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Singularities of Differentiable Maps

Volume II Monodromy and Asymptotic Integrals

  • Editors
  • V. I. Arnold
  • S. M. Gusein-Zade
  • A. N. Varchenko

Part of the Monographs in Mathematics book series (MMA, volume 83)

Table of contents

  1. Front Matter
    Pages I-VIII
  2. The topological structure of isolated critical points of functions

    1. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 1-8
    2. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 9-28
    3. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 29-66
    4. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 67-113
    5. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 114-138
    6. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 139-167
  3. Oscillatory integrals

    1. Front Matter
      Pages 169-267
    2. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 170-214
    3. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 215-232
    4. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 233-262
    5. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 263-267
  4. Integrals of holomorphic forms over vanishing cycles

    1. Front Matter
      Pages 269-463
    2. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 270-295
    3. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 296-315
    4. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 316-350
    5. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 394-440
    6. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 441-463
  5. Back Matter
    Pages 465-488

About this book

Introduction

The present. volume is the second volume of the book "Singularities of Differentiable Maps" by V.1. Arnold, A. N. Varchenko and S. M. Gusein-Zade. The first volume, subtitled "Classification of critical points, caustics and wave fronts", was published by Moscow, "Nauka", in 1982. It will be referred to in this text simply as "Volume 1". Whilst the first volume contained the zoology of differentiable maps, that is it was devoted to a description of what, where and how singularities could be encountered, this volume contains the elements of the anatomy and physiology of singularities of differentiable functions. This means that the questions considered in it are about the structure of singularities and how they function. Another distinctive feature of the present volume is that we take a hard look at questions for which it is important to work in the complex domain, where the first volume was devoted to themes for which, on the whole, it was not important which field (real or complex) we were considering. Such topics as, for example, decomposition of singularities, the connection between singularities and Lie algebras and the asymptotic behaviour of different integrals depending on parameters become clearer in the complex domain. The book consists of three parts. In the first part we consider the topological structure of isolated critical points of holomorphic functions. We describe the fundamental topological characteristics of such critical points: vanishing cycles, distinguished bases, intersection matrices, monodromy groups, the variation operator and their interconnections and method of calculation.

Keywords

Calculation Lie Volume algebra behavior bifurcation character classification differential equation holomorphic function integral operator physiology singularity themes

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