Euler calculus: Difference between revisions
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{{for|numerical analysis of ordinary differential equations|Euler's method}} |
{{for|numerical analysis of ordinary differential equations|Euler's method}} |
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'''Euler calculus''' is a methodology from applied [[algebraic topology]] and [[integral geometry]] that integrates [[constructible function]]s and more recently [[definable|definable function]]s<ref>Baryshnikov, 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.</ref> by integrating with respect to the [[euler characteristic]] as a finitely-additive [[measure theory|measure]]. It was introduced independently by [[Pierre Schapira (mathematician)|Pierre Schapira]]<ref>Schapira, P. "Cycles Lagrangiens, fonctions constructibles et applications", Seminaire EDP, Publ. Ecole Polytechnique (1988/89)</ref><ref>Schapira, P. [http://linkinghub.elsevier.com/retrieve/pii/002240499190131K Operations on constructible functions], ''J. Pure Appl. Algebra'' 72, 1991, 83–93.</ref><ref>Schapira, Pierre. [http://people.math.jussieu.fr/~schapira/respapers/TomoLN.pdf Tomography of constructible functions], Applied Algebra, Algebraic Algorithms and Error-Correcting Codes Lecture Notes in Computer Science, 1995, Volume 948/1995, 427–435, |
'''Euler calculus''' is a methodology from applied [[algebraic topology]] and [[integral geometry]] that integrates [[constructible function]]s and more recently [[definable|definable function]]s<ref>Baryshnikov, 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.</ref> by integrating with respect to the [[euler characteristic]] as a finitely-additive [[measure theory|measure]]. It was introduced independently by [[Pierre Schapira (mathematician)|Pierre Schapira]]<ref>Schapira, P. "Cycles Lagrangiens, fonctions constructibles et applications", Seminaire EDP, Publ. Ecole Polytechnique (1988/89)</ref><ref>Schapira, P. [http://linkinghub.elsevier.com/retrieve/pii/002240499190131K Operations on constructible functions], ''J. Pure Appl. Algebra'' 72, 1991, 83–93.</ref><ref>Schapira, Pierre. [http://people.math.jussieu.fr/~schapira/respapers/TomoLN.pdf Tomography of constructible functions], 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}}</ref> and [[Oleg Viro]]<ref>Viro, O. [http://www.springerlink.com/index/m5366nl51165k332.pdf Some integral calculus based on Euler characteristic], ''Lecture Notes in Math.'', vol. 1346, Springer-Verlag, 1988, 127–138.</ref> in 1988, and is useful for enumeration problems in [[computational geometry]] and [[sensor network]]s.<ref>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.</ref> |
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==See also== |
==See also== |
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Revision as of 09:41, 5 July 2012
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. It was introduced independently by Pierre Schapira[2][3][4] and Oleg Viro[5] in 1988, and is useful for enumeration problems in computational geometry and sensor networks.[6]
See also
References
- ^ Baryshnikov, Y.; Ghrist, R. Euler integration for definable functions, Proc. National Acad. Sci., 107(21), 9525–9530, 25 May 2010.
- ^ 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, 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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