Aspects of higher degree forms with symmetries

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dc.contributor.advisorHughes, Kenneth Ren_ZA
dc.contributor.authorOmar, Mohammed Rafiqen_ZA
dc.date.accessioned2016-01-02T04:38:52Z
dc.date.available2016-01-02T04:38:52Z
dc.date.issued1996en_ZA
dc.descriptionBibliography: pages 113-119.en_ZA
dc.description.abstractIn Chapter One we develop a basis for studying higher degree alternating forms. The concepts and results we present are mostly obvious analogues of Harrison's treatment of higher degree symmetric forms. We explain antisymmetrization; discuss the derivative of an alternating form and its corresponding anticommutative polynomial; define alternating spaces and their direct sum; establish decomposition and cancellation results for alternating spaces; and construct a Witt-Grothendieck group of alternating spaces. In Chapter Two we discuss hyperbolic alternating space. We compute the centre, algebraic isometry group and its corresponding Lie algebra, and prove a descent result. There are important parallels with Keet's results for hyperbolic symmetric spaces, as well as significant differences, especially in the methods we employ. In Chapter Three we develop a framework for the study of two aspects of forms of general Young symmetry type: their hyperbolics, and a generalization of the Weil-Siegel duality between symmetric and alternating bilinear forms. We introduce notions like nondegeneracy, derivative of a form, and derivative and integral symmetry types, and are then able to construct a hyperbolic space which is cofinal for spaces equipped with a form of the same symmetry type, and show that symmetry types are Siegel duals in our generalized sense if they have the same derivative symmetry type. In Chapter Four we present a few results and observations concerning nondegeneracytype conditions on symmetric forms. These include: an extension of Harrison's proof that nonsingularity implies nonzero Hessian to forms of arbitrary degree; a discussion of s-nondegeneracy and s-regularity; and a relation between a strong nondegeneracy condition on forms of even degree and the catalecticant, a classical invariant.en_ZA
dc.identifier.apacitationOmar, M. R. (1996). <i>Aspects of higher degree forms with symmetries</i>. (Thesis). University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics. Retrieved from http://hdl.handle.net/11427/16109en_ZA
dc.identifier.chicagocitationOmar, Mohammed Rafiq. <i>"Aspects of higher degree forms with symmetries."</i> Thesis., University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 1996. http://hdl.handle.net/11427/16109en_ZA
dc.identifier.citationOmar, M. 1996. Aspects of higher degree forms with symmetries. University of Cape Town.en_ZA
dc.identifier.ris TY - Thesis / Dissertation AU - Omar, Mohammed Rafiq AB - In Chapter One we develop a basis for studying higher degree alternating forms. The concepts and results we present are mostly obvious analogues of Harrison's treatment of higher degree symmetric forms. We explain antisymmetrization; discuss the derivative of an alternating form and its corresponding anticommutative polynomial; define alternating spaces and their direct sum; establish decomposition and cancellation results for alternating spaces; and construct a Witt-Grothendieck group of alternating spaces. In Chapter Two we discuss hyperbolic alternating space. We compute the centre, algebraic isometry group and its corresponding Lie algebra, and prove a descent result. There are important parallels with Keet's results for hyperbolic symmetric spaces, as well as significant differences, especially in the methods we employ. In Chapter Three we develop a framework for the study of two aspects of forms of general Young symmetry type: their hyperbolics, and a generalization of the Weil-Siegel duality between symmetric and alternating bilinear forms. We introduce notions like nondegeneracy, derivative of a form, and derivative and integral symmetry types, and are then able to construct a hyperbolic space which is cofinal for spaces equipped with a form of the same symmetry type, and show that symmetry types are Siegel duals in our generalized sense if they have the same derivative symmetry type. In Chapter Four we present a few results and observations concerning nondegeneracytype conditions on symmetric forms. These include: an extension of Harrison's proof that nonsingularity implies nonzero Hessian to forms of arbitrary degree; a discussion of s-nondegeneracy and s-regularity; and a relation between a strong nondegeneracy condition on forms of even degree and the catalecticant, a classical invariant. DA - 1996 DB - OpenUCT DP - University of Cape Town LK - https://open.uct.ac.za PB - University of Cape Town PY - 1996 T1 - Aspects of higher degree forms with symmetries TI - Aspects of higher degree forms with symmetries UR - http://hdl.handle.net/11427/16109 ER - en_ZA
dc.identifier.urihttp://hdl.handle.net/11427/16109
dc.identifier.vancouvercitationOmar MR. Aspects of higher degree forms with symmetries. [Thesis]. University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 1996 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/16109en_ZA
dc.language.isoengen_ZA
dc.publisher.departmentDepartment of Mathematics and Applied Mathematicsen_ZA
dc.publisher.facultyFaculty of Scienceen_ZA
dc.publisher.institutionUniversity of Cape Town
dc.subject.otherMathematics and Applied Mathematicsen_ZA
dc.titleAspects of higher degree forms with symmetriesen_ZA
dc.typeDoctoral Thesis
dc.type.qualificationlevelDoctoral
dc.type.qualificationnamePhDen_ZA
uct.type.filetypeText
uct.type.filetypeImage
uct.type.publicationResearchen_ZA
uct.type.resourceThesisen_ZA
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