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

Bazalii B. V., Krasnoshchok M. V.

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 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.

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

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 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.

### Hopficity and Co-Hopficity in Soluble Groups

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 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.

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

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 56, № 10. - pp. 1342 – 1356

We prove theorems that generalize the Skitovich-Darmois theorem to the case where independent random variables ξ_{j}, *j* = 1, 2, ..., *n*, *n* ≥ 2, take values in a locally compact Abelian group and the coefficients α_{j} and β_{j} of the linear forms *L* _{1} = α_{1}ξ_{1} + ... + α_{n}ξ_{n} and *L* _{2} = β_{1}ξ_{1} + ... + β_{n}ξ_{n} are automorphisms of this group.

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

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 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.

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

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 56, № 10. - pp. 1369-1377

We prove a generalization of the Brody reparametrization lemma.

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

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 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 $q > p > 0$.

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

Kapustyan O. A., Sukretna A. V.

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 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.

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

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 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.

### On Exponential Sums Related to the Circle Problem

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 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 4*l* are coprime, and give some applications (almost periodicity, limit distribution, an asymptotic mean-square formula, and *O*- and Ω-estimates for the error term).

### Groups with Few Nonmodular Subgroups

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 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.

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

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 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.

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

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 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.