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wiki_2300_chunk_0 | Petrovsky lacuna | In mathematics, a Petrovsky lacuna, named for the Russian mathematician I. G. Petrovsky, is a region where the fundamental solution of a linear hyperbolic partial differential equation vanishes. They were studied by Petrovsky (1945) who found topological conditions for their existence. Petrovsky's work was generalized ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2301_chunk_0 | Pfister form | In mathematics, a Pfister form is a particular kind of quadratic form, introduced by Albrecht Pfister in 1965. In what follows, quadratic forms are considered over a field F of characteristic not 2. For a natural number n, an n-fold Pfister form over F is a quadratic form of dimension 2n that can be written as a tensor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2302_chunk_0 | PV number | In mathematics, a Pisot–Vijayaraghavan number, also called simply a Pisot number or a PV number, is a real algebraic integer greater than 1, all of whose Galois conjugates are less than 1 in absolute value. These numbers were discovered by Axel Thue in 1912 and rediscovered by G. H. Hardy in 1919 within the context of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2303_chunk_0 | Plücker formula | In mathematics, a Plücker formula, named after Julius Plücker, is one of a family of formulae, of a type first developed by Plücker in the 1830s, that relate certain numeric invariants of algebraic curves to corresponding invariants of their dual curves. The invariant called the genus, common to both the curve and its ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2304_chunk_0 | Dirichlet-to-Neumann operator | In mathematics, a Poincaré–Steklov operator (after Henri Poincaré and Vladimir Steklov) maps the values of one boundary condition of the solution of an elliptic partial differential equation in a domain to the values of another boundary condition. Usually, either of the boundary conditions determines the solution. Thus... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2305_chunk_0 | Poisson algebra | In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central in the study of quantum groups. Manifolds with a Poisson algebra structur... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2306_chunk_0 | Poisson ring | In mathematics, a Poisson ring is a commutative ring on which an anticommutative and distributive binary operation {\displaystyle } satisfying the Jacobi identity and the product rule is defined. Such an operation is then known as the Poisson bracket of the Poisson ring. Many important operations and results of symple... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2307_chunk_0 | Poisson superbracket | In mathematics, a Poisson superalgebra is a Z2-graded generalization of a Poisson algebra. Specifically, a Poisson superalgebra is an (associative) superalgebra A with a Lie superbracket : A ⊗ A → A {\displaystyle :A\otimes A\to A} such that (A, ) is a Lie superalgebra and the operator : A → A {\displaystyle :A\to A} i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2308_chunk_0 | Poisson superbracket | This gives the classical dynamics of fermion fields and classical spin-1/2 particles. The other is to define an antibracket algebra instead. This is used in the BRST and Batalin-Vilkovisky formalism. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2309_chunk_0 | Poisson–Lie group | In mathematics, a Poisson–Lie group is a Poisson manifold that is also a Lie group, with the group multiplication being compatible with the Poisson algebra structure on the manifold. The infinitesimal counterpart of a Poisson–Lie group is a Lie bialgebra, in analogy to Lie algebras as the infinitesimal counterparts of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2310_chunk_0 | Q-category | In mathematics, a Q-category or almost quotient category is a category that is a "milder version of a Grothendieck site." A Q-category is a coreflective subcategory. The Q stands for a quotient. The concept of Q-categories was introduced by Alexander Rosenberg in 1988. The motivation for the notion was its use in nonco... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2311_chunk_0 | Q-matrix | In mathematics, a Q-matrix is a square matrix whose associated linear complementarity problem LCP(M,q) has a solution for every vector q. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2312_chunk_0 | Ramanujan prime | In mathematics, a Ramanujan prime is a prime number that satisfies a result proven by Srinivasa Ramanujan relating to the prime-counting function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2313_chunk_0 | Ramanujan–Sato series | In mathematics, a Ramanujan–Sato series generalizes Ramanujan’s pi formulas such as, 1 π = 2 2 99 2 ∑ k = 0 ∞ ( 4 k ) ! k ! 4 26390 k + 1103 396 4 k {\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{99^{2}}}\sum _{k=0}^{\infty }{\frac {(4k)! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2314_chunk_0 | Ramanujan–Sato series | }{k!^{4}}}{\frac {26390k+1103}{396^{4k}}}} to the form 1 π = ∑ k = 0 ∞ s ( k ) A k + B C k {\displaystyle {\frac {1}{\pi }}=\sum _{k=0}^{\infty }s(k){\frac {Ak+B}{C^{k}}}} by using other well-defined sequences of integers s ( k ) {\displaystyle s(k)} obeying a certain recurrence relation, sequences which may be express... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2315_chunk_0 | Ramsey cardinal | In mathematics, a Ramsey cardinal is a certain kind of large cardinal number introduced by Erdős & Hajnal (1962) and named after Frank P. Ramsey, whose theorem establishes that ω enjoys a certain property that Ramsey cardinals generalize to the uncountable case. Let <ω denote the set of all finite subsets of κ. A cardi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2316_chunk_0 | Raynaud surface | In mathematics, a Raynaud surface is a particular kind of algebraic surface that was introduced in William E. Lang (1979) and named for Michel Raynaud (1978). To be precise, a Raynaud surface is a quasi-elliptic surface over an algebraic curve of genus g greater than 1, such that all fibers are irreducible and the fibr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2317_chunk_0 | Redheffer matrix | In mathematics, a Redheffer matrix, often denoted A n {\displaystyle A_{n}} as studied by Redheffer (1977), is a square (0,1) matrix whose entries aij are 1 if i divides j or if j = 1; otherwise, aij = 0. It is useful in some contexts to express Dirichlet convolution, or convolved divisors sums, in terms of matrix prod... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2318_chunk_0 | Relevance vector machine | In mathematics, a Relevance Vector Machine (RVM) is a machine learning technique that uses Bayesian inference to obtain parsimonious solutions for regression and probabilistic classification. The RVM has an identical functional form to the support vector machine, but provides probabilistic classification. It is actuall... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2319_chunk_0 | Riccati differential equation | In mathematics, a Riccati equation in the narrowest sense is any first-order ordinary differential equation that is quadratic in the unknown function. In other words, it is an equation of the form y ′ ( x ) = q 0 ( x ) + q 1 ( x ) y ( x ) + q 2 ( x ) y 2 ( x ) {\displaystyle y'(x)=q_{0}(x)+q_{1}(x)\,y(x)+q_{2}(x)\,y^{2... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2320_chunk_0 | Riemann relations | In mathematics, a Riemann form in the theory of abelian varieties and modular forms, is the following data: A lattice Λ in a complex vector space Cg. An alternating bilinear form α from Λ to the integers satisfying the following Riemann bilinear relations:the real linear extension αR:Cg × Cg→R of α satisfies αR(iv, iw)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2321_chunk_0 | Rectangle method | In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lines on a graph, but also the length of curves and other approximations. The... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2322_chunk_0 | Rectangle method | This approach can be used to find a numerical approximation for a definite integral even if the fundamental theorem of calculus does not make it easy to find a closed-form solution. Because the region by the small shapes is usually not exactly the same shape as the region being measured, the Riemann sum will differ fro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2323_chunk_0 | Flat metric | In mathematics, a Riemannian manifold is said to be flat if its Riemann curvature tensor is everywhere zero. Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles of a triangle add up to 180°. The universal cover of a complete flat manif... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2324_chunk_0 | Riemann–Roch theorem for smooth manifolds | In mathematics, a Riemann–Roch theorem for smooth manifolds is a version of results such as the Hirzebruch–Riemann–Roch theorem or Grothendieck–Riemann–Roch theorem (GRR) without a hypothesis making the smooth manifolds involved carry a complex structure. Results of this kind were obtained by Michael Atiyah and Friedri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2325_chunk_0 | Vector lattice | In mathematics, a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice. Riesz spaces are named after Frigyes Riesz who first defined them in his 1928 paper Sur la décomposition des opérations fonctionelles linéaires. Riesz spaces have wid... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2326_chunk_0 | Ringel–Hall algebra | In mathematics, a Ringel–Hall algebra is a generalization of the Hall algebra, studied by Claus Michael Ringel (1990). It has a basis of equivalence classes of objects of an abelian category, and the structure constants for this basis are related to the numbers of extensions of objects in the category. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2327_chunk_0 | Rota–Baxter algebra | In mathematics, a Rota–Baxter algebra is an associative algebra, together with a particular linear map R which satisfies the Rota–Baxter identity. It appeared first in the work of the American mathematician Glen E. Baxter in the realm of probability theory. Baxter's work was further explored from different angles by Gi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2328_chunk_0 | Schauder dimension | In mathematics, a Schauder basis or countable basis is similar to the usual (Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This makes Schauder bases more suitable for the analysis of infinite-dimension... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2329_chunk_0 | Schur-concave function | In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function f: R d → R {\displaystyle f:\mathbb {R} ^{d}\rightarrow \mathbb {R} } that for all x , y ∈ R d {\displaystyle x,y\in \mathbb {R} ^{d}} such that x {\displaystyle x} is majorized by y {\displays... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2330_chunk_0 | Schwartz–Bruhat function | In mathematics, a Schwartz–Bruhat function, named after Laurent Schwartz and François Bruhat, is a complex valued function on a locally compact abelian group, such as the adeles, that generalizes a Schwartz function on a real vector space. A tempered distribution is defined as a continuous linear functional on the spac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2331_chunk_0 | Seifert matrix | In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) is an orientable surface whose boundary is a given knot or link. Such surfaces can be used to study the properties of the associated knot or link. For example, many knot invariants are most easily calculated using a Seifert surface. Se... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2332_chunk_0 | Seifert matrix | Specifically, let L be a tame oriented knot or link in Euclidean 3-space (or in the 3-sphere). A Seifert surface is a compact, connected, oriented surface S embedded in 3-space whose boundary is L such that the orientation on L is just the induced orientation from S. Note that any compact, connected, oriented surface w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2333_chunk_0 | Severi variety (Hilbert scheme) | In mathematics, a Severi variety is an algebraic variety in a Hilbert scheme that parametrizes curves in projective space with given degree and geometric genus and at most node singularities. Its dimension is 3d + g − 1. It is a theorem that Severi varieties are algebraic varieties, i.e. it is irreducible. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2334_chunk_0 | Severi–Brauer variety | In mathematics, a Severi–Brauer variety over a field K is an algebraic variety V which becomes isomorphic to a projective space over an algebraic closure of K. The varieties are associated to central simple algebras in such a way that the algebra splits over K if and only if the variety has a rational point over K. Fra... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2335_chunk_0 | Severi–Brauer variety | They represent (at least if K is a perfect field) Galois cohomology classes in H1(PGLn), where PGLn is the projective linear group, and n is the dimension of the variety V. There is a short exact sequence 1 → GL1 → GLn → PGLn → 1of algebraic groups. This implies a connecting homomorphism H1(PGLn) → H2(GL1)at the level ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2336_chunk_0 | Sheffer sequence | In mathematics, a Sheffer sequence or poweroid is a polynomial sequence, i.e., a sequence (pn(x): n = 0, 1, 2, 3, ...) of polynomials in which the index of each polynomial equals its degree, satisfying conditions related to the umbral calculus in combinatorics. They are named for Isador M. Sheffer. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2337_chunk_0 | Shintani zeta function | In mathematics, a Shintani zeta function or Shintani L-function is a generalization of the Riemann zeta function. They were first studied by Takuro Shintani (1976). They include Hurwitz zeta functions and Barnes zeta functions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2338_chunk_0 | Smarandache–Wellin number | In mathematics, a Smarandache–Wellin number is an integer that in a given base is the concatenation of the first n prime numbers written in that base. Smarandache–Wellin numbers are named after Florentin Smarandache and Paul R. Wellin. The first decimal Smarandache–Wellin numbers are: 2, 23, 235, 2357, 235711, 23571113... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2339_chunk_0 | Sobolev space | In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2340_chunk_0 | Generalized Mersenne prime | In mathematics, a Solinas prime, or generalized Mersenne prime, is a prime number that has the form f ( 2 m ) {\displaystyle f(2^{m})} , where f ( x ) {\displaystyle f(x)} is a low-degree polynomial with small integer coefficients. These primes allow fast modular reduction algorithms and are widely used in cryptography... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2341_chunk_0 | Stanley sequence | In mathematics, a Stanley sequence is an integer sequence generated by a greedy algorithm that chooses the sequence members to avoid arithmetic progressions. If S {\displaystyle S} is a finite set of non-negative integers on which no three elements form an arithmetic progression (that is, a Salem–Spencer set), then the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2342_chunk_0 | Stanley–Reisner ring | In mathematics, a Stanley–Reisner ring, or face ring, is a quotient of a polynomial algebra over a field by a square-free monomial ideal. Such ideals are described more geometrically in terms of finite simplicial complexes. The Stanley–Reisner ring construction is a basic tool within algebraic combinatorics and combina... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2343_chunk_0 | Stone algebra | In mathematics, a Stone algebra, or Stone lattice, is a pseudo-complemented distributive lattice such that a* ∨ a** = 1. They were introduced by Grätzer & Schmidt (1957) and named after Marshall Harvey Stone. Boolean algebras are Stone algebras, and Stone algebras are Ockham algebras. Examples: The open-set lattice of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2344_chunk_0 | Suslin algebra | In mathematics, a Suslin algebra is a Boolean algebra that is complete, atomless, countably distributive, and satisfies the countable chain condition. They are named after Mikhail Yakovlevich Suslin.The existence of Suslin algebras is independent of the axioms of ZFC, and is equivalent to the existence of Suslin trees ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2345_chunk_0 | Suslin set | In mathematics, a Suslin representation of a set of reals (more precisely, elements of Baire space) is a tree whose projection is that set of reals. More generally, a subset A of κω is λ-Suslin if there is a tree T on κ × λ such that A = p. By a tree on κ × λ we mean here a subset T of the union of κi × λi for all i ∈ ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2346_chunk_0 | Suslin tree | In mathematics, a Suslin tree is a tree of height ω1 such that every branch and every antichain is at most countable. They are named after Mikhail Yakovlevich Suslin. Every Suslin tree is an Aronszajn tree. The existence of a Suslin tree is independent of ZFC, and is equivalent to the existence of a Suslin line (shown ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2347_chunk_0 | Suslin tree | The diamond principle, a consequence of V=L, implies that there is a Suslin tree, and Martin's axiom MA(ℵ1) implies that there are no Suslin trees. More generally, for any infinite cardinal κ, a κ-Suslin tree is a tree of height κ such that every branch and antichain has cardinality less than κ. In particular a Suslin ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2348_chunk_0 | Sylvester domain | In mathematics, a Sylvester domain, named after James Joseph Sylvester by Dicks & Sontag (1978), is a ring in which Sylvester's law of nullity holds. This means that if A is an m by n matrix, and B is an n by s matrix over R, then ρ(AB) ≥ ρ(A) + ρ(B) – nwhere ρ is the inner rank of a matrix. The inner rank of an m by n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2349_chunk_0 | Sylvester matrix | In mathematics, a Sylvester matrix is a matrix associated to two univariate polynomials with coefficients in a field or a commutative ring. The entries of the Sylvester matrix of two polynomials are coefficients of the polynomials. The determinant of the Sylvester matrix of two polynomials is their resultant, which is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2350_chunk_0 | Takiff algebra | In mathematics, a Takiff algebra is a Lie algebra over a truncated polynomial ring. More precisely, a Takiff algebra of a Lie algebra g over a field k is a Lie algebra of the form g/(xn+1) = g⊗kk/(xn+1) for some positive integer n. Sometimes these are called generalized Takiff algebras, and the name Takiff algebra is u... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2351_chunk_0 | Tamari lattice | The number of elements in a Tamari lattice for a sequence of n + 1 objects is the nth Catalan number Cn. The Tamari lattice can also be described in several other equivalent ways: It is the poset of sequences of n integers a1, ..., an, ordered coordinatewise, such that i ≤ ai ≤ n and if i ≤ j ≤ ai then aj ≤ ai (Huang &... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2352_chunk_0 | Tannakian duality | In mathematics, a Tannakian category is a particular kind of monoidal category C, equipped with some extra structure relative to a given field K. The role of such categories C is to approximate, in some sense, the category of linear representations of an algebraic group G defined over K. A number of major applications ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2353_chunk_0 | Tannakian duality | The pattern of the theory is that of Grothendieck's Galois theory, which is a theory about finite permutation representations of groups G which are profinite groups. The gist of the theory is that the fiber functor Φ of the Galois theory is replaced by a tensor functor T from C to K-Vect. The group of natural transform... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2354_chunk_0 | Tate vector space | In mathematics, a Tate vector space is a vector space obtained from finite-dimensional vector spaces in a way that makes it possible to extend concepts such as dimension and determinant to an infinite-dimensional situation. Tate spaces were introduced by Alexander Beilinson, Boris Feigin, and Barry Mazur (1991), who na... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2355_chunk_0 | Thue equation | In mathematics, a Thue equation is a Diophantine equation of the form ƒ(x,y) = r,where ƒ is an irreducible bivariate form of degree at least 3 over the rational numbers, and r is a nonzero rational number. It is named after Axel Thue, who in 1909 proved that a Thue equation can have only finitely many solutions in inte... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2356_chunk_0 | Tschirnhaus transformation | In mathematics, a Tschirnhaus transformation, also known as Tschirnhausen transformation, is a type of mapping on polynomials developed by Ehrenfried Walther von Tschirnhaus in 1683.Simply, it is a method for transforming a polynomial equation of degree n ≥ 2 {\displaystyle n\geq 2} with some nonzero intermediate coeff... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2357_chunk_0 | Tschirnhaus transformation | , a n − 1 ′ {\displaystyle a'_{1},...,a'_{n-1}} , are exactly zero. For example, finding a substitutionfor a cubic equation of degree n = 3 {\displaystyle n=3} ,such that substituting x = x ( y ) {\displaystyle x=x(y)} yields a new equationsuch that a 1 ′ = 0 {\displaystyle a'_{1}=0} , a 2 ′ = 0 {\displaystyle a'_{2}=0... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2358_chunk_0 | Tutte–Grothendieck invariant | In mathematics, a Tutte–Grothendieck (TG) invariant is a type of graph invariant that satisfies a generalized deletion–contraction formula. Any evaluation of the Tutte polynomial would be an example of a TG invariant. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2359_chunk_0 | Verlinde formula | In mathematics, a Verlinde algebra is a finite-dimensional associative algebra introduced by Erik Verlinde (1988), with a basis of elements φλ corresponding to primary fields of a rational two-dimensional conformal field theory, whose structure constants Nνλμ describe fusion of primary fields. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2360_chunk_0 | Vitali set | In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by Giuseppe Vitali in 1905. The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vitali sets, and their existence depends on the axiom of choice. In 1970, Rob... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2361_chunk_0 | Vogan diagram | In mathematics, a Vogan diagram, named after David Vogan, is a variation of the Dynkin diagram of a real semisimple Lie algebra that indicates the maximal compact subgroup. Although they resemble Satake diagrams they are a different way of classifying simple Lie algebras. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2362_chunk_0 | Voronoi decomposition | In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed there is a corresponding region, called a Voronoi cell, consisting of all p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2363_chunk_0 | Voronoi formula | In mathematics, a Voronoi formula is an equality involving Fourier coefficients of automorphic forms, with the coefficients twisted by additive characters on either side. It can be regarded as a Poisson summation formula for non-abelian groups. The Voronoi (summation) formula for GL(2) has long been a standard tool for... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2364_chunk_0 | Walsh matrix | In mathematics, a Walsh matrix is a specific square matrix of dimensions 2n, where n is some particular natural number. The entries of the matrix are either +1 or −1 and its rows as well as columns are orthogonal, i.e. dot product is zero. The Walsh matrix was proposed by Joseph L. Walsh in 1923. Each row of a Walsh ma... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2365_chunk_0 | Walsh matrix | The Walsh matrices are a special case of Hadamard matrices. The naturally ordered Hadamard matrix is defined by the recursive formula below, and the sequency-ordered Hadamard matrix is formed by rearranging the rows so that the number of sign changes in a row is in increasing order. Confusingly, different sources refer... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2366_chunk_0 | Weierstrass point | In mathematics, a Weierstrass point P {\displaystyle P} on a nonsingular algebraic curve C {\displaystyle C} defined over the complex numbers is a point such that there are more functions on C {\displaystyle C} , with their poles restricted to P {\displaystyle P} only, than would be predicted by the Riemann–Roch theore... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2367_chunk_0 | Weierstrass point | Any other case is a Weierstrass point. A Weierstrass gap for P {\displaystyle P} is a value of k {\displaystyle k} such that no function on C {\displaystyle C} has exactly a k {\displaystyle k} -fold pole at P {\displaystyle P} only. The gap sequence is 1 , 2 , … , g {\displaystyle 1,2,\dots ,g} for a non-Weierstrass p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2368_chunk_0 | Weierstrass point | For a Weierstrass point it contains at least one higher number. (The Weierstrass gap theorem or Lückensatz is the statement that there must be g {\displaystyle g} gaps.) For hyperelliptic curves, for example, we may have a function F {\displaystyle F} with a double pole at P {\displaystyle P} only. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2369_chunk_0 | Weierstrass point | In general if the gap sequence is a , b , c , … {\displaystyle a,b,c,\dots } the weight of the Weierstrass point is ( a − 1 ) + ( b − 2 ) + ( c − 3 ) + … . {\displaystyle (a-1)+(b-2)+(c-3)+\dots .} This is introduced because of a counting theorem: on a Riemann surface the sum of the weights of the Weierstrass points is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2370_chunk_0 | Weierstrass point | Therefore, there are (at most) 2 ( g + 1 ) {\displaystyle 2(g+1)} of them. The 2 g + 2 {\displaystyle 2g+2} ramification points of the ramified covering of degree two from a hyperelliptic curve to the projective line are all hyperelliptic Weierstrass points and these exhausts all the Weierstrass points on a hyperellipt... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2371_chunk_0 | Weyl sequence | In mathematics, a Weyl sequence is a sequence from the equidistribution theorem proven by Hermann Weyl:The sequence of all multiples of an irrational α, 0, α, 2α, 3α, 4α, ... is equidistributed modulo 1.In other words, the sequence of the fractional parts of each term will be uniformly distributed in the interval [0, 1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2372_chunk_0 | Whittaker function | In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by Whittaker (1903) to make the formulas involving the solutions more symmetric. More generally, Jacquet (1966, 1967) introduced Whittaker functions of reductive groups... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2373_chunk_0 | Whittaker function | {\displaystyle {\frac {d^{2}w}{dz^{2}}}+\left(-{\frac {1}{4}}+{\frac {\kappa }{z}}+{\frac {1/4-\mu ^{2}}{z^{2}}}\right)w=0.} It has a regular singular point at 0 and an irregular singular point at ∞. Two solutions are given by the Whittaker functions Mκ,μ(z), Wκ,μ(z), defined in terms of Kummer's confluent hypergeometr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2374_chunk_0 | Whittaker function | {\displaystyle W_{\kappa ,\mu }\left(z\right)=\exp \left(-z/2\right)z^{\mu +{\tfrac {1}{2}}}U\left(\mu -\kappa +{\tfrac {1}{2}},1+2\mu ,z\right).} The Whittaker function W κ , μ ( z ) {\displaystyle W_{\kappa ,\mu }(z)} is the same as those with opposite values of μ, in other words considered as a function of μ at fixe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2375_chunk_0 | Wieferich pair | In mathematics, a Wieferich pair is a pair of prime numbers p and q that satisfy pq − 1 ≡ 1 (mod q2) and qp − 1 ≡ 1 (mod p2)Wieferich pairs are named after German mathematician Arthur Wieferich. Wieferich pairs play an important role in Preda Mihăilescu's 2002 proof of Mihăilescu's theorem (formerly known as Catalan's ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2376_chunk_0 | Young symmetrizer | In mathematics, a Young symmetrizer is an element of the group algebra of the symmetric group, constructed in such a way that, for the homomorphism from the group algebra to the endomorphisms of a vector space V ⊗ n {\displaystyle V^{\otimes n}} obtained from the action of S n {\displaystyle S_{n}} on V ⊗ n {\displayst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2377_chunk_0 | Zariski geometry | In mathematics, a Zariski geometry consists of an abstract structure introduced by Ehud Hrushovski and Boris Zilber, in order to give a characterisation of the Zariski topology on an algebraic curve, and all its powers. The Zariski topology on a product of algebraic varieties is very rarely the product topology, but ri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2378_chunk_0 | Zinbiel algebra | In mathematics, a Zinbiel algebra or dual Leibniz algebra is a module over a commutative ring with a bilinear product satisfying the defining identity: ( a ∘ b ) ∘ c = a ∘ ( b ∘ c ) + a ∘ ( c ∘ b ) . {\displaystyle (a\circ b)\circ c=a\circ (b\circ c)+a\circ (c\circ b).} Zinbiel algebras were introduced by Jean-Louis Lo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2379_chunk_0 | Zinbiel algebra | The name was proposed by Jean-Michel Lemaire as being "opposite" to Leibniz algebra.In any Zinbiel algebra, the symmetrised product a ⋆ b = a ∘ b + b ∘ a {\displaystyle a\star b=a\circ b+b\circ a} is associative. A Zinbiel algebra is the Koszul dual concept to a Leibniz algebra. The free Zinbiel algebra over V is the t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2380_chunk_0 | Balanced matrix | In mathematics, a balanced matrix is a 0-1 matrix (a matrix where every entry is either zero or one) that does not contain any square submatrix of odd order having all row sums and all column sums equal to 2. Balanced matrices are studied in linear programming. The importance of balanced matrices comes from the fact th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2381_chunk_0 | 4-ball (mathematics) | In a one-dimensional space, a ball is a line segment. In other contexts, such as in Euclidean geometry and informal use, sphere is sometimes used to mean ball. In the field of topology the closed n {\displaystyle n} -dimensional ball is often denoted as B n {\displaystyle B^{n}} or D n {\displaystyle D^{n}} while the o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2382_chunk_0 | Band (algebra) | In mathematics, a band (also called idempotent semigroup) is a semigroup in which every element is idempotent (in other words equal to its own square). Bands were first studied and named by A. H. Clifford (1954). The lattice of varieties of bands was described independently in the early 1970s by Biryukov, Fennemore and... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2383_chunk_0 | Countable base | In mathematics, a base (or basis; PL: bases) for the topology τ of a topological space (X, τ) is a family B {\displaystyle {\mathcal {B}}} of open subsets of X such that every open set of the topology is equal to the union of some sub-family of B {\displaystyle {\mathcal {B}}} . For example, the set of all open interva... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2384_chunk_0 | Countable base | Many important topological definitions such as continuity and convergence can be checked using only basic open sets instead of arbitrary open sets. Some topologies have a base of open sets with specific useful properties that may make checking such topological definitions easier. Not all families of subsets of a set X ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2385_chunk_0 | Countable base | Under some conditions detailed below, a family of subsets will form a base for a (unique) topology on X {\displaystyle X} , obtained by taking all possible unions of subfamilies. Such families of sets are very frequently used to define topologies. A weaker notion related to bases is that of a subbase for a topology. Ba... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2386_chunk_0 | Base-orderable matroid | In mathematics, a base-orderable matroid is a matroid that has the following additional property, related to the bases of the matroid. For any two bases A {\displaystyle A} and B {\displaystyle B} there exists a feasible exchange bijection, defined as a bijection f {\displaystyle f} from A {\displaystyle A} to B {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2387_chunk_0 | Algebraic operation | An algebraic operation may also be defined simply as a function from a Cartesian power of a set to the same set.The term algebraic operation may also be used for operations that may be defined by compounding basic algebraic operations, such as the dot product. In calculus and mathematical analysis, algebraic operation ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2388_chunk_0 | Semi-algebraic sets | In mathematics, a basic semialgebraic set is a set defined by polynomial equalities and polynomial inequalities, and a semialgebraic set is a finite union of basic semialgebraic sets. A semialgebraic function is a function with a semialgebraic graph. Such sets and functions are mainly studied in real algebraic geometry... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2389_chunk_0 | Blending function | In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors. In numerical analysis and app... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2390_chunk_0 | Bi-directional delay line | In mathematics, a bi-directional delay line is a numerical analysis technique used in computer simulation for solving ordinary differential equations by converting them to hyperbolic equations. In this way an explicit solution scheme is obtained with highly robust numerical properties. It was introduced by Auslander in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2391_chunk_0 | Bi-directional delay line | It was then found that it could be used as an efficient numerical technique for numerically insulating different parts of a simulation model in each times step. It is used in the HOPSAN simulation package (Krus et al. 1990). It is also known as the Transmission Line Modelling (TLM) from an independent development by Jo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2392_chunk_0 | Bialgebra | In mathematics, a bialgebra over a field K is a vector space over K which is both a unital associative algebra and a counital coassociative coalgebra. The algebraic and coalgebraic structures are made compatible with a few more axioms. Specifically, the comultiplication and the counit are both unital algebra homomorphi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2393_chunk_0 | Bialgebra | Similar bialgebras are related by bialgebra homomorphisms. A bialgebra homomorphism is a linear map that is both an algebra and a coalgebra homomorphism. As reflected in the symmetry of the commutative diagrams, the definition of bialgebra is self-dual, so if one can define a dual of B (which is always possible if B is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2394_chunk_0 | Bidiagonal matrix | In mathematics, a bidiagonal matrix is a banded matrix with non-zero entries along the main diagonal and either the diagonal above or the diagonal below. This means there are exactly two non-zero diagonals in the matrix. When the diagonal above the main diagonal has the non-zero entries the matrix is upper bidiagonal. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2395_chunk_0 | Bidiagonal matrix | When the diagonal below the main diagonal has the non-zero entries the matrix is lower bidiagonal. For example, the following matrix is upper bidiagonal: ( 1 4 0 0 0 4 1 0 0 0 3 4 0 0 0 3 ) {\displaystyle {\begin{pmatrix}1&4&0&0\\0&4&1&0\\0&0&3&4\\0&0&0&3\\\end{pmatrix}}} and the following matrix is lower bidiagonal: (... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2396_chunk_0 | Bijective relation | In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2397_chunk_0 | Bijective relation | For infinite sets, the picture is more complicated, leading to the concept of cardinal number—a way to distinguish the various sizes of infinite sets. A bijective function from a set to itself is also called a permutation, and the set of all permutations of a set forms the symmetric group. Bijective functions are essen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2398_chunk_0 | Radical of a quadratic space | In mathematics, a bilinear form is a bilinear map V × V → K on a vector space V (the elements of which are called vectors) over a field K (the elements of which are called scalars). In other words, a bilinear form is a function B: V × V → K that is linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2399_chunk_0 | Separate continuity | In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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