The Page-R{\'e}nyi parking process

In the Page parking (or packing) model on a discrete interval (also known as the discrete R{\'e}nyi packing problem or the unfriendly seating problem), cars of length two successively park uniformly at random on pairs of adjacent places, until only isolated places remain. We give a probabilistic proof of the (known) fact that the proportion of the interval covered by cars goes to 1-exp(-2) , when the length of the interval goes to infinity. We obtain some new consequences, and also study a version of this process defined on the infinite line

On the algebraic numbers computable by some generalized Ehrenfest urns

This article deals with some stochastic population protocols, motivated by theoretical aspects of distributed computing. We modelize the problem by a large urn of black and white balls from which at every time unit a fixed number of balls are drawn and their colors are changed according to the number of black balls among them. When the time and the number of balls both tend to infinity the proportion of black balls converges to an algebraic number. We prove that, surprisingly enough, not every algebraic number can be "computed" this way

Molecular gas in absorption and emission along the line of sight to W31C G10.62-0.38

We used the ARO 12m antenna to observe emission from the J=1-0 lines of carbon monoxide, \hcop\ and HNC and the J=2-1 line of CS toward and around the continuum peak used for absorption studies and we compare them with CH, HNC, C\p\ and other absorption spectra from PRISMAS. We develop a kinematic analysis that allows a continuous description of the spectral properties and relates them to viewing geometry in the Galaxy. As for CH, HF, C\p, \hcop\ and other species observed in absorption, mm-wave emission in CO, \hcop, HNC and CS is continuous over the full velocity range expected for material between the Sun and W31 4.95 kpc away. CO emission is much stronger than average in the Galactic molecular ring and the mean \HH\ density derived from CH, 4 \pccc \la 2$$ \la 10 \pccc at 4 \la R \la 6.4 kpc, is similarly elevated. The CO-\HH\ conversion factor falls in a narrow range \XCO\ = 1-2\times10^{20}~\HH\ \pcc~({\rm K}-\kms)^{-1} if the emitting gas is mostly on the near side of the sub-central point, as we suggest. The brightnesses of \hcop, HNC, and CS are comparable (0.83\%, 0.51\% and 1.1\% respectively relative to CO) and have no variation in galactocentric radius with respect to CO. Comparison of the profile-averaged \hcop\ emission brightness and optical depth implies local densities n(H) \approx 135\pm25\pccc with most of excitation of \hcop\ from electrons. At such density, a consistent picture of the \HH-bearing gas, accounting also for the CO emission, has a volume filling factor 3\% and a 5 pc clump or cloud size.Comment: Accepted to A&

Construction of a short path in high dimensional First Passage Percolation

For First Passage Percolation in Z^d with large d, we construct a path connecting the origin to {x_1 =1}, whose passage time has optimal order \log d/d. Besides, an improved lower bound for the "diagonal" speed of the cluster combined with a result by Dhar (1988) shows that the limiting shape in FPP with exponential passage times (and thus that of Eden model) is not the euclidian ball in dimension larger than 35

Stellar Feedback in the ISM Revealed by Wide-Field Far-Infrared Spectral-Imaging

The radiative and mechanical interaction of stars with their environment drives the evolution of the ISM and of galaxies as a whole. The far-IR emission (lambda ~30 to 350 microns) from atoms and molecules dominates the cooling of the warm gas in the neutral ISM, the material that ultimately forms stars. Far-IR lines are thus the most sensitive probes of stellar feedback processes, and allow us to quantify the deposition and cycling of energy in the ISM. While ALMA (in the (sub)mm) and JWST (in the IR) provide astonishing sub-arcsecond resolution images of point sources and their immediate environment, they cannot access the main interstellar gas coolants, nor are they designed to image entire star-forming regions (SFRs). Herschel far-IR photometric images of the interstellar dust thermal emission revealed the ubiquitous large-scale filamentary structure of SFRs, their mass content, and the location of thousands of prestellar cores and protostars. These images, however, provide a static view of the ISM: not only they dont constrain the cloud dynamics, moreover they cannot reveal the chemical composition and energy transfer within the cloud, thus giving little insight into the regulation process of star formation by stellar feedback. In this white paper we emphasize the need of a space telescope with wide-field spectral-imaging capabilities in the critical far-IR domain.Comment: White Paper submitted to the Astro 2020 Decadal Survey on Astronomy and Astrophysics (National Academies of Science, Engineering, and Medicine