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.

Zeta and L-functions of elliptic curves

In questa tesi si studiano alcune proprietà fondamentali delle funzioni Zeta e L associate ad una curva ellittica. In particolare, si dimostra la razionalità della funzione Zeta e l'ipotesi di Riemann per due famiglie specifiche di curve ellittiche. Si studia poi il problema dell'esistenza di un prolungamento analitico al piano complesso della funzione L di una curva ellittica con moltiplicazione complessa, attraverso l'analisi diretta di due casi particolari.

Description and Prediction of Freezing of Gait in Parkinson's Disease from Wearable Inertial Measurements Units

Parkinson's disease (PD) is a neuro-degenerative disorder, the second most common after Alzheimer's disease. After diagnosis, treatments can help to relieve the symptoms, but there is no known cure for PD. PD is characterized by a combination of motor and no-motor dysfunctions. Among the motor symptoms there is the so called Freezing of Gait (FoG). The FoG is a phenomenon in PD patients in which the feet stock to the floor and is difficult for the patient to initiate movement. FoG is a severe problem, since it is associated with falls, anxiety, loss of mobility, accidents, mortality and it has substantial clinical and social consequences decreasing the quality of life in PD patients. Medicine can be very successful in controlling movements disorders and dealing with some of the PD symptoms. However, the relationship between medication and the development of FoG remains unclear. Several studies have demonstrated that visual or auditory rhythmical cuing allows PD patients to improve their motor abilities. Rhythmic auditory stimulation (RAS) was shown to be particularly effective at improving gait, specially with patients that manifest FoG. While RAS allows to reduce the time and the effects of FoGs occurrence in PD patients after the FoG is detected, it can not avoid the episode due to the latency of detection. An improvement of the system would be the prediction of the FoG. This thesis was developed following two main objectives: (1) the finding of specifics properties during pre FoG periods different from normal walking context and other walking events like turns and stops using the information provided by the inertial measurements units (IMUs) and (2) the formulation of a model for automatically detect the pre FoG patterns in order to completely avoid the upcoming freezing event in PD patients. The first part focuses on the analysis of different methods for feature extraction which might lead in the FoG occurrence.

Elementary properties of Modular curves and Modular forms

Lo scopo di questa tesi è introdurre in breve le prime proprietà delle curve modulari e delle forme modulari, per poi mostrarne alcune applicazioni archetipiche. Per farlo, dopo aver richiamato alcune nozioni utili nel primo capitolo, sviluppiamo, nel secondo capitolo, la teoria di base delle curve modulari compatte come superfici di Riemann, calcolandone il genere nel caso dei sottogruppi principali di congruenza. Dunque, nel terzo capitolo, dopo un estesa trattazione dell'esempio delle forme modulari rispetto al gruppo modulare, viene calcolata la dimensione degli spazi delle forme intere e delle forme cuspidali rispetto a un sottogruppo di indice finito del gruppo modulare. Questo capitolo si conclude con tre esempi di applicazione della teoria esposta, tra i quali spiccano la dimostrazione del Grande Teorema di Picard e del Teorema dei quattro quadrati di Jacobi.