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# Department of Applied Mathematics

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- ItemA bio-economic application to the Cape Rock Lobster resource using a delay difference modelling approach(Operations Research Society of South Africa, 2004) Roos, E.
Show more In many species, like the Cape Rock Lobster (Jasus lalandii), the life cycles of males and females differ. This may motivate the use of two-sex models in a stock-assessment analysis. It is also true for this resource, that juveniles do not reach sexual maturity immediately. Therefore a delay-difference model is appropriate. In this study we follow a bio-economic approach and use a two-sex delay-difference model to determine a maximum economic yield strategy. Thus we determine an economic optimum steady state solution at which to harvest this resource subject to the biological constraints of the species.Show more - ItemCommunity OR and OR for development : a South African perspective(Operations Research Society of South Africa, 1999) Fourie, Philip
Show more An overview is given of Community Operations Research and of the connection between OR and development. The RDP is the main framework for development in South Africa, and its present state is described. Some suggestions are made as to ways in which ORSSA could support the RDP and development in South Africa.Show more - ItemDie fassinerende meganika van hoppende fietswiele(LitNet, 2016-05-26) Theron, Willem; Maritz, Milton
Show more Die gedrag van eksentries-belaaide wiele of hoepels het tydens die afgelope twee dekades aandag geniet, veral met betrekking tot sulke wiele se geneigdheid om te hop. Hier word ’n oorsig gegee oor die verskillende aspekte wat in die vorige artikels bespreek is, beginnende met J.E. Littlewood se voorbeeld van ’n massalose hoepel in 1953, die foto van ’n hoppende hoelahoepel in T. Pritchett se artikel in 1999, asook ander aspekte, soos die effek wat wrywing op die beweging het. Feitlik al die gepubliseerde numeriese resultate is gegee vir beweging van ’n hoepel op ’n horisontale vlak; in hierdie artikel word ’n wiel wat afrol op ’n skuins vlak beskou. Die parameters vir ’n werklike fietswiel wat belas is deur ’n stuk lood op die rand vas te smelt, soos getoon in die foto in figuur 1, word in die wiskundige modelle gebruik. Resultate word gegee vir drie modelle van die wiel, naamlik vir ’n starre wiel, ’n lineer elastiese ˆ wiel sonder demping, en ’n lineer elastiese wiel met viskeuse demping. Onderskeid word ˆ tussen sentrifugale hoppe en elastiese hoppe gemaak. Hier word vir die eerste keer gekyk na meervoudige hoppe wat veroorsaak word deur die impuls wat op die wiel uitgeoefen word wanneer dit die vlak tref na ’n hop. Daar word gevind dat die tweede en later hoppe se groottes buitengewoon sensitief is vir klein aanpassings in die waardes van die parameters. Video’s van die werklike beweging van die wiel wat teen ’n steil pad afrol, word ingesluit. Die hoofresultaat is rekenaar-gegenereerde animasies wat poog om hierdie bewegings so goed as moontlik na te boots.Show more - ItemGeometric versus kinetic modelling approach for characterizing porous metal foams(WIT Press, 2019) Mare, Esmari; Woudberg, Sonia
Show more Knowledge of the geometric and kinematic parameters of porous foams are of great importance since it is used in a wide variety of industrial multiphase flow applications that require optimal functionality, e.g. gas filters, heat exchangers and catalyst supports. The large external surface area and high porosity of metal foams provide good chemical resistance, enhanced heat and mass transfer properties and low pressure drops. Four generic geometric models will be considered to characterize the metal foam geometry, namely the cubic unit cell, tetrakaidecahedron, dodecahedron and rectangular representative unit cell (RUC) models, as well as three kinetic approaches from the literature in order to predict the specific surface area (SSA). Two sets of experimental data from the literature will then be compared to the SSA model predictions of the geometric approach and to the SSA values obtained from the kinetic approach. A comparative analysis reveals that the most geometrically complex tetrakaidecahedron model indeed provides the best correspondence with the experimental data for the SSAs, followed by the geometrically simplest RUC model. The latter model, in addition, provides accurate results for the kinetic approach. The advantage of the RUC model is that it is the only geometric model that provides both a geometric and kinetic approach, and, as a result of its relatively simple geometry it is geometrically adaptable towards anisotropy. The Klinkenberg effect will also be considered to determine the influence on the predictions of the SSAs dependency on the permeability coefficients for different fluid phases.Show more - ItemMinimum density hyperplanes(Journal of Machine Learning Research, 2016) Pavlidis, Nicos G.; Hofmeyr, David P.; Tasoulis, Sotiris K.
Show more Associating distinct groups of objects (clusters) with contiguous regions of high probability density (high-density clusters), is central to many statistical and machine learning approaches to the classification of unlabelled data. We propose a novel hyperplane classifier for clustering and semi-supervised classification which is motivated by this objective. The proposed minimum density hyperplane minimises the integral of the empirical probability density function along it, thereby avoiding intersection with high density clusters. We show that the minimum density and the maximum margin hyperplanes are asymptotically equivalent, thus linking this approach to maximum margin clustering and semi-supervised support vector classifiers. We propose a projection pursuit formulation of the associated optimisation problem which allows us to find minimum density hyperplanes efficiently in practice, and evaluate its performance on a range of benchmark data sets. The proposed approach is found to be very competitive with state of the art methods for clustering and semi-supervised classification.Show more - ItemA note on exact solutions of the logistic map(AIP Publishing, 2020-03) Maritz, Milton F
Show more The logistic map, whose iterations lead to period doubling and chaos as the control parameter, is increased and has three cases of the control parameter where exact solutions are known. In this paper, we show that general solutions also exist for other values of the control parameter. These solutions employ a special function, not expressible in terms of known analytical functions. A method of calculating this function numerically is proposed, and some graphs of this function are given, and its properties are discussed. The logistic map is often studied as a model of the period doubling route to chaos as its control parameter is increased. For three particular values of the control parameter, exact solutions are known. In particular, for one of these values, when the solution is chaotic, an exact solution in terms of the square of the sine function is known. In this study, a general solution for the logistic map for all values of the control parameter in a continuous range is proposed. This solution employs a special function, not expressible in terms of known analytical functions. A method of calculating this function numerically, as well as calculating its inverse numerically is proposed and some of its properties are discussed. This study, therefore, contributes to the field of nonlinear dynamics by introducing a novel tool for visualization and for investigation. The technique may be extendable to other nonlinear maps.Show more - ItemOffline signature verification using the discrete Radon transform and a hidden Markov model(Hindawi, 2004) Coetzer, J.; Herbst, B. M.; Du Preez, J. A.
Show more We developed a system that automatically authenticates offline handwritten signatures using the discrete Radon transform (DRT) and a hidden Markov model (HMM). Given the robustness of our algorithm and the fact that only global features are considered, satisfactory results are obtained. Using a database of 924 signatures from 22 writers, our system achieves an equal error rate (EER) of 18% when only high-quality forgeries (skilled forgeries) are considered and an EER of 4.5% in the case of only casual forgeries. These signatures were originally captured offline. Using another database of 4800 signatures from 51 writers, our system achieves an EER of 12.2% when only skilled forgeries are considered. These signatures were originally captured online and then digitally converted into static signature images. These results compare well with the results of other algorithms that consider only global features.Show more - ItemOn the number of increasing trees with label repetitions(Elsevier, 2019) Bodini, Olivier; Genitrini, Antoine; Gittenberger, Bernhard; Wagner, Stephan
Show more We study the asymptotic number of certain monotonically labeled increasing trees arising from a generalized evolution process. The main difference between the presented model and the classical model of binary increasing trees is that the same label can appear in distinct branches of the tree. In the course of the analysis we develop a method to extract asymptotic information on the coefficients of purely formal power series. The method is based on an approximate Borel transform (or, more generally, Mittag-Leffler transform) which enables us to quickly guess the exponential growth rate. With this guess the sequence is then rescaled and a singularity analysis of the generating function of the scaled counting sequence yields accurate asymptotics. The actual analysis is based on differential equations and a Tauberian argument. The counting problem for trees of size n exhibits interesting asymptotics involving powers of n with irrational exponents.Show more - ItemOn the(d)–chromatic number of a complete balanced multipartite graph(Operations Research Society of South Africa, 2007) Burger, A. P.; Nieuwoudt, I.; Van Vuuren, J. H.
Show more In this paper we solve (approximately) the problem of finding the minimum number of colours with which the vertices of a complete, balanced, multipartite graph G may be coloured such that the maximum degrees of all colour class induced subgraphs are at most some specified integer d 2 N. The minimum number of colours in such a colouring is referred to as the (d)–chromatic number of G. The problem of finding the (d)–chromatic number of a complete, balanced, multipartite graph has its roots in an open graph theoretic characterisation problem and has applications conforming to the generic scenario where users of a system are in conflict if they require access to some shared resource. These conflicts are represented by edges in a so–called resource access graph, where vertices represent the users. An efficient resource access schedule is an assignment of the users to a minimum number of groups (modelled by means of colour classes) where some threshold d of conflict may be tolerated in each group. If different colours are associated with different time periods in the schedule, then the minimum number of groupings in an optimal resource access schedule for the above set of users is given by the (d)–chromatic number of the resource access graph. A complete balanced multipartite resource access graph represents a situation of maximum conflict between members of different user groups of the system, but where no conflict occurs between members of the same user group (perhaps due to an allocation of diverse duties to the group members).Show more - ItemOperations research and development(Operations Research Society of South Africa, 2007) Fourie, P.
Show more The paper begins with a discussion of the concepts “operations research” and “development.” An overview is next given of the RDP (Reconstruction and Development Programme), which still embodies the vision of development in South Africa associated with the political transition in 1994. The efforts of ORSSA (the Operations Research Society of South Africa) to help with the implementation of the RDP are described. The international OR (Operations Research) community has been involved in various ways with the promotion of development, especially through IFORS (the International Federation of Operational Research Societies), and these are reviewed. Sustainable development, which deals specifically with a long-term view of development, is mentioned briefly. Lastly some suggestions are made as to how ORSSA and its members could help to promote development in South Africa. In the Appendix a summary is given of a paper by Gerhard Geldenhuys in which he analyzes the needs identified by the RDP as well as relevant OR methods.Show more - Itemscikit-image : image processing in Python(PeerJ, 2014-06-19) Van der Walt, Stefan; Schonberger, Johannes L.; Nunez-Iglesias, Juan; Boulogne, Francois; Warner, Joshua D.; Yager, Neil; Gouillart, Emmanuelle; Yu, Tony
Show more scikit-image is an image processing library that implements algorithms and utilities for use in research, education and industry applications. It is released under the liberal Modified BSD open source license, provides a well-documented API in the Python programming language, and is developed by an active, international team of collaborators. In this paper we highlight the advantages of open source to achieve the goals of the scikit-image library, and we showcase several real-world image processing applications that use scikit-image. More information can be found on the project homepage, http://scikit-image.org.Show more