Investigations have been carried out via Monte Carlo simulation of simple, inhomogeneous fluids into two important quantities of colloidal systems; the depletion force and the disjoin'': ing pressure. Simulations of a hard-sphere solvent confined to the annular wedge formed between a spherical hard colloid and a planar hard wall were performed in order to shed light on the recently discovered disagreement of several results for the depletion force in the nanD-colloidal regime at solvent density pa3 > 0.6. Emphasis is placed on attempting to understand the limits of validity in terms of colloid size for the Derjaguin approximation applied to depletion forces and fundamental-measures-theory density functional theory (FMT-DFT), and the manner in which the depletion force scales between these two results at intermediate colloid sizes. The depletion force was evaluated via an exact statistical mechanical sum rule requiring only knowledge of the integral of the one-body density of solvent at the planar hard wall from the apex of the wedge to a large distance from the colloid. Simulations were performed for a colloid/solvent size ratio of 8 = 20 for several colloid-wall separations, h, between physical contact and the hard-sphere solvent diameter, a, at pa3 = 0.764, the results for the depletion force appearing to be consistent with a recently proposed theoretic model suggesting a (curiously non-analytic) 8-1/ 2 correction to the linear scaling behaviour of the depletion force with colloid size between the FMT-DFT and Derjaguin results, with the Derjaguin result valid in the large colloidal limit 8 --t 00 and FMT-DFT only as colloid size approaches solvent size. Further simulations, restricted to h = a for 8 = 10, 30, 50 and 100 though reveal that at least for this special separation 8-1/ 2 scaling does not hold, suggesting that to confirm scaling behaviour requires simulations over the entire range 0 ~ h ~ a for several values of 8. The disjoining pressure profile has been simulated through the three-phase contact line formed between the liquid-vapour interface of a square-well fluid at bulk liquid-vapour coexistence and a planar, square-well wall for three different depths of the wall-fluid potential. The disjoining pressure is found to follow a smooth, downward curve across the contact line that is well fit by a Gaussian. The simulation method used to make these disjoining pressure measurements has been validated using a statistical mechanical sum rule linking the integral of the disjoining pres- . . sure across the contact line to the liquid-vapour surface tension and macroscopic Young's contact angle, both measured from the interface far from three-phase contact.

Prospect theory, which proposes both reference-dependent utilityand probability weighting, has proved to be the most successful ofseveral approaches to generalising nonnative expected utility theory. It has been widely adopted in the social sciences to explain risk behaviour, with applications ranging from infrahuman, pre-industrial, financial, corporate and political decision-making. This wide, and growing, range of applications demands effective econometric analysis ofthe prospect theoretical framework. A major advance has been Prelec's approach using a weak consistency condition known as compound invariance (Cn as an axiomatic basis for developing exponential functions that describe probability weighting. Combined .with a standardly adopted constant relative risk aversion (CRRA) utility function this leads to a CI+CRRA model. Prelec proposl?s two versions ofthis model, one in which probability weighting is described by two (a probability sensitivity and a probability attractiveness) parameter and a one-parameter version in which the probability attractiveness parameter is axiomatically eliminated. One mathematical convenience ofthis version ofthe model is that it may be linearised via a double log transfonnation to provide risk decision descriptive parameters from the plot's intercept ~d gradient of certainty equivalence ~aluations. The research uses standard laboratory gambles (under - , varied incentive and problem presentation conditions) as the basis oftesting this model and the predictions ofprospect theory relating to risk attitude, pattern ofprobability weighting, gain-loss risk attitude reflection, four-fold inversion ofrisk attitude, loss aversion and framing effects. The research, involving over 12,000 individual prospect valuations generally finds strongly for prospect theory predictions. However, the two-parameter version of the CI+CRRA model performs far better as a predictor ofsubjects' prospect valuation than the one-parameter version. This is largely due to the increasing subadditivity ofvaluation as outcome value increases. Consequently the results draw into question the multiplicative assumption in prospect theory that outcome value and probability weighting are psychometrically independent. The dissertation concludes with considerations on how the model might be developed and applied to ecological risk decision-making studies in the future.

The thesis first examines conditions on topological spaces - especially dense normality and kappa normality - that are known to be similar to but weaker than classical normality, and it extends the range of examples thaUllustrate them. Arising from this study, general methodologies are then developed and applied that permit the determination of whether a range of topological spaces - generated in specific ways - satisfy such conditions. Identification and axiomatising of a central concept, closely related to the classical notion of proximity and here called a 'separator', leads on to the study of topologies defined through separators and to the transfer of techniques from the initial specific examples to the general setting. It is established that these techniques are particularly effective when the underlying set carries a vector space structure; in particular, in this casea proof or disproof of dense normality - which is traditionally a relatively difficult problem - is easily obtained. Amore general view is derived of how to determine near-normality conditions for a wide variety of topological spaces. Amongst the by-products of the investigation is an iterative procedure of some independent interest (here termed 'echo sequence') that seeks to determine whether, in a given pair of disjoint closed sets, there are subsets on whkh the difficulty of separating them by neighbourhoods, as reqiJired by normality conditions, focuses. Results are obtained on the circumstances in which this focus of difficulty is empty: in which case, the determination of normality is simplified.

The study of the sandwich classification theorem of unitary hyperbolic groups over commutative form ring (R, A) (in the sense of Bak) is naturally inspired by the sandwich classification theorem of general linear groups, which is initialed by Bak in a manuscript in 1967. In Chapter 1, we briefly review the history of the developing of normal and subnormal structure problems in the setting of general linear group and unitary group, as well as the co~nectionwith other problems. The proofs of sandwich. classification theorem and the structure theorem of subnormal subgroups of general linear group are given in details in Chapter 2. The brief introduction of Bak's form ring and unitary groups, and their fundamental properties are reproduced in Chapter 3. In chapter 4, we study the main theorem of the thesis, i.e., sandwich classification theorem for unitary groups. Furthermore the properties of two important classes of subgroups are studied, i.e., the subnormal subgroups of unitary. groups and mixed commutator subgroups of unitary groups. The final Chapter contains a weaker sandwich theorem on non-stable unitary groups and its proof.

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Accretion discs are fluid-dynamical entities which surround many black holes. Observations reveal that these systems exhibit variability on a range of time scales. This thesis investigates phenomena occurring in black-hole accretion discs which are likely to induce high-frequency quasi-periodic variability. Two classes of pseudo-relativistic theoretical models are investigated. The first is based on the stability of transonic accretion flows and its connection to a disc instability that takes the form of propagating waves (viscous overstability). The time-dependent study looks at the conditions under which the transition between subsonic disc-like accretion, which occurs at large radii, and the supersonic flow characteristic of the immediate vicinity of the black hole is stable. In agreement with previous findings, results indicate that the system reaches a steady state for low viscosity. Above that threshold the transonic solutions are unstable to viscous overstability. The overstable inertial-acoustic waves appear to be excited near the maximum of the epicyclic frequency and are global in the sense that their frequency is maintained for a wide range of radii. The second class of models looks at accretion-disc oscillations which are trapped due to the non-monotonic variation of the epicyclic frequency in relativistic flows. In particular, it focuses on inertial waves trapped below the maximum of the epicyclic frequency which are excited in deformed, warped or eccentric, discs. The excitation mechanism involves a non-linear coupling between the global deformation, an intermediate wave and the inertial mode and results, under a variety of conditions, in growth of the latter. Excitationis only effective when global deformations are capable of reaching the inner disc with non-negligible amplitude. With that in mind, the conditions favourable to the propagation of warped and eccentric modes from the outer to the inner regions are analysed. Another aspect that is taken into account is the influence of a transonic background, ignored in the coupling calculations, on the propagation of modes in the disc. It is found that, under certain conditions, inertial waves may be severely affected or destroyed in this background. On the other hand, results indicate that the decay rate of inertial waves due to the presence of the radial inflow is small in sufficiently thin discs. In this case, the coupling mechanism can still work to excite trapped inertial modes.

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