Abstract
In this paper, we apply the following four power indices to the Portuguese Parliament: Shapley-Shubik index, Banzhaf index, Deegan-Packel index and Public Good Index. We also present the main concepts related with simple games and discuss the features of each power index by means of their axiomatic characterizations.

Abstract
The coordinate change eigenvalues for the MacKay and van Zeijts period doubling renormalisation operator for bimodal 1D maps are derived. They are found in numerical computations of the spectrum at all the periodic orbits of renormalisation of period up to five.

Abstract
We compute the analytic expression of the probability distributions F(AEX,+) and F(AEX,-) of the normalized positive and negative AEX (Netherlands) index daily returns r(t). Furthermore, we define the alpha re-scaled AEX daily index positive returns r(t)(alpha) and negative returns (-r(t))(alpha), which we call, after normalization, the alpha positive fluctuations and alpha negative fluctuations. We use the Kolmogorov-Smirnov statistical test as a method to find the values of alpha that optimize the data collapse of the histogram of the alpha fluctuations with the Bramwell-Holdsworth-Pinton (BHP) probability density function. The optimal parameters that we found are alpha(+) = 0.46 and alpha(-) = 0.43. Since the BHP probability density function appears in several other dissimilar phenomena, our result reveals a universal feature of stock exchange markets.

Abstract
We give an elementary proof of existence and uniqueness of Gibbs states for Holder weight systems on subshifts of finite type. This uses a notion of duality for such subshifts. The approach of Paterson [2] is used to construct a measure with a prescribed Jacobian and the duality is used to produce an invariant measure from this.

Abstract

Abstract