# Volume 46, № 10, 1994

### Institute of Mathematics of The Ukrainian National Academy of Sciences: 60 years of development

Mitropolskiy Yu. A., Samoilenko A. M., Strok V. V.

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1291–1303

In this brief historical essay, we describe main stages of the formation and development of the Institute of Mathematics of the Ukrainian National Academy of Sciences from its foundation in 1934 till now. Our attention is mainly focused on the achievements of its leading scientists and main directions of mathematical researches carried out in the Institute of Mathematics.

### Well-posedness of many-dimensional Darboux problems for degenerating hyperbolic equations

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1304–1311

Forthe equation $$\sum^{m}_{i=t}t^{k_i}U_{x_i,x_i} - U_n + \sum^{m}_{i=t} a_i(x,t)U_{x_i} + b(x, t)u_t + c(x,t)u = 0,$$ $$k_i = \text{const} ≥ 0,\; i = l ..... m, x = (x_1,..., x_m),\; m_>2,$$ we find a many-dimensional analog of the well-known "Gellerstedt condition" $$ a_i(x,t) = O(1)t^{\alpha},\; i = 1,..., m,\, \alpha >\frac{k_1}{2} - 2.$$ We prove that if this condition is satisfied, then the Darboux problems are uniquely solvable.

### On local limit values of subharmonic and holomorphic functions

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1312–1317

We prove theorems analogous to the local maximum principle and theorems on relation between limit sets, which can be used for studying singular sets.

### On the best polynomial approximation of entire transcendental functions in Banach spaces. II

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1318–1322

We study the behavior of the best approximations $E_n(f)_{E′_p}$ of entire transcendental functions $f(z)$ of the order $ρ = 0$ by polynomials of at most $n$ th degree in the metric of the space $E′_p(Ω),\, p ≥ 1$. In particular, we describe the relationship between the best approximations $E_n(f)E′_p$ and the logarithmic order $ρ_L$ and type $σ_L$ of the function $f(z)$.

### Weighted pseudoinversion of matrices with singular weights

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1323–1327

Weighted pseudoinverse matrices with singular weights are represented in terms of the coefficients of characteristic polynomials of certain square matrices. By using the expression obtained, we construct limit representations of weighted pseudoinverse matrices with singular weights.

### Generalized moment representations, biorthogonal polynomials, and Padé approximants

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1328–1335

By using the method of generalized moment representations and certain properties of biorthogonal polynomials, we establish new invariance properties of the Padé approximants.

### On the ?-differentiability of mappings of Banach spaces

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1336–1342

We study the monogeneity conditions for $ℝ$-differentiable mappings of domains of a Banach space and establish criteria of $ℂ$-differentiability of a mapping at a point.

### Separation of variables in two-dimensional wave equations with potential

Fushchich V. I., Revenko I. V., Zhdanov R. Z.

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1343–1361

The paper is devoted to solution of a problem of separation of variables in the wave equation $u_{tt} - u_{xx} + V(x)u = 0$. We give a complete classification of potentials $V(x)$ for which this equation admits a nontrivial separation of variables. Furthermore, we obtain all coordinate systems that provide separability of the equation considered.

### Boundary-layer averaging for standard systems with lag

Efendiev V. V., Zheltikov V. P.

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1362–1368

The $k$-th-order asymptotic solution of a standard system with lag is constructed along trajectories calculated according to the averaging scheme of A. N. Filatov. If the perturbation parameter $ε ≪ 1$, then the use of the step method for finding the solution is connected with cumbersome calculations because the number of required steps is inversely proportional to $ε$. We suggest another approach in which the step method is used only $k$ times for $t \in [0,k]$ and justify the asymptotic method.

### Solution of Luzin's category problem on the existence of a primitive function

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1369–1374

N. N. Luzin's problem on the existence of a primitive function is considered. We present a more precise formulation of the solution of this problem obtained by Landis.

### On the optimal reconstruction of the values of operators

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1375–1381

For a continuous operator $А:\; X \rightarrow Y,$ we formulate the problem of the optimal renewal of values $Аx,\; x \in X$ by decreasing the uncertainty domain by using an information $\mu_k(x),\; k = 1, 2, ...,$, where $\mu_k$ are continuous functionals, defined on the space $X$. Specific results are obtained for some integral operators in functional spaces.

### On the existence and properties of periodic solutions of discrete difference equations

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1382–1387

We establish conditions for the existence of periodic solutions for a broad class of nonlinear difference equations with discrete argument.

### Averaging of aperiodic problems of control over fracture surfaces

Plotnikov V. A., Slobodyanyuk O. E., Zverkova T. S.

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1388–1392

An averaging method is justified for standard aperiodic systems with parameters and discontinuous right-hand sides under the condition of aperiodicity and applied to the problems of control over fracture surfaces.

### On topological groups with weak minimality (maximality) condition for noncompact subgroups

Poletskikh V. M., Shestakov S. S.

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1393–1398

We present the structure of a locally compact solvable*p*-group satisfying the weak minimality (maximality) condition for noncompact subgroups. As a consequence, we obtain the structure of a locally compact prosolvable*p*-group satisfying the minimality (maximality) condition for noncompact sub-groups. We also construct an example to demonstrate that these results are not true for arbitrary inductively compact locally compact totally disconnected solvable groups.

### Investigation of a dynamical system in a neighborhood of an invariant toroidal manifold in the general case

Bazhura B. P., Samoilenko A. M.

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1399–1408

A dynamical system is studied in the neighborhood of an invariant toroidal manifold for the most general relationship between the dimensionality of the phase space and the dimensionality of the manifold.

### On the 75th birthday of Vyacheslav Alekseevich Dobrovol'skii

Bogolyubov A. N., Tamrazov P. M., Valeyev K. G.

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1409

### BestL1-approximations of classes $W_1^r$ by Splines from $W_1^r$

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1410–1413

We obtain the exact values of the best $L_1$-approximations of the classes $W_1^r$ of periodic functions by periodic polynomial splines of degree $r$ and defect 1 with equidistant knots that belong to the class $W_1^r$.

### Generalized Euler equations in quaternions

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1414–1416

The quaternion analog of the Euler dynamic equations obtained in [1–3] is generalized to the case of an arbitrary trihedron connected with a body.

### Approximate method for the solution of the generalized Dirichlet problem

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1417–1420

We extend the method for approximate solution of classical boundary-value problems for the Laplace equation suggested in [1–3] to the case of the Poisson equation with generalized functions on the right-hand side of the equation and in the boundary conditions.

### Central limit theorem for stochastically additive functionals of ergodic chains

Moskal'tsova N. V., Shurenkov V. M.

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1421–1423

A central limit theorem is proved for stochastically additive functional of ergodic Markov chains.

### On the rate of convergence of an unstable solution of a stochastic differential equation

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1424–1427

We study the rate of convergence of the process $ξ(tT)/\sqrt{T}$ to the process $w(t)/σ$ as $T → ∞$, where $ξ(t)$ is a solution of the stochastic differential equationd $ξ(t) = a(ξ(t))dt + σ(ξ(t))dw(t)$.

### On a hypothesis concerning Hamiltonians of canonical systems

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1428–1431

We present a classification of canonical systems and show that, under certain conditions, Hamiltonians belonging to the same class are linearly similar.