2019
Том 71
№ 11

# Volume 56, № 10, 2004

Article (Russian)

### Classical Solvability of the First Initial Boundary-Value Problem for a Nonlinear Strongly Degenerate Parabolic Equation

Ukr. Mat. Zh. - 2004. - 56, № 10. - pp. 1299-1320

We prove the existence of a classical solution global in time for the first initial boundary-value problem for a nonlinear strongly degenerate parabolic equation.

Article (Ukrainian)

### Stability of Solutions of a Quasilinear Index-2 Tractable Differential Algebraic Equation by the Lyapunov Second Method

Ukr. Mat. Zh. - 2004. - 56, № 10. - pp. 1321-1334

The Lyapunov second method is an important tool in the qualitative theory of ordinary differential equations. In this paper, we consider the behavior of solutions of quasilinear index-2 tractable differential algebraic equations. Using the Lyapunov second method, we prove sufficient conditions for the stability of zero solution of such equations.

Article (English)

### Hopficity and Co-Hopficity in Soluble Groups

Ukr. Mat. Zh. - 2004. - 56, № 10. - pp. 1335-1341

We show that a soluble group satisfying the minimal condition for its normal subgroups is co-hopfian and that a torsion-free finitely generated soluble group of finite rank is hopfian. The latter property is a consequence of a stronger result: in a minimax soluble group, the kernel of an endomorphism is finite if and only if its image is of finite index in the group.

Article (Ukrainian)

### On the Skitovich-Darmois Theorem on Abelian Groups

Ukr. Mat. Zh. - 2004. - 56, № 10. - pp. 1342 – 1356

We prove theorems that generalize the Skitovich-Darmois theorem to the case where independent random variables ξj, j = 1, 2, ..., n, n ≥ 2, take values in a locally compact Abelian group and the coefficients αj and βj of the linear forms L 1 = α1ξ1 + ... + αnξn and L 2 = β1ξ1 + ... + βnξn are automorphisms of this group.

Article (Ukrainian)

### Separately Continuous Functions on Products and Their Dependence on ℵ Coordinates

Ukr. Mat. Zh. - 2004. - 56, № 10. - pp. 1357-1369

We investigate necessary and sufficient conditions on topological products X = ∏s ∈ s X s and Y = ∏t ∈ T Y t for every separately continuous function f: X × Y → ℝ to be dependent on at most ℵ coordinates with respect to a certain coordinate.

Article (English)

### A Higher-Dimensional Version of the Brody Reparametrization Lemma

Ukr. Mat. Zh. - 2004. - 56, № 10. - pp. 1369-1377

We prove a generalization of the Brody reparametrization lemma.

Article (Russian)

### Best Approximations of $q$-Ellipsoids in Spaces $S_{ϕ}^{p,μ}$

Ukr. Mat. Zh. - 2004. - 56, № 10. - pp. 1378-1383

We find exact values of the best approximations and basic widths of $q$-ellipsoids in the spaces $S_{ϕ}^{p,μ}$ for $q > p > 0$.

Article (Ukrainian)

### Approximate Averaged Synthesis of the Problem of Optimal Control for a Parabolic Equation

Ukr. Mat. Zh. - 2004. - 56, № 10. - pp. 1384–1394

For a problem of optimal control for a parabolic equation, in the case of bounded control, we construct and justify an approximate averaged control in the form of feedback.

Article (Ukrainian)

### Diffusive Lotka-Volterra System: Lie Symmetries and Exact and Numerical Solutions

Ukr. Mat. Zh. - 2004. - 56, № 10. - pp. 1395-1404

We present a complete description of Lie symmetries for the nonlinear diffusive Lotka-Volterra system. The results are used for the construction of exact solutions of the Lotka-Volterra system, which, in turn, are used for solving the corresponding nonlinear boundary-value problems with zero Neumann conditions. The analytic results are compared with the results of computation based on the finite-element method. We conclude that the obtained exact solutions play an important role in solving Neumann boundary-value problems for the Lotka-Volterra system.

Article (English)

### On Exponential Sums Related to the Circle Problem

Ukr. Mat. Zh. - 2004. - 56, № 10. - pp. 1405-1418

Let r(n) count the number of representations of a positive integer n as a sum of two integer squares. We prove a truncated Voronoi-type formula for the twisted Mobius transform $$\mathop \sum \limits_{n \leqslant x} \;\,r(n)\;\exp \left( {2\pi i\frac{{nk}}{{4l}}} \right),$$ where k and l are positive integers such that k and 4l are coprime, and give some applications (almost periodicity, limit distribution, an asymptotic mean-square formula, and O- and Ω-estimates for the error term).

Brief Communications (English)

### Groups with Few Nonmodular Subgroups

Ukr. Mat. Zh. - 2004. - 56, № 10. - pp. 1419-1423

Let G be a Tarski-free group such that the join of all nonmodular subgroups of G is a proper subgroup in G. It is proved that G contains a finite normal subgroup N such that the quotient group G/N has a modular subgroup lattice.

Brief Communications (Russian)

### On Cubic Operators Defined on Finite-Dimensional Simplexes

Ukr. Mat. Zh. - 2004. - 56, № 10. - pp. 1424-1433

We introduce the concept of cubic operator. For one class of cubic operators defined on finite-dimensional simplexes, a complete description of the behavior of their trajectories is given. The convergence of Cesaro means is established.

Brief Communications (English)

### On Generalized Hardy Sums $s_5(h, k)$

Ukr. Mat. Zh. - 2004. - 56, № 10. - pp. 1434–1440

The aim of this paper is to study generalized Hardy sums $s_5(h, k)$. By using mediants and the adjacent difference of Farey fractions, we establish a relationship between $s_5(h, k)$ and Farey fractions. Using generalized Dedekind sums and a generalized periodic Bernoulli function, we define generalized Hardy sums $s_5(h, k)$. A relationship between $s_5(h, k)$ and the Hurwitz zeta function is established. By using the definitions of Lambert series and cotπz, we establish a relationship between $s_5(h, k)$ and Lambert series.