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We introduce partial $r$-Bell polynomials in three combinatorial Hopf algebras. We prove a factorization formula for the generating functions which is a consequence of the Zassenhauss formula., Comment: 7 pages

International audience; The time discretization of gradient flows in metric spaces uses variants of the celebrated implicit Euler-type scheme of Jordan, Kinderlehrer and Otto. We propose in this Note a different approach which allows to construct two second-order in time numerical schemes. In a metric space framework, we show that the schemes are well defined and prove the convergence for one of them under some regularity assumptions. For the particular case of a Fokker–Planck gradient flow in the Wasserstein space, we obtain (theoretically and numerically) the second-order convergence.; La discrétisation temporelle des flots de gradient dans des espaces métriques utilise des variantes du schéma d’Euler implicite issu du travail séminal de Jordan, Kinderlehrer et Otto. Nous proposons dans cette Note une approche différente permettant de construire deux schémas numériques d’ordre deux en temps. Dans le cadre d’un espace métrique, nous montrons que les schémas sont bien définis et prouvons la convergence de l’un d’entre eux sous des hypothèses de régularité. Pour le cas particulier d’un flot de gradient Fokker–Planck dans l’espace de Wasserstein, nous obtenons (théoriquement et numériquement) la convergence à l’ordre deux.

International audience; We use the formalism of the Rényi entropies to establish the symmetry range of extremal functions in a family of subcriti-cal Caffarelli-Kohn-Nirenberg inequalities. By extremal functions we mean functions which realize the equality case in the inequalities, written with optimal constants. The method extends recent results on critical Caffarelli-Kohn-Nirenberg inequalities. Using heuristics given by a nonlinear diffusion equation, we give a variational proof of a symmetry result, by establishing a rigidity theorem: in the symmetry region, all positive critical points have radial symmetry and are therefore equal to the unique positive, radial critical point, up to scalings and multiplications. This result is sharp. The condition on the parameters is indeed complementary of the condition which determines the region in which symmetry breaking holds as a consequence of the linear instability of radial optimal functions. Compared to the critical case, the subcritical range requires new tools. The Fisher information has to be replaced by Rényi entropy powers, and since some invariances are lost, the estimates based on the Emden-Fowler transformation have to be modified.

In this note, we characterize the solution of a system of elliptic integro-differential equations describing a phe-notypically structured population subject to mutation, selection and migration. Generalizing an approach based on Hamilton-Jacobi equations, we identify the dominant terms of the solution when the mutation term is small (but nonzero). This method was initially used, for different problems from evolutionary biology, to identify the asymptotic solutions, while the mutations vanish, as a sum of Dirac masses. A key point is a uniqueness property related to the weak KAM theory. This method allows to go further than the Gaussian approximation commonly used by biologists and is an attempt to fill the gap between the theories of adaptive dynamics and quantitative genetics., Comment: arXiv admin note: substantial text overlap with arXiv:1612.06193

Resume On modelise un solide elastique fragile (pouvant developper des fractures) comme limite d'un modele d'endommagement et proposons une methode numerique afin de determiner l'evolution quasi-statique d'une fracture dans l'esprit des travaux de Francfort et Marigo [3] et Allaire et al. [1] , [2] .

We give conditions under which the intersection between two attracting immediate basins boundaries of a rational map contains at least one periodic point.

In this work, we will establish local in time dispersive estimates for solutions to the model-case Dirichlet wave equation inside a cylindrical convex domain Ω ⊂ R 3 with a smooth boundary ∂ Ω ≠ ∅ . Let us recall that dispersive estimates are key ingredients to prove Strichartz estimates. Nonoptimal Strichartz estimates for waves inside an arbitrary domain Ω have been proved by Blair–Smith–Sogge [1] , [2] . Better estimates in strictly convex domains have been obtained in [4] . Our case of cylindrical domains is an extension of the result of [4] in the case where the curvature radius ≥0 depends on the incident angle and vanishes in some directions.

We address the topology of the set of singularities of a solution to a Hamilton–Jacobi equation. For this, we will apply the idea of the first two authors (Cannarsa and Cheng, Generalized characteristics and Lax–Oleinik operators: global result, preprint, arXiv:1605.07581 , 2016) to use the positive Lax–Oleinik semi-group to propagate singularities.

International audience; This paper introduces the notion of a metric for a simplex. Such a metric is a key ingredient when solving a PDE system by means of a Finite Element Method. Both mesh generation methods and error estimates (as far as interpolation error is considered) are concerned. Moreover, this metric gives information about the geometry of the corresponding element, typically about its shape quality in any Euclidean space.; Ce papier introduit la notion de métrique liée à un élément géométrique simplicial (un simplexe). Cette métrique est utilisée dans la résolution par la méthode des éléments finis d'équations aux dérivées partielles, à la fois pour la génération de maillages et les estimateurs d'erreur basés sur l'erreur d'interpolation. En outre, cette métrique renseigne sur la géométrie de l'élément et, en particulier, sur sa qualité en forme dans un espace euclidien quelconque.

We consider the energy critical semilinear heat equation ∂ t u = Δ u + | u | 4 d − 2 u , x ∈ R d in dimension d ≥ 3 . We propose a self-contained proof of the stability of solutions u blowing-up in finite time with type-I ODE blow-up ‖ u ‖ L ∞ ∼ κ ( T − t ) d − 2 4 , T > 0 , κ : = ( d − 2 4 ) d − 2 4 which adapts to the energy critical case the proof of Fermanian, Merle, Zaag [4] .

Nous montrons que les \'equations du rep\`ere mobile des surfaces de Bonnet conduisent \`a une paire de Lax matricielle isomonodromique d'ordre deux pour la sixi\`eme \'equation de Painlev\'e. We show that the moving frame equations of Bonnet surfaces can be extrapolated to a second order, isomonodromic matrix Lax pair of the sixth Painlev\'e equation., Comment: in French. Formula (14) added. Two paragraphs added p 5-6. Old ref 1 removed, ref 7 added. To appear, Comptes rendus Acad. sciences Paris

In the perspective of estimating main effects of model inputs, two approaches are studied to iteratively construct replicated designs based on Sobol' sequences. Space-filling properties of the resulting designs are studied based on two criteria.

The goal of this note is to announce certain results in orbit equivalence theory, especially concerning the approximation of p.m.p. standard equivalence relations by increasing sequences of sub-relations, with application to the behavior of the Bernoulli percolation on Cayley graphs at the threshold p u .

C. R. Math. Acad. Sci. Paris.http://dx.doi.org/10.1016/j.crma.2016.06.007Open access; International audience; In this Note, we present a new version of the vector penalty-projection splitting method described in [1] (Angot et al., 2012) for the fast numerical computation of incompressible flows with variable density and viscosity. We show that the velocity correction can be made completely independent on the mass density ρ. Hence, this step is purely kinematic using the fast Helmholtz-Hodge decompositions proposed in [2] (Angot et al., 2013). Then, it is shown that the dynamic step of pressure gradient correction can be fast and locally consistent on edge-based generalized MAC-type unstructured meshes which naturally verify the compatibility condition in the proposed discrete setting. By the way, a new accurate front-tracking Lagrangian-advection technique is also introduced for multiphase flows. This new method preserves the fully vector formulation of both the prediction and correction steps of the original scheme, the primary unknowns being (v, Grad p) and ρ by advection, since the pressure Neumann-Poisson problem remains eliminated. The efficiency of the present method is demonstrated through numerical results on sharp test cases.; Résumé. Une méthode cinématique de pénalité-projection vectorielle pour l' écoulement incompressiblè à densité variable. On présente dans cette Note une nouvelle version de la méthode de splitting par pénalité-projection vectorielle décrite dans [1] pour le calcul des écoulements incompressibles à masse volumique et viscosité variables. Le principal résultat est de rendre la correction vectorielle de vitesse complètement indépendante de la masse volumique ρ. Cette étape devient donc purement cinématique et correspond à une décomposition rapide de Helmholtz-Hodge proposée dans [2]. On montre que l' étape dynamique de correction du gradient de pression peut être rapide et localement consistante sur des maillages généralisés de type MAC non structurés. Version française abrégée. L'idée clé pour la construction de cette nouvelle méthode (K-VPP ε) repose sur les décompositions rapides de Helmholtz-Hodge en domaine borné proposées dans [2] et analysées dans [3]. L'étape de correction vectorielle en vitesse v est rendue purement cinématique. Elle correspond à une méthode de projection à divergence nulle approchée pour obtenir la composante irrotationnelle $\widehat{v} := Grad φ$ du champ de vitesse prédit $\widetilde{v}$. L'étape dynamique de correction du gradient de pression est alors obtenue en introduisant le concept de densité inertielle $\bar{ρ}$ calculée sur chaque arête du maillage principal et reliant les sommets des inconnues scalaires p, φ et ρ. On montre qu'il existe $\bar{ρ}$ satisfaisant l'équation de consistance locale : $Grad(\bar{ρ} φ) = ρ Grad φ = ρ \widehat{v}$, et permettant de recouvrer un gradient de pression entre les sommets. En pratique, comme la condition de compatibilité associée est naturellement satisfaite pour les maillages de type MAC, le calcul de $\bar{ρ}$ est effectué facilement et de façon rapide à partir de ρ et d'une fonction de partage α, tandis que φ est reconstruit à partir de son gradient connu sur les arêtes.

International audience; Mould calculus is a powerful combinatorial tool that often provides some explicit formulae when there are no other available computational methods. It has a well-known interpretation/dictionary in terms of Hopf algebras. But this dictionary does not provide any equivalent of formal moulds. Thus, we present here such an interpretation and give a generic way to prove mould symmetries of formal moulds.; Le calcul moulien est un outil combinatoire puissant, qui fournit souvent des formules explicites, alors que d'autres moyens de calcul n'aboutissent pas. Il en existe une interprétation/un dictionnaire en termes d'algèbres de Hopf. Mais ce dictionnaire n'a pas été développé jusqu'aux moules formels. Nous présentons ici une telle interprétation et donnons alors une méthode générique permettant de prouver les symétries de moules formels.

In this paper we consider a punctured Riemann surface endowed with a Hermitian metric which equals the Poincar\'e metric near the punctures and a holomorphic line bundle which polarizes the metric. We show that the Bergman kernel can be localized around the singularities and its local model is the Bergman kernel of the punctured unit disc endowed with the standard Poincar\'e metric. As a consequence, we obtain an optimal uniform estimate of the supremum norm of the Bergman kernel, involving a fractional growth order of the tensor power., Comment: 42 pages, 2 figures; v.2 is a final update to agree with the published paper

International audience; This note introduces reflective tomography in a mathematical framework. The effect of the filtered backprojection on reflective-kind projections is studied: a reflective projection is defined, tomographic filtering of such a projection is analysed, and so is the filtered backprojection. The results emphasize the role of the contrasts: we get a decomposition in which the contributions of the discontinuities and of the tangential variations are enlightened.

We provide a significant extension of the Hyperboloidal Foliation Method introduced by the authors in 2014 in order to establish global existence results for systems of quasilinear wave equations posed on a curved space, when wave equations and Klein–Gordon equations are coupled. This method is based on a ( 3 + 1 ) foliation (of the interior of a future light cone in Minkowski spacetime) by spacelike and asymptotically hyperboloidal hypersurfaces. In the new formulation of the method, we succeed to cover wave-Klein–Gordon systems containing “strong interaction” terms at the level of the metric, and then generalize our method in order to establish a new existence theory for the Einstein equations of general relativity. Following pioneering work by Lindblad and Rodnianski on the Einstein equations in wave coordinates, we establish the nonlinear stability of Minkowski spacetime for self-gravitating massive scalar fields.

In this note, we prove the non-degeneracy of the Kobayashi–Eisenman volume measure of a singular directed varieties (X,V), i.e. the Kobayashi measure hyperbolicity of (X,V), as long as the canonical sheaf KV of V is big in the sense of Demailly.

This note addresses the following shape matching problem: given a 'template' shape, numerically described by means of a computational mesh, and a 'target' shape, known only via a signed distance function to its boundary, we aim at deforming iteratively the mesh of the template shape into a computational mesh of the target shape. To achieve this goal, we rely on techniques from shape optimization. Under the sole assumption that both shapes share the same topology, the desired transformation is realized as a sequence of elastic displacements, which are obtained by minimizing an energy functional based on the distance between the two shapes. The proposed method has been implemented in a finite elements setting and numerical examples in two and three dimensions are presented to illustrate its efficiency.; Dans cette note, nous nous intéressons au problème d'appariement de formes suivant : étant donné une forme de référence, représentée numériquement par un maillage de calcul, et une forme cible, connue seulement par l'intermédiaire de la fonction de distance signée a celle-ci, notre objectif consiste à déformer itérativement le maillage de la forme de référence en un maillage de la forme cible. Pour ce faire, nous nous appuyons sur des techniques d'optimisation de formes. Sous l'hypothèse que les deux formes ont la même topologie, la transformation cherchée s'obtient comme une suite de déplacements élastiques, solutions d'un problème de minimisation d'une énergie basée sur la distance entre les formes. La méthode a été implémentée en deux et trois dimensions d'espace et nous présentons des exemples numériques permettant d'apprécier son efficacité.

Resume Dans cette Note, nous etudions l'estimation non parametrique de la fonction de regression, lorsque la variable reponse et la covariable sont fonctionnelles. Nous construisons un estimateur local lineaire de l'operateur de regression, et nous evaluons son erreur d'estimation. Ensuite, nous demontrons sa convergence presque complete et uniforme. La vitesse de convergence obtenue est exprimee en fonction de la probabilite des petites boules de la covariable et en fonction de la fonction d'entropie de l'ensemble sur lequel la convergence uniforme est obtenue.

Conférence invitée; International audience; In this talk the expression NURBS is meant with the general meaning of GeometricallyContinuous Piecewise Quasi-Chebyshevian NURBS, that is, rational B-splines built fromthe largest class $\mathcal C$ of spline spaces which can be used for design. A spline space in $\mathcal C$ hasdifferent Quasi Extended Chebyshev spaces (QEC-spaces) as section-spaces, andwe allow connection matrices at the knots. Moreover, as usual for design, we must requirethe presence of blossoms. We recently achieved a recursive constructive characterisationof the class $\mathcal C$. The important part of this characterisation consists in provingthat a spline space in $\mathcal C$ can automatically be based on infinitely many possible PiecewiseQEC-spaces.Interpreted in an appropriate way, the first step of this construction can be viewed as theconstruction of all rational spline spaces based on a spline space in $\mathcal C$. This guarantees thatany such rational spline space belongs in turn to the class $\mathcal C$ . It thus possesses blossoms aswell as B-spline bases (NURBS in the sense explained earlier).The classical NURBS are thus examples of parametrically continuous splines in theclass $\mathcal C$. Compared to their polynomial counterparts, one major interest of introducing themwas the shape effects permitted by the parameters defining them. Now, a natural questionarises: is it worthwhile building NURBS when the class $\mathcal C$ already provides us with sucha great variety of shape parameters (coming either from the section-spaces or from theconnection matrices) and of exactly represented curves, and when this does not increase theclass $\mathcal C$ ?

We present a method of the fictitious domain type for the Poisson–Dirichlet problem. The computational mesh is obtained from a background (typically uniform Cartesian) mesh by retaining only the elements intersecting the domain where the problem is posed. The resulting mesh does not thus fit the boundary of the problem domain. Several finite element methods (XFEM, CutFEM) adapted to such meshes have been recently proposed. The originality of the present article consists in avoiding integration over the elements cut by the boundary of the problem domain, while preserving the optimal convergence rates, as confirmed by both the theoretical estimates and the numerical results.

Resume Nous utilisons des resultats classiques sur les zeros des fonctions L de Dirichlet pour prouver la non-nullite de deux determinants analogues au determinant de Maillet. Nous en deduisons un theoreme d'extension pour les isometries de Lee et euclidienne des codes lineaires sur un corps premier.

International audience; In this note we propose a general procedure to construct exact sequences on non-conforming function spaces and we show how this construction can be used to derive a proper discrete Gauss law for structure-preserving discontinuous Galerkin (DG) approximations to the time-dependent 2d Maxwell equations.; Dans cette note nous proposons un procédé général pour construire des suites exactes sur des espaces non-conformes, et nous montrons comment ce procédé peut servir à écrire une loi de Gauss discrète convenable dans le cadre d’approximations Galerkin discontinues (DG) des équations de Maxwell temporelles en 2d.

A closed smooth differential (q,q)-form θ in a complex projective manifold X embedded in the projective space Pn gives rise to an “extension” to the projective space as a closed current. This means that the intersection number of this current with the direct image by the canonical injection of X of a smooth test form on X is equal to the value of θ on this test form, and that its restriction to X is obtained by means of the blow up with center X. We define in this way an integral transform given by a kernel which is a closed current in Pn×X, and which is used to bring an answer to a problem of Grothendieck. This transform is still defined for an algebraic cycle of codimension q instead of the differential (q,q)-form. We obtain a characterization of the complete intersections by the positivity of the transform. Moreover, we obtain a version of the Lefschetz theorem on the hyperplane section stating that the cohomology class arises from the hyperplane section when the intersection with X is transverse outside the hyperplane section.

International audience; We introduce and study differential-reflection operators$\Lambda_{A, \varepsilon}$ acting on smooth functions defined on $\mathbb R.$ Here $A$ is a Sturm-Liouville function with additional hypotheses and $\varepsilon\in \mathbb R.$ For special pairs $(A,\varepsilon),$ we recover Dunkl's, Heckman's and Cherednik's operators (in one dimension). As, by construction, the operators $\Lambda_{A, \varepsilon}$ are mixture of ${\rm d}/{\rm d}x$and reflection operators, we prove the existence of an operator $V_{A,\varepsilon}$ so that $\Lambda_{A, \varepsilon}\circ V_{A,\varepsilon}=V_{A,\varepsilon}\circ {\rm d}/{\rm d}x.$ The positivity of the intertwining operator $V_{A,\varepsilon}$ is also established. Via the eigenfunctions of $\Lambda_{A,\varepsilon},$ we introduce a generalized Fourier transform $\mathcal F_{A,\varepsilon}.$ For $-1\leq \varepsilon\leq 1 $ and $0 < p \leq \frac{2}{1+\sqrt{1-\varepsilon^2}},$ we develop an $L^p$-Fourier analysis for $\mathcal F_{A,\varepsilon},$ and then we prove an $L^p$-Schwartz space isomorphism theorem for $\mathcal F_{A,\varepsilon}$.Details of this paper will be given in another article.

We study the positivity of cohomology classes related to the convergence problem of the inverse σ k -flow, according to a conjecture proposed by Lejmi and Szekelyhidi.

We construct contact forms with constant $Q^\prime$-curvature on compact three-dimensional CR manifolds which admit a pseudo-Einstein contact form and satisfy some natural positivity conditions. These contact forms are obtained by minimizing the CR analogue of the $II$-functional from conformal geometry. Two crucial steps are to show that the $P^\prime$-operator can be regarded as an elliptic pseudodifferential operator and to compute the leading order terms of the asymptotic expansion of the Green's function for $\sqrt{P^\prime}$., Comment: 36 pages

In this note, we use the theory of Toeplitz operators to give an asymptotic formula for the holomorphic analytic torsion forms associated with a family of holomorphic vector bundles that are the direct image of L ⊗ p , where L is a line bundle. To obtain the asymptotics, we make a positivity assumption along the fibers on L . This result is the family version of the results of Bismut and Vasserot on the holomorphic torsion.

International audience; We give an example of a stochastic Hamilton-Jacobi equation du = H(Du)dξ which has an infinite speed of propagation as soon as the driving signal ξ is not of bounded variation.; Nous présentons un exemple d'équation d'Hamilton–Jacobi stochastique du=H(Du)dξ dont la vitesse de propagation est infinie dès que le signal ξ n'est pas à variation bornée.

We investigate the transitivity properties of the group of morphisms generated by Vieta involutions on the solutions in congruences to the Markoff equation as well as to other Markoff type affine cubic surfaces. These are dictated by the finite Q ¯ orbits of these actions and these can be determined effectively. The results are applied to give forms of strong approximation for integer points, and to sieving, on these surfaces.

International audience; Nous considérons une classe de schémas Lagrange-Projection sur maillage décalé pour résoudre les équations d’Euler 1D. Les schémas proposés, formulés en énergie interne, sont d’ordre élevé en espace comme en temps, utilisables quelle que soit l’équation d’état. La présence des termes non-conservatifs est gérée en ajoutant une discrétisation de l’équation de l’énergie cinétique ainsi qu'une procédure de synchronisation entre l’énergie cinétique et l'impulsion. Cette procédure assure de résoudre le système formulé en énergie totale et permet au niveau discret de préserver globalement l’énergie totale ainsi qu'une bonne capture des chocs. Une extension nD sur grille cartésienne décalée de type C avec splitting directionnel d’ordre élevé est proposée. Des résultats numériques sont présentés jusqu’à l’ordre 8.

International audience; We compute in this article the KK-theory of A-valued semi-circular systems thanks to tools developed by Pimsner (see [1]) to study generalized Toeplitz algebras.; On calcule dans cet article la KK-théorie de systèmes semi-circulaires A-valués à l'aide d'outils développés par Pimsner (voir [1]) pour étudier les algèbres de Toeplitz généralisées.

International audience; We prove that there cannot exist square-integrable nonzero solutions to the Helmholtz equation in an axisymmetric conical domain whose vertex angle is greater than π. This implies in particular the absence of eigenvalues embedded in the essential spectrum of a large class of partial differential operators which coincide with the Laplacian in the conical domain.

We present a new proof of Thurston's theorem that the unit ball of a seminorm on $\mathbf{R}^d$ taking integer values on $\mathbf{Z}^d$ is a polyhedra defined by finitely many inequalities with integer coefficients., Comment: Bilingual french-english note; 3 pages, 1 figure

International audience; Euler published a formula that now reads View the MathML source∑1∞λ(n)n=0, λ being the completely multiplicative function equal to −1 on the prime numbers. Thus View the MathML source(λ(n)n) is an example of a CMO function (completely multiplicative with sum 0). We extend this formula by considering λ as defined on Beurling's generalized prime numbers and integers, according to Diamond's condition on generalized primes, which implies a regular distribution of the generalized integers (théorème 3). As an application, we show how to contruct a CMO function carried by a set of integers whose counting function is of the form Dxa(1+o(1))Dxa(1+o(1))(x→∞)(x→∞), for any given a between 0 and 1 (théorème 1).; Euler a publié une formule que nous écrivons aujourd'hui View the MathML source∑1∞λ(n)n=0, λ étant la fonction complètement multiplicative qui vaut −1 sur les nombres premiers. Ainsi, View the MathML source(λ(n)n) est un exemple de fonction CMO (complètement multiplicative à somme nulle). Nous étendons cette formule au cas où λ est définie sur des nombres premiers et entiers généralisés de Beurling, suivant la condition sur les premiers généralisés donnée par Diamond pour assurer la régularité de la distribution des entiers généralisés (théorème 3). En guise d'application, nous indiquons comment construire, pour tout a entre 0 et 1, une fonction CMO dont la distribution du support est de la forme Dxa(1+o(1))Dxa(1+o(1))(x→∞)(x→∞) (théorème 1).

In this note we give embeddings and other properties of Besov spaces, as well as give spectral and Fourier multiplier theorems, in the setting of graded Lie groups. We also present a Nikolskii type inequality and the Littlewood-Paley theorem that play a role in this analysis and are also of interest on their own. To cite this article: Cardona, D., Ruzhansky, M., C. R. Acad. Sci. Paris, Ser. I 340 (2005).

We announce a Fourier restriction result for general polynomial curves in R d . Measuring the Fourier restriction with respect to the affine arclength measure of the curve, we obtain a universal bound for the class of all polynomial curves of bounded degree. Our method relies on establishing a geometric inequality for general polynomial curves which is of interest in its own right. There are applications of this geometric inequality to other problems in euclidean harmonic analysis. To cite this article: S. Dendrinos, J. Wright, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

We study the near diagonal asymptotic expansion of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle over a compact symplectic manifold. We show how to compute the coefficients of the expansion by recurrence and give a closed formula for the first two of them. As consequence, we calculate the density of states function of the Bochner-Laplacian and establish a symplectic version of the convergence of the induced Fubini-Study metric. We also discuss generalizations of the asymptotic expansion for non-compact or singular manifolds as well as their applications. Our approach is inspired by the analytic localization techniques of Bismut-Lebeau., 48 pages. Add two references on the Hermitian scalar curvature

In this Note, we present the weak convergence of additive functionals of processes with locally independent increments and Markov switching in Levy and Poisson approximation schemes. The singular perturbation problem for the generators of switched processes is used to prove the semimartingales' predictable characteristics convergence.

International audience; We investigate the Klein-Gordon equation in the past causal domain of a de Sitter brane imbedded in an anti-de Sitter bulk. We solve the global mixed hyperbolic problem. We prove that any finite energy solution can be expressed as a Kaluza Klein tower that is a superposition of free fields in the Steady State Universe, of which we study the asymptotic behaviors. We show that the leading term of a gravitational fluctuation is a massless graviton, i.e. the de Sitter brane is linearly stable.

Resume Nous etudions les plongements de Carleson sur les espaces de Hardy H p ou p > 1 . Nous caracterisons leur appartenance a la classe des operateurs r-sommants quelle que soit la valeur de r ≥ 1 . Ceci generalise completement des resultats anterieurs et resout un probleme ouvert depuis les annees 1970. Cette caracterisation est de nature differente suivant la valeur du couple ( p , r ) . Ces resultats englobent le cas des operateurs de composition, et des operateurs de composition a poids.

We prove that a linear growth graph has finitely many horofunctions. This provides a short and simple proof that any finitely generated infinite group of linear growth is virtually cyclic., 4 pages. Published in C. R. Math. Acad. Sci. Paris at http://www.sciencedirect.com/science/article/pii/S1631073X16302060

International audience; Mathai, Melrose, and Singer introduced the notion of projective elliptic operators on manifolds equipped with an Azumaya bundle. In this note we compute the equivariant index of transversally elliptic operators that are the pullback of projective elliptic operators on the trivialization of the Azumaya bundle. It encompasses the fractional index formula of projective elliptic operator by Mathai-Melrose-Singer.

Consider an action of a connected compact Lie group on a compact complex manifold $M$, and two equivariant vector bundles $L$ and $E$ on $M$, with $L$ of rank 1. The purpose of this paper is to establish holomorphic Morse inequalities \`{a} la Demailly for the invariant part of the Dolbeault cohomology of tensor powers of $L$ twisted by $E$. To do so, we define a moment map $\mu$ by the Kostant formula and we define the reduction of $M$ under a natural hypothesis on $\mu^{-1}(0)$. Our inequalities are given in term of the curvature of the bundle induced by $L$ on this reduction.

Recently A. Vasseur and C. Yu have proved (see A. Vasseur, C. Yu, Existence of global weak solutions for 3D degenerate compressible Navier–Stokes equations, arXiv:1501.06803 , 2015) the existence of global entropy-weak solutions to the compressible Navier–Stokes equations with viscosities ν ( ϱ ) = μ ϱ and λ ( ϱ ) = 0 and a pressure law under the form p ( ϱ ) = a ϱ γ with a > 0 and γ > 1 constants. In this note, we propose a non-trivial relative entropy for such system in a periodic box and give some applications. This extends, in some sense, results with constant viscosities recently initiated by E. Feireisl, B.J. Jin and A. Novotny in [J. Math. Fluid Mech. (2012)]. We present some mathematical results related to the weak–strong uniqueness, the convergence to a dissipative solution to compressible or incompressible Euler equations. As a by-product, this mathematically justifies the convergence of solutions to a viscous shallow-water system to solutions to the inviscid shallow-water system.

International audience; This note provides a description of the parareal method, a numerical section to assess the performance of the method for American contracts in the scalar case computed by LSMC and parallelized by parareal time decomposition with two or more levels. It contains also a convergence proof for the two levels pa- rareal Monte-Carlo method when the coarse grid solution is computed by an Euler explicit scheme with time step ∆t > δt, the time step used for the Euler scheme at the fine grid level. Hence the theorem provides a tool to analyze also the multilevel parareal method.; La méthode pararéelle pour les options américaines. Dans cette note la méthode pararéelle est introduite pour l'algorithme LSMC de Longstaff-Schartz pour calculer des options américaines sur une machineparalì ele. Dans une section numérique les performances de la méthode sont données dans le cas scalairè a deux niveaux d'abord puis multi-niveaux. Un théor eme de convergence est aussí enoncé lorsque la méthode d'Euler explicite est utilisée avec un pas de temps ∆t > δt le pas de temps de la grille fine. Une estimation est obtenue qui permet d'analyser la méthode pararéelle multi-niveaux. Abstract The Parareal Algorithm for American Options. This note provides a description of the parareal method, a numerical section to assess the performance of the method for American contracts in the scalar case computed by LSMC and parallelized by parareal time decomposition with two or more levels. It contains also a convergence proof for the two levels parareal Monte-Carlo method when the coarse grid solution is computed by an Euler explicit scheme with time step ∆t > δt, the time step used for the Euler scheme at the fine grid level. Hence the theorem provides a tool to analyze also the multilevel parareal method.

Lev A. Borisov has shown that the class of the affine line is a zero divisor in the Grothendieck ring of algebraic varieties over complex numbers. We improve the final formula by removing a factor., Comptes Rendus Math\'ematique, Elsevier Masson, 2016

Resume Dans un probleme de regression avec variable explicative fonctionnelle, on s'interesse a la selection des points les plus informatifs. Un modele parcimonieux de type non parametrique ainsi qu'une procedure de choix de variables basee sur une pre-selection par depistage sont proposes, et des resultats asymptotiques sont etablis concernant a la fois la selection des points informatifs et l'estimation des parametres du modele.