This is a survey of higher-dimensional Kleinian groups, i.e., discrete isometry groups of the hyperbolic n-space H for n ≥ 4. Our main emphasis is on the topological and geometric aspects of higher-dimensional Kleinian groups and their contrast with the discrete groups of isometry of H. To the memory of Sasha Reznikov

Abstract Let and $H$?> be locally compact groups with fixed two-sided invariant Haar measures. A polyhomomorphism $G\rightarrowtail H$?> is a closed subgroup $R\subset G\times measure, whose marginals on are dominated by the measures . can regarded as multi-valued map sending points to sets equipped ‘uniform’ For two polyhomomorphisms ${G\rightarrowtail H}$?> $H\rightarrowtail K$?...

In this paper, we give a probabilistic counterpart of  Mazur-Ulam theorem  in probabilistic normed groups. We show, under some conditions, that every surjective isometry between two probabilistic normed groups is a homomorphism.

An atomic decomposition is proved for Banach spaces which satisfy some affine geometric axioms compatible with notions from the quantum mechanical measuring process. This is then applied to yield, under appropriate assumptions, geometric characterizations, up to isometry, of the unit ball of the dual space of a JB∗-triple, and up to complete isometry, of one-sided ideals in C∗-algebras.

In this article we prove the problem on isometric mappings of S posed by Th. M. Rassias. We prove that any map f : S → S, p ≥ n > 1, preserving two angles θ and mθ (mθ < π, m > 1) is an isometry. With the assumption of continuity we prove that any map f : S → S preserving an irrational angle is an isometry.

In this paper we investigatet the efficiency of the Orthogonal Matching Pursuit for random dictionaries. We concentrate on dictionaries satisfying Restricted Isometry Property. We introduce a stronger Homogenous Restricted Isometry Property which is satisfied with overwhelming probability for random dictionaries used in compressed sensing. We also present and discuss some open problems about OMP.