Triangular Gross-Pitaevskii breathers and Damski-Chandrasekhar shock waves

The recently proposed map [arXiv:2011.01415] between the hydrodynamic equations governing the two-dimensional triangular cold-bosonic breathers [Phys. Rev. X 9, 021035 (2019)] and the high-density zero-temperature triangular free-fermionic clouds, both trapped harmonically, perfectly explains the former phenomenon but leaves uninterpreted the nature of the initial ($t=0$) singularity. This singularity is a density discontinuity that leads, in the bosonic case, to an infinite force at the cloud edge. The map itself becomes invalid at times $t<0$. A similar singularity appears at $t = T/4$, where $T$ is the period of the harmonic trap, with the Fermi-Bose map becoming invalid at $t > T/4$. Here, we first map -- using the scale invariance of the problem -- the trapped motion to an untrapped one. Then we show that in the new representation, the solution [arXiv:2011.01415] becomes, along a ray in the direction normal to one of the three edges of the initial cloud, a freely propagating one-dimensional shock wave of a class proposed by Damski in [Phys. Rev. A 69, 043610 (2004)]. There, for a broad class of initial conditions, the one-dimensional hydrodynamic equations can be mapped to the inviscid Burgers' equation, a nonlinear transport equation. More specifically, under the Damski map, the $t=0$ singularity of the original problem becomes, verbatim, the initial condition for the wave catastrophe solution found by Chandrasekhar in 1943 [Ballistic Research Laboratory Report No. 423 (1943)]. At $t=T/8$, our interpretation ceases to exist: at this instance, all three effectively one-dimensional shock waves emanating from each of the three sides of the initial triangle collide at the origin, and the 2D-1D correspondence between the solution of [arXiv:2011.01415] and the Damski-Chandrasekhar shock wave becomes invalid.Comment: 13 pages, 2 figures. Submission to SciPos

Exposure of young children to household water lead in the Montreal area (Canada): the potential influence of winter-to-summer changes in water lead levels on children's blood lead concentration

Drinking water represents a potential source of lead exposure. The purpose of the present study was to estimate the magnitude of winter-to-summer changes in household water lead levels (WLLs), and to predict the impact of these variations on BLLs in young children. A study was conducted from September, 2009 to March, 2010 in 305 homes, with a follow-up survey carried out from June to September 2011 in a subsample of 100 homes randomly selected. The first 1-L sample was drawn after 5 min of flushing, followed by a further 4 consecutive 1-L samples after 30 min of stagnation. Non-linear regression and general linear mixed models were used for modelling seasonal effects on WLL. The batchrun mode of Integrated Exposure Uptake Biokinetic (IEUBK) model was used to predict the impact of changes in WLL on children's blood lead levels (BLLs). The magnitude of winter-to-summer changes in average concentrations of lead corresponded to 6.55 mug/L in homes served by lead service lines (LSL+ homes) and merely 0.30 mug/L in homes without lead service lines. For stagnant samples, the value reached 10.55 mug/L in 'LSL+ homes' and remained very low (0.36 mug/L) in 'LSL- homes'. The change in the probability of BLLs >/=5 mug/dL due to winter-to-summer changes in WLL was increased from /=5 mug/dL in young children during warm months was reduced by at least 40% by flushing tap-water