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.
- ItemComputational and analytical methods for constructing a multilevel model for human glucose metabolism(Stellenbosch : Stellenbosch University, 2022-03) Green, Kathleen Alice; Snoep, Jacob Leendert; Van Niekerk, David Douglas ; Cang, Hui; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.ENGLISH ABSTRACT: Glucose metabolism is carefully regulated in humans to ensure that homeosta- sis is maintained. Disruptions in the multiple processes involved, or the inabil- ity to sustain adequate glucose concentrations, can cause various metabolic complications that can become life threatening. These complications can present as a result of diseases such as Type 2 diabetes (hyperglycaemia), or severe malaria (hypoglycaemia). In the context of malaria, two key metabolic indicators for poor chance of survival are hypoglycaemia (low plasma glu- cose concentrations) and lactic acidosis (high plasma lactate concentrations). Currently, it is understood that these conditions are the result of various clin- ical complications, and the extent to which the malaria parasite Plasmodium falciparum contributes to them is unknown. This contribution could be a consequence of the accelerated glycolytic flux, brought about by the parasite increasing the glucose demand and lactate production, once it has invaded the host’s red blood cells, a hypothesis that is tested in this thesis using a mathematical modelling approach. We used a new approach to building a whole body glucose metabolism model that is well-grounded in a large number of clinical studies following a thorough literature review to obtain clinical data on glucose metabolism. This model is parametrised using data from 49 different studies, and 74 figures that have been successfully reproduced between different softwares. The model construction is performed using a specialised package for model merging called Hierarchical Model Composition [1]. This model consists of several different organs that contribute to glucose metabolism in humans with a specific compartment that was incorporated to describe red blood cell metabolism. In addition to the reference model built for glucose metabolism in a healthy individual, we extend the model to represent malaria patients by explicitly modelling parasitaemia via the inclusion of a detailed mathematical model for Plasmodium falciparum into the red blood cell compartment. The multilevel model for malaria reveals that a 13% parasite burden leads to hypoglycaemia, but lactic acidosis as is observed in malaria patients, is not induced. Patient data and sensitivity analysis is used for initial model validations and identification of potential treatment targets in the parasite’s glycolytic pathway. The multilevel model is large (303 variables) which makes it difficult to anal- yse. Therefore we developed a flexible model reduction technique that can aid in the simplification of the multilevel model through selection of the relevant enzyme mechanisms, while retaining the whole body descriptions on the higher level. This reduction method applies a combination of structural and kinetic modification to the original model, and was tested on different modelling struc- tures and kinetics occurring in biochemical pathways. Thereafter, the method is extended to biological applications which show how multiple model simpli- fications for different inhibitor titration studies can be investigated starting from a single model description and performing various selections of reactions or species. During model merging we encountered logistical challenges such as unit con- version and the use of unique identifiers that are generic for merging different modules into a single model. Our solution was to use automated approaches as much as possible as developed in the Systems Biology community, and to code additional solutions for our automated workflow. This work highlights the benefits of utilising automated approaches, as well as combining differ- ent computational and analytical techniques from different disciplines, during model construction, validation and analysis. By making these models, all datasets, and simulation experiment descriptions available on JWS Online [2], FAIRDOMHub [3], and PK database [4], we envisage that future improve- ments and extensions can be implemented in a systematic way owing to the modular structure of the model, and the transparency and reproducibility of the construction process.
- 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.