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11 
Distances in and between graphs.Bean, Timothy Jackson. January 1991 (has links)
Aspects of the fundamental concept of distance are investigated in this
dissertation. Two major topics are discussed; the first considers metrics
which give a measure of the extent to which two given graphs are removed
from being isomorphic, while the second deals with Steiner distance in
graphs which is a generalization of the standard definition of distance in
graphs.
Chapter 1 is an introduction to the chapters that follow. In Chapter
2, the edge slide and edge rotation distance metrics are defined. The edge
slide distance gives a measure of distance between connected graphs of the same order and size, while the edge rotation distance gives a measure of distance between graphs of the same order and size. The edge slide and
edge rotation distance graphs for a set S of graphs are defined and investigated.
Chapter 3 deals with metrics which yield distances between graphs
or certain classes of graphs which utilise the concept of greatest common
subgraphs. Then follows a discussion on the effects of certain graph operations on some of the metrics discussed in Chapters 2 and 3. This chapter also considers bounds and relations between the metrics defined in Chapters 2 and 3 as well as a partial ordering of these metrics.
Chapter 4 deals with Steiner distance in a graph. The Steiner distance
in trees is studied separately from the Steiner distance in graphs in general.
The concepts of eccentricity, radius, diameter, centre and periphery are generalised under Steiner distance. This final chapter closes with an algorithm which solves the Steiner problem and a Heuristic which approximates the solution to the Steiner problem. / Thesis (M.Sc.)University of Natal, 1991.

12 
Dissipative gravitating systems.Fleming, Darryl. January 2011 (has links)
In this thesis we investigate the effect of shear on radiating stars undergoing gravitational
collapse. The interior spacetime is described by the most general spherically
symmetric line element in the absence of rotation. The energy momentum tensor for
the stellar interior is taken to be an anisotropic
fluid with heat
flux. The thermodynamics
of a relativistic
fluid is reviewed for the Eckart and causal theories. Since the star
is radiating energy to the exterior in the form of a radial heat
flux, the atmosphere is
described by Vaidya's outgoing solution. We provide the matching conditions required
for the continuity of the momentum
flux across the boundary, which determines the
temporal evolution junction conditions for the metric functions. We provide a general
method to obtain shearing solutions of the Einstein field equations describing a radiating,
collapsing sphere. A particular exact solution satisfying the boundary condition
and field equations is found. The validity of this specific model is investigated by employing
a causal heat transport equation which yields the temperature profile within
the stellar core. The energy conditions are studied and yield interesting features of this
particular model which are absent in the shearfree case. / Thesis (M.Sc.)University of KwaZuluNatal, Durban, 2011.

13 
Normalitylike properties, paraconvexity and selections.Makala, Narcisse Roland Loufouma. January 2012 (has links)
In 1956, E. Michael proved his famous convexvalued selection theorems for l.s.c. mappings
de ned on spaces with higher separation axioms (paracompact, collectionwise
normal, normal and countably paracompact, normal, and perfectly normal), [39]. In
1959, he generalized the convexvalued selection theorem for mappings de ned on paracompact
spaces by replacing \convexity" with \ paraconvexity", for some xed constant
0 < 1 (see, [42]). In 1993, P.V. Semenov generalized this result by replacing
with some continuous function f : (0;1) ! [0; 1) (functional paraconvexity) satisfying
a certain property called (PS), [63]. In this thesis, we demonstrate that the classical
Michael selection theorem for l.s.c. mappings with a collectionwise normal domain can
be reduced only to compactvalued mappings modulo Dowker's extension theorem for
such spaces. The idea used to achieve this reduction is also applied to get a simple
direct proof of that selection theorem of Michael's. Some other possible applications
are demonstrated as well. We also demonstrate that the paraconvexvalued and the
functionallyparaconvex valued selection theorems remain true for C 0
(Y )valued mappings
de ned on collectionwise normal spaces, where is an in nite cardinal number.
Finally, we prove that these theorems remain true for C (Y )valued mappings de ned
on PFnormal spaces; and we provide a general approach to such selection theorems. / Thesis (Ph.D.)University of KwaZuluNatal, Westville, 2012.

14 
Mathematical analysis of preexposure prophylaxis on HIV infection.Afassinou, Komi. January 2012 (has links)
We develop a mathematical model which seeks to assess the impact of HIV PreExposure
Prophylaxis (PrEP) on the prevalence and incidence of HIV infection. Mathematical analysis
of the model is carried out to establish the threshold conditions that determine the stability of
the steady states. Numerical simulations are performed to gain insight into the use and e cacy
of PrEP. Results from our model reveal that the basic reproduction number is a function of the
rate at which individuals use PrEP and the rate at which PrEP protects individuals from HIV
infection. Furthermore, strategies where either PrEP awareness or PrEP e cacy was low show
potential loopholes that can lead to more complications than bene ts. The best strategies
revealed by our results is that a high level of awareness and high PrEP e cacy are needed. / Thesis (M.Sc.)University of KwaZuluNatal, Westville, 2012.

15 
Degree theory in nonlinear functional analysis.Pillay, Paranjothi. 21 October 2013 (has links)
The objective of this dissertation is to expand on the proofs and concepts of Degree Theory, dealt with in chapters 1 and 2 of Deimling [28], to make it more readable and
accessible to anyone who is interested in the field. Chapter 1 is an introduction and contains the basic requirements for the subsequent chapters. The remaining chapters aim at defining a llvalued map D (the degree) on the set M = {(F, Ω, y) / Ω C X open, F : Ὠ → X, y ɇ F(∂Ω)} (each time, the elements of M satisfying extra conditions) that satisfies :
(D1) D(I, Ω, y) = 1 if y Є Ω.
(D2) D(F, Ω, y) = D(F, Ω1 , y) + D(F, Ω2, y) if Ω1 and Ω2 are disjoint open subsets of Ω o such that y ɇ F(Ὠ \ Ω1 U Ω2 ).
(D3) D(I  H(t, .), Ω, y(t)) is independent of t if H : J x Ὠ →X and y : J → X.
An important property that follows from these three properties is (D4) F1(y) ≠ Ø if D(F, Ω, y) ≠ 0.
This property ensures that equations of the form Fx = y have solutions if D(F, Ω, y) ≠ 0.
Another property that features in these chapters is the Borsuk property which gives us conditions under which the degree is odd and hence nonzero. / Thesis (M.Sc.)University of DurbanWestville, 1989.

16 
On spectral torsion theories.Uworwabayeho, Alphonse. January 2003 (has links)
The purpose of this thesis is to investigate how "spect ralness" properties of a torsion theory T on R  Mod are reflected by properties of the ring R and its ring of quotients R,.. The development of "spectral" torsion theory owes much to Zelmanowitz [50] and GomezPardo [23] . GomezPardo proved that there exists a bijective correspondence between the set of spectral torsion theories on R  Modand rings of quotients of R that are Von Neumann regular and left selfinjective. Chapter 1 is concerning with the notation used in the thesis and a summary of main results which are needed for understanding the sequel. Chapter 2 is concerned with the construction of a maximal ring of quotients of an arbitrary ring R by using the
notion of denseness and relative injective hull. In Chapter 3, we survey the three equivalent ways of formulating Torsion Theory: by means of preradical functors on the category R Mod, pairs of torsion / torsionfree classes and topologizing filters on rings. We shall show that Golan's approach to Torsion Theory via equivalence classes of injectives; and Dickson's one (as presented by Stenstrom) are equivalent. With a torsion theory T defined on RMod we associate R,. a ring of quotients of R.
The full subcategory (R, T)  Mod of R Mod whose objects are the Ttorsionfree rinjective left Rmodules is a Grothendieck category called the quotient category of R  Mod with respect to T. A left R,.module that is rtorsionfree Tinjective as a left Rmodule is injective if and only if it is injective as a left Rmodule (Proposition 3.6.4). Because of its use in the sequel , particular attention is paid to the lattice isomorphism that exists between the lattice of .rpure submodules of a left Rmodule M and the lattice of subobjects of the quotient module M; in the category (R , T)  Mod. Chapter 4 introduces the definition of a spectral torsion theory: a Vll torsion theory r on R  Mod is said to be spectral if the Grothendieck category (R, r)  Mod is spectral. Using the notion of relative essential submodule, one can construct a spectral torsion theory from an arbitrary torsion theory on R  Mod.
We shall show how an investigation of a general spectral torsion theory on R  Mod reduces to the Goldie torsion theory on R/tT (R)  Mod. Moreover, we shall exhibit necessary and sufficient conditions for R; to be a regular left selfinjective ring (Theorem 4.2.10). In Chapter 5, after constructing the torsion functor Soce() which is associated with the pseudocomplement r.l of r in R  tors, we show how semiartinian rings can be characterized by means of spectral torsion theories: if a spectral torsion theory r on R  Mod is generated by the class of rtorsion simple left Rmodules or, equivalently, cogenerated by the class of rtorsionfree simple left Rmodules, then R is a left semiartinian ring (Proposition 5.3.2). Chapter 6 gives Zelmanowitz' important result [50]: R; is a semisimple artinian ring if and only if the torsion theory r is spectral and the associated left Gabriel topology has a basis of finitely generated left ideals. We also exhibit results due to M.J.
Arroyo and J. Rios ([4] and [5]) which illustrate how spectral torsion theories can be used to describe when R; is (1) prime regular and left selfinjective, (2) a left full linear ring, and (3) a direct product of left full linear rings. We also study the relationship between the flatness of the ring of quotients R; and the r coherence of the ring R when r is a spectral torsion theory. It is proved that if r is a spectral torsion theory on R  Mod then the following conditions are equivalent: (1) R is left rcoherent; (2) (Rr)R is flat; (3) every right Rrmodule is flat as a right Rmodule (Proposition 6.3.9). This result is an extension of Cateforis' results. / Thesis (M.Sc)  University of Natal, Pietermaritzburg, 2003

17 
Long time behaviour of population models.Namayanja, Proscovia. January 2010 (has links)
Nonnegative matrices arise naturally in population models. In this thesis, we look at the theory
of such matrices and we study the PerronFrobenius type theorems regarding their spectral
properties. We use these theorems to investigate the asymptotic behaviour of solutions to
continuous time problems arising in population biology. In particular, we provide a description
of longtime behaviour of populations depending on the nature of the associated matrix. Finally,
we describe a few applications to population biology. / Thesis (M.Sc.)University of KwaZuluNatal, Westville, 2010.

18 
Real options : duopoly dynamics with more than one source of randomness.MacKenzie, Natalie. January 2009 (has links)
The valuation of real options has been of interest for some time. Recently,
the model has been revised to include more than one source of randomness,
e.g. Paxson and Pinto (2005). In this dissertation, we present a
model with more than one diffusion process to analyze strategic interaction
in a duopolistic framework. We consider a complete market where the
profit per unit and the number of units sold are assumed to evolve according
to distinct, but possibly correlated, geometric Brownian motions, and
aim to extend Paxson and Pinto’s research to a wider context by adjusting
the model to include the effect of the covariance between the stochastic
factors. In particular, we present results in both the preemptive and non
preemptive equilibrium case pertaining to the follower’s and leader’s value
function. We also investigate the consequences for the model in relation to
traditional net present value theory, and include an analysis of the comparative
static relationships that exist between the parameters. We then conclude
with a chapter that extends our twovariable model to three sources
of randomness  first by allowing the investment cost to be modelled as a
random onceoff payment, and then by considering it to be a stochastically
variable ongoing cost.
Keywords
Real options, complete markets, more than one stochastic process, competitive
games, duopoly. / Thesis (M.Sc.)University of KwaZuluNatal, Westville, 2009.

19 
Locally finite nearness frames.Naidoo, Inderasan. January 1998 (has links)
The concept of a frame was introduced in the midsixties by Dowker and
Papert. Since then frames have been extensively studied by several authors,
including Banaschewski, Pultr and Baboolal to mention a few. The idea of a
nearness was first introduced by H. Herrlich in 1972 and that of a nearness
frame by Banaschewski in the late eighties. T. Dube made a fairly detailed
study of the latter concept.
The purpose of this thesis is to study the property of local finiteness and
metacompactness in the setting of nearness frames. J. W. Carlson studied
these ideas (including Lindelof and Pervin nearness structures) in the realm
of nearness spaces.
The first four chapters are a brief overview of frame theory culminating in
results concerning regular, completely regular, normal and compact frames.
In chapter five we provide the definitions for various nearness frames: Pervin,
Lindelof , Locally Finite and Metacompact to mention a few. A particular
locally finite nearness structure, denoted by µLF, is studied in detail. It
is defined to be the nearness structure on a regular frame L generated by
the family of all locally finite covers on the frame L. Also, a particular
metacompact nearness structure, denoted by µPF, is studied in detail. It is
defined to be the nearness structure on a regular frame L generated by the
family of all pointfinite covers of the frame L. Various theorems related the
above nearness frames and these nearness structures are obtained. / Thesis (M.Sc.)University of DurbanWestvile, 1998.

20 
Completion of uniform and metric frames.Murugan, Umesperan Goonaselan. January 1996 (has links)
The term "frame" was introduced by C H Dowker, who studied them in
a long series of joint papers with D Papert Strauss. J R Isbell , in a path breaking
paper [1972] pointed out the need to introduce separate terminology
for the opposite of the category of Frames and coined the term "locale". He
was the progenitor of the idea that the category of Locales is actually more
convenient in many ways than the category of Frames. In fact, this proves
to be the case in one of the approaches adopted in this thesis.
Sublocales (quotient frames) have been studied by several authors, notably
Dowker and Papert [1966] and Isbell [1972]. The term "sublocale" is due to
Isbell, who also used "part " to mean approximately the same thing. The use
of nuclei as a tool for studying sublocales (as is used in this thesis) and the
term "nucleus" itself was initiated by H Simmons [1978] and his student D
Macnab [1981].
Uniform spaces were introduced by Weil [1937]. Isbell [1958] studied algebras
of uniformly continuous functions on uniform spaces. In this thesis, we
introduce the concept of a uniform frame (locale) which has attracted much
interest recently and here too Isbell [1972] has some results of interest. The
notion of a metric frame was introduced by A Pultr [1984]. The main aim of
his paper [11] was to prove metrization theorems for pointless uniformities.
This thesis focuses on the construction of completions in Uniform Frames and
Metric Frames. Isbell [6] showed the existence of completions using a frame
of certain filters. We describe the completion of a frame L as a quotient of the
uniformly regular ideals of L, as expounded by Banaschewski and Pultr[3].
Then we give a substantially more elegant construction of the completion of a
uniform frame (locale) as a suitable quotient of the frame of all downsets of L.
This approach is attributable to Kriz[9]. Finally, we show that every metric
frame has a unique completion, as outlined by Banaschewski and Pultr[4].
In the main, this thesis is a standard exposition of known, but scattered
material.
Throughout the thesis, choice principles such as C.D.C (Countable Dependent
Choice) are used and generally without mention. The treatment of category
theory (which is used freely throughout this thesis) is not selfcontained.
Numbers in brackets refer to the bibliography at the end of the thesis. We
will use 0 to indicate the end of proofs of lemmas, theorems and propositions.
Chapter 1 covers some basic definitions on frames , which will be utilized in
subsequent chapters. We will verify whatever we need in an endeavour to
enhance clarity. We define the categories, Frm of frames and frame homomorphisms,
and Lac the category of locales and frame morphisms. Then we
explicate the adjoint situation that exists between Frm and Top , the category
of topological spaces and continuous functions. This is followed by
an introduction to the categories, RegFrm of all regular frames and frame
homomorphisms, and KRegFrm the category of compact regular frames and
their homomorphisms. We then present the proofs of two very important
lemmas in these categories. Finally, we define the compactification of and a
congruence on a frame.
In Chapter 2 we recall some basic definitions of covers, refinements and star
refinements of covers. We introduce the notion of a uniform frame and define
certain mappings (morphisms) between uniform frames (locales) . In the
terminology of Banaschewski and Kriz [9] we define a complete uniform
frame and the completion of a uniform frame.
The aim of Chapter 3 is twofold : first, to construct the compact regular
coreflection of uniform frames , that is, the frame counterpart of the Samuel
Compactification of uniform spaces [12] , and then to use it for a description
of the completion of a uniform frame as an alternative to that previously
given by Isbell[6].
The main purpose of Chapter 4 is to provide another description of uniform
completion in frames (locales), which is in fact even more straightforward
than the original topological construction. It simply consists of writing down
generators and defining relations. We provide a detailed examination of the
main result in this section, that is, a uniform frame L is complete of each
uniform embedding f : (M,UM) t (L,UL) is closed, where UM and UL
denote the uniformities on the frames M and L respectively.
Finally, in Chapter 5, we introduce the notions of a metric diameter and a
metric frame. Using the fact that every metric frame is a uniform frame and
hence has a uniform completion, we show that every metric frame L has a
unique completion : CL  L. / Thesis (M.Sc.)University of DurbanWestville, 1996.

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