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In Models of Mathematical Biology and Elsewhere
AUTHOR: Adam Bobrowski
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.
• Arises from the author's long-held fascination with phenomena accompanying convergence of semigroups
• New results appear for the first time in book form
• Illustrates how real-world phenomena lead to interesting mathematical insight and new theorems
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.
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