In Genome-Wide Association Studies (GWAS) the aim is to look for associationbetween genetic markers and phenotype (disease). For each genetic marker weperform an hypothesis test. Since the number of markers is high (in the order of hundred thousands), we use multiple hypothesis tests. One popular strategy in multippel testing is to estimate an effective number of independent tests, and then use methods based on independent tests to control the total type I error. The focus of this thesis has been to study different methods for estimating the effective number of independent tests. The methods are applied to a large data set on bipolar disorder and schizophrenia in Norwegian individuals from the TOP study at the University of Oslo and Oslo University Hospital (OUS). A key featureof these methods is the correlation between the genetic markers. The methodsconsidered in this thesis are based on either haplotype or genotype correlation andone focus of this thesis has been to study the difference between haplotype andgenotype correlation.

The objective of this study is to classify lithology/fluid(LF) variables along depth profiles. The classification is done by a Bayesian inversion method to obtain the posterior probability density functions(PDFs) for the LF classes at every depth, given data in form of petrophysical variables or elastic properties. In this way we determine the most probable lithology/fluid profile. A stationary Markov chain prior model will be used to model the continuity of the LF classes a priori. The likelihood relates the LF classes to data. A statistical rock-physics forward model is used to relate the petrophysical variables to elastic attributes. This will be done for synthetic test data inspired by a North Sea sandstone reservoir and for real test data in form of a well log from the North Sea. Data for the synthetic case is either the petrophysical variables or the elastic properties. For the real data is only the elastic properties considered.

This thesis aims to implement time integrators in the FEniCS framework. More specifically, the thesis focuses on selecting suitable time integrators, implement these and verify that the implementation works by applying them to various relevant test problems. This work resulted in a module for FEniCS, named Gryphon. The thesis is divided into four parts.The first part builds a theoretical framework which will motivate why singly diagonally implicit Runge-Kutta methods with an explicit first stage (ESDIRKs) should be considered for solving stiff ordinary differential equations (ODEs). It will also be shown how an ESDIRK method can be utilized to solve time dependent partial differential equations (PDEs) by solving the semidiscretized system arising from first applying a finite element method. We will restrict our attention to PDEs which either give rise to a pure ODE system or a DAE (differential-algebraic equation) system of index 1.The second part discusses the implementation of Gryphon, focusing on why such a module is useful and how the source code is structured.The third part is devoted to numerical experiments on the ESDIRK solvers implemented in Gryphon. The experiments will establish convergence and give some run-time statistics for various ESDIRK schemes. We will also see that L-stability is a favorable trait when working with stiff equations, by comparing an ESDIRK method to the trapezoidal rule. It will also be verified that the step size selectors implemented in Gryphon behaves as expected. As test problems we consider the heat equation, the Fisher-Kolmogorov equation, the Gray-Scott equations, the Fitzhugh-Nagumo equations and the Cahn-Hilliard equations.The fourth part is a user manual for Gryphon. All the parameters which can be changed by the user are explained. The manual also includes example code for solving the heat equation, the Gray-Scott equations and the Cahn-Hilliard equation, to get the reader starting on solving their own problems.

This thesis summarizes an existing protocol, that we have chosen to call the Token Key Agreement protocol. It then goes on to introduce two new protocols we have chosen to name the Symmetric Key Agreement protocol and the Asymmetric Key Agreement protocol. We are working within the UC framework, and as such introduce ideal functionalities and protocol descriptions for the protocols. For the first protocol we also introduce a simulated adversary. Further, the paper includes an overview of the security offered by the three protocols.

We give a review of the Smart-Vercauteren fully homomorphic encryp-tion scheme presented in 2010. The scheme follows Craig Gentry’sblueprint of first defining a somewhat homomorphic encryption scheme,and prove that it is bootstrappable. This is then used to create the fullyhomomorphic scheme. Compared to the original paper by Smart andVercauteren, we give a more comprehensive background, and explainsthe concepts of the scheme more in detail. This text is therefore wellsuited for readers who find Smart and Vercauteren’s paper too brief.

In this thesis we present the theoretical background for the two-point flux-approximation method; (TPFA), mimetic discretisation methods, and the multipoint flux approximation method; (MPFA). Theoretical arguments concerning monotonicity and the fact that loss of monotonicity may lead to oscillations and nonphysical cycles in the flux field are also discussed. TPFA is only consistent for $mathbf{K}$-orthogonal grids. Multipoint flux approximation methods and mimetic discretisation methods are consistent, even for grids that are not K-orthogonal, but sometimes they lead to solutions containing cycles in the flux field. These cycles may cause problems for some transport solvers and diminish the efficiency of others, and to try to cure this problem, we present two hybrid methods. The first is a hybrid mimetic method applying TPFA in the vertical direction and mimetic discretisation in the plane. The second hybrid method is the hybrid MPFA method applying TPFA in the vertical direction and MPFA in the plane. We present results comparing the accuracy of the methods and the number of cycles obtained by the different methods. The results obtained shows that the hybrid methods are more accurate than TPFA, and for specific cases they have less cycles than the original full methods.

This paper extends standard GARCH models of volatility with realized measures for the realized GARCH framework. A key feature of the realized GARCH framework is the measurement equation that relates the observed realized measure to latent volatility. We pay special attention to linear and log-linear realized GARCH models. Moreover, the framework enhance the joint modeling of returns and realized measures of volatility. An empirical application with ICE Brent Crude Oil future front month contracts shows that a realized GARCH specification improves the empirical fit substantially relative to a standard GARCH model. The estimates give weak evidence for a skewed student's t distribution for the standardized error term and the leverage function shows a clear negative asymmetry between today's return and tomorrow's volatility.

We present a method for finding the optimal decision on Random Variables in a graphical model. Upper and lower bounds on the exact value for each decision are used to reduce the complexity of the algorithm, while we still ensure that the decision chosen actually represents the exact optimal choice. Since the highest lower bound value is also a lower bound on the value of the optimal decision, we rule out any candidate with an upper bound of lower value than the highest lower bound. By this strategy, we try to reduce the number of candidates to a number we can afford to do exact calculations on.We generate five Bayesian Networks with corresponding value functions, and apply our strategy to these. The bounds on the values are obtained by use of an available computer program, where the complexity is controlled by an input constant. We study the number of decisions accepted for different values of this input constant. From the first Network, we learn that the bounds does not work well unless we split the calculations into parts for different groups of the nodes. We observe that this splitting works well on the next three Networks, while the last Network illustrates how the method fails when we add more edges to the graph. We realize that our improved strategy is successful on sparse graphs, while the method is unsuccessful when we increase the density of edges among the nodes.

In this thesis we study time series containing pressure measurements from a three phase flow pipeline at the Ekofisk oil field. The pipeline transports a mixture of oil, water and gas from $15$ wells for approximately 2.5km to a production facility. Our aim is to develop techniques that allow the selection and (to some extent) prediction of "non-standard" behavior in the system (sharp pressure changes and other type of instabilities). To advice this aim we perform a scalewise decomposition of the input signal/time series and investigate the behavior of each scale separately. We introduce the Sliding Window Wavelet Transform (SWWT) method. The method evaluate the variability on different scales within the time interval of a characteristic length (a window) and then trace these characteristics as the window slides in time.We use the discrete wavelet transform (DWT) in order to obtain the scalewise decomposition within the window. Using orthonormal discrete wavelets, we show that the variability of such sequences can be decomposed into their corresponding scales. Based on this, a thresholding algorithm is applied, characterizing the state of the system any given time. The results we find are promising and we show that different parameters in the thresholding algorithm extracts different types of special events. We also show that in some cases, this approach allows to predict special events before they really occur.While we investigate one particular system in this thesis, the procedures developed can be applied to other complicated systems where instability in system parameters is important.

The Cauchy problem for a two-component Hunter-Saxton equation, begin{align*}(u_t+uu_x)_x&=frac{1}{2}u_x^2+frac{1}{2}rho^2,rho_t+(urho)_x) &= 0,end{align*}on $mathbb{R}times[0,infty)$ is studied. Conservative and dissipative weak solutions are defined and shown to exist globally. This is done by explicitly solving systems of ordinary differential equation in the Lagrangian coordinates, and using these solutions to construct semigroups of conservative and dissipative solutions.