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Mosc. Math. J., 2003, Volume 3, Number 3, Pages 1097–1112 (Mi mmj123)  

This article is cited in 3 scientific papers (total in 3 papers)

On the topology of singularities of Maxwell sets

V. D. Sedykh

Gubkin Russian State University of Oil and Gas

Abstract: We determine new conditions for the coexistence of corank-one singularities of the Maxwell set of a generic family of smooth functions with respect to taking global minima (or maxima) in cases when this set does not have more complicated singularities. In particular, the Euler number of every odd-dimensional manifold of singularities of a given type is a linear combination of the Euler numbers of even-dimensional manifolds of singularities of higher codimensions. The coefficients of this combination are universal numbers (that is, they do not depend on the family and depend only on the classes of singularities).
We obtain these conditions as a corollary to the general coexistence conditions for corank 1 singularities of generic wave fronts which were found recently by the author. As an application, we obtain many-dimensional generalizations of the classical Bose formula relating the number of supporting curvature circles for a smooth closed convex generic plane curve to the number of supporting circles which are tangent to this curve at three points.

Key words and phrases: Families of smooth functions, global minima and maxima, Maxwell sets, corank-one singularities of smooth functions, Euler number, convex curves, supporting hyperspheres.

DOI: https://doi.org/10.17323/1609-4514-2003-3-3-1097-1112

Full text: http://www.ams.org/.../abst3-3-2003.html
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Bibliographic databases:

MSC: Primary 58C05, 58K30; Secondary 53A04
Received: June 26, 2002
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Citation: V. D. Sedykh, “On the topology of singularities of Maxwell sets”, Mosc. Math. J., 3:3 (2003), 1097–1112

Citation in format AMSBIB
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\by V.~D.~Sedykh
\paper On the topology of singularities of Maxwell sets
\jour Mosc. Math.~J.
\yr 2003
\vol 3
\issue 3
\pages 1097--1112
\mathnet{http://mi.mathnet.ru/mmj123}
\crossref{https://doi.org/10.17323/1609-4514-2003-3-3-1097-1112}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2078575}
\zmath{https://zbmath.org/?q=an:1065.58029}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000208594300016}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Yomdin Y., “Generic singularities of surfaces”, Singularity Theory, 2007, 357–375  crossref  mathscinet  zmath  isi
    2. Haviv D., Yomdin Y., “Uniform approximation of near-singular surfaces”, Theoretical Computer Science, 392:1–3 (2008), 92–100  crossref  mathscinet  zmath  isi
    3. Houston K., van Manen M., “A Bose type formula for the internal medial axis of an embedded manifold”, Differential Geometry and Its Applications, 27:2 (2009), 320–328  crossref  mathscinet  zmath  isi
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