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


Part of New Mathematical Monographs

  • Date Published: July 2016
  • availability: Available
  • format: Hardback
  • isbn: 9781107137431

£ 141.00

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About the Authors
  • 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
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    Reviews & endorsements

    '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

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    Product details

    • Date Published: July 2016
    • format: Hardback
    • isbn: 9781107137431
    • length: 454 pages
    • dimensions: 235 x 158 x 12 mm
    • weight: 0.79kg
    • contains: 60 b/w illus. 9 colour illus. 160 exercises
    • availability: Available
  • Table of Contents

    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

  • Author

    Adam Bobrowski, Politechnika Lubelska, Poland
    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.

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