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We study a class of pairs of Lie algebras $(\g, \g_1)$ that we call Cartan pairs ; here $\g$ is semisimple and $\g_1$ is a reductive in $\g$ subalgebra. For these pairs, which generalize symmetric ones, we have standardly defined Cartan subspaces, and consequently the set of restricted roots $\Sigma(\g, \a)$. We prove that there are infinitely many interesting nonsymmetric Cartan pairs. Next we prove that every pair of the well known Brylinski-Kostant list of shared orbit pairs is a Cartan pair. As a continuation of the previous research we obtained some further useful and clarifying results and examples related to Cartan pairs and Cartan subspaces.

We study the problem asking which ideals in Noetherian rings have the Artin-Rees property, and want to obtain new ideals with this property from those for which we know that satisfy it. We show that the class $ARRingc$, of all (Noetherian) Artin-Rees rings in which every prime ideal is moreover completely prime, is closed under localization at an arbitrary denominator set. We discuss and illustrate our results on two particular examples: the enveloping algebra of $mathfrak{sl}(2)$ and the three-dimensional Heisenberg Lie algebra.

We consider a class P of pairs (g, g_1) of K-Lie algebras $g_1\subseteq g$ satisfying certain "rigidity conditions" ; here K is a field of characteristic 0, and g_1 is reductive. We provide some further evidence that P contains a number of nonsymmetric pairs that are worth studying ; e.g., in some branching problems, and for the purposes of the geometry of orbits. In particular, for an infinite series (g, g_1)=(sl(n+1), sl(2)) we show that it is in P, and precisely describe a g_1-module structure of the Killing-orthogonal p(n) of g_1 in g. Using this and the Kostant's philosophy concerning the exponents for (complex) Lie algebras, we obtain two more results. First ; suppose K is algebraically closed, g is semisimple all of whose factors are classical, and s is a principal TDS. Then (g, s) belongs to P. Second ; suppose (g, g_1) is a pair satisfying certain condition (C), and there exists a semisimple $s\subseteq g_1$ such that (g, s) is from P (e.g., s is a principal TDS). Then (g, g_1) is from P as well. Finally, given a pair (g, g_1), we have some useful observations concerning the relationship between the coadjoint orbits corresponding to g and g_1, respectively.

We explain in more details our work ''A generalized global Cartan decomposition: a basic example'', by providing some new arguments, remarks and examples. In particular we establish some auxiliary results describing the symplectic Lie algebra sp(n) and its Killing-orthogonal p(n) in sl(2n) so that these will be convenient for our computations.

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U ovom radu govorimo o kvadratnim proširenjima polja racionalnih brojeva, i to prvenstveno realnim kvadratnim proširenjima. Polazeći od definicije i elementarnih pojmova algebre, pokazujemo karakteristike i konstrukcije općenitih kvadratnih proširenja. Nadalje definiramo ključne pojmove algebarski broj i algebarskog cijelog broja i navodimo bitne rezultate vezane za njih. Na kraju navodimo neke strukturne rezultate za prsten cijelih brojeva kao što su jedinice, ideali, prosti ideali i faktorizacija. In this thesis we consider quadratic field extensions of the field of rational numbers, and in particular real quadratic extensions. We begin by definitions and basic concepts of algebra, and then we characterize quadratic field extensions in general case. Furthermore we introduce the key notions of an algebraic number and algebraic integer, and for them we present some relevant results. Finally we prove certain structural results for the rings of integers of real quadratic extensions; e.g, some concerning units, ideals, prime ideals and factorization.

U ovom radu govorimo o funkcijama izvodnicama i njihovoj primjeni pri određivanju eksplicitne formule za opći član niza. Na samom početku dajemo jedan motivacijski primjer u kojem je potrebno odrediti formulu za opći član niza gdje uvodimo i definiramo običnu funkciju izvodnicu. Nadalje, popisujemo sve popratne teoreme koje primjenjujemo u raznim primjerima kombinatoričkih problema. Na isti način uvodimo eksponencijalnu funkciju izvodnicu. Na kraju promatramo funkciju izvodnicu u teoriji brojeva te pomoću nje dolazimo do rekurzivne formule za Bernoullijeve brojeve. In this graduate thesis we are talking about generating functions and their use with determining the explicit formula for the general sequence member. Right at the beginning we will show a motivational example in which it’s necessary to determine the general formula for the general sequence member where we introduce and define the ordinary generating function. Furthermore, we will describe all following theorems which will be applied in different examples of combinatorial problems. In the same way we will introduce the exponential generating function. At the end we will observe the generating function in number theory and use it to get to the recursive formula for Bernoulli numbers.

U ovom diplomskom radu proučavamo teoriju kategorija s posebnim fokusom na kartezijanski zatvorene kategorije, jednostavno tipizirani \(\lambda-\)račun i opisujemo vezu između njih koja je poznata kao Curry-Howard-Lambekov izomorfizam. Dan je uvod u teoriju kategorija, te kroz brojne primjere definiramo osnovne pojmove nužne za razumijevanje Curry-Howard-Lambekov izomorfizma. Koristimo programski jezik Haskell za demonstraciju praktične primjene teorija kategorija i lakše razumijevanje nekih koncepata. Definirane su kartezijanski zatvorene kategorije, vrsta kategorije koja ima ekspresivnu moć jednaku tipiziranom \(\lambda-\)računu i pokazano je nekoliko primjera kartezijanski zatvorenih kategorija. Definirano je proširenje jednostavno tipiziranog \(\lambda-\)računa s pravilima dedukcije i osnovni pojmovi vezani za tipizirani \(\lambda-\)račun. U posljednjem poglavlju je opisana konstrukcija kojom se pokazala veza između tipiziranog \(\lambda-\)računa i kartezijanski zatvorenih kategorija. Rad završavamo s primjerom kojim se pokazuje razmjenjivost funkcija u \(\lambda-\)-računu s projekcijama u kartezijanski zatvorenim kategorijama. Dajemo kratki ostvrt na prednosti kategorijskog pristupa u programiranju pozivajući se na nekoliko najčešće korištenih programskih paketa. In this master thesis we study category theory with special focus on Cartesian closed categories and simply typed \(\lambda-\)calculus. We describe the relationship between them, known as Curry-Howard-Lambek isomorphism. An introduction to the category theory and neccesary terms required for understanding the Curry-Howard-Lambek isomorphism are given. Programming language Haskell is used to demonstrate practical applications of category theory. We define Cartesian closed categories, type of category which has expressive power equivalent to the simply typed \(\lambda-\)calculus and demonstrate few examples of Cartesian closed categories. In the last chapter we define a construction that describes relationship between simply typed \(\lambda-\)calculus and Cartesian closed categories. Master thesis is finalized with an example which demonstrates interchangeability of functions in simply typed \(\lambda-\)caluclus with projections in Cartesian closed categories. A short overview with advantages of categorical approach to computation is given, demonstrating practicality with few well known programming libraries.

U ovom radu govorimo o skupu cijelih brojeva, točnije rečeno o njegovim prostim elementima. Polazeći od definicije i elementarnih pojmova teorije brojeva, pokazujemo neke od osnovnih rezultata, te karakterizacija tog skupa. Nadalje, navodimo razne varijacije dokaza beskonačnosti skupa prostih brojeva, gdje pritom koristimo neke od zanimljivih analitičkih pristupa. Na kraju proučavamo vezu skupa prostih brojeva s nizom Fermatovih, Mersennovih i Fibonaccijevih brojeva, te pomoću tih nizova i njihovih svojstava, ponovno pokazujemo tvrdnju da je skup prostih brojeva beskonačan. In this diploma thesis we consider the set of integers, or better said its prime elements. Starting from definitions and basic concepts of number theory, we are showing some of the main results, and characterizations of this set. Furthermore, we consider variations of proof referred to infinity of the set of prime numbers, where while we are using some of interesting analytical approaches. Finally, we are exploring the relation of the set of prime numbers with a sequences of Fermat, Mersenne and Fibonacci numbers, and using these sequences and their properties, again showing the claim that the set of prime numbers is infinite.

U računalnoj znanosti se dugo vremena čekalo na deterministički algoritam koji u polinomnom vremenu može odrediti je li neki prirodan broj prost. Agrawal-Kayal-Saxenin algoritam je prvi algoritam za koji se dokazalo da određuje prostost nekog broja u polinomnom vremenu, tj. navedena trojica matematičara su pokazali da jezik PRIMES pripada klasi složenosti P. Cilj ovog diplomskog rada bio je pokazati da jezik PRIMES pripada klasi P, tj. predstaviti Agrawal-Kayal-Saxenov algoritam, dokazati njegovu korektnost i odrediti njegovu složenost. U prvom, uvodnom poglavlju diplomskog rada dali smo bitne definicije i rezultate iz teorije brojeva, teorije složenosti i algebre. Te definicije i rezultati su nam bili potrebni da bi u drugom, glavnom dijelu diplomskog rada mogli dokazati korektnost Agrawal-Kayal-Saxeninog algoritam i pokazati da pripada klasi složenosti P. Također smo u prvom poglavlju opisali algoritme sa slučajnim elementima te smo prošli kroz povijest problema PRIMES. Glavni dio diplomskog rada smo podijelili u dva dijela. U prvom dijelu, AKS algoritam dokaz korektnosti, dali smo motivaciju za algoritam koja se temeljila na Pascalovu trokutu, te smo predstavili Agrawal-Kayal-Saxenin algoritam i kroz niz rezultata pokazali smo da je korektan. U drugom dijelu smo uz pomoć raznih algoritama, koji su nam olakšali dokazivanje, pokazali da je Agrawal-Kayal-Saxenin algoritam polinomne složenosti. Deterministic algorithm for primality test was long time expected thing in computer science. Agrawal-Kayal- Saxena algorithm is the first algorithm which in polynomial time determines whether a given number is prime, namely the threesome has shown that language PRIMES is in complexity class P. The main goal of this work is to show that language PRIMES is in complexity class P, namely introduce Agrawal-Kayal-Saxena primality test, prove its correctness and determine its complexity. In the first, introductory chapter of this work we have gave essential definitions and results from number theory, complexity theory and algebra. We need this definitions and results for our second, main chapter, so that we can prove correctness of Agrawal-Kayal-Saxena primality test and show that it is in complexity class P. In the introductory chapter we have also described randomized algorithms and we have gave walk through the histroy of PRIMES problem. The main part of this work we have divided into two parts. We have mentioned Pascal triangle as a motivation for the AKS algorithm. We have also introduced Agrawal-KayalSaxena primality test and with help of various results we have proved its correctness. In the second part of main chapter, with help of various algorithms we have proved that AgrawalKayal-Saxena primality test has polynomial complexity.