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Algebraic Structures and Operator Calculus: Volume I: Representations and Probability Theory(241 Mathematics and Its Applications)

Algebraic Structures and Operator Calculus: Volume I: Representations and Probability Theory(241 Mathematics and Its Applications)


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About the Book

This is the first of three volumes which present, in an original way, some of the most important tools of applied mathematics, in areas such as probability theory, operator calculus, representation theory, and special functions, used in solving problems in mathematics, physics and computer science. Volume 1 deals with probability theory in connection with group representation. Volume 2 will consider applications to computer science, and Volume 3 will deal largely with infinite-dimensional representations of Lie algebras and groups. This first volume - "Representations and Probability Theory" - presents an introduction to Lie algebras and Lie groups which emphasises the connections with probability theory and representation theory. The book contains an introduction and seven chapters which treat, respectively, noncommutative algebra, hypergeometric functions, probability and Fock spaces, moment systems, Bernoulli processes/systems, and matrix elements. Each chapter contains exercises which range in difficulty from easy to advanced. The text is written so as to be suitable for self-study for both beginning graduate students and researchers. For students, teachers and researchers with an interest in algebraic structures and operator calculus.

Table of Contents:
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.


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Product Details
  • ISBN-13: 9780792321163
  • Publisher: Kluwer Academic Publishers
  • Publisher Imprint: Kluwer Academic Publishers
  • Height: 235 mm
  • No of Pages: 226
  • Returnable: Y
  • Sub Title: Volume I: Representations and Probability Theory
  • ISBN-10: 0792321162
  • Publisher Date: 31 Jan 1993
  • Binding: Hardback
  • Language: English
  • Returnable: Y
  • Series Title: 241 Mathematics and Its Applications
  • Width: 155 mm


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Algebraic Structures and Operator Calculus: Volume I: Representations and Probability Theory(241 Mathematics and Its Applications)
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Algebraic Structures and Operator Calculus: Volume I: Representations and Probability Theory(241 Mathematics and Its Applications)
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