Dans cette note, nous considérons l'invariance par symétrie des mesures harmoniques et la symétrie asymptotique de la fonction de Green associées aux variétés compactes de courbure négative.

Let $M (k,G)$ be the moduli space of based gauge equivalence classes of $G$-instantons on $S^4$ with instanton number $k$. $M(k, G)$ has the Uhlenbeck completion $\overline{M} (k,G) = \bigcup_{q=0}^k \mathrm{SP}^q (\mathbb R^4) \times M (k-q,G)$, where $\mathrm{SP}^q (\mathbb R^4)$ denotes the $q$-fold symmetric product of $\mathbb R^4$. Let $X (k,G)$ be the first two strata of the completion: $X (k,G) = M (k,G) \cup \mathbb R^4 \times M (k-1,G)$. In this paper we study the homology of $X (k,G)$ for $G= SU(n)$ or $Sp(n)$, and relate this to the homology of a certain homotopy theoretic fibre.

In this paper we prove the existence, uniqueness and asymptotic behaviour of global regular solutions of the mixed problem for the Kirchhoff nonlinear model given by the hyperbolic-parabolic equation $$(\rho_1u_t)_t + \rho_2u_t - M\Bigl(t,\int_{\alpha(t)}^{\beta(t)} |u_x|^2dx\Bigr)u_{xx} = f \quad\text{in $\hat Q$},$$ where $\hat Q = \bigl\{(x,t)\in \Bbb R^2\bigl| \alpha(t) < x < \beta(t), 0 < t < \infty\bigr\}$ is a noncylindrical domain of $\Bbb R^2$ and $\beta(\cdot)$, $\alpha(\cdot)$ are positive functions such that $$\displaystyle\lim_{t\to\infty}(\beta(t) - \alpha(t)) = + \infty.$$ The real function $M(\cdot,\cdot)$ is such that $M(t,\lambda) \geq m_0 > 0$ $\forall (t,\lambda)\in [0,\infty[\times[0,\infty[$, while $\rho_1(\cdot)$, $\rho_2(\cdot)$ are given functions which satisfy some appropriate conditions.

We define and determine the spectrum of order one of the d'Alembertian operator. We resolve the Fredholm alternative and the nonlinear problem at resonance.

The paper deals with the local and global nonexistence of weak solutions to a class of weakly coupled systems of parabolic inequalities with a fractional power of the Laplacian. Our results include nonexistence results as well as necessary conditions for the local and global solvability.

We consider a family of minimal sequences on a $3$-symbol alphabet with complexity $2n+1$, which satisfy a special combinatorial property. These sequences were originally defined by P. Arnoux and G. Rauzy as a generalization of the binary sturmian sequences. We prove that the dynamical system associated to each of these sequences of this family, can be realized as a dynamical system defined on a geodesic lamination on the hyperbolic disk. This is a generalization of the results shown in a previous paper of the author. We also show some applications of these results.

The notion of a subgeometry partition of a finite projective space $% PG(2m-1,q^{2})$ by $PG(m-1,q^{2})$'s and $PG(2m-1,q)$'s or a partition of $% PG(2m,q^{2})$ by $PG(2m,q)$'s is generalized to quasi-subgeometry partitions of $PG(2m-1,q^{d})$ by $PG(dm/e-1,q^{e})$'s for a set of divisors $e$ of $d$ and, partitions of $PG(2m,q^{2d})$ by $PG(d(2m+1)/f-1,q^{f})$'s for a set of divisors $f$ of $d$. In all cases, there are associated vector space spreads that are unions of `fans'.

More generally, in the arbitrary dimensional case, a complete theory of quasi-subgeometry partitions of $PG(V-1,D)$ corresponding to generalized spreads admitting $D^{\ast }$ as a fixed-point-free collineation group is obtained. When $D$ is a quadratic extension of a base field, `subgeometry' partitions are obtain.

Let $V_i$, $0\le i \le s$, $s \ge 1$, be complex Banach spaces. Assume $V_i$ infinite-dimensional for $1\le i \le s$, $V_0$ finite-dimensional and $V_0 \ne 0$. Let $\pi : {\bf {P}}(V_0)\times \dots \times {\bf {P}}(V_s) \to {\bf {P}}(V_0)$ be the projection. Let $X$ be a closed analytic subset of finite codimension of ${\bf {P}}(V_0)\times \dots \times {\bf {P}}(V_s)$. Here we prove that $\pi (X)$ is a closed analytic subset of ${\bf {P}}(V_0)$ with the following universal property. For every finite-dimensional reduced analytic space $Y$ and every holomorphic map $f: X \to Y$ there is a unique holomorphic map $g: \pi (X) \to Y$ such that $f = g\circ (\pi \vert X)$.

The paper deals with the local Cauchy problem for first order partial functional differential systems. A general class of difference methods is constructed. The convergence of explicit difference schemes is proved by means of consistency and stability arguments. It is assumed that the given functions satisfy nonlinear estimates of Perron type with respect to functional variables. Differential systems with deviated variables and differential integral problems can be obtained from a general case by specializing the given operators. The results are illustrated by numerical examples.

In this paper we study a boundary value problem for functional differential equations of second order. Applying a quasilinearization technique we obtain two monotone sequences showing that they converge to the unique solution and this convergence is superlinear.

L'objet de ce travail est l'approximation dans l'espace d'énergie des suites de solutions de l'équation $$ \partial^2_t u-div(A(x).\nabla_xu)+|u|^{p-1}u=0\quad\mbox{dans}\quad \mathbb R_t\times \mathbb R^d_x $$ où $d\geq 3$,\quad$1 <p\leq p_c:=\frac{d+2}{d-2}$ et $A$ est une fonction régulière à valeurs dans l'espace des matrices $d\times d$ définies positives, valant l'identité en dehors d'un compact. Nous démontrons des résultats analogues à ceux de P. Gérard concernant le cas constant ($A(x)\equiv Id$).

This work is devoted to the description of bounded energy sequences of solutions to the equation $$ \partial^2_t u-div(A(x).\nabla_xu)+|u|^{p-1}u=0\quad\mbox{in}\quad \mathbb R_t\times \mathbb R^d_x $$ where $d\geq 3$,\quad$1 <p\leq p_c:=\frac{d+2}{d-2}$ and $A$ is a regular function, valued in the space of $d\times d$ positive definite matrix, which is the identity outside a compact set of $\mathbb R^d$. We obtain the same results as P. Gérard about constant case ($A(x)\equiv Id$).