Euler calculus is a methodology from applied algebraic topology and integral geometry that integrates constructible functions and more recently definable functions 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. It was introduced independently by Pierre Schapira and Oleg Viro in 1988, and is useful for enumeration problems in computational geometry and sensor networks.
See also
References
- 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 .
- 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
- Ghrist, Robert. Euler Calculus video presentation, June 2009. published 30 July 2009.
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