# Volume 58, № 6, 2006

### On the dynamical equations of a system of linearly coupled nonlinear oscillators

Ukr. Mat. Zh. - 2006. - 58, № 6. - pp. 723–729

We consider a system of differential equations that describes the dynamics of an infinite chain of linearly coupled nonlinear oscillators. Some results concerning the existence and uniqueness of global solutions of the Cauchy problem are obtained.

### On the Schur theorem for n-ary groups

Ukr. Mat. Zh. - 2006. - 58, № 6. - pp. 730–741

We prove *n*-ary analogs of the well-known Schur theorem on the finiteness of a commutator subgroup of a group whose center is of finite index.

### Structure of a permutable Munn semigroup of finite rank

Ukr. Mat. Zh. - 2006. - 58, № 6. - pp. 742–746

A semigroup any two congruences of which commute as binary relations is called a permutable semigroup. We describe the structure of a permutable Munn semigroup of finite rank.

### Extremal problems dual to the Gauss variational problem

Ukr. Mat. Zh. - 2006. - 58, № 6. - pp. 747–764

We formulate and solve extremal problems of potential theory that are dual to the Gauss variational problem but, unlike the latter, are always solvable. Statements on the compactness of classes of solutions and the continuity of extremals are also established.

### On Frobenius groups with noninvariant factor *SL*_{ 2}(3)

Kozulin S. N., Senashov V. I., Shunkov V. P.

Ukr. Mat. Zh. - 2006. - 58, № 6. - pp. 765–777

We obtain a criterion for the unsimplicity of an infinite group containing the infinite class of the Frobenius groups $L_g = \langle a, g^{-1} a g\rangle$ with complement $SL_2 ( 3 )$.

### Convergence of the Galerkin method for a wave equation with singular right-hand side

Ukr. Mat. Zh. - 2006. - 58, № 6. - pp. 778–786

We consider analogs of the Galerkin method for a linear wave equation of the fifth order with generalized functions on the right-hand side. Theorems on the convergence of an approximate method, depending on the order of singularity of the right-hand side, are proved.

### Mel’nikov-Samoilenko adiabatic stability problem

Ukr. Mat. Zh. - 2006. - 58, № 6. - pp. 787–803

We develop a symplectic method for the investigation of invariant submanifolds of nonautonomous Hamiltonian systems and ergodic measures on them. The so-called Mel’nikov-Samoilenko problem for the case of adiabatically perturbed completely integrable oscillator-type Hamiltonian systems is studied on the basis of a new construction of “ virtual” canonical transformations.

### Generalized solutions of mixed problems for first-order partial functional differential equations

Ukr. Mat. Zh. - 2006. - 58, № 6. - pp. 804–828

A theorem on the existence of solutions and their continuous dependence upon initial boundary conditions is proved. The method of bicharacteristics is used to transform the mixed problem into a system of integral functional equations of the Volterra type. The existence of solutions of this system is proved by the method of successive approximations using theorems on integral inequalities. Classical solutions of integral functional equations lead to generalized solutions of the original problem. Differential equations with deviated variables and differential integral problems can be obtained from the general model by specializing given operators.

### Julia lines of entire functions of slow growth

Ukr. Mat. Zh. - 2006. - 58, № 6. - pp. 829–834

We obtain sufficient conditions under which the Julia lines of entire functions of slow growth do not have finite exceptional values.

### On the uniqueness of a solution of the problem with oblique derivative for the equation Δ ^{n} *v* = 0

Ukr. Mat. Zh. - 2006. - 58, № 6. - pp. 835–841

We prove the uniqueness of a solution of the problem with oblique derivative for the equation Δ ^{n} *v* = 0.

### Problem of interpolation of functions by two-dimensional continued fractions

Ukr. Mat. Zh. - 2006. - 58, № 6. - pp. 842–851

We investigate the problem of interpolation of functions of two real variables by two-dimensional continued fractions.

### On the modularity of a lattice of τ-closed totally saturated formations of finite groups

Ukr. Mat. Zh. - 2006. - 58, № 6. - pp. 852–858

We study τ-closed totally saturated formations of finite groups.

### On calculation of integrals over spherical domains

Ukr. Mat. Zh. - 2006. - 58, № 6. - pp. 859–864

We construct cubature formulas for the computation of integrals over spherical domains containing less nodes as compared with known ones.