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Zonal Polynomial
In mathematics, a zonal polynomial is a multivariate symmetric homogeneous polynomial. The zonal polynomials form a basis of the space of symmetric polynomials. They appear as zonal spherical functions of the Gelfand pairs (S_,H_n) (here, H_n is the hyperoctahedral group) and (Gl_n(\mathbb), O_n), which means that they describe canonical basis of the double class algebras \mathbb _n \backslash S_ / H_n/math> and \mathbb _d(\mathbb)\backslash M_d(\mathbb)/O_d(\mathbb)/math>. They are applied in multivariate statistics. The zonal polynomials are the \alpha=2 case of the C normalization of the 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 .... References * Robb Muirhead, ''Aspects of Multivariate Statistical Theory'', John Wiley & Sons, Inc., New York, 1984. {{algebr ... [...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 points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Polynomial
In mathematics, a symmetric polynomial is a polynomial in variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, is a ''symmetric polynomial'' if for any permutation of the subscripts one has . Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients, since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting. From this point of view the elementary symmetric polynomials are the most fundamental symmetric polynomials. A theorem states that any symmetric polynomial can be expressed in terms of elementary symmetric polynomials, which implies that every ''symmetric'' polynomial expression in the roots of a monic polynomial can alternatively be given as a polynomial expression in the coefficients of the polynomial. Symmetric polynomials also form an interesting structure by themselve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial x^3 + 3 x^2 y + z^7 is not homogeneous, because the sum of exponents does not match from term to term. The function defined by a homogeneous polynomial is always a homogeneous function. An algebraic form, or simply form, is a function defined by a homogeneous polynomial. A binary form is a form in two variables. A ''form'' is also a function defined on a vector space, which may be expressed as a homogeneous function of the coordinates over any basis. A polynomial of degree 0 is always homogeneous; it is simply an element of the field or ring of the coefficients, usually called a constant or a scalar. A form of degree 1 is a linear form. A form of degree 2 is a quadratic fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis (algebra)
In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to . The elements of a basis are called . Equivalently, a set is a basis if its elements are linearly independent and every element of is a linear combination of elements of . In other words, a basis is a linearly independent spanning set. A vector space can have several bases; however all the bases have the same number of elements, called the ''dimension'' of the vector space. This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. Definition A basis of a vector space over a field (such as the real numbers or the complex numbers ) is a linearly independent subset of that spans . This ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zonal Spherical Function
In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group ''G'' with compact subgroup ''K'' (often a maximal compact subgroup) that arises as the matrix coefficient of a ''K''-invariant vector in an irreducible representation of ''G''. The key examples are the matrix coefficients of the '' spherical principal series'', the irreducible representations appearing in the decomposition of the unitary representation of ''G'' on ''L''2(''G''/''K''). In this case the commutant of ''G'' is generated by the algebra of biinvariant functions on ''G'' with respect to ''K'' acting by right convolution. It is commutative if in addition ''G''/''K'' is a symmetric space, for example when ''G'' is a connected semisimple Lie group with finite centre and ''K'' is a maximal compact subgroup. The matrix coefficients of the spherical principal series describe precisely the spectrum of the corresponding C* algebra generated by the biinvariant funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gelfand Pair
In mathematics, a Gelfand pair is a pair ''(G,K)'' consisting of a Group (mathematics), group ''G'' and a subgroup ''K'' (called an Euler subgroup of ''G'') that satisfies a certain property on restricted representations. The theory of Gelfand pairs is closely related to the topic of Zonal spherical function, spherical functions in the classical theory of special functions, and to the theory of Riemannian symmetric spaces in differential geometry. Broadly speaking, the theory exists to abstract from these theories their content in terms of harmonic analysis and representation theory. When ''G'' is a finite group the simplest definition is, roughly speaking, that the ''(K,K)''-double cosets in ''G'' commute. More precisely, the Hecke algebra of a finite group, Hecke algebra, the algebra of functions on ''G'' that are invariant under translation on either side by ''K'', should be commutative for the convolution on ''G''. In general, the definition of Gelfand pair is roughly that the r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multivariate Statistics
Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable. Multivariate statistics concerns understanding the different aims and background of each of the different forms of multivariate analysis, and how they relate to each other. The practical application of multivariate statistics to a particular problem may involve several types of univariate and multivariate analyses in order to understand the relationships between variables and their relevance to the problem being studied. In addition, multivariate statistics is concerned with multivariate probability distributions, in terms of both :*how these can be used to represent the distributions of observed data; :*how they can be used as part of statistical inference, particularly where several different quantities are of interest to the same analysis. Certain types of problems involving multivariate data, for example simple linear regression an ... [...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] |
Homogeneous Polynomials
In mathematics, a homogeneous polynomial, sometimes called wikt:quantic, quantic in older texts, is a polynomial whose nonzero terms all have the same Degree of a polynomial, degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial x^3 + 3 x^2 y + z^7 is not homogeneous, because the sum of exponents does not match from term to term. The function defined by a homogeneous polynomial is always a homogeneous function. An algebraic form, or simply form, is a function (mathematics), function defined by a homogeneous polynomial. A binary form is a form in two variables. A ''form'' is also a function defined on a vector space, which may be expressed as a homogeneous function of the coordinates over any basis (linear algebra), basis. A polynomial of degree 0 is always homogeneous; it is simply an element of the field (mathematics), field or ring (mathematics), ring of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
