No abstract uploaded!

No abstract uploaded!

Dans ce papier, nous nous intéressons au problème suivant : soit$a$_{$N$}une suite de points d’une hypersurface algèbrique convergeant vers a∈H ; supposons que pour tout$N$, il existe un germe de disque holomorphe en$a$_{$N$}, γ_{$N$}, contenu dans$H$. Alors, d’après des résultats classiques (voir [5] et [6]) le point$a$n’est pas un point de 1-type fini et donc, il existe un germe de disque holomorphe en a vivant dans$H$(voir [5]). Dans ce qui suit, nous proposons une méthode effective pour reconstruire, à partir de la suite$a$_{$N$}et des γ_{$N$}, un germe d’ensemble analytique en$a$contenu dans$H$.

We prove that torsion-free G2 structures are (weakly) dynamically stable along the Laplacian flow for closed G2 structures. More precisely, given a torsion-free G2 structure φ on a compact 7-manifold, the Laplacian flow with initial value cohomologous and sufficiently close to φ will converge to a torsion-free G2 structure which is in the orbit of φ under diffeomorphisms isotopic to the identity. We deduce, from fundamental work of Joyce [18], that the Laplacian flow starting at any closed G2 structure with sufficiently small torsion will exist for all time and converge to a torsion-free G2 structure.

Staggered grid techniques have been applied successfully to many problems. A distinctive advantage is that physical laws arising from the corresponding partial differential equations are automatically preserved. Recently, a staggered discontinuous Galerkin (SDG) method was developed for the convectiondiffusion equation. In this paper, we are interested in solving the steady state convectiondiffusion equation with a small diffusion coefficient . It is known that the exact solution may have large gradient in some regions and thus a very fine mesh is needed. For convection dominated problems, that is, when is small, exact solutions may contain sharp layers. In these cases, adaptive mesh refinement is crucial in order to reduce the computational cost. In this paper, a new SDG method is proposed and the proof of its stability is provided. In order to construct an adaptive mesh refinement strategy for this new