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On any closed manifold (Mn, g) of dimension n∈ { 4 ,5 } we exhibit new blow-up configurations for perturbations of a purely critical stationary Schrödinger equation. We construct positive solutions which blow-up as the sum of two isolated bubbles, one of which concentrates at a point ξ where the potential k of the equation satisfies k(ξ)>n-24(n-1)Sg(ξ),where Sg is the scalar curvature of (Mn, g). The latter condition requires the bubbles to blow-up at different speeds and forces us to work at an elevated precision. We take care of this by performing a construction which combines a priori asymptotic analysis methods with a Lyapounov–Schmidt reduction., SCOPUS: ar.j, info:eu-repo/semantics/published

In this paper, we study the mixed dispersion fourth order nonlinear Helmholtz equation $\Delta^2 u -\beta \Delta u + \alpha u= \Gamma|u|^{p-2} u$ in $\mathbb R^N$ for positive, bounded and $\mathbb Z^N$-periodic functions $\Gamma$. Using the dual method of Evequoz and Weth, we find solutions to this equation and establish some of their qualitative properties.

We study singular radially symmetric solution of the stationary Keller-Segel equation, that is, an elliptic equation with exponential nonlinearity, which is super-critical in dimension N ≥ 3 . The solutions are unbounded at the origin and we show that they describe the asymptotics of bifurcation branches of regular solutions. It is shown that for any ball and any k ≥ 0 , there is a singular solution that satisfies Neumann boundary condition and oscillates at least k times around the constant equilibrium. Moreover, we prove that in dimension 3 ≤ N ≤ 9 there are regular solutions satisfying Neumann boundary conditions that are close to singular ones when the value at the origin is close to infinity. Hence, it follows that there exist regular solutions on any ball with arbitrarily fast oscillations. For generic radii, we show that the bifurcation branches of regular solutions oscillate in the bifurcation plane when 4 ≤ N ≤ 9 and approach to a singular solution. In dimension N > 10 , we show that the Morse index of the singular solution is finite, and therefore the existence of regular solutions with fast oscillations is not expected.

We consider the electrostatic Born–Infeld energy∫RN(1-1, SCOPUS: ar.j, info:eu-repo/semantics/published

We study the mixed dispersion fourth order nonlinear Schrödinger equation i∂ tψ γ Δ 2ψ +β Δ ψ +, SCOPUS: ar.j, info:eu-repo/semantics/published

In this paper, we study the existence of solutions to the mixed dispersion nonlinear Schrödinger equation γΔ2u − Δu + αu =, SCOPUS: ar.j, info:eu-repo/semantics/published

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The essay offers an account of Alfred Marshall’s approach to mathematics and aims to deepen our understanding on his methodological considerations and views on Time. Contrary to what is commonly believed, 19th century mathematics developed in a manner similar to the evolution of political and economic ideas i.e. through passionate intellectual debates between opposing ideological parts. Time and the seek for its true meaning, constitutes a peculiar aspect of this controversy that the Cambridge Mathematical Tradition offers vital space for reflection. It is due to this educational background that Marshall separates between ‘t’ and ‘T’ and develops his skepticism against analytical reasoning. His thoughts on real time processes and on translating mathematical statements to economic language, advocates in favor of his unique intellectual depth. Research is textual, based on primary as well as secondary sources. L’essai offre un compte rendu de l’approche d’Alfred Marshall aux mathématiques et vise à approfondir notre compréhension de ses considérations méthodologiques et de ses vues sur le Temps. Contrairement à ce que l'on croit, les Mathématiques du 19e siècle se sont développées de manière similaire à l’évolution des idées politiques et économiques, c’est-à-dire par le biais de débats intellectuels passionnés entre des camps opposés. Le Temps et la recherche de son vrai sens, constitue un aspect particulier de cette controverse dont la Tradition Mathématique de Cambridge offre un espace vital pour la réflexion. Cela est dû à cette formation que Marshall sépare entre ‘t’ et ‘T’ et développe son scepticisme contre le raisonnement analytique. Ses réflexions sur les processus en temps réel et sur la traduction des déclarations mathématiques au langage économique, plaide en faveur de sa profondeur intellectuelle unique. La recherche est textuelle, basée sur des sources primaires aussi bien que secondaires.

This study proposes a new framework to tackle the uncertainty that prevails in the exploitation of renewable resources. It deals with the question of how to guarantee both a minimum multi-species harvest and the renewal of resources when their evolutions are uncertain. The problem is twofold: to decentralize a global constraint (on a multi-species harvest) into local constraints (on the resources of the different species) and, then, to use a "tychastic" approach necessitating only the forecasts of the lowers bounds of the resource growth rates. This study, formulated as a "tychastic" regulated system with viability constraints, departs from stochastic approaches generally used to deal with uncertain situations. It provides the time dependent harvesting rule allowing to always comply with a minimum harvest objective and resources replenishment thresholds whatever happens and a tychastic measure of risk viability in terms of minimum resources initially required. To solve this problem involving global and local constraints a new method that decentralizes the constraints has been devised. An example is presented whose numerical results are obtained thanks to a dedicated software using mathematical and algorithmic tools of viability theory.

We consider the homogenization of a coupled system of PDEs describing flows in highly heterogeneous porous media. Due to the coupling, the effective coefficients always depend on the slow variable, even in the simple case when the porosity is taken purely periodic. Therefore the most important part of the computational time for the numerical simulation of such flows is dedicated to the determination of these coefficients. We propose a new numerical algorithm based on Reduced Basis techniques, which significantly improves the computational performances., info:eu-repo/semantics/published

International audience; Dans la région du Sud-Ouest du Niger, des bas fourneaux à usage unique ont été identifiés lors d'une prospection archéologique. À Saga Gorou, dans les environs de Niamey, un inventaire exhaustif de 4 504 bas fourneaux permet de proposer une analyse spatiale au sein d'un bassin versant, de 32 km², représentatif du paysage sahélien. Les sites métallurgiques de Bilfouda et de Garbel ont les plus fortes concentrations de bas fourneaux. Chacun d'eux a été cartographié avec précision à partir de photographies aériennes à basse altitude. Six dates radiocarbone, obtenues sur ces sites, sont comprises entre le iv e et le ix e siècles. Une analyse spatiale à partir d'outils mathématiques (classification ascendante hiérarchique, k-moyennes, arbre couvrant de poids minimum) a mis en évidence des regroupements imbriqués de bas fourneaux. Quatre niveaux d'organisation, du local au plus global, ont été identifiés : la batterie, l'ensemble, le site, et le complexe. Chacun est défini par un nombre de bas fourneaux, une superficie, une durée de fonctionnement et un facteur déterminant le contrôle de la production. Au sein de la batterie, les bas fourneaux sont disposés en forme de fer à cheval ou en lignes plus ou moins rectilignes. Cette disposition est liée à une gestion maîtrisée de l'espace de travail et semble régie par des rituels socioculturels dépendant de la famille ethnique des métallurgistes. Abstract: In southwestern Niger, single-use furnaces were identified during archaeological prospecting. An exhaustive inventory of 4,504 bloomery furnaces from Saga Gorou, near Niamey, allowed us to propose a spatial analysis of catchment area of 32 km², representing the Sahelian landscape. The smelting sites of Bilfouda and Garbel have the highest concentrations of bloomery furnaces; each was mapped accurately from low-altitude aerial photographs. Six radiocarbon dates were obtained, ranging from the fourth to the ninth century. A spatial analysis with a series of mathematical tools (Ward's method, k-means and minimum spanning tree) showed an organization of bloomery furnaces where groups were nested. Four levels from local to global were identified: battery, cluster, site and complex. Each was defined by a number of bloomery furnaces, an area, an operating time and control of production parameters. In a battery, bloomery furnaces can be arranged in a horseshoe shape or in a more or less straight line. This organization of bloomery furnaces could be linked to the socio-cultural context and reflects good workspace management. Mots-clés : Afrique, bas fourneaux, espace de travail, métallurgie du fer, organisation spatiale, Sahel, I er millénaire AD.

We consider the boundary value problem {-Δu+u-λeu=0,u>0inB1(0)∂νu=0on∂B1(0),whose solutions correspond to steady states of the Keller–Segel system for chemotaxis. Here B1(0) is the unit disk, ν the outer normal to ∂B1(0) ,and λ> 0 is a parameter. We show that, provided λ is sufficiently small, there exists a family of radial solutions uλ to this system which blow up at the origin and concentrate on ∂B1(0) ,as λ→ 0. These solutions satisfy limλ→0uλ(0), SCOPUS: ar.j, info:eu-repo/semantics/published

Consider an elliptic operator in divergence form with symmetric coefficients.If the diffusion coefficients are periodic, the Bloch theorem allows one to diagonalize the elliptic operator, which is key to the spectral properties of the elliptic operator and the usual starting point for the study of its long-time homogenization.When the coefficients are not periodic (say, quasi-periodic, almost periodic, or random with decaying correlations at infinity), the Bloch theorem does not hold and both the spectral properties and the long-time behavior of the associatedoperator are unclear.At low frequencies, we may however consider a formal Taylor expansion of Bloch waves (whether they exist or not) based on correctors in elliptic homogenization.The associated Taylor-Bloch waves diagonalize the elliptic operator up to an error term (an "eigendefect"), which we express with the help of a new family of extended correctors.We use the Taylor-Bloch waves with eigendefects to quantify the transport properties and homogenization error over large timesfor the wave equation in terms of the spatial growth of these extended correctors.On the one hand, this quantifies the validity of homogenization over large times (both for the standard homogenized equation and higher-order versions).On the other hand, this allows us to prove asymptotic ballistic transport of classical waves at low energies for almost periodic and random operators., Comment: Annales Scientifiques de l'{\'E}cole Normale Sup{\'e}rieure, Elsevier Masson, In press

Efficient water resources management is an issue of major importance in the field of sustainable development, especially in the agricultural sector which represents the main consumer through irrigations. Therefore irrigation management is an important and innovating area which was the subject of several research and studies. Modeling, and more particularly, the Agent-Based Modeling (ABM), allows better representing the multiplicity of these actors, the diversity of their roles and their interactions. The main reason why we chose the agent technology in the field of gravity irrigation systems, is the complexity to manage in real-time the water distribution operations those arrive asynchronously and dynamically and to be reactive and adaptive to the dynamic and unpredictable events that characterizes the field (mainly rainy advents). Our objectives are mainly located on two levels. The first one, concerns the gravity irrigation modeling by a multi-agent technology and the agent modeling through AML language. The second one focuses on the irrigations scheduling optimization using an evolutionary algorithm. Comparisons between schedules before and after optimization are made and the results shows that our approach can be considered as an efficient tool for planning irrigation schedules by considering crops water needs.

This paper is concerned with the mean-field limit for the gradient flow evolution of particle systems with pairwise Riesz interactions, as the number of particles tends to in-finity. Based on the method introduced by Serfaty [Mean-Field Limits of the Gross-Pitaevskii and Parabolic Ginzburg-Landau equations, preprint, arXiv:1507.03821, 2015] in the context of the Ginzburg-Landau vortices, using regularity and stability properties of the limiting equation, we prove a mean-field limit result in dimensions 1 and 2 in cases for which this problem was still open., SCOPUS: ar.j, info:eu-repo/semantics/published

We show that certain linear elliptic equations (and systems) in divergence form with almost periodic coefficients have bounded, almost periodic correctors. This is proved under a new condition we introduce which quantifies the almost periodic assumption and includes (but is not restricted to) the class of smooth, quasiperiodic coefficient fields which satisfy a Diophantine-type condition previously considered by Kozlov (Mat Sb (N.S), 107(149):199–217, 1978). The proof is based on a quantitative ergodic theorem for almost periodic functions combined with the new regularity theory recently introduced by Armstrong and Shen (Pure Appl Math, 2016) for equations with almost periodic coefficients. This yields control on spatial averages of the gradient of the corrector, which is converted into estimates on the size of the corrector itself via a multiscale Poincaré-type inequality., SCOPUS: ar.j, info:eu-repo/semantics/published

We build upon recent advances on the distributional aspect of Stein's method to propose a novel and flexible technique for computing Stein operators for random variables that can be written as products of independent random variables. We show that our results are valid for a wide class of distributions including normal, beta, variance-gamma, generalized gamma and many more. Our operators are $k$th degree differential operators with polynomial coefficients; they are straightforward to obtain even when the target density bears no explicit handle. As an application, we derive a new formula for the density of the product of $k$ independent symmetric variance-gamma distributed random variables., 20 pages

The mathematical relations between the regular Coulomb function Fηℓ(ρ) and the irregular Coulomb functions H±ηℓ(ρ) and Gηℓ(ρ) are obtained in the complex plane of the variables η and ρ for integer or half-integer values of ℓ. These relations, referred to as “connection formulas,” form the basis of the theory of Coulomb wave functions and play an important role in many fields of physics, especially in the quantum theory of charged particle scattering. As a first step, the symmetry properties of the regular function Fηℓ(ρ) are studied, in particular, under the transformation ℓ ↦ −ℓ − 1, by means of the modified Coulomb function Φηℓ(ρ), which is entire in the dimensionless energy η−2 and the angular momentum ℓ. Then, it is shown that, for integer or half-integer ℓ, the irregular functions H±ηℓ(ρ) and Gηℓ(ρ) can be expressed in terms of the derivatives of Φη,ℓ(ρ) and Φη,−ℓ−1(ρ) with respect to ℓ. As a consequence, the connection formulas directly lead to the description of the singular structures of H±ηℓ(ρ)and Gηℓ(ρ) at complex energies in their whole Riemann surface. The analysis of the functions is supplemented by novel graphical representations in the complex plane of η−1., SCOPUS: ar.j, info:eu-repo/semantics/published

Real analysis is pervasive to many applications, if only because it is a suitable tool for modeling physical or socio-economical systems. As such, its support is warranted in proof assistants, so that the users have a way to formally verify mathematical theorems and correctness of critical systems. The Coq system comes with an axiomatization of standard real numbers and a library of theorems on real analysis. Unfortunately, this standard library is lacking some widely used results. For instance, the definitions of integrals and derivatives are based on dependent types, which make them cumbersome to use in practice. This thesis first describes various state-of-the-art libraries available in proof assistants. To palliate the inadequacies of the Coq standard library, we have designed a user-friendly formalization of real analysis: Coquelicot. An easier way of writing formulas and theorem statements is achieved by relying on total functions in place of dependent types for limits, derivatives, integrals, power series, and so on. To help with the proof process, the library comes with a comprehensive set of theorems that cover not only these notions, but also some extensions such as parametric integrals and asymptotic behaviors. Moreover, an algebraic hierarchy makes it possible to apply some of the theorems in a more generic setting, such as complex numbers or matrices. Coquelicot is a conservative extension of the classical analysis of Coq's standard library and we provide correspondence theorems between the two formalizations. We have exercised the library on several use cases: in an exam at university entry level, for the definitions and properties of Bessel functions, and for the solution of the one-dimensional wave equation.; L'analyse réelle a de nombreuses applications car c'est un outil approprié pour modéliser de nombreux phénomènes physiques et socio-économiques. En tant que tel, sa formalisation dans des systèmes de preuve formelle est justifié pour permettre aux utilisateurs de vérifier formellement des théorèmes mathématiques et l'exactitude de systèmes critiques. La bibliothèque standard de Coq dispose d'une axiomatisation des nombres réels et d'une bibliothèque de théorèmes d'analyse réelle. Malheureusement, cette bibliothèque souffre de nombreuses lacunes. Par exemple, les définitions des intégrales et des dérivées sont basées sur les types dépendants, ce qui les rend difficiles à utiliser dans la pratique. Cette thèse décrit d'abord l'état de l'art des différentes bibliothèques d'analyse réelle disponibles dans les assistants de preuve. Pour pallier les insuffisances de la bibliothèque standard de Coq, nous avons conçu une bibliothèque facile à utiliser : Coquelicot. Une façon plus facile d'écrire les formules et les théorèmes a été mise en place en utilisant des fonctions totales à la place des types dépendants pour écrire les limites, dérivées, intégrales et séries entières. Pour faciliter l'utilisation, la bibliothèque dispose d'un ensemble complet de théorèmes couvrant ces notions, mais aussi quelques extensions comme les intégrales à paramètres et les comportements asymptotiques. En plus, une hiérarchie algébrique permet d'appliquer certains théorèmes dans un cadre plus générique comme les nombres complexes pour les matrices. Coquelicot est une extension conservative de l'analyse classique de la bibliothèque standard de Coq et nous avons démontré les théorèmes de correspondance entre les deux formalisations. Nous avons testé la bibliothèque sur plusieurs cas d'utilisation : sur une épreuve du Baccalauréat, pour les définitions et les propriétés des fonctions de Bessel ainsi que pour la solution de l'équation des ondes en dimension 1.

We propose a preconditioning technique that is applicable in a "black box" fashion to linear systems arising from second order scalar elliptic PDEs discretized by finite differences or finite elements with nodal basis functions. This technique is based on an algebraic multilevel scheme with coarsening by aggregation. We introduce a new aggregation method which, for the targeted class of applications, produces semicoarsening effects whenever desirable, while the number of nodes is decreased by a factor of about 4 at each level, regardless of the problem at hand. Moreover, the number of nonzero entries per row in the successive coarse grid matrices remains approximately constant, ensuring small set up cost and modest memory requirements. This aggregation technique can be used in an algebraic multigrid (AMG)-like framework, but better results are obtained with an algebraic multilevel scheme based on a block approximate factorization of the matrix. In this scheme, the block pivot corresponding to fine grid nodes is approximated by a modified incomplete LU (MILU) factorization. To enhance robustness and avoid any potential breakdown, the coarsening process is refined by recasting as "coarse" fine grid nodes for which the corresponding pivot in this MILU factorization would be negative or too small. Numerical results display the efficiency, the scalability, and the robustness of the resulting pre-conditioner on a wide set of discrete scalar PDE problems, ranging from the two-dimensional Poisson equation to three-dimensional convection-diffusion problems with high Reynolds number and strongly varying convection. © 2006 Society for Industrial and Applied Mathematics., SCOPUS: cp.j, info:eu-repo/semantics/published