# Volume 63, № 4, 2011

### Conditions of smoothness for the distribution density of a solution of a multidimensional linear stochastic differential equation with levy noise

Ukr. Mat. Zh. - 2011. - 63, № 4. - pp. 435-447

A sufficient condition is obtained for smoothness of the density of distribution for a multidimensional Levy-driven Ornstein-Uhlenbeck process, i.e., a solution to a linear stochastic differential equation with Levy noise.

### On the mean value of the function $\overline{S}_k(n)$

Ukr. Mat. Zh. - 2011. - 63, № 4. - pp. 448-458

An asymptotic formula is constructed for a mean value of the function $\overline{S}_k(n)$ which is dual to the Smarandache function $S_k(n)$. $O$- and $\Omega$-estimates for the second moment of the remainder term are obtained.

### On hyperholomorphic functions of the space variable

Ukr. Mat. Zh. - 2011. - 63, № 4. - pp. 459-465

For quaternionic-differentiable functions of a spatial variable, we prove a theorem on an integral over a closed surface. This theorem is an analog of the Cauchy theorem from complex analysis.

### On some criteria of convexity for compact sets

Tkachuk M. V., Vyhovs'ka I. Yu., Zelinskii Yu. B.

Ukr. Mat. Zh. - 2011. - 63, № 4. - pp. 466-471

We establish some criteria of convexity of compact sets in the Euclidean space. Analogs of these results are extended to complex and hypercomplex cases.

### On positive solutions of one class of evolutionary inclusions of the subdifferential type

Kapustyan O. V., Shklyar T. B.

Ukr. Mat. Zh. - 2011. - 63, № 4. - pp. 472-480

Sufficient conditions of the existence of a nonnegative solution are obtained for an evolution inclusion of subdifferential type with multivalued non-Lipschitz perturbation. Under the additional condition of dissipativity, the existence of the global attractor in the class of nonnegative functions is proved.

### Asymptotic behavior of generalized quasiisometries at a point

Kovtonyuk D. A., Salimov R. R.

Ukr. Mat. Zh. - 2011. - 63, № 4. - pp. 481-488

We consider $Q$-homeomorphisms with respect to the $p$-modulus. An estimate for a measure of a ball image is obtained under such mappings and the asymptotic behavior at zero is investigated.

### On some imbedding relations between certain sequence spaces

Ukr. Mat. Zh. - 2011. - 63, № 4. - pp. 489-501

In the present paper, we introduce the sequence space $l^{\lambda}_p$ of non-absolute type which is a $p$-normed space and a $BK$-space in the cases of $0 < p < 1$ and $0 < p < 1$ i $1 \leq p < \infty$, respectively. Further, we derive some imbedding relations and construct the basis for the space $l^{\lambda}_p$, where $1 \leq p < \infty$.

### Properties of a certain product of submodules

Heidari S., Nikandish R., Nikmehr M. J.

Ukr. Mat. Zh. - 2011. - 63, № 4. - pp. 502-512

Let $R$ be a commutative ring with identity, $M$ an $R$-module and $K_1,..., K_n$ submodules of $M$. In this article, we construct an algebraic object, called product of $K_1,..., K_n$. We equipped this structure with appropriate operations to get an $R(M)$-module. It is shown that $R(M)$-module $M^n = M... M$ and $R$-module $M$ inherit some of the most important properties of each other. For example, we show that $M$ is a projective (flat) $R$-module if and only if $M^n$ is a projective (flat) $R(M)$-module.

### Resonance elliptic variational inequalities with discontinuous nonlinearities of linear growth

Ukr. Mat. Zh. - 2011. - 63, № 4. - pp. 513-522

We consider resonance elliptic variational inequalities with second-order differential operators and discontinuous nonlinearity of linear grows. The theorem on the existence of a strong solution is obtained. The initial problem is reduced to the problem of the existence of a fixed point in a compact multivalued mapping and then, with the use of the Leray - Schauder method, the existence of the fixed point is established.

### On the reconstruction of the variation of the metric tensor of a surface on the basis of a given variation of christoffel symbols of the second kind under infinitesimal deformations of surfaces in the euclidean space $E_3$

Ukr. Mat. Zh. - 2011. - 63, № 4. - pp. 523-530

We investigate the problem of reconstruction of variation of a metric tensor of a surface on the basis of given variation of the sekond-kind Christoffel symbols for infinitesimal deformations of surfaces in the Euclidean space $E_3$.

### Landau-Kolmogorov problem for a class of functions absolutely monotone on a finite interval

Ukr. Mat. Zh. - 2011. - 63, № 4. - pp. 531-548

We solve the Landau - Kolmogorov problem for the class of functions absolutely monotone on a finite interval. For this class of functions, a new exact additive inequalities of the Kolmogorov type are obtained.

### Best $m$-term approximation of the classes $B ^{r}_{\infty, \theta}$ of functions of many variables by polynomials in the haar system

Ukr. Mat. Zh. - 2011. - 63, № 4. - pp. 549-555

We obtain the exact-order estimate for the best $m$-term approximation of the classes $B ^{r}_{\infty, \theta}$ of periodic functions of many variables by polynomials with respect to the Haar system in the metric of the space $L_q,\quad 1 < q < \infty$.

### On uniqueness theorems for holomorphic curves sharing hypersurfaces without counting multiplicity

Ukr. Mat. Zh. - 2011. - 63, № 4. - pp. 556-565

We prove some uniqueness theorems for algebraically nondegenerate holomorphic curves sharing hyper-surfaces ignoring multiplicity.

### On the fredholm theory of a planar problem with shift for a pair of functions

Lysenko Z. M., Matviyuk L. V., Nechaev L. V., Shvets V. T.

Ukr. Mat. Zh. - 2011. - 63, № 4. - pp. 566-571

We obtain necessary and sufficient conditions of the Fredholm properties and the formula for the calculation of index of a planar problem with shift and conjugation for a pair of functions.

### Multidimensional random motion with uniformly distributed changes of direction and Erlang steps

Pogorui A. О., Rodriguez-Dagnino R. M.

Ukr. Mat. Zh. - 2011. - 63, № 4. - pp. 572-577

In this paper we study transport processes in $\mathbb{R}^n,\quad n \geq 1$, having non-exponential distributed sojourn times or non-Markovian step durations. We use the idea that the probabilistic properties of a random vector are completely determined by those of its projection on a fixed line, and using this idea we avoid many of the difficulties appearing in the analysis of these problems in higher dimensions. As a particular case, we find the probability density function in three dimensions for 2-Erlang distributed sojourn times.