Electronic Theses and Dissertations (Masters)

Permanent URI for this collection

https://hdl.handle.net/10539/38015

Browse

Browsing Electronic Theses and Dissertations (Masters) by School "School of Mathematics"

Now showing 1 - 9 of 9
  • Thumbnail Image
    Item
    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
  • Thumbnail Image
    Item
    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
  • Thumbnail Image
    Item
    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.
  • Thumbnail Image
    Item
    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.
  • Thumbnail Image
    Item
    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.
  • Thumbnail Image
    Item
    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)
  • Thumbnail Image
    Item
    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
  • Thumbnail Image
    Item
    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.
  • Thumbnail Image
    Item
    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.