(k+1)-sums versus k-sums

A $k$-sum of a set $A\subseteq \mathbb{Z}$ is an integer that may be expressed as a sum of $k$ distinct elements of $A$. How large can the ratio of the number of $(k+1)$-sums to the number of $k$-sums be? Writing $k\wedge A$ for the set of $k$-sums of $A$ we prove that $$\frac{|(k+1)\wedge A|}{|k\wedge A|}\, \le \, \frac{|A|-k}{k+1}$$ whenever $|A|\ge (k^{2}+7k)/2$. The inequality is tight -- the above ratio being attained when $A$ is a geometric progression. This answers a question of Ruzsa.Comment: 5 page

An assessment of failure to rescue derived from routine NHS data as a nursing sensitive patient safety indicator (report to Policy Research Programme)

Objectives: This study aims to assess the potential for deriving 2 mortality based failure to rescue indicators and a proxy measure, based on exceptionally long length of stay, from English hospital administrative data by exploring change in coding practice over time and measuring associations between failure to rescue and factors which would suggest indicators derived from these data are valid.Design: Cross sectional observational study of routinely collected administrative data.Setting: 146 general acute hospital trusts in England.Participants: Discharge data from 66,100,672 surgical admissions (1997 to 2009).Results: Median percentage of surgical admissions with at least one secondary diagnosis recorded increased from 26% in 1997/8 to 40% in 2008/9. The failure to rescue rate for a hospital appears to be relatively stable over time: inter-year correlations between 2007/8 and 2008/9 were r=0.92 to r=0.94. No failure to rescue indicator was significantly correlated with average number of secondary diagnoses coded per hospital. Regression analyses showed that failure to rescue was significantly associated (p<0.05) with several hospital characteristics previously associated with quality including staffing levels. Higher medical staffing (doctors + nurses) per bed and more doctors relative to the number of nurses were associated with lower failure to rescue. Conclusion: Coding practice has improved, and failure to rescue can be derived from English administrative data. The suggestion that it is particularly sensitive to nursing is not clearly supported. Although the patient population is more homogenous than for other mortality measures, risk adjustment is still required

What can the left learn from Friedrich Hayek?

An engagement with Hayek does not mean a capitulation to the market, writes Simon Griffiths. Instead it can provide several sophisticated insights for the contemporary left, in particular on knowledge, the spontaneous order, and freedom. The left’s discovery of Hayek is also significant as an example of how ideologies, such as socialism or liberalism, can be transformed over time

Invasion percolation on the Poisson-weighted infinite tree

We study invasion percolation on Aldous' Poisson-weighted infinite tree, and derive two distinct Markovian representations of the resulting process. One of these is the $\sigma\to\infty$ limit of a representation discovered by Angel et al. [Ann. Appl. Probab. 36 (2008) 420-466]. We also introduce an exploration process of a randomly weighted Poisson incipient infinite cluster. The dynamics of the new process are much more straightforward to describe than those of invasion percolation, but it turns out that the two processes have extremely similar behavior. Finally, we introduce two new "stationary" representations of the Poisson incipient infinite cluster as random graphs on $\mathbb {Z}$ which are, in particular, factors of a homogeneous Poisson point process on the upper half-plane $\mathbb {R}\times[0,\infty)$.Comment: Published in at http://dx.doi.org/10.1214/11-AAP761 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Quenched Voronoi percolation

We prove that the probability of crossing a large square in quenched Voronoi percolation converges to 1/2 at criticality, confirming a conjecture of Benjamini, Kalai and Schramm from 1999. The main new tools are a quenched version of the box-crossing property for Voronoi percolation at criticality, and an Efron-Stein type bound on the variance of the probability of the crossing event in terms of the sum of the squares of the influences. As a corollary of the proof, we moreover obtain that the quenched crossing event at criticality is almost surely noise sensitive.Comment: 21 pages, 2 figure

Responses to the new right: the engagement of the British left with the work of Friedrich Hayek, 1989-1997

This is an examination of the context, content and significance of the surprising engagement of the British left with the arguments of Friedrich Hayek (1899-1992), one of the most influential theorists of the new right and an important influence on leading figures in the Conservative Government elected in the UK in 1979. The thesis examines in detail the engagement by four thinkers on the British left with Hayek's work: David Miller, Raymond Plant, Andrew Gamble and Hilary Wainwright. Its chronological parameters are the collapse of the Berlin Wall in 1989 and the election of ‘New Labour’ in the UK in 1997. Important contextual factors behind this engagement include the rise and fall of the British Conservative Party, the difficulties of statist forms of socialism and Hayek's own death. The engagement with Hayek's work provides a case study that demonstrates changes in political themes, in particular, the decline of statist forms of socialism with the left's embrace of the market and individual freedom, the decline in support for the paternalistic state and the search for more ‘feasible’ alternatives. I argue that the British left's engagement with Hayek is part of a wider intellectual break that constitutes the end of a ‘short twentieth century’ in political thought, and that the political landscape is now dominated by two strands of the liberal tradition. As such, the research will be of importance to anyone seeking a clearer understanding of recent changes in political thought and to the shape of the contemporary political landscape