A theorem concerning uniform estimates of oscillatory integrals when the phase is a function of two variables

In the article it is shown that an oscillatory integral with phase depending on two variables has an upper estimate for small parameters which is proportional to the value of the integral when the parameters are zero. An analogous uniform estimate is obtained for volumes. V. I. Arnold's conjecture about the semicontinuity of the height for functions of two variables is also proved.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 10, pp. 150–169, 1984.The author is grateful to V. I. Arnol'd for posing the problem and for his advice with the work, and also to A. N. Varchenko for numerous valuable discussions.     

The First Power Mean of the Inversion of L-Functions and General Kloostermann Sums

 The main purpose of this paper is using the estimate for classical Kloostermann sums and E. Bombieri–A. I. Vinogradov’s important work to study the first power mean of the inversion of Dirichlet L-functions with the weight of general Kloostermann sums, and give an interesting asymptotic formula.     

The First Power Mean of the Inversion of L-Functions and General Kloostermann Sums

 The main purpose of this paper is using the estimate for classical Kloostermann sums and E. Bombieri–A. I. Vinogradov’s important work to study the first power mean of the inversion of Dirichlet L-functions with the weight of general Kloostermann sums, and give an interesting asymptotic formula. Received 30 March 2001; in revised form 12 November 2001     

Un résultat de stabilité pour la récupération d'un paramètre du système de la viscoélasticité 3D

In this Note, we prove a Carleman's estimate for the integro-differential hyperbolic system of the viscoelasticity problem and we use this estimate to obtain a stability result for the inverse problem of recovering a viscoelastic coefficient from a unique internal measure. To cite this article: M. de Buhan, A. Osses, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Carleman estimates for second-order hyperbolic systems in anisotropic cases and applications. Part II: an inverse source problem

In this paper, we consider Carleman-type estimate and consider an inverse problem for second order hyperbolic systems in an anisotropic case. In the previous Part I paper, we established a Carleman-type estimate for hyperbolic systems in which the coefficient matrices satisfy suitable conditions. We apply a Carleman estimate in the previous Part I paper to an inverse source problem for second-order hyperbolic systems in an anisotropic case and prove an estimate of the Hölder type.     

Variational principle and asymptotic behavior of the density of states for random pseudodifferential operators

A variational principle (similar to Glazman's lemma) is proved for the density of the states of random self-adjoint hypoelliptic operators (of positive or negative order). Some results from the index theory of random elliptic operators are used in the proof. As an application, the asymptotic behavior of the density of the states of such operators and an estimate of the remainder are obtained by the method of the approximate spectral projection.The authors are grateful to V. I. Bezyaev for numerous useful discussions.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 11, pp. 98–117, 1986.     

Estimate for the difference of the first derivatives of partial sums arising by non-selfadjoint Sturm-Liouville operators

In this paper we prove an exact estimate for the difference of the first derivatives of the partial sums arising by non-selfadjoint Sturm-Liouville operators. Our result extends to that of [3] for the case when the eigenvalues may be arbitrary complex numbers. The present proof is based on an efficient method due to V. A. Il'in [6], [7].The author is indebted to Dr. I. Joó and Dr. V. Komornik for their valuable remarks during the preparation of this paper.     

Estimate for the norm of matrix-valued functions

New precise estimates for a norm of matrix-valued functions are obtained. These estimates improve I. Gel'fand's and G. Shilov's estimate for regular functions of matrices and carleman's estimate for resolvents. Applications to differential equations and to spectrum perturbations are considered.     

On the ground state energy for a magnetic Schrödinger operator and the effect of the de Gennes boundary condition

Motivated by the Ginzburg–Landau theory of superconductivity, we estimate the ground state energy of a magnetic Schrödinger operator with de Gennes boundary condition in the semi-classical limit and we study the localization of the corresponding ground states. We exhibit cases when the de Gennes boundary condition has a strong effect on this localization. To cite this article: A. Kachmar, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

与I-超滤相关的基数不变量

我们对自然数ω上的每一个理想I引入了一个新的基数不变量non^**(I). 我们证明相应的I-超滤的兼纳存在性可以用non^**(I)与连续统c的等式来刻画. 具体地, 我们有如下的结果:(1) 如果non^**(I)=c, 那么任何一个由小于c个集合生成的滤子都包含在某个I-超滤中.(2) 存在一个滤子刚好可以由non^**(I)个集合生成, 但不包含在任何一个I-超滤中.(3) 任何一个由小于non^**(I)个集合生成的超滤一定是I-超滤.以上的结果是相应的P-点和Ramsey超滤的经典结论的一个推广. 我们将对一些具体的理想, 确定non^**(I)的大小, 具体地, 我们得到了non^**(Fin×Fin)=∂, non^**(εD)=cov(M)以及对不满足Fin-BW性质的理想I, 都有non^**(I)>∂.     

Estimation fonctionnelle de la densité de l'opérateur de transition d'un processus de Markov à temps discretFunctional estimate of Markov transition operator density: Discrete time case

In this study, we build a kernel estimate of the transition operator density for some discrete time continuous states Markov processes which satisfy some general conditions. Assumptions are not restrictive and results can be used on transition Markov operator, viewed as an endomorphism on L^p, p∈[1,∞[, as well as on its adjoint. The main result deals with convergence rate of built kernel estimate. To cite this article: A. Laksaci, A. Yousfate, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1035–1038.     

Bernstein–Frechet inequalities for the parameter of the first order autoregressive process

The autoregressive process takes an important part in predicting problems leading to decision making. In practice, we use the least squares method to estimate the parameter of the autoregressive process. In the case of the first order autoregressive process, we know that the least squares estimator converges in probability to the unknown parameter θ. In this Note, we show that the least squares estimator converges almost completely to θ and so we construct the inequalities of type Bernstein–Frechet for the coefficient of the first order autoregressive process. Using these inequalities a confidence interval is then obtained. To cite this article: A. Dahmani, M. Tari, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

On the geometry of null cones in Einstein-vacuum spacetimes

In this paper we study the geometry of null cones in smooth Einstein vacuum spacetimes. We provide the L^∞$L^{\infty }$ estimate for the trace of the null second fundamental form, as well as estimates for other geometric quantities. This paper is based on the work of Klainerman and Rodnianski [S. Klainerman, I. Rodnianski, Causal geometry of Einstein-vacuum spacetimes with finite curvature flux, Invent. Math. 159 (3) (2005) 437–529; S. Klainerman, I. Rodnianski, Sharp trace theorems for null hypersurfaces on Einstein metrics with finite curvature flux, Geom. Funct. Anal. 16 (1) (2006) 164–229; S. Klainerman, I. Rodnianski, A geometric Littlewood–Paley theory, Geom. Funct. Anal. 16 (1) (2006) 126–163].     

Variation of the instability index on stationary curves of elliptic equations depending on a parameter

The set of solutions of the equation A(u, )=0 in the case of general position consists of smooth curves. Solutions of a quasilinear elliptic equation with large instability index are constructed and an estimate obtained for the number of turning points of the solution curves.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 12, pp. 47–58, 1987.     

Convergence rate for predictive recursion estimation of finite mixtures

Predictive recursion (PR) is a fast stochastic algorithm for nonparametric estimation of mixing distributions in mixture models. It is known that the PR estimates of both the mixing and mixture densities are consistent under fairly mild conditions, but currently very little is known about the rate of convergence. Here I first investigate asymptotic convergence properties of the PR estimate under model misspecification in the special case of finite mixtures with known support. Tools from stochastic approximation theory are used to prove that the PR estimates converge, to the best Kullback-Leibler approximation, at a nearly root-n rate. When the support is unknown, PR can be used to construct an objective function which, when optimized, yields an estimate of the support. I apply the known-support results to derive a rate of convergence for this modified PR estimate in the unknown support case, which compares favorably to known optimal rates.     

The lower estimate for linear positive operators,I

A method to prove lower estimates for linear operators is introduced. As a result the best lower estimate for certain convolution operators, for the multivariate Bernstein-Durrmeyer operators in part I and the Bernstein polynomial operators in part II (see [10]), are obtained.Communicated by Hubert Berens     

Jackson-type inequality on the sphere

In a recent paper, I introduced new moduli of smoothness for functions on the sphere which did not use averages and, as a result, had some interesting properties. The direct, Jackson-type, estimate of the best approximation by spherical harmonics using the new moduli will be proved here. Equivalence with the appropriate K-functionals will be given. Relations with the moduli used earlier will be shown and used to prove new results for these moduli.     

Time decay of maxwell—klein—gordon equations in 4—dimensional minkowski space

Abstract. In this paper I derive a gauge invariant decay estimate of the solutions of massive Maxwell—Klein—Gordon fields equations in the 4—dimensional Minkowski space, provided that the initial energy of the system is bounded. This estimate implies that the Klein—Gordon field decays to zero in the local L² norm. I also show that the local energy decays. The proof is based on gauge invariant energy identities.     

A twisted theorem of Chebotarev

The classical function field version of Chebotarev's Theorem follows from the Lang–Weil estimate and an ‘untwisting’ trick. We obtain an analogue in the framework of difference schemes, using Hrushovski's twisted Lang–Weil estimate. To cite this article: I. Toma?i?, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Maximum likelihood estimation for density as an infinite-dimensional Gaussian shift

A new approach is suggested for the nonparametric estimation of the unknown distribution density under the assumption of the bounded variation of the true density. As an estimator there occurs a statistic based on the application of the maximum likelihood method to the estimation of an infinite-dimensional shifi of a Gaussian process with a known correlation function. The quality of the obtained estimate is investigated.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 177, pp. 6–7, 1989.