Let dk,?(n) be the function number of divisors of the integer n?1, in arithmetic progressions {?+mk}, with 1???k and ?,k coprime, and let F(n;k,?) defined as follows: In this Note, we study and give the structure of dk,?-superior, highly composite numbers, which generalize those defined by S. Ramanujan. We prove that F(n;k,?) reaches its maximum among these numbers. We give it explicitly for k=2,…,13. This generalizes the study of Nicolas and Robin, in which the case k=1 is treated. To cite this article: A. Derbal, A. Smati, C. R. Acad. Sci. Paris, Ser. I 339 (2004).相似文献
Soit la fonction somme des diviseurs unitaires du nombre entier n dans la progression arithmétique définie, pour , par : . Dans cette Note nous établissons un théorème sur le comportement relatif de cette fonction et de son ordre maximal qui sera explicitement déterminé et nous donnons des majorations effectives de . Pour citer cet article : A. Derbal, C. R. Acad. Sci. Paris, Ser. I 342 (2006).相似文献
We consider a real Gaussian process X with unknown smoothness where the mean-square derivative is supposed to be Hölder continuous in quadratic mean. First, from the discrete observations , we study reconstruction of , , with , a piecewise polynomial interpolation of degree . We show that the mean-square error of interpolation is a decreasing function of r but becomes stable as soon as . Next, from an interpolation-based empirical criterion, we derive an estimator of and prove its strong consistency by giving an exponential inequality for . Finally, we prove the strong convergence of toward with a similar rate as in the case ‘ known’. To cite this article: D. Blanke, C. Vial, C. R. Acad. Sci. Paris, Ser. I 343 (2006).相似文献
Let be the exponent that measures how a non-quadratic real number ξ and its square can be simultaneously approximated by rational numbers with the same denominator. Davenport and Schmidt have proved that is always between the golden ratio γ and 2. Roy, and after him Bugeaud and Laurent, have constructed numbers ξ such that . Their method involves infinite words with many palindrome prefixes. In this text, we define new exponents of approximation that allow us to obtain, to some extent, a characterization of the values obtained by these authors. To cite this article: S. Fischler, C. R. Acad. Sci. Paris, Ser. I 339 (2004).相似文献
We present in this note a posteriori error estimates based on the postprocessing technique for the reduced model of flow in fractured porous media, introduced and analysed by V. Martin, J. Jaffré, and J. Roberts. This model is approximated by the Raviart–Thomas finite elements of lowest order. In this type of approximation, the velocity is well approximated. A postprocessing of the pressure appears to be necessary since it does not belong to . We give an upper bound for the error in the energy norm, with some indicators that are expressed in terms of the reconstruction of the pressure. Numerical results show that all indicators converge to zero when the mesh size goes to zero, with the same speed as the error. One of these indicators can be interpreted as both a discretization indicator and an indicator of the reduced model validity. 相似文献
We consider a composite material constituted of carbon or glass fibres included in a resin which becomes solid when it is heated up (reaction of reticulation). The mathematical modelling of the cure process is given by a kinetic equation describing the evolution of the reaction of reticulation coupled with the heat equation. The geometry of the composite material is periodic, with a small period . First we prove the existence and uniqueness of a solution by using Schauder's fixed point theorem. Then, by using an asymptotic expansion, we derive the homogenized problem which describes the macroscopic behaviour of the material. We prove the convergence of the solution of the problem to the solution of the homogenized problem when ? tends to zero and we obtain an error estimate in a case of weak non-linearity. Finally we solve numerically the homogenized problem. To cite this article: S. Meliani et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).相似文献
A theorem of Baker says that a function F entire on ?d such that F(?d) ? ? and increasing slower (in a precise sense) than \(2^{z_1 + \cdots + z_d } \) is necessarily a polynomial. This is a multivariate generalisation of the celebrated theorem of Pólya (case d = 1). Using the theory of analytic functionals with non-compact carrier, Yoshino proved a general theorem dealing with the growth of arithmetic analytic functions, which implies that the conclusion of Baker’s theorem holds if F is only assumed to be holomorphic on the domain
The case d = 1 was also treated in a different way by Gel’fond and Pólya by means of the characteristic function of Carlson-Nörlund. This function was introduced to bound in a nearly optimal way the growth of holomorphic functions of one variable that can be expanded in a Newton interpolation series in the half-plane
that can be expanded in multiple Newton series. These considerations enable us to improve Gel’fond-Pólya’s and Yoshino’s theorems, in particular, to remove or to weaken certain of their technical conditions.
In this Note, we prove a theorem on the extension of a negative (or positive) plurisubharmonic current T (i.e. such that ) with condition on the slices with respect to some coordinates. This theorem generalizes a result proved by El Mir–Ben Messaoud relative to d-closed positive currents with a condition on slices. The method consists first of proving a Chern–Levine–Nirenberg inequality for a positive (or negative) psh current, which is a generalization of results obtained by Bedford–Taylor, Demailly and Sibony for d-closed positive currents. Also we prove an Oka type inequality for positive psh currents, thereby generalizing former results by Ben Messaoud–El Mir concerning positive currents with a negative . To cite this article: M. Toujani, H. Ben Messaoud, C. R. Acad. Sci. Paris, Ser. I 339 (2004).相似文献
Recent works are devoted to the rational smoothness of varieties of representations for quivers, notably those of Robert Bédard, Ralf Schiffler and Philippe Caldéro. Their main result is the equivalence for a variety of representations of a quiver of Dynkin type, between This Note gives a different approach to the question and a strengthened version of Bédard–Schiffler–Caldéro's equivalences. To cite this article: A. Arabia, C. R. Acad. Sci. Paris, Ser. I 338 (2004).相似文献
The Lévy transform of a Brownian motion B is the Brownian motion ; denote by Bn the Brownian motion obtained from B by iterating n times the Lévy transform. We establish that the set of all instants t such that Btn=0 for some n, is a.s. dense in the time-axis . To cite this article: M. Malric, C. R. Acad. Sci. Paris, Ser. I 336 (2003).相似文献
Let G be a non-Abelian, connected, nilpotent Lie group. Then there exist and such that , contrary to what happens for the group . Moreover, the set of zero divisors is a total subset of . This result is first proven for the Heisenberg group where it is based on the existence of non-trivial Schwartz functions f satisfying for . To cite this article: J. Ludwig et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).相似文献
In this paper we prove a global well-posedness result for the following Cauchy problem: where the initial data are compactly supported, 1?α<5, , . To cite this article: N. Visciglia, C. R. Acad. Sci. Paris, Ser. I 338 (2004).相似文献
Let be a closed, non compact C2 manifold and be a C2 function definable in an o-minimal structure. We prove that the flow of the gradient field of f with respect to the induced riemannian metric on V embeds a non singular asymptotic critical level of f into a typical level of f. We apply this result to complex polynomials. To cite this article: D. D'Acunto, C. R. Acad. Sci. Paris, Ser. I 337 (2003).相似文献
We study the problem of the nonparametric estimation of a probability density in . Expressing the mean integrated squared error in the Fourier domain, we show that it is close to the mean squared error in the Gaussian sequence model. Then, applying a modified version of Stein's blockwise method, we obtain a linear monotone oracle inequality and a kernel oracle inequality. As a consequence, the proposed estimator is sharp minimax adaptive (i.e. up to a constant) on a scale of Sobolev classes of densities. To cite this article: Ph. Rigollet, C. R. Acad. Sci. Paris, Ser. I 340 (2005).相似文献