Average prime-pair counting formula

Taking $r>0$, let $\pi_{2r}(x)$ denote the number of prime pairs $(p, p+2r)$ with $p\le x$. The prime-pair conjecture of Hardy and Littlewood (1923) asserts that $\pi_{2r}(x)\sim 2C_{2r} {\rm li}_2(x)$ with an explicit constant $C_{2r}>0$. There seems to be no good conjecture for the remainders $\omega_{2r}(x)=\pi_{2r}(x)- 2C_{2r} {\rm li}_2(x)$ that corresponds to Riemann's formula for $\pi(x)-{\rm li}(x)$. However, there is a heuristic approximate formula for averages of the remainders $\omega_{2r}(x)$ which is supported by numerical results.Comment: 26 pages, 6 figure

The hexagonal versus the square lattice

We establish Schmutz Schaller's conjecture that the hexagonal lattice is `better' than the square lattice. Schmutz Schaller (Bulletin of the AMS 35 (1998), p. 201), motivated by considerations from hyperbolic geometry, conjectured that in dimensions 2 to 8 the best known lattice sphere packings have `maximal lengths' and goes on to write: "In dimension 2 the conjecture means in particular that the hexagonal lattice is `better' than the square lattice. More precisely, let 0<h_1<h_2<... be the positive integers, listed in ascending order, which can be written as h_i=x^2+3y^2 for integers x and y. Let 0<q_1<q_2<... be the positive integers, listed in ascending order, which can be written as q_i=x^2+y^2 for integers x and y. Then the conjecture is that q_i<=h_i for i=1,2,3,..." Our proof requires computational prime number theory in combination with methods from a preprint of the first author (to appear in Math. Comp.), arXiv:math.NT/0112100.Comment: 24 pages, 6 figures, 2 table

Direct patterning of oxides by pulsed laser stencil deposition

This thesis describes a detailed study of the application of stencil technology in the patterning of epitaxial oxide thin films by pulsed laser deposition (PLD). Stencil patterning has been applied in thin film sub-micron patterning of metals successfully for decades since it has several advantages over lithography techniques. It is a single processing step technique which can be applied to many different types of surfaces. The stencil patterning process does not utilize any solvents which makes it a favored technique for patterning metals on fragile and/or organic materials. However, for successful stencil patterning and unlimited (re)use of stencils, several issues need to be solved. The main issues that limit the re-usability of stencils are clogging of the apertures and deformation of the stencil caused by stress induced by the deposited material

The relationship between chest X-ray findings, bacterial load and treatment-related outcomes in persons with extensively drug resistant TB

Includes bibliographical references.Multi-drug resistant tuberculosis (MDR-TB) was first recognized in the 1990s with an increase in caseload of eighty-two percent between 2000 and 2007. Extensively drug-resistant tuberculosis (XDR-TB), a more difficult and more expensive form of TB to treat with poorer outcomes, emerged in South Africa in 2006. The prevalence of XDR-TB is likely to be underestimated in South Africa as a result of incomplete detection and notification. In peri-urban areas like Khayelitsha where there are high rates of HIV, TB and poverty, the prevalence of MDR-TB is estimated at 51/100 000. A significant proportion of these cases are indeed undetected pre-XDR-TB (MDR and resistance to either a fluoroquinolone or a 2nd line injectable drug) and XDR (MDR and resistance to both fluoroquinolones and any one of the 2nd line injectable drugs) cases with inadequate access to drug sensitivity testing. Treatment outcomes of XDR-TB have been variable with countries like Peru showing a 60% overall cure (or completed treatment) rate, and studies in KwaZulu Natal in South Africa showing much poorer outcomes. The reasons for the poor outcomes in XDR-TB remain unclear. We are continuing to investigate the role of strain-type and several other factors including nutritional status, degree of drug resistance, HIV status and drug regimens in determining outcomes. There is a paucity of literature describing the chest X-ray (CXR) findings in patients with XDR-TB, and whether disease extent is related to treatment outcomes and the evolution of resistance remains unclear. It has been shown that patients with radiological extensive drug-sensitive TB have higher initial sputum mycobacterial loads and take a longer time to sputum conversion than those without . The extent of disease on the CXR at baseline has been used as a tool to inform and predict the need for infection control measures, treatment duration, and outcomes. The time-to-positivity (TTP) of mycobacterium tuberculosis in a liquid medium culture has become a validated indicator of bacterial sputum load and indeed a surrogate bio-marker of treatment response to anti-tuberculosis drugs. The relationships between mycobacterial sputum load, radiological disease and treatment outcomes have been studied in drug-sensitive TB, but little is known about XDR-TB