Vol. 44 No. 2 (222) (2010)

In the paper for any $\rho\geq 1$ and an arbitrary increasing sequence of positive numbers

$\{\lambda_j\}_0^\infty$, the systems of operators and functions are introduced: $$\{L_\infty^{\frac{n}{\rho}}\}_0^\infty,~\{\varphi_n(x)\}_0^\infty,~x \in [0, +\infty),~L_\infty^{\frac{0}{\rho}}f \equiv f,~L_\infty^{\frac{n}{\rho}}f \equiv \prod\limits_{j=0}^{n-1}\left(D_\infty^{\frac{1}{\rho}}+\lambda_j\right)f, n\geq1,$$ where $D_\infty^{\frac{n}{\rho}}f\equiv D_\infty^{\frac{1}{\rho}} D_\infty^{\frac{n-1}{\rho}} f\left(1-\alpha=\dfrac{1}{\rho}\right)$; $\varphi_0(x)=e^{-\lambda_0^\rho x}$, $\varphi_n(x)=\sum\limits_{k=0}^n C_k^{(n)}e^{-\lambda_k^\rho x}$, $C_k^{(n)}=\left( \prod\limits_{j=0, (j\neq k)}^n\left(\lambda_j-\lambda_k\right)\right)^{-1}$. Some properties of these systems are investigated, as well as specific differential equations of fractional order are solved. Finally, for some classes of functions Taylor–McLaurens type formulas are obtained.

In the present paper solvability of a class of boundary problems associated with the anisotropic Helmholtz–Shrodinger equation in the upper and lower semiplanes of Sobolev spaces is studied. The first and second type boundary conditions are assumed to hold on the line y=0. Solvability of these boundary problems reduces to solvability of Riman–Hilbert boundary problem. The solvability analysis is based on the factorization problem of some matrix-function.

A mixed problem for the equation

                                                $ Lu\equiv (t^{\alpha}u'')''+\alpha u=f $   (1)

is considered. Firstly, the weighted Sobolev spaces $W_\alpha^2,~ W_{\alpha}^2(0),~W_{\alpha}^2(b)$ and the generalized solution to equation (1) are defined. Next, the existence and uniqueness of the generalized solution for the mixed problem is studied, as well as the description of the spectrum of corresponding operator is given.

In the present paper some properties of one random process arising in many limit theorems of risk theory are investigated. Connection formulas with stable distributions and with one class of integral transforms are found. The asymptotical properties of this law are studied.

Quantitative characterization of randomly roving agents in wireless sensor networks (WSN) is studied. Above the formula simplifications, regarding the known results, it is shown that the basic agent model is stochastically equivalent to a similar simpler model. Then a formula for frequencies is achieved in terms of combinatorial second kind Stirling numbers. At allows to justify the roving agents quantitative characteristics.

In the present paper the independence number of generalized cycles product is investigated. A method for constructing the maximal independent set in the product graph is presented. The method is particularly based on a specific combinatorial problem, which is also solved in the paper. The main result generalizes the similar fact known for odd cycles [6].

The minimal number of systems of linear equations with n unknowns over a finite field $F_q$,  such that the union of all solutions of the systems forms an exact cover for a given subset in $F_q^n$, is the complexity of a linearized covering. An upper bound for the complexity for “almost all” subsets in $F_q^n$ is presented.

Cosmological models of the Universe in the framework of Jordan–Brans– Dicke theory in the Einstein frame in the case of scalar field domination as well as in the presence of cosmological constant $\wedge$ with matter, which is described by barometric equation of state $P=\alpha \epsilon$ ( $P$ is the pressure, $\epsilon$ is the energy density of matter), are considered. The analysis of obtained results in the light of observational data are done. It is shown that the contribution of the scalar field and $\wedge$-member ( $\wedge>0$ ) in the case of $q=-1/2$ ($q$ is the “decelerating” parameter) compensate each other. In addition, the situation leads to Einstein’s theory.

Let $\cdot, *$, be idempotent operations on the set A, and * be a commutative operation. We prove, that if the pair $(\cdot, *)$ satisfies the generalized entropic property, then $(\cdot, *)$ is entropic.

In this work we prove that for an arbitrary odd $n\geq 1003$ there exist two words $u( x, y), v (x, y)$, almost every images of which in free Burnside group $B (m, n)$  are independent.

In this paper we suggest a method for determining the refractive index of X-ray radiation for a one-layer medium, having a fibrous and granular structure. It is shown that the refractive index can be calculated as the ratio of intensities, measured at the location of the substance on the path of the beam between the two-crystal X-ray spectrometer and behind it.