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.

Testing future weak lensing surveys through simulations of observations

Weak lensing experiments such as the future ESA-accepted mission Euclid aim to measure cosmological parameters with unprecedented accuracy. It is important to assess the precision that can be obtained in these measurements by applying analysis software on mock images that contain many sources of noise present in the real data. In this Thesis, we show a method to perform simulations of observations, that produce realistic images of the sky according to characteristics of the instrument and of the survey. We then use these images to test the performances of the Euclid mission. In particular, we concentrate on the precision of the photometric redshift measurements, which are key data to perform cosmic shear tomography. We calculate the fraction of the total observed sample that must be discarded to reach the required level of precision, that is equal to 0.05(1+z) for a galaxy with measured redshift z, with different ancillary ground-based observations. The results highlight the importance of u-band observations, especially to discriminate between low (z < 0.5) and high (z ~ 3) redshifts, and the need for good observing sites, with seeing FWHM < 1. arcsec. We then construct an optimal filter to detect galaxy clusters through photometric catalogues of galaxies, and we test it on the COSMOS field, obtaining 27 lensing-confirmed detections. Applying this algorithm on mock Euclid data, we verify the possibility to detect clusters with mass above 10^14.2 solar masses with a low rate of false detections.

Finite-element discretization of 3D energy-transport equations for semiconductors

In this thesis a mathematical model was derived that describes the charge and energy transport in semiconductor devices like transistors. Moreover, numerical simulations of these physical processes are performed. In order to accomplish this, methods of theoretical physics, functional analysis, numerical mathematics and computer programming are applied. After an introduction to the status quo of semiconductor device simulation methods and a brief review of historical facts up to now, the attention is shifted to the construction of a model, which serves as the basis of the subsequent derivations in the thesis. Thereby the starting point is an important equation of the theory of dilute gases. From this equation the model equations are derived and specified by means of a series expansion method. This is done in a multi-stage derivation process, which is mainly taken from a scientific paper and which does not constitute the focus of this thesis. In the following phase we specify the mathematical setting and make precise the model assumptions. Thereby we make use of methods of functional analysis. Since the equations we deal with are coupled, we are concerned with a nonstandard problem. In contrary, the theory of scalar elliptic equations is established meanwhile. Subsequently, we are preoccupied with the numerical discretization of the equations. A special finite-element method is used for the discretization. This special approach has to be done in order to make the numerical results appropriate for practical application. By a series of transformations from the discrete model we derive a system of algebraic equations that are eligible for numerical evaluation. Using self-made computer programs we solve the equations to get approximate solutions. These programs are based on new and specialized iteration procedures that are developed and thoroughly tested within the frame of this research work. Due to their importance and their novel status, they are explained and demonstrated in detail. We compare these new iterations with a standard method that is complemented by a feature to fit in the current context. A further innovation is the computation of solutions in three-dimensional domains, which are still rare. Special attention is paid to applicability of the 3D simulation tools. The programs are designed to have justifiable working complexity. The simulation results of some models of contemporary semiconductor devices are shown and detailed comments on the results are given. Eventually, we make a prospect on future development and enhancements of the models and of the algorithms that we used.

Locating the source of volcanic tremor at stromboli volcano, italy

We have developed a method for locating sources of volcanic tremor and applied it to a dataset recorded on Stromboli volcano before and after the onset of the February 27th 2007 effusive eruption. Volcanic tremor has attracted considerable attention by seismologists because of its potential value as a tool for forecasting eruptions and for better understanding the physical processes that occur inside active volcanoes. Commonly used methods to locate volcanic tremor sources are: 1) array techniques, 2) semblance based methods, 3) calculation of wave field amplitude. We have choosen the third approach, using a quantitative modeling of the seismic wavefield. For this purpose, we have calculated the Green Functions (GF) in the frequency domain with the Finite Element Method (FEM). We have used this method because it is well suited to solve elliptic problems, as the elastodynamics in the Fourier domain. The volcanic tremor source is located by determining the source function over a regular grid of points. The best fit point is choosen as the tremor source location. The source inversion is performed in the frequency domain, using only the wavefield amplitudes. We illustrate the method and its validation over a synthetic dataset. We show some preliminary results on the Stromboli dataset, evidencing temporal variations of the volcanic tremor sources.

The gaseous halo of the Milky Way

In the last decade, sensitive observations have revealed that disc galaxies are surrounded by multiphase gaseous halos produced by the circulation of gas from the discs to the environment and vice-versa. This Thesis is a study of the gaseous halo of the Milky Way carried out via the modelling of the HI emission and the available absorption-line data. We fitted simple kinematical models to the HI LAB Survey and found that the Galaxy has a massive (~3x10^8 Mo) HI halo extending a few kiloparsecs above the plane. This layer rotates more slowly than the disc and shows a global inflow motion, a kinematics similar to that observed in the HI halos of nearby galaxies. We built a dynamical model of the galactic fountain to reproduce the properties of this layer. In this model, fountain clouds are ejected from the disc by SN feedback and - as suggested by hydrodynamical simulations - triggers the cooling of coronal gas, which is entrained by the cloud wakes and accretes onto the disc when the clouds fall back. For a proper choice of the parameters, the model reproduces well the HI data and predicts an accretion of coronal gas onto the disc at a rate of 2 Mo/yr. We extended this model to the warm-hot component of the halo, showing that most of the ion absorption features observed towards background sources are consistent with being produced in the turbulent wakes that lag behind the fountain clouds. Specifically, the column densities, positions, and velocities of the absorbers are well reproduced by our model. Finally, we studied the gas content of galaxies extracted from a cosmological N-body+SPH simulation, and found that an HI halo with the forementioned properties is not observed, probably due ti the relatively low resolution of the simulations.

Harnack inequalities in sub-Riemannian settings

In the present thesis, we discuss the main notions of an axiomatic approach for an invariant Harnack inequality. This procedure, originated from techniques for fully nonlinear elliptic operators, has been developed by Di Fazio, Gutiérrez, and Lanconelli in the general settings of doubling Hölder quasi-metric spaces. The main tools of the approach are the so-called double ball property and critical density property: the validity of these properties implies an invariant Harnack inequality. We are mainly interested in the horizontally elliptic operators, i.e. some second order linear degenerate-elliptic operators which are elliptic with respect to the horizontal directions of a Carnot group. An invariant Harnack inequality of Krylov-Safonov type is still an open problem in this context. In the thesis we show how the double ball property is related to the solvability of a kind of exterior Dirichlet problem for these operators. More precisely, it is a consequence of the existence of some suitable interior barrier functions of Bouligand-type. By following these ideas, we prove the double ball property for a generic step two Carnot group. Regarding the critical density, we generalize to the setting of H-type groups some arguments by Gutiérrez and Tournier for the Heisenberg group. We recognize that the critical density holds true in these peculiar contexts by assuming a Cordes-Landis type condition for the coefficient matrix of the operator. By the axiomatic approach, we thus prove an invariant Harnack inequality in H-type groups which is uniform in the class of the coefficient matrices with prescribed bounds for the eigenvalues and satisfying such a Cordes-Landis condition.

Resolved stellar populations in milky way and andromeda's new satellites

This thesis concerns the study of the variable stars and resolved stellar populations in four recently discovered dSphs, namely, Hercules and Ursa Major I (UMa I), which are UFD satellites of the MW; Andromeda XIX (And XIX) and Andromeda XXI (And XXI), which are satellites of M31. The main aim is to obtain detailed informations on the properties (age, metallicity, distance, and Oosterhoff type) of the stellar populations in these galaxies, to compare them with those of other satellites around the MW and M31, both ''classical'' dSphs and UFDs. The observables used to achieve these goals are the pulsating variables, especially the RR Lyrae stars, and the color magnitude diagram (CMD) of the resolved stellar populations. In particular, for UMa I, we combined B, V time-series observations from four different ground-based telescopes (Cassini, TLS, TT1 and Subaru) and for Hercules, we used archival data acquired with the Advanced Camera for Surveys (ACS) on board the HST. We used, instead B and V times-series photometry obtained with the Large Binocular Telescope (LBT) for And XIX and And XXI .

Sparse Signal Representation of Ultrasonic Signals for Structural Health Monitoring Applications

Assessment of the integrity of structural components is of great importance for aerospace systems, land and marine transportation, civil infrastructures and other biological and mechanical applications. Guided waves (GWs) based inspections are an attractive mean for structural health monitoring. In this thesis, the study and development of techniques for GW ultrasound signal analysis and compression in the context of non-destructive testing of structures will be presented. In guided wave inspections, it is necessary to address the problem of the dispersion compensation. A signal processing approach based on frequency warping was adopted. Such operator maps the frequencies axis through a function derived by the group velocity of the test material and it is used to remove the dependence on the travelled distance from the acquired signals. Such processing strategy was fruitfully applied for impact location and damage localization tasks in composite and aluminum panels. It has been shown that, basing on this processing tool, low power embedded system for GW structural monitoring can be implemented. Finally, a new procedure based on Compressive Sensing has been developed and applied for data reduction. Such procedure has also a beneficial effect in enhancing the accuracy of structural defects localization. This algorithm uses the convolutive model of the propagation of ultrasonic guided waves which takes advantage of a sparse signal representation in the warped frequency domain. The recovery from the compressed samples is based on an alternating minimization procedure which achieves both an accurate reconstruction of the ultrasonic signal and a precise estimation of waves time of flight. Such information is used to feed hyperbolic or elliptic localization procedures, for accurate impact or damage localization.

Constraints on neutrino mass fraction using Redshift Space Distortions

Redshift Space Distortions (RSD) are an apparent anisotropy in the distribution of galaxies due to their peculiar motion. These features are imprinted in the correlation function of galaxies, which describes how these structures distribute around each other. RSD can be represented by a distortions parameter $\beta$, which is strictly related to the growth of cosmic structures. For this reason, measurements of RSD can be exploited to give constraints on the cosmological parameters, such us for example the neutrino mass. Neutrinos are neutral subatomic particles that come with three flavours, the electron, the muon and the tau neutrino. Their mass differences can be measured in the oscillation experiments. Information on the absolute scale of neutrino mass can come from cosmology, since neutrinos leave a characteristic imprint on the large scale structure of the universe. The aim of this thesis is to provide constraints on the accuracy with which neutrino mass can be estimated when expoiting measurements of RSD. In particular we want to describe how the error on the neutrino mass estimate depends on three fundamental parameters of a galaxy redshift survey: the density of the catalogue, the bias of the sample considered and the volume observed. In doing this we make use of the BASICC Simulation from which we extract a series of dark matter halo catalogues, characterized by different value of bias, density and volume. This mock data are analysed via a Markov Chain Monte Carlo procedure, in order to estimate the neutrino mass fraction, using the software package CosmoMC, which has been conveniently modified. In this way we are able to extract a fitting formula describing our measurements, which can be used to forecast the precision reachable in future surveys like Euclid, using this kind of observations.

Galaxy evolution in the VIMOS public extragalactic redshift survey (VIPERS)

In this work I present the first measurements of the galaxy stellar mass function (GSMF) from the first public release of the VIPERS catalogue, containing ∼55,000 objects. First, I present the survey design, its scientific goal, the redshift measurements and validation. Then, I provide details about the estimate of galaxy stellar masses, star formation rates, and other physical quantities. I derive the GSMF of different galaxy types (e.g. active and passive galaxies) and as a function of the environment (defined through the local galaxy density contrast). These estimates represent new observational evidence useful to characterise the mechanism of galaxy evolution.

Identificazione di possibili unita' di stock di Anguilla anguilla mediante analisi microchimica e morfometrica degli otoliti

L’anguilla europea, è una specie eurialina catadroma con un complesso ciclo biologico: l’area di riproduzione, unica, si trova molto distante da quella di distribuzione. La specie necessita di una gestione dello stock a fini conservazionistici. Il problema è europeo: lo stock è unico, distribuito in Europa e nell’Africa settentrionale, si riproduce in Atlantico ed è panmittico. C’è preoccupazione per il declino del reclutamento e delle catture di adulti. Lo scopo del progetto è di individuare possibili unità di stock nella penisola italiana. La ricerca è basata sullo studio degli otoliti mediante analisi morfometrica e microchimica. I contorni degli otoliti sono sottoposti ad analisi ellittica di Fourier per individuare eventuali gruppi. Gli otoliti sono stati levigati per effettuare: letture d’età, indagini microstrutturali al SEM delle fasi larvali, analisi microchimiche LA-ICP-MS del nucleo, studiarne l’origine e valutare l’ambiente di sviluppo. Le indagini morfometriche mostrano evidenti pattern ontogenetici, ma non legati ocorrelati alla località, sesso o anno di nascita. Le indagini microstrutturali hanno evidenziano l’alto contenuto organico nucleare, un pattern comune di crescita ed eventi chiave delle fasi larvali, con una media di 212 anelli giornalieri. La microchimica rivela che le larve si sviluppano in acque salate fino alla metamorfosi, poi migrano verso acque meno salate. Le analisi su campioni nati nello stesso anno, evidenziano due gruppi: individui di rimonta naturale e individui di ripopolamento. I profili nucleo bordo evidenziano la permanenza a salinità intermedie degli adulti. L’attività di ricerca si è dimostrata proficua dal punto di vista tecnico con la messa a punto di protocolli innovativi e con forti ricadute sulla riduzione dei tempi e costi d’analisi. Il debole segnale di possibili unità di stock andrà verificato in futuro mediante analisi più dettagliate discriminando meglio la storia di ogni singolo individuo.

Constraining mass and shape of galaxy clusters through large scale structures

The mass estimation of galaxy clusters is a crucial point for modern cosmology, and can be obtained by several different techniques. In this work we discuss a new method to measure the mass of galaxy clusters connecting the gravitational potential of the cluster with the kinematical properties of its surroundings. We explore the dynamics of the structures located in the region outside virialized cluster, We identify groups of galaxies, as sheets or filaments, in the cluster outer region, and model how the cluster gravitational potential perturbs the motion of these structures from the Hubble fow. This identification is done in the redshift space where we look for overdensities with a filamentary shape. Then we use a radial mean velocity profile that has been found as a quite universal trend in simulations, and we fit the radial infall velocity profile of the overdensities found. The method has been tested on several cluster-size haloes from cosmological N-body simulations giving results in very good agreement with the true values of virial masses of the haloes and orientation of the sheets. We then applied the method to the Coma cluster and even in this case we found a good correspondence with previous. It is possible to notice a mass discrepancy between sheets with different alignments respect to the center of the cluster. This difference can be used to reproduce the shape of the cluster, and to demonstrate that the spherical symmetry is not always a valid assumption. In fact, if the cluster is not spherical, sheets oriented along different axes should feel a slightly different gravitational potential, and so give different masses as result of the analysis described before. Even this estimation has been tested on cosmological simulations and then applied to Coma, showing the actual non-sphericity of this cluster.

Black Hole and Galaxy Growth over Cosmic Time: the Chandra COSMOS Legacy Survey

The study of supermassive black hole (SMBH) accretion during their phase of activity (hence becoming active galactic nuclei, AGN), and its relation to the host-galaxy growth, requires large datasets of AGN, including a significant fraction of obscured sources. X-ray data are strategic in AGN selection, because at X-ray energies the contamination from non-active galaxies is far less significant than in optical/infrared surveys, and the selection of obscured AGN, including also a fraction of heavily obscured AGN, is much more effective. In this thesis, I present the results of the Chandra COSMOS Legacy survey, a 4.6 Ms X-ray survey covering the equatorial COSMOS area. The COSMOS Legacy depth (flux limit f=2x10^(-16) erg/s/cm^(-2) in the 0.5-2 keV band) is significantly better than that of other X-ray surveys on similar area, and represents the path for surveys with future facilities, like Athena and X-ray Surveyor. The final Chandra COSMOS Legacy catalog contains 4016 point-like sources, 97% of which with redshift. 65% of the sources are optically obscured and potentially caught in the phase of main BH growth. We used the sample of 174 Chandra COSMOS Legacy at z>3 to place constraints on the BH formation scenario. We found a significant disagreement between our space density and the predictions of a physical model of AGN activation through major-merger. This suggests that in our luminosity range the BH triggering through secular accretion is likely preferred to a major-merger triggering scenario. Thanks to its large statistics, the Chandra COSMOS Legacy dataset, combined with the other multiwavelength COSMOS catalogs, will be used to answer questions related to a large number of astrophysical topics, with particular focus on the SMBH accretion in different luminosity and redshift regimes.

Picard-Fuchs equations of dimensionally regulated Feynman integrals

Zusammenfassung In der vorliegenden Arbeit besch¨aftige ich mich mit Differentialgleichungen von Feynman– Integralen. Ein Feynman–Integral h¨angt von einem Dimensionsparameter D ab und kann f¨ur ganzzahlige Dimension als projektives Integral dargestellt werden. Dies ist die sogenannte Feynman–Parameter Darstellung. In Abh¨angigkeit der Dimension kann ein solches Integral divergieren. Als Funktion in D erh¨alt man eine meromorphe Funktion auf ganz C. Ein divergentes Integral kann also durch eine Laurent–Reihe ersetzt werden und dessen Koeffizienten r¨ucken in das Zentrum des Interesses. Diese Vorgehensweise wird als dimensionale Regularisierung bezeichnet. Alle Terme einer solchen Laurent–Reihe eines Feynman–Integrals sind Perioden im Sinne von Kontsevich und Zagier. Ich beschreibe eine neue Methode zur Berechnung von Differentialgleichungen von Feynman– Integralen. ¨ Ublicherweise verwendet man hierzu die sogenannten ”integration by parts” (IBP)– Identit¨aten. Die neue Methode verwendet die Theorie der Picard–Fuchs–Differentialgleichungen. Im Falle projektiver oder quasi–projektiver Variet¨aten basiert die Berechnung einer solchen Differentialgleichung auf der sogenannten Griffiths–Dwork–Reduktion. Zun¨achst beschreibe ich die Methode f¨ur feste, ganzzahlige Dimension. Nach geeigneter Verschiebung der Dimension erh¨alt man direkt eine Periode und somit eine Picard–Fuchs–Differentialgleichung. Diese ist inhomogen, da das Integrationsgebiet einen Rand besitzt und daher nur einen relativen Zykel darstellt. Mit Hilfe von dimensionalen Rekurrenzrelationen, die auf Tarasov zur¨uckgehen, kann in einem zweiten Schritt die L¨osung in der urspr¨unglichen Dimension bestimmt werden. Ich beschreibe außerdem eine Methode, die auf der Griffiths–Dwork–Reduktion basiert, um die Differentialgleichung direkt f¨ur beliebige Dimension zu berechnen. Diese Methode ist allgemein g¨ultig und erspart Dimensionswechsel. Ein Erfolg der Methode h¨angt von der M¨oglichkeit ab, große Systeme von linearen Gleichungen zu l¨osen. Ich gebe Beispiele von Integralen von Graphen mit zwei und drei Schleifen. Tarasov gibt eine Basis von Integralen an, die Graphen mit zwei Schleifen und zwei externen Kanten bestimmen. Ich bestimme Differentialgleichungen der Integrale dieser Basis. Als wichtigstes Beispiel berechne ich die Differentialgleichung des sogenannten Sunrise–Graphen mit zwei Schleifen im allgemeinen Fall beliebiger Massen. Diese ist f¨ur spezielle Werte von D eine inhomogene Picard–Fuchs–Gleichung einer Familie elliptischer Kurven. Der Sunrise–Graph ist besonders interessant, weil eine analytische L¨osung erst mit dieser Methode gefunden werden konnte, und weil dies der einfachste Graph ist, dessen Master–Integrale nicht durch Polylogarithmen gegeben sind. Ich gebe außerdem ein Beispiel eines Graphen mit drei Schleifen. Hier taucht die Picard–Fuchs–Gleichung einer Familie von K3–Fl¨achen auf.

Suche nach Dunkler Materie in Galaxien und Galaxienhaufen mit dem Neutrinoteleskop IceCube

Trotz zahlreicher Hinweise auf die Existenz von Dunkler Materie, konnten entsprechende Teilchen bisher nicht nachgewiesen werden. Eine große Anzahl an Experimenten wird durchgeführt, um die Eigenschaften möglicher Kandidatenteilchen zu untersuchen. Eine Strategie ist die Suche nach einem Neutrinosignal aus den Annihilationen von schwach wechselwirkenden massiven Teilchen (WIMPs) in Regionen mit hoher Dichte von Dunkler Materie. Mögliche Zielobjekte dieser Suchen sind die Erde, die Sonne, das Zentrum und der Halo der Milchstraße sowie entfernte Objekte, die einen hohen Anteil Dunkler Materie aufweisen.In der vorliegenden Arbeit wird die erste Suche nach einem Neutrinosignal von Zwerggalaxien, größeren Galaxien und Galaxienhaufen beschrieben. Da kein Signal nachgewiesen wurde, konnten obere Grenzen auf den Annihilationsquerschnitt von WIMPs gesetzt werden. Die stärksten Grenzen wurden aus der Beobachtung des Virgo-Haufens unter der Annahme einer großen Signalverstärkung durch Unterstrukturen in der Dichteverteilung abgeleitet. Für WIMP-Massen oberhalb von einigen TeV ist das Ergebnis vergleichbar mit Grenzen, die aus der Suche mit Gammateleskopen abgeleitet wurden. Für den direkten Annihilationskanal in zwei Neutrinos konnte der Wirkungsquerschnitt stärker eingeschränkt werden, als in bisherigen Analysen.