# Volume 45, № 2, 1993

### On N. N. Bogolyubov's works in classical and quantum statistical mechanics

Mitropolskiy Yu. A., Petrina D. Ya.

Ukr. Mat. Zh. - 1993. - 45, № 2. - pp. 155–201

A review of N. N. Bogolyubov's works in classical and quantum statistical mechanics is presented.

### Periodic solutions of strongly nonlinear systems with nonclassical right-hand side in the case of a family of generating solutions

Ukr. Mat. Zh. - 1993. - 45, № 2. - pp. 202–208

The problem of the existence of periodic solutions to differential equations with pulse effects on the surfaces and to differential equations with discontinuous right-hand sides close to arbitrary nonlinear ones is studied. The existence of a family of periodic solutions to generating equations is assumed.

### Description of bilateral ideals in a class of noncommutative rings. I

Ukr. Mat. Zh. - 1993. - 45, № 2. - pp. 209–220

For generalized Weil algebras of degree 1 with base Dedekind ring, bilateral ideals are classified. The (noncommutative) algebras, in which the product of ideals is permutable and any proper ideal is uniquely decomposed into the product of prime ideals, are described.

### Weakly nonlinear boundary-value problems for differential systems with pulse influence

Boichuk О. A., Khrashchevskaya R. F.

Ukr. Mat. Zh. - 1993. - 45, № 2. - pp. 221–225

The coefficient sufficient conditions for the existence of solutions and the iteration algorithm of constructing these solutions are obtained for weakly nonlinear boundary-value problems for systems of ordinary differential equations with pulse influence in the general case in which the number of boundary conditions does not coincide with the order of the differential system. The equation is derived for generating amplitudes of these boundary-value problems. This equation determines the amplitude of a solution, which can be regarded as generating for the required solution, and gives necessary conditions for the existence of this solution.

### On the structure of sets of $ σ$-monogeneity for continuous functions

Ukr. Mat. Zh. - 1993. - 45, № 2. - pp. 226–232

The notion of the sets of σ-monogeneity for continuous functions is introduced which makes it possible to study pseudo-analytic properties of these functions. The theorem on the structure of these sets is proved.

### On equicontinuous factors of linear extensions of minimal dynamical systems

Ukr. Mat. Zh. - 1993. - 45, № 2. - pp. 233–238

The concept of the equicontinuous factor of the linear extension of a minimal transformation group is introduced and investigated. It is shown that a subset of motions, bounded and distal with respect to the extension, forms a maximal equicontinuous subsplitting of the linear extension. As a consequence, any distal linear extension has a nontrivial equicontinuous invariant subsplitting. The linear extensions without exponential dichotomy possess similar subsplittings if the Favard condition is satisfied. The same statement holds for linear extensions with the property of recurrent motions additivity provided that at least one nonzero motion of this sort exists.

### On periodic and bounded solutions of the operator Riccati equation

Ukr. Mat. Zh. - 1993. - 45, № 2. - pp. 239–242

Sufficient conditions for the existence of periodic and bounded solutions of the operator Riccati equation are presented.

### $k$-Snakes in finite-order interpolational classes of functions

Ukr. Mat. Zh. - 1993. - 45, № 2. - pp. 243-250

$k$-snakes are a generalization of the polynomials with the least deviation from zero underk-constraints. We prove the existence and uniqueness of $k$-snakes in finite-order interpolational classes and their continuous differentiability with respect to a parameter.

### Semimartingales with values on groups and Lie algebras

Ukr. Mat. Zh. - 1993. - 45, № 2. - pp. 251–257

We establish a one-to-one correspondence between the processes taking values on the Lie group and its Lie algebra. This correspondence preserves the main properties of the processes: semimartingale property with respect to a certain flow, independence of increments, and continuity.

### On passive and active algorithms of reconstruction of functions

Ukr. Mat. Zh. - 1993. - 45, № 2. - pp. 258–264

We consider passive and active algorithms of reconstruction of functions, satisfying the condition $|f(t′) − f(t″)| ≤ |t′ − t″|^{α},\; 0 < α ≤ 1,$ according to their values $f(t)$ at the points of the interval $[a, b]$. An active algorithm is presented which guarantees, for monotonic functions from the above-mentioned class with $0 < α < 1$, a higher order of error in $C [a, b]$ than can be attained by any passive algorithm.

### On asymptotics of the potential of a countable ergodic Markov chain

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

Ukr. Mat. Zh. - 1993. - 45, № 2. - pp. 265–269

For a class of functions $f$, the convergence in Abel's sense is proved for the potential $\sum_{n⩾o}P^nf(i) of a uniform ergodic Markov chain in a countable phase space. Several corollaries are obtained which are useful from the point of view of the possible application to CLT (the central limit theorem) for Markov chains. In particular, we establish the condition equivalent to the boundedness of the second moment for the time of the first return into the state.

### On perturbation of semi-Noether operators in incomplete spaces. I

Ukr. Mat. Zh. - 1993. - 45, № 2. - pp. 270–278

The perturbation problem is considered for a semi-Noether operator under minimal assumptions imposed on given spaces.

### On regularity of linear systems with a degenerate matrix by the derivative

Symокоп V. G., Trofimchuk E. P.

Ukr. Mat. Zh. - 1993. - 45, № 2. - pp. 279–286

New sufficient conditions for the existence of invariant manifolds are proved for linear systems with degenerate matrix by the derivative.

### Minimal dynamical system with given $D$-function and topological entropy

Ukr. Mat. Zh. - 1993. - 45, № 2. - pp. 287–292

The $D$-function is a new topological invariant introduced by the author in [3] to classify the minimal dynamical system and to generalize Sharkovskii's theorem on the coexistence of periodic orbits. We show that the $D$-function and the topological entropy are independent.

### Nonlocal ansatze and solutions of a nonlinear system of heat-conduction equations

Amerov T. K., Fushchich V. I., Serov N. I.

Ukr. Mat. Zh. - 1993. - 45, № 2. - pp. 293–302

By a nonlocal substitution, a nonlinear system of heat-conduction equations is reduced to a scalar nonlinear heat-conduction equation. The Lie and conditional invariance of the scalar equation is used to find nonlocal ansatze which reduce the original system to systems of ordinary differential equations.

### Jubilees Mykola I. Shkil' (on his 60th birthday)

Ukr. Mat. Zh. - 1993. - 45, № 2. - pp. 303-304