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Classifications of Singularities and Catastrophes

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Part of the Encyclopaedia of Mathematical Sciences book series (volume 5)

Abstract

At first glance, the most natural classification principle is classifying by codimension beginning with small codimensions. To classify objectsup to codimension ⩽ k” means to represent the entire space of objects studied as a finite union of submanifolds of codimensions not greater than k (called classes) and a remainder of codimension ⩾k + 1 so that within each class an object’s properties that are of interest to us do not change. Then all objects in typical, no more than k-parameter families, belong to our classes: the remaining ones may be avoided by a small perturbation of the family.

Keywords

Bifurcation Diagram Weyl Group Coxeter Group Catastrophe Theory Reflection Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 5a.
    The coefficient 11/ 10, forgotten in Khesin’s article, is necessary. Normal forms of the gradient families correspond for example to the values \( p = \pm 1{\text{ and}} - 1/2{\text{for }}D_4^ + \), and to the value \( p = 1{\text{ for }}D_4^ - \). Google Scholar
  2. 6.
    The investigations of this situation in catastrophe theory (R. Thorn (1972), L. Dara (1975) and F. Takens (1976)) basically repeat the work of M. Cibrario (1932), and A.A. Shestakov, and A.V. Pkhakadze (1959) carried out decades earlier.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteMoscowRussia

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