# Volume 68, № 11, 2016

### Laplacian with respect to the measure on a Riemannian manifold and the Dirichlet problem. II

Bogdanskii Yu. V., Potapenko A. Yu.

Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1443-1449

We propose the $L^2$ -version of Laplacian with respect to measure on an (infinite-dimensional) Riemannian manifold. The Dirichlet problem for equations with proposed Laplacian is solved in a part of the Rimannian manifold of a certain class.

### Almost periodic solutions of systems with delay and nonfixed times of impulsive action

Dvornyk A. V., Tkachenko V. I.

Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1450-1466

We study the existence and asymptotic stability of piecewise continuous almost periodic solutions for systems of differential equations with delay and nonfixed times of impulsive action that can be regarded as mathematical models of neural networks.

### Estimations of the Laplace – Stieltjes integrals

Dobushovs’kyi M. S., Sheremeta M. M.

Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1467-1482

We study the Laplace – Stieltjes integrals with an arbitrary abscissa of convergence. The lower and upper estimates for these integrals are established. The accumulated results are used to deduce the relationships between the growth of the integral and the maximum of the integrand.

### General proximal point algorithm for monotone operators

Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1483-1492

We introduce a new general proximal point algorithm for an infinite family of monotone operators in a real Hilbert space. We establish strong convergence of the iterative process to a common zero point of the infinite family of monotone operators. Our result generalizes and improves numerous results in the available literature.

### I. Approximative properties of biharmonic Poisson integrals in the classes $W^r_{\beta} H^{\alpha}$

Kalchuk I. V., Kharkevych Yu. I.

Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1493-1504

We deduce asymptotic equalities for the least upper bounds of approximations of functions from the classes $W^r_{\beta} H^{\alpha}$, and $H^{\alpha}$ by biharmonic Poisson integrals in the uniform metric.

### Asymptotically independent estimators in a structural linear model with measurement errors

Kukush A. G., Shklyar S. V., Tsaregorodtsev Ya. V.

Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1505-1517

We consider a structural linear regression model with measurement errors. A new parameterization is proposed, in which the expectation of the response variable plays the role of a new parameter instead of the intercept. This enables us to form three groups of asymptotically independent estimators in the case where the ratio of variances of the errors is known and two groups of this kind if the variance of the measurement error in the covariate is known. In this case, it is not assumed that the errors and the latent variable are normally distributed.

### Sufficient conditions under which the solutions of general parabolic initial-boundaryvalue problems are classical

Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1518-1527

We establish new sufficient conditions under which the generalized solutions of initial-boundary-value problems for the linear parabolic differential equations of any order with complex-valued coefficients are classical. These conditions are formulated in the terms of belonging of the right-hand sides of this problem to certain anisotropic H¨ormander spaces. In the definition of classical solution, its continuity on the line connecting the lateral surface with the base of the cylinder (in which the problem is considered) is not required.

### Two-dimensional Coulomb dynamics of two and three equal negative charges in the field of two equal fixed positive charges

Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1528-1539

Periodic and bounded for positive time solutions of the planar Coulomb equation of motion for two and three identical negative charges in the field of two equal fixed positive charges are found. The systems possess equilibrium configurations to which the found bounded solutions converge in the infinite time limit. The periodic solutions are obtained with the help of the Lyapunov center theorem.

### Hypersurfaces with nonzero constant Gauss – Kronecker curvature in $M^{n+1}(±1)$

Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1540-1551

We study hypersurfaces in a unit sphere and in a hyperbolic space with nonzero constant Gauss – Kronecker curvature and two distinct principal curvatures one of which is simple. Denoting by $K$ the nonzero constant Gauss – Kronecker curvature of hypersurfaces, we obtain some characterizations of the Riemannian products $S^{n-1}(a) \times S^1(\sqrt{1 - a^2}),\quad$ $a^2 = 1/\left(1 + K^{\frac{2}{n - 2}}\right)$ or $S^{n-1}(a) \times H^1(- \sqrt{1 + a^2}),\quad$ $a^2 = 1/\left(K^{\frac{2}{n - 2}} - 1\right)$.

### A construction of regular semigroups with quasiideal regular *-transversals

Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1552-1560

Let $S$ be a semigroup and let “$\ast$ ” be a unary operation on S satisfying the following identities: $$xx^{\ast} x = x, x^{\ast} xx^{\ast} = x^{\ast},\; x^{\ast \ast \ast} = x^{\ast},\; (xy^{\ast} )^{\ast} = y^{\ast \ast} x^{\ast},\; (x^{\ast} y)^{\ast} = y^{\ast} x^{\ast \ast}.$$ Then S\ast = \{ x\ast | x \in S\} is called a regular \ast -transversal of $S$ in the literatures. We propose a method for the construction of regular semigroups with quasiideal regular $\ast$ -transversals based on the use of fundamental regular semigroups and regular $\ast$ -semigroups.

### On the growth of meromorphic solutions of difference equation

Chen Zong-Xuan, Lan Shuang-Ting

Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1561-1570

We estimate the order of growth of meromorphic solutions of some linear difference equations and study the relationship between the exponent of convergence of zeros and the order of growth of the entire solutions of linear difference equations.

### Complete classification of finite semigroups for which the inverse monoid of local automorphisms is a permutable semigroup

Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1571-1578

A semigroup $S$ is called permutable if $\rho \circ \sigma = \sigma \circ \rho$. for any pair of congruences $\rho, \sigma$ on $S$. A local automorphism of semigroup $S$ is defined as an isomorphism between two of its subsemigroups. The set of all local automorphisms of the semigroup $S$ with respect to an ordinary operation of composition of binary relations forms an inverse monoid of local automorphisms. We present a complete classification of finite semigroups for which the inverse monoid of local automorphisms is permutable.

### On the equivalence of some perturbations of the operator of multiplication by the independent variable

Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1579-1584

We study the conditions of equivalence of two operators obtained as perturbations of the operator of multiplication by the independent variable by certain Volterra operators in the space of functions analytic in an arbitrary domain of the complex plane starlike with respect to the origin.