On the Degree of Standard Geometric Predicates for Line Transversals in 3D

International audienceIn this paper we study various geometric predicates for determining the existence of and categorizing the configurations of lines in 3D that are transversal to lines or segments. We compute the degrees of standard procedures of evaluating these predicates. The degrees of some of these procedures are surprisingly high (up to 168), which may explain why computing line transversals with finite-precision floating-point arithmetic is prone to error. Our results suggest the need to explore alternatives to the standard methods of computing these quantities

From invariants to predicates: example of line transversals to lines

International audienceThis work explores a method that reduces the design of evaluation strategies for geometric predicates to the computation of polynomial invariants of a group action. We apply it to the classical problem of counting line transversals to lines in the real 3-dimensional projective space and capture polynomials previously obtained by more pedestrian approaches.Ce travail explore une méthode où l'on réduit la recherche d'une stratégie d'évaluation polynomiale pour un prédicat géométrique au calcul d'invariants polynomiaux d'une action de groupe. On utilise cette méthode pour compter le nombre de droites transversales à des droites dans l'espace projectif réel de dimension 3 et on obtient des polynômes précédemment obtenus par des approches plus pédestres

Qualitative Symbolic Perturbation: a new geometry-based perturbation framework

In a classical Symbolic Perturbation scheme,degeneracies are handled by substituting some polynomials in$\varepsilon$ to the input of a predicate. Instead of a singleperturbation, we propose to use a sequence of (simpler)perturbations. Moreover, we look at their effects geometricallyinstead of algebraically; this allows us to tackle cases that werenot tractable with the classical algebraic approach.Avec les méthodes de perturbations symboliques classiques,les dégénérescences sont résolues en substituant certains polynômes en $\varepsilon$ aux entrées du prédicat.Au lieu d'une seule perturbation compliquée, nous proposons d'utiliser unesuite de perturbation plus simple. Et nous regardons les effets deces perturbations géométriquement plutôt qu'algébriquementce qui permet de traiter des cas inatteignables par les méthodesalgébriques classiques

On groups and initial segments in nonstandard models of Peano Arithmetic

This thesis concerns M-finite groups and a notion of discrete measure in models of Peano Arithmetic. First we look at a measure construction for arbitrary non-M-finite sets via suprema and infima of appropriate M-finite sets. The basic properties of the measures are covered, along with non-measurable sets and the use of end-extensions. Next we look at nonstandard finite permutations, introducing nonstandard symmetric and alternating groups. We show that the standard cut being strong is necessary and sufficient for coding of the cycle shape in the standard system to be equivalent to the cycle being contained within the external normal closure of the nonstandard symmetric group. Subsequently the normal subgroup structure of nonstandard symmetric and alternating groups is given as a result analogous to the result of Baer, Schreier and Ulam for infinite symmetric groups. The external structure of nonstandard cyclic groups of prime order is identified as that of infinite dimensional rational vector spaces and the normal subgroup structure of nonstandard projective special linear groups is given for models elementarily extending the standard model. Finally we discuss some applications of our measure to nonstandard finite groups

Algorithms for polycyclic-by-finite groups

A set of fundamental algorithms for computing with polycyclic-by-finite groups is presented here. Polycyclic-by-finite groups arise naturally in a number of contexts; for example, as automorphism groups of large finite soluble groups, as quotients of finitely presented groups, and as extensions of modules by groups. No existing mode of representation is suitable for these groups, since they will typically not have a convenient faithful permutation representation. A mixed mode is used to represent elements of such a group; utilising a polycyclic presentation or a power-conjugate presentation for the elements of the normal subgroup, and a permutation representation for the elements of the quotient

Collection of abstracts of the 24th European Workshop on Computational Geometry

International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop

Symmetry and complexity in propositional reasoning

We establish computational complexity results for a number of simple problem formulations connecting group action and prepositional formulas. The results are discussed in the context of complexity results arising from established work in the area of automated reasoning techniques which exploit symmetry

Vapnik-Chervonenkis density in some theories without the independence property, I

We recast the problem of calculating Vapnik-Chervonenkis (VC) density into one of counting types, and thereby calculate bounds (often optimal) on the VC density for some weakly o-minimal, weakly quasi-o-minimal, and $P$-minimal theories.Comment: 59