Mathematical Criteria Research Articles

Банківська екосистема: новий понятійний апарат банківської науки

The essence and structure of the banking system of Ukraine were largely regulated by law in the 1990s. Many changes have taken place since then. Today there are questions about changing the conceptual tools of banking science in terms of new understanding and formation of the banking ecosystem. The article is devoted to the study of basic concepts of banking science. The aims of the study are to formulate a new concept of “banking ecosystem” from the standpoint of open systems theory, to improve infrastructural components of the banking system of Ukraine. The research methodology is based on the use of open systems theory, a dialectical approach, systematic and comparative analysis, generalization of concepts, theory of open systems in the natural sciences. The information base is the legislation and regulations regulating banking activities in Ukraine, scientific works of domestic and foreign scientists, Internet resources. Results. The analysis reveals that the legislative wording has shortcomings in the definition of responsible regulators on the part of the state. The author's vision of concept “banking ecosystem” from the standpoint of open economic environment, the formation of incoming and outgoing financial flows and relationships is proposed. The list of participants of the modern infrastructure of the banking system of Ukraine is formulated. It is argued that the banking system of Ukraine should be studied using chaos theory and open systems theory, when the impact of the cooperative effect of joint behaviour of players cannot be fully predicted and there exists a synergistic effect. The mathematical approach and criteria for the openness index of the banking system and for the synergistic effect of the interaction between participants in the banking system is offered.

Evaluation of supply chain routes based on control engineering and DEA considering traffic congestion and uncertainty

In this paper, a mathematical model is proposed to assess the flow distribution based on the traffic criterion. In the mathematical model, the resilience and stability criteria are developed based on the concepts of control engineering and optimal control. The results show that the route stability is dependent on the traffic parameter. Then, considering several factors including stability, safety, resilience, risk-averse investment, the environment and noise, a decision-making model is presented to evaluate the performance of the flow distribution routes during crises. Goal programming and data-oriented modelling are used to solve the mathematical model. Data envelopment analysis (DEA) is used as the data-oriented model to evaluate the performance of scenarios under uncertainty. Furthermore, a decision-making methodology is proposed for ranking the routes. The results show that increasing beta can lead to lower desirability of a route. Finally, a managerial analysis is presented based on efficiency, risk, and PEST measures.

Gravitational clearing of natural satellite orbits

Abstract The distribution of diameters and orbital distances from the parent body of 156 named moons of the planets in the Solar System is not random. All 11 moons with diameters larger than $1\,000\,\mathrm{km}$ are positioned between $400\,000\,\mathrm{km}$ and 4 million km from the parent, whereas the far more numerous small moons are distributed on both sides of this central region and are largely absent from the region in between. This small-satellite ‘exclusion region’ is particularly evident for the gas giants since they have multiple satellites spanning a wide range of distances from the parent. Application of mathematical criteria analogous to those that have been used to help define the ‘gravitational clearing’ of planetary orbits around the Sun suggests that the absence of small satellites in this region around the planets may be a result (atleast in part) of gravitational clearing by the large moons present at these distances from the parent. The most significant exception to the observed diameter-distance distribution—Hyperion, on Saturn—is attributed to its 3:4 orbital resonance with Titan, while other obvious exceptions are the Trojan satellites of Saturn’s moons Tethys and Dione. The smallest satellite diameter that seems necessary for clearing of its ‘sphere of influence’ is around $400\,\mathrm{km}$ .

RESOLVABILITY OF ELECTORAL SYSTEMS AS A CRITERION OF THEIR FAIRNESS

Abstract: the legal content of the mathematical criterion of resolvability of electoral systems is considered. The connection of this rule with the principle of fair elections has been established. As it turned out, almost all electoral systems used in elections in Russia do not fully or partially comply with this provision. It is proposed to abandon the use of unscientific rules (the method of seniority, drawing lots) to determine the winner in elections in case of equality of votes among several candidates. It is proposed to use a number of preferential methods that meet this criterion and allow for fair elections in one round. Ultimately, taking into account the criterion of resolvability in the development of electoral systems will increase the legitimacy of elections as a scientific method of decision-making in social choice in a democratic society.

Assessing the quality of budget risk management of the subjects of the Russian Federation

Subject. The article explores indicators for budget risk management in the subjects of the Russian Federation. Objectives. The aim is to develop a system of indicators to assess the quality of budget risk management, to create a methodology for quality management assessment and test it on the case of 85 subjects, to build a rating for the quality of budget risk management in 2020. Methods. The study rests on creating a model for assessing the quality of budget risk management on the basis of 18 mathematical and statistical criteria and their testing on budget data from 85 federal subjects for 2020, and the deductive approach. Results. I formulated and justified indicators of analysis that demonstrate the level of readiness of the budget of Russian subjects to accept budget risks. The proposed indicators are logically structured into groups and detailed by data source for evaluation. The system of indicators is incorporated into the existing methodology created by the Ministry of Finance of the Russian Federation. The results of the analysis are presented as a rating of subjects of the Russian Federation based on the aggregate assessment of the quality of budget risk management. Conclusions. The offered methodology may serve as a supplement to the methodology for assessing the quality of regional finance management created by the Ministry of Finance of the Russian Federation. It may also help evaluate decisions on proactive risk management and mitigation of consequences of risk events in regional budget utilization.

Criterion for the occurrence of the macro destruction and formation of breaking during metal deformation

When generalizing the geometrically nonlinear law of Murnaghan elasticity to plasticity, a formally mathematical criterion was introduced for deformational macrofracture (macrocrack appearance) associated with an increase in elastic and plastic anisotropy as a failure cause. The use of the double potentiality of the governing equations in stresses and their velocities made it possible to obtain the reliable information on the structure of the deviatory section of the yield surface, the existence of which is a classical hypothesis in solid mechanics. The normal vector to the surface of the deviatory section is selected from two mutually orthogonal eigenvectors of the constructed operator. There are two families of regular concave surfaces, and a section surface is formed by joining the parts of two representatives of the families at singular points. To select normal vectors, the obtained ratio for them is used for isotropy. In connection with the considered problem of a double simple shift, it is established that multiple eigenvalues appear for the both normal vectors. To unambiguously determine the normal vector at a regular point, it is necessary to exclude the presence of multiple eigenvalues for the both normal vectors at the same time. At a singular point, the appearance of a multiple eigenvalue of one of the normal vectors is still unacceptable. These two conditions are necessary and sufficient to validate the governing equations of the generalized Murnaghan model. Otherwise, a macrocrack occurs. The theoretical construction is supported by the developed software complexes.

Smoothness metrics for reaching performance after stroke. Part 1: which one to choose?

BackgroundSmoothness is commonly used for measuring movement quality of the upper paretic limb during reaching tasks after stroke. Many different smoothness metrics have been used in stroke research, but a ‘valid’ metric has not been identified. A systematic review and subsequent rigorous analysis of smoothness metrics used in stroke research, in terms of their mathematical definitions and response to simulated perturbations, is needed to conclude whether they are valid for measuring smoothness. Our objective was to provide a recommendation for metrics that reflect smoothness after stroke based on: (1) a systematic review of smoothness metrics for reaching used in stroke research, (2) the mathematical description of the metrics, and (3) the response of metrics to simulated changes associated with smoothness deficits in the reaching profile.MethodsThe systematic review was performed by screening electronic databases using combined keyword groups Stroke, Reaching and Smoothness. Subsequently, each metric identified was assessed with mathematical criteria regarding smoothness: (a) being dimensionless, (b) being reproducible, (c) being based on rate of change of position, and (d) not being a linear transform of other smoothness metrics. The resulting metrics were tested for their response to simulated changes in reaching using models of velocity profiles with varying reaching distances and durations, harmonic disturbances, noise, and sub-movements. Two reaching tasks were simulated; reach-to-point and reach-to-grasp. The metrics that responded as expected in all simulation analyses were considered to be valid.ResultsThe systematic review identified 32 different smoothness metrics, 17 of which were excluded based on mathematical criteria, and 13 more as they did not respond as expected in all simulation analyses. Eventually, we found that, for reach-to-point and reach-to-grasp movements, only Spectral Arc Length (SPARC) was found to be a valid metric.ConclusionsBased on this systematic review and simulation analyses, we recommend the use of SPARC as a valid smoothness metric in both reach-to-point and reach-to-grasp tasks of the upper limb after stroke. However, further research is needed to understand the time course of smoothness measured with SPARC for the upper limb early post stroke, preferably in longitudinal studies.

Mathematical model of coal self-heating in a stack

The article presents a mathematical model of coal self-heating in the stack in which the heat exchange and gas exchange processes are described by a system of two non-linear differential equations of the second order with respect to the temperature t of coal self-heating and the volume fraction C of oxygen in the voids of the stack with boundary and initial conditions. The differential equations took into account that self-heating of coal in the stack and appearance of spontaneous combustion are observed in a relatively small layer adjacent to the surface of its contact with the air and called the zone of oxygen influence. In the mathematical model, the influence on the process of coal self-heating of parameter F- specific heat release power was taken into account, which in addition characterises the stability of coal during storage. When compiling the differential equations, such physical parameters as thermal conductivity, diffusion coefficient, specific heat capacity of coal in the stack, bulk density, thermal effect of oxidation, stack voidness, temperature coefficient of exponential growth of heat release power were also used. For numerical implementation of the mathematical model, dimensionless variables and criteria were introduced, which allowed us to apply the net method. Analysis of the obtained results allowed to get: change in the stack temperature profiles with time; change in the stack oxygen concentration profiles with time; influence on the stack temperature profile of the specific heat release power; influence on the stack temperature profile of the parameter characterizing exponential growth of heat release intensity with temperature increase. It has been determined that the dynamics of coal self-heating in the stack is mostly influenced by the Lykov criterion, proportional to the diffusion coefficient, and the Nusselt criterion related to the effective thermal conductivity and to the effective thermal diffusivity of coal. The obtained results suggest that self-heating in the stack is due on the one hand to intensive penetration of air oxygen and on the other hand to a weakened heat transfer. Self-heating and the transition of self-heating into ignition are associated with the occurrence of turbulent diffusion in the stack, arising from increased thermal blowing, whose impact can be enhanced by directing it perpendicular to the surface of the stack.

Tertiary lymphoid structures (TLS) identification and density assessment on H&E-stained digital slides of lung cancer.

Tertiary lymphoid structures (TLS) are ectopic aggregates of lymphoid cells in inflamed, infected, or tumoral tissues that are easily recognized on an H&E histology slide as discrete entities, distinct from lymphocytes. TLS are associated with improved cancer prognosis but there is no standardised method available to quantify their presence. Previous studies have used immunohistochemistry to determine the presence of specific cells as a marker of the TLS. This has now been proven to be an underestimate of the true number of TLS. Thus, we propose a methodology for the automated identification and quantification of TLS, based on H&E slides. We subsequently determined the mathematical criteria defining a TLS. TLS regions were identified through a deep convolutional neural network and segmentation of lymphocytes was performed through an ellipsoidal model. This methodology had a 92.87% specificity at 95% sensitivity, 88.79% specificity at 98% sensitivity and 84.32% specificity at 99% sensitivity level based on 144 TLS annotated H&E slides implying that the automated approach was able to reproduce the histopathologists’ assessment with great accuracy. We showed that the minimum number of lymphocytes within TLS is 45 and the minimum TLS area is 6,245μm2. Furthermore, we have shown that the density of the lymphocytes is more than 3 times those outside of the TLS. The mean density and standard deviation of lymphocytes within a TLS area are 0.0128/μm2 and 0.0026/μm2 respectively compared to 0.004/μm2 and 0.001/μm2 in non-TLS regions. The proposed methodology shows great potential for automated identification and quantification of the TLS density on digital H&E slides.

СВЯЗЬ АНТИБАКТЕРИАЛЬНОЙ АКТИВНОСТИ ВИДА РАСТЕНИЙ С ХИМИЧЕСКИМИ СОЕДИНЕНИЯМИ РАЗНЫХ ГРУПП: СТАТИСТИЧЕСКИЙ ПОДХОД

The article deals with the question of the statistical attraction of chemical compounds that have an antibacterial effect and belong to various groups (terpenoids, phenols, flavonoids, quinones, alkaloids, lactones, coumarins, iridoids, xanthones) to the same types of plants. To assess the occurrence of species containing an antibacterial compound of a certain group in a set of species containing an antibacterial compound of another group (in comparison with the species of the flora of the Russian Federation outside this set), the student’s mathematical and statistical criterion was used. It is shown that in 12 combinations of groups ("terpenoids-phenols", "terpenoids-flavonoids", "terpenoids-quinones", "terpenoids-coumarins", "phenols-flavonoids". "Phenols-quinones", "phenols-coumarins", "flavonoids-coumarins", "flavonoids-iridoids", "flavonoids-xanthones", "lactones-coumarins") the occurrence of species containing an antibacterial compound related to one element of the combination is significantly increased in the set of species containing an antibacterial compound related to another element. It is noted that these statistical relationships are created by species belonging to 19 families of the flora of the Russian Federation (Iridaceae, Ranunculaceae, Chenopodyaceae, Ericaceae, Pyrolaceae, Salicaceae, Brassicaceae, Moraceae, Crassulaceae, Rosaceae, Fabaceae, Apiaceae, Valerianaceae,Viburnaceae,Rubiacae,Gentianaceae, Scrophulariaceae, Lamiaceae, Asteracaeae).

Neural network surgery: Combining training with topology optimization

With ever increasing computational capacities, neural networks become more and more proficient at solving complex tasks. However, picking a sufficiently good network topology usually relies on expert human knowledge. Neural architecture search aims to reduce the extent of expertise that is needed. Modern architecture search techniques often rely on immense computational power, or apply trained meta-controllers for decision making. We develop a framework for a genetic algorithm that is both computationally cheap and makes decisions based on mathematical criteria rather than trained parameters. It is a hybrid approach that fuses training and topology optimization together into one process. Structural modifications that are performed include adding or removing layers of neurons, with some re-training applied to make up for any incurred change in input–output behaviour. Our ansatz is tested on several benchmark datasets with limited computational overhead compared to training only the baseline. This algorithm can achieve a significant increase in accuracy (as compared to a fully trained baseline), rescue insufficient topologies that in their current state are only able to learn to a limited extent, and dynamically reduce network size without loss in achieved accuracy. On standard ML datasets, accuracy improvements compared to baseline performance can range from 20% for well performing starting topologies to more than 40% in case of insufficient baselines, or reduce network size by almost 15%.

Machine learning assisted empirical formula augmentation

Tuning the martensite transformation temperature through composition design has become an important way to broaden the applicable temperature range of shape memory alloys (SMAs). The empirical formula based on traditional statistics is a key reference for composition design. Due to the lack of experimental data, a large deviation may exist among the prediction results from the empirical formulas obtained by different data sources. In present work, we proposed an augmentation strategy of empirical formula based on a machine learning method to build the relationship between martensite transformation start temperature (Ms) and compositions in Cu-Al-based SMA system. A series of ML models were established by physical and chemical features and a Gaussian radial basis kernel function support vector machine (SVR.rbf) model was screened out based on mathematical and domain knowledge criteria. An augmented empirical formula of Ms as the function of compositions was fitted based on the abundant augmented dataset combined experimental data with predicted data by the SVR.rbf model. Compared with previous empirical formula fitted by small experimental dataset, the accuracy and robustness of the augmented empirical formula was significantly improved without additional experimental cost. This strategy offers a recipe to build empirical formula based on a small experimental dataset.

Gaia Early Data Release 3

Context. The unprecedented astrometric precision of the Gaia mission relies on accurate estimates of the locations of sources in the Gaia data stream. This is ultimately performed by point spread function (PSF) fitting, which in turn requires an accurate reconstruction of the PSF, including calibrations of all the major dependences. These include a strong colour dependence due to Gaia’s broad G band and a strong time dependence due to the evolving contamination levels and instrument focus. Accurate PSF reconstruction is also important for photometry. Gaia Early Data Release 3 (EDR3) will, for the first time, use a PSF calibration that models several of the strongest dependences, leading to signficantly reduced systematic errors. Aims. We describe the PSF model and calibration pipeline implemented for Gaia EDR3, including an analysis of the calibration results over the 34 months of data. We include a discussion of the limitations of the current pipeline and directions for future releases. This will be of use both to users of Gaia data and as a reference for other precision astrometry missions. Methods. We develop models of the 1D line spread function (LSF) and 2D PSF profiles based on a linear combination of basis components. These are designed for flexibility and performance, as well as to meet several mathematical criteria such as normalisation. We fit the models to selected primary sources in independent time ranges, using simple parameterisations for the colour and other dependences. Variation in time is smoothed by merging the independent calibrations in a square root information filter, with resets at certain mission events that induce a discontinuous change in the PSF. Results. The PSF calibration shows strong time and colour dependences that accurately reproduce the varying state of the Gaia astrometric instrument. Analysis of the residuals reveals both the performance and the limitations of the current models and calibration pipeline, and indicates the directions for future development. Conclusions. The PSF modelling and calibration carried out for Gaia EDR3 represents a major step forwards in the data processing and will lead to reduced systematic errors in the core mission data products. Further significant improvements are expected in the future data releases.

Analysis of Student's Mathematic Communication Ability in Solving Problems of Pattern Procedures in Pythagoras Theorem

This study aims at knowing and describing students' mathematical communication skills in problem-solving on Pythagorean theorem material based on Polya's procedure. The Polya's problem-solving phase consists of the stage of understanding problems, planning problem solving, implementing problem-solving plans, and re-examining. The subject of this study were 6 students from eighth-grade of Junior High School which was selected based on good, sufficient, and poor mathematical communication criteria. Moreover, this study employed a descriptive qualitative method where the data was collected by using a test and interview. The results of the study revealed that the students with good mathematical communication capabilities criteria were able to solve the problems based on Polya's procedure. Meanwhile, students with sufficient criteria of mathematical communication capability unable to solve the problem based on Polya's procedure at the stage of making plans by altering mathematical information in the form of picture, students did not realize the need for design actions first, students just go ahead and didn 't see the results of completion that have been made. Additionally, students with poor mathematical communication capabilities were unable to solve the problems based on Polya's procedure, students were not meet at the stage of making plans by changing mathematical information in the form of picture, making plans by using and selecting mathematical information or images to solve the problems, and the students did not look back at the completed results.

Physicomechanical and design characteristics of surface high-entropy alloys

We investigated the phase composition of high-entropy shape memory alloys CoCuTiZrHf, NiCoTiZrHf, NiCuTiZrHf, ZrCuNiCoTi, TiNiZrHfCoCu after HVOF, thermomechanical treatment and thermal cycling. We have also developed criteria for calculating the energy capacity, enthalpy, entropy, theoretical strength of high-entropy alloys, taking into account the chemical composition and structural-phase state according to the results of X-ray phase analysis. We proposed a criterion for assessing the damageability of high-entropy alloys. The article describes calculations of energy consumption, entropy, enthalpy, theoretical strength of high-entropy alloys CoCuTiZrHf, NiCoTiZrHf, NiCuTiZrHf, ZrCuNiCoTi, TiNiZrHfCoCu both after HVOF and after thermomechanical treatment. After full-scale mechanical tests of surface-modified layers of high-entropy materials for bending with rotation, we obtained experimental values of cyclic durability and plotted fatigue curves. The calculated assessment of the cyclic durability was carried out according to the developed mathematical criteria. We calculated the damageability of high-entropy alloys and constructed the graphical curves of the dependence of the damageability on the cyclic life.

Toward Real-Time Volcano Seismic Events’ Classification: A New Approach Using Mathematical Morphology and Similarity Criteria

This work proposes a new approach based on a suit combination of mathematical morphology and similarity criteria techniques to classify long-period and volcano-tectonic seismic events of the Cotopaxi volcano. The proposed method explores the seismic signal domain to compute a new feature space based on the edges map of the seismic events pattern represented in the gray-level spectrogram images, which is used to feed a set of similarity-based classifiers. The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L_{2}$ </tex-math></inline-formula> -norm was selected as the best metric to be implemented by the proposed method. In terms of classification performance, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L_{2}$ </tex-math></inline-formula> -norm was statistically superior in the D1 data set (seismic events with overlapped signals of nonvolcanic origin) and similar in the D2 data set (events without overlapped signals) with respect to the other metrics, reaching accuracy mean scores of 93.34% and 96.88%, respectively. These results demonstrated that the computed edges map feature space is a better environment for separating both seismic events compared with the original gray-level space. Regarding the execution time, total time (TT) and time per-sample (TS) did not exceed 0.388 and 0.002 s during the training stage, respectively. During the testing stage, a TS of no more than 0.012 s was achieved. Finally, its execution time is faster, and the algorithm complexity is lower compared with the state-of-the-art methods, which makes it a practical and beneficial scheme to implement for real-time seismic events’ classification.

Using linkage logic theory to control dynamics of a gene regulatory network of a chordate embryo

Linkage logic theory provides a mathematical criterion to control network dynamics by manipulating activities of a subset of network nodes, which are collectively called a feedback vertex set (FVS). Because many biological functions emerge from dynamics of biological networks, this theory provides a promising tool for controlling biological functions. By manipulating the activity of FVS molecules identified in a gene regulatory network (GRN) for fate specification of seven tissues in ascidian embryos, we previously succeeded in reproducing six of the seven cell types. Simultaneously, we discovered that the experimentally reconstituted GRN lacked information sufficient to reproduce muscle cells. Here, we utilized linkage logic theory as a tool to find missing edges in the GRN. Then, we identified a FVS from an updated version of the GRN and confirmed that manipulating the activity of this FVS was sufficient to induce all seven cell types, even in a multi-cellular environment. Thus, linkage logic theory provides tools to find missing edges in experimentally reconstituted networks, to determine whether reconstituted networks contain sufficient information to fulfil expected functions, and to reprogram cell fate.

ЗАСТОСУВАННЯ АВТОРЕГРЕСІЙНИХ МОДЕЛЕЙ ДЛЯ ПРОГНОЗУВАННЯ РІВНЯ БЕЗРОБІТТЯ В УКРАЇНІ

For any country one of the most important social and economic problems is an unemployment. In today's conditions this problem is one of the main economic problems in Ukraine and poses a real threat to the state and social well-being. The main problem of unemployment is caused by the fact that the unemployed face the loss of qualification, social status and the lowering of the standard of living. Due to the decline in income of the population, through the loss of work, the demand for goods and services on the domestic market is decreasing, tax revenues to the state budget are decreasing, social pressure and criminality are growing. The level of unemployment is an indicator of social processes of the state, a characteristic of stability and confidence in the future of the country. The creation of an adequate predict of the level of unemployment taking into account the accidental nature of the problem provides the choice of management strategy in the employment sphere, taking into account the peculiarities of the economic situation, priorities of social development, makes it possible to assess its current state, trends and changes, as well as to take appropriate management decisions in the employment sphere. To combat this economic phenomenon, state support for business, such as subsidies for retaining employees at their workplaces, can be implemented. In general, at the level of the state it is necessary to develop new measures of a strategic nature in order not to fight against unemployment, but to prevent it. The work examines the main statistical methods of predict, which are based on the data of one time series. The peculiarities of using trend models for predicts are examined. It is noted that in the current conditions of computer software usage, the choice of trend formations for predicting is essentially simplified: different trend forms can be produced for the same time series and the one which best describes the output series by mathematical criteria can be selected. This work is devoted to the study of the problem of predict the level of unemployment in Ukraine. It is suggested to make predicts on the basis of autoregressive models of time series. In the work the models of autoregression, autoregression with a coveted average and autoregression with a trend are examined in detail. As a result, the information and analytical system for modeling and forecasting of financial processes was created. The method of prediction on the basis of auto-regressive time series model that we reviewed consists in creating a model for predicting the future events (predicting the level of unemployment) ґrunning on the known events of the past, and predicting the future data before they will be measured. The found average absolute volumetric forecast error (MAPE) and Tale coefficient allowed us to conclude that the proposed model is appropriate for making short-term forecasts of unemployment.

ПРЕДЛОЖЕНИЕ ПО РЕФОРМИРОВАНИЮ ИЗБИРАТЕЛЬНОЙ СИСТЕМЫ РОССИЙСКОЙ ФЕДЕРАЦИИ

Introduction: the article analyzes the extent to which electoral systems meet different (mathematical and judicial) evaluation criteria. Purpose: to identify the fairest algorithm for mandate distribution when electing deputies and officials. Methods: apart from general scientific methods, we extensively used special scientific methods including comparative legal and systematic ones. Results: we have defined the range of mathematical criteria (the absolute majority rule, the Condorcet paradox, the principle of proportional representation of the parties, etc.) and legal criteria (the provisions of the principles of equal, free, and fair election). Preferential voting systems have been found to meet mathematical and judicial criteria most of all. A number of preferential voting systems meet the Condorcet paradox and ensure proportional representation of the parties. During such an election, free and equal participation of independent and party candidates is ensured. Many other systems of majoritarian, proportional, and mixed types do not meet the abovementioned criteria. Unfortunately, in particular cases the Condorcet paradox does not allow determining the candidate to be elected. Consequently, we suggest combining the algorithms developed by Tideman, Kemeny, Young, and Schulze in order to find the ‘strongest path’, or the most reliable way of determining the candidates’ positions in the election. Conclusions: the preferential voting system developed by us is universal and can be used while electing both officers (the president, the governor) and deputies. If it was implemented, electoral legislation would be standardized, as well as the operation of election committees. Such a unification would allow improving the level of citizens' electoral rights protection.

Using the Mathematical Criterion of Gas Pipeline Failure to Determine the Remaining Operation Life

During the years of operation, underground pipelines deteriorate because of aging, wear, damage, and destruction. A comprehensive inspection of the main pipeline revealed various defects on a certain pipeline route section. To identify the remaining operation life of the pipeline, a number of hydraulic tensile strength tests were performed using the cyclic loading method. The authors propose to calculate the minimum allowable wall thickness capable of withstanding the maximum internal pressure, as an alternative to cyclic hydraulic tests. The strength calculations were performed using the well-known mathematical strength criteria proposed by K.V. Zakharov and I.I. Goldenblatt and V.A. Kopnov.