Vi vil utforske bruken av adeler i matematisk fysikk.

The level-set method is an implicit interface capturing method that can be used in two or more dimensions. The method is popular e.g. in computer graphics, and as here, in simulations of two-phase flow. The motivation for the simulations performed here is to obtain a better understanding of the complex two-phase flow phenomena ocurring in heat exchangers used for liquefaction of natural gas, including the study of droplet-film interactions and coalescence.One of the main advantages of the level-set method is that it handles changes in the interface topology in a natural way. In the present work, the calculation of the curvature and normal vectors of an interface represented by the level-set method is considered. The curvature and normal vectors are usually calculated using central-difference stencils, but this standard method fails when the interface undergoesa topological change, e.g. when two droplets collide and merge. Several methodshave previously been developed to handle this problem. In the present work,a new method is presented, which is a development on existing methods. The newmethod handles more general cases than previous methods. In contrast to someprevious methods, the present method retains the implicit formulation and can easily be extended to three-dimensional simulations, as demonstrated in this work.Briefly, the new method consists in extracting one or more local level sets forbodies close to the grid point considered, reinitializing these local level setsto remove kinks, and using these to calculate the curvature and normal vector atthe grid point considered. For the curvature, multiple values are averaged,while for the normal vector, the one corresponding to the closest interface isselected.With this new method, several two-phase flow simulations are performed that arerelevant for understanding the liquefaction of natural gas. The new methodenables simulations that are more general than previous ones. A two-dimensionalsimulation was performed of a 0.6 mm diameter methanol droplet falling through air and merging with a deep pool of methanol. The new method gave good results in this case, but unphysical oscillations in the pressure field rendered this result unsuitablefor comparison with experimental results.Several similar cases with significantly lower density differences between thetwo fluids were also considered; in these cases, the pressure field behavedphysically, but the results are less applicable to the understanding of naturalgas liquefication, and better suited for validation of the new method. Inparticular, an axisymmetric simulation of a 0.11 mm diameter water droplet in decanemerging with a deep pool of water has been considered. The results of thissimulation show a very close agreement with experimental data. Attempts werealso made to simulate a larger droplet, but in this case finer grids were neededthan what could be achieved here due to the computational time cost of gridrefinement. Purely geometrical results are also presented in order to validate the results of the new method, and three-dimensional results are given for a static interface configuration, demonstrating that the method is easily extended to higher dimensions.

This thesis is focused on solving the Darcy flow problem using divergence-free isogeometric methods, and comparing these results to the ones obtained using traditional finite element methods with Taylor Hood elements. A short introduction to B-splines is given, and a chapter is also about using repeated knots in the knot vectors to obtain a discontinuous basis for the finite element method. This can be useful when dealing with varying permeabilities.

The least squares method has been applied to estimate parameters inan aerodynamic model of a simulated aircraft, using data that can beexpected to available from sensors on an Unmanned Aerial Vehicle. Acombination of two non-linear state observers have been implemented toestimate wind data such as angle of attack, sideslip and dynamic pressure.Simulations have confirmed that the observers are able to estimete thewind data using noisy sensor measurements. Parameter estimation havebeen demonstrated with both measured and estimated wind data.

The Maximum Entropy Theory of Ecology (METE), developed by John Harte, presents an entirely new method of making inferences in ecology. The method is based on the established mathematical procedure of Maximum Information Entropy (MaxEnt), developed by Edwin T. Jaynes, and is used to derive a range of important relationships in macroecology. The Maximum Entropy Production (MEP) principle is a more recent theory. This principle was used by Paltridge to successfully predict the climate on Earth in 1975. It has been suggested that this principle can be used for predicting the evolution of ecosystems over time in the framework of METE. This idea is at the very frontier of Harte's theory. This thesis investigates the hypothesis that the information entropy defined in METE is described by the MEP principle.I show that the application of the MEP principle to the information entropy in METE leads to a range of conceptual and mathematical difficulties. I show that the initial hypothesis alone cannot predict the time rate of change, but that it does predict that the number of individual organisms and the total metabolic rate of an ecosystem will continue to grow indefinitely, whereas the number of species will approach one.I also conduct a thorough review of the MEP literature and discuss the possibility of an application of the MEP principle to METE based on analogies. I also study a proof of the MEP principle published by Dewar in 2003 and 2005 in order to investigate the possibility of an application based on first principles. I conclude that the MEP principle has a low probability of success if applied directly to the information entropy in METE.One of the most central relationships derived in METE is the expected number of species in a plot of area $A$. I conduct a numerical simulation in order to study the variance of the actual number of species in a collection of plots. I then suggest two methods to be used for comparison between predictions and observations in METE.I also conduct a numerical study of selectied stability properties of Paltridge's climate model and conclude that none of these can explain the observed MEP state in nature.

It is of interest to construct windows that can spread the transmitted light in a specified manner. The Kirchhoff approximation in the geometrical optical limit in combination with a chosen general form of the window surfaces yields the profile for the window surfaces, letting us specify how the light should be spread.The probability distribution function for the slopes of a window surface consisting of joined line segments was implemented in the simulation software Maxwell1D. Simulations show the feasibility of such windows. However they do not respond well when subject to light incident with another angle of incidence than the angle in mind.By using Snell's law to compensate for using a simpler system, the time needed for the simulations can be greatly reduced while simultaneously obtaining a higher accuracy in the results.

An inverse statistics analysis of one minute stock quotes from 492 large Europeancompanies has revealed the existence of a gain-loss asymmetry in thefollowing index. The gain-loss asymmetry differs from that observed for dailyclosure prices of the Dow Jones Industrial Average [38], as the probability ofthe optimal investment horizon for a gain is higher than that of a loss. Forindividual stocks, the gain-loss asymmetry was observed to only appear forsignificantly larger return-levels. To the best of our knowledge, this is thefirst time such an analysis has been performed on high-frequency data.A principal component analysis was done by performing an eigenvalue decompositionof the correlation matrix from a sliding time-window. The firstprincipal component was observed to describe the market excellently. Its correspondingeigenvalue was observed to be significantly larger than theoreticalpredictions from random matrix theory, implying that the eigenvalue carriesinformation common to all stocks. Using this eigenvalue as an index measuringthe collectivity in the market has revealed the existence of collectivetrends that appear to be stronger during falling than rising markets. Thishas been observed for two different datasets, the above described one minutestock quotes and daily closure prices from 29 stocks composing the DJIA lateFebruary 2008. The observation is in accordance with results of Balogh etal. [40], and provides further support to the speculation of Johansen et al.[37] that a difference in collective trends is the reason behind the gain-lossasymmetry observed in indexes and not for individual stocks for the samereturn-level.The key idea behind the fear factor model of Donangelo et al. [42] has beenstrongly supported by the observation that collective trends appear to bestronger during sharp index drops. As the collectivity increment has beenobserved to be dependent on the size of the index drop, it is suggested thatthe model should incorporate also individual fear factors for economic sectors,in addition to the global fear factor governing the market as a whole. Periodsexhibiting a rising index positively correlated to the strength of collectivityhas indicated the presence of an optimism factor that also should be incorporatedin the fear factor model [42], forcing stocks to rise synchronously.

Energivariasjon i fullt utviklet turbulens er studert. I en tidsserie for tur- bulens energi er det oppdaget en asymmetri mellom positive og negative energiendringer. Invers statistikk metode gir en mulighet til å studere denne asymmetrien nærmere.For analyse i denne oppgaven er turbulent strømning generert ved bruk av GOY skallmodell. Skallmodellen er en tilnærmingmodell til Navier-Stokes likninger for strømningens bevegelse. Modellen er tidligere vist å gi realistiske verdier for energi og hastighet for en turbulens strømning.Ved bruk av forward statistikk og invers statistikk på turbulens ener- gi er det funnet en asymmetri i energiendringen. Det er vist at en negativ energivariasjon høyst sannsynlig inntreffer før en positiv energivariasjon av samme størrelse. Tidsforskjellen er empirisk funnet til å følge sammenhen- gen δτδE ∼ δE0,749, hvor τδE er forventningstiden for en energivariasjon δE. I tillegg er det funnet en positiv trend i tidsserien for energi, som sier at ener- giendring etter korte tidsintervall høyst sannsynlig er positiv. Det er gjort et forsøk på å beskrive denne asymmetrien og opprinnelsen av dette fenomenet er drøftet.

The main focus of this master thesis work has been to image InAs emph{quantum dots} (QDs) using emph{atomic force microscopy} (AFM), to identify and evaluate various image processing methods used to estimate the volume of the InAs QDs. The InAs QDs studied in this thesis work, had been deposited on GaAs substrates, using solid-source emph{molecular beam epitaxy} (MBE) before the thesis work started. The total QD volume was determined for all samples, using eight different estimation methods. The purpose of estimating the total QD volume, was to compare the total volume to the deposited volume.Previous studies on similar samples, have indicated that the total volume can be larger than the deposited volume during MBE growth. This discrepancy is explained by incorporation of Ga from the substrate during growth. This was not observed in this thesis work. One possible explanation is that the samples have oxidized; resulting in a lower measured height.In addition, the relationship between atomic steps, defects and the appearance of large QDs were studied. parTwo series of samples were studied: one in which the QD growth temperature was varied and one in which the amount of deposited InAs was varied. The total QD volumes were found to increase with the QD growth temperature and the deposited InAs thickness. Square-shaped defects and contours of 2D islands were observed in nearly all samples. Higher/multiple terraces seem to be related to regions of higher QD density.

This thesis investigates the electrodeposition process of zinc-, copper-, silver-, and silver copperoxides at cathodic and anodic voltages. Silver copper oxide has been successfully electrodepositedon a substrate of PtSi from a pH 12 dilute solution of copper nitrate, silver nitrate and sodiumhydroxide at 0.9V vs a silver metal cathode. This film was confirmed to be polycrystalline AgCuO2by EDS and XRD studies. Zinc oxide and copper oxide were deposited on gold substrates from their respective nitrates. The zinc oxide deposition was confirmed polycrystalline in XRD and had a band gap between 3.2eV and 3.5eV measured by optical reflectance. The copper oxide appeared polycrystalline in SEM but only amorphous signal was achieved in XRD, the material had a band gap of around 2eV. Despite many attempts, clean silver oxide was not successfully deposited. These materials may all be suitable for solar cells applications.