Exposé succinct d’une démonstration du théorème de division pour les fonctions $p$ fois continûment différentiables $(p\le \infty )$, donnant pour les classes du quotient et du reste les meilleurs résultats possibles lorsque $p$ est fini.

These lectures will focus on those properties of maximal monotone operators which are valid in arbitrary real Banach spaces.

In this work we consider a class of germs of singularities of integrable 1-forms in $R^n$ which are structurally stable in class $C^r$ ($r\ge 2$ if $n=3$, $r\ge 4$ if $n\ge 4$), whose 1-jet is zero at the singularity. In this class the stability depends essentially on the fact that the perturbations allowed are integrable.

Let $G:{ℂ}^n\times {ℂ}^r\to ℂ$ be a holomorphic family of functions. If $\Lambda \subset {ℂ}^n\times {ℂ}^r$, ${\pi }_r:{ℂ}^n\times {ℂ}^r\to {ℂ}^r$ is an analytic variety then   $Q_{\Lambda }\left(G\right)=(x,u)\in {ℂ}^n\times {ℂ}^r:G(·,u)hasacriticalpointin\Lambda \cap {\pi }_r^{-1}\left(u\right)$is a natural generalization of the bifurcation variety of G. We investigate the local structure of $Q_{\Lambda }\left(G\right)$ for locally trivial deformations of $\Lambda ₀={\pi }_r^{-1}\left(0\right)$. In particular, we construct an algorithm for determining logarithmic stratifications provided G is versal.

Let M be a complete Riemannian manifold, M ∈ ℕ and p ≥ 1. We prove that almost everywhere on x = (x1,...,xN) ∈ MN for Lebesgue measure in MN, the measure $\mu \left(x\right)=N{\sum }_{k=1}^N{x}_k$ μ ( x ) = 1 N ∑ k = 1 N δ x k has a uniquep–mean ep(x). As a consequence, if X = (X1,...,XN) is a MN-valued random variable with absolutely continuous law, then almost surely μ(X(ω)) has a unique p–mean. In particular if (Xn)n ≥ 1 is an independent sample of an absolutely continuous law in M, then the process ep,n(ω) = ep(X1(ω),...,Xn(ω)) is...

Suppose M is a noncompact connected n-manifold and ω is a good Radon measure of M with ω(∂M) = 0. Let ℋ(M,ω) denote the group of ω-preserving homeomorphisms of M equipped with the compact-open topology, and ${\mathscr{H}}_E(M,\omega )$ the subgroup consisting of all h ∈ ℋ(M,ω) which fix the ends of M. S. R. Alpern and V. S. Prasad introduced the topological vector space (M,ω) of end charges of M and the end charge homomorphism $c^{\omega }:{\mathscr{H}}_E(M,\omega )\to (M,\omega )$, which measures for each $h\in {\mathscr{H}}_E(M,\omega )$ the mass flow toward ends induced by h. We show that the map $c^{\omega }$ has...