Theory Of Random Functions Research Articles

A model of the radar situation of the coastal vessel traffic control system

Since the elements of the ship traffic control system at sea (VTS) are geographically distributed, and the observed spaces have well-defined zones, the radar facilities (radars) that make up their technological basis are evaluated by the ability to detect, classify (recognize), and ensure tracking of ships at a certain range in the spatial zone. The solution of these problems is probabilistic in nature, and the indicators are represented by random functions. However, due to the complexity of accounting for random factors in such stochastic control systems, calculations of their efficiency indicators are usually carried out deterministically for the "average" values of random parameters. The development of a mathematical model for evaluating the effectiveness of VTS in its ability to detect marine targets is a scientific task, the solution of which is possible using elements of the theory of random functions and processes.

Electromagnetohydrodynamic (EMHD) Flow in a Microchannel with Random Surface Roughness.

This study investigates the effect of small random transverse wall roughness on electromagnetohydrodynamic (EMHD) flow is in a microchannel, employing the perturbation method based upon stationary random function theory. An exact solution of a random corrugation function ξ, which is a measure of the flow rate deviated from the case without the roughness of two plates, is obtained by integrating the spectral density. After the sinusoidal, triangular, rectangular, and sawtooth functions that satisfy the Dirichlet condition are expanded into the Fourier sine series, the spectral density of the sine function is used to represent the corrugation function. Interestingly, for sinusoidal roughness, the final expression of the corrugation function is in good agreement with our previous work. Results show that no matter the shape of the wall roughness, the flow rate always decreases due to the existence of wall corrugation. Variations of the corrugation function and the flow rate strongly depend on fluid wavenumber λ and Hartmann number Ha. Finally, the flow resistance is found to become small, and the flow rate increases with roughness that is in phase (θ = 0) compared with the one that is out of phase (θ = π).

Prediction error analysis for intelligent management and predictive diagnostics systems

Random signal prediction is efficient for intelligent management and predictive diagnostics systems. Aim. The paper aims to analyse the error of random signal prediction. To develop recommendations for the selection of random signal extrapolator parameters. Methods. The paper uses the mathematics of the theory of random functions, formalization adopted in the theory of pulse systems, mathematical description of extrapolators with Chebyshev polynomials orthogonal over a set of equally spaced points. The coefficients of the predicting polynomial are selected according to the minimal least squares. Results. The paper describes the mathematical model of the extrapolator. Design ratios were obtained for prediction error assessments. The maximum and prediction interval-averaged relative mean square error of extrapolation were defined. The authors analyse the error of extrapolation of random processes defined by the sum of a centred stationary random process and a deterministic time function. Based on diverse calculations, recommendations were defined that allow selecting the parameters of the extrapolator (degree of the extrapolating polynomial, number of test points that precede the prediction interval, discretisation interval of the predicting function) under the specified input signal models. Conclusion. The use of extrapolators based on Chebyshev polynomials orthogonal on a set of equally spaced points and the least square method allows implementing a procedure for calculating predicted values of a random process with the required accuracy. Under the specified models of the predicting signal, a method was developed that allows selecting the extrapolator’s parameters (order, number of points involved in the generation of the prediction, sample spacing) for the purpose of ensuring the required accuracy.

Modeling variability of infiltration tests in ephemeral stream beds as a random function for uncertainty quantification

Infiltration processes are highly variable in space and time, and therefore, building reliable hydrological models without considering the variability is questionable. In this research, we propose a methodology that can systematically handle the variability in the infiltration process. The methodology is based on the theory of random functions in a dimensionless formalism that allows the derivation of a generalized model from the observed infiltration test data. The Monte Carlo technique is utilized to generate hypothetical infiltration tests that carputer the characteristics of the real tests. The methodology is applied to a case study in ephemeral stream beds located in Al Madinah Al Munawarah Province in Saudi Arabia. The measurements are made by the double-ring infiltrometer. Beta distribution fits the dimensionless cumulative infiltration relatively well at a 1% significant level at all times, and therefore, it can be used to model the uncertainty in hydrological modeling. High variability is observed in infiltration tests at the early time (a platykurtic distribution with high dispersion); however, it decreases at the late time (Leptokurtic distribution with low dispersion) since the infiltration reaches a steady infiltration. Some extreme tests show different behavior from the fourteen tests that cannot be captured by the model and therefore need special treatment.

Method of Detecting Radio Signals using Means of Covert by Obtaining Information on the basis of Random Signals Model

The article presents a developed method for determining random radio signals. Random signals can be signals from hidden means of obtaining information.The signal is considered as a random process. The description of such signals is based on probability theory and the theory of random functions. In the practice of analysis of random radio signals, static methods based on the theory of stationary random functions have become widespread. The existing models of a stationary random process are not adequate for a large number of random processes, especially for processes that are dynamic in nature and are observed during a finite time interval.The developed method allows to detect signals from hidden means of obtaining information with greater efficiency. The novelty of the method is to determine the deviation of the main parameters of the signals from the function of the sample. The method is based on determining the function of a sample of signals of a given radio range. The function of the sample is obtained as the implementation of the smoothing function of a random process, by the method of least squares using the principle of sliding smoothing.It is proposed to determine the signals of the means of covert receipt of information by the instantaneous deviation of the parameters of random signals. To determine the signals of the means of covert obtaining information, it is proposed to determine the deviation of the amplitude of the random signals from the amplitude of the signals of the sample function, then, if necessary, to determine the deviation of the signal phases. So, the effectiveness of the method is achieved by determining two parameters of the deviation of the amplitude and phase. This makes it possible to detect random signals with a higher probability.Determining the function of the sample of the required radio band significantly increases the probability of determining random signals, by reducing the scanning time of a given radio band, by excluding known signals from the additional software analysis of a given radio band.

Design and analysis of the mechanical properties of controllable porous scaffolds for bone tissue engineering.

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On the asymptotic behavior of the Diaconis–Freedman chain in a multi-dimensional simplex

AbstractWe give a setting of the Diaconis–Freedman chain in a multi-dimensional simplex and consider its asymptotic behavior. By using techniques from random iterated function theory and quasi-compact operator theory, we first give some sufficient conditions which ensure the existence and uniqueness of an invariant probability measure and, in particular cases, explicit formulas for the invariant probability density. Moreover, we completely classify all behaviors of this chain in dimension two. Some other settings of the chain are also discussed.

ПРИМЕНЕНИЕ НЕЙРОСЕТЕВОГО ПОДХОДА И МЕТОДА НА ОСНОВЕ ТЕОРИИ СЛУЧАЙНЫХ ФУНКЦИЙ ДЛЯ КЛАССИФИКАЦИИ ОБЪЕКТОВ ПО ИХ АКУСТИЧЕСКИМ ИЗЛУЧЕНИЯМ

The article describes the results of the classification of objects by their acoustic emissions using two methods: a neural network approach, as well as a method based on the theory of random functions. The counts of the SPM assessment act as an informative feature.

Stochastic dynamical behavior of COVID-19 model based on secondary vaccination.

This paper mainly studies the dynamical behavior of a stochastic COVID-19 model. First, the stochastic COVID-19 model is built based on random perturbations, secondary vaccination and bilinear incidence. Second, in the proposed model, we prove the existence and uniqueness of the global positive solution using random Lyapunov function theory, and the sufficient conditions for disease extinction are obtained. It is analyzed that secondary vaccination can effectively control the spread of COVID-19 and the intensity of the random disturbance can promote the extinction of the infected population. Finally, the theoretical results are verified by numerical simulations.

Random polynomials: Central limit theorems for the real roots

The number of real roots has been a central subject in the theory of random polynomials and random functions since the fundamental papers of Littlewood, Offord, and Kac in the 1940s. The main task here is to determine the limiting distribution of this random variable. In 1974, Maslova famously proved a central limit theorem (CLT) for the number of real roots of Kac polynomials. It has remained the only limiting theorem available for the number of real roots for more than four decades. In this paper, using a new approach, we derive a general CLT for the number of real roots of a large class of random polynomials with coefficients growing polynomially. Our result both generalizes and strengthens Maslova’s theorem.

Method of Detecting Radio Signals using Means of Covert by Obtaining Information on the basis of Random Signals Model

The article presents a developed method for determining random radio signals. Random signals can be signals from hidden means of obtaining information. The signal is considered as a random process. The description of such signals is based on probability theory and the theory of random functions. In the practice of analysis of random radio signals, static methods based on the theory of stationary random functions have become widespread. The existing models of a stationary random process are not adequate for a large number of random processes, especially for processes that are dynamic in nature and are observed during a finite time interval. The developed method allows to detect signals from hidden means of obtaining information with greater efficiency. The novelty of the method is to determine the deviation of the main parameters of the signals from the function of the sample. The method is based on determining the function of a sample of signals of a given radio range. The function of the sample is obtained as the implementation of the smoothing function of a random process, by the method of least squares using the principle of sliding smoothing. It is proposed to determine the signals of the means of covert receipt of information by the instantaneous deviation of the parameters of random signals. To determine the signals of the means of covert obtaining information, it is proposed to determine the deviation of the amplitude of the random signals from the amplitude of the signals of the sample function, then, if necessary, to determine the deviation of the signal phases. So, the effectiveness of the method is achieved by determining two parameters of the deviation of the amplitude and phase. This makes it possible to detect random signals with a higher probability. Determining the function of the sample of the required radio band significantly increases the probability of determining random signals, by reducing the scanning time of a given radio band, by excluding known signals from the additional software analysis of a given radio band.

Study of structure of flows of a technological apparatus using the theory of random functions

The object of research is the structure of flows in the absorber of hydrogen chloride. One of the most problematic areas in the study of flow hydrodynamics in chemical-technological devices are both technological and technical difficulties, when the device is exposed to random disturbances and/or the supply of a standard indicator is impossible due to a violation of the technological regulations. A method for studying the hydrodynamic structure of flows in a shelf absorber of hydrogen chloride of the «Korobon-KA» type (Germany) in the normal operation of a chemical apparatus using the theory of random functions is proposed. An industrial experiment was carried out on the operating equipment to determine the input and output concentrations of the components of the gas flow. The absorber of hydrogen chloride is considered as a one-dimensional object, at the input of which a random function acts – the concentration of hydrogen chloride in the input stream, and at the output there is a random variable – the concentration of hydrogen chloride in the output stream. The method for determining hydrogen chloride and chlorine in a gas stream is based on the absorption of chlorine by a solution of potassium iodide, followed by titration of the released iodine with sodium thiosulfate. In parallel, portions of acid were sampled at the inlet and outlet, and then the density and temperature of the hydrochloric acid solutions were determined. An algorithm for calculating the impulse function estimates is developed. The obtained experimental data are smoothed. As a result of processing the experimental data, autocorrelation and cross-correlation functions were obtained, the Wiener-Hopf equation was solved, and the impulse weight function was obtained. Having calculated the moments of the obtained impulse weight function, it was proved that the structure of flows in the «Korobon-KA» absorber can be satisfactorily described by the ideal displacement model. The calculations were carried out in software environments MathCAD, Matlab. According to the results obtained, the proposed method for determining the hydrodynamic structure of flows will find application in the study of chemical-technological devices, when the object is exposed to random disturbances and the supply of a standard indicator is impossible due to violation of technological regulations. This makes it possible to find the parameters of flow hydrodynamics in the apparatus in the mode of its normal operation.

ОЦІНКА БАГАТОРІЧНОЇ МІНЛИВОСТІ СЕРЕДНЬОГО РІЧНОГО СТОКУ ВОДИ РІЧОК БАСЕЙНУ ПРИП’ЯТІ В МЕЖАХ УКРАЇНИ ТА ЙОГО РОЗРАХУНКОВІ ХАРАКТЕРИСТИКИ У ФАЗИ ВОДНОСТІ

The aim of the study was to analyze the long-term variability of the average annual water runoff of the rivers of the Pripyat basin within Ukraine and to assess its calculated characteristics in the high-water and low-water phases of water content.The research of the cyclical nature of long-term fluctuations of river water flow, which contributes to the establishment of long-term dynamics of water content and potential (forecast) changes – is a topical issue of modern hydrological research. To describe the long-term variability of river water flow and its structure (cycles and phases of water content) the most effective method is stochastic, based on mathematical statistics, the theory of random variables and functions, probability theory. Autocorrelation analysis, various statistical criteria (homogeneity, series, series lengths), total and difference integral curves, probability theory, correlations, statistical estimation of probable errors, etc. were used to identify stochastic regularities of long-term variability. According to the results of the study of long-term variability of the average annual water runoff of the rivers of the Pripyat basin within Ukraine, it is established that cycles with periods of 29±2 years have high reliability and indicate stability of periods of low (10±2 years) and high water content. 17±2 years). According to the identified stochastic patterns, it is assumed that by 2025-26 it is necessary to expect the continuation of the low-water phase of water, then with the duration of 16-17 years the high-water phase will begin and from 2044-45 there will be low water again until 2055-56. According to the proposed regression equations between the average annual water discharge for a long-term period and their average values during the high-water and low-water phases of water content (with very significant approximation coefficients) and the obtained transition coefficients, it is possible to establish the calculated characteristics of the average annual flow of water of various availability in the high-water and low-water phases of water content, thereby giving their forecast estimates.

Refined assessment of strength of concrete pavement probabilistic method

The object of research is the method of calculating the strength of cement concrete airfield pavement. According to the current regulatory documents, the calculation of the strength of the coating is carried out according to the first limit state, taking into account the internal forces arising in the slab from the impact of the wheel load and temperature. The repeated impact of loads from the wheels of the main supports of various types of air is replaced by the influence of the main support of the heaviest aircraft, causing the greatest bending moment in the design section of the plate. However, the calculations do not take into account the stochastic nature of such factors as the take-off weight of aircraft, the plate thickness, the bed ratio of the elastic foundation, the probability of the combined effect of operating loads from the main bearing wheels and temperature in the calculated section of the plate, as well as the probability distribution of loads from the aircraft spectrum across the width of the airfield elements. The article proposes a method for taking into account the probabilistic-statistical variability of operating loads, physical and mechanical properties of materials and the geometric characteristics of the plate, recommended when developing a new regulatory framework for the design of rigid airfield pavements. The method of statistical linearization of random functions of several random arguments was chosen as the research method. Using the mathematical apparatus of the theory of random functions, the statistical coefficient of the working conditions of the plate is analytically determined, and a numerical analysis of the obtained analytical solution is carried out. It has been established that with the same unfavorable combination of variability of basic arguments, the calculated absolute value of the statistical coefficient of the working conditions of the slab is more influenced by the coefficient of variation of the slab thickness than the coefficient of variation of the strength of cement concrete in bending. The condition for calculating the required thickness of the hard airfield slab of monolithic cement concrete is determined.

Stochastic Process Simulation of Soil Displacement in Calculations of Seismic Resistant Buildings

The solution to the problem of finding the statistical characteristics for a random process of seismic displacement of the foundation soil is given. Analytical methods of the theory of random functions and the method of statistical tests are used. The initial data for solving this problem are the characteristics of the random process of accelerating the soil of the base. The preliminary adjustment of the original accelerogram used for the subsequent calculation is shown. The implementations (accelerogram and seismogram) of the considered processes are constructed for two types of spectral acceleration densities in the form of a polyextreme function and an averaged function with one extreme. The obtained results are intended for carrying out probabilistic calculations of seismic resistant buildings as nonlinear systems and assessing their level of reliability

COMPLEXING IMAGES IN A TWO-CHANNEL OPTICAL-ELECTRONIC SYSTEM AS A METHOD OF REDUCING THE PROBABILITY OF DISAPPOINTING AN AUTOMATIC SURVEILLANCE OF LOW CONTRAST OBJECTS

The paper discusses the work of the correlation algorithm for automatically tracking objects of interest in a two-channel optical-electronic system in a complex background-target environment and the presence of intentional interference. Conditions in which the contrast of the desired object in both channels is negligible but not equal to zero are considered difficult in the paper. Intentional interference refers to masking interference of natural and artificial origin, which helps to reduce the contrast of the object in both channels. By two-channel means an optical-electronic system, which includes channels of the visible and infrared ranges. It is believed that the multi-spectral images of both channels are reduced to a single time and scale, which allows them to be integrated using various methods. The purpose of this paper is to prove that the likelihood of disruption of automatic tracking is reduced when complexing the source images of the visible and infrared ranges, when the contrast of the desired object in both channels is negligible. For research the mathematical apparatus of the theory of random functions and simulation with subsequent statistical data processing are used. It is shown that the probability of disruption, characterized by the ratio of the correlation coefficients of two fragments of images, one of which contains the desired object, and the second not, depends on both the correlation coefficients and the values of their mean square deviations. The simulation shows that the breakdown of tracking occurs both when the mean square deviations of these correlation coefficients are equal, and at their different values, moreover, an increase in their difference increases the probability of a breakdown. The article shows that the likelihood of a breakdown in a two-channel optoelectronic system will decrease when using two channels, compared to working only in the visible or infrared channel. The obtained results substantiate the promise of using image complexing in multichannel systems of automatic tracking of objects in a complex background-target environment and the presence of deliberate interference.

Mean-Square Non-Local Stability of Ship in Storm Conditions of Operation

Abstract The purpose of the paper is to create a method for studying nonlocal stability in the mean and in the mean square of the ship, positioned on the beam of an intensive wind–waves mode, which is based on the use of the correlation theory of random functions close to continuous Markov processes. With the help of this method and the integral formula of event probability, a method for determining the reliability indicator of the ship in respect of the existing wind–waves excitations of the operating area is formed. An example of investigating the nonlinear motion of the ship, determining its local and nonlocal stability in the first approximation of the theory of considered random functions, is given. Such approximation uses correlation theory with models of acting excitations represented by the generalised derivatives of the Wiener process. Moreover, special attention is paid to reflecting the connection of the proposed methods for investigating the ship stability under constantly acting random excitations with the traditional methods of studying ship stability at small and large inclinations. The established connection defines the proposed methods as a development of the traditional methods of ship stability deterministic theory during the transition to its formation in the class of random functions, with the addition to these methods of the missing link of determining the level of reliability of ships towards the acting wind–waves excitations of the operation area.

Modeling 3D soil lithotypes variability through geostatistical data fusion of CPT parameters

The Cone Penetration Test (CPT) measures enable to recognize vertical lithological sequence at each investigated point. From the tip resistance qc and sleeve resistance fs profiles, the Soil Behavior Type index ISBT has been calculated, in order to identify the lithotypes alongside depth. The present study focuses on the combination of different variables to provide a lithological and mechanical subsoil characterization. The main objectives of the paper are: (1) to model the 3D spatial variability structure of the soil lithotypes and mechanical properties using qc and fs profiles; 2) to evaluate the uncertainties of the estimates for designing purposes. 182 CPTs were collected in a 900 km2 area (corresponding to a subsoil volume of about 12 km3) located in the study site in the Bologna province (Italy). The study area is made up of fine-grained soils, silt and clay mixtures that are intercalated at different depths by sandy and gravelly soils. These variations of each soil fraction affect the engineering properties of these alluvial deposits. For 3D modeling, two geostatistical methods, Ordinary Kriging (stationary method), and Intrinsic Random Function theory (non-stationary method) have been used. The results show that the non-stationary method allows to obtain more reliable qc and fs predicted values. The final stochastic mechanical and lithological model enables engineers and geologists to detect the emerging of fan and paleochannels bodies where mean resistance values can abruptly change in terms of bearing capacity, liquefaction potential and static and dynamic settlement occurrence.

Analysis and evaluation of technogenic risk of technological equipment of metallurgical enterprises

The results of the theoretical review and research for the evaluation of technogenic risk of technological equipment of metallurgical enterprises are presented. The hazards and the risk of accidents or catastrophes of the bearing structure of a bridge crane are considered. A well-known technique based on the theory of random functions and processes is used. Theoretical calculations are based on known methods of probability theory. A relatively new numerical (test) calculation of the evaluation of technogenic risk for elements of technological equipment of metallurgical enterprises is made. The analysis and evaluation of technogenic risk will allow one to make correct management decisions.

Expansion of Wave Rays and Fronts in Media with Inhomogeneous Structure

In the article on the principles of Fermet, Huygens obtaine the differential equations in the form of Hamilton, which describe the ray trajectories and wave fronts in inhomogeneous media. Established that the vector of Poynting-Umov's determining the direction of energy propagation in inhomogeneous medium is coincident with the vector tangent to the ray. In the second part of the article established that the equations of the theory of rays' propagation in inhomogeneous media have the form of equations of nonlinear dynamics and describe the emergence of deterministic chaos in the geometry of rays for a wide variety of types of heterogeneous structures. In this case, the rays behave randomly and their description you must go to the description based on the theory of random functions and fields. In the third part of the paper is considered a model which is equivalent to the random medium and the calculation of the coordinates of the ray (the mathematical expectation and correlation functions). Understanding of these characteristics gives information about the behavior of the trajectories of the rays for these models of media. The description of the behavior of rays on the basis of the equations of statistical mechanics is discussed in the article for functions of Markov's type.