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＜電子ブック＞
Algebraic Structures and Operator Calculus : Volume I: Representations and Probability Theory / by P. Feinsilver, René Schott
(Mathematics and Its Applications ; 241)

版	1st ed. 1993.
出版者	(Dordrecht : Springer Netherlands : Imprint: Springer)
出版年	1993
本文言語	英語
大きさ	IX, 226 p : online resource
著者標目	*Feinsilver, P author Schott, René author SpringerLink (Online service)
件　名	LCSH:Nonassociative rings LCSH:Topological groups LCSH:Lie groups LCSH:Probabilities LCSH:Special functions LCSH:Operator theory FREE:Non-associative Rings and Algebras FREE:Topological Groups and Lie Groups FREE:Probability Theory FREE:Special Functions FREE:Operator Theory
一般注記	I. General remarks -- II. Some notations -- III. Gamma and beta functions -- IV. Numbering -- V. References -- VI. Polynomials: notations and formulas -- VII. Exercises and examples -- 1 Introductory Noncommutative Algebra -- I. Representations -- II. Heisenberg-Weyl algebra -- III. sl(2) algebra -- IV. Splitting formulas -- V. Exercises and examples -- 2 Hypergeometric functions -- I. Notations -- II. Generating function for 2F1 -- III. General formulation of CVPS and transformation formulas -- IV. Formulas related to HW algebra -- V. Formulas related to sl(2) algebra -- VI. Exercises and examples -- 3 Probability and Fock Spaces -- I. Trace functionals: probability and operators -- II. Fock spaces -- III. Tensor products -- IV. Exercises and examples -- 4 Moment Systems -- I. Moment generating functions and convolution -- II. Moment systems -- III. Radial moment systems -- IV. Holomorphic canonical variables -- V. Exercises and examples -- 5 Bernoulli Processes -- I. Bernoulli systems: general structure -- II. Binomial process and Krawtchouk polynomials -- III. Negative binomial process and Meixner polynomials -- IV. Continuous binomial process and Meixner-Pollaczek polynomials -- V. Poisson process and Poisson-Charlier polynomials -- VI. Exponential process and Laguerre polynomials -- VII. Brownian motion and Hermite polynomials -- VIII. Canonical moments -- IX. Exercises and examples -- 6 Bernoulli Systems -- I. Expansions, Rodrigues, and Riccati -- II. w-kernel -- III. X operator -- IV. Generating functions -- V. Riccati equations and Bernoulli systems -- VI. Reproducing kernels -- VII. Exercises -- 7 Matrix Elements -- I. Matrix elements -- II. Addition formulas and Riccati equations -- III. Coherent states and coherent state representations -- IV. Addition formulas for matrix elements of the group -- V. Exercises.-References This series presents some tools of applied mathematics in the areas of proba­ bility theory, operator calculus, representation theory, and special functions used currently, and we expect more and more in the future, for solving problems in math­ ematics, physics, and, now, computer science. Much of the material is scattered throughout available literature, however, we have nowhere found in accessible form all of this material collected. The presentation of the material is original with the authors. The presentation of probability theory in connection with group represen­ tations is new, this appears in Volume I. Then the applications to computer science in Volume II are original as well. The approach found in Volume III, which deals in large part with infinite-dimensional representations of Lie algebras/Lie groups, is new as well, being inspired by the desire to find a recursive method for calcu­ lating group representations. One idea behind this is the possibility of symbolic computation of the matrix elements. In this volume, Representations and Probability Theory, we present an intro­ duction to Lie algebras and Lie groups emphasizing the connections with operator calculus, which we interpret through representations, principally, the action of the Lie algebras on spaces of polynomials. The main features are the connection with probability theory via moment systems and the connection with the classical ele­ mentary distributions via representation theory. The various systems of polynomi­ als that arise are one of the most interesting aspects of this study HTTP:URL=https://doi.org/10.1007/978-94-011-1648-0

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