L'exposé portera sur les calculs de flows maximaux dans les graphes avec l'algorithme de Ford-Fulkerson, puis de l'utilisation de ces flows pour la segmentation d'image. Je parlerais aussi des différentes implémentations de ces algorithmes pour améliorer leur efficacité ou modifier leur comportement par rapport aux images choisies. Je terminerai par quelques exemples concrets de certaines implémentations que j'aurais pu essayer et conclurais.
Many categorical models of linear logic with fixed points arise as total categories over the category Rel of sets and relations. They are form ∫Q, the Grothendieck category for a functor Q : Rel -> Pos. We will define the concepts of fixed points and Grothendieck category and then we give a result to lift functor from the base category to the total category and studies also how to lift fixed points. In particular, the category of coalgebras for the lifted functor is a total category, and when Q factors through SLatt, the category of posets with joins and maps that preserve them, we found the same result.
Let ϒ be a generalised cone in Rd. Roughly speaking, Yaglom limit describes the behaviour of the process conditioned not to exit the cone, or, in other words, not to become extinct or not to be absorbed. In the talk we will discuss the existence of this limit for a class of (not necessarily symmetric) α-stable Lévy processes living in the cone ϒ. To this end, we will use the so-called Martin kernel at inifinty - the invariant function for the killed semigroup - to obtain the so-called entrance law from the origin, which we also call the self-similar solution. Using this approach, for the isotropic case we will also obtain the large-time asymptotics for the killed semigroup and provide several examples of our resutts.
Graph embedding approaches attempt to project graphs into geometric entities, {\em i.e.}, manifolds. At its core, the idea is that the geometric properties of the projected manifolds are helpful in the inference of graph properties. However, the choice of the embedding manifold is critical and, if incorrectly performed, can lead to misleading interpretations due to incorrect geometric inference. We argue that the classical embedding techniques cannot lead to correct geometric interpretation as the microscopic details, {\em e.g.}, curvature at each point, of manifold, that are needed to derive geometric properties in Riemannian geometry methods are not available, and we cannot evaluate the impact of the underlying space on geometric properties of shapes that lie on them. We advocate that for doing correct geometric interpretation the embedding of a graph should be done over regular constant curvature manifolds. To this end, we present an embedding approach, the discrete Ricci flow graph embedding (dRfge) based on the discrete Ricci flow that adapts the distance between nodes in a graph so that the graph can be embedded onto a constant curvature manifold that is homogeneous and isotropic, {\em i.e.}, all directions are equivalent and distances comparable, resulting in correct geometric interpretations. A major contribution of this paper is that for the first time, we prove the convergence of discrete Ricci flow to a constant curvature and stable distance metric over the edges. A drawback of using the discrete Ricci flow is the high computational complexity that prevented its usage in large-scale graph analysis. Another contribution of our work is a new algorithmic solution that makes it feasible to calculate the Ricci flow for graphs of up to 50k nodes, and beyond. The intuitions behind the discrete Ricci flow make it possible to obtain new insights into the structure of large-scale graphs.
The aim of the mini course is to give a self-consistent introduction into the basic theory about Stochastic Differential Equations (SDE) driven by the Brownian motion. The following topics are planed:
(a) Brownian motion
(b) Stochastic integration (Itô integral) with respect to the Brownian motion
(c) Ito’s formula
(d) Existence and uniqueness of solutions to SDEs under Lipschitz conditions
(e) Feynman-Kac theory for parabolic PDEs
(f) A remark on weak solutions and SDEs under non-Lipschitz conditions
The aim of the mini course is to give a self-consistent introduction into the basic theory about Stochastic Differential Equations (SDE) driven by the Brownian motion. The following topics are planed:
(a) Brownian motion
(b) Stochastic integration (Itô integral) with respect to the Brownian motion
(c) Ito’s formula
(d) Existence and uniqueness of solutions to SDEs under Lipschitz conditions
(e) Feynman-Kac theory for parabolic PDEs
(f) A remark on weak solutions and SDEs under non-Lipschitz conditions
The aim of the mini course is to give a self-consistent introduction into the basic theory about Stochastic Differential Equations (SDE) driven by the Brownian motion. The following topics are planed:
(a) Brownian motion
(b) Stochastic integration (Itô integral) with respect to the Brownian motion
(c) Ito’s formula
(d) Existence and uniqueness of solutions to SDEs under Lipschitz conditions
(e) Feynman-Kac theory for parabolic PDEs
(f) A remark on weak solutions and SDEs under non-Lipschitz conditions
When discretizing partial differential equations, one can choose local (finite differences, volumes, elements) or global (spectral) methods. The most common spectral basis is built on trigonometric polynomials, i.e. Fourier series. It constrains the boundary conditions to be periodic and has been an important tool in physics, used for instance to study theoretical scalings of turbulence. While spectral methods show "exponential convergence" for smooth functions, large DNS simulations also become too expensive for e.g. when reaching very large Reynolds numbers. In practice, it is possible to solve a coarser version of the DNS by removing the largest wavenumbers in spectral space (cut-off) and modeling transfers at the smallest (sub-grid) scales instead. The definition of such a model has been an open problem for a long time and classical ones are either too diffusive or unstable. Machine learning started to be an interesting alternative few years ago and people quickly found that learning a model that performs better on a priori (instantaneous) metrics is possible. We have shown that in order to lead to stable simulations in a posteriori tests, the temporal dimension must be taken into account during the learning process. This problem has now been largely explored with periodic boundary conditions, but when it comes to spectral methods with orthogonal polynomials and fixed boundaries, new challenges appear.
In this work in collaboration with T. Iguchi, we show that the Saint-Venant equations in 2D with a partially submerged obstacle is well-posed. To do so, we show that the problem is equivalent to the usual Saint-Venant equations in an external domain, with additionnal non-standard boundary conditions because they are not local in space and time. These conditions do not fit into any category of dissipativity for which the hyperbolic theory is well posed, but we introduce a new class of well-posed hyperbolic boundary problems: that of weakly dissipative boundary conditions. We then show that our system belongs to this class and is therefore well posed.
J'estimerai la croissance asymptotique de l'espérance mathématique de l'aire des amibes des courbes planes complexes aléatoires. Cela nécessitera, étant donnée une collection de bi-disques de taille inverse à la racine carrée du degré, de minorer la probabilité que l'un de ces bi-disques soit une carte de sous-variété d'une courbe plane. Il s'agit d'un travail en collaboration avec Ali Ulaş Özgür Kişisel.
Implicative algebras, recently discovered by Miquel, are combinatorial structures unifying classical and intuitionistic realizability as well as forcing. In this talk we introduce implicative assemblies as sets valued in the separator of an underlying implicative algebra. Given a fixed implicative algebra A, implicative assemblies over A organise themselves in a category AsmA with tracked set-theoretical functions as morphisms. We show that AsmA is a quasitopos with a natural numbers object (NNO).
Mass movements and delta collapses are significant sources of tsunamis in lacustrine environments, impacting human societies enormously. Paleotsunamis studies play an essential role in understanding historical events and their consequences, along with their return periods. This study investigates a paleotsunami induced by a subaqueous mass movement during the Younger Dryas to Early Holocene transition, ca. 11,700 years ago in Lake Aiguebelette (NW Alps, France). Utilizing high‐resolution seismic and bathymetric surveys associated with sedimentological, geochemical, and magnetic analyses, we uncovered a paleotsunami triggered by a seismically induced mass transport deposit. Numerical simulations of mass movement have been conducted using a visco‐plastic Herschel‐Bulkeley rheological model and corresponding tsunami wave modeled with dispersive and nondispersive models. Our findings reveal for the first time that dispersive effects may be negligible for subaqueous landslides in a relatively small lake. This research reconstructs a previously unreported paleotsunami event and enhances our understanding of tsunami dynamics in lacustrine environments.
Among 3-dimensional sets of given Newtonian capacity, which shape minimizes the q-th moment (q>0) of electrostatic equilibrium measure? One readily shows it is the ball. But what if the set is confined to the plane? A centered disk is then the natural minimizer, yet the proof is quite different and involves a cylindrical variant of Baernstein’s star-function. The approach succeeds when 0 <q <= 2. Higher moments (q>2) remain a tantalizing open problem, as do the analogous questions for Riesz equilibrium measures.
Note: this talk does not assume any previous knowledge about capacities.
(Joint work with Carrie Clark, Univ. of Illinois Urbana–Champaign.)
In 2020, Parusinski and Rond proved that every algebraic set $V \subset \mathbb{R}^n$ is homeomorphic to a $\bar{\mathbb{Q}}^r$-algebraic set $V' \subset \mathbb{R}^n$, where $\bar{\mathbb{Q}}^r$ denotes the field of real algebraic numbers. Latter very general result motivates the following open problem: $\mathbb{Q}$-algebraicity problem: (Parusinski, 2021) Is every algebraic set $V \subset \mathbb{R}^n$ homeomorphic to some $\mathbb{Q}$-algebraic set $V' \subset \mathbb{R}^m$, with $m \ge n$? The aim of the talk is to introduce above open problem and to explain how our new approximation techniques over $\mathbb{Q}$ allowed us to provide some classes of real algebraic sets that positively answer the $\mathbb{Q}$-algebraicity problem.
Comment un résultat de catégories a inspiré une solution aux problèmes de performance de Darcs. Darcs est un système de contrôle de versions sorti en 2005, basé sur des patchs, ce qui le rendait extrêmement simple à utiliser et particulièrement rigoureux, en particulier dans sa gestion des conflits. Seul problème, il prenait parfois un temps exponentiel en la taille de l'histoire pour son opération la plus courante (appliquer des patchs). Je vous expliquerai comment nous avons résolu le problème, en utilisant des catégories et des structures de données purement fonctionnelles, pour concevoir un algo en temps logarithmique pour tous les cas (sauf un cas dégénéré en temps linéaire).
Le résultat est un système déterministe (ce qui le distingue de tous les autres outils de contrôle de versions), facile à apprendre et à utiliser, tout en passant à des échelles qu'aucun autre système de contrôle de versions distribué ne peut atteindre.
The topic of the talk is existence of weak solutions to the pressureless Navier-Stokes system with nonlocal attraction--repulsion forces. We construct the solutions on the whole three-dimensional space, assuming that the viscosity coefficients are density-dependent. For the nonlocal term it is further assumed that the interaction kernel has the quadratic growth at infinity and almost quadratic singularity at zero. The main point of the construction is the derivation of the analogs of the Bresch--Desjardins and Mellet--Vasseur estimates in the nonlocal setting.