Melotti, Paul Ramassamy, Sanjay Thevenin, Paul

We characterize the topological configurations of points and lines that may arise when placing n points on a circle and drawing the n perpendicular bisectors of the sides of the corresponding convex cyclic n-gon. We also provide exact and asymptotic formulas describing a random realizable configuration, obtained either by sampling the points unifor...

Dureisseix, David

This is an English version of an article published as: David Dureisseix, Recyclez ! Le Pli 162:20,2021, the journal of the Mouvement Français des Plieurs de Papier (MFPP), the French paperfoldingassociation.

Daw, Lara Seuret, Stéphane

The macroscopic Hausdorff dimension Dim H (E) of a set E ⊂ R d was introduced by Barlow and Taylor to quantify a "fractal at large scales" behavior of unbounded, possibly discrete, sets E. We develop a method based on potential theory in order to estimate this dimension in R d. Then, we apply this method to obtain Marstrand-like projection theorems...

Chalopin, Jérémie Changat, Manoj Chepoi, Victor Jacob, Jeny

The main goal of this note is to provide a First-Order Logic with Betweenness (FOLB) axiomatization of the main classes of graphs occurring in Metric Graph Theory, in analogy to Tarski's axiomatization of Euclidean geometry. We provide such an axiomatization for weakly modular graphs and their principal subclasses (median and modular graphs, bridge...

Petitjean, Michel

An object is chiral when its symmetry group contains no indirect isometry. It can be difficult to classify isometries as direct or indirect, except in the Euclidean case. We classify them with the help of outer semidirect products of isometry groups, in particular in the case of an affine space defined over a finite-dimensional real quadratic space...

Boulanger, Adrien Glorieux, Olivier

In this article we define and study a stochastic process on Galoisian covers of compact manifolds. The successive positions of the process are defined recursively by picking a point uniformly in the Dirichlet domain of the previous one. We prove a theorem à la Kesten for such a process: the escape rate of the random walk is positive if and only if ...

Jean, Frédéric Jerhaoui, Othmane Zidani, Hasnaa

In this article, we study an optimal control problem on a compact Riemannian manifold M with imperfect information on the initial state of the system. The lack of information is modelled by a Borel probability measure along which the initial state is distributed. The state space of this problem is the space of Borel probability measures over M. We ...

Bay, Thierry Cattiaux-Huillard, Isabelle Saini, Laura

This paper deals with the construction of the Algebraic Trigonometric Pythagorean Hodograph (ATPH) cubic Hermite interpolant and analyzes the existence and characterizations of solutions according to the tangents at both ends and a global shape parameter denoted α. Since this degree of freedom can be used for adjustments, we study how the curve evo...

Chalopin, Jérémie Chepoi, Victor Giocanti, Ugo

In this paper, we investigate the graphs in which all balls are convex and the groups acting on them geometrically (which we call CB-graphs and CB-groups). These graphs have been introduced and characterized by Soltan and Chepoi (1983) and Farber and Jamison (1987). CB-graphs and CB-groups generalize systolic (alias bridged) and weakly systolic gra...

Chalopin, Jérémie Chepoi, Victor Genevois, Anthony Hiroshi, Hirai Osajda, Damian

Helly graphs are graphs in which every family of pairwise intersecting balls has a non-empty intersection. This is a classical and widely studied class of graphs. In this article we focus on groups acting geometrically on Helly graphs -- Helly groups. We provide numerous examples of such groups: all (Gromov) hyperbolic, CAT(0) cubical, finitely pre...