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3044 results in Probability theory and stochastic processes

Page 30 of 153

Semigroups of Linear Operators
  • With Applications to Analysis, Probability and Physics
  • David Applebaum
  • Published online:
    27 July 2019
    Print publication:
    15 August 2019
  • The theory of semigroups of operators is one of the most important themes in modern analysis. Not only does it have great intellectual beauty, but also wide-ranging applications. In this book the author first presents the essential elements of the theory, introducing the notions of semigroup, generator and resolvent, and establishes the key theorems of Hille–Yosida and Lumer–Phillips that give conditions for a linear operator to generate a semigroup. He then presents a mixture of applications and further developments of the theory. This includes a description of how semigroups are used to solve parabolic partial differential equations, applications to Levy and Feller–Markov processes, Koopmanism in relation to dynamical systems, quantum dynamical semigroups, and applications to generalisations of the Riemann–Liouville fractional integral. Along the way the reader encounters several important ideas in modern analysis including Sobolev spaces, pseudo-differential operators and the Nash inequality.

The Random Matrix Theory of the Classical Compact Groups
  • Elizabeth S. Meckes
  • Published online:
    12 July 2019
    Print publication:
    01 August 2019
  • This is the first book to provide a comprehensive overview of foundational results and recent progress in the study of random matrices from the classical compact groups, drawing on the subject's deep connections to geometry, analysis, algebra, physics, and statistics. The book sets a foundation with an introduction to the groups themselves and six different constructions of Haar measure. Classical and recent results are then presented in a digested, accessible form, including the following: results on the joint distributions of the entries; an extensive treatment of eigenvalue distributions, including the Weyl integration formula, moment formulae, and limit theorems and large deviations for the spectral measures; concentration of measure with applications both within random matrix theory and in high dimensional geometry; and results on characteristic polynomials with connections to the Riemann zeta function. This book will be a useful reference for researchers and an accessible introduction for students in related fields.

  • 1 - Simple counting

  • from Part I - Counting
  • Benedict Gross, Harvard University, Massachusetts, Joe Harris, Harvard University, Massachusetts, Emily Riehl, The Johns Hopkins University, Maryland
  • Book:
    Fat Chance
    Published online:
    24 May 2019
    Print publication:
    13 June 2019, pp 3-10
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Page 30 of 153