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List Of Things Named After Issai Schur
This is a list of things named after Issai Schur. * Frobenius–Schur indicator * Herz–Schur multiplier *Jordan–Schur theorem *Lehmer–Schur algorithm * Schur algebra *Schur class * Schur's conjecture *Schur complement method *Schur complement *Schur-convex function *Schur decomposition *Schur functor * Schur index * Schur's inequality * Schur's lemma (from Riemannian geometry) *Schur's lemma * Schur module *Schur multiplier ** Schur cover *Schur orthogonality relations *Schur polynomial * Schur product *Schur product theorem *Schur test * Schur's property * Schur's theorem ** Schur's number *Schur–Horn theorem *Schur–Weyl duality *Schur–Zassenhaus theorem {{DEFAULTSORT:List of things named after Issai Schur Schur, Issai Issai Schur (10 January 1875 – 10 January 1941) was a Russian mathematician who worked in Germany for most of his life. He studied at the University of Berlin. He obtained his doctorate in 1901, became lecturer in 1903 and, after a stay at th ...< ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Issai Schur
Issai Schur (10 January 1875 – 10 January 1941) was a Russian mathematician who worked in Germany for most of his life. He studied at the University of Berlin. He obtained his doctorate in 1901, became lecturer in 1903 and, after a stay at the University of Bonn, professor in 1919. As a student of Ferdinand Georg Frobenius, he worked on group representations (the subject with which he is most closely associated), but also in combinatorics and number theory and even theoretical physics. He is perhaps best known today for his result on the existence of the Schur decomposition and for his work on group representations ( Schur's lemma). Schur published under the name of both I. Schur, and J. Schur, the latter especially in ''Journal für die reine und angewandte Mathematik''. This has led to some confusion. Childhood Issai Schur was born into a Jewish family, the son of the businessman Moses Schur and his wife Golde Schur (née Landau). He was born in Mogilev on the Dnieper Riv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur's Lemma
In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if ''M'' and ''N'' are two finite-dimensional irreducible representations of a group ''G'' and ''φ'' is a linear map from ''M'' to ''N'' that commutes with the action of the group, then either ''φ'' is invertible, or ''φ'' = 0. An important special case occurs when ''M'' = ''N'', i.e. ''φ'' is a self-map; in particular, any element of the center of a group must act as a scalar operator (a scalar multiple of the identity) on ''M''. The lemma is named after Issai Schur who used it to prove the Schur orthogonality relations and develop the basics of the representation theory of finite groups. Schur's lemma admits generalisations to Lie groups and Lie algebras, the most common of which are due to Jacques Dixmier and Daniel Quillen. Representation theory of groups Representation theory is the study of homomo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur–Weyl Duality
Schur–Weyl duality is a mathematical theorem in representation theory that relates irreducible finite-dimensional representations of the general linear and symmetric groups. It is named after two pioneers of representation theory of Lie groups, Issai Schur, who discovered the phenomenon, and Hermann Weyl, who popularized it in his books on quantum mechanics and classical groups as a way of classifying representations of unitary and general linear groups. Schur–Weyl duality can be proven using the double centralizer theorem. Description Schur–Weyl duality forms an archetypical situation in representation theory involving two kinds of symmetry that determine each other. Consider the tensor space : \mathbb^n\otimes\mathbb^n\otimes\cdots\otimes\mathbb^n with ''k'' factors. The symmetric group ''S''''k'' on ''k'' letters acts on this space (on the left) by permuting the factors, : \sigma(v_1\otimes v_2\otimes\cdots\otimes v_k) = v_\otimes v_\otimes\cdots\otimes v_. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur–Horn Theorem
In mathematics, particularly linear algebra, the Schur–Horn theorem, named after Issai Schur and Alfred Horn, characterizes the diagonal of a Hermitian matrix with given eigenvalues. It has inspired investigations and substantial generalizations in the setting of symplectic geometry. A few important generalizations are Kostant's convexity theorem, Atiyah–Guillemin–Sternberg convexity theorem, Kirwan convexity theorem. Statement Theorem. Let \mathbf=\_^N and \mathbf=\_^N be two sequences of real numbers arranged in a non-increasing order. There is a Hermitian matrix with diagonal values \_^N and eigenvalues \_^N if and only if : \sum_^n d_i \leq \sum_^n \lambda_i \qquad n=1,2,\ldots,N and : \sum_^N d_i= \sum_^N \lambda_i. Polyhedral geometry perspective Permutation polytope generated by a vector The permutation polytope generated by \tilde = (x_1, x_2,\ldots, x_n) \in \mathbb^n denoted by \mathcal_ is defined as the convex hull of the set \. Here S_n denotes the sy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur%27s Theorem
In discrete mathematics, Schur's theorem is any of several theorems of the mathematician Issai Schur. In differential geometry, Schur's theorem is a theorem of Axel Schur. In functional analysis, Schur's theorem is often called Schur's property, also due to Issai Schur. Ramsey theory In Ramsey theory, Schur's theorem states that for any partition of the positive integers into a finite number of parts, one of the parts contains three integers ''x'', ''y'', ''z'' with :x + y = z. For every positive integer ''c'', ''S''(''c'') denotes the smallest number ''S'' such that for every partition of the integers \ into ''c'' parts, one of the parts contains integers ''x'', ''y'', and ''z'' with x + y = z. Schur's theorem ensures that ''S''(''c'') is well-defined for every positive integer ''c''. The numbers of the form ''S''(''c'') are called Schur's number. Folkman's theorem generalizes Schur's theorem by stating that there exist arbitrarily large sets of integers, all of whose n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur's Property
In mathematics, Schur's property, named after Issai Schur, is the property of normed spaces that is satisfied precisely if weak convergence of sequences entails convergence in norm. Motivation When we are working in a normed space ''X'' and we have a sequence (x_) that converges weakly to x, then a natural question arises. Does the sequence converge in perhaps a more desirable manner? That is, does the sequence converge to x in norm? A canonical example of this property, and commonly used to illustrate the Schur property, is the \ell_1 sequence space In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural .... Definition Suppose that we have a normed space (''X'', , , ·, , ), x an arbitrary member of ''X'', and (x_) an arbitrary sequence in the space. We say that ''X'' has Schur's proper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur Test
In mathematical analysis, the Schur test, named after German mathematician Issai Schur, is a bound on the L^2\to L^2 operator norm of an integral operator in terms of its Schwartz kernel (see Schwartz kernel theorem). Here is one version. Let X,\,Y be two measurable spaces (such as \mathbb^n). Let \,T be an integral operator with the non-negative Schwartz kernel \,K(x,y), x\in X, y\in Y: :T f(x)=\int_Y K(x,y)f(y)\,dy. If there exist real functions \,p(x)>0 and \,q(y)>0 and numbers \,\alpha,\beta>0 such that : (1)\qquad \int_Y K(x,y)q(y)\,dy\le\alpha p(x) for almost all \,x and : (2)\qquad \int_X p(x)K(x,y)\,dx\le\beta q(y) for almost all \,y, then \,T extends to a continuous operator T:L^2\to L^2 with the operator norm : \Vert T\Vert_ \le\sqrt. Such functions \,p(x), \,q(y) are called the Schur test functions. In the original version, \,T is a matrix and \,\alpha=\beta=1. Common usage and Young's inequality A common usage of the Schur test is to take \,p(x)=q( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur Product Theorem
In mathematics, particularly in linear algebra, the Schur product theorem states that the Hadamard product of two positive definite matrices is also a positive definite matrix. The result is named after Issai Schur (Schur 1911, p. 14, Theorem VII) (note that Schur signed as J. Schur in ''Journal für die reine und angewandte Mathematik''.) We remark that the converse of the theorem holds in the following sense. If M is a symmetric matrix and the Hadamard product M \circ N is positive definite for all positive definite matrices N, then M itself is positive definite. Proof Proof using the trace formula For any matrices M and N, the Hadamard product M \circ N considered as a bilinear form acts on vectors a, b as : a^* (M \circ N) b = \operatorname\left(M^\textsf \operatorname\left(a^*\right) N \operatorname(b)\right) where \operatorname is the matrix trace and \operatorname(a) is the diagonal matrix having as diagonal entries the elements of a. Suppose M and N are posit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hadamard Product (matrices)
In mathematics, the Hadamard product (also known as the element-wise product, entrywise product or Schur product) is a binary operation that takes two matrices of the same dimensions and produces another matrix of the same dimension as the operands, where each element is the product of elements of the original two matrices. It is to be distinguished from the more common matrix product. It is attributed to, and named after, either French mathematician Jacques Hadamard or German Russian mathematician Issai Schur. The Hadamard product is associative and distributive. Unlike the matrix product, it is also commutative. Definition For two matrices and of the same dimension , the Hadamard product A \circ B (or A \odot B) is a matrix of the same dimension as the operands, with elements given by :(A \circ B)_ = (A \odot B)_ = (A)_ (B)_. For matrices of different dimensions ( and , where or ), the Hadamard product is undefined. Example For example, the Hadamard product for a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur Polynomial
In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in ''n'' variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In representation theory they are the characters of polynomial irreducible representations of the general linear groups. The Schur polynomials form a linear basis for the space of all symmetric polynomials. Any product of Schur polynomials can be written as a linear combination of Schur polynomials with non-negative integral coefficients; the values of these coefficients is given combinatorially by the Littlewood–Richardson rule. More generally, skew Schur polynomials are associated with pairs of partitions and have similar properties to Schur polynomials. Definition (Jacobi's bialternant formula) Schur polynomials are indexed by integer partitions. Given a partition , where , and each is a non-negative integer, the functions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur Orthogonality Relations
In mathematics, the Schur orthogonality relations, which were proven by Issai Schur through Schur's lemma, express a central fact about representations of finite groups. They admit a generalization to the case of compact groups in general, and in particular compact Lie groups, such as the rotation group SO(3). Finite groups Intrinsic statement The space of complex-valued class functions of a finite group G has a natural inner product: :\left \langle \alpha, \beta\right \rangle := \frac\sum_ \alpha(g) \overline where \overline means the complex conjugate of the value of \beta on ''g''. With respect to this inner product, the irreducible characters form an orthonormal basis for the space of class functions, and this yields the orthogonality relation for the rows of the character table: :\left \langle \chi_i, \chi_j \right \rangle = \begin 0 & \mbox i \ne j, \\ 1 & \mbox i = j. \end For g, h \in G, applying the same inner product to the columns of the character table yields: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
