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master's thesis
Izračunljivost u euklidskom prostoru
Zagreb: University of Zagreb, Faculty of Science, 2017. urn:nbn:hr:217:775307
Babojelić, Renato
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Babojelić, R. (2017). Izračunljivost u euklidskom prostoru (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:775307

Babojelić, Renato. "Izračunljivost u euklidskom prostoru." Master's thesis, University of Zagreb, Faculty of Science, 2017. https://urn.nsk.hr/urn:nbn:hr:217:775307

Babojelić, Renato. "Izračunljivost u euklidskom prostoru." Master's thesis, University of Zagreb, Faculty of Science, 2017. https://urn.nsk.hr/urn:nbn:hr:217:775307

Babojelić, R. (2017). 'Izračunljivost u euklidskom prostoru', Master's thesis, University of Zagreb, Faculty of Science, accessed 03 November 2024, https://urn.nsk.hr/urn:nbn:hr:217:775307

Babojelić R. Izračunljivost u euklidskom prostoru [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2017 [cited 2024 November 03] Available at: https://urn.nsk.hr/urn:nbn:hr:217:775307

R. Babojelić, "Izračunljivost u euklidskom prostoru", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2017. Available at: https://urn.nsk.hr/urn:nbn:hr:217:775307

Metadata
TitleIzračunljivost u euklidskom prostoru
AuthorRenato Babojelić
MentorZvonko Iljazović (mentor)
Committee memberZvonko Iljazović (predsjednik povjerenstva)
Committee memberVedran Čačić (član povjerenstva)
Committee memberFilip Najman (član povjerenstva)
Committee memberMatija Kazalicki (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2017-03-03, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
U radu se proučava izračunljivost metričkih prostora i skupova u metričkim prostorima. Uvode se pojmovi izračunljive točke i izračunljivog niza. Pokazuje se da je euklidski prostor izračunljiv metrički prostor. Definiramo izračunljiv skup izračunljivog metričkog prostora i rekruzivno prebrojiv skup u takvom prostoru. Dolazimo do rezultata kako je svaki izračunljiv skup rekurzivno prebrojiv, te kako svaki rekurzivno prebrojiv, neprazan i potpun skup sadrži gust izračunljiv niz. Promatramo prostor kompaktnih skupova \((\mathscr{K}, d_{H})\) nekog izračunljivog metričkog prostora \((X, d, \alpha)\) uz Hausdorffovu metriku. Pokazujemo kako je \((\mathscr{K}, d_{H}, \Lambda)\) jedan izračunljiv metrički prostor, te kako su svi izračunljivi skupovi u prostoru \((X, d, \alpha)\) upravo izračunljive točke prostora \((\mathscr{K}, d_{H}, \Lambda)\) i obratno.
Abstract (english)
The thesis examines the computability of metric spaces and sets in metric spaces. We introduce the terms of computable point and computable sequence. We prove the Euclidean space is a computable metric space. Furthermore, we define a computable set in a computable metric space and recursively computable set in such a space. We go to prove that every computable set is recursively enumerable and every recursively enumerable, non-empty and complete set contains a dense computable sequence. Further we go to examine a space of compact sets \((\mathscr{K}, d_{H})\) of a computable metric space \((X, d, \alpha)\) considering Hausdorff metric. We show \((\mathscr{K}, d_{H}, \Lambda)\) is a computable metric space, and that all computable metric sets of \((X, d, \alpha)\) space are exactly the computable points of \((\mathscr{K}, d_{H}, \Lambda)\), and that the converse also holds.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:775307
Study programmeTitle: Computer Science and Mathematics
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra računarstva i matematike (magistar/magistra računarstva i matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2019-03-01 13:16:33