**Full Volume**/ Весь том (UA) [ (PDF) ]**Cover**/ Титул (UA) [ (PDF) ]**Editorial Board**/ Редакційна колегія (UA).*P. 2.*[ (PDF) ]**Contents / Зміст (UA)**.*P.3.*[ (PDF) ]**O.O. Novikov, O.G. Rovenska, Yu.A. Kozachenko,**Approximation of classes of Poisson integrals by Fejer sums.*P. 4-12.*[ Abstract ] [ Full-text available (PDF) ]**Yu.V. Romashov, E.V. Povolotskii,**Influence of the temperature state on the damageability due to the creep of claddings of cylindrical fuel elements.*P. 13-28.*[ Abstract ] [ Full-text available (PDF) ]**V.V. Karieva, S.V. Lvov,**Mathematical model of liver regeneration processes: homogeneous approximation.*P. 29-41.*[ Abstract ] [ Full-text available (PDF) ]**M.V. Goncharenko, L.O. Khilkova,**Homogenized conductivity tensor and absorption function of a locally periodic porous medium,*P. 42-60.*[ Abstract ] [ Full-text available (PDF) ]**A.A. Makarov, D.A. Levkin,**The boundary-value problem in the layer for evolution pseudodifferential equations with integral condition,*P. 61-68.*[ Abstract ] [ Full-text available (PDF) ]

For upper bounds of the deviations of Fejer sums taken over classes of periodic functions that admit analytic extensions to a fixed strip of the complex plane, we obtain asymptotic equalities. In certain cases, these equalities give a solution of the corresponding Kolmogorov-Nikolsky problem.

*Keywords:* asymptotic equality; Poisson integrals; Fejer sums.

2010 Mathematics Subject Classification: 42А10. [ Full-text available (PDF) ] Top of the page.

This paper deals with the deformation and damageability of the fuel cladding of nuclear reactors, taking into account the creep and the temperature fields across the thickness. Mathematical models and quantitative estimates for durability of the fuel cladding, obtaining using computer simulations, are presented.

*Keywords:* damageability; creep; fuel cladding; durability; computer simulation.

2010 Mathematics Subject Classification: 74S99; 74R99. [ Full-text available (PDF) ] Top of the page.

This paper deals with the rules and the mechanisms regulation of liver regeneration. The generalized mathematical model was developed. This model has an explicit dependence on the control parameters. To solve this problem there were accepted such assumptions: homogeneous approximation; small toxic factors.

*Keywords:* mathematical model; liver regeneration; homogeneous approximation.

2010 Mathematics Subject Classification: 92C37; 65C20. [ Full-text available (PDF) ] Top of the page.

*Short abstract:* We consider a problem describing the process of stationary diffusion in a locally-periodic
porous medium with nonlinear absorption at the boundary. We base on a work, in which this problem
considered in a wider class of perforated domains. We obtain explicit formulas for the effective
characteristics of the medium: a conductivity tensor and a function of absorption.

*Extended abstract:* We study a process of stationary diffusion in locally-periodic porous
media with nonlinear absorption at the pore boundary. This process is described by a boundary-value
problem for an elliptic equation considered in a complex perforated domain, with a nonlinear third
boundary condition on the perforation boundary. In view of the smallness of the local scale of
porosity of the media and the complexity of the perforated domain, the direct solution of such
boundary-value problems is almost impossible. Therefore, a natural approach in this situation
is to study the asymptotic behavior of the solution when the microstructure scale tends to 0,
and the transition to the homogenized macroscopic model of the process. Our earlier papers were
devoted to homogenization the diffusion equation in a wide class of non-periodically perforated
domains: strongly-connected domains, which includes locally-periodically perforated domains.
In these works, an homogenized model was obtained, the coefficients of which are expressed
in terms of “mesoscopic” (local energy) characteristics of the media, which are determined
in small cubes, the size of which, however, are much larger than the microstructure scale.
In these papers, convergence theorems were proved under the conditions of the existence of
limiting densities of "mesoscopic" characteristics, the fulfillment of which is generally
difficult to show, but in a number of specific situations this can be done. In this paper,
we show the fulfillment of these conditions and, by studing them, we obtain explicit
formulas for the effective characteristics of the locally-periodic porous medium:
a conductivity tensor and a function of absorption.

*Keywords:* homogenization; stationary diffusion; non-linear
third boundary value problem; locally periodic porous medium.

2010 Mathematics Subject Classification: 35G65; 35Q80. [ Full-text available (PDF) ] Top of the page.

The boundary-value problem for evolutionary pseudodifferential equations with integral condition is examined. The conditions of correctness of this problem in L. Schwartz spaces were obtained, and the existence of correct boundary-value problem for any mentioned class equation was proved.

*Keywords:* pseudo-differential equations; boundary-value problem; Fourier transform; Schwartz space.

2010 Mathematics Subject Classification: 35S10. [ Full-text available (PDF) ] Top of the page.

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