Randomness in Competitions

We study the effects of randomness on competitions based on an elementary random process in which there is a finite probability that a weaker team upsets a stronger team. We apply this model to sports leagues and sports tournaments, and compare the theoretical results with empirical data. Our model shows that single-elimination tournaments are efficient but unfair: the number of games is proportional to the number of teams N, but the probability that the weakest team wins decays only algebraically with N. In contrast, leagues, where every team plays every other team, are fair but inefficient: the top $\sqrt{N}$ of teams remain in contention for the championship, while the probability that the weakest team becomes champion is exponentially small. We also propose a gradual elimination schedule that consists of a preliminary round and a championship round. Initially, teams play a small number of preliminary games, and subsequently, a few teams qualify for the championship round. This algorithm is fair and efficient: the best team wins with a high probability and the number of games scales as $N^{9/5}$, whereas traditional leagues require N^3 games to fairly determine a champion.Comment: 10 pages, 8 figures, reviews arXiv:physics/0512144, arXiv:physics/0608007, arXiv:cond-mat/0607694, arXiv:physics/061221

Ricci Solitons on Lorentzian Manifolds with Large Isometry Groups

We show that Lorentzian manifolds whose isometry group is of dimension at least $\frac{1}{2}n(n-1)+1$ are expanding, steady and shrinking Ricci solitons and steady gradient Ricci solitons. This provides examples of complete locally conformally flat and symmetric Lorentzian Ricci solitons which are not rigid

Generic Absorbing Transition in Coevolution Dynamics

We study a coevolution voter model on a network that evolves according to the state of the nodes. In a single update, a link between opposite-state nodes is rewired with probability $p$, while with probability $1-p$ one of the nodes takes its neighbor's state. A mean-field approximation reveals an absorbing transition from an active to a frozen phase at a critical value $p_c=\frac{\mu-2}{\mu-1}$ that only depends on the average degree $\mu$ of the network. The approach to the final state is characterized by a time scale that diverges at the critical point as $\tau \sim |p_c-p|^{-1}$. We find that the active and frozen phases correspond to a connected and a fragmented network respectively. We show that the transition in finite-size systems can be seen as the sudden change in the trajectory of an equivalent random walk at the critical rewiring rate $p_c$, highlighting the fact that the mechanism behind the transition is a competition between the rates at which the network and the state of the nodes evolve.Comment: 5 pages, 4 figure

Spiral structure of the Third Galactic Quadrant and the solution to the Canis Major debate

With the discovery of the Sagittarius dwarf spheroidal (Ibata et al. 1994), a galaxy caught in the process of merging with the Milky Way, the hunt for other such accretion events has become a very active field of astrophysical research. The identification of a stellar ring-like structure in Monoceros, spanning more than 100 degrees (Newberg et al. 2002), and the detection of an overdensity of stars in the direction of the constellation of Canis Major (CMa, Martin et al. 2004), apparently associated to the ring, has led to the widespread belief that a second galaxy being cannibalised by the Milky Way had been found. In this scenario, the overdensity would be the remaining core of the disrupted galaxy and the ring would be the tidal debris left behind. However, unlike the Sagittarius dwarf, which is well below the Galactic plane and whose orbit, and thus tidal tail, is nearly perpendicular to the plane of the Milky Way, the putative CMa galaxy and ring are nearly co-planar with the Galactic disk. This severely complicates the interpretation of observations. In this letter, we show that our new description of the Milky Way leads to a completely different picture. We argue that the Norma-Cygnus spiral arm defines a distant stellar ring crossing Monoceros and the overdensity is simply a projection effect of looking along the nearby local arm. Our perspective sheds new light on a very poorly known region, the third Galactic quadrant (3GQ), where CMa is located.Comment: 5 pages, 2 figures. Quality of Fig 1 has been degraded to make it smaller. Original fig. available on request. accepted for publication in MNRAS letter

The pressure distribution in thermally bistable turbulent flows

We present a systematic numerical study of the effect of turbulent velocity fluctuations on the thermal pressure distribution in thermally bistable flows. The simulations employ a random turbulent driving generated in Fourier space rather than star-like heating. The turbulent fluctuations are characterized by their rms Mach number M and the energy injection wavenumber, k_for. Our results are consistent with the picture that as either of these parameters is increased, the local ratio of turbulent crossing time to cooling time decreases, causing transient structures in which the effective behavior is intermediate between the thermal-equilibrium and adiabatic regimes. As a result, the effective polytropic exponent gamma_ef ranges between ~0.2 to ~1.1. The fraction of high-density zones with P>10^4 Kcm^-3 increases from roughly 0.1% at k_for=2 and M=0.5 to roughly 70% for k_for=16 and M=1.25. A preliminary comparison with the pressure measurements of Jenkins (2004) favors our case with M=0.5 and k_for=2. In all cases, the dynamic range of the pressure summed over the entire density range, typically spans 3-4 orders of magnitude. The total pressure histogram widens as the Mach number is increased, and develops near-power-law tails at high (resp.low) pressures when gamma_ef<~ 0.5 (resp. gamma_ef>~ 1), which occurs at k_for=2 (resp.k_for=16) in our simulations. The opposite side of the pressure histogram decays rapidly, in an approx. lognormal form. Our results show that turbulent advection alone can generate large pressure scatters, with power-law high-P tails for large-scale driving, and provide validation for approaches attempting to derive the shape of the pressure histogram through a change of variable from the known form of the density histogram, such as that performed by MacLow et al.(2004).Comment: to be published in Ap

Ignition characteristics of the iron-based alloy UNS S66286 in pressurized oxygen

The development of ignition and combustion in pressurized oxygen atmospheres was studied for the iron based alloy UNS S66286. Ignition of the alloy was achieved by heating the top surface of a cylindrical specimen with a continuous-wave CO2 laser. Two heating procedures were used. In the first, laser power was adjusted to maintain an approximately linear increase in surface temperature. In the second, laser power was periodically increased until autoheating (self-heating) was established. It was found that the alloy would autoheat to destruction from temperatures below the solidus temperature. In addition endothermic events occurred as the alloy was heated, many at reproducible temperatures. Many endothermic events occurred prior to abrupt increases in surface temperature and appeared to accelerate the rate of increase in specimen temperature to rates greater than what would be expected from increased temperature alone. It is suggested that the source of these endotherms may increase the oxidation rate of the alloy. Ignition parameters are defined and the temperatures at which these parameters occur are given for the oxygen pressure range of 1.72 to 13.8 MPa (25 to 2000 psia)

Ignition characteristics of the nickel-based alloy UNS N07718 in pressurized oxygen

The development of ignition and combustion in pressurized oxygen atmospheres was studied for the nickel based alloy UNS N07718. Ignition of the alloy was achieved by heating the top. It was found that the alloy would autoheat to destruction from temperatures below the solidus temperature. In addition, endothermic events occurred as the alloy was heated, many at reproducible temperatures. Many endothermic events occurred prior to abrupt increases in surface temperature and appeared to accelerate the rate of increase in specimen temperature. It appeared that the source of some endotherms may increase the oxidation rate of the alloy. Ignition parameters are defined and the temperatures at which these parameters occur are given for the oxygen pressure range of 1.72 to 13.8 MPa (250 to 2000 psia)