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Singularities of Differentiable Maps, Volume 2

Monodromy and Asymptotics of Integrals

  • Book
  • © 2012

Overview

Authors:
  1. V.I. Arnold

    1. Russian Academy of Sciences, Moscow, Russia

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  2. S.M. Gusein-Zade

    1. Moscow State University, Moscow, Russia

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  3. A.N. Varchenko

    1. , Department Mathematics, University of North Carolina, Chapel Hill, USA

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  • Affordable reprint of a classic monograph written by experts in the field
  • Useful for a wide range of applications across disciplines in fields such as differential equations, dynamical systems, optimal control, and optics
  • Suitable for a broad audience of mathematicians, post-graduates, and specialists in the areas of mechanics, physics, technology, and other sciences dealing with the theory of singularities of differentiable maps
  • Includes supplementary material: sn.pub/extras

Part of the book series: Modern Birkhäuser Classics (MBC)

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Table of contents (15 chapters)

  1. Front Matter

    Pages i-x
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  2. The topological structure of isolated critical points of functions

    1. Front Matter

      Pages 1-8
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  3. The Topological Structure of Isolated Critical Points of Functions

    1. Elements of the theory of Picard-Lefschetz

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 9-28

    2. The topology of the non-singular level set and the variation operator of a singularity

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 29-66

    3. The bifurcation sets and the monodromy group of a singularity

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 67-113

    4. The intersection matrices of singularities of functions of two variables

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 114-138

    5. The intersection forms of boundary singularities and the topology of complete intersections

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 139-167

  4. Oscillatory integrals

    1. Front Matter

      Pages 169-169
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  5. Oscillatory Integrals

    1. Discussion of results

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 170-214

    2. Elementary integrals and the resolution of singularities of the phase

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 215-232

    3. Asymptotics and Newton polyhedra

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 233-262

    4. The singular index, examples

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 263-267

  6. Integrals of holomorphic forms over vanishing cycles

    1. Front Matter

      Pages 269-269
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  7. Integrals of Holomorphic forms over Vanishing cycles

    1. The simplest properties of the integrals

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 270-295

    2. Complex oscillatory integrals

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 296-315

    3. Integrals and differential equations

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 316-350

    4. The coefficients of series expansions of integrals, the weight and Hodge filtrations and the spectrum of a critical point

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 351-393

    5. The mixed Hodge structure of an isolated critical point of a holomorphic function

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 394-440

    6. The period map and the intersection form

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 441-463

  8. Back Matter

    Pages 465-492
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Keywords

About this book

​​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.

Authors and Affiliations

  • Russian Academy of Sciences, Moscow, Russia

    V.I. Arnold

  • Moscow State University, Moscow, Russia

    S.M. Gusein-Zade

  • , Department Mathematics, University of North Carolina, Chapel Hill, USA

    A.N. Varchenko

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