Abstract: Publication date: Available online 2 August 2017 Source:Comptes Rendus Mathematique Author(s): Teng Huang In this note, we prove an L n 2 -energy gap result for Yang–Mills connections on a principal G-bundle over a compact manifold without using the Lojasiewicz–Simon gradient inequality ([2] Theorem 1.1).

Abstract: Publication date: Available online 31 July 2017 Source:Comptes Rendus Mathematique Author(s): Marie-Claude Arnaud, Patrice Le Calvez We introduce a notion of Denjoy sub-system that generalizes that of the Aubry–Mather set. For such systems, we prove a result similar to Denjoy theorem (non-existence of C 2 Denjoy sub-systems), and study their Lyapunov exponents.

Abstract: Publication date: Available online 31 July 2017 Source:Comptes Rendus Mathematique Author(s): Pierre Cantin An extension of the well-posedness analysis of the scalar and the vector advection–reaction problem in Banach graph spaces of power p ∈ ( 1 , ∞ ) is proposed. This analysis is based on the sign of the associated Friedrichs tensor, taking positive, null or reasonably negative values.

Abstract: Publication date: Available online 31 July 2017 Source:Comptes Rendus Mathematique Author(s): Aihua Fan We prove that fully oscillating sequences are orthogonal to multiple ergodic realizations of affine maps of zero entropy on compact Abelian groups. It is more than what Sarnak's conjecture requires for these dynamical systems.

Abstract: Publication date: Available online 27 July 2017 Source:Comptes Rendus Mathematique Author(s): Saulius Norvidas We discuss a uniqueness property of the characteristic function of an absolutely continuous probability measure. Our study is initiated by the question posed by N.G. Ushakov: is it true that, for any interval [ a , b ] ⊂ R , 0 ∉ [ a , b ] , there exists a characteristic function f such that f ≢ e − t 2 / 2 , but f ( t ) = e − t 2 / 2 for all t ∈ [ a , b ] '

Abstract: Publication date: Available online 21 July 2017 Source:Comptes Rendus Mathematique Author(s): Tushar Das, Lior Fishman, David Simmons, Mariusz Urbański We establish a new connection between metric Diophantine approximation and the parametric geometry of numbers by proving a variational principle facilitating the computation of the Hausdorff and packing dimensions of many sets of interest in Diophantine approximation. In particular, we show that the Hausdorff and packing dimensions of the set of singular m × n matrices are both equal to m n ( 1 − 1 m + n ) , thus proving a conjecture of Kadyrov, Kleinbock, Lindenstrauss, and Margulis as well as answering a question of Bugeaud, Cheung, and Chevallier. Other applications include computing the dimensions of the sets of points witnessing conjectures of Starkov and Schmidt.

Abstract: Publication date: Available online 21 July 2017 Source:Comptes Rendus Mathematique Author(s): Mário Bessa In this note, we prove the flowbox theorem for divergence-free Lipschitz vector fields.

Abstract: Publication date: Available online 21 July 2017 Source:Comptes Rendus Mathematique Author(s): Samuel Le Fourn We present results on the integral points of some modular varieties. These results are based on a generalisation of the so-called Runge's method to higher dimensions, which will be explained first. In particular, we obtain an explicit result for the Siegel modular variety A 2 ( 2 ) .

Abstract: Publication date: Available online 10 July 2017 Source:Comptes Rendus Mathematique Author(s): Christos Sourdis We prove that entire, complex valued solutions to the Ginzburg–Landau system with positive real and imaginary parts are constant in any spatial dimension. This property was shown very recently only in the planar case.

Abstract: Publication date: Available online 10 July 2017 Source:Comptes Rendus Mathematique Author(s): Jean-François Bony, Setsuro Fujiié, Thierry Ramond, Maher Zerzeri In the framework of semiclassical resonances, we make more precise the link between the polynomial estimates of the extension of the resolvent and the propagation of the singularities through the trapped set. This approach makes it possible to eliminate infinity and to concentrate the study near the trapped set. It has allowed us in previous papers to obtain the asymptotic of resonances in various geometric situations.

Abstract: Publication date: Available online 10 July 2017 Source:Comptes Rendus Mathematique Author(s): Antoine Clais In this note, we use some combinatorial modulus on the boundary of a right-angled hyperbolic building to control its conformal dimension. The lower bound obtained is optimal in the case of Fuchsian buildings.

Abstract: Publication date: Available online 4 July 2017 Source:Comptes Rendus Mathematique Author(s): Enchao Bi, Zhenhan Tu The Cartan–Hartogs domains are defined as a class of Hartogs-type domains over irreducible bounded symmetric domains. For a Cartan–Hartogs domain Ω B ( μ ) endowed with the natural Kähler metric g ( μ ) , Zedda conjectured that the coefficient a 2 of the Rawnsley's ε-function expansion for the Cartan–Hartogs domain ( Ω B ( μ ) , g ( μ ) ) is constant on Ω B ( μ ) if and only if ( Ω B ( μ ) , g ( μ ) ) is biholomorphically isometric to the complex hyperbolic space. In this paper, following Zedda's argument, we give a geometric proof of the Zedda's conjecture by computing the curvature tensors of the Cartan–Hartogs domain ( Ω B ( μ ) , g ( μ ) ) .

Abstract: Publication date: Available online 4 July 2017 Source:Comptes Rendus Mathematique Author(s): Yoshikazu Nagata We give a sufficient condition for complex manifolds for automorphism groups to become Lie groups. As an application, we see that the automorphism group of any strictly pseudoconvex domain or finite-type pseudoconvex domain has a Lie group structure.

Abstract: Publication date: Available online 3 July 2017 Source:Comptes Rendus Mathematique Author(s): Mark Malamud, Hagen Neidhardt, Vladimir Peller In this note, we study the problem of evaluating the trace of f ( T ) − f ( R ) , where T and R are contractions on a Hilbert space with trace class difference, i.e. T − R ∈ S 1 , and f is a function analytic in the unit disk D . It is well known that if f is an operator Lipschitz function analytic in D , then f ( T ) − f ( R ) ∈ S 1 . The main result of the note says that there exists a function ξ (a spectral shift function) on the unit circle T of class L 1 ( T ) such that the following trace formula holds: trace ( f ( T ) − f ( R ) ) = ∫ T f ′ ( ζ ) ξ ( ζ ) d ζ , whenever T and R are contractions with T − R ∈ S 1 , and f is an operator Lipschitz function analytic in D .

Abstract: Publication date: Available online 28 June 2017 Source:Comptes Rendus Mathematique Author(s): Bao-Wei Wang, Jun Wu, Jian Xu In this note, we consider the metric theory of the dynamical covering problems on the triadic Cantor set K . More precisely, let T x = 3 x ( mod 1 ) be the natural map on K , μ the standard Cantor measure and x 0 ∈ K a given point. We consider the size of the set of points in K which can be well approximated by the orbit { T n x 0 } n ≥ 1 of x 0 , namely the set D ( x 0 , φ ) : = { y ∈ K : T n x 0 − y < φ ( n ) for infinitely many n ∈ N } , where φ is a positive function defined on N . It is shown that for μ almost all x 0 ∈ K , the Hausdorff measure of D ( x 0 , φ ) is either zero or full depending upon the convergence or divergence of a certain series. Among the proof, as a byproduct, we obtain an inhomogeneous counterpart of Levesley, Salp and Velani's work on a Mahler's question about the Diophantine approximation on the Cantor set K .

Abstract: Publication date: Available online 28 June 2017 Source:Comptes Rendus Mathematique Author(s): Liangang Ma For a real x ∈ ( 0 , 1 ) ∖ Q , let x = [ a 1 ( x ) , a 2 ( x ) , ⋯ ] be its continued fraction expansion. Denote by T n ( x ) : = max { a k ( x ) : 1 ≤ k ≤ n } the maximum partial quotient up to n. For any real α ∈ ( 0 , ∞ ) , γ ∈ ( 0 , ∞ ) , let F ( γ , α ) : = { x ∈ ( 0 , 1 ) ∖ Q : lim n → ∞ T n ( x ) e n γ = α } . For a set E ⊂ ( 0 , 1 ) ∖ Q , let dim H E be its Hausdorff dimension. Recently, Lingmin Liao and Michal Rams showed that dim H F ( γ , α ) = { 1 i f γ ∈ ( 0 , 1 / 2 ) 1 / 2 i f γ ∈ ( 1 / 2 , ∞ ) for any α ∈ ( 0 , ∞ ) . In this paper, we show that dim H F ( 1 / 2 , α ) = 1 / 2 for any α ∈ ( 0 , ∞ ) following Liao and Rams' method, which supplements their result.

Abstract: Publication date: Available online 27 June 2017 Source:Comptes Rendus Mathematique Author(s): Francesco Della Pietra, Nunzia Gavitone, Gianpaolo Piscitelli We prove an optimal lower bound for the best constant in a class of weighted anisotropic Poincaré inequalities.

Abstract: Publication date: Available online 20 June 2017 Source:Comptes Rendus Mathematique Author(s): Pierre Cardaliaguet, Panagiotis E. Souganidis We prove, under some assumptions, the existence of correctors for the stochastic homogenization of “viscous” possibly degenerate Hamilton–Jacobi equations in stationary ergodic media. The general claim is that, assuming knowledge of homogenization in probability, correctors exist for all extreme points of the convex hull of the sublevel sets of the effective Hamiltonian. Even when homogenization is not a priori known, the arguments imply the existence of correctors and, hence, homogenization in some new settings. These include positively homogeneous Hamiltonians and, hence, geometric-type equations including motion by mean curvature, in radially symmetric environments and for all directions. Correctors also exist and, hence, homogenization holds for many directions for nonconvex Hamiltonians and general stationary ergodic media.

Abstract: Publication date: Available online 20 June 2017 Source:Comptes Rendus Mathematique Author(s): Vivette Girault, L. Ridgway Scott We modify an argument of Renardy proving existence and regularity for a subset of a class of models of non-Newtonian fluids suggested by Oldroyd, including the upper-convected and lower-convected Maxwellian models. We suggest an effective method for solving these models, which can provide a variational formulation suitable for finite element computation.

Abstract: Publication date: Available online 16 June 2017 Source:Comptes Rendus Mathematique Author(s): Siming He, Eitan Tadmor We study the systems of Euler equations that arise from agent-based dynamics driven by velocity alignment. It is known that smooth solutions to such systems must flock, namely the large-time behavior of the velocity field approaches a limiting “flocking” velocity. To address the question of global regularity, we derive sharp critical thresholds in the phase space of initial configuration that characterizes the global regularity and hence the flocking behavior of such two-dimensional systems. Specifically, we prove for that a large class of sub-critical initial conditions such that the initial divergence is “not too negative” and the initial spectral gap is “not too large”, global regularity persists for all time.

Abstract: Publication date: Available online 16 June 2017 Source:Comptes Rendus Mathematique Author(s): Stéphane Ballet, Nicolas Baudru, Alexis Bonnecaze, Mila Tukumuli The Chudnovsky algorithm for the multiplication in extensions of finite fields provides a bilinear complexity uniformly linear with respect to the degree of the extension. Recently, Randriambololona has generalized the method, allowing asymmetry in the interpolation procedure and leading to new upper bounds on the bilinear complexity. In this note, we describe the construction of this asymmetric method without derived evaluation. To do this, we translate this generalization into the language of algebraic function fields and we give a strategy of construction and implementation.

Abstract: Publication date: Available online 13 June 2017 Source:Comptes Rendus Mathematique Author(s): Oana Ivanovici, Gilles Lebeau The purpose of this note is to prove dispersive estimates for the wave and the Schrödinger equations outside strictly convex obstacles in R d . If d = 3 , we show that, for both equations, the linear flow satisfies the (corresponding) dispersive estimates as in R 3 . In higher dimensions d ≥ 4 and if the domain is the exterior of a ball in R d , we show that losses in dispersion do appear and this happens at the Poisson spot.

Abstract: Publication date: Available online 13 June 2017 Source:Comptes Rendus Mathematique Author(s): Mohammed Attouch, Ali Laksaci, Fatima Rafaa We consider the problem of the local linear estimation of the regression operator when the regressor is functional. We construct an estimator by the kNN method and we study its asymptotic properties. Precisely, we establish the almost complete consistency of this estimator with rate both pointwise and uniform on the number of neighbor cases.

Abstract: Publication date: Available online 12 June 2017 Source:Comptes Rendus Mathematique Author(s): Sylvain Carpentier It has been proved by Sokolov that Krichever–Novikov equation's hierarchy is hamiltonian for the Hamiltonian operator H 0 = u x ∂ − 1 u x and possesses two weakly non-local recursion operators of degrees 4 and 6, L 4 and L 6 . We show here that H 0 , L 4 H 0 and L 6 H 0 are compatible Hamiltonians operators for which the Krichever–Novikov equation's hierarchy is hamiltonian.

Abstract: Publication date: Available online 8 June 2017 Source:Comptes Rendus Mathematique Author(s): Benzaghou Benali, Mokhfi Siham We commence by giving a generalisation of Pulita exponential series. We then use these series to establish an analog of the trace formula for Witt vector rings.

Abstract: Publication date: Available online 7 June 2017 Source:Comptes Rendus Mathematique Author(s): Martial Agueh, Guillaume Carlier The notion of Wasserstein barycenters is a natural way to interpolate between several probability measures, useful in various applied settings like image processing or machine learning. We conjecture that such barycenters obey a central limit theorem which we prove in some (very) particular cases.

Abstract: Publication date: Available online 5 June 2017 Source:Comptes Rendus Mathematique Author(s): Gonçalo Tabuada Making use of the recent theory of noncommutative mixed motives, we prove that the Voevodsky's mixed motive of a quadric fibration over a smooth curve is Kimura-finite.

Abstract: Publication date: Available online 31 May 2017 Source:Comptes Rendus Mathematique Author(s): Mustapha Raïssouli, Mohammad Sal Moslehian, Shigeru Furuichi Let A and B be two accretive operators. We first introduce the weighted geometric mean of A and B together with some related properties. Afterwards, we define the relative entropy as well as the Tsallis entropy of A and B. The present definitions and their related results extend those already introduced in the literature for positive invertible operators.

Abstract: Publication date: Available online 31 May 2017 Source:Comptes Rendus Mathematique Author(s): Camille Carvalho, Lucas Chesnel, Patrick Ciarlet We consider a class of eigenvalue problems involving coefficients changing sign on the domain of interest. We describe the main spectral properties of these problems according to the features of the coefficients. Then, under some assumptions on the mesh, we explain how one can use classical finite element methods to approximate the spectrum as well as the eigenfunctions while avoiding spurious modes. We also prove localisation results of the eigenfunctions for certain sets of coefficients.

Abstract: Publication date: Available online 30 May 2017 Source:Comptes Rendus Mathematique Author(s): Robert O'Connor We present a successive constraint approach that makes it possible to cheaply solve large-scale linear matrix inequalities for a large number of parameter values. The efficiency of our method is made possible by an offline/online decomposition of the workload. Expensive computations are performed beforehand, in the offline stage, so that the problem can be solved very cheaply in the online stage. We also extend the method to approximate solutions to semidefinite programming problems.

Abstract: Publication date: Available online 30 May 2017 Source:Comptes Rendus Mathematique Author(s): Philippe G. LeFloch, Jean-Marc Mercier For multi-dimensional Fokker–Planck–Kolmogorov equations, we propose a numerical method which is based on a novel localization technique. We present extensive numerical experiments that demonstrate its practical interest for finance applications. In particular, this approach allows us to treat calibration and valuation problems, as well as various risk measure computations.

Abstract: Publication date: Available online 26 May 2017 Source:Comptes Rendus Mathematique Author(s): Xavier Blanc, Éric Cancès, Mi-Song Dupuy In Kohn–Sham electronic structure computations, wave functions have singularities at nuclear positions. Because of these singularities, plane wave expansions give a poor approximation of the eigenfunctions. The PAW (projector augmented-wave) method circumvents this issue by replacing the original eigenvalue problem by a new one with the same eigenvalues, but smoother eigenvectors. Here a slightly different method, called VPAW (variational PAW), is proposed and analyzed. This new method allows for a better convergence with respect to the number of plane waves. Some numerical results on an idealized case corroborate this efficiency.

Abstract: Publication date: Available online 18 May 2017 Source:Comptes Rendus Mathematique Author(s): Claude Bardos, Kim Dang Phung We study the unique continuation property for the neutron transport equation and for a simplified model of the Fokker–Planck equation in a bounded domain with absorbing boundary condition. An observation estimate is derived. It depends on the smallness of the mean free path and the frequency of the velocity average of the initial data. The proof relies on the well-known diffusion approximation under convenience scaling and on the basic properties of this diffusion. Eventually, we propose a direct proof for the observation at one time of parabolic equations. It is based on the analysis of the heat kernel.

Abstract: Publication date: Available online 10 May 2017 Source:Comptes Rendus Mathematique Author(s): Raphaël Ponge In this note, we produce explicit quasi-isomorphisms computing the cyclic homology of crossed-product algebras associated with group actions on manifolds. We obtain explicit relationships with equivariant cohomology. On the way, we extend the results of the first part to the setting of group actions on locally convex algebras.

Abstract: Publication date: Available online 9 May 2017 Source:Comptes Rendus Mathematique Author(s): Khadijeh Baghaei, Ali Khelghati In this paper, we study the chemotaxis system: { u t = ∇ ⋅ ( ξ ∇ u − χ u ∇ v ) , x ∈ Ω , t > 0 , v t = Δ v − u v , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R n , n ≥ 1 , with smooth boundary. Here, ξ and χ are some positive constants. We prove that the classical solutions to the above system are uniformly in-time-bounded provided that: ‖ v 0 ‖ L ∞ ( Ω ) < { 1 χ ξ 2 ( n + 1 ) [ π + 2 arctan ( ( 1 − ξ ) 2 2 ( n + 1 ) ξ ) ] , if 0 < ξ < 1 , π χ 2 ( n + 1 ) , if ξ = 1 , 1 χ ξ 2 ( n + 1 ) [ π − 2 arctan ( ( ξ − 1 ) 2 2 ( n + 1 ) ξ ) ] , if ξ > 1 . In the case of ξ = 1 , the recent results show that the classical solutions are global and bounded provided that 0 < ‖ v 0 ‖ L ∞ ( Ω ) ≤ 1 6 ( n PubDate: 2017-05-12T17:34:58Z

Abstract: Publication date: Available online 9 May 2017 Source:Comptes Rendus Mathematique Author(s): Michael Ruzhansky, Durvudkhan Suragan, Nurgissa Yessirkegenov In this paper, we give an extension of the classical Caffarelli–Kohn–Nirenberg inequalities with respect to the range of parameters. We also establish best constants for large families of parameters. Moreover, we also obtain anisotropic versions of these inequalities which can be conveniently formulated in the language of Folland and Stein's homogeneous groups. We also establish sharp Hardy type inequalities in L p , 1 < p < ∞ , with superweights.

Abstract: Publication date: Available online 9 May 2017 Source:Comptes Rendus Mathematique Author(s): Raphaël Ponge In this note we produce explicit quasi-isomorphisms computing the cyclic homology of crossed-product algebras.

Abstract: Publication date: Available online 9 May 2017 Source:Comptes Rendus Mathematique Author(s): Cagri Sert The aim of this note is to announce some results about the probabilistic and deterministic asymptotic properties of linear groups. The first one is the analogue, for norms of random matrix products, of the classical theorem of Cramér on large deviation principles (LDP) for sums of iid real random variables. In the second result, we introduce a limit set describing the asymptotic shape of the powers S n = { g 1 . … . g n g i ∈ S } of a subset S of a semisimple linear Lie group G (e.g., SL ( d , R ) ). This limit set has applications, among others, in the study of large deviations.

Abstract: Publication date: Available online 9 May 2017 Source:Comptes Rendus Mathematique Author(s): Ibrahim Ekren, Igor Kukavica, Mohammed Ziane We address the long-time behavior of solutions to damped dispersive stochastic partial differential equations, namely the KdV equation and the nonlinear Schrödinger equation on the whole space. We prove that the transition semigroup is Feller and establish the existence of an invariant measure using the asymptotic compactness property of the transition semigroup and the Aldous criterion.

Abstract: Publication date: Available online 2 May 2017 Source:Comptes Rendus Mathematique Author(s): Grégoire Allaire, Charles Dapogny, Alexis Faure, Georgios Michailidis The purpose of this article is to introduce a new functional of the domain, to be used in shape optimization problems as a means to enforce the constructibility of shapes by additive manufacturing processes. This functional aggregates the self-weights of all the intermediate structures of the shape appearing in the course of its layer-by-layer assembly. Its mathematical analysis is performed and an algorithm is proposed to accelerate the significant computational effort entailed by the implementation of these ideas. Eventually, a numerical validation and a concrete example are discussed.

Abstract: Publication date: Available online 26 April 2017 Source:Comptes Rendus Mathematique Author(s): Nabiullah Khan, Talha Usman, Junesang Choi Many extensions and variants of the so-called Apostol-type polynomials have recently been investigated. Motivated mainly by those works and their usefulness, we aim to introduce a new class of Apostol-type Laguerre–Genocchi polynomials associated with the modified Milne–Thomson's polynomials introduced by Derre and Simsek and investigate its properties, including, for example, various implicit formulas and symmetric identities in a systematic manner. The new family of polynomials introduced here, being very general, contains, as its special cases, many known polynomials. So the properties and identities presented here reduce to yield those results of the corresponding known polynomials.