Pub Date : 2020-01-01DOI: 10.17721/2706-9699.2020.1.01
A. Bomba, S. Baranovsky
This paper proposes modification of the simplest model of the infectious disease in the conditions of pharmacotherapy taking into account influence of small-scale spatial distributed diffusion influences. The singular disturbed model problem with time-delay is reduced to a sequence of problems without time-delay for which the corresponding representations of the asymptotic expansions of solutions are constructed. We present the results of numerical experiments that characterize the influence of spatial distributed diffusion «redistributions» of infectious disease factors on the development of the process on condition of pharmacotherapy. The decrease in the maximum level of concentration of pathogenic antigens in the locus of infection due to their diffusion «redistribution» is illustrated.
{"title":"MODELING SMALL-SCALE SPATIAL DISTRIBUTED INFLUENCES ON THE DYNAMICS OF INFECTIOUS DISEASE ON CONDITION OF PHARMACOTHERAPY","authors":"A. Bomba, S. Baranovsky","doi":"10.17721/2706-9699.2020.1.01","DOIUrl":"https://doi.org/10.17721/2706-9699.2020.1.01","url":null,"abstract":"This paper proposes modification of the simplest model of the infectious disease in the conditions of pharmacotherapy taking into account influence of small-scale spatial distributed diffusion influences. The singular disturbed model problem with time-delay is reduced to a sequence of problems without time-delay for which the corresponding representations of the asymptotic expansions of solutions are constructed. We present the results of numerical experiments that characterize the influence of spatial distributed diffusion «redistributions» of infectious disease factors on the development of the process on condition of pharmacotherapy. The decrease in the maximum level of concentration of pathogenic antigens in the locus of infection due to their diffusion «redistribution» is illustrated.","PeriodicalId":40347,"journal":{"name":"Journal of Numerical and Applied Mathematics","volume":"37 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87849488","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 : 2020-01-01DOI: 10.17721/2706-9699.2020.2.04
G. Sandrakov, A. Hulianytskyi
Initial boundary value problems for nonstationary equations of diffusion and filtration in weakly porous media are considered. Assertions about the solvability of such problems and the corresponding homogenized problems with convolutions are given. These statements are proved for general initial data and inhomogeneous initial conditions and are generalizations of classical results on the solvability of initial-boundary value problems for the heat equation. The proofs use the methods of a priori estimates and the well-known Agranovich–Vishik method, developed to study parabolic problems of general type.
{"title":"SOLVABILITY OF HOMOGENIZED PROBLEMS WITH CONVOLUTIONS FOR WEAKLY POROUS MEDIA","authors":"G. Sandrakov, A. Hulianytskyi","doi":"10.17721/2706-9699.2020.2.04","DOIUrl":"https://doi.org/10.17721/2706-9699.2020.2.04","url":null,"abstract":"Initial boundary value problems for nonstationary equations of diffusion and filtration in weakly porous media are considered. Assertions about the solvability of such problems and the corresponding homogenized problems with convolutions are given. These statements are proved for general initial data and inhomogeneous initial conditions and are generalizations of classical results on the solvability of initial-boundary value problems for the heat equation. The proofs use the methods of a priori estimates and the well-known Agranovich–Vishik method, developed to study parabolic problems of general type.","PeriodicalId":40347,"journal":{"name":"Journal of Numerical and Applied Mathematics","volume":"61 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89573669","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 : 2020-01-01DOI: 10.17721/2706-9699.2020.2.05
O. Kharkov, Yana Vedel, V. V. Semenov
The paper develops the theory of existence and necessary optimality conditions for optimal control problems with a vector quality criterion for systems with distributed parameters and generalized impacts. The concept of $(K, e, epsilon)$-approximate efficiency is investigated. Necessary conditions for $(K, e, epsilon)$-approximate efficiency of admissible controls in the form of variational inclusions are proved. Methods for solving problems of vector optimization of linear distributed systems with generalized control are proposed. Convergence of algorithms with errors is proved.
{"title":"METHODS FOR PROBLEMS OF VECTOR GENERALIZED OPTIMAL CONTROL OF SYSTEMS WITH DISTRIBUTED PARAMETERS","authors":"O. Kharkov, Yana Vedel, V. V. Semenov","doi":"10.17721/2706-9699.2020.2.05","DOIUrl":"https://doi.org/10.17721/2706-9699.2020.2.05","url":null,"abstract":"The paper develops the theory of existence and necessary optimality conditions for optimal control problems with a vector quality criterion for systems with distributed parameters and generalized impacts. The concept of $(K, e, epsilon)$-approximate efficiency is investigated. Necessary conditions for $(K, e, epsilon)$-approximate efficiency of admissible controls in the form of variational inclusions are proved. Methods for solving problems of vector optimization of linear distributed systems with generalized control are proposed. Convergence of algorithms with errors is proved.","PeriodicalId":40347,"journal":{"name":"Journal of Numerical and Applied Mathematics","volume":"24 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81259986","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 : 2020-01-01DOI: 10.17721/2706-9699.2020.1.02
Yana Vedel, S. Denisov, V. Semenov
In this paper, we consider bilevel problem: variational inequality problem over the set of solutions the equilibrium problems. To solve this problem, an iterative algorithm is proposed that combines the ideas of a two-stage proximal method and iterative regularization. For monotone bifunctions of Lipschitz type and strongly monotone Lipschitz continuous operators, the theorem on strong convergence of sequences generated by the algorithm is proved.
{"title":"ALGORITHM FOR VARIATIONAL INEQUALITY PROBLEM OVER THE SET OF SOLUTIONS THE EQUILIBRIUM PROBLEMS","authors":"Yana Vedel, S. Denisov, V. Semenov","doi":"10.17721/2706-9699.2020.1.02","DOIUrl":"https://doi.org/10.17721/2706-9699.2020.1.02","url":null,"abstract":"In this paper, we consider bilevel problem: variational inequality problem over the set of solutions the equilibrium problems. To solve this problem, an iterative algorithm is proposed that combines the ideas of a two-stage proximal method and iterative regularization. For monotone bifunctions of Lipschitz type and strongly monotone Lipschitz continuous operators, the theorem on strong convergence of sequences generated by the algorithm is proved.","PeriodicalId":40347,"journal":{"name":"Journal of Numerical and Applied Mathematics","volume":"69 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74123542","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 : 2020-01-01DOI: 10.17721/2706-9699.2020.2.03
M. Savkina
In the paper in case heteroscedastic independent deviations a regression model whose function has the form $f(x) = ax^2+bx+c$, where $a$, $b$ and $c$ are unknown parameters, is studied. Approximate values (observations) of functions $f(x)$ are registered at equidistant points of a line segment. The theorem which is proved at the paper gives a sufficient condition on the variance of the deviations at which the Aitken estimation of parameter $a$ coincides with its estimation of the LS in the case of odd number of observation points and bisymmetric covariance matrix. Under this condition, the Aitken and LS estimations of $b$ and $c$ will not coincide. The proof of the theorem consists of the following steps. First, the original system of polynomials is simplified: we get the system polynomials of the second degree. The variables of both systems are unknown variances of deviations, each of the solutions of the original system gives a set variances of deviations at which the estimations of Aitken and LS parameter a coincide. In the next step the solving of the original system polynomials is reduced to solving an equation with three unknowns, and all other unknowns are expressed in some way through these three. At last it is proved that there are positive unequal values of these three unknowns, which will be the solution of the obtained equation. And all other unknowns when substituting in their expression these values will be positive.
{"title":"SUFFICIENT CONDITION FOR COINCIDENCE OF THE LS AND AITKEN ESTIMATIONS OF PARAMETER OF QUADRATIC REGRESSION IN CASE HETEROSCEDASTIC DEVIATIONS","authors":"M. Savkina","doi":"10.17721/2706-9699.2020.2.03","DOIUrl":"https://doi.org/10.17721/2706-9699.2020.2.03","url":null,"abstract":"In the paper in case heteroscedastic independent deviations a regression model whose function has the form $f(x) = ax^2+bx+c$, where $a$, $b$ and $c$ are unknown parameters, is studied. Approximate values (observations) of functions $f(x)$ are registered at equidistant points of a line segment. The theorem which is proved at the paper gives a sufficient condition on the variance of the deviations at which the Aitken estimation of parameter $a$ coincides with its estimation of the LS in the case of odd number of observation points and bisymmetric covariance matrix. Under this condition, the Aitken and LS estimations of $b$ and $c$ will not coincide. The proof of the theorem consists of the following steps. First, the original system of polynomials is simplified: we get the system polynomials of the second degree. The variables of both systems are unknown variances of deviations, each of the solutions of the original system gives a set variances of deviations at which the estimations of Aitken and LS parameter a coincide. In the next step the solving of the original system polynomials is reduced to solving an equation with three unknowns, and all other unknowns are expressed in some way through these three. At last it is proved that there are positive unequal values of these three unknowns, which will be the solution of the obtained equation. And all other unknowns when substituting in their expression these values will be positive.","PeriodicalId":40347,"journal":{"name":"Journal of Numerical and Applied Mathematics","volume":"21 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86533549","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 : 2020-01-01DOI: 10.17721/2706-9699.2020.1.03
V. Dudar, V. Semenov
We present a general method for analysis of convolutional layers under geometric transformations of the input that are linear with respect to pixel values. We also describe the algorithm for finding all possible types of behaviours of the output of convolutional layers under geometric transformations of the input. We also present a general method for construction of convolutional architectures with desired behaviour under geometric transformations of the input.
{"title":"CONSTRUCTION OF NEURAL ARCHITECTURES WITH DESIRED BEHAVIOUR UNDER GEOMETRIC TRANSFORMATIONS OF THE INPUT","authors":"V. Dudar, V. Semenov","doi":"10.17721/2706-9699.2020.1.03","DOIUrl":"https://doi.org/10.17721/2706-9699.2020.1.03","url":null,"abstract":"We present a general method for analysis of convolutional layers under geometric transformations of the input that are linear with respect to pixel values. We also describe the algorithm for finding all possible types of behaviours of the output of convolutional layers under geometric transformations of the input. We also present a general method for construction of convolutional architectures with desired behaviour under geometric transformations of the input.","PeriodicalId":40347,"journal":{"name":"Journal of Numerical and Applied Mathematics","volume":"35 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84013820","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 : 2019-01-01DOI: 10.17721/2706-9699.2019.3.01
S. Aldashev
Two-dimensional spectral problems for elliptic equations are well studied, and their multidimensional analogs, as far as the author knows, are little studied. This is due to the fact that in the case of three or more independent variables there are difficulties of a fundamental nature, since the method of singular integral equations, which is very attractive and convenient, used for two-dimensional problems, cannot be used here because of the lack of any complete theory of multidimensional singular integral equations. The theory of multidimensional spherical functions, on the contrary, has been adequately and fully studied. In the cylindrical domain of Euclidean space, for a single class of multidimensional elliptic equations, the spectral Poincare problem. The solution is sought in the form of an expansion in multidimensional spherical functions. The existence and uniqueness theorems of the solution are proved. Conditions for unique solvability of the problem are obtained, which essentially depend on the height of the cylinder.
{"title":"A CRITERION FOR THE UNIQUE SOLVABILITY OF THE POINCARE SPECTRAL PROBLEM IN A CYLINDRICAL DOMAIN FOR ONE CLASS OF MULTIDIMENSIONAL ELLIPTIC EQUATIONS","authors":"S. Aldashev","doi":"10.17721/2706-9699.2019.3.01","DOIUrl":"https://doi.org/10.17721/2706-9699.2019.3.01","url":null,"abstract":"Two-dimensional spectral problems for elliptic equations are well studied, and their multidimensional analogs, as far as the author knows, are little studied. This is due to the fact that in the case of three or more independent variables there are difficulties of a fundamental nature, since the method of singular integral equations, which is very attractive and convenient, used for two-dimensional problems, cannot be used here because of the lack of any complete theory of multidimensional singular integral equations. The theory of multidimensional spherical functions, on the contrary, has been adequately and fully studied. In the cylindrical domain of Euclidean space, for a single class of multidimensional elliptic equations, the spectral Poincare problem. The solution is sought in the form of an expansion in multidimensional spherical functions. The existence and uniqueness theorems of the solution are proved. Conditions for unique solvability of the problem are obtained, which essentially depend on the height of the cylinder.","PeriodicalId":40347,"journal":{"name":"Journal of Numerical and Applied Mathematics","volume":"70 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85151297","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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