Scientific Notes of

Taurida National V.I. Vernadsky University

Taurida National V.I. Vernadsky University

Series

Physical and Mathematical Sciences

**Vol. 14 (53), No. 1. 2001 **

**Mathematics.**

**Anashkin О.V.**

Lyapumov-Krasovskii Functions in the Theory of Stability
for Differential Equations with a Finite Delay

*Sufficient conditions for the stability of the zero solution of a retarded functional differential equation are obtained. Theorems on stability are given in the form of conditions
for existence of the auxiliary functional of Lyapunov-Krasovskii type. The functional are
non-monotone alone the solutions.*

**Baygozina J.Yu.**

Auxiliary Operators T_{μλ} and B_{α} in Space with Indefinite
Metric

*In the proposed article the auxiliary operators playing the important role in the construction of the characteristic function of regular extensions of Hermite operators in the space
with in definite metric, have been sdudied, and their basic properties have been proved.*

**Batyr El'dar**

Small Movements of a Double Pendulum with the Cavities Containing an Ideal Incompressible Fluid

*In this paper we considered a linear problem of hydrodynamics connected with small
movements of a double pendulum with cavities containing an ideal incompressible fluid. The
existence theorem of solutions of the Cauchy problem is formulated. The properties of normal
oscillations are described. The known N.E. Zhukovsky's theorem is transferred on a case of
movement of a double pendulum.*

**Вelan Е.P.**

Bifurcation of Periodic Solutions in Parabolic Problem with Transformed Argument *Nonlinear parabolic equations on the circle with a rotation transformation of the angle
variable is considered. The problem of stability and quantity of the running waves in the
neighborhood of a stationary solution is solved.*

**Wronsky В.М.**

On Proper Oscillations of Compressible Stratified Fluid

*In the article the problem on small motions and proper oscillations of stratified fluid in the
bounded region. It was shown that the spectre consists of a countable set of eigenvalues and
a part of continuous spectre. Asymptotic formulas for eigenvalues were obtained. Correct
solvability of Cauchy problem has been proven.*

**Donskoy V.I. **

The Asymptotic Estimation of a Number of Binary Decision Trees

*The asymptotic estimation d(n, k, μ) ≡ (μ — 1 )![k(k — 1)] ^{μ-1}n(n — 1) ^{μ-2}, n — ∞ is found,
where d(n, к, μ) is a number of Binary Decision Trees'with μ leaf, n Boolean variables, and
k classes of decisions.*

**Zakora D.A.**

Problem on Normal Oscillations of a Partially Dissipative Hydrosystem

*Some statements on p-basis property. Riesz' basis, property, multiple completeness of part
оf root elements, asymptotic behavior for eigenvalues are obtained.*

**Ivanov Yu.B.**

Generalized Solutions of a Spectral Boundary Problem for the
Differential System in the Shallow Water Theory

*A spectral differential boundary problem of Tree oscillations in rotating ideal liquid is
considered. With relation to the differential system a selfadjoint operator polynomial pencil
of the third degree with unbounded opertor coefficients is derived. The generalized solutions
of the boundary differential problem are regarded as solutions of the spectral problem for
that operator pencil.*

**Karpenko I.I. **

Characteristic Functions of Hermitian Operators

*In the present paper it's investigated properties of dissipative and accumulative extensions
of hermitian operators with a noncleuse domain, which have finite and equal defect numbers.
On this class of hermitian operators we generalize the notion of characteristic function,
suggested by A.N.Kochubey for symmetric operators with dense domain. It's proved the
criterion of unitary equivalence of hermitian operators.*

**Krivoruchko А.I.**

On a Cross-Ratio of a Quadruple of Linear Spans of Orbits of
Directions of Simmetries of Infinite Reflection Group

*Let G be an infinite reflection group acting on a non-cylindrical algebraic suface in n-dimensional real vector space and having four linear spans of G-orbits of directions of sym-metries.
All possible values of the cross-ratio of these linear spans as well as corresponding groups
G and their rings of invariants are obtained.*

**Kuzhel А.V.**

Boundary Value Space of Symmetric Operator

*The work is deviced the viewing of the main results, relays to the theory of boundary
value spaces with the equal defect numbers.*

**Moskaleva Yu.P.**

Generalization of Trace Formula for Case of K ^{r}_{+} (M)-Operators

*In this paper, the formula of trace is generalized by case of unbounded nonself-adjoint dissipative operators.*

**Orlov I.V.**

Almost Everywhere Convergence as a Convergence in Nonlinear
Inductive Scale of Locally Convex Spaces

*It is shown that, almost everywhere convergence may be considered as a convergence in
the inductive scale of locally convex spaces. The properties of the created scale are studied.*

**Papkov S.О., Chekhov V.N.**

Regular Infinite Systems of the Algebraic Equations in Case Long Period of Strains of a Prism

*In a paper we offer the theorem about, sufficient condition of existence of a. nonzero limit
of solution of a, infinite system of the linear algebraic equations with noil-negative matrix.
With help of this theorem, the stress condition of an elastic prism is investigated in the field
of low frequencies.*

**Persidskii S.K., Zhuravlev V.V. **

Some Theorems On Uniform Asymptotic
Stability of Solutions of System of Finite Difference Equations

*In this article two new theorems on a uniform asymptotic stability for finite difference
system are considered. In these theorems the first finite differences of appropriate Lyapunov
functions satisfy the condition Δ V _{m}≤0.*

**Rudenko L.I.**

On Development of Approaches to Decision Making with Incomplete Information

*Decision making linear models with incomplete information and their classification by
types of primary information are described. Approaches to solving of such problems based
on synthesis algorithmes are considered.*