# Volume 48, № 10, 1996

### On the complexity of boundary integral equations with analytic coefficients with logarithmic singularities

Ukr. Mat. Zh. - 1996. - 48, № 10. - pp. 1299-1310

We find the exact order of the ε-complexity of weakly singular integral equations with periodic and analytic coefficients of logarithmic singularities. This class of equations includes boundary equations for outer boundary-value problems for the two-dimensional Helmholtz equation.

### Widths of sets of functions of discrete variable

Velikin V. L., Velikina Yu. V.

Ukr. Mat. Zh. - 1996. - 48, № 10. - pp. 1311-1320

We obtain exact values of Kolmogorov and linear widths of arbitrary dimension for sets of functions of discrete variable with bounded difference of a given order.

### Estimate of the modulus of continuity of a cauchy-type integral in a domain and on its boundary

Ukr. Mat. Zh. - 1996. - 48, № 10. - pp. 1321-1328

We estimate the modulus of continuity of a Cauchy-type integral in a closed domain and its limit values on the boundary in the case where the boundary of the domain is an arbitrary closed rectifiable Jordan curve.

### On two statements of “The scottish book” concerning a ring of bounded polynomial functionals on banach spaces

Ukr. Mat. Zh. - 1996. - 48, № 10. - pp. 1329-1336

We prove an infinite-dimensional version of the Hilbert theorem about zeros (according to “The Scottish Book”). We study topological properties of the set of zeros of a continuous polynomial functional and establish necessary and sufficient conditions for this set to cut the space.

### Structure of finite nondispersible groups each nonmetacyclic subgroup of which is normal

Ukr. Mat. Zh. - 1996. - 48, № 10. - pp. 1337-1341

We determine the structure of finite minimal nondispersible groups each nonmetacyclic subgroup of which is normal (Theorem 2) and describe all finite nondispersible groups each nonmetacyclic subgroup of which is normal (Theorem 3).

### Structure of separative dedekind groups

Ukr. Mat. Zh. - 1996. - 48, № 10. - pp. 1342-1351

We describe groups such that all their subgroups that do not belong to a certain proper subgroup are normal. We also solve the separate problem of description of such groups with normal non-Abelian subgroups.

### Stability of large-scale discrete systems under structural perturbations

Martynyuk A. A., Miladzhanov V. G., Muminov M. M.

Ukr. Mat. Zh. - 1996. - 48, № 10. - pp. 1352-1362

By using the matrix Lyapunov function, we establish conditions of (uniform) stability and (uniform) asymptotic stability of a large-scale discrete system under structural perturbations.

### On the navier-stokes equation with the additional condition $u_1^1 = u^3 = 0$

Popovich R. O., Popovich V. O.

Ukr. Mat. Zh. - 1996. - 48, № 10. - pp. 1363-1374

We study the Navier-Stokes equation with the additional condition $u_1^1 = u^3 = 0$. In certain cases, solutions are represented in a closed form. In other cases, the investigated system reduces to simpler systems of partial differential equations. We study the symmetry properties of these systems and construct classes of their particular solutions.

### Asymptotic behavior of solutions of pulse systems with small parameter and markov switchings. I. Uniform boundedness of solutions

Sverdan M. L., Tsar’kov E. F., Yasinsky V. K.

Ukr. Mat. Zh. - 1996. - 48, № 10. - pp. 1375-1385

We consider pulse systems with Markov switchings. We study the problems of uniform boundedness of solutions of these systems and the stability of the systems with respect to the limit equation.

### Stability of semi-markov evolution systems and its application in financial mathematics

Biirdeinyi A. G., Svishchuk A. V.

Ukr. Mat. Zh. - 1996. - 48, № 10. - pp. 1386-1401

We study the problem of stability of semi-Markov evolution systems and its application in financial mathematics.

### On the gyroscopic stabilization of conservative systems

Ukr. Mat. Zh. - 1996. - 48, № 10. - pp. 1402-1408

We consider conservative systems with gyroscopic forces and prove theorems on stability and instability of equilibrium states for such systems. These theorems can be regarded as a generalization of the Kelvin theorem to nonlinear systems.

### On the exponential dichotomy of linear difference equations

Ukr. Mat. Zh. - 1996. - 48, № 10. - pp. 1409-1416

We consider a system of linear difference equations*x* ^{n+1} =A (n)x^{n} in an*m*-dimensional real or complex space*Vsum* with det*A(n)* = 0 for some or all*n* ε*Z*. We study the exponential dichotomy of this system and prove that if the sequence {*A(n)*} is Poisson stable or recurrent, then the exponential dichotomy on the semiaxis implies the exponential dichotomy on the entire axis. If the sequence {*A (n)*} is almost periodic and the system has exponential dichotomy on the finite interval {*k*, ...,*k* +*T*},*k* ε*Z*, with sufficiently large*T*, then the system is exponentially dichotomous on*Z*.

### One condition of complementability in groups

Chernikov N. S., Malan’ina G. A.

Ukr. Mat. Zh. - 1996. - 48, № 10. - pp. 1417-1425

We consider groups satisfying the following condition: Any subgroup of such a group that can be complemented in a larger subgroup can also be complemented in the entire group. A complete description of such groups is obtained under some weak conditions of finiteness.

### Weighted singular decomposition and weighted pseudoinversion of matrices

Ukr. Mat. Zh. - 1996. - 48, № 10. - pp. 1426-1430

For a rectangular real matrix, we obtain a decomposition in weighted singular numbers. On this basis, we obtain a representation of a weighted pseudoinverse matrix in terms of weighted orthogonal matrices and weighted singular numbers.

### Restrictions on free actions of the alternating group $A_6$ on products of spheres

Ukr. Mat. Zh. - 1996. - 48, № 10. - pp. 1431-1434

We prove that the alternating group $A_6$ cannot freely act on $(S^n)^5$ We give an example of free action of the alternating group $A_4$ on $(S^n)^3$.

### On the factorization of polynomial matrices over the domain of principal ideals

Ukr. Mat. Zh. - 1996. - 48, № 10. - pp. 1435-1439

We consider the problem of decomposition of polynomial matrices over the domain of principal ideals into a product of factors of lower degrees with given characteristic polynomials. We establish necessary and, under certain restrictions, sufficient conditions for the existence of the required factorization.