The behaviour of electrons in nanostructures such as quantum wells is of interest for the design of new electronic and electro-optic devices, and also for exploration of basic many-body physics. This thesis develops and tests improved methods for describing such electronic behaviour. The system used for this work was the parabolic quantum well (PQW), an important special system which has recently attracted much experimental and theoretical attention. We firstly report self-consistent nonlinear groundstate solutions of the Poisson equation together with the Thomas-Fermi (TF) hydrostatic equations. In contrast to most previous solutions, all the electron density profiles were inhomogeneous and continuous. We also added a von Weizsacker term with and without the exchange/exchange-correlation to the above treatment, using a novel numerical approach allowing for wider electron gases than previously possible. We also report for the first time the effects of spatially varying effective mass and dielectric function in theories of this type. To investigate infrared response of these systems, we apply new hydrodynamic theories recently proposed by Dobson. By using this type of theory, we simultaneously satisfy the Harmonic Potential Theorem (extended generalized Kohn theorem) and obtain the correct 2D plasmon dispersion, as well as obtaining the correct spacing of standing plasmons. Other inhomogeneous hydrodynamic theories do not achieve this. We also showed analytically an exact solution for a plasmon mode at the Kohn frequency in addition to one found in the Harmonic Potential Theorem. An open hydrodynamic theory was then developed based on this type of mode. Numerical application of Kohn Frequency Theorem theory was shown and the results were compared with other existing hydrodynamic theories.

<p>Solar electricity is a promising energy technology for the future, and by using reflectors for concentrating solar radiation onto photovoltaic cells, the cost per produced kWh can be significantly reduced. The optical efficiency of a concentrating system determines the fraction of the incident energy that is transferred to the cells and depends on the optical properties of the system components. In this thesis, low-concentrating photovoltaic and photovoltaic-thermal systems with two-dimensional parabolic reflectors were studied and optimised, and a new biaxial model for the incidence angle dependence of the optical efficiency was proposed.</p><p>Concentration of light generally results in high cell temperatures, and the uneven irradiance distribution on cells with parabolic reflectors leads to high local currents and temperatures, which reduce fill-factor and voltage. Cooling the cells by means of water increases the voltage and makes it possible to utilize the thermal energy. The performance of a 4X concentrating photovoltaic-thermal system was evaluated. If operated at 50°C, this system would produce 250 kWh_electrical and 800 kWh_thermal per m² cell area and year. Optical performance can be increased by 20% by using better reflectors and anti-reflectance glazing.</p><p>Low-concentrating photovoltaic systems for façade-integration were studied and optimised for maximum annual electricity production. The optimisation was based on measured short-circuit currents versus solar altitude. Measurements were performed outdoors and in a solar simulator. It was found that the use of 3X parabolic reflectors increases the annual electricity production by more than 40%. High solar reflectance is crucial to system performance but by using a low-angle scattering reflector, the fill-factor and power are increased due to a more even irradiance on the modules.</p><p>Long-term system performance depends on the durability of the components. The optical properties and degradation of reflector materials were assessed using spectrophotometry, angular resolved scatterometry, Fresnel modelling, optical microscopy, and surface profilometry before and after ageing. The degradation of reflectors was found to be strongly dependent on material composition and environmental conditions. Back surface mirrors, all-metal reflectors, and polymer-metal laminates degraded in different ways, and therefore accelerated ageing must be tailored for testing of different types of reflector materials. However, new types of reflector laminates showed a potential for increasing the cost-effectiveness of low-concentrating solar energy systems.</p>

<p>This thesis consists of four papers and a summary. The common topic of the included papers are the pricing equations of path-dependent options. Various properties of barrier options and American options are studied, such as convexity of option prices, the size of the continuation region in American option pricing and pricing formulas for turbo warrants. In Paper I we study the effect of model misspecification on barrier option pricing. It turns out that, as in the case of ordinary European and American options, this is closely related to convexity properties of the option prices. We show that barrier option prices are convex under certain conditions on the contract function and on the relation between the risk-free rate of return and the dividend rate. In Paper II a new condition is given to ensure that the early exercise feature in American option pricing has a positive value. We give necessary and sufficient conditions for the American option price to coincide with the corresponding European option price in at least one diffusion model. In Paper III we study parabolic obstacle problems related to American option pricing and in particular the size of the non-coincidence set. The main result is that if the boundary of the set of points where the obstacle is a strict subsolution to the differential equation is C¹-Dini in space and Lipschitz in time, there is a positive distance, which is uniform in space, between the boundary of this set and the boundary of the non-coincidence set. In Paper IV we derive explicit pricing formulas for turbo warrants under the classical Black-Scholes assumptions.</p>

<p>In recent years, there has been a great interest towards the higher order numerical methods for singularly perturbed problems. As compared to their lower order counterparts, they provide better accuracy with fewer mesh points. Construction and/or implementation of direct higher order methods is usually very complicated. Thus a natural choice is to use some convergence acceleration techniques, e.g., Richardson extrapolation, defect correction, etc. In this thesis, we will consider various classes of problems described by singularly perturbed ordinary and partial differential equations. For these problems, we design some novel numerical methods and attempt to increase their accuracy as well as the order of convergence. We also do the same for existing numerical methods in some instances. We &macr / nd that, even though the Richardson extrapolation technique always improves the accuracy, it does not perform equally well when applied to different methods for certain classes of problems. Moreover, while in some cases it improves the order of convergence, in other cases it does not. These issues are discussed in this thesis for linear and nonlinear singularly perturbed ODEs as well as PDEs. Extrapolation techniques are analyzed thoroughly in all the cases, whereas the limitations of the defect correction approach for certain problems is indicated at the end of the thesis</p>

This thesis consists of four papers and a summary. The common topic of the included papers are the pricing equations of path-dependent options. Various properties of barrier options and American options are studied, such as convexity of option prices, the size of the continuation region in American option pricing and pricing formulas for turbo warrants. In Paper I we study the effect of model misspecification on barrier option pricing. It turns out that, as in the case of ordinary European and American options, this is closely related to convexity properties of the option prices. We show that barrier option prices are convex under certain conditions on the contract function and on the relation between the risk-free rate of return and the dividend rate. In Paper II a new condition is given to ensure that the early exercise feature in American option pricing has a positive value. We give necessary and sufficient conditions for the American option price to coincide with the corresponding European option price in at least one diffusion model. In Paper III we study parabolic obstacle problems related to American option pricing and in particular the size of the non-coincidence set. The main result is that if the boundary of the set of points where the obstacle is a strict subsolution to the differential equation is C1-Dini in space and Lipschitz in time, there is a positive distance, which is uniform in space, between the boundary of this set and the boundary of the non-coincidence set. In Paper IV we derive explicit pricing formulas for turbo warrants under the classical Black-Scholes assumptions.

The initial stage of liquid phase sintering, involving liquid penetration, agglomerate separation, particle spreading and growth has been investigated in experiments using tungsten heavy alloys. The particle composites used were produced by hot isostatic pressing (HIP) of pure powder mixtures of W-Ni-Fe-(Co). By using different HIP temperatures, volume fractions of tungsten, alloying elements like Cobalt and Sulphur or excluding Iron from the matrix, liquid penetration, agglomerate separation and particle growth conditions were affected. The investigations were performed mainly under microgravity (sounding rockets or parabolic trajectories by airplanes) but at high tungsten particle fractions, short sintering times or at infiltration of solid pure tungsten, they were performed at normal gravity. The liquid penetration of the tungsten agglomerates is explained by initial wetting under non-equilibrium conditions, due to the reaction between the liquid matrix and the particles, and a decrease of interfacial energy. The dissolving of tungsten gives a pressure drop in the penetrating liquid and a driving force for the liquid movement by a suggested parabolic penetration model. For cold worked tungsten, a penetration theory was proposed, where an internal stress release in the penetrated tungsten grains creates space for the advancing liquid. The spreading of the tungsten agglomerates is explained by an interagglomerate melt swelling due to a Kirkendall effect. The liquid matrix undergoes a volume increase since the diffusion rates of Ni-Fe are higher than for W and initial concentration gradients of W and Ni, Fe exists. The suggested model by Kirkendall are also used for an analysis of the interaction behaviour between solid particles and a solidification front and inclusion behaviour in iron base alloys during teeming and deoxidation. The average tungsten particles size decrease initially since part of the tungsten particles is dissolved when the non-equilibrium matrix phase is melting. When equilibrium is reached, the tungsten particles grow in accordance with the Ostwald ripening process by an approximately 1/3 power law. Larger particle fraction of particles showed a higher growth rate, due to shorter diffusion distances between the particles. Cobalt, Sulphur and absence of iron in the matrix were found to increase the growth rate of the tungsten particles due to a higher surface tension between the solid tungsten particles and the matrix melt. / QC 20100528

Solar electricity is a promising energy technology for the future, and by using reflectors for concentrating solar radiation onto photovoltaic cells, the cost per produced kWh can be significantly reduced. The optical efficiency of a concentrating system determines the fraction of the incident energy that is transferred to the cells and depends on the optical properties of the system components. In this thesis, low-concentrating photovoltaic and photovoltaic-thermal systems with two-dimensional parabolic reflectors were studied and optimised, and a new biaxial model for the incidence angle dependence of the optical efficiency was proposed. Concentration of light generally results in high cell temperatures, and the uneven irradiance distribution on cells with parabolic reflectors leads to high local currents and temperatures, which reduce fill-factor and voltage. Cooling the cells by means of water increases the voltage and makes it possible to utilize the thermal energy. The performance of a 4X concentrating photovoltaic-thermal system was evaluated. If operated at 50°C, this system would produce 250 kWhelectrical and 800 kWhthermal per m2 cell area and year. Optical performance can be increased by 20% by using better reflectors and anti-reflectance glazing. Low-concentrating photovoltaic systems for façade-integration were studied and optimised for maximum annual electricity production. The optimisation was based on measured short-circuit currents versus solar altitude. Measurements were performed outdoors and in a solar simulator. It was found that the use of 3X parabolic reflectors increases the annual electricity production by more than 40%. High solar reflectance is crucial to system performance but by using a low-angle scattering reflector, the fill-factor and power are increased due to a more even irradiance on the modules. Long-term system performance depends on the durability of the components. The optical properties and degradation of reflector materials were assessed using spectrophotometry, angular resolved scatterometry, Fresnel modelling, optical microscopy, and surface profilometry before and after ageing. The degradation of reflectors was found to be strongly dependent on material composition and environmental conditions. Back surface mirrors, all-metal reflectors, and polymer-metal laminates degraded in different ways, and therefore accelerated ageing must be tailored for testing of different types of reflector materials. However, new types of reflector laminates showed a potential for increasing the cost-effectiveness of low-concentrating solar energy systems.

This thesis discusses and develops one approach to solve parabolic partial differential equations with random input data. The stochastic problem is firstly transformed into a parametrized one by using finite dimensional noise assumption and the truncated Karhunen-Loeve expansion. The approach, Monte Carlo discontinuous Galerkin (MCDG) method, randomly generates $M$ realizations of uncertain coefficients and approximates the expected value of the solution by averaging M numerical solutions. This approach is applied to two numerical examples. The first example is a two-dimensional parabolic partial differential equation with random convection term and the second example is a benchmark problem coupling flow and transport equations. I first apply polynomial kernel principal component analysis of second order to generate M realizations of random permeability fields. They are used to obtain M realizations of random convection term computed from solving the flow equation. Using this approach, I solve the transport equation M times corresponding to M velocity realizations. The MCDG solution spreads toward the whole domain from the initial location and the contaminant does not leave the initial location completely as time elapses. The results show that MCDG solution is realistic, because it takes the uncertainty in velocity fields into consideration. Besides, in order to correct overshoot and undershoot solutions caused by the high level of oscillation in random velocity realizations, I solve the transport equation on meshes of finer resolution than of the permeability, and use a slope limiter as well as lower and upper bound constraints to address this difficulty. Finally, future work is proposed.

We try to explain the mathematical theory of thin liquid film evolution. We start with introducing physical processes in which thin film evolution plays an important role. Derivation of the classical thin film equation and existing mathematical theory in the literature are also introduced. To explain the thin film evolution we derive a new family of degenerate parabolic equations. We prove results on existence, uniqueness, long time behavior, regularity and support properties of solutions for this equation. At the end of the thesis we consider the classical thin film Cauchy problem on the whole real line for which we use asymptotic equipartition to show H^1(R) convergence of solutions to the unique self-similar solution.

A class of Lagrange-Newton-SQP methods is investigated for optimal control problems governed by semilinear parabolic initial- boundary value problems. Distributed and boundary controls are given, restricted by pointwise upper and lower bounds. The convergence of the method is discussed in appropriate Banach spaces. Based on a weak second order sufficient optimality condition for the reference solution, local quadratic convergence is proved. The proof is based on the theory of Newton methods for generalized equations in Banach spaces.