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We establish sufficient conditions under which the harmonic measure distribution functions hn of a sequence of domains Dn converge pointwise to the distribution function h of the limiting domain D, at all points of continuity of h. In the case of a model example, we establish this convergence of the distribution functions. Here, the value of the function h(r) gives the harmonic measure of the part of the boundary of the domain that lies within distance r of a fixed basepoint in the domain, thus relating the geometry of the domain to the behaviour of Brownian motion in the domain.

This paper considers a network of energy harvesting wireless nodes transmitting simultaneously in a Gaussian interference channel and investigates a distributed power allocation algorithm that maximizes the sum-rate. The power consumption model is based on a series of step functions that allow to model, among others, radio frequency circuits being on/off and the startup power consumption of the transmitter. After showing that the sum-rate maximization problem is nonsmooth, nonconvex, and NP-hard, the Iterative Smooth and Convex approximation Algorithm (ISCA) is proposed, which successively approximates the step functions by proper smooth functions to obtain a sequence of smooth nonconvex problems that can be solved by means of the successive convex approximation method. It is demonstrated that the ISCA distributedly converges to a stationary solution of the sum-rate maximization problem. For the particular case of point to point communications, the numerical results show that the ISCA is able to avoid bad stationary solutions, performing close to the globally optimal solution. The performance of the ISCA is also evaluated in the interference channel and with real solar energy harvesting data.

Objective: The study investigates the effect on cooperation in multiplayer games, when the population from which all individuals are drawn is structured - i.e. when a given individual is only competing with a small subset of the entire population. Method: To optimize the focus on multiplayer effects, a class of games were chosen for which the payoff depends nonlinearly on the number of cooperators - this ensures that the game cannot be represented as a sum of pair-wise interactions, and increases the likelihood of observing behaviour different from that seen in two-player games. The chosen class of games are named "threshold games", and are defined by a threshold, $M > 0$, which describes the minimal number of cooperators in a given match required for all the participants to receive a benefit. The model was studied primarily through numerical simulations of large populations of individuals, each with interaction neighbourhoods described by various classes of networks. Results: When comparing the level of cooperation in a structured population to the mean-field model, we find that most types of structure lead to a decrease in cooperation. This is both interesting and novel, simply due to the generality and breadth of relevance of the model - it is likely that any model with similar payoff structure exhibits related behaviour. More importantly, we find that the details of the behaviour depends to a large extent on the size of the immediate neighbourhoods of the individuals, as dictated by the network structure. In effect, the players behave as if they are part of a much smaller, fully mixed, population, which we suggest an expression for., in PLOS ONE, 4th Feb 2016