Mastergradsoppgaver i matematikk
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A boundary integral approach to the modeling of surface waves in a wave tank
(Master thesis; Mastergradsoppgave, 20200614)Boundary integral equations (BIEs) are used to model surface waves in a wave tank. Assuming an ideal fluid, the velocity of the fluid can be considered as a potential flow and be modeled by the Laplace equation on the domain. The domain in this case will be a section of a wave channel with an incoming wave from the right, a rigid bottom, a reflective wall on the right and a time varying surface that ... 
A bidirectional pulse propagation model for extreme nonlinear optics: derivation and implementation.
(Master thesis; Mastergradsoppgave, 20200513)With growing capabilities of highintensity laser beams to generate ultrashort pulses of light, the simulation of pulse propagation in nonlinear media is expected to catch up with the frontline experimental setups. Among the challenges of nonlinear material response modeling is the ability to capture the backscatter effect  a phenomenon inherently elusive for the wellestablished methods of ... 
Joint Invariants of Symplectic and Contact Lie Algebra Actions
(Master thesis; Mastergradsoppgave, 20200623)By restricting generating functions of infinitesimal symmetries of symplectic and contact vector spaces to quadratic forms, we obtain a finitedimensional Lie subalgebra, consisting of vector fields isomorphic to the linear symplectic or conformal symplectic algebra. This allows us to look for joint invariants of the diagonal action of g on product manifolds. We find an explicit recipe for creating ... 
Differential Invariants of Symplectic and Contact Lie Algebra Actions
(Master thesis; Mastergradsoppgave, 20200623)In this thesis we consider the equivalence problem for symplectic and conformal symplectic group actions on submanifolds and functions. We solve the equivalence problem for general submanifolds by means of computing differential invariants and describing all the invariants of the associated group action by appealing to the LieTresse theorem. 
The Four Faces of Hyperelliptic curves
(Master thesis; Mastergradsoppgave, 20200513)In this thesis we will look at elliptic and hyperelliptic curves. There are three abelian groups that are isomorphic to hyperelliptic curves. The Jacobian of hyperelliptic curves, the ideal class group and the form class group, will all be defined and given abelian group structure. We will give an algorithm for point addition and point doubling done exclusively in the jacobian of the curve. ... 
Group Cohomology and Extensions
(Master thesis; Mastergradsoppgave, 20190831)The goal of this thesis is to classify all extensions where the kernel has order p^s and the cokernel has order p^t, p is a prime, and 1 ≤ s,t ≤ 2. We determine (up to weak congruence) the different combinations of kernel, cokernel and operators, and for each case, calculate the second cohomology group. By comparing resolutions, we get an explicit correspondence between the second cohomology group ... 
Mathematics of Viral Infections: A Review of Modeling Approaches and A CaseStudy for Dengue Dynamics
(Master thesis; Mastergradsoppgave, 20180920)In this thesis we use mathematical models to study the mechanisms by which diseases spread. Transmission dynamics is modelled by the class of SIR models, where the abbreviation stands for susceptible (S), infected (I) and recovered (R). These models are also called compartmental models, and they serve as the basic mathematical framework for understanding the complex dynamics of infectious diseases. ... 
Modelling high intensity laser pulse propagation in air using the modified Kortewegde Vries equation
(Master thesis; Mastergradsoppgave, 20180601)Ultrafast laser pulse experiments and applications are entering a phase that challenges the validity of mathematical models utilised to model longer pulses in nonlinear optics. This thesis aims to propose a possible mathematical model for high intensity laser pulse propagation in air through a multiple scales expansion of Maxwell’s equations and discuss a method on how to solve the corresponding ... 
Modern climateeconomic models and climate policies
(Master thesis; Mastergradsoppgave, 20180515)The problem of climate change is one of the most discussed problems nowadays. The global warming has an unquestionable influence on the economic growth of the different countries, and, consequently, on the whole world economics. The climate economics thus is an actual topic to study. Moreover, it is important to predict how the climate will change over the next century and which resulting outcomes ... 
Almost affine codes and matroids
(Master thesis; Mastergradsoppgave, 20170515)In this thesis we study various types of block codes, like linear, mutltilinear, almost affine codes. We also look at how these codes can be described by associated matroids. In addition we look at flags (chains) of codes and see how their behavior can be described using demimatroids. We also introduce weight polynomials for almost affine codes. 
On the EwaldOseen scattering formulation for linear and nonlinear transient wave scattering
(Master thesis; Mastergradsoppgave, 20170516)In this thesis work we develop and apply EOS formulation to three scattering problems: two of them are 1D problems and one is 2D. The first chapter comprises EOS formulations and numerical implementation for 1D scattering problems. Also in this chapter we use different numerical methods to solve test problem and choose the most stable and accurate method for solving of the given 1D problems. ... 
A complex contour based perfectly matched layer applied to a pattern generating model equation.
(Master thesis; Mastergradsoppgave, 20170217)The observable universe consists of several non equilibrium systems that generate spatiotemporal behaviour in the form of various patterns. As the elementary laws of physics and chemistry are unable to explain the pattern forming behaviour of such systems, scientists have turned to desktop experiments and model equations to gain further insight. The model equations that generate numerical solutions ... 
Separable representations of the Poisson, Helmholtz and complex Helmholtz kernels
(Master thesis; Mastergradsoppgave, 20170215)For high accuracy applications of integral operators in higher dimensions the complexity of operation and storage usually grows exponentially with dimensions. One method that has proven successful for handling these difficulties are the separation of the integral kernels as linear combinations of products of onedimensional kernels, commonly referred to as separation of variables. In this ... 
Insulating the Vacuum. Calculating the Casimir force using the boundary integral method with von Neumann boundary conditions
(Master thesis; Mastergradsoppgave, 20160815)In 2012, a new method for calculating the Casimir force between compact objects was developed, expressing the force in terms of a boundary integral equation. The case of perfectly conducting objects with Dirichlet boundary conditions in two dimensions was treated by Isak Kilen. The method was later extended to three dimensions by Karl Øyvind Mikalsen. The contribution of this thesis will be to ... 
Modelling lasermatter interactions using resonant states
(Master thesis; Mastergradsoppgave, 20160512)Studying how light interacts with materials has become important for many technological applications from optical communication to developing of new materials. Therefore scientists have always tried to improve their understanding of these effects. The primary goal has always been to microscopically describe the pertinent processes. This paper provides a brief introduction into the interactions of ... 
High frequency financial time series prediction: machine learning approach
(Master thesis; Mastergradsoppgave, 20160513)Machine learning is a rapidly evolving subfield of computer science. It has enormous amount of applications. One of the application domains is financial data analysis. Machine learning was usually applied for analysis and forecasting of daily financial time series. Availability of high frequency financial data became another challenge with its own specifics and difficulties. Regressors, being a ... 
Modelling the evolution of ideal, infinite domain patterns, on a finite domain using a Perfectly Matched Layer
(Master thesis; Mastergradsoppgave, 20160128)The SwiftHohenberg equation is an evolution equation which can produce a Pattern, or a patternlike picture, to be more precise. For example, it could be used to model some simple natural patterns, like stripes and rolls that one may observe in a RayleighBenard convection experiment. But for any pattern formation obtained by an evolution equation to look ideal, we have to consider this equation ... 
The Unidirectional Pulse Propagation Equation for Cylindrical Vector modes
(Master thesis; Mastergradsoppgave, 20150727)A new model for the unidirectional pulse propagation equations (UPPE) was developed by Per Jacobsen[1], this model is based on the assumption of cylindrical vector (CV) modes. The model will be strong for CV type electrical eld representations where only a few modes will be excited. In this thesis we will investigate the model further. The model will be implemented as a pseudo spectral method where ... 
Arctic tipping points
(Master thesis; Mastergradsoppgave, 20150515)The Arctic is warming much faster than the entire planet, and this causes severe melting of sea ice. However, the climate of different regions of the Earth is interconnected, and changes in the amount of ice in the Arctic can dramatically affect the climate across the whole planet. Some scientists claim that a possible tipping point is the event of the icefree Arctic Ocean in summer. Certain ... 
Tipping points and crises in financial markets
(Master thesis; Mastergradsoppgave, 20150515)Electricity spot markets and other financial markets are complex systems, and it is difficult to forecast their behaviour, especially uncontrolled and unmanageable situations, such as power crises and deflation of financial bubbles. An energy crisis is any price rise in the supply of energy resources to an economy. It has undesirable consequences, occasionally irreversible. The most known of these ...