Evaluation of binomial double sums involving absolute values

We show that double sums of the form $$\sum_{i,j=-n} ^{n} |i^sj^t(i^k-j^k)^\beta| \binom {2n} {n+i} \binom {2n} {n+j}$$ can always be expressed in terms of a linear combination of just four functions, namely $\binom {4n}{2n}$, ${\binom {2n}n}^2$, $4^n\binom {2n}n$, and $16^n$, with coefficients that are rational in $n$. We provide two different proofs: one is algorithmic and uses the second author's computer algebra package Sigma; the second is based on complex contour integrals. In many instances, these results are extended to double sums of the above form where $\binom {2n}{n+j}$ is replaced by $\binom {2m}{m+j}$ with independent parameter $m$.Comment: AmS-LaTeX, 42 pages; substantial revision: several additional and more general results, see Proposition 11 and Theorems 15-1

A Substantial Population of Low Mass Stars in Luminous Elliptical Galaxies

The stellar initial mass function (IMF) describes the mass distribution of stars at the time of their formation and is of fundamental importance for many areas of astrophysics. The IMF is reasonably well constrained in the disk of the Milky Way but we have very little direct information on the form of the IMF in other galaxies and at earlier cosmic epochs. Here we investigate the stellar mass function in elliptical galaxies by measuring the strength of the Na I doublet and the Wing-Ford molecular FeH band in their spectra. These lines are strong in stars with masses <0.3 Msun and weak or absent in all other types of stars. We unambiguously detect both signatures, consistent with previous studies that were based on data of lower signal-to-noise ratio. The direct detection of the light of low mass stars implies that they are very abundant in elliptical galaxies, making up >80% of the total number of stars and contributing >60% of the total stellar mass. We infer that the IMF in massive star-forming galaxies in the early Universe produced many more low mass stars than the IMF in the Milky Way disk, and was probably slightly steeper than the Salpeter form in the mass range 0.1 - 1 Msun.Comment: To appear in Natur

Exoplanet Catalogues

One of the most exciting developments in the field of exoplanets has been the progression from 'stamp-collecting' to demography, from discovery to characterisation, from exoplanets to comparative exoplanetology. There is an exhilaration when a prediction is confirmed, a trend is observed, or a new population appears. This transition has been driven by the rise in the sheer number of known exoplanets, which has been rising exponentially for two decades (Mamajek 2016). However, the careful collection, scrutiny and organisation of these exoplanets is necessary for drawing robust, scientific conclusions that are sensitive to the biases and caveats that have gone into their discovery. The purpose of this chapter is to discuss and demonstrate important considerations to keep in mind when examining or constructing a catalogue of exoplanets. First, we introduce the value of exoplanetary catalogues. There are a handful of large, online databases that aggregate the available exoplanet literature and render it digestible and navigable - an ever more complex task with the growing number and diversity of exoplanet discoveries. We compare and contrast three of the most up-to-date general catalogues, including the data and tools that are available. We then describe exoplanet catalogues that were constructed to address specific science questions or exoplanet discovery space. Although we do not attempt to list or summarise all the published lists of exoplanets in the literature in this chapter, we explore the case study of the NASA Kepler mission planet catalogues in some detail. Finally, we lay out some of the best practices to adopt when constructing or utilising an exoplanet catalogue.Comment: 14 pages, 6 figures. Invited review chapter, to appear in "Handbook of Exoplanets", edited by H.J. Deeg and J.A. Belmonte, section editor N. Batalh

Tea and coffee consumption in relation to vitamin D and calcium levels in Saudi adolescents

Background Coffee and tea consumption was hypothesized to interact with variants of vitamin D-receptor polymorphisms, but limited evidence exists. Here we determine for the first time whether increased coffee and tea consumption affects circulating levels of 25-hydroxyvitamin D in a cohort of Saudi adolescents. Methods A total of 330 randomly selected Saudi adolescents were included. Anthropometrics were recorded and fasting blood samples were analyzed for routine analysis of fasting glucose, lipid levels, calcium, albumin and phosphorous. Frequency of coffee and tea intake was noted. 25-hydroxyvitamin D levels were measured using enzyme-linked immunosorbent assays. Results Improved lipid profiles were observed in both boys and girls, as demonstrated by increased levels of HDL-cholesterol, even after controlling for age and BMI, among those consuming 9–12 cups of coffee/week. Vitamin D levels were significantly highest among those consuming 9–12 cups of tea/week in all subjects (p-value 0.009) independent of age, gender, BMI, physical activity and sun exposure. Conclusion This study suggests a link between tea consumption and vitamin D levels in a cohort of Saudi adolescents, independent of age, BMI, gender, physical activity and sun exposure. These findings should be confirmed prospectively

Search algorithms as a framework for the optimization of drug combinations

Combination therapies are often needed for effective clinical outcomes in the management of complex diseases, but presently they are generally based on empirical clinical experience. Here we suggest a novel application of search algorithms, originally developed for digital communication, modified to optimize combinations of therapeutic interventions. In biological experiments measuring the restoration of the decline with age in heart function and exercise capacity in Drosophila melanogaster, we found that search algorithms correctly identified optimal combinations of four drugs with only one third of the tests performed in a fully factorial search. In experiments identifying combinations of three doses of up to six drugs for selective killing of human cancer cells, search algorithms resulted in a highly significant enrichment of selective combinations compared with random searches. In simulations using a network model of cell death, we found that the search algorithms identified the optimal combinations of 6-9 interventions in 80-90% of tests, compared with 15-30% for an equivalent random search. These findings suggest that modified search algorithms from information theory have the potential to enhance the discovery of novel therapeutic drug combinations. This report also helps to frame a biomedical problem that will benefit from an interdisciplinary effort and suggests a general strategy for its solution.Comment: 36 pages, 10 figures, revised versio

Cognitive function in a randomized trial of evolocumab

Inga Stuķēna as well as a complete list of investigators is provided in the Supplementary Appendix, available at NEJM.org. https://www.nejm.org/doi/suppl/10.1056/NEJMoa1701131/suppl_file/nejmoa1701131_appendix.pdf Funding Information: (Funded by Amgen; EBBINGHAUS ClinicalTrials.gov number, NCT02207634.) Supported by Amgen. We thank Sarah T. Farias, Ph.D., at UC Davis Health for providing the English-language and translated versions of the Everyday Cognition (ECog) tool. Publisher Copyright: Copyright © 2017 Massachusetts Medical Society.BACKGROUND: Findings from clinical trials of proprotein convertase subtilisin–kexin type 9 (PCSK9) inhibitors have led to concern that these drugs or the low levels of low-density lipoprotein (LDL) cholesterol that result from their use are associated with cognitive deficits. METHODS: In a subgroup of patients from a randomized, placebo-controlled trial of evolocumab added to statin therapy, we prospectively assessed cognitive function using the Cambridge Neuropsychological Test Automated Battery. The primary end point was the score on the spatial working memory strategy index of executive function (scores range from 4 to 28, with lower scores indicating a more efficient use of strategy and planning). Secondary end points were the scores for working memory (scores range from 0 to 279, with lower scores indicating fewer errors), episodic memory (scores range from 0 to 70, with lower scores indicating fewer errors), and psychomotor speed (scores range from 100 to 5100 msec, with faster times representing better performance). Assessments of cognitive function were performed at baseline, week 24, yearly, and at the end of the trial. The primary analysis was a noninferiority comparison of the mean change from baseline in the score on the spatial working memory strategy index of executive function between the patients who received evolocumab and those who received placebo; the noninferiority margin was set at 20% of the standard deviation of the score in the placebo group. RESULTS: A total of 1204 patients were followed for a median of 19 months; the mean (±SD) change from baseline over time in the raw score for the spatial working memory strategy index of executive function (primary end point) was −0.21±2.62 in the evolocumab group and −0.29±2.81 in the placebo group (P<0.001 for noninferiority; P=0.85 for superiority). There were no significant between-group differences in the secondary end points of scores for working memory (change in raw score, −0.52 in the evolocumab group and −0.93 in the placebo group), episodic memory (change in raw score, −1.53 and −1.53, respectively), or psychomotor speed (change in raw score, 5.2 msec and 0.9 msec, respectively). In an exploratory analysis, there were no associations between LDL cholesterol levels and cognitive changes. CONCLUSIONS: In a randomized trial involving patients who received either evolocumab or placebo in addition to statin therapy, no significant between-group difference in cognitive function was observed over a median of 19 months.publishersversionPeer reviewe