2017
Том 69
№ 5

All Issues

Volume 68, № 2, 2016

Article (Ukrainian)

The coefficients of power expansion and $a$-points of an entire function with Borel exceptional value

Andrusyak I. V., Filevych P. V.

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 2. - pp. 147-155

For entire functions with Borel exceptional values, we establish the relationship between the rate of approaching $\infty$ for the sequence of their $a$-points and the rate of approaching 0 for the sequence of their Taylor coefficients.

Article (Ukrainian)

Sequential closure of the space of jointly continuous functions in the space of separately continuous functions

Maslyuchenko V. K., Voloshyn H. A.

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 2. - pp. 156-161

Given compact spaces $X$ and $Y$, we study the space $S(X \times Y )$ of separately continuous functions $f : X \times Y \rightarrow R$ endowed with the locally convex topology generated by the seminorms $|| f||^x = \mathrm{max}_{y \in Y} |f(x, y)|,\; x \in X$, and $|| f||_y = \mathrm{max}_{x \in X} |f(x, y)|,\; y \in Y$. Under the assumption that the compact space $X$ is metrizable, we prove that a separately continuous function $f : X \times Y \rightarrow R$ is the limit of a sequence $(f_n)^{\infty}_{n=1}$ of jointly continuous function $f_n : X \times Y \rightarrow R$ in $S(X \times Y )$ provided that the set $D(f)$ of discontinuity points of $f$ has countable projections on $X$.

Article (Russian)

Initial-boundary-value problem for a semilinear parabolic equation with nonlinear nonlocal boundary conditions

Gladkov A. L., Kavitova T. V.

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 2. - pp. 162-174

We consider an initial-boundary-value problem for a semilinear parabolic equation with nonlinear nonlocal boundary conditions. We prove comparison principle, establish the existence of a local solution, and study the problem of uniqueness and nonuniqueness.

Article (English)

Lehmer sequences in finite groups

Deveci Ö., Karaduman E.

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 2. - pp. 175-182

We study the Lehmer sequences modulo $m$. Moreover, we define the Lehmer orbit and the basic Lehmer orbit of a 2-generator group $G$ for a generating pair $(x, y) \in G$ and examine the lengths of the periods of these orbits. Furthermore, we obtain the Lehmer lengths and the basic Lehmer lengths of the Fox groups $G_{1,t}$ for $t \geq 3$.

Article (English)

Generalized derivations and commuting additive maps on multilinear polynomials in prime rings

De Filippis V., Dhara B., Scudo G.

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 2. - pp. 183-201

Let $R$ be a prime ring with characteristic different from $2, U$ be its right Utumi quotient ring, $C$ be its extended centroid, $F$ and $G$ be additive maps on $R$ , $f(x_1, ..., x_n)$ be a multilinear polynomial over $C$, and $I$ be a nonzero right ideal of $R$ . We obtain information about the structure of $R$ and describe the form of $F$ and $G$ in the following cases: $$(1) [(F^2 + G)(f(r_1, ..., r_n)), f(r_1, ..., r_n)] = 0$$ for all $r_1, . . . , r_n \in R$, where $F$ and $G$ are generalized derivations of $R$ ; $$(2) [(F^2 + G)(f(r_1, ..., r_n)), f(r_1, ..., r_n)] = 0$$for all $r_1, ..., r_n \in I$, where $F$ and $G$ are derivations of $R$.

Article (Russian)

Preacyclicity over the rings with infinite fields of residues

Zainalov B. R.

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 2. - pp. 202-216

We consider a simplicial scheme, its geometric realization, and the required properties of the simplicial scheme of unimodal frames and prove both a sufficiently strong theorem on acyclicity for the simplicial scheme of unimodal frames and a theorem stating that the first nontrivial group of homologies is generated by standard cycles over the rings with infinite fields of residues.

Article (Ukrainian)

Topological conjugate piecewise linear unimodal mappings of an interval into itself

Kirichenko V. V., Plakhotnyk M. V.

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 2. - pp. 217-226

Let $f, g : [0, 1] \rightarrow [0, 1]$ be a pair of continuous piecewise linear unimodal mappings and let $f$ be defined as follows: $f(x) = 2x$ for $x \leq 1/2$ and $f(x) = 2 - 2x$ for $x > 1/2$. Also let $h : [0, 1] \rightarrow [0, 1]$ be a piecewise differentiable homeomorphism such that $fh = hg$. Then $h$ is piecewise linear and the mapping $f$ completely determines $g$ and $h$, together with the ascending or descending monotone parts of $g$.

Article (Russian)

Sharp Remez-type inequalities for differentiable periodic functions, polynomials and splines

Kofanov V. A.

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 2. - pp. 227-240

For any $\omega > 0,\; \beta \in (0, 2\omega)$, and any measurable set $B \in I_d := [0, d],\; \mu B = \beta$, we obtain the following sharp inequality of the Remez type: $$||x||_{\infty} \leq \frac{3||\varphi||_{\infty} - \varphi \biggl(\frac{\omega - \beta}2 \biggr)}{||\varphi||_{\infty} + \varphi \biggl(\frac{\omega - \beta}2 \biggr)} ||x||_{L_{\infty}(I_d\setminus B)}$$ on the set $S_{\varphi} (\omega )$ of functions $x$ with minimal period $d (d \geq 2\omega)$ and a given sine-shaped $2\omega$ -periodic comparison function $\varphi$. In particular, we prove the sharp Remez-type inequalities on the Sobolev spaces of differentiable periodic functions. We also obtain inequalities of the indicated type on the spaces of trigonometric polynomials and polynomial splines.

Article (Ukrainian)

Inverse problem in the space of generalized functions

Lopushanskaya G. P., Lopushanskyi A. O., Rapita V.

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 2. - pp. 241-253

For a linear nonhomogeneous diffusion equation with fractional derivative of order $\beta \in (0, 2)$ with respect to time, we establish a unique solvability of the inverse problem of determination of a pair of functions: the generalized solution u (classical as a function of time) of the first boundary-value problem for the indicated equation with given generalized functions on the right-hand sides and the unknown (depending on time) continuous coefficient of the minor term of the equation under the overdetermination condition $$\bigl( u(\cdot , t), \varphi_0(\cdot ) \bigr) = F(t), t \in [0, T].$$ Here, $F$ is a given continuous function and $(u(\cdot , t), \varphi_0(\cdot ))$ is the value of the unknown generalized function u on a given test function $\varphi_0$ for any $t \in [0, T]$.

Article (Ukrainian)

Trees as set levels for pseudoharmonic functions in the plane. II

Polulyakh E. O.

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 2. - pp. 254-270

Let $T$ be a forest formed by finitely many locally finite trees. Let $V_0$ be the set of all vertices of $T$ of degree 1. We propose a sufficient condition for the image of an embedding $\Psi : T \setminus V_0 \rightarrow R^2$ to be a level set of a pseudoharmonic function.

Article (Ukrainian)

Relationship between normalized tensors of two regular networks on the surfaces in the Euclidean space $E_3$

Potapenko I. V.

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 2. - pp. 271-277

We establish the relationship between the normalized tensors of two regular networks on the surfaces in the Euclidean space $E_3$.

Article (Ukrainian)

Generalized kernels of the Toeplitz type for exponentially convex functions

Chernobai O. B.

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 2. - pp. 278-288

We prove an integral representation for the generalized kernels of the Toeplitz type connected with exponentially convex functions but not with positive-definite functions.