Scientific Notes of

Taurida National V.I. Vernadsky University

Taurida National V.I. Vernadsky University

Series

Physical and Mathematical Sciences

**Wronsky В.М.**

On Problem on Small Motions and Proper Oscillations
of System "Fluid-Gas" in the Bounded Region

*In the article the problem on small motions and proper oscillations of system "fluid-gas"in the bounded region is considered. It was show spectrum consists of a
countable set of eigenvalues. Asymptotic formulas for eigenvalues were obtained.*

**Kopachevsky N.D., Zakora D.A.**

On Spectral Problem Connected
with the Second Order Integro-Differential Equation

*The Cauchy problem
*

**Karpenko I.I., Sukhtaev А.I., Tyshkevich D.L.**

On One Approach for
Differentiability of Functions of Quaternion Variables

*In this paper the new approach for differentiability of functions defined on
region of the real quaternion skew-field is suggested. It's proved different criteria
for functions of a quaternion variable to be differentiable.*

**Krivoruchko А.І. **

On the Non-Centroaffine Groups Generated by
Reflections with Respect to the Straight Lines

*The basic polynomial invariants of a transformation group H of the affine space
are found in the case when H satisfies the following conditions: *

*H is the non-centroaffine group generated by affine reflections with respect to straight lines;**H acts on some non-cylindrical algebraic surface.*

**Lyakh A.M.**

Space Deformations and their Application to
Phytoplankton Shape Modeling

*The theory of Bezier curves, pathes and volumes is investigated, and two
methods of 3D space deformation, — Free Form Deformation and Axial Deformation, which are used for phytoplankton cells shape modeling, are considered. The
method of 3D phytoplankton models building, on the basis of a photographic
images, is described. These models are used for accurate calculating phytoplankton
volumes and surfaces area.*

**Muratov М.A., Rubshtain В.A. **

On One Class of Maximal Invariant
Subspaces of Continuous Operator in Indefinite Inner Product Space

*We study analogs of Dominated Ergodic Theorems in rearrangement invariant
spaces of a measurable functions for sequence absolute contraction on the positive
semiaxis.*

**Pavlov E.A.**

On Birkof-Hinchin Theorem

*In this paper Birkof-Hinchin theorem is generalized on ideal Banach functional
spaces.*

**Orlov I.V. **

Sufficient Conditions of Extremum and If-Extremum in the
Product of Two Nuclear LCS (General Case)

*In the work the sufficient conditions of extremum and compact extremum in the
product of two nuclear LCS for the case of non-commuting second partial deri-vatives are proposed. In particular, the sufficient conditions of compact extremum
for Euler-Lagrange functional are proved.*

**Tretyakov D.V. **

On the Orthogonality Relation in Hilbert Space and
Some Properties of Additive Operators

*The special inner product and orthogonality relation in complex Hilbert space
were defined. With the help of this product it's defined and studied involution and
different classes of operators on the bimodulne of all continued additive operators.
The classification of spectrum of bounded additive operator is proposed too.*

**Tyshkevich D.L.**

On One Class of Maximal Invariant Subspaces of Continuous Operator in Indefinite Inner Product Space

*In this paper we study the structure of maximal reducing subspaces of continuous linear operator in indefinite inner product space inducing unitary operator.*

**Tikhonov A.S.**

Property of Maximality for Spectral Components (Case of
Multiply Connected Domains)

*The aim of this paper is to show that operators closed to normal ones with spectrum on a curve possess certain maximal invariant subspaces, which are
related to the interior and the exterior of the curve. We generalize the corresponding results (for the real axis and the unit circle) established by Naboko and
Makarov.*