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Local class field theory via Lubin-Tate theoryMohamed, Adam 12 1900 (has links)
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.
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Growth optimal portfolios and real world pricingRamarimbahoaka, Dimbinirina 12 1900 (has links)
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.
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Modelling of flow through porous packing elements of a CO2 absorption towerRautenbach, Christo 12 1900 (has links)
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.
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Interpolatory refinement pairs with properties of symmetry and polynomial fillingGavhi, Mpfareleni Rejoyce 03 1900 (has links)
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.
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A kernel to support computer-aided verification of embedded softwareGrobler, Leon D 03 1900 (has links)
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.
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Geometric actions of the absolute Galois groupJoubert, Paul 03 1900 (has links)
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.
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The discrete pulse transform and applicationsDu Toit, Jacques Pierre 03 1900 (has links)
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.
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Cardinal spline wavelet decomposition based on quasi-interpolation and local projectionAhiati, Veroncia Sitsofe 03 1900 (has links)
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.
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Endomorphism rings of hyperelliptic JacobiansKriel, Marelize 03 1900 (has links)
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.
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A numerical and analytical investigation into non-Hermitian HamiltoniansWessels, Gert Jermia Cornelus 03 1900 (has links)
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.
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