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wiki_2000_chunk_0 | Vitale's random Brunn–Minkowski inequality | In mathematics, Vitale's random Brunn–Minkowski inequality is a theorem due to Richard Vitale that generalizes the classical Brunn–Minkowski inequality for compact subsets of n-dimensional Euclidean space Rn to random compact sets. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2001_chunk_0 | Viète's formula | In mathematics, Viète's formula is the following infinite product of nested radicals representing twice the reciprocal of the mathematical constant π: It can also be represented as: The formula is named after François Viète, who published it in 1593. As the first formula of European mathematics to represent an infinite... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2002_chunk_0 | Viète's formula | The formula can be derived as a telescoping product of either the areas or perimeters of nested polygons converging to a circle. Alternatively, repeated use of the half-angle formula from trigonometry leads to a generalized formula, discovered by Leonhard Euler, that has Viète's formula as a special case. Many similar ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2003_chunk_0 | Voigt notation | In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig of old ideas o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2004_chunk_0 | Voigt notation | Nomenclature may vary according to what is traditional in the field of application. For example, a 2×2 symmetric tensor X has only three distinct elements, the two on the diagonal and the other being off-diagonal. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2005_chunk_0 | Voigt notation | Thus it can be expressed as the vector ⟨ x 11 , x 22 , x 12 ⟩ {\displaystyle \langle x_{11},x_{22},x_{12}\rangle } .As another example: The stress tensor (in matrix notation) is given as σ = . {\displaystyle {\boldsymbol {\sigma }}=\left.} In Voigt notation it is simplified to a 6-dimensional vector: σ ~ = ( σ x x , σ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2006_chunk_0 | Voigt notation | {\displaystyle {\tilde {\sigma }}=(\sigma _{xx},\sigma _{yy},\sigma _{zz},\sigma _{yz},\sigma _{xz},\sigma _{xy})\equiv (\sigma _{1},\sigma _{2},\sigma _{3},\sigma _{4},\sigma _{5},\sigma _{6}).} The strain tensor, similar in nature to the stress tensor—both are symmetric second-order tensors --, is given in matrix for... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2007_chunk_0 | Voigt notation | Its representation in Voigt notation is ϵ ~ = ( ϵ x x , ϵ y y , ϵ z z , γ y z , γ x z , γ x y ) ≡ ( ϵ 1 , ϵ 2 , ϵ 3 , ϵ 4 , ϵ 5 , ϵ 6 ) , {\displaystyle {\tilde {\epsilon }}=(\epsilon _{xx},\epsilon _{yy},\epsilon _{zz},\gamma _{yz},\gamma _{xz},\gamma _{xy})\equiv (\epsilon _{1},\epsilon _{2},\epsilon _{3},\epsilon _{... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2008_chunk_0 | Vojta's conjecture | In mathematics, Vojta's conjecture is a conjecture introduced by Paul Vojta (1987) about heights of points on algebraic varieties over number fields. The conjecture was motivated by an analogy between diophantine approximation and Nevanlinna theory (value distribution theory) in complex analysis. It implies many other ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2009_chunk_0 | Volterra's function | In mathematics, Volterra's function, named for Vito Volterra, is a real-valued function V defined on the real line R with the following curious combination of properties: V is differentiable everywhere The derivative V ′ is bounded everywhere The derivative is not Riemann-integrable. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2010_chunk_0 | Vopěnka's principle | According to Pudlák (2013, p. 204), Vopěnka's principle was originally intended as a joke: Vopěnka was apparently unenthusiastic about large cardinals and introduced his principle as a bogus large cardinal property, planning to show later that it was not consistent. However, before publishing his inconsistency proof he... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2011_chunk_0 | Waldspurger's theorem | In mathematics, Waldspurger's theorem, introduced by Jean-Loup Waldspurger (1981), is a result that identifies Fourier coefficients of modular forms of half-integral weight k+1/2 with the value of an L-series at s=k/2. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2012_chunk_0 | Ward's conjecture | In mathematics, Ward's conjecture is the conjecture made by Ward (1985, p. 451) that "many (and perhaps all?) of the ordinary and partial differential equations that are regarded as being integrable or solvable may be obtained from the self-dual gauge field equations (or its generalizations) by reduction". | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2013_chunk_0 | Watt's curve | In mathematics, Watt's curve is a tricircular plane algebraic curve of degree six. It is generated by two circles of radius b with centers distance 2a apart (taken to be at (±a, 0)). A line segment of length 2c attaches to a point on each of the circles, and the midpoint of the line segment traces out the Watt curve as... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2014_chunk_0 | Watt's curve | It arose in connection with James Watt's pioneering work on the steam engine. The equation of the curve can be given in polar coordinates as r 2 = b 2 − 2 . {\displaystyle r^{2}=b^{2}-\left^{2}.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2015_chunk_0 | Weber's theorem (Algebraic curves) | In mathematics, Weber's theorem, named after Heinrich Martin Weber, is a result on algebraic curves. It states the following. Consider two non-singular curves C and C′ having the same genus g > 1. If there is a rational correspondence φ between C and C′, then φ is a birational transformation. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2016_chunk_0 | Wedderburn theorem | In mathematics, Wedderburn's little theorem states that every finite division ring is a field. In other words, for finite rings, there is no distinction between domains, division rings and fields. The Artin–Zorn theorem generalizes the theorem to alternative rings: every finite alternative division ring is a field. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2017_chunk_0 | Weibel's conjecture | In mathematics, Weibel's conjecture gives a criterion for vanishing of negative algebraic K-theory groups. The conjecture was proposed by Charles Weibel (1980) and proven in full generality by Kerz, Strunk & Tamme (2018) using methods from derived algebraic geometry. Previously partial cases had been proven by Morrow (... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2018_chunk_0 | Weil's criterion | In mathematics, Weil's criterion is a criterion of André Weil for the Generalized Riemann hypothesis to be true. It takes the form of an equivalent statement, to the effect that a certain generalized function is positive definite. Weil's idea was formulated first in a 1952 paper. It is based on the explicit formulae of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2019_chunk_0 | Weil's criterion | A single statement thus combines statements on the complex zeroes of all Dirichlet L-functions. Weil returned to this idea in a 1972 paper, showing how the formulation extended to a larger class of L-functions (Artin-Hecke L-functions); and to the global function field case. Here the inclusion of Artin L-functions, in ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2020_chunk_0 | Weingarten function | In mathematics, Weingarten functions are rational functions indexed by partitions of integers that can be used to calculate integrals of products of matrix coefficients over classical groups. They were first studied by Weingarten (1978) who found their asymptotic behavior, and named by Collins (2003), who evaluated the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2021_chunk_0 | Weisner's method | In mathematics, Weisner's method is a method for finding generating functions for special functions using representation theory of Lie groups and Lie algebras, introduced by Weisner (1955). It includes Truesdell's method as a special case, and is essentially the same as Rainville's method. ... Weisner's group-theoretic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2022_chunk_0 | Weitzenböck's inequality | In mathematics, Weitzenböck's inequality, named after Roland Weitzenböck, states that for a triangle of side lengths a {\displaystyle a} , b {\displaystyle b} , c {\displaystyle c} , and area Δ {\displaystyle \Delta } , the following inequality holds: a 2 + b 2 + c 2 ≥ 4 3 Δ . {\displaystyle a^{2}+b^{2}+c^{2}\geq 4{\sq... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2023_chunk_0 | Welch bounds | In mathematics, Welch bounds are a family of inequalities pertinent to the problem of evenly spreading a set of unit vectors in a vector space. The bounds are important tools in the design and analysis of certain methods in telecommunication engineering, particularly in coding theory. The bounds were originally publish... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2024_chunk_0 | Weyl's lemma (Laplace equation) | In mathematics, Weyl's lemma, named after Hermann Weyl, states that every weak solution of Laplace's equation is a smooth solution. This contrasts with the wave equation, for example, which has weak solutions that are not smooth solutions. Weyl's lemma is a special case of elliptic or hypoelliptic regularity. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2025_chunk_0 | Wilkie's theorem | In mathematics, Wilkie's theorem is a result by Alex Wilkie about the theory of ordered fields with an exponential function, or equivalently about the geometric nature of exponential varieties. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2026_chunk_0 | Wilson polynomials | In mathematics, Wilson polynomials are a family of orthogonal polynomials introduced by James A. Wilson (1980) that generalize Jacobi polynomials, Hahn polynomials, and Charlier polynomials. They are defined in terms of the generalized hypergeometric function and the Pochhammer symbols by p n ( t 2 ) = ( a + b ) n ( a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2027_chunk_0 | Wirtinger's representation and projection theorem | In mathematics, Wirtinger's representation and projection theorem is a theorem proved by Wilhelm Wirtinger in 1932 in connection with some problems of approximation theory. This theorem gives the representation formula for the holomorphic subspace H 2 {\displaystyle \left.\right.H_{2}} of the simple, unweighted holomor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2028_chunk_0 | Wirtinger's representation and projection theorem | If F ( z ) {\displaystyle \left.\right.\left.F(z)\right.} is of the class L 2 {\displaystyle \left.\right.L^{2}} on | z | < 1 {\displaystyle \left.\right.|z|<1} , i.e. ∬ | z | < 1 | F ( z ) | 2 d S < + ∞ , {\displaystyle \iint _{|z|<1}|F(z)|^{2}\,dS<+\infty ,} where d S {\displaystyle \left.\right.dS} is the area eleme... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2029_chunk_0 | Wirtinger's representation and projection theorem | The last formula gives a form for the orthogonal projection from L 2 {\displaystyle \left.\right.L^{2}} to H 2 {\displaystyle \left.\right.H_{2}} . Besides, replacement of F ( ζ ) {\displaystyle \left.\right.F(\zeta )} by f ( ζ ) {\displaystyle \left.\right.f(\zeta )} makes it Wirtinger's representation for all f ( z )... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2030_chunk_0 | Wirtinger's representation and projection theorem | Later, after the 1950s, a degree of the Cauchy kernel was called reproducing kernel, and the notation A 0 2 {\displaystyle \left.\right.A_{0}^{2}} became common for the class H 2 {\displaystyle \left.\right.H_{2}} . In 1948 Mkhitar Djrbashian extended Wirtinger's representation and projection to the wider, weighted Hil... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2031_chunk_0 | Witt vector cohomology | In mathematics, Witt vector cohomology was an early p-adic cohomology theory for algebraic varieties introduced by Serre (1958). Serre constructed it by defining a sheaf of truncated Witt rings Wn over a variety V and then taking the inverse limit of the sheaf cohomology groups Hi(V, Wn) of these sheaves. Serre observe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2032_chunk_0 | Wolstenholme's theorem | In mathematics, Wolstenholme's theorem states that for a prime number p ≥ 5 {\displaystyle p\geq 5} , the congruence ( 2 p − 1 p − 1 ) ≡ 1 ( mod p 3 ) {\displaystyle {2p-1 \choose p-1}\equiv 1{\pmod {p^{3}}}} holds, where the parentheses denote a binomial coefficient. For example, with p = 7, this says that 1716 is one... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2033_chunk_0 | Wolstenholme's theorem | In 1819, Charles Babbage showed the same congruence modulo p2, which holds for p ≥ 3 {\displaystyle p\geq 3} . An equivalent formulation is the congruence ( a p b p ) ≡ ( a b ) ( mod p 3 ) {\displaystyle {ap \choose bp}\equiv {a \choose b}{\pmod {p^{3}}}} for p ≥ 5 {\displaystyle p\geq 5} , which is due to Wilhelm Ljun... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2034_chunk_0 | Wolstenholme's theorem | A prime that satisfies the congruence modulo p4 is called a Wolstenholme prime (see below). As Wolstenholme himself established, his theorem can also be expressed as a pair of congruences for (generalized) harmonic numbers: 1 + 1 2 + 1 3 + ⋯ + 1 p − 1 ≡ 0 ( mod p 2 ) , and {\displaystyle 1+{1 \over 2}+{1 \over 3}+\dots... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2035_chunk_0 | Young's convolution inequality | In mathematics, Young's convolution inequality is a mathematical inequality about the convolution of two functions, named after William Henry Young. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2036_chunk_0 | Young's inequality for products | In mathematics, Young's inequality for products is a mathematical inequality about the product of two numbers. The inequality is named after William Henry Young and should not be confused with Young's convolution inequality. Young's inequality for products can be used to prove Hölder's inequality. It is also widely use... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2037_chunk_0 | Young's lattice | In mathematics, Young's lattice is a lattice that is formed by all integer partitions. It is named after Alfred Young, who, in a series of papers On quantitative substitutional analysis, developed the representation theory of the symmetric group. In Young's theory, the objects now called Young diagrams and the partial ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2038_chunk_0 | Zahorski theorem | In mathematics, Zahorski's theorem is a theorem of real analysis. It states that a necessary and sufficient condition for a subset of the real line to be the set of points of non-differentiability of a continuous real-valued function, is that it be the union of a Gδ set and a G δ σ {\displaystyle {G_{\delta }}_{\sigma ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2039_chunk_0 | Zeckendorf's theorem | In mathematics, Zeckendorf's theorem, named after Belgian amateur mathematician Edouard Zeckendorf, is a theorem about the representation of integers as sums of Fibonacci numbers. Zeckendorf's theorem states that every positive integer can be represented uniquely as the sum of one or more distinct Fibonacci numbers in ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2040_chunk_0 | Zeckendorf's theorem | Such a sum is called the Zeckendorf representation of N. The Fibonacci coding of N can be derived from its Zeckendorf representation. For example, the Zeckendorf representation of 64 is 64 = 55 + 8 + 1.There are other ways of representing 64 as the sum of Fibonacci numbers 64 = 55 + 5 + 3 + 1 64 = 34 + 21 + 8 + 1 64 = ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2041_chunk_0 | Zolotarev polynomials | In mathematics, Zolotarev polynomials are polynomials used in approximation theory. They are sometimes used as an alternative to the Chebyshev polynomials where accuracy of approximation near the origin is of less importance. Zolotarev polynomials differ from the Chebyshev polynomials in that two of the coefficients ar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2042_chunk_0 | Quantization condition | In physics, the connection is the fundamental physical object. One of the fundamental observations in the theory of characteristic classes in algebraic topology is that many homotopical structures of nontrivial principal bundles may be expressed as an integral of some polynomial over any connection over it. Note that a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2043_chunk_0 | Quantization condition | The transition function maps the strip to G, and the different ways of mapping a strip into G are given by the first homotopy group of G. So in the G-bundle formulation, a gauge theory admits Dirac monopoles provided G is not simply connected, whenever there are paths that go around the group that cannot be deformed to... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2044_chunk_0 | Taut submanifold | In mathematics, a (compact) taut submanifold N of a space form M is a compact submanifold with the property that for every q ∈ M {\displaystyle q\in M} the distance function L q: N → R , L q ( x ) = dist ( x , q ) 2 {\displaystyle L_{q}:N\to \mathbf {R} ,\qquad L_{q}(x)=\operatorname {dist} (x,q)^{2}} is a perfect Mo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2045_chunk_0 | Monge-Ampere equation | In mathematics, a (real) Monge–Ampère equation is a nonlinear second-order partial differential equation of special kind. A second-order equation for the unknown function u of two variables x,y is of Monge–Ampère type if it is linear in the determinant of the Hessian matrix of u and in the second-order partial derivati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2046_chunk_0 | Monge-Ampere equation | The most complete results so far have been obtained when the equation is elliptic. Monge–Ampère equations frequently arise in differential geometry, for example, in the Weyl and Minkowski problems in differential geometry of surfaces. They were first studied by Gaspard Monge in 1784 and later by André-Marie Ampère in 1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2047_chunk_0 | Leibniz algebra | In mathematics, a (right) Leibniz algebra, named after Gottfried Wilhelm Leibniz, sometimes called a Loday algebra, after Jean-Louis Loday, is a module L over a commutative ring R with a bilinear product satisfying the Leibniz identity , c ] = ] + , b ] . {\displaystyle ,c]=]+,b].\,} In other words, right multiplic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2048_chunk_0 | Leibniz algebra | Conversely any Lie algebra is obviously a Leibniz algebra. In this sense, Leibniz algebras can be seen as a non-commutative generalization of Lie algebras. The investigation of which theorems and properties of Lie algebras are still valid for Leibniz algebras is a recurrent theme in the literature. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2049_chunk_0 | Leibniz algebra | For instance, it has been shown that Engel's theorem still holds for Leibniz algebras and that a weaker version of Levi-Malcev theorem also holds.The tensor module, T(V) , of any vector space V can be turned into a Loday algebra such that = a 1 ⊗ ⋯ a n ⊗ x for a 1 , … , a n , x ∈ V . {\displaystyle =a_{1}\otimes \cdot... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2050_chunk_0 | Leibniz algebra | They attracted interest after Jean-Louis Loday noticed that the classical Chevalley–Eilenberg boundary map in the exterior module of a Lie algebra can be lifted to the tensor module which yields a new chain complex. In fact this complex is well-defined for any Leibniz algebra. The homology HL(L) of this chain complex i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2051_chunk_0 | Leibniz algebra | If L is the Lie algebra of (infinite) matrices over an associative R-algebra A then Leibniz homology of L is the tensor algebra over the Hochschild homology of A. A Zinbiel algebra is the Koszul dual concept to a Leibniz algebra. It has defining identity: ( a ∘ b ) ∘ c = a ∘ ( b ∘ c ) + a ∘ ( c ∘ b ) . {\displaystyle (... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2052_chunk_0 | *-algebra | In mathematics, a *-ring is a ring with a map *: A → A that is an antiautomorphism and an involution. More precisely, * is required to satisfy the following properties: (x + y)* = x* + y* (x y)* = y* x* 1* = 1 (x*)* = xfor all x, y in A. This is also called an involutive ring, involutory ring, and ring with involution.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2053_chunk_0 | *-algebra | One can define a sesquilinear form over any *-ring. Also, one can define *-versions of algebraic objects, such as ideal and subring, with the requirement to be *-invariant: x ∈ I ⇒ x* ∈ I and so on. *-rings are unrelated to star semirings in the theory of computation. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2054_chunk_0 | 2-valued morphism | In mathematics, a 2-valued morphism is a homomorphism that sends a Boolean algebra B onto the two-element Boolean algebra 2 = {0,1}. It is essentially the same thing as an ultrafilter on B, and, in a different way, also the same things as a maximal ideal of B. 2-valued morphisms have also been proposed as a tool for un... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2055_chunk_0 | 3-step group | In mathematics, a 3-step group is a special sort of group of Fitting length at most 3, that is used in the classification of CN groups and in the Feit–Thompson theorem. The definition of a 3-step group in these two cases is slightly different. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2056_chunk_0 | 4-manifold | In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2057_chunk_0 | 5-manifold | In mathematics, a 5-manifold is a 5-dimensional topological manifold, possibly with a piecewise linear or smooth structure. Non-simply connected 5-manifolds are impossible to classify, as this is harder than solving the word problem for groups. Simply connected compact 5-manifolds were first classified by Stephen Smale... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2058_chunk_0 | Bailey pair | In mathematics, a Bailey pair is a pair of sequences satisfying certain relations, and a Bailey chain is a sequence of Bailey pairs. Bailey pairs were introduced by W. N. Bailey (1947, 1948) while studying the second proof Rogers 1917 of the Rogers–Ramanujan identities, and Bailey chains were introduced by Andrews (198... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2059_chunk_0 | Baire measure | In mathematics, a Baire measure is a measure on the σ-algebra of Baire sets of a topological space whose value on every compact Baire set is finite. In compact metric spaces the Borel sets and the Baire sets are the same, so Baire measures are the same as Borel measures that are finite on compact sets. In general Baire... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2060_chunk_0 | Banach bundle (non-commutative geometry) | In mathematics, a Banach bundle is a fiber bundle over a topological Hausdorff space, such that each fiber has the structure of a Banach space. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2061_chunk_0 | Banach bundle | In mathematics, a Banach bundle is a vector bundle each of whose fibres is a Banach space, i.e. a complete normed vector space, possibly of infinite dimension. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2062_chunk_0 | Barnes integral | In mathematics, a Barnes integral or Mellin–Barnes integral is a contour integral involving a product of gamma functions. They were introduced by Ernest William Barnes (1908, 1910). They are closely related to generalized hypergeometric series. The integral is usually taken along a contour which is a deformation of the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2063_chunk_0 | Barnes zeta function | In mathematics, a Barnes zeta function is a generalization of the Riemann zeta function introduced by E. W. Barnes (1901). It is further generalized by the Shintani zeta function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2064_chunk_0 | Batalin–Vilkovisky formalism | In mathematics, a Batalin–Vilkovisky algebra is a graded supercommutative algebra (with a unit 1) with a second-order nilpotent operator Δ of degree −1. More precisely, it satisfies the identities | a b | = | a | + | b | {\displaystyle |ab|=|a|+|b|} (The product has degree 0) | Δ ( a ) | = | a | − 1 {\displaystyle |\De... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2065_chunk_0 | Beatty sequence | In mathematics, a Beatty sequence (or homogeneous Beatty sequence) is the sequence of integers found by taking the floor of the positive multiples of a positive irrational number. Beatty sequences are named after Samuel Beatty, who wrote about them in 1926. Rayleigh's theorem, named after Lord Rayleigh, states that the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2066_chunk_0 | Beurling zeta function | In mathematics, a Beurling zeta function is an analogue of the Riemann zeta function where the ordinary primes are replaced by a set of Beurling generalized primes: any sequence of real numbers greater than 1 that tend to infinity. These were introduced by Beurling (1937). A Beurling generalized integer is a number tha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2067_chunk_0 | Bianchi group | The quotient space M d = P S L 2 ( O d ) ∖ H 3 {\displaystyle M_{d}=PSL_{2}({\mathcal {O}}_{d})\backslash \mathbb {H} ^{3}} is a non-compact, hyperbolic 3-fold with finite volume, which is also called Bianchi orbifold. An exact formula for the volume, in terms of the Dedekind zeta function of the base field Q ( − d ) {... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2068_chunk_0 | Finitary boolean function | In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually {true, false}, {0,1} or {-1,1}). Alternative names are switching function, used especially in older computer science literature, and truth function (or logical function), used in logic. Boolean func... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2069_chunk_0 | Finitary boolean function | A Boolean function with multiple outputs, f: { 0 , 1 } k → { 0 , 1 } m {\displaystyle f:\{0,1\}^{k}\to \{0,1\}^{m}} with m > 1 {\displaystyle m>1} is a vectorial or vector-valued Boolean function (an S-box in symmetric cryptography).There are 2 2 k {\displaystyle 2^{2^{k}}} different Boolean functions with k {\displays... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2070_chunk_0 | Finitary boolean function | . . , x k {\displaystyle x_{1},...,x_{k}} , and two propositional formulas are logically equivalent if and only if they express the same Boolean function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2071_chunk_0 | Boolean matrix | In mathematics, a Boolean matrix is a matrix with entries from a Boolean algebra. When the two-element Boolean algebra is used, the Boolean matrix is called a logical matrix. (In some contexts, particularly computer science, the term "Boolean matrix" implies this restriction.) Let U be a non-trivial Boolean algebra (i.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2072_chunk_0 | Boolean matrix | Intersection, union, complementation, and containment of elements is expressed in U. Let V be the collection of n × n matrices that have entries taken from U. Complementation of such a matrix is obtained by complementing each element. The intersection or union of two such matrices is obtained by applying the operation ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2073_chunk_0 | Boolean matrix | The product of two Boolean matrices is expressed as follows: According to one author, "Matrices over an arbitrary Boolean algebra β satisfy most of the properties over β0 = {0, 1}. The reason is that any Boolean algebra is a sub-Boolean algebra of β 0 S {\displaystyle \beta _{0}^{S}} for some set S, and we have an isom... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2074_chunk_0 | Boolean rings | In mathematics, a Boolean ring R is a ring for which x2 = x for all x in R, that is, a ring that consists only of idempotent elements. An example is the ring of integers modulo 2. Every Boolean ring gives rise to a Boolean algebra, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to ex... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2075_chunk_0 | Wightman functional | In mathematics, a Borchers algebra or Borchers–Uhlmann algebra or BU-algebra is the tensor algebra of a vector space, often a space of smooth test functions. They were studied by H. J. Borchers (1962), who showed that the Wightman distributions of a quantum field could be interpreted as a state, called a Wightman funct... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2076_chunk_0 | Borel isomorphism | In mathematics, a Borel isomorphism is a measurable bijective function between two standard Borel spaces. By Souslin's theorem in standard Borel spaces (which says that a set that is both analytic and coanalytic is necessarily Borel), the inverse of any such measurable bijective function is also measurable. Borel isomo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2077_chunk_0 | Borel algebra | In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel. For a topological space X, the collection of all Borel ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2078_chunk_0 | Borel algebra | Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called a Borel measure. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2079_chunk_0 | Borel algebra | Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory. In some contexts, Borel sets are defined to be generated by the compact sets of the topological space, rather than the open sets. The two definitions are equivalent for many well-behaved spaces, including all Hausdorff... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2080_chunk_0 | Bose–Mesner algebra | In mathematics, a Bose–Mesner algebra is a special set of matrices which arise from a combinatorial structure known as an association scheme, together with the usual set of rules for combining (forming the products of) those matrices, such that they form an associative algebra, or, more precisely, a unitary commutative... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2081_chunk_0 | Bost–Connes system | In mathematics, a Bost–Connes system is a quantum statistical dynamical system related to an algebraic number field, whose partition function is related to the Dedekind zeta function of the number field. Bost & Connes (1995) introduced Bost–Connes systems by constructing one for the rational numbers. Connes, Marcolli &... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2082_chunk_0 | Brauer algebra | In mathematics, a Brauer algebra is an associative algebra introduced by Richard Brauer in the context of the representation theory of the orthogonal group. It plays the same role that the symmetric group does for the representation theory of the general linear group in Schur–Weyl duality. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2083_chunk_0 | Buekenhout geometry | In mathematics, a Buekenhout geometry or diagram geometry is a generalization of projective spaces, Tits buildings, and several other geometric structures, introduced by Buekenhout (1979). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2084_chunk_0 | Bézout matrix | In mathematics, a Bézout matrix (or Bézoutian or Bezoutiant) is a special square matrix associated with two polynomials, introduced by James Joseph Sylvester (1853) and Arthur Cayley (1857) and named after Étienne Bézout. Bézoutian may also refer to the determinant of this matrix, which is equal to the resultant of the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2085_chunk_0 | Böhmer integral | In mathematics, a Böhmer integral is an integral introduced by Böhmer (1939) generalizing the Fresnel integrals. There are two versions, given by C ( x , α ) = ∫ x ∞ t α − 1 cos ( t ) d t {\displaystyle \displaystyle C(x,\alpha )=\int _{x}^{\infty }t^{\alpha -1}\cos(t)\,dt} S ( x , α ) = ∫ x ∞ t α − 1 sin ( t ) d t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2086_chunk_0 | C0-semigroup | In mathematics, a C0-semigroup, also known as a strongly continuous one-parameter semigroup, is a generalization of the exponential function. Just as exponential functions provide solutions of scalar linear constant coefficient ordinary differential equations, strongly continuous semigroups provide solutions of linear ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2087_chunk_0 | CR manifold | In mathematics, a CR manifold, or Cauchy–Riemann manifold, is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a complex vector space, or more generally modeled on an edge of a wedge. Formally, a CR manifold is a differentiable manifold M together with a preferred ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2088_chunk_0 | Caccioppoli set | In mathematics, a Caccioppoli set is a set whose boundary is measurable and has (at least locally) a finite measure. A synonym is set of (locally) finite perimeter. Basically, a set is a Caccioppoli set if its characteristic function is a function of bounded variation. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2089_chunk_0 | Cantor algebra | In mathematics, a Cantor algebra, named after Georg Cantor, is one of two closely related Boolean algebras, one countable and one complete. The countable Cantor algebra is the Boolean algebra of all clopen subsets of the Cantor set. This is the free Boolean algebra on a countable number of generators. Up to isomorphism... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2090_chunk_0 | Cantor algebra | The complete Cantor algebra is the complete Boolean algebra of Borel subsets of the reals modulo meager sets (Balcar & Jech 2006). It is isomorphic to the completion of the countable Cantor algebra. (The complete Cantor algebra is sometimes called the Cohen algebra, though "Cohen algebra" usually refers to a different ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2091_chunk_0 | Cantor cube | )By a theorem of Schepin, these four properties characterize Cantor cubes; any space satisfying the properties is homeomorphic to a Cantor cube. In fact, every AE(0) space is the continuous image of a Cantor cube, and with some effort one can prove that every compact group is AE(0). It follows that every zero-dimension... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2092_chunk_0 | Cantor space | In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the" Cantor space. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2093_chunk_0 | Carleman matrix | In mathematics, a Carleman matrix is a matrix used to convert function composition into matrix multiplication. It is often used in iteration theory to find the continuous iteration of functions which cannot be iterated by pattern recognition alone. Other uses of Carleman matrices occur in the theory of probability gene... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2094_chunk_0 | Carlyle circle | In mathematics, a Carlyle circle is a certain circle in a coordinate plane associated with a quadratic equation; it is named after Thomas Carlyle. The circle has the property that the solutions of the quadratic equation are the horizontal coordinates of the intersections of the circle with the horizontal axis. Carlyle ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2095_chunk_0 | Carnot group | In mathematics, a Carnot group is a simply connected nilpotent Lie group, together with a derivation of its Lie algebra such that the subspace with eigenvalue 1 generates the Lie algebra. The subbundle of the tangent bundle associated to this eigenspace is called horizontal. On a Carnot group, any norm on the horizonta... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2096_chunk_0 | Cartan algebra | In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra h {\displaystyle {\mathfrak {h}}} of a Lie algebra g {\displaystyle {\mathfrak {g}}} that is self-normalising (if ∈ h {\displaystyle \in {\mathfrak {h}}} for all X ∈ h {\displaystyle X\in {\mathfrak {h}}} , then Y ∈ h {\displaysty... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2097_chunk_0 | Cartan algebra | In a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero (e.g., C {\displaystyle \mathbb {C} } ), a Cartan subalgebra is the same thing as a maximal abelian subalgebra consisting of elements x such that the adjoint endomorphism ad ( x ): g → g {\displaystyle \operatorna... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2098_chunk_0 | Casimir invariant | In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator, which is a Casimir element of the three-dimensional rotation group. Mo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2099_chunk_0 | Cauchy initial value problem | In mathematics, a Cauchy (French: ) boundary condition augments an ordinary differential equation or a partial differential equation with conditions that the solution must satisfy on the boundary; ideally so as to ensure that a unique solution exists. A Cauchy boundary condition specifies both the function value and no... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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