Vol. 50 No. 2 (240) (2016)

An analogue of the Lindemann–Weierstrass theorem in differential setting for the differential equation $Dy = aDx$ is proved, where a is a non-constant parameter.

We study the Hilbert boundary value problem in the half-plane, when the boundary function is continuous on the real axis. It was proved that this problem is Noetherian and the solutions of the corresponding homogeneous problem are determined in explicit form.

Banach spaces $h_\infty (\varPhi)$, $h_0 (\varPhi)$ and $h^1(\eta) $ of functions, pluriharmonic in the unit ball in $\mathbb{C}^n$, depending on weight function $\varPhi$ and weighting measure $\eta$ are introduced. The question we consider is: for given $\varPhi$ we find a finite positive Borel measure $\eta$ on $[0,1)$ such that $h^1(\eta)^* $ $\thicksim$ $h_\infty (\varPhi)$ and $h_0 (\varPhi)^*$ $\thicksim$ $h^1(\eta) $.

In this paper we introduce the concept of $P_1$ property of sequences, consisting of positive integers and prove two criteria revealing this property. First one deals with rather slow increasing sequences while the second one works for those sequences of positive integers which satisfy certain number theoretic condition. Additionally, we prove the unboundedness of common divisors of distinct terms of sequences of the form $(2^{2^n} +d)^\infty_ {n=1}$ for integers d≠1.

The present work completes a research started in the papers [1–3]. Based on the results obtained in the previous papers, here we give a definitive solution to the problem of the Moore–Penrose inversion of singular upper bidiagonal matrices.

In this paper a new representation of the Riemann function  in the disc $U(2,1)$ is obtained:$\zeta (z) = \dfrac{1}{z-1} + \displaystyle\sum_{n=0}^\infty(-1)^n\alpha_n(z-2)^n,$ where the coefficients $\alpha_k$ are real numbers tending to zero. Hence is obtained $\gamma=\displaystyle\lim_{m\rightarrow\infty}\left[\displaystyle\sum_{k=0}^n\dfrac{\zeta^{(k)}(2)}{k!}-n\right],$ where $\gamma$  is the Euler-Mascheroni constant.

In this paper the arithmetical functions with indeterminate values of arguments are regarded. It is known that every $\lambda$-definable arithmetical function with indeterminate values of arguments is monotonic and computable. The $\lambda$-definability of every computable, monotonic, 1-ary arithmetical function with indeterminate values of arguments is proved. For computable, monotonic, $k$-ary, $k \geq 2$, arithmetical functions with indeterminate values of arguments, the so-called diagonal property is defined. It is proved that every computable, monotonic, $k$-ary, $k \geq 2$, arithmetical function with indeterminate values of arguments, which has the diagonal property, is not $\lambda$-definable. It is proved that for any $k \geq 2$; the problem of $\lambda$-definability for computable, monotonic, $k$-ary arithmetical functions with indeterminate values of arguments is algorithmic unsolvable. It is also proved that the problem of diagonal property of such functions is algorithmic unsolvable, too.

In this work phenomenon of resonant transfer of the excited energy between quantum dipole emitters (QDE) by the localized plasmon polaritons (LPP) is investigated. In system the molecules or semiconductor quantum dots act as QDEs and the nanometer size metal particles (MNP) care LPPs. The dependence of the frequency of Rabi oscillations on system parameters (the distance between MNP and QDE, the radius of MNP and dielectric permittivity of the surrounding medium) was determined. Conditions when the period of Rabi oscillations is considerably shorter than system relaxation time were defined.

Since transition of the fuel used in NPP from initial enrichment of 3.6% to 3.82% a necessity arises of revaluation of the security of transportation containers of spent nuclear fuel and based on it appropriate design modifications. To ensure subcriticality of containers borated steel insets are used. In this paper the results of study of optimal spatial configuration of borated and regular stainless steel guide sleeves configurations that allow meeting regulatory requirements on criticality safety. The model of the SNF transport cask was developed by KENO-VI code of SCALE 6.1 package. Isotopic composition of WWER-440 SNF was calculated by ORIGEN-S code of SCALE 6.1 package.

This calculation describes an accident to analyze and study the emergency instructions in case of fracture in the airtight zone of the first circuit. Sensitivity analysis was performed to determine the extent of the fracture, when a highpressure feeder pump remains 100 N/cm² higher than the saturation pressure in case of emergency pressure of the circuit (the pressure drop allows the main circulating pumps work). The conventional diameter of the fracture is 29 mm and the schedule of the development of events is given in the study.

In the paper some properties of Lebesgue constants $\big\{L_n(W)\big\}_{n=1}^{\infty}$ of Vilenkin system are investigated. Non almost convergence property for the sequence $\left\{\dfrac{L_n(W)}{\log_2 n}\right\}_{n=2}^{\infty}$ is obtained.