Note on a partition limit theorem for rank and crank

If L is a partition of n, the rank of L is the size of the largest part minus the number of parts. Under the uniform distribution on partitions, Bringmann, Mahlburg, and Rhoades showed that the rank statistic has a limiting distribution. We identify the limit as the difference between two independent extreme value distributions and as the distribution of B(T) where B(t) is standard Brownian motion and T is the first time that an independent three-dimensional Brownian motion hits the unit sphere. The same limit holds for the crank.Comment: 3 page

Network reachability of real-world contact sequences

We use real-world contact sequences, time-ordered lists of contacts from one person to another, to study how fast information or disease can spread across network of contacts. Specifically we measure the reachability time -- the average shortest time for a series of contacts to spread information between a reachable pair of vertices (a pair where a chain of contacts exists leading from one person to the other) -- and the reachability ratio -- the fraction of reachable vertex pairs. These measures are studied using conditional uniform graph tests. We conclude, among other things, that the network reachability depends much on a core where the path lengths are short and communication frequent, that clustering of the contacts of an edge in time tend to decrease the reachability, and that the order of the contacts really do make sense for dynamical spreading processes.Comment: (v2: fig. 1 fixed

Discovery of a stellar companion to the nearby solar-analogue HD 104304

Sun-like stars are promising candidates to host exoplanets and are often included in exoplanet surveys by radial velocity (RV) and direct imaging. In this paper we report on the detection of a stellar companion to the nearby solar-analogue star HD 104304, which previously was considered to host a planetary mass or brown dwarf companion. We searched for close stellar and substellar companions around extrasolar planet host stars with high angular resolution imaging to characterize planet formation environments. The detection of the stellar companion was achieved by high angular resolution measurements, using the "Lucky Imaging" technique at the ESO NTT 3.5m with the AstraLux Sur instrument. We combined the results with VLT/NACO archive data, where the companion could also be detected. The results were compared to precise RV measurements of HD 104304, obtained at the Lick and Keck observatories from 2001-2010. We confirmed common proper motion of the binary system. A spectral type of M4V of the companion and a mass of 0.21 M_Sun was derived. Due to comparison of the data with RV measurements of the unconfirmed planet candidate listed in the Extrasolar Planets Encyclopaedia, we suggest that the discovered companion is the origin of the RV trend and that the inclination of the orbit of approximately 35 degrees explains the relatively small RV signal.Comment: 4 pages, 4 PNG figures, use aa.cls, accepted for publication in Astronomy & Astrophysic

Fully Analyzing an Algebraic Polya Urn Model

This paper introduces and analyzes a particular class of Polya urns: balls are of two colors, can only be added (the urns are said to be additive) and at every step the same constant number of balls is added, thus only the color compositions varies (the urns are said to be balanced). These properties make this class of urns ideally suited for analysis from an "analytic combinatorics" point-of-view, following in the footsteps of Flajolet-Dumas-Puyhaubert, 2006. Through an algebraic generating function to which we apply a multiple coalescing saddle-point method, we are able to give precise asymptotic results for the probability distribution of the composition of the urn, as well as local limit law and large deviation bounds.Comment: LATIN 2012, Arequipa : Peru (2012

High-contrast Imaging with Spitzer: Deep Observations of Vega, Fomalhaut, and epsilon Eridani

Stars with debris disks are intriguing targets for direct imaging exoplanet searches, both due to previous detections of wide planets in debris disk systems, as well as commonly existing morphological features in the disks themselves that may be indicative of a planetary influence. Here we present observations of three of the most nearby young stars, that are also known to host massive debris disks: Vega, Fomalhaut, and eps Eri. The Spitzer Space Telescope is used at a range of orientation angles for each star, in order to supply a deep contrast through angular differential imaging combined with high-contrast algorithms. The observations provide the opportunity to probe substantially colder bound planets (120--330 K) than is possible with any other technique or instrument. For Vega, some apparently very red candidate point sources detected in the 4.5 micron image remain to be tested for common proper motion. The images are sensitive to ~2 Mjup companions at 150 AU in this system. The observations presented here represent the first search for planets around Vega using Spitzer. The upper 4.5 micron flux limit on Fomalhaut b could be further constrained relative to previous data. In the case of eps Eri, planets below both the effective temperature and the mass of Jupiter could be probed from 80 AU and outwards, although no such planets were found. The data sensitively probe the regions around the edges of the debris rings in the systems where planets can be expected to reside. These observations validate previous results showing that more than an order of magnitude improvement in performance in the contrast-limited regime can be acquired with respect to conventional methods by applying sophisticated high-contrast techniques to space-based telescopes, thanks to the high degree of PSF stability provided in this environment.Comment: 11 pages, 12 figures, accepted for publication in A&

The first order convergence law fails for random perfect graphs

We consider first order expressible properties of random perfect graphs. That is, we pick a graph $G_n$ uniformly at random from all (labelled) perfect graphs on $n$ vertices and consider the probability that it satisfies some graph property that can be expressed in the first order language of graphs. We show that there exists such a first order expressible property for which the probability that $G_n$ satisfies it does not converge as $n\to\infty$.Comment: 11 pages. Minor corrections since last versio

Spatially resolved spectroscopy of the exoplanet HR 8799 c

HR 8799 is a multi-planet system detected in direct imaging, with three companions known so far. Here, we present spatially resolved VLT/NACO 3.88--4.10 micron spectroscopy of the middle planet, HR 8799 c, which has an estimated mass of ~10 Mjup, temperature of ~1100 K and projected separation of 38 AU. The spectrum shows some differences in the continuum from existing theoretical models, particularly longwards of 4 microns, implying that detailed cloud structure or non-equilibrium conditions may play an important role in the physics of young exoplanetary atmospheres.Comment: 4 pages, 4 figures, accepted for publication in ApJ Letter

Critical behaviour of combinatorial search algorithms, and the unitary-propagation universality class

The probability P(alpha, N) that search algorithms for random Satisfiability problems successfully find a solution is studied as a function of the ratio alpha of constraints per variable and the number N of variables. P is shown to be finite if alpha lies below an algorithm--dependent threshold alpha\_A, and exponentially small in N above. The critical behaviour is universal for all algorithms based on the widely-used unitary propagation rule: P[ (1 + epsilon) alpha\_A, N] ~ exp[-N^(1/6) Phi(epsilon N^(1/3)) ]. Exponents are related to the critical behaviour of random graphs, and the scaling function Phi is exactly calculated through a mapping onto a diffusion-and-death problem.Comment: 7 pages; 3 figure