A priority based rule for use during the process of scheduling trains oper- ating on a single track railway line was proposed by the Norwegian railway operator and owner, Jernbaneverket. The purpose of this study is to inves- tigate the effect of the suggested scheduling rule on the scheduled waiting times suffered by trains operating on a segment of the railway line. It is shown that the scheduling rule, under certain limiting assumptions, can be studied in the setting of queuing theory and that it has properties in common with a theoretical priority discipline combining two well docu- mented priority rules. The main part of this study is the development and analysis of a threshold based, combined preemptive/non-preemptive priority discipline. Under the combined discipline, preemptions are allowed during the early stage of processing only. Theoretical expressions for flow-times of jobs passing through the queuing system are reached through detailed studies of the non-preemptive and the preemptive priority discipline. The relationship between the suggested priority based scheduling rule and the theoretical, combined priority discipline is finally illustrated by sim- ulations. When adjusted for actual time spent by trains on traversing the line segment, the steady state solution for flow-times obtained from queuing theory yields an accurate expression for the trains’ average scheduled wait- ing times. The scheduling problem can in fact be modeled accurately by an M/G/1 queue under the combined priority discipline.

In this thesis I have studied how to estimate parameters in an extreme value model with Markov Chain Monte Carlo (MCMC) given a data set. This is done with synthetic Gaussian time series generated by spectral densities, called spectrums, with a "box" shape. Three different spectrums have been used. In the acceptance probability in the MCMC algorithm, the likelihood have been built up by dividing the time series into blocks consisting of a constant number of points. In each block, only the maximum value, i.e. the extreme value, have been used. Each extreme value will then be interpreted as independent. Since the time series analysed are generated the way they are, there exists theoretical values for the parameters in the extreme value model. When the MCMC algorithm is tested to fit a model to the generated data, the true parameter values are already known. For the first and widest spectrum, the method is unable to find estimates matching the true values for the parameters in the extreme value model. For the two other spectrums, I obtained good estimates for some block lengths, others block lengths gave poor estimates compared to the true values. Finally, it looked like an increasing block length gave more accurate estimates as the spectrum became more narrow banded. A final simulation on a time series generated by a narrow banded spectrum, disproved this hypothesis.

Analyzing latent Gaussian models by using approximate Bayesian inference methods has proven to be a fast and accurate alternative to running time consuming Markov chain Monte Carlo simulations. A crucial part of these methods is the use of a Gaussian approximation, which is commonly found using an asymptotic expansion approximation. This study considered an alternative method for making a Gaussian approximation, the expectation propagation (EP) algorithm, which is known to be more accurate, but also more computationally demanding. By assuming that the latent field is a Gaussian Markov random field, specialized algorithms for factorizing sparse matrices was used to speed up the EP algorithm. The approximation methods were then compared both with regards to computational complexity and accuracy in the approximations. The expectation propagation algorithm was shown to provide some improvements in accuracy compared to the asymptotic expansion approximation when tested on a binary logistic regression model. However, tests of computational time requirement for computing approximations in simple examples show that the EP algorithm is as much as 15-20 times slower than the alternative method.

Today there is an immense production of data, and the need for better methods to analyze data is ever increasing. Topology has many features and good ideas which seem favourable in analyzing certain datasets where statistics is starting to have problems. For example, we see this in datasets originating from microarray experiments. However, topological methods cannot be directly applied on finite point sets coming from such data, or atleast it will not say anything interesting. So, we have to modify the data sets in some way such that we can work on them with the topological machinery. This way of applying topology may be viewed as a kind of discrete version of topology. In this thesis we present some ways to construct simplicial complexes from a finite point cloud, in an attempt to model the underlying space. Together with simplicial homology and persistent homology and barcodes, we obtain a tool to uncover topological features in finite point clouds. This theory is tested with a Java software package called JPlex, which is an implementation of these ideas. Lastly, a method called Mapper is covered. This is also a method for creating simplicial complexes from a finite point cloud. However, Mapper is mostly used to create low dimensional simplicial complexes that can be easily visualized, and structures are then detected this way. An implementation of the Mapper method is also tested on a self made data set.

The main task of this assignment is to filter out noise from a series of radar images and to carry out short term precipitation forecasts. It is important that the final routine is performed online, yielding new forecasts as radar images arrive with time. The data available is a time series arriving at a one hour ratio, from the Rissa radar located in Sør Trøndelag. Gaussian radial basis functions are introduced to create the precipitation field, whose movement is solely governed by its velocity field, called advection. By performing discretization forward in time, from the basis given by the differential advection equation, prior distributions can be obtained for both basis functions and advection. Assuming normal distributed radar errors, the basis functions and advection are conditioned on associating radar images, which in turn can be taken into the prior distributions, yielding new forecasts. A modification to the model, labeling the basis functions either active or inactive, enable the process of birth and death of new rain showers. The preferred filtering technique is a joint MCMC sampler, but we make some approximations, sampling from a single MCMC sampler, to successfully implement an online routine. The model yield good results on synthetic data. In the real data situation the filtered images are satisfying, and the forecast images are approximately predicting the forthcoming precipitation. The model removes statistical noise efficiently and obtain satisfying predictions. However, due to the approximation in the MCMC algorithm used, the variance is somewhat underestimated. With some further work with the MCMC update scheme, and given a higher frequency of incoming data, it is the authors belief that the model can be a very useful tool in short term precipitation forecasting. Using gauge data to estimate the radar errors, and merging online gauge data with incoming radar images using block-Kriging, will further improve the estimates.

For small error the isogeometric analysis is more efficient than the finite element method. The condition number is lower in the isogeometric analysis than the finite element method. The isogeometric analysis produce general and robust implementation methods. The isogeometric analysis basis has higher continuity than the finite element method basis, and is more suitable to represent different geometries.

We present the noncommutative version of the Polly Cracker cryptosystem, which is more promising than the commutative version. This is partly because many of the ideals in a free (noncommutative) algebra have an infinite Gröbner basis, which can be used as the public key in the cryptosystem. We start with a short brief of the commutative case which ends with the conclusion that the existence of "intelligent" linear algebra attacks ensures that such cryptosystems are left insecure. Further, we see that it is hard to prove that noncommutative ideals have an infinite reduced Gröbner basis for all admissible orders. Nevertheless, in chapter 4 we consider some ideals for which it seems infeasible to realize a finite Gröbner basis. These are considered further in a cryptographic setting, and there will be shown that one class of ideals seems more promising than the others with respect to encountering attacks on the cryptosystem. In fact, at the end of this thesis we are proposing a way of constructing a cryptosystem based on this class of ideals, such that any linear algebra attack will not be successful. However, many of the results are on experimental level, so there remains a serious amount of research in order to conclude that we have found a secure cryptosystem.

A preliminary study of the papers ``A Cortical Based Model of Perceptual Completion in the Roto-Translation Space'' and ``Minimal Surfaces in the Roto-Translation Group with Applications to a Neuro-Biological Image Completion Model'' is done. The first one, written by Citti and Sarti, describe a perceptual completion model where a part of the visual cortex is modelled using a sub-Riemannian geometry on the Lie group SE(2). The second one, written by Hladky and Pauls, describe a model which completes the interior of a circular hole by spanning the lifted boundary by a minimal surface, presuming such a surface exists. These surfaces are solutions of occluded visual data as described by Citti and Sarti. Based on the models above, we propose a new model. The lifted boundary of an arbitrary hole is spanned by a surface consisting of geodesics between points with matching Dirichlet boundary values. All the three models are based on the sub-Riemannian geometry for the roto-translational space introduced by Citti and Sarti. The basic theory of sub-Riemannian geometries, including the derivation of some flows and operators in this degenerate space, is described. The models are implemented, and numerical results are presented.

It is a well known fact that the value of an option on an asset following a Levy jump-process, can be found by solving a Partial Integro-Differential Equation (PIDE). In this project, two new schemes are presented to solve these kinds of PIDEs when the underlying Levy process is of infinite activity. The infinite activity jump-process leads to a singular Levy measure, which has important numerical ramifications and needs to be handled with care. The schemes presented calculate the non-local integral operator via a fast Fourier transform (FFT), and an explicit/implicit operator splitting scheme of the local/global operators is performed. Both schemes will be of 2nd order on a regular Levy measure, but the singularity degrades convergence to lie in between 1st and 2nd order depending on the singularity strength. On the logarithmically transformed PIDE, the schemes are proven to be consistent, monotone and stable in $L^infty$, hence convergent by Barles-Perthame Souganidis.

In this thesis we will explore the possibilities of making a finite element solver for partial differential equations using the isogeometric framework established by Hughes et al. Whereas general B-splines and NURBS only allow for tensor product refinement, a new technology called T-splines will open for true local refinement. We will give an introduction into T-splines along with B-splines and NURBS on which they are built, presenting as well a refinement algorithm which will preserve the exact geometry of the T-spline and allow for more control points in the mesh. For the solver we will apply a residual-based a posteriori error estimator to identify elements which contribute the most to the error, which in turn allows for a fully automatic adaptive refinement scheme. The performance of the T-splines is shown to be superior on problems which contains singularities when compared with more traditional splines. Moreover the T-splines along with a posteriori error estimators are shown to have a very positive effect on badly parametrized models, as it seem to make the solution grid independent of the original parametrization.