Manifolds of Differentiable Densities

We develop a family of infinite-dimensional (non-parametric) manifolds of probability measures. The latter are defined on underlying Banach spaces, and have densities of class $C_b^k$ with respect to appropriate reference measures. The case $k=\infty$, in which the manifolds are modelled on Fréchet spaces, is included. The manifolds admit the Fisher-Rao metric and, unusually for the non-parametric setting, Amari's $\alpha$-covariant derivatives for all $\alpha\in\R$. By construction, they are $C^\infty$-embedded submanifolds of particular manifolds of finite measures. The statistical manifolds are dually ($\alpha=\pm 1$) flat, and admit mixture and exponential representations as charts. Their curvatures with respect to the $\alpha$-covariant derivatives are derived. The likelihood function associated with a finite sample is a continuous function on each of the manifolds, and the $\alpha$-divergences are of class $C^\infty$

Regression-based variance reduction approach for strong approximation schemes

In this paper we present a novel approach towards variance reduction for discretised diffusion processes. The proposed approach involves specially constructed control variates and allows for a significant reduction in the variance for the terminal functionals. In this way the complexity order of the standard Monte Carlo algorithm ($\varepsilon^{-3}$) can be reduced down to $\varepsilon^{-2}\sqrt{\left|\log(\varepsilon)\right|}$ in case of the Euler scheme with $\varepsilon$ being the precision to be achieved. These theoretical results are illustrated by several numerical examples.Comment: arXiv admin note: text overlap with arXiv:1510.0314

A Class of Non-Parametric Statistical Manifolds modelled on Sobolev Space

We construct a family of non-parametric (infinite-dimensional) manifolds of finite measures on Rd. The manifolds are modelled on a variety of weighted Sobolev spaces, including Hilbert-Sobolev spaces and mixed-norm spaces. Each supports the Fisher-Rao metric as a weak Riemannian metric. Densities are expressed in terms of a deformed exponential function having linear growth. Unusually for the Sobolev context, and as a consequence of its linear growth, this "lifts" to a nonlinear superposition (Nemytskii) operator that acts continuously on a particular class of mixed-norm model spaces, and on the fixed norm space W²'¹ i.e. it maps each of these spaces continuously into itself. It also maps continuously between other fixed-norm spaces with a loss of Lebesgue exponent that increases with the number of derivatives. Some of the results make essential use of a log-Sobolev embedding theorem. Each manifold contains a smoothly embedded submanifold of probability measures. Applications to the stochastic partial differential equations of nonlinear filtering (and hence to the Fokker-Planck equation) are outlined

Passive water control at the surface of a superhydrophobic lichen

Some lichens have a super-hydrophobic upper surface, which repels water drops, keeping the surface dry but probably preventing water uptake. Spore ejection requires water and is most efficient just after rainfall. This study was carried out to investigate how super-hydrophobic lichens manage water uptake and repellence at their fruiting bodies, or podetia. Drops of water were placed onto separate podetia of Cladonia chlorophaea and observed using optical microscopy and cryo-scanning-electron microscopy (cryo-SEM) techniques to determine the structure of podetia and to visualise their interaction with water droplets. SEM and optical microscopy studies revealed that the surface of the podetia was constructed in a three-level structural hierarchy. By cryo-SEM of water-glycerol droplets placed on the upper part of the podetium, pinning of the droplet to specific, hydrophilic spots (pycnidia/apothecia) was observed. The results suggest a mechanism for water uptake, which is highly sophisticated, using surface wettability to generate a passive response to different types of precipitation in a manner similar to the Namib Desert beetle. This mechanism is likely to be found in other organisms as it offers passive but selective water control

Prognostic indicators in adults hospitalized with falciparum malaria in Western Thailand.

Background: Severe malaria remains a major cause of death and morbidity amongst adults in the Asiatic tropics. Methods: A retrospective analysis of the clinical and laboratory data of 988 adult patients, hospitalized with Plasmodium falciparum malaria and prospectively recruited to malaria studies in western Thailand between 1986 and 2002, was performed to assess the factors associated with a fatal outcome. Different severity scores and classifications for defining severe malaria were compared and, using multiple logistic regression, simple models for predicting mortality developed. Results: The proportion of patients fulfilling the WHO 2000 definition of severe malaria was 78.1%, and their mortality was 10%. Mortality in patients given parenteral artesunate or artemether (16/317, 5%) was lower than in those given parenteral quinine (59/442, 13%) (P < 0.001). Models using parameter sets based on WHO 1990, 2000 and Adapted AQ criteria plus blood smear parasite-stage assessment gave the best mortality prediction. A malaria prognostic index (MPI), derived from the dataset using five clinical or laboratory variables gave similar prognostic accuracy. Conclusions: The mortality of severe malaria in adults has fallen and the switch from quinine to artesunate has probably been an important contributor. Prognostic indices based on WHO 2000 definitions, and other simpler indices based on fewer variables, provide clinically useful predictions of outcome in Asian adults with severe malaria

Solar-type dynamo behaviour in fully convective stars without a tachocline

In solar-type stars (with radiative cores and convective envelopes), the magnetic field powers star spots, flares and other solar phenomena, as well as chromospheric and coronal emission at ultraviolet to X-ray wavelengths. The dynamo responsible for generating the field depends on the shearing of internal magnetic fields by differential rotation. The shearing has long been thought to take place in a boundary layer known as the tachocline between the radiative core and the convective envelope. Fully convective stars do not have a tachocline and their dynamo mechanism is expected to be very different, although its exact form and physical dependencies are not known. Here we report observations of four fully convective stars whose X-ray emission correlates with their rotation periods in the same way as in Sun-like stars. As the X-ray activity - rotation relationship is a well-established proxy for the behaviour of the magnetic dynamo, these results imply that fully convective stars also operate a solar-type dynamo. The lack of a tachocline in fully convective stars therefore suggests that this is not a critical ingredient in the solar dynamo and supports models in which the dynamo originates throughout the convection zone.Comment: 6 pages, 1 figure. Accepted for publication in Nature (28 July 2016). Author's version, including Method