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1 
Centre manifold theory with an application in population modelling.Phongi, Eddy Kimba. January 2009 (has links)
There are basically two types of variables in population modelling, global and local variables. The former describes the behavior of the entire population while the latter describes the behavior of individuals within this population. The description of the population using local variables is more detailed, but it is also computationally costly. In many cases to study the dynamics of this population, it is sufficient to focus only on global variables. In applied sciences, to achieve this, the method of aggregation of variables is used. One of methods used to mathematically justify variables aggregation is the centre manifold theory. In this dissertation we provide detailed proofs of basic results of the centre manifold theory and discuss some examples of applications in population modelling. / Thesis (M.Sc.)University of KwaZuluNatal, Westville, 2009.

2 
Interative approaches to convex feasibility problems.Pillay, Paranjothi. January 2001 (has links)
Solutions to convex feasibility problems are generally found by iteratively constructing
sequences that converge strongly or weakly to it. In this study, four types
of iteration schemes are considered in an attempt to find a point in the intersection
of some closed and convex sets.
The iteration scheme Xn+l = (1  λn+1)y + λn+1Tn+lxn is first considered for infinitely
many nonexpansive maps Tl , T2 , T3 , ... in a Hilbert space. A result of Shimizu
and Takahashi [33] is generalized, and it is shown that the sequence of iterates converge
to Py, where P is some projection. This is further generalized to a uniformly
smooth Banach space having a weakly continuous duality map. Here the iterates
converge to Qy, where Q is a sunny nonexpansive retraction. For this same iteration
scheme, with finitely many maps Tl , T2, ... , TN , a complementary result to a result of
Bauschke [2] is proved by introducing a new condition on the sequence of parameters
(λn). The iterates converge to Py, where P is the projection onto the intersection
of the fixed point sets of the Tis. Both this result and Bauschke's result [2] are then
generalized to a uniformly smooth Banach space, and to a reflexive Banach space
having a weakly continuous duality map and having Reich's property. Now the iterates
converge to Qy, where Q is the unique sunny nonexpansive retraction onto the
intersection of the fixed point sets of the Tis.
For a random map r : N {I, 2, ... ,N}, the iteration scheme xn+l = Tr(n+l)xn
is considered. In a finite dimensional Hilbert space with Tr(n) = Pr(n) , the iterates
converge to a point in the intersection of the fixed point sets of the PiS. In an arbitrary
Banach space, under certain conditions on the mappings, the iterates converge to a
point in the intersection of the fixed point sets of the Tis.
For the scheme xn+l = (1 λn+l)xn+λn+lTr(n+l)xn, in a finite dimensional Hilbert
space the iterates converge to a point in the intersection of the fixed point sets of the
Tis, and in an infinite dimensional Hilbert space with the added assumption that the
random map r is quasicyclic, then the iterates converge weakly to a point in the
intersection of the fixed point sets of the Tis.
Lastly, the minimization of a convex function θ is considered over some closed and
convex subset of a Hilbert space. For both the case where θ is a quadratic function
and for the general case, first the unique fixed points of some maps Tλ are shown
to converge to the unique minimizer of θ and then an algorithm is proposed that
converges to this unique minimizer. / Thesis (Ph.D.)University of DurbanWestville, 2001.

3 
Aspects of graph vulnerability.Day, David Peter. January 1994 (has links)
This dissertation details the results of an investigation into, primarily, three aspects of graph vulnerability namely, lconnectivity, Steiner Distance hereditatiness and functional isolation. Following the introduction in Chapter one, Chapter two focusses on the lconnectivity of graphs and introduces the concept of the strong lconnectivity of digraphs. Bounds on this latter parameter are investigated and then the lconnectivity function of particular types of graphs, namely caterpillars and complete multipartite graphs as well as the strong lconnectivity function of digraphs, is explored. The chapter concludes with an examination of extremal graphs with a given lconnectivity. Chapter three investigates Steiner distance hereditary graphs. It is shown that if G is 2Steiner distance hereditary, then G is kSteiner distance hereditary for all k≥2. Further, it is shown that if G is kSteiner distance hereditary (k≥ 3), then G need not be (k  l)Steiner distance hereditary. An efficient algorithm for determining the Steiner distance of a set of k vertices in a kSteiner distance hereditary graph is discussed and a characterization of 2Steiner distance hereditary graphs is given which leads to an efficient algorithm for testing whether a graph is 2Steiner distance hereditary. Some general properties about the cycle structure of kSteiner distance hereditary graphs are established and are then used to characterize 3Steiner distance hereditary graphs. Chapter four contains an investigation of functional isolation sequences of supply graphs. The concept of the Ranked supply graph is introduced and both necessary and sufficient conditions for a sequence of positive nondecreasing integers to be a functional isolation sequence of a ranked supply graph are determined. / Thesis (Ph.D.)University of Natal, 1994.

4 
Aspects of distance and domination in graphs.Smithdorf, Vivienne. January 1995 (has links)
The first half of this thesis deals with an aspect of domination; more specifically, we
investigate the vertex integrity of ndistancedomination in a graph, i.e., the extent
to which ndistancedomination properties of a graph are preserved by the deletion
of vertices, as well as the following: Let G be a connected graph of order p and let
oi S s;:; V(G). An Sndistancedominating set in G is a set D s;:; V(G) such that
each vertex in S is ndistancedominated by a vertex in D. The size of a smallest
Sndominating set in G is denoted by I'n(S, G). If S satisfies I'n(S, G) = I'n(G),
then S is called an ndistancedominationforcing set of G, and the cardinality of a
smallest ndistancedominationforcing set of G is denoted by On(G). We investigate
the value of On(G) for various graphs G, and we characterize graphs G for which
On(G) achieves its lowest value, namely, I'n(G), and, for n = 1, its highest value,
namely, p(G). A corresponding parameter, 1](G), defined by replacing the concept
of ndistancedomination of vertices (above) by the concept of the covering of edges
is also investigated.
For k E {a, 1, ... ,rad(G)}, the set S is said to be a kradiusforcing set if, for each
v E V(G), there exists Vi E S with dG(v, Vi) ~ k. The cardinality of a smallest
kradiusforcing set of G is called the kradiusforcing number of G and is denoted
by Pk(G). We investigate the value of Prad(G) for various classes of graphs G,
and we characterize graphs G for which Prad(G) and Pk(G) achieve specified values.
We show that the problem of determining Pk(G) is NPcomplete, study the
sequences (Po(G),Pl(G),P2(G), ... ,Prad(G)(G)), and we investigate the relationship
between Prad(G)(G) and Prad(G)(G + e), and between Prad(G)(G + e) and the connectivity
of G, for an edge e of the complement of G.
Finally, we characterize integral triples representing realizable values of the triples
b,i,p), b,l't,i), b,l'c,p), b,l't,p) and b,l't,l'c) for a graph. / Thesis (Ph.D.Mathematics and Applied Mathematics)University of Natal, 1995.

5 
Graph and digraph embedding problems.Maharaj, Hiren. January 1996 (has links)
This thesis is a study of the symmetry of graphs and digraphs by considering certain
homogeneous embedding requirements.
Chapter 1 is an introduction to the chapters that follow. In Chapter 2 we present
a brief survey of the main results and some new results in framing number theory.
In Chapter 3, the notions of frames and framing numbers is adapted to digraphs. A
digraph D is homogeneously embedded in a digraph H if for each vertex x of D and
each vertex y of H, there exists an embedding of D in H as an induced subdigraph
with x at y. A digraph F of minimum order in which D can be homogeneously
embedded is called a frame of D and the order of F is called the framing number of
D. We show that that every digraph has at least one frame and, consequently, that
the framing number of a digraph is a well defined concept. Several results involving
the framing number of graphs and digraphs then follow. Analogous problems to those
considered for graphs are considered for digraphs.
In Chapter 4, the notions of edge frames and edge framing numbers are studied.
A nonempty graph G is said to be edge homogeneously embedded in a graph H if for
each edge e of G and each edge f of H, there is an edge isomorphism between G and
a vertex induced subgraph of H which sends e to f. A graph F of minimum size
in which G can be edge homogeneously embedded is called an edge frame of G and
the size of F is called the edge framing number efr(G) of G. We also say that G is
edge framed by F. Several results involving edge frames and edge framing numbers
of graphs are presented.
For graphs G1 and G2 , the framing number fr(G1 , G2 ) (edge framing number
ef r(GI, G2 )) of G1 and G2 is defined as the minimum order (size, respectively) of a
graph F such that Gj (i = 1,2) can be homogeneously embedded in F. In Chapter 5
we study edge framing numbers and framing number for pairs of cycles. We also
investigate the framing number of pairs of directed cycles. / Thesis (Ph.D.)University of Natal, Pietermaritzburg, 1996.

6 
Aspects of distance measures in graphs.Ali, Patrick Yawadu. January 2011 (has links)
In this thesis we investigate bounds on distance measures, namely, Steiner diameter and radius, in terms of other graph parameters. The thesis consists of four chapters. In Chapter 1, we define the most significant terms used throughout the thesis, provide an underlying motivation
for our research and give background in relevant results. Let G be a connected graph of order p and S a nonempty set of vertices of G. Then the Steiner distance d(S) of S is the minimum size of a connected subgraph of G whose vertex set contains S. If n is an integer, 2 ≤ n ≤ p, the Steiner ndiameter, diamn(G), of G is the maximum Steiner distance of any nsubset of vertices of G. In Chapter 2, we give a bound on diamn(G) for a graph G in terms of the order of G and the minimum degree of G. Our result implies a bound on the ordinary diameter by Erdös, Pach, Pollack
and Tuza. We obtain improved bounds on diamn(G) for K3free graphs and C4free graphs. In Chapter 3, we prove that, if G is a 3connected plane graph of order p and maximum face length l then the radius of G does not exceed p/6 + 5l/6 + 5/6. For constant l, our bound improves on a bound by Harant. Furthermore we extend these results to 4 and 5connected planar graphs. Finally, we complete our study in Chapter 4 by providing an upper bound on diamn(G) for a maximal planar graph G. / Thesis (Ph.D.)University of KwaZuluNatal, Westville, 2011.

7 
Application of the wavelet transform for sparse matrix systems and PDEs.Karambal, Issa. January 2009 (has links)
We consider the application of the wavelet transform for solving sparse matrix systems and partial differential equations. The first part is devoted to the theory and algorithms of wavelets. The second part is concerned with the sparse representation of matrices and wellknown operators. The third part is directed to the application of wavelets to partial differential equations, and to sparse linear systems resulting from differential equations. We present several numerical examples and simulations for the above cases. / Thesis (M.Sc.)University of KwaZuluNatal, Westville, 2009.

8 
Distance measures in graphs and subgraphs.Swart, Christine Scott. January 1996 (has links)
In this thesis we investigate how the modification of a graph affects various
distance measures. The questions considered arise in the study of how the
efficiency of communications networks is affected by the loss of links or nodes.
In a graph C, the distance between two vertices is the length of a shortest
path between them. The eccentricity of a vertex v is the maximum distance
from v to any vertex in C. The radius of C is the minimum eccentricity of a
vertex, and the diameter of C is the maximum eccentricity of a vertex. The
distance of C is defined as the sum of the distances between all unordered
pairs of vertices.
We investigate, for each of the parameters radius, diameter and distance
of a graph C, the effects on the parameter when a vertex or edge is removed or
an edge is added, or C is replaced by a spanning tree in which the parameter is
as low as possible. We find the maximum possible change in the parameter
due to such modifications. In addition, we consider the cases where the
removed vertex or edge is one for which the parameter is minimised after
deletion.
We also investigate graphs which are critical with respect to the radius or
diameter, in any of the following senses: the parameter increases when any
edge is deleted, decreases when any edge is added, increases when any vertex
is removed, or decreases when any vertex is removed. / Thesis (M.Sc.)University of Natal, 1996.

9 
On the integrity of domination in graphs.Smithdorf, Vivienne. January 1993 (has links)
This thesis deals with an investigation of the integrity of domination in a.graph, i.e., the extent to
which domination properties of a graph are preserved if the graph is altered by the deletion of
vertices or edges or by the insertion of new edges.
A brief historical introduction and motivation are provided in Chapter 1. Chapter 2 deals with kedge(
domination)critical graphs, i.e., graphsG such that )'(G) = k and )'(G+e) < k for all e E
E(G). We explore fundamental properties of such graphs and their characterization for small
values of k. Particular attention is devoted to 3edgecritical graphs.
In Chapter 3, the changes in domination number brought aboutby vertex removal are investigated.
\
Parameters )'+'(G) (and "((G)), denoting the smallest number of vertices of G in a set 5 such that
)'(G5) > )'(G) ()'(G 5) < )'(G), respectively), are investigated, as are'kvertexcritical graphs G
(with )'(G) = k and )'(Gv) < k for all v E V(O)). The existence of smallest'dominationforcing
sets of vertices of graphs is considered.
The bondage number 'Y+'(G), i.e., the smallest number of edges of a graph G in a set F such that
)'(G F) > )'(0), is investigated in Chapter 4, as are associated extremal graphs. Graphs with
dominating sets or domination numbers that are insensitive to the removal of an arbitrary edge are
considered, with particular reference to such graphs of minimum size.
Finally, in Chapter 5, wediscuss ndominating setsD of a graph G (such that each vertex in GD
is adjacent to at least n vertices in D) and associated parameters. All chapters but the first and
fourth contain a listing of unsolved problems and conjectures. / Thesis (M.Sc.)University of Natal, 1993.

10 
The preliminary group classification of the equation utt = f(x,ux)uxx + g(x, ux)Narain, Ojen Kumar. January 1995 (has links)
We study the class of partial differential equations Utt = f(x, ux)uxx +
g(x, u x), with arbitrary functions f(x, u x) and g(x, u x), from the point of view
of group classification. The principal Lie algebra of infinitesimal symmetries
admitted by the whole class is threedimensional. We use the method of preliminary
group classification to obtain a classification of these equations with
respect to a onedimesional extension of the principal Lie algebra and then a
countabledimensional subalgebra of their equivalence algebra. Each of these
equations admits an additional infinitesimal symmetry. L.V. Ovsiannikov [9]
has proposed an algorithm to construct efficiently the optimal system of an
arbitrary decomposable Lie algebra. We use this algorithm to construct an
optimal system of subalgebras of all dimensionalities (from onedimensional
to six dimensional) of a sevendimensional solvable Lie algebra. / Thesis (M.Sc)University of DurbanWestville, 1995.

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