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Nonlinear Perron–Frobenius Theory

$131.00 (C)

Part of Cambridge Tracts in Mathematics

  • Date Published: June 2012
  • availability: Available
  • format: Hardback
  • isbn: 9780521898812

$ 131.00 (C)

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About the Authors
  • In the past several decades the classical Perron–Frobenius theory for nonnegative matrices has been extended to obtain remarkably precise and beautiful results for classes of nonlinear maps. This nonlinear Perron–Frobenius theory has found significant uses in computer science, mathematical biology, game theory and the study of dynamical systems. This is the first comprehensive and unified introduction to nonlinear Perron–Frobenius theory suitable for graduate students and researchers entering the field for the first time. It acquaints the reader with recent developments and provides a guide to challenging open problems. To enhance accessibility, the focus is on finite dimensional nonlinear Perron–Frobenius theory, but pointers are provided to infinite dimensional results. Prerequisites are little more than basic real analysis and topology.

    • The first systematic text on the subject, by authors who are among the key developers of the field
    • Useful to researchers in nonlinear operator theory, matrix analysis, dynamical systems theory and nonlinear analysis
    • Assumes little more than basic real analysis and topology
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    Reviews & endorsements

    "In their introduction the authors state that "The main purpose of this book is to give a systematic self-contained introduction to nonlinear Perron-Frobenius theory and to provide a guide to various challenging open problems." They have achieved their aim excellently. After a discussion of the linear theory of cone preserving maps, the book turns to its main theme, nonlinear Perron–Frobenius theory in finite dimension. Of particular importance is the linking of this theory to that of non-expansive maps in various metrics. Applications are presented, for example to dynamical systems and diagonal scaling of matrices. In its various incarnations, Perron–Frobenius theory has had a deep influence over 100 years on many parts of pure and applied mathematics. An exposition of the finite-dimensional nonlinear theory from a specific point of view is a valuable and timely addition to the literature."
    Hans Schneider, University of Wisconsin, Madison

    "This textbook is a carefully arranged journey through large parts of this beautiful theory, which has seen various contributions by the authors in the past. The material is accessible with little more than a basic knowledge of linear algebra, real analysis and some topology. The book is self-contained, all results are proven very rigorously, and where appropriate, the evolution of results is explained and framed in the historical context. I recommend this book very warmly and without any reservations to anyone interested in nonlinear Perron-Frobenius theory."
    Bjorn S. Ruffer, Mathematical Reviews

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    Product details

    • Date Published: June 2012
    • format: Hardback
    • isbn: 9780521898812
    • length: 336 pages
    • dimensions: 234 x 156 x 21 mm
    • weight: 0.62kg
    • contains: 15 b/w illus.
    • availability: Available
  • Table of Contents

    1. What is nonlinear Perron–Frobenius theory?
    2. Non-expansiveness and nonlinear Perron–Frobenius theory
    3. Dynamics of non-expansive maps
    4. Sup-norm non-expansive maps
    5. Eigenvectors and eigenvalues of nonlinear cone maps
    6. Eigenvectors in the interior of the cone
    7. Applications to matrix scaling problems
    8. Dynamics of subhomogeneous maps
    9. Dynamics of integral-preserving maps
    Appendix A. The Birkhoff–Hopf theorem
    Appendix B. Classical Perron–Frobenius theory
    Notes and comments
    List of symbols

  • Authors

    Bas Lemmens, University of Kent, Canterbury
    Bas Lemmens is a Lecturer in Mathematics at the University of Kent, Canterbury. His research interests lie in nonlinear operator theory, dynamical systems theory and metric geometry. He is one of the key developers of nonlinear Perron–Frobenius theory.

    Roger Nussbaum, Rutgers University, New Jersey
    Roger Nussbaum is a Professor of Mathematics at Rutgers University. His research interests include nonlinear differential-delay equations, the theory of nonlinear positive operators and fixed point theory and its applications. He has published extensively on nonlinear Perron–Frobenius theory.

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