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wiki_1600_chunk_0 | Gabriel's theorem | In mathematics, Gabriel's theorem, proved by Pierre Gabriel, classifies the quivers of finite type in terms of Dynkin diagrams. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1601_chunk_0 | Galois cohomology | In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group G associated to a field extension L/K acts in a natural way on some abelian groups, for example those constructed directly from L, but al... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1602_chunk_0 | Solvable by radicals | In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand. Galois int... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1603_chunk_0 | Solvable by radicals | This widely generalizes the Abel–Ruffini theorem, which asserts that a general polynomial of degree at least five cannot be solved by radicals. Galois theory has been used to solve classic problems including showing that two problems of antiquity cannot be solved as they were stated (doubling the cube and trisecting th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1604_chunk_0 | Gaussian brackets | In mathematics, Gaussian brackets are a special notation invented by Carl Friedrich Gauss to represent the convergents of a simple continued fraction in the form of a simple fraction. Gauss used this notation in the context of finding solutions of the indeterminate equations of the form a x = b y ± 1 {\displaystyle ax=... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1605_chunk_0 | Gauss elimination | In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used to compute the rank of a matrix, the determinant of a square matrix, and... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1606_chunk_0 | Gauss elimination | To perform row reduction on a matrix, one uses a sequence of elementary row operations to modify the matrix until the lower left-hand corner of the matrix is filled with zeros, as much as possible. There are three types of elementary row operations: Swapping two rows, Multiplying a row by a nonzero number, Adding a mul... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1607_chunk_0 | Gauss elimination | This final form is unique; in other words, it is independent of the sequence of row operations used. For example, in the following sequence of row operations (where two elementary operations on different rows are done at the first and third steps), the third and fourth matrices are the ones in row echelon form, and the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1608_chunk_0 | Gaussian measure | In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space Rn, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are named after the German mathematician Carl Friedrich Gauss. One reason why Gaussian ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1609_chunk_0 | Ultraspherical polynomials | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1610_chunk_0 | 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 multilinear alternating forms on the Lie algebra of smooth ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1611_chunk_0 | 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–Schneider theorem, noting that where i is the imaginar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1612_chunk_0 | 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 lies in Z*(G), which is the inver... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1613_chunk_0 | 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 |
wiki_1614_chunk_0 | 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 \displaystyle \sigma _{\lambda }=\det(\sigma _{\lambda _{i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1615_chunk_0 | 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 every symmetric polynomial is a polynomial in the elementary symmetr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1616_chunk_0 | Goldie ring | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1617_chunk_0 | Goldie ring | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1618_chunk_0 | Noncommutative algebra | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1619_chunk_0 | Noncommutative algebra | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1620_chunk_0 | 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 rational function of n); then necessarily a(n)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1621_chunk_0 | 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 a problem regarding Banach spaces. The r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1622_chunk_0 | 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 discovered the principal idea of the method. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1623_chunk_0 | 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 |
wiki_1624_chunk_0 | 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 |
wiki_1625_chunk_0 | 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 classical setting of field theory, an alternative perspective to that... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1626_chunk_0 | 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 proved of the type G ≅ Aut(Φ),the latter being the group of automorphis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1627_chunk_0 | 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 called k-connected if its dimension is greater than k and the complement of every clo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1628_chunk_0 | 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 insight was that many of the elementary... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1629_chunk_0 | 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, fixing the point 0, maps every disk |z| < ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1630_chunk_0 | Grönwall's inequality | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1631_chunk_0 | Grönwall's inequality | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1632_chunk_0 | Grönwall's inequality | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1633_chunk_0 | 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 |
wiki_1634_chunk_0 | 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}=1.} Göbel's sequence starts with 1, 1, 2, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1635_chunk_0 | 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 systems that are provable in that system but whose shortest pro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1636_chunk_0 | 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 in Peano arithmetic). In this argument, Peano arithmetic ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1637_chunk_0 | 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 shortest proof has length at least A(1000), where A(0)=1 and ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1638_chunk_0 | 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, 1932). Riesz (1938, 1949) showed that this can be interpr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1639_chunk_0 | 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 assuming that f (t) can be represented by its Taylor ser... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1640_chunk_0 | 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, interpolates the factorial and extends it to real and complex... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1641_chunk_0 | 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 vectors. In geometrical terms, when restricted to real nu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1642_chunk_0 | 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 |
wiki_1643_chunk_0 | 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 generalized by Anatoly Maltsev and Ber... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1644_chunk_0 | 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 consideration of the Mordell equation in the theor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1645_chunk_0 | 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 constant c(ε) depending on ε such that for any inte... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1646_chunk_0 | 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 finite collection of sets has a transversal—that is, whether ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1647_chunk_0 | 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 1936. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1648_chunk_0 | 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 straight-line drawings for every planar graph. For this ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1649_chunk_0 | 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 {\displaystyle f} is not a constant function. If on... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1650_chunk_0 | 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 special case of it defined in terms of the asymptotic behavior of a zo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1651_chunk_0 | 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. Serrin (1955), and J. Moser (1961, 1964) generali... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1652_chunk_0 | 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 F: C n → C {\displaystyle F:{\textbf {C}}^{n}\to {\textbf {C}}} is a function which is analytic i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1653_chunk_0 | 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 easier to prove and in this form is known as Osgood's lemma. There is no analogue of this theorem f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1654_chunk_0 | 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 be continuous. A counterexample in two dimensions is given by f ( x , y ) = x y x 2 + y 2 . {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1655_chunk_0 | 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, and of a cube is 3. That is, for sets of points... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1656_chunk_0 | 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 zero, of a line is one, etc., and that irregul... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1657_chunk_0 | 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 direction, so that its length increases to N=... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1658_chunk_0 | 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 uniformly bounded monotone functions. It is named for the Aus... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1659_chunk_0 | 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 implies compactness of a tight family of measures. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1660_chunk_0 | 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 handled by examining the equation in the com... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1661_chunk_0 | 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 p. More generally, if a polynomial... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1662_chunk_0 | 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 of a function F ( x ) {\displaystyle F(x)} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1663_chunk_0 | 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 ⌋ . {\displaystyle \sum _{k=0}^{n-1}\left\lfloor x+{\frac ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1664_chunk_0 | Hilbert Space | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1665_chunk_0 | Hilbert Space | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1666_chunk_0 | Hilbert Space | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1667_chunk_0 | Hilbert Space | 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1668_chunk_0 | Hilbert's 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 relates algebraic sets to ideals in polynomial rings over alg... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1669_chunk_0 | Hilbert's 14th 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 functions in n variables, k(x1, ..., xn ) over k.Consider no... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1670_chunk_0 | 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 classical geometry (Euclidean, hyperbolic and elliptic), with th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1671_chunk_0 | 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 given by Ukrainian mathematician Aleksei Pogorelov in 1973. In... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1672_chunk_0 | 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 inconsistencies. As a solution, Hilbert propose... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1673_chunk_0 | 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 with a computable set of axioms which is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1674_chunk_0 | 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 the ones given in Hilbert (1900), which include a second or... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1675_chunk_0 | 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 modern algebraic geometry. The two other theorems are Hilbe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1676_chunk_0 | 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 relations, after at most n steps. Hilbert's syzygy theorem i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1677_chunk_0 | 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 important ingredient in the derivation of HX preconditioners in t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1678_chunk_0 | 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 as one of the top ten breakthroughs in computational science in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1679_chunk_0 | 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 differential form that vanishes unde... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1680_chunk_0 | 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 connection to the study of algebraic c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1681_chunk_0 | 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 characteristic 0 is naturally isomorphic (via restriction)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1682_chunk_0 | 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 the Euler's number. The first few terms of this sequence... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1683_chunk_0 | 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 |
wiki_1684_chunk_0 | Hua's lemma | In mathematics, Hua's lemma, named for Hua Loo-keng, is an estimate for exponential sums. It states that if P is an integral-valued polynomial of degree k, ε {\displaystyle \varepsilon } is a positive real number, and f a real function defined by f ( α ) = ∑ x = 1 N exp ( 2 π i P ( x ) α ) , {\displaystyle f(\alpha )... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1685_chunk_0 | Hudde's rules | In mathematics, Hudde's rules are two properties of polynomial roots described by Johann Hudde. 1. If r is a double root of the polynomial equation a 0 x n + a 1 x n − 1 + ⋯ + a n − 1 x + a n = 0 {\displaystyle a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x+a_{n}=0} and if b 0 , b 1 , … , b n − 1 , b n {\displaystyle b_{0},b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1686_chunk_0 | Hudde's rules | This definition is a form of the modern theorem that if r is a double root of ƒ(x) = 0, then r is a root of ƒ '(x) = 0.2. If for x = a the polynomial a 0 x n + a 1 x n − 1 + ⋯ + a n − 1 x + a n {\displaystyle a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x+a_{n}} takes on a relative maximum or minimum value, then a is a root ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1687_chunk_0 | Hudde's rules | This definition is a modification of Fermat's theorem in the form that if ƒ(a) is a relative maximum or minimum value of a polynomial ƒ(x), then ƒ '(a) = 0, where ƒ ' is the derivative of ƒ.Hudde was working with Frans van Schooten on a Latin edition of La Géométrie of René Descartes. In the 1659 edition of the transla... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1688_chunk_0 | Hurwitz's automorphisms theorem | In mathematics, Hurwitz's automorphisms theorem bounds the order of the group of automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus g > 1, stating that the number of such automorphisms cannot exceed 84(g − 1). A group for which the maximum is achieved is called a Hurwitz... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1689_chunk_0 | Hurwitz's automorphisms theorem | The theorem is named after Adolf Hurwitz, who proved it in (Hurwitz 1893). Hurwitz's bound also holds for algebraic curves over a field of characteristic 0, and over fields of positive characteristic p>0 for groups whose order is coprime to p, but can fail over fields of positive characteristic p>0 when p divides the g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1690_chunk_0 | Euclidean Hurwitz algebra | In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism into ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1691_chunk_0 | Euclidean Hurwitz algebra | The theory of composition algebras has subsequently been generalized to arbitrary quadratic forms and arbitrary fields. Hurwitz's theorem implies that multiplicative formulas for sums of squares can only occur in 1, 2, 4 and 8 dimensions, a result originally proved by Hurwitz in 1898. It is a special case of the Hurwit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1692_chunk_0 | Christiaan Huygens | In mathematics, Huygens mastered the methods of ancient Greek geometry, particularly the work of Archimedes, and was an adept user of the analytic geometry and infinitesimal techniques of Descartes and Fermat. His mathematical style can be best described as geometrical infinitesimal analysis of curves and of motion. Dr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1693_chunk_0 | Christiaan Huygens | Huygens brought this type of geometrical analysis to a close, as more mathematicians turned away from classical geometry to the calculus for handling infinitesimals, limit processes, and motion.Huygens was moreover able to fully employ mathematics to answer questions of physics. Often this entailed introducing a simple... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1694_chunk_0 | Christiaan Huygens | Huygens favoured axiomatic presentations of his results, which require rigorous methods of geometric demonstration: although he allowed levels of uncertainty in the selection of primary axioms and hypotheses, the proofs of theorems derived from these could never be in doubt. Huygens's style of publication exerted an in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1695_chunk_0 | Christiaan Huygens | In demanding such mathematical tractability and precision, Huygens set an example for eighteenth-century scientists such as Johann Bernoulli, Jean le Rond d'Alembert, and Charles-Augustin de Coulomb.Although never intended for publication, Huygens made use of algebraic expressions to represent physical entities in a ha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1696_chunk_0 | Hölder summation | In mathematics, Hölder summation is a method for summing divergent series introduced by Hölder (1882). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1697_chunk_0 | Hölder's theorem | In mathematics, Hölder's theorem states that the gamma function does not satisfy any algebraic differential equation whose coefficients are rational functions. This result was first proved by Otto Hölder in 1887; several alternative proofs have subsequently been found.The theorem also generalizes to the q {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1698_chunk_0 | Ingleton's inequality | In mathematics, Ingleton's inequality is an inequality that is satisfied by the rank function of any representable matroid. In this sense it is a necessary condition for representability of a matroid over a finite field. Let M be a matroid and let ρ be its rank function, Ingleton's inequality states that for any subset... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1699_chunk_0 | Itô's lemma | In mathematics, Itô's lemma or Itô's formula (also called the Itô-Doeblin formula, especially in the French literature) is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process. It serves as the stochastic calculus counterpart of the chain rule. It can be heurist... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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