Integrazione p-adica e teorema di Igusa

Questa tesi si prefigge lo scopo di dimostrare il teorema di Igusa. Inizia introducendo algebricamente i numeri p-adici e ne dà una rappresentazione grafica. Sviluppa poi un integrale definito dalla misura di Haar, invariante per traslazione e computa alcuni esempi. Utilizza il blow up come strumento per la risoluzione di alcuni integrali ed enuncia un'applicazione del teorema di Hironaka sulla risolubilità delle singolarità. Infine usa questi risultati per dimostrare il teorema di Igusa.

Keller-Segel system, chemotaxis and applications to a mathematical model for Alzheimer's disease

In questa tesi viene presentato il modello di Keller-Segel per la chemiotassi, un sistema di tipo parabolico-ellittico che appare nella descrizione di molti fenomeni in ambito biologico e medico. Viene mostrata l'esistenza globale della soluzione debole del modello, per dati iniziali sufficientemente piccoli in dimensione N>2. La scelta di dati iniziali abbastanza grandi invece può causare il blow-up della soluzione e viene mostrato sotto quali condizioni questo si verifica. Infine il modello della chemiotassi è stato applicato per descrivere una fase della malattia di Alzheimer ed è stata effettuata un'analisi di stabilità del sistema.

Analysis of evening and morning transitions in complex terrain: MATERHORN campaign 2013

A field study of thermal circulation over very gentle slope is described for a specific day characterised by weak synoptic conditions. The emphasis was on morning and evening transitions, but measurements cover the entire day; therefore a brief analysis is performed to represent the general thermal circulation pattern. Both transition periods are characterised by complex dynamic behaviours. During evening transition, the upslope flow has got through a stagnation condition characterised by wind velocity U<0:5 m=s. Only when the stagnating air has become negative buoyant, the flow is allowed to pour downslope like a slab. Some features of front formation has been found during the transition development, such as delay time of downslope flow start up along the slope, and the presence of positive turbulent kinetic energy at the onset of the motion. Eventually the observed evening transition has followed a mixed mechanisms, with features from different models. Therefore the Rayleigh number seems not to be a good criterion by which parametrise evening transition itself. Morning transition is characterised by destruction of nocturnal temperature inversion and the onset of upslope flow. Inversion destruction can be described in terms of CBL growth at surface and inversion decent from the top of the layer. KH has found to be a good indicator of inversion breakup, if used as parameter to study the inversion breakup in terms of temperature reversal. After the inversion breakup, buoyancy and mechanical productions supply the flow with the necessary energy to start the upslope wind. More quantitative analysis are provided by the study of stability parameters and turbulent kinetic energy budgets. Gradient Richardson number has been used in this terms, finding that a mixed SBL-CBL behaviour dominates the most of the observed layers. Tke budget has shown high turbulent behaviour during morning transition while the evening transition has developed entirely in laminar condition, apart from short intermittent turbulent events.