Étude asymptotique locale de l'estimateur par ondelettesA law of the iterated logarithm for wavelet density estimator

We state a pointwise central limit theorem for the linear wavelet density estimator in a more general setting than the result of Wu [12]. Furthermore, we also give a pointwise law of the iterated logarithm for this density estimator. Our proof of the law of the iterated logarithm uses the results of Mason [9] on the asymptotic behavior of the tail empirical process. To cite this article: A. Massiani, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 553–556.     

Asymptotic distribution of a Pickands-type estimator of the extreme-value index

One of the main goals of extreme-value analysis is to estimate the probability of rare events given a sample from an unknown distribution. The upper tail behavior of this distribution is described by the extreme-value index ξ. The aim of this Note is to establish the asymptotic distribution of the estimator of $\xi \in \mathbb{R}$ introduced in Gardes and Girard [A Pickands-type estimator of the extreme-value index, Technical Report LMC-IMAG, RR-1063, 2004]. We also give its rate of convergence in some typical situations. To cite this article: L. Gardes, S. Girard, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Waves in a thin and periodically oscillating medium

We study the asymptotic behavior of the spectrum of an elliptic operator with periodically oscillating coefficients, in a thin domain, with vanishing Dirichlet conditions. Two cases are treated: the case where the periodicity of the oscillations and the thickness of the domain have the same order of magnitude and the case where the oscillations have a frequency much greater than the thickness of the domain. A physical motivation can be to understand the behavior of the probability density associated to the wave function of a particle confined to a very thin domain, with periodically varying characteristics. To cite this article: R. Ferreira, M.L. Mascarenhas, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

A formula for the Mahler measure of axy+bx+cy+d

We introduce a formula for the Mahler measure of axy+bx+cy+d with complex coefficients a,b,c, and d and give examples which demonstrate a connection with L-functions. We then prove a generalization of Maillot's formula when the coefficients are real. Next, we discuss operations on the coefficients which fix the Mahler measure. Finally, we prove an alternate formulation of the main result in order to calculate the Mahler measure of a two-parameter family of polynomials in three variables.     

Asymptotics of Green functions and Martin boundaries for elliptic operators with periodic coefficients

We give the asymptotics at infinity of a Green function for an elliptic equation with periodic coefficients on R^d. Basic ingredients in establishing the asymptotics are an integral representation of the Green function and the saddle point method. We also completely determine the Martin compactification of R^d with respect to an elliptic equation with periodic coefficients by using the exact asymptotics at infinity of the Green function.     

Large gains in the St. Petersburg game

We consider the accumulated gains of geometric size in the St. Petersburg game and study the logarithmic tail asymptotics of their distribution. To cite this article: G. Stoica, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Comportement à très basses énergies de la densité d'états intégrée du modèle d'Anderson non borné inférieurement

We obtain a complete asymptotic expansion of the integrated density of states of the unbounded Anderson model at low energies. We also study the evolution of this asymptotic when the decay of the tail of the distribution of the random potential increases. To cite this article: O. Saad, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On a property of tan x

In a paper of Hirzebruch (Ann. of Math.60 (1954), 213–235) the following number-theoretical question occurs in a problem (Problem 5, p. 217): Determine those coefficients in the Taylor expansion of tan xwhose reciprocals are inZ. We give here the complete solution.In 1960, Adams solved Problem 5 along algebraic-topological lines so that the question about tan x was no longer important for this problem.     

Propagation de flammes gazeuses dans la limite d'une diffusion massique nullePropagation of gaseous flames in the limit of small mass diffusion coefficients

In this Note, we study a thermo-diffusive model for the propagation of 1D gaseous flames for complex chemistry with open graph. We use a new approach in order to prove the existence of a traveling wave and we study the behavior of the solution in the limit of zero mass diffusion coefficients: we show the continuity of the wave velocity and of the temperature and mass fraction profiles. To cite this article: F. Laurent et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 405–410.     

Effets d'anisotropie par homogénéisation dans un problème à frontière libre

The Elrod–Adams model is a pressure–saturation formulation which takes into account cavitation phenomena in thin fluid films mechanics. We study the asymptotic behavior of the model, using the two-scale convergence technique, in devices such as journal bearings. Interest is highly motivated by the roughness effects of the surfaces. Anisotropic effects appear on the coefficients, which is usual, but also on the saturation function, since various saturation functions appear through the homogenization process. To cite this article: G. Bayada et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

The tail of a quantum spin network

The tail of a sequence \(\{P_n(q)\}_{n \in \mathbb {N}}\) of formal power series in \(\mathbb {Z}[[q]]\) is the formal power series whose first n coefficients agree up to a common sign with the first n coefficients of \(P_n\). This paper studies the tail of a sequence of admissible trivalent graphs with edges colored n or 2n. We use local skein relations to understand and compute the tail of these graphs. We also give product formulas for the tail of such trivalent graphs. Furthermore, we show that our skein theoretic techniques naturally lead to a proof for the Andrews–Gordon identities for the two-variable Ramanujan theta function as well to corresponding identities for the false theta function.     

Tail dependence between order statistics

In this work, we introduce the s,k-extremal coefficients for studying the tail dependence between the s-th lower and k-th upper order statistics of a normalized random vector. If its margins have tail dependence then so do their order statistics, with the strength of bivariate tail dependence decreasing as two order statistics become farther apart. Some general properties are derived for these dependence measures which can be expressed via copulas of random vectors. Its relations with other extremal dependence measures used in the literature are discussed, such as multivariate tail dependence coefficients, the coefficient η of tail dependence, coefficients based on tail dependence functions, the extremal coefficient ?, the multivariate extremal index and an extremal coefficient for min-stable distributions. Several examples are presented to illustrate the results, including multivariate exponential and multivariate Gumbel distributions widely used in applications.     

On linear forms with algebraic coefficients and diophantine equations

In my paper, [Man. Math.18 (1976), Satz 1.1] I proved a result on simultaneous diophantine inequalities for p-adic linear forms with algebraic coefficients. In this paper I shall generalize this result and give a necessary and sufficient criterion for the estimation of a product of complex and p-adic linear forms with algebraic coefficients, implying a theorem of Schmidt, [Math. Ann.191 (1971), Satz 1]. Using this estimate I shall obtain the p-adic generalization of Schmidt's theorems on diophantine equations of norm form type [Ann. of Math.96 (1972)].     

The Critical Case of the Cramer-Lundberg Theorem on the Asymptotic Tail Behavior of the Maximum of a Negative Drift Random Walk

We study the asymptotic tail behavior of the maximum M = max{0,S _n ,n ≥ = 1} of partial sums S _n = ξ₁ + ? + ξ _n of independent identically distributed random variables ξ₁,ξ₂,... with negative mean. We consider the so-called Cramer case when there exists a β > 0 such that E e ^βξ1 = 1. The celebrated Cramer-Lundberg approximation states the exponential decay of the large deviation probabilities of M provided that Eξ₁ e ^βξ1 is finite. In the present article we basically study the critical case Eξ₁ e ^βξ1 = ∞.     

Accélération de schémas d'intégration temporelle pour les équations différentielles algébriques linéaires

We propose an approach to accelerate implicit schemes associated to linear differential-algebraic equations with constant coefficients. To cite this article: M. Al Sayed Ali, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

On the classification of conditionally integrable evolution systems in (1 + 1) dimensions

We generalize earlier results of Fokas and Liu and find all locally analytic (1 + 1)-dimensional evolution equations of order n that admit an N-shock-type solution with N ≤ n + 1. For this, we develop a refinement of the technique from our earlier work, where we completely characterized all (1+1)-dimensional evolution systems u _t = F (x, t, u, ?u/?x,..., ?ⁿ u/? x ⁿ) that are conditionally invariant under a given generalized (Lie-Bäcklund) vector field Q(x, t, u, ?u/?x,..., ?^k u/?x ^k)?/?u under the assumption that the system of ODEs Q = 0 is totally nondegenerate. Every such conditionally invariant evolution system admits a reduction to a system of ODEs in t, thus being a nonlinear counterpart to quasi-exactly solvable models in quantum mechanics.     

Tail asymptotics for dependent subexponential differences

We study the asymptotic behavior of ?(X ? Y > u) as u → ∞, where X is subexponential, Y is positive, and the random variables X and Y may be dependent. We give criteria under which the subtraction of Y does not change the tail behavior of X. It is also studied under which conditions the comonotonic copula represents the worst-case scenario for the asymptotic behavior in the sense of minimizing the tail of X ? Y. Some explicit construction of the worst-case copula is provided in other cases.     

On the longest service time in a busy period of the M⧸G⧸1 queue

This paper considers the supremum m of the service times of the customers served in a busy period of the M?G?1 queueing system. An implicit expression for the distribution m(w) of m is derived. This expression leads to some bounds for m(w), while it can also be used to obtain numerical results. The tail behaviour of m(w) is investigated, too. The results are particularly useful in the analysis of a class of tandem queueing systems.     

Influence coefficients for variational integral equations

We compute exact formulas for the influence coefficients deriving from the finite element discretization of integral equation methods. We consider the case of the Newtonian potential and plane triangles of the lower degree. To cite this article: M. Lenoir, C. R. Acad. Sci. Paris, Ser. I 343 (2006).