Thesis (PhD (Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: Classical formulations model belief revision as a deterministic process. Under certain circumstances, the process may have more than one outcome, which suggests that belief revision is non-deterministic instead. Representations exist that model belief revision in either format, and for both formats there are axiom schemes that determine whether the representation is in fact a belief revision process. Although the axiom scheme for the non-deterministic case generalises that of the deterministic case, both schemes entail that all of the beliefs held by an agent are affected by new information, which is perhaps unintuitive. Rather, one may consider that belief revision should be local, with beliefs only affected if the new information is pertinent to them. We approach the problem of belief revision from the standpoint that it is local and non-deterministic, and the purpose and contribution of this dissertation is the development of a topological framework with which to model belief revision in this manner. / AFRIKAANSE OPSOMMING: Geloofshersiening word gewoonlik as ’n deterministiese proses voorgestel. Meer as een uitkoms mag bestaan vir verskeie omstandighede, wat aandui dat die proses liewer nie-deterministies van aard is. Beide die gevalle word deur aksiomaskemas gereguleer, en die aksiomas vir die nie-deterministiese geval veralgemeen dié van die deterministiese geval. Albei aksiomaskemas stipuleer, miskien onintuïtief, dat alle gelowe van ’n agent deur die nuwe informasie geaffekteer word. ’n Beter metode is dat net daardie gelowe waarvoor die nuwe informasie toepaslik is geaffekteer word. Ons benader die probleem van geloofshersiening uit die standpunt dat dit lokaal en nie-deterministies is, en die doel en bydrae van hierdie proefskrif is dus die ontwikkeling van ’n topologiese raamwerk waarmee ons geloofshersiening op hierdie manier kan voorstel.

Prof. Marcel Wild completed his PhD with Zurick University and this is a copy of the original works / The original works can be found at http://www.hbz.uzh.ch/ / ABSTRACT: A linear representation of a modular lattice L is a homomorphism from L into the lattice Sub(V) of all subspaces of a vector space V. The representation theory of lattices was initiated by the Darmstadt school (Wille, Herrmann, Poguntke, et al), to large extent triggered by the linear representations of posets (Gabriel, Gelfand-Ponomarev, Nazarova, Roiter, Brenner, et al). Even though posets are more general than lattices, none of the two theories encompasses the other. In my thesis a natural type of finite lattice is identified, i.e. triangle lattices, and their linear representation theory is advanced. All of this was triggered by a more intricate setting of quadratic spaces (as opposed to mere vector spaces) and the question of how Witt’s Theorem on the congruence of finite-dimensional quadratic spaces lifts to spaces of uncountable dimensions. That issue is dealt with in the second half of the thesis.

Thesis (PhD)--Stellenbosch University, 2012. / Please refer to full text for abstract.

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.

Thesis (PhD)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: This thesis deals with a function eld analog of the André-Oort conjecture. The (classical) André-Oort conjecture concerns the distribution of special points on Shimura varieties. In our case we consider the André-Oort conjecture for special points in the product of Drinfeld modular varieties. We in particular manage to prove the André- Oort conjecture for subvarieties in a product of two Drinfeld modular surfaces under a characteristic assumption. / AFRIKAANSE OPSOMMING: Hierdie tesis handel van 'n funksieliggaam analoog van die André-Oort Vermoeding. Die (Klassieke) André-Oort Vermoeding het betrekking tot die verspreiding van spesiale punte op Shimura varietiete. Ons geval beskou ons die André-Oort Vermoeding vir spesiale punte op die produk Drinfeldse modulvarietiete. In die besonders, bewys ons die André-Oort Vermoeding vir ondervarieteite van 'n produk van twee Drinfeldse modulvarietiete, onderhewig aan 'n karakteristiek-aanname.

Thesis (DSc (Mathematical Sciences))--University of Stellenbosch, 2007. / ENGLISH ABSTRACT: (Math symbols have changed) Wavelet and subdivision techniques have developed, over the last two decades, into powerful mathematical tools, for example in signal analysis and geometric modelling. Both wavelet and subdivision analysis are based on the concept of a matrix–refinable function, i.e. a finitely supported matrix function which is self-replicating in the sense that it can be expressed as a linear combination of the integer shifts of its own dilation with factor 2: F = TAF = å k∈Z F(2 ・ −k)Ak. The coefficients Ak, k ∈ Z of d × d matrices, of this linear combination constitute the so-called matrix- mask sequence. Wavelets are in fact constructed as a specific linear combination of the integer shifts of the 2-dilation of a matrix- refinable function cf. [2; 9], whereas the convergence of the associated matrix- subdivision scheme c0 = c, cr+1 = SAcr, r ∈ Z+, SA : c = (ck : k ∈ Z) 7→ SAc = å ℓ∈Z Ak−2ℓ cℓ : k ∈ Z ! , subject to the necessary condition that rank := dim   \ ǫ∈{0,1} n y ∈ Rd : Qǫy = y o   > 0, Qǫ := å j∈Z Aǫ+2j, ǫ ∈ {0, 1}, ( cf. [26]) , implies the existence of a finitely supported matrix- function which is refinable with respect to the mask coefficients defining the refinement equation and the subdivision scheme. Throughout this thesis, we investigate in time–domain for a given matrix mask sequence, the related issues of the existence of a matrix–refinable function and the convergence of the corresponding matrix– cascade algorithm, and finally we apply some results to the particular research area of Hermite interpolatory subdivision schemes. The dissertation is organized as follows: In order to provide a certain flexibility or freedom over the project, we established in Chapter 1 the equivalence relation between the matrix cascade algorithm and the matrix subdivision scheme, subject to a well defined class of initial iterates. Despite the general noncommutativity of matrices, we make use in the full rank case Qǫ = I, ǫ ∈ {0, 1}, of a symbol factorization, to develop in Chapter 2 some useful tools, yielding a convergence result which comes as close to the scalar case as possible: we obtained a concrete sufficient condition on the mask sequence based on the matrix version of the generating function introduced in [3, page 22] for existence and convergence. Whilst the conjecture on nonnegative masks was confirmed in 2005 by Zhou [29], our result on scalar case provided a progress for general mask sequences. We then applied to obtain a new one-parameter family of refinable functions which includes the cardinal splines as a special case, as well as corresponding convergent subdivision schemes. With the view to broaden the class of convergent matrix-masks, we replaced in chapter 3 the full rank condition by the rank one condition Qǫu = u, ǫ ∈ {0, 1}, u := (1, . . . , 1)T, then improved the paper by Dubuc and Merrien [13] by using the theory of rank subdivision schemes by Micchelli and Sauer [25; 26], and end up this improvement with a generalization of [13, Theorem 13, p.8] in to the context of rank subdivision schemes. In Chapter 4, we translated the concrete convergence criteria of the general theory from Theorem 3.2, based on the r-norming factor introduced in [13, Definition 6, p.6], into the context of rank, factorization and spectral radius (cf. [26]), and presented a careful analysis of the relationship between the two concepts. We then proceed with generalizations and improvements: we classified the matrix cascade algorithms in term of rank = 1, 2, . . . , d, and provided a complete characterization of each class with the use of a more general r−norming factor namely τ(r)-norming factor. On the other hand, we presented numerical methods to determine, if possible, the convergence of each class of matrix cascade algorithms. In both the scalar and matrix cases above, we also obtained explicitly the geometric constant appearing in the estimate for the geometric convergence of thematrix-cascade algorithm iterates to the matrix- refinable function. This same geometric convergence rate therefore also holds true for the corresponding matrix–cascade algorithm. Finally, in Chapter 5, we apply the theory and algorithms developed in Chapter 4 to the particular research area of Hermite interpolatory subdivision schemes: we provided a new convergence criterium, and end up with new convergence ranges of the parameters’ values of the famous Hermite interpolatory subdivision scheme with two parameters, due to Merrien [23]. / AFRIKAANSE OPSOMMING :(Wiskundige simbole het verander) Golfie en subdivisietegnieke het oor die afgelope twee dekades ontwikkel in kragtige wiskundige gereedskap, byvoorbeeld in seinanalise en geometriesemodellering. Beide golfie en subdivisie analise is gebaseer op die konsep van ’n matriks-verfynbare funksie; oftewel ’n eindig-ondersteunde matriksfunksie F wat selfreproduserend is in die sin dat dit uitgedruk kan word as ’n lineêre kombinasie van die heelgetalskuiwe van F se eie dilasie met faktor 2: F = Σ F(2 · −α)A(α), met A(α), α ∈ Z, wat aandui die sogenaamde matriks-masker ry. Golfies kan dan gekonstrueer word as ’n spesifieke lineêre kombinasie van die funksie ry {F(2 · −α) : α ∈ Z} (sien [2; 9]), terwyl die konvergensie van die ooreenstemmende matriks-subdivisie skema cº = c, cr+1 =(Σ β∈Z A(α − 2β) cr(β) : α ∈ Z ! , r ∈ Z+, onderhewig aan die nodige voorwaarde dat rank := dim   \ ǫ∈{0,1} n y ∈ Rd : Qǫy = y o   > 0, Qǫ := å α∈Z A(ǫ + 2α), ǫ ∈ {0, 1}, (sien [27]) die bestaan impliseer van ’n eindig-ondersteunde matriksfunksie F wat verfynbaar ismet betrekking tot diemaskerko¨effisi¨entewat die subdivisieskema definieer, en in terme waarvan die limietfunksie F van die subdivisieskema uitgedruk kan word as F = å α∈Z F(· − α)c(α). Ons hoofdoel hier is om , in die tydgebied, en vir ’n gegewematriks-masker ry, die verwante kwessies van die bestaan van ’nmatriks-verfynbare funksie en die konvergensie van die ooreenstemmende matriks-kaskade algoritme, en matriks-subdivisieskema, te ondersoek, en om uiteindelik sommige van ons resultate toe te pas op die spesifieke kwessie van die konvergensie van Hermite interpolerende subdivisieskemas. Summary v Eerstens, in Hoofstuk 1, ondersoek ons die verwantskap tussen matriks-kaskade algoritmes en matriks-subdivisie skemas, met verwysing na ’n goedgedefinieerde klas van begin-iterate. Vervolgens beskou ons die volle rang geval Qǫ = I, ǫ ∈ {0, 1}, om, in Hoofstuk 2, nuttige gereedskap te ontwikkel, en wat daarby ’n konvergensie resultaat met ’n sterk konneksie ten opsigte van die skalaar-geval oplewer. Met die doelstelling om ons klas van konvergente matriks-maskers te verbreed, vervang ons, in Hoofstuk 3, die volle rang voorwaarde met die rang een voorwaarde Qǫu = u, ǫ ∈ {0, 1}, u := (1, . . . , 1)T, en verkry ons dan ’n verbetering op ’n konvergensieresultaat in die artikel [14] deur Dubuc en Merrien, deur gebruik te maak van die teorie van rang subdivisieskemas van Micchelli en Sauer [26; 27], waarna ons die resultaat [14, Stelling 13, page 8] na die konteks van rang subdivisieskemas veralgemeen. InHoofstuk 4 herlei ons die konkrete konvergensie kriteria van Stelling 3.2, soos gebaseer op die r-normerende faktor gedefinieer in [14, Definisie 6, page 6] , na die konteks van rang, faktorisering en spektraalradius (sien [27]), en gee ons ’n streng analise van die verwantskap tussen die twee konsepte. Verder stel ons dan bekend ’n nuwe klassifikasie van matriks-kaskade algoritmes ten opsigte van rang, en verskaf ons ’n volledige karakterisering van elke klasmet behulp van ’nmeer algemene r-normerende faktor, nl. die τ(r)-normerende faktor. Daarby gee ons doeltreffende numeriesemetodes vir die implementering van ons teoretiese resultate. Ons verkry ook eksplisiet die geometriese konstante wat voorkom in die afskatting van die geometriese konvergensie van die matriks-kaskade algoritme iterate na die matriks-verfynbare funksie. Ten slotte, in Hoofstuk 5, pas ons die teorie en algoritmes ontwikkel in Hoofstuk 4 toe om die konvergensie van Hermite-interpolerende subdivisieskemas te analiseer. Spesifiek lei ons ’n nuwe konvergensie kriterium af, wat ons dan toepas om nuwe konvergensie gebiede vir die parameter waardes te verkry vir die beroemde Hermite interpolerende subdivisieskema met twee parameters, soos toegeskryf aan Merrien [24].

Thesis (PhD)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: Rank 2 Drinfeld modular forms have been studied for more than 30 years, and while it is known that a higher rank theory could be possible, higher rank Drinfeld modular forms have only recently been de ned. In 1988 Gekeler published [Ge2] in which he studies the coe cients of rank 2 Drinfeld modular forms. The goal of this thesis is to perform a similar study of the coe cients of higher rank Drinfeld modular forms. The main results are that the coe cients themselves are (weak) Drinfeld modular forms, a product formula for the discriminant function, the rationality of certain naturally de ned modular forms, and the computation of some Hecke eigenforms and their eigenvalues. / AFRIKAANSE OPSOMMING: Drinfeld modulêre vorme van rang 2 word al vir meer as 30 jaar bestudeer en alhoewel dit lankal bekend is dat daar Drinfeld modulêre vorme van hoër rang moet bestaan, is die de nisie eers onlangs vasgepen. In 1988 het Gekeler die artikel [Ge2] gepubliseer waarin hy die koeffisiënte van Fourier reekse van rang 2 Drinfeld modulêre vorme bestudeer. Die doel van hierdie proefskrif is om dieselfde studie vir Drinfeld modulêre vorme van hoër rang uit te voer. Die hoofresultate is dat die koeffi siënte self (swak) Drinfeld modulêre vorme is, `n produk formule vir die diskriminant funksie, die feit dat sekere natuurlik gede finiëerde modulêre vorme rasionaal is, en die vasstelling van Hecke eievorme en hul eiewaardes.

Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: The interest rates market has changed dramatically since the 2007 credit crunch with the explosion of basis spreads between rates of different tenors and currencies. Consequently, the classical replication of FRA rates with spot LIBOR rates is no longer valid. Moreover, the 2007 credit crunch yields a separation between the curve used for discounting and the forward or projection curves that estimate all future cash-fl ows. Another impact of the credit crunch in risk management is that market participants have started to give more importance to the difference between collateralized and uncollateralized trades. Nowadays, the wide spread use of collateral, especially in swap contracts, has made the overnight index swap (OIS) rate the appropriate benchmark for discounting collateralized trades. Inspired by the seminal works of Mercurio (2010a,b), Kijima et al. (2008), Fujii et al. (2011), Bianchetti (2010b), with the contributions of other authors, and motivated by the evolution of the interest rates market and models, this thesis examines a new framework that uses multiple-curves to value interest rate derivatives which is compatible with the current market practice. Firstly, we discuss the roots of the 2007 credit crunch and its serious consequences for pricing interest rate derivatives. We underscore the necessity of a multiple-curve pricing framework for interest rate derivatives. This is followed by a discussion on the importance of collateralization and OIS discounting in pricing Over-The-Counter (OTC) derivatives. The central part of the thesis discusses the modern theoretical framework and the practical implementation of the multiple curve pricing method. We present a bootstrapping algorithm used to construct and fit the multiple-yield curves to market prices of plainvanilla contracts. Secondly, starting with the single-currency economy, the extended version of the LIBOR Market Model, developed by Mercurio (2010a,b), which proposes a joint model of FRA rates, implied forward rates and their corresponding spread is investigated. Analogously, the extended version of short-rate model in a multiple-curve setup and in the presence of basis spread, proposed by Kijima et al. (2008), is presented and discussed. This work provides a detailed analysis of these extensions and the corresponding closed formulae for liquid products such as caps and swaptions. Finally, in the multiple-currencies case, the HJM model with stochastic basis spreads, introduced by Fujii et al. (2011), consistent with the foreign exchange and cross-currency swaps markets that includes the effect of collateralization is examined thoroughly. / AFRIKAANSE OPSOMMING: Die rentekoers mark het dramaties verander sedert die 2007 krediet krisis met 'n ontplo ng van basisverspreidings tussen koerse van verskillende looptye ("tenor") en geldeenhede. As gevolg, is die klassieke replikasie van FRA koerse met LIBOR sigkoerse nie langer geldig nie. Verder het die 2007 kredietkrisis 'n skeiding veroorsaak tussen die kromme wat gebruik word vir diskontering en die voorwaardse of vooruitskattings krommes wat toekomstige kontantvloei voorspel. 'n Verdere impak van die kredietkrisis in risikobestuur is dat mark deelnemers begin het om meer klem te lê op verskille tussen aangevulde en onaangevulde handel. Deesdae, met die algemene gebruik van kollaterale sekuriteit, veral in ruiltransaksiekontrakte, is die oornagse indeks ruiltransaksie (overnight index swap, OIS) koers die geskikte maatstaf om aangevulde handel te diskonteer. Geïnspireer deur die gedagteryke werk van Mercurio (2010a,b), Kijima et al. (2008), Fujii et al. (2011), Bianchetti (2010b), met bydrae van menige outeurs, en gemotiveer deur die evolusie van die rentekoers markte en modelle, ondersoek hierdie tesis 'n nuwe raamwerk wat multikrommes gebruik om rentekoers afgeleide effekte te waardeer wat versoenbaar is met die lopende mark praktyk. Eerstens, bespreek ons die oorsake van die 2007 kredietkrisis en die ernstige nagevolge vir die waardering van rentekoers afgeleide effekte. Ons beklemtoon die noodsaaklikheid van 'n multikromme waarderings raamwerk vir rentekoers afgeleide effekte. Dit word gevolg deur 'n bespreking oor die belangrikheid van aanvulling en OIS diskontering in die waardering van oor-die-toonbank (over-the-counter, OTC) effekte. Die teoretiese raamwerk en die praktiese implimentering van die multikromme waarderings metode word bespreek. Ons stel ook ten toon 'n skoenlus ("bootstrapping") algoritme wat gebruik kan word om meervoudige opbrengs krommes saam te stel en die dan te pas op mark pryse van vanielje kontrakte. Tweedens, met 'n enkel geldeenheid ekonomie as beginpunt, word die uitgebreide weergawe van die LIBOR Mark Model (ontwikkel deur Mercurio (2010a,b), wat 'n gesamentlike model van FRA koerse voorstel), geïmpliseerde termyn koerse en hul ooreenstemmende verspreiding bestudeer. Ooreenkomstig word die uitgebreide weergawe van die kort koers model in 'n multikromme opset en in die aanwesigheid van basisspreiding (voorgestel deur Kijima et al. (2008)) uiteengesit en bespreek. Hierdie werk verskaf 'n uitvoerige analise van hierdie uitbreidings en die ooreenstemmende geslote formules vir vloeibare produkte soos perke en ruiltransaksie opsies. Ten slotte, in die multi-geldeenheid geval, word die HJM model met stogastiese basisverspreiding (voorgestel deur Fujii et al. (2011)), nie-strydig met buitelandse valuta en kruisvaluta ruiltransaksie markte wat die effekte van aanvulling insluit word deuglik bestudeer.

Thesis (MSc (Mathematics))--Stellenbosch University, 2008. / In this thesis we study a standard example of refinable functions, that is, functions which can be reproduced by the integer shifts of their own dilations. Using the cardinal B-spline as an introductory example, we prove some of its properties, thereby building a basis for a later extension to the vector setting. Defining a subdivision scheme associated to the B-spline refinement mask, we then present the proof of a well-known convergence result. Subdivision is a powerful tool used in computer-aided geometric design (CAGD) for the generation of curves and surfaces. The basic step of a subdivision algorithm consists of starting with a given set of points, called the initial control points, and creating new points as a linear combination of the previous ones, thereby generating new control points. Under certain conditions, repeated applications of this procedure yields a continuous limit curve. One important goal of this thesis is to study a particular extension of scalar subdivision to matrix subdivision ...

Thesis (MSc (Mathematics))--Stellenbosch University, 2008. / This is an exposition of the explicit approach to Local Class Field Theory due to J. Tate and J. Lubin. We mainly follow the treatment given in [15] and [25]. We start with an informal introduction to p-adic numbers. We then review the standard theory of valued elds and completion of those elds. The complete discrete valued elds with nite residue eld known as local elds are our main focus. Number theoretical aspects for local elds are considered. The standard facts about Hensel's lemma, Galois and rami cation theory for local elds are treated. This being done, we continue our discussion by introducing the key notion of relative Lubin-Tate formal groups and modules. The torsion part of a relative Lubin-Tate module is then used to generate a tower of totally rami ed abelian extensions of a local eld. Composing this tower with the maximal unrami ed extension gives the maximal abelian extension: this is the local Kronecker-Weber theorem. What remains then is to state and prove the theorems for explicit local class eld theory and end our discussion.