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.

Synchronization and Detection Techniques for Navigation and Communication Systems

Synchronization is a key issue in any communication system, but it becomes fundamental in the navigation systems, which are entirely based on the estimation of the time delay of the signals coming from the satellites. Thus, even if synchronization has been a well known topic for many years, the introduction of new modulations and new physical layer techniques in the modern standards makes the traditional synchronization strategies completely ineffective. For this reason, the design of advanced and innovative techniques for synchronization in modern communication systems, like DVB-SH, DVB-T2, DVB-RCS, WiMAX, LTE, and in the modern navigation system, like Galileo, has been the topic of the activity. Recent years have seen the consolidation of two different trends: the introduction of Orthogonal Frequency Division Multiplexing (OFDM) in the communication systems, and of the Binary Offset Carrier (BOC) modulation in the modern Global Navigation Satellite Systems (GNSS). Thus, a particular attention has been given to the investigation of the synchronization algorithms in these areas.

Valorizzazione di cartografia storica attraverso moderne tecniche geomatiche: recupero metrico, elaborazione e consultazione in ambiente digitale

“Cartographic heritage” is different from “cartographic history”. The second term refers to the study of the development of surveying and drawing techniques related to maps, through time, i.e. through different types of cultural environment which were background for the creation of maps. The first term concerns the whole amount of ancient maps, together with these different types of cultural environment, which the history has brought us and which we perceive as cultural values to be preserved and made available to many users (public, institutions, experts). Unfortunately, ancient maps often suffer preservation problems of their analog support, mostly due to aging. Today, metric recovery in digital form and digital processing of historical cartography allow preserving map heritage. Moreover, modern geomatic techniques give us new chances of using historical information, which would be unachievable on analog supports. In this PhD thesis, the whole digital processing of recovery and elaboration of ancient cartography is reported, with a special emphasis on the use of digital tools in preservation and elaboration of cartographic heritage. It is possible to divide the workflow into three main steps, that reflect the chapter structure of the thesis itself: • map acquisition: conversion of the ancient map support from analog to digital, by means of high resolution scanning or 3D surveying (digital photogrammetry or laser scanning techniques); this process must be performed carefully, with special instruments, in order to reduce deformation as much as possible; • map georeferencing: reproducing in the digital image the native metric content of the map, or even improving it by selecting a large number of still existing ground control points; this way it is possible to understand the projection features of the historical map, as well as to evaluate and represent the degree of deformation induced by the old type of cartographic transformation (that can be unknown to us), by surveying errors or by support deformation, usually all errors of too high value with respect to our standards; • data elaboration and management in a digital environment, by means of modern software tools: vectorization, giving the map a new and more attractive graphic view (for instance, by creating a 3D model), superimposing it on current base maps, comparing it to other maps, and finally inserting it in GIS or WebGIS environment as a specific layer. The study is supported by some case histories, each of them interesting from the point of view of one digital cartographic elaboration step at least. The ancient maps taken into account are the following ones: • three maps of the Po river delta, made at the end of the XVI century by a famous land-surveyor, Ottavio Fabri (he is single author in the first map, co-author with Gerolamo Pontara in the second map, co-author with Bonajuto Lorini and others in the third map), who wrote a methodological textbook where he explains a new topographical instrument, the squadra mobile (mobile square) invented and used by himself; today all maps are preserved in the State Archive of Venice; • the Ichnoscenografia of Bologna by Filippo de’ Gnudi, made in the 1702 and today preserved in the Archiginnasio Library of Bologna; it is a scenographic view of the city, captured in a bird’s eye flight, but also with an icnographic value, as the author himself declares; • the map of Bologna by the periti Gregorio Monari and Antonio Laghi, the first map of the city derived from a systematic survey, even though it was made only ten years later (1711–1712) than the map by de’ Gnudi; in this map the scenographic view was abandoned, in favor of a more correct representation by means of orthogonal projection; today the map is preserved in the State Archive of Bologna; • the Gregorian Cadastre of Bologna, made in 1831 and updated until 1927, now preserved in the State Archive of Bologna; it is composed by 140 maps and 12 brogliardi (register volumes). In particular, the three maps of the Po river delta and the Cadastre were studied with respect to their acquisition procedure. Moreover, the first maps were analyzed from the georeferencing point of view, and the Cadastre was analyzed with respect to a possible GIS insertion. Finally, the Ichnoscenografia was used to illustrate a possible application of digital elaboration, such as 3D modeling. Last but not least, we must not forget that the study of an ancient map should start, whenever possible, from the consultation of the precious original analogical document; analysis by means of current digital techniques allow us new research opportunities in a rich and modern multidisciplinary context.