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Algorithmic randomness on computable metric spaces and hyperspaces
In this text we shall be focusing on generalizing Martin-Löf randomness to computable metric spaces with arbitrary measure (for examples of this type of generalization see Gács [14], Rojas and Hoyrup [15]. The aim of this generalization is to define algorithmic randomness on the hyperspace of non-em...
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| Format: | Thesis |
| Language: | English |
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Department of Mathematics and Applied Mathematics
2016
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| _version_ | 1867614407518846976 |
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| access_status_str | Open Access |
| author | Birch, Thomas |
| author2 | Brattka, Vasco |
| author_browse | Birch, Thomas Brattka, Vasco |
| author_facet | Brattka, Vasco Birch, Thomas |
| author_sort | Birch, Thomas |
| collection | Thesis |
| description | In this text we shall be focusing on generalizing Martin-Löf randomness to computable metric spaces with arbitrary measure (for examples of this type of generalization see Gács [14], Rojas and Hoyrup [15]. The aim of this generalization is to define algorithmic randomness on the hyperspace of non-empty compact subsets of a computable metric space, the study of which was first proposed by Barmpalias et al. [16] at the University of Florida in their work on the random closed subsets of the Cantor space. Much work has been done in the study of random sets with authors such as Diamondstone and Kjos-Hanssen [17] continuing the Florida approach, whilst others such as Axon [18] and Cenzer and Broadhead [19] have been studying the use of capacities to define hyperspace measures for use in randomness tests. Lastly in section 6.4 we shall be looking at the work done by Hertling and Weihrauch [13] on universal randomness tests in effective topological measure spaces and relate their results to randomness on computable metric measure spaces and in particular to the randomness of compact sets in the hyperspace of non-empty compact subsets of computable metric spaces. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/22093 |
| institution | University of Cape Town (South Africa) |
| language | eng |
| last_indexed | 2026-06-10T12:51:33.401Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2016 |
| publishDateRange | 2016 |
| publishDateSort | 2016 |
| publisher | Department of Mathematics and Applied Mathematics |
| publisherStr | Department of Mathematics and Applied Mathematics |
| record_format | dspace |
| source_str | UCTD — University of Cape Town Open Access Repository |
| spelling | oai:open.uct.ac.za:11427/22093 Algorithmic randomness on computable metric spaces and hyperspaces Birch, Thomas Brattka, Vasco Mathematics In this text we shall be focusing on generalizing Martin-Löf randomness to computable metric spaces with arbitrary measure (for examples of this type of generalization see Gács [14], Rojas and Hoyrup [15]. The aim of this generalization is to define algorithmic randomness on the hyperspace of non-empty compact subsets of a computable metric space, the study of which was first proposed by Barmpalias et al. [16] at the University of Florida in their work on the random closed subsets of the Cantor space. Much work has been done in the study of random sets with authors such as Diamondstone and Kjos-Hanssen [17] continuing the Florida approach, whilst others such as Axon [18] and Cenzer and Broadhead [19] have been studying the use of capacities to define hyperspace measures for use in randomness tests. Lastly in section 6.4 we shall be looking at the work done by Hertling and Weihrauch [13] on universal randomness tests in effective topological measure spaces and relate their results to randomness on computable metric measure spaces and in particular to the randomness of compact sets in the hyperspace of non-empty compact subsets of computable metric spaces. 2016-10-04T10:09:10Z 2016-10-04T10:09:10Z 2012 Master Thesis Masters MSc http://hdl.handle.net/11427/22093 eng application/pdf Department of Mathematics and Applied Mathematics Faculty of Science University of Cape Town |
| spellingShingle | Mathematics Birch, Thomas Algorithmic randomness on computable metric spaces and hyperspaces |
| thesis_degree_str | Master's |
| title | Algorithmic randomness on computable metric spaces and hyperspaces |
| title_full | Algorithmic randomness on computable metric spaces and hyperspaces |
| title_fullStr | Algorithmic randomness on computable metric spaces and hyperspaces |
| title_full_unstemmed | Algorithmic randomness on computable metric spaces and hyperspaces |
| title_short | Algorithmic randomness on computable metric spaces and hyperspaces |
| title_sort | algorithmic randomness on computable metric spaces and hyperspaces |
| topic | Mathematics |
| url | http://hdl.handle.net/11427/22093 |
| work_keys_str_mv | AT birchthomas algorithmicrandomnessoncomputablemetricspacesandhyperspaces |