Euler calculus

{{for|numerical analysis of ordinary differential equations|Euler's method}}

Euler calculus is a methodology from applied algebraic topology and integral geometry that integrates constructible functions and more recently definable functionsBaryshnikov, Y.; Ghrist, R. [http://www.pnas.org/content/107/21/9525.full Euler integration for definable functions], Proc. National Acad. Sci., 107(21), 9525–9530, 25 May 2010. 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.{{cite arXiv |last=McTague |first=Carl |author-link= |eprint=1511.00257 |title=A New Approach to Euler Calculus for Continuous Integrands |class=math.DG |date=1 Nov 2015 }} It was introduced independently by Pierre SchapiraSchapira, P. "Cycles Lagrangiens, fonctions constructibles et applications", Seminaire EDP, Publ. Ecole Polytechnique (1988/89)Schapira, P. [http://linkinghub.elsevier.com/retrieve/pii/002240499190131K Operations on constructible functions], J. Pure Appl. Algebra 72, 1991, 83–93.Schapira, Pierre. [http://people.math.jussieu.fr/~schapira/respapers/TomoLN.pdf Tomography of constructible functions] {{Webarchive|url=https://web.archive.org/web/20111005073342/http://people.math.jussieu.fr/~schapira/respapers/TomoLN.pdf |date=2011-10-05 }}, 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}} and Oleg ViroViro, O. [https://doi.org/10.1007%2FBFb0082775 Some integral calculus based on Euler characteristic], Lecture Notes in Math., vol. 1346, Springer-Verlag, 1988, 127–138. in 1988, and is useful for enumeration problems in computational geometry and sensor networks.Baryshnikov, Y.; Ghrist, R. [http://repository.upenn.edu/cgi/viewcontent.cgi?article=1002&context=grasp_papers Target enumeration via Euler characteristic integrals], SIAM J. Appl. Math., 70(3), 825–844, 2009.