The heterogeneous multiscale method (HMM) was proposed by E and Engquist and is considered to be an efficient method for problems with multiple time scales. We give a short introduction to the HMM for multiscale problems in general, before we restrict our work to HMM schemes developed for stiff ODEs, based on results found by Engquist et al. HMM provides an efficient and systematic way to move between the macroscopic and microscopic model in a problem having multiscale physics. By taking advantage of scale separation in multiscale problems, the HMM approximates the macroscopic variables of the solution without fully resolving the microscopic solution. This introduces computational savings as the total number of evaluations needed for convergence are significantly reduced.We test the features of the HMM on the spinning top. The governing equations of the top produces a highly oscillatory solution as the top spins fast. Despite this fast oscillating nature, we would intuitively expect some slow behavior of the top, for instance the inclination from the vertical axis or the circulation of the top around the vertical axis. We find a set of slow variables of the spinning top, and show that the HMM provides an accurate solution of the macroscopic variables of the top, with a significant gain in computational cost compared to standard solvers.We also study the spinning top subjected to a vibrational external force and find a set of slow variables, which can be approximated accurately with HMM. Finally, we find an averaged equation to the spinning top subjected to a vertical vibrating force. This analysis is based on the Modulated Fourier expansion.The work of this thesis is an extension of my semester project, and we emphasize that the theory part of this thesis is partially from this work.

The overall aim of this project was to produce homogeneously sized polysaccharide microparticles and apply these and similar sized particles as probes for investigation of mucin layers as a model for a biological barrier. Small polysaccharide particles have many applications, e.g. within the medical field of drug delivery. In this study a microfluidic system was developed to produce alginate beads, which can be used in drug delivery systems. Different designs, continuous phases and concentrations were tested in order to find an optimal system. Beads in the size range of 10 µm were produced using a device with T-shaped design and three inlets. An electrostatic bead generator was also used to make alginate beads, however the beads produced were too large to be used in the experiments with the mucin layers.One of the many challenges when working with drug delivery systems is the mucus barrier protecting the epithelial cells. In this study a model mucus barrier was made by immobilizing mucins, the glycoprotein responsible for the physical properties of the barrier. A procedure for fluorescence labelling of polystyrene beads with quantum dots was developed, and penetration of these beads into the model barrier was measured with total internal reflection fluorescence (TIRF) microscopy. In TIRF the excitation field intensity decays exponentially, and the emitted fluorescence intensity from the beads gives an indication of the distance between the beads and the surface. Measurements performed on mucin layers of different concentrations indicate that mucin concentrations above 0,5 mg/ml will result in a layer too thick or too dense to give a intensity signal. At mucin concentration 0,05 mg/ml fluorescence was observable in TIRF, and it was clearly weaker than for the control with a bead directly on a glass surface. This indicates that the beads hover over the surface due to the mucin layers, and show that it is in principle possible to measure the penetration depth of beads into mucin layer using TIRF. To simulate the condition in the lungs of cystic fibrosis (CF) patients, the mucin layers were incubated with alginate. Measurements were performed to see how this affected the penetration of the beads into the layer. A weaker fluorescent signal was obtained for these samples in TIRF, which suggests that there has been interaction between mucin and alginate. It was in addition investigated how different concentrations of G-blocks in the solution affected the penetration into the mucin-alginate layer. These testes were carried out using both TIRF and atomic force microscopy (AFM) nanoindentation experiments. The TIRF measurements were inconclusive, while the nanoindentation experiments showed decreased interaction between mucin-alginate layer and a bead.

We study scalar waves scattered from self-affine and Gaussian correlated surfaces. The simulations are performed using rigorous simulation of the integral equations derived from the Helmholtz equation, describing a scalar wave above a non-penetrable surface with a hard wall or free surface boundary condition.An incident, Gaussian shaped beam is scattered from the surface, and the full angular distribution of the scattered intensity is obtained. Self-affine and Gaussian correlated random surfaces are generated, and the resulting scattered intensity is averaged over a large number of surfaces (in the order $N_s=3000$), using the ergodicity of the surface.Compared with analytical calculation of the scattered intensity in the Kirchhoff approximation, our approach gives similar results for less rough surfaces. Compared with simulations of electromagnetic waves scattered from a perfect conductor, without recording the polarisation of the scattered light, our simulations give similar results when using a hard wall boundary condition.We observe phenomena such as specular scattering for less rough surfaces, diffuse forward scattering for more rough surfaces and enhanced backscattering for surfaces where waves scattered multiple times by the surface roughness gives a large contribution to the scattered intensity.

The behaviour of particles in turbulent flow, with emphasis on particle deposition on both the frontside and backside of a cylinder, was investigated by means of Direct Numerical Simulations. One-way coupling between the fluid and the particles was applied. Simulations of turbulence forced at small, intermediate and large scales were run on a three-dimensional domain. The turbulence was used as inlet on a two-dimensional flow domain, where a Lagrangian tracker was used to compute the particle motions. The Reynolds numbers used were Rec = 421 and Rec = 1685. For intermediate Stokes numbers, or particle sizes, the number of particles deposited increased when the Reynolds number was raised. The presence of turbulence lead to a further increase in the deposition for these Stokes numbers, compared to the deposition in laminar flow. The increase was at its highest for large scale forced turbulence. The increased deposition of intermediate Stokes numbers on the frontside of the cylinder in turbulent flow was found to be related to the variance of the effective Stokes number, resulting from the fluctuating nature of the turbulent velocity. The deposition of particles with small Stokes numbers was also altered by the presence of turbulence, but this could not be explained by the variance of Stokes number. The mechanism leading to deposition of these smallest particles is related to the turbulent eddies close to the boundary layer of the cylinder, and will need further study. At the backside of the cylinder, the increased deposition at raised Reynolds number and at the presence of turbulence was related to the magnitude of vorticity. Furthermore, preferential concentration of particles in turbulence was observed in the Rec = 1685 cases, demonstrating the effect of the vorticity magnitude being sustained throughout the domain.

Psoriasis er en kronisk, inflammatorisk hudsykdom som gir tykke skjelldannelser på en rød hudoverflate. Sykdommen rammer 2-3% av den europeiske befolkningen. Ultrafiolett (UV) stråling som bruk i behandling av psoriasis er effektivt og naturlig. Solens UV-stråling vil forbedre de fleste av pasientenes psoriasisflekker. Samtidig som solen har en positiv effekt, vil overeksponering kunne forårsake direkte DNA-skade i hudcellene som igjen kan forårsake hudkreft. I den forbindelse vil det være viktig å finne mengden UV som vil gi størst klinisk utbytte for psoriasisflekkene, samtidig som den skader den omkringliggende friske huden minst mulig. Hypotesen bak, er at en optimal strålemengde også vil gi det beste kliniske resultatet. Oppgaven vil utforske UV-dose til 24 pasienter som deltok på en treukers klimabehandling på Gran Canaria i november 2010. Under oppholdet ble den maksimalt tilgjengelige UV-intensiteten målt kontinuerlig fra taket av behandlingssenteret. Dette ble samkjørt med at pasientene notert når og hvor lenge de hadde vært eksponert for solen i dagbøker. I tillegg bar hver pasient UV-dosimetre, ett på håndleddet og ett festet til badetøyet. Denne oppgaven skulle sammenstille de to målemetodene for å finne UV-dosen for hver pasients hud. Sammenligninger av de UV-doser funnet med de to metodene skulle også utføres og resultatene av målingene ble vurdert.Gjennomsnittlig individuell kumulativ dose for (målt med UVB-Biometeret i samsvar med dagbøkene) var på 216,527,8 SED. Målt individuell kumulativ gjennomsnittsdose (målt med dosimetrene) var på 107,225,6 SED og 63,521,6 SED for henholdsvis dosimeteret på håndleddet og det på badetøyet. Soleksponering i tre uker førte til en gjennomsnittlig Psoriasis Area and Severity Index (PASI)-reduksjon på 77,416,3%. Ingen signifikante verdier ble funnet som kan forutsi utfallet for psoriasispasientene. Det ble heller ikke påvist noen sammenheng mellom tid eksponert for solen og UV-dose eller dosen absorbert av huden i forhold til PASI- reduksjon.Dosimeteret festet til håndleddet fikk i gjennomsnitt 50% av den maksimalt tilgjengelige intensiteten. Den makstilgjengelige intensiteten vil måles for et horisontalt plan, og alle andre målinger med dosimetre på ulike områder av kroppen, kan sees på som en fraksjon av dette. Hvor på kroppen psoriasisflekken befinner seg vil ha mye å si for hvor høy UV-dose den faktisk vil få. Forskjellen mellom de ulike områdene vil være stor, i hovedsak grunnet ulik vinkling til forhold til solen. Dosen dosimetrene gir vil derfor bare være veiledende for hvor stor dose de ulike psoriasisflekkene faktisk fikk, avhengig av hvor på kroppen de var i forhold til dosimeteret.62,5% av pasientene oppnådde PASI75 (PASI-reduksjon75%). Studien ble gjennomført i november som er en av månedene med lavest UV-indeks. Pasienter som får klimabehandling andre måneder av året med langt høyere UVI vil kunne få for høy eksponering for ultrafiolett stråling.

During the past few decades, the power sectors of several countries have been substantially reorganized, and liberalized markets for trading of electricity have been established. In this report, evidence from six electricity markets are studied in order to identify characteristics of electricity prices. The market structures, statistical quantities, as well as long-term dependence, are investigated. Detrended fluctuation analysis and the average wavelet coefficient method are employed in order to estimate the Hurst exponent, which quantifies the presence of long-termed dependence. Since it is concluded that the price series are periodic on several time scales, all characteristics are investigated for both the original and deseasonalised versions of the time series. In particular, it is confirmed that the electricity prices are volatile, but that a considerable amount of the volatility is caused by the daily and weekly periodicities. Furthermore, the characteristic return distributions, volatility clustering and price spikes are analysed.

This is the Master’s Thesis of Eirik Timo Bøe Vilpponen, and was writtenat the Department of Physics at the Norwegian University of Science andTechnology (NTNU). The thesis is twofold, the first part being the comple-tion of a project undertaken during the fall of 2010. The project concernedthe setup of a solar cell current-voltage characterization lab [4] at the sameDept. of Physics.The second part is an analysis of the current-voltage characteristics ofNTNU-made quantum-dot intermediate band solar cells. A strong focus isput on exploring the possibility of utilizing the Two-Diode Model as a meansof analysis. The two-diode model has been found to have a good potential forthis use, but that it requires a specially made curve fitting computer programto fulfill this potential.

In this thesis the Minimum Ignition Energy, in a hydrogen-air system, is studied by Direct Numerical Simulations (DNS) in a program called the Pencil Code. The heat source used to achieve ignition is modeled by a Gaussian temperature distribution. Three different geometries of the heat source are looked upon, one with spherical geometry in three dimensions, one with cylindrical geometry in two dimensions and the last in one dimension. The results show that the dimensionality of the heat source has a strong impact on ignition.In addition, a new simpler zero dimensional simulation method is proposed with the goal of replicating the results from the Pencil Code. This method needs less calculation power, and uses ignition delay time data together with the heat equation to simulate ignition. The model has proven itself useful since it reproduces the Pencil Code results very well.

In this project we have first and foremost been comparing the performance of the ACER method with the POT method in the prediction of extreme values from the heavy tailed distributions; especially for data from the energy markets. The energy market is an exciting dynamic market where small singularities can make large differences in the price. Therefore it is very important and challenging to analyse and make predictions in this market. We have also analysed a dataset which is not from the energy market, to compare and see the main differences between the two markets. We have also taken in consideration of removing the return value for the dates of maturity to see whether this will have any influence on the results.The main concept of the POT method is to find a threshold, $u$, and let the excesses be distributed by the Generalised Pareto Distribution. Whilst for the ACER method, we assume a specific shape of the tail, which in this project was of the kind Fréchet. We have done this analysis for five different data sets where two of them have been considered with and without their expiration dates. We have also filtrated the data sets with an AR-GARCH filter, and then used the POT and ACER on the residuals from the process. We have found out that both methods are not greatly influenced by the filtration, but we see the tendency of the POT method predicting a heavier tail than the ACER method. Further on, we can say that there are no significant large effects of removing the return values for the dates of maturity. Lastly, the data sets from the energy market prove themselves much more heavy tailed than for the data set from Norsk Hydro.

We study persistent homology, methods in discrete differential geometry and discrete Morse theory. Persistent homology is applied to computational biology and range image analysis. Theory from differential geometry is used to define curvature estimates of triangulated hypersurfaces. In particular, a well-known method for triangulated surfacesis generalised to hypersurfaces of any dimension. The thesis concludesby discussing a discrete analogue of Morse theory.