The variance conjecture on projections of the cube

We prove that the uniform probability measure $\mu$ on every $(n-k)$-dimensional projection of the $n$-dimensional unit cube verifies the variance conjecture with an absolute constant $C$ $$\textrm{Var}_\mu|x|^2\leq C \sup_{\theta\in S^{n-1}}{\mathbb E}_\mu\langle x,\theta\rangle^2{\mathbb E}_\mu|x|^2,$$ provided that $1\leq k\leq\sqrt n$. We also prove that if $1\leq k\leq n^{\frac{2}{3}}(\log n)^{-\frac{1}{3}}$, the conjecture is true for the family of uniform probabilities on its projections on random $(n-k)$-dimensional subspaces

On the limits of engine analysis for cheating detection in chess

The integrity of online games has important economic consequences for both the gaming industry and players of all levels, from professionals to amateurs. Where there is a high likelihood of cheating, there is a loss of trust and players will be reluctant to participate — particularly if this is likely to cost them money. Chess is a game that has been established online for around 25 years and is played over the Internet commercially. In that environment, where players are not physically present “over the board” (OTB), chess is one of the most easily exploitable games by those who wish to cheat, because of the widespread availability of very strong chess-playing programs. Allegations of cheating even in OTB games have increased significantly in recent years, and even led to recent changes in the laws of the game that potentially impinge upon players’ privacy. In this work, we examine some of the difficulties inherent in identifying the covert use of chess-playing programs purely from an analysis of the moves of a game. Our approach is to deeply examine a large collection of games where there is confidence that cheating has not taken place, and analyse those that could be easily misclassified. We conclude that there is a serious risk of finding numerous “false positives” and that, in general, it is unsafe to use just the moves of a single game as prima facie evidence of cheating. We also demonstrate that it is impossible to compute definitive values of the figures currently employed to measure similarity to a chess-engine for a particular game, as values inevitably vary at different depths and, even under identical conditions, when multi-threading evaluation is used

Microscopic entropy of the three-dimensional rotating black hole of BHT massive gravity

Asymptotically AdS rotating black holes for the Bergshoeff-Hohm-Townsend (BHT) massive gravity theory in three dimensions are considered. In the special case when the theory admits a unique maximally symmetric solution, apart from the mass and the angular momentum, the black hole is described by an independent "gravitational hair" parameter, which provides a negative lower bound for the mass. This bound is saturated at the extremal case and, since the temperature and the semiclassical entropy vanish, it is naturally regarded as the ground state. The absence of a global charge associated with the gravitational hair parameter reflects through the first law of thermodynamics in the fact that the variation of this parameter can be consistently reabsorbed by a shift of the global charges, giving further support to consider the extremal case as the ground state. The rotating black hole fits within relaxed asymptotic conditions as compared with the ones of Brown and Henneaux, such that they are invariant under the standard asymptotic symmetries spanned by two copies of the Virasoro generators, and the algebra of the conserved charges acquires a central extension. Then it is shown that Strominger's holographic computation for general relativity can also be extended to the BHT theory; i.e., assuming that the quantum theory could be consistently described by a dual conformal field theory at the boundary, the black hole entropy can be microscopically computed from the asymptotic growth of the number of states according to Cardy's formula, in exact agreement with the semiclassical result.Comment: 10 pages, no figure

Procinura, P. aemula

Number of Pages: 4Integrative BiologyGeological Science