Convergence of One-Parameter Operator Semigroups: In Models of Mathematical Biology and Elsewhere

by Adam Bobrowski

Overview

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.

Product Details

ISBN-13:	9781107137431
Publisher:	Cambridge University Press
Publication date:	07/14/2016
Series:	New Mathematical Monographs , #30
Pages:	454
Product dimensions:	6.22(w) x 9.25(h) x 0.47(d)

About the Author

Adam Bobrowski is a professor and Chairman of the Department of Mathematics at Lublin University of Technology, Poland. He has authored over 50 scientific papers and two books, Functional Analysis for Probability and Stochastic Processes and An Operator Semigroup in Mathematical Genetics.

Table of Contents

Preface; 1. Semigroups of operators; Part I. Regular Convergence: 2. The first convergence theorem; 3. Example - boundary conditions; 4. Example - a membrane; 5. Example - sesquilinear forms; 6. Uniform approximation of semigroups; 7. Convergence of resolvents; 8. (Regular) convergence of semigroups; 9. Example - a queue; 10. Example - elastic boundary; 11. Example - membrane again; 12. Example - telegraph; 13. Example - Markov chains; 14. A bird's-eye view; 15. Hasegawa's condition; 16. 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, stablization; Part II. Irregular Convergence: 26. First examples; 27. Example - genetic drift; 28. The nature of irregular convergence; 29. Convergence under perturbations; 30. Stein's model; 31. Uniformly holomorphic semigroups; 32. Asymptotic behavior of semigroups; 33. Fast neurotransmitters; 34. Fast neurotransmitters II; 35. Diffusions on graphs and Markov chains; 36. Semilinear equations; 37. Coagulation-fragmentation equation; 38. Homogenization theorem; 39. Shadow systems; 40. Kinases; 41. Uniformly differentiable semigroups; 42. Kurtz's theorem; 43. A singularly perturbed Markov chain; 44. A Tikhonov-type theorem; 45. Fast motion and frequent jumps; 46. Gene regulation and gene expression; 47. Some non-biological models; 48. Convex combinations of generators; 49. Dorroh and Volkonskii theorems; 50. Convex combinations in biology; 51. Recombination; 52. Recombination (continued); 53. Khasminskii's example; 54. Comparing semigroups; 55. Asymptotic analysis; 56. Greiner's theorem; 57. Fish dynamics; 58. Emergence of transmission conditions; 59. Emergence of transmission conditions II; Part III. Convergence of Cosine Families: 60. Regular convergence; 61. Cosines converge in a regular way; Part IV. Appendices: 62. Laplace transform; 63. Measurability implies continuity; References; Index.

Editorial Reviews

'This book is excellent in many respects. It is beautifully written, it contains many new and clever arguments, and it is a long, connected story told by a masterful storyteller. … Operator semigroup theory continues to grow and thrive and new and unexpected applications continue to lead to new theory. There is a large textbook/monograph literature including the early book by Hille and by Hille and Phillips, and later books by, alphabetically, Cialdea and Maz’ya, Davies, Dunford and Schwartz, Engel and Nagel, Fattorini, Goldstein, Kato, Krein, Lax, Nagel et al., Pazy, and Yosida. Bobrowski’s book stands with these as books which contain information about theory and applications which could not be found elsewhere at the time of publication. Bobrowski’s superb exposition and his wide scope and new applications will keep the semigroup community busy. We can all be grateful.' Jerome A. Goldstein, Semigroup Forum

From the Publisher