Non-PORC behaviour of a class of descendant pp-groups

We prove that the number of immediate descendants of order $p^10$ of $G_p$ is not PORC (Polynomial On Residue Classes) where $G_p$ is the $p$-group of order $p^9$ defined by du Sautoy's nilpotent group encoding the elliptic curve $y^2=x^3-x$. This has important implications for Higman's PORC conjecture

Duality, Phase Structures and Dilemmas in Symmetric Quantum Games

Symmetric quantum games for 2-player, 2-qubit strategies are analyzed in detail by using a scheme in which all pure states in the 2-qubit Hilbert space are utilized for strategies. We consider two different types of symmetric games exemplified by the familiar games, the Battle of the Sexes (BoS) and the Prisoners' Dilemma (PD). These two types of symmetric games are shown to be related by a duality map, which ensures that they share common phase structures with respect to the equilibria of the strategies. We find eight distinct phase structures possible for the symmetric games, which are determined by the classical payoff matrices from which the quantum games are defined. We also discuss the possibility of resolving the dilemmas in the classical BoS, PD and the Stag Hunt (SH) game based on the phase structures obtained in the quantum games. It is observed that quantization cannot resolve the dilemma fully for the BoS, while it generically can for the PD and SH if appropriate correlations for the strategies of the players are provided.Comment: 12 pages, 14 figure

The recreational value of beaches in the Nelson Mandela Bay area, South Africa

Using beach visitation data collected via the administration of a questionnaire to 226 respondents, this paper estimates a random utility model of beach recreation. The relative value of selected attributes of beaches is estimated, and the recreational values of lost access to four Blue Flag beaches in the Nelson Mandela Bay area, namely Kings beach, Humewood beach, Hobie beach and Wells Estate beach, respectively are calculated to be R44.73, R24.61, R37.85 and R2.68 per person, per trip.

Coalitions in the quantum Minority game: classical cheats and quantum bullies

In a one-off Minority game, when a group of players agree to collaborate they gain an advantage over the remaining players. We consider the advantage obtained in a quantum Minority game by a coalition sharing an initially entangled state versus that obtained by a coalition that uses classical communication to arrive at an optimal group strategy. In a model of the quantum Minority game where the final measurement basis is randomized, quantum coalitions outperform classical ones when carried out by up to four players, but an unrestricted amount of classical communication is better for larger coalition sizes.Comment: 12 pages, 1 figur