In this thesis, we discuss some of the results that were proven by Fabes and Stroock in 1984. Our main purpose is to give a self-contained presentation of the proof of this results. The first result is on the existence of a “reverse H ̈older inequality” for the Green’s function. We utilize the work of Muckenhoupt on the reverse Ho ̈lder inequality and its connection to the A∞ class to establish a comparability property for the Green’s functions. Additionally, we discuss some of the underlying preliminaries. In that, we prove the Alexandrov-Bakelman-Pucci estimate, give a treatment to the Ap and A∞ classes of Muckenhoupt, and establish two intrinsic lemmas on the behavior of Green’s function.

Actively-managed funds have recently come under fire as it has been determined that they consistently underperform passive funds. Benchmarking, and the constraints placed on actively-managed funds, are standard practices within the industry, but research suggests that these constraints negatively affect fund performance. This research paper explores the effectiveness of actively-managed funds in relation to their benchmark indices, in terms of tracking errors and weighting constraints. This is done by qualifying the effect of these constraints on the performance of hypothetically constructed portfolios in relation to the FTSE / JSE Top 40 Index. The results are presented graphically and show that tracking error limits did, as expected, limit the possible upside returns of these funds. It was found however, that the tracking error constraints had a much greater effect on limiting downside risk than they had on limiting upside effects. Weighting limitations did not have a single universal effect on the simulated portfolios’ performance but affect performance in conjunction with tracking error limits. It was concluded that for the hypothetically constructed portfolios for the period studied, constraints did not affect the possible upside return to such a magnitude that the constraints themselves could account for the underperformance of actively managed funds and they had an overall positive effect on performance. / Mini Dissertation (MBA)--University of Pretoria, 2018. / Gordon Institute of Business Science (GIBS) / MBA / Unrestricted

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FJRW-theory is a recent advancement in singularity theory arising from physics. The FJRW-theory is a graded vector space constructed from a quasihomogeneous weighted polynomial and symmetry group, but it has been conjectured that the theory only depends on the weights of the polynomial and the group. In this thesis, I prove this conjecture using Poincaré polynomials and Koszul complexes. By constructing the Koszul complex of the state space, we have found an expression for the Poincaré polynomial of the state space for a given polynomial and associated group. This Poincaré polynomial is defined over the representation ring of a group in order for us to take G-invariants. It turns out that the construction of the Koszul complex is independent of the choice of polynomial, which proves our conjecture that two different polynomials with the same weights will have isomorphic FJRW rings as long as the associated groups are the same.

This dissertation investigates the interactive or joint influence of autocorrelative processes (autoregressive-AR, moving average-MA, and autoregressive moving average-ARMA) and sample weights present in a longitudinal panel data set. Specifically, to what extent are the sample estimates influenced when autocorrelation (which is usually present in a panel data having correlated observations and errors) and sample weights (complex sample design feature used in longitudinal data having multi-stage sampling design) are modeled versus when they are not modeled or either one of them is taken into account. The current study utilized a Monte Carlo simulation design to vary the type and magnitude of autocorrelative processes and sample weights as factors incorporated in growth or latent curve models to evaluate the effect on sample latent curve estimates (mean intercept, mean slope, intercept variance, slope variance, and intercept slope correlation). Various latent curve models with weights or without weights were specified with an autocorrelative process and then fitted to data sets having either the AR, MA or ARMA process. The relevance and practical importance of the simulation results were ascertained by testing the joint influence of autocorrelation and weights on the Early Childhood Longitudinal Study for Kindergartens (ECLS-K) data set which is a panel data set having complex sample design features. The results indicate that autocorrelative processes and weights interact with each other as sources of error to a statistically significant degree. Accounting for just the autocorrelative process without weights or utilizing weights while ignoring the autocorrelative process may lead to bias in the sample estimates particularly in large-scale datasets in which these two sources of error are inherently embedded. The mean intercept and mean slope of latent curve models without weights was consistently underestimated when fitted to data sets having AR, MA or ARMA process. On the other hand, the intercept variance, intercept slope, and intercept slope correlation were overestimated for latent curve models with weights. However, these three estimates were not accurate as the standard errors associated with them were high. In addition, fit indices, AR and MA estimates, parsimony of the model, behavior of sample latent curve estimates, and interaction effects between autocorrelative processes and sample weights should be assessed for all the models before a particular model is deemed as most appropriate. If the AR estimate is high and MA estimate is low for a LCAR model than the other models that are fitted to a data set having sample weights and the fit indices are in the acceptable cut-off range, then the data set has a higher likelihood of having an AR process between the observations. If the MA estimate is high and AR estimate is low for a LCMA model than the other models that are fitted to a data set having sample weights and the fit indices are in the acceptable cut-off range, then the data set has a higher likelihood of having an MA process between the observations. If both AR and MA estimates are high for a LCARMA model than the other models that are fitted to a data set having sample weights and the fit indices are in the acceptable cut-off range, then the data set has a higher likelihood of having an ARMA process between the observations. The results from the current study recommends that biases from both autocorrelation and sample weights needs to be simultaneously modeled to obtain accurate estimates. The type of autocorrelation (AR, MA or ARMA), magnitude of autocorrelation, and sample weights influences the behavior of estimates and all the three facets should be carefully considered to correctly interpret the estimates especially in the context of measuring growth or change in the variable(s) of interest over time in large-scale longitudinal panel data sets.

This study investigates and quantifies the effect of different specifications of the spatial weights matrix (��) on estimates and inferences in the context of a regression model using the lattice perspective with polygon-type data. The study also investigates an alternative to the specification of �� by estimating a spatial variance-covariance matrix based on known features of the spatial data. Previous literature has addressed the a priori construction of �� and selection criteria but assumes point-type data. This study’s primary contribution is the setup of a true and known benchmark that allows the comparison of the different specifications of ��. This is accomplished by using a disaggregate point-type data generating process which is then aggregated into polygon-type data. Monte Carlo simulations show that current specifications of �� used in maximum likelihood estimation for the spatial error model perform poorly. Additionally, the estimated spatial variance-covariance matrix outperforms the traditional specifications of ��.

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No / Beamforming is used to achieve diversity gain. It is a technique in diversity to remedy power penalty due to channel fading. The beam-weights for multi-antenna system are evaluated for two different approaches. These weights are then used to weight the signal beams of each transmit antenna branch. Results reveal that by exploiting the transmit channel substructures exposed by the singular vector decomposition algorithm, the capacity of a multi-antenna system can be enhanced.

No / The paper explores some of the political assumptions implicit in the use of social cost-benefit analysis, and the role of economic planners using this analysis. It is argued that the recent emphasis on income distribution in project planning techniques rests on questionable assumptions about the nature and intentions of governments. These techniques can be thought of as attempts to redistribute income ‘through the back door’. Some possible justification for the use of income distribution weights in project planning is given, but it is concluded that they are very much second best to direct macro-economic policies and popular participation in planning.