In memory of Academician Nikolai Aleksandrovich Borisevich (1923-2015)

This article is concerned with studying the classical solution of the boudary problem for the third-order nonhomogeneous nonstrictly hyperbolic equation. The equation is defined in the half-band of two independent variables. There are Cauchy’s conditions at the bottom of the domain and Dirichlet’s conditions at side boundaries. Using the method of characteristics, the analytic solution of the considered problem is written. The uniqueness of the solution is proved. 

Тhе implicit iteration method for solution of the first-kind operator equations with a non-negative self-adjoint bounded operator in the Hilbert space is proposed. Convergence of the method is proved in the case of an a posteriori choice of the regularization parameter in the usual norm of the Hilbert space, supposing that not only the right hand-side of the equation, but also the operator have errors. Тhе estimates of a method error and an a posteriori stop moment are obtained. 

This article considers the Lagrange interpolation processes, in which as interpolation nodes the zeros of the first-kind rational Chebyshev – Markov functions are chosen. The Lebesgue constant in the case of real poles is estimated. 

The method for approximate evaluation of matrix valued functional integrals with a small parameter b in the differential equation for the transition function is proposed. This method is based on the approximation of a chronologically ordered exponent by usual exponents. 

The linear proportional transformation of two-dimensional matrices was developed. This transformation was implemented as a computer program and was applied to recognize handwritten Cyrillic letters. The proposed transformation showed higher recognition accuracy in comparison with other transformations. 

The main result of the article is the theorem stating that in the universal covering group of any n-ary group all classes of Post’s equivalence determined as m-neutral sequences form the (k + 1)-ary group that at m = n is isomorphic to the n-ary subgroup of all identities of the n-ary group. Among the consequences of the theorem is a well-known result, according to which in any n-ary group the set of all the identities, if it is not empty, is a characteristic n-ary subgroup lying in the center of the n-ary group. 

Expressions for the variables of the mass center and the relative motion of two material particles in the three-dimensional Lobachevsky space and in the three-dimensional sphere with a time-dependent radius curvature are defined in terms of biquaternions. By using the action of the two particles in the above spaces, we have found that the problem of separation of the mass center and relative motion coordinates of this system reduces to the problem in the spaces of a completely constant curvature. 

The article is concerned with constructing the exact solutions for the Dirac – Kahler wave equation in the non-relativistic approximation for the simplest non-Euclidean geometrical model – hyperbolic Lobachevsky space. For the minimum value of the conserved angular momentum j = 0, the radial equations reduce to those solved in the elementary function. For greater values j = 1, 2, 3, ...,  the radial equations reduce to a couple of the four-order differential equations that are solved with the help of the factorization method. The general solution to each four-order order equation, involving four fundamental solutions, is constructed. The obtained solutions of the Dirac – Kahler wave equation cannot be solved in terms of the Pauli solutions on the background of Lobachevsky space, therefore any fermion interpretation for the Dirac – Kahler particle cannot be used. 

The collinear acousto-optic transformation of Bessel light beams into two ring beams of internal conical refraction has been considered. Using the Fourier transformation method, the expressions for power flows of ring diffracted beams of internal conical refraction are found in closed form. For a potassium biftalate crystal, the dependence of the diffraction efficiency on the ultrasound intensity and interaction durability is calculated. Based on this interaction, the creation of collinear convertible AO filters with a narrow width of a transmission band of about 0.1 is proposed. 

The laser ablation efficiency of eye cornea for UV (213 nm) and combined IR+UV (213 nm + 1444 nm) ablation treatment in safe conditions, preventing laser coagulation at 1444 nm, was investigated. It is shown that the IR+UV combination, when, at the first stage of cornea treatment, an IR laser pulse at 1444 nm, falling into a water absorption band with an absorption coefficient of about 31 cm–1, dehydrates a superficial corneal layer and at the second stage, in about 100 ms, a UV laser pulse at 213 nm ablates a thin (about 1–3 mm) collagen layer, gives promising results. Due to the controlled dehydration of the corneal surface, the process of 213 nm + 1444 nm laser ablation ensures a better reproducibility and a higher productivity. 

The article studies the effect of electron radiation with energy of 4 MeV on the characteristics of high-voltage (4.5 kV) silicon diode p-n-structures based on nuclear-doped silicon. During the research the dose dependences of the correlations between the static and dynamic characteristics of the p-n structures were built, and the radiation damage coefficient of minority carrier lifetime Kτ was determined. Using the DLTS-spectroscopy method, the formation in the band-gap of basic n-Si of six energy levels of radiation defects affecting the characteristics of irradiated p-n-structures was determined. 

We present the modification of the line method for calculation of the parameters of guided modes of straight and bent three-core microstructured fibers. On the basis of the method proposed, we analyzed the effective indices, radiation loss coefficients and a transverse field distribution of the fundamental modes in such fibers as a function of their structure (air holes diameter and separation, core separation) and bend value. 

A new version of the Fuzzy C-means clustering method, which we have called the Fuzzy C-means with the Gibbs Penalty (GPFCM), is introduced. The algorithm is designed to cluster gray scale, color and multichannel images. The specificity of the algorithm is to use the additive Gibbs penalty function in order to control the local smoothness of cluster representations of images in the regions of image homogeneity. The local smoothness of the solutions found by the GPFCM depends on the values of the coefficients of the penalty function. In turn, these coefficients can be taken as functions of magnitude and direction of the image gradient. Unlike the standard Fuzzy C-means clustering algorithm, which is unstable with respect to image noise and distortions, the proposed GPFCM allows building robust cluster approximations of noisy and corrupted pictures. Algorithm properties are illustrated on real images. Experiments have shown that on the one hand, the algorithm properly represents the homogeneous image regions as separate clusters and, on the other hand, it prevents the merging of different homogeneous areas. 

It is proposed to use an optimality box of the job permutation π as a measure of the stability of an optimal permutation π for processing a set of jobs in the single-stage processing system respecting to variations of the processing times of the jobs. For randomly generated problems, experimental comparison is made of the relative dimensions and the volumes of the optimality box for permutation and relative errors of the objective function with the optimal value of the objective function calculated for the actual processing jobs. 

We proposed a diagnostic nomogram (map) for hemodynamic state classification, based on linear regression modeling of blood pressure (BP) parameters and Data Mining algorithm. Mapping the vector of the BP characteristics, which coordinates are the coefficients of linear regression of the systolic, diastolic on pulse pressure, the pathological changes of hemodynamics inherent in hypertension, prehypertension or clinically latent hemodynamic disturbances can be diagnosed. The normal physiology of cardiovascular interaction corresponds to the vector location within certain limits on the nomogram. We have investigated the factors influencing the accuracy of the diagnostics in a clinical setting. 

Some properties of solitons in negative selfocs are considered.