A note on the stability number of an orthogonality graph

We consider the orthogonality graph Omega(n) with 2^n vertices corresponding to the 0-1 n-vectors, two vertices adjacent if and only if the Hamming distance between them is n/2. We show that the stability number of Omega(16) is alpha(Omega(16))= 2304, thus proving a conjecture by Galliard. The main tool we employ is a recent semidefinite programming relaxation for minimal distance binary codes due to Schrijver. As well, we give a general condition for Delsarte bound on the (co)cliques in graphs of relations of association schemes to coincide with the ratio bound, and use it to show that for Omega(n) the latter two bounds are equal to 2^n/n.Comment: 10 pages, LaTeX, 1 figure, companion Matlab code. Misc. misprints fixed and references update

Seasonal variations of water and sediment quality parameters in endorheic reed pans on the Mpumalanga Highveld

The Mpumalanga Lakes District consists of approximately 320 pans, of which less than 3% are classified as reed pans. There is limited information available on reed pans and as a result they are at risk of various anthropogenic activities, for example mining and agriculture. Four reed pans were selected and assessed to determine seasonal trends of a variety of water and sediment quality parameters. The study took place over one seasonal cycle from 2008–2009; samples were collected seasonally to account for various hydrological extremes. Water samples were collected and their nutrient and chlorophyll a concentrations were determined, while various other water quality parameters were measured in situ. Sediment samples were analysed for physical and chemical properties, namely, grain size and organic carbon content. The seasonal changes in concentrations of As, Cr, Cu, Fe, Pb, Mn, Mo, Ni, Se, Sr, U and Zn were also analysed within the surface water and sediment. Increased nutrient concentrations within the water were evident during spring and summer at some of the sites, which influenced other water quality variables, e.g., dissolved oxygen and pH. Seasonal trends in metal concentrations were influenced by the prevailing environmental conditions (e.g., rainfall) experienced at the selected sites as well as physical and chemical properties (e.g., grain size and organic carbon content). This study showed distinct seasonal variability of water and sediment quality parameters in endorheic reed pans on the Mpumalanga Highveld. There is a need for further studies on all of the different types of pans in terms of their water and sediment quality. This type of information will allow for a sound and defensible scientific basis for the assessment of likely impacts (e.g., eutrophication), the evaluation of the significance of these impacts, and the design of remedial and preventative measures

Patterns of alcohol use on a South African university campus: the findings of two annual drinking surveys

While alcohol continues to be abused on university campuses around the world, the precise situation on South African campuses is unknown. This paper attempts to address this gap by reporting the results of two annual surveys of alcohol consumption amongst students at Rhodes University, the smallest tertiary institution in South Africa, with an annual intake of approximately 6000 students. The results of a survey using the Alcohol Use Disorders Identification Test (AUDIT) suggests that alcohol consumption amongst these students is a public health concern, and likely to result in serious medical and social consequences. KEYWORDS: Alcohol use, binge drinking, South Africa, university students, AUDI

A linear programming reformulation of the standard quadratic optimization problem

The problem of minimizing a quadratic form over the standard simplex is known as the standard quadratic optimization problem (SQO). It is NP-hard, and contains the maximum stable set problem in graphs as a special case. In this note, we show that the SQO problem may be reformulated as an (exponentially sized) linear program (LP). This reformulation also suggests a hierarchy of polynomial-time solvable LP’s whose optimal values converge finitely to the optimal value of the SQO problem. The hierarchies of LP relaxations from the literature do not share this finite convergence property for SQO, and we review the relevant counterexamples.Accepted versio

Improved bounds for the crossing numbers of K_m,n and K_n

It has been long--conjectured that the crossing number cr(K_m,n) of the complete bipartite graph K_m,n equals the Zarankiewicz Number Z(m,n):= floor((m-1)/2) floor(m/2) floor((n-1)/2) floor(n/2). Another long--standing conjecture states that the crossing number cr(K_n) of the complete graph K_n equals Z(n):= floor(n/2) floor((n-1)/2) floor((n-2)/2) floor((n-3)/2)/4. In this paper we show the following improved bounds on the asymptotic ratios of these crossing numbers and their conjectured values: (i) for each fixed m >= 9, lim_{n->infty} cr(K_m,n)/Z(m,n) >= 0.83m/(m-1); (ii) lim_{n->infty} cr(K_n,n)/Z(n,n) >= 0.83; and (iii) lim_{n->infty} cr(K_n)/Z(n) >= 0.83. The previous best known lower bounds were 0.8m/(m-1), 0.8, and 0.8, respectively. These improved bounds are obtained as a consequence of the new bound cr(K_{7,n}) >= 2.1796n^2 - 4.5n. To obtain this improved lower bound for cr(K_{7,n}), we use some elementary topological facts on drawings of K_{2,7} to set up a quadratic program on 6! variables whose minimum p satisfies cr(K_{7,n}) >= (p/2)n^2 - 4.5n, and then use state--of--the--art quadratic optimization techniques combined with a bit of invariant theory of permutation groups to show that p >= 4.3593.Comment: LaTeX, 18 pages, 2 figure

Transcriptional Control of the Human Aggrecan Gene

Osteoarthritis (OA) is a degenerative joint disease and is the leading cause of physical disability in industrialised nations. Cartilage has received the most attention in the study of OA due to articular chondrocytes acting as potential instigators of disease. These cells are responsible for the anabolic-catabolic balance required for matrix maintenance due to their ability to synthesise the structural components of the extra-cellular matrix along with matrix-degrading proteases. In order to better understand the mechanism by which this balance is shifted in OA, it would be useful to investigate the roles of the genes that are expressed in these cells. One of the most powerful tools to do this would be an inducible and chondrocyte-specific system that utilises a cartilage-specific promoter, such as the aggrecan promoter. Aggrecan, one of the major structural components of cartilage, is a large aggregating chondroitin sulphate proteoglycan. This PhD set out to further our understanding of the transcriptional regulatory mechanisms that govern the chondrocyte-specific expression of the gene. A series of constructs containing various combinations of non-coding aggrecan DNA (mostly upstream of the transcriptional start site and including the first untranslated exon) were used to generate transgenic mouse embryos and an adult line in which the cartilage-specific expression of the transgene indicated the presence of cis-regulatory elements. These studies have identified three putatively important regions. One serves as a basal/core promoter at and around the transcriptional start site (TSS) (containing what could possibly be a non-essential 92bp sequence depending on the developmental stage of the mouse). Another located 10kb upstream of the TSS can serve as an enhancer, and finally another located within a 7kb region further upstream. One or more of these elements seem to respond to injury as indicated by increased promoter activity in the OA-induced knees of the adult line containing all three regulatory regions. Using a combination of our data alongside published work (Han and Lefebvre, 2008), a cartilage-specific Cre recombinase line was generated and tested for specificity and continual gene expression under the induction of OA. The work presented in this thesis can now be used to contribute to knowledge on the pathogenesis of OA

Some geochemical constraints upon models for the crystallization of the upper critical zone-main zone interval, northwestern Bushveld complex

Ratios between elements Mg, Fe, Co, Cr, Ni, V, and Sc are consistently different in mafic rocks of the upper critical zone, and those above the Bastard unit. Within the 300 m section above the Merensky Reef, 87Sr/86Sr ratios increase from c.0.7063 to c.0.7087, irrespective of rock type. Decoupling of Mg/(Mg + Fe2+) ratios and the Ca contents of plagioclase, and wide variations in the proportions of anorthosite within the Bastard, Merensky, and Merensky Footwall units, are inconsistent with anorthosite formation by simple fractional crystallization of magma batches of limited volume

Perceptions and knowledge about the acquired immunodeficiency syndrome among students in university residences

Using an anonymous questionnaire to obtain baseline data on sexual behaviour and knowledge of the acquired immunodeficiency syndrome (AIDS) among students in university residences, the following information was obtained: Knowledge of AIDS was found to be high, although misconceptions regarding transmission of the virus were prevalent. Mosquito bites (15%) and the donating of blood (31%) were incorrectly identified as methods of transmission of the virus. Deep kissing was recognised by only 27% of the students as a possible method of transmission while 22% and 13% of the students, respectively, failed to identify the sharing of razor blades and blood transfusions as possible means of contracting HIV infection. The students' knowledge had not greatly affected sexual behaviour. Two-thirds of the respondents had previously had sexual contact; 38% .were sexually active at the time of the study, and of these 74% had engaged in unprotected intercourse with casual partners in the previous 6 months. There was a negative attitude towards condom use and over 80% of sexually active students did not perceive themselves to be at risk of contracting AIDS. Campus Health (82%), State health services (51%), and public advertisements (60%) were preferred sources of AIDS information. Newspapers/ magazines (80%) and leaflets (69%) were identified as the preferred media. In reality, significantly fewer students obtained their information from Campus Health (15%; P < 0,05), State health services (19%; P < 0,05) and leaflets (44%; P < 0,05). It can be concluded that education programmes should be developed by credible organisations to ensure that an awareness of AIDS results in appropriate sexual behaviour