Vol. 54 No. 1 (251) (2020)

In this article we define two new integral geometric concepts: conditional moments of the chord length distribution of a convex domain and conditional moments of the distance distribution of two independent uniformly distributed points in a convex domain. We also found a relation between these two concepts.

In this paper we obtain some conditions for the existence of locally-balanced 2-partitions with an open (with a closed) neighborhood of some classes of graphs. In particular, we give necessary conditions for the existence of locallybalanced 2-partitions of even and odd graphs. We also obtain some results on the existence of locally-balanced 2-partitions of rook’s graphs and powers of cycles. In particular, we prove that if \(m,n \geq 2\), then the graph \(K_m \Box K_n\) has a locally-balanced 2-partition with a closed neighborhood if and only if m and n are even. Moreover, all our proofs are constructive and provide polynomial time algorithms for constructing the required 2-partitions.

In this paper a universal function \(U \in L^1 [0,1)^2\), which with respect to the double Walsh system has universal property in the sense of modification, is constructed.

We constract the exact solution of the Dirichlet problem in the Sobolev space for two-dimensional elliptic equation considered on the half-plane.

We consider sequences $ {\lbrace \rho_j \rbrace}_{0}^{\infty} $ $ (\rho_0 \mathclose{=} 1, \rho_j \mathclose{\geq} 1) $, $ {\lbrace \alpha_j \rbrace}_{0}^{\infty} $ $ (\alpha_0 \mathclose{=} 1, \alpha_j \mathclose{=} 1 \mathclose{-} (1/\rho_j )) $, $ {\lbrace W_j (x) \rbrace}_{0}^{\infty} \mathclose{\in} W $, where

$$ W \mathclose{=} \lbrace {\lbrace W_j (x) \rbrace}_{0}^{\infty} / W_0 (x) \mathclose{\equiv} 1, W_j (x) \mathclose{>} 0, {W}_{j}^{\prime} (x) \mathclose{\leq} 0, W_j (x) \mathclose{\in} C^\infty [0,a] \rbrace, $$

$ C^\infty [0,a] $ is the class of functions of infinitely differentiable. For such sequences we introduce systems of operators $ {\lbrace {A}_{a,n}^{\ast} f \rbrace}_{0}^{\infty} $, $ {\lbrace \tilde{A}_{a,n}^{\ast} f \rbrace}_{0}^{\infty} $ and functions $ {\lbrace {U}_{a,n} (x) \rbrace}_{0}^{\infty} $, $ {\lbrace {\Phi}_{n} (x,t) \rbrace}_{0}^{\infty} $. For a certain class of functions a generalization of Taylor–Maclaurin type formulae was obtained. We also introduce the concept of $ \langle \rho_j,W_j \rangle $ generalized completely monotone functions and establish a theorem on their representation.

In this paper we consider semi-infinite flat waveguides with different boundary conditions on the planes and on the edges that bound the waveguide. The possibility of localizing shear waves in the vicinity of the junction of neighbouring parts of a semi-infinite flat waveguide is established.

This work is dedicated to the study of the energy resolution of electrons using simulated data of “Drell-Yan” process with \( e^{-}e^{+} \) yield in \(pp\)‑interactions at \( \sqrt{s}=13\ TeV \). The work contains a brief description of event modeling and electron reconstruction from the simulated data. The energy resolution was estimated from the distribution of the ratio of the transverse energies of reconstructed and generated electrons. Two approaches used in calculations. In the first approach, the standard definition of energy resolution was used (Full width at half maximum – FWHM), and in the second, the root mean square of distribution was taken as a describing value for energy resolution.

In this work we study the propagation of light in a twist-oriented pleochroic nematic liquid crystal with pleochroic dye. The dependence of the intensity of transmitted light on the azimuthal angle of the linearly polarized incoming light has been investigated experimentally. As a result, the role of absorption anisotropy has been identified.

An important task of modeling and testing radar systems is to synchronize the order of the sub-degree of the radiated signals between the two ports of the transmitters. The simulation of synchronization of two NI PXI 5840 vector transceivers was performed with the separation of the power of the local oscillator and the clock generator. The simulation was done in the LabVIEW software environment using the developed program.

One of the major interaction mechanisms between carbon nanotubes (CNT) and DNA is $ \pi \mbox{-} \pi $ stacking, which is a van der Waals type of interaction. Attraction between the polycyclic aromatic surface elements of CNTs and the aromatic nucleotides of DNA leads to reversible adsorption (physisorption) between them. We propose to describe the ssDNA adsorption on CNT using zipper model and address the connection between the thermodynamics of adsorption and the flexibility of ssDNA.

The synthesis of new photo-orienting liquid crystal polymers made it possible to create a completely new type of optical elements consisting of ultrathin layers with a spatially structured orientation of the optical axis. In this paper the optical elements based on centrally symmetric periodic structures with cylindrical orientations of molecules are described. These elements are implemented using the recording method that provides a smooth change of optical axis in a thin film of a liquid crystal polymer. The optical elements on the base of described structures have new functionalities and may be assigned to the class of optical elements of a new generation.