Volume 48, № 12, 1996
Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1587-1588
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 $AG(1, 1)$.
Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1602-1603
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.
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.
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.
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.
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.
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.
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.
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.
Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1704-1707
We prove that the group of rational numbersQ is absolutely decomposable.
Asymptotic behavior of solutions of pulse systems with small parameter and markov switchings. II. Weak convergence of solutions
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.
Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1721
Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1722-1727