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wiki_1300_chunk_0 | K-synchronized sequence | In mathematics and theoretical computer science, a k-synchronized sequence is an infinite sequence of terms s(n) characterized by a finite automaton taking as input two strings m and n, each expressed in some fixed base k, and accepting if m = s(n). The class of k-synchronized sequences lies between the classes of k-au... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1301_chunk_0 | Semiautomaton | In mathematics and theoretical computer science, a semiautomaton is a deterministic finite automaton having inputs but no output. It consists of a set Q of states, a set Σ called the input alphabet, and a function T: Q × Σ → Q called the transition function. Associated with any semiautomaton is a monoid called the char... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1302_chunk_0 | Set constraint | In mathematics and theoretical computer science, a set constraint is an equation or an inequation between sets of terms. Similar to systems of (in)equations between numbers, methods are studied for solving systems of set constraints. Different approaches admit different operators (like "∪", "∩", "\", and function appli... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1303_chunk_0 | Automatic sequence | In mathematics and theoretical computer science, an automatic sequence (also called a k-automatic sequence or a k-recognizable sequence when one wants to indicate that the base of the numerals used is k) is an infinite sequence of terms characterized by a finite automaton. The n-th term of an automatic sequence a(n) is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1304_chunk_0 | Analysis of Boolean functions | In mathematics and theoretical computer science, analysis of Boolean functions is the study of real-valued functions on { 0 , 1 } n {\displaystyle \{0,1\}^{n}} or { − 1 , 1 } n {\displaystyle \{-1,1\}^{n}} (such functions are sometimes known as pseudo-Boolean functions) from a spectral perspective. The functions studie... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1305_chunk_0 | Amplituhedron | In mathematics and theoretical physics (especially twistor string theory), an amplituhedron is a geometric structure introduced in 2013 by Nima Arkani-Hamed and Jaroslav Trnka. It enables simplified calculation of particle interactions in some quantum field theories. In planar N = 4 supersymmetric Yang–Mills theory, al... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1306_chunk_0 | Twistor space | In mathematics and theoretical physics (especially twistor theory), twistor space is the complex vector space of solutions of the twistor equation ∇ A ′ ( A Ω B ) = 0 {\displaystyle \nabla _{A'}^{(A}\Omega _{^{}}^{B)}=0} . It was described in the 1960s by Roger Penrose and Malcolm MacCallum. According to Andrew Hodges,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1307_chunk_0 | Noether's second theorem | In mathematics and theoretical physics, Noether's second theorem relates symmetries of an action functional with a system of differential equations. The action S of a physical system is an integral of a so-called Lagrangian function L, from which the system's behavior can be determined by the principle of least action.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1308_chunk_0 | Noether's second theorem | Noether's second theorem is sometimes used in gauge theory. Gauge theories are the basic elements of all modern field theories of physics, such as the prevailing Standard Model. The theorem is named after Emmy Noether. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1309_chunk_0 | One particle Hilbert space | In mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative ( E ≥ 0 ) {\displaystyle ~(~E\geq 0~)~} energy irreducible unitary representations of the Poincaré group which have either finite or zero mass eigenvalues. (Since this group is noncompact, these unitary representation... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1310_chunk_0 | One particle Hilbert space | The Casimir invariants of the Poincaré group are C 1 = P μ P μ , {\displaystyle ~C_{1}=P^{\mu }\,P_{\mu }~,} (Einstein notation) where P is the 4-momentum operator, and C 2 = W α W α , {\displaystyle ~C_{2}=W^{\alpha }\,W_{\alpha }~,} where W is the Pauli–Lubanski pseudovector. The eigenvalues of these operators serve ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1311_chunk_0 | One particle Hilbert space | The physically relevant representations may thus be classified according to whether m > 0 ; {\displaystyle ~m>0~;} m = 0 {\displaystyle ~m=0~} but P 0 > 0 ; {\displaystyle ~P_{0}>0~;\quad } or whether m = 0 {\displaystyle ~m=0~} with P μ = 0 , for μ = 0 , 1 , 2 , 3 . {\displaystyle ~P^{\mu }=0~,{\text{ for }}\mu =0,1,2... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1312_chunk_0 | One particle Hilbert space | For the first case Note that the eigenspace (see generalized eigenspaces of unbounded operators) associated with P = ( m , 0 , 0 , 0 ) {\displaystyle ~P=(m,0,0,0)~} is a representation of SO(3).In the ray interpretation, one can go over to Spin(3) instead. So, massive states are classified by an irreducible Spin(3) uni... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1313_chunk_0 | One particle Hilbert space | This is the double cover of SE(2) (see projective representation). We have two cases, one where irreps are described by an integral multiple of 1/2 called the helicity, and the other called the "continuous spin" representation. For the third case The only finite-dimensional unitary solution is the trivial representatio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1314_chunk_0 | Gerstenhaber algebra | In mathematics and theoretical physics, a Gerstenhaber algebra (sometimes called an antibracket algebra or braid algebra) is an algebraic structure discovered by Murray Gerstenhaber (1963) that combines the structures of a supercommutative ring and a graded Lie superalgebra. It is used in the Batalin–Vilkovisky formali... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1315_chunk_0 | Bifundamental representation | In mathematics and theoretical physics, a bifundamental representation is a representation obtained as a tensor product of two fundamental or antifundamental representations. For example, the MN-dimensional representation (M,N) of the group S U ( M ) × S U ( N ) {\displaystyle SU(M)\times SU(N)} is a bifundamental repr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1316_chunk_0 | Locally compact quantum group | In mathematics and theoretical physics, a locally compact quantum group is a relatively new C*-algebraic approach toward quantum groups that generalizes the Kac algebra, compact-quantum-group and Hopf-algebra approaches. Earlier attempts at a unifying definition of quantum groups using, for example, multiplicative unit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1317_chunk_0 | Pseudo-Euclidean space | In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional real n-space together with a non-degenerate quadratic form q. Such a quadratic form can, given a suitable choice of basis (e1, …, en), be applied to a vector x = x1e1 + ⋯ + xnen, giving which is called the scalar square of the vect... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1318_chunk_0 | Representation theory of Lie groups | In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vector space. Representations play an important role in the study of continuous ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1319_chunk_0 | Associative superalgebra | In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading. The prefix super- comes from the theory of supersymmetry in theoretical phy... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1320_chunk_0 | Associative superalgebra | Superalgebras and their representations, supermodules, provide an algebraic framework for formulating supersymmetry. The study of such objects is sometimes called super linear algebra. Superalgebras also play an important role in related field of supergeometry where they enter into the definitions of graded manifolds, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1321_chunk_0 | Supermatrix | In mathematics and theoretical physics, a supermatrix is a Z2-graded analog of an ordinary matrix. Specifically, a supermatrix is a 2×2 block matrix with entries in a superalgebra (or superring). The most important examples are those with entries in a commutative superalgebra (such as a Grassmann algebra) or an ordinar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1322_chunk_0 | Antisymmetric tensor | In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged. The index subset must generally either be all covariant or all contravariant. For example, holds when the tensor is antisymmetric wit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1323_chunk_0 | Invariant differential operator | In mathematics and theoretical physics, an invariant differential operator is a kind of mathematical map from some objects to an object of similar type. These objects are typically functions on R n {\displaystyle \mathbb {R} ^{n}} , functions on a manifold, vector valued functions, vector fields, or, more generally, se... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1324_chunk_0 | Deformed Hermitian Yang–Mills equation | In mathematics and theoretical physics, and especially gauge theory, the deformed Hermitian Yang–Mills (dHYM) equation is a differential equation describing the equations of motion for a D-brane in the B-model (commonly called a B-brane) of string theory. The equation was derived by Mariño-Minasian-Moore-Strominger in ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1325_chunk_0 | Braid statistics | In mathematics and theoretical physics, braid statistics is a generalization of the spin statistics of bosons and fermions based on the concept of braid group. While for fermions (Bosons) the corresponding statistics is associated to a phase gain of π {\displaystyle \pi } ( 2 π {\displaystyle 2\pi } ) under the exchang... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1326_chunk_0 | Fusion rules | In mathematics and theoretical physics, fusion rules are rules that determine the exact decomposition of the tensor product of two representations of a group into a direct sum of irreducible representations. The term is often used in the context of two-dimensional conformal field theory where the relevant group is gene... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1327_chunk_0 | Quasi-periodic motion | In mathematics and theoretical physics, quasiperiodic motion is in rough terms the type of motion executed by a dynamical system containing a finite number (two or more) of incommensurable frequencies.That is, if we imagine that the phase space is modelled by a torus T (that is, the variables are periodic like angles),... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1328_chunk_0 | Resummation | In mathematics and theoretical physics, resummation is a procedure to obtain a finite result from a divergent sum (series) of functions. Resummation involves a definition of another (convergent) function in which the individual terms defining the original function are re-scaled, and an integral transformation of this n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1329_chunk_0 | Resummation | Feynman and H. Kleinert. In quantum mechanics it was extended to any order here, and in quantum field theory here. See also Chapters 16–20 in the textbook cited below. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1330_chunk_0 | Eguchi–Hanson space | In mathematics and theoretical physics, the Eguchi–Hanson space is a non-compact, self-dual, asymptotically locally Euclidean (ALE) metric on the cotangent bundle of the 2-sphere T*S2. The holonomy group of this 4-real-dimensional manifold is SU(2). The metric is generally attributed to the physicists Tohru Eguchi and ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1331_chunk_0 | Eguchi–Hanson space | The even dimensional space of dimension d {\displaystyle d} can be described using complex coordinates w i ∈ C d / 2 {\displaystyle w_{i}\in \mathbb {C} ^{d/2}} with a metric g i j ¯ = ( 1 + ρ d r d ) 2 / d , {\displaystyle g_{i{\bar {j}}}={\bigg (}1+{\frac {\rho ^{d}}{r^{d}}}{\bigg )}^{2/d}{\bigg },} where ρ {\displa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1332_chunk_0 | Induced metric | In mathematics and theoretical physics, the induced metric is the metric tensor defined on a submanifold that is induced from the metric tensor on a manifold into which the submanifold is embedded, through the pullback. It may be determined using the following formula (using the Einstein summation convention), which is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1333_chunk_0 | Quantum group | In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras), compact matrix quantum groups (which are structures on unital separable C*... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1334_chunk_0 | Quantum group | The same term is also used for other Hopf algebras that deform or are close to classical Lie groups or Lie algebras, such as a "bicrossproduct" class of quantum groups introduced by Shahn Majid a little after the work of Drinfeld and Jimbo. In Drinfeld's approach, quantum groups arise as Hopf algebras depending on an a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1335_chunk_0 | Zeta function regularization | In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent sums or products, and in particular can be used to define determinants and traces of some self-adjoint operators. The technique is now commonly applied to proble... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1336_chunk_0 | Circular surface | In mathematics and, in particular, differential geometry a circular surface is the image of a map ƒ: I × S1 → R3, where I ⊂ R is an open interval and S1 is the unit circle, defined by f ( t , θ ) := γ ( t ) + r ( t ) u ( t ) cos θ + r ( t ) v ( t ) sin θ , {\displaystyle f(t,\theta ):=\gamma (t)+r(t){\mathbf {u} }(... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1337_chunk_0 | Dini derivative | In mathematics and, specifically, real analysis, the Dini derivatives (or Dini derivates) are a class of generalizations of the derivative. They were introduced by Ulisse Dini, who studied continuous but nondifferentiable functions. The upper Dini derivative, which is also called an upper right-hand derivative, of a co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1338_chunk_0 | Dini derivative | The lower Dini derivative, f′−, is defined by f − ′ ( t ) = lim inf h → 0 + f ( t ) − f ( t − h ) h , {\displaystyle f'_{-}(t)=\liminf _{h\to {0+}}{\frac {f(t)-f(t-h)}{h}},} where lim inf is the infimum limit. If f is defined on a vector space, then the upper Dini derivative at t in the direction d is defined by f + ′ ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1339_chunk_0 | Monge array | In mathematics applied to computer science, Monge arrays, or Monge matrices, are mathematical objects named for their discoverer, the French mathematician Gaspard Monge. An m-by-n matrix is said to be a Monge array if, for all i , j , k , ℓ {\displaystyle \scriptstyle i,\,j,\,k,\,\ell } such that 1 ≤ i < k ≤ m and 1 ≤ ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1340_chunk_0 | Sarason interpolation theorem | In mathematics complex analysis, the Sarason interpolation theorem, introduced by Sarason (1967), is a generalization of the Caratheodory interpolation theorem and Nevanlinna–Pick interpolation. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1341_chunk_0 | Antifundamental representation | In mathematics differential geometry, an antifundamental representation of a Lie group is the complex conjugate of the fundamental representation, although the distinction between the fundamental and the antifundamental representation is a matter of convention. However, these two are often non-equivalent, because each ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1342_chunk_0 | Number bond | In mathematics education at primary school level, a number bond (sometimes alternatively called an addition fact) is a simple addition sum which has become so familiar that a child can recognise it and complete it almost instantly, with recall as automatic as that of an entry from a multiplication table in multiplicati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1343_chunk_0 | Number bond | The term "number bond" is also used to refer to a pictorial representation of part-part-whole relationships, often found in the Singapore mathematics curriculum. Number bonds consist of a minimum of 3 circles that are connected by lines. The “whole” is written in the first circle and its “parts” are written in the adjo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1344_chunk_0 | Finite mathematics | In mathematics education, Finite Mathematics is a syllabus in college and university mathematics that is independent of calculus. A course in precalculus may be a prerequisite for Finite Mathematics. Contents of the course include an eclectic selection of topics often applied in social science and business, such as fin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1345_chunk_0 | Mathematical manipulative | Examples of common manipulatives include number lines, Cuisenaire rods; fraction strips, blocks, or stacks; base ten blocks (also known as Dienes or multibase blocks); interlocking linking cubes (such as Unifix); construction sets (such as Polydron and Zometool); colored tiles or tangrams; pattern blocks; colored count... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1346_chunk_0 | Number sentence | In mathematics education, a number sentence is an equation or inequality expressed using numbers and mathematical symbols. The term is used in primary level mathematics teaching in the US, Canada, UK, Australia, New Zealand and South Africa. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1347_chunk_0 | Procept | Examples of such notations are: 3 + 4 {\displaystyle 3+4}: refers to the process of adding as well as the outcome of the process. ∑ n = 0 ∞ ( a n ) {\displaystyle \sum _{n=0}^{\infty }(a_{n})}: refers to the process of summing an infinite sequence, and to the outcome of the process. f ( x ) = 3 x + 2 {\displaystyle f(x... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1348_chunk_0 | Differential and integral calculus | In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits. The word calculus is Latin for "small pebble" (the diminutive of calx, meaning "stone"), a meaning which still persists in medicine. Because such pebbles were used for c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1349_chunk_0 | Precalculus | In mathematics education, precalculus is a course, or a set of courses, that includes algebra and trigonometry at a level which is designed to prepare students for the study of calculus, thus the name precalculus. Schools often distinguish between algebra and trigonometry as two separate parts of the coursework. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1350_chunk_0 | Van Hiele model | In mathematics education, the Van Hiele model is a theory that describes how students learn geometry. The theory originated in 1957 in the doctoral dissertations of Dina van Hiele-Geldof and Pierre van Hiele (wife and husband) at Utrecht University, in the Netherlands. The Soviets did research on the theory in the 1960... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1351_chunk_0 | Van Hiele model | Pierre van Hiele published Structure and Insight in 1986, further describing his theory. The model has greatly influenced geometry curricula throughout the world through emphasis on analyzing properties and classification of shapes at early grade levels. In the United States, the theory has influenced the geometry stra... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1352_chunk_0 | Process graph | In mathematics graph theory a process graph or P-graph is a directed bipartite graph used in workflow modeling. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1353_chunk_0 | Single-entry single-exit | In mathematics graph theory, a single-entry single-exit (SESE) region in a given graph is an ordered edge pair. For example, with the ordered edge pair, (a, b) of distinct control-flow edges a and b where: a dominates b b postdominates a Every cycle containing a also contains b and vice versa.where a node x is said to ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1354_chunk_0 | Characterization of probability distributions | In mathematics in general, a characterization theorem says that a particular object – a function, a space, etc. – is the only one that possesses properties specified in the theorem. A characterization of a probability distribution accordingly states that it is the only probability distribution that satisfies specified ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1355_chunk_0 | Characterization of probability distributions | C = F − 1 B , {\displaystyle X\in {\mathcal {A}},\mathbf {F} X\in {\mathcal {B}}\Leftrightarrow X\in {\mathcal {C}},i.e.{\mathcal {C}}=\mathbf {F} ^{-1}{\mathcal {B}},} where F − 1 B {\displaystyle \mathbf {F} ^{-1}{\mathcal {B}}} denotes the complete inverse image of B {\displaystyle {\mathcal {B}}} in A {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1356_chunk_0 | Characterization of probability distributions | "Memoryless" means that if X {\displaystyle X} is a random variable with such a distribution, then for any numbers 0 < y < x {\displaystyle 0 x ∣ X > y ) = Pr ( X > x − y ) {\displaystyle \Pr(X>x\mid X>y)=\Pr(X>x-y)} . Verification of conditions of characterization theorems in practice is possible only with some error ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1357_chunk_0 | Characterization of probability distributions | That is why there arises the following natural question. Suppose that the conditions of the characterization theorem are fulfilled not exactly but only approximately. May we assert that the conclusion of the theorem is also fulfilled approximately? The theorems in which the problems of this kind are considered are call... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1358_chunk_0 | Cocurvature | In mathematics in the branch of differential geometry, the cocurvature of a connection on a manifold is the obstruction to the integrability of the vertical bundle. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1359_chunk_0 | Bracket ring | In mathematics invariant theory, the bracket ring is the subring of the ring of polynomials k generated by the d-by-d minors of a generic d-by-n matrix (xij). The bracket ring may be regarded as the ring of polynomials on the image of a Grassmannian under the Plücker embedding.For given d ≤ n we define as formal variab... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1360_chunk_0 | Normal convergence | In mathematics normal convergence is a type of convergence for series of functions. Like absolute-convergence, it has the useful property that it is preserved when the order of summation is changed. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1361_chunk_0 | Nyström method | In mathematics numerical analysis, the Nyström method or quadrature method seeks the numerical solution of an integral equation by replacing the integral with a representative weighted sum. The continuous problem is broken into n {\displaystyle n} discrete intervals; quadrature or numerical integration determines the w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1362_chunk_0 | Nyström method | This discrete problem may be ill-conditioned, depending on the original problem and the chosen quadrature rule. Since the linear equations require O ( n 3 ) {\displaystyle O(n^{3})} operations to solve, high-order quadrature rules perform better because low-order quadrature rules require large n {\displaystyle n} for a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1363_chunk_0 | Neuman–Sándor mean | In mathematics of special functions, the Neuman–Sándor mean M, of two positive and unequal numbers a and b, is defined as: M ( a , b ) = a − b 2 arsinh ( a − b a + b ) {\displaystyle M(a,b)={\frac {a-b}{2\operatorname {arsinh} \left({\frac {a-b}{a+b}}\right)}}} This mean interpolates the inequality of the unweighted ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1364_chunk_0 | Cycle shape | In mathematics texts it is customary to denote permutations using lowercase Greek letters. Commonly, either α {\displaystyle \alpha } and β , {\displaystyle \beta ,} or σ , τ {\displaystyle \sigma ,\tau } and π {\displaystyle \pi } are used.Permutations can be defined as bijections from a set S onto itself. All permuta... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1365_chunk_0 | Cycle shape | {\displaystyle \pi \sigma \neq \sigma \pi .} As a bijection from a set to itself, a permutation is a function that performs a rearrangement of a set, and is not an arrangement itself. An older and more elementary viewpoint is that permutations are the arrangements themselves. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1366_chunk_0 | Function field sieve | In mathematics the Function Field Sieve is one of the most efficient algorithms to solve the Discrete Logarithm Problem (DLP) in a finite field. It has heuristic subexponential complexity. Leonard Adleman developed it in 1994 and then elaborated it together with M. D. Huang in 1999. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1367_chunk_0 | Function field sieve | Previous work includes the work of D. Coppersmith about the DLP in fields of characteristic two. The discrete logarithm problem in a finite field consists of solving the equation a x = b {\displaystyle a^{x}=b} for a , b ∈ F p n {\displaystyle a,b\in \mathbb {F} _{p^{n}}} , p {\displaystyle p} a prime number and n {\di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1368_chunk_0 | Goodwin–Staton integral | In mathematics the Goodwin–Staton integral is defined as: G ( z ) = ∫ 0 ∞ e − t 2 t + z d t {\displaystyle G(z)=\int _{0}^{\infty }{\frac {e^{-t^{2}}}{t+z}}\,dt} It satisfies the following third-order nonlinear differential equation： 4 w ( z ) + 8 z d d z w ( z ) + ( 2 + 2 z 2 ) d 2 d z 2 w ( z ) + z d 3 d z 3 w ( z ) ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1369_chunk_0 | Jacobian ideal | In mathematics the Jacobian ideal or gradient ideal is the ideal generated by the Jacobian of a function or function germ. Let O ( x 1 , … , x n ) {\displaystyle {\mathcal {O}}(x_{1},\ldots ,x_{n})} denote the ring of smooth functions in n {\displaystyle n} variables and f {\displaystyle f} a function in the ring. The ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1370_chunk_0 | Korovkin approximation | In mathematics the Korovkin approximation is a convergence statement in which the approximation of a function is given by a certain sequence of functions. In practice a continuous function can be approximated by polynomials. With Korovkin approximations one comes a convergence for the whole approximation with examinati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1371_chunk_0 | Markov theorem | In mathematics the Markov theorem gives necessary and sufficient conditions for two braids to have closures that are equivalent knots or links. The conditions are stated in terms of the group structures on braids. Braids are algebraic objects described by diagrams; the relation to topology is given by Alexander's theor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1372_chunk_0 | Markov theorem | describes the elementary moves generating the equivalence relation on braids given by the equivalence of their closures. More precisely Markov's theorem can be stated as follows: given two braids represented by elements β n , β m ′ {\displaystyle \beta _{n},\beta _{m}'} in the braid groups B n , B m {\displaystyle B_{n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1373_chunk_0 | Mott polynomials | In mathematics the Mott polynomials sn(x) are polynomials introduced by N. F. Mott (1932, p. 442) who applied them to a problem in the theory of electrons. They are given by the exponential generating function e x ( 1 − t 2 − 1 ) / t = ∑ n s n ( x ) t n / n ! . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1374_chunk_0 | Mott polynomials | {\displaystyle e^{x({\sqrt {1-t^{2}}}-1)/t}=\sum _{n}s_{n}(x)t^{n}/n!.} Because the factor in the exponential has the power series 1 − t 2 − 1 t = − ∑ k ≥ 0 C k ( t 2 ) 2 k + 1 {\displaystyle {\frac {{\sqrt {1-t^{2}}}-1}{t}}=-\sum _{k\geq 0}C_{k}\left({\frac {t}{2}}\right)^{2k+1}} in terms of Catalan numbers C k {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1375_chunk_0 | Mott polynomials | 2 n ∑ n = l 1 + l 2 + ⋯ + l k C ( l 1 − 1 ) / 2 C ( l 2 − 1 ) / 2 ⋯ C ( l k − 1 ) / 2 {\displaystyle s_{n}(x)=(-1)^{k}{\frac {n! }{k!2^{n}}}\sum _{n=l_{1}+l_{2}+\cdots +l_{k}}C_{(l_{1}-1)/2}C_{(l_{2}-1)/2}\cdots C_{(l_{k}-1)/2}} ,according to the general formula for generalized Appell polynomials, where the sum is over... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1376_chunk_0 | Mott polynomials | {\displaystyle s_{n}(x)={\frac {(-1)^{n}n(n-1)(n-2)}{2^{n}}}.} By differentiation the recurrence for the first derivative becomes s ′ ( x ) = − ∑ k = 0 ⌊ ( n − 1 ) / 2 ⌋ n ! ( n − 1 − 2 k ) ! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1377_chunk_0 | Padovan cuboid spiral | This pattern continues, forming in succession a 2x2x3 cuboid, a 2x3x4 cuboid etc. Joining the diagonals of the exposed end of each new added cuboid creates a spiral (seen as the black line in the figure). The points on this spiral all lie in the same plane.The cuboids are added in a sequence that adds to the face in th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1378_chunk_0 | Quintuple product identity | In mathematics the Watson quintuple product identity is an infinite product identity introduced by Watson (1929) and rediscovered by Bailey (1951) and Gordon (1961). It is analogous to the Jacobi triple product identity, and is the Macdonald identity for a certain non-reduced affine root system. It is related to Euler'... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1379_chunk_0 | Differential calculus over commutative algebras | In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms. Instances of this are: The whole topological information of a smooth manifold M {\displa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1380_chunk_0 | Differential calculus over commutative algebras | More generally, a linear differential operator of order k, sending sections of a vector bundle E → M {\displaystyle E\rightarrow M} to sections of another bundle F → M {\displaystyle F\rightarrow M} is seen to be an R {\displaystyle \mathbb {R} } -linear map Δ: Γ ( E ) → Γ ( F ) {\displaystyle \Delta :\Gamma (E)\to \Ga... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1381_chunk_0 | Differential calculus over commutative algebras | Replacing the real numbers R {\displaystyle \mathbb {R} } with any commutative ring, and the algebra C ∞ ( M ) {\displaystyle C^{\infty }(M)} with any commutative algebra the above said remains meaningful, hence differential calculus can be developed for arbitrary commutative algebras. Many of these concepts are widely... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1382_chunk_0 | Discrete least squares meshless method | In mathematics the discrete least squares meshless (DLSM) method is a meshless method based on the least squares concept. The method is based on the minimization of a least squares functional, defined as the weighted summation of the squared residual of the governing differential equation and its boundary conditions at... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1383_chunk_0 | Division polynomial | In mathematics the division polynomials provide a way to calculate multiples of points on elliptic curves and to study the fields generated by torsion points. They play a central role in the study of counting points on elliptic curves in Schoof's algorithm. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1384_chunk_0 | Elliptic rational function | In mathematics the elliptic rational functions are a sequence of rational functions with real coefficients. Elliptic rational functions are extensively used in the design of elliptic electronic filters. (These functions are sometimes called Chebyshev rational functions, not to be confused with certain other functions o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1385_chunk_0 | Elliptic rational function | Rational elliptic functions are identified by a positive integer order n and include a parameter ξ ≥ 1 called the selectivity factor. A rational elliptic function of degree n in x with selectivity factor ξ is generally defined as: R n ( ξ , x ) ≡ c d ( n K ( 1 / L n ( ξ ) ) K ( 1 / ξ ) c d − 1 ( x , 1 / ξ ) , 1 / L n (... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1386_chunk_0 | Estimation lemma | In mathematics the estimation lemma, also known as the ML inequality, gives an upper bound for a contour integral. If f is a complex-valued, continuous function on the contour Γ and if its absolute value |f (z)| is bounded by a constant M for all z on Γ, then | ∫ Γ f ( z ) d z | ≤ M l ( Γ ) , {\displaystyle \left|\int ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1387_chunk_0 | Estimation lemma | Out of all the maximum |f (z)|s for the segments, there will be an overall largest one. Hence, if the overall largest |f (z)| is summed over the entire path then the integral of f (z) over the path must be less than or equal to it. Formally, the inequality can be shown to hold using the definition of contour integral, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1388_chunk_0 | Finite Fourier transform | In mathematics the finite Fourier transform may refer to either another name for discrete-time Fourier transform (DTFT) of a finite-length series. E.g., F.J.Harris (pp. 52–53) describes the finite Fourier transform as a "continuous periodic function" and the discrete Fourier transform (DFT) as "a set of samples of the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1389_chunk_0 | Finite Fourier transform | In actual implementation, that is not two separate steps; the DFT replaces the DTFT. So J.Cooley (pp. 77–78) describes the implementation as discrete finite Fourier transform.or another name for the Fourier series coefficients.or another name for one snapshot of a short-time Fourier transform. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1390_chunk_0 | Let expression | In mathematics the let expression is described as the conjunction of expressions. In functional languages the let expression is also used to limit scope. In mathematics scope is described by quantifiers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1391_chunk_0 | Let expression | From this definition the following standard definition of a let expression, as used in a functional language may be derived. x ∉ FV ( y ) ⟹ ( let x: x = y in z ) = z = ( λ x . z ) y {\displaystyle x\not \in \operatorname {FV} (y)\implies (\operatorname {let} x:x=y\operatorname {in} z)=z=(\lambda x.z)\ y} For sim... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1392_chunk_0 | Polynomial basis | In mathematics the monomial basis of a polynomial ring is its basis (as a vector space or free module over the field or ring of coefficients) that consists of all monomials. The monomials form a basis because every polynomial may be uniquely written as a finite linear combination of monomials (this is an immediate cons... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1393_chunk_0 | Negative sign | In mathematics the one-sided limit x → a+ means x approaches a from the right (i.e., right-sided limit), and x → a− means x approaches a from the left (i.e., left-sided limit). For example, 1/x → + ∞ {\displaystyle \infty } as x → 0+ but 1/x → − ∞ {\displaystyle \infty } as x → 0−. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1394_chunk_0 | Signal-to-noise statistic | In mathematics the signal-to-noise statistic distance between two vectors a and b with mean values μ a {\displaystyle \mu _{a}} and μ b {\displaystyle \mu _{b}} and standard deviation σ a {\displaystyle \sigma _{a}} and σ b {\displaystyle \sigma _{b}} respectively is: D s n = ( μ a − μ b ) ( σ a + σ b ) {\displaystyle ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1395_chunk_0 | Spin group | As a Lie group, Spin(n) therefore shares its dimension, n(n − 1)/2, and its Lie algebra with the special orthogonal group. For n > 2, Spin(n) is simply connected and so coincides with the universal cover of SO(n). The non-trivial element of the kernel is denoted −1, which should not be confused with the orthogonal tran... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1396_chunk_0 | Symmetrization methods | In mathematics the symmetrization methods are algorithms of transforming a set A ⊂ R n {\displaystyle A\subset \mathbb {R} ^{n}} to a ball B ⊂ R n {\displaystyle B\subset \mathbb {R} ^{n}} with equal volume vol ( B ) = vol ( A ) {\displaystyle \operatorname {vol} (B)=\operatorname {vol} (A)} and centered at the ori... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1397_chunk_0 | Symmetrization methods | From this many other isoperimetric problems sprung and other symmetrization algorithms. For example, Rayleigh's conjecture is that the first eigenvalue of the Dirichlet problem is minimized for the ball (see Rayleigh–Faber–Krahn inequality for details). Another problem is that the Newtonian capacity of a set A is minim... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1398_chunk_0 | Synchrotron function | In mathematics the synchrotron functions are defined as follows (for x ≥ 0): First synchrotron function F ( x ) = x ∫ x ∞ K 5 3 ( t ) d t {\displaystyle F(x)=x\int _{x}^{\infty }K_{\frac {5}{3}}(t)\,dt} Second synchrotron function G ( x ) = x K 2 3 ( x ) {\displaystyle G(x)=xK_{\frac {2}{3}}(x)} where Kj is the modifie... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1399_chunk_0 | Theory of | In mathematics the use of the term theory is different, necessarily so, since mathematics contains no explanations of natural phenomena, per se, even though it may help provide insight into natural systems or be inspired by them. In the general sense, a mathematical theory is a branch of or topic in mathematics, such a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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