The time to extinction for an SIS-household-epidemic model

We analyse a stochastic SIS epidemic amongst a finite population partitioned into households. Since the population is finite, the epidemic will eventually go extinct, i.e., have no more infectives in the population. We study the effects of population size and within household transmission upon the time to extinction. This is done through two approximations. The first approximation is suitable for all levels of within household transmission and is based upon an Ornstein-Uhlenbeck process approximation for the diseases fluctuations about an endemic level relying on a large population. The second approximation is suitable for high levels of within household transmission and approximates the number of infectious households by a simple homogeneously mixing SIS model with the households replaced by individuals. The analysis, supported by a simulation study, shows that the mean time to extinction is minimized by moderate levels of within household transmission

Strict inequalities of critical values in continuum percolation

We consider the supercritical finite-range random connection model where the points $x,y$ of a homogeneous planar Poisson process are connected with probability $f(|y-x|)$ for a given $f$. Performing percolation on the resulting graph, we show that the critical probabilities for site and bond percolation satisfy the strict inequality $p_c^{\rm site} > p_c^{\rm bond}$. We also show that reducing the connection function $f$ strictly increases the critical Poisson intensity. Finally, we deduce that performing a spreading transformation on $f$ (thereby allowing connections over greater distances but with lower probabilities, leaving average degrees unchanged) {\em strictly} reduces the critical Poisson intensity. This is of practical relevance, indicating that in many real networks it is in principle possible to exploit the presence of spread-out, long range connections, to achieve connectivity at a strictly lower density value.Comment: 38 pages, 8 figure

Phase Transitions on Nonamenable Graphs

We survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph. Most attention is devoted to transitive graphs and trees

Numerical Study of a Field Theory for Directed Percolation

A numerical method is devised for study of stochastic partial differential equations describing directed percolation, the contact process, and other models with a continuous transition to an absorbing state. Owing to the heightened sensitivity to fluctuationsattending multiplicative noise in the vicinity of an absorbing state, a useful method requires discretization of the field variable as well as of space and time. When applied to the field theory for directed percolation in 1+1 dimensions, the method yields critical exponents which compare well against accepted values.Comment: 18 pages, LaTeX, 6 figures available upon request LC-CM-94-00

Andragogy : a theoretical overview on learning theories that impact on benchmarking blended learning at the Central Univerity of Technology, Free State

Published ArticleBlended learning has shown enormous growth worldwide during the last decade. Blended learning initiatives expanded rapidly, as technological improvement and a greater demand from learners arose as a result of the need for life-long learning and changing demands in the world of work. Many higher education institutions use blended learning as an alternative for or additional to conventional face-to-face education. The changing needs of learners, and especially adults, forced institutions to upgrade their traditional courses and increasingly make use of technologically enhanced courses. The research problem encompasses the fact that the growth of blended learning poses many problems to higher education institutions, as the delivery by means of technology is complex. Many higher learning institutions were not prepared adequately to deliver education by using blended learning and did not have the required systems in place. The greatest concern was that staff was not familiar with the technology or its use and that the particular demands of staff arrangements to engage in blended learning programs were not taken into account. It appears that institutions did not plan or have systems and guidelines in place to implement programs at a distance using blended learning technology. Another concern was that most of the programs held traditional approaches and has not been adapted to suit the particular characteristics and needs of distance education by taking the planning, development and review of such programs into consideration. As part of ongoing research at the Central University of Technology, Free State appropriate research had to be undertaken. Consequently the following research question emanated from the research problem: Within the context of higher education in South Africa, what framework underpinned by andragogy, national education policies and blended learning theory, will benchmark and support the implementation of blended learning at the Central University of Technology, Free State

Andragogy : a theoretical overview on learning theories that impact on benchmarking blended learning at the Central Univerity of Technology, Free State

Published ArticleBlended learning has shown enormous growth worldwide during the last decade. Blended learning initiatives expanded rapidly, as technological improvement and a greater demand from learners arose as a result of the need for life-long learning and changing demands in the world of work. Many higher education institutions use blended learning as an alternative for or additional to conventional face-to-face education. The changing needs of learners, and especially adults, forced institutions to upgrade their traditional courses and increasingly make use of technologically enhanced courses. The research problem encompasses the fact that the growth of blended learning poses many problems to higher education institutions, as the delivery by means of technology is complex. Many higher learning institutions were not prepared adequately to deliver education by using blended learning and did not have the required systems in place. The greatest concern was that staff was not familiar with the technology or its use and that the particular demands of staff arrangements to engage in blended learning programs were not taken into account. It appears that institutions did not plan or have systems and guidelines in place to implement programs at a distance using blended learning technology. Another concern was that most of the programs held traditional approaches and has not been adapted to suit the particular characteristics and needs of distance education by taking the planning, development and review of such programs into consideration. As part of ongoing research at the Central University of Technology, Free State appropriate research had to be undertaken. Consequently the following research question emanated from the research problem: Within the context of higher education in South Africa, what framework underpinned by andragogy, national education policies and blended learning theory, will benchmark and support the implementation of blended learning at the Central University of Technology, Free State

Self-avoiding walks and connective constants

The connective constant $\mu(G)$ of a quasi-transitive graph $G$ is the asymptotic growth rate of the number of self-avoiding walks (SAWs) on $G$ from a given starting vertex. We survey several aspects of the relationship between the connective constant and the underlying graph $G$. $\bullet$ We present upper and lower bounds for $\mu$ in terms of the vertex-degree and girth of a transitive graph. $\bullet$ We discuss the question of whether $\mu\ge\phi$ for transitive cubic graphs (where $\phi$ denotes the golden mean), and we introduce the Fisher transformation for SAWs (that is, the replacement of vertices by triangles). $\bullet$ We present strict inequalities for the connective constants $\mu(G)$ of transitive graphs $G$, as $G$ varies. $\bullet$ As a consequence of the last, the connective constant of a Cayley graph of a finitely generated group decreases strictly when a new relator is added, and increases strictly when a non-trivial group element is declared to be a further generator. $\bullet$ We describe so-called graph height functions within an account of "bridges" for quasi-transitive graphs, and indicate that the bridge constant equals the connective constant when the graph has a unimodular graph height function. $\bullet$ A partial answer is given to the question of the locality of connective constants, based around the existence of unimodular graph height functions. $\bullet$ Examples are presented of Cayley graphs of finitely presented groups that possess graph height functions (that are, in addition, harmonic and unimodular), and that do not. $\bullet$ The review closes with a brief account of the "speed" of SAW.Comment: Accepted version. arXiv admin note: substantial text overlap with arXiv:1304.721

Sonic injection into a PGM Peirce-Smith converter: CFD modelling and industrial trials

Peirce-Smith converters (PSCs) are extensively used in the copper, nickel, and platinum group metals industries. The typical converting operation involves lateral purging of air into molten matte through a bank of tuyeres. This blowing operation occurs at low pressure from the blowers, resulting in a bubbling regime that is considered inefficient from both a process and an energy utilization perspective. Inherent drawbacks also include recurrent tuyere blockage, tuyere punching, and low oxygen efficiency. Western Platinum embarked on a full-scale industrial evaluation of generating a jetting regime by using sonic injection. Prior to industrial- scale tests, a numerical assessment to ascertain the feasibility of implementing sonic injection into a PSC was conducted. The work included flow characterization at high-pressure injection achieving sonic velocity at the tuyere exit. The 2D and 3D simulations of the three-phase system were carried out using the volume of fluid method together with the RKE turbulence model to account for the multiphase and turbulent nature of the flow. This paper discusses the key findings in understanding plume extension, velocity distribution, shear wall stress analysis, and phase distribution characteristics in the system. Plant trials are also discussed with reference to the commercial aspects of a full-scale implementation of sonic injection in the smelter

Validation of training priorities of engineers in the platinum mining industry in South Africa

Despite a significant need, there is very little South African research on the validation of the generic training priorities of engineers within the platinum mining industry. The purpose of this study was to validate the generic training priorities of engineers in the platinum mining industry in three provinces in South Africa. For this study, a quantitative research design was used, a four-point Likert-type questionnaire was designed, and a purposive sample was used. The results of this study indicated that theoretical knowledge, technical skills, management skills, and engineering principles are generic training priorities