It is long known that any expansion, M, of the field of real numbers that defines N (the set of all natural numbers) also defines every real Borel set, hence also every real projective set (in the sense of descriptive set theory). Thus, one can easily ask questions about the definable sets of M that turn out to be independent of ZFC (e.g., whether every definable set is Lebesgue measurable). This leads naturally to wondering what can be said about its definable sets if M does not define N. Philipp Hieronymi (Urbana-Champaign) and I have recently obtained a result that can be stated loosely as: M avoids defining N if and only if all metric dimensions commonly encountered in geometric measure theory, fractal geometry and analysis on metric spaces coincide with topological dimension on all images of closed definable sets under definable continuous maps. I will make this statement precise (assuming essentially no knowledge of model theory or dimension theory), explain its significance, and give some easy (yet striking) corollaries and applications.
I will report on some recent progress on optimization problems involving the first Dirichlet eigenvalue and the torsional rigidity. This is joint work with G. Buttazzo, B. Velichkov and with C. Trombetti, C. Nitsch, V. Ferone.
The M2Disco (Multiresolution, Discrete and Combinatorial Models) research team aims at proposing new combinatorial, discrete and multiresolution models to analyse and manage various types of data such as images, 3D volumes and 3D meshes, represented as Digital Sur- faces (ie subset of Zn). One of their project called PALSE foam requires the computation of the shortest path between two points on a manifold. We are proposing the study of two algorithm for computing such a dis- tance, but also providing metric embedding inside a Discrete Exterior Calculus structure (DEC). We performs various tests regarding the two algorithms, but also con rm through experience DEC's operators con- vergence using suitable metric. This work is the base of both a research project named CoMeDiC (Convergent Metrics for Digital Calculus) and Ph.D. project.
L'équation Landau est une caricature ``diffusive'' de l'équation de Boltzmann, décrivant la densité de particules interagissant lors de chocs. Nous allons nous intéresser ici à l'approximation particulaire de l'équation de Landau dans le cas Maxwellien ou sphère dure et montrerons comment établir la propriété de propagation du chaos soit le fait que la loi d'une particule approche la solution de l'équation de Landau et que deux particules typiques sont presque indépendantes. (en collaboration avec F. Bolley (P6) et N. Fournier (P6))
In joint work with Raf Cluckers, we propose a conjecture for exponential sums which generalizes both a conjecture by Igusa and a local variant by Denef and Sperber, in particular, it is without the homogeneity condition on the polynomial in the phase, and with new predicted uniform behavior. The exponential sums have summation sets consisting of integers modulo p^m lying p-adically close to y, and the proposed bounds are uniform in p, y, and m. We give evidence for the conjecture, by showing uniform bounds in p, y, and in some values for m. On the way, we prove new bounds for log-canonical thresholds which are closely related to the bounds predicted by the conjecture.
Il existe de très nombreuses façons de représenter et discrétiser une courbe ou une surface, en raison notamment des applications envisagées et des modes d'acquisitions des données (nuages de points, approximations volumiques, triangulations...). Le but de cet exposé sera de présenter un cadre commun pour l'approximation des surfaces, dans l'esprit de la théorie géométrique de la mesure, à travers la notion de varifold discret. Ce cadre nous a notamment permis de dégager une notion de courbure moyenne discrète (à une échelle donnée) unifiée dont on présentera les propriétés de convergence et qu'on illustrera numériquement sur des nuages de points.
Dans cet exposé nous présenterons les bases d’un modèle de température pour les matériaux ferromagnétiques. Dans un premier temps nous ferons le lien entre différentes échelles de description à température nulle des matériaux ferromagnétiques. Nous irons de l’échelle microscopique des atomes aux noyaux localisés sur des points, à l’échelle mésoscopique du micromagnétisme. Dans un second temps nous nous focaliserons sur l’échelle microscopique perturbée par un champ extérieur aléatoire modélisant les effets thermiques.
In this talk we introduce and study a new property of infinite words: An infinite word x on an alphabet A is said to be it self-shuffling, if x admits factorizations: $x=prod_{i=1}^infty U_iV_i=prod_{i=1}^infty U_i=prod_{i=1}^infty V_i$ with $U_i,V_i in A^*$. In other words, there exists a shuffle of x with itself which reproduces x. This property of infinite words is shown to be an intrinsic property of the word and not of its language (set of factors). For instance, every aperiodic uniformly recurrent word contains a non self-shuffling word in its shift orbit closure. On the other hand, we build an infinite word such that no word in its shift orbit closure is self-shuffling. We prove that many important and well studied words are self-shuffling: This includes the Thue-Morse word and all Sturmian words (except those of the form aC where a ∈ {0,1} and C is a characteristic Sturmian word). We further establish a number of necessary conditions for a word to be self-shuffling, and show that certain other important words (including the paper-folding word and infinite Lyndon words) are not self-shuffling. One important feature of self-shuffling words is its morphic invariance: The morphic image of a self-shuffling word is again self-shuffling. This provides a useful tool for showing that one word is not the morphic image of another. In addition to its morphic invariance, this new notion has other unexpected applications: For instance, as a consequence of our characterization of self-shuffling Sturmian words, we recover a number theoretic result, originally due to Yasutomi, on a classification of pure morphic Sturmian words in the orbit of the characteristic. Finally, we provide a positive answer to a recent question by T. Harju whether square-free self-shuffling words exist and discuss self-shufflings in a shift orbit closure. E. Charlier, T. Kamae, S. Puzynina, L. Q. Zamboni: Infinite self-shuffling words. J. Comb. Theory, Ser. A 128: 1-40 (2014) M. Müller, S. Puzynina, M. Rao: On Shuffling of Infinite Square-Free Words. Electr. J. Comb. 22(1): P1.55 (2015)
We propose a notion of morphisms between tree automata based on game semantics. Morphisms are winning strategies on a synchronous restriction of the linear implication between acceptance games. This leads to split indexed categories, with substitution based on a suitable notion of synchronous tree function. By restricting to tree functions issued from maps on alphabets, this gives a fibration of tree automata. We then discuss the (fibrewise) monoidal structure issued from the synchronous product of automata. We also discuss how a variant of the usual projection operation on automata leads to an existential quantification in the fibered sense. Our notion of morphism is correct, in the sense that it respects language inclusion, and in a weaker sense also complete.
In this talk we present the starting mechanical model of the lamellipodial actin-cytoskeleton meshwork. The model is derived starting from the microscopic description of mechanical properties of laments and cross-links and also of the life-cycle of cross-linker molecules]. We introduce a simplified system of equations that accounts for adhesions created by a single point on which we apply a force. We present the adimensionalisation that led to a singular limit that motivated our mathematical study. Then we explain the mathe- matical setting and results already published. In the last part we present the latest developments : we give results for the fully coupled system with unbounded non-linear o-rate. This leads to two possible regimes : under certain hypotheses on the data there is global existence, out of this range we are able to prove blow-up in nite time.