Format:
Online-Ressource (X, 492 p. 83 illus, digital)
ISBN:
9780817683436
Series Statement:
Modern Birkhäuser Classics
Content:
Part I. The topological structure of isolated critical points of functions -- Introduction -- Elements of the theory of Picard-Lefschetz -- The topology of the non-singular level set and the variation operator of a singularity -- The bifurcation sets and the monodromy group of a singularity -- The intersection matrices of singularities of functions of two variables -- The intersection forms of boundary singularities and the topology of complete intersections -- Part II. Oscillatory integrals -- Discussion of results -- Elementary integrals and the resolution of singularities of the phase -- Asymptotics and Newton polyhedra -- The singular index, examples -- Part III. Integrals of holomorphic forms over vanishing cycles -- The simplest properties of the integrals -- Complex oscillatory integrals -- Integrals and differential equations -- The coefficients of series expansions of integrals, the weighted and Hodge filtrations and the spectrum of a critical point -- The mixed Hodge structure of an isolated critical point of a holomorphic function -- The period map and the intersection form -- References -- Subject Index.
Content:
Originally published in the 1980s, Singularities of Differentiable Maps: Monodromy and Asymptotics of Integrals was the second of two volumes that together formed a translation of the authors' influential Russian monograph on singularity theory. This uncorrected softcover reprint of the work brings its still-relevant content back into the literature, making it available—and affordable—to a global audience of researchers and practitioners. While the first volume of this title, subtitled Classification of Critical Points, Caustics and Wave Fronts, contained the zoology of differentiable maps—that is, 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 here are about the structure of singularities and how they function. In the first part the authors consider the topological structure of isolated critical points of holomorphic functions: vanishing cycles; distinguished bases; intersection matrices; monodromy groups; the variation operator; and their interconnections and method of calculation. The second part is devoted to the study of the asymptotic behavior of integrals of the method of stationary phase, which is widely met within applications. The third and last part deals with integrals evaluated over level manifolds in a neighborhood of the critical point of a holomorphic function. This monograph is suitable for mathematicians, researchers, postgraduates, and specialists in the areas of mechanics, physics, technology, and other sciences dealing with the theory of singularities of differentiable maps.
Note:
Description based upon print version of record
,
Singularities of Differentiable Maps; Preface; Contents; Part I The topological structure of isolated critical points of functions; Introduction; Chapter 1 Elements of the theory of Picard-Lefschetz; 1.1 The monodromy and variation operators; 1.2 Vanishing cycles and the monodromy group; 1.3 The Picard-Lefschetz Theorem; Chapter 2 The topology of the non-singular level set and the variation operator of a singularity; 2.1 The non-singular level set of a singularity; 2.2 Vanishing Cycles and the Monodromy Group of a singularity; 2.3 The Variation Operator and Seifert Form of a Singularity
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2.4 Proof of the Picard-Lefschetz theorem2.5 The intersection matrix of a singularity; 2.6 Change of basis; 2. 7 The Variation Operator and the Intersection Matrix of a "Direct Sum" of Singularities; 2.8 The stabilisation of a singularity; 2.9 Example; Chapter 3 The bifurcation sets and the monodromy group of a singularity; 3.1 The bifurcation sets of a singularity; Examples.; 3.2 The connectedness of the D-diagram and the irreducibility' of the classical monodromy operator of a singularity; 3.3 The bifurcation sets of simple singularities
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3.4 The p =constant stratum and the topological type of singularity3.6 The monodromy group and distinguished bases of simple singularities; 3.7 The polar curve and the intersection matrix of a singularity; 3.8 Intersection fonns of unimodal and bimodal singularities; Chapter 4 The intersection matrices of singularities of functions of two variables; 4.1 Intersection matrices of real singularities; 4.2 Germs of complex curves and singularities of functions of two variables; 4.3 The resolution of singularities of functions of two variables and the construction of their real perturbations
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4.4 Partial diagonalisation of the quadratic form of a singularityChapter 5 The intersection forms of boundary singularities and the topology of complete intersections; 5.1 Singularities with the action of finite groups; 5.2 Singularities of functions on manifolds with boundary; 5.3 The topology of complete intersections; 5.4 Singularities of projections onto a line; Part II Oscillatory integrals; Chapter 6 Discussion of results; 6.1 Examples and defmitions; 6.1.1 Oscillatory integrals and shortwave oscillations; 6.1.2.; 6.1.3 Fresnel integrals; 6.1.4 Caustics
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6.1.5 Asymptotic oscillatory integrals near caustics6.1.6 Oscillatory integrals in a halfspace; 6.1.7 Light, dark and twilight zones; 6.1.8.; 6.1.9 The oscillation index and the singular index; 6.1.10 Tables of singular indices; 6.2 Formulation of the results; 6.2.1 The Newton polyhedron; 6.2.2 Nondegeneracy of the principal part; 6.2.3 The distance to a polyhedron and the remoteness of a polyhedron; 6.2.4 Formulation of the main results; 6.3 The resolution of a singularity; 6.4 Asymptotics of volumes; 6.4.1 The Gelfand-Leray form; 6.4.2 The volume of an infralevel set
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6.4.3 The area of a level surface
Additional Edition:
ISBN 9780817683429
Additional Edition:
Buchausg. u.d.T. Arnolʹd, V. I., 1937 - 2010 Singularities of differentiable maps ; 2: Monodromy and asymptotics of integrals Boston : Birkhäuser, 1988 ISBN 0817631852
Additional Edition:
ISBN 3764331852
Additional Edition:
ISBN 9780817683429
Language:
English
Subjects:
Mathematics
DOI:
10.1007/978-0-8176-8343-6
URL:
Volltext
(lizenzpflichtig)
Author information:
Arnolʹd, V. I. 1937-2010
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