Euler integration over definable functions

We extend the theory of Euler integration from the class of constructible functions to that of "tame" R-valued functions (definable with respect to an o-minimal structure). The corresponding integral operator has some unusual defects (it is not a linear operator); however, it has a compell...

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Bibliographic Details
Published in:Proceedings of the National Academy of Sciences - PNAS 2010-05, Vol.107 (21), p.9525-9530
Main Authors: Baryshnikov, Yuliy, Ghrist, Robert
Format: Article
Language:English
Subjects:
Algebra
Approximation
Critical points
Data smoothing
Eulers equations
Fourier Bessel transformations
Integrands
Mathematical functions
Mathematical integrals
Models, Theoretical
Morse theory
Noise
Physical Sciences
Sensors
Triangulation
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Summary:We extend the theory of Euler integration from the class of constructible functions to that of "tame" R-valued functions (definable with respect to an o-minimal structure). The corresponding integral operator has some unusual defects (it is not a linear operator); however, it has a compelling Morse-theoretic interpretation. In addition, it is an advantageous setting in which to integrate in applications to diffused and noisy data in sensor networks.
ISSN:0027-8424
1091-6490
DOI:10.1073/pnas.0910927107