A moment symbolic representation of Lévy processes with applications

By using a symbolic method, known in the literature as the classical umbral calculus, a symbolic representation of Lévy processes is given and a new family of time-space harmonic polynomials with respect to such processes, which includes and generalizes the exponential complete Bell polynomials, is introduced. The usefulness of time-space harmonic polynomials with respect to Lévy processes is that it is a martingale the stochastic process obtained by replacing the indeterminate x of the polynomials with a Lévy process, whereas the Lévy process does not necessarily have this property. Therefore to find such polynomials could be particularly meaningful for applications. This new family includes Hermite polynomials, time-space harmonic with respect to Brownian motion, Poisson-Charlier polynomials with respect to Poisson processes, Laguerre and actuarial polynomials with respect to Gamma processes , Meixner polynomials of the first kind with respect to Pascal processes, Euler, Bernoulli, Krawtchuk, and pseudo-Narumi polynomials with respect to suitable random walks. The role played by cumulants is stressed and brought to the light, either in the symbolic representation of Lévy processes and their infinite divisibility property, either in the generalization, via umbral Kailath-Segall formula, of the well-known formulae giving elementary symmetric polynomials in terms of power sum symmetric polynomials. The expression of the family of time-space harmonic polynomials here introduced has some connections with the so-called moment representation of various families of multivariate polynomials. Such moment representation has been studied here for the first time in connection with the time-space harmonic property with respect to suitable symbolic multivariate Lévy processes. In particular, multivariate Hermite polynomials and their properties have been studied in connection with a symbolic version of the multivariate Brownian motion, while multivariate Bernoulli and Euler polynomials are represented as powers of multivariate polynomials which are time-space harmonic with respect to suitable multivariate Lévy processes.

The spatial representation of time

Numerosi studi mostrano che gli intervalli temporali sono rappresentati attraverso un codice spaziale che si estende da sinistra verso destra, dove gli intervalli brevi sono rappresentati a sinistra rispetto a quelli lunghi. Inoltre tale disposizione spaziale del tempo può essere influenzata dalla manipolazione dell’attenzione-spaziale. La presente tesi si inserisce nel dibattito attuale sulla relazione tra rappresentazione spaziale del tempo e attenzione-spaziale attraverso l’uso di una tecnica che modula l’attenzione-spaziale, ovvero, l’Adattamento Prismatico (AP). La prima parte è dedicata ai meccanismi sottostanti tale relazione. Abbiamo mostrato che spostando l’attenzione-spaziale con AP, verso un lato dello spazio, si ottiene una distorsione della rappresentazione di intervalli temporali, in accordo con il lato dello spostamento attenzionale. Questo avviene sia con stimoli visivi, sia con stimoli uditivi, nonostante la modalità uditiva non sia direttamente coinvolta nella procedura visuo-motoria di AP. Questo risultato ci ha suggerito che il codice spaziale utilizzato per rappresentare il tempo, è un meccanismo centrale che viene influenzato ad alti livelli della cognizione spaziale. La tesi prosegue con l’indagine delle aree corticali che mediano l’interazione spazio-tempo, attraverso metodi neuropsicologici, neurofisiologici e di neuroimmagine. In particolare abbiamo evidenziato che, le aree localizzate nell’emisfero destro, sono cruciali per l’elaborazione del tempo, mentre le aree localizzate nell’emisfero sinistro sono cruciali ai fini della procedura di AP e affinché AP abbia effetto sugli intervalli temporali. Infine, la tesi, è dedicata allo studio dei disturbi della rappresentazione spaziale del tempo. I risultati ci indicano che un deficit di attenzione-spaziale, dopo danno emisferico destro, provoca un deficit di rappresentazione spaziale del tempo, che si riflette negativamente sulla vita quotidiana dei pazienti. Particolarmente interessanti sono i risultati ottenuti mediante AP. Un trattamento con AP, efficace nel ridurre il deficit di attenzione-spaziale, riduce anche il deficit di rappresentazione spaziale del tempo, migliorando la qualità di vita dei pazienti.

On the renormalization flow representation of field theory and some applications

In this thesis we discuss a representation of quantum mechanics and quantum and statistical field theory based on a functional renormalization flow equation for the one-particle-irreducible average effective action, and we employ it to get information on some specific systems.

Sparse Signal Representation of Ultrasonic Signals for Structural Health Monitoring Applications

Assessment of the integrity of structural components is of great importance for aerospace systems, land and marine transportation, civil infrastructures and other biological and mechanical applications. Guided waves (GWs) based inspections are an attractive mean for structural health monitoring. In this thesis, the study and development of techniques for GW ultrasound signal analysis and compression in the context of non-destructive testing of structures will be presented. In guided wave inspections, it is necessary to address the problem of the dispersion compensation. A signal processing approach based on frequency warping was adopted. Such operator maps the frequencies axis through a function derived by the group velocity of the test material and it is used to remove the dependence on the travelled distance from the acquired signals. Such processing strategy was fruitfully applied for impact location and damage localization tasks in composite and aluminum panels. It has been shown that, basing on this processing tool, low power embedded system for GW structural monitoring can be implemented. Finally, a new procedure based on Compressive Sensing has been developed and applied for data reduction. Such procedure has also a beneficial effect in enhancing the accuracy of structural defects localization. This algorithm uses the convolutive model of the propagation of ultrasonic guided waves which takes advantage of a sparse signal representation in the warped frequency domain. The recovery from the compressed samples is based on an alternating minimization procedure which achieves both an accurate reconstruction of the ultrasonic signal and a precise estimation of waves time of flight. Such information is used to feed hyperbolic or elliptic localization procedures, for accurate impact or damage localization.