A generalization of Hausdorff dimension applied to Hilbert cubes and Wasserstein spaces

A Wasserstein spaces is a metric space of sufficiently concentrated probability measures over a general metric space. The main goal of this paper is to estimate the largeness of Wasserstein spaces, in a sense to be precised. In a first part, we generalize the Hausdorff dimension by defining a family of bi-Lipschitz invariants, called critical parameters, that measure largeness for infinite-dimensional metric spaces. Basic properties of these invariants are given, and they are estimated for a naturel set of spaces generalizing the usual Hilbert cube. In a second part, we estimate the value of these new invariants in the case of some Wasserstein spaces, as well as the dynamical complexity of push-forward maps. The lower bounds rely on several embedding results; for example we provide bi-Lipschitz embeddings of all powers of any space inside its Wasserstein space, with uniform bound and we prove that the Wasserstein space of a d-manifold has "power-exponential" critical parameter equal to d.Comment: v2 Largely expanded version, as reflected by the change of title; all part I on generalized Hausdorff dimension is new, as well as the embedding of Hilbert cubes into Wasserstein spaces. v3 modified according to the referee final remarks ; to appear in Journal of Topology and Analysi

1420 MHz Continuum Absorption Towards Extragalactic Sources in the Galactic Plane

We present a 21-cm emission-absorption study towards extragalactic sources in the Canadian Galactic Plane Survey (CGPS). We have analyzed HI spectra towards 437 sources with S > 150 mJy, giving us a source density of 0.6 sources per square degree at arcminute resolution. We present the results of a first analysis of the HI temperatures, densities, and feature statistics. Particular emphasis is placed on 5 features with observed spin temperatures below 40 K. We find most spin temperatures in the range from 40 K to 300 K. A simple HI two-component model constrains the bulk of the cold component to temperatures (T_c) between 40 K and 100 K. T_c peaks in the Perseus arm region and clearly drops off with Galactocentric radius, R, beyond that. The HI density follows this trend, ranging from a local value of 0.4 cm^{-3} to less than 0.1 cm^{-3} at R = 20 kpc. We find that HI emission alone on average traces about 75% of the total HI column density, as compared to the total inferred by the emission and absorption. Comparing the neutral hydrogen absorption to CO emission no correlation is found in general, but all strong CO emission is accompanied by a visible HI spectral feature. Finally, the number of spectral HI absorption features per kpc drop off exponentially with increasing R.Comment: 13 pages, 13 figures, Accepted for March 2004 Ap

Relationship between cardiovascular risk and lipid testing in one health care system: a retrospective cohort study.

BackgroundThe US Preventive Services Taskforce (USPSTF) recommends routine lipid screening beginning age 35 for men [1]. For women age 20 and older, as well as men age 20-34, screening is recommended if cardiovascular risk factors are present. Prior research has focused on underutilization but not overuse of lipid testing. The objective is to document over- and under-use of lipid testing in an insured population of persons at low, moderate and high cardiovascular disease (CVD) risk for persons not already on statins.MethodsThe study is a retrospective cohort study that included all adults without prior CVD who were continuously enrolled in a large integrated healthcare system from 2005 to 2010. Measures included lipid test frequency extracted from administrative data and Framingham cardiovascular risk equations applied using electronic medical record data. Five year lipid testing patterns were examined by age, sex and CVD risk. Generalized linear models were used to estimate the relative risk for over testing associated with patient characteristics.ResultsAmong males and females for whom testing is not recommended, 35.8 % and 61.5 % received at least one lipid test in the prior 5 years and 8.4 % and 24.4 % had two or more. Over-testing was associated with age, race, comorbidity, primary care use and neighborhood income. Among individuals at moderate and high-risk (not already treated with statins) and for whom screening is recommended, between 21.4 % and 25.1 % of individuals received no screening in the prior 5 years.ConclusionsBased on USPSTF lipid screening recommendations, this study documents substantial over-testing among individuals with low CVD risk and under-testing among individuals with moderate to high-risk not already on statins. Opportunity exists to better focus lipid screening efforts appropriate to CVD risk

Fitting Together the HI Absorption and Emission in the SGPS

In this paper we study 21-cm absorption spectra and the corresponding emission spectra toward bright continuum sources in the test region (326deg< l < 333 deg) of the Southern Galactic Plane Survey. This survey combines the high resolution of the Australia Telescope Compact Array with the full brightness temperature information of the Parkes single dish telescope. In particular, we focus on the abundance and temperature of the cool atomic clouds in the inner galaxy. The resulting mean opacity of the HI, , is measured as a function of Galactic radius; it increases going in from the solar circle, to a peak in the molecular ring of about four times its local value. This suggests that the cool phase is more abundant there, and colder, than it is locally. The distribution of cool phase temperatures is derived in three different ways. The naive, ``spin temperature'' technique overestimates the cloud temperatures, as expected. Using two alternative approaches we get good agreement on a histogram of the cloud temperatures, T(cool), corrected for blending with warm phase gas. The median temperature is about 65 K, but there is a long tail reaching down to temperatures below 20 K. Clouds with temperatures below 40 K are common, though not as common as warmer clouds (40 to 100 K). Using these results we discuss two related quantities, the peak brightness temperature seen in emission surveys, and the incidence of clouds seen in HI self-absorption. Both phenomena match what would be expected based on our measurements of and T(cool).Comment: 50 pages, 20 figure

GSH23.0-0.7+117, a neutral hydrogen shell in the inner Galaxy

GSH23.0-0.7+117 is a well-defined neutral hydrogen shell discovered in the VLA Galactic Plane Survey (VGPS). Only the blueshifted side of the shell was detected. The expansion velocity and systemic velocity were determined through the systematic behavior of the HI emission with velocity. The center of the shell is at (l,b,v)=(23.05,-0.77,+117 km/s). The angular radius of the shell is 6.8', or 15 pc at a distance of 7.8 kpc. The HI mass divided by the volume of the half-shell implies an average density n_H = 11 +/- 4 cm^{-3} for the medium in which the shell expanded. The estimated age of GSH23.0-0.7+117 is 1 Myr, with an upper limit of 2 Myr. The modest expansion energy of 2 * 10^{48} erg can be provided by the stellar wind of a single O4 to O8 star over the age of the shell. The 3 sigma upper limit to the 1.4 GHz continuum flux density (S_{1.4} < 248 mJy) is used to derive an upper limit to the Lyman continuum luminosity generated inside the shell. This upper limit implies a maximum of one O9 star (O8 to O9.5 taking into account the error in the distance) inside the HI shell, unless most of the incident ionizing flux leaks through the HI shell. To allow this, the shell should be fragmented on scales smaller than the beam (2.3 pc). If the stellar wind bubble is not adiabatic, or the bubble has burst (as suggested by the HI channel maps), agreement between the energy and ionization requirements is even less likely. The limit set by the non-detection in the continuum provides a significant challenge for the interpretation of GSH23.0-0.7+117 as a stellar wind bubble. A similar analysis may be applicable to other Galactic HI shells that have not been detected in the continuum.Comment: 18 pages, 6 figures. Figures 1 and 4 separately in GIF format. Accepted for publication in Astrophysical Journa