}
+ 2\hskip -3mm\sum_{(n-1)\geq j>k\geq1}\hskip -3mm {{(1+T_jT_k)^2-(T_j-T_k)^2}\over {(1+T_j^2)(1+T_k^2))}} A_{j}A_{k} \label{E_2.9a}
\end{equation}
Hence collecting the $T_n$ terms in (\ref{E_2.9a}), we can write $B_{n}^{2}$ as a quadratic expression in $T_{n}$
\begin{equation}
Q\triangleq (1+T_n^2)(B_{n}^{2}-b_{n}^{2})=aT_{n}^{2}+4A_{n}\bar{S}%
_{n}T_{n}+(I_{n}+2A_{n}\bar{C}_{n})\ . \label{E_2.10}
\end{equation}
The probability on the LHS of (\ref{2.6}) is the multiple integral of the
pdf over the regions where the quadratic (\ref{E_2.10}) is non-positive. Thus distinguishing between the cases
where $a$ or $\Delta_Q$ are positive or negative, we have that $Q$ is non-positive only in the union of three disjoint regions
within the space $\{\mathbf{a},{\mathbf{t}}\}$, namely the regions
$\{\Delta_Q>0, a>0, T_{n}\in\lbrack\bar{a}_{n},\bar{b}_{n}]\}$, $\{\Delta_Q>0, a<0,$ $T_{n}\notin\lbrack\bar{b}_{n},\bar{a}_{n}]\}$, and $\{\Delta_Q<0, a<0\}$
, where $\bar{a}_{n}$ and $\bar{b}_{n}$ are the roots of $Q$, and $\bar b_n$ is the larger root in the first region, and vice-versa in the second. Hence
\begin{align}
P(B_n \leq b_n) =&\int_{-\infty}^\infty\hskip -3mm d{\bf a} \Biggl\{ \int_{-\infty}^{\infty}\hskip -3mm d\bar {\bf t} {\bf 1}_{\{\Delta_Q>0, a>0\}} \int_{\bar a_n}^{\bar b_n}\hskip -3mm dt_n f_{{\bf A},{\bf T}} ( {\bf a, t})
+ \int_{-\infty}^{\infty}\hskip -3mm d\bar {\bf t} {\bf 1}_{\{\Delta_Q>0, a<0\}}
\Bl\{ \int_{-\infty}^{\bar b_n}\hskip -3mm +\int_{\bar a_n}^{\infty} \Br\} dt_n f_{{\bf A},{\bf T}} ( {\bf a, t})\cr
&\shs + \int_{-\infty}^{\infty}\hskip -3mm d\bar {\bf t} {\bf 1}_{\{\Delta_Q<0, a<0 \}} f_{{\bf A},\bar {\bf T}} ( {\bf a}, \bar{\bf t}) \Biggr\}\label{E_2.10b}
\end{align}
The second term in (\ref{E_2.10b}), which we denote by $I_2$, can be written as
\begin{align}
I_2
&=\int_{-\infty}^{\infty}\hskip -3mm d{\bf a} \Biggl\{ \int_{-\infty}^{\infty}\hskip -3mm d\bar {\bf t} {\bf 1}_{\{\Delta_Q>0, a<0 \}} \int_{-\infty}^{\infty} dt_n f_{{\bf A},{\bf T}} ({\bf a, t})
- \int_{-\infty}^{\infty}\hskip -3mm d\bar {\bf t} {\bf 1}_{\{\Delta_Q>0, a<0 \}} \int_{\bar b_n}^{\bar a_n}\hskip -3mm dt_n f_{{\bf A},{\bf T}} ( {\bf a, t})\Biggr\} \cr
&=\int_{-\infty}^{\infty}\hskip -3mm d{\bf a} \Biggl\{ \int_{-\infty}^{\infty}\hskip -3mm d\bar {\bf t} {\bf 1}_{\{\Delta_Q>0, a<0 \}} f_{{\bf A},\bar {\bf T}} ( {\bf a}, \bar{\bf t})
+ \int_{-\infty}^{\infty}\hskip -3mm d\bar {\bf t} {\bf 1}_{\{\Delta_Q>0, a<0 \}} \int_{\bar a_n}^{\bar b_n}\hskip -3mm dt_n f_{{\bf A},{\bf T}} ( {\bf a, t}) \Biggr\} \label{E_2.10c}
\end{align}
Substituting the RHS of (\ref{E_2.10c}), into (\ref{E_2.10b}) for $I_2$, and combining the integrals, we obtain (\ref{2.6}) and the proof of the Theorem is complete \bull
\vskip 3mm}\def\xvs{\vskip 2mm {\it Remark}\ \ Note that in the regions where $a=0$\ and/or\ ${\Delta } }\def\del{{\delta} }\def\lam{{\lambda } _Q=0$, the probability measures are zero, hence these cases need not be included in the regions of integration.
\bvs\subsection{Proof of Theorem {\rm 2 (The EGED formula )} }
\vskip 3mm}\def\xvs{\vskip 2mm The proof is by a method which we call {\it The} ($n-1$)-{\it Conditioning Method}, where we calculate the cdf of $B_n^2$ by conditioning on the value of the random vector $(\bf A, \bar {\bf\F})$, where $\bar {\bf\F}$ denotes the vector $({\bf \F}_1,...,{\bf \F}_{n-1})$. We obtain
\begin{equation}
P\bigl}\def\br{\bigr}\def\Bl{\Bigl}\def\Br{\Bigr(B^2_n\le b_n^2\br)
=\int_{-\infty}^{\infty}\hskip -3mm d{\bf a}\int_{-\pi}^{\pi} d{\it \bar \F}\ P\Bl (\cos(\F_n-\theta _{n-1})\le \psi_n / {\bf a}, {\it \bar \F}\Br) f_{_{_{ {\bf A},\bar {\bf \F}}}}({\bf a}, {\it \bar \F})\ , \label{E_2.11}
\end{equation}
\noin where we have used the EST formula (\ref{3.1}), ${\it \bar\F}$ denotes the vector $(\phi _1,...,\phi _{n-1})$, and without loss of generality $a_{n}$ is assumed to be positive (see the
remark below), and
\begin{equation}
\psi_n\den \Bl( {b_n^2-b_{n-1}^2-a_n^2\over 2a_n b_{n-1}}\Br)\ . \label{E_2.12}
\end{equation}
\noin Taking inverse $\cos$ within the probability integrand of (\ref{E_2.11}), while noting that $(\F_n-\theta _{n-1})\in (-2\pi, 2\pi)$, we obtain
\begin{equation} P\bigl}\def\br{\bigr}\def\Bl{\Bigl}\def\Br{\Bigr(B^2_n\le b_n^2\br) = \int_{-\infty}^{\infty} \hskip -3mm d{\bf a} \int_{-\pi}^{\pi}\hskip -3mm d{\it \bar \F} P\Bl (\vert }\def\Ve{\Vert \cos^{-1} \bar\psi_n\vert }\def\Ve{\Vert +\vert }\def\Ve{\Vert \theta _{n-1}\vert }\def\Ve{\Vert \le \vert }\def\Ve{\Vert \F_n\vert }\def\Ve{\Vert \le 2\pi-\vert }\def\Ve{\Vert \cos^{-1} \bar\psi_n\vert }\def\Ve{\Vert +\vert }\def\Ve{\Vert \theta _{n-1}\vert }\def\Ve{\Vert / {\bf a}, {\it \bar \F}\Br) f_{_{_{{\bf A},\bar {\bf \F}}}}({\bf a}, {\it \bar \F}) \label{E_2.13}
\end{equation}
\vskip -3mm\noin where $\F_n$ and $\theta _{n-1}$ carry the same sign, and the function
\begin{equation}
\bar\psi_n =\left\{
\begin{array}{cc}
\psi_n & |\psi_n|\leq 1
\\
\hbox{sign} (\psi_n) & \text{otherwise}\ ,
\end{array}
\right. \notag
\end{equation}
\noin ensures that the $\cos^{-1}$ function is well-defined. Note that, as required by (\ref{E_2.11}), when $\psi_{n}\ge 1$ or $\le -1$, the RHS of (\ref{E_2.13}) takes the values 1 and $0$ respectively. Now writing (\ref{E_2.13}) in terms of the conditional cdf $F_{_{\F_n/{\bf A},\bar{\bf \F}}}(\phi_n)$ of $\F_n$, given that $({\bf A},\bar {\bf \F})=({\bf a}, {\it \bar \F})$, which we simply denote by $\tilde F_n$, and letting $C_k(\bar\psi_n)\den \vert }\def\Ve{\Vert \cos^{-1}\bar \psi_n\vert }\def\Ve{\Vert -2k\pi$, we obtain
\begin{equation}
P\bigl}\def\br{\bigr}\def\Bl{\Bigl}\def\Br{\Bigr(B^2_n\le b_n^2\br)
=\int_{-\infty}^{\infty}\hskip -3mm d{\bf a}\int_{-\pi}^{\pi}\hskip -3mm d{\it \bar \F} \Bl [\sum_{k=0}^1 \bigl}\def\br{\bigr}\def\Bl{\Bigl}\def\Br{\Bigr\{\tilde F_n(-C_k(\bar\psi_n)+\theta _{n-1}) -\tilde F_n(C_k(\bar\psi_n)+\theta _{n-1})\br\} \Br ]
f_{_{_{{\bf A}, \bar {\bf \F}}}} ({\bf a}, {{\it \bar \F}} )\ . \label{E_2.14}
\end{equation}
\noin While noting that, $f_{B_n}(b_n)=2b_n f_{B_n^2}(b_n^2)$, we differentiate (\ref{E_2.14}) with respect to $b_n^2$ to get the pdf. The derivative of $\bar\psi_n$ w.r.t. $b_n^2$ yields the function $\tilde{\bf 1}\den {\bf 1}_{\bigl}\def\br{\bigr}\def\Bl{\Bigl}\def\Br{\Bigr\{\vert }\def\Ve{\Vert \psi_n\vert }\def\Ve{\Vert \le 1\br\}}$ into the integrand, which is followed by $\tilde f_n$, the conditional pdf of $\F_n$
\begin{equation}
f_{B_n}(b_n) =2 b_n \int_{-\infty}^{\infty}\hskip -3mm d{\bf a}\int_{-\pi}^{\pi}\hskip -3mm d{\it \bar \F} \ \tilde {\bf 1} \Bl [ \sum_{k=0}^1 \bigl}\def\br{\bigr}\def\Bl{\Bigl}\def\Br{\Bigr\{\tilde f_n(-C_k(\psi_n)+\theta _{n-1})
+ \tilde f_n(C_k(\psi_n)+\theta _{n-1})\br \} \Br] V(\psi_n) f_{_{_{{\bf A}, \bar {\bf \F}}}} ({\bf a}, {{\it \bar \F}} ) \label{E_2.15}
\end{equation}
\noin where $V(\psi_n)\den (2a_n b_{n-1} \sqrt { 1- {\psi}_n^2 }\br)^{-1} =- {\partial \vert }\def\Ve{\Vert \cos^{-1}(\psi_n)\vert }\def\Ve{\Vert \over\partial b^2_n}$. Note that $\bar\psi_n$ no longer needs to replace $\psi_n$ within the integrands in (\ref{E_2.15}), since now the function $\tilde {\bf 1}$ forces the integrand to zero when $\psi_n$ is outside the range $[-1,1]$, which is as required by the fact that the probability sum in (\ref{E_2.14}) takes the constant values of 1 or 0, for all $\psi_n\notin [-1,1]$ (see also (\ref{E_2.11})), hence its derivative within the integrand of (\ref{E_2.15}) should be zero in this region. The product of $\tilde f_n$ and the marginal pdf $f_{_{_{{\bf A}, \bar {\bf \F}}}} ({\bf a}, {{\it \bar \F}} )$ in the integrand of (\ref{E_2.15}), gives the joint pdf $f_{ {\bf A},{\bf \F} } ({\bf a},{\it \F})$, hence we have
\begin{align} f_{B_n}(b_n)
&=2 b_n \int_{-\infty}^{\infty}\hskip -3mm d{\bf a}\int_{-\pi}^{\pi}\hskip -3mm d{\it \bar \F} \ \tilde {\bf 1} \Bl [ \sum_{k=0}^1 \bigl}\def\br{\bigr}\def\Bl{\Bigl}\def\Br{\Bigr\{ f_{ {\bf A},{\bf \F} } ({\bf a},{{\it \bar \F}},-C_k(\psi_n)+\theta _{n-1})
+ f_{ {\bf A},{\bf \F} } ({\bf a},{{\it \bar \F}},C_k(\psi_n)+\theta _{n-1}) \br \} \Br] V(\psi_n) \cr
&=2 b_n \int_{-\infty}^{\infty}\hskip -3mm d{\bf a}\int_{-\pi}^{\pi}\hskip -3mm d{\it \F}\ \tilde {\bf 1} \Bl [ \sum_{k=0}^1 \Bl\{ {\bf 1}_{\bigl}\def\br{\bigr}\def\Bl{\Bigl}\def\Br{\Bigr\{\phi _n=-C_k(\psi_n)+\theta _{n-1}\br\}}
+ {\bf 1}_{\bigl}\def\br{\bigr}\def\Bl{\Bigl}\def\Br{\Bigr\{\phi _n=C_k(\psi_n)+\theta _{n-1}\br\}}\Br\}\Br ] V(\psi_n)f_{ {\bf A},{\bf \F} } ({\bf a},{\it \F})\ . \label{E_2.16}
\end{align}
\noin Finally writing (\ref{E_2.16}) in terms of the expectation with respect to the given joint law $f_{_{ {\bf A},{\bf \F} }} ({\bf a},{\it \F})$ we obtain (\ref{3.3}) and the proof of Theorem 2 is complete.\hfill \bull
\vskip 3mm}\def\xvs{\vskip 2mm\textit{Remark} \ \ \ Note that identical steps prove Theorem 2 for the region of integration where $a_{n}$ is negative, as 1 minus the probability in the integrand
of (\ref{E_2.14}) is obtained, which after differentiation gives the same result with a negative sign, which changes to the positive sign, by replacing $a_{n}$ with $-|a_n|$ in front of the function $V(\psi_n)$. Hence, for all $a_{n}$, (\ref{E_2.16}) holds with, $|a_{n}|$ replacing $a_{n}$ in front of $V(\psi_n)$, hence formula (\ref{3.3}) follows.
\vfill\eject
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train/arxiv
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BkiUc3fxK4tBVgEFUslc
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\section{Introduction}
Cold atoms in highly excited Rydberg states with a principal quantum number $n\gg1$ are of interest for basic research and practical applications [1]. In particular, spectroscopy of cold atoms opens up new possibilities for improving the measurement accuracy of the energy and spectroscopy parameters of atoms due to eliminating the Doppler effect [2-4], while the cold Rydberg atoms themselves can be used for implementing logic gates of a quantum computer [5,6].
In this kind of research, an important role is played by the line width of optical and microwave transitions between Rydberg states, because this line width determines the spectral resolution in spectroscopic measurements and the coherence time when performing quantum operations. In the absence of external electromagnetic fields, the ultimate line width is equal to the inverse lifetime of Rydberg states [7,8] and amounts to 1$-$10 kHz for $n>30$. However, this line width is hard to observe experimentally, because Rydberg states have a hyperfine structure (hundreds of kilohertz for \textit{nS} states and tens of kilohertz for \textit{nP} states at $n\sim 40$ [3,4]). The ultimate resolution is also restricted by the finite interaction time $t_{0} $ with resonance radiation, which gives a Fourier width of resonances $\sim 1/t_{0} $. In addition, experimental investigations have revealed a number of problems associated with the broadening of spectral lines in external electromagnetic fields.
First, in magneto-optical traps (MOTs), an inhomogeneous quadrupole magnetic field is used for trapping and cooling atoms. The typical values of the field gradient are 10$-$15 G/cm; therefore, the variation of the magnetic field over a cloud of cold atoms of $\sim1$ mm in size is about 1 G. This field acts on Rydberg atoms and splits their levels by 1$-$3 MHz due to the Zeeman effect. To reduce this effect on line broadening, one applies either a short-term switching off of the magnetic field during measurements [3,4], or a localization of a small excitation volume of Rydberg atoms near the center of the cloud of cold atoms, where the field vanishes [9]. This allows one to reduce the contribution of the inhomogeneity of the MOT magnetic field to the resonance line width to 10-100 kHz. It was also pointed out that, when studying two-photon microwave transitions between Rydberg states with identical magnetic structures, it is possible to observe narrow resonances even without switching off the magnetic field, because the energy levels of these transitions are identically shifted by the magnetic field [2].
\begin{figure*}
\includegraphics[scale=0.8]{Fig1}
\caption{\label{Fig1} (\textit{a}) Scheme of the experiment with cold Rb Rydberg atoms in a magneto-optical trap. (\textit{b}) Scheme of the three-step laser excitation of Rydberg \textit{nP} states in Rb atoms. (\textit{c}) Timing diagram of the pulses in experiments with selective field ionization (SFI) of Rydberg atoms. (\textit{d}) Histogram of the amplified output pulses of the channeltron VEU-6. The observed peaks correspond to 1$-$5 electrons detected from Rydberg atoms by SFI.}
\end{figure*}
Second, transition lines in cold Rydberg atoms are broadened by spurious electromagnetic fields, which always exist in the experiments. Rydberg atoms exhibit the highest sensitivity to spurious electric fields, because the polarizability of Rydberg levels grows as $n^{7} $ [1]. Electric fields of a few mV/cm may be enough to shift and split the transition levels by several or even tens of megahertz. If a spurious field is spatially homogeneous, it can be compensated for with the use of a system of additional electrodes by minimizing the Stark shifts and line broadening [3,4]. If the field is inhomogeneous, then it can be compensated for only partially even in a small interaction volume; therefore, the transition lines experience residual broadening.
Third, experiments with cold Rydberg atoms in a MOT revealed a spontaneous evolution of these atoms into an ultracold plasma upon reaching a certain critical density, which depends on \textit{n} [10]. For the density of atoms of higher than $10^9$ cm$^{-3}$, plasma is formed within a few microseconds in spite of the low kinetic energy of the atoms. Primary charged particles can be produced either due to the photoionization of Rydberg atoms by laser and background thermal radiation, or as a result of collisions of cold Rydberg atoms in the case of attracting potential between them [11]. From the viewpoint of precision spectroscopy, it is important that, in the absence of an extracting electric field, charged particles of the ultracold plasma stay near the Rydberg atoms for a long time. The inhomogeneous electric field of these particles may lead to the broadening and shift of spectral lines in the Rydberg atoms due to the Stark effect [12].
In the present study, we investigated the effect of cold photoions on the spectra of the F\"orster energy-exchange resonance Rb(37\textit{P})+Rb(37\textit{P})$\rightarrow$Rb(37\textit{S})+Rb(38\textit{S}) and microwave transitions $nP\to n'S,\; n'D$ between Rydberg states of cold rubidium atoms in a MOT. In our experiments, we have revealed that one of the main sources of line broadening is the inhomogeneous electric field of cold photoions that are generated under the excitation of Rb(\textit{nP}) Rydberg atoms by broadband pulsed laser radiation. By using the spectroscopy of microwave transitions and exchange interactions, we implemented a nondestructive spectroscopic method for the diagnostics of an ultracold plasma in a gas of cold rubidium Rydberg atoms in a MOT. The applicability of this method was first demonstrated in [12]. In contrast to [12], in the present experiments we obtain higher sensitivity to weak electric fields from a small amount of photoions and higher spatial resolution owing to the localization of the excitation volume in the geometry of tightly focused crossed laser beams.
\section{EXPERIMENTAL SETUP}
A detailed description of the experimental setup was given in our paper [9]. The experiments are carried out with cold Rb atoms trapped in a MOT, which is shown in Fig.~1(a). The trap consists of a vacuum chamber with optical windows, a heated source of Rb atoms, anti-Helmholtz coils for creating a three-dimensional gradient of a magnetic field of 10$-$15 G/cm at the center of the MOT, and a laser cooling system based on two semiconductor lasers with external cavities tuned to a wavelength of 780 nm. The atoms are cooled by three orthogonal pairs of light waves. The cooling laser is tuned with a red detuning of $\sim$20 MHz to the closed transition $5S_{1/2} (F=3)\to 5P_{3/2} (F=4)$ of the $^{85} {\rm Rb}$ isotope, and the repumping laser to the transition $5S_{1/2} (F=2)\to 5P_{3/2} (F=3)$. After the fine tuning of the wavelengths of the lasers, a cloud of cold atoms with a size of 0.5$-$1 mm and temperature of 100$-$300 $\mu $K arises at the center of the trap. In our experiments, about $10^6$ atoms of $^{85} {\rm Rb}$ are trapped in the MOT, which corresponds to a number density of $10^9$ cm$^{-3}$.
The excitation and detection of Rydberg atoms occur in the space between two stainless-steel plates with holes of 10 mm in diameter at the centers [Fig.~1(a)]. To form a homogeneous electric field, the holes are closed by optically transparent (85\% transparency) metal meshes. The distance between the plates is 10 mm. The electric field is used for the spectroscopy of the Stark effect and for detecting Rydberg atoms by the selective field ionization (SFI) method [1]. The electrons generated upon ionization are accelerated by the electric field, fly through the upper mesh, and are directed by a deflecting electrode into the input horn of a channel electron multiplier VEU-6. Pulse signals from the output of this multiplier are processed by a fast analog-to-digital converter, a box-car integrator, and a computer. This allows one to monitor the number of atoms and the populations of the Rydberg states in a wide range of the principal quantum number \textit{n}.
The excitation of cold Rb atoms to the Rydberg states \textit{nP} (\textit{n}=30-60) is performed by a three-step scheme [Fig.1(b)]. The first step $5S_{1/2} \to 5P_{3/2} $ is excited by a cooling laser operating in a continuous-wave mode. At the second step $5P_{3/2} \to 8S_{1/2} $ the radiation of a pulsed Rhodamine 6G dye laser (with a wavelength of 615 nm) is used at a pulse repetition rate of 5 kHz. The 8\textit{S} state has a lifetime of 160 ns and rapidly decays into lower lying \textit{P} states, including the 6\textit{P} state. In turn, this state has a lifetime of 110 ns and rapidly populates the state 6\textit{S}\,, which has a lifetime of 50 ns. At the third step the Rydberg \textit{nP} states are excited from the 6\textit{S} state by the radiation of a pulsed titanium-sapphire laser with a wavelength of 743 nm. The width of both laser pulses is about 50 ns and the lasers themselves are time synchronized. The radiation of the second- and third-step lasers is focused onto a cloud of cold atoms in the crossed-beam geometry [Fig.~1(a)] with waist diameters of 9$-$10 $\mu $m. In the region of intersection of the focused beams, an effective excitation volume of Rydberg atoms with a size of 20$-$30 $\mu $m is formed, which depends on the relative position of the waists and on whether or not the transitions are saturated.
The diagnostics of cold Rb Rydberg atoms in a MOT is carried out by the method of microwave spectroscopy [1]. A backward-wave-tube oscillator G4-142 is used as a source of microwave radiation. The frequency of the oscillator is locked to a quartz frequency synthesizer and is swept within 53$-$80 GHz range at a line width of 20$-$50~kHz. The radiation is fed through the MOT window. The spectra of microwave transitions give information on the presence of external electric and magnetic fields and on their spatial distribution, because the excitation volume can be shifted within the cold atoms cloud.
The time diagram of signals in the detection system is shown in Fig.~1(c). After every laser pulse that excites a part of cold atoms to the initial \textit{nP} Rydberg state, atoms freely interact with each other or with microwave radiation during a period of $t_0=$1$-$10 $\mu $s. Then a ramp of the ionizing electric field with a rise time of about 2~$\mu $s is switched on. Depending on the state of a Rydberg atom, the ionization occurs at different instants of time after a laser pulse. Then a pulsed ionization signal is detected at the output of VEU-6 by means of two gates that correspond to the initial \textit{nP} and final $n'L$ states of a Rydberg atom. The number of electrons detected from a single laser pulse is determined by the number of Rydberg atoms in the excitation region and by the detection efficiency of VEU-6 [13]. Figure 1(d) shows a histogram of the amplitudes of output signals of VEU-6. The histogram shows several peaks that correspond to different numbers (from 1 to 5) of detected Rydberg atoms. After every laser pulse the data acquisition system measures the amplitude of the output signal of the VEU-6 in both detection channels (for the initial and final states); then, by a previously measured histogram, the system determines the number of atoms detected in each channel, and upon accumulating data from $10^3-10^5$ laser pulses sorts the signals according to the number of atoms and calculates the probability of transition from the initial to the final Rydberg state.
\section{SPECTROSCOPY OF A F\"ORSTER RESONANCE IN THE PRESENCE OF COLD PHOTOIONS}
A F\"orster resonance, or the resonance energy-exchange transfer, occurs due to the dipole-dipole interaction between neighboring Rydberg atoms and is observed in the case of two or more atoms when the energy intervals of the up and down transitions from the initial state are equal [1]. In our experiments, we used the 37$P_{3/2}$ state as the initial state; for this state, the resonance interaction of two atoms occurs according to the scheme [9]
\noindent
\begin{equation} \label{Eq1}
\begin{array}{l} {{\rm Rb}(37P_{3/2} )+{\rm Rb}(37P_{3/2} )\to } \\ \\{{\rm Rb}(37S_{1/2} )+{\rm Rb}(38S_{1/2} ).} \end{array}
\end{equation}
\noindent The exact energy resonance for this process is achieved by the Stark tuning of Rydberg levels in the electric field [Fig.~2(a)]. As a result of interaction, one of the atoms goes to the lower state 37$S_{1/2}$ and the other atom simultaneously goes to the higher state 38$S_{1/2}$. The interaction may involve not only two but a larger number of atoms located in the excitation volume, provided that they are situated sufficiently closely.
\begin{figure*}
\includegraphics[scale=0.8]{Fig2}
\caption{\label{Fig2} (\textit{a}) Scheme of tuning the double Stark resonance $37S_{1/2} -37P_{3/2} -38S_{1/2} $ in Rb atoms in the electric field \textit{E}=1.8-2.0~V/cm. (\textit{b}) Records of the spectra of the F\"orster resonance Rb(37\textit{P})+Rb(37\textit{P})$\rightarrow$Rb(37\textit{S})+Rb(38\textit{S}) at $\pi $-polarization of the exciting laser and selective detection of 1$-$5 Rydberg atoms. The vertical arrows indicate the calculated positions of the three resonances. (\textit{c}) The same as in (\textit{b}) for $\sigma $-polarization of the exciting laser.}
\end{figure*}
To calculate the evolution of the populations of Rydberg states, we solve a quantum-mechanical problem for a quasimolecule formed by \textit{N} interacting Rydberg atoms [14,15]. In [15] we have shown that for a weak dipole-dipole interaction of frozen Rydberg atoms the evolution of the population of the final state 37\textit{S} in each atom and the line shape of the F\"orster resonance are described by the expression
\noindent
\begin{equation} \label{Eq2}
\rho _{N} (t_{0} )\approx \frac{1}{N} \, \; \frac{V_{N}^{2} }{V_{N}^{2} +\Delta ^{2} /4} \sin ^{2} \left(t_{0} \; \sqrt{V_{N}^{2} +\Delta ^{2} /4} \right),
\end{equation}
\noindent where $V_{N} $ is the total energy of the dipole-dipole interaction of all pairs of atoms (in the frequency scale), $\Delta =\left(2E_{37P} -E_{37S} -E_{38S} \right)/\hbar $ is the electric-field controlled detuning from the exact energy resonance, $t_{0} $ is the interaction time, and $1/N$ is the normalization constant. This formula is similar to the formula describing the Rabi oscillations in a two-level atom; as $V_{N} \to 0$, the line shape is given by the Fourier spectrum of a square pulse of width $1/t_{0} $. Accurate numerical calculations have shown that Eq.\eqref{Eq2} gives correct results for $\rho _{N} (t_{0} )<0.1$.
The change in the states of atoms is detected by the SFI method by measuring the population of the final state 37$S_{1/2}$. The distinctive feature of our experiments is the possibility of determining the number \textit{N} of Rydberg atoms and their states after every laser pulse. As a result, we measure the following signals:
\begin{equation} \label{Eq3}
S_{N} =\frac{n_{N} (37S)}{n_{N} (37P)+n_{N} (37S)+n_{N} (38S)}\; .
\end{equation}
\noindent Here $n_{N} (nL)$ is the total number of Rydberg atoms in state \textit{nL} that are detected during the measurement time in the case of \textit{N} Rydberg atoms. In fact, the signal $S_N$ represents the mean probability of transition in each atom after the interaction with \textit{N}$-$1 surrounding atoms; i.e., $S_{N} =\rho _{N} $ if an ideal SFI detector with a detection efficiency $\eta =1$ is used. The efficiency of our detection system is 65\%; therefore, actually $S_N$ represents a mixture of signals $\rho _{i} $ from a larger number of actually excited atoms $i\ge N$ [14].
\begin{figure*}
\includegraphics[scale=0.6]{Fig3}
\caption{\label{Fig3} Experimental records of the spectrum of the F\"orster resonance Rb(37\textit{P})+Rb(37\textit{P})$\rightarrow$Rb(37\textit{S})+Rb(38\textit{S}) in cold Rb atoms for 1$-$3 detected Rydberg atoms at $\pi $-polarization of the exciting laser radiation: (\textit{a}) In the presence of cold photoions, which appeared at laser excitation of the 37\textit{P} Rydberg state. The electric field of the photoions broadens the F\"orster resonance. (\textit{b}) Additional electric pulse (5 V/cm, 2~$\mu $s) at the moment of laser excitation rapidly extracts photoions. A narrow F\"orster resonance in the field 1.79 V/cm is observed.}
\end{figure*}
Since the state $37P_{3/2} $ is split in the electric field into two sublevels, $|M_{J} |=1/2$ and $|M_{J} |=3/2$, where $M_J$ is a moment projection onto the direction of the electric field, in general three F\"orster resonances should be observed for the electric fields of 1.79, 1.89, and 2.0 V/cm (these values are obtained from the numerically calculated Stark diagram of Rydberg levels). In our first paper [9] we could not resolve these resonances due to the broadening and overlapping of individual resonances due to the inhomogeneity of the electric field and spurious electromagnetic fields, as well as due to the weakness of interatomic interaction in the excitation volume with a size of $\sim$100 $\mu $m.
In the subsequent experiments, the excitation volume was first reduced by making a small separation of the waists of two exciting laser beams, so that they only slightly overlapped, forming a reduced effective excitation volume with a size of 30$-$40 $\mu $m. In spite of the fact that such a technique reduces the average number of atoms in the excitation volume and degrades the signal-to-noise ratio, we could record individual components of the F\"orster resonance. These records are shown in Figs.~2(b) and 2(c) for $\pi $ and $\sigma $ polarizations of the exciting laser radiation. In the case of $\pi $ polarization, only the Stark sublevel $37P_{3/2} (|M_{J} |=1/2)$ is excited from the $6S_{1/2} $ state; therefore, only a single F\"orster resonance is observed in a field of 1.79 V/cm. In the case of $\sigma $ polarization, both Stark sublevels $|M_{J} |=1/2$ and $|M_{J} |=3/2$ are excited, and the population of the latter sublevel turns out to be three times higher; therefore, the F\"orster resonance in a field of 1.79 V/cm is three times weaker, and one can clearly observe only resonances in the fields of 1.89 and 2.9 V/cm over the background of noise. In this experiment, the width of each resonance was 30$-$40 mV/cm, which corresponds to 3$-$4 MHz when recalculated to the frequency scale. Thus, the spectral resolution was still several times poorer than that calculated by Eq.~\eqref{Eq2}, which should be about 1 MHz with regard to the unresolved hyperfine structure of Rydberg levels (about 0.7 MHz in total) and the finite interaction time (0.3 MHz for an interaction time of 3 $\mu $s). This means that other sources of line broadening listed in the Introduction contribute to the width of the resonances.
To date, we have undertaken a number of efforts that have allowed us to reduce the excitation volume to 20$-$30~$\mu $m by tighter focusing the laser beams and more carefully superimposing their waists. We have also undertaken special measures to eliminate spurious electric fields associated with ground loops and stray fields in the detection system. This has allowed us to substantially reduce the effect of inhomogeneity of the electric field and improve the spectral resolution when detecting F\"orster resonances [14].
To analyze additional sources of line broadening, we recorded a single F\"orster resonance (at $\pi $ polarization of laser radiation) with a smaller step in the electric field scale and with a larger accumulation time to improve the signal-to-noise ratio (Fig.~3). The F\"orster resonance was observed in a weak dc electric field, which was varied near the value of 1.79 V/cm. The interaction time $t_{0} =3$ $\mu $s was defined by the time interval between a laser pulse and the instant when a strong ionizing field for SFI is switched on. Some of the records showed a strongly asymmetric broadening of the F\"orster resonance for higher values of the electric field [Fig.~3(a)]. While the resonance amplitude strongly depended on the number of detected Rydberg atoms, its width and shape were constant. This indicates that the asymmetric broadening is attributed to an inhomogeneous electric field of unknown nature, rather than to the interaction between Rydberg atoms.
According to our previous experiments [9], the measured inhomogeneity of the electric field formed by the plates of the detection system in the excitation volume of 100 $\mu $m in size was not greater than 0.5\%, which amounts to 10 mV/cm for a field of 2 V/cm. Since in our present experiment we have even smaller excitation volume of 20$-$30~$\mu $m in size, the field inhomogeneity cannot be greater than 2$-$3~mV/cm. However, the broadening of resonances shown in Fig.~3(a) amounts to 50 mV/cm. The only source of such an inhomogeneous electric field can be the charged particles situated directly in the excitation volume.
It was suggested that the source of broadening of the F\"orster resonance is the inhomogeneous electric field of photoions generated under a pulsed excitation of Rb Rydberg atoms by a broadband laser radiation via the three-step scheme shown in Fig.~1(b). It was observed in the experiments that laser pulses used on the second and third steps produced a strong photoionization signal during a laser pulse if one additionally applies an extracting electric field of higher than 10 V/cm. Since photoionization does not require resonance radiation, all photons of laser pulses take part in the photoionization, whereas only resonance photons (the absorption line width $\sim$5 MHz) from the entire spectrum of the pulsed lasers (the line width of the lasers $\sim$10 GHz) take part in the excitation of Rydberg states. The photoionization signal turns out to be comparable with, or even greater than, the signals from the Rydberg atoms. The wavelengths of the lasers of the second and third steps allowed one to photoionize all the states above the 6\textit{S} state that are populated under a spontaneous decay of the state 8\textit{S}\,. Therefore, it seems impossible to calculate the total photoionization probability because of the unknown distribution of populations over these levels and their time dynamics.
Photoionization gives rise to the appearance of free electrons and cold photoions Rb$^+$. A weak electric field of about 1.79 V/cm applied to observe the F\"orster resonance extracts photoelectrons from the excitation volume of 20 $\mu $m in a time of about 1 ns, whereas the extraction of cold photoions takes about 0.5 $\mu $s. This time period is comparable with the interaction time $t_{0} =3$ $\mu $s of atoms with each other. Due to the Stark effect, the cold ions lead to the deviation of the frequencies of atomic transitions during the interaction time. The asymmetric broadening of the F\"orster resonance in Fig.~3(a) is about 50 mV/cm and it immediately yields the mean electric field of photoions, which start to move to the negatively charged plate of the detection system under our external field. Thus, the F\"orster resonance spectroscopy allows one to measure the mean field of an ultracold plasma in a gas of cold Rydberg atoms.
Our method for detecting charged particles allows one to measure also the number (from 0 to 10) of photoions produced after each laser pulse. The probability of generation of a certain number of ions is described by the Poisson statistics. Figure 3(a) shows that the left wing of the F\"orster resonance is not broadened; this may imply that the signal of this wing corresponds to the cases when photoions are not generated. The asymmetric signal on the right wing corresponds to the cases when photoions are first generated and then are slowly moved by the control electric field to one side. The possibility of the direct measurement of the mean electric field and of the number of photons is of interest for studying the initial stage of formation of ultracold plasma.
To check the assumption on the presence of photoions, we suggested that during a laser pulse one should apply an additional electric-field pulse of amplitude 5 V/cm and duration of 2 $\mu $s; this pulse should extract photoions in a time of 0.2$-$0.3 $\mu $s [Fig.~3(b)]. During this pulse, atoms are far-detuned from the F\"orster resonance and do not interact with each other, while photoions are completely removed from the excitation volume. After the end of the pulse, the electric field decreases to the resonance value, and then atoms interact with each other during $t_{0} =3$ $\mu $s. The experimental record of the F\"orster resonance demonstrated in Fig.~3(b) shows that photoions indeed cease to affect the line shape and that the resonance becomes nearly symmetric (the residual asymmetry is associated with the exponential transient process during switching off the extracting pulse). For a one-atom signal $S_{1} $ the resonance width decreases to $16.4\pm 0.3$ mV/cm, which corresponds to $1.94\pm 0.04$ MHz on the frequency scale. For multiatom signals the resonance width turns out to be larger due to the larger energy of interatomic interactions. A detailed analysis of the shape of the F\"orster resonance as a function of the number of detected Rydberg atoms was carried out in [14,15]; therefore, here we do not discuss this issue.
\section{SPECTROSCOPY OF MICROWAVE TRANSITIONS IN THE PRESENCE OF COLD PHOTOIONS}
To absolutely calibrate the electric field strength in experiments on spectroscopy of the F\"orster resonance ${\rm Rb}(37P_{3/2} )+{\rm Rb}(37P_{3/2} )\to {\rm Rb}(37S_{1/2} )+{\rm Rb}(38S_{1/2} )$, we used a resonant microwave radiation. The exact calibration was needed to measure and compensate for the spurious electric fields in the detection system. As can be seen from Fig.~2(a), for the calibration the frequency of the microwave field should be tuned to the frequency of the exact double resonance $37S_{1/2} -37P_{3/2} -38S_{1/2} $ that arises in an electric field of 1.79$-$2.0 V/cm. When the electric field strength is varied, the measured signal of the population of the state $37S_{1/2} $ should exhibit a resonance similar to the F\"orster resonance in Fig.~3; however, the amplitude of this resonance is determined by the intensity of the microwave field rather than by the energy of interatomic interactions. To prevent additional broadening of the resonance due to these interactions, one should consider a one-atom signal $S_{1} $\,.
Figure 4 shows the records of such resonances for a microwave field frequency of 81.072 GHz and for $\pi $ polarization of the exciting laser radiation. In the presence of photoions [Fig.~4(a)], the resonance peak at 1.79~V/cm is broadened asymmetrically. Moreover, a forbidden component arises in the spectrum in a field of 2.0 V/cm, that can only be observed if the Stark sublevel $37P_{3/2} (|M_{J} |)=3/2$ is populated. However, in the case of $\pi $ polarization of the laser radiation this sublevel should not be populated; therefore, one should conclude that the electric field of photoions in this experiment had a transverse component with respect to the dc electric field in the detection system. The transverse field lead to the precession of the magnetic moment and ultimately lead to the population of the state $37P_{3/2} (|M_{J} |)=3/2$, as we pointed out in [16]. This fact is confirmed by the record of a microwave resonance in the absence of photoions [Fig.~4(b)]. In this case, one observes a single narrow peak in a field of 1.79 V/cm, which was used for the absolute calibration of the electric field strength in the detection system in our experiments.
\begin{figure*}
\includegraphics[scale=0.6]{Fig4}
\caption{\label{Fig4} Calibration of the position of the F\"orster resonance in a microwave field with the 81.072 GHz frequency at $\pi $-polarization of the exciting laser radiation: (\textit{a}) In the presence of cold photoions, which appeared at laser excitation of the 37\textit{P} Rydberg state. The electric field of the photoions broadens the microwave resonance and causes an appearance of the forbidden component in the field 2.0 V/cm. (\textit{b}) Additional electric pulse (5 V/cm, 2~$\mu $s) at the moment of laser excitation rapidly extracts photoions. A single narrow microwave resonance in the field 1.79 V/cm is observed.}
\end{figure*}
We also carried out experiments on the spectroscopy of one-photon microwave transitions 37\textit{P}$-$37\textit{S}\,, 40\textit{P}$-$40\textit{S}\,, 45\textit{P}$-$43\textit{D}\,, 50\textit{P}$-$50\textit{D}\,, 55\textit{P}$-$55\textit{D}\,, and 60\textit{P}$-$61\textit{D} between Rydberg states of cold Rb atoms in a MOT. The choice of these transitions was motivated by the fact that their frequencies fall into the tuning range (50$-$80 GHz) of the G-142 microwave generator. The aim of the experiments was to study the possibility of excitation and diagnostics of high Rydberg states for further investigations in the field of quantum information, as well as to improve the values of quantum defects and transition frequencies in Rb Rydberg atoms. To this end, we carried out a detailed analysis of the line shapes of microwave resonances.
As already discussed in the Introduction, the ultimate spectral resolution in experiments on the spectroscopy of cold Rydberg atoms is determined by their finite lifetime or finite interaction time between atoms and microwave radiation. Since the interaction time is usually limited to the values of 1$-$10 $\mu $s and the lifetimes are greater than tens of microseconds for $n>30$ [7,8], the ultimate line width is mainly determined by the Fourier width of microwave pulses and amounts to 0.1$-$1 MHz. The probability of a one-photon transition and its line shape for a square microwave pulse of duration $t_{0} $ are described by the well-known formula
\begin{equation} \label{Eq4}
\rho (t_{0} )\approx \frac{\Omega ^{2} }{\Omega ^{2} +\delta ^{2} } \sin ^{2} \left(\frac{t_{0} }{2} \sqrt{\Omega ^{2} +\delta ^{2} } \right),
\end{equation}
\noindent where $\Omega $ is the Rabi frequency and $\delta $ is the frequency detuning from the exact resonance. This formula describes Rabi oscillations in a two-level atom, and for $\Omega \to 0$ the line shape is given by the Fourier spectrum of a square pulse of width $1/t_{0} $. In our experiments, the interaction time was chosen equal to $t_{0} =3$ $\mu $s, which corresponds to a Fourier width of 0.33 MHz. However, in the first experiments, we observed approximately equal line widths for all microwave transitions, which was 2$-$3 MHz irrespective of the interaction time. Moreover, the measured transition frequencies were appreciably shifted with respect to the calculated values obtained with the use of quantum defects from [2].
As the main mechanism of the shift and broadening of microwave transition lines, we considered the Stark effect in spurious electric fields. To study this phenomenon, we applied an additional small dc voltage of either polarity to the plates of the detection system that formed a homogeneous electric field in the vertical direction; this small dc field was used to compensate for the dc component of the spurious electric field. We found an unaccounted electric field of strength 100$-$150 mV/cm in the detection system, which penetrated into the excitation region from the deflecting electrode through the metal mesh of the detection system [see Fig.~1(a)]. The compensation of the vertical component of this field allowed us to significantly reduce the shifts and broadenings of microwave resonances to values of 2 MHz. As an example, Fig.~5 shows the spectra of the microwave resonance $37P_{3/2} \to 37S_{1/2} $ for various values of the compensating dc voltage \textit{U}. The figure also presents the resulting electric field calculated by the shift of the resonance with respect to the calculated position. Note that even for the minimum shift of the microwave resonance Fig.~5 shows broadening due to the inhomogeneous magnetic field of the MOT. Later, this broadening was eliminated by adjusting the position of the excitation volume to the point of zero magnetic field [9].
\begin{figure}
\includegraphics[scale=0.7]{Fig5}
\caption{\label{Fig5} Spectrum of the microwave resonance $37P_{3/2} \to 37S_{1/2} $ between Rydberg states of Rb atoms at various values of the compensating voltage \textit{U} in the detection system. Also shown are the values of the resulting vertical component of the electric field calculated from the shift of the resonance with respect to the theoretical position (dashed line).}
\end{figure}
Moreover, often additional spurious electric fields appeared to the end of the experiments; these fields were presumably attributed to the deposition of Rb atoms on the metal parts of the detection system and to the rise of a contact potential difference. The elimination of these fields required a periodic adjustment of the compensating voltage in the detection system. This phenomenon was earlier observed in a number of other publications on microwave spectroscopy of Rydberg atoms [17].
Figure 6(a) shows the spectra of the microwave transitions $37P_{3/2} \to 37S_{1/2} $, $45P_{3/2} \to 43D_{3/2} $, and $50P_{3/2} \to 50D_{3/2} $ obtained after the adjustment of the position of the excitation volume to a zero magnetic field point and after the compensation of the vertical component of the spurious electric field, but without the extracting pulse for photoions. The arrows indicate the calculated positions of the resonances according to the quantum defects from [2]. One can see that the measured transition frequencies differ from the calculated ones by at most 1~MHz. Nevertheless, the observed resonance widths are still rather large and amount to 1.5$-$2 MHz, which provides evidence for the existence of a time-dependent electric field of photoions generated under laser excitation, just as in the case of the spectroscopy of the F\"orster resonance.
\begin{figure*}
\includegraphics[scale=0.6]{Fig6}
\caption{\label{Fig6} Spectra of various microwave resonances between Rydberg states of Rb atoms: (\textit{a}) In the presence of cold photoions, which appeared at laser excitation of the initial \textit{nP} Rydberg state. The electric field of the photoions noticeably broadens the microwave resonances. (\textit{b}) Additional electric pulse (5 V/cm, 2~$\mu $s) at the moment of laser excitation rapidly extracts photoions, resulting in a narrowing of the microwave resonances. The arrows indicate the calculated positions of the resonances.}
\end{figure*}
The application of an additional extracting pulse for photoions allowed us to substantially reduce the width of microwave resonances [Fig.~6(b)] to 1.1 MHz for the transition $37P_{3/2} \to 37S_{1/2} $, to 0.64 MHz for the transition $45P_{3/2} \to 43D_{3/2} $, and to 0.38 MHz for $50P_{3/2} \to 50D_{3/2} $. In the case of the transition $37P_{3/2} \to 37S_{1/2} $ one should take into account that in the $^{85}$Rb isotope this transition originally has a hyperfine structure with a total width of about 0.44 MHz [9]. In addition, the power broadening by a microwave field also contributes to the total width of each resonance, because the Rabi frequency for the investigated resonance amounts to $\Omega /(2\pi )=$0.1$-$0.2~MHz. With regard to these sources of line broadening, the widths of microwave transitions in Fig.~6(b) are close to the least possible ones. Our experiments have shown that a necessary condition for this is a fast removal of photoions from the interaction region.
In spite of the fact the ultimate line width is reached in Fig.~6(b), the measured transition frequencies remain shifted with respect to the calculated values by $-$0.23~MHz for the transition $37P_{3/2} \to 37S_{1/2} $, by $-$0.73~MHz for the transition $45P_{3/2} \to 43D_{3/2} $, and by +0.81~MHz for the transition $50P_{3/2} \to 50D_{3/2} $, whereas the absolute accuracy of frequency measurements in our experiments is 0.1$-$0.2 MHz. This means that there is an uncompensated transverse electric field in the detection system. Applying a compensating voltage to the plates of the detection system, we can compensate only for the vertical component of the spurious electric field. The transverse field component, which arises due to the penetration of the field of the deflecting electrode or due to the contact potentials, remains uncompensated. This is indirectly confirmed by the fact that for the transitions $37P_{3/2} \to 37S_{1/2} $ and $45P_{3/2} \to 43D_{3/2} $ the shift is to lower frequencies, whereas for the transition $50P_{3/2} \to 50D_{3/2} $ the shift is to higher frequencies, in full agreement with the signs of the polarizability differences of these levels in a weak electric field. If our measurements had a systematic error, all the resonances would be shifted to the same side and by the same value. Moreover, along with the calculated values of the polarizabilities of Rydberg levels, the above shifts of the transition frequencies give the same value of the spurious transverse electric field, equal to $0.12\pm 0.2$ V/cm, which also indirectly confirms our interpretation of these shifts.
We can conclude that, to increase the accuracy of spectroscopic measurements on microwave transitions in cold Rydberg atoms, one should, in addition to removing photoions, carefully compensate for spurious electric fields in all coordinates. To this end, one should equip the detection system with additional electrodes that would allow one to vary the sign and the magnitude of the transverse compensating field in the interaction region. In [4], such compensation, combined with fast switching off the magnetic field of a MOT for the interaction time with microwave radiation, allowed the authors to observe resonances with a width of about 30 kHz.
In conclusion, we note that the method of microwave spectroscopy of Rydberg atoms was first applied to measuring weak electric fields in ultracold plasma of Rb atoms in [12]. In that study, the magnetic field of a MOT was not switched off; therefore, a one-photon microwave resonance had a width of 5 MHz, and the limit sensitivity of measuring the microscopic field of the plasma was 0.1~V/cm. To observe the shift of the microwave resonance, one needed about 10$^4$ Rb ions, because the size of the excitation volume was greater than 100 $\mu $m. The distinctive feature of our experiments is the use of a narrower resonance at the point of zero magnetic field in a MOT and thereby a higher sensitivity to weak electric fields (of about 10 mV/cm), the possibility of detecting microscopic fields form a small number (1$-$10) of photoions, and the higher spatial resolution (20$-$30 $\mu $m) due to the localization of the excitation volume in the geometry of tightly focused crossed laser beams.
\section {CONCLUSIONS}
Based on the experimental data obtained, we can conclude that one of the fundamental sources of line broadening in ensembles of cold Rydberg atoms are photoions that are generated under the excitation of Rydberg states by broadband pulsed laser radiation. Since photoionization does not require resonance radiation, all the photons of pulsed lasers take part in photoionization, whereas only resonance photons take part in the excitation of Rydberg states; therefore, the photoionization probability turns out to be comparable with the probability of excitation of Rydberg atoms.
Photoionization gives rise to cold photoions that may stay in the excitation volume for a long time. Due to the Stark effect, the presence of cold photoions leads to the deviation of the frequency of atomic transitions during the interaction time, thus leading to asymmetric broadening of the F\"orster and microwave resonances. On the other hand, asymmetric broadening allows one nondestructively measure the mean field of ultracold plasma in a gas of cold Rydberg atoms. Moreover, our method for detecting charged particles makes it possible to measure the number of photoions generated in every laser pulse. The possibility of direct measurement of the mean electric field and the number of photoions may be of interest when studying various stages of formation of ultracold plasma in ensembles of Rydberg atoms.
The obtained data on the line shapes of microwave and F\"orster resonances in Rydberg atoms can also be used for developing various methods of narrowing the lines and increasing the measurement accuracy of transition frequencies. For example, a method to eliminate the unwanted generation of photoions consists in using a lower power narrow-band continuous-wave laser radiation to excite Rydberg states. In our case, a decrease in the line width of laser radiation from 10 GHz to 1 MHz would lead to a 10$^4$ time decrease in the generation probability of photoions, whereby the latter will hardly affect the observed spectra. This will allow one to analyze other possible sources of formation of ultracold plasma in dense ensembles of cold Rydberg atoms, such as the photoionization of Rydberg atoms by background thermal radiation [18], the Penning ionization under dipole-dipole interaction of Rydberg atoms [19], and the photoionization of cold atoms by laser radiation [20].
\begin{acknowledgments}
This work was supported by the Russian Foundation for Basic Research (Grant Nos. 10-02-00133, 09-02-90427, and 09-02-92428), by the programs of the Russian Academy of Sciences and of the Siberian Branch of the Russian Academy of Sciences, by the grants of the President of the Russian Federation (project Nos. MK-6386.2010.2 and MK-3727.2011.2), by the Dynasty Foundation, and by the project FP7-PEOPLE-2009-IRSES "COLIMA."
\end{acknowledgments}
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\section{Introduction}
\label{sec:intro}
Continued interest in multi-agent reinforcement learning (MARL) has yielded a variety of algorithms over the years, from Minmax-Q~\cite{minimaxq} and Nash-Q~\cite{nashq} during its initial study to the more recent ones such as MADDPG~\cite{maddpg}, COMA~\cite{coma}, and IA2C~\cite{ia2c}. While the early methods mostly generalized Q-learning~\cite{qlearning} to multiagent settings, the later methods utilize the actor-critic schema with centralized or decentralized actor and critic components. The neural network representations of the actor and critic components allow these methods, which by default target settings with perfectly observed states, to expand to partial observability by maintaining a moving window of past observations. While these methods have demonstrated good performance on the standard MARL problem domains, the RL does not practically scale beyond a handful of interacting agents.
Multiagent planning frameworks such as DEC- and I-POMDPs~\cite{decpomdp,ipomdp} faced a similar hurdle of scaling in a meaningful way to many agents. A key insight -- that many domains exhibit the {\em action anonymity} structure~\cite{Jovanovic88:Anonymous} -- helped mitigate this curse of many agents afflicting planning. More specifically, it is the number of agents that perform the various actions which matters to the reasoning rather than the respective identities of the agents performing the actions; in other words, joint action permutations are equivalent. Modeling this invariance enables the planning complexity to drop from being exponential in the number of agents to polynomial thereby facilitating multiagent planning for thousands of agents~\cite{Sonu17:Anonymous,Varakantham14:Decentralized}.
In this paper, we aim to bring this insight to MARL and scale the learning to many-agent settings. An existing MARL approach that implicitly presumes action anonymity (perhaps, without being aware of this presumption) suggests using the average action vector, obtained as the mean of one-hot encodings of actions of the other agents~\cite{MF}. {\em This mean-field value is a near-optimal approximation under action anonymity when the multiagent system can be decomposed into pairwise interactions.} Our first contribution is an elucidation of the approximation that the mean-field necessitates and, in response, a general technique that represents other agents' behavior under action anonymity using {\em action configurations} without loss in value. This new method does not require approximating agent interactions as pairwise ones.
We integrate both the mean-field approximation and the generic action configuration based representations in two recent MARL approaches: MADDPG and IA2C. Our second contribution is a comprehensive comparison of the scalable representations on a recently introduced cooperative-competitive \textsf{Organization}{} domain~\cite{ia2c} that can organically scale to many agents and exhibits action anonymity. We show that interaction topologies exist where the two representations yield identical values, which is also the optimal joint behavior. However, several topologies also exist where the learned behaviors and values differ, and that the mean-field converges to suboptimal policies. Finally, our third contribution is a demonstration of MARL under partial observability in settings that contain up to a hundred agents.
\section{Background}
\label{sec:background}
We briefly describe the domain for modeling an organization in the next subsection, and follow it up with a review of the recently introduced {\em interactive A2C} (IA2C) method for learning in mixed cooperative-competitive settings under partial observability.
\subsection{The Organization domain}
\label{subsec:org}
The \textsf{Organization}{} domain~\cite{ia2c} is a partially observable multi-agent domain, modeling a typical business organization that features a mix of individual competition with cooperation to improve the financial health of the organization. There are 5 (hidden) states ($s\in S_f$) corresponding to various levels of financial health, which map to 3 observations ($o\in O_f$) that an agent can receive, as shown in Fig.~\ref{fig:state} (left). An agent 0 has 3 action choices ($a_0\in A_0$), viz., {\sf self} (self-interest), {\sf group} (cooperative), and {\sf balance}, the latter benefiting both the group and the individual. Agent 0's reward is comprised of an individual (competitive) component, $R_0$ that depends on the agent's action ($a_0$), and a group (cooperative) component, $R_G$ that depends on the joint action ($\mathbf{a}$). Let,
\begin{equation}
R_0^t\leftarrow R_0(s^t,a_0^t), \ \ R_G^t\leftarrow R_G(s^t,\mathbf{a}^t)
\label{eqn:R1}
\end{equation}
A key feature of this domain is an additional {\em history-dependent} reward component, $R_{-1}$, that models a bonus payoff based on the organization's previous year performance, specifically a fraction $\phi$ of the previous reward, given by
\begin{align}
R_{-1}^t = \phi(\sum\nolimits_iR_i^{t-1}+R_G^{t-1}).
\label{eqn:R2}
\end{align}
Joint actions are determined by the {\em number} of agents picking self actions compared to that picking group actions, and this affects the state transitions as shown in Fig.~\ref{fig:state} (right). On the one hand, {\sf self} action yields a higher individual reward to an agent than {\sf balance} and {\sf group} actions, it also damages the financial health of the organization if too many agents act in a self-interested manner. On the other hand, {\sf group} action improves the financial health at the expense of individual reward. Thus, with the objective of optimizing $\mathbb{E}_{trajectories}\left [\sum_t \gamma^t(R_G^t + R_i^t + R_{-1}^t)\right ]$, agent $i$ needs to balance greed with group welfare to optimize long-term payoff. He, Banerjee, and Doshi~\cite{ia2c} presents the optimal policy where all agents pick {\sf self} when $observation = many$, and {\sf group} when $observation = meager$, but $\#group=\#self+1$ when $observation=several$. It also presents an I-POMDP formulation of this domain, where the history-dependent reward is explicitly made a (additional) feature of the state to preserve the Markov property. This is elaborated in the next subsection.
\begin{figure}
\includegraphics[width=\linewidth]{orgfig.pdf}
\caption{\small States, observations, and transition dynamics of the \textsf{Organization}{} domain. The exact specification of the domain is given in the Appendix available in the supplement.}
\label{fig:state}
\end{figure}
\subsection{Interactive A2C for RL in mixed settings}
\label{subsec:IA2C}
Interactive advantage actor critic (IA2C)~\cite{ia2c} is a decentralized actor-critic method designed for egocentric RL in interactive partially observable Markov decision processes (I-POMDP). I-POMDPs are a generalization of POMDPs~\cite{pomdp} to sequential decision-making in multi-agent environments~\cite{ipomdp,Doshi12:Decision}. We first discuss the I-POMDP formulation of \textsf{Organization}{}, followed by the IA2C algorithm.
In order to capture the history-dependent reward in \textsf{Organization}{}, an extra state feature $S_r$ is introduced. An I-POMDP for agent $0$ in this environment with $N$ other agents is defined as,
\[
\text{I-POMDP}_0 = \langle IS_0, A, T_0, O_0, Z_0, R_0, OC_0 \rangle
\]
\noindent $\bullet~IS_0$ denotes the interactive state space, $IS_0=S_f\times S_r\times\prod_{j=1}^N M_j$. This includes the physical (financial) state $S_f$, and the previous-step reward as an additional state feature $S_r$, as well as models of the other agent $M_j$, which may be intentional (ascribing beliefs, capabilities and preferences) or subintentional~\cite{intention}. Examples of the latter are probability distributions and finite state machines. In this paper, we ascribe subintentional models to the other agents, $m_j=\langle \pi_j,h_j\rangle$, $m_j \in M_j$, where $\pi_j$ is $j$'s policy and $h_j$ is its action-observation history.\\
$\bullet~A = A_0 \times \prod_{j=1}^N A_j$ is the set of joint actions of all agents. Let $\mathbf{a}_{-0}$ denote the joint actions of $N$ other agents, $\mathbf{a}_{-0}\in \prod_{j=1}^N A_j$\\
$\bullet~R_0$ defines the reward function for agent $0$,
\begin{align}
R_0(\langle s_f,s_r\rangle,a_0,\mathbf{a}_{-0}) = {\cal R}_0(s_f, a_0, \mathbf{a}_{-0}) + \phi \cdot s_r.
\end{align}
${\cal R}_0(s_f, a_0, \mathbf{a}_{-0})$ is the sum of agent $0$'s current individual and group rewards from \textsf{Organization}{}.\\
$\bullet~T_0$ represents the transition function,
\begin{align}
T_0(\langle s_f,s_r\rangle, a_0, \mathbf{a}_{-0}, \langle s_f',s_r'\rangle)
= \left\{ \begin{array}{ll} T(s_f, a_0, \mathbf{a}_{-0}, s_f'), & \text{if }s_r'={\cal R}_0(s_f, a_0, \mathbf{a}_{-0}) +\phi\cdot s_r\\
0 & \text{otherwise}\end{array}\right.
\end{align}
where $T$ is the \textsf{Organization}{}'s transition function. The transition function is defined over the physical states and excludes the other agent's models. This is a consequence of the model non-manipulability assumption, which states that an agent's actions do not directly influence the other agent's models.\\
$\bullet~\Omega_0$ is the set of agent $0$'s {\em private} observations.\\
$\bullet~W_0: A \times \Omega_0 \rightarrow [0,1]$ is the private observation function.\\
$\bullet~O_0 = O_f \times O_r$ is the set of agent $0$'s {\em public} observations, where $O_f$ informs about the physical state and $O_r=S_r$, allowing the agent to observe the past reward.\\
$\bullet~Z_0$ is the observation function,
\begin{align}
&Z_0(a_0, \mathbf{a}_{-0}, \langle s_f, s_r\rangle, \langle s_f',s_r'\rangle, \langle o_f',o_r'\rangle)
\nonumber\\
&= \small \left\{ \begin{array}{ll} Z(a_0, \mathbf{a}_{-0}, s_f, s_f', o_f'), & \text{if }(s_r'={\cal R}_0(s_f,a_0, \mathbf{a}_{-0}) +\phi\cdot
s_r) \land (o_r'=s_r')\\
0 & \text{otherwise}\end{array}\right.
\end{align}
The observation function is defined over the physical state space only as a consequence of the model non-observability assumption, which states that other's model parameters may not be observed directly.\\
$\bullet~OC_0$ is the subject agent's optimality criterion, which may be a finite horizon $H$ or a discounted infinite horizon where the discount factor $\gamma \in (0,1)$.
The subject agent's belief is a distribution over the interactive state space, $b_0\in\Delta(S_f\times S_r\times \prod_{j=1}^N M_j)$. However, this can be factorized as
\begin{equation}
b_0(\langle s_f,s_r\rangle,m_1,\ldots, m_N)=b_0(\langle s_f,s_r\rangle)~b_0(m_1|\langle s_f,s_r\rangle)\ldots b_0(m_N|\langle s_f,s_r\rangle)
\label{eqn:belief-factorization}
\end{equation}
under the assumption that the other agents' models are conditionally independent given the state. Let $\mathbf{a}_{-0}=a_1\ldots,a_N$. Given the agent's prior belief $b_0$, action $a_0$, as well as its public and private observations $o_0'$, $\omega_0'$, the agent updates its belief over agent $j$'s model for $m_j'=\langle \pi_j',h_j'\rangle$ as
\begin{align}
b_0'(m_j'|b_0,a_0,o_0',\omega_0')\propto & \sum_{\mathbf{a}_{-0}}
\left ( \prod_{k=1}^N\sum_{m_k \in M_k} b_0(m_k)~Pr(a_k|m_k) \right )~W_0(a_0,\mathbf{a}_{-0},\omega_0')~\nonumber\\
& \times \delta_K(\pi_j',\pi_j)~\delta_K(APPEND(h_j,\langle a_j,o_f' \rangle),h_j')\label{eq:model-bu}
\end{align}
where $\delta_K$ is the Kronecker delta function and APPEND returns a string with the second argument appended to its first.
In IA2C, each agent has its own critic and actor neural network, the former mapping individual observations to joint action values in terms of the agent's own reward function, $Q_0(\langle o_f,o_r\rangle,a_0,\mathbf{a}_{-0})$, and the latter mapping individual observations to individual action probabilities, $\pi_{0,\boldsymbol{\theta}}(a_0|\langle o_f,o_r\rangle)$, $\boldsymbol{\theta}$ is its set of parameters. IA2C estimates advantages as
\[A_0(\langle o_f,o_r\rangle,a_0,\hat{\mathbf{a}}_{-0}) = avg\left [r + \gamma Q_0(\langle o_f', o_r'\rangle,a_0',\hat{\mathbf{a}}_{-0}') - Q_0(\langle o_f,o_r\rangle,a_0,\hat{\mathbf{a}}_{-0})\right ]\]
while the actor's gradient is estimated as
\[avg[\nabla_{\boldsymbol{\theta}}\log\pi_{0,\boldsymbol{\theta}}(a_0|\langle o_f,o_r\rangle)~A_0(\langle o_f,o_r\rangle,a_0,\hat{\mathbf{a}}_{-0})]\]
where $r,\langle o_f',o_r'\rangle$ and $a_0'$ are samples, $\hat{\mathbf{a}}_{-0}$ and $\hat{\mathbf{a}}_{-0}'$ are {\em predicted} actions, and the $avg$ is taken over sampled trajectories. In contrast with previous multi-agent deep RL algorithms, IA2C does not require direct exchange of information among RL agents, such as actions and/or gradients. Rather, agents predict each others' actions using their dynamic beliefs over models, updated using their noisy private observations via a belief filter integrated into the critic network to constitute joint actions. Agent $0$'s belief about agent $j$'s model $m_j' = (\pi_j', h_j')$ is updated based on its prior belief over $m_j = (\pi_j, h_j)$, as given in Eq.~\ref{eq:model-bu}.
\section{Many-agent reinforcement learning}
\label{sec:mang-agent}
The primary challenge in scaling multi-agent RL to many agents is the exponential growth of the joint action space. However, if the population is {\em homogeneous} in that all the agents have the same action space ($A_0 = A_1 = \ldots = A_N$) and the domain exhibits the {\em action anonymity} property, which means that both the dynamics and the rewards depend on the count distribution of actions in the population, while not needing the agents' identities, then we may potentially scale.
\subsection{Mean field approximation}
Under the conditions of population homogeneity and action anonymity, Yang et al.~\cite{MF}, inspired by the mean field theory~\cite{MFT}, utilizes the {\em mean action} of the immediate neighborhood in place of the global joint action. Note that an action can be represented as a one-hot encoding with each component indicating one of the possible actions: $a = [a^1, ..., a^{|A|}]$. The mean action $\bar{a}_0$ is calculated based on the neighborhood of agent 0 as:
\begin{align*}
\bar{a}_0 = \frac{1}{Ng_0} \sum\nolimits_{j \in Ng(0)} a_j
\end{align*}
where agent $j$ is a neighbor of 0 and $Ng(0)$ is the index set of the neighboring agents of $0$ with size $Ng_0 = |Ng(0)|$. The one-hot action for each neighbor $j$ can be expressed in terms of the sum of $\bar{a}_0$ and a small fluctuation $\Delta a_{0,j}$ as, $a_j = \bar{a}_0 + \Delta a_{0,j}$.
Importantly, the replacement of the other agents' joint action by the mean action is made possible by factorizing the joint action Q-function for agent $i$ using only the pairwise local interactions:
\begin{align}
Q_0(s, a_0, \mathbf{a}_{-0}) & \approx \frac{1}{Ng_0}\sum\nolimits_{j \in Ng(0)} Q_0(s, a_0, a_j)
\approx Q_0(s, a_0, \bar{a}_0).
\label{eq:MF}
\end{align}
Yang et al.~\cite{MF} notes that the second approximation in~\eqref{eq:MF} is not significant under the conditions of a twice-smooth Q-function and the previously mentioned population homogeneity.
However, we show that the pairwise interaction basis for the mean-field approximation could be a source of significant error. To demonstrate this, consider an instance of the \textsf{Organization}{} domain in which the worker-agent interactions take the shape of a diamond topology, as shown in Fig.~\ref{fig:diamond}. For illustration, we consider a single-shot interaction between the agents. The topology allows us to factorize the joint action Q-function, which can be written as:
\begin{align}
Q(a_1,a_2,a_3,a_4) = Q_1(a_1,a_2) + Q_2(a_2,a_3) + Q_3(a_3,a_4)
+ Q_4(a_1,a_4).
\label{eq:potential}
\end{align}
Next, we may use Guestrin, Koller, and Parr's well-known value factorization technique~\cite{factor} to obtain the joint action for the agents which optimizes Eq.~\ref{eq:potential} given these potentials. Briefly, optimizing out agent 4's action $a_4$ yields $e_4(a_1,a_3) = \max_{a_4}[Q_3(a_3,a_4) + Q_4(a_1,a_4)]$. Then, $a_3$ can be optimized out as $e_3(a_1,a_2) = \max_{a_3}[Q_2(a_2,a_3) + e_4(a_1,a_3)]$ followed by optimizing out $a_2$:
$e_2(a_1) = \max_{a_2}[Q_1(a_1,a_2) + e_3(a_1,a_2)]$. The optimal action for agent 1 is then $\arg\max_{a_1}e_2(a_1)$. Subsequently, we may obtain the optimal actions for the other three agents from $e_2$, $e_3$, and $e_4$.
\begin{wrapfigure}{r}{1.5in}
\centerline{\includegraphics[width=1.25in]{diamond.png}}
\caption{\small The \textsf{Organization}{} domain with 4 worker-agents exhibiting a diamond interaction graph.}
\label{fig:diamond}
\end{wrapfigure}
Recall the reward function of an individual agent in the \textsf{Organization}{} domain from Eqs.~\ref{eqn:R1},~\ref{eqn:R2} in Section~\ref{subsec:org}. Then, $Q_1(a_1,a_2)$ is the potential function of the interaction between agents 1 and 2. Equation~\ref{eq:MF} suggests that $Q_1$ be obtained as, $Q_1(a_1,a_2) = R_1(a_1) + R_2(a_2) + R_G(a_1,a_2)$; here, the group component $R_G$ of the reward function is limited to the pair of agents participating in the interaction. The remaining Q-potentials on the right-hand side of Eq.~\ref{eq:potential} are obtained similarly. As we show in the Appendix, the value factorization procedure now yields two optimal joint actions $\langle${\sf self}, {\sf group}, {\sf group}, {\sf group}$\rangle$ and $\langle${\sf group}, {\sf self}, {\sf self}, {\sf group}$\rangle$ both yielding a total reward of 12.~\footnote{Alternately, Eq.~\ref{eq:MF} suggests that we obtain an agent-centric reward using the mean action of its neighborhood to represent others. In other words, for agent 1 obtain $Q_1(a_1,\bar{a}_1) = R_1(a_1) + R_G(a_1,\bar{a}_1)$, and analogously for the other agents. This allows us to construct a normal-form game with the egocentric Q-values contingent on the joint actions as the payoffs for each agent. A search for the Nash equilibrium of this game yields the joint action $\langle${\sf self}, {\sf group}, {\sf self}, {\sf group}$\rangle$ yielding a total reward of 12.}
But, the \textsf{Organization}{} actually does better and the above joint actions are not the optimal solution. Relaxing the pairwise interaction constraint, let us obtain the potential $Q_1$ as: $Q_1(a_1,a_2) = R_1(a_1) + R_2(a_2) + R_G(a_1,a_2,a_3,a_4)$. Notice that the group component of the reward is relaxed to include all agents in the organization. Applying Guestrin, Koller, and Parr's value factorization technique as before involves optimizing out $a_4$: $e_4(a_1,a_3) = \max_{a_4}[Q_3(a_3,a_4) + Q_4(a_1,a_4)]$. Notice that action $a_2$ appearing in the group component $R_G(a_1,a_2,a_3,a_4)$ included in both $Q_3$ and $Q_4$ remains unspecified and we set its value as the one that maximizes $Q_3(a_3,a_4) + Q_4(a_1,a_4)$. Similarly, in the next step of optimizing out $a_3$ as $e_3(a_1,a_2) = \max_{a_3}[Q_2(a_2,a_3) + e_4(a_1,a_3)]$, the unspecified action $a_4$ included in the group component of $Q_2(a_2,a_3)$ is the one that maximizes $Q_2$. Moving forward, we optimize out $a_2$ as, $e_2(a_1) = \max_{a_2}[Q_1(a_1,a_2) + e_3(a_1,a_2)]$, while choosing $a_3$ and $a_4$ in $Q_1(a_1,a_2)$, which maximize the $Q_1$ potential, leading to the final: $\arg\max_{a_1}e_2(a_1)$. All other agents' actions are obtained analogously. This procedure yields the globally optimal joint action $\langle${\sf self}, {\sf balance}, {\sf group}, {\sf group}$\rangle$ and a total reward of 13.
\subsection{Many-agent RL using action configurations}
\label{subsec:configuration}
The mean field action offers a scalable way to model agent populations under the conditions of population homogeneity and action anonymity. But, its use is based on value approximations as we demonstrated in the previous section; our experiments reveal that this approximation can be costly. We present an alternate way to model agent populations that is both scalable and lossless under the same population conditions, and which has been effective in scaling decision-theoretic planning.
\subsubsection{Action configurations}
We begin by defining the concept of a configuration and characterize its properties.
\begin{defn}[Configuration] Define a configuration denoted by $\mathcal{C}{}$ as a vector of counts of the distinct actions performed by the agents, $\mathcal{C}{}^\mathbf{a}{} = \langle \#a^1, \#a^2, \ldots, \#a^{|A|} \rangle$, where $\#a^1$ denotes the count of an action $a^1$ in the joint action $\mathbf{a}$. Denote by $\boldsymbol{\mathcal{C}}$ the set of all configurations.
\end{defn}
\begin{defn}[Projection] Define a projection function $\delta$ as a mapping $\delta: A \rightarrow \boldsymbol{\mathcal{C}}$, which maps a joint action to its corresponding configuration.
\end{defn}
For example, $\delta$ projects the joint action $\mathbf{a} = \langle$ {\sf self}, {\sf self}, {\sf group}, {\sf group} $\rangle$ in the \textsf{Organization}{} domain instance of Fig.~\ref{fig:diamond} to the configuration vector, $\mathcal{C}{}^\mathbf{a}= \langle 2, 2, 0 \rangle$, where the first component of the vector gives the count of action {\sf self}, the second gives the count of {\sf group}, while the third component gives the count of the {\sf balance} action.
Observe that $\delta$ is a many-one mapping as multiple distinct joint actions, which are permutations of each other, yield the same configuration vector. In other words, for any $s_f$, $s_r$, $a_0$, $s_f'$, $s_r'$, $\mathbf{a}_{-0}$, and a permutation of $\mathbf{a}_{-0}$ denoted as $\mathbf{\dot{a}}_{-0}$, we have:
\centerline{$T_0(\langle s_f, s_r \rangle, a_0, \mathbf{a_{-0}},\langle s_f', s_r' \rangle) = T_0(\langle s_f, s_r \rangle, a_0, \mathbf{\dot{a}_{-0}},\langle s_f', s_r' \rangle) = T_0(\langle s_f, s_r \rangle, a_0, \mathcal{C}^\mathbf{a_{-0}},\langle s_f', s_r' \rangle)$,}
\centerline{$Z_0(a_0, \mathbf{a_{-0}}, \langle s_f, s_r \rangle, \langle s_f', s_r' \rangle, \langle o_f', o_r' \rangle) = Z_0(a_0, \mathbf{\dot{a}_{-0}}, \langle s_f, s_r \rangle, \langle s_f', s_r' \rangle, \langle o_f', o_r' \rangle)$} \centerline{$= Z_0(a_0, \mathcal{C}^\mathbf{a_{-0}},\langle s_f, s_r \rangle, \langle s_f', s_r' \rangle, \langle o_f', o_r' \rangle)$,}
\centerline{$W_0(a_0, \mathbf{a_{-0}}, \omega_o') = W_0(a_0, \mathbf{\dot{a}_{-0}},\omega_o') = W_0(a_0, \mathcal{C}^\mathbf{a_{-0}},\omega_o')$,~~~ \textit{and}}
\centerline{$R_0(\langle s_f, s_r \rangle, a_0, \mathbf{a_{-0}}) = R_0(\langle s_f, s_r \rangle, a_0, \mathbf{\dot{a}_{-0}}) = R_0(\langle s_f, s_r \rangle, a_0, \mathcal{C}^\mathbf{a_{-0}})$}
where $\delta(\mathbf{a_{-0}})=\delta(\mathbf{\dot{a}_{-0}})=\mathcal{C}^\mathbf{a_{-0}}$. As such, we may not recover the original joint action back from the configuration -- a direct consequence of the action anonymity property. The above equivalences due to permutation invariance in the environment's dynamics and the agent's observation capabilities and preferences naturally lead to the following property of the Q-function:
\centerline{$Q_0(\langle o_f, o_r \rangle,a_0,\mathbf{a}_{-0}) = Q_0(\langle o_f, o_r \rangle ,a_0,\mathbf{\dot{a}}_{-0}) = Q_0(\langle o_f, o_r \rangle,a_0,\mathcal{C}^\mathbf{a_{-0}})$.}
Subsequently, the advantage function $A_0(\langle o_f, o_r \rangle,a_0,\mathbf{a_{-0}})$ is also rewritten with the projection to the configuration. {\em A key advantage of using configurations is that the space of vectors of action counts is polynomial in the number of agents in comparison to the exponential growth of the joint action space as the number of agents grows.}
We adapt the IA2C method of Section~\ref{subsec:IA2C} to include action configurations to enable many-agent RL in partially-observable settings, and label this new method as IA2C$^{++}$. IA2C's belief filter is modified and a new dynamic programming module is prepended to the belief filter in the critic.
\begin{algorithm} [!t]
\caption{\small Computing configuration distribution $Pr(\mathcal{C}|b_0(M_1), b_0(M_2), \ldots, b_0(M_N))$}
\label{alg}
\small
\begin{algorithmic}
\REQUIRE $\langle b_0(M_1), b_0(M_2), \ldots, b_0(M_N) \rangle$
\ENSURE $P_N$, which is the distribution $Pr(\boldsymbol{\mathcal{C}}^\mathbf{a_{-0}})$ represented as a trie.
\STATE Initialize $c^{a_i}_0 \leftarrow (0,\dots,0)$, $P_0[c^{a_i}_0] \leftarrow 1.0$
\FOR {$k = 1$ to $N$}
\STATE Initialize $P_k$ to be an empty trie
\FOR {$c^{a_i}_{k-1}$ from $P_{k-1}$}
\FOR {$a^{a_i}_k \in A^{a_i}_k$ such that $\pi^{a_i}_k(a^{a_i}_k) > 0$}
\STATE $c^{a_i}_k \leftarrow c^{a_i}_{k-1}$
\IF {$a^{a_i}_k \neq \emptyset$}
\STATE $c^{a_i}_k(a^{a_i}_k) \overset{+}{\leftarrow} 1$
\ENDIF
\IF {$P_k[c^{a_i}_k]$ does not exist}
\STATE $P_k[c^{a_i}_k] \leftarrow 0$
\ENDIF
\STATE $P_k[c^{a_i}_k] \overset{+}{\leftarrow} P_{k-1}[c^{a_i}_{k-1}] \times \pi^{a_i}_k(a^{a_i}_k)$
\ENDFOR
\ENDFOR
\ENDFOR
\RETURN $P_N$
\end{algorithmic}
\label{alg:dp}
\end{algorithm}
\subsubsection{Belief update with configurations}
Equation~\ref{eq:model-bu} in Section~\ref{subsec:IA2C} gives the update of agent 0's belief over {\em one} other agent's possible models $M_j$, and this is performed for each other agent $j= \{1, 2, \ldots, N\}$ -- growing linearly in $N$. Joint action in the private observation function $W_0$ is now replaced by the configuration, as introduced previously. But, this also necessitates an additional term in the equation as we show below.
\begin{small}
\begin{align}
b_0'(m_j'|b_0,a_0,o_0',\omega_0')\propto & \sum\limits_{m_j \in M_j} b_0(m_j) \sum\limits_{a_j} Pr(a_j|m_j) ~\sum\limits_{\mathcal{C} \in \boldsymbol{\mathcal{C}}^\mathbf{a_{-0}}} Pr(\mathcal{C}|b_0(M_1), b_0(M_2),\nonumber\\
&~~ \ldots, b_0(M_N))~W_0(a_0,\mathcal{C},\omega_0')~\delta_K(\pi_j,\pi_j')\delta_K(APPEND(h_j, \langle a_j,o_f' \rangle),h_j').
\label{eq:model-bu-cfg}
\end{align}
\end{small}
Here, $Pr(\mathcal{C}|b_0(M_1), b_0(M_2), \ldots, b_0(M_N))$ is the probability of a configuration in the distribution over the set of configurations $\boldsymbol{\mathcal{C}}^\mathbf{a_{-0}}$. The distribution is obtained from agent 0's factored beliefs over the models of each other agent using a known dynamic programming procedure~\cite{configuration} that is outlined in Algorithm~\ref{alg:dp}. The algorithm takes as input just $N$ beliefs each of size $|M_j|$ compared to a single large belief of exponential size $|M_j|^N$, which is a benefit of the belief factorization shown in Section~\ref{subsec:IA2C}.
\subsubsection{IA2C$^{++}$ architecture}
We illustrate IA2C$^{++}$'s architecture in the schematic of Fig.~\ref{fig:network}. It consists of two main components: the actor and the critic. The expression in~\eqref{eq:gradient} below gives the actor's revised gradient, which is updated as the subject agent 0 interacts with the environment:
\begin{align}
avg \left [ \nabla_{\boldsymbol{\theta}} ~log ~\pi_{0,\boldsymbol{\theta}}(a_0|\langle o_f, o_r \rangle) ~A_0(\langle o_f, o_r \rangle, a_0,\mathcal{C}^\mathbf{a_{-0}}) \right ].
\label{eq:gradient}
\end{align}
Notice that the actions of the other agents traditionally appearing in the advantage function are now replaced with its projected configuration $\mathcal{C}^\mathbf{a_{-0}} (= \delta(\mathbf{a_{-0}}))$. This new advantage function is computed by the critic as:
\begin{align*}
A_0(\langle o_f, o_r \rangle, a_0, \mathcal{C}^\mathbf{a_{-0}}) = avg \left [r + \gamma ~Q_0(\langle o_f', o_r' \rangle, a_0', \mathcal{C}^\mathbf{a_{-0}'}) - Q_0(\langle o_f, o_r \rangle, a_0, \mathcal{C}^\mathbf{a_{-0}}) \right ]
\end{align*}
where $r$, $\langle o_f', o_r' \rangle$, and $a_0'$ are samples, $\mathbf{a_{-0}}$ and $\mathbf{a_{-0}'}$ are the {\em predicted} most-likely joint actions of the other agents for the current and next step, respectively, replaced by their corresponding configurations, and $avg$ is taken over the sampled trajectories. An agent $j$'s predicted action for the next time step is obtained by first sampling its model from the updated $b_0'(m_j')$, where the update occurs as per Eq.~\ref{eq:model-bu-cfg}. The sampled model yields an action distribution from which the action is sampled. This procedure is performed for each other agent and the corresponding configuration is obtained.
\begin{figure}[t!]
\centerline{\includegraphics[width=4.5in]{architecture.pdf}}
\caption{\small The belief filter in the critic utilizes the distribution over configurations computed using dynamic programming to update the agent's belief over models. Predicted actions are projected to their corresponding configurations and used in obtaining the advantage value.}
\label{fig:network}
\vspace{-0.05in}
\end{figure}
Thus, the actor network forwards the public and private observations from the environment to the belief filter in the critic. The belief filter first runs the dynamic programming procedure in Algorithm~\ref{alg:dp}, uses the output distribution over configurations to then update agent 0's belief over other agents' models, and predicts their actions. The projection operator yields the corresponding current and next time-step configurations, all of which is sent to the critic neural net for gradient-based updating and then to the advantage module for computing the advantage function. The latter is sent back to the actor component for its gradient update.
We implement the actor neural network with one input layer for the observations, two hidden layers one with tanh and the other with ReLU activation, followed by the output layer. The critic network consists of one input layer for the observations, one hidden layer with tanh activation followed by the output layer. All layers are fully connected to the next layer.
\section{Experiments}
\label{sec:experiments}
We instantiate IA2C and MADDPG~\cite{maddpg} with mean-field approximation and configuration, and label their scalable versions as IA2C$^{++}$(MF), IA2C$^{++}$(CF), MADDPG$^{++}$(MF), and MADDPG$^{++}$(CF), respectively. We conduct experiments using these four methods on \textsf{Organization}{} with five different graph structures of agent connectivity. Our code is available on GitLab and will be publicly released upon publication. The original \textsf{Organization}{} presumes a fully connected structure (Fig.~\ref{fig:full}) between agents, which models a group of employees working on the same project. A tree structure (Fig.~\ref{fig:tree}) models the common hierarchical structure of an organization. We study additional structures that accommodate loopy connections, specifically the lattice (Fig.~\ref{fig:lattice}) and circle topologies (Fig.~\ref{fig:circle}). Furthermore, from a macro perspective, we can model an organization with its subsidiary units using a star structure (Fig.~\ref{fig:star}).
\begin{figure}[!t]
\centerline{
\subfigure[\small Connected]{\label{fig:full}\includegraphics[width=0.16\textwidth]{full.png}}
\subfigure[\small Tree]{\label{fig:tree}\includegraphics[width=0.32\textwidth]{tree.png}}
\subfigure[\small Lattice]{\label{fig:lattice}\includegraphics[width=0.16\textwidth]{lattice.png}}
\subfigure[\small Circle]{\label{fig:circle}\includegraphics[width=0.16\textwidth]{circle.png}}
\subfigure[\small Star]{\label{fig:star}\includegraphics[width=0.16\textwidth]{star.png}}}
\caption{\small Interaction topologies for the \textsf{Organization}{} domain with 27 agents. We experiment with (a) fully connected, (b) tree, (c) lattice, (d) circle, and (e) star structures. Each dot denotes an agent.}
\vspace{-0.05in}
\end{figure}
The five structures differ by the number of neighborhoods and number of agents in each neighborhood. For example, in the fully connected structure, all agents share one single neighborhood; on the contrary, each agent forms a neighborhood with its left and right neighbors in the circle structure. We demonstrate that the various neighborhoods indeed impact the performance of the mean-field based methods. It is important to note that as \textsf{Organization}{}'s reward function depends on joint actions (hence the number of agents) but not on the graph structure, the globally optimal value and policy are also dependent on the number of agents and not on the graph structure. Consequently, the varying graph structures do not impact the value and policy learned by the configuration based methods.
Table~\ref{tbl:policyvalue} lists the values of converged policies from the four methods. The two configuration based methods (IA2C$^{++}$(CF) and MADDPG$^{++}$(CF)) always converge to the optimal policy no matter the structure, so we show only one column for them. Furthermore, IA2C$^{++}$(MF) and MADDPG$^{++}$(MF) converge to the same policies for any given structure and number of agents, so we show one column for each structure. In the fully connected structure, IA2C$^{++}$(MF) and MADDPG$^{++}$(MF) also converge to optimal policies, as the pairwise neighborhood basis of the mean field encompasses all other agents and induces no approximation.
However, in the tree, lattice, circle, and star structures, the mean-field instantiations are approximations and do not converge to optimal policies. Moreover, MADDPG$^{++}$(MF) and IA2C$^{++}$(MF) converged to better policies in lattice and circle structure than star and tree structures. This is because star and tree structures contain many neighborhoods of only two agents, making it harder for the mean-field approximation to coordinate across many small neighborhoods without knowing the actions outside the neighborhoods.
\begin{table}[ht!]
\caption{\small Policy value comparison between the configuration and mean-field based methods on the fully-connected, tree, lattice, circle, and star interaction topologies in the \textsf{Organization}{} domain. Both MADDPG$^{++}$ and IA2C$^{++}$ converge to the same policies but differ in run times. While the configuration instantiation consistently converged to the optimal policy in all topologies, mean-field approximation converged to the optimal policy in the fully connected structure only.}
\label{tbl:policyvalue}
\centerline{
\begin{small}
\begin{tabular}{|c||c|c|c|c|c|c|}
\hline
\multirow{2}{*}{\shortstack[l]{\# of \\ agents}}& {Configuration} & \multicolumn{5}{c|}{Mean field approximation} \\
\cline{2-7}
& {\bf All} & \textbf{Full} & \textbf{Tree} & \textbf{Lattice} & \textbf{Circle} & \textbf{Star} \\
\hline
27 & 3,180 & 3,180 & 1,620 & 2,820 & 2,700 & 1,740 \\
\hline
40 & 4,710 & 4,710 & 2,400 & 3,840 & 3,960 & 2,520 \\
\hline
60 & 7,110 & 7,110 & 3,600 & 6,000 & 6,000 & 3,720 \\
\hline
80 & 9,510 & 9,510 & 4,800 & 7,650 & 7,920 & 4,920 \\
\hline
100 & 11,910 & 11,910 & 6,000 & 9,450 & 9,960 & 6,120 \\
\hline
\end{tabular}
\end{small}}
\vspace{-0.05in}
\end{table}
Figure~\ref{fig:timecompare} shows the run time of MADDPG$^{++}$ and IA2C$^{++}$ with mean-field approximation and configuration instantiations. All experiments are run on a Linux platform with 2.3GHz quad-core i7 processor with 8GB memory. We observe that in general the mean-field approximation instantiations converge {\em faster} and require less episodes to train than their configuration based counterparts. On average, the mean-field approximation instantiations require 43.35\% less episodes to converge than the configuration instantiations across all topologies and numbers of agents. This can be viewed as a trade-off against mean field's convergence to suboptimal policies. Among the various interaction topologies, the fully connected and lattice structures consume the most time to converge due to their larger neighborhoods.
Furthermore, IA2C$^{++}$(MF) and IA2C$^{++}$(CF) (Fig.~\ref{fig:comp2}) are faster to converge than MADDPG$^{++}$(MF) and MADDPG$^{++}$(CF) (Fig.~\ref{fig:comp1}), consistent with the observation in He, Banerjee, and Doshi~\cite{ia2c} that IA2C converges faster than MADDPG. This is a benefit of the belief filter that can predict the other agents' actions more accurately than the maximum likelihood estimation of MADDPG.
\begin{figure}
\centering
\subfigure[\small MADDPG$^{++}$($\cdot$)]{\label{fig:comp1}\includegraphics[width=0.45\textwidth]{MADDPG.png}}
\subfigure[\small IA2C$^{++}$($\cdot$)]{\label{fig:comp2}\includegraphics[width=0.45\textwidth]{IA2C.png}}
\caption{\small Run time comparison between mean-field approximation and configuration instantiations on MADDPG$^{++}$ and IA2C$^{++}$. Run times are the average of three runs. Notice that MADDPG$^{++}$($\cdot$) generally takes more time to converge than IA2C$^{++}$($\cdot$). All experiments are run on a Linux platform with 2.3GHz quad-core i7 processor with 8GB memory. The discount factor is 0.9 and the learning rates for the actor and critic networks are 0.001 and 0.005, respectively. }
\label{fig:timecompare}
\vspace{-0.1in}
\end{figure}
\section{Related work}
\label{sec:relatedworks}
\vspace{-0.05in}
Mean-field Q learning (MF-Q)~\cite{MF} is a Q-learning RL algorithm that scales to many agents. If the agents are indistinguishable and independent from each other, they are represented as a single virtual agent who performs a mean action. The Q-value of the state and joint action is approximated by the Q-value of the state and mean action. A mean-field actor-critic (MF-AC) is also presented, but experiments show that MF-AC rarely improves on MF-Q. Both MF-Q and MF-AC are tested on domains with only one neighborhood or with several neighborhoods that are nearly fully isolated from each other. In contrast, we evaluate MADDPG$^{++}$ and IA2C$^{++}$ on domains with multiple neighborhoods that are connected to each other.
In contrast to our use of action configurations, Verma, Varakantham, and Lau~\cite{de} utilize state configurations under anonymity in the form of counts of agents located in various zones (the agents are taxis). Other agents' actions in the transition and reward functions are now replaced by state configurations without loss of information in the taxi domain. Algorithms that extend deep Q-networks and A2C to these settings are presented. However, the state is perfectly observed obviating the need for distributions over configurations and the experiments were mostly limited to just 20 agents operating in about 100 zones.
\section{Concluding remarks}
\label{sec:conclusion}
\vspace{-0.05in}
We presented new scalable instantiations of existing actor-critic MARL algorithms built on the notion of action anonymity that allows joint actions to be replaced by action configurations. Owing to a polynomial-time dynamic programming approach for computing distributions over configurations based on model beliefs, our instantiations of recent algorithms -- IA2C and MADDPG -- are able to scale polynomially with the number of agents, unlike the original algorithms. When compared to a recent scalable alternative for MARL that uses mean-field approximation, we found the latter to be significantly faster than using configurations in the cooperative-competitive \textsf{Organization}{} domain. However, this advantage in run time comes at a debilitating cost. The mean-field approximation was unable to learn optimal policies in all but the fully-connected interaction topology of agents. On the other hand, the configuration based approach learned optimal policies in all of the topology structures. Varying topologies are a natural feature of organizations, and indeed of many MARL domains, therefore the ability of the configuration based approach to learn optimal policies in a topology-independent way while being able to scale sets it apart in the sparse field of scalable MARL algorithms. Our future work involves investigating other mixed cooperative-competitive domains where MARL can help learn optimal behaviors.
\clearpage
\bibliographystyle{abbrv}
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\section{I. Introduction}
Heavy-ion collisions induced by radioactive beam at intermediate
energies play a significant role to extract the information of
nuclear equation of state (EoS) of isospin asymmetric nuclear matter
under extreme conditions, such as high density, high temperature and large isospin asymmetry etc.
Besides nucleonic observables such as
rapidity distribution and flow of free nucleons or light clusters
(such as deuteron, triton and alpha etc.), also mesons emitted from
the reaction zone can be probes of the hot and dense nuclear matter.
The energy per nucleon in the isospin asymmetric nuclear matter is
usually expressed as
$E(\rho,\delta)=E(\rho,\delta=0)+E_{\textrm{sym}}(\rho)\delta^{2}+\textsc{O}(\delta^{2})$
in terms of baryon density $\rho=\rho_{n}+\rho_{p}$, relative
neutron excess $\delta=(\rho_{n}-\rho_{p})/(\rho_{n}+\rho_{p})$,
energy per nucleon in a symmetric nuclear matter $E(\rho,\delta=0)$
and bulk nuclear symmetry energy
$E_{\textrm{sym}}=\frac{1}{2}\frac{\partial^{2}E(\rho,\delta)}{\partial
\delta^{2}}\mid_{\delta=0}$. In general, two different forms have
been predicted by some microscopical or phenomenological many-body
approaches. One is the symmetry energy increases monotonically with the baryon
density, and the other is the symmetry energy increases initially up
to a supra-saturation density and then decreases at higher
densities. The difference of the symmetry energy at supra-saturation densities predicted by transport models is huge.
It is not only in understanding the reaction dynamics, the high-density behavior of the symmetry energy also has an important application in astrophysics,
such as the structure of neutron star, the cooling of protoneutron stars, the nucleosynthesis
during supernova explosion of massive stars etc \cite{St05}.
With the establishment of high-energy radioactive beam
facilities in the world, such as the CSR (IMP in Lanzhou, China),
FAIR (GSI in Darmstadt, Germany), RIKEN (Japan), SPIRAL2 (GANIL in
Caen, France) and FRIB (MSU, USA) \cite{Li08}, the high-density
behavior of the symmetry energy can be studied more detail
experimentally in the near future.
Based on recent analysis of experimental data associated
with transport models, a symmetry energy of the form
$E_{\textrm{sym}}(\rho)\approx 31.6(\rho/\rho_{0})^{\gamma}$ MeV
with $\gamma=0.69-1.05$ was extracted for densities between
0.1$\rho_{0}$ and 1.2$\rho_{0}$ \cite{Li08,Ch05}. Theoretically, the symmetry
energy at supra-saturation densities can be investigated by
analyzing isospin sensitive observables that are emitted in the high-density region, such as
the neutron/proton ratio of emitted nucleons at mid-rapidity, $\pi^{-}/\pi^{+}$,
$\Sigma^{-}/\Sigma^{+}$ and $K^{0}/K^{+}$ ratios etc \cite{Li08}. Heavy-ion collisions around 1A GeV
energies can reach a 2-3$\rho_{0}$ nuclear matter. The emission of pion
in heavy-ion collisions in the region 1 A GeV is especially sensitive as a probe
of symmetry energy at supra-saturation densities. Recently, a
very soft symmetry energy at supra-saturation densities was pointed out
through fitting the FOPI data \cite{Re07} by using IBUU04 model \cite{Xi09}. The inverse results
were also reported by the Catania group with the RBUU model, in which a hard symmetry energy in
the high-density region resulted in the large values of the $\pi^{-}/\pi^{+}$ and $K^{0}/K^{+}$ ratios \cite{Fe06,Pr10}.
Further investigations of the pion emissions in the 1 A GeV region are still
necessary by improving transport models or developing some new
approaches. The ImIQMD model has been successfully applied to treat
dynamics in heavy-ion fusion reactions near Coulomb barrier and also to describe the capture of two heavy colliding nuclides
\cite{Fe05,Fe08}. Some further improvements of the ImIQMD model have been performed in order to
investigate the pion dynamics in heavy-ion collisions \cite{Fe09,Fe10}.
A systematic comparison of the ImIQMD results and the available experimental data is performed in this work. Calculations are focused on the pion production in heavy-ion collisions around 1A GeV energies. Some observables sensitive to the high-density behavior of the symmetry energy are investigated and discussed. The paper is organized as follows. In Sec. II we give a detailed description of the ImIQMD model. Calculated
results of pion dynamics in heavy-ion collisions and constraining the symmetry energy at supra-saturation densities
are given in Sec. III. In Sec. IV conclusions are discussed.
\section{II. Model description}
The same as the QMD \cite{Ai91} or IQMD model \cite{Ha98,Ch98}, the wave function for each
nucleon in ImIQMD is represented by a Gaussian wave packet
\begin{eqnarray}
\psi_{i}(\mathbf{r},t)=&& \frac{1}{(2\pi
L)^{3/4}}\exp\left[-\frac{(\mathbf{r}-\mathbf{r}_{i}(t))^{2}}{4L}\right] \nonumber \\
&& \times \exp\left(\frac{i\mathbf{p}_{i}(t)\cdot\mathbf{r}}{\hbar}\right).
\end{eqnarray}
Here $\mathbf{r}_{i}(t)$, $\mathbf{p}_{i}(t)$ are the centers of the
$i$th nucleon in the coordinate and momentum space, respectively.
The $L$ is the square of the Gaussian wave packet width, which
depends on the mass number of nucleus. The total N-body wave function is
assumed as the direct product of the coherent states, where the
anti-symmetrization is neglected. After performing Wigner transformation for Eq.
(1), we get the Wigner density as
\begin{equation}
f(\mathbf{r},\mathbf{p},t)=\sum_{i}f_{i}(\mathbf{r},\mathbf{p},t)
\end{equation}
with
\begin{eqnarray}
f_{i}(\mathbf{r},\mathbf{p},t)=&& \frac{1}{(\pi\hbar)^{3}}\exp \nonumber \\
&& \times \left[-
\frac{(\mathbf{r}-\mathbf{r}_{i}(t))^{2}}{2L}-\frac{(\mathbf{p}-\mathbf{p}_{i}(t))^{2}\cdot
2L}{\hbar^{2}}\right].
\end{eqnarray}
The density distributions in coordinate and momentum space are given
by
\begin{eqnarray}
\rho(\mathbf{r},t)=&& \int f(\mathbf{r},\mathbf{p},t) d\mathbf{p} \nonumber \\
=&& \sum_{i}\frac{1}{(2\pi L)^{3/2}}\exp\left[-
\frac{(\mathbf{r}-\mathbf{r}_{i}(t))^{2}}{2L}\right],
\end{eqnarray}
\begin{eqnarray}
g(\mathbf{p},t)=&& \int f(\mathbf{r},\mathbf{p},t) d\mathbf{r} \nonumber \\
=&& \sum_{i}\left(\frac{2L}{\pi\hbar^{2}}\right)^{3/2}\exp\left[-
\frac{(\mathbf{p}-\mathbf{p}_{i}(t))^{2}\cdot 2L}{\hbar^{2}}\right],
\end{eqnarray}
respectively, where the sum runs over all nucleons in the reaction systems.
The time evolutions of the baryons and pions in
the system under the self-consistently generated mean-field are
governed by Hamilton's equations of motion, which read as
\begin{eqnarray}
\dot{\mathbf{p}}_{i}=-\frac{\partial H}{\partial\mathbf{r}_{i}},
\quad \dot{\mathbf{r}}_{i}=\frac{\partial
H}{\partial\mathbf{p}_{i}}.
\end{eqnarray}
Here we omit the shell correction part in the Hamiltonian $H$ as
described in Ref. \cite{Fe08}. The Hamiltonian of baryons consists
of the relativistic energy, the effective interaction potential energy and
the momentum dependent part as follows:
\begin{equation}
H_{B}=\sum_{i}\sqrt{\textbf{p}_{i}^{2}+m_{i}^{2}}+U_{int}+U_{mom}.
\end{equation}
Here the $\textbf{p}_{i}$ and $m_{i}$ represent the momentum and the
mass of the baryons.
The effective interaction potential is composed of the Coulomb
interaction and the local interaction
\begin{equation}
U_{int}=U_{Coul}+U_{loc}.
\end{equation}
The Coulomb interaction potential is calculated by
\begin{equation}
U_{Coul}=\frac{1}{2}\sum_{i,j,j\neq
i}\frac{e_{i}e_{j}}{r_{ij}}erf(r_{ij}/\sqrt{4L})
\end{equation}
where the $e_{j}$ is the charged number including protons and
charged resonances. The $r_{ij}=|\mathbf{r}_{i}-\mathbf{r}_{j}|$ is
the relative distance of two charged particles.
The local interaction potential energy is derived directly from the Skyrme
energy-density functional and expressed as
\begin{equation}
U_{loc}=\int V_{loc}(\rho(\mathbf{r}))d\mathbf{r}.
\end{equation}
The local potential energy-density functional reads
\begin{eqnarray}
V_{loc}(\rho)=&& \frac{\alpha}{2}\frac{\rho^{2}}{\rho_{0}}+
\frac{\beta}{1+\gamma}\frac{\rho^{1+\gamma}}{\rho_{0}^{\gamma}}+
\frac{g_{sur}}{2\rho_{0}}(\nabla\rho)^{2} \nonumber \\
&& + \frac{g_{sur}^{iso}}{2\rho_{0}}[\nabla(\rho_{n}-\rho_{p})]^{2} \nonumber \\
&& + \left(a_{sym}\frac{\rho^{2}}{\rho_{0}}+b_{sym}\frac{\rho^{1+\gamma}}{\rho_{0}^{\gamma}}+
c_{sym}\frac{\rho^{8/3}}{\rho_{0}^{5/3}}\right)\delta^{2} \nonumber \\
&& + g_{\tau}\rho^{8/3}/\rho_{0}^{5/3},
\end{eqnarray}
where the $\rho_{n}$, $\rho_{p}$ and $\rho=\rho_{n}+\rho_{p}$ are
the neutron, proton and total densities, respectively, and the
$\delta=(\rho_{n}-\rho_{p})/(\rho_{n}+\rho_{p})$ is the isospin
asymmetry. Here, all the terms in the Skyrme energy functional are included in the model besides the spin-orbit coupling. The coefficients $\alpha$, $\beta$, $\gamma$, $g_{sur}$,
$g_{sur}^{iso}$, $g_{\tau}$ are related to the Skyrme parameters
$t_{0}, t_{1}, t_{2}, t_{3}$ and $x_{0}, x_{1}, x_{2}, x_{3}$ as \cite{Fe08},
\begin{eqnarray}
&& \frac{\alpha}{2}=\frac{3}{8}t_{0}\rho_{0}, \quad
\frac{\beta}{1+\gamma}=\frac{t_{3}}{16}\rho_{0}^{\gamma}, \\
&& \frac{g_{sur}}{2}=\frac{1}{64}(9t_{1}-5t_{2}-4x_{2}t_{2})\rho_{0}, \\
&& \frac{g_{sur}^{iso}}{2}=-\frac{1}{64}[3t_{1}(2x_{1}+1)+t_{2}(2x_{2}+1)]\rho_{0}, \\
&& g_{\tau}=\frac{3}{80}\left(\frac{3}{2}\pi^{2}\right)^{2/3}(3t_{1}+5t_{2}+4x_{2}t_{2})\rho_{0}^{5/3}.
\end{eqnarray}
The parameters of the potential part in the bulk symmetry
energy term are also derived directly from Skyrme energy-density
parameters as
\begin{eqnarray}
&& a_{sym}=-\frac{1}{8}(2x_{0}+1)t_{0}\rho_{0}, \quad b_{sym}=-\frac{1}{48}(2x_{3}+1)t_{3}\rho_{0}^{\gamma}, \nonumber \\
&& c_{sym}=-\frac{1}{24}\left(\frac{3}{2}\pi^{2}\right)^{2/3}\rho_{0}^{5/3}[3t_{1}x_{1}-t_{2}(5x_{2}+4)].
\end{eqnarray}
The momentum dependent term in the Hamiltonian is taken as the same form in Ref. \cite{Ai87} and expressed as
\begin{equation}
U_{mom}=\frac{\delta}{2}\sum_{i,j,j\neq
i}\frac{\rho_{ij}}{\rho_{0}}[\ln(\epsilon(\textbf{p}_{i}-\textbf{p}_{j})^{2}+1)]^{2},
\end{equation}
with
\begin{equation}
\rho_{ij}=\frac{1}{(4\pi L)^{3/2}}\exp\left[
-\frac{(\textbf{r}_{i}-\textbf{r}_{j})^{2}}{4L}\right],
\end{equation}
which does not distinguish between protons and neutrons. The parameters $\delta$ and
$\epsilon$ were determined by fitting the real part of the
proton-nucleus optical potential as a function of incident energy from the experimental elastic scattering data.
The last two terms in Eq. (11) originate from the momentum-dependent part of the Skyrme interaction which, in principle, should contribute to the nucleon effective mass \cite{Br85,Ch97}. However, in the ImIQMD model their contribution is taken into account in the potential energy-density functional only and not in the momentum-dependent interaction given by the term of Eq. (17) solely. Therefore, the effective (Landau) mass in the model is fixed as $m_{\infty}^{\ast}=\left(\frac{1}{m}+\frac{1}{|\textbf{p}|}|\frac{dU_{mom}}{d\textbf{p}}|\right)^{-1}$ with the free mass $m$ at Fermi momentum $\textbf{p}=\textbf{p}_{F}$ \cite{Da05}, which produces the value of $m_{\infty}^{\ast}/m=0.77$ for all sets of the model parameters.
\begin{table*}
\caption{\label{tab:table3}ImIQMD parameters and properties of symmetric nuclear
matter for Skyrme effective interactions after the inclusion of the
momentum dependent interaction with parameters $\delta$=1.57 MeV and
$\epsilon$=500 c$^{2}$/GeV$^{2}$.}
\begin{ruledtabular}
\begin{tabular}{ccccccccc}
&Parameters &SkM* &Ska &SIII &SVI &SkP &RATP &SLy6 \\
\hline
&$\alpha$ (MeV) &-325.1 &-179.3 &-128.1 &-123.0 &-357.7 &-250.3 &-296.7 \\
&$\beta$ (MeV) &238.3 &71.9 &42.2 &51.6 &286.3 &149.6 &199.3 \\
&$\gamma$ &1.14 &1.35 &2.14 &2.14 &1.15 &1.19 &1.14 \\
&$g_{sur}$(MeV fm$^{2}$) &21.8 &26.5 &18.3 &14.1 &19.5 &25.6 &22.9 \\
&$g_{sur}^{iso}$(MeV fm$^{2}$)&-5.5 &-7.9 &-4.9 &-3.0 &-11.3 &0.0 &-2.7 \\
&$g_{\tau}$ (MeV) &5.9 &13.9 &6.4 &1.1 &0.0 &11.0 &9.9 \\
&$C_{sym}$ (MeV) &30.1 &33.0 &28.2 &27.0 &30.9 &29.3 &32.0 \\
&$a_{sym}$ (MeV) &62.4 &29.8 &38.9 &42.9 &94.0 &79.3 &130.6 \\
&$b_{sym}$ (MeV) &-38.3 &-5.9 &-18.4 &-22.0 &-63.5 &-58.2 &-123.7 \\
&$c_{sym}$ (MeV) &-6.4 &-3.0 &-3.8 &-5.5 &-13.0 &-4.1 &12.8 \\
&$\rho_{\infty}$ (fm$^{-3}$)&0.16 &0.155 &0.145 &0.144 &0.162 &0.16 &0.16 \\
&$K_{\infty}$ (MeV) &215 &262 &353 &366 &200 &239 &230 \\
\end{tabular}
\end{ruledtabular}
\end{table*}
In Table 1 we list the ImIQMD parameters related to several typical
Skyrme forces after including the momentum dependent interaction.
The parameters $\alpha$, $\beta$ and $\gamma$ are redetermined in
order to reproduce the binding energy ($E_{B}$=-16 MeV) of symmetric
nuclear matter at saturation density $\rho_{0}$ and to satisfy the saturation properties by
relation $\frac{\partial E/A}{\partial\rho}\mid _{\rho=\rho_{0}}$=0
for a given incompressibility. Combined Eq. (16) with the kinetic
energy part, the symmetry energy per nucleon in the ImIQMD model is
given by
\begin{eqnarray}
E_{sym}(\rho)=&& \frac{1}{3}\frac{\hbar^{2}}{2m}\left(\frac{3}{2}\pi^{2}\rho\right)^{2/3}+
a_{sym}\frac{\rho}{\rho_{0}}+b_{sym}\left(\frac{\rho}{\rho_{0}}\right)^{\gamma} \nonumber \\
&& + c_{sym}\left(\frac{\rho}{\rho_{0}}\right)^{5/3}.
\end{eqnarray}
More clearly compared with other transport models, such as IBUU04, RBUU, and IQMD etc, the bulk symmetry
energy can be expressed as
\begin{equation}
E_{sym}(\rho)=\frac{1}{3}\frac{\hbar^{2}}{2m}\left(\frac{3}{2}\pi^{2}\rho\right)^{2/3}+
\frac{1}{2}C_{pot}\left(\frac{\rho}{\rho_{0}}\right)^{\gamma_{s}}.
\end{equation}
The value $\gamma_{s}=1$ is used in the IQMD model \cite{Ha98,Ch98}. In
Fig. 1 we show a comparison of the nuclear symmetry energy at the situations of
different Skyrme forces SkP, SLy6, Ska and SIII calculated by
Eq. (19), and also the cases in Eq. (20) of $\gamma_{s}$= 0.5 (soft), 1 (linear), 2 (hard), 3 (superhard) with the coefficient $C_{pot}$=38 MeV, which have the value of the symmetry energy $E_{sym}(\rho_{0})$=31.5 MeV at the saturation nuclear density $\rho_{0}$=0.165 fm$^{-3}$. The parameter SLy6 is used in the calculations by default.
\begin{figure}
\includegraphics[width=8 cm]{esym}
\caption{\label{fig:epsart} The density dependence of the nuclear
symmetry energy for different Skyrme forces SkP, Sly6, Ska and SIII (supersoft),
and for the cases of soft, linear, hard and superhard trends.}
\end{figure}
The pion is created by the decay of the resonances $\triangle$(1232)
and N*(1440) which are produced in inelastic NN scattering. The
reaction channels are given as follows:
\begin{eqnarray}
&& NN \leftrightarrow N\triangle, \quad NN \leftrightarrow NN^{\ast}, \quad NN
\leftrightarrow \triangle\triangle, \nonumber \\
&& \Delta \leftrightarrow N\pi, \quad N^{\ast} \leftrightarrow N\pi, \quad NN \rightarrow NN\pi (\textrm{s-state}).
\end{eqnarray}
The cross sections of each channel to produce resonances are
parameterized by fitting the data calculated with the one-boson
exchange model \cite{Hu94}. In the 1 A GeV region, there are mostly
$\Delta$ resonances which disintegrate into a $\pi$ and a nucleon,
however, the $N^{\ast}$ yet gives considerable contribution to the
high energetic pion yield. The energy and momentum dependent decay
width is used in the calculation \cite{Fe09}.
Analogously to baryons, the evolution of pions is also determined by the Hamiltonian, which is given by
\begin{eqnarray}
H_{\pi}&& = \sum_{i=1}^{N_{\pi}}\left( V_{i}^{\textrm{Coul}} + \omega(\textbf{p}_{i},\rho_{i}) \right) \nonumber \\
&& = \sum_{i=1}^{N_{\pi}}\left( V_{i}^{\textrm{Coul}} + \sqrt{\textbf{p}_{i}^{2}+m_{\pi}^{2}}
+ \textrm{Re}V_{\pi}^{opt}(\textbf{p}_{i},\rho_{i}) \right),
\end{eqnarray}
where the $\textbf{p}_{i}$ and $m_{\pi}$ represent the momentum and
the mass of the pions. The Coulomb interaction is given by
\begin{equation}
V_{i}^{\textrm{Coul}}=\sum_{j=1}^{N_{B}}\frac{e_{i}e_{j}}{r_{ij}},
\end{equation}
where the $N_{\pi}$ and $N_{B}$ are the total numbers of pions and
baryons including charged resonances. The pion optical potential $\textrm{Re}V_{\pi}^{opt}$ originates
from the medium effects in the hot and dense nuclear matter. In the calculation, we can also choose that of the vacuum, i.e.,
the $\textrm{Re}V_{\pi}^{opt}$ is set equal zero. The influence of the pionic mean field in heavy-ion collisions on the transverse momentum distribution
was investigated by using a phenomenological ansatz and a microscopic approach based on the $\Delta$-hole model by Fuchs \emph{et al.} \cite{Fu97}.
Here we use the phenomenological ansatz suggested by Gale and Kapusta \cite{Ga87}. Then the dispersion relation reads
\begin{eqnarray}
\omega(\textbf{p}_{i},\rho_{i})=\sqrt{(|\textbf{p}_{i}|-p_{0})^{2}+m_{0}^{2}}-U, \\
U=\sqrt{p_{0}^{2}+m_{0}^{2}}-m_{\pi}, \\
m_{0}=m_{\pi}+6.5(1-x^{10})m_{\pi}, \\
p_{0}^{2}=(1-x)^{2}m_{\pi}^{2}+2m_{0}m_{\pi}(1-x).
\end{eqnarray}
The phenomenological medium dependence on the baryon density is introduced via the coefficient $x(\rho_{i})=\exp(-a(\rho_{i}/\rho_{0}))$ with the parameter
$a=0.154$ and the saturation density $\rho_{0}$ in nuclear matter. Influence of the in-medium effects on the charged
pion ratio is also investigated in Ref. \cite{Xu10}.
\section{III. Results and discussions}
\subsection{A. Charge distributions}
Nuclear multifragmentation is a common phenomenon observed in
heavy-ion collisions at intermediate energies, in which one can
learn more about the properties of the fermionic nuclear matter,
such as the compressibility, the liquid-gas phase transition and
also the isospin asymmetric EoS. As a test of the ImIQMD model, we
calculated the charge distributions in multifragmentation reactions
of the system $^{197}$Au+$^{197}$Au at different incident energies
labeled in Fig. 2. The fragment multiplicity decreases with the
atomic number and the trend is more rapidly at the higher incident
energy The available experimental data \cite{De98,Re97} can
reproduced rather well besides the energy at 35 A MeV in the region of the intermediate mass fragments (IMFs) (Z=3-15).
The failure description of the production of the IMFs at several tens A MeV is due to the roughly treatment of fermionic nature in the dynamical evolution of nucleons.
These reactions are well described by the Statistical Multifragmentation Model (SMM) in which one assumes the formation of a thermal source and uses the statistical approach for its breakup \cite{Ag96}. Combination of transport models and SMM can reproduce the experimental data at fermionic energies. We constructed the fragments with the coalescence model, in which nucleons of the reaction system are
considered to belong to a cluster in the phase space with the
relative momentum smaller than $P_{0}$ and with the relative
distance smaller than $R_{0}$ (here $P_{0}$=200 MeV/c and
$R_{0}$=2.4 fm).
\begin{figure*}
\includegraphics{chargd}
\caption{\label{fig:wide} (Color online) Comparison of the
distribution of charged fragments in central $^{197}$Au+$^{197}$Au
collisions with the available experimental data at different
incident energies.}
\end{figure*}
\subsection{B. Production of the total pion and the $\pi^{-}/\pi^{+}$ yields}
Production of pion in heavy-ion collisions in the region of 1A GeV
is a primary product in nucleon-nucleon inelastic scattering and
mainly produced by the decay of the resonance $\triangle$(1232). So
it has enough yields to measure pion in experiments to get the
information of the EoS. Dynamics of the pion emission calculated by
transport models is helpful for understanding the experimental
observables. We calculated the time evolution of the multiplicities
of the total $\pi$, $\Delta(1232)$, $N^{\ast}(1440)$ and the ratio
$\rho/\rho_{0}$ in central $^{197}$Au+$^{197}$Au collisions at 1A
GeV incident energy as shown in Fig. 3. One can see that pions are
mainly in the domain at supra-saturation densities of compressed
nuclear matter larger than the normal density $\rho_{0}$. The
production of pions is also influenced by the Fermi motion of
baryons in the vicinity of the threshold energies. Figure 4 is a
comparison of the measured total pion multiplicity by the FOPI
collaboration in the reaction $^{40}$Ca+$^{40}$Ca for head on
collisions \cite{Re07} and the results calculated by the ImIQMD
model for Skyrme parameters SkP, SLy6, Ska and SIII in the left
panel, which correspond to different modulus of the
incompressibility as listed in table 1, and the ratio of the
calculated results to the experimental data at different incident
energies. The total multiplicity of pion is mainly determined by the
cross sections of the channels $NN \leftrightarrow N\triangle$. The
ImIQMD model with four Skyrme parameters predicts rather well the
total yields at higher incident energies, but slightly overestimates
the values near threshold energies, which may be influenced by the
in-medium cross sections. In the calculation we use the in-vacuum
cross sections of nucleon-nucleon elastic and inelastic collisions.
Reasonable consideration of the in-medium inelastic collisions in
producing $\Delta$ and $N^{\ast}$ is still an open problem in
transport models, which have been performed in Giessen-BUU model
\cite{La01}.
\begin{figure}
\includegraphics[width=8 cm]{time}
\caption{\label{fig:epsart} Time evolution of the total $\pi$, $\Delta(1232)$, $N^{\ast}(1440)$ and the ratio $\rho/\rho_{0}$
in the reaction $^{197}$Au+$^{197}$Au for central collisions at incident energy 1A GeV.}
\end{figure}
\begin{figure*}
\includegraphics{caca}
\caption{\label{fig:wide} (Color online) The excitation functions of
the total pion multiplicities and the ratio of the calculated
results with different Skyrme parameters to the experimental data in
the reaction $^{40}$Ca+$^{40}$Ca for head on collisions.}
\end{figure*}
The $\pi^{-}$/$\pi^{+}$ ratio a sensitive probe to extract the
high-density behavior of the symmetry energy per energy. Using the
isobar model \cite{St86}, one gets the ratio $\pi^{-}$/$\pi^{+}$=2.09 for pions
from the $\Delta$ resonance, and $\pi^{-}$/$\pi^{+}$=1.8 from the N*
for the system $^{132}$Sn+$^{124}$Sn. These relations
are determined by the Clebsch-Gordan coefficients and globally valid, i.e. independent of the pion energy. On the
other hand, the statistical model predicts that the
$\pi^{-}$/$\pi^{+}$ ratio is sensitive to the difference in the
chemical potentials of neutrons and protons by the relation
$\pi^{-}/\pi^{+}\propto \exp [2(\mu_{n}-\mu_{p})/T]=\exp[8\delta
E_{sym}(\rho)/T]$, where the $T$ is the nuclear temperature \cite{Be80}.
The observed energy dependence of the $\pi^{-}$/$\pi^{+}$ ratio is
due to the re-scattering and absorption process of pions and
nucleons in the mean field of the compressed nuclear matter. We use
the free absorption cross sections in collisions of pions and
nucleons by fitting the experimental data. The branch ratio of the
charged $\pi$ and $\pi^{0}$ is determined by the Clebsch-Gordan
coefficients with the decay of the resonances $\triangle$(1232) and
N*(1440). The $\pi^{-}/\pi^{+}$ ratio is sensitive to the stiffness
of the symmetry energy at the lower incident energies. The
compressed nuclear matter with central density about two times of
the normal density is formed in heavy-ion collisions in the 1 A GeV
region. To extract more information of the symmetry energy in
heavy-ion collisions from the pion production, in Fig. 5 we
calculated the $\pi^{-}/\pi^{+}$ ratios as a function of the N/Z of
the Sn isotopes in the reactions $^{100}$Sn+$^{112}$Sn,
$^{112}$Sn+$^{112}$Sn, $^{112}$Sn+$^{124}$Sn, $^{124}$Sn+$^{124}$Sn
and $^{132}$Sn+$^{124}$Sn in the left panel, and the excitation
functions of the $\pi^{-}$/$\pi^{+}$ ratios with the force SLy6 for the coefficients by Eqs. (12)-(15) besides the symmetry energy term. Here, the symmetry energy is treated as the cases without the term or with but different stiffness of the potential part which corresponds to superhard ($\gamma_{s}$=3), hard
($\gamma_{s}$=2), linear ($\gamma_{s}$=1), soft ($\gamma_{s}$=0.5)
and supersoft (SIII). The symmetry energy reduces the
$\pi^{-}/\pi^{+}$ ratio for neutron-rich systems at the lower
incident energies because of its attractive force on protons and
opposite for neutrons in the dynamical evolution. So the probabilities of neutron-neutron collisions decrease after including the symmetry energy, which lead to the reduction of $\pi^{-}$ production. The ratio increases with the stiffness of the symmetry energy, but decreases for the case of the superhard symmetry energy. The phenomena can be explained from the fact that although the symmetry energy reduces the
N/Z ratio in the high-density region. A hard symmetry energy repulses neutrons of the colliding partners to flow the high-density region and
enhances the domain of the supra-saturation density. Therefore, the neutron-neutron collisions in the case of the hard symmetry energy take place in a larger region at supra-saturation densities than a soft one. A too hard symmetry energy also reduces the value of $\pi^{-}$/$\pi^{+}$ ratio because of the lower probabilities of the neutron-neutron collisions. The decrease of the $\pi^{-}$/$\pi^{+}$ ratio with the incident energy is mainly
owing to the production of pions from secondary nucleon-nucleon
collisions, such as a neutron converts a proton by producing
$\pi^{-}$. Subsequent collisions of the energetic proton can convert
again to neutron by producing $\pi^{+}$. The $\pi^{-}$/$\pi^{+}$ ratios of different situation of the symmetry energy at high incident energies (E$_{lab}>$1.0A GeV) are close each other and approach the value calculated by using the isobar model. The larger
$\pi^{-}$/$\pi^{+}$ ratio in the case of the hard symmetry energy was also
reported in Refs \cite{Fe06,Pr10}. Shown in Fig. 6 is the profile of pion production without the symmetry energy and with increasing the stiffness of the potential part of the symmetry energy in Eq. (20). The symmetry energy reduces the $\pi^{-}$ production owing to the decrease of the N/Z ratio in the high-density region. Influence of the stiffness of the symmetry energy on $\pi^{+}$ is negligible.
\begin{figure*}
\includegraphics{sniso}
\caption{\label{fig:wide} (Color online) The $\pi^{-}$/$\pi^{+}$
yields as functions of the neutron over proton N/Z of reaction
systems in collisions of Sn isotopes at energy E$_{lab}$=0.4A GeV and of the incident energy for the
reaction $^{132}$Sn+$^{124}$Sn for the cases of different stiffness of the symmetry energy and without symmetry energy.}
\end{figure*}
\begin{figure}
\includegraphics[width=8 cm]{evls}
\caption{\label{fig:epsart} Time evolution of the pion production for the cases of the linear, hard and superhard symmetry energy and without the symmetry energy in the reaction $^{132}$Sn+$^{124}$Sn at incident energy E$_{lab}$=0.4A GeV.}
\end{figure}
\subsection{C. Influence of the in-medium pion dispersion relation}
Propagation of pion in the compressed hadronic matter is a
complicated process that is involved in the interaction of pions and
nucleons or pions and resonances. Furthermore, the pion in-medium
effect has a non-negligible influence on the charged pion ratios.
Calculations based on the QMD model show that the transverse
momentum spectrum is not influenced with an inclusion of the
$\Delta$-hole potential \cite{Fu97}. Opposite in the BUU
calculations with the $\Delta$-hole potential, the production of
pion is enhanced at the low kinetic energies \cite{Xi93}. In Fig. 7
we show the influence of the Coulomb potential and the symmetry
energy in the mean field as well as the pion optical potential in
the nuclear medium on the distributions of the transverse momenta of
the charged and neutral pion production in the left and right
panels, respectively, and also compared with the experimental data by FOPI collaboration \cite{Pe97}. We get the in-medium pion dispersion relation
by using the phenomenological approach in Eqs (24)-(27). The
propagation of pion in the nuclear matter is governed by the Coulomb
interaction and the pion optical potential. Here we use the linear
($\gamma_{s}$=1) symmetry energy and the potential parameters in Eqs (12)-(15) are got from the Skyrme force SLy6 in table 1.
The evolutions of produced pions in the baryonic matter are governed by the optical potential and also the Coulomb interaction of the charged pions and baryons.
One can see that the Coulomb potential and the symmetry energy as
shown in Eqs (9) and (20) have minor influence on the $p_{t}$
spectrum, especially for the $\pi^{+}$ and $\pi^{0}$ production. The Coulomb interaction
reduces the high-momentum $\pi^{-}$ production. The optical potential enhances the high-momentum pion
production and also the $\pi^{+}$ in the region of low $p_{t}$ comparing with the cases of the Coulomb potential and
the symmetry energy. The same phenomena is observed in the QMD
calculations for the $\pi^{0}$ transverse momentum spectrum \cite{Fu97}. The
behavior can be understood by the strong attractive potential which
enforces the pions to follow the trajectories of nucleons. Most
pions get bound by the stopped participant matter resulting in an enhancement of the low $p_{t}$ yield.
On the other hand, pions which are bound by the spectator matter are driven out
to high transverse momenta by the nucleonic flow and thus the high $p_{t}$ range is strongly enhanced. The different trends
of the $\pi^{-}$ and $\pi^{+}$ are due to the Coulomb interaction of
the charged pions and baryons. Calculations without the pion potential underpredict the high $p_{t}$ pion production is related to the missed resonances higher than N$^{\ast}$(1440). The high-$p_{t}$ pion yields are described without introducing pion potential by the calculations of Ref. \cite{La01} which takes into account an extended set of baryonic resonances with masses below 2 GeV together with the in-medium reduced NN$\rightarrow$NR cross sections.
\begin{figure*}
\includegraphics{trans}
\caption{\label{fig:wide} Comparison of the transverse momentum distributions of
charged pions and $\pi^{0}$ in the reaction $^{197}$Au+$^{197}$Au
at incident energy E$_{lab}$= 1A GeV for the cases of including all,
parts and none of the Coulomb, symmetry energy and the pion optical
potential with the available experimental data \cite{Pe97}.}
\end{figure*}
Figure 8 is a comparison of the longitudinal rapidity distribution with the same cases as shown
in Fig. 7 for the reaction $^{197}$Au+$^{197}$Au collisions at
incident energy E$_{lab}$= 1A GeV. A wider distribution is observed for the $\pi^{-}$
in the case of the optical potential. It is resulted from the fact that the optical potential enhances the high-energy
pions especially for the $\pi^{-}$ owing to the strongly attractive
potentials. The optical potential underpredicts the mid-rapidity charged pion production. The symmetry energy reduces the mid-rapidity $\pi^{-}$ production. The influence of the Coulomb potential and the symmetry
energy on the $\pi^{+}$ spectra is slightly. We also
calculated the total pion production and the $\pi^{-}$/$\pi^{+}$
yields as a function of the incident energies at the cases of the
influence of the Coulomb, symmetry energy and pion optical
potential, and also compared with the FOPI data \cite{Re07} as shown in Fig. 9. One can
see that the total pion multiplicity is not changed obviously by including any one or all parts of these
terms. However, the $\pi^{-}$/$\pi^{+}$ ratio is sensitive to the Coulomb potential and the symmetry energy owing to their sensitive dependence of the isospin degree of freedom of colliding systems, and weakly depends on the optical potential. The lower values of the $\pi^{-}$/$\pi^{+}$ ratios after including the in-medium pion dispersion at near threshold energies are consistent with the analysis in Ref. \cite{Xu10}.
Reasonable inclusion of the in-medium pion optical potential in the transport models is still an open problem, which is of importance not only in analyzing
experimental data of the pion spectrum, also in constraining the high-density behavior of the nuclear symmetry energy.
\begin{figure*}
\includegraphics{rap}
\caption{\label{fig:wide} The same as in Fig. 7, but for the rapidity
distributions.}
\end{figure*}
\begin{figure*}
\includegraphics{opt}
\caption{\label{fig:wide} (Color online) The excitation functions of
the calculated pion multiplicity and the $\pi^{-}$/$\pi^{+}$ ratios
in the reaction $^{197}$Au+$^{197}$Au for head on collisions with
including all, parts and none of the Coulomb, symmetry energy and
in-medium pion optical potential, and compared with the FOPI data \cite{Re07}.}
\end{figure*}
\subsection{D. Flow distributions}
It is well known that the EoS at extreme conditions, the
in-medium properties and the nuclear dynamics in heavy-ion collisions have been widely
studied through the analysis of collective flow in experimentally associated with transport models, such as nucleonic flow, light
particles flow, and meson flow etc \cite{Re07,Gu84,Ra99}. It is possible to reconstruct the reaction plane
with flow analysis and hence to study azimuthal correlations. The analysis of the pion collective flow would be possible to get the information of
the dense isospin asymmetric nuclear matter formed in heavy-ion collisions. The
directed flow $v_{1}$ is related to the azimuth $\phi$ as
$v_{1}=\langle p_{x}/p_{t} \rangle = \langle \cos\phi \rangle$, and the elliptic flow
$v_{2}=\langle (p_{x}/p_{t})^{2}-(p_{y}/p_{t})^{2} \rangle = \langle \cos(2\phi) \rangle$, where
the angle brackets indicate averaging over all simulation events and the transverse momentum $p_{t}=\sqrt{p_{x}^{2}+p_{y}^{2}}$.
We compared the directed flows of the charged pions with protons and also with the FOPI data \cite{Re07} as a function of the longitudinal rapidity in
the $^{197}$Au+$^{197}$Au reaction at incident energy E$_{lab}$=1.5A GeV as shown in Fig. 10. The rapidity is scaled by the incident projectile in the center of mass (c.m.) frame as the relation $y_{0}=y/y_{p}$. The experimental data are selected in terms of the scaled impact parameter $b_{0}=b/b_{\textrm{max}}$ from the measured differential cross sections using a geometrical sharp-cut approximation with $b_{\textrm{max}}=1.15(A_{P}^{1/3}+A_{T}^{1/3})$. We should note that the experimental data are sharply cut in the transverse momentum range 1.0$<u_{t0}<$4.2 scaled by projectile four-velocity in the c.m. frame $u_{p}$ as the relation $u_{t0}=u_{t}/u_{p}$ with the transverse four-velocity $u_{t}$. Calculations are performed in the near central collisions with impact parameters b=2-4 fm and in the peripheral collisions with b=8-10 fm with the linear symmetry energy and without
including the pion optical potential, but the Coulomb interaction of the charged pions and baryons is considered. The well-known 'S-shape' is clear
in the distributions of the directed flows. The diagrams rotate clockwise with increasing the impact parameter. Calculated distribution trends are consistent with the experimental results in both cases. Antiflow of $\pi^{+}$ comparing the proton at larger impact parameter appears in both of the experiments and the calculations. The phenomena is caused by the shadowing effect of participant nucleons in heavy-ion collisions and was also investigated by BUU model \cite{Li94} and IQMD \cite{Ba95}.
The difference of the flow of the charged pions is due to the Coulomb interactions. In Fig. 11 we present the transverse momentum distributions of the
directed flow of charged pions and compared the recent FOPI data \cite{Re07} in the rapidity bin $-1.8<y_{0}<0$. The ImIQMD model predicts the directed flow disappears with increasing the transverse momentum ($u_{t0}>$4). Calculated distribution trends are consistent with the experimental results in both cases, although the theoretical and experimental impact parameter ranges do not precisely match.
\begin{figure*}
\includegraphics{dirfl}
\caption{\label{fig:wide} (Color online) Comparison the
directed flow of charged pions and protons with the FOPI data in the $^{197}$Au+$^{197}$Au reaction at incident energy E$_{lab}$=1.5A
GeV for the cases of near central and peripheral collisions.}
\end{figure*}
\begin{figure*}
\includegraphics{dirtrans}
\caption{\label{fig:wide} (Color online) Transverse momentum dependence of the
directed flow of charged pions for the system $^{197}$Au+$^{197}$Au at incident energy E$_{lab}$=1.5A
GeV for the cases of near central and peripheral collisions, and compared with the FOPI data within the rapidity bin $-1.8<y_{0}<0$.}
\end{figure*}
The elliptic flow in the $^{197}$Au+$^{197}$Au reaction is also investigated with the same framework of the directed flow as shown in Fig. 12. The flow spectra can be well reproduced in the near central collisions. However, the calculations overpredict the experimental data in the peripheral collisions owing to the different definition of the impact parameters. The structures of the flow distributions are similar. To explore more information of the isospin effect, we calculated the flow
difference of the charged pions in Fig. 13, for that the isospin difference is defined by $Dv_{1}=v_{1}^{\pi^{+}}-v_{1}^{\pi^{-}}$ for the directed flow and $Dv_{2}=v_{2}^{\pi^{+}}-v_{2}^{\pi^{-}}$ for the elliptic flow. One can see that the Coulomb potential plays a significant role in the distribution of the flow difference. The influence of the symmetry energy on the distribution is slightly for the directed flow, and relatively apparent for the elliptic flow. But the symmetry energy at any stiffness does not change the distribution structure. The dependence of flow distributions of the charged pions on the Coulomb potential and on the stiffness of the symmetry energy was also investigated by the UrQMD model and the same conclusions were reported \cite{Li06}.
\begin{figure*}
\includegraphics{elpfl}
\caption{\label{fig:wide} (Color online) The same as in Fig. 10, but for the elliptic flow.}
\end{figure*}
\begin{figure*}
\includegraphics{direlp}
\caption{\label{fig:wide} (Color online) Rapidity dependence of the
flow difference of the $\pi^{+}$ and $\pi^{-}$ for the system
$^{197}$Au+$^{197}$Au at incident energy E$_{lab}$=1.5A GeV and
impact parameters b=8-10 fm.}
\end{figure*}
\subsection{E. Polar anisotropy}
\begin{figure*}
\includegraphics{polards}
\caption{\label{fig:wide} (Color online) Polar angle distributions of $\pi^{-}$ mesons in the reaction $^{197}$Au+$^{197}$Au at incident energy E$_{lab}$=1.5A GeV for different collision centralities (left panel) and stiffness of the symmetry energy (right panel).}
\end{figure*}
The analysis of the polar or azimuthal angle distributions of pion emission in heavy-ion collisions is useful to extract the information of the hot and dense nuclear matter. Anisotropic distribution of the polar angle of pion in center of mass system was reported by a number of experiments \cite{Re07,Ho96}. Isotropic pion emission is often assumed in the thermal model analysis of the compressed hadronic matter \cite{Av03}. We calculated the polar angle distributions of the $\pi^{-}$ in the reaction $^{197}$Au+$^{197}$Au at different collision centralities and also at the cases of the different stiffness of the symmetry energy as shown in Fig. 14. One can see that the anisotropy is pronounced in peripheral heavy-ion collisions, which is consistent with the experimental observable \cite{Re07}. The symmetry energies of the different stiffness have a minor influence on the polar angle distributions. Relatively, a wider distribution in the case of supersoft symmetry energy appears. The polar angle anisotropies result from the fact that pions are produced and emitted close to the surface of the reaction zone and they can escape without absorption by nuclear medium. An ellipsoid configuration in the reaction zone is formed in peripheral heavy-ion collisions and a significant fraction in the total pions are produced near the surface, so the large polar angle anisotropies are observed.
\section{IV. Conclusions}
The dynamics of the pion production in heavy-ion collisions in the
region of 1A GeV energies is investigated systematically by using the ImIQMD
model. The calculated total pions are consistent with the different Skyrme parameters SkP, SLy6, Ska, SIII which correspond to the
different modulus of incompressibility of symmetric nuclear matter, and the available experimental data are reproduced rather well.
The excitation functions of the $\pi^{-}/\pi^{+}$ ratio are sensitive to the stiffness of the symmetry energy. The difference is
pronounced at near threshold energies and for neutron-rich systems. Calculations show that the ratio does not monotonically depend on the stiffness of the symmetry energy at supra-saturation densities. The influence of the in-medium pion dispersion relation on the distributions of transverse momentum and longitudinal rapidity is pronounced. But it has a minor contribution on the excitation functions of the total pion production and the $\pi^{-}/\pi^{+}$ ratio. Calculations of the structure of the directed and elliptic flows are similar to the available experimental data. The flow distributions are sensitive to the Coulomb potential in the mean field, but weakly depend on the different cases of the symmetry energies. Polar angle anisotropies are pronounced in the peripheral collisions and the influence of the stiffness of the symmetry energy on the polar angle distribution is weak.
\section{Acknowledgements}
This work was supported by the National Natural Science Foundation
of China under Grant 10805061; the Special Foundation of the
President Fund; the West Doctoral Project of Chinese Academy of
Sciences; and the Major State Basic Research Development Program
under Grant 2007CB815000.
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\section{Introduction}
\label{sec_introduction}
In a recent series of papers \cite{hb1,hb2,hb3,hb4,assise}, we have
considered some theoretical aspects of the dynamics and thermodynamics
of systems with weak long-range interactions
\cite{dauxois,assisetot}. In these systems, the interaction potential
$u(r)$ decays with a rate slower than $1/r^d$ at large distances,
where $d$ is the dimension of space (these potentials are sometimes
called ``non-integrable''). As a result, any particle feels a
potential dominated by interactions with far away particles (i.e. the
interaction is not restricted to nearest neighbours) and the energy is
{\it non-additive}. This can lead to striking properties (absent in
systems with short-range interactions) such as inequivalence of
statistical ensembles and negative specific heats in the
microcanonical ensemble. On the other hand, the usual thermodynamic
limit $N\rightarrow +\infty$ with $N/V$ fixed is not relevant for
these systems and must be reconsidered. If we write the potential of
interaction as $u(|{\bf r}-{\bf r}'|)=k\tilde{u}(|{\bf r}-{\bf r}'|)$
where $k$ is the coupling constant, then the appropriate thermodynamic
limit for weak long-range interactions corresponds to $N\rightarrow
+\infty$ in such a way that the coupling constant $k\sim
1/N\rightarrow 0$ and the volume $V\sim 1$. In that limit, we have an
{\it extensive} scaling of the energy $E\sim N$ and entropy $S\sim N$
(while the temperature $T\sim 1$), but the system remains
fundamentally {\it non-additive}. Other equivalent combinations of the
parameters are possible to define the thermodynamic limit as discussed
in
\cite{hb1,assise} and in various contributions of \cite{assisetot}.
For systems with weak long-range interactions, it is often claimed
that the mean field approximation is a very good approximation and
that it becomes exact in the proper thermodynamic limit $N\rightarrow
+\infty$. In fact, this is true only if we are far from a critical
point. Close to a critical point, the fluctuations become large and
cannot be ignored. In that case, the two-body correlation function
does not factor out in a product of two one-body distribution
functions and the mean field approximation breaks down. It is
therefore highly desirable to derive stochastic kinetic equations that
go beyond the mean field approximation and that take full account of
fluctuations. This is the main object of the present paper.
In our previous studies, we have distinguished two types of systems:
Hamiltonian and Brownian. Hamiltonian systems with long-range
interactions are {\it isolated} and evolve at fixed energy. The
dynamics of the particles is described by $N$ coupled deterministic
Newton equations. Since the energy is conserved, the relevant
statistical ensemble is the microcanonical ensemble. The evolution of
the $N$-body distribution function is governed by the Liouville
equation and the statistical equilibrium state is described by the
microcanonical distribution. Examples of such systems are provided by
stellar systems
\cite{saslawbook,bt,hut,paddy,review}, two-dimensional
vortices \cite{km,sommeria,tabeling,houches} and the Hamiltonian Mean
Field (HMF) model
\cite{ar,cvb}; see also the important list of references
in these papers. Brownian systems with long-range interactions, on
the other hand, are {\it dissipative} and evolve at fixed
temperature. The particles are subject to their mutual long-range
interactions but they experience, in addition, a friction force and a
stochastic force which mimic the interaction with a thermal bath
(that is due to {\it short-range} interactions). Therefore, the
dynamics of the particles is described by $N$ coupled stochastic
Langevin equations. The temperature is defined through the Einstein
relation as the ratio between the diffusion coefficient and the
friction coefficient, and it measures the strength of the stochastic
force. Since the temperature is fixed, the relevant statistical
ensemble is the canonical ensemble. The evolution of the $N$-body
distribution function is governed by the Fokker-Planck equation and
the statistical equilibrium state is described by the canonical
distribution. Examples of such systems are provided by
self-gravitating Brownian particles
\cite{crs,virial2}, bacterial populations experiencing chemotaxis
\cite{ks,ng,bio,nfp} and the Brownian Mean Field (BMF) model \cite{cvb,bco}.
Systems with long-range interactions have a very peculiar dynamics and
thermodynamics \cite{dauxois,assisetot}. When the interaction is attractive,
there exists a critical point separating a spatially homogeneous
(gaseous) phase from a spatially inhomogeneous (clustered) phase. In
the microcanonical ensemble (MCE), the ``clustered'' phase appears
below a critical energy $E_{c}$ and in the canonical ensemble (CE), it
appears below a critical temperature $T_{c}$. The homogeneous phase
may still exist for $E<E_{c}$ or $T<T_{c}$ but is dynamically and
thermodynamically unstable. One fundamental illustration of this type
of phase transitions corresponds to the famous Jeans instability in
astrophysics \cite{jeans,bt}. For the Jeans problem \footnote{As is
well-known, the standard Jeans analysis assumes that an infinite and
homogeneous self-gravitating system is a steady state of the
hydrodynamical equations (the so-called Euler-Poisson system), which
is not correct. This inconsistency in the stability analysis is
referred to as the {\it Jeans swindle}
\cite{bt}. However, the Jeans treatment can be justified in cosmology
\cite{peebles} if we take into account the expansion of the universe
because this creates a sort of ``neutralizing background'' in the
comoving frame expanding with the system allowing for infinite and
homogeneous steady states.}, the critical temperature $T_{c}=+\infty$
so that the system is always in the ``clustered'' phase: this
corresponds to the universe that we know, filled of galaxies. A
spatially homogeneous distribution of matter is always unstable to
sufficiently large wavelengths, above the Jeans length
$k_{J}^{-1}=(4\pi Gn m^2\beta)^{-1/2}$, and this leads to
gravitational collapse and clustering. If we formally consider a
screened Newtonian interaction (attractive Yukawa potential) with
screening length $k_{0}^{-1}$, we find that the phase transition
occurs at a finite critical temperature $T_{c}=4\pi Gnm^2/k_{0}^2$
separating a homogeneous phase from a clustered phase \cite{hb1}. For
$T>T_{c}$ the homogeneous phase is stable and for $T<T_{c}$ it becomes
unstable to wavenumbers $k<k_{m}\equiv
k_{0}(T_c/T-1)^{1/2}$. Alternatively, we can consider a spatially
inhomogeneous self-gravitating gas in a spherical box of radius
$R$. In that case, there exists a critical energy $E_{c}=-0.335GM^2/R$
in MCE (discovered by Antonov
\cite{antonov,lbw} for stellar systems) and a critical temperature
$T_{c}=GMm/(2.52k_B R)$ in CE (discovered by Emden \cite{emden} for
isothermal stars and recently emphasized by the author for molecular
clouds in contact with a thermal bath \cite{aaiso} and for
self-gravitating Brownian particles
\cite{crs}) below which the system passes from a slightly inhomogeneous
gaseous phase to a highly inhomogeneous condensed phase \footnote{In
order to have a well-defined condensed phase, one has to introduce a
small-scale regularization of the gravitational potential. More
fundamentally, we can invoke quantum mechanics (Pauli exclusion
principle) and consider the case of self-gravitating fermions
(e.g. white dwarfs, neutron stars and fermion balls)
\cite{review}.}. Interestingly, the analogue of these ``gravitational phase
transitions'' also takes place in the context of the chemotaxis of
bacterial populations in biology \cite{jeansbio,jeansbio2}. In these
systems, an infinite and uniform distribution of cells {\it is} a
steady state of the equations of motion (the so-called Keller-Segel model and
its generalizations) so there is no ``Jeans swindle''. Furthermore, the
screened Yukawa potential enters naturally in the problem and the
screening length has a clear physical interpretation as it takes into
account the degradation of the secreted chemical. Another illustration
of this type of phase transitions is given by the study of toy models
like the Hamiltonian Mean Field (HMF) and the Brownian Mean Field
(BMF) models
\cite{ar,cvb}. For these systems, there exists a critical energy
$E_{c}=kM^2/(8\pi)$ and a critical temperature $T_{c}=kM/(4\pi)$
separating a spatially homogeneous (non-magnetized) phase from a
spatially inhomogeneous (magnetized) phase. The magnetization plays
the role of the order parameter and the phase transition is second
order in that case
\cite{ar,cvb}. We expect the above-mentioned types of
phase transitions, and their generalizations
\cite{klein,ellis,martzel,bb,touchette,review,hb1}, to occur for a
large class of mean field systems with different potentials of
interaction.
The critical point separating the homogeneous phase from the
inhomogeneous phase is often called a {\it spinodal point}. The
instability threshold of the homogeneous phase can be obtained from
different classical methods: (i) by studying the sign of the second
order variations of the thermodynamical potential (entropy in MCE or
free energy in CE) and determining when the spatially uniform
distribution becomes an unstable saddle point (see, e.g., Sec. 4.4. of
Paper I or Appendix C of \cite{nfp}) (ii) by studying the bifurcation
(from homogeneous to inhomogeneous) of the solutions of the meanfield
integrodifferential equation (I-19) determining the statistical
equilibrium state (see, e.g., Sec. 2.3 of Paper I and Appendix C of
\cite{nfp}) (iii) by studying the linear and nonlinear dynamical
stability of the homogeneous state with respect to kinetic equations:
Vlasov, Euler, Landau, Kramers, Smoluchowski... (see, e.g., Paper II and
\cite{jeansbio2,nfp}). In the
homogeneous phase, the one-body distribution function is trivial (it
is spatially uniform with a Maxwellian velocity distribution) and the
state of the system is usually characterized by the two-body
distribution function. For weak long-range interactions, the two-body
correlation function $h(|{\bf r}_{1}-{\bf r}_{2}|)$ can be obtained
from the equilibrium BBGKY-hierarchy at the order $O(1/N)$ by
neglecting the three-body correlation function that is of order
$O(1/N^2)$
\cite{hb1}. This is similar to the Debye-H\"uckel approximation in
plasma physics. It is found however that, for large but fixed $N$, the
Fourier transform of the correlation function {diverges} at the
critical point (or at the instability threshold) so that the
mean field approximation breaks down close to the critical point. This
implies that the phase transition should take place {\it strictly
before} the critical point (or strictly before the instability
threshold) predicted by mean field theory. In
\cite{hb1,cvb}, we have reached this conclusion from the study of the
equilibrium BBGKY hierarchy. In the present work, we would like to
complement our previous studies by developing a theory of fluctuations
starting directly from the stochastic kinetic equation governing
the evolution of the fluctuating density field. In this paper, we restrict
ourselves to the case of Brownian systems.
The paper is organized as follows. In Sec. \ref{sec_overdamped}, we
consider a gas of Brownian particles in interaction in an overdamped
limit where inertial effects are neglected. In Sec. \ref{sec_ea},
using a BBGKY-like hierarchy, we give the deterministic kinetic
equation (\ref{ea5}) satisfied by the smooth density profile and the
Smoluchowski equation (\ref{ea7}) resulting from a mean field
approximation. In Sec. \ref{sec_ex}, we give the stochastic kinetic
equation (\ref{ex2}) satisfied by the exact density distribution
(expressed in terms of $\delta$-functions) and, averaging over the
noise, we recover the equation obtained from the BBGKY-like
hierarchy. In Sec. \ref{sec_cg}, we give the stochastic kinetic
equation (\ref{cg2}) satisfied by the coarse-grained density
distribution obtained by averaging the exact density distribution over
a small spatio-temporal window (thus keeping track of fluctuations).
In Appendix \ref{sec_ll}, we provide another derivation of this
equation by using the general theory of fluctuations exposed by Landau
\& Lifshitz \cite{ll}. In Sec. \ref{sec_ch}, we derive generalized
stochastic Cahn-Hilliard equations in the limit of short-range
interactions. In Sec. \ref{sec_f}, starting from the stochastic
Smoluchowski equation (\ref{f1})-(\ref{f2}), we develop a theory of
fluctuations for Brownian particles with weak long-range
interactions. Specifically, we study the correlations $\langle
\delta\hat{\rho}_{k}(t)\delta\hat{\rho}_{k'}(t+\tau)\rangle$ of the Fourier
transform of the density fluctuations and show that it behaves like
$A(k) e^{\sigma(k)\tau}$. In the stable regime, the decay rate
$\sigma(k)<0$ of the temporal correlations coincides with the decay
rate of the perturbations $\delta\hat{\rho}_{k}(t)\sim e^{\sigma(k)t}$
governed by the deterministic mean field Smoluchowski equation
(without noise). On the other hand, for $T<T_{c}$, the amplitude of
the correlation function $A(k)$ diverges as we approach the
instability threshold $k\rightarrow k_m$, suggesting that the
instability takes place sooner than predicted by mean field theory
(for $T>T_{c}$, considering the ``dangerous'' mode $k=k_{*}$, the
correlation function diverges as we approach the critical point
$T\rightarrow T_{c}^{+}$). These results clearly demonstrate that
fluctuations cannot be ignored close to a critical point. In
Sec. \ref{sec_inertial}, we generalize our results to the case of an
inertial model of particles in interaction including a friction force
and a stochastic force. The overdamped model is recovered in a strong
friction limit. Again, the correlations of the density fluctuations
diverge as we approach the instability threshold but, in sharp
contrast, the correlations of the velocity fluctuations remain finite
at this threshold. In Sec. \ref{sec_mm}, we derive a generalized
Smoluchowski equation taking into account memory effects and make
contact with the Cattaneo model and the telegraph equation. In
Sec. \ref{sec_chemo}, we derive a stochastic model of chemotaxis
generalizing the deterministic mean field Keller-Segel model by
keeping track of fluctuations. Finally, in Sec. \ref{sec_ps}, we
extend our study in phase space taking full account of the inertia of
the particles. We derive a stochastic Kramers equation and make the
connection with the diffusive and hydrodynamic models of the previous
sections.
\section{The overdamped case}
\label{sec_overdamped}
\subsection{The smooth density distribution}
\label{sec_ea}
We consider an overdamped system of $N$ Brownian particles in interaction whose dynamics is governed by the coupled stochastic equations (see Paper II):
\begin{equation}
\label{ea1}
\frac{d{\bf r}_{i}}{dt}=-\mu m^2\nabla_i U+\sqrt{2D_{*}}{\bf R}_{i}(t),
\end{equation}
where $\mu=1/(m\xi)$ is the mobility ($\xi$ denotes the friction
coefficient), $U({\bf r}_{1},...,{\bf r}_{N})=\sum_{i<j}u(|{\bf
r}_{i}-{\bf r}_{j}|)$ is the potential of interaction, $D_{*}$ is the
diffusion coefficient and ${\bf R}_{i}(t)$ is a white noise such that
$\langle {\bf R}_{i}(t)\rangle={\bf 0}$ and $\langle
R_{i}^{\alpha}(t)R_{j}^{\beta}(t')\rangle=\delta_{ij}\delta_{\alpha\beta}\delta(t-t')$
where $i=1,...,N$ refer to the particles and $\alpha=1,...,d$ to the
coordinates of space. We assume that the particles interact via a
binary potential $u(|{\bf r}_{i}-{\bf r}_{j}|)$ depending only on the
absolute distance between the particles. As discussed in the
Introduction, this stochastic model can describe self-gravitating
Brownian particles \cite{crs,virial2}, bacterial populations experiencing
chemotaxis \cite{ng,bio,nfp} (see also Sec. \ref{sec_chemo}) or toy
models like the BMF model
\cite{cvb}. The $N$-body distribution $P_{N}({\bf r}_{1},...,{\bf
r}_{N},t)$ is solution of the Fokker-Planck equation
\begin{eqnarray}
\label{ea2}
\frac{\partial P_{N}}{\partial t}
=\sum_{i=1}^{N}\frac{\partial}{\partial {\bf r}_{i}}\cdot \left (D_{*}\frac{\partial P_{N}}{\partial {\bf r}_{i}}+\mu m^2 P_{N}\frac{\partial U}{\partial {\bf r}_{i}}\right ).
\end{eqnarray}
The stationary solution of this equation is the Gibbs canonical distribution
\begin{eqnarray}
\label{ea3}
P_{N}=\frac{1}{Z}e^{-\beta m^2 U},
\end{eqnarray}
provided that the inverse temperature $\beta=1/(k_{B}T)$ is related to the mobility and the diffusion coefficient through the Einstein relation
\begin{eqnarray}
\label{ea4}
\beta=\frac{\mu}{D_{*}}.
\end{eqnarray}
From the Fokker-Planck equation (\ref{ea2}), we can construct a
BBGKY-like hierarchy \cite{hb2}. The exact first equation of the
hierarchy is
\begin{equation}
\label{ea5} {\partial P_{1}\over\partial t}=
{\partial\over\partial {\bf r}_{1}}\cdot \biggl\lbrack D_{*} {\partial
P_{1}\over\partial {\bf r}_{1}}+\mu m^{2}(N-1)\int
{\partial {u}_{12}\over\partial {\bf r}_{1}}P_{2}
d{\bf r}_{2}\biggr\rbrack.
\end{equation}
The two-body distribution function can be written as
\begin{equation}
\label{ea5v}
P_{2}({\bf r}_{1},{\bf r}_{2},t)=P_{1}({\bf r}_{1},t)P_{1}({\bf r}_{2},t)+P'_{2}({\bf r}_{1},{\bf r}_{2},t),
\end{equation}
where $P'_{2}({\bf r}_{1},{\bf r}_{2},t)$ is the correlation function.
Let us consider a weak long-range potential of interaction $u({\bf
r}_{12})=k\tilde{u}({\bf r}_{12})$ in a proper thermodynamic limit
$N\rightarrow +\infty$ in such a way that $k\sim 1/N$ and $V\sim 1$.
Using scaling arguments, it can be shown that $P_{2}'=O(1/N)$ except
close to a critical point (see \cite{hb1,assise} for details).
Therefore, if we are far from a critical point and if $N$ is
sufficiently large, we can make the mean field approximation
$P_{2}({\bf r}_{1},{\bf r}_{2},t)\simeq P_{1}({\bf r}_{1},t)
P_{1}({\bf r}_{2},t)$ and we obtain
\begin{eqnarray}
\label{ea6}
\frac{\partial P_{1}}{\partial t}=
{\partial\over\partial {\bf r}_{1}}\cdot \biggl\lbrack D_{*} {\partial
P_{1}\over\partial {\bf r}_{1}}+\mu m^{2}NP_{1}({\bf r},t)\int
{\partial {u}_{12}\over\partial {\bf r}_{1}}P_{1}({\bf r}_{2},t)
d{\bf r}_{2}\biggr\rbrack.
\end{eqnarray}
Introducing the smooth density distribution $\rho({\bf r},t)=NmP_{1}({\bf r},t)$, this equation can be rewritten as
\begin{eqnarray}
\label{ea7}
\frac{\partial\rho}{\partial t}({\bf r},t)=D_{*}\Delta\rho({\bf r},t)
+\mu m\nabla \cdot \left (\rho({\bf r},t)\nabla\int \rho({\bf r}',t)u({\bf r}-{\bf r}')d{\bf r}'\right ).
\end{eqnarray}
It can be put in the form of a mean field Smoluchowski equation
\begin{eqnarray}
\label{ea8}
\frac{\partial\rho}{\partial t}=D_{*}\Delta\rho
+\mu m \nabla\cdot (\rho\nabla\Phi),
\end{eqnarray}
where $\Phi({\bf r},t)$ is a smooth potential produced by the
particles themselves
\begin{eqnarray}
\label{ea9}
\Phi({\bf r},t)= \int \rho({\bf r}',t)u({\bf r}-{\bf r}')\, d{\bf r}'.
\end{eqnarray}
If we introduce the mean field Boltzmann free energy functional
\begin{eqnarray}
\label{ea12}
F=E-TS=\frac{1}{2}\int \rho\Phi \, d{\bf r}+k_{B}T\int \frac{\rho}{m}\ln \frac{\rho}{m}\, d{\bf r},
\end{eqnarray}
we can rewrite the mean field Smoluchowski equation (\ref{ea8}) in the form
\begin{eqnarray}
\label{ea13}
\frac{\partial\rho}{\partial t}=\nabla\cdot \left\lbrack \frac{1}{\xi}\rho({\bf r},t)\nabla\frac{\delta F}{\delta\rho}\right\rbrack.
\end{eqnarray}
This equation satisfies an $H$-theorem appropriate to the canonical ensemble
\begin{eqnarray}
\label{ea13a}
\dot F=\int \frac{\delta F}{\delta\rho}\frac{\partial\rho}{\partial t}d{\bf r}=\int \frac{\delta F}{\delta\rho}\nabla\cdot \left\lbrack \frac{1}{\xi}\rho\nabla\frac{\delta F}{\delta\rho}\right\rbrack d{\bf r}=-\int\frac{1}{\xi}\rho \left (\nabla\frac{\delta F}{\delta\rho}\right )^{2} d{\bf r}\le 0.
\end{eqnarray}
The steady solution of the mean field Smoluchowski equation (\ref{ea8}) or (\ref{ea13}) corresponds to a uniform $\mu=\delta F/\delta\rho$ leading to the mean field Boltzmann distribution
\begin{eqnarray}
\label{ea10}
\rho=Ae^{-\beta m\Phi},
\end{eqnarray}
where $\Phi({\bf r})$ is given by Eq. (\ref{ea9}).
Finally, we note that the mean field Smoluchowski equation (\ref{ea7}) can be written in Fourier space as
\begin{eqnarray}
\label{ea11}
\frac{\partial\hat{\rho}}{\partial t}({\bf k},t)
=-D_{*} k^2 \hat{\rho}({\bf k},t)- (2\pi)^d\mu m\int {\bf k}\cdot {\bf k}'\hat{\rho}({\bf k}-{\bf k}',t)\hat{u}({\bf k}')\hat{\rho}({\bf k}',t)d{\bf k}'.
\end{eqnarray}
For weak long-range potentials of interaction, the mean field
approximation usually provides a good and useful description of the
system as a first approach \footnote{It is often advocated that
long-range potentials of interaction exhibit lack of temperedness and
stability. Some potentials, like the cosine potential in the HMF model
are well-behaved. By contrast, the gravitational potential is singular
and, strictly speaking, there is no statistical equilibrium state (no
global maximum of entropy in MCE and no global minimum of free energy
in CE). However, there exist long-lived metastable states (local
maxima of entropy or local minima of free energy) that can be
adequately described by the mean field approximation
\cite{review}. The formation of a Dirac peak in CE, which can be
viewed as the ``equilibrium state'' of the system, can also be
described by the mean field Smoluchowski-Poisson system
\cite{post}. By contrast, the formation of binary stars in MCE
requires going beyond the mean field approximation.}. We must,
however, recall its domain of validity: (i) first, it assumes that the
number of particles in the system is large (mathematically speaking it
is valid in the limit $N\rightarrow +\infty$). Therefore, we can
expect deviations from mean field theory due to finite $N$
effects. These deviations will become manifest for sufficiently large
times (see discussion at the end of Sec. \ref{sec_cg}). (ii) Close to
a critical point $T\rightarrow T_{c}$, the correlation function
diverges (see
\cite{hb1} and Secs. \ref{sec_f} and \ref{sec_examples}). Typically, we expect a scaling of the form
$P_{2}'\sim N^{-1}(T-T_{c})^{-1}$ so that the limits $N\rightarrow
+\infty$ and $T\rightarrow T_{c}$ do not commute (see Sec. 2.7 in
\cite{cvb}). Therefore, even for large $N$, the mean field
approximation is expected to break down close to a critical point
because $P_{2}'$ is not necessarily small (the mean field
approximation is valid for $N\gg (T-T_{c})^{-1}$, which requires
larger and larger particle numbers as $T\rightarrow T_{c}$). In the two
cases (i) and (ii) mentioned above, we must take fluctuations into
account and consider stochastic kinetic equations as discussed in the
sequel.
\subsection{The exact density distribution}
\label{sec_ex}
The exact density distribution of the particles is expressed as a sum
of Dirac distributions in the form
\begin{equation}
\label{ex1}
\rho_{d}({\bf r},t)=m \sum_{i=1}^{N}\delta({\bf r}-{\bf r}_{i}(t)).
\end{equation}
It was shown by Dean \cite{dean} (see also \cite{kk,mt,arb}) that the exact density field satisfies a stochastic
equation of the form
\begin{eqnarray}
\label{ex2}
\frac{\partial\rho_d}{\partial t}({\bf r},t)=D_{*}\Delta\rho_{d}({\bf r},t)
+\mu m\nabla \cdot \left (\rho_{d}({\bf r},t)\nabla\int \rho_{d}({\bf r}',t)u({\bf r}-{\bf r}')d{\bf r}'\right )\nonumber\\
+\nabla \cdot \left (\sqrt{2D_{*} m\rho_{d}({\bf r},t)}{\bf R}({\bf r},t)\right ),\qquad
\end{eqnarray}
where ${\bf R}({\bf r},t)$ is a Gaussian random field such that
$\langle {\bf R}({\bf r},t)\rangle={\bf 0}$ and $\langle
R^{\alpha}({\bf r},t)R^{\beta}({\bf
r}',t')\rangle=\delta_{\alpha\beta}\delta({\bf r}-{\bf
r}')\delta(t-t')$. Note that the noise is multiplicative
\cite{dean}. Introducing the exact potential
\begin{eqnarray}
\label{ex3}
\Phi_{d}({\bf r},t)= \int \rho_{d}({\bf r}',t)u({\bf r}-{\bf r}')\, d{\bf r}',
\end{eqnarray}
the stochastic equation (\ref{ex2}) can be rewritten as
\begin{eqnarray}
\label{ex4}
\frac{\partial\rho_d}{\partial t}=D_{*}\Delta\rho_{d}
+\mu m \nabla\cdot (\rho_{d}\nabla\Phi_{d})+\nabla\cdot (\sqrt{2D_{*}m\rho_{d}}{\bf R}).
\end{eqnarray}
For $D_{*}=0$ and $\mu\neq 0$, we get the exact deterministic
equation
\begin{eqnarray}
\label{ex5}
\frac{\partial\rho_d}{\partial t}=\mu m \nabla\cdot (\rho_{d}\nabla\Phi_{d}).
\end{eqnarray}
If we introduce the discrete Boltzmann free energy functional
\begin{eqnarray}
\label{ex6}
F_{d}=E_{d}-TS_{d}=\frac{1}{2}\int \rho_{d}\Phi_{d} \, d{\bf r}+k_{B}T\int \frac{\rho_{d}}{m}\ln \frac{\rho_{d}}{m}\, d{\bf r},
\end{eqnarray}
we can rewrite the stochastic equation (\ref{ex4}) in the form
\begin{eqnarray}
\label{ex7}
\frac{\partial\rho_{d}}{\partial t}=\nabla\cdot \left\lbrack \frac{1}{\xi}\rho_{d}({\bf r},t)\nabla\frac{\delta F_{d}}{\delta\rho_{d}}\right\rbrack+\nabla\cdot \left (\sqrt{\frac{2k_{B}T\rho_{d}}{\xi}}{\bf R}\right ).
\end{eqnarray}
This equation can be viewed as a Langevin equation for the field
${\rho}_d({\bf r},t)$. From this equation, it can be shown that
the probability of the density distribution $W[\rho_{d},t]$ is
governed by the Fokker-Planck equation
\begin{eqnarray}
\label{ex7add}
\frac{\partial W[\rho_d,t]}{\partial t}=-\int \frac{\delta}{\delta\rho_{d}({\bf r},t)}\left\lbrace \nabla\cdot \rho_{d}({\bf r},t)\nabla\left\lbrack D_{*}\frac{\delta}{\delta\rho_{d}}+\mu\frac{\delta F_{d}}{\delta\rho_{d}}\right \rbrack W[\rho_d,t]\right\rbrace d{\bf r}.
\end{eqnarray}
At equilibrium, we get the Boltzmann distribution $W[\rho_{d}]\propto e^{-\beta (F_{d}[\rho_{d}]-\mu\int \rho_{d}d{\bf r})}$ \cite{kk,dean,mt,arb}.
If we average Eq. (\ref{ex2}) over the noise, we find that the evolution of
the smooth density field $\rho({\bf r},t)=\langle\rho_{d}\rangle$ is
governed by an equation of the form
\begin{eqnarray}
\label{ex8}
\frac{\partial\rho}{\partial t}({\bf r},t)
=D_{*}\Delta\rho({\bf r},t)+\mu m\nabla \cdot \int \langle \rho_{d}({\bf r},t)\rho_{d}({\bf r}',t)\rangle \nabla u({\bf r}-{\bf r}')d{\bf r}'.
\end{eqnarray}
Using the identity (see Appendix \ref{sec_cf}):
\begin{eqnarray}
\label{ex9}
\langle \rho_d({\bf r},t) \rho_{d}({\bf r}',t)\rangle=Nm^2 P_{1}({\bf r},t)\delta({\bf r}-{\bf r}')
+N(N-1)m^2 P_{2}({\bf r},{\bf r}',t),
\end{eqnarray}
and assuming that $\nabla u({\bf 0})={\bf 0}$, we find that Eq. (\ref{ex8})
coincides with the exact equation (\ref{ea5}) of the BBGKY-like
hierarchy giving the evolution of the one-body distribution
function. Furthermore, if we make the mean field approximation
$\langle \rho_d({\bf r},t)
\rho_{d}({\bf r}',t)\rangle\simeq \rho({\bf r},t) \rho({\bf r}',t)$,
we recover the mean field Smoluchowski equation (\ref{ea8}).
\subsection{The coarse-grained density distribution}
\label{sec_cg}
Equation (\ref{ea5}) (or equivalently Eq. (\ref{ex8})) for the
ensemble averaged density field $\rho({\bf r},t)$ is a deterministic
equation since we have averaged over the noise. In contrast,
Eq. (\ref{ex2}) for the exact density field $\rho_{d}({\bf r},t)$ is a
stochastic equation taking into account fluctuations. However, it is
not very useful in practice since the field $\rho_{d}({\bf r},t)$ is a
sum of Dirac distributions, not a regular function. Therefore, it is
easier to directly solve the stochastic equations (\ref{ea1}) rather
than the equivalent Eq. (\ref{ex2}). Following \cite{arb}, we can keep
track of fluctuations while avoiding the problem of $\delta$-functions
by defining a ``coarse-grained'' density field $\overline{\rho}({\bf
r},t)$ obtained by averaging the exact density field on a
spatio-temporal window of finite resolution. The ``coarse-grained''
density field satisfies a stochastic equation of the form
\begin{eqnarray}
\label{cg1}
\frac{\partial\overline{\rho}}{\partial t}({\bf r},t)=D_{*}\Delta\overline{\rho}({\bf r},t)
+\mu m\nabla \cdot \left (\int \overline{\rho}^{(2)}({\bf r},{\bf r}',t) \nabla u({\bf r}-{\bf r}')d{\bf r}'\right )\nonumber\\
+\nabla \cdot \left (\sqrt{2D_{*} m\overline{\rho}({\bf r},t)}{\bf R}({\bf r},t)\right ).\qquad
\end{eqnarray}
where $\overline{\rho}^{(2)}({\bf r},{\bf r}',t)$ is a two-body
correlation function. For a weak long-range potential of interaction
and for a sufficiently small spatio-temporal window, we propose to
make the approximation $\overline{\rho}^{(2)}({\bf r},{\bf
r}',t)\simeq
\overline{\rho}({\bf r},t)\overline{\rho}({\bf r}',t)$. In that case, we obtain
a stochastic equation of the form
\begin{eqnarray}
\label{cg2}
\frac{\partial\overline{\rho}}{\partial t}({\bf r},t)=D_{*}\Delta\overline{\rho}({\bf r},t)
+\mu m \nabla \cdot \left (\overline{\rho}({\bf r},t)\int \overline{\rho}({\bf r}',t) \nabla u({\bf r}-{\bf r}')d{\bf r}'\right )\nonumber\\
+\nabla \cdot \left (\sqrt{2D_{*} m\overline{\rho}({\bf r},t)}{\bf R}({\bf r},t)\right ).\qquad
\end{eqnarray}
Introducing the smooth potential
\begin{eqnarray}
\label{cg3}
\overline{\Phi}({\bf r},t)= \int \overline{\rho}({\bf r}',t)u({\bf r}-{\bf r}')\, d{\bf r}',
\end{eqnarray}
the stochastic equation (\ref{cg2}) can be rewritten
\begin{eqnarray}
\label{cg4}
\frac{\partial\overline{\rho}}{\partial t}=D_{*}\Delta\overline{\rho}
+\mu m \nabla\cdot (\overline{\rho}\nabla\overline{\Phi})+\nabla\cdot (\sqrt{2D_{*}m\overline{\rho}}\ {\bf R}).
\end{eqnarray}
This will be called the {\it stochastic Smoluchowski equation} for the
smoothed-out density field $\overline{\rho}({\bf r},t)$. This
equation is intermediate between Eqs. (\ref{ea7}) and (\ref{ex2}). It
keeps track of fluctuations while dealing with a continuous density field
instead of a sum of $\delta$-functions. This equation will be central
in the rest of the paper. We will see that it can reproduce the
equilibrium density correlation function (\ref{f26}) that was obtained in
Paper I from the equilibrium BBGKY-like hierarchy
\cite{hb1}. Therefore, it represents an improvement with respect to
the mean field Smoluchowski equation (\ref{ea7}). We stress that this equation is physically distinct from Eq. (\ref{ex2}). In Appendix
\ref{sec_ll}, we propose an alternative derivation of Eq. (\ref{cg4}) by
using the general theory of fluctuations exposed by Landau \&
Lifshitz \cite{ll}.
If we introduce the coarse-grained Boltzmann free energy functional
\begin{eqnarray}
\label{cg5}
F_{c.g.}=E_{c.g.}-TS_{c.g.}=\frac{1}{2}\int \overline{\rho}\overline{\Phi} \, d{\bf r}+k_{B}T\int \frac{\overline{\rho}}{m}\ln \frac{\overline{\rho}}{m}\, d{\bf r},
\end{eqnarray}
we can write the stochastic equation (\ref{cg4}) in the form
\begin{eqnarray}
\label{cg6}
\frac{\partial\overline{\rho}}{\partial t}=\nabla\cdot \left\lbrack \frac{1}{\xi}\overline{\rho}({\bf r},t)\nabla\frac{\delta F_{c.g.}}{\delta\overline{\rho}}\right\rbrack+\nabla\cdot \left (\sqrt{\frac{2k_{B}T\overline{\rho}}{\xi}}{\bf R}\right ).
\end{eqnarray}
This equation can be viewed as a Langevin equation for the field
$\overline{\rho}({\bf r},t)$. The evolution of the probability of the
density distribution $W[\overline{\rho},t]$ is governed by a
Fokker-Planck equation of the form (\ref{ex7add}) where $F_{d}$ and
$\rho_{d}$ are replaced by $F_{c.g.}$ and $\overline{\rho}$. At
equilibrium, we have $W[\overline{\rho}]\propto e^{-\beta
(F_{c.g.}[\overline{\rho}]-\mu\int \overline{\rho}d{\bf r})}$. For
$N\rightarrow +\infty$, the equilibrium distribution
$W[\overline{\rho}]$ is strongly peaked around the {\it global}
minimum of $F_{c.g.}[\overline{\rho}]$ at fixed mass. However, the
system can remain trapped in a metastable state (local minimum of
$F_{c.g.}[\overline{\rho}]$) for a very long time. Let us be more
precise. If we ignore the noise term, Eq. (\ref{cg6}) reduces to
Eq. (\ref{ea13}). In that case, the system tends to a steady state
that is a {minimum} (global or local) of the free energy functional
$F_{c.g.}[\overline{\rho}]$ at fixed mass (maxima or saddle points of
free energy are linearly dynamically unstable with respect to mean
field Fokker-Planck equations
\cite{nfp}). If the free energy admits several local minima, the
selection of the steady state will depend on a notion of {\it basin of
attraction}. Without noise, the system remains on a minimum of free
energy forever. Now, in the presence of noise, the fluctuations can
induce a {\it dynamical phase transition} from one minimum to the
other. Thus, accounting correctly for fluctuations is very important
when there exists metastable states. The probability of transition
scales as $e^{-\Delta F/k_{B}T}$ where $\Delta F$ is the barrier of
free energy between two minima. Therefore, in an infinite time, the
system will explore all the minima and will spend most time in the
global minimum for which $\Delta F$ is the largest. Now, for systems
with long-range interactions, the barrier of free energy $\Delta F$
scales as $N$ so that the probability of escape from a local minimum
is very small and behaves as $e^{-N}$. Therefore, even if the global
minimum is in principle the most probable state, metastable states are
highly robust in practice since their lifetime scales like $e^{N}$. They
are thus fully relevant for $N\gg 1$ \cite{meta}. These interesting
features (basin of attraction, dynamical phase transitions,
metastability,...) would be interesting to study in more detail in
the case of systems with long-range interactions. Some results in this
direction have been reported in \cite{monaghan,ko,meta} in the
gravitational case.
\subsection{Generalized Cahn-Hilliard equations}
\label{sec_ch}
Let us assume that $u(|{\bf r}-{\bf r}'|)$ is a short-range potential
of interaction and that the preceding equation (\ref{cg4}) remains valid (to
simplify the notations, we drop the bar on the coarse-grained
fields). Setting ${\bf q}={\bf r}'-{\bf r}$ and writing
\begin{equation}
\label{ch1}
{\Phi}({\bf r},t)=\int u(q){\rho}({\bf r}+{\bf q},t)d{\bf q},
\end{equation}
we can Taylor expand ${\rho}({\bf r}+{\bf q},t)$ to second order in ${\bf q}$
so that
\begin{equation}
\label{ch2}
{\rho}({\bf r}+{\bf q},t)={\rho}({\bf r},t)+\sum_{i}\frac{\partial{\rho}}{\partial x_{i}}q_{i}+\frac{1}{2}\sum_{i,j}\frac{\partial^{2}{\rho}}{\partial x_{i}\partial x_{j}}q_{i}q_{j}.
\end{equation}
Substituting this expansion in Eq. (\ref{ch1}), we obtain \cite{lemou}:
\begin{equation}
\label{ch3}
{\Phi}({\bf r},t)=-a \rho({\bf r},t)
-\frac{b}{2}\Delta\rho({\bf r},t),
\end{equation}
with $a=-S_{d}\int_{0}^{+\infty} u(q) q^{d-1} dq$ and $b=-\frac{1}{d}
S_{d}\int_{0}^{+\infty} u(q) q^{d+1} dq$. Note that $l=(b/a)^{1/2}$
has the dimension of a length corresponding to the range of the
interaction. Substituting Eq. (\ref{ch3}) in Eq. (\ref{cg5}), we can
put the free energy in the form
\begin{eqnarray}
\label{ch5}
F_{c.g.}\lbrack{\rho}\rbrack=\int \left\lbrack \frac{1}{2} (\nabla{\rho})^{2}+V({\rho})\right\rbrack d{\bf r},
\end{eqnarray}
where $V$ is the effective potential
\begin{equation}
\label{ch6}
V(\rho)=-\frac{a}{b}\rho^{2}+\frac{2k_{B}T}{m b} \rho\ln\rho+V_{0}.
\end{equation}
In that case, Eq. (\ref{cg6}) can be rewritten
\begin{equation}
\label{ch7}
\frac{\partial{\rho}}{\partial t}=-A\nabla\cdot \left\lbrack {\rho}\nabla \left (\Delta{\rho}-V'({\rho})\right)\right\rbrack+\nabla\cdot \left (\sqrt{\frac{2k_{B}T{\rho}}{\xi}}{\bf R}\right ),
\end{equation}
with $A=b/(2\xi)$. Substituting Eq. (\ref{ch6}) in Eq. (\ref{ch7}) we explicitly obtain
\begin{equation}
\label{ch8}
\xi\frac{\partial{\rho}}{\partial t}=\frac{k_{B}T}{m}\Delta{\rho}-\frac{a}{2}\Delta{\rho}^{2}-\frac{b}{2}\nabla\cdot ({\rho}\nabla(\Delta{\rho}))+\nabla\cdot \left (\sqrt{{2k_{B}T\xi{\rho}}}\, {\bf R}\right ).
\end{equation}
Without the noise term, the steady state of Eq. (\ref{cg6}), (\ref{ch7}) or (\ref{ch8}) corresponds to a uniform $\mu=\delta F/\delta\rho$ leading to
\begin{equation}
\label{ch9}
\Delta{\rho}=V'({\rho})-\mu=-\frac{2a}{b}{\rho}+\frac{2k_{B}T}{mb}\ln{\rho}+\frac{2k_{B}T}{mb}-\mu.
\end{equation}
In particular, at $T=0$, Eq. (\ref{ch8}) reduces to
\begin{equation}
\label{ch10}
\xi\frac{\partial{\rho}}{\partial t}=-\frac{a}{2}\Delta{\rho}^{2}-\frac{b}{2}\nabla\cdot ({\rho}\nabla(\Delta{\rho})),
\end{equation}
and its steady state is solution of the Helmholtz equation
\begin{equation}
\label{ch11}
\Delta{\rho}+\frac{2a}{b}{\rho}=-\mu.
\end{equation}
Morphologically, Eq. (\ref{cg6}) with Eq. (\ref{ch5}), or equivalently
Eq. (\ref{ch7}), is similar to the stochastic Cahn-Hilliard equation
\cite{bray} for model B (conserved dynamics):
\begin{eqnarray}
\label{ch12}
\xi\frac{\partial\rho}{\partial t}=\Delta \frac{\delta F}{\delta\rho}+\sqrt{2\xi k_{B}T}\nabla\cdot {\bf R}, \qquad F\lbrack{\rho}\rbrack=\int \left\lbrack \frac{1}{2} (\nabla{\rho})^{2}+V({\rho})\right\rbrack d{\bf r},
\end{eqnarray}
or explicitly
\begin{equation}
\label{ch13}
\xi\frac{\partial\rho}{\partial t}=-\Delta (\Delta\rho-V'(\rho))+\sqrt{2\xi k_{B}T}\nabla\cdot {\bf R}.
\end{equation}
There are, however, crucial differences between Eqs. (\ref{ch7}) and
(\ref{ch13}). First, the presence of the density $\rho({\bf r},t)$ in
the deterministic current and in the noise term. Secondly, in the usual
Cahn-Hilliard problem, the potential has a double-well shape of the
typical form $V(\rho)=(1-\rho^2)^2$ leading to a phase separation
while, in the present case, the potential (\ref{ch6}) is of a
different nature.
\subsection{Theory of fluctuations}
\label{sec_f}
Let us return to the stochastic Smoluchowski equation (\ref{cg6})
satisfied by the coarse-grained density distribution that we write in
the form (for convenience, we drop the bars on the coarse-grained
fields):
\begin{eqnarray}
\label{f1}
\frac{\partial\rho}{\partial t}
=\nabla\cdot \left\lbrack \frac{1}{\xi}\left (\frac{k_{B}T}{m}\nabla\rho+\rho\nabla\Phi\right )\right\rbrack+\nabla\cdot \left (\sqrt{\frac{2k_{B}T\rho}{\xi}}{\bf R}\right ),
\end{eqnarray}
\begin{eqnarray}
\label{f2}
\Phi({\bf r},t)= \int \rho({\bf r}',t)u({\bf r}-{\bf r}')\, d{\bf r}'.
\end{eqnarray}
We wish to study the fluctuations of the density around an infinite
and homogeneous equilibrium distribution. To that purpose, we consider
small perturbations $\delta\rho({\bf r},t)$ and $\delta\Phi({\bf
r},t)$ around the steady state $\rho({\bf r})=\rho$, $\Phi({\bf
r})=\Phi$ with $\Phi=\rho\int u(x)d{\bf x}$. The linearized equations
for the perturbations are
\begin{eqnarray}
\label{f3}
\xi\frac{\partial\delta\rho}{\partial t}
=\frac{k_{B}T}{m}\Delta \delta\rho +\rho\Delta\delta\Phi+ \sqrt{2k_{B}T\xi\rho}\nabla\cdot {\bf R},
\end{eqnarray}
\begin{eqnarray}
\label{f4}
\delta\Phi({\bf r},t)= \int \delta\rho({\bf r}',t)u({\bf r}-{\bf r}')\, d{\bf r}'.
\end{eqnarray}
We now decompose the perturbations in Fourier modes in the form
\begin{eqnarray}
\label{f5}
\delta\rho({\bf r},t)=\int\delta\hat{\rho}({\bf k},\omega)e^{i({\bf k}\cdot {\bf r}-\omega t)}d{\bf k}d\omega,
\end{eqnarray}
and similar expressions for $\delta\Phi({\bf r},t)$ and ${\bf R}({\bf
r},t)$. Taking the Fourier transform of Eqs. (\ref{f3}) and
(\ref{f4}), we obtain the algebraic equations
\begin{eqnarray}
\label{f6}
-i\xi \omega \delta\hat{\rho}_{k\omega}=-\frac{k_{B}T}{m}k^{2}\delta\hat{\rho}_{k\omega}-\rho k^2 \delta\hat{\Phi}_{k\omega}+\sqrt{2k_{B}T\xi\rho}\nabla\cdot \hat{\bf R}_{k\omega},
\end{eqnarray}
\begin{eqnarray}
\label{f7}
\delta\hat{\Phi}_{k\omega}=(2\pi)^{d}\hat{u}(k)\delta\hat{\rho}_{k\omega},
\end{eqnarray}
where we have used the fact that the integral in Eq. (\ref{f4}) is a
convolution. Solving for $\delta\hat{\rho}_{k\omega}$, we obtain
\begin{eqnarray}
\label{f8}
\left\lbrack \frac{k_{B}T}{m}k^{2}+(2\pi)^{d}\hat{u}(k)k^2\rho-i\xi\omega\right\rbrack\delta\hat{\rho}_{k\omega}=\sqrt{2k_{B}T\xi\rho} \ i k^{\mu}\hat{R}_{k\omega}^{\mu},
\end{eqnarray}
where the correlations of the Fourier transform of the noise are given by
\begin{eqnarray}
\label{f9}
\langle \hat{R}^{\mu}_{k\omega}\hat{R}^{\nu}_{k'\omega'}\rangle = \frac{1}{(2\pi)^{d+1}}\delta^{\mu\nu}\delta({\bf k}+{\bf k}')\delta(\omega+\omega').
\end{eqnarray}
Without noise (${\bf R}={\bf 0}$), Eq. (\ref{f8}) gives the dispersion
relation (see Paper II) associated with the mean field Smoluchowski equation
(\ref{ea8}), i.e.:
\begin{eqnarray}
\label{f10}
Z(k,\omega)\equiv \frac{k_{B}T}{m}k^{2}+(2\pi)^{d}\hat{u}(k)k^2\rho-i\xi\omega=0.
\end{eqnarray}
The function $Z(k,\omega)$ plays a role similar to the dielectric
function in plasma physics. The perturbation evolves exponentially
rapidly like $\delta\hat{\rho}_{k}(t)\propto
e^{\sigma(k)t}$ with a rate
given by $\sigma(k)=-\omega_{0}^{2}(k)/\xi$ where
\begin{eqnarray}
\label{f11}
\omega_{0}^{2}(k)\equiv \frac{k_{B}T}{m}k^{2}+(2\pi)^{d}\hat{u}(k)k^2\rho.
\end{eqnarray}
Thus, we find that the spatially homogeneous phase is stable with respect to the mean field Smoluchowski equation (\ref{ea8}) if $\omega_{0}^{2}(k)> 0$, i.e.
\begin{eqnarray}
\label{f12}
1+(2\pi)^{d}\beta \rho m \hat{u}(k)>0,
\end{eqnarray}
for all modes $k$ and unstable (to some modes) otherwise. If $\hat{u}>0$, the homogeneous phase is always stable. Otherwise, a necessary condition of
instability is that
\begin{eqnarray}
\label{f13}
k_{B}T<k_{B}T_{c}\equiv (2\pi)^{d}\rho m|\hat{u}(k)|_{max}.
\end{eqnarray}
If this condition is fulfilled, the range of unstable wavenumbers is determined by
\begin{eqnarray}
\label{f14}
(2\pi)^{d}|\hat{u}(k)|>\frac{k_{B}T}{\rho m}.
\end{eqnarray}
Some explicit examples of potentials of interaction, and the
corresponding conditions of instability, are given in Paper I and in
\cite{jeansbio2}.
Let us now determine the correlations of the fluctuations around a
stable equilibrium homogeneous distribution in the presence of
noise. If we take the noise into account (${\bf R}\neq {\bf 0}$), the
Fourier transform of the density fluctuations is given by
\begin{eqnarray}
\label{f15}
\delta\hat{\rho}_{k\omega}=\frac{\sqrt{2k_{B}T\xi\rho} \ i k^{\mu}\hat{R}_{k\omega}^{\mu}}{Z(k,\omega)}.
\end{eqnarray}
Therefore, the correlations of the fluctuations in Fourier space are
\begin{eqnarray}
\label{f16}
\left\langle \delta\hat{\rho}_{k\omega}\delta\hat{\rho}_{k'\omega'}\right\rangle=\frac{-2k_{B}T\xi\rho k^{\mu}k^{'\nu}\langle \hat{R}_{k\omega}^{\mu}\hat{R}_{k'\omega'}^{\nu}\rangle}{Z(k,\omega)Z(k',\omega')}.
\end{eqnarray}
Using Eq. (\ref{f9}), we get
\begin{eqnarray}
\label{f17}
\langle \delta\hat{\rho}_{k\omega}\delta\hat{\rho}_{k'\omega'}\rangle=\frac{1}{(2\pi)^{d+1}}\frac{2k_{B}T\xi\rho k^{2}}{|Z(k,\omega)|^{2}}\delta({\bf k}+{\bf k}')\delta(\omega+\omega'),
\end{eqnarray}
or, more explicitly,
\begin{eqnarray}
\label{f18}
\langle \delta\hat{\rho}_{k\omega}\delta\hat{\rho}_{k'\omega'}\rangle=\frac{1}{(2\pi)^{d+1}}\frac{2k_{B}T\xi\rho k^{2}}{\left\lbrack \frac{k_{B}T}{m}k^{2}+(2\pi)^{d}\hat{u}(k)k^2\rho\right\rbrack^{2}+\xi^{2}\omega^{2}}\delta({\bf k}+{\bf k}')\delta(\omega+\omega').
\end{eqnarray}
The temporal correlation function of the Fourier components of the
density fluctuations is given by
\begin{eqnarray}
\label{f19}
\langle \delta\hat{\rho}_{k}(t)\delta\hat{\rho}_{k'}(t+\tau)\rangle=\int \langle \delta\hat{\rho}_{k\omega}\delta\hat{\rho}_{k'\omega'}\rangle e^{i\omega t}e^{i\omega'(t+\tau)}d\omega d\omega'.
\end{eqnarray}
Using Eq. (\ref{f18}), the integral on $\omega'$ is trivial and yields
\begin{eqnarray}
\label{f20}
\langle \delta\hat{\rho}_{k}(t)\delta\hat{\rho}_{k'}(t+\tau)\rangle=\frac{1}{(2\pi)^{d+1}}2k_{B}T\xi\rho k^{2}\delta({\bf k}+{\bf k}')\int_{-\infty}^{+\infty} d\omega\frac{e^{i\omega\tau}}{|Z(k,\omega)|^{2}}.
\end{eqnarray}
The integral on $\omega$ can be performed by using the Cauchy residue
theorem. The poles of the integrand are the zeros of the functions
$Z(k,\omega)$ and $Z(k,\omega)^{*}$, i.e. they are solutions of the
dispersion relation (\ref{f10}) and its complex conjugate. If
$\omega_{0}^{2}(k)>0$, which is required for the stability of the
homogeneous phase, the integrand has a single pole in the upper-half
plane at $\omega=i\omega_{0}^{2}(k)/\xi$ and the residue is
$e^{-\omega_{0}^{2}(k)\tau/\xi}/\lbrack
2i\xi\omega_{0}^{2}(k)\rbrack$. Therefore, we obtain
\begin{eqnarray}
\label{f21}
\langle \delta\hat{\rho}_{k}(t)\delta\hat{\rho}_{k'}(t+\tau)\rangle=\frac{1}{(2\pi)^{d}}\frac{k_{B}T\rho k^{2}}{\omega_{0}^{2}(k)}\delta({\bf k}+{\bf k}')e^{-\omega_{0}^{2}(k)\tau/\xi},
\end{eqnarray}
or, more explicitly,
\begin{eqnarray}
\label{f22}
\langle \delta\hat{\rho}_{k}(t)\delta\hat{\rho}_{k'}(t+\tau)\rangle=\frac{1}{(2\pi)^{d}}\frac{k_{B}T\rho }{\frac{k_{B}T}{m}+(2\pi)^{d}\hat{u}(k)\rho}\delta({\bf k}+{\bf k}') e^{-\left\lbrack \frac{k_{B}T}{m}k^2+(2\pi)^{d}\hat{u}(k)k^2\rho\right\rbrack \tau/\xi}.
\end{eqnarray}
This formula is one of the most important results of this paper. We
shall come back to its physical interpretation in
Sec. \ref{sec_examples}. The equal time correlation function (corresponding
to $\tau=0$) is given by
\begin{eqnarray}
\label{f23}
\langle \delta\hat{\rho}_{k}\delta\hat{\rho}_{k'}\rangle=\frac{1}{(2\pi)^{d}} \frac{\rho m}{1+(2\pi)^{d}\beta \rho m\hat{u}(k)} \delta({\bf k}+{\bf k}').
\end{eqnarray}
In the absence of any interaction between the particles ($u=0$), we get
\begin{eqnarray}
\label{f24}
\langle \delta\hat{\rho}_{k}\delta\hat{\rho}_{k'}\rangle=\frac{1}{(2\pi)^{d}}\rho m {\delta({\bf k}+{\bf k}')},
\end{eqnarray}
which corresponds to the standard result \cite{ll}:
\begin{eqnarray}
\label{f24bis}
\langle (\delta\rho)^2\rangle=\frac{m\rho}{\Delta V}.
\end{eqnarray}
On the other hand, using Eq. (\ref{f23}) and the identity (see
Appendix \ref{sec_cf}):
\begin{eqnarray}
\label{f25}
\langle \delta\hat{\rho}_{k}\delta\hat{\rho}_{k'}\rangle=\frac{1}{(2\pi)^{d}}\rho m \left\lbrack 1+(2\pi)^{d}n\hat{h}({\bf k})\right\rbrack \delta({\bf k}+{\bf k}'),
\end{eqnarray}
we find that the Fourier transform of the correlation function is
\begin{eqnarray}
\label{f26}
\hat{h}_{eq}({\bf k})=\frac{-\beta m^2 \hat{u}({k})}{1+(2\pi)^{d}\beta n m^2 \hat{u}({k})}.
\end{eqnarray}
This is precisely the result (I-54) obtained in Paper I by analysing
the second equation of the equilibrium BBGKY-like hierarchy and neglecting
the three-body correlation function. Therefore, the stochastic Smoluchowski
equation (\ref{f1})-(\ref{f2}) is able to reproduce the equilibrium
two-body correlation function. On the other hand, from
Eqs. (\ref{f22}) and (\ref{f25}), the Fourier transform of the
equilibrium temporal correlation function is
\begin{eqnarray}
\label{f27}
\hat{h}({\bf k},t,t+\tau)=\hat{h}_{eq}({\bf k}) e^{-\left\lbrack \frac{k_{B}T}{m}k^2+(2\pi)^{d}\hat{u}(k)k^2\rho\right\rbrack \tau/\xi}.
\end{eqnarray}
This can be compared to the out-of-equilibrium temporal evolution of
the equal-time spatial correlation function $\hat{h}({\bf k},t)$ given
by Eq. (II-165) of Paper II. Note that the condition of stability
(\ref{f12}) is implied by Eq. (\ref{f22}), Eq. (\ref{f27}),
Eq. (II-165) and Eq. (\ref{f23}). This completes the discussion given
in Sec. 4.2 of Paper I.
\subsection{Specific examples}
\label{sec_examples}
The physical content of formula (\ref{f21}) is very
instructive. First, we note that the temporal correlation function of
the Fourier components of the density fluctuations decreases
exponentially rapidly with a decay rate
$\sigma(k)=-\omega_{0}^{2}(k)/\xi$ that coincides with the decay rate
of a perturbation of the density governed by the deterministic mean field
Smoluchowski equation (\ref{ea8}), i.e. without noise
\footnote{Similarly, in Sec. 2.9 of Paper II, we noted that, for Hamiltonian
systems with long-range interactions, the Fourier modes of the
temporal correlation function of the force decay exponentially rapidly
with a decay rate that coincides with the decay rate of a perturbation
of the distribution function governed by the Vlasov equation.}.
According to this temporal factor, or according to the mean field
theory based on the Smoluchowski equation (\ref{ea8}), the modes
satisfying the inequality $\omega_{0}^{2}(k)\ge 0$ should be
stable. For $T\le T_{c}$, the threshold of instability corresponds to
the wavenumber(s) $k=k_{m}$ where $k_{m}$ is defined by
$\omega_{0}^{2}(k_{m})=0$. Now, we note that
the amplitude of the fluctuations in formula (\ref{f21}) behaves like
$\omega_{0}^{2}(k)^{-1}$, so that it {\it diverges} as we approach the
instability threshold $k\rightarrow k_{m}$. On the other hand, if we denote by
$k_{m}^{*}$ the value of the critical wavenumber at $T=T_c$ satisfying
$1+(2\pi)^{d}\beta_{c}\rho m \hat{u}(k_{m}^{*})=0$ and if we consider
the mode $k=k_{m}^{*}$ in Eq. (\ref{f22}), we get for $T\ge T_{c}$:
\begin{eqnarray}
\label{f22b}
\langle \delta\hat{\rho}_{k}(t)\delta\hat{\rho}_{k'}(t+\tau)\rangle=\frac{1}{(2\pi)^{d}}\frac{T\rho m}{T-T_c}\delta({\bf k}+{\bf k}') e^{-k_{B}(T-T_c)(k_{m}^{*})^{2}\tau/(\xi m)}.
\end{eqnarray}
This formula clearly shows that the correlation function diverges at
the critical point $T=T_{c}$ for the ``dangerous''mode
$k=k_{m}^{*}$. These results imply that the mean field approximation
breaks down close to the critical point (or close to the instability
threshold) and that the instability triggering the phase transition
can occur {\it sooner} than what is predicted by mean field theory
(i.e. from the stability analysis of the mean field Smoluchowski
equation). Some results in this direction have been reported in
\cite{monaghan,ko,meta} in the gravitational case.
Let us consider specific examples for illustration (we use the
notations of Paper I). For the BMF model (the Brownian version of the
HMF model) \cite{cvb}, there exists a critical temperature
$T_{c}=kM/(4\pi)$. Considering the linear dynamical stability of a
spatially homogeneous distribution with respect to the mean field
Smoluchowski equation (\ref{disp3}), we find that
$\omega_{0}^{2}(n)=Tn^2+2\pi\hat{u}_{n}\rho n^2$ where $\hat{u}_{n}$
is given by Eq. (\ref{disp5}). The modes $n\neq
\pm 1$ decay exponentially rapidly as $e^{-Tn^{2}t/\xi}$ so they are
always stable. By contrast, the modes $n=\pm 1$ evolve in time like
$e^{-(T-T_c)t/\xi}$. For $T>T_{c}$, the perturbation is damped and for
$T<T_{c}$ the perturbation has exponential growth. In that case, the
homogeneous phase is unstable to the $n=\pm 1$ modes (see Appendix
\ref{sec_disp}). According to formula (\ref{f21}), the correlation
function of the density fluctuations can be written for the stable
modes $n\neq \pm 1$:
\begin{eqnarray}
\label{exa1}
\langle\delta\hat{\rho}_{n}(t)\delta\hat{\rho}_{m}(t+\tau)\rangle=\frac{M}{4\pi^{2}n^{2}}e^{-Tn^{2}\tau/\xi}\delta_{n,-m},
\end{eqnarray}
and for the ``dangerous modes'' $n=\pm 1$:
\begin{eqnarray}
\label{exa2}
\langle\delta\hat{\rho}_{\pm 1}(t)\delta\hat{\rho}_{m}(t+\tau)\rangle=\frac{M}{4\pi^{2}}\frac{T}{T-T_{c}}e^{-(T-T_{c})\tau/\xi}\delta_{m,\mp 1}.
\end{eqnarray}
This simple toy model, where the potential of interaction is
restricted to one Fourier mode, is very interesting for pedagogical
purposes because it clearly illustrates the discussion given
above. Considering the temporal factor in Eq. (\ref{exa2}), we see
that the correlations decay for $T>T_{c}$ with the rate given by mean field
theory. However, as we approach the critical temperature from above
($T\rightarrow T_{c}^{+}$), the amplitude of the fluctuations diverges
like $(T-T_{c})^{-1}$ implying that the mean field approximation
breaks down \footnote{This implies that the limits $N\rightarrow
+\infty$ (mean field) and $T\rightarrow T_{c}$ do not commute.} and
that the phase transition should occur for $T$ {\it strictly} above
$T_{c}$. We had also reached this conclusion in \cite{cvb} from the
study of the equilibrium BBGKY-like hierarchy.
Let us now consider the attractive Yukawa potential \cite{hb1}. A
detailed stability analysis of the homogeneous phase with respect to
the mean field Smoluchowski equation (and generalizations) has been
performed in \cite{jeansbio2}. There exists a critical temperature
$k_{B}T_{c}=S_{d}G\rho m/k_{0}^{2}$ depending on the screening length
$k_{0}^{-1}$. Furthermore, in the linear regime, the perturbation
evolves exponentially rapidly as $\delta\hat{\rho}_{k}(t)\propto
e^{\sigma(k)t}$ with a rate
\begin{eqnarray}
\label{exa3re}
\sigma(k)=-\frac{\omega_{0}^{2}(k)}{\xi}=\frac{k_{B}T}{m\xi}\frac{k^{2}}{k^{2}+k_{0}^{2}}\lbrack k_{0}^{2}(T_{c}/T-1)-k^2\rbrack.
\end{eqnarray}
For $T>T_{c}$, the homogeneous phase is always stable and
for $T<T_{c}$ it is unstable to wavenumbers $k<k_{m}(T)\equiv
k_{0}(T_c/T-1)^{1/2}$. Taking into account the fluctuations, formula
(\ref{f21}) shows that the correlation function of the density fluctuations is
\begin{eqnarray}
\label{exa3}
\langle \delta\hat{\rho}_{k}(t)\delta\hat{\rho}_{k'}(t+\tau)\rangle=\frac{\rho m}{(2\pi)^{d}} \frac{k^{2}+k_{0}^{2}}{k^{2}+k_{0}^{2}(1-T_{c}/T)}\delta({\bf k}+{\bf k}')e^{-\frac{k_{B}T}{m}\frac{k^{2}}{k^{2}+k_{0}^{2}}\lbrack k^{2}+k_{0}^{2}(1-T_{c}/T)\rbrack \frac{\tau}{\xi}}.
\end{eqnarray}
Considering the mode $k=k_{m}^{*}=0$, we see that the correlation function
diverges like $(1-T_c/T)^{-1}$ as we approach the critical temperature
$T\rightarrow T_{c}^{+}$. On the other hand, for $T<T_{c}$, we see
that the amplitude diverges like $(k^{2}-k_{m}^{2})^{-1}$ as we approach
the critical wavenumber $k\rightarrow k_{m}^{+}(T)$. This is
particularly true for the gravitational interaction ($k_0=0$) for
which Eq. (\ref{exa3}) reduces to
\begin{eqnarray}
\label{exa4}
\langle \delta\hat{\rho}_{k}(t)\delta\hat{\rho}_{k'}(t+\tau)\rangle=\frac{\rho m}{(2\pi)^{d}} \frac{k^{2}}{k^{2}-k_{J}^2}\delta({\bf k}+{\bf k}')e^{-\frac{k_{B}T}{m}(k^{2}-k_{J}^{2})\frac{\tau}{\xi}},
\end{eqnarray}
where $k_{J}=(S_{d}Gm\beta\rho)^{1/2}$ is the Jeans
wavenumber. According to the standard Jeans analysis \cite{jeans}, the
homogeneous phase is stable against perturbations with wavenumbers
$k>k_{J}$ and it becomes unstable for $k\le k_{J}$. However, the
divergence of the correlation function as $k\rightarrow k_{J}^{+}$
suggests that the gravitational instability will take place {\it
sooner}, i.e. for smaller wavelengths than the Jeans length. This
conclusion was previously reached by Monaghan \cite{monaghan} on the
basis of a hydrodynamical model of self-gravitating system
incorporating viscosity and fluctuations.
\section{The inertial model}
\label{sec_inertial}
In this section, we generalise the previous results to the case of a
stochastic model taking into account inertial effects. This
corresponds to the damped Euler equations with a long-range potential
of interaction and a stochastic forcing. This generalization allows us
to study the correlations of the fluctuations of the velocity field
and their behaviour close to the critical point. The stochastic
Smoluchowski equation (\ref{f1})-(\ref{f2}) is recovered in a strong
friction limit $\xi\rightarrow +\infty$ by neglecting the inertial
term (l.h.s.) in Eq. (\ref{dc2}). The stochastic damped Euler
equations can find applications in certain biological models of
chemotaxis where inertial effects are relevant
\cite{gamba,filbet,bio} (see also Sec. \ref{sec_chemo}).
\subsection{The density correlations}
\label{sec_dc}
We consider the stochastic damped Euler equations
\begin{eqnarray}
\label{dc1}
\frac{\partial \rho}{\partial t}+\nabla\cdot (\rho {\bf u})=0,
\end{eqnarray}
\begin{eqnarray}
\label{dc2}
\rho\left \lbrack \frac{\partial {\bf u}}{\partial t}+({\bf u}\cdot \nabla){\bf u}\right \rbrack=-\frac{k_{B}T}{m}\nabla\rho-\rho\nabla\Phi-\xi\rho {\bf u}-\sqrt{2k_{B}T\xi\rho}\, {\bf R}({\bf r},t),
\end{eqnarray}
\begin{eqnarray}
\label{dc3}
\Phi({\bf r},t)=\int u({\bf r}-{\bf r}')\rho({\bf r}',t)\, d{\bf r}',
\end{eqnarray}
where the quantities have their usual meaning. Without the stochastic
term (${\bf R}={\bf 0}$), we recover the damped Euler equations
introduced in
\cite{gen} (see also \cite{virial2,bio,jeansbio2,nfp}). With the stochastic term, we obtain a more general model
taking into account fluctuations. The form of the noise is justified
in Appendix \ref{sec_ll} using the general theory of fluctuations of
Landau \& Lifshitz \cite{ll}. For $\xi=0$, we get the usual Euler
equations and for $\xi\rightarrow +\infty$, neglecting the inertial
term (l.h.s.) in Eq. (\ref{dc2}) and substituting the resulting
expression $\xi\rho {\bf u}\simeq
-(k_{B}T/{m})\nabla\rho-\rho\nabla\Phi-\sqrt{2k_{B}T\xi\rho}{\bf R}$
in the continuity equation (\ref{dc1}), we recover the stochastic
Smoluchowski equation (\ref{f1})-(\ref{f2}). Note that Eq. (\ref{dc2})
can be written in terms of the free energy (\ref{ea12}) as
\begin{eqnarray}
\label{mm5}
\rho\left \lbrack \frac{\partial {\bf u}}{\partial t}+({\bf u}\cdot \nabla){\bf u}\right \rbrack=-\rho\nabla \frac{\delta F}{\delta\rho}-\xi\rho {\bf u}-\sqrt{2k_{B}T\xi\rho}\, {\bf R}({\bf r},t).
\end{eqnarray}
Considering small
perturbations around a uniform distribution with $\rho({\bf r})=\rho$,
$\Phi({\bf r})=\Phi$ and ${\bf u}={\bf 0}$, we find that the
linearized equations for the perturbations are
\begin{eqnarray}
\label{dc4}
\frac{\partial \delta\rho}{\partial t}+\rho \nabla\cdot {\bf u}=0,
\end{eqnarray}
\begin{eqnarray}
\label{dc5}
\rho\frac{\partial {\bf u}}{\partial t}=-\frac{k_{B}T}{m}\nabla\delta\rho-\rho\nabla\delta\Phi-\xi\rho {\bf u}-\sqrt{2k_{B}T\xi\rho}\, {\bf R}({\bf r},t),
\end{eqnarray}
\begin{eqnarray}
\label{dc6}
\delta\Phi({\bf r},t)=\int u({\bf r}-{\bf r}')\delta\rho({\bf r}',t)\, d{\bf r}'.
\end{eqnarray}
They can be combined to give
\begin{eqnarray}
\label{dc7}
\frac{\partial^{2} \delta\rho}{\partial t^2}+\xi\frac{\partial\delta\rho}{\partial t}
=\frac{k_{B}T}{m}\Delta \delta\rho +\rho\Delta\delta\Phi+ \sqrt{2k_{B}T\xi\rho}\nabla\cdot {\bf R}.
\end{eqnarray}
If we decompose the perturbations in Fourier modes of the form $e^{i({\bf k}\cdot {\bf r}-\omega t)}$, we obtain the system of algebraic equations
\begin{eqnarray}
\label{dc8}
-i\omega\delta\hat{\rho}_{k\omega}+i\rho {\bf k}\cdot \hat{\bf u}_{k\omega}=0,
\end{eqnarray}
\begin{eqnarray}
\label{dc9}
-i\omega\rho \hat{\bf u}_{k\omega}=-\frac{k_{B}T}{m}i{\bf k}\delta\hat{\rho}_{k\omega}-i{\bf k}\rho\delta\hat{\Phi}_{k\omega}-\xi\rho \hat{\bf u}_{k\omega}-\sqrt{2k_{B}T\xi\rho}\, \hat{\bf R}_{k\omega},
\end{eqnarray}
\begin{eqnarray}
\label{dc10}
\delta\hat{\Phi}_{k\omega}=(2\pi)^{d}\hat{u}(k)\delta\hat{\rho}_{k\omega}.
\end{eqnarray}
Solving for the density perturbation, we get
\begin{eqnarray}
\label{dc11}
\left\lbrack {\frac{k_{B}T}{m}k^{2}+(2\pi)^{d}\hat{u}(k)\rho k^2-\omega(\omega+i\xi)}\right\rbrack\delta\hat{\rho}_{k\omega}={\sqrt{2k_{B}T\xi\rho} \ i k^{\mu}\hat{R}_{k\omega}^{\mu}}.
\end{eqnarray}
Without noise (${\bf R}={\bf 0}$), Eq. (\ref{dc11}) gives the
dispersion relation associated with the mean field
damped Euler equation \cite{virial2}, i.e.
\begin{eqnarray}
\label{dc12}
Z(k,\omega)\equiv \frac{k_{B}T}{m}k^{2}+(2\pi)^{d}\hat{u}(k)\rho k^2-\omega(\omega+i\xi)=0.
\end{eqnarray}
Using the definition (\ref{f11}), the dispersion relation can be rewritten
\begin{eqnarray}
\label{dc14}
\omega^{2}+i\xi\omega-\omega_{0}^{2}(k)=0,
\end{eqnarray}
with solutions
\begin{eqnarray}
\label{dc15}
\omega(k)=\frac{-i\xi\pm\sqrt{-\xi^{2}+4\omega_{0}^{2}(k)}}{2}\equiv -i\frac{\xi}{2}\pm\Omega(k).
\end{eqnarray}
The perturbation evolves in time as $e^{-i\omega(k) t}$. If
$\omega_{0}^{2}(k)<0$, the perturbation grows exponentially rapidly in
time with a rate $\lbrack
-\xi+\sqrt{\xi^{2}+4|\omega_{0}^{2}|}\rbrack/2$. If
$\omega_{0}^{2}(k)>\xi^{2}/4$, the perturbation decays exponentially
rapidly in time with a rate $-\xi/2$ and oscillates with a pulsation
$\sqrt{4\omega_{0}^{2}-\xi^{2}}/2$. If
$0<\omega_{0}^{2}(k)<\xi^{2}/4$, the perturbation decays exponentially
rapidly in time with a rate $\lbrack
-\xi+\sqrt{\xi^{2}-4\omega_{0}^{2}}\rbrack/2$ without
oscillating. Therefore, according to mean field theory, the system is
stable iff $\omega_{0}^{2}(k)>0$ for all $k$. This returns the
condition (\ref{f12}) studied in Sec. \ref{sec_f}. For $T<T_{c}$, the
onset of instability corresponds to the wavenumber(s) $k_{m}$ such that
$\omega_{0}(k_{m})=0$.
We now consider stable modes ($\omega_{0}^{2}(k)>0$) and study the
correlations of the density fluctuations in the presence of
noise. Repeating the steps of Sec. \ref{sec_f} going from
Eq. (\ref{f15}) to Eq. (\ref{f20}), the density correlation function
can be written
\begin{eqnarray}
\label{dc16}
\left\langle \delta\hat{\rho}_{k}(t)\delta\hat{\rho}_{k'}(t+\tau)\right\rangle=\frac{1}{(2\pi)^{d+1}} 2k_{B}T\xi\rho k^{2}I(k,\tau)\delta({\bf k}+{\bf k}'),
\end{eqnarray}
where $I(k,\tau)$ is the integral
\begin{eqnarray}
\label{dc17}
I=\int_{-\infty}^{+\infty}f(\omega)e^{i\omega\tau}d\omega,\qquad f(\omega)=\frac{1}{|Z(k,\omega)|^{2}}.
\end{eqnarray}
We can evaluate the integral with the Cauchy residue theorem. The
poles of $f(\omega)$ are the zeros of the functions $Z(k,\omega)$ and
$Z(k,\omega)^{*}$, i.e. they are solutions of the dispersion relation
(\ref{dc14}) and its complex conjugate. If $\omega_{0}^{2}(k)>0$, the function
$f(\omega)$ has two simple poles in the upper-half plane at
$\omega=i\xi/2+\Omega$ and $\omega=i\xi/2-\Omega$. The residues of
$f(\omega)e^{i\omega\tau}$ at $\omega=i\xi/2+\Omega$ and at
$\omega=i\xi/2-\Omega$ are
\begin{eqnarray}
\label{dc18}
\frac{e^{-\frac{\xi}{2}\tau}e^{i\Omega\tau}}{i\xi (i\xi+2\Omega)(2\Omega)}, \qquad \frac{e^{-\frac{\xi}{2}\tau}e^{-i\Omega\tau}}{(i\xi-2\Omega)i\xi (-2\Omega)}.
\end{eqnarray}
Using the Cauchy residue theorem, and recalling that
\begin{eqnarray}
\label{dc19}
\Omega^{2}=\omega_{0}^{2}-\frac{1}{4}\xi^{2},
\end{eqnarray}
we obtain after simplification
\begin{eqnarray}
\label{dc20}
I=\frac{\pi}{\xi\omega_{0}^{2}}e^{-\frac{\xi}{2}\tau}\left\lbrack \cos(\Omega\tau)+\frac{\xi}{2\Omega}\sin(\Omega\tau)\right\rbrack,
\end{eqnarray}
where $\Omega$ can be complex. In conclusion, the density correlation
function for the inertial model is
\begin{eqnarray}
\label{dc21}
\left\langle \delta\hat{\rho}_{k}(t)\delta\hat{\rho}_{k'}(t+\tau)\right\rangle=\frac{1}{(2\pi)^{d}}\frac{k_{B}T\rho k^2}{\omega_{0}^{2}(k)}e^{-\frac{\xi}{2}\tau}\left\lbrack \cos(\Omega\tau)+\frac{\xi}{2\Omega}\sin(\Omega\tau)\right\rbrack \delta({\bf k}+{\bf k}').
\end{eqnarray}
The interpretation of this formula is similar to the one given in
Sec. \ref{sec_examples}. In particular, we see that the amplitude of
the fluctuations diverges when $k\rightarrow k_{m}$, corresponding to
$\omega_{0}^{2}(k)\rightarrow 0$. Thus, instability should set in
slightly before we reach the range of unstable wavenumbers
(\ref{f14}). In particular, the amplitude diverges at the critical
point $T\rightarrow T_{c}^{+}$ for the ``dangerous'' wavenumber
$k_{m}^{*}$. For $\xi\rightarrow 0$ (Euler), we can make the
approximation $\Omega\simeq \omega_{0}$, and we obtain
\begin{eqnarray}
\label{dc22}
\left\langle \delta\hat{\rho}_{k}(t)\delta\hat{\rho}_{k'}(t+\tau)\right\rangle=\frac{1}{(2\pi)^{d}}\frac{k_{B}T\rho k^2}{\omega_{0}^{2}(k)}e^{-\frac{\xi}{2}\tau}\cos(\omega_{0}\tau) \delta({\bf k}+{\bf k}').
\end{eqnarray}
For $\xi\rightarrow +\infty$ (Smoluchowski), we can make the
approximations
\begin{eqnarray}
\label{dc23}
\Omega\simeq \frac{\xi}{2}i\left (1-\frac{2\omega_{0}^{2}}{\xi^{2}}\right ),\quad e^{-\frac{\xi}{2}\tau} \cos(\Omega\tau)\simeq \frac{1}{2}e^{-\frac{\omega_{0}^{2}}{\xi}\tau},\quad e^{-\frac{\xi}{2}\tau} \sin(\Omega\tau)\simeq -\frac{1}{2i}e^{-\frac{\omega_{0}^{2}}{\xi}\tau},
\end{eqnarray}
and we recover the result of Eq. (\ref{f21}) obtained in the
overdamped limit.
\subsection{The velocity correlations}
\label{sec_v}
Let us now turn to the correlations of the velocity fluctuations. From
Eq. (\ref{dc9}), the Fourier components of the velocity fluctuations
satisfy the relation
\begin{eqnarray}
\label{v1}
\rho(\xi-i\omega)\hat{\bf u}_{k\omega}=-\frac{\omega_{0}^{2}}{k^2}i{\bf k}\delta\hat{\rho}_{k\omega}-\sqrt{2k_{B}T\xi\rho}\, \hat{\bf R}_{k\omega}.
\end{eqnarray}
Therefore, the velocity correlations can be expressed as
\begin{eqnarray}
\label{v2}
\rho^2 (\xi-i\omega)(\xi-i\omega')\left\langle \hat{u}_{k\omega}^{\mu}\hat{u}_{k'\omega'}^{\nu}\right\rangle=-\frac{\omega_{0}^{4}}{k^2 k^{'2}}k^{\mu}k^{'\nu}\left\langle \delta\hat{\rho}_{k\omega}\delta\hat{\rho}_{k'\omega'}\right\rangle +2k_{B}T\xi\rho \langle \hat{R}_{k\omega}^{\mu}\hat{R}_{k'\omega'}^{\nu}\rangle\nonumber\\
+\frac{\omega_{0}^{2}}{k^2}i k^{\mu} \sqrt{2k_{B}T\xi\rho} \langle \delta\hat{\rho}_{k\omega}\hat{R}_{k'\omega'}^{\nu}\rangle+\frac{\omega_{0}^{'2}}{k^{'2}}i k^{'\nu} \sqrt{2k_{B}T\xi\rho} \langle \delta\hat{\rho}_{k'\omega'}\hat{R}_{k\omega}^{\mu}\rangle.
\end{eqnarray}
Using the relation
\begin{eqnarray}
\label{v3}
\delta\hat{\rho}_{k\omega}=\frac{i\sqrt{2k_{B}T\xi\rho}k^{\alpha}\hat{R}_{k\omega}^{\alpha}}{Z(k,\omega)},
\end{eqnarray}
and Eq. (\ref{f9}), we get
\begin{eqnarray}
\label{v4}
\rho^2 (\xi^{2}+\omega^{2})\left\langle \hat{u}_{k\omega}^{\mu}\hat{u}_{k'\omega'}^{\nu}\right\rangle=-2k_{B}T\xi\rho\frac{\omega_{0}^{4}}{k^4}\frac{k^{\mu}k^{\nu}}{|Z(k,\omega)|^{2}}k^{\alpha}k^{\beta}\langle \hat{R}_{k\omega}^{\alpha}\hat{R}_{k'\omega'}^{\beta}\rangle +2k_{B}T\xi\rho \langle \hat{R}_{k\omega}^{\mu}\hat{R}_{k'\omega'}^{\nu}\rangle\nonumber\\
- 2k_{B}T\xi\rho\frac{\omega_{0}^{2}}{k^2} \frac{ k^{\mu}k^{\alpha}}{Z(k,\omega)} \langle \hat{R}_{k\omega}^{\alpha}\hat{R}_{k'\omega'}^{\nu}\rangle- 2k_{B}T\xi\rho \frac{\omega_{0}^{2}}{k^2} \frac{k^{\nu}k^{\alpha}}{Z(k,\omega)^{*}} \langle \hat{R}_{k'\omega}^{\alpha}\hat{R}_{k\omega}^{\mu}\rangle,\qquad
\end{eqnarray}
or, more explicitly,
\begin{eqnarray}
\label{v5}
\rho^2 (\xi^{2}+\omega^{2})\left\langle \hat{u}_{k\omega}^{\mu}\hat{u}_{k'\omega'}^{\nu}\right\rangle=\frac{1}{(2\pi)^{d+1}}2k_{B}T\xi\rho \delta({\bf k}+{\bf k}')\delta(\omega+\omega')\nonumber\\
\times \left\lbrace -\frac{\omega_{0}^{4}}{k^2}\frac{k^{\mu}k^{\nu}}{|Z(k,\omega)|^{2}}+\delta^{\mu\nu} -\frac{\omega_{0}^{2}}{k^2} \frac{ k^{\mu}k^{\nu}}{Z(k,\omega)} -\frac{\omega_{0}^{2}}{k^2} \frac{ k^{\mu}k^{\nu}}{Z(k,\omega)^{*}}\right\rbrace.
\end{eqnarray}
Finally, using the identity
\begin{eqnarray}
\label{v6}
\frac{1}{Z(k,\omega)}+\frac{1}{Z(k,\omega)^{*}}=\frac{2{\rm Re}(Z)}{|Z(k,\omega)|^{2}}=\frac{2(\omega_{0}^{2}-\omega^{2})}{|Z(k,\omega)|^{2}},
\end{eqnarray}
the foregoing relation can be rewritten
\begin{eqnarray}
\label{v7}
\left\langle \hat{u}_{k\omega}^{\mu}\hat{u}_{k'\omega'}^{\nu}\right\rangle=\frac{1}{(2\pi)^{d+1}}\frac{2k_{B}T\xi}{\rho}\frac{1}{\xi^{2}+\omega^{2}} \delta({\bf k}+{\bf k}')\delta(\omega+\omega')\nonumber\\
\times \left\lbrace -\frac{3\omega_{0}^{4}}{k^2}\frac{k^{\mu}k^{\nu}}{|Z(k,\omega)|^{2}}+\frac{2\omega_{0}^{2}\omega^{2}}{k^2}\frac{k^{\mu}k^{\nu}}{|Z(k,\omega)|^{2}}+\delta^{\mu\nu}\right\rbrace.
\end{eqnarray}
Taking the inverse Fourier transform in $\omega$-space of this
relation, we find that the temporal correlations of the velocity
fluctuations are given by
\begin{eqnarray}
\label{v8}
\left\langle \hat{u}_{k}^{\mu}(t)\hat{u}_{k'}^{\nu}(t+\tau)\right\rangle=\frac{1}{(2\pi)^{d+1}}\frac{2k_{B}T\xi}{\rho}\delta({\bf k}+{\bf k}')
\left\lbrace -{3\omega_{0}^{4}}\frac{k^{\mu}k^{\nu}}{k^2}K+2\omega_{0}^{2}\frac{k^{\mu}k^{\nu}}{k^{2}}K'+K''\delta^{\mu\nu}\right\rbrace,
\end{eqnarray}
where we have introduced the integrals
\begin{eqnarray}
\label{v9}
K=\int_{-\infty}^{+\infty}d\omega\frac{e^{i\omega\tau}}{(\xi^{2}+\omega^{2}){|Z(k,\omega)|^{2}}},\quad K'=\int_{-\infty}^{+\infty}d\omega\frac{\omega^2 e^{i\omega\tau}}{(\xi^{2}+\omega^{2}){|Z(k,\omega)|^{2}}},
\end{eqnarray}
\begin{eqnarray}
\label{v10}
K''=\int_{-\infty}^{+\infty}d\omega\frac{e^{i\omega\tau}}{\xi^{2}+\omega^{2}}.
\end{eqnarray}
These integrals are easily calculated with the Cauchy residue theorem. After simplification, we obtain
\begin{eqnarray}
\label{v11}
\left\langle \hat{u}_{k}^{\mu}(t)\hat{u}_{k'}^{\nu}(t+\tau)\right\rangle=\frac{1}{(2\pi)^{d}}\frac{k_{B}T}{\rho}\delta({\bf k}+{\bf k}') \biggl\lbrace {\omega_{0}^{2}}\frac{k^{\mu}k^{\nu}}{k^2}\frac{1}{\omega_{0}^{2}+2\xi^{2}}e^{-\frac{\xi}{2}\tau}\biggl\lbrack\left (\frac{2\xi^{2}}{\omega_{0}^{2}}-1\right )\cos(\Omega\tau)\nonumber\\
-\frac{\xi}{\Omega} \left (\frac{7}{2}+\frac{\xi^{2}}{\omega_{0}^{2}}\right )\sin(\Omega\tau)-\left (3+\frac{2\xi^{2}}{\omega_{0}^{2}}\right ) e^{-\frac{\xi}{2}\tau}\biggr\rbrack+ e^{-{\xi}\tau}\delta^{\mu\nu}\biggr\rbrace.
\end{eqnarray}
We note that, at variance with the density correlation function (\ref{dc21}),
the velocity correlation function does {\it not} diverge when
$k\rightarrow k_{m}$. Indeed, for $\omega_{0}^{2}=0$, we have
\begin{eqnarray}
\label{v11b}
\left\langle \hat{u}_{k}^{\mu}(t)\hat{u}_{k'}^{\nu}(t+\tau)\right\rangle=\frac{1}{(2\pi)^{d}}\frac{k_{B}T}{\rho}e^{-\xi\tau}\delta({\bf k}+{\bf k}')\delta^{\mu\nu}.
\end{eqnarray}
On the other hand, taking $\tau=0$ in Eq. (\ref{v11}), we obtain the equal time velocity correlation function
\begin{eqnarray}
\label{v12}
\left\langle \hat{u}_{k}^{\mu}(t)\hat{u}_{k'}^{\nu}(t)\right\rangle=\frac{1}{(2\pi)^{d}}\frac{k_{B}T}{\rho}\delta({\bf k}+{\bf k}') \biggl ( \delta^{\mu\nu}-\frac{4\omega_{0}^{2}}{\omega_{0}^{2}+2\xi^{2}}\frac{k^{\mu}k^{\nu}}{k^2} \biggr ).
\end{eqnarray}
Contracting the indices, we get
\begin{eqnarray}
\label{v13}
\left\langle \hat{{\bf u}}_{k}(t)\cdot \hat{{\bf u}}_{k'}(t)\right\rangle=\frac{1}{(2\pi)^{d}}\frac{k_{B}T}{\rho}\delta({\bf k}+{\bf k}') \frac{2d\xi^{2}-(4-d)\omega_{0}^{2}}{\omega_{0}^{2}+2\xi^{2}}.
\end{eqnarray}
For $\xi\rightarrow 0$ (Euler), Eq. (\ref{v11}) can be simplified into
\begin{eqnarray}
\label{v14}
\left\langle \hat{u}_{k}^{\mu}(t)\hat{u}_{k'}^{\nu}(t+\tau)\right\rangle=\frac{1}{(2\pi)^{d}}\frac{k_{B}T}{\rho}\delta({\bf k}+{\bf k}') \biggl\lbrace - \frac{k^{\mu}k^{\nu}}{k^2}e^{-\frac{\xi}{2}\tau}\left\lbrack \cos(\omega_{0}\tau)+3 e^{-\frac{\xi}{2}\tau}\right\rbrack+ e^{-{\xi}\tau}\delta^{\mu\nu}\biggr\rbrace.
\end{eqnarray}
Alternatively, for $\xi\rightarrow +\infty$ (Smoluchowski), we get
\begin{eqnarray}
\label{v15}
\left\langle \hat{u}_{k}^{\mu}(t)\hat{u}_{k'}^{\nu}(t+\tau)\right\rangle=-\frac{1}{(2\pi)^{d}}\frac{3k_{B}T}{\rho}\delta({\bf k}+{\bf k}') \frac{\omega_{0}^{2}}{\xi^{2}} \frac{k^{\mu}k^{\nu}}{k^2}e^{-\frac{\omega_{0}^2}{\xi}\tau}.
\end{eqnarray}
This result can be obtained directly from the study of the stochastic Smoluchowski equation in Sec. \ref{sec_f} by defining the velocity field
\begin{eqnarray}
\label{v16}
\xi\rho {\bf u}\equiv -\frac{k_{B}T}{m}\nabla\rho-\rho\nabla\Phi-\sqrt{2k_{B}T\xi\rho}\, {\bf R}.
\end{eqnarray}
Note that the velocity correlations in Eq. (\ref{v15}) tend to zero when $k\rightarrow k_m$ contrary to Eq. (\ref{v11b}). This shows that the limits $\xi\rightarrow +\infty$ and $k\rightarrow k_{m}$ do not commute.
\subsection{Specific examples}
\label{sec_se}
Let us discuss specific examples by restricting ourselves, for
brevity, to the overdamped limit (\ref{v15}). For the stochastic BMF
model, we get for the stable modes $n\neq \pm 1$:
\begin{eqnarray}
\label{se1}
\left\langle \hat{u}_{n}(t)\hat{u}_{m}(t+\tau)\right\rangle=-\frac{3T^2n^2}{M\xi^{2}}\delta_{m,-n}e^{-Tn^2\tau/\xi},
\end{eqnarray}
and for the ``dangerous'' modes $n=\pm 1$:
\begin{eqnarray}
\label{se2}
\left\langle \hat{u}_{\pm 1}(t)\hat{u}_{m}(t+\tau)\right\rangle=-\frac{3T}{M\xi^{2}}\delta_{m,\mp 1}(T-T_{c})e^{-(T-T_c)\tau/\xi}.
\end{eqnarray}
The velocity correlations tend to zero when $T\rightarrow
T_{c}^{+}$. For the attractive Yukawa potential, we get
\begin{eqnarray}
\label{se3}
\left\langle \hat{u}_{k}^{\mu}(t)\hat{u}_{k'}^{\nu}(t+\tau)\right\rangle=-\frac{3(k_{B}T)^2}{(2\pi)^{d}\xi^{2}\rho m} \frac{k^2+k_{0}^{2}(1-T_c/T)}{k^2+k_0^2} {k^{\mu}k^{\nu}}e^{-\frac{k_{B}T}{m}\frac{k^{2}}{k^{2}+k_{0}^{2}}\lbrack k^{2}+k_{0}^{2}(1-T_{c}/T)\rbrack \frac{\tau}{\xi}}\delta({\bf k}+{\bf k}').\nonumber\\
\end{eqnarray}
For $T\le T_{c}$, the amplitude tends to
zero as we approach the critical wavenumber $k\rightarrow
k_{m}^{+}(T)$. For the gravitational
interaction ($k_0=0$), Eq. (\ref{se3}) reduces to
\begin{eqnarray}
\label{se4}
\left\langle \hat{u}_{k}^{\mu}(t)\hat{u}_{k'}^{\nu}(t+\tau)\right\rangle=-\frac{3(k_{B}T)^2}{(2\pi)^{d}\xi^{2}\rho m} (k^2-k_{J}^{2})\frac{k^{\mu}k^{\nu}}{k^2}e^{-\frac{k_{B}T}{m}(k^{2}-k_{J}^{2}) \frac{\tau}{\xi}}\delta({\bf k}+{\bf k}').
\end{eqnarray}
The amplitude goes to zero as $k\rightarrow k_{J}^{+}$. The fact that
the velocity correlation function does not diverge when $k\rightarrow
k_{J}^{+}$ was previously observed by Monaghan \cite{monaghan} with his
hydrodynamic model.
\subsection{Stochastic model with memory}
\label{sec_mm}
The stochastic damped Euler equations (\ref{dc1})-(\ref{dc2}) can be rewritten
\begin{eqnarray}
\label{mm1}
\frac{\partial \rho}{\partial t}+\nabla\cdot (\rho {\bf u})=0,
\end{eqnarray}
\begin{eqnarray}
\label{mm2}
\frac{\partial}{\partial t}(\rho {\bf u})+\nabla (\rho {\bf u}\otimes {\bf u})=-\frac{k_{B}T}{m}\nabla\rho-\rho\nabla\Phi-\xi\rho {\bf u}-\sqrt{2k_{B}T\xi\rho}\, {\bf R}({\bf r},t).
\end{eqnarray}
If we neglect the inertial term (l.h.s.) in Eq. (\ref{mm2}) and
substitute the resulting expression for $\rho {\bf u}$ in
Eq. (\ref{mm1}), we obtain the stochastic Smoluchowski equation
(\ref{f1}). This is valid in a strong friction limit $\xi\rightarrow
+\infty$. We can obtain a more general model taking into account some
memory effects. If we neglect only the nonlinear term $\nabla (\rho {\bf u}\otimes {\bf u})$ in Eq. (\ref{mm2}), we get
\begin{eqnarray}
\label{mm3}
\frac{\partial}{\partial t}(\rho {\bf u})=-\frac{k_{B}T}{m}\nabla\rho-\rho\nabla\Phi-\xi\rho {\bf u}-\sqrt{2k_{B}T\xi\rho}\, {\bf R}({\bf r},t).
\end{eqnarray}
Taking the time derivative of Eq. (\ref{mm1}) and substituting
Eq. (\ref{mm3}) in the resulting expression, we obtain a simplified
stochastic model keeping track of memory effects
\begin{eqnarray}
\label{mm4}
\frac{\partial^{2}\rho}{\partial t^{2}}+\xi\frac{\partial\rho}{\partial t}
=\nabla\cdot \left (\frac{k_{B}T}{m}\nabla\rho+\rho\nabla\Phi\right )+\nabla\cdot \left (\sqrt{{2k_{B}T\xi\rho}}{\bf R}\right ).
\end{eqnarray}
In terms of the free energy (\ref{ea12}), we have
\begin{eqnarray}
\label{mm6}
\frac{\partial^{2}\rho}{\partial t^{2}}+\xi\frac{\partial\rho}{\partial t}
=\nabla\cdot \left (\rho\nabla \frac{\delta F}{\delta\rho}\right ) +\nabla\cdot \left (\sqrt{{2k_{B}T\xi\rho}}{\bf R}\right ).
\end{eqnarray}
This equation, which is second order in time, is analogous to the {\it
telegraph equation} which generalizes the diffusion equation by
introducing memory effects. We note that in the linear regime $|{\bf
u}|\ll 1$ considered in Sec. \ref{sec_inertial}, the nonlinear term
$\nabla (\rho {\bf u}\otimes {\bf u})$ in Eq. (\ref{mm2}) is
negligible so that Eq. (\ref{mm4}) can be justified rigorously from
the damped Euler equations in this regime. This implies that the
theory of fluctuations that we have developed in
Sec. \ref{sec_inertial} directly applies to the stochastic equation
(\ref{mm4}). In particular, the linearization of Eq. (\ref{mm4})
around a homogeneous distribution returns Eq. (\ref{dc7}). However,
the stochastic Smoluchowski equation with memory (\ref{mm4}) may also
be relevant in the nonlinear regime as a heuristic equation. Indeed,
although we have neglected the nonlinear term $\nabla (\rho {\bf
u}\otimes {\bf u})$ in Eq. (\ref{mm2}), we have kept the full
nonlinearities in the right hand side. Therefore, Eq. (\ref{mm4}) is a
semi-linear
model intermediate between the fully nonlinear
hydrodynamical model (\ref{mm1})-(\ref{mm2}) and the linearized
hydrodynamical model (\ref{dc7}).
Finally, we note that Eq. (\ref{mm6}) can be viewed as a form of stochastic
Cattaneo model. The deterministic Smoluchowski equation can be written
as a continuity equation $\partial_{t}\rho=-\nabla\cdot {\bf J}$ where
the current ${\bf J}=-(1/\xi)\rho\nabla\mu$ is proportional to the gradient of a
chemical potential $\mu=\delta F/\delta\rho$ \cite{nfp}. This is
similar to Fick's law for the diffusion of particles or to Fourier's
law for the diffusion of heat. In the context of heat conduction,
Cattaneo \cite{cattaneo} has proposed a modification of Fourier's law
in order to describe heat conduction with finite speed. He assumed
that the current is not instantaneously equal to the gradient
$\nabla\mu$ but relaxes to it with a time constant $1/\tau$. In the present situation, this would lead to a model of the form
\begin{eqnarray}
\label{mm7}
\frac{\partial \rho}{\partial t}+\nabla\cdot {\bf J}=0,
\end{eqnarray}
\begin{eqnarray}
\label{mm8}
\tau \frac{\partial {\bf J}}{\partial t}+{\bf J}=-\frac{1}{\xi}\rho\nabla \frac{\delta F}{\delta\rho} -\sqrt{\frac{2k_{B}T\rho}{\xi}}{\bf R},
\end{eqnarray}
where we have included the stochastic term for completeness. These
equations are equivalent to the semi-linear model formed by
Eqs. (\ref{mm1}) and (\ref{mm3}) if we set ${\bf J}=\rho {\bf u}$ and
$\tau=1/\xi$. For $\tau=0$, we recover the stochastic Smoluchowski
equation (\ref{cg6}). More generally, taking the time derivative of Eq. (\ref{mm7})
and using Eq. (\ref{mm8}), we obtain
\begin{eqnarray}
\label{mm9}
\tau\frac{\partial^{2}\rho}{\partial t^{2}}+\frac{\partial\rho}{\partial t}=\nabla\cdot \left (\frac{1}{\xi}\rho\nabla \frac{\delta F}{\delta\rho}\right ) +\nabla\cdot \left (\sqrt{\frac{2k_{B}T\rho}{\xi}}{\bf R}\right ),
\end{eqnarray}
which coincides with Eq. (\ref{mm6}) provided that we take $\tau=1/\xi$.
\subsection{Application to chemotaxis}
\label{sec_chemo}
In this section, we briefly mention the application of the preceding
results to the problem of chemotaxis in biology \cite{murray}. A more
detailed discussion is given in a specific paper \cite{chemo}
with complements and amplification. The standard Keller-Segel (KS) model
\cite{ks} of chemotaxis can be viewed as a form of mean
field Smoluchowski equation
\cite{nfp}. It describes the diffusion of bacteria (or other
chemotactic species) in the concentration gradient of a chemical
produced by the particles themselves. As we have seen in this paper,
the correlation function diverges close to a critical point. In that
case, the mean field approximation breaks down and the fluctuations
must be taken into account. Fluctuations also play an important role
when the particle number $N$ is small and when there exist metastable
states (local minima of free energy). In that case,
fluctuations can trigger dynamical phase transitions from one state to
the other (see Sec. \ref{sec_cg}). For these different reasons, it is
important to derive a chemotactic model going beyond the mean field
approximation and taking into account fluctuations.
We start from a microscopic model of chemotaxis where the dynamics of the
particles is governed by $N$
coupled stochastic equations of the form
\begin{eqnarray}
\label{c1}
\frac{d{\bf r}_{i}}{dt}=\chi\nabla c_{d}({\bf r}_{i}(t),t)+\sqrt{2D_{*}}{\bf R}_{i}(t),
\end{eqnarray}
\begin{eqnarray}
\label{c2}
\frac{\partial c_{d}}{\partial t}=-kc_{d}+D_{c}\Delta c_{d}+h\sum_{i=1}^{N}\delta({\bf r}-{\bf r}_{i}(t)),
\end{eqnarray}
where ${\bf r}_{i}(t)$, with $i=1,...,N$, denote the positions of the
particles and $c_{d}({\bf r},t)$ is the exact field of secreted
chemical. In these equations, $\chi$ and $D_{*}$ represent the
mobility and the diffusion coefficient of the organisms and $k$, $h$
and $D_{c}$ represent the degradation rate, the production rate and
the diffusion coefficient of the secreted chemical. By extending
Dean's approach (and the results of Sec. \ref{sec_cg}) to the case of
chemotactic species, we obtain a stochastic Keller-Segel model of
chemotaxis:
\begin{eqnarray}
\label{c17}
\frac{\partial {\rho}}{\partial t}({\bf r},t)=D_{*}\Delta{\rho}({\bf r},t)-\chi \nabla\cdot (\rho({\bf r},t)\nabla c({\bf r},t))
+\nabla \cdot \left (\sqrt{2D_{*}{\rho}({\bf r},t)}{\bf R}({\bf r},t)\right ),
\end{eqnarray}
\begin{eqnarray}
\label{c18}
\frac{\partial c}{\partial t}({\bf r},t)=-kc({\bf r},t)+D_{c}\Delta c({\bf r},t)+h\rho({\bf r},t),
\end{eqnarray}
generalizing the deterministic mean field Keller-Segel model. This
model fully takes into account the effect of fluctuations
\footnote{{\it Note added:} Until now, fluctuations have been ignored by
people working on chemotaxis. Therefore, our paper is the first
attempt to include fluctuations in the Keller-Segel model. However,
after submission of this paper [arXiv:0803.0263], a paper by Tailleur
\& Cates [arXiv:0803.1069] came out on a related subject. These
authors also consider the effect of fluctuations in the motion of
bacteria. They derive transport coefficients from microscopic models
but do not take into account the long-range interaction between
bacteria due to chemotaxis. Alternatively, in our approach, the
coefficients $D_*$ and $\chi$ appearing in the Langevin equations are
phenomenological coefficients but chemotaxis is fully taken into
account. Therefore, these two independent studies are complementary to
each other.}. On the other hand, there exists situations in biology
where inertial effects must be taken into account \cite{gamba}. In
that case, parabolic models like the Keller-Segel model must be
replaced by hyperbolic models similar to hydrodynamic equations
\cite{gamba,filbet,bio}. By extending the results of Sec. \ref{sec_inertial}, we obtain a hydrodynamic model of chemotaxis taking into account inertial
effects and fluctuations in the form:
\begin{eqnarray}
\label{c19}
\frac{\partial \rho}{\partial t}+\nabla\cdot (\rho {\bf u})=0,
\end{eqnarray}
\begin{eqnarray}
\label{c20}
\frac{\partial}{\partial t}(\rho {\bf u})+\nabla (\rho {\bf u}\otimes {\bf u})=-\xi D_{*}\nabla\rho+\rho\nabla c-\xi\rho {\bf u}-\sqrt{2D_{*}\xi^2\rho}\, {\bf R}({\bf r},t),
\end{eqnarray}
coupled to the field equation (\ref{c18}). In the strong friction limit
$\xi\rightarrow +\infty$ where the inertial term in Eq. (\ref{c20}) can be
neglected, it returns the stochastic KS model (\ref{c17}) with
$\chi=1/\xi$. On the other hand, if we only neglect the term $\nabla (\rho {\bf u}\otimes {\bf u})$
in Eq. (\ref{c20}) like in Sec. \ref{sec_mm}, we obtain a
stochastic equation of the form
\begin{eqnarray}
\label{c21}
\chi\frac{\partial^{2}\rho}{\partial t^{2}}+\frac{\partial\rho}{\partial t}
=\nabla\cdot (D_{*}\nabla\rho-\chi\rho\nabla c)+\nabla\cdot \left (\sqrt{2D_{*}\rho}{\bf R}\right ),
\end{eqnarray}
It can
be viewed as a stochastic Cattaneo model of chemotaxis (or a stochastic
telegraph equation).
\section{The stochastic Kramers equation }
\label{sec_ps}
In this section, we generalize the results of
Secs. \ref{sec_ea}-\ref{sec_cg} in phase space. This is the rigorous
way to take into account inertial effects and fluctuations in the
problem. The motion of the Brownian particles is described by $N$
coupled stochastic Langevin equations of the form (see Paper I):
\begin{eqnarray}
\label{ps0}
\frac{d{\bf r}_{i}}{dt}={\bf v}_{i},
\end{eqnarray}
\begin{eqnarray}
\label{ps0b}
\frac{d{\bf v}_{i}}{dt}=-\xi {\bf v}_{i}-m\nabla_{i}U({\bf r}_{1},...,{\bf r}_{N})+\sqrt{2D}{\bf R}_{i}(t).
\end{eqnarray}
The friction coefficient $\xi$ and the diffusion
coefficient $D$ are related to each other by the Einstein relation
$\xi=D\beta m$ where $\beta=1/(k_{B}T)$ is the inverse temperature
\cite{hb1}. In the strong friction limit $\xi\rightarrow +\infty$, we
can neglect the inertial term in Eq. (\ref{ps0b}) and we obtain the overdamped
equations (\ref{ea1}) of Sec. \ref{sec_ea} with $\mu=1/(\xi m)$ and $D_{*}=D/\xi^2$.
Extending Dean's approach \cite{dean} in
phase space, we find that the exact distribution function $f_{d}({\bf
r},{\bf v},t)=m\sum_{i=1}^{N}\delta({\bf r}-{\bf r}_{i}(t))\delta({\bf
v}-{\bf v}_{i}(t))$ expressed in terms of $\delta$-functions satisfies
a stochastic equation of the form
\begin{eqnarray}
\label{ps1}
\frac{\partial f_d}{\partial t}+{\bf v}\cdot \frac{\partial f_d}{\partial {\bf r}}-\nabla\Phi_d\cdot \frac{\partial f_d}{\partial {\bf v}}=\frac{\partial}{\partial {\bf v}}\cdot \left ( D\frac{\partial f_d}{\partial {\bf v}}+\xi f_d {\bf v}\right )+\frac{\partial}{\partial {\bf v}}\cdot (\sqrt{2Dmf_d}{\bf Q}({\bf r},{\bf v},t)),
\end{eqnarray}
where ${\bf Q}({\bf r},{\bf v},t)$ is a Gaussian random field such
that $\langle {\bf Q}({\bf r},{\bf v},t)\rangle={\bf 0}$ and $\langle
Q_{\alpha}({\bf r},{\bf v},t)Q_{\beta}({\bf r}',{\bf v}',t')\rangle\\
=\delta_{\alpha\beta}\delta({\bf r}-{\bf r}')\delta({\bf v}-{\bf
v}')\delta(t-t')$ and $\Phi_d({\bf r},t)$ is defined by Eq. (\ref{ex3}). If we average over the noise, we obtain
\begin{eqnarray}
\label{ps1av}
\frac{\partial f}{\partial t}+{\bf v}\cdot \frac{\partial f}{\partial {\bf r}}-\frac{\partial}{\partial {\bf v}}\cdot \int d{\bf r}'d{\bf v}' [\nabla u({\bf r}-{\bf r}')] \langle f_{d}({\bf r},{\bf v},t)f_{d}({\bf r}',{\bf v}',t)\rangle=\frac{\partial}{\partial {\bf v}}\cdot \left ( D\frac{\partial f}{\partial {\bf v}}+\xi f{\bf v}\right ).
\end{eqnarray}
Using $f=NmP_{1}$ and the identity
\begin{eqnarray}
\label{idnw}
\langle f_{d}({\bf r},{\bf v},t)f_{d}({\bf r}',{\bf v}',t)\rangle=Nm^2 P_{1}({\bf r},{\bf v},t)\delta ({\bf r}-{\bf r'})\delta ({\bf v}-{\bf v'})+N(N-1)m^2 P_{2}({\bf r},{\bf v},{\bf r}',{\bf v}',t),
\end{eqnarray}
we find that Eq. (\ref{ps1av}) is equivalent to Eq. (II-139) obtained
from the BBGKY-like hierarchy. If we implement a mean field
approximation $\langle f_{d}({\bf r},{\bf v},t)f_{d}({\bf r}',{\bf
v}',t)\rangle \simeq f({\bf r},{\bf v},t)f({\bf r}',{\bf v}',t)$, we
obtain the mean field Kramers equation \cite{hb2}:
\begin{eqnarray}
\label{ps1mf}
\frac{\partial f}{\partial t}+{\bf v}\cdot \frac{\partial f}{\partial {\bf r}}-\nabla\Phi\cdot \frac{\partial f}{\partial {\bf v}}=\frac{\partial}{\partial {\bf v}}\cdot \left ( D\frac{\partial f}{\partial {\bf v}}+\xi f {\bf v}\right ),
\end{eqnarray}
where $\Phi({\bf r},t)$ is defined by Eq. (\ref{ea9}). Finally, we can
heuristically propose a stochastic kinetic equation for the evolution
of the coarse-grained distribution function $\overline{f}({\bf r},{\bf
v},t)$ obtained by averaging $f_{d}({\bf r},{\bf v},t)$ over a small
spatio-temporal window. This leads to the stochastic Kramers equation
\begin{eqnarray}
\label{ps1cg}
\frac{\partial \overline{f}}{\partial t}+{\bf v}\cdot \frac{\partial \overline{f}}{\partial {\bf r}}-\nabla\overline{\Phi}\cdot \frac{\partial \overline{f}}{\partial {\bf v}}=\frac{\partial}{\partial {\bf v}}\cdot \left ( D\frac{\partial \overline{f}}{\partial {\bf v}}+\xi \overline{f} {\bf v}\right )+\frac{\partial}{\partial {\bf v}}\cdot (\sqrt{2Dm\overline{f}}{\bf Q}({\bf r},{\bf v},t)),
\end{eqnarray}
where $\overline{\Phi}({\bf r},t)$ is defined by Eq. (\ref{cg3}).
This equation keeps track of fluctuations but applies to a continuous
distribution function instead of a sum of Dirac distributions. An
altnernative derivation of this equation is proposed in Appendix
\ref{sec_ll} using the general theory of fluctuations of Landau \&
Lifshitz \cite{ll}.
Let us now try to make the link with the parabolic and hydrodynamic
models considered in Secs. \ref{sec_overdamped} and
\ref{sec_inertial}. Taking the hydrodynamic moments on the stochastic
Kramers equation (\ref{ps1}) and proceeding as in \cite{gen,virial2}, we
obtain
\begin{eqnarray}
\label{ps2}
\frac{\partial\rho}{\partial t}+\nabla\cdot (\rho {\bf u})=0,
\end{eqnarray}
\begin{eqnarray}
\label{ps3}
\rho\left (\frac{\partial u_{i}}{\partial t}+u_{j}\frac{\partial u_{i}}{\partial x_{j}}\right )=-\frac{\partial P_{ij}}{\partial x_{j}}-\rho\frac{\partial \Phi}{\partial x_{i}}-\xi\rho u_{i}-\int \sqrt{2Dmf}Q_{i}d{\bf v},
\end{eqnarray}
where $\rho({\bf r},t)=\int f d{\bf v}$ is the density, ${\bf u}({\bf r},t)=(1/\rho)\int f{\bf v}d{\bf v}$ is the local velocity, ${\bf w}={\bf v}-{\bf u}({\bf r},t)$ is the relative velocity and $P_{ij}=\int f w_{i}w_{j}d{\bf v}$ is the pressure tensor. Defining ${\bf g}({\bf r},t)=\int \sqrt{2Dmf}{\bf Q}d{\bf v}$, it is clear that ${\bf g}$ is a Gaussian noise and that its correlation function is
\begin{eqnarray}
\label{ps4}
\langle g_{i}({\bf r},t)g_{j}({\bf r}',t')\rangle=2Dm\int \sqrt{f({\bf r},{\bf v},t)f({\bf r}',{\bf v}',t')}\langle Q_{i}({\bf r},{\bf v},t)Q_{j}({\bf r}',{\bf v}',t')\rangle d{\bf v}d{\bf v}'\nonumber\\
=2Dm\delta_{ij}\delta({\bf r}-{\bf r}')\delta(t-t')\int f({\bf r},{\bf v},t) d{\bf v}=2Dm\delta_{ij}\delta({\bf r}-{\bf r}')\delta(t-t')\rho({\bf r},t).
\end{eqnarray}
Therefore, the equation for the momentum (\ref{ps3}) can be rewritten
\begin{eqnarray}
\label{ps5}
\rho\left (\frac{\partial u_{i}}{\partial t}+u_{j}\frac{\partial u_{i}}{\partial x_{j}}\right )=-\frac{\partial P_{ij}}{\partial x_{j}}-\rho\frac{\partial \Phi}{\partial x_{i}}-\xi\rho u_{i}-\sqrt{2Dm\rho}{R}_{i}({\bf r},t).
\end{eqnarray}
This equation is not closed since the pressure tensor depends on the
next order moment of the velocity. If, following \cite{gen,virial2},
we make a local thermodynamic equilibrium (L.T.E.) approximation
$f_{LTE}({\bf r},{\bf v},t)\simeq (\beta m/{2\pi})^{d/2}
\rho({\bf r},t) e^{-\beta m w^2/2}$ to compute the
pressure tensor, we find that $P_{ij}\simeq
(k_{B}T/m)\rho\delta_{ij}$. In that case, Eqs. (\ref{ps2}) and
(\ref{ps5}) yield the stochastic damped Euler equations
(\ref{dc1})-(\ref{dc2}). We recall, however, that there is no rigorous
justification for this local thermodynamic equilibrium
approximation. Therefore, it does not appear possible to rigorously
derive the damped Euler equations (\ref{dc1})-(\ref{dc2}) from the
Kramers equation (\ref{ps1cg}). Alternatively, if we consider the
strong friction limit $\xi\rightarrow +\infty$ for fixed $\beta$,
leading to $D=\xi/(\beta m)\rightarrow +\infty$, the first term in the
r.h.s. of Eq. (\ref{ps1}) implies that $f({\bf r},{\bf v},t)\simeq
(\beta m/{2\pi})^{d/2}
\rho({\bf r},t) e^{-\beta m v^2/2}+O(1/\xi)$, ${\bf u}=O(1/\xi)$ and $P_{ij}=(k_{B}T/m)\rho\delta_{ij}+O(1/\xi)$. To leading order in $1/\xi$, Eq. (\ref{ps5}) becomes
\begin{eqnarray}
\label{ps6}
\rho {\bf u}\simeq -\frac{1}{\xi}\left (\frac{k_{B}T}{m}\nabla\rho+\rho\nabla\Phi+\sqrt{2Dm\rho}{\bf R}({\bf r},t)\right ).
\end{eqnarray}
Inserting Eq. (\ref{ps6}) in the continuity equation (\ref{ps2}) and
defining $\mu=1/(\xi m)$ and $D_{*}=D/\xi^{2}=k_{B}T/(\xi m)$, we
obtain the stochastic Smoluchowski equation (\ref{cg4}). This equation
can thus be derived from Eq. (\ref{ps1cg}) in the limit
$\xi\rightarrow +\infty$.
\section{Conclusion}
In this paper, we have developed a theory of fluctuations for a system
of Brownian particles with weak long-range interactions. Starting from
the {\it stochastic} Smoluchowski equation (\ref{f1})-(\ref{f2}),
justified in Appendix \ref{sec_ll} from the Landau \& Lifshitz general
theory, we have obtained a simple formula (\ref{f22}) for the temporal
correlation function of the Fourier components of the density
fluctuations at equilibrium (for an infinite and homogeneous
distribution). This formula shows that the correlations decay in time
with the same damping rate as the one obtained from the study of the
normal modes of the {\it deterministic} Smoluchowski equation
(\ref{ea8}), without noise. Furthermore, the amplitude of the
correlation function diverges at the critical point $T_{c}$ (or at the
instability threshold $k=k_{m}$) leading to a failure of the mean
field approximation in that case. As a result, the limits
$N\rightarrow +\infty$ and $T\rightarrow T_{c}$ do not commute and the
instability occurs strictly before the critical point as discussed in
\cite{monaghan,ko,meta} for gravitational systems. In future works, we
shall extend this theory of fluctuations to more general
models. Indeed, the method developed in this paper can be generalized
to any type of kinetic equations including fluctuations. In
particular, the structure of formula (\ref{f20}) where $Z(k,\omega)$
is a sort of ``dielectric function'' obtained from the linearized kinetic
equation without noise, has a general scope.
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\section{Introduction and Related Work}
In the last years, there has been an increasing interest in the domain of Machine Unlearning. Several approaches were presented, which apply to different models and come with different assumptions and restrictions. So far, these approaches cover unlearning in decision trees and random forests \cite{brophy2021machine,schelter2021hedgecut}, linear models such as logistic regression \cite{aldaghri2021coded,golatkar2020eternal,guo2019certified}, neural networks \cite{bourtoule2021machine,golatkar2021mixed,golatkar2020eternal,graves2020amnesiac,guo2019certified} and even Markov Chain Monte Carlo \cite{fu2022knowledge,nguyen2022markov}. While most approaches focus on forgetting\footnote{We use the terms unlearning and forgetting interchangeably.} in a single model, there also exist works that deal with federated models instead \cite{wang2021federated}. The vast majority of Machine Unlearning algorithms are motivated by legal requirements such as the General Data Protection Regulation (GDPR) or the California Consumer Privacy Act. However, it is worth mentioning that Machine Unlearning is not limited to this use case. In \cite{liu2022backdoor}, Liu et al., for instance, utilize forgetting in order to remove backdoors that were induced into a model. Since the field of Machine Unlearning is rather young, we believe that applications in many other domains will likely arise soon, e.g., model revision, continual learning and bias correction.
When it comes to evaluation, the existing approaches can be divided into three categories. First, there are those approaches that provably guarantee perfect unlearning \cite{aldaghri2021coded,bourtoule2021machine,brophy2021machine,chen2022recommendation,golatkar2020eternal,neel2021descent,ullah2021machine}, and therefore do not need any evaluation. However, they often come with strong assumptions, making them only applicable in some specific scenarios. Others like the SISA (Sharded, Isolated, Sliced, Aggregated) training approach \cite{bourtoule2021machine} are generally applicable but come with considerable performance losses compared to a regularly trained model. Second, several approaches give guarantees on how well the forgetting works \cite{fu2022knowledge,gupta2021adaptive,neel2021descent,sekhari2021remember}. All of them argue about guarantees in the sense of differential privacy, i.e., the scrubbed model cannot be distinguished from a retrained model up to a certain probability. In the context of Machine Unlearning, this property is usually referred to as certified removal and was first introduced by Guo et al. \cite{guo2019certified}. Just as for the perfect forgetting approaches, they often come with strong assumptions.
Finally, we have those approaches that are neither perfect nor give guarantees \cite{golatkar2021mixed,golatkar2020eternal,graves2020amnesiac,liu2022backdoor,schelter2021hedgecut,wang2022efficiently,wang2021federated}, but evaluate the success of unlearning purely empirical or compare the resulting model with an actual retrained model. The latter might be interesting from a theoretical point of view but is inapplicable in practice. However, measuring the success of forgetting purely empirical might give us a good hint whether unlearning works. A general approach for empirically evaluating the success of an unlearning algorithm is through adversarial attacks. Evaluation solely through adversarial attacks, however, comes with some clear disadvantages.
On the one hand, performing different attacks is rather expensive since it requires training the corresponding attack models. Concretely, if the costs for forgetting \textit{and} evaluation exceed the costs for retraining from scratch, then the evaluation is not suitable in practice. On the other hand, we know that a failed attack is in no case a guarantee that unlearning really removed the information about the sensitive data. An attack might also fail due to other aspects such as badly chosen hyper-parameters, poorly estimated model parameters or the type and structure of the attack model. So to say, a failed attack on the scrubbed model can be seen as necessary but not as sufficient. Another point is that there might be new attacks in the future that can successfully gain sensitive information from the model, even though none of the currently existing attacks were able to do so. Nonetheless, the major advantage of those approaches is that they only come with few or even no assumptions, which makes them generally applicable, e.g., for deep neural networks.
Given all of the concerns mentioned above, we utter the necessity of an evaluation metric that captures how well an unlearning algorithm removed the information about the sensitive data points and, at the same time, is theoretically profound and efficient enough for practical use.
The contributions of this work are the following:
\begin{enumerate}
\item The definition of an evaluation metric that measures the success of Machine Unlearning algorithms. This metric is based on epistemic uncertainty and also allows an information theoretical interpretation.
\item We give a theoretical upper bound for the metric that is more efficient to compute than the actual metric, which is particularly important for larger models and datasets.
\item In the experiments, we investigate our evaluation metric on three forgetting approaches and discuss that it is useful for those approaches that change the model parameters in the direction of the retrained model.
\end{enumerate}
The rest of this paper is structured as follows. In \Cref{sec:preliminaries}, we will give a short introduction to the unlearning algorithms used in this work. We will then present our evaluation metric in \Cref{sec:metric}, including its intuition, derivation, and theoretical upper bound. Afterward, we outline the experimental setup and state the results in \Cref{sec:experiments}. Finally, we discuss the results and give possible research directions for future work in \Cref{sec:conclusion}.
\section{Unlearning Algorithms} \label{sec:preliminaries}
In the following, we give a short overview of the forgetting approaches used in this work.
For this we assume an already trained model with parameters $\theta$ trained on a dataset $D = D_r \cup D_f$ with $D_f \cap D_r = \emptyset$. $D_f$ states the target data that should be forgotten and $D_r$ states the remaining data.
\subsection{Retraining}\label{sec:retraining}
In many works \cite{aldaghri2021coded,bourtoule2021machine,fu2022knowledge,golatkar2020eternal,guo2019certified,gupta2021adaptive,neel2021descent,wang2021federated}, retraining the model from scratch on the remaining data $D_r$ is considered the optimal solution, since it does not use the sensitive data points $D_f$ during training and achieves high performance on $D_r$. However, for large models and datasets retraining from scratch is computationally expensive, which is why this is often considered impractical. The fact that retraining is considered the optimal solution is deeply connected to the idea of certified removal (CR) \cite{guo2019certified} or differential privacy \cite{dwork2014algorithmic}, respectively. A forgetting algorithm $\mathcal{U}$ is considered an $\epsilon$-CR \cite{guo2019certified}, iff $\forall \mathcal{T} \subseteq \mathcal{H}, D \subseteq \mathcal{X}$ and $x \in D$:
\begin{equation} \label{eq:ecr}
e^{- \epsilon} \leq \frac{P(\mathcal{U} ( \mathcal{A}(D), D, x) \in \mathcal{T})}{P(\mathcal{A}(D \setminus \{x\}) \in \mathcal{T})} \leq e^\epsilon,
\end{equation}
where $\mathcal{A}$ is considered a learning algorithm, $\mathcal{H}$ the hypothesis space, $\mathcal{X}$ the data space, and $x$ an arbitrary data point from our dataset $D$. \Cref{eq:ecr} states that for any hypothesis space, dataset, and data point, the chance of obtaining the same result via forgetting and retraining should be equal with a tolerated margin of $\epsilon$.
\subsection{Amnesiac Unlearning}
Amnesiac Unlearning \cite{graves2020amnesiac} is an imperfect forgetting algorithm without guarantees that removes the influence of the target data points by actually reverting all parameter updates they are related to. To implement this, it is necessary to keep track of the parameter updates $\Delta_{\theta_{e, b}}$ in all training epochs $e$ and for each batch $b$. Additionally, the set of data points $D_{e,b}$ relating to the updates must be provided. Given the parameter updates and the related data points, Amnesiac Unlearning subtracts all the updates $\Delta_{\theta_{e, b}}$ from the model parameters, where $D_{e,b}$ and the target data points $D_f$ have at least one data point in common:
\begin{equation}
\mathcal{U}_{AU}(\theta, D_f) = \theta - \sum_{e=1}^{E}\sum_{b=1}^{B} \mathbb{1}[ D_f \cap D_{e, b} \neq \emptyset ] \Delta_{\theta_{e, b}}.
\end{equation}
After forgetting, the updates that were subtracted and the corresponding data points no longer need to be held up.
Also note that for Amnesiac Unlearning iterative and batch forgetting are equivalent, since $\forall D_f, D'_f \subseteq D$:
\begin{equation} \label{eq:amnesiac_equiv}
\mathcal{U}_{AU}(\mathcal{U}_{AU}(\theta, D_f), D'_f) = \mathcal{U}_{AU}(\theta, D_f \cup D'_f ).
\end{equation}
\subsection{Fisher Forgetting}
Fisher Forgetting \cite{golatkar2020eternal} follows a different approach by hiding the difference between the given model and a model that could have been obtained by retraining from scratch. For this, Fisher Forgetting assumes both models to be close already, such that a normal distribution can describe the difference between them with variance $\sigma_h^2$. The goal is to add normal distributed noise to the parameters to hide this difference. The choice of the covariance matrix for this normal distributed noise is the key element of this approach. Here, we will only give its definition and a short explanation. For further details, we refer to the original work by Golatkar et al. \cite{golatkar2020eternal}.
Fisher Forgetting is defined as
\begin{equation} \label{eq:fisher_forgetting}
\mathcal{U}_{FF}(\theta, D_f) = \theta + \alpha^{\frac{1}{4}}F^{-\frac{1}{4}} n,
\end{equation}
where $n \sim \mathcal{N}(0, I)$ is standard normal distributed noise, $F$ is the approximated Fisher Information matrix (see \Cref{eq:fim} below) of the model w.r.t. the remaining data and $\alpha = \lambda \sigma_h^2$. The hyper-parameter $\lambda$ trades off the loss on the remaining data and the difference between the scrubbed and a retrained model. This trade-off directly originates from their Forgetting Lagrangian \cite{golatkar2020eternal}. $\sigma_h^2$ is also treated as a hyper-parameter because it is generally unknown and cannot be computed efficiently. Since $\lambda$ and $\sigma_h^2$ always occur together as a multiplied factor, they are combined into a single hyper-parameter $\alpha$. Just as Amnesiac Unlearning, Fisher Forgetting is an imperfect unlearning approach without any guarantees. However, it is worth mentioning that it originates from generalizing a perfect forgetting approach, namely Optimal Quadratic Scrubbing \cite{golatkar2020eternal}.
\section{Measuring the Success of Forgetting} \label{sec:metric}
When it comes to evaluating Machine Unlearning algorithms, there are three aspects that are of importance.
\begin{enumerate}
\item The scrubbed model should contain as little information as possible about the target data $D_f$. (Efficacy)
\item The scrubbed model should still perform well on the remaining data $D_r$. (Fidelity)
\item The computation of the scrubbed model should be more efficient than retraining the prior model from scratch on the remaining data $D_r$. (Efficiency)
\end{enumerate}
These three aspects form the foundation for evaluating Machine Unlearning algorithms and are widely agreed on as they are stated explicitly or implicitly in many works in the domain \cite{bourtoule2021machine,brophy2021machine,chen2022recommendation,fu2022knowledge,golatkar2021mixed,golatkar2020eternal,graves2020amnesiac,schelter2021hedgecut,sekhari2021remember,ullah2021machine,wang2021federated}. Here, we make use of the terminology as stated by Warnecke et al. in \cite{warnecke2021machine}.
Instead of evaluating all three aspects with a single metric, we argue that it is more reasonable to evaluate them separately. While this might seem obvious for the efficiency, it is not that clear for the efficacy and the fidelity. For example, Golatkar et al. formalize the unlearning problem by their Forgetting Lagrangian \cite{golatkar2020eternal}, which incorporates both the performance on the remaining data and the difference between the scrubbed and the retrained model. However, the efficacy and the fidelity should strictly be evaluated separately. Otherwise, it would be possible to receive a better evaluation by only improving the performance on the remaining data without scrubbing any information about the target data. Thus, a metric that combines both aspects might be misleading in its interpretation.
Since there are plenty of ways to evaluate the fidelity of a model, e.g. by accuracy, we will focus on evaluating the efficacy in this work.
\subsection{Evaluating Forgetting via Epistemic Uncertainty}
Our evaluation metric is both motivated by information theory and epistemic uncertainty. Since the efficacy informally describes the goal to minimize residual information about the target data, we want to start by giving a more precise formalization of the term of information in this context. We think that the most suitable notion of information is that of the Fisher Information matrix (FIM) \cite{schervish2012theory}
\begin{equation}\label{eq:fim}
\mathcal{I}(\theta; D) = \mathbb{E}_\theta \left[ \sum_{x,y \in D} \frac{-\partial^2 \log p_\theta(y | x)}{\partial \theta \partial \theta^T} \right] \; ,
\end{equation}
since it already measures the amount of information the model parameters $\theta$ carry about the dataset $D$. For practical reasons, we further assume an empirical approximation of the FIM, namely the Levenberg-Marquart approximation \cite{martens2014new}
\begin{equation}\label{eq:fim_approx}
\mathcal{I}(\theta; D) \simeq \frac{1}{|D|} \sum_{x, y \in D} \left(\frac{\partial \log p_\theta(y|x)}{\partial \theta}\right)^2.
\end{equation}
Approximating the FIM is often necessary in practice, since the computation of the second derivatives w.r.t. all parameter combinations and for each data point separately is computationally expensive, i.e. $\mathcal{O}(|D| \cdot F + |D| \cdot G \cdot |\theta|^2)$. Here $F$ and $G$ denote the runtime complexities for the model inference $p_\theta(y|x)$ and the gradient computations, respectively. The FIM might not even fit in memory for larger models such as deep neural networks. Keeping in mind that forgetting and its evaluation must be more efficient than retraining from scratch, computing the whole FIM is not reasonable.
Note that the approximation in \Cref{eq:fim_approx}, which can be done in $\mathcal{O}(|D| \cdot F + |D| \cdot G \cdot |\theta|)$, will give us a diagonal matrix or a vector of length $|\theta|$, respectively. For all $i \in [1, |\theta|]$ the $i$-th entry states the amount of information $\theta_i$ carries about the dataset $D$. By computing the trace of $\mathcal{I}(\theta; D)$, we obtain an overall information value
\begin{equation}\label{eq:uncertainty}
\imath(\theta; D) = tr(\mathcal{I}(\theta; D)) = \frac{1}{|D|} \sum_{i = 1}^{|\theta|} \sum_{x, y \in D} \left(\frac{\partial \log p_\theta(y|x)}{\partial \theta_i}\right)^2.
\end{equation}
The trace of the FIM can not only be seen as the total amount of information, but also as the epistemic uncertainty of $\theta$ with respect to $D$ \cite{hullermeier2021aleatoric}. Thus, $\imath(\theta; D)$ can be interpreted in the following ways:
\begin{enumerate}
\item The amount of information the model parameters carry about the given dataset.
\item The degree of how much the model parameters can vary, while still describing the dataset equally well.
\item The epistemic uncertainty of the model parameters with respect to the dataset.
\end{enumerate}
Given the information respectively uncertainty score in \Cref{eq:uncertainty}, we see that it is also sufficient to compute the FIM diagonal only since none of the other matrix entries are required. We argue that scrubbing information from a model should always increase the epistemic uncertainty and decrease the amount of information about the target data. Otherwise, a Streisand effect \cite{jansen2015streisand} may occur. Therefore, we introduce the efficacy score
\begin{equation}\label{eq:efficacy}
\text{efficacy}(\theta; D) = \begin{cases}
\frac{1}{\imath(\theta; D)}, &\text{if } \imath(\theta; D) > 0 \\
\infty, &\text{otherwise}
\end{cases}
\end{equation}
as a measure for how much information the model exposes. The model converges to the optimum of 0 with increasing uncertainty.
However, we argue that in most cases, even exact unlearning algorithms or retraining will not achieve an efficacy score of (approximately) 0 due to the generalization from the remaining data points. Although it is hard to tell what the actual optimum efficacy is, the efficacy score is still helpful to compare the results of multiple unlearning algorithms since we still know that the lower the score, the higher the uncertainty about the target data.
With an increasing number of target data points and model parameters, the efficacy score becomes more computationally expensive since it requires the computation of the gradients w.r.t. the model parameters for each target data point separately. In the following, we address this problem by defining a theoretical upper bound for the efficacy score. For this, we first define the cross-entropy loss
\begin{equation}
\mathcal{L}(\theta, D) = \frac{1}{|D|} \sum_{x, y \in D} -\log p_\theta(y | x)
\end{equation}
with gradients
\begin{equation}
\nabla \mathcal{L}(\theta, D) = \frac{1}{|D|} \sum_{x, y \in D} \frac{-\partial \log p_\theta(y | x)}{\partial \theta}
\end{equation}
and $\ell_2$-norm of the gradients
\begin{equation}
\| \nabla \mathcal{L}(\theta, D) \|_2 = \left(\sum_{\theta_i \in \theta} \frac{1}{|D|^2} \left(\sum_{x, y \in D} \frac{\partial \log p_\theta(y | x)}{\partial \theta_i}\right)^2\right)^{\frac{1}{2}}.
\end{equation}
Then, \Cref{th:uncertainty_lower_bound} shows that the uncertainty is lower bounded by the squared gradient norm of the cross-entropy loss.
\begin{theorem}\label{th:uncertainty_lower_bound}
Let $\mathcal{L}(\theta, D)$ be the cross-entropy loss.
The squared gradient norm of the cross-entropy loss forms a lower bound for the information score:
\begin{equation}
\| \nabla \mathcal{L}(\theta, D) \|_2^2 \leq \imath(\theta; D)
\end{equation}
\end{theorem}
\begin{proof}[\Cref{th:uncertainty_lower_bound}]
\begin{align*}
\imath(\theta; D) &= \frac{1}{|D|} \sum_{\theta_i \in \theta} \sum_{x,y \in D} \left(\frac{\partial \log p_\theta(y|x)}{\partial \theta_i}\right)^2 & \text{(\Cref{eq:uncertainty})}\\
&\geq \frac{1}{|D|} \sum_{\theta_i \in \theta} \frac{1}{|D|} \left(\sum_{x,y \in D} \frac{\partial \log p_\theta(y|x)}{\partial \theta_i}\right)^2 & \text{(*)}\\
&= \sum_{\theta_i \in \theta} \frac{1}{|D|^2} \left(\sum_{x,y \in D} \frac{\partial \log p_\theta(y|x)}{\partial \theta_i}\right)^2 \\
&= \| \nabla \mathcal{L}(\theta, D) \|_2^2
\end{align*}
(*) Cauchy-Schwarz inequality \cite{steele2004cauchy}
\end{proof}
From the uncertainty lower bound in \Cref{th:uncertainty_lower_bound}, we directly obtain the theoretical upper bound for the efficacy score (\Cref{lem:efficacy_upper_bound}).
\begin{lemma}\label{lem:efficacy_upper_bound}
Let $\mathcal{L}(\theta, D)$ be the cross-entropy loss.
If $\imath(\theta; D) > 0$, then the efficacy score is upper bounded by
\begin{equation}
\textnormal{efficacy}(\theta; D) \leq \frac{1}{\| \nabla \mathcal{L}(\theta, D) \|_2^2}.
\end{equation}
\end{lemma}
In contrast to the efficacy score, its upper bound is far less computationally expensive since it only requires computing the gradients w.r.t. the parameters once, instead of $|D|$ times. Thus, the efficacy upper bound can be computed in $\mathcal{O}(|D| \cdot F + G \cdot |\theta|)$ in contrast to $\mathcal{O}(|D| \cdot F + |D| \cdot G \cdot |\theta|)$ for the efficacy.
\section{Experiments} \label{sec:experiments}
Our experiments aim to study the practical usefulness of the efficacy score and its theoretical upper bound stated in \Cref{lem:efficacy_upper_bound}. For this, we formulate the following two hypotheses:
\begin{enumerate}
\item Applying a forgetting algorithm to a trained model always reduces the efficacy score for the target data.
\item A lower efficacy score always implies a less successful adversarial attack.
\end{enumerate}
In order to test the above hypotheses, we consider the following experimental setup.
\subsection{Experimental Setup}
We define a simple neural network architecture, consisting of five fully connected layers with $N$, 512, 256, 128, and 10 neurons per layer, respectively. $N$ denotes the input dimension that varies depending on the input data. All layers but the output layer are ReLU activated. We use the softmax function for the last layer to obtain an output distribution over all classes. For the datasets, we use the well-known MNIST \cite{lecun1998gradient} and CIFAR10 \cite{krizhevsky2009learning} datasets for image classification. We only use a small subset of both datasets to reduce the applied forgetting algorithms' runtime and memory consumption. Thus, we only use the first 100 samples for each class. All training examples are transformed to greyscale images, and pixel values are normalized to $[0, 1]$.
We train the model on the MNIST dataset for 50 epochs with a learning rate of 0.1 and a batch size of 32 using standard stochastic gradient descent (SGD). For CIFAR10, we train the model for 200 epochs with a learning rate of 0.1 and a batch size of 64, again using standard SGD. Each training is repeated 20 times with different random initializations, giving us a total number of 40 pre-trained models.
For the target data we want to forget, we chose a fixed class for both datasets: Class 3 for MNIST and class 8 for CIFAR10. Both classes are chosen arbitrarily. Furthermore, we define a percentage value for how many data points of the target class we would like to forget. In all experiments the percentage values are 0.01, 0.1, 0.25, 0.5, 0.8 and 1, respectively.
As a baseline for how successful an adversarial attack should be at most after forgetting, we perform a membership inference attack (MIA) \cite{shokri2017membership} on all 40 pre-trained models for each of the above percentage values. This leads to a total number of 240 model attacks. The goal of a MIA is to predict whether a data point is part of the training dataset of the target model. Therefore, the attack model only has black-box access to the target model. Here we make use of the MIA implementation in IBM's Adversarial Robustness Toolbox \cite{art2018}. Each attack model is trained with default parameters on the same training set as its target model.
We consider three different approaches for forgetting algorithms: Retraining, Fisher Forgetting, and Amnesiac Unlearning. Thereby, retraining is considered the optimal solution (see \Cref{sec:retraining}). All forgetting approaches are applied to each pre-trained model using the same target classes and percentage values as the MIAs. This again leads to a total of applying each forgetting approach 240 times. Retraining is always performed with the same model initialization and hyper-parameters as the original training. Our implementation of Fisher Forgetting exactly follows the implementation details stated in the extended version of \cite{golatkar2020eternal}, which slightly differs from the definition of Fisher Forgetting given in \Cref{eq:fisher_forgetting}. The implementation details are purely empirical and arose from the experiments in which these led to better results.
Finally, we want to note that all our experiments were implemented using PyTorch \cite{paszke2019pytorch} and are publicly available on GitHub\footnote{Code will be publicly available on acceptance but is included in the additional resources for review.}.
\subsection{Results}
In the following, we present our experimental results on the MNIST dataset. We omit our experimental results on CIFAR10, since they do not provide any additional insights. The observation for both datasets are identical. \footnote{Plots and tables of the CIFAR10 experiments are included in the appendix for review.}
First of all, we take a look at the efficacy values of the pre-trained models in comparison to the efficacy after forgetting (\Cref{fig:efficacy}). We observe that the efficacy increases on average for the pre-trained models and varies more with the fewer data points we take into account. For a single data point of the target class, i.e., the percentage equals 0.01, the observed efficacy values range from $10^{-7}$ to $10^{13}$, which exceeds the lowest and highest values for all other percentages. Note that for reasons of readability, the efficacy values for a single data point are omitted in \Cref{fig:efficacy}.
Altogether training increases the efficacy of the target data, as can be seen in \Cref{fig:efficafy_comparison}.
\begin{figure}
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[scale=0.43]{img/efficacy_ub_pre_trained_mnist_zoom.png}
\caption{Pre-trained}
\end{subfigure}\hfill
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[scale=0.43]{img/efficacy_ub_retraining_mnist_zoom.png}
\caption{Retraining}
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[scale=0.43]{img/efficacy_ub_amnesiac_mnist_zoom.png}
\caption{Amnesiac Unlearning}
\label{fig:efficacy_amnesiac}
\end{subfigure}\hfill
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[scale=0.43]{img/efficacy_ub_fisher_mnist_zoom.png}
\caption{Fisher Forgetting}
\label{fig:efficacy_fisher}
\end{subfigure}
\caption{Distributions of efficacy scores (solid lines) and upper bounds (dashed lines) over all pre-trained models trained on the MNIST dataset (a) before and (b)-(d) after forgetting. Each distribution corresponds to a percentage of the target class. For reasons of readability we omit the percentage of 0.01. Both axes are log scaled.}
\label{fig:efficacy}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.6]{img/efficacy_comparison.png}
\caption{Efficacy comparison w.r.t. the whole target class before training (Initial), after training (Pre-trained) and after forgetting (Retraining, Amnesiac Unlearning, Fisher Forgetting). Both axes are log scaled.}
\label{fig:efficafy_comparison}
\end{figure}
Forgetting about the target data decreases the efficacy in all our experiments, except for forgetting a single data point only. Here the efficacy almost remains unchanged, and in the case of Amnesiac Unlearning, we were even able to observe an increase over the pre-trained models. When forgetting more than just a single data point, we see that the efficacy continually decreases the more data we forget from our target class. This holds for both retraining from scratch and Fisher Forgetting. Interestingly enough, we observe the opposite behavior for Amnesiac Unlearning.
In comparison to the pre-trained model, the efficacy values decrease, but a higher percentage yields an increase in the efficacy (\Cref{fig:efficacy_amnesiac}). Amnesiac Unlearning forgets about target data points by reverting those updates from training, which they directly influence, so this behavior is expected. With an increasing number of target data points, the number of updates that must be reverted increases as well. Therefore, the model further develops in the direction of the initial model. As a consequence, the efficacy values converge towards those of the initial model as well (see \Cref{fig:efficacy_amnesiac,fig:efficafy_comparison}). Finally, we see that Fisher Forgetting only slightly decreases the efficacy independent of the target data percentage. This holds for all values of $\alpha$ (see \Cref{eq:fisher_forgetting}) we tried in our experiments and can be traced back to clipping the values in the Fisher approximation, which is one of the implementation details mentioned in \cite{golatkar2020eternal}. This also leads to the fact that the scrubbed model can still classify all target data points correctly, which is undesirable in general. At the same time, we observe that the accuracy of the target data points decreases for the other approaches the higher the percentage (\Cref{tab:acc}).
For the efficacy upper bounds (\Cref{fig:efficacy}), we observe that both the distribution shapes and the relations between the different distributions are preserved. This shows that the efficacy upper bound is suitable for comparing multiple forgetting results even though the absolute values are an order of magnitude larger than the actual efficacies.
\begingroup
\setlength{\tabcolsep}{10pt}
\begin{table}
\centering
\begin{tabular}{c c c c c}
\toprule
\multirow{2}{*}{Model} & \multirow{2}{*}{$p$} & \multicolumn{3}{c}{Accuracy} \\
&& $D_r$ & $D_f$ & $D_{test}$ \\ \midrule
Pre-trained & * & 1.00 ± 0.00 & 1.00 ± 0.00 & 0.87 ± 0.00 \\ \midrule
\multirow{6}{*}{Retraining} & 1 & 1.00 ± 0.00 & 0.00 ± 0.00 & 0.80 ± 0.00 \\
& 0.8 & 1.00 ± 0.00 & 0.57 ± 0.05 & 0.84 ± 0.00 \\
& 0.5 & 1.00 ± 0.00 & 0.89 ± 0.02 & 0.86 ± 0.00 \\
& 0.25 & 1.00 ± 0.00 & 0.96 ± 0.01 & 0.87 ± 0.00 \\
& 0.1 & 1.00 ± 0.00 & 1.00 ± 0.00 & 0.87 ± 0.00 \\
& 0.01 & 1.00 ± 0.00 & 1.00 ± 0.00 & 0.87 ± 0.00 \\ \midrule
\multirow{6}{*}{Amnesiac Unlearning} & 1 & 0.13 ± 0.03 & 0.00 ± 0.00 & 0.11 ± 0.02 \\
& 0.8 & 0.12 ± 0.03 & 0.00 ± 0.00 & 0.11 ± 0.02 \\
& 0.5 & 0.11 ± 0.01 & 0.00 ± 0.00 & 0.10 ± 0.01 \\
& 0.25 & 0.16 ± 0.07 & 0.00 ± 0.00 & 0.15 ± 0.07 \\
& 0.1 & 0.19 ± 0.08 & 0.00 ± 0.00 & 0.18 ± 0.07 \\
& 0.01 & 0.60 ± 0.21 & 0.50 ± 0.50 & 0.53 ± 0.18 \\ \midrule
Fisher Forgetting & * & 1.00 ± 0.00 & 1.00 ± 0.00 & 0.87 ± 0.00 \\ \toprule
\end{tabular}
\caption{Mean accuracy and standard deviation of all MNIST models on the remaining data $D_r$, the target data $D_f$ and the test data $D_{test}$. $p$ denotes the percentage of the target data and $*$ indicates that the accuracy values are the same over all percentages.}
\label{tab:acc}
\end{table}
\endgroup
Next, we take a look at the mean probabilities of the membership inference attacks performed on the pre-trained and the scrubbed models as shown in \Cref{fig:mean_mia}.
We compute the mean probability over all target data points for each attack. The effectiveness of the MIA on the pre-trained model thereby forms the baseline, which is not to be exceeded. Otherwise, forgetting would expose information about the target data rather than removing it. On the other hand, we have a second baseline, which is given by the effectiveness of the MIA on the retrained model since the target data points were not used during training. However, due to correlations and generalization, an attack might still be successful, especially if not all data points of the target class were to be forgotten.
Our results show that forgetting always decreases the mean MIA probabilities and that the more data points we forget, the lower they get. Here we want to point out that this also holds for Amnesiac Unlearning, even though it increases the efficacy with increasing target data points.
\begin{figure}[ht]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[scale=0.43]{img/mia_pre_trained_mnist_zoom.png}
\caption{Pre-trained}
\end{subfigure}\hfill
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[scale=0.43]{img/mia_retrained_mnist_zoom.png}
\caption{Retraining}
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[scale=0.43]{img/mia_amnesiac_mnist_zoom.png}
\caption{Amnesiac Unlearning}
\end{subfigure}\hfill
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[scale=0.43]{img/mia_fisher_mnist_zoom.png}
\caption{Fisher Forgetting}
\end{subfigure}
\caption{Distributions of membership inference attack mean probabilities over all pre-trained models trained on the MNIST dataset (a) before and (b)-(d) after forgetting. Each distribution corresponds to a percentage of the target class. Both axes are log scaled.}
\label{fig:mean_mia}
\end{figure}
In \Cref{fig:efficacy_mia}, we illustrate how efficacy relates to the effectiveness of the MIAs. The efficacy values and the MIA mean probabilities do not vary much for the pre-trained model, even though the efficacy values are much larger when computed for a single data point only.
For retraining and Fisher Forgetting, we observe that with an increased efficacy score, the mean probabilities of the MIAs also become larger. Just as for the efficacy values themselves, we observe the opposite effect for forgetting via Amnesiac Unlearning. Here lower efficacies yield higher average MIA probabilities. Again, this can be traced back to the model converging to the initial model rather than to the pre-trained model as more data points are forgotten. Also, note that the efficacy varies over a large range of values for single data points while having similar MIA mean probabilities.
\begin{figure}[ht]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[scale=0.43]{img/efficacy_mia_pre_trained_mnist.png}
\caption{Pre-trained}
\end{subfigure}\hfill
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[scale=0.43]{img/efficacy_mia_retraining_mnist.png}
\caption{Retraining}
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[scale=0.43]{img/efficacy_mia_amnesiac_mnist.png}
\caption{Amnesiac Unlearning}
\end{subfigure}\hfill
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[scale=0.43]{img/efficacy_mia_fisher_mnist.png}
\caption{Fisher Forgetting}
\end{subfigure}
\caption{Log-log plot showing the relation between the efficacy and the membership inference attack mean probability (a) before and (b)-(d) after forgetting.}
\label{fig:efficacy_mia}
\end{figure}
With the results given above, we revisit our hypotheses. Even though most experiments give evidence for the first hypothesis, we have to reject it due to our experiments on Amnesiac Unlearning. Here, forgetting first decreases the efficacy, but the more data points we forget, the more the efficacy increases. This is especially problematic since there is no difference between batch forgetting and iterative forgetting for Amnesiac Unlearning (\Cref{eq:amnesiac_equiv}). Thus, forgetting all target points consecutively increases the efficacy. Likewise, we have to reject our second hypothesis. Our results from forgetting through retraining and Fisher Forgetting support this hypothesis. However, the results from forgetting through Amnesiac Unlearning show the exact opposite behaviour.
\section{Conclusion and Future Work} \label{sec:conclusion}
In this work, we presented a metric for evaluating the success of forgetting algorithms and formulated two hypotheses, which we tested in our experiments. This metric has many advantages compared to existing evaluation methods since it can be computed efficiently, allows an information theoretical interpretation, does not require retraining the model, and follows cognitive considerations regarding the relationship between uncertainty and forgetting. Moreover, for its theoretical upper bound, it even preserves the relation of the efficacy distributions over all percentage values. All of those advantages are crucial for practical applications.
Our results clearly show that the evaluation of forgetting algorithms is a complex task that cannot simply be solved by measuring accuracy or performing adversarial attacks. Decreasing the accuracy of the target data points and preventing adversarial attacks from being successful can be seen as necessary conditions but are not sufficient to guarantee that sensitive information was really scrubbed from the model. The former can be observed in our experiments using Fisher Forgetting (\Cref{tab:acc}, \Cref{fig:mean_mia}), while the latter can easily be illustrated using a small example. Consider a neural network consisting of a feature extraction part, followed by a classifier part. Reinitializing the classifier part would degenerate the model's performance. However, since the feature extraction part has not changed, the overall model still contains information about the training data.
Even though we had to reject both hypotheses, we see that both retraining from scratch as the optimal solution and Fisher Forgetting give evidence for them. Merely our results of using Amnesiac Unlearning lead to a rejection of the hypotheses. As a crucial difference of the forgetting algorithms leading to this rejection, we identified the direction in which forgetting changes the model parameters. While retraining is considered the optimal solution and Fisher Forgetting aims to blur the difference between the model and the retrained model by adding noise, Amnesiac Unlearning lets the model converge towards the initial model before training (\Cref{fig:efficafy_comparison,fig:efficacy_amnesiac}). Thus, the directions in which the parameters are changed are fundamentally different. This implies that multiple evaluation metrics might be necessary that depend on the way the algorithms remove information from models. Therefore, we claim that the here presented efficacy metric is a first step in the direction of more general evaluation metrics for Machine Unlearning.
Given the insights from this work, we want to do a larger survey on evaluating Machine Unlearning algorithms in the future, where we categorize the algorithms depending on how and in which direction the model parameters are updated. In this context, we will further study the here presented efficacy metric for those algorithms that aim to obtain a model close to a retrained one, since the results of our experiments look quite promising. Finally, finding the relation between the efficacy and certified removal is also an important direction for future work, since this will allow relating the metric to privacy guarantees in the sense of differential privacy.
\bibliographystyle{splncs04}
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\section{Introduction} \label{sec:intro}
The Standard Model (SM) of particle physics has achieved its triumph after the discovery of a 125 GeV scalar resonance at the Large Hadron Collider (LHC)~\cite{Aad:2012tfa,Chatrchyan:2012xdj} experiment.
Nevertheless, the quest for new physics is yet to accomplish. Possibility of having an exotic particle in the realm of current collider reach still can not be discarded and it may
just need a closer look at the present LHC data. In the second run of the LHC with increased energy and luminosity, a greater amount of data is already
available for more detailed and intricate analysis.
Among many, a charged scalar particle is one of the appealing exotic candidates in particle phenomenology and has been
searched for long.
Most of the beyond Standard Model (BSM) scenarios render singly charged scalar candidate and it has always remained an important probe for the models with SM scalar sector
extensions~\cite{Cheung:2002gd, Dev:2013ff, Coleppa:2014cca, Bhattacharyya:2014oka, Han:2015hba,Mitra:2016wpr,Arbey:2017gmh,Cen:2018okf,Guchait:2018nkp,Ghosh:2017pxl,
Abbas:2018pfp,Ghosh:2018jpa,Babu:2019mfe}.
In this paper, we will talk about probing the Georgi-Machacek (GM)
model~\cite{Georgi:1985nv,Chanowitz:1985ug,Gunion:1989ci}, a variant of scalar triplet extension of the SM, through the searches of a singly-charged scalar at the current and future LHC run. The usual Higgs triplet model (HTM) is favored for the explanation on the neutrino mass generation but
it suffers from electroweak (EW) $\rho$-parameter constraint.
The GM model, on the other hand, consists of two scalar triplets, one real and one complex, thus, preserving the custodial SU(2) symmetry and
lifting the bound on the triplet vacuum expectation value (vev).
This large triplet vev induces a large mixing between the GM and the SM sector leading to interesting search processes for the non standard scalars.
In particular, the charged scalar of the GM model couples to the SM fermions with a strength directly proportional to the vev $(v_t)$ and a significantly large $v_t$ enhances the detection
possibility of such charged scalar at the collider. Such decay processes are highly suppressed in the HTM
and can thus be an important probe of the GM model.
Besides the two CP-even scalars, the lightest of which resembles the SM-like Higgs with mass 125 GeV,
there lies seven scalar particles in the GM particle spectrum arising from the custodial triplet and fiveplet respectively.
The singly-charged scalar of the custodial fiveplet ($H_5^\pm$), however, has no tree-level interaction with the SM fermions.
The singly charged scalar of the custodial triplet ($H_3^\pm$) can have significant tree level coupling to SM fermion-anti-fermion pair
due to the large overlap with the Higgs field corresponding to the triplet vev.
In particular, $H_3^{+}$ may acquire a considerable branching ratio (BR) to the $t\bar b$ channel in the kinematically possible mass region.
Recently, the ATLAS collaboration~\cite{Aaboud:2018cwk} has reported their latest analysis on charged Higgs searches with the data from the 13 TeV run of the LHC
with 36 $fb^{-1}$ luminosity
and has provided a model independent bound on the production cross-section times the BR for charged Higgs decaying into $t\bar b$ mode.
Following their analysis, in this paper, we aim to put constraints on the GM model parameter space in the physical basis in terms of
the scalar masses and mixing angles. This limit will definitely put an upper bound on $v_t$ for a particular charged Higgs mass corresponding to
its BR.
On the other hand, the theoretical constraints on the GM scalar potential from EW vacuum stability and perturbative
unitarity at tree-level already limits $v_t$ from the lower end. Simultaneously, the charged scalar masses are also restricted from above due to
the unitarity constraints. We show the bounds for two variants of GM potential, with and without the trilinear terms where the potential with
vanishing trilinear term results in a more stringent bound.
Additionally, the latest LHC Higgs data also cause a severe constraint in the model parameter space.
Therefore, a combination of theoretical constraint, Higgs data and the limit from charged Higgs analysis by ATLAS will put
a definitive restriction on the GM model. In this study, we have given a comprehensive description in this regard.
Furthermore, as a future endeavor, we show how the region allowed by the above
bounds can be probed at the future run of the LHC. For this, we propose the other competent decay channel of $H_3^\pm \to W^\pm h$~\cite{Coleppa:2014cca},
$h$ being the 125 GeV Higgs resonance. This decay mode is dominant for large neutral scalar mixing angle $\alpha$ and can be
an effective probe in the region where the ATLAS limit is relaxed. It should be mentioned that though the GM model have been studied
extensively in the post-Higgs discovery era~\cite{Kanemura:2012rs,Chiang:2012cn,Hartling:2014aga,Hartling:2014zca,Chiang:2014bia,
Chiang:2015amq,Chang:2017niy,Degrande:2017naf,Blasi:2017xmc,Biswas:2018jun,Das:2018vkv,Chiang:2018cgb,
Banerjee:2019gmr}, implication of charged Higgs searches in association with the latest Higgs data to explore the model
parameter space has not been discussed so far.
The paper is organized in the following manner. In sec.~\ref{sec:model}, we introduce the model briefly and explain
the theoretical as well as constraint from Higgs data on the model parameter space. In sec.~\ref{sec:ATLAS}, we explain
the production and decay of the charged Higgs state $H_3^\pm$ for various parameter space and portray the
limits provided by the ATLAS analysis. In sec.~\ref{sec:collider}, we demonstrate our analysis for the future probe of the model
at the LHC
and, in sec.~\ref{sec:results} we show our findings collectively. Finally, we conclude in sec.~\ref{sec:summary}.
\section{The Model } \label{sec:model}
Here, we present a brief description of the scalar potential of the GM model.
in addition to the SM particle content, the scalar sector of the
GM model~\cite{Georgi:1985nv,Chanowitz:1985ug,Gunion:1989ci} consists of one real $SU(2)_L$ triplet $\xi$
and one complex $SU(2)_L$ triplet $\chi$ with hypercharges $Y=0$ and $Y=2$ forming a bi-triplet $X$.
Now, the most general scalar potential for the GM model can be written as~\cite{Hartling:2014zca,Das:2018vkv}
\begin{eqnarray}
V(\Phi,X) &=& \frac{\mu_{\phi}^2}{2} \Tr(\Phi^\dagger \Phi) + \frac{\mu_{X}^2}{2} \Tr(X^\dagger X) + \lambda_1[\Tr(\Phi^\dagger \Phi)]^2 +
\lambda_2 \Tr(\Phi^\dagger \Phi)\Tr(X ^\dagger X) \nonumber \\
&& + \lambda_3 \Tr(X^\dagger X X^\dagger X) + \lambda_4 [\Tr(X^\dagger X)]^2 - \lambda_5 \Tr(\Phi^\dagger \tau_a \Phi \tau_b)\Tr(X^\dagger t_a X t_b) \nonumber \\
&& - M_1\Tr(\Phi^\dagger \tau_a \Phi \tau_b)\left( U X U^\dagger \right)_{ab} - M_2\Tr(X^\dagger t_a X t_b)\left( U X U^\dagger \right)_{ab} \,,
\label{e:potential}
\end{eqnarray}
where, $\tau_a \equiv \sigma_a/2$, ($a=1,2,3$) with $\sigma_a$'s being
the Pauli matrices and $t_a$'s are the generators of triplet representation of
$SU(2)$ which are expressed as
\begin{eqnarray}
t_1 = \frac{1}{\sqrt{2}}\left(\begin{array}{ccc}
0 & 1 & 0 \\
1 & 0 & 1 \\
0 & 1 & 0 \\
\end{array}\right)\,, \qquad
t_2 = \frac{1}{\sqrt{2}}\left(\begin{array}{ccc}
0 & -i & 0 \\
i & 0 & -i \\
0 & i & 0 \\
\end{array}\right)\,, \qquad
t_3 =\left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & -1 \\
\end{array}\right)\,.
\label{su2_tripgen}
\end{eqnarray}
Furthermore, the matrix $U$ in the trilinear terms of \Eqn{e:potential} is given by,
\begin{eqnarray}
U = \frac{1}{\sqrt{2}} \left(\begin{array}{ccc}
-1 & 0 & 1 \\
-i & 0 & -i \\
0 & \sqrt{2} & 0 \\
\end{array}\right) \,.
\label{matU}
\end{eqnarray}
It is worth mentioning here that the absence of these trilinear term would enhance the
symmetry of the potential. Moreover, it should be noted that the GM potential has
no explicit CP violation since there are no non-Hermitian terms.
Now, the bi-doublet $\Phi$ and the bi-triplet $X$ are represented as
\begin{eqnarray}
\label{e:fields}
\Phi = \left(\begin{array}{cc}
\phi^{0*} & \phi^+ \\
-\phi^- & \phi^0 \\
\end{array}\right) \,, \qquad
X= \left(\begin{array}{ccc}
\chi^{0*} & \xi^+ & \chi^{++} \\
-\chi^- & \xi^0 & \chi^+ \\
\chi^{--} & -\xi^- & \chi^0 \\
\end{array}\right) \,.
\end{eqnarray}
After the electroweak symmetry breaking (EWSB), the neutral components of the bi-doublet and the bi-triplet can be expanded around their vacuum expectation values(vev) as
$\phi^0 = \frac{1}{\sqrt{2}} (v_d + h_d + i \eta_d)$, $\xi^0 = (v_t + h_\xi )$ and $\chi^0 = (v_t + \frac{h_\chi + i \eta_\chi}{\sqrt{2}})$.
The equality in vevs to the real and the complex triplets
corresponds to the preserved custodial symmetry of the potential. The EW vev
in terms of the vevs of the scalar multiplets turns out to be
\begin{eqnarray}
\sqrt{v_d^2 + 8 v_t^2} = v = 246~{\rm GeV} \,.
\label{e:custodial}
\end{eqnarray}
Thus, there will be two independent minimization conditions corresponding to the two vevs of the
bi-doublet and the bi-triplet ($v_d$ and $v_t$) which can be used to extract the bilinear
coefficients of the potential $\mu_{\phi}^2$ and $\mu_X^2$ as
\begin{subequations}
\begin{eqnarray}
\mu_{\phi}^2 &=& - 4 \lambda_1 v_d^2 - 3\left(2\lambda_2 -\lambda_5\right) v_t^2 + \frac{3}{2} M_1 v_t \,, \\
%
\mu_{X}^2 &=& -\left(2\lambda_2 -\lambda_5\right) v_d^2 - 4 \left(
\lambda_3+ 3\lambda_4 \right)v_t^2 + \frac{M_1 v_d^2}{4v_t} + 6 M_2 v_t \,.
\end{eqnarray}
\label{e:bilinears}
\end{subequations}
The scalar potential consists of a custodial quintuplet $(H_5^{++},H_5^+,H_5^0, H_5^-, H_5^{--})$
of common mass $m_5$ and a custodial triplet $(H_3^+,H_0, H_3^-)$ of mass $m_3$. Alongside,
there are two CP-even scalars that are custodial singlets, namely $h$ and $H$ with masses
$m_h$ and $m_H$ respectively. The neutral scalar mixing angle $\alpha$ diagonalizes
the CP even sector to obtain the mass eigenstates $h$ and $H$. Hereby, we refrain ourselves from giving a detailed description
of the diagonalization procedure and we refer the reader to Refs.~\cite{Chiang:2012cn,Hartling:2014zca}.
We only show the mass eigenstate for the charged scalars and the custodial singlets.
\begin{subequations}
\begin{eqnarray}
\label{e:ch}
H_5^\pm &=& \frac{1}{\sqrt{2}} \left(\chi^\pm - \xi^\pm \right) \,, \qquad
H_3^\pm = -\sin\beta~ \phi^\pm + \cos \beta \frac{1}{\sqrt{2}} \left(\chi^\pm + \xi^\pm \right) \,, \\
\label{e:alpha}
h &=& \cos\alpha~ h_d + \sin\alpha \left(\sqrt{\frac{1}{3}}h_\xi + \sqrt{\frac{2}{3}}h_\chi \right) \,, \\
H &=& -\sin\alpha ~h_d + \cos\alpha \left(\sqrt{\frac{1}{3}}h_\xi + \sqrt{\frac{2}{3}}h_\chi \right) \,,
\end{eqnarray}
\end{subequations}
where, $h_d,h_\chi,h_\xi$ are the fields corresponding to the neutral components of the doublet and the triplets
after the electroweak symmetry breaking. The angle $\alpha$ represents the neutral mixing angle while $\tan\beta$ is defined as
\begin{eqnarray}
\tan\beta = \frac{2\sqrt{2} v_t}{v_d} \,.
\end{eqnarray}
In total, there are nine independent parameters in the GM scalar potential with
two bilinears ($\mu_{\phi}^2$ and $\mu_X^2$), five quartic couplings ($\lambda_i$,
$i=1,\dots, 5$) and two trilinear couplings ($M_1$ and $M_2$). Among these,
the bilinears have already been traded in favor of the vevs, $v_d$ and $v_t$
as in \Eqn{e:bilinears}. Besides the trilinear terms ($M_1,M_2$), the five quartic couplings can now be
exchanged with the four physical scalar masses,
$m_5$, $m_3$, $m_H$ and $m_h$ and the mixing angle, $\alpha$.
It should be mentioned here that, $h$ is the lightest among all the $CP$-even scalars
corresponding to the Higgs-like scalar discovered at the LHC with mass $m_h \approx 125$~GeV.
Below, we present the relation between the $\lambda_i$-s with the physical
masses and mixings~\cite{Das:2018vkv}.
\begin{subequations}
\begin{eqnarray}
\lambda_1 &=& \frac{1}{8 v^2 \cos^2 \beta}\left(m_h^2 \cos^2 \alpha +m_H^2 \sin^2 \alpha\right) \,, \label{lambda1} \\
\lambda_2 &=&\frac{1}{12 v^2 \cos \beta \sin \beta}\left( \sqrt{6}\left(m_h^2 - m_H^2\right) \sin 2 \alpha + 12 m_3^2 \sin\beta \cos\beta - 3 \sqrt{2}v \cos\beta M_1 \right) \,, \label{lambda2}\\
\lambda_3 &=& \frac{1}{v^2 \sin^2 \beta}\left( m_5^2 - 3 m_3^2 \cos^2\beta + \sqrt{2}v \cos\beta \cot \beta M_1 - 3 \sqrt{2} v \sin\beta M_2 \right) \,, \label{lambda3} \\
\lambda_4 &=& \frac{1}{6 v^2 \sin^2 \beta}\Big( 2 m_H^2 \cos^2 \alpha + 2 m_h^2 \sin^2\alpha - 2 m_5^2
+6 \cos^2\beta m_3^2 - 3\sqrt{2} v \cos\beta \cot \beta M_1 \nonumber \\
&& + 9 \sqrt{2} v \sin \beta M_2 \Big) \,, \label{lambda4} \\
\lambda_5 &=& \frac{2 m_3^2}{v^2} -\frac{ \sqrt{2} M_1}{ v \sin \beta} \,. \label{lambda5}
\end{eqnarray}
\label{e:masstolam}
\end{subequations}
\subsection{Theoretical Constraint and Higgs data}\label{subsec:theocon}
In this section, we discuss the consequences of theoretical
constraints, such as the electroweak stability and the perturbative
unitarity on the GM scalar potential as well as the parameter constraints coming from the
latest Higgs data. We will thus use \Eqn{e:masstolam} to
translate the scalar couplings in terms of the physical masses and mixings.
\begin{figure}[ht!]
\centering
\includegraphics[width=8cm,height=6cm]{m3_vt_notri.png}
\includegraphics[width=8cm,height=6cm]{m3_vt_tri.png} \\
\includegraphics[width=8cm,height=6cm]{m3_m5_notri.png}
\includegraphics[width=8cm,height=6cm]{m3_m5_tri.png}
\caption{\em Allowed regions in the $v_t$-$m_3$ plane (upper panel) and in the $m_3$-$m_5$ plane (lower panel) from
theoretical constraints for zero (left panel) and non-zero (right panel) trilinear couplings.}
\label{f:theo}
\end{figure}
We first consider the theoretical constraints on the model parameter space. In Fig.~\ref{f:theo}, we show the region allowed simultaneously by the electroweak vacuum stability and
the perturbative unitarity bounds on the scalar quartic couplings~\cite{Aoki:2007ah,Hartling:2014zca}. For illustration purpose, we
have shown the allowed region in the $v_t$-$m_3$ plane and in the $m_3$-$m_5$ plane. In doing so, we choose two different
possibilities for the GM scalar potential, with and without the trilinear couplings. For nonzero trilinear couplings, we choose
$M_1=M_2=100~\rm GeV$ as a benchmark.
As is evident from the plots that the theoretical constraints
impose a lower bound on the triplet vev $v_t$ as well as an upper bound on the mass of the custodial triplet $m_3$ and custodial fiveplet $m_5$
when the scalar potential does not contain any trilinear terms, i.e. $M_1=M_2=0$. It also draws
a correlation between $m_3$ and $m_5$ as can be seen from the lower panel of Fig.~\ref{f:theo}.
These limits are actually a direct consequence of the unitarity condition on the scalar quartics as has been discussed in detail in Ref~\cite{Das:2018vkv}.
Briefly, the triplet is required to obtain a reasonable vev $v_t \gtrsim 27 \rm GeV$ to get a simultaneous solution of all the unitarity conditions.
The limits can however be lifted by the introduction of the non-zero trilinear terms in the potential as can be viewed from the right panel of Fig.~\ref{f:theo}.
The reason is the additional contribution of the nonzero trilinear couplings to the scalar quartic couplings, given in Eq.~(\ref{e:masstolam}),
that helped to alleviate the bounds on the masses and the vev $v_t$. Moreover,
with non-zero trilinear terms, the limit for small or vanishing $v_t$ can be achieved for large non standard scalar masses $m_3 \simeq m_5 \simeq m_H \gg m_h$ and
for $M_1 \simeq {\cal O}(v_t)$, this limit is often referred to as the {\it decoupling limit} \cite{Hartling:2014zca}.
\begin{figure}[ht]
\centering
\includegraphics[width=6cm,height=6cm]{higgs_data.png}
\caption{\em Model parameter space allowed in the ($\sin\alpha$-$v_t$) plane from the latest Higgs data (shaded blue)~\cite{CMS:2018lkl,ATLAS:2018doi} for a benchmark scenario $m_3(m_5)=300(400)$ {\rm GeV}
allowed by the theoretical constraints. The dot-dashed (red) and dashed (black) lines correspond to the bound coming from theoretical constraint for vanishing and non vanishing trilinear term
of the GM potential respectively.}
\label{f:hdata}
\end{figure}
Next, one should also consider the bounds on the model parameter space placed by the updated 13 TeV LHC Higgs data~\cite{CMS:2018lkl,ATLAS:2018doi}.
Here, it is to be mentioned that the lightest among all the CP-even states $(m_h)$ corresponds to the 125 GeV SM-like Higgs
in our model scenario. Therefore, its tree-level couplings to the SM fermions and vector bosons must be matched with the
LHC data. Additionally, the charged scalar particles coming from both the custodial multiplets of masses $m_3$ and $m_5$
contributes to the loop-induced Higgs to diphoton decay mode. Therefore, the observed limit from $h\to\gamma\gamma$
decay~\cite{CMS:2018lkl,ATLAS:2018doi} should also be considered for lighter charged Higgs masses. In fact, the combined limits from Higgs decay to
fermions, weak gauge bosons together with di-photon channel severely constraint the model parameter space as shown in Fig.~\ref{f:hdata}.
To implement the current limit, we define the Higgs coupling modifiers as the
ratio of the theoretical model prediction to the SM value, i.e,
\begin{eqnarray}
\kappa_i = \frac{g_{{hii}(GM)}}{g_{{hii}(SM)}}, \quad i=f,W^{\pm},Z,\gamma \,.
\end{eqnarray}
The model predictions for the coupling modifiers are well-known~\cite{Das:2018vkv} and it depend on the mixing angles
$\alpha$ and $\beta$ for a fixed $m_3$ and $m_5$.
In Fig.~\ref{f:hdata}, we show the allowed region in the
$(\sin \alpha-v_t)$ plane from the latest 13 TeV LHC Higgs result~\cite{CMS:2018lkl,ATLAS:2018doi}. The blue shaded region in the figure is allowed by all
Higgs channels including the loop induced diphoton mode. For the diphoton decay mode, we have fixed the
charged scalar masses at $m_3=300~\rm GeV$ and $m_5=400~\rm GeV$. The benchmark of such mass points are followed
by the maximum allowed value by the theoretical constraints, as is evident from Fig.~\ref{f:theo}. Now, in Fig.~\ref{f:hdata}, the solid(red) and dashed(black) lines
further corresponds to the equivalent theoretical limits on $v_t > 35 (10)~\rm GeV$ for the two cases, $M_1=M_2=0 (100~\rm GeV)$ respectively with $m_3=300~\rm GeV$ following the upper panel of Fig.\ref{f:theo}. Obviously, the limit is more relaxed in the non vanishing trilinear case, allowing $v_t$
as low as around 10 GeV. The most important point in Fig.~\ref{f:hdata}, is however, the region allowed
by the Higgs data. It shows a clear preference towards larger $\sin\alpha$ at large $v_t$ allowing only $\sin\alpha > 0.3$
for $v_t \gtrsim 30~\rm GeV$. This has notable consequences on the charged Higgs decay modes as we will explain in our subsequent analysis.
\section{Charged Higgs production and decay}\label{sec:ATLAS}
As mentioned earlier, the GM model contains two pairs of singly charged Higsses, one from each of the custodial triplet $(H_3^\pm)$ and the custodial
quintuplet $(H_5^\pm)$. However, the quintuplet component $H_5^\pm$ does not couple to SM quarks since it has a pure triplet origin, as can also be seen
from Eq.~(\ref{e:ch}). On the other hand, the triplet candidate $H_3^\pm$ can mix with SM quarks through doublet mixing. Therefore, in this article,
we will venture the possibility of limiting or probing the GM model from direct searches of the triplet charged candidate, namely, $H_3^\pm$.
To begin with, we show the decay branching ratio of this particle for various mass regions. In Fig.~\ref{f:h3br}, we depict the BR of $(H_3^\pm)$ of mass
$m_3$ for two different choices of $v_t = (30,45)~\rm GeV$ and each for four different values of $\sin\alpha = (0.001,0.01,0.1,1.0))$, represented by four different
color variations. It must be mentioned that our choice of $v_t$ relies on the parameter space available after satisfying the theoretical constraints, as described in Sec.~\ref{subsec:theocon}.
Although, the non-zero trilinear scenario allows much lower $v_t$,
the BRs has no dependence on
the parameter values $M_1$ and $M_2$. Therefore, in our following analysis, we will mainly focus on $v_t \gtrsim 30~\rm GeV$.
It should however, be mentioned that we have not imposed the Higgs data while plotting Figs. \ref{f:h3br}, \ref{f:h3tb} and \ref{f:h3wh}.
\begin{figure}[ht!]
\centering
\includegraphics[width=8cm,height=6cm]{h3_br_vt30.pdf} ~
\includegraphics[width=8cm,height=6cm]{h3_br_vt45.pdf}
\caption{\em The branching ratio of charged Higgs $H_3^\pm$ for two different choices of the triplet vev, $v_t = 30$ GeV (left panel) and $v_t = 45$ GeV (right panel).
The box and circular points define the two distinct decay modes of the charged scalar, $H_3^\pm \to tb$ and $H_3^\pm \to W^{\pm}h$ channel respectively while the
four different colors (red, green, magenta, blue) corresponds to four different orders of magnitude for $\sin\alpha$ (1,0.1,0.01,0.001). }
\label{f:h3br}
\end{figure}
Let us first briefly understand the nature of Fig.~\ref{f:h3br}. As one can see, $H_3^\pm$ has two dominant decay channels, $H_3^+ \to t \bar{b}$ and $H_3^+ \to W^+ h$ (similarly for the anti-particle), $h$ being the 125 GeV resonance.
It is clear from Fig.~\ref{f:h3br}, that as the mass of
$H_3$ reaches 205 GeV (threshold to produce $W^{\pm} h$), the $BR( H_3^{\pm} \rightarrow W^{\pm} h)$ increases considerably with an increase in $\sin\alpha$
reducing the $BR(H_3^{\pm} \rightarrow t b)$. This turns out to be the crucial point for the charged Higgs searches in this model. As we showed in Sec.~\ref{subsec:theocon} that the current Higgs data preferred larger $\sin\alpha$ for larger $v_t$. Hence, the combined limit of
theoretical constraints with the Higgs data points towards an inclination of a lighter charged Higgs mass $m_3 < 300~\rm GeV$ and a larger mixing angle in the CP-even sector. From this viewpoint and referring to Fig.~\ref{f:h3br}, one can interpret that probing the charged Higgs candidate in this model would be more probable in $W^\pm h$ channel instead of the quark final state.
Nevertheless, it is mandatory to validate the model parameter space first with the current limit on charged Higgs decaying to $tb$ final state, as reported by the ATLAS collaboration~\cite{Aaboud:2018cwk} with the 13 TeV data. There is a similar analysis by CMS collaboration lately~\cite{CMS:2019yat}, however there limits for low charged Higgs mass $m_3 < 500 ~\rm GeV$ are weaker and hence we will only consider the ATLAS bound.
In Fig.~\ref{f:h3tb}, we draw the limits on the charged Higgs production cross sec times its BR to $tb$ channel, as observed in experiment (further details on cross section measurement are given in section \ref{sec:collider}). The observed limit on the effective cross section from ATLAS for a range of charged Higgs masses has been shown for three distinct triplet vev $v_t = 30,~45~ \&~ 60~\rm GeV$ with two different $\sin\alpha \equiv (0.1,1)$ values. As anticipated, even for low charged Higgs masses
$m_3 \lesssim 300 ~\rm GeV$, the null result from observed data can only exclude $v_t > 45~\rm GeV$ for $\sin\alpha = 0.1$. The limit is much relaxed for higher mixing angles.
\begin{figure}[ht!]
\centering
\includegraphics[width=8cm,height=6cm]{h3_tb_sa_p1.pdf} ~
\includegraphics[width=8cm,height=6cm]{h3_tb_sa_1.pdf}
\caption{\em The production cross-section times the branching ratio for the charged Higgs decaying to $tb$ channel is shown for two different values of $\sin\alpha$,
0.1 (left panel) and 1.0 (right panel). The three colored lines, orange(dot-dashed), blue(dotted) and magenta(dashed) stands for three different values of triplet vev $v_t = 30,~45~ \&~ 60 $ GeV
respectively. The solid black line corresponds to the current limit from ATLAS analysis~\cite{Aaboud:2018cwk}.}
\label{f:h3tb}
\end{figure}
Following the above findings, in the next section we will carry out a simple signal-to-background (cut-based) analysis to predict the efficiency of probing the model parameter space still allowed by theoretical constraints and the present experimental observation made for both the Higgs and the charged Higgs.
\section{Analysis}\label{sec:collider}
In this section, we address the question that we posed in the last section, i.e. how efficiently we can probe the parameter space of GM model allowed by the experimental and
theoretical bounds together. Charged Higgs decaying into $W^{+}h$ channel is the signal that we will be interested in. The single production of the charged Higgs can however
be made either in the 4-flavor or in the 5-flavor scheme, where the production processes will be $p p \to t \bar b H_3^+$ and $p p \to t H_3^+$ respectively.
In our subsequent analysis, we will work in the 5-flavor scheme.
\begin{figure}[ht!]
\centering
\includegraphics[width=8cm,height=6cm]{ppth3_wh_p1.pdf} ~
\includegraphics[width=8cm,height=6cm]{ppth3_wh_1.pdf}
\caption{\em The production cross section times branching ratio plot for charged Higgs decaying to $W^\pm h$ channel for two values of $\sin\alpha =0.1$ (left panel)
and $\sin\alpha =1.0$ (right panel). The color coding is same as in Fig.~\ref{f:h3tb}.}
\label{f:h3wh}
\end{figure}
In Fig.~\ref{f:h3wh}, we show the production cross section times the BR for $\sin\alpha=1, 0.1$.
Following our previous line of argument, a direct comparison between Fig.~\ref{f:h3tb} and Fig.~\ref{f:h3wh} shows
that for larger $\sin\alpha$, the $W^{+}h$ decay mode for the charged higgs imparts larger effective cross-section.
Hence, we may expect to discover or restrict the model in its full parameter region with larger efficiency
in the future run of the LHC probing the specified channel. In the following, we will give a simple cut-based
analysis to give a prediction of probing the GM model at the future LHC run from the charged Higgs searches through
$W^{+}h$ final state.
Therefore, we consider the following signal channel:
\begin{eqnarray}\label{e:process}
p p \to t H^-_3 \to b W^+ W^- h \to 3b + 2\ell + E_T \hspace{-1.0em}/\;\: \,
\end{eqnarray}
where, we assume that the two $W$ bosons decays leptonically and the Higgs decay yields two b-tagged jets.
Thus as a final state we look for 3 b-tagged jets, two opposite signed leptons and missing transverse energy.
The irreducible SM background contribution will come from $t\bar t j$ with at least one extra hard jet, $t\bar t h$ and $t \bar t V, V={W^{\pm},Z}$ processes.
It should be noted here that the $t\bar t $ plus jets background has the largest cross section among all and in turn it is the most dominant
background in this analysis.
Below we present our cut-based analysis for the signal-to-background event ratio computation.
For our analysis, we choose two benchmark points in consistent with all the above limits, and the relevant parameter values are shown in Table.~\ref{tab:bp}.
In passing, we would like to mention that our benchmark value for $m_5$ is consistent with the latest bound from ATLAS collaboration~\cite{Aaboud:2018qcu}
where a doubly-charged Higgs mass between 200 and 220 GeV is excluded at 95\% CL from same-sign $W$ boson signal search.
The model has been implemented in {\tt FeynRules}~\cite{Alloul:2013bka}
to obtain the UFO model files required for event generation in madgraph.
We generate both the signal and SM backgrounds events at the Leading Order (LO)
in {\tt Madgraph5(v2.3.3)}~\cite{Alwall:2014hca} using the {\tt NNPDF3.0} parton distributions~\cite{Ball:2014uwa}.
The parton showering and hadronisation is done using the built-in {\tt Pythia}
\cite{Sjostrand:2006za} in the Madgraph.
The showered events are then passed through {\tt Delphes}(v3)~\cite{deFavereau:2013fsa}
for the detector simulation where the jets are constructed
using the anti-$K_{T}$ jet algorithm with minimum jet formation radius $\Delta R = 0.4$.
The isolated leptons are considered to be separated from the jets and other leptons by $\Delta R_{\ell i} \gtrsim 0.4, i =j,\ell$.
For the background processes with
hard jets, proper MLM matching scheme \cite{Hoche:2006ph} has been chosen.
\begin{table}
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
& $\sin\alpha$ & $v_t$ & $m_3$ & $m_5$ & $\sigma_{prod}$\\
& & (in GeV) & (in GeV) & (in GeV) & (in fb) \\
\hline
BP1 & 0.6 & 35 & 250 & 300 & 14.80\\ \hline
BP2 & 0.5 & 45 & 300 & 350 & 18.53\\
\hline
\end{tabular}
\caption{\it Benchmark points valid by all constraints and the corresponding production cross-section of our signal. }
\label{tab:bp}
\end{table}
To generate our signal and background events, we employ the following pre-selection cuts.
\begin{eqnarray}
p_T(j,b) &>& 30 ~{\rm GeV}\,; \quad |\eta(j)| < 4.7 \,; \quad |\eta(b)| < 2.5 \,, \nonumber \\
p_T(\ell) &>& 10 ~{\rm GeV}\,, \quad |\eta(\ell)| <2.5 \,.
\label{basic_cut}
\end{eqnarray}
The $b$-jets are tagged with the $p_T$-dependent $b$-tag efficiency following the medium criteria of Ref.~\cite{Sirunyan:2017ezt}
which has an average 75\% tagging efficiency of the $b$-jets with $50~{\rm GeV} < p_T < 200~{\rm GeV}$ and 1\% mis-tagging
efficiency for light jets.
Additionally, we propose the following selection cuts to dis-entangle the signal from the SM backgrounds that would
enhance the signal significance.
\begin{itemize}
\item {\bf Selecting jets and leptons:}To affirm that our signal has 3 $b$ jets, we select 3 $b$ jets with $p_T(b) >$ 30 GeV
while rejecting any fourth $b$ jet with $p_T >$ 35 GeV. In addition to this, we demand two opposite sign leptons with
$p_T(\ell) >$ 10 GeV.
\item {\bf Selecting $E_T \hspace{-1.0em}/\;\:$:} Both for signal and backgrounds, the missing energy($E_T \hspace{-1.0em}/\;\:$) comes from the
decay of $W^{\pm}$ boson. From the left panel of Fig.~\ref{fig:dist}, one can see that the distributions for signal
and backgrounds are peaking around similar $E_T \hspace{-1.0em}/\;\:$ value. However, the SM background has a large tail of the distribution.
We, thus, impose an upper bound of $E_T \hspace{-1.0em}/\;\: < 175$ GeV as an optimum cut to enhance the
signal over background ratio.
\item {\bf Higgs mass reconstruction:} In our signal, two $b$-tagged jets among the three are produced by the decay of the
Higgs. Therefore, the invariant mass of the two $b$-jets are expected to peak around 125 GeV. On the other hand,
only the $t\bar t h$ can yield similar peak structure for the $M_{b\bar b}$. The right panel of Fig.~\ref{fig:dist}
depicts the nature of $M_{b\bar b}$ distribution for both the signal and background. It is to be mentioned that
the pair of b-jets is selected after rejecting any third b-jet that has the minimum azimuthal angle $\phi$. This follows
from the argument that the third $b$-jet is a remnant of direct top decay in our signal and therefore expected to
lie closer to the beam direction than the other two. Following Fig.~\ref{fig:dist}, we select
$100 ~\rm GeV < M_{b\bar{b}} < 150 ~\rm GeV$ for significant background reduction. This cut turns out to be
the most effective to reduce the large $t\bar t j$ background events.
\end{itemize}
\begin{figure}[htbp!]
\centering
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[width=0.95\linewidth]{met.pdf}
\caption{}\label{fig:met}
\end{subfigure}
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[width=0.95\linewidth]{invmass.pdf}
\caption{}\label{fig:invmass}
\end{subfigure}
\caption{\textit{ Normalized distribution of the (left panel) Missing transverse energy $(E_T \hspace{-1.0em}/\;\:)$ and
(right panel) the invariant mass of $b\bar{b}$ pair
after the basic kinematical acceptance cuts(Eq.~\ref{basic_cut}) for the benchmark BP1.}}
\label{fig:dist}
\end{figure}
The cut-flow is presented in Table.~\ref{tab:signal} for our two chosen benchmarks (BP1, BP2) of Table.~\ref{tab:bp}. The signal significance is
defined as $\mathcal{S} = \frac{S}{\sqrt{S + B}}$, $S$ and $B$ being the number of signal and background events
after all the cuts respectively.
As is evident from Table.~\ref{tab:signal}, one can reach more than $5\sigma$ significance at the High Luminosity run of LHC (HL-LHC)
with 3000 $fb^{-1}$ luminosity and more than $3\sigma$ significance with only 1000 $fb^{-1}$ in the
specified parameter space of this model with this particular signal channel.
\begin{table}[ht!]
\centering
\footnotesize
\begin{tabular}{|p{3cm}|c|c|c|p{1.7cm}|}
\cline{2-5}
\multicolumn{1}{c|}{}& \multicolumn{3}{|c|}{Effective cross-section after the cut(fb)} & Significance reach at \\ \cline{1-4}
SM-background
& Preselection cuts &$E_T \hspace{-1.0em}/\;\: $ cut & Invmass cut & 3(1) $ab^{-1}$
\\ \cline{1-4}
$t\bar{t}+$jet &124.41 & 119.52 &27.17 & integrated \\ \cline{1-4}
$t\bar{t}h$ & 1.05 & 0.97 & 0.32 & luminosity \\ \cline{1-4}
$t\bar{t}W^{\pm}$ & 0.21 & 0.19 & 0.049 & at 14 TeV \\ \cline{1-4}
$t\bar{t}Z$ & 0.84 & 0.79 & 0.21 & LHC \\ \hline \hline
${\rm Total~SM~Background}$ & 126.51 & 121.49 & 27.75 & --\\ \cline{1-5} \hline
\multicolumn{1}{|c|}{$M_{H_{3^{\pm}}}=250 ~\rm GeV$}& 1.49 & 1.42 & 0.52 & 5.37 (3.10) \\ \hline
\multicolumn{1}{|c|}{$M_{H_{3^{\pm}}}=300 ~\rm GeV$}& 1.87 & 1.78 & 0.65 & 6.68 (3.86) \\ \hline
\end{tabular
\caption{\it No. of events obtained after each cut for both signal ($\ell^+ \ell^- + 3b + E_T \hspace{-1.0em}/\;\: $) and background and the significance obtained for $3 (1)~ ab^{-1}$ i$ $ntegrated luminosity at $\rm 14~TeV$ LHC.}
\label{tab:signal}
\end{table}
It is worth mentioning here that we did not consider the systematic uncertainty in background prediction while
calculating the signal significance. At the LHC Run-II, the current systematic uncertainty on our dominant SM background process $t\bar t + \rm {jet}$
can be considered to be around 10\% \cite{ATLAS:2018doi}. Inclusion of this uncertainty appropriately\cite{timmurphy.org} yield the
signal significance for our becnhmark points for $M_{H_{3^{\pm}}}=250(300) ~\rm GeV$ respectively as 5.16(2.98)$\sigma$ and 6.46(3.73)$\sigma$
at 3(1)$\rm {ab^{-1}}$ luminosity. Therefore the inclusion of uncertainty will only make $\sim$4\% deviation from our quoted significance reach given in Table.~\ref{tab:signal}.
In fact, even within a conservative approximation of 20\% systematic uncertainty, the significance reach for our first signal point remains around $\approx 5\sigma$.
Moreover, it should also be noted that the actual systematic uncertainty
with the future high luminosity run of LHC at $\rm 14~TeV$ is currently unknown.
Considering the future systematic uncertainty to be much lower than the current value \cite{PaganGriso:2642427},
our prediction on signal significance will not differ much from the realistic result and is useful for an preliminary understanding.
\section{Results}\label{sec:results}
Let us now present our findings in a comprehensive manner.
In Fig.~\ref{f:combined}, we show our complete results spanned in the model parameters $\{\sin\alpha-v_t\}$ plane for two particular choices of
$m_3$ and $m_5$ masses corresponding to our benchmark points in Table~\ref{tab:bp}. The region allowed by the latest LHC Higgs data~\cite{CMS:2018lkl,ATLAS:2018doi} and the
bounds from the theoretical constraints for the two distinct cases of trilinear terms follows the same color combinations as in Fig.~\ref{f:hdata}.
On top of it, we show the bounds imposed by the ATLAS analysis for 13 TeV data on charged Higgs searches in $tb$ channel~\cite{Aaboud:2018cwk} which is shown by the
gray shaded region bounded by the solid black line.
As can be seen from the Fig.~\ref{f:hdata}, this bound can only restrict a small parameter space in association with the Higgs data. Quantitatively,
for charged Higgs masses around 250 GeV, $v_t > 50 ~\rm GeV$ can only be discarded. For such $v_t$, the bound on $\sin\alpha$ though comes solely from the Higgs data.
As already mentioned, for smaller $v_t$, Higgs data only allows large $\sin\alpha$ which can only be probed by the other significant decay mode of the charged scalar to
$Wh$ channel. Hence, we present the $3\sigma$ and $5\sigma$ discovery reach at the HL-LHC with
3 $ab^{-1}$ luminosity from our proposed signal channel for charged Higgs decaying to $W^{\pm}~h$ mode, depicted as solid (orange) and dashed (green)
lines respectively. The areas above those lines shaded in Orange and Green describe the parameter
space that can be probed at the future HL-LHC with more than $3\sigma$ and $5\sigma$ signal significance respectively. To illustrate further, one can see that a
5$\sigma$ discovery is possible for $v_t \gtrsim $ 31(35) GeV for $m_3 =$250(300) GeV with 3 $ab^{-1}$ integrated luminosity.
In a nutshell, Fig.~\ref{f:combined}
shows the complete allowed parameter space of GM model from theoretical constraints, latest LHC Higgs data and charged Higgs searches altogether, while
simultaneously showing the future reach of charged Higgs searches for the unbounded parameter space.
\begin{figure}[ht!]
\centering
\includegraphics[width=8cm,height=6cm]{combined_250.png} ~
\includegraphics[width=8cm,height=6cm]{combined_300.png}
\caption{\em The combined results on the model parameter space in the $(\sin\alpha-v_t)$ plane for charged Higgs masses $m_3 = 250$ GeV (left panel) and
$m_3 = 300$ GeV (right panel). The color coding for Higgs data and theoretical constraint is
same as in Fig.~\ref{f:hdata}. The gray shaded region with black solid boundary corresponds to the current ATLAS limit from charged Higgs analysis~\cite{Aaboud:2018cwk}.
The Green(dashed) and Orange(Solid) lines are respectively the $5\sigma$ and $3\sigma$ signal significance reach at 14 TeV LHC with 3 $ab^{-1}$ luminosity
in the charged Higgs decaying to $Wh$ mode.}
\label{f:combined}
\end{figure}
Before concluding, we should remark on the limit imposed by the CMS doubly charged Higgs search analysis at 13 TeV LHC with 35 $\rm fb^{-1}$ luminosity \cite{Sirunyan:2017ret}.
The 95\% C.L. upper limit has been given on the GM model based on the searches made for Vector Boson fusion production of a doubly charged Higgs decaying into a pair of same-sign W boson.
For a 200-300 $\rm GeV$ doubly charged Higgs, the limit is translated as an exclusion of parameter space above $v_t > 30$ GeV at 95\% C.L. This already is quite in tension with the model scenario
with vanishing trilinear terms ($M_1=M_2=0$) where the theoretical bounds only allow $v_t >30$ GeV.
Therefore, following our prediction on charged Higgs decaying to $W h$ mode in the future high luminosity run, a non-observance of such a signal would definitely seal the fate of the model scenario for $M_1=M_2=0$.
\section{Summary }\label{sec:summary}
To summarise, we consider the GM model as a preferred variant of scalar triplet extension of the SM that preserves the custodial symmetry allowing for a larger
triplet vev unlike the usual HTMs. Such large triplet vev leads to interesting search prospects for the singly charged scalar of the model that can decay to SM fermion
with large coupling. The latest LHC search result on charged Higgs decaying to $tb$ channel thus impose an upper bound on the triplet vev. On the other hand,
the theoretical constraints on the model parameter restricts the triplet vev from the lower end and can be as strong as 30 GeV for vanishing triliinear term of the
GM potential. Moreover, the theoretical constraints entails an upper bound on the triplet and quintuplet masses which puts significant upper bound on
the singly and doubly charged Higgs mass. Additionally, the non-standard triplet scalars mixing will modify the SM-like Higgs coupling to the SM fermions and
gauge bosons. Therefore, it is mandatory to match the model prediction for the SM-like Higgs coupling with the latest LHC Higgs data.
In the first part of this paper, we show the complete limit on the model parameter space from both the theoretical constraints as well as the 13 TeV LHC Higgs results.
We show that these two constraints together put a stringent bound on the model parameter space in neutral mixing angle $\sin\alpha$ and triplet vev $v_t$ plane.
As shown, a large positive $\sin\alpha \gtrsim 0.1(0.3)$ with $v_t>10 (30)~\rm GeV$ is preferred by these bounds for the two distinct cases of vanishing and non-vanishing trilinear terms.
In the second part of our paper, we intended to combine these limits with the latest bound
from LHC on the charged scalar searches to $tb$ channel. We show that this limit excludes $v_t > 50$ GeV for $\sin\alpha \lesssim 0.4$ for a 250 GeV charged Higgs
in the region allowed by the theoretical constraints and the Higgs data.
We also described that for a larger $\sin\alpha$, the singly charged Higgs predominantly decays to $Wh$ channel.
Therefore, we showed a simple cut-based signal-to-background analysis for the singly charged Higgs decaying to $Wh$ mode. Our analysis revealed that this search channel can probe
the unbounded region of the model parameter by all the above-mentioned constraints at $5\sigma$ signal significance at the future high luminosity run of the LHC.
\section*{Acknowledgements}
IS would like to thank Michihisa Takeuchi and Satoshi Shirai for useful discussions.
The work of IS was supported by World Premier International Research Center Initiative (WPI), MEXT, Japan.
NG would like to acknowledge the Council of Scientific and Industrial Research (CSIR), Government of India
for financial support.
SG would like to thank the University Grants Commission, Government of India, for a research fellowship.
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