Scientific Notes of

Taurida National V.I. Vernadsky University

Taurida National V.I. Vernadsky University

Series

Physical and Mathematical Sciences

**Anaphiyev A.S. **

Delta-Boolean Functions and their Properties *In this article we consider the new class of Delta-Boolean functions. Delta-Boolean
functions and their arguments may take values 0,1 or Δ. The definitions of the degree of
uncertainty, relative and absolute essential variable degrees were introduced for Delta-Boolean functions. These definitions are necessary to research the problem of Boolean
functions reconstruction. Also we consider some properties of Delta-Boolean functions. *

**Batyr El'dar **

Small Movements of a System of the Consistently
Connected Bodies with the Cavities Containing an Ideal Incompressible Fluid

*In this paper we considered a linear problem of hydrodynamics connected with small
movements of a system n bodies. The system is a circuit of the consistently connected
bodies. Each of bodies of such circuit is a gyrostat. 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 system
n bodies.*

**Belan Е.Р.**

Bifurcation of Periodic Solutions for Nonlinear Parabolic
Functional Differentional Equations

*This paper is devoted to bifurcation of equilibria for nonlinear parabolic functional
differential equations with transformation of space variables. Theorem on bifurcation of
periodic solutions, stability, asymptotic form is proved by the method of center manifolds.*

**Donskoy V.l., Ilchenko A.V.**

On Statistical Substantiation for Classes
of Object Separation Possibility in Empirical Samples *The approach for substantiation of ability of classification based on the hypothesis
of more than one mode existence in pair distance distribution is considered. The
sufficient condition of nonrandomness of two modes is proved and statistical criterion of
classification is given.*

**Ivanov Yu.B.**

Approximate Equations for Low-Frequency Oscillating Free
Surface of Rotating Two-Layer Liquid

*For mathematical simulation of low-frequency oscillating free surface of rotating two-layer liquid a polinomial operator pencil is used. It is presupposed that density p _{1} of
an upper layer close to density p_{2} of an underlying layer with p_{1} < p_{2}. In that case
the mathematical model has a small parameter. Equations for the zero and the first
approximations in parameter-series expansion are derived.*

**Karpenko Irina**

About One Approach to Constructing of Boundary
Value Space of J— Hermitian Operators

*In this paper we considered some properties of defect subspaces of J-Hermitian linear
relations in the space with an indefinite metric. These results are used for definition and
study of the boundary value space for the J-Hermitian operator with nondense domain.
In particular, different properties of the boundary operators are obtained.*

**Kyrychenko A.A.**

On Linear Combinations of Orthoprojectors

*We investigate n-tiples of the projectors P _{1},P_{2}...,P_{n} in the complex Hilbert space H satisfying the condition ∑^{n}_{i=1} α_{i}P_{i}=I_{H}(here I_{H} is identity operator, α_{i}>0) and their application to description of the self-adjoint operators having two points of the spectrum and representable as a sum of three orthoprojectors.*

**Kovalenko А., Маrуаnіn В., Smolich V.**

Queueing Process of Type
(M/G/1/1) *Stationary probabilities of system described by non-Markovian stochastic process are
derived. The authors avoid the traditional technique of embedded Markov chains and
suggest their own technique.*

**Krivoruchko А.I.**

On a Mutual Disposition of Four Linear Spans of Symmetry
Directions of an Infinite Reflection Group

*Let G be an infinite reflection group acting on a non-cylindrical algebraic hypersuface
in n-dimensional real vector space.
Some new restrictions on a mutual disposition of four linear spans of symmetry
directions of a group G with pairwise 0-dimensional intersections are obtained.*

**Makhina G.A.**

Neural Network Architecture for Weekly Defined
Optimization Problem with Linear Restrictions *The neural network architecture for the inequality constrained weakly defined
optimization problem is introduced in this paper. The objective function is reconstructed
in the form of neural network using the initial precedent information. Inclusion of
inequality restrictions into the problem leads to the extended architecture of neural
network, which is mathematically based on the logarithmic barrier functions approach.*

**Muratov М.А.**

Different Kind of Convergence in the Rings of the Measurable
Operators *In the paper, relation between convergence in measure, convergence almost everywhere end bilaterally convergence almost everywhere in the ring of measurable operators,
affiliated to a finite von Neumann algebra is studied.*

**Myagkov V.l.**

Characteristic Property of Mutual Transformation on the
Plane

*Two curves γ and γ* in a Euclidean plane are called mutual with the mutuality center
at the point Q if there is mutually one-to-one correspondence between the curves, where
two conditions
are fulfilled.
Here M and M* are corresponding points of the curves, , are vectors of normal
of γ and γ* curves at the corresponding points M and M*.
The transition from an original curve γ onto the mutual curve γ* is called mutuality
transformation H with the mutuality center at Q point.
The simpler proof of the theorem has been found:
The class of real disintegrated curves of the second order is invariable concerning
mutuality transformation H.*

**Orlov I.V.**

Young Theorem for Normal Derivatives and Functional
Extremums in the Products of Nuclear Spaces

*The classical sufficient conditions for extremum of two variables function are
generalized for the case of twice normally ditferentiable function in the product of the
nuclear spaces.*

**Rudnitsky О.I.**

About one Class of the Wild Groups of the Skew Symmetries,
which have Four Orbits of Directions of Symmetry *Let G be the infinite group generated by skew reflections with respect to hyperplanes
in an euclidean space and have four infinite G-orbits of the directions of symmetry. The
conditions of existence of the groups G, if the linear spans of G-orbits of the directions
of symmetry have three different straight lines of intersection, are obtained.*

**Starkov P.A.**

On the Operator Pencil of the Transmission Problem for
Helmholtz Equation

*In the work we consider the general case for the spectral problem, which is appear in
the operator approach to the transmission problem for Helmholtz equation.*

**Tikhonov A.S.**

"Curved" Conservative Systems and their Applications *In this paper the notion of "curved"linear conservative systems is introduced. It is
shown that such systems: 1) naturally appear for problems relating to functional models;
2) allow an informative definition of the transfer function; 3) have interesting applications
to perturbation theory, scattering theory, and for extreme factorizations of operator-
valued functions.*

**Tyshkevich D.L.**

On Classes of Adjointable Operators in General Indefinite
Inner Product Spaces

*General approach for definition of operation of conjugation of linear operators in
general indefinite inner product spaces (including degenerate spaces) is suggested. Some
simple consequences of introduced definitions are considered.*

**Tsvetkov D.O.**

On Small Motions of a Stratified Fluid

*The problem on small motions of viscous fluid, which density in a equilibrium state has
stable stratification is investigated on the base of approach connected with applications
of operator matrix theory with unbounded entries. The theorem on strong solvability of
initial boundary value problem is proved.*

Small Transverse Oscillations of a Visco-Elastic Beam with a Load at the End