Philosophiae Doctor - PhD (Mathematics)

Permanent URI for this collectionhttps://hdl.handle.net/10566/19489

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Codes, graphs and designs related to iterated line graphs of complete graphs
(University of the Western Cape, 2011) Kumwenda, Khumbo
In this thesis, we describe linear codes over prime fields obtained from incidence designs of iterated line graphs of complete graphs Li(Kn) where i = 1,2. In the binary case, results are extended to codes from neighbourhood designs of the line graphs Li+l(Kn) using certain elementary relations. Codes from incidence designs of complete graphs, Kn' and neighbourhood designs of their line graphs, £1(Kn) (the so-called triangular graphs), have been considered elsewhere by others. We consider codes from incidence designs of Ll(Kn) and L2(Kn), and neighbourhood designs of L2(Kn) and L3(Kn). In each case, the basic parameters of the codes are determined. Further, we introduce a family of vertex-transitive graphs Rn that are embeddable into the strong product Ll(Kn) ~ K2' of triangular graphs and K2' a class that at first sight may seem unnatural but, on closer look, is a repository of graphs rich with combinatorial structures. For instance, unlike most regular graphs considered here and elsewhere that only come with incidence and neighbourhood designs, Rn also has what we have termed as 6-cycle designs. These are designs in which the point set contains vertices of the graph and every block contains vertices of a 6-cycle in the graph. Also, binary codes from incidence matrices of these graphs have other minimum words in addition to incidence vectors of the blocks. In addition, these graphs have induced subgraphs isomorphic to the family Hn of complete porcupines (see Definition 4.11). We describe codes from incidence matrices of Rn and Hn and determine their parameters. The discussion is concluded with a look at complements of Rn and Hn, respectively denoted by Rn and Hn. Among others, the complements rn are contained in the union of the categorical product Ll(Kn) x Kn' and the categorical product £1(Kn) x Kn (where £1(Kn) is the complement of the iii triangular graph £1(Kn)). As with the other graphs, we have also considered codes from the span of incidence matrices of Rn and Hn and determined some of their properties. In each case, automorphisms of the graphs, designs and codes have been determined. For the codes from incidence designs of triangular graphs, embeddings of Ll(Kn) x K2 and complements of complete porcupines, we have exhibited permutation decoding sets (PD-sets) for correcting up to terrors where t is the full error-correcting capacity of the codes. For the remaining codes, we have only been able to determine PD-sets for which it is possible to correct a fraction of t-errors (partial permutation decoding). For these codes, we have also determined the number of errors that can be corrected by permutation decoding in the worst-case.
Construction and analysis of efficient numerical methods to solve mathematical models of TB and HIV co-infection
(University of the Western Cape, 2011) Ahmed, Hasim Abdalla Obaid
The global impact of the converging dual epidemics of tuberculosis (TB) and human immunodeficiency virus (HIV) is one of the major public health challenges of our time, because in many countries, human immunodeficiency virus (HIV) and mycobacterium tuberculosis (TB) are among the leading causes of morbidity and mortality. It is found that infection with HIV increases the risk of reactivating latent TB infection, and HIV-infected individuals who acquire new TB infections have high rates of disease progression. Research has shown that these two diseases are enormous public health burden, and unfortunately, not much has been done in terms of modeling the dynamics of HIV-TB co-infection at a population level. In this thesis, we study these models and design and analyze robust numerical methods to solve them. To proceed in this direction, first we study the sub-models and then the full model. The first sub-model describes the transmission dynamics of HIV that accounts for behavior change. The impact of HIV educational campaigns is also studied. Further, we explore the effects of behavior change and different responses of individuals to educational campaigns in a situation where individuals may not react immediately to these campaigns. This is done by considering a distributed time delay in the HIV sub-model. This leads to Hopf bifurcations around the endemic equilibria of the model. These bifurcations correspond to the existence of periodic solutions that oscillate around the equilibria at given thresholds. Further, we show how the delay can result in more HIV infections causing more increase in the HIV prevalence. Part of this study is then extended to study a co-infection model of HIV-TB. A thorough bifurcation analysis is carried out for this model. Robust numerical methods are then designed and analyzed for these models. Comparative numerical results are also provided for each model.
Topogenous orders and their applications on lattices
(University of the Western Cape, 2024) Iragi, Bakulikira Claude
In his influential book, [Cs´a63] ´A. Cs´asz´ar developed the well-known theory of syntopogenous structures on a set. His intention was to create a comprehensive framework that simultaneously encompasses the study of topological, proximal, and uniform structures. In the same monograph, he demonstrated independently, along with Pervin [Per62], that every topological space possesses a compatible quasi-uniformity. A similar observation was noted for a uniform space, provided the topological space is completely regular. On the other hand, Herrlich in [Her74a] introduced the concept of “nearness” with the aim of unifying various topological structures. This Ph.D. thesis aims to investigate topogenous orders and their generalizations, such as quasi-uniformities, syntopogenous structures, on complete lattices which extend and generalize existing literature in this field. We explore the study of quasi-uniformities through the lens of syntopogenous structures, and establish a Galois connection between these two constructs. Furthermore, we provide conditions under which certain Cs´asz´ar structures are order isomorphic to quasi-uniformities on a complete lattice.As Cs´asz´ar structures are deeply rooted in pointfree topology, our research naturally extends into the realm of frames. We establish a correspondence between pre-nearness and Cs´asz´ar structures. In line with these ideas, we also delve into the relationship between pre-uniformities and entourage quasi-uniformities in the context of frames.
The Smarandache vertices of the annihilation graphs of commutator posets and lattices with respect to an element and an ideal
(University of the Western Cape, 2024) Siame Happy
A vertex a in a simple graph G is said to be a Smarandache vertex (or S-vertex for short) provided that there exist three distinct vertices x, y, and b (all different from a) in G such that x—a, a—b, and b—y are edges in G, but there is no edge between x and y. In this interdisciplinary subject, we investigate the interplay between the algebraic properties of the commutator posets and lattices and their associated annihilation graphs with respect to an element [resp. an ideal] using the notion of the Smarandache vertices. Actually, AGz(L) (the annihilation graph of the commutator poset [lattice] L with respect to an element z ∈ L) [resp. AGI(L) (the annihilation graph of the commutator poset [lattice] L with respect to an ideal I ⊆ L, where AGI(L) is an extension of AGz(L) from an element to an ideal of L)] is a widely generalized context for the study of the zerodivisor type (annihilating-ideal) graphs, where the vertices of the graphs are not elements/ideals of a commutative ring, but elements of an abstract ordered set [lattice] (imitating the lattice of ideals of a ring), equipped with a commutative (not necessarily associative) binary operation (imitating the product of ideals of a ring). We discuss when AGz(L) [resp. AGI(L)] is a complete r-partite graph together with some of its other graph-theoretic properties. We investigate the interplay between some (order-) lattice-theoretic properties of L and graphtheoretic properties of its associated graph AGz(L) [resp. AGI(L)]. We provide some examples to show that some conditions are not superfluous assumptions. We prove and show by an example that the class of lower sets of a commutator poset L is properly contained in the class of m-ideals of L [i.e. multiplicatively absorptive ideals (sets) of L that are defined by commutator operation].

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