Qualified difference sets are a class of combinatorial configuration. The sets are related to the residue difference sets that were first discussed in detail in 1953 by Emma Lehmer. Qualified difference sets consist of a set of residues modulo an integer v and they possess attractive properties that suggest potential applications in areas such as image formation, signal processing and aperture synthesis. This thesis outlines the theory behind qualified difference sets and gives conditions for the existence and nonexistence of these sets in various cases. A special case of the qualified difference sets is the qualified residue difference sets. These consist of the set of nth power residues of certain types of prime. Necessary and sufficient conditions for the existence of qualified residue difference sets are derived and the precise conditions for the existence of these sets are given for n = 2, 4 and 6. Qualified residue difference sets are proved nonexistent for n = 8, 10, 12, 14 and 18. A generalisation of the qualified residue difference sets is introduced. These are the qualified difference sets composed of unions of cyclotomic classes. A cyclotomic class is defined for an integer power n and the results of an exhaustive computer search are presented for n = 4, 6, 8, 10 and 12. Two new families of qualified difference set were discovered in the case n = 8 and some isolated systems were discovered for n = 6, 10 and 12. An explanation of how qualified difference sets may be implemented in physical applications is given and potential applications are discussed.

We propose an improved technique to calculate the linear response of a single and multiple plates models due to ocean waves. The single plate model is the basis for the multiple plates model which we take to be a periodic array of identical plates. For the single plate model we solve the plate displacement by the Finite Element Method (FEM) and the water potential by the Boundary Element Method (BEM). The displacement is expanded in terms of the basis functions of the FEM. The boundary integral equation representing the potential is approximated by these basis functions. The resulting integral operator involving the free-surface Green's function is solved using an elementary integration scheme. Results are presented for the single plate model. We then use the same technique to solve for the periodic array of plates problem because the single and the periodic array plates model differ only in the expression of the Green's function. For the periodic array plate model the boundary integral equation for the potential involves a periodic Green's function which can be obtained by taking an infinite sum of the free-surface Green's function for the single plate model. The solution for the periodic array plate is derived in the same way as the single plate model. From this solution we then calculate the waves scattered by this periodic array.

Thesis (MSc (Mathemathical Sciences))--Stellenbosch University, 2009. / We have the familiar Kuratowski-Mr owka theorem in topology, where compactness is characterised by a closure and a projection-map (X is compact i p : X Y ! Y is a closed mapping, for any space Y , i.e. p(A) = p(A) A X Y ). Using this as our starting point, we generalise compactness to a categorical setting. We then generalise even further to "asymmetric" compactness. Then we discuss a functional approach to compactness, where we do not explicitly mention closure operators. All this provides economical proofs as well as applications in di erent areas of mathematics.

Thesis (MSc (Mathematical Sciences. Computer Science))--University of Stellenbosch, 2009. / Since the late 1990s, the Robot Soccer World Cup has been used as a testing ground for new technology in the eld of robotic design and arti cial intelligence. This research initiative pits two teams of robots against each other in a game of soccer. It is hoped that the technology gained will enable the construction of a fully autonomous team of robot players to play a normal soccer game against a human team by the year 2050. In robot soccer matches, as in real soccer matches, inferring an opponent's strategy can give a team a major advantage. One important aspect of a team's strategy is the formation the team uses. Knowing the formations that an opposing team tends to take, enables a team to prepare appropriate countermeasures. This thesis will investigate methods to extract formation information from a completed soccer game. The results show that these methods can be used to infer a classical team formation, as well as other distinguishing characteristics of the players, such as which areas on the eld the players tend to occupy, or the players' movement patterns - both valuable items of information for a future opposition team.

<div> <br></div><div> We consider the behavior of zeros of random polynomials of the from</div><div> \begin{equation*}</div><div> P_{n,m}(z) := \eta_0\varphi_m^{(m)}(z) + \eta_1 \varphi_{m+1}^{(m)}(z) + \cdots + \eta_n \varphi_{n+m}^{(m)}(z)</div><div> \end{equation*}</div><div> as \( n\to\infty \), where \( m \) is a non-negative integer (most of the work deal with the case \( m =0 \) ), \( \{\eta_n\}_{n=0}^\infty \) is a sequence of i.i.d. Gaussian random variables, and \( \{\varphi_n(z)\}_{n=0}^\infty \) is a sequence of orthonormal polynomials on the unit circle \( \mathbb T \) for some Borel measure \( \mu \) on \( \mathbb T \) with infinitely many points in its support. Most of the work is done by manipulating the density function for the expected number of zeros of a random polynomial, which we call the intensity function.</div>

Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: In this thesis we describe the creation of an embodied conversational agent which exhibits the behavioural traits of a child who has Asperger Syndrome. The agent is rule-based, rather than arti cially intelligent, for which we give justi cation. We then describe the design and implementation of the agent, and pay particular attention to the interaction between emotion, personality and social context. A 3D demonstration program shows the typical output to conform to Asperger-like answers, with corresponding emotional responses. / AFRIKAANSE OPSOMMING: In hierdie tesis beskryf ons die ontwerp en implementasie van 'n gestaltegespreksagent wat die gedrag van 'n kind met Asperger se sindroom uitbeeld. Ons regverdig die besluit dat die agent reël-gebaseerd is, eerder as 'n ware skynintelligensie implementasie. Volgende beskryf ons die wisselwerking tussen emosies, persoonlikheid en sosiale konteks en hoe dit inskakel by die ontwerp en implementasie van die agent. 'n 3D demonstrasieprogram toon tipiese ooreenstemmende Asperger-agtige antwoorde op vrae, met gepaardgaande emosionele reaksies.

Thesis (MSc (Mathematical Sciences))--Stellenbosch University, 2008. / Class field theory describes the abelian extensions of a given field K in terms of various class groups of K, and can be viewed as one of the great successes of 20th century number theory. However, the main results in class field theory are pure existence results, and do not give explicit constructions of these abelian extensions. Such explicit constructions are possible for a variety of special cases, such as for the field Q of rational numbers, or for quadratic imaginary fields. When K is a global function field, however, there is a completely explicit description of the abelian extensions of K, utilising the theory of sign-normalised Drinfeld modules of rank one. In this thesis we give detailed survey of explicit class field theory for rational function fields over finite fields, and of the fundamental results needed to master this topic.

<div>Given a parametrized family of Iterated Function System (IFS) we give sufficient conditions for a parameter on the boundary of the connectedness locus, M, to be accessible from the complement of M.</div><div>Moreover, we provide a few examples of such parameters and describe how they are connected to Misiurewicz parameter in the Mandelbrot set, i.e. the connectedness locus of the quadratic family z^2+c.<br></div>

The purpose of fisheries management is to achieve a sustainable development of the activity, so that future generations can also benefit from the resource. However, the optimal harvesting strategy usually maximizes an economically important objective function formed by the harvester which can lead to the extinction of the resource population. Therefore, sustainability has been far more difficult to achieve than is commonly thought; fish populations are becoming increasingly limited and catches are declining due to overexploitation. The aim of this research is to determine an optimal harvesting strategy which fulfills the economic objective of the harvester while maintaining the population density over a pre-specified minimum viable level throughout the harvest. We develop and investigate the harvesting model in both deterministic and stochastic settings. We first employ the Expected Net Present Value approach and determine the optimal harvesting policy using various optimization techniques including optimal control theory and dynamic programming. Next we use real options theory, model fish harvesting as a real option, and compute the value of the harvesting opportunity which also yields the optimal harvesting strategy. We further extend the stochastic problem to include price elasticity of demand and present results for di¤erent values of the coefficient of elasticity.

Microplitis croceipes, one of the nectar feeding parasitoid wasps, has been found to associatively learn chemical cues through feeding. The experiments on M. croceipes are performed and recorded by a Sony camcorder in the USDA-ARS Biological Control Laboratory in Tifton, GA, USA. The experimental videos have shown that M. croceipes can respond to Coffee odour in this study. Their detection capabilities and the behaviour of M. croceipes with different levels of coffee odours were studied. First, the data that are related to trained M. croceipes behaviour was extracted from the experimental videos and stored in a Microsoft Excel database. The extracted data represent the behaviour of M. croceipes trained to 0.02g and then exposed to 0.001g, 0.005g, 0.01g, 0.02g and 0.04g of coffee. Secondly, indices were developed to uniquely characterise the behaviour of trained M. croceipes under different coffee concentrations. Thirdly, a preliminary model and its parameters were developed to classify the response of trained wasps when exposed to these five different coffee odours. In summary, the success of this thesis demonstrates the usefulness of data analysis for interpreting experimental data, developing indices, as well as understanding the design principles of a simple model based on trained wasps.