Pub Date : 2023-01-01DOI: 10.7546/giq-26-2023-1-25
Stoil Donev
{"title":"Dynamical Coherence and Strain-Deformation Curvature View on Gravity","authors":"Stoil Donev","doi":"10.7546/giq-26-2023-1-25","DOIUrl":"https://doi.org/10.7546/giq-26-2023-1-25","url":null,"abstract":"","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135611065","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 : 2023-01-01DOI: 10.7546/giq-25-2023-47-72
Ying-Qiu Gu
Hypercomplex number systems are vector algebras with the definition of multiplication and division of vectors, satisfying the associativity and distributive law. In this paper, some new types of hypercomplex numbers and their fundamental properties are introduced, the Clifford algebra formalisms of hydrodynamics and gauge field equations are established, and some novel consistent conditions helpful to understand the properties of solutions to nonlinear physical equations are derived. The coordinate transformation and covariant derivatives of hypercomplex numbers are also discussed. The basis elements of the hypercomplex numbers have group-like properties and satisfy a structure equation $A^2=nA$. The hypercomplex number system integrates the advantages of algebra, geometry and analysis, and provides a unified, standard and elegant language and tool for scientific theories and engineering technology, so it is easy to learn and use. The description of mathematical, physical and engineering problems by hypercomplex numbers is of neat formalism, symmetric structure and standard derivation, which is especially suitable for the efficient processing of the higher dimensional complicated systems.
超复数系统是具有向量乘法和除法定义的向量代数，满足结合律和分配律。本文介绍了几种新型的超复数及其基本性质，建立了流体力学和规范场方程的Clifford代数形式，导出了一些有助于理解非线性物理方程解的性质的新的一致性条件。讨论了超复数的坐标变换和协变导数。超复数的基元具有类群性质，满足结构方程$ a ^2=n a $。超复数系统综合了代数、几何和分析的优点，为科学理论和工程技术提供了统一、标准和优雅的语言和工具，因此易于学习和使用。超复数对数学、物理和工程问题的描述具有整洁的形式、对称的结构和标准的推导，特别适用于高维复杂系统的高效处理。
{"title":"Clifford Algebras, Hypercomplex Numbers and Nonlinear Equations in Physics","authors":"Ying-Qiu Gu","doi":"10.7546/giq-25-2023-47-72","DOIUrl":"https://doi.org/10.7546/giq-25-2023-47-72","url":null,"abstract":"Hypercomplex number systems are vector algebras with the definition of multiplication and division of vectors, satisfying the associativity and distributive law. In this paper, some new types of hypercomplex numbers and their fundamental properties are introduced, the Clifford algebra formalisms of hydrodynamics and gauge field equations are established, and some novel consistent conditions helpful to understand the properties of solutions to nonlinear physical equations are derived. The coordinate transformation and covariant derivatives of hypercomplex numbers are also discussed. The basis elements of the hypercomplex numbers have group-like properties and satisfy a structure equation $A^2=nA$. The hypercomplex number system integrates the advantages of algebra, geometry and analysis, and provides a unified, standard and elegant language and tool for scientific theories and engineering technology, so it is easy to learn and use. The description of mathematical, physical and engineering problems by hypercomplex numbers is of neat formalism, symmetric structure and standard derivation, which is especially suitable for the efficient processing of the higher dimensional complicated systems.","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82533792","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 : 2023-01-01DOI: 10.7546/giq-25-2023-95-116
C. Mladenova, I. Mladenov
Despite the longstanding interest in the shapes of the eggs since the ancient time till nowadays, the available parametric descriptions in the modern literature are given only via purely empirical formulas without any clear relationships with their measurable physical parameters. Here we present a geometrical model of the eggs based on Perseus spirics which were known as well since the ancient time but their analytical parameterizations were absent in the meantime. Such parameterizations have been found recently and the present work is based on the idea to use the spirics as a geometrical model of the egg's shapes. Explicit formulas for the volume, surface area and the curvatures of the eggs are derived from the first principles and these have been compared with the available empirical formulas and experimental data.
{"title":"Geometry of the Ovoids: Reptilian Eggs and Similar Symmetric Forms","authors":"C. Mladenova, I. Mladenov","doi":"10.7546/giq-25-2023-95-116","DOIUrl":"https://doi.org/10.7546/giq-25-2023-95-116","url":null,"abstract":"Despite the longstanding interest in the shapes of the eggs since the ancient time till nowadays, the available parametric descriptions in the modern literature are given only via purely empirical formulas without any clear relationships with their measurable physical parameters. Here we present a geometrical model of the eggs based on Perseus spirics which were known as well since the ancient time but their analytical parameterizations were absent in the meantime. Such parameterizations have been found recently and the present work is based on the idea to use the spirics as a geometrical model of the egg's shapes. Explicit formulas for the volume, surface area and the curvatures of the eggs are derived from the first principles and these have been compared with the available empirical formulas and experimental data.","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75364861","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 : 2023-01-01DOI: 10.7546/giq-26-2023-39-52
Mardon Pardabaev, Firdavs Almuratov
{"title":"On the Spectrum of the Discrete Bilaplacian with Rank-One Perturbation","authors":"Mardon Pardabaev, Firdavs Almuratov","doi":"10.7546/giq-26-2023-39-52","DOIUrl":"https://doi.org/10.7546/giq-26-2023-39-52","url":null,"abstract":"","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135611068","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 : 2023-01-01DOI: 10.7546/giq-25-2023-1-45
R. G. Calvet
{"title":"On the Dynamics of the Solar System III: Perihelion Precession and Eccentricity Variation","authors":"R. G. Calvet","doi":"10.7546/giq-25-2023-1-45","DOIUrl":"https://doi.org/10.7546/giq-25-2023-1-45","url":null,"abstract":"","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83031136","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 : 2023-01-01DOI: 10.7546/giq-25-2023-73-94
V. Kovalchuk, I. Mladenov
In this article, we consider deformed spheres as a new reference model for the geoid, alternatively to the classical ellipsoidal one. The parametrization of deformed spheres is furnished through the incomplete elliptic integrals. From the other side, the solutions for geodesics on those surfaces are given entirely via elementary analytical functions, contrary to the case of ellipsoids of revolution. We explicitly described algorithms (all necessary computational steps) for the solution of the direct and inverse geodetic problems on the deformed spheres. Finally, we presented a few illustrative numerical solutions of the inverse geodetic problems for two conceptual cases of near and far points. It had turned out that even in the non-optimized case we obtained the good agreement with the predictions of the World Geodetic System 1984's ellipsoidal reference model.
{"title":"Explicit Solutions for Geodetic Problems on the Deformed Sphere as Reference Model for the Geoid","authors":"V. Kovalchuk, I. Mladenov","doi":"10.7546/giq-25-2023-73-94","DOIUrl":"https://doi.org/10.7546/giq-25-2023-73-94","url":null,"abstract":"In this article, we consider deformed spheres as a new reference model for the geoid, alternatively to the classical ellipsoidal one. The parametrization of deformed spheres is furnished through the incomplete elliptic integrals. From the other side, the solutions for geodesics on those surfaces are given entirely via elementary analytical functions, contrary to the case of ellipsoids of revolution. We explicitly described algorithms (all necessary computational steps) for the solution of the direct and inverse geodetic problems on the deformed spheres. Finally, we presented a few illustrative numerical solutions of the inverse geodetic problems for two conceptual cases of near and far points. It had turned out that even in the non-optimized case we obtained the good agreement with the predictions of the World Geodetic System 1984's ellipsoidal reference model.","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88074630","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 : 2022-03-05DOI: 10.7546/giq-23-2022-39-57
Daniele Corradetti, A. Marrani, David Chester, Raymond Aschheim
In this work we present a useful way to introduce the octonionic projective and hyperbolic plane $mathbb{O}P^{2}$ through the use of Veronese vectors. Then we focus on their relation with the exceptional Jordan algebra $mathfrak{J}_{3}^{mathbb{O}}$ and show that the Veronese vectors are the rank-one elements of the algebra. We then study groups of motions over the octonionic plane recovering all real forms of $text{G}_{2}$, $text{F}_{4}$ and $text{E}_{6}$ groups and finally give a classification of all octonionic and split-octonionic planes as symmetric spaces.
{"title":"Octonionic Planes and Real Forms of $G_2$, $F_4$ and $E_6$","authors":"Daniele Corradetti, A. Marrani, David Chester, Raymond Aschheim","doi":"10.7546/giq-23-2022-39-57","DOIUrl":"https://doi.org/10.7546/giq-23-2022-39-57","url":null,"abstract":"In this work we present a useful way to introduce the octonionic projective and hyperbolic plane $mathbb{O}P^{2}$ through the use of Veronese vectors. Then we focus on their relation with the exceptional Jordan algebra $mathfrak{J}_{3}^{mathbb{O}}$ and show that the Veronese vectors are the rank-one elements of the algebra. We then study groups of motions over the octonionic plane recovering all real forms of $text{G}_{2}$, $text{F}_{4}$ and $text{E}_{6}$ groups and finally give a classification of all octonionic and split-octonionic planes as symmetric spaces.","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"172 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82939664","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}
We discuss the relationship between (co)homology groups and categorical diagonalization. We consider the category of chain complexes in the category of finite-dimensional vector spaces over a fixed field. For a fixed chain complex with zero maps as an object, a chain map from the object to another chain complex is defined, and the chain map introduce a mapping cone. The fixed object is isomorphic to the (co)homology groups of the codomain of the chain map if and only if the chain map is injective to the cokernel of differentials of the codomain chain complex and the mapping cone is homotopy equivalent to zero. On the other hand, it was found that the fixed object can be regarded as a categorified eigenvalue of the chain complex in the context of the categorical diagonalization, recently. It is found that (co)homology groups are constructed as the eigenvalue of a chain complex.
{"title":"(Co)Homology Groups and Categorified Eigenvalues","authors":"Jumpei Gohara, Yuji Hirota, Keisui Ino, Akifumi Sako","doi":"10.7546/giq-23-2022-59-74","DOIUrl":"https://doi.org/10.7546/giq-23-2022-59-74","url":null,"abstract":"We discuss the relationship between (co)homology groups and categorical diagonalization. We consider the category of chain complexes in the category of finite-dimensional vector spaces over a fixed field. For a fixed chain complex with zero maps as an object, a chain map from the object to another chain complex is defined, and the chain map introduce a mapping cone. The fixed object is isomorphic to the (co)homology groups of the codomain of the chain map if and only if the chain map is injective to the cokernel of differentials of the codomain chain complex and the mapping cone is homotopy equivalent to zero. On the other hand, it was found that the fixed object can be regarded as a categorified eigenvalue of the chain complex in the context of the categorical diagonalization, recently. It is found that (co)homology groups are constructed as the eigenvalue of a chain complex.","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82663727","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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