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El. knyga: Discrete Painleve Equations

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Discrete Painleve equations are nonlinear difference equations, which arise from translations on crystallographic lattices. The deceptive simplicity of this statement hides immensely rich mathematical properties, connecting dynamical systems, algebraic geometry, Coxeter groups, topology, special functions theory, and mathematical physics.

This book necessarily starts with introductory material to give the reader an accessible entry point to this vast subject matter. It is based on lectures that the author presented as principal lecturer at a Conference Board of Mathematical Sciences and National Science Foundation conference in Texas in 2016. Instead of technical theorems or complete proofs, the book relies on providing essential points of many arguments through explicit examples, with the hope that they will be useful for applied mathematicians and physicists.
Preface v
Chapter 1 Introduction
1(16)
1.1 Background
1(3)
1.2 A brief history of the Painleve equations
4(3)
1.3 Recurrence Relations
7(2)
1.4 Three types of difference equations
9(1)
1.5 Birational Maps
10(1)
1.6 Identification of discrete Painleve equations
11(6)
Chapter 2 A dynamical systems approach
17(8)
2.1 A prototypical example
17(2)
2.2 Vector fields
19(2)
2.3 Vector bundles
21(2)
2.4 Dynamics of a discrete Painleve equation
23(2)
Chapter 3 Initial value spaces
25(20)
3.1 The Weierstrass cubic pencil
25(2)
3.2 Blowing up a point
27(2)
3.3 Resolution of singularities
29(3)
3.4 Resolution of base points
32(4)
3.5 Biquadratic pencils and QRT maps
36(2)
3.6 Geometric properties of the initial value space
38(7)
Chapter 4 Foliated initial value spaces
45(14)
4.1 A perturbation of the Weierstrass pencil
45(9)
4.2 A discrete foliation
54(5)
Chapter 5 Cremona mappings
59(16)
5.1 Cremona isometries
60(1)
5.2 Symmetry actions
61(3)
5.3 Translations
64(2)
5.4 The case (A2 + A1)(1)
66(4)
5.5 Construction of a discrete Painleve equation
70(5)
Chapter 6 Asymptotic analysis
75(12)
6.1 Asymptotics of an additive discrete Painleve equation
76(4)
6.2 Asymptotics of a multiplicative discrete Painleve equation
80(7)
Chapter 7 Lax pairs
87(8)
7.1 Background
87(2)
7.2 Lax Pair for dPI
89(2)
7.3 Lax Pairs for qPIII and qPVI
91(1)
7.4 Lax Pairs for qPIV
92(3)
Chapter 8 Riemann-Hilbert problems
95(10)
8.1 Basic definitions and notation
95(2)
8.2 Additive discrete linear problems
97(3)
8.3 Multiplicative discrete linear problems
100(5)
Appendix A Foliations and vector bundles
105(2)
A.1 Foliations
105(1)
A.2 Vector bundles
105(2)
Appendix B Projective spaces
107(8)
B.1 The two dimensional projective plane
107(2)
B.2 Homogeneous Polynomials
109(2)
B.3 General Algebraic Curves
111(1)
B.4 Resolutions and Intersection Theory
112(3)
Appendix C Reflection groups
115(4)
C.1 Cartan matrices
115(1)
C.2 Root systems and reflection groups
116(1)
C.3 Affine Weyl Groups
117(1)
C.4 Coxeter Relations
118(1)
Appendix D Lists of discrete Painleve equations
119(6)
D.1 List of additive discrete Painleve equations
119(2)
D.2 List of q-discrete Painleve Equations
121(2)
D.3 The elliptic discrete Painleve Equation
123(2)
Appendix E Asymptotics of discrete equations
125(12)
E.1 Introduction
125(3)
E.2 Ordinary Differential Equations
128(3)
E.3 Ordinary Difference Equations
131(6)
Bibliography 137(8)
Index 145
Nalini Joshi, University of Sydney, Australia.
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