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.
- ItemModelling multi-species co-occurrence patterns and processes(Stellenbosch : Stellenbosch University, 2022-04) Lagat, Vitalis Kimutai; Cang, Hui; Guillaume, Latombe; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.ENGLISH ABSTRACT: The structure of ecological communities is determined by the interplay among a range of processes, such as biotic interactions, abiotic filters, and disper- sal. Their effects can be detected by examining patterns of co-occurrence between different species. Using species-by-site matrices, null models that are based on permutations under constraints on row or column sums, have been widely used for comparing the observed values of co-occurrence met- rics (e.g., C-score and the natural metric) against null model expectations. This allows to detect significant signals of species association or dissocia- tion, from which the type of biotic interactions between species (e.g., facil- itative or antagonistic) can be inferred. In such a permutation-based null model test, the levels of co-occurrence between randomly paired species are often pooled to obtain a sampling distribution. However, the level of co-occurrence for three or more species are ignored, which could reflect functional guilds or motifs composed of multiple species within the com- munity. Null model tests without considering multi-species co-occurrence could often lead to false negatives (Type II error) in detecting non-random forces at play. Moreover, variations of co-occurrence have been explored by many models with covariates reflecting between-site environmental filters and distance decay of similarity. This, however, does not allow us to explic- itly explore the role of biotic interactions that could give rise to the observed co-occurrence patterns. An R software package for performing null model testing of multi-species co-occurrence patterns is currently lacking. This dis- sertation focuses on addressing all the above challenges. First, we propose a multi-species co-occurrence index that measures the number of sites jointly occupied by three or more species simultaneously, with the pairwise metric of co-occurrence as a special case for order two. We identify nine archetypes of species co-occurrence and show the majority of real communities con- form to six of these archetypes. Second, we develop a statistical model (gen- eralised B-spline modelling) that can use trait variations among species as a niche-based force and encounter rate as a neutral force to explain the la- tent interaction strength structure. This method decomposes each predictor into a linear combination of B-splines that allow to measure the local sen- sitivity of joint occupancy along the full range of the predictor’s variation. The generalised B-spline modelling can explain the observed co-occurrence and joint occupancy at different orders of joint occupancy. Finally, we im- plement the proposed multi-species co-occurrence index and the associated generalised B-spline modelling in the multi-species co-occurrence (msco) R package for null model testing of multi-species interactions and interference with covariates.
- 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.