# Archive

**Vol. 22 (61), No. 1. 2009**

**Mathematics. Mechanics. Informatics and Cybernetics.**

**O. A. Andronova **

Initial-Boundary and Spectral Problems with Surface and Internal Dissipation of an Energy

*There was considered generalization of the initial-boundary value problem with surface and internal dissipation of an energy. There was formulated the abstract problem on the basis of the abstract Green formula for the triple of Hilbert spaces and the trace operator. We proved the theorem on strong solvability of the considered problem. There were discussed spectral problems with internal dissipation of an energy at its different intensity. The elementary properties of the solutions of the problem were presented.*

**Yu. Artamonov**

Variation Problems with the Condition of Isoperimetric Type on Movable Boundary

*In this paper the applicability of Lagrange's multipliers method for isoperimetric type variational problems with movable boundary is shown. The obtained results are applied for finding the peak of energy form of integral operator with movable boundary. *

**E. V. Bozhonok **

Simple Sufficient Conditions of KSmoothness of the Basic Variational Functional in Sobolev Space W1 2

*The simple sufficient conditions of K–smoothness of the basic variational functional in Sobolev space W1 2 are obtained. The connection between these conditions and the classical growth conditions for integrand is investigated. The examples are considered.*

**O. A. Dudik**

Operational Approach to the Problem on Small Motions and Normal Oscillations of a Pendulum with a Cavity Fully Filled with a System of Capillary Viscous Fluids

*In the work, we consider the problem on small motions and normal oscillations of a pendulum with a cavity fully filled with a system of capillary viscous fluids. The theorem on strong solvability of the investigated hydrosystem is proved. Asymptotic behavior of eigenvalues is established. The theorem on basisity by Abel–Lidsky of the system of root functions and the inversion of Lagrange theorem on stability are proved. *

**D. A. Zakora **

Small Motions and Normal Oscillations of an Ideal Relaxing Fluid

*The problem on normal oscillations of an ideal relaxing fluid filling a rotating container is investigated. The operator pencil corresponding to spectral problem is obtained. For this operator pencil localization of spectrum, discreteness of spectrum and essential spectrum are investigated. Asymptotic formulas for all branches of spectrum are obtained. The double completeness with finite defect for the system of an eigne elements and associate elements is proved. *

**A. I. Krivoruchko**

On the Infinite Reflection Groups with Two Linear Spans of Orbits of Symmetry Directions

*The invariants of infinite reflection groups having two linear spans of orbits of symmetry directions are obtained. The linear classification of infinite reflection groups acting on non-cylindrical algebraic surfaces and having two linear spans of orbits of symmetry directions is given.*

**M. A. Muratov, Yu. S. Pashkova, B. A. Rubshtein**

Dominated Ergodic Theorems Hold in Lorenz Spaces

*In the present work we study conditions under which Dominated Ergodic Theorems hold in Lorenz spaces for a positive contraction on positive semiaxis. The method's of the rearrangements invariant spaces was used. *

**A. M. Pogrebitskaya, S.I. Smirnova**

On the Problem of Accuracy of Analytical Hybrid Solution for the Nonlinear Thermal Emission Problem

*In this paper the estimate of the part of an analytical hybrid solution for the nonlinear second order differential equation, that arises while describing the mathematical model of the thermal emission problem, is obtained. *

**F. S. Stonyakin**

K-Property of Radon-Nikodym for Frechet Spaces

*In this paper the new properties of compact absolutely continuity and K-property of Radon-Nikodym for mappings into locally convex spaces are considered. It is proved that each Frechet space possesses K-property of Radon-Nikodym. The differentiability almost everywhere of each strong compact absolutely continuous mapping in the topology of some subspace, generated by absolutely convex compact set is asserted. The generalized Lagrange formula with compact estimation for differentiable mappings into Frechet spaces is obtained.*