Constructible numbers are the numbers that can be constructed by using compass and straightedge in a finite sequence. They can be produced from natural numbers using only addition, subtraction, multiplication, division, and square root operations. These operations can be repeated, which creates more complicated expressions for a mathematical object. Calculation by computers only gives an approximation of the exact value, which could lead to a loss of accuracy. An alternative to approximation is exact arithmetic, which is the computation to find an exact value without rounding errors. In this thesis, we have presented a method of computing with the exact value of constructible numbers, specifically the rational numbers and its field extension as well as repeated field extension, through the basic operations. However, we only limit our implementation to the quadratic polynomial and the operations between two numbers of the same extension field. Future work on polynomials with higher degrees and algorithms to include operations with numbers from different extension fields and expression of number as an element of a new extension field remains to be done.

We study admissible transformations and solve group classification problems for various classes of linear and nonlinear Schrödinger equations with an arbitrary number n of space variables. The aim of the thesis is twofold. The first is the construction of the new theory of uniform seminormalized classes of differential equations and its application to solving group classification problems for these classes. Point transformations connecting two equations (source and target) from the class under study may have special properties of semi-normalization. This makes the group classification of that class using the algebraic method more involved. To extend this method we introduce the new notion of uniformly semi-normalized classes. Various types of uniform semi-normalization are studied: with respect to the corresponding equivalence group, with respect to a proper subgroup of the equivalence group as well as the corresponding types of weak uniform semi-normalization. An important kind of uniform semi-normalization is given by classes of homogeneous linear differential equations, which we call uniform semi-normalization with respect to linear superposition of solutions. The class of linear Schrödinger equations with complex potentials is of this type and its group classification can be effectively carried out within the framework of the uniform semi-normalization. Computing the equivalence groupoid and the equivalence group of this class, we show that it is uniformly seminormalized with respect to linear superposition of solutions. This allow us to apply the version of the algebraic method for uniformly semi-normalized classes and to reduce the group classification of this class to the classification of appropriate subalgebras of its equivalence algebra. To single out the classification cases, integers that are invariant under equivalence transformations are introduced. The complete group classification of linear Schrödinger equations is carried out for the cases n = 1 and n = 2. The second aim is to study group classification problem for classes of generalized nonlinear Schrödinger equations which are not uniformly semi-normalized. We find their equivalence groupoids and their equivalence groups and then conclude whether these classes are normalized or not. The most appealing classes are the class of nonlinear Schrödinger equations with potentials and modular nonlinearities and the class of generalized Schrödinger equations with complex-valued and, in general, coefficients of Laplacian term. Both these classes are not normalized. The first is partitioned into an infinite number of disjoint normalized subclasses of three kinds: logarithmic nonlinearity, power nonlinearity and general modular nonlinearity. The properties of the Lie invariance algebras of equations from each subclass are studied for arbitrary space dimension n, and the complete group classification is carried out for each subclass in dimension (1+2). The second class is successively reduced into subclasses until we reach the subclass of (1+1)-dimensional linear Schrödinger equations with variable mass, which also turns out to be non-normalized. We prove that this class is mapped by a family of point transformations to the class of (1+1)-dimensional linear Schrödinger equations with unique constant mass.

In this bachelor's thesis we will enter the world of projective geometry and algebraic geometry. The main part of this thesis focuses on describing projective spaces and their properties. We use the gained knowledge of projective spaces to then study projective varieties, algebraic geometry and Bézouts theorem.

Matrices whose rows (or columns) consists of monomials of sequential powers are called Vandermonde matrices and can be used to describe several useful concepts and have properties that can be helpful for solving many kinds of problems. In this thesis we will discuss this matrix and some of its properties as well as a generalization of it and how it can be applied to curve fitting discharge current for the purpose of ensuring electromagnetic compatibility. In the first chapter the basic theory for later chapters is introduced. This includes the Vandermonde matrix and some of its properties, history, applications and generalizations, interpolation and regression problems, optimal experiment design and modelling of electrostatic discharge currents with the purpose to ensure electromagnetic compatibility. The second chapter focuses on finding the extreme points for the determinant for the Vandermonde matrix on various surfaces including spheres, ellipsoids, cylinders and tori. The extreme points are analysed in three dimensions or more. The third chapter discusses fitting a particular model called the p-peaked Analytically Extended Function (AEF) to data taken either from a standard for electromagnetic compatibility or experimental measurements. More specifically the AEF will be fitted to discharge currents from the IEC 62305-1 and IEC 61000-4-2 standards for lightning protection and electrostatic discharge immunity as well as some experimentally measured data of similar phenomena.

Noncommutative geometry has over the past four of decades grown into a rich field of study. Novel ideas and concepts are rapidly being developed, and a notable application of the theory outside of pure mathematics is quantum theory. This thesis will focus on a derivation-based approach to noncommutative geometry using the framework of real calculi, which is a rather direct approach to the subject. Due to their direct nature, real calculi are useful when studying classical concepts in Riemannian geometry and how they may be generalized to a noncommutative setting. This thesis aims to shed light on algebraic aspects of real calculi by introducing a concept of morphisms of real calculi, which enables the study of real calculi on a structural level. In particular, real calculi over matrix algebras are discussed both from an algebraic and a geometric perspective.Morphisms are also interpreted geometrically, giving a way to develop a noncommutative theory of embeddings. As an example, the noncommutative torus is minimally embedded into the noncommutative 3-sphere. / Ickekommutativ geometri har under de senaste fyra decennierna blivit ett etablerat forskningsområde inom matematiken. Nya idéer och koncept utvecklas i snabb takt, och en viktig fysikalisk tillämpning av teorin är inom kvantteorin. Denna avhandling kommer att fokusera på ett derivationsbaserat tillvägagångssätt inom ickekommutativ geometri där ramverket real calculi används, vilket är ett relativt direkt sätt att studera ämnet på. Eftersom analogin mellan real calculi och klassisk Riemanngeometri är intuitivt klar så är real calculi användbara när man undersöker hur klassiska koncept inom Riemanngeometri kan generaliseras till en ickekommutativ kontext. Denna avhandling ämnar att klargöra vissa algebraiska aspekter av real calculi genom att introducera morfismer för dessa, vilket möjliggör studiet av real calculi på en strukturell nivå. I synnerhet diskuteras real calculi över matrisalgebror från både ett algebraiskt och ett geometriskt perspektiv. Morfismer tolkas även geometriskt, vilket leder till en ickekommutativ teori för inbäddningar. Som ett exempel blir den ickekommutativa torusen minimalt inbäddad i den ickekommutativa 3-sfären.

Deligne and Howe described polarized abelian varieties over finite fields in terms of finitely generated free Z-modules satisfying a list of easy to state axioms. In this thesis we address the problem of developing an effective algorithm to compute isomorphism classes of (principally) polarized abelian varieties over a finite field, together with their automorphism groups. Performing such computations requires the knowledge of the ideal classes (both invertible and non-invertible) of certain orders in number fields. Hence we describe a method to compute the ideal class monoid of an order and we produce concrete computations in dimension 2, 3 and 4.

Inom linja ̈r algebra har varje vektorrum ett s ̊a kallat dualrum, vilket är ett vektorrum bestående av alla linjära funktioner från det ursprungliga rummet till sin kropp. Att beräkna dimensionen av ett dualrum tillhörande ett ändlig-dimensionellt vektorrum är relativt enkelt, för oändlig-dimensionella vektorrum är det mer komplicerat. Den sats vi ska diskutera, Erdős–Kaplansky Satsen, ämnar lösa den frågan med påståendet att ett dualrum tillhörande ett oändlig-dimensionellt vektorrum har dimension lika med sin kardinalitet.

We study partially ordered sets of prime ideals as found in commutative Noetherian rings. These structures, commonly known as prime spectra, have long been a popular topic in the field of commutative algebra. As a consequence, there are many related questions that remain unanswered. Among them is the question of what partially ordered sets appear as Spec(A) of some Noetherian ring A, asked by Kaplansky during the 1950's. As a partial case of Kaplansky's question, we consider finite posets that are ring spectra of commutative Noetherian rings. Specifically, we show that finite spectra of such rings are always order-isomorphic to a bipartite graph. However, the most significant undertaking of this study is that of devising a constructive methodology for finding a ring with prime spectrum that is order-isomorphic to an arbitrary bipartite graph. As a result, we prove that any complete bipartite graph is order-isomorphic to the prime spectrum of some ring of essentially finite type over the field of rational numbers. Moreover, a series of potential generalizations and extensions are proposed to further enhance the constructive methodology. Ultimately, the results of this study constitute an original contribution and perspective on questions related to commutative ring spectra.

Innehållet i studien handlar om att kategorisera olika typer av fel som elever i åk 1 på gymnasiet gör i algebra. Data utgörs av 80 elevprov skrivna av elever på samhällsvetenskapsprogrammet och VVS- och fastighetsprogrammet läsåret 2017/2018 och 2018/2019. Uppgifterna som eleverna har fått göra är lösa ekvationer, förenkla uttryck, räkna värdet av ett uttryck samt problemlösning. Elevernas svar har analyserats och kategoriserats i sex feltyper: 1. Förståelsefel, 2. Procedurfel, 3. Modelleringsfel eller problemlösningsfel, 4. Resonemangsfel, 5. Redovisningsfel eller kommunikationsfel, 6. Övriga fel. I resultatet preseneteras varje feltyp illustrerad med elevexempel. Med tidigre forskning som utgångspunkt identifieras och diskuteras vilka missuppfattningar och svårigheter som kan vara den bakomliggande orsaken till att eleverna gjort dessa fel.  Några exempel på orsaker är att eleverna inte uppfattar variabelns (x) symboliska värde, förstår inte variablers generella beteckning (a och b), att variabeln kan representera en siffra, eleverna övergeneraliserar, förstår inte räkning med negativa tal, kan inte hantera aritmetik, förstår inte likhetstecknets betydelse, har oeffektiv resonemang (gissar, testar sig fram), samt skriver av uppgiften fel.

The research field of graded ring theory is a rich area of mathematics with many connections to e.g. the field of operator algebras. In the last 15 years, algebraists and operator algebraists have defined algebraic analogues of important operator algebras. Some of those analogues are rings that come equipped with a group grading. We want to reach a better understanding of the graded structure of those analogue rings. Among group graded rings, the strongly graded rings stand out as being especially well-behaved. The development of the general theory of strongly graded rings was initiated by Dade in the 1980s and since then numerous structural results have been established for strongly graded rings.  In this thesis, we study the class of epsilon-strongly graded rings which was recently introduced by Nystedt, Öinert and Pinedo. This class is a natural generalization of the well-studied class of unital strongly graded rings. Our aim is to lay the foundation for a general theory of epsilon-strongly graded rings generalizing the theory of strongly graded rings. This thesis is based on three articles. The first two articles mainly concern structural properties of epsilon-strongly graded rings. In the first article, we investigate a functorial construction called the induced quotient group grading. In the second article, using results from the first article, we generalize the Hilbert Basis Theorem for strongly graded rings to epsilon-strongly graded rings and apply it to Leavitt path algebras.  In the third article, we study the graded structure of algebraic Cuntz-Pimsner rings. In particular, we obtain a partial classification of unital strongly, epsilon-strongly and nearly epsilon-strongly graded Cuntz-Pimsner rings up to graded isomorphism.