Diskretna interpretacija Riemannovega upodobitvenega izreka : delo diplomskega seminarja

Kišek, Anja (author), Kuzman, Uroš (mentor)

V diplomskem delu bomo obravnavali diskretno interpretacijo Riemannovega upodobitvenega izreka oziroma alternativni postopek iskanja biholomorfizma med poljubno, pravo, enostavno povezano podmnožico kompleksne ravnine in enotskim diskom. Ta bo temeljil na dejstvu, da konformna preslikava na infinitezimalni ravni krožnice preslika v krožnice. Natančneje, predstavili bomo metodo polnjenja s krožnicami in z njeno pomočjo definirali zaporedje diskretnih preslikav, ki jih bomo zvezno razširili na triangulacijo obeh območij. Izkazalo se bo, da v limiti dobimo biholomorfno preslikavo iz Riemannovega upodobitvenega izreka.

• Language: Slovenian
• Keywords: konformne preslikave, kvazikonformne preslikave, polnjenje s krožnicami
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2018
• PID: 20.500.12556/RUL-103786
• UDC: 517.5
• COBISS.SI-ID: 18455641
• Publication date in RUL: 26.09.2018

KIŠEK, Anja, 2018, Diskretna interpretacija Riemannovega upodobitvenega izreka : delo diplomskega seminarja [online]. Bachelor’s thesis. [Accessed 14 April 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=103786

Secondary language

• Language: English
• Title: The Discrete Interpretation of The Riemann Mapping Theorem

Abstract

In this thesis we will observe the Riemann mapping theorem in an alternative way through the theory of discrete analytic functions. The fact that conformal mapping sends infinitesimal circles to circles will be used to construct biholomorphism between non-empty simply connected open subset of the complex plane and the open unit disk. We will describe a method called circle packing, which will help us to define a sequence of discrete mappings which can be continuously extended to a triangulation of both domains. Finally, we will prove that this sequence converges to a conformal mapping, which conicides with the one from the Riemann mapping theorem.

• Keywords: conformal mappings, quasiconformal mappings, circle packing