Search

In this chapter, a statistical measure of complexity and the Fisher-Shannon information product are introduced and their properties are discussed. These measures are based on the interplay between the Shannon information, or a function of it, and the separation of the set of accessible states to a system from the equiprobability distribution, i.e. the disequilibrium or the Fisher information, respectively. Different applications in discrete and continuous systems are shown. Some of them are concerned with quantum systems, from prototypical systems such as the H-atom, the harmonic oscillator and the square well to other ones such as He-like ions, Hooke’s atoms or just the periodic table. In all of them, these statistical indicators show an interesting behavior able to discern and highlight some conformational properties of those systems.

The Principle of Indifference (PI) – roughly, the claim that equipossibility entails equiprobability – has long been regarded with suspicion by philosophers and scientists alike. This hostility has a twofold motivation. On one hand, there are the well-known inconsistencies (Bertrand paradoxes) to which the use of the principle leads. On the other, there are some general metaphysical and epistemological worries regarding the principle’s support for the idea that the occurrence of events in the physical world should follow ‘the directives of human ignorance’, as Reichenbach ([1949]/1971, 354) put it. These concerns naturally led philosophers to design strategies to eliminate PI from probabilistic reasoning. In this paper I examine two such strategies and argue that they are not successful. This conclusion, however, should not count so much as an endorsement of PI, but rather as an indication that another approach to explicating its role in probabilistic reasoning is needed.

Many people believe the “Nugget Effect” is the result of rap: in-situ changes in the concentration of a given constituent interest, taking place on a very small scale. They are right however, there is much more to it. Indeed, the “Nugget Effect is the result of at least seven types of variability: 1 The true in-situ, small-scale, random variability; 2 The variability introduced by Constitution Heterogenei during all sampling and subsampling stages, which is function of fragment size and sample or subsample weight 3 The variability introduced by small-scale Distributiol Heterogeneity during all sampling and subsampling stages which is a function of transient segregation as soon a the material has been broken up; 4 The variability introduced by any deviation from an isotropic module of observation ensuring sampling equiprobability in all relevant dimensions, during al: sampling and subsampling stages; 5 The variability introduced by selectivity and poor recovery during all sampling and subsampling stages; 6 The variability introduced by contamination, losses, alteration of physical or chemical properties, and huma errors; and 7 The variability introduced by the analytical procedure

A concern central to any grammar of a language is that of distinguishing those combinations of its elements which constitute the observed sequences. Clearly, not all combinations of elements occur; in English, the phoneme sequence /ls/ does not occur after pause, the word sequence by at here is not an admissible sentence. This redundancy, or departure from equiprobability of occurrence of its elements, is crucial for characterizing language structure, i.e. stating those combinations of its elements which can occur as against those which cannot. Natural language cannot avail itself of any external metalanguage to designate its elements or couch its description; the very possibility of constructing linguistic elements rests on the non-occurrence of various combinations1. It follows that the description of the occurences of a language can only be given in the same language (as a natural language, e.g. English contains its grammar in English as one of its sublanguages) or in another (as a grammar of French in English). The grammatical description already makes use of the same kinds of elements and of their possibilities of combination which are to be defined. Insofar as language structure is descriptively the resultant of a succession of constraints on possible combinations,2 it is essential that grammatical statement of each restriction not contribute to the redundancy it attempts to characterize.3

Among the numerous approaches proposed for the analysis of spatial processes, the autocorrelation index (Geary, 1954) introduced by Cliff & Ord (1973) possesses remarkable properties. For any given variable (presence/absence, cover, quantitative measures of abundance) and random distribution of sampling points in space, this index tests the null hypothesis of absence of correlation for values recorded at two neighbouring points, using a non parametric model for equiprobability of N ! attributions of N numerical values at N sampling points.

There are now 10 years that the theory of random sequences revived beginning with a stimulating paper of Kolmogorov in 1965. In this paper Kolmogorov proposed a definition of finite random sequences with respect to the equiprobability distribution. Those binary sequences x were considered to be random for which the minimal length of a description for x (program complexity of x) differed little from the length of x. The idea that randomness is related to the minimal length of descriptions was independently developed in Chaffin (1966). By introducing the concept of a universal algorithm Kolmogorov gave a rigorous definition of program complexity which no more depends on the choice of a special machine (or algorithm).