Scientific Notes of

Taurida National V.I. Vernadsky University

Taurida National V.I. Vernadsky University

Series

Physical and Mathematical Sciences

The Sufficient and Necessary Conditions of Extremum of Functionals in Product of Nuclear Locally Convex Spaces in Case of Many Variables

**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 Ρ_{4,abo,τ}_{}

*In the paper irreducible nonequivalent *-representation *-algebras Ρ _{4,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. |

About of the Behaviour of Solutions of Discrete Equations

**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.*