For any manifold, we describe the notion of geometric distribution on it, in terms of its first neighbourhood of the diagonal. In these “combinatorial” terms, we state the Frobenius Integrability Theorem, and use it to give a combinatorial proof of the Ambrose–Singer Theorem on connections in principal bundles. The consideration of the k’th neighbourhood of the diagonal of a manifold M , M(k) ⊆...

This work introduces the Geometric Machine (GM) – a computational model for the construction and representation of concurrent and non-deterministic processes, preformed in a synchronized way, with infinite memory whose positions are labelled by the points of a geometric space. The ordered structure of the GM model is based on Girard's Coherence Spaces. Starting with a coherence space of element...

We consider the possibility of designing an election method that eliminates the incentives for a voter to rank any other candidate equal to or ahead of his or her sincere favorite. We refer to these methods as satisfying the " Strong Favorite Betrayal Criterion " (SFBC). Methods satisfying our strategic criteria can be classified into four categories, according to their geometrical properties. ...

We have been interested in a visual servoing system for several years. Visual servoing is based on an array of measurements taken from a set of images and used each time as an error function to compute a control vector. This is applied to the system (robot and camera) and makes it move in order to reach a desired situation, at the end of the task, directly depicted in the image. The originality...

We study the first order form of the NS string sigma model allowing for worldsheet couplings corresponding on the target space to a bi-vector, a two-form and an inverse metric. Lifting the topological sector of this action to three dimensions produces several Wess-Zumino like terms which encode the bi-vector generalization of the Courant bracket. This bracket may be familiar to physicists throu...

An important problem in geometric reasoning is to find the configuration of a collection of geometric bodies so as to satisfy a set of given constraints. Recently, it has been suggested that this problem can be solved efficiently by symbolically reasoning about geometry using a degrees of f r e e d o m analysis. The approach employs a set of specialized routines called plan fragments that speci...

There is a prevalent myth in Computer Vision that the gradient weighted algebraic distance, the so-called “Sampson Error,” is a first order approximation of the distance from a point to a curve or surface. In truth, however, it is the exact geometric distance to the first order approximation of the curve. The linguistic difference is subtle, but mathematically, the two statements are at odds. I...

In this paper, we prove that the first eigenvalues of −∆+ cR (c ≥ 1 4 ) are nondecreasing under the Ricci flow. We also prove the monotonicity under the normalized Ricci flow for the cases c = 1/4 and r ≤ 0. 1. First eigenvalue of −∆+ cR Let M be a closed Riemannian manifold, and (M,g(t)) be a smooth solution to the Ricci flow equation ∂ ∂t gij = −2Rij on 0 ≤ t < T . In [Cao07], we prove that a...