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Positive Operator Semigroups: From Finite to Infinite Dimensions 1st ed. 2017 [Kietas viršelis]

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  • Formatas: Hardback, 364 pages, aukštis x plotis: 235x155 mm, weight: 6978 g, XVIII, 364 p., 1 Hardback
  • Serija: Operator Theory: Advances and Applications 257
  • Išleidimo metai: 20-Feb-2017
  • Leidėjas: Birkhauser Verlag AG
  • ISBN-10: 331942811X
  • ISBN-13: 9783319428116
Kitos knygos pagal šią temą:
Positive Operator Semigroups: From Finite to Infinite Dimensions 1st ed. 2017Didesnis paveikslėlis
  • Formatas: Hardback, 364 pages, aukštis x plotis: 235x155 mm, weight: 6978 g, XVIII, 364 p., 1 Hardback
  • Serija: Operator Theory: Advances and Applications 257
  • Išleidimo metai: 20-Feb-2017
  • Leidėjas: Birkhauser Verlag AG
  • ISBN-10: 331942811X
  • ISBN-13: 9783319428116
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9783319428116.html
Raktažodžiai:
This book gives a gentle but up-to-date introduction into the theory of operator semigroups (or linear dynamical systems), which can be used with great success to describe the dynamics of complicated phenomena arising in many applications. Positivity is a property which naturally appears in physical, chemical, biological or economic processes. It adds a beautiful and far reaching mathematical structure to the dynamical systems and operators describing these processes.  In the first part, the finite dimensional theory in a coordinate-free way is developed, which is difficult to find in literature. This is a good opportunity to present the main ideas of the Perron-Frobenius theory in a way which can be used in the infinite dimensional situation. Applications to graph matrices, age structured population models and economic models are discussed.  The infinite dimensional theory of positive operator semigroups with their spectral and asymptotic theory is developed in the second part.

Recent applications illustrate the theory, like population equations, neutron transport theory, delay equations or flows in networks. Each chapter is accompanied by a large set of exercises, many of them with solutions. An up-to-date bibliography and a detailed subject index help the interested reader.  The book is intended primarily for graduate and master students. The finite dimensional part, however, can be followed by an advanced bachelor with a solid knowledge of linear algebra and calculus.

1 An Invitation to Positive Matrices.- 2 Functional Calculus.- 3 Powers of Matrices.- 4 Matrix Exponential Function.- 5 Positive Matrices.- 6 Applications of Positive Matrices.- 7 Positive Matrix Semigroups and Applications.- 8 Positive Linear Systems.- 9 Banach Lattices.- 10 Positive Operators.- 11 Operator Semigroups.- 12 Generation Properties.- 13 Spectral Theory for Positive Semigroups I.- 14 Spectral Theory for Positive Semigroups II.- 15 An application to linear transport equations.- Appendices.- Index.

Recenzijos

The book covers a remarkable range of material, from basic to relatively advanced, and its coherent and streamlined exposition of both the classical theory and modern applications makes it a very useful addition to the literature . Overall, the book works very well as a modern introduction to the theory of positive operator semigroups, and its broad scope and application-driven approach are likely to make it a standard reference work for newcomers to the subject and experts alike. (David Seifert, zbMATH 1420.47001, 2019) The book gathers diverse material around the subject of positive operator semigroups and illustrates it well with a large number of evolution equations, to give in the end an original contribution to the literature. Each section contains notes and remarks, and ends with exercises. Mathematicians from a graduate student level on should find this book useful. (Christoph Kriegler, Mathematical Reviews, February,2018)

Foreword vii
Preface xv
Part I Finite Dimensions
1 An Invitation to Positive Matrices
1.1 Motivating Examples
3(3)
1.2 Convergence
6(6)
1.3 Notes and Remarks
12(1)
1.4 Exercises
12(3)
2 Functional Calculus
2.1 Polynomials
15(3)
2.2 Smooth Functions
18(6)
2.3 Spectral Theory
24(5)
2.4 Notes and Remarks
29(1)
2.5 Exercises
29(2)
3 Powers of Matrices
3.1 The Coordinate Sequences
31(2)
3.2 The Spectral Radius
33(2)
3.3 Asymptotics
35(5)
3.4 Notes and Remarks
40(1)
3.5 Exercises
41(2)
4 The Matrix Exponential Function
4.1 Main Properties
43(3)
4.2 Coordinate Functions
46(2)
4.3 The Spectral Bound
48(1)
4.4 Asymptotics
49(4)
4.5 Notes and Remarks
53(1)
4.6 Exercises
53(2)
5 Positive Matrices
5.1 Positivity
55(4)
5.2 Irreducibility
59(4)
5.3 Imprimitivity
63(4)
5.4 Notes and Remarks
67(1)
5.5 Exercises
68(1)
6 Applications of Positive Matrices
6.1 Motivating Examples Revisited
69(5)
6.2 The Google Matrix
74(2)
6.3 Age-structured Population Models
76(3)
6.4 Notes and Remarks
79(1)
6.5 Exercises
79(2)
7 Positive Matrix Semigroups and Applications
7.1 Positive Semigroups
81(4)
7.2 The Competitive Market Model
85(1)
7.3 Queueing Models
85(3)
7.4 Disease Transition Models
88(1)
7.5 Discrete Maximum Principles
89(1)
7.6 Notes and Remarks
90(1)
7.7 Exercises
90(3)
8 Positive Linear Systems
8.1 Externally and Internally Positive Systems
93(6)
8.2 Controllability
99(4)
8.3 Stabilization
103(1)
8.4 Notes and Remarks
104(1)
8.5 Exercises
104(5)
Part II Infinite Dimensions
9 A Crash Course on Operator Semigroups
9.1 Exponential Functions
109(3)
9.2 Motivation for Generalizations
112(3)
9.3 Basic Properties
115(3)
9.4 The Infinitesimal Generator
118(6)
9.5 Multiplication Semigroups
124(3)
9.6 Gaussian Semigroup
127(4)
9.7 Resolvent of a Generator
131(4)
9.8 Adjoint Semigroups
135(2)
9.9 Notes and Remarks
137(1)
9.10 Exercises
138(3)
10 Banach Lattices and Positive Operators
10.1 Ordered Function Spaces
141(4)
10.2 Vector Lattices
145(2)
10.3 Banach Lattices
147(4)
10.4 Sublattices and Ideals
151(3)
10.5 Complexification of Real Banach Lattices
154(1)
10.6 Positive Operators
155(3)
10.7 Positive Exponential Functions
158(4)
10.8 Notes and Remarks
162(1)
10.9 Exercises
162(3)
11 Generation Properties
11.1 The Hille--Yosida Generation Theorem
165(3)
11.2 Bounded Perturbations
168(3)
11.3 Positive Contraction Semigroups
171(7)
11.4 Positive Minimum Principle
178(1)
11.5 Notes and Remarks
179(1)
11.6 Exercises
179(2)
12 Spectral Theory for Positive Semigroups
12.1 Asymptotic Stability of Semigroups
181(4)
12.2 The Spectral Bound for Positive Semigroups
185(7)
12.3 The Identity ω0(T) = s(A) for Positive Semigroups
192(2)
12.4 Notes and Remarks
194(1)
12.5 Exercises
195(2)
13 Unbounded Positive Perturbations
13.1 Unbounded Dispersive Perturbations
197(3)
13.2 Miyadera Perturbations
200(3)
13.3 Positive Perturbations in L1
203(6)
13.4 Notes and Remarks
209(1)
13.5 Exercises
210(3)
Part III Advanced Topics and Applications
14 Advanced Spectral Theory and Asymptotics
14.1 Spectral Decomposition
213(6)
14.2 Periodic Semigroups
219(3)
14.3 Irreducible Semigroups
222(8)
14.4 Asymptotic Behavior
230(2)
14.5 Notes and Remarks
232(1)
14.6 Exercises
233(2)
15 Positivity and Delay Equations
15.1 Abstract Delay Equations
235(6)
15.2 Gearhart's Theorem
241(4)
15.3 Stability of Delay Equations
245(6)
15.4 Notes and Remarks
251(1)
15.5 Exercises
252(1)
16 Koopman Semigroups
16.1 Ordinary Differential Equations and Semiflows
253(4)
16.2 Koopman Semigroups
257(6)
16.3 Applications of Koopman Semigroups
263(3)
16.4 Notes and Remarks
266(1)
16.5 Exercises
266(3)
17 Linear Boltzmann Transport Equations with Scattering
17.1 The Reactor Problem
269(1)
17.2 The One-dimensional Reactor Problem
270(5)
17.3 Notes and Remarks
275(1)
17.4 Exercises
276(3)
18 Transport Problems in Networks
18.1 The Model and the Associated Abstract Cauchy Problem
279(5)
18.2 The Simple Case cj = 1
284(4)
18.3 The General Case
288(6)
18.4 Vertex Control in Networks
294(5)
18.5 Notes and Remarks
299(1)
18.6 Exercises
299(4)
19 Population Equations with Diffusion
19.1 The Mathematical Model
303(2)
19.2 Hille--Yosida Operators and Extrapolated Semigroups
305(2)
19.3 Spectral Properties of Perturbed Hille--Yosida Operators
307(5)
19.4 Evolution Equations with Boundary Perturbations
312(6)
19.5 Back to the Population Equation
318(6)
19.6 Notes and Remarks
324(1)
Appendix Background Material from Linear Algebra and Functional Analysis
325(22)
A.1 Basic Linear Algebra
325(2)
A.2 Reducing Subspaces and Projections
327(2)
A.3 Interpolation Polynomials
329(1)
A.4 Function Spaces
330(2)
A.5 The Strong Operator Topology
332(1)
A.6 Some Classical Theorems
333(2)
A.7 Riemann Integral
335(1)
A.8 Dual Spaces and Adjoint Operators
336(2)
A.9 Spectrum, Essential Spectrum, and Compact Operators
338(4)
A.10 Bochner Integral, Laplace and Fourier Transforms
342(2)
A.11 Distributions and Sobolev Spaces
344(3)
Bibliography 347(10)
Index 357
Andrįs Bįtkai was born in Budapest, Hungary, in 1972, received his PhD in 2000 in Tübingen and is currently associate professor of mathematics at the Eötvös Lorįnd University Budapest. He is mainly interested in the theory and applications of operator semigroup theory, in paricular in applications to delay equations.He is the author of a research monograph and over 25 research papers, and the editor of the Open Mathematics journal. He was a fellow of the Alexander-von Humboldt Stiftung, held several Marie-Curie postdoctoral fellowships and and the Alexits prize of the Hungarian Academy of Sciences.



Marjeta Kramar Fijav was born in 1973 in Ljubljana, Slovenia. She received her PhD in Mathematics in 2004 at University of Ljubljana and has been associate professor of mathematics there since 2013. Her primary interest is in linear algebra and operator theory, in particular operator semigroups, evolution equations, and dynamical networks.



Abdelaziz Rhandi was born in Casablanca, Morocco, in 1964. He received the  first Ph.D degree from the University of Besanēon (France) and the second Ph.D degree from the University of Tübingen (Germany). Presently, he is full professor of analysis at the University of Salerno (Italy). His main interest is in applied functional analysis and partial differential equations. He is the author of more than 50 publications, and the editor of the journals Semigroup Forum and Positivity. He was the winner of the 2006 HP Technology for Teaching Higher Education and a fellow of the Alexander-von Humboldt Stiftung.