Pub Date : 2022-11-01DOI: 10.35634/2226-3594-2022-60-05
A. Sukhinov, A. Chistyakov, A. Atayan, I. Kuznetsova, V. Litvinov, A. Nikitina
The paper considers 2D and 3D models of transport of suspended particles, taking into account the following factors: movement of aqueous medium; variable density depending on the suspension concentration; multicomponent character of suspension; changes in bottom geometry as a result of suspension sedimentation. The approximation of the three-dimensional diffusion-convection equation is based on splitting schemes into two-dimensional and one-dimensional problems. In this work, we use discrete analogues of convective and diffusion transfer operators in the case of partial cell occupancy. The geometry of the computational domain is described based on the occupancy function. The difference scheme used is a linear combination of the Upwind and Standard Leapfrog difference schemes with weight coefficients obtained by minimizing the approximation error. This scheme is designed to solve the problem of impurity transfer at large grid Peclet numbers. Based on the results of numerical experiments, conclusions are drawn about the advantage of the 3D model of multicomponent suspension transport in comparison with the 2D model. Computational experiments have been performed to simulate the process of sedimentation of a multicomponent suspension, as well as its effect on the bottom topography and changes in its composition.
{"title":"Mathematical model of process of sedimentation of multicomponent suspension on the bottom and changes in the composition of bottom materials","authors":"A. Sukhinov, A. Chistyakov, A. Atayan, I. Kuznetsova, V. Litvinov, A. Nikitina","doi":"10.35634/2226-3594-2022-60-05","DOIUrl":"https://doi.org/10.35634/2226-3594-2022-60-05","url":null,"abstract":"The paper considers 2D and 3D models of transport of suspended particles, taking into account the following factors: movement of aqueous medium; variable density depending on the suspension concentration; multicomponent character of suspension; changes in bottom geometry as a result of suspension sedimentation. The approximation of the three-dimensional diffusion-convection equation is based on splitting schemes into two-dimensional and one-dimensional problems. In this work, we use discrete analogues of convective and diffusion transfer operators in the case of partial cell occupancy. The geometry of the computational domain is described based on the occupancy function. The difference scheme used is a linear combination of the Upwind and Standard Leapfrog difference schemes with weight coefficients obtained by minimizing the approximation error. This scheme is designed to solve the problem of impurity transfer at large grid Peclet numbers. Based on the results of numerical experiments, conclusions are drawn about the advantage of the 3D model of multicomponent suspension transport in comparison with the 2D model. Computational experiments have been performed to simulate the process of sedimentation of a multicomponent suspension, as well as its effect on the bottom topography and changes in its composition.","PeriodicalId":42053,"journal":{"name":"Izvestiya Instituta Matematiki i Informatiki-Udmurtskogo Gosudarstvennogo Universiteta","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77873075","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 : 2022-11-01DOI: 10.35634/2226-3594-2022-60-04
P. Lebedev, A. Lempert, A. Kazakov
The paper deals with the problem of constructing the thinnest covering for a convex set by a set of similar elements. As a distance between two points, we use the shortest time it takes to achieve one point from another, and the boundary of each covering circle is an isochron. Such problems arise in applications, particularly in sonar and underwater surveillance systems. To solve the problems of covering with such circles and balls, we previously proposed algorithms based both on variational principles and geometric methods. The purpose of this article is to construct coverings when the characteristics of the medium change over time. We propose a computational algorithm based on the theory of wave fronts and prove the statement about its properties. Illustrative calculations are performed.
{"title":"Algorithms of optimal covering of 2D sets with dynamical metrics","authors":"P. Lebedev, A. Lempert, A. Kazakov","doi":"10.35634/2226-3594-2022-60-04","DOIUrl":"https://doi.org/10.35634/2226-3594-2022-60-04","url":null,"abstract":"The paper deals with the problem of constructing the thinnest covering for a convex set by a set of similar elements. As a distance between two points, we use the shortest time it takes to achieve one point from another, and the boundary of each covering circle is an isochron. Such problems arise in applications, particularly in sonar and underwater surveillance systems. To solve the problems of covering with such circles and balls, we previously proposed algorithms based both on variational principles and geometric methods. The purpose of this article is to construct coverings when the characteristics of the medium change over time. We propose a computational algorithm based on the theory of wave fronts and prove the statement about its properties. Illustrative calculations are performed.","PeriodicalId":42053,"journal":{"name":"Izvestiya Instituta Matematiki i Informatiki-Udmurtskogo Gosudarstvennogo Universiteta","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72449443","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 : 2022-11-01DOI: 10.35634/2226-3594-2022-60-06
A. Chernov
It is well known that using the Tikhonov regularization method for solving operator equations of the first kind one has to minimize a regularized residual functional. The minimizer is determined from so called Euler equation which in finite-dimensional case and at its discretization is written as a one-parametric (depending on the regularization parameter) system of linear algebraic equations of special form. Here, there exist various ways of choosing the regularization parameter. In particular, in the frame of principle of generalized residual, it is necessary to solve the corresponding equation of generalized residual with respect to the regularization parameter. And it implies (when solving this equation numerically), in turn, multifold solving a one-parametric system of linear algebraic equations for arbitrary value of the parameter. In this paper we obtain an explicit simple and effective formula of solution to a one-parametric system for an arbitrary value of the parameter. We give an example of computations by above-mentioned formula and also an example of numerical solution of the Fredholm integral equation of the first kind under usage of this formula which substantiates its effectiveness.
{"title":"On explicit expression of the solution to the regularizing by Tikhonov optimization problem in terms of the regularization parameter in the finite-dimensional case","authors":"A. Chernov","doi":"10.35634/2226-3594-2022-60-06","DOIUrl":"https://doi.org/10.35634/2226-3594-2022-60-06","url":null,"abstract":"It is well known that using the Tikhonov regularization method for solving operator equations of the first kind one has to minimize a regularized residual functional. The minimizer is determined from so called Euler equation which in finite-dimensional case and at its discretization is written as a one-parametric (depending on the regularization parameter) system of linear algebraic equations of special form. Here, there exist various ways of choosing the regularization parameter. In particular, in the frame of principle of generalized residual, it is necessary to solve the corresponding equation of generalized residual with respect to the regularization parameter. And it implies (when solving this equation numerically), in turn, multifold solving a one-parametric system of linear algebraic equations for arbitrary value of the parameter. In this paper we obtain an explicit simple and effective formula of solution to a one-parametric system for an arbitrary value of the parameter. We give an example of computations by above-mentioned formula and also an example of numerical solution of the Fredholm integral equation of the first kind under usage of this formula which substantiates its effectiveness.","PeriodicalId":42053,"journal":{"name":"Izvestiya Instituta Matematiki i Informatiki-Udmurtskogo Gosudarstvennogo Universiteta","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87088235","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 : 2022-11-01DOI: 10.35634/2226-3594-2022-60-02
I. Zykov
The paper considers the problem of approximate construction of reachability sets for a linear control system, when the control action is constrained simultaneously by geometric and several integral constraints. A variant of the transition from a continuous to a discrete system is proposed by uniformly dividing the time interval and replacing the controls at the step of dividing them with their mean values. The convergence of the reachability set of the approximating system to the reachability set of the original system in the Hausdorff metric is proved as the discretization step tends to zero, and an estimate is obtained for the rate of convergence. An algorithm for constructing the boundary of reachable sets based on solving a family of conic programming problems is proposed. Numerical simulation has been carried out.
{"title":"Approximate calculation of reachable sets for linear control systems with different control constraints","authors":"I. Zykov","doi":"10.35634/2226-3594-2022-60-02","DOIUrl":"https://doi.org/10.35634/2226-3594-2022-60-02","url":null,"abstract":"The paper considers the problem of approximate construction of reachability sets for a linear control system, when the control action is constrained simultaneously by geometric and several integral constraints. A variant of the transition from a continuous to a discrete system is proposed by uniformly dividing the time interval and replacing the controls at the step of dividing them with their mean values. The convergence of the reachability set of the approximating system to the reachability set of the original system in the Hausdorff metric is proved as the discretization step tends to zero, and an estimate is obtained for the rate of convergence. An algorithm for constructing the boundary of reachable sets based on solving a family of conic programming problems is proposed. Numerical simulation has been carried out.","PeriodicalId":42053,"journal":{"name":"Izvestiya Instituta Matematiki i Informatiki-Udmurtskogo Gosudarstvennogo Universiteta","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84107937","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 : 2022-05-01DOI: 10.35634/2226-3594-2022-59-06
B. Samatov, A. Akbarov, B. I. Zhuraev
In the paper, a pursuit-evasion differential game is considered when controls of the players are subject to differential constraints in the form of Grönwall's integral inequality. The strategy of parallel pursuit (briefly, $Pi$-strategy) was introduced and used by L.A. Petrosyan to solve simple pursuit problems under phase constraints on the states of the players in the case when control functions of both players are chosen from the class $L_infty$. In the present work, the $Pi$-strategy is constructed for a simple pursuit problem in the cases when control functions of both players are chosen from different classes of the Grönwall type constraints, and sufficient conditions of capture and optimal capture time are found in these cases. To solve the evasion problem, we suggest a control function for the Evader and find sufficient conditions of evasion. In addition, an attainability domain of the players is constructed and its conditions of embedding in respect to time are given. Results of this work continue and extend the works of R. Isaacs, L.A. Petrosyan, B.N. Pshenichnyi, A.A. Chirii, A.A. Azamov and other researchers, including the authors.
{"title":"Pursuit-evasion differential games with Gr-constraints on controls","authors":"B. Samatov, A. Akbarov, B. I. Zhuraev","doi":"10.35634/2226-3594-2022-59-06","DOIUrl":"https://doi.org/10.35634/2226-3594-2022-59-06","url":null,"abstract":"In the paper, a pursuit-evasion differential game is considered when controls of the players are subject to differential constraints in the form of Grönwall's integral inequality. The strategy of parallel pursuit (briefly, $Pi$-strategy) was introduced and used by L.A. Petrosyan to solve simple pursuit problems under phase constraints on the states of the players in the case when control functions of both players are chosen from the class $L_infty$. In the present work, the $Pi$-strategy is constructed for a simple pursuit problem in the cases when control functions of both players are chosen from different classes of the Grönwall type constraints, and sufficient conditions of capture and optimal capture time are found in these cases. To solve the evasion problem, we suggest a control function for the Evader and find sufficient conditions of evasion. In addition, an attainability domain of the players is constructed and its conditions of embedding in respect to time are given. Results of this work continue and extend the works of R. Isaacs, L.A. Petrosyan, B.N. Pshenichnyi, A.A. Chirii, A.A. Azamov and other researchers, including the authors.","PeriodicalId":42053,"journal":{"name":"Izvestiya Instituta Matematiki i Informatiki-Udmurtskogo Gosudarstvennogo Universiteta","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89256566","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 : 2022-05-01DOI: 10.35634/2226-3594-2022-59-07
V. Sumin, M. I. Sumin
Regularization of the classical optimality conditions - the Lagrange principle and the Pontryagin maximum principle - in a convex optimal control problem subject to functional equality and inequality constraints is considered. The controlled system is described by a linear functional-operator equation of second kind of the general form in the space $L_2^m$. The main operator on the right-hand side of the equation is assumed to be quasi-nilpotent. The objective functional to be minimized is strongly convex. The derivation of the regularized classical optimality conditions is based on the use of the dual regularization method. The main purpose of the regularized Lagrange principle and regularized Pontryagin maximum principle is to stably generate minimizing approximate solutions in the sense of J. Warga. As an application of the results obtained for the general linear functional-operator equation of second kind, two examples of concrete optimal control problems related to a system of delay equations and to an integro-differential transport equation are discussed.
{"title":"On regularization of the Lagrange principle in the optimization problems for linear distributed Volterra type systems with operator constraints","authors":"V. Sumin, M. I. Sumin","doi":"10.35634/2226-3594-2022-59-07","DOIUrl":"https://doi.org/10.35634/2226-3594-2022-59-07","url":null,"abstract":"Regularization of the classical optimality conditions - the Lagrange principle and the Pontryagin maximum principle - in a convex optimal control problem subject to functional equality and inequality constraints is considered. The controlled system is described by a linear functional-operator equation of second kind of the general form in the space $L_2^m$. The main operator on the right-hand side of the equation is assumed to be quasi-nilpotent. The objective functional to be minimized is strongly convex. The derivation of the regularized classical optimality conditions is based on the use of the dual regularization method. The main purpose of the regularized Lagrange principle and regularized Pontryagin maximum principle is to stably generate minimizing approximate solutions in the sense of J. Warga. As an application of the results obtained for the general linear functional-operator equation of second kind, two examples of concrete optimal control problems related to a system of delay equations and to an integro-differential transport equation are discussed.","PeriodicalId":42053,"journal":{"name":"Izvestiya Instituta Matematiki i Informatiki-Udmurtskogo Gosudarstvennogo Universiteta","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73793497","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 : 2022-05-01DOI: 10.35634/2226-3594-2022-59-08
A. Chernov
For finite-dimensional mathematical programming problems (approximating problems) being obtained by a parametric approximation of control functions in lumped optimal control problems with functional equality constraints, we introduce concepts of rigidity and flexibility for a system of constraints. The rigidity in a given admissible point is treated in the sense that this point is isolated for the admissible set; otherwise, we call a system of constraints as flexible in this point. Under using a parametric approximation for a control function with the help of quadratic exponentials (Gaussian functions) and subject to some natural hypotheses, we establish that in order to guarantee the flexibility of constraints system in a given admissible point it suffices to increase the dimension of parameter space in the approximating problem. A test of our hypotheses is illustrated by an example of the soft lunar landing problem.
{"title":"On flexibility of constraints system under approximation of optimal control problems","authors":"A. Chernov","doi":"10.35634/2226-3594-2022-59-08","DOIUrl":"https://doi.org/10.35634/2226-3594-2022-59-08","url":null,"abstract":"For finite-dimensional mathematical programming problems (approximating problems) being obtained by a parametric approximation of control functions in lumped optimal control problems with functional equality constraints, we introduce concepts of rigidity and flexibility for a system of constraints. The rigidity in a given admissible point is treated in the sense that this point is isolated for the admissible set; otherwise, we call a system of constraints as flexible in this point. Under using a parametric approximation for a control function with the help of quadratic exponentials (Gaussian functions) and subject to some natural hypotheses, we establish that in order to guarantee the flexibility of constraints system in a given admissible point it suffices to increase the dimension of parameter space in the approximating problem. A test of our hypotheses is illustrated by an example of the soft lunar landing problem.","PeriodicalId":42053,"journal":{"name":"Izvestiya Instituta Matematiki i Informatiki-Udmurtskogo Gosudarstvennogo Universiteta","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78995304","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 : 2022-05-01DOI: 10.35634/2226-3594-2022-59-05
N. Petrov
In a finite-dimensional Euclidean space, the problem of pursuit by a group of pursuers of two evaders described by a system of the form $$dot z_{ij} = u_i - v,quad u_i, v in V $$ is considered. It is assumed that all evaders use the same control. The pursuers use counterstrategies based on information about the initial positions and control history of the evaders. The set of admissible controls $V$ is unit ball centered at zero, target sets are the origin. The goal of the pursuers' group is to capture at least one evader by two pursuers or to capture two evaders. In terms of initial positions and game parameters a sufficient condition for the capture is obtained. In the study, the method of resolving functions is used as a basic one, which allows obtaining sufficient conditions for the solvability of the approach problem in some guaranteed time.
在有限维欧几里得空间中，考虑了由形式为$$dot z_{ij} = u_i - v,quad u_i, v in V $$的系统描述的两个逃避者的一组追逐者的追逐问题。假设所有逃避者使用相同的控制。追捕者根据关于逃避者的初始位置和控制历史的信息使用对抗策略。容许控制的集合$V$是以零为中心的单位球，目标集合是原点。追捕者组的目标是被两个追捕者捕获至少一个逃避者或捕获两个逃避者。在初始位置和博弈参数方面，得到了捕获的充分条件。在本研究中，采用函数解析法作为基本方法，可以在一定保证时间内得到逼近问题可解的充分条件。
{"title":"On one simple pursuit problem of two rigidly coordinated evaders","authors":"N. Petrov","doi":"10.35634/2226-3594-2022-59-05","DOIUrl":"https://doi.org/10.35634/2226-3594-2022-59-05","url":null,"abstract":"In a finite-dimensional Euclidean space, the problem of pursuit by a group of pursuers of two evaders described by a system of the form $$dot z_{ij} = u_i - v,quad u_i, v in V $$ is considered. It is assumed that all evaders use the same control. The pursuers use counterstrategies based on information about the initial positions and control history of the evaders. The set of admissible controls $V$ is unit ball centered at zero, target sets are the origin. The goal of the pursuers' group is to capture at least one evader by two pursuers or to capture two evaders. In terms of initial positions and game parameters a sufficient condition for the capture is obtained. In the study, the method of resolving functions is used as a basic one, which allows obtaining sufficient conditions for the solvability of the approach problem in some guaranteed time.","PeriodicalId":42053,"journal":{"name":"Izvestiya Instituta Matematiki i Informatiki-Udmurtskogo Gosudarstvennogo Universiteta","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78001912","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 : 2022-05-01DOI: 10.35634/2226-3594-2022-59-04
M. Ibrahim, V. Pimenov
We consider a system of two space-fractional superdiffusion equations with functional general delay and Neumann boundary conditions. For this problem, an analogue of the Crank-Nicolson method is constructed, based on the shifted Grünwald-Letnikov formulas for approximating fractional Riesz derivatives with respect to a spatial variable and using piecewise linear interpolation of discrete prehistory with extrapolation by continuation to take into account the delay effect. With the help of the Gershgorin theorem, the solvability of the difference scheme and its stability are proved. The order of convergence of the method is obtained. The results of numerical experiments are presented.
{"title":"Numerical method for system of space-fractional equations of superdiffusion type with delay and Neumann boundary conditions","authors":"M. Ibrahim, V. Pimenov","doi":"10.35634/2226-3594-2022-59-04","DOIUrl":"https://doi.org/10.35634/2226-3594-2022-59-04","url":null,"abstract":"We consider a system of two space-fractional superdiffusion equations with functional general delay and Neumann boundary conditions. For this problem, an analogue of the Crank-Nicolson method is constructed, based on the shifted Grünwald-Letnikov formulas for approximating fractional Riesz derivatives with respect to a spatial variable and using piecewise linear interpolation of discrete prehistory with extrapolation by continuation to take into account the delay effect. With the help of the Gershgorin theorem, the solvability of the difference scheme and its stability are proved. The order of convergence of the method is obtained. The results of numerical experiments are presented.","PeriodicalId":42053,"journal":{"name":"Izvestiya Instituta Matematiki i Informatiki-Udmurtskogo Gosudarstvennogo Universiteta","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91001280","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 : 2022-05-01DOI: 10.35634/2226-3594-2022-59-01
A. Mohammed Yousif, Qusay S. A. Al-Zamil
In this paper, we determine the components of the Weyl tensor of almost contact metric (ACR-) manifold of class $C_{12}$ on associated G-structure (AG-structure) space. As an application, we prove that the conformally flat ACR-manifold of class $C_{12}$ with $n>2$ is an $eta$-Einstein manifold and conclude that it is an Einstein manifold such that the scalar curvature $r$ has provided. Also, the case when $n=2$ is discussed explicitly. Moreover, the relationships among conformally flat, conformally symmetric, $xi$-conformally flat and $Phi$-invariant Ricci tensor have been widely considered here and consequently we determine the value of scalar curvature $r$ explicitly with other applications. Finally, we define new classes with identities analogously to Gray identities and discuss their connections with class $C_{12}$ of ACR-manifold.
{"title":"On Weyl tensor of ACR-manifolds of class $C_{12}$ with applications","authors":"A. Mohammed Yousif, Qusay S. A. Al-Zamil","doi":"10.35634/2226-3594-2022-59-01","DOIUrl":"https://doi.org/10.35634/2226-3594-2022-59-01","url":null,"abstract":"In this paper, we determine the components of the Weyl tensor of almost contact metric (ACR-) manifold of class $C_{12}$ on associated G-structure (AG-structure) space. As an application, we prove that the conformally flat ACR-manifold of class $C_{12}$ with $n>2$ is an $eta$-Einstein manifold and conclude that it is an Einstein manifold such that the scalar curvature $r$ has provided. Also, the case when $n=2$ is discussed explicitly. Moreover, the relationships among conformally flat, conformally symmetric, $xi$-conformally flat and $Phi$-invariant Ricci tensor have been widely considered here and consequently we determine the value of scalar curvature $r$ explicitly with other applications. Finally, we define new classes with identities analogously to Gray identities and discuss their connections with class $C_{12}$ of ACR-manifold.","PeriodicalId":42053,"journal":{"name":"Izvestiya Instituta Matematiki i Informatiki-Udmurtskogo Gosudarstvennogo Universiteta","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75601660","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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