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Hilbert–Burch theorem Summary Hilbert–Burch_theorem In mathematics, the Hilbert–Burch theorem describes the structure of some free resolutions of a quotient of a local or graded ring in the case that the quotient has projective dimension 2. Hilbert (1890) proved a version of this theorem for polynomial rings, and Burch... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hilbert–Mumford criterion Summary Hilbert–Mumford_criterion In mathematics, the Hilbert–Mumford criterion, introduced by David Hilbert and David Mumford, characterizes the semistable and stable points of a group action on a vector space in terms of eigenvalues of 1-parameter subgroups (Dieudonné & Carrell 1970, 1971, p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hilbert-Pólya conjecture Summary Hilbert–Pólya_conjecture In mathematics, the Hilbert–Pólya conjecture states that the non-trivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator. It is a possible approach to the Riemann hypothesis, by means of spectral theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hilbert–Smith conjecture Summary Hilbert–Smith_conjecture In mathematics, the Hilbert–Smith conjecture is concerned with the transformation groups of manifolds; and in particular with the limitations on topological groups G that can act effectively (faithfully) on a (topological) manifold M. Restricting to G which are ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hilbert–Smith conjecture Summary Hilbert–Smith_conjecture It is considered by some to be a better formulation of Hilbert's fifth problem, than the characterisation in the category of topological groups of the Lie groups often cited as a solution. In 1997, Dušan Repovš and Evgenij Ščepin proved the Hilbert–Smith conject... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hilbert–Speiser theorem Summary Hilbert–Speiser_theorem In mathematics, the Hilbert–Speiser theorem is a result on cyclotomic fields, characterising those with a normal integral basis. More generally, it applies to any finite abelian extension of Q, which by the Kronecker–Weber theorem are isomorphic to subfields of cy... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hilbert–Speiser theorem Summary Hilbert–Speiser_theorem This result was introduced by Hilbert (1897, Satz 132, 1998, theorem 132) in his Zahlbericht and by Speiser (1916, corollary to proposition 8.1). In cases where the theorem states that a normal integral basis does exist, such a basis may be constructed by means of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hilbert–Speiser theorem Summary Hilbert–Speiser_theorem For a field K contained in it, the field trace can be used to construct such a basis in K also (see the article on Gaussian periods). Then in the case of n squarefree and odd, Q(ζn) is a compositum of subfields of this type for the primes p dividing n (this follow... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hill differential equation Summary Hill_differential_equation In mathematics, the Hill equation or Hill differential equation is the second-order linear ordinary differential equation d 2 y d t 2 + f ( t ) y = 0 , {\displaystyle {\frac {d^{2}y}{dt^{2}}}+f(t)y=0,} where f ( t ) {\displaystyle f(t)} is a periodic functio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hirzebruch–Riemann–Roch theorem Summary Hirzebruch–Riemann–Roch_theorem In mathematics, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch's 1954 result generalizing the classical Riemann–Roch theorem on Riemann surfaces to all complex algebraic varie... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hitchin system Summary Hitchin_fibration In mathematics, the Hitchin integrable system is an integrable system depending on the choice of a complex reductive group and a compact Riemann surface, introduced by Nigel Hitchin in 1987. It lies on the crossroads of algebraic geometry, the theory of Lie algebras and integrab... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hitchin system Summary Hitchin_fibration (The Garnier system is the classical limit of the Gaudin model. In turn, the Schlesinger equations are the classical limit of the Knizhnik–Zamolodchikov equations). Almost all integrable systems of classical mechanics can be obtained as particular cases of the Hitchin system or ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hjelmslev transformation Summary Hjelmslev_transformation In mathematics, the Hjelmslev transformation is an effective method for mapping an entire hyperbolic plane into a circle with a finite radius. The transformation was invented by Danish mathematician Johannes Hjelmslev. It utilizes Nikolai Ivanovich Lobachevsky's... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hjelmslev transformation Summary Hjelmslev_transformation Lobachevsky observes, using a combination of his 16th and 23rd theorems, that it is a fundamental characteristic of hyperbolic geometry that there must exist a distinct angle of parallelism for any given line length. Let us say for the length AE, its angle of pa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hjelmslev transformation Summary Hjelmslev_transformation Consequently, any line drawn perpendicular to base AE between A and E must necessarily cross line AH at some finite distance. Johannes Hjelmslev discovered from this a method of compressing an entire hyperbolic plane into a finite circle. The method is as follow... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hjelmslev transformation Summary Hjelmslev_transformation This point H thus mapped must fall between A and E. By applying this process for every line within the plane, the infinite hyperbolic space thus becomes contained and planar. Hjelmslev's transformation does not yield a proper circle however. The circumference of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hjelmslev transformation Summary Hjelmslev_transformation Likewise, when this transformation is extended in all three dimensions, it is referred to as a Hjelmslev Ball. There are a few properties that are retained through the transformation which enable valuable information to be ascertained therefrom, namely: The imag... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hjelmslev transformation Summary Hjelmslev_transformation Any angle with the center of the transformation as its vertex will be preserved. The image of any straight line will be a finite straight line segment. Likewise, the point order is maintained throughout a transformation, i.e. if B is between A and C, the image o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hochschild–Mostow group Summary Hochschild–Mostow_group In mathematics, the Hochschild–Mostow group, introduced by Hochschild and Mostow (1957), is the universal pro-affine algebraic group generated by a group. == References == | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hodge vector bundle Summary Hodge_bundle In mathematics, the Hodge bundle, named after W. V. D. Hodge, appears in the study of families of curves, where it provides an invariant in the moduli theory of algebraic curves. Furthermore, it has applications to the theory of modular forms on reductive algebraic groups and st... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hodge conjecture Summary Hodge_conjecture In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties. In simple terms, the Hodge conjecture asserts that the basic topologi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hodge conjecture Summary Hodge_conjecture It was formulated by the Scottish mathematician William Vallance Douglas Hodge as a result of a work in between 1930 and 1940 to enrich the description of de Rham cohomology to include extra structure that is present in the case of complex algebraic varieties. It received littl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hodge index theorem Summary Hodge_index_theorem In mathematics, the Hodge index theorem for an algebraic surface V determines the signature of the intersection pairing on the algebraic curves C on V. It says, roughly speaking, that the space spanned by such curves (up to linear equivalence) has a one-dimensional subspa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hodge index theorem Summary Hodge_index_theorem The dimension of D is finite and is usually denoted by ρ(V). The Hodge index theorem says that the subspace spanned by H in D has a complementary subspace on which the intersection pairing is negative definite. Therefore, the signature (often also called index) is (1,ρ(V)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hodge index theorem Summary Hodge_index_theorem The abelian group of divisor classes up to algebraic equivalence is now called the Néron-Severi group; it is known to be a finitely-generated abelian group, and the result is about its tensor product with the rational number field. Therefore, ρ(V) is equally the rank of t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hodge index theorem Summary Hodge_index_theorem This result was proved in the 1930s by W. V. D. Hodge, for varieties over the complex numbers, after it had been a conjecture for some time of the Italian school of algebraic geometry (in particular, Francesco Severi, who in this case showed that ρ < ∞). Hodge's methods w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hodge dual Summary Hodge_duality In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the Hodge dual of the ele... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hodge dual Summary Hodge_duality Generalizing this to an n-dimensional vector space, the Hodge star is a one-to-one mapping of k-vectors to (n – k)-vectors; the dimensions of these spaces are the binomial coefficients ( n k ) = ( n n − k ) {\displaystyle {\tbinom {n}{k}}={\tbinom {n}{n-k}}} . The naturalness of the sta... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hodge–de Rham spectral sequence Summary Hodge–de_Rham_spectral_sequence In mathematics, the Hodge–de Rham spectral sequence (named in honor of W. V. D. Hodge and Georges de Rham) is an alternative term sometimes used to describe the Frölicher spectral sequence (named after Alfred Frölicher, who actually discovered it).... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mutually recursive Mathematical functions Mutually_recursive > Examples > Mathematical functions In mathematics, the Hofstadter Female and Male sequences are an example of a pair of integer sequences defined in a mutually recursive manner. Fractals can be computed (up to a given resolution) by recursive functions. This... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Holomorphic Lefschetz fixed-point formula Summary Holomorphic_Lefschetz_fixed-point_formula In mathematics, the Holomorphic Lefschetz formula is an analogue for complex manifolds of the Lefschetz fixed-point formula that relates a sum over the fixed points of a holomorphic vector field of a compact complex manifold to ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Honda–Tate theorem Summary Honda–Tate_theorem In mathematics, the Honda–Tate theorem classifies abelian varieties over finite fields up to isogeny. It states that the isogeny classes of simple abelian varieties over a finite field of order q correspond to algebraic integers all of whose conjugates (given by eigenvalues... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hopf decomposition Summary Hopf_decomposition In mathematics, the Hopf decomposition, named after Eberhard Hopf, gives a canonical decomposition of a measure space (X, μ) with respect to an invertible non-singular transformation T:X→X, i.e. a transformation which with its inverse is measurable and carries null sets ont... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hopf lemma Summary Hopf_lemma In mathematics, the Hopf lemma, named after Eberhard Hopf, states that if a continuous real-valued function in a domain in Euclidean space with sufficiently smooth boundary is harmonic in the interior and the value of the function at a point on the boundary is greater than the values at ne... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hopf lemma Summary Hopf_lemma In the special case of the Laplacian, the Hopf lemma had been discovered by Stanisław Zaremba in 1910. In the more general setting for elliptic equations, it was found independently by Hopf and Olga Oleinik in 1952, although Oleinik's work is not as widely known as Hopf's in Western countr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Horrocks construction Summary Horrocks_construction In mathematics, the Horrocks construction is a method for constructing vector bundles, especially over projective spaces, introduced by Geoffrey Horrocks (1964, section 10). His original construction gave an example of an indecomposable rank 2 vector bundle over 3-dim... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Humbert polynomials Summary Humbert_polynomials In mathematics, the Humbert polynomials πλn,m(x) are a generalization of Pincherle polynomials introduced by Humbert (1921) given by the generating function ( 1 − m x t + t m ) − λ = ∑ n = 0 ∞ π n , m λ ( x ) t n {\displaystyle \displaystyle (1-mxt+t^{m})^{-\lambda }=\sum... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Rational Hurewicz theorem Summary Hurewicz_theorem In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results of Henri Poincaré. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hurwitz class number Summary Hurwitz_class_number In mathematics, the Hurwitz class number H(N), introduced by Adolf Hurwitz, is a modification of the class number of positive definite binary quadratic forms of discriminant –N, where forms are weighted by 2/g for g the order of their automorphism group, and where H(0) ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hurwitz problem Summary Hurwitz_problem In mathematics, the Hurwitz problem (named after Adolf Hurwitz) is the problem of finding multiplicative relations between quadratic forms which generalise those known to exist between sums of squares in certain numbers of variables. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hurwitz Zeta function Summary Hurwitz_Zeta_function In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables s with Re(s) > 1 and a ≠ 0, −1, −2, … by ζ ( s , a ) = ∑ n = 0 ∞ 1 ( n + a ) s . {\displaystyle \zeta (s,a)=\sum _{n=0}^{\infty }{\frac {1}{(n+a)^... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hurwitz Zeta function Summary Hurwitz_Zeta_function This series is absolutely convergent for the given values of s and a and can be extended to a meromorphic function defined for all s ≠ 1. The Riemann zeta function is ζ(s,1). The Hurwitz zeta function is named after Adolf Hurwitz, who introduced it in 1882. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hutchinson metric Summary Hutchinson_metric In mathematics, the Hutchinson metric otherwise known as Kantorovich metric is a function which measures "the discrepancy between two images for use in fractal image processing" and "can also be applied to describe the similarity between DNA sequences expressed as real or com... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hénon map Summary Hénon_map In mathematics, the Hénon map, sometimes called Hénon–Pomeau attractor/map, is a discrete-time dynamical system. It is one of the most studied examples of dynamical systems that exhibit chaotic behavior. The Hénon map takes a point (xn, yn) in the plane and maps it to a new point { x n + 1 =... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hénon map Summary Hénon_map For the classical values the Hénon map is chaotic. For other values of a and b the map may be chaotic, intermittent, or converge to a periodic orbit. An overview of the type of behavior of the map at different parameter values may be obtained from its orbit diagram. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hénon map Summary Hénon_map The map was introduced by Michel Hénon as a simplified model of the Poincaré section of the Lorenz model. For the classical map, an initial point of the plane will either approach a set of points known as the Hénon strange attractor, or diverge to infinity. The Hénon attractor is a fractal, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ihara zeta function Summary Ihara_zeta_function In mathematics, the Ihara zeta function is a zeta function associated with a finite graph. It closely resembles the Selberg zeta function, and is used to relate closed walks to the spectrum of the adjacency matrix. The Ihara zeta function was first defined by Yasutaka Iha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ihara zeta function Summary Ihara_zeta_function Jean-Pierre Serre suggested in his book Trees that Ihara's original definition can be reinterpreted graph-theoretically. It was Toshikazu Sunada who put this suggestion into practice in 1985. As observed by Sunada, a regular graph is a Ramanujan graph if and only if its I... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ikeda lift Summary Ikeda_lift In mathematics, the Ikeda lift is a lifting of modular forms to Siegel modular forms. The existence of the lifting was conjectured by W. Duke and Ö. Imamoḡlu and also by T. Ibukiyama, and the lifting was constructed by Ikeda (2001). It generalized the Saito–Kurokawa lift from modular forms... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ince polynomials Summary Ince_polynomial In mathematics, the Ince equation, named for Edward Lindsay Ince, is the differential equation w ′ ′ + ξ sin ( 2 z ) w ′ + ( η − p ξ cos ( 2 z ) ) w = 0. {\displaystyle w^{\prime \prime }+\xi \sin(2z)w^{\prime }+(\eta -p\xi \cos(2z))w=0.\,} When p is a non-negative integer, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Incomplete polylogarithm Summary Incomplete_polylogarithm In mathematics, the Incomplete Polylogarithm function is related to the polylogarithm function. It is sometimes known as the incomplete Fermi–Dirac integral or the incomplete Bose–Einstein integral. It may be defined by: Li s ( b , z ) = 1 Γ ( s ) ∫ b ∞ x s − ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Incomplete polylogarithm Summary Incomplete_polylogarithm Expanding about z=0 and integrating gives a series representation: Li s ( b , z ) = ∑ k = 1 ∞ z k k s Γ ( s , k b ) Γ ( s ) {\displaystyle \operatorname {Li} _{s}(b,z)=\sum _{k=1}^{\infty }{\frac {z^{k}}{k^{s}}}~{\frac {\Gamma (s,kb)}{\Gamma (s)}}} where Γ(s) ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Itô isometry Summary Itô_isometry In mathematics, the Itô isometry, named after Kiyoshi Itô, is a crucial fact about Itô stochastic integrals. One of its main applications is to enable the computation of variances for random variables that are given as Itô integrals. Let W: × Ω → R {\displaystyle W:\times \Omega \to \... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Itô isometry Summary Itô_isometry In other words, the Itô integral, as a function from the space L a d 2 ( × Ω ) {\displaystyle L_{\mathrm {ad} }^{2}(\times \Omega )} of square-integrable adapted processes to the space L 2 ( Ω ) {\displaystyle L^{2}(\Omega )} of square-integrable random variables, is an isometry of no... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Iverson bracket Summary Iverson_bracket In mathematics, the Iverson bracket, named after Kenneth E. Iverson, is a notation that generalises the Kronecker delta, which is the Iverson bracket of the statement x = y. It maps any statement to a function of the free variables in that statement. This function is defined to t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Iverson bracket Summary Iverson_bracket That is, for any property P ( k ) {\displaystyle P(k)} of the integer k {\displaystyle k} , one can rewrite the restricted sum ∑ k: P ( k ) f ( k ) {\displaystyle \sum _{k:P(k)}f(k)} in the unrestricted form ∑ k f ( k ) ⋅ {\displaystyle \sum _{k}f(k)\cdot } . With this conventio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Iwahori–Hecke algebra Summary Hecke_algebra_of_a_Coxeter_group In mathematics, the Iwahori–Hecke algebra, or Hecke algebra, named for Erich Hecke and Nagayoshi Iwahori, is a deformation of the group algebra of a Coxeter group. Hecke algebras are quotients of the group rings of Artin braid groups. This connection found ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Iwasawa algebra Summary Iwasawa_algebra In mathematics, the Iwasawa algebra Λ(G) of a profinite group G is a variation of the group ring of G with p-adic coefficients that take the topology of G into account. More precisely, Λ(G) is the inverse limit of the group rings Zp(G/H) as H runs through the open normal subgroup... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Iwasawa decomposition Summary Iwasawa_decomposition In mathematics, the Iwasawa decomposition (aka KAN from its expression) of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (QR decomposition, a consequence of Gram–Schmi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Adams conjecture Summary Adams_conjecture In mathematics, the J-homomorphism is a mapping from the homotopy groups of the special orthogonal groups to the homotopy groups of spheres. It was defined by George W. Whitehead (1942), extending a construction of Heinz Hopf (1935). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
JSJ decomposition Summary JSJ_decomposition In mathematics, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem: Irreducible orientable closed (i.e., compact and without boundary) 3-manifolds have a unique (up to isotopy) minimal collection of disjoint... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Jack polynomials Summary Jack_function In mathematics, the Jack function is a generalization of the Jack polynomial, introduced by Henry Jack. The Jack polynomial is a homogeneous, symmetric polynomial which generalizes the Schur and zonal polynomials, and is in turn generalized by the Heckman–Opdam polynomials and Mac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Jacobian curve Summary Jacobian_curve In mathematics, the Jacobi curve is a representation of an elliptic curve different from the usual one defined by the Weierstrass equation. Sometimes it is used in cryptography instead of the Weierstrass form because it can provide a defence against simple and differential power an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Jacobi elliptic cosine Summary Jacobi_elliptic_functions In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While trigonometric func... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Jacobi elliptic cosine Summary Jacobi_elliptic_functions The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions of complex analysis to be defined and/or understood. They were introduced by Carl Gustav Jakob Jacobi (1829). Carl Fried... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Jacobi group Summary Jacobi_group In mathematics, the Jacobi group, introduced by Eichler & Zagier (1985), is the semidirect product of the symplectic group Sp2n(R) and the Heisenberg group R1+2n. The concept is named after Carl Gustav Jacob Jacobi. Automorphic forms on the Jacobi group are called Jacobi forms. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Jacobi identities Summary Jacobi_identity In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associative property, any order of e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Jacobi identities Summary Jacobi_identity The cross product a × b {\displaystyle a\times b} and the Lie bracket operation {\displaystyle } both satisfy the Jacobi identity. In analytical mechanics, the Jacobi identity is satisfied by the Poisson brackets. In quantum mechanics, it is satisfied by operator commutators o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Jacobi triple product identity Summary Jacobi_triple_product_identity In mathematics, the Jacobi triple product is the mathematical identity: ∏ m = 1 ∞ ( 1 − x 2 m ) ( 1 + x 2 m − 1 y 2 ) ( 1 + x 2 m − 1 y 2 ) = ∑ n = − ∞ ∞ x n 2 y 2 n , {\displaystyle \prod _{m=1}^{\infty }\left(1-x^{2m}\right)\left(1+x^{2m-1}y^{2}\ri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Jacobi zeta function Summary Jacobi_zeta_function In mathematics, the Jacobi zeta function Z(u) is the logarithmic derivative of the Jacobi theta function Θ(u). It is also commonly denoted as z n ( u , k ) {\displaystyle zn(u,k)} Θ ( u ) = Θ 4 ( π u 2 K ) {\displaystyle \Theta (u)=\Theta _{4}\left({\frac {\pi u}{2K}}\r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Jacobian conjecture Summary Jacobian_conjecture In mathematics, the Jacobian conjecture is a famous unsolved problem concerning polynomials in several variables. It states that if a polynomial function from an n-dimensional space to itself has Jacobian determinant which is a non-zero constant, then the function has a p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Jacobian conjecture Summary Jacobian_conjecture The Jacobian conjecture is notorious for the large number of attempted proofs that turned out to contain subtle errors. As of 2018, there are no plausible claims to have proved it. Even the two-variable case has resisted all efforts. There are currently no known compellin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Jacobian of a curve Summary Jacobian_of_a_curve In mathematics, the Jacobian variety J(C) of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of C, hence an abelian variety. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Jacobi–Anger expansion Summary Jacobi-Anger_expansion In mathematics, the Jacobi–Anger expansion (or Jacobi–Anger identity) is an expansion of exponentials of trigonometric functions in the basis of their harmonics. It is useful in physics (for example, to convert between plane waves and cylindrical waves), and in sign... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Jacobi–Anger expansion Summary Jacobi-Anger_expansion {\textstyle i^{2}=-1.} Substituting θ {\textstyle \theta } by θ − π 2 {\textstyle \theta -{\frac {\pi }{2}}} , we also get: e i z sin θ ≡ ∑ n = − ∞ ∞ J n ( z ) e i n θ . {\displaystyle e^{iz\sin \theta }\equiv \sum _{n=-\infty }^{\infty }J_{n}(z)\,e^{in\theta }.} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Jacobsthal number Summary Jacobsthal_number In mathematics, the Jacobsthal numbers are an integer sequence named after the German mathematician Ernst Jacobsthal. Like the related Fibonacci numbers, they are a specific type of Lucas sequence U n ( P , Q ) {\displaystyle U_{n}(P,Q)} for which P = 1, and Q = −2—and are de... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Jacquet functor Summary Jacquet_module In mathematics, the Jacquet module is a module used in the study of automorphic representations. The Jacquet functor is the functor that sends a linear representation to its Jacquet module. They are both named after Hervé Jacquet. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
James embedding Summary James_embedding In mathematics, the James embedding is an embedding of a real, complex, or hyperbolic projective space into a sphere, introduced by Ioan James (1958, 1959). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Jessen–Wintner theorem Summary Jessen–Wintner_theorem In mathematics, the Jessen–Wintner theorem, introduced by Jessen and Wintner (1935), asserts that a random variable of Jessen–Wintner type, meaning the sum of an almost surely convergent series of independent discrete random variables, is of pure type. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
John ellipsoid Summary John_ellipsoid In mathematics, the John ellipsoid or Löwner-John ellipsoid E(K) associated to a convex body K in n-dimensional Euclidean space Rn can refer to the n-dimensional ellipsoid of maximal volume contained within K or the ellipsoid of minimal volume that contains K. Often, the minimal vo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Johnson scheme Summary Johnson_scheme In mathematics, the Johnson scheme, named after Selmer M. Johnson, is also known as the triangular association scheme. It consists of the set of all binary vectors X of length ℓ and weight n, such that v = | X | = ( ℓ n ) {\displaystyle v=\left|X\right|={\binom {\ell }{n}}} . Two v... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Johnson–Lindenstrauss lemma Summary Johnson–Lindenstrauss_lemma In mathematics, the Johnson–Lindenstrauss lemma is a result named after William B. Johnson and Joram Lindenstrauss concerning low-distortion embeddings of points from high-dimensional into low-dimensional Euclidean space. The lemma states that a set of poi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Johnson–Lindenstrauss lemma Summary Johnson–Lindenstrauss_lemma Much of the data stored and manipulated on computers, including text and images, can be represented as points in a high-dimensional space (see vector space model for the case of text). However, the essential algorithms for working with such data tend to be... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Jordan–Chevalley decomposition Summary Jordan_decomposition_in_a_Lie_algebra In mathematics, the Jordan–Chevalley decomposition, named after Camille Jordan and Claude Chevalley, expresses a linear operator as the sum of its commuting semisimple part and its nilpotent part. The multiplicative decomposition expresses an ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Jordan–Pólya number Summary Jordan–Pólya_number In mathematics, the Jordan–Pólya numbers are the numbers that can be obtained by multiplying together one or more factorials, not required to be distinct from each other. For instance, 480 {\displaystyle 480} is a Jordan–Pólya number because 480 = 2 ! ⋅ 2 ! ⋅ 5 ! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Jordan–Pólya number Summary Jordan–Pólya_number {\displaystyle 480=2!\cdot 2!\cdot 5!} . Every tree has a number of symmetries that is a Jordan–Pólya number, and every Jordan–Pólya number arises in this way as the order of an automorphism group of a tree. These numbers are named after Camille Jordan and George Pólya, w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Jordan's theorem on finite linear groups Summary Jordan's_theorem_on_finite_linear_groups In mathematics, the Jordan–Schur theorem also known as Jordan's theorem on finite linear groups is a theorem in its original form due to Camille Jordan. In that form, it states that there is a function ƒ(n) such that given a finit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Jordan's theorem on finite linear groups Summary Jordan's_theorem_on_finite_linear_groups 12n(π(n+1)+1)where π(n) is the prime-counting function. This was subsequently improved by Hans Frederick Blichfeldt who replaced the 12 with a 6. Unpublished work on the finite case was also done by Boris Weisfeiler. Subsequently,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Jucys–Murphy element Summary Jucys–Murphy_element In mathematics, the Jucys–Murphy elements in the group algebra C {\displaystyle \mathbb {C} } of the symmetric group, named after Algimantas Adolfas Jucys and G. E. Murphy, are defined as a sum of transpositions by the formula: X 1 = 0 , X k = ( 1 k ) + ( 2 k ) + ⋯ + (... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Kadison–Singer problem Summary Kadison–Singer_problem In mathematics, the Kadison–Singer problem, posed in 1959, was a problem in functional analysis about whether certain extensions of certain linear functionals on certain C*-algebras were unique. The uniqueness was proved in 2013. The statement arose from work on the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Kadison–Singer problem Summary Kadison–Singer_problem Kadison, Singer, and most later authors believed the statement to be false, but, in 2013, it was proven true by Adam Marcus, Daniel Spielman and Nikhil Srivastava, who received the 2014 Pólya Prize for the achievement. The solution was made possible by a reformulati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Kallman–Rota inequality Summary Kallman–Rota_inequality In mathematics, the Kallman–Rota inequality, introduced by Kallman & Rota (1970), is a generalization of the Landau–Kolmogorov inequality to Banach spaces. It states that if A is the infinitesimal generator of a one-parameter contraction semigroup then ‖ A f ‖ 2 ≤... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Kalmanson combinatorial conditions Summary Kalmanson_combinatorial_conditions In mathematics, the Kalmanson combinatorial conditions are a set of conditions on the distance matrix used in determining the solvability of the traveling salesman problem. These conditions apply to a special kind of cost matrix, the Kalmanso... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Kantor double Summary Kantor_double In mathematics, the Kantor double is a Jordan superalgebra structure on the sum of two copies of a Poisson algebra. It is named after Isaiah Kantor, who introduced it in Kantor (1990). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Kantorovich inequality Summary Kantorovich_inequality In mathematics, the Kantorovich inequality is a particular case of the Cauchy–Schwarz inequality, which is itself a generalization of the triangle inequality. The triangle inequality states that the length of two sides of any triangle, added together, will be equal ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Kantorovich inequality Summary Kantorovich_inequality (See vector space, inner product, and normed vector space for other examples of how the basic ideas inherent in the triangle inequality—line segment and distance—can be generalized into a broader context.) More formally, the Kantorovich inequality can be expressed t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Kardar–Parisi–Zhang equation Summary Kardar–Parisi–Zhang_equation In mathematics, the Kardar–Parisi–Zhang (KPZ) equation is a non-linear stochastic partial differential equation, introduced by Mehran Kardar, Giorgio Parisi, and Yi-Cheng Zhang in 1986. It describes the temporal change of a height field h ( x → , t ) {\d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Kardar–Parisi–Zhang equation Summary Kardar–Parisi–Zhang_equation In one spatial dimension, the KPZ equation corresponds to a stochastic version of Burgers' equation with field u ( x , t ) {\displaystyle u(x,t)} via the substitution u = − λ ∂ h / ∂ x {\displaystyle u=-\lambda \,\partial h/\partial x} . Via the renormal... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Karoubi conjecture Summary Karoubi_conjecture In mathematics, the Karoubi conjecture is a conjecture by Max Karoubi (1979) that the algebraic and topological K-theories coincide on C* algebras spatially tensored with the algebra of compact operators. It was proved by Andrei Suslin and Mariusz Wodzicki (1990, theorem 6,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
KdV hierarchy Summary KdV_hierarchy In mathematics, the KdV hierarchy is an infinite sequence of partial differential equations which contains the Korteweg–de Vries equation. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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