Doctoral Degrees (Mathematical Sciences)
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- ItemDomination and generalized domination in ordered Banach algebras(Stellenbosch : Stellenbosch University, 2024-03) Rabearivony, Andriamahazosoa Dimbinantenaina; Mouton, Sonja; Benjamin, Ronalda; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.ENGLISH ABSTRACT: Let A be a complex unital Banach algebra. An algebra cone in A is a non-empty subset C of A containing the unit of A and which is closed under addition, multiplication and non-negative scalar multiplication. Any algebra cone C in A corresponds to a partial ordering in A satisfying C = {a ∈ A : 0 ≤ a}. The pair (A, C) is called an ordered Banach algebra (OBA). In this dissertation, we investigate the domination (resp. generalized dom- ination) problems in OBAs, stated as follows: given an OBA (A, C) and two elements a, b ∈ A such that 0 ≤ a ≤ b (resp. ±a ≤ b), under which conditions do we have that a property of b is inherited by a? In 2014, Mouton and Muzundu investigated the domination problem for ergodic elements, where an element a ∈ A is called ergodic if the sequence (Ln 1 aᵏ) is convergent. While their theorem is an outstanding extension k=1 n of an operator theoretic result, the assumption of weak monotonicity of the spectral radius in the quotient algebra is quite strong, limiting its applicability to the regular operators on a Dedekind complete Banach lattice. Our first main result in this dissertation (Theorem 2.3.10) provides a partial solution to this problem in the form of an ergodic domination-type theorem without assuming weak monotonicity of the spectral radius in the quotient algebra. Furthermore, since their result was only for the domination problem, we extend it, as well as most of the other existing domination results, to generalized domination results. (See, in particular, Theorems 3.10.5, 3.12.6 and 3.13.8 regarding ergodicity.) These two contributions provide (partial) answers to two open questions in the survey paper [42] of Mouton and Raubenheimer.
- ItemExploring some categorical aspects of foundational concepts in algebraic geometry(Stellenbosch : Stellenbosch University, 2024-03) Mgani, Damas Karmel; Marques, Sophie; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.ENGLISH ABSTRACT: This thesis examines the foundational concepts of algebraic geometry, with a partic- ular emphasis on elucidating its categorical aspects. Our primary contribution lies in the comprehensive exploration of the gluing property across diverse categories. Throughout this exploration, we introduce a categorical framework for gluing, fea- turing two pivotal constructs: the gluing index category and the gluing data functor. This framework not only provides a unified methodology applicable to (pre)sheaves on sites, (locally) ringed topological spaces and schemes but also paves the way for potential future extensions into new categories. Furthermore, our research focuses on the separation property of (pre)sheaves, presenting a categorical description of separafication through the introduction of stalk sheaves associated with a presheaf. We also investigate the concept of sheafi- fication, aiming to understand if a sheaf can be defined as a composition of limits within the category of (pre)sheaves. While we successfully achieve this goal at a local level, it presents captivating prospects for further inquiry. In addition to these original contributions, this thesis presents an extensive and meticulous exploration of the fundamental principles of algebraic geometry, with a central emphasis on category theory. This part includes intricate details and formalities not readily accessible in existing literature on algebraic geometry.
- ItemCoherent loop states and their applications in geometric quantization(Stellenbosch : Stellenbosch University, 2024-03) Nzaganya, Nzaganya Edson; Bartlett, Bruce; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.ENGLISH ABSTRACT: In the first part of this study, we study coherent loop states (also known as Bohr‑ Sommerfeld states) on 𝑆², with application to the representation theory of 𝑆𝑈(2). These states offer a precise bridge between the classical and quantum descriptions of angular momentum. We show that they recover the usual basis of angular mo‑ mentum eigenstates used in physics, and give a self‑contained proof of the asymp‑ totics of their inner products. As an application, we use these states to derive Little‑ john and Yu’s geometric formula for the asymptotics of the Wigner matrix elements. In the second part of this thesis, we consider coherent loop states on a general Riemann surface 𝑀. We show that for quasi‑regular polarizations of 𝑀, the second derivatives of the Bergman kernel on the diagonal of 𝑀 can be computed precisely in terms of the Kähler form of 𝑀. Therefore, the asymptotics of the inner product of coherent loop states can be computed using the complex stationary phase principle. This gives an alternative proof, for quasi‑regular polarized Riemann surfaces, of a variant of a result of Borthwick, Paul and Uribe.
- ItemStudy of cyclotomic extensions of degree power of 2 and classification of radical extensions up to isomorphism(Stellenbosch : Stellenbosch University, 2024-03) Mrema, Elizabeth; Marques, Sophie; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.In this thesis, we gain a deeper understanding of cyclotomic extensions of degree powers of 2 and the classification of radical extensions (both separable and insepa‑ rable) up to isomorphism. Our main results about cyclotomic extensions of degree power of 2 describe their Galois structures, their degrees, their subextensions, their tower decompositions, and the minimal polynomials of some traces of root of unity generating all their subsextensions over an arbitrary base field. In exploring these as‑ pects, we discover two important invariants 𝓁𝑝∞ and 𝜈𝑝∞ where 𝑝 is a prime number, holding essential information about cyclotomic extensions of degree 2 and those gen‑ erated by primitive (2𝑒)𝑡ℎ roots of unity where 𝑒 ∈ ℕ. In our quest to provide explicit expressions for the coefficients of the minimal polynomials of the subextensions of cyclotomic extensions generated by primitive (2𝑒)𝑡ℎ root of unity, we discover fasci‑ nating characterizations, some of which are linked to the well‑known Catalan num‑ bers solving Combinatorial problems using field theory. Building upon the insights gained from our exploration of cyclotomic extensions, we provide a comprehensive classification of separable and inseparable radical ex‑ tensions up to isomorphism. In order to have a global understanding of these exten‑ sions up to isomorphism, we exhibit a meaningful parameterization of the set of iso‑ morphic radical extensions into moduli spaces involving the action of some groups.
- ItemAn analysis of security protocols for lightweight systems(Stellenbosch : Stellenbosch University, 2022-04) Kamkuemah, Martha Ndeyapeuomagano; Sanders, Jeff; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.ENGLISH SUMMARY: Security is hard to maintain in distributed systems especially for communicating agents restricted to lightweight computations, as in the Internet of Things, which struggle to implement strong cryptographic security. A methodology is developed for specifying and reasoning algebraically about security in such systems which combines epistemic logic and a state-based formalism. The knowledge modality K is used to define a uthentication a nd s ecrecy i n t erms o f w hat e ach agent knows. Operations are defined a s s tate t ransitions. Having g ained c onfidence in our methodology by applying it to the benchmark case studies Needham-Schroeder and Diffie-Hellman protocols, we then apply it to the contemporary examples Signal and Long-Range Wide-Area Network protocols. A mitigation is proposed and verified for a Long-Range Wide-Area Network.