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Title:On a continuation of quaternionic and octonionic logarithm along curves and the winding number
Authors:ID Gentili, Graziano (Author)
ID Prezelj, Jasna (Author)
ID Vlacci, Fabio (Author)
Files:URL URL - Source URL, visit https://www.sciencedirect.com/science/article/pii/S0022247X24001410
 
.pdf PDF - Presentation file, download (720,93 KB)
MD5: 67E62C9CA46C7537CC66F52581457EAB
 
Language:English
Typology:1.01 - Original Scientific Article
Organization:Logo IMFM - Institute of Mathematics, Physics, and Mechanics
Abstract:This paper deals with the problem of finding a continuous extension of the hypercomplex (quaternionic or octonionic) logarithm along (quaternionic or octonionic) paths which avoid the origin. The main difficulty depends upon this fact: while a branch of the complex logarithm can be defined in a small open neighbourhood of a strictly negative real point, no continuous branch of the hypercomplex logarithm can be defined in any open set which contains a strictly negative real point. To overcome this difficulty, we use the logarithmic manifold: in general, the existence of a lift of a path to this manifold is not guaranteed and, indeed, the problem of lifting a path to the logarithmic manifold is completely equivalent to the problem of finding a continuation of the hypercomplex logarithm along this path. The second part of the paper scrutinizes the existence of a notion of winding number (with respect to the origin) for hypercomplex loops that avoid the origin, even though it is known that the definition of winding number for such loops is not natural in Rn when n is greater than 2. The surprise is that, in the hypercomplex setting, the new definition of winding number introduced in this paper can be given and has full meaning for a large class of hypercomplex loops (untwisted loops with companion that avoid the origin). Finally an original but rather natural notion of homotopy for these hypercomplex loops (the c-homotopy) is presented and it is proved to be suitable to comply with the intrinsic geometrical meaning of the winding number for this class of loops, namely, two such hypercomplex loops are c-homotopic if, and only if, they have the same winding number.
Keywords:hypercomplex logarithm, continuation of the hypercomplex logarithm along paths, winding number
Publication status:Published
Publication version:Version of Record
Publication date:01.08.2024
Year of publishing:2024
Number of pages:25 str.
Numbering:Vol. 536, iss. 1, [article no.] 128219
PID:20.500.12556/DiRROS-18310 New window
UDC:517.5
ISSN on article:0022-247X
DOI:10.1016/j.jmaa.2024.128219 New window
COBISS.SI-ID:187512579 New window
Publication date in DiRROS:04.03.2024
Views:710
Downloads:319
Metadata:XML DC-XML DC-RDF
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GENTILI, Graziano, PREZELJ, Jasna and VLACCI, Fabio, 2024, On a continuation of quaternionic and octonionic logarithm along curves and the winding number. Journal of mathematical analysis and applications [online]. 2024. Vol. 536, no. 1,  128219. [Accessed 6 April 2025]. DOI 10.1016/j.jmaa.2024.128219. Retrieved from: https://dirros.openscience.si/IzpisGradiva.php?lang=eng&id=18310
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Record is a part of a journal

Title:Journal of mathematical analysis and applications
Shortened title:J. math. anal. appl.
Publisher:Elsevier
ISSN:0022-247X
COBISS.SI-ID:3081231 New window

Document is financed by a project

Funder:ARIS - Slovenian Research and Innovation Agency
Funding programme:Javna agencija za znanstvenoraziskovalno in inovacijsko dejavnost Republike Slovenije
Project number:P1-0291-2022
Name:Analiza in geometrija
Funder:ARIS - Slovenian Research and Innovation Agency
Funding programme:Javna agencija za znanstvenoraziskovalno in inovacijsko dejavnost Republike Slovenije
Project number:N1-0237-2022
Name:Holomorfne parcialne diferencialne relacije
Funder:ARIS - Slovenian Research and Innovation Agency
Funding programme:Javna agencija za znanstvenoraziskovalno in inovacijsko dejavnost Republike Slovenije
Project number:J1-3005-2021
Name:Kompleksna in geometrijska analiza
Funder:Other - Other funder or multiple funders
Funding programme:GNSAGA
Project number:INdaM
Name:Hypercomplex function theory and applications
Funder:Other - Other funder or multiple funders
Funding programme:MIUR, Italy
Project number:FOE 2014
Name:Splines for accUrate NumeRics: adaptIve models for Simulation Environments

Licences

License:CC BY 4.0, Creative Commons Attribution 4.0 International
Link:http://creativecommons.org/licenses/by/4.0/
Description:This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

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