In recent research, several results from the theory of Riemann surfaces have been related to similar results on graphs. For instance, the Riemann–Roch theorem for Riemann surfaces has an analogue...Show moreIn recent research, several results from the theory of Riemann surfaces have been related to similar results on graphs. For instance, the Riemann–Roch theorem for Riemann surfaces has an analogue for graphs, and the theory of (principal) divisors can also be applied to graphs. In that fashion, gonality has been defined for graphs as well. It turns out that the gonality of a graph is also related to winning strategies in certain chip-firing games on graphs, which have been studied since the 1980s. In this thesis, the most important results and techniques are constructed in Section 3, most notably lower and upper bounds on the gonality. Some examples of graphs and their gonalities can be found in Section 4. Finally we will conduct an in-depth analysis of an algorithm for computing the reduced divisor in Section 5. We also briefly discuss algorithms for other, related tasks, such as computing the rank of a divisor and computing the gonality of a graph. This thesis gives an overview of some of the recent developments in the area of graph gonality, aimed at undergraduates in their final year as well as graduate students in mathematics. Some prior knowledge of graph theory and group theory might be useful, but in general the text is accessible for anyone with a general mathematical background (naive set theory, linear algebra).Show less
A model based on random processes for radiative transfer is introduced and investigated. Based on the ideas of SimpleX, for a given point distribution of an astronomical object, the points are...Show moreA model based on random processes for radiative transfer is introduced and investigated. Based on the ideas of SimpleX, for a given point distribution of an astronomical object, the points are being connected with a number of nearest neighbours. On the edges of the resulting graph, probabilities are assigned that reflect the way how radiation is being transferred throughout the object. Here only diffuse transfer is considered, i.e. all radiation is divided equally among the neighbours. The idea of adding sources and sinks is introduced and used. All of this eventually results in a large, sparse matrix, and replaces the question of solving the radiation equation to finding the stationary distribution vector, the eigenvector of the matrix corresponding to eigenvalue one. Various results from Markov theory, the Perron Frobenius theorem and numerical linear algebra are used to find this solution. As it turns out, our matrix is irreducible and aperiodic, so that the dominant eigenvalue equals one and the corresponding eigenvector is the only eigenvector that has positive entries. Hence we know that a stationary distribution vector exists and is unique. By means of the Arnoldi algorithm one can compute this eigenvector, along with other eigenvectors and eigenvalues. As an example, a given point distribution describing the collision of two small galaxies is taken. The model is investigated by experimenting with the parameters of the model, i.e., including the number of nearest neighbours, the luminosity of the sources and the location of the sinks, to see if we get any physically true results. As it turns out, taking 20 nearest neighbours and a certain allocation of the sinks gives a physcially satisfying result. Finally, one scheme for absorption by the points in the distribution is considered. The conclusion of this thesis is that this model can be used to model radiative transfer, although one needs to work more on the absorption aspect, and there is still work to be done on various other types of radiative transfer.Show less
Context. To get fully acquainted with Voronoi diagrams, Delaunay triangulations and the relationship between the two. To investigate a computational side of these tessellations and an application...Show moreContext. To get fully acquainted with Voronoi diagrams, Delaunay triangulations and the relationship between the two. To investigate a computational side of these tessellations and an application in astronomy in the form of modeling the Cosmic Web. Aims. First of all, to present and understand Brown’s algorithm, its tools, benefits and drawbacks. Secondly, to familiarize ourselves with modern views on and models of the Cosmic Web, and one of the new interesting tools used, namely the Delaunay Tessellation Field Estimator (DTFE). In particular, we are interested in accessing the quality of its reconstructions quantitatively. Methods. Obvious key concepts are Voronoi diagrams and Delaunay triangulations. For the computational component inversion and complexity analysis are of importance. For the astronomical component, various sampling methods and Fourier transforms come into play. Results. It seems that Brown’s algorithm has clear benefits in higher dimensional computations, but for two and three dimensions there may be better alternatives. Even though the DTFE reconstructions of the Cosmic Web appear to be visually satisfying, it appears that it is actually very sensitive to Poisson noise in the point distribution and in principle, minor effects may seriously distort the actual underlying continuous distribution. Greater care needs to be taken to access this further. There are many further research topics open here.Show less
