Euler calculus
Euler calculus is a methodology from applied algebraic topology and integral geometry that integrates constructible functions and more recently definable functions[1] by integrating with respect to the Euler characteristic as a finitely-additive measure. In the presence of a metric, it can be extended to continuous integrands via the Gauss–Bonnet theorem.[2] It was introduced independently by Pierre Schapira[3][4][5] and Oleg Viro[6] in 1988, and is useful for enumeration problems in computational geometry and sensor networks.[7]
See also
[edit]References
[edit]- ^ Baryshnikov, Y.; Ghrist, R. Euler integration for definable functions, Proc. National Acad. Sci., 107(21), 9525–9530, 25 May 2010.
- ^ McTague, Carl (1 Nov 2015). "A New Approach to Euler Calculus for Continuous Integrands". arXiv:1511.00257 [math.DG].
- ^ Schapira, P. "Cycles Lagrangiens, fonctions constructibles et applications", Seminaire EDP, Publ. Ecole Polytechnique (1988/89)
- ^ Schapira, P. Operations on constructible functions, J. Pure Appl. Algebra 72, 1991, 83–93.
- ^ Schapira, Pierre. Tomography of constructible functions Archived 2011-10-05 at the Wayback Machine, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes Lecture Notes in Computer Science, 1995, Volume 948/1995, 427–435, doi:10.1007/3-540-60114-7_33
- ^ Viro, O. Some integral calculus based on Euler characteristic, Lecture Notes in Math., vol. 1346, Springer-Verlag, 1988, 127–138.
- ^ Baryshnikov, Y.; Ghrist, R. Target enumeration via Euler characteristic integrals, SIAM J. Appl. Math., 70(3), 825–844, 2009.
- Van den Dries, Lou. Tame Topology and O-minimal Structures, Cambridge University Press, 1998. ISBN 978-0-521-59838-5
- Arnold, V. I.; Goryunov, V. V.; Lyashko, O. V. Singularity Theory, Volume 1, Springer, 1998, p. 219. ISBN 978-3-540-63711-0
External links
[edit]- Ghrist, Robert. Euler Calculus video presentation, June 2009. published 30 July 2009.
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