Euler integration over definable functions

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.Comment: 6 page

Positive Alexander Duality for Pursuit and Evasion

Considered is a class of pursuit-evasion games, in which an evader tries to avoid detection. Such games can be formulated as the search for sections to the complement of a coverage region in a Euclidean space over a timeline. Prior results give homological criteria for evasion in the general case that are not necessary and sufficient. This paper provides a necessary and sufficient positive cohomological criterion for evasion in a general case. The principal tools are (1) a refinement of the Cech cohomology of a coverage region with a positive cone encoding spatial orientation, (2) a refinement of the Borel-Moore homology of the coverage gaps with a positive cone encoding time orientation, and (3) a positive variant of Alexander Duality. Positive cohomology decomposes as the global sections of a sheaf of local positive cohomology over the time axis; we show how this decomposition makes positive cohomology computable as a linear program.Comment: 19 pages, 6 figures; improvements made throughout: e.g. positive (co)homology generalized to arbitrary degrees; Positive Alexander Duality generalized from homological degrees 0,1; Morse and smoothness conditions generalized; illustrations of positive homology added. minor corrections in proofs, notation, organization, and language made throughout. variant of Borel-Moore homology now use