Tecniche avanzate per l'analisi di sensitività globale: applicazione alla contaminazione delle acque sotterranee.

The thesis is framed within the field of the stochastic approach to flow and transport themes of solutes in natural porous materials. The methodology used to characterise the uncertainty associated with the modular predictions is completely general and can be reproduced in various contexts. The theme of the research includes the following among its main objectives: (a) the development of a Global Sensitivity Analysis on contaminant transport models in the subsoil to research the effects of the uncertainty of the most important parameters; (b) the application of advanced techniques, such as Polynomial Chaos Expansion (PCE), for obtaining surrogate models starting from those which conduct traditionally developed analyses in the context of Monte Carlo simulations, characterised by an often not negligible computational burden; (c) the analyses and the understanding of the key processes at the basis of the transport of solutes in natural porous materials using the aforementioned technical and analysis resources. In the complete picture, the thesis looks at the application of a Continuous Injection transport model of contaminants, of the PCE technique which has already been developed and applied by the thesis supervisors, by way of numerical code, to a Slug Injection model. The methodology was applied to the aforementioned model with original contribution deriving from surrogate models with various degrees of approximation and developing a Global Sensitivity Analysis aimed at the determination of Sobol’ indices.

Il teorema di Chebyshev

Dopo aver introdotto alcune nozioni della teoria della probabilità, ho esposto il teorema di Chebyshev ed alcuni teoremi ad esso collegati. Ho infine analizzato un'applicazione legata alle strategie d'investimento.

The Poincaré polynomial for the Hilbert scheme of points of C^2

Questo lavoro prende in esame lo schema di Hilbert di punti di C^2, il quale viene descritto assieme ad alcune sue proprietà, ad esempio la sua struttura hyper-kahleriana. Lo scopo della tesi è lo studio del polinomio di Poincaré di tale schema di Hilbert: ciò che si ottiene è una espressione del tipo serie di potenze, la quale è un caso particolare di una formula molto più generale, nota con il nome di formula di Goettsche.

Multi-scalar critical theories: first steps for a non perturbative functional renormalization group analysis

In this work the fundamental ideas to study properties of QFTs with the functional Renormalization Group are presented and some examples illustrated. First the Wetterich equation for the effective average action and its flow in the local potential approximation (LPA) for a single scalar field is derived. This case is considered to illustrate some techniques used to solve the RG fixed point equation and study the properties of the critical theories in D dimensions. In particular the shooting methods for the ODE equation for the fixed point potential as well as the approach which studies a polynomial truncation with a finite number of couplings, which is convenient to study the critical exponents. We then study novel cases related to multi field scalar theories, deriving the flow equations for the LPA truncation, both without assuming any global symmetry and also specialising to cases with a given symmetry, using truncations based on polynomials of the symmetry invariants. This is used to study possible non perturbative solutions of critical theories which are extensions of known perturbative results, obtained in the epsilon expansion below the upper critical dimension.