Doctoral Degrees (Mathematical Sciences)
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- 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.
- ItemInvestigations on the Wigner derivative and on an integral formula for the quantum 6j symbols(Stellenbosch : Stellenbosch University, 2022-04) Ranaivomanana, Valimbavaka Hosana; Bartlett, Bruce; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.ENGLISH SUMMARY: wo separate studies are done in this thesis: 1. TheWigner derivative is the partial derivative of dihedral angle with respect to opposite edge length in a tetrahedron, all other edge lengths remaining fixed. We compute the inverse Wigner derivative for spherical tetrahedra, namely the partial derivative of edge length with respect to opposite dihedral angle, all other dihedral angles remaining fixed. We show that the inverse Wigner derivative is actually equal to theWigner derivative. 2. We investigate a conjectural integral formula for the quantum 6j symbols suggested by Bruce Bartlett. For that we consider the asymptotics of the integral and compare it with the known formula for the asymptotics of the quantum 6j symbols due to Taylor and Woodward. Taylor and Woodward’s formula can be rewritten as a sum of two quantities: ins and bound. The asymptotics of the integral splits into an interior and boundary contribution. We successfully compute the interior contribution using the stationary phase method. The result is indeed quite similar to although not exactly the same as ins. Though we expect the boundary contribution to be similar to bound, the computation is left for future work.
- ItemThe definable (p,q)-theorem for dense pairs of certain geometric structures(Stellenbosch : Stellenbosch University, 2021-12) Rakotonarivo, Tsinjo Odilon; Boxall, Gareth John; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Division Mathematics.ENGLISH ABSTRACT: The definable (p, q)-conjecture is a model-theoretic version of a (p, q)- theorem in combinatorics, which was expressed in the form of a question by A. Chernikov and P. Simon in 2015. Researchers have proved that the property holds for certain classes of structures. Based on those existing results, the main objective of the present thesis is to show that the definable (p, q)-conjecture holds for a dense pair of geometric distal structures that satisfies the following condition: algebraic closure and definable closure are the same in sense of the original geometric structure. Independently, we also explore a different approach to prove that under some conditions, the definable (p, q)-conjecture holds in certain cases for dense pairs of real closed ordered fields.
- ItemComplexity and stability of mutualistic local networks and meta-networks(Stellenbosch : Stellenbosch University, 2021-03) Nnakenyi, Chinenye Assumpta; Hui, Cang; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Division Computer Science.; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Division Computer Science.ENGLISH ABSTRACT: Biotic interactions, either in local networks or in meta-networks, are ubiquitous in nature. Species interact with other species of different interaction strengths in the ecosystem. For example, mutualistic interactions, whereby species benefit from each other, have been found to play a significant role in the function and structure of ecological communities. Previous empirical and theoretical studies have shown the vital contribution of mutualistic interactions in maintaining diversity amidst perturbations from the environment. Such perturbations affect the species and their interactions, exerting pressure on the ecosystem. However, it is unclear how the strengths of species interactions affect species abundances in the communities, and understanding the mechanism behind the complexity and stability of mutualistic meta-networks and local networks remains a challenge to be addressed. In this thesis, using a random matrix approach, I found that the stability criteria of a block-structured network or matrix is obtained from max( r1; r2) m < 0, where m is derived from the expectation of the diagonal elements of the matrix, while r1 and r2 are derived from the off-diagonal elements of the matrix when the expectation of the off-diagonal elements is different from zero and equal to zero respectively. Also, using a Lotka-Volterra model of mixed interaction types in different proportions, that describes the dynamics of species abundances, I found that species abundances are determined more by the species’ sensitivities to the interaction pressures from their partners than by species’ impacts on their partners. Besides, the abundances of the rarest species was found to be a good indicator of the resilience of the communities. Even when modelling real mutualistic local networks using a modified Lotka-Volterra model that incorporates adaptive interaction switching (AIS) and environmental variables, I found that the AIS could destabilise the local networks. However, to explain the emergence of nestedness and modularity in those networks, I found AIS to be a key driving mechanism behind community nestedness, with the environmental variables playing a secondary role in explaining nestedness and modularity. Finally, using a competition-mutualism model of meta-networks, I showed the role of dispersal and the role of mutualism to the complexity and stability of the networks. I found that incorporating mutualism in the model of meta-networks is crucial to the functioning of the meta-networks, as mutualism increases the stability of the meta-networks, increases the total abundance of species, decreases unevenness in the species abundances, and increases nestedness more than in the model without mutualism. Also, I showed that dispersal is a strong stabilising factor for the meta-networks. Importantly, dispersal heterogeneity between local networks drives the changes in total abundance, unevenness, and compositional similarity of species in the meta-networks and the local networks, irrespective of the dispersal heterogeneity across species. That is dispersal heterogeneity between the local networks decreases total abundance, increases unevenness and decreases compositional similarity in the meta-networks and local networks. Knowledge about the dispersal rates between local networks and across species is crucial to understand the complexity and stability of the local and meta-networks. Hence, these findings have contributed to the stability and complexity of ecological networks, at both local and regional scales, which is relevant for the management and conservation of interaction networks with the objective of preserving the species functions and services in the ecosystem.
- ItemTheoretical study of variable viscosity nanofluids flow in microchannels(Stellenbosch : Stellenbosch University, 2020-12) Monaledi, Ramotjaki; Makinde, Oluwole Daniel; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Division Mathematics.ENGLISH ABSTRACT: The study of fluid flow and heat transfer through a microchannel is an important research area due to its wide applications in engineering and industrial processes. Some practical applications include problems dealing with cooling, lubrication of porous bearings, petroleum technology, ground water hydrology, drainage and purification processes. A nanofluid is the suspension of nanoparticles in a base fluid. Nanofluids are capable of heat transfer enhancement due to their high thermal conductivity. For practical applications of nanofluids, research in nanofluids convection is significant. Due to their enhanced properties, nanofluids can be used in the deficiency of technical and biomedical applications such as nanofluid coolant in electronics cooling, vehicle cooling and transformer cooling. This study considered the detailed analysis of both single and two-phase Couette and Poiseuille flow behaviour and heat transfer using this innovative fluid as working fluid through a microchannel. Useful results for the velocity, temperature, nanoparticles concentration profiles, skin friction and Nusselt number were obtained and discussed quantitatively. The effects of important governing flow parameters on the entire flow structure were examined. In this thesis, a more realistic modified Buongiorno’s nanofluid model is proposed and utilized to examine the impact of nanoparticles’ injection and distribution on inherent irreversibility in a microchannel Poiseuille flow of nanofluid with variable properties. The governing nonlinear differential equations are obtained and tackled numerically using the shooting method coupled with the Runge-Kutta-Fehlberg integration scheme. Graphical results showing the effects of the pertinent parameters on the nanofluid velocity, temperature, nanoparticles concentration, skin friction, Nusselt number, Sherwood number, entropy generation rate and Bejan number are presented and discussed quantitatively.