We prove two theorems of reduction of cocycles taking values in the group of diffeomorphisms of the circle. They generalize previous results obtained by the author concerning rigidity for smooth actions on the circle of Kazhdan groups and higher rank lattices.

We consider a mathematical model for the process of contact between a viscoelastic body and a reactive foundation. The material is viscoelastic with internal state variable which may describe the damage of the system. We establish a variational formulation for the model and prove the existence and uniqueness result of the weak solution. Finally we prove a dependence result with respect to the data.

We study convergent sequences of Baumslag-Solitar groups in the space of marked groups. We prove that $BS(\mathfrak m,\mathfrak n) \to \F_2$ for $|\mathfrak m|,|\mathfrak n| \to \infty$ and $BS(1,\mathfrak n) \to \mathbb{Z}\wr\mathbb{Z}$ for $|\mathfrak n| \to \infty$. For $\mathfrak m$ fixed, $|\mathfrak m| \geqslant 2$, we show that the sequence $(BS(\mathfrak m,\mathfrak n))_{\mathfrak n}$ is not convergent and characterize many convergent subsequences. Moreover if $X_\mathfrak m$ is the set of $BS(\mathfrak m,\mathfrak n)$'s for $\mathfrak n$ relatively prime to $\mathfrak m$ and $|\mathfrak n| \geqslant 2$, then the map $BS(\mathfrak m,\mathfrak n) \mapsto \mathfrak n$ extends continuously on $\overline{X_\mathfrak m}$ to a surjection onto invertible $\mathfrak m$-adic integers.

We introduce a new class of algebras which extends classical $A^{*}$ -algebras to the $p$-normed case with generalized involution. We give results concerning the symmetry and the $C^{*}$-algebra structure in such algebras.

It is well-known that many transitive continuous self-maps have a dense orbit. On the other hand several interesting transitive dynamical systems present some kind of discontinuity. Motivated by a result of A. Peris, we investigate under which conditions a discontinuous transitive map has a dense orbit.

For any commutative ring $R$, we introduce a group attached to $R$, the {\em Brauer-Galois group of $R$}, defined to be the subgroup of the Brauer group of $R$ consisting of the classes of the Azumaya $R$-algebras which can be represented, via Brauer equivalence, by a Galois extension of $R$. We compute this group for some particular commutative rings.

It is the aim of this paper to obtain the generalized Hyers-Ulam stability result for a mixed type of cubic and additive functional equation \begin{eqnarray*} &&f\Big(\Big(\sum_{i=1}^{l}x_i\Big) +x_{l+1}\Big)+f\Big(\Big(\sum_{i=1}^{l}x_i\Big) -x_{l+1}\Big)+2\sum_{i=1}^{l}f(x_i)\\ &&\qquad \qquad =2f\Big(\sum_{i=1}^{l}x_i\Big)+\sum_{i=1}^{l}[f(x_i +x_{l+1})+f(x_i -x_{l+1})] \end{eqnarray*} for all $(x_1,\cdots,x_l, x_{l+1}) \in X^{l+1},$ where $l\ge 2.$

Let $n$ be an integer $\geqslant 2$. A group $G$ is called generalized $n$-abelian if it admits a {\em positive polynomial} endomorphism of degree $n$, that is if there exist $n$ elements $a_1, a_2, \dots, a_n$ of $G$ such that the function $\varphi: x\mapsto x^{a_1}x^{a_2}\cdots x^{a_n}$ is an endomorphism of $G$. In this paper we give some sufficient conditions for a generalized $n$-abelian group to be abelian. In particular, we show that every group admitting a positive polynomial monomorphism of degree 3 is abelian.

Soit $n$ un entier $\geqslant 2$. Un groupe $G$ est dit $n$-abélien généralisé s'il admet un endomorphisme {\em polynomial positif} de degré $n$, c'est à dire s'il existe $n$ éléments $a_1, a_2 \dots, a_n$ de $G$ tels que l'application $\varphi: x\mapsto x^{a_1}x^{a_2}\cdots x^{a_n}$ soit un endomorphisme de $G$. Dans cet article, on donne quelques conditions suffisantes que doit vérifier un groupe n-abélien généralisé pour qu'il soit abélien. En particulier, on montre que tout groupe admettant un monomorphisme polynomial positif de degré 3 est abélien.

Given a duality $\langle E, F\rangle$, a dual strong sequence is a sequence of bidual enlargements of $F$ in the algebraic dual $E^\ast$ of $E$. In this article, we investigate the bounded sets generated by a dual strong sequence and related associated topologies.

Let $G$ be a locally compact group and let $\omega$ be a weight function on $G$. Under a very mild assumption on $\omega$, we show that $L^1(G,\omega)$ is (2n+1)-weakly amenable for every $n\in \mathbb Z^+$. Also for every odd $n\in\mathbb{N}$ we show that $\h^2(L^1(G,\omega),(L^1(G,\omega))^{(n)})$ is a Banach space.

In the present paper, we deal with a large class of Banach algebras known as Lau algebras. It is well-known that if ${\frak A}$ is a left amenable Lau algebra, then any $f\in {\frak A}$ such that $|fg|=|f|g$ for all $g\in {\frak A}$ with $g\geq 0$ is a scalar multiple of a positive element in ${\frak A}$. We show that this result remains valid for the group algebra $\ell^1(G)$ of any, not necessarily amenable, discrete group $G$. We also give an example which shows that the result is, in general, not true without the hypothesis of left amenability of ${\frak A}$. This resolves negatively an open problem raised by F. Ghahramani and A. T. Lau.

In this paper, we consider the problem (${\cal P}_{\lambda}$) in the setting of a weighted Sobolev space $W^{1, p}(\Omega, \omega)$, where $\omega$ is a weight function defined on the unbounded domain $\Omega$. The study is based on the variational methods and critical point theory. We show the existence of at least two nonnegative solutions, one with negative energy, the other one with energy which changes sign at a certain value of the positive parameter $\lambda$.

In this note, we prove that all subgroups of $\textnormal{SL}(2,R)$ have the Haagerup Property if $R$ is a commutative reduced ring. This is based on the case when $R$ is a field, recently established by Guentner, Higson, and Weinberger. As an application, residually free groups have the Haagerup Property.

A proof of a sufficient condition for a strongly continuous semigroup $\{ T(t)\}_{t\ge 0}$ on a Banach space $X$ to be uniformly exponentially stable is given. This result is a simplification of an earlier theorem by van Neerven, and concludes that a semigroup is uniformly exponentially stable provided $\sup\nolimits_{||x||\le 1}J(||T(\cdot)x||)<\infty$ ; here $J$ is a certain nonlinear functional with certain natural properties. A non-autonomous version of this theorem for evolution families is also given. This implies the well-known Datko-Pazy and Rolewicz Theorems. This result is connected to the uniform asymptotic stability of the well-posed linear and non-autonomous abstract Cauchy problem \begin{equation*} \left\{ \begin{array}{rcl} \dot{u}(t)& = & A(t)u(t),\quad t\geq s\geq 0, \\ u(s) & = & x\in X. \end{array} \right. \end{equation*}

An existence result to infinite boundary-value problem is proved via Schauder-Tychonoff fixed point theorem.

We prove that there exist non-archimedean (n.a.) locally convex spaces without basic orthogonal sequences, and even without Schauder basic sequences. Among other things any n.a. Köthe space with the weak topology has no basic orthogonal sequence. On the other hand, we show that the strong dual of any infinite-dimensional n.a. polar Fréchet space and any n.a. LF-space have basic orthogonal sequences.