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Hypergeometric Function Of A Matrix Argument
In mathematics, the hypergeometric function of a matrix argument is a generalization of the classical hypergeometric series. It is a function defined by an infinite summation which can be used to evaluate certain multivariate integrals. Hypergeometric functions of a matrix argument have applications in random matrix theory. For example, the distributions of the extreme eigenvalues of random matrices are often expressed in terms of the hypergeometric function of a matrix argument. Definition Let p\ge 0 and q\ge 0 be integers, and let X be an m\times m complex symmetric matrix. Then the hypergeometric function of a matrix argument X and parameter \alpha>0 is defined as : _pF_q^(a_1,\ldots,a_p; b_1,\ldots,b_q;X) = \sum_^\infty\sum_ \frac\cdot \frac \cdot C_\kappa^(X), where \kappa\vdash k means \kappa is a partition of k, (a_i)^_ is the generalized Pochhammer symbol, and C_\kappa^(X) is the "C" normalization of the Jack function. Two matrix arguments If X and Y are two m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Matrix Theory
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables. Many important properties of physical systems can be represented mathematically as matrix problems. For example, the thermal conductivity of a lattice can be computed from the dynamical matrix of the particle-particle interactions within the lattice. Applications Physics In nuclear physics, random matrices were introduced by Eugene Wigner to model the nuclei of heavy atoms. Wigner postulated that the spacings between the lines in the spectrum of a heavy atom nucleus should resemble the spacings between the eigenvalues of a random matrix, and should depend only on the symmetry class of the underlying evolution. In solid-state physics, random matrices model the behaviour of large disordered Hamiltonians in the mean-field approximation. In quantum chaos, the Bohigas–Giannoni–Schmit (BGS) conjecture asser ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition (number Theory)
In number theory and combinatorics, a partition of a positive integer , also called an integer partition, is a way of writing as a summation, sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition (combinatorics), composition.) For example, can be partitioned in five distinct ways: : : : : : The order-dependent composition is the same partition as , and the two distinct compositions and represent the same partition as . A summand in a partition is also called a part. The number of partitions of is given by the partition function . So . The notation means that is a partition of . Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials and of the symmetric group and in group representation, group representation theory in genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Pochhammer Symbol
In mathematics, the generalized Pochhammer symbol of parameter \alpha>0 and partition \kappa=(\kappa_1,\kappa_2,\ldots,\kappa_m) generalizes the classical Pochhammer symbol, named after Leo August Pochhammer, and is defined as :(a)^_\kappa=\prod_^m \prod_^ \left(a-\frac+j-1\right). It is used in multivariate analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 .... References * Gamma and related functions Factorial and binomial topics {{numtheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jack Function
In mathematics, the Jack function is a generalization of the Jack polynomial, introduced by Henry Jack. The Jack polynomial is a homogeneous, symmetric polynomial which generalizes the Schur and zonal polynomials, and is in turn generalized by the Heckman–Opdam polynomials and Macdonald polynomials. Definition The Jack function J_\kappa^(x_1,x_2,\ldots,x_m) of an integer partition \kappa, parameter \alpha, and arguments x_1,x_2,\ldots,x_m can be recursively defined as follows: ; For ''m''=1 : : J_^(x_1)=x_1^k(1+\alpha)\cdots (1+(k-1)\alpha) ; For ''m''>1: : J_\kappa^(x_1,x_2,\ldots,x_m)=\sum_\mu J_\mu^(x_1,x_2,\ldots,x_) x_m^\beta_, where the summation is over all partitions \mu such that the skew partition \kappa/\mu is a horizontal strip, namely : \kappa_1\ge\mu_1\ge\kappa_2\ge\mu_2\ge\cdots\ge\kappa_\ge\mu_\ge\kappa_n (\mu_n must be zero or otherwise J_\mu(x_1,\ldots,x_)=0) and : \beta_=\frac, where B_^\nu(i,j) equals \kappa_j'-i+\alpha(\kappa_i-j+1) if \kappa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Exponential
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group. Let be an real or complex matrix. The exponential of , denoted by or , is the matrix given by the power series e^X = \sum_^\infty \frac X^k where X^0 is defined to be the identity matrix I with the same dimensions as X. The above series always converges, so the exponential of is well-defined. If is a 1×1 matrix the matrix exponential of is a 1×1 matrix whose single element is the ordinary exponential of the single element of . Properties Elementary properties Let and be complex matrices and let and be arbitrary complex numbers. We denote the identity matrix by and the zero matrix by 0. The matrix exponential satisfies the foll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
