# Volume 48, № 12, 1996

### Wilhelm Illich Fushchych (on his 60th birthday)

Mitropolskiy Yu. A., Nikitin A. G., Parasyuk O. S., Samoilenko A. M.

Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1587-1588

### Galilei-invariant higher-order equations of burgers and korteweg-de vries types

Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1589-1601

We describe nonlinear Galilei-invariant higher-order equations of Burgers and Korteweg-de Vries types. We study symmetry properties of these equations and construct new nonlinear extensions for the Galilei algebra $AG(1, 1)$.

### Aleksandr Nikolaevich Sharkovsky (on his 60th birthday)

Berezansky Yu. M., Fedorenko V. V., Kolyada S. F., Romanenko O. Yu., Sivak A. G., Vereikina M. B.

Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1602-1603

### From one-dimensional to infinite-dimensional dynamical systems: Ideal turbulence

Romanenko O. Yu., Sharkovsky O. M.

Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1604-1627

There is a very short chain that joins dynamical systems with the simplest phase space (real line) and dynamical systems with the “most complicated” phase space containing random functions, as well. This statement is justified in this paper. By using “simple” examples of dynamical systems (one-dimensional and two-dimensional boundary-value problems), we consider notions that generally characterize the phenomenon of turbulence—first of all, the emergence of structures (including the cascade process of emergence of coherent structures of decreasing scales) and self-stochasticity.

### Full cascades of simple periodic orbits on the interval

Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1628-1637

Any continuous interval map of type greater than 2∞ is shown to have what we call a full cascade of simple periodic orbits. This is used to prove that, for maps of any types, the existence of such a full cascade is equivalent to the existence of an infinite ω-limit set. For maps of type 2∞, this is equivalent to the existence of a (period doubling) solenoid. Hence, any map of type 2∞ which is either piecewise monotone (with finite number of pieces) or continuously differentiable has both a full cascade of simple periodic orbits and a solenoid.

### Bernstein inequality under averaging of elliptic systems in periodic random media

Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1638-1650

We construct the exponential Bernstein inequality for normed fluctuations of a solution of the Dirichlet problem with rapidly oscillating periodic random coefficients with respect to a solution of the averaged Dirichlet problem.

### Basic identities for additive continuously distributed sequences

Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1651-1660

For an additive sequence ξ(*n*), we establish basic factorization identities and express the distributions of limiting Junctionals (extremum values of ξ(*n*), the time and value of the first jump over a fixed level, etc.) in terms of the components of factorization.

### Differential-geometric structures in operads

Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1661-1669

We describe certain structures of formal differential geometry in terms of the theory of operads and introduce group structures, Lie-algebra structures, exponential mappings, and an analog of the de Rham complex.

### On capacity characteristics of condensers

Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1670-1682

We introduce and study a one-parameter family of capacity characteristics of condensers in ℝ^{ p },*p* ≥ 3, that contains some known capacities as elements extremal with respect to the parameter. We establish new relations between the capacity characteristics of condensers and sets.

### Complexity of approximation problems

Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1683-1694

We consider some aspects of optimal encoding and renewal related to the problem of complexity of the ε-definition of functions posed by Kolmogorov in 1962. We present some estimates for the ε-complexity of the problem of renewal of functions in the uniform metric and Hausdorff metric.

### Potential fields with axial symmetry and algebras of monogenic functions of a vector variable. II

Mel'nichenko I. P., Plaksa S. A.

Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1695-1703

We obtain a new representation of potential and flow functions for spatial potential solenoidal fields with axial symmetry. We study principal algebraic-analytic properties of monogenic functions of a vector variable with values in an infinite-dimensional Banach algebra of even Fourier series and describe the relationship between these functions and the axially symmetric potential and Stokes flow function. The suggested method for the description of the above-mentioned fields is an analog of the method of analytic functions in the complex plane for the description of plane potential fields.

### Absolute decomposability of the group of rational numbers

Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1704-1707

We prove that the group of rational numbers**Q** is absolutely decomposable.

### Asymptotic behavior of solutions of pulse systems with small parameter and markov switchings. II. Weak convergence of solutions

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

Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1708-1720

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

### To the memory of Dmytro Ivanovych Martynyuk

Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1721

### Index of volume 48 of "Ukrainian Mathematical Journal"

Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1722-1727