Discrete Moments Of Time Research Articles

On the problem of controlling a second-order nonlinear system by means of discrete control under disturbance

The problem of bringing a trajectory to a neighborhood of zero under disturbance is considered in terms of a differential pursuit game. The dynamics are described by a nonlinear autonomous system of second-order differential equations. The set of values of the pursuer's controls is finite, and that of the evader (disturbance) is compact. The goal of the control, that is, the goal of the pursuer, is to bring, within a finite time, the trajectory to any predetermined neighborhood of zero, regardless of the actions of the disturbance. To construct the control, the pursuer knows only the phase coordinates and the value of the velocity at some discrete moments of time and the choice of the disturbance control is unknown. Conditions are obtained for the existence of a set of initial positions, from each point of which a capture occurs in the specified sense. Moreover, this set contains a certain neighborhood of zero. The winning control is constructed constructively and has an additional property specified in the theorem. In addition, an estimate of the time required to bring the speed from one given point to the neighborhood of another given point under disturbance conditions was obtained.

Numerical modeling of surface subsidence due to compaction of soil with fine inclusions

A mathematical model of filtration consolidation of an inhomogeneous soil mass was formed taking into account the change in the size of the area during the compaction process. The inhomogeneity is considered as the presence of fine inclusions (geobarriers) the physical and mechanical characteristics of which differ from those of the main soil. From a mathematical viewpoint, the model is described by a one-phase Stefan problem that has a kinematic boundary condition on the upper moving boundary as its component. The purpose of the research is to find out the effect of fine inclusion on the dynamics of subsidence of the soil surface in the process of compaction. The change in the dimensions of the solution area is physically determined by the change in the volume of the pores of the porous medium in the process of dissipating excess pressure. If the permeability of the geobarrier is low, it affects the dynamics of consolidation processes and, accordingly, the magnitude of subsidence. Finite element solutions of the initial-boundary value problem for the nonlinear parabolic equation in the heterogeneous region with the conjugation condition of non-ideal contact were found. Numerical time discretization methods, a method for determining the change in the position of the upper boundary at discrete moments of time, and an algorithm for determining the physical and mechanical characteristics of a porous medium depending on the degree of consolidation are given. A number of test examples were considered, and the effect of a thin inclusion on the dynamics of the change in the position of the upper boundary of the problem solution area was investigated.

A Generalized Stacked Reinforcement Learning Method for Sampled Systems

A common setting of reinforcement learning (RL) is a Markov decision process (MDP) in which the environment is astochastic discrete-time dynamical system. Whereas MDPs are suitable in such applications as video-games or puzzles, physical systems are time-continuous. A general variant of RL is of digital format, where updates of the value (or cost) and policy are performed at discrete moments in time. The agent-environment loop then amounts to a sampled system, whereby sample-and-hold is a specific case. In this paper, we propose and benchmark two RL methods suitable for sampled systems. Specifically, we hybridize model-predictive control (MPC) with critics learning the optimal Q- and value (or cost-to-go) function. Optimality is analyzed and performance comparison is done in an experimental case study with a mobile robot.

Integrated approach to analyzing the stress-strain state of the tubbing erector manipulator mechanism based on the SolidWorks program

Purpose. The research purpose is todevelop and substantiate a method for computer analysis of the stress-strain state that occurs in the tubbing erector manipulator mechanism with several degrees of freedom, using modern computing complexes based on the finite-element method. Methods. The research uses software complexes, with the help of which a calculation-theoretical apparatus has been created, which is intended for calculating dynamic and static loads that occur in the elements of the tubbing erector manipulator mechanism. The results of modeling the stress-strain state in the parts and nodes of this mechanism, obtained using the SolidWorks Motion and SolidWorks Simulation programs, have been compared. The errors in these results have been estimated. Findings. The effectiveness of the proposed analysis method, based on the combination of SolidWorks applications, in particular, SolidWorks Motion and SolidWorks Simulation, has been proven on the example of modeling the most complex design of the UT62 tubbing erector mechanism.The essence of the method is that initially, using the SolidWorks Motion application, the laws of motion of the motors are set. The tubbing erecting process itself is divided into discrete moments of time with the selection of parts in which the stress fields are calculated and with the determination of the maximum equivalent stress, using the Simulation Setup tool. The change in these stresses during the erecting cycle is analyzed, and if it is necessary to refine the data, the discrete time step of the mechanism motion is reduced. Specific information about the stress-strain state of the part is obtained in the SolidWorks Simulation application, using data imported from the SolidWorks Motion program about gravitational and inertial forces that occur in the tubbing erector mechanism. Originality. The scientific novelty of the developed method of computer analysis for the mechanism of the manipulator of the UT62 type tubing stacker is manifested in a comprehensive approach to modeling the stress-strain state, which includes the integration of data from SolidWorks Motion and SolidWorks Simulation programs for accurate determination of stresses, taking into account both gravitational forces and dynamic loads, which ensures high accuracy and complexity of calculations in real operating conditions of the mechanism. Practical implications. The proposed method can be used for modeling the motion of mechanisms of different complexity, in particular, in the design of domestic tubbing erectors.

Substantiating a method for computer analysis of the stress-strain state of the tubbing erector manipulator mechanism

Purpose. To develop and substantiate a method for computer analysis of the stress-strain state that occurs in the tubbing erector manipulator mechanism with several degrees of freedom, using modern computing complexes based on the finite-element method. Methodology. The research uses software complexes, with the help of which a calculation-theoretical apparatus has been created, which is intended for calculating dynamic and static loads that occur in the elements of the tubbing erector manipulator mechanism. The results of modeling the stress-strain state in the parts and nodes of this mechanism, obtained using the SolidWorks Motion and SolidWorks Simulation programs, have been compared. The errors in these results have been estimated. Findings. The effectiveness of the proposed analysis method, based on the combination of SolidWorks applications, in particular, SolidWorks Motion and SolidWorks Simulation, has been proven on the example of modeling the most complex design of the UT62 tubbing erector mechanism. Originality. For the first time, a method for computer analysis of the stress-strain state that occurs in the tubbing erector manipulator mechanism of the UT62 type has been developed and substantiated. The essence of the method is that initially, using the SolidWorks Motion application, the laws of motion of the motors are set. The tubbing erecting process itself is divided into discrete moments of time with the selection of parts in which the stress fields are calculated and with the determination of the maximum equivalent stress, using the Simulation Setup tool. The change in these stresses during the erecting cycle is analyzed, and if it is necessary to refine the data, the discrete time step of the mechanism motion is reduced. Specific information about the stress-strain state of the part is obtained in the SolidWorks Simulation application, using data imported from the SolidWorks Motion program about gravitational and inertial forces that occur in the tubbing erector mechanism. Practical value. The proposed method can be used for modeling the motion of mechanisms of different complexity, in particular, in the design of domestic tubbing erectors.

Hyper Coupled Map Lattices for Hiding Multiple Images

The concept of a hyper coupled map lattice (CML) is presented in this paper. The complexity of the lattice is increased not by adding another spatial dimension of the lattice but by replacing scalar nodal variables by multidimensional square matrices of iterative variables. The proposed scheme exploits the nonlinear effects of the spatiotemporal divergence induced by nilpotent nodal matrices to generate separate secret images at different discrete moments of time during the evolution of the CML. The time variable plays the primary role in the decoding stage of the scheme. The carrying capacity of the proposed scheme is n − 1 different dichotomous digital images, where n is the dimension of the nilpotent nodal matrices. Computational experiments are used to demonstrate the efficacy of the proposed scheme.

Dynamic grain models via fast heuristics for diagram representations

ABSTRACT The present paper introduces a mathematical model for analysing dynamic grain growth. In particular, we show how the characteristic measurements grain volumes, centroids, and central second-order moments at discrete moments in time can be quickly turned into a continuous description of the grain growth process in terms of geometric diagrams (which largely generalize the well-known Voronoi and Laguerre tessellations). We give a theoretical analysis of common optimization-free heuristics in terms of discriminant analysis and evaluate the computational behaviour of our algorithm on real-world data.

Toward a Super-COP? Timing, Temporality, and Rethinking World Climate Governance

Toward a Super-COP? Timing, Temporality, and Rethinking World Climate Governance

Defining state-transition diagrams as part of technical state monitoring

State-transition diagrams are given for the situation of monitoring that takes into consideration explicit and hidden failures.Aim. To generate state-transition diagrams that are used in the development of the dependability model and associated calculation methods under conditions of periodic monitoring.Methods. The method is based on the classification of failures by detection, i.e., explicit and hidden failures. Accordingly, facility condition inspection may be continuous and/or periodical. Periodical inspection is conducted at equal intervals. The state-transition diagram is generated taking into consideration the relations between the states and events, i.e., each state can be the cause of a certain event and, at the same time, a state is the consequence of an event. Similarly, each event is the cause of a state change, while at the same time being a consequence of a state. Within the time between two periodic inspections, continuous-time transitions take place due to failures. This process is described by the theory of continuous-time Markov processes. Upon the completion of a periodic inspection operation, the up state ensues if an operable item is subjected to inspection. If an item proves to be down, it is submitted to maintenance. Such transitions in discrete moments in time are described by semi-Markov processes. For a better understanding of the information set forth in the paper, a list is provided of the used terminology referring to state standards.Results. Under the adopted failure modes within a single inspection period, state transitions are described in continuous time, while after the inspection operation they are described in discrete time. Within each inspection period, a system of differential equations is defined and solved based on the assumption of initial up state. Based on the deduced probabilities, the probabilities of a period without failures and with a failure are calculated along with the mean time of up and down states within the period with a failure. Should failures be detected, maintenance state is initiated. The probability of such transition is within the theory of semi-Markov processes. Based on the probabilities of the semi-Markov process, the mean failure-free periods are calculated. Such model adequately reflects the dependability processes of operated equipment. If the method is not fully observed, the dependability indicators may take significantly different values. Three examples of state-transition diagrams are given: taking into account only explicit failures, only hidden failures and both.Discussion and conclusions. The presented approach with a constant (regular) inspection period allows building models that adequately reflect processes occurring in actual systems. All operations involved in the implementation of simulated processes are performed based on the theory of continuous-time Markov processes and semi-Markov processes. The operations are performed in matrix form. Such method enables mathematical operations using computers. The presented approach can be used for improving dependability models of technical systems.

СРАВНИТЕЛЬНЫЙ АНАЛИЗ ЦЕНТРАЛИЗОВАННОГО И ДЕЦЕНТРАЛИЗОВАННОГО АЛГОРИТМОВ ДВИЖЕНИЯ СТРОЕМ БЛА МУЛЬТИКОПТЕРНОГО ТИПА

The development of robotics makes their group application relevant for solving varioustasks. The effectiveness of performing the tasks of detecting and determining the coordinates ofobjects by a group of robots significantly depends on the accuracy of maintaining a given formation.In this regard, the task of determining motion planning algorithms that ensure the greatestaccuracy of maintaining a given formation is of practical interest. This article is devoted to thestudy of the accuracy of maintaining the formation of a multicopter-type UAV group using a centralizedmotion planning algorithm and a decentralized algorithm. The centralized algorithm usesa master UAV, which transmits its coordinates to the slave UAVs. Based on the coordinates obtainedand the given framework of the formation, the guided UAVs plan their movement. In a decentralizedsystem, neighboring UAV groups transmit their coordinates to each other, on the basisof which the movement of a separate UAV is planned. The accuracy of the control system is investigateddepending on the errors of the navigation system and the frequency of updating data on theposition of the leading or neighboring UAVs. It is assumed that the group's UAVs determine their coordinates in discrete moments of time using an external navigation system. Centralized anddecentralized algorithms are worked out by the same motion control system. The algorithms areinvestigated in this article by numerical modeling methods. In the process of simulation, models ofkinematics, dynamics and actuators are taken into account, as well as models for the formation oferrors in the navigation system. It is shown that the de-centralized algorithm of group motionplanning provides higher accuracy compared to the centralized algorithm. However, the technicalimplementation of a decentralized algorithm is more complicated from the point of view of organizinga group communication system. In a centralized system, data transmission from the masterUAV to the slave should be implemented. In a decentralized system, it is required to implementnetwork communication.

Adaptive efficient estimation for generalized semi-Markov big data models

In this paper we study generalized semi-Markov high dimension regression models in continuous time, observed at fixed discrete time moments. The generalized semi-Markov process has dependent jumps and, therefore, it is an extension of the semi-Markov regression introduced in Barbu et al. (Stat Inference Stoch Process 22:187–231, 2019a). For such models we consider estimation problems in nonparametric setting. To this end, we develop model selection procedures for which sharp non-asymptotic oracle inequalities for the robust risks are obtained. Moreover, we give constructive sufficient conditions which provide through the obtained oracle inequalities the adaptive robust efficiency property in the minimax sense. It should be noted also that, for these results, we do not use neither sparse conditions nor the parameter dimension in the model. As examples, regression models constructed through spherical symmetric noise impulses and truncated fractional Poisson processes are considered. Numerical Monte-Carlo simulations confirming the theoretical results are given in the supplementary materials.

Реалізація прийняття рішень організаційного управління на основі онтології діяльності

The complexity of the tasks of making decisions in organizational management is stipulated by the extremely large dimension of the management space, the weak structure of the tasks being solved due to their uniqueness, the uncertainty of conditions, and the changeability of the goals (aspects) of management. The ontological approach provides a range of advantages in solving the problem of organizational management automatization. The use of the activity ontology makes it possible to significantly expand the scope of formalized methods application to organ-izational multi-aspect management. Due to the ontology, the detailing of managerial alterna-tives is increased, and the factors of the performers’ activity and the multi-aspect nature of management are taken into account. The article considers the interpretation of the process of forming managerial alternatives as a process of the threefold grouping of elements of the activi-ty ontology and discrete moments of time. The method of distributed formation of an ontologi-cal network with the grouping of its elements using expert methods, implemented in the prac-tice of designing corporate ACS, is described. The topical scientific tasks of the development of the scientific and methodological apparatus of organizational management based on ontology are formulated. These tasks include the formation of decision-making spaces that are invariant to aspects of management, the concretization of characteristics-invariants and reconfiguration of procedural regulations concerning the actual aspect of management, synthesis of alternatives based on the ontology elements. Some approaches to solving the formulated tasks are shown. The formulation of the problem of synthesizing a system of procedures with given input and output parameters is presented as a problem of tensor transformation of an initial multi-coil electrical network (library of procedures) into a network with the given parameters. A tensor equation is proposed. Its solution allows determining the values of voltages on the intermediate coils of the original electrical network which forms the structural connections for the synthe-sized procedural regulations.

Continuous–discrete cell mass distribution observer design using impulsive ISS Lyapunov functions

The problem of estimating the cell–mass distribution in a yeast cell fermentation process based on non–periodic discrete–time cell mass distribution measurements is addressed. Given that the underlying problem can be cast into a nonlinear error propagation subsystem coupled with an impulsive (i.e., at discrete moments in time) stabilization scheme for the observation error, the design problem is addressed within the framework of impulsive ISS Lyapunov theory. Using a finite–dimensional approximation of the cell population balance equations modeling the cell mass distribution evolution over time, the proposed approach yields conditions on the maximum time between measurements and the resulting ISS gain with respect to the measurement noise disturbance. In order to make the resulting global conditions more accessible and easier to verify in a real process application, the obtained conditions are cast into an additional application–oriented approximation that is amenable to tuning for a given process at hand. The approach is illustrated and tested using experimental data from a yeast batch fermentation.

Method of description for the dynamics of the signal delay change in discrete time with changing aircraft position in space

The article presents an innovative approach to the description of the aircraft range parameter in discrete time when simulating the process of its repositioning in space. A method of describing the dynamics of changes in signal delay in discrete time when the aircraft is changing its spatial position is proposed. Such a model adequately describes the change in signal delay in discrete time. The direction of estimation for the adequacy of radio signal delay change simulation in algorithms of optimum filtration is defined in accordance with the aerodynamic properties of the aircraft. Simulation of the signal delay dynamic change is carried out (it is completely described by the dynamics of change in the distance to the aircraft). The transformation stages of simulation data for the initial model in continuous time with realization of the standard Gaussian random numbers are justified. Information on the simulation data transformation taking into account the correlation matrix of discrete white noise is provided. A method of calculating the transition matrix through the Laplace transform is proposed. The scientific-applied direction of research is determined – it lies in the development of a method for the legitimate representation for the mathematical model of aircraft’s changing range in discrete time within one-dimensional space: the longitudinal and the transverse dimensions. This approach takes into account the continuous description for a system of stochastic differential equations. A comprehensive algorithm for modeling the values of discrete white noise on modern computer equipment and calculating the dynamics of the aircraft range parameter changes is proposed. This algorithm allows to correctly form the a priori information about the change of the vector parameters of the aircraft spatial position at discrete moments of time. As a result, it was shown that the use of the obtained information in the optimal filtering algorithms minimizes the error when determining distance to the aircraft and, accordingly, allows to increase the accuracy and adequacy of the signal delay simulation in discrete time. The results of this research can be used in modernization of the existing models and development of promising on-board radar stations, integrated rangefinders, systems of radio technical reconnaissance and electronic warfare systems, as well as in technical implementation of aircraft flight simulation systems.

Self-organizing Strategies in Game of Agent Movement

In this paper, a stochastic game model of self-organization of strategies of stochastic game of mobile agents in the form of cyclic behavioral patterns, which consist of coordinated strategies for moving agents in a limited discrete space, is developed. The behavioral pattern of a multi-agent system is a visualized form of orderly movement of agents that arises from their initial chaotic movement during the learning of a stochastic game. The mobility of multi-step stochastic game agents is ensured by the fact that in discrete moments of time they randomly, simultaneously and independently choose their own pure strategy of moving in one of the possible directions. Current player payments are defined as loss functions that depend on the strategies of neighboring players. These functions are formed from the penalty for irregular spacing of agents in a limited discrete space and the penalty for collisions when moving agents. Random selection of players’ strategies aims to minimize their average loss functions. The generation of sequences of pure strategies is performed by a discrete distribution based on the vectors of mixed strategies. The elements of the vectors of mixed strategies are the conditional probabilities of choosing the appropriate pure displacement strategies. Mixed strategies change over time, adaptively taking into account the value of current losses. This provides an increase in the probability of choosing those strategies that lead to a decrease in the functions of average losses. The dynamics of the vectors of mixed strategies is determined by the Markov recurrent method, for the construction of which a stochastic approximation of the modified condition of complementary non- rigidity, which is valid at Nash equilibrium points, is performed, and a projection operator for expandable unit epsilon simplex is applied. The convergence of the recurrent game method is ensured by compliance with the fundamental conditions and limitations of stochastic approximation. The stochastic game begins with untrained mixed strategies that set a chaotic picture of moving agents. During the learning of the stochastic game, the vectors of mixed strategies are purposefully changed so as to ensure an orderly, conflict-free movement of agents. As a result of computer simulation of stochastic game, cyclic patterns of self-organization of mobile agents on the surface of a discrete torus and within a rectangular region on a plane are obtained. The reliability of the experimental studies was confirmed by the similarity of the obtained patterns of moving agents for different sequences of random variables. The results of the study are proposed to be used in practice for the construction of distributed systems with elements of self-organization, solving various flow and transport problems and collective decision-making in conditions of uncertainty.

Stochastic multi-step saddle point process minimax control parametric synthesis

The actual research problem of operations is development of methods of increase of a management efficiency by processes of the conflict nature. Article is devoted to development of methods of increase of a management efficiency by such processes. The purpose of article is the substantiation of the approach to parametrical synthesis of optimum control by multi-step stochastic minimax processes and procedures of the numerical analysis of likelihood dynamic characteristics of process. Formalization of process consists in definition of its type, a vector of phase coordinates and corresponding restrictions, the task of set of the actions sold by each the parties, efficiency of each action, control parameters (varied parameters of process) which task of values each of the parties influences a course of process, control restrictions, criteria of efficiency of the parties expressed through elements of a vector of phase coordinates. Discrete final stochastic process is considered. Change of phase coordinates occurs during the discrete moments of time, named steps of process. Phase coordinates depend on values of two groups of control parameters (controls of the counteracting parties). Within the limits of the modern theory of optimization of stochastic systems procedure of synthesis of optimum control is realized two-phase. At the first stage with use of analytical methods the structure of optimum control is determined. For these purposes the simplified determined model of process can to be used. At the second stage parametrical control optimization with use of algorithmic methods and computing procedures statistical linearization is carried out. Dynamics of process is described vectorial finite-difference equation. It is necessary to distinguish cases when there is saddle a point and when saddle the point is absent. Parametrical synthesis of optimum control is possible only in the first case. It is considered three basic variants of the equation: the linear equation; the nonlinear equation with optimum controls on border of a range of definition; the nonlinear equation with optimum controls inside of a range of definition. For the first variant there is an effective algorithm of parametrical synthesis of optimum control. For the second variant of synthesis of optimum control it is possible, but the algorithm is not effective. For the third variant to determine optimum managements it is not possible. Procedure statistical linearization is offered. Procedure consists in generation of set of realizations of the casual process set by the vector equation, calculation of optimum control for each concrete realization and the further statistical processing of the received results. The process described by the piecewise linear vector equation, is a special case of nonlinear process. At that it keeps property of independence of optimum control from coordinates of process. It provides expansion of a scope of effective computing procedure of synthesis of optimum control on a new class of piecewise linear processes. Property of a constancy process Hamiltonian can be used as criterion of correctness of calculation of optimum control in concrete cases. Application of the offered procedure provides use of methods of statistical modelling for the decision of tasks of the analysis of dynamics of the conflict and synthesis of optimum control in view of nonlinearity of functions of losses of the parties, dependence of efficiency of means used by them on the random factors formalized in the form of stochastic functions with various likelihood distributions, and also uncertainty concerning actions of the opponent.

Case Study of Self-Organization Processes in Information System Caching Components

Most modern Information Systems (IS) are designed with Object-Relational Mapping components (ORM). Such components bring down the number of queries to the database server and therefore increase the system performance. Caching mechanisms in software components are complex dynamic systems of open type. A simulation model of cache-application interaction has been created to assess the critical modes of the system. The authors suggest accumulating the cache elements states during the application run at discrete moments of time and present them as multi-variable time series. This work also suggests a method for reconstruction of phase-plane portrait of the system with multidimensional dynamics of the cache elements states. The article shows self-organization processes in caching components of information systems and illustrates the variability of system states for various initial conditions followed by transition to steady-state conditions. In particular, the paper illustrates dissipative structures as well as deterministic chaos with the complete determinism of queries in simulated information systems.

ОСОБЛИВОСТІ ВСТАНОВЛЕННЯ ВЕЛИЧИНИ МЕТРИКИ, ЩО ЗАСТОСОВУЄТЬСЯ ДЛЯ ІДЕНТИФІКАЦІЇ ПОДІБНОСТІ МАРШРУТІВ СУДЕН УСИСТЕМІ ВИСВІТЛЕННЯ НАДВОДНОЇ ОБСТАНОВКИ

The information component of the modern model of border protection in the maritime area is implemented using the integrated information and telecommunication system of the Marine Guard "Gart-12". The use of this system allows to obtain information about the current position of ships, port of departure and port of destination, type of cargo and other additional data. When choosing a vessel in the surface lighting system, it is possible to view the previous points of its route, obtained at certain discrete moments of time. The latter is necessary to establish possible signs of violation of the rules of the border regime by violators of border legislation. In this case, the visualization of the route is carried out using a linear approximation, and a possible violation of the rules of the border regime is established by comparing the route of an individual vessel and a cluster of routes connecting the departure and destination of the vessel. However, the use of linear approximation in the formation of a continuous approximation route of the vessel is questionable and unconfirmed. The influence of the type of approximation in the construction of a continuous route of the vessel through a set of discrete points of the vessel's location at fixed moments of time on the value of the metric used to establish the similarity of the route of two different vessels is investigated. The study involved: the formalization of the studied problem; analysis of the possibility of using linear, piecewise-square approximation, approximation using Lagrange interpolation polynomial and Newton's interpolation polynomial, as well as spline interpolation for its solution; calculation of metrics to establish the degree of similarity of ship routes; testing the hypothesis about the feasibility of applying a linear approximation based on a comparison of the results using different approximation methods. Software-algorithmic implementation of the solution of the researched problem, carried out by the authors, allows to automate the process of hypothesis testing and is the basis for further study of the influence of the type of approximation on the shape of the ship route cluster.

THEORETICAL STUDY OF HEAT CONDUCTION IN MULTI-LAYER SPHERICAL BODY WITH PHASE TRANSFORMATION IN LAYERS

The objective of the study is to build a simple yet informative mathematical model that describes the heat conduction in a spherical multi-layer body with phase transformation in the layers. The numerical scheme based on the theory of Markov chains is proposed to solve this problem numerically. The radial sector of the body is divided into finite number of spherical perfectly mixed cells of different volume, which form a chain of cells. The heat exchange between the cells is described with the heat conduction matrix, the entries of which depend on the local thermo-physical properties of material in the cells (heat conduction coefficient, density, specific heat capacity). These properties can vary from one cell to another and with time. The outer cell of the chain can exchange with heat with outside environment, the temperature of which can vary with time. The state of the process is observed in discrete moments of time separated by small but finite transition duration. If the temperature of a cell reaches the value, the phase transformation begins at which, the evolution of the cell thermal and phase state is described with the corresponding kinetic equation of the phase transformation. The process of melting and solidification is used as the example to verify the qualitative predictability of the model. The graphs of temperature distribution evolution and diagrams of phase content distribution in a multi-layer spherical body are presented. The obtained results on the evolution of the thermal and phase state of the ball have no contradiction to the physical sense of the process. The proposed algorithm has very low computational time (1-3 min for one regime). The other processes of the phase transformation can be easily implemented in the model, for instance, drying, exothermic and endothermic chemical reactions, etc.

Space-time prisms on a sphere with applications to long-distance movement

ABSTRACT We study large-scale movement on the surface of the earth that is recorded at discrete moments in time and are interested in the uncertainty between recorded locations. Hereto, we define and study space-time prisms for movement on the surface of the earth (modelled as a sphere). We give description of these prisms and their spatial projection onto the sphere (the so-called ‘potential path area’) with respect to spherical coordinates, given by latitude and longitude, and Cartesian coordinates. We also study space-time prisms and their spatial projections on the Mercator map projection. Finally, we show how these concepts can be applied to intercontinental movement. Possible applications include the flight of birds during the migration seasons, the trail of sea or land mammals over longer periods of time and the path of adrift containers on the oceans.