Laboratoire de Mathématiques Raphaël Salem

We give a comprehensive presentation of the periodic unfolding method for perforated domains, both when the unit hole is a compact subset of the open unit cell and when this is impossible to achieve. In order to apply the method to boundary-value problems with nonhomogeneous Neumann conditions on the boundaries of the holes, the properties of the boundary unfolding operator are also extensively studied. The paper concludes with applications to such problems and examples of reiterated unfolding.

The one-dimensional nearest neighbor asymmetric simple exclusion process has been used as a microscopic approximation for the Burgers equation. This equation has travelling wave solutions. In this paper we show that those solutions have a microscopic structure. More precisely, we consider the simple exclusion process with rate $p$ (respectively, $q = 1 - p)$ for jumps to the right (left), $\frac{1}{2} < p \leq 1$, and we prove the following results: There exists a measure $\mu$ on the space of configurations approaching asymptotically the product measure with densities $\rho$ and $\lambda$ to the left and right of the origin, respectively, $\rho < \lambda$, and there exists a random position $X(t) \in \mathbb{Z}$, such that, at time $t$, the system "as seen from $X(t)$," remains distributed according to $\mu$, for all $t \geq 0$. The hydrodynamical limit for the simple exclusion process with initial measure $\mu$ converges to the travelling wave solution of the inviscid Burgers equation. The random position $X(t)/t$ converges strongly to the speed $\nu = (1 - \lambda - \rho)(p - q)$ of the travelling wave. Finally, in the weakly asymmetric hydrodynamical limit, the stationary density profile converges to the travelling wave solution of the Burgers equation.

microRNAs are noncoding RNAs which downregulate a large number of target mRNAs and modulate cell activity. Despite continued progress, bioinformatics prediction of microRNA targets remains a challenge since available software still suffer from a lack of accuracy and sensitivity. Moreover, these tools show fairly inconsistent results from one another. Thus, in an attempt to circumvent these difficulties, we aggregated all human results of four important prediction algorithms (miRanda, PITA, SVmicrO, and TargetScan) showing additional characteristics in order to rerank them into a single list. Instead of deciding which prediction tool to use, our method clearly helps biologists getting the best microRNA target predictions from all aggregated databases. The resulting database is freely available through a webtool called miRabel which can take either a list of miRNAs, genes, or signaling pathways as search inputs. Receiver operating characteristic curves and precision-recall curves analysis carried out using experimentally validated data and very large data sets show that miRabel significantly improves the prediction of miRNA targets compared to the four algorithms used separately. Moreover, using the same analytical methods, miRabel shows significantly better predictions than other popular algorithms such as MBSTAR, miRWalk, ExprTarget and miRMap. Interestingly, an F-score analysis revealed that miRabel also significantly improves the relevance of the top results. The aggregation of results from different databases is therefore a powerful and generalizable approach to many other species to improve miRNA target predictions. Thus, miRabel is an efficient tool to guide biologists in their search for miRNA targets and integrate them into a biological context.

We consider a discrete-time semi-Markov process, with a finite state space. Taking a censored history, we obtain empirical estimators for the semi-Markov kernel, semi-Markov transition function, reliability and availability. We study the strong consistency and the asymptotic normality for each estimator.

Most of the work on the Boltzmann equation is based on the Grad'sangular cutoff assumption. Even though the smoothing effect from thesingular cross-section without the angular cutoff corresponding tothe grazing collision is expected, there is no general mathematicaltheory especially for the spatially inhomogeneous case. As a furtherstudy on the problem in the spatially homogeneous situation, in thispaper, we will prove the Gevrey smoothing property of the solutionsto the Cauchy problem for Maxwellian molecules without angularcutoff by using pseudo-differential calculus. Furthermore, we applysimilar analytic techniques for the Sobolev space regularity to thenonlinear equation, and prove the smoothing property of solutionsfor the spatially homogeneous nonlinear Boltzmann equation with theDebye-Yukawa potential.

International audience

The periodic unfolding method was introduced in [4] by D. Cioranescu, A. Damlamian and G. Griso for the study of classical periodic homogenization. The main tools are the unfolding operator and a macro-micro decomposition of functions which allows to separate the macroscopic and microscopic scales.In this paper, we extend this method to the homogenization in domains with holes,introducing the unfolding operator for functions defined on periodically perforated domains as well as a boundary unfolding operator. As an application, we study the homogenization of some elliptic problems with a Robin condition on the boundary of the holes, proving convergence and corrector results.

The spatially homogeneous Boltzmann equation without angular cutoff is discussed on the regularity of solutions for the modified hard potential and Debye-Yukawa potential. When the angular singularity of the cross section is moderate, any weak solution having the finite mass, energy and entropy lies in the Sobolev space of infinite order for any positive time, while for the general potentials, it lies in the Schwartz space if it has moments of arbitrary order. The main ingredients of the proof are the suitable choice of the mollifiers composed of pseudo-differential operators and the sharp estimates of the commutators of the Boltzmann collision operator and pseudo-differential operators. The method developed here also provides some new estimates on the collision operator.

An old result by Shearer relates the Lovász local lemma with the independent set polynomial on graphs, and consequently, as observed by Scott and Sokal, with the partition function of the hard-core lattice gas on graphs. We use this connection and a recent result on the analyticity of the logarithm of the partition function of the abstract polymer gas to get an improved version of the Lovász local lemma. As an application we obtain tighter bounds on conditions for the existence of Latin transversal matrices.

In this paper we describe the asymptotic behavior of a problem depending on a small parameter ε&gt;0 and modelling the stationary heat diffusion in a two-component conductor. The flow of heat is proportional to the jump of the temperature field, due to a contact resistance on the interface. More precisely, we give an homogenization result for the stationary heat equation with oscillating coefficients in a domain [Formula: see text] of ℝ n , where [Formula: see text] is connected and [Formula: see text] is union of ε-periodic disconnected inclusions of size ε. These two sub-domains of Ω are separated by a contact surface Γ ε , on which we prescribe the continuity of the conormal derivatives and a jump of the solution proportional to the conormal derivative, by means of a function of order ε γ . We describe the limit problem for γ&gt;-1. The two cases -1&lt;γ≤1 (Theorem 2.1) and γ&gt;1 (Theorem 2.2) need to be treated separately, because of different a priori estimates.

We prove new L 2-estimates and regularity results for generalized porous media equations “shifted by” a function-valued Wiener path. To include Wiener paths with merely first spatial (weak) derivates we introduce the notion of “ζ-monotonicity” for the non-linear function in the equation. As a consequence we prove that stochastic porous media equations have global random attractors. In addition, we show that (in particular for the classical stochastic porous media equation) this attractor consists of a random point.

In this paper we combine concepts from Riemannian optimization [P.-A. Absil, R. Mahony, and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, 2008] and the theory of Sobolev gradients [J. W. Neuberger, Sobolev Gradients and Differential Equation, 2nd ed., Springer, 2010] to derive a new conjugate gradient method for direct minimization of the Gross--Pitaevskii energy functional with rotation. The conservation of the number of particles in the system constrains the minimizers to lie on a manifold corresponding to the unit $L^2$ norm. The idea developed here is to transform the original constrained optimization problem into an unconstrained problem on this (spherical) Riemannian manifold, so that fast minimization algorithms can be applied as alternatives to more standard constrained formulations. First, we obtain Sobolev gradients using an equivalent definition of an $H^1$ inner product which takes into account rotation. Then, the Riemannian gradient (RG) steepest descent method is derived based on projected gradients and retraction of an intermediate solution back to the constraint manifold. Finally, we use the concept of the Riemannian vector transport to propose a Riemannian conjugate gradient (RCG) method for this problem. It is derived at the continuous level based on the “optimize-then-discretize” paradigm instead of the usual “discretize-then-optimize” approach, as this ensures robustness of the method when adaptive mesh refinement is performed in computations. We evaluate various design choices inherent in the formulation of the method and conclude with recommendations concerning selection of the best options. Numerical tests carried out in the finite-element setting based on Lagrangian piecewise quadratic space discretization demonstrate that the proposed RCG method outperforms the simple gradient descent RG method in terms of rate of convergence. While on simple problems a Newton-type method implemented in the Ipopt library exhibits a faster convergence than the RCG approach, the two methods perform similarly on more complex problems requiring the use of mesh adaptation. At the same time the RCG approach has far fewer tunable parameters. Finally, the RCG method is extensively tested by computing complicated vortex configurations in rotating Bose--Einstein condensates, a task made challenging by large values of the nonlinear interaction constant and the rotation rate as well as by strongly anisotropic trapping potentials.

We investigate a stationary random coefficient autoregressive process. Using renewal type arguments tailor-made for such processes, we show that the stationary distribution has a power-law tail. When the model is normal, we show that the model is in distribution equivalent to an autoregressive process with ARCH errors. Hence, we obtain the tail behavior of any such model of arbitrary order.

We rephrase the conditions from the Chowla and the Sarnak conjectures in abstract setting, that is, for sequences in $\{-1,0,1\}^{{\mathbb{N}^*}}$, and introduce several natural generalizations. We study the relationships between these properties and other notions from topological dynamics and ergodic theory.

Multicomponent Bose-Einstein condensates exhibit an intriguing variety of nonlinear structures. In recent theoretical work [C. Qu, L. P. Pitaevskii, and S. Stringari, Phys. Rev. Lett. 116, 160402 (2016)], the notion of magnetic solitons has been introduced. Here we examine a variant of this concept in the form of vector dark-antidark solitary waves in multicomponent Bose-Einstein condensates (BECs). We first provide concrete experimental evidence for such states in an atomic BEC and subsequently illustrate the broader concept of these states, which are based on the interplay between miscibility and intercomponent repulsion. Armed with this more general conceptual framework, we expand the notion of such states to higher dimensions presenting the possibility of both vortex-antidark states and ring-antidark-ring (dark soliton) states. We perform numerical continuation studies, investigate the existence of these states, and examine their stability using the method of Bogoliubov--de Gennes analysis. Dark-antidark and vortex-antidark states are found to be stable for broad parametric regimes. In the case of ring dark solitons, where the single-component ring state is known to be unstable, the vector entity appears to bear a progressively more and more stabilizing role as the intercomponent coupling is increased.

In this paper, we apply the periodic unfolding method in perforated domains introduced by Cioranescu et al. (Portugal. Math. 63(4) (2006), 467–496) to a class of elliptic problems with nonlinear conditions on the boundary of the holes. A new compactness result will play a major role in the homogenization process and provides correctors without using any extension operator.

We study a lattice field model which qualitatively reflects the phenomenon of Anderson localization and delocalization for real symmetric band matrices. In this statistical mechanics model, the field takes values in a supermanifold based on the hyperbolic plane. Correlations in this model may be described in terms of a random walk in a highly correlated random environment. We prove that in three or more dimensions the model has a ‘diffusive’ phase at low temperatures. Localization is expected at high temperatures. Our analysis uses estimates on non-uniformly elliptic Green’s functions and a family of Ward identities coming from internal supersymmetry.

\n We give a short overview of recent results on a specific class of Markov process: the\n Piecewise Deterministic Markov Processes (PDMPs). We first recall the definition of these\n processes and give some general results. On more specific cases such as the TCP model or a\n model of switched vector fields, better results can be proved, especially as regards long\n time behaviour. We continue our review with an infinite dimensional example of neuronal\n activity. From the statistical point of view, these models provide specific challenges: we\n illustrate this point with the example of the estimation of the distribution of the\n inter-jumping times. We conclude with a short overview on numerical methods used for\n simulating PDMPs.\n

Abstract: We study a lattice field model which qualitatively reflects the phenomenon of Anderson localization and delocalization for real symmetric band matrices. In this statistical mechanics model, the field takes values in a supermanifold based on the hyperbolic plane. Correlations in this model may be described in terms of a random walk in a highly correlated random environment. We prove that in three or more dimensions the model has a ‘diffusive ’ phase at low temperatures. Localization is expected at high temperatures. Our analysis uses estimates on non-uniformly elliptic Green’s functions and a family of Ward identities coming from internal supersymmetry. 1

In this paper, we investigate the effect of emergency signs on evacuation dynamics under smoke conditions. We assume that in a smoky hall the visual field of pedestrians is limited to a certain range, and they do not know the exact location of the exit. In this kind of evacuation process, we analyze the influence of emergency signs on movement direction and speed, and the herd behavior of pedestrians. In the analysis, we divide the emergency signs into two types: the wall signs (WS) and the ground signs (GS). Then, we analyze the variation of pedestrian behavior when they encounter the WS, the GS, and the exit in the evacuation process. Combined with the analysis results, we build our improved model based on the social force model. In the simulation, we study the evacuation process in the case of WS and GS. According to the result of the simulation, we consider that the effect of the emergency signs on herd behavior and the desired speed is an important factor to improve evacuation efficiency. We find that, from the perspective of evacuation time, the evacuation in the case of WS is more efficient, but from the perspective of the interaction between pedestrians, the evacuation in the case of GS presents less security risk. Finally, we explore how to design a mixed layout scheme of WS and GS.