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master's thesis
Teorija slučajnih matrica i Riemannova zeta - funkcija
2019. urn:nbn:hr:126:678939
Behin, Andrea
Josip Juraj Strossmayer University of Osijek
Department of Mathematics

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Behin, A. (2019). Teorija slučajnih matrica i Riemannova zeta - funkcija (Master's thesis). Osijek: Josip Juraj Strossmayer University of Osijek, Department of Mathematics. Retrieved from https://urn.nsk.hr/urn:nbn:hr:126:678939

Behin, Andrea. "Teorija slučajnih matrica i Riemannova zeta - funkcija." Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2019. https://urn.nsk.hr/urn:nbn:hr:126:678939

Behin, Andrea. "Teorija slučajnih matrica i Riemannova zeta - funkcija." Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2019. https://urn.nsk.hr/urn:nbn:hr:126:678939

Behin, A. (2019). 'Teorija slučajnih matrica i Riemannova zeta - funkcija', Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, accessed 15 November 2024, https://urn.nsk.hr/urn:nbn:hr:126:678939

Behin A. Teorija slučajnih matrica i Riemannova zeta - funkcija [Master's thesis]. Osijek: Josip Juraj Strossmayer University of Osijek, Department of Mathematics; 2019 [cited 2024 November 15] Available at: https://urn.nsk.hr/urn:nbn:hr:126:678939

A. Behin, "Teorija slučajnih matrica i Riemannova zeta - funkcija", Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, Osijek, 2019. Available at: https://urn.nsk.hr/urn:nbn:hr:126:678939

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Metadata
TitleTeorija slučajnih matrica i Riemannova zeta - funkcija
Title (english)Random Matrix Theory and the Riemann Zeta-Function
AuthorAndrea Behin
MentorIvica Martinjak (mentor)
Committee memberDomagoj Ševerdija (predsjednik povjerenstva)
Committee memberIvica Martinjak (član povjerenstva)
Committee memberIvan Matić (član povjerenstva)
GranterJosip Juraj Strossmayer University of Osijek
Department of Mathematics
Osijek
Defense date and country2019-05-07, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Other Mathematical Disciplines
Abstract
Teorija slučajnih matrica je grana matematike motivirana ostalim područjima matematike i fizike poput kvantne mehanike, linearne algebre i matematičke statistike. Na početku rada definirali smo slučajne matrice i naveli motivaciju za njihovo uvođenje. Posebno smo razmotrili svojstvene vrijednosti takvih matrica i njihova svojstva te opisali Gaussov model slučajnih matrica za koji je moguće dati eksplicitno rješenje za zajedničku funkciju gustoće njezinih svojstvenih vrijednosti. Nadalje, definirali smo Riemannovu zeta-funkciju ζ(s) i naveli njezina osnovna svojstva. Riemannova slutnja, koja do danas nije dokazana niti opovrgnuta, govori nam da sve nultočke od ζ(s) imaju realan dio jednak 1/2 i važna je zbog povezanosti s distribucijom prostih brojeva, ali i jer mnogi rezultati u teoriji brojeva njezinu istinitost uzimaju kao polaznu pretpostavku. U radu smo opisali povezanost svojstvenih vrijednosti slučajnih matrica s nultčkama Riemannove zeta-funkcije. Na kraju rada na primjeru smo opisali primjenu teorije slučajnih matrica na financijske podatke za filtriranje matrice kros-korelacija te u problemu optimizacije portfelja.
Abstract (english)
Random matrix theory is a branch of mathematics motivated by many other disciplines of mathematics and physics such as quantum mechanics, linear algebra and mathematical statistics. This text begins with a definition of random matrix and the motivation for its appearance in science. We considered the eigenvalues of such matrices and their properties and described Gaussian unitary ensemble for which it is possible to give an explicit formula of the joint probability density function of its eigenvalues. In addition, we define Riemann zeta-function ζ(s) and give its basic properties. Riemann hypothesis, which has not been proved or denied so far, tells us that the real part of all of the zeros of ζ(s) equals 1/ 2 and is important because of the relationship with the distribution of primes and because many results in the number theory take its truthfulness as a starting point. In this paper we have described the connection of the eigenvalues of random matrices with zeros of Riemann’s zeta-function. Finally, we discuss an example of approach to cross-correlations in financial data to determine which part of cross-correlation matrix is just noise and which is the real signal and also in portfolio optimization problem.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:126:678939
Study programmeTitle: Mathematics; specializations in: Financial and Statistical Mathematics, Mathematics and Computer Science, Industrial and Applied Mathematics
Course: Financial and Statistical Mathematics
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2019-05-08 07:45:55