Singularities of Differentiable Maps, Volume 1

Classification of Critical Points, Caustics and Wave Fronts

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Author: V.I. Arnold,S.M. Gusein-Zade,Alexander N. Varchenko

Publisher: Springer Science & Business Media

ISBN: 0817683402

Category: Mathematics

Page: 282

View: 6208

​Singularity theory is a far-reaching extension of maxima and minima investigations of differentiable functions, with implications for many different areas of mathematics, engineering (catastrophe theory and the theory of bifurcations), and science. The three parts of this first volume of a two-volume set deal with the stability problem for smooth mappings, critical points of smooth functions, and caustics and wave front singularities. The second volume describes the topological and algebro-geometrical aspects of the theory: monodromy, intersection forms, oscillatory integrals, asymptotics, and mixed Hodge structures of singularities. The first volume has been adapted for the needs of non-mathematicians, presupposing a limited mathematical background and beginning at an elementary level. With this foundation, the book's sophisticated development permits readers to explore more applications than previous books on singularities.

Singularities of Differentiable Maps

Volume I: The Classification of Critical Points Caustics and Wave Fronts

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Author: V.I. Arnold,A.N. Varchenko,S.M. Gusein-Zade

Publisher: Springer Science & Business Media

ISBN: 1461251540

Category: Mathematics

Page: 396

View: 4524

... there is nothing so enthralling, so grandiose, nothing that stuns or captivates the human soul quite so much as a first course in a science. After the first five or six lectures one already holds the brightest hopes, already sees oneself as a seeker after truth. I too have wholeheartedly pursued science passionately, as one would a beloved woman. I was a slave, and sought no other sun in my life. Day and night I crammed myself, bending my back, ruining myself over my books; I wept when I beheld others exploiting science fot personal gain. But I was not long enthralled. The truth is every science has a beginning, but never an end - they go on for ever like periodic fractions. Zoology, for example, has discovered thirty-five thousand forms of life ... A. P. Chekhov. "On the road" In this book a start is made to the "zoology" of the singularities of differentiable maps. This theory is a young branch of analysis which currently occupies a central place in mathematics; it is the crossroads of paths leading from very abstract corners of mathematics (such as algebraic and differential geometry and topology, Lie groups and algebras, complex manifolds, commutative algebra and the like) to the most applied areas (such as differential equations and dynamical systems, optimal control, the theory of bifurcations and catastrophes, short-wave and saddle-point asymptotics and geometrical and wave optics).

Singularities of Differentiable Maps, Volume 2

Monodromy and Asymptotics of Integrals

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Author: Elionora Arnold,S.M. Gusein-Zade,Alexander N. Varchenko

Publisher: Springer Science & Business Media

ISBN: 0817683437

Category: Mathematics

Page: 492

View: 365

​​​The present volume is the second in a two-volume set entitled Singularities of Differentiable Maps. While the first volume, subtitled Classification of Critical Points and originally published as Volume 82 in the Monographs in Mathematics series, contained the zoology of differentiable maps, that is, it was devoted to a description of what, where, and how singularities could be encountered, this second volume concentrates on elements of the anatomy and physiology of singularities of differentiable functions. The questions considered are about the structure of singularities and how they function.

Homogeneous Structures on Riemannian Manifolds

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Author: F. Tricerri,L. Vanhecke

Publisher: Cambridge University Press

ISBN: 0521274893

Category: Mathematics

Page: 125

View: 5510

The central theme of this book is the theorem of Ambrose and Singer, which gives for a connected, complete and simply connected Riemannian manifold a necessary and sufficient condition for it to be homogeneous. This is a local condition which has to be satisfied at all points, and in this way it is a generalization of E. Cartan's method for symmetric spaces. The main aim of the authors is to use this theorem and representation theory to give a classification of homogeneous Riemannian structures on a manifold. There are eight classes, and some of these are discussed in detail. Using the constructive proof of Ambrose and Singer many examples are discussed with special attention to the natural correspondence between the homogeneous structure and the groups acting transitively and effectively as isometrics on the manifold.

Real and Complex Singularities

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Author: W. L. Marar

Publisher: CRC Press

ISBN: 9780582277809

Category: Mathematics

Page: 224

View: 1477

Much progress has been made recently in a number of areas by the application of new geometrical methods arising from advances in singularity theory. This collection of invited papers presented at the 3rd International Workshop on Real and Complex Singularities, held in August 1994 at ICMSC-USP (Sao Carlos), documents the geometric study of singularities and its applications.

Theorems on Regularity and Singularity of Energy Minimizing Maps

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Author: Leon Simon

Publisher: Birkhäuser

ISBN: 3034891938

Category: Mathematics

Page: 152

View: 5643

The aim of these lecture notes is to give an essentially self-contained introduction to the basic regularity theory for energy minimizing maps, including recent developments concerning the structure of the singular set and asymptotics on approach to the singular set. Specialized knowledge in partial differential equations or the geometric calculus of variations is not required; a good general background in mathematical analysis would be adequate preparation.

Topology of Singular Fibers of Differentiable Maps

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Author: Osamu Saeki

Publisher: Springer

ISBN: 3540446486

Category: Mathematics

Page: 154

View: 8124

The volume develops a thorough theory of singular fibers of generic differentiable maps. This is the first work that establishes the foundational framework of the global study of singular differentiable maps of negative codimension from the viewpoint of differential topology. The book contains not only a general theory, but also some explicit examples together with a number of very concrete applications. This is a very interesting subject in differential topology, since it shows a beautiful interplay between the usual theory of singularities of differentiable maps and the geometric topology of manifolds.