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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] |
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] |
Integer Partition
In number theory and combinatorics, a partition of a positive integer , also called an integer partition, is a way of writing as a 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.) 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 theory in general. Examples The seven partitions of 5 are: * 5 * 4 + 1 * 3 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonality, orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials. The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials, and the Legendre polynomials as special cases. The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by Pafnuty Chebyshev, P. L. Chebyshev and was pursued by Andrey Markov, A. A. Markov and Thomas Joannes Stieltjes, T. J. Stieltjes. They appear in a wide variety of fields: numerical analysis (Gaussian quadrature, quadrature rules), probability theory, representation theory (of Lie group, Lie groups, quantum group, quantum groups, and re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * |
Mathematics Of Computation
''Mathematics of Computation'' is a bimonthly mathematics journal focused on computational mathematics. It was established in 1943 as ''Mathematical Tables and other Aids to Computation'', obtaining its current name in 1960. Articles older than five years are available electronically free of charge. Abstracting and indexing The journal is abstracted and indexed in Mathematical Reviews, Zentralblatt MATH, Science Citation Index, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... of 2.417. References External links * Delayed open access journals English-language journals Mathematics journals Publications ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Young Tableau
In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties. Young tableaux were introduced by Alfred Young, a mathematician at Cambridge University, in 1900. They were then applied to the study of the symmetric group by Georg Frobenius in 1903. Their theory was further developed by many mathematicians, including Percy MacMahon, W. V. D. Hodge, G. de B. Robinson, Gian-Carlo Rota, Alain Lascoux, Marcel-Paul Schützenberger and Richard P. Stanley. Definitions ''Note: this article uses the English convention for displaying Young diagrams and tableaux''. Diagrams A Young diagram (also called a Ferrers diagram, particularly when represented using dots) is a finite collection of boxes, or cells, arranged in left-justified rows, with the row lengths in non-increasing o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Macdonald Polynomials
In mathematics, Macdonald polynomials ''P''λ(''x''; ''t'',''q'') are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987. He later introduced a non-symmetric generalization in 1995. Macdonald originally associated his polynomials with weights λ of finite root systems and used just one variable ''t'', but later realized that it is more natural to associate them with affine root systems rather than finite root systems, in which case the variable ''t'' can be replaced by several different variables ''t''=(''t''1,...,''t''''k''), one for each of the ''k'' orbits of roots in the affine root system. The Macdonald polynomials are polynomials in ''n'' variables ''x''=(''x''1,...,''x''''n''), where ''n'' is the rank of the affine root system. They generalize many other families of orthogonal polynomials, such as Jack polynomials and Hall–Littlewood polynomials and Askey–Wilson polynomials, which in turn include most of the named 1-va ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Young Diagram
In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties. Young tableaux were introduced by Alfred Young, a mathematician at Cambridge University, in 1900. They were then applied to the study of the symmetric group by Georg Frobenius in 1903. Their theory was further developed by many mathematicians, including Percy MacMahon, W. V. D. Hodge, G. de B. Robinson, Gian-Carlo Rota, Alain Lascoux, Marcel-Paul Schützenberger and Richard P. Stanley. Definitions ''Note: this article uses the English convention for displaying Young diagrams and tableaux''. Diagrams A Young diagram (also called a Ferrers diagram, particularly when represented using dots) is a finite collection of boxes, or cells, arranged in left-justified rows, with the row lengths in non-increasing o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Macdonald Polynomial
In mathematics, Macdonald polynomials ''P''λ(''x''; ''t'',''q'') are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987. He later introduced a non-symmetric generalization in 1995. Macdonald originally associated his polynomials with weights λ of finite root systems and used just one variable ''t'', but later realized that it is more natural to associate them with affine root systems rather than finite root systems, in which case the variable ''t'' can be replaced by several different variables ''t''=(''t''1,...,''t''''k''), one for each of the ''k'' orbits of roots in the affine root system. The Macdonald polynomials are polynomials in ''n'' variables ''x''=(''x''1,...,''x''''n''), where ''n'' is the rank of the affine root system. They generalize many other families of orthogonal polynomials, such as Jack polynomials and Hall–Littlewood polynomials and Askey–Wilson polynomials, which in turn include most of the named 1-va ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henry Jack
Henry Jack FRSE (6 July 1917 – 5 January 1978) was a Scottish mathematician at University College Dundee. The Jack polynomials are named after him. His research dealt with the development of analytic methods to evaluate certain integrals over matrix spaces. His most famous paper relates his integrals to classes of symmetric polynomials important in the theory of the representation of the symmetric group. He discovered a new, natural basis for the symmetric polynomials. Life He was born at Menzieshill near Dundee on 6 July 1917, the son of Henry Jack, a farmer. He was christened at Benvie church. He was educated at the High School of Dundee then studied Mathematics at the University of Edinburgh (graduating MA in 1940). During the Second World War he served as a meteorologist with the RAF then resumed studies at Cambridge University, gaining a further BA degree in 1949. In 1950 he was appointed as a lecturer in mathematics at University College Dundee, then a college of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heckman–Opdam Polynomials
In mathematics, Heckman–Opdam polynomials (sometimes called Jacobi polynomials) ''P''λ(''k'') are orthogonal polynomials in several variables associated to root systems. They were introduced by . They generalize Jack polynomials when the roots system is of type ''A'', and are limits of Macdonald polynomials In mathematics, Macdonald polynomials ''P''λ(''x''; ''t'',''q'') are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987. He later introduced a non-symmetric generalization in 1995. Macdonald origi ... ''P''λ(''q'', ''t'') as ''q'' tends to 1 and (1 − ''t'')/(1 − ''q'') tends to ''k''. Main properties of the Heckman–Opdam polynomials have been detailed by Siddhartha Sahi A new formula for weight multiplicities and characters, Theorem 1.3. about Heckman–Opdam polynomials, Siddhartha Sahi References * * * * {{DEFAULTSORT:Heckman-Opdam polynomials Orthogonal polynomials ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
