School of Mathematics (ETDs)

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An Essay on Branching Time Logics
(University of the Witwatersrand, Johannesburg, 2024) Marais, Chantel
In this thesis we investigate the Priorian logics of a variety of classes of trees. These classes of trees are divided in to irreflexive and reflexive trees, and each of these has a number of subclasses, for example, dense irreflexive trees, discrete reflexive trees, irreflexive trees with branches isomorphic to the natural numbers, etc. We find finite axiomatisations for the logics of these different classes of trees and show that each logic is sound and strongly / weakly complete with respect to the respective class of trees. The methods use to show completeness vary from adapting some known constructions for specific purposes, including unravelling and bulldozing, building a network step-by-step, filtering through a finite set of formulas, as well as using some new processes, namely refining the filtration and unfolding. Once the logics have been shown to be sound and complete with respect to the different classes of trees, we also show that most of these logics are decidable, using methods that include the finite model property, mosaics and conservative extensions. Lastly, we give a glimpse into the available research on other languages used to study branching time structures, including the Peircean and Ockhamist languages, and languages that include additional modal operators like “since” and “until”
Analysis of some convergence results for inertial variational inequalities problem and its application
(University of the Witwatersrand, Johannesburg, 2023) Kunene, Thembinkosi Eezy
Some core aspects of nonlinear analysis, which is a major branch of mathematics, are the optimization problems, fixed point theory and its applications. These concepts, that is, optimization theory, fixed point theory and its applications are widely applied in several fields of science such as networking, inventory control, engineering, economics, policy modelling, transportation and mathematical sciences to mention but a few. Due to its relevance to different fields, the theory of optimization and fixed point has been a popular field of research for a long time. Given its expansive nature, researchers continue to make new discoveries and advancements, contributing to its enduring significance across various disciplines. The goal of this dissertation is to explore some convergence iterative methods for approximating optimization problems. We propose a new modified projection and contraction algorithm for approximating solutions of a variational inequality problem involving a quasi-monotone and Lipschitz continuous mapping in real Hilbert spaces. We incorporate the technique of two-step inertial into a single projection and contraction method and prove a weak convergence theorem for the proposed algorithm. The weak convergence theorem proved requires neither the prior knowledge of the Lipschitz constant nor the weak sequential continuity of the associated mapping. Under additional strong pseudomonotonicity, the R-linear convergence rate of the two-step inertial algorithm is presented. Finally, some numerical examples are given to illustrate the effectiveness and competitiveness of the proposed algorithm in comparison with some existing algorithms in the literature
Chromatic Polynomials and Certain Classes of Graphs
(University of the Witwatersrand, Johannesburg, 2023) Maphakela, Lesiba Joseph; Mphako-Banda, G.
The chromatic polynomial of a graph has been widely studied in the literature. The focus of this research is on exploring the chromatic polynomial of specific graphs that result from the application of a join operation. The chromatic polynomial of a graph can be expressed in various forms; power form, tree form, factorial form and cycle form. The expressions in various forms, such as power form, tree form, and factorial form, have been subject to comprehensive investigation. However, it should be noted that the cycle form presents relative gaps that necessitate further exploration. This work builds upon the existing literature by engaging in a discussion of the coefficients of the chromatic polynomial of a graph expressed in cycle form. To achieve this objective, we commence by presenting the general formula of the chromatic polynomial in cycle form. Following this, we introduce an algorithm that computes the chromatic polynomial of a graph in cycle form. Additionally, we outline a method for converting the chromatic polynomial of a graph from its tree form into the cycle form. Furthermore, we determine the values of the first and second terms of the chromatic polynomial in its cycle form. This research also complements the well established knowledge of the chromatic polynomial of graphs resulting from the application of a join operation. Of particular interest, we explore the joins of various classes of graphs, including the join of a null graph, N1 with a graph G, which is known as the vertex join of graph G. Building upon this framework, we extend our analysis to encompass the join of a null graph, N2, with graph G. Similarly, we present results pertaining to the join of a complete graph, Kn, with a graph G. Significantly, we conduct a thorough comparative analysis of the chromatic equivalence class among these derived classes of graphs. Lastly, we discuss the chromatic uniqueness of these derived classes of graphs, alongside introducing variations to these derived graphs by deleting their edges and subgraphs.
Convergence Results for Inertial Regularized Bilevel Variational Inequality Problems
(University of the Witwatersrand, Johannesburg, 2024) Okorie, Kalu Okam; Okeke, Chibueze Christian
In this dissertation, we introduce and study the inertial forward-reflected-backward method for approximating a solution of bilevel variational inequality problems. Our proposed method involves a single projection onto a feasible set, one functional evaluation and adopts the inertial extrapolation term. These features make our algorithm cost-effective and efficient, which is desirable when the cost operator and the feasible set have a complex structure. We incorporate the regularization technique in our method and establish that the sequences generated by our method converge strongly to a solution of the bilevel variational inequality problem studied in this work; furthermore, we modified our method by replacing the stepsizes and projection onto a feasible set with a self-adaptive non-monotonic stepsizes and projection onto a constructive halfspace, respectively. The non-monotonic stepsizes ensure that our method performs without the previous detail of the Lipschitz constant, and the projection onto a constructive halfspace is cheap since its computation is through an explicit formula. These adjustments in our method ensure an improved performance, cheap computation and easy implementation of our method. We show the strong convergence result of the iterative sequences. Lastly, we give numerical experiments comparing the performance of the proposed methods with existing methods
Distance measures, independence number and chromatic number
(University of the Witwatersrand, Johannesburg, 2023-03) Moholane, Letlhogonolo; Jonck, Betsie; Mukwembi, Simon
There are numerous parameters in graph theory. In this dissertation, we pay a special attention to average distance, independence number, average eccentricity, order and the chromatic number of a graph. In 1975, Doyle and Graver proved an upper bound on the average distance with respect to the order of the graph. This gave rise to studies that focus on upper and lower bounds on average distance in terms of other graph parameters. Approximately, three decades after Doyle and Graver proved their result, Dankelmann, Goddard, and Swart in 2004 produced a study that gave an upper bound on average eccentricity in terms of minimum degree and order of the graph, initiating studies that focus on giving bounds on average eccentricity with respect to other known graph parameters. In this dissertation, we investigate bounds on average eccentricity and on average distance. We give upper bounds on average eccentricity in terms of independence number of the graph and order of the graph. Then, we present bounds on average eccentricity when order and chromatic number of the graph are prescribed. The second part of the dissertation is dedicated to presenting upper bounds on average distance with respect to independence number and order of the graph, and again, in terms of chromatic number and order of the graph.
Enumeration of binary strings and applications to compositions and partitions
(University of the Witwatersrand, Johannesburg, 2023) Raphadu, Maleka
In this dissertation we first introduce binary strings and give a historical background. Then we discus some techniques for enumerating restricted sets of binary strings ,with several example . We employ mainly the symbolic method and recursive techniques, among others, to obtain our results. A chapter is devoted to a discussion of some published case studies on bit string enumerations which are relevant to our project. Then we consider how the study of binary string may facilitate the enumeration of selected classes of compositions and integer partitions.
On polarity-based semantics for non-distributive modal logics
(University of the Witwatersrand, Johannesburg, 2023) Clingman, R.; Conradie, Willem
This masters study builds upon recent research in polarity-based semantics for non-distributive modal logics (NDMLs). Current formulations of polarity-based semantics for NDML impose compatibility requirements on additional relations of polarity-based frames, hindering applicability of the semantics, as arbitrary frames need not be compatible. In this study we develop a polarity-based semantics for NDML with modalites that are, in general, neither normal nor distributive and without the imposition of compatibility requirements. We provide a sound and complete axiomatization of this logic. The second half of the thesis focuses on a special class of enriched polarities, those who are in a sense liftings of Kripke frames. The compatibility of these liftings combined with the intuitive nature of the underlying Kripke frames makes for a useful case study in which to explore p-morphisms between enriched polarities, and enriched polarity-based models, from a relational perspective.
On the Geometry of Lightlike Hypersurfaces
(University of the Witwatersrand, Johannesburg, 2022) Khoza, Jabulile; Ssekajja, Samuel
The study of lightlike geometry in semi-Riemannian manifolds was introduced by A. Bejancu and K. L. Duggal in their book [3]. The book laid a foundation for the study of lightlike submanifolds by providing some constructions on such subspaces. In this research, we focus on the study of lightlike hypersurfaces of semi-Riemannian manifolds, i.e. submanifolds of codimension one. In this line, we prove necessary and sufficient conditions for a lightlike hypersurface of a semi-Riemannian space form to be totally umbilical, screen conformal and screen almost conformal (see Theorem 2.5.2, 2.5.4 and 2.5.5)
Relaxed Inertial Algorithm for Solving Equilibrium Problems
(University of the Witwatersrand, Johannesburg, 2024) Elijah, Nwakpa Chidi
In this dissertation, we propose and study two relaxed inertial methods for solving equilibrium problems. In our first proposed method, we establish that the generated sequence of our proposed method weakly converges to a solution of the equilibrium problems. We apply this proposed method to variational inequality and fixed point problems. Further- more, a modification of the first method leads us to our second iterative method. Again, we established that the sequence generated by this method converges strongly to a solution of the equilibrium problems. Our proposed methods involve self-adaptive stepsizes and hence, do not require the fore knowledge of the Lipschitz constants for implementation. In each of our proposed methods, the convergence is established when the associated cost bifunction is pseudomonotone and satisfies the Lipschitz-type condition
Resolvability of groups
(University of the Witwatersrand, Johannesburg, 2020) Ndhlalane, Mororiseng
A topological group is called resolvable if it can be partitioned into two dense subsets. A group is absolutely resolvable if it can be partitioned into two subsets dense in any nondescript group topology. The aim of this dissertation is to give a unified exposition of some major results about resolvability of groups. In particular, we show that; 1. Every countable nondescript topological group not containing an open Boolean subgroup is resolvable, 2. Every infinite Abelian group not containing an infinite Boolean subgroup is absolutely resolvable.
Tableaux and Decision Procedures for Many-Valued Modal Logics
(University of the Witwatersrand, Johannesburg, 2024) Axelrod, Guy Ross
The aim of this dissertation is to present results expanding on the work done by Melvin Fitting in [22] and [24]. In [22], Fitting introduces a framework of many-valued modal logics, where modal formulas are interpreted via generalized Kripke models in which both the propositional valuation and the accessibility relation take on values from some Heyting algebra of truth values. For a fixed arbitrary finite Heyting algebra, H, [24] presents a signed semantic tableau system that is sound and complete with respect to all H-frames. We go on to consider the many-valued generalizations of frame properties such as reflexivity and transitivity (as presented in [39]) and give parameterized tableau systems which are sound and complete with respect to classes of H-frames satisfying such properties. Further, a prefixed tableau system is introduced, which allows us to define an intuitive decision procedure deciding the logics of the above- mentioned H-frame classes, as well as logics of H-frames satisfying generalized symmetry properties, which cannot be captured by Fitting’s unprefixed systems. Further, they allow us to derive finite frame properties. Such a decision procedure has been implemented, and is available on GitHub.
The role of invariants in obtaining exact solutions of differential equations
(University of the Witwatersrand, Johannesburg, 2024) Ahmed, Mogahid Mamoon Abkar; Kara, A.H.
We show here that variational and gauge symmetries have additional appli- cations to the integrability of differential equations. We present a general method to construct first integrals for some classes. In particular, we present a broad class of diffusion type equations, viz., the Fisher Kolmorov and Fitzhugh Nagumo equations, which satisfy the Painlev´e properties of their respective travelling wave forms and solitons. It is then shown how a study of invari- ance properties and conservation laws is used to ‘twice’ reduce the equations to solutions. We further constructing the first integrals of a large class of the well-known second-order Painlev´e equations. In some cases, variational and gauge symmetries have additional applications following a known Lagrangian in which case the first integral is obtained by Noether’s theorem. Generally, it is more convenient to adopt the ‘multiplier’ approach to find the first integrals. The main chapters of this thesis have either been published or submitted for publication in accredited journals. The contents of Chapters 2, 3 and 5 has been published ([54], [55]). All computations were done either by hand or Maple

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