Distribution of Gaussian Process Arc Lengths

We present the first treatment of the arc length of the Gaussian Process (GP) with more than a single output dimension. GPs are commonly used for tasks such as trajectory modelling, where path length is a crucial quantity of interest. Previously, only paths in one dimension have been considered, with no theoretical consideration of higher dimensional problems. We fill the gap in the existing literature by deriving the moments of the arc length for a stationary GP with multiple output dimensions. A new method is used to derive the mean of a one-dimensional GP over a finite interval, by considering the distribution of the arc length integrand. This technique is used to derive an approximate distribution over the arc length of a vector valued GP in $\mathbb{R}^n$ by moment matching the distribution. Numerical simulations confirm our theoretical derivations.Comment: 10 pages, 4 figures, Accepted to The 20th International Conference on Artificial Intelligence and Statistics (AISTATS

Mean age gradient and asymmetry in the star formation history of the Small Magellanic Cloud

We derive the star formation history in four regions of the Small Magellanic Cloud (SMC) using the deepest VI color-magnitude diagrams (CMDs) ever obtained for this galaxy. The images were obtained with the Advanced Camera for Surveys onboard the Hubble Space Telescope and are located at projected distances of 0.5-2 degrees from the SMC center, probing the main body and the wing of the galaxy. We derived the star-formation histories (SFH) of the four fields using two independent procedures to fit synthetic CMDs to the data. We compare the SFHs derived here with our earlier results for the SMC bar to create a deep pencil-beam survey of the global history of the central SMC. We find in all the six fields observed with HST a slow star formation pace from 13 to 5-7 Gyr ago, followed by a ~ 2-3 times higher activity. This is remarkable because dynamical models do not predict a strong influence of either the LMC or the Milky Way (MW) at that time. The level of the intermediate-age SFR enhancement systematically increases towards the center, resulting in a gradient in the mean age of the population, with the bar fields being systematically younger than the outer ones. Star formation over the most recent 500 Myr is strongly concentrated in the bar, the only exception being the area of the SMC wing. The strong current activity of the latter is likely driven by interaction with the LMC. At a given age, there is no significant difference in metallicity between the inner and outer fields, implying that metals are well mixed throughout the SMC. The age-metallicity relations we infer from our best fitting models are monotonically increasing with time, with no evidence of dips. This may argue against the major merger scenario proposed by Tsujimoto and Bekki 2009, although a minor merger cannot be ruled out.Comment: 30 pages, 16 figures, accepted for publication in Ap

Collective modes and ballistic expansion of a Fermi gas in the BCS-BEC crossover

We evaluate the frequencies of collective modes and the anisotropic expansion rate of a harmonically trapped Fermi superfluid at varying coupling strengths across a Feshbach resonance driving a BCS-BEC crossover. The equations of motion for the superfluid are obtained from a microscopic mean-field expression for the compressibility and are solved within a scaling ansatz. Our results confirm non-monotonic behavior in the crossover region and are in quantitative agreement with current measurements of the transverse breathing mode by Kinast {\it et al.} [Phys. Rev. Lett. {\bf 92}, 150402 (2004)] and of the axial breathing mode by Bartenstein {\it et al.} [Phys. Rev. Lett. {\bf 92}, 203201 (2004)].Comment: 4 pages and 4 figures; proof version with more extensive discussions on the comparison between theoretical results and experimental findings; to appear in Phys. Rev. Lett. (Nov. 2004

The habitability of a stagnant-lid Earth

Plate tectonics is a fundamental component for the habitability of the Earth. Yet whether it is a recurrent feature of terrestrial bodies orbiting other stars or unique to the Earth is unknown. The stagnant lid may rather be the most common tectonic expression on such bodies. To understand whether a stagnant-lid planet can be habitable, i.e. host liquid water at its surface, we model the thermal evolution of the mantle, volcanic outgassing of H$_2$O and CO$_2$, and resulting climate of an Earth-like planet lacking plate tectonics. We used a 1D model of parameterized convection to simulate the evolution of melt generation and the build-up of an atmosphere of H$_2$O and CO$_2$ over 4.5 Gyr. We then employed a 1D radiative-convective atmosphere model to calculate the global mean atmospheric temperature and the boundaries of the habitable zone (HZ). The evolution of the interior is characterized by the initial production of a large amount of partial melt accompanied by a rapid outgassing of H$_2$O and CO$_2$. At 1 au, the obtained temperatures generally allow for liquid water on the surface nearly over the entire evolution. While the outer edge of the HZ is mostly influenced by the amount of outgassed CO$_2$, the inner edge presents a more complex behaviour that is dependent on the partial pressures of both gases. At 1 au, the stagnant-lid planet considered would be regarded as habitable. The width of the HZ at the end of the evolution, albeit influenced by the amount of outgassed CO$_2$, can vary in a non-monotonic way depending on the extent of the outgassed H$_2$O reservoir. Our results suggest that stagnant-lid planets can be habitable over geological timescales and that joint modelling of interior evolution, volcanic outgassing, and accompanying climate is necessary to robustly characterize planetary habitability

Genomic imbalances are confined to non-proliferating cells in paediatric patients with acute myeloid leukaemia and a normal or incomplete karyotype

Copyright @ 2011 Ballabio et al.Leukaemia is often associated with genetic alterations such as translocations, amplifications and deletions, and recurrent chromosome abnormalities are used as markers of diagnostic and prognostic relevance. However, a proportion of acute myeloid leukaemia (AML) cases have an apparently normal karyotype despite comprehensive cytogenetic analysis. Based on conventional cytogenetic analysis of banded chromosomes, we selected a series of 23 paediatric patients with acute myeloid leukaemia and performed whole genome array comparative genome hybridization (aCGH) using DNA samples derived from the same patients. Imbalances involving large chromosomal regions or entire chromosomes were detected by aCGH in seven of the patients studied. Results were validated by fluorescence in situ hybridization (FISH) to both interphase nuclei and metaphase chromosomes using appropriate bacterial artificial chromosome (BAC) probes. The majority of these copy number alterations (CNAs) were confirmed by FISH and found to localize to the interphase rather than metaphase nuclei. Furthermore, the proliferative states of the cells analyzed by FISH were tested by immunofluorescence using an antibody against the proliferation marker pKi67. Interestingly, these experiments showed that, in the vast majority of cases, the changes appeared to be confined to interphase nuclei in a non-proliferative status.This work was supported by a grant from Leukaemia Research UK (grant no. 0253). SJLK and RR were supported by the NIHR Biomedical Research Centre, Oxford, with funding from the Department of Health’s NIHR Biomedical Research Centres funding schemeThis article is available through the Brunel Open Access Publishing Fund