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Parametrizing finite order automorphisms of power series ringsBasson, Dirk (Dirk Johannes) 12 1900 (has links)
Thesis (MSc (Mathematics))--University of Stellenboswch, 2010. / ENGLISH ABSTRACT: In the work of Green and Matignon it was shown that the Oort-Sekiguchi
conjecture is equivalent to a local question of lifting automorphisms of power
series rings. The Oort-Sekiguchi conjecture asks when an algebraic curve
in characteristic p can be lifted to a relative curve in characteristic 0, while
keeping the same automorphism group. The local formulation asks when an
automorphism of a power series ring over a field k of characteristic p can be
lifted to an automorphism of a power series ring over a discrete valuation
ring with residue field k of the same order as the original automorphism.
This thesis looks at the local formulation and surveys many of the results
for this case. At the end it presents a new theorem giving a Hensel's Lemma
type sufficient condition under which lifting is possible. / AFRIKAANSE OPSOMMING: Green en Matignon het bewys dat die Oort-Sekiguchi vermoede ekwivalent
is aan `n lokale vraag oor of outomorfismes van magsreeksringe gelig kan
word. Die Oort-Sekiguchi vermoede vra of `n algebra ese kromme in karakteristiek
p gelig kan word na `n relatiewe kromme in karakteristiek 0, terwyl
dit dieselfde outomorfisme groep behou. Die lokale vraag vra wanneer
`n outomorfisme van `n magsreeksring oor `n liggaam k van karakteristiek
p gelig kan word na `n outomorfisme van `n magsreeksring oor `n diskrete
waarderingsring met residuliggaam k, terwyl dit dieselfde orde behou as die
aanvanklike outomorfisme.
Hierdie tesis fokus op die lokale vraag en bied `n opsomming van baie
bekende resultate vir hierdie geval. Aan die einde word `n nuwe stelling
aangebied wat voorwaardes stel waaronder hierdie vraag positief beantwoord
kan word.
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Modelling of nonlinear dynamic systems : using surrogate data methodsConradie, Tanja 12 1900 (has links)
Thesis (MSc)--Stellenbosch University, 2000. / ENGLISH ABSTRACT: This study examined nonlinear modelling techniques as applied to dynamic systems, paying
specific attention to the Method of Surrogate Data and its possibilities. Within the field of
nonlinear modelling, we examined the following areas of study: attractor reconstruction, general
model building techniques, cost functions, description length, and a specific modelling
methodology. The Method of Surrogate Data was initially applied in a more conventional
application, i.e. testing a time series for nonlinear, dynamic structure. Thereafter, it was used in a
less conventional application; i.e. testing the residual vectors of a nonlinear model for
membership of identically and independently distributed (i.i.d) noise.
The importance of the initial surrogate analysis of a time series (determining whether the apparent
structure of the time series is due to nonlinear, possibly chaotic behaviour) was illustrated. This
study confrrmed that omitting this crucial step could lead to a flawed conclusion.
If evidence of nonlinear structure in the time series was identified, a radial basis model was
constructed, using sophisticated software based on a specific modelling methodology. The model
is an iterative algorithm using minimum description length as the stop criterion. The residual
vectors of the models generated by the algorithm, were tested for membership of the dynamic
class described as i.i.d noise. The results of this surrogate analysis illustrated that, as the model
captures more of the underlying dynamics of the system (description length decreases), the
residual vector resembles Li.d noise. It also verified that the minimum description length
criterion leads to models that capture the underlying dynamics of the time series, with the residual
vector resembling Li.d noise. In the case of the "worst" model (largest description length), the
residual vector could be distinguished from Li.d noise, confirming that it is not the "best" model.
The residual vector of the "best" model (smallest description length), resembled Li.d noise,
confirming that the minimum description length criterion selects a model that captures the
underlying dynamics of the time series.
These applications were illustrated through analysis and modelling of three time series: a time
series generated by the Lorenz equations, a time series generated by electroencephalograhpic
signal (EEG), and a series representing the percentage change in the daily closing price of the
S&P500 index. / AFRIKAANSE OPSOMMING: In hierdie studie ondersoek ons nie-lineere modelleringstegnieke soos toegepas op dinamiese
sisteme. Spesifieke aandag word geskenk aan die Metode van Surrogaat Data en die
moontlikhede van hierdie metode. Binne die veld van nie-lineere modellering het ons die
volgende terreine ondersoek: attraktor rekonstruksie, algemene modelleringstegnieke,
kostefunksies, beskrywingslengte, en 'n spesifieke modelleringsalgoritme. Die Metode and
Surrogaat Data is eerstens vir 'n meer algemene toepassing gebruik wat die gekose tydsreeks vir
aanduidings van nie-lineere, dimanise struktuur toets. Tweedens, is dit vir 'n minder algemene
toepassing gebruik wat die residuvektore van 'n nie-lineere model toets vir lidmaatskap van
identiese en onafhanlike verspreide geraas.
Die studie illustreer die noodsaaklikheid van die aanvanklike surrogaat analise van 'n tydsreeks,
wat bepaal of die struktuur van die tydsreeks toegeskryf kan word aan nie-lineere, dalk chaotiese
gedrag. Ons bevesting dat die weglating van hierdie analise tot foutiewelike resultate kan lei.
Indien bewyse van nie-lineere gedrag in die tydsreeks gevind is, is 'n model van radiale
basisfunksies gebou, deur gebruik te maak van gesofistikeerde programmatuur gebaseer op 'n
spesifieke modelleringsmetodologie. Dit is 'n iteratiewe algoritme wat minimum
beskrywingslengte as die termineringsmaatstaf gebruik. Die model se residuvektore is getoets vir
lidmaatskap van die dinamiese klas wat as identiese en onafhanlike verspreide geraas bekend
staan. Die studie verifieer dat die minimum beskrywingslengte as termineringsmaatstaf weI
aanleiding tot modelle wat die onderliggende dinamika van die tydsreeks vasvang, met die
ooreenstemmende residuvektor wat nie onderskei kan word van indentiese en onafhanklike
verspreide geraas nie. In die geval van die "swakste" model (grootse beskrywingslengte), het die
surrogaat analise gefaal omrede die residuvektor van indentiese en onafhanklike verspreide
geraas onderskei kon word. Die residuvektor van die "beste" model (kleinste
beskrywingslengte), kon nie van indentiese en onafhanklike verspreide geraas onderskei word nie
en bevestig ons aanname.
Hierdie toepassings is aan die hand van drie tydsreekse geillustreer: 'n tydsreeks wat deur die
Lorenz vergelykings gegenereer is, 'n tydsreeks wat 'n elektroenkefalogram voorstel en derdens,
'n tydsreeks wat die persentasie verandering van die S&P500 indeks se daaglikse sluitingsprys
voorstel.
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Spectrum preserving linear mappings between Banach algebrasWeigt, Martin 04 1900 (has links)
Thesis (MSc)--University of Stellenbosch, 2003. / ENGLISH ABSTRACT: Let A and B be unital complex Banach algebras with identities 1 and I'
respectively. A linear map T : A -+ B is invertibility preserving if Tx is
invertible in B for every invertible x E A. We say that T is unital if Tl = I'.
IfTx2 = (TX)2 for all x E A, we call T a Jordan homomorphism. We examine
an unsolved problem posed by 1. Kaplansky:
Let A and B be unital complex Banach algebras and T : A -+ B a unital
invertibility preserving linear map. What conditions on A, Band T imply
that T is a Jordan homomorphism?
Partial motivation for this problem are the Gleason-Kahane-Zelazko Theorem
(1968) and a result of Marcus and Purves (1959), these also being special
instances of the problem. We will also look at other special cases answering
Kaplansky's problem, the most important being the result stating that if A
is a von Neumann algebra, B a semi-simple Banach algebra and T : A -+ B
a unital bijective invertibility preserving linear map, then T is a Jordan
homomorphism (B. Aupetit, 2000).
For a unital complex Banach algebra A, we denote the spectrum of x E A
by Sp (x, A). Let a(x, A) denote the union of Sp (x, A) and the bounded
components of <C \ Sp (x, A). We denote the spectral radius of x E A by
p(x, A).
A unital linear map T between unital complex Banach algebras A and
B is invertibility preserving if and only if Sp (Tx, B) C Sp (x, A) for all
x E A. This leads one to consider the problems that arise when, in turn,
we replace the invertibility preservation property of T in Kaplansky's problem
with Sp (Tx, B) = Sp (x, A) for all x E A, a(Tx, B) = a(x, A) for all
x E A, and p(Tx, B) = p(x, A) for all x E A. We will also investigate
some special cases that are solutions to these problems. The most important
of these special cases says that if A is a semi-simple Banach algebra, B a
primitive Banach algebra with minimal ideals and T : A -+ B a surjective
linear map satisfying a(Tx, B) = a(x, A) for all x E A, then T is a Jordan
homomorphism (B. Aupetit and H. du T. Mouton, 1994). / AFRIKAANSE OPSOMMING: Gestel A en B is unitale komplekse Banach algebras met identiteite 1 en I'
onderskeidelik. 'n Lineêre afbeelding T : A -+ B is omkeerbaar-behoudend
as Tx omkeerbaar in B is vir elke omkeerbare element x E A. Ons sê dat T
unitaal is as Tl = I'. As Tx2 = (TX)2 vir alle x E A, dan noem ons T 'n
Jordan homomorfisme. Ons ondersoek 'n onopgeloste probleem wat deur I.
Kaplansky voorgestel is:
Gestel A en B is unitale komplekse Banach algebras en T : A -+ B is 'n
unitale omkeerbaar-behoudende lineêre afbeelding. Watter voorwaardes op
A, B en T impliseer dat T 'n Jordan homomorfisme is?
Gedeeltelike motivering vir hierdie probleem is die Gleason-Kahane-Zelazko
Stelling (1968) en 'n resultaat van Marcus en Purves (1959), wat terselfdertyd
ook spesiale gevalle van die probleem is. Ons salook na ander spesiale gevalle
kyk wat antwoorde lewer op Kaplansky se probleem. Die belangrikste van
hierdie resultate sê dat as A 'n von Neumann algebra is, B 'n semi-eenvoudige
Banach algebra is en T : A -+ B 'n unitale omkeerbaar-behoudende bijektiewe
lineêre afbeelding is, dan is T 'n Jordan homomorfisme (B. Aupetit,
2000).
Vir 'n unitale komplekse Banach algebra A, dui ons die spektrum van
x E A aan met Sp (x, A). Laat cr(x, A) die vereniging van Sp (x, A) en die
begrensde komponente van <C \ Sp (x, A) wees. Ons dui die spektraalradius
van x E A aan met p(x, A).
'n Unitale lineêre afbeelding T tussen unit ale komplekse Banach algebras
A en B is omkeerbaar-behoudend as en slegs as Sp (Tx, B) c Sp (x, A) vir
alle x E A. Dit lei ons om die probleme te beskou wat ontstaan as ons die
omkeerbaar-behoudende eienskap van T in Kaplansky se probleem vervang
met Sp (Tx, B) = Sp (x, A) vir alle x E A, O"(Tx, B) = O"(x, A) vir alle
x E A en p(Tx, B) = p(x, A) vir alle x E A, onderskeidelik. Ons salook
'n paar spesiale gevalle van hierdie probleme ondersoek. Die belangrikste
van hierdie spesiale gevalle sê dat as A 'n semi-eenvoudige Banach algebra
is, B 'n primitiewe Banach algebra met minimale ideale is, en T : A -+ B
'n surjektiewe lineêre afbeelding is sodanig dat O"(Tx, B) = O"(x, A) vir alle
x E A, dan is T 'n Jordan homomorfisme (B. Aupetit en H. du T. Mouton,
1994).
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Interpolatory refinable functions, subdivision and waveletsHunter, Karin M. 03 1900 (has links)
Thesis (DSc (Mathematical Sciences))--University of Stellenbosch, 2005. / Subdivision is an important iterative technique for the efficient generation of curves and
surfaces in geometric modelling. The convergence of a subdivision scheme is closely connected
to the existence of a corresponding refinable function. In turn, such a refinable
function can be used in the multi-resolutional construction method for wavelets, which
are applied in many areas of signal analysis.
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On the computation of freely generated modular latticesSemegni, Jean Yves 12 1900 (has links)
Thesis (PhD (Mathematical Sciences))--Stellenbosch University, 2008 / Please refer to full text for abstract.
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Contributions to centralizers in matrix ringsMarais, Magdaleen Suzanne 12 1900 (has links)
Thesis (PhD (Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: THE concept of a k-matrix in the full 2 2 matrix ring M2(R=hki), where R is an arbitrary unique
factorization domain (UFD) and k is an arbitrary nonzero nonunit in R, is introduced. We obtain
a concrete description of the centralizer of a k-matrix bB in M2(R=hki) as the sum of two subrings S1
and S2 ofM2(R=hki), where S1 is the image (under the natural epimorphism fromM2(R) toM2(R=hki))
of the centralizer in M2(R) of a pre-image of bB, and where the entries in S2 are intersections of certain
annihilators of elements arising from the entries of bB. Furthermore, necessary and sufficient conditions
are given for when S1 S2, for when S2 S1 and for when S1 = S2. It turns out that if R is a principal
ideal domain (PID), then every matrix in M2(R=hki) is a k-matrix for every k. However, this is not the
case in general if R is a UFD. Moreover, for every factor ring R=hki with zero divisors and every n > 3
there is a matrix for which the mentioned concrete description is not valid. Finally we provide a formula
for the number of elements of the centralizer of bB in case R is a UFD and R=hki is finite. / AFRIKAANSE OPSOMMING: DIE konsep van ’n k-matriks in die volledige 2 2 matriksring M2(R=hki), waar R ’n willekeurige
unieke faktoriseringsgebied (UFG) en k ’n willekeurige nie-nul nie-inverteerbare element in R
is, word bekendgestel. Ons verkry ’n konkrete beskrywing van die sentraliseerder van ’n k-matriks bB
in M2(R=hki) as die som van twee subringe S1 en S2 van M2(R=hki), waar S1 die beeld (onder die
natuurlike epimorfisme van M2(R) na M2(R=hki)) van die sentraliseerder in M2(R) van ’n trubeeld
vanbB is, en die inskrywings van S2 die deursnede van sekere annihileerders van elemente afkomstig van
die inskrywings van bB is. Verder word nodige en voldoende voorwaardes gegee vir wanneer S1 S2,
vir wanneer S2 S1 en vir wanneer S1 = S2. Dit blyk dat as R ’n hoofideaalgebied (HIG) is, dan is elke
matriks in M2(R=hki) ’n k-matriks vir elke k. Dit is egter nie in die algemeen waar indien R ’n UFG is
nie. Meer nog, vir elke faktorring R=hki met nuldelers en elke n > 3 is daar ’n matriks waarvoor die
bogenoemde konkrete beskrywing nie geldig is nie. Laastens word ’n formule vir die aantal elemente
van die sentraliseerder van bB verskaf, indien R ’n UFG en R=hki eindig is.
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Spectral theory in commutatively ordered banach algebrasMuzundu, Kelvin 12 1900 (has links)
Thesis (PhD)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: See full text. / AFRIKAANSE OPSOMMING: Sien volteks.
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Exact solutions for spherical relativistic models.Nyonyi, Yusuf. January 2011 (has links)
In this thesis we study relativistic models of gravitating
uids with heat
ow and electric
charge. Firstly, we derive the model of a charged shear-free spherically symmetric cosmological
model with heat
ow. The solution of the Einstein-Maxwell equations of the system
is governed by the pressure isotropy condition. This condition is a highly nonlinear partial
di erential equation. We analyse this master equation using Lie's group theoretic approach.
The Lie symmetry generators that leave the equation invariant are found. We provide exact
solutions to the gravitational potentials using the rst symmetry admitted by the equation.
Our new exact solutions contain the earlier results of Msomi et al (2011) without charge.
Using the second symmetry we are able to reduce the order of the master equation to a rst
order highly nonlinear di erential equation.
Secondly, we study a shear-free spherically symmetric cosmological model with heat
ow
in higher dimensions. We establish the Einstein eld equations and nd the governing
pressure isotropy condition. We use an algorithm due to Deng (1989) to provide several
new classes of solutions to the model. The four-dimensional case is contained in our general
result. Solutions due to Bergmann (1981), Maiti (1982), Modak (1984) and Sanyal and Ray
(1984) for the four-dimensional case are regained. We also establish a new class of solutions
that contains the results of Deng (1989) from four dimensions. / Thesis (M.Sc.)-University of KwaZulu-Natal, Durban, 2011.
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On the theory and examples of group extensions.Rodrigues, Bernardo Gabriel. January 1999 (has links)
The work described in this dissertation was largely motivated by the aim of producing a survey on the theory of group extensions. From the broad scope of the theory of group extensions we single out two aspects to discuss, namely the study of the split and the non-split cases and give examples of both. A great part of this dissertation is dedicated to the study of split extensions. After setting the background theory for the study of the split extensions we proceed in exploring the ramifications of this concept within the development of the group structure and consequently investigate well known products which are its derived namely the holomorph, and the wreath product. The theory of group presentations provides in principle the necessary tools that permit the description of a group by means of its generators and relators. Through this knowledge we give presentations for the groups of order pq,p2q and p3. Subsequently using a classical result of Gaschutz we investigate the split extensions of
non-abelian groups in which the normal subgroup is either a non-abelian normal nilpotent group or a non-abelian normal solvable group. We also study other cases of split extensions such as the affine subgroups of the general linear and the symplectic groups. It is expected that some of the results obtained will provide a theoretical algorithm to describe these affine subgroups. A particular case of the non-split extensions is discussed as the Frattini extensions. In fact a simplest example of a Frattini extension is a non-split extension in which the kernel of an epimorphism e is an irreducible G-module. / Thesis (M.Sc.)-University of Natal, Pietermaritzburg, 1999.
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Parameters related to fractional domination in graphs.Erwin, D. J. January 1995 (has links)
The use of characteristic functions to represent well-known sets in graph theory such as dominating, irredundant, independent, covering and packing sets - leads naturally to fractional versions of these sets and corresponding fractional parameters. Let S be a dominating set of a graph G and f : V(G)~{0,1} the characteristic function of that set. By first translating the restrictions which define a dominating set from a set-based to a function-based form, and then allowing the function f to map the vertex set to the unit closed interval, we obtain the fractional generalisation of the dominating set S. In chapter 1, known domination-related parameters and their fractional generalisations are introduced, relations between them are investigated, and Gallai type results are derived. Particular attention is given to graphs with symmetry and to products of graphs. If instead of replacing the function f : V(G)~{0,1} with a function which maps the vertex set to the unit closed interval we introduce a function f' which maps the vertex set to {0, 1, ... ,k} (where k is some fixed, non-negative integer) and a corresponding change in the restrictions on the dominating set, we obtain a k-dominating function. In chapter 2 corresponding k-parameters are considered and are related to the classical and fractional parameters. The calculations of some well known fractional parameters are expressed as optimization problems involving the k- parameters. An e = 1 function is a function f : V(G)~[0,1] which obeys the restrictions that (i) every non-isolated vertex u is adjacent to some vertex v such that f(u)+f(v) = 1, and every isolated vertex w has f(w) = 1. In chapter 3 a theory of e = 1 functions and parameters is developed. Relationships are traced between e = 1 parameters and those previously introduced, some Gallai type results are derived for the e = 1
parameters, and e = 1 parameters are determined for several classes of graphs. The e = 1 theory is applied to derive new results about classical and fractional domination parameters. / Thesis (M.Sc.)-University of Natal, 1995.
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