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

Determining the Best Sensing Coverage for 2-Dimensional Acoustic Target Tracking

Distributed acoustic target tracking is an important application area of wireless sensor networks. In this paper we use algebraic geometry to formally model 2-dimensional acoustic target tracking and then prove its best degree of required sensing coverage. We present the necessary conditions for three sensing coverage to accurately compute the spatio-temporal information of a target object. Simulations show that 3-coverage accurately locates a target object only in 53% of cases. Using 4-coverage, we present two different methods that yield correct answers in almost all cases and have time and memory usage complexity of Θ(1). Analytic 4-coverage tracking is our first proposed method that solves a simultaneous equation system using the sensing information of four sensor nodes. Redundant answer fusion is our second proposed method that solves at least two sets of simultaneous equations of target tracking using the sensing information of two different sets of three sensor nodes, and fusing the results using a new customized formal majority voter. We prove that 4-coverage guarantees accurate 2-dimensional acoustic target tracking under ideal conditions

Neural Connectivity with Hidden Gaussian Graphical State-Model

The noninvasive procedures for neural connectivity are under questioning. Theoretical models sustain that the electromagnetic field registered at external sensors is elicited by currents at neural space. Nevertheless, what we observe at the sensor space is a superposition of projected fields, from the whole gray-matter. This is the reason for a major pitfall of noninvasive Electrophysiology methods: distorted reconstruction of neural activity and its connectivity or leakage. It has been proven that current methods produce incorrect connectomes. Somewhat related to the incorrect connectivity modelling, they disregard either Systems Theory and Bayesian Information Theory. We introduce a new formalism that attains for it, Hidden Gaussian Graphical State-Model (HIGGS). A neural Gaussian Graphical Model (GGM) hidden by the observation equation of Magneto-encephalographic (MEEG) signals. HIGGS is equivalent to a frequency domain Linear State Space Model (LSSM) but with sparse connectivity prior. The mathematical contribution here is the theory for high-dimensional and frequency-domain HIGGS solvers. We demonstrate that HIGGS can attenuate the leakage effect in the most critical case: the distortion EEG signal due to head volume conduction heterogeneities. Its application in EEG is illustrated with retrieved connectivity patterns from human Steady State Visual Evoked Potentials (SSVEP). We provide for the first time confirmatory evidence for noninvasive procedures of neural connectivity: concurrent EEG and Electrocorticography (ECoG) recordings on monkey. Open source packages are freely available online, to reproduce the results presented in this paper and to analyze external MEEG databases