The Hitchin map for one-nodal base curves of compact type

Studying moduli spaces of semistable Higgs bundles (E, \phi) of rank n on a smooth curve C, a key role is played by the spectral curve X (Hitchin), because an important result by Beauville-Narasimhan-Ramanan allows us to study isomorphism classes of such Higgs bundles in terms of isomorphism classes of rank-1 torsion-free sheaves on X. This way, the generic fibre of the Hitchin map, which associates to any semistable Higgs bundle the coefficients of the characteristic polynomial of \phi, is isomorphic to the Jacobian of X. Focusing on rank-2 Higgs data, this construction was extended by Barik to the case in which the curve C is reducible, one-nodal, having two smooth components. Such curve is called of compact type because its Picard group is compact. In this work, we describe and clarify the main points of the construction by Barik and we give examples, especially concerning generic fibres of the Hitchin map. Referring to Hausel-Pauly, we consider the case of SL(2,C)-Higgs bundles on a smooth base curve, which are such that the generic fibre of the Hitchin map is a subvariety of the Jacobian of X, the Prym variety. We recall the description of special loci, called endoscopic loci, such that the associated Prym variety is not connected. Then, letting G be an affine reductive group having underlying Lie algebra so(4,C), we consider G-Higgs bundles on a smooth base curve. Starting from the construction by Bradlow-Schaposnik, we discuss the associated endoscopic loci. By adapting these studies to a one-nodal base curve of compact type, we describe the fibre of the SL(2,C)-Hitchin map and of the G-Hitchin map, together with endoscopic loci. In the Appendix, we give an interpretation of generic spectral curves in terms of families of double covers.

A new method for the estimation of the reverberation time from measured room impulse responses with application to Italian opera houses

The Schroeder's backward integration method is the most used method to extract the decay curve of an acoustic impulse response and to calculate the reverberation time from this curve. In the literature the limits and the possible improvements of this method are widely discussed. In this work a new method is proposed for the evaluation of the energy decay curve. The new method has been implemented in a Matlab toolbox. Its performance has been tested versus the most accredited literature method. The values of EDT and reverberation time extracted from the energy decay curves calculated with both methods have been compared in terms of the values themselves and in terms of their statistical representativeness. The main case study consists of nine Italian historical theatres in which acoustical measurements were performed. The comparison of the two extraction methods has also been applied to a critical case, i.e. the structural impulse responses of some building elements. The comparison underlines that both methods return a comparable value of the T30. Decreasing the range of evaluation, they reveal increasing differences; in particular, the main differences are in the first part of the decay, where the EDT is evaluated. This is a consequence of the fact that the new method returns a “locally" defined energy decay curve, whereas the Schroeder's method accumulates energy from the tail to the beginning of the impulse response. Another characteristic of the new method for the energy decay extraction curve is its independence on the background noise estimation. Finally, a statistical analysis is performed on the T30 and EDT values calculated from the impulse responses measurements in the Italian historical theatres. The aim of this evaluation is to know whether a subset of measurements could be considered representative for a complete characterization of these opera houses.

The XENON1T experiment: Monte Carlo background estimation and sensitivity curves study

Despite the scientific achievement of the last decades in the astrophysical and cosmological fields, the majority of the Universe energy content is still unknown. A potential solution to the “missing mass problem” is the existence of dark matter in the form of WIMPs. Due to the very small cross section for WIMP-nuleon interactions, the number of expected events is very limited (about 1 ev/tonne/year), thus requiring detectors with large target mass and low background level. The aim of the XENON1T experiment, the first tonne-scale LXe based detector, is to be sensitive to WIMP-nucleon cross section as low as 10^-47 cm^2. To investigate the possibility of such a detector to reach its goal, Monte Carlo simulations are mandatory to estimate the background. To this aim, the GEANT4 toolkit has been used to implement the detector geometry and to simulate the decays from the various background sources: electromagnetic and nuclear. From the analysis of the simulations, the level of background has been found totally acceptable for the experiment purposes: about 1 background event in a 2 tonne-years exposure. Indeed, using the Maximum Gap method, the XENON1T sensitivity has been evaluated and the minimum for the WIMP-nucleon cross sections has been found at 1.87 x 10^-47 cm^2, at 90% CL, for a WIMP mass of 45 GeV/c^2. The results have been independently cross checked by using the Likelihood Ratio method that confirmed such results with an agreement within less than a factor two. Such a result is completely acceptable considering the intrinsic differences between the two statistical methods. Thus, in the PhD thesis it has been proven that the XENON1T detector will be able to reach the designed sensitivity, thus lowering the limits on the WIMP-nucleon cross section by about 2 orders of magnitude with respect to the current experiments.

Potential Analysis for Hypoelliptic Second Order PDEs with Nonnegative Characteristic Form

In this Thesis we consider a class of second order partial differential operators with non-negative characteristic form and with smooth coefficients. Main assumptions on the relevant operators are hypoellipticity and existence of a well-behaved global fundamental solution. We first make a deep analysis of the L-Green function for arbitrary open sets and of its applications to the Representation Theorems of Riesz-type for L-subharmonic and L-superharmonic functions. Then, we prove an Inverse Mean value Theorem characterizing the superlevel sets of the fundamental solution by means of L-harmonic functions. Furthermore, we establish a Lebesgue-type result showing the role of the mean-integal operator in solving the homogeneus Dirichlet problem related to L in the Perron-Wiener sense. Finally, we compare Perron-Wiener and weak variational solutions of the homogeneous Dirichlet problem, under specific hypothesis on the boundary datum.

Circadian rhythms, sleep and autonomic function in progressive supranuclear palsy: characteristic features and reciprocal interactions

Background: Progressive supranuclear palsy (PSP) is a rare neurodegenerative condition. The aims of this study were to evaluate the association between sleep, the circadian system and autonomic function in a cohort of PSP patients. Methods: Patients with PSP diagnosed according to consensus criteria were recruited prospectively and retrospectively and performed the following tests: body core temperature (BcT), sleep-wake cycle, systolic and diastolic blood pressure (SBP, DBP) continuous monitoring for 48 h under controlled environmental conditions; cardiovascular reflex tests (CRTs). The analysis of circadian rhythmicity was performed with the single cosinor method. For state-dependent analysis, the mean value of variables in each sleep stage was calculated as well as the difference to the value in wake. Results: PSP patients presented a reduced total duration of night sleep, with frequent and prolonged awakenings. During daytime, patients had very short naps, suggesting a state of profound sleep deprivation across the 24-h. REM sleep behaviour disorder was found in 15%, restless legs syndrome in 46%, periodic limb movements in 52% and obstructive sleep apnea in 54%. BcT presented the expected fall during night-time, however, compared to controls, mean values during day and night were higher. However BcT state-dependent modulation was maintained. Increased BcT could be attributed to an inability to properly reduce sympathetic activity favoured by the sleep deprivation. At CRTs, PSP presented mild cardiovascular adrenergic impairment and preserved cardiovagal function. 14% had non-neurogenic orthostatic hypotension. Only 2 PSP presented the expected BP dipping pattern, possibly as a consequence of sleep disruption. State-dependent analysis showed a partial loss of the state-dependent modulation for SBP. Discussion: This study showed that PSP presented abnormalities of sleep, circadian rhythms and cardiovascular autonomic function that are likely to be closely linked one to another.

Landslide susceptibility in the Belt and Road Initiative

The Belt and Road Initiative (BRI) is a project launched by the Chinese Government whose main goal is to connect more than 65 countries in Asia, Europe, Africa and Oceania developing infrastructures and facilities. To support the prevention or mitigation of landslide hazards, which may affect the mainland infrastructures of BRI, a landslide susceptibility analysis in the countries involved has been carried out. Due to the large study area, the analysis has been carried out using a multi-scale approach which consists of mapping susceptibility firstly at continental scale, and then at national scale. The study area selected for the continental assessment is the south-Asia, where a pixel-based landslide susceptibility map has been carried out using the Weight of Evidence method and validated by Receiving Operating Characteristic (ROC) curves. Then, we selected the regions of west Tajikistan and north-east India to be investigated at national scale. Data scarcity is a common condition for many countries involved into the Initiative. Therefore in addition to the landslide susceptibility assessment of west Tajikistan, which has been conducted using a Generalized Additive Model and validated by ROC curves, we have examined, in the same study area, the effect of incomplete landslide dataset on the prediction capacity of statistical models. The entire PhD research activity has been conducted using only open data and open-source software. In this context, to support the analysis of the last years an open-source plugin for QGIS has been implemented. The SZ-tool allows the user to make susceptibility assessments from the data preprocessing, susceptibility mapping, to the final classification. All the output data of the analysis conducted are freely available and downloadable. This text describes the research activity of the last three years. Each chapter reports the text of the articles published in international scientific journal during the PhD.

Statistical learning of random probability measures

The study of random probability measures is a lively research topic that has attracted interest from different fields in recent years. In this thesis, we consider random probability measures in the context of Bayesian nonparametrics, where the law of a random probability measure is used as prior distribution, and in the context of distributional data analysis, where the goal is to perform inference given avsample from the law of a random probability measure. The contributions contained in this thesis can be subdivided according to three different topics: (i) the use of almost surely discrete repulsive random measures (i.e., whose support points are well separated) for Bayesian model-based clustering, (ii) the proposal of new laws for collections of random probability measures for Bayesian density estimation of partially exchangeable data subdivided into different groups, and (iii) the study of principal component analysis and regression models for probability distributions seen as elements of the 2-Wasserstein space. Specifically, for point (i) above we propose an efficient Markov chain Monte Carlo algorithm for posterior inference, which sidesteps the need of split-merge reversible jump moves typically associated with poor performance, we propose a model for clustering high-dimensional data by introducing a novel class of anisotropic determinantal point processes, and study the distributional properties of the repulsive measures, shedding light on important theoretical results which enable more principled prior elicitation and more efficient posterior simulation algorithms. For point (ii) above, we consider several models suitable for clustering homogeneous populations, inducing spatial dependence across groups of data, extracting the characteristic traits common to all the data-groups, and propose a novel vector autoregressive model to study of growth curves of Singaporean kids. Finally, for point (iii), we propose a novel class of projected statistical methods for distributional data analysis for measures on the real line and on the unit-circle.

Evaluation and management of demersal stocks in the Adriatic: the case of common sole (Solea solea, L.) in the northern and central Adriatic Sea

This PhD thesis aimed to assess the status of common sole, one of the main commercial stocks in the Adriatic Sea, using a mix of conventional and innovative techniques to provide more reliable estimates of stock status compared to past advice. First, a meta-analysis was carried out using data-poor assessment model to analyze the whole catch assemblage of rapido fishery. The outcomes were used to estimate rebuilding time and forecast catches under different harvest control rule scenarios, with a reduction of 20% of fishing effort being suggested as a way to allow most of the species to recover to sustainable levels. Secondly, an ensemble of data-rich assessment models was developed to better incorporate uncertainty by using alternative hypotheses of main parameters. This was the first time an ensemble of models has been used in the Mediterranean to provide management advice. Consistent with data-poor analysis results, the ensemble outcomes indicated that the common sole stock was showing a recovering trend probably due to the effective management actions underway in the area rather than the moderate effort reduction according to the actual management plan. Moreover, back-calculation measurements were used to fit and compare monophasic and biphasic growth curves through the use of non-linear mixed effects models. The analyses revealed that the fitting of the biphasic curve was superior, confirming the theory that growth in size would decrease as a consequence of reproductive effort. A stock assessment simulation showed how the use of the monophasic pattern would result in a critical overestimation of biomass that could lead to a greater risk of overfishing. As a final step, a simulation-testing procedure was applied to determine the best performing reference points using stock-specific characteristic. The procedure could be routinely adopted to increase transparency in reference points calculation enhancing the credibility of scientific advice.