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We study the equilibrium configurations for several fluid-structure interaction problems. The fluid is confined in a 2D unbounded channel that contains a body, free to move inside the channel with rigid motions (transversal translations and rotations). The motion of the fluid is generated by a Poiseuille inflow/outflow at infinity and governed by the stationary Navier-Stokes equations. For a model where the fluid is the air and the body represents the cross-section of a suspension bridge, therefore also subject to restoring elastic forces, we prove that for small inflows there exists a unique equilibrium position, while for large inflows we numerically show the appearance of additional equilibria. A similar uniqueness result is also obtained for a discretized 3D bridge, consisting in a finite number of cross-sections interacting with the adjacent ones. The very same model, but without restoring forces, is used to describe the mechanism of the Leonardo da Vinci ferry, which is able to cross a river without engines. We numerically determine the optimal orientation of the ferry that allows it to cross the river in minimal time., SCOPUS: ar.j, info:eu-repo/semantics/published

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We consider the prescribed mean curvature equation for entire spacelike hypersurfaces in the Lorentz-Minkowski space, namely -div(∇u1, SCOPUS: ar.j, info:eu-repo/semantics/published

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This work is concerned with the mathematical analysis of the bulk rheology of random suspensions of rigid particles settling under gravity in viscous fluids. Each particle generates a fluid flow that in turn acts on other particles and hinders their settling. In an equilibrium perspective, for a given ensemble of particle positions, we analyze both the associated mean settling speed and the velocity fluctuations of individual particles. In the 1970s, Batchelor gave a proper definition of the mean settling speed, a 60-year-old open problem in physics, based on the appropriate renormalization of long-range particle contributions. In the 1980s, a celebrated formal calculation by Caflisch and Luke suggested that velocity fluctuations in dimension d= 3 should diverge with the size of the sedimentation tank, contradicting both intuition and experimental observations. The role of long-range self-organization of suspended particles in form of hyperuniformity was later put forward to explain additional screening of this divergence in steady-state observations. In the present contribution, we develop the first rigorous theory that allows to justify all these formal calculations of the physics literature., SCOPUS: ar.j, info:eu-repo/semantics/published

This work is devoted to the asymptotic behavior of eigenvalues of an elliptic operator with rapidly oscillating random coefficients on a bounded domain with Dirichlet boundary conditions. A sharp convergence rate is obtained for eigenvalues towards those of the homogenized problem, as well as a quantitative two-scale expansion result for eigenfunctions. Next, a quantitative central limit theorem is established for fluctuations of isolated eigenvalues; more precisely, a pathwise characterization of eigenvalue fluctuations is obtained in terms of the so-called homogenization commutator, in parallel with the recent fluctuation theory for the solution operator., SCOPUS: ar.j, info:eu-repo/semantics/published

A shape invariant nonseparable and nondiagonalizable two-dimensional model with quadratic complex interaction, first studied by Cannata, Ioffe, and Nishnianidze, is re-examined with the purpose of exhibiting its hidden algebraic structure. The two operators A+ and A−, coming from the shape invariant supersymmetrical approach, where A+ acts as a raising operator while A− annihilates all wavefunctions, are completed by introducing a novel pair of operators B+ and B−, where B− acts as the missing lowering operator. These four operators then serve as building blocks for constructing gl(2) generators, acting within the set of associated functions belonging to the Jordan block corresponding to a given energy eigenvalue. This analysis is extended to the set of Jordan blocks by constructing two pairs of bosonic operators, finally yielding an sp(4) algebra, as well as an osp(1/4) superalgebra. Hence, the hidden algebraic structure of the model is very similar to that known for the two-dimensional real harmonic oscillator., SCOPUS: ar.j, info:eu-repo/semantics/published

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Motivated by the long-time transport properties of quantum waves in weakly disordered media, the present work puts random Schrödinger operators into a new spectral perspective. Based on a stationary random version of a Floquet type fibration, we reduce the description of the quantum dynamics to a fibered family of abstract spectral perturbation problems on the underlying probability space. We state a natural resonance conjecture for these fibered operators: in contrast with periodic and quasiperiodic settings, this would entail that Bloch waves do not exist as extended states but rather as resonant modes, and this would justify the expected exponential decay of time correlations. Although this resonance conjecture remains open, we develop new tools for spectral analysis on the probability space, and in particular, we show how ideas from Malliavin calculus lead to rigorous Mourre type results: we obtain an approximate dynamical resonance result and the first spectral proof of the decay of time correlations on the kinetic timescale. This spectral approach suggests a whole new way of circumventing perturbative expansions and renormalization techniques., SCOPUS: ar.j, info:eu-repo/semantics/published

This note is concerned with the global well-posedness of nonlinear Schrödinger equations in the continuum with spatially homogeneous random initial data., SCOPUS: ar.j, info:eu-repo/semantics/published

We consider the best constant in a critical Sobolev inequality of second order.We show non-rigidity for the optimizers above a certain threshold, namely, we prove that the best constant is achieved by a nonconstant solution of the associated fourth order elliptic problem under Neumann boundary conditions.Our arguments rely on asymptotic estimates of the Rayleigh quotient.We also show rigidity below another threshold., SCOPUS: ar.j, info:eu-repo/semantics/published

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In this paper, we prove existence of multiple non-radial solutions to the Hardy-Sobolev equation [Formula presented] where N≥3, s∈[0,2), [Formula presented] and [Formula presented]. We extend results of E.N. Dancer, F. Gladiali, M. Grossi (2017) [12] where only the case s=0 is considered. The results specially rely on a careful analysis of the kernel of the linearized operator. Moreover, thanks to monotonicity properties of the solutions, we separate two branches of non-radial solutions., SCOPUS: ar.j, info:eu-repo/semantics/published

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We consider Hardy-Sobolev nonlinear equations on domains with singularities. We introduced this problem in Cheikh-Ali [4]. Under a local geometric hypothesis, namely that the generalized mean curvature is negative (see (7) below), we proved the existence of extremals for the relevant Hardy-Sobolev inequality for large dimensions. In the present paper, we tackle the question of small dimensions that was left open. We introduce a “mass”, that is a global quantity, the positivity of which ensures the existence of extremals in small dimensions. As a byproduct, we prove the existence of solutions to a perturbation of the initial equation via the Mountain-Pass Lemma., SCOPUS: ar.j, info:eu-repo/semantics/inPress

A polymer-chain network is a collection of interconnected polymer-chains, made themselves of the repetition of a single pattern called a monomer. Our first main result establishes that, for a class of models for polymer-chain networks, the thermodynamic limit in the canonical ensemble yields a hyperelastic model in continuum mechanics. In particular, the discrete Helmholtz free energy of the network converges to the infimum of a continuum integral functional (of an energy density depending only on the local deformation gradient) and the discrete Gibbs measure converges (in the sense of a large deviation principle) to a measure supported on minimizers of the integral functional. Our second main result establishes the small temperature limit of the obtained continuum model (provided the discrete Hamiltonian is itself independent of the temperature), and shows that it coincides with the Γ -limit of the discrete Hamiltonian, thus showing that thermodynamic and small temperature limits commute. We eventually apply these general results to a standard model of polymer physics from which we derive nonlinear elasticity. We moreover show that taking the Γ -limit of the Hamiltonian is a good approximation of the thermodynamic limit at finite temperature in the regime of large number of monomers per polymer-chain (which turns out to play the role of an effective inverse temperature in the analysis)., SCOPUS: ar.j, info:eu-repo/semantics/published

The paper presents a comparison of the two languages Python and R related to the classification tools and demonstrates the differences in their syntax and graphical output. It indicates the functionality of R and Python packages {dendextend} and scipy.cluster as effective tools for the dendrogram modelling by the algorithms of sorting and ranking datasets. R and Python programming languages have been tested on a sample dataset including marine geological measurements. The work aims to detect how bathymetric data change along the 25 bathymetric profiles digitized across the Mariana Trench. The methodology includes performed hierarchical cluster analysis with dendrograms and plotted clustermap with marginal dendrograms. The statistical libraries include Matplotlib, SciPy, NumPy, Pandas by Python and {dendextend}, {pvclust}, {magrittr} by R. The dendrograms were compared by the model-simulated clusters of the bathymetric ranges. The results show three distinct groups of the profiles sorted by the elevation ranges with maximal depths detected in a group of profiles 19-21. The dendrogram visualization in a cluster analysis demonstrates the effective representation of the data sorting, grouping and classifying by the machine learning algorithms. The programming codes presented in this study enable to sort a dataset in a similar research aimed to group data based on the similarity of attributes. Effective visualization by dendrograms is a useful modelling tool for the geospatial management where data ranking is required. Plotting dendrograms by R, comparing to Python, presented functional and sophisticated algorithms, refined design control and fine graphical data output. The interdisciplinary nature of this work consists in application of the coding algorithms for spatial data analysis., info:eu-repo/semantics/published

Cocoa is an important cash crop in Central and West Africa, especially in Cameroon. In this part of the world, cacao production is impacted by several diseases, like the Cocoa Swollen Shoot Virus (CSSV) and the black pod disease, and several pests, like Miridae, Distantiella theobroma and Sahlbergella singularis, causing significant damage to pods and vegetative parts of the cocoa tree, and thus impacting the cocoa production. However, the damage and losses associated with this pest remain difficult to estimate due in particular to the bio/ecology of this species (camouflage, low Statistical models have shown to be ineffective in describing these dynamics. That is why we have developed several Mathematical models to describe the time evolution dynamics of mirids, including the effect of different control methods. After an introductory chapter where we recall the biology and ecology of mirids, we develop, analyze and study a compartmental cooperative periodic and nonperiodic model in chapter 2. Then, considering time developments and sexual maturation duration for females, we develop and study a delayed cooperative model (with and without periodic parameters). In this latter model, we consider different control methods, including chemical control (insecticides) and semi-chemical control (sexual confusion and trapping). Through our numerical simulations, we recover recommendation given by mirid control organizations for the use of insecticides and show that chemical treatment can be replaced efficiently by mating disrupting and trapping. We also derive a global sensitivity analysis highlighting the importance of some key parameters. Then, based on the previous results, we develop, in chapter 3, a more complex delay model, modeling mating disrupting and trapping, using the piecewise-smooth system approach. Our analysis shows the existence of two thresholds based on mirid's biological parameters: one under which the control has no effect on established populations, and

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Consider a colloidal suspension of rigid particles in a steady Stokes flow. In a celebrated work, Einstein argued that in the regime of dilute particles the system behaves at leading order like a Stokes fluid with some explicit effective viscosity. In the present contribution, we rigorously define a notion of effective viscosity, regardless of the dilute regime assumption. More precisely, we establish a homogenization result for when particles are distributed according to a given stationary and ergodic random point process. The main novelty is the introduction and analysis of suitable corrector equations., SCOPUS: ar.j, info:eu-repo/semantics/published

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We consider strongly coupled competitive elliptic systems that arise in the study of two-component Bose-Einstein condensates. As the coupling parameter tends to infinity, solutions that remain uniformly bounded are known to converge to a segregated limiting profile, with the difference of its components satisfying a limit scalar PDE. In the case of radial symmetry, under natural non-degeneracy assumptions on a solution of the limit problem, we establish by a perturbation argument its persistence as a solution to the elliptic system., SCOPUS: ar.j, DecretOANoAutActif, info:eu-repo/semantics/published

Several types of generically-nondegenerate Poisson structures can be effectively studied as symplectic structures on naturally associated Lie algebroids. Relevant examples of this phenomenon include log-, elliptic, bk-, scattering and elliptic-log Poisson structures. In this paper we discuss topological obstructions to the existence of such Poisson structures, obtained through the characteristic classes of their associated symplectic Lie algebroids. In particular we obtain the full obstructions for surfaces to carry such Poisson structures., SCOPUS: ar.j, DecretOANoAutActif, info:eu-repo/semantics/published

We study the Dirichlet problem for the following prescribed mean curvature PDE {-div∇v1+, SCOPUS: ar.j, DecretOANoAutActif, info:eu-repo/semantics/published

We consider a linear elliptic system in divergence form with random coefficients and study the random fluctuations of large-scale averages of the field and the flux of the solution operator. In the context of the random conductance model, we developed in a previous work a theory of fluctuations based on the notion of homogenization commutator: we proved that the two-scale expansion of this special quantity is accurate at leading order in the fluctuation scaling when averaged on large scales (as opposed to the two-scale expansion of the solution operator taken separately) and that the large-scale fluctuations of the field and the flux of the solution operator can be recovered from those of the commutator. This implies that the large-scale fluctuations of the commutator of the corrector drive all other large-scale fluctuations to leading order, which we refer to as the pathwise structure of fluctuations in stochastic homogenization. In the present contribution we extend this result in two directions: we treat continuum elliptic (possibly non-symmetric) systems and allow for strongly correlated coefficient fields (Gaussian-like with a covariance function that can display an arbitrarily slow algebraic decay at infinity). Our main result shows in this general setting that the two-scale expansion of the homogenization commutator is still accurate to leading order when averaged on large scales, which illustrates the robustness of the pathwise structure of fluctuations., SCOPUS: ar.j, info:eu-repo/semantics/published

The epidermis is the first barrier of protection of living organisms against external attacks. It is constantly renewed throughout life, through a process called "homeostasis", which ensures that every cell lost on its surface is replaced by new ones. Recent studies have shown that this balance is ensured by a hierarchy of stem cells (SC) and progenitors that perform 3 types of cell divisions, each having a fixed probability. Although the epidermis has been extensively studied during homeostasis, little is known about the cellular dynamics taking place when the epidermis must expand its surface. Are these probabilities of division immutable or can they change? In this project, we focused on two conditions of epidermal expansion: postnatal growth and wound healing. Using the mouse tail epidermis as a model, we show that the re-epithelialization after a wound is achieved via the formation of two transient compartments that are spatially and molecularly distinct :a leading edge and a proliferative hub. We show that the leading edge cells have a specific transcriptional signature that is independent of their quiescent state and we propose new markers not previously described. Using the technique of "lineage tracing", coupled with clonal analysis and mathematical modeling, we highlight the proliferation dynamics of SCs and progenitors during healing. We show that different populations of cells residing in different compartments, the hair follicle infundibulum and the interfollicular epidermis, acquire a similar dynamics and re-activate their SC while the progenitors increase their rate of proliferation without changing their division probabilities. This similar proliferation dynamics in two compartments of the epidermis suggests that division probabilities are not dictated by the cell of origin. Interestingly, cell dynamics is different during postnatal growth. Using lineage tracing, clonal analysis and single-cell transcriptional analysis, we demonstrate that the post-n, L’épiderme est la première barrière de protection des organismes vivants contre des attaques extérieures. Il est constamment renouvelé au cours de la vie, via un processus appelé « homeostasie », qui assure que chaque cellule perdue à sa surface soit remplacée par de nouvelles. Des études récentes ont montré que cet équilibre était assuré par une hiérarchie de cellules souches (CS) et de progéniteurs qui réalisent 3 types de divisions cellulaires, chaque type de division ayant une probabilité fixe. Bien que l’épiderme ait été intensivement étudié durant l’homeostasie, peu de choses sont connues concernant la dynamique cellulaire prenant place lors de phénomènes où l’épiderme doit grandir. Ces probabilités de division sont-elles immuables ou peuvent-elles au contraire changer ?Dans ce projet, nous nous sommes intéressés à deux conditions d’expansion de l’épiderme :la croissance post-natale et la cicatrisation des plaies. En utilisant l’épiderme de la queue de souris comme modèle, nous montrons que la ré-épithélialisation d’une plaie est réalisée via la formation de deux compartiments cellulaires transitoires distincts spatialement et du point de vue moléculaire :un front de migration et un centre prolifératif. Nous montrons que les cellules du front de migration ont une signature transcriptionnelle spécifique qui est indépendante de leur état de quiescence et proposons de nouveaux marqueurs non décrits auparavant. En utilisant la technique du « lineage tracing », couplée à une analyse clonale et à de la modélisation mathématique, nous mettons en évidence la dynamique de prolifération des CS et des progéniteurs lors de la cicatrisation. Nous montrons que différentes populations de cellules résidant dans des compartiments différents, l’infundibulum du follicule pileux et l’épiderme interfolliculaire, acquièrent une dynamique similaire et ré-activent leur CS tandis que les progéniteurs augmentent leur taux de prolifération sans changer leur probabilité de division. Cett, Doctorat en Sciences biomédicales et pharmaceutiques (Pharmacie), info:eu-repo/semantics/nonPublished

We study radial sign-changing solutions of a class of fully nonlinear elliptic Dirichlet problems in a ball, driven by the extremal Pucci's operators and with a power nonlinear term. We first determine a new critical exponent related to the existence or nonexistence of such solutions. Then we analyze the asymptotic behavior of the radial nodal solutions as the exponents approach the critical values, showing that new concentration phenomena occur. Finally we define a suitable weighted energy for these solutions and compute its limit value., SCOPUS: ar.j, info:eu-repo/semantics/published

We give necessary and sufficient conditions for the existence of positive radial solutions for a class of fully nonlinear uniformly elliptic equations posed in the complement of a ball in RN, and equipped with homogeneous Dirichlet boundary conditions., SCOPUS: ar.j, info:eu-repo/semantics/published

Four quantities are fundamental in homogenization of elliptic systems in divergence form and in its applications: the field and the flux of the solution operator (applied to a general deterministic right-hand side), and the field and the flux of the corrector. Homogenization is the study of the large-scale properties of these objects. In case of random coefficients, these quantities fluctuate and their fluctuations are a priori unrelated. Depending on the law of the coefficient field, and in particular on the decay of its correlations on large scales, these fluctuations may display different scalings and different limiting laws (if any). In this contribution, we identify another crucial intrinsic quantity, motivated by H-convergence, which we refer to as the homogenization commutator and is related to variational quantities first considered by Armstrong and Smart. In the simplified setting of the random conductance model, we show what we believe to be a general principle, namely that the homogenization commutator drives at leading order the fluctuations of each of the four other quantities in a strong norm in probability, which is expressed in form of a suitable two-scale expansion and reveals the pathwise structure of fluctuations in stochastic homogenization. In addition, we show that the (rescaled) homogenization commutator converges in law to a Gaussian white noise, and we analyze to which precision the covariance tensor that characterizes the latter can be extracted from the representative volume element method. This collection of results constitutes a new theory of fluctuations in stochastic homogenization that holds in any dimension and yields optimal rates. Extensions to the (non-symmetric) continuum setting are also discussed, the details of which are postponed to forthcoming works., SCOPUS: ar.j, info:eu-repo/semantics/published

We consider linear elliptic equations in divergence form with stationary random coefficients of integrable correlations. We characterize the fluctuations of a macroscopic observable of a solution to relative order d2, where d is the spatial dimension; the fluctuations turn out to be Gaussian. As for previous work on the leading order, this higher-order characterization relies on a pathwise proximity of the macroscopic fluctuations of a general solution to those of the (higher-order) correctors, via a (higher-order) two-scale expansion injected into the “homogenization commutator”, thus confirming the scope of this notion. This higher-order generalization sheds a clearer light on the algebraic structure of the higher-order versions of correctors, flux correctors, two-scale expansions, and homogenization commutators. It reveals that in the same way as this algebra provides a higher-order theory for microscopic spatial oscillations, it also provides a higher-order theory for macroscopic random fluctuations, although both phenomena are not directly related. We focus on the model framework of an underlying Gaussian ensemble, which allows for an efficient use of (second-order) Malliavin calculus for stochastic estimates. On the technical side, we introduce annealed Calderón–Zygmund estimates for the elliptic operator with random coefficients, which conveniently upgrade the known quenched large-scale estimates., SCOPUS: ar.j, info:eu-repo/semantics/published