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International audience; We study here the approximation by a finite-volume scheme of a heat equation forced by a Lipschitz continuous multiplicative noise in the sense of Itô. More precisely, we consider a discretization which is semi-implicit in time and a two-point flux approximation scheme (TPFA) in space. We adapt the method based on the theoremof Prokhorov to obtain a convergence in distribution result, then Skorokhod'srepresentation theorem yields the convergence of the scheme towards a martingalesolution and the Gyöngy-Krylov argument is used to prove convergence in probabilityof the scheme towards the unique variational solution of our parabolic problem.

We solve the St.Venant torsion problem for an infinite cylindrical rod whose behaviour is described by a family of isotropic generalized continua, including the relaxed micromorphic and classical micromorphic model. The results can be used to determine the material parameters of these models. Special attention is given to the possible nonphysical stiffness singularity for a vanishing rod diameter, since slender specimens are in general described as stiffer., 41 pages, 29 figures

International audience; Nowadays is clearer and clearer that peculiar micromechanical effects can only be described through non local and higher-order models in the framework of elastic wave propagation. In particular, the study of finite-size solids is of great interest since for such cases these effects are magnified. The two peculiar micromechanical effects here studied are the so called \textitsize effects, which shows how, for a given wave load, the scattered filed changes as the ratio between the size of the unit cell and the dimension of the finite domain changes, and the presence of the so called \textitband gaps, which are frequencies ranges for which the propagation of waves in a solid is inhibited. A relaxed micromorphic model is here proposed [1]-[2] in order to mimic both these micromechanical effects with an elastic continuous material. The comparison in the mechanical response for several frequencies between a finite-size domain for a given microstructure ([2]) and both the micromorphic and Cauchy solids equivalent to this microstructured material acts as a reliability test for this model. \em References \beginthebibliography99 \vspace-5pt \bibitemd'Ago Marco Valerio d'Agostino, Gabriele Barbagallo, Ionel-Dumitrel Ghiba, Bernhard Eidel, Patrizio Neff, Angela Madeo. Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model. \textitJournal of Elasticity, pp 1-31, 2019. \vspace-7pt \bibitemalexis Alexios Aivaliotis, Domenico Tallarico, Marco-Valerio d'Agostino, Ali Daouadji, Patrizio Neff and Angela Madeo. Frequency- and angle-dependent scattering of finite-size meta-structures via the relaxed micromorphic model. \textitSubmitted. Acknowledgements: Angela Madeo acknowledge funding from the French Research Agency ANR, ?METASMART? (ANR-17CE08-0006). Angela Madeo and Gianluca Rizzi acknowledges support from IDEXLYON in the framework of the ?Programme Investissement d?Avenir? ANR-16-IDEX- 0005.

We show the existence of global minimizers for a geometrically nonlinear isotropic elastic Cosserat 6-parameter shell model. The proof of the main theorem is based on the direct methods of the calculus of variations using essentially the convexity of the energy in the nonlinear strain and curvature measures. We first show the existence of the solution for the theory including $O(h^{5})$ terms. The form of the energy allows us to show the coercivity for terms up to order $O(h^{5})$ and the convexity of the energy. Secondly, we consider only that part of the energy including $O(h^{3})$ terms. In this case the obtained minimization problem is not the same as those previously considered in the literature, since the influence of the curved initial shell configuration appears explicitly in the expression of the coefficients of the energies for the reduced two-dimensional variational problem and additional mixed bending-curvature and curvature terms are present. While in the theory including $O(h^{5})$ the conditions on the thickness $h$ are those considered in the modelling process and they are independent of the constitutive parameter, in the $O(h^{3})$ -case the coercivity is proven under some more restrictive conditions on the thickness $h$ .

International audience; In this paper, a stochastic nonlinear evolution system under Neumann boundary conditions is investigated. Precisely, we are interested in finding an existence and uniqueness result for a system of random heat equation coupled with a Barenblatt's type equation with a multiplicative stochastic force in the sense of Itô. To do so, we investigate in a first step the case of an additive noise through a semi-implicit in time discretization of the system. This allows us to show the well-posedness of the system in the additive case. In a second step, the derivation of continuous dependence estimates of the solution with respect to the data allows us to show the desired existence and uniqueness result for the multiplicative case.

We present a new geometrically nonlinear Cosserat shell model incorporating effects up to order $O(h^{5})$ in the shell thickness $h$ . The method that we follow is an educated 8-parameter ansatz for the three-dimensional elastic shell deformation with attendant analytical thickness integration, which leads us to obtain completely two-dimensional sets of equations in variational form. We give an explicit form of the curvature energy using the orthogonal Cartan-decomposition of the wryness tensor. Moreover, we consider the matrix representation of all tensors in the derivation of the variational formulation, because this is convenient when the problem of existence is considered, and it is also preferential for numerical simulations. The step by step construction allows us to give a transparent approximation of the three-dimensional parental problem. The resulting 6-parameter isotropic shell model combines membrane, bending and curvature effects at the same time. The Cosserat shell model naturally includes a frame of orthogonal directors, the last of which does not necessarily coincide with the normal of the surface. This rotation-field is coupled to the shell-deformation and augments the well-known Reissner-Mindlin kinematics (one independent director) with so-called in-plane drill rotations, the inclusion of which is decisive for subsequent numerical treatment and existence proofs. As a major novelty, we determine the constitutive coefficients of the Cosserat shell model in dependence on the geometry of the shell which are otherwise difficult to guess.

For $1< p we prove an $L^{p}$-version of the generalized trace-free Korn inequality for incompatible tensor fields $P$ in $W^{1,p}_0(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$. More precisely, let $\Omega \subset \mathbb {R}^{3}$ be a bounded Lipschitz domain. Then there exists a constant $c>0$ such that \[ \lVert{ P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\leq c\,\left(\lVert{\operatorname{dev} \operatorname{sym} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})} + \lVert{ \operatorname{dev} \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\right) \] holds for all tensor fields $P\in W^{1,p}_0(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$, i.e., for all $P\in W^{1,p} (\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$ with vanishing tangential trace $P\times \nu =0$ on $\partial \Omega$ where $\nu$ denotes the outward unit normal vector field to $\partial \Omega$ and $\operatorname {dev} P : = P -\frac 13 \operatorname {tr}(P) {\cdot } {\mathbb {1}}$ denotes the deviatoric (trace-free) part of $P$. We also show the norm equivalence \begin{align*} &\lVert{ P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}+\lVert{ \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\\ &\quad\leq c\,\left(\lVert{P}\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})} + \lVert{ \operatorname{dev} \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\right) \end{align*} for tensor fields $P\in W^{1,p}(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$. These estimates also hold true for tensor fields with vanishing tangential trace only on a relatively open (non-empty) subset $\Gamma \subseteq \partial \Omega$ of the boundary.

We consider the cylindrical bending problem for an infinite plate as modelled with a family of generalized continuum models, including the micromorphic approach. The models allow to describe length scale effects in the sense that thinner specimens are comparatively stiffer. We provide the analytical solution for each case and exhibit the predicted bending stiffness. The relaxed micromorphic continuum shows bounded bending stiffness for arbitrary thin specimens, while classical micromorphic continuum or gradient elasticity as well as Cosserat models [35] exhibit unphysical unbounded bending stiffness for arbitrary thin specimens. This finding highlights the advantage of using the relaxed micromorphic model, which has a definite limit stiffness for small samples and which aids in identifying the relevant material parameters., Comment: 50 pages, 28 figures (38 pictures)

Let X be a smooth proper variety over the quotient field of a Henselian discrete valuation ring with algebraically closed residue field of characteristic p. We show that for any coherent sheaf E on X, the index of X divides the Euler-Poincar\'e characteristic \chi(X,E) if p=0 or p>dim(X)+1. If 0dim(X)+1). When p=0, such statements also have implications for the possible multiplicities of singular fibers in degenerations of complex projective varieties., Comment: 20 pages; final version

In the present contribution we show that the relaxed micromorphic model is the only non-local continuum model which is able to account for the description of band-gaps in metamaterials for which the kinetic energy accounts separately for micro and macro-motions without considering a micro-macro coupling. Moreover, we show that when adding a gradient inertia term which indeed allows for the description of the coupling of the vibrations of the microstructure to the macroscopic motion of the unit cell, other enriched continuum models of the micromorphic type may allow the description of the onset of band-gaps. Nevertheless, the relaxed micromorphic model proves to be yet the most effective enriched continuum model which is able to describe multiple band-gaps in non-local metamaterials., Comment: arXiv admin note: text overlap with arXiv:1607.07385

International audience; This panel considers theoretical rather than practical frameworks for the alignment of history of mathematics and mathematics education. Naturally, such a theoretical framework will have practical implications, but its main task is to set out a certain line of questioning. First of all, it must also ask, even if only implicitly, what it really means to be a theoretical framework in the first place. But, principally, it must ask the ends and the value, and, ultimately, the meaning of history of mathematics in mathematics education. Though all the authors agree that in any theoretical framework, history of mathematics, as such, must be taken as the starting point. That said, no claim is made that there must be a single theoretical framework. Nevertheless, each of the three parts of this paper emphasize similar themes. One of these is the relationship between readers and texts: what do readers bring to texts? How are they affected by the texts? How are their own mathematics selves shaped by their engagement with historical material?

Given a smooth and separated K(pi,1) variety X over a field k, we associate a "cycle class" in etale cohomology with compact supports to any continuous section of the natural map from the arithmetic fundamental group of X to the absolute Galois group of k. We discuss the algebraicity of this class in the case of curves over p-adic fields, and deduce in particular a new proof of Stix's theorem according to which the index of a curve X over a p-adic field k must be a power of p as soon as the natural map from the arithmetic fundamental group of X to the absolute Galois group of k admits a section. Finally, an etale adaptation of Beilinson's geometrization of the pronilpotent completion of the topological fundamental group allows us to lift this cycle class in suitable cohomology groups., Comment: 19 pages; final version (added MSC codes and keywords)