### Browsing by Subject "Mathematics"

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- ItemOpen AccessA finite-difference solution of solute transport through a membrane bioreactor(2015) Godongwana, B; Solomons, D; Sheldon, M SThe current paper presents a theoretical analysis of the transport of solutes through a fixed-film membrane bioreactor (MBR), immobilised with an active biocatalyst. The dimensionless convection-diffusion equation with variable coefficients was solved analytically and numerically for concentration profiles of the solutes through the MBR. The analytical solution makes use of regular perturbation and accounts for radial convective flow as well as axial diffusion of the substrate species. The Michaelis-Menten (or Monod) rate equation was assumed for the sink term, and the perturbation was extended up to second-order. In the analytical solution only the first-order limit of the Michaelis-Menten equation was considered; hence the linearized equation was solved. In the numerical solution, however, this restriction was lifted. The solution of the nonlinear, elliptic, partial differential equation was based on an implicit finite-difference method (FDM). An upwind scheme was employed for numerical stability. The resulting algebraic equations were solved simultaneously using the multivariate Newton-Raphson iteration method. The solution allows for the evaluation of the effect on the concentration profiles of (i) the radial and axial convective velocity, (ii) the convective mass transfer rates, (iii) the reaction rates, (iv) the fraction retentate, and (v) the aspect ratio.
- ItemOpen Access
- ItemOpen AccessAn algorithmic approach to continuous location(1995) Chiang, Y B; Becker, Ronald IWe survey the p-median problem and the p-centre problem. Then we investigate two new techniques for continuous optimal partitioning of a tree T with n - 1 edges, where a nonnegative rational valued weight is associated with each edge. The continuous Max-Min tree partition problem (the continuous Min-Max tree partition problem) is to cut the edges in p - 1 places, so as to maximize (respectively minimize) the weight of the lightest (respectively heaviest) resulting subtree. Thus the tree is partitioned into approximately equal components. For each optimization problem, an inefficient implementation of the algorithm is given, which runs in pseudo-polynomial time, using a previously developed algorithm and a construction. We then derive from it a much faster algorithm using a top-down greedy technique, which runs in polynomial time. The algorithms have a variety of applications among others to highway and pipeline maintenance.
- ItemOpen AccessAlgorithmic randomness on computable metric spaces and hyperspaces(2012) Birch, Thomas; Brattka, VascoIn this text we shall be focusing on generalizing Martin-Löf randomness to computable metric spaces with arbitrary measure (for examples of this type of generalization see Gács [14], Rojas and Hoyrup [15]. The aim of this generalization is to define algorithmic randomness on the hyperspace of non-empty compact subsets of a computable metric space, the study of which was first proposed by Barmpalias et al. [16] at the University of Florida in their work on the random closed subsets of the Cantor space. Much work has been done in the study of random sets with authors such as Diamondstone and Kjos-Hanssen [17] continuing the Florida approach, whilst others such as Axon [18] and Cenzer and Broadhead [19] have been studying the use of capacities to define hyperspace measures for use in randomness tests. Lastly in section 6.4 we shall be looking at the work done by Hertling and Weihrauch [13] on universal randomness tests in effective topological measure spaces and relate their results to randomness on computable metric measure spaces and in particular to the randomness of compact sets in the hyperspace of non-empty compact subsets of computable metric spaces.
- ItemOpen AccessAmalgamation in varieties of algebras(1995) Jacobs, David Frank; Brink, Chris HOne of the most successful approaches to research in universal algebra has been the study of varieties, initiated by Garett Birkhoff in the 1930's. Examples of varieties include many classes of algebras such as groups, semigroups, lattices and Boolean algebras. In 1927, O. Schreier showed that for any set of extensions of a given group, there is another extension of that group that in some sense contains all other extensions in the set. This property of groups, known as the amalgamation property was generalized to a universal-algebraic setting by R. Fralsse in 1954, and the important question as to which varieties satisfied the amalgamation property arose. While some of the answers to this question were positive (such as for the varieties of lattices, distributive lattices and Boolean algebras), many common varieties such as the variety of semigroups and all non-distributive modular lattice varieties were shown to fail to satisfy the amalgamation property. In the light of these negative results, attempts were made to "localize" this property from the variety to its individual members, the most successful being the notions of amalgamation base and amalgamation class, first introduced by George Gratzer and Henry Lakser in 1971. Investigations into the nature of the amalgamation classes of varieties that fail to satisfy the amalgamation property were carried out in the 1970's and 1980's by among others, Clifford Bergman and Henry Rose, the main focus being congruence distributive varieties, of which lattice varieties form the prime example. The topic of amalgamation has also been studied in fields as diverse as topology, logic and the theory of field extensions. In this dissertation, however, I will focus on the more algebraic results concerning amalgamation. My aim is to present a selection of these results, using as examples varieties of groups, semigroups, lattices and Heyting algebras, in a universal-algebraic framework that is (more or less) self-contained and uniform in its notation. Bibliography: pages 142-150.
- ItemOpen AccessAn analysis of the theory- and employment-demands on mathematics for electrical engineering programmes at technikons(1989) Swanepoel, Jonathan EA preliminary study indicated a degree of dissatisfaction with the present mathematics curriculum at technikons amongst academic staff members of technikons as well as members of the electrical-engineering industry. The hypothesis of this study is that the present mathematics curricula for electrical-engineering at technikons are not fully compatible with the demands emanating from the theoretical and the industrial-training (in-service or workplace) components of the training of electrical-engineering technicians. The intent of this study is, firstly, to propose a framework of thought supporting engineering-mathematics curriculum-change in the context of electrical-engineering programmes as offered at technikons. The actual formulation of the syllabus content is supported by a curriculum-change model which takes cognisance of both the theoretical-demands and the workplace- demands in accordance with the aims of co-operative education espoused by technikons in South Africa. Secondly, a literature study of relevant past research leads to the development of a research methodology sympathetic to the present philosophy of technikon education for engineering-technicians in the country. The research methodology involves, firstly, a questionnaire response from practising engineers and technicians. Secondly, it involves the gathering of suggestions from technikon academic staff and the analysis thereof by a work-group representative of all technikons, and led by the researcher. Thirdly, seventy-nine (79) reference-texts to the electrical-engineering programmes (study-levels 1 to 4), offered at the Peninsula Technikon, were analysed for its mathematical content. The research findings supports the hypothesis. The thesis culminates in set of recommendations with regard to the applicability and composition of mathematics syllabi for electrical-engineering programmes at technikons.
- ItemOpen AccessAn application of climate change ensemble averaging methods to fisheries management(2018) Vrancken, Candysse Amy Louise; Butterworth, Doug S; Kunzi, Hans-PeterFisheries management has a long history, having started gradually in the middle of the 19th century. As the awareness of the importance of managing fisheries increased, the need for fisheries scientists and for methodical research into the dynamics of fish populations arose. The methods and science used to manage fisheries have continued to grow and improve since then. Discussions related to climate change have also been ongoing for a long time, resulting in the field of climate change science which focuses on the various components of the natural environment. Both fisheries management and climate change science have similar main objectives, and overall methods for achieving those objectives. Those similarities are explored using the example of the South African hake fishery (South Africa’s most valuable fishery) where, as in other fisheries, problems can occur when an equally weighted average over different assessment models (also termed Operating Models, or OMs) in an ensemble is used. In order to address these problems, a number of methods used in climate change to address similar problems are applied in the case of the hake fishery. The aim is to determine the impact that the application of the model ensemble approaches used in climate change would have had on the recent results obtained from the use of an equally weighted model ensemble average in developing and selecting a management procedure for the South African hake resource. The particular intent of these methods is to reduce any “bias” arising from the use of many models that are rather similar when computing such averages. Chapter 1 contains a brief introduction to the work. Chapter 2 provides a review of a sample of model ensemble types that are used in fisheries management and/or climate change science. Chapter 3 contains a brief history of the management of the South African hake fishery, as well as a detailed description of the components that make up the management procedure used to manage the resource (OMP-2014). The remaining chapters of the dissertation present the data and methods (Chapter 4), results and discussion (Chapter 5) and the conclusions, along with an outline of possible future work (Chapter 6). It seems that weighting the OMs in the Reference Set (RS) ensemble for the South African hake fishery using the climate change model weighting methods, among others, and taking model (dis-)similarity and the goodness of fit of the models to the data into account, would not have had a major impact on the results obtained from using the equally weighted model RS adapted in the development and selection of the OMP-2014 for the resource. Since the time taken for a resource below MSYL to recover to MSYL is a key consideration in the very important Marine Stewardship Council certification process, the impact of different weighting schemes for the RS is of interest in this context. All except one (which is not a recommended method) of the unequally weighted approaches result in projections of the spawning biomass of the more depleted deep-water hake species (M. paradoxus) reaching MSYL at the same time, or a year later, than the equally weighted OMP-2014 projection. The differences arising from the different weighting approaches are therefore not substantial for the South African hake fishery. Hence, although these climate change weighting methods can be applied in this fisheries management context, the weights they produced did not add considerable value to the equally weighted average method used currently for South African hake. However, the object here was only to illustrate these approaches. It could well be that for other fisheries, the weighting scheme could have a greater effect on the eventual results and decisions. The weighting of individual models in an ensemble continues to be of increasing interest in many different fields, including fisheries management and climate change. A continued investigation into other weighting methods that may impact the selection of OMPs for the South African hake and other fisheries is certainly warranted.
- ItemOpen AccessAn analysis of gender related differences in performance and attitudes of participants in the 1997 UCT Mathematics competition(1998) Tucker, Diane Jean; Webb, J HIn this study, possible gender differences in and attitudes towards mathematics will be investigated. As a sample, the candidates taking part in the individual competition in the University of Cape Town Mathematics Competition will be used. This sample has been chosen since it appears that even though the gender related differences in performance that are reported are often very small, the differences are often more apparent at the upper end of the ability scale. Since the University of Cape Town Mathematics Competition attracts entries from candidates of wide ranging ability, a number of investigations can be done. The investigations that will be carried out. included statistical analyses of a number of different categories in mathematics (algebra, arithmetic, geometry and problem solving), various sub-categories and special categories; questions that have been repeated in more than one question paper will also be investigated for any patterns in performance (in terms of maturity in mathematics). Since learners engaging in mathematical activities (including participating in mathematics competitions) are affected by external and internal influences on their perception and attitudes towards mathematics, it was felt that an investigation into the relationship between performance in mathematics and attitudes towards mathematics was important. Gender related differences in attitudes towards mathematics will also be investigated. The results of this study will show that, where statistically significant differences in performance exist, these differences are in fact very small. The results of the attitudes questionnaire demonstrate that there is a statistically significant correlation between attitudes and performance in mathematics and that there exists small, yet statistically significant differences in attitudes towards mathematics.
- ItemOpen AccessAn approach to coincidence theory through universal covering spaces(1973) Harvey, Duncan Reginald Arthur; Schlagbauer, HThe close relationship between the theory of fixed points and the theory of coincidences of maps is well known. This presentation is aimed at recording one of the less well documented approaches to fixed point theory as extended to the more general situation of coincidences. The approach referred to is that by way of the Universal Covering Spaces. The existing theory of coincidences is geometrically well realised in this setting and after some consideration, the necessary extensions and generalizations of the techniques as utilized in fixed point theory lead to an appealing conceptual notion of "essentiality of coincidence classes". Many hints have been made in the literature (see [1] and "On the sharpness of the Δ₂ and Δ₁ Nielsen Numbers" by Robin Brooks, J.Reine Angew. Math. 259, (1973), 101-108.) that lifts of mappings and the theory of fibres and related topics lend themselves to coincidence theory. It is the intention of this presentation to follow some of the basic properties through this approach and to show, wherever it is thought desirable, the ties between this and two of the existing approaches - for example, in the definition of the Nielsen Number, which is fundamental to both fixed point theory and coincidence theory.
- ItemOpen AccessBaer sum systems and generalized extensions(1972) Sheridan, Michael John; Hardie, K A
- ItemOpen AccessBisigma frames(1999) Matutu, P; Gilmour, Christopher Robert Anderson; Brümmer, G C LWe introduce and investigate the concept of a bi σ-frame. The cozero part of a biframe, itself a bi σ-frame, is defined and used to construct the compact regular, and regular Lindelöf coreflections for biframes. Pseudocompactness for biframes is defined in a natural way and characterised in terms of the cozero part. Finally we obtain the σ-frame analogue of the result that characterises the stably continuous frames in terms of the compact regular biframes.
- ItemOpen AccessBoolean ultrapowers(2000) Fish, Washiela; Rose, HenryThe Boolean ultrapower construction is a generalisation of the ordinary ultrapower construction in that an arbitrary complete Boolean algebra replaces the customary powerset Boolean algebra. B. Koppelberg and S. Koppelberg [1976] show that the class of ordinary ultrapowers is properly contained in the class of Boolean ultrapowers thereby justifying the development of a theory for Boolean ultrapowers. This thesis is an exploration into the strategies whereby and the conditions under which aspects of the theory of ordinary ultrapowers can be extended to the theory of Boolean ultrapowers. Mansfield [1971] shows that a finitely iterated Boolean ultrapower is isomorphic to a single Boolean ultrapower under certain conditions. Using a different approach and under somewhat different conditions, Ouwehand and Rose [1998] show that the result also holds for K-bounded Boolean ultrapowers. Mansfield [1971] also proves a Boolean version of the Keisler-Shelah theorem. By redefining the notion of a K-good ultrafilter on a Boolean algebra, Benda [1974] obtains a complete generalisation of a theorem of Keisler which states that an ultrapower is K-saturated iff the ultrafilter is K-good. Potthoff [1974] defines the notion of a limit Boolean ultrapower and shows that, as is the case for ordinary ultrapowers, the complete extensions of a model are characterised by its limit Boolean ultrapowers. Upon the discovery by Frayne, Morel and Scott [1962] of an ultrapower of a simple group which is not simple, Burris and Jeffers [1978] investigate necessary and sufficient conditions for a Boolean ultrapower to be simple, or subdirectly irreducible, provided that the language is countable. Finally, Jipsen, Pinus and Rose [2000] extend the notion of the Rudin-Keisler ordering to ultrafilters on complete Boolean algebras, and prove that by using this definition, Blass' Characterisation Theorem can be generalised for Boolean ultrapowers.
- ItemOpen AccessA categorial study of initiality in uniform topology(1971) Brümmer, Guillaume C L; Hardie, K AThis thesis consists of two chapters, of which the first presents a categorial study of the concept of initiality (also known as projective generation) and the second gives applications in the theory of uniform and quasi-uniform spaces.
- ItemOpen AccessChallenges in the hunt for dark energy dynamics(2008) Hlozek, Renée; Bassett, BruceOne of the greatest challenges in modern cosmology is determining the origin of the observed acceleration of the Universe. The 'dark energy' believed to supply the negative pressure responsible for this cosmic acceleration remains elusive despite over a decade of investigation. Hunting for deviation from the 'vanilla' cosmological model, ACDM, and detecting dynamics with redshift in the equation of state remains a key research area, with many challenges. We introduce some of the challenges in the search for such dark energy dynamics. We illustrate that under the assumption of well-motivated scaling models for dark energy dynamics early universe constraints on the dark energy density imply that these models will be essentially indistinguishable from ACDM for the next decade. After introducing the Fisher Matrix formalism, we derive the Fisher Flex test as a measure of whether the assumption of Gaussianity in the likelihood is incorrect for parameter estimation. This formalism is general for any cosmological survey. Lastly, we study the degeneracies between dark energy and curvature and matter in a non-parametric approach, and show that incorrectly assuming values of cosmological components can exactly mimic dark energy dynamics. We connect to the parametric approach by showing how these uncertainties also degrade constraints on the dark energy parameters in an assumed functional form for w. Improving the accuracy of surveys and experiments to search for possible signatures of dark energy dynamics is the focus of much attention in contemporary cosmology; we highlight challenges in the hunt for dark energy dynamics.
- ItemOpen AccessChaotic behavior and energy polarisation in flatband lattice models(2023) Cheong, Su Ho; Skokos, CharalamposFlatbands (FBs) in lattice systems correspond to dispersionless energy bands cre- ated through what is called destructive interference, a phenomenon caused by the existence of lattice symmetries, resulting in compactly localised wave functions to just a few lattice sites. The existence of FBs in materials entails high sensitivity to the initial conditions of the system, especially with regard to disorder and nonlinear- ity. In this study, we numerically investigate the wave packet dynamics and chaotic behaviour of a simple tight-binding system exhibiting FBs, the so-called stub lattice model. Initially we show the existence of a FB and of two dispersive frequency bands for this model and identify three different dynamical regimes, namely the weak chaos, strong chaos, and self trapping regimes. Using symplectic integration techniques, we evolve in time, t, initially localised wave packets for these regimes and quantify their spreading through the computation of the wave packets second moment, m2, while the extent of localisation is characterised using the wave packets participation number. We show that, for both the weak and strong chaos regimes, the spreading of wave packets, which is characterised by a power law increases of m2 (respectively ∝ t0.33 and ∝ t0.5), is a chaotic process whose maximum Lyapunov exponent respectively decreases ∝ t−0.25 and ∝ t−0.3. By decreasing the system's disorder strength we alter the width of the bandgap, something which does not ap- pear to affect the spreading dynamics in the weak chaos regime, while our results do not lead to clear conclusions in the case of strong chaos. Furthermore, we find that particular disorder configurations which preserve the FB do not alter the wave packet spreading dynamics for the weak chaos regime. Finally, we observe that the wave packets norm distributions at the subsites of the stub lattice unit cells reach equilibrium if the evolution time is sufficiently long
- ItemOpen AccessComplex algebras, varieties and games(2003) Schalekamp, Hendrick; Jipsen, PComplex algebras have proven very useful in presenting the modern day logician with a tool to approach a wide variety of problems in the field of algebraic logic. This dissertation is intended as an exploration of various approaches to the study of complex algebras. In particular we will take a look at the logical and semantic views of complex algebras, as well as logical games involving these algebras.
- ItemOpen AccessCongruence frames of frames and k-frames(2015) Manuell, Graham Richard; Frith, John; Gilmour, Christopher Robert AndersonWe describe the congruence lattices of frames and k-frames. We look at the role that congruence biframes play in the category of strictly zero-dimensional biframes and discuss some reflections and coreflections of congruence frames.
- ItemOpen AccessCongruences on lattices (with application to amalgamation)(1996) Laing, Lyneve; Rose, HenryWe present some aspects of congruences on lattices. An overview of general results on congruence distributive algebras is given in Chapter 1 and in Chapter 2 we examine weak projections; including Dilworth's characterization of congruences on lattices and a finite basis theorem for lattices. The outstanding problem of whether congruence lattices of lattices characterize distributive algebraic lattices is discussed in Chapter 3 and we look at some of the partial results known to date. The last chapter (Chapter 6) characterizes the amalgamation class of a variety B generated by a B-lattice, B, as the intersection of sub direct products of B, 2-congruence extendible members of B and 2-chain limited members of B. To this end we consider 2-congruence extendibility in Chapter 4 and n-chain limited lattices in Chapter 5. Included in Chapter 4 is the result that in certain lattice varieties the amalgamation class is contained in the class of 2-congruence extendible members of the variety. A final theorem in Chapter 6 states that the amalgamation class of a B-lattice variety is a Horn class.
- ItemOpen AccessCountable inductive limits(1972) Martens, Eric; Webb, John HInductive systems and inductive limits have by now become fairly well established in the general theory of topological vector spaces. It is a branch of Functional Analysis which is receiving a reasonable amount of attention by modern mathematicians. It is of course a very interesting subject of its own accord, but is also useful in solving problems and proving theorems which one does not suspect are intimately related to it. As an example we can consider the proof of the non-existence of a countably infinite dimensional metrisable barrelled space.
- ItemOpen AccessThe Delta-Nielsen number in products(1973) Mordant, Ian; Schlagbauer, HIn 1967 Robert F. Brown derived a formula which relates the Nielsen number N(f) of a fibre map f to the Nielsen numbers N(f),(fb), where f,fb are induced by f. This work is concerned to prove an analogous result for the Δ-Nielsen number, N(f,g,Δ). In Chapter I we introduce the set of coincidences of two maps f,g: X->Γ,f(f,g) = {xϵX: f(x)=g(x)}. We partition this set into equivalence classes by means of the equivalence relation of fixed end-point homotopy and then study some of the geometry of the equivalence classes. We then proceed to introduce the Δ-Nielsen number N(f,g,Δ) by means of an index, which we show satisfies the axioms of Brooks [1969] for a coincidence index. Thereafter we show N(f,g,Δ) to be a homotopy invariant. In Chapter II we introduce the class of fibre spaces. By restricting ourselves to fibre spaces which are products of closed, finitely triangulable manifolds, we derive an analogous formula for coincidences as Brown has for fixed points. Some suggestions for a complete analogue conclude the work.