Research Outputs (Economics)http://hdl.handle.net/10500/41452015-08-05T04:29:51Z2015-08-05T04:29:51ZNonstandard analysis, fractal properties and Brownian motionPotgieter, Paulhttp://hdl.handle.net/10500/187832015-07-08T01:05:49Z2009-03-01T00:00:00ZNonstandard analysis, fractal properties and Brownian motion
Potgieter, Paul
Using Loeb measure theory, it is possible to construct Lebesgue or even Wiener measure as a hyperfinite counting measure. In this paper I explore an extension of the idea to Hausdorff measure, which yields a hyperfinite formulation of Hausdorff dimension. In cases where the problem of dimension can be interpreted as an equivalent problem on a hyperfinite time line, this can lead to a simple and intuitively satisfying proof. For instance, I shall later consider certain properties of Brownian motion, and present nonstandard proofs which are somewhat easier than the original, and also seem to obey certain statistical “rules of thumb”. I now provide a short overview of the necessary nonstandard analysis as well as the standard formulation of Hausdorff dimension, since the nonstandard version will follow the same style and notation.
2009-03-01T00:00:00ZThe Fourier dimension of Brownian limsup fractalsPotgieter, Paulhttp://hdl.handle.net/10500/187822015-07-08T01:03:14Z2013-01-01T00:00:00ZThe Fourier dimension of Brownian limsup fractals
Potgieter, Paul
Robert Kaufman’s proof that the set of rapid points of Brownian motion has a Fourier dimension equal to its Hausdorff dimension was first published in 1974. A study of the proof of the original paper revealed several gaps in the arguments and a slight inaccuracy in the main theorem. This paper presents a new version of the construction and incorporates some recent results in order to establish a corrected version of Kaufman’s theorem. The method of proof can then be extended to show that functionally determined rapid points of Brownian motion also form Salem sets for absolutely continuous functions of finite energy.
2013-01-01T00:00:00ZThe rapid points of a complex oscillationPotgieter, Paulhttp://hdl.handle.net/10500/187712015-07-04T01:01:19Z2012-01-01T00:00:00ZThe rapid points of a complex oscillation
Potgieter, Paul
By considering a counting-type argument on Brownian sample paths, we prove
a result similar to that of Orey and Taylor on the exact Hausdorff dimension of the rapid
points of Brownian motion. Because of the nature of the proof we can then apply the
concepts to so-called complex oscillations (or algorithmically random Brownian motion),
showing that their rapid points have the same dimension.
2012-01-01T00:00:00ZDoes governance matter in infrastructure: evidence from Sub-Saharan AfricaAkanbi, Olusegun Ayodelehttp://hdl.handle.net/10500/119492014-06-24T10:37:58Z2013-01-01T00:00:00ZDoes governance matter in infrastructure: evidence from Sub-Saharan Africa
Akanbi, Olusegun Ayodele
This study empirically examines the pattern of physical infrastructure that would unlock the productive potential of sub-Saharan Africa. The estimations are carried out in a panel of 21 selected sub-Saharan African countries over the period 2000 to 2010 using the Two-Stage Least Square (TSLS) estimation techniques. The infrastructure variable is constructed based on the three physical infrastructure stock (roads, telecommunication and electricity) using the Principal Component Analysis (PCA). Whilst governance is measured using the worldwide governance indicators. The results conform to the findings of existing literature that real output, government capital expenditure, external balance, and inflation are significant determinants of physical infrastructure in sub-Saharan Africa. The distinctive feature of the study is the significant role played by governance in explaining physical infrastructure. The results from the panel estimations revealed that investment in governance/institutional structures is a necessary first step in providing quality infrastructure stock and hence, a pro-poor long-term economic growth for the region. Therefore, in modelling physical infrastructure, it is imperative to incorporate the important role played by governance.
2013-01-01T00:00:00Z