Theses and Dissertations (Mathematical Sciences)
http://hdl.handle.net/10500/3017
2015-05-22T08:25:02ZA Lie symmetry analysis of the heat equation through modified one-parameter local point transformation
http://hdl.handle.net/10500/18414
A 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.
2014-08-01T00:00:00ZA Lie symmetry analysis of the Black-Scholes Merton finance model through modified local one-parameter transformations
http://hdl.handle.net/10500/18410
A Lie symmetry analysis of the Black-Scholes Merton finance model through modified local one-parameter transformations
Masebe, Tshidiso Phanuel
The thesis presents a new method of Symmetry Analysis of the Black-Scholes Merton
Finance Model through modi ed Local one-parameter transformations. We determine
the symmetries of both the one-dimensional and two-dimensional Black-Scholes
equations through a method that involves the limit of in nitesimal ! as it approaches
zero. The method is dealt with extensively in [23]. We further determine an invariant
solution using one of the symmetries in each case. We determine the transformation
of the Black-Scholes equation to heat equation through Lie equivalence transformations.
Further applications where the method is successfully applied include working out
symmetries of both a Gaussian type partial di erential equation and that of a di erential
equation model of epidemiology of HIV and AIDS. We use the new method to
determine the symmetries and calculate invariant solutions for operators providing
them.
2014-09-01T00:00:00ZThe application and empirical comparison of item parameters of Classical Test Theory and Partial Credit Model of Rasch in performance assessments
http://hdl.handle.net/10500/18362
The application and empirical comparison of item parameters of Classical Test Theory and Partial Credit Model of Rasch in performance assessments
Mokilane, Paul Moloantoa
This study empirically compares the Classical Test Theory (CTT) and the Partial Credit Model
(PCM) of Rasch focusing on the invariance of item parameters. The invariance concept which is
the consequence of the principle of specific objectivity was tested in both CTT and PCM using the
results of learners who wrote the National Senior Certificate (NSC) Mathematics examinations in
2010. The difficulty levels of the test items were estimated from the independent samples of learn-
ers. The same sample of learners used in the calibration of the difficulty levels of the test items in
the PCM model were also used in the calibration of the difficulty levels of the test items in CTT
model. The estimates of the difficulty levels of the test items were done using RUMM2030 in the
case of PCM while SAS was used in the case of CTT. RUMM2030 and SAS are both the statistical
softwares. The analysis of variance (ANOVA) was used to compare the four different design groups
of test takers. In cases where the ANOVA showed a significant difference between the means of the
design groups, the Tukeys groupings was used to establish where the difference came from.
The research findings were that the test items' difficulty parameter estimates based on the CTT theoretical framework were not invariant across the different independent sample groups. The over-
all findings from this study were that the CTT theoretical framework was unable to produce item
difficulty invariant parameter estimates. The PCM estimates were very stable in the sense that for
most of the items, there was no significant difference between the means of at least three design
groups and the one that deviated from the rest did not deviate that much. The item parameters of
the group that was representative of the population (proportional allocation) and the one where the
same number of learners (50 learners) was taken from different performance categories did not differ
significantly for all the items except for item 6.6 in examination question paper 2. It is apparent
that for the test item parameters to be invariant of the group of test takers in PCM, the group of
test takers must be heterogeneous and each performance category needed to be big enough for the proper calibration of item parameters.
The higher values of the estimated item parameters in CTT were consistently found in the sample
that was dominated by the high proficient learners in Mathematics ("bad") and the lowest values
were consistently calculated in the design group that was dominated by the less proficient learners. This phenomenon was not apparent in the Rasch model.
2014-05-01T00:00:00ZAperture photometry at the Unisa observatory; system evaluation via light curves of eclipsing binaries
http://hdl.handle.net/10500/18321
Aperture photometry at the Unisa observatory; system evaluation via light curves of eclipsing binaries
Pieters, Vanessa
The University of South Africa has built a small observatory for teaching and
research. The 35cm Schmidt-Cassegrain reflector and the SSP-5A photometer with
Johnson UBV filters were used to evaluate the local possibilities for aperture
photometry. Extinction and transformation coefficients and zero points were
determined with the RPHOT software package from standard star photometry. To
determine the system limits, UBV light curves were obtained for four eclipsing
binaries, ranging between magnitudes 7 and 13. In B the limit is 12,5, in U it is
12,0 and in V only 11,0 with large scatter even for bright objects. The rapidly
varying sky background in an urban site may be a major cause. A related problem
is the large, fixed photometer diaphragm, aggravating the bright sky situation.
Possible solutions to this and other problems were suggested. The system can be
used fruitfully, especially for single filter photometry of periodic and nonperiodic
variables.
2015-01-01T00:00:00Z