動(dòng)力系統(tǒng)VIII:奇異理論II—應(yīng)用

前言

要使我國(guó)的數(shù)學(xué)事業(yè)更好地發(fā)展起來，需要數(shù)學(xué)家淡泊名利并付出更艱苦地努力。另一方面，我們也要從客觀上為數(shù)學(xué)家創(chuàng)造更有利的發(fā)展數(shù)學(xué)事業(yè)的外部環(huán)境，這主要是加強(qiáng)對(duì)數(shù)學(xué)事業(yè)的支持與投資力度，使數(shù)學(xué)家有較好的工作與生活條件，其中也包括改善與加強(qiáng)數(shù)學(xué)的出版工作。從出版方面來講，除了較好較快地出版我們自己的成果外，引進(jìn)國(guó)外的先進(jìn)出版物無疑也是十分重要與必不可少的。從數(shù)學(xué)來說，施普林格（springer）出版社至今仍然是世界上最具權(quán)威的出版社?？茖W(xué)出版社影印一批他們出版的好的新書，使我國(guó)廣大數(shù)學(xué)家能以較低的價(jià)格購(gòu)買，特別是在邊遠(yuǎn)地區(qū)工作的數(shù)學(xué)家能普遍見到這些書，無疑是對(duì)推動(dòng)我國(guó)數(shù)學(xué)的科研與教學(xué)十分有益的事。這次科學(xué)出版社購(gòu)買了版權(quán)，一次影印了23本施普林格出版社出版的數(shù)學(xué)書，就是一件好事，也是值得繼續(xù)做下去的事情。大體上分一下，這23本書中，包括基礎(chǔ)數(shù)學(xué)書5本，應(yīng)用數(shù)學(xué)書6本與計(jì)算數(shù)學(xué)書12本，其中有些書也具有交叉性質(zhì)。這些書都是很新的，2000年以后出版的占絕大部分，共計(jì)16本，其余的也是1990年以后出版的。這些書可以使讀者較快地了解數(shù)學(xué)某方面的前沿，例如基礎(chǔ)數(shù)學(xué)中的數(shù)論、代數(shù)與拓?fù)淙荆际怯稍擃I(lǐng)域大數(shù)學(xué)家編著的“數(shù)學(xué)百科全書”的分冊(cè)。對(duì)從事這方面研究的數(shù)學(xué)家了解該領(lǐng)域的前沿與全貌很有幫助。按照學(xué)科的特點(diǎn)，基礎(chǔ)數(shù)學(xué)類的書以“經(jīng)典”為主，應(yīng)用和計(jì)算數(shù)學(xué)類的書以“前沿”為主。這些書的作者多數(shù)是國(guó)際知名的大數(shù)學(xué)家，例如《拓?fù)鋵W(xué)》一書的作者諾維科夫是俄羅斯科學(xué)院的院士，曾獲“菲爾茲獎(jiǎng)”和“沃爾夫數(shù)學(xué)獎(jiǎng)”。這些大數(shù)學(xué)家的著作無疑將會(huì)對(duì)我國(guó)的科研人員起到非常好的指導(dǎo)作用。當(dāng)然，23本書只能涵蓋數(shù)學(xué)的一部分，所以，這項(xiàng)工作還應(yīng)該繼續(xù)做下去。更進(jìn)一步，有些讀者面較廣的好書還應(yīng)該翻譯成中文出版，使之有更大的讀者群?？傊?，我對(duì)科學(xué)出版社影印施普林格出版社的部分?jǐn)?shù)學(xué)著作這一舉措表示熱烈的支持，并盼望這一工作取得更大的成績(jī)。

內(nèi)容概要

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.

作者簡(jiǎn)介

作者：(俄羅斯)阿諾德 (Arnol'd.V.L)

書籍目錄

ForewordChapter 1. Classification of Functions and Mappings  1. Functions on a Manifold with Boundary    1.1. Classification of Functions on a Manifold with a Smooth Boundary    1.2. Versal Deformations and Bifurcation Diagrams    1.3. Relative Homology Basis    1.4. Intersection Form    1.5. Duality of Boundary Singularities    1.6. Functions on a Manifold with a Singular Boundary  2. Complete Intersections    2.1.  Start of the Classification    2.2.  Critical and Discriminant Sets    2.3.  The Nonsingular Fiber    2.4.  Relations Between the'Tyurina and Milnor Numbers    2.5.  Adding a Power of a New Variable    2.6.  Relative Monodromy    2.7.  Dynkin Diagrams    2.8.  Parabolic and Hyperbolic Singularities    2.9.  Vector Fields on a Quasihomogeneous Complete Intersection    2.10. The Space of a Miniversal Deformation of a Quasihomogeneous Singularity    2.11. Topological Triviality of Versal Deformations  3. Projections and Left-Right Equivalence    3.1. Projections of Space Curves onto the Plane    3.2. Singularities of Projections of Surfaces onto the Plane    3.3. Projections of Complete Intersections    3.4. Projections onto the Line    3.5. Mappings of the Line into the Plane    3.6. Mappings of the Plane into Three-Space  4. Nonisolated Singularities of Functions    4.1. Transversal Type of a Singularity    4.2. Realization    4.3. Topology of the Nonsingular Fiber    4.4. Series of Isolated Singularities    4.5. The Number of Indices of a Series    4.6. Functions with a One-Dimensional Complete Intersection as Critical Set and with Transversal Type A1  5. Vector Fields Tangent to Bifurcation Varieties    5.1. Functions on Smooth Manifolds    5.2. Projections onto the Line    5.3. Isolated Singularities of Complete Intersections    5.4. The Equation of a Free Divisor  6. Divergent and Cyclic Diagrams of Mappings    6.1. Germs of Smooth Functions    6.2. Envelopes    6.3. Holomorphic DiagramsChapter 2. Applications of the Classification of Critical Points of Functions  1. Legendre Singularities    1.1. Equidistants    1.2. Projective Duality    1.3. Legendre Transformation    1.4. Singularities of Pedals and Primitives    1.5. The Higher-Dimensional Case  2. Lagrangian Singularities    2.1. Caustics    2.2. The Manifold of Centers    2.3. Caustics of Systems of Rays    2.4. The Gauss Map    2.5. Caustics of Potential Systems of Noninteracting Particles    2.6. Coexistence of Singularities  3. Singularities of Maxwell Sets    3.1. Maxwell Sets    3.2. Metamorphoses of Maxwell Sets    3.3. Extended Maxwell Sets    3.4. Complete Maxwell Set Close to the Singularity A5    3.5. The Structure of Maxwell Sets Close to the Metamorphosis As    3.6. Enumeration of the Connected Components of Spaces of    Nondegenerate Polynomials  4. Bifurcations of Singular Points of Gradient Dynamical Systems    4.1. Thorn's Conjecture    4.2. Singularities of Corank One    4.3. Guckenheimer's Counterexample    4.4. Three-Parameter Families of Gradients    4.5. Normal Forms of Gradient Systems D4    4.6. Bifurcation Diagrams and Phase Portaits of Standard Families.    4.7. Multiparameter FamiliesChapter 3. Singularities of the Boundaries of Domains of Function Spaces  1. Boundary of Stability    1.1. Domains of Stability    1.2. Singularities of the Boundary of Stability in Low-Dimensional Spaces    1.3. Stabilization Theorem    1.4. Finiteness Theorem  2. Boundary of Ellipticity    2.1. Domains of Ellipticity    2.2. Stabilization Theorems    2.3. Boundaries of Ellipticity and Minimum Functions    2.4. Singularities of the Boundary of Ellipticity in Low-Dimensional Spaces  3. Boundary of Hyperbolicity    3.1. Domain of Hyperbolicity    3.2. Stabilization Theorems    3.3. Local Hyperbolicity    3.4. Local Properties of Domains of Hyperbolicity  4. Boundary of the Domain of Fundamental Systems    4.1. Domain of Fundamental Systems and the Bifurcation Set    4.2. Singularities of Bifurcation Sets of Generic Three-Parameter Families    4.3. Bifurcation Sets and Schubert Cells    4.4. Normal Forms    4.5. Duality    4.6. Bifurcation Sets and Tangential Singularities    4.7. The Group of Transformations of Sets and Finite Determinacy    4.8. Bifurcation Diagrams of Flattenings of Projective Curves  5. Linear Differential Equations and Complete Flag ManifoldsChapter 4. Applications of Ramified Integrals and Generalized Picard- Lefschetz Theories  1. Newton's Theorem on Nonintegrability    1.1.  Newton's Theorem and Archimedes's Example    1.2.  Multi-dimensional Newton Theorem (Even Case)    1.3.  Obstructions to Inegrability in the Odd-Dimensional Case    1.4.  Newton's Theorem for Nonconvex Domains    1.5.  The Case of Nonsmooth Domains    1.6.  Homological Formulation and the General Statement ofthe Problem    1.7.  Localization and Lowering the Dimension in the Calculation of Monondromy    1.8.  General Construction of the Variation Operators    1.9.  The "Cap" Element    1.10. Ramification of Cycles Close to Nonsingular Points    1.11. Ramification Close to Individual Singularities    1.12. Stabilization of Monodromy Close to Strata of Positive Dimension    1.13. Ramification Around the Asymptotic Directions and Monodromy of Boundary Singularities    1.14. Pham's Formulas    1.15. Problems, Conjectures, Complements  2. Ramification of Solutions of Hyperbolic Equations    2.1.  Hyperbolic Operators and Hyperbolic Polynomials    2.2.  Wave Front of a Hyperbolic Operator    2.3.  Singularities of Wave Fronts and Generating Functions    2.4.  Lacunas, Sharpness, Diffusion    2.5.  Sharpness and Diffusion Close to the Simplest Singularities of Wave Fronts    2.6.  The Herglotz-Petrovskii-Leray Integral Formula    2.7.  The Petrovskii Criterion    2.8.  Local Petrovskii Criterion    2.9.  Local Petrovskii Cycle    2.10. C∞-Inversion of the Petrovskii Criterion, Stable Singularities of Fronts and Sneaky Diffusion    2.11. Normal Forms of Nonsharpness Close to Singularities of Wave Fronts    2.12. Construction of Leray and Petrovskii Cycles for Strictly Hyperbolic Polynomials    2.13. Problems  3. Integrals of Ramified Forms and Monodromy of Homology with Nontrivial Coefficients    3.1. The Hypergeometric Function of Gauss    3.2. Homology of Local Systems    3.3. Meromorphy of the Integral of the Function Pλ    3.4. The Integral of the Function pλ as a Function of P    3.5. Monodromy and Linear Independence of Hypergeometric Functions    3.6. Twisted Picard-Lefschetz Theory of Isolated Singularities of Smooth Functions and Representations of Hecke AlgebrasChapter 5. Deformations of Real Singularities and Local Petrovskii Lacunas  1. Local Petrovskii Cycles and their Properties    1.1. Definition of Local Petrovskii Cycles    1.2. Complex Conjugation    1.3. Boundary of the Petrovskii Class    1.4. Computation of Petrovskii Cocycles in Terms of Vanishing Cycles    1.5. Stabilization  2. Local Lacunas for Concrete Singularities    2.1. Local Lacunas for Singularities that are Stably Equivalent to Extrema    2.2. The Number of Local Lacunas for the Tabulated Singularit  3. Complements of Discriminants of Real Singularities    3.1. Components of the Complement of the Discriminant of Simple Singularities    3.2. A Regular Search Algorithm for Morse Decompositions of Singularities    3.3. Remarks on the Realization of the Algorithm    3.4. Problems and PerspectivesBibliographyAuthor IndexSubject Index

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