Mastergradsoppgaver i matematikkhttps://hdl.handle.net/10037/2252024-03-28T09:31:12Z2024-03-28T09:31:12ZOn elementary particles as representations of the Poincaré groupMartínez Marín, Pauhttps://hdl.handle.net/10037/316942023-11-07T10:35:31Z2023-08-14T00:00:00ZMartínez Marín, Pau<br />
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 structures of quantum mechanics, together with Wigner’s theorem on symmetries. With respect to special relativity, we will present the Minkowski spacetime as an affine space an derive its group of automorphisms, the Poincaré group. We will finally talk about representations of Lie groups and define an elementary particle to be an irreducible projective representation of the Poincaré group.<br />
2023-08-14T00:00:00ZOn elementary particles as representations of the Poincaré groupMartínez Marín, PauThis 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 structures of quantum mechanics, together with Wigner’s theorem on symmetries. With respect to special relativity, we will present the Minkowski spacetime as an affine space an derive its group of automorphisms, the Poincaré group. We will finally talk about representations of Lie groups and define an elementary particle to be an irreducible projective representation of the Poincaré group.UiT Norges arktiske universitetUiT The Arctic University of NorwayThe, DennisMaster thesisMastergradsoppgaveCodes, matroids and derived matroidsKnutsen, Teodor Dahlhttps://hdl.handle.net/10037/295672023-07-06T06:04:39Z2023-05-15T00:00:00ZKnutsen, Teodor Dahl<br />
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 matroid was dependent both on the matroid, and a specific representation of this matroid. The construction was generalized again by Freij-Hollandi, Jurrius and Kuznetsova to encompass all matroids. This thesis will prove that the rank of the derived matroid constructed by FJK is equal to the corank of the matroid for a large class of matroids, and the Vámos matroid is given as an example where the rank of the derived matroid is strictly smaller than the corank of the matroid. Further, some generalizations of these constructions to lattices and q-matroids are given, in addition to a generalization of the construction by Longyear to all matroids. A software library was developed alongside this thesis to calculate properties of matroids and derived matroids, and detail of this software will be given.<br />
2023-05-15T00:00:00ZCodes, matroids and derived matroidsKnutsen, Teodor DahlThis 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 matroid was dependent both on the matroid, and a specific representation of this matroid. The construction was generalized again by Freij-Hollandi, Jurrius and Kuznetsova to encompass all matroids. This thesis will prove that the rank of the derived matroid constructed by FJK is equal to the corank of the matroid for a large class of matroids, and the Vámos matroid is given as an example where the rank of the derived matroid is strictly smaller than the corank of the matroid. Further, some generalizations of these constructions to lattices and q-matroids are given, in addition to a generalization of the construction by Longyear to all matroids. A software library was developed alongside this thesis to calculate properties of matroids and derived matroids, and detail of this software will be given.UiT Norges arktiske universitetUiT The Arctic University of NorwayJohnsen, TrygveMastergradsoppgaveMaster thesisMurnaghan-Nakayama Rule The Explanation and Usage of the AlgorithmSandal, Eliashttps://hdl.handle.net/10037/295662023-07-06T06:04:38Z2023-05-15T00:00:00ZSandal, Elias<br />
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 rule is a combinatorial method for computing character values of irreducible representations of symmetric groups. This makes this rule an important part of representation theory. One of the versions of this rule is stated in the recursive Murnaghan-Nakayama rule. Where, in this version, we can use border strips and diagrams to calculate the character values of representations on a given composition. This algorithm is quite fast in these calculations.
The Murnaghan-Nakayama rule can also be considered a central algorithm in representation theory over symmetric groups. It is a fascinating and powerful algorithm that has a strong connection to both combinatorics and representation theory.<br />
2023-05-15T00:00:00ZMurnaghan-Nakayama Rule The Explanation and Usage of the AlgorithmSandal, EliasCharacter 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 rule is a combinatorial method for computing character values of irreducible representations of symmetric groups. This makes this rule an important part of representation theory. One of the versions of this rule is stated in the recursive Murnaghan-Nakayama rule. Where, in this version, we can use border strips and diagrams to calculate the character values of representations on a given composition. This algorithm is quite fast in these calculations.
The Murnaghan-Nakayama rule can also be considered a central algorithm in representation theory over symmetric groups. It is a fascinating and powerful algorithm that has a strong connection to both combinatorics and representation theory.UiT Norges arktiske universitetUiT The Arctic University of NorwayRiener, CordianMastergradsoppgaveMaster thesisAn Energy Balance Model on an Infinite LineElvevold, Askhttps://hdl.handle.net/10037/276042022-11-30T07:16:38Z2022-06-01T00:00:00ZElvevold, Ask<br />
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 symmetry breaking, possible infinitely many solutions for a specific solar irradiation constant, bistability and in some cases tristability. Unconnected bifurcation diagrams were also found for some choices of unsymmetric heat diffusion function.<br />
2022-06-01T00:00:00ZAn Energy Balance Model on an Infinite LineElvevold, AskThe 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 symmetry breaking, possible infinitely many solutions for a specific solar irradiation constant, bistability and in some cases tristability. Unconnected bifurcation diagrams were also found for some choices of unsymmetric heat diffusion function.UiT Norges arktiske universitetUiT The Arctic University of NorwayPer, JakobsenMaster thesisMastergradsoppgaveEffects of feedbacks for an energy balance model on a circleBrynjulfsen, Synnehttps://hdl.handle.net/10037/268212022-09-16T06:48:11Z2022-05-31T00:00:00ZBrynjulfsen, Synne<br />
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. Segmenting the circle into water and continent areas adds additional feedback mechanisms, whose effects are investigated through the bifurcation diagram form in different cases, and the stability of the bifurcation branches. It is found that additional feedback mechanisms result in smaller, step by step, drop offs in stationary state temperature when varying the solar constant.<br />
2022-05-31T00:00:00ZEffects of feedbacks for an energy balance model on a circleBrynjulfsen, SynneA 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. Segmenting the circle into water and continent areas adds additional feedback mechanisms, whose effects are investigated through the bifurcation diagram form in different cases, and the stability of the bifurcation branches. It is found that additional feedback mechanisms result in smaller, step by step, drop offs in stationary state temperature when varying the solar constant.UiT Norges arktiske universitetUiT The Arctic University of NorwayJakobsen, Per KristenMastergradsoppgaveMaster thesisKilling Tensors in Koutras-McIntosh SpacetimesSteneker, Wijnandhttps://hdl.handle.net/10037/255362022-06-22T10:04:34Z2022-05-15T00:00:00ZSteneker, Wijnand<br />
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 a Killing tensor is actually an overdetermined linear first order partial differential equation. We shall study the Killing equation using methods from the geometric theory of PDEs. More precisely, we use Cartan's prolongation method to prove the (non)existence of Killing tensors in several Koutras-McIntosh spacetimes. A subclass of the Koutras--McIntosh spacetimes are the conformally flat pp-waves. We show that a generic conf. flat pp-wave has an irreducible Killing 2-tensor, which reproves a result obtained by Keane and Tupper using a different method. Moreover, we prove in particular examples of pp-waves that all Killing tensors of
degree 3 and 4 are reducible.
We then study the Wils metric, another subclass of the Koutras-McIntosh spacetimes. This metric has a univariate function as its parameter. By using Cartan's prolongation method we deduce the explicit form of the function for which the Wils metric admits a Killing vector, and for which a Killing 2-tensor. This existence result for a Killing vector makes a statement by Koutras and McIntosh more precise. Finally, we show in particular examples of a Wils metric that all Killing 3- and 4-tensors are reducible.<br />
2022-05-15T00:00:00ZKilling Tensors in Koutras-McIntosh SpacetimesSteneker, WijnandThis 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 a Killing tensor is actually an overdetermined linear first order partial differential equation. We shall study the Killing equation using methods from the geometric theory of PDEs. More precisely, we use Cartan's prolongation method to prove the (non)existence of Killing tensors in several Koutras-McIntosh spacetimes. A subclass of the Koutras--McIntosh spacetimes are the conformally flat pp-waves. We show that a generic conf. flat pp-wave has an irreducible Killing 2-tensor, which reproves a result obtained by Keane and Tupper using a different method. Moreover, we prove in particular examples of pp-waves that all Killing tensors of
degree 3 and 4 are reducible.
We then study the Wils metric, another subclass of the Koutras-McIntosh spacetimes. This metric has a univariate function as its parameter. By using Cartan's prolongation method we deduce the explicit form of the function for which the Wils metric admits a Killing vector, and for which a Killing 2-tensor. This existence result for a Killing vector makes a statement by Koutras and McIntosh more precise. Finally, we show in particular examples of a Wils metric that all Killing 3- and 4-tensors are reducible.UiT Norges arktiske universitetUiT The Arctic University of NorwayKruglikov, BorisMastergradsoppgaveMaster thesisWachpress Conjecture Restricted To Arrangements Of Three ConicsSchena, Alessandrohttps://hdl.handle.net/10037/255352022-06-22T10:04:33Z2022-05-15T00:00:00ZSchena, Alessandro<br />
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 real projective plane. The polycons of lowest degree for which the conjecture has not been proven completely yet are those which arise from arrangements of three conics. We state the current knowledge on the veracity of the conjecture on the polycons of this family. Throughout the thesis we view real rational polycons as positive geometries which encode both differential and algebraic properties in their unique canonical form.<br />
2022-05-15T00:00:00ZWachpress Conjecture Restricted To Arrangements Of Three ConicsSchena, AlessandroThis 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 real projective plane. The polycons of lowest degree for which the conjecture has not been proven completely yet are those which arise from arrangements of three conics. We state the current knowledge on the veracity of the conjecture on the polycons of this family. Throughout the thesis we view real rational polycons as positive geometries which encode both differential and algebraic properties in their unique canonical form.UiT Norges arktiske universitetUiT The Arctic University of NorwayRiener, CordianMastergradsoppgaveMaster thesisOn the effects of symmetry in the energy balance on a sphereSamuelsberg, Akselhttps://hdl.handle.net/10037/255342022-06-22T10:04:30Z2022-05-15T00:00:00ZSamuelsberg, Aksel<br />
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, an energy balance model on a sphere, and we highlight a possible factor in the behavioral discrepancy between many simple, low-dimensional models and earth system models. A hypothesis is put forward and investigated regarding the effects of symmetry and the presence of multiple equilibria in this simple model. A novel application of boundary integral methods in the context of energy balance models is presented, and used to find semi-analytical solutions to the stationary energy balance equation. The hypothesis is eventually refuted and a new hypothesis is formulated based on the evidence presented: Symmetry violations in energy balance models cause the steady state solutions to become more similar. This is discussed in light of earth system models and how a similar dynamic would make the detection of multiple equilibra challenging in earth system models.<br />
2022-05-15T00:00:00ZOn the effects of symmetry in the energy balance on a sphereSamuelsberg, AkselSimple 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, an energy balance model on a sphere, and we highlight a possible factor in the behavioral discrepancy between many simple, low-dimensional models and earth system models. A hypothesis is put forward and investigated regarding the effects of symmetry and the presence of multiple equilibria in this simple model. A novel application of boundary integral methods in the context of energy balance models is presented, and used to find semi-analytical solutions to the stationary energy balance equation. The hypothesis is eventually refuted and a new hypothesis is formulated based on the evidence presented: Symmetry violations in energy balance models cause the steady state solutions to become more similar. This is discussed in light of earth system models and how a similar dynamic would make the detection of multiple equilibra challenging in earth system models.UiT Norges arktiske universitetUiT The Arctic University of NorwayJakobsen, Per KristenRypdal, MartinMastergradsoppgaveMaster thesisIndividual-Based Modeling of COVID-19 Vaccine StrategiesSkagseth, Håvard Mikalhttps://hdl.handle.net/10037/225722021-09-17T06:04:13Z2021-06-01T00:00:00ZSkagseth, Håvard Mikal<br />
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 is to prioritze old people over the people with the most contacts.<br />
2021-06-01T00:00:00ZIndividual-Based Modeling of COVID-19 Vaccine StrategiesSkagseth, Håvard MikalCOVID-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 is to prioritze old people over the people with the most contacts.UiT Norges arktiske universitetUiT The Arctic University of NorwayRypdal, MartinMaster thesisMastergradsoppgaveReal Plane Algebraic CurvesGonzález García, Pedrohttps://hdl.handle.net/10037/216912021-07-02T08:16:23Z2021-06-18T00:00:00ZGonzález García, Pedro<br />
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 contributions to this problem, which have been obtained within
the last century. The main aim of the thesis is to clarify these and to make them more
accessible.
Chapter 1 gives a brief introduction into the study of real plane algebraic curves. The
exposition of this chapter builds on the standard knowledge which are normally obtained
in an undergraduate course of algebraic curves, which usually focus only on complex plane
algebraic curves. In Chapter 2, several topological properties of real plane curves are
developed. The main statements here can be mostly established from Bezout's theorem
and its consequences. The main result presented in this chapter is Harnack's inequality
and the classi cation of the curves until degree ve. The goal of Chapter 3 is to prove
Petrowski's inequalities using Morse theoretic results along with the original arguments
which appeared in Petrowski's manuscript. Chapter 4 presents results arising from the
complexi cation of a real plane curve. Finally, Chapter 5 mainly presents results from
Smith theory. In particular, this allows to see how Smith's inequality generalizes Harnack's
inequality which were presented in Chapter 2 to higher dimensions.<br />
2021-06-18T00:00:00ZReal Plane Algebraic CurvesGonzález García, PedroThis 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 contributions to this problem, which have been obtained within
the last century. The main aim of the thesis is to clarify these and to make them more
accessible.
Chapter 1 gives a brief introduction into the study of real plane algebraic curves. The
exposition of this chapter builds on the standard knowledge which are normally obtained
in an undergraduate course of algebraic curves, which usually focus only on complex plane
algebraic curves. In Chapter 2, several topological properties of real plane curves are
developed. The main statements here can be mostly established from Bezout's theorem
and its consequences. The main result presented in this chapter is Harnack's inequality
and the classi cation of the curves until degree ve. The goal of Chapter 3 is to prove
Petrowski's inequalities using Morse theoretic results along with the original arguments
which appeared in Petrowski's manuscript. Chapter 4 presents results arising from the
complexi cation of a real plane curve. Finally, Chapter 5 mainly presents results from
Smith theory. In particular, this allows to see how Smith's inequality generalizes Harnack's
inequality which were presented in Chapter 2 to higher dimensions.UiT Norges arktiske universitetUiT The Arctic University of NorwayRiener, CordianMaster thesisMastergradsoppgave