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master's thesis
Topološki i diskretni pristup problemu podjele ogrlice
2024. urn:nbn:hr:166:956102
Tadić, Robert
University of Split
Faculty of Science
Department of Mathematics

Cite this document

Tadić, R. (2024). Topološki i diskretni pristup problemu podjele ogrlice (Master's thesis). Split: University of Split, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:166:956102

Tadić, Robert. "Topološki i diskretni pristup problemu podjele ogrlice." Master's thesis, University of Split, Faculty of Science, 2024. https://urn.nsk.hr/urn:nbn:hr:166:956102

Tadić, Robert. "Topološki i diskretni pristup problemu podjele ogrlice." Master's thesis, University of Split, Faculty of Science, 2024. https://urn.nsk.hr/urn:nbn:hr:166:956102

Tadić, R. (2024). 'Topološki i diskretni pristup problemu podjele ogrlice', Master's thesis, University of Split, Faculty of Science, accessed 14 November 2024, https://urn.nsk.hr/urn:nbn:hr:166:956102

Tadić R. Topološki i diskretni pristup problemu podjele ogrlice [Master's thesis]. Split: University of Split, Faculty of Science; 2024 [cited 2024 November 14] Available at: https://urn.nsk.hr/urn:nbn:hr:166:956102

R. Tadić, "Topološki i diskretni pristup problemu podjele ogrlice", Master's thesis, University of Split, Faculty of Science, Split, 2024. Available at: https://urn.nsk.hr/urn:nbn:hr:166:956102

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Metadata
TitleTopološki i diskretni pristup problemu podjele ogrlice
Title (english)Topological and discrete approach to the necklace splitting problem
AuthorRobert Tadić
MentorTanja Vojković (mentor)
Committee memberTanja Vojković (predsjednik povjerenstva)
Committee memberAna Laštre (član povjerenstva)
Committee memberAljoša Šubašić (član povjerenstva)
GranterUniversity of Split
Faculty of Science
(Department of Mathematics)
Split
Defense date and country2024-07-04, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
Cilj ovog rada je riješiti neke varijante problema podjele ogrlice koristeći različite strategije. U tu svrhu smo prvo formalizirali problem, odnosno uveli definiciju ogrlice i pravedne podjele ogrlice i iskazali teorem o egzistenciji pravedne podjele. Prvi pristup koji smo koristili za rješavanje problema je topološki. Definirali smo homotopiju, fundamentalnu grupu i odredili fundamentalnu grupu kružnice s ciljem dokazivanja Borsuk-Ulam teorema. Konačno, iskoristili smo Borsuk-Ulam teorem za dokazivanje egzistencije pravedne podjele ogrlice. Nadalje, promotrili smo neke posebne slučajeve problema i dokazali ih direktno i u tom procesu iskoristili određene rezultate iz teorije grafova. Naposlijetku, definirali smo kubične komplekse i promatrali kubična preslikavanja. Konstruktivno smo dokazali Ky-Fanov teorem i primjenili konstrukciju iz dokaza na rješavanje problema pravedne podjele ogrlice.
Abstract (english)
Goal of this paper is to solve some variants of necklace-splitting problem using different strategies. In this context, we firstly formalized the problem, i.e. introduced the definition of the necklace and fair division of the necklace and stated the theorem on the existence of fair division. The first approach we used to solve the problem was topological. We defined a homotopy, a fundamental group and determined a fundamental group of a circle with the aim of proving the Borsuk-Ulam theorem. Finally, we used the Borsuk-Ulam theorem to prove the existence of a fair division of the necklace. Furthermore, we looked at some special cases of the problem and proved them directly and in the process used certain results from graph theory. Finally, we defined cubical complexes and observed cubical mappings. We constructively proved the Ky-Fan theorem and applied the construction from the proof to solve the problem of fair division of the necklace.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:166:956102
Study programmeTitle: Mathematics; specializations in: specialization in Pure Mathematics, specialization in Computer Science, specialization in Education, specialization in Statistics and Computer Science
Course: specialization in Pure Mathematics
Study programme type: university
Study level: graduate
Academic / professional title: sveučilišni/a magistar/magistra matematike (sveučilišni/a magistar/magistra matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2024-07-17 12:39:07