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Comparitive analysis of bornology in the categories of frolicher spaces and topological spaces
(2024) Mahudu, Ben Moditi
This thesis seeks to introduce the concept of bornology to the theory of Fr¨olicher spaces. Bornologies are induced from the Fr¨olicher structure, Fr¨olicher topology and the canonical topology on the underlying set of Fr¨olicher space. In each case the bornologies are compared in a general Fr¨olicher space, subspace, product, coproduct and quotient. The Fr¨olicher topology refers to the topology induced from structure functions of the Fr¨olicher space. The bornology induced from the Fr¨olicher structure is induced from the structure functions of Fr¨olicher space. An initial bornology is canonically induced on the underlying set of Fr¨olicher subspace and product, and a final bornology is induced canonically on the underlying set of Fr¨olicher coproduct and quotient. For Fr¨olicher subspace and product the initial bornology is finer than the bornology induced from structure functions. Dually, the final bornology is coarser than the bornology induced from structure functions for Fr¨olicher coproduct and quotient. Relatively-compact and compact bornologies are induced from the Fr¨olicher topology and the canonical topology on the underlying set of Fr¨olicher space. For each of Fr¨olicher subspace, product, coproduct and quotient, that is, the objects in the category of Fr¨olicher spaces under the study of this thesis, there are two topologies - the canonical topology induced from the underlying set and the Fr¨olicher topology. Subsequently there are two relatively-compact and compact bornologies, induced from these topologies, for each of the mentioned objects. The bornological comparison between the relatively-compact bornologies and the bornological comparison between the compact bornologies, for each object, is determined by the comparison of these topologies, that is, the comparison between the Fr¨olicher topology and the canonical topology on the underlying set.
Continued fractions and cryptographic applications
(2022) Sookraj, Priyanka
In this dissertation, Michael J. Wiener’s proposed attack on short secret exponents used in the RSA cryptosystem is studied and thus an application of continued fractions in the cryptanalysis is highlighted. Furthermore, some variants and improvements to the attack proposed by A. Dujella, M. Bunder and J. Tonien are studied.
Enumerations and bijections of Dyck paths
(2024) Mohlala, Derrick
A Dyck path is a non-negative lattice path with the steps u = (1, 1) and d = (1, −1) such that the path starts at the origin and ends on the x-axis. In this research we consider some bijections that Dyck paths have with certain Catalan objects: bargraphs, d-ary trees, Motzkin paths and other Dyck paths. We apply the bijections to derive relationships that arise between the statistics of the Dyck paths and the Catalan objects, and subsequently show the enumerations of Dyck paths with regard to these statistics. The statistics that we consider include: the semiperimeter minus the number of peaks of the corresponding bargraph, the semilength and size k of the downward step d = (1, −k) of the k-Dyck path, the semilength, the size k of the downward step d = (1, −k) and the lower bound y = −t of the kt-Dyck path, the number of hills, odd rises, even rises, returns and semilength of the Dyck path, and lastly the number of centred, left and right tunnels, centred multitunnels and semilength of the Dyck path. Finally, we apply several techniques of the symbolic method to derive the enumeration of cornerless Motzkin paths, bargraphs and kt-Dyck paths.
Inverse operations on tensor products of matrices
(2022) van der Merwe, Francesca
A variety of products of matrices arise by considering different algebraic structures – for example, linear transformations (matrix multiplication), product vector spaces which leads to entrywise products (known as the Hadamard or Schur product) and bilinear transformations (tensor products). Inverse operations of linear transformations have been extensively studied in the literature, but inverse tensor products are less well known. This dissertation considers these inverse operations from different perspectives, by focusing on characterising such operations and examining certain desirable properties. Primarily, by abiding to an algebraic perspective, quotients of vector spaces of (ms) × (nt) matrices are considered by characterising linear quotient functions. Requirements for such functions to satisfy desirable properties, in addition to linear properties, are considered. Additional quotients, which do not appear in the literature, are derived. Multiplicative (monoidal) quotients are also considered. These quotients only exist on restricted structures, and their limitations are briefly examined. Lastly, by relaxing the requirement for a purely algebraic quotient and finitedimensional spaces, an analytic approach is considered by assessing a least squares minimisation of objects on reproducing kernel Hilbert spaces. In this method, Tikhonov regularisation is employed to ensure boundedness in obtaining inverse operations
Mathematical analysis of graphene grid structures with defects
(2022) Hlawe, Phinda K.
In this dissertation, we explore the work by M. Archibald, S. Currie and M. Nowaczyk in their paper “Finding the hole in a wall.” In this paper, the authors solve the inverse problem of locating the position of a single vacancy break using lengths of closed paths on an infinite hexagonal grid structure. In order to do this they transform the infinite hexagonal grid structure that models graphene into a brick wall structure. When a single vacancy break occurs, polygons of odd length are introduced into the grid structure. First, we explore lemmas that state which polygon the closed paths of shortest odd length circumnavigate. We then use these to provide a rigorous proof of the two main theorems in “Finding the hole in a wall”. These depend on the region that the path originates from in the brick walls and tell us what the path is congruent to modulo 4. Finally, we study the algorithm for determining the exact position of the defect, and sometimes, provide alternative formulae for locating the defect. When this is the case, we show the formulas are equivalent to those in their work. Also provided are potential future studies in this area.
On identities of Euler type and partitions with initial repetitions
(2024) Mugwangwavari, Beaullah
In this research, we explore partition functions related to identities of Euler type and partitions with initial repetitions. On identities of Euler type, we mainly focus on identities due to P. A. MacMahon, G. E. Andrews and M. V. Subbarao. We give new bijections for partition theorems due to MacMahon and Andrews and provide a generalization of Subbarao’s finitization theorem. For the original Euler’s identity which gave rise to identities of Euler type, Andrews and M. Merca found a new combinatorial interpretation for the total number of even parts in all partitions of n into distinct parts. We genaralize their result and establish more variations with connections to some of the work of S. Fu and D. Tang. By conjugating partitions into distinct parts, we obtain partitions without gaps. Andrews extended the notion of partitions without gaps to partitions with initial repetitions. We study partitions with initial repetitions and give several Legendre theorems. These theorems provide partition-theoretic interpretations of some well known q-series identities of Rogers-Ramanujan type. We further deduce parity formulas for partition functions associated with this class of partitions.
On the expressivity of the many-valued interval-based temporal logics
(2022) Ledwaba, Lesibana Andries
Interval-Based Temporal Logics take intervals over linear orders as the primary objects of temporal analysis. The are 13 relations between the intervals known as Allen’s Relations on a linear order. We use Allen’s relation as the accessibility relation between intervals and interpret the interval structures as Kripke frames. One can think of Interval-Based Temporal Logics in a Many-Valued Interval setting where propositional variables are not just true or false but they are true or false to some extent and this extent we take as members of an algebra of truth values. Moreover, intervals can be taken to arise from many-valued linear orders. In this thesis we consider the interdefinability of modalities in the many valued interval setting. We define truth preserving morphisms that allow us to characterize the expressivity of many-valued interval-based temporal logic (MVIBTL). We use bismimulation as our primary truth preserving morphism and characterize which of the MVIBTL modalities are expressible in terms of the other modalities
On the Rogers-Ramanujan identities and partition congruences
(2024) Seleka, Phodiso
In this dissertation, we study the Rogers-Ramanujan identities and partition congruences. The original Rogers-Ramanujan identities are proved analytically via Bailey’s construction. To the related Rogers-Ramanujan identities, our approach to the proof is via partition analysis and Bailey’s construction. The summation part of these Rogers-Ramanujan identities is established using partition analysis. Our work on congruences starts with a revisit of the popular Ramanujan’s congruences for the unrestricted partition function. The Atkin-Swinnerton-Dyer congruences for moduli 5 and 7 are obtained and the Ramanujan’s most beautiful identity is proved. The proof techniques for Ramanujan’s most beautiful identity are extended to another version in modulo 7. As part of the contribution to knowledge, a recurrence formula for the parity of the number of 2-color partitions of 2n in which one of the colors appears only in parts that are even is derived.
Sequence alignment using the seeding heuristics method
(2022) Oyekanmi, Adedoyin
In this dissertation, we studied Guillaume J. Filion’s paper on utilizing analytic combinatorics to calculate the probability that a read of size (s = k) has no seed (P(S)). First, we computed the weighted generating functions of all reads and reads that have no seed by constructing sequences of combinatorial objects using transfer graphs and transfer matrices as formulated by Filion. Then, we extended his logic to calculate the probability that a seed is present in a read (1 − P(S)) by singularity analysis, after obtaining the probability that a read has no seed. This involved extracting coefficients of weighted generating functions and solving complex polynomials which is the analytic combinatorics approach. Finally, we use recurrence approach (on Mathematica) to find more precise probabilities than what was provided by Filion for some practical examples.
Stochastic modelling of the spread of infectious diseases
(2022) Docrat, Raeesa
Background The spread of infectious diseases is a world-wide problem that has a greater impact on low-income countries. Mathematical modelling is a useful tool to better understand these diseases and to plan prevention and interventions. These models can either be deterministic or stochastic. Generally, deterministic models are used as they are easier to understand and predict average behaviour. Stochastic models are probabilistic models which incorporate random fluctuations over time. Stochastic models allow the calculation of probabilities and likelihoods of possible progressions of an infectious disease. These models can be built to include many aspects and features of infectious diseases. One such inclusion is the spatial nature inherent in the spread of infectious diseases. The inclusion of a spatial component for the spread of infectious diseases is needed, especially in South Africa, where it is spatially heterogeneous, to accurately analyse and efficiently allocate resources to where it is most needed. Methods This thesis begins with a review of both deterministic and stochastic compartmental models to illustrate the uses of these models in the context of disease modelling. The spatial autocorrelation of the spread of TB in South Africa is then studied. The results, although intuitive, confirm the existence of spatial autocorrelation and hence the need for models which accounts for the spatial v heterogeneity. A spatio-stochastic model in discrete time and space is then developed using binomial chain models as a basis. The spatial and temporal differences are modelled in the number of contacts a person would encounter. The spatio-stochastic models are then further extended to account for different modelling scenarios: closed spatial populations, spatial populations with interactions, but no migrations and finally a spatial model with migrations and interactions from other spatial units. This developed model is then compared to existing models in a simulation study. Lastly, the developed model is applied to a measles data set. Results The simulation study illustrates the differences in the outcome of different cases of the spatio-stochastic model. The disease develops and progresses very differently, depending on the modelling case chosen. This highlights the importance of choosing the correct model. The simulation study demonstrated the differences between other binomial chain based models, namely the ReedFrost and the Tuckwell Williams model. The inclusion of spatial heterogeneity produced outcomes that varied from the two non-spatial models. Lastly, the application of the model to a measles data set confirms that the inclusion of the spatial component allows a better fit of the models to the data. Conclusion The spatio-stochastic models are flexible and can be tailored to the dynamics of a particular disease’s spatial spread. This aspect makes these models very useful in understanding and analysing the spatial spread of the infectious disease. The developed spatio-stochastic model can be applied to a wide variety of cases. The fact that it is stochastic allows not only the ability to predict average behaviour, but also to calculate probabilities related to vital questions in planning and prevention. Questions related to the probability of outbreak in a particular area or which areas are likely to be hot spot areas can now be addressed. The modelling of the spread of infectious diseases using a spatial model, like the model developed in this thesis, is important to make informed decisions on how to stop and prevent further infection. The spatial models can provide information on sub-populations which can be used to allocate resources accordingly. It is also useful in identifying the effects of different interventions like vaccines. Due to globalisation and the level of connection in society, the spatio-stochastic modelling of infectious diseases is vital.
Symmetry analysis of geometries in general relativity and mathematical models in quantitative finance
(2024) Obaidullah, Usaamah
This thesis may be divided into two themes. The first, consists of a mathematical analysis of selected models in cosmology, while the second, is in the field of quantitative finance. The pp-wave spacetime, Bianchi I spacetime, and the Bianchi II spacetime are the three universes that we examine. The latter two are studied in the framework of f(R) theory of gravity, a plausible substitute for general relativity and a solution to the dark energy problem. We apply the Killing and homothetic vector fields to divide the spacetimes into classes or categories. Subsequently, potential functions are established using the geometry of the point symmetries of the space, while Noether’s theorem provides the first integrals connected to each isometry. We take advantage of the geometric fact that the homothetic algebra of spacetimes yields the Noether point symmetries of geodesic Lagrangians. For the Bianchi spacetimes, the Wheeler-DeWitt equations are derived by the quantisation of the spacetime Lagrangians, and from the Lie point symmetries it admits, we sequentially find invariant solutions to solve for the universe’s wave function. Finally, exact solutions to the field equations are also found. The research comprising of the second theme, investigates two nonlinear partial differential equations used in derivative pricing for financial markets. The point symmetries, invariant solutions, and conversation laws of these equations are found. In our analysis, we vary certain variables that change the nonlinearity of the models and thus give us unique symmetries and solutions. With various parameter settings, graphical solutions are investigated.
The algebra and geometry of using continued fractions for approximating real and complex numbers
(2022) Mennen, Carminda Margaretha
Through geometric analysis we forge a new interpretation of the link between nearest integer continued fractions and the Farey tessellation of hyperbolic space. Instead of just truncating the continued fraction to generate approximations, we focus on • how to parse the product of maps derived from the nearest integer continued fraction into a product of parabolic and elliptic Mobius maps and ¨ • on the collection of points on which these maps act. It turns out that we need to set apart the elliptic maps that permute a collection of six vertices, three values in R, namely ∞, 0 and 1, and three values in C, namely i, 1 + i and 1+i 2 . The action of the remaining parabolic maps on the same six vertices results in the creation of a sequence of nested Farey quadrilaterals, containing the target, whose boundaries are based in the Schmidt arrangement formed by the Farey sets and dual Farey sets of Schmidt.
The ARS algorithm and invariance analysis of ordinary difference equations
(2022) Kubayi, Jollet Truth
This dissertation will be divided into two parts. The first part will involve the use of symmetries to find exact solutions of higher order difference equations. The second part will deal with important ordinary differential equations, in the search for singularities and integrability testing. We will analyse a Painlev´e equation, Ivey’s equation, the higher-order Lane-Emden type equation and a class of third-order boundary flow equations.
The bipartite ramsey number of cycles
(2021) Tivane, Amukelani
The Ramsey number R(H1, H2) of two graphs H1 and H2 is the smallest positive integer n for which every red-blue coloring of the complete graph Kn of order n results in a subgraph of Kn isomorphic to H1 all of whose edges are colored red (called a red H1), or a subgraph of Kn isomorphic to H2 all of whose edges are colored blue (called a blue H2). The s-bipartite Ramsey number bs(H1, H2) of two bipartite graphs H1 and H2 is the smallest positive integer t, with t ≥ s, such that every red-blue coloring of the complete bipartite graph Ks,t results in a red H1 or a blue H2. For s = t, the sbipartite Ramsey number is known as the bipartite Ramsey number, which we denote by b(H1, H2). In this dissertation we investigate b(C2m, C2n) and bs(C2m, C2n) for m ≥ 2 and n ≥ 2.
The Genus of a Nilpotent R-Powered Group
(2022) Ginster, Carl
In [14], Mislin define the genus G(N) of a finitely generated nilpotent group N to be the set of isomorphism classes of finitely generated nilpotent groups M such that the localizations Mp and Np are isomorphic at every prime p. In [6], Hilton and Mislin define an abelian group structure on the genus set G(N) of a finitely generated nilpotent group N with finite commutator subgroups. Let χ0 be the class of finitely generated groups with finite commutator subgroup. For a χ0-group G, the non cancellation set of G, denoted by χ(G), is the set of isomorphism classes of groups H such that G×Z ∼= H ×Z. Warfield, in [19], proved that, if N is a nilpotent χo-group, then G(N) = χ(N). In [20], the author showed that, for a χo-group G the non- cancellation set χ(G) has a group structure similar to the group structure on the Mislin genus of a nilpotent χo-group. Let R be a binomial ring. The nilpotent R-powered group, first introduced by P. Hall in [5], is a nilpotent group G extended by a binomial ring R. Many results that are found in the theory of nilpotent groups carry over to the class of nilpotent R-powered groups. In particular, Majewicz and Zyman in [13], showed that the P-localization of a nilpotent R-powered group G, for a set of primes P in R can be obtained. We study the genus of a finitely R-generated nilpotent R-powered group. We show that the for any two finitely R-generated nilpotent R-powered groups G and H and some finitely R-powered abelian group A, if G(G × A) = G(H × A) then we have G(G) = G(H)
Threshold functions of colorings of random graphs
(2024) Lucas, Kim
Threshold Functions of Colorings of Random Graphs By Elizabeth Jonck, Kim Lucas, and Ronald Maartens. Let Fn,p be the set of all graphs with n vertices, m edges and with probability p = p(n) of an edge occurring independently. Each graph G ∈ Fn,p has probability P[G] = p m(1 − p) ( n 2)−m of occurring. This is called the Binomial Random Graph Model, denoted G(n, p). Let G now be a connected graph. A rainbow colored graph is when every two vertices of V (G) are connected by a path where each edge has a unique color. If Q is an increasing property, then a function t = t(n) is called the threshold function for Q if (i) p << t so that limn→∞(P[G has property Q]) = 0, and (ii) p >> t so that limn→∞(P[G has property Q]) = 1. A function f(n) is called the sharp threshold function for the property Q if there exists constants C and c such that G satis es Q almost surely for p ≥ Cf(n), and G almost surely does not satisfy Q for p ≤ cf(n). In this dissertation, we investigate threshold functions and sharp threshold functions of random graphs to be rainbow colored.
X and Y-coordinates of Pell equation of some special forms
(2024) Zottor, Faith Shadow
This work primarily characterises the values of d in Pell equation X 2 −dY 2 = ±1 that have at least some specified number of the sequence of X or Y solutions belonging to some interesting sequence of positive integers.

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