# Volume 51, № 9, 1999

### Complete asymptotic decomposition of the sojourn probability of a diffusion process in thin domains with moving boundaries

Ukr. Mat. Zh. - 1999. - 51, № 9. - pp. 1155–1164

We investigate a diffusion process ξ(*t*) with absorption defined in a thin domain*D* _{ ε }=*{(x,t)∶εG* _{1} *(t) *

_{2}

*(t), t≥0}*. We obtain the complete decomposition of the sojourn probability of ξ(

*t*) in

*D*

_{ε}with respect to ε→0.

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### On the product of an abelian group and a nilpotent group

Ukr. Mat. Zh. - 1999. - 51, № 9. - pp. 1165–1171

We study the structure of the product of an Abelian group and a nilpotent group. Conditions for the existence of a normal subgroup in one of the factors are given. These conditions generalize the known results on the product of two Abelian groups. The statements obtained are used to describe the structure of a product of an infinite cyclic subgroup and a periodic nilpotent subgroup.

### Properties of the likelihood ratio for semimartingales with deterministic triplets in the parametric case

Lin'kov Yu. N., Shevlyakov A. Yu.

Ukr. Mat. Zh. - 1999. - 51, № 9. - pp. 1172–1180

We consider semimartingales with deterministic discontinuous triplets. We obtain properties of the like-lihood ratio for the parametric case in terms of the Hellinger processes.

### On the Sobolev problem in the complete scale of Banach spaces

Ukr. Mat. Zh. - 1999. - 51, № 9. - pp. 1181–1192

In a bounded domain*G* with boundary ∂*G* that consists of components of different dimensions, we consider an elliptic boundary-value problem in complete scales of Banach spaces. The orders of boundary expressions are arbitrary; they are pseudodifferential along ∂*G*. We prove the theorem on a complete set of isomorphisms and generalize its application. The results obtained are true for elliptic Sobolev problems with a parameter and parabolic Sobolev problems as well as for systems with the Douglis-Nirenberg structure.

### Relationship between the asymptotic behavior of exponents of a multidimensional exponential series and the asymptotic behavior of its coefficients in a neighborhood of singular points

Ukr. Mat. Zh. - 1999. - 51, № 9. - pp. 1193–1200

We study the relation of the asymptotic behavior of the coefficients of multidimensional exponential series to the asymptotic behavior of its sum by using the*R*-order of the growth*p* _{ QR } *(a* _{1},...,*a* _{ n }) in an octant*Q(a* _{1},...,*a* _{ n }).

### Convergence of distributions of integral functionals of extremal random functions

Ukr. Mat. Zh. - 1999. - 51, № 9. - pp. 1201–1209

We study the convergence of distributions of integral functionals of random processes of the form*U* _{ n } *(t)*=*b* _{ n }(*Z* _{ n }(*t*)-*a* _{ n } G(*t*)),*t*⃛*T*, where {*X=X(t), t*⃛*T*} is a random process,*X* _{ n },*n*≥1, are independent copies of*X*, and*Z* _{ n } *(t)*=max_{1≤k≤n } *X* _{ k } *(t)*.

### A diffusion process on a plane with membranes on two intersecting straight lines

Ukr. Mat. Zh. - 1999. - 51, № 9. - pp. 1210–1216

We construct a generalized diffusion process in ℝ^{2} with the unit diffusion matrix and a drift vector with δ-functions concentrated on two intersecting straight lines.

### Theory of quadratic estimates of variance

Ukr. Mat. Zh. - 1999. - 51, № 9. - pp. 1217–1231

A class of quadratic estimates is constructed for the second-order moment and variance of a random variable. These estimates are found on the basis of sample values obtained by simple sampling. The best quadratic estimates are found for the second-order moment and variance in the case of known mathematical expectation. The exactness of biased and unbiased estimates of variance is investigated in the case of unknown mathematical expectation.

### Nonlocal Neumann problem for a degenerate parabolic equation

Ukr. Mat. Zh. - 1999. - 51, № 9. - pp. 1232–1243

In the spaces of classical functions with power weight, we prove the existence and uniqueness of a solution of the nonlocal Neumann problem for nonuniformly parabolic equations without restrictions on the power order of coefficient degeneration. We find an estimate of the solution of this problem in the spaces considered.

### The theory of the numerical-analytic method: Achievements and new trends of development. VII

Ronto M. I., Samoilenko A. M., Trofimchuk S. I.

Ukr. Mat. Zh. - 1999. - 51, № 9. - pp. 1244–1261

For the numerical-analytic method suggested by A. M. Samoilenko in 1965, we analyze the application to abstract differential equations, implicit equations, and control problems.

### A stability criterion for periodic solutions of two-dimensional discontinuous dynamical systems

Ukr. Mat. Zh. - 1999. - 51, № 9. - pp. 1262–1266

We establish stability conditions for periodic solutions of two-dimensional systems of ordinary differential equations with pulse influence. We study the properties of the jump operator for such systems.

### Coincidence criteria for the kernel of a function and the kernel of its integral almost positive means

Ukr. Mat. Zh. - 1999. - 51, № 9. - pp. 1267–1275

We establish necessary and sufficient conditions for a point A of the Knopp kernel*K(f)* of a function*f* to belong to the kernel*K(M)* of a function*M(t)*:=∫_{ S } *fd*μ_{ t }, where the so-called almost positive measures μ_{ t } determine a regular method of summation. In particular, this gives coincidence criteria for the kernels*K(f)* and*K(M)*.

### Stability of one difference equation with positive coefficients

Khusainov D. Ya., Nikiforova N. S.

Ukr. Mat. Zh. - 1999. - 51, № 9. - pp. 1276–1280

We consider a linear homogeneous difference equation of order*n* with positive coefficients, which is presented in the normal form. We obtain necessary and sufficient conditions for the stability and asymptotic stability.

### On one method of factorization of polynomials

Ukr. Mat. Zh. - 1999. - 51, № 9. - pp. 1281–1286

We propose and justify a numerical method of factorization of polynomials with complex coefficients. We construct and algorithm of factorization of polynomials with real coefficients into real factors in the case of multiple roots.

### On the stability of invariant sets of a system of autonomous differential equations under constantly acting perturbations

Ignat'ev A. O., Konosevich В. I.

Ukr. Mat. Zh. - 1999. - 51, № 9. - pp. 1287–1291

We consider a system of ordinary autonomous differential equations that has an invariant set. We obtain sufficient conditions for the stability of this system under constantly acting perturbations.

### On invariant tori of countable systems of differential equations with delay

Elnazarov A. A., Samoilenko A. M.

Ukr. Mat. Zh. - 1999. - 51, № 9. - pp. 1292–1295

We obtain conditions for the existence and smoothness of invariant tori of countable systems of differential equations with delay.

### International conference on the theory of approximation of functions and its applications dedicated to the memory of V. K. Dzyadyk

Romanyuk A. S., Serdyuk A. S., Stepanets O. I.

Ukr. Mat. Zh. - 1999. - 51, № 9. - pp. 1296–1297