We discuss some aspects of the extension to continuous systems of a statistical measure of complexity introduced by López-Ruiz, Mancini, and Calbet [Phys. Lett. A 209, 321 (1995)]. In general, the extension of a magnitude from the discrete to the continuous case is not a trivial process and requires some kind of choice. In the present study, several possibilities appear available. One of them is examined in detail. Some interesting properties desirable for any magnitude of complexity are discovered on this particular extension.

Abstract-Since Chavez proposed the highpass filtering procedure to fuse multispectral and panchromatic images, several fusion methods have been developed based on the same principle: to extract from the panchromatic image spatial detail information to later inject it into the multispectral one. In this paper, we present new fusion alternatives based on the same concept, using the multiresolution wavelet decomposition to execute the detail extraction phase and the intensity-hue-saturation (IHS) and principal component analysis (PCA) procedures to inject the spatial detail of the panchromatic image into the multispectral one. The multiresolution wavelet decomposition has been performed using both decimated and undecimated algorithms and the resulting merged images compared both spectral and spatially. These fusion methods, as well as standard IHS-, PCA-, and wavelet-based methods have been used to merge Systeme Pour l'Observation de la Terre (SPOT) 4 XI and SPOT 4 M images with a ratio 4 : 1. We have estimated the validity of each fusion method by analyzing, visually and quantitatively, the quality of the resulting fused images. The methodological approaches proposed in this paper result in merged images with improved quality with respect to those obtained by standard IHS, PCA, and standard wavelet-based fusion methods. For both proposed fusion methods, better results are obtained when an undecimated algorithm is used to perform the multiresolution wavelet decomposition.Index Terms-Decimated wavelet transform, intensity-hue-saturation (IHS) transform, image-fusion, multiresolution analysis, principal component analysis (PCA), undecimated wavelet transform.

In the present paper we show an striking relationship between the existence of solutions of a generalization of the Abel functional equation, and the Scott-Suppes representability of semiordered structures. We analyze this relationship and, in addition, we study other functional equations related to the numerical representability of semiordered structures.

We analyze various models introduced in social choice to aggregate individual preferences. We show that on the basis of most of these models there is a system of functional equations such that, in many cases, the origin of impossibility results in a social choice model is the non-existence of a solution for the corresponding system. Among the functional equations considered, we pay a particular attention to general means and associativity, proving that the existence of an associative bivariate mean is equivalent to the existence of a semilatticial partial order. This key result allows us to explain how the knowledge of associative bivariate means can be used to solve social choice paradoxes. In our analysis we deal both with crisp and fuzzy settings.