ON NON-REVERSIBLE CIRCUIT CHAINS

This book gives a comprehensive and self-contained introduction to the theory of symmetric Markov processes and symmetric quasi-regular Dirichlet forms. In a detailed and accessible manner, the book covers the essential elements and applications of the theory of symmetric Markov processes, including recurrence/transience criteria, probabilistic potential theory, additive functional theory, and time change theory. The book develops the theory in a general framework of symmetric quasi-regular Dirichlet forms in a unified manner with that of regular Dirichlet forms, emphasizing the role of extended Dirichlet spaces and the rich interplay between the probabilistic and analytic aspects of the theory. It then addresses the latest advances in the theory, presented here for the first time in any book. Topics include the characterization of time-changed Markov processes in terms of Douglas integrals and a systematic account of reflected Dirichlet spaces, and the important roles such advances play in the boundary theory of symmetric Markov processes. This book is an ideal resource for researchers and practitioners, and can also serve as a textbook for advanced graduate students. It includes examples, appendixes, and exercises with solutions.

In the theory of Markov processes discussed in Chapter 4 we dealt with the motion of a single particle. In physical phenomena, and more generally in natural phenomena, particles may disappear suddenly and may multiply. In this chapter we will discuss problems of particle extinction (annihilation) and multiplication. First of all, we will deal with particle extinction in the theory of Markov processes. We will define the lifetime of a particle and let the motion terminate at the lifetime. Next, given a Markov process with finite lifetime, we will discuss the converse process of gluing one path to another to extend the life span. We will show that this “piecing together method” can be applied to the multiplication of particles: one lets new particles start when the life of the observed particle is exhausted. The stochastic processes defined in this way are called branching Markov processes. We note that the extinction and multiplication of particles were traditionally dealt with in quantum field theory.

Purpose of the study. The aim of the research is to substantiate the analytical provisions of engineering management of the state of independent elements of plant engineering. The practical result will be the formation of the initial provisions of Smart-technology of engineering management to change the state of independent elements of plant engineering. Research methods. The methodological basis of the work is the generalization and analysis of known scientific results regarding the engineering management of the state of independent elements of plant engineering, and the use of a systems approach. Analytical method and comparative analysis was used to form a scientific problem, determine the purpose and set research objectives. The main provisions of the theory of systems engineering, methodology of systems analysis, theory of matrices, differential calculations and research on the theory of graphs, the theory of Markov processes, Bernoulli's theorem was used to create analytical models. The results of the study. The article considers the solutions that allowed substantiate the differential equations that characterize changes in the state of agricultural machinery in the complex of plant engineering, similar to which in the theory of Markov processes is called equations of reproduction and death. The analytical paper uses the ergodic property of Markov processes, according to which the existence of a limit or established regime of stationary probabilistic process of changing the state of agricultural machinery in the complex of plant engineering systems is established. The article considers the probability models of changes in the state of agricultural machinery in the aggregate of plant engineering systems, which tend to the limit values and do not depend on the initial distribution of . The solution of these models for the regime, we can find the probabilities of , where , of all possible states of agricultural machinery in the aggregate of plant engineering. This was used to determine the level of technical readiness of agricultural machinery. According to Bernoulli's theorem, it is established that with increasing number of experiments, namely state transitions in the analyzed case, the frequency of the event converges in probability with the probability of the event, and for the regime. Conclusions. As a result of research: - new scientifically sound technical and technological solutions and developments were obtained, which are aimed at substantiating the analytical provisions of engineering management of independent elements of plant engineering, which will serve as the starting point of Smart technology of engineering management. - it was confirmed that the intensity of recovery, in addition to the number of faulty agricultural machinery in the structure of crop production technologies, also depends on the nature of faults and the type of troubleshooting, and the type of engineering management tools. - the analytical provisions for determining the relationship between the coefficient of technical readiness of agricultural machinery, the reliability of agricultural machinery and the power of engineering management in the structure of crop production technologies are substantiated. Key words: restoration, coefficient of technical readiness, agricultural machine, reliability, system engineering.

The mathematical base of stochastic labor markets is the theory of Markov processes, and the uncertainty is its indivisible part. In this paper, Markov processes are used to calculate the equilibrium position, the time needed to reach it, natural rate of unemployment, transition probabilities and first passage time. While the theories of uncertainty give explanation why workers transit, identify the market anomalies and best fitted Markov model. The findings showed that the semi-Markov labor model better fits Switzerland data. Furthermore, the time needed to reach the equilibrium position is 4.6 years and the right number of employed workers, unemployed and inactive workers maintain the highest rate of uncertainty reduction is 67.16 %, 1.31 % and 31.54 % respectively, where 1.31% is the natural rate of unemployment. What is also important to point out is that the percent of employed workers in Switzerland is expected to decrease from 79.4% to 67.16%, while the percent of inactive workers is projected to increase significantly, from 15.9%to 31.54%. Workers who are expected to transit to an Inactive state in the future and stay there for a longer time are the older works (above 45 years). In other words, the Switzerland labor market is directed toward its “bad” equilibrium. In the end, the demographic structure is considered as one of the main factors for sustainable growth. Therefore, the government is suggested to control the population growth and put into practice new-innovative youth policies,becausethe traditional pro-family policies implemented to encourage bigger families has failed to increase the fertility rates to expected levels.

We develop a Markov process theory of charged particle scattering from stationary, transverse, magnetic waves. We examine approximations that lead to quasilinear theory, in particular the resonant diffusion approximation. We find that, when appropriate, the resonant diffusion approximation simplifies the result of the weak turbulence approximation without significant further restricting the regime of applicability. We also explore a theory generated by expanding drift and diffusion rates in terms of a presumed small correlation time. This small correlation time expansion leads to results valid for relatively small pitch angle and large wave energy density—a regime that may govern pitch angle scattering of high-energy electrons into the geomagnetic loss cone.

Purpose. To develop a concept for a model for managing the risk of workplace fatigue, taking into account both the intensity of the negative impact of fatigue factors on the employee in the workplace and the intensity of their recovery from such impact to prevent incidents in organizations and industrial accidents and disasters at high-risk facilities. Method. An analysis of scientific literature — to update the purpose and definition of research tasks; methods of the theory of Markov processes — for a formalized description of the random processes of the development of the state of fatigue in the employee and their recovery from the state of fatigue during the work shift. Results. The proposed conceptual models allow the organization to manage the risk of fatigue by adjusting the duration of work and rest periods, depending on the characteristics of the parameters of the intensity of the negative impact of fatigue factors on the employee and the recovery of their body from such an impact. By changing the characteristics of the specified parameters, it is possible to determine the duration of the work periods that maximize the time an employee can work without entering a state of fatigue and the duration of the rest periods that minimize the time required for full recovery. Originality. A conceptual approach for managing the risk of fatigue is proposed, by establishing dependencies between the duration of work and rest periods and the characteristics of the parameters of the intensity of the negative impact of fatigue factors on the worker and their recovery from such an impact, based on the application of the theory of Markov processes. Practical value. The application of the proposed models allows increasing the level of labor productivity in the organization, by determining such durations of work and rest schedules, which provide the opportunity for employees to perform labor activities during the maximum possible time period of the work shift, without reaching a state of fatigue.

At first there was the Markov property. The theory of stochastic processes, which can be considered as an exten- sion of probability theory, allows the modeling of the evolution of systems through the time. It cannot be properly understood just as pure mathemat- ics, separated from the body of experience and examples that have brought it to life. The theory of stochastic processes entered a period of intensive develop- ment, which is not finished yet, when the idea of the Markov property was brought in. Not even a serious study of the renewal processes is possible without using the strong tool of Markov processes. The modern theory of Markov processes has its origins in the studies by A. A: Markov (1856-1922) of sequences of experiments connected in a chain and in the attempts to describe mathematically the physical phenomenon known as Brownian mo- tion. Later, many generalizations (in fact all kinds of weakenings of the Markov property) of Markov type stochastic processes were proposed. Some of them have led to new classes of stochastic processes and useful applications. Let us mention some of them: systems with complete connections [90, 91, 45, 86]; K-dependent Markov processes [44]; semi-Markov processes, and so forth. The semi-Markov processes generalize the renewal processes as well as the Markov jump processes and have numerous applications, especially in relia- bility.

It is known that the theory of Markov processes is a rapidly developing field with numerous applications to many branches of mathematics and physics, biology, and so on. But there are some physical models which cannot be described by such processes. One of such models is related to population genetics. These processes are called quadratic stochastic processes (q.s.p.). In the present paper, we associate to given q.s.p. two kind of processes, which call marginal processes. Note that one of them is Markov process. We prove that such kind of processes uniquely define q.s.p. Moreover, we provide a construction of nontrivial examples of q.s.p. Weak ergodicity of q.s.p. is also studied in terms of the marginal processes.

The Markovianity within a wind turbine array boundary layer is explored for data taken in a wind tunnel containing a model wind turbine array. A stochastic analysis of the data is carried out using Markov chain theory. The data were obtained via hot-wire anemometry thus providing point velocity statistics. The theory of Markovian processes is applied to obtain a statistical description of longitudinal velocity increments inside the turbine wake using conditional probability density functions. It is found that two and three point conditional probability density functions are similar for scale differences larger than the Taylor micro-scale. This result is quantified by use of the Wilcoxon rank-sum test which verifies that this relationship holds independent of initial scale selection outside of the near-wake region behind a wind turbine. Furthermore, at the locations which demonstrate Markovian properties there is a well defined inertial sub-range which follows Kolmogorv's -5/3 scaling behavior. Results indicate an existence of Markovian properties at scales on the order of the Taylor micro-scale, λ for most locations in the wake. The exception being directly behind the tips of the rotor and the hub where the complex turbulent interactions characteristic of the near-wake demonstrate influence upon the Markov process. The presence of a Markov process in the remaining locations leads to characterization of the multi-point statistics of the wind turbine wakes using the most recent states of the flow.

Accounting for possible risks that affect the successful implementation of construction projects is one of the most important tasks in planning and managing construction. The main problem for risk accounting is that they arise under the influence of random factors under conditions of high uncertainty. Therefore, the vast majority of risk accounting methods in construction are based on probabilistic models. Given the fact that recently the level of uncertainty in almost all areas of economic activity has increased significantly, it is necessary to develop new approaches to risk analysis that would respond as accurately and quickly as possible to emerging threats. Aim. The purpose of the study is to develop a mathematical model based on the theory of random Markov processes, which allows, in a probabilistic approach, to analyze the negative impact of adverse factors on the implementation of construction projects in the dynamics of their receipt under conditions of high uncertainty in order to increase the efficiency of construction work. Materials and methods. The risk assessment model of the influence of unfavorable factors on a construction project, presented in the paper, is based on the theory of Markov random processes, which makes it possible to assess the probability of the implementation of possible threats with varying degrees of damage, and to calculate risks during construction work. The model assumes that threats that cause different levels of damage to a construction project arrive at random times and require different times to eliminate them. To implement the model, a system of differential equations is given, which is solved by numerical methods. An analysis of the solution under various conditions of threats is carried out. The influence of negative factors on the construction project during its long-term implementation is considered separately. The analysis of the influence of temporal and probabilistic parameters of the task on the degree of risks in the implementation of construction projects was carried out. Results. The relevance of developing a risk assessment model for the implementation of construction projects is shown, goals and objectives for scientific research are set. The ways of solving the set tasks are given and substantiated. A mathematical model has been developed that makes it possible to estimate the probabilities of finding a construction project in various states, differing in different degrees of risk during the implementation of the project. On the basis of numerical methods, the practical implementation of the model is given for various parameters characteri¬zing the influence of negative factors on the progress of the construction project. The results of the practical implementation of the model are analyzed, recommendations are given for using the model in practice. Conclusion. The presented dynamic risk assessment model for the implementation of construction projects can serve as the basis for building an optimal control system for the progress of construction work and making decisions on the organization of construction activities.

Using the control problem of a two-state Markov process in discrete time as an example, we consider the basic stages concerning the application of theory of conditional Markov processes to synthesize optimal algorithms of the control of stochastic systems. It is assumed that the control changes the statistical properties of the states of a controlled plant. The numerical method for solving the problem and the results of solving particular example are presented. The special features of the solution of this problem compared to the well-known problem in continuous time are discussed.

Parrondo’s games are essentially Markov games. They belong to the same class as Snakes and Ladders. The important distinguishing feature of Parrondo’s games is that the transition probabilities may vary in time. It is as though “snakes,” “ladders” and “dice” were being added and removed while the game was still in progress. Parrondo’s games are not homogeneous in time and do not necessarily settle down to an equilibrium. They model non-equilibrium processes in physics.We formulate Parrondo’s games as an inhomogeneous sequence of Markov transition operators, with rewards. Parrondo’s “paradox” is shown to be equivalent to saying that the expected value of the reward, from the whole process, is not a linear function of the Markov operators. When we say that a game is “winning” or “losing” then we must be careful to include the whole process in our definition of the word “game.” Specifically, we must include the time varying probability vector in our calculations.We give practical rules for calculating the expected value of the return from sequences of Parrondo’s games. We include a worked example and a comparison between the theory and a simulation.We apply visualization techniques, from physics and engineering, to an inhomogeneous Markov process and show that the limiting set or “attractor” of this process has fractal geometry. This is in contrast to the relevant theory for homogeneous Markov processes where the stable, equilibrium limiting set is a single point in the state space.We show histograms of simulations and describe methods for calculating the capacity dimension and the moments of the fractal attractors. We indicate how to construct optimal forms of Parrondo’s games and describe a symmetrical family of games which includes the optimal form, as a limiting case.We investigate the fractal geometry of the attractors for this symmetrical family of games. The resulting geometry is very interesting, even beautiful.

A mini-tutorial on measure-valued Markov processes and nonlinear martingale problems—As a reply to McCauley's comment

The aim of the studyis a probabilistic description of the functioning of the cognitive system, taking into account its internal logic and interaction with the external environment.Such concepts of cognitive theory as sensory imaginative representations, models, systems are the most common, so the attempt to formalize them is by obtaining the most common results. One of the key concepts of cognitive theory is Gestalt, which is understood in this work as a kind of holistic perception of the sensual image, as well as the sensual image. Formalization (mathematical description) of Gestalt, as well as other concepts of cognitive theory meets the natural difficulties associated with the uncertainty of these concepts. On the other hand, there are well-developed mathematical models of behavior of quite specific organizational systems, allowing obtaining meaningful results. In this regard, the mathematical description of a wide class of cognitive systems, not limited to the specific content of their functioning, is an urgent task. In this study, it is assumed that sensory images occur at random times and affect the cognitive system with certain probabilities. In this regard, one of the adequate mathematical tools are, apparently, probability-theoretic methods, in particular, the application of the theory of Markov processes. The methodof research within the framework of the adopted model is the application of the theory of Markov processes developing at fixed points in time, i.e. Markov chains. It is believed that the functioning of the cognitive system is described by abstract probabilities of changes in the system states. This approach allows formalizing the processes of representation of sensory images in the cognitive system, taking into account both the internal logic of the system and the interaction of the system with the outside world. The main attention is paid to the study of the influence on the behavior of the system external to her sensual images.As a resultof the study shows that the inclusion of the interactions of the system is achieved by introducing the stochastic matrix of probabilities of the system response to external influences. Taking into account the well-developed theory of Markov chains, analytical expressions for the probabilities of the system in each of the possible states are obtained. The influence on the behavior of the system of elements of the matrix of probability reactions of the system is investigated, the corresponding graphs are presented. The asymptotic behavior of the system is studied with an unlimited increase in the number of steps that change the state of the system, as well as the average characteristics of the system. It is noted that the presented description is formal, operates only with probabilistic characteristics of the system and does not take into account specific signals that can enter the system from its sensors, and generally sensitive elements. In this regard, the further development of the model may be associated with the assessment of the probability of the system response to external influences, taking into account the characteristics of these specific signals, as well as the development of optimal algorithms for decision-making about the presence or absence of impacts on the system from the outside world.

The goal of the paper is to include non-stationary random vibration excitation into the "classic" statistical solution of vehicles. Other vehicle parameters are the deterministic functions. The non-stationary random function will be modelled by changeable speed of the vehicle and vertical unevenness of track. We shall assume evolutionary Gaussian process. The dynamics of railway vehicle motion will be analysed by the Monte Carlo simulation and the theory of Markov processes.