|
Harish-Chandra's Ξ Function
In mathematical harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ..., Harish-Chandra's ''Ξ'' function is a special spherical function on a semisimple Lie group, studied by . Harish-Chandra used it to define Harish-Chandra's Schwartz space. gives a detailed description of the properties of ''Ξ''. Definition :\Xi(g)=\int_Ka(kg)^\rho dk, where *''K'' is a maximal compact subgroup of a semisimple Lie group with Iwasawa decomposition ''G''=''NAK'' *''g'' is an element of ''G'' *''ρ'' is a Weyl vector *''a''(''g'') is the element ''a'' in the Iwasawa decomposition ''g''=''nak'' References * * {{DEFAULTSORT:Harish-Chandra's Xi function Harmonic analysis Representation theory Special functions ... [...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] |
Abstract Harmonic Analysis
Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis). In the past two centuries, it has become a vast subject with applications in areas as diverse as number theory, representation theory, signal processing, quantum mechanics, tidal analysis and neuroscience. The term "harmonics" originated as the Ancient Greek word ''harmonikos'', meaning "skilled in music". In physical eigenvalue problems, it began to mean waves whose frequencies are integer multiples of one another, as are the frequencies of the harmonics of music notes, but the term has been generalized beyond its original meaning. The classical Fourier transform on R''n'' is still an area of ongoing research, particularly concerning Fourier transformation on more general objects such as tempered d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Function
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by consensus, and thus lacks a general formal definition, but the List of mathematical functions contains functions that are commonly accepted as special. Tables of special functions Many special functions appear as solutions of differential equations or integrals of elementary functions. Therefore, tables of integrals usually include descriptions of special functions, and tables of special functions include most important integrals; at least, the integral representation of special functions. Because symmetries of differential equations are essential to both physics and mathematics, the theory of special functions is closely related to the theory of Lie groups and Lie algebras, as well as certain topics in mathematical physics. Sy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semisimple Lie Group
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of characteristic 0. For such a Lie algebra \mathfrak g, if nonzero, the following conditions are equivalent: *\mathfrak g is semisimple; *the Killing form, κ(x,y) = tr(ad(''x'')ad(''y'')), is non-degenerate; *\mathfrak g has no non-zero abelian ideals; *\mathfrak g has no non-zero solvable ideals; * the radical (maximal solvable ideal) of \mathfrak g is zero. Significance The significance of semisimplicity comes firstly from the Levi decomposition, which states that every finite dimensional Lie algebra is the semidirect product of a solvable ideal (its radical) and a semisimple algebra. In particular, there is no nonzero Lie algebra that is both solvable and semisimple. Semisimple Lie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harish-Chandra's Schwartz Space
In mathematical abstract harmonic analysis, Harish-Chandra's Schwartz space is a space of functions on a semisimple Lie group whose derivatives are rapidly decreasing, studied by . It is an analogue of the Schwartz space on a real vector space, and is used to define the space of tempered distributions on a semisimple Lie group. Definition The definition of the Schwartz space uses Harish-Chandra's Ξ function and his ''σ'' function. The ''σ'' function is defined by : \sigma(x)=\, X\, for ''x''=''k'' exp ''X'' with ''k'' in ''K'' and ''X'' in ''p'' for a Cartan decomposition ''G'' = ''K'' exp ''p'' of the Lie group ''G'', where , , ''X'', , is a ''K''-invariant Euclidean norm on ''p'', usually chosen to be the Killing form. . The Schwartz space on ''G'' consists roughly of the functions all of whose derivatives are rapidly decreasing compared to ''Ξ''. More precisely, if ''G'' is connected then the Schwartz space consists of all smooth functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iwasawa Decomposition
In mathematics, the Iwasawa decomposition (aka KAN from its expression) of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (QR decomposition, a consequence of Gram–Schmidt orthogonalization). It is named after Kenkichi Iwasawa, the Japanese mathematician who developed this method. Definition *''G'' is a connected semisimple real Lie group. * \mathfrak_0 is the Lie algebra of ''G'' * \mathfrak is the complexification of \mathfrak_0 . *θ is a Cartan involution of \mathfrak_0 * \mathfrak_0 = \mathfrak_0 \oplus \mathfrak_0 is the corresponding Cartan decomposition * \mathfrak_0 is a maximal abelian subalgebra of \mathfrak_0 *Σ is the set of restricted roots of \mathfrak_0 , corresponding to eigenvalues of \mathfrak_0 acting on \mathfrak_0 . *Σ+ is a choice of positive roots of Σ * \mathfrak_0 is a nilpotent Lie algebra given as the sum of the root spaces of Σ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Vector
In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the character of an irreducible representation of a semisimple Lie algebra. In Weyl's approach to the representation theory of connected compact Lie groups, the proof of the character formula is a key step in proving that every dominant integral element actually arises as the highest weight of some irreducible representation. Important consequences of the character formula are the Weyl dimension formula and the Kostant multiplicity formula. By definition, the character \chi of a representation \pi of ''G'' is the trace of \pi(g), as a function of a group element g\in G. The irreducible representations in this case are all finite-dimensional (this is part of the Peter–Weyl theorem); so the notion of trace is the usual one from linear alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Academic Press
Academic Press (AP) is an academic book publisher founded in 1941. It was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier. Academic Press publishes reference books, serials and online products in the subject areas of: * Communications engineering * Economics * Environmental science * Finance * Food science and nutrition * Geophysics * Life sciences * Mathematics and statistics * Neuroscience * Physical sciences * Psychology Psychology is the scientific study of mind and behavior. Psychology includes the study of conscious and unconscious phenomena, including feelings and thoughts. It is an academic discipline of immense scope, crossing the boundaries betwe ... Well-known products include the '' Methods in Enzymology'' series and encyclopedias such as ''The International Encyclopedia of Public Health'' and the ''Encyclopedia of Neuroscience''. See also * Akademische Ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Analysis
Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis). In the past two centuries, it has become a vast subject with applications in areas as diverse as number theory, representation theory, signal processing, quantum mechanics, tidal analysis and neuroscience. The term " harmonics" originated as the Ancient Greek word ''harmonikos'', meaning "skilled in music". In physical eigenvalue problems, it began to mean waves whose frequencies are integer multiples of one another, as are the frequencies of the harmonics of music notes, but the term has been generalized beyond its original meaning. The classical Fourier transform on R''n'' is still an area of ongoing research, particularly concerning Fourier transformation on more general objects such as tempered ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.png "Click for more on -> Abstract Harmonic Analysis")
