# Volume 56, № 12, 2004

### Orthogonal Approach to the Construction of the Theory of Generalized Functions of Infinitely Many Variables and the Poisson Analysis of White Noise

Berezansky Yu. M., Tesko V. A.

Ukr. Mat. Zh. - 2004. - 56, № 12. - pp. 1587-1615

We develop an orthogonal approach to the construction of the theory of generalized functions of infinitely many variables (without using Jacobi fields) and apply it to the construction and investigation of the Poisson analysis of white noise.

### Variational Ultraparabolic Inequalities

Lavrenyuk S. P., Protsakh N. P.

Ukr. Mat. Zh. - 2004. - 56, № 12. - pp. 1616-1628

In a bounded domain of the space ℝ^{ n +2}, we consider variational ultraparabolic inequalities with initial condition. We establish conditions for the existence and uniqueness of a solution of this problem. As a special case, we establish the solvability of mixed problems for some classes of nonlinear ultraparabolic equations with nonclassical and classical boundary conditions.

### Stochastic Dynamics and Hierarchy for the Boltzmann Equation with Arbitrary Differential Scattering Cross Section

Ukr. Mat. Zh. - 2004. - 56, № 12. - pp. 1629-1653

The stochastic dynamics for point particles that corresponds to the Boltzmann equation with arbitrary differential scattering cross section is constructed. We derive the stochastic Boltzmann hierarchy the solutions of which outside the hyperplanes of lower dimension where the point particles interact are equal to the product of one-particle correlation functions, provided that the initial correlation functions are products of one-particle correlation functions. A one-particle correlation function satisfies the Boltzmann equation. The Kac dynamics in the momentum space is obtained.

### Approximation of Densities of Absolutely Continuous Components of Measures in a Hilbert Space Using the Ornstein-Uhlenbeck Semigroup

Ukr. Mat. Zh. - 2004. - 56, № 12. - pp. 1654-1664

We study the behavior of measures obtained as a result of the action of the Ornstein-Uhlenbeck semigroup *T* _{ t } associated with the Gaussian measure μ on an arbitrary probability measure ν in a separable Hilbert space as *t* → 0+. We prove that the densities of the parts of *T* _{ t }ν absolutely continuous with respect to μ converge in the measure μ to the density of the part of ν absolutely continuous with respect to μ. For a finite-dimensional space, we prove the convergence of these densities μ-almost everywhere. In the infinite-dimensional case, we give sufficient conditions for almost-everywhere convergence. We also consider conditions on the absolute continuity of *T* _{ t }ν with respect to μ in terms of the coefficients of the expansion of *T* _{ t }ν in a series in Hermite polynomials (an analog of the Ito- Wiener expansion) and the connection with finite absolute continuity.

### Rate of Convergence of Positive Series

Ukr. Mat. Zh. - 2004. - 56, № 12. - pp. 1665-1674

We investigate the rate of convergence of series of the form $$F(x) = \mathop \sum \limits_{n = 0}^{ + \infty } \;a_n e^{x\lambda _n + \tau (x)\beta _n } ,\quad a_n \geqslant 0,\quad n \geqslant 1,\quad a_0 = 1$$ where λ = (λ_{n}), 0 = λ_{0} < λ_{n} ↑ + ∞, *n* → + ∞, β = {β_{n}: *n* ≥ 0} ⊂ ℝ_{+}, and τ(*x*) is a nonnegative function nondecreasing on [0; +∞), and $$F(x) = \mathop \sum \limits_{n = 0}^{ + \infty } \;a_n f(x\lambda _n ),\quad a_n \geqslant 0,\quad n \geqslant 1,\quad a_0 = 1,$$ where the sequence λ = (λ_{n}) is the same as above and *f* (*x*) is a function decreasing on [0; +∞) and such that *f* (0) = 1 and the function ln *f*(*x*) is convex on [0; +∞).

### Academician V. Ya. Bunyakovs'kyi (on 200-th anniversary of his birthday)

Mel'nik V. S., Mel'nyk O. M., Samoilenko A. M.

Ukr. Mat. Zh. - 2004. - 56, № 12. - pp. 1675-1683

### On Some Consequences of the Equation for the Markov Renewal Function of a Semi-Markov Process

Ukr. Mat. Zh. - 2004. - 56, № 12. - pp. 1684-1690

We obtain chains of equations that relate the sojourn times of a semi-Markov process in a set of states to its Markov renewal function. We use the mathematical apparatus of the theory of Markov and semi-Markov processes.

### Estimate of the Remainder of the Best Quadratic Approximation of Differentiable Functions by Polynomials

Ukr. Mat. Zh. - 2004. - 56, № 12. - pp. 1691-1698

We establish lower and upper bounds for the quantity $$C_m^q (W^r ,x) = \mathop {\sup }\limits_{f \in W^r } \left| {f(x) - T_m (x,f)} \right|,$$ , where $$T_m (x,f) = \frac{2}{q}\mathop \sum \limits_{l = 0}^{q - 1} \;f(x_l )D_m (x - x_l ),\quad q \in \mathbb{N},\quad q > 2m,\quad x_l = \frac{{2\pi l}}{q},\quad l = 0,\;1,\;...\;,\;q - 1,$$ , and *D* _{ m }(*t*) is the Dirichlet kernel, for the class *W* ^{ r } of 2π-periodic functions, whose *r*th derivative satisfies the condition |*f* ^{ r }(*x*)| ≤ 1.

### On the Stabilization of a Solution of the Cauchy Problem for One Class of Integro-Differential Equations

Kulinich G. L., Kushnirenko S. V.

Ukr. Mat. Zh. - 2004. - 56, № 12. - pp. 1699 – 1706

We consider a solution of the Cauchy problem *u*(*t, x*), *t* > 0, *x* ∈ *R* ^{2}, for one class of integro-differential equations. These equations have the following specific feature: the matrix of the coefficients of higher derivatives is degenerate for all *x*. We establish conditions for the existence of the limit lim_{ t→∞} *u*(*t, x*) = *v*(*x*) and represent the solution of the Cauchy problem in explicit form in terms of the coefficients of the equation.

### On Homeomorphisms Realized by Certain Partial Differential Operators

Ukr. Mat. Zh. - 2004. - 56, № 12. - pp. 1707-1716

For a sufficiently broad class of partial differential operators, we prove a theorem on homeomorphisms. Applications of this theorem to some classical operators are considered.

### Elementary Reduction of Matrices over Right 2-Euclidean Rings

Ukr. Mat. Zh. - 2004. - 56, № 12. - pp. 1717 – 1721

We introduce a concept of noncommutative (right) 2-Euclidean ring. We prove that a 2-Euclidean ring is a right Hermite ring, a right Bezout ring, and a *GE* _{ n }-ring. It is shown that an arbitrary right unimodular string of length not less than 3 over a right Bezout ring of stable rank possesses an elementary diagonal reduction. We prove that a right Bezout ring of stable rank 1 is a right 2-Euclidean ring.

### The international conference „International workshop on analysis and its applications"

Samoilenko A. M., Shevchuk I. A., Stepanets O. I.

Ukr. Mat. Zh. - 2004. - 56, № 12. - pp. 1722

### Index of volume 56 of „Ukrainian Mathematical Journal"

Ukr. Mat. Zh. - 2004. - 56, № 12. - pp. 1723-1728