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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 Jacobi polynomials. They are named after Leopold Gegenbauer.
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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 multilinear alternating forms on the Lie algebra of smooth vector fields where the latter is given the C ∞ {\displaystyle C^{\infty }} topology.
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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–Schneider theorem, noting that where i is the imaginary unit. Since −i is algebraic but not rational, eπ is transcendental.
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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.
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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 lies in Z*(G), which is the inverse image in G of the center of G/O(G). This generalizes the Brauer–Suzuki theorem (and the proof uses the Brauer–Suzuki theorem to deal with some small cases).
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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.
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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.
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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 \displaystyle \sigma _{\lambda }=\det(\sigma _{\lambda _{i}+j-i})_{1\leq i,j\leq r}} where σλ is the Schubert class of a partition λ. Giambelli's formula is a consequence of Pieri's formula. The Porteous formula is a generalization to morphisms of vector bundles over a variety.
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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, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 1, ... (sequence A090822 in the OEIS)The sequence is similar in definition to the Kolakoski sequence, but instead of counting the longest run of single terms, the sequence counts the longest run of blocks of terms of any length. Gijswijt's sequence is known for its remarkably slow rate of growth. For example, the first 4 appears at the 220th term, and the first 5 appears near the 10 10 23 {\displaystyle 10^{10^{23}}} rd term.
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Giraud subcategory
Summary
Giraud_subcategory
In mathematics, Giraud subcategories form an important class of subcategories of Grothendieck categories. They are named after Jean Giraud.
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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 every symmetric polynomial is a polynomial in the elementary symmetric polynomials, from polynomials to smooth functions.
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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 chain condition on right annihilators of subsets of R. Goldie's theorem states that the semiprime right Goldie rings are precisely those that have a semisimple Artinian right classical ring of quotients. The structure of this ring of quotients is then completely determined by the Artin–Wedderburn theorem. In particular, Goldie's theorem applies to semiprime right Noetherian rings, since by definition right Noetherian rings have the ascending chain condition on all right ideals.
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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 semiprime principal right ideal ring is isomorphic to a finite direct sum of prime principal right ideal rings. Every prime principal right ideal ring is isomorphic to a matrix ring over a right Ore domain.
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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 called "finite rank") as a right module over itself, and satisfies the ascending chain condition on right annihilators of subsets of R. Goldie's theorem states that the semiprime right Goldie rings are precisely those that have a semisimple Artinian right classical ring of quotients. The structure of this ring of quotients is then completely determined by the Artin–Wedderburn theorem. In particular, Goldie's theorem applies to semiprime right Noetherian rings, since by definition right Noetherian rings have the ascending chain condition on all right ideals.
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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 consequence of Goldie's theorem, again due to Goldie, is that every semiprime principal right ideal ring is isomorphic to a finite direct sum of prime principal right ideal rings. Every prime principal right ideal ring is isomorphic to a matrix ring over a right Ore domain.
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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 rational function of n); then necessarily a(n) is itself a hypergeometric term, and given the formula for a(n) Gosper's algorithm finds that for S(n).
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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 \displaystyle \ell _{n}(x,\lambda )=e^{-n\lambda }\sum _{k}(1-e^{\lambda })^{k}{\binom {n}{k}}{\binom {x}{k}}=e^{-n\lambda }{}_{2}F_{1}(-n,-x;1;1-e^{\lambda })}
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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 a problem regarding Banach spaces. The result was subsequently generalised by Bartošová, Kwiatkowska, and Lupini.
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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 discovered the principal idea of the method.
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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.
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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.
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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 ==
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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.
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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.
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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 classical setting of field theory, an alternative perspective to that of Emil Artin based on linear algebra, which became standard from about the 1930s. The approach of Alexander Grothendieck is concerned with the category-theoretic properties that characterise the categories of finite G-sets for a fixed profinite group G. For example, G might be the group denoted Z ^ {\displaystyle {\hat {\mathbb {Z} }}} , which is the inverse limit of the cyclic additive groups Z/nZ — or equivalently the completion of the infinite cyclic group Z for the topology of subgroups of finite index. A finite G-set is then a finite set X on which G acts through a quotient finite cyclic group, so that it is specified by giving some permutation of X. In the above example, a connection with classical Galois theory can be seen by regarding Z ^ {\displaystyle {\hat {\mathbb {Z} }}} as the profinite Galois group Gal(F/F) of the algebraic closure F of any finite field F, over F. That is, the automorphisms of F fixing F are described by the inverse limit, as we take larger and larger finite splitting fields over F. The connection with geometry can be seen when we look at covering spaces of the unit disk in the complex plane with the origin removed: the finite covering realised by the zn map of the disk, thought of by means of a complex number variable z, corresponds to the subgroup n.Z of the fundamental group of the punctured disk.
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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 proved of the type G ≅ Aut(Φ),the latter being the group of automorphisms (self-natural equivalences) of Φ. An abstract classification of categories with a functor to the category of sets is given, by means of which one can recognise categories of G-sets for G profinite. To see how this applies to the case of fields, one has to study the tensor product of fields. In topos theory this is a part of the study of atomic toposes.
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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 called k-connected if its dimension is greater than k and the complement of every closed subset of dimension less than k is connected.It is a local analogue of Bertini's theorem.
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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 insight was that many of the elementary facts relating cohomology on X and Y were formal consequences of a small number of axioms. These axioms hold in many cases completely unrelated to the original context, and therefore the formal consequences also hold. The six operations formalism has since been shown to apply to contexts such as D-modules on algebraic varieties, sheaves on locally compact topological spaces, and motives.
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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, fixing the point 0, maps every disk |z| < r onto a starlike domain for r ≤ tanh π/4. The largest r for which this is true is called the radius of starlikeness of the function.
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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 equation. There are two forms of the lemma, a differential form and an integral form. For the latter there are several variants. Grönwall's inequality is an important tool to obtain various estimates in the theory of ordinary and stochastic differential equations.
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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 his name as Gronwall in his scientific publications after emigrating to the United States.
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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 inequality. Other variants and generalizations can be found in Pachpatte, B.G. (1998).
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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.
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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}=1.} Göbel's sequence starts with 1, 1, 2, 3, 5, 10, 28, 154, 3520, 1551880, ... (sequence A003504 in the OEIS)The first non-integral value is x43.
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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 systems that are provable in that system but whose shortest proof is unimaginably long. For example, the statement: "This statement cannot be proved in Peano arithmetic in fewer than a googolplex symbols"is provable in Peano arithmetic (PA) but the shortest proof has at least a googolplex symbols, by an argument similar to the proof of Gödel's first incompleteness theorem: If PA is consistent, then it cannot prove the statement in fewer than a googolplex symbols, because the existence of such a proof would itself be a theorem of PA, a contradiction. But simply enumerating all strings of length up to a googolplex and checking that each such string is not a proof (in PA) of the statement, yields a proof of the statement (which is necessarily longer than a googolplex symbols).
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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 in Peano arithmetic). In this argument, Peano arithmetic can be replaced by any more powerful consistent system, and a googolplex can be replaced by any number that can be described concisely in the system. Harvey Friedman found some explicit natural examples of this phenomenon, giving some explicit statements in Peano arithmetic and other formal systems whose shortest proofs are ridiculously long (Smoryński 1982).
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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 shortest proof has length at least A(1000), where A(0)=1 and A(n+1)=2A(n). The statement is a special case of Kruskal's theorem and has a short proof in second order arithmetic. If one takes Peano arithmetic together with the negation of the statement above, one obtains an inconsistent theory whose shortest known contradiction is equivalently long.
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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, 1932). Riesz (1938, 1949) showed that this can be interpreted as taking the meromorphic continuation of a convergent integral. If the Cauchy principal value integral exists, then it may be differentiated with respect to x to obtain the Hadamard finite part integral as follows: Note that the symbols C {\displaystyle {\mathcal {C}}} and H {\displaystyle {\mathcal {H}}} are used here to denote Cauchy principal value and Hadamard finite-part integrals respectively.
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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 assuming that f (t) can be represented by its Taylor series about t = x. For details, see Ang (2013). (Note that the term − f (x)/2(1/b − x − 1/a − x) in the second equivalent definition above is missing in Ang (2013) but this is corrected in the errata sheet of the book.) Integral equations containing Hadamard finite part integrals (with f (t) unknown) are termed hypersingular integral equations. Hypersingular integral equations arise in the formulation of many problems in mechanics, such as in fracture analysis.
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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, interpolates the factorial and extends it to real and complex numbers in a different way than Euler's gamma function. It is defined as: H ( x ) = 1 Γ ( 1 − x ) d d x { ln ⁡ ( Γ ( 1 2 − x 2 ) Γ ( 1 − x 2 ) ) } , {\displaystyle H(x)={\frac {1}{\Gamma (1-x)}}\,{\dfrac {d}{dx}}\left\{\ln \left({\frac {\Gamma ({\frac {1}{2}}-{\frac {x}{2}})}{\Gamma (1-{\frac {x}{2}})}}\right)\right\},} where Γ(x) denotes the classical gamma function. If n is a positive integer, then: H ( n ) = Γ ( n ) = ( n − 1 ) ! {\displaystyle H(n)=\Gamma (n)=(n-1)!}
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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 vectors. In geometrical terms, when restricted to real numbers, it bounds the volume in Euclidean space of n dimensions marked out by n vectors vi for 1 ≤ i ≤ n in terms of the lengths of these vectors ||vi||. Specifically, Hadamard's inequality states that if N is the matrix having columns vi, then | det ( N ) | ≤ ∏ i = 1 n ‖ v i ‖ . {\displaystyle \left|\det(N)\right|\leq \prod _{i=1}^{n}\|v_{i}\|.} If the n vectors are non-zero, equality in Hadamard's inequality is achieved if and only if the vectors are orthogonal.
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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.
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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 generalized by Anatoly Maltsev and Bernhard Neumann to a non-commutative setting). They allow for arbitrary exponents of the indeterminate so long as the set supporting them forms a well-ordered subset of the value group (typically Q {\displaystyle \mathbb {Q} } or R {\displaystyle \mathbb {R} } ). Hahn series were first introduced, as groups, in the course of the proof of the Hahn embedding theorem and then studied by him in relation to Hilbert's second problem.
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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 consideration of the Mordell equation in the theory of integer points on elliptic curves.
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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|}}.}
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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 no δ > 0 is there a constant C such that |y2 - x3| > C|x|1/2 + δ whenever y2 ≠ x3. In 1965, Davenport proved an analogue of the above conjecture in the case of polynomials: if f(t) and g(t) are nonzero polynomials over C such that g(t)3 ≠ f(t)2 in C, then deg ⁡ ( g ( t ) 2 − f ( t ) 3 ) ≥ 1 2 deg ⁡ f ( t ) + 1.
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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 constant c(ε) depending on ε such that for any integers x and y for which y2 ≠ x3, | y 2 − x 3 | > c ( ε ) x 1 / 2 − ε . {\displaystyle |y^{2}-x^{3}|>c(\varepsilon )x^{1/2-\varepsilon }.}
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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 4478849284284020423079182 - 58538865167812233 = -1641843, for which compatibility with Hall's conjecture would require C to be less than .0214 ≈ 1/50, so roughly 10 times smaller than the original choice of 1/5 that Hall suggested. The weak form of Hall's conjecture would follow from the ABC conjecture.
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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.
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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 finite collection of sets has a transversal—that is, whether an element can be chosen from each set without repetition. Hall's condition is that for any group of sets from the collection, the total unique elements they contain is at least as large as the number of sets in the group. The graph theoretic formulation answers whether a finite bipartite graph has a perfect matching—that is, a way to match each vertex from one group uniquely to an adjacent vertex from the other group. Hall's condition is that any subset of vertices from one group has a neighbourhood of equal or greater size.
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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 1936.
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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 straight-line drawings for every planar graph. For this reason, a drawing with integer edge lengths is also known as an integral Fáry embedding. Despite much subsequent research, Harborth's conjecture remains unsolved.
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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 {\displaystyle f} is not a constant function. If one defines I ( r ) = 1 2 π ∫ 0 2 π | f ( r e i θ ) | d θ {\displaystyle I(r)={\frac {1}{2\pi }}\int _{0}^{2\pi }\!\left|f(re^{i\theta })\right|\,d\theta } for 0 < r < R , {\displaystyle 0
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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 special case of it defined in terms of the asymptotic behavior of a zonal spherical function of a Lie group, and Harish-Chandra (1970) introduced a more general c-function called Harish-Chandra's (generalized) C-function. Gindikin and Karpelevich (1962, 1969) introduced the Gindikin–Karpelevich formula, a product formula for Harish-Chandra's c-function.
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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 closure property is crucial for many inductive arguments in representation theory of Lie groups, whereas the classes of semisimple or connected semisimple Lie groups are not closed in this sense.
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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. Serrin (1955), and J. Moser (1961, 1964) generalized Harnack's inequality to solutions of elliptic or parabolic partial differential equations. Such results can be used to show the interior regularity of weak solutions. Perelman's solution of the Poincaré conjecture uses a version of the Harnack inequality, found by R. Hamilton (1993), for the Ricci flow.
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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 F: C n → C {\displaystyle F:{\textbf {C}}^{n}\to {\textbf {C}}} is a function which is analytic in each variable zi, 1 ≤ i ≤ n, while the other variables are held constant, then F is a continuous function. A corollary is that the function F is then in fact an analytic function in the n-variable sense (i.e. that locally it has a Taylor expansion).
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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 easier to prove and in this form is known as Osgood's lemma. There is no analogue of this theorem for real variables.
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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 be continuous. A counterexample in two dimensions is given by f ( x , y ) = x y x 2 + y 2 . {\displaystyle f(x,y)={\frac {xy}{x^{2}+y^{2}}}.} If in addition we define f ( 0 , 0 ) = 0 {\displaystyle f(0,0)=0} , this function has well-defined partial derivatives in x {\displaystyle x} and y {\displaystyle y} at the origin, but it is not continuous at origin. (Indeed, the limits along the lines x = y {\displaystyle x=y} and x = − y {\displaystyle x=-y} are not equal, so there is no way to extend the definition of f {\displaystyle f} to include the origin and have the function be continuous there.)
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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, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions.
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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 dimension is a dimensional number associated with a metric space, i.e. a set where the distances between all members are defined. The dimension is drawn from the extended real numbers, R ¯ {\displaystyle {\overline {\mathbb {R} }}} , as opposed to the more intuitive notion of dimension, which is not associated to general metric spaces, and only takes values in the non-negative integers.
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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 zero, of a line is one, etc., and that irregular sets can have noninteger Hausdorff dimensions. For instance, the Koch snowflake shown at right is constructed from an equilateral triangle; in each iteration, its component line segments are divided into 3 segments of unit length, the newly created middle segment is used as the base of a new equilateral triangle that points outward, and this base segment is then deleted to leave a final object from the iteration of unit length of 4.
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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 direction, so that its length increases to N=SD. This equation is easily solved for D, yielding the ratio of logarithms (or natural logarithms) appearing in the figures, and giving—in the Koch and other fractal cases—non-integer dimensions for these objects. The Hausdorff dimension is a successor to the simpler, but usually equivalent, box-counting or Minkowski–Bouligand dimension.
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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 in R n {\displaystyle \mathbb {R} ^{n}} or, more generally, in any metric space. The zero-dimensional Hausdorff measure is the number of points in the set (if the set is finite) or ∞ if the set is infinite. Likewise, the one-dimensional Hausdorff measure of a simple curve in R n {\displaystyle \mathbb {R} ^{n}} is equal to the length of the curve, and the two-dimensional Hausdorff measure of a Lebesgue-measurable subset of R 2 {\displaystyle \mathbb {R} ^{2}} is proportional to the area of the set.
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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.
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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.
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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 not a square, then it has a solution if it has a solution in an extension of odd degree.
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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 uniformly bounded monotone functions. It is named for the Austrian mathematician Eduard Helly.
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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 implies compactness of a tight family of measures.
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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 handled by examining the equation in the completions of the rational numbers: the real numbers and the p-adic numbers. A more formal version of the Hasse principle states that certain types of equations have a rational solution if and only if they have a solution in the real numbers and in the p-adic numbers for each prime p.
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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 p. More generally, if a polynomial factors modulo p into two coprime polynomials, this factorization can be lifted to a factorization modulo any higher power of p (the case of roots corresponds to the case of degree 1 for one of the factors). By passing to the "limit" (in fact this is an inverse limit) when the power of p tends to infinity, it follows that a root or a factorization modulo p can be lifted to a root or a factorization over the p-adic integers. These results have been widely generalized, under the same name, to the case of polynomials over an arbitrary commutative ring, where p is replaced by an ideal, and "coprime polynomials" means "polynomials that generate an ideal containing 1". Hensel's lemma is fundamental in p-adic analysis, a branch of analytic number theory. The proof of Hensel's lemma is constructive, and leads to an efficient algorithm for Hensel lifting, which is fundamental for factoring polynomials, and gives the most efficient known algorithm for exact linear algebra over the rational numbers.
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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.
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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 of a function F ( x ) {\displaystyle F(x)} is The inverse Hermite transform is given by
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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 ) {\displaystyle A_{n,k}=\prod _{\begin{smallmatrix}1\leq j\leq n\\j\neq k\end{smallmatrix}}\cot(a_{k}-a_{j})} (in particular, A1,1, being an empty product, is 1). Then cot ⁡ ( z − a 1 ) ⋯ cot ⁡ ( z − a n ) = cos ⁡ n π 2 + ∑ k = 1 n A n , k cot ⁡ ( z − a k ) . {\displaystyle \cot(z-a_{1})\cdots \cot(z-a_{n})=\cos {\frac {n\pi }{2}}+\sum _{k=1}^{n}A_{n,k}\cot(z-a_{k}).} The simplest non-trivial example is the case n = 2: cot ⁡ ( z − a 1 ) cot ⁡ ( z − a 2 ) = − 1 + cot ⁡ ( a 1 − a 2 ) cot ⁡ ( z − a 1 ) + cot ⁡ ( a 2 − a 1 ) cot ⁡ ( z − a 2 ) . {\displaystyle \cot(z-a_{1})\cot(z-a_{2})=-1+\cot(a_{1}-a_{2})\cot(z-a_{1})+\cot(a_{2}-a_{1})\cot(z-a_{2}).\,} == Notes and references ==
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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 ⌋ . {\displaystyle \sum _{k=0}^{n-1}\left\lfloor x+{\frac {k}{n}}\right\rfloor =\lfloor nx\rfloor .}
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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 representations Sm Sn C2 and Sn Sm C2 of GL2 are isomorphic.
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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 finite alphabet, then there exist indices i < j {\displaystyle i
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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 and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz.
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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 Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications.
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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 functions, and Hardy spaces of holomorphic functions. Geometric intuition plays an important role in many aspects of Hilbert space theory.
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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 aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to an orthonormal basis, in analogy with Cartesian coordinates in classical geometry. When this basis is countably infinite, it allows identifying the Hilbert space with the space of the infinite sequences that are square-summable. The latter space is often in the older literature referred to as the Hilbert space.
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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 relates algebraic sets to ideals in polynomial rings over algebraically closed fields. This relationship was discovered by David Hilbert, who proved the Nullstellensatz in his second major paper on invariant theory in 1893 (following his seminal 1890 paper in which he proved Hilbert's basis theorem).
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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 functions in n variables, k(x1, ..., xn ) over k.Consider now the k-algebra R defined as the intersection R := K ∩ k . {\displaystyle R:=K\cap k\ .}
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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 by Zariski in 1954). Then in 1959 Masayoshi Nagata found a counterexample to Hilbert's conjecture. The counterexample of Nagata is a suitably constructed ring of invariants for the action of a linear algebraic group.
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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 classical geometry (Euclidean, hyperbolic and elliptic), with those axioms of congruence that involve the concept of the angle dropped, and `triangle inequality', regarded as an axiom, added. If one assumes the continuity axiom in addition, then, in the case of the Euclidean plane, we come to the problem posed by Jean Gaston Darboux: "To determine all the calculus of variation problems in the plane whose solutions are all the plane straight lines.
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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 given by Ukrainian mathematician Aleksei Pogorelov in 1973. In 1976, Armenian mathematician Rouben V. Ambartzumian proposed another proof of Hilbert's fourth problem.
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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 inconsistencies. As a solution, Hilbert proposed to ground all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent. Hilbert proposed that the consistency of more complicated systems, such as real analysis, could be proven in terms of simpler systems.
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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 with a computable set of axioms which is capable of expressing arithmetic can never be complete: it is possible to construct a statement that can be shown to be true, but that cannot be derived from the formal rules of the system. In his second theorem, he showed that such a system could not prove its own consistency, so it certainly cannot be used to prove the consistency of anything stronger with certainty. This refuted Hilbert's assumption that a finitistic system could be used to prove the consistency of itself, and therefore could not prove everything else.
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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 the ones given in Hilbert (1900), which include a second order completeness axiom. In the 1930s, Kurt Gödel and Gerhard Gentzen proved results that cast new light on the problem. Some feel that Gödel's theorems give a negative solution to the problem, while others consider Gentzen's proof as a partial positive solution.
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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 modern algebraic geometry. The two other theorems are Hilbert's basis theorem that asserts that all ideals of polynomial rings over a field are finitely generated, and Hilbert's Nullstellensatz, which establishes a bijective correspondence between affine algebraic varieties and prime ideals of polynomial rings. Hilbert's syzygy theorem concerns the relations, or syzygies in Hilbert's terminology, between the generators of an ideal, or, more generally, a module.
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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 relations, after at most n steps. Hilbert's syzygy theorem is now considered to be an early result of homological algebra. It is the starting point of the use of homological methods in commutative algebra and algebraic geometry.
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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 important ingredient in the derivation of HX preconditioners in two and three dimensions is the so-called regular decomposition, which decomposes a Sobolev space function into a component of higher regularity and a scalar or vector potential. The key to the success of HX preconditioners is the discrete version of this decomposition, which is also known as HX decomposition. The discrete decomposition decomposes a discrete Sobolev space function into a discrete component of higher regularity, a discrete scale or vector potential, and a high-frequency component.
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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 as one of the top ten breakthroughs in computational science in recent years. Researchers from Sandia, Los Alamos, and Lawrence Livermore National Labs use this algorithm for modeling fusion with magnetohydrodynamic equations. Moreover, this approach will also be instrumental in developing optimal iterative methods in structural mechanics, electrodynamics, and modeling of complex flows.
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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 field, and extended to algebras over more general rings by Henri Cartan and Samuel Eilenberg (1956).
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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 differential form that vanishes under the Laplacian operator of the metric. Such forms are called harmonic. The theory was developed by Hodge in the 1930s to study algebraic geometry, and it built on the work of Georges de Rham on de Rham cohomology.
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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 connection to the study of algebraic cycles. While Hodge theory is intrinsically dependent upon the real and complex numbers, it can be applied to questions in number theory. In arithmetic situations, the tools of p-adic Hodge theory have given alternative proofs of, or analogous results to, classical Hodge theory.
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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 Arakelov and W. V. D. Hodge.
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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 characteristic 0 is naturally isomorphic (via restriction) to the d2-dimensional space of functions on the d-torsion points. It is called a 'comparison theorem' as it is an analogue for Arakelov theory of comparison theorems in cohomology relating de Rham cohomology to singular cohomology of complex varieties or étale cohomology of p-adic varieties. In Mochizuki (1999) and Mochizuki (2002a) he pointed out that arithmetic Kodaira–Spencer map and Gauss–Manin connection may give some important hints for Vojta's conjecture, ABC conjecture and so on; in 2012, he published his Inter-universal Teichmuller theory, in which he didn't use Hodge-Arakelov theory but used the theory of frobenioids, anabelioids and mono-anabelian geometry.
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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 the Euler's number. The first few terms of this sequence are 1 , 2 , 1 , 2 , 1 , 2 , 1 , 2 , 1 , 2 , 1 , 3 , 1 , 2 , 2 , 2 , 1 , 2 , 1 , 3 , 2 , 2 , 1 , 4 {\displaystyle 1,2,1,2,1,2,1,2,1,2,1,3,1,2,2,2,1,2,1,3,2,2,1,4} (sequence A226898 in the OEIS).
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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.