http://hdl.handle.net/10500/3016 Sun, 19 May 2013 09:00:33 GMT 2013-05-19T09:00:33Z http://hdl.handle.net/10500/8603 Bydraes tot die oplossing van die veralgemeende knapsakprobleem Venter, Geertien In this thesis contributions to the solution of the generalised knapsack problem are given and discussed. Attention is given to problems with functions that are calculable but not necessarily in a closed form. Algorithms and test problems can be used for problems with closed-form functions as well. The focus is on the development of good heuristics and not on exact algorithms. Heuristics must be investigated and good test problems must be designed. A measure of convexity for convex functions is developed and adapted for concave functions. A test problem generator makes use of this measure of convexity to create challenging test problems for the concave, convex and mixed knapsack problems. Four easy-to-interpret characteristics of an S-function are used to create test problems for the S-shaped as well as the generalised knapsack problem. The in uence of the size of the problem and the funding ratio on the speed and the accuracy of the algorithms are investigated. When applicable, the in uence of the interval length ratio and the ratio of concave functions to the total number of functions is also investigated. The Karush-Kuhn-Tucker conditions play an important role in the development of the algorithms. Suf- cient conditions for optimality for the convex knapsack problem with xed interval lengths is given and proved. For the general convex knapsack problem, the key theorem, which contains the stronger necessary conditions, is given and proved. This proof is so powerful that it can be used to proof the adapted key theorems for the mixed, S-shaped and the generalised knapsack problems as well. The exact search-lambda algorithm is developed for the concave knapsack problem with functions that are not in a closed form. This algorithm is used in the algorithms to solve the mixed and S-shaped knapsack problems. The exact one-step algorithm is developed for the convex knapsack problem with xed interval length. This algorithm is O(n). The general convex knapsack problem is solved by using the pivot algorithm which is O(n2). Optimality cannot be proven but in all cases the optimal solution was found and for all practical reasons this problem will be considered as being concluded. A good heuristic is developed for the mixed knapsack problem. Further research can be done on this heuristic as well as on the S-shaped and generalised knapsack problems. Text in Afikaans Wed, 06 Feb 2013 00:00:00 GMT http://hdl.handle.net/10500/8603 2013-02-06T00:00:00Z http://hdl.handle.net/10500/7046 Radio astronomy techniques : the use of radio instruments from single dish radio telescopes to radio interferometers De Witt, Aletha New radio telescopes under development, will significantly enhance the capabilities of radio astronomy in the Southern Hemisphere. South Africa, in particular, is actively involved in the development of a new array (MeerKAT) as well as in the expansion of existing very long baseline interferometer arrays in the south. Participation in these new developments demands a thorough understanding of radio astronomy techniques, and data analysis, and this thesis focusses on two projects with the aim of gaining such experience. The Southern Hemisphere very long baselines array is not well served with calibrator sources and there are significant gaps in the present calibrator distribution on the sky. An adequately dense, well distributed, set of strong, compact calibrator or reference sources is needed. With this in mind, observations using the Southern Hemisphere long baseline array were conducted to investigate a sample of candidate calibrator sources. The compactness of the sources was investigated and new potential calibrators have been identified. Single antenna radio spectroscopy of OH masers has identified sources of 1720 MHz emission associated with supernova remnants at the shock interface between the expanding supernova remnant and a molecular cloud. Models indicate that these masers are shock excited and can only be produced under tight physical constraints. Out ows from newly-formed stars create nebulous regions known as Herbig-Haro objects when they interact with the surrounding medium, and these regions are potentially similar to those seen in supernova remnants. If conditions behind the shock fronts of Herbig-Haro objects are able to support 1720-MHz OH masers they could be a useful diagnostic tool for star formation. A survey toward Herbig-Haro objects using a single-dish radio telescope did detect 1720-MHz OH lines in emission, but neither their spectral signature nor follow-up observations with the Very Large Array showed evidence of maser emission. Thu, 01 Mar 2012 00:00:00 GMT http://hdl.handle.net/10500/7046 2012-03-01T00:00:00Z http://hdl.handle.net/10500/5810 ARIMA forecasts of the number of beneficiaries of social security grants in South Africa Luruli, Fululedzani Lucy The main objective of the thesis was to investigate the feasibility of accurately and precisely fore- casting the number of both national and provincial bene ciaries of social security grants in South Africa, using simple autoregressive integrated moving average (ARIMA) models. The series of the monthly number of bene ciaries of the old age, child support, foster care and disability grants from April 2004 to March 2010 were used to achieve the objectives of the thesis. The conclusions from analysing the series were that: (1) ARIMA models for forecasting are province and grant-type spe- ci c; (2) for some grants, national forecasts obtained by aggregating provincial ARIMA forecasts are more accurate and precise than those obtained by ARIMA modelling national series; and (3) for some grants, forecasts obtained by modelling the latest half of the series were more accurate and precise than those obtained from modelling the full series. Thu, 01 Dec 2011 00:00:00 GMT http://hdl.handle.net/10500/5810 2011-12-01T00:00:00Z http://hdl.handle.net/10500/4905 Koliha–Drazin invertibles form a regularity Smit, Joukje Anneke The axiomatic theory of ` Zelazko defines a variety of general spectra where specified axioms are satisfied. However, there arise a number of spectra, usually defined for a single element of a Banach algebra, that are not covered by the axiomatic theory of ` Zelazko. V. Kordula and V. M¨uller addressed this issue and created the theory of regularities. Their unique idea was to describe the underlying set of elements on which the spectrum is defined. The axioms of a regularity provide important consequences. We prove that the set of Koliha-Drazin invertible elements, which includes the Drazin invertible elements, forms a regularity. The properties of the spectrum corresponding to a regularity are also investigated. Mon, 01 Nov 2010 00:00:00 GMT http://hdl.handle.net/10500/4905 2010-11-01T00:00:00Z