Review: The subject of the research is the mathematical model in the form of recurrent integral relation system that describes the distribution of unit intervals where at least one random stream event with the arbitrary distribution function has happened. The author of the article examines numerous aspects of the numerical implementation of this system such as the Laplace transformation access method, numerical integration near discontinuity points, stability of calculations, validity check results, and particularities of real number machine arithmetic. Trub pays special attention to the connection between calculation data and semantics of the applied problem which solution is represented by these data. The methodology of the research is based on the probability theory (distribution types and qualities), numerical mathematics methods (numerical integration, interpolation, Laplace transformation), software implementation of the mathematical model and computing experiment conduction. The main conclusions of this research is the validity and numerical implementability of the mathematical model created by the author of the article as well as substantiation of the numerical solution for arbitrary distribution of the random stream events distribution. The novelty of the research is caused by the fact that the author develops a numerical solution of the bitmap-indices distribution for Weibull distribution, gamma distribution, logarithmically normal distribution, etc., and analyzes dependencies of different kinds such as the indices density function and average number of indices for the specified interval length. 

Keywords: binomial distribution, Simpson method, stability of calculations, discontinuity point, quadrature expressions, numerical integration, numerical methods, Laplace transformation, probability density function, improper integral

Review: The subject of the research is the graphical models for solving the problem of minimizing the completion time of competing jobs in batch planning. Great interest in such problems is explained by numerous applications that arise in the theory of graphs, in the planning of production and transport routes, in the calculation of distribution, etc. In general, there is no effective algorithm for solving these problems. Particular attention is paid to the use of the coloring of the conflict graph, which makes it possible to build estimates of the minimum execution time for all jobs. The main research methods were methods of graph theory and computational experiment. Particular attention was paid to the graph coloring. Based on the coloring of the conflict graph, two-sided estimates of the minimum execution time of competing tasks for batch planning are constructed. All the estimates found are attainable. They are linear combinations of time slots for executing work packets. The coefficients of the linear combination are expressed through the different characteristics of the conflict graph, its subgraphs (for example, the chromatic number, the density of the graph).

Keywords: completion time, full subgraph, chromatic number of graph, minimal coloring, conflict graph, graph theoretic model, batch planning, competing jobs, minimum time, work schedule

Review: The subject of the present research is the theoretical research of Hilbert curve properties that allow to arrange an efficient access to image pixels during slider filter image processing by using Hilbert curve as a image point traversal order. The effect is achieved by means of reducing the number of repeated memory operations during adjacent image pixels and equally distributing restored pixels at horizontal and vertical steps between image pixels. The research method is the theoretical analysis. The validity of theoretical results is proved by previous experimental researches the authors of the present article refer to. The novelty of the research is caused by the fact that the authors give a theoretical substantiation of previous experimental researches. The authors give their evaluation of a number of memory operations that can be avoided during image point traverse along Hilbert curve depending on the size of a sliding filter. The authors also carry out a comparison with other continuous traversal orders (such as horizontal steps between pixels from the point of view of counting distribution of readings during filtration. 

Keywords: graphical computing, continuous traversal order, memory access pattern, memory bandwidth, high performance, theoretical analysis, sliding image filter, image processing, Hilbet curve, image memory layout

Review: The subject of the research is the morphometric properties of Al2O3 (aluminum oxide) micro-disperse system. The authors pay special attention to the fractional composition and particle sphericity factor for defined particle groups constituting the disperse system. The authors state that image analysis allows to use the sphericity factor for evaluating elements of the particle surface combined with the interpretation of a real geometrical shape of individual particles. This allows to better define the dispersion factor as an important property of a disperse system because this parameter may be a measure of substance activity in a disperse system. Digital images for analysis ere obtained using a scanning electronic microscope with a high resolution S3400N HITACHI (made in Japan). Images were analyzed using a universal software ImageJ-FiJi. The object of the research was the powder Al2O3 of Sigms-Aldrich brand (article 265497, α-phase, particle average size is 10 micron). The powder was made at mills with spherical grinding bodies. The main conclusion of the research is that image analysis is a powerful tool that can be used to define physical properties of disperse systems. For the first time in the academic literature the authors focus on analyzing angles created by particle brim peaks. By calculating the dispersion factor, the authors have demonstrated that data obtained during particle image analysis allows to increase efficiency in defining the real factor of particle dispersity with the average validity of 5%. 

Keywords: Aluminium oxide, disperse systems, surface texture, granulometry, sphericity factor, morphology, image analysis, dispersion factor, microscopy, metrology

Review: Improvement of the quality of mechanical parts and products is one of the high-priority tasks. This task is achieved by increasing accuracy, automation of control and implementation of a single concept of part accuracy assurance at all stages of life cycle. The tool to solve this task is a universal algorithm created by the authors to construct and visualise a geometrical model of a real part. The scientific novelty of the resaerch is caused by the fact that the authors take into account information capacity of elements, coordinate planes and axes and materialized sets of parts bases. The practical important of the algorithm is proved by the results of using it as a method of regulating dimensional and geometrical accuracy of a mechanical part. In the process of building a geometrical model, the authors have used the modular principle where a possible number of freedom degrees deprived by each element of the module is known beforehand. Geometrical model's properties, notes and recommendations are input in the form of a matrix into the data base. The geometrical model starts with the image of a generalized coordinate frame OXYZ on a one-three view contour drawing of a mechanical part. The number and image of coordinates and initial calibration accuracy of each main base are determined by the number and image of movements that haven't been spent by the base for creating a generalized coordinate system. Angular coordinate sizes and graduation errors are calculated starting from coordinate axes with the 4 and 2 information capacity and marked on projections perpendicular to turning point axes. Linear coodinate sizes and their graduation errors are calculated along relevant coordinate axes together with the reference coordinate values including zero values. The surface of elements which bases materialize coordinate systems and executive surfaces have an inclined form that is visualized on the model as the main wavy lines tangent to the elements. The main feature of this approach which also reflects the novelty thereof is that it allows to objectively evaluate all necessay geometrical characteristics of a mechanical part based on the principle of unity of bases. The algorithm offered by the authors may form the basis for developing the automated quality management system in the processes of mechanical part design and production. 

Keywords: accuracy, geometrical properties, coordinate plane, coordinate axes, automation, control, visualisation, set of bases, algorithm, geometrical model