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The class number one problem in function fieldsHarper, John-Paul 12 1900 (has links)
Thesis (MComm)--Stellenbosch University, 2003. / ENGLISH ABSTRACT: In this dissertation I investigate the class number one problem in function fields. More
precisely I give a survey of the current state of research into extensions of a rational function
field over a finite field with principal ring of integers. I focus particularly on the quadratic
case and throughout draw analogies and motivations from the classical number field situation.
It was the "Prince of Mathematicians" C.F. Gauss who first undertook an in depth study of
quadratic extensions of the rational numbers and the corresponding rings of integers. More
recently however work has been done in the situation of function fields in which the arithmetic
is very similar.
I begin with an introduction into the arithmetic in function fields over a finite field and
prove the analogies of many of the classical results. I then proceed to demonstrate how the
algebra and arithmetic in function fields can be interpreted geometrically in terms of curves
and introduce the associated geometric language. After presenting some conjectures, I proceed
to give a survey of known results in the situation of quadratic function fields. I present also
a few results of my own in this section. Lastly I state some recent results regarding arbitrary
extensions of a rational function field with principal ring of integers and give some heuristic
results regarding class groups in function fields. / AFRIKAANSE OPSOMMING: In hierdie tesis ondersoek ek die klasgetal een probleem in funksieliggame. Meer spesifiek
ondersoek ek die huidige staat van navorsing aangaande uitbreidings van 'n rasionale funksieliggaam
oor 'n eindige liggaam sodat die ring van heelgetalle 'n hoofidealgebied is. Ek kyk in
besonder na die kwadratiese geval, en deurgaans verwys ek na die analoog in die klassieke
getalleliggaam situasie. Dit was die beroemde wiskundige C.F. Gauss wat eerste kwadratiese
uitbreidings van die rasionale getalle en die ooreenstemende ring van heelgetalle in diepte ondersoek
het. Onlangs het wiskundiges hierdie probleme ook ondersoek in die situasie van
funksieliggame oor 'n eindige liggaam waar die algebraïese struktuur baie soortgelyk is.
Ek begin met 'n inleiding tot die rekenkunde in funksieliggame oor 'n eindige liggaam en
bewys die analogie van baie van die klassieke resultate. Dan verduidelik ek hoe die algebra in
funksieliggame geometries beskou kan word in terme van kurwes en gee 'n kort inleiding tot
die geometriese taal. Nadat ek 'n paar vermoedes bespreek, gee ek 'n oorsig van wat alreeds
vir quadratiese funksieliggame bewys is. In hierdie afdeling word 'n paar resultate van my
eie ook bewys. Dan vermeld ek 'n paar resultate aangaande algemene uitbreidings van 'n
rasionale funksieliggaam oor 'n eindige liggaam waar die van ring heelgetalle 'n hoofidealgebied
is. Laastens verwys ek na 'n paar heurisitiese resultate aangaande klasgroepe in funksieliggame.
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Investigating the simultaneous effect of age and temperature on the population dynamics of female tsetse fliesElama Ameh, Josephine, Ochigbo, Josephine Elanma 12 1900 (has links)
Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: Age and temperature are two factors that affect mortality in adult tsetse flies. Both are found
to be very important, but the simultaneous effect of these factors on the mortality rate have
not been studied. This study seeks to address this, with an application to a population of
female tsetse, using a model based on partial differential equations. Adult mortality is agedependent
and is modelled as the sum of two exponentials, with four parameters (coefficients
of each exponential): numerical analysis of a population model with this mortality structure
predicts exponential growth. Analysis of each of the parameters through parameter variation
shows that two of these parameters control the mortality of the nulliparous (ages 0 − 10
days) flies only while the other two only take care of flies of mature ages. Measurement of
the impact of these parameters on the mortality of tsetse of different ages by the normalized
forward sensitivity index method is also carried out. This is followed by fitting the model
based on the age-dependent mortality along with a constant tsetse birth rate to data representing
the catches of female Glossina pallidipes at Rekomitjie Research station, Zimbabwe.
Considering a three parameter adult tsetse mortality, parameter analysis shows the effect of
one of the parameters to affect the mortality of flies of all ages while a second controls only the
mature tsetse flies of reproductive ages. A further analysis resulted in the estimate of these
parameters as functions of temperature, thereby leading to the establishment of an age and
temperature-dependent adult tsetse mortality. Using data for the daily average temperature
records obtained in 1981 on Antelope Island, Lake Kariba, Zimbabwe, daily changes in the
pupal duration (adult tsetse birth rate) changes negatively with temperature change. Incorporating
this (temperature-dependent ) birth rate into the model, together with the established
age and temperature-dependent adult mortality, the adult tsetse population dynamics is explored
numerically. The latter model is then fitted to population data of female Glossina
morsitans morsitans obtained from the same Island and for the same period as used for
the temperature data. The data suggests peak tsetse population to be in the month of July
and lowest in the month of December. The first quarter of the year is predicted to be most
favorable for breeding tsetse while the second, showed a period of stable growth rate and a
time of tsetse abundance. In addition, the dynamics with both age and temperature showed a
non-uniform daily population growth contrary to that with age effect only. This study has enhanced
our understanding of tsetse population dynamics for age and temperature-dependent
adult mortality with temperature-dependent pupal duration and suggests the period of tsetse
abundance on Antelope Island. / AFRIKAANSE OPSOMMING: Geen opsomming in Afrikaans.
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On towers of function fields over finite fieldsLotter, Ernest Christiaan 03 1900 (has links)
Thesis (PhD (Mathematical Sciences))--University of Stellenbosch, 2007. / Explicit towers of algebraic function fields over finite fields are studied
by considering their ramification behaviour and complete splitting. While
the majority of towers in the literature are recursively defined by a single
defining equation in variable separated form at each step, we consider
towers which may have different defining equations at each step and with
arbitrary defining polynomials.
The ramification and completely splitting loci are analysed by directed
graphs with irreducible polynomials as vertices. Algorithms are exhibited
to construct these graphs in the case of n-step and -finite towers.
These techniques are applied to find new tamely ramified n-step towers
for 1 n 3. Various new tame towers are found, including a family
of towers of cubic extensions for which numerical evidence suggests that
it is asymptotically optimal over the finite field with p2 elements for each
prime p 5. Families of wildly ramified Artin-Schreier towers over small
finite fields which are candidates to be asymptotically good are also considered
using our method.
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Off-line signature verificationCoetzer, Johannes 03 1900 (has links)
Thesis (PhD (Mathematical Sciences))--University of Stellenbosch, 2005. / A great deal of work has been done in the area of off-line signature verification over the
past two decades. Off-line systems are of interest in scenarios where only hard copies of
signatures are available, especially where a large number of documents need to be authenticated.
This dissertation is inspired by, amongst other things, the potential financial
benefits that the automatic clearing of cheques will have for the banking industry.
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On the analysis of refinable functions with respect to mask factorisation, regularity and corresponding subdivision convergenceDe Wet, Wouter de Vos 12 1900 (has links)
Thesis (PhD (Mathematical Sciences))--University of Stellenbosch, 2007. / We study refinable functions where the dilation factor is not always assumed to be 2. In
our investigation, the role of convolutions and refinable step functions is emphasized as a
framework for understanding various previously published results. Of particular importance
is a class of polynomial factors, which was first introduced for dilation factor 2 by
Berg and Plonka and which we generalise to general integer dilation factors.
We obtain results on the existence of refinable functions corresponding to certain reduced
masks which generalise similar results for dilation factor 2, where our proofs do not
rely on Fourier methods as those in the existing literature do.
We also consider subdivision for general integer dilation factors. In this regard, we extend
previous results of De Villiers on refinable function existence and subdivision convergence
in the case of positive masks from dilation factor 2 to general integer dilation factors.
We also obtain results on the preservation of subdivision convergence, as well as on the
convergence rate of the subdivision algorithm, when generalised Berg-Plonka polynomial
factors are added to the mask symbol.
We obtain sufficient conditions for the occurrence of polynomial sections in refinable
functions and construct families of related refinable functions.
We also obtain results on the regularity of a refinable function in terms of the mask
symbol factorisation. In this regard, we obtain much more general sufficient conditions
than those previously published, while for dilation factor 2, we obtain a characterisation of
refinable functions with a given number of continuous derivatives.
We also study the phenomenon of subsequence convergence in subdivision, which explains
some of the behaviour that we observed in non-convergent subdivision processes
during numerical experimentation. Here we are able to establish different sets of sufficient
conditions for this to occur, with some results similar to standard subdivision convergence,
e.g. that the limit function is refinable. These results provide generalisations of the corresponding
results for subdivision, since subsequence convergence is a generalisation of
subdivision convergence. The nature of this phenomenon is such that the standard subdivision
algorithm can be extended in a trivial manner to allow it to work in instances where
it previously failed.
Lastly, we show how, for masks of length 3, explicit formulas for refinable functions can
be used to calculate the exact values of the refinable function at rational points.
Various examples with accompanying figures are given throughout the text to illustrate
our results.
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Polynomial containment in refinement spaces and wavelets based on local projection operatorsMoubandjo, Desiree V. 03 1900 (has links)
Dissertation (PhD)--University of Stellenbosch, 2007. / ENGLISH ABSTRACT: See full text for abstract / AFRIKAANSE OPSOMMING: Sien volteks vir opsomming
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Refinable functions with prescribed values at the integersGavhi, Mpfareleni Rejoyce 03 1900 (has links)
Thesis (PhD)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: See full text / AFRIKAANSE OPSOMMING: Sien volteks
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Limit theorems for integer partitions and their generalisationsRalaivaosaona, Dimbinaina 03 1900 (has links)
Thesis (PhD)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: Various properties of integer partitions are studied in this work, in particular
the number of summands, the number of ascents and the multiplicities of
parts. We work on random partitions, where all partitions from a certain
family are equally likely, and determine moments and limiting distributions of
the different parameters.
The thesis focuses on three main problems: the first of these problems is
concerned with the length of prime partitions (i.e., partitions whose parts are
all prime numbers), in particular restricted partitions (i.e., partitions where
all parts are distinct). We prove a central limit theorem for this parameter
and obtain very precise asymptotic formulas for the mean and variance.
The second main focus is on the distribution of the number of parts of a
given multiplicity, where we obtain a very interesting phase transition from
a Gaussian distribution to a Poisson distribution and further to a degenerate
distribution, not only in the classical case, but in the more general context of
⋋-partitions: partitions where all the summands have to be elements of a given
sequence ⋋ of integers.
Finally, we look into another phase transition from restricted to unrestricted
partitions (and from Gaussian to Gumbel-distribution) as we study
the number of summands in partitions with bounded multiplicities. / AFRIKAANSE OPSOMMING: Verskillende eienskappe van heelgetal-partisies word in hierdie tesis bestudeer,
in die besonder die aantal terme, die aantal stygings en die veelvoudighede
van terme. Ons werk met stogastiese partisies, waar al die partisies in ’n
sekere familie ewekansig is, en ons bepaal momente en limietverdelings van die
verskillende parameters.
Die teses fokusseer op drie hoofprobleme: die eerste van hierdie probleme
gaan oor die lengte van priemgetal-partisies (d.w.s., partisies waar al die terme
priemgetalle is), in die besonder beperkte partisies (d.w.s., partisies waar al
die terme verskillend is). Ons bewys ’n sentrale limietstelling vir hierdie parameter
en verkry baie presiese asimptotiese formules vir die gemiddelde en die
variansie.
Die tweede hooffokus is op die verdeling van die aantal terme van ’n gegewe
veelvoudigheid, waar ons ’n baie interessante fase-oorgang van ’n normaalverdeling
na ’n Poisson-verdeling en verder na ’n ontaarde verdeling verkry, nie
net in die klassieke geval nie, maar ook in die meer algemene konteks van sogenaamde
⋋-partities: partisies waar al die terme elemente van ’n gegewe ry ⋋ van heelgetalle moet wees.
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A topological framework for modeling belief revisionJeftha, Lindsey Craig 12 1900 (has links)
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.
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Dreieckverbande : lineare und quadratische darstellungstheorie / Triangle lattices : linear and quadratic representation theoryWild, Marcel Wolfgang 05 1900 (has links)
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.
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