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master's thesis
Konstrukcija formi koje dopuštaju kompoziciju
Zagreb: University of Zagreb, Faculty of Science, 2022. urn:nbn:hr:217:198205
Markač, Kristijan
University of Zagreb
Faculty of Science
Department of Mathematics

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Markač, K. (2022). Konstrukcija formi koje dopuštaju kompoziciju (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:198205

Markač, Kristijan. "Konstrukcija formi koje dopuštaju kompoziciju." Master's thesis, University of Zagreb, Faculty of Science, 2022. https://urn.nsk.hr/urn:nbn:hr:217:198205

Markač, Kristijan. "Konstrukcija formi koje dopuštaju kompoziciju." Master's thesis, University of Zagreb, Faculty of Science, 2022. https://urn.nsk.hr/urn:nbn:hr:217:198205

Markač, K. (2022). 'Konstrukcija formi koje dopuštaju kompoziciju', Master's thesis, University of Zagreb, Faculty of Science, accessed 14 January 2025, https://urn.nsk.hr/urn:nbn:hr:217:198205

Markač K. Konstrukcija formi koje dopuštaju kompoziciju [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2022 [cited 2025 January 14] Available at: https://urn.nsk.hr/urn:nbn:hr:217:198205

K. Markač, "Konstrukcija formi koje dopuštaju kompoziciju", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2022. Available at: https://urn.nsk.hr/urn:nbn:hr:217:198205

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Metadata
TitleKonstrukcija formi koje dopuštaju kompoziciju
Title (english)Construction of forms that allow composition
AuthorKristijan Markač
MentorZrinka Franušić (mentor)
Committee memberZrinka Franušić (predsjednik povjerenstva)
Committee memberMaro Cvitan (član povjerenstva)
Committee memberŽeljka Milin Šipuš (član povjerenstva)
Committee memberBoris Muha (član povjerenstva)
Committee memberMihael Makek (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2022-09-29, 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
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:198205
Study programmeTitle: 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 resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
RepositoryRepository of the Faculty of Science
Created on2022-11-02 13:20:07

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