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Vol. 23 (62), No. 2. 2010
Physics and Mathematics Sciences.

Anashkin O.V., Shulgin A.V.
Influence of Delay on Stability of the Quadratic Difference Equation
The problem of stability of a quadratic difference equation with constant delay is considered. Coefficients of the equation are quasi-periodic functions of a discrete argument. The study is carried out by constructing the so-called perturbed Lyapunov function. We obtain sufficient conditions for stability of the equation in the form of a numeric index, a formula which shows the dependence on coefficients and the magnitude of the delay.

Akhramovich M.V., Orlov I.V.
Variant of Universal Banach-Steinhaus Type Theorem for Arbitrary Locally Convex Spaces
A Banach-Steinhaus type theorem and the corresponding equicontinuity principle for the arbitrary complete separable locally convex spaces are received. Some applications of the results above are considered. 

Баtyr E.I.
Small Movements and Normal Oscillations of a System of Three Connected Bodies with the Cavities, Filled by an Ideal Incompressible Fluid.batyr
In this paper we considered an initial-boundary value and spectral problems about thesmall movements of a system of three bodies. The system is a circuit of the consistently connected hard bodies. Each of the bodies of such circuit is a gyrostat. The existencetheorem of solutions of the Cauchy problem is formulated. The properties of normaloscillations are described. The known N.E. Zhukovsky’s theorem is transferred on a caseof a movement of three bodies.

Bozhonok E.V.
Conditions of Existence of Nonlocal Compact Extrema of Variational Functionals in Sobolev Space H1
The minimal pseudoquadratic representation of W2K2(z) class integrands ofvariation functionals in H1 is derived. A general form of variational functional satisfying stationary form of Legendre–Jacobi condition is described. Theconditions of existence of nonlocal K–extremum at zero of variation functionalin space H1 are obtained. Some examples are considered.

Vronsky B.M., Kopachevsky N.D.
Asymptotic of Some Operator-Function is Investigated

Donskoy V.I.
VC-Dimension Estimations of the Basic Algorithms of Empirical Generalization for Pattern Recognition Problems Obtained by the pVCD Method
In this paper, VC-Dimension estimations for Decision Trees, Neural Networks, DNF, Function Compositions obtained by pVCD method are presented. The pVCD method is based on the Kolmogorov’ approach to complexity definition.

Dudik O.A.
On Normal Oscillations of a Pendulum with a Cavity Partially Filled with a Capillary Viscous Fluid
The problem on normal oscillations of a pendulum with a cavity partially filled with a capillary viscous fluid is investigated in the work. Spectral properties are studied, the theorem on basisity of the system of root elements is proved. The Lagrange theorem on nonstability is proved.

Karpenko I.I.
Non-Hermitian Self-Adjoint Quaternionic Matrixes
In this paper there considered matrices which are self-adjoint concerning the non-Hermitian involution in the real algebra of quaternions (a– self-adjoint matrices). Analog of Takagi decomposition for this matrix class is received. As applications the theorem about similarity of any quaternionic matrix and some a–self-adjoint matrix is proved and properties of the finite-dimensional quaternionic module with the inner product are investigated.

Muratov M.A., Pashkova Yu.S., Rubshteyin B.A.
Order Ergodic Theorems in Orlicz Spaces
In this work we study the order convergence of Cesáro averages in Orlicz spaces. We investigate the case, when the considered measure is infinite. The problems of order convergence is connected with both Dominated and Individual Ergodic Theorems. In particular, the classical Dominated and Individual Ergodic Theorems for spaces Lp and Zygmund classes L logr L are obtained as the partial cases. The method’s of the rearrangements invariant spaces and of Ergodic Theory are used.

Pogrebitskaya A.M., Smirnova S.I.
On Estimate of Exactness of Analytical Hybrid Solution for the Heat Transfer Problem
In the paper, the analytical estimate of the part of an double hybrid WKB- Galerkin solution for the nonlinear second order di?erential equation, that describes the mathematical model of the heat transfer process, is presented. 

Rudnitsky O.I., Romanenko I.A.
Algebras of Polynomials Pogorelov of the Octahedral Groups
The algebras POm of the Pogorelov polynomials of the octahedral groups Om generated by reflections on the unitary plane are investigated. The degrees of a generators of the algebra PO46 of the Pogorelov polynomials of the group O46 are found. 

Stonyakin F.S.
Strong Compact Properties and Limit Form of Radon-Nikodym Property for Vector Measures
In this paper new properties for vector measures with values in locally convex spaces, namely the strong compact variation, the strong compact absolute continuity, the universal compact and limit forms of the Radon-Nikodym property are investigated. It is proved that each Frechet space possesses the universal compact and limit forms of the Radon-Nikodym property for vector measures. Some applications are considered. 

Tyshkevich D.L.
On Commensurability of Bimodules of the Special Types
In the paper the conditions for commensurability (i.e. coincidence of the left and right dimensions) of free bimodules of the special types over IBN–rings. The results may be useful for studying of bimodules over algebras of hypercomplex numbers.