# Volume 52, № 4, 2000

### Global small solutions of the cauchy problem for a semilinear system of equations of thermoelasticity

Ukr. Mat. Zh. - 2000. - 52, № 4. - pp. 435-446

For a semilinear system of equations of thermoelasticity, we establish a theorem on the existence and uniqueness of global solutions in a multidimensional space under the condition that the initial data are sufficiently small. We also obtain estimates for the decrease of solutions as time increases.

### On new realizations of the poincare groups *P* (1,2) and *P*(2, 2)

Ukr. Mat. Zh. - 2000. - 52, № 4. - pp. 447-462

We classify realizations of the Poincare groups *P* (1, 2) and *P* (2, 2) in the class of local Lie groups of transformations and obtain new realizations of the Lie algebras of infinitesimal operators of these groups.

### A brief survey of scientific results of E. A. Storozhenko

Kashin B. S., Korneichuk N. P., Shevchuk I. A., Ul'yanov P. L.

Ukr. Mat. Zh. - 2000. - 52, № 4. - pp. 463-473

We present a survey of the scientific results obtained by E. A. Storozhenko and related results of her disciples and give brief information about the seminar on the theory of functions held under her guidance.

### Methods for derivation of the stochastic Boltzmann hierarchy

Ukr. Mat. Zh. - 2000. - 52, № 4. - pp. 474-491

We consider different methods for the derivation of the stochastic Boltzmann hierarchy corresponding to the stochastic dynamics that is the Boltzmann-Grad limit of the Hamiltonian dynamics of hard spheres. Solutions of the stochastic Boltzmann hierarchy are the Boltzmann-Grad limit of solutions of the BBGKY hierarchy of hard spheres in the entire phase space. A new concept of reduced distribution functions corresponding to the stochastic dynamics are introduced. They take into account the contribution of the hyperplanes of lower dimension where stochastic point particles interact with one another. The solutions of the Boltzmann equation coincide with one-particle distribution functions of the stochastic Boltzmann hierarchy and are represented by integrals over the hyperplanes where the stochastic point particles interact with one another.

### Dirichlet problem for axisymmetric potential fields in a disk of the meridian plane. I

Ukr. Mat. Zh. - 2000. - 52, № 4. - pp. 492-511

We develop new methods for the solution of boundary-value problems in the meridian plane of an antisymmetric potential solenoidal field with regard for the nature and specific features of axisymmetric problems. We determine the solutions of the Dirichlet problems for an axisymmetric potential and the Stokes flow function in a disk in an explicit form.

### On approximation of functions from below by splines of the best approximation with free nodes

Ukr. Mat. Zh. - 2000. - 52, № 4. - pp. 512-523

Let M be the set of functions integrable to the power β=(*r*+1+1/*p*)^{-1}. We obtain asymptotically exact lower bounds for the approximation of individual functions from the set M by splines of the best approximation of degree *r*and defect *k* in the metric of *L* _{p}.

### Generalization of one problem of stochastic geometry and related measure-valued processes

Ukr. Mat. Zh. - 2000. - 52, № 4. - pp. 524-534

We prove a functional limit theorem for the measure of a domain in which the values of a time-dependent random field do not exceed a given level. We illustrate this theorem by a geometric model.

### Mariya Evdokimovna Temchenko

Ukr. Mat. Zh. - 2000. - 52, № 4. - pp. 535-536

### Logarithmic derivatives of diffusion measures in a Hilbert space

Ukr. Mat. Zh. - 2000. - 52, № 4. - pp. 537-543

For the logarithmic derivative of transition probability of a diffusion process in a Hilbert space, we construct a sequence of vector fields on Riemannian *n*-dimensional manifolds that converge to this derivative.

### Extremal problems in a class of mappings with bounded integral characteristics

Ukr. Mat. Zh. - 2000. - 52, № 4. - pp. 544-547

We consider mappings with bounded integral characteristics. We construct extremal mappings of plane rings realizing the minimum of these characteristics.

### Approximation of a bounded solution of one difference equation with unbounded operator coefficient by solutions of the corresponding boundary-value problems

Gorodnii M. F., Romanenko V. N.

Ukr. Mat. Zh. - 2000. - 52, № 4. - pp. 548-552

We investigate the problem of approximation of a bounded solution of a difference analog of the differential equation $$x^{(m)}(t) + A_1x^{(m-1)}(t) + ... + A_{m-1}x'(t)) = Ax(t) +f(0), t \in R$$
by solutions of the corresponding boundary-value problems. Here, *A* is an unbounded operator in a Banach space *B*, {*A* _{1},...,*A* _{ m-1}} ⊂*L*(*B*) and *f*:ℝ→*B* is a fixed function.

### On the generating function of the time of first hitting the boundary by a semicontinuous difference of independent renewal processes with discrete time

Ukr. Mat. Zh. - 2000. - 52, № 4. - pp. 553-561

For a semicontinuous difference of two independent renewal processes, we find the generating function of the time of first hitting the boundary.

### A note on the theorems of paley and stein

Ukr. Mat. Zh. - 2000. - 52, № 4. - pp. 562-566

We show that, under the conditions of the theorems of Paley and Stein, a system with bounded norms in $L_{∞}$ can be replaced by a system with bounded norms in the space of functions of bounded mean oscillation.

### Green function of a parabolic boundary-value problem and the optimization problem

Ukr. Mat. Zh. - 2000. - 52, № 4. - pp. 567-571

We establish necessary and sufficient conditions for the choice of optimal control over systems de-scribed by a general parabolic problem with restricted internal control.

### Solution of one boundary-value problem for a hyperbolic equation of the second order

Ukr. Mat. Zh. - 2000. - 52, № 4. - pp. 572-573

We find conditions for the existence of the classical solution of the boundary-value problem *u* _{ tt }-*u* _{ xx }= *f*(*x,t*), *u*(0,t)=*u*(π, *t*)=0, *u*(*x*, 0)=*u*(*x*, 2π).

### Second international Algebraic Conference in Ukraine dedicated to the memory of Professor L. A. Kaluzhnin

Kirichenko V. V., Sushchanskii V. I.

Ukr. Mat. Zh. - 2000. - 52, № 4. - pp. 574-576