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.

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.

1st and 2nd Generation Ethanol from Biomass Crops

In chapter 1 and 2 calcium hydroxide as impregnation agent before steam explosion of sugarcane bagasse and switchgrass, respectively, was compared with auto-hydrolysis, assessing the effects on enzymatic hydrolysis and simultaneous saccharification and fermentation (SSF) at high solid concentration of pretreated solid fraction. In addition, anaerobic digestion of pretreated liquid fraction was carried out, in order to appraise the effectiveness of calcium hydroxide before steam explosion in a more comprehensive way. In As water is an expensive input in both cultivation of biomass crops and subsequent pretreatment, Chapter 3 addressed the effects of variable soil moisture on biomass growth and composition of biomass sorghum. Moreover, the effect of water stress was related to the characteristics of stem juice for 1st generation ethanol and structural carbohydrates for 2nd generation ethanol. In the frame of chapter 1, calcium hydroxide was proven to be a suitable catalyst for sugarcane bagasse before steam explosion, in order to enhance fibre deconstruction. In chapter 2, effect of calcium hydroxide on switchgrass showed a great potential when ethanol was focused, whereas acid addition produced higher methane yield. Regarding chapter 3, during crop cycle the amount of cellulose, hemicellulose and AIL changed causing a decrease of 2G ethanol amount. Biomass physical and chemical properties involved a lower glucose yield and concentration at the end of enzymatic hydrolysis and, consequently, a lower 2G ethanol concentration at the end of simultaneous saccharification and fermentation, proving that there is strong relationship between structure, chemical composition, and fermentable sugar yield. The significantly higher concentration of ethanol at the early crop stage could be an important incentive to consider biomass sorghum as second crop in the season, to be introduced into some agricultural systems, potentially benefiting farmers and, above all, avoiding the exacerbation of the debate about fuel vs food crops.