Combinatorial Games with a Pass: A dynamical systems approach

By treating combinatorial games as dynamical systems, we are able to address a longstanding open question in combinatorial game theory, namely, how the introduction of a "pass" move into a game affects its behavior. We consider two well known combinatorial games, 3-pile Nim and 3-row Chomp. In the case of Nim, we observe that the introduction of the pass dramatically alters the game's underlying structure, rendering it considerably more complex, while for Chomp, the pass move is found to have relatively minimal impact. We show how these results can be understood by recasting these games as dynamical systems describable by dynamical recursion relations. From these recursion relations we are able to identify underlying structural connections between these "games with passes" and a recently introduced class of "generic (perturbed) games." This connection, together with a (non-rigorous) numerical stability analysis, allows one to understand and predict the effect of a pass on a game.Comment: 39 pages, 13 figures, published versio

The Albedo, Size, and Density of Binary Kuiper Belt Object (47171) 1999 TC36

We measured the system-integrated thermal emission of the binary Kuiper Belt Object 1999 TC36 at wavelengths near 24 and 70 microns using the Spitzer space telescope. We fit these data and the visual magnitude using both the Standard Thermal Model and thermophysical models. We find that the effective diameter of the binary is 405 km, with a range of 350 -- 470 km, and the effective visible geometric albedo for the system is 0.079 with a range of 0.055 -- 0.11. The binary orbit, magnitude contrast between the components, and system mass have been determined from HST data (Margot et al., 2004; 2005a; 2005b). Our effective diameter, combined with that system mass, indicate an average density for the objects of 0.5 g/cm3, with a range 0.3 -- 0.8 g/cm3. This density is low compared to that of materials expected to be abundant in solid bodies in the trans-Neptunian region, requiring 50 -- 75% of the interior of 1999 TC36 be taken up by void space. This conclusion is not greatly affected if 1999 TC36 is ``differentiated'' (in the sense of having either a rocky or just a non-porous core). If the primary is itself a binary, the average density of that (hypothetical) triple system would be in the range 0.4 -- 1.1 g/cm3, with a porosity in the range 15 -- 70%.Comment: ApJ, in press (May, 2006

Spontaneous Polarisation Build up in a Room Temperature Polariton Laser

We observe the build up of strong (~50%) spontaneous vector polarisation in emission from a GaN-based polariton laser excited by short optical pulses at room temperature. The Stokes vector of emitted light changes its orientation randomly from one excitation pulse to another, so that the time-integrated polarisation remains zero. This behaviour is completely different to any previous laser. We interpret this observation in terms of the spontaneous symmetry breaking in a Bose-Einstein condensate of exciton-polaritons