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Selberg's Integral
In mathematics, the Selberg integral is a generalization of Euler beta function to ''n'' dimensions introduced by Atle Selberg. Selberg's integral formula When Re(\alpha) > 0, Re(\beta) > 0, Re(\gamma) > -\min \left(\frac 1n , \frac, \frac\right), we have : \begin S_ (\alpha, \beta, \gamma) & = \int_0^1 \cdots \int_0^1 \prod_^n t_i^(1-t_i)^ \prod_ , t_i - t_j , ^\,dt_1 \cdots dt_n \\ & = \prod_^ \frac \end Selberg's formula implies Dixon's identity In mathematics, Dixon's identity (or Dixon's theorem or Dixon's formula) is any of several different but closely related identities proved by A. C. Dixon, some involving finite sums of products of three binomial coefficients, and some evaluating ... for well poised hypergeometric series, and some special cases of Dyson's conjecture. This is a corollary of Aomoto. Aomoto's integral formula Aomoto proved a slightly more general integral formula. With the same conditions as Selberg's formula, : \int_0^1 \cdots \int_0^1 \left( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^(1-t)^\,dt for complex number inputs z_1, z_2 such that \operatorname(z_1), \operatorname(z_2)>0. The beta function was studied by Leonhard Euler and Adrien-Marie Legendre and was given its name by Jacques Binet; its symbol is a Greek capital beta. Properties The beta function is symmetric, meaning that \Beta(z_1,z_2) = \Beta(z_2,z_1) for all inputs z_1 and z_2.. Specifically, see 6.2 Beta Function. A key property of the beta function is its close relationship to the gamma function: : \Beta(z_1,z_2)=\frac A proof is given below in . The beta function is also closely related to binomial coefficients. When (or , by symmetry) is a positive integer, it follows from the definition of the gamma function that : \Beta(m,n) =\fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atle Selberg
Atle Selberg (14 June 1917 – 6 August 2007) was a Norwegian mathematician known for his work in analytic number theory and the theory of automorphic forms, and in particular for bringing them into relation with spectral theory. He was awarded the Fields Medal in 1950 and an honorary Abel Prize in 2002. Early years Selberg was born in Langesund, Norway, the son of teacher Anna Kristina Selberg and mathematician Ole Michael Ludvigsen Selberg. Two of his three brothers, Sigmund Selberg, Sigmund and Henrik Selberg, Henrik, were also mathematicians. His other brother, Arne Selberg, Arne, was a professor of engineering. While he was still at school he was influenced by the work of Srinivasa Ramanujan and he found an exact analytical formula for the Partition function (number theory), partition function as suggested by the works of Ramanujan; however, this result was first published by Hans Rademacher. He studied at the University of Oslo and completed his PhD, doctorate in 1943 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dixon's Identity
In mathematics, Dixon's identity (or Dixon's theorem or Dixon's formula) is any of several different but closely related identities proved by A. C. Dixon, some involving finite sums of products of three binomial coefficients, and some evaluating a hypergeometric sum. These identities famously follow from the MacMahon Master theorem, and can now be routinely proved by computer algorithms . Statements The original identity, from , is :\sum_^(-1)^^3 =\frac. A generalization, also sometimes called Dixon's identity, is :\sum_(-1)^k = \frac where ''a'', ''b'', and ''c'' are non-negative integers . The sum on the left can be written as the terminating well-poised hypergeometric series :_3F_2(-2a,-a-b,-a-c;1+b-a,1+c-a;1) and the identity follows as a limiting case (as ''a'' tends to an integer) of Dixon's theorem evaluating a well-poised 3''F''2 generalized hypergeometric series at 1, from : :\;_3F_2 (a,b,c;1+a-b,1+a-c;1)= \frac . This holds for Re(1 + ''a'' − ''b'' &m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dyson's Conjecture
In mathematics, the Dyson conjecture is a conjecture about the constant term of certain Laurent polynomials, proved independently in 1962 by Wilson and Gunson. Andrews generalized it to the q-Dyson conjecture, proved by Zeilberger and Bressoud and sometimes called the Zeilberger–Bressoud theorem. Macdonald generalized it further to more general root systems with the Macdonald constant term conjecture, proved by Cherednik. Dyson conjecture The Dyson conjecture states that the Laurent polynomial :\prod_(1-t_i/t_j)^ has constant term :\frac. The conjecture was first proved independently by and . later found a short proof, by observing that the Laurent polynomials, and therefore their constant terms, satisfy the recursion relations :F(a_1,\dots,a_n) = \sum_^nF(a_1,\dots,a_i-1,\dots,a_n). The case ''n'' = 3 of Dyson's conjecture follows from the Dixon identity. and used a computer to find expressions for non-constant coefficients of Dyson's Laurent poly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Mathematical Physics
The ''Journal of Mathematical Physics'' is a peer-reviewed journal published monthly by the American Institute of Physics devoted to the publication of papers in mathematical physics. The journal was first published bimonthly beginning in January 1960; it became a monthly publication in 1963. The current editor is Jan Philip Solovej from University of Copenhagen The University of Copenhagen (, KU) is a public university, public research university in Copenhagen, Copenhagen, Denmark. Founded in 1479, the University of Copenhagen is the second-oldest university in Scandinavia, after Uppsala University. .... Its 2018 Impact Factor is 1.355 Abstracting and indexing This journal is indexed by the following services: 2013. References External ...
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Root System
In mathematics, a root system is a configuration of vector space, vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups (and some analogues such as algebraic groups) and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory (such as singularity theory). Finally, root systems are important for their own sake, as in spectral graph theory. Definitions and examples As a first example, consider the six vectors in 2-dimensional Euclidean space, R2, as shown in the image at the right; call them roots. These vectors Li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
