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

We consider the hp-version of the discontinuous Galerkin (DG) time-stepping method for linear parabolic problems with non-symmetric elliptic spatial operators. We derive new analyticity estimates for the exact solutions by means of semigroup techniques. These estimates allow us to show that the hp-DG time-stepping method can resolve start-up singularities at exponential rates of convergence.

We analyse the structure of locally minimising closed (n−1)-currents in an n-dimensional Riemannian manifold M. In particular, we prove that such currents are measured laminations by (possibly singular) minimal hypersurfaces. We use ideas from the theory of codimension one singular foliations to decompose these currents. The results are used to investigate the stable norm on $\text{H}{}_{\mathrm{n-1}}\text{(M,}\text{R}\text{)}$.

Using a criterion for invariance of canonical analysis, we propose a new method for variable selection for linear regression models with random covariates. The convergence of this method is established. A consistent test for the validity of a selected submodel is then proposed.

We revisit fractional step projection methods for solving the Navier–Stokes equations. We study a variant of pressure-correction methods and introduce a new class of velocity-correction methods. We prove stability and $\text{O}\text{(δt}{}^2\text{)}$ convergence in the L² norm of the velocity for both variants. We also prove $\text{O}\text{(δt}{}^{\mathrm{3/2}}\text{)}$ convergence in the H¹ norm of the velocity and the L² norm of the pressure. We show that the new family of projection methods can be related to a set of methods introduced in [4,3]. As a result, this Note provides the first rigorous proof of stability and convergence of the methods introduced in [4,3].

We are interested in the approximation of invariant subspaces of large non-Hermitian matrices by the Rayleigh–Ritz procedure. Despite its nonoptimality, this procedure is widely used. We justify, in some sense, its use and derive an a priori error bound that extends Saad's result obtained for eigenvectors in the Hermitian case.

The work is devoted to reaction–diffusion–convection problems in unbounded cylinders. We study the Fredholm property and properness of the corresponding elliptic operators and define the topological degree. Together with analysis of the spectrum of the linearized operators it allows us to study bifurcations of solutions and to prove existence of convective waves. Finally, we make some conclusions about the possible appearance of a “convective instability”.

In this paper, we establish a limit theorem for the local behavior of compound empirical processes based on two independent sequences of independent and identically distributed random variables. Our proofs rely on Poisson approximation methods.

In the theory of $\text{O}$-modules with an integrable logarithmic connection in the context of log schemes (over a field of characteristic zero), one of the first problems is that, contrary to the classical case, an object of these categories which is $\text{O}$-coherent is not necessarily locally free. We present some sufficient conditions for the local freeness, based essentially on the notion of residues of a log connection. Then we handle the problem of stability of local freeness under derived direct image for morphisms of log schemes; we prove a generalization of the Deligne–Illusie results on degeneration of the “Hodge to de Rham” spectral sequence, local freeness, compatibility with base change.