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

We consider Darcy's equations with variable permeability coefficient in a two- or three-dimensional domain. We propose a finite volume scheme, which turns out to be equivalent to a finite element problem, and we derive optimal a priori error estimates. We describe error indicators and prove that they provide an appropriate tool for mesh adaptivity, since estimates allow to compare them with the error.

We introduce here the notion of p-adic ideal of p-rank d and the one of p-adic radical of p-rank d of an ideal by analogy with the real case. We use it to give another proof of the p-adic Nullstellensatz.

We consider a regular oblique derivative problem for a linear parabolic operator $\text{P}$ with VMO principal coefficients. Its unique strong solvability is proved in [15], when $\text{P}\text{u∈}\text{L}{}^{\mathrm{p}}\text{(Q}{}_{\mathrm{T}}\text{)}$. Our goal here is to show that the solution belongs to the parabolic Morrey space W_p,λ^2,1(Q_T), when $\text{P}\text{u∈}\text{L}{}^{\mathrm{p,\lambda }}\text{(Q}{}_{\mathrm{T}}\text{)}$, p∈(1,∞), λ∈(0,n+2), and Q_T is a cylinder in $\text{R}{}_{\mathrm{+}}{}^{\mathrm{n+1}}$. The a priori estimates of the solution are derived through L^p,λ estimates for singular and nonsingular integral operators.

We give a new definition of the lower pointwise dimension associated with a Borel probability measure with respect to a general Carathéodory–Pesin structure. Then we show that the spectrum of the measure coincides with the essential supremum of the lower pointwise dimension. We provide an example coming from dynamical systems.

We give a very simple proof of a theorem of Eliashberg and Gromov implying that intersection between the conormal bundles νM and νN of two proper submanifolds M, N of $\text{R}{}^{\mathrm{n}}$ is persistent under compactly supported Hamiltonian deformations of νM and νN.

Let F be a neutral to the right, random distribution function on [0,+∞[, with a stationary subordinator. We introduce new linear functionals of the increments of F. Each of them is distributed like the unique fixed point of a random affine system. For gamma subordinators, we prove, building on earlier work, that their densities involve linear combinations of hypergeometric functions. We apply these ideas to the random sampling of alternating renewal processes.

Using lambda coordinates from Teichmüller theory, we study the action of the Thompson group T on a relative Teichmüller space, which is defined in terms of piecewise projective homeomorphisms. As an application, we give a geometric interpretation of the homology equivalence between BT and the free loop space $\text{L}\text{S}{}^3$.

We apply a duality method for solving a nonconvex optimization problem. We construct an algorithm converging to a ∂-critical point and we establish the relationship with critical point of the primal problem. The application of this method to a nonlinear Stokes problem leads to a weak solution.

Let G be a topological locally compact group. The aim of this Note is a contribution to the study of the existence problem for square integrable continuous and unitary representations for G. One of our main results (Theorem 6.2) will give a necessary and sufficient condition for the existence of the discrete series for G. Our approach is based on the notions of units and bounded elements in L²(G) introduced by R. Godement in [6]. We perform a study of these notions. A particular attention is paid to the case of pure units. We associate to each pure unit a transform called Plancherel transform. We characterize the pure units with the use of their Plancherel transforms. We develop new methods giving a new proof to the well known theorem of Bargmann (see [3]) in the case of Lorentz groups, the Harish-Chandra theorem (see [5]) in the case of semi-simple Lie groups and a well known theorem of Duflo and Moore (see [4]) in the case of general nonunimodular locally compact groups. Our methods allow us to give an explicit expression of the formal operator introduced in [4] by Duflo and Moore.

Let $\text{φ:=(f,g):}\text{C}{}^2\text{→}\text{C}{}^2$ with f and g polynomial mappings. We establish the connection that exists between the Newton polygon of the curve which is the union of the discriminant and of the non-proper locus of φ and the topology of the links at infinity of the curves f⁻¹(0) and g⁻¹(0).