Pub Date : 2020-06-25DOI: 10.5772/intechopen.92869
F. Bulnes
In this research we studied t he tensor product on derived categories of Étale sheaves with transfers considering as fundamental, the tensor product of categories X ⊗ Y ¼ X (cid:2) Y , on the category Cor k , (finite correspondences category) by under-standing it to be the product of the underlying schemes on k . Although, to this is required to build a total tensor product on the category PST( k ), where this construction will be useful to obtain generalizations on derived categories using pre-sheaves and contravariant and covariant functors on additive categories to define the exactness of infinite sequences and resolution of spectral sequences. Some concrete applications are given through a result on field equations solution.
{"title":"Derived Tensor Products and Their Applications","authors":"F. Bulnes","doi":"10.5772/intechopen.92869","DOIUrl":"https://doi.org/10.5772/intechopen.92869","url":null,"abstract":"In this research we studied t he tensor product on derived categories of Étale sheaves with transfers considering as fundamental, the tensor product of categories X ⊗ Y ¼ X (cid:2) Y , on the category Cor k , (finite correspondences category) by under-standing it to be the product of the underlying schemes on k . Although, to this is required to build a total tensor product on the category PST( k ), where this construction will be useful to obtain generalizations on derived categories using pre-sheaves and contravariant and covariant functors on additive categories to define the exactness of infinite sequences and resolution of spectral sequences. Some concrete applications are given through a result on field equations solution.","PeriodicalId":189982,"journal":{"name":"Advances on Tensor Analysis and their Applications","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131704790","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 : 2020-05-13DOI: 10.5772/intechopen.92274
C. Ruscitti, Laura B. Langoni, Augusto Melgarejo
The method of Riemannian geometry is fruitful in equilibrium thermodynamics. From the theory of fluctuations it has been possible to construct a metric for the space of thermodynamic equilibrium states. Inspired by these geometric elements, we will discuss the geometric-differential approach of nonequilibrium systems. In particular we will study the geometric aspects from the knowledge of the macroscopic potential associated with the Uhlenbeck-Ornstein (UO) nonequilibrium process. Assuming the geodesic curve as an optimal path and using the affine connection, known as α -connection, we will study the conditions under which a diffusive process can be considered optimal. We will also analyze the impact of this behavior on the entropy of the system, relating these results with studies of instabilities in diffusive processes.
{"title":"Differential Geometry and Macroscopic Descriptions in Nonequilibrium Process","authors":"C. Ruscitti, Laura B. Langoni, Augusto Melgarejo","doi":"10.5772/intechopen.92274","DOIUrl":"https://doi.org/10.5772/intechopen.92274","url":null,"abstract":"The method of Riemannian geometry is fruitful in equilibrium thermodynamics. From the theory of fluctuations it has been possible to construct a metric for the space of thermodynamic equilibrium states. Inspired by these geometric elements, we will discuss the geometric-differential approach of nonequilibrium systems. In particular we will study the geometric aspects from the knowledge of the macroscopic potential associated with the Uhlenbeck-Ornstein (UO) nonequilibrium process. Assuming the geodesic curve as an optimal path and using the affine connection, known as α -connection, we will study the conditions under which a diffusive process can be considered optimal. We will also analyze the impact of this behavior on the entropy of the system, relating these results with studies of instabilities in diffusive processes.","PeriodicalId":189982,"journal":{"name":"Advances on Tensor Analysis and their Applications","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121062779","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 : 2020-03-21DOI: 10.5772/intechopen.91779
D. Condurache
In this chapter, using the ring properties of dual number algebra, vector and tensor calculus, a computing method for the higher-order acceleration vector field properties in general rigid body motion is proposed. The higher-order acceleration field of a rigid body in a general motion is uniquely determined by higher-order time derivative of a dual twist. For the relative kinematics of rigid body motion, equations that allow the determination of the higher-order acceleration vector field are given, using an exponential Brockett-like formula in the dual Lie algebra. In particular cases, the properties for velocity, acceleration, jerk, and jounce fields are given. This approach uses the isomorphism between the Lie algebra of the rigid displacements se (3), of the Special Euclidean group, S 3 , and the Lie algebra of dual vectors. The results are coordinate free and in a closed
{"title":"Higher-Order Kinematics in Dual Lie Algebra","authors":"D. Condurache","doi":"10.5772/intechopen.91779","DOIUrl":"https://doi.org/10.5772/intechopen.91779","url":null,"abstract":"In this chapter, using the ring properties of dual number algebra, vector and tensor calculus, a computing method for the higher-order acceleration vector field properties in general rigid body motion is proposed. The higher-order acceleration field of a rigid body in a general motion is uniquely determined by higher-order time derivative of a dual twist. For the relative kinematics of rigid body motion, equations that allow the determination of the higher-order acceleration vector field are given, using an exponential Brockett-like formula in the dual Lie algebra. In particular cases, the properties for velocity, acceleration, jerk, and jounce fields are given. This approach uses the isomorphism between the Lie algebra of the rigid displacements se (3), of the Special Euclidean group, S 3 , and the Lie algebra of dual vectors. The results are coordinate free and in a closed","PeriodicalId":189982,"journal":{"name":"Advances on Tensor Analysis and their Applications","volume":"147 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133581495","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 : 2020-03-11DOI: 10.5772/INTECHOPEN.91284
D. Nikushchenko, V. Pavlovsky
In the current chapter, some applications of tensor analysis to fluid dynamics are presented. Governing equations of fluid motion and energy are obtained and analyzed. We shall discuss about continuity equation, equation of motion, and mechanical energy transport equation and four forms of energy equation. Finally, we shall talk about the divergence from transfer equations of different parameters of motion. The tensor form of equations has advantages over the component form: these are, first, compact writing of equations and, second, independency from reference frames, etc. Moreover, it allows to obtain new forms of equations on the basis of governing ones easily.
{"title":"Fluid Motion Equations in Tensor Form","authors":"D. Nikushchenko, V. Pavlovsky","doi":"10.5772/INTECHOPEN.91284","DOIUrl":"https://doi.org/10.5772/INTECHOPEN.91284","url":null,"abstract":"In the current chapter, some applications of tensor analysis to fluid dynamics are presented. Governing equations of fluid motion and energy are obtained and analyzed. We shall discuss about continuity equation, equation of motion, and mechanical energy transport equation and four forms of energy equation. Finally, we shall talk about the divergence from transfer equations of different parameters of motion. The tensor form of equations has advantages over the component form: these are, first, compact writing of equations and, second, independency from reference frames, etc. Moreover, it allows to obtain new forms of equations on the basis of governing ones easily.","PeriodicalId":189982,"journal":{"name":"Advances on Tensor Analysis and their Applications","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130028944","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 : 2020-03-05DOI: 10.5772/intechopen.89906
Pınar Kirezli Uludağ
One of the most known alternative gravitational theories is Brans-Dicke (BD) theory. The theory offers a new approach by taking a scalar field ϕ instead of Newton ’ s gravitational constant G. Solutions of the theory are under consideration and results are discussed in many papers. Stationary, axially symmetric solutions become important because gravitational field of celestial objects can be described by such solutions. Since obtaining exact solutions of BD is not an easy task, some solution-generating techniques are proposed. In this context, some solutions of Einstein general relativity, such as black hole or wormhole solutions, are discussed in BD theory. Indeed, black hole solutions in BD theory are not fully understood yet. Old and new such solutions and their analysis will be reviewed in this chapter.
{"title":"Brans-Dicke Solutions of Stationary, Axially Symmetric Spacetimes","authors":"Pınar Kirezli Uludağ","doi":"10.5772/intechopen.89906","DOIUrl":"https://doi.org/10.5772/intechopen.89906","url":null,"abstract":"One of the most known alternative gravitational theories is Brans-Dicke (BD) theory. The theory offers a new approach by taking a scalar field ϕ instead of Newton ’ s gravitational constant G. Solutions of the theory are under consideration and results are discussed in many papers. Stationary, axially symmetric solutions become important because gravitational field of celestial objects can be described by such solutions. Since obtaining exact solutions of BD is not an easy task, some solution-generating techniques are proposed. In this context, some solutions of Einstein general relativity, such as black hole or wormhole solutions, are discussed in BD theory. Indeed, black hole solutions in BD theory are not fully understood yet. Old and new such solutions and their analysis will be reviewed in this chapter.","PeriodicalId":189982,"journal":{"name":"Advances on Tensor Analysis and their Applications","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130656576","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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