Dispersion and Strichartz estimates for the Liouville equation

We consider the Liouville equation associated to a metric g and we prove dispersion and Strichartz estimates for the solution of this equation in terms of the geometry of the trajectories associated to g. In particular, we obtain global Strichartz estimates in time for metrics where dispersion estimate is false even locally in time. We also study the analogy between Strichartz estimates obtained for the Liouville equation and the Schrödinger equation with variable coefficients. To cite this article: D. Salort, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

General formulas for the smoothed analysis of condition numbers

We provide estimates on the volume of tubular neighborhoods around a subvariety Σ of real projective space, intersected with a disk of radius σ. The bounds are in terms of σ, the dimension of the ambient space, and the degree of equations defining Σ. We use these bounds to obtain smoothed analysis estimates for some conic condition numbers. To cite this article: P. Bürgisser et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Error estimates for three-dimensional Stokes problem with non-standard boundary conditions

This work is devoted to the optimal and a posteriori error estimates of the Stokes problem with some non-standard boundary conditions in three dimensions. The variational formulation is decoupled into a system for the velocity and a Poisson equation for the pressure. The velocity is approximated with curl conforming finite elements and the pressure with standard continuous elements. Next, we establish optimal a posteriori estimates.     

Interior error estimate for periodic homogenization

The aim of this Note is to give interior error estimates for problems in periodic homogenization, by using the periodic unfolding method. The interior error estimates are obtained by transposition without any supplementary hypothesis of regularity on correctors. This error is of order ?. To cite this article: G. Griso, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Efficiencies for window estimates of the parameters of the Cauchy distribution

Higgins and Tichenor [Appl. Math. and Comp. 3 (1977), 113-126] considered “window estimates” of location and reciprocal scale parameters for a general class of distributions and showed them to be asymptotically efficient for the Cauchy distribution. In this study, efficiencies of these estimates for the Cauchy distribution are investigated for small and moderate sample sizes by Monte Carlo methods. For n?40, window estimates of location are nearly optimal, and for n?20, they compare favorably with other easy-to-compute estimates. Window estimates of reciprocal scale are very good even for small samples and are nearly optimal for n?10. Thus, window estimates appear to have high efficiency for moderate as well as large sample sizes. Approximate normality is also investigated. The estimate of location converges rapidly to normality, whereas the estimate of reciprocal scale does not.     

On convergence of LAD estimates in autoregression with infinite variance

The least absolute deviation estimates L(N), from N data points, of the autoregressive constants a = (a₁, …, a_q)′ for a stationary autoregressive model, are shown to have the property that N^σ(L(N) ? a) converge to zero in probability, for $\text{σ <}\text{1}\text{α}$, where the disturbances are i.i.d., attracted to a stable law of index α, 1 ≤ α < 2, and satisfy some other conditions.     

An integral estimate and the equivalent norms on F(p,q, s,k) spaces in the unit ball

In this article, the authors give a typical integral's bidirectional estimates for all cases. At the same time, several equivalent characterizations on the F(p, q, s, k) space in the unit ball of Cⁿ are given.     

Growth estimates for solutions to initial-boundary value problems in viscoelasticity

For the abstract Volterra integro-differential equation u_tt ? Nu + ∝_?∞^t K(t ? τ) u(τ) dτ = 0 in Hilbert space, with prescribed past history u(τ) = U(τ), ? ∞ < τ < 0, and associated initial data u(0) = f, u_t(0) = g, we establish conditions on K(t), ? ∞ < t < + ∞ which yield various growth estimates for solutions u(t), belonging to a certain uniformly bounded class, as well as lower bounds for the rate of decay of solutions. Our results are interpreted in terms of solutions to a class of initial-boundary value problems in isothermal linear viscoelasticity.     

Remarks on estimating upper limit of norm of delayed connection weight matrix in the study of global robust stability of delayed neural networks

The question of estimating the upper limit of the norm ∥B∥₂ of the delayed connection weight matrix B, which is a key step in some recently reported global robust stability criteria for delayed neural networks (DNNs), is considered. An estimate of the upper limit of ∥B∥₂ was previously given by Cao, Huang and Qu. More recently Singh has presented an alternative estimate. Presently it is shown that an estimate of the upper limit of ∥B∥₂ may be found in some cases, which would be an improvement over each of the above-mentioned two estimates. Some observations concerning the determination of the least conservative upper limit of ∥B∥₂ are presented.     

Scaling-sharp dispersive estimates for the Korteweg–de Vries group

We prove weighted estimates on the linear KdV group, which are scaling sharp. This kind of estimates is in the spirit of that used to prove small data scattering for the generalized KdV equations. To cite this article: R. Côte, L. Vega, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

P-convergence des TRA estimateurs : modèle MA(q)P-convergence of the TRA estimates: the MA(q) model

The TRA estimates are introduced in the case of a moving average model in the presence of additive outliers. We establish, under some hypothesis, the consistency of these estimates. To cite this article: N. Berrahou, K. El Himdi, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 549–552.     

Marches aléatoires sur certains groupes unimodulaires p-adiques

We give central and off-diagonal estimates for the transition kernels corresponding to simple random walks on certain unimodular solvable p-adic groups. To cite this article: S. Mustapha, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Phragmén-Lindelöf Theorems and p-harmonic Measures for Sets Near Low-dimensional Hyperplanes

We prove estimates of a p-harmonic measure, p∈(n?m,∞], for sets in Rⁿ which are close to an m-dimensional hyperplane Λ?Rⁿ, m∈[0,n?1]. Using these estimates, we derive results of Phragmén-Lindelöf type in unbounded domains Ω?Rⁿ?Λ for p-subharmonic functions. Moreover, we give local and global growth estimates for p-harmonic functions, vanishing on sets in Rⁿ, which are close to an m-dimensional hyperplane.     

Regularity of averages over hypersurfaces

Averages over smooth measures on smooth compact hypersurfaces inR ⁿ are studied. With assumptions on the decay of the Fourier transform of the measure we obtain mixed norm estimates for these means, for exampleL ^p estimates of multiparameter maximal functions over compact hypersurfaces.     

Local gradient estimates of solutions to some conformally invariant fully nonlinear equations

We establish local gradient estimates to solutions of general conformally invariant fully nonlinear second order elliptic equations. To cite this article: Y.Y. Li, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Rank procedures for some two-population multivariate extended classification problems

Given independent samples from three multivariate populations with cumulative distribution functions F⁽¹⁾(x), F⁽²⁾(x), and F⁽⁰⁾(x) = θF⁽¹⁾(x) + (1 ? θ)F⁽²⁾(x), where 0 ≤ θ ≤ 1 is unknown, the three-action problem involving decision as to whether the value of θ is high, low, or intermediate, is considered. A class of consistent procedures based on the relative spacing of three sample averages of linearly compounded rank scores is formulated. The asymptotic operating characteristics of the procedures when F⁽¹⁾ and F⁽²⁾ come close together are studied and the best choice of the compounding coefficients in terms of these considered. The consequence of using estimates of the best coefficients on the asymptotic operating characteristics is also examined.     

Équation des ondes sur les espaces symétriques riemanniens

We prove that the solutions of the homogeneous wave equation on Riemannian symmetric spaces have dispersion properties and we deduce Strichartz type estimates for these solutions. To cite this article: A. Hassani, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Étude de la consistance d'un algorithme stochastique dans un cadre mélangeant

This Note is devoted to the study of the consistency of a stochastic algorithm having for object the estimates of a parameter under strong mixing assumption. To cite this article: A. Ait Saidi, A. Dahmani, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Equivalence estimates for a class of singular perturbation problems

We give some equivalence estimates on the solution of a singular perturbation problem that represents, among other models, the Koiter and Naghdi shell models. Two of the estimates apply to intermediate shell problems and the third is for membrane/shear dominated shells. From these equivalences, many known and some new sharp estimates on the solutions of the singular perturbation problems easily follow. To cite this article: S. Zhang, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Convergence analysis of finite element methods for H(curl; Ω)-elliptic interface problems

In this article we investigate the analysis of a finite element method for solving H(curl; ??)-elliptic interface problems in general three-dimensional polyhedral domains with smooth interfaces. The continuous problems are discretized by means of the first family of lowest order Nédélec H(curl; ??)-conforming finite elements on a family of tetrahedral meshes which resolve the smooth interface in the sense of sufficient approximation in terms of a parameter ?? that quantifies the mismatch between the smooth interface and the triangulation. Optimal error estimates in the H(curl; ??)-norm are obtained for the first time. The analysis is based on a ??-strip argument, a new extension theorem for H ¹(curl; ??)-functions across smooth interfaces, a novel non-standard interface-aware interpolation operator, and a perturbation argument for degrees of freedom for H(curl; ??)-conforming finite elements. Numerical tests are presented to verify the theoretical predictions and confirm the optimal order convergence of the numerical solution.