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List Of Things Named After David Hilbert
David Hilbert (1862–1943), a mathematician, is the eponym of all of the things (and topics) listed below. Mathematics and physics *Brouwer–Hilbert controversy *Einstein–Hilbert action * Einstein–Hilbert equations * Hilbert algebra * Hilbert C*-module *Hilbert basis (linear programming) *Hilbert class field *Hilbert cube *Hilbert curve **Hilbert curve scheduling * Hilbert field *Hilbert function *Hilbert manifold *Hilbert matrix *Hilbert metric *Hilbert modular form *Hilbert modular variety *Hilbert–Mumford criterion *Hilbert number * Hilbert plane *Hilbert polynomial **Hilbert series *Hilbert ring *Hilbert–Samuel function *Hilbert projection theorem *Hilbert R-tree * Hilbert reciprocity *Hilbert scheme *Hilbert space ** Hilbert dimension **Projective Hilbert space **Reproducing kernel Hilbert space **Rigged Hilbert space ** Semi-Hilbert space *Hilbert spectrum *Hilbert symbol *Hilbert system *Hilbert transform **Hilbert spectroscopy **Hilbert–Huang transform ** Hilber ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly proof theory). Hilbert adopted and defended Georg Cantor's set theory and transfinite numbers. In 1900, he presented a collection of problems that set the course for much of the mathematical research of the 20th century. Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic. Life Early life and edu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Modular Variety
In mathematics, a Hilbert modular surface or Hilbert–Blumenthal surface is an algebraic surface obtained by taking a quotient of a product of two copies of the upper half-plane by a Hilbert modular group. More generally, a Hilbert modular variety is an algebraic variety obtained by taking a quotient of a product of multiple copies of the upper half-plane by a Hilbert modular group. Hilbert modular surfaces were first described by using some unpublished notes written by David Hilbert about 10 years before. Definitions If ''R'' is the ring of integers of a real quadratic field, then the Hilbert modular group SL2(''R'') acts on the product ''H''×''H'' of two copies of the upper half plane ''H''. There are several birationally equivalent surfaces related to this action, any of which may be called Hilbert modular surfaces: *The surface ''X'' is the quotient of ''H''×''H'' by SL2(''R''); it is not compact and usually has quotient singularities coming from points with no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus 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 defines 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. 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 under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus 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 defines 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. 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 under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Scheme
In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbert polynomials. The basic theory of Hilbert schemes was developed by . Hironaka's example shows that non-projective varieties need not have Hilbert schemes. Hilbert scheme of projective space The Hilbert scheme \mathbf(n) of \mathbb^n classifies closed subschemes of projective space in the following sense: For any locally Noetherian scheme , the set of -valued points :\operatorname(S, \mathbf(n)) of the Hilbert scheme is naturally isomorphic to the set of closed subschemes of \mathbb^n \times S that are flat over . The closed subschemes of \mathbb^n \times S that are flat over can informally be thought of as the families of subschemes of projective space parameterized by . Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reciprocity Law
In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials f(x) with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an irreducible polynomial f(x) = x^2 + ax + b splits into linear terms when reduced mod p. That is, it determines for which prime numbers the relationf(x) \equiv f_p(x) = (x-n_p)(x-m_p) \text (\text p)holds. For a general reciprocity lawpg 3, it is defined as the rule determining which primes p the polynomial f_p splits into linear factors, denoted \text\. There are several different ways to express reciprocity laws. The early reciprocity laws found in the 19th century were usually expressed in terms of a power residue symbol (''p''/''q'') generalizing the quadratic reciprocity symbol, that describes when a prime number is an ''n''th power residue modulo another prime, and gave a relation between (''p''/''q'') and (''q''/''p''). Hilbert refo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert R-tree
Hilbert R-tree, an R-tree variant, is an index for multidimensional objects such as lines, regions, 3-D objects, or high-dimensional feature-based parametric objects. It can be thought of as an extension to B+-tree for multidimensional objects. The performance of R-trees depends on the quality of the algorithm that clusters the data rectangles on a node. Hilbert R-trees use space-filling curves, and specifically the Hilbert curve, to impose a linear ordering on the data rectangles. There are two types of Hilbert R-trees: one for static databases, and one for dynamic databases. In both cases Hilbert space-filling curves are used to achieve better ordering of multidimensional objects in the node. This ordering has to be "good", in the sense that it should group "similar" data rectangles together, to minimize the area and perimeter of the resulting minimum bounding rectangles (MBRs). Packed Hilbert R-trees are suitable for static databases in which updates are very rare or in which th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Projection Theorem
In mathematics, the Hilbert projection theorem is a famous result of convex analysis that says that for every vector x in a Hilbert space H and every nonempty closed convex C \subseteq H, there exists a unique vector m \in C for which \, c - x\, is minimized over the vectors c \in C; that is, such that \, m - x\, \leq \, c - x\, for every c \in C. Finite dimensional case Some intuition for the theorem can be obtained by considering the first order condition In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information abou ... of the optimization problem. Consider a finite dimensional real Hilbert space H with a subspace C and a point x. If m \in C is a or of the function N : C \to \R defined by N(c) := \, c - x\, (which is the same as the minimum point of c \mapsto \, c - x\, ^2), then deriv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert–Samuel Function
In commutative algebra the Hilbert–Samuel function, named after David Hilbert and Pierre Samuel,H. Hironaka, Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I. Ann. of Math. 2nd Ser., Vol. 79, No. 1. (Jan., 1964), pp. 109-203. of a nonzero finitely generated module M over a commutative Noetherian local ring A and a primary ideal I of A is the map \chi_^:\mathbb\rightarrow\mathbb such that, for all n\in\mathbb, :\chi_^(n)=\ell(M/I^M) where \ell denotes the length over A. It is related to the Hilbert function of the associated graded module \operatorname_I(M) by the identity : \chi_M^I (n)=\sum_^n H(\operatorname_I(M),i). For sufficiently large n, it coincides with a polynomial function of degree equal to \dim(\operatorname_I(M)), often called the Hilbert-Samuel polynomial (or Hilbert polynomial).Atiyah, M. F. and MacDonald, I. G. ''Introduction to Commutative Algebra''. Reading, MA: Addison–Wesley, 1969. Examples For the ring of form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Ring
In algebra, a Hilbert ring or a Jacobson ring is a ring such that every prime ideal is an intersection of primitive ideals. For commutative rings primitive ideals are the same as maximal ideals so in this case a Jacobson ring is one in which every prime ideal is an intersection of maximal ideals. Jacobson rings were introduced independently by , who named them after Nathan Jacobson because of their relation to Jacobson radicals, and by , who named them Hilbert rings after David Hilbert because of their relation to Hilbert's Nullstellensatz. Jacobson rings and the Nullstellensatz Hilbert's Nullstellensatz of algebraic geometry is a special case of the statement that the polynomial ring in finitely many variables over a field is a Hilbert ring. A general form of the Nullstellensatz states that if ''R'' is a Jacobson ring, then so is any finitely generated ''R''-algebra ''S''. Moreover, the pullback of any maximal ideal ''J'' of ''S'' is a maximal ideal ''I'' of ''R'', and ''S/J'' i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Series
In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homogeneous components of the algebra. These notions have been extended to filtered algebras, and graded or filtered modules over these algebras, as well as to coherent sheaves over projective schemes. The typical situations where these notions are used are the following: * The quotient by a homogeneous ideal of a multivariate polynomial ring, graded by the total degree. * The quotient by an ideal of a multivariate polynomial ring, filtered by the total degree. * The filtration of a local ring by the powers of its maximal ideal. In this case the Hilbert polynomial is called the Hilbert–Samuel polynomial. The Hilbert series of an algebra or a module is a special case of the Hilbert–Poincaré series of a graded vector space. The Hilbert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Polynomial
In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homogeneous components of the algebra. These notions have been extended to filtered algebras, and graded or filtered modules over these algebras, as well as to coherent sheaves over projective schemes. The typical situations where these notions are used are the following: * The quotient by a homogeneous ideal of a multivariate polynomial ring, graded by the total degree. * The quotient by an ideal of a multivariate polynomial ring, filtered by the total degree. * The filtration of a local ring by the powers of its maximal ideal. In this case the Hilbert polynomial is called the Hilbert–Samuel polynomial. The Hilbert series of an algebra or a module is a special case of the Hilbert–Poincaré series of a graded vector space. The Hilbert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
