The Feynman-Kac formula and applications to PDEs

The thesis presents a probabilistic approach to the theory of semigroups of operators, with particular attention to the Markov and Feller semigroups. The first goal of this work is the proof of the fundamental Feynman-Kac formula, which gives the solution of certain parabolic Cauchy problems, in terms of the expected value of the initial condition computed at the associated stochastic diffusion processes. The second target is the characterization of the principal eigenvalue of the generator of a semigroup with Markov transition probability function and of second order elliptic operators with real coefficients not necessarily self-adjoint. The thesis is divided into three chapters. In the first chapter we study the Brownian motion and some of its main properties, the stochastic processes, the stochastic integral and the Itô formula in order to finally arrive, in the last section, at the proof of the Feynman-Kac formula. The second chapter is devoted to the probabilistic approach to the semigroups theory and it is here that we introduce Markov and Feller semigroups. Special emphasis is given to the Feller semigroup associated with the Brownian motion. The third and last chapter is divided into two sections. In the first one we present the abstract characterization of the principal eigenvalue of the infinitesimal generator of a semigroup of operators acting on continuous functions over a compact metric space. In the second section this approach is used to study the principal eigenvalue of elliptic partial differential operators with real coefficients. At the end, in the appendix, we gather some of the technical results used in the thesis in more details. Appendix A is devoted to the Sion minimax theorem, while in appendix B we prove the Chernoff product formula for not necessarily self-adjoint operators.

Synthesis, reactivity and applications of 3,5-dimethyl-4-nitroisoxazole derivatives

3,5-dimethyl-4-nitroisoxazole derivatives are useful synthetic intermediates as the isoxazole nucleus chemically behaves as an ester, but establish better-defined interactions with chiral catalysts and lability of its N-O aromatic bond can unveil other groups such as 1,3-dicarbonyl compounds or carboxylic acids. In the present work, these features are employed in a 3,5-dimethyl-4-nitroisoxazole based synthesis of the γ-amino acid pregabalin, a medication for the treatment of epilepsy and neuropatic pain, in which this moiety is fundamental for the enantioselective formation of a chiral center by interaction with doubly-quaternized cinchona phase-transfer catalysts, whose ability of asymmetric induction will be investigated. Influence of this group in cinchona-derivatives catalysed stereoselective addition and Darzens reaction of a mono-chlorinated 3,5-dimethyl-4-nitroisoxazole and benzaldehyde will also be investigated.

Jarrow-Yildirim model for inflation: theory and applications

This thesis deals with inflation theory, focussing on the model of Jarrow & Yildirim, which is nowadays used when pricing inflation derivatives. After recalling main results about short and forward interest rate models, the dynamics of the main components of the market are derived. Then the most important inflation-indexed derivatives are explained (zero coupon swap, year-on-year, cap and floor), and their pricing proceeding is shown step by step. Calibration is explained and performed with a common method and an heuristic and non standard one. The model is enriched with credit risk, too, which allows to take into account the possibility of bankrupt of the counterparty of a contract. In this context, the general method of pricing is derived, with the introduction of defaultable zero-coupon bonds, and the Monte Carlo method is treated in detailed and used to price a concrete example of contract. Appendixes: A: martingale measures, Girsanov's theorem and the change of numeraire. B: some aspects of the theory of Stochastic Differential Equations; in particular, the solution for linear EDSs, and the Feynman-Kac Theorem, which shows the connection between EDSs and Partial Differential Equations. C: some useful results about normal distribution.

Lateral tires characterization: testing, analytical models and applications for aeronautic purpose

The lateral characteristics of tires in terms of lateral forces as a function of sideslip angle is a focal point in the prediction of ground loads and ground handling aircraft behavior. However, tests to validate such coefficients are not mandatory to obtain Aircraft Type Certification and so they are not available for ATR tires. Anyway, some analytical values are implemented in ATR calculation codes (Flight Qualities in-house numerical code and Loads in-house numerical code). Hence, the goal of my work is to further investigate and validate lateral tires characteristics by means of: exploitation and re-parameterization of existing test on NLG tires, implementation of easy-handle model based on DFDR parameters to compute sideslip angles, application of this model to compute lateral loads on existing flight tests and incident cases, analysis of results. The last part of this work is dedicated to the preliminary study of a methodology to perform a test to retrieve lateral tire loads during ground turning with minimum requirements in terms of aircraft test instrumentation. This represents the basis for future works.

Mean curvature flow in SE (2) and applications to visual perception

Our goal in this thesis is to provide a result of existence of the degenerate non-linear, non-divergence PDE which describes the mean curvature flow in the Lie group SE(2) equipped with a sub-Riemannian metric. The research is motivated by problems of visual completion and models of the visual cortex.

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.

Statistical mechanics of polymer models: rigorous results and applications

Monomer-dimer models are amongst the models in statistical mechanics which found application in many areas of science, ranging from biology to social sciences. This model describes a many-body system in which monoatomic and diatomic particles subject to hard-core interactions get deposited on a graph. In our work we provide an extension of this model to higher-order particles. The aim of our work is threefold: first we study the thermodynamic properties of the newly introduced model. We solve analytically some regular cases and find that, differently from the original, our extension admits phase transitions. Then we tackle the inverse problem, both from an analytical and numerical perspective. Finally we propose an application to aggregation phenomena in virtual messaging services.

Synthesis, Characterization, and Applications of a Novel Cu(II)-MOF Based on a Propargylcarbamate-Functionalized Isophthalate Ligand

This work describes the synthesis of a propargylcarbamate-functionalized isophthalate ligand and its use in the solvothermal preparation of a new copper(II)-based metal organic framework named [Cu(1,3-YBDC)]ˑxH2O (also abbreviated as Cu-MOF. The characterization of this compound was performed using several complementary techniques such as infrared (ATR-FTIR) and Raman spectroscopy, X-ray powder diffraction spectroscopy (PXRD), scanning electron microscopy (SEM), X-ray photoelectron spectroscopy (XPS), atomic absorption spectroscopy (AAS) as well as thermal and surface area measurements. Synchrotron X-ray diffraction analysis revealed that this MOF contains a complex network of 5-substituted isophthalate anions bound to Cu(II) centers, arranged in pairs within paddlewheel (or “Chinese lantern”) structure with a short Cu…Cu distance of 2.633 Å. Quite unexpectedly, the apical atom in the paddlewheel structure belongs to the carbamate carbonyl oxygen atom. Such extra coordination by the propargylcarbamate groups drastically reduces the MOF porosity, a feature that was also confirmed by BET measurements. Indeed, its surface area was determined to be low (14.5 ± 0.8 m2/g) as its total pore volume (46 mm3/g). Successively the Cu-MOF was treated with HAuCl4 with the aim of studying the ability of the propargylcarbamate functionality to capture the Au(III) ion and reduce it to Au(0) to give gold nanoparticles (AuNPs). The overall amount of gold retained by the Cu-MOF/Au was determined by AAS while the amount of gold and its oxidation state on the surface of the MOF was studied by XPS. A glassy carbon (GC) electrode was drop-casted with a Cu-MOF suspension to electrochemically characterize the material through cyclic voltammetry (CV) and electrochemical impedance spectroscopy (EIS). The performance of the modified electrodes towards nitrite oxidation was tested by CV and chronoamperometry.

The road to augmented reality: a comprehensive study of hardware and applications over the last ten years

Augmented reality has been growing extensively over the years in all aspects and multiple fields. My aim in this paper is to present a comprehensive study on augmented reality(AR) hardware and its applications from early developments to the possible future trends. Particularly my research is more focused on last 11 years(2012-2022), where I systematically reviewed 30 research papers per year to get a clear knowledge on trends of AR. A total of 330 publications were reviewed and grouped according to their application. The review's main contribution is to show the entire landscape of AR research and to provide a broad view of how it has evolved. Along with various AR glasses history and specifications are presented in detail. In the penultimate chapter I explained my methodology of research following my analysis from the past to the present along with my thoughts for the future. To conclude my study, In the final chapter I made some statements about possible future with AR, VR and XR(extended reality).

Low Power instrument Transformer: technologies and applications

Instrument transformers serve an important role in the protection and isolation of AC electrical systems for measurements of different electrical parameters like voltage, current, power factor, frequency, and energy. As suggested by name these transformers are used in connection with suitable measuring instruments like an ammeter, wattmeter, voltmeter, and energy meters. We have seen how higher voltages and currents are transformed into lower magnitudes to provide isolation between power networks, relays, and other instruments. Reducing transient, suppressing electrical noises in sensitive devices, standardization of instruments and relays up to a few volts and current. Transformer performance directly affects the accuracy of power system measurements and the reliability of relay protection. We classified transformers in terms of purpose, insulating medium, Voltage ranges, temperature ranges, humidity or environmental effect, indoor and outdoor use, performance, Features, specification, efficiency, cost analysis, application, benefits, and limitations which enabled us to comprehend their correct use and selection criteria based on our desired requirements. We also discussed modern Low power instrument transformer products that are recently launched or offered by renowned companies like Schneider Electric, Siemens, ABB, ZIV, G&W etc. These new products are innovations and problem solvers in the domain of measurement, protection, digital communication, advance, and commercial energy metering. Since there is always some space for improvements to explore new advantages of Low power instrument transformers in the domain of their wide linearity, high-frequency range, miniaturization, structural and technological modification, integration, smart frequency modeling, and output prediction of low-power voltage transformers.

Caratterizzazione meccanica di calcestruzzi fibrorinforzati con fibre d'acciaio

L’utilizzo di materiali compositi come i calcestruzzi fibrorinforzati sta diventando sempre più frequente e diffuso. Tuttavia la scelta di nuovi materiali richiede una approfondita analisi delle loro caratteristiche e dei loro comportamenti. I vantaggi forniti dall’aggiunta di fibre d’acciaio ad un materiale fragile, quale il calcestruzzo, sono legati al miglioramento della duttilità e all'aumento di assorbimento di energia. L’aggiunta di fibre permette quindi di migliorare il comportamento strutturale del composito, dando vita ad un nuovo materiale capace di lavorare non solo a compressione ma anche in piccola parte a trazione, ma soprattutto caratterizzato da una discreta duttilità ed una buona capacità plastica. Questa tesi ha avuto come fine l’analisi delle caratteristiche di questi compositi cementizi fibrorinforzati. Partendo da prove sperimentali classiche quali prove di trazione e compressione, si è arrivati alla caratterizzazione di questi materiali avvalendosi di una campagna sperimentale basata sull’applicazione della norma UNI 11039/2003. L’obiettivo principale di questo lavoro consiste nell’analizzare e nel confrontare calcestruzzi rinforzati con fibre di due diverse lunghezze e in diversi dosaggi. Studiando questi calcestruzzi si è cercato di comprendere meglio questi materiali e trovare un riscontro pratico ai comportamenti descritti in teorie ormai diffuse e consolidate. La comparazione dei risultati dei test condotti ha permesso di mettere in luce differenze tra i materiali rinforzati con l’aggiunta di fibre corte rispetto a quelli con fibre lunghe, ma ha anche permesso di mostrare e sottolineare le analogie che caratterizzano questi materiali fibrorinforzati. Sono stati affrontati inoltre gli aspetti legati alle fasi della costituzione di questi materiali sia da un punto di vista teorico sia da un punto di vista pratico. Infine è stato sviluppato un modello analitico basato sulla definizione di specifici diagrammi tensione-deformazione; i risultati di questo modello sono quindi stati confrontati con i dati sperimentali ottenuti in laboratorio.