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Ultraspherical polynomials Summary Gegenbauer_polynomial 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 J... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gelfand–Fuks cohomology Summary Gelfand–Fuks_cohomology 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 mul... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gelfond's constant Summary Gelfond's_constant 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–Schneide... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gelfond's constant Summary Gelfond's_constant 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Z* theorem Summary Z*_theorem 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 lie... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
ZJ theorem Summary ZJ_theorem 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Geronimus polynomials Summary Geronimus_polynomials In mathematics, Geronimus polynomials may refer to one of the several different families of orthogonal polynomials studied by Yakov Lazarevich Geronimus. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Giambelli's formula Summary Giambelli's_formula 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 \displayst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gijswijt's sequence Summary Gijswijt's_sequence 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Giraud subcategory Summary Giraud_subcategory In mathematics, Giraud subcategories form an important class of subcategories of Grothendieck categories. They are named after Jean Giraud. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Glaeser's composition theorem Summary Glaeser's_composition_theorem 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Goldie ring Summary Goldie's_theorem 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 ch... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Goldie ring Summary Goldie's_theorem 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 semipri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Noncommutative algebra Goldie's theorem Non-commutative_ring_theory > Important theorems > Goldie's theorem 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 c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Noncommutative algebra Goldie's theorem Non-commutative_ring_theory > Important theorems > Goldie's theorem 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 cons... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gosper's algorithm Summary Gosper's_algorithm 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gottlieb polynomials Summary Gottlieb_polynomials 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 \displays... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gowers' theorem Summary Gowers'_theorem 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dandelin–Gräffe method Summary Dandelin–Gräffe_method 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 discove... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dandelin–Gräffe method Summary Dandelin–Gräffe_method 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gram's theorem Summary Gram's_theorem 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gray's conjecture Summary Gray's_conjecture 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 == | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Green's identities Summary Green's_identities 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gromov-Hausdorff convergence Summary Gromov–Hausdorff_convergence 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Grothendieck's Galois theory Summary Grothendieck's_Galois_theory 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 cla... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Grothendieck's Galois theory Summary Grothendieck's_Galois_theory 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 prove... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Grothendieck's connectedness theorem Summary Grothendieck's_connectedness_theorem 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 cal... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Six operations Summary Six_operations 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 i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Grunsky's theorem Summary Grunsky's_theorem 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Grönwall's inequality Summary Grönwall's_lemma 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 equat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Grönwall's inequality Summary Grönwall's_lemma 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Grönwall's inequality Summary Grönwall's_lemma 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 inequali... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gårding's inequality Summary Gårding's_inequality 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Göbel's sequence Summary Göbel's_sequence 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}=... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gödel's speed-up theorem Summary Gödel's_speed-up_theorem 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 syste... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gödel's speed-up theorem Summary Gödel's_speed-up_theorem 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gödel's speed-up theorem Summary Gödel's_speed-up_theorem 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 sho... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hadamard regularization Summary Hadamard_regularization 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, 1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hadamard regularization Summary Hadamard_regularization 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hadamard's gamma function Summary Hadamard's_gamma_function 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, i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hadamard's inequality Summary Hadamard's_inequality 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 vecto... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hadamard's lemma Summary Hadamard's_lemma 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hahn series Summary Hahn_series 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 genera... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hall's conjecture Summary Hall's_conjecture 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 consid... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hall's conjecture Summary Hall's_conjecture 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|}}.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hall's conjecture Summary Hall's_conjecture 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 n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hall's conjecture Summary Hall's_conjecture {\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 constan... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hall's conjecture Summary Hall's_conjecture 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 447884928... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hall's conjecture Summary Hall's_conjecture 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hall's marriage theorem Summary Hall's_marriage_theorem 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 finit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hanner's inequalities Summary Hanner's_inequalities 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Harborth's conjecture Summary Harborth's_conjecture 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 stra... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hardy's theorem Summary Hardy's_theorem 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 {\displayst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Harish-Chandra's c-function Summary Harish-Chandra's_c-function 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 specia... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Harish-Chandra class Summary Harish-Chandra_class 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 closur... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Harnack's inequality Summary Harnack's_inequality 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.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hartogs's theorem on separate holomorphicity Summary Hartogs's_theorem_on_separate_holomorphicity 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hartogs's theorem on separate holomorphicity Summary Hartogs's_theorem_on_separate_holomorphicity 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 e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hartogs's theorem on separate holomorphicity Summary Hartogs's_theorem_on_separate_holomorphicity 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Capacity dimension Summary Capacity_dimension 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, a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Capacity dimension Summary Capacity_dimension 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Capacity dimension Summary Capacity_dimension 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Capacity dimension Summary Capacity_dimension 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hausdorff measure Summary Hausdorff_measure 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hausdorff measure Summary Hausdorff_measure Thus, the concept of the Hausdorff measure generalizes the Lebesgue measure and its notions of counting, length, and area. It also generalizes volume. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hausdorff measure Summary Hausdorff_measure 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Heegner's lemma Summary Heegner's_lemma 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 n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Helly's selection theorem Summary Helly's_selection_theorem 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 u... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Helly's selection theorem Summary Helly's_selection_theorem 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hasse principle Summary Hasse_principle 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 hand... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hensel lemma Summary Hensel_lemma 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hermite number Summary Hermite_number In mathematics, Hermite numbers are values of Hermite polynomials at zero argument. Typically they are defined for physicists' Hermite polynomials. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hermite transform Summary Hermite_transform 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hermite's cotangent identity Summary Hermite's_cotangent_identity 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 ) {\dis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hermite's identity Summary Hermite's_identity 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 ⌋ . {\displa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hermite reciprocity Summary Hermite_reciprocity 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 representa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Higman's lemma Summary Higman's_lemma 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 fin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hilbert Space Summary Hilbert_spaces 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 an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hilbert Space Summary Hilbert_spaces 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 Neuman... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hilbert Space Summary Hilbert_spaces 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 func... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hilbert Space Summary Hilbert_spaces 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hilbert's Nullstellensatz Summary Weak_Nullstellensatz 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 relate... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hilbert's 14th problem Summary Hilbert's_fourteenth_problem 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hilbert's 14th problem Summary Hilbert's_fourteenth_problem 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hilbert's fourth problem Summary Hilbert's_fourth_problem 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 classi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hilbert's fourth problem Summary Hilbert's_fourth_problem "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 give... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hilbert program Summary Hilbert_program 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 inconsi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hilbert program Summary Hilbert_program 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 w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hilbert's second problem Summary Hilbert's_second_problem 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 t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hilbert's syzygy theorem Summary Hilbert's_syzygy_theorem 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 m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hilbert's syzygy theorem Summary Hilbert's_syzygy_theorem 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 re... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hiptmair–Xu preconditioner Summary Hiptmair–Xu_preconditioner 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 imp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hiptmair–Xu preconditioner Summary Hiptmair–Xu_preconditioner 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 a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hochschild cohomology Summary Hochschild_cohomology 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 fie... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Harmonic form Summary Harmonic_form 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Harmonic form Summary Harmonic_form 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 co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hodge–Arakelov theory Summary Hodge–Arakelov_theory 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 Arak... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hodge–Arakelov theory Summary Hodge–Arakelov_theory 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 charac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hooley's delta function Summary Hooley's_delta_function 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hopf conjecture Summary Hopf_conjecture In mathematics, Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to Heinz Hopf. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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