{"title":"真实三维Clifford代数中多向量的对数","authors":"Artūras Acus, Adolfas Dargys","doi":"10.15388/namc.2024.29.33535","DOIUrl":null,"url":null,"abstract":"Closed form expressions for a logarithm of general multivector (MV) in basis-free form in real geometric algebras (GAs) Clp,q are presented for all n = p + q = 3. In contrast to logarithm of complex numbers (isomorphic to Cl0,1), 3D logarithmic functions, due to appearance of two double angle arc tangent functions, allow to include two sets of sheets characterized by discrete coefficients. Formulas for generic and special cases of individual blades and their combinations are provided.","PeriodicalId":49286,"journal":{"name":"Nonlinear Analysis-Modelling and Control","volume":null,"pages":null},"PeriodicalIF":2.6000,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Logarithm of multivector in real 3D Clifford algebras\",\"authors\":\"Artūras Acus, Adolfas Dargys\",\"doi\":\"10.15388/namc.2024.29.33535\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Closed form expressions for a logarithm of general multivector (MV) in basis-free form in real geometric algebras (GAs) Clp,q are presented for all n = p + q = 3. In contrast to logarithm of complex numbers (isomorphic to Cl0,1), 3D logarithmic functions, due to appearance of two double angle arc tangent functions, allow to include two sets of sheets characterized by discrete coefficients. Formulas for generic and special cases of individual blades and their combinations are provided.\",\"PeriodicalId\":49286,\"journal\":{\"name\":\"Nonlinear Analysis-Modelling and Control\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.6000,\"publicationDate\":\"2023-11-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Nonlinear Analysis-Modelling and Control\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.15388/namc.2024.29.33535\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinear Analysis-Modelling and Control","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.15388/namc.2024.29.33535","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 1
摘要
在实几何代数(GAs) Clp,q中，对于所有n = p + q = 3，给出了无基形式的一般多向量(MV)的对数的封闭表达式。与复数的对数(同构于Cl0,1)相反，三维对数函数由于两个双角弧切函数的出现，允许包含两组以离散系数为特征的片。给出了单个叶片及其组合的一般和特殊情况的公式。
Logarithm of multivector in real 3D Clifford algebras
Closed form expressions for a logarithm of general multivector (MV) in basis-free form in real geometric algebras (GAs) Clp,q are presented for all n = p + q = 3. In contrast to logarithm of complex numbers (isomorphic to Cl0,1), 3D logarithmic functions, due to appearance of two double angle arc tangent functions, allow to include two sets of sheets characterized by discrete coefficients. Formulas for generic and special cases of individual blades and their combinations are provided.
期刊介绍：
The scope of the journal is to provide a multidisciplinary forum for scientists, researchers and engineers involved in research and design of nonlinear processes and phenomena, including the nonlinear modelling of phenomena of the nature. The journal accepts contributions on nonlinear phenomena and processes in any field of science and technology.
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