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.

El Niño southern oscillation and its effect on fog oases along the Peruvian and Chilean coastal deserts

Fog oases, locally named Lomas, are distributed in a fragmented way along the western coast of Chile and Peru (South America) between ~6°S and 30°S following an altitudinal gradient determined by a fog layer. This fragmentation has been attributed to the hyper aridity of the desert. However, periodically climatic events influence the ‘normal seasonality’ of this ecosystem through a higher than average water input that triggers plant responses (e.g. primary productivity and phenology). The impact of the climatic oscillation may vary according to the season (wet/dry). This thesis evaluates the potential effect of climate oscillations, such as El Niño Southern Oscillation (ENSO), through the analysis of vegetation of this ecosystem following different approaches: Chapters two and three show the analysis of fog oasis along the Peruvian and Chilean deserts. The objectives are: 1) to explain the floristic connection of fog oases analysing their taxa composition differences and the phylogenetic affinities among them, 2) to explore the climate variables related to ENSO which likely affect fog production, and the responses of Lomas vegetation (composition, productivity, distribution) to climate patterns during ENSO events. Chapters four and five describe a fog-oasis in southern Peru during the 2008-2010 period. The objectives are: 3) to describe and create a new vegetation map of the Lomas vegetation using remote sensing analysis supported by field survey data, and 4) to identify the vegetation change during the dry season. The first part of our results show that: 1) there are three significantly different groups of Lomas (Northern Peru, Southern Peru, and Chile) with a significant phylogenetic divergence among them. The species composition reveals a latitudinal gradient of plant assemblages. The species origin, growth-forms typologies, and geographic position also reinforce the differences among groups. 2) Contradictory results have emerged from studies of low-cloud anomalies and the fog-collection during El Niño (EN). EN increases water availability in fog oases when fog should be less frequent due to the reduction of low-clouds amount and stratocumulus. Because a minor role of fog during EN is expected, it is likely that measurements of fog-water collection during EN are considering drizzle and fog at the same time. Although recent studies on fog oases have shown some relationship with the ENSO, responses of vegetation have been largely based on descriptive data, the absence of large temporal records limit the establishment of a direct relationship with climatic oscillations. The second part of the results show that: 3) five different classes of different spectral values correspond to the main land cover of Lomas using a Vegetation Index (VI). The study case is characterised by shrubs and trees with variable cover (dense, semi-dense and open). A secondary area is covered by small shrubs where the dominant tree species is not present. The cacti area and the old terraces with open vegetation were not identified with the VI. Agriculture is present in the area. Finally, 4) contrary to the dry season of 2008 and 2009 years, a higher VI was obtained during the dry season of 2010. The VI increased up to three times their average value, showing a clear spectral signal change, which coincided with the ENSO event of that period.

Toward a Magnitude Scale for Low Frequency Seismicity The LP energy for Campi Flegrei caldera

The energy released during a seismic crisis in volcanic areas is strictly related to the physical processes in the volcanic structure. In particular Long Period seismicity, that seems to be related to the oscillation of a fluid-filled crack (Chouet , 1996, Chouet, 2003, McNutt, 2005), can precedes or accompanies an eruption. The present doctoral thesis is focused on the study of the LP seismicity recorded in the Campi Flegrei volcano (Campania, Italy) during the October 2006 crisis. Campi Flegrei Caldera is an active caldera; the combination of an active magmatic system and a dense populated area make the Campi Flegrei a critical volcano. The source dynamic of LP seismicity is thought to be very different from the other kind of seismicity ( Tectonic or Volcano Tectonic): it’s characterized by a time sustained source and a low content in frequency. This features implies that the duration–magnitude, that is commonly used for VT events and sometimes for LPs as well, is unadapted for LP magnitude evaluation. The main goal of this doctoral work was to develop a method for the determination of the magnitude for the LP seismicity; it’s based on the comparison of the energy of VT event and LP event, linking the energy to the VT moment magnitude. So the magnitude of the LP event would be the moment magnitude of a VT event with the same energy of the LP. We applied this method to the LP data-set recorded at Campi Flegrei caldera in 2006, to an LP data-set of Colima volcano recorded in 2005 – 2006 and for an event recorded at Etna volcano. Experimenting this method to lots of waveforms recorded at different volcanoes we tested its easy applicability and consequently its usefulness in the routinely and in the quasi-real time work of a volcanological observatory.

Estimation of Quasi-Rational DSGE Models

Small-scale dynamic stochastic general equilibrium have been treated as the benchmark of much of the monetary policy literature, given their ability to explain the impact of monetary policy on output, inflation and financial markets. One cause of the empirical failure of New Keynesian models is partially due to the Rational Expectations (RE) paradigm, which entails a tight structure on the dynamics of the system. Under this hypothesis, the agents are assumed to know the data genereting process. In this paper, we propose the econometric analysis of New Keynesian DSGE models under an alternative expectations generating paradigm, which can be regarded as an intermediate position between rational expectations and learning, nameley an adapted version of the "Quasi-Rational" Expectatations (QRE) hypothesis. Given the agents' statistical model, we build a pseudo-structural form from the baseline system of Euler equations, imposing that the length of the reduced form is the same as in the `best' statistical model.