NEW DOCTORAL DEGREES Elliptic curves of large rank over quadratic fields

In this thesis, we study the construction of elliptic curves of positive and relatively large rank with a fixed torsion group over quadratic fields. If $K$ is a quadratic field and $E|K$ an elliptic curve over the field $K$, the possible torsion groups are: begin{itemize} item[-] $Z/nZ$, where $1le nle 16$, $n=18$, item[-] $Z/2Ztimes Z/2mZ$, where $1le mle 6$, item[-] $Z/3Z times Z/3kZ$, where $k=1,2$, $K=Q(sqrt{-3})$, item[-] $Z/4Z times Z/4Z$, for $K=Q(i)$. end{itemize} First we construct curves of relatively large rank with torsion groups $Z/4Ztimes Z/4Z$, $Z/3Ztimes Z/6Z$ and $Z/3Ztimes Z/3Z$. For the first two groups, curves of rank at least 3 were already known, and for the third group, the already known result was that there exists a curve with rank $ge2$. In the thesis we construct an elliptic curve over $Q(i)$ with torsion group $Z/4Ztimes Z/4Z$ and rank equal to 7 and a family of elliptic curves with the same torsion and rank $ge 2$. In the case of elliptic curves over the quadratic field $Q(sqrt{-3})$, we construct an elliptic curve with torsion group $Z/3Ztimes Z/3Z$ and rank equal to 7 and an elliptic curve with torsion group $Z/3Ztimes Z/6Z$ and rank equal to 6. Mestre\u27s conditional upper bound for rank plays a significant role in the methods we use, so it is studied in more detail. For torsion groups $Z/2Ztimes Z/10Z$, $Z/2Ztimes Z/12Z$, $Z/15Z$, $Z/11Z$ and $Z/14Z$ we study elliptic curves of positive rank over the quadratic field $Q(sqrt{d})$, where the absolute value of the discriminant of the quadratic field is minimal. Specifically, we construct curves of positive rank with torsion groups $Z/2Ztimes Z/10Z$, $Z/2Ztimes Z/12Z$ and $Z/15Z$ over the fields $Q(sqrt{-2})$, $Q(sqrt{13})$ and $Q(sqrt{-7})$, respectively, and we determine curves of conditionally positive rank over the fields $Q(sqrt{-7})$ and $Q(sqrt{3})$ for torsion groups $Z/11Z$ and $Z/14Z$, respectively. We also determine new families of elliptic curves with torsion groups $Z/2Ztimes Z/10Z$ and $Z/2Ztimes Z/12Z$. Additionally, we construct elliptic curves with a fixed torsion group and maximal rank over quadratic fields $Q(sqrt{d})$, $|d|le 10^{100}$, where $|d|$ is minimal, and we give an overview of current results

Project GAMMA

Projekt GAMMA je Erasmus+ projekt na kojemu je Odjel za matematiku Sveučilišta u Osijeku koordinator. U radu predstavljamo projekt i njegove rezultate, opisujemo aktivnosti važne za razvoj rezultata i najavljujemo službene diseminacijske aktivnosti.The project GAMMA is an Erasmus+ project coordinated by the Department of Mathematics, University of Osijek. The paper presents the project and its results, describes the activities important for the development of the results, and announces the official dissemination activities

Applications of the Euclidean algorithm

U članku pokazujemo kako se Euklidov algoritam može iskoristiti u rješavanju linearnih diofantskih jednadžbi i dobivene rezultate primjenjujemo na razne vrste zadataka. Osim toga, dovodimo u vezu Euklidov algoritam i razvoj racionalnog broja u verižni razlomak te navodimo neka od svojstava ovakvih razvoja.In this article, we show how the Euclidean algorithm can be used for solving linear Diophantine equations and we apply the obtained results to various types of tasks. We also show that the Euclidean algorithm and expansion of a rational number to a continued fraction are closely related and we discuss some properties of those expansions

Eulerova funkcija

Eulerova funkcija je jedna od najvažnijih funkcija u teoriji brojeva. U članku ćemo izvesti formulu za određivanje vrijednosti ove funkcije te ćemo dokazati neka od njezinih svojstava. Na primjerima ćemo pokazati neke od primjena dobivenih rezultata, a na kraju ćemo opisati nekoliko problema koji su usko vezani uz Eulerovu funkciju

Multimedijsko učenje i vrednovanje matematičkih panoa na primjeru GeoGebre

Pod utjecajem naglog razvoja Interneta te različitih programa, posebice interaktivnih matematičkih računalnih softvera, mijenjale su se i metode uključivanja računala u obrazovni proces. Na primjerima izrade matematičkih panoa iz nastavnih cjelina linearne funkcije, kvadratne funkcije, eksponencijalne i logaritamske funkcije, potrošačkog kredita i kalkulacije prikazana je metoda učenja i vrednovanja učeničkih radova uz pomoć multimedije. U istraživanju je sudjelovalo 420 učenika početnih razreda srednje strukovne škole Splitsko-dalmatinske županije. Eksperimentalno je istraživanje pokazalo da višekratna upotreba računalnih strategija i pragmatična upotreba multimedijske didaktike može značajno utjecati na odgojno-obrazovni proces učenja. Rezultati istraživanja prikazuju kako je primjenom matematičkog programa GeoGebra, alata za e-učenje na praktičnim zadacima kroz istraživanje i kreativno prikazivanje rezultata, olakšano usvajanje gradiva. Time se povećava motivacija za učenje matematike, ali i razvoj digitalnih kompetencija kod učenika i nastavnika

Degradation of inkjet ink by greensand and ultrasonic sonification

The study describes the degradation of inkjet ink at low frequency ultrasound (US) and greensand to compare their reactivity. Environmental sonochemistry is a rapidly growing area and an example of the advanced oxidation process (AOP) that deals with the destruction of organic species in aqueous solutions. Greensand is a granular material coated with a thin layer of manganese dioxide (MnO2) which is among the strongest natural oxidants. In our study magenta inkjet water-based printing ink was dissolved in distilled water and the solutions obtained after degradation were analysed in terms of total organic compound (TOC) and absorption curves in the visible spectra. Also used for the process monitoring was high performance liquid chromatography (HPLC). The efficiency of discoloration is significantly affected by the effluent pH. The efficiency of discolouration was higher when the pH of initial solution was 2 with respect to the initial solution pH of 5.5. In all solutions, irrespective of the initial pH value and the processing method the oxidation of polyhydric alcohols occurs. Although the decomposition is significant, surface peaks resulting from HPLC analysis are very small. Decolourization is closely related to the cleavage of the – C=C and -N=N- bonds, and oxidation of polyhydric alcohol to the formation of monosaccharides, carboxylic acids or other low molecular weight compounds with a lesser number of unsaturated double bonds. These compounds have low UV absorbance or they absorb below 200 nm and therefore their detection is impossible. Thus, the obtained total organic compound results indicate a small degree of mineralization. The effectiveness of the low-frequency ultrasound (20 kHz) oxidation is similar to the effectiveness of oxidation by greendsand

Concentration of nutritional important minerals in Croatian goat and cow milk and some dairy products made of these

The concentration of the minerals (Ca, Mg and P) and trace elements (Zn, Fe) were determined in goat and cow’s dairy products. The aim of this work was to determine the concentrations of mentioned minerals and trace elements in fermented dairy products made of goat milk, as well as in East Croatia traditional White Slice goat cheese. Obtained results show that goat milk and dairy products from goat milk had higher concentration of Mg and Fe than these of cow milk. Goat milk and dairy products from goat milk also had higher concentration of P, whereas the concentration of Ca was equally in goat and cow milk. However, significantly lower concentrations of Zn in goat milk and goat milk products were determined. Levels of analyzed major and trace minerals were higher in fermented dairy products and cheeses than in liquid milk. The levels of major and trace minerals in White Slice cheese were greater than those in fermented milk products. High content of phosphorus in White Slice goat cheese than in White Slice cow cheese was determined