Maximum principle, mean value operators and quasi boundedness in non-euclidean settings

This work deals with some classes of linear second order partial differential operators with non-negative characteristic form and underlying non- Euclidean structures. These structures are determined by families of locally Lipschitz-continuous vector fields in RN, generating metric spaces of Carnot- Carath´eodory type. The Carnot-Carath´eodory metric related to a family {Xj}j=1,...,m is the control distance obtained by minimizing the time needed to go from two points along piecewise trajectories of vector fields. We are mainly interested in the causes in which a Sobolev-type inequality holds with respect to the X-gradient, and/or the X-control distance is Doubling with respect to the Lebesgue measure in RN. This study is divided into three parts (each corresponding to a chapter), and the subject of each one is a class of operators that includes the class of the subsequent one. In the first chapter, after recalling “X-ellipticity” and related concepts introduced by Kogoj and Lanconelli in [KL00], we show a Maximum Principle for linear second order differential operators for which we only assume a Sobolev-type inequality together with a lower terms summability. Adding some crucial hypotheses on measure and on vector fields (Doubling property and Poincar´e inequality), we will be able to obtain some Liouville-type results. This chapter is based on the paper [GL03] by Guti´errez and Lanconelli. In the second chapter we treat some ultraparabolic equations on Lie groups. In this case RN is the support of a Lie group, and moreover we require that vector fields satisfy left invariance. After recalling some results of Cinti [Cin07] about this class of operators and associated potential theory, we prove a scalar convexity for mean-value operators of L-subharmonic functions, where L is our differential operator. In the third chapter we prove a necessary and sufficient condition of regularity, for boundary points, for Dirichlet problem on an open subset of RN related to sub-Laplacian. On a Carnot group we give the essential background for this type of operator, and introduce the notion of “quasi-boundedness”. Then we show the strict relationship between this notion, the fundamental solution of the given operator, and the regularity of the boundary points.

MCAO for Extremely Large Telescopes: the cases of LBT and E-ELT

Several MCAO systems are under study to improve the angular resolution of the current and of the future generation large ground-based telescopes (diameters in the 8-40 m range). The subject of this PhD Thesis is embedded in this context. Two MCAO systems, in dierent realization phases, are addressed in this Thesis: NIRVANA, the 'double' MCAO system designed for one of the interferometric instruments of LBT, is in the integration and testing phase; MAORY, the future E-ELT MCAO module, is under preliminary study. These two systems takle the sky coverage problem in two dierent ways. The layer oriented approach of NIRVANA, coupled with multi-pyramids wavefront sensors, takes advantage of the optical co-addition of the signal coming from up to 12 NGS in a annular 2' to 6' technical FoV and up to 8 in the central 2' FoV. Summing the light coming from many natural sources permits to increase the limiting magnitude of the single NGS and to improve considerably the sky coverage. One of the two Wavefront Sensors for the mid- high altitude atmosphere analysis has been integrated and tested as a stand- alone unit in the laboratory at INAF-Osservatorio Astronomico di Bologna and afterwards delivered to the MPIA laboratories in Heidelberg, where was integrated and aligned to the post-focal optical relay of one LINC-NIRVANA arm. A number of tests were performed in order to characterize and optimize the system functionalities and performance. A report about this work is presented in Chapter 2. In the MAORY case, to ensure correction uniformity and sky coverage, the LGS-based approach is the current baseline. However, since the Sodium layer is approximately 10 km thick, the articial reference source looks elongated, especially when observed from the edge of a large aperture. On a 30-40 m class telescope, for instance, the maximum elongation varies between few arcsec and 10 arcsec, depending on the actual telescope diameter, on the Sodium layer properties and on the laser launcher position. The centroiding error in a Shack-Hartmann WFS increases proportionally to the elongation (in a photon noise dominated regime), strongly limiting the performance. To compensate for this effect a straightforward solution is to increase the laser power, i.e. to increase the number of detected photons per subaperture. The scope of Chapter 3 is twofold: an analysis of the performance of three dierent algorithms (Weighted Center of Gravity, Correlation and Quad-cell) for the instantaneous LGS image position measurement in presence of elongated spots and the determination of the required number of photons to achieve a certain average wavefront error over the telescope aperture. An alternative optical solution to the spot elongation problem is proposed in Section 3.4. Starting from the considerations presented in Chapter 3, a first order analysis of the LGS WFS for MAORY (number of subapertures, number of detected photons per subaperture, RON, focal plane sampling, subaperture FoV) is the subject of Chapter 4. An LGS WFS laboratory prototype was designed to reproduce the relevant aspects of an LGS SH WFS for the E-ELT and to evaluate the performance of different centroid algorithms in presence of elongated spots as investigated numerically and analytically in Chapter 3. This prototype permits to simulate realistic Sodium proles. A full testing plan for the prototype is set in Chapter 4.

Analisi della dinamica e composizione chimica del particolato atmosferico mediante un modello alla mesoscala

The vertical profile of aerosol in the planetary boundary layer of the Milan urban area is studied in terms of its development and chemical composition in a high-resolution modelling framework. The period of study spans a week in summer of 2007 (12-18 July), when continuous LIDAR measurements and a limited set of balloon profiles were collected in the frame of the ASI/QUITSAT project. LIDAR observations show a diurnal development of an aerosol plume that lifts early morning surface emissions to the top of the boundary layer, reaching maximum concentration around midday. Mountain breeze from Alps clean the bottom of the aerosol layer, typically leaving a residual layer at around 1500-2000 m which may survive for several days. During the last two days under analysis, a dust layer transported from Sahara reaches the upper layers of Milan area and affects the aerosol vertical distribution in the boundary layer. Simulation from the MM5/CHIMERE modelling system, carried out at 1 km horizontal resolution, qualitatively reproduced the general features of the Milan aerosol layer observed with LIDAR, including the rise and fall of the aersol plume, the residual layer in altitude and the Saharan dust event. The simulation highlighted the importance of nitrates and secondary organics in its composition. Several sensitivity tests showed that main driving factors leading to the dominance of nitrates in the plume are temperature and gas absorption process. A modelling study turn to the analysis of the vertical aerosol profiles distribution and knowledge of the characterization of the PM at a site near the city of Milan is performed using a model system composed by a meteorological model MM5 (V3-6), the mesoscale model from PSU/NCAR and a Chemical Transport Model (CTM) CHIMERE to simulate the vertical aerosol profile. LiDAR continuous observations and balloon profiles collected during two intensive campaigns in summer 2007 and in winter 2008 in the frame of the ASI/QUITSAT project have been used to perform comparisons in order to evaluate the ability of the aerosol chemistry transport model CHIMERE to simulate the aerosols dynamics and compositions in this area. The comparisons of model aerosols with measurements are carried out over a full time period between 12 July 2007 and 18 July 2007. The comparisons demonstrate the ability of the model to reproduce correctly the aerosol vertical distributions and their temporal variability. As detected by the LiDAR, the model during the period considered, predicts a diurnal development of a plume during the morning and a clearing during the afternoon, typically the plume reaches the top of the boundary layer around mid day, in this time CHIMERE produces highest concentrations in the upper levels as detected by LiDAR. The model, moreover can reproduce LiDAR observes enhancement aerosols concentrations above the boundary layer, attributing the phenomena to dust out intrusion. Another important information from the model analysis regard the composition , it predicts that a large part of the plume is composed by nitrate, in particular during 13 and 16 July 2007 , pointing to the model tendency to overestimates the nitrous component in the particular matter vertical structure . Sensitivity study carried out in this work show that there are a combination of different factor which determine the major nitrous composition of the “plume” observed and in particular humidity temperature and the absorption phenomena are the mainly candidate to explain the principal difference in composition simulated in the period object of this study , in particular , the CHIMERE model seems to be mostly sensitive to the absorption process.

Analisi climatica ad alta risoluzione delle precipitazioni sul nord Italia (1961-2005)

L’obiettivo di questo lavoro di tesi è di ottenere un’analisi climatica giornaliera ad alta risoluzione della precipitazione sul territorio del nord Italia realizzata con tecniche di controllo statistico, di analisi e di strumenti di descrizione dei risultati presentati nella recente letteratura. A tal fine, sono stati utilizzati i dati dell’Archivio ARCIS. In seguito alle fasi di controllo qualità, omogeneità e sincronicità i dati sono stati utilizzati per realizzare un’analisi giornaliera su grigliato regolare a 10 km di risoluzione utile alla rappresentazione della variabilità spazio-temporale della precipitazione sul Nord Italia per il periodo 1961-2005. I risultati di tale analisi mettono in evidenza dei valori medi di precipitazione annuale abbastanza intensi sulla parte centrale dell’arco Alpino, con massimi (oltre 2000 mm) sull’estremità orientale e sull’Appennino Ligure. Valori minimi (500 – 600 mm) sono osservati lungo le aree prospicienti il fiume Po, in Val d’Aosta ed in Alto Adige. La corrispondente analisi del trend temporale indica la presenza di lievi cali statisticamente significativi solo in aree limitate del territorio. In coerenza con questi risultati, la variazione nel tempo della precipitazione annuale mediata su tutto il territorio mette in evidenza un’intensa variabilità decennale, ma solo una lieve flessione lineare sull’intero periodo. Il numero annuo di giorni piovosi ed il 90° percentile della precipitazione giornaliera presentano invece trend lineari un po’ più pronunciati. In particolare, sul periodo considerato si nota un calo del numero di giorni piovosi su gran parte del territorio e solo su alcune aree del territorio un aumento dell’intensità del 90° percentile, sia a scala annuale che stagionale. Nell’ultima parte di questo lavoro è stato realizzato uno studio della relazione fra la forzante climatica e l’evoluzione della morfologia dell’Appennino Emiliano-Romagnolo. I risultati mostrano che a parità di quota, di pendenza e di litologia, la franosità è influenzata dalle precipitazioni.