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E-raamat: Nonlinear Perron-Frobenius Theory

(University of Kent, Canterbury), (Rutgers University, New Jersey)
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Nonlinear Perron-Frobenius TheorySuurem pilt
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In the past several decades the classical PerronFrobenius theory for nonnegative matrices has been extended to obtain remarkably precise and beautiful results for classes of nonlinear maps. This nonlinear PerronFrobenius 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 PerronFrobenius 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 PerronFrobenius theory, but pointers are provided to infinite dimensional results. Prerequisites are little more than basic real analysis and topology.

Arvustused

'In their introduction the authors state that 'the main purpose of this book is to give a systematic self-contained introduction to nonlinear PerronFrobenius theory and to provide a guide to various challenging open problems'. They have achieved their aim excellently.' Hans Schneider, University of Wisconsin, Madison 'Undoubtedly, this remarkable book will be of interest to all specialists in nonlinear analysis and its applications. Certainly, any mathematical library ought to carry this book.' Peter Zabreiko, Zentralblatt MATH '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 PerronFrobenius theory.' Bjorn S. Ruffer, Mathematical Reviews

Muu info

Guides the reader through the nonlinear PerronFrobenius theory, introducing them to recent developments and challenging open problems.
Preface ix
1 What is nonlinear Perron--Frobenius theory?
1(25)
1.1 Classical Perron--Frobenius theory
1(6)
1.2 Cones and partial orderings
7(4)
1.3 Order-preserving maps
11(2)
1.4 Subhomogeneous maps
13(4)
1.5 Topical maps
17(6)
1.6 Integral-preserving maps
23(3)
2 Non-expansiveness and nonlinear Perron--Frobenius theory
26(32)
2.1 Hilbert's and Thompson's metrics
26(8)
2.2 Polyhedral cones
34(4)
2.3 Lorentz cones
38(5)
2.4 The cone of positive-semidefinite symmetric matrices
43(2)
2.5 Completeness
45(3)
2.6 Convexity and geodesics
48(7)
2.7 Topical maps and the sup-norm
55(1)
2.8 Integral-preserving maps and the l1-norm
56(2)
3 Dynamics of non-expansive maps
58(23)
3.1 Basic properties of non-expansive maps
58(9)
3.2 Fixed-point theorems for non-expansive maps
67(3)
3.3 Horofunctions and horoballs
70(4)
3.4 A Denjoy--Wolff type theorem
74(3)
3.5 Non-expansive retractions
77(4)
4 Sup-norm non-expansive maps
81(15)
4.1 The size of the ω-limit sets
81(4)
4.2 Periods of periodic points
85(5)
4.3 Iterates of topical maps
90(6)
5 Eigenvectors and eigenvalues of nonlinear cone maps
96(24)
5.1 Extensions of order-preserving maps
96(4)
5.2 The cone spectrum
100(6)
5.3 The cone spectral radius
106(6)
5.4 Eigenvectors corresponding to the cone spectral radius
112(3)
5.5 Continuity of the cone spectral radius
115(3)
5.6 A Collatz--Wielandt formula
118(2)
6 Eigenvectors in the interior of the cone
120(41)
6.1 First principles
120(7)
6.2 Perturbation method
127(7)
6.3 Bounded invariant sets
134(3)
6.4 Uniqueness of the eigenvector
137(9)
6.5 Convergence to a unique eigenvector
146(8)
6.6 Means and their eigenvectors
154(7)
7 Applications to matrix scaling problems
161(22)
7.1 Matrix scaling: a fixed-point approach
161(5)
7.2 The compatibility condition
166(7)
7.3 Special DAD theorems
173(3)
7.4 Doubly stochastic matrices: the classic case
176(4)
7.5 Scaling to row stochastic matrices
180(3)
8 Dynamics of subhomogeneous maps
183(29)
8.1 Iterations on polyhedral cones
183(5)
8.2 Periodic orbits in polyhedral cones
188(8)
8.3 Denjoy--Wolff theorems for cone maps
196(8)
8.4 A Denjoy--Wolff theorem for polyhedral cones
204(8)
9 Dynamics of integral-preserving maps
212(43)
9.1 Lattice homomorphisms
212(5)
9.2 Periodic orbits of lower semi-lattice homomorphisms
217(8)
9.3 Periodic points and admissible arrays
225(16)
9.4 Computing periods of admissible arrays
241(7)
9.5 Maps on the whole space
248(7)
Appendix A The Birkhoff--Hopf theorem
255(29)
A.1 Preliminaries
255(3)
A.2 Almost Archimedean cones
258(1)
A.3 Projective diameter
259(4)
A.4 The Birkhoff--Hopf theorem: reduction to two dimensions
263(6)
A.5 Two-dimensional cones
269(4)
A.6 Completion of the proof of the Birkhoff--Hopf theorem
273(6)
A.7 Eigenvectors of cone-linear maps
279(5)
Appendix B Classical Perron--Frobenius theory
284(16)
B.1 A general version of Perron's theorem
284(5)
B.2 The finite-dimensional Krein--Rutman theorem
289(1)
B.3 Irreducible linear maps
290(1)
B.4 The peripheral spectrum
291(9)
Notes and comments 300(7)
References 307(12)
List of symbols 319(2)
Index 321
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 PerronFrobenius theory. 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 PerronFrobenius theory.
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