Erdos-Turan with a moving target, equidistribution of roots of reducible quadratics, and Diophantine quadruples

A Diophantine $m$-tuple is a set $A$ of $m$ positive integers such that $ab+1$ is a perfect square for every pair $a,b$ of distinct elements of $A$. We derive an asymptotic formula for the number of Diophantine quadruples whose elements are bounded by $x$. In doing so, we extend two existing tools in ways that might be of independent interest. The Erd\H os-Tur\'an inequality bounds the discrepancy between the number of elements of a sequence that lie in a particular interval modulo 1 and the expected number; we establish a version of this inequality where the interval is allowed to vary. We also adapt an argument of Hooley on the equidistribution of solutions of polynomial congruences to handle reducible quadratic polynomials.Comment: 24+epsilon page

The baseline intracluster entropy profile from gravitational structure formation

The radial entropy profile of the hot gas in clusters of galaxies tends to follow a power law in radius outside of the cluster core. Here we present a simple formula giving both the normalization and slope for the power-law entropy profiles of clusters that form in the absence of non-gravitational processes such as radiative cooling and subsequent feedback. It is based on seventy-one clusters drawn from four separate cosmological simulations, two using smoothed-particle hydrodynamics (SPH) and two using adaptive-mesh refinement (AMR), and can be used as a baseline for assessing the impact of non-gravitational processes on the intracluster medium outside of cluster cores. All the simulations produce clusters with self-similar structure in which the normalization of the entropy profile scales linearly with cluster temperature, and these profiles are in excellent agreement outside of 0.2 r_200. Because the observed entropy profiles of clusters do not scale linearly with temperature, our models confirm that non-gravitational processes are necessary to break the self-similarity seen in the simulations. However, the core entropy levels found by the two codes used here significantly differ, with the AMR code producing nearly twice as much entropy at the centre of a cluster.Comment: Accepted to MNRAS, 8 pages, 9 figure

Tactics Against Sexual Harassment: The Role of Backfire

To oppose sexual harassment, it is useful to understand tactics commonly used by perpetrators. A useful approach to tactics is through the concept of backfire: if an action is perceived as unjust and information about it is communicated to receptive audiences, it has the capacity to cause outrage and consequently backfire on the perpetrator. Perpetrators regularly use five types of tactics to inhibit outrage: (1) cover-up of the action; (2) devaluation of the target; (3) reinterpretation of the events; (4) use of official channels to give the appearance of justice; and (5) intimidation and bribery of targets, witnesses and others. These tactics are regularly used against targets of sexual harassment. The deployment of these tactics is illustrated through the case of Anita Hill, who in 1991 accused US Supreme Court nominee Clarence Thomas of sexual harassment. An analysis using the backfire framework offers guidance on effective ways of deterring and countering sexual harassment

CBS domains form energy-sensing modules whose binding of adenosine ligands is disrupted by disease mutations

CBS domains are defined as sequence motifs that occur in several different proteins in all kingdoms of life. Although thought to be regulatory, their exact functions have been unknown. However, their importance was underlined by findings that mutations in conserved residues within them cause a variety of human hereditary diseases, including (with the gene mutated in parentheses): Wolff-Parkinson-White syndrome (γ2 subunit of AMP-activated protein kinase); retinitis pigmentosa (IMP dehydrogenase-1); congenital myotonia, idiopathic generalized epilepsy, hypercalciuric nephrolithiasis, and classic Bartter syndrome (CLC chloride channel family members); and homocystinuria (cystathionine β-synthase). AMP-activated protein kinase is a sensor of cellular energy status that is activated by AMP and inhibited by ATP, but the location of the regulatory nucleotide-binding sites (which are prime targets for drugs to treat obesity and diabetes) was not characterized. We now show that tandem pairs of CBS domains from AMP-activated protein kinase, IMP dehydrogenase-2, the chloride channel CLC2, and cystathionine β-synthase bind AMP, ATP, or S-adenosyl methionine,while mutations that cause hereditary diseases impair this binding. This shows that tandem pairs of CBS domains act, in most cases, as sensors of cellular energy status and, as such, represent a newly identified class of binding domain for adenosine derivatives

Development of the Dual Aerodynamic Nozzle Model for the NTF Semi-Span Model Support System

The recent addition of a dual flow air delivery system to the NASA Langley National Transonic Facility was experimentally validated with a Dual Aerodynamic Nozzle semi-span model. This model utilized two Stratford calibration nozzles to characterize the weight flow system of the air delivery system. The weight flow boundaries for the air delivery system were identified at mildly cryogenic conditions to be 0.1 to 23 lbm/sec for the high flow leg and 0.1 to 9 lbm/sec for the low flow leg. Results from this test verified system performance and identified problems with the weight-flow metering system that required the vortex flow meters to be replaced at the end of the test