This thesis examines misspecifed log-location-scale regression models. Particularily how the models' Cox–Snell residuals can be used to infer the functional form of possibly misspecified covariates in the regression. Two different methods are considered. One is using a transformation of the expected value of the residuals. The second is based on estimating the hazard rate function of the residuals using the covariate order method. Simulations and computations in the statistical computing environment R are used to obtain relevant and illustrative results. The conclusion is that both methods are able to recover the functional form of a misspecified covariate, but the covariate order method is best when high levels of censoring are introduced. The Kullback–Leibler theory, applied to misspecified regression models, is a part of the basis for the investigations. The thesis shows that a theoretical approach to this theory is consistent with the methods used in R.

Our main objective for this thesis is to present and discuss the linear mixed effects model and, in particular, the different possible covariance structures for the random effects and the residuals. The linear mixed effects model is widely used in biology and medical research.We use data from a diet intervention study where the aim was to investigate the difference between a diet rich in carbohydrates and a diet rich in fat and protein. Data from $32$ participants were available. A series of biomarkers were measured before and after both diets, giving repeated measurements from each participant across time and diet.We have studied different linear mixed effects models varying in covariance structure for the random effects and the residuals. Further, we have focused on a thorough treatment of statistical contrasts. The contrasts of interest in this study are estimates of the effect of the two diets and the difference in effect between the two diets, and is especially relevant to biologists and medical researchers. Statistically, there is no common agreement on how degrees of freedom should be calculated when testing contrasts. We will show that using different parameter coding for a between-subject factor in the same model, yield different results.The linear mixed effects model allows complex structures in correlated data to be modeled, and so it is important to look at the implied marginal variance-covariance matrix to understand the structure. We have calculated the empirical variance-covariance matrix of the data, and compared it to the estimated implied marginal variance-covariance matrix, in an attempt to get a more thorough understanding of the covariance structures for the random effects and the residuals.The estimated implied marginal variance-covariance matrix have also been used to estimate the intraclass correlations.Finally, we have fitted the linear mixed effects model using the Bayesian approach, integrated nested Laplace approximations (INLA), and compared the results to the results of the frequentist approach.

This report concerns preliminary studies of fingering and fracturing in gelling materials, with the aim of identifying how the phenomena of fingering transitions into fracturing as the breaking medium gradually becomes more solid-like.

This thesis seeks to develop a better understanding of the analysis of gene expression to find the amount of transcript in a sample. The mainstream method used is called Polymerase Chain Reaction (PCR) and it exploits the DNA's ability to replicate. The comparative CT method estimate the starting fluorescence level f0 by assuming constant amplification in each PCR cycle, and it uses the fluorescence level which has risen above a certain threshold. We present a generalization of this method, where different threshold values can be used. The main aim of this thesis is to evaluate a new method called the Enzymological method. It estimates f0 by considering a cycle dependent amplification and uses a larger part of the fluorescence curves, than the two CT methods. All methods are tested on dilution series, where the dilution factors are known. In one of the datasets studied, the Clusterin dilution-dataset, we get better estimates from the Enzymological method compared to the two CT methods.

The subject of this thesis is the study of the creation of fault lines in a random lattice, provoked by the successive failure of optimal paths. Using the recently developed Optimal Path Cracked model, we investigate how central characteristics of the successive optimal paths evolve as the lattice breaks down, and how this progression of characteristics depends on the magnitude of disorder imparted on the lattice. We then see how the OPC model, while originally proposed in the context of the shortest path problem, can be generalized to alternate optimal path problems, namely the minimax problem and the widest path problem. It is shown that for a given lattice, the minimax OPC is equal to the the backbone of the shortest OPC. The widest path OPC, although constituting a distinct object on any lattice, is shown to scale with lattice size in the same manner as the minimax OPC and the backbone of the shortest path OPC; with the fundamental process behind it being closely related to the minimax OPC process. Lastly, we explain the connection between the OPC process and a variety of other phenomena which have previously been shown to exhibit similar scaling behavior. We show how the OPC process for the widest path problem can be reduced to the shortest path problem on the dual lattice using the limit of very high disorder, the so-called ultrametric limit, and how an algorithm based on invasion percolation can be used as a quicker method of finding an OPC.

This thesis studies variational problems invariant under a Lie group transformation, and invariant discretizations of these. In chapters two and three, a general method for creating symplectic integrators preserving certain classes of variational symmetries of first order Lagrangians is developed and demonstrated. In chapters four and five, it is assumed that the discrete Lagrangian is invariant under a certain group action, and the Euler--Lagrange equations for the variational problem are expressed in the invariants of the group action.

We discuss numerical quadratures in one and two dimensions, which is followed by a discussion regarding the differentiation of general operators in Banach spaces. In addition, we discuss the standard and fractional Sobolev spaces, and prove several properties for these spaces.We show that the operator splitting methods of the Godunov type and Strang type applied to the viscous Burgers’ equation, and the Korteweg-de Vries (KdV) equation (and other equations), have the correct convergence in the Sobolev spaces. In the proofs we use the new framework originally introduced in [11].We investigate the Godunov method and Strang method numerically for the viscous Burgers’ equation and the KdV equation, and present different numerical methods for the subequations from the splitting. We numerically check the convergence rates for the split step size, in addition with other aspects for the numerical methods. We find that the operator splitting methods work well numerically for the two equations. For the viscous Burgers’ equation, we find that several combination of numerical solvers for the subequations work well on the test problems, while we for the KdV equation find only one combination of numerical solvers which works well on all test problems.

In this master thesis, thin lms of ZnO doped with various cobalt concentra-tions have been grown using Pulsed Laser Deposition (PLD). This growthtechnique is preferred because it is able to grow thin lms from a targetmaterial without changing the elemental ratios from the target. The targetsusing during deposition are sintered pellets of ZnO doped with cobalt, withdoping concentrations of 10, 20 and 30 %.The material was chosen because of its potential for Intermediate BandSolar Cells (IBSC). IBSC is a new concept which aims to create solar cellswhich have higher eciencies than the solar cells available today. From de-tailed balance theory, the concept of intermediate band solar cells shows apotential of a 86 % eciency limit under perfect conditions, which is remark-able compared to the conventional silicon solar cell devices on the markettoday with a eciency limit of 41 %.A series of ZnO thin lms of 10, 20 and 30 % cobalt concentration weredeposited by PLD on a sapphire substrate. These lms were, together with asimilar series containing 1,2 and 5 % cobalt concentration deposited on bothsilicon and sapphire substrates, characterized using Scanning Electron Mi-croscopy (SEM), Energy Dispersive X-ray (EDX) Microanalysis, X-ray Pho-toelectron Spectroscopy (XPS), Hall measurements, X-ray diraction (XRD)and optical transmission spectroscopy.In addition, an attempt to make Schottky diodes of the cobalt dopedZnO thin lms and a p-IB-n junction using GaN:Mg and ZnO:Al as the p-and n- emitter was done. These intermediate band solar cell devices weregoing to be studied by recording dark and illuminated current-voltage (I-V)characteristics.The focus in the study has been to determine which cobalt doping con-centration gives the desired properties for use of ZnO:Co in an intermediateband solar cell device, and to study how a solar cell device with an inter-mediate band material can be realized. The experiments were carried outat NTNU NanoLab, NTNU Department of Physics, NTNU Department ofElectronics and Telecommunications and SINTEF Oslo.Optical transmission spectroscopy and XPS showed that the cobalt inthe ZnO:Co lms were Co2+. This indicates that cobalt is substituting forzink in the ZnO lattice. XRD showed the crystal structure to be c-orientedwith reasonable order. The carrier concentration in the ZnO:Co lms were inthe order of 1018- 1019, decreasing with increasing cobalt concentration. Theresistivity in the samples increased with increasing doping concentration.Investigation of the GaN:Mg wafer showed that it did not have the desiredproperties for device fabrication as it showed n-type conductivity insteadof p-type. The Schottky diodes made of Au-ZnO:Co and Pt-ZnO:Co did not show diode behavior because of surface states in ZnO:Co making ohmiccontacts.

This master's thesis concerns the electrical activity and lifetime in compensated multicrystalline silicon wafers used for solar cell production.Resistivity profiles across grain boundaries have been obtained by a Four Point Probe (FPP). Profiles have been investigated in relation to minority carrier lifetime acquired by Microwave Photo Conductance Decay (uW-PCD).It has been found that a two-step process consisting of pre-annealing at either 600C or at 900C followed by phosphorus diffusion (P) gettering will increase the electrical activity of crystalline defects. It has been proposed that a P gettering step should follow directly after annealing for a better dissolution of metallic precipitates. Introduced defects in the material as a consequence of both pre-annealing at 900$^circ$C and of resistivity measurements before gettering, have possibly enhanced the phosphorus diffusion depth in the gettering process. The higher concentration of phosphorus has lead to an augmented lifetime in the material. Metallic impurity precipitation at defects, affecting the electrical activity and the minority carrier recombination rate, has been observed. A good correlation between grain structure, resistivity- and lifetime profiles has thus been established.

In this Master's thesis we have used data from eight different locations in Norway to estimate the distributions of extreme water levels. The locations used are Oslo, Heimsjø, Honningsvåg, Narvik, Harstad, Tregde, Andenes and Viker, and four different methods have been used for the estimation. We use two established and widely used methods, the Annual Maxima Method (AMM) and the Peaks-over-Threshold (POT) method. The AMM uses the observed annual maxima from each location, while the POT method uses the exceedances of some high threshold in relation to the data. In addition, we also use two approaches that are less known and less used, the Revised Joint Probability Method (RJPM) and the Average Conditional Exceedance Rate (ACER) method. In the former, a distribution for extreme surges is found and used together with information from numerical tide predictions. We also take into account the interaction between tides and surges where applicable. The ACER method uses only the series of measured sea levels, but tries to account for dependence by a cascade of conditioning probabilities. We find that for return periods up to 20 years the return levels are usually very similar for all methods except RJPM, which seems to underestimate these values. The data intensive methods of POT, RJPM and ACER seem to agree for the 200 year levels, while particularly the Gumbel method seems to overshoot the levels in comparison to the other methods.