# Volume 51, № 4, 1999

### On the best $L_1$-approximations of functional classes by splines under restrictions imposed on their derivatives

Babenko V. F., Parfinovych N. V.

Ukr. Mat. Zh. - 1999. - 51, № 4. - pp. 435-444

We find the exact asymptotics ($n → ∞$) of the best $L_1$-approximations of classes $W_1^r$ of periodic functions by splines $s ∈ S_{2n, r∼-1}$ ($S_{2n, r∼-1}$ is a set of $2π$-periodic polynomial splines of order $r−1$, defect one, and with nodes at the points $kπ/n,\; k ∈ ℤ$) such that $V_0^{2π} s^{( r-1)} ≤ 1+ɛ_n$, where $\{ɛ_n\}_{n=1}^{ ∞}$ is a decreasing sequence of positive numbers such that $ɛ_n n^2 → ∞$ and $ɛ_n → 0$ as $n → ∞$.

### On a form of the scattering matrix for ρ-perturbations of an abstract wave equations

Ukr. Mat. Zh. - 1999. - 51, № 4. - pp. 445–457

We present the definition of ρ-perturbations of an abstract wave equation. As a special case, this definition involves perturbations with compact support for the classical wave equation. We construct the scattering matrix for equations of such a type.

### On the stability of solutions of a quasilinear uncertain system

Ukr. Mat. Zh. - 1999. - 51, № 4. - pp. 458–465

We generalize the Lyapunov direct method, which can be used for establishing new conditions of the uniform asymptotic stability of solutions of an uncertain system with respect to an invariant moving set.

### Discrete dynamical systems with invariant asymptotically stable toroidal manifold

Samoilenko A. M., Slyusarchuk V. V.

Ukr. Mat. Zh. - 1999. - 51, № 4. - pp. 466–471

We obtain conditions for asymptotic stability of quasiperiodic trajectories of discrete dynamical systems in the case of infinite-dimensional Banach space.

### Sufficient conditions for the almost layer finiteness of groups

Ukr. Mat. Zh. - 1999. - 51, № 4. - pp. 472–485

We prove a theorem that describes almost layer-finite groups in the class of conjugatively biprimitive-finite groups.

### On the existence and uniqueness of a solution of the problem of uniform *SK*-spline-interpolation

Ukr. Mat. Zh. - 1999. - 51, № 4. - pp. 486–492

We establish necessary and sufficient conditions for the existence and uniqueness of generalized interpolating*SK*-splines with a uniform distribution of interpolation points.

### Approximate properties of the Zygmund method

Ukr. Mat. Zh. - 1999. - 51, № 4. - pp. 493–518

We give a review of results on approximate properties of Zygmund sums and their generalizations.

### Limit theorems in the theory of multipoint boundary-value problems

Nedokis V. A., Teplinsky Yu. V.

Ukr. Mat. Zh. - 1999. - 51, № 4. - pp. 519–531

We present a reduction of a countable system of differential equations with countably-point boundary conditions to the case of a finite-dimensional multipoint boundary-value problem. We separately consider the case of a linear system.

### Optimal control over nonlinear stochastic systems

Ukr. Mat. Zh. - 1999. - 51, № 4. - pp. 532–541

The synthesis of optimal control over nonlinear stochastic systems that are described by the Itô equations is reduced to the solution of recurrence relations derived from the Bellman stochastic equation.

### A functional limit theorem of the type of the law of large numbers for random reliefs

Ukr. Mat. Zh. - 1999. - 51, № 4. - pp. 542–552

For a random function dependent on time and on a point of a space with measure, we find an asymptotic expression for the measure of the region in which values of the function do not exceed a given level.

### Two theorems in the theory of summation of numerical series by lower triangular positive monotone matrices

Ukr. Mat. Zh. - 1999. - 51, № 4. - pp. 553-555

For so-called monotone matrices, we establish necessary and sufficient conditions for the summation with them of some divergent sequences of 0 and 1 up to zero or some unbounded sequences of nonnegative numbers.

### Optimization of a system of linear differential equations with random coefficients

Dzhalladova I. A., Valeyev K. G.

Ukr. Mat. Zh. - 1999. - 51, № 4. - pp. 556–561

We consider a system of differential equations with controls that are linearly contained in the right-hand sides. We establish a necessary condition for the optimal control that minimizes a quadratic functional.

### Conditions for one-valued solvability of nonlinear stationary heat-conduction problems

Ukr. Mat. Zh. - 1999. - 51, № 4. - pp. 562–567

We establish conditions for existence and uniqueness of nonnegative solutions of nonlinear stationary heat-conduction problems, the Dirichlet, problem and the Neumann one, with regard for the dependence of the heat-conduction coefficient and inner heat sources on temperature.

### Isomorphisms of combinatorial block decompositions of three-dimensional manifolds

Ukr. Mat. Zh. - 1999. - 51, № 4. - pp. 568–571

For three-dimensional manifolds with the structure of a combinatorial block complex, we construct an invariant that allows one to verify the existence of isomorphisms, between these manifolds. For complexes of small dimensionality, we solve the problem on the possibility of extending the isomorphisms of subcomplexes to those of complexes.

### $T_0$-group and its place in the theory of groups

Ukr. Mat. Zh. - 1999. - 51, № 4. - pp. 572-576

A class of $T_0$-groups is characterized which is closely associated with free Burnside groups with odd period not less than 665. Examples based on the well-known Adyan and Olshanskii constructions are given. In addition, the place of a finite group in the class of all groups is indicated.