# Volume 54, № 4, 2002

### Mykhailo Iosypovych Yadrenko (On His 70th Birthday)

Buldygin V. V., Korolyuk V. S., Kozachenko Yu. V., Mitropolskiy Yu. A., Perestyuk N. A., Portenko N. I., Samoilenko A. M., Skorokhod A. V.

Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 435-438

### On Entire Functions Belonging to a Generalized Class of Convergence

Gal' Yu. M., Mulyava O. M., Sheremeta M. M.

Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 439-446

In terms of Taylor coefficients and distribution of zeros, we describe the class of entire functions *f* defined by the convergence of the integral \(\int\limits_{r_0 }^\infty {\frac{{\gamma (\ln M_{f} (r))}}{{r^{\rho + 1} }}} dr\) , where γ is a slowly increasing function.

### System $G|G^κ|1$ with Batch Service of Calls

Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 447-465

For the queuing system *G*|*G* ^{κ}|1 with batch service of calls, we determine the distributions of the following characteristics: the length of a busy period, the queue length in transient and stationary modes of the queuing system, the total idle time of the queuing system, the output stream of served calls, etc.

### On One Property of a Regular Markov Chain

Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 466-471

We prove that if a certain row of the transition probability matrix of a regular Markov chain is subtracted from the other rows of this matrix and then this row and the corresponding column are deleted, then the spectral radius of the matrix thus obtained is less than 1. We use this property of a regular Markov chain for the construction of an iterative process for the solution of the Howard system of equations, which appears in the course of investigation of controlled Markov chains with single ergodic class and, possibly, transient states.

### Iterative Method for the Solution of Linear Equations with Restrictions

Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 472-482

We propose a new approach to the investigation of linear equations with restrictions. For the problem considered, we establish consistency conditions and justify the application of an iterative method.

### Varieties of Groups with Invariant Centralizers of Subgroups

Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 483-491

We present a structural description of free groups and some critical subgroups of a given variety.

### Topological Equivalence of Morse–Smale Vector Fields with beh2 on Three-Dimensional Manifolds

Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 492-500

For the Morse–Smale vector fields with beh2 on three-dimensional manifolds, we construct complete topological invariants: diagram, minimal diagram, and recognizing graph. We prove a criterion for the topological equivalence of these vector fields.

### On Invariant Tori of Itô Stochastic Systems

Samoilenko A. M., Stanzhitskii A. N.

Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 501-513

By using the Green–Samoilenko function, we establish conditions for the existence of invariant sets of Itô stochastic systems that are extensions of dynamical systems on a torus.

### On Impulsive Lotka–Volterra Systems with Diffusion

Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 514-526

We study a two-dimensional Lotka–Volterra system with diffusion and impulse action at fixed moments of time. We establish conditions for the permanence of the system. In the case where the coefficients of the system are periodic in *t* and independent of the space variable *x*, we obtain conditions for the existence and uniqueness of periodic solutions of the system.

### On the Spatial and Temporal Behavior in Dynamics of Porous Elastic Mixtures

Ciarlettci M., Iovane G., Passarella F.

Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 527-544

In this paper, we study the spatial and temporal behavior of dynamic processes in porous elastic mixtures. For the spatial behavior, we use the time-weighted surface power function method in order to obtain a more precise determination of the domain of influence and establish spatial-decay estimates of the Saint-Venant type with respect to time-independent decay rate for the inside of the domain of influence. For the asymptotic temporal behavior, we use the Cesáro means associated with the kinetic and strain energies and establish the asymptotic equipartition of the total energy. A uniqueness theorem is proved for finite and infinite bodies, and we note that it is free of any kind of *a priori* assumptions on the solutions at infinity.

### Application of the Numerical-Analytic Method to Systems of Differential Equations with Parameter

Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 545-554

The numerical-analytic method is applied to systems of differential equations with parameter under the assumption that the corresponding functions satisfy the Lipschitz conditions in matrix notation. We also obtain several existence results for problems with deviations of an argument.

### Analog of the Krein Formula for Resolvents of Normal Extensions of a Prenormal Operator

Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 555-562

We prove a formula that relates resolvents of normal operators that are extensions of a certain prenormal operator. This formula is an analog of the Krein formula for resolvents of self-adjoint extensions of a symmetric operator. We describe properties of the defect subspaces of a prenormal operator.

### Homoclinic Points for a Singularly Perturbed System of Differential Equations with Delay

Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 563-567

We obtain a representation of the integral manifold of a system of singularly perturbed differential-difference equations with periodic right-hand side. We show that, under certain conditions imposed on the right-hand side, the Poincaré map for the perturbed system has a transversal homoclinic point.

### On Regularity of Certain Linear Expansions of Dynamical Systems on a Torus

Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 568-574

We investigate the problem of the existence of the Green–Samoilenko function for linear expansions of dynamical systems on a torus of the form $$\frac{{d\phi }}{{dt}} = a(\phi ),{\text{ }}C(\phi )\frac{{d\phi }}{{dt}} + \frac{1}{2}\dot C(\phi )x = A(\phi )x,$$ where *C*(ϕ) ∈ *C*′(*T* _{m}; *a*) is a nondegenerate symmetric matrix.

### International scientific conference "New approaches to the solution of differential equations"

Ptashnik B. I., Samoilenko A. M.

Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 575-576