# Volume 67, № 7, 2015

### Classification of Finite Commutative Semigroups for Which the Inverse Monoid of Local Automorphisms is a ∆-Semigroup

Ukr. Mat. Zh. - 2015. - 67, № 7. - pp. 867-873

A semigroup $S$ is called a ∆-semigroup if the lattice of its congruences forms a chain relative to the inclusion. A local automorphism of the semigroup $S$> is called an isomorphism between its two subsemigroups. The set of all local automorphisms of the semigroup $S$ relative to the ordinary operation of composition of binary relations forms an inverse monoid of local automorphisms. We present a classification of finite commutative semigroups for which the inverse monoid of local automorphisms is a ∆-semigroup.

### On the Derived Length of a Finite Group with Complemented Subgroups of Order $p^2$

Knyagina V. N., Monakhov V. S.

Ukr. Mat. Zh. - 2015. - 67, № 7. - pp. 874–881

It is shown that a finite group with complemented subgroups of order $p^2$ is soluble for all $p$ and its derived length does not exceed 4.

### The First Passage Time and Estimation of the Number of Level-Crossings for a Telegraph Process

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

Ukr. Mat. Zh. - 2015. - 67, № 7. - pp. 882-889

It is a well-known result that almost all sample paths of a Brownian motion or Wiener process *{W*(*t*)*}* have infinitely many zero-crossings in the interval (0*, δ*) for * δ >* 0. Under the Kac condition, the telegraph process weakly converges to the Wiener process. We estimate the number of intersections of a level or the number of level-crossings for the telegraph process. Passing to the limit under the Kac condition, we also obtain an estimate of the level-crossings for the Wiener process.

### Dynamical Bifurcation of Multifrequency Oscillations in a Fast-Slow System

Parasyuk I. O., Repeta B. V., Samoilenko A. M.

Ukr. Mat. Zh. - 2015. - 67, № 7. - pp. 890-915

We study a dynamical analog of bifurcations of invariant tori for a system of interconnected fast phase variables and slowly varying parameters. It is shown that, in this system, due to the slow evolution of the parameters, we observe the appearance of transient processes (from the damping process to multifrequency oscillations) asymptotically close to motions on the invariant torus.

### Order Estimates for the Best Orthogonal Trigonometric Approximations of the Classes of Convolutions of Periodic Functions of Low Smoothness

Serdyuk A. S., Stepanyuk T. A.

Ukr. Mat. Zh. - 2015. - 67, № 7. - pp. 916–936

We establish order estimates for the best uniform orthogonal trigonometric approximations on the classes of $2π$-periodic functions whose $(ψ, β)$-derivatives belong to unit balls in the spaces $L_p,\; 1 ≤ p < ∞$, in the case where the sequence $ψ(k)$ is such that the product $ψ(n)n^{1/p}$ may tend to zero slower than any power function and $∑^{∞}_{k=1} ψ^{p′}(k)k^{p′−2} < ∞$ for $1 < p < ∞,\; 1\p+1\p′ = 1$, or $∑^{∞}_{k=1} ψ(k) < ∞$ for $p = 1$. Similar estimates are also established in the $L_s$-metrics, $1 < s ≤ ∞$, for the classes of summable $(ψ, β)$-differentiable functions such that $‖f_{β}^{ψ} ‖1 ≤ 1$.

### Estimation of the Accuracy of Finite-Element Petrov–Galerkin Method in Integrating the One-Dimensional Stationary Convection-Diffusion-Reaction Equation

Ukr. Mat. Zh. - 2015. - 67, № 7. - pp. 937-961

The accuracy and convergence of the numerical solutions of a stationary one-dimensional linear convection-diffusion-reaction equation (with Dirichlet boundary conditions) by the Petrov–Galerkin finiteelement method with piecewise-linear basis functions and piecewise-quadratic weighting functions are analyzed and the accuracy estimates of the method are obtained in certain norms depending on the choice of the collection of stabilization parameters of weight functions.

### Optimal Control over Moving Sources in the Heat Equation

Ukr. Mat. Zh. - 2015. - 67, № 7. - pp. 962-972

We study the problem of optimal control over the processes described by the heat equation and a system of ordinary differential equations. For the problem of optimal control, we prove the existence and uniqueness of solutions, establish sufficient conditions for the Fréchet differentiability of the purpose functional, deduce the expression for its gradient, and obtain necessary conditions of optimality in the form of an integral maximum principle.

### Two Theorems of Complex Analysis

Ukr. Mat. Zh. - 2015. - 67, № 7. - pp. 973-980

We prove two fundamental theorems of multidimensional complex analysis by the methods of this analysis without using the theory of subharmonic functions. As a single violation, we can mention the use of Green’s formula.

### On Fundamental Theorems for Holomorphic Curves on the Annuli

Ukr. Mat. Zh. - 2015. - 67, № 7. - pp. 981-994

We prove some fundamental theorems for holomorphic curves on the annuli crossing a finite set of fixed hyperplanes in the general position in $ℙ_n (ℂ)$ with ramification.

### Motornyi Vitalii Pavlovych (on his 75th birthday)

Babenko V. F., Davydov O. V., Kofanov V. A., Parfinovych N. V., Pas'ko A. N., Romanyuk A. S., Ruban V. I., Samoilenko A. M., Shevchuk I. A., Shumeiko A. A., Timan M. P., Trigub R. M., Vakarchuk S. B., Velikin V. L.

Ukr. Mat. Zh. - 2015. - 67, № 7. - pp. 995-999

### On Stability of the Cauchy Equation on Solvable Groups

Ukr. Mat. Zh. - 2015. - 67, № 7. - pp. 1000-1005

The notion of $(ψ, γ)$-stability was introduced in [V. A. Faiziev, Th. M. Rassias, and P. K. Sahoo, *Trans. Amer. Math. Soc.*, 354, 4455 (2002)]. It was shown that the Cauchy equation $f (xy) = f (x) + f (y)$ is $(ψ, γ)$-stable both on any Abelian group and on any meta-Abelian group. In [V. A. Faiziev and P. K. Sahoo, *Publ. Math. Debrecen*, 75, 6 (2009)], it was proved that the Cauchy equation is $(ψ, γ)$-stable on step-two solvable groups and step-three nilpotent groups. In the present paper, we prove a more general result and show that the Cauchy equation is $(ψ, γ)$-stable on solvable groups.

### On a Factorizable Group with Large Cyclic Subgroups in Factors

Ukr. Mat. Zh. - 2015. - 67, № 7. - pp. 1006-1008

We prove the supersolvability of a finite factorizable group $G = G_1 G_2 ...G_n$ with pairwise permutable factors each of which has a cyclic subgroup of odd order $H_i$ and $|G_i : H_i | ≤ 2$.