We develop geometric optimisation on the manifold of Hermitian positive definite (HPD) matrices. In particular, we consider optimising two types of cost functions: (i) geodesically convex (g-convex); and (ii) log-nonexpansive (LN). G-convex functions are nonconvex in the usual euclidean sense, but convex along the manifold and thus allow global optimisation. LN functions may fail to be even g-convex, but still remain globally optimisable due to their special structure. We develop theoretical tools to recognise and generate g-convex functions as well as cone theoretic fixed-point optimisation algorithms. We illustrate our techniques by applying them to maximum-likelihood parameter estimation for elliptically contoured distributions (a rich class that substantially generalises the multivariate normal distribution). We compare our fixed-point algorithms with sophisticated manifold optimisation methods and obtain notable speedups. 1 To our knowledge the name "geometric optimisation" has not been previously attached to g-convex and cone theoretic HPD matrix optimisation, though several scattered examples do exist. Our theorems offer a formal starting point for recognising HPD geometric optimisation problems. arXiv:1312.1039v3 [math.FA] 12 Dec 2014 2 programming has enjoyed great success across a spectrum of applications-see e.g., the survey of Boyd et al. [11]; we hope this paper helps conic geometric optimisation gain wider exposure.Perhaps the best known conic geometric optimisation problem is computation of the Karcher (Fréchet) mean of a set of HPD matrices, a topic that has attracted great attention within matrix theory [7,8,25,48], computer vision [16], radar imaging [41, Part II], medical imaging [17, 52]-we refer the reader to the recent book [41] for additional applications and references. Another basic geometric optimisation problem arises as a subroutine in image search and matrix clustering [18].Conic geometric optimisation problems also occur in several other areas: statistics (covariance shrinkage) [15], nonlinear matrix equations [31], Markov decision processes and more broadly in the fascinating areas of nonlinear Perron-Frobenius theory [32].As a concrete illustration of our ideas, we discuss the task of maximum likelihood estimate (mle) for elliptically contoured distributions (ECDs) [13,21,37]-see §5. We use ECDs to illustrate our theory, not only because of their instructive value but also because of their importance in a variety of applications [42]. OutlineThe main focus of this paper is on recognising and constructing certain structured nonconvex functions of HPD matrices. In particular, Section 2 studies the class of geodesically convex functions, while Section 4 introduces "log-nonexpansive" functions. We present a limited-memory BFGS algorithm in Section 3, where we also present a derivation for the parallel transport, which, we could not find elsewhere in the literature. Even though manifold optimisation algorithms apply to both classes of functions, for log-nonexpansive functions we advance...

Arsenic trioxide is effective as first-line treatment for APL. Results of arsenic trioxide combination therapy with chemotherapy/ATRA requires further study.

The aim of this study was to evaluate genotoxicity and oxidative stress in workers who formulate organophosphorus (OP) pesticides. In this survey, blood leukocytes and erythrocytes of a group of 21 pesticide formulating workers and an equal number of control subjects were examined for genotoxicity and oxidative stress parameters. The mean comet tail length and mean comet length were used to measure DNA damage. Lipid peroxidation level, catalase, superoxide dismutase (SOD) and glutathione peroxidase activities in erythrocytes were analysed as biomarkers of oxidative stress. In addition, the acetylcholinesterase activity was measured as a biomarker of toxicity. The average duration of employment of workers in the factory was 97 months. Results indicated that chronic exposure (multiple5dose, greater than or equal to 6 months duration) to OP pesticides was associated with increased activities of catalase, SOD and glutathione peroxidase in erythrocytes. The level of lipid peroxidation and acetylcholinesterase activity did not show any significant differences between the two groups. The results also indicated that chronic exposure to OP pesticides was associated with increased DNA damage. It is concluded that human chronic exposure to OP pesticides may result in stimulated antioxidant enzymes and increased DNA damage in the absence of depressed acetylcholinesterase levels. Routine genotoxicity monitoring concomitant to acetylcholinesterase activity in workers occupationally exposed to OP insecticides is suggested.

We present the results of the Gravitational LEnsing Accuracy Testing 2008 (GREAT08) Challenge, a blind analysis challenge to infer weak gravitational lensing shear distortions from images. The primary goal was to stimulate new ideas by presenting the problem to researchers outside the shear measurement community. Six GREAT08 Team methods were presented at the launch of the Challenge and five additional groups submitted results during the 6-month competition. Participants analyzed 30 million simulated galaxies with a range in signal-to-noise ratio, point spread function ellipticity, galaxy size and galaxy type. The large quantity of simulations allowed shear measurement methods to be assessed at a level of accuracy suitable for currently planned future cosmic shear observations for the first time. Different methods perform well in different parts of simulation parameter space and come close to the target level of accuracy in several of these. A number of fresh ideas have emerged Results of the GREAT08 Challenge 2045 as a result of the Challenge including a re-examination of the process of combining information from different galaxies, which reduces the dependence on realistic galaxy modelling. The image simulations will become increasingly sophisticated in future GREAT Challenges, meanwhile the GREAT08 simulations remain as a benchmark for additional developments in shear measurement algorithms.A clump of matter induces a curvature in space-time which causes the trajectory of a light ray to appear bent. This effect, known as gravitational lensing, is analogous to light passing through a sheet of glass of varying thickness such as a bathroom window. In both cases, the light-emitting objects appear distorted. Making assumptions about the intrinsic (original) shapes of the emitting objects allows us to infer information about the intervening material. In cosmology, we learn about the distribution of matter by studying the shapes of distant galaxies. In the vast majority of cases, the distortion varies very little as a function of position on the galaxy image, and it can be approximated by a matrix distortion. This regime is known as weak gravitational lensing or cosmic shear when applied to large numbers of randomly selected distant galaxies.Gravitational attraction of ordinary matter and dark matter is expected to slow the expansion of the Universe, causing the expansion to decelerate. However, multiple lines of evidence now show that the present-day expansion of the Universe seems instead to be accelerating. The main explanations explored in the literature are that (i) Einstein's cosmological constant is non-zero, (ii) the vacuum energy is small but non-negligible, (iii) the Universe is filled with some new fluid, dubbed dark energy or (iv) the laws of general relativity are wrong at large distances. Possibilities (i) and (ii) can be subsumed within item (iii) because they look like a dark energy fluid with equation of state p = wρc 2 , where w = −1. To find out more about the nature of dark energy or modifications...