Long path and cycle decompositions of even hypercubes

We consider edge decompositions of the $n$-dimensional hypercube $Q_n$ into isomorphic copies of a given graph $H$. While a number of results are known about decomposing $Q_n$ into graphs from various classes, the simplest cases of paths and cycles of a given length are far from being understood. A conjecture of Erde asserts that if $n$ is even, $\ell < 2^n$ and $\ell$ divides the number of edges of $Q_n$, then the path of length $\ell$ decomposes $Q_n$. Tapadia et al.\ proved that any path of length $2^mn$, where $2^m<n$, satisfying these conditions decomposes $Q_n$. Here, we make progress toward resolving Erde's conjecture by showing that cycles of certain lengths up to $2^{n+1}/n$ decompose $Q_n$. As a consequence, we show that $Q_n$ can be decomposed into copies of any path of length at most $2^{n}/n$ dividing the number of edges of $Q_n$, thereby settling Erde's conjecture up to a linear factor

Polychromatic Colorings on the Integers

We show that for any set $S\subseteq \mathbb{Z}$, $|S|=4$ there exists a 3-coloring of $\mathbb{Z}$ in which every translate of $S$ receives all three colors. This implies that $S$ has a codensity of at most $1/3$, proving a conjecture of Newman [D. J. Newman, Complements of finite sets of integers, Michigan Math. J. 14 (1967) 481--486]. We also consider related questions in $\mathbb{Z}^d$, $d\geq 2$.Comment: 16 pages, improved presentatio

The importance of episodic accretion for low-mass star formation

A star acquires much of its mass by accreting material from a disc. Accretion is probably not continuous but episodic. We have developed a method to include the effects of episodic accretion in simulations of star formation. Episodic accretion results in bursts of radiative feedback, during which a protostar is very luminous, and its surrounding disc is heated and stabilised. These bursts typically last only a few hundred years. In contrast, the lulls between bursts may last a few thousand years; during these lulls the luminosity of the protostar is very low, and its disc cools and fragments. Thus, episodic accretion enables the formation of low-mass stars, brown dwarfs and planetary-mass objects by disc fragmentation. If episodic accretion is a common phenomenon among young protostars, then the frequency and duration of accretion bursts may be critical in determining the low-mass end of the stellar initial mass function.Comment: To appear in the Astrophysical Journal. Press release available at: http://www.astro.cf.ac.uk/pub/Dimitrios.Stamatellos/News/News.html Full resolution paper available at http://stacks.iop.org/0004-637X/730/3

Modelling circumstellar discs with 3D radiation hydrodynamics

We present results from combining a grid-based radiative transfer code with a Smoothed Particle Hydrodynamics code to produce a flexible system for modelling radiation hydrodynamics. We use a benchmark model of a circumstellar disc to determine a robust method for constructing a gridded density distribution from SPH particles. The benchmark disc is then used to determine the accuracy of the radiative transfer results. We find that the SED and the temperature distribution within the disc are sensitive to the representation of the disc inner edge, which depends critically on both the grid and SPH resolution. The code is then used to model a circumstellar disc around a T-Tauri star. As the disc adjusts towards equilibrium vertical motions in the disc are induced resulting in scale height enhancements which intercept radiation from the central star. Vertical transport of radiation enables these perturbations to influence the mid-plane temperature of the disc. The vertical motions decay over time and the disc ultimately reaches a state of simultaneous hydrostatic and radiative equilibrium.Comment: MNRAS accepted; 15 pages; 17 figures, 4 in colou

A retrospective analysis of geriatric trauma patients: venous lactate is a better predictor of mortality than traditional vital signs

BACKGROUND: Traditional vital signs (TVS), including systolic blood pressure (SBP), heart rate (HR) and their composite, the shock index, may be poor prognostic indicators in geriatric trauma patients. The purpose of this study is to determine whether lactate predicts mortality better than TVS. METHODS: We studied a large cohort of trauma patients age ≥ 65 years admitted to a level 1 trauma center from 2009-01-01 - 2011-12-31. We defined abnormal TVS as hypotension (SBP < 90 mm Hg) and/or tachycardia (HR > 120 beats/min), an elevated shock index as HR/SBP ≥ 1, an elevated venous lactate as ≥ 2.5 mM, and occult hypoperfusion as elevated lactate with normal TVS. The association between these variables and in-hospital mortality was compared using Chi-square tests and multivariate logistic regression. RESULTS: There were 1987 geriatric trauma patients included, with an overall mortality of 4.23% and an incidence of occult hypoperfusion of 20.03%. After adjustment for GCS, ISS, and advanced age, venous lactate significantly predicted mortality (OR: 2.62, p < 0.001), whereas abnormal TVS (OR: 1.71, p = 0.21) and SI ≥ 1 (OR: 1.18, p = 0.78) did not. Mortality was significantly greater in patients with occult hypoperfusion compared to patients with no sign of circulatory hemodynamic instability (10.67% versus 3.67%, p < 0.001), which continued after adjustment (OR: 2.12, p = 0.01). CONCLUSIONS: Our findings demonstrate that occult hypoperfusion was exceedingly common in geriatric trauma patients, and was associated with a two-fold increased odds of mortality. Venous lactate should be measured for all geriatric trauma patients to improve the identification of hemodynamic instability and optimize resuscitative efforts