# Volume 45, № 8, 1993

### Pseudoanalyticity of continuous functions with the ?-preservation of angles

Ukr. Mat. Zh. - 1993. - 45, № 8. - pp. 1051–1057

The following theorem is proved: Every continuous function satisfying the condition $K'_{\sigma}$ is pseudo-analytic. The condition $K'_{\sigma}$ is a generalization of the Men'shov condition, well known in the theory of analytic functions.

### On the one-dimensional two-phase inverse Stefan problems

Ukr. Mat. Zh. - 1993. - 45, № 8. - pp. 1058–1065

New formulations of the inverse nonstationary Stefan problems are considered:

(a) for $x ∈ [0,1]$ (the inverse problem IP_1;

(b) for $x ∈ [0, β(t)]$ with a degenerate initial condition (the inverse problem IP_{β}).

Necessary conditions for the existence and uniqueness of a solution to these problems are formulated. On the first phase $\{x ∈ [0, y(t)]\}$, the solution of the inverse problem is found in the form of a series; on the second phase $\{x ∈ [y(t), 1]$ or $x ∈ [y(t), β (t)]\}$, it is found as a sum of heat double-layer potentials. By representing the inverse problem in the form of two connected boundary-value problems for the heat conduction equation in the domains with moving boundaries, it can be reduced to the integral Volterra equations of the second kind. An exact solution of the problem IPβ is found for the self similar motion of the boundariesx=y(t) andx=β(t).

### Inverse problems for the heat-conduction equation with nonlocal boundary conditions

Ukr. Mat. Zh. - 1993. - 45, № 8. - pp. 1066–1071

Conditions under which the time-dependent temperature conductivity coefficient is determined uniquely are established in the case where the boundary conditions and the overdetermination conditions are non local.

### Asymptotic analysis of a bounded control in optimal elliptic problems

Ukr. Mat. Zh. - 1993. - 45, № 8. - pp. 1072–1083

An asymptotics of a bounded control is constructed and substantiated for a singularly perturbed optimal elliptic problem.

### On the definition of singular bilinear forms and singular linear operators

Karwowski W., Koshmanenko V. D.

Ukr. Mat. Zh. - 1993. - 45, № 8. - pp. 1084–1089

We analyze various definitions of the concepts of a singular operator and a singular bilinear form and propose the most suitable ones. We also study the simplest properties of these objects.

### Convergence near a point and the Arzela-Ascoli-type theorems

Ukr. Mat. Zh. - 1993. - 45, № 8. - pp. 1090–1095

Theorems are proved giving necessary and sufficient conditions for the convergence of a sequence of continuous (differentiable) functions to a continuous (differentiable) function. The concepts of convergence near a point and equipotential convergence near a point are introduced. These concepts are introduced locally; on a segment, they are equivalent to the quasiuniform convergence and to the uniform convergence of a sequence of functions, respectively.

### On a class of hybrid integral transformations (Bessel-Fourier-Bessel-...-Fourier-Bessel) on the polar axis with $2n$ junction points

Ukr. Mat. Zh. - 1993. - 45, № 8. - pp. 1096–1103

The hybrid integral transformations (Bessel-Fourier-Bessel-...-Fourier-Bessel) are constructed on the polar axis with $2n$ junction points by using the method of a delta-shaped sequence regarded as a Dirichlet kernel. The principal identity of the integral transformation of a differential operator is obtained.

### Operator methods in the problem of estimating the asymptotics of the time of the first hit for the birth and death process

Ukr. Mat. Zh. - 1993. - 45, № 8. - pp. 1104–1108

Estimates are established for the exponential asymptotics of the time of the first hit of a certain level $n$ for the birth and death process.

### The third mixed problem for the Sonin equation in a half space

Ukr. Mat. Zh. - 1993. - 45, № 8. - pp. 1109–1114

We consider follwing mixed boundary-value problem: $$\begin{array}{*{20}c} {u'_t (t,R) + xu'_y (t,R) + yu'_z (t,R) = u''_{x^2 } (t,R) + f(t,R)} \\ {in \Pi _T = \{ (t,R),0< t \leqslant T,R = (x,y,z),R \in E_3 ,0< x\} ,} \\ {u(0,R) = u_0 (R),u'_x (t,0,y,z) + \beta (t)u(t,0,y,z) = g(t,y,z).} \\ \end{array}$$ A solution of this problem is obtained in the form of a potential.

### On the periodic solutions of the second-order wave equations. V

Khoma G. P., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1993. - 45, № 8. - pp. 1115–1121

It is established that the linear problem $u_{tt} - a^2 u_{xx} = g(x, t),\quad u(0, t) = u(\pi, t),\quad u(x, t + T) = u(x, t)$ is always solvable in the space of functions $A = \{g:\; g(x, t) = g(x, t + T) = g(\pi - x, t) = -g(-x, t)\}$ provided that $aTq = (2p - 1)\pi, \quad (2p - 1, q) = 1$, where $p, q$ are integers. To prove this statement, an explicit solution is constructed in the form of an integral operator which is used to prove the existence of a solution to aperiodic boundary value problem for nonlinear second order wave equation. The results obtained can be employed in the study of solutions to nonlinear boundary value problems by asymptotic methods.

### An analog of the rolle theorem for differential operators and $L$-spline interpolation

Ukr. Mat. Zh. - 1993. - 45, № 8. - pp. 1122–1128

An analog of the Rolle theorem is established for linear differential operators with continuous periodic coefficients. By using this result, exact values of the deviations of interpolational $L$-splines are obtained on certain classes of functions given by a linear differential operator.

### On subharmonic extension and extension in the Hardy-Orlicz classes

Riihettiaus J., Tamrazov P. M.

Ukr. Mat. Zh. - 1993. - 45, № 8. - pp. 1129–1139

The paper contains a generalization of the results obtained earlier concerning the subharmonic extension of functions and the extension of functions in the Hardy-Orlicz classes. We give the unified proofs of these results.

### Smoothness of generalized solutions of the third boundary-value problem for an elliptic differential-difference equation

Ukr. Mat. Zh. - 1993. - 45, № 8. - pp. 1140–1150

Unlike the case of elliptic differential equations, generalized solutions of elliptic differential-difference equations may be nonsmooth on an entire domain $Q$, only preserving smoothness on certain subdomains $Q_r \subset Q$. The conditions are considered under which the generalized solutions of the third boundary-value problem remain smooth on the boundaries of the neighboring subdomains $Q_r$.

### Handle decompositions of simply connected five-manifolds. I

Ukr. Mat. Zh. - 1993. - 45, № 8. - pp. 1151–1156

Handle decompositions of simply connected smooth or piecewise linear five-manifolds are considered. The basic concepts and constructions necessary for proving our main result are introduced.

### On the solution of a quasilinear differential system with periodic coefficients

Ukr. Mat. Zh. - 1993. - 45, № 8. - pp. 1157–1161

The existence of a partial solution is proved for a quasilinear differential system whose coefficients are representable by a trigonometric series with slowly varying coefficients and frequency. The solution obtained has the same structure.

### On a coupled system of abstract differential equations similar to the thermoelasticity equations

Ukr. Mat. Zh. - 1993. - 45, № 8. - pp. 1162–1165

The study of a model system of differential equations arising from the dynamical problems of thermo elasticity is continued. The case of shifts of the general type is investigated. We employ the “commutant method” based on the properties of the operator $\Delta(A, B) = AB - BA$.

### Eigenfunction expansion for symmetric multiparameter problems

Konstantinov A. Yu., Stadnyuk A. G.

Ukr. Mat. Zh. - 1993. - 45, № 8. - pp. 1166–1169

An abstract theorem on the expansion in generalized eigenvectors of multiparameter problems for Hermitian operators is established. The applications to differential operators are considered.

### Regular linear extensions of dynamical systems on a torus

Ukr. Mat. Zh. - 1993. - 45, № 8. - pp. 1170-1173

We consider certain structures of linear extensions of the dynamical systems on a torus having a single Green's function.

### On the calculation of certain singular integrals

Kovalenko I. L., Litvin A. I., Simonzhenkov S. D.

Ukr. Mat. Zh. - 1993. - 45, № 8. - pp. 1174–1176

A method for the approximate calculation of Cauchy-type integrals with logarithmic singularities is proposed. It is based on the expansion of a function $f(x)$ in a series in the Chebyshev polynomials.

### On the composition of operators in vector spaces and their Noether property

Ukr. Mat. Zh. - 1993. - 45, № 8. - pp. 1177–1180

Under the minimal assumptions imposed on given spaces, the sufficient conditions are established, under which the composition $BA$ of operators $A$ and $B$ has the Noether property and is normally solvable. Similar conditions, guaranteeing that the operator $A$ is normally solvable or possesses the Noether property, are obtained for the operators $B$ and $AВ$ .

### A method for finding a common linear divisor of the matrix polynomials over an arbitrary field

Ukr. Mat. Zh. - 1993. - 45, № 8. - pp. 1181–1183

Conditions are found for existence of a common linear unital divisor with a given characteristic polynomial of regular matrix polynomials over an arbitrary field. The result obtained is also applied to finding a common solution of matrix polynomial equations.