published_or_final_version / Mathematics / Master / Master of Philosophy

Quantum mechanics is a theory developed to explain both particle and wave-like properties of small matter such as light and electrons. The consequences of the theory can be counter-intuitive but lead to mathematical and physical theory rich in fascinating phenomena and challenging questions. This dissertation investigates the nature of quantum systems in Bernoulli distributed random potentials for systems on the one dimensional lattice {0, 1, ..., L, L+1} ⊂ Z in the large system limit L → ∞. For single particle systems, the behavior of the low energy states is shown to be approximated by systems where positive potential is replaced by infinite potential. The approximate shape of these states is described, the asymptotics of their eigenvalues are calculated in the large system limit L → ∞, and a Lifschitz tail estimate on the sparsity of low energy states is proven. For interacting multi-particle systems, a Lieb-Liniger model with Bernoulli distributed potential is studied in the Gross-Pitaevskii approximation. First, to investigate localization in these settings, a general inequality is proven to bound from below the support of the mean-field. The bound depends on the per particle energy, number of particles, and interaction strength. Then, the ground state for the one-dimensional lattice with Bernoulli potential is studied in the large system limit. Specifically, the case where the product of interaction strength and particle density is near zero is considered to investigate whether localization can be recovered.

The mean value engine model (MVEM) is a mathematical model derived from basic physical principles such as conservation of mass and energy equations. Although the MVEM is based on some simplified assumptions and time averaged combustion engine parameters, it models the engine with a reasonable approximation and gives a satisfactory amount of information about the physics of the fluid energy passing through an engine system. MVEM can predict an engine’s main external variables such as crankshaft speed and manifold pressure, and important internal variables, such as volumetric and thermal efficiencies. Usually, the differential equations used in MVEM will predict fuel film flow, manifold pressure, and crankshaft speed. Because of its simplicity and short simulation time, the MVEM is widely used for engine control development. A mean value engine based on mathematical and parametric equations has recently been developed in the new MapleSim software. The model consists of three main components: the throttle body, the manifold, and the engine. The new MVEM uses combinations of causal and acausal components along with lookup tables and parametric equations. Adjusting the parameters allows the model to be used for new engines of interest. The model is forward-looking and so benefits from both Maple’s powerful mathematical tool and Modelica’s modern equation-based language. A set of throttle angle and mass flow data is used to find the throttle angle function, and to validate the throttle mass flow rates obtained from the model and the experiment.

HIV/Aids Relative Survival and Mean Residual Life Analysis BY XINJIAN ZHANG Under the Direction of Gengsheng (Jeff) Qin and Ruiguang (Rick) Song ABSTRACT Generalized linear models with Poisson error were applied to investigate HIV/AIDS relative survival. Relative excess risk for death within 3 years after HIV/AIDS diagnosis was significantly higher for non-Hispanic blacks, American Indians and Hispanics compared with Whites. Excess hazard for death was also higher in men injection drug users compared with men who have sex with men (MSM). The relative excess hazard of old HIV/AIDS patients is significantly higher compared with younger patients. When CD4 increased, the relative excess hazard decreased; while with the increase of HIV viral load, the relative excess hazard decreased. This is the first study to use national wide data to examine the significance of HIV viral load as a determinant risk factor of disease progression after HIV infection; The mean residual lie needs to be further analyzed. INDEX WORDS: Human Immunodeficiency Virus (HIV), Acquired Immunodeficiency Syndrome (AIDS), Survival, Mean residual life (MRL).

Empowered by linear precoding and decoding, a spatially multiplexed multiple-input multiple-output (MIMO) system becomes a convenient framework to offer high data rate, diversity and interference management. While most of the current precoding/decoding designs have assumed perfect channel state information (CSI) at the receiver, and sometimes even at the transmitter, in this thesis we design the precoder and decoder with imperfect CSI at both the transmit and the receive sides, and investigate the joint impact of channel estimation errors and channel correlation on system structure and performance. The mean-square error (MSE) related performance metrics are used as the design criteria. We begin with the minimum total MSE precoding/decoding design for a single-user MIMO system assuming imperfect CSI at both ends. Here the CSI includes the channel estimate and channel correlation information. The structures of the optimum precoder and decoder are determined. Compared to the perfect CSI case, linear filters are added to the transceiver structure to improve system robustness against imperfect CSI. The effects of channel estimation error and channel correlation are quantified by simulations. With imperfect CSI at both ends, the exact capacity expression for a single-user MIMO channel is difficult to obtain. Instead, a tight capacity lower-bound is used for system design. The optimum structure of the transmit covariance matrix for the lower-bound has not been found in the existing literature. By transforming the transmitter design into a joint precoding/decoding design problem, we derive the expression of the optimum transmit covariance matrix. The close relationship between the maximum mutual information design and the minimum total MSE design is also discovered assuming imperfect CSI. For robust multiuser MIMO communications, minimum average sum MSE transceiver (precoder-decoder pairs) design problems are formulated for both the uplink and the downlink, assuming imperfect channel estimation and channel correlation at the base station (BS). We propose improved iterative algorithms based on the associated Karush-Kuhn-Tucker (KKT) conditions. Under the assumption of imperfect CSI, an uplink--downlink duality in average sum MSE is proved. As an alternative for the uplink optimization, a sequential semidefinite programming (SDP) method is proposed. Simulation results are provided to corroborate the analysis. / Thesis (Ph.D, Electrical & Computer Engineering) -- Queen's University, 2008-07-25 10:53:45.175

In this thesis we study three different problems: convex ancient solutions to the power-of-mean curvature flow; Sharp inequalities; regularity results in some applications of optimal transportation. The second chapter is devoted to the power-of-mean curvature flow; We prove some estimates for convex ancient solutions (the existence time for the solution starts from -\infty) to the power-of-mean curvature flow, when the power is strictly greater than \frac{1}{2}. As an application, we prove that in two dimension, the blow-down of an entire convex translating solution, namely u_{h}=\frac{1}{h}u(h^{\frac{1}{1+\alpha}}x), locally uniformly converges to \frac{1}{1+\alpha}|x|^{1+\alpha} as h\rightarrow\infty. The second application is that for generalized curve shortening flow (convex curve evolving in its normal direction with speed equal to a power of its curvature), if the convex compact ancient solution sweeps the whole space \mathbb{R}^{2}, it must be a shrinking circle. Otherwise the solution must be defined in a strip region. In the first section of the third chapter, we prove a one-parameter family of sharp conformally invariant integral inequalities for functions on the $n$-dimensional unit ball. As a limiting case, we obtain an inequality that generalizes Carleman's inequality for harmonic functions in the plane to poly-harmonic functions in higher dimensions. The second section represents joint work with Tobias Weth and Rupert Frank; the main result is that, one can always put a sharp remainder term on the righthand side of the sharp fractional sobolev inequality. In the first section of the final chapter, under some suitable condition, we prove that the solution to the principal-agent problem must be C^{1}. The proof is based on a perturbation argument. The second section represents joint work with Emanuel Indrei; the main result is that, under (A3S) condition on the cost and c-convexity condition on the domains, the free boundary in the optimal partial transport problem is C^{1,\alpha}.

In this thesis we study three different problems: convex ancient solutions to the power-of-mean curvature flow; Sharp inequalities; regularity results in some applications of optimal transportation. The second chapter is devoted to the power-of-mean curvature flow; We prove some estimates for convex ancient solutions (the existence time for the solution starts from -\infty) to the power-of-mean curvature flow, when the power is strictly greater than \frac{1}{2}. As an application, we prove that in two dimension, the blow-down of an entire convex translating solution, namely u_{h}=\frac{1}{h}u(h^{\frac{1}{1+\alpha}}x), locally uniformly converges to \frac{1}{1+\alpha}|x|^{1+\alpha} as h\rightarrow\infty. The second application is that for generalized curve shortening flow (convex curve evolving in its normal direction with speed equal to a power of its curvature), if the convex compact ancient solution sweeps the whole space \mathbb{R}^{2}, it must be a shrinking circle. Otherwise the solution must be defined in a strip region. In the first section of the third chapter, we prove a one-parameter family of sharp conformally invariant integral inequalities for functions on the $n$-dimensional unit ball. As a limiting case, we obtain an inequality that generalizes Carleman's inequality for harmonic functions in the plane to poly-harmonic functions in higher dimensions. The second section represents joint work with Tobias Weth and Rupert Frank; the main result is that, one can always put a sharp remainder term on the righthand side of the sharp fractional sobolev inequality. In the first section of the final chapter, under some suitable condition, we prove that the solution to the principal-agent problem must be C^{1}. The proof is based on a perturbation argument. The second section represents joint work with Emanuel Indrei; the main result is that, under (A3S) condition on the cost and c-convexity condition on the domains, the free boundary in the optimal partial transport problem is C^{1,\alpha}.

The UK has a commitment to reduce greenhouse gases by at least 80% from 1990 levels by 2050. This will see the proportion of energy generated in the UK from renewable resources such as wind, solar, marine and bio-fuels is increasing and likely to dominate the future energy market over the next few decades. However, it is unclear what effect future physical climate changes could have on the long term average energy output characteristics of individual renewable energy technologies that may dominate the low carbon energy technologies. It is also unclear how these changes to individual technologies will affect a diverse portfolio of electricity generation technologies. This thesis explores the influence of climate change on renewable electricity generation portfolios and energy security in the UK, with the aim of determining if climate change will affect renewable energy resource in such a way that may leave future low carbon generation portfolios sub-optimal. The research allows long term renewable resource variability to be reflected within models of the costs and risks associated with different electricity generation technologies and using Mean Variance Portfolio Theory (MVPT), it explores the influence of climate change on renewable energy portfolios and energy security in the UK. The scope of this study has a considerable range spanning from renewable resources through to the sensitivity of an optimal portfolio mix of generation technologies to climate change. In brief, the objectives were as follows: Characterise the variability of renewable energy resources and electricity generation output from renewable technology in the UK, in particular solar PV, on and offshore wind, for future climate scenarios for the 2050s and 2080s. Characterise the variability of electricity generation costs and explore the effect of climate change scenarios on generation costs and risk by examining the cost-risk balance of current and potential future low carbon electricity generation technology portfolios. The outcome saw distinctive changes in solar, wind, wave and hydro resource. The changes were largely negative, except in the case of solar, which increased. Levelised costs decreased for solar PV but increased for the technologies with negative resource changes. Evident changes in optimal portfolio mixes were observed and explored.

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Three different groups of materials were chosen to investigate the uniaxial compaction of particulate solids. Dentritic and cubic sodium chloride were chosen as plastically deforming, dicalcium phosphcte and sugar as fragmentary and styrocell, homopolymer and copolyrinier as non-compactable materials. The uniaxial compaction of the materials was continuously followed by measurement of the applied force, the force transmitted radially to the die wall and the upper punch displacement. The data obtained was presented in the form of Mohr circles, stress pathways (shear-mean compaction stress planes) and a three dimensional representation in mean compaction stress, shear stress and volume change. The yield loci evaluated from Mohr circles and shear-mean compaction stress relationships of compactable and non-compactable materials were found to be similar in shape. The unloading stress profiles were however more informative. All unloading shear-mean compaction stress curves of the compactable materials cross the mean compaction stress axis to give negative values of shear stress and reach a minimum value of ^t_min' which was material and compaction pressure dependent. The unloading curves of non-compactable materials gave approximately zero shear. The parameters evaluated from the characteristic stress profiles were correlated to the tensile strength and hardness of compacts. Mathematical expressions have been proposed for the shear-mean compaction stress relationships of the materials investigated. The materials were characterised before and after compaction in terms of specific surface area, porosity and mechanical strength of compacts with compaction pressure.