On a topic chosen by the student from a list of proposals. The project entails progress reports, a written report, an oral presentation and the preparation of a conference poster. Students meet with the coordinator from time to time to be taught generic skills and to discuss progress on the projects. Honneursstudente kies aan die begin van die jaar 'n onderwerp en werk selfstandig, onder leiding van sy/haar projekadviseur, aan die gekose probleem. Aan die einde van die jaar word 'n verslag ingehandig en 'n kort mondelinge voordrag gelewer. Die module behels ook dat studente generiese navorsingsvaardighede aanleer.

General numerical methods for solving flow equations; finite difference/volume methods; procedures for the simulation of diffusive and convective processes; boundary values; solving algorithms such as the SIMPLE range; introduction to CFX. Algemene numeriese metodes vir die oplos van vloeivergelykings; eindige-verskil/volume metodes; prosedures vir simulasie van diffusiewe en konvektiewe prosesse; randwaardes; algoritmes soos die SIMPLE-reeks; inleiding tot CFX.

Focus on numerical methods for matrix computations. Effective solution of square linear systems, least squares problems, the eigenvalue problem. Direct and iterative methods, special attention to sparse matrices and structured matrices. Numerical instability and ill-conditioning. Model problems from partial differential equations and image processing. Fokus op numeriese metodes vir matriksbewerkings. Effektiewe oplos van vierkantige lineêre stelsels, kleinste-kwadrate probleme, die eiewaarde probleem. Direkte en iteratiewe metodes, klem op yl matrikse en matrikse met struktuur. Numeriese onstabiliteit en sleg-geaardheid. Modelprobleme uit parsiële differensiaalvergelykings en beeldverwerking.

Broad introduction to graph theory. Problems such as enumeration of graphs; optimal paths in networks; optimal spanning trees; centres and medians; planarity; vertex and edge colouring; Eulerian graphs and Hamiltonicity; tournaments; domination and independence; and Ramsey theory. Breë inleiding tot grafiekteorie. Probleme soos enumerasie van grafieke; optimale paaie in netwerke; optimale spanbome; senters en mediane; planariteit; punt- en lynkleuring; Euler-grafieke en Hamilton-grafieke; toernooie; dominasie en onafhanklikheid; en Ramsey-teorie.

Differential and integral calculus of volume averages in two phase media and its use in the mathematical modelling of transport processes in porous media; the rectangular unit cell model. Differensiaal- en integraalrekening van volumegemiddeldes in twee-fasige media; die modellering van vloeiprossesse in poreuse media; die reghoekige eenheidselmodel.

Basic grey-scale transformations and image enhancement techniques in the spatial domain; Fourier analysis in two dimensions and image enhancement techniques in the Fourier domain; image restoration; morphological filters; image compression techniques; image segmentation, representation, description and recognition. Basiese grysskaal-transformasies en beeldverbeteringstegnieke in die fisiese ruimte; Fourier-analise in twee dimensies en beeldverbeteringstegnieke in die Fourier-ruimte; die herstel van beelde; morfologiese filters; tegnieke vir beeldkompaktering; beeldsegmentasie, -voorstelling, -beskrywing en herkenning.

Following up from AM244, this module goes deeper into the use of Ordinary Differential Equations and Difference Equations for modelling problems in the applied sciences. Most of such problems need to be described with nonlinear terms, making the dynamic behaviour of these models quite varied and sometimes counter-intuitive. For example, dynamics can converge to stationary points (equilibria), but also periodic or quasi-periodic (limit cycles and tori), and deterministically chaotic orbits (strange attractors). Different initial conditions of the system can lead to different types of such asymptotic behaviours. Additionally, the number and type of these attractors can change with model parameters through bifurcations. Most examples of applications will be presented in the field of biology (ecology, evolution, epidemiology), but also in environmental (exploitation of natural resources such as fish stocks) and social sciences (love dynamics)

Most modelling problems lead to mathematical equations that cannot be solved explicitly. The only recourse in this case is to solve numerically, or to use analytical methods to generate approximate solutions. The latter family of methods forms the focal point of this module. The approximate solution of transcendental equations and differential equations will be discussed, as well as the asymptotic evaluation of integrals that depend on a large or small parameter. Applications include nonlinear oscillators in mechanics, boundary-layer problems in fluids, and a derivation of Stirling's approximation to the factorial function. Numerical methods will be used as a check on the accuracy of the analytical methods. Die meeste modelleringsprobleme lei na wiskundige vergelykings wat nie eksplisiet opgelos kan word nie. Die enigste toevlug in so 'n geval is om numeries op te los, of om analitiese metodes te gebruik om benaderde oplossings te konstrueer. Die metodes in laasgenoemde klas vorm die fokuspunt van hierdie module. Benaderde oplossings tot transendentale vergelykings asook differensiaalvergelykings sal bespreek word, asook die asimptotiese afskatting van integrale met 'n groot of klein parameter. Toepasssings sluit in nie-lineêre ossilators in meganika, randlaagprobleme in vloeistowwe, en 'n herleiding van Stirling se benadering tot die fakulteit-funksie. Numeriese metodes sal gebruik word as bevestiging van die akkuraatheid van die analitiese metodes.

Introduction to Markov processes and their applications for modelling randomly evolving (stochastic) systems. Theory: Markov chains in discrete and continuous time, random walks, Brownian motion, stochastic differential equations, stochastic calculus, large deviations. Applications: Word statistics, population dynamics, particle systems, diffusions, noise-perturbed dynamical systems, stochastic control, finance. The course also has practicals on simulation methods (in MATLAB or Python).

The first part of the module covers the basics of image processing, feature detection and matching, projective geometry, perspective transformations, robust model estimation with RANSAC, the pinhole camera model, and stereo vision for depth estimation. The second part focuses on deep neural networks, and specifically, convolutional neural networks, which are typically used for image classification, object detection, and image segmentation. Relevant machine learning concepts, including training by stochastic gradient descent, are also introduced.
Die eerste deel van die module dek die basiese beginsels van beeldverwerking, kenmerkopsporing en -ooreenstemming, projektiewe meetkunde, perspektieftransformasies, robuuste modelberaming met RANSAC, die gaatjiekameramodel, en stereovisie vir diepteberaming. Die tweede deel fokus op diep neurale netwerke, en spesifiek konvolusionele neurale netwerke, wat tipies gebruik word vir beeldklassifikasie, objekopsporing en beeldsegmentering. Relevante masjienleer- konsepte, insluitend opleiding deur stogastiese gradiëntafdaling, word ook bekendgestel.

Development of a physical understanding of the mathematical concepts associated with general and Cartesian tensor analysis; introductory differential geometry; curvilinear coordinate systems; coordinate transformations; development of a sound foundation for advanced mathematical modelling in scientific and engineering research environments. Ontwikkeling van 'n fisiese begrip van die wiskundige konsepte van algemene en Cartesiese tensoranalise; inleidende differensiaalmeetkunde; kromlynige koodinaatstelsels; assetransformasies; vorming van 'n stewige grondslag vir gevorderde wiskundige modellering in wetenskaplike- en ingenieursomgewings.

The module covers: basic option concepts and markets (calls, puts, payoffs, arbitrage and put-call parity), probability and simple asset price models, the Black–Scholes framework (PDE, formulas and risk-neutral valuation), option sensitivities (Greeks), numerical methods for pricing (root-finding for implied volatility, Monte Carlo simulation, binomial trees and finite difference methods), selected exotic options and discrete-time American options, and volatility estimation and variance-reduction techniques. Die module dek: basiese opsiekonsepte en -markte (koopopsies, verkoopopsies, uitbetalings, arbitrage en verkoop-kooppariteit), waarskynlikheids- en eenvoudige bateprysmodelle, die Black-Scholes-raamwerk (PDE, formules en risiko-neutrale waardasie), opsiesensitiwiteite (Grieke), numeriese metodes vir prysbepaling (wortelvind vir geïmpliseerde wisselvalligheid, Monte Carlo-simulasie, binomiale bome en eindige verskilmetodes), geselekteerde eksotiese opsies en diskrete-tyd Amerikaanse opsies, en wisselvalligheidsberaming en variansieverminderingstegnieke.

Focus on numerical methods for matrix computations. Effective solution of square linear systems, least squares problems, the eigenvalue problem. Direct and iterative methods, special attention to sparse matrices and structured matrices. Numerical instability and ill-conditioning. Model problems from partial differential equations and image processing. Fokus op numeriese metodes vir matriksbewerkings. Effektiewe oplos van vierkantige lineêre stelsels, kleinste-kwadrate probleme, die eiewaarde probleem. Direkte en iteratiewe metodes, klem op yl matrikse en matrikse met struktuur. Numeriese onstabiliteit en sleg-geaardheid. Modelprobleme uit parsiële differensiaalvergelykings en beeldverwerking.