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Vol. 18 (57), No. 1. 2005
Mathematics. Mechanics. Informatics and Cybernetics.

Bozhonok Е.V.
The Sufficient and Necessary Conditions of Extremum of Functionals in Product of Nuclear Locally Convex Spaces in Case of Many Variables
In the work both sufficient, and necessary conditions of local extremum and compact extremum for functionals in n variables in product of nuclear spaces are proposed. The applications to integral functionals in n variables are received.

Ivanov Yu.B.
Forced Oscillations of Homogeneous Rotating Liquid in a Bounded Vessel
The problem of forced oscillations of homogeneous rotating liquid in a bounded vessel with variable depth and inclined walls is formulated in a form of nonhomogeneous equation related to a polynomial operator pencil. The equation has a frequency of forced oscillations as a parameter. The numerical solution of the problem may be obtained with a method of iteration. The convergence of iterations is proved.

Komissarenko Е.V., Krivouchko А.I.
On the Invariants of Infinite Reflection Groups with Four Linear Spans of Orbits of Directions of Symmetries
All rational invariants of infinite reflection group acting on a non-cylindrical algebraic hypersurface in n-dimensional real vector space and having four linear spans of orbits of directions of symmetries, anyone three of which form the direct sum, are found.

Kudryashov Yu.L.
Isomorphism of Distinct Representations of J-Self-Adjoint Dilatations of Linear Operator
In the paper isomorphism of spectral and broadcasted representations of J-self-adjoint dilatation are constructed.

Lebedeva Е.V.
A Selection Rule of Discrete Information for Solving Ill-Posed Problems
A projection regularized method is constructed for solving operator equations of the first kind with coefficients given inexactly. It is proved that the method is optimal on some set of smooth solutions and economical in the sense of volume of used discrete information.

Moskaleva Yu.P.
About * - Representations of Algebra Ρ4,abo,τ
In the paper irreducible nonequivalent *-representation *-algebras Ρ4,abo,τ are described.

Muratov М.А.
*- Algebra т - Measurable Operators Associated by Semi-Finite von Neumann Algebra
In the paper the *-algebra т - measurable operators associated by semi-finite von Neumann algebra M are considered.

Orlov I.V.
Q - Continuous and S - Continuous Mappings of Topological Spaces

Persidskiy S.K.
Zhuravlev V.V.
About of the Behaviour of Solutions of Discrete Equations
It is shown that combined difference equations second numbers smoothness conditions do not guarantee uniqueness of its solutions, this is the inherent difference between differential and difference equations systems, which has not always been taken into account in scientific literature. Sufficient unique solutions living conditions are substantiated for autonomous difference system, that allowed to consider behaviour of trajectory of linear system of difference equations with constant coefficients on plane, depending on such systems general properties.

Starkov  P.A.
Examples of Multicomponent Transmition Problems
Possible generalizations related to the increase of the number of interface regions (two or more) and complication of their mutual borders were described.

Temnenko V.A.
Mathematical Theory of the Perfect Test
The model of the "Perfect Test"(PT) was introduced. PT is the triple {I, f, c}, where I - infinite item-bank of test items with beforehand known indexes of difficulty p (0< p < 1), and f(p) and c(p) - arbitrarily chosen function of the difficulties of the test and price function of the test, satisfying conditions of normalization and to some other conditions described in the paper. An idea about neutral and ceiling tests is entered. Standardized Euler functions of difficulty of the test are described.

Tyshkevich D.L.
On Orthogonalization of Countable Systems of Vectors in Indefinite Inner Product Spaces
Sufficient condition for countable system of vectors in indefinite inner product spaces to be orthogonalized is obtained.

Chemishov K.I.
Reducibility of Singular Linear Relations
Certain questions of the spectral theory of closed linear relations constructing on the ordered pair of linear operators and acting from one Banach space to another isomorphic Banach space are considered, when points 0, oo are in the point spectrum of the relation. A construction of sequences of subspaces, when their stabilization takes place, is carried out.

Denk R., Volevich L.R.
A New Class of Parabolic Problems Connected with Newton's Polygon
A new class of boundary value problems for parabolic operators is introduced. We discuss some linearized free boundary problems not satisfying the classical parabolicity condition. It is shown that they belong to this class and by means of the Newton polygon method the nontrivial two-sided estimates of these problems are found.

Tikhonov A.S.
Selfadjointness of Cauchy Singular Integral Operator
We extend Krupnik's criterion of selfadjointness of the Cauchy singular integral operator to the case of finite-connected domains. The main aim of the paper is to present a new proof for the criterion.