# Volume 68, № 2, 2016

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

Andrusyak I. V., Filevych P. V.

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.

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

Maslyuchenko V. K., Voloshyn H. A.

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 $X$.

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

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.

### Lehmer sequences in finite groups

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 $t \geq 3$.

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

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

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 $R$.

### Preacyclicity over the rings with infinite fields of residues

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.

### Topological conjugate piecewise linear unimodal mappings of an interval into itself

Kirichenko V. V., Plakhotnyk M. V.

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 $g$.

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

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.

### Inverse problem in the space of generalized functions

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

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 $t \in [0, T]$.

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

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.

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

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 $E_3$.

### Generalized kernels of the Toeplitz type for exponentially convex functions

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.