School of Mathematics (ETDs)

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Browsing School of Mathematics (ETDs) by SDG "SDG-4: Quality education"

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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”
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    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.
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    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
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    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
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    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