2019
Том 71
№ 10

# Volume 71, № 8, 2019

Article (Ukrainian)

### Isometry of the subspaces of solutions of systems of differential equations to the spaces of real functions

Ukr. Mat. Zh. - 2019. - 71, № 8. - pp. 1011-1027

UDC 517.5
We determine the subspaces of solutions of the systems of Laplace and heat-conduction differential equations isometric to the corresponding spaces of real functions determined on the set of real numbers.

Article (Ukrainian)

### Deformations in the general position of the optimal functions on oriented surfaces with boundary

Ukr. Mat. Zh. - 2019. - 71, № 8. - pp. 1028-1039

UDC 516.91
It is considered simple functions with non-degenerated singularities on smooth compact oriented surfaces with the boundary. Authors describe a connection between optimality and polarity of Morse functions, $m$-functions and $mm$-functions on smooth compact oriented connected surfaces. The concept of an equipped Kronrod – Reeb graph is used to define a deformation in general position. Also, it is obtained the whole list of deformations of simple functions of one of abovedescribed class on torus, 2-dimensional disc with the boundary and on connected sum of two toruses.

Article (English)

### Application of the infinite matrix theory to the solvability of sequence spaces inclusion equations with operators

Ukr. Mat. Zh. - 2019. - 71, № 8. - pp. 1040-1052

UDC 517.9
Given any sequence $a = (a_n)_n \geq 1$ of positive real numbers and any set $E$ of complex sequences, we write $E_a$ for the set of all sequences $y=(y_{n})_{n \geq 1}$ such that y/a = $y/a=(y_{n}/a_{n})_{n\geq 1}\in E.$ In particular, $c_{a}$ denotes the set of all sequences $y$ such that $y/a$ converges. We deal with sequence spaces inclusion equations (SSIE) of the form $F\subset E_{a}+F_{x}'$ with $e\in F$ and explicitly find the solutions of these SSIE when $a=(r^{n})_{n\geq 1},$ $F$ is either $c$ or $s_{1},$ and $E,$ $F'$ are any sets $c_{0},$ $c,$ $s_{1},$ $\ell_{p},$ $w_{0},$ and $w_{\infty }.$ Then we determine the sets of all positive sequences satisfying each of the SSIE $c\subset D_{r}\ast (c_{0})_{\Delta }+c_{x}$ and $c\subset D_{r}\ast (s_{1})_{\Delta}+c_{x},$ where $\Delta$ is the operator of the first difference defined by $\Delta _{n}y=y_{n}-y_{n-1}$ for all $n\geq 1$ with $y_{0}=0.$ Then we solve the SSIE $c\subset D_{r}\ast E_{C_{1}}+s_{x}^{(c)}$ with $E\in \{ c,s_{1}\}$ and $s_{1}\subset D_{r}\ast(s_{1}) _{C_{1}}+s_{x},$ where $C_{1}$, is the Cesaro operator defined by $(C_{1}) _{n}y=n^{-1} \displaystyle \sum \nolimits_{k=1}^{n}y_{k}$ for all $y$. We also deal with the solvability of the sequence spaces equations (SSE) associated with the previous SSIE and defined as $D_{r}\ast E_{C_{1}}+s_{x}^{(c)}=c$ with $E\in \{c_{0},c, s_{1}\}$ and $D_{r}\ast E_{C_{1}}+s_{x}=s_{1}$ with $E\in \{ c,s_{1}\}.$

Article (Russian)

### On the stability of a program manifold of control systems with variable coefficients

Ukr. Mat. Zh. - 2019. - 71, № 8. - pp. 1053-1063

UDC 517.956
We study the absolute stability of the program manifold of basic control systems with variable coefficients and stationary nonlinearities. The conditions of stability of basic systems are investigated in a neighborhood of a given program manifold. The nonlinearities satisfy the conditions of local quadratic relationship. Sufficient conditions for the absolute stability of the program manifold with respect to a given vector function are established by constructing the Lyapunov function. A method used to select the Lyapunov matrix is specified.

Article (English)

### Existence of nonnegative solutions for a fractional parabolic equation in the whole space

Ukr. Mat. Zh. - 2019. - 71, № 8. - pp. 1064-1072

UDC 517.9
We study existence of nonnegative solutions for a parabolic problem $\dfrac{\partial u}{\partial t} = - (-\triangle)^{\frac{\alpha}{2}}u + \dfrac{c}{|x|^{\alpha}}u$ in $\mathbb{R}^{d}\times (0, T).$ Here $0<\alpha<\min(2,d),$ $(-\triangle)^{\frac{\alpha}{2}}$ is the fractional Laplacian on $\mathbb{R}^{d}$ and $\mathbb{R}^{d}$ and $u_{0}\in L^{2}(\mathbb{R}^{d}).$

Article (Russian)

### On the Lebesgue constants

Ukr. Mat. Zh. - 2019. - 71, № 8. - pp. 1073-1081

UDC 517.5
We give the solution of a classical problem of approximation theory on sharp asymptotic of the Lebesgue constants or norms of the Fourier – Laplace projections on the real spheres $\mathbb{S}^{d},$ complex $\mathrm{P}^{d}(\mathbb{C})$ and quaternionic $\mathrm{P}^{d}(\mathbb{H})$ projective spaces, and the Cayley elliptic plane $\mathrm{P}^{16}(\mathrm{Cay}).$ In particular, these results extend sharp asymptotic found by Fejer in the case of $\mathbb{S}^{1}$ in 1910 and by Gronwall in 1914 in the case of $\mathbb{S}^{2}$ .

Article (English)

### The $\Box_b$-heat equation on finite type CR manifolds with comparable Levi form

Ukr. Mat. Zh. - 2019. - 71, № 8. - pp. 1082-1101

UDC 517.9
The main purpose of this paper is to study the initial-value problems for the heat equations associated with the operator $\Box_b$ on compact CR manifolds of finite type. The critical component of our analysis is the condition called $D^{\epsilon}(q)$ and introduced by K. D. Koenig [Amer. J. Math. -- 2002. -- {\bf 124}. -- P. 129--197]. Actually, it states that the $\min\{q, n-1-q\}$th smallest eigenvalue of the Levi form is comparable with the largest eigenvalue of the Levi form.

Article (Ukrainian)

### Estimates of some approximating characteristics of the classes of periodic functions of one and many variables

Ukr. Mat. Zh. - 2019. - 71, № 8. - pp. 1102-1115

UDC 517.5
We obtain the exact-order estimates for some approximating characteristics of the classes $\mathbb{W}^{\boldsymbol{r}}_{p,\boldsymbol{\alpha}}$ and $\mathbb{B}^{\boldsymbol{r}}_{p,\theta}$ of periodic functions of one and many variables in the norm of the space $B_{\infty, 1}.$

Article (Russian)

### On the dynamics of a quasistrictly non-Volterra quadratic stochastic operator

Ukr. Mat. Zh. - 2019. - 71, № 8. - pp. 1116-1122

UDC 517.98
We find all fixed and periodic points for a quasistrictly non-Volterra quadratic stochastic operator on the two-dimensional simplex. The description of the limit set of trajectories for this operator is presented.

Article (Russian)

### On the existence, uniqueness, and nonexistence of solutions of one boundary-value problem for a semilinear hyperbolic equation

Ukr. Mat. Zh. - 2019. - 71, № 8. - pp. 1123-1132

UDC 517.956.35
We consider a boundary-value problem for a semilinear hyperbolic equation with iterated multidimensional wave operator in the principal part. The theorems on existence, uniqueness, and nonexistence of solutions of this problem are established.

Brief Communications (English)

### On the Merkulov construction of $A_{ \infty}$ -(co)algebras

Ukr. Mat. Zh. - 2019. - 71, № 8. - pp. 1133-1140

UDC 512.5
The aim of this short note is to complete some aspects of a theorem proved by S. Merkulov in [Int. Math. Res. Not. IMRN. – 1999. – 3. – P. 153 – 167] (Theorem 3.4), as well as to provide a complete proof of the dual result for dg coalgebras.

Brief Communications (Russian)

### On the correct definition of the flow of a discontinuous solenoidal vector field

Ukr. Mat. Zh. - 2019. - 71, № 8. - pp. 1141-1149

UDC 517.51
We prove inequalities connecting a flow through the $(n- 1)$-dimensional surface $S$ of a smooth solenoidal vector field with its $L^{p}(U)$-norm ($U$ is an $n$-dimensional domain that contains $S$). On the basis of these inequalities, we propose a correct definition of the flow through the surface $S$ of a discontinuous solenoidal vector field $f \in L^{p}(U)$ (or, more precisely, of the class of vector fields that are equal almost everywhere with respect to the Lebesque measure).

Brief Communications (Ukrainian)

### On the maximal unipotent subgroups of a special linear group over commutative ring

Ukr. Mat. Zh. - 2019. - 71, № 8. - pp. 1150-1156

UDC 512.64
We prove that all maximal unipotent subgroups of a special linear group over commutative ring with identity (such that the factor ring of its modulo primitive radical is a finite direct sum of Bezout domains) are pairwise conjugated and describe one maximal unipotent subgroup of the general linear group (and of a special linear group) over an arbitrary commutative ring with identity.