We give necessary and sufficient conditions on a ring $R$ and an endomorphism $\sigma$ of $R$ for the skew power series ring $R[[x; \sigma]]$ to be right duo right Bézout. In particular, we prove that $R[[x; \sigma]]$ is right duo right Bézout if and only if $R[[x; \sigma]]$ is reduced right distributive if and only if $R[[x; \sigma]]$ is right duo of weak dimension less than or equal to $1$ if and only if $R$ is $\aleph_0$-injective strongly regular and $\sigma$ is bijective and idempotent-stabilizing, extending to skew power series rings the Brewer-Rutter-Watkins characterization of commutative Bézout power series rings.

Let $\Gamma$ be a graph endowed with a reversible Markov kernel $p$, and $P$ the associated operator, defined by $Pf(x)=\sum_y p(x,y)f(y)$. Denote by $\nabla$ the discrete gradient. We give necessary and/or sufficient conditions on $\Gamma$ in order to compare $\left\Vert \nabla f\right\Vert_{p}$ and $\left\Vert (I-P)^{1/2}f\right\Vert_{p}$ uniformly in $f$ for $1<p<+\infty$. These conditions are different for $p<2$ and $p>2$. The proofs rely on recent techniques developed to handle operators beyond the class of Calderón-Zygmund operators. For our purpose, we also prove Littlewood-Paley inequalities and interpolation results for Sobolev spaces in this context, which are of independent interest.

We associate a Cantor dynamical system to a non-properly ordered Bratteli diagram. Group valued edge labellings $\lambda$ of a Bratteli diagram $B$ give rise to a skew-product Bratteli diagram $B(\lambda)$ on which the group acts. The quotient by the group action of the associated dynamics can be a nontrivial extension of the dynamics of $B$. We exhibit a Bratteli diagram for this quotient and construct a morphism to $B$ with unique path lifting property. This is shown to be an isomorphism for the dynamics if a property ``loops lifting to loops'' is satisfied by $B(\lambda )\to B$.

We study the Dirichlet problem for a class of nonlinear parabolic equations with nonstandard anisotropic growth conditions. Equations of this class generalize the evolutional $p(x,t)$-Laplacian. We prove theorems of existence and uniqueness of weak solutions in suitable Orlicz-Sobolev spaces, derive global and local in time $L^{\infty}$ bounds for the weak solutions.

An element of a group is said to be reversible if it is conjugate to its inverse. We characterise the reversible elements in the group of diffeomorphisms of the real line, and in the subgroup of order preserving diffeomorphisms.

We characterize the contractions that are similar to the backward shift in the Hardy space $H^2$. This characterization is given in terms of the geometry of the eigenvector bundles of the operators.

Let the group $G=AB$ be a product of two $\pi$-decomposable subgroups $A=O_\pi(A) \times O_{\pi'}(A)$ and $B=O_\pi(B) \times O_{\pi'}(B)$ where $\pi$ is a set of primes. The authors conjecture that $O_\pi(A)O_\pi(B)=O_\pi(B)O_\pi(A)$ if $\pi$ is a set of odd primes. In this paper it is proved that the conjecture is true if $A$ and $B$ are soluble. A similar result with certain additional restrictions holds in the case $2 \in \pi$. Moreover, it is shown that the conjecture holds if $O_{\pi'}(A)$ and $O_{\pi'}(B)$ have coprime orders.

In the setting of a manifold with doubling property satisfying a Gaussian upper estimate of the heat kernel, one gives a characterization of the lower Gaussian estimate in terms of certain Hölder inequalities.

In this work we prove that if $S$ is a dual bounded sequence of points in the unit ball ${\mathbb{B}}$ of ${\mathbb{C}}^{n}$ for the Hardy space $H^{p}({\mathbb{B}})$, then $S$ is $H^{s}({\mathbb{B}})$ interpolating with the linear extension property for any $s\in [1,p[$.

We give a cohomological characterization of Gromov relative hyperbolicity. As an application we prove a converse to the combination theorem for graphs of relatively hyperbolic groups given in F. Gautero, "Geodesics in trees of hyperbolic and relatively hyperbolic groups," Preprint. We build upon and follow the ideas of the work of S. M. Gersten about the same topics in the classical Gromov hyperbolic setting.

We establish a strengthening of Jordan separation, to the setting of maps $f\colon X \rightarrow S^{n+1}$, where $X$ is not necessarily a manifold, and $f$ is not necessarily injective.