Normal Distribution Law Research Articles

Geochemical multifractal modeling of soil and stream sediment data applied to gold prospectivity mapping of the Pitangui Greenstone Belt, northwest of Quadrilátero Ferrífero, Brazil

Geochemical mapping has become essential to prospective assessments in both brownfield and greenfield exploration targeting. During the last two decades, several techniques have been applied to the identification of mineralization's geochemical signature. Among these techniques, Hierarchical Clustering (HC) and Principal Component Analysis (PCA) have been widely used. Other techniques, like Multifractal Spectrum (MS) and Local Singularity Mapping (LSM), are efficient in the generation of models suitable to determine the type of database statistical distribution and mapping zones of geochemical enrichment and depletion. We combined these techniques for soil and stream sediment geochemistry and generated two prospective geochemical models for the gold deposits from the Pitangui Greenstone Belt (PGB), located at northwest of Quadrilátero Ferrífero – Brazil. The results of PCA and HC showed a geochemical association of As-Bi-Mo-Sb with gold mineralization in the PGB. According with descriptive statistical analysis, these elements do not display normal or log-normal statistical distribution, which allowed the application of multifractal models on the geochemical mapping. MS technique revealed that all elements associated with gold have a multifractal distribution. In addition, geochemistry data shows that high contents of elements measured in the stream sediment analysis displays greater dispersion when compared to elements of soil geochemistry data, due to their mobility in environmental conditions. We used LSM technique to map geochemical enrichment of the elements associated with gold. The maps were evaluated using the Ground Truth Analysis (GTA) and the results showed a good performance in the identification of prospective zones. The anomalies identified in both geochemical maps showed a strong correlation between geochemical enrichment zones and geological factors that favors gold mineralization. These results bring a new perspective on the distribution of geochemical anomalies using, for the first time, the multifractal geochemical modeling on the PGB. The geochemical maps point to new targets for mineral exploration and contribute significantly for the prospective models of the region.

ОЦЕНКА ПОВРЕЖДАЕМОСТИ АВТОСЦЕПНОГО УСТРОЙСТВА ГРУЗОВЫХ ВАГОНОВ В ЭКСПЛУАТАЦИИ

The paper objective is to estimate the damage to the automatic coupling device of freight cars in operation. A statistical analysis of auto-coupling breaks on roads and a statistical estimation of nondestructive testing of auto-coupling device parts are carried out. During the analysis, it is found that 68% of breaks occur due to the presence of cracks in the body of the automatic coupler, 60% of breaks of the automatic couplers occur in areas where the track has a slope, including the mountain pass areas and 77% of breaks occur in the train traction mode. The scientific novelty is in obtaining a logarithmically normal law of the crack length distribution in the coupling body and traction clamp during the operation of the car in the overhaul period. The most likely crack formation in the body of the automatic coupling during the operation of the car between scheduled repairs is 41 mm long, in the traction clamp it is 63 mm. The practical significance of the work is in the fact that the obtained results can be used in making recommendations and organization of technical measures to reduce the cases of auto-coupling breaks and increase the quality of repair and maintenance of the auto-coupling device.

Improvement of the methodology for determining unproductive loss of time in the disbandment system

The purpose of the work is to improve the methodology for determining inefficient time losses in the distribution system of the sorting station. Systematic analysis, analytical, graphic modeling, the law of normal distribution and methods for comparing the performance of wagon standing times were used. Unproductive time losses in the disbandment system were determined by the elements of the time spent by wagons at the sorting station. In the sorting station distribution system, the methodology for determining the time of stay and inefficient time losses of transit recyclable cars has been improved using the law of normal distribution. The analysis of unproductive time losses in the receiving and rolling stock of the sorting station by the actual average daily and average monthly values of the elements of unproductive time losses allows not only to purposefully identify methods to reduce the time spent by wagons, but also to carry out an accurate assessment of the work of the station staff, as well as to adjust the norm of the time spent by wagons.

Production estimate of technological operations duration in railway transport

Purpose: To stabilize the departure of trains on schedule, to do this, to justify the size of the factor of unevenness taken into account when normalizing the duration of the main technological operations with the compositions of container block trains for departure at the junction stations. Methods: From the existing automated systems, statistical data were collected and processed using MS Excel software, and the mathematical expectation, standard deviation, and dispersion of the duration of operations with the block train at the departure station were determined. The hypothesis about the normal law of distribution of the random variable of the duration of each of the main operations at the station was tested by the “chi-square” goodness-of-fit test. Results: The distribution of the random variable of the duration of operations at the station is described using the normal distribution law; a technological schedule for processing the composition of the container block train on the tracks of the railway junction station was built. Practical significance: Practical recommendations will reduce the technological norms for the processing time of a container block train on the tracks of a railway junction station. As a result, the processing time for block trains on departure on the station tracks will be reduced from 154 minutes to 132 minutes, which will save time by 22 minutes.

ОСОБЛИВОСТІ ЗВ’ЯЗКУ КОНЦЕНТРАЦІЙ ГЕРМАНІЮ ІЗ ВМІСТОМ ТОКСИЧНИХ ЕЛЕМЕНТІВ Й СІРКИ ЗАГАЛЬНОЇ У ВУГІЛЬНОМУ ПЛАСТІ С5В ШАХТИ «ТЕРНІВСЬКА»

Purpose. The purpose of the work is to establish and analyze the relationship between germanium concentrations, toxic elements and the content of total sulfur in the coal seam c5в of the «Ternivska» mine field of the Pavlohrad-Petropavlivka area of Donbas. Methodology. The factual basis of the work was the results of germanium, beryllium, fluorine, mercury, arsenic and total sulfur analyzes in the coal seam c5в. The content of Ge, Be, and F was determined by quantitative emission spectral analysis, Hg and As by atomic absorption analysis. Using the STATISTICA 13.3 and IBM SPSS Statistics 22 programs, the values of the main descriptive statistical indicators were calculated. Correlation and regression analysis was carried out using the methods implemented in Micromine program and their analysis was carried out in geological concepts. Results. It is substantiated that the minimum contents of at least As, Hg, Ge and total sulfur in the coal seam c5в of the "Ternivska" mine were jointly accumulated at the syngenetic stage of its formation. The simultaneous increase in the content of Ge and Be in the area of their low and abnormally low concentrations is caused by the joint accumulation of these elements at the syngenetic stage of coal seam formation and their increase in the area of maximum contents is caused by the manifestation of the empirical regularity of the «Zilbermints law». It is proved that the distribution of the values of Ge content, toxic elements and total sulfur in the coal seam c5в of the «Ternivska» mine differ from the Gaussian-Laplace and lognormal distributions, in all cases the polymodality of the distribution is fixed and the kernel of the distribution density is shifted to the left. Scientific novelty of the results of the conducted research consists in establishing the non-compliance of the samples of all considered elements with the normal or lognormal distribution law, while in all cases the polymodality of the distribution of indicators is recorded, which is confirmed by analytical calculations of the correspondence of the empirical distributions of the studied parameters of the Gaussian distribution using the Kolmogorov-Smirnov, Shapiro-Uilk criteria, Lilliefors and Pearson's chi-squared agreement. It is proved that the forms of finding of the investigated elements in coal with their concentrations around Clark and in coal with increased contents differ significantly. The influence of syngenetic and epigenetic processes on the nature of the relationship between Ge concentrations, toxic elements and the content of total sulfur in the coal seam c5в of the «Ternivska» mine was revealed. Practical significance of the results of the work consists in establishing the presence of a very low correlation between the concentrations of Ge and toxic elements and total sulfur, therefore, the extraction of germanium from the coal seam will not be accompanied by their significant accumulation during technological processes. It has been proven that for a more realistic assessment of the central tendency of the content of Ge and toxic elements and total sulfur, instead of the values of the arithmetic mean, it is necessary to use the median values. Key words: germanium, beryllium, mercury, arsenic, total sulfur, coal seam, frequency histograms, regression analysis, correlation analysis, polymodality of distribution.

Features of spectral vibration diagnostics of traction power transformers in high-speed motion

This paper uses spectral analysis methods to develop a mathematical model and algorithms for evaluating the diagnosed signals corresponding to certain defects of windings and magnetic circuits of traction power transformers. A distinctive feature of the study is at certain frequencies of spectral dependences, taking into account the effects associated with defect formation that have a different cause of their manifestation, for example, wide-field fluctuations and changes in the shape and width of spectral peaks. The presence of random fluctuations caused the application of the normal Gaussian distribution law of the measured values of the spectra. To solve the problem of identifying the diagnosed spectra corresponding to certain defects of the power transformer, algorithms for identifying the vector of diagnostic features based on the method of statistical recognition theory for large amounts of information are proposed.

To the study of 10D100 diesel engines of operating diesel locomotives by the fuel supply advance angle

The results of the justification for the distribution of diesel engines 10D100 are presented, taking into account the value of the actual advance angle of fuel supply at the nominal operating mode of the diesel generator set of operating diesel locomotives. A methodology for studying the dynamics of changes in the empirical and theoretical distribution of quantities is proposed, lean on the basic principles of probability theory and mathematical statistics, taking into account the criteria of agreement of Pearson and A. N. Kolmogorov's. It has been proven that the distribution of the studied diesel locomotive diesel engines according to the actual fuel supply advance angle corresponds to the law of normal distribution. It is recommended to continue these studies for partial load conditions, covering a larger number of operating diesel locomotives on railway sections of different complexity.

Методика оптимизации вскрышных работ с использованием системы автоматизированного позиционирования бульдозера

The article discusses the development of methods for assessing mineral losses in the development of alluvial gold deposits as well as the optimization and reduction of their quantity. It is proposed to improve the quality of loss accounting by introducing a system of automated determination of the position of equipment during the development of the deposit. Statistical processing of the experimental data of the bulldozer positioning accuracy with the use of GNSS systems was carried out in the course of the research. Spatial positioning accuracy of the bulldozer blade was determined and was tested the hypothesis of subjecting the positioning errors to the normal law of distribution. Based on the results obtained a model of potential losses depending on the bulldozer's levelling and the thickness of the «protective jacket» - the unexcavated waste rock left over the expected boundary of the productive layer's roof was derived. Based on the generated model of losses, dependencies were derived to optimize the value of the "protective jacket" from the position of minimizing losses relative to the amount of potential gold losses and additional costs of transportation and flushing of waste rock.

Методика оценки однородности створов мониторинга рек по оптической характеристике водной поверхности — индексу мутности

Preservation and improvement of aquatic recourses ecological state is an important state goal. The goal demands monitoring of river discharge and hydrochemical parameters of the river. Only the data allow evaluating of ecologically reasonable anthropogenic impact on an aquatic ecosystem. It is evident that homogeneity of river water at a monitoring river station is the requirement of the monitoring data representativity. Of course, regular control of the homogeneity by each of the identifiable hydrochemical parameters is extremely time-consuming task. The paper presents methodology for check of homogeneity of a monitoring river station by means of Earth remote sensing data analysis. The methodology includes calculation of the Normalized Difference Turbidity Index (NDTI) on the base of Sentinel-2 satellite images in geographical information systems ArcGIS and QGIS within buffer zone of a river section, calculation of distances between cell centers of the NDTI matrix and the initial vertex of the river section (the riverside), the results export in Microsoft Excel files, statistical analysis of the NDTI values homogeneity along the river section. The homogeneity analysis is made by means of the Student and Fisher tests when the NDTI distributions belong the normal law of distribution and by the tests of Wilcoxon-Mann-Whitney and Siegel-Tukey otherwise. The data processing is performed by means a number of workflow models and Python-language programs, a Lazarus-language program, and a number of Mathcad-language programs. The tools are elaborated by Victor Tretyakov. An example of the methodology application is provided in the paper. There is considered evaluation of NDTI homogeneity at the Neva River section located downstream from the Slavyanka River outlet into the Neva River. Of course, the NDTI homogeneity cannot ensure homogeneity of the river section for all the hydrochemical parameters. However, absence of the NDTI homogeneity may serve as a mark of the stream flow non-homogeneity at the river section.

Application of the method of mathematical statistics to study the wear of parts of modern machines

This article describes the method of mathematical statistics for studying the wear of parts of modern machines. It is examined how the service life of the parts is established. It was also assumed that the statistical data on the size of worn parts obey the law of normal distribution. The figure presented in the paper illustrates the scheme for determining the inter-repair service life of the part for this case according to statistical data. An example is considered that allows to establish the norm of the maintenance period of the support roller of a modern tractor VT-150. This article is recommended for agricultural specialists, researchers, teachers, postgraduates, undergraduates and students of agricultural universities in the field of training “Agroengineering”.

STUDY OF DURABILITY IN THE CONNECTION “CONE-TUNGSTEN CARBIDE INSERT” FOR TENSION DEPENDENCE

During and after the technological operation of pressing these factors significantly determine the residual stress states of tungsten carbide inserts, as well as in the areas near the holes of the cones’ crowns of the cone. The stressed state of the inserts and the magnitude of the stress concentration determine their fatigue strength of them and the durability of the inserted carbide rock-blasting equipment of the cones in the area where the inserts the cone. Dimensional analysis was done for the inserts and holes in the cones of bit and it was established that size errors of component links of dimensional chains are distributed according to laws that are similar to the normal law of distribution of random variables. The test was carried out according to the criteria of Pearson and Kolmogorov. The influence of various random and systematic factors results to the dispersion of dimensions on the dimensional processing of conjugated surfaces. The amount of tension is a function of the dimensions for mating surfaces during assembly operations. Dispersion of dimensions for the component links of the dimensional chain of a press connection causes the formation of a practical field of dispersion of the closing link. Accuracy control for the closing link in the dimensional chain is impossible if there are no direct measurements of the constituent links. A mathematical model of the process of assembling "cone-tungsten carbide insert" joints was created. For this, the methods of mathematical statistics were applied in the study. It was established that the tension values have a greater influence on the force of pressing inserts into the holes of drill steels 14 NiCrMo1, when its hardness value is HRC 59-60 compared to HRC 48-50. This dependence has a linear character within the limits of the studied tension values. The established relationships make it possible to reasonably and most accurately form selective groups of inserts and mark holes for them. This method significantly reduces losses in the production of drilling tools.

Technological proposal for the attention of the risk in the management of the work of a railway station with a port

The purpose of the article is to definean improved technology for managing the operation of a railway station with a port when carrying out container transportationwith the attention of the financial risk.To achieve this goal, the authors used methods of statistical data analysis of dynamically changing parameters, methods of mathematical statistics, probability theory, methods of optimization and combinatorial analysis.The article formalizes the technology of managing the work of a railway station with a port in the form of a mathematical model.The objective function of the model is based on the sum of operating costs and takes into account the risk of financial losses of the port station due to the occurrence of risky events that threaten the stable and safe operation of railway transport in the conditions of martial law. For representation of the level of financial risk, the use of the density function of the normal distribution law is proposed. The application of the proposed mathematical model will allow to determine the optimal sizeof a batch of large-tonnage containers intended for loading onto a shipwith minimal operating costs.

A model of random pulsations of radiant heat flow from a flammable liquid fire

The object of the study is the process of liquid combustion in a tank or in a spill. Unlike the standard approach that assumes the shape of the flame is constant, random pulsations of the flame due to the turbulent mode of liquid combustion are considered. The pulsations lead to the random nature of the mutual radiation coefficient and the temperature of the radiating surface of the flame. This leads to a random value of the radiant heat flux density from the fire. Using the central limit theorem allows justifying the assumption about the normal law of the distribution of the radiant heat flux density, the mutual radiation coefficient and the temperature of the radiating surface of the flame. The assumption of a normal distribution law allows calculating the mathematical expectation of the heat flux density. It is shown that the average value of the heat flux density increases with an increase in the dispersion of the temperature of the radiating surface and the coefficient of mutual radiation, as well as with an increase in the correlation coefficient between them. It means that neglecting random flame pulsations can lead to underestimates of the average heat flux density from a fire. The dispersion of the radiant heat flux density was found and it was shown that it increases with the growth of the dispersion of the flame temperature and the mutual radiation coefficient. The standard deviation of the heat flux density can be more than 40 % of its average value if the standard deviations of the flame temperature and the mutual radiation coefficient are up to 10 % of their average values. The obtained results can be used to clarify the thermal effect of a liquid fire on neighboring objects.

Оценка вероятности отказа при эксплуатации сборного режущего инструмента со сменными многогранными пластинами методами непараметрической статистики

The article discusses the issues of durability of a prefabricated cutting tool in the process of metal cutting, as well as the risk of loss of its operability. In the course of assessing the probability of tool failure in violation of the established wear criterion, the results of a production experiment are used when roughing the Shaft part made of a rod with a diameter of ø 58 mm made of HN77TYUR alloy. The processing of the part is carried out with a prefabricated cutting tool with replaceable polyhedral plates of round and square shape, having a special type of alloys: VK6m, VK8 and T15K6. To evaluate the results obtained and describe their distribution, the authors use the Shapiro–Wilk agreement criterion to check the correlation between the initial data and the corresponding assumption of their normal distribution. Since the obtained production test data do not obey the normal distribution law, the nonparametric Parsen–Rosenblatt method is used for further analysis.

Formation of input information arrays for computer simulation of welded trusses behavior under thermal force effects

A methodological approach is proposed for detecting the behavior of welded structures under thermomechanical influences with high reliability of results using computer modeling experiments. A series of samples were produced from a batch of rolled angle bars with dimensions of 25x25x4. By applying standardized methods on the certified electro-hydraulic testing complex STM-100, experimental investigations of the mechanical characteristics of VSt3ps steel were conducted both at room temperature and within the range of technological and fire temperatures. The obtained results were subjected to statistical analysis, and an input data array for the ultimate strength and yield strength of VSt3ps steel was formed within the temperature range from +20 to +450°C. It was found that the research results follow a normal distribution law. The scattering of values for the ultimate strength is 2.1%, and for the yield strength is 2.6%, which is significantly lower than the information provided by Ukrainian National Standard and quality certificates of the material.

MULTIFACTOR REGRESSION MODEL FOR PREDICTION OF CHRONIC RHINOSINUSITIS RECURRENCE.

The aim: To propose an approach to forecasting the risk of chronic rhinosinusitis recurrence based on multivariate regression analysis for effective diagnosis and carrying out treatment and preventive measures. Materials and methods: 104 patients aged 18 to 80, including 58 women and 46 men, diagnosed with chronic rhinosinusitis were examined. Results: To build a multifactorial regression model for predicting the recurrence of chronic rhinosinusitis, probable factors of the occurrence of the disease were selected. 14 possible factors were analyzed using multivariate regression analysis. 13 risk factors were selected for predicting recurrence of chronic rhinosinusitis with a significance level of less than 0.05. Histograms of the residual deviations of predicting the recurrence of chronic rhinosinusitis were obtained, which are distributed symmetrically, and a normal-probability straight line is presented, on which there are no systematic deviations. The given results confirm the statistical hypothesis that the residual deviations correspond to the normal distribution law. Residual deviations relative to the predicted values are scattered chaotically, which indicates the absence of dependence on the predicted values of the risk of recurrence of chronic rhinosinusitis. The value of the coefficient of determination was calculated, which is 0.988, which gives grounds to claim that 98.8% of the factors are taken into account in the model for predicting the recurrence of chronic rhinosinusitis and its high reliability and acceptability in general. Conclusions: The proposed model makes it possible to predict in advance potential complications and the possibility of recurrence of the studied disease.

Математична модель емпіричного закону розподілу виходу з ладу силових трансформаторів через пошкодження обмоток

The work is devoted to the identification of the law of distribution according to statistical data of failure of power transformers of 6–10 kV due to damage to their windings for a three-year period: from 2018 to 2020. According to statistical data, it has been established that there is a tendency to reduce damage to power transformers by almost two times. But the number of these failures still remains quite large. The main cause of power transformers failure is damage to their windings, which accounts for 82.17 % of other causes. Therefore, the statistics that characterize the failure of power transformers due to damage to the windings were taken for research. It was established that in 2018 and 2019, the largest number of damages to power transformer windings occurred in April—July with a peak in June, and in 2020, the maximum damages were recorded in November and December. The prerequisite for building mathematical models for predicting the failure of power transformers due to the damage to the windings in order to ensure the necessary number of spare parts for their quick recovery is the study of the nature of the failure of transformers. The following methods were used to solve the tasks: passive experiment — to collect information about the failure of power transformers; statistical analysis — to build an empirical law of distribution. After analyzing the view of the histogram of the variation series of the sample of failure of power transformers due to damage to the windings, a hypothesis was put forward about the logarithmic normal distribution law of the failure of transformers due to damage to the windings. Pearson's c2-test was used to check it. As a result of the study, the proposed hypothesis was confirmed. Using Student's tables and the c2-distribution table with a confidence probability of g2 = 0.98, confidence intervals were determined for statistical estimates of the mean value, variance and root mean square deviation, which characterize the failure of power transformers due to damage to the windings. With further research, it will be possible to build mathematical models for predicting the failure of power transformers due to damage to the windings.

Building a mathematical model for assessing the hazard of emergency situations and the formation of types of causes for rolling stock derailments

The technique provides an analytical criterion for selecting the one set of numerical data most involved in an emergency out of several ones which are the relative deviations of the main factors from the critical ones in transport. The criterion is based on finding the theoretical left boundary of the interval of each set of deviations. The sets under study are reduced to normal distribution laws using a well-known method. The analysis uses basic numerical characteristics similar to those used in probability theory and mathematical statistics. The proposed method is based on finding the central theoretical moments up to the third order inclusive. Explanatory demonstration examples are provided.

Formation of highly dispersed droplets with harmless effect on the environment and the results of their mathematical and statistical processing

In this article, given density, diameter, and dispersion of high-dispersion drops in the process of processing plant leaves with an improved cutter were studied. Digital drip cards 9950-0028, which are liquid sensitive, were used and analysed using Deposit Scan software. As a result of the mathematical-statistical processing of the data collected, the number of lines of variation, the diameter of the largest and smallest drops, the arithmetic mean, the standard deviation, and the coefficient of variation were determined. On the volumetric dimensions of the droplets, an integral curve line of the curve and distribution of the differential function of the Gaussian distribution law was constructed. It was observed in the tests that the limit of distribution by volume measurement of drops was equal to 53-149 mkm. With the agro technical reduction of the size of the drops that is with an increase in the quality of the spray, the working fluid consumption can be saved, and the negative effect on other plants in the environment, on products consumed by the people, can be reduced.

Особливості розвитку ринку банківських послуг України: ефект розміру банку та закон Гібрата

Quantitative description of the development of the banking services market is important for making optimal decisions in the regulation of the banking sector. The importance of the question lies in the fact that such a description makes it possible to take into account the transformation of banks, sources of risks, to identify the main trends or regularities in the process of evolution. The aim of this article is to test Gibrat's law as a hypothesis for the development of the banking services market of Ukraine, by an empirical experiment where all existing banks are tracked over time on an annual basis and the distribution of banks by the size of assets or income is constructed annually. The work assumes the possibility of establishing a statistical size (assets or income) distribution of banks with certain average values and variances according to the lognormal distribution of banks. The research methodology is based on statistical analysis and verification of the correspondence of the empirical size distribution of banks to the logarithmically normal one provided by Gibrat's law. The study period covers the banking sector from 2014 to 2023 and includes all operating banks. The information base for the analysis was the annual official reporting data (report on the financial state) of banks, published by the National Bank of Ukraine and on the banks' official websites. The data set defines the assets and income of all banks and each bank separately, annually. It has been proven that Gibrat's law is not fulfilled for banks of Ukraine, the size distributions do not obey the Gaussian distribution law. The rate of growth of banks is not random (as it is assumed by Gibrat's hypothesis). The trend of gradual approximation of income distributions to the lognormal type and the potential possibility of implementation of Gibrat's law for the period of 2022–2023 has been revealed.