Symplectic Regularization of Binary Collisions in the Circular N+2 Sitnikov Problem

We present a brief overview of the regularizing transformations of the Kepler problem and we relate the Euler transformation with the symplectic structure of the phase space of the N-body problem. We show that any particular solution of the N-body problem where two bodies have rectilinear dynamics can be regularized by a linear symplectic transformation and the inclusion of the Euler transformation into the group of symplectic local diffeomorphisms over the phase space. As an application we regularize a particular configuration of the circular N+2 Sitnikov problem.Comment: 23 pages, 5 figures. References to algorithmic regularization included, changes in References and small typographic corrections. Accepted in J. of Phys. A: Math. Theor 44 (2011) 265204 http://stacks.iop.org/1751-8121/44/26520

Sigma theory for Bredon modules

We develop new invariants similar to the Bieri-Strebel-Neumann-Renz invariants but in the category of Bredon modules (with respect to the class of the finite subgroups of G). We prove that for virtually soluble groups of type FP_{\infty} and finite extension of the Thompson group F the new invariants coincide with the classical ones

ROAM: a Rich Object Appearance Model with Application to Rotoscoping

Rotoscoping, the detailed delineation of scene elements through a video shot, is a painstaking task of tremendous importance in professional post-production pipelines. While pixel-wise segmentation techniques can help for this task, professional rotoscoping tools rely on parametric curves that offer the artists a much better interactive control on the definition, editing and manipulation of the segments of interest. Sticking to this prevalent rotoscoping paradigm, we propose a novel framework to capture and track the visual aspect of an arbitrary object in a scene, given a first closed outline of this object. This model combines a collection of local foreground/background appearance models spread along the outline, a global appearance model of the enclosed object and a set of distinctive foreground landmarks. The structure of this rich appearance model allows simple initialization, efficient iterative optimization with exact minimization at each step, and on-line adaptation in videos. We demonstrate qualitatively and quantitatively the merit of this framework through comparisons with tools based on either dynamic segmentation with a closed curve or pixel-wise binary labelling