>Magister Scientiae - MSc / This research undertakes to investigate both long-term and short-term investor overreaction on the JSE Limited (JSE) over the period from 1 January 2002 to 31 December 2009. The period covers the restructuring and reform of the JSE in the early 2000s to the end of global financial market crisis in late 2008/2009, which can be regarded as a complete economic cycle. The performances of the winner and loser portfolios are evaluated by assessing their cumulative abnormal returns (CAR) over a 24-month holding period. The test results show no evidence of mean reversion for winner and loser portfolios formed based on prior returns of 12 months or less. However, test results show evidence of significant mean reversion for the winner and loser portfolios constructed based on their prior 24 months and 36 months returns. In addition, the study reveals that the mean reversion is more significant for longer-formation-period portfolios as well as for longer holding periods. The examination of the cumulative loser-winner spreads obtained from the contrarian portfolios based on the constituents’ prior 24 month and 36 month returns indicates that the contrarian returns increase for portfolios formed between 2004 and 2006, and declines thereafter towards the end of the examination period. The deterioration of contrarian returns coincides with the subprime mortgage crisis in 2007 and the subsequent global financial crisis in 2008. This evidence suggests that the degree of mean reversion on the JSE is positively correlated to the South African business cycle.

The Johannesburg Stock Exchange market was tested for the existence of the random walk hypothesis using All Share Index (ALSI) and time series data for the period between 2000 and 2011. The traditionally used methods, the unit root tests and autocorrelation test were employed first and they all confirmed that during the period under consideration, the JSE price index followed the random walk process. In addition, the ARIMA model was built and it was found that the ARIMA ( 1, 1, 1) was the model that best fitted the data in question. Furthermore, residual tests to help determine whether the residuals of the estimated equation show random walk process in the series were done. It was found that the ALSI resembles series that follow random walk hypothesis with strong evidence of RWH indicated in the conducted forecasting tests which showed vast variance between forecasted values and actual indicating little or no forecasting strength in the series. To further validate the findings in this research, the variance ratio test was conducted under heteroscedasticity and it also strongly corroborated that the existence of a random walk process cannot be rejected in the JSE. It was concluded that since the returns follow the random walk hypothesis, it can be said that JSE is efficient in the weak form level of the EMH and therefore opportunities of making excess returns based on out- performing the market is ruled out and is merely a game of chance. In other words, it will be of no use to choose stocks based on information about recent trends in stock prices.

Rotational-translational decoupling, a phenomenon commonly observed in supercooled liquids, has been a topic of great interest. Despite its prevalence, the underlying cause of this phenomenon, often attributed to dynamic heterogeneity, has not been conclusively elucidated. This thesis investigates and evaluates how dynamic heterogeneity may lead to this decoupling using simultaneous single-molecule rotational and translational measurements. In the experimental study, single molecule fluorescence imaging experiments are performed on the ideal probe N,N’-dipentyl-3,4,9,10-perylenedicarboximide in high molecular weight polystyrene near its glass transition temperature. A novel trajectory linking method based on hierarchical clustering is developed to facilitate single molecule tracking even in imaging data where specific molecules cannot be observed visually for a substantial number of frames. This linking algorithm then allows molecules to be localized over full movies, such that rotational and translational measurements can be compared over comparable timespans. The investigation of translational dynamics using such long trajectories, which was not previously achieved, reveals that both rotational-translational decoupling and translational enhancement persist on the single molecule level, supporting the hypothesis that temporally heterogeneous dynamics experienced by the probe molecules is a contributing factor in observed rotational-translational breakdown in both ensemble and single molecule studies. A tendency towards dynamical convergence between subgroups with fast and slow dynamics is observed, demonstrating temporal heterogeneity at the single molecule level. In comparison to rotational dynamics, translational dynamics was discovered to have a longer lifetime. Other key observations facilitated by the linked trajectory analysis include that apparent diffusion coefficient of probe molecules decreases with longer observation time, a finding inconsistent with normal diffusive behavior. To investigate the origin of this anomalous slowing in single molecule studies existing alongside the observed overall enhancement in translational motion, temporally heterogeneous models with multiple types of correlation were studied via simulations. The results emphasize the critical role that bias in translational and rotational measurements can play when investigating and observing dynamic heterogeneity, as nearly all models including dynamic heterogeneity show increasing diffusion coefficient with increasing number of dynamic environments explored. Strikingly, translational enhancement is evident in single molecule translational simulations even when slow dynamics are reinforced via positive correlation in the models. A comparison of the diffusion coefficient evolution between simulations and experiments reveals that the sub-diffusive continuous time random walk model is the most plausible candidate to account for the set of observations seen in experiment.

Thesis (MSc)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: We study random walks on nite graphs. The reader is introduced to general Markov chains before we move on more specifically to random walks on graphs. A random walk on a graph is just a Markov chain that is time-reversible. The main parameters we study are the hitting time, commute time and cover time. We nd novel formulas for the cover time of the subdivided star graph and broom graph before looking at the trees with extremal cover times. Lastly we look at a connection between random walks on graphs and electrical networks, where the hitting time between two vertices of a graph is expressed in terms of a weighted sum of e ective resistances. This expression in turn proves useful when we study the cover cost, a parameter related to the cover time. / AFRIKAANSE OPSOMMING: Ons bestudeer toevallige wandelings op eindige gra eke in hierdie tesis. Eers word algemene Markov kettings beskou voordat ons meer spesi ek aanbeweeg na toevallige wandelings op gra eke. 'n Toevallige wandeling is net 'n Markov ketting wat tyd herleibaar is. Die hoof paramaters wat ons bestudeer is die treftyd, pendeltyd en dektyd. Ons vind oorspronklike formules vir die dektyd van die verdeelde stergra ek sowel as die besemgra ek en kyk daarna na die twee bome met uiterste dektye. Laastens kyk ons na 'n verband tussen toevallige wandelings op gra eke en elektriese netwerke, waar die treftyd tussen twee punte op 'n gra ek uitgedruk word in terme van 'n geweegde som van e ektiewe weerstande. Hierdie uitdrukking is op sy beurt weer nuttig wanneer ons die dekkoste bestudeer, waar die dekkoste 'n paramater is wat verwant is aan die dektyd.

Ce travail est consacré à certaines classes de systèmes dynamiques ergodiques, munis d'une mesure invariante infinie, telles que des applications de l'intervalle avec un point fixe neutre ou des marches aléatoires. Le comportement asymptotique des sommes de Birkhoff d'observables d'intégrale non nulle est assez bien connu, pour peu que le système ait une certaine forme d'hyperbolicité. Une situation particulièrement intéressante est celle des tours au-dessus d'une application Gibbs-Markov. Nous cherchons dans ce contexte à étudier le cas d'observables d'intégrale nulle. Nous obtenons ainsi une forme de théorème central limite pour des systèmes dynamiques munis d'une mesure infinie. Après avoir introduit l'ensemble des notions nécessaires, nous adaptons des résultats de E. Csáki et A. Földes sur les marches aléatoires au cas des applications Gibbs-Markov. Les théorèmes d'indépendance asymptotique qui en découlent forment le cœur de cette thèse, et permettent de démontrer un théorème central limite généralisé. Quelques variations sur l'énoncé de ce théorème sont obtenues. Ensuite, nous abordons les processus en temps continu, tels que des semi-flots et des flots. Un premier travail consiste à étudier les propriété en temps grand du temps de premier retour et du temps local pour des extensions de systèmes dynamiques, ce qui se fait par des méthodes spectrales. Enfin, par réductions successives, nous pouvons obtenir une version du théorème central limite pour des flots périodiques, et en particulier le flot géodésique sur le fibré tangent unitaire de certaines variétés périodiques hyperboliques. / This work is focused on some classes of ergodic dynamical systems endowed with an infinite invariant measure, such as transformations of the interval with a neutral fixed point or random walks. The asymptotic behavior of the Birkhoff sums of observables with a non-zero integral is well known, as long as the system shows some kind of hyperbolicity. The towers over a Gibbs-Markov map are especially interesting. In this context, we aim to study the case of observables whose integral is zero. We get the equivalent of a central limit theorem for some dynamical systems endowed with an infinite measure. After we introduce the necessary definitions, we adapt some results by E. Csáki and A. Földes on random walks to the case of Gibbs-Markov maps. We derive a theorem on the asymptotic independence of Birhoff sums, which is the core of this thesis, and from this point we work out a generalised central limit theorem. We also prove a few variations on this generalised central limit theorem. Then, we study dynamical systems in continuous time, such as semi-flows and flows. We first work on the asymptotic properties of the first return time and the local time for extensions of dynamical systems; this is done by spectral methods. Finally, step by step, we extend our generalised central limit theorem to cover some periodic flows, and in particular the geodesic flow on the unitary tangent bundle of some hyperbolic periodic manifolds.

University of Macau / Faculty of Science and Technology. / Department of Computer and Information Science

This thesis studies connections between disorder type in tree polymers and the branching random walk and presents an application to swarm site-selection. Chapter two extends results on tree polymers in the infinite volume limit to critical strong disorder. Almost sure (a.s.) convergence in the infinite volume limit is obtained for weak disorder by standard theory on multiplicative cascades or the branching random walk. Chapter three establishes results for a simple branching random walk in connection with a related tree polymer. A central limit theorem (CLT) is shown to hold regardless of polymer disorder type, and a.s. connectivity of the support is established in the asymmetric case. Chapter four contains a model for site-selection in honeybee swarms. Simulations demonstrate a trade-off between speed and accuracy, and strongly suggest that increasing the quorum threshold at which the process terminates usually improves decision performance. / Graduation date: 2013

Thesis (Ph.D.) - Carleton University, 2005. / Includes bibliographical references (p. 168-173). Also available in electronic format on the Internet.

Thèse (M.Eng.) -- Université du Québec à Chicoutimi, 2004. / Bibliogr.: f. [70]-76. Document électronique également accessible en format PDF. CaQCU

Thesis (Ph. D.)--University of Missouri-Columbia, 2002. / Typescript. Vita. Includes bibliographical references (leaves 79-81). Also available on the Internet.