Chapter 1 contains most of the background material for this thesis. In Chapter 2 we provide a formal method for determining degeneration for algebras of finite representation type. In Chapter 3 we give an alternative procedure for finding algebraic equations determining the orbit closure of a representation.

e will evaluate forward contracts in the electricity market. A thorough presentation of stochastic analysis for processes with discontinuous paths are provided, and some results concerning these from mathematical finance are stated. Using a Feynman-Kac-type theorem by Pham we derive a partial integro-differential equation giving the forward price from the spot dynamics taken from Geman and Roncoroni. This spot model is regime switching, so we get two equations. These equations are then attempted solved numerically. We suggest the following approach: When implementing boundary-conditions numerically we use values obtained from a Monte Carlo simulation of the spot dynamics to calibrate the boundary.

The main result in this thesis is a new generalization of Selberg's inequality in Hilbert spaces with a proof. In Chapter 1 we define Hilbert spaces and give a proof of the Cauchy-Schwarz inequality and the Bessel inequality. As an example of application of the Cauchy-Schwarz inequality and the Bessel inequality, we give an estimate for the dimension of an eigenspace of an integral operator. Next we give a proof of Selberg's inequality including the equality conditions following [Furuta]. In Chapter 2 we give selected facts on positive semidefinite matrices with proofs or references. Then we use this theory for positive semidefinite matrices to study inequalities. First we give a proof of a generalized Bessel inequality following [Akhiezer,Glazman], then we use the same technique to give a new proof of Selberg's inequality. We conclude with a new generalization of Selberg's inequality with a proof. In the last section of Chapter 2 we show how the matrix approach developed in Chapter 2.1 and Chapter 2.2 can be used to obtain optimal frame bounds. We introduce a new notation for frame bounds.

We give a brief account of the general analytic theory of continued fractions and state and prove the Lorentzen bestness theorem. We investigate the possibility of a new proof of the Lorentzen bestness theorem and gives a related convergence theorem together with a conjecture. We explore some connections between the limit-periodic continued fractions and other parts of mathematics and we give a few suggestions of topics suitable for further research.

In this paper we investigate some partial orders used in representation theory of algebras. Let $K$ be a commutative ring, $Lambda$ a finitely generated $K$-algebra and $d$ a natural number. We then study partial orders on the set of isomorphism classes of $Lambda$-modules of length $d$. The orders degeneration, virtual degeneration and hom-order are discussed. The main purpose of the paper is to study the relation $leq_n$ constructed by considering the ranks of $ntimes n$-matrices over $Lambda$ as $K$-endomorphisms on $M^n$ for a $Lambda$-module $M$. We write $Mleq_n N$ when for any $ntimes n$-matrix the rank with respect to $M$ is greater than or equal to the rank with respect to $N$. We study these relations for various algebras and determine when $leq_n$ is a partial order.

We present a complete solution to the 4 subspace problem in the generality of an algebraically closed eld. We do this by means of Auslander-Reiten theory and give the Auslander-Reiten quiver of the extended $D_4$ Dynkin diagram. We also give a geometric interpretation when two congurations of four lines through the origin in the plane are equivalent.

This master's thesis explores the area of combinatorics concerned with counting mathematical objects with regards to symmetry. Two main theorems in this field are Burnside's Lemma and P'{o}lya's Enumeration Theoremfootnote{P'{o}lya's Enumeration Theorem is also known as Redfield--P'{o}lya's Theorem.}. Both theorems yield a formula that will count mathematical objects with regard to a group of symmetries. Burnside's Lemma utilizes the concept of orbits to count mathematical objects with regard to symmetry. As a result of the Burnside Lemma's reliance on orbits, implementation of the lemma can be computationally heavy. In comparison, P'{o}lya's Enumeration Theorem's use of the cycle index of a group eases the computational burden. In addition, P'{o}lya's Enumeration Theorem allows for the introduction of weights allowing the reader to tackle more complicated problems. Building from basic definitions taken from abstract algebra a presentation of the theory leading up to P'{o}lya's Enumeration Theorem is given, complete with proofs. Examples are given throughout to illustrate these concepts. Applications of this theory are present in the enumeration of graphs and chemical compounds.

Using the LLL-algorithm to find small roots of polynomial equations using lattice basis reduction.

In this paper we present the first efficient point counting algorithm due to Schoof, before giving a significant improvement due to Elkies. In the final section I give Satoh's algorithm which is even faster and has paved the way for the field of $p$-adic point counting.

Sammendrag I kapittel 1 gjøres klart hvilke forkunnskaper som er nødvendige for å kunne forstå og ha utbytte av å lese denne oppgaven. Det henvises også til litteratur som kan brukes som oppslagsverk hvis noen begreper under lesningen skulle være ukjente. I kapittel 2 klargjøres sentrale begreper som fikspunkt, kontraksjon og kompletthet. Dernest presenteres Banachs fikspunktteorem med bevis. Kapitlet peker også mot ulike anvendelser av teoremet. Kapittel 3 handler i sin helhet om den velkjente Newtons metode. I sin enkelthet demonstrerer den hvordan vi kan finne tilnærmede løsninger ved iterasjoner. Arbeid med denne metoden er derfor nyttig for forståelse av Banachs fikspunktteorem og anvendelser av teoremet i mer abstrakte sammenhenger. I kapittel 4 viser vi hvordan vi kan finne en tilnærmet løsning av en partiell differensialligning ved hjelp Jacobi-iterasjon. For å demonstrere metoden tar vi utgangspunkt i et velkjent varmeledningsproblem. Vi bruker Banachs fikspunktteorem for å vise at metoden kan brukes i vårt tilfelle. Dette leder oss inn i flere disipliner av matematikken, og vi får derved demonstrert ulike teknikker for å løse denne type problemer. I kapittel 5 dreier det seg om differensialligninger og integralligninger. Som vi skal se har Picards entydighets- og eksistensteorem nær tilknytning til Banachs fikspunktteorem. Vi vil få demonstrert hvordan metoden kan brukes til å finne tilnærmede løsninger og i noen tilfeller eksakte løsninger av ordinære differensialligninger. Vi gir også et enkelt eksempel på bruk av Banachs fikspunktteorem i forhold til en spesiell type integralligning, Fredholm-ligninger. I kapittel 6 bruker vi Banachs fikspunktteorem til å bevise det inverse funksjonsteoremet. I kapittel 7 er siktemålet å sette The Scottish Café inn i en kulturhistorisk sammenheng. Vi begynner med å skissere et historisk bakteppe hvor vi søker å gi en viss oversikt over Polens historie. Behandlingen av jøder i Polen fram til 1940 blir et hovedtema. Dernest beskriver vi historien til The Scottish Café. Vi gir også en vurdering av hvilke faktorer som hadde betydning for utviklingen av miljøet. Noe av hovedkonklusjonen er at utviklinv vi gen av de sterke matematikkmiljøene i Polen i mellomkrigstiden langt på vei kan sees på som en del av nasjonsbyggingen i Polen. Utviklingen av det spesielle og særegne miljøet i The Scottish Café kan derimot sees som en reaksjon på antisemittismen i det polske samfunnet. I kapittel 8 finner vi noen refleksjoner over det vi har arbeidet med.