Masters Degrees (Mathematical Sciences)
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- ItemMassivelly parallel modular algorithms for the image of rational maps(Stellenbosch : Stellenbosch University, 2024-03) Rakotoarisoa, Hobihasina Patrick; Basson, Dirk; Bohm, Janko; Marais, Magdaleen; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.ENGLISH ABSTRACT: Modular methods are a tool which can be applied in computer algebra to signifi‑ cantly improve the performance of algorithms in characteristic 0 by addressing the problem of intermediate coefficient growth. Computations are done simul‑ taneously over multiple finite fields by reducing the input data, applying the algorithm under consideration in positive characteristic, and then lifting the modular results to the rationals via Chinese remaindering and the Farey map. Even in the existence of bad primes, error tolerance of this process ensures that for a sufficiently large set of good primes the approach terminates with the cor‑ rect answer. The method is clearly parallel and has the potential to scale across multiple computers. It has been applied for various use cases, for example, for the computation of Gröbner bases. In this thesis, we provide a generic modular approach which is applicable to polynomial data structures arising from com‑ mutative algebra and algebraic geometry, such as modules, varieties, and ratio‑ nal maps. Moreover, we develop a massively parallel framework for modular computations, which we model in terms of Petri nets. We give an implementa‑ tion relying on the SINGULAR/GPI‑SPACE framework.
- ItemCanonical connections in Riemannian and Hermitian geometry(Stellenbosch : Stellenbosch University, 2024-03) Sarah, Brian; Bartlett, Bruce; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.ENGLISH ABSTRACT: This thesis presents explicit calculations of three naturally occurring connec- tions in Riemannian and Hermitian geometry. Namely, the Levi-Civita con- nection and the ambient connection in Riemannian geometry, and the Chern connection and the ambient connection in Hermitian geometry. Precisely, we show that the Chern connection and the ambient connection are equal on the tautological line bundle over CP¹. Next, we show that the Levi-Civita con- nection and the ambient connection are equal on the tangent bundle of the two-sphere. Finally, we compute the Chern connection on the tangent bundle of the two-sphere regarded as a Hermitian holomorphic line bundle and show that it is equal to the Levi-Civita connection on the tangent bundle of the two-sphere.
- ItemThe 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.
- ItemThe 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.
- ItemPositive 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.