Pub Date : 2021-05-01DOI: 10.34229/1028-0979-2021-3-3
V. Stoyan, S. Voloshchuk
Complex problems of three-dimensional elasticity theory for thick elastic plates with arbitrary geometry of their lateral surface are solved. Analytical dependencies of the components of the elastic dynamic displacements’ field of the plate’s inner points from the boundary-surface external-dynamic disturbing factors, defined by continuous functions or their values’ vectors, are constructed. It is assumed, that these disturbances have a classically defined powerful character, or are specified by a certain number of differential transformations of the field’s components of the plate’s dynamic displacement points. The absence of quantitative and qualitative restrictions on the determined transformations of the initial-boundary problems of the considered plates’ dynamics makes it incorrect and unsolvable by methods of classical and computational mathematics. The methodology of root-mean square mathematical modeling of discretely and continuously specified observations for the initial-boundary plate’s condition by the system of modeling functions and their values’ vectors is proposed in the paper. Constructed in this way field’s components of spatial-dynamic displacements of the plate’s points, precisely satisfying classical Lyame equation, with the available information on its initial-boundary condition, are agreed according to the root-mean square criterion. The problem of the obtained solutions’ ambiguity is investigated, their accuracy evaluation in accordance with the information on the external- dynamic condition of the investigated plate is conducted. The plate’s dynamics in the particular mode, for cases of information lack on external- dynamic influences on it and under the conditions of its geometric background according to spatial coordinates. The computer realization of the obtained mathematical results is engineeringly simple and can be easily implemented with the help of well-known methods of computational mathematics.
{"title":"ON THREE-DIMENSIONAL INITIAL-BOUNDARY PROBLEMS OF THICK ELASTIC PLATES’ DYNAMICS","authors":"V. Stoyan, S. Voloshchuk","doi":"10.34229/1028-0979-2021-3-3","DOIUrl":"https://doi.org/10.34229/1028-0979-2021-3-3","url":null,"abstract":"Complex problems of three-dimensional elasticity theory for thick elastic plates with arbitrary geometry of their lateral surface are solved. Analytical dependencies of the components of the elastic dynamic displacements’ field of the plate’s inner points from the boundary-surface external-dynamic disturbing factors, defined by continuous functions or their values’ vectors, are constructed. It is assumed, that these disturbances have a classically defined powerful character, or are specified by a certain number of differential transformations of the field’s components of the plate’s dynamic displacement points. The absence of quantitative and qualitative restrictions on the determined transformations of the initial-boundary problems of the considered plates’ dynamics makes it incorrect and unsolvable by methods of classical and computational mathematics. The methodology of root-mean square mathematical modeling of discretely and continuously specified observations for the initial-boundary plate’s condition by the system of modeling functions and their values’ vectors is proposed in the paper. Constructed in this way field’s components of spatial-dynamic displacements of the plate’s points, precisely satisfying classical Lyame equation, with the available information on its initial-boundary condition, are agreed according to the root-mean square criterion. The problem of the obtained solutions’ ambiguity is investigated, their accuracy evaluation in accordance with the information on the external- dynamic condition of the investigated plate is conducted. The plate’s dynamics in the particular mode, for cases of information lack on external- dynamic influences on it and under the conditions of its geometric background according to spatial coordinates. The computer realization of the obtained mathematical results is engineeringly simple and can be easily implemented with the help of well-known methods of computational mathematics.","PeriodicalId":54874,"journal":{"name":"Journal of Automation and Information Sciences","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47347600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-01DOI: 10.34229/1028-0979-2021-3-2
Mikhail Petrik, A. Chikrii, I. Mudrik
The foundations of mathematical modeling and identification of parameters of heterogeneous abnormal neurological movements (ANM) in multicomponent neuro-biosystems with cognitive feedback have been developed. Based on the methods of integral transformations and spectral analysis developed by the authors for heterogeneous media, a new approach to the construction of hybrid models of wave signal propagation is proposed that describes unwanted tremors of the patient's arm (T-object) as a result of an unconstrained contraction of skeletal muscles due to the cognitive effects of a certain group of neural nodes in the cortex cerebral (CC). A hybrid model of a neuro-biosystem is developed, which describes the state and behavior, namely, the segment-by-segment description of 3D elements of the ANM trajectories of the T-object, taking into account the matrix of cognitive influences of the groups of neuro nodes of the CC. On the basis of hybrid integral Fourier transforms a high-speed analytical vector solution of the model is obtained, which describes the elements of the trajectories on each AND-segment. A new method for calculating of hybrid spectral function, spectral values and matrix of cognitive influences of CC neuronodes is proposed, which determine hybrid integral transformation of solution construction. New non-classical problems of multi-parameter identification of neuro-feedback systems in heterogeneous media based on minimization of the residual functional between observation trajectories and their model analogs are formulated and solved. High-performance algorithms of the amplitude-frequency characteristics identifying of a feedback-system in analytical expressions for the gradients of the residual functional have been constructed, which allow parallel-computations on multicore computers. Computer modeling and identification of ANM trajectories of the studied neuro-feedback-system have been performed.
{"title":"SIMULATION AND PARAMETERS-IDENTIFICATION METHODS OF HETEROGENEOUS ABNORMAL NEUROLOGICAL MOVEMENTS IN MULTICOMPONENT NEURO-BIOSYSTEMS WITH COGNITIVE FEEDBACK","authors":"Mikhail Petrik, A. Chikrii, I. Mudrik","doi":"10.34229/1028-0979-2021-3-2","DOIUrl":"https://doi.org/10.34229/1028-0979-2021-3-2","url":null,"abstract":"The foundations of mathematical modeling and identification of parameters of heterogeneous abnormal neurological movements (ANM) in multicomponent neuro-biosystems with cognitive feedback have been developed. Based on the methods of integral transformations and spectral analysis developed by the authors for heterogeneous media, a new approach to the construction of hybrid models of wave signal propagation is proposed that describes unwanted tremors of the patient's arm (T-object) as a result of an unconstrained contraction of skeletal muscles due to the cognitive effects of a certain group of neural nodes in the cortex cerebral (CC). A hybrid model of a neuro-biosystem is developed, which describes the state and behavior, namely, the segment-by-segment description of 3D elements of the ANM trajectories of the T-object, taking into account the matrix of cognitive influences of the groups of neuro nodes of the CC. On the basis of hybrid integral Fourier transforms a high-speed analytical vector solution of the model is obtained, which describes the elements of the trajectories on each AND-segment. A new method for calculating of hybrid spectral function, spectral values and matrix of cognitive influences of CC neuronodes is proposed, which determine hybrid integral transformation of solution construction. New non-classical problems of multi-parameter identification of neuro-feedback systems in heterogeneous media based on minimization of the residual functional between observation trajectories and their model analogs are formulated and solved. High-performance algorithms of the amplitude-frequency characteristics identifying of a feedback-system in analytical expressions for the gradients of the residual functional have been constructed, which allow parallel-computations on multicore computers. Computer modeling and identification of ANM trajectories of the studied neuro-feedback-system have been performed.","PeriodicalId":54874,"journal":{"name":"Journal of Automation and Information Sciences","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49256090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-01DOI: 10.34229/0572-2691-2021-2-1
A. L. Gurin, Irina S. Grashchenko, L. Savchenko
We consider one method of solving the problem of mathematical safe on certain graphs called parametric. Its gist consist in denoting some variables, corresponding to graph vertices, by certain parameters. Other unknown variables are expressed through these parameters. Then unknown variables chosen in special way are compared and the mentioned parameters are found by solving additional system of equations for these parameters. Dimension of this system is equal to the number of parameters. Solution to the problem i.e. all unknown variables of the original system, are found by solving additional system of equations. In the paper this method is described on specially chosen examples. The method is demonstrated by solving the mathematical safe problem on the graphs of «chain», «ladder» and «window» types that showed its efficiency. Besides special attention is paid to special cases when solution does not exist. This occurs in the cases when the weighed sum of system equations is not divisable without remainder to its modulo. In such cases, to find solution the initial state of the vector b is corrected in such a way that the weighted sum of equations satisfies the above mentioned condition. Then solution of the problem is performed according to the general method scheme.
{"title":"PARAMETRIC METHOD OF SOLVING PROBLEMS OF MATHEMATICAL SAFE ON GRAPHS","authors":"A. L. Gurin, Irina S. Grashchenko, L. Savchenko","doi":"10.34229/0572-2691-2021-2-1","DOIUrl":"https://doi.org/10.34229/0572-2691-2021-2-1","url":null,"abstract":"We consider one method of solving the problem of mathematical safe on certain graphs called parametric. Its gist consist in denoting some variables, corresponding to graph vertices, by certain parameters. Other unknown variables are expressed through these parameters. Then unknown variables chosen in special way are compared and the mentioned parameters are found by solving additional system of equations for these parameters. Dimension of this system is equal to the number of parameters. Solution to the problem i.e. all unknown variables of the original system, are found by solving additional system of equations. In the paper this method is described on specially chosen examples. The method is demonstrated by solving the mathematical safe problem on the graphs of «chain», «ladder» and «window» types that showed its efficiency. Besides special attention is paid to special cases when solution does not exist. This occurs in the cases when the weighed sum of system equations is not divisable without remainder to its modulo. In such cases, to find solution the initial state of the vector b is corrected in such a way that the weighted sum of equations satisfies the above mentioned condition. Then solution of the problem is performed according to the general method scheme.","PeriodicalId":54874,"journal":{"name":"Journal of Automation and Information Sciences","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49602515","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-01DOI: 10.34229/1028-0979-2021-2-11
M. Rakushev
To predict the motion of spacecrafts, a numerical-analytical method for integrating the differential equation of the orbital motion of a spacecraft stabilized by the Baumgart differential method is proposed. The stabilization of the differential equation of motion by the Baumgart method is carried out according to the energy of the spacecraft. Stabilization is carried out to reduce the influence of the Lyapunov instability on the accumulation of numerical errors in the integration of the differential equation, which is effective when conducting a long-term numerical prediction of the motion of spacecraft. Integration of the stabilized equation is based on differential Taylor transformations. Computational schemes with a constant step and an integration order are considered, as well as schemes with adaptation by an integration step and order. For adaptive schemes, the results of forecasting the motion of spacecraft according to the criterion “accuracy-computational complexity» for a given relative error of integration with respect to integration phase variables and spacecraft energy are presented. It is shown that both options require setting various internal adaptation parameters, but they have comparable efficiency. Recommendations are proposed on the use of the developed method for integrating energy-stabilized equations for predicting the motion of spacecraft in the near space in the Greenwich rectangular coordinate system.
{"title":"A METHOD FOR PREDICTING ENERGY-STABILIZED MOTION OF SPACECRAFT BASED ON DIFFERENTIAL TAYLOR TRANSFORMATIONS","authors":"M. Rakushev","doi":"10.34229/1028-0979-2021-2-11","DOIUrl":"https://doi.org/10.34229/1028-0979-2021-2-11","url":null,"abstract":"To predict the motion of spacecrafts, a numerical-analytical method for integrating the differential equation of the orbital motion of a spacecraft stabilized by the Baumgart differential method is proposed. The stabilization of the differential equation of motion by the Baumgart method is carried out according to the energy of the spacecraft. Stabilization is carried out to reduce the influence of the Lyapunov instability on the accumulation of numerical errors in the integration of the differential equation, which is effective when conducting a long-term numerical prediction of the motion of spacecraft. Integration of the stabilized equation is based on differential Taylor transformations. Computational schemes with a constant step and an integration order are considered, as well as schemes with adaptation by an integration step and order. For adaptive schemes, the results of forecasting the motion of spacecraft according to the criterion “accuracy-computational complexity» for a given relative error of integration with respect to integration phase variables and spacecraft energy are presented. It is shown that both options require setting various internal adaptation parameters, but they have comparable efficiency. Recommendations are proposed on the use of the developed method for integrating energy-stabilized equations for predicting the motion of spacecraft in the near space in the Greenwich rectangular coordinate system.","PeriodicalId":54874,"journal":{"name":"Journal of Automation and Information Sciences","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47941057","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-01DOI: 10.34229/1028-0979-2021-2-7
A. Lakeyev, V. Rusanov, A. Banshchikov
The analytical conditions (necessary and sufficient) are defined for the solvability of the problem of differential realization of a continuous beam of controlled trajectory curves in the class of bilinear nonautonomous ordinary differential equations (with delay and without it) of the second order in a real separable Hilbert space. The problem under consideration belongs to the type of inverse problems for an additive combination of nonstationary linear and bilinear operators of evolution equations in an infinite-dimensional Hilbert space. The meta-language of this theory is the constructions of tensor products of Hilbert spaces, the structures of lattices with ortho-complementation, and the functional apparatus of the nonlinear Rayleigh-Ritz operator. It is shown that in the case of a finite bundle of trajectories, the presence of a sublinearity-type property of this operator allows us to obtain sufficient conditions for the existence of such realizations. Along the way, the topological-metric conditions for the continuity of the projectivization of the nonlinear Rayleigh-Ritz functional operator with the calculation of the fundamental group of its image are justified. The results obtained provide the motivation for the development of a qualitative theory of nonlinear structural identification of higher-order multi-linear differential models (e.g. for processes, induced by the «brain–machine» interface-platform of the type of Neuralink).
{"title":"TO A TENSOR ANALYSIS OF THE SOLVABILITY OF THE PROBLEM OF REALIZATION OF A SECOND-ORDER BILINEAR SYSTEM WITH DELAY","authors":"A. Lakeyev, V. Rusanov, A. Banshchikov","doi":"10.34229/1028-0979-2021-2-7","DOIUrl":"https://doi.org/10.34229/1028-0979-2021-2-7","url":null,"abstract":"The analytical conditions (necessary and sufficient) are defined for the solvability of the problem of differential realization of a continuous beam of controlled trajectory curves in the class of bilinear nonautonomous ordinary differential equations (with delay and without it) of the second order in a real separable Hilbert space. The problem under consideration belongs to the type of inverse problems for an additive combination of nonstationary linear and bilinear operators of evolution equations in an infinite-dimensional Hilbert space. The meta-language of this theory is the constructions of tensor products of Hilbert spaces, the structures of lattices with ortho-complementation, and the functional apparatus of the nonlinear Rayleigh-Ritz operator. It is shown that in the case of a finite bundle of trajectories, the presence of a sublinearity-type property of this operator allows us to obtain sufficient conditions for the existence of such realizations. Along the way, the topological-metric conditions for the continuity of the projectivization of the nonlinear Rayleigh-Ritz functional operator with the calculation of the fundamental group of its image are justified. The results obtained provide the motivation for the development of a qualitative theory of nonlinear structural identification of higher-order multi-linear differential models (e.g. for processes, induced by the «brain–machine» interface-platform of the type of Neuralink).","PeriodicalId":54874,"journal":{"name":"Journal of Automation and Information Sciences","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44453233","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-01DOI: 10.34229/1028-0979-2021-2-4
Sergey Sapegin, Gennady Riabtsev, E. Zubareva, I. Basantsov
The article solves the problem of low efficiency of traditional methods for organizing the expert interaction in e-government decision support systems. It is shown that the activity of a highly qualified expert is still insufficiently organized, systematized and informationally provided, although expert assessments significantly affect the effectiveness, efficiency and economy of decisions. The requirements for automated expert systems are formulated. The features of the interaction of experts with each other and with the decision maker are taken into account. The place of automated expert systems in e-government decision support systems has been identified. It is shown that qualified users (experts who are able to evaluate (transform) the data, to generate variants of decision, to establish the rules for choosing the best) are an integral "element" of decision support systems architecture. At the same time, automated expert systems are mandatory component of decision support systems. An expert interoperability software tool, the PsycheaEXPERTUS automated system, is developed and implemented. Testing of this system confirmed that the use of such systems in e-government will simplify expert procedures, reduce unproductive time spent on organizing and conducting face-to-face consultations, and increase the efficiency of attracting specialists to make public policy, when the uncertainty of the environment and resource constraints exist. Further research is planned to be aimed at overcoming the psychological unpreparedness of decision makers for the widespread use of expert systems in Ukraine.
{"title":"TOOL FOR EFFECTIVE EXPERT INTEROPERABILITY IN E-GOVERNMENT DECISION SUPPORT SYSTEMS","authors":"Sergey Sapegin, Gennady Riabtsev, E. Zubareva, I. Basantsov","doi":"10.34229/1028-0979-2021-2-4","DOIUrl":"https://doi.org/10.34229/1028-0979-2021-2-4","url":null,"abstract":"The article solves the problem of low efficiency of traditional methods for organizing the expert interaction in e-government decision support systems. It is shown that the activity of a highly qualified expert is still insufficiently organized, systematized and informationally provided, although expert assessments significantly affect the effectiveness, efficiency and economy of decisions. The requirements for automated expert systems are formulated. The features of the interaction of experts with each other and with the decision maker are taken into account. The place of automated expert systems in e-government decision support systems has been identified. It is shown that qualified users (experts who are able to evaluate (transform) the data, to generate variants of decision, to establish the rules for choosing the best) are an integral \"element\" of decision support systems architecture. At the same time, automated expert systems are mandatory component of decision support systems. An expert interoperability software tool, the PsycheaEXPERTUS automated system, is developed and implemented. Testing of this system confirmed that the use of such systems in e-government will simplify expert procedures, reduce unproductive time spent on organizing and conducting face-to-face consultations, and increase the efficiency of attracting specialists to make public policy, when the uncertainty of the environment and resource constraints exist. Further research is planned to be aimed at overcoming the psychological unpreparedness of decision makers for the widespread use of expert systems in Ukraine.","PeriodicalId":54874,"journal":{"name":"Journal of Automation and Information Sciences","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45064779","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-01DOI: 10.34229/1028-0979-2021-2-2
V. Romanenko, Y. Milyavsky, Georgy Kantsedal
In this paper a cognitive map (CM) of the cryptocurrency in the financial market has been developed based on causal relationships. This is a weighted directed graph with nodes reflecting: the cryptocurrency rate, the volume of cryptocurrency trading, the number of cryptocurrency users, the volume of capitalization, the volume of investments, the volume of speculation in cryptocurrency, supply and demand for cryptocurrency, indirect profit, level of confidence in cryptocurrency, variance of the cryptocurrency rate, integral level of risks when using cryptocurrency. On the basis of the CM, a dynamic model of CM impulse processes is developed in the form of a system of difference equations (Roberts equations). The external control vector for the CM impulse process is selected. It is formed by varying the resources of the following CM nodes coordinates: cryptocurrency supply, trading, capitalization, investment and speculation volumes. Controls are designed in a closed-loop control system of the impulse process based on the automatic control theory methods and are implemented by a decision-maker. A closed-loop control system for the CM impulse process is implemented, which includes a multidimensional discrete state controller synthesized on the basis of the modal control method, which generates the selected control vector and directly affects respective CM nodes by varying their coordinates. The paper solves the problem of a stabilization system design for the unstable transient process of the cryptocurrency rate based on the modal control method. An identification system for the CM weights based on the recurrent least squares method has been developed. The options of modal control with two, three and five controls in the CM with 12 nodes are investigated.
{"title":"AN ADAPTIVE SYSTEM FOR STABILIZING UNSTABLE CRYPTOCURRENCY RATE BASED ON A COGNITIVE MAP IMPULSE PROCESS MODEL","authors":"V. Romanenko, Y. Milyavsky, Georgy Kantsedal","doi":"10.34229/1028-0979-2021-2-2","DOIUrl":"https://doi.org/10.34229/1028-0979-2021-2-2","url":null,"abstract":"In this paper a cognitive map (CM) of the cryptocurrency in the financial market has been developed based on causal relationships. This is a weighted directed graph with nodes reflecting: the cryptocurrency rate, the volume of cryptocurrency trading, the number of cryptocurrency users, the volume of capitalization, the volume of investments, the volume of speculation in cryptocurrency, supply and demand for cryptocurrency, indirect profit, level of confidence in cryptocurrency, variance of the cryptocurrency rate, integral level of risks when using cryptocurrency. On the basis of the CM, a dynamic model of CM impulse processes is developed in the form of a system of difference equations (Roberts equations). The external control vector for the CM impulse process is selected. It is formed by varying the resources of the following CM nodes coordinates: cryptocurrency supply, trading, capitalization, investment and speculation volumes. Controls are designed in a closed-loop control system of the impulse process based on the automatic control theory methods and are implemented by a decision-maker. A closed-loop control system for the CM impulse process is implemented, which includes a multidimensional discrete state controller synthesized on the basis of the modal control method, which generates the selected control vector and directly affects respective CM nodes by varying their coordinates. The paper solves the problem of a stabilization system design for the unstable transient process of the cryptocurrency rate based on the modal control method. An identification system for the CM weights based on the recurrent least squares method has been developed. The options of modal control with two, three and five controls in the CM with 12 nodes are investigated.","PeriodicalId":54874,"journal":{"name":"Journal of Automation and Information Sciences","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41440888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-01DOI: 10.34229/1028-0979-2021-2-8
Yu. I. Kharkevich
In most cases, solutions to problems of the motion of a system of interacting material points are reduced to either ordinary differential equations or partial differential equations. One of the solutions of this type of equations is the so-called generalized Poisson integrals, which in partial cases turn into the well-known Abel-Poisson integrals or biharmonic Poisson integrals. A number of results is known on the approximation of various classes of differentiable periodic and nonperiodic functions by the mentioned above integrals (the so-called Kolmogorov-Nikol’skii problem in the terminology of A.I. Stepanets). Nevertheless, there is a significant drawback practically in all of the solved Kolmogorov-Nikol’skii problems for both Abel-Poisson integrals and Poisson biharmonic integrals from the mathematical modeling (computational experiment) point of view. The core point here is that in most of the previously solved Kolmogorov-Nikol’skii problems for both Abel-Poisson integrals and Poisson biharmonic integrals only the leading and remainder terms of the approximation were obtained, that can significantly affect the accuracy of the computational experiment. In the present paper we obtain exact equalities for approximation of functions from the Sobolev classes by their generalized Poisson integrals. Consequently, the theorem proved in this paper is a generalization and refinement of previously known results characterizing the approximation properties of Abel-Poisson integrals and biharmonic Poisson integrals on the classes of differentiable periodic functions. A peculiarity of the solved in this work problem of approximation for the generalized Poisson integral on the classes of differentiable functions is that the result obtained is successfully written using the well-known Akhiezer-Krein-Favard constants. This fact substantially increases the accuracy of the mathematical modeling result (computational experiment) for a real process described using the generalized Poisson integral. These results can further significantly expand the scope of application of the Kolmogorov-Nikol’skii problems to mathematical modeling.
{"title":"EXACT EQUALITIES FOR APPROXIMATION OF FUNCTIONS FROM THE SOBOLEV CLASS BY THEIR GENERALIZED POISSON INTEGRALS","authors":"Yu. I. Kharkevich","doi":"10.34229/1028-0979-2021-2-8","DOIUrl":"https://doi.org/10.34229/1028-0979-2021-2-8","url":null,"abstract":"In most cases, solutions to problems of the motion of a system of interacting material points are reduced to either ordinary differential equations or partial differential equations. One of the solutions of this type of equations is the so-called generalized Poisson integrals, which in partial cases turn into the well-known Abel-Poisson integrals or biharmonic Poisson integrals. A number of results is known on the approximation of various classes of differentiable periodic and nonperiodic functions by the mentioned above integrals (the so-called Kolmogorov-Nikol’skii problem in the terminology of A.I. Stepanets). Nevertheless, there is a significant drawback practically in all of the solved Kolmogorov-Nikol’skii problems for both Abel-Poisson integrals and Poisson biharmonic integrals from the mathematical modeling (computational experiment) point of view. The core point here is that in most of the previously solved Kolmogorov-Nikol’skii problems for both Abel-Poisson integrals and Poisson biharmonic integrals only the leading and remainder terms of the approximation were obtained, that can significantly affect the accuracy of the computational experiment. In the present paper we obtain exact equalities for approximation of functions from the Sobolev classes by their generalized Poisson integrals. Consequently, the theorem proved in this paper is a generalization and refinement of previously known results characterizing the approximation properties of Abel-Poisson integrals and biharmonic Poisson integrals on the classes of differentiable periodic functions. A peculiarity of the solved in this work problem of approximation for the generalized Poisson integral on the classes of differentiable functions is that the result obtained is successfully written using the well-known Akhiezer-Krein-Favard constants. This fact substantially increases the accuracy of the mathematical modeling result (computational experiment) for a real process described using the generalized Poisson integral. These results can further significantly expand the scope of application of the Kolmogorov-Nikol’skii problems to mathematical modeling.","PeriodicalId":54874,"journal":{"name":"Journal of Automation and Information Sciences","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44635783","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-01DOI: 10.34229/1028-0979-2021-2-3
A. Nakonechny, Grigory Kudin, Petr N. Zinko, T. Zinko
Linear estimation of observations in conditions of various types of interference in order to obtain unbiased estimates is the subject of research in numerous scientific publications. The problem of linear regression analysis in conditions when the elements of vector observations are known matrices that allow small deviations from the calculated ones was studied in previous publications of the authors. Using the technology of pseudo inverse operators, as well as the perturbation method, the problem was solved under the condition that linearly independent matrices are subject to small perturbations. The parameters of the linear estimates were presented in the form of expansions in a small parameter. Over the past decades, solving linear estimation problems under uncertainty has been carried out within the framework of the well-known minimax estimation method. Formally, the problems that arise in this direction are solved in the presence of some spaces for unknown observation parameters, as well as spaces to which observation errors may belong. The coefficients of the linear estimates are determined in the process of optimizing the guaranteed mean-square error of the desired estimate. Thus, the subject of scientific research can be problems of linear estimation of unknown rectangular matrices based on observations from errors with unknown correlation matrices of errors: unknown matrices belong to some bounded set, correlation matrices of random perturbations of the observation vector are unknown, but it is possible to assume cases when they belong to one or another defined bounded set. Some formulations of problems of linear estimation of observations are investigated in the proposed publication. The problem of linear estimation for a vector of observations of a special form is considered, the components of which are known rectangular matrices that are subject to small perturbations. Variants of the problem statement are proposed, which allow obtaining an analytical solution in the first approximation of a small parameter. A test example is presented.
{"title":"GUARANTEED ROOT-MEAN-SQUARE ESTIMATES OF LINEAR MATRIX TRANSFORMATIONS UNDER CONDITIONS OF STATISTICAL UNCERTAINTY","authors":"A. Nakonechny, Grigory Kudin, Petr N. Zinko, T. Zinko","doi":"10.34229/1028-0979-2021-2-3","DOIUrl":"https://doi.org/10.34229/1028-0979-2021-2-3","url":null,"abstract":"Linear estimation of observations in conditions of various types of interference in order to obtain unbiased estimates is the subject of research in numerous scientific publications. The problem of linear regression analysis in conditions when the elements of vector observations are known matrices that allow small deviations from the calculated ones was studied in previous publications of the authors. Using the technology of pseudo inverse operators, as well as the perturbation method, the problem was solved under the condition that linearly independent matrices are subject to small perturbations. The parameters of the linear estimates were presented in the form of expansions in a small parameter. Over the past decades, solving linear estimation problems under uncertainty has been carried out within the framework of the well-known minimax estimation method. Formally, the problems that arise in this direction are solved in the presence of some spaces for unknown observation parameters, as well as spaces to which observation errors may belong. The coefficients of the linear estimates are determined in the process of optimizing the guaranteed mean-square error of the desired estimate. Thus, the subject of scientific research can be problems of linear estimation of unknown rectangular matrices based on observations from errors with unknown correlation matrices of errors: unknown matrices belong to some bounded set, correlation matrices of random perturbations of the observation vector are unknown, but it is possible to assume cases when they belong to one or another defined bounded set. Some formulations of problems of linear estimation of observations are investigated in the proposed publication. The problem of linear estimation for a vector of observations of a special form is considered, the components of which are known rectangular matrices that are subject to small perturbations. Variants of the problem statement are proposed, which allow obtaining an analytical solution in the first approximation of a small parameter. A test example is presented.","PeriodicalId":54874,"journal":{"name":"Journal of Automation and Information Sciences","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42474608","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-01DOI: 10.34229/1028-0979-2021-2-5
M. Mamatov, A. Zunnunov, Egamberdi Esonov
The paper is devoted to the study of the problem of constructing a pursuit strategy in simple differential games of many persons with phase constraints in the state of the players, in the sense of getting into a certain neighborhood of the evader. The game takes place in -dimensional Euclidean space on a convex compact set. The pursuit problem is considered when the number of pursuing players is , that is, less than , in the sense of — captures. A structure for constructing pursuit controls is proposed, which will ensure the completion of the game in a finite time. An upper bound is obtained for the game time for the completion of the pursuit. An auxiliary problem of simple pursuit on a unit cube in the first orthant is considered, and strategies of pursuing players are constructed to complete the game with special initial positions. The results obtained are used to solve differential games with arbitrary initial positions. For this task, a structure for constructing a pursuit strategy is proposed that will ensure the completion of the game in a finite time. The generalization of the problem in the sense of complicating the obstacle is also considered. A more general problem of simple pursuit on a cube of arbitrary size in the first orthant is considered. With the help of the proposed strategies, the possibilities of completing the pursuit are proved and an estimate of the time is obtained. As a consequence of this result, lower and upper bounds are obtained for the pursuit time in a game with ball-type obstacles. Estimates are obtained for the pursuit time when the compact set is an arbitrarily convex set. The concept of a convex set in a direction relative to a section, which is not necessarily convex, is defined. And in it the problem of simple pursuit in a differential game of many players is studied and the possibilities of completing the pursuit using the proposed strategy are shown. The time of completion of the pursuit of the given game is estimated from above.
{"title":"EVALUATION OF THE PURSUIT TIME IN DIFFERENTIAL GAMES OF MANY PLAYERS ON A CONVEX COMPACT","authors":"M. Mamatov, A. Zunnunov, Egamberdi Esonov","doi":"10.34229/1028-0979-2021-2-5","DOIUrl":"https://doi.org/10.34229/1028-0979-2021-2-5","url":null,"abstract":"The paper is devoted to the study of the problem of constructing a pursuit strategy in simple differential games of many persons with phase constraints in the state of the players, in the sense of getting into a certain neighborhood of the evader. The game takes place in -dimensional Euclidean space on a convex compact set. The pursuit problem is considered when the number of pursuing players is , that is, less than , in the sense of — captures. A structure for constructing pursuit controls is proposed, which will ensure the completion of the game in a finite time. An upper bound is obtained for the game time for the completion of the pursuit. An auxiliary problem of simple pursuit on a unit cube in the first orthant is considered, and strategies of pursuing players are constructed to complete the game with special initial positions. The results obtained are used to solve differential games with arbitrary initial positions. For this task, a structure for constructing a pursuit strategy is proposed that will ensure the completion of the game in a finite time. The generalization of the problem in the sense of complicating the obstacle is also considered. A more general problem of simple pursuit on a cube of arbitrary size in the first orthant is considered. With the help of the proposed strategies, the possibilities of completing the pursuit are proved and an estimate of the time is obtained. As a consequence of this result, lower and upper bounds are obtained for the pursuit time in a game with ball-type obstacles. Estimates are obtained for the pursuit time when the compact set is an arbitrarily convex set. The concept of a convex set in a direction relative to a section, which is not necessarily convex, is defined. And in it the problem of simple pursuit in a differential game of many players is studied and the possibilities of completing the pursuit using the proposed strategy are shown. The time of completion of the pursuit of the given game is estimated from above.","PeriodicalId":54874,"journal":{"name":"Journal of Automation and Information Sciences","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46153240","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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