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

In this Note, we prove existence of trajectories to – possibly multivalued – differential equations remaining in a given closed set of constraints. It is a consequence of behaviour of solutions on the boundary of the set of constraints.

The Ericksen–Leslie model of nematic liquid crystals is a coupled system between the Navier–Stokes and the Ginzburg–Landau equations. We show here the local well-posedness for this problem for any initial data regular enough, by a fixed point approach relying on some weak continuity properties in a suitable functional setting. By showing the existence of an appropriate local Lyapunov functional, we also give sufficient conditions for the global existence of the solution, and some stability conditions.

Let k be an algebraically closed field of characteristic zero. Let K be either a function field in two variables over k or the fraction field of a 2-dimensional, excellent, strictly henselian local domain with residue field k. We show that linear algebraic groups over such a field K satisfy most properties familiar in the context of number fields: finiteness of R-equivalence, Hasse principle for complete homogeneous spaces.

We answer a question of D. Huybrechts about the Kähler cone of a compact hyperkähler manifold. More precisely, we show how the methods he uses to describe the closure of this cone do in fact extend to get the following description: the Kähler cone of a hyperkähler manifold is the set of elements of the positive cone attached to the canonical quadratic form which are positive on the rational curves.

In this paper we give generalizations of Bartle–Graves theorem [2]. We define some functional spaces which take their values in quotient spaces.

We propose a new technique for enhancing the results coming from computational simulations with a hyperbolic system. The process works only on the extracted data obtained after an interruption of the program, and is not depending neither on the mathematical model nor on the numerical scheme, as long as the model is a hyperbolic one and the scheme is diffusive enough. The numerical tests show a good sharpening of the discontinuities, on less than two cells, and also a good precision on the rarefaction waves and the conservation of some quantities, such as the mass.

We determine a subset in $\text{R}{}^2$ and a measure on this set which allow to construct coupled non-localized solutions of the KP-I equation, which are connected by the change of variables (x,t)↦(−x,−t), and split into asymptotic solitons as t→∞ in the neighbourhood of the leading edge of the solutions. The solitons corresponding to each of the solutions have different amplitudes and lines of constant phase.

The notion of a matched pair of Leibniz algebroids is introduced and it is shown that a Nambu–Jacobi structure of order n, n>2, over a manifold M defines a matched pair of Leibniz algebroids. As a consequence, one deduces that the vector bundle $\text{⋀}{}^{\mathrm{n-1}}\text{(}\text{T}{}^{\mathrm{\ast }}\text{M)⊕⋀}{}^{\mathrm{n-2}}\text{(}\text{T}{}^{\mathrm{\ast }}\text{M)→M}$ is a Leibniz algebroid. Finally, if M is orientable, the modular class of M is defined as a cohomology class of order 1 with respect to this Leibniz algebroid.

Let f₀ be a surface diffeomorphism such that the maximal invariant set in an open set V is the union of a horseshoe and a quadratic tangency between the stable and unstable foliations of this horseshoe. We assume that the dimension of the horseshoe is larger than but close to one. We announce that, for most diffeomorphisms f close to f₀, the maximal f-invariant set in V is a non-uniformly hyperbolic horseshoe, with dynamics of the same type as met in Hénon attractors.

In the setting of additive models, this paper studies kernel marginal integration estimates. These estimates are used for the construction of a test of additivity. We propose to show the asymptotic normality of a test statistic under conditions of absolute regularity.