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# Doctoral Degrees (Mathematical Sciences)

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- ItemAn algebraic framework for reasoning about privacy(Stellenbosch : University of Stellenbosch, 2016-03) Rajaona, Solofomampionona Forunat; Sanders, J. W.; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Division Mathematics.
Show more ENGLISH ABSTRACT: In this thesis, we study a formal programming language and algebraic tech-niques to analyse computational systems that considers data confidentiality and hidden computations. The reasoning techniques are based on the refinement of programs (Back and von Wright, Carroll Morgan). The underlying logic is a first-order S 5 n epistemic logic that distinguish b etween o bjects and concepts – of the family of Melvin Fitting’s First Order Intensional Logic. We give a relational semantics and a weakest-precondition semantics to prove the soundness of programming laws. The laws for confidentiality r efinement ex-tends those of Carroll Morgan’s Shadow Knows refinement c alculus, whereas the laws for reasoning about knowledge derives mostly from the Public An-nouncement Logic. As applications for knowledge dynamics, we study the classical puzzles of the Three Wise Men and the Muddy Children by means of the programming laws; and as an application for reasoning about confiden-tiality and anonymity, we give a sketch of formal analysis of the Anonymous Cocaine Auction Protocol.Show more - ItemAn analogue of the Andre-Oort conjecture for products of Drinfeld modular surfaces(Stellenbosch : Stellenbosch University, 2013-03) Karumbidza, Archie; Breuer, Florian; Keet, A. P.; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
Show more ENGLISH ABSTRACT: This thesis deals with a function eld analog of the André-Oort conjecture. The (classical) André-Oort conjecture concerns the distribution of special points on Shimura varieties. In our case we consider the André-Oort conjecture for special points in the product of Drinfeld modular varieties. We in particular manage to prove the André- Oort conjecture for subvarieties in a product of two Drinfeld modular surfaces under a characteristic assumption.Show more - 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.
Show more 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.Show more - ItemAnalysis of the rolling motion of loaded hoops(Stellenbosch : University of Stellenbosch, 2008-03) Theron, Willem F. D.; Maritz, M. F.; University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences. Applied Mathematics.
Show more This dissertation contains a detailed report on the results of a research project on the behaviour of a dynamical system consisting of a hoop to which a heavy particle is fixed at the rim. This loaded hoop rolls on a rough surface while remaining in the vertical plane. The motion of the hoop consists of various, possibly alternating, phases consisting of rolling without slipping, spinning or skidding motion and in some cases ends by hopping off the surface. A general mathematical model is developed, consisting of a system of second order ordinary differential equations, one for each of the three degrees of freedom. Analytic solutions are obtained in some cases; otherwise numerical solutions are used. Three specific applications of the general model are dealt with. In the first application the problem of massless hoops is investigated. The main emphasis is on the somewhat controversial question of what happens after the normal reaction becomes zero in a position where the particle is still moving downwards. A new result shows that the hoop can continue to move horizontally in a motion defined as skimming. The second application deals with rigid hoops and a large number of detailed results are presented. Classification schemes for the different types of behaviour are introduced and summarised in the form of phase diagrams. Some emphasis is placed on the rather amazing number of different patterns of motion that can be obtained by varying the parameters. In the third application two elastic models are analysed, with the primary purpose of explaining one aspect of the reported behaviour of experimental hoops, namely hopping while the particle is moving downwards. A chapter on experimental models rounds off the project.Show more - ItemAnalysis of tree spectra(Stellenbosch : Stellenbosch University, 2018-12) Dadedzi, Kenneth; Wagner, Stephan; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Division Mathematics.
Show more ENGLISH ABSTRACT : We study the set of eigenvalues (spectrum) of the adjacency matrix, Laplacian matrix and the distance matrix of trees. In particular, we focus on the distribution of eigenvalues in the spectra of large random trees. The families of random trees considered in this work are simply generated trees and increasing trees. We prove that attaching several copies (two or more) of a tree H to vertices in a tree T “forces" certain real numbers into the adjacency, Laplacian or distance spectrum of the resulting tree. With this construction of forcing subtrees, we prove that the mean proportion of an eigenvalue α in the spectrum of a large random tree is at least the mean number of occurrences of a specific forcing subtree in the large random tree. This gives us explicit lower bounds on the asymptotic mean multiplicity of eigenvalues for different families of random trees. We prove that the mean proportion of an eigenvalue α in the spectrum of the adjacency matrix and Laplacian matrix of a large simply generated tree can be obtained by solving a system of functional equations. We provide an algorithm to solve this system numerically for a given eigenvalue. For instance, using this algorithm, we show that on average approximately 1.4%, 2.1%, 2.5% and 3.3% of the spectrum of a large pruned binary tree, labelled rooted tree, plane tree and pruned ternary tree respectively consist of the eigenvalue 1. Further, we provide explicit formulas for computing the mean proportion of the eigenvalue 0 in the spectrum of a large simply generated tree. We also study the spectra of recursive trees and binary increasing trees. We show that the distribution of the eigenvalue 0 (and other eigenvalues) in these random trees satisfies a central limit theorem. We also compute the mean and variance of the multiplicity of the eigenvalue 0 in the spectrum of a large recursive tree and binary increasing tree. The final chapter deals with related, but somewhat different questions. Given a rooted tree T with leaves v1, v2, . . . , vn, we define the ancestral matrix C(T) of T to be the n × n matrix for which the entry in the i-th row, j-th column is the level (distance from the root) of the first common ancestor of vi and vj . We study properties of this matrix, in particular regarding its spectrum: we obtain several upper and lower bounds for the eigenvalues in terms of other tree parameters. We also find a combinatorial interpretation for the coefficients of the characteristic polynomial of C(T), and show that for dary trees, a specific value of the characteristic polynomial is independent of the precise shape of the tree.Show more - ItemApplying the MDCT to image compression(Stellenbosch : University of Stellenbosch, 2009-03) Muller, Rikus; Herbst, B. M.; Hunter, K. M.; University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences. Applied Mathematics.
Show more The replacement of the standard discrete cosine transform (DCT) of JPEG with the windowed modifed DCT (MDCT) is investigated to determine whether improvements in numerical quality can be achieved. To this end, we employ an existing algorithm for optimal quantisation, for which we also propose improvements. This involves the modelling and prediction of quantisation tables to initialise the algorithm, a strategy that is also thoroughly tested. Furthermore, the effects of various window functions on the coding results are investigated, and we find that improved quality can indeed be achieved by modifying JPEG in this fashion.Show more - ItemArithmetic of carlitz polynomials(Stellenbosch : Stellenbosch University, 2014-12) Bamunoba, Alex Samuel; Keet, Arnold; Stellenbosch University. Faculty of Economic and Management Sciences. Department of Africa Centre for HIV/AIDS Management.
Show more ENGLISH ABSTRACT: See pdf for abstractShow more - ItemAspects of the pre- and post-selection classification performance of discriminant analysis and logistic regression(Stellenbosch : Stellenbosch University, 1997-12) Louw, Nelmarie; Le Roux, N. J.; Steel, S. J.; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
Show more ENGLISH ABSTRACT: Discriminani analysis and logistic regression are techniques that can be used to classify entities of unknown origin into one of a number of groups. However, the underlying models and assumptions for application of the two techniques differ. In this study, the two techniques are compared with respect to classification of entities. Firstly, the two techniques were compared in situations where no data dependent variable selection took place. Several underlying distributions were studied: the normal distribution, the double exponential distribution and the lognormal distribution. The number of variables, sample sizes from the different groups and the correlation structure between the variables were varied to' obtain a large number of different configurations. .The cases of two and three groups were studied. The most important conclusions are: "for normal and double' exponential data linear discriminant analysis outperforms logistic regression, especially in cases where the ratio of the number of variables to the total sample size is large. For lognormal data, logistic regression should be preferred, except in cases where the ratio of the number of variables to the total sample size is large. " Variable selection is frequently the first step in statistical analyses. A large number of potenti8.Ily important variables are observed, and an optimal subset has to be selected for use in further analyses. Despite the fact that variable selection is often used, the influence of a selection step on further analyses of the same data, is often completely ignored. An important aim of this study was to develop new selection techniques for use in discriminant analysis and logistic regression. New estimators of the postselection error rate were also developed. A new selection technique, cross model validation (CMV) that can be applied both in discriminant analysis and logistic regression, was developed. ."This technique combines the selection of variables and the estimation of the post-selection error rate. It provides a method to determine the optimal model dimension, to select the variables for the final model and to estimate the post-selection error rate of the discriminant rule. An extensive Monte Carlo simulation study comparing the CMV technique to existing procedures in the literature, was undertaken. In general, this technique outperformed the other methods, especially with respect to the accuracy of estimating the post-selection error rate. Finally, pre-test type variable selection was considered. A pre-test estimation procedure was adapted for use as selection technique in linear discriminant analysis. In a simulation study, this technique was compared to CMV, and was found to perform well, especially with respect to correct selection. However, this technique is only valid for uncorrelated normal variables, and its applicability is therefore limited. A numerically intensive approach was used throughout the study, since the problems that were investigated are not amenable to an analytical approach.Show more - ItemAutomated program generation : bridging the gap between model and implementation(Stellenbosch : Stellenbosch University, 2012-02) Bezuidenhout, Johannes Abraham; Geldenhuys, Jaco; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Institute for Applied Computer Science.
Show more ENGLISH ABSTRACT: The general goal of this thesis is the investigation of a technique that allows model checking to be directly integrated into the software development process, preserving the benefits of model checking while addressing some of its limitations. A technique was developed that allows a complete executable implementation to be generated from an enhanced model specification. This included the development of a program, the Generator, that completely automates the generation process. In addition, it is illustrated how structuring the specification as a transitions system formally separates the control flow from the details of manipulating data. This simplifies the verification process which is focused on checking control flow in detail. By combining this structuring approach with automated implementation generation we ensure that the verified system behaviour is preserved in the actual implementation. An additional benefit is that data manipulation, which is generally not suited to model checking, is restricted to separate, independent code fragments that can be verified using verification techniques for sequential programs. These data manipulation code segments can also be optimised for the implementation without affecting the verification of the control structure. This technique was used to develop a reactive system, an FTP server, and this experiment illustrated that efficient code can be automatically generated while preserving the benefits of model checking.Show more - ItemAutomatic acquisition of two-level morphological rules(Stellenbosch : Stellenbosch University, 1999-02) Theron, Pieter Zacharias; Cloete, Ian; Stellenbosch University. Faculty of Sciences. Dept. of Mathematical Sciences.
Show more ENGLISH SUMMARY: There are numerous applications for computational systems with a natural language processing capability. All these applications, which include free-text information retrieval, machine-translation and computer-assisted language learning, require a detailed and correctly structured database (or lexicon) of language information on all the levels of language analysis (phonology, morphology, syntax, semantics, etc.). To hand-code this information can be time-consuming and error prone. An alternative approach is to attempt the automation of the lexicon construction process. The contribution of this thesis is to present a method to automatically construct rule sets for the morphological and phonological levels of language analysis. The particular computational morphological framework used is that of two-level morphology. The lexicon, which contains the language specific information of two-level analyzers/ generators, consists of two components: (1) A morphotactic description of the words to be processed, as well as (2) a set of two-level phonological (or spelling) rules. The input to the acquisition process is source-target word pairs, where the target is an inflected form of the source word. It is assumed that the target word is formed from the source through the optional addition of a prefix and/or a suffix. There are two phases in the acquisition process: (1) segmentation of the target into morphemes and (2) determination of the optimal two-level rule set with minimal discerning contexts. In phase one, an acyclic deterministic finite state automaton (ADFSA) is constructed from string edit sequences of the input pairs. Segmentation of the words into morphemes is achieved through viewing the ADFSA as a directed acyclic graph (DAG) and applying heuristics using properties of the DAG as well as the elementary string edit operations. For phase two, the determination of the optimal rule set is made possible with a novel representation of rule contexts, with morpheme boundaries added, in a new DAG. We introduce the notion of a delimiter edge. Delimiter edges are used to select the correct two-level rule type as well as to extract minimal discerning rule contexts from the DAG. To illustrate the language independence of an acquired rule set, results are presented for English adjectives, Xhosa noun locatives, Afrikaans noun plurals and Spanish adjectives. Furthermore, it is shown how rules are acquired from thousands of input source target word pairs. Finally, the excellent generalization of an acquired rule set is shown by applying a slightly manually modified rule set to previously unseen words. The recognition accuracy on unseen words was 98.9% while the generation accuracy was 97.8%.Show more - ItemBinary closure operators(Stellenbosch : Stellenbosch University, 2016-03) Abdalla, Abdurahman Masoud; Janelidze, Zurab; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences (Mathematics)
Show more ENGLISH ABSTRACT : In this thesis we provide a new foundation to categorical closure operators, using more elementary binary closure operators on posets. The original goal of the thesis was to study a categorical closure operator in terms of the family of closure operators on the posets of subobjects. However, this does not allow to express hereditariness, which is an important property of a categorical closure operator. Representing instead a categorical closure operator in terms of the family of binary closure operators on the posets of subobjects, xes this problem. Moreover, the structure of a binary closure operator on a poset is self-dual, unlike that of a unary closure operator or that of a categorical closure operator, and this duality has a useful application in the study of properties of closure operators on categories, where it groups properties of categorical closure operators in dual pairs, and allows to unify results which relate these properties to each other.Show more - ItemBivariate wavelet construction based on solutions of algebraic polynomial identities(Stellenbosch : Stellenbosch University, 2012-03) Van der Bijl, Rinske; De Villiers, J. M.; Chui, C. K.; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
Show more ENGLISH ABSTRACT: Multi-resolution analysis (MRA) has become a very popular eld of mathematical study in the past two decades, being not only an area rich in applications but one that remains lled with open problems. Building on the foundation of re nability of functions, MRA seeks to lter through levels of ever-increasing detail components in data sets { a concept enticing to an age where development of digital equipment (to name but one example) needs to capture more and more information and then store this information in di erent levels of detail. Except for designing digital objects such as animation movies, one of the most recent popular research areas in which MRA is applied, is inpainting, where \lost" data (in example, a photograph) is repaired by using boundary values of the data set and \smudging" these values into the empty entries. Two main branches of application in MRA are subdivision and wavelet analysis. The former uses re nable functions to develop algorithms with which digital curves are created from a nite set of initial points as input, the resulting curves (or drawings) of which possess certain levels of smoothness (or, mathematically speaking, continuous derivatives). Wavelets on the other hand, yield lters with which certain levels of detail components (or noise) can be edited out of a data set. One of the greatest advantages when using wavelets, is that the detail data is never lost, and the user can re-insert it to the original data set by merely applying the wavelet algorithm in reverse. This opens up a wonderful application for wavelets, namely that an existent data set can be edited by inserting detail components into it that were never there, by also using such a wavelet algorithm. In the recent book by Chui and De Villiers (see [2]), algorithms for both subdivision and wavelet applications were developed without using Fourier analysis as foundation, as have been done by researchers in earlier years and which have left such algorithms unaccessible to end users such as computer programmers. The fundamental result of Chapter 9 on wavelets of [2] was that feasibility of wavelet decomposition is equivalent to the solvability of a certain set of identities consisting of Laurent polynomials, referred to as Bezout identities, and it was shown how such a system of identities can be solved in a systematic way. The work in [2] was done in the univariate case only, and it will be the purpose of this thesis to develop similar results in the bivariate case, where such a generalization is entirely non-trivial. After introducing MRA in Chapter 1, as well as discussing the re nability of functions and introducing box splines as prototype examples of functions that are re nable in the bivariate setting, our fundamental result will also be that wavelet decomposition is equivalent to solving a set of Bezout identities; this will be shown rigorously in Chapter 2. In Chapter 3, we give a set of Laurent polynomials of shortest possible length satisfying the system of Bezout identities in Chapter 2, for the particular case of the Courant hat function, which will have been introduced as a linear box spline in Chapter 1. In Chapter 4, we investigate an application of our result in Chapter 3 to bivariate interpolatory subdivision. With the view to establish a general class of wavelets corresponding to the Courant hat function, we proceed in the subsequent Chapters 5 { 8 to develop a general theory for solving the Bezout identities of Chapter 2 separately, before suggesting strategies for reconciling these solution classes in order to be a simultaneous solution of the system.Show more - ItemA categorical approach to lattice-like structures(Stellenbosch : Stellenbosch University, 2018-12) Hoefnagel, Michael Anton; Janelidze, Zurab; Gray, J.; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Division Mathematics.
Show more ENGLISH ABSTRACT : This thesis is a first step in a categorical approach to lattice-like structures. Its central notion, that of a majority category, relates to the category of lattices, in a similar way as Mal’tsev categories relate to the category of groups. This notion provides a context in which to establish categorical counterparts of various lattice-theoretic results. Surprisingly, many categories of a geometric nature naturally possess the dual property; namely, they are comajority categories. We show that several characterizations of varieties admitting a majority term, extend to characterizations of regular majority categories. These characterizations then show how majority categories relate to other well known notions in the literature, such as arithmetical and protoarithmetical categories. The most interesting results, from the point of view of the author, are those that concern decomposition and factorization. For example, every subobject of a finite product of objects in a regular majority category is uniquely determined by its two-fold projections – which can be seen as a certain subobject decomposition property. One of the main points of the thesis proves that in a regular majority category, every product of directly-irreducible objects is unique.Show more - ItemCentrality in random trees(Stellenbosch : Stellenbosch University, 2017-12) Durant, Kevin; Wagner, Stephan; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
Show more ENGLISH ABSTRACT : We consider two notions of centrality—namely, the betweenness centrality of a node and whether or not it is a centroid—in families of simply generated and increasing trees. Both of these concepts are defined in terms of paths within a tree: the betweenness centrality of a node v is the sum, over pairs of nodes, of the proportions of shortest paths that pass through v; and v is a centroid (there can be at most two) if it minimises the sum of the distances to the other nodes in the tree. We find that betweenness centrality in a large, random simply generated tree is generally linear in the size n of the tree, and that due to the tall, thin nature of simply generated trees, the probability of a random node having quadratic-order betweenness centrality decreases as n increases. This leads to a kth moment of order n2k−(1/2) for the betweenness centrality of a root node, even though a limiting distribution arises upon linearly rescaling the betweenness centrality. The class of labelled subcritical graphs, which are tree-like in structure, behave similarly. Betweenness centrality in a random increasing tree is also usually linear, except for nodes near to the root of the tree, which typically have centralities of order n2. The kth moment of the betweenness centrality of any node with a fixed label is thus of order n2k, but once again the distribution of the betweenness centrality of a random node converges to a limit when scaled by 1/n. To complement known results involving centroid nodes in simply generated trees, we also derive limiting distributions, along with limits of moments, for the depth, label, and subtree size of the centroid nearest to the root in an increasing tree. The first two of these distributions are concentrated around the root, while the latter is a combination of a point measure at 1 and a decreasing density on [1/2, 1). In addition, we show that the distributions of the maximum betweenness centrality in simply generated and increasing trees converge, upon suitable rescalings, to limiting distributions, and that the probability of the centroid attaining maximal betweenness centrality tends in both cases to a limiting constant.Show more - ItemCombinatorics of oriented trees and tree-like structures(Stellenbosch : Stellenbosch University, 2015-03) Okoth, Isaac Owino; Wagner, Stephan; Stellenbosch University. Faculty of Science. Department of Mathematical Sciences.
Show more ENGLISH ABSTRACT : In this thesis, a number of combinatorial objects are enumerated. Du and Yin as well as Shin and Zeng (by a different approach) proved an elegant formula for the number of labelled trees with respect to a given in degree sequence, where each edge is oriented from a vertex of lower label towards a vertex of higher label. We refine their result to also take the number of sources (vertices of in degree 0) or sinks (vertices of out degree 0) into account. We find formulas for the mean and variance of the number of sinks or sources in these trees. We also obtain a differential equation and a functional equation satisfied by the generating function for these trees. Analogous results for labelled trees with two marked vertices, related to functional digraphs, are also established. We extend the work to count reachable vertices, sinks and leaf sinks in these trees. Among other results, we obtain a counting formula for the number of labelled trees on n vertices in which exactly k vertices are reachable from a given vertex v and also the average number of vertices that are reachable from a specified vertex in labelled trees of order n. In this dissertation, we also enumerate certain families of set partitions and related tree-like structures. We provide a proof for a formula that counts connected cycle-free families of k set partitions of {1, . . . , n} satisfying a certain coherence condition and then establish a bijection between these families and the set of labelled free k-ary cacti with a given vertex-degree distribution. We then show that the formula also counts coloured Husimi graphs in which there are no blocks of the same colour that are incident to one another. We extend the work to count coloured oriented cacti and coloured cacti. Noncrossing trees and related tree-like structures are also considered in this thesis. Specifically, we establish formulas for locally oriented noncrossing trees with a given number of sources and sinks, and also with given indegree and outdegree sequences. The work is extended to obtain the average number of reachable vertices in these trees. We then generalise the concept of noncrossing trees to find formulas for the number of noncrossing Husimi graphs, cacti and oriented cacti. The study is further extended to find formulas for the number of bicoloured noncrossing Husimi graphs and the number of noncrossing connected cycle-free pairs of set partitions.Show more - ItemComparative analysis of predictive equations for transfer processes in different porous structures(Stellenbosch : Stellenbosch University, 2012-12) Woudberg, Sonia; Du Plessis, Jean Prieur; Smit, G. J. F.; Rewitzky, I. M.; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
Show more ENGLISH ABSTRACT: Research on transfer processes in various types of porous media has become important for the optimization of high technology engineering devices and processes. In this study the micro-structural parameters of different types of porous media, namely granular media, foamlike media and fibre beds, are characterized and quantified. Existing analytical modelling procedures for the three different types of porous media have been unified and refined to improve their predictive capabilities. Deterministic equations are proposed for predicting the streamwise pressure gradient, permeability and inertial coefficient of each type of porous medium. The equations are applicable over the entire porosity range and steady laminar flow regime and well suited as drag models in numerical computations. It is shown that the improved granular model can be regarded as qualitative and quantitative proof of the extensively used semi-empirical Ergun equation. The proposed model is used to provide physical meaning to the empirical coefficients. An Ergun-type equation is also proposed for foamlike media by remodelling the interstitial geometric configuration and accompanying flow conditions. The range of applicability of the existing foam model has been extended by incorporating the effect of developing flow in the pressure drop prediction. An equation is proposed in which the variation in the cross-sectional shape of the fibres can be incorporated into the interstitial form drag coefficient used in the foam model. This serves as an improvement on the constant value previously used. The existing foam model is also adapted to account for anisotropy resulting from compression. Two case studies are considered, namely compression of a non-woven glass fibre filter and compression of a soft polyester fibre material. The significant effect of compression on permeability is illustrated. In each case study the permeability values range over more than an order of magnitude for the narrow porosity ranges involved. The pressure drop prediction of the foam model is furthermore adapted to account for the combined effects of compression and developing flow. The newly proposed model diminishes the significant over-prediction of the existing foam model. An equation is furthermore proposed for predicting the permeability of Fontainebleau sandstones in which the effect of blocked throats is accounted for. Lastly, equations are proposed for predicting diffusivity ratios of unconsolidated arrays of squares and cubes. The prediction of the diffusivity ratio proposed in the present study, as opposed to model predictions from the literature, takes into account diffusion that may take place in stagnant fluid volumes. It is shown that a specific weighted average model proposed in the literature is not adequate to predict the diffusivity ratio of fully staggered arrays of squares, since it is shown not to be applicable to rectangular unit cells. Instead a new weighted average model is proposed which is applicable over the entire porosity range and for both staggered and non-staggered arrays of solid squares and cubes. The proposed weighted average model provides satisfactory agreement with experimental data from the literature and numerical data generated in the present study.Show more - 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.
Show more 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.Show more - ItemConcrete foundations of the theory of Noetherian forms(Stellenbosch : Stellenbosch University, 2019-12) Van Niekerk, Francois Koch; Janelidze, Zurab; Gray, James; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
Show more ENGLISH ABSTRACT: This thesis concerns certain investigations in abstract algebra that bring together the ideas of the category of algebraic structures and the lattice of substructures. A central notion in such investigation is that of a noetherian form. Originally, noetherian forms were introduced to provide a self-dual axiomatic context for establishing homomorphism theorems for (non-abelian) group-like structures. It is known that the form of “subobjects” over any variety is a noetherian form exactly when the variety is semi-abelian. An unexpected result in this thesis is that there is a noetherian form over any variety. In particular, this shows that the context of a noetherian form is much wider than originally thought. One of the aims of the thesis is to explore methods of constructing new noetherian forms out of existing forms; the mentioned result is obtained as an application of one of these constructions. Another aim is to show how the self-dual analogue of products in noetherian forms, called “biproducts” (first introduced in the author’s MSc thesis), are related to products. Finally, in this thesis we study the notion of an n-complemented lattice. This notion arose from studying subgroup lattices of finite abelian groups.Show more - ItemContributions to centralizers in matrix rings(Stellenbosch : University of Stellenbosch, 2010-12) Marais, Magdaleen Suzanne; Van Wyk, L.; University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences.
Show more ENGLISH ABSTRACT: THE concept of a k-matrix in the full 2 2 matrix ring M2(R=hki), where R is an arbitrary unique factorization domain (UFD) and k is an arbitrary nonzero nonunit in R, is introduced. We obtain a concrete description of the centralizer of a k-matrix bB in M2(R=hki) as the sum of two subrings S1 and S2 ofM2(R=hki), where S1 is the image (under the natural epimorphism fromM2(R) toM2(R=hki)) of the centralizer in M2(R) of a pre-image of bB, and where the entries in S2 are intersections of certain annihilators of elements arising from the entries of bB. Furthermore, necessary and sufficient conditions are given for when S1 S2, for when S2 S1 and for when S1 = S2. It turns out that if R is a principal ideal domain (PID), then every matrix in M2(R=hki) is a k-matrix for every k. However, this is not the case in general if R is a UFD. Moreover, for every factor ring R=hki with zero divisors and every n > 3 there is a matrix for which the mentioned concrete description is not valid. Finally we provide a formula for the number of elements of the centralizer of bB in case R is a UFD and R=hki is finite.Show more - ItemContributions to the analysis of approximate counting(Stellenbosch : Stellenbosch University, 2016-03) Prodinger, Helmut; Wagner, Stephan; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences (Mathematics)
Show more ENGLISH ABSTRACT : Approximate Counting is a classical technique with very challenging questions re- lated to its performance analysis. It is also somewhat similar to parameters around Digital Search trees. Surprising links to q-analysis and the theory of partitions exist. The author has contributed to the analysis during the last decades; the relevant pa- pers have been collected in this thesis. Some emphasis is on a recent development, namely, to introduce a parameter m (m counters instead of one).Show more