2018
Том 70
№ 6

# Volume 70, № 4, 2018

Article (Russian)

### Surfaces generated by the real and imaginary parts of analytic functions: $A$-deformations occurring independently or simultaneously

Ukr. Mat. Zh. - 2018. - 70, № 4. - pp. 447-463

It is proved that the surfaces generated by the real and imaginary parts of analytic functions allow nontrivial infinitesimal areal deformations of certain three types. The fields of displacements are explicitly expressed in all three cases. Given surfaces are rigid with respect to infinitesimal bendings of each type.

Article (Ukrainian)

### On the approximation and growth of entire harmonic functions in $R_n$

Ukr. Mat. Zh. - 2018. - 70, № 4. - pp. 464-470

We establish a criterion of extendability of harmonic function in a ball of the $n$-dimensional space to harmonic entire function and study the growth of entire harmonic function in terms of the best approximation of these function by harmonic polynomials.

Article (Ukrainian)

### Value distribution of differential-difference polynomials of meromorphic functions

Ukr. Mat. Zh. - 2018. - 70, № 4. - pp. 471-480

We obtain the results on the deficiencies of differential-difference polynomials. These results can be regarded as differentialdifference analogs of some classical theorems on differential polynomials. In particular, an exact estimate of the deficiency of some differential-difference polynomials is presented. We also give examples showing that these results are best possible in a certain sense.

Article (Ukrainian)

### Mitigation method for a round plate under the action of mass forces

Ukr. Mat. Zh. - 2018. - 70, № 4. - pp. 481-494

We develop a refined approximate method for the analytic investigation of the stress-strain states of orthotropic plates. The efficiency of the method is confirmed by comparing the exact and approximate solutions of the problem of bending of a circular plate.

Article (Russian)

### On the criteria of transversality and disjointness of nonnegative selfadjoint extensions of nonnegative symmetric operators

Ukr. Mat. Zh. - 2018. - 70, № 4. - pp. 495-505

We propose criteria of transversality and disjointness for the Friedrichs and Krein extensions of a nonnegative symmetric operator in terms of the vectors $\{ \varphi j , j \in J\}$ that form a Riesz basis of the defect subspace. The criterion is applied to the Friedrichs and Krein extensions of the minimal Schr¨odinger operator $\scr A$ d with point potentials. We also present a new proof of the fact that the Friedrichs extension of the operator $\scr A$ d is a free Hamiltonian.

Article (Ukrainian)

### Limit theorems for the maximum of sums of independent random processes

Ukr. Mat. Zh. - 2018. - 70, № 4. - pp. 506-518

We study the conditions for the weak convergence of the maximum of sums of independent random processes in the spaces $C[0, 1]$ and $L_p$ and present examples of applications to the analysis of statistics of the type $\omega 2$.

Article (English)

### Mechanical systems with singular equilibria and the Coulomb dynamics of three charges

Ukr. Mat. Zh. - 2018. - 70, № 4. - pp. 519-533

We consider mechanical systems for which the matrices of second partial derivatives of the potential energies at equilibria have zero eigenvalues. It is assumed that their potential energies are holomorphic functions in these singular equilibrium states. For these systems, we prove the existence of proper bounded (for positive time) solutions of the Newton equation of motion convergent to the equilibria in the infinite-time limit. These results are applied to the Coulomb systems of three point charges with singular equilibrium in a line.

Article (English)

### The Drazin inverses of infinite triangular matrices and their linear preservers

Ukr. Mat. Zh. - 2018. - 70, № 4. - pp. 534-548

We consider the ring of all infinite $(N \times N)$ upper triangular matrices over a field F. We give a description of elements that are Drazin invertible in this ring. In the case where $F$ is such that $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}(F) \not = 2$ and $| F| > 4$, we find the form of linear preservers for the Drazin inverses.

Article (English)

### A note on the coefficient estimates for some classes of $p$ -valent functions

Ukr. Mat. Zh. - 2018. - 70, № 4. - pp. 549-563

We obtain estimates of the Taylor – Maclaurin coefficients of some classes of p-valent functions. This problem was initially studied by Aouf in the paper “Coefficient estimates for some classes of p-valent functions” (Internat. J. Math. and Math. Sci. – 1988. – 11. – P. 47 – 54). The proof given by Aouf was found to be partially erroneous. We propose the correct proof of this result.

Article (Ukrainian)

### Estimates of the best bilinear approximations for the classes of (ψ,β)-differentiable periodic multivariate functions

Ukr. Mat. Zh. - 2018. - 70, № 4. - pp. 564-573

Order estimates are obtained for the best bilinear approximations of 2d-variable functions $f(x y),\; x, y \in \pi_d,\; \pi_d =\prod^d_{j=1} [ \pi , \pi ]$, formed by $d$-variable functions $f(x) \in L^{\psi}_{\beta} ,p$ by the shifts of their argument $x \in \pi_d$ by all possible values of $y \in \pi_d$ in the space $L_{q_1,q_2} (\pi 2d)$. The results include various relations between the parameters $p, q_1$ and $q_2$.

Article (Ukrainian)

### Best approximation of the functions from anisotropic Nikol’skii – Besov classes defined in $R^d$

Ukr. Mat. Zh. - 2018. - 70, № 4. - pp. 574-582

We establish the exact-order estimates for the best approximations of the functions from anisotropic Nikol’skii – Besov classes of functions of several variables by entire functions in the Lebesgue spaces.

Brief Communications (English)

### A corrigendum to “Hereditary properties between a ring and its maximal subrings”

Ukr. Mat. Zh. - 2018. - 70, № 4. - pp. 583-584

Let $R$ be a commutative ring with identity. In [2] (Proposition 3.1), Azarang proved that if $R$ is an integral domain and $S$ is a maximal subring of $R$, and is integrally closed in $R$, then $\mathrm{d}\mathrm{i}\mathrm{m}(S) = 1$ implies that $\mathrm{d}\mathrm{i}\mathrm{m}(R) = 1$ if and only if $(S : R) = 0$. An example is given which shows the above mentioned proposition is not correct.