In this note, we study primitive symmetric spaces, namely discrete symmetric spaces whose main group acts primitively. We give three families of examples of such spaces, and we prove that every primitive symmetric space belongs to one of these families.

In this note we show that the algebraic parameters of a linear translation generalized quadrangle are not restricted. This is done with a free construction of fourgonal families on vector spaces. Secondly we prove that a compact translation generalized quadrangle can only have the topological parameters $(1,t)$, $(2,2)$, $(3,4t)$ or $(7,8t)$ for $t\in\mathbb N$. This is achieved by determining the possible dimensions of the elements of continuous partial spreads which satisfy a certain planarity condition.

Nous commençons par donner une axiomatisation de caractère géométrique de la théorie des corps ordonnés différentiellement clos (notée $CODF$) introduite par M.Singer en 1978. Nous nous basons pour cela sur une construction similaire de D. Pierce et A. Pillay concernant la théorie des corps différentiellement clos de caractéristique zéro (notée $DCF_0$). Nous remarquons ensuite qu'un principe de {\it relèvement différentiel} permet de montrer que $CODF$ n'a pas la propriété d'indépendance (en d'autres termes, ses ensembles définissables ont une dimension de Vapnik-Chervonenkis finie). La méthode de preuve utilisée ici peut s'appliquer à d'autres exemples de théories de corps différentiels.

We first propose a ``geometrical'' axiomatization for the theory of closed ordered differential fields (denoted CODF) introduced by M.Singer in 1978. This axiomatization is the analogue of the Pierce-Pillay axiomatization for the theory of differentially closed fields of characteristic zero . We also remark that a {\it differential lifting} principle can be used to prove that $CODF$ has not the independence property (this result gives new examples of V-C classes of definable sets). The proof used here can be generalized to other examples of theories of differential fields.

Let $q$ be a power of a prime $p \neq 3.$ We characterize the following two sets of polynomials: $M(q)=\{P \in {\bf F}_{q}[t]$ such that $P$ is a strict sum of cubes in ${\bf F}_{q}[t]\}$ and $S(q)=\{P \in {\bf F}_{q}[t]$ such that $P$ is a strict sum of cubes and squares in ${\bf F}_{q}[t]\}.$ Let $g(3,{\bf F}_{q}[t])=g \geq 0$ be the minimal integer such that every $P \in M(q)$ is a strict sum of $g$ cubes. Similarly let $g_1(3,2,{\bf F}_{q}[t])=g$ be the minimal integer such that every $P \in S(q)$ is a strict sum of $g$ cubes and a square. Our main result is:\begin{itemize} \item[i)] $4 \leq g(3,{\bf F}_{q}[t]) \leq 9\,\,\,$ for $q \in \{2,4\}.$ \item[ii)] $3 \leq g_1(3,2,{\bf F}_{q}[t]) \leq 4\,\,\,$ for $q =4.$ \end{itemize}

We prove two results concerning the generalized Fermat equation $x^4 + y^4 = z^p$. In particular we prove that the First Case is true if $p \neq 7$

The transitive partial parallelisms of deficiency one in $PG(3,q)$ are classified under mild assumptions on the corresponding group.

We prove that, given an index $m$, if every Pietsch integral $m$-homo\-geneous polynomial on a Banach space $E$ is nuclear, then $E$ is Asplund. The converse was proved by Alencar.

It is proved that if $k$ and $d$ are positive integers such that the product of any two distinct elements of the set \[ \{F_{2k},\, 5F_{2k},\, 4F_{2k+2},\, d\} \] increased by $4$ is a perfect square, than $d=4L_{2k}F_{4k+2}$. This is a generalization of the results of Kedlaya, Mohanty and Ramasamy for $k=1$.

In this short note, we prove a characteristic property of nest algebras. This property is closely related to the extreme point structure of nest algebras.

In this paper we study a scalar Neumann problem driven by the ordinary p-Lapacian and a nonsmooth potential. The nonlinearity exhibits an asymmetric behavior. Namely growth restriction is imposed in one direction only (either the positive direction or the negative direction). Using a variational approach based on the nonsmooth critical point theory for locally Lipschitz function, we prove the existence of a solution.

We study two nonlinear degenerate eigenvalue problems on $\mathbb R^N$. For the first problem we prove the existence of a positive eigenvalue while for the second we show the existence of a continuous family of eigenvalues. Our approach is based on standard tools in the critical point theory combined with adequate variational methods. We also apply an idea developed recently by Szulkin and Willem.

Dans cet article, on améliore la proposition 2.2.3 énoncée par B. Adam ce qui nous permet de réduire de huit à cinq au maximum le nombre de choix d'un point extrémal adjacent à 1 de deuxième espèce pour toute les formes d'écriture du vecteur isotrope d'une forme quadratique.\\ On donne également le développement par l'algorithme de Voronoi de la première famille paramétrée infinie de corps de nombres cubiques introduite par C. Levesque et G. Rhin et on obtient ainsi l'unité fondamentale de ces corps.

In this paper, we improve the proposition 2.2.3 studied by B. Adam, which allows to reduce from eight to five at most the number of choices for a minimal point adjacent to 1 of second kind for all forms of writing an isotropic vector of a quadratic form.\\ We also give the Voronoi algorithm expansion of the first infinite family of cubic number fields introduced by C. Levesque and G. Rhin and so we obtain a fundamental unit of these fields.

We consider linear partial differential operators with constant coefficients $P$ and show that the inclusion of the Gevrey classes $G^d_P$ defined by the iterates of $P$ in some multianisotropic Gevrey classes implies a growth condition on the symbol of $P$. Under the hypothesis of hypoellipticity, the converse implication is also true. These results are also related to the regular weight of hypoellipticity, that gives a precise description of the growth of the symbol of $P$ with respect to its derivatives.