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Program: BScHons Wiskunde en Wiskundeonderwys Toegepaste Analise

Code Faculty Duration Credits Download
02240184 Fakulteit Natuur- en Landbouwetenskappe Minimum duur van studie: 1 jaar Totale krediete: 137

Programinligting

Hierdie inligting is slegs in Engels beskikbaar.

The programme consists of seven honours modules (five modules of 15 credits each from the Department of Mathematics and Applied Mathematics and two modules of 16 credits each from the Department of Science, Mathematics and Technology Education) as well as the compulsory research project (30 credits). Elective modules should be selected according to the prerequisites of these modules.
Candidates are required to familiarise themselves with the General Regulations regarding the maximum period of registration and other requirements for honours degrees.

Toelatingsvereistes

A BSc in Mathematics, Applied Mathematics or equivalent degree with at least a 60% average in the final year Mathematics or Applied Mathematics subjects. The final year should include at least four of the following third-year level modules or equivalent: partial differential equations, dynamical systems (ordinary differential equations), real analysis, complex analysis, numerical analysis and continuum mechanics. In the selection procedure the candidate's complete undergraduate academic record will be considered.

Bevordering tot volgende studiejaar

The progress of all honours candidates is monitored biannually by the postgraduate coordinator/head of department. A candidate’s study may be terminated if the progress is unsatisfactory or if the candidate is unable to finish his/her studies during the prescribed period.

Slaag met lof

The BScHons degree is awarded with distinction to a candidate who obtains a weighted average of at least 75% in all the prescribed modules and a minimum of 65% in any one module.

Minimum krediete: 137

Fundamentele modules

Kernmodules

  • Module-inhoud:

    Onderrig- en leerperspektiewe in wiskunde. Hierdie eenheid fokus op huidige kwessies in wiskunde-onderrig, byvoorbeeld: Aard van wiskundige kennis in die opvoedkunde; leerteorieë in wiskundeonderwys; gebruik van tegnologie in wiskunde-onderwys; navorsing in die klaskamer; geslag; taal; kultuur (Etno-wiskunde).
    Wiskunde in konteks: vooruitsigte en uitdagings. Hierdie eenheid fokus op die rol van wiskunde in verskillende kontekste (beroep- en alledaagse situasies ingesluit): Aard van wiskunde – wiskunde as ‘n menslike aktiwiteit; rasionale vir die leer van wiskunde; die teorie van realistiese wiskunde-onderwys; inhouds- en konteksgedrewe aanslag in wiskunde; wiskundige geletterdheid; kennisoordrag: uitdagings – skoolwiskunde vs realiteit.

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  • Module-inhoud:

    Die aard van onderwyskundige navorsing: navorsingskonteks, wetenskap, navorsings-etiek, waarheid, rasionaliteit, subjektiwiteit en objektiwiteit. Kwantitatiewe en kwalitatiewe navorsingsbenaderings, navorsingsontwerpe en data-insamelingstegnieke. Verskeidenheid benaderings in kwalitatiewe navorsing, insluitend: gevallestudies, historiese navorsing, etnografiese en aksienavorsing. Basiese konsepte en beginsels van kwantitatiewe navorsing. Statistiese tegnieke in die onderwysnavorsingsproses. Opnamemetodologie en vraelysontwerp. Klassifikasie en grafiese voorstelling van data. Beskrywende metings. Statistiese inferensie. Dataverwerkingsprosedures. Parametriese versus nieparametriese toetse. Enkele toetsstatistiek (bv. F-toetse, en T–toetse).

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  • Module-inhoud:

    *Hierdie inligting is slegs in Engels beskikbaar.

    An introduction to the basic mathematical objects of linear functional analysis will be presented. These include metric spaces, Hilbert spaces and Banach spaces. Subspaces, linear operators and functionals will be discussed in detail. The fundamental theorems for normed spaces: The Hahn-Banach theorem, Banach-Steinhaus theorem, open mapping theorem and closed graph theorem. Hilbert space theory: Riesz' theorem, the basics of projections and orthonormal sets.

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  • Module-inhoud:

    *Hierdie inligting is slegs in Engels beskikbaar.

    An analysis as well as an implementation (including computer programs) of methods are covered. Numerical linear algebra: Direct and iterative methods for linear systems and matrix eigenvalue problems: Iterative methods for nonlinear systems of equations. Finite difference method for partial differential equations: Linear elliptic, parabolic, hyperbolic and eigenvalue problems. Introduction to nonlinear problems. Numerical stability, error estimates and convergence are dealt with.

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  • Module-inhoud:

    *Hierdie inligting is slegs in Engels beskikbaar.

    Classical optimisation:  Necessary and sufficient conditions for local minima.  Equality constraints and Lagrange multipliers.  Inequality constraints and the Kuhn-Tucker conditions.  Application of saddle point theorems to the solutions of the dual problem.  One-dimensional search techniques.  Gradient methods for unconstrained optimisation.  Quadratically terminating search algorithms.  The conjugate gradient method.  Fletcher-Reeves.  Second order variable metric methods:  DFP and BFCS.  Boundary following and penalty function methods for constrained problems.   Modern multiplier methods and sequential quadratic programming methods.  Practical design optimisation project.

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  • Module-inhoud:

    *Hierdie inligting is slegs in Engels beskikbaar.

    Field-theoretic and material models of mathematical physics. The Friedrichs-Sobolev spaces. Energy methods and Hilbert spaces, weak solutions – existence and uniqueness. Separation of variables, Laplace transform, eigenvalue problems and eigenfunction expansions. The regularity theorems for elliptic forms (without proofs) and their applications. Weak solutions for the heat/diffusion and related equations.

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Keusemodules

  • Module-inhoud:

    *Hierdie inligting is slegs in Engels beskikbaar.

    A selection of special topics will be presented that reflects the expertise of researchers in the Department. The presentation of a specific topic is contingent on student numbers. Consult the website of the Department of Mathematics and Applied Mathematics for more details.

    Sien meer

  • Module-inhoud:

    *Hierdie inligting is slegs in Engels beskikbaar.

    Measure and integration theory: The Caratheodory extension procedure for measures defined on a ring, measurable functions, integration with respect to a measure on a σ-ring, in particular the Lebesgue integral, convergence theorems and Fubini's theorem.
    Probability theory: Measure theoretic modelling, random variables, expectation values and independence, the Borel-Cantelli lemmas, the law of large numbers. L¹-theory, L²-theory and the geometry of Hilbert space, Fourier series and the Fourier transform as an operator on L², applications of Fourier analysis to random walks, the central limit theorem.

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  • Module-inhoud:

    *Hierdie inligting is slegs in Engels beskikbaar.

    Study of main principles of analysis in the context of their applications to modelling, differential equations and numerical computation. Specific principles to be considered are those related to mathematical biology, continuum mechanics and mathematical physics as presented in the modules WTW 772, WTW 787 and WTW 776, respectively.

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  • Module-inhoud:

    *Hierdie inligting is slegs in Engels beskikbaar.

    An analysis as well as an implementation (including computer programs) of methods is covered. Introduction to the theory of Sobolev spaces. Variational and weak formulation of elliptic, parabolic, hyperbolic and eigenvalue problems. Finite element approximation of problems in variational form, interpolation theory in Sobolev spaces, convergence and error estimates.

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  • Module-inhoud:

    *Hierdie inligting is slegs in Engels beskikbaar.

    Mathematical modelling of Random walk. Conditional expectation and Martingales. Brownian motion and other Lévy processes. Stochastic integration. Ito's Lemma. Stochastic differential equations. Application to finance.

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  • Module-inhoud:

    *Hierdie inligting is slegs in Engels beskikbaar.

    This module aims at using advanced undergraduate mathematics and rigorously applying mathematical methods to concrete problems in various areas of natural science and engineering.
    The module will be taught by several lecturers from UP, industry and public sector. The content of the module may vary from year to year and is determined by relevant focus areas within the Department. The list of areas from which topics to be covered will be selected, includes: Systems of differential equations; dynamical systems; discrete structures; Fourier analysis; methods of optimisation; numerical methods; mathematical models in biology, finance, physics, etc.

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  • Module-inhoud:

    *Hierdie inligting is slegs in Engels beskikbaar.

    Analysis of spatial versus material description of motion. Conservation laws. Derivation of stress tensors. Analysis of finite strain and rate of deformation tensors. Stress and strain invariants. Energy. Linear and nonlinear constitutive equations. Applications to boundary value problems in elasticity and fluid mechanics.

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Die inligting wat hier verskyn, is onderhewig aan verandering en kan na die publikasie van hierdie inligting gewysig word.. Die Algemene Regulasies (G Regulasies) is op alle fakulteite van die Universiteit van Pretoria van toepassing. Dit word vereis dat elke student volkome vertroud met hierdie regulasies sowel as met die inligting vervat in die Algemene Reëls sal wees. Onkunde betrefffende hierdie regulasies en reels sal nie as ‘n verskoning by oortreding daarvan aangebied kan word nie.

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