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內容簡介

　　This volume of the Encyclopaedia is devoted to applications of singularity theory in mathematics and physics. The authors Arnol'd，Vasil'ev， Goryunov and Lyashko study bifurcation sets arising in various contexts such as the stability of singular points of dynamical systems， boundaries of the domains of ellipticity and hyperbolicity of partial differential equations， boundaries of spaces of oscillating linear equations with variable coefficients and boundaries of fundamental systems of solutions.
　　The book also treats applications of the following topics： functions on manifolds with boundary， projections of complete intersections， caustics， wave fronts， evolvents， maximum functions， shock waves， Petrovskij lacunas and generalizations of Newton's topological proof that Abelian integrals are transcendental.
　　The book contains a list of open problems， conjectures and directions for future research.
　　It will be of great interest for mathematicians and physicists as a reference and research aid.

內頁插圖

目錄

Singularity Theory II Classification and Applications
V.I Arnol'd，V.V Goryunov，O.V Lyashko，V.A Vasil'ev
Translated from the Russian by J.S Joel
Contents
Foreword
Chapter 1. Classification of Functions and Mappings 8
1. Functions on a Manifold with Boundary 8
1.1. Classification of Functions on a Manifold with a Smooth Boundary 8
1.2. Versal Deformations and Bifurcation Diagrams 11
1.3. Relative Homology Basis 14
1.4. Intersection Form 14
1.5. Duality of Boundary Singularities 17
1.6. Functions on a Manifold with a Singular Boundary 17
2. Complete Intersections 20
2.1. Start of the Classification 21
2.2. Critical and Discriminant Sets 24
2.3. The Nonsingular Fiber 26
2.4. Relations Between the Tyurina and Milnor Numbers 28
2.5. Adding a Power of a New Variable 29
2.6. Relative Monodromy 29
2.7. Dynkin Diagrams 30
2.8. Parabolic and Hyperbolic Singularities 31
2.9. Vector Fields on a Quasihomogeneous Complete Intersection 33
2.10. The Space of a Miniversal Deformation of a Quasihomogeneous Singularity 35
2.11. Topological Triviality of Versal Deformations 36
3. Projections and Left-Right Equivalence 37
3.1. Projections of Space Curves onto the Plane 38
3.2. Singularities of Projections of Surfaces onto the Plane 39
3.3. Projections of Complete Intersections 43
3.4. Projections onto the Line 47
3.5. Mappings of the Line into the Plane 57
3.6. Mappings of the Plane into Three-Space 59
4. Nonisolated Singularities of Functions 65
4.1. Transversal Type of a Singularity 65
4.3. Topology of the Nonsingular Fiber 66
4.4. Series of Isolated Singularities 67
4.5. The Number of Indices of a Series 68
4.6. Functions with a One-Dimensional Complete Intersection as Critical Set and with Transversal Type Ai 69
5. Vector Fields Tangent to Bifurcation Varieties 79
5.1. Functions on Smooth Manifolds 79
5.2. Projections onto the Line 81
5.3. Isolated Singularities of Complete Intersections 82
5.4. The Equation of a Free Divisor 84
6. Divergent and Cyclic Diagrams of Mappings 84
6.1. Germs of Smooth Functions 85
6.2. Envelopes 85
6.3. Holopmorphic Diagrams 87
Chapter 2. Applications of the Classification of Critical Points of Functions 88
1. Legendre Singularities 88
1.1. Equidistants 89
1.2. Projective Duality 90
1.3. Legendre Transformation 90
1.4. Singularities of Pedals and Primitives 91
1.5. The Higher-Dimensional Case 91
2. Lagrangian Singularities 92
2.1. Caustics 92
2.2. The Manifold of Centers 93
2.3. Caustics of Systems of Rays 94
2.4. The Gauss Map 95
2.5. Caustics of Potential Systems of Noninteracting Particles 95
2.6. Coexistence of Singularities 97
3. Singularities of Maxwell Sets 98
3.1. Maxwell Sets 98
3.2. Metamorphoses of Maxwell Sets 100
3.3. Extended Maxwell Sets 103
3.4. Complete Maxwell Set Close to the Singularity As 106
3.5. The Structure of Maxwell Sets Close to the Metamorphosis As 110
3.6. Enumeration of the Connected Components of Spaces of Nondegenerate Polynomials 112
4. Bifurcations of Singular Points of Gradient Dynamical Systems 113
4.1. Thom's Conjecture 114
4.2. Singularities of Corank One 115
4.3. Guckenheimer's Counterexample 116
4.4. Three-Parameter Families of Gradients 117
4.5. Normal Forms of Gradient Systems D4 118
4.6. Bifurcation Diagrams and Phase Portaits of Standard Families 118
4.7. Multiparameter Families 120
Chapter 3. Singularities of the Boundaries of Domains of Function Spaces 121
1. Boundary of Stability 122
1.1. Domains of Stability 122
1.2. Singularities of the Boundary of Stability in Low-Dimensional Spaces 122
1.3. Stabilization Theorem 123
1.4. Finiteness Theorem 124
2. Boundary of Ellipticity 124
2.1. Domains of Ellipticity 124
2.2. Stabilization Theorems 124
2.3. Boundaries of Ellipticity and Minimum Functions 125
2.4. Singularities of the Boundary of Ellipticity in Low-Dimensional Spaces 126
3. Boundary of Hyperbolicity 127
3.1. Domain of Hyperbolicity 127
3.2. Stabilization Theorems 127
3.3. Local Hyperbolicity 128
3.4. Local Properties of Domains of Hyperbolicity 129
4. Boundary of the Domain of Fundamental Systems 131
4.1. Domain of Fundamental Systems and the Bifurcation Set 131
4.2. Singularities of Bifurcation Sets of
Generic Three-Parameter Families 132
4.3. Bifurcation Sets and Schubert Cells 136
4.4. Normal Forms 140
4.5. Duality 141
4.6. Bifurcation Sets and Tangential Singularities 142
4.7. The Group of Transformations of Sets and Finite Determinacy. 143
4.8. Bifurcation Diagrams of Flattenings of Projective Curves 145
S 5. Linear Differential Equations and Complete Flag Manifolds 146
Chapter 4. Applications of Ramified Integrals and Generalized Picard-Lefschetz Theories 149
1. Newton's Theorem on Nonintegrability 150
1.1. Newton's Theorem and Ar

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動力係統VIII 奇異理論II：應用 [Dynamical Systems Ⅷ: Singularity Theory Ⅱ:Applications] 下載 mobi epub pdf txt 電子書

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