# Volume 66, № 10, 2014

### Asymptotic Solutions of the Dirichlet Problem for the Heat Equation at a Characteristic Point

Antoniouk A. Vict., Kiselev O. M., Tarkhanov N. N.

Ukr. Mat. Zh. - 2014. - 66, № 10. - pp. 1299–1317

The Dirichlet problem for the heat equation in a bounded domain $G ⊂ ℝ^{n+1}$ is characteristic because there are boundary points at which the boundary touches a characteristic hyperplane $t = c$, where c is a constant. For the first time, necessary and sufficient conditions on the boundary guaranteeing that the solution is continuous up to the characteristic point were established by Petrovskii (1934) under the assumption that the Dirichlet data are continuous. The appearance of Petrovskii’s paper was stimulated by the existing interest to the investigation of general boundary-value problems for parabolic equations in bounded domains. We contribute to the study of this problem by finding a formal solution of the Dirichlet problem for the heat equation in a neighborhood of a cuspidal characteristic boundary point and analyzing its asymptotic behavior.

### Boundary-Value Problem for a Degenerate High-Odd-Order Equation

Ukr. Mat. Zh. - 2014. - 66, № 10. - pp. 1318–1331

We consider a boundary-value problem for a degenerate high-odd-order equation. The uniqueness of the solution is shown by the method of energy integrals. The solution is constructed by the method of separation of variables. In this case, we get the eigenvalue problem for a degenerate even-order ordinary differential equation. The existence of eigenvalues is proved by means of reduction to the integral equation.

### On One Minkowski–Radon Problem and Its Generalizations

Savost’yanova I. M., Volchkov V. V.

Ukr. Mat. Zh. - 2014. - 66, № 10. - pp. 1332–1347

We study functions on a sphere with zero weighted means over the circles of fixed radius. A description of these functions is obtained in the form of series in special functions.

### Linear Methods for Summing Fourier Series and Approximation in Weighted Lebesgue Spaces with Variable Exponents

Ukr. Mat. Zh. - 2014. - 66, № 10. - pp. 1348–1356

In the present work, we study the estimates for the periodic functions of linear operators constructed on the basis of their Fourier series in weighted Lebesgue spaces with variable exponent and Muckenhoupt weights. In this case, the obtained estimates depend on the sequence of the best approximation in weighted Lebesgue spaces with variable exponent.

### Notes on the Uniqueness and Value Sharing for Meromorphic Functions Concerning Differential Polynomials

Ukr. Mat. Zh. - 2014. - 66, № 10. - pp. 1357–1366

We study the problem of uniqueness of meromorphic functions concerning differential polynomials, and obtain some results. These results improve the results obtained earlier by Li [J. Sichuan Univ. (Natural Science Edition), 45, 21–24 (2008)] and Dyavanal [J. Math. Anal. Appl., 374, 335–345 (2011)].

### Dirichlet Problems for Harmonic Functions in Half Spaces

Ukr. Mat. Zh. - 2014. - 66, № 10. - pp. 1367–1378

In our paper, we prove that if the positive part $u^{+}(x)$ of a harmonic function $u(x)$ in a half space satisfies the condition of slow growth, then its negative part $u^{-}(x)$ can also be dominated by a similar growth condition. Moreover, we give an integral representation of the function $u(x)$. Further, a solution of the Dirichlet problem in the half space for a rapidly growing continuous boundary function is constructed by using the generalized Poisson integral with this boundary function.

### Stability Criteria and Localization of the Matrix Spectrum in Terms of Trace Functions

Ukr. Mat. Zh. - 2014. - 66, № 10. - pp. 1379–1386

New necessary and sufficient conditions for the asymptotic stability and localization of the spectra of linear autonomous systems are proposed by using the matrix trace functions. The application of these conditions is reduced to the solution of two scalar inequalities for a symmetric positive-definite matrix. As a corollary, for linear control systems, we present a procedure aimed at the construction of the set of stabilizing measurable output feedbacks.

### Quasiperiodic Extremals of Nonautonomous Lagrangian Systems on Riemannian Manifolds

Ukr. Mat. Zh. - 2014. - 66, № 10. - pp. 1387–1406

The paper deals with a quasiperiodically excited natural Lagrangian system on a Riemannian manifold. We find sufficient conditions under which this system has a weak Besicovitch quasiperiodic solution minimizing the averaged Lagrangian. It is proved that this solution is indeed a twice continuously differentiable uniformly quasiperiodic function, and the corresponding system in variations is exponentially dichotomous on the real axis.

### Topological Classification of the Oriented Cycles of Linear Mappings

Rybalkina T. V., Sergeychuk V. V.

Ukr. Mat. Zh. - 2014. - 66, № 10. - pp. 1407–1413

We consider oriented cycles of linear mappings over the fields of real and complex numbers. the problem of their classification to within the homeomorphisms of spaces is reduced to the problem of classification of linear operators to within the homeomorphisms of spaces studied by N. Kuiper and J. Robbin in 1973.

### Well-Posedness of the Dirichlet and Poincaré Problems for the Wave Equation in a Many-Dimensional Domain

Ukr. Mat. Zh. - 2014. - 66, № 10. - pp. 1414–1419

We determine a many-dimensional domain in which the Dirichlet and Poincaré problems for the wave equation are uniquely solvable.

### On the Third Moduli of Continuity

Bezkryla S. I., Chaikovs'kyi A. V., Nesterenko A. N.

Ukr. Mat. Zh. - 2014. - 66, № 10. - pp. 1420-1424

An inequality for the third uniform moduli of continuity is proved. This inequality implies that an arbitrary 3-majorant is not necessarily a modulus of continuity of order 3.

### On Weakly (μ, λ)-Open Functions

Ukr. Mat. Zh. - 2014. - 66, № 10. - pp. 1425–1430

We study some characterizations and properties of almost (μ, λ)-open functions. Some conditions are presented under which an almost (μ, λ)-open function is equivalent to a (μ, λ)-open function.

### Factorizations of Finite Groups into $r$-Soluble Subgroups with Given Embeddings

Knyagina V. N., Tyutyanov V. N.

Ukr. Mat. Zh. - 2014. - 66, № 10. - pp. 1431–1435

Let $X$ be a subset of the set of positive integers. A subgroup $H$ of a group $G$ is called $X$-subnormal in $G$ if there exists a chain of subgroups $H = H_0 ⊆ H_1 ⊆ … ⊆ H_n = G$ such that $|H_i : H_{i-1}| ∈ X$ for all $i$. We study the solubility and $r$ -solubility of a finite group $G = AB$ with some restrictions imposed on the subgroups $A$ and $B$ and on the set $X$ .

### Remarks on Certain Identities with Derivations on Semiprime Rings

Baydar N., Fošner A., Strašek R.

Ukr. Mat. Zh. - 2014. - 66, № 10. - pp. 1436–1440

Let $n$ be a fixed positive integer, let $R$ be a $(2n)!$ -torsion-free semiprime ring, let $\alpha$ be an automorphism or an anti-automorphism of $R$, and let $D_1 , D_2 : R → R$ be derivations. We prove the following result: If $(D_1^2 (x) + D_2(x))^n ∘ α(x)^n = 0 $ holds for all $x Є R$, then $D_1 = D_2 = 0$. The same is true if $R$ is a 2-torsion free semiprime ring and F(x) ° β(x) = 0 for all x ∈ R, where $F(x) = (D_1^2 (x) + D_2(x)) ∘ α(x),\; x ∈ R$, and $β$ is any automorphism or antiautomorphism on $R$.