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$H$.

The study of Hamiltonian graphs began with Dirac’s classic result in 1952. This was followed by that of Ore in 1960. In 1984 Fan generalized both these results with the following result: If$G$is a 2-connected graph of order$n$and max{$d$($u$),$d$($v$)}≥$n$/2 for each pair of vertices$u$and$v$with distance$d$($u$,$v$)=2, then$G$is Hamiltonian. In 1991 Faudree–Gould–Jacobson–Lesnick proved that if$G$is a 2-connected graph and |$N$($u$)∪$N$($v$)|+δ($G$)≥$n$for each pair of nonadjacent vertices$u$,$v$∈$V$($G$), then$G$is Hamiltonian. This paper generalizes the above results when$G$is 3-connected. We show that if$G$is a 3-connected graph of order$n$and max{|$N$($x$)∪$N$($y$)|+$d$($u$),|$N$($w$)∪$N$($z$)|+$d$($v$)}≥$n$for every choice of vertices$x$,$y$,$u$,$w$,$z$,$v$such that$d$($x$,$y$)=$d$($y$,$u$)=$d$($w$,$z$)=$d$($z$,$v$)=$d$($u$,$v$)=2 and where$x$,$y$and$u$are three distinct vertices and$w$,$z$and$v$are also three distinct vertices (and possibly |{$x$,$y$}∩{$w$,$z$}| is 1 or 2), then$G$is Hamiltonian.

We prove the existence of steady periodic capillary water waves on flows with arbitrary vorticity distributions. They are symmetric two-dimensional waves whose profiles are monotone between crest and trough.

In this paper we generalize an interpolation result due to J.-O. Strömberg and A. Torchinsky to the case of one-sided Hardy spaces. This generalization is important in the study of the weak type (1,1) for lateral strongly singular operators. We shall need an atomic decomposition in which for every atom there exists another atom supported contiguously at its right. In order to obtain this decomposition we have developed a rather simple technique to break up an atom into a sum of others atoms.

Let $A=-(\nabla-i\vec{a})^2+V$ be a magnetic Schrödinger operator acting on$L$^{2}($R$^{$n$}),$n$≥1, where $\vec{a}=(a_1,\cdots,a_n)\in L^2_{\rm loc}$ and 0≤$V$∈$L$^{1}_{loc}. Following [1], we define, by means of the area integral function, a Hardy space$H$^{1}_{$A$}associated with$A$. We show that Riesz transforms (∂/∂$x$_{$k$}-$i$$a$_{$k$})$A$^{-1/2}associated with$A$, $k=1,\cdots,n$ , are bounded from the Hardy space$H$^{1}_{$A$}into$L$^{1}. By interpolation, the Riesz transforms are bounded on$L$^{$p$}for all 1<$p$≤2.