The Asymptotic distribution of circles in the orbits of Kleinian groups

Let P be a locally finite circle packing in the plane invariant under a non-elementary Kleinian group Gamma and with finitely many Gamma-orbits. When Gamma is geometrically finite, we construct an explicit Borel measure on the plane which describes the asymptotic distribution of small circles in P, assuming that either the critical exponent of Gamma is strictly bigger than 1 or P does not contain an infinite bouquet of tangent circles glued at a parabolic fixed point of Gamma. Our construction also works for P invariant under a geometrically infinite group Gamma, provided Gamma admits a finite Bowen-Margulis-Sullivan measure and the Gamma-skinning size of P is finite. Some concrete circle packings to which our result applies include Apollonian circle packings, Sierpinski curves, Schottky dances, etc.Comment: 31 pages, 8 figures. Final version. To appear in Inventiones Mat

Marginal contribution stochastic games for dynamic resource allocation

We develop a new formalism for solving team Markov decision processes (MDPs), called marginal–contribution stochastic games (MCSGs). In MCSGs, each agent’s utility for a state transition is given by its marginal contribution to the team value function so that utilities differ between agents, and sparse interaction between them is naturally exploited. We prove that a MCSG admits a potential function and show that the locally optimal solutions, including the global optimum, correspond to the Nash equilibria of the game. We go on to show that any Nash equilibrium of a dynamic resource allocation problem with monotone submodular resource functions in MCSG form has a price of anarchy of > 1/2. Finally, we characterize a class of distributed algorithms for MCSGs

Makers, Owners and Users of Music Sources Before 1600: Sources of Identity

print Share/Save/Bookmark Leaving aside the traditional view of early music sources as a means of access to medieval and Renaissance repertoires, this anthology focuses instead on the people who commissioned, made, owned and used music books, and on their reasons for so doing. The chapters in this volume were presented, in much shorter form, at a conference held at the University of Sheffield in 2013. The aim of the event was to leave aside the traditionally dominant view of early music sources as a means of access to medieval and Renaissance repertoires, focussing instead on the people who commissioned, made, owned and used music books, and on their reasons for so doing. In the terms proposed by a recent study of art patronage in the period, what was the ‘payoff’ enjoyed by individuals and groups who created and deployed such objects