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Remarks on the singularity indices of Arnol'd and Berry

Published online by Cambridge University Press:  14 November 2011

David Chillingworth  and
Carmen Romero-Fuster
David Chillingworth
Affiliation:
Department of Mathematics, University of Southampton, Southampton SO9 5NH
Carmen Romero-Fuster
Affiliation:
Department of Mathematics, University of Southampton, Southampton SO9 5NH
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Synopsis

We give a coordinate-free algebraic interpretation for singularity indices of quasihomogeneous functions and their unfoldings, as first exploited systematically by Berry in statistical optics. We also make some observations on geometric interpretation, on formulae for indices, and on the difficulties of rigorous treatment for non-quasihomogeneous functions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 94 , Issue 3-4 , 1983 , pp. 339 - 350
Copyright
Copyright © Royal Society of Edinburgh 1983

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References

1Arnol'd, V. I.. Integrals of rapidly oscillating functions and singularities of projections of Lagrangian manifolds. Funkcional. Anal, i Priložen 6 (1972), No. 3, 61–62; Functional Anal. Appl. 6 (1972), 222–224.Google Scholar
2Arnol'd, V. I.. Normal forms for functions near degenerate critical points, the Weyl groups of Ak, Dk, Ek and Lagrangian singularities. Funkcional. Anal, i Priložen. 6 (1972), No. 4, 3–25; Functional Anal. Appl. 6 (1972), 254–272.Google Scholar
3Arnol'd, V. I.. Remarks on the stationary phase method and Coxeter numbers. Russian Math. Surveys 28 (1973), 19–48.Google Scholar
4Arnol'd, V. I.. Normal forms for functions in the neighbourhood of degenerate critical points. Russian Math. Surveys 29 (1974), 10–50.Google Scholar
5Arnol'd, V. I.. Critical points of smooth functions and their normal forms. Russian Math. Surveys 30 (1975), 1–75. (Note; references [3, 4, 5] are among those collected in: Arnol, V. I.'d. Singularity Theory. London Math. Soc. Lecture Notes 53, Cambridge University Press, 1981).CrossRefGoogle Scholar
6Berry, M. V.. Focussing and twinkling: critical exponents from catastrophes in non-Gaussian random short waves. J. Phys. A 10 (1977), 2061–2081.CrossRefGoogle Scholar
7Berry, M. V.. Universal power-law tails for singularity-dominated strong fluctuations. J. Phys. A 15 (1982), 2735–2749.CrossRefGoogle Scholar
8Berry, M. V. and Upstill, C.. Catastrophe optics: morphologies of caustics and their diffraction patterns. Progr. Optics 18 (1980), 257–346.Google Scholar
9Bourbaki, N.. Groupes et Algèbres de Lie (Paris: Hermann, 1971).Google Scholar
10Bröcker, Th. and Lander, L.. Differentiable Germs and Catastrophes. London Math. Soc. Lecture Notes 17 (Cambridge University Press, 1975).CrossRefGoogle Scholar
11Dangelmayr, G.. Analytic singularities in dispersive wave phenomena based on topological singularities of dispersion relations. Preprint, Inst. Informat. Sci., Univ. Tubingen, 1982.CrossRefGoogle Scholar
12Dangelmayr, G.. Singularities of dispersion relations. Preprint, Inst. Informat. Sci., Univ. Tübingen, 1982.Google Scholar
13Dangelmayr, G. and Güttinger, W.. Topological approach to remote sensing. Geophys. J. Roy. Astronom. Soc. 71 (1982), 79–126.CrossRefGoogle Scholar
14Dangelmayr, G. and Veit, W.. Semiclassical approximation of path integrals on and near caustics in terms of catastrophes. Ann. Physics 118 (1979), 108–138.Google Scholar
15Duistermaat, J. J.. Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 28 (1974), 207–281.CrossRefGoogle Scholar
16Gibson, C. G.. Singular Points of Smooth Mappings. Research Notes in Mathematics 25 (London: Pitman, 1979).Google Scholar
17Guillemin, V. and Schaeffer, D.. Remarks on a paper of D. Ludwig. Bull. Amer. Math. Soc. 79 (1973), 382–385.CrossRefGoogle Scholar
18Guillemin, V. and Sternberg, S.. Geometric Asymptotics. Amer. Math. Soc. Surveys 14 (Providence, R.I.: Amer. Math. Soc, 1977).Google Scholar
19Hörmander, L.. Fourier integral operators, I. Acta Math. 127 (1971), 79–183.Google Scholar
20Looijenga, E.. A period mapping for certain semi-universal deformations. Compositio Math. 30 (1975), 299–316.Google Scholar
21Malgrange, B.. Integrales asymptotiques et monodromie. Ann. Sci. Ecole Norm. Sup. Ser. 4, 7 (1974), 405–430.Google Scholar
22Magnus, R.. The transformation of vector-functions and its use in bifurcation theory. Preprint, Dept. Math., Univ. Iceland, 1981.Google Scholar
23Matsumura, H.. Commutative Algebra (New York: Benjamin, 1970).Google Scholar
24Poston, T. and Stewart, I.. Catastrophe Theory and its Applications (London: Pitman, 1978).Google Scholar
25Saito, K.. Quasihomogene isolierte Singularitaten von Hyperflachen. Invent. Math. 14 (1971), 123–142.CrossRefGoogle Scholar
26Slodowy, P.. Einige Bemerkungen zur Entfaltung symmetrischer Funktionen. Math. Z. 158 (1978), 157–170.CrossRefGoogle Scholar
27Thorn, R.. Topological models in biology. Topology 8 (1969), 313–335.Google Scholar
28Thorn, R.. Stabilité Structurelle et Morphogenese (Reading, Mass.: Benjamin, 1972) (English trans, by Fowler, D. H., 1975).Google Scholar
29Varchenko, A. N.. Newton polyhedra and estimation of oscillating integrals. Funkcional. Anal, i Priložen. 10 (1976), 13–38; Functional Anal. Appl. 10 (1976), 175–196.Google Scholar
30Wall, C. T. C.. A note on symmetry of singularities. Bull. London Math. Soc. 12 (1980), 169–175.CrossRefGoogle Scholar
31Wall, C. T. C.. A second note on symmetry of singularities. Bull. London Math. Soc. 12 (1980), 347–354.Google Scholar
32Zeeman, E. C.. Catastrophe Theory: Selected Papers 1972–1977 (Reading, Mass.: Addison-Wesley, 1977).Google Scholar