# Volume 54, № 2, 2002

### Mykola Ivanovych Portenko (On His 60th Birthday)

Dorogovtsev A. A., Kopytko B.I., Korolyuk V. S., Mitropolskiy Yu. A., Samoilenko A. M., Skorokhod A. V., Sytaya G. N.

Ukr. Mat. Zh. - 2002. - 54, № 2. - pp. 147-148

### Properties of a Subclass of Avakumović Functions and Their Generalized Inverses

Buldygin V. V., Klesov O. I., Steinebach J. G.

Ukr. Mat. Zh. - 2002. - 54, № 2. - pp. 149-169

We study properties of a subclass of ORV functions introduced by Avakumović and provide their applications for the strong law of large numbers for renewal processes.

### Equations with Random Gaussian Operators with Unbounded Mean Value

Ukr. Mat. Zh. - 2002. - 54, № 2. - pp. 170-177

We consider an equation in a Hilbert space with a random operator represented as a sum of a deterministic, closed, densely defined operator and a Gaussian strong random operator. We represent a solution of an equation with random right-hand side in terms of stochastic derivatives of solutions of an equation with deterministic right-hand side. We consider applications of this representation to the anticipating Cauchy problem for a stochastic partial differential equation.

### Measure-Valued Markov Processes and Stochastic Flows

Ukr. Mat. Zh. - 2002. - 54, № 2. - pp. 178-189

We consider a new class of Markov processes in the space of measures with constant mass. We present the construction of such processes in terms of probabilities that control the motion of individual particles. We study additive functionals of such processes and give examples related to stochastic flows with interaction.

### On One Case of Existence of Homogeneous Solutions

Ukr. Mat. Zh. - 2002. - 54, № 2. - pp. 190-194

We present necessary and sufficient conditions for the existence of a homogeneous solution for a class of partial differential equations with a homogeneous random perturbation in a Banach space.

### Stability of a Dynamical System with Semi-Markov Switchings under Conditions of Stability of the Averaged System

Chabanyuk Ya. M., Korolyuk V. S.

Ukr. Mat. Zh. - 2002. - 54, № 2. - pp. 195-204

We establish additional stability conditions on the rate of a dynamical system with semi-Markov switchings and on the Lyapunov function for the averaged system.

### Elementary Representations of the Group $B_0^ℤ$ of Upper-Triangular Matrices Infinite in Both Directions. I

Ukr. Mat. Zh. - 2002. - 54, № 2. - pp. 205-216

We define so-called “elementary representations” $T_p^{R,µ},\; p ∈ ℤ$, of the group $B_0^ℤ$ of finite upper-triangular matrices infinite in both directions by using quasi-invariant measures on certain homogeneous spaces and give a criterion for the irreducibility and equivalence of the representations constructed. We also give a criterion for the irreducibility of the tensor product of finitely many and infinitely many elementary representations.

### Malliavin Calculus for Functionals with Generalized Derivatives and Some Applications to Stable Processes

Ukr. Mat. Zh. - 2002. - 54, № 2. - pp. 216-226

We introduce the notion of a generalized derivative of a functional on a probability space with respect to some formal differentiation. We establish a sufficient condition for the existence of the distribution density of a functional in terms of its generalized derivative. This result is used for the proof of the smoothness of the distribution of the local time of a stable process.

### Stroock–Varadhan Theorem for Flows Generated by Stochastic Differential Equations with Interaction

Ukr. Mat. Zh. - 2002. - 54, № 2. - pp. 227-236

We prove a theorem that characterizes the support of a flow generated by a system of stochastic differential equations with interaction.

### On Homotopic Equivalence of Fibering into Tori and Total Space. Case of Nonempty Boundary

Ukr. Mat. Zh. - 2002. - 54, № 2. - pp. 237-248

We present a criterion that indicates the case where a smooth compact 4-manifold with irreducible boundary is homotopically equivalent to the total space of a fibering into two-dimensional closed aspherical surfaces over a two-dimensional aspherical surface with boundary.

### On Stability of Integral Sets of Impulsive Differential Systems

Chernikova O. S., Perestyuk N. A.

Ukr. Mat. Zh. - 2002. - 54, № 2. - pp. 249-257

We introduce the notion of stability of integral sets of impulsive differential systems of general form (with nonfixed times of impulse influence). We establish conditions sufficient for the stability of an integral set.

### Problems of Transmission with Inhomogeneous Principal Conjugation Conditions and High-Accuracy Numerical Algorithms for Their Discretization

Deineka V. S., Sergienko I. V.

Ukr. Mat. Zh. - 2002. - 54, № 2. - pp. 258-275

We construct new problems of transmission and high-accuracy computational algorithms for their discretization.

### Generalized Lindelöf Finiteness Conditions for the λ-Type of a Subharmonic Function

Kondratyuk A. A., Kondratyuk Ya. V.

Ukr. Mat. Zh. - 2002. - 54, № 2. - pp. 276-179

We establish a finiteness criterion for the λ-type of a subharmonic function. In the case where λ(*r*) = *r* ^{ρ} *L*(*r*), ρ, where *L* is a slowly varying function, this criterion coincides with the Lindelöf criterion.

### On Locally Linearly Convex Domains

Ukr. Mat. Zh. - 2002. - 54, № 2. - pp. 280-284

We construct a counterexample to the hypothesis on global linear convexity of locally linearly convex domains with everywhere smooth boundary. We refine the theorem on the topological classification of linearly convex domains with smooth boundary.

### On the Impossibility of Stabilization of Solutions of a System of Linear Deterministic Difference Equations by Perturbations of Its Coefficients by Stochastic Processes of “White-Noise” Type

Ukr. Mat. Zh. - 2002. - 54, № 2. - pp. 285-288

We consider the problem of mean-square stabilization of solutions of a system of linear deterministic difference equations with discrete time by perturbations of its coefficients by a stochastic “white-noise” process. The answer is negative and is based on the analysis of the corresponding matrix algebraic Sylvester equation introduced earlier by the author in the theory of stability of stochastic systems. At the same time, we answer the same question for a vector matrix system of linear difference equations with continuous time and for a vector matrix system of differential equations.