Some remarks on Bell's Inequality tests

We emphasize the difficulties of an experiment that can definitely discriminate between local realistic hidden variables theories and quantum mechanics using the Bell CHSH inequalities and a real measurement apparatus. In particular we analyze some examples in which the noise in real instruments can alter the experimental results, and the nontrivial problem to find a real "fair sample" of particles to test the inequalities.Comment: 12 pages, Latex, 2 figures, to be published in Europhysics Letter

A dynamical symmetry breaking model in Weyl space

The dynamical process following the breaking of Weyl geometry to Riemannian geometry is considered by studying the motion of de Sitter bubbles in a Weyl vacuum. The bubbles are given in terms of an exact, spherically symmetric thin shell solution to the Einstein equations in a Weyl-Dirac theory with a time-dependent scalar field of the form beta = f(t)/r. The dynamical solutions obtained lead to a number of possible applications. An important feature of the thin shell model is the manner in which beta provides a connection between the interior and exterior geometries since information about the exterior geometry is contained in the boundary conditions for beta.Comment: 18 pages, RevTex, to be published in J. Math. Phy

The history and the meaning of Einstein’s Principle of Equivalence

We review from a historical and a didactic point of view the Equivalence Principle, which was considered by Einstein as the corner stone of his new theory of Gravitation: the General Relativity. Before and after the enormous success of his theory, this principle was the subject of studies and discussions. Still today, after more than one century, the debate about its interpretation, application and generalization is very fertile. Einstein soon understood the revolutionary significance of his idea and defined it as “the happiest thought of my life”

Classification of plant communities and fuzzy diversity of vegetation systems

After stressing the need to keep separated the concept of variability and/or inequality and dissimilarity from that of diversity, it is suggested that diversity of a system should be measured primarily by the number of different classes (K) we can define in it (richness) by classification or identification processes. An index d, ranging between 0 and 1, that summarizes the similarity pattern within the system, can be used if necessary to transform K to a “fuzzy” diversity number, according to the idea that the higher is the similarity within the system the lower should be its diversity. Another index, r, is proposed to measure the “loss” of diversity due to similarity within the system, an index that fits the concept of “redundancy”. Since every diversity vector may be interpreted as a crisp symmetric similarity matrix, of which the Gini-Simpson’s index is the average dissimilarity, while the index of Shannon is the entropy of its eigenvalues, the index d can be chosen to quantify one among the following similarities: a) the overall average similarity of the classes considering the within classes similarity equal to 1 and the between classes similarity equal to 0 (crisp similarity pattern): this is coincident with the evenness of the proportion of importance of the classes, b) the average similarity between the classes without considering evenness, or c) the combination of the two similarities (similarity between the classes and evenness). In these last two cases, the similarity between the classes is characterizing the similarity pattern of a system in a fuzzy way (fuzzy diversity). It is stressed that the diversity of vegetation systems may be of two complementary types: plant individual-based diversity and plant community-based diversity. If we assume that each plant community type corresponds to one habitat then habitat diversity (or niche width) can be calculated for each class of plant individuals according to the number of classes of plant communities in which we can find it. Habitat diversity can be used to measure the indicator value of species or other classes of plant individuals and of plant communities. In this last case, we have to consider the distribution of plant communities in classes defined by environmental factors. It is suggested that the terminology alpha, beta, gamma diversity can be useful only if used to distinguish types of diversity in vegetation systems: alpha diversity = plant individual based diversity, gamma diversity = the union of alpha diversities, beta diversity = plant community based diversity. Thanks to the availability of mathematical tools, it is concluded that rather than being worried about measuring diversity it would be more fruitful to worry about why we are willing to measure it