Vjerojatnosne i funkcionalne varijante Szemerédijeve leme o regularnosti

master's thesis

• Author: Filip Bosnić
• Mentor: Vjekoslav Kovač (mentor)
• Committee member: Vjekoslav Kovač (predsjednik povjerenstva)
• Committee member: Mirko Primc (član povjerenstva)
• Committee member: Pavle Pandžić (član povjerenstva)
• Committee member: Ilja Gogić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2016-09-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Ovaj rad daje dokaze funkcionalno analitičkih i vjerojatnosnih varijanti Szemerédijeve leme o regularnosti i međusobno ih uspoređuje. Dokaz funkcionalne varijante bazira se na Hahn-Banachovom teoremu o separaciji, dok je dokaz vjerojatnosne varijante elementaran i koristi samo osnove teorije vjerojatnosti. Kroz cijeli rad motivaciju pruža Szemerédijev teorem o aritmetičkim nizovima. Posebni slučaj ovog teorema, Rothov teorem, dokazan je korištenjem vjerojatnosne varijante leme o regularnosti.

Abstract (english)

This paper gives proofs of functional analytic and probabilistic variants of the Szemerédi regularity lemma and compares them with each other. The proof of the functional variant is based on the Hahn-Banach separation theorem, while the proof of the probabilistic variant is completely elementary and uses only basic probability theory. Throughout the thesis, Szemerédi’s theorem on arithmetic progressions provides the main source of the motivation. A special case of this theorem, Roth’s theorem, is proved using the probabilistic variant of regularity lemma.

• Keywords: Szemerédijeva lema o regularnosti; Hahn-Banachov teorem o separaciji; Szemerédijev teorem o aritmetičkim nizovima; Rothov teorem
• Keywords (english): Szemerédi regularity lemma; Hahn-Banach separation theorem; Szemerédi’s theorem on arithmetic progressions; Roth’s theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:810935
• Study programme: Title: Pure Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2017-06-07 12:45:42