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Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: The aim of this thesis is to study some methods used in solving exponential Diophan- tine equations. There is no generic method or algorithm that can be used in solving all Diophantine equations. The main focus for our study will be solving the exponential Dio- phantine equations using the modular approach and the linear forms in two logarithms approach. / AFRIKAANSE OPSOMMING: Die doel van hierdie tesis is om sommige metodes te bestudeer om sekere Diophantiese vergelykings op te los. Daar is geen metode wat alle Diophantiese vergelykings kan oplos nie. Die fokus van os studie is hoofsaaklik om eksponensiele Diophantiese vergelykings op te los met die modul^ere metode en met die metode van line^ere vorms in twee logaritmes.

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Deterministic diffusion is studied in simple, parameter-dependent dynamical systems. The diffusion coefficient is often a fractal function of the control parameter, exhibiting regions of scaling and self-similarity. Firstly, the concepts of chaos and deterministic diffusion are introduced in the context of dynamical systems. The link between deterministic diffusion and physical diffusion is made via random walk theory. Secondly, parameter-dependent diffusion coefficients are analytically derived by solving the Taylor-Green-Kubo formula. This is done via a recursion relation solution of fractal 'generalised Takagi functions'. This method is applied to simple one-dimensional maps and for the first time worked out fully analytically. The fractal parameter dependence of the diffusion coefficient is explained via Markov partitions. Linear parameter dependence is observed which in some cases is due to ergodicity breaking. However, other cases are due to a previously unobserved phenomenon called the 'dominating-branch' effect. A numerical investigation of the two-dimensional 'sawtooth map' yields evidence for a possible fractal structure. Thirdly, a study of different techniques for approximating the diffusion coefficient of a parameter-dependent dynamical system is then performed. The practicability of these methods, as well as their capability in exposing a fractal structure is compared. Fourthly, an analytical investigation into the dependence of the diffusion coefficient on the size and position of the escape holes is then undertaken. It is shown that varying the position has a strong effect on diffusion, whilst the asymptotic regime of small-hole size is dependent on the limiting behaviour of the escape holes. Finally, an exploration of a method which involves evaluating the zeros of a system's dynamical zeta function via the weighted Milnor-Thurston kneading determinant is performed. It is shown how to relate the diffusion coefficient to a zero of the dynamical zeta function before analytically deriving the diffusion coefficient via the kneading determinant.

Previous work on two-treatment comparisons has shown that the use of optimal response-adaptive randomisation with group sequential analysis can allocate more patients to the better-performing treatment while preserving the overall type I error rate. The sequence of test statistics for this adaptive design asymptotically satisfi es the canonical joint distribution. The overall type I error rate can be controlled by utilising the error-spending approach. However, previous work focused on immediate responses. The application of the adaptive design to censored survival responses is investigated and different optimal response-adaptive randomised procedures compared. For a maximum duration trial, the information level at the fi nal look is usually unpredictable. An approximate information time is defi ned. Several treatments are often compared in a clinical trial nowadays. The adaptive design generalised to multi-arm clinical trials is studied. First, a global test is considered. The joint distribution of the sequence of test statistics no longer has the canonical distribution. However, the joint distribution can be derived, since the test statistic is a quadratic form of independent normal variables. Existing critical boundaries are based on normal responses and known variances with equal allocation and equal increments in information. Our results show that these boundaries can be used approximately for designs with other types of responses, unequal variances or unbalanced allocation. If the global null hypothesis is rejected, then pairwise comparisons are conducted at the current and subsequent looks to investigate which treatment effects differ. This is an analogue of Fisher's least signi cant difference method that can control the family-wise error rate. The adaptive design can target any optimal allocation to achieve some optimality criterion, and allows dropping of inferior treatments at interim looks, which can be unequally spaced in information time. Optimal allocation proportions after dropping arms are described. The power is not adversely affected by unbalanced allocation.