Geometrijska zeta funkcija fraktalne strune

master's thesis

• Title (english): The geometric zeta function of the fractal string
• Author: Ivan Sinčić
• Mentor: Goran Radunović (mentor)
• Committee member: Goran Radunović (predsjednik povjerenstva)
• Committee member: Goran Muić (član povjerenstva)
• Committee member: Filip Najman (član povjerenstva)
• Committee member: Pavle Goldstein (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2024-02-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu prezentiramo početak teorije geometrije obične fraktalne strune te ju motiviramo proučavanjem Dirichletovih svojstvenih vrijednosti. U prvom poglavlju uvodimo nekoliko načina pridruživanja dimenzije fraktalnoj struni i dokazujemo njihova osnovna svojstva. Zatim analiziramo geometrijsku zeta funkciju i kompleksne dimenzije fraktalne strune. U drugom poglavlju obrađujemo poseban slučaj sebi-sličnih fraktalnih struna te ih povezujemo sa sebi-sličnim skupovima i na kraju navodimo njihove strukturne teoreme. U trećem poglavlju proučavamo valnu jednadžbu i svojstva Dirichletovih svojstvenih vrijednosti za određene rubne uvjete te analiziramo njihovu asimptotiku za jednostavne domene, motivirajući promatranje domena s fraktalnim rubom. Na kraju poglavlja definiramo spektralnu zeta funkciju strune, povezujemo ju s Riemannovom zeta funkcijom, za koju nalazimo meromorfno proširenje na kompleksnoj ravnini te dokazujemo Weylov asimptotski zakon za fraktalnu strunu.

Abstract (english)

In this thesis, we present the start of the theory of fractal string geometry and motivate it by studying Dirichlet eigenvalues. In the first chapter, we look at some of the ways to assign a dimension to a fractal string and prove basic properties of each one. Next, we analyze geometric zeta function and complex dimensions of a fractal string. In the second chapter, we study the case of self-similar fractal strings by connecting them with self-similar sets and then state their structural theorems. In the third chapter we examine the wave equation and properties of Dirichlet eigenvalues, after which we look at their asymptotic behaviour for simple domains, motivating the study of domains with fractal boundary. In the end, spectral zeta function of a fractal string is defined and connected to the Riemann zeta function, which we find the meromorphic continuation on the complex plane for, and then prove Weyl’s asymptotic law for fractal strings.

• Keywords: Dirichletove svojstvene vrijednosti; sebi-slične fraktalne strune; Riemannova zeta funkcija
• Language: croatian
• URN:NBN: urn:nbn:hr:217:015021
• Study programme: Title: Theoretical Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar matematike (sveučilišni magistar matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-03-14 11:38:28