Euler integration over definable functions

TL;DR

This paper extends Euler integration to a broader class of tame real-valued functions using o-minimal structures, providing a Morse-theoretic interpretation and applications in sensor network data analysis.

Contribution

It introduces a new framework for Euler integration over definable functions, with a Morse-theoretic perspective and potential for numerical analysis in uncertain data contexts.

Findings

The integral operator is non-linear but Morse-theoretically meaningful.

Applicable to numerical analysis of incomplete and uncertain sensor data.

Provides a new mathematical foundation for Euler integration in tame structures.

Abstract

We extend the theory of Euler integration from the class of constructible functions to that of "tame" real-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, we show that it is an appropriate setting in which to do numerical analysis of Euler integrals, with applications to incomplete and uncertain data in sensor networks.