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wiki_5900_chunk_0 | Hitchin system | 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 integrable system theory. It also plays an import... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5901_chunk_0 | Hitchin system | (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 their common generalization defined by Bo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5902_chunk_0 | 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 23rd theorem from his work Geometrical Investigations on ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5903_chunk_0 | 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 parallelism is angle BAF. This being the case, line AH and E... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5904_chunk_0 | 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 follows: for any angle of parallelism, draw from its line AE a p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5905_chunk_0 | Hodge vector 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 string theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5906_chunk_0 | 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 topological information like the number of holes i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5907_chunk_0 | 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 subspace on which it is positive definite (not uniquel... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5908_chunk_0 | 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)-1). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5909_chunk_0 | 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 the Néron-Severi group (which can have a non-triv... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5910_chunk_0 | 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 were the topological ones brought in by Lefschetz... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5911_chunk_0 | Hodge dual | 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 element. This map was introduced by ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5912_chunk_0 | Hodge dual | 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 star operator means it can play a ro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5913_chunk_0 | 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 a sum over its Dolbeault cohomology groups. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5914_chunk_0 | 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 of the Frobenius endomorphism on the first co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5915_chunk_0 | 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 nearby points inside the domain,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5916_chunk_0 | 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-dimensional projective space, and generalizes to give e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5917_chunk_0 | 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 _{n=0}^{\infty }\pi _{n,m}^{\lambda }(x)t^{n}} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5918_chunk_0 | Rational 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 |
wiki_5919_chunk_0 | 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)^{s}}}.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5920_chunk_0 | 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 |
wiki_5921_chunk_0 | 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 complex genomic signals". | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5922_chunk_0 | 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, smooth in one direction and ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5923_chunk_0 | 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 Ihara in the 1960s in the context of discrete subgr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5924_chunk_0 | 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 Ihara zeta function satisfies an analogue of the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5925_chunk_0 | Ince polynomials | 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, it has polynomial solutions called Ince p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5926_chunk_0 | 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 − 1 e x / z − 1 d x . {\displaystyle \operatorname {Li} _{s}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5927_chunk_0 | 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) is the gamma function and Γ(s,x) is the upper incomplete g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5928_chunk_0 | 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 normed vector spaces with respect to... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5929_chunk_0 | 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 take the value 1 for the values of the va... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5930_chunk_0 | 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 convention, f ( k ) {\displaystyle f(k)} does not... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5931_chunk_0 | Iwahori–Hecke algebra | 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 a spectacular application in Vaughan Jones' construction of new... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5932_chunk_0 | 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 subgroups of G. Commutative Iwasawa algebras wer... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5933_chunk_0 | 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–Schmidt orthogonalization). It is named after Kenkichi Iw... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5934_chunk_0 | 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 disjointly embedded incompressible tori such that ea... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5935_chunk_0 | Jack polynomials | 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 Macdonald polynomials. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5936_chunk_0 | 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 analysis style (SPA) attacks; it is poss... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5937_chunk_0 | Jacobi identities | 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 on a Hilbert space and equivalently in the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5938_chunk_0 | 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}\right)\left(1+{\frac {x^{2m-1}}{y^{2}}}\right)=\sum _{n=-\infty }^{\inft... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5939_chunk_0 | 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}}\right)} Z ( u ) = ∂ ∂ u ln Θ ( u ) {\displaystyle... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5940_chunk_0 | 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 polynomial inverse. It was first conjectured in 1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5941_chunk_0 | 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 compelling reasons for believing the conjecture to be tru... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5942_chunk_0 | 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 signal processing (to describe FM signals). This identity ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5943_chunk_0 | 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 |
wiki_5944_chunk_0 | 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 points in a high-dimensional space can be embedded into a space of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5945_chunk_0 | 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 become bogged down very quickly as dimension increases. It is ther... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5946_chunk_0 | 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 finite subgroup G of the group GL(n, C) of invertible n-by-n complex matrices, there is a subgr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5947_chunk_0 | 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, Michael Collins, using the classification of finite simple groups, showed that in the fin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5948_chunk_0 | 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 ) + ⋯ + ( k − 1 k ) , k = 2 , … , n . {\displaystyle X_{1}=... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5949_chunk_0 | 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 foundations of quantum mechanics done by Paul Dirac i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5950_chunk_0 | 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 ≤ 4 ‖ f ‖ ‖ A 2 f ‖ . {\displaystyle \|Af\|^{2}\leq 4\|f\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5951_chunk_0 | 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 Kalmanson matrix, and are named after Kenneth Kalmanson. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5952_chunk_0 | 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 |
wiki_5953_chunk_0 | 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 to or greater than the length of the third side. In si... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5954_chunk_0 | 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 this way: Let p i ≥ 0 , 0 < a ≤ x i ≤ b for i = 1 , … ,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5955_chunk_0 | 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 ) {\displaystyle h({\vec {x}},t)} with spatial coordinate x → {\display... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5956_chunk_0 | 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 renormalization group, the KPZ equation is conjectured to be the field the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5957_chunk_0 | 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, 1992). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5958_chunk_0 | 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 |
wiki_5959_chunk_0 | Kervaire semi-characteristic | In mathematics, the Kervaire semi-characteristic, introduced by Michel Kervaire (1956), is an invariant of closed manifolds M of dimension 4 n + 1 {\displaystyle 4n+1} taking values in Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } , given by k ( M ) = ∑ i = 0 2 n dim H 2 i ( M , R ) mod 2 {\displaystyle k(M)=\su... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5960_chunk_0 | Khinchin integral | In mathematics, the Khinchin integral (sometimes spelled Khintchine integral), also known as the Denjoy–Khinchin integral, generalized Denjoy integral or wide Denjoy integral, is one of a number of definitions of the integral of a function. It is a generalization of the Riemann and Lebesgue integrals. It is named after... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5961_chunk_0 | Khintchine inequality | In mathematics, the Khintchine inequality, named after Aleksandr Khinchin and spelled in multiple ways in the Latin alphabet, is a theorem from probability, and is also frequently used in analysis. Heuristically, it says that if we pick N {\displaystyle N} complex numbers x 1 , … , x N ∈ C {\displaystyle x_{1},\dots ,x... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5962_chunk_0 | Kirby calculus | In mathematics, the Kirby calculus in geometric topology, named after Robion Kirby, is a method for modifying framed links in the 3-sphere using a finite set of moves, the Kirby moves. Using four-dimensional Cerf theory, he proved that if M and N are 3-manifolds, resulting from Dehn surgery on framed links L and J resp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5963_chunk_0 | Klein bottle | In mathematics, the Klein bottle () is an example of a non-orientable surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. More formally, the Klein bottle is a two-dimensional manifold on which one cannot define... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5964_chunk_0 | Kneser's theorem (differential equations) | In mathematics, the Kneser theorem can refer to two distinct theorems in the field of ordinary differential equations: the first one, named after Adolf Kneser, provides criteria to decide whether a differential equation is oscillating or not; the other one, named after Hellmuth Kneser, is about the topology of the set ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5965_chunk_0 | Kodaira embedding theorem | In mathematics, the Kodaira embedding theorem characterises non-singular projective varieties, over the complex numbers, amongst compact Kähler manifolds. In effect it says precisely which complex manifolds are defined by homogeneous polynomials. Kunihiko Kodaira's result is that for a compact Kähler manifold M, with a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5966_chunk_0 | Kodaira embedding theorem | The converse that projective manifolds are Hodge manifolds is more elementary and was already known. Kodaira also proved (Kodaira 1963), by recourse to the classification of compact complex surfaces, that every compact Kähler surface is a deformation of a projective Kähler surface. This was later simplified by Buchdahl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5967_chunk_0 | Kodaira vanishing theorem | In mathematics, the Kodaira vanishing theorem is a basic result of complex manifold theory and complex algebraic geometry, describing general conditions under which sheaf cohomology groups with indices q > 0 are automatically zero. The implications for the group with index q = 0 is usually that its dimension — the numb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5968_chunk_0 | Kodaira–Spencer map | In mathematics, the Kodaira–Spencer map, introduced by Kunihiko Kodaira and Donald C. Spencer, is a map associated to a deformation of a scheme or complex manifold X, taking a tangent space of a point of the deformation space to the first cohomology group of the sheaf of vector fields on X. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5969_chunk_0 | Koenigs function | In mathematics, the Koenigs function is a function arising in complex analysis and dynamical systems. Introduced in 1884 by the French mathematician Gabriel Koenigs, it gives a canonical representation as dilations of a univalent holomorphic mapping, or a semigroup of mappings, of the unit disk in the complex numbers i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5970_chunk_0 | Kolmogorov continuity theorem | In mathematics, the Kolmogorov continuity theorem is a theorem that guarantees that a stochastic process that satisfies certain constraints on the moments of its increments will be continuous (or, more precisely, have a "continuous version"). It is credited to the Soviet mathematician Andrey Nikolaevich Kolmogorov. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5971_chunk_0 | Kolmogorov extension theorem | In mathematics, the Kolmogorov extension theorem (also known as Kolmogorov existence theorem, the Kolmogorov consistency theorem or the Daniell-Kolmogorov theorem) is a theorem that guarantees that a suitably "consistent" collection of finite-dimensional distributions will define a stochastic process. It is credited to... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5972_chunk_0 | Konhauser polynomials | In mathematics, the Konhauser polynomials, introduced by Konhauser (1967), are biorthogonal polynomials for the distribution function of the Laguerre polynomials. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5973_chunk_0 | Kontorovich–Lebedev transform | In mathematics, the Kontorovich–Lebedev transform is an integral transform which uses a Macdonald function (modified Bessel function of the second kind) with imaginary index as its kernel. Unlike other Bessel function transforms, such as the Hankel transform, this transform involves integrating over the index of the fu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5974_chunk_0 | Kontorovich–Lebedev transform | Laguerre previously studied a similar transform regarding Laguerre function as: g ( y ) = ∫ 0 ∞ f ( x ) e − x L y ( x ) d x {\displaystyle g(y)=\int _{0}^{\infty }f(x)e^{-x}L_{y}(x)\,dx} f ( x ) = ∫ 0 ∞ g ( y ) Γ ( y ) L y ( x ) d y . {\displaystyle f(x)=\int _{0}^{\infty }{\frac {g(y)}{\Gamma (y)}}L_{y}(x)\,dy.} Erdél... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5975_chunk_0 | Kontsevich quantization formula | In mathematics, the Kontsevich quantization formula describes how to construct a generalized ★-product operator algebra from a given arbitrary finite-dimensional Poisson manifold. This operator algebra amounts to the deformation quantization of the corresponding Poisson algebra. It is due to Maxim Kontsevich. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5976_chunk_0 | Korteweg de Vries equation | In mathematics, the Korteweg–De Vries (KdV) equation is a partial differential equation (PDE) which serves as a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an integrable PDE and exhibits many of the expected behaviors for an integrable PDE, such as a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5977_chunk_0 | Koszul complex | In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul (see Lie algebra cohomology). It turned out to be a useful general construction in homological algebra. As a tool, its homology can be used to tell when a set of elements of a (local) ring is an ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5978_chunk_0 | Kronecker delta function | In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: or with use of Iverson brackets: For example, δ 12 = 0 {\displaystyle \delta _{12}=0} because 1 ≠ 2 {\displaystyle 1\neq ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5979_chunk_0 | Kronecker delta function | In linear algebra, the n × n {\displaystyle n\times n} identity matrix I {\displaystyle \mathbf {I} } has entries equal to the Kronecker delta: where i {\displaystyle i} and j {\displaystyle j} take the values 1 , 2 , ⋯ , n {\displaystyle 1,2,\cdots ,n} , and the inner product of vectors can be written as Here the Eucl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5980_chunk_0 | Krull–Schmidt theorem | In mathematics, the Krull–Schmidt theorem states that a group subjected to certain finiteness conditions on chains of subgroups, can be uniquely written as a finite direct product of indecomposable subgroups. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5981_chunk_0 | Krylov–Bogolyubov theorem | In mathematics, the Krylov–Bogolyubov theorem (also known as the existence of invariant measures theorem) may refer to either of the two related fundamental theorems within the theory of dynamical systems. The theorems guarantee the existence of invariant measures for certain "nice" maps defined on "nice" spaces and we... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5982_chunk_0 | Kubilius model | In mathematics, the Kubilius model relies on a clarification and extension of a finite probability space on which the behaviour of additive arithmetic functions can be modeled by sum of independent random variables.The method was introduced in Jonas Kubilius's monograph Tikimybiniai metodai skaičių teorijoje (published... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5983_chunk_0 | Kuramoto–Sivashinsky equation | In mathematics, the Kuramoto–Sivashinsky equation (also called the KS equation or flame equation) is a fourth-order nonlinear partial differential equation. It is named after Yoshiki Kuramoto and Gregory Sivashinsky, who derived the equation in the late 1970s to model the diffusive–thermal instabilities in a laminar fl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5984_chunk_0 | Kuratowski and Ryll-Nardzewski measurable selection theorem | In mathematics, the Kuratowski–Ryll-Nardzewski measurable selection theorem is a result from measure theory that gives a sufficient condition for a set-valued function to have a measurable selection function. It is named after the Polish mathematicians Kazimierz Kuratowski and Czesław Ryll-Nardzewski.Many classical sel... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5985_chunk_0 | Kuratowski–Ulam theorem | In mathematics, the Kuratowski–Ulam theorem, introduced by Kazimierz Kuratowski and Stanislaw Ulam (1932), called also the Fubini theorem for category, is an analog of Fubini's theorem for arbitrary second countable Baire spaces. Let X and Y be second countable Baire spaces (or, in particular, Polish spaces), and let A... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5986_chunk_0 | Kuratowski–Ulam theorem | comeager) in Y } {\displaystyle \{x\in X:A_{x}{\text{ is meager (resp. comeager) in }}Y\}} is comeager in X, where A x = π Y {\displaystyle A_{x}=\pi _{Y}} , where π Y {\displaystyle \pi _{Y}} is the projection onto Y.Even if A does not have the Baire property, 2. follows from 1. Note that the theorem still holds (per... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5987_chunk_0 | Kurosh problem | In mathematics, the Kurosh problem is one general problem, and several more special questions, in ring theory. The general problem is known to have a negative solution, since one of the special cases has been shown to have counterexamples. These matters were brought up by Aleksandr Gennadievich Kurosh as analogues of t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5988_chunk_0 | Kurosh problem | A special case is whether or not every nil algebra is locally nilpotent. For PI-algebras the Kurosh problem has a positive solution. Golod showed a counterexample to that case, as an application of the Golod–Shafarevich theorem. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5989_chunk_0 | Kurosh problem | The Kurosh problem on group algebras concerns the augmentation ideal I. If I is a nil ideal, is the group algebra locally nilpotent? There is an important problem which is often referred as the Kurosh's problem on division rings. The problem asks whether there exists an algebraic (over the center) division ring which i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5990_chunk_0 | Functional equation (L-function) | In mathematics, the L-functions of number theory are expected to have several characteristic properties, one of which is that they satisfy certain functional equations. There is an elaborate theory of what these equations should be, much of which is still conjectural. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5991_chunk_0 | Lagrange number | In mathematics, the Lagrange numbers are a sequence of numbers that appear in bounds relating to the approximation of irrational numbers by rational numbers. They are linked to Hurwitz's theorem. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5992_chunk_0 | Lagrange reversion theorem | In mathematics, the Lagrange reversion theorem gives series or formal power series expansions of certain implicitly defined functions; indeed, of compositions with such functions. Let v be a function of x and y in terms of another function f such that v = x + y f ( v ) {\displaystyle v=x+yf(v)} Then for any function g,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5993_chunk_0 | Lagrange reversion theorem | {\displaystyle g(v)=g(x)+\sum _{k=1}^{\infty }{\frac {y^{k}}{k! }}\left({\frac {\partial }{\partial x}}\right)^{k-1}\left(f(x)^{k}g'(x)\right).} If g is the identity, this becomes v = x + ∑ k = 1 ∞ y k k ! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5994_chunk_0 | Lagrange reversion theorem | ( ∂ ∂ x ) k − 1 ( f ( x ) k ) {\displaystyle v=x+\sum _{k=1}^{\infty }{\frac {y^{k}}{k! }}\left({\frac {\partial }{\partial x}}\right)^{k-1}\left(f(x)^{k}\right)} In which case the equation can be derived using perturbation theory. In 1770, Joseph Louis Lagrange (1736–1813) published his power series solution of the im... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5995_chunk_0 | Lagrangian Grassmannian | In mathematics, the Lagrangian Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space V. Its dimension is 1/2n(n + 1) (where the dimension of V is 2n). It may be identified with the homogeneous space U(n)/O(n),where U(n) is the unitary group and O(n) the orthogonal group. Followin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5996_chunk_0 | Variational bicomplex | In mathematics, the Lagrangian theory on fiber bundles is globally formulated in algebraic terms of the variational bicomplex, without appealing to the calculus of variations. For instance, this is the case of classical field theory on fiber bundles (covariant classical field theory). The variational bicomplex is a coc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5997_chunk_0 | Laguerre form | In mathematics, the Laguerre form is generally given as a third degree tensor-valued form, that can be written as, L = ( w 1 ) 2 D a 11 + 2 w 1 w 2 D a 12 + ( w 2 ) 2 D a 22 {\displaystyle {\mathfrak {L}}=(w^{1})^{2}Da_{11}+2w^{1}w^{2}Da_{12}+(w^{2})^{2}Da_{22}} . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5998_chunk_0 | Laguerre functions | In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation: which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer. Sometimes the name Laguerre polynomials is u... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5999_chunk_0 | Laguerre functions | More generally, a Laguerre function is a solution when n is not necessarily a non-negative integer. The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form These polynomials, usually denoted L0, L1, …, are a polynomial sequence which may be defined by the Rodrigues fo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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