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c_34nnh1or0u2k | 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 (1968, p. 944) proved a more general version. Several other authors later rediscovered and published variations of this theorem. Eisenbud (1995, theorem 20.15) gives a statement and proof. | Hilbert–Burch theorem |
c_1h67n382v5mh | 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.58). | Hilbert–Mumford criterion |
c_2m0ejxg9zutb | 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. | Hilbert-Pólya conjecture |
c_chawq290z9in | 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 locally compact and have a continuous, faithful group action on M, it states that G must be a Lie group. Because of known structural results on G, it is enough to deal with the case where G is the additive group Zp of p-adic integers, for some prime number p. An equivalent form of the conjecture is that Zp has no faithful group action on a topological manifold. The naming of the conjecture is for David Hilbert, and the American topologist Paul A. Smith. | Hilbert–Smith conjecture |
c_bmkldw67a2tc | 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 conjecture for groups acting by Lipschitz maps on a Riemannian manifold using the covering, fractal and cohomological dimension theory. In 1999, Gaven Martin extended their dimension-theoretic argument to quasiconformal actions on a Riemannian manifold and gave applications concerning unique analytic continuation for Beltrami systems.In 2013, John Pardon proved the three-dimensional case of the Hilbert–Smith conjecture. | Hilbert–Smith conjecture |
c_tcn0to9xe3on | 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 cyclotomic fields. Hilbert–Speiser Theorem. A finite abelian extension K/Q has a normal integral basis if and only if it is tamely ramified over Q.This is the condition that it should be a subfield of Q(ζn) where n is a squarefree odd number. | Hilbert–Speiser theorem |
c_ajc1jnw2dmy5 | 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 Gaussian periods. For example if we take n a prime number p > 2, Q(ζp) has a normal integral basis consisting of all the p-th roots of unity other than 1. | Hilbert–Speiser theorem |
c_n26cpedagdmy | 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 follows from a simple argument on ramification). This decomposition can be used to treat any of its subfields. Cornelius Greither, Daniel R. Replogle, and Karl Rubin et al. (1999) proved a converse to the Hilbert–Speiser theorem: Each finite tamely ramified abelian extension K of a fixed number field J has a relative normal integral basis if and only if J =Q.There is an elliptic analogue of the theorem proven by Anupam Srivastav and Martin J. Taylor (1990). It is now called the Srivastav-Taylor theorem (1996). | Hilbert–Speiser theorem |
c_c21jm0dyojfl | 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 function by minimal period π {\displaystyle \pi } . By these we mean that for all t {\displaystyle t} f ( t + π ) = f ( t ) , {\displaystyle f(t+\pi )=f(t),} and ∫ 0 π f ( t ) d t = 0 , {\displaystyle \int _{0}^{\pi }f(t)\,dt=0,} and if p {\displaystyle p} is a number with 0 < p < π {\displaystyle 0 | Hill differential equation |
c_jlbfauxbodaj | 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 varieties of higher dimensions. The result paved the way for the Grothendieck–Hirzebruch–Riemann–Roch theorem proved about three years later. | Hirzebruch–Riemann–Roch theorem |
c_mu1npyupdz5f | 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 important role in the geometric Langlands correspondence over the field of complex numbers through conformal field theory. A genus zero analogue of the Hitchin system, the Garnier system, was discovered by René Garnier somewhat earlier as a certain limit of the Schlesinger equations, and Garnier solved his system by defining spectral curves. | Hitchin system |
c_qle782rl53t3 | (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 Bottacin and Markman in 1994. | Hitchin system |
c_oehx3nhbgjcv | 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 the Theory of Parallels. | Hjelmslev transformation |
c_hzk910qwb3ya | 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 EJ will be hyperparallel, and therefore will never meet. | Hjelmslev transformation |
c_i6cr8dr4m6mv | 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 perpendicular to the other ray; using that cutoff length, e.g., AH, as the radius of a circle, "map" the point H onto the line AE. | Hjelmslev transformation |
c_mswq85doqo87 | 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 the circle created does not have a corresponding location within the plane, and therefore, the product of a Hjelmslev transformation is more aptly called a Hjelmslev Disk. | Hjelmslev transformation |
c_9moxoh2t7a52 | 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 image of a circle sharing the center of the transformation will be a circle about this same center. As a result, the images of all the right angles with one side passing through the center will be right angles. | Hjelmslev transformation |
c_oa227ys7fvxx | 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 of B will be between the image of A and the image of C. The image of a rectilinear angle is a rectilinear angle. | Hjelmslev transformation |
c_q68ya7w0urqi | In mathematics, the Hochschild–Mostow group, introduced by Hochschild and Mostow (1957), is the universal pro-affine algebraic group generated by a group. == References == | Hochschild–Mostow group |
c_k3wm5r7xi2n0 | 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. | Hodge vector bundle |
c_ahl3voqmqasq | 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 in certain geometric spaces, complex algebraic varieties, can be understood by studying the possible nice shapes sitting inside those spaces, which look like zero sets of polynomial equations. The latter objects can be studied using algebra and the calculus of analytic functions, and this allows one to indirectly understand the broad shape and structure of often higher-dimensional spaces which can not be otherwise easily visualized. More specifically, the conjecture states that certain de Rham cohomology classes are algebraic; that is, they are sums of Poincaré duals of the homology classes of subvarieties. | Hodge conjecture |
c_w30kbvkry3p9 | 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 little attention before Hodge presented it in an address during the 1950 International Congress of Mathematicians, held in Cambridge, Massachusetts. The Hodge conjecture is one of the Clay Mathematics Institute's Millennium Prize Problems, with a prize of $1,000,000US for whoever can prove or disprove the Hodge conjecture. | Hodge conjecture |
c_kp0nc92pecfu | 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 uniquely determined), and decomposes as a direct sum of some such one-dimensional subspace, and a complementary subspace on which it is negative definite. In a more formal statement, specify that V is a non-singular projective surface, and let H be the divisor class on V of a hyperplane section of V in a given projective embedding. Then the intersection H ⋅ H = d {\displaystyle H\cdot H=d\ } where d is the degree of V (in that embedding). Let D be the vector space of rational divisor classes on V, up to algebraic equivalence. | Hodge index theorem |
c_cwfikgsi0qvz | 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). | Hodge index theorem |
c_mbfijfqmkl2d | 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-trivial torsion subgroup, on occasion). | Hodge index theorem |
c_hjjedmrx17q0 | 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. The result holds over general (algebraically closed) fields. | Hodge index theorem |
c_0wjioz0f424c | 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 W. V. D. Hodge. For example, in an oriented 3-dimensional Euclidean space, an oriented plane can be represented by the exterior product of two basis vectors, and its Hodge dual is the normal vector given by their cross product; conversely, any vector is dual to the oriented plane perpendicular to it, endowed with a suitable bivector. | Hodge dual |
c_bwxyj6huvq5i | 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 role in differential geometry, when applied to the cotangent bundle of a pseudo-Riemannian manifold, and hence to differential k-forms. This allows the definition of the codifferential as the Hodge adjoint of the exterior derivative, leading to the Laplace–de Rham operator. This generalizes the case of 3-dimensional Euclidean space, in which divergence of a vector field may be realized as the codifferential opposite to the gradient operator, and the Laplace operator on a function is the divergence of its gradient. An important application is the Hodge decomposition of differential forms on a closed Riemannian manifold. | Hodge dual |
c_m1h1lskhdyj6 | 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). This spectral sequence describes the precise relationship between the Dolbeault cohomology and the de Rham cohomology of a general complex manifold. On a compact Kähler manifold, the sequence degenerates, thereby leading to the Hodge decomposition of the de Rham cohomology. | Hodge–de Rham spectral sequence |
c_c0v9w2p50hwq | 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 can sometimes be done more elegantly via mutually recursive functions; the Sierpiński curve is a good example. | Mutually recursive |
c_lryd537xd8yl | 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. | Holomorphic Lefschetz fixed-point formula |
c_vskf67s3q5l2 | 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 cohomology group or Tate module) have absolute value √q. Tate (1966) showed that the map taking an isogeny class to the eigenvalues of the Frobenius is injective, and Taira Honda (1968) showed that this map is surjective, and therefore a bijection. | Honda–Tate theorem |
c_9yb17seesx67 | 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 onto null sets. Up to null sets, X can be written as a disjoint union C ∐ D of T-invariant sets where the action of T on C is conservative and the action of T on D is dissipative. Thus, if τ is the automorphism of A = L∞(X) induced by T, there is a unique τ-invariant projection p in A such that pA is conservative and (I–p)A is dissipative. | Hopf decomposition |
c_a9t4ttg8wt4f | 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, then the derivative of the function in the direction of the outward pointing normal is strictly positive. The lemma is an important tool in the proof of the maximum principle and in the theory of partial differential equations. The Hopf lemma has been generalized to describe the behavior of the solution to an elliptic problem as it approaches a point on the boundary where its maximum is attained. | Hopf lemma |
c_6caq9lqz5f3l | 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 countries. There are also extensions which allow domains with corners. | Hopf lemma |
c_9v249vydyhqh | 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 examples of vector bundles of higher ranks over other projective spaces. The Horrocks construction is used in the ADHM construction to construct instantons over the 4-sphere. | Horrocks construction |
c_fvx82cpirs7c | 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}} Boas & Buck (1958, p.58). | Humbert polynomials |
c_n5hg9pe9z051 | 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. | Rational Hurewicz theorem |
c_14lxfgdsq6b3 | 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) = –1/12. Zagier (1975) showed that the Hurwitz class numbers are coefficients of a mock modular form of weight 3/2. | Hurwitz class number |
c_punb0tpsrs9l | 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. | Hurwitz problem |
c_f4ondvsx8tie | 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}}}.} | Hurwitz Zeta function |
c_63y50pasv18x | 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. | Hurwitz Zeta function |
c_6cpiaq359nm4 | 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". | Hutchinson metric |
c_9uvl3i961f25 | 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 = 1 − a x n 2 + y n y n + 1 = b x n . {\displaystyle {\begin{cases}x_{n+1}=1-ax_{n}^{2}+y_{n}\\y_{n+1}=bx_{n}.\end{cases}}} The map depends on two parameters, a and b, which for the classical Hénon map have values of a = 1.4 and b = 0.3. | Hénon map |
c_4vc9m6xi86da | 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. | Hénon map |
c_cohnt0xga75r | 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 a Cantor set in another. Numerical estimates yield a correlation dimension of 1.21 ± 0.01 or 1.25 ± 0.02 (depending on the dimension of the embedding space) and a Box Counting dimension of 1.261 ± 0.003 for the attractor of the classical map. | Hénon map |
c_62eelv05qiiy | 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 subgroups of the two-by-two p-adic special linear group. | Ihara zeta function |
c_mtp7zntwroa5 | 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 Riemann hypothesis. | Ihara zeta function |
c_a1n8k96s2el5 | 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 of weight 2k to genus 2 Siegel modular forms of weight k + 1. | Ikeda lift |
c_npku3dqxhhf9 | 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 polynomials. In particular, when p = 1 , η ± ξ = 1 {\displaystyle p=1,\eta \pm \xi =1} , then it has a closed-form solution w ( z ) = C e − i z ( e 2 i z ∓ 1 ) {\displaystyle w(z)=Ce^{-iz}(e^{2iz}\mp 1)} where C {\displaystyle C} is a constant. | Ince polynomials |
c_2238ofs608vs | 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}(b,z)={\frac {1}{\Gamma (s)}}\int _{b}^{\infty }{\frac {x^{s-1}}{e^{x}/z-1}}~dx.} | Incomplete polylogarithm |
c_toj65izz7on0 | 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 gamma function. Since Γ(s,0)=Γ(s), it follows that: Li s ( 0 , z ) = Li s ( z ) {\displaystyle \operatorname {Li} _{s}(0,z)=\operatorname {Li} _{s}(z)} where Lis(.) is the polylogarithm function. | Incomplete polylogarithm |
c_qjdqsvflni6k | 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 \mathbb {R} } denote the canonical real-valued Wiener process defined up to time T > 0 {\displaystyle T>0} , and let X: × Ω → R {\displaystyle X:\times \Omega \to \mathbb {R} } be a stochastic process that is adapted to the natural filtration F ∗ W {\displaystyle {\mathcal {F}}_{*}^{W}} of the Wiener process. Then E = E , {\displaystyle \operatorname {E} \left=\operatorname {E} \left,} where E {\displaystyle \operatorname {E} } denotes expectation with respect to classical Wiener measure. | Itô isometry |
c_6396seh0cf67 | 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 the norms induced by the inner products ( X , Y ) L a d 2 ( × Ω ) := E ( ∫ 0 T X t Y t d t ) {\displaystyle {\begin{aligned}(X,Y)_{L_{\mathrm {ad} }^{2}(\times \Omega )}&:=\operatorname {E} \left(\int _{0}^{T}X_{t}\,Y_{t}\,\mathrm {d} t\right)\end{aligned}}} and ( A , B ) L 2 ( Ω ) := E ( A B ) . {\displaystyle (A,B)_{L^{2}(\Omega )}:=\operatorname {E} (AB).} As a consequence, the Itô integral respects these inner products as well, i.e. we can write E = E {\displaystyle \operatorname {E} \left=\operatorname {E} \left} for X , Y ∈ L a d 2 ( × Ω ) {\displaystyle X,Y\in L_{\mathrm {ad} }^{2}(\times \Omega )} . | Itô isometry |
c_8z6l6uxhalap | 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 variables for which the statement is true, and takes the value 0 otherwise. It is generally denoted by putting the statement inside square brackets: In other words, the Iverson bracket of a statement is the indicator function of the set of values for which the statement is true. The Iverson bracket allows using capital-sigma notation without restriction on the summation index. | Iverson bracket |
c_jm27bi6ourd3 | 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 need to be defined for the values of k for which the Iverson bracket equals 0; that is, a summand f ( k ) {\displaystyle f(k)} must evaluate to 0 regardless of whether f ( k ) {\displaystyle f(k)} is defined. The notation was originally introduced by Kenneth E. Iverson in his programming language APL, though restricted to single relational operators enclosed in parentheses, while the generalisation to arbitrary statements, notational restriction to square brackets, and applications to summation, was advocated by Donald Knuth to avoid ambiguity in parenthesized logical expressions. | Iverson bracket |
c_yodjh4t29afv | 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 invariants of knots. Representations of Hecke algebras led to discovery of quantum groups by Michio Jimbo. Michael Freedman proposed Hecke algebras as a foundation for topological quantum computation. | Iwahori–Hecke algebra |
c_aovylcuxh61m | 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 were introduced by Iwasawa (1959) in his study of Zp extensions in Iwasawa theory, and non-commutative Iwasawa algebras of compact p-adic analytic groups were introduced by Lazard (1965). | Iwasawa algebra |
c_p7oi3aszitvn | 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 Iwasawa, the Japanese mathematician who developed this method. | Iwasawa decomposition |
c_au1ov07eve31 | 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). | Adams conjecture |
c_q1bgziqwem3d | 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 each component of the 3-manifold obtained by cutting along the tori is either atoroidal or Seifert-fibered.The acronym JSJ is for William Jaco, Peter Shalen, and Klaus Johannson. The first two worked together, and the third worked independently. | JSJ decomposition |
c_ht44ak7k9hd5 | 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. | Jack polynomials |
c_4iupihkdpcpl | 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 possible, indeed, to use the general addition formula also for doubling a point on an elliptic curve of this form: in this way the two operations become indistinguishable from some side-channel information. The Jacobi curve also offers faster arithmetic compared to the Weierstrass curve. The Jacobi curve can be of two types: the Jacobi intersection, that is given by an intersection of two surfaces, and the Jacobi quartic. | Jacobian curve |
c_6g884grd65b7 | 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 functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation sn {\displaystyle \operatorname {sn} } for sin {\displaystyle \sin } . | Jacobi elliptic cosine |
c_1swxmpb4rnoy | 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 Friedrich Gauss had already studied special Jacobi elliptic functions in 1797, the lemniscate elliptic functions in particular, but his work was published much later. | Jacobi elliptic cosine |
c_26370321nlge | 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. | Jacobi group |
c_by99kviqy1l7 | 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 evaluation gives the same result (parentheses in a multiple product are not needed). The identity is named after the German mathematician Carl Gustav Jacob Jacobi. | Jacobi identities |
c_yh9wwywz3o2o | 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 phase space formulation of quantum mechanics by the Moyal bracket. | Jacobi identities |
c_05yoxgc7yr15 | 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 }^{\infty }x^{n^{2}}y^{2n},} for complex numbers x and y, with |x| < 1 and y ≠ 0. It was introduced by Jacobi (1829) in his work Fundamenta Nova Theoriae Functionum Ellipticarum. The Jacobi triple product identity is the Macdonald identity for the affine root system of type A1, and is the Weyl denominator formula for the corresponding affine Kac–Moody algebra. | Jacobi triple product identity |
c_uuwuewao7fig | 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 Z(u)={\frac {\partial }{\partial u}}\ln \Theta (u)} = Θ ′ ( u ) Θ ( u ) {\displaystyle ={\frac {\Theta '(u)}{\Theta (u)}}} Z ( ϕ | m ) = E ( ϕ | m ) − E ( m ) K ( m ) F ( ϕ | m ) {\displaystyle Z(\phi |m)=E(\phi |m)-{\frac {E(m)}{K(m)}}F(\phi |m)} Where E, K, and F are generic Incomplete Elliptical Integrals of the first and second kind. Jacobi Zeta Functions being kinds of Jacobi theta functions have applications to all their relevant fields and application. | Jacobi zeta function |
c_u3vmmpbmvwnl | 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 1939 by Ott-Heinrich Keller, and widely publicized by Shreeram Abhyankar, as an example of a difficult question in algebraic geometry that can be understood using little beyond a knowledge of calculus. | Jacobian conjecture |
c_hg886mf8bz9n | 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 true, and according to van den Essen there are some suspicions that the conjecture is in fact false for large numbers of variables (indeed, there is equally also no compelling evidence to support these suspicions). The Jacobian conjecture is number 16 in Stephen Smale's 1998 list of Mathematical Problems for the Next Century. | Jacobian conjecture |
c_n7dlkg2itksw | 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. | Jacobian of a curve |
c_f6rukgmwnuqi | 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 is named after the 19th-century mathematicians Carl Jacobi and Carl Theodor Anger. The most general identity is given by: e i z cos θ ≡ ∑ n = − ∞ ∞ i n J n ( z ) e i n θ , {\displaystyle e^{iz\cos \theta }\equiv \sum _{n=-\infty }^{\infty }i^{n}\,J_{n}(z)\,e^{in\theta },} where J n ( z ) {\displaystyle J_{n}(z)} is the n {\displaystyle n} -th Bessel function of the first kind and i {\displaystyle i} is the imaginary unit, i 2 = − 1. | Jacobi–Anger expansion |
c_1qvvi6736gaw | {\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 }.} Using the relation J − n ( z ) = ( − 1 ) n J n ( z ) , {\displaystyle J_{-n}(z)=(-1)^{n}\,J_{n}(z),} valid for integer n {\displaystyle n} , the expansion becomes: e i z cos θ ≡ J 0 ( z ) + 2 ∑ n = 1 ∞ i n J n ( z ) cos ( n θ ) . {\displaystyle e^{iz\cos \theta }\equiv J_{0}(z)\,+\,2\,\sum _{n=1}^{\infty }\,i^{n}\,J_{n}(z)\,\cos \,(n\theta ).} | Jacobi–Anger expansion |
c_noubzkpdnetz | 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 defined by a similar recurrence relation: in simple terms, the sequence starts with 0 and 1, then each following number is found by adding the number before it to twice the number before that. The first Jacobsthal numbers are: 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, … (sequence A001045 in the OEIS)A Jacobsthal prime is a Jacobsthal number that is also prime. The first Jacobsthal primes are: 3, 5, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243, … (sequence A049883 in the OEIS) | Jacobsthal number |
c_vfq144tnssa8 | 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. | Jacquet functor |
c_cpv3fi0stbpl | 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). | James embedding |
c_1vft9c6otedt | 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. | Jessen–Wintner theorem |
c_fbkhyg3v5aph | 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 volume ellipsoid is called the Löwner ellipsoid, and the maximal volume ellipsoid is called the John ellipsoid (although John worked with the minimal volume ellipsoid in its original paper). One can also refer to the minimal volume circumscribed ellipsoid as the outer Löwner-John ellipsoid, and the maximum volume inscribed ellipsoid as the inner Löwner-John ellipsoid. | John ellipsoid |
c_k3x6b3px2sek | 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 vectors x, y ∈ X are called ith associates if dist(x, y) = 2i for i = 0, 1, ..., n. The eigenvalues are given by p i ( k ) = E i ( k ) , {\displaystyle p_{i}\left(k\right)=E_{i}\left(k\right),} q k ( i ) = μ k v i E i ( k ) , {\displaystyle q_{k}\left(i\right)={\frac {\mu _{k}}{v_{i}}}E_{i}\left(k\right),} where μ i = ℓ − 2 i + 1 ℓ − i + 1 ( ℓ i ) , {\displaystyle \mu _{i}={\frac {\ell -2i+1}{\ell -i+1}}{\binom {\ell }{i}},} and Ek(x) is an Eberlein polynomial defined by E k ( x ) = ∑ j = 0 k ( − 1 ) j ( x j ) ( n − x k − j ) ( ℓ − n − x k − j ) , k = 0 , … , n . {\displaystyle E_{k}\left(x\right)=\sum _{j=0}^{k}(-1)^{j}{\binom {x}{j}}{\binom {n-x}{k-j}}{\binom {\ell -n-x}{k-j}},\qquad k=0,\ldots ,n.} == References == | Johnson scheme |
c_4keujd0jr302 | 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 much lower dimension in such a way that distances between the points are nearly preserved. The map used for the embedding is at least Lipschitz, and can even be taken to be an orthogonal projection. The lemma has applications in compressed sensing, manifold learning, dimensionality reduction, and graph embedding. | Johnson–Lindenstrauss lemma |
c_7kb3ljko67lu | 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 therefore desirable to reduce the dimensionality of the data in a way that preserves its relevant structure. The Johnson–Lindenstrauss lemma is a classic result in this vein. Also, the lemma is tight up to a constant factor, i.e. there exists a set of points of size m that needs dimension Ω ( log ( m ) ε 2 ) {\displaystyle \Omega \left({\frac {\log(m)}{\varepsilon ^{2}}}\right)} in order to preserve the distances between all pairs of points within a factor of ( 1 ± ε ) {\displaystyle (1\pm \varepsilon )} . | Johnson–Lindenstrauss lemma |
c_8h6klbf0jdbq | 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 invertible operator as the product of its commuting semisimple and unipotent parts. The decomposition is easy to describe when the Jordan normal form of the operator is given, but it exists under weaker hypotheses than the existence of a Jordan normal form. Analogues of the Jordan-Chevalley decomposition exist for elements of linear algebraic groups, Lie algebras, and Lie groups, and the decomposition is an important tool in the study of these objects. | Jordan–Chevalley decomposition |
c_a9arxrnw51xj | 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 ! | Jordan–Pólya number |
c_doxqj7nyzlgn | {\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, who both wrote about them in the context of symmetries of trees.These numbers grow more quickly than polynomials but more slowly than exponentials. As well as in the symmetries of trees, they arise as the numbers of transitive orientations of comparability graphs and in the problem of finding factorials that can be represented as products of smaller factorials. | Jordan–Pólya number |
c_ywshwtybcufn | 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 subgroup H of G with the following properties: H is abelian. H is a normal subgroup of G. The index of H in G satisfies (G: H) ≤ ƒ(n).Schur proved a more general result that applies when G is not assumed to be finite, but just periodic. Schur showed that ƒ(n) may be taken to be ((8n)1/2 + 1)2n2 − ((8n)1/2 − 1)2n2.A tighter bound (for n ≥ 3) is due to Speiser, who showed that as long as G is finite, one can take ƒ(n) = n! | Jordan's theorem on finite linear groups |
c_nhnlqyl3momu | 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 finite case, one can take ƒ(n) = (n + 1)! when n is at least 71, and gave near complete descriptions of the behavior for smaller n. | Jordan's theorem on finite linear groups |
c_4altjhqia9nn | 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}=0,~~~X_{k}=(1\;k)+(2\;k)+\cdots +(k-1\;k),~~~k=2,\dots ,n.} They play an important role in the representation theory of the symmetric group. | Jucys–Murphy element |
c_b5egew4dnhde | 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 in the 1940s and was formalized in 1959 by Richard Kadison and Isadore Singer. The problem was subsequently shown to be equivalent to numerous open problems in pure mathematics, applied mathematics, engineering and computer science. | Kadison–Singer problem |
c_6lsl1du50ure | 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 reformulation provided by Joel Anderson, who showed in 1979 that his "paving conjecture", which only involves operators on finite-dimensional Hilbert spaces, is equivalent to the Kadison–Singer problem. Nik Weaver provided another reformulation in a finite-dimensional setting, and this version was proved true using random polynomials. | Kadison–Singer problem |
c_yo2nr7c1oly2 | 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\|\|A^{2}f\|.} | Kallman–Rota inequality |
c_qj2sns2f8swi | 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. | Kalmanson combinatorial conditions |
c_0xnbl75ghf4m | 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). | Kantor double |
c_hhpcakmuttja | 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 simplest terms, the Kantorovich inequality translates the basic idea of the triangle inequality into the terms and notational conventions of linear programming. | Kantorovich inequality |
c_bn9vuu2vrc40 | (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 , … , n . {\displaystyle p_{i}\geq 0,\quad 0 | Kantorovich inequality |
c_d4ha0ehrsf1q | 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 → {\displaystyle {\vec {x}}} and time coordinate t {\displaystyle t}: ∂ h ( x → , t ) ∂ t = ν ∇ 2 h + λ 2 ( ∇ h ) 2 + η ( x → , t ) . {\displaystyle {\frac {\partial h({\vec {x}},t)}{\partial t}}=\nu \nabla ^{2}h+{\frac {\lambda }{2}}\left(\nabla h\right)^{2}+\eta ({\vec {x}},t)\;.} Here, η ( x → , t ) {\displaystyle \eta ({\vec {x}},t)} is white Gaussian noise with average ⟨ η ( x → , t ) ⟩ = 0 {\displaystyle \langle \eta ({\vec {x}},t)\rangle =0} and second moment ⟨ η ( x → , t ) η ( x → ′ , t ′ ) ⟩ = 2 D δ d ( x → − x → ′ ) δ ( t − t ′ ) , {\displaystyle \langle \eta ({\vec {x}},t)\eta ({\vec {x}}',t')\rangle =2D\delta ^{d}({\vec {x}}-{\vec {x}}')\delta (t-t'),} ν {\displaystyle \nu } , λ {\displaystyle \lambda } , and D {\displaystyle D} are parameters of the model, and d {\displaystyle d} is the dimension. | Kardar–Parisi–Zhang equation |
c_gct3hm0xzjuy | 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 theory of many surface growth models, such as the Eden model, ballistic deposition, and the weakly asymmetric single step solid on solid process (SOS) model. A rigorous proof has been given by Bertini and Giacomin in the case of the SOS model. | Kardar–Parisi–Zhang equation |
c_t682qi0w34jg | 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). | Karoubi conjecture |
c_tcfa7pc5bd1a | In mathematics, the KdV hierarchy is an infinite sequence of partial differential equations which contains the Korteweg–de Vries equation. | KdV hierarchy |
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