**Full Volume**/ Весь том (UA) [ (PDF) ]**Cover**/ Титул (UA) [ (PDF) ]**Editorial Board**/ Редакційна колегія (UA).*P. 2.*[ (PDF) ]**Contents / Зміст (UA)**.*P.3.*[ (PDF) ]**Ya. Rybalko, D. Shepelsky,**Riemann-Hilbert approach for the integrable nonlocal nonlinear Schr\"odinger equation with step-like initial data.*P. 4-16.*[ Abstract ] [ Full-text available (PDF) ]**V.A. Rezunenko,**Diffraction of the field of vertical electric dipole on the spiral conductive sphere in the presence of a cone.*P. 17-26.*[ Abstract ] [ Full-text available (PDF) ]**S.M. Chuiko, Ya.V. Kalinichenko,**On the regularization of the Cauchy problem for a system of linear difference equations.*P. 27-34.*[ Abstract ] [ Full-text available (PDF) ]**D.N. Andreieva, S.Yu. Ignatovich,**On constructing single-input non-autonomous systems of full rank.*P. 35-43.*[ Abstract ] [ Full-text available (PDF) ]**H.N. Solovyova, N.N. Kizilova,**Mathematical modeling of bioactive arterial wall.*P. 44-57.*[ Abstract ] [ Full-text available (PDF) ]**S.V. Zhuchenko,**Numerical simulation of the thermodynamics of a fast neutron reactor.*P. 58-83.*[ Abstract ] [ Full-text available (PDF) ]

We study the Cauchy problem for the integrable nonlocal nonlinear Schr\"odinger (NNLS) equation \[ iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0 \] with a step-like initial data: $q(x,0)=o(1)$ as $x\to-\infty$ and $q(x,0)=A+o(1)$ as $x\to\infty$, where $A>0$ is an arbitrary constant. We develop the inverse scattering transform method for this problem in the form of the Riemann-Hilbert approach and obtain the representation of the solution of the Cauchy problem in terms of the solution of an associated Riemann-Hilbert-type analytic factorization problem, which can be efficiently used for further studying the properties of the solution, including the large time asymptotic behavior.

*Keywords:* nonlocal nonlinear Schr\"odinger equation; inverse scattering transform method; Riemann-Hilbert problem.

2010 Mathematics Subject Classification: 35Q55; 35Q15. [ Full-text available (PDF) ] Top of the page.

The problem of diffraction of a vertical electric dipole field on a spiral conductive sphere and a cone has been solved. By the method of regularization of the matrix operator of the problem, an infinite system of linear algebraic equations of the second kind with a compact matrix operator in Hilbert space $\ell_2$ is obtained. Some limiting variants of the problem statement are considered.

*Keywords:* spiral conductive sphere; cone; vertical electric dipole; regularization method; system of equations of the second kind.

2010 Mathematics Subject Classification: 78A40; 78A45; 35A22; 97N42. [ Full-text available (PDF) ] Top of the page.

*Short abstract:* An original regularization scheme for the Cauchy problem for a linear singular system of difference equations is proposed.

*Extended abstract:*
The article proposes unusual regularization conditions as well as a scheme for finding solutions of the linear Cauchy problem for a system of difference equations in the critical case, significantly using the Moore-Penrose matrix pseudo-inversion technology.
The problem posed in the article continues the study of the regularization conditions for linear Noetherian boundary value problems in the critical case given in the monographs by S.G. Krein, N.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina, A.M. Samoilenko and A.A. Boichuk.
The general case is studied in which a linear bounded operator corresponding to a homogeneous part of a linear Cauchy problem has no inverse. In the article, a generalized Green operator is constructed and the type of a linear perturbation of a regularized linear Cauchy problem for a system of difference equations in the critical case is found.
The proposed regularization conditions, as well as the scheme for finding solutions to linear Cauchy problems for a system of difference equations in the critical case, are illustrated in details with examples.
In contrast to the earlier articles of the authors, the regularization problem for a linear Cauchy problem for a system of difference equations in the critical case has been resolved constructively, and sufficient conditions has been obtained for the existence of a solution to the regularization problem.

*Keywords:* regularization scheme; Cauchy problem, linear difference equations; pseudoinverse matrices.

2010 Mathematics Subject Classification: 15A24; 34В15; 34C25. [ Full-text available (PDF) ] Top of the page.

For a nonlinear system of differential equations $\dot x=f(x)$, a method of constructing a system of full rank $\dot x=f(x)+g(x)u$ is studied for vector fields of the class $C^k$, $1\le k < \infty$, in the case when $f(x)\not=0$. A method for constructing a non-autonomous system of full rank is proposed in the case when the vector field $f(x)$ can vanish.

*Keywords:* nonlinear control system; accessible system; system of full rank; non-autonomous system;
the straightening theorem for vector fields.

2010 Mathematics Subject Classification: 93B10; 93C10. [ Full-text available (PDF) ] Top of the page.

Biological tissues and their artificial substitutes are composed by different fibers and possess complex viscoelastic properties. Here the most popular 3-element and 5-element rheological models of human soft tissues as viscoelastic bodies are considered accounting for the time delay between the load and mechanical respond of the material. The obtained data compared to the experimental curves got on the vessel wall and heart tissues.

*Keywords:* active biomaterials; visoelastic fluids; rheology; mathematical modelling.

2010 Mathematics Subject Classification: 35K05; 76Z05. [ Full-text available (PDF) ] Top of the page.

*Short abstract:*
The article offers an algorithm of solving two dimensional initial-boundary value
problem with computational modelling of fast proceeding thermodynamical processes,
that appears in the cassette that consist of several fuel elements, and adjacent collectors.
What is more there are some results of computing experiments that were conducted by
using author's PC program.

*Extended abstract:* The article deals with one reactors design, which,
under the International Forum, are attributed to the 4th generation of the GIF-IV
(Generation IV International Forum) of fast neutron reactors with a helium coolant
and a closed fuel cycle (GFR). Although the use of helium as a coolant in reactors
of this type and has great advantages in comparison with other coolants, for example,
CO2 gas, however, due to the great difficulties encountered in the implementation of
such a project, only prototypes of similar reactors are currently implemented.
Due to the complexity of gas flow in the collectors and backfill, the averaged
flow of the coolant is considered throughout the proposed mathematical model.
It is assumed that the averaged flow is symmetric everywhere relative to the common
axis of the cylinders forming the annular domain, and, consequently, is axisymmetric,
that is, two-dimensional. One such annular cylindrical cavity will be called a fuel
element. The mathematical model of a cassette of several such fuel elements
connected by common distributed and gathering collectors is considered in the article.
The algorithm for solving the arising non-stationary initial-boundary value problem
is proposed in the article, as well as the results of some computational experiments
that are obtained using the PC program, compiled and debugged by the author
of the article. The experiments were carried out both for one fuel element,
and for cassettes of 2, 3 and 4 fuel elements. The algorithm for solving the arising
non-stationary initial-boundary value problem is proposed in the article, as well
as the results of some computational experiments that are obtained using the PC program,
compiled and debugged by the author of the article.

*Keywords:* cassette of fuel elements; distributive and collapsible collector;
helium coolant-moderator; turbulent flows; recurrent streams; emergency situation;
trajectories of free whirlwinds.

2010 Mathematics Subject Classification: 76W05. [ Full-text available (PDF) ] Top of the page.

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