registration, lectures and tea breaks
TRANSCRIPT
Registration, lectures and tea breaks
Registration is on Monday 2 April 2007 from 08:00to 08:50 in the foyer of the general engineering build-ing (on the corner of Banghoek and Joubert roads,marked A on the map below). All tea breaks willalso be held in the foyer of the general engineeringbuilding.
Lunches
Lunch will be at Die Bloukamer, in the Neelsie student center (marked B on the map) and are included for allconference participants.
Reception
Drinks and light snacks will be served between 08h00 and 20h00 at Katjiepiering Restaurant in the Stellenboschbotanical gardens (marked C on the map).
Conference Dinner
The conference dinner is on Tuesday, 03 April 2007 atRestaurant Barrique at Vredenheim Estate. We willleave from the parking in front of the general engi-neering building at 18:00 to arrive at the restaurantat 18:30. Those who need a lift and those who cano�er a lift must please meet in front of the general en-gineering building. Note that the conference dinneris not included in the student registration fee.
Details of email and internet facilities will be announced.
Monday 2 April 1
Monday 2 April
OPENING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
First-Order System LL* (FOSLL*) for Maxwell's equations in 3D with edge singularities by Tom
Manteu�el . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
MORNING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Explicit two-step peer methods by Weiner, Rüdiger . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Numerical Analysis of Variational Inequalities by Daya Reddy . . . . . . . . . . . . . . . . . . . . . . 2
PARALLEL I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Comparison of linear beam theories by Anneke Labuschagne . . . . . . . . . . . . . . . . . . . . . . . 3
The Usefulness of the Euler-Bernoulli Theory for the Vibrations of a Beam by Inger Plaskitt . . . . . 3
Vibrations of a beam between obstacles: convergence and simulation of a fully discretized approxi-mation by Y. Dumont . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
PARALLEL II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Application of singular perturbation methods to the solution of polynomial equations by D P Mason 4
High frequency induced instability in some non linear numerical schemes by S.W. Schoombie . . . . 4
Criticism of Asymptotic Global Error Expansion and a New Theory of Extrapolation Methods byGennady Kulikov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
PARALLEL III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Elementary stable and dissipative numerical schemes for dynamical systems by Jean M-S Lubuma . . 4
Apllication of the method af additional conditions to the Neuman problem for the Laplace equationin the angle domain by Neugebauer Igor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
PARALLEL IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Maximum Likelihood Estimation Using Square-Root Covariance Filter by Maria V. Kulikova . . . . 5
The E�ciency of an Adaptive Algorithm for the Simulation of Di�usions by Jack Urombo . . . . . . 5
2 Monday 2 April
FIRST-ORDER SYSTEM LL* (FOSLL*)FOR MAXWELL'S EQUATIONS IN 3DWITH EDGE SINGULARITIES
Tom Manteu�el (University of Colorado at Boul-der, USA)
The L2-norm version of �rst-order system least squares(FOSLS) attempts to reformulate a given system ofpartial di�erential equations so that applying a least-squares principle yields a functional whose bilinear partis H1-elliptic. This means that the minimization processamounts to solving a loosely coupled system of ellipticscalar equations. An unfortunate limitation of the L2-norm FOSLS approach is that this product H1 equiva-lence generally requires su�cient smoothness of the orig-inal problem. Inverse-norm FOSLS overcomes this lim-itation, but at a substantial loss of real e�ciency. TheFOSLL* approach described here is a promising alterna-tive that is based on recasting the original problem as aminimization principle involving the adjoint equations.This talk provides a theoretical foundation for theFOSLL* methodology and application to the eddy currentform of Maxwell's equations. It is shown that singular-ities due to discontinuous coe�cients are easily treated.However, singularities due to reentrant edges require afurther modi�cation. A partially weighted norm is usedonly on the slack equations. The solution retains optimalorder accuracy and the resulting linear systems are easilysolved by multigrid methods.Comparison is made to the curlcurl formulation and theweighted regularization approach. The FOSLL* is shownto have equal or better accuracy, obtained at a smallercost. Numerical examples are presented that support thetheory.
EXPLICIT TWO-STEP PEER METH-ODS
Weiner, Rüdiger (University of Halle)
Podhaisky, Helmut (University of Halle)
Schmitt, Bernhard A. (University of Marburg)
Over the last years two-step peer methods for sti� andnonsti� di�erential equations have been intensively stud-ied, both for sequential and parallel implementation.Linearly-implicit and implicit two-step peer methods com-bine good stability properties with a high stage order,making them attractive for very sti� problems. We givean overview about construction, stability and order re-sults of these methods. In more detail we discuss s-stage explicit two-step peer methods for nonsti� di�er-ential equations. We give order conditions and presenta construction principle for methods of order p=s. By aspecial choice of the parameters we obtain optimal zerostability. An additional condition allows to obtain su-perconvergence of order p=s+1 for variable stepsizes. Wepresent numerical results for explicit two-step methods onsome well-known test problems and compare our methodswith the state of the art codes DOPRI5 and DOP853.The results show the e�ciency of the two-step methods.Furthermore we consider a special subclass of these meth-ods, where all s stages can be computed in parallel withs processors.
NUMERICAL ANALYSIS OF VARIA-TIONAL INEQUALITIES
Daya Reddy (University of Cape Town, SouthAfrica)
Variational inequalities arise in a great many applicationsin the physical sciences and engineering, often in the con-text of weak or variational formulations of problems posedon convex sets rather than on subspaces of functions, andof problems involving di�erential inclusions rather thanequations. The aim of this talk is, �rst, to present anoverview of some typical elliptic variational inequalitiesarising in physics and engineering. The second and mainobjective of the talk is to explore the properties of �niteelement approximations of particular classes of these vari-ational inequalities, and to construct, analyse and illus-trate recently developed algorithms that are well suitedto the solution of such problems.
Parallel II 3
COMPARISON OF LINEAR BEAMTHEORIES
Anneke Labuschagne (University of Pretoria)[email protected]
Three models for a cantilever beam based on di�erentlinear theories are considered; the Euler-Bernoulli, Timo-shenko and two-dimensional elasticity theory. Using nat-ural frequencies and modes of vibration as a yardstick,the conclusion is that the Timoshenko theory is close tothe two-dimensional theory for modes of practical impor-tance, but that the applicability of the Euler-Bernoullitheory is seriously limited.
THE USEFULNESS OF THE EULER-BERNOULLI THEORY FOR THE VI-BRATIONS OF A BEAM
Inger Plaskitt (University of Pretoria)[email protected]
The well known Euler-Bernoulli theory for the vibration ofa beam was improved by Timoshenko in 1921. Since thennumerous studies have yielded convincing results that theTimoshenko theory is indeed a great improvement. Nev-ertheless, the Euler-Bernoulli theory is still widely usedand the question is when this can be justi�ed. Since bothmathematical models are linear, comparison of naturalfrequencies is usually considered to be su�cient. How-ever, what is really required is to determine the di�erencebetween solutions resulting from the same disturbance.We discuss the complications involved in such an investi-gation and present some results.
VIBRATIONS OF A BEAM BETWEENOBSTACLES: CONVERGENCE ANDSIMULATION OF A FULLY DIS-CRETIZED APPROXIMATION
Y. Dumont (University of Reunion Island(France))
L. Paoli (University of Saint Etienne (France))
Yves Dumont (University of Saint Etienne(France))
We study the motion of an elastic beam which is clampedat its left end to a vibrating support and which can movefreely at its right end between two rigid obstacles. Thecontact is modelled with Signorini's complementary con-ditions for the displacement and the shear stress. For thisin�nite dimensional impact problem we propose a familyof fully discretized approximations dealing directly withthe unilateral boundary conditions, which convergence isproved. The results given here are also valid in the case ofa beam oscillating between two longitudinal rigid obsta-cles. Some numerical results are presented and comparedto the approximate solutions obtained with a penalty ap-proach.
4 Monday 2 April
APPLICATION OF SINGULAR PER-TURBATION METHODS TO THE SO-LUTION OF POLYNOMIAL EQUA-TIONS
D P Mason (School of Computational and Ap-plied Mathematics, University of the Witwater-srand, Johannesburg, Sout)[email protected]
Two applications of singular perturbation methods to thesolution of polynomial equations will be described. The�rst application is to the in�ation of a thin-walled spher-ical shell in �nite elasticity. As the extension increasesand if a condition is satis�ed, the in�ation pressure �rstincreases, reaches a maximum value and then decreasesto a minimum value before again increasing steadily. In-�ation with a decrease in pressure is referred to as snapbuckling. The stretch at the turning points of the pres-sure satis�es a fourth degree polynomial equation with asmall parameter multiplying the term of highest degree.The perturbation solution of this fourth degree polyno-mial equation is described. The second application is toslow viscous �ow past a liquid drop. The Reynolds num-ber for which standing eddies �rst appear downstream ofthe liquid drop satis�es a cubic equation. Although thesmall parameter does not multiply the highest degree termit is a singular perturbation problem. The perturbationsolution of the cubic equation for the Reynolds number isdescribed.
HIGH FREQUENCY INDUCED INSTA-BILITY IN SOME NON LINEAR NU-MERICAL SCHEMES
S.W. Schoombie (University of the Free State)
E Mare (University of Pretoria)
When applying Nystrom methods to the van der Pol equa-tion, a high frequency mode sooner or later causes severeinstabilies. The onset of these can be delayed by decreas-ing the time step. Thus, when the purpose of the compu-tation is to �nd the limit cycle, there is an e�ective upperlimit to the time step, which gets lower with higher or-der schemes. This phenomenon is not limited to the vander Pol equation, but can also be observed in a predatorprey system with a limit cycle. Equations like the Du�ngequation seems to be free of this instability for the secondorder scheme.
CRITICISM OF ASYMPTOTICGLOBAL ERROR EXPANSION AND ANEW THEORY OF EXTRAPOLATIONMETHODS
Gennady Kulikov (University of the Witwater-srand, Johannesburg)[email protected]
In this paper we discuss existence of the asymptotic globalerror expansion for numerical solutions obtained fromgeneral one-step methods applied to ordinary di�erentialequations. The asymptotic global error expansion was dis-covered independently by Henrici, Gragg and Stetter in1962, 1964 and 1965, respectively. It is an important the-oretical background for extrapolation methods. We drawattention to some �aws in that theory and show that suchan expansion is likely to fail to work in practice. Thereforewe give another substantiation for extrapolation methods.The Richardson extrapolation technique is a key means toexplain how extrapolation methods perform. Addition-ally, we prove that the Aitken-Neville algorithm worksfor any one-step method of an arbitrary order s undersuitable smoothness.
ELEMENTARY STABLE AND DISSI-PATIVE NUMERICAL SCHEMES FORDYNAMICAL SYSTEMS
Jean M-S Lubuma (University of Pretoria)[email protected]
We analyze non-standard �nite di�erence schemes thathave no spurious �xed-points compared to the dy-namical system under consideration, the linear stabil-ity/instability property of the �xed-points being the samefor both the schemes and the continuous system. For morecomplex systems, which are dissipative, we design schemesthat replicate this property.
Parallel IV 5
APLLICATION OF THE METHODAF ADDITIONAL CONDITIONS TOTHE NEUMAN PROBLEM FOR THELAPLACE EQUATION IN THE ANGLEDOMAIN
Neugebauer Igor (Department of Applied Math-ematics, National University of Rwanda)[email protected]
The method of additional conditions can be used inthe classical boundary value problems of mathematicalphysics which can give some non-physical solutions withsingularities. There were considered the classical Gri�thsproblem for linear elastic plane under tension. The pre-sented method gave the �nite stresses near the tip of acrack which value corresponds to the information aboutexperiments of Gri�ths and Inglis. There was consid-ered the classical Dirichlet problem for Laplace equationin the angle domain. The obtained �rst approximate so-lution gave the nonsingular solution in the domain withobtuse angles where usually exist the singular solution ofthe standard Dirichlet problem. The obtained solutiondepends continuously on the angle of the domain. Herein this talk will be considered the Neuman problem forthe Laplace equation in the angle domain. The obtained�rst approximation of the solution gives the nonsingularsolution in the domain with the obtuse angle. This solu-tion depends continuously on the angle of the considereddomain.
MAXIMUM LIKELIHOOD ESTIMA-TION USING SQUARE-ROOT COVARI-ANCE FILTER
Maria V. Kulikova (University of the Witwater-srand, Johannesburg)[email protected]
In this paper we consider a discrete-time linear stochas-tic system and discuss the problem of Maximum Likeli-hood Estimation (MLE) of unknown system's parameters.MLE requires maximization of the Likelihood Functionthat is often done by using a gradient approach startingfrom some a priori values. Thus, to apply a gradient-search optimization technique, computation of the Log-Likelihood Gradient (LLG) is necessary. This problemleads to implementation of a Kalman �lter (and its deriva-tive with respect to each parameter), which is known tobe unstable. As a consequence, any method of evalua-tion of LLG based upon the conventional Kalman �lterimplementation inherits its pitfalls and, hence, cannotbe considered as numerically stable. Here, we discussthe problem of e�cient evaluation of LLG and presentsome new square-root algorithms based upon the covari-ance forms of Square-Root Filters. By constructing ourmethods, we add a new feature to the underlying �ltersthat allows the Log-Likelihood Function and its gradient(with respect to unknown parameters) to be computedsimultaneously. Thus, these new algorithms are ideal forsimultaneous state estimation and parameter identi�ca-tion. With the theoretical results we give examples ofill-conditioned problems and show that our methods out-perform the conventional approach for accuracy.
THE EFFICIENCY OF AN ADAPTIVEALGORITHM FOR THE SIMULATIONOF DIFFUSIONS
Jack Urombo (Department of Mathematical Sci-ences, Faculty of Engineering Sciences, HarareInstitue of Technology)[email protected]
We develop an adaptive method for the simulation ofsolutions of stochastic di�erential equations. The algo-rithm controls the approximation error and minimises theamount of computational work.We test the algorithm on both implicit and non implicitschemes for a certain class of stocahstic di�erential equa-tions.
6 Tuesday 3 April
Tuesday 3 April
POROUS MEDIA I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Reconstruction of porous media and transport on the pore scale by Pierre Adler . . . . . . . . . . . 7
Transient �ow through porous barriers. by Beric Skews . . . . . . . . . . . . . . . . . . . . . . . . . . 7
APPROXIMATION THEORY I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Scattered Data Approximation by Radial Basis Functions by Armin Iske . . . . . . . . . . . . . . . . 8
The Highlight Theorem in the Digital Pulse Transform by Dirk Laurie . . . . . . . . . . . . . . . . . 8
Approximation Problems around the Discrete Pulse Transform by Carl Rohwer . . . . . . . . . . . . 8
POROUS MEDIA II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
On the de�nition of hydrodynamical entrance lengths in straight ducts by S Woudberg . . . . . . . . 8
Permeability prediction of Fontainebleau sandstone by S Woudberg . . . . . . . . . . . . . . . . . . . 9
APPROXIMATION THEORY II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
LULU operators for functions of continuous argument by Carl Rohwer . . . . . . . . . . . . . . . . . 9
Wavelets based on local projection operators I by Johan de Villiers . . . . . . . . . . . . . . . . . . . 9
Wavelets based on local projection operators II by Désirée Moubandjo . . . . . . . . . . . . . . . . . 10
POROUS MEDIA III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Computer-analytic solution of subsonic irrotational �ow over a sphere by G.D. Thiart . . . . . . . . 10
On the tortuosity of 2D porous media. by M. Cloete . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
APPROXIMATION THEORY III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Subdivision Convergence and Hermite Interpolation by Guy Blaise Dongmo . . . . . . . . . . . . . . 11
Generalised smoothing factors in re�nement mask symbols by Deter de Wet . . . . . . . . . . . . . . 11
PARALLEL I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Application of Discrete-Velocity Models in Practical Flow Computations by Dr. Mapundi Kondwani
Banda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Singularity cancellation quadrature scheme for curvilinear triangular domains by Matthys M. Botha . 11
Porous Media I 7
RECONSTRUCTION OF POROUS ME-DIA AND TRANSPORT ON THE PORESCALE
Pierre Adler (University of Paris VI (Sisyphe),France)
The quantitative study of real porous media on the porescale necessitates two steps, namely to simulate realisticmedia and to calculate the desired transport properties inthese structures. This procedure and this point of viewhave now invaded the literature thanks to the simultane-ous development of a powerful measurement device themicrotomograph.The realistic simulation of porous media was initiated in1990 with the reconstruction of a Fontainebleau sand-stone. This standard reconstruction procedure will bebrie�y recalled. Its advantages and limitations will bepointed out. Then, it will be shown how this techniquecan be used to generate media with complex structuressuch as media with variable porosity, with several lengthsscales... Random packings will also be addressed.In order to calculate the transport properties, the rele-vant local partial di�erential equations must be solved,for instance the resolution of the Stokes equation yieldsthe permeability of the medium. Elementary exampleswill be addressed to demonstrate the methodology.Then, the complex case of multiphase �ows will be de-tailed with deposition of a solute. These �uids containmost of the times solutes which may deposit onto thesolid walls. This deposition onto the walls may have twoconsequences that we have studied separately. In the �rstcase, the solid phase is increased by deposition; porosityis progressively decreased until clogging occurs and stopsthe �ow and the deposition process; during this stage allthe quantities such as the absolute permeability and therelative permeabilities vary as function of time; such asituation is common in formation damage.To conclude, various recent studies will be mentioned.
TRANSIENT FLOW THROUGHPOROUS BARRIERS.
Beric Skews (University of the Witwatersrand)[email protected]
A programme of experimental studies on shock wave in-duced transient �ow through porous materials has beenundertaken. Materials tested range from permeable grids,textiles, rigid porous blocks, and elastic porous foams, forboth head-on and inclined cases of shock wave impact.Examples of some of these will be illustrated. Interestinge�ects occur for the foam and textile cases if positionedon or adjacent to a wall, in that signi�cant wall pressureampli�cation can occur. Many experiments have beenconducted and numerical models formulated in the pastin order to elucidate the processes occurring. A recent re-view of these has been given in [1]. The numerical modelsgive results showing a reasonably close physical represen-tation of the experimentally determined motion, althoughthey rely on complex models and computational methodsand nearly always require empirical adjustment to matchthe experiments. There appears to be a need for a simpleanalytical model in order to extract the primary physicalfactors of importance rather than including the multiplic-ity of foam properties in empirical numerical and physicalexperiments.A simple model is proposed to describe the main featuresfor the case of a collapsing foam block. The complextwo-phase interactions can be simpli�ed using the exper-imental evidence that the foam face may be treated as acontact surface, that the length of the compacted foamplug grows linearly in time, and that the gas pressurepro�le in the region between the head of the plug and theundisturbed material may be treated as being pseudo-stationary. These simpli�cations enable the prominentfeatures of the compression process to be predicted for avariety of foam types.References1. Skews BW, Levy A, Levi-Hevroni D. Shock wave prop-agation in porous materials. In: Handbook of ShockWaves. Ben-Dor G, Igra O, Elperin T. (Eds.), AcademicPress. 545-596, 2001
8 Tuesday 3 April
SCATTERED DATA APPROXIMATIONBY RADIAL BASIS FUNCTIONS
Armin Iske (Department of Mathematics, Uni-versity of Hamburg)[email protected]
Radial basis functions are powerful methods for approx-imation of scalar-valued and vector-valued multivariatefunctions from scattered data. This talk �rst surveys basicfeatures of scattered data approximation by radial basisfunctions, before recent advances concerning the method'sstability, local approximation properties, adaptivity, ande�cient implementation are explained. In this discussion,special emphasis is placed on reconstruction by polyhar-monic splines. In particular, the suitability of polyhar-monic spline reconstruction for multiscale simulation bymeshfree particle methods is shown. Supporting numeri-cal examples arising from real-world application scenariosare presented.
THE HIGHLIGHT THEOREM IN THEDIGITAL PULSE TRANSFORM
Dirk Laurie (University of Stellenbosch)
Carl Rohwer (University of Stellenbosch)
In 2005, Carl Rohwer, with co-author J P Harper, gavea talk at SANUM entitled "The Highlight Conjecture inLULU-transforms". In this talk, the conjecture is proved.Reference[Roh2004] C H Rohwer, Nonlinear Smmothing and Mul-tiresolution Analysis. Birkhäuser, Basel, 2004.
APPROXIMATION PROBLEMS AROUNDTHE DISCRETE PULSE TRANSFORM
Carl Rohwer (University of Stellenbosch)[email protected]
The Discrete Pulse Transform (DPT) yields a Multireso-lution Decomposition of a sequence into resolution lev-els on the edge of sets of locally monotone sequences.This transform meets the requirements, as interpretedby Meyer, for a long standing representation problem ofMarr. There are several interesting non-standard Approx-imation problems connected with the success of the DPT.Some crucial results are explained and open problems arepresented.
ON THE DEFINITION OF HYDRODY-NAMICAL ENTRANCE LENGTHS INSTRAIGHT DUCTS
S Woudberg (Applied Mathematics, StellenboschUniversity)
JP du Plessis (Applied Mathematics, Stellen-bosch University)
J Legrand (GEPEA, Universite de Nantes,France)
In the mathematical modelling of transfer processes anddispersion in micro-�uidic devices and porous media theconcept of piecewise straight duct sections is of consider-able importance. In each section �ow is developing froma certain inlet velocity pro�le towards a fully developedpro�le. Due to the short sections the �ow will be mostlyin the developing stage, rendering it an important �eld ofstudy. In this contribution some views on currently usedde�nitions of entrance lengths and an alternate de�nitionin terms of asymptotic solutions will be discussed.
Approximation Theory II 9
PERMEABILITY PREDICTION OFFONTAINEBLEAU SANDSTONE
S Woudberg (University of Stellenbosch)
JP du Plessis (University of Stellenbosch)
The microscopic geometry of Fontainebleau sandstone isapproximated in the literature as either being predomi-nantly granular or consolidated with tube-like pores. Twopore-scale models, based on rectangular geometry, areused to predict the permeability of Fontainebleau sand-stone and the results compared to experimental data. Theoccurrence of blocked throats at very low porosities is ac-counted for in the analytical modelling procedure by theintroduction of a percolation threshold porosity and theapplication of an asymptote matching technique. It isfound that although both granular and foamlike modelsyield plausible results, the granular model appears to besomewhat better, at least for the sets of data considered.The Klinkenberg correction is derived analytically and in-corporated into the models to relate gas and liquid per-meabilities.
LULU OPERATORS FOR FUNCTIONSOF CONTINUOUS ARGUMENT
Carl Rohwer (University of Stellenbosch)
Roumen Anguelov (University of Pretoria)
The LULU operators, fundamental in the nonlinear mul-tiresolution analysis of sequences, are extended to func-tions de�ned on a a real interval. We show that the ex-tended operators replicate the essential properties of theirdiscrete counterparts. More precisely, they form a fullyordered semi-group of four elements, preserve the localtrend and the total variation.
WAVELETS BASED ON LOCAL PRO-JECTION OPERATORS I
Johan de Villiers (University of Stellenbosch)
Désirée Moubandjo (University of Stellenbosch)
The study of re�nement pairs (a , φ), with a denoting a re-�nement mask, and where φ is the corresponding re�nablefunction, is of fundamental importance in both subdivi-sion and wavelet analysis, as used extensively in the ap-plication areas of, respectively, geometric modelling andsignal analysis.We start here with a given re�nement pair (a , φ), andbuild a theory based entirely on time-domain methods, toeventually yield a wavelet decomposition technique with�nite algorithms, and with the ability to e�ciently detectlocal irregularities (or non-smoothness) in a given signal.In part I, we prove that, for every resolution level r ∈ Z,the presence of zero of order N at −1 in the correspondingre�nement mask symbol A guarantees that all polynomi-als of degree ≤ N − 1 are contained in the re�nementspace V (r) spanned by the integer shifts of φ(2r·). In theprocess, a fundamental identity for the φ-commutator op-erator, as well as a generalised Marsden identity, are ob-tained. Our results hold for a larger class of re�nementpairs than was established before by means of Fouriertransform methods where φ was assumed to also satisfyproperties like integer shift independence, Riesz-stabilityand/or bounded variation. A quasi-interpolation operatorQr mapping , for every r ∈ Z, real-valued functions on Rinto V (r), such that polynomials in ΠN−1 are reproduced, is then explicitly constructed.
10 Tuesday 3 April
WAVELETS BASED ON LOCAL PRO-JECTION OPERATORS II
Désirée Moubandjo (University of Stellenbosch)
Johan de Villiers (University of Stellenbosch)
Following the settings of part I, we characterise a locallinear projection operator sequence {Pr : r ∈ Z}, wherePr : V (r+1) → V (r) , r ∈ Z, by means of the Lau-rent polynomial solution Λ of a certain Bezout identitybased on the re�nement mask symbol A. In particu-lar, it is shown that, provided A possesses no symmet-ric zeros, and A(0) 6= 0, there does indeed always existsuch a Laurent polynomial Λ of minimal length. Next,we de�ne the error space sequence {W (r) : r ∈ Z} byW (r) = {f − Prf : f ∈ V (r+1)} , r ∈ Z, and construc-tively, by means of solving more Bezout identities, showthe existence of a �nitely supported function ψ ∈ V (1)
such that, for every r ∈ Z, W (r) is spanned by the inte-ger shifts of ψ(2r·). According to our de�nition, we thencall ψ a wavelet. The wavelet decomposition algorithmbased on the quasi-interpolation operator Qr, the projec-tion operator Pr, and the wavelet ψ, is then based on�nite sequences, and is shown to possess, for a given sig-nal f , the essential property of yielding relatively smallwavelet coe�cients in regions where the support intervalof ψ(2r · −j) overlaps with a CN -smooth region of f . Fi-nally, we show how our theory can be used to deduce theknown results on the orthonormal Daubechies wavelets.
COMPUTER-ANALYTIC SOLUTIONOF SUBSONIC IRROTATIONAL FLOWOVER A SPHERE
G.D. Thiart (Dept of Mechanical and Mecha-tronic Engineering, Stellenbosch University)[email protected]
The analytic solution of the subsonic irrotational �owover a sphere, as a series expansion of velocity potentialin terms of even powers of Mach number (the Rayleigh-Janzen method), is available in the literature, but onlyup to the second-order term. The corresponding streamfunction solution is only available for the zeroth-orderterm. The availability of more accurate analytical so-lutions would be valuable for the purpose of benchmark-ing in the development of numerical solution methods.An algorithm is developed for the implementation of theRayleigh-Janzen solution method in a digital computerprogram. The computer implementation hinges on a de-rived type that implements a series solution, which wasemployed in a Fortran 95 computer program to computethe velocity potential and stream function solutions forspeci�c heat ratios ranging from 1.0 to 2.0. Results show-ing the relationship between speci�c heat ratio and criticalMach number are presented. These results were computedby means of the twelfth-order velocity potential solution,which is shown to be su�ciently accurate for practicalpurposes. It is also shown that the value of critical Machnumber for subsonic �ow over a sphere for a speci�c heatratio of 1.4 that has been accepted up to the present timeis approximately 2% high.
ON THE TORTUOSITY OF 2D POROUSMEDIA.
M. Cloete (University of Stellenbosch)
J.P. Du Plessis (University of Stellenbosch)
It is necessary to develop a good understanding of thegeometrical properties of porous media, because it playsan important role in the analysis of the transport processthrough any particular medium. Among these proper-ties is tortuosity (loosely, a measure of the waviness ofa bundle of pathlines (or streamlines for stationary �ow)through a porous medium). Tortuosity is a very vagueconcept and there exist many confusing interpretationsfor it. Also, the inverse of tortuosity will be considered.Similarities and di�erences regarding the de�nitions forthe average straightness of pathlines, expressed in litera-ture, are examined. A new de�nition, allowing di�erentchannel widths in the streamwise and the transverse �owdirections, for the tortuosity is derived from �rst princi-ples.
Parallel I 11
SUBDIVISION CONVERGENCE ANDHERMITE INTERPOLATION
Guy Blaise Dongmo (University of Stellenbosch)[email protected]
We present, separately for the scalar and vector cases, suf-�cients conditions on the Laurent polynomial mask sym-bol and the initial iterate for cascade algorithm conver-gence, and therefore also for re�nable function existenceand subdivision convergence. Moreover, we obtain ex-plicitly the geometric constant appearing in the estimatefor the geometric convergence of the cascade iterates tothe corresponding re�nable function. For the vector case,we consider an application of our result to the issue ofHermite subdivision convergence. For the scalar case, weapply our result to a one-parameter family of mask sym-bols to obtain a re�nable function existence and subdivi-sion convergence parameter interval, yielding in particularalso masks with at least one negative coe�cient.
GENERALISED SMOOTHING FAC-TORS IN REFINEMENT MASK SYM-BOLS
Deter de Wet (University of Stellenbosch)[email protected]
We show that traditional smoothing factors of the form(1+z) can be replaced by more general factors allowing usto establish new su�cient conditions concerning the reg-ularity of re�nable functions. We can also show the exis-tence of polynomial sections in re�nable functions whichare not polynomials on some other sections in the sup-port of the function. The results are derived for a generalinteger dilation factor at least 2.
APPLICATION OF DISCRETE-VELOCITY MODELS IN PRACTICALFLOW COMPUTATIONS
Dr. Mapundi Kondwani Banda (University ofKwaZulu-Natal (PMB))[email protected]
Discrete-velocity models of mesoscopic kinetic equationsare used to design algorithms for �ow computations. Oneof the most common type of a discrete velocity model isthe Lattice Boltzmann approach. In the low Mach num-ber limit of the lattice Boltzmann type models the incom-pressible Navier-Stokes equations are obtained. The sameapproach is extended to ShallowWater �ow. To derive the�ow equations from discrete-velocity models asymptoticanalysis based on the Chapman-Enskog analysis or di�usescaling is applied. Moreover, in the course of this anal-ysis, the discrete-velocity models reduce to a relaxationsystem of equations for macroscopic �ow variables whichcan be discretized using higher-order non-oscillatory up-wind relaxation schemes. These approaches o�er an alter-native to the traditional solvers. The implementation ofboth these approaches is simple. Numerical computationsare carried out on various practical test problems includ-ing shallow water �ow, incompressible �ow with radiativeheat transfer and turbulent thermal �ows.
SINGULARITY CANCELLATIONQUADRATURE SCHEME FOR CURVI-LINEAR TRIANGULAR DOMAINS
Matthys M. Botha (University of Stellenbosch)[email protected]
Surface integral equations are routinely solved by bound-ary element methods (BEM's), for scattering analysis.Typically, discretization involves creating a mesh of tri-angular elements approximating the scatterer's surface,upon which basis functions are de�ned. Geometrically,these elements can be either rectilinear or curvilinear,with the latter generally yielding more accurate results.In the BEM, the product of the Green function (typicallywith order 1/R singularity) with any given basis functionis routinely integrated over the support of the basis func-tion, for wide ranging observation points. These integralscan be very di�cult to evaluate in the near-singularitycase. Recently, a new singularity cancellation quadraturescheme has been proposed by Wilton et al., for the numer-ical integration of 1/R near-singularities on rectilinear tri-angles. In this presentation, the scheme will be extendedto the general curvilinear case. Also, a theoretical erroranalysis of the extended scheme as well as standard Gaus-sian product rule integration will be conducted. Finally,numerical results will be presented.
12 Tuesday 3 April
Tuesday 3 April
MORNING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
From Tsunamis to Insect Wakes: Solitary Waves in Free Surface Flows by Paul Milewski . . . . . . . 13
Multi-scale modelling of biochemical systems by the chemical master equation by Kevin Burrage . . 13
Models for blink cycles of the human tear �lm by Richard J. Braun . . . . . . . . . . . . . . . . . . . 14
PARALLEL I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Approximations of nonlinear phenomena arising in angular deviation of light rays that emerge fromprisms by Shirley Abelman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Unstructured High Order Nodal Finite Elements to solve the Bidomain Equations by Nicolas Jeannequin 14
PARALLEL II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
A Hydra of Singularities and harmonic subseries by Thomas Schmelzer . . . . . . . . . . . . . . . . . 15
A Non-Polynomial Spline Based Family of Methods For Solution Of One-Dimensional Di�usion Equa-tion by Siraj-ul-Islam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
PARALLEL III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
A cubic heat balance integral method for one-dimensional melting of a �nite thickness layer by Dr.
Sarah Mitchell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Mathematical Modelling of Metal Sample Treatment with Cooling System by I. Fedotov . . . . . . . 16
An algorithm for approximate Renyi dimensions from data on multifractal densities. by Henri Laurie 16
PARALLEL IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Detecting Discontinuities from Spectral Data, Polynomial Filtering and the Practical Resolution ofthe Gibbs Phenomenon by Alvaro R. de Pierro . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Tracking with active appearance models by McElory Ho�mann . . . . . . . . . . . . . . . . . . . . . 17
Increased Resolution methods by Stéfan van der Walt . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Morning 13
FROM TSUNAMIS TO INSECTWAKES: SOLITARY WAVES IN FREESURFACE FLOWS
Paul Milewski (University of Wisconsin, Madi-son, USA)
We discuss the various solitary waves that can occur onthe surface of water. Solitary waves are waves that decayaway from their center and propagate in a given directionwithout changing their shape. These range from the cel-ebrated solitons of the Korteweg-de Vries equation whichoccur under the right depth conditions, to newly discov-ered three dimensional solitary waves occurring on scaleswhere surface tension is important. Examples of obser-vations of solitary waves in a variety of settings will beshown.
MULTI-SCALE MODELLING OF BIO-CHEMICAL SYSTEMS BY THE CHEM-ICAL MASTER EQUATION
Kevin Burrage (University of Queensland)[email protected]
Biochemical reactions underlying genetic regulation areoften modelled as a continuous-time, discrete-state,Markov process, and the evolution of the associated prob-ability density is described by the so-called chemical mas-ter equation (CME). However the CME is typically dif-�cult to solve, since the state-space involved can be verylarge or even countably in�nite. Until now this hasmeant that the direct computation of the probability den-sity function (pdf) associated with chemical kinetics thattakes into account intrinsic noise has not been computa-tionally feasible. Recently a �nite state projection method(FSP) that truncates the state-space has been suggestedby Munsky and Khammash and shown to be e�ective onsmall scale problems. Presented in this talk is a KrylovFSP algorithm based on a combination of state-spacetruncation and inexact matrix-vector product routines.This allows larger-scale models to be studied and solu-tions for larger �nal times to be computed in a realisticexecution time.We compare the e�ciency of this technique with trajec-torial approaches such as the Stochastic Simulation Al-gorithm and also show how a strong trajectory can becomputed directly from the pdf computed by the CME.Numerical results indicate that this new approach can befast and is extendable to large biological models.This is joint work with Markus Hegland (ANU), ShevMacnamara (UQ), and Roger B. Sidje (UQ).
14 Tuesday 3 April
MODELS FOR BLINK CYCLES OF THEHUMAN TEAR FILM
Richard J. Braun (Mathematical Sciences, Uni-versity of Delaware, USA)
A. Heryudono, T. Driscoll, L.P. Cook (Mathe-matical Sciences, University of Delaware, USA)
P.E. King-Smith (College of Optometry, Univer-sity of Delaware, USA)
Every time one blinks, a tear �lm is left on the front of theeye. After a brief introduction to the structure of the tear�lm and the physiology of the anterior eye, we discuss re-cent models for the tear �lm formation and evolution overone or more blink cycles. The models employ high-orderlubrication-type equations on a moving domain; both �-nite di�erence and a spectral-based methods have beenapplied to solving the problem. We �nd the latter out-performs the latter, but not dramatically. These modelsare already contributing to our understanding of the tear�lm behavior. For example, many blinks are partial inthe sense that eyelids don't fully close; results show thatif the eyelids close to about 1/8 of the fully open width,that is enough to fully refresh the tear �lm. Quantitativecomparison with in vivo tear �lm thickness measurementsfor half blinks will be discussed as well.
APPROXIMATIONS OF NONLINEARPHENOMENA ARISING IN ANGULARDEVIATION OF LIGHT RAYS THATEMERGE FROM PRISMS
Shirley Abelman (University of the Witwater-srand, Johannesburg)
Herven Abelman (University of the Witwater-srand, Johannesburg)
Monochromatic light rays incident from some directionson a glass prism emerge from the prism with their di-rection changed. For many thick prisms emerging lightrays are obscured at a boundary. The purpose of thistalk is to show that particular light ray deviations canbe approximated by polynomials of varying degree overa domain of incident angles. The angles of deviation de-pend on the apex angle, the direction of incidence withrespect to the prism and the material of the prism. Fora prism in air, the incident direction is allowed to varyfor a chosen range of apex angles. For each apex an-gle value and each incident direction, the correspondingray deviation values are calculated. The theoretical equa-tions for the extremes of angular deviation are nonlinearand awkward to use. Because of their ease of applicationand goodness of �t, polynomials of varying degree andnature are chosen to approximate these nonlinear equa-tions. Graphical comparisons are made between these ap-proximating polynomial equations and the correspondingexact nonlinear extrema of angular deviation equations.We show that these cumbersome nonlinear equations canvery con�dently be replaced by their much simpler speci�cpolynomial least-squares approximating equations. Themost accurate and easily computed of these approximat-ing equations can then more readily be used in furthercomputations.
UNSTRUCTURED HIGH ORDERNODAL FINITE ELEMENTS TO SOLVETHE BIDOMAIN EQUATIONS
Nicolas Jeannequin (Computing Laboratory, Ox-ford University)[email protected]
In this talk we will describe an explicit method for theconstruction of nodes that are suitable for polynomial in-terpolation on simplices. These nodes show a performanceon par with Fekete and electrostatic nodes up to 12th or-der, but their very simple construction makes them muchmore desirable. I will then present a novel application ofthese nodes: a pseudo-spectral scheme on unstructuredsimplices to solve the bidomain equations. The bidomainmodel describes the macroscale electrical activity of car-diac tissue and is coupled to a cellular model, such as theLuo-Rudy '91 model.
Parallel III 15
A HYDRA OF SINGULARITIES ANDHARMONIC SUBSERIES
Thomas Schmelzer (Oxford University Comput-ing Laboratory, Balliol College)[email protected]
In the tradition of the SIAM 100-Digit challenge I will at-tack two problems that have been posed by Brian Daviesand Folkmar Bornemann. Davies cooked up a rathernasty oscillating integral as a problem that would in-volve serious di�culties both analytically and numeri-cally. Bornemann raised the question about the limit ofthe harmonic series when omitting all terms that containthe substring �42�.
A NON-POLYNOMIAL SPLINE BASEDFAMILY OF METHODS FOR SOLU-TION OF ONE-DIMENSIONAL DIFFU-SION EQUATION
Siraj-ul-Islam (University of Engineering & Tech-nology Peshawar (NWFP), Pakistan)[email protected]
In this paper a new approach based on non-polynomialspline approximation to the one-dimensional heat conduc-tion equation is discussed. This approach correspondsto the cubic spline approximation by Papamichael &Whiteman and to a �nitedi�erence scheme consideredby Saulyev. The non-polynomial spline space consideredin this paper consists of polynomial and a trigonomet-ric part. Free parameter introduced in the trigonometricpart of the non-polynomial spline is used to improve theaccuracy of the scheme, while the trigonometric part ofthe non-polynomial spline function compensates for lossof smoothness inherited by polynomial splines. Also thealgorithm developed by the use of non-polynomial splinesleads to the generalization of the algorithm developed byPapamichael & Whiteman which is based on cubic splinesand the unconditionally stable �nite-di�erence schemeslike Laasonen formula, Crank-Nicolson scheme and Cran-dalls implicit formula. The spline approximation pro-duces at each time level a spline function which may beused to obtain the solution at any point in the range ofthe space variable. The procedure to �nd best values ofthe parameters is shown and problems from the litera-ture for both homogenous and non-homogenous cases areconsidered to test the practical usefulness of the method
16 Tuesday 3 April
A CUBIC HEAT BALANCE INTEGRALMETHOD FOR ONE-DIMENSIONALMELTING OF A FINITE THICKNESSLAYER
Dr. Sarah Mitchell (The University of CapeTown)
Professor Tim Myers (The University of CapeTown)
This work concerns the one-dimensional melting of a �nitethickness layer. A series solution describes the tempera-ture in the melt regions. In the solid region the thermalboundary layers are approximated by a cubic polynomial.Results are compared with the exact solution for a semi-in�nite block, and shown to agree to within less than 1%.The method is then applied to a situation where no ana-lytical solution is available. A �nite thickness frozen solidis placed on a warm substrate in a warm environment:initially the base of the solid heats to the melting tem-perature when a single melted region develops and subse-quently a second melting front appears on the top bound-ary. We also present an example relevant to heating anice layer from below, which occurs with de-icing systems.
MATHEMATICAL MODELLING OFMETAL SAMPLE TREATMENT WITHCOOLING SYSTEM
I. Fedotov (TUT)
T. Fedotova (WITS)
M. Shatalov (CSIR)
One of the most e�ective method of metal surface ther-mal strengthening is their laser treatment. Very often thechoice of treatment mode is a result of length and veryexpensive industrial testing including a big number of pa-rameters which are changing at various levels. Creationof a mathematical model of that process allows to signif-icantly decrease the expenses and cut the time of processoptimization of laser thermal treatment of metal samples.In the present work the authors present a mathemati-cal model of laser treatment with cooling based on three-dimensional heat transfer equation using Greens function.For Greens function we used the source method which hasthe property of rapid convergence at small time of transferprocess. It is shown that the combination of laser heatingmethod and synchronous cooling improves the microstruc-ture of the treated metal layer through cutting the timeof austenite to martensite conversion to minimum.
AN ALGORITHM FOR APPROXIMATERENYI DIMENSIONS FROM DATA ONMULTIFRACTAL DENSITIES.
Henri Laurie (UCT)
Edith Perrier (IRD)
The basic intuition about multifractals is that they aredensities with detail at all spatial scales. We de�ne thespectrum of Renyi dimensions D(q) of a multifracta, givenan algorithm for calculating D(q) from density data, andtest the algorithm on a density for which D(q) is ana-lytically known. If time permits we will brie�y discuss anapplication to data on species richness in Cape Proteaceaeand southern African birds.
Parallel IV 17
DETECTING DISCONTINUITIESFROM SPECTRAL DATA, POLYNO-MIAL FILTERING AND THE PRAC-TICAL RESOLUTION OF THE GIBBSPHENOMENON
Alvaro R. de Pierro (State University of Camp-inas, Department of Applied Mathematics)[email protected]
Many practical problems require the reconstruction of apiecewise smooth function from its Fourier coe�cients(Nuclear Magnetic Resonance, Fourier inversion in Com-puted Tomography, Fourier methods for ConservationLaws Di�erential Equations and many others). It is wellknown that the convergence of the Fourier series dependson the function's smoothness. In particular, discontinu-ities give rise to large oscillations close to the jumps, theso called Gibbs phenomenon. Filtering in the Fourier do-main and Gegenbauer expansions have been proposed byseveral authors (Gotlieb, Shu, Vandeven, Solomono�) tosolve the problem. All these methods assume a preciseknowledge of the jumps location and amplitude. Fastmethods for this task have been based on the conjugateFourier series and concentration kernels (A. Gelb and E.Tadmor, J. Sci. Comput., 28, 2-3, 2006 and referencestherein). Recently, Wei, Martinez and De Pierro (Appl.Comput. Harmon. Anal. 2007) presented a new approachfor edge detection from Fourier coe�cients, based on anew family of polynomial �lters (M. Wei, A.R. De Pierroand J. Yin, IEEE Trans. on Signal Proc., 53, 1, 136-146,2005 and references therein) that proves to be much moreaccurate and stable than previous approaches. In this talkwe brie�y describe the new polynomial and spline based�lters introduced by Wei, De Pierro, Martinez and Yin,their application to develop methods for edge detectionfrom spectral data and how to use them for the practicalresolution of the Gibbs phenomenon. We illustrate thenew approach with numerical experiments.
TRACKING WITH ACTIVE APPEAR-ANCE MODELS
McElory Ho�mann (Applied Mathematics, Uni-versity of Stellenbosch)[email protected]
Active appearance models (AAMs) provide a frameworkfor tracking the shape and texture of an object. However,deterministic tracking with AAMs is not robust. For thisreason, AAMs are combined with a particle �lter in or-der to increase the robustness. This combination can beachieved in two ways: a direct combination or by combin-ing AAMs with active contours. Encouraging results areobtained.
INCREASED RESOLUTION METHODS
Stéfan van der Walt (University of Stellenbosch)[email protected]
A polygon-based interpolation algorithm is presented foruse in stacking RAW CCD images. The algorithm im-proves on linear interpolation in this scenario by closelydescribing the underlying geometry. 25 frames are stackedin a comparison. When stacking images, it is requiredthat these images are accurately aligned. We presenta novel implementation of the log-polar transform thatovercomes its prohibitively expensive computation, result-ing in fast, robust image registration. This is demon-strated by registering and stacking CCD frames of starstaken by a telescope.
18 Tuesday 3 April
Attendees
Name Surname A�liation Email
1 Chama Abdoulkadri Africa Institute for Mathematical Sciences [email protected]
2 Shirley Abelman University of the Witwatersrand [email protected]
3 Pierre Adler University of Paris VI (Sisyphe), France [email protected]
4 Saheed Akindeinde Africa Institute for Mathematical Sciences [email protected]
5 Mapundi Banda University of KwaZulu-Natal [email protected]
6 Matthys M. Botha University of Stellenbosch [email protected]
7 Ilyes Boulkabeit Africa Institute for Mathematical Sciences [email protected]
8 Richard Braun University of Delaware, USA [email protected]
9 Bernard Broens University of Stellenbosch [email protected]
10 Kevin Burrage University Queensland [email protected]
11 Vasco Chikwasha Harare Institute of Technology, Zimbabwe [email protected]
12 Richard Clark University of Stellenbosch [email protected]
13 Marèt Cloete University of Stellenbosch [email protected]
14 Hanno Coetzer University of Stellenbosch [email protected]
15 Alvaro de Pierro State University of Campinas [email protected]
16 Johan de Villiers University of Stellenbosch [email protected]
17 Deter de Wet University of Stellenbosch [email protected]
18 Blaise Dongmo University of Stellenbosch [email protected]
19 Elsa du Plessis University of Stellenbosch [email protected]
20 Prieur du Plessis University of Stellenbosch [email protected]
21 Wilna du Toit University of Stellenbosch [email protected]
22 Yves Dumont University of Reunion Island [email protected]
23 Igor Fedotov University of the Witwatersrand [email protected]
24 Tatiana Fedotova University of the Witwatersrand [email protected]
25 Andreas Hafver University of Stellenbosch [email protected]
26 Ben Herbst University of Stellenbosch [email protected]
27 McElory Ho�mann University of Stellenbosch [email protected]
28 Karin Hunter University of Stellenbosch [email protected]
29 Armin Iske University of Hamburg, Germany [email protected]
30 Siraj-ul-Islam Islam University of Engineering and Technolgy, Pak-istan
31 Nicolas Jeannequin Oxford University [email protected]
32 Sarah Jane Johnston University of the Witwatersrand [email protected]
33 Margaret Kajotoni Africa Institute for Mathematical Sciences [email protected]
34 Damian Kajunguri Africa Institute for Mathematical Sciences [email protected]
35 Gennady Kulikov University of the Witwatersrand [email protected]
Parallel IV 19
36 Maria Kulikova University of the Witwatersrand [email protected]
37 Anneke Labuschagne University of Pretoria [email protected]
38 Dirk Laurie University of Stellenbosch [email protected]
39 Henri Laurie University of Cape Town [email protected]
40 Jean M-S Lubuma University of Pretoria [email protected]
41 Roland Makala Africa Institute for Mathematical Sciences [email protected]
42 Tom Manteu�el University of Colorado at Boulder, USA [email protected]
43 Milton Maritz University of Stellenbosch [email protected]
44 David P Mason University of the Witwatersrand [email protected]
45 Paul Milewski Mathematics, Univ. of Wisconsin [email protected]
46 Sarah Mitchell University of Cape Town [email protected]
47 Hisham Mohamed Africa Institute for Mathematical Sciences [email protected]
48 Ebrahim Momoniat University of the Witwatersrand [email protected]
49 Désirée Moubandjo University of Stellenbosch [email protected]
50 Neil Muller iThemba labs [email protected]
51 Dana Murray University of the Free State [email protected]
52 Patrick Muthivhi Unknown [email protected]
53 Tim Myers University of Cape Town [email protected]
54 Igor Neugebauer National University of Rwanda [email protected]
55 Jean-Medard Ngnotchouye Africa Institute for Mathematical Sciences [email protected]
56 Fernando Nieuwveldt University of Stellenbosch [email protected]
57 Joy Okonkwo Africa Institute for Mathematical Sciences [email protected]
58 Eric Okyere Africa Institute for Mathematical Sciences [email protected]
59 Carel Olivier University of Stellenbosch [email protected]
60 Eddy Kimba Phongi Africa Institute for Mathematical Sciences [email protected]
61 Inger Plaskitt University of Pretoria [email protected]
62 Doriano Pougaza Africa Institute for Mathematical Sciences [email protected]
63 Barton Pro�tt University of Stellenbosch [email protected]
64 Fabi Rabarison Africa Institute for Mathematical Sciences [email protected]
65 Dimbinaina Ralaivaosaona Africa Institute for Mathematical Sciences [email protected]
66 Natacha Ranariboana Africa Institute for Mathematical Sciences [email protected]
67 Christo Rautenbach University of Stellenbosch [email protected]
68 Paul Raza�mandimby Africa Institute for Mathematical Sciences [email protected]
69 Daya Reddy University of Cape Town [email protected]
70 Carl Rohwer University of Stellenbosch [email protected]
20 Tuesday 3 April
71 Chris Rohwer University of Stellenbosch [email protected]
72 Thomas Schmelzer Oxford University [email protected]
73 Schalk Schoombie University of the Free State [email protected]
74 Wynand Singels University of Stellenbosch [email protected]
75 Beric Skews University of the Witwatersrand [email protected]
76 Joseph Ssebuliba Africa Institute for Mathematical Sciences [email protected]
77 Albert Swart University of Stellenbosch [email protected]
78 Eric-Martial Takougang Africa Institute for Mathematical Sciences [email protected]
79 Antoine Tambue Africa Institute for Mathematical Sciences [email protected]
80 Gerrie Thiart University of Stellenbosch [email protected]
81 Zola Nigussa Urgessa Africa Institute for Mathematical Sciences [email protected]
82 Jack Urombo Harare Institute of Technology, Zimbabwe [email protected]
83 Rinske van der Bijl University of Stellenbosch [email protected]
84 Stefan van der Walt University of Stellenbosch [email protected]
85 Barry van Wyk University of Stellenbosch [email protected]
86 André Weideman University of Stellenbosch [email protected]
87 'Rüdiger' Weiner Martin-Luther University Halle-Wittenberg,Germany
88 Gert Wessels University of Stellenbosch [email protected]
89 Sonia Woudberg University of Stellenbosch [email protected]
90 Dessalegn Yizengaw Africa Institute for Mathematical Sciences [email protected]
Index
Abelman, 14Adler, 7
Banda, 11Botha, 11Braun, 14Burrage, 13
Cloete, 10
De Plessis, 8De Villiers, 9De Wet, 11Dongmo, 11Dumont, 3
Fedotov, 16
Ho�mann, 17
Igor, 5Iske, 8
Jeannequin, 14
Kulikov, 4Kulikova, 5
Labuschagne, 3Laurie, 8, 16Lubuma, 4
Manteu�el, 2Mason, 4Milewski, 13Mitchell, 16Moubandjo, 10
Pierro, 17Plaskitt, 3
Reddy, 2Rohwer, 8, 9
Ruediger, 2
Schmelzer, 15Schoombie, 4Siraj-ul-Islam, 15Skews, 7
Thiart, 10
Urombo, 5
Van der Walt, 17
Woudberg, 9
21