Algorithmic randomness on computable metric spaces and hyperspaces

 

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dc.contributor.advisor Brattka, Vasco en_ZA
dc.contributor.author Birch, Thomas en_ZA
dc.date.accessioned 2016-10-04T10:09:10Z
dc.date.available 2016-10-04T10:09:10Z
dc.date.issued 2012 en_ZA
dc.identifier.citation Birch, T. 2012. Algorithmic randomness on computable metric spaces and hyperspaces. University of Cape Town. en_ZA
dc.identifier.uri http://hdl.handle.net/11427/22093
dc.description.abstract 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. en_ZA
dc.language.iso eng en_ZA
dc.subject.other Mathematics en_ZA
dc.title Algorithmic randomness on computable metric spaces and hyperspaces en_ZA
dc.type Master Thesis
uct.type.publication Research en_ZA
uct.type.resource Thesis en_ZA
dc.publisher.institution University of Cape Town
dc.publisher.faculty Faculty of Science en_ZA
dc.publisher.department Department of Mathematics and Applied Mathematics en_ZA
dc.type.qualificationlevel Masters
dc.type.qualificationname MSc en_ZA
uct.type.filetype Text
uct.type.filetype Image
dc.identifier.apacitation Birch, T. (2012). <i>Algorithmic randomness on computable metric spaces and hyperspaces</i>. (Thesis). University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics. Retrieved from http://hdl.handle.net/11427/22093 en_ZA
dc.identifier.chicagocitation Birch, Thomas. <i>"Algorithmic randomness on computable metric spaces and hyperspaces."</i> Thesis., University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 2012. http://hdl.handle.net/11427/22093 en_ZA
dc.identifier.vancouvercitation Birch T. Algorithmic randomness on computable metric spaces and hyperspaces. [Thesis]. University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 2012 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/22093 en_ZA
dc.identifier.ris TY - Thesis / Dissertation AU - Birch, Thomas AB - 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. DA - 2012 DB - OpenUCT DP - University of Cape Town LK - https://open.uct.ac.za PB - University of Cape Town PY - 2012 T1 - Algorithmic randomness on computable metric spaces and hyperspaces TI - Algorithmic randomness on computable metric spaces and hyperspaces UR - http://hdl.handle.net/11427/22093 ER - en_ZA


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