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Capelli Identity
In mathematics, Capelli's identity, named after , is an analogue of the formula det(''AB'') = det(''A'') det(''B''), for certain matrices with noncommuting entries, related to the representation theory of the Lie algebra \mathfrak_n. It can be used to relate an invariant ''ƒ'' to the invariant Ω''ƒ'', where Ω is Cayley's Ω process. Statement Suppose that ''x''''ij'' for ''i'',''j'' = 1,...,''n'' are commuting variables. Write ''E''ij for the polarization operator :E_ = \sum_^n x_\frac. The Capelli identity states that the following differential operators, expressed as determinants, are equal: : \begin E_+n-1 & \cdots &E_& E_ \\ \vdots& \ddots & \vdots&\vdots\\ E_ & \cdots & E_+1&E_ \\ E_ & \cdots & E_& E_ +0\end = \begin x_ & \cdots & x_ \\ \vdots& \ddots & \vdots\\ x_ & \cdots & x_ \end \begin \frac & \cdots &\frac \\ \vdots& \ddots & \vdots\\ \frac & \cdots &\frac \end. Both sides are differential operators. The determinant on the left has non-com ... [...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] |
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 ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duke Mathematical Journal
''Duke Mathematical Journal'' is a peer-reviewed mathematics journal published by Duke University Press. It was established in 1935. The founding editors-in-chief were David Widder, Arthur Coble, and Joseph Miller Thomas Joseph Miller Thomas (16 January 1898 – 1979) was an American mathematician, known for the Thomas decomposition of algebraic and differential systems. Thomas received his Ph.D., supervised by Frederick Wahn Beal, from the University of Pennsylva .... The first issue included a paper by Solomon Lefschetz. Leonard Carlitz served on the editorial board for 35 years, from 1938 to 1973. The current managing editor is Richard Hain (Duke University). Impact According to the journal homepage, the journal has a 2018 impact factor of 2.194, ranking it in the top ten mathematics journals in the world. References External links * Mathematics journals Duke University, Mathematical Journal Publications established in 1935 Multilingual journals English-language jo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Group
In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras), compact matrix quantum groups (which are structures on unital separable C*-algebras), and bicrossproduct quantum groups. Despite their name, they do not themselves have a natural group structure, though they are in some sense 'close' to a group. The term "quantum group" first appeared in the theory of quantum integrable systems, which was then formalized by Vladimir Drinfeld and Michio Jimbo as a particular class of Hopf algebra. The same term is also used for other Hopf algebras that deform or are close to classical Lie groups or Lie algebras, such as a "bicrossproduct" class of quantum groups introduced by Shahn Majid a little after the work of Drinfeld and Jimbo. In Drinfeld's approach, quantum groups arise as Hopf algebras depe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Superalgebra
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2 grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the ''even'' elements of the superalgebra correspond to bosons and ''odd'' elements to fermions (but this is not always true; for example, the BRST supersymmetry is the other way around). Definition Formally, a Lie superalgebra is a nonassociative Z2-graded algebra, or ''superalgebra'', over a commutative ring (typically R or C) whose product , · called the Lie superbracket or supercommutator, satisfies the two conditions (analogs of the usual Lie algebra axioms, with grading): Super skew-symmetry: : ,y-(-1)^ ,x\ The super Jacobi identity: :(-1)^ ,_z.html"_;"title=",_[y,_z">,_[y,_z_+_(-1)^ ,_[y,_z_+_(-1)^[y,_[z,_x.html"_;"title=",_z.html"_;"title=",_[y,_z">,_[y,_z_+_(-1)^[y,_[z,_x">,_z.html"_;"title=",_[y,_z">,_[y,_z_+_(-1)^[y,_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Lie Group
In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symmetric spaces. Together with the commutative Lie group of the real numbers, \mathbb, and that of the unit-magnitude complex numbers, U(1) (the unit circle), simple Lie groups give the atomic "blocks" that make up all (finite-dimensional) connected Lie groups via the operation of group extension. Many commonly encountered Lie groups are either simple or 'close' to being simple: for example, the so-called "special linear group" SL(''n'') of ''n'' by ''n'' matrices with determinant equal to 1 is simple for all ''n'' > 1. The first classification of simple Lie groups was by Wilhelm Killing, and this work was later perfected by Élie Cartan. The final classification is often referred to as Killing-Cartan classification. Definition Unfortun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proc
Proc may refer to: * Proč, a village in eastern Slovakia * '' Proč?'', a 1987 Czech film * procfs or proc filesystem, a special file system (typically mounted to ) in Unix-like operating systems for accessing process information * Protein C (PROC) * Proc, a term in video game terminology * Procedures or process, in the programming language ALGOL 68 * People's Republic of China, the formal name of China China, officially the People's Republic of China (PRC), is a country in East Asia. It is the world's most populous country, with a population exceeding 1.4 billion, slightly ahead of India. China spans the equivalent of five time zones and ... * the official acronym for the Canadian House of Commons Standing Committee on Procedure and House Affairs {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herbert Westren Turnbull
Prof Herbert Westren Turnbull FRS FRSE LLD (31 August 1885 – 4 May 1961) was an English mathematician. From 1921 to 1950 he was Regius Professor of Mathematics at the University of St Andrews. Life He was born in the Tettenhall district, on the outskirts of Wolverhampton on 31 August 1885, the eldest of five sons of William Peveril Turnbull, HM Inspector of Schools. He was educated at Sheffield Grammar School then studied Mathematics at Cambridge University graduating MA. After serving as lecturer at St. Catharine's College, Cambridge (1909), the University of Liverpool (1910), and the University of Hong Kong (1912), Turnbull became master at St. Stephen's College in Hong Kong (1911–15), and warden of the University Hostel (1913–15). He was a Fellow at St John's College, Oxford (1919–26), and from 1921 held a chair of mathematics at the University of St Andrews. In 1922 he was elected a Fellow of the Royal Society of Edinburgh. His proposers were Arthur Crichton Mitc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zeilberger
Zeilberger ( he, ציילברגר) may refer to: * Doron Zeilberger (born 1950), an Israeli mathematician ** Wilf–Zeilberger pair In mathematics, specifically combinatorics, a Wilf–Zeilberger pair, or WZ pair, is a pair of functions that can be used to certify certain combinatorial identities. ... ** Zeilberger-Bressoud theorem * Johann Zeilberger (1831–1881), Austrian politician * Rabbi Binyamin Zeilberger {{surname, Zeilberger, Tsaylberger German-language surnames Jewish surnames Yiddish-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alan Sokal
Alan David Sokal (; born January 24, 1955) is an American professor of mathematics at University College London and professor emeritus of physics at New York University. He works in statistical mechanics and combinatorics. He is a critic of postmodernism, and caused the Sokal affair in 1996 when his deliberately nonsensical paper was published by Duke University Press's ''Social Text''. He also co-authored a paper criticizing the critical positivity ratio concept in positive psychology. Academic career Sokal received his BA from Harvard College in 1976 and his PhD from Princeton University in 1981. He was advised by Arthur Wightman. In the summers of 1986, 1987, and 1988, Sokal taught mathematics at the National Autonomous University of Nicaragua, when the Sandinistas were heading the elected government. Research interests Sokal's research lies in mathematical physics and combinatorics. In particular, he studies the interplay between these fields based on questions arising in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrei Okounkov
Andrei Yuryevich Okounkov (russian: Андре́й Ю́рьевич Окунько́в, ''Andrej Okun'kov'') (born July 26, 1969) is a Russian mathematician who works on representation theory and its applications to algebraic geometry, mathematical physics, probability theory and special functions. He is currently a professor at the University of California, Berkeley and the academic supervisor of HSE International Laboratory of Representation Theory and Mathematical Physics. In 2006, he received the Fields Medal "for his contributions to bridging probability, representation theory and algebraic geometry.""Information about Andrei Okounkov, Fields Medal winner" ICM Press Release Education and career He received his doctorate at |
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award honours the Canadian mathematician John Charles Fields. The Fields Medal is regarded as one of the highest honors a mathematician can receive, and has been described as the Nobel Prize of Mathematics, although there are several major differences, including frequency of award, number of awards, age limits, monetary value, and award criteria. According to the annual Academic Excellence Survey by ARWU, the Fields Medal is consistently regarded as the top award in the field of mathematics worldwide, and in another reputation survey conducted by IREG in 2013–14, the Fields Medal came closely after the Abel Prize as the second most prestigious international award in mathematics. The prize includes a monetary award which, since 2006, has bee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
