for a bounded linear operator on hilbert space we define a sequence of the so-called weakly extremal vectors‎. ‎we study the properties of weakly extremal vectors and show that the orthogonality equation is valid for weakly extremal vectors‎. ‎also we show that any quasinilpotent operator $t$ has an hypernoncyclic vector‎, ‎and so $t$ has a nontrivial hyperinvariant subspace‎.

‎This paper presents two main results that the singular values of the Hadamard product of normal matrices $A_i$ are weakly log-majorized by the singular values of the Hadamard product of $|A_{i}|$ and the singular values of the sum of normal matrices $A_i$ are weakly log-majorized by the singular values of the sum of $|A_{i}|$‎. ‎Some applications to these inequalities are also given‎. ‎In addi...

Answering a question of D.Dikranjan and I.Protasov we prove that each infinite group G contains a weakly P-small subset A ⊂ G which is not P-small. This means that for every n there are n pairwise disjoint translation copies of A in G but there is no infinitely many such disjoint copies. In this paper we answer a question of D.Dikranjan and I.Protasov concerning the relation between P-small and...

Let $WS(X, d)$ be the class of ultrametric spaces which are weakly similar to space $(X, d)$. The main results paper completely describe for equality $$ \rho(x, y) = f(d(\Phi(x), \Phi(y))) holds every $(Y, \rho) \in WS(X, d)$, weak similarity $\Phi \colon Y \to X$, and all $x$, $y Y$ with some (pseudoultrametric) preserving function $f$ depending on $\Phi$.

We study the complexity of approximation for a weighted counting constraint satisfaction problem #CSP(F). In the conservative case, where F contains all unary functions, a classification is known for the Boolean domain. We give a classification for problems with general finite domain. We define weak log-modularity and weak log-supermodularity, and show that #CSP(F) is in FP if F is weakly log-m...

Let X be a reflexive Banach space which has a weakly sequentially continuous duality mapping. In this paper, we consider the following viscosity approximation sequence x n λ n fx n 1−λ n T n x n , where λ n ∈ 0, 1, {T n } is a uniformly asymptotically regular sequence, and f is a weakly contractive mapping. Strong convergence of the sequence {x n } is proved.