Diagonal and Low-Rank Matrix Decompositions, Correlation Matrices, and Ellipsoid Fitting

In this paper we establish links between, and new results for, three problems that are not usually considered together. The first is a matrix decomposition problem that arises in areas such as statistical modeling and signal processing: given a matrix $X$ formed as the sum of an unknown diagonal matrix and an unknown low rank positive semidefinite matrix, decompose $X$ into these constituents. The second problem we consider is to determine the facial structure of the set of correlation matrices, a convex set also known as the elliptope. This convex body, and particularly its facial structure, plays a role in applications from combinatorial optimization to mathematical finance. The third problem is a basic geometric question: given points $v_1,v_2,...,v_n\in \R^k$ (where $n > k$) determine whether there is a centered ellipsoid passing \emph{exactly} through all of the points. We show that in a precise sense these three problems are equivalent. Furthermore we establish a simple sufficient condition on a subspace $U$ that ensures any positive semidefinite matrix $L$ with column space $U$ can be recovered from $D+L$ for any diagonal matrix $D$ using a convex optimization-based heuristic known as minimum trace factor analysis. This result leads to a new understanding of the structure of rank-deficient correlation matrices and a simple condition on a set of points that ensures there is a centered ellipsoid passing through them.Comment: 20 page

HIV/AIDS at a South African University: Investigating the role of Walter Sisulu University's prevention role players and student behaviour at the Institute for Advanced Tooling

This thesis investigated perceptions of HIV/AIDS at the Walter Sisulu University (WSU), situated in the Eastern Cape, South Africa. The study focused on understanding opinions related to HIV/AIDS using data derived from interviews with twenty HIV/AIDS key role players from across the entire institution, and twenty students at the Institute for Advanced Tooling (IAT), a postgraduate section of the Mechanical Engineering Department in the Faculty of Science, Engineering and Technology (FSET) in Chiselhurst, East London. A key concern of the study was to examine the way in which local cultural beliefs and practices may shape understandings in relation to HIV/AIDS and to help inform more sensitive prevention campaigns in the future. The study, methodologically utilising Denzin’s concept of interpretative interactionism,and Giddens’ structuration theory, found that there was no single ‘cultural belief’ regarding health or related issues, but that cultural beliefs were always expressed in personal and contextual ways. The investigation into nutrition, health and general well-being and perceived causes of HIV/AIDS revealed that research informants subscribed to cultural beliefs for different reasons in personal constructs, and the study concluded that cultural issues surrounding these factors would need to be socially debated in intervention efforts. Perceptions of gender, as the most significant factor, were highly contested, with differing beliefs expressed regarding female sexual agency. It was further stated by research informants that the ABC approach lacked contextual consideration of environmental factors. The thesis therefore argues that for effective preventative action, there is a necessity, firstly, for a health-enabling environment to be created that includes strategies for alleviating nutritional deficiencies in a culturally contextual fashion. Second, based on the data, it was established that in relation to respondents’ orientation towards cultural beliefs, social HIV/AIDS debate programmes should be introduced in conjunction with health officials, the community and traditional healers in a peer-oriented approach. It was thirdly established that this approach should include addressing contextual factors from a ‘lived experience’ perspective, and that social positioning by the WSU should occur towards certain social issues (for example, constructions of gender) identified within this study

Sensory testing in leprosy:Comparison of ballpoint pen and monofilaments

The 10 g monofilament has been replaced by the ballpoint pen in routine sensory testing of nerves in leprosy control in Ethiopia. Results of sensory testing between the ballpoint pen and different monofilaments on hands and feet were compared. Ballpoint pen underdiagnosis of loss of sensation was defined to occur when the pen was felt and the monofilament was not. Differences were evaluated both for individual test points (test point level) and for the test points of extremities collectively (extremity level). An extremity (either a hand or a foot) was defined as having sensory nerve function impairment (SNFI) if a supplying nerve had SNFI, which was the case when sensation was absent in two or more test points in the area supplied by that nerve. At test point level, the percentages with ballpoint pen underdiagnosis relative to the 2, 10, 20 and 50 g monofilaments were 40, 21, 9 and 7%, respectively, in the hands, and 47, 30, 15 and 7% in the feet. Ballpoint pen underdiagnosis percentages of SNFI at extremity level were 32, 18, 8 and 9% in the hands, and 37, 26, 14 and 6% in the feet. The risk of ballpoint pen underdiagnosis appears to be higher in extremities without visible damage. In conclusion, substantial levels of underdiagnosis of sensory loss with the ballpoint pen were observed. However, the consequences for the prognosis of treatment with corticosteroids in patients with the more subtle sensation loss noted here need to be established. Development and testing of guidelines is a prerequisite for the use of the ballpoint pen

Performance Prediction of Helical-Type Seawater MHD Power Generator for Enlargement

Immunogenetics and cellular immunology of bacterial infectious disease

Polymorphisms in Toll-like receptor 4 ( TLR4 ) are associated with protection against leprosy

Accumulating evidence suggests that polymorphisms in Toll-like receptors (TLRs) influence the pathogenesis of mycobacterial infections, including leprosy, a disease whose manifestations depend on host immune responses. Polymorphisms in TLR2 are associated with an increased risk of reversal reaction, but not susceptibility to leprosy itself. We examined whether polymorphisms in TLR4 are associated with susceptibility to leprosy in a cohort of 441 Ethiopian leprosy patients and 197 healthy controls. We found that two single nucleotide polymorphisms (SNPs) in TLR4 (896G>A [D299G] and 1196C>T [T399I]) were associated with a protective effect against the disease. The 896GG, GA and AA genotypes were found in 91.7, 7.8 and 0.5% of leprosy cases versus 79.9, 19.1 and 1.0% of controls, respectively (odds ratio [OR] = 0.34, 95% confidence interval [CI] 0.20-0.57, P < 0.001, additive model). Similarly, the 1196CC, CT and TT genotypes were found in 98.1, 1.9 and 0% of leprosy cases versus 91.8, 7.7 and 0.5% of controls, respectively (OR = 0.16, 95% CI 0.06--.40, P < 0.001, dominant model). We found that Mycobacterium leprae stimulation of monocytes partially inhibited their subsequent response to lipopolysaccharide (LPS) stimulation. Our data suggest that TLR4 polymorphisms are associated with susceptibility to leprosy and that this effect may be mediated at the cellular level by the modulation of TLR4 signalling by M. lepra