Convergence of one-parameter operator semigroups : in models of mathematical biology and elsewhere / Adam Bobrowski.

Format:

Book

Author/Creator:

Bobrowski, Adam, author.

Series:

New mathematical monographs ; 30.

New mathematical monographs ; 30

Language:

English

Subjects (All):

Operator theory.

Semigroups of operators.

Physical Description:

1 online resource (xiv, 438 pages) : digital, PDF file(s).

Place of Publication:

Cambridge : Cambridge University Press, 2016.

Summary:

This book presents a detailed and contemporary account of the classical theory of convergence of semigroups and its more recent development treating the case where the limit semigroup, in contrast to the approximating semigroups, acts merely on a subspace of the original Banach space (this is the case, for example, with singular perturbations). The author demonstrates the far-reaching applications of this theory using real examples from various branches of pure and applied mathematics, with a particular emphasis on mathematical biology. The book may serve as a useful reference, containing a significant number of new results ranging from the analysis of fish populations to signaling pathways in living cells. It comprises many short chapters, which allows readers to pick and choose those topics most relevant to them, and it contains 160 end-of-chapter exercises so that readers can test their understanding of the material as they go along.

Contents:

Cover; Half title; Series; Title; Copyright; Dedication; Contents; Preface; 1 Semigroups of Operators and Cosine Operator Functions; Part I Regular Convergence; 2 The First Convergence Theorem; 3 Continuous Dependence on Boundary Conditions; 4 Semipermeable Membrane; 5 Convergence of Forms; 6 Uniform Approximation of Semigroups; 7 Convergence of Resolvents; 8 (Regular) Convergence of Semigroups; 9 A Queue in Heavy Traffic; 10 Elastic Brownian Motions; 11 Back to the Membrane; 12 Telegraph with Small Parameter; 13 Minimal Markov Chains; 14 Outside of the Regularity Space: A Bird's-Eye View

15 Hasegawa's Condition16 Blackwell's Example; 17 Wright's Diffusion; 18 Discrete-Time Approximation; 19 Discrete-Time Approximation: Examples; 20 Back to Wright's Diffusion; 21 Kingman's n-Coalescent; 22 The Feynman-Kac Formula; 23 The Two-Dimensional Dirac Equation; 24 Approximating Spaces; 25 Boundedness, Stabilization; Part II Irregular convergence; 26 First Examples; 27 Extremely Strong Genetic Drift; 28 The Nature of Irregular Convergence; 29 Irregular Convergence Is Preserved Under Bounded Perturbations; 30 Stein's Model; 31 Uniformly Holomorphic Semigroups

32 Asymptotic Behavior of Semigroups33 Fast Neurotransmitters; 34 Fast Neurotransmitters II; 35 From Diffusions on Graphs to Markov Chains and Back Again; 36 Semilinear Equations, Early Cancer Modeling; 37 Coagulation-Fragmentation Equation; 38 Homogenization Theorem; 39 Shadow Systems; 40 Kinases; 41 Uniformly Differentiable Semigroups; 42 Kurtz's Singular Perturbation Theorem; 43 A Singularly Perturbed Markov Chain; 44 A Tikhonov-Type Theorem; 45 Fast Motion and Frequent Jumps Theorems for Piecewise Deterministic Processes; 46 Models of Gene Regulation and Gene Expression

47 Oligopolies, Manufacturing Systems, and Climate Changes48 Convex Combinations of Feller Generators; 49 The Dorroh Theorem and the Volkonskii Formula; 50 Convex Combinations in Biological Models; 51 Recombination; 52 Recombination (Continued); 53 Averaging Principle of Freidlin and Wentzell: Khasminskii's Example; 54 Comparing Semigroups; 55 Relations to Asymptotic Analysis; 56 Greiner's Theorem; 57 Fish Population Dynamics and Convex Combination of Boundary Conditions; 58 Averaging Principle of Freidlin and Wentzell: Emergence of Transmission Conditions

59 Averaging Principle Continued: L[sup(1)]-SettingPart III Convergence of cosine families; 60 Regular Convergence of Cosine Families; 61 Cosines Converge in a Regular Way; Part IV Appendixes; 62 Appendix A: Representation Theorem for the Laplace Transform; 63 Appendix B: Measurable Cosine Functions Are Continuous; References; Index

Notes:

Title from publisher's bibliographic system (viewed on 04 Jul 2016).

Includes bibliographical references and index.

ISBN:

1-316-55379-5

1-316-55407-4

1-316-55435-X

1-316-55463-5

1-316-55575-5

1-316-48066-6