# Volume 64, № 2, 2012

### Value-sharing problem for *p*-adic meromorphic functions and their difference operators and difference polynomials

Ukr. Mat. Zh. - 2012. - 64, № 2. - pp. 147-164

We discuss the value-sharing problem, versions of the Hayman conjecture, and the uniqueness problem for *p*-adic
meromorphic functions and their difference operators and difference polynomials.

### A result on generalized derivations on right ideals of prime rings

Ukr. Mat. Zh. - 2012. - 64, № 2. - pp. 165-175

Let $R$ be a prime ring of characteristic not 2 and let $I$ be a nonzero right ideal of $R$. Let $U$ be the right Utumi quotient ring of $R$ and let $C$ be the center of $U$. If $G$ is a generalized derivation of $R$ such that $[[G(x), x], G(x)] = 0$ for all $x \in I$, then $R$ is commutative or there exist $a, b \in U$ such that $G(x) = ax + xb$ for all $x \in R$ and one of the following assertions is true: $$(1)\quad (a - \lambda)I = (0) = (b + \lambda)I \;\;\text{for some}\; \lambda \in C,$$ $$(2)\quad (a - \lambda)I = (0) \;\;\text{for some}\; \lambda \in C \;\;\text{and}\; b \in C.$$

### Classification of finite commutative semigroups for which the inverse monoid of local automorphisms is permutable

Ukr. Mat. Zh. - 2012. - 64, № 2. - pp. 176-184

We give a classification of finite commutative semigroups for which the inverse monoid of local automorphisms is permutable.

### Vector bundles over noncommutative nodal curves

Ukr. Mat. Zh. - 2012. - 64, № 2. - pp. 185-199

We describe vector bundles over a class of noncommutative curves, namely, over noncommutative nodal curves of string type and of almost string type. We also prove that, in other cases, the classification of vector bundles over a noncommutative curve is a wild problem.

### On Agarwal - Pang-type integral inequalities

Ukr. Mat. Zh. - 2012. - 64, № 2. - pp. 199-209

We establish some new Agarwal – Pang-type inequalities involving second-order partial derivatives. Our results in special cases yield some of interrelated results and provide new estimates for inequalities of this type.

### Recognition of the groups $L_5(4)$ and $U_4(4)$ by the prime graph

Darafsheh M. R., Nosratpour P.

Ukr. Mat. Zh. - 2012. - 64, № 2. - pp. 210-217

Let $G$ be a finite group. The prime graph of $G$ is the graph $\Gamma(G)$ whose vertex set is the set $\Pi(G)$ of all prime divisors of the order $|G|$ and two distinct vertices $p$ and $q$ of which are adjacent by an edge if $G$ has an element of order $pq$. We prove that if $S$ denotes one of the simple groups $L_5(4)$ and $U_4(4)$ and if $G$ is a finite group with $\Gamma(G) = \Gamma(S)$, then $G$ has a $G$ normal subgroup $N$ such that $\Pi(N) \subseteq \{2, 3, 5\}$ and $\cfrac GN \cong S$.

### On equalities involving integrals of the logarithm of the Riemann ζ-function and equivalent to the Riemann hypothesis

Beltraminelli S., Merlini D., Sekatskii S. K.

Ukr. Mat. Zh. - 2012. - 64, № 2. - pp. 218-228

Using the generalized Littlewood theorem about a contour integral involving the logarithm of an analytical function, we show how an infinite number of integral equalities involving integrals of the logarithm of the Riemann ζ-function and equivalent to the Riemann hypothesis can be established and present some of them as an example. It is shown that all earlier known equalities of this type, viz., the Wang equality, Volchkov equality, Balazard-Saias-Yor equality, and an equality established by one of the authors, are certain particular cases of our general approach.

### Investigation of solutions of boundary-value problems with essentially infinite-dimensional elliptic operator

Ukr. Mat. Zh. - 2012. - 64, № 2. - pp. 229-236

We consider Dirichlet problems for the Poisson equation and linear and nonlinear equations with essentially infinite-dimensional elliptic operator (of the Laplace -Levy type). The continuous dependence of solutions on boundary values and sufficient conditions for increasing the smoothness of solutions are investigated.

### Boundary-value problems for a nonlinear hyperbolic equation with divergent part and Levy Laplacian

Ukr. Mat. Zh. - 2012. - 64, № 2. - pp. 237-244

We propose an algorithm for the solution of the boundary-value problem $U(0,x) = u_0,\;\; U(t, 0) = u_1$ and the external boundary-value problem $U(0, x) = v_0, \;\;U(t, x) |_{\Gamma} = v_1, \;\; \lim_{||x||_H \rightarrow \infty} U(t, x) = v_2$ for the nonlinear hyperbolic equation $$\frac{\partial}{\partial t}\left[k(U(t,x))\frac{\partial U(t,x)}{\partial t}\right] = \Delta_L U(t,x)$$ with divergent part and infinite-dimensional Levy Laplacian $\Delta_L$.

### On Shiba - Waterman space

Ukr. Mat. Zh. - 2012. - 64, № 2. - pp. 245-252

We give a necessary and sufficient condition for the inclusion of $\Lambda BV^{(p)}$ in the classes $H^q_{\omega}$.

### Canonical form of polynomial matrices with all identical elementary divisors

Ukr. Mat. Zh. - 2012. - 64, № 2. - pp. 253-267

The problem of reducing polynomial matrices to the canonical form by using semiscalar equivalent transformations is studied. A class of polynomial matrices is singled out, for which the canonical form with respect to semiscalar equivalence is indicated. This form enables one to solve the classification problem for collections of matrices over a field up to similarity.

### Control of linear dynamical systems by time transformations

Ukr. Mat. Zh. - 2012. - 64, № 2. - pp. 268-274

Necessary and sufficient conditions for the controllability of solutions of linear inhomogeneous integral equations are obtained.

### On the complexity of the ideal of absolute null sets

Ukr. Mat. Zh. - 2012. - 64, № 2. - pp. 275-276

Answering a question of Banakh and Lyaskovska, we prove that for an arbitrary countable infinite amenable group $G$ the ideal of sets having $\mu$-measure zero for every Banach measure $\mu$ on $G$ is an $F_{\sigma \delta}$ subset of $\{0,1\}^G$.

### Spectral problem for discontinuous integro-differential operator

Ukr. Mat. Zh. - 2012. - 64, № 2. - pp. 277-282

A representation of solutions of a discontinuous integro-differential operator is obtained. The asymptotic behavior of the eigenvalues and eigenfunctions of this operator is described.

### Diagonalizability of matrices over a principal ideal domain

Ukr. Mat. Zh. - 2012. - 64, № 2. - pp. 283-288

A square matrix is said to be diagonalizable if it is similar to a diagonal matrix. We establish necessary and sufficient conditions for the diagonalizability of matrices over a principal ideal domain.