Masters Degrees (Mathematical Sciences)

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    The nonvanishing of almost-prime twists of modular L-functions
    (Stellenbosch : Stellenbosch University, 2023-11) Andrianarisoa, Tolotranirina Gabriel; Ralaivaosaona, Dimbinaina; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH ABSTRACT: 𝐿 functions are special types of Dirichlet series which often hold fundamen tal arithmetic information. Hence, they are among the most important objects in analytic number theory. In this thesis, we consider the so called Hecke 𝐿 function 𝐿(𝑠, 𝑓, 𝜒𝑑) associated to a given normalized holomorphic newform 𝑓 twisted by the Kronecker symbol 𝜒𝑑. It is well known that the twisted 𝐿(𝑠, 𝑓, 𝜒𝑑) converges absolutely for Re(𝑠) > 1 and admits a functional equation which extends it analytically to the whole complex plane. The value of 𝐿(𝑠, 𝑓, 𝜒𝑑) at 𝑠 = 1/2 is of special interest. For instance, if the form 𝑓 parametrizes a twisted elliptic curve 𝐸 of given rank 𝑟 ≥ 0, then the Birch Swinnerton Dyer conjecture asserts that 𝑟 is precisely the order of vanishing of 𝐿(𝑠, 𝑓, 𝜒𝑑) at 𝑠 = 1/2. In this work, we ϐix a holomorphic newform 𝑓 of weight at least 2, level 𝑁 with trivial nebentype and consider the family of twisted 𝐿 functions 𝐿(𝑠, 𝑓, 𝜒𝑑) where 𝑑 is any fundamental discriminant with (𝑑, 𝑁) = 1. Using an adapta tion of a method by Iwaniec, we prove that there are inϐinitely many funda mental discriminants 𝑑 such that 𝐿(1/2, 𝑓, 𝜒𝑑) ≠ 0. In addition, following an idea outlined by Hoffstein and Luo, using combinatorial sieve, we prove that the same holds for inϐinitely many almost prime fundamental discriminants 𝑑 with at most 84 prime factors. Further improvement of this result, which relies on properties of some multiple Dirichlet series, is also discussed in this work. Under some assumptions on certain weight factors, it is possible to reduce the number 84 to just 4.
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    The categorical and algebraic aspects of near-modules and near-vector spaces
    (Stellenbosch : Stellenbosch University, 2023-03) Moore, Daniella; Janelidze, Zurab; Marques, Sophie; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH SUMMARY: In this thesis we generalize some results from vector spaces to near-vector spaces in the sense of J. André. In particular, we establish the First Isomorphism Theorem, which leads us to proving that the category of near-vector spaces is an abelian category. We also include an algebraic proof of the non-trivial fact that a subspace of a near-vector space is itself a near-vector space. Other algebraic and categorical properties of near-vector spaces are also obtained.
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    Positive weighted koopman semigroups on banach lattice modules
    (Stellenbosch : Stellenbosch University, 2023-03) Olabiyi, Tobi David; Heymann, Retha; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH SUMMARY: In this thesis, we introduce the notion of a positive weighted semigroup representation on a Banach lattice module over a group representation on a commutative Banach lattice algebra. One main theme of this work is the following: for topological dynamics, we obtain the abstract representation of the lattice of continuous sections vanishing at infinity of a topological Banach lattice bundle (over a locally compact space Ω) as a structure which we call an AM m-lattice module over C0(Ω) on which every positive weighted semigroup representation over the Koopman group representation on C0(Ω) is isomorphic to a positive weighted Koopman semigroup representation induced by a unique positive semiflow on the underlying topological Banach lattice bundle (over the continuous flow on the base space Ω). And as a result, every positive dynamical Banach lattice bundle can be assigned uniquely to a certain positive dynamical m-lattice module and vice versa, which is the Gelfand-type theorem that we proved. In order to do this, we, in particular, establish the following two categories of (i) Banach lattice modules and their dynamics; and (ii) Banach lattice bundles and their dynamics. We pay special attention to the case of a topological positive R+-dynamical Banach lattice bundle by which we obtain the corresponding C0-semigroup of positive weighted Koopman operators, and using the theory of strongly continuous semigroup of positive operators, we obtain results pertaining to properties of the generator, and spectral theory of this positive semigroup.
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    On extensivity of morphisms in general categories
    (Stellenbosch : Stellenbosch University, 2023-03) Theart, Emma; Hoefnagel, Michael; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH SUMMARY: The notion of an extensive category captures a fundamental property of the category of sets, namely, that coproducts are disjoint and universal. This property may be restricted in several ways, one of which is with respect to morphisms in a category. The resulting notion of “extensive morphism” is the central notion of this thesis. An object is then called “mono-extensive” if every monomorphism into it is extensive. We explore these notions in categories which are far from being extensive. The category Set of pointed sets, for instance, in not extensive (since it is pointed), but a morphism in Set is extensive if and only if it has trivial kernel. In the category of finitely generated abelian groups, we show that a group G is mono-extensive if and only if it is cyclic. This leads to an open question about the category of abelian groups: is an abelian group G mono-extensive if and only if it is locally cyclic? We establish various theoretical results, one of the main results being a characterisation of coextensive categories: a Barr-exact category with global support is coextensive if and only if its monomorphisms are coextensive.
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    Population-level considerations for the treatment of lethal diseases in resource-limited settings
    (Stellenbosch : Stellenbosch University, 2023-03) Sereo, Tumelo Donald; Pulliam, Juliet RC; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH ABSTRACT: Diseases such as Severe Acute Respiratory Syndrome Coronavirus (SARS-CoV) and Ebola Virus disease (EVD), continue to challenge health systems worldwide. At the onset of outbreaks of unknown pathogens, there is often no cure, treatment or vaccine available to limit their impact. As the outbreak unfolds, randomised controlled trials are conducted, usually in patients with severe disease, to investigate candidate treatments. Often, several treatments end up being effective, raising the question of which one is most optimal to deploy. While clinical trials focus on individual-level outcomes, population-level outcomes are often more important for public health decision-making. This study answers two questions: First, when can a hypothetical treatment that increases hospital stay duration and probability of survival be used to improve the population-level mortality outcomes under constrained hospital capacity? Second, when is it preferable to invest in treatments versus beds, in a limited resource setting? We developed a transmission dynamic model, parameterised separately for SARS-CoV2 and Ebola, to address the questions posed. For the first question, we ran the model for baseline (no treatment) and treatment scenarios defined by the probability of surviving and duration of hospital stay. We used cumulative mortality as the metric to compare the population-level outcomes. The model shows that there is a substantial region of parameter space in which it is beneficial to use hypothetical treatments that increase probability of surviving and hospital stay duration. For the second question, we performed a cost-minimization analysis to examine when it is preferable to invest in treatments versus beds, in a limited resource setting. The model identified the number of additional beds that would be needed to obtain approximately the same outcomes compared to what is expected with existing treatments. We found that is it preferable to invest in additional beds rather than the existing treatments when the cost per course of treatment is greater than a threshold that depends on the drug under consideration. We estimated that this threshold is around R5 000 for existing SARS-CoV-2 drugs but higher for available Ebola therapies.