# Volume 64, № 1, 2012

### On the solution of a boundary-value problem for a third-order equation with multiple characteristics

Ukr. Mat. Zh. - 2012. - 64, № 1. - pp. 3-13

We consider the first boundary-value problem for the third-order equation with multiple characteristics $u_{x x x} - u_{y y} = f (x,y)$ in the domain $D = \{ ( x , y ) : 0 < x < p, 0 < y < l\}$ The uniqueness of a solution is proved by the energy-integral method, and the solution is constructed in explicit form with the use of the Green function.

### On modules over group rings of nilpotent groups

Ukr. Mat. Zh. - 2012. - 64, № 1. - pp. 13-23

We study an $\mathbf{R}G$-module $A$, where $\mathbf{R}$ is a ring, $A/C_A(G)$ is not a minimax $\mathbf{R}$-module, $C_A(G) = 1$, and $G$ is a nilpotent group. Let $\mathfrak{L}_{nm}(G)$ be the system of all subgroups $H \leq G$ such that the quotient modules $A/C_A(G)$ are not minimax $\mathbf{R}$-modules. We investigate a $\mathbf{R}G$ - module $A$ such that $\mathfrak{L}_{nm}(G)$ satisfies either the weak minimal condition or the weak maximal condition as an ordered set. It is proved that a nilpotent group $G$ that satisfies these conditions is a minimax group.

### On one Dubinin extreme problem

Ukr. Mat. Zh. - 2012. - 64, № 1. - pp. 24-31

We obtain a particular solution of the known conjecture of V. N. Dubinin about nonoverlapping domains on a complex area.

### On the theory of $\mathcal{PT}$-symmetric operatorss

Ukr. Mat. Zh. - 2012. - 64, № 1. - pp. 32-49

This article develops a general theory of $\mathcal{PT}$-symmetric operators. Special attention is given to $\mathcal{PT}$-symmetric quasi-self-adjoint extensions of symmetric operator with deficiency indices 〈 2, 2 〉. For these extensions, the possibility of their interpretation as self-adjoint operators in Krein spaces is investigated, and a description of nonreal eigenvalues is given. These abstract results are applied to the Schrodinger operator with Coulomb potential on the real axis.

### Stability of motion of nonlinear systems with fuzzy characteristics of parameters

Martynyuk A. A., Martynyuk-Chernienko Yu. A.

Ukr. Mat. Zh. - 2012. - 64, № 1. - pp. 50-70

We investigate the stability of a stationary solution of a fuzzy dynamical system by a generalized Lyapunov direct method.

### Problem with nonlocal condition on the free boundary

Ukr. Mat. Zh. - 2012. - 64, № 1. - pp. 71-80

We investigate the one-phase Florin problem for a parabolic equation with nonlocal condition. Theorems one the existence and uniqueness of a solution are proved, and a priori estimates for the solution are obtained.

### On the generalized convolution for $F_c$, $F_c$, and $K - L$ integral transforms

Ukr. Mat. Zh. - 2012. - 64, № 1. - pp. 81-91

We study new generalized convolutions $f \overset{\gamma}{*} g$ with weight function $\gamma(y) = y$ for the Fourier cosine, Fourier sine, and Kontorovich-Lebedev integral transforms in weighted function spaces with two parameters $L(\mathbb{R}_{+}, x^{\alpha} e^{-\beta x} dx)$. These generalized convolutions satisfy the factorization equalities $$F_{\left\{\frac SC\right\}} (f \overset{\gamma}{*} g)_{\left\{\frac 12\right\}}(y) = y (F_{\left\{\frac SC\right\}} f)(y) (K_{sy}g) \quad \forall y > 0$$ We establish a relationship between these generalized convolutions and known convolutions, and also relations that associate them with other convolution operators. As an example, we use these new generalized convolutions for the solution of a class of integral equations with Toeplitz-plus-Hankel kernels and a class of systems of two integral equations with Toeplitz-plus-Hankel kernels.

### $S\Phi$-Supplemented subgroups of finite groups

Ukr. Mat. Zh. - 2012. - 64, № 1. - pp. 92-99

We call $H$ an $S\Phi$-supplemented subgroup of a finite group $G$ if there exists a subnormal subgroup $T$ of $G$ such that $G = HT$ and $H \bigcap T \leq \Phi(H)$, where $\Phi(Н)$ is the Frattini subgroup of $H$. In this paper, we characterize the $p$-nilpotency and supersolubility of a finite group $G$ under the assumption that every subgroup of a Sylow $p$-subgroup of $G$ with given order is $S\Phi$-supplemented in $G$. Some results about formations are also obtained.

### Iteration process for multiple Rogers-Ramanujan identities

Ukr. Mat. Zh. - 2012. - 64, № 1. - pp. 100-125

Replacing the monomials by an arbitrary sequence in the recursive lemma found by Bressoud (1983), we establish several general transformation formulas from unilateral multiple basic hypergeometric series to bilateral univariate ones, which are then used for the derivation of numerous multiple series identities of Rogers-Ramanujan type.

### Commutative domains of elementary divisors and some properties of their elements

Ukr. Mat. Zh. - 2012. - 64, № 1. - pp. 126-139

We study commutative domains of elementary divisors from the viewpoint of investigation of the structure of invertible matrices that reduce a given matrix to the diagonal form. Some properties of elements of these domains are indicated. We establish conditions, close to the stable-rank conditions, under which a commutative Bezout domain is a domain of ´ elementary divisors.

### Best approximation of periodic functions of several variables from the classes $MB^{\omega}_{p,\theta}$

Ukr. Mat. Zh. - 2012. - 64, № 1. - pp. 140-144

We obtain exact-order estimates for the best approximation of periodic functions of several variables from the classes $MB^{\omega}_{p,\theta}$ by trigonometric polynomials with the "numbers" of harmonics from graded hyperbolic crosses in the metric of the space $L_q$.