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c_de6x15mjj559 | In mathematics, Gegenbauer polynomials or ultraspherical polynomials C(α)n(x) are orthogonal polynomials on the interval with respect to the weight function (1 − x2)α–1/2. They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials. They are named after Leopold Gegenbaue... | Ultraspherical polynomials |
c_ivktcf8b0ufv | In mathematics, Gelfand–Fuks cohomology, introduced in (Gel'fand & Fuks 1969–70), is a cohomology theory for Lie algebras of smooth vector fields. It differs from the Lie algebra cohomology of Chevalley-Eilenberg in that its cochains are taken to be continuous multilinear alternating forms on the Lie algebra of smooth ... | Gelfand–Fuks cohomology |
c_53lg61anu7it | In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is eπ, that is, e raised to the power π. Like both e and π, this constant is a transcendental number. This was first established by Gelfond and may now be considered as an application of the Gelfond–Schneider theorem, noting that where i is the imaginar... | Gelfond's constant |
c_o6abhfupmug7 | The constant was mentioned in Hilbert's seventh problem. A related constant is 2√2, known as the Gelfond–Schneider constant. The related value π + eπ is also irrational. | Gelfond's constant |
c_mc02l8jvth3o | In mathematics, George Glauberman's Z* theorem is stated as follows: Z* theorem: Let G be a finite group, with O(G) being its maximal normal subgroup of odd order. If T is a Sylow 2-subgroup of G containing an involution not conjugate in G to any other element of T, then the involution lies in Z*(G), which is the inver... | Z* theorem |
c_14k897elxckg | In mathematics, George Glauberman's ZJ theorem states that if a finite group G is p-constrained and p-stable and has a normal p-subgroup for some odd prime p, then Op′(G)Z(J(S)) is a normal subgroup of G, for any Sylow p-subgroup S. | ZJ theorem |
c_o8tfag41kw72 | In mathematics, Geronimus polynomials may refer to one of the several different families of orthogonal polynomials studied by Yakov Lazarevich Geronimus. | Geronimus polynomials |
c_qktd4znxlxph | In mathematics, Giambelli's formula, named after Giovanni Giambelli, expresses Schubert classes in terms of special Schubert classes, or Schur functions in terms of complete symmetric functions. It states σ λ = det ( σ λ i + j − i ) 1 ≤ i , j ≤ r {\displaystyle \displaystyle \sigma _{\lambda }=\det(\sigma _{\lambda _{i... | Giambelli's formula |
c_otqyu0744vul | In mathematics, Gijswijt's sequence (named after Dion Gijswijt by Neil Sloane) is a self-describing sequence where each term counts the maximum number of repeated blocks of numbers in the sequence immediately preceding that term. The sequence begins with: 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 1, ... ... | Gijswijt's sequence |
c_hev6v9i8qoce | In mathematics, Giraud subcategories form an important class of subcategories of Grothendieck categories. They are named after Jean Giraud. | Giraud subcategory |
c_78eig0ixxtrf | In mathematics, Glaeser's theorem, introduced by Georges Glaeser (1963), is a theorem giving conditions for a smooth function to be a composition of F and θ for some given smooth function θ. One consequence is a generalization of Newton's theorem that every symmetric polynomial is a polynomial in the elementary symmetr... | Glaeser's composition theorem |
c_hdqx7nird2dc | In mathematics, Goldie's theorem is a basic structural result in ring theory, proved by Alfred Goldie during the 1950s. What is now termed a right Goldie ring is a ring R that has finite uniform dimension (="finite rank") as a right module over itself, and satisfies the ascending chain condition on right annihilators o... | Goldie ring |
c_sx5g6htpzj43 | This is sufficient to guarantee that a right-Noetherian ring is right Goldie. The converse does not hold: every right Ore domain is a right Goldie domain, and hence so is every commutative integral domain. A consequence of Goldie's theorem, again due to Goldie, is that every semiprime principal right ideal ring is isom... | Goldie ring |
c_igedsuqz6pj7 | In mathematics, Goldie's theorem is a basic structural result in ring theory, proved by Alfred Goldie during the 1950s. What is now termed a right Goldie ring is a ring R that has finite uniform dimension (also called "finite rank") as a right module over itself, and satisfies the ascending chain condition on right ann... | Noncommutative algebra |
c_0c66k9wovzg7 | This is sufficient to guarantee that a right-Noetherian ring is right Goldie. The converse does not hold: every right Ore domain is a right Goldie domain, and hence so is every commutative integral domain. A consequence of Goldie's theorem, again due to Goldie, is that every semiprime principal right ideal ring is isom... | Noncommutative algebra |
c_5mn5j8mbume6 | In mathematics, Gosper's algorithm, due to Bill Gosper, is a procedure for finding sums of hypergeometric terms that are themselves hypergeometric terms. That is: suppose one has a(1) + ... + a(n) = S(n) − S(0), where S(n) is a hypergeometric term (i.e., S(n + 1)/S(n) is a rational function of n); then necessarily a(n)... | Gosper's algorithm |
c_buq7xk4l6kpf | In mathematics, Gottlieb polynomials are a family of discrete orthogonal polynomials introduced by Morris J. Gottlieb (1938). They are given by ℓ n ( x , λ ) = e − n λ ∑ k ( 1 − e λ ) k ( n k ) ( x k ) = e − n λ 2 F 1 ( − n , − x ; 1 ; 1 − e λ ) {\displaystyle \displaystyle \ell _{n}(x,\lambda )=e^{-n\lambda }\sum _{k}... | Gottlieb polynomials |
c_icxxtgtkqbmn | In mathematics, Gowers' theorem, also known as Gowers' Ramsey theorem and Gowers' FINk theorem, is a theorem in Ramsey theory and combinatorics. It is a Ramsey-theoretic result about functions with finite support. Timothy Gowers originally proved the result in 1992, motivated by a problem regarding Banach spaces. The r... | Gowers' theorem |
c_5wqk43m532dg | In mathematics, Graeffe's method or Dandelin–Lobachesky–Graeffe method is an algorithm for finding all of the roots of a polynomial. It was developed independently by Germinal Pierre Dandelin in 1826 and Lobachevsky in 1834. In 1837 Karl Heinrich Gräffe also discovered the principal idea of the method. | Dandelin–Gräffe method |
c_1o2nyhc9pfuu | The method separates the roots of a polynomial by squaring them repeatedly. This squaring of the roots is done implicitly, that is, only working on the coefficients of the polynomial. Finally, Viète's formulas are used in order to approximate the roots. | Dandelin–Gräffe method |
c_ui52mlvi8goj | In mathematics, Gram's theorem states that an algebraic set in a finite-dimensional vector space invariant under some linear group can be defined by absolute invariants. (Dieudonné & Carrell 1970, p. 31). It is named after J. P. Gram, who published it in 1874. | Gram's theorem |
c_37zepgxg4yk0 | In mathematics, Gray's conjecture is a conjecture made by Brayton Gray in 1984 about maps between loop spaces of spheres. It was later proved by John Harper. == References == | Gray's conjecture |
c_j15ijyeb0jtt | In mathematics, Green's identities are a set of three identities in vector calculus relating the bulk with the boundary of a region on which differential operators act. They are named after the mathematician George Green, who discovered Green's theorem. | Green's identities |
c_k13tnuf2a3vh | In mathematics, Gromov–Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence. | Gromov-Hausdorff convergence |
c_gvkevf49movm | In mathematics, Grothendieck's Galois theory is an abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the setting of algebraic geometry. It provides, in the classical setting of field theory, an alternative perspective to that... | Grothendieck's Galois theory |
c_kjsf899nw74t | The theory of Grothendieck, published in SGA1, shows how to reconstruct the category of G-sets from a fibre functor Φ, which in the geometric setting takes the fibre of a covering above a fixed base point (as a set). In fact there is an isomorphism proved of the type G ≅ Aut(Φ),the latter being the group of automorphis... | Grothendieck's Galois theory |
c_iq4pqmb8rclq | In mathematics, Grothendieck's connectedness theorem , states that if A is a complete Noetherian local ring whose spectrum is k-connected and f is in the maximal ideal, then Spec(A/fA) is (k − 1)-connected. Here a Noetherian scheme is called k-connected if its dimension is greater than k and the complement of every clo... | Grothendieck's connectedness theorem |
c_f6ukxrfthnsf | In mathematics, Grothendieck's six operations, named after Alexander Grothendieck, is a formalism in homological algebra, also known as the six-functor formalism. It originally sprang from the relations in étale cohomology that arise from a morphism of schemes f: X → Y. The basic insight was that many of the elementary... | Six operations |
c_wy4w7xlrom05 | In mathematics, Grunsky's theorem, due to the German mathematician Helmut Grunsky, is a result in complex analysis concerning holomorphic univalent functions defined on the unit disk in the complex numbers. The theorem states that a univalent function defined on the unit disc, fixing the point 0, maps every disk |z| < ... | Grunsky's theorem |
c_fhaku40dv2ai | In mathematics, Grönwall's inequality (also called Grönwall's lemma or the Grönwall–Bellman inequality) allows one to bound a function that is known to satisfy a certain differential or integral inequality by the solution of the corresponding differential or integral equation. There are two forms of the lemma, a differ... | Grönwall's inequality |
c_ln36kluw44bm | In particular, it provides a comparison theorem that can be used to prove uniqueness of a solution to the initial value problem; see the Picard–Lindelöf theorem. It is named for Thomas Hakon Grönwall (1877–1932). Grönwall is the Swedish spelling of his name, but he spelled his name as Gronwall in his scientific publica... | Grönwall's inequality |
c_3zvslylqlbxe | The inequality was first proven by Grönwall in 1919 (the integral form below with α and β being constants).Richard Bellman proved a slightly more general integral form in 1943.A nonlinear generalization of the Grönwall–Bellman inequality is known as Bihari–LaSalle inequality. Other variants and generalizations can be f... | Grönwall's inequality |
c_k3pyby698mwc | In mathematics, Gårding's inequality is a result that gives a lower bound for the bilinear form induced by a real linear elliptic partial differential operator. The inequality is named after Lars Gårding. | Gårding's inequality |
c_40l5x0j1ehyq | In mathematics, Göbel's sequence is a sequence of rational numbers defined by the recurrence relation x n = 1 + x 0 2 + x 1 2 + ⋯ + x n − 1 2 n , {\displaystyle x_{n}={\frac {1+x_{0}^{2}+x_{1}^{2}+\cdots +x_{n-1}^{2}}{n}},\!\,} with starting value x 0 = 1. {\displaystyle x_{0}=1.} Göbel's sequence starts with 1, 1, 2, ... | Göbel's sequence |
c_uia03sn0i4qa | In mathematics, Gödel's speed-up theorem, proved by Gödel (1936), shows that there are theorems whose proofs can be drastically shortened by working in more powerful axiomatic systems. Kurt Gödel showed how to find explicit examples of statements in formal systems that are provable in that system but whose shortest pro... | Gödel's speed-up theorem |
c_bskmc5n5q4dq | The statement has a short proof in a more powerful system: in fact the proof given in the previous paragraph is a proof in the system of Peano arithmetic plus the statement "Peano arithmetic is consistent" (which, per the incompleteness theorem, cannot be proved in Peano arithmetic). In this argument, Peano arithmetic ... | Gödel's speed-up theorem |
c_hucaxeka72zm | For example, the statement "there is an integer n such that if there is a sequence of rooted trees T1, T2, ..., Tn such that Tk has at most k+10 vertices, then some tree can be homeomorphically embedded in a later one"is provable in Peano arithmetic, but the shortest proof has length at least A(1000), where A(0)=1 and ... | Gödel's speed-up theorem |
c_mwky44qs7iz4 | In mathematics, Hadamard regularization (also called Hadamard finite part or Hadamard's partie finie) is a method of regularizing divergent integrals by dropping some divergent terms and keeping the finite part, introduced by Hadamard (1923, book III, chapter I, 1932). Riesz (1938, 1949) showed that this can be interpr... | Hadamard regularization |
c_rv4hk51eyjha | The Hadamard finite part integral above (for a < x < b) may also be given by the following equivalent definitions: The definitions above may be derived by assuming that the function f (t) is differentiable infinitely many times at t = x for a < x < b, that is, by assuming that f (t) can be represented by its Taylor ser... | Hadamard regularization |
c_rt8b841h8ava | In mathematics, Hadamard's gamma function, named after Jacques Hadamard, is an extension of the factorial function, different from the classical gamma function (it is an instance of a pseudogamma function.) This function, with its argument shifted down by 1, interpolates the factorial and extends it to real and complex... | Hadamard's gamma function |
c_1i9w1he4gf49 | In mathematics, Hadamard's inequality (also known as Hadamard's theorem on determinants) is a result first published by Jacques Hadamard in 1893. It is a bound on the determinant of a matrix whose entries are complex numbers in terms of the lengths of its column vectors. In geometrical terms, when restricted to real nu... | Hadamard's inequality |
c_w4zmsczqwxeu | In mathematics, Hadamard's lemma, named after Jacques Hadamard, is essentially a first-order form of Taylor's theorem, in which we can express a smooth, real-valued function exactly in a convenient manner. | Hadamard's lemma |
c_7jl7wn4e1okc | In mathematics, Hahn series (sometimes also known as Hahn–Mal'cev–Neumann series) are a type of formal infinite series. They are a generalization of Puiseux series (themselves a generalization of formal power series) and were first introduced by Hans Hahn in 1907 (and then further generalized by Anatoly Maltsev and Ber... | Hahn series |
c_100i3pol76ce | In mathematics, Hall's conjecture is an open question, as of 2015, on the differences between perfect squares and perfect cubes. It asserts that a perfect square y2 and a perfect cube x3 that are not equal must lie a substantial distance apart. This question arose from consideration of the Mordell equation in the theor... | Hall's conjecture |
c_o6w04crmz08m | The original version of Hall's conjecture, formulated by Marshall Hall, Jr. in 1970, says that there is a positive constant C such that for any integers x and y for which y2 ≠ x3, | y 2 − x 3 | > C | x | . {\displaystyle |y^{2}-x^{3}|>C{\sqrt {|x|}}.} | Hall's conjecture |
c_u1qtimu5lvbg | Hall suggested that perhaps C could be taken as 1/5, which was consistent with all the data known at the time the conjecture was proposed. Danilov showed in 1982 that the exponent 1/2 on the right side (that is, the use of |x|1/2) cannot be replaced by any higher power: for no δ > 0 is there a constant C such that |y2 ... | Hall's conjecture |
c_mrd2mue6wesu | {\displaystyle \deg(g(t)^{2}-f(t)^{3})\geq {\frac {1}{2}}\deg f(t)+1.} The weak form of Hall's conjecture, stated by Stark and Trotter around 1980, replaces the square root on the right side of the inequality by any exponent less than 1/2: for any ε > 0, there is some constant c(ε) depending on ε such that for any inte... | Hall's conjecture |
c_dri5zd536plg | The original, strong, form of the conjecture with exponent 1/2 has never been disproved, although it is no longer believed to be true and the term Hall's conjecture now generally means the version with the ε in it. For example, in 1998, Noam Elkies found the example 4478849284284020423079182 - 58538865167812233 = -1641... | Hall's conjecture |
c_98qal53jf6mp | A generalization to other perfect powers is Pillai's conjecture. The table below displays the known cases with r = x / | y 2 − x 3 | > 1 {\displaystyle r={\sqrt {x}}/|y^{2}-x^{3}|>1} . Note that y can be computed as the nearest integer to x3/2. | Hall's conjecture |
c_xxj4d4hb21b2 | In mathematics, Hall's marriage theorem, proved by Philip Hall (1935), is a theorem with two equivalent formulations. In each case, the theorem gives a necessary and sufficient condition for an object to exist: The combinatorial formulation answers whether a finite collection of sets has a transversal—that is, whether ... | Hall's marriage theorem |
c_ke6ku344iuqv | In mathematics, Hanner's inequalities are results in the theory of Lp spaces. Their proof was published in 1956 by Olof Hanner. They provide a simpler way of proving the uniform convexity of Lp spaces for p ∈ (1, +∞) than the approach proposed by James A. Clarkson in 1936. | Hanner's inequalities |
c_bkcnwq5boz3x | In mathematics, Harborth's conjecture states that every planar graph has a planar drawing in which every edge is a straight segment of integer length. This conjecture is named after Heiko Harborth, and (if true) would strengthen Fáry's theorem on the existence of straight-line drawings for every planar graph. For this ... | Harborth's conjecture |
c_lu2872pxtf1s | In mathematics, Hardy's theorem is a result in complex analysis describing the behavior of holomorphic functions. Let f {\displaystyle f} be a holomorphic function on the open ball centered at zero and radius R {\displaystyle R} in the complex plane, and assume that f {\displaystyle f} is not a constant function. If on... | Hardy's theorem |
c_eg1ur9200ehv | In mathematics, Harish-Chandra's c-function is a function related to the intertwining operator between two principal series representations, that appears in the Plancherel measure for semisimple Lie groups. Harish-Chandra (1958a, 1958b) introduced a special case of it defined in terms of the asymptotic behavior of a zo... | Harish-Chandra's c-function |
c_pa1l2eda5ltb | In mathematics, Harish-Chandra's class is a class of Lie groups used in representation theory. Harish-Chandra's class contains all semisimple connected linear Lie groups and is closed under natural operations, most importantly, the passage to Levi subgroups. This closure property is crucial for many inductive arguments... | Harish-Chandra class |
c_yxc4i99z1a9z | In mathematics, Harnack's inequality is an inequality relating the values of a positive harmonic function at two points, introduced by A. Harnack (1887). Harnack's inequality is used to prove Harnack's theorem about the convergence of sequences of harmonic functions. J. Serrin (1955), and J. Moser (1961, 1964) generali... | Harnack's inequality |
c_tinjuh39rvb6 | In mathematics, Hartogs's theorem is a fundamental result of Friedrich Hartogs in the theory of several complex variables. Roughly speaking, it states that a 'separately analytic' function is continuous. More precisely, if F: C n → C {\displaystyle F:{\textbf {C}}^{n}\to {\textbf {C}}} is a function which is analytic i... | Hartogs's theorem on separate holomorphicity |
c_pjmf27new3xp | Therefore, 'separate analyticity' and 'analyticity' are coincident notions, in the theory of several complex variables. Starting with the extra hypothesis that the function is continuous (or bounded), the theorem is much easier to prove and in this form is known as Osgood's lemma. There is no analogue of this theorem f... | Hartogs's theorem on separate holomorphicity |
c_pvf3jzbwscrn | If we assume that a function f: R n → R {\displaystyle f\colon {\textbf {R}}^{n}\to {\textbf {R}}} is differentiable (or even analytic) in each variable separately, it is not true that f {\displaystyle f} will necessarily be continuous. A counterexample in two dimensions is given by f ( x , y ) = x y x 2 + y 2 . {\disp... | Hartogs's theorem on separate holomorphicity |
c_0eaibr58xjir | In mathematics, Hausdorff dimension is a measure of roughness, or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points... | Capacity dimension |
c_3n1u2kxtmoop | Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of dimensions for highly irregular or "rough" sets, this dimension is also commonly referred to as the Hausdorff–Besicovitch dimension. More specifically, the Hausdorff dimension is a dimensional number associated ... | Capacity dimension |
c_4wj5duebdfia | In mathematical terms, the Hausdorff dimension generalizes the notion of the dimension of a real vector space. That is, the Hausdorff dimension of an n-dimensional inner product space equals n. This underlies the earlier statement that the Hausdorff dimension of a point is zero, of a line is one, etc., and that irregul... | Capacity dimension |
c_piiuxu6clqcj | That is, after the first iteration, each original line segment has been replaced with N=4, where each self-similar copy is 1/S = 1/3 as long as the original. Stated another way, we have taken an object with Euclidean dimension, D, and reduced its linear scale by 1/3 in each direction, so that its length increases to N=... | Capacity dimension |
c_wrgwg6xo71jl | In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that assigns a number in to each set in R n {\displaystyle \mathbb {R} ^{n}} or,... | Hausdorff measure |
c_o63zqmd95skf | Thus, the concept of the Hausdorff measure generalizes the Lebesgue measure and its notions of counting, length, and area. It also generalizes volume. | Hausdorff measure |
c_qjno1jndsmh5 | In fact, there are d-dimensional Hausdorff measures for any d ≥ 0, which is not necessarily an integer. These measures are fundamental in geometric measure theory. They appear naturally in harmonic analysis or potential theory. | Hausdorff measure |
c_tkxgcksokefg | In mathematics, Heegner's lemma is a lemma used by Kurt Heegner in his paper on the class number problem. His lemma states that if y 2 = a 4 x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 {\displaystyle y^{2}=a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}} is a curve over a field with a4 not a square, then it has a solution if i... | Heegner's lemma |
c_dmadnczv2e6x | In mathematics, Helly's selection theorem (also called the Helly selection principle) states that a uniformly bounded sequence of monotone real functions admits a convergent subsequence. In other words, it is a sequential compactness theorem for the space of uniformly bounded monotone functions. It is named for the Aus... | Helly's selection theorem |
c_m1t5e13a7icc | A more general version of the theorem asserts compactness of the space BVloc of functions locally of bounded total variation that are uniformly bounded at a point. The theorem has applications throughout mathematical analysis. In probability theory, the result implies compactness of a tight family of measures. | Helly's selection theorem |
c_twxz50hve4iy | In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number. This is handled by examining the equation in the com... | Hasse principle |
c_20b4pglf5bbc | In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a prime number p, then this root can be lifted to a unique root modulo any higher power of p. More generally, if a polynomial... | Hensel lemma |
c_r3afgtrx9sdm | In mathematics, Hermite numbers are values of Hermite polynomials at zero argument. Typically they are defined for physicists' Hermite polynomials. | Hermite number |
c_euv0xl5z79nh | In mathematics, Hermite transform is an integral transform named after the mathematician Charles Hermite, which uses Hermite polynomials H n ( x ) {\displaystyle H_{n}(x)} as kernels of the transform. This was first introduced by Lokenath Debnath in 1964.The Hermite transform of a function F ( x ) {\displaystyle F(x)} ... | Hermite transform |
c_8ze5z01co4ls | In mathematics, Hermite's cotangent identity is a trigonometric identity discovered by Charles Hermite. Suppose a1, ..., an are complex numbers, no two of which differ by an integer multiple of π. Let A n , k = ∏ 1 ≤ j ≤ n j ≠ k cot ( a k − a j ) {\displaystyle A_{n,k}=\prod _{\begin{smallmatrix}1\leq j\leq n\\j\neq ... | Hermite's cotangent identity |
c_5s9d8f78ywud | In mathematics, Hermite's identity, named after Charles Hermite, gives the value of a summation involving the floor function. It states that for every real number x and for every positive integer n the following identity holds: ∑ k = 0 n − 1 ⌊ x + k n ⌋ = ⌊ n x ⌋ . {\displaystyle \sum _{k=0}^{n-1}\left\lfloor x+{\frac ... | Hermite's identity |
c_qojwy4qkx3rt | In mathematics, Hermite's law of reciprocity, introduced by Hermite (1854), states that the degree m covariants of a binary form of degree n correspond to the degree n covariants of a binary form of degree m. In terms of representation theory it states that the representations Sm Sn C2 and Sn Sm C2 of GL2 are isomorphi... | Hermite reciprocity |
c_u332futxi7n7 | In mathematics, Higman's lemma states that the set of finite sequences over a finite alphabet, as partially ordered by the subsequence relation, is well-quasi-ordered. That is, if w 1 , w 2 , … {\displaystyle w_{1},w_{2},\ldots } is an infinite sequence of words over some fixed finite alphabet, then there exist indices... | Higman's lemma |
c_l8hjapbocuj4 | In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spac... | Hilbert Space |
c_em50qf34oca5 | They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term Hilbert space for t... | Hilbert Space |
c_33qu85vdyaod | The success of Hilbert space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean vector spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphi... | Hilbert Space |
c_1jq14w4lam34 | Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a linear subspace or a subspace (the analog of "dropping the altitude" of a triangle) plays a significant role in optimization problems and other aspects of the theory. An element of ... | Hilbert Space |
c_cdza1b2bwslp | In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros", or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry. It relates algebraic sets to ideals in polynomial rings over alg... | Hilbert's Nullstellensatz |
c_v4zftsdvrz4i | In mathematics, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems proposed in 1900, asks whether certain algebras are finitely generated. The setting is as follows: Assume that k is a field and let K be a subfield of the field of rational functions in n variables, k(x1, ..., xn ) over k.Consider no... | Hilbert's 14th problem |
c_nioe2iftiazs | Hilbert conjectured that all such algebras are finitely generated over k. Some results were obtained confirming Hilbert's conjecture in special cases and for certain classes of rings (in particular the conjecture was proved unconditionally for n = 1 and n = 2 by Zariski in 1954). Then in 1959 Masayoshi Nagata found a c... | Hilbert's 14th problem |
c_j27imkj97187 | In mathematics, Hilbert's fourth problem in the 1900 list of Hilbert's problems is a foundational question in geometry. In one statement derived from the original, it was to find — up to an isomorphism — all geometries that have an axiomatic system of the classical geometry (Euclidean, hyperbolic and elliptic), with th... | Hilbert's fourth problem |
c_4jfhs4qgsuub | "There are several interpretations of the original statement of David Hilbert. Nevertheless, a solution was sought, with the German mathematician Georg Hamel being the first to contribute to the solution of Hilbert's fourth problem.A recognized solution was given by Ukrainian mathematician Aleksei Pogorelov in 1973. In... | Hilbert's fourth problem |
c_qy4pitfhc3mx | In mathematics, Hilbert's program, formulated by German mathematician David Hilbert in the early 1920s, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies. As a solution, Hilbert propose... | Hilbert program |
c_93fnuwfns8xo | Ultimately, the consistency of all of mathematics could be reduced to basic arithmetic. Gödel's incompleteness theorems, published in 1931, showed that Hilbert's program was unattainable for key areas of mathematics. In his first theorem, Gödel showed that any consistent system with a computable set of axioms which is ... | Hilbert program |
c_08y99xs2kwi1 | In mathematics, Hilbert's second problem was posed by David Hilbert in 1900 as one of his 23 problems. It asks for a proof that the arithmetic is consistent – free of any internal contradictions. Hilbert stated that the axioms he considered for arithmetic were the ones given in Hilbert (1900), which include a second or... | Hilbert's second problem |
c_50un7pleaae8 | In mathematics, Hilbert's syzygy theorem is one of the three fundamental theorems about polynomial rings over fields, first proved by David Hilbert in 1890, which were introduced for solving important open questions in invariant theory, and are at the basis of modern algebraic geometry. The two other theorems are Hilbe... | Hilbert's syzygy theorem |
c_2yrp1vgvql0p | As the relations form a module, one may consider the relations between the relations; the theorem asserts that, if one continues in this way, starting with a module over a polynomial ring in n indeterminates over a field, one eventually finds a zero module of relations, after at most n steps. Hilbert's syzygy theorem i... | Hilbert's syzygy theorem |
c_caev2pcua5c2 | In mathematics, Hiptmair–Xu (HX) preconditioners are preconditioners for solving H ( curl ) {\displaystyle H(\operatorname {curl} )} and H ( div ) {\displaystyle H(\operatorname {div} )} problems based on the auxiliary space preconditioning framework. An important ingredient in the derivation of HX preconditioners in t... | Hiptmair–Xu preconditioner |
c_ybew4uoa8yqh | HX preconditioners have been used for accelerating a wide variety of solution techniques, thanks to their highly scalable parallel implementations, and are known as AMS and ADS precondition. HX preconditioner was identified by the U.S. Department of Energy as one of the top ten breakthroughs in computational science in... | Hiptmair–Xu preconditioner |
c_izbcnpp5zso8 | In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by Gerhard Hochschild (1945) for algebras over a field, and extended to algebras over more general rings... | Hochschild cohomology |
c_1gjuqb5ypg38 | In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold M using partial differential equations. The key observation is that, given a Riemannian metric on M, every cohomology class has a canonical representative, a differential form that vanishes unde... | Harmonic form |
c_qdf3c0n4ghx7 | It has major applications in two settings: Riemannian manifolds and Kähler manifolds. Hodge's primary motivation, the study of complex projective varieties, is encompassed by the latter case. Hodge theory has become an important tool in algebraic geometry, particularly through its connection to the study of algebraic c... | Harmonic form |
c_t3p77mgxj985 | In mathematics, Hodge–Arakelov theory of elliptic curves is an analogue of classical and p-adic Hodge theory for elliptic curves carried out in the framework of Arakelov theory. It was introduced by Mochizuki (1999). It bears the name of two mathematicians, Suren Arakelov and W. V. D. Hodge. | Hodge–Arakelov theory |
c_t53jsv1fidpx | The main comparison in his theory remains unpublished as of 2019. Mochizuki's main comparison theorem in Hodge–Arakelov theory states (roughly) that the space of polynomial functions of degree less than d on the universal extension of a smooth elliptic curve in characteristic 0 is naturally isomorphic (via restriction)... | Hodge–Arakelov theory |
c_4pc9nnhvlvn3 | In mathematics, Hooley's delta function ( Δ ( n ) {\displaystyle \Delta (n)} ), also called Erdős--Hooley delta-function, defines the maximum number of divisors of n {\displaystyle n} in {\displaystyle } for all u {\displaystyle u} , where e {\displaystyle e} is the Euler's number. The first few terms of this sequence... | Hooley's delta function |
c_efc0ou0upalf | In mathematics, Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to Heinz Hopf. | Hopf conjecture |
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