Convergence of One-Parameter Operator Semigroups: In Models of Mathematical Biology and Elsewhere (New Mathematical Monographs 30)

Convergence of One-Parameter Operator Semigroups: In Models of Mathematical Biology and Elsewhere (New Mathematical Monographs 30)

By: Adam Bobrowski (author)Hardback

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Description

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.

About 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.

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.

Product Details

  • ISBN13: 9781107137431
  • Format: Hardback
  • Number Of Pages: 454
  • ID: 9781107137431
  • weight: 790
  • ISBN10: 1107137438

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