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norskMastergradsoppgaver i matematikk
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On elementary particles as representations of the Poincaré group
(Master thesis; Mastergradsoppgave, 2023-08-14)This thesis is concerned with the definition of elementary particles as irreducible projective unitary representations of the Poincaré group. During the contents of this work, we will introduce the relevant prerequisites and results. Concerning differential geometry, we will discuss smooth manifolds, Lie groups and Lie algebras. About quantum mechanics, we will introduce Hilbert spaces and the basic ... -
Codes, matroids and derived matroids
(Mastergradsoppgave; Master thesis, 2023-05-15)This thesis first introduces some theory on coding theory and matroids, and properties that are shared between these, and then we will investigate derived matroids. In 1979 Longyear made a construction of derived matroids for binary matroids, which illuminates "dependencies among dependencies". The construction was later generalized to representable matroids by Oxley and Wang, where the derived ... -
Murnaghan-Nakayama Rule The Explanation and Usage of the Algorithm
(Mastergradsoppgave; Master thesis, 2023-05-15)Character values are not the easiest to calculate, so it is important to find good algorithms that can help ease these calculations. In the 20th century, the two mathematicians Murnaghan and Nakayama developed a rule that calculates character values for partitions on some computations. This rule has later been given the name The Murnaghan-Nakayama rule, after these two authors. The Murnaghan-Nakayama ... -
An Energy Balance Model on an Infinite Line
(Master thesis; Mastergradsoppgave, 2022-06-01)The thesis is an expansion of the work Gerald R. North did in 1975 on an energy balance climate model. By considering a similar model on an infinite line, and allowing the heat diffusion coefficient to vary on the line, more complicated behaviour arose from the model. Much of Norths work was recreated on the infinite lines, but a lot of new discoveries were made. Among what was found were spontaneous ... -
Effects of feedbacks for an energy balance model on a circle
(Mastergradsoppgave; Master thesis, 2022-05-31)A simple, North like, energy balance model on a circle is studied using boundary formulations derived with Green's functions for the stationary case, and a pseudo-spectral and finite difference solution for the time dependent case. The bifurcation software Auto-07p is also applied. The boundary formulation solution can be solved analytically in most cases, and is solved for bifurcation diagrams. ... -
Killing Tensors in Koutras-McIntosh Spacetimes
(Mastergradsoppgave; Master thesis, 2022-05-15)This thesis is concerned with the (non)existence of Killing Tensors in Koutras-McIntosh spacetimes. Killing tensors are of particular interest in general relativity, because these correspond to conserved quantities for the geodesic motion. For instance, Carter found such a conserved quantity in the Kerr metric which he used to explicitly integrate the geodesic equations. The equation defining ... -
Wachpress Conjecture Restricted To Arrangements Of Three Conics
(Mastergradsoppgave; Master thesis, 2022-05-15)This thesis discusses Wachpress conjecture restricted to arrangements of three conics. Wachpress conjectured the existence of a set of barycentric coordinates, namely Wachpress coordinates, on all polycons. Barycentric coordinates are very useful in many different fields as they can be used to define a finite element approximation scheme with linear precision. This thesis focuses on the conjecture ... -
On the effects of symmetry in the energy balance on a sphere
(Mastergradsoppgave; Master thesis, 2022-05-15)Simple climate models have gathered much attention as they have suggested the possibility of abrupt climate change associated with tipping points. Several simple climate models are found to have multiple equilibria, but in most cases similar equilibria do not appear or become too difficult to find in complex, fully coupled earth system models. In this thesis, we investigate a simple climate model, ... -
Individual-Based Modeling of COVID-19 Vaccine Strategies
(Master thesis; Mastergradsoppgave, 2021-06-01)COVID-19 is a respiratory disease with influenza-like symptoms originating from Wuhan, China, towards the end of 2019. There has been developed multiple vaccines to contain the virus and to protect the most vulnerable people in society. In this thesis we look at two different vaccination strategies to prevent most deaths and years of life lost. We conclude that the safest and most consistent strategy ... -
Real Plane Algebraic Curves
(Master thesis; Mastergradsoppgave, 2021-06-18)This master thesis studies several properties of real plane algebraic curves, focusing on the case of even degree. The question of the relative positions of the connected components of real plane algebraic curves originates in Hilbert's sixteenth problem which, despite its prominence, is still open in the case of higher degree curves. The goal of this thesis is an exposition of fundamental ... -
Symmetric Ideals
(Mastergradsoppgave; Master thesis, 2021-05-14)Polynomials appear in many different fields such as statistics, physics and optimization. However, when the degrees or the number of variables are high, it generally becomes quite difficult to solve polynomials or to optimize polynomial functions. An approach that can often be helpful to reduce the complexity of such problems is to study symmetries in the problems. A relatively new field, that has ... -
Ice-albedo tipping points in a diffusive energy-balance model with land and ocean
(Mastergradsoppgave; Master thesis, 2021-01-20)The ice-albedo feedback is associated with the nonlinearity in the climate system, due to the sudden change in albedo between ice-free and ice-covered surfaces. This nonlinearity can potentially cause abrupt and dramatic shifts in the climate, referred to as tipping points. It is also believed that this mechanism has contributed significantly to the precipitous losses of Arctic sea ice, which have ... -
A boundary integral approach to the modeling of surface waves in a wave tank
(Master thesis; Mastergradsoppgave, 2020-06-14)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, 2020-05-13)With growing capabilities of high-intensity laser beams to generate ultra-short pulses of light, the simulation of pulse propagation in nonlinear media is expected to catch up with the front-line experimental setups. Among the challenges of nonlinear material response modeling is the ability to capture the back-scatter effect - a phenomenon inherently elusive for the well-established methods of ... -
Joint Invariants of Symplectic and Contact Lie Algebra Actions
(Master thesis; Mastergradsoppgave, 2020-06-23)By restricting generating functions of infinitesimal symmetries of symplectic and contact vector spaces to quadratic forms, we obtain a finite-dimensional 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, 2020-06-23)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 Lie-Tresse theorem. -
The Four Faces of Hyperelliptic curves
(Master thesis; Mastergradsoppgave, 2020-05-13)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, 2019-08-31)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 Case-Study for Dengue Dynamics
(Master thesis; Mastergradsoppgave, 2018-09-20)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 Korteweg-de Vries equation
(Master thesis; Mastergradsoppgave, 2018-06-01)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 ...
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