Masters Degrees (Mathematical Sciences)

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Browsing Masters Degrees (Mathematical Sciences) by browse.metadata.advisor "Boxall, Gareth John"

Now showing 1 - 5 of 5
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    Algebraic points in tame expansions of fields
    (Stellenbosch : Stellenbosch University, 2021-12) Harrison-Migochi, Andrew; Boxall, Gareth John; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH ABSTRACT: We investigate the behaviour of algebraic points in several expansions of the real, complex and p-adic fields. We build off the work of Eleftheriou, Günaydin and Hieronymi in [17] and [18] to prove a Pila-Wilkie result for a p-adic subanalytic structure with a predicate for either a dense elementary substructure or a dense dcl-independent set. In the process we prove a structure theorem for p-minimal structures with a predicate for a dense independent set. We then prove quantifier reduction results for the complex field with a predicate for the singular moduli and the real field with an exponentially transcendental power function and a predicate for the algebraic numbers using a Schanuel property proved by Bays, Kirby and Wilkie [5]. Finally we adapt a theorem by Ax [2] about exponential fields, key to the proof of the Schanuel property for power functions, to power functions.
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    Geometry of Complex Polynomials: On Sendov's Conjecture
    (Stellenbosch : Stellenbosch University, 2016-12) Chalebgwa, Taboka Prince; Boxall, Gareth John; Breuer, Florian; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences
    ENGLISH ABSTRACT : Sendov’s conjecture states that if all the zeroes of a complex polynomial P(z) of degree at least two lie in the unit disk, then within a unit distance of each zero lies a critical point of P(z). In a paper that appeared in 2014, Dégot proved that, for each α ε (0, 1), there is an integer N such that for any polynomial P(z) with degree greater than N, P(a) = 0 and all zeroes inside the unit disk, the disk │z- α│ ≤ 1 contains a critical point of P(z). Basing on this result, we derive an explicit formula N(a) for each α ε (0, 1) and, furthermore, obtain a uniform bound N for all a ε [α,β] where 0 < α < β < 1. This addresses the questions posed in Dégot’s paper.
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    Imaginaries in dense pairs of real-closed fields
    (Stellenbosch : Stellenbosch University, 2017-03) Rakotonarivo, Tsinjo Odilon; Boxall, Gareth John; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences
    ENGLISH ABSTRACT : Imaginaries are definable equivalence classes, which play an important role in model theory. In this thesis, we are interested in imaginaries of dense pairs of real-closed fields. More precisely, we consider the following problem: is acleq equal to dcleq in dense pairs of real-closed fields? To answer this question, we first present some results about real-closed fields, which are basically completeness, quantifier elimination and elimination of imaginaries. Then, we concentrate on the completeness and near model-completeness for the theory of dense pairs of real-closed fields. And finally, we present the key point of the thesis. Namely, we demonstrate that acleq(∅) = dcleq(∅) but there exists A such that acleq(A) 6= dcleq(A)
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    On the definable generalized Bohr compactification of SL(2,Qp)
    (Stellenbosch : Stellenbosch University, 2018-12) Pillay, Nathan Lingamurthi; Boxall, Gareth John; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Division Mathematics.
    ENGLISH ABSTRACT : This paper provides an overview of existing knowledge regarding the socalled definable generalized Bohr compactification of the group SL(2, Qp) of 2 × 2 matrices with determinant 1 and entries in Qp. The (open) question of whether this definable generalized Bohr compactification coincides with the Ellis group of the action of SL(2, Qp) on its type space is also studied in detail. This includes a discussion on the topologies associated with the space of complete types over Qp concentrating on SL(2, Qp), as well as an investigation of the possibility of first-countability of this type space.
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    Ultraproducts and Los’s Theorem: A Category-Theoretic Analysis
    (Stellenbosch : Stellenbosch University, 2017-03) Chimes, Mark Jonathan; Boxall, Gareth John; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences
    ENGLISH ABSTRACT : Ultraproducts are an important construction in model theory, especially as applied to algebra. Given some family of structures of a certain type, an ultraproduct of this family is a single structure which, in some sense, captures the important aspects of the family, where “important” is defined relative to a set of sets called an ultrafilter, which encodes which subfamilies are considered “large”. This follows from Lo´s’s Theorem, namely, the Fundamental Theorem of Ultraproducts, which states that every first-order sentence is true of the ultraproduct if, and only if, there is some “large” subfamily of the family such that it is true of every structure in this subfamily. In this dissertation, ultraproducts are examined both from the standard model-theoretic, as well as from the category-theoretic view. Some potential problems with the categorytheoretic definition of ultraproducts are pointed out, and it is argued that these are not as great an issue as first perceived. A general version of Lo´s’s Theorem is shown to hold for category-theoretic ultraproducts in general. This makes use of the concept of injectivity of a (compact) tree, which is intended to generalize truth of first-order formulae (under given assignments of variables), and, in the category of relational structures, corresponds exactly to first-order formulae. This type of thinking leads to a means of characterizing fields in the category of rings, and a new proof that every ultraproduct of fields is a field, which takes place entirely in the category of rings (along with the inclusion of the category of fields). Finally, the family of all (category-theoretic) ultraproducts on a given family is shown to arise from the “codensity monad" of the functor which includes the category of finite families into the category of families. In this sense, it is shown that ultraproducts are a rather natural construction category-theoretically speaking.