Geometric investigations of a vorticity model equation

This article consists of a detailed geometric study of the one-dimensional vorticity model equation $$\omega_{t} + u\omega_{x} + 2\omega u_{x} = 0, \qquad \omega = H u_{x}, \qquad t\in\mathbb{R},\; x\in S^{1}\,,$$ which is a particular case of the generalized Constantin-Lax-Majda equation. Wunsch showed that this equation is the Euler-Arnold equation on $\operatorname{Diff}(S^{1})$ when the latter is endowed with the right-invariant homogeneous $\dot{H}^{1/2}$-metric. In this article we prove that the exponential map of this Riemannian metric is not Fredholm and that the sectional curvature is locally unbounded. Furthermore, we prove a Beale-Kato-Majda-type blow-up criterion, which we then use to demonstrate a link to our non-Fredholmness result. Finally, we extend a blow-up result of Castro-C\'ordoba to the periodic case and to a much wider class of initial conditions, using a new generalization of an inequality for Hilbert transforms due to C\'ordoba-C\'ordoba.Comment: 30 pages; added references; corrected typo

Isometric Immersions and the Waving of Flags

In this article we propose a novel geometric model to study the motion of a physical flag. In our approach a flag is viewed as an isometric immersion from the square with values into $\mathbb R^3$ satisfying certain boundary conditions at the flag pole. Under additional regularity constraints we show that the space of all such flags carries the structure of an infinite dimensional manifold and can be viewed as a submanifold of the space of all immersions. The submanifold result is then used to derive the equations of motion, after equipping the space of isometric immersions with its natural kinetic energy. This approach can be viewed in a similar spirit as Arnold's geometric picture for the motion of an incompressible fluid.Comment: 25 pages, 1 figur

Further evidence on the link between health care spending and health outcomes in England

This report describes results from research funded by the Health Foundation under its Quest for Quality and Improved Performance (QQuIP) initiative. It builds on our earlier report for the Health Foundation – The link between health care spending and health outcomes: evidence from English programme budgeting data – that took advantage of the availability of a major new dataset to examine the relationship between health care expenditure and mortality rates for two disease categories (cancer and circulation problems) across 300 English Primary Care Trusts. Our results are useful from a number of perspectives. Scientifically, they confirm our previous findings that health care has an important impact on health across a range of conditions, suggesting that those results were robust across programmes of care and across years. From a policy perspective, these results can help set priorities by informing resource allocation across a larger number of programmes of care. They also add further evidence to help NICE decide whether its current QALY threshold is at the right level.

Measuring x-ray polarization in the presence of systematic effects: Known background

The prospects for accomplishing x-ray polarization measurements of astronomical sources have grown in recent years, after a hiatus of more than 37 years. Unfortunately, accompanying this long hiatus has been some confusion over the statistical uncertainties associated with x-ray polarization measurements of these sources. We have initiated a program to perform the detailed calculations that will offer insights into the uncertainties associated with x-ray polarization measurements. Here we describe a mathematical formalism for determining the 1- and 2-parameter errors in the magnitude and position angle of x-ray (linear) polarization in the presence of a (polarized or unpolarized) background. We further review relevant statistics-including clearly distinguishing between the Minimum Detectable Polarization (MDP) and the accuracy of a polarization measurement.Comment: 12 pages, 4 figures, for SPIE conference proceeding

On understanding the figures of merit for detection and measurement of x-ray polarization

The prospects for accomplishing X-ray polarization measurements appear to have grown in recent years after a more than 35-year hiatus. Unfortunately, this long hiatus has brought with it some confusion over the statistical uncertainties associated with polarization measurements of astronomical sources. The heart of this confusion stems from a misunderstanding (or potential misunderstanding) of a standard figure of merit-the minimum detectable polarization (MDP)-that one of us introduced many years ago. We review the relevant statistics, and quantify the differences between the MDP and the uncertainty of an actual polarization measurement. We discuss the implications for future missions.Comment: 5 pages, 2 figures, to be presented at SPIE conference 7732 (paper 13), corrected typo

The link between health care spending and health outcomes for the new English Primary Care Trusts

English programme budgeting data have yielded major new insights into the link between health care spending and health outcomes. This paper updates two recent studies that have used programme budgeting data for 295 Primary Care Trusts (PCTs) in England to examine the link between spending and outcomes for several programmes of care. We use the same economic model employed in the two previous studies. It focuses on a decision maker who must allocate a fixed budget across programmes of care so as to maximize social welfare given a health production function for each programme. Two equations – a health outcome equation and an expenditure equation – are estimated for each programme (data permitting). The two previous studies employed expenditure data for 2004/05 and 2005/06 for 295 health authorities and found that in several care programmes – cancer, circulation problems, respiratory problems, gastro-intestinal problems, trauma burns and injury, and diabetes – expenditure had the anticipated negative effect on the mortality rate. Each health outcome equation was used to estimate the marginal cost of a life year saved. In 2006/07 the number of PCTs in England was reduced to 152, largely through a series of mergers. In addition, several changes were made to the methods employed to construct the programme budgeting data. This paper employs updated budgeting and mortality data for the new 152 PCTs to re-estimate health production and expenditure functions, and also presents updated estimates of the marginal cost of a life year saved in each programme. Although there are some differences, the results obtained are broadly similar to those presented in our two previous studies.

The Link Between Health Care Spending and Health Outcomes: Evidence from English Programme Budgeting Data

This report describes preliminary results from research funded by the Health Foundation under its Quest for Quality and Improved Performance (QQuIP) initiative.

Value for money in the English NHS: Summary of the evidence

The extent to which the English National Health Service secures value for money for taxpayers has become a central issue of political and public debate. Questions include: how much expenditure growth has been made available to the NHS? on what has that money been spent? what improvements in the volume and quality of health care have been secured? and what are the implications for productivity? There has been a flurry of research activity designed to address these and similar questions. This report seeks to bring together this research in a concise format and draws some tentative conclusions about recent productivity changes in the NHS. It finds that there is considerable evidence of growth in both the volume and quality of NHS activity. However, this has not in general kept pace with the growth in expenditure. On most measures, therefore, NHS productivity is either static or declining. However, the report highlights a large number of unresolved methodological issues that make it hard to draw any definitive conclusions. We conclude that the measurement of NHS productivity change makes an important contribution to national debate. However, there remains considerable scope for improving both the data and the methods underlying current estimates.

The Many Faces of Heterogeneous Ice Nucleation: Interplay Between Surface Morphology and Hydrophobicity

What makes a material a good ice nucleating agent? Despite the importance of heterogeneous ice nucleation to a variety of fields, from cloud science to microbiology, major gaps in our understanding of this ubiquitous process still prevent us from answering this question. In this work, we have examined the ability of generic crystalline substrates to promote ice nucleation as a function of the hydrophobicity and the morphology of the surface. Nucleation rates have been obtained by brute-force molecular dynamics simulations of coarse-grained water on top of different surfaces of a model fcc crystal, varying the water-surface interaction and the surface lattice parameter. It turns out that the lattice mismatch of the surface with respect to ice, customarily regarded as the most important requirement for a good ice nucleating agent, is at most desirable but not a requirement. On the other hand, the balance between the morphology of the surface and its hydrophobicity can significantly alter the ice nucleation rate and can also lead to the formation of up to three different faces of ice on the same substrate. We have pinpointed three circumstances where heterogeneous ice nucleation can be promoted by the crystalline surface: (i) the formation of a water overlayer that acts as an in-plane template; (ii) the emergence of a contact layer buckled in an ice-like manner; and (iii) nucleation on compact surfaces with very high interaction strength. We hope that this extensive systematic study will foster future experimental work aimed at testing the physiochemical understanding presented herein.Comment: Main + S