Optimization results for a generalized coupon collector problem

We study in this paper a generalized coupon collector problem, which consists in analyzing the time needed to collect a given number of distinct coupons that are drawn from a set of coupons with an arbitrary probability distribution. We suppose that a special coupon called the null coupon can be drawn but never belongs to any collection. In this context, we prove that the almost uniform distribution, for which all the non-null coupons have the same drawing probability, is the distribution which stochastically minimizes the time needed to collect a fixed number of distinct coupons. Moreover, we show that in a given closed subset of probability distributions, the distribution with all its entries, but one, equal to the smallest possible value is the one, which stochastically maximizes the time needed to collect a fixed number of distinct coupons. An computer science application shows the utility of these results.Comment: arXiv admin note: text overlap with arXiv:1402.524

Substellar Objects in Nearby Young Clusters VII: The substellar mass function revisited

The abundance of brown dwarfs (BDs) in young clusters is a diagnostic of star formation theory. Here we revisit the issue of determining the substellar initial mass function (IMF), based on a comparison between NGC1333 and IC348, two clusters in the Perseus star-forming region. We derive their mass distributions for a range of model isochrones, varying distances, extinction laws and ages, with comprehensive assessments of the uncertainties. We find that the choice of isochrone and other parameters have significant effects on the results, thus we caution against comparing IMFs obtained using different approaches. For NGC1333, we find that the star/BD ratio R is between 1.9 and 2.4, for all plausible scenarios, consistent with our previous work. For IC348, R is between 2.9 and 4.0, suggesting that previous studies have overestimated this value. Thus, the star forming process generates about 2.5-5 substellar objects per 10 stars. The derived star/BD ratios correspond to a slope of the power-law mass function of alpha=0.7-1.0 for the 0.03-1.0Msol mass range. The median mass in these clusters - the typical stellar mass - is between 0.13-0.30Msol. Assuming that NGC1333 is at a shorter distance than IC348, we find a significant difference in the cumulative distribution of masses between the two clusters, resulting from an overabundance of very low mass objects in NGC1333. Gaia astrometry will constrain the cluster distances better and will lead to a more definitive conclusion. Furthermore, ratio R is somewhat larger in IC348 compared with NGC1333, although this difference is still within the margins of error. Our results indicate that environments with higher object density may produce a larger fraction of very low mass objects, in line with predictions for brown dwarf formation through gravitational fragmentation of filaments falling into a cluster potential.Comment: 16 pages, 4 figures, accepted for publication in Ap

Substellar Objects in Nearby Young Clusters (SONYC) VI: The planetary-mass domain of NGC1333

Within the SONYC - Substellar Objects in Nearby Young Clusters - survey, we investigate the frequency of free-floating planetary-mass objects (planemos) in the young cluster NGC1333. Building upon our extensive previous work, we present spectra for 12 of the faintest candidates from our deep multi-band imaging, plus seven random objects in the same fields, using MOIRCS on Subaru. We confirm seven new sources as young very low mass objects (VLMOs), with Teff of 2400-3100K and mid-M to early-L spectral types. These objects add to the growing census of VLMOs in NGC1333, now totaling 58. Three confirmed objects (one found in this study) have masses below 15 MJup, according to evolutionary models, thus are likely planemos. We estimate the total planemo population with 5-15 MJup in NGC1333 is <~8. The mass spectrum in this cluster is well approximated by dN/dM ~ M^-alpha, with a single value of alpha = 0.6+/-0.1 for M<0.6Msol, consistent with other nearby star forming regions, and requires alpha <~ 0.6 in the planemo domain. Our results in NGC1333, as well as findings in several other clusters by ourselves and others, confirm that the star formation process extends into the planetary-mass domain, at least down to 6 MJup. However, given that planemos are 20-50 times less numerous than stars, their contribution to the object number and mass budget in young clusters is negligible. Our findings disagree strongly with the recent claim from a microlensing study that free-floating planetary-mass objects are twice as common as stars - if the microlensing result is confirmed, those isolated Jupiter-mass objects must have a different origin from brown dwarfs and planemos observed in young clusters.Comment: 14 pages, 11 figures. Updated version after proof corrections, additional comment in Sect. 5.