# Volume 66, № 11, 2014

### Mixed Boundary-Value Problem for Linear Second-Order Nondivergent Parabolic Equations with Discontinuous Coefficients

Guliyev A. F., Ismayilova S. H.

Ukr. Mat. Zh. - 2014. - 66, № 11. - pp. 1443-1462

The mixed boundary-value problem is considered for linear second-order nondivergent parabolic equations with discontinuous coefficients satisfying the Cordes conditions. The one-valued strong (almost everywhere) solvability of this problem is proved in the space $Ŵ_p^{2,1}$, where $p$ belongs to the same segment containing point 2.

### Estimation of the Reachable Set for the Problem of Vibrating Kirchhoff Plate

Ukr. Mat. Zh. - 2014. - 66, № 11. - pp. 1463-1476

We consider a dynamical system with distributed parameters for the description of controlled vibrations of a Kirchhoff plate without polar moment of inertia. A class of optimal controls corresponding to finite-dimensional approximations is used to study the reachable set. Analytic estimates for the norm of these control functions are obtained depending on the boundary conditions. These estimates are used to study the reachable set for the infinite-dimensional system. For a model with incommensurable frequencies, an estimate of the reachable set is obtained under the condition of power decay of the amplitudes o generalized coordinates.

### A Generalization of Lifting Modules

Ukr. Mat. Zh. - 2014. - 66, № 11. - pp. 1477–1484

We introduce the notion of $I$ -lifting modules as a proper generalization of the notion of lifting modules and present some properties of this class of modules. It is shown that if $M$ is an $I$ -lifting direct projective module, then $S/▽$ is regular and $▽ = \text{Jac} S$, where $S$ is the ring of all $R$-endomorphisms of $M$ and $▽ = \{ϕ ∈ S | Im ϕ ≪ M\}$. Moreover, we prove that if $M$ is a projective $I$ -lifting module, then $M$ is a direct sum of cyclic modules. The connections between $I$ -lifting modules and dual Rickart modules are presented.

### Complete Volterra Integrodifferential Equations of the Second Order Unsolved with Respect to the Higher Derivative

Kopachevskii N. D., Semkina E. V.

Ukr. Mat. Zh. - 2014. - 66, № 11. - pp. 1485-1497

We prove the theorem on solvability of the Cauchy problem for the complete Volterra integrodifferential linear equations of the second order in a Hilbert space. For a complete equation, we consider three main classes of equations depending on the ordering of operator coefficients.

### Approximation of Functions on the Sphere by Linear Methods

Ukr. Mat. Zh. - 2014. - 66, № 11. - pp. 1498-1511

We study some problems of approximation of functions by the linear methods of summation of their Fourier–Laplace series.

### Analysis of the Set of Trajectories of Fuzzy Equations of Perturbed Motion

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

Ukr. Mat. Zh. - 2014. - 66, № 11. - pp. 1512–1527

The paper presents a new approach to the investigation of the first-order fuzzy initial-value problems. We use different versions of the comparison principle to establish conditions for the existence of solutions of a set of differential equations.

### Strongly Semicommutative Rings Relative to a Monoid

Ukr. Mat. Zh. - 2014. - 66, № 11. - pp. 1528–1539

For a monoid *M,* we introduce strongly *M*-semicommutative rings obtained as a generalization of strongly semicommutative rings and investigate their properties. We show that if *G* is a finitely generated Abelian group, then *G* is torsion free if and only if there exists a ring *R* with *|R| ≥* 2 such that *R* is strongly *G*-semicommutative.

### Best Approximations for the Cauchy Kernel on the Real Axis

Chaichenko S. O., Savchuk V. V.

Ukr. Mat. Zh. - 2014. - 66, № 11. - pp. 1540–1549

We compute the values of the best approximations for the Cauchy kernel on the real axis $ℝ$ by some subspaces from $L_q (ℝ)$. This result is applied to the evaluation of the sharp upper bounds for pointwise deviations of certain interpolation operators with interpolation nodes in the upper half plane and certain linear means of the Fourier series in the Takenaka–Malmquist system from the functions lying in the unit ball of the Hardy space $H_p,\; 2 ≤ p < ∞$.

### Big Picard Theorem for Meromorphic Mappings with Moving Hyperplanes in $P_n (C)$

Ukr. Mat. Zh. - 2014. - 66, № 11. - pp. 1550–1562

We present some extension theorems in the style of the Big Picard theorem for meromorphic mappings of $C_m$ into $P_n (C)$ with a few moving hyperplanes.

### Averaging of Impulsive Differential Inclusions with Fuzzy Right-Hand Sides

Ukr. Mat. Zh. - 2014. - 66, № 11. - pp. 1563–1577

We substantiate the possibility of application of the method of averaging on a finite interval to impulsive differential inclusions with fuzzy right-hand sides containing a small parameter. In the case of periodic right-hand sides, it is shown that the estimate can be improved.

### On the Dirichlet Kernels with Respect to Certain Special Representative Product Systems

Ukr. Mat. Zh. - 2014. - 66, № 11. - pp. 1578–1584

The Fourier analysis uses the calculations with kernel functions from the very beginning. The maximal values of the *n* th Dirichlet kernels divided by *n* for the Walsh–Paley, “classical” Vilenkin, and some other systems are 1. In the paper, we deal with some more general systems and use the accumulated results to develop the methods aimed at determination of the properties of specific systems. In these cases, the situation with \( \frac{D_n}{n} \) may be different.