# Volume 47, № 1, 1995

### On the 75th birthday of Nikolai Pavlovich Korneichuk, corresponding member of the ukrainian academy of sciences

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 3-3

### Essential self-adjointness of Dirichlet operators of Gibbs measures on infinite products of manifolds

Antoniouk A. Val., Antoniouk A. Vict., Kondratiev Yu. G.

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 4–11

We obtain the conditions of essential self-adjointness of Dirichlet operators of Gibbs measures with essential domains consisting of smooth cylindrical functions. It is proved that certain spaces of smooth functions are invariant under the action of the semigroup of the Dirichlet operator.

### Some duality relations for local splines

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 12–19

We obtain duality relations for local periodic cubic and parabolic splines of minimal defect and establish some of their corollaries.

### Boundary-value problem for a mixed parabolic-hyperbolic equation of the third order

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 20–29

The unique solvability of a certain boundary-value problem is proved for a mixed parabolic-hyperbolic type equation of the third order.

### Integration of a class of systems of differential equations by using a contour integral

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 30–38

For systems of*q* linear differential equations of*n*th order with polynomial matrix coefficients, a fundamental family of formal solutions defined in a certain sector of a complex plane is constructed by using the Laplace contour integral. For large positive values of an independent variable, the asymptotic representations of indicated solutions are obtained.

### On the nilpotency class of a multiplicative group of a modular group algebra of a dihedral 2-group

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 39–45

It is proved that the wreath product of a second-order group and the commutant of a dihedral group is imbedded into a multiplicative group of a modular group algebra of a dihedral group of order 2^{ n }. This implies that the nilpotency class of the multiplicative group is equal to 2^{ n−2}, i.e., to the order of the commutant of the dihedral group.

### Limits on the real line of symmetric spaces on segments

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 46–55

In the same way as the known spaces*M* _{ p },M _{ p }, and*I* _{ p } are constructed on the basis of the space*L* _{ p }(−1, 1), we construct the corresponding “limit” spaces*M* _{ E },M _{ E }, and*I* _{ E }on the real line on the basis of a symmetric function space*E* on a segment and study some of their Banach properties.

### Existence and properties of local times for Markov random fields

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 56–63

Random fields that have the “coordinatewise” Markov property are considered. The notions of an excessive function, a potential, and a continuous additive functional are introduced. Sufficient conditions for the existence of local time as a special form of continuous additive functional are formulated, and the uniqueness of this time to within a multiplicative constant is proved.

### Matrix convexity

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 64–69

The notion of a convex set is generalized. In the definition of ordinary convexity, sums of products of vectors and numbers are used. In the generalization considered in this paper, the role of numbers is played by matrices; this is why we call it “matrix convexity.”

### Wild problems in the theory of representations of *-algebras generated by generatrices and relations

Piryatinskaya A. Yu., Samoilenko Yu. S.

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 70–78

Two definitions of wild *-algebras are considered from the point of view of the theory of representations. We present methods for proving the fact that a *-algebra is wild. It is also shown that if a algebra is*p*-wild, then it is*f*-wild. The converse statement is not true.

### On the best approximations and Kolmogorov widths of besov classes of periodic functions of many variables

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 79–92

Order estimates are obtained for the best approximations of the classes*B* _{1, θ} ^{ r } in the space*L* _{ q }with 1<*q*<∞ and classes*B* _{∞, θ} ^{ r } in a uniform metric. The behavior of Kolmogorov widths of the classes*B* _{ p, θ} ^{ r } ,1<*p*≤∞, in the metric of L_{∞} is studied.

### On regularity of generalized solutions of the third boundary-value problem for an elliptic difference-differential equation

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 93–104

Unlike the case of elliptic differential equations, generalized solutions of elliptic difference-differential equations may be not smooth in a domain*Q* but remain smooth only in certain subdomains*Q* _{ r }⊂*Q* Conditions are considered which are necessary and sufficient for generalized solutions of the third boundary-value problem to preserve smoothness on the boundary of adjacent subdomains*Q* _{ r }.

### On inequalities for norms of intermediate derivatives on a finite interval

Babenko V. F., Kofanov V. A., Pichugov S. A.

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 105–107

For functions*f* which have an absolute continuous (*n*−1)th derivative on the interval [0, 1], it is proved that, in the case of*n*>4, the inequality $$\left\| {f^{(n - 2)} } \right\|_\infty \leqslant 4^{n - 2} (n - 1) ! \left\| f \right\|_\infty + \left\| {f^{(n)} } \right\|_\infty /2$$ holds with the exact constant 4^{ n−2}(n−1)!.

### On exact constants in Jackson-type inequalities in the space $L^2$

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 108-110

The exact dependence of constants in Jackson-type inequalities on the rate of convergence of the Diophantine approximations of certain numbers is obtained.

### Characteristic subgroups of the Jonquere group over a field of characteristic zero

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 111–113

The lattice of characteristic subgroups of the Jonquere group over a field of characteristic zero is described.

### Weak invariance principle for solutions of stochastic recurrence equations in a banach space

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 114–117

We show that, in a Banach space, continuous random processes constructed by using solutions of the difference equation*X* _{ n }=*A* _{ n } *X* _{ n+1}+*V* _{ n }, n=1, 2,..., converge in distribution to a solution of the corresponding operator equation.

### Remark on the central limit theorem for ergodic chains

Moskal'tsova N. V., Shurenkov V. M.

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 118-120

We obtain sufficient conditions that should be imposed on a function*f* in order that, for ergodic Markov chains, the sum $$\frac{1}{{\sqrt n }} \sum\limits_{k = 0}^{n - 1} { f(X_k )} $$ be asymptotically normal.

### On the symmetry and exact solutions of a certain transport equation

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 121–125

Both the Lie and*Q*-conditional symmetry of a certain linear transport equation are studied and classes of its exact solutions are obtained.

### Generalized Dicke model as an integrable dynamical system inverse to the nonlinear Schrödinger equation

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 126–128

We prove that a dynamical system obtained by the space-time inversion of the nonlinear Schrödinger equation is equivalent to a generalized Dicke model. We study the complete Liouville integrability of the obtained dynamical system.

### Attractors of dynamical systems with control: topology of purposeful formation

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 129–132

The definitions of homogeneous and mosaic attractors of codimension one are given. A topological method for their purposeful formation by using the feedback control laws of controlled dynamical systems is suggested.

### On the asymptotic behavior of certain infinite-dimensional recurrence sequences

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 133-137

Under certain conditions imposed on an operator $B$, we obtain criteria of boundedness of sequences $x_{n+1} = A x_n + Bb_n,\; n > 0$, for any $x_0$ and bounded $\{b_n, \; n ≥ 0\}$ in infinite-dimensional spaces. The results are given in terms of spectral properties of the operator $A$.

### A. M. Samoilenko's numerical-analytic method without determining equation

Kovalenko O. V., Trofimchuk E. P.

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 138-140

We suggest a modification of A. M. Samoilenko's numerical-analytic method for investigating the problem $dx/dt=f(t,x), \mathfrak{L}(x) = d$ (here $\mathfrak{L}(x): C([0, T], R^n) \rightarrow R^n$ is a linear continuous operator} in which it is not necessary to solve an additional determining equation.

### Matrix infinite-dimensional analog of the Wiener theorem on local inversion of Fourier transforms

Degtyar' S. V., Shurenkov V. M.

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 141-145

A matrix infinite-dimensional analog of the Wiener theorem on local inversion of Fourier transforms is proved.