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Analysis of Functions of Split-Complex, Multicomplex, and Split-Quaternionic Variables and Their Associated Conformal Geometries

Title: Analysis of Functions of Split-Complex, Multicomplex, and Split-Quaternionic Variables and Their Associated Conformal Geometries.
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Name(s): Emanuello, John Anthony, author
Nolder, Craig, professor directing dissertation
Tabor, Samuel Lynn, university representative
Case, Bettye Anne, committee member
Quine, J. R. (John R.), committee member
Florida State University, degree granting institution
College of Arts and Sciences, degree granting college
Department of Mathematics, degree granting department
Type of Resource: text
Genre: Text
Issuance: monographic
Date Issued: 2015
Publisher: Florida State University
Place of Publication: Tallahassee, Florida
Physical Form: computer
online resource
Extent: 1 online resource (133 pages)
Language(s): English
Abstract/Description: The connections between algebra, geometry, and analysis have led the way for numerous results in many areas of mathematics, especially complex analysis. Considerable effort has been made to develop higher dimensional analogues of the complex numbers, such as Clifford algebras and Multicomplex numbers. These rely heavily on geometric notions, and we explore the analysis which results. This is what is called hyper-complex analysis. This dissertation explores the most prominent of these higher dimensional analogues and highlights a many of the relevant results which have appeared in the last four decades, and introduces new ideas which can be used to further the research of this discipline. Indeed, the objects of interest are Clifford algebras, the algebra of the Multicomplex numbers, and functions which are valued in these algebras and lie in the kernels of linear operators. These lead to prominent results in Clifford analysis and multicomplex analysis which can be viewed as analogues of complex analysis. Additionally, we explain the link between Clifford algebras and conformal geometry. We explore two low dimensional examples, namely the split-complex numbers and split-quaternions, and demonstrate how linear fractional transformations are conformal mappings in these settings.
Identifier: FSU_migr_etd-9326 (IID)
Submitted Note: A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
Degree Awarded: Spring Semester, 2015.
Date of Defense: March 27, 2015.
Bibliography Note: Includes bibliographical references.
Advisory Committee: Craig A. Nolder, Professor Directing Dissertation; Samuel Tabor, University Representative; Bettye Anne Case, Committee Member; John R. Quine, Committee Member.
Subject(s): Mathematics
Persistent Link to This Record: http://purl.flvc.org/fsu/fd/FSU_migr_etd-9326
Owner Institution: FSU

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Emanuello, J. A. (2015). Analysis of Functions of Split-Complex, Multicomplex, and Split-Quaternionic Variables and Their Associated Conformal Geometries. Retrieved from http://purl.flvc.org/fsu/fd/FSU_migr_etd-9326