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Nonlinear Perron-Frobenius theory /
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,...
| Clasificación: | Libro Electrónico |
|---|---|
| Otros Autores: | , |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cambridge :
Cambridge University Press,
2012.
|
| Colección: | Cambridge tracts in mathematics ;
189. |
| Temas: | |
| Acceso en línea: | Texto completo |
MARC
| LEADER | 00000cam a2200000Mi 4500 | ||
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| 245 | 0 | 0 | |a Nonlinear Perron-Frobenius theory / |c Bas Lemmens, Roger Nussbaum. |
| 260 | |a Cambridge : |b Cambridge University Press, |c 2012. | ||
| 300 | |a 1 online resource (xii, 323 pages) : |b illustrations, tables | ||
| 336 | |a text |b txt |2 rdacontent | ||
| 337 | |a computer |b c |2 rdamedia | ||
| 338 | |a online resource |b cr |2 rdacarrier | ||
| 490 | 1 | |a Cambridge tracts in mathematics ; |v 189 | |
| 500 | |a Title from publishers bibliographic system (viewed 09 May 2012). | ||
| 504 | |a Includes chapter notes and comments, bibliographical references (pages 307-318), list of symbols, and index. | ||
| 505 | 0 | |6 880-01 |a 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. | |
| 520 | |a 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. | ||
| 546 | |a English. | ||
| 590 | |a eBooks on EBSCOhost |b EBSCO eBook Subscription Academic Collection - Worldwide | ||
| 650 | 0 | |a Non-negative matrices. | |
| 650 | 0 | |a Eigenvalues. | |
| 650 | 0 | |a Eigenvectors. | |
| 650 | 0 | |a Algebras, Linear. | |
| 650 | 6 | |a Matrices non-négatives. | |
| 650 | 6 | |a Valeurs propres. | |
| 650 | 6 | |a Vecteurs. | |
| 650 | 6 | |a Algèbre linéaire. | |
| 650 | 7 | |a MATHEMATICS |x Differential Equations. |2 bisacsh | |
| 650 | 7 | |a MATHEMATICS |x Algebra |x Linear. |2 bisacsh | |
| 650 | 7 | |a Álgebra lineal |2 embne | |
| 650 | 0 | 7 | |a Matrices no negativas |2 embucm |
| 650 | 7 | |a Algebras, Linear |2 fast | |
| 650 | 7 | |a Eigenvalues |2 fast | |
| 650 | 7 | |a Eigenvectors |2 fast | |
| 650 | 7 | |a Non-negative matrices |2 fast | |
| 700 | 1 | |a Lemmens, Bas. | |
| 700 | 1 | |a Nussbaum, Roger D., |d 1944- | |
| 776 | 0 | 8 | |i Print version: |z 9780521898812 |
| 830 | 0 | |a Cambridge tracts in mathematics ; |v 189. | |
| 856 | 4 | 0 | |u https://ebsco.uam.elogim.com/login.aspx?direct=true&scope=site&db=nlebk&AN=443672 |z Texto completo |
| 880 | 0 | 0 | |6 505-01/(S |g Machine generated contents note: |g 1. |t What is nonlinear Perron--Frobenius theory-- |g 1.1. |t Classical Perron--Frobenius theory -- |g 1.2. |t Cones and partial orderings -- |g 1.3. |t Order-preserving maps -- |g 1.4. |t Subhomogeneous maps -- |g 1.5. |t Topical maps -- |g 1.6. |t Integral-preserving maps -- |g 2. |t Non-expansiveness and nonlinear Perron--Frobenius theory -- |g 2.1. |t Hilbert's and Thompson's metrics -- |g 2.2. |t Polyhedral cones -- |g 2.3. |t Lorentz cones -- |g 2.4. |t cone of positive-semidefinite symmetric matrices -- |g 2.5. |t Completeness -- |g 2.6. |t Convexity and geodesics -- |g 2.7. |t Topical maps and the sup-norm -- |g 2.8. |t Integral-preserving maps and the l1-norm -- |g 3. |t Dynamics of non-expansive maps -- |g 3.1. |t Basic properties of non-expansive maps -- |g 3.2. |t Fixed-point theorems for non-expansive maps -- |g 3.3. |t Horofunctions and horoballs -- |g 3.4. |t Denjoy--Wolff type theorem -- |g 3.5. |t Non-expansive retractions -- |g 4. |t Sup-norm non-expansive maps -- |g 4.1. |t size of the ω-limit sets -- |g 4.2. |t Periods of periodic points -- |g 4.3. |t Iterates of topical maps -- |g 5. |t Eigenvectors and eigenvalues of nonlinear cone maps -- |g 5.1. |t Extensions of order-preserving maps -- |g 5.2. |t cone spectrum -- |g 5.3. |t cone spectral radius -- |g 5.4. |t Eigenvectors corresponding to the cone spectral radius -- |g 5.5. |t Continuity of the cone spectral radius -- |g 5.6. |t Collatz--Wielandt formula -- |g 6. |t Eigenvectors in the interior of the cone -- |g 6.1. |t First principles -- |g 6.2. |t Perturbation method -- |g 6.3. |t Bounded invariant sets -- |g 6.4. |t Uniqueness of the eigenvector -- |g 6.5. |t Convergence to a unique eigenvector -- |g 6.6. |t Means and their eigenvectors -- |g 7. |t Applications to matrix scaling problems -- |g 7.1. |t Matrix scaling: a fixed-point approach -- |g 7.2. |t compatibility condition -- |g 7.3. |t Special DAD theorems -- |g 7.4. |t Doubly stochastic matrices: the classic case -- |g 7.5. |t Scaling to row stochastic matrices -- |g 8. |t Dynamics of subhomogeneous maps -- |g 8.1. |t Iterations on polyhedral cones -- |g 8.2. |t Periodic orbits in polyhedral cones -- |g 8.3. |t Denjoy--Wolff theorems for cone maps -- |g 8.4. |t Denjoy--Wolff theorem for polyhedral cones -- |g 9. |t Dynamics of integral-preserving maps -- |g 9.1. |t Lattice homomorphisms -- |g 9.2. |t Periodic orbits of lower semi-lattice homomorphisms -- |g 9.3. |t Periodic points and admissible arrays -- |g 9.4. |t Computing periods of admissible arrays -- |g 9.5. |t Maps on the whole space -- |g Appendix |t A Birkhoff--Hopf theorem -- |g A.1. |t Preliminaries -- |g A.2. |t Almost Archimedean cones -- |g A.3. |t Projective diameter -- |g A.4. |t Birkhoff--Hopf theorem: reduction to two dimensions -- |g A.5. |t Two-dimensional cones -- |g A.6. |t Completion of the proof of the Birkhoff--Hopf theorem -- |g A.7. |t Eigenvectors of cone-linear maps -- |g Appendix B |t Classical Perron--Frobenius theory -- |g B.1. |t general version of Perron's theorem -- |g B.2. |t finite-dimensional Krein--Rutman theorem -- |g B.3. |t Irreducible linear maps -- |g B.4. |t peripheral spectrum. |
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