This thesis investigates the representation of a stochastic epidemic process as a directed random graph; we use this representation to impute the missing information in final size data to make Bayesian statistical inference about the model parameters using MCMC techniques. The directed random graph representation is analysed, in particular its behaviour under the condition that the epidemic has a given final size. This is used to construct efficient updates for MCMC algorithms. The MCMC method is extended to include two-level mixing models and two-type models, with a general framework given for an arbitrary number of levels and types. Partially observed epidemics, that is, where the number of susceptibles is unknown or where only a subset of the population is observed, are analysed. The method is applied to several well known data sets and comparisons are made with previous results. Finally, the method is applied to data of an outbreak of Equine Influenza (H3N8) at Newmarket in 2003, with a comparison to another analysis of the same data. Practical issues of implementing the method are discussed and are overcome using parallel computing (GNU OpenMP) and arbitrary precision arithmetic (GNU MPFR).

In this thesis we treat three problems from the theory and applications of random walks. The first question we tackle is from the theory of the optimal stopping of random walks. We solve the infinite-horizon optimal stopping problem for a class of reward functions admitting a representation introduced in Boyarchenko and Levendorskii [1], and obtain closed expressions for the expected reward and optimal stopping time. Our methodology is a generalization of an early paper by Darling et al. [2] and is based on probabilistic techniques: in particular a path decomposition related to the Wiener-Hopf factorization. Examples from the literature and perturbations are treated to demonstrate the flexibility of our approach. The second question is related to the path structure of lattice random walks. We obtain the exact asymptotics of the variance of the self- intersection local time Vn which counts the number of times the paths of a random walk intersect themselves. Our approach extends and improves upon that of Bolthausen [3], by making use of complex power series. In particular we state and prove a complex Tauberian lemma, which avoids the assumption of monotonicity present in the classical Tauberian theorem. While a bound of order 0(n2) has previously been claimed in the literature ([3], [4]) we argue that existing methods only show the tipper bound O(n2 log n), unless extra conditions are imposed to ensure monotonicity of the underlying sequence. Using the complex Tauberian approach we show that Var (Vn ) Cn2, thus settling a long-standing misunderstanding. Finally, in the last chapter, we prove a functional central limit theorem for one-dimensional random walk in random scenery, a result conjectured in 1979 by Kesten and Spitzer [5]. Essentially random walk in random scenery is the process defined by the partial suins of a collection of random variables (the random scenery), sampled by a random walk. We show that for Z-valued random walk attracted to the symmetric Cauchy law, and centered random scenery with second moments, a functional central limit theorem holds, thus proving the Kesten and Spitzer [5] conjecture which had remained open since 1979. Our proof makes use of tile asymptotic results obtained in the Chapter 3.

In this thesis, three stochastic epidemic models for intervention for emerging diseases are considered. The models are variants of real-time, responsive intervention, based upon observing diagnosed cases and targeting intervention towards individuals they have infected or are likely to have infected, be they housemates or named contacts. These models are: (i) a local tracing model for a disease spreading amongst a community of households, wherein intervention (vaccination and/or isolation) is directed towards housemates of diagnosed individuals, (ii) a contact tracing model for a disease spreading amongst a homogeneously-mixing population, with isolation of traced contacts of a diagnosed individual, (iii) a local tracing and contact tracing model for a disease spreading amongst a community of households, with intervention directed towards housemates of both diagnosed and traced individuals. These are quantified by deriving threshold parameters that determine whether the disease will infect a few individuals or a sizeable proportion of the population, as well as probabilities for such events occurring.

We consider various portfolio optimization problems when the stock prices follow jump-diusion processes. In the first part the classical optimal consumption-investment problem is considered. The investor's goal is to maximize utility from consumption and terminal wealth over a finite investment horizon. We present results that modify and extend the duality approach that can be found in Kramkov and Schachermayer (1999). The central result is that the optimal trading strategy and optimal equivalent martingale measure can be determined as a solution to a system of non-linear equations. In another problem a benchmark process is introduced, which the investor tries to outperform. The benchmark can either be a generic jump-diusion process or, as a special case, a wealth process of a trading strategy. Similar techniques as in the first part of the thesis can be applied to reach a solution. In the special case that the benchmark is a wealth process, the solution can be deduced from the first part's consumption-investment problem via a transform of the parameters. The benchmark problem presented here gives a dierent approach to benchmarks as in, for instance, Browne (1999b) or Pra et al. (2004). It is also, as far as the author is aware, the first time that martingale methods are employed for this kind of problem. As a side effect of our analysis some interesting relationships to Platen's benchmark approach (cf. Platen (2006)) and change of numeraire techniques (cf. German et al. (1995)) can be observed. In the final part of the thesis the set of trading strategies in the previous two problems are restricted to constraints. These constraints are, for example, a prohibition of shortselling or the restriction on the number of assets. Conditions are provided under which a solution to the two problems can still be found. This extends the work of Cvitanic and Karatzas (1993) to jump diffusions where the initial market set-up is incomplete.

Carbon dioxide dissolution into water is a ubiquitous chemical process on earth, and having a full understanding of this process is becoming ever more important as we seek to understand the consequences of 250 years of exponentially-increasing anthropogenic C02 emissions to the atmosphere since the start of the Industrial Revolution. We examine the dissolution of C02 into water in a number of contexts. First, we analyse what happens to a range of chemical species dissolved in water following an injection of additional C02. We consider the well-mixed problem, and use the method of matched asymptotic expansions to obtain new expressions for the changes in the species' concentrations with time, the new final chemical equilibrium, and the time scales over which this equilibrium is reached, as functions of time, the parameters and the initial condition. These results can be used to help predict the changes in the pH and concentrations of dissolved carbonic species that will occur in the oceans as a result of anthropogenic C02 emissions, and in saline aquifer formations after pumping C02 deep underground. Second, we consider what happens deep underground in a saline aquifer when C02 has been pumped in, spreads through the pore space, and dissolves into the resident water, when advection, diffusion, and chemical reaction have varying levels of relative importance. We examine the length scales over which the dissolved C02 will spread out through an individual pore, ahead of a spreading drop of C02, and the concentrations of the different chemical species within the pore, in the steady-state case. Finally, some experiments have been carried out to investigate the effect of an injection of gaseous C02 on the chemical composition and pH of a saturated limestone aquifer formation. As the C02 enters the soil, it dissolves into the water, and we model the changes in the chemical composition of the water/limestone mixture with time.

This thesis determines the response of Left-Handed Media (LHM) to surface effects. A LHM half-space with a roughened interface, modelled by a graded index boundary, is shown to give rise to an analytical solution for the propagation of electromagnetic radiation through this inhomogeneous layer. Significant field localization is generated within the layer, caused by the coherent superposition of evanescent waves. The localization is shown to greatly deteriorate transmission when losses are present. The addition of a second interface to the LHM, creating a perfect lens configuration, allows for the exploration of evanescent mode propagation through a perfect lens with roughened boundaries. The effects of the field localisations at the boundaries serves to diminish the resolving capability of the lens. Specifically the layers produce an effect that is qualitatively similar to nonlinearly enhanced dissipation. Ray-optics is used to analyse negative refraction through a roughened interface, prescribed by Gaussian statistics. This shows that rays can focus at smaller distances from the interface due to the negative refractive effects. Moreover, a new reflection mechanism is shown to exist for LHM. Consequently an impedance matched configuration involving LHM (such as the perfect lens) with a roughened interface can still display reflection. A physical-optics approach is used to determine the mean intensity and fluctuations of a wave passing into a half-space of LHM through a roughened interface in two ways. Firstly through the perturbation analysis of Rice theory which shows that the scattered field evolves from a real Gaussian process near the surface into a complex Gaussian process as distance into the second media increases. Secondly through large-scale Monte-Carlo simulations that show that illuminating a roughened interface between air and a LHM produces a regime for enhanced focussing of light close to the boundary, generating caustics that are brighter, fluctuate more, and cause Gaussian speckle at distances closer to the interface than in right-handed matter.

Random graphs and networks are of great importance in any fields including mathematics, computer science, statistics, biology and sociology. This research aims to develop statistical theory and methods of statistical inference for random graphs in novel directions. A major strand of the research is the development of conditional goodness-of-fit tests for random graph models and for random block graph models. On the theoretical side, this entails proving a new conditional central limit theorem for a certain graph statistics, which are closely related to the number of two-stars and the number of triangles, and where the conditioning is on the number of edges in the graph. A second strand of the research is to develop composite likelihood methods for estimation of the parameters in exponential random graph models. Composite likelihood methods based on edge data have previously been widely used. A novel contribution of the thesis is the development of composite likelihood methods based on more complicated data structures. The goals of this PhD thesis also include testing the numerical performance of the novel methods in extensive simulation studies and through applications to real graphical data sets.

A research project submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in partial fulfilment of the requirements for the degree of Masters in Science Education. 2015 / The Curriculum Assessment Policy Statements (CAPS) re-introduced some mathematics topics such as probability. An immediate effect of this re-introduction is that most teachers and learners were not well equipped to deal with this topic. To at least begin addressing this problem, this research explored the errors and misconceptions that learners have when solving probability problems using different representations. The study draws from Nesher’s (1987) theory of errors and misconceptions as well as Sfard’s (2007) theory of commognition in explaining representations and prevalence of errors in learning mathematics. Twenty two Grade 10 learners wrote probability tasks after which their scripts were analysed for errors. Six of those learners were interviewed on the errors they made in solving probability problems with different representations. The findings reveal five main categories of errors and misconceptions. These are: (1) difficulty with construction of visual representations; (2) improper distinction between simple and compound events; (3) application of inappropriate routines; (4) errors associated with familiarity; and, (5) misinterpreted language. The findings also showed that inappropriate choice of representations was caused by misinterpretation of probability terminology. Concurring with Zahner and Corter (2010) the researcher found that learners made a multitude of errors if they constructed and used their own probability representations. Further, learners committed fewer errors where the task provided representations. Results also show that learners were most confident in using tree diagram representations even though they struggled to construct them from scratch. Most learners avoided Venn diagrams, outcome listings and matrix representations even though they would be the most useful in answering the questions. As a result many errors and misconceptions resulted when learners tried to use these representations. The study recommends that teachers take time to discuss probability terminology and the use of different representations with their learners. This promotes both the conceptual and procedural knowledge of probability. Also, to reduce learners’ errors and misconceptions on the topic, teachers need to scaffold the construction of representations by providing partially constructed representations and gradually encourage learners to construct their own probability representations.

陳浩琛. / Thesis (M.A.)--香港中文大學硏究院哲學部. / Manuscript. / Includes bibliographical references: leaves 17-30 (3d. group) / Chen Haochen. / Thesis (M.A.)--Xianggang Zhong wen da xue yan jiu yuan zhe xue bu. / 序言 --- p.I-III / Chapter 第一部份 --- 歸納法的問題及其解答 / Chapter 第1章 --- 歸納法的問題 / Chapter 1.1 --- 歸納的跳躍 --- p.1-2 / Chapter 1.2 --- 科學方法及歸納推論 --- p.2-5 / Chapter 1.3 --- 休謨吊詭 --- p.5-6 / Chapter 第2章 --- 休謨吊詭的解答 / Chapter 2.1 --- 基本原則的進路 --- p.7-13 / Chapter 2.2 --- 自相支持的歸納論証的進路 --- p.13-21 / Chapter 2.3 --- 消解的進路 --- p.21-28 / Chapter 2.4 --- 日常語言分析的進路 --- p.29-34 / Chapter 2.5 --- 實效証立的進路 --- p.34-39 / Chapter 第二部份 --- 概率的形式系統及其語意解釋 / 概要 --- p.40-41 / Chapter 第3章 --- 概率的形式系統 / Chapter 3.1 --- 離散空間中之概率 --- p.42-46 / Chapter 3.2 --- 條件概率 --- p.46-49 / Chapter 3.3 --- 具努里試驗 --- p.50-51 / Chapter 3.4 --- 中間項 --- p.51-55 / Chapter 3.5 --- 大數律 --- p.55-58 / Chapter 3.6 --- 連續空間中之概率 --- p.58-64 / Chapter 第4章 --- 語意解釋的判準 --- p.65-67 / Chapter 第5章 --- 古典解釋 --- p.68-75 / Chapter 第6章 --- 頻率解釋 / Chapter 6.1 --- 頻率極限 --- p.76-78 / Chapter 6.2 --- 有限序列的頻率解釋 --- p.78-80 / Chapter 6.3 --- 頻率的極限是否存在 --- p.80-88 / Chapter 6.4 --- 趨同規則 --- p.88-97 / Chapter 6.5 --- 單個事件 --- p.97-101 / Chapter 第7章 --- 邏輯解釋 / Chapter 7.1 --- 邏輯解釋概要 --- p.102-105 / Chapter 7.2 --- 結構描述 --- p.105-110 / Chapter 7.3 --- 歸納邏輯(概率邏輯)的公理系統 --- p.111-116 / Chapter 7.4 --- 適當性條件 --- p.116-123 / Chapter 7.5 --- C－函數的決定 --- p.123-136 / Chapter 7.6 --- 邏輯概率在決定實際判斷上的應用 --- p.136-142 / 餘論 --- p.143-145 / 註目 --- p.I-XI / 附錄 --- p.XII-XVI / 參考書目 --- p.XVII-XXX

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