Konstrukcija formi koje dopuštaju kompoziciju

master's thesis

• Title (english): Construction of forms that allow composition
• Author: Kristijan Markač
• Mentor: Zrinka Franušić (mentor)
• Committee member: Zrinka Franušić (predsjednik povjerenstva)
• Committee member: Maro Cvitan (član povjerenstva)
• Committee member: Željka Milin Šipuš (član povjerenstva)
• Committee member: Boris Muha (član povjerenstva)
• Committee member: Mihael Makek (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-09-29, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Kažemo da \(\textit{forma g}:\mathbb{R}^n\rightarrow\mathbb{R} \textit{ dopušta kompoziciju}\) ako vrijedi \(g(x)g(y)=g(x*y)\) za sve \(x,y\in \mathbb{R}^n\), pri čemu je s x ∗ y označen neki bilinearan produkt \(\mathbb{R}^n \times \mathbb{R}^n\). Primjer jednostavne cjelobrojne linearne forme koja dopušta kompoziciju je preslikavanje \(f(x,y)=x^2-dy^2,\ x,y\in \mathbb{Z}\). Forma f dopušta kompoziciju jer vrijedi dobro poznati identitet \((x_1^2-dy_1^2)(x_2^2-dy_2^2)=(x_1x_2-dy_1y_2)^2-d(x_1y_2-x_2y_1)^2\), koji se koristi za generiranje rješenja Pellove jednadžbe \(x_1^2-dy_1^2=1\), gdje \(d\in \mathbb{N}\) nije potpun kvadrat. U ovom radu opisujemo tehniku kojom se mogu konstruirati primjeri formi koje dopuštaju kompoziciju, a koja za to koristi algebru matrica.

Abstract (english)

We say that a \(\textit{forma g}:\mathbb{R}^n\rightarrow\mathbb{R} \textit{ admits composition }\) if \(g(x)g(y)=g(x*y)\) for all \(x,y\in \mathbb{R}^n\), where x ∗ y is a bilinear product \(\mathbb{R}^n \times \mathbb{R}^n\). An example of a simple integer form admitting composition is a function \(f(x,y)=x^2-dy^2,\ x,y\in \mathbb{Z}\). The form f admits composition because of the well known identity \((x_1^2-dy_1^2)(x_2^2-dy_2^2)=(x_1x_2-dy_1y_2)^2-d(x_1y_2-x_2y_1)^2\), which is used to generate solutions to Pell’s equation \(x_1^2-dy_1^2=1\), where \(d\in \mathbb{N}\) is not a perfect square. In this paper we describe a technique, which uses matrix algebras, by which one can construct examples of forms admitting composition.

• Keywords: jednostavna cjelobrojna linearna forma; preslikavanje; Pellova jednadžba; algebra matrica
• Keywords (english): simple integer form; Pell's equation; matrix algebras
• Language: croatian
• URN:NBN: urn:nbn:hr:217:198205
• Study programme: Title: Mathematics and Physics Education; specializations in: Mathematics and Physics Education; Course: Mathematics and Physics Education; Study programme type: university; Study level: integrated undergraduate and graduate; Academic / professional title: magistar/magistra edukacije matematike i fizike (magistar/magistra edukacije matematike i fizike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2022-11-02 13:20:07