The Covariant Galileon Model is an extension of General Relativity constructed by adding an extra scalar degree of freedom to it. The mathematical background of the model is therefore also found in...Show moreThe Covariant Galileon Model is an extension of General Relativity constructed by adding an extra scalar degree of freedom to it. The mathematical background of the model is therefore also found in differential geometry. The equations of motion of the model can be derived from its Lagrangian. Using the ADM formalism and tools from differential geometry the EFT functions of the model are then found. Numerical solutions to the model are given for two different sets of parameters and for variations of the the present day matter density and the Hubble constant at the start of the simulation. From these it is concluded that a more thorough Monte Carlo simulation of the model is a useful tool for further analysis of the model. Furthermore more research is needed for a better interpretation of the found solutions to the model.Show less
In this thesis, we will investigate the transformation of electromagnetic fields under conformal maps. When a conformal map is applied to such a field, the resulting field is again a valid...Show moreIn this thesis, we will investigate the transformation of electromagnetic fields under conformal maps. When a conformal map is applied to such a field, the resulting field is again a valid electromagnetic field. Even when the conformal map is complex, i.e. it mixes real and complex points of space, the resulting field is valid. To better understand complex conformal maps, we introduce Dirac spinors and Twistor space. Using these concepts, we find a nicer expression for a — possibly complex — conformal transformation. This could ease the calculation of the transformed electromagnetic field.Show less
Abstract De kunstenaar Richard Serra heeft een verweerd stalen kunstwerk gemaakt [5] waarin hij twee ellipsen, die in verschillende parallelle vlakken liggen en gedraaid zijn ten opzichte van...Show moreAbstract De kunstenaar Richard Serra heeft een verweerd stalen kunstwerk gemaakt [5] waarin hij twee ellipsen, die in verschillende parallelle vlakken liggen en gedraaid zijn ten opzichte van elkaar, met een vlak heeft omwikkeld. Hierdoor worden puntenparen op de ellipsen verbonden door middel van rechte lijnen. Dit beeld, genaamd Torqued Ellipse, is door Bas Edixhoven geanalyseerd en hij heeft laten zien dat het werk gemodelleerd kan worden als het reële deel van een deelverzameling van een algebraïsche verzameling die we Serra’s oppervlak noemen. Het oppervlak leeft in de complexe projectieve ruimte P3 pCq en in dit oppervlak vinden we een tweede beeld dat het eerste beeld snijdt in de twee ellipsen en dat we de Siamese Tweeling noemen. We zullen zien dat onze modellen van de beelden eenvoudig en op veel verschillende manieren als oppervlakken geparametriseerd kunnen worden in de R3 . Het doel zal zijn om een parametrisatie te maken, waarin de algebraïsche structuur van Serra’s oppervlak is terug te vinden, dat wil zeggen een parametrisatie die aan de ene kant een familie rechte lijnen en aan de andere kant een familie algebraïsche krommen is. Om deze parametrisatie te bereiken, inbedden we de R3 in de P3 pCq en laten we zien dat Serra’s oppervlak een P1 pCq-bundel vormt, wat ons een parametrisatie van de Torqued Ellipse en de Siamese Tweeling oplevert. Vervolgens construeren we een werking van een groep van bepaalde morfismen op de verzameling van P1 pCq-bundels over Serra’s oppervlak. De groep bevat een bijzonder element, dat onze P1 pCq-bundel trivialiseert en dit levert opnieuw een parametrisatie van de Torqued Ellipse en de Siamese Tweeling. Deze parametrisatie is, zoals gewenst, degene die de algebraïsche krommen op het oppervlak weergeeft. Via de volgende link kunnen we een aantal interactieve afbeeldingen openen van het oppervlak en de krommen erop, die (beter dan de statische afbeeldingen) de drie-dimensionale structuur van het beeld naar voren zal brengen:Show less
We will study a certain random partition on the vertex set of a graph that has recently been introduced. To do this we will analyse a related two-point correlation. The random partition has an...Show moreWe will study a certain random partition on the vertex set of a graph that has recently been introduced. To do this we will analyse a related two-point correlation. The random partition has an exact sampling algorithm and it has recently been used to design novel algorithms for network datasets. In statistical physics there is also interest in this object as it is related to the random cluster model and the uniform spanning tree. In the preliminary study these objects have been analysed for complete graphs, which are dense. The goal of this thesis is to study the random partition and the two-point correlation on tree structures, which are sparse graphs. We will prove two theorems that give formulas for the computation of two point correlations on trees, and we will do a detailed analysis of star graphs. Through this analysis we will find out what values of an associated tuning parameter controlling the size of the blocks in the random partition can uncover information about the underlying structure of the star graph.Show less
In 1993 Häggström [3] derived a characterisation of the uniform spanning tree on the {0, 1}×Z lattice, or shorthand 2-lattice. He proposed a way to extend to the {0, ...m − 1} × Z lattice, or...Show moreIn 1993 Häggström [3] derived a characterisation of the uniform spanning tree on the {0, 1}×Z lattice, or shorthand 2-lattice. He proposed a way to extend to the {0, ...m − 1} × Z lattice, or shorthand m-lattice. We propose a characterisation for the m-lattice that is slightly more compact, but the main ideas are the same. We use a different representation of spanning trees, or rather it is a representation for so called special forests, namely a sequence of letters and partitions. A special forest may be viewed as a spanning tree in the making. We have found a characterisation of the uniform spanning tree on the m-lattice. The results may be extended to so called repetitive graphs. Others have used the same representation of special forests to find a recurrence relation for the number of spanning trees and of Hamilton cycles of various (finite) repetitive graphs. Finally we discuss the matrix tree theorem and the Markov chain tree theorem. Combining these two theorems immediately proves a formula for the stationary distribution of an irreducible, aperiodic Markov chain.Show less
In this work, we apply the theory of group representations to the study of degeneracies of modes in photonic crystals. After a rigorous mathematical description of the problem, we proceed to study...Show moreIn this work, we apply the theory of group representations to the study of degeneracies of modes in photonic crystals. After a rigorous mathematical description of the problem, we proceed to study induced representations, leading to a way of determining their irreducibility. We then apply this theory to symmetry groups of photonic crystals, allowing predictions on the degeneracies of photonic modes. The theory is exemplified in a number of numerical simulations of two-dimensional crystals with various wallpaper groups as their symmetry group. We find degeneracies of modes in highly symmetric structures that can be removed by breaking the symmetry. Removing these symmetries generally leads to the formation of gaps and photonic bands with low group velocity, or ‘slow light’. The use of representation theory thus comprises a novel design principle for photonic crystals. In several cases we find so called ‘accidental degeneracies’ that are not easily predicted by our mathematical framework. Further research on this issue needs to be conducted in order to achieve a robust design principle.Show less
