In the systems of digital VLSI design (Very Large Integrated Circuits), the BDD (Binary Decision Diagram) is used for VLSI verification, as well as for technologically independent optimization as the first stage in the synthesis of logic circuits in various technological bases. The BDD is an acyclic graph defining a Boolean function or a system of Boolean functions. Each vertex of this graph corresponds to the complete or reduced Shannon expansion formula. When BDD representation for systems of Boolean functions is constructed, it is possible to perform additional logical optimization based on the proposed method of searching for algebraic representations of cofactors (subfunctions) of the same BDD level in the form of a disjunction, conjunction either exclusive-or of cofactors of the same level or lower levels of BDD. A directed BDD graph for a system of functions is constructed on the basis of Shannon expansion of all component functions of the system by the same permutation of variables. The method allows to reduce the number of literals by replacing the Shannon expansion formulas with simpler formulas that are disjunctions or conjunctions of cofactors, and to reduce the number of literals in specifying a system of Boolean functions. The number of literals in algebraic multilevel representations of systems of fully defined Boolean functions is the main optimization criterion in the synthesis of combinational circuits from librarian logic elements.

The orthogonal disjunctive normal forms (DNFs) of Boolean functions have wide applications in the logical design of discrete devices. The problem of DNF orthogonalization is to get for a given function such a DNF that any two its terms would be orthogonal, i. e. the conjunction of them would be equal identically to zero. An approach to solve the problem using the means of graph theory is suggested. The approach is proposed by representation of the function as perfect DNF. Obtaining all the intervals of the Boolean space where the given function has value 1 is supposed, and the intersection graph of those intervals is considered. Two methods to obtain a minimum orthogonal DNF are considered. One of them reduces the problem toward finding out the smallest dominating set in the graph by covering its vertices with their closed neighborhoods, the other - to obtain the maximum independent set by lexicographic enumeration. It is shown how the suggested approach can be extended on incompletely specified Boolean functions.

The study of measures of reliability of the hardware part of the office cluster was carried out on the example of the cluster SKIF-GEO-Office RB (further as “cluster”) developed within the framework of scientific and technical program "SKIF-NEDRA" (2015-2018, Program of the Union State of Russia and Belarus). The cluster components are located in a small rack on the basis of full Tower "Aerocool Expredator Black" type case.

The basic architectural principles implemented in the cluster, the composition, structural and functional scheme of the cluster are given. The methodological support for calculating the reliability of the cluster, based on previous studies of the authors, and its structural scheme of reliability is justified. The choice of the main measures of reliability of the cluster core and the set of computing facilities is justified and formulas of calculation of these measures are given. The analysis of the consequences of failures of component parts of the cluster is carried out.

A mathematical model of reliability (state graph) of the set of computing facilities of cluster is proposed, which allows to derive formulas for calculating the average value of the time-to-failure and time-to-interruption of cluster. The estimation of the reliability of the cluster as a whole, based on the calculation of measures of reliability on the reference data on the reliability of components as well as on the operation of supercomputers of the family SKIF. The measures of reliability of the cluster are calculated.

The article provides the analysis of the advantages and disadvantages of distance education, the specifics of educational process and the use of educational content for this form of education. The problems of assessing the quality of educational content for the distance educational process and its effectiveness are determined. The structure of the developed learning management system "Scorina" and the mechanisms for creating educational content, the learning process, attestation of students and the collection of progress statistics are described. A mathematical model of educational process of learning management system "Scorina", and approach to assessing the knowledge of students are presented.

The implementation of functional diagram and the algorithm of the module for the intelligent analysis of the quality of educational content of learning management system "Scorina" is proposed. The module is designed to identify the shortcomings in educational content and to improve the materials that could cause the difficulties for students when studying certain issues or topics of the disciplines are studied based on statistical data of student performance. The connection of specific parts of educational content with specific questions of control tests is implemented through special labels created during the development of the content of the studied disciplines. The mining algorithm allows to identify and exclude from the statistical sample the information of the students whose academic performance characteristics are not objectively related to the quality of educational content.

When measuring low-power optical signals in single-photon communication channels, the receiving modules must ensure the minimum probability of erroneous data recording. In this regard, it is advisable to use photon counters. They are highly sensitive, but are characterized by data recording errors. The aim of the work was to determine the effect of intensity of optical signal on the probability of erroneous data recording in a single-photon communication channel with a receiving module based on a photon counter.

Based on the results of experimental studies, the dependences of the probability of erroneous registration of binary symbols «0» P_{err 0} on the intensity of the optical signal J₀ , used to transmit these symbols for various supply voltages of the avalanche photodetector U, were obtained. It was found that with an increase of intensity of the optical signal, the dependences P_{err 0}(J₀) first decrease, reaching their lowest value, and then increase. The performed assessment showed that the lowest probability of erroneous registration the binary symbols «0» for the investigated communication channel is equal to 17,56 • 10^-2 and corresponds to J₀ = 52,41 • 10^-2 rel. units and U = 52,54 V.

An algorithm of automatic detection and tracking the moving objects for the use in equipment on board of unmanned aerial vehicles is considered. The developed algorithm is based on a tracking specially selected points for a certain period. Tracked points are selected from the areas on the current frame, where the pixel intensity differs from the intensities of the same pixels in previous frames, aligned with the current frame using projective transformation. If the displacement of the tracked points is not fixed on several adjacent frames, they are being deleted, and new points from the areas presumably belonging to moving objects in the current frame are added instead. On each frame the points similar by the location and shape of trajectories of movement are combined into groups that presumably correspond to moving objects. Objects are tracked by comparing the groups of moving points with the points of neighboring frames. Groups of moving points from neighboring frames are matched if they contain a large number of common tracked points. The algorithm allows simultaneous tracking of more than 20 objects in real time. The indication of objects as moving occurs only if during the time of its tracking it has shifted a considerable distance. The algorithm has a low percentage of false detections of moving objects, it detects well small objects and is capable reliably to accompany moving objects.