Comptes Rendus de l'Académie des Sciences - Series I - Mathematics最新文献

We obtain some new results in the theory of linear forms in logarithms on a commutative algebraic group, defined over $\text{Q}$. We generalize recent progress of S. David and N. Hirata [1]. In particular, we achieve an optimal linear independence measure in the height of the linear form and a measure more precise than Hirata's [4] for parameters related to the logarithms. The proof rests on Baker's method and a new argument of arithmetical nature.

In this paper, we extend to anisotropic case with variable coefficients the boundary stabilization result obtained in [2]. We use as a main tool local coordinates in the expression of boundary integrals. Our conditions are purely geometrical and few restrictive.

We consider complex-valued solutions u_ε of the Ginzburg–Landau on a smooth bounded simply connected domain $\text{Ω}$ of $\text{R}{}^{\mathrm{N}}$, N⩾2 (here ε is a parameter between 0 and 1). We assume that u_ε=g_ε on $\text{∂Ω}$, where |g_ε|=1 and g_ε is uniformly bounded in $\text{H}{}^{\mathrm{1/2}}\text{(∂Ω)}$. We also assume that the Ginzburg–Landau energy E_ε(u_ε) is bounded by M₀|logε|, where M₀ is some given constant. We establish, for every 1⩽p<N/(N−1), uniform W^1,p bounds for u_ε (independent of ε). These types of estimates play a central role in the asymptotic analysis of u_ε as ε→0.

We introduce a class of four dimensional Lorentzian manifolds with closed curves of null type or timelike. We investigate some global problems for the wave equation: global Cauchy problem and asymptotic completeness of the wave operators for the chronological but non-causal metrics; uniqueness of solution with data on a changing type hypersurface; existence of resonant states; scattering by a violation of the chronology; poles of the scattering matrix.

We prove the NP-hardness of two problems. The first is the well-known minimal realization problem in the max-plus semiring. The second problem (Pisot's problem) is the problem of determining if a given integer linear recurrent sequence has a zero coefficient.

Let M be a compact connected Riemannian manifold, and fix x,y∈M. For a sufficiently small constant R>0, Poincaré inequalities w.r.t. pinned Wiener measure with time parameter T>0 are proven on the sets $\text{Ω}{}^{\mathrm{R,N}}{}_{\mathrm{x,y}}$, $\text{N∈}\text{N}$, consisting of all continuous paths ω:[0,1]→M such that ω(0)=x, ω(1)=y, and d(ω(s),ω(t))<R if s,t∈[(i−1)/N,i/N] for some integer i. Moreover, the asymptotic behaviour of the best constants in the Poincaré inequalities as T goes to 0 is studied. It turns out that the asymptotic depends crucially on the Riemannian metric on M and, in particular, on the geodesics contained in $\text{Ω}{}^{\mathrm{R,N}}{}_{\mathrm{x,y}}$. Key ingredients in the proofs are a bisection argument, estimates for finite-dimensional spectral gaps, and a crucial variance estimate by Malliavin and Stroock.

We propose a mathematical theory for the refocusing properties observed in time-reversal experiments, where classical waves propagate through a medium, are recorded in time, then time-reversed and sent back into the medium. The salient feature of such experiments is that the refocusing quality of the time-reversed reemitted signals is greatly enhanced when the underlying medium is heterogeneous. Based on the Wigner transform formalism, we show that random media indeed greatly improve refocusing. We analyze two different types of random media, where in the limit of high frequencies, the Wigner transform satisfies a random Liouville equation or a linear transport equation.

Signorini's law of unilateral contact and Coulomb's friction law constitute a simple and useful framework for the analysis of unilateral frictional contact problems of a linearly elastic body with a rigid support. For quasi-static, monotone-loadings, the discrete dual formulation of this problem leads to a quasi-variational inequality, whose unknowns, after condensation, are the normal and tangential contact forces at nodes of the initial contact area. A new block-relaxation solution technique is proposed here. At the typical iteration step, shown to be a contraction for small friction coefficients, two quadratic programming problems are solved one after the other: the former is a friction problem with given normal forces, the latter is a unilateral contact problem with prescribed tangential forces. The contraction principle is used to establish the well-posedness of the discrete formulation, to prove the convergence of the algorithm, and to obtain an estimate of the convergence rate.

For simply connected nilpotent Lie groups G, we show that limit laws of infinitesimal triangular systems Δ of symmetric probability measures on G are infinitely divisible even if Δ is not commutative. The same holds also if the measures of Δ are supported by some fixed discrete subgroup Γ⊂G. Furthermore, we give a weakening of Wehn's conditions for the accompanying laws theorem in the case of discrete subgroups of exponential Lie groups.

We study random walks on $\text{Z}$ in a stationary random medium, defined by an ergodic dynamical system, when the possible jumps are {0,±1,±2}. We give a simple proof of a recurrence criterion of the same kind as Key's [6]. We show that the intermediate Lyapunov exponent involved in the criterion is always simple.