Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-23T11:55:34.576Z Has data issue: false hasContentIssue false
  • English
  • Français

Computational Aspects of Classifying Singularities

Published online by Cambridge University Press:  01 February 2010

N. P. Kirk
N. P. Kirk
Affiliation:
Department of Mathematical Sciences, The University of Liverpool, P.O. Box 147, Liverpool L69 3BX, [email protected], http://www.liv.ac.uk/Maths/

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A Maple package which performs the symbolic algebra central to problems in local singularity theory is described. This is a generalisation of previous projects, which dealt only with problems in elementary catastrophe theory. Applications to specific problems are described, and a survey given of the powerful techniques from singularity theory that are used by the package. A description of the underlying algorithm is given, and some of the more important computational aspects discussed. The package, user manual and installation instructions are available in the appendices to this article.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 3 , 2000 , pp. 207 - 228
Copyright
Copyright © London Mathematical Society 2000

References

1. Arnold, V. I., ‘Critical points of smooth functions and their normal forms’, Russian Math. Surveys 30 (1975) 1–75.Google Scholar
2. Bayer, D. and Stillman, M., ‘Macaulay: a system for computation in algebraic geometry and commutative algebra’ (1982–1994). Source and object code available from the authors for Unix and Macintosh computers: http://www.math.uiuc.edu/Macaulay2.Google Scholar
3. Bruce, J. W., ‘Generic geometry and duality’, Singularities, Lille 1991, London Math. Soc. Lecture Note Ser. 201 (ed. Brasselet, J.-P., Cambridge University Press, 1994).Google Scholar
4. Bruce, J. W., Kirk, N. P. and Du Plessis, A. A., ‘Complete transversals and the classification of singularities’, Nonlinearity 10 (1997) 253–275.Google Scholar
5. Bruce, J. W., Kirk, N. P. and West, J. M., ‘Classification of map-germs from surfaces to four-space’, Preprint, University of Liverpool, 1995.Google Scholar
6. Bruce, J. W., Du Plessis, A. A. and Wall, C. T. C., ‘Determinacy and unipotency’, Invent. Math. 88 (1987) 521–554.CrossRefGoogle Scholar
7. Char, B. W., Geddes, K. O., Gonnet, G. H., Leong, B. L., Monagan, M. B. and Watt, S. M., Maple V language reference manual (Springer-Verlag and Waterloo Maple Publishing, 1991).Google Scholar
8. Cowell, R. G. and Wright, F. J., ‘CATFACT:computer algebraic tools for applications of catastrophe theory’, Proc. EUROCAL '87, European Conference on Computer Algebra, Leipzig, GDR, 1987, Lecture Notes in Comput. Sci. 378 (ed. Davenport, J. H., Springer, 1989)71–80.Google Scholar
9. Damon, J. N., ‘The unfolding and determinacy theorems for subgroups of and ’ Mem. Amer. Math. Soc. 306 (1984). Also in Singularities, Proc. Symp. Pure Math. 40 (part 1) (American Mathematical Society, Providence, RI, 1983) 233–254.Google Scholar
10. Fox, L., An introduction to numerical linear algebra (Clarendon Press, Oxford, 1964).Google Scholar
11. Greuel, G.-M., Pfister, G. and Schoenemann, H., ‘The SINGULAR project’, Department of Mathematics, University of Kaiserslautern. Latest information available from http://www.singular.uni-kl.de.Google Scholar
12. Hawes, W., ‘Multi-dimensional motions of the plane and space’, Ph.D. thesis, University of Liverpool, 1994.Google Scholar
13. Hobbs, C. A. and Kirk, N. P., ‘On the classification and bifurcation of multigerms of maps from surfaces to 3-space’, Math. Scand. To appear.Google Scholar
14. Houston, K. and Kirk, N. P., ‘On the classification and geometry of corank-1 mapgerms from 3-space to 4-space’, Singularity theory, Proceedings of the European Singularities Conference in honour of C. T. C. Wall on the occasion of his 60th birthday (ed.Bruce, J. W. and Mond, D. M. Q., Cambridge University Press, 1999) 325’351.Google Scholar
15. Kirk, N. P., ‘Computational aspects of singularity theory’, Ph.D. thesis, University of Liverpool, 1993.Google Scholar
16. Kirk, N. P., ‘Transversal: a Maple package for singularity theory; user manual, Version 3.1’, Preprint, University of Liverpool, 1998. Available with the Transversalpackage via ftp. (See also Appendix A, Appendix B and Appendix C above.)Google Scholar
17. Martinet, J., Singularities of smooth functions and maps, London Math. Soc. Lecture Note Ser. 58 (Cambridge University Press, 1982).Google Scholar
18. Millington, K. and Wright, F. J., ‘Algebraic computations in elementary catastrophe theory’, Proc. EUROCAL '85, European Conference on Computer Algebra, Linz, Austria, 1985, Lecture Notes in Comput. Sci. 204 (ed. Caviness, B. F., Springer, 1985) 116–125.Google Scholar
19. Mond, D. M. Q., ‘On the classification of germs of maps from R2 to R3’, Proc. London Math. Soc. (3) 50 (1985) 333–369.CrossRefGoogle Scholar
20. Du Plessis, A. A., ‘On the determinacy of smooth map-germs’, Invent. Math. 58 (1980) 107–160.Google Scholar
21. Poston, T. and Stewart, I. N., Catastrophe theory and its applications (Pitman, 1978).Google Scholar
22. Ratcliffe, D., ‘A classification of map-germs C2, 0 → C3, 0 up to -equivalence’, Preprint, University of Warwick, 1994.Google Scholar
23. Rieger, J. H., ‘Families of maps from the plane to the plane’, J. London Math. Soc. (2)36(1987)351–369.CrossRefGoogle Scholar
24. Tari, F., ‘Recognition of K-singularities of functions’, Experiment. Math. 1 (1992) 225–229.CrossRefGoogle Scholar
25. Wall, C. T. C., ‘Finite determinacy of smooth map-germs’, Bull. London Math. Soc. 13 (1981) 481–539.Google Scholar
26. West, J. M., ‘The differential geometry of the crosscap’, Ph.D. thesis, University of Liverpool, 1995.Google Scholar
Supplementary material: File

JCM 3 Kirk Appendix A Part 1

Kirk Appendix A Part 1

Download JCM 3 Kirk Appendix A Part 1(File)
File 16.4 KB
Supplementary material: PDF

JCM 3 Kirk Appendix A Part 2

Kirk Appendix A Part 2

Download JCM 3 Kirk Appendix A Part 2(PDF)
PDF 356.1 KB