Stochastic Calculus has been applied to the problem of pricing financial derivatives since 1973 when Black and Scholes published their famous paper "The Pricing of Options and Corporate Liabilities" in the Joumal of Political Economy. The purpose of this thesis is to show the mathematical principles underlying the methods applied to finance and to present a new model of the stock price process. As part of this paper, we present proofs of Ito's Formula and Girsanov's Theorem which are frequently used in financial applications. We demonstrate the application of these theorems to calculating the fair price of a European call option. There are two methods that result in the same price: the risk neutral valuation and the Black-Scholes partial differential equation. A new model of the stock price process is presented in Section 4. This model was inspired by the model of Cox and Ross published in 1976. We develop the model such that a martingale measure will exist for the present value of the stock price. We fit data to the traditional geometric Brownian motion model and the new model and compare the resulting prices. The data fit some stocks well, but in some cases the new model provided a better fit. The price of a European call is calculated for both models for several different stocks.

<p>Applied topology is a rapidly growing discipline aiming at using ideas coming from algebraic topology to solve problems in the real world, including analyzing point cloud data, shape analysis, etc. Semi-algebraic geometry deals with studying properties of semi-algebraic sets that are subsets of $\mathbb{R}^n$ and defined in terms of polynomial inequalities. Semi-algebraic sets are ubiquitous in applications in areas such as modeling, motion planning, etc. Developing efficient algorithms for computing topological invariants of semi-algebraic sets is a rich and well-developed field.</p> <p>However, applied topology has thrown up new invariants---such as persistent homology and barcodes---which give us new ways of looking at the topology of semi-algebraic sets. In this thesis, we investigate the interplay between these two areas. We aim to develop new efficient algorithms for computing topological invariants of semi-algebraic sets, such as persistent homology, and to develop new mathematical tools to make such algorithms possible.</p>

<div>The integral closure $\overline{I}$ of an ideal $I$ in a ring $R$ consists of all elements $x \in R$ that are integral over $I$. If $R$ is an algebra over an infinite field $k$, one can define general elements of $I=\left( x_{1},\ldots,x_{n}\right)$ as $x_{\alpha}=\sum_{i=1}^{n}\alpha_{i}x_{i}$ with $(\alpha_{1},\ldots,\alpha_{n})$ belonging to a Zariski-open subset of $k^{n}$. </div><div><br></div><div>We prove that for any ideal $I$ of height at least $2$ in a local, equidimensional excellent algebra over a field of characteristic zero, the integral closure specializes with respect to a general element of $I$. That is, we show that $\overline{I}/(x)=\overline{I/(x)}$. </div><div><br></div><div>In a Noetherian local ring $(R,m)$ of dimension $d$, one has a sequence of ideals approximating the integral closure of $I$ for $I$ an $m$-primary ideal. The ideals </div><div>\[ I \subseteq I_{\{d\}} \subseteq \cdots \subseteq I_{\{1\}} \subseteq I_{\{0\}}=\overline{I}\]</div><div>are the coefficient ideals of $I$. The $i^{\text{th}}$ coefficient ideal $I_{\{i\}}$ of $I$ is the largest ideal containing $I$ and integral over $I$ for which the first $i+1$ Hilbert coefficients of $I$ and $I_{\{i\}}$ coincide. </div><div><br></div><div>With a goal of understanding how coefficient ideals behave under specialization by general elements, we turn to the case of monomial ideals in polynomial rings over a field. A consequence of the specialization of the integral closure is that the $i^{\text{th}}$ coefficient ideal specializes when the $i^{\text{th}}$ coefficient ideal coincides with the integral closure. To this end, we give a formula for first coefficient ideals of $m$-primary monomial ideals generated in one degree in $2$ variables in order to describe when $I_{\{1\}}=\overline{I}$. In the $2$-dimensional case, we characterize the behavior of all coefficient ideals with respect to specialization by general elements. </div><div><br></div><div>In the $d$-dimensional case for $d \geq 3$, we give a characterization of when $I_{\{1\}}=\overline{I}$ for $m$-primary monomial ideals generated in one degree. In the final chapter, we give an application to the core, by characterizing when $\core(I)=\adj(I^{d})$ for such ideals.</div><div><br></div>

<p>Proofs are notoriously difficult. While the challenges students face when working on proofs are well-documented, more research is needed on what students do when working on proofs, especially in the context of linear algebra. My research focuses on student work on proofs in linear algebra through the lens of Tall's worlds of mathematics: the embodied world, the symbolic world, and the formal world. The embodied world consists of graphs, diagrams, and their properties. The symbolic world contains operations, formulas, and calculations. The formal world consists of axioms, formal definitions, and formal proofs. I conducted task-based interviews with linear algebra students in which they determined if given proofs were valid and then constructed their own proofs for different statements. In different interviews, I encouraged participants to use different worlds of mathematics. Through this study, I hoped to gain some understanding of how approaches to proof constructions and validations within the different worlds of mathematics affect students' personal proof constructions and validations. I also sought to understand what participants viewed as challenging or helpful about each world of mathematics with regards to proofs.</p> <p><br></p> <p>I found that participants often chose a world of mathematics based on the given topic rather than their preferred world or my encouragement to use a given world. Encouraging participants to use the embodied world resulted in their using generic examples. Encouraging participants to use the symbolic or formal worlds had little effect, likely due to participants' views of the symbolic and formal worlds which differed from my views of the symbolic and formal worlds. Participants said the embodied world was helpful for developing understanding, but felt limited by its specificity. A challenge of the symbolic world was the level of precision needed and the large number of variables. Participants viewed the formal world as helpful for proofs and its generality, but logic was a challenge. Reflecting on the study, participants said that the three-world framework was helpful for organizing their thoughts when working on problems.</p>

Thesis (MSc)--University of Stellenbosch, 2002. / See article for abstract

Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: We investigate the application of weighted tree transducers to correcting grammatical errors in natural language. Weighted finite-state transducers (FST) have been used successfully in a wide range of natural language processing (NLP) tasks, even though the expressiveness of the linguistic transformations they perform is limited. Recently, there has been an increase in the use of weighted tree transducers and related formalisms that can express syntax-based natural language transformations in a probabilistic setting. The NLP task that we investigate is the automatic correction of grammar errors made by English language learners. In contrast to spelling correction, which can be performed with a very high accuracy, the performance of grammar correction systems is still low for most error types. Commercial grammar correction systems mostly use rule-based methods. The most common approach in recent grammatical error correction research is to use statistical classifiers that make local decisions about the occurrence of specific error types. The approach that we investigate is related to a number of other approaches inspired by statistical machine translation (SMT) or based on language modelling. Corpora of language learner writing annotated with error corrections are used as training data. Our baseline model is a noisy-channel FST model consisting of an n-gram language model and a FST error model, which performs word insertion, deletion and replacement operations. The tree transducer model we use to perform error correction is a weighted top-down tree-to-string transducer, formulated to perform transformations between parse trees of correct sentences and incorrect sentences. Using an algorithm developed for syntax-based SMT, transducer rules are extracted from training data of which the correct version of sentences have been parsed. Rule weights are also estimated from the training data. Hypothesis sentences generated by the tree transducer are reranked using an n-gram language model. We perform experiments to evaluate the performance of different configurations of the proposed models. In our implementation an existing tree transducer toolkit is used. To make decoding time feasible sentences are split into clauses and heuristic pruning is performed during decoding. We consider different modelling choices in the construction of transducer rules. The evaluation of our models is based on precision and recall. Experiments are performed to correct various error types on two learner corpora. The results show that our system is competitive with existing approaches on several error types. / AFRIKAANSE OPSOMMING: Ons ondersoek die toepassing van geweegde boomoutomate om grammatikafoute in natuurlike taal outomaties reg te stel. Geweegde eindigetoestand outomate word suksesvol gebruik in ’n wye omvang van take in natuurlike taalverwerking, alhoewel die uitdrukkingskrag van die taalkundige transformasies wat hulle uitvoer beperk is. Daar is die afgelope tyd ’n toename in die gebruik van geweegde boomoutomate en verwante formalismes wat sintaktiese transformasies in natuurlike taal in ’n probabilistiese raamwerk voorstel. Die natuurlike taalverwerkingstoepassing wat ons ondersoek is die outomatiese regstelling van taalfoute wat gemaak word deur Engelse taalleerders. Terwyl speltoetsing in Engels met ’n baie hoë akkuraatheid gedoen kan word, is die prestasie van taalregstellingstelsels nog relatief swak vir meeste fouttipes. Kommersiële taalregstellingstelsels maak oorwegend gebruik van reël-gebaseerde metodes. Die algemeenste benadering in onlangse navorsing oor grammatikale foutkorreksie is om statistiese klassifiseerders wat plaaslike besluite oor die voorkoms van spesifieke fouttipes maak te gebruik. Die benadering wat ons ondersoek is verwant aan ’n aantal ander benaderings wat geïnspireer is deur statistiese masjienvertaling of op taalmodellering gebaseer is. Korpora van taalleerderskryfwerk wat met foutregstellings geannoteer is, word as afrigdata gebruik. Ons kontrolestelsel is ’n geraaskanaal eindigetoestand outomaatmodel wat bestaan uit ’n n-gram taalmodel en ’n foutmodel wat invoegings-, verwyderings- en vervangingsoperasies op woordvlak uitvoer. Die boomoutomaatmodel wat ons gebruik vir grammatikale foutkorreksie is ’n geweegde bo-na-onder boom-na-string omsetteroutomaat geformuleer om transformasies tussen sintaksbome van korrekte sinne en foutiewe sinne te maak. ’n Algoritme wat ontwikkel is vir sintaksgebaseerde statistiese masjienvertaling word gebruik om reëls te onttrek uit die afrigdata, waarvan sintaksontleding op die korrekte weergawe van die sinne gedoen is. Reëlgewigte word ook vanaf die afrigdata beraam. Hipotese-sinne gegenereer deur die boomoutomaat word herrangskik met behulp van ’n n-gram taalmodel. Ons voer eksperimente uit om die doeltreffendheid van verskillende opstellings van die voorgestelde modelle te evalueer. In ons implementering word ’n bestaande boomoutomaat sagtewarepakket gebruik. Om die dekoderingstyd te verminder word sinne in frases verdeel en die soekruimte heuristies besnoei. Ons oorweeg verskeie modelleringskeuses in die samestelling van outomaatreëls. Die evaluering van ons modelle word gebaseer op presisie en herroepvermoë. Eksperimente word uitgevoer om verskeie fouttipes reg te maak op twee leerderkorpora. Die resultate wys dat ons model kompeterend is met bestaande benaderings op verskeie fouttipes.

Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: Flow and reactive transport of chemical species is a very common phenomenon that occurs in natural and artificial systems. However in this study, the topic is related to acid mine drainage in the South African mining environment. Due to the hazards associated with acid mine drainage, prevention or treatment of mine effluent water before discharging to receiving waters and other environments is a necessity. A new time-dependent mathematical model is developed for a passive treatment method, based on multi-scale modelling of the coupled physico-chemical processes such as diffusion, convection, reactions and filtration, that are involved in the treatment process. The time-dependent model is simulated on a two-dimensional domain using finite volume discretization to obtain chemical species distributions. / AFRIKAANSE OPSOMMING: Vloei en reagerende transport van chemiese spesies is ’n baie algemene verskynsel wat in natuurlike en kunsmatige stelsels plaasvind. In hierdie studie is die onderwerp egter verwant aan suurmyndreinering in die Suid-Afrikaanse mynbou-omgewing. As gevolg van die gevare wat verband hou met suurmyndreinering, is die voorkoming of die behandeling van die afval-mynwater voor dit in opvangswaters en ander omgewings beland ’n noodsaaklikheid. ’n Nuwe tydafhanklike wiskundige model vir ’n passiewe behandelingsmetode is ontwikkel. Dit is gebaseer op die multi-skaal modulering van gekoppelde fisies-chemiese prosesse soos diffusie, konveksie, reaksies en filtrasie, wat by die behandelingsproses betrokke is. Die tydafhanklike model word gesimuleer op ’n twee-dimensionele domein met behulp van eindige volume diskretisasie om die verspreiding van chemiese spesies te bepaal.

Thesis (PhD)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: Nurses, midwives and clinical officers referred to as Mid-level Practioners (MLPs) play an important role in the health care system especially in rural Africa. With particular reference to rural Uganda, due to the large shortage of doctors, MLPs handle most of the duties usually meant for doctors, at health centre IV(s). From 2009 to 2011, two training interventions of MLPs were performed at 36 sites in Uganda by the Integrated Infectious Disease Capacity Building Evaluation (IDCAP). The two interventions were: Integrated Management of Infectious Diseases (IMID) and On-site Support Services (OSS) which aimed at improving MLPs’ case management for four diseases: HIV, TB, pneumonia and malaria. In this thesis, we have developed three mathematical models to investigate the effect of the two training interventions on these infectious diseases. All the models are formulated using systems of ordinary differential equations which are structured in three age groups: [0, 5), [5, 14) and [14, 50). We explored the effect of the two training interventions in the context of malaria-pneumonia, HIV-TB co-infections and the four diseases together. Our analysis shows that: i) For malaria-pneumonia, both IMID and the combination of IMID and OSS reduce the number of cases, deaths and prevalence of disease but have no effect on the incident episodes of disease. ii) Results from the HIVTB model propose that HIV and TB testing are important steps in quality of health care and are capable of offsetting slightly negative effects of reduction in ART enrollment and provision of treatment. iii) The HIV-TB-malaria-pneumonia (HTMP) model concurs with the results of the first two models and its results demonstrate that high coverage levels of the training interventions increase the positive effects that the interventions have on mortality and morbidity. Overall, our results suggest that training of MLPs is much more effective for the short term duration diseases such as malaria and pneumonia, where the baseline values for most of the performance indicators are ≥ 0.6, but not so much for long term duration diseases such as HIV and TB, whose baseline values for most of the performance indicators are < 0.6. The results further highlight that problems such as case detection and drug stock-outs need to be addressed in order for training to have substantial impact, especially in instances where the performance indicator proportions are low. / AFRIKAANSE OPSOMMING: Verpleegsters, vroedvroue en kliniese beamptes wat gesamentlik na verwys word as midvlak praktisyns (MVPs) , speel n belangrike rol in die gesondheidsorg sisteem, veral in landelike dele van Afrika. Met spesifieke verwysing na gesondheid sentrums in Uganda, waar daar te min dokters is, hanteer MVPs die meeste van die pligte wat eintlik deur dokters verrig moet word. Vanaf 2009 tot 2011 is twee opleidingsprogramme vir MVPs by 36 fasiliteite in Uganda deur die Integrated Infectious Disease Capacity Building Evaluation (IDCAP) organisasie aangebied. Die twee programme staan bekend as: Integrated Management of Infectious Diseases (IMID) and On-site Support Services (OSS). Beide die programme stel ten doel om die MVPs se pasint bestuur vir die siektes MIV, tuberkulose (TB), longontsteking en malaria te verbeter. Drie wiskundige modelle word in hierdie tesis ontwikkel om die effek van die opleidingsprogramme op hierdie oordraagbare siektes te ondersoek. Al die modelle word geformuleer deur gebruik te maak van stelsels van gewone differensiaal vergelykings wat gestruktureer is in drie ouderdomsgroepe: [0, 5), [5, 14) en [14, 50). Die effek van die opleidings programme word in die konteks van longontstekingmalaria mede-infeksie, MIV- TB mede-infeksie en al vier siektes gelyk, ondersoek. Die analise wys dat: i) Vir longontsteking-malaria mede-infeksie het beide IMID en die kombinasie van IMID en OSS die aantal siekte-gevalle, sterftes en die prevalensie van die siektes verminder, maar het geen effek op die insidensie van siekte-gevalle nie. ii) Resultate van die MIV-TB model dui aan dat MIV en TB toetsing n belangrike aspek van die gehalte van sorg is en dat dit die effense negatiewe effek van die afname in ART inskrywing en voorsiening van behandeling, teenstaan. iii) Die MIV-TB-longontsteking-malaria model (HTMP) stem ooreen met die resultate van die bogenoemde twee modelle en demonstreer dat ho dekking van die opleidingsprogramme die positiewe effek van die programme op mortaliteit en morbiditeit verhoog. In geheel stel die resultate van hierdie studie voor dat die opleiding van MVPs baie meer effektief is vir die korttermyn siektes soos malaria en longontsteking waarvoor die meeste van die beginwaardes van die prestasie-aanwysers ≥ 0.6 is, maar nie soveel vir lang-termyn siektes soos MIV en TB waarvoor die meeste van die beginwaarde van die prestasie-aanwysers < 0.6 is. Die resultate dui verder aan dat opleiding nie voldoende is wanneer die prestasie-aanwysers < 0.6 is nie en dat probleme soos die opsporing van siekte-gevalle en n gebrek aan medisyne by die klinieke aangespreek moet word vir opleiding om aansienlike impak te hê.

Living cells are made up of networks of interacting genes, proteins and other bio-molecules. Simple interactions between network components in forms of feedback regulations can lead to complex collective dynamics. A key task in cell biology is to gain a thorough understanding of the dynamics of intracellular systems and processes. In this thesis, a combined approach of mathematical modelling, computational simulation and analytical techniques, has been used to obtain a deeper insight into the dynamical aspects of a variety of feedback systems commonly encountered in cells. These systems range from model system with detailed available molecular knowledge to general regulatory motifs with varying network structures. Deterministic as well as stochastic modelling techniques have been employed, depending primarily on the specific questions asked. The first part of the thesis focuses on dissecting the principles behind the regulatory design of the Tryptophan Operon system in Escherichia coli. It has evolved three negative feedback loops, namely repression, attenuation and enzyme inhibition, as core regulator mechanisms to control the intracellular level of tryptophan amino acid, which is taken up for protein synthesis. Despite extensive experimental knowledge, the roles of these seemingly redundant loops remain unclear from a dynamical point of view. We aim to understand why three loops, rather than one, have evolved. Using a large-scale perturbation/response analysis through modelling and simulations and novel metrics for transient dynamics quantification, it has been revealed that the multiple negative feedback loops employed by the tryptophan operon are not redundant. In fact, they have evolved to concertedly give rise to a much more efficient, adaptive and stable system, than any single mechanism would provide. Since even the full topology of feedback interactions within a network is insufficient to determine its behavioural dynamics, other factors underlying feedback loops must be characterised to better predict system dynamics. In the second part of the thesis, we aim to derive these factors and explore how they shape system dynamics. We develop an analytical approach for stability and bifurcation analysis and apply it to class of feedback systems commonly encountered in cells. Our analysis showed that the strength and the Hill coefficient of a feedback loop play key role in determining the dynamics of the system carrying the loop. Not only that, the position of the loop was also found to be crucial in this decision. The analytical method we developed also facilitates parameter sensitivity analysis in which we investigate how the production and degradation rates affect system dynamics. We find that these rates are quite different in the way they shape up system behaviour, with the degradation rates exhibiting a more intricate manner. We demonstrated that coupled-loop systems display greater complexity and a richer repertoire of behaviours in comparison with single-loop ones. Different combinations of the feedback strengths of individual loops give rise to different dynamical regimes. The final part of the thesis aims to understand the effects of molecular noise on dynamics of specific systems, in this case the Tryptophan Operon. We developed two stochastic models for the system and compared their predictions to those given by the deterministic model. By means of simulations, we have shown that noise can induce oscillatory behaviour. On the other hand, incorporating noise in an oscillatory system can alter the characteristics of oscillation by shifting the bifurcation point of certain parameters by a substantial amount. Measurement of fluctuations reveals that that noise at the transcript level is most significant while noise at the enzyme level is smallest. This study highlights that noise should not be neglected if we want to obtain a complete understanding of the dynamic behaviour of cells.

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.