Thesis (MSc)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: There is an ever increasing wealth of information that is being generated regarding biological systems, in particular information on the interactions and dependencies of genes and their regulatory process. It is thus important to be able to attach functional understanding to this wealth of information. Mathematics can potentially provide the tools needed to generate the necessary abstractions to model the complex system of gene interaction. Here the problem of uncovering gene interactions is cast in several contexts, namely uncovering gene interaction patterns using statistical dependence, cooccurrence as well as feature enrichment. Several techniques have been proposed in the past to solve these, with various levels of success. Techniques have ranged from supervised learning, clustering analysis, boolean networks to dynamical Bayesian models and complex system of di erential equations. These models attempt to navigate a high dimensional space with challenging degrees of freedom. In this work a number of approaches are applied to hypothesize a gene interaction network structure. Three di erent models are applied to real biological data to generate hypotheses on putative biological interactions. A cluster-based analysis combined with a feature enrichment detection is initially applied to a Vitis vinifera dataset, in a targetted analysis. This model bridges a disjointed set of putatively co-expressed genes based on signi cantly associated features, or experimental conditions. We then apply a cross-cluster Markov Blanket based model, on a Saccharomyces cerevisiae dataset. Here the disjointed clusters are bridged by estimating statistical dependence relationship across clusters, in an un-targetted approach. The nal model applied to the same Saccharomyces cerevisiae dataset is a non-parametric Bayesian method that detects probeset co-occurrence given a local background and inferring gene interaction based on the topological network structure resulting from gene co-occurance. In each case we gather evidence to support the biological relevance of these hypothesized interactions by investigating their relation to currently established biological knowledge. The various methods applied here appear to capture di erent aspects of gene interaction, in the datasets we applied them to. The targetted approach appears to putatively infer gene interactions based on functional similarities. The cross-cluster-analysis-based methods, appear to capture interactions within pathways. The probabilistic-co-occurrence-based method appears to generate modules of functionally related genes that are connected to potentially explain the underlying experimental dynamics. / AFRIKAANSE OPSOMMING: Daar is 'n toenemende rykdom van inligting wat gegenereer word met betrekking tot biologiese stelsels, veral inligting oor die interaksies en afhanklikheidsverhoudinge van gene asook hul regulatoriese prosesse. Dit is dus belangrik om in staat te wees om funksionele begrip te kan heg aan hierdie rykdom van inligting. Wiskunde kan moontlik die gereedskap verskaf en die nodige abstraksies bied om die komplekse sisteem van gene interaksies te modelleer. Hier is die probleem met die beraming van die interaksies tussen gene benader uit verskeie kontekste uit, soos die ontdekking van patrone in gene interaksie met behulp van statistiese afhanklikheid , mede-voorkoms asook funksie verryking. Verskeie tegnieke is in die verlede voorgestel om hierdie probleem te benader, met verskillende vlakke van sukses. Tegnieke het gewissel van toesig leer , die groepering analise, boolean netwerke, dinamiese Bayesian modelle en 'n komplekse stelsel van di erensiaalvergelykings. Hierdie modelle poog om 'n hoë dimensionele ruimte te navigeer met uitdagende grade van vryheid. In hierdie werk word 'n aantal benaderings toegepas om 'n genetiese interaksie netwerk struktuur voor te stel. Drie verskillende modelle word toegepas op werklike biologiese data met die doel om hipoteses oor vermeende biologiese interaksies te genereer. 'n Geteikende groeperings gebaseerde analise gekombineer met die opsporing van verrykte kenmerke is aanvanklik toegepas op 'n Vitis vinifera datastel. Hierdie model verbind disjunkte groepe van vermeende mede-uitgedrukte gene wat gebaseer is op beduidende verrykte kenmerke, hier eksperimentele toestande . Ons pas dan 'n tussen groepering Markov Kombers model toe, op 'n Saccharomyces cerevisiae datastel. Hier is die disjunkte groeperings ge-oorbrug deur die beraming van statistiese afhanklikheid verhoudings tussen die elemente in die afsondelike groeperings. Die nale model was ons toepas op dieselfde Saccharomyces cerevisiae datastel is 'n nie- parametriese Bayes metode wat probe stelle van mede-voorkommende gene ontdek, gegee 'n plaaslike agtergrond. Die gene interaksie is beraam op grond van die topologie van die netwerk struktuur veroorsaak deur die gesamentlike voorkoms gene. In elk van die voorgenome gevalle word ons hipotese vermoedelik ondersteun deur die beraamde gene interaksies in terme van huidige biologiese kennis na te vors. Die verskillende metodes wat hier toegepas is, modelleer verskillende aspekte van die interaksies tussen gene met betrekking tot die datastelle wat ons ondersoek het. In die geteikende benadering blyk dit asof ons vermeemde interaksies beraam gebaseer op die ooreenkoms van biologiese funksies. Waar die a eide gene interaksies moontlik gebaseer kan wees op funksionele ooreenkomste tussen die verskeie gene. In die analise gebaseer op die tussen modelering van gene groepe, blyk dit asof die verhouding van gene in bekende biologiese substelsels gemodelleer word. Dit blyk of die model gebaseer op die gesamentlike voorkoms van gene die verband tussen groepe van funksionele verbonde gene modelleer om die onderliggende dinamiese eienskappe van die experiment te verduidelik.

Thesis (PhD (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2007. / A structure-from-motion implementation for enclosed environments is presented. The various aspects covered include a discussion on optimised luminance computations—a technique to compute an optimally weighted luminance that maintains a greatest amount of data fidelity. Furthermore a visual engine is created that forms the basis of data input for reconstruction purposes; such an inexpensive solution is found to offer realistic environments along with precise control of scene and camera elements. A motion estimation system provides tracking information of scene elements and an unscented Kalman filter is used as depth estimator. The elements are combined into an accurate reconstructor for enclosed environments.

Thesis (PhD (Applied Mathematics))--University of Stellenbosch, 2009. / ENGLISH ABSTRACT: In many applications, an accurate, robust and fast tracker is needed, for example in surveillance, gesture recognition, tracking lips for lip-reading and creating an augmented reality by embedding a tracked object in a virtual environment. In this dissertation we investigate the viability of a tracker that combines the accuracy of active appearancemodels with the robustness of the particle lter (a stochastic process)—we call this combination the PFAAM. In order to obtain a fast system, we suggest local optimisation as well as using active appearance models tted with non-linear approaches. Active appearance models use both contour (shape) and greyscale information to build a deformable template of an object. ey are typically accurate, but not necessarily robust, when tracking contours. A particle lter is a generalisation of the Kalman lter. In a tutorial style, we show how the particle lter is derived as a numerical approximation for the general state estimation problem. e algorithms are tested for accuracy, robustness and speed on a PC, in an embedded environment and by tracking in ìD. e algorithms run real-time on a PC and near real-time in our embedded environment. In both cases, good accuracy and robustness is achieved, even if the tracked object moves fast against a cluttered background, and for uncomplicated occlusions. / AFRIKAANSE OPSOMMING: ’nAkkurate, robuuste en vinnige visuele-opspoorderword in vele toepassings benodig. Voorbeelde van toepassings is bewaking, gebaarherkenning, die volg van lippe vir liplees en die skep van ’n vergrote realiteit deur ’n voorwerp wat gevolg word, in ’n virtuele omgewing in te bed. In hierdie proefskrif ondersoek ons die lewensvatbaarheid van ’n visuele-opspoorder deur die akkuraatheid van aktiewe voorkomsmodellemet die robuustheid van die partikel lter (’n stochastiese proses) te kombineer—ons noem hierdie kombinasie die PFAAM. Ten einde ’n vinnige visuele-opspoorder te verkry, stel ons lokale optimering, sowel as die gebruik van aktiewe voorkomsmodelle wat met nie-lineêre tegnieke gepas is, voor. Aktiewe voorkomsmodelle gebruik kontoer (vorm) inligting tesamemet grysskaalinligting om ’n vervormbaremeester van ’n voorwerp te bou. Wanneer aktiewe voorkomsmodelle kontoere volg, is dit normaalweg akkuraat,maar nie noodwendig robuust nie. ’n Partikel lter is ’n veralgemening van die Kalman lter. Ons wys in tutoriaalstyl hoe die partikel lter as ’n numeriese benadering tot die toestand-beramingsprobleem afgelei kan word. Die algoritmes word vir akkuraatheid, robuustheid en spoed op ’n persoonlike rekenaar, ’n ingebedde omgewing en deur volging in ìD, getoets. Die algoritmes loop intyds op ’n persoonlike rekenaar en is naby intyds op ons ingebedde omgewing. In beide gevalle, word goeie akkuraatheid en robuustheid verkry, selfs as die voorwerp wat gevolg word, vinnig, teen ’n besige agtergrond beweeg of eenvoudige okklusies ondergaan.

Thesis (PhD (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2004. / Suppose a lottery scheme consists of randomly selecting an unordered winning n-subset from a universal set of m numbers, while a player participates in the scheme by purchasing a playing set of any number of unordered n-subsets from the same universal set prior to a winning draw, and is awarded a prize if ...

ENGLISH ABSTRACT: Determining the (classical) chromatic number of a graph (i.e. finding the smallest number of colours with which the vertices of a graph may be coloured so that no two adjacent vertices receive the same colour) is a well known combinatorial optimization problem and is widely encountered in scheduling problems. Since the late 1960s the notion of the chromatic number has been generalized in several ways by relaxing the restriction of independence of the colour classes. / AFRIKAANSE OPSOMMING: Die bepaling van die (klassieke) chromatiese getal van ’n grafiek (naamlik die kleinste aantal kleure waarmee die punte van ’n grafiek gekleur kan word sodat geen twee naasliggende punte dieselfde kleur ontvang nie) is ’n bekende kombinatoriese optimeringsprobleem wat wyd in skeduleringstoepassings te¨egekom word. Sedert die laat 1960s is die definisie van die chromatiese getal op verskeie maniere veralgemeen deur die vereiste van onafhanklikheid van die kleurklasse te verslap. / Thesis (DPhil)--Stellenbosch University, 2007.

Thesis (PhD)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: Research on transfer processes in various types of porous media has become important for the optimization of high technology engineering devices and processes. In this study the micro-structural parameters of different types of porous media, namely granular media, foamlike media and fibre beds, are characterized and quantified. Existing analytical modelling procedures for the three different types of porous media have been unified and refined to improve their predictive capabilities. Deterministic equations are proposed for predicting the streamwise pressure gradient, permeability and inertial coefficient of each type of porous medium. The equations are applicable over the entire porosity range and steady laminar flow regime and well suited as drag models in numerical computations. It is shown that the improved granular model can be regarded as qualitative and quantitative proof of the extensively used semi-empirical Ergun equation. The proposed model is used to provide physical meaning to the empirical coefficients. An Ergun-type equation is also proposed for foamlike media by remodelling the interstitial geometric configuration and accompanying flow conditions. The range of applicability of the existing foam model has been extended by incorporating the effect of developing flow in the pressure drop prediction. An equation is proposed in which the variation in the cross-sectional shape of the fibres can be incorporated into the interstitial form drag coefficient used in the foam model. This serves as an improvement on the constant value previously used. The existing foam model is also adapted to account for anisotropy resulting from compression. Two case studies are considered, namely compression of a non-woven glass fibre filter and compression of a soft polyester fibre material. The significant effect of compression on permeability is illustrated. In each case study the permeability values range over more than an order of magnitude for the narrow porosity ranges involved. The pressure drop prediction of the foam model is furthermore adapted to account for the combined effects of compression and developing flow. The newly proposed model diminishes the significant over-prediction of the existing foam model. An equation is furthermore proposed for predicting the permeability of Fontainebleau sandstones in which the effect of blocked throats is accounted for. Lastly, equations are proposed for predicting diffusivity ratios of unconsolidated arrays of squares and cubes. The prediction of the diffusivity ratio proposed in the present study, as opposed to model predictions from the literature, takes into account diffusion that may take place in stagnant fluid volumes. It is shown that a specific weighted average model proposed in the literature is not adequate to predict the diffusivity ratio of fully staggered arrays of squares, since it is shown not to be applicable to rectangular unit cells. Instead a new weighted average model is proposed which is applicable over the entire porosity range and for both staggered and non-staggered arrays of solid squares and cubes. The proposed weighted average model provides satisfactory agreement with experimental data from the literature and numerical data generated in the present study. / AFRIKAANSE OPSOMMING: Navorsing op oordragsprosesse in verskeie tipes poreuse media het belangrik geword vir die optimisering van ho¨e-tegnologie ingenieurstoestelle- en prosesse. In hierdie studie word die mikro-struktuur parameters van verskillende tipes poreuse media, naamklik korrelagtige media, sponsatige media en veselbeddens gekarakteriseer en gekwantifiseer. Bestaande analitiese modelleringsprosedures vir die drie verskillende tipes poreuse media is verenig en verfyn om die voorspelbare bekwaamheid daarvan te verbeter. Deterministiese vergelykings is voorgestel vir die voorspelling van die stroomsgewyse gradi¨ent, permeabiliteit en inersi¨ele ko¨effisi¨ent van elke tipe poreuse medium. Die vergelykings is geldig oor die hele porositeitsgrens en gestadigde laminˆere vloeigrens en goed geskik as weerstandsmodelle in numeriese berekeninge. Dit is aangetoon dat die verbeterde korrelmodel beskou kan word as kwalitatiewe en kwantitatiewe bewys van die ekstensiewe gebruikte semi-empiriese Ergun vergelyking. Die voorgestelde model is gebruik om fisiese betekenis aan die empiriese ko¨effisi¨ente te gee. ’n Ergun-tipe vergelyking is ook voorgestel vir sponsagtige media deur hermodellering van die tussenruimtelike geometriese konfigurasie en gepaardgaande vloeikondisies. Die grense van toepaslikheid van die bestaande sponsmodel is uitgebrei deur die inkorporering van die effek van ontwikkelende vloei in die voorspelling van die drukval. ’n Vergelyking is voorgestel waarin die variasie in die deursnit vorm van die vesels ingesluit is in die sponsmodel. Dit dien as verbetering op die konstante waarde wat voorheen gebruik is. Die bestaande sponsmodel is ook aangepas om voorsiening te maak vir anisotropie as gevolg van kompressie. Twee gevallestudies is oorweeg, naamlik kompressie van ’n nie-geweefde glasvesel filter en kompressie van ’n sagte polyester veselmateriaal. Die beduidende effek van kompressie op permeabiliteit is aangetoon. In elke gevallestudie strek die permeabiliteit waardes oor meer as ’n grootte orde vir die skrale porositeitgrense betrokke. Die drukvalvoorspelling van die sponsmodel is verder aangepas om voorsiening te maak vir die gekombineerde effekte van kompressie en ontwikkelende vloei. Die nuwe voorgestelde model verminder die beduidende oor-voorspelling van die bestaande sponsmodel. ’n Vergelyking is verder voorgestel vir die voorspelling van die permeabiliteit van Fontainebleau sandsteen waarin die effek van geblokte porie¨e in ag geneem is. Laastens is vergelykings voorgestel vir die voorspelling van die diffusiwiteitsverhoudings van nie-konsoliderende rangskikkings van vierkante en kubusse. Die diffusiwiteitsverhouding voorspel in die huidige studie, teenoor modelvoorspellings vanaf die literatuur, neem diffusie in ag wat plaasvind in die stagnante vloeistofvolumes. Dit is aangetoon dat ’n geweegde gemiddelde model, voorgestel in die literatuur, nie in staat is om die diffusiwiteitsverhouding van ten volle verspringende rangskikkings van vierkante te voorspel nie, aangesien dit nie toepaslik is vir reghoekige eenheidselle nie. ’n Nuwe geweegde model is in plaas daarvan voorgestel wat toepaslik is oor die hele porositeitsgrens en vir beide verspringende en nieverspringende rangskikkings van soliede vierkante en kubusse. Die voorgestelde geweegde gemiddelde model bied bevredigende ooreenstemming met eksperimentele data uit die literatuur en numeriese data gegenereer in die huidige studie.

Solving the fluid-structure interaction (FSI) problems is particularly challenging. This is because the coupling of the fluid and structure may require different solvers in different points of the solution domain, and with different mesh requirements. In this thesis, a partitioned approach is considered. Two solvers are employed to deal with each part of the problem (fluid and structure), where the interaction process is realized via exchanging information from the fluid-structure interface in a staggered fashion. One of the advantages of this approach is that we can take advantage of the existing algorithms that have been used for solving fluid or structural problems, which leads a reduction in the code development time, Hou (2012). However, it requires careful implementation so that spurious results in terms of stability and accuracy can be avoided. We found that most fluid-structure interaction computations through a staggered approach are based on at most second order time integration methods. In this thesis we studied the performance of some high order fluid and structure dynamic methods, when applied in a staggered approach to an FSI problem in a structure prediction way by combining predictors with time integration schemes to obtain stable schemes. Nonlinear Euler equations for gas dynamics were investigated and the analysis was realized through the piston problem. An adapted one-dimensional high order finite volume WENO₃ scheme for nonlinear hyperbolic conservation laws-Dumbser (2007a), Dumbser et al. 2007b)-was considered and a numerical flux was proposed. The numerical results of the proposed method show the non-oscillatory property when compared with traditional numerical methods such as the Local Lax-Friedrichs. So far to our knowledge, the WENO₃₋ as proposed in this work- has not been applied to FSI problems. Thus, it was proposed to discretize the fluid domain in space, and in order to adapt it to a moving mesh was reformulated to couple with an Arbitrary Lagrangian Eulerian (ALE) approach. To integrate in time the structure we started by using Newmark schemes as well as the trapezoidal-rule backward differentiation formulae of order 2 (TR − BDF2). Two study cases were carried out by taking into account the transient effects on the fluid behaviour. In the first case, we only consider the structural mass in the dynamic coupled system and in the second case, a quasi-steady fluid was considered. In order to test the performance of the structural solvers, simulations were carried out, firstly, without the contribution of fluid mass, and then a comparative study of the performance of various structure solvers in a staggered approach framework were realized in order to study the temporal accuracy for the partitioned fluid-structure interaction coupling. For a quasi-steady fluid case, the oscillation frequency of the coupled system was successfully estimated using the TR-BDF2 scheme, and the coupled system was solved for various Courant numbers in a structural predictor fashion. The results showed better performance of the TR-BDF2 scheme. Newmark’s schemes as well as the TR-BDF2 are only second order accuracy. However, the Newmark (average acceleration) is traditionally preferred by researchers as a structure solver in a staggered approach for FSI problems, although higher order schemes do exist. van Zuijlen (2004), in his partitioned algorithm proposed the explicit singly diagonal implicit Runge-Kutta (ESDIRK) family of schemes of order 3 to 5 to integrate both fluid and structure. Therefore in this work, these schemes were considered and applied as structural solvers. Their performance was studied through numerical experiments, and comparisons were realized with the performance of the traditional Newmark’s schemes. The results show that although their computational cost is high, they present a high order of accuracy. / Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2013.

Partial exchangeability is the fundamental building block in the subjective approach to the probability of multi-type sequences which replaces the independence concept of the objective theory. The aim of this thesis is to present some theory for partially exchangeable sequences of random variables based on well-known results for exchangeable sequences. The reader is introduced to the concepts of partially exchangeable events, partially exchangeable sequences of random variables and partially exchangeable o-fields, followed by some properties of partially exchangeable sequences of random variables. Extending de Finetti's representation theorem for exchangeable random variables to hold for multi-type sequences, we obtain the following result to be used throughout the thesis: There exists a o-field, conditional upon which, an infinite partially exchangeable sequence of random variables behaves like an independent sequence of random variables, identically distributed within types. Posing (i) a stronger requirement (spherical symmetry) and (ii) a weaker requirement (the selection property) than partial exchangeability on the infinite multi-type sequence of random variables, we obtain results related to de Finetti's representation theorem for partially exchangeable sequences of random variables. Regarding partially exchangeable sequences as mixtures of independent and identically distributed (within types) sequences, we (i) give three possible expressions for the directed random measures of the partially exchangeable sequence and (ii) look at three possible expressions for the o-field mentioned in de Finetti's representation theorem. By manipulating random measures and using de Finetti's representation theorem, we point out some concrete ways of constructing partially exchangeable sequences. The main result of this thesis follows by extending de Finetti's represen. tation theorem in conjunction with the Chatterji principle to obtain the following result: Given any a.s. limit theorem for multi-type sequences of independent random variables, identically distributed within types, there exists an analogous theorem satisfied by all partially exchangeable sequences and by all sub-subsequences of some subsequence of an arbitrary dependent infinite multi-type sequence of random variables, tightly distributed within types. We finally give some limit theorems for partially exchangeable sequences of random variables, some of which follow from the above mentioned result. / Thesis (Ph.D.)-University of Natal, Durban, 1991.

Wadley’s problem frequently emerges in dosage-mortality data and is one in which the number of surviving organisms is observed but the number initially treated is unknown. Data in this setting are also often overdispersed, that is the variability within the data exceeds that described by the distribution modelling it. The aim of this thesis is to explore distributions that can accommodate overdispersion in a Wadley’s problem setting. Two methods are essentially considered. The first considers adapting the beta-binomial and multiplicative binomial models that are frequently used for overdispersed binomial-type data to a Wadley’s problem setting. The second strategy entails modelling Wadley’s problem with a distribution that is suitable for modelling overdispersed count data. Some of the distributions introduced can be used for modelling overdispersed count data as well as overdispersed doseresponse data from a Wadley context. These models are compared using goodness of fit tests, deviance and Akaike’s Information Criterion and their properties are explored. / Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2009.

According to the needs, real systems can be modeled at various level of resolution. It can be detailed interactions at the individual level (or at microscopic level) or a sample of the system (or at mesoscopic level) and also by averaging over mesoscopic (structural) states; that is, at the level of interactions between subsystems of the original system (or at macroscopic level). With the microscopic study one can get a detailed information of the interaction but at a cost of heavy computational work. Also sometimes such a detailed information is redundant. On the other hand, macroscopic analysis, computationally less involved and easy to verify by experiments. But the results obtained may be too crude for some applications. Thus, the mesoscopic level of analysis has been quite popular in recent years for studying real systems. Here we will focus on structured population models where we can observe various level of organization such as individual, a group of population, or a community. Due to fast movement of the individual compare of the other demographic processes (like death and birth), the problem is multiple-scale. There are various methods to handle multiple-scale problem. In this work we will follow asymptotic analysis ( or more precisely compressed Chapman–Enskog method) to approximate the microscopic model by the averaged one at a given level of accuracy. We also generalize our model by introducing reducible migration structure. Along with this, considering age dependency of the migration rates and the mortality rates, the thesis o ers improvement of the existing literature. / Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2011.