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latex-hard-2line-split

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	\begin{align}\left(\begin{array}{cc} \mu_1 & 0 \\ 0 & \mu_2 \\ \end{array} \right)\left(\begin{array}{cc} 0 & d_1 \\ d_2 & 0 \\ \end{array} \right)\left(\begin{array}{cc}\bar{\mu_1}^{-1} & 0 \\0 & \bar{\mu_2}^{-1} \\\end{array}\right) =\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} \right).\end{align}
	\begin{align}f_M=\left(\begin{array}{c}f^1\\\Lambda f^{1}.\end{array}\right).\end{align*}
	\begin{align}A\|_{I_\mu} \;=\; \left( \begin{array}{cccc} \mu & 1 & \cdots & 0 \\0 & \mu & \cdots & 0 \\ \vdots & \vdots & \ddots & 1 \\ 0 & 0 & \cdots & \mu \end{array} \right) \;.\end{align}
	\begin{align}\begin{array}{c}V(z_{12}^{\prime})=U^{-1}(z_{1};g)V(z_{12})U(z_{2};g)\,,\\{}\\V(z_{21}^{\prime})=U^{-1}(z_{2};g)V(z_{21})U(z_{1};g)\,.\end{array}\end{align}
	\begin{align}\begin{array}{rcl}{\cal R}(u,c)&=&\displaystyle\frac{u-icP}{u+ic}\\begin{align}4mm] &=&\displaystyle\frac{1}{u+ic}\left[ \begin{array}{cccc} u-ic& 0 & 0 & 0 \\ 0 & u &-ic& 0 \\ 0 &-ic& u & 0 \\ 0 & 0 & 0 &u-ic \end{array}\right]~,\end{array}\end{align*}
	\begin{align}R(u_1 -u_2 ) t^{c}_{b}(u_1 )\otimes t^{b}_{a}(u_2 )=\sum_{d} W\left( \left. \begin{array}{cc} c & d \\ b & a \end{array} \right\| u_1 -u_2 \right) t^{d}_{a}(u_1 )\otimes t^{c}_{d}(u_2 ). \end{align}
	\begin{align}\begin{array}{lcr}\sum\limits_{j}q^j_a(\|X_j\|^2-\|Y_j\|^2)\eta_j^+\overline{\eta}_j^+&=r^2_a\eta_a^+\overline{\eta}_a^+ \qquad &(a)\\\sum\limits_{j}q^j_a(X_j\overline{Y}_j)\eta_j^+{\eta}_j^+&=0 \qquad &(b)\\\sum\limits_{j}q^j_a(\overline {X}_j{Y_j})\overline{\eta}_j^+\overline{\eta}_j^+&=0. \qquad &(c)\end{array}\end{align}
	\begin{align} \mathcal{S}_{N} = \begin{cases} \{(G, N)\}, & {\rm if} ~G/N~{\rm cyclic;}\\ \emptyset, & {\rm otherwise.} \end{cases} \end{align}
	\begin{align} d_{i}:= \begin{cases}1,~{\rm if} -1 \in \langle r_{i}\rangle; \\ 2, ~{\rm otherwise}.\end{cases}\end{align}
	\begin{align}\begin{array}{r c l}\frac{df}{dt}&=&\frac{\partial f}{\partial t}+\{f,H_T\}\\ & & \\&=&\frac{\partial f}{\partial t}+\{f,H_T\}_D+\{f,\chi_s\}\left(\Delta^{-1}\right)^{ss'}\{\chi_{s'},H_T\}\approx\frac{\partial f}{\partial t}+\{f,H_T\}_D\ .\end{array}\end{align}
	\begin{align}F=\left(\begin{array}{ccccc}0 & 0 & 0 & 0 & 0 \\0 & 0 & f & 0 & 0 \\0 & -f & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0\end{array}\right)\end{align}
	\begin{align}\langle g_{xy} \rangle =\left(\begin{array}{cccc}b & 0 & 0 & 0\\0& b &0&0\\0&0&\langle g_{33} \rangle &\langle g_{34}\rangle\\0&0&\langle g_{43} \rangle &\langle g_{44} \rangle\end{array}\right)\end{align}
	\begin{align}\epsilon=\left\{\begin{array}{ll} 1&\mbox{if $\omega=\pi/2$}\;,\\-1&\mbox{if $\omega=3\pi/2$}\;.\end{array}\right.\end{align}
	\begin{align} \begin{aligned}A_N &= A_1(w_{N-1})\;\! A_{N-1} = \prod_{i=0}^{N-1} A_1(w_{N-i-1}) \\D_N &= D_1(w_{N-1})\;\! D_{N-1} = \prod_{i=0}^{N-1} D_1(w_{N-i-1}) \\begin{align}0.3em]C_N &= C_1(w_{N-1})\;\! A_{N-1} + D_1(w_{N-1})\;\! C_{N-1} \\&= \sum_{i=0}^{N-1} D_i(w_{N-1-i})\, C_1(w_{N-1-i}) \, A_{N-1-i} . \\\end{aligned} \end{align*}
	\begin{align}\left( \begin{array}{cc} u_{0} m & u_{0} b \\ u_{0} b^{\dag} & u_{0} n \end{array} \right)\end{align}
	\begin{align}u^{35,35}_i(y,z)=\left\{\begin{array}{rrrl}- 5 b_i^{35,35}(y,z) \ & \ \ \ \ \ if& \ \ \ \ \ \ \ \ \ \ \ \ \|5 b_i^{35,35}(y,z)\|\leq 1, \\-1 \ & \ \ \ \ \ if& \ \ \ \ \ \ \ \ \ \ \ \ \ -5 b_i^{35,35}(y,z) < -1,\\1 \ & \ \ \ \ \ if& \ \ \ \ \ \ \ \ \ \ \ \ \-5 b_i^{35,35}(y,z) > 1,\\\end{array}\right\},\ \ \ i=1,2,\end{align}
	\begin{align}\begin{pmatrix}A^{-1}_N & {\bf 0} \\begin{align}0.4em]- D^{-1}_N \;\! C_N \;\! A_N^{-1} & D^{-1}_N\end{pmatrix}\begin{pmatrix}u \\1\end{pmatrix}\end{align*}
	\begin{align}\tilde\partial_{i}(g_{0}, \dots, g_{n}) = \begin{cases}(g_{0}, \dots, g_{i-1}, g_{i}g_{i+1}, g_{i+2}, \dots, g_{p}) & i=0, \dots p-1, \\(g_{0}, \dots, g_{p-1}) & i=p.\end{cases}\end{align}
	\begin{align}U = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right)~.\end{align}
	\begin{align}g^{\mu \nu} = \left( \begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right) \ , \qquad\epsilon^{\mu \nu} = \left( \begin{array}{rr} 0 & -1 \\ 1 & 0 \end{array}\right)\end{align}
	\begin{align}\left(\begin{array}{cc}\hat h+\Sigma & \beta \Delta ^{\dagger }\beta C \\-C\,\Delta & \hat h+\Sigma\end{array}\right) \left(\begin{array}{c}u \\v\end{array}\right) =E\left(\begin{array}{c}u \\v\end{array}\right) \end{align}
	\begin{align}f(x)=\begin{cases} \frac{3}{2} x+\frac{3}{2} & x < -1\\ 0 & -1 \leqslant x \leqslant 1\\ \frac{3}{2} x-\frac{3}{2} & x > 1\end{cases}\end{align}
	\begin{align}u_{0,l}(r)=\left( \begin{array}{l}\phi _{0,l,1}(r) \\ 0\end{array}\right) ,{\rm \;}{}l\geq 1;\;\;\;u_{0}^{II}(r)=\left( \begin{array}{l}\phi _{0,1}^{ir}(r) \\ 0\end{array}\right) ,\;{}l=0. \end{align}
	\begin{align}\begin{array}{lcl}{\cal L}_A^{2n+1} a &=& e^{i\phi} \rho^{2n+1}a^\dagger \\{\cal L}_A^{2n} a &=& \rho^{2n}a \\ \end{array}\end{align}
	\begin{align}\widetilde{A} = \left(\begin{array}{cc} A & 0 \\ 0 & 0\end{array}\right).\end{align}
	\begin{align}\begin{array}{lcr}D^{++}=u^{+ \alpha}{\partial\over{\partial u^{-\alpha}}};\qquad D^{--}=u^{- \alpha}{\partial\over{\partial u^{+\alpha}}}\\2D^{++}=[D^0,D^{++}];\qquad -2D^{--}=[D^0,D^{--}]\\D^0=[D^{++},D^{--}]=u^{+ \alpha}{\partial\over{\partial u^{+\alpha}}}-u^{- \alpha}{\partial\over{\partial u^{-\alpha}}}\end{array}\end{align}
	\begin{align}\left\{\begin{array}{lcl}\Sigma &=& r^2 + a^2\cos^2\theta\\\Delta &=& r^2 + a^2 + Q^2 - 2Mr\;. \end{array}\right.\end{align}
	\begin{align} z_t = \begin{pmatrix} {x}_t \\ {y}_t \end{pmatrix} =U_t({T},{D})\, \begin{pmatrix} {x}_1^\prime \\ {y}_1^\prime \end{pmatrix}- {D}\, U_{t-1}({T},{D})\, \begin{pmatrix} {x}_0^\prime \\ {y}_0^\prime \end{pmatrix}+ \begin{pmatrix} {x}^ \\ {y}^* \end{pmatrix}\end{align*}
	\begin{align}\begin{array}{l}\lbrack q^{-},\alpha _0^{+}]=i\,, \\\left[ \alpha _n^{-}\,,\alpha _m^{+}\right] =n\,\delta _{n+m,0}\,\,\,, \\\left[ s_n\,,s_m\right] =\left( \frac k2-1\right) \,n\,\delta _{n+m,0}\,\,\,,\\\left[ L_n^{\prime }\,,L_m^{\prime }\right] =(n-m)L_{n+m}^{\prime }\end{array}\end{align}
	\begin{align}B=\left\|\begin{array}{cc}\alpha & \alpha +s \\-\alpha +s & -\alpha\end{array}\right\| ,\end{align}
	\begin{align}\begin{array}{c}a^{A}=a^{A}~~~~b^{A}=b^{A}~~~~\omega_{AB}{}^{}=\omega_{AB}~~~~\lambda^{}=\lambda~~~~T^{\dagger}=-T\\{}\\\bar{\rho}_{i}=\rho^{i\dagger}\gamma^{0}~~~~~~~\bar{\varepsilon}_{i}=\varepsilon^{i\dagger}\tilde{\gamma}^{0}\end{array}\end{align}
	\begin{align}[h_{j,j+1}, S^{\pm}] = [h_{j,j+1}, S^{z}] = 0\ \ \\ \forall j = 1,\ldots ,N-1 \end{align}
	\begin{align}= \{\chi^{\mu\nu}\} +\left(\begin{array}{cccc}((p^2 - q^2)a + 2pqb) & (pa + qb) & (pb -qa) & ((p^2 - q^2)a + 2pqb) \\(pa + qb) &0 & 0 & (pa + qb) \\(pb -qa) & 0 & 0 & (pb -qa) \\((p^2 - q^2)a + 2pqb) &( pa + qb) &( pb -qa) & ((p^2 - q^2)a + 2pqb)\end{array}\right).\end{align}
	\begin{align} C \, = \, \begin{pmatrix} T & -D \\ 1 & 0 \end{pmatrix}.\end{align}
	\begin{align}\begin{array}{l}a_{ij}^{{\bf 4}}= \frac{1}{g}\sum\limits_{\gamma=1}^{r}r_{\gamma }\chi_{\gamma }^{R_M \oplus {\bf 1}^2 }\chi_{\gamma }^{(i)}\chi_{\gamma }^{(j)} \\a_{ij}^{{\bf 6}}=\frac{1}{g}\sum\limits_{\gamma=1}^{r}r_{\gamma }\chi_{\gamma }^{R_M \oplus R_M \oplus {\bf 1}^2 }\chi_{\gamma }^{(i)}\chi_{\gamma }^{(j)} \\\end{array}\end{align}
	\begin{align}f(x)=\begin{cases} a_1x & x < 0\\ a_2 x & x \geqslant 0.\end{cases}\end{align}
	\begin{align}\Bigl(X_{AB}^{ab}(x,y)\Bigr)=\left(\begin{array}{cc}0&-\,m^2\,\delta^{ab}\\\delta^{ab}&\frac{1}{2}\,g\,f^{abc}\,(D_i\pi^i)^c\end{array}\right)\,\delta(x-y)\,.\end{align}
	\begin{align}F_{k}(x,1-x)=\left\{\begin{array}{l}\left[x(1-x)\right]^{\beta+k}\\\left[x(1-x)\right]^{\beta+k}(2x-1)\ ,\end{array}\right.\end{align}
	\begin{align}Z^{IJ} = \left(\begin{array}{cc}Z_1 \epsilon & 0 \\ 0 &Z_2\epsilon\end{array}\right) \ , \quad {\rm where}\quad \epsilon = \left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right).\end{align}
	\begin{align}(\Psi_l,\Psi_l) = \|U_{\rho_l}(k_0)\|^{-1}&\int_{U_{\rho_l}(k_0)} dk \int_{F_k^n} \Big\|\phi\Big(k;p_1,\ldots,p_n \Big)\Big\|^2\,dp\\& \times \int_{F_{K-k}^{N-n}}\Big\|\psi\Big(K-k;p_{n+1},\ldots,p_N\Big)\Big\|^2\,dp=1.\end{align}
	\begin{align}\left\{ \begin{array}{rcl}3F^{2}(A) +F^{2}(B) & = & 0\, ,\\& & \\\sqrt{3}F(A)-s(\alpha){}^{\star}F(B) & = & 0\, .\\ \end{array}\right.\end{align}
	\begin{align}& A_{i}^{\top}\lambda^{}\in\partial f_{i}(x_{i}^{}), i=1,2,\cdots,N, \\& \sum\limits_{i=1}^{N}A_{i}x_{i}^{}-b=0. \end{align*}
	\begin{align}\delta^{(1)} S = - \frac{1}{12} \sum_{\langle ijk \rangle} \bar{\epsilon}_1 \Gamma_{\mu} \theta_{[ \, i} \bar{\theta}_j \Gamma^{\mu} \theta_{k \, ]}. \\\end{align}
	\begin{align}{\Phi}=\frac{1}{2}\left(\begin{array}{c}-A_{Ij}E^{j}_{\nu}\\D_{ij}E^{j}_{\nu}\\E^{i}_{\nu}\\\end{array}\right)={\frac{1}{2}}\left(\begin{array}{ccc}{\cal E}_{I}^{M}&0&0\\0&T_{i}^{\mu}&0\\0&0&T_{i}^{\mu}\\\end{array}\right){\hat \Phi}\end{align}
	\begin{align}-\div{\lvert \nabla u \rvert^{p-2} \nabla u} = f \;\; \Omega, \\ u-g \in W^1_0(L_p(\Omega)), \end{align}
	\begin{align}R\to \left\{\begin{array}{ll}\frac\zeta{\sqrt{2n^3}}, & (r\to\infty) \\ \frac\zeta{\sqrt{2n'n^2}}. & (r= 0)\end{array}\right.\end{align}
	\begin{align}\left( \begin{array}{c} R\\ G\\ B \end{array} \right) \; \; \; , \; \; \;j\left( \begin{array}{c} R\\ G\\ B \end{array} \right) \; \; \; \; \; .\end{align}
	\begin{align}\tau \rightarrow \frac{a\tau +b}{c\tau +d} \;\; , \;\;\;\;\; \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \in Sl(2;Z)\end{align}
	\begin{align} L_i^{(x)}(u)=\left(\begin{array}{cc} x_i & 1 \\ x_iX_i+\beta_i+\lambda & X_i \end{array}\right )\;\;.\end{align}
	\begin{align}\begin{array}{c}\partial_m\psi_1=\partial_m\psi_2=0\\\psi_0=\psi_{12}=0\end{array}\end{align}
	\begin{align}u \in B^{\sigma}_{\tau}(L_{\tau}(\Omega)) 0 < \sigma < \begin{cases} 2 & 1<p \leq 2,\\1 + \frac{1}{p-1} & 2 < p < \infty\\\end{cases} \frac{1}{\tau} = \frac{\sigma}{2} + \frac{1}{p}.\end{align}
	\begin{align}\mathcal{M}^{ab}=\left( \begin{array}{ll}f^{abi}\varphi ^{i} & \delta ^{ab}\partial ^{2} \\ -\delta ^{ab}\partial ^{2} & -f^{abi}\overline{\varphi }^{i}\end{array}\right) \;. \end{align}
	\begin{align}(VII)' \hspace{7mm} \Psi_{\bf D} =\left(\begin{array}{c}a_1 +{\rm j}b_1 \\a_2 +{\rm j}b_2 \\a_3 +{\rm j}b_3 \\a_4 +{\rm j}b_4\end{array}\right) \hspace{5mm} \leftrightarrow \hspace{5mm}\Psi^{SR}_{\bf C} =\left(\begin{array}{c}-b_1+{\rm i}b_3 \\-b_2+{\rm i}b_4 \\-a_1+{\rm i}a_3 \\-a_2+{\rm i}a_4\end{array}\right)\end{align}
	\begin{align}\Psi\equiv\left(\begin{array}{c}\psi_1\\\psi_2\end{array}\right)=\left(\begin{array}{c}\gamma_5\psi_2^\\-\gamma_5\psi_1^\end{array}\right)\, ,\end{align}
	\begin{align} 0&=-v_t - (\mathbf{L}^r +\mathbf{L}^Y +\mathbf{L}^{S})v - r x v_x \\ & -\sup_{\theta} \left\{\theta (\mu\!-\!r)v_x \!+\!\theta \sigma^2 y s^{2 \beta-1} v_{xs} \!+\!\frac12 \theta^2\sigma^2 y s^{2\beta-2} \right\} \end{align}
	\begin{align} 0&=-v_t - (\mathbf{L}^Y +\mathbf{L}^{S})v \\ & -\sup_{\theta} \left\{\theta \cdot \mu v_x \!+\!\theta \sigma ^2 y s^{2 \beta-1} v_{xs} \!+\!\frac12(\theta^2\sigma^2 y s^{2\beta-2} \right\} \end{align}
	\begin{align} \begin{array}{c} h^{m(s)}=\varphi^{m(s)} + \eta^{m(2)}\,\chi^{m(s-2)}\;;\;{\varphi_l}^{lm(s-2)} =0, \\ \epsilon^{m(s-1)}=\varepsilon^{m(s)} + \eta^{m(2)}\, \zeta^{m(s-3)}\;;\;{\varepsilon_l}^{lm(s-3)} =0, \end{array}\end{align}
	\begin{align}\begin{array}{rcl}S_{{\rm IIB}} & = & -T_{M_2}l\int d^2\xi\sqrt{\|{\tilde g}_{ij}-\frac{1}{{\tilde k}^2}(\partial_i \varrho+{\tilde B}^{(1)}_i)(\partial_j \varrho+{\tilde B}^{(1)}_j)\|}\\& & \\& &-\frac{T_{M_2}l}{2}\int d^2\xi\epsilon^{ij}\left[\tilde{B}^{(2)}{}_{ij}+2\partial_{i}\varrho\ \left({\tilde A}^{(1)}{}_{i}+m\pi\alpha^{\prime} b_{j}\right)\right]\, .\\\end{array}\end{align}
	\begin{align}\delta A_{\mu\nu} = \epsilon^_\nu \psi_\mu - \epsilon_\mu \psi^_\nu\\ \delta \psi_\mu = i \epsilon_\nu A_{\mu\nu}\\ \delta \psi^_\mu = i \epsilon^_\nu A_{\nu\mu}\\\end{align}
	\begin{align}\begin{array}{c} F_{09} = F_{0i} = F_{9i} = 0, \\\end{array}\end{align}
	\begin{align} \frac{1}{\beta_L}\mathbb{E}\bigg(G^{\Lambda}(z_L;n,n)\bigg) &\leq \frac{1}{Imz}~\pi\bigg(1+\frac{k}{2}\bigg)S_\mu\bigg(\frac{2~Imz_L}{k}\bigg),~~\Lambda=C_p,~\Lambda_L\\&\leq C ~\bigg(\frac{2~\beta_L^{-1}~Imz}{k}\bigg)^{\alpha}\\&\leq C ~(2L+1)^{-d},~~(\mathrm{since} ~~ \beta_L=(2L+1)^{d/\alpha}).\end{align}
	\begin{align} \lim_{n\to\infty}\mathbb{E}\bigg(\sum_p^{N_{L_n}^d}\eta^{\omega}_{p,E}(I)\bigg) =\lim_{n\to\infty}\mathbb{E}\big(\zeta^{\omega}_{L_n,E}(I)\big) =\gamma_{E,I}\\\end{align}
	\begin{align}H^0 = \left(\begin{array}{cc}-\partial_z^2+4 & 0 \\0 & -\partial_z^2 +4\end{array}\right),\end{align}
	\begin{align}L_{i_2}^{(2)}(x):=\left\{\begin{array}{llll}L_i^{(0)},~~~\ i_1=1,...7,\\\frac{d}{dt}(\lambda^{(0) A}_iL^{(0)}_i),~~~i_2 = 7 +A ,~A=1,2,3, \\\frac{d}{dt} ( \Omega_4^{(0)} ), ~~~i_2 = 11, \\\frac{d}{dt} ( \lambda_{i_1}^{(1) B} L_{i_1}^{(1)}),~~~i_2 = 11+B , ~B=1,2.\end{array}\right.\end{align}
	\begin{align}2\left(\begin{array}{cc}0&\partial_z\\\partial_{\bar{z}}&0\end{array}\right)\left(\begin{array}{c}\;^0\chi_1(z,\bar{z})\\\;^0\chi_2(z,\bar{z})\end{array}\right)=0 .\end{align}
	\begin{align}\left (\begin{array}{cc} A & B \\ C & D \end{array} \right)^{-1} =\left (\begin{array}{cc} (A - BD^{-1}C)^{-1} & -A^{-1}B (D - CA^{-1}B)^{-1}\\ -D^{-1}C (A - BD^{-1}C)^{-1} & (D - CA^{-1}B)^{-1} \end{array} \right).\end{align}
	\begin{align} M_{NSM}^2 =g^2 v^2 \left(\begin{array}{ccc} 2 & +1 & +1 \\ +1 & 2 & +1 \\ +1 & +1 & 2 \\ \end{array} \right)~.~\,\end{align}
	\begin{align} (x,g,y)(y,h,z)&=(x,gh,z) \\ (x,g,y)^{-1}&=(y,g^{-1},x).\end{align}
	\begin{align}\gamma_{ab}=\left(\begin{array}{cc}\rho_{[a}\rho_{b]}^{T}&0\\ 0&\rho_{[a}^{T}\rho_{b]}\end{array}\right)\,.\end{align}
	\begin{align}\left(\begin{array}{ccc} \gamma^{00}&\gamma^{01}&\gamma^{02}\\ \gamma^{10}&\gamma^{11}&\gamma^{12}\\ \gamma^{20}&\gamma^{21}&\gamma^{22} \end{array}\right)= \left(\begin{array}{ccc} \eta&\eta^\prime&\eta^\prime\\ \eta^\prime&\eta&\eta^\prime\\ \eta^\prime&\eta^\prime&\eta \end{array}\right)\end{align}
	\begin{align} E_{g}^{2} &=V_{b}^{}E_{a}V_{b}V_{b}^{}E_{a}V_{b}\\ & = V_{b}^{}E_{a}E_{b}E_{a}V_{b} \\ &=V_{b}^{}E_{b}E_{a}^{2}V_{b} ~~~\textrm{( Since $E_{a}$ and $E_{b}$ commute)} \\ &=V_{b}^{}E_{a}V_{b} \\ &=E_{g}.\end{align*}
	\begin{align}\begin{array}{ccccc} & SO(2n) & \subset & SO(2n+1) & R_M = {\bf (2n+1)} \\ & SU(3) & \subset & G_2 & R_M = {\bf 7} \\\tilde{R}_M = {\bf 9} & SO(9) & \subset & F_4 & R_M = {\bf 26} \end{array} \quad .\end{align}
	\begin{align} E_{g_{1}}E_{g_{2}}&=V_{b_{1}^{'}}^{}E_{a_{1}^{'}}V_{b_{1}^{'}}V_{b_{1}^{'}}^{}E_{a_{2}^{'}}V_{b_{1}^{'}} \\ & = V_{b_{1}^{'}}^{}E_{a_{2}^{'}}V_{b_{1}^{'}}V_{b_{1}^{'}}^{}E_{a_{1}^{'}}V_{b_{1}^{'}} \\ & = E_{g_{2}}E_{g_{1}}.\end{align}
	\begin{align}E_{g_{1}}&=V_{c}^{}V_{ab}V_{ab}^{}V_{c} \\ &= V_{c}^{}V_{b}(V_{a}V_{a}^{})V_{b}^{}V_{c} \\ & \leq V_{c}^{}V_{b}V_{b}^{}V_{c} \\ & \leq E_{g_{2}}.\end{align*}
	\begin{align}r = \left\{ \begin{array}{cc} -(q^2)^\alpha & (-1<r<0) \\ (q^2)^\alpha & (0<r<1) \end{array} \right., \qquad q = -e^{-\pi K'/K}.\end{align}
	\begin{align}W_{g}W_{h}&=V_{b_1}^{}V_{a_1}V_{a_1}^{}V_{c_1} \\ &=V_{b_1}^{}E_{a_1}V_{b_{1}}V_{b_{1}}^{}V_{c_1} (\textrm{~Since $E_{a_1}$ and $E_{b_1}$ commute}) \\ &=E_{g}W_{hg}\end{align}
	\begin{align}E(j) = \left\{\begin{array}{ll}0 & \mbox{if the path has an extremum at ($x$-position) $j$} \\\frac{1}{2} j\qquad & \mbox{otherwise.}\end{array}\right.\end{align}
	\begin{align} (\widetilde{\chi}a)(T)&=\widetilde{\chi}(V_{a}^{}TV_{a}) \\ &=\chi(V_{a}V_{a}^{}TV_{a}V_{a}^{})\\ &= \chi(E_{a})\chi(T)\chi(E_{a}) \\ &= 1_{A}(a)\chi(T)1_{A}(a) \\ & = \chi(T).\end{align*}
	\begin{align}\Psi = (1-{\rho \over 4} x^+ x^-)^{1 \over 2} \left( \begin{array}{c} e^{-{\theta \over 2}} e^{\imath v}p(x^-) \\ e^{{\theta \over 2}} e^{\imath u}f(x^+) \\ \end{array} \right)\end{align}
	\begin{align} \lim \inf A_{n} & = \{ x \in X: \lim \sup d(x,A_{n}) = 0 \}, \textrm{~and~} \\ \lim\sup A_{n} & = \{x \in X: \lim \inf d(x,A_{n})=0 \}.\end{align}
	\begin{align}M_{1}= \left( \begin{array}{cccc} 1 & 2 & 0 & 2 \\ -2 & -2 & -2 & -1 \\ 1 & 0 & 1 & 0 \\ 2 & 1 & 2 & 0 \end{array} \right) \, .\end{align}
	\begin{align} \phi_{n}(1)&= \chi \Big(\int_{b \in supp(f_{n})}f_{n}(b)E_{b}db \Big) \\ & = \int_{b \in supp(f_{n})}f_{n}(b)1_{A}(b)db \\ & = \int_{b \in supp(f_{n})}f_{n}(b)db ~ (\textrm{ Since $supp(f_{n}) \subset Int(A)$}) \\ & =1.\end{align}
	\begin{align}& \|\phi_{n}(\alpha_{a}(E_{\psi}))-\chi(E_{\psi})\| \\&= \| \phi_{n}(\alpha_{a}(E_{\psi}))-\chi(E_{f_{n}}E_{\psi})\|\\& \leq \int_{b \in U_{n}}f_{n}(b)\Big(\int \|\Delta(b^{-1}a)^{m}\psi(g_{1}b^{-1}a,\cdots,g_{m}b^{-1}a)-\psi(g_{1},g_{2},\cdots,g_{m})\| dg_{1} \cdots dg_{m} \Big)db \\ & \leq \epsilon \int _{b \in U_{n}}f_{n}(b) \\& \leq \epsilon.\end{align}
	\begin{align}\#_{n}=\left\{\begin{array}{ll}1~~~~&\mbox{for}~n=3\,,\\{}&{}\\4n^{2}-12n~~~~~~~&\mbox{for}~n\geq 4\,.\end{array}\right.\end{align}
	\begin{align}< \pi(F)\xi,\xi> &= \int F(x) d\mu_{\xi,\xi} (x)\\ & = \int f(g)1_{Xg}(x)dg d \mu_{\xi,\xi}(x) \\ & = \int f(g) \Big (\int 1_{Xg}(x)d\mu_{\xi,\xi}(x) \Big) dg \\ & = \int f(g) < \pi(1_{Xg\cap X})\xi,\xi> dg \\ &= \int f(g)<E_{g}\xi,\xi> dg. \end{align}
	\begin{align}L_1=\left( \begin{array}{ccccc}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0\end{array}\right) ,\,\,\,L_2=\left( \begin{array}{ccccc}0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0\end{array}\right)\end{align}
	\begin{align}\begin{array}{ll} [h_i,h_j]=0,& [e_i,f_j]=\delta_{ij}h_i, \\\mbox{$[$} h_{i},e_{j}\mbox{$]$}= A_{ij}e_{j}, &\mbox{$[$} h_{i},f_{j}\mbox{$]$}=-A_{ij}f_{j},\end{array}\end{align}
	\begin{align}(\Gamma^{0})^{\underline\alpha}_{~~\underline\beta}= \left(\begin{array}{cc}0 & -1\\1 & 0\end{array}\right)\,, \quad(\Gamma^{1})^{\underline\alpha}_{~~\underline\beta}= \left(\begin{array}{cc}-1 & 0\\0 & 1\end{array}\right)\, , \quad \Gamma^2=\Gamma^0\Gamma^1=\left(\begin{array}{cc}0 & -1\\-1 & 0\end{array}\right) \,.\end{align}
	\begin{align}\left\{ \begin{array}{l}\partial ^{\mu }V_{\mu }\left( x\right) =0 \\ \partial ^{\mu }A_{\mu }\left( x\right) =2miP\left( x\right) .\end{array}\right.\end{align}
	\begin{align}\begin{array}{ll}L^{-1}(T,X,Y,Z)=(\cosh t\cos\frac{r_{+}-r_{-}}{2},&\sinh t\cos\frac{r_{+}+r_{-}}{2}, \\& \sinh t\sin\frac{r_{+}+r_{-}}{2},-\cosh t\sin\frac{r_{+}-r_{-}}{2}).\end{array}\end{align}
	\begin{align}\partial_{0} J_{1} - \partial_{x} J_{0} = \frac{1}{2}(\partial_{0} L_{1} \\ - \partial_{x} L_{0}) + \frac{1}{2} (\partial_{0} L_{0} -\partial_{x} L_{1})g + \\ \frac{1}{2} (L_{0}^{2} - L_{1}^{2} )g \;\;\; ,\end{align}
	\begin{align} v(\psi\cdot g_{uuv}) &=g_v\cdot g_{uuv}=(g_v\cdot g_{uv})_u-g_{uv}\cdot g_{uv} =\frac{(g_v\cdot g_{v})_{uu}}2-g_{uv}\cdot g_{uv} \\ &=\frac{(f_v\cdot f_{v})_{uu}}2-g_{uv}\cdot g_{uv} =v^2\frac{(\phi\cdot \phi)_{uu}}2-v^2 \psi_{u}\cdot \psi_{u}, \end{align}
	\begin{align}\begin{array}{c}G_{T}(z^{\prime\prime})=G_{T}(z^{\prime\prime}_{1})G_{T}(z_{s}(z^{\prime};u))\,,\\{}\\G_{T}(z^{\prime})=G_{T}(z_{1})^{-1}G_{T}(z)\,,\end{array}\end{align}
	\begin{align}\|\| T - T.O.E. \|\| = \left( \begin{array}{c} \mbox{\# of unexplained} \\ \mbox{bits of information} \end{array} \right)\end{align}
	\begin{align}(c^{\pm}(\alpha),c^{\pm}(\beta))=\left\{\begin{array}{ccc}(\cosh\frac{\alpha_{\pm}}{2},\cosh\frac{\beta_{\pm}}{2})&for&{\cal M}_{S}^{(\pm)}\\(\quad1\quad,\quad1\quad)&for&{\cal M}_{N}^{(\pm)}\\(\cos\frac{\rho_{\pm}}{2},\cos\frac{\sigma_{\pm}}{2})&for&{\cal M}_{T}^{(\pm)}.\end{array}\right. \end{align}
	\begin{align} \begin{alignedat}{2} g(u,0)&=\sigma(u), &&r(u,0)=\dot\sigma(u),\\ \psi(u,0)&=X^+_\sigma(t) \qquad&&(\mbox{resp. }\,\, \psi(u,0)=X^-_\sigma(t) ). \end{alignedat} \end{align}
	\begin{align}\begin{array}{rl}A\circ_{X}B & = (AX)(X^{ \dagger }B) \\ \\& = (AX^{2/3}X^{1/3})(X^{-1/3}X^{-2/3}B) \\ \\& = X^{1/3}[(X^{-1/3}AX^{2/3})(X^{-2/3}B)] \\ \\& = X^{1/3}[(X^{-1/3}AX^{1/3}X^{1/3})(X^{-1/3}X^{-1/3}B)] \\ \\& = X^{1/3}\{[(X^{-1/3}AX^{1/3})(X^{-1/3}BX^{1/3})]X^{-1/3}\} \\\\ & = X^{1/3}[(X^{-1/3}AX^{1/3})(X^{-1/3}BX^{1/3})]X^{-1/3}. \\\end{array}\end{align}
	\begin{align}\begin{array}{c}-a+\alpha =\frac \lambda {\lambda -\mu }2\alpha _{+}-\frac \mu {\lambda -\mu}2\alpha _{-}, \\-a-\alpha =\frac \mu {\lambda -\mu }2\alpha _{+}-\frac \lambda {\lambda -\mu}2\alpha _{-}.\end{array}\end{align}
	\begin{align}\begin{array}{c}(\dot{x},\dot{y})=i(a\alpha\zeta^{\alpha-1},b\beta\zeta^{\beta-1})= (A\zeta^{\alpha-1},B\zeta^{\beta-1})\\(i\dot{y}^,-i\dot{x}^)=(iB^(\zeta^)^{\beta-1},-iA^(\zeta^)^{\alpha-1})\end{array}\end{align}
	\begin{align}\begin{array}{l}\mathbf{E}(\mathbf{x},t) =\displaystyle\frac{1}{4\pi}\int\rho_e(\mathbf{x}',t-r)\frac{\mathbf{r}}{r^3}\,d^3x'\\~\\+\displaystyle\frac{1}{4\pi}\int\mathbf{J}_m(\mathbf{x}',t-r)\,{\bf\times}\,\frac{\mathbf{r}}{r^3}\,d^3x'-\displaystyle\frac{1}{4\pi}\int\frac{1}{r}\frac{\partial\mathbf{J}_m(\mathbf{x}',t-r)}{\partial t}\,d^3x' ,\end{array}\end{align}

End of preview. Expand in Data Studio

LaTeX Hard 2-Line Dataset (Split)

Dataset ini adalah subset dari OleehyO/latex-formulas yang telah dikurasi untuk formula LaTeX "hard" dengan 2+ baris.

Dataset Details

Source Dataset: OleehyO/latex-formulas (config: cleaned_formulas)
Subset: hard_2line - Formula dengan 2 atau lebih baris (mengandung \\)
Total Samples: 99,305
Seed: 42

Splits

Split	Samples	Percentage
Train	69,513	70%
Validation	19,861	20%
Test	9,931	10%

Usage

from datasets import load_dataset

# Load all splits
dataset = load_dataset("wulanbhai/latex-hard-2line-split")

# Load specific split
train_data = load_dataset("wulanbhai/latex-hard-2line-split", split="train")
val_data = load_dataset("wulanbhai/latex-hard-2line-split", split="validation")
test_data = load_dataset("wulanbhai/latex-hard-2line-split", split="test")

Curation Process

Dataset ini dikurasi menggunakan regex pattern untuk mendeteksi formula multi-baris:

Pattern: (?<!\\)\\\\(?![A-Za-z]) - break baris nyata (bukan escaped)
Environment blocks: matrix, align, gather, dll.

Generation Details

Generated: 2025-10-28T04:42:49+00:00
Curation script: latex-hard-2line-matrix-indices

License

Indices only - no images or original LaTeX content redistributed. Refer to source dataset for licensing.

Downloads last month: 83

Size of downloaded dataset files:

841 MB

Size of the auto-converted Parquet files:

841 MB

Number of rows:

99,305

image imagewidth (px) 42 746	latex_formula stringlengths 36 422
	\begin{align}\left(\begin{array}{cc} \mu_1 & 0 \\ 0 & \mu_2 \\ \end{array} \right)\left(\begin{array}{cc} 0 & d_1 \\ d_2 & 0 \\ \end{array} \right)\left(\begin{array}{cc}\bar{\mu_1}^{-1} & 0 \\0 & \bar{\mu_2}^{-1} \\\end{array}\right) =\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} \right).\end{align}
	\begin{align}f_M=\left(\begin{array}{c}f^1\\\Lambda f^{1}.\end{array}\right).\end{align*}
	\begin{align}A\|_{I_\mu} \;=\; \left( \begin{array}{cccc} \mu & 1 & \cdots & 0 \\0 & \mu & \cdots & 0 \\ \vdots & \vdots & \ddots & 1 \\ 0 & 0 & \cdots & \mu \end{array} \right) \;.\end{align}
	\begin{align}\begin{array}{c}V(z_{12}^{\prime})=U^{-1}(z_{1};g)V(z_{12})U(z_{2};g)\,,\\{}\\V(z_{21}^{\prime})=U^{-1}(z_{2};g)V(z_{21})U(z_{1};g)\,.\end{array}\end{align}
	\begin{align}\begin{array}{rcl}{\cal R}(u,c)&=&\displaystyle\frac{u-icP}{u+ic}\\begin{align}4mm] &=&\displaystyle\frac{1}{u+ic}\left[ \begin{array}{cccc} u-ic& 0 & 0 & 0 \\ 0 & u &-ic& 0 \\ 0 &-ic& u & 0 \\ 0 & 0 & 0 &u-ic \end{array}\right]~,\end{array}\end{align*}
	\begin{align}R(u_1 -u_2 ) t^{c}_{b}(u_1 )\otimes t^{b}_{a}(u_2 )=\sum_{d} W\left( \left. \begin{array}{cc} c & d \\ b & a \end{array} \right\| u_1 -u_2 \right) t^{d}_{a}(u_1 )\otimes t^{c}_{d}(u_2 ). \end{align}
	\begin{align}\begin{array}{lcr}\sum\limits_{j}q^j_a(\|X_j\|^2-\|Y_j\|^2)\eta_j^+\overline{\eta}_j^+&=r^2_a\eta_a^+\overline{\eta}_a^+ \qquad &(a)\\\sum\limits_{j}q^j_a(X_j\overline{Y}_j)\eta_j^+{\eta}_j^+&=0 \qquad &(b)\\\sum\limits_{j}q^j_a(\overline {X}_j{Y_j})\overline{\eta}_j^+\overline{\eta}_j^+&=0. \qquad &(c)\end{array}\end{align}
	\begin{align} \mathcal{S}_{N} = \begin{cases} \{(G, N)\}, & {\rm if} ~G/N~{\rm cyclic;}\\ \emptyset, & {\rm otherwise.} \end{cases} \end{align}
	\begin{align} d_{i}:= \begin{cases}1,~{\rm if} -1 \in \langle r_{i}\rangle; \\ 2, ~{\rm otherwise}.\end{cases}\end{align}
	\begin{align}\begin{array}{r c l}\frac{df}{dt}&=&\frac{\partial f}{\partial t}+\{f,H_T\}\\ & & \\&=&\frac{\partial f}{\partial t}+\{f,H_T\}_D+\{f,\chi_s\}\left(\Delta^{-1}\right)^{ss'}\{\chi_{s'},H_T\}\approx\frac{\partial f}{\partial t}+\{f,H_T\}_D\ .\end{array}\end{align}
	\begin{align}F=\left(\begin{array}{ccccc}0 & 0 & 0 & 0 & 0 \\0 & 0 & f & 0 & 0 \\0 & -f & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0\end{array}\right)\end{align}
	\begin{align}\langle g_{xy} \rangle =\left(\begin{array}{cccc}b & 0 & 0 & 0\\0& b &0&0\\0&0&\langle g_{33} \rangle &\langle g_{34}\rangle\\0&0&\langle g_{43} \rangle &\langle g_{44} \rangle\end{array}\right)\end{align}
	\begin{align}\epsilon=\left\{\begin{array}{ll} 1&\mbox{if $\omega=\pi/2$}\;,\\-1&\mbox{if $\omega=3\pi/2$}\;.\end{array}\right.\end{align}
	\begin{align} \begin{aligned}A_N &= A_1(w_{N-1})\;\! A_{N-1} = \prod_{i=0}^{N-1} A_1(w_{N-i-1}) \\D_N &= D_1(w_{N-1})\;\! D_{N-1} = \prod_{i=0}^{N-1} D_1(w_{N-i-1}) \\begin{align}0.3em]C_N &= C_1(w_{N-1})\;\! A_{N-1} + D_1(w_{N-1})\;\! C_{N-1} \\&= \sum_{i=0}^{N-1} D_i(w_{N-1-i})\, C_1(w_{N-1-i}) \, A_{N-1-i} . \\\end{aligned} \end{align*}
	\begin{align}\left( \begin{array}{cc} u_{0} m & u_{0} b \\ u_{0} b^{\dag} & u_{0} n \end{array} \right)\end{align}
	\begin{align}u^{35,35}_i(y,z)=\left\{\begin{array}{rrrl}- 5 b_i^{35,35}(y,z) \ & \ \ \ \ \ if& \ \ \ \ \ \ \ \ \ \ \ \ \|5 b_i^{35,35}(y,z)\|\leq 1, \\-1 \ & \ \ \ \ \ if& \ \ \ \ \ \ \ \ \ \ \ \ \ -5 b_i^{35,35}(y,z) < -1,\\1 \ & \ \ \ \ \ if& \ \ \ \ \ \ \ \ \ \ \ \ \-5 b_i^{35,35}(y,z) > 1,\\\end{array}\right\},\ \ \ i=1,2,\end{align}
	\begin{align}\begin{pmatrix}A^{-1}_N & {\bf 0} \\begin{align}0.4em]- D^{-1}_N \;\! C_N \;\! A_N^{-1} & D^{-1}_N\end{pmatrix}\begin{pmatrix}u \\1\end{pmatrix}\end{align*}
	\begin{align}\tilde\partial_{i}(g_{0}, \dots, g_{n}) = \begin{cases}(g_{0}, \dots, g_{i-1}, g_{i}g_{i+1}, g_{i+2}, \dots, g_{p}) & i=0, \dots p-1, \\(g_{0}, \dots, g_{p-1}) & i=p.\end{cases}\end{align}
	\begin{align}U = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right)~.\end{align}
	\begin{align}g^{\mu \nu} = \left( \begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right) \ , \qquad\epsilon^{\mu \nu} = \left( \begin{array}{rr} 0 & -1 \\ 1 & 0 \end{array}\right)\end{align}
	\begin{align}\left(\begin{array}{cc}\hat h+\Sigma & \beta \Delta ^{\dagger }\beta C \\-C\,\Delta & \hat h+\Sigma\end{array}\right) \left(\begin{array}{c}u \\v\end{array}\right) =E\left(\begin{array}{c}u \\v\end{array}\right) \end{align}
	\begin{align}f(x)=\begin{cases} \frac{3}{2} x+\frac{3}{2} & x < -1\\ 0 & -1 \leqslant x \leqslant 1\\ \frac{3}{2} x-\frac{3}{2} & x > 1\end{cases}\end{align}
	\begin{align}u_{0,l}(r)=\left( \begin{array}{l}\phi _{0,l,1}(r) \\ 0\end{array}\right) ,{\rm \;}{}l\geq 1;\;\;\;u_{0}^{II}(r)=\left( \begin{array}{l}\phi _{0,1}^{ir}(r) \\ 0\end{array}\right) ,\;{}l=0. \end{align}
	\begin{align}\begin{array}{lcl}{\cal L}_A^{2n+1} a &=& e^{i\phi} \rho^{2n+1}a^\dagger \\{\cal L}_A^{2n} a &=& \rho^{2n}a \\ \end{array}\end{align}
	\begin{align}\widetilde{A} = \left(\begin{array}{cc} A & 0 \\ 0 & 0\end{array}\right).\end{align}
	\begin{align}\begin{array}{lcr}D^{++}=u^{+ \alpha}{\partial\over{\partial u^{-\alpha}}};\qquad D^{--}=u^{- \alpha}{\partial\over{\partial u^{+\alpha}}}\\2D^{++}=[D^0,D^{++}];\qquad -2D^{--}=[D^0,D^{--}]\\D^0=[D^{++},D^{--}]=u^{+ \alpha}{\partial\over{\partial u^{+\alpha}}}-u^{- \alpha}{\partial\over{\partial u^{-\alpha}}}\end{array}\end{align}
	\begin{align}\left\{\begin{array}{lcl}\Sigma &=& r^2 + a^2\cos^2\theta\\\Delta &=& r^2 + a^2 + Q^2 - 2Mr\;. \end{array}\right.\end{align}
	\begin{align} z_t = \begin{pmatrix} {x}_t \\ {y}_t \end{pmatrix} =U_t({T},{D})\, \begin{pmatrix} {x}_1^\prime \\ {y}_1^\prime \end{pmatrix}- {D}\, U_{t-1}({T},{D})\, \begin{pmatrix} {x}_0^\prime \\ {y}_0^\prime \end{pmatrix}+ \begin{pmatrix} {x}^ \\ {y}^* \end{pmatrix}\end{align*}
	\begin{align}\begin{array}{l}\lbrack q^{-},\alpha _0^{+}]=i\,, \\\left[ \alpha _n^{-}\,,\alpha _m^{+}\right] =n\,\delta _{n+m,0}\,\,\,, \\\left[ s_n\,,s_m\right] =\left( \frac k2-1\right) \,n\,\delta _{n+m,0}\,\,\,,\\\left[ L_n^{\prime }\,,L_m^{\prime }\right] =(n-m)L_{n+m}^{\prime }\end{array}\end{align}
	\begin{align}B=\left\|\begin{array}{cc}\alpha & \alpha +s \\-\alpha +s & -\alpha\end{array}\right\| ,\end{align}
	\begin{align}\begin{array}{c}a^{A}=a^{A}~~~~b^{A}=b^{A}~~~~\omega_{AB}{}^{}=\omega_{AB}~~~~\lambda^{}=\lambda~~~~T^{\dagger}=-T\\{}\\\bar{\rho}_{i}=\rho^{i\dagger}\gamma^{0}~~~~~~~\bar{\varepsilon}_{i}=\varepsilon^{i\dagger}\tilde{\gamma}^{0}\end{array}\end{align}
	\begin{align}[h_{j,j+1}, S^{\pm}] = [h_{j,j+1}, S^{z}] = 0\ \ \\ \forall j = 1,\ldots ,N-1 \end{align}
	\begin{align}= \{\chi^{\mu\nu}\} +\left(\begin{array}{cccc}((p^2 - q^2)a + 2pqb) & (pa + qb) & (pb -qa) & ((p^2 - q^2)a + 2pqb) \\(pa + qb) &0 & 0 & (pa + qb) \\(pb -qa) & 0 & 0 & (pb -qa) \\((p^2 - q^2)a + 2pqb) &( pa + qb) &( pb -qa) & ((p^2 - q^2)a + 2pqb)\end{array}\right).\end{align}
	\begin{align} C \, = \, \begin{pmatrix} T & -D \\ 1 & 0 \end{pmatrix}.\end{align}
	\begin{align}\begin{array}{l}a_{ij}^{{\bf 4}}= \frac{1}{g}\sum\limits_{\gamma=1}^{r}r_{\gamma }\chi_{\gamma }^{R_M \oplus {\bf 1}^2 }\chi_{\gamma }^{(i)}\chi_{\gamma }^{(j)} \\a_{ij}^{{\bf 6}}=\frac{1}{g}\sum\limits_{\gamma=1}^{r}r_{\gamma }\chi_{\gamma }^{R_M \oplus R_M \oplus {\bf 1}^2 }\chi_{\gamma }^{(i)}\chi_{\gamma }^{(j)} \\\end{array}\end{align}
	\begin{align}f(x)=\begin{cases} a_1x & x < 0\\ a_2 x & x \geqslant 0.\end{cases}\end{align}
	\begin{align}\Bigl(X_{AB}^{ab}(x,y)\Bigr)=\left(\begin{array}{cc}0&-\,m^2\,\delta^{ab}\\\delta^{ab}&\frac{1}{2}\,g\,f^{abc}\,(D_i\pi^i)^c\end{array}\right)\,\delta(x-y)\,.\end{align}
	\begin{align}F_{k}(x,1-x)=\left\{\begin{array}{l}\left[x(1-x)\right]^{\beta+k}\\\left[x(1-x)\right]^{\beta+k}(2x-1)\ ,\end{array}\right.\end{align}
	\begin{align}Z^{IJ} = \left(\begin{array}{cc}Z_1 \epsilon & 0 \\ 0 &Z_2\epsilon\end{array}\right) \ , \quad {\rm where}\quad \epsilon = \left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right).\end{align}
	\begin{align}(\Psi_l,\Psi_l) = \|U_{\rho_l}(k_0)\|^{-1}&\int_{U_{\rho_l}(k_0)} dk \int_{F_k^n} \Big\|\phi\Big(k;p_1,\ldots,p_n \Big)\Big\|^2\,dp\\& \times \int_{F_{K-k}^{N-n}}\Big\|\psi\Big(K-k;p_{n+1},\ldots,p_N\Big)\Big\|^2\,dp=1.\end{align}
	\begin{align}\left\{ \begin{array}{rcl}3F^{2}(A) +F^{2}(B) & = & 0\, ,\\& & \\\sqrt{3}F(A)-s(\alpha){}^{\star}F(B) & = & 0\, .\\ \end{array}\right.\end{align}
	\begin{align}& A_{i}^{\top}\lambda^{}\in\partial f_{i}(x_{i}^{}), i=1,2,\cdots,N, \\& \sum\limits_{i=1}^{N}A_{i}x_{i}^{}-b=0. \end{align*}
	\begin{align}\delta^{(1)} S = - \frac{1}{12} \sum_{\langle ijk \rangle} \bar{\epsilon}_1 \Gamma_{\mu} \theta_{[ \, i} \bar{\theta}_j \Gamma^{\mu} \theta_{k \, ]}. \\\end{align}
	\begin{align}{\Phi}=\frac{1}{2}\left(\begin{array}{c}-A_{Ij}E^{j}_{\nu}\\D_{ij}E^{j}_{\nu}\\E^{i}_{\nu}\\\end{array}\right)={\frac{1}{2}}\left(\begin{array}{ccc}{\cal E}_{I}^{M}&0&0\\0&T_{i}^{\mu}&0\\0&0&T_{i}^{\mu}\\\end{array}\right){\hat \Phi}\end{align}
	\begin{align}-\div{\lvert \nabla u \rvert^{p-2} \nabla u} = f \;\; \Omega, \\ u-g \in W^1_0(L_p(\Omega)), \end{align}
	\begin{align}R\to \left\{\begin{array}{ll}\frac\zeta{\sqrt{2n^3}}, & (r\to\infty) \\ \frac\zeta{\sqrt{2n'n^2}}. & (r= 0)\end{array}\right.\end{align}
	\begin{align}\left( \begin{array}{c} R\\ G\\ B \end{array} \right) \; \; \; , \; \; \;j\left( \begin{array}{c} R\\ G\\ B \end{array} \right) \; \; \; \; \; .\end{align}
	\begin{align}\tau \rightarrow \frac{a\tau +b}{c\tau +d} \;\; , \;\;\;\;\; \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \in Sl(2;Z)\end{align}
	\begin{align} L_i^{(x)}(u)=\left(\begin{array}{cc} x_i & 1 \\ x_iX_i+\beta_i+\lambda & X_i \end{array}\right )\;\;.\end{align}
	\begin{align}\begin{array}{c}\partial_m\psi_1=\partial_m\psi_2=0\\\psi_0=\psi_{12}=0\end{array}\end{align}
	\begin{align}u \in B^{\sigma}_{\tau}(L_{\tau}(\Omega)) 0 < \sigma < \begin{cases} 2 & 1<p \leq 2,\\1 + \frac{1}{p-1} & 2 < p < \infty\\\end{cases} \frac{1}{\tau} = \frac{\sigma}{2} + \frac{1}{p}.\end{align}
	\begin{align}\mathcal{M}^{ab}=\left( \begin{array}{ll}f^{abi}\varphi ^{i} & \delta ^{ab}\partial ^{2} \\ -\delta ^{ab}\partial ^{2} & -f^{abi}\overline{\varphi }^{i}\end{array}\right) \;. \end{align}
	\begin{align}(VII)' \hspace{7mm} \Psi_{\bf D} =\left(\begin{array}{c}a_1 +{\rm j}b_1 \\a_2 +{\rm j}b_2 \\a_3 +{\rm j}b_3 \\a_4 +{\rm j}b_4\end{array}\right) \hspace{5mm} \leftrightarrow \hspace{5mm}\Psi^{SR}_{\bf C} =\left(\begin{array}{c}-b_1+{\rm i}b_3 \\-b_2+{\rm i}b_4 \\-a_1+{\rm i}a_3 \\-a_2+{\rm i}a_4\end{array}\right)\end{align}
	\begin{align}\Psi\equiv\left(\begin{array}{c}\psi_1\\\psi_2\end{array}\right)=\left(\begin{array}{c}\gamma_5\psi_2^\\-\gamma_5\psi_1^\end{array}\right)\, ,\end{align}
	\begin{align} 0&=-v_t - (\mathbf{L}^r +\mathbf{L}^Y +\mathbf{L}^{S})v - r x v_x \\ & -\sup_{\theta} \left\{\theta (\mu\!-\!r)v_x \!+\!\theta \sigma^2 y s^{2 \beta-1} v_{xs} \!+\!\frac12 \theta^2\sigma^2 y s^{2\beta-2} \right\} \end{align}
	\begin{align} 0&=-v_t - (\mathbf{L}^Y +\mathbf{L}^{S})v \\ & -\sup_{\theta} \left\{\theta \cdot \mu v_x \!+\!\theta \sigma ^2 y s^{2 \beta-1} v_{xs} \!+\!\frac12(\theta^2\sigma^2 y s^{2\beta-2} \right\} \end{align}
	\begin{align} \begin{array}{c} h^{m(s)}=\varphi^{m(s)} + \eta^{m(2)}\,\chi^{m(s-2)}\;;\;{\varphi_l}^{lm(s-2)} =0, \\ \epsilon^{m(s-1)}=\varepsilon^{m(s)} + \eta^{m(2)}\, \zeta^{m(s-3)}\;;\;{\varepsilon_l}^{lm(s-3)} =0, \end{array}\end{align}
	\begin{align}\begin{array}{rcl}S_{{\rm IIB}} & = & -T_{M_2}l\int d^2\xi\sqrt{\|{\tilde g}_{ij}-\frac{1}{{\tilde k}^2}(\partial_i \varrho+{\tilde B}^{(1)}_i)(\partial_j \varrho+{\tilde B}^{(1)}_j)\|}\\& & \\& &-\frac{T_{M_2}l}{2}\int d^2\xi\epsilon^{ij}\left[\tilde{B}^{(2)}{}_{ij}+2\partial_{i}\varrho\ \left({\tilde A}^{(1)}{}_{i}+m\pi\alpha^{\prime} b_{j}\right)\right]\, .\\\end{array}\end{align}
	\begin{align}\delta A_{\mu\nu} = \epsilon^_\nu \psi_\mu - \epsilon_\mu \psi^_\nu\\ \delta \psi_\mu = i \epsilon_\nu A_{\mu\nu}\\ \delta \psi^_\mu = i \epsilon^_\nu A_{\nu\mu}\\\end{align}
	\begin{align}\begin{array}{c} F_{09} = F_{0i} = F_{9i} = 0, \\\end{array}\end{align}
	\begin{align} \frac{1}{\beta_L}\mathbb{E}\bigg(G^{\Lambda}(z_L;n,n)\bigg) &\leq \frac{1}{Imz}~\pi\bigg(1+\frac{k}{2}\bigg)S_\mu\bigg(\frac{2~Imz_L}{k}\bigg),~~\Lambda=C_p,~\Lambda_L\\&\leq C ~\bigg(\frac{2~\beta_L^{-1}~Imz}{k}\bigg)^{\alpha}\\&\leq C ~(2L+1)^{-d},~~(\mathrm{since} ~~ \beta_L=(2L+1)^{d/\alpha}).\end{align}
	\begin{align} \lim_{n\to\infty}\mathbb{E}\bigg(\sum_p^{N_{L_n}^d}\eta^{\omega}_{p,E}(I)\bigg) =\lim_{n\to\infty}\mathbb{E}\big(\zeta^{\omega}_{L_n,E}(I)\big) =\gamma_{E,I}\\\end{align}
	\begin{align}H^0 = \left(\begin{array}{cc}-\partial_z^2+4 & 0 \\0 & -\partial_z^2 +4\end{array}\right),\end{align}
	\begin{align}L_{i_2}^{(2)}(x):=\left\{\begin{array}{llll}L_i^{(0)},~~~\ i_1=1,...7,\\\frac{d}{dt}(\lambda^{(0) A}_iL^{(0)}_i),~~~i_2 = 7 +A ,~A=1,2,3, \\\frac{d}{dt} ( \Omega_4^{(0)} ), ~~~i_2 = 11, \\\frac{d}{dt} ( \lambda_{i_1}^{(1) B} L_{i_1}^{(1)}),~~~i_2 = 11+B , ~B=1,2.\end{array}\right.\end{align}
	\begin{align}2\left(\begin{array}{cc}0&\partial_z\\\partial_{\bar{z}}&0\end{array}\right)\left(\begin{array}{c}\;^0\chi_1(z,\bar{z})\\\;^0\chi_2(z,\bar{z})\end{array}\right)=0 .\end{align}
	\begin{align}\left (\begin{array}{cc} A & B \\ C & D \end{array} \right)^{-1} =\left (\begin{array}{cc} (A - BD^{-1}C)^{-1} & -A^{-1}B (D - CA^{-1}B)^{-1}\\ -D^{-1}C (A - BD^{-1}C)^{-1} & (D - CA^{-1}B)^{-1} \end{array} \right).\end{align}
	\begin{align} M_{NSM}^2 =g^2 v^2 \left(\begin{array}{ccc} 2 & +1 & +1 \\ +1 & 2 & +1 \\ +1 & +1 & 2 \\ \end{array} \right)~.~\,\end{align}
	\begin{align} (x,g,y)(y,h,z)&=(x,gh,z) \\ (x,g,y)^{-1}&=(y,g^{-1},x).\end{align}
	\begin{align}\gamma_{ab}=\left(\begin{array}{cc}\rho_{[a}\rho_{b]}^{T}&0\\ 0&\rho_{[a}^{T}\rho_{b]}\end{array}\right)\,.\end{align}
	\begin{align}\left(\begin{array}{ccc} \gamma^{00}&\gamma^{01}&\gamma^{02}\\ \gamma^{10}&\gamma^{11}&\gamma^{12}\\ \gamma^{20}&\gamma^{21}&\gamma^{22} \end{array}\right)= \left(\begin{array}{ccc} \eta&\eta^\prime&\eta^\prime\\ \eta^\prime&\eta&\eta^\prime\\ \eta^\prime&\eta^\prime&\eta \end{array}\right)\end{align}
	\begin{align} E_{g}^{2} &=V_{b}^{}E_{a}V_{b}V_{b}^{}E_{a}V_{b}\\ & = V_{b}^{}E_{a}E_{b}E_{a}V_{b} \\ &=V_{b}^{}E_{b}E_{a}^{2}V_{b} ~~~\textrm{( Since $E_{a}$ and $E_{b}$ commute)} \\ &=V_{b}^{}E_{a}V_{b} \\ &=E_{g}.\end{align*}
	\begin{align}\begin{array}{ccccc} & SO(2n) & \subset & SO(2n+1) & R_M = {\bf (2n+1)} \\ & SU(3) & \subset & G_2 & R_M = {\bf 7} \\\tilde{R}_M = {\bf 9} & SO(9) & \subset & F_4 & R_M = {\bf 26} \end{array} \quad .\end{align}
	\begin{align} E_{g_{1}}E_{g_{2}}&=V_{b_{1}^{'}}^{}E_{a_{1}^{'}}V_{b_{1}^{'}}V_{b_{1}^{'}}^{}E_{a_{2}^{'}}V_{b_{1}^{'}} \\ & = V_{b_{1}^{'}}^{}E_{a_{2}^{'}}V_{b_{1}^{'}}V_{b_{1}^{'}}^{}E_{a_{1}^{'}}V_{b_{1}^{'}} \\ & = E_{g_{2}}E_{g_{1}}.\end{align}
	\begin{align}E_{g_{1}}&=V_{c}^{}V_{ab}V_{ab}^{}V_{c} \\ &= V_{c}^{}V_{b}(V_{a}V_{a}^{})V_{b}^{}V_{c} \\ & \leq V_{c}^{}V_{b}V_{b}^{}V_{c} \\ & \leq E_{g_{2}}.\end{align*}
	\begin{align}r = \left\{ \begin{array}{cc} -(q^2)^\alpha & (-1<r<0) \\ (q^2)^\alpha & (0<r<1) \end{array} \right., \qquad q = -e^{-\pi K'/K}.\end{align}
	\begin{align}W_{g}W_{h}&=V_{b_1}^{}V_{a_1}V_{a_1}^{}V_{c_1} \\ &=V_{b_1}^{}E_{a_1}V_{b_{1}}V_{b_{1}}^{}V_{c_1} (\textrm{~Since $E_{a_1}$ and $E_{b_1}$ commute}) \\ &=E_{g}W_{hg}\end{align}
	\begin{align}E(j) = \left\{\begin{array}{ll}0 & \mbox{if the path has an extremum at ($x$-position) $j$} \\\frac{1}{2} j\qquad & \mbox{otherwise.}\end{array}\right.\end{align}
	\begin{align} (\widetilde{\chi}a)(T)&=\widetilde{\chi}(V_{a}^{}TV_{a}) \\ &=\chi(V_{a}V_{a}^{}TV_{a}V_{a}^{})\\ &= \chi(E_{a})\chi(T)\chi(E_{a}) \\ &= 1_{A}(a)\chi(T)1_{A}(a) \\ & = \chi(T).\end{align*}
	\begin{align}\Psi = (1-{\rho \over 4} x^+ x^-)^{1 \over 2} \left( \begin{array}{c} e^{-{\theta \over 2}} e^{\imath v}p(x^-) \\ e^{{\theta \over 2}} e^{\imath u}f(x^+) \\ \end{array} \right)\end{align}
	\begin{align} \lim \inf A_{n} & = \{ x \in X: \lim \sup d(x,A_{n}) = 0 \}, \textrm{~and~} \\ \lim\sup A_{n} & = \{x \in X: \lim \inf d(x,A_{n})=0 \}.\end{align}
	\begin{align}M_{1}= \left( \begin{array}{cccc} 1 & 2 & 0 & 2 \\ -2 & -2 & -2 & -1 \\ 1 & 0 & 1 & 0 \\ 2 & 1 & 2 & 0 \end{array} \right) \, .\end{align}
	\begin{align} \phi_{n}(1)&= \chi \Big(\int_{b \in supp(f_{n})}f_{n}(b)E_{b}db \Big) \\ & = \int_{b \in supp(f_{n})}f_{n}(b)1_{A}(b)db \\ & = \int_{b \in supp(f_{n})}f_{n}(b)db ~ (\textrm{ Since $supp(f_{n}) \subset Int(A)$}) \\ & =1.\end{align}
	\begin{align}& \|\phi_{n}(\alpha_{a}(E_{\psi}))-\chi(E_{\psi})\| \\&= \| \phi_{n}(\alpha_{a}(E_{\psi}))-\chi(E_{f_{n}}E_{\psi})\|\\& \leq \int_{b \in U_{n}}f_{n}(b)\Big(\int \|\Delta(b^{-1}a)^{m}\psi(g_{1}b^{-1}a,\cdots,g_{m}b^{-1}a)-\psi(g_{1},g_{2},\cdots,g_{m})\| dg_{1} \cdots dg_{m} \Big)db \\ & \leq \epsilon \int _{b \in U_{n}}f_{n}(b) \\& \leq \epsilon.\end{align}
	\begin{align}\#_{n}=\left\{\begin{array}{ll}1~~~~&\mbox{for}~n=3\,,\\{}&{}\\4n^{2}-12n~~~~~~~&\mbox{for}~n\geq 4\,.\end{array}\right.\end{align}
	\begin{align}< \pi(F)\xi,\xi> &= \int F(x) d\mu_{\xi,\xi} (x)\\ & = \int f(g)1_{Xg}(x)dg d \mu_{\xi,\xi}(x) \\ & = \int f(g) \Big (\int 1_{Xg}(x)d\mu_{\xi,\xi}(x) \Big) dg \\ & = \int f(g) < \pi(1_{Xg\cap X})\xi,\xi> dg \\ &= \int f(g)<E_{g}\xi,\xi> dg. \end{align}
	\begin{align}L_1=\left( \begin{array}{ccccc}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0\end{array}\right) ,\,\,\,L_2=\left( \begin{array}{ccccc}0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0\end{array}\right)\end{align}
	\begin{align}\begin{array}{ll} [h_i,h_j]=0,& [e_i,f_j]=\delta_{ij}h_i, \\\mbox{$[$} h_{i},e_{j}\mbox{$]$}= A_{ij}e_{j}, &\mbox{$[$} h_{i},f_{j}\mbox{$]$}=-A_{ij}f_{j},\end{array}\end{align}
	\begin{align}(\Gamma^{0})^{\underline\alpha}_{~~\underline\beta}= \left(\begin{array}{cc}0 & -1\\1 & 0\end{array}\right)\,, \quad(\Gamma^{1})^{\underline\alpha}_{~~\underline\beta}= \left(\begin{array}{cc}-1 & 0\\0 & 1\end{array}\right)\, , \quad \Gamma^2=\Gamma^0\Gamma^1=\left(\begin{array}{cc}0 & -1\\-1 & 0\end{array}\right) \,.\end{align}
	\begin{align}\left\{ \begin{array}{l}\partial ^{\mu }V_{\mu }\left( x\right) =0 \\ \partial ^{\mu }A_{\mu }\left( x\right) =2miP\left( x\right) .\end{array}\right.\end{align}
	\begin{align}\begin{array}{ll}L^{-1}(T,X,Y,Z)=(\cosh t\cos\frac{r_{+}-r_{-}}{2},&\sinh t\cos\frac{r_{+}+r_{-}}{2}, \\& \sinh t\sin\frac{r_{+}+r_{-}}{2},-\cosh t\sin\frac{r_{+}-r_{-}}{2}).\end{array}\end{align}
	\begin{align}\partial_{0} J_{1} - \partial_{x} J_{0} = \frac{1}{2}(\partial_{0} L_{1} \\ - \partial_{x} L_{0}) + \frac{1}{2} (\partial_{0} L_{0} -\partial_{x} L_{1})g + \\ \frac{1}{2} (L_{0}^{2} - L_{1}^{2} )g \;\;\; ,\end{align}
	\begin{align} v(\psi\cdot g_{uuv}) &=g_v\cdot g_{uuv}=(g_v\cdot g_{uv})_u-g_{uv}\cdot g_{uv} =\frac{(g_v\cdot g_{v})_{uu}}2-g_{uv}\cdot g_{uv} \\ &=\frac{(f_v\cdot f_{v})_{uu}}2-g_{uv}\cdot g_{uv} =v^2\frac{(\phi\cdot \phi)_{uu}}2-v^2 \psi_{u}\cdot \psi_{u}, \end{align}
	\begin{align}\begin{array}{c}G_{T}(z^{\prime\prime})=G_{T}(z^{\prime\prime}_{1})G_{T}(z_{s}(z^{\prime};u))\,,\\{}\\G_{T}(z^{\prime})=G_{T}(z_{1})^{-1}G_{T}(z)\,,\end{array}\end{align}
	\begin{align}\|\| T - T.O.E. \|\| = \left( \begin{array}{c} \mbox{\# of unexplained} \\ \mbox{bits of information} \end{array} \right)\end{align}
	\begin{align}(c^{\pm}(\alpha),c^{\pm}(\beta))=\left\{\begin{array}{ccc}(\cosh\frac{\alpha_{\pm}}{2},\cosh\frac{\beta_{\pm}}{2})&for&{\cal M}_{S}^{(\pm)}\\(\quad1\quad,\quad1\quad)&for&{\cal M}_{N}^{(\pm)}\\(\cos\frac{\rho_{\pm}}{2},\cos\frac{\sigma_{\pm}}{2})&for&{\cal M}_{T}^{(\pm)}.\end{array}\right. \end{align}
	\begin{align} \begin{alignedat}{2} g(u,0)&=\sigma(u), &&r(u,0)=\dot\sigma(u),\\ \psi(u,0)&=X^+_\sigma(t) \qquad&&(\mbox{resp. }\,\, \psi(u,0)=X^-_\sigma(t) ). \end{alignedat} \end{align}
	\begin{align}\begin{array}{rl}A\circ_{X}B & = (AX)(X^{ \dagger }B) \\ \\& = (AX^{2/3}X^{1/3})(X^{-1/3}X^{-2/3}B) \\ \\& = X^{1/3}[(X^{-1/3}AX^{2/3})(X^{-2/3}B)] \\ \\& = X^{1/3}[(X^{-1/3}AX^{1/3}X^{1/3})(X^{-1/3}X^{-1/3}B)] \\ \\& = X^{1/3}\{[(X^{-1/3}AX^{1/3})(X^{-1/3}BX^{1/3})]X^{-1/3}\} \\\\ & = X^{1/3}[(X^{-1/3}AX^{1/3})(X^{-1/3}BX^{1/3})]X^{-1/3}. \\\end{array}\end{align}
	\begin{align}\begin{array}{c}-a+\alpha =\frac \lambda {\lambda -\mu }2\alpha _{+}-\frac \mu {\lambda -\mu}2\alpha _{-}, \\-a-\alpha =\frac \mu {\lambda -\mu }2\alpha _{+}-\frac \lambda {\lambda -\mu}2\alpha _{-}.\end{array}\end{align}
	\begin{align}\begin{array}{c}(\dot{x},\dot{y})=i(a\alpha\zeta^{\alpha-1},b\beta\zeta^{\beta-1})= (A\zeta^{\alpha-1},B\zeta^{\beta-1})\\(i\dot{y}^,-i\dot{x}^)=(iB^(\zeta^)^{\beta-1},-iA^(\zeta^)^{\alpha-1})\end{array}\end{align}
	\begin{align}\begin{array}{l}\mathbf{E}(\mathbf{x},t) =\displaystyle\frac{1}{4\pi}\int\rho_e(\mathbf{x}',t-r)\frac{\mathbf{r}}{r^3}\,d^3x'\\~\\+\displaystyle\frac{1}{4\pi}\int\mathbf{J}_m(\mathbf{x}',t-r)\,{\bf\times}\,\frac{\mathbf{r}}{r^3}\,d^3x'-\displaystyle\frac{1}{4\pi}\int\frac{1}{r}\frac{\partial\mathbf{J}_m(\mathbf{x}',t-r)}{\partial t}\,d^3x' ,\end{array}\end{align}