Department of Mathematical Scienceshttp://hdl.handle.net/10500/30162015-07-07T04:42:45Z2015-07-07T04:42:45ZModellering van afhanklikheid in die lineêre model : 'n meteorologiese toepassingNieuwoudt, Reinahttp://hdl.handle.net/10500/187372015-06-25T01:00:15Z1997-06-01T00:00:00ZModellering van afhanklikheid in die lineêre model : 'n meteorologiese toepassing
Nieuwoudt, Reina
As deel van die weermodifikasie-eksperiment in Bethlehem, Suid-Afiika, is 'n reenmeternetwerk
geinstalleer, en word die neerslagwaardes R; wat by 43 reenmeterstasies waargeneem is, vergelyk
met die waargenome radar reflektiwiteit Z;. Alhoewel radar ruimtelike en tydskontinue metings van
reflektiwiteit bied wat onmiddellik by een sentrale punt beskikbaar is, is die akkuraatheid van radar
om reenval te meet onseker as gevolg van verskeie potensiele foute in die omskakeling van
reflektiwiteit na reenval. Dit word aanvaar dat reenmeters akkurate puntwaarnemings van reenval
gee en daar bestaan eenstemmigheid dat die kombinasie van die twee metodes beter is as enigeen
van die metodes afsonderlik. In hierdie studie ondersoek ek die toepassing van die veralgemeende
lineere model as 'n beramingstegniek.
Vorige studies gebruik die log-log transformasie, d. w.s. logZ = logA + b(logR) van die Z = ARb
verwantskap om die koeffisiente A en b met behulp van kleinste-kwadrate-regressie te bepaal.
Die implisiete aanname hiermee is dat die foute ongekorreleerd is.
Met die inverse verwantskap R = czd d.w.s. logR = logC + d(logZ) neem ek aan dat die
waarnemings nie onafhanklik is nie sodat die regressiekoeffisiente bereken word met behulp van
die metode van die veralgemeende lineere model. Om die ruimtelike afhanklikheid van die reenmeterwaarnemings
te modelleer, word eksperimentele variogramme uit die data bereken en gepas
met teoretiese variogramme wat gebruik word om die variansie-kovariansiematriks te vu!.
"Gemiddeld" vaar hierdie metode beter as gewone regressie vir analises wat reenmeters wat verder
as 45km vanaf die radarstel is, insluit.
Residu-stipping wys dat die afstand van die meter vanaf die radarstel as 'n afsonderlike onafhanklike
veranderlike in die regressievergelyking ingesluit behoort te word, d.w.s. die beraming
verbeter met logR = 3-0 + a,(logZ) + a2(afstand). Hierdie meervoudige regressiemodel stem ooreen
met die teoretiese model van Smith en Krajewski omdat e -- afstand as 'n praktiese manifestasie van
die foutproses [e.,, (ij)] beskou kan word. Omdat E(ez) = eE<ZJ e'"a' as Z 'n lognormaalverdeling het, kan die sydigheid wat ontstaan
wanneer antilogaritmes geneem word, reggestel word deur die beraamde reenval met e112
"' te
vermenigvuldig.
Die studie !ewer 'n bydrae met die afleiding van 'n beramingstegniek wat die beraming van
neerslag uit radar betekenisvol verbeter.; In a study of a rain-gauge network that was installed for a weather modification experiment in
Bethlehem, South Africa, precipitation values R; observed at 43 gauging stations are compared to
the observed radar reflectivity Z;. Although radar provides spatial and temporal measurements of
reflectivity that are immediately available at one location, the accuracy of radar estimation of
rainfall is uncertain due to various potential errors in the conversion from reflectivity to rainfall.
Rain-gauges are assumed to give accurate point measurements of rainfall and there is general
agreement that the combination of systems is better than either system alone. In this study I
explore the application of the general linear model as an estimation technique.
Previous studies have used the log-log transform, i.e. logZ = logA + b(logR) of the Z = ARb
relation, and applied least-squares regression analysis to determine the coefficients A and b. This
implicitly assumes that the disturbances are uncorrelated.
Working with the inverse relation R = czd i.e. logR = logC + d(logZ) and assuming that the
observations are not independent we compute the regression coefficients using generalised least
squares. To model the spatial dependence of the rain-gauge observations we compute
experimental variograms from the data and fit them with theoretical variograms which are then
used to fill the variance-covariance matrix. "On average" this method performs better than
ordinary regression for the analyses that included rain-gauges further than 45km from the radar
set.
Residual plotting revealed that distance of the rain-gauge from the radar set should be included as
a separate independent variable in the regression equation, i.e. logR = ao + a1(logZ) + a1(distance)
improved the estimation of rainfall as it performs better than ordinary regression. This multiple
regression model agrees with the theoretical model of Smith and Krajewski in the sense that
e "'distance is a practical manifestation of the error process [ e,, (ij)].
Showing that E( ez) = el!.(!.) e 112
"' if Z has a lognormal distribution, the bias when taking antilogs can be removed by multiplying estimated rainfall by e1
'
2a'.
The contribution of this study is the derivation of an estimation technique which significantly
improves the estimation of rainfall from radar
Text in Afrikaans, abstract in Afrikaans and English
1997-06-01T00:00:00ZThe differential geometry of the fibres of an almost contract metric submersionTshikunguila, Tshikuna-Matambahttp://hdl.handle.net/10500/186222015-06-30T07:13:47Z2013-10-01T00:00:00ZThe differential geometry of the fibres of an almost contract metric submersion
Tshikunguila, Tshikuna-Matamba
Almost contact metric submersions constitute a class of Riemannian submersions whose
total space is an almost contact metric manifold. Regarding the base space, two types
are studied. Submersions of type I are those whose base space is an almost contact
metric manifold while, when the base space is an almost Hermitian manifold, then the
submersion is said to be of type II.
After recalling the known notions and fundamental properties to be used in the
sequel, relationships between the structure of the fibres with that of the total space
are established. When the fibres are almost Hermitian manifolds, which occur in the
case of a type I submersions, we determine the classes of submersions whose fibres
are Kählerian, almost Kählerian, nearly Kählerian, quasi Kählerian, locally conformal
(almost) Kählerian, Gi-manifolds and so on. This can be viewed as a classification of
submersions of type I based upon the structure of the fibres.
Concerning the fibres of a type II submersions, which are almost contact metric
manifolds, we discuss how they inherit the structure of the total space.
Considering the curvature property on the total space, we determine its corresponding
on the fibres in the case of a type I submersions. For instance, the cosymplectic
curvature property on the total space corresponds to the Kähler identity on the fibres.
Similar results are obtained for Sasakian and Kenmotsu curvature properties.
After producing the classes of submersions with minimal, superminimal or umbilical
fibres, their impacts on the total or the base space are established. The minimality of
the fibres facilitates the transference of the structure from the total to the base space.
Similarly, the superminimality of the fibres facilitates the transference of the structure
from the base to the total space. Also, it is shown to be a way to study the integrability
of the horizontal distribution.
Totally contact umbilicity of the fibres leads to the asymptotic directions on the total
space.
Submersions of contact CR-submanifolds of quasi-K-cosymplectic and
quasi-Kenmotsu manifolds are studied. Certain distributions of the under consideration
submersions induce the CR-product on the total space.
2013-10-01T00:00:00ZThe use of computer simulations for cognitive load change and acquisition of knowledge and skills in geometrical opticsKaheru, Sam James Murungihttp://hdl.handle.net/10500/186092015-07-06T05:40:05Z2014-06-01T00:00:00ZThe use of computer simulations for cognitive load change and acquisition of knowledge and skills in geometrical optics
Kaheru, Sam James Murungi
The aim of the study was to compare the effects of the use of interactive computer simulations for cognitive load change of grade 11 learners in the acquisition of knowledge and a science process skill in geometrical optics. Both the use of computer simulations and traditional teaching was teacher centred. The study was done in a rural area in South Africa, in the Limpopo Province in the district of Vhembe. The theoretical framework was based on the information processing model. Within the non-equivalent quasi experimental design a switching replications design study was used whereby 105 learners in four schools took part. This study found that in terms of the acquisition of knowledge, female learners gained more by the use of simulations than their male counterparts. No significant effect was found in the acquisition of the skill when computer simulations were used. Initial reduction of cognitive load was found when simulations were used and with time this increased. Experienced educators reduced the cognitive load through use of their knowledge and expertise and their role needs to be highlighted. Further studies are suggested to study the effect of a learner centred approach on decreasing the cognitive load and its effect on the acquisition of knowledge and skills.
2014-06-01T00:00:00ZA Lie symmetry analysis of the heat equation through modified one-parameter local point transformationAdams, Conny Molatlhegihttp://hdl.handle.net/10500/184142015-04-23T06:48:44Z2014-08-01T00:00:00ZA Lie symmetry analysis of the heat equation through modified one-parameter local point transformation
Adams, Conny Molatlhegi
Using a Lie symmetry group generator and a generalized form of Manale's formula
for solving second order ordinary di erential equations, we determine new symmetries
for the one and two dimensional heat equations, leading to new solutions. As
an application, we test a formula resulting from this approach on thin plate heat
conduction.
2014-08-01T00:00:00Z