Title Nonlinear Perron-Frobenius theory / Bas Lemmens, Roger Nussbaum

Published Cambridge : Cambridge University Press, 2012

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Description 1 online resource (xii, 323 pages) : illustrations, tables Series Cambridge tracts in mathematics ; 189 Cambridge tracts in mathematics ; 189. Contents 880-01 Cover; CAMBRIDGE TRACTS IN MATHEMATICS; GENERAL EDITORS; Title; Copyright; Contents; Preface; 1 What is nonlinear Perron-Frobenius theory?; 1.1 Classical Perron-Frobenius theory; 1.2 Cones and partial orderings; 1.3 Order-preserving maps; 1.4 Subhomogeneous maps; 1.5 Topical maps; 1.6 Integral-preserving maps; 2 Non-expansiveness and nonlinear Perron-Frobenius theory; 2.1 Hilbert's and Thompson's metrics; 2.2 Polyhedral cones; 2.3 Lorentz cones; 2.4 The cone of positive-semidefinite symmetric matrices; 2.5 Completeness; 2.6 Convexity and geodesics; 2.7 Topical maps and the sup-norm 880-01/(S Machine generated contents note: 1. What is nonlinear Perron--Frobenius theory-- 1.1. Classical Perron--Frobenius theory -- 1.2. Cones and partial orderings -- 1.3. Order-preserving maps -- 1.4. Subhomogeneous maps -- 1.5. Topical maps -- 1.6. Integral-preserving maps -- 2. Non-expansiveness and nonlinear Perron--Frobenius theory -- 2.1. Hilbert's and Thompson's metrics -- 2.2. Polyhedral cones -- 2.3. Lorentz cones -- 2.4. cone of positive-semidefinite symmetric matrices -- 2.5. Completeness -- 2.6. Convexity and geodesics -- 2.7. Topical maps and the sup-norm -- 2.8. Integral-preserving maps and the l1-norm -- 3. Dynamics of non-expansive maps -- 3.1. Basic properties of non-expansive maps -- 3.2. Fixed-point theorems for non-expansive maps -- 3.3. Horofunctions and horoballs -- 3.4. Denjoy--Wolff type theorem -- 3.5. Non-expansive retractions -- 4. Sup-norm non-expansive maps -- 4.1. size of the ω-limit sets -- 4.2. Periods of periodic points -- 4.3. Iterates of topical maps -- 5. Eigenvectors and eigenvalues of nonlinear cone maps -- 5.1. Extensions of order-preserving maps -- 5.2. cone spectrum -- 5.3. cone spectral radius -- 5.4. Eigenvectors corresponding to the cone spectral radius -- 5.5. Continuity of the cone spectral radius -- 5.6. Collatz--Wielandt formula -- 6. Eigenvectors in the interior of the cone -- 6.1. First principles -- 6.2. Perturbation method -- 6.3. Bounded invariant sets -- 6.4. Uniqueness of the eigenvector -- 6.5. Convergence to a unique eigenvector -- 6.6. Means and their eigenvectors -- 7. Applications to matrix scaling problems -- 7.1. Matrix scaling: a fixed-point approach -- 7.2. compatibility condition -- 7.3. Special DAD theorems -- 7.4. Doubly stochastic matrices: the classic case -- 7.5. Scaling to row stochastic matrices -- 8. Dynamics of subhomogeneous maps -- 8.1. Iterations on polyhedral cones -- 8.2. Periodic orbits in polyhedral cones -- 8.3. Denjoy--Wolff theorems for cone maps -- 8.4. Denjoy--Wolff theorem for polyhedral cones -- 9. Dynamics of integral-preserving maps -- 9.1. Lattice homomorphisms -- 9.2. Periodic orbits of lower semi-lattice homomorphisms -- 9.3. Periodic points and admissible arrays -- 9.4. Computing periods of admissible arrays -- 9.5. Maps on the whole space -- Appendix A Birkhoff--Hopf theorem -- A.1. Preliminaries -- A.2. Almost Archimedean cones -- A.3. Projective diameter -- A.4. Birkhoff--Hopf theorem: reduction to two dimensions -- A.5. Two-dimensional cones -- A.6. Completion of the proof of the Birkhoff--Hopf theorem -- A.7. Eigenvectors of cone-linear maps -- Appendix B Classical Perron--Frobenius theory -- B.1. general version of Perron's theorem -- B.2. finite-dimensional Krein--Rutman theorem -- B.3. Irreducible linear maps -- B.4. peripheral spectrum Summary 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 Notes Title from publishers bibliographic system (viewed 09 May 2012) Bibliography Includes chapter notes and comments, bibliographical references (pages 307-318), list of symbols, and index Notes English Subject Non-negative matrices. Eigenvalues. Eigenvectors. Algebras, Linear. MATHEMATICS -- Differential Equations. MATHEMATICS -- Algebra -- Linear. Álgebra lineal Matrices no negativas Algebras, Linear Eigenvalues Eigenvectors Non-negative matrices Form Electronic book Author Lemmens, Bas. Nussbaum, Roger D., 1944- ISBN 9781139026079 1139026070 9780521898812 0521898811 1280877952 9781280877957 9781139376822 1139376829 9781139379687 1139379682 9781139375399 1139375393 1107226341 9781107226340 9786613719263 6613719269 1139378252 9781139378253 1139371401 9781139371407

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