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Contributions to the theory of near-vector spaces, their geometry, and hyperstructures
Thesis (PhD)--Stellenbosch University, 2022.
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| Format: | Thesis |
| Language: | en_ZA |
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Stellenbosch : Stellenbosch University
2022
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| _version_ | 1867613790344839168 |
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| access_status_str | Open Access |
| author | Rabie, Jacques |
| author2 | Howell, Karin-Therese |
| author_browse | Howell, Karin-Therese Rabie, Jacques |
| author_facet | Howell, Karin-Therese Rabie, Jacques |
| author_sort | Rabie, Jacques |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (PhD)--Stellenbosch University, 2022. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/125917 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:41:44.687Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2022 |
| publishDateRange | 2022 |
| publishDateSort | 2022 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/125917 Contributions to the theory of near-vector spaces, their geometry, and hyperstructures Rabie, Jacques Howell, Karin-Therese Stellenbosch University. Faculty of Science. Dept. of Applied Mathematics. Near-vector spaces Nearaffine spaces Incidence geometry -- Mathematical models Hypergroups Hyper near-vector spaces Geometry, Differential Decomposition theorem -- Mathematical models UCTD Thesis (PhD)--Stellenbosch University, 2022. ENGLISH ABSTRACT: This thesis expands on the theory and application of near-vector spaces — in particular, the underlying geometry of near-vector spaces is studied, and the theory of near-vector spaces is applied to hyperstructures. More specifically, a near-linear space is defined and some properties of these spaces are proved. It is shown that by adding some axioms, the nearaffine space, as defined by André, i s obtained. A correspondence is shown between subspaces of nearaffine spaces generated by near-vector spaces, and the cosets of subspaces of the corresponding near-vector space. As a highlight, some of the geometric results are used to prove an open problem in near-vector space theory, namely that a non-empty subset of a near-vector space that is closed under addition and scalar multiplication is a subspace of the near-vector space. The geometric work of this thesis is concluded with a first look into the projections of nearaffine s paces, a branch of the geometry that contains interesting avenues for future research. Next the theory of hyper near-vector spaces is developed. Hyper near-vector spaces are defined having similar properties to André’s near-vector space. Important concepts, including independence, the notion of a basis, regularity, and subhyperspaces are defined, and an analogue of the Decomposition Theorem, an important theorem in the study of near-vector spaces, is proved for these spaces. AFRIKAANS OPSOMMING: Hierdie tesis bou op die teorie en toepassing van naby-vektorruimtes — besonderlik word die onderliggende meetkunde van naby-vektorruimtes bestudeer en die teorie van naby-vektorruimtes word toegepas op hiperstrukture. Spesifiek work ’n naby-lineêre ruimte gedefinieer en sommige eienskappe van hier- die ruimtes word bewys. Dit word bewys dat, deur sekere aksiomas by te las, die naby-affiene ruimte, soos gedefinieer deur André, verkry w ord. ’n Verwantskap tus- sen die deelruimtes van naby-affiene ruimtes gegenereer deur naby-vektorruimtes en die resklasse van die deelruimtes van die verwante naby-vektorruimte word be- wys. As ’n hoogtepunt word van die meetkundige resultate gebruik om ’n oop probleem op te los in naby-vektorruimteteorie, naamlik dat ’n nie-leë deelversa- meling van ’n naby-vektorruimte wat geslote is onder optelling en skalaarverme- nigvuldiging ’n deelruimte is van die naby-vektorruimte. Die meetkundige werk in dié tesis sluit af met ’n eerste bestudering van projeksies van naby-affiene ruimtes, ’n tak in die meetkunde wat interessante toekomstige navorsingsrigtings bevat. Volgende word die teorie agter hiper naby-vektorruimtes ontwikkel. Hiper naby- vektorruimtes word gedefinieer s oortgelyk a an A ndré s e n aby-vektorruimte. Be- langrike konsepte, insluitent onafhanklikheid, die begrip van ’n basis, regulêriteit en hiper-deelruimtes word gedefinieer e n ’ n analoog van die Ontbindingstelling, belangrik in die teorie van naby-vektorruimtes, word bewys vir hierdie ruimtes. Doctoral 2022-10-25T19:51:56Z 2023-01-16T12:41:12Z 2022-10-25T19:51:56Z 2023-01-16T12:41:12Z 2022-12 Thesis http://hdl.handle.net/10019.1/125917 en_ZA Stellenbosch University vi, 84 pages application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Near-vector spaces Nearaffine spaces Incidence geometry -- Mathematical models Hypergroups Hyper near-vector spaces Geometry, Differential Decomposition theorem -- Mathematical models UCTD Rabie, Jacques Contributions to the theory of near-vector spaces, their geometry, and hyperstructures |
| title | Contributions to the theory of near-vector spaces, their geometry, and hyperstructures |
| title_full | Contributions to the theory of near-vector spaces, their geometry, and hyperstructures |
| title_fullStr | Contributions to the theory of near-vector spaces, their geometry, and hyperstructures |
| title_full_unstemmed | Contributions to the theory of near-vector spaces, their geometry, and hyperstructures |
| title_short | Contributions to the theory of near-vector spaces, their geometry, and hyperstructures |
| title_sort | contributions to the theory of near vector spaces their geometry and hyperstructures |
| topic | Near-vector spaces Nearaffine spaces Incidence geometry -- Mathematical models Hypergroups Hyper near-vector spaces Geometry, Differential Decomposition theorem -- Mathematical models UCTD |
| url | http://hdl.handle.net/10019.1/125917 |
| work_keys_str_mv | AT rabiejacques contributionstothetheoryofnearvectorspacestheirgeometryandhyperstructures |