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International audience; We give a full characterization of nonsmooth Lyapunov pairs for perturbed sweeping processes under very general hypotheses. As a consequence, we provide an existence result and a criterion for weak invariance for perturbed sweeping processes. Moreover, we characterize Lyapunov pairs for gradient complementarity dynamical systems.

This thesis is dedicated to the study of differential inclusions involving normal cones of nonregular sets in Hilbert spaces. In particular, we are interested in the sweeping process and its variants. The sweeping process is a constrained differential inclusion involving normal cones which appears naturally in several applications such as elastoplasticity, electrical circuits, hysteresis, crowd motion, etc.This work is divided conceptually in three parts: Study of positively alpha-far sets, existence results for differential inclusions involving normal cones and characterizations of Lyapunov pairs for the sweeping process. In the first part (Chapter 2), we investigate the class of positively alpha-far sets. This class of nonregular sets is very general and includes convex, uniformly prox-regular and uniformly subsmooth sets, among others. It turns out that this class is the best suited to the study of differential inclusions involving normal cones.In the second part (Chapter 3 to the first part of Chapter 8), we provide several existence results for the sweeping process and its variants. In order to do that, we consider three approaches: The Catching-up algorithm, the Galerkin-like method and the Moreau-Yosida regularization. The first method is the most classic in the study of differential inclusions involving normal cones. We used it in the case where the set considered is fixed.The second method (Galerkin-like) consists in approximating the original problem by projecting the state into a finite-dimensional Hilbert space, but not the velocity. Approximate problems always have a solution and, under some compactness conditions, we prove that they converge strongly pointwisely (up to a subsequence) to a solution of the original differential inclusion. Moreover, it is shown that this method is well adapted to deal with differential inclusions involving normal cones, by providing general existence results for the generalized sweeping process. As a result, existence of solutions for the first order and second order sweeping process is obtained. Furthermore, this method is used to show existence of solutions of the perturbed sweeping process with nonlocal initial conditions.The third method is the Moreau-Yosida regularization technique which consists in approximating a given differential inclusion by a penalized one, depending on a positive parameter and then to pass to the limit when the parameter goes to zero. This method is used to deal with state-dependent sweeping processes governed by uniformly subsmooth sets.Finally, in the third part (Second part of Chapter 8 and Chapter 9), we give some characterizations of weak Lyapunov pairs and weak invariance for the perturbed sweeping process with uniformly subsmooth sets., Esta tesis está dedicada al estudio de inclusiones diferenciales con conos normales de conjuntos no regulares en espacios de Hilbert. En particular, nos interesa el proceso de arrastre y sus variantes. El proceso de arrastre es una inclusión diferencial restringida con conos normales que aparece naturalmente en varias aplicaciones tales como elastoplasticidad, histéresis, circuitos eléctricos, movimiento de multitudes, etc.Este trabajo está dividido conceptualmente en tres partes: Estudio de los conjuntos ''alpha-far'', existencia de soluciones para las inclusiones diferenciales con conos normales y caracterizaciones de los pares de Lyapunov para el proceso de arrastre en espacios de Hilbert separable.En la primera parte (Capítulo 2), investigamos la clase de conjuntos positivamente ''alpha-far''. Esta clase de conjuntos no regulares es muy general e incluye los conjuntos convexos, uniformemente prox-regulares y uniformemente sub-lisos, entre otros. Esta clase de conjuntos es la mejor adaptada al estudio de inclusiones diferenciales con conos normales.En la segunda parte (Capítulo 3 hasta la primera parte del Capítulo 8), se entregan varios resultados de existencia para el proceso de arrastre y sus variantes. Para ello, consideramos tres enfoques: el algoritmo de rectificación (Catching-up algorithm), el método de tipo Galerkin y la regularización de Moreau-Yosida. El primer método es el más clásico en el estudio de inclusiones diferenciales gobernadas por conos normales. Aquí es utilizado en el caso donde el conjunto considerado es fijo. El segundo método (de tipo Galerkin) consiste en aproximar el problema original proyectando el estado sobre un espacio de Hilbert de dimensión finita, pero no la velocidad. Los problemas aproximados siempre tienen una solución y, bajo ciertas condiciones de compacidad, se demuestra que ellos convergen fuertemente (salvo subsucesión) a una solución de la inclusión diferencial original. Más aún, se muestra que este método está bien adaptado para tratar inclusiones diferenciales con conos normales, proporcionando resultados generales de existencia para el proceso de arrastre generalizado. En consecuencia, se obtiene la existencia de soluciones para el proceso de arrastre de primer y segundo orden. Adicionalmente, este método es utilizado para mostrar la existencia de soluciones del proceso de arrastre con condiciones iniciales no locales.El tercer método es la técnica de regularización de Moreau-Yosida que consiste en aproximar una inclusión diferencial por una penalizada, en función de un parámetro positivo, para luego pasar al límite cuando el parámetro tiende a cero. Este método es utilizado para tratar el proceso de arrastre dependiente del estado gobernado por conjuntos uniformemente sub-lisos.Finalmente, en la tercera parte (segunda parte del Capítulo 8 y Capítulo 9), se proporcionan algunas caracterizaciones de los pares de Lyapunov débiles y la invariancia débil para el proceso de arrastre perturbado con conjuntos uniformemente sub-lisos., Cette thèse est consacrée à l'étude des inclusions différentielles sur les espaces de Hilbert séparables avec des cônes normaux à des ensembles non réguliers. En particulier, nous nous sommes intéressés à l'étude des processus de rafle et de ses variantes. Le processus de rafle est une inclusion différentielle contrainte avec des cônes normaux qui apparaissent naturellement dans plusieurs applications telles que l'elastoplasticité, les circuits électriques, l'hystérésis, le mouvement de foule, etc.Ce travail est divisé conceptuellement en trois parties : Étude des ensembles positivement ''alpha-far'', existence de solutions pour les inclusions différentielles avec des cônes normaux et caractérisations des paires de Lyapunov pour le processus de rafle dans des espaces de Hilbert séparables.Dans la première partie (Chapitre 2), nous étudions la classe d'ensembles positivement ''alpha-far''. Cette classe d'ensembles non réguliers est très générale et comprend des ensembles convexes, uniformément prox-réguliers et uniformément sous-lisses, entre autres. Il se trouve que cette classe est la mieux adaptée à l'étude des inclusions différentielles avec des cônes normaux.Dans la deuxième partie (Chapitre 3 à la première partie du Chapitre 8), nous fournissons plusieurs résultats d'existence pour le processus de rafle et ses variantes. Pour cela, nous considérons trois approches : L'algorithme de rattrapage (Catching-up algorithm), la méthode de type Galerkin et la régularisation de Moreau-Yosida.La première méthode est la plus classique dans l'étude des inclusions différentiels gouvernées par des cônes normaux. On l'a utilisé dans le cas où l'ensemble considéré est fixe.La deuxième méthode (de type Galerkin) consiste à approcher le problème original en projetant l'état sur un espace de Hilbert de dimension finie, mais pas la vitesse. Les problèmes approchés ont toujours une solution et, sous certaines conditions de compacité, on montre qu'ils convergent fortement (via une sous-suite) vers une solution de l'inclusion différentielle initiale. En outre, on montre que cette méthode est bien adaptée pour traiter les inclusions différentielles avec des cônes normaux, en fournissant des résultats généraux d'existence pour le processus de rafle généralisé. En conséquence, l'existence de solutions pour le processus de rafle de premier ordre et de deuxième ordre est obtenue. En plus, cette méthode est utilisée pour montrer l'existence de solutions du processus de rafle perturbé avec des conditions initiales non locales.La troisième méthode est la technique de régularisation de Moreau-Yosida qui consiste à approcher une inclusion différentielle donnée par une pénalisée, en fonction d'un paramètre positif, puis passer à la limite lorsque le paramètre tend vers zéro. Cette méthode est utilisée pour traiter les processus de rafle dépendants de l'état régis par des ensembles uniformément sous-lisses.Finalement, dans la troisième partie (Deuxième partie du Chapitre 8 et Chapitre 9), on fournit des caractérisations des paires de Lyapunov faibles et l'invariance faible pour le processus de rafle perturbé avec des ensembles uniformément sous-lisses.

International audience; In many natural environments, water resources are not directly usable because their quality does not fulfill the drinking water norms, and therefore treatment or control needs to be applied. It could also happen that the quality of these resource is temporarily acceptable but impacted by its extraction. This is typically the case of coastal lagoons that communicate with the sea. Depending on the level of water in the lagoon, the communication with the sea increases its salt concentration and then one may want to maximize the water pumping within giving bound on the salt concentration, and preserving water volume in the lagoon for its sustainability. In this work, we model the problem as a non-autonomous optimal control problem under state constraints, distinguising two seasons (dry and humid) in the year. The state of the system describes the salt and volume concentration of a coastal lagoon connected to the sea, and the constraints represent bounds on the salt concentration and water volume of the lagoon. The control variable is the output flow rate from the lagoon and the objective the maximization of the total water extracted during one year. Instead of deriving necessary conditions of optimal control under state constraints (i.e. Maximum Principle under state constraints), we first characterize the viability kernel over the year period, that is the set of initial conditions for which there exists a (time-varying) control such that the corresponding trajectory satisfies the constraints over one year. Then, we study the optimal control within this set, expliciting for any initial condition an optimal control policy, as a state feedback, maximizing the pumped water over one year. This problem has been motivated by the case study of the Tunquen lagoon in Chile, but the modelling approach and the methodolgy could be extended to similar situations.

International audience; The existence and the convergence (up to a subsequence) of the Moreau-Yosida regularization for the state-dependent sweeping process with nonregular (subsmooth and positively alpha-far) sets are established. Then, by a reparametrization technique, the existence of solutions for bounded variation continuous state-dependent sweeping processes with nonregular (subsmooth and positively alpha-far) sets is proved. An application to vector hysteresis is discussed, where it is shown that the Play operator with positively alpha-far sets is well defined for bounded variation continuous inputs.

International audience; In this paper we present a new method to solve differential inclusions in Hilbert spaces. This method is a Galerkin-like method where we approach the original problem by projecting the state into a $n$-dimensional Hilbert space but not the velocity. We prove that the approached problem always has a solution and that, under some compactness conditions, the approached problems have a subsequence which converges strongly pointwisely to a solution of the original differential inclusion. We apply this method to the generalized perturbed sweeping process governed by nonregular sets (equi-uniformly subsmooth or positively $\alpha$-far). This differential inclusion includes Moreau's sweeping process, the state-dependent sweeping process, and second-order sweeping process for which we give very general existence results.

International audience; We consider the general class of positively alpha-far sets, introduced in [29], which contains strictly the class of uniformly prox-regular sets and the class of uniformly subsmooth sets. We provide some conditions to assure the uniform subsmoothness, and thus the positive alpha-farness, of the inverse images under a differentiable mapping. Then, we take advantage of the properties of this class to study the generalized perturbed sweeping process{-(x)over dot(t) is an element of F(t, x(t))+ g(x(t))N (C(t), h(x(t))) a.e. t is an element of[T-0, T], x(T-0) = x(0) is an element of h(-1) (C(T-0)), where g : X -> L(Y; X), h: X -> Y are two functions, X, Y are two separable Hilbert spaces and the sets C(t) belong to the class of positively alpha-far sets. This differential inclusion includes the classical perturbed sweeping process as well as complementarity dynamical systems. Our study is achieved by approximating the given differential inclusion with maximally perturbed differential inclusions which, under certain compactness conditions, converges to an absolutely continuous solution. Moreover, this approach allows us to get existence for evolution inclusions of the form {-(x)over dot(t) is an element of partial derivative f(t, x(t)) + F(t, x(t)) a.e. t is an element of[T-0, T], x(T-0) = x(0), where [T-0, T] is a fixed interval with 0 RU {+infinity} is a lower semi-continuous function, not necessarily convex. Here partial derivative f(t, .) denotes the Clarke subdifferential of the function f(t, .) and F : [T-0, T] x X paired right arrows X is a perturbation term.

We estimate the lowest eigenvalue in the gap of a Dirac operator with mass in terms of a Lebesgue norm of the potential. Such a bound is the counterpart for Dirac operators of the Keller estimates for the Schrödinger operator, which are equivalent to Gagliardo-Nirenberg-Sobolev interpolation inequalities. Domain, self-adjointness, optimality and critical values of the norms are addressed, while the optimal potential is given by a Dirac equation with a Kerr nonlinearity. A new critical bound appears, which is the smallest value of the norm of the potential for which eigenvalues may reach the bottom of the gap in the essential spectrum. Most of our result are established in the Birman-Schwinger reformulation of the problem.

International audience; In this paper we study one-dimensional BSDE's whose coefficient f is monotonic in y and non-Lipschitz in z. We obtain a general existence result when f has at most quadratic growth in z and is bounded. We study the special case f (t, y, z) = vertical bar z vertical bar(p) where p is an element of (1, 2]. Finally, we study the case f has a linear growth in z, general growth in y and is not necessarily bounded.

International audience

International audience; We give the explicit solution of the optimal control problem which consists in minimizing the epidemic peak in the SIR model when the control is an attenuation factor of the infectious rate, subject to a L 1 constraint on the control which represents a budget constraint. The optimal strategy is given as a feedback control which consists in a singular arc maintaining the infected population at a constant level until the immunity threshold is reached, and no intervention outside the singular arc. We discuss and compare this strategy with the one that minimizes the peak when fixing the duration of a single intervention, as already proposed in the literature. Numerical simulations illustrate the benefits of the proposed control.

In this paper we deal with solving efficiently 2D linear parabolic singularly perturbed systems of convection–diffusion type. We analyze only the case of a system of two equations where both of them feature the same diffusion parameter. Nevertheless, the method is easily extended to systems with an arbitrary number of equations which have the same diffusion coefficient. The fully discrete numerical method combines the upwind finite difference scheme, to discretize in space, and the fractional implicit Euler method, together with a splitting by directions and components of the reaction–convection–diffusion operator, to discretize in time. Then, if the spatial discretization is defined on an appropriate piecewise uniform Shishkin type mesh, the method is uniformly convergent and it is first order in time and almost first order in space. The use of a fractional step method in combination with the splitting technique to discretize in time, means that only tridiagonal linear systems must be solved at each time level of the discretization. Moreover, we study the order reduction phenomenon associated with the time dependent boundary conditions and we provide a simple way of avoiding it. Some numerical results, which corroborate the theoretical established properties of the method, are shown. This research was partially supported by the project MTM2014-52859-P and by the Aragón Government and European Social Fund, Spain (group E24–17R ).

An automata network (AN) is a network of entities, each holding a state from a finite set and related by a graph structure called an interaction graph. Each node evolves according to the states of its neighbors in the interaction graph, defining a discrete dynamical system. This thesis work explores two main questions : a) what is the link between dynamical and computational properties of an AN ? and b) what is the impact of the interaction graph topology on the global dynamics of an AN ?.In order to tackle the first question a notion of computational complexity of an AN family is defined in terms of the computational complexity of decision problems related to the dynamics of the network. On the other hand, dynamical complexity of a particular AN family is defined in terms of the existence of attractors of exponential period. A strong link between these two last definitions is presented in terms of the notion of simulation between AN families. In this context, complexity is characterized from a localized standpoint by studying the existence of structures called coherent gadgets which satisfy two properties : i) they can locally interact in a coherent way as dynamical systems and ii) they are capable of simulating a finite set of functions defined over a fixed finite set.Finally, the second question is addressed in the context of a well-known family called freezing automata networks. An AN is freezing if there is an order on states such that the state evolution of any node is non-decreasing in any orbit. A general model checking problem capturing many classical decision problems is presented. In addition, when three graph parameters, the maximum degree, the treewidth and the alphabet size are bounded, a fast-parallel algorithm that solves general model checking problem is presented. Moreover, it is shown that the latter problem is unlikely to be fixed-parameter tractable on the treewidth parameter as well as on the alphabet size when considered as single parameters.; Un réseau d’automates (RA) est un réseau d’entités (les automates) en interaction. Ces automates ont un nombre fini d’états possibles et sont reliés les uns aux autres par une structure de graphe appelée graphe d’interaction. Chaque automate évolue au cours du temps discret en fonction des états de ses voisins dans le graphe d’interaction, ce qui définit un système dynamique. Ce travail de thèse explore deux questions principales : a) quel est le lien entre les propriétés dynamiques et calculatoires d’un RA ? et b) quel est l’impact de la topologie du graphe d’interaction sur la dynamique globale d’un RA ?.Pour aborder la première question, une notion de complexité calculatoire est définie au regard de problèmes de décision liés à la dynamique des RA. De même, une notion de complexité dynamique est définie en termes de l’existence d’attracteurs de période exponentielle. Un lien fort entre ces deux définitions est présenté qui met en exergue le concept de simulation entre familles de RA. Dans ce contexte, la complexité se caractérise d’un point de vue localisé en étudiant l’existence de structures appelées gadgets qui satisfont deux propriétés : i) ils peuvent interagir localement de manière cohérente comme des systèmes dynamiques et ii) ils sont capables de simuler un ensemble fini de fonctions définies sur un ensemble fini.La deuxième question est quant à elle abordée dans le contexte des RA “freezing”. Un RA est “freezing” s’il y a un ordre sur les états de telle sorte que l’évolution de l’état de n’importe quel automate ne diminue pas quelle que soit l’orbite. Un problème général de model-checking capturant de nombreux problèmes de décision classiques est présenté. De plus, lorsque trois paramètres de graphe, le degré maximum, la largeur arborescente et la taille de l’alphabet sont bornés, un algorithme parallèle efficace résolvant le problème mentionné est donné. De plus, il est montré que ce problème est peu susceptible d’être FPT (fixed-parameter tractable) lorsque le paramètre de largeur arborescente ou celui de taille de l’alphabet sont considérés comme unique paramètre.

Versión actualizada del trabajo presentado a las Jornadas sobre Herramientas Difusas para el Razonamiento No Canónico (III HARMONIC 2018). organizadas por la Universidad de Cádiz en Jimena de la Frontera (Cádiz) del 1 al 4 de noviembre de 2018. A mathematical model is presented to define new inclusions, intersections and unions as alternatives to the usual ones between crisp and fuzzy subsets. In particular, the concept of “the non-trivial content of the empty set ∅” is analyzed. This proposed model is based on the interconnection of two consolidated mathematical concepts in specialized literature: One, (which belongs to the field of image processing using Mathematical Morphology techniques), is that of order of activity and that we use here in the general context of lattices (L, ≤) and in particular in that of Boolean Algebras. The other consists of a version in distributive lattices (L, ≤) of the symmetric difference operator Δ, a classic concept in Set Theory. The utility of the model is illustrated in the following fields: analysis of risk maps, (areas of avalanches, risk of fires, landslides, earthquakes, ...), as well as maps with contour lines: (isochrons, isotherms, salinity , rainfall, intensity of earthquakes, ...). Also in data pre-processing for “Data Mining” tasks and in “Data Analysis with Uncertainty”. A special section is dedicated to the application of the model in Digital Image Processing using Mathematical Morphology techniques. Finally, it is justified that the model can be useful in other fields such as Analysis of Formal Concepts, Probability and in theoretical contexts such as Topology.

International audience

Sport activities have benefited in recent years from the progressive adoption of different technological assets in order to improve individual as well as group training, collect different statistics or enhance the spectator experiences. The progressive adoption of Internet of Things paradigms can also be considered within the scope of sport activities, providing high levels of user interactivity as well as enabling cloud-based data storage and processing. In this work, a system for monitoring biophysical, kinematic and environmental parameters within the development of basketball training is presented. A set of on-body nodes with multiple sensors and wireless body area network capabilities have been designed, implemented and tested under real training conditions during a match. Wireless channel analysis results have been obtained with the aid of in house implemented deterministic 3D ray launching algorithm, providing accurate coverage/capacity estimations in relation with human body consideration in the field as well as in the stadium. Measurement results give relevant information in relation with individual player characteristics as well as with team characteristics, providing a flexible tool to improve training development of basketball. This work was supported in part by the Ministerio de Ciencia, Innovación y Universidades, Spain, (MCIU)/Agencia Estatal de Investigación (AEI)/Fondo Europeo de Desarrollo Regional (FEDER)/European Union under Grant RTI2018-095499-B-C31 (IoTrain), and in part by the European Union's Horizon 2020 Research and Innovation Program (Stardust-Holistic and Integrated Urban Model for Smart Cities) under Grant 774094.

We consider conservative cross-diffusion systems for two species where individual motion rates depend linearly on the local density of the other species. We develop duality estimates and obtain stability and approximation results. We first control the time evolution of the gap between two bounded solutions by means of its initial value. As a by product, we obtain a uniqueness result for bounded solutions valid for any space dimension, under a smallness assumption. Using a discrete counterpart of our duality estimates, we prove the convergence of random walks with local repulsion in one dimensional discrete space to cross-diffusion systems. More precisely, we prove sharp quantitative estimates for the gap between the stochastic process and the cross-diffusion system. We complete this study with a rough but general estimate and convergence results, when the population and the number of sites become large., Comment: 40 pages

We consider the problem of studying the simulation capabilities of the dynamics of arbitrary networks of finite states machines. In these models, each node of the network takes two states 0 (passive) and 1 (active). The states of the nodes are updated in parallel following a local totalistic rule, i.e., depending only on the sum of active states. Four families of totalistic rules are considered: linear or matrix defined rules (a node takes state 1 if each of its neighbours is in state 1), threshold rules (a node takes state 1 if the sum of its neighbours exceed a threshold), isolated rules (a node takes state 1 if the sum of its neighbours equals to some single number) and interval rule (a node takes state 1 if the sum of its neighbours belong to some discrete interval). We focus in studying the simulation capabilities of the dynamics of each of the latter classes. In particular, we show that totalistic automata networks governed by matrix defined rules can only implement constant functions and other matrix defined functions. In addition, we show that t by threshold rules can generate any monotone Boolean functions. Finally, we show that networks driven by isolated and the interval rules exhibit a very rich spectrum of boolean functions as they can, in fact, implement any arbitrary Boolean functions. We complement this results by studying experimentally the set of different Boolean functions generated by totalistic rules on random graphs.

International audience; Turing machines have been studied as dynamical systems for more than two decades, first formalized by Kurka, proposing a topological dynamical system named Turing machine with moving tape (TMT). It was conjectured that every TMT has at least one periodic point. Nowadays, there are several examples of aperiodic Turing machines, disproving Kurka’s conjecture. Moreover, one of these machines, named SMART, has other interesting properties like reversibility, completeness, aperiodicity, topological minimality, among others. This machine has four states and worksover an alphabet of three symbols. In this research, we study the dynamical properties of BinSmart, a 2-symbols reconstruction of the main dynamic of SMART machine. This machine results to be aperiodic, topologically minimal (therefore transitive) but not time-symmetric, as it is not a direct translation of the original machine. We also prove that its t-shift is a primitive substitution.

We consider the stationary Boussinesq system with non-homogeneous Dirichlet boundary conditions in a bounded domain Ω⊂R3 of class C1,1 with a possibly disconnected boundary. We prove the ex...

This is an author-created, un-copyedited version of an article accepted for publication/published in Nonlinearity. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at https://doi.org/10.1088/1361-6544/aab591. We investigate the dynamics of resonant Hamiltonians with n degrees of freedom to which we attach a small perturbation. Our study is based on the geometric interpretation of singular reduction theory. The flow of the Hamiltonian vector field is reconstructed from the cross sections corresponding to an approximation of this vector field in an energy surface. This approximate system is also built using normal forms and applying reduction theory obtaining the reduced Hamiltonian that is defined on the orbit space. Generically, the reduction is of singular character and we classify the singularities in the orbit space, getting three different types of singular points. A critical point of the reduced Hamiltonian corresponds to a family of periodic solutions in the full system whose characteristic multipliers are approximated accordingly to the nature of the critical point. The authors are partially supported by Projects MTM 2011-28227-C02-01 of the Ministry of Science and Innovation of Spain, MTM 2014-59433-C2-1-P of the Ministry of Economy and Competitiveness of Spain and by the Charles Phelps Taft Foundation.

In this paper, we propose a numerical method for the solution of time-dependent flow problems in mixed form. Such problems can be efficiently approximated on hierarchical grids, obtained from an unstructured coarse triangulation by using a regular refinement process inside each of the initial coarse elements. If these elements are considered as subdomains, we can formulate a non-overlapping domain decomposition method based on the lowest-order Raviart-Thomas elements, properly enhanced with Lagrange multipliers on the boundaries of each subdomain (excluding the Dirichlet edges). A suitable choice of mixed finite element spaces and quadrature rules yields a cell-centered scheme for the pressures with a local 10-point stencil. The resulting system of differential-algebraic equations is integrated in time by the Crank-Nicolson method, which is known to be a stiffly accurate scheme. As a result, we obtain independent subdomain linear systems that can be solved in parallel. The behaviour of the algorithm is illustrated on a variety of numerical experiments., Comment: 35 pages, 12 figures, 5 tables

In previous papers [6–8,10], we derived convergent and asymptotic expansions of solutions of second order linear differential equations with a large parameter. In those papers we generalized and developed special cases not considered in Olver’s theory [Olver, 1974]. In this paper we go one step forward and consider linear differential equations of the third order: y ′′′ +aΛ2y′ +bΛ3y = f(x)y′ +g(x)y, with a, b ∈ C fixed, f′ and g continuous, and Λ a large positive parameter. We propose two different techniques to handle the problem: (i) a generalization of Olver’s method and (ii) the transformation of the differential problem into a fixed point problem from which we construct an asymptotic sequence of functions that converges to the unique solution of the problem. Moreover, we show that this second technique may also be applied to nonlinear differential equations with a large parameter. As an application of the theory, we obtain new convergent and asymptotic expansions of the Pearcey integral P(x, y) for large |x|. The Ministerio de Econom´ıa y Competitividad (REF. MTM2014-52859-P) is acknowledged by its financial support.

We consider the incomplete beta function Bz(a, b) in the maximum domain of analyticity of its three variables: a, b, z ∈ C, −a /∈ N, z /∈ [1, ∞). For 0. The expansions are accompanied by realistic error bounds. Some numerical experiments show the accuracy of the approximations. This research was supported by Ministerio de Economía, Industria y Competitividad, Gobierno de España, project MTM2017-83490-P, Gobierno de Aragón and European Social Fund (group E24-17R).

The multiscale hybrid-mixed (MHM) method is extended to the Stokes and Brinkman equations with highly heterogeneous coefficients. The approach is constructive. We first propose an equivalent dual-hybrid formulation of the original problem using a coarse partition of the heterogeneous domain. Faces may be not aligned with jumps in the data. Then, the exact velocity and the pressure are characterized as the solution of a global face problem and the solutions of local independent Stokes (or Brinkman) problems at the continuous level. Owing to this decomposition, the one-level MHM method stems from the standard Galerkin approach for the Lagrange multiplier space. Basis functions are responsible for upscaling the unresolved scales of the medium into the global formulation. They are the exact solution of the local problems with prescribed Neumann boundary conditions on faces driven by the Lagrange multipliers. We make the MHM method effective by adopting the unusual stabilized finite element method to solve the local problems approximately. As such, equal-order interpolation turns out to be an option for the velocity, the pressure and the Lagrange multipliers. The numerical solutions share the important properties of the continuum, such as local equilibrium with respect to external forces and local mass conservation. Several academic and highly heterogeneous tests infer that the method achieves super-convergence for the velocity as well optimal convergence for the pressure and also for the stress tensor in their natural norms.

This is a post-peer-review, pre-copyedit version of an article published in Advances in Computational Mathematics. The final authenticated version is available online at: https://doi.org/10.1007/s10444-017-9543-y We consider the Bessel functions Jν (z) and Yν (z) for ν > −1/2 and z ≥ 0. We derive a convergent expansion of Jν (z) in terms of the derivatives of (sin z)/z, and a convergent expansion of Yν (z) in terms of derivatives of (1−cos z)/z, derivatives of (1 − e−z)/z and (2ν, z). Both expansions hold uniformly in z in any fixed horizontal strip and are accompanied by error bounds. The accuracy of the approximations is illustrated with some numerical experiments. This research was supported by the Spanish Ministry of "Economía y Competitividad", project MTM2014-52859-P. The Universidad Pública de Navarra is acknowledged by its financial support.

International audience; Naor, Parter, and Yogev [SODA 2020] recently designed a compiler for automatically translating standard centralized interactive protocols to distributed interactive protocols, as introduced by Kol, Oshman, and Saxena [PODC 2018]. In particular, by using this compiler, every linear-time algorithm for deciding the membership to some fixed graph class can be translated into a $\mathsf{dMAM}(O(\log n))$ protocol for this class, that is, a distributed interactive protocol with $O(\log n)$-bit proof size in $n$-node graphs, and three interactions between the (centralizer) computationally-unbounded but non-trustable prover Merlin, and the (decentralized) randomized computationally-limited verifier Arthur. As a corollary, there is a $\mathsf{dMAM}(O(\log n))$ protocol for the class of planar graphs, as well as for the class of graphs with bounded genus. We show that there exists a distributed interactive protocol for the class of graphs with bounded genus performing just a single interaction, from the prover to the verifier, yet preserving proof size of $O(\log n)$ bits. This result also holds for the class of graphs with bounded demi-genus, that is, graphs that can be embedded on a non-orientable surface of bounded genus. The interactive protocols described in this paper are actually proof-labeling schemes, i.e., a subclass of interactive protocols, previously introduced by Korman, Kutten, and Peleg [PODC 2005]. In particular, these schemes do not require any randomization from the verifier, and the proofs may often be computed a priori, at low cost, by the nodes themselves. Our results thus extend the recent proof-labeling scheme for planar graphs by Feuilloley et al. [PODC 2020], to graphs of bounded genus, and to graphs of bounded demigenus.

We present a penalization parameter method for obstacle identification in an incompressible fluid flow for a modified version of the Oseen equations. The proposed method consists in adding a high resistance potential to the system such that some subset of its boundary support represents the obstacle. This allows to work in a fixed domain and highly simplify the solution of the inverse problem via some suitable cost functional. Existence of minimizers and first and second order optimality conditions are derived through the differentiability of the solutions of the Oseen equation with respect to the potential. Finally, several numerical experiments using Navier–Stokes flow illustrate the applicability of the method, for the localization of a bi-dimensional cardiac valve from MRI and ultrasound flow type imaging data.

Let X, Y be asymmetric normed spaces and Lc(X, Y) the convex cone of all linear continuous operators from X to Y. It is known that in general, Lc(X, Y) is not a vector space. The aim of this note is to prove, using the Baire category theorem, that if Lc(X, Y) is a vector space for some asymmetric normed space Y , then X is isomorphic to its associated normed space (the converse is true for every asymmetric normed space Y and is easy to establish). For this, we introduce an index of symmetry of the space X denoted c(X) $\in$ [0, 1] and we give the link between the index c(X) and the fact that Lc(X, Y) is in turn an asymmetric normed space for every asymmetric normed space Y. Our study leads to a topological classification of asymmetric normed spaces.

International audience; In order to prevent the propagation of human diseases transmitted by mosquitoes (as dengue or zika), a solution is to release mosquitoes infected by Wolbachia. In this study, we model the release and the propagation over time and space of such infected mosquitoes in a population of uninfected ones. The aim of this study is to investigate the best location in space of the release to ensure invasion by the infected mosquitoes.; Afin de prévenir la propagation de maladies transmises à l'homme par les moustiques (comme la dengue ou le zika), une solution consiste à relâcher des moustiques infectés par la bactérie Wolbachia. Dans cette étude, nous modélisons le relâcher et la propagation dans le temps et l'espace de ces moustiques infectés dans une population de moustiques hôtes non infectés. Le but de cette étude est d'étudier le meilleur emplacement dans l'espace des relâcher afin d'assurer l'invasion par les moustiques infectés.

International audience; This work extends the general form of the Multiscale Hybrid-Mixed (MHM) method for the second-order Laplace (Darcy) equation to general non-conforming polygonal meshes. The main properties of the MHMmethod, i.e., stability, optimal convergence, and local conservation, are proven independently of the geometryof the elements used for the first level mesh. More precisely, it is proven that piecewise polynomials of degree $k$ and $k+1$, $k \geq 0$, for the Lagrange multipliers (flux), along with continuous piecewise polynomial interpolations of degree $k+1$ posed on second-level sub-meshes are stable if the latter is fine enough with respect to the mesh for the Lagrange multiplier. We provide an explicit sufficient condition for this restriction. Also, we prove that the error converges with order $k+1$ and $k+2$ in the broken $H^1$ and $L^2$ norms, respectively, under usual regularity assumptions, and that such estimates also hold for non-convex; or even non-simply connected elements. Numerical results confirm the theoreticalfindings and illustrate the gain that the use of multiscale functions provides.

The significant growth of wireless communications systems in the last years has led to the adoption of a wide range of applications not only for the general public but, also, including utilities and administrative authorities. In this context, the notable expansion of new services for smart cities requires, in some specific cases, the construction of underground tunnels in order to enable the maintenance and operation works of utilities, as well as to reduce the visual impact within the city center. One of the main challenges is that, inherently, underground service tunnels lack coverage from exterior wireless communication systems, which can be potentially dangerous for maintenance personnel working within the tunnels. Accordingly, wireless coverage should be deployed within the underground installation in order to guarantee real-time connectivity for safety maintenance, remote surveillance or monitoring operations. In this work, wireless channel characterization for complex urban tunnel environments was analyzed based on the assessment of LoRaWAN and ZigBee technologies operating at 868 MHz. For that purpose, a real urban utility tunnel was modeled and simulated by means of an in-house three-dimensional ray-launching (3D-RL) code. The utility tunnel scenario is a complex and singular environment in terms of radio wave propagation due to the limited dimensions and metallic elements within it, such as service trays, user pathways or handrails, which were considered in the simulations. The simulated 3D-RL algorithm was calibrated and verified with experimental measurements, after which, the simulation and measurement results showed good agreement. Besides, a complete wireless sensor network (WSN) deployment within the tunnels was presented, providing remote cloud data access applications and services, allowing infrastructure security and safety work conditions. The obtained results provided an adequate radio planning approach for the deployment of wireless systems in complex urban utility scenarios, with optimal coverage and enhanced quality of service. This research was funded in part by the School of Engineering and Sciences at Tecnológico de Monterrey, in part by Project RTI2018-095499-B-C31, funded by Ministerio de Ciencia, Innovación y Universidades, Gobierno de España (MCIU/AEI/FEDER, UE) and, in part, by the European Union's Horizon 2020 research and Innovation program under grant agreement N◦774094 (Stardust-Holistic and Integrated Urban Model for Smart Cities).

Comision Nacional de Investigacion Cientifica y Tecnologica (CONICYT) CONICYT FONDECYT 1160567 1160204 Basal Program from CONICYT-Chile CMM-AFB 170001 Comision Nacional de Investigacion Cientifica y Tecnologica (CONICYT) PFCHA/Doctorado Nacional/2017-21170249 LabEx NUMEV incorporated into the I-Site MUSE

In this paper we study the family of freezing cellular automata (FCA) in the context of asynchronous updating schemes. A cellular automaton is called freezing if there exists an order of its states, and the transitions are only allowed to go from a lower to a higher state. A cellular automaton is asynchronous if at each time-step only one cell is updated. Given configuration, we say that a cell is unstable if there exists a sequential updating scheme that changes its state. In this context, we define the problem AsyncUnstability, which consists in deciding if a cell is unstable or not. In general AsyncUnstability is in NP, and we study in which cases we can solve the problem by a more efficient algorithm. We begin showing that AsyncUnstability is in NL for any one-dimensional FCA. Then we focus on the family of life-like freezing CA (LFCA), which is a family of two-dimensional two-state FCA that generalize the freezing version of the game of life, known as life without death. We study the complexity of AsyncUnstability for all LFCA in the triangular and square grids, showing that almost all of them can be solved in NC, except for one rule for which the problem is NP-complete.

We study the capillary retraction of a Newtonian semi-infinite liquid filament through analytical methods. We derive a long-time asymptotic-state expansion for the filament profile using a one-dimensional free-surface slender cylindrical flow model based on the three-dimensional axisymmetric Navier-Stokes equations. The analysis identifies three distinct length and time scale regions in the retraction domain: a steady filament section, a growing spherical blob, and an intermediate matching zone. We show that liquid filaments naturally develop travelling capillary waves along their surface and a neck behind the blob. We analytically prove that the wavelength of the capillary waves is approximately 3.63 times the filament's radius at the inviscid limit. Additionally, the waves' asymptotic wavelength, decay length, and the minimum neck size are analysed in terms of the Ohnesorge number. Finally, our findings are compared with previous results from the literature and numerical simulations in Basilisk obtaining a good agreement. This analysis provides a full picture of the recoiling process going beyond the classic result of the velocity of retraction found by Taylor and Culick., Comment: 12 pages, 5 figures

We extend, to Raviart-Thomas finite elements of any degree, two methods for the construction of basis of the space of divergence-free functions that are well established in the case of degree one. The first one computes directly a basis of the kernel of the divergence operator whereas the second one computes a basis of the image of the curl operator that, if the boundary of the domain is not connected, is completed with a basis of the second de Rham cohomology group (namely, the space of divergence-free functions that are not curls). When using the lower order Whitney elements on a tetrahedral mesh, the degrees of freedom are supported on the vertices, edges, faces and tetrahedra of the mesh respectively and, from Stokes theorem, the matrices describing the differential operators gradient, curl and divergence are the transposed of the connectivity matrices of the mesh. This allows the use of a tree-cotree splitting of associated oriented graphs to efficiently construct a basis of either the kernel of the divergence or the image of the curl operator. We prove that these two properties hold true also for r > 0 when using as degrees of freedom a particular realization, based on Berstein polynomials, of the moments. In this work we analyze in detail the second method, the one based on the identification of a basis of the space of the curls of Nédélec finite elements. (The first one has been analyzed in [5].)

International audience; We revisit the optimal control problem of maximizing biogas production in continuous bio-processes in two directions: 1. over an infinite horizon, 2. with sub-optimal controllers independent of the time horizon. For the first point, we identify a set of optimal controls for the problems with an averaged reward and with a discounted reward when the discount factor goes to 0 and we show that the value functions of both problems are equal. For the finite horizon problem, our approach relies on a framing of the value function by considering a different reward for which the optimal solution has an explicit optimal feedback that is time-independent. In particular, we show that this technique allows us to provide explicit bounds on the sub-optimality of the proposed controllers. The various strategies are finally illustrated on Haldane and Contois growth functions.

Wide low-mass substellar companions are known to be very rare among low-mass stars, but appear to become increasingly common with increasing stellar mass. However, B-type stars, which are the most massive stars within ~150 pc of the Sun, have not yet been examined to the same extent as AFGKM-type stars in that regard. In order to address this issue, we launched the ongoing B-star Exoplanet Abundance Study (BEAST) to examine the frequency and properties of planets, brown dwarfs, and disks around B-type stars in the Scorpius-Centaurus (Sco-Cen) association; we also analyzed archival data of B-type stars in Sco-Cen. During this process, we identified a candidate substellar companion to the B9-type spectroscopic binary HIP 79098 AB, which we refer to as HIP 79098 (AB)b. The candidate had been previously reported in the literature, but was classified as a background contaminant on the basis of its peculiar colors. Here we demonstrate that the colors of HIP 79098 (AB)b are consistent with several recently discovered young and low-mass brown dwarfs, including other companions to stars in Sco-Cen. Furthermore, we show unambiguous common proper motion over a 15-year baseline, robustly identifying HIP 79098 (AB)b as a bona fide substellar circumbinary companion at a 345+/-6 AU projected separation to the B9-type stellar pair. With a model-dependent mass of 16-25 Mjup yielding a mass ratio of, 9 pages, 6 figures, accepted for publication in A&A

International audience; The aim of this paper is to prove existence results for a class of sweeping processes in Hilbert spaces by using the catching-up algorithm. These processes are governed by ball-compact non autonomous sets. Moreover, a full characterization of nonsmooth Lyapunov pairs is obtained under very general hypotheses. We also provide a criterion for weak invariance. Some applications to hysteresis and crowd motion are given.

International audience; In this paper, we study the stationary Stokes and Navier-Stokes equations with non-homogeneous Navier boundary condition in a bounded domain Ω ⊂ R 3 of class C 1,1 from the viewpoint of the behavior of solutions with respect to the friction coefficient α. We first prove the existence of a unique weak solution (and strong) in W 1,p (Ω) (and W 2,p (Ω)) of the linear problem for all 1 < p < ∞ considering minimal regularity of α, using some inf-sup condition concerning the rotational operator. Furthermore, we deduce uniform estimates of the solutions for large α which enables us to obtain the strong convergence of Stokes solutions with Navier slip boundary condition to the one with no-slip boundary condition as α → ∞. Finally, we discuss the same questions for the non-linear system. Résumé Dans cette note, nousétudions leséquations stationnaires de Stokes et de Navier-Stokes avec une condition aux limites non homogène de Navier dans un domaine borné Ω ⊂ R 3 de classe C 1,1 , dont le comporte-ment des solutions par rapport au coefficient de friction α. Nous prouvons, d'abord dans le cas linéaire, l'existence d'une solution faible (et d'une solution forte) unique dans W 1,p (Ω) (et W 2,p (Ω)) pour tout 1 < p < ∞ en supposant α le moins régulier possible et en utilisant une condition inf-sup concernant l'opérateur rotationnel. De plus, nous déduisons des estimations uniformes des solutions pour α grand qui nous permettent d'obtenir la convergence forte des solutions de Stokes avec la condition de glissement vers les solutions vérifiant la condition d'adhérence lorsque α → ∞. Finalement, nousétudions les mêmes questions pour le système non linéaire. Pour citer cet article : P. Acevedo, C. Amrouche, C. Conca, A. Ghosh, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Faraday waves are a classic example of a system in which an extended pattern emerges under spatially uniform forcing. Motivated by systems in which uniform excitation is not plausible, we study both experimentally and theoretically the effect of heterogeneous forcing on Faraday waves. Our experiments show that vibrations restricted to finite regions lead to the formation of localized subharmonic wave patterns and change the onset of the instability. The prototype model used for the theoretical calculations is the parametrically driven and damped nonlinear Schr\"odinger equation, which is known to describe well Faraday-instability regimes. For an energy injection with a Gaussian spatial profile, we show that the evolution of the envelope of the wave pattern can be reduced to a Weber-equation eigenvalue problem. Our theoretical results provide very good predictions of our experimental observations provided that the decay length scale of the Gaussian profile is much larger than the pattern wavelength., Comment: 10 pages, 9 figures, Accepted

We consider a class of low Reynolds number swimmers, of prolate spheroidal shape, which can be seen as simplified models of ciliated microorganisms. Within this model, the form of the swimmer does not change, the propelling mechanism consisting in tangential displacements of the material points of swimmer's boundary. Using explicit formulas for the solution of the Stokes equations at the exterior of a translating prolate spheroid the governing equations reduce to a system of ODE's with the control acting in some of its coefficients (bilinear control system). The main theoretical result asserts the exact controllability of the prolate spheroidal swimmer. In the same geometrical situation, we consider the optimal control problem of maximizing the efficiency during a stroke and we prove the existence of a maximum. We also provide a method to compute an approximation of the efficiency by using explicit formulas for the Stokes system at the exterior of a prolate spheroid, with some particular tangential velocities at the fluid-solid interface. We analyze the sensitivity of this efficiency with respect to the eccentricity of the considered spheroid and show that for small positive eccentricity, the efficiency of a prolate spheroid is better than the efficiency of a sphere. Finally, we use numerical optimization tools to investigate the dependence of the efficiency on the number of inputs and on the eccentricity of the spheroid. The ``best'' numerical result obtained yields an efficiency of $30.66\%$ with $13$ scalar inputs. In the limiting case of a sphere our best numerically obtained efficiency is of $30.4\%$, whereas the best computed efficiency previously reported in the literature is of $22\%$.

We study some of the main features of Fractional Step Runge–Kutta–Nyström methods when they are used to integrate Initial–Boundary Value Problems of second order in time, in combination with a suitable spatial discretization. We focus our attention on the order reduction phenomenon, which appears if classical boundary conditions are taken at the internal stages. This drawback is specially hard when time dependent boundary conditions are considered. In this paper we present an efficient technique, very simple and computationally cheap, which allows us to avoid the order reduction; such technique consists in modifying the boundary conditions for the internal stages of the method. M.J. Moreta had financial support from MTM 2015-66837-P. B. Bujanda had financial support from TEC 2013-45585-C2-1-R. J.C. Jorge had financial support from MTM 2014-52859.

A boolean network is a map \begin{document}$ F:\{0,1\}^n \to \{0,1\}^n $\end{document} that defines a discrete dynamical system by the subsequent iterations of \begin{document}$ F $\end{document} . Nevertheless, it is thought that this definition is not always reliable in the context of applications, especially in biology. Concerning this issue, models based in the concept of adding asynchronicity to the dynamics were propose. Particularly, we are interested in a approach based in the concept of delay. We focus in a specific type of delay called firing memory and it effects in the dynamics of symmetric (non-directed) conjunctive networks. We find, in the caseis in which the implementation of the delay is not uniform, that all the complexity of the dynamics is somehow encapsulated in the component in which the delay has effect. Thus, we show, in the homogeneous case, that it is possible to exhibit attractors of non-polynomial period. In addition, we study the prediction problem consisting in, given an initial condition, determinate if a fixed coordinate will eventually change its state. We find again that in the non-homogeneous case all the complexity is determined by the component that is affected by the delay and we conclude in the homogeneous case that this problem is PSPACE-complete.

In a previous paper by Ferreira and L��pez [Journal of Mathematical Analysis and Applications, 298(1), 2004], the authors derived an asymptotic expansion of the Lerch's transcendent $��(z,s,a)$ for large $\vert a\vert$, valid for $\mathrm{Re}(a)>0$, $\mathrm{Re}(s)>0$ and $z\in\mathbb{C}\setminus[1,\infty)$. In this paper we study the special case $z\ge 1$ not covered in the previous result, deriving a complete asymptotic expansion of the Lerch's transcendent $��(z,s,a)$ for $z > 1$ and $\mathrm{Re}(s)>0$ as $\mathrm{Re}(a)$ goes to infinity. We also show that when $a$ is a positive integer, this expansion is convergent for $\mathrm{Re}(z) \ge 1$. As a corollary, we get a full asymptotic expansion for the sum $\sum_{n=1}^{m} z^{n}/n^{s}$ for fixed $z >1 $ as $m \to \infty$. Some numerical results show the accuracy of the approximation., 12 page

International audience; We consider the motion of a rigid body in a viscoplastic material. This material is modeled by the 3D Bingham equations, and the Newton laws govern the displacement of the rigid body. Our main result is the existence of a weak solution for the corresponding system. The weak formulation is an inequality (due to the plasticity of the fluid), and it involves a free boundary (due to the motion of the rigid body). We approximate it by regularizing the convex terms in the Bingham fluid and by using a penalty method to take into account the presence of the rigid body.

One of the simplest deterministic mathematical model for the spread of an epidemic disease is the so-called SI system made of two Ordinary Differential Equations. It exhibits simple dynamics: a bifurcation parameter \(\mathscr {T}_0\) yielding persistence of the disease when \(\mathscr {T}_0 > 1\), else extinction occurs. A natural question is whether this gentle dynamic can be disturbed by spatial diffusion. It is straightforward to check it is not feasible for linear/nonlinear diffusions. When cross diffusion is introduced for suitable choices of the parameter data set this persistent state of the ODE model system becomes linearly unstable for the resulting initial and no-flux boundary value problem. On the other hand “natural” weak solutions can be defined for this initial and no-flux boundary value problem and proved to exist provided nonlinear and cross diffusivities satisfy some constraints. These constraints are not fully met for the parameter data set yielding instability. A remaining open question is: to which solutions does this apply? Periodic behaviors are observed for a suitable range of cross diffusivities.

This Ph.D. thesis aims to obtain and to develop some mathematical models to understand some aspects of the dynamics of heterogeneous granular fluids. More precisely, the expected result is to develop three models, one where the dynamics of the granular material is modeled using a mixture theory approach, and the other two, where we consider the granular fluid is modeled using a multiphase approach involving rigid structures and fluids. More precisely : • In the first model, we obtained a set of equations based on the mixture theory using homogenization tools and a thermodynamic procedure. These equations reflect two essential properties of granular fluids : the viscous nature of the interstitial fluid and a Coulomb-type of behavior of the granular component. With our equations, we study the problem of a dense granular heterogeneous flow, composed by a Newtonian fluid and a solid component in the setting of the Couette flow between two infinite cylinders. • In the second model, we consider the motion of a rigid body in a viscoplastic material. The 3D Bingham equations model this material, and the Newton laws govern the displacement of the rigid body. Our main result is the existence of a weak solution for the corresponding system. • In the third model, we consider the motion of a perfect heat conductor rigid body in a heat conducting Newtonian fluid. The 3D Fourier-Navier-Stokes equations model the fluid, and the Newton laws and the balance of internal energy model the rigid body. Our main result is the existence of a weak solution for the corresponding system. The weak formulation is composed by the balance of momentum and the balance of total energy equation which includes the pressure of the fluid, and it involves a free boundary (due to the motion of the rigid body). To obtain an integrable pressure, we consider a Navier slip boundary condition for the outer boundary and the mutual interface; Cette thèse vise à obtenir et à développer des modèles mathématiques pour comprendre certains aspects de la dynamique des fluides granulaires hétérogènes. Plus précisément, le résultat attendu consiste à développer trois modèles. Nous supposons dans un premier temps que la dynamique du matériau granulaire est modélisée à l’aide d’une approche fondée sur la théorie du mélange. D’autre part, pour les deux modèles restant, nous considérons que le fluide granulaire est modélisé à l’aide d’une approche multiphase associant des structures et des fluides rigides. Plus exactement : • Dans le premier modèle, nous avons obtenu un ensemble d’équations basées sur la théorie du mélange en utilisant des outils d’homogénéisation et une procédure thermodynamique. Ces équations reflètent deux propriétés essentielles des fluides granulaires : la nature visqueuse du fluide interstitiel et un comportement de type Coulomb de la composante granulaire. Avec nos équations, nous étudions le problème de Couette entre deux cylindres infinis d’un écoulement hétérogène granulaire dense, composé d’un fluide newtonien et d’une composante solide. • Dans le deuxième modèle, nous considérons le mouvement d’un corps rigide dans un matériau viscoplastique. Les équations 3D de Bingham modélisent ce matériau et les lois de Newton régissent le déplacement du corps rigide. Notre résultat principal est d’établir l’existence d’une solution faible pour le système correspondant. • Dans le troisième modèle, nous considérons le mouvement d’un corps rigide conducteur thermique parfait dans un fluide newtonien conducteur de la chaleur. Les équations 3D de Fourier-Navier-Stokes modélisent le fluide, tandis que les lois de Newton et l’équilibre de l’énergie interne modélisent le déplacement du corps rigide. Notre principal objectif dans cette partie est de prouver l’existence d’une solution faible pour le système correspondant. La formulation faible est composée de l’équilibre entre la quantité du mouvement et l’équation de l’énergie totale, qui inclut la pression du fluide, et implique une limite libre due au mouvement du corps rigide. Pour obtenir une pression intégrable, nous considérons une condition au limite de glissement de Navier pour la limite extérieure et l’interface mutuelle

This Ph.D. thesis aims to obtain and to develop some mathematical models to understand some aspects of the dynamics of heterogeneous granular fluids. More precisely, the expected result is to develop three models, one where the dynamics of the granular material is modeled using a mixture theory approach, and the other two, where we consider the granular fluid is modeled using a multiphase approach involving rigid structures and fluids. More precisely : • In the first model, we obtained a set of equations based on the mixture theory using homogenization tools and a thermodynamic procedure. These equations reflect two essential properties of granular fluids : the viscous nature of the interstitial fluid and a Coulomb-type of behavior of the granular component. With our equations, we study the problem of a dense granular heterogeneous flow, composed by a Newtonian fluid and a solid component in the setting of the Couette flow between two infinite cylinders. • In the second model, we consider the motion of a rigid body in a viscoplastic material. The 3D Bingham equations model this material, and the Newton laws govern the displacement of the rigid body. Our main result is the existence of a weak solution for the corresponding system. • In the third model, we consider the motion of a perfect heat conductor rigid body in a heat conducting Newtonian fluid. The 3D Fourier-Navier-Stokes equations model the fluid, and the Newton laws and the balance of internal energy model the rigid body. Our main result is the existence of a weak solution for the corresponding system. The weak formulation is composed by the balance of momentum and the balance of total energy equation which includes the pressure of the fluid, and it involves a free boundary (due to the motion of the rigid body). To obtain an integrable pressure, we consider a Navier slip boundary condition for the outer boundary and the mutual interface; Cette thèse vise à obtenir et à développer des modèles mathématiques pour comprendre certains aspects de la dynamique des fluides granulaires hétérogènes. Plus précisément, le résultat attendu consiste à développer trois modèles. Nous supposons dans un premier temps que la dynamique du matériau granulaire est modélisée à l’aide d’une approche fondée sur la théorie du mélange. D’autre part, pour les deux modèles restant, nous considérons que le fluide granulaire est modélisé à l’aide d’une approche multiphase associant des structures et des fluides rigides. Plus exactement : • Dans le premier modèle, nous avons obtenu un ensemble d’équations basées sur la théorie du mélange en utilisant des outils d’homogénéisation et une procédure thermodynamique. Ces équations reflètent deux propriétés essentielles des fluides granulaires : la nature visqueuse du fluide interstitiel et un comportement de type Coulomb de la composante granulaire. Avec nos équations, nous étudions le problème de Couette entre deux cylindres infinis d’un écoulement hétérogène granulaire dense, composé d’un fluide newtonien et d’une composante solide. • Dans le deuxième modèle, nous considérons le mouvement d’un corps rigide dans un matériau viscoplastique. Les équations 3D de Bingham modélisent ce matériau et les lois de Newton régissent le déplacement du corps rigide. Notre résultat principal est d’établir l’existence d’une solution faible pour le système correspondant. • Dans le troisième modèle, nous considérons le mouvement d’un corps rigide conducteur thermique parfait dans un fluide newtonien conducteur de la chaleur. Les équations 3D de Fourier-Navier-Stokes modélisent le fluide, tandis que les lois de Newton et l’équilibre de l’énergie interne modélisent le déplacement du corps rigide. Notre principal objectif dans cette partie est de prouver l’existence d’une solution faible pour le système correspondant. La formulation faible est composée de l’équilibre entre la quantité du mouvement et l’équation de l’énergie totale, qui inclut la pression du fluide, et implique une limite libre due au mouvement du corps rigide. Pour obtenir une pression intégrable, nous considérons une condition au limite de glissement de Navier pour la limite extérieure et l’interface mutuelle

Stochastic Partial Differential Equations are an essential tool for the analysis of scaling limits of a diverse array of microscopic models coming from other fields such as physics and chemistry.This type of equations correspond to classical partial differential equations to which one has added a random forcing which is typically very irregular ; the most basic example is perhaps the Stochastic Heat Equation, one of whose versions is studied in this thesis. The roughness of the potential turns the analysis of solutions to these probles a lot more difficult than the classic case. In fact, there are cases where solutions can be understood only in the sense of distributions, i.e. as generalised functions. There are some critical cases, such the Kardar-Parisi-Zhang (KPZ)equation where, even though the solutions can be shown to be continuous (even Hölder continuous) they are not regular enough so that some non-linear terms appearing in this equation are well defined. In the last 20 years certain techniques have been developed for the analysis of these equations, among which there is the theory of Rough Paths by T. Lyons (1998), their branched version introduced by M. Gubinelli (2010) and more recently the theory of Regularity Structures of M. Hairer (2014) and for which he was awarded the Fields Medal in 2014. All these techniques have as main idea that of renormalisation, coming from physics. In particular, Wick renormalisation plays an essential role in Regularity Structures. In this work we develop Wick products and polynomials from a Hopf-algebraic point of view, inspired by G.-C. Rota's Umbral Calculus. We also explore the general theory of Rough Paths and in particular in their branched version, where we show some new results in the direction of incorporating an analogue of Wick renormalisation as found in Hairer's Regularity Structures. Finally, the semi-discrete multi-layer polymer model, introduced by I. Corwin and A. Hammond (2014) is studied. We show the convergence of its partition function towards a stochastic process known as (the solution to) "the multi-layer Stochastic Heat Equation" introduced by N. O'Connell and J. Warren (2011) some years earlier. We remark that at the time of writing of this work there were no results allowing to interpret this last process as the solution to a singular SPDE as is the case, for example, for the KPZ equation. This was one of the main sources of inspiration of this work.; Les Équations aux Dérivées Partielles Stochastiques (SPDE en anglais) sont un outil essentiel pour l'analyse des limites d'échelle de divers modèles microscopiques provenant d'autres aires des sciences telles que la physique ou la chimie. Ce type d'équations corresponds à une équations aux dérivées partielles classique à laquelle on a ajouté un terme de force externe aléatoire qui est très irrégulier ; l'exemple le plus basique est peut-être l'Équation de la Chaleur Stochastique, qui est étudiée dans une de ses version dans cette thèse. L'irrégularité du potentiel fait que l'analyse des solutions à ce problème soit beaucoup plus compliqué que dans le cas classique. En fait, il y a des cas où les solutions ne peuvent être définies que dans le sens des distributions (ou des fonctions généralisées). Il y a des cas critiques comme l'équations de Kardar-Parisi-Zhang (KPZ) dans une dimension d'espace où, bien que ces solutions sont Hölderiennes, elles ne sont pas assez régulières pour définir quelques termes non linéaires qu'y apparaissent. Durant les 20 dernières années des techniques pour l'analyse de ces équations ont été développées parmi lesquelles il y a les chemins rugueux géométriques de T. Lyons (1998), les chemins rugueux ramifiés de M. Gubinelli (2010) et plus récemment les structures de régularité de M. Hairer (2014), pour laquelle ce dernier à remporté la médaille Fields en 2014. Toutes ces techniques ont comme idée centrale le concept de renormalisation (provenant de la physique). En particulier, la renormalisation de Wick joue un rôle essentiel dans la théorie des structures de régularité. Dans ce travail nous développons les produits et polynômes de Wick d'un point de vue algébrique inspiré du "calcul ombrale" de G.-C. Rota. Nous explorons aussi la théorie générale des chemins rugueux et leur version ramifié en particulier, où nous donnons des résultats nouveaux dans la direction d'incorporer un analogue de la renormalisation de Wick existant dans la théorie de Hairer. Finalement, le modèle de polymère semidiscret multicouches, introduit par I. Corwin et A. Hammond (2014) est étudié. Nous montrons la convergence de sa fonction de partition vers le processus stochastique appelé "la solution de l'équation de la chaleur stochastique multicouches" par N. O'Connell et J. Warren (2011) quelques années avant. Nous remarquons qu'au moment de rédaction de cette thèse il n'existait aucun résultat permettant interpréter ce dernier processus dans le continuum comme la solution d'une SPDE singulière comme dans le cas, par exemple, de l'équation de KPZ. Ceci a été une des principales sources d'inspiration pour ce travail.

For every connected graph G , a subgraph H of G is isometric if the distance between any two vertices in H is the same in H as in G . A distance-preserving elimination ordering of G is a total ordering of its vertex-set V ( G ) , denoted ( v 1 , v 2 , … , v n ) , such that any subgraph G i = G ∖ ( v 1 , v 2 , … , v i ) with 1 ≤ i n is isometric. This kind of ordering has been introduced by Chepoi in his study on weakly modular graphs (Chepoi, 1998). We prove that it is NP-complete to decide whether such ordering exists for a given graph — even if it has diameter at most 2. Then, we prove on the positive side that the problem of computing a distance-preserving ordering when there exists one is fixed-parameter-tractable in the treewidth. Lastly, we describe a heuristic in order to compute a distance-preserving ordering when there exists one that we compare to an exact exponential time algorithm and to an ILP formulation for the problem.