Doctoral Degrees (Applied Mathematics)


Recent Submissions

Now showing 1 - 5 of 6
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    Contributions to the theory of near-vector spaces, their geometry, and hyperstructures
    (Stellenbosch : Stellenbosch University, 2022-12) Rabie, Jacques; Howell, Karin-Therese; Stellenbosch University. Faculty of Science. Dept. of Applied Mathematics.
    ENGLISH ABSTRACT: This thesis expands on the theory and application of near-vector spaces — in particular, the underlying geometry of near-vector spaces is studied, and the theory of near-vector spaces is applied to hyperstructures. More specifically, a near-linear space is defined and some properties of these spaces are proved. It is shown that by adding some axioms, the nearaffine space, as defined by André, i s obtained. A correspondence is shown between subspaces of nearaffine spaces generated by near-vector spaces, and the cosets of subspaces of the corresponding near-vector space. As a highlight, some of the geometric results are used to prove an open problem in near-vector space theory, namely that a non-empty subset of a near-vector space that is closed under addition and scalar multiplication is a subspace of the near-vector space. The geometric work of this thesis is concluded with a first look into the projections of nearaffine s paces, a branch of the geometry that contains interesting avenues for future research. Next the theory of hyper near-vector spaces is developed. Hyper near-vector spaces are defined having similar properties to André’s near-vector space. Important concepts, including independence, the notion of a basis, regularity, and subhyperspaces are defined, and an analogue of the Decomposition Theorem, an important theorem in the study of near-vector spaces, is proved for these spaces.
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    Path planning for wheeled mobile robots using an optimal control approach
    (Stellenbosch : Stellenbosch University, 2019-12) Matebese, Belinda Thembisa; Banda, Mapundi K.; Withey, Daniel; Brink, Willie; Stellenbosch University. Faculty of Science. Department of Mathematical Sciences (Applied Mathematics).
    ENGLISH ABSTRACT: The capability and practical use of wheeled mobile robots in real-world applications have resulted in them being a topic of recent interest. These systems are most prevalent because of their simple design and ease to control. In many cases, they also have an ability to move around in an environment without any human intervention. A main stream of research for wheeled mobile robots is that of planning motions of the robot under nonholonomic constraints. A typical motion planning problem is to find a feasible path in the configuration space of the mobile robot that starts at the given initial state and reaches the desired goal state while satisfying robot kinematic or dynamic constraints. A variety of methods have been used to solve various aspects of the motion planning problem. Depending on the desired quality of the solution, an optimal path is often sought. In this dissertation, optimal control is employed to obtain optimal collision-free paths for two-wheeled mobile robots and manipulators mounted on wheeled mobile platforms from an initial state to a goal state while avoiding obstacles. Obstacle avoidance is mathematically modelled using the potential field technique. The optimal control problem is then solved using an indirect method approach. This approach employs Pontryagin’s minimum principle where analytical solutions for optimality conditions are derived. Solving the optimality condition leads to two sets of differential equations that have to be solved simultaneously and whose conditions are given at different times. This set of equations is known as a two-point boundary value problem (TPBVP) and can be solved using numerical techniques. An indirect method, namely Leapfrog, is then implemented to solve the TPBVP. The Leapfrog method begins with a feasible trajectory, which is divided into smaller subdivisions where the local optimal controls are solved. The locally optimal trajectories are added and following a certain scheme of updating the number of subdivisions, the algorithm ends with the generation of an optimal trajectory along with the corresponding cost. An advantage of using the Leapfrog method is that it does not depend on the provision of good initial guesses along a path. In addition, the solution provided by the method satisfies both boundary conditions at every step. Moreover, in each iteration the paths generated are feasible and their cost decreases asymptotically. To illustrate the effectiveness of the algorithm numerically, a quadratic cost with the control objective of steering the mobile robot from an initial state to a final state while avoiding obstacles is minimized. Simulations and numerical results are presented for environments with and without obstacles. A comparison is made between the Leapfrog method and the BVP4C optimization algorithm, and also the kinodynamic-RRT algorithm. The Leapfrog method shows value for continued development as a path planning method since it initializes easily, finds kinematically feasible paths without the need of post processing and where other techniques may fail. To our knowledge the work presented here is the first application of the Leapfrog method to find optimal trajectories for motion planning on a two-wheeled mobile robot and mobile manipulator.
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    Writer-independent handwritten signature verification
    (Stellenbosch : Stellenbosch University, 2015-12) Swanepoel, Jacques Philip; Coetzer, Johannes; Stellenbosch University. Faculty of Science. Department of Mathematical Sciences (Applied Mathematics).
    AFRIKAANSE OPSOMMING : In hierdie verhandeling stel ons 'n nuwe strategie vir outomatiese handtekening-verifikasie voor. Die voorgestelde raamwerk gebruik 'n skrywer-onafhanklike benadering tot handtekening- modellering en is dus in staat om bevraagtekende handtekeninge, wat aan enige skrywer behoort, te bekragtig, op voorwaarde dat minstens een outentieke voorbeeld vir vergelykingsdoeleindes beskikbaar is. Ons ondersoek die tradisionele statiese geval (waarin 'n bestaande pen-op-papier handtekening vanuit 'n versyferde dokument onttrek word), asook die toenemend gewilde dinamiese geval (waarin handtekeningdata outomaties tydens ondertekening m.b.v. gespesialiseerde elektroniese hardeware bekom word). Die statiese kenmerk-onttrekkingstegniek behels die berekening van verskeie diskrete Radontransform (DRT) projeksies, terwyl dinamiese handtekeninge deur verskeie ruimtelike en temporele funksie-kenmerke in die kenmerkruimte voorgestel word. Ten einde skryweronafhanklike handtekening-ontleding te bewerkstellig, word hierdie kenmerkstelle na 'n verskil-gebaseerde voorstelling d.m.v. 'n geskikte digotomie-transformasie omgeskakel. Die klassikasietegnieke, wat vir handtekeking-modellering en -verifikasie gebruik word, sluit kwadratiese diskriminant-analise (KDA) en steunvektormasjiene (SVMe) in. Die hoofbydraes van hierdie studie sluit twee nuwe tegnieke, wat op die bou van 'n robuuste skrywer-onafhanklike handtekeningmodel gerig is, in. Die eerste, 'n dinamiese tydsverbuiging digotomie-transformasie vir statiese handtekening-voorstelling, is in staat om vir redelike intra-klas variasie te kompenseer, deur die DRT-projeksies voor vergelyking nie-lineêr te belyn. Die tweede, 'n skrywer-spesieke verskil-normaliseringstrategie, is in staat om inter-klas skeibaarheid in die verskilruimte te verbeter deur slegs streng relevante statistieke tydens die normalisering van verskil-vektore te beskou. Die normaliseringstrategie is generies van aard in die sin dat dit ewe veel van toepassing op beide statiese en dinamiese handtekening-modelkonstruksie is. Die stelsels wat in hierdie studie ontwikkel is, is spesi ek op die opsporing van hoë-kwaliteit vervalsings gerig. Stelselvaardigheid-afskatting word met behulp van 'n omvattende eksperimentele protokol bewerkstellig. Verskeie groot handtekening-datastelle is oorweeg. In beide die statiese en dinamiese gevalle vaar die voorgestelde SVM-gebaseerde stelsel beter as die voorgestelde KDA-gebaseerde stelsel. Ons toon ook aan dat die stelsels wat in hierdie studie ontwikkel is, die meeste bestaande stelsels wat op dieselfde datastelle ge evalueer is, oortref. Dit is selfs meer belangrik om daarop te let dat, wanneer hierdie stelsels met bestaande tegnieke in die literatuur vergelyk word, ons aantoon dat die gebruik van die nuwe tegnieke, soos in hierdie studie voorgestel, konsekwent tot 'n statisties beduidende verbetering in stelselvaardigheid lei.
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    Video Surveillance Incorporating Pan-Tilt-Zoom Cameras
    (Stellenbosch : Stellenbosch University, 2015-03) Holtzhausen, Petrus Jacobus; Herbst, B. M.; Crnojevic, Vladimir; Brink, Willie; Stellenbosch University. Faculty of Science. Department of Applied Mathematics
    ENGLISH ABSTRACT : When trespassers target businesses and homes, outdoor spaces are typically the first point of illegal entry. Camera systems can help secure these environments, but typically many cameras are needed to cover large areas. In practice most camera systems are only used to review events after they have happened. It is however possible to do much more, and we explore the paradigm of active monitoring where cameras detect trespassers and give visual verification of the alarm. This detection needs to be resilient to weather effects and other environmental noise, while running in real-time on high resolution video sequences. Our goal is to replace multiple static cameras with a single Pan-Tilt-Zoom (PTZ) camera that can monitor expansive terrain. These cameras can survey, detect and zoom in on objects of interest. We develop and implement a robust, real-time algorithm based on an interaction framework between an illumination invariant and a color based background model. We also developed and implemented a novel technique where we use optical flow motion vectors to determine the size and shape of the spatial Gaussian kernels in non parametric models. Although computationally more expensive we demonstrate these more sophisticated models can be more robust. These ideas are adapted for PTZ cameras, exploiting their pan, tilt and zoom capabilities. We apply background modeling on panorama images that inform PTZ camera movement. By this we discuss the construction of a system that secures perimeters using zooming camera analytics.
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    Pore-scale modelling of transport phenomena in homogeneous porous media
    (Stellenbosch : Stellenbosch University, 1999-12) Diedericks, Gerhardus Petrus Jacobus; Du Plessis, J. P.; Stellenbosch University. Faculty of Science. Dept. of Mathematics.
    ENGLISH ABSTRACT: The main purpose of this study is to develop deterministic, process-based models of incompressible Newtonian flow and electrical c01iduction in homogeneous, anisotropic porous media. The foundation of the models is provided by the volume averaging theory which is used to obtain the macroscopic balance equations for momentum transport and electrical conduction. These volume averaged equations contain, amongst others, integral terms over the fluid-solid surface area where the integrands are related to the microscopic fluxes of the transport quantities. The closure modelling is conducted by employing a pore-scale model which requires explicit assumptions regarding the mean geometric properties of the porous medium microstructure and accounts for the configuration of the fluid-solid surface area. The pore-scale model also provides an estimate of the microscopic flow paths. The average geometry of different anisotropic materials, namely two types of foamlike materials, granular porous media and fibre beds, is captured in representative unit cells which form the core of the physical pore-scale model. This particular type of closure modelling further requires a direct transformation of microscopic fluxes to the macroscopic level. It is indicated, in context of the volume averaging theory, that microscopic fluxes may be estimated by the respective macroscopic channel average fluxes. The transformation of the microscopic flux to the channel average flux is accomplished through the flux related tortuosity tensor. New definitions for the tortuosity and lineality as second-order tensors are proposed for porous media in general. Novel names, semantically in line with the respective physical meanings, are proposed for these quantities. It is shown that the definitions produce results which conform with several other published results and are applicable to anisotropic media. Application of the modelling technique to Newtonian flow results in momentum transport equations valid for both the Darcy and Forchheimer flow regimes. The coefficients appearing in these equations are expressed in terms of fluid properties and measurable geometric features of the porous medium. The predictions of the anisotropic foamlike materials are validated against experimental pressure gradient measurements for flow through a high porosity, anisotropic knitted wire mesh rolled up to form a cylindrical plug. The predictions compare reasonably well with the experimental results. The modelling approach is also applied to electrical conduction in anisotropic porous media saturated with an electrically conductive fluid. A macroscopic form of Ohm's law is derived as well as deterministic expressions for the formation factor. The formation factor predictions for isotropic porous media are compared to several experimental measurements as well as to semi-empirical expressions. The predictions compare favourably to the measurements.