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Mathematical Foundations of the State Lumping of Large Systems : Mathematics and Its Applications - Vladimir S. Koroliuk

Mathematical Foundations of the State Lumping of Large Systems

Mathematics and Its Applications

Hardcover Published: 31st August 1993
ISBN: 9780792324133
Number Of Pages: 278

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This volume is devoted to theoretical results which formalize the concept of state lumping: the transformation of evolutions of systems having a complex (large) phase space to those having a simpler (small) phase space. The theory of phase lumping has aspects in common with averaging methods, projection formalism, stiff systems of differential equations, and other asymptotic theorems. Numerous examples are presented in this book from the theory and applications of random processes, and statistical and quantum mechanics which illustrate the potential capabilities of the theory developed.
The volume contains seven chapters. Chapter 1 presents an exposition of the basic notions of the theory of linear operators. Chapter 2 discusses aspects of the theory of semigroups of operators and Markov processes which have relevance to what follows. In Chapters 3--5, invertibly reducible operators perturbed on the spectrum are investigated, and the theory of singularly perturbed semigroups of operators is developed assuming that the perturbation is subordinated to the perturbed operator. The case of arbitrary perturbation is also considered, and the results are presented in the form of limit theorems and asymptotic expansions. Chapters 6 and 7 describe various applications of the method of phase lumping to Markov and semi-Markov processes, dynamical systems, quantum mechanics, etc. The applications discussed are by no means exhaustive and this book points the way to many more fruitful applications in various other areas.
For researchers whose work involves functional analysis, semigroup theory, Markov processes and probability theory.

Classes of Linear Operatorsp. 1
Basic notionsp. 1
Closed and closable operatorsp. 4
Normally solvable operatorsp. 12
Invertibly reducible operatorsp. 18
Pseudo-resolventsp. 29
Semigroups of Operators and Markov Processesp. 33
Basic notionsp. 33
Infinitesimal operators of ergodic Markov processesp. 35
Holomorphic semigroups with invertibly reducible infinitesimal operatorsp. 43
Semigroups of operators uniformly and strongly ergodic at the infinityp. 46
"Generating" operators of ergodic semi-Markov processesp. 53
Abstract potential operatorsp. 57
Examples of invertibly reducible operatorsp. 65
Perturbations of Invertibly Reducible Operatorsp. 81
Eigen-projectors and eigen-operatorsp. 81
Inversion of an invertibly reducible operator perturbed on the spectrump. 90
Resolvents of singularly perturbed semigroupsp. 98
Limit theorems and asymptotic expansions for resolvents of singularly perturbed semigroupsp. 103
Limit theorems and asymptotic expansions for resolvents of singularly perturbed semigroups. The case of s ] 2p. 109
Singular Perturbations of Holomorphic Semigroupsp. 119
Principal problems. The method of Vishyk-Lyusternik-Vasilyevap. 119
Structure of singularly perturbed semigroupsp. 121
Regular lumped approximations to solutions of singularly perturbed equationsp. 127
Asymptotic Expansions and Limit Theoremsp. 131
Strong limits of singularly perturbed semigroups. Resolvent approachp. 131
Asymptotic analysis of singularly perturbed semigroups. The case of s = 1p. 134
Asymptotic analysis of singularly perturbed semigroupsp. 153
Asymptotic Phase Lumping of Markov and Semi-Markov Processesp. 167
Limit theoremsp. 167
Asymptotic phase lumping. The case of s = 1p. 181
Some examplesp. 198
Asymptotic phase lumping. The case of s [actual symbol not reproducible] 2p. 202
Classification of processes admitting asymptotic phase lumpingp. 213
Limit theorems and asymptotic theorems for additive functionalsp. 214
Applications of the Theory of Singularly Perturbed Semigroupsp. 241
Tikhonov systems of differential equationsp. 241
Nonrelativistic limit of the Dirac operatorp. 246
Hydrodynamic limit for the linearized Boltzmann equationp. 254
Referencesp. 267
Subject Indexp. 277
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780792324133
ISBN-10: 0792324137
Series: Mathematics and Its Applications
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 278
Published: 31st August 1993
Publisher: Springer
Country of Publication: NL
Dimensions (cm): 23.5 x 15.5  x 1.7
Weight (kg): 1.3