id
int32 0
100k
| text
stringlengths 20
3.54k
| source
stringlengths 1
104
| similarity
float32 0.8
0.91
|
|---|---|---|---|
200
|
"Is evolution compatible with Christianity or other religious systems?" "Are there laws of biology like the laws of physics? "Ideas drawn from philosophical ontology and logic are being used by biologists in the domain of bioinformatics.
|
Philosophy of biology
| 0.858011
|
201
|
Since many students who apply to graduate programs in biochemistry do so during the first half of their fourth year, the scope of most questions is largely that of the first three years of a standard American undergraduate biochemistry curriculum. A sampling of test item content is given below:
|
GRE Biochemistry, Cell and Molecular Biology Test
| 0.857973
|
202
|
The direct sum of the division ring of quaternions with itself is denoted H ⊕ H {\displaystyle \mathbb {H} \oplus \mathbb {H} } . The product of two elements ( a ⊕ b ) {\displaystyle (a\oplus b)} and ( c ⊕ d ) {\displaystyle (c\oplus d)} is a c ⊕ b d {\displaystyle ac\oplus bd} in this direct sum algebra. Proposition: The algebra of split-biquaternions is isomorphic to H ⊕ H . {\displaystyle \mathbb {H} \oplus \mathbb {H} .}
|
Split-biquaternion
| 0.857918
|
203
|
In biology, translation is the process in living cells in which proteins are produced using RNA molecules as templates. The generated protein is a sequence of amino acids. This sequence is determined by the sequence of nucleotides in the RNA. The nucleotides are considered three at a time.
|
MRNA translation
| 0.85783
|
204
|
Later, René Descartes (17th century) introduced the modern notation (for example, the use of x—see below) and showed that the problems occurring in geometry can be expressed and solved in terms of algebra (Cartesian geometry).Equally important as the use or lack of symbolism in algebra was the degree of the equations that were addressed. Quadratic equations played an important role in early algebra; and throughout most of history, until the early modern period, all quadratic equations were classified as belonging to one of three categories. x 2 + p x = q {\displaystyle x^{2}+px=q} x 2 = p x + q {\displaystyle x^{2}=px+q} x 2 + q = p x {\displaystyle x^{2}+q=px} where p {\displaystyle p} and q {\displaystyle q} are positive.
|
Syncopated algebra
| 0.857786
|
205
|
He suggests that resolving the first problem will require introducing some material model and employing the theory of elasticity. 1910: Theodor Kaluza points out that there is nothing inherently paradoxical about the static and disk-riding observers obtaining different results for the circumference. This does however imply, Kaluza argues, that "the geometry of the rotating disk" is non-euclidean.
|
Ehrenfest paradox
| 0.857695
|
206
|
The subset consisting of O and I (highlighted in red in the tables at the right) is also a field, known as the binary field F2 or GF(2). In the context of computer science and Boolean algebra, O and I are often denoted respectively by false and true, and the addition is then denoted XOR (exclusive or). In other words, the structure of the binary field is the basic structure that allows computing with bits.
|
Field (mathematics)
| 0.85766
|
207
|
The primary algebra embodies a point noted by Huntington in 1933: Boolean algebra requires, in addition to one unary operation, one, and not two, binary operations. Hence the seldom-noted fact that Boolean algebras are magmas. (Magmas were called groupoids until the latter term was appropriated by category theory.) To see this, note that the primary algebra is a commutative: Semigroup because primary algebra juxtaposition commutes and associates; Monoid with identity element , by virtue of J0.Groups also require a unary operation, called inverse, the group counterpart of Boolean complementation.
|
Boundary algebra
| 0.857601
|
208
|
Similarly, a normal subgroup N of G is said to be a maximal normal subgroup (or maximal proper normal subgroup) of G if N < G and there is no normal subgroup K of G such that N < K < G. We have the following theorem: Theorem: A normal subgroup N of a group G is a maximal normal subgroup if and only if the quotient G/N is simple.
|
Maximal subgroup
| 0.857578
|
209
|
Below Level 1 0 – 175 Tasks at this level require the respondents to carry out simple processes such as counting, sorting, performing basic arithmetic operations with whole numbers or money, or recognizing common spatial representations in concrete, familiar contexts where the mathematical content is explicit with little or no text or distractors.Level 1 176 – 225 Tasks at this level require the respondent to carry out basic mathematical processes in common, concrete contexts where the mathematical content is explicit with little text and minimal distractors. Tasks usually require one-step or simple processes involving counting, sorting, performing basic arithmetic operations, understanding simple percent's such as 50%, and locating and identifying elements of simple or common graphical or spatial representations.Level 2 226 – 275 Tasks at this level require the respondent to identify and act on mathematical information and ideas embedded in a range of common contexts where the mathematical content is fairly explicit or visual with relatively few distractors. Tasks tend to require the application of two or more steps or processes involving calculation with whole numbers and common decimals, percents and fractions; simple measurement and spatial representation; estimation; and interpretation of relatively simple data and statistics in texts, tables and graphs.Level 3 – 276 – 325 Tasks at this level require the respondent to understand mathematical information that may be less explicit, embedded in contexts that are not always familiar and represented in more complex ways. Tasks require several steps and may involve the choice of problem-solving strategies and relevant processes.
|
Programme for the International Assessment of Adult Competencies
| 0.857471
|
210
|
This produces a characteristic 9:7 ratio of pigmented to unpigmented plants. Complementary epistasis in contrast produces an unpigmented plant if and only if the genotype is cc and dd, and the characteristic ratio is 15:1 between pigmented and unpigmented plants.Classical genetics considered epistatic interactions between two genes at a time. It is now evident from molecular genetics that all gene loci are involved in complex interactions with many other genes (e.g., metabolic pathways may involve scores of genes), and that this creates epistatic interactions that are much more complex than the classic two-locus models.
|
Autosomal recessive disorder
| 0.857466
|
211
|
NMT has been widely applied in plant biology in fields such as abiotic/biotic stress, plant nutrition, plant growth and development, plant/microbe interaction, plant defense, photosynthesis, signal transduction research, and more. Roots are commonly measured, in addition to many other plant samples such as leaf tissue, root hairs, guard cells, salt gland cells, mesophyll cells, and condensed organelles like chloroplasts and vacuoles. NMT can help identify plants that are more resistant to stressors like salt, temperature, drought, and disease. It’s also a useful tool for studying plant nutrition absorption and regulation mechanisms in ways such as monitoring rates of nutrient uptake at the root surface.
|
Non-invasive micro-test technology
| 0.857388
|
212
|
The same thing is believed to occur in laboratory measurements. Parallel plate geometry is used in laboratory measurements without corona generation. A stainless steel cup holds the sample.
|
Electrostatic filter
| 0.857387
|
213
|
Searching for the real or rational solutions are much more difficult problems that are not considered in this article. The set of solutions is not always finite; for example, the solutions of the system x ( x − 1 ) = 0 x ( y − 1 ) = 0 {\displaystyle {\begin{aligned}x(x-1)&=0\\x(y-1)&=0\end{aligned}}} are a point (x,y) = (1,1) and a line x = 0. Even when the solution set is finite, there is, in general, no closed-form expression of the solutions (in the case of a single equation, this is Abel–Ruffini theorem).
|
Polynomial system of equations
| 0.857347
|
214
|
The second branch in the sub-hierarchy considers the skills associated with advanced substitution. Attribute A7 requires the examinee to substitute numbers into algebraic expressions. The complexity of attribute A7 relative to attribute A4 lies in the concurrent management of multiple pairs of numbers and multiple equations.
|
Attribute hierarchy method
| 0.857326
|
215
|
JEE(Advanced) is conducted in two shifts of three hours each - Paper-1 and Paper-2, both the papers consist questions from three major subjects- Physics, Chemistry and Mathematics. Unlike most of the other exams, the type and the number of questions being asked in the paper varies from year to year, with an average of about 36-38 questions asked from each subject across both the shifts, For example, the 2021 JEE Advanced paper had 38 questions (19 questions in Paper-1 and the next 19 in Paper-2) from each of the three subjects. Every subject had questions divided into 4 main sections- 1. Section 1 consisted of 4 Single correct-multiple choice questions with marking scheme of +4 for every correct answer, 0 if question is left unanswered and -1 for every wrong answer.
|
Joint Entrance Examination – Advanced
| 0.857305
|
216
|
When the bottleneck is localized, optimization usually starts with a rethinking of the algorithm used in the program. More often than not, a particular algorithm can be specifically tailored to a particular problem, yielding better performance than a generic algorithm. For example, the task of sorting a huge list of items is usually done with a quicksort routine, which is one of the most efficient generic algorithms.
|
Software optimization
| 0.857269
|
217
|
The problems on the competition range from basic algebra to precalculus and are within the grasp of a high schooler's mathematical knowledge. The contest contains concepts from a variety of topics, including geometry and combinatorics.
|
Michigan Mathematics Prize Competition
| 0.857243
|
218
|
The P versus NP problem is a major unsolved problem in theoretical computer science. In informal terms, it asks whether every problem whose solution can be quickly verified can also be quickly solved. The informal term quickly, used above, means the existence of an algorithm solving the task that runs in polynomial time, such that the time to complete the task varies as a polynomial function on the size of the input to the algorithm (as opposed to, say, exponential time). The general class of questions for which some algorithm can provide an answer in polynomial time is "P" or "class P".
|
P versus NP problem
| 0.857202
|
219
|
AP Physics 1 is an algebra-based, introductory college-level physics course that includes mechanics topics such as motion, force, momentum, energy, harmonic motion, and rotation; The College Board published a curriculum framework that includes seven big ideas on which the AP Physics 1 and 2 courses are based, along with "enduring understandings" students are expected to acquire within each of the big ideas. :Questions for the exam are constructed with direct reference to items in the curriculum framework. Student understanding of each topic is tested with reference to multiple skills—that is, questions require students to use quantitative, semi-quantitative, qualitative, and experimental reasoning in each content area.
|
AP Physics 1
| 0.857159
|
220
|
In astronomy, the stellar classification of stars and their place on the Hertzsprung–Russell diagram are based, in part, upon their surface temperature, known as effective temperature. The photosphere of the Sun, for instance, has an effective temperature of 5772 K as adopted by IAU 2015 Resolution B3. Digital cameras and photographic software often use colour temperature in K in edit and setup menus. The simple guide is that higher colour temperature produces an image with enhanced white and blue hues. The reduction in colour temperature produces an image more dominated by reddish, "warmer" colours.
|
Kelvin
| 0.857158
|
221
|
In mathematics, an algebraic structure consists of a nonempty set A (called the underlying set, carrier set or domain), a collection of operations on A (typically binary operations such as addition and multiplication), and a finite set of identities, known as axioms, that these operations must satisfy. An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures. For instance, a vector space involves a second structure called a field, and an operation called scalar multiplication between elements of the field (called scalars), and elements of the vector space (called vectors). Abstract algebra is the name that is commonly given to the study of algebraic structures.
|
Structure (algebraic)
| 0.857104
|
222
|
The second orbit allows eight electrons, and when it is full the atom is neon, again inert. The third orbital contains eight again, except that in the more correct Sommerfeld treatment (reproduced in modern quantum mechanics) there are extra "d" electrons. The third orbit may hold an extra 10 d electrons, but these positions are not filled until a few more orbitals from the next level are filled (filling the n=3 d orbitals produces the 10 transition elements). The irregular filling pattern is an effect of interactions between electrons, which are not taken into account in either the Bohr or Sommerfeld models and which are difficult to calculate even in the modern treatment.
|
Sommerfeld–Wilson quantization
| 0.857078
|
223
|
Sequencing is used in molecular biology to study genomes and the proteins they encode. Information obtained using sequencing allows researchers to identify changes in genes and noncoding DNA (including regulatory sequences), associations with diseases and phenotypes, and identify potential drug targets.
|
DNA sequence
| 0.857068
|
224
|
From Lagrange's theorem we know that any non-trivial subgroup of a group with 6 elements must have order 2 or 3. In fact the two cyclic permutations of all three blocks, with the identity, form a subgroup of order 3, index 2, and the swaps of two blocks, each with the identity, form three subgroups of order 2, index 3. The existence of subgroups of order 2 and 3 is also a consequence of Cauchy's theorem. The first-mentioned is { (), (RGB), (RBG) }, the alternating group A3.
|
Nonabelian group of order 6
| 0.857004
|
225
|
The spectral radiance in janskys per steradian can be converted to a brightness temperature, useful in radio and microwave astronomy. Starting with Planck's law, we see B ν = 2 h ν 3 c 2 1 e h ν / k T − 1 . {\displaystyle B_{\nu }={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{h\nu /kT}-1}}.} This can be solved for temperature, giving T = h ν k ln ( 1 + 2 h ν 3 B ν c 2 ) .
|
Jansky
| 0.856972
|
226
|
The dimensionless constants that arise in the results obtained, such as the C in the Poiseuille's Law problem and the κ {\displaystyle \kappa } in the spring problems discussed above, come from a more detailed analysis of the underlying physics and often arise from integrating some differential equation. Dimensional analysis itself has little to say about these constants, but it is useful to know that they very often have a magnitude of order unity. This observation can allow one to sometimes make "back of the envelope" calculations about the phenomenon of interest, and therefore be able to more efficiently design experiments to measure it, or to judge whether it is important, etc.
|
Dimensional units
| 0.856971
|
227
|
--gc:arc – Automatic reference counting (ARC) with move semantics optimizations, offers a shared heap. It offers fully deterministic performance for hard realtime systems. Reference cycles may cause memory leaks: these may be dealt with by manually annotating {.acyclic.}
|
Nim (programming language)
| 0.856935
|
228
|
One DNA or RNA molecule differs from another primarily in the sequence of nucleotides. Nucleotide sequences are of great importance in biology since they carry the ultimate instructions that encode all biological molecules, molecular assemblies, subcellular and cellular structures, organs, and organisms, and directly enable cognition, memory, and behavior. Enormous efforts have gone into the development of experimental methods to determine the nucleotide sequence of biological DNA and RNA molecules, and today hundreds of millions of nucleotides are sequenced daily at genome centers and smaller laboratories worldwide. In addition to maintaining the GenBank nucleic acid sequence database, the National Center for Biotechnology Information (NCBI, https://www.ncbi.nlm.nih.gov) provides analysis and retrieval resources for the data in GenBank and other biological data made available through the NCBI web site.
|
Nucleic acids
| 0.856883
|
229
|
In elementary algebra, the quadratic formula is a formula that provides the two solutions to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as completing the square. Given a general quadratic equation of the form whose discriminant b 2 − 4 a c {\displaystyle b^{2}-4ac} is positive, with x representing an unknown, with a, b and c representing constants, and with a ≠ 0, the quadratic formula is: where the plus–minus symbol "±" indicates that the quadratic equation has two solutions. Written separately, they become: Each of these two solutions is also called a root (or zero) of the quadratic equation.
|
Quadratic formula
| 0.856707
|
230
|
For example, the force between two separated electric charges with spherical symmetry (in the vacuum of classical electromagnetism) is given by Coulomb's law: F C = 1 4 π ε 0 q 1 q 2 r 2 {\displaystyle F_{\text{C}}={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q_{1}q_{2}}{r^{2}}}} Here, q1 and q2 are the charges, r is the distance between their centres, and the value of the constant fraction 1 / ( 4 π ε 0 ) {\displaystyle 1/(4\pi \varepsilon _{0})} (known as the Coulomb constant, ke) is approximately 9 × 109 N⋅m2⋅C−2. Likewise, ε0 appears in Maxwell's equations, which describe the properties of electric and magnetic fields and electromagnetic radiation, and relate them to their sources. In electrical engineering, ε0 itself is used as a unit to quantify the permittivity of various dielectric materials.
|
Permittivity constant
| 0.85669
|
231
|
Some non-prime numbers n are such that every group of order n is cyclic. One can show that n = 15 is such a number using the Sylow theorems: Let G be a group of order 15 = 3 · 5 and n3 be the number of Sylow 3-subgroups. Then n3 ∣ {\displaystyle \mid } 5 and n3 ≡ 1 (mod 3). The only value satisfying these constraints is 1; therefore, there is only one subgroup of order 3, and it must be normal (since it has no distinct conjugates).
|
Sylow p-subgroup
| 0.856656
|
232
|
In 2003, Anthony Trewavas led a study to see how the roots interact with one another and study their signal transduction methods. He was able to draw similarities between water stress signals in plants affecting developmental changes and signal transductions in neural networks causing responses in muscle. Particularly, when plants are under water stress, there are abscisic acid dependent and independent effects on development. This brings to light further possibilities of plant decision-making based on its environmental stresses.
|
Plant intelligence
| 0.856618
|
233
|
The concept can also be applied to more general systems of equations, such as systems of polynomial equations or partial differential equations. In the case of the systems of polynomial equations, it may happen that an overdetermined system has a solution, but that no one equation is a consequence of the others and that, when removing any equation, the new system has more solutions. For example, ( x − 1 ) ( x − 2 ) = 0 , ( x − 1 ) ( x − 3 ) = 0 {\displaystyle (x-1)(x-2)=0,(x-1)(x-3)=0} has the single solution x = 1 , {\displaystyle x=1,} but each equation by itself has two solutions.
|
Over-determined system
| 0.856614
|
234
|
There is a significant runtime overhead in calling a function, including passing the arguments, branching to the subprogram, and branching back to the caller. The overhead often includes saving and restoring certain processor registers, allocating and reclaiming call frame storage, etc.. In some languages, each function call also implies automatic testing of the function's return code or the handling of exceptions that it may raise. A significant source of overhead in object-oriented languages is the intensively used dynamic dispatch for method calls. There are some seemingly obvious optimizations of procedure calls that cannot be applied if the procedures may have side effects.
|
Subroutine call
| 0.856456
|
235
|
Positional Assembly Constructing materials an atom or molecule at a time. Protein Design The design and construction of new proteins; an enabling technology for nanotechnology. Protein folding The process by which proteins acquire their functional, preordained, three-dimensional structure after they emerge, as linear polymers of amino acids, from the ribosome. Proteomics The term proteome refers to all the proteins expressed by a genome, and thus proteomics involves the identification of proteins in the body and the determination of their role in physiological and pathophysiological functions.
|
Glossary of nanotechnology
| 0.856438
|
236
|
3. The converse of the theorem need not hold. For example, R2 and R3 have isomorphic fundamental groups but are still not homeomorphic.
|
Induced homomorphism
| 0.856358
|
237
|
GUIDE-Seq (Genome-wide, Unbiased Identification of DSBs Enabled by Sequencing) is a molecular biology technique that allows for the unbiased in vitro detection of off-target genome editing events in DNA caused by CRISPR/Cas9 as well as other RNA-guided nucleases in living cells. Similar to LAM-PCR, it employs multiple PCRs to amplify regions of interest that contain a specific insert that preferentially integrates into double-stranded breaks. As gene therapy is an emerging field, GUIDE-Seq has gained traction as a cheap method to detect the off-target effects of potential therapeutics without needing whole genome sequencing.
|
GUIDE-Seq
| 0.856345
|
238
|
A well-known lower bound for unstructured sorting, in the decision tree model, is based on the factorial number of sorted orders that an unstructured list may have. Because each comparison can at best reduce the number of possible orderings by a factor of two, sorting requires a number of comparisons at least equal to the binary logarithm of the factorial, which is n log 2 n − O ( n ) {\displaystyle n\log _{2}n-O(n)} . Early work on X + Y {\displaystyle X+Y} sorting followed a similar approach by asking how many different sorted orderings are possible for this problem, and proving that this number is at most O ( n 8 n ) {\displaystyle O(n^{8n})} . However, because its binary logarithm is at most 8 n log 2 n + O ( 1 ) {\displaystyle 8n\log _{2}n+O(1)} , much smaller than the known time bounds for X + Y {\displaystyle X+Y} sorting, this method can only lead to weak lower bounds on the number of comparisons.The proof of this bound relates X + Y {\displaystyle X+Y} sorting to the complexity of an arrangement of hyperplanes in high-dimensional geometry.
|
X + Y sorting
| 0.856342
|
239
|
The laboratory infrastructure and computing resources used to perform this experiment on the MinION nanopore sequencer would be available in most molecular laboratories around the world. Using this system, the concept of portability, and thus sequencing influenza viruses in the clinic or field is now tenable.“ In a paper and accompanying editorial published in October 2015, a group of MinION users wrote, “At the time of this writing, around a dozen reports have emerged recounting utility of the MinION for de novo sequencing of viral, bacterial, and eukaryotic genomes.”. In March 2016 the company announced a chemistry upgrade to ‘R9’, using the protein nanopore CsgG in collaboration with the lab of Han Remaut (VIB/Vrije Universiteit Brussel).
|
Oxford Nanopore Technologies
| 0.856193
|
240
|
The number of left cosets of H in G is equal to the number of right cosets of H in G. This common value is called the index of H in G and is usually denoted by . Cosets are a basic tool in the study of groups; for example, they play a central role in Lagrange's theorem that states that for any finite group G, the number of elements of every subgroup H of G divides the number of elements of G. Cosets of a particular type of subgroup (a normal subgroup) can be used as the elements of another group called a quotient group or factor group. Cosets also appear in other areas of mathematics such as vector spaces and error-correcting codes.
|
Right coset
| 0.856164
|
241
|
Very simple algebra skills are often taught to young children. For example, a student might convert an addition equation such as 19 + 15 = ? into a simpler equation, 20 + 14 = ?
|
NCTM standards
| 0.856114
|
242
|
Similar to point masses, in electromagnetism physicists discuss a point charge, a point particle with a nonzero electric charge. The fundamental equation of electrostatics is Coulomb's law, which describes the electric force between two point charges. Another result, Earnshaw's theorem, states that a collection of point charges cannot be maintained in a static equilibrium configuration solely by the electrostatic interaction of the charges. The electric field associated with a classical point charge increases to infinity as the distance from the point charge decreases towards zero, which suggests that the model is no longer accurate in this limit.
|
Point-like particle
| 0.856104
|
243
|
The teacher can steer the students either through careful design of the questions (this rules out barometer questions), or through guiding the students to the desired choices. In case of the original barometer question, the examiner may explicitly say that the problem has more than one solution, insist on applying the laws of physics, or give them the "ending point" of the solution: "How did I discover that the building was 410 feet in height with only a barometer? "Herson used the Calandra account as an illustration of the difference between academic tests and assessment in education. Tests, even the ones designed for reliability and validity, are useful, but they are not sufficient in real-world education.Sanders interpreted Calandra's story as a conflict between perfection and optimal solutions: "We struggle to determine a 'best' answer, when a simple call to a building superintendent (the resource man) would quickly provide adequate information."
|
Barometer question
| 0.856095
|
244
|
Intuitively, though, it is plausible that if the theorem holds for a single point charge then it would also hold for two opposite point charges connected together. In particular, it would hold in the limit where the distance between the charges is decreased to zero while maintaining the dipole moment – that is, it would hold for an electric dipole. But if the theorem holds for an electric dipole, then it will also hold for a magnetic dipole, since the (static) force/energy equations take the same form for both electric and magnetic dipoles.
|
Earnshaw's Theorem
| 0.85609
|
245
|
Each calculation is able to be done with a certain trick or shortcut that makes the calculations easier. The high school exam includes calculus and other difficult topics in the questions also with the same rules applied as to the middle school version. It is well known that the grading for this event is particularly stringent as errors such as writing over a line or crossing out potential answers are considered as incorrect answers.General Mathematics is a 50-question exam that students are given only 40 minutes to solve.
|
Texas Math and Science Coaches Association
| 0.856076
|
246
|
The maximum number of electrons that can be placed in a subshell is given by 2(2l + 1). This gives two electrons in an s subshell, six electrons in a p subshell, ten electrons in a d subshell and fourteen electrons in an f subshell. The numbers of electrons that can occupy each shell and each subshell arise from the equations of quantum mechanics, in particular the Pauli exclusion principle, which states that no two electrons in the same atom can have the same values of the four quantum numbers.
|
Electron shell configuration
| 0.856036
|
247
|
There are two rounds in the Junior Section: a written round (Round 1) and an invitational round (Round 2). The paper in Round 1 comprises 5 multiple-choice questions, each with five options, and 20 short answer questions. The Junior section is geared towards Lower Secondary students, and topics tested include number theory, combinatorics, geometry, algebra, and probability. Beginning in 2006, a second round was added, based on the Senior Invitational Round, in the form of a 5-question, 3-hour long paper requiring full-length solutions. Only the top 10% of students from Round 1 are eligible to take Round 2.
|
Singapore Mathematical Olympiad
| 0.85586
|
248
|
If Theorem 1 holds, and φ is not satisfiable in any structure, then ¬φ is valid in all structures and therefore provable, thus φ is refutable and Theorem 2 holds. If on the other hand Theorem 2 holds and φ is valid in all structures, then ¬φ is not satisfiable in any structure and therefore refutable; then ¬¬φ is provable and then so is φ, thus Theorem 1 holds.
|
Original proof of Gödel's completeness theorem
| 0.855688
|
249
|
Rabin–Karp string-search algorithm radix quicksort radix sort ragged matrix Raita algorithm random-access machine random number generation randomization randomized algorithm randomized binary search tree randomized complexity randomized polynomial time randomized rounding randomized search tree Randomized-Select random number generator random sampling range (function) range sort Rank (graph theory) Ratcliff/Obershelp pattern recognition reachable rebalance recognizer rectangular matrix rectilinear rectilinear Steiner tree recurrence equations recurrence relation recursion recursion termination recursion tree recursive (computer science) recursive data structure recursive doubling recursive language recursively enumerable language recursively solvable red–black tree reduced basis reduced digraph reduced ordered binary decision diagram (ROBDD) reduction reflexive relation regular decomposition rehashing relation (mathematics) relational structure relative performance guarantee relaxation relaxed balance rescalable restricted universe sort result cache Reverse Colussi Reverse Factor R-file Rice's method right rotation right-threaded tree root root balance rooted tree rotate left rotate right rotation rough graph RP R+-tree R*-tree R-tree run time
|
List of terms relating to algorithms and data structures
| 0.855684
|
250
|
An ideal constant-force spring is a spring for which the force it exerts over its range of motion is a constant, that is, it does not obey Hooke's law. In reality, "constant-force springs" do not provide a truly constant force and are constructed from materials which do obey Hooke's law. Generally constant-force springs are constructed as a rolled ribbon of spring steel such that the spring is in a rolled up form when relaxed. As the spring is unrolled, the material coming off the roll (un)bends from the radius of the roll into a straight line between the reel and the load. Because the material tension-stiffness of the straight section is orders of magnitude greater than the bending stiffness of the ribbon, the straight section does not stretch significantly, the restoring force comes primarily from the deformation of the portion of the ribbon near the roll. Because the geometry of that region remains nearly constant as the spring unrolls (with material coming off the roll joining the curved section, and material in the curved section joining the straight section), the resulting force is nearly constant.
|
Constant-force spring
| 0.855635
|
251
|
The goal of complexity analysis in this model is to find time bounds that depend only on n and not on the actual size of the input values or the machine words. In modeling integer computation, it is necessary to assume that machine words are limited in size, because models with unlimited precision are unreasonably powerful (able to solve PSPACE-complete problems in polynomial time). The transdichotomous model makes a minimal assumption of this type: that there is some limit, and that the limit is large enough to allow random access indexing into the input data.As well as its application to integer sorting, the transdichotomous model has also been applied to the design of priority queues and to problems in computational geometry and graph algorithms.
|
Transdichotomous model
| 0.855605
|
252
|
Therefore, it is a basic tool for the homotopy continuation method described below. Optimization is rarely used for solving polynomial systems, but it succeeded, circa 1970, in showing that a system of 81 quadratic equations in 56 variables is not inconsistent. With the other known methods, this remains beyond the possibilities of modern technology, as of 2022.
|
Polynomial system of equations
| 0.855598
|
253
|
Calculators are not allowed. The problems chosen are from various areas of secondary school mathematics, broadly classifiable as geometry, number theory, algebra, and combinatorics. They require no knowledge of higher mathematics such as calculus and analysis, and solutions are often elementary.
|
International Mathematical Olympiad
| 0.855541
|
254
|
The three letters ABC can be reordered into ABC, ACB, BAC, BCA, CAB, CBA, forming in total 6 (factorial of 3) elements. The group operation is composition of these reorderings, and the identity element is the reordering operation that leaves the order unchanged. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group S N {\displaystyle \mathrm {S} _{N}} for a suitable integer N {\displaystyle N} , according to Cayley's theorem.
|
Group (mathematics)
| 0.855531
|
255
|
Computational advances have enabled cheaper and faster sequencing. Research has focused on combinatorial chemistry, genomic mining, omic technologies, and high throughput screening. As the cost per genetic test decreases, the development of personalized drug therapies will increase.
|
Extensive metabolizer
| 0.855511
|
256
|
From the viewpoint of the experimental physics, the requirement that the marks be made simultaneously is redundant for a stationary object with constant shape and size, and can in this case be dropped from such definition. Since the rod is stationary in K0, the distance between the marks is the proper length of the rod regardless of the time lapse between the two markings. On the other hand, it is not the proper distance between the marking events if the marks are not made simultaneously in K0."
|
Proper length
| 0.855466
|
257
|
During the lessons Four Team Selection Tests are held which are considered the four parts of the Selection Competition above 15.5 and are called Euclides. All the student have the same test. In each of the competition five students are eliminated. So after the fourth competition the six member of national team for IMO and BMO and the four runners-up are selected. In every competition or test there are four problem usually covering geometry, number theory, algebra, and combinatorics (elementary level) and last four hours each.
|
Cyprus Mathematical Society
| 0.855455
|
258
|
DNA sequencing is the process of determining the nucleic acid sequence – the order of nucleotides in DNA. It includes any method or technology that is used to determine the order of the four bases: adenine, guanine, cytosine, and thymine. The advent of rapid DNA sequencing methods has greatly accelerated biological and medical research and discovery.Knowledge of DNA sequences has become indispensable for basic biological research, DNA Genographic Projects and in numerous applied fields such as medical diagnosis, biotechnology, forensic biology, virology and biological systematics. Comparing healthy and mutated DNA sequences can diagnose different diseases including various cancers, characterize antibody repertoire, and can be used to guide patient treatment.
|
DNA sequence
| 0.855444
|
259
|
This variable represents the difference between the two sides of the inequality and is assumed to be non-negative. For example, the inequalities x 2 + 2 x 3 ≤ 3 − x 4 + 3 x 5 ≥ 2 {\displaystyle {\begin{aligned}x_{2}+2x_{3}&\leq 3\\-x_{4}+3x_{5}&\geq 2\end{aligned}}} are replaced with x 2 + 2 x 3 + s 1 = 3 − x 4 + 3 x 5 − s 2 = 2 s 1 , s 2 ≥ 0 {\displaystyle {\begin{aligned}x_{2}+2x_{3}+s_{1}&=3\\-x_{4}+3x_{5}-s_{2}&=2\\s_{1},\,s_{2}&\geq 0\end{aligned}}} It is much easier to perform algebraic manipulation on inequalities in this form. In inequalities where ≥ appears such as the second one, some authors refer to the variable introduced as a surplus variable.
|
Pivot operations
| 0.855429
|
260
|
An associative quasigroup is either empty or is a group, since if there is at least one element, the invertibility of the quasigroup binary operation combined with associativity implies the existence of an identity element, which then implies the existence of inverse elements, thus satisfying all three requirements of a group. The following construction is due to Hans Zassenhaus. On the underlying set of the four-dimensional vector space F4 over the 3-element Galois field F = Z/3Z define(x1, x2, x3, x4) ∗ (y1, y2, y3, y4) = (x1, x2, x3, x4) + (y1, y2, y3, y4) + (0, 0, 0, (x3 − y3)(x1y2 − x2y1)). Then, (F4, ∗) is a commutative Moufang loop that is not a group.More generally, the nonzero elements of any division algebra form a quasigroup with the operation of multiplication in the algebra.
|
Quasigroup
| 0.855376
|
261
|
Referring back to the pictorial representation of Ratio and Algebra hierarchy, an item can be constructed to measure the skills described in each of the attributes. For example, attribute A1 includes the most basic arithmetic operation skills, such as addition, subtraction, multiplication, and division of numbers. An item that measures this skill could be the following: examinees are presented with the algebraic expression 4 ( t + u ) + 3 = 19 {\displaystyle 4(t+u)+3=19} , and asked to solve for (t + u). For this item, examinees need to subtract 3 from 19 and then divide 16 by 4.
|
Attribute hierarchy method
| 0.855326
|
262
|
The slope of a line is a number that describes both the direction and the steepness of the line. Slope is often denoted by the letter m.Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged during the 3rd century BC from applications of geometry to astronomical studies. The slope is studied in grade 8.
|
Elementary mathematics
| 0.855319
|
263
|
Other applications include medical image classification, genomic sequence analysis, as well as protein structure classification and prediction. Deep learning has been applied to regulatory genomics, variant calling and pathogenicity scores. Natural language processing and text mining have helped to understand phenomena including protein-protein interaction, gene-disease relation as well as predicting biomolecule structures and functions.
|
Machine learning in bioinformatics
| 0.855227
|
264
|
Saha had previously reached the following conclusion on the subject: It will be admitted from what has gone before that the temperature plays the leading role in determining the nature of the stellar spectrum. Too much importance must not be attached to the figures given, for the theory is only a first attempt for quantitatively estimating the physical processes taking place at high temperature. We have practically no laboratory data to guide us, but the stellar spectra may be regarded as unfolding to us, in an unbroken sequence, the physical processes succeeding each other as the temperature is continually varied from 3000 K to 40,000 K. Saha also invented an instrument to measure the weight and pressure of solar rays and helped establish several scientific institutions, including the Physics Department at Allahabad University and the Institute of Nuclear Physics in Kolkata.
|
Meghnad Saha
| 0.855222
|
265
|
However, in some cases a compiler optimization is possible, namely performing escape analysis and proving that escape is not possible, and thus the object can be allocated on the stack; this is significant in Java. In this case object destruction will occur promptly – possibly even during the variable's lifetime (before the end of its scope), if it is unreachable. A complex case is the use of an object pool, where objects may be created ahead of time or reused, and thus apparent creation and destruction may not correspond to actual creation and destruction of an object, only (re)initialization for creation and finalization for destruction. In this case both creation and destruction may be nondeterministic.
|
Object creation
| 0.855214
|
266
|
Equations involving linear or simple rational functions of a single real-valued unknown, say x, such as 8 x + 7 = 4 x + 35 or 4 x + 9 3 x + 4 = 2 , {\displaystyle 8x+7=4x+35\quad {\text{or}}\quad {\frac {4x+9}{3x+4}}=2\,,} can be solved using the methods of elementary algebra.
|
Equation solving
| 0.855124
|
267
|
Automated DNA sequencing techniques have revolutionized the field of molecular biology – generating vast amounts of DNA sequence data. However, the sequence data is produced at a significantly higher rate than can be manually processed (i.e. interpreting the trace data to produce the sequence data), thereby creating a bottleneck.
|
Phred base calling
| 0.855113
|
268
|
There are several theorems and conjectures regarding the worst-case runtime for performing a sequence S of m accesses in a splay tree containing n elements.
|
Splay tree
| 0.855104
|
269
|
The multiple choice section is scored by computer, with a correct answer receiving 1 point, with omitted and incorrect answers not affecting the raw score. This total is multiplied by 1.2 to calculate the adjusted multiple-choice score.The free response section is hand-graded by hundreds of educators each June. The raw score is then added to the adjusted multiple choice score to receive a composite score. This total is compared to a composite-score scale for that year's exam and converted into an AP score of 1 to 5. For the Calculus BC exam, an AB sub-score is included in the score report to reflect their proficiency in the fundamental topics of introductory calculus. The AB sub-score is based on the correct number of answers for questions pertaining to AB-material only.
|
AP Calculus
| 0.855103
|
270
|
Curves on the graph can be traced by a cursor, where local maximum, local minimum, and intersections of the curves can be solved by a graph solver tool. Conics having their axes parallel to x-axis or y-axis can be plotted through a separate application. This means that the calculators cannot plot other conic/quadratic equations A x 2 + B x y + C y 2 + D x + E y + F = 0 {\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0} where B {\displaystyle B} is not equal to 0. The Algebra FX series cannot plot any implicit function of two variables, while the Texas Instrument's TI-89 series can.In addition to the simple graphing tool, the calculators also provide a dynamic graph mode, where real-time changes of the graphs, affected by changes in variables and coefficients, are visualized.
|
Casio Algebra FX Series
| 0.855085
|
271
|
Let Z {\displaystyle Z} be the set of integers and ' − {\displaystyle -} ' be the binary operation 'subtraction'. Then the algebraic structure ( Z , − ) {\displaystyle (Z,-)} obeys the following properties: x − x = 0 {\displaystyle x-x=0} x − 0 = x {\displaystyle x-0=x} 0 − ( x − y ) = y − x {\displaystyle 0-(x-y)=y-x}
|
BF-algebra
| 0.854971
|
272
|
The third problem is the clinical utility of personal genome kits and associated risks, and the benefits of introducing them into clinical practices.People need to be educated on interpreting their results and what they should be rationally taking from the experience. Concerns about customers misinterpreting health information was one of the reasons for the 2013 shutdown by the FDA of 23&Me's health analysis services. It is not only the average person who needs to be educated in the dimensions of their own genomic sequence but also professionals, including physicians and science journalists, who must be provided with the knowledge required to inform and educate their patients and the public. Examples of such efforts include the Personal Genetics Education Project (pgEd), the Smithsonian collaboration with NHGRI, and the MedSeq, BabySeq and MilSeq projects of Genomes to People, an initiative of Harvard Medical School and Brigham and Women's Hospital. A major use of personal genomics outside the realm of health is that of ancestry analysis (see Genetic Genealogy), including evolutionary origin information such as neanderthal content.
|
Genome analysis
| 0.854935
|
273
|
Many computer science problems are contained in NP, like decision versions of many search and optimization problems.
|
NP class
| 0.854919
|
274
|
That gives rise to 3 equivalent classes, from the code review itself. To solve the input problem, we take refuge in the inequation z m i n ≤ x + y ≤ z m a x {\displaystyle z_{min}\leq x+y\leq z_{max}} we note that there is a fixed size of Integer (computer science) hence, the z can be replaced with:- INT_MIN ≤ x + y ≤ INT_MAXand with x ∈ { INT_MIN , ... , INT_MAX } and y ∈ { INT_MIN , ... , INT_MAX } The values of the test vector at the strict condition of the equality that is INT_MIN = x + y and INT_MAX = x + y are called the boundary values, Boundary-value analysis has detailed information about it.
|
Equivalence partitioning
| 0.854872
|
275
|
As long as they are not twisted beyond their elastic limit, torsion springs obey an angular form of Hooke's law: τ = − κ θ {\displaystyle \tau =-\kappa \theta \,} where τ {\displaystyle \tau \,} is the torque exerted by the spring in newton-meters, and θ {\displaystyle \theta \,} is the angle of twist from its equilibrium position in radians κ {\displaystyle \kappa \,} is a constant with units of newton-meters / radian, variously called the spring's torsion coefficient, torsion elastic modulus, rate, or just spring constant, equal to the change in torque required to twist the spring through an angle of 1 radian.The torsion constant may be calculated from the geometry and various material properties. It is analogous to the spring constant of a linear spring. The negative sign indicates that the direction of the torque is opposite to the direction of twist. The energy U, in joules, stored in a torsion spring is: U = 1 2 κ θ 2 {\displaystyle U={\frac {1}{2}}\kappa \theta ^{2}}
|
Torsion spring
| 0.854829
|
276
|
An almost perfect blackbody spectrum is exhibited by the cosmic microwave background radiation. Hawking radiation is the hypothetical blackbody radiation emitted by black holes, at a temperature that depends on the mass, charge, and spin of the hole. If this prediction is correct, black holes will very gradually shrink and evaporate over time as they lose mass by the emission of photons and other particles. A black body radiates energy at all frequencies, but its intensity rapidly tends to zero at high frequencies (short wavelengths). For example, a black body at room temperature (300 K) with one square meter of surface area will emit a photon in the visible range (390–750 nm) at an average rate of one photon every 41 seconds, meaning that, for most practical purposes, such a black body does not emit in the visible range.The study of the laws of black bodies and the failure of classical physics to describe them helped establish the foundations of quantum mechanics.
|
Thermal cavity
| 0.854784
|
277
|
In computer science, heapsort is a comparison-based sorting algorithm. Heapsort can be thought of as an improved selection sort: like selection sort, heapsort divides its input into a sorted and an unsorted region, and it iteratively shrinks the unsorted region by extracting the largest element from it and inserting it into the sorted region. Unlike selection sort, heapsort does not waste time with a linear-time scan of the unsorted region; rather, heap sort maintains the unsorted region in a heap data structure to more quickly find the largest element in each step.Although somewhat slower in practice on most machines than a well-implemented quicksort, it has the advantage of a more favorable worst-case O(n log n) runtime (and as such is used by Introsort as a fallback should it detect that quicksort is becoming degenerate). Heapsort is an in-place algorithm, but it is not a stable sort.
|
Heap Sort
| 0.854777
|
278
|
Applications of group theory abound. Almost all structures in abstract algebra are special cases of groups. Rings, for example, can be viewed as abelian groups (corresponding to addition) together with a second operation (corresponding to multiplication). Therefore, group theoretic arguments underlie large parts of the theory of those entities.
|
Group Theory
| 0.854764
|
279
|
Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%). These concepts have been given an axiomatic mathematical formalization in probability theory, which is used widely in areas of study such as statistics, mathematics, science, finance, gambling, artificial intelligence, machine learning, computer science, game theory, and philosophy to, for example, draw inferences about the expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of complex systems.
|
Probability
| 0.85476
|
280
|
: 21 Rather, each mathematical theory describes its objects by some of their properties, precisely those that are put as axioms at the foundations of the theory. : 20 Distances and angles cannot appear in theorems of projective geometry, since these notions are neither mentioned in the axioms of projective geometry nor defined from the notions mentioned there. The question "what is the sum of the three angles of a triangle" is meaningful in Euclidean geometry but meaningless in projective geometry.
|
List of mathematical spaces
| 0.854753
|
281
|
In genetics, a three-point cross is used to determine the loci of three genes in an organism's genome. An individual heterozygous for three mutations is crossed with a homozygous recessive individual, and the phenotypes of the progeny are scored. The two most common phenotypes that result are the parental gametes; the two least common phenotypes that result come from a double crossover in gamete formation. By comparing the parental and double-crossover phenotypes, the geneticist can determine which gene is located between the others on the chromosome.
|
Three-point cross
| 0.854752
|
282
|
In algebraic form we have two equations: a + b + c + d = 1a + b + c + d + e + f + g = 1Here a, b, c, and d correspond to the top four grayed cells in Figure 6. The cell with Δ is represented by the variable f, and the other two grayed cells are marked as e and g. If we set f = 1, then a = 0, b = 0, c = 0, d = 0, e = 0, g = 0. The first equation above will have the left hand side equal to 0 while the right hand side has 1.
|
Board puzzles with algebra of binary variables
| 0.854741
|
283
|
Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication, and conjugation in groups. An operation of arity two that involves several sets is sometimes also called a binary operation. For example, scalar multiplication of vector spaces takes a scalar and a vector to produce a vector, and scalar product takes two vectors to produce a scalar. Such binary operations may be called simply binary functions. Binary operations are the keystone of most algebraic structures that are studied in algebra, in particular in semigroups, monoids, groups, rings, fields, and vector spaces.
|
Binary operation
| 0.854728
|
284
|
Robbins Equation: n(n(x + y) + n(x + n(y))) = x,do (1), (2), and (4) form a basis for Boolean algebra? Calling (1), (2), and (4) a Robbins algebra, the question then becomes: Is every Robbins algebra a Boolean algebra? This question (which came to be known as the Robbins conjecture) remained open for decades, and became a favorite question of Alfred Tarski and his students.
|
Boolean lattice
| 0.854726
|
285
|
However, this alternative definition includes one exceptional structure of order 2 which fails to satisfy various basic theorems (such as x ⋅ 0 = 0 {\displaystyle x\cdot 0=0} for all x {\displaystyle x} ). Thus it is much more convenient, and more usual, to use the axioms in the form given above. The difference is that A4 requires 1 to be an identity for all elements, A4* only for non-zero elements. The exceptional structure can be defined by taking an additive group of order 2, and defining multiplication by x ⋅ y = x {\displaystyle x\cdot y=x} for all x {\displaystyle x} and y {\displaystyle y} .
|
Near-field (mathematics)
| 0.854713
|
286
|
(from the Turing–Wittgenstein debate) Do Gödel's incompleteness theorems imply that physical theories will always be incomplete? (from Stephen Hawking) Is mathematics invented or discovered? (millennia-old question, raised among others by Mario Livio)
|
Relationship between mathematics and physics
| 0.854688
|
287
|
Now, through bioinformatics, there are computer programs that can in some cases predict and model the structure of proteins. These programs use the chemical properties of amino acids and structural properties of known proteins to predict the 3D model of sample proteins. This also allows scientists to model protein interactions on a larger scale. In addition, biomedical engineers are developing methods to factor in the flexibility of protein structures to make comparisons and predictions.
|
Protein analysis
| 0.854652
|
288
|
These rules are summarized as follows: Example: The quantum numbers used to refer to the outermost valence electrons of a carbon (C) atom, which are located in the 2p atomic orbital, are; n = 2 (2nd electron shell), ℓ = 1 (p orbital subshell), mℓ = 1, 0, −1, ms = 1/2 (parallel spins). Results from spectroscopy indicated that up to two electrons can occupy a single orbital. However two electrons can never have the same exact quantum state nor the same set of quantum numbers according to Hund's rules, which addresses the Pauli exclusion principle. A fourth quantum number, which represented spin with two possible values, was added as an ad hoc assumption to resolve the conflict; this supposition would later be explained in detail by relativistic quantum mechanics and from the results of the renowned Stern–Gerlach experiment.
|
Quantum numbers
| 0.854591
|
289
|
The analog of Lagrange's theorem for double cosets is false. This means that the size of a double coset need not divide the order of G. For example, let G = S3 be the symmetric group on three letters, and let H and K be the cyclic subgroups generated by the transpositions (1 2) and (1 3), respectively. If e denotes the identity permutation, then H e K = H K = { e , ( 12 ) , ( 13 ) , ( 132 ) } .
|
Double coset
| 0.854561
|
290
|
Targeted analysis sequencing (sometimes called target amplicon sequencing) (TAS) is a next-generation DNA sequencing technique focusing on amplicons and specific genes. It is useful in population genetics since it can target a large diversity of organisms. The TAS approach incorporates bioinformatics techniques to produce a large amount of data at a fraction of the cost involved in Sanger sequencing. TAS is also useful in DNA studies because it allows for amplification of the needed gene area via PCR, which is followed by next-gen sequencing platforms.
|
Targeted analysis sequencing
| 0.854549
|
291
|
For example: a) Things would be different if Microsoft were located in Georgia.In sentence (a), a shallow information extraction system might infer wrongly that Microsoft's headquarters was located in Georgia. While as humans, we understand from the sentence that Microsoft office was never in Georgia. b) The National Institute for Psychology in Israel was established in May 1971 as the Israel Center for Psychobiology by Prof. Joel.In sentence (b), a shallow system could wrongly infer that Israel was established in May 1971.
|
Deep linguistic processing
| 0.854487
|
292
|
It has in the past been combined with array technology (ChIP-chip) with some success. Next gen sequencing can also be applied in this area. Methylation immunoprecipitation (MeDIP) can also be performed and also on arrays. The ability to learn more about methylation and TF binding sites on a genome wide scale is a valuable resource and could teach us much about disease and molecular biology in general.
|
ABI Solid Sequencing
| 0.85446
|
293
|
In computer science, splaysort is an adaptive comparison sorting algorithm based on the splay tree data structure.
|
Splaysort
| 0.854435
|
294
|
Such an overdetermined system has no solution in general (that is if the coefficients are not specific). If it has a finite number of solutions, this number is at most 53 = 125, by Bézout's theorem. However, it has been shown that, for the case of the singular points of a surface of degree 6, the maximum number of solutions is 65, and is reached by the Barth surface.
|
Polynomial system of equations
| 0.854415
|
295
|
This is the defining arithmetical identity of group theory, in boundary notation. The primary algebra differs from an abelian group in two ways: From A2, it follows that ≠ . If the primary algebra were a group, = would hold, and one of a = or a = a would have to be a primary algebra consequence. Note that and are mutual primary algebra complements, as group theory requires, so that | ¯ | ¯ | ¯ = | ¯ {\displaystyle {\overline {\ {\overline {\ {\overline {\ \ |}}\ {\Big |}}}\ {\Bigg |}}}={\overline {\ \ |}}} is true of both group theory and the primary algebra; C2 most clearly demarcates the primary algebra from other magmas, because C2 enables demonstrating the absorption law that defines lattices, and the distributive law central to Boolean algebra.Both A2 and C2 follow from B's being an ordered set.
|
Boundary algebra
| 0.854396
|
296
|
In computer science, smoothsort is a comparison-based sorting algorithm. A variant of heapsort, it was invented and published by Edsger Dijkstra in 1981. Like heapsort, smoothsort is an in-place algorithm with an upper bound of O(n log n), but it is not a stable sort. The advantage of smoothsort is that it comes closer to O(n) time if the input is already sorted to some degree, whereas heapsort averages O(n log n) regardless of the initial sorted state.
|
Smoothsort
| 0.854355
|
297
|
In Euclidean geometry, it is possible to associate a set of coordinates to each point in space, for example by an orthogonal grid. This method allows one to characterize geometric figures by equations. A plane in three-dimensional space can be expressed as the solution set of an equation of the form a x + b y + c z + d = 0 {\displaystyle ax+by+cz+d=0} , where a , b , c {\displaystyle a,b,c} and d {\displaystyle d} are real numbers and x , y , z {\displaystyle x,y,z} are the unknowns that correspond to the coordinates of a point in the system given by the orthogonal grid. The values a , b , c {\displaystyle a,b,c} are the coordinates of a vector perpendicular to the plane defined by the equation.
|
Equation
| 0.854324
|
298
|
If mathematicians sometimes talk about a field with one element, this abstract and somewhat mysterious mathematical object is not a field. In categories where the multiplicative identity must be preserved by morphisms, but can equal to zero, the {0} object can exist. But not as initial object because identity-preserving morphisms from {0} to any object where 1 ≠ 0 do not exist. For example, in the category of rings Ring the ring of integers Z is the initial object, not {0}. If an algebraic structure requires the multiplicative identity, but neither its preservation by morphisms nor 1 ≠ 0, then zero morphisms exist and the situation is not different from non-unital structures considered in the previous section.
|
Zero vector space
| 0.854312
|
299
|
For example, a multiple-choice exam designed to assess knowledge of key concepts in natural selection does not meet a number of standards of quality control. One problem with the exam is that the two members of each of several pairs of parallel items, with each pair designed to measure exactly one key concept in natural selection, sometimes have very different levels of difficulty. Another problem is that the multiple-choice exam overestimates knowledge of natural selection as reflected in student performance on a diagnostic essay exam and a diagnostic oral exam, two instruments with reasonably good construct validity. Although scoring concept inventories in the form of essay or oral exams is labor-intensive, costly, and difficult to implement with large numbers of students, such exams can offer a more realistic appraisal of the actual levels of students' conceptual mastery as well as their misconceptions. Recently, however, computer technology has been developed that can score essay responses on concept inventories in biology and other domains, promising to facilitate the scoring of concept inventories organized as (transcribed) oral exams as well as essays.
|
Concept inventory
| 0.854277
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.