# Volume 63, № 2, 2011

### Optimization of approximate integration of set-valued functions monotone with respect to inclusion

Ukr. Mat. Zh. - 2011. - 63, № 2. - pp. 147-155

The best quadrature formula is found for the class of convex-valued functions defined on the interval [0, 1] and monotone with respect to an inclusion.

### On the dirichlet problem for an improperly elliptic equation

Burskii V. P., Kirichenko E. V.

Ukr. Mat. Zh. - 2011. - 63, № 2. - pp. 156-164

The solvability of the inhomogeneous Dirichlet problem in a bounded domain for scalar improperly elliptic differential equation with complex coefficients is investigated. We study a model case where the unit disk is chosen as a domain and the equation does not contain lowest terms. We prove that the problem has a unique solution in the Sobolev space for special classes of Dirichlet data that are spaces of functions with exponential decrease of the Fourier coefficients.

### Free dimonoids

Ukr. Mat. Zh. - 2011. - 63, № 2. - pp. 165-175

We characterize the least semilattice congruence on the free dimonoid and prove that the free dimonoid is a semilattice of *s*-simple subdimonoids each being a rectangular band of subdimonoids.

### Estimates for the approximate characteristics of the classes $B_{p, \theta}^{\Omega}$ of periodic functions of two variables with given majorant of mixed moduli of continuity

Ukr. Mat. Zh. - 2011. - 63, № 2. - pp. 176-186

Order estimates are obtained for approximation $B_{p, \theta}^{\Omega}$ of the classes of periodic functions of two variables in the space $L_q$ by operators of orthogonal projection as well as by linear operators subjected to some conditions.

### Quasipoint spectral measures in the theory of dynamical systems of conflict

Ukr. Mat. Zh. - 2011. - 63, № 2. - pp. 187-199

In the framework of dynamical picture of interacting physical systems, the notion of a spectral measure with quasipoint spectrum is introduced. It is shown that, under conflict interaction with point measures, only quasipoint singularly continuous measures are admitted for the transformation into measures with purely point spectrum.

### Group classification of quasilinear elliptic-type equations. II. Invariance under solvable Lie algebras

Ukr. Mat. Zh. - 2011. - 63, № 2. - pp. 200-215

The problem of the group classification of quasilinear elliptic-type equations in a two-dimensional space is considered. The list of all equations of this type, which admit the solvable Lie algebras of symmetry operators, is obtained. The results of this paper along with results obtained by the authors earlier give a complete solution of the problem of the group classification of quasilinear elliptic-type equations.

### Relatively thin and sparse subsets of groups

Ukr. Mat. Zh. - 2011. - 63, № 2. - pp. 216-225

Let $G$ be a group with identity $e$ and let $\mathcal{I}$ be a left-invariant ideal in the Boolean algebra $\mathcal{P}_G$ of all subsets of $G$. A subset $A$ of $G$ is called $\mathcal{I}$-thin if $gA \bigcap A \in \mathcal{I}$ for every $g \in G \ \{e\}$. A subset $A$ of $G$ is called $\mathcal{I}$-sparse if, for every infinite subset $S$ of $G$, there exists a finite subset $F \subset S$ such that $\bigcap_{g \in F}gA \in F$. An ideal $\mathcal{I}$ is said to be thin-complete (sparse-complete) if every $\mathcal{I}$-thin ($\mathcal{I}$-sparse) subset of $G$ belongs to $\mathcal{I}$. We define and describe the thin-completion and the sparse-completion of an ideal in $\mathcal{P}_G$.

### Analytic criterion for linear convexity of Hartogs domains with smooth boundary in $H^2$

Ukr. Mat. Zh. - 2011. - 63, № 2. - pp. 226-236

We establish a criterion of the local linear convexity of sets in the two-dimensional quaternion space $H^2$, that are similar to the bounded Hartogs domains with smooth boundaries in the two-dimensional complex space $C^2$.

### Some problems of the linear theory of systems of ordinary differential equations

Ukr. Mat. Zh. - 2011. - 63, № 2. - pp. 237-269

We consider problems of the linear theory of systems of ordinary differential equations related to the investigation of invariant hyperplanes of these systems, the notion of equivalence for these systems, and the Floquet – Lyapunov theory for periodic systems of linear equations. In particular, we introduce the notion of equivalence of systems of linear differential equations of different orders, propose a new formula of the Floquet form for periodic systems, and present the application of this formula to the introduction of amplitude-phase coordinates in a neighborhood of a periodic trajectory of a dynamical system.

### On the holomorphic solutions of Hamiltonian equations of motion of point charges

Ukr. Mat. Zh. - 2011. - 63, № 2. - pp. 270-280

The Maxwell - Lorenz system of an electromagnetic field interacting with charged particles (point charges) is considered in the Darwin approximation which is characterized by the Lagrangian and Hamiltonian of the particles both uncoupled with the field. The solution of the equation of motion of the particles with the approximated Darwin Hamiltonian is found on a finite time interval with the use of the Cauchy theorem. Components of this solution are represented as holomorphic functions of time.

### Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation

Ukr. Mat. Zh. - 2011. - 63, № 2. - pp. 281-288

We propose generalized forms of ultraexponential and infralogarithm functions introduced and studied by the author earlier and present two classes of special functions, namely, ultraexponential and infralogarithm $f$-type functions. As a result of present investigation, we obtain general solution of the Abel equation $\alpha (f(x)) = \alpha (x) + 1$ under some conditions on a real function $f$ and prove a new completely different uniqueness theorem for the Abel equation stating that the infralogarithm $f$-type function is its unique solution. We also show that the infralogarithm $f$-type function is an essentially unique solution of the Abel equation. Similar theorems are proved for the ultraexponential $f$-type functions and their functional equation $\beta(x) = f(\beta(x − 1))$ which can be considered as dual to the Abel equation. We also solve certain problem being unsolved before, study some properties of two considered functional equations and some relations between them.