Hill's Spectral Curves and the Invariant Measure of the Periodic KdV Equation

This paper analyses the periodic spectrum of Schr\"odinger's equation $-f''+qf=\lambda f$ when the potential is real, periodic, random and subject to the invariant measure $\nu_N^\beta$ of the periodic KdV equation. This $\nu_N^\beta$ is the modified canonical ensemble, as given by Bourgain ({Comm. Math. Phys.} {166} (1994), 1--26), and $\nu_N^\beta$ satisfies a logarithmic Sobolev inequality. Associated concentration inequalities control the fluctuations of the periodic eigenvalues $(\lambda_n)$. For $\beta, N>0$ small, there exists a set of positive $\nu_N^\beta$ measure such that $(\pm \sqrt{2(\lambda_{2n}+\lambda_{2n-1})})_{n=0}^\infty$ gives a sampling sequence for Paley--Wiener space $PW(\pi )$ and the reproducing kernels give a Riesz basis. Let $(\mu_j)_{j=1}^\infty$ be the tied spectrum; then $(2\sqrt{\mu_j}-j)$ belongs to a Hilbert cube in $\ell^2$ and is distributed according to a measure that satisfies Gaussian concentration for Lipschitz functions. The sampling sequence $(\sqrt{\mu_j})_{j=1}^\infty$ arises from a divisor on the spectral curve, which is hyperelliptic of infinite genus. The linear statistics $\sum_j g(\sqrt{\lambda_{2j}})$ with test function $g\in PW(\pi)$ satisfy Gaussian concentration inequalities.Comment: 34 page

Dust Abundance Variations in the Magellanic Clouds: Probing the Lifecycle of Metals with All-Sky Surveys

Observations and modeling suggest that the dust abundance (gas-to-dust ratio, G/D) depends on (surface) density. The variations of the G/D provide constraints on the timescales for the different processes involved in the lifecycle of metals in galaxies. Recent G/D measurements based on Herschel data suggest a factor 5---10 decrease in the dust abundance between the dense and diffuse interstellar medium (ISM) in the Magellanic Clouds. However, the relative nature of the Herschel measurements precludes definitive conclusions on the magnitude of those variations. We investigate the variations of the dust abundance in the LMC and SMC using all-sky far-infrared surveys, which do not suffer from the limitations of Herschel on their zero-point calibration. We stack the dust spectral energy distribution (SED) at 100, 350, 550, and 850 microns from IRAS and Planck in intervals of gas surface density, model the stacked SEDs to derive the dust surface density, and constrain the relation between G/D and gas surface density in the range 10---100 \Msu pc$^{-2}$ on $\sim$ 80 pc scales. We find that G/D decreases by factors of 3 (from 1500 to 500) in the LMC and 7 (from 1.5$\times 10^4$ to 2000) in the SMC between the diffuse and dense ISM. The surface density dependence of G/D is consistent with elemental depletions and with simple modeling of the accretion of gas-phase metals onto dust grains. This result has important implications for the sub-grid modeling of galaxy evolution, and for the calibration of dust-based gas mass estimates, both locally and at high-redshift.Comment: 20 pages, 14 figure

Managing pregnancy in inflammatory rheumatological diseases

Historically, pregnancy in women with many inflammatory rheumatic diseases was not considered safe and was discouraged. Combined care allows these pregnancies to be managed optimally, with the majority of outcomes being favorable. Disease activity at the time of conception and anti-phospholipid antibodies are responsible for most complications. Disease flares, pre-eclampsia, and thrombosis are the main maternal complications, whereas fetal loss and intrauterine growth restriction are the main fetal complications. Antirheumatic drugs used during pregnancy and lactation to control disease activity are corticosteroids, hydroxychloroquine, sulphasalzine, and azathioprine. Vaginal delivery is possible in most circumstances, with cesarean section being reserved for complications