Indicatori di correlazione e di disordine basati sul concetto di entropia

The aim of this work is to carry out an applicative, comparative and exhaustive study between several entropy based indicators of independence and correlation. We considered some indicators characterized by a wide and consolidate literature, like mutual information, joint entropy, relative entropy or Kullback Leibler distance, and others, more recently introduced, like Granger, Maasoumi and racine entropy, also called Sρ, or utilized in more restricted domains, like Pincus approximate entropy or ApEn. We studied the behaviour of such indicators applying them to binary series. The series was designed to simulate a wide range of situations in order to characterize indicators limit and capability and to identify, case by case, the more useful and trustworthy ones. Our target was not only to study if such indicators were able to discriminate between dependence and independence because, especially for mutual information and Granger, Maasoumi and Racine, that was already demonstrated and reported in literature, but also to verify if and how they were able to provide information about structure, complexity and disorder of the series they were applied to. Special attention was paid on Pincus approximate entropy, that is said by the author to be able to provide information regarding the level of randomness, regularity and complexity of a series. By means of a focused and extensive research, we furthermore tried to clear the meaning of ApEn applied to a couple of different series. In such situation the indicator is named in literature as cross-ApEn. The cross-ApEn meaning and the interpretation of its results is often not simple nor univocal and the matter is scarcely delved into by literature, thereby users can easily leaded up to a misleading conclusion, especially if the indicator is employed, as often unfortunately it happens, in uncritical manner. In order to plug some cross-ApEn gaps and limits clearly brought out during the experimentation, we developed and applied to the already considered cases a further indicator we called “correspondence index”. The correspondence index is perfectly integrated into the cross-ApEn computational algorithm and it is able to provide, at least for binary data, accurate information about the intensity and the direction of an eventual correlation, even not linear, existing between two different series allowing, in the meanwhile, to detect an eventual condition of independence between the series themselves.

Cosmic-Lab: Optical companions to binary Millisecond Pulsars

Millisecond Pulsars (MSPs) are fast rotating, highly magnetized neutron stars. According to the "canonical recycling scenario", MSPs form in binary systems containing a neutron star which is spun up through mass accretion from the evolving companion. Therefore, the final stage consists of a binary made of a MSP and the core of the deeply peeled companion. In the last years, however an increasing number of systems deviating from these expectations has been discovered, thus strongly indicating that our understanding of MSPs is far to be complete. The identification of the optical companions to binary MSPs is crucial to constrain the formation and evolution of these objects. In dense environments such as Globular Clusters (GCs), it also allows us to get insights on the cluster internal dynamics. By using deep photometric data, acquired both from space and ground-based telescopes, we identified 5 new companions to MSPs. Three of them being located in GCs and two in the Galactic Field. The three new identifications in GCs increased by 50% the number of such objects known before this Thesis. They all are non-degenerate stars, at odds with the expectations of the "canonical recycling scenario". These results therefore suggest either that transitory phases should also be taken into account, or that dynamical processes, as exchange interactions, play a crucial role in the evolution of MSPs. We also performed a spectroscopic follow-up of the companion to PSRJ1740-5340A in the GC NGC 6397, confirming that it is a deeply peeled star descending from a ~0.8Msun progenitor. This nicely confirms the theoretical expectations about the formation and evolution of MSPs.