Vol. 49 No. 3 (238) (2015)

In this paper a formula for a translation invariant measure of planes intersecting a n-dimensional convex body in terms of curvatures of 2-dimensional projections of the body was found. The paper also gives a new simple proof of the representation for the support function of an origin symmetric 3-dimensional convex body, which was obtained by means of a stochastic approximation of the convex body.

The present paper is a direct continuation of the paper [1]. Here we start our study of the Moore–Penrose inversion problem for upper bidiagonal matrices with any arrangement of one or more zeros on the main diagonal. In the paper we obtain some preliminary results, which will be used in subsequent, third part of the study.

It is well-known that exactly $N-1$ n-independent nodes uniquely determine the curve of degree n passing through them, where $N=\dfrac{1}{2}(n+1)(n+2).$ It was proved in [1], that at least $N-4$ number of n-independent nodes are needed to determine the curve of degree $n-1$ uniquely. The paper has also posed a conjecture concerning the analogous problem for general degree $k\le n$. In the present paper the conjecture is proved, establishing that the minimal number of n-independent nodes uniquely determining the curve of degree $k\le n$ is equal to $\dfrac{(k-1)(2n+4-k)}{2}+2.$

We prove the boundedness of Bergman type integral operators in mixed norm spaces over the unit ball of ${\mathbb R}^n$. Bounded harmonic projections are found in the mixed norm and Lipschitz spaces. Corresponding Forelli–Rudin type theorems are proved.

The approach proposed in this paper uses ideas and instruments of gauging theory to handle nonlinear problems in economics, nonlinear dynamics and various technological issues. We see nonlinearity of gauging as hardly a technical issue arising while solving the variation problem, but a structural feature of an intrinsically nonlinear space shaped by its gauge-based connections. Under these assumptions, the solution of the variation problem is shown as invariant to transformations of the vector field which present the initial source of nonlinear disturbances.

A total coloring of a graph $G$ is a coloring of its vertices and edges such that no adjacent vertices, edges, and no incident vertices and edges obtain the same color. An interval total $t$-coloring of a graph $G$ is a total coloring of $G$ with colors $1,2,\ldots,t$ such that all colors are used, and the edges incident to each vertex $v$ together with $v$ are colored by $d_G(v)+1$ consecutive colors, where $d_G(v)$ is the degree of a vertex $v$ in $G$. In this paper we describe some methods for constructing of graphs that have no interval total coloring.

In the paper laser induced thermomechanical flow and oscillations in hybrid oriented nematic liquid crystals (NLC) are theoretically predicted. The effect is conditioned by the hydrodynamic flow tendency to reduce the curvature of the “flexible ribbon” of hybrid NLC.

In the present work the thermodynamic studies of mitoxantrone complexformation with four-stranded poly(G) have been carried out. The binding thermodynamic parameters, the binding constant K and the number of nucleotides corresponding to one binding site n, were determined. It was shown that mitoxantrone is bound stronger with four-stranded poly(G), then it is bound with double-stranded helix. It was also shown that due to difficulties of the intercalation of mitoxantrone molecules into four-stranded structure is bound mainly by external binding mode with [poly(G)]4 and at the interaction saturation mitoxantrone bound to much more guanines ($n\cong 10$) per one molecule. 

The problem is considered of a dynamic reflection of a plane monochromatic X-ray wave at the superlattice with a stacking fault between the layers when absorption is taken into account. It is shown, that taking absorption into account reduces the reflectivity in the direction of satellites. At the same time, the closer stacking fault is to the crystalline surface, the less impact it has on the reflectivity.

The efficiency of detection of laser radiation in the magnetized transparent ferromagnetic materials was investigated. The nonlinear interaction of electromagnetic waves in the ferromagnetic for the different forms of the magnetization curve of a ferromagnetic in MatLab environment was modeled. It is shown that the detection of amplitude modulated electromagnetic radiation can be obtained as a result of such nonlinear interaction. The magnitude and sign of detected signal greatly depends on the shape of the magnetization curve and on the applied external magnetic field. The simulation results for the samples of monocrystalline yttrium iron garnet are well correlated with the experimental results.

In the paper correspondence between the monotonic functions and the set of irredundant checking tests of binary tables is formed. The maximum number of irredundant checking tests of binary table is determined.