Mean curvature flow in SE (2) and applications to visual perception

Our goal in this thesis is to provide a result of existence of the degenerate non-linear, non-divergence PDE which describes the mean curvature flow in the Lie group SE(2) equipped with a sub-Riemannian metric. The research is motivated by problems of visual completion and models of the visual cortex.

The Feynman-Kac formula and applications to PDEs

The thesis presents a probabilistic approach to the theory of semigroups of operators, with particular attention to the Markov and Feller semigroups. The first goal of this work is the proof of the fundamental Feynman-Kac formula, which gives the solution of certain parabolic Cauchy problems, in terms of the expected value of the initial condition computed at the associated stochastic diffusion processes. The second target is the characterization of the principal eigenvalue of the generator of a semigroup with Markov transition probability function and of second order elliptic operators with real coefficients not necessarily self-adjoint. The thesis is divided into three chapters. In the first chapter we study the Brownian motion and some of its main properties, the stochastic processes, the stochastic integral and the Itô formula in order to finally arrive, in the last section, at the proof of the Feynman-Kac formula. The second chapter is devoted to the probabilistic approach to the semigroups theory and it is here that we introduce Markov and Feller semigroups. Special emphasis is given to the Feller semigroup associated with the Brownian motion. The third and last chapter is divided into two sections. In the first one we present the abstract characterization of the principal eigenvalue of the infinitesimal generator of a semigroup of operators acting on continuous functions over a compact metric space. In the second section this approach is used to study the principal eigenvalue of elliptic partial differential operators with real coefficients. At the end, in the appendix, we gather some of the technical results used in the thesis in more details. Appendix A is devoted to the Sion minimax theorem, while in appendix B we prove the Chernoff product formula for not necessarily self-adjoint operators.

Keller-Segel system, chemotaxis and applications to a mathematical model for Alzheimer's disease

In questa tesi viene presentato il modello di Keller-Segel per la chemiotassi, un sistema di tipo parabolico-ellittico che appare nella descrizione di molti fenomeni in ambito biologico e medico. Viene mostrata l'esistenza globale della soluzione debole del modello, per dati iniziali sufficientemente piccoli in dimensione N>2. La scelta di dati iniziali abbastanza grandi invece può causare il blow-up della soluzione e viene mostrato sotto quali condizioni questo si verifica. Infine il modello della chemiotassi è stato applicato per descrivere una fase della malattia di Alzheimer ed è stata effettuata un'analisi di stabilità del sistema.

DTN discovery and routing: from space applications to terrestrial networks.

L'argomento di questa tesi è l'architettura di rete Delay-/Disruption-Tolerant Networking (DTN), progettata per operare nelle reti “challenged”, dove la suite di protocolli TCP/IP risulta inefficace a causa di lunghi ritardi di propagazione del segnale, interruzioni e disturbi di canale, ecc. Esempi di reti “challenged” variano dalle reti interplanetarie alle Mobile Ad-Hoc Networks (MANETs). Le principali implementazioni dell'architettura DTN sono DTN2, implementazione di riferimento, e ION, sviluppata da NASA JPL per applicazioni spaziali. Una grande differenza tra reti spaziali e terrestri è che nello spazio i movimenti dei nodi sono deterministici, mentre non lo sono per i nodi mobili terrestri, i quali generalmente non conoscono la topologia della rete. Questo ha portato allo sviluppo di diversi algoritmi di routing: deterministici per le reti spaziali e opportunistici per quelle terrestri. NASA JPL ha recentemente deciso di estendere l'ambito di applicazione di ION per supportare anche scenari non deterministici. Durante la tesi, svolta presso NASA JPL, mi sono occupato di argomenti diversi, tutti finalizzati a questo obiettivo. Inizialmente ho testato la nuova implementazione dell'algoritmo IP Neighbor Discovery (IPND) di ION, corretti i bug e prodotta la documentazione ufficiale. Quindi ho contribuito ad integrare il Contact Graph Routing (CGR) di ION nel simulatore DTN “ONE” utilizzando la Java Native Interface (JNI) come ponte tra il codice Java di ONE e il codice C di ION. In particolare ho adattato tutte le librerie di ION necessarie per far funzionare CGR all'interno dell'ambiente di ONE. Infine, dopo aver analizzato un dataset di tracce reali di nodi mobili, ho contribuito a progettare e a sviluppare OCGR, estensione opportunistica del CGR, quindi ne ho curato l'integrazione in ONE. I risultati preliminari sembrano confermare la validità di OCGR che, una volta messo a punto, può diventare un valido concorrente ai più rinomati algoritmi opportunistici.

The use of remote sensing imagery for monitoring recovery in the aftermath of a natural disaster: the Hurricane Katrina Case

Although Recovery is often defined as the less studied and documented phase of the Emergency Management Cycle, a wide literature is available for describing characteristics and sub-phases of this process. Previous works do not allow to gain an overall perspective because of a lack of systematic consistent monitoring of recovery utilizing advanced technologies such as remote sensing and GIS technologies. Taking into consideration the key role of Remote Sensing in Response and Damage Assessment, this thesis is aimed to verify the appropriateness of such advanced monitoring techniques to detect recovery advancements over time, with close attention to the main characteristics of the study event: Hurricane Katrina storm surge. Based on multi-source, multi-sensor and multi-temporal data, the post-Katrina recovery was analysed using both a qualitative and a quantitative approach. The first phase was dedicated to the investigation of the relation between urban types, damage and recovery state, referring to geographical and technological parameters. Damage and recovery scales were proposed to review critical observations on remarkable surge- induced effects on various typologies of structures, analyzed at a per-building level. This wide-ranging investigation allowed a new understanding of the distinctive features of the recovery process. A quantitative analysis was employed to develop methodological procedures suited to recognize and monitor distribution, timing and characteristics of recovery activities in the study area. Promising results, gained by applying supervised classification algorithms to detect localization and distribution of blue tarp, have proved that this methodology may help the analyst in the detection and monitoring of recovery activities in areas that have been affected by medium damage. The study found that Mahalanobis Distance was the classifier which provided the most accurate results, in localising blue roofs with 93.7% of blue roof classified correctly and a producer accuracy of 70%. It was seen to be the classifier least sensitive to spectral signature alteration. The application of the dissimilarity textural classification to satellite imagery has demonstrated the suitability of this technique for the detection of debris distribution and for the monitoring of demolition and reconstruction activities in the study area. Linking these geographically extensive techniques with expert per-building interpretation of advanced-technology ground surveys provides a multi-faceted view of the physical recovery process. Remote sensing and GIS technologies combined to advanced ground survey approach provides extremely valuable capability in Recovery activities monitoring and may constitute a technical basis to lead aid organization and local government in the Recovery management.