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.

Thesis (MSc (Mathematics))--Stellenbosch University, 2008. / In the Benchmark Approach to Finance, it has been shown that by taking the Growth Optimal Portfolio as numéraire, a candidate for a pricing derivatives formula under the real world probability can be given. This result allows us to price in an incomplete financial market model. The result comes from two different approaches. In the first approach we use the supermartingale property of portfolios in units of the benchmark portfolio which leads to the fact that an equivalent measure is not needed. In the second approach the numéraire property of the Growth Optimal Portfolio is used. The numéraire portfolio defines an equivalent martingale measure and by change of measure using the Radon-Nikodým derivative, a real world pricing formula is derived which is the same as the one given by the first approach stated above.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2009. / ENGLISH ABSTRACT: Packed beds are widely used in industry to improve the total contact area between two substances in a multiphase process. The process typically involves forced convection of liquid or gas through either structured or dumped solid packings. Applications of such multiphase processes include mass transfer to catalyst particles forming the packed bed and the adsorption of gases or liquids on the solid packing. An experimental study on the determination of air flow pressure drops over different packingmaterialswas carried out at the Telemark University College in Porsgrunn,Norway. The packed bed consisted of a cylindrical column of diameter 0.072m and height 1.5m, filled with different packingmaterials. Air was pumped vertically upwards through a porous distributor to allow for a uniform inlet pressure. Resulting pressure values were measured at regular height intervals within the bed. Due to the geometric nature of a Raschig ring packing wall effects, namely the combined effects of extra wall shear stress due to the column surface and channelling due to packing adjacent to a solid column surface, were assumed to be negligible. Several mathematical drag models exist for packed beds of granular particles and an important question arises as to whether they can be generalized in a scientific manner to enhance the accuracy of predicting the drag for different kinds of packing materials. Problems with the frequently used Ergun equation, which is based on a tubular model for flow between granules and then being empirically adjusted, will be discussed. Some theoretical models that improve on the Ergun equation and their correlation with experimental work will be discussed. It is shown that a particular pore-scale model, that allows for different geometries and porosities, is superior to the Ergun equation in its predictions. Also important in the advanced models is the fact that it could take into account anomalies such as dead zones where no fluid transport is present and surfaces that do neither contribute to shear stress nor to interstitial form drag. The overall conclusion is that proper modelling of the dynamical situation present in the packing can provide drag models that can be used with confidence in a variety of packed bed applications. / AFRIKAANSE OPSOMMING: Gepakte materiaal strukture word in die industrie gebruik om die kontak area tussen twee stowwe in meervoudige faseprosesse te vergroot. Die proses gaan gewoonlik gepaard met geforseerde konveksie van ’n vloeistof of ’n gas deur gestruktureerde of lukrake soliede gepakte strukture. Toepassings van sulke meervoudige faseprossese sluit onder andere in die massa-oordrag na katalisator partikels wat die gepakte struktuur vorm of die absorpsie van gasse of vloeistowwe op die soliede gepakte elemente. ’n Eksperimentele ondersoek oor die drukval van veskillende gepakte elemente in ’n kolom is gedoen by die Telemark University College in Porsgrunn, Noorweë. Die gepakte struktuur het bestaan uit ’n kolommet ’n diameter van 0.072m en ’n hoogte van 1.5m. Lug is vertikaal opwaarts gepomp deur ’n poreuse plaat wat gesorg het vir ’n benaderde uniforme snelheidsprofiel. Die druk is toe op intervalle deur die poreuse struktuur gemeet. In die studie is die effekte van die eksterne wande, nl. die bydrae van die wand se wrywing en die vorming van kanale langs die kolom wand, as weglaatbaar aanvaar. Daar bestaan baie wiskundige dempingsmodelle vir gepakte strukture wat uit korrels saamgestel is. ’n Belangrike vraag kan dus gevra word, of laasgenoemde modelle veralgemeen kan word op ’n wetenskaplike manier om die demping deur verskillende gepakte strukture akkuraat te kan voorspel. Probleme wat ontstaan het met die wel bekende Ergun vergelyking, wat gebaseer is op ’n kapillêre model en wat toe verder aangepas is deur empiriese resultate van uniforme sfere, sal bespreek word. Teoretiesemodelle wat verbeteringe op die Ergun vergelyking voorstel sal bespreek word en vergelyk word met eksperimentele data. Daar word ook gewys dat ’n spesifieke porie-skaal model, wat aanpasbaar is vir verskillende geometrieë en porositeite, in baie gevalle beter is as die Ergun vergelyking. ’n Ander baie belangrike aspek van gevorderde modelle is die moontlikheid om stagnante gebiede in die gepakte strukture in ag te neem. Laasgenoemde gebiede sal die totale kontak area sowel as die intermediêre vorm demping verlaag. Die gevolgtrekking is dat wanneer deeglike modulering van dinamiese situasies in die industrie gedoen word kan dempings modelle met vertroue op ’n verskeidenheid gepakte strukture toegepas word.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2008. / Subdivision techniques have, over the last two decades, developed into a powerful tool in computer-aided geometric design (CAGD). In some applications it is required that data be preserved exactly; hence the need for interpolatory subdivision schemes. In this thesis,we consider the fundamentals of themathematical analysis of symmetric interpolatory subdivision schemes for curves, also with the property of polynomial filling up to a given odd degree, in the sense that, if the initial control point sequence is situated on such a polynomial curve, all the subsequent subdivision iterates fills up this curve, for it to eventually also become also the limit curve. A subdivision scheme is determined by its mask coefficients, which we find convenient to mathematically describe as a bi-infinite sequnce a with finite support. This sequence is in one-to-one correspondence with a corresponding Laurent polynomial A with coefficients given by the mask sequence a. After an introductory Chapter 1 on notation, basic definitions, and an overview of the thesis, we proceed in Chapter 2 to separately consider the issues of interpolation, symmetry and polynomial filling with respect to a subdivision scheme, eventually leading to a definition of the class Am,n of mask symbols in which all of the above desired properties are combined. We proceed in Chapter 3 to deduce an explicit characterization formula for the classAm,n, in the process also showing that its optimally local member is the well-known Dubuc–Deslauriers (DD) mask symbol Dm of order m. In fact, an alternative explicit characterization result appears in recent work by De Villiers and Hunter, in which the authors characterized mask symbols A ∈Am,n as arbitrary convex combinations of DD mask symbols. It turns out that Am,m = {Dm}, whereas the class Am,m+1 has one degree of freedom, which we interpret here in the formof a shape parameter t ∈ R for the resulting subdivision scheme. In order to investigate the convergence of subdivision schemes associated with mask symbols in Am,n, we first introduce in Chapter 4 the concept of a refinement pair (a,φ), consisting of a finitely-supported sequence a and a finitelysupported function φ, where φ is a refinable function in the sense that it can be expressed as a finite linear combination, as determined by a, of the integer shifts of its own dilation by factor 2. After presenting proofs of a variety of properties satisfied by a given refinement pair (a,φ), we next introduce the concept of an interpolatory refinement pair as one for which the refinable function φ interpolates the delta sequence at the integers. A fundamental result is then that the existence of an interpolatory refinement pair (a,φ) guarantees the convergence of the interpolatory subdivision scheme with subdivision mask a, with limit function © expressible as a linear combination of the integer shifts of φ, and with all the subdivision iterates lying on ©. In Chapter 5, we first present a fundamental result byMicchelli, according to which interpolatory refinable function existence is obtained for mask symbols in Am,n if the mask symbol A is strictly positive on the unit circle in complex plane. After showing that the DD mask symbol Dm satisfies this sufficient property, we proceed to compute the precise t -interval for such positivity on the unit circle to occur for the mask symbols A = Am(t |·) ∈Am,m+1. Also, we compare our numerical results with analogous ones in the literature. Finally, in Chapter 6, we investigate the regularity of refinable functions φ = φm(t |·) corresponding to mask symbols Am(t |·). Using a standard result fromthe literature in which a lower bound on the Hölder continuity exponent of a refinable function φ is given explicitly in terms of the spectral radius of a matrix obtained from the corresponding mask sequence a, we compute this lower bound for selected values of m.

Thesis (MSc (Mathematical Sciences)--University of Stellenbosch, 2006. / Formal methods, such as model checking, have the potential to improve the reliablility of software. Abstract models of systems are subjected to formal analysis, often showing subtle defects not discovered by traditional testing.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2006. / This thesis gives an introduction to some of the ideas originating from A. Grothendieck's 1984 manuscript Esquisse d'un programme. Most of these ideas are related to a new geometric approach to studying the absolute Galois group over the rationals by considering its action on certain geometric objects such as dessins d'enfants (called stick figures in this thesis) and the fundamental groups of certain moduli spaces of curves. I start by defining stick figures and explaining the connection between these innocent combinatorial objects and the absolute Galois group. I then proceed to give some background on moduli spaces. This involves describing how Teichmuller spaces and mapping class groups can be used to address the problem of counting the possible complex structures on a compact surface. In the last chapter I show how this relates to the absolute Galois group by giving an explicit description of the action of the absolute Galois group on the fundamental group of a particularly simple moduli space. I end by showing how this description was used by Y. Ihara to prove that the absolute Galois group is contained in the Grothendieck-Teichmuller group.

Thesis (MSc (Mathematical Sciences))--University of Stellenbosch, 2007. / Data analysis frequently involves the extraction (i.e. recognition) of parts that are important at the expense of parts that are deemed unimportant. Many mathematical perspectives exist for performing these separations, however no single technique is a panacea as the de nition of signal and noise depends on the purpose of the analysis. For data that can be considered a sampling of a smooth function with added 'well-behaved' noise, linear techniques tend to work well. When large impulses or discontinuities are present, a non-linear approach becomes necessary. The LULU operators, composed using the simplest rank selectors, are non-linear operators that are comparable to the well-known median smoothers, but are computationally e cient and allow a conceptually simple description of behaviour. De ned using compositions of di erent order LULU operators, the discrete pulse transform (dpt) allows the interpretation of sequences in terms of pulses of di erent scales: thereby creating a multi-resolution analysis. These techniques are very di erent from those of standard linear analysis, which renders intuitions regarding their behaviour somewhat undependable. The LULU perspective and analysis tools are investigated with a strong emphasis on practical applications. The LULU smoothers are known to separate signal and noise ef- ciently: they are idempotent and co-idempotent. Sequences are smoothed by mapping them into smoothness classes; which is achieved by the removal, in a consistent manner, of block-pulses. Furthermore, these operators preserve local trend (i.e. they are fully trend preserving). Di erences in interpretation with respect to Fourier and Wavelet decompositions are also discussed. The dpt is de ned, its implications are investigated, and a linear time algorithm is discussed. The dpt is found to allow a multi-resolution measure of roughness. Practical sequence processing through the reconstruction of modi ed pulses is possible; in some cases still maintaining a consistent multi-resolution interpretation. Extensions to two-dimensions is discussed, and a technique for the estimation of standard deviation of a random distribution is presented. These tools have been found to be e ective in the analysis and processing of sequences and images. The LULU tools are an useful alternative to standard analysis methods. The operators are found to be robust in the presence of impulsive and more 'well-behaved' noise. They allow the fast design and deployment of specialized detection and processing algorithms, and are possibly very useful in creating automated data analysis solutions.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2009. / Wavelet decomposition techniques have grown over the last two decades into a powerful tool in signal analysis. Similarly, spline functions have enjoyed a sustained high popularity in the approximation of data. In this thesis, we study the cardinal B-spline wavelet construction procedure based on quasiinterpolation and local linear projection, before specialising to the cubic B-spline on a bounded interval. First, we present some fundamental results on cardinal B-splines, which are piecewise polynomials with uniformly spaced breakpoints at the dyadic points Z/2r, for r ∈ Z. We start our wavelet decomposition method with a quasi-interpolation operator Qm,r mapping, for every integer r, real-valued functions on R into Sr m where Sr m is the space of cardinal splines of order m, such that the polynomial reproduction property Qm,rp = p, p ∈ m−1, r ∈ Z is satisfied. We then give the explicit construction of Qm,r. We next introduce, in Chapter 3, a local linear projection operator sequence {Pm,r : r ∈ Z}, with Pm,r : Sr+1 m → Sr m , r ∈ Z, in terms of a Laurent polynomial m solution of minimally length which satisfies a certain Bezout identity based on the refinement mask symbol Am, which we give explicitly. With such a linear projection operator sequence, we define, in Chapter 4, the error space sequence Wr m = {f − Pm,rf : f ∈ Sr+1 m }. We then show by solving a certain Bezout identity that there exists a finitely supported function m ∈ S1 m such that, for every r ∈ Z, the integer shift sequence { m(2 · −j)} spans the linear space Wr m . According to our definition, we then call m the mth order cardinal B-spline wavelet. The wavelet decomposition algorithm based on the quasi-interpolation operator Qm,r, the local linear projection operator Pm,r, and the wavelet m, is then based on finite sequences, and is shown to possess, for a given signal f, the essential property of yielding relatively small wavelet coefficients in regions where the support interval of m(2r · −j) overlaps with a Cm-smooth region of f. Finally, in Chapter 5, we explicitly construct minimally supported cubic B-spline wavelets on a bounded interval [0, n]. We also develop a corresponding explicit decomposition algorithm for a signal f on a bounded interval. ii Throughout Chapters 2 to 5, numerical examples are provided to graphically illustrate the theoretical results.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2005. / The aim of this thesis is to study the unital subrings contained in associative algebras arising as the endomorphism algebras of hyperelliptic Jacobians over finite fields. In the first part we study associative algebras with special emphasis on maximal orders. In the second part we introduce the theory of abelian varieties over finite fields and study the ideal structures of their endomorphism rings. Finally we specialize to hyperelliptic Jacobians and study their endomorphism rings.

Thesis (MSc (Physical and Mathematical Analysis))--University of Stellenbosch, 2009. / In this thesis we aim to show that the Schr odinger equation, which is a boundary eigenvalue problem, can have a discrete and real energy spectrum (eigenvalues) even when the Hamiltonian is non-Hermitian. After a brief introduction into non-Hermiticity, we will focus on solving the Schr odinger equation with a special class of non-Hermitian Hamiltonians, namely PT - symmetric Hamiltonians. PT -symmetric Hamiltonians have been discussed by various authors [1, 2, 3, 4, 5] with some of them focusing speci cally on obtaining the real and discrete energy spectrum. Various methods for solving this problematic Schr odinger equation will be considered. After starting with perturbation theory, we will move on to numerical methods. Three di erent categories of methods will be discussed. First there is the shooting method based on a Runge-Kutta solver. Next, we investigate various implementations of the spectral method. Finally, we will look at the Riccati-Pad e method, which is a numerical implemented analytical method. PT -symmetric potentials need to be solved along a contour in the complex plane. We will propose modi cations to the numerical methods to handle this. After solving the widely documented PT -symmetric Hamiltonian H = p2 􀀀(ix)N with these methods, we give a discussion and comparison of the obtained results. Finally, we solve another PT -symmetric potential, illustrating the use of paths in the complex plane to obtain a real and discrete spectrum and their in uence on the results.