Quantum systems with a finite Hilbert space, where position x and momen- tum p take values in Z(d) (integers modulo d), are studied. An analytic representation of finite quantum systems is considered. Quantum states are represented by analytic functions on a torus. This function has exactly d zeros, which define uniquely the quantum state. The analytic function of a state can be constructed using its zeros. As the system evolves in time, the d zeros follow d paths on the torus. Examples of the paths ³n(t) of the zeros, for various Hamiltonians, are given. In addition, for given paths ³n(t) of the d zeros, the Hamiltonian is calculated. Furthermore, periodic finite quantum systems are considered. Special cases where M of the zeros follow the same path are also studied, and general ideas are demonstrated with several ex- amples. Examples of the path with multiplicity M = 1; 2; 3; 4; 5 are given. It is evidenced within the study that a small perturbation of the initial values of the zeros splits a path with multiplicity M into M different paths.

published_or_final_version / Chemistry / Doctoral / Doctor of Philosophy

published_or_final_version / Chemistry / Master / Master of Science

PCBs are persistent organic pollutants that have intentionally and unintentionally (through contamination) been added to mineral insulating oil to improve its insulating and cooling properties within electrical equipment. The Stockholm Convention on Persistent Organic Pollutants (2001) orders the eradication of PCBs from use by 2025. Sodium based dechlorination is a PCB destruction process that is non-thermal, relatively cost effective and allows for the recovery of a reusable end product. A comparative benefits study, contained in chapter 2 of this dissertation, describes the increased financial and environmental impact associated with incinerating large volumes of PCB contaminated mineral oil. The results of the comparative analysis indicate a cost ratio of 1: 2.5, in favour of sodium dechlorination. In addition to the financial benefit, the sodium based PCB dechlorination process is versatile and can be either batched or skid mounted and is typically combined with an oil regeneration step, allowing for transformers to be treated onsite and whilst energised. Eskom is currently considering obtaining the mobile dechlorination unit for the purpose of conducting dechlorination and regeneration on its PCB contaminated transformers while energised. Mineral insulating oil is considered a strategic asset within most industries. Eskom uses mainly uninhibited mineral oil in its older transformers and the effects of PCB dechlorination on the natural inhibitor content of the oil is uncertain. The objective of this study was to investigate the effects of sodium dechlorination on the oxidation stability and thereby indirectly the natural inhibitor content of uninhibited naphthenic based mineral oil. The study involved the dechlorination, regeneration and subsequent chemical analyses of PCB contaminated oils in the PCB ranges <50ppm, 50 to 500 ppm and >500 ppm as stipulated by the Stockholm Convention on Persistent Organic Pollutants (2001). The study confirmed the reduction in oxidation stability and thereby the natural inhibitor content of the mineral oil after sodium dechlorination. Based on the results obtain a preliminary algorithm was established to predict the reduction in oxidation stability after sodium dechlorination, as a function of the PCB concentration prior to dechlorination. This will provide an indirect indication of the rate of natural inhibitor depletion of the oil, based on its exposure to the sodium dechlorination reagents and process conditions. / Thesis (M.Sc.Eng.)-University of KwaZulu-Natal, Durban, 2010.

In this thesis we extend a method of Hall $[30, 34]$ which he used to show the existence of large gaps between consecutive zeros, on the critical line, of the Riemann zeta-function $zeta(s)$. Our modification involves introducing an "amplifier" and enables us to show the existence of gaps between consecutive zeros, on the critical line at height $T,$ of $zeta(s)$ of length at least $2.766 x (2pi/log{T})$. To handle some integral-calculations, we use the article $[44]$ by Hughes and Young. Also, we show that Hall's strategy can be applied not only to $zeta(s),$ but also to Dirichlet $L$-functions $L(s,chi),$ where $chi$ is a primitive Dirichlet character. This also enables us to use stronger integral-results, the article $[14]$ by Conrey, Iwaniec and Soundararajan is used. An unconditional result here about large gaps between consecutive zeros, on the critical line, of some Dirichlet $L$-functions $L(s,chi),$ with $chi$ being an even primitive Dirichlet character, is found. However, we will need to use the Generalised Riemann Hypothesis to make sense of the average gap-length between such zeros. Then the gaps, whose existence we show, have a length of at least $3.54$ times the average.

This thesis investigates the topological reconstruction problem, which is inspired by the reconstruction conjecture in graph theory. We ask how much information about a topological space can be recovered from the homeomorphism types of its point-complement subspaces. If the whole space can be recovered up to homeomorphism, it is called reconstructible. In the first part of this thesis, we investigate under which conditions compact spaces are reconstructible. It is shown that a non-reconstructible compact metrizable space must contain a dense collection of 1-point components. In particular, all metrizable continua are reconstructible. On the other hand, any first-countable compactification of countably many copies of the Cantor set is non-reconstructible, and so are all compact metrizable h-homogeneous spaces with a dense collection of 1-point components. We then investigate which non-compact locally compact spaces are reconstructible. Our main technical result is a framework for the reconstruction of spaces with a maximal finite compactification. We show that Euclidean spaces &reals;ⁿ and all ordinals are reconstructible. In the second part, we show that it is independent of ZFC whether the Stone-&Ccaron;ech remainder of the integers, &omega;&ast;, is reconstructible. Further, the property of being a normal space is consistently non-reconstructible. Under the Continuum Hypothesis, the compact Hausdorff space &omega;&ast; has a non-normal reconstruction, namely the space &omega;&ast;&bsol;&lcub;p&rcub; for a P-point p of &omega;&ast;. More generally, the existence of an uncountable cardinal &kappa; satisfying &kappa; = &kappa;^&lt;&kappa; implies that there is a normal space with a non-normal reconstruction. The final chapter discusses the Stone-&Ccaron;ech compactification and the Stone-&Ccaron;ech remainder of spaces &omega;&ast;&bsol;&lcub;x&rcub;. Assuming the Continuum Hypothesis, we show that for every point x of &omega;&ast;, the Stone-&Ccaron;ech remainder of &omega;&ast;&lcub;x&rcub; is an &omega;₂-Parovi&ccaron;enko space of cardinality 2^{2^c} which admits a family of 2^c disjoint open sets. This implies that under 2^c = &omega;₂, the Stone-&Ccaron;ech remainders of &omega;&ast;&bsol;&lcub;x&rcub; are all homeomorphic, regardless of which point x gets removed.

This study aims to describe how researchers may incorporate four different generalisation methods (working hypothesis, analytic generalisation, critical case and naturalistic generalisation) to aid them in properly generaliseing from case study results. It was found that critical cases can be used in combination with working hypothesis to determine the falsifiability of the hypotheses. Working hypothesis and analytic generalisation complement each other by making it possible for researchers to generalise to both theory and new hypotheses (i.e. both inductively and deductively). Lastly, naturalistic and analytic generalisation enables a double-ended generalisation, where both the reader and the scientist themself generalise to specific situations and over-all theory, respectively.

<p>The central idea of this dissertation is to interpret certain invariants constructed from Laplace spectral data on a compact Riemannian manifold as regularized integrals of closed differential forms on the space of Riemannian metrics, or more generally on a space of metrics on a vector bundle. We apply this idea to both the Ray-Singer analytic torsion</p><p>and the eta invariant, explaining their dependence on the metric used to define them with a Stokes' theorem argument. We also introduce analytic multi-torsion, a generalization of analytic torsion, in the context of certain manifolds with local product structure; we prove that it is metric independent in a suitable sense.</p> / Dissertation

Performance measurement of Non-Profit Organizations (NPOs) is of increasing importance for aid agencies, policy-makers and donors. A widely used benchmark for measuring the efficiency of NPOs is the overhead cost ratio, consisting of the total money spent on administration and fundraising relative to the budget. Donors generally favor a lower overhead cost ratio as it ensures that more Money directly reaches beneficiaries. Unlike fundraising expenses, administrative costs do not contribute to advertising the actions of an NPO even though they account for a significant proportion of overhead cost. Reducing administrative expenses is a logical consequence from a financial viewpoint, but might negatively affect NPOs through the resulting administrative capacities. This phenomenon is known as "Nonprofit Starvation Cycle" This work provides an analytical framework for analyzing NPO decision making concerning administrative costs. The paper provides answers to important research questions on the optimal level of administrative spending, the influencing factors and the effects of available information on NPOs. The research shows that focusing on financial performance measurements can result in reduced utility created for NPOs. Less transparency often leads to increased utility for NPOs, but more transparency can increase NPOs' utility if the information available exceeds a certain threshold. Fluctuating donations are challenging for NPOs' planning and may Impact administrative capacities negatively.

The phenomenon of panic is one of the most unedifying experiences to inflict the human condition. It is a globally-recognised problem regularly encountered in psychotherapeutic practice. Whilst it is thought that distressing psychological and social (‘psychosocial’) problems might help account for this experience, the precise role they play - particularly in first onset - remains difficult to fathom. For example, whilst there is evidence to suggest that stress related to an individual’s family and work life, marital circumstances, age and gender appear linked with initial episodes of panic, these and many associated stressors people endure remain largely under-researched. Following an inquiry aim that recognises the social construction of reality, this research offers an insight into my first experience of panic attack (my being both a panic sufferer and psychotherapist). The aim was to identify an ‘invisible cord’ (e.g. a series of causally linked stressful life events) related to my panic. These events are typically thought to be found in the twelve months prior to first onset and hold important clues to an individual’s recovery. Hence my research question was: ‘What sense can be made of the invisible cord of events leading to my first experience of panic attack’? Using analytical autoethnographic methods to guide this research, significant personal events were discovered and are presented here in the findings. The earliest events uncovered would stretch back far longer than twelve months; with a series of five scenarios plotted from childhood to my mid-forties. To ensure that this research remained an exercise in critical thinking, each event was then examined alongside broader psychosocial theory and frameworks; offering a connected analysis of this first attack and contingent factors. A summary follows, ‘pulling together’ aspects of this undertaking and offering implications for practice. For example, having only made visible elements of my stressful cord by means of the analytical methods at my disposal (including use of collage and timelines) I suggest that such tools might routinely help other panic sufferers in retracing their past. Equally, in learning that my (often confused) Christian faith was implicated in this panic, I advance that we, as therapists, must remain vigilant to matters of client spirituality: noting that traditional forms of religious guidance are receding in an increasingly sceptical UK society. The thesis concludes with a personal reflection that aims to facilitate a deeper understanding of my research journey.