# Volume 60, № 7, 2008

### Problem of uniqueness of an element of the best nonsymmetric *L*_{ 1}-approximation of continuous functions with values in *KB* -spaces

Babenko V. F., Tkachenko M. E.

Ukr. Mat. Zh. - 2008. - 60, № 7. - pp. 867 – 878

### On the behavior of a simple-layer potential for a parabolic equation on a Riemannian manifold

Ukr. Mat. Zh. - 2008. - 60, № 7. - pp. 879–891

On a Riemannian manifold of nonpositive sectional curvature (Cartan-Hadamard-type manifold), we consider a parabolic equation. The second boundary-value problem for this equation is set in a bounded domain whose surface is a smooth submanifold. We prove that the gradient of the simple-layer potential for this problem has a jump when passing across the submanifold, similarly to its behavior in a Euclidean space.

### On the uniqueness of a solution of the inverse problem for a simple-layer potential

Ukr. Mat. Zh. - 2008. - 60, № 7. - pp. 892–899

We prove the uniqueness of solution of the inverse problem of single-layer potential
for star-shaped smooth surfaces in the case of the metaharmonic equation Δ*v* - *K*² *v* = 0. For the Laplace equation, a similar statement is not true.

### Solution of a second-order Poincaré-Perron-type equation and differential equations that can be reduced to it

Ukr. Mat. Zh. - 2008. - 60, № 7. - pp. 900–917

The analytical solution of the second-order difference Poincare–Perron equation is presented. This enables us to construct in the explicit form a solution of the differential equation $$t^2(A_1t^2 + B_1t + C_1)u'' + t(A_2t^2 + B_2t + C_2)u' + (A_3t^2 + B_3t + C_3)u = 0 $$ The solution of the equation is represented in terms of two hypergeometric functions and one new special function. As a separate case, the explicit solution of the Heun equation is obtained, and polynomial solutions of this equation are found.

### On some properties of solutions of quasilinear degenerate equations

Ukr. Mat. Zh. - 2008. - 60, № 7. - pp. 918–936

For the quasilinear equations div *A*(*x, u, ∇u*) = 0 with degeneracy ω(*x*) from the Muckenhaupt *A _{p }*-class, we prove the Harnack inequality, an estimate of the Holder norm, and a sufficient test for the regularity of boundary points of the Wiener type.

### Common periodic trajectories of two mappings

Ukr. Mat. Zh. - 2008. - 60, № 7. - pp. 937–948

For a map *f* ∈ *C ^{r }*(

*I, I*),

*r*> 0, we consider the problem of the existence of a map close to it and having the common periodic trajectories of given periods with

*f*.

### Higher-order parabolic variational inequality in unbounded domains

Ukr. Mat. Zh. - 2008. - 60, № 7. - pp. 949–968

We prove the existence and uniqueness of a solution of a nonlinear parabolic variational inequality in an unbounded domain without conditions at infinity. In particular, the initial data may infinitely increase at infinity, and a solution of the inequality is unique without any restrictions on its behavior at infinity.

### Comparison theorems for some nonsymmetric classes of functions

Motornaya O. V., Motornyi V. P.

Ukr. Mat. Zh. - 2008. - 60, № 7. - pp. 969–975

We prove comparison theorems of the Kolmogorov type for some nonsymmetric classes of functions.

### Approximation of Poisson integrals by one linear approximation method in uniform and integral metrics

Ukr. Mat. Zh. - 2008. - 60, № 7. - pp. 976–982

We obtain asymptotic equalities for the least upper bounds of approximations of classes of Poisson integrals of periodic functions by a linear approximation method of special form in the metrics of the spaces *C* and *L _{p }*.

### On the construction of cubature formulas invariant under dihedral groups

Ukr. Mat. Zh. - 2008. - 60, № 7. - pp. 983–991

We study cubature formulas invariant under the dihedral group of order 16*p*.

### Classification of topologies on finite sets using graphs

Adamenko N. P., Velichko I. G.

Ukr. Mat. Zh. - 2008. - 60, № 7. - pp. 992–996

With the use of digraphs, topologies on finite sets are studied. On this basis, a new classification of such topologies is proposed.
Some properties of *T*_{0}-topologies on finite sets are proved.
In particular, it is proved that, in *T*_{0}-topologies, there exist open sets containing arbitrary number of
elements that does not exceed the cardinality of the set itself.

### Generators and relations for wreath products

Drozd Yu. A., Skuratovskii R. V.

Ukr. Mat. Zh. - 2008. - 60, № 7. - pp. 997–999

Generators and defining relations for wreath products of groups are given. Under a certain condition (conormality of generators), they are minimal.

### First eigenvalue of the Laplace operator and mean curvature

Ukr. Mat. Zh. - 2008. - 60, № 7. - pp. 1000–1003

The main theorem of this paper states a relation between the first nonzero eigenvalue of the Laplace operator and the squared norm of mean curvature in irreducible compact homogeneous manifolds under spatial conditions. This statement has some consequences presented in the remainder of paper.

### Estimation of the product of inner radii of partially nonoverlapping domains

Ukr. Mat. Zh. - 2008. - 60, № 7. - pp. 1004–1008

We present new results on the maximization of products of positive powers of inner radii of some special domain systems in the extended complex plane $\overline{{\mathbb C}}$ with respect to points of finite sets such that any two distinct points $z_1, z_2 \in {\mathbb C}\setminus \{0\}$ of such set belong to different rays emerging from the origin.