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Mathematics - Geometry & Topology | Dynamical Systems VIII - Singularity Theory II. Applications

Dynamical Systems VIII

Singularity Theory II. Applications

Arnol'd, V.I. (Ed.)

Translated by Joel, J.S.

Original Russian edition published by Publisher VINITI, Moscow, 1989

1993, VI, 238 p.

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In the first volume of this survey (Arnol'd et al. (1988), hereafter cited as "EMS 6") we acquainted the reader with the basic concepts and methods of the theory of singularities of smooth mappings and functions. This theory has numerous applications in mathematics and physics; here we begin describing these applica­ tions. Nevertheless the present volume is essentially independent of the first one: all of the concepts of singularity theory that we use are introduced in the course of the presentation, and references to EMS 6 are confined to the citation of technical results. Although our main goal is the presentation of analready formulated theory, the readerwill also come upon some comparatively recent results, apparently unknown even to specialists. We pointout some of these results. 2 3 In the consideration of mappings from C into C in§ 3. 6 of Chapter 1, we define the bifurcation diagram of such a mapping, formulate a K(n, 1)-theorem for the complements to the bifurcation diagrams of simple singularities, give the definition of the Mond invariant N in the spirit of "hunting for invariants", and we draw the reader's attention to a method of constructing the image of a mapping from the corresponding function on a manifold with boundary. In§ 4. 6 of the same chapter we introduce the concept of a versal deformation of a function with a nonisolated singularity in the dass of functions whose critical sets are arbitrary complete intersections of fixed dimension.

Content Level » Research

Keywords » Bifurkationsmengen - Maxwellmannigfaltigkeiten - Singularitäten von Funktionen und Abbildungen - Singularitäten von Rändern von Differentialgleichungen - bifurcation sets - generalized Picard-Lefschetz theory - manifold - maximum - maxwell manifolds - singularitie

Related subjects » Algebra - Analysis - Geometry & Topology - Theoretical, Mathematical & Computational Physics

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Singularity Theory II Classification and Applications.

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