Scientific Notes of

Taurida National V.I. Vernadsky University

Taurida National V.I. Vernadsky University

Series

Physical and Mathematical Sciences

**Zakora D.A.**

Оn Some Quasilinear Nonstationary Equation

*Some quasilinear nonstationary equation with hounded operator coefficients
depending of time and specific quadratic nonlinearitv is investigated. The theorem
on global solvability of the Cauchy problem is proved. Conditions for boundedness
of solutions are obtained.*

**Karpenko І.I., Omel'chenko P.V.**

Structure of a Linear Operator in
a Finite-Dimensional Quaternionic Hilbert Space

*In this paper the structure of a linear operator acting in a finite-dimensional
quaternionic Hilbert space is studied. The main result is that an analogue of
decomposition of complex linear space in the direct sum of the algebraic eigenspaces has place for such operators. The method of the construction of minimal invariant cyclic subspaces is suggested. Other results concern the spectral
decomposition of normal operators in a finite-dimensional quaternionic space.*

**Krivoruchko А.I.**

On the Groups Generated by Reflections with
Respect to the Skew Straight Lines

*The basic polynomial invariants of a transformation group H of the affine space
V are found in the case when H satisfies the following conditions: a) H acts
on some non-cylindrical algebraic surface F ⊂ V; b) H is generated by affine
reflections with respect, to straight lines, some two of which are skew.*

**Muratov М.А., Pashkova Yu.S., Rubshtein В.A. **

Dominated Ergodic Theorem for Sequence Positive Contraction in Rearrangement Invariant Spaces of Measurable Functions

*We study conditions under which Dominated Ergodic Theorems hold in
rearrangement invariant spaces for sequence positive contraction.*

**Orlov I.V. **

Theorems on Homomorphism, Open Mappings and Closed
Graph in Inductive Scales of Spaces

*Banach theorems on homomorphism, open mappings and closed graph are
transferred to linear operators in inductive scales of spaces. The results are applied
to the operators acting in inductive limits.*

**Orlov I.V., Tolstopyatov Ya.V. **

Operators in the Double Scales of
Spaces and Regularity Condition

*In the work linear continuous operators in double scales of spaces are investigated. The extension conditions for operators in unary scales up to the operators
in double scales are obtained. In particular, new regularity conditions for classical
linear operators are obtained.*

**Starkov P.A. **

Exclusive Values of the Fixed Parameter for Transmition
Problems

*Given article finishes studying properties of decisions of an abstract spectral
problem, which was obtained during researching the transmition problem. Last
case of values of the fixed parameter λ unexplored earlier by the author, namely its
exclusive values is considered. In work theorems of a spectrum and basis of eigen
elements of a researched abstract spectral problem are formulated and proved.*

**Tikhonov A.S.**

An Extreme Factorizations of J-Contractive-Valued
Functions

*Using the notion of abstract weak subspace, unified approach to extreme factorizations of J-contractive-valued functions is presented.*

**Tyshkevich D.L.**

Nonorthoextendable Subspaces of Indefinite Inner Product Spaces

*In this paper we study structure and properties of so-called nonorthoextendable subspaces of indefinite inner product spaces, i.e. subspaces ӿ _{1} satisfying relation ӿ _{1}^{[⊥]}⊆ ӿ_{1}. Subspaces of such type naturally appear in study of maximal orthogonal systems in indefinite inner product spaces.
*

**Tsvetkov D.О.**

Problem on Normal Oscillations of a Partially Dissipative Hydrosystem

*There was investigated the problem on normal oscillations of some partially dissipative hydrosystem. The spectrum of normal oscillations and other questions are
studied.*

**Yakovlev А.V. **

Small Movements of a Rigid Body with a Cavity Fulfilled
by a Visco-Elastic Fluid

*In the paper, there was investigated two-dimentional initial boundary value
problem on small movements of a rigid body with a cavity fulfilled by a visco-
elastic fluid. Under some assumptions/ the theorem on strong solvability of the
problem on arbitrary interval of time is proved.*

**Kruglyak S.A., Popovych S.V., Samoilenko Yu.S.**

Representations of *-Algebras Associated with Dynkin Graphs and Horn's Problem

*The connection between *-representations of *-algebras associated with graphs
locally scalar graph representations and algebraic Horn's problem is studied. The
solution of algebraic Horn's problem for Dynkin graph D _{4} is given.*

**Donskoy V.l.**

On Metrical Properties of the Shortest Empirical
Conjunctive Regularities

*In this paper, it is shown that utilization of the shortest empirical conjunctive
regularities as elementary classifier for Pattern Recognition problem, satisfying
the compactness hypothesis, can reduce to considerable errors.*