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.

Antipodal random sequences with prescribed second-order statistics: application to Compressive Sensing and UWB system based on DS-CDMA

It is usual to hear a strange short sentence: «Random is better than...». Why is randomness a good solution to a certain engineering problem? There are many possible answers, and all of them are related to the considered topic. In this thesis I will discuss about two crucial topics that take advantage by randomizing some waveforms involved in signals manipulations. In particular, advantages are guaranteed by shaping the second order statistic of antipodal sequences involved in an intermediate signal processing stages. The first topic is in the area of analog-to-digital conversion, and it is named Compressive Sensing (CS). CS is a novel paradigm in signal processing that tries to merge signal acquisition and compression at the same time. Consequently it allows to direct acquire a signal in a compressed form. In this thesis, after an ample description of the CS methodology and its related architectures, I will present a new approach that tries to achieve high compression by design the second order statistics of a set of additional waveforms involved in the signal acquisition/compression stage. The second topic addressed in this thesis is in the area of communication system, in particular I focused the attention on ultra-wideband (UWB) systems. An option to produce and decode UWB signals is direct-sequence spreading with multiple access based on code division (DS-CDMA). Focusing on this methodology, I will address the coexistence of a DS-CDMA system with a narrowband interferer. To do so, I minimize the joint effect of both multiple access (MAI) and narrowband (NBI) interference on a simple matched filter receiver. I will show that, when spreading sequence statistical properties are suitably designed, performance improvements are possible with respect to a system exploiting chaos-based sequences minimizing MAI only.

Permutation classes, sorting algorithms, and lattice paths

A permutation is said to avoid a pattern if it does not contain any subsequence which is order-isomorphic to it. Donald Knuth, in the first volume of his celebrated book "The art of Computer Programming", observed that the permutations that can be computed (or, equivalently, sorted) by some particular data structures can be characterized in terms of pattern avoidance. In more recent years, the topic was reopened several times, while often in terms of sortable permutations rather than computable ones. The idea to sort permutations by using one of Knuth’s devices suggests to look for a deterministic procedure that decides, in linear time, if there exists a sequence of operations which is able to convert a given permutation into the identical one. In this thesis we show that, for the stack and the restricted deques, there exists an unique way to implement such a procedure. Moreover, we use these sorting procedures to create new sorting algorithms, and we prove some unexpected commutation properties between these procedures and the base step of bubblesort. We also show that the permutations that can be sorted by a combination of the base steps of bubblesort and its dual can be expressed, once again, in terms of pattern avoidance. In the final chapter we give an alternative proof of some enumerative results, in particular for the classes of permutations that can be sorted by the two restricted deques. It is well-known that the permutations that can be sorted through a restricted deque are counted by the Schrӧder numbers. In the thesis, we show how the deterministic sorting procedures yield a bijection between sortable permutations and Schrӧder paths.

Permutation-based stochastic ordering using pairwise comparisons

The topic of this work concerns nonparametric permutation-based methods aiming to find a ranking (stochastic ordering) of a given set of groups (populations), gathering together information from multiple variables under more than one experimental designs. The problem of ranking populations arises in several fields of science from the need of comparing G>2 given groups or treatments when the main goal is to find an order while taking into account several aspects. As it can be imagined, this problem is not only of theoretical interest but it also has a recognised relevance in several fields, such as industrial experiments or behavioural sciences, and this is reflected by the vast literature on the topic, although sometimes the problem is associated with different keywords such as: "stochastic ordering", "ranking", "construction of composite indices" etc., or even "ranking probabilities" outside of the strictly-speaking statistical literature. The properties of the proposed method are empirically evaluated by means of an extensive simulation study, where several aspects of interest are let to vary within a reasonable practical range. These aspects comprise: sample size, number of variables, number of groups, and distribution of noise/error. The flexibility of the approach lies mainly in the several available choices for the test-statistic and in the different types of experimental design that can be analysed. This render the method able to be tailored to the specific problem and the to nature of the data at hand. To perform the analyses an R package called SOUP (Stochastic Ordering Using Permutations) has been written and it is available on CRAN.

Generatore di entropia basato su dinamica caotica con interfaccia usb

È impossibile implementare sorgenti autenticamente casuali su hardware digitale. Quindi, storicamente, si è fatto ampio uso di generatori di numeri pseudo-casuali, evitando così i costi necessari per la progettazione di hardware analogico dedicato. Tuttavia, le sorgenti pseudo-casuali hanno proprietà (riproducibilità e periodicità) che si trasformano in vulnerabilità, nel caso in cui vengano adottate in sistemi di sicurezza informatica e all’interno di algoritmi crittografici. Oggi la richiesta di generatori di numeri autenticamente casuali è ai suoi massimi storici. Alcuni importanti attori dell’ICT sviluppato proprie soluzioni dedicate, ma queste sono disponibili solo sui sistemi moderni e di fascia elevata. È quindi di grande attualità rendere fruibili generatori autenticamente casuali per sistemi già esistenti o a basso costo. Per garantire sicurezza e al tempo stesso contenere i costi di progetto è opportuno pensare ad architetture che consentano di riusare parti analogiche già disponibili. Particolarmente interessanti risultano alcune architetture che, grazie all’utilizzo di dinamiche caotiche, consentono di basare buona parte della catena analogica di elaborazione su ADC. Infatti, tali blocchi sono ampiamente fruibili in forma integrata su architetture programmabili e microcontrollori. In questo lavoro, si propone un’implementazione a basso costo ed elevata flessibilità di un architettura basata su un ADC, inizialmente concepita all’Università di Bologna. La riduzione di costo viene ottenuta sfruttando il convertitore già presente all’interno di un microcontrollore. L’elevata flessibilità deriva dal fatto che il microcontrollore prescelto mette a disposizione una varietà di interfacce di comunicazione, tra cui quella USB, con la quale è possibile rendere facilmente fruibili i numeri casuali generati. Quindi, l’intero apparato comprende solo un microcontrollore e una minima catena analogica di elaborazione esterna e può essere interfacciato con estrema facilità ad elaboratori elettronici o sistemi embedded. La qualità della proposta, in termini di statistica delle sequenze casuali generate, è stata validata sfruttando i test standardizzati dall’U.S. NIST.

Implementazione di un codice di calcolo random close per un propellente solido

L’impacchettamento risulta essere importante in molti settori industriali, come il settore minerario, farmaceutico e soprattutto il settore spaziale, in quanto permette di massimizzare il grado di riempimento del propellente solido di un razzo ottenendo prestazioni migliori e notevoli vantaggi economici. Il lavoro di tesi presentato nel seguente elaborato consiste nello studio dell’impacchettamento casuale, in particolare il caso Random Close Packing, di un propellente solido; per fare ciò è stato implementato un codice in ambiente C++ presso l’hangar della Scuola di Ingegneria ed Architettura con sede a Forlì. L’obiettivo principale era quello di trovare la granulometria delle particelle di perclorato di ammonio e delle particelle di alluminio tali da minimizzare gli spazi lasciati vuoti dalle particelle stesse.

Random lattice models

This thesis deals with three different physical models, where each model involves a random component which is linked to a cubic lattice. First, a model is studied, which is used in numerical calculations of Quantum Chromodynamics.In these calculations random gauge-fields are distributed on the bonds of the lattice. The formulation of the model is fitted into the mathematical framework of ergodic operator families. We prove, that for small coupling constants, the ergodicity of the underlying probability measure is indeed ensured and that the integrated density of states of the Wilson-Dirac operator exists. The physical situations treated in the next two chapters are more similar to one another. In both cases the principle idea is to study a fermion system in a cubic crystal with impurities, that are modeled by a random potential located at the lattice sites. In the second model we apply the Hartree-Fock approximation to such a system. For the case of reduced Hartree-Fock theory at positive temperatures and a fixed chemical potential we consider the limit of an infinite system. In that case we show the existence and uniqueness of minimizers of the Hartree-Fock functional. In the third model we formulate the fermion system algebraically via C*-algebras. The question imposed here is to calculate the heat production of the system under the influence of an outer electromagnetic field. We show that the heat production corresponds exactly to what is empirically predicted by Joule's law in the regime of linear response.

Crystallization and morphology of the PLLA phase within random poly (L-lactide-ran-ɛ-caprolactone)

This work has mainly focused on the poly (L-lactide) (PLLA) which is a material for multiple applications with performances comparable to those of petrochemical polymers (PP, PS, PET, etc. ...), readily recyclable and also compostable. However, PLLA has certain shortcomings that limit its applications. It is a brittle, hard polymer with a very low elongation at break, hydrophobic, exhibits low crystallization kinetics and takes a long time to degrade. The properties of PLLA may be modified by copolymerization (random, block, and graft) of L-lactide monomers with other co-monomers. In this thesis it has been studied the crystallization and morphology of random copolymers poly (L-lactide-ran-ε-caprolactone) with different compositions of the two monomers since the physical, mechanical, optical and chemical properties of a material depend on this behavior. Thermal analyses were performed by differential scanning calorimetry (DSC) and thermogravimetry (TGA) to observe behaviors due to the different compositions of the copolymers. The crystallization kinetics and morphology of poly (L-lactide-ran-ε-caprolactone) was investigated by polarized light optical microscopy (PLOM) and differential scanning calorimetry (DSC). Their thermal behavior was observed with crystallization from melt. It was observed that with increasing amounts of PCL in the copolymer, there is a decrease of the thermal degradation. Studies on the crystallization kinetics have shown that small quantities of PCL in the copolymer increase the overall crystallization kinetics and the crystal growth rate which decreases with higher quantities of PCL.

Studio e implementazione di un modello random-close per propellenti solidi

L'obiettivo di questo lavoro di tesi è quello di implementare un codice di calcolo, attraverso l'algoritmo di Lubachevsky-Stillinger, in modo da poter prevedere la frazione volumetrica occupata dalle particelle solide che costituiscono il grain negli endoreattori a propellente solido. Particolare attenzione verrà rivolta al problema dell'impacchettamento sferico random (Random-Close Packing) che tale algoritmo cerca di modellare, e le ipotesi per cui tale modellazione può essere applicata al tipo di problema proposto. Inoltre saranno descritte le procedure effettuate per l'ottenimento dei risultati numerici delle simulazioni e la loro motivazione, oltre ai limiti del modello utilizzato e alle migliorie apportate per un'esecuzione più efficiente e veloce.

Analisi della sicurezza del protocollo WPS ed offline cracking basato su generatori di numeri pseudo-casuali a bassa entropia

In questa tesi si analizza in parte il Wi-Fi Protected Setup (WPS), standard creato con l’obiettivo di configurare in maniera semplice, sicura e veloce reti Wi-Fi. Dopo avere introdotto lo standard ne vengono presentate le vulnerabilità progettuali ed anche quelle relative a problematiche implementative. In seguito viene mostrato il programma che si è sviluppato per eseguire una delle tipologie di attacchi presentati.

Fluctuation properties in random walks on networks and simple integrate and fire models

In questa tesi si è studiato l’insorgere di eventi critici in un semplice modello neurale del tipo Integrate and Fire, basato su processi dinamici stocastici markoviani definiti su una rete. Il segnale neurale elettrico è stato modellato da un flusso di particelle. Si è concentrata l’attenzione sulla fase transiente del sistema, cercando di identificare fenomeni simili alla sincronizzazione neurale, la quale può essere considerata un evento critico. Sono state studiate reti particolarmente semplici, trovando che il modello proposto ha la capacità di produrre effetti "a cascata" nell’attività neurale, dovuti a Self Organized Criticality (auto organizzazione del sistema in stati instabili); questi effetti non vengono invece osservati in Random Walks sulle stesse reti. Si è visto che un piccolo stimolo random è capace di generare nell’attività della rete delle fluttuazioni notevoli, in particolar modo se il sistema si trova in una fase al limite dell’equilibrio. I picchi di attività così rilevati sono stati interpretati come valanghe di segnale neurale, fenomeno riconducibile alla sincronizzazione.

Sulla Pseudo-Telepatia Quantistica

La tesi descrive il fenomeno della pseudo-telepatia quantistica; vengono spiegati quali avvenimenti storici hanno portato alla nascita di questa teoria. Attraverso vari esempi, in un primo momento si cerca di far capire cosa sia realmente il fenomeno analizzato, successivamente verrà dimostrato che la pseudo-telepatia è la migliore strategia esistente per la risoluzione di alcune particolari tipologie di compiti.