Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

Zajíček's theorem characterizes the sets of non-differentiability of convex functions.
Its statement is sharp, elegant and tremendously powerful.
We provide the proof in a simple case and give some applications.

Based on the mechanical viewpoint that living tissues present a fluid-like behaviour, Partial Differential Equations models inspired by fluid dynamics are nowadays well established as one of the main mathematical tools for the macroscopic description of tissue growth. Depending on the type of tissue, these models link the pressure to the velocity field using either Brinkman’s law (visco-elastic models) or Darcy’s law (porous-medium equations (PME)).

A cellular automaton is monotonous if it preserves the order on configurations inherited from an order on

states. If at first sight, the significantly sparse aspect of that product order lead us to believe that mono-

tonicity would not be that restrictive, the fact that cellular automata are characterised by a local rule

applied shift-invariantly makes the impact of this constraint more remarkable than expected. After intro-

ducing the basic concepts of our work, we first state general results about monotonous cellular automata

Underground cavities known as "marnières", typically consisting of a cavity 1 to 3 meters high and connected to the surface by a vertical shaft 20 to 40 meters deep and 1 to 2 meters in diameter, were historically dug by farmers—especially during the 19th century—to extract chalk used as fertilizer.

Urban air quality is a major issue today. Pollutant concentrations, such as NO2, must be monitored to ensure that they do not exceed dangerous thresholds. Two recent techniques help map pollutant concentrations on a small scale. First, deterministic physicochemical models take into account the street network and calculate concentration estimates on a grid, providing a map. On the other hand, the advent of new low-cost technologies allows monitoring organizations to densify measurement networks.

Résumé : Un sous-shift $X$ est le rétracté d'un autre sous-shift $Y \supset X$ si le morphisme de plongement $emb : X \to Y$ est split-monique dans la catégorie des sous-shifts (i.e. a pour inverse à gauche une fonction par blocs, ou "rétraction"). Pour caractériser cette notion, nous introduisons la notion de "contractibilité", un renforcement de la propriété d'irréducibilité forte, demandant en plus que les collages soient donnés par une fonction par blocs.

The study of diffusion problems with sign changing coefficients increased recently because these problems appear in several areas of physics, 
for example in electromagnetism. Thus some mathematical and  numerical investigations have been performed for domains of the plane or of the space.
 Similar questions were recently considered on graphs. 
Hence the goal of this talk is to present some results on arbitrary compact graphs with Kirchhoff interior vertex conditions and, as much as possible, 
variable coefficients.

In this talk, I will present to you a dummy proof talk intoducing our model to study the adaptation of a diffusing population facing two different dynamics.  On  one hand, the population growth  is  time and space dependent, thus modelling strong heterogeneities of the environment. On the other hand, the environment itself is dynamic. It can both change size and shift over time.  The reasons for such moving range boundaries could be the consequences of flooding, forest fire, etc.