International Scientific Journals
of Scientific Technical Union of Mechanical Engineering "Industry 4.0"

The article presents the results of researches that are a continuation of analyzes of the correlation properties of generating functions intended for the formation of derivatives of Walsh functions. The derivatives of the Walsh functions are designed to obtain pseudorandom sequences (PRS) used in the formation of noise-like signals in multichannel data transmission systems with channel division according to the shape (code) of the signal. The analysis performed showed that the correlation properties of the derivatives of the Walsh functions depend on the type of generating functions, which were the modified Barker codes – direct and inverse composite Barker codes and de Bruijn sequences. It was justified the advantage of using these signals in the development of CDMA systems in order to reduce the interference level of multiple access and to protect against unauthorized access.

Topological Data Analysis (TDA) is relatively new filed of Applied Mathematics that emerged rapidly last years. The main tool of Topological Data Analysis is Persistent Homology. Persistent Homology provides some topological characteristics of the datasets. In this paper we will discuss classification of digital images using their topological signatures computed with Persistent Homology. We will experiment on the Fashion-MNIST dataset. Using Topological Data Analysis, the classification was improved.

It is known that a large number of algorithms of the Monte Carlo method and experiment planning are based on the choice of a
certain probability distribution ρ.
This probability distribution ρ is given on a measurable space (X, B). And a given measurable space (X, B) has a density ρ(x) = dρ/dν by
some σ-finite measure v on (X,B).
When choosing a probability distribution ρ, the problem of solving the problem of finding the optimal density ρ arises.
As a result of solving the tasks, an explicit form of the Least squares Method of unknown parameters and variance was obtained. The
criterion of D – optimality is considered.
The D-optimal plans considered in this article are well known due to an important class of efficiency functions. To compare plans in terms of
D-optimality, the effectiveness of an arbitrary plan relative to the optimal plan is determined.
Thus, this article is devoted to the analysis of methods for constructing D-optimal experimental plans, where the basic object is a piecewise
constant function.

In the present work, we study the regularities of the influence of the type of distorting algorithm on the result of evaluating the image quality by the Full-Reference method in the presence of subjective quality assessments. As an example, we used the TID2013 database with 3000 images distorted by 24 types of algorithms and subjective mean square scores (MOS) quality ratings. An image quality score based on the Weibull distribution model and the usual PSNR similarity measure is applied. It is shown that the applied distorting algorithms are classified into two types – normal, leading to results consistent with the Human Visual System, and “anomalous”, the corresponding quality estimates of which are disordered or chaotic.

In this study, we solve linear singularly perturbed two-point boundary value problems applying the collocation method based on mapped Chebyshev polynomials and simply computed collocation points. The proposed approach generates well-conditioned collocation matrices and produces highly accurate results if a suitable mapping function is used. Numerical results for a problem with a steep interior layer are presented.

In this paper, the infinite numbers and functions are introduced and applied in modeling physical and mathematical systems and processes, where infinity somehow appears, i.e. infinite ladder electrical networks and systems instability analysis, series of numbers and limits calculation. Infinite numbers are defined as limits of complex functions that tend to infinity. Using these numbers it is possible to calculate the extended Laplace transform across the whole frequency spectrum as well as the extended bilateral Laplace transform, where the corresponding integral does not converge. Furthermore, the derivatives/integrals of the infinite number functions are determined. Using infinite numbers certain mathematical problems can be analysed and calculated, as well as problems in Physics and Engineering, where infinity appears, can be easily modeled and solved.

The paper stress-strain state of the homogeneous isotropic body bounded by coordinate lines of the parabolic coordinate system is studied, when on parabolic border normal or tangential stress is given. Analytical solution is obtained by the method of separation of variables. Using the MATLAB software, the numerical results are obtained of some specific problems and relevant graphs are presented.

A paper recently published explains the differential equations for deflection of beams under bending, including the deflection due to transversal force [1]. The present article contains derivation of the main equations, according to the mentioned approach, for deformations of a simply supported beam that is symmetrically loaded with two forces, also known as four-point bending. These deformations are rotation and deflection of the neutral line due to bending moment and transversal force. For thick beams, deflection due to the transverse force is more than 1% from deflection caused by the bending moment. Special attention was paid to the third-point loading test. The presented model is applicable for calculating deflection due to bending moment and transverse force for both thin and thick beams.

The technological progress in computer technologies gave rise to new possibilities and progresses for numerical and iterative methods. As being one of the computational studies, computational fluid dynamics is highly related to today’s advances. There are various types of methods and algorithms developed to model complex phenomena of fluid flow. In this study, we will introduce a new, still in development stage, CFD code with a pre-processor and a solver. Our research is focused on developing and studying a CFD code for mainly internal flows. Laminar and turbulent 2-dimensional flows can be analyzed using the software. The code is equipped with a graphical user interface (GUI) to make it simple to use. The GUI has the all-necessary components to define and analyze a fluid flow problem. We used an open-source post processor in order to visualize flow data and linked it to GUI, so the resulting software is a complete CFD package. The entire software is written using Python which has an easy code structure and rich code libraries. In order to decrease the time for convergence, code is modified with Numba and Cython libraries. To confirm accuracy of the solver, various basic test cases from the literature such as backward facing step flow, impinging jet flow, flow across a square cylinder, lid driven cavity flow are tested for both laminar and turbulent flows and the results was described in detail.

The Energy method is widely applied to determine the critical loads in elastic systems. A widely used variant of the method applies the Rayleigh-Ritz approach where the approximation of the buckling mode of the column is a function that satisfies the boundary conditions. A numerical example of a two-storey column is considered. An important aspect in the problem is the solution the complex integrals that emerge during the solution process. That problem could be overcome by the use of math software. The investigated column is hinged at its both ends and has an additional lateral support in the middle. It is loaded with a compressive distributed load alongside its length.

The paper is devoted to distance exams, as in that the focus is on mathematical disciplines. Some advantages are standing out, as well as significant disadvantages have been discussed. Current modern tools are considered, which are applied both in the preparation of students and in distance exams.

The definitions of Hermitian splines with nonlinear expressions in the links are given. Formulas for the parameters of Hermitian splines with exponential-power links with four parameters are derived. The definition of the balance approximation of functions by splines is given. The formula for the balance approximation error for Hermitian splines with four parameters are obtained. The error of approximation of functions by polynomial Hermitian splines is compared with the error of approximation of functions by Hermitian splines with exponential-power expressions in the links