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,...
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| Other Authors: | , |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Cambridge :
Cambridge University Press,
2012.
|
| Series: | Cambridge tracts in mathematics ;
189. |
| Subjects: | |
| Online Access: | DE-862 DE-863 |
| 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. |
| Item Description: | Title from publishers bibliographic system (viewed 09 May 2012). |
| Physical Description: | 1 online resource (xii, 323 pages) : illustrations, tables |
| Bibliography: | Includes chapter notes and comments, bibliographical references (pages 307-318), list of symbols, and index. |
| 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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| 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 | ||
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| 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. | ||
| 650 | 0 | |a Non-negative matrices. |0 http://id.loc.gov/authorities/subjects/sh85092227 | |
| 650 | 0 | |a Eigenvalues. |0 http://id.loc.gov/authorities/subjects/sh85041389 | |
| 650 | 0 | |a Eigenvectors. |0 http://id.loc.gov/authorities/subjects/sh85041390 | |
| 650 | 0 | |a Algebras, Linear. |0 http://id.loc.gov/authorities/subjects/sh85003441 | |
| 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. |0 http://id.loc.gov/authorities/names/n2012008334 | |
| 700 | 1 | |a Nussbaum, Roger D., |d 1944- |1 https://id.oclc.org/worldcat/entity/E39PBJrm49pkGv9JmP7hGHkMT3 |0 http://id.loc.gov/authorities/names/n78044209 | |
| 776 | 0 | 8 | |i Print version: |z 9780521898812 |
| 830 | 0 | |a Cambridge tracts in mathematics ; |v 189. |0 http://id.loc.gov/authorities/names/n42005726 | |
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| 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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Record in the Search Index
| DE-BY-FWS_katkey | ZDB-4-EBA-ocn794731490 |
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| _version_ | 1829094728278212608 |
| adam_text | |
| any_adam_object | |
| author2 | Lemmens, Bas Nussbaum, Roger D., 1944- |
| author2_role | |
| author2_variant | b l bl r d n rd rdn |
| author_GND | http://id.loc.gov/authorities/names/n2012008334 http://id.loc.gov/authorities/names/n78044209 |
| author_facet | Lemmens, Bas Nussbaum, Roger D., 1944- |
| author_sort | Lemmens, Bas |
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| callnumber-search | QA188 .L456 2012 |
| callnumber-sort | QA 3188 L456 42012 |
| callnumber-subject | QA - Mathematics |
| collection | ZDB-4-EBA |
| contents | 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. |
| ctrlnum | (OCoLC)794731490 |
| dewey-full | 512.5 512/.5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.5 512/.5 |
| dewey-search | 512.5 512/.5 |
| dewey-sort | 3512.5 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Electronic eBook |
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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.</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="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. 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| id | ZDB-4-EBA-ocn794731490 |
| illustrated | Illustrated |
| indexdate | 2025-04-11T08:37:44Z |
| institution | BVB |
| isbn | 9781139026079 1139026070 9780521898812 0521898811 1280877952 9781280877957 9781139376822 1139376829 9781139379687 1139379682 9781139375399 1139375393 1107226341 9781107226340 9786613719263 6613719269 1139378252 9781139378253 1139371401 9781139371407 |
| language | English |
| oclc_num | 794731490 |
| open_access_boolean | |
| owner | MAIN DE-862 DE-BY-FWS DE-863 DE-BY-FWS |
| owner_facet | MAIN DE-862 DE-BY-FWS DE-863 DE-BY-FWS |
| physical | 1 online resource (xii, 323 pages) : illustrations, tables |
| psigel | ZDB-4-EBA FWS_PDA_EBA ZDB-4-EBA |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | Cambridge University Press, |
| record_format | marc |
| series | Cambridge tracts in mathematics ; |
| series2 | Cambridge tracts in mathematics ; |
| spelling | Nonlinear Perron-Frobenius theory / Bas Lemmens, Roger Nussbaum. Cambridge : Cambridge University Press, 2012. 1 online resource (xii, 323 pages) : illustrations, tables text txt rdacontent computer c rdamedia online resource cr rdacarrier Cambridge tracts in mathematics ; 189 Title from publishers bibliographic system (viewed 09 May 2012). Includes chapter notes and comments, bibliographical references (pages 307-318), list of symbols, and index. 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. 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. English. Non-negative matrices. http://id.loc.gov/authorities/subjects/sh85092227 Eigenvalues. http://id.loc.gov/authorities/subjects/sh85041389 Eigenvectors. http://id.loc.gov/authorities/subjects/sh85041390 Algebras, Linear. http://id.loc.gov/authorities/subjects/sh85003441 Matrices non-négatives. Valeurs propres. Vecteurs. Algèbre linéaire. MATHEMATICS Differential Equations. bisacsh MATHEMATICS Algebra Linear. bisacsh Álgebra lineal embne Matrices no negativas embucm Algebras, Linear fast Eigenvalues fast Eigenvectors fast Non-negative matrices fast Lemmens, Bas. http://id.loc.gov/authorities/names/n2012008334 Nussbaum, Roger D., 1944- https://id.oclc.org/worldcat/entity/E39PBJrm49pkGv9JmP7hGHkMT3 http://id.loc.gov/authorities/names/n78044209 Print version: 9780521898812 Cambridge tracts in mathematics ; 189. http://id.loc.gov/authorities/names/n42005726 505-01/(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. |
| spellingShingle | Nonlinear Perron-Frobenius theory / Cambridge tracts in mathematics ; 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. Non-negative matrices. http://id.loc.gov/authorities/subjects/sh85092227 Eigenvalues. http://id.loc.gov/authorities/subjects/sh85041389 Eigenvectors. http://id.loc.gov/authorities/subjects/sh85041390 Algebras, Linear. http://id.loc.gov/authorities/subjects/sh85003441 Matrices non-négatives. Valeurs propres. Vecteurs. Algèbre linéaire. MATHEMATICS Differential Equations. bisacsh MATHEMATICS Algebra Linear. bisacsh Álgebra lineal embne Matrices no negativas embucm Algebras, Linear fast Eigenvalues fast Eigenvectors fast Non-negative matrices fast |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85092227 http://id.loc.gov/authorities/subjects/sh85041389 http://id.loc.gov/authorities/subjects/sh85041390 http://id.loc.gov/authorities/subjects/sh85003441 |
| title | Nonlinear Perron-Frobenius theory / |
| title_auth | Nonlinear Perron-Frobenius theory / |
| title_exact_search | Nonlinear Perron-Frobenius theory / |
| title_full | Nonlinear Perron-Frobenius theory / Bas Lemmens, Roger Nussbaum. |
| title_fullStr | Nonlinear Perron-Frobenius theory / Bas Lemmens, Roger Nussbaum. |
| title_full_unstemmed | Nonlinear Perron-Frobenius theory / Bas Lemmens, Roger Nussbaum. |
| title_short | Nonlinear Perron-Frobenius theory / |
| title_sort | nonlinear perron frobenius theory |
| topic | Non-negative matrices. http://id.loc.gov/authorities/subjects/sh85092227 Eigenvalues. http://id.loc.gov/authorities/subjects/sh85041389 Eigenvectors. http://id.loc.gov/authorities/subjects/sh85041390 Algebras, Linear. http://id.loc.gov/authorities/subjects/sh85003441 Matrices non-négatives. Valeurs propres. Vecteurs. Algèbre linéaire. MATHEMATICS Differential Equations. bisacsh MATHEMATICS Algebra Linear. bisacsh Álgebra lineal embne Matrices no negativas embucm Algebras, Linear fast Eigenvalues fast Eigenvectors fast Non-negative matrices fast |
| topic_facet | Non-negative matrices. Eigenvalues. Eigenvectors. Algebras, Linear. Matrices non-négatives. Valeurs propres. Vecteurs. Algèbre linéaire. MATHEMATICS Differential Equations. MATHEMATICS Algebra Linear. Álgebra lineal Matrices no negativas Algebras, Linear Eigenvalues Eigenvectors Non-negative matrices |
| work_keys_str_mv | AT lemmensbas nonlinearperronfrobeniustheory AT nussbaumrogerd nonlinearperronfrobeniustheory |