The weak null condition for Einstein's equations

We show that Einstein's equations of General Relativity expressed in wave coordinates satisfy a ‘weak null condition’. In a forthcoming article we will use this to prove a global existence result for Einstein's equations in wave coordinates with small initial data. To cite this article: H. Lindblad, I. Rodnianski, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On Schur's complement

A new proof is given for Haynsworth's formula for Schur's complement.     

On Guerra's broken replica-symmetry bound

Consider a random Hamiltonian $\text{H}{}_{\mathrm{N}}\text{(}\text{σ}\text{→}\text{)}$ for $\text{σ}\text{→}\text{∈Σ}{}_{\mathrm{N}}\text{={0,1}}{}^{\mathrm{<mtext>N</mtext>}}\text{.}$ We assume that the family $\text{(H}{}_{\mathrm{N}}\text{(}\text{σ}\text{→}\text{))}$ is jointly Gaussian centered and that for $\text{σ}\text{→}{}^1\text{,}\text{σ}\text{→}{}^2\text{∈Σ}{}_{\mathrm{N}}\text{,}$$\text{N}{}^{\mathrm{?1}}\text{EH}{}_{\mathrm{N}}\text{(}\text{σ}\text{→}{}^1\text{)H}{}_{\mathrm{N}}\text{(}\text{σ}\text{→}{}^2\text{)}$ =ξ(N^?1∑_i?Nσ¹_iσ²_i) for a certain function ξ on $\text{R}$. F. Guerra proved the remarkable fact that the free energy of the system with Hamiltonian $\text{H}{}_{\mathrm{N}}\text{(}\text{σ}\text{→}\text{)+h∑}{}_{\mathrm{i?N}}\text{σ}{}_{\mathrm{i}}$ is bounded below by the free energy of the Parisi solution provided that ξ is convex on $\text{R}$. We prove that this fact remains (asymptotically) true when the function ξ is only assumed to be convex on $\text{R}{}^{\mathrm{+}}$. This covers in particular the case of the p-spin interaction model for any p. To cite this article: M. Talagrand, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Reiter's condition (P2) and hypergroup representations

Let K be a hypergroup with Haar measure. We relate Reiter's condition $(P_2)$ to positive definite measures and associated representations, induced representations from a subgroup of K and convolution operators on $L_2(K)$. To cite this article: L. Pavel, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Gradient bounds for solutions of semilinear parabolic equations without Bernstein's quadratic condition

We establish gradient estimates for bounded solutions of semilinear parabolic equations, where the nonlinearity only satisfies one-sided quadratic upper growth assumptions, instead of the classical (two-sided) Bernstein's condition. This extends a recent work of Al. and Ar. Tersenov (Indiana Univ. Math. J. 50 (2001) 1899–1913), where results of this kind were obtained for radial solutions in a ball, by a different technique. To cite this article: J.-Ph. Bartier, Ph. Souplet, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On the choice of signs for householder's matrices

The numerical construction of Householder's matrices of the form I-2ww^H is known to be a problem with two distinct solutions; more precisely, the actual construction of such a matrix in a given context involves a choice of sign, and it is widely believed that only one alternative is correct, the other one leading to possible numerical unstabilities. This paper shows that the numerical stability of the process depends not on the chosen sign itself but only on the implementation of the actual computations; as well-conditioned approach for the non-classical case is presented and illustrated by a numerical example. Both signs are thus equally correct and there seems to be no reason at all why a specific sign should be prefered to the other.     

On Zienkiewicz–Zhu error estimators for Maxwell's equations

We consider a posteriori Zienkiewicz–Zhu (ZZ) type error estimators for the Maxwell equations. The main tool is the use of appropriate recovered values of the electric field and its curl. To cite this article: S. Nicaise, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

On the controllability for Maxwell's equations in specific media

In this Note we study the problem of exact controllability of the Maxwell's equations in specific media with two different models, on the one hand the so-called Drude–Born–Fedorov model, in the time domain, and on the other hand a simplified bilinear medium.For the first one we prove the non approximate controllability whereas for the second one we are able to prove the exact controllability under the usual conditions of the wave equation. To cite this article: P. Courilleau, T. Horsin, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Kronecker's theorem and Lehmer's problem for polynomials in several variables

Mahler defined the measure of a polynomial in several variables to be the geometric mean of the modulus of the polynomial averaged over the torus. The classical theorem of Kronecker which characterizes monic polynomials with integer coefficients all of whose roots are inside the unit disk can be regarded as characterizing those polynomials of one variable whose measure is exactly 1. Here this result is generalized to polynomials in several variables. The method employed also gives easy generalizations of recent results of Schinzel and Dobrowolski on Lehmer's problem.     

分布视角下的异质性群体收入分配格局研究

本文运用反映不同收入群体异质性的混合分布测度方法估计收入分布,对收入分布拟合效果的评判方法进行了探讨,在利用假设检验思想从收入变动角度给出一种收入群体界定标准的基础上,从规模上刻画我国收入分配格局,并运用夏普里分解规则对影响格局形成的宏观因素进行了解构。对CHNS(1989-2011)住户调查数据实证分析的结果表明,近年来城乡居民收入分布逐渐呈多峰特征且城镇更为明显;城镇和农村居民不同收入群体间的相互作用相当,低收入群体内部收入的增长可有效促进自身规模的缩小,中收入群体内部收入的增长虽同样有助于低收入群体规模的缩小但对自身群体规模变动影响不大,高收入群体内部收入的增长则将有助于自身规模的增大,而各群体内收入分配差距的扩大多会阻碍"橄榄型"格局的形成。     

On Poincaré's isoperimetric problem for simple closed geodesics

We show in the context of integral currents that Poincaré's isoperimetric variational problem for simple closed geodesics on ovaloids has a smooth extremal C without self-intersection. Provided the smooth Riemannian metric on the ovaloid M in question does not deviate too far from constant curvature, we further show that (i) this extremal C is connected and so is the desired non-trivial geodesic of shortest length on M and (ii) C is close (in the sense of Hausdorff distance) to a great circle.     

On Reid's characterization of the ternary matroids

In this paper we prove a stronger version of a result of Ralph Reid characterizing the ternary matroids (i.e., the matroids representable over the field of 3 elements, GF(3)). In particular, we prove that a matroid is ternary if it has no seriesminor of type L_n for n ≥ 5 (n cells and n circuits, each of size n ? 1), and no series-minor of type $\text{L}{}_5{}^1$ (dual of L₅), BII (Fano matroid) or BI (dual of type BII). The proof we give does not assume Reid's theorem. Rather we give a direct proof based on the methods (notably the homotopy theorem) developed by Tutte for proving his characterization of regular matroids. Indeed, the steps involved in our proof closely parallel Tutte's proof, but carrying out these steps now becomes much more complicated.     

Maharam's problem

We solve Maharam's problem [D. Maharam, An algebraic characterization of measure algebras, Ann. Math. 48 (1947) 154–167. [3]], also known as the Control Measure Problem. We construct a non-zero exhaustive submeasure on the algebra of clopen sets of the Cantor set that is not absolutely continuous with respect to a measure. To cite this article: M. Talagrand, C. R. Acad. Sci. Paris, Ser. I 342 (2006).