Maximum Principle for Elliptic and Parabolic Equations

Nel primo capitolo si riporta il principio del massimo per operatori ellittici. Sarà considerato, in un primo momento, l'operatore di Laplace e, successivamente, gli operatori ellittici del secondo ordine, per i quali si dimostrerà anche il principio del massimo di Hopf. Nel secondo capitolo si affronta il principio del massimo per operatori parabolici e lo si utilizza per dimostrare l'unicità delle soluzioni di problemi ai valori al contorno.

Il problema di Dirichlet per equazioni lineari ellittiche in forma di divergenza

The purpose of this dissertation is to prove that the Dirichlet problem in a bounded domain is uniquely solvable for elliptic equations in divergence form. The proof can be achieved by Hilbert space methods based on generalized or weak solutions. Existence and uniqueness of a generalized solution for the Dirichlet problem follow from the Fredholm alternative and weak maximum principle.

A maximum entropy approach to disturbed ecosystems

The current climate crisis requires a comprehensive understanding of biodiversity to acknowledge how ecosystems’ responses to anthropogenic disturbances may result in feedback that can either mitigate or exacerbate global warming. Although ecosystems are dynamic and macroecological patterns change drastically in response to disturbance, dynamic macroecology has received insufficient attention and theoretical formalisation. In this context, the maximum entropy principle (MaxEnt) could provide an effective inference procedure to study ecosystems. Since the improper usage of entropy outside its scope often leads to misconceptions, the opening chapter will clarify its meaning by following its evolution from classical thermodynamics to information theory. The second chapter introduces the study of ecosystems from a physicist’s viewpoint. In particular, the MaxEnt Theory of Ecology (METE) will be the cornerstone of the discussion. METE predicts the shapes of macroecological metrics in relatively static ecosystems using constraints imposed by static state variables. However, in disturbed ecosystems with macroscale state variables that change rapidly over time, its predictions tend to fail. In the final chapter, DynaMETE is therefore presented as an extension of METE from static to dynamic. By predicting how macroecological patterns are likely to change in response to perturbations, DynaMETE can contribute to a better understanding of disturbed ecosystems’ fate and the improvement of conservation and management of carbon sinks, like forests. Targeted strategies in ecosystem management are now indispensable to enhance the interdependence of human well-being and the health of ecosystems, thus avoiding climate change tipping points.

Probabilistic Envelope Curves for Extreme Rainfall Events - Curve Inviluppo Probabilistiche per Precipitazioni Estreme

A regional envelope curve (REC) of flood flows summarises the current bound on our experience of extreme floods in a region. RECs are available for most regions of the world. Recent scientific papers introduced a probabilistic interpretation of these curves and formulated an empirical estimator of the recurrence interval T associated with a REC, which, in principle, enables us to use RECs for design purposes in ungauged basins. The main aim of this work is twofold. First, it extends the REC concept to extreme rainstorm events by introducing the Depth-Duration Envelope Curves (DDEC), which are defined as the regional upper bound on all the record rainfall depths at present for various rainfall duration. Second, it adapts the probabilistic interpretation proposed for RECs to DDECs and it assesses the suitability of these curves for estimating the T-year rainfall event associated with a given duration and large T values. Probabilistic DDECs are complementary to regional frequency analysis of rainstorms and their utilization in combination with a suitable rainfall-runoff model can provide useful indications on the magnitude of extreme floods for gauged and ungauged basins. The study focuses on two different national datasets, the peak over threshold (POT) series of rainfall depths with duration 30 min., 1, 3, 9 and 24 hrs. obtained for 700 Austrian raingauges and the Annual Maximum Series (AMS) of rainfall depths with duration spanning from 5 min. to 24 hrs. collected at 220 raingauges located in northern-central Italy. The estimation of the recurrence interval of DDEC requires the quantification of the equivalent number of independent data which, in turn, is a function of the cross-correlation among sequences. While the quantification and modelling of intersite dependence is a straightforward task for AMS series, it may be cumbersome for POT series. This paper proposes a possible approach to address this problem.

Studio e messa a punto del modello maximum-slope per l'analisi dei parametri perfusionali da sequenza di immagini tomografiche

Miglioramento delle prestazioni del modello mono-compartimentale del maximum slope dovuto all'introduzione di sistemi per l'eliminazione degli outliers.

Absolute Hodge cycles and Deligne's principle b

I cicli di Hodge assoluti sono stati utilizzati da Deligne per dividere la congettura di Hodge in due sotto-congetture. La prima dice che tutte le classi di Hodge su una varietà complessa proiettiva liscia sono assolute, la seconda che le classi assolute sono algebriche. Deligne ha dato risposta affermativa alla prima sottocongettura nel caso delle varietà abeliane. La dimostrazione si basa su due teoremi, conosciuti rispettivamente come Principio A e Principio B. In questo lavoro vengono presentate la teoria delle classi di Hodge assolute e la dimostrazione del Principio B.